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Title: Encyclopaedia Britannica, 11th Edition, Volume 8, Slice 4 - "Diameter" to "Dinarchus"
Author: Various
Language: English
As this book started as an ASCII text book there are no pictures available.
Copyright Status: Not copyrighted in the United States. If you live elsewhere check the laws of your country before downloading this ebook. See comments about copyright issues at end of book.

*** Start of this Doctrine Publishing Corporation Digital Book "Encyclopaedia Britannica, 11th Edition, Volume 8, Slice 4 - "Diameter" to "Dinarchus"" ***

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Transcriber's notes:

(1) Numbers following letters (without space) like C2 were originally
      printed in subscript. Letter subscripts are preceded by an
      underscore, like C_n.

(2) Characters following a carat (^) were printed in superscript.

(3) Side-notes were relocated to function as titles of their respective
      paragraphs.

(4) Letters topped by Macron are represented as [=x].

(5) dP stands for the partial-derivative symbol, or curled 'd'.

(6) [oo] stands for the infinity symbol, and [int] for the integral
       symbol.

(7) The following typographical errors have been corrected:

    Article DIAMOND: "If this be so the form of the diamond is really
      the tetrahedron (and the various figures derived symmetrically from
      it) and not the octahedron". 'octahedron' amended from
      'octadehron'.

    Article DIARY: "diaries began to be largely written in England,
      although in most cases without any idea of even eventual
      publication". 'largely' amended from 'largly'.

    Article DICOTYLEDONS: "The arrangement of the conducting tissue in
      the stem is characteristic; a transverse section of the very young
      stem shows a number of distinct conducting strands". 'number'
      amended from 'nunber'.

    Article DIEKIRCH: "It remained more or less fortified until the
      beginning of the 19th century when the French during their
      occupation levelled the old walls, and substituted the avenues of
      trees that now encircle the town". 'or' amended from 'for'.

    Article DIFFERENCES, CALCULUS OF: "as the second difference of u_n,
      and therefore as corresponding to the value x_n"; 'difference'
      amended from 'dfference'.

    Article DINAJPUR: "a town (with a population in 1901 of 13,430) and
     district of British India, in the Rajshahi division of Eastern
     Bengal and Assam". 'British' amended from 'Britsh'.



          ENCYCLOPAEDIA BRITANNICA

  A DICTIONARY OF ARTS, SCIENCES, LITERATURE
           AND GENERAL INFORMATION

              ELEVENTH EDITION


           VOLUME VIII, SLICE IV

           Diameter to Dinarchus



ARTICLES IN THIS SLICE:


  DIAMETER                             DIEDENHOFEN
  DIAMOND                              DIEKIRCH
  DIAMOND NECKLACE, THE AFFAIR OF THE  DIELECTRIC
  DIANA                                DIELMANN, FREDERICK
  DIANA MONKEY                         DIEMEN, ANTHONY VAN
  DIANE DE FRANCE                      DIEPENBECK, ABRAHAM VAN
  DIANE DE POITIERS                    DIEPPE
  DIAPASON                             DIERX, LÉON
  DIAPER                               DIES, CHRISTOPH ALBERT
  DIAPHORETICS                         DIEST
  DIAPHRAGM                            DIESTERWEG, FRIEDRICH ADOLF WILHELM
  DIARBEKR                             DIET
  DIARRHOEA                            DIETARY
  DIARY                                DIETETICS
  DIASPORE                             DIETRICH, CHRISTIAN WILHELM ERNST
  DIASTYLE                             DIETRICH OF BERN
  DIATOMACEAE                          DIEZ, FRIEDRICH CHRISTIAN
  DIAULOS                              DIEZ
  DIAVOLO, FRA                         DIFFERENCES, CALCULUS OF
  DIAZ, NARCISSE VIRGILIO              DIFFERENTIAL EQUATION
  DIAZ, PORFIRIO                       DIFFLUGIA
  DIAZ DE NOVAES, BARTHOLOMEU          DIFFRACTION OF LIGHT
  DIAZO COMPOUNDS                      DIFFUSION
  DIAZOMATA                            DIGBY, SIR EVERARD
  DIBDIN, CHARLES                      DIGBY, SIR KENELM
  DIBDIN, THOMAS FROGNALL              DIGBY, KENELM HENRY
  DIBDIN, THOMAS JOHN                  DIGENES ACRITAS, BASILIUS
  DIBRA                                DIGEST
  DIBRUGARH                            DIGESTIVE ORGANS
  DICAEARCHUS                          DIGGES, WEST
  DICE                                 DIGIT
  DICETO, RALPH DE                     DIGITALIS
  DICEY, EDWARD                        DIGNE
  DICHOTOMY                            DIGOIN
  DICK, ROBERT                         DIJON
  DICK, THOMAS                         DIKE
  DICKENS, CHARLES JOHN HUFFAM         DIKKA
  DICKINSON, ANNA ELIZABETH            DILAPIDATION
  DICKINSON, JOHN                      DILATATION
  DICKSON, SIR ALEXANDER               DILATORY
  DICKSON, SIR JAMES ROBERT            DILEMMA
  DICOTYLEDONS                         DILETTANTE
  DICTATOR                             DILIGENCE
  DICTIONARY                           DILKE, SIR CHARLES WENTWORTH
  DICTYOGENS                           DILL
  DICTYS CRETENSIS                     DILLEN, JOHANN JAKOB
  DICUIL                               DILLENBURG
  DIDACHE, THE                         DILLENS, JULIEN
  DIDACTIC POETRY                      DILLINGEN
  DIDEROT, DENIS                       DILLMANN, CHRISTIAN AUGUST
  DIDIUS SALVIUS JULIANUS, MARCUS      DILLON, ARTHUR RICHARD
  DIDO                                 DILLON, JOHN
  DIDON, HENRI                         DILUVIUM
  DIDOT                                DIME
  DIDRON, ADOLPHE NAPOLÉON             DIMENSION
  DIDYMI                               DIMITY
  DIDYMIUM                             DINAJPUR
  DIDYMUS                              DINAN
  DIDYMUS CHALCENTERUS                 DINANT
  DIE (town of France)                 DINAPUR
  DIE (datum)                          DINARCHUS
  DIEBITSCH, HANS KARL FRIEDRICH ANTON



DIAMETER (from the Gr. [Greek: dia], through, [Greek: metron],
measure), in geometry, a line passing through the centre of a circle or
conic section and terminated by the curve; the "principal diameters" of
the ellipse and hyperbola coincide with the "axes" and are at right
angles; "conjugate diameters" are such that each bisects chords parallel
to the other. The diameter of a quadric surface is a line at the
extremities of which the tangent planes are parallel. Newton defined the
diameter of a curve of any order as the locus of the centres of the mean
distances of the points of intersection of a system of parallel chords
with the curve; this locus may be shown to be a straight line. The word
is also used as a unit of linear measurement of the magnifying power of
a lens or microscope.

In architecture, the term is used to express the measure of the lower
part of the shaft of a column. It is employed by Vitruvius (iii. 2) to
determine the height of a column, which should vary from eight to ten
diameters according to the intercolumniation: and it is generally the
custom to fix the lower diameter of the shaft by the height required and
the Order employed. Thus the diameter of the Roman Doric should be about
one-eighth of the height, that of the Ionic one-ninth, and of the
Corinthian one-tenth (see ORDER).



DIAMOND, a mineral universally recognized as chief among precious
stones; it is the hardest, the most imperishable, and also the most
brilliant of minerals.[1] These qualities alone have made it supreme as
a jewel since early times, and yet the real brilliancy of the stone is
not displayed until it has been faceted by the art of the lapidary
(q.v.); and this was scarcely developed before the year 1746. The
consummate hardness of the diamond, in spite of its high price, has made
it most useful for purposes of grinding, polishing and drilling.
Numerous attempts have been made to manufacture the diamond by
artificial means, and these attempts have a high scientific interest on
account of the mystery which surrounds the natural origin of this
remarkable mineral. Its physical and chemical properties have been the
subject of much study, and have a special interest in view of the
extraordinary difference between the physical characters of the diamond
and those of graphite (blacklead) or charcoal, with which it is
chemically identical, and into which it can be converted by the action
of heat or electricity. Again, on account of the great value of the
diamond, much of the romance of precious stones has centred round this
mineral; and the history of some of the great diamonds of historic times
has been traced through many extraordinary vicissitudes.

The name [Greek: Adamas], "the invincible," was probably applied by the
Greeks to hard metals, and thence to corundum (emery) and other hard
stones. According to Charles William King, the first undoubted
application of the name to the diamond is found in Manilius (A.D.
16),--_Sic Adamas_, _punctum lapidis_, _pretiosior auro_,--and Pliny
(A.D. 100) speaks of the rarity of the stone, "the most valuable of
gems, known only to kings." Pliny described six varieties, among which
the Indian, having six pointed angles, and also resembling two pyramids
(_turbines_, whip-tops) placed base to base, may probably be identified
as the ordinary octahedral crystal (fig. 1). The "diamond" (_Yahalom_)
in the breastplate of the high priest (Ex. xxxix. 11) was certainly some
other stone, for it bore the name of a tribe, and methods of engraving
the true diamond cannot have been known so early. The stone can hardly
have become familiar to the Romans until introduced from India, where it
was probably mined at a very early period. But one or other of the
remaining varieties mentioned by Pliny (the Macedonian, the Arabian, the
Cyprian, &c.) may be the true diamond, which was in great request for
the tool of the gem-engraver. Later Roman authors mentioned various
rivers in India as yielding the _Adamas_ among their sands. The name
_Adamas_ became corrupted into the forms _adamant_, _diamaunt_,
_diamant_, _diamond_; but the same word, owing to a medieval
misinterpretation which derived it from _adamare_ (compare the French
word _aimant_), was also applied to the lodestone.

Like all the precious stones, the diamond was credited with many
marvellous virtues; among others the power of averting insanity, and of
rendering poison harmless; and in the middle ages it was known as the
"pietra della reconciliazione," as the peacemaker between husband and
wife.

_Scientific Characters._--The majority of minerals are found most
commonly in masses which can with difficulty be recognized as aggregates
of crystalline grains, and occur comparatively seldom as distinct
crystals; but the diamond is almost always found in single crystals,
which show no signs of previous attachment to any matrix; the stones
were, until the discovery of the South African mines, almost entirely
derived from sands or gravels, but owing to the hardness of the mineral
it is rarely, if ever, water-worn, and the crystals are often very
perfect. The crystals belong to the cubic system, generally assuming the
form of the octahedron (fig. 1), but they may, in accordance with the
principles of crystallography, also occur in other forms symmetrically
derived from the octahedron,--for example, the cube, the 12-faced figure
known as the rhombic dodecahedron (fig. 2), or the 48-faced figure known
as the hexakis-octahedron (fig. 3), or in combinations of these. The
octahedron faces are usually smooth; most of the other faces are rounded
(fig. 4). The cube faces are rough with protruding points. The cube is
sometimes found in Brazil, but is very rare among the S. African stones;
and the dodecahedron is perhaps more common in Brazil than elsewhere.
There is often a furrow running along the edges of the octahedron, or
across the edges of the cube, and this indicates that the apparently
simple crystal may really consist of eight individuals meeting at the
centre; or, what comes to the same thing, of two individuals
interpenetrating and projecting through each other. If this be so the
form of the diamond is really the tetrahedron (and the various figures
derived symmetrically from it) and not the octahedron. Fig. 5 shows how
the octahedron with furrowed edge may be constructed from two
interpenetrating tetrahedra (shown in dotted lines). If the grooves be
left out of account, the large faces which have replaced each
tetrahedron corner then make up a figure which has the aspect of a
simple octahedron. Such regular interpenetrations are known in
crystallography as "twins." There are also twins of diamond in which two
octahedra (fig. 6) are united by contact along a surface parallel to an
octahedron face without interpenetration. On account of their
resemblance to the twins of the mineral spinel (which crystallizes in
octahedra) these are known as "spinel twins." They are generally
flattened along the plane of union. The crystals often display
triangular markings, either elevations or pits, upon the octahedron
faces; the latter are particularly well defined and have the form of
equilateral triangles (fig. 7). They are similar to the "etched figures"
produced by moistening an octahedron of alum, and have probably been
produced, like them, by the action of some solvent. Similar, but
somewhat different markings are produced by the combustion of diamond in
oxygen, unaccompanied by any rounding of the edges.

[Illustration: FIG. 1.]

[Illustration: FIG. 2.]

[Illustration: FIG. 3.]

[Illustration: FIG. 4.]

[Illustration: FIG. 5.]

[Illustration: FIG. 6.]

[Illustration: FIG. 7.]

Diamond possesses a brilliant "adamantine" lustre, but this tends to be
greasy on the surface of the natural stones and gives the rounded
crystals somewhat the appearance of drops of gum. Absolutely colourless
stones are not so common as cloudy and faintly coloured specimens; the
usual tints are grey, brown, yellow or white; and as rarities, red,
green, blue and black stones have been found. The colour can sometimes
be removed or changed at a high temperature, but generally returns on
cooling. It is therefore more probably due to metallic oxides than to
hydrocarbons. Sir William Crookes has, however, changed a pale yellow
diamond to a bluish-green colour by keeping it embedded in radium
bromide for eleven weeks. The black coloration upon the surface produced
by this process, as also by the electric bombardment in a vacuum tube,
appears to be due to a conversion of the surface film into graphite.
Diamond may break with a conchoidal fracture, but the crystals always
cleave readily along planes parallel to the octahedron faces: of this
property the diamond cutters avail themselves when reducing the stone to
the most convenient form for cutting; a sawing process, has, however,
now been introduced, which is preferable to that of cleavage. It is the
hardest known substance (though tantalum, or an alloy of tantalum now
competes with it) and is chosen as 10 in the mineralogist's scale of
hardness; but the difference in hardness between diamond (10) and
corundum (9) is really greater than that between corundum (9) and talc
(1); there is a difference in the hardness of the different faces; the
Borneo stones are also said to be harder than those of Australia, and
the Australian harder than the African, but this is by no means certain.
The specific gravity ranges from 3.56 to 3.50, generally about 3.52. The
coefficient of expansion increases very rapidly above 750°, and
diminishes very rapidly at low temperatures; the maximum density is
attained about -42° C.

The very high refractive power (index = 2.417 for sodium light) gives
the stone its extraordinary brilliancy; for light incident within a
diamond at a greater angle than 24½° is reflected back into the stone
instead of passing through it; the corresponding angle for glass is
40½°. The very high dispersion (index for red light = 2.402, for blue
light = 2.460) gives it the wonderful "fire" or display of spectral
colours. Certain absorption bands at the blue end of the spectrum are
supposed to be due to rare elements such as samarium. Unlike other cubic
crystals, diamond experiences a diminution of refractive index with
increase of temperature. It is very transparent for Röntgen rays,
whereas paste imitations are opaque. It is a good conductor of heat, and
therefore feels colder to the touch than glass and imitation stones. The
diamond has also a somewhat greasy feel. The specific heat increases
rapidly with rising temperature up to 60° C., and then more slowly.
Crystals belonging to the cubic system should not be birefringent unless
strained; diamond often displays double refraction particularly in the
neighbourhood of inclusions, both liquid and solid; this is probably due
to strain, and the spontaneous explosion of diamonds has often been
observed. Diamond differs from graphite in being a bad conductor of
electricity: it becomes positively electrified by friction. The
electrical resistance is about that of ordinary glass, and is diminished
by one-half during exposure by Röntgen rays; the dielectric constant
(16) is greater than that which should correspond to the specific
gravity.

The phosphorescence produced by friction has been known since the time
of Robert Boyle (1663); the diamond becomes luminous in a dark room
after exposure to sunlight or in the presence of radium; and many stones
phosphoresce beautifully (generally with a pale green light) when
subjected to the electric discharge in a vacuum tube. Some diamonds are
more phosphorescent than others, and different faces of a crystal may
display different tints. The combustibility of the diamond was predicted
by Sir Isaac Newton on account of its high refractive power; it was
first established experimentally by the Florentine Academicians in 1694.
In oxygen or air diamond burns at about 850°, and only continues to do
so if maintained at a high temperature; but in the absence of oxidising
agents it may be raised to a much higher temperature. It is, however,
infusible at the temperature of the electric arc, but becomes converted
superficially into graphite. Experiments on the combustion of diamond
were made by Smithson Tennant (1797) and Sir Humphry Davy (1816), with
the object of proving that it is pure carbon; they showed that burnt in
oxygen it yields exactly the same amount of carbon dioxide as that
produced by burning the same weight of carbon. Still more convincing
experiments were made by A. Krause in 1890. Similarly Guyton de Morveau
showed that, like charcoal, diamond converts soft iron into steel.
Diamond is insoluble in acid and alkalis, but is oxidised on heating
with potassium bichromate and sulphuric acid.

Bort (or Boart) is the name given to impure crystals or fragments
useless for jewels; it is also applied to the rounded crystalline
aggregates, which generally have a grey colour, a rough surface, often a
radial structure, and are devoid of good cleavage. They are sometimes
spherical ("shot bort"). Carbonado or "black diamond," found in Bahia
(also recently in Minas Geraes), is a black material with a minutely
crystalline structure somewhat porous, opaque, resembling charcoal in
appearance, devoid of cleavage, rather harder than diamond, but of less
specific gravity; it sometimes displays a rude cubic crystalline form.
The largest specimen found (1895) weighed 3078 carats. Both bort and
carbonado seem to be really aggregates of crystallized diamond, but the
carbonado is so nearly structureless that it was till recently regarded
as an amorphous modification of carbon.

_Uses of the Diamond._--The use of the diamond for other purposes than
jewelry depends upon its extreme hardness: it has always been the only
material used for cutting or engraving the diamond itself. The
employment of powdered bort and the lapidary's wheel for faceting
diamonds was introduced by L. von Berquen of Bruges in 1476. Diamonds
are now employed not only for faceting precious stones, but also for
cutting and drilling glass, porcelain, &c,; for fine engraving such as
scales; in dentistry for drilling; as a turning tool for electric-light
carbons, hard rubber, &c.; and occasionally for finishing accurate
turning work such as the axle of a transit instrument. For these tools
the stone is actually shaped to the best form: it is now electroplated
before being set in its metal mount in order to secure a firm fastening.
It is also used for bearings in watches and electric meters. The best
glaziers' diamonds are chosen from crystals such that a natural curved
edge can be used. For rock drills, and revolving saws for stone cutting,
either diamond, bort or carbonado is employed, set in steel tubes, disks
or bands. Rock drilling is the most important industrial application;
and for this, owing to its freedom from cleavage, the carbonado is more
highly prized than diamond; it is broken into fragments about 3 carats
in weight; and in 1905 the value of carbonado was no less than from £10
to £14 a carat. It has been found that the "carbons" in drills can
safely be subjected to a pressure of over 60 kilograms per square
millimetre, and a speed of 25 metres per second. A recent application of
the diamond is for wire drawing; a hole tapering towards the centre is
drilled through a diamond, and the metal is drawn through this. No other
tool is so endurable, or gives such uniform thickness of wire.

_Distribution and Mining._--The most important localities for diamonds
have been: (1) India, where they were mined from the earliest times till
the close of the 19th century; (2) South America, where they have been
mined since the middle of the 18th century; and (3) South Africa, to
which almost the whole of the diamond-mining industry has been
transferred since 1870.

  _India._--The diamond is here found in ancient sandstones and
  conglomerates, and in the river gravels and sands derived from them.
  The sandstones and conglomerates belong to the Vindhyan formation and
  overlie the old crystalline rocks: the diamantiferous beds are well
  defined, often not more than 1 ft. in thickness, and contain pebbles
  of quartzite, jasper, sandstone, slate, &c. The mines fall into five
  groups situated on the eastern side of the Deccan plateau about the
  following places (beginning from the south), the first three being in
  Madras. (1) Chennur near Cuddapah on the river Pennar. (2) Kurnool
  near Baneganapalle between the rivers Pennar and Kistna. (3) Kollar
  near Bezwada on the river Kistna. (4) Sambalpur on the river Mahanadi
  in the Central Provinces. (5) Panna near Allahabad, in Bundelkhand.
  The mining has always been carried on by natives of low caste, and by
  primitive methods which do not differ much from those described by the
  French merchant Jean Baptiste Tavernier (1605-1689), who paid a
  prolonged visit to most of the mines between 1638 and 1665 as a
  dealer in precious stones. According to his description shallow pits
  were sunk, and the gravel excavated was gathered into a walled
  enclosure where it was crushed and water was poured over it, and it
  was finally sifted in baskets and sorted by hand. The buying and
  selling was at that period conducted by young children. In more modern
  times there has been the same excavation of shallow pits, and
  sluicing, sifting and sorting, by hand labour, the only machinery used
  being chain pumps made of earthen bowls to remove the water from the
  deeper pits.

  At some of the Indian localities spasmodic mining has been carried on
  at different periods for centuries, at some the work which had been
  long abandoned was revived in recent times, at others it has long been
  abandoned altogether. Many of the large stones of antiquity were
  probably found in the Kollar group, where Tavernier found 60,000
  workers in 1645 (?), the mines having, according to native accounts,
  been discovered about 100 years previously. Golconda was the fortress
  and the market for the diamond industry at this group of mines, and so
  gave its name to them. The old mines have now been completely
  abandoned, but in 1891 about 1000 carats were being raised annually in
  the neighbourhood of Hyderabad. The Sambalpur group appear to have
  been the most ancient mines of all, but they were not worked later
  than 1850. The Panna group were the most productive during the 19th
  century. India was no doubt the source of all the large stones of
  antiquity; a stone of 67-3/8 carats was found at Wajra Karur in the
  Chennur group in 1881, and one of 210½ carats at Hira Khund in 1809.
  Other Indian localities besides those mentioned above are Simla, in
  the N.W. Provinces, where a few stones have been found, and a district
  on the Gouel and the Sunk rivers in Bengal, which V. Ball has
  identified with the Soumelpour mentioned by Tavernier. The mines of
  Golconda and Kurnool were described as early as 1677 in the twelfth
  volume of the _Philosophical Transactions_ of the Royal Society. At
  the present time very few Indian diamonds find their way out of the
  country, and, so far as the world's supply is concerned, Indian mining
  of diamonds may be considered extinct. The first blow to this industry
  was the discovery of the Brazilian mines in Minas Geraes and Bahia.

  _Brazil._---Diamonds were found about 1725 at Tejuco (now Diamantina)
  in Minas Geraes, and the mining became important about 1740. The chief
  districts in Minas Geraes are (1) Bagagem on the W. side of the Serra
  da Mata da Corda; (2) Rio Abaete on the E. side of the same range;
  these two districts being among the head waters of the Rio de San
  Francisco and its tributaries; (3) Diamantina, on and about the
  watershed separating the Rio de San Francisco from the Rio
  Jequitinhonha; and (4) Grao Mogul, nearly 200 m. to the N.E. of
  Diamantina on the latter river.

  The Rio Abaete district was worked on a considerable scale between
  1785 and 1807, but is now abandoned. Diamantina is at present the most
  important district; it occupies a mountainous plateau, and the
  diamonds are found both on the plateau and in the river valleys below
  it. The mountains consist here of an ancient laminated micaceous
  quartzite, which is in parts a flexible sandstone known as
  itacolumite, and in parts a conglomerate; it is interbedded with
  clay-slate, mica-schist, hornblende-schist and haematite-schist, and
  intersected by veins of quartz. This series is overlain unconformably
  by a younger quartzite of similar character, and itself rests upon the
  crystalline schists. The diamond is found under three conditions: (1)
  in the gravels of the present rivers, embedded in a ferruginous
  clay-cemented conglomerate known as _cascalho_; (2) in terraces
  (gupiarras) in a similar conglomerate occupying higher levels in the
  present valleys; (3) in plateau deposits in a coarse surface
  conglomerate known as _gurgulho_, the diamond and other heavy minerals
  being embedded in the red clay which cements the larger blocks. Under
  all these three conditions the diamond is associated with fragments of
  the rocks of the country and the minerals derived from them,
  especially quartz, hornstone, jasper, the polymorphous oxide of
  titanium (rutile, anatase and brookite), oxides and hydrates of iron
  (magnetite, ilmenite, haematite, limonite), oxide of tin, iron
  pyrites, tourmaline, garnet, xenotime, monazite, kyanite, diaspore,
  sphene, topaz, and several phosphates, and also gold. Since the heavy
  minerals of the _cascalho_ in the river beds are more worn than those
  of the terraces, it is highly probable that they have been derived by
  the cutting down of the older river gravels represented by the
  terraces; and since in both deposits the heavy minerals are more
  abundant near the heads of the valleys in the plateau, it is also
  highly probable that both have really been derived from the plateau
  deposit. In the latter, especially at São João da Chapada, the
  minerals accompanying the diamond are scarcely worn at all; in the
  terraces and the river beds they are more worn and more abundant; the
  terraces, therefore, are to be regarded as a first concentration of
  the plateau material by the old rivers; and the _cascalho_ as a second
  concentration by the modern rivers. The mining is carried on by
  negroes under the supervision of overseers; the _cascalho_ is dug out
  in the dry season and removed to a higher level, and is afterwards
  washed out by hand in running water in shallow wooden basins
  (_bateas_). The terraces can be worked at all seasons, and the
  material is partly washed out by leading streams on to it. The washing
  of the plateau material is effected in reservoirs of rain water.

  It is difficult to obtain an estimate of the actual production of the
  Minas Geraes mines, for no official returns have been published, but
  in recent years it has certainly been rivalled by the yield in Bahia.
  The diamond here occurs in river gravels and sands associated with
  the same minerals as in Minas Geraes; since 1844 the richest mines
  have been worked in the Serra de Cincora, where the mountains are
  intersected by the river Paraguassu and its tributaries; it is said
  that there were as many as 20,000 miners working here in 1845, and it
  was estimated that 54,000 carats were produced in Bahia in 1858. The
  earlier workings were in the Serra de Chapada to the N.W. of the mines
  just mentioned. In 1901 there were about 5000 negroes employed in the
  Bahia mines; methods were still primitive; the _cascalho_ was dug out
  from the river beds or tunnelled out from the valley side, and washed
  once a week in sluices of running water, where it was turned over with
  the hoe, and finally washed in wooden basins and picked over by hand;
  sometimes also the diamantiferous material is scooped out of the bed
  of the shallow rivers by divers, and by men working under water in
  caissons. It is almost exclusively in the mines of Bahia, and in
  particular in the Cincora district, that the valuable carbonado is
  found. The carbonado and the diamond have been traced to an extensive
  hard conglomerate which occurs in the middle of the sandstone
  formation. Diamonds are also mined at Salobro on the river Pardo not
  far inland from the port of Canavieras in the S.E. corner of Bahia.
  The enormous development of the South African mines, which supplied in
  1906, about 90% of the world's produce, has thrown into the shade the
  Brazilian production; but the _Bulletin_ for Feb. 1909 of the
  International Bureau of American Republics gave a very confident
  account of its future, under improved methods.

  _South Africa._---The first discovery was made in 1867 by Dr W. G.
  Atherstone, who identified as diamond a pebble obtained from a child
  in a farm on the banks of the Orange river and brought by a trader to
  Grahamstown; it was bought for £500 and displayed in the Paris
  Exhibition of that year. In 1869 a stone weighing 83½ carats was found
  near the Orange river; this was purchased by the earl of Dudley for
  £25,000 and became famous as the "Star of South Africa." A rush of
  prospectors at once took place to the banks of the Orange and Vaal
  rivers, and resulted in considerable discoveries, so that in 1870
  there was a mining camp of no less than 10,000 persons on the "River
  Diggings." In the River Diggings the mining was carried on in the
  coarse river gravels, and by the methods of the Brazilian negroes and
  of gold placer-miners. A diggers' committee limited the size of claims
  to 30 ft. square, with free access to the river bank; the gravel and
  sand were washed in cradles provided with screens of perforated metal,
  and the concentrates were sorted by hand on tables by means of an iron
  scraper.

  But towards the close of 1870 stones were found at Jagersfontein and
  at Dutoitspan, far from the Vaal river, and led to a second great rush
  of prospectors, especially to Dutoitspan, and in 1871 to what is now
  the Kimberley mine in the neighbourhood of the latter. At each of
  these spots the diamantiferous area was a roughly circular patch of
  considerable size, and in some occupied the position of one of those
  depressions or "pans" so frequent in S. Africa. These "dry diggings"
  were therefore at first supposed to be alluvial in origin like the
  river gravels; but it was soon discovered that, below the red surface
  soil and the underlying calcareous deposit, diamonds were also found
  in a layer of yellowish clay about 50 ft. thick known as "yellow
  ground." Below this again was a hard bluish-green serpentinous rock
  which was at first supposed to be barren bed-rock; but this also
  contained the precious stone, and has become famous, under the name of
  "blue ground," as the matrix of the S. African diamonds. The yellow
  ground is merely decomposed blue ground. In the Kimberley district
  five of these round patches of blue ground were found within an area
  little more than 3 m. in diameter; that at Kimberley occupying 10
  acres, that at Dutoitspan 23 acres. There were soon 50,000 workers on
  this field, the canvas camp was replaced by a town of brick and iron
  surrounded by the wooden huts of the natives, and Kimberley became an
  important centre.

  It was soon found that each mine was in reality a huge vertical funnel
  or crater descending to an unknown depth, and filled with
  diamantiferous blue ground. At first each claim was an independent pit
  31 ft. square sunk into the blue ground; the diamantiferous rock was
  hoisted by bucket and windlass, and roadways were left across the pit
  to provide access to the claims. But the roadways soon fell in, and
  ultimately haulage from the claims could only be provided by means of
  a vast system of wire ropes extending from a triple staging of
  windlasses erected round the entire edge of the mine, which had by
  this time become a huge open pit; the ropes from the upper windlasses
  extended to the centre, and those from the lower tier to the sides of
  the pit; covering the whole mass like a gigantic cobweb. (See Plate
  II. fig. 12.) The buckets of blue ground were hauled up these ropes by
  means of horse whims, and in 1875 steam winding engines began to be
  employed. By this time also improved methods in the treatment of the
  blue ground were introduced. It was carried off in carts to open
  spaces, where an exposure of some weeks to the air was found to
  pulverize the hard rock far more efficiently than the old method of
  crushing with mallets. The placer-miner's cradle and rocking-trough
  were replaced by puddling troughs stirred by a revolving comb worked
  by horse power; reservoirs were constructed for the scanty
  water-supply, bucket elevators were introduced to carry away the
  tailings; and the natives were confined in compounds. For these
  improvements co-operation was necessary; the better claims, which in
  1872 had risen from £100 to more than £4000 in value, began to be
  consolidated, and a Mining Board was introduced.


  PLATE I.

  [Illustration: FIG. 9.--DE BEERS MINE, 1874.]

  [Illustration: FIG. 10.--KIMBERLEY MINE, 1874.]

  [Illustration: FIG. 11.--DE BEERS MINE, 1873. (From photographs by C.
  Evans.)]


  PLATE II.

  [Illustration: _Fig. 12._--KIMBERLEY MINE, 1874.]

  [Illustration: _Fig. 13._--KIMBERLEY MINE, 1902. (From Photographs by
  C. Evans.)]


  In a very few years, however, the open pit mining was rendered
  impossible by the mud rushes, by the falls of the masses of barren
  rock known as "reef," which were left standing in the mine, and by
  landslips from the sides, so that in 1883, when the pit had reached a
  depth of about 400 ft., mining in the Kimberley crater had become
  almost impossible. By 1889, in the whole group of mines, Kimberley,
  Dutoitspan, De Beers and Bultfontein, open pit working was practically
  abandoned. Meanwhile mining below the bottom of the pits by means of
  shafts and underground tunnels had been commenced; but the full
  development of modern methods dates from the year 1889 when Cecil
  Rhodes and Alfred Beit, who had already secured control of the De
  Beers mine, acquired also the control of the Kimberley mine, and
  shortly afterwards consolidated the entire group in the hands of the
  De Beers Company. (See KIMBERLEY.)

  The scene of native mining was now transferred from the open pit to
  underground tunnels; the vast network of wire ropes (Plate II. fig.
  12) with their ascending and descending buckets disappeared, and with
  it the cosmopolitan crowd of busy miners working like ants at the
  bottom of the pit. In place of all this, the visitor to Kimberley
  encounters at the edge of the town only a huge crater, silent and
  apparently deserted, with no visible sign of the great mining
  operations which are conducted nearly half a mile below the surface.
  The aspect of the Kimberley pit in 1906 is shown in fig. 13 of Plate
  II., which may be compared with the section of fig. 8.

  In fig. 13, Plate II., the sequence of the basalt, shale and melaphyre
  is clearly visible on the sides of the pit; and fig. 8 shows how the
  crater or "pipe" of blue ground has penetrated these rocks and also
  the underlying quartzite. The workings at De Beers had extended into
  the still more deeply seated granite in 1906. Figure 9, Plate I.,
  shows the top of the De Beers' crater with basalt overlying the shale.
  Figure 8 also explains the modern system of mining introduced by
  Gardner Williams. A vertical shaft is sunk in the vicinity of the
  mine, and from this horizontal tunnels are driven into the pipe at
  different levels separated by intervals of 40 ft. Through the blue
  ground itself on each level a series of parallel tunnels about 120 ft.
  apart are driven to the opposite side of the pipe, and at right angles
  to these, and 36 ft. apart, another series of tunnels. When the
  tunnels reach the side of the mine they are opened upwards and
  sideways so as to form a large chamber, and the overlying mass of blue
  ground and débris is allowed to settle down and fill up the gallery.
  On each level this process is carried somewhat farther back than on
  the level below (fig. 8); material is thus continually withdrawn from
  one side of the mine and extracted by means of the rock shaft on the
  opposite side, while the superincumbent débris is continually sinking,
  and is allowed to fall deeper on the side farthest from the shaft as
  the blue ground is withdrawn from beneath it. In 1905 the main shaft
  had been sunk to a depth of 2600 ft. at the Kimberley mine.

  For the extraction and treatment of the blue ground the De Beers
  Company in its great winding and washing plant employs labour-saving
  machinery on a gigantic scale. The ground is transferred in trucks to
  the shaft where it is automatically tipped into skips holding 96 cubic
  ft. (six truck loads); these are rapidly hoisted to the surface, where
  their contents are automatically dumped into side-tipping trucks, and
  these in turn are drawn away in a continual procession by an endless
  wire rope along the tram lines leading to the vast "distributing
  floors." These are open tracts upon which the blue ground is spread
  out and left exposed to sun and rain until it crumbles and
  disintegrates, the process being hastened by harrowing with steam
  ploughs; this may require a period of three or six months, or even a
  year. The stock of blue ground on the floors at one time in 1905 was
  nearly 4,500,000 loads. The disintegrated ground is then brought back
  in the trucks and fed through perforated cylinders into the washing
  pans; the hard blue which has resisted disintegration on the floors,
  and the lumps which are too big to pass the cylindrical sieves, are
  crushed before going to the pans. These are shallow cylindrical
  troughs containing muddy water in which the diamonds and other heavy
  minerals (concentrates) are swept to the rim by revolving toothed
  arms, while the lighter stuff escapes near the centre of the pan. The
  concentrates are then passed over sloping tables (pulsator) and shaken
  to and fro under a stream of water which effects a second
  concentration of the heaviest material.

  Until recently the final separation of the diamond from the
  concentrates was made by hand picking, but even this has now been
  replaced by machinery, owing to the remarkable discovery that a
  greased surface will hold a diamond while allowing the other heavy
  minerals to pass over it. The concentrates are washed down a sloping
  table of corrugated iron which is smeared with grease, and it is found
  that practically all the diamonds adhere to the table, and the other
  minerals are washed away. At the large and important Premier mine in
  the Transvaal the Elmore process, used in British Columbia and in
  Wales for the separation of metallic ores, has been also introduced.
  In the Elmore process oil is employed to float off the materials which
  adhere to it, while the other materials remain in the water, the oil
  being separated from the water by centrifugal action. The other
  minerals found in the concentrates are pebbles and fragments of
  pyrope, zircon, cyanite, chrome-diopside, enstatite, a green pyroxene,
  mica, ilmenite, magnetite, chromite, hornblende, olivine, barytes,
  calcite and pyrites.

  In all the S. African mines the diamonds are not only crystals of
  various weights from fractions of a carat to 150 carats, but also
  occur as microscopic crystals disseminated through the blue ground. In
  spite of this, however, the average yield in the profitable mines is
  only from 0.2 carat to 0.6 carat per load of 1600 lb., or on an average
  about 1½ grs. per ton. The annual output of diamonds from the De Beers
  mines was valued in 1906 at nearly £5,000,000; the value per carat
  ranging from about 35s. to 70s.

  [Illustration: SECTION OF KIMBERLEY MINE LOOKING EAST

  From Gardner Williams's _Diamond Mines of South Africa_. FIG. 8.]

  Pipes similar to those which surround Kimberley have been found in
  other parts of S. Africa. One of the best known is that of
  Jagersfontein, which was really the first of the dry diggings
  (discovered in 1870). This large mine is near Fauresmith and 80 m. to
  the south of Kimberley. In 1905 the year's production from the Orange
  River Colony mines was more than 320,000 carats, valued at £938,000.
  But by far the largest of all the pipes hitherto discovered is the
  Premier mine in the Transvaal, about 300 m. to the east of Kimberley.
  This was discovered in 1902 and occupies an area of about 75 acres. In
  1906 it was being worked as a shallow open mine; but the description
  of the Kimberley methods given above is applicable to the washing
  plant at that time being introduced into the Premier mine upon a very
  large scale. Comparatively few of the pipes which have been discovered
  are at all rich in diamonds, and many are quite barren; some are
  filled with "hard blue" which even if diamantiferous may be too
  expensive to work.

  The most competent S. African geologists believe all these remarkable
  pipes to be connected with volcanic outbursts which occurred over the
  whole of S. Africa during the Cretaceous period (after the deposition
  of the Stormberg beds), and drilled these enormous craters through all
  the later formations. With the true pipes are associated dykes and
  fissures also filled with diamantiferous blue ground. It is only in
  the more northerly part of the country that the pipes are filled with
  blue ground (or "kimberlite"), and that they are diamantiferous; but
  over a great part of Cape Colony have been discovered what are
  probably similar pipes filled with agglomerates, breccias and tuffs,
  and some with basic lavas; one, in particular, in the Riversdale
  Division near the southern coast, being occupied by a melilite-basalt.
  It is quite clear that the occurrence of the diamond in the S. African
  pipes is quite different from the occurrences in alluvial deposits
  which have been described above. The question of the origin of the
  diamond in S. Africa and elsewhere is discussed below.

  The River Diggings on the Vaal river are still worked upon a small
  scale, but the production from this source is so limited that they are
  of little account in comparison with the mines in the blue ground. The
  stones, however, are good; since they differ somewhat from the
  Kimberley crystals it is probable that they were not derived from the
  present pipes. Another S. African locality must be mentioned;
  considerable finds were reported in 1905 and 1906 from gravels at
  Somabula near Gwelo in Rhodesia where the diamond is associated with
  chrysoberyl, corundum (both sapphire and ruby), topaz, garnet,
  ilmenite, staurolite, rutile, with pebbles of quartz, granite,
  chlorite-schist, &c. Diamond has also been reported from kimberlite
  "pipes" in Rhodesia.

  _Other Localities._--In addition to the South American localities
  mentioned above, small diamonds have also been mined since their
  discovery in 1890 on the river Mazaruni in British Guiana, and finds
  have been reported in the gold washings of Dutch Guiana. Borneo has
  possessed a diamond industry since the island was first settled by the
  Malays; the references in the works of Garcia de Orta, Linschoten, De
  Boot, De Laet and others, to Malacca as a locality relate to Borneo.
  The large Borneo stone, over 360 carats in weight, known as the Matan,
  is in all probability not a diamond. The chief mines are situated on
  the river Kapuas in the west and near Bandjarmassin in the south-east
  of the island, and the alluvial deposits in which they occur are
  worked by a small number of Chinese and Malays. Australia has yielded
  diamonds in alluvial deposits near Bathurst (where the first discovery
  was made in 1851) and Mudgee in New South Wales, and also near Bingara
  and Inverell in the north of the colony. At Mount Werong a stone
  weighing 29 carats was found in 1905. At Ruby Hill near Bingara they
  were found in a breccia filling a volcanic pipe. At Ballina, in New
  England, diamonds have been found in the sea sand. Other Australian
  localities are Echunga in South Australia; Beechworth, Arena and
  Melbourne in Victoria; Freemantle and Nullagine in Western Australia;
  the Palmer and Gilbert rivers in Queensland. These have been for the
  most part discoveries in alluvial deposits of the goldfields, and the
  stones were small. In Tasmania also diamonds have been found in the
  Corinna goldfields. Europe has produced few diamonds. Humboldt
  searched for them in the Urals on account of the similarity of the
  gold and platinum deposits to those of Brazil, and small diamonds were
  ultimately found (1829) in the gold washings of Bissersk, and later at
  Ekaterinburg and other spots in the Urals. In Lapland they have been
  found in the sands of the Pasevig river. Siberia has yielded isolated
  diamonds from the gold washings of Yenisei. In North America a few
  small stones have been found in alluvial deposits, mostly auriferous,
  in Georgia, N. and S. Carolina, Kentucky, Virginia, Tennessee,
  Wisconsin, California, Oregon and Indiana. A crystal weighing 23¾
  carats was found in Virginia in 1855, and one of 21¼ carats in
  Wisconsin in 1886. In 1906 a number of small diamonds were discovered
  in an altered peridotite somewhat resembling the S. African blue
  ground, at Murfreesboro, Pike county, Arkansas. Considerable interest
  attaches to the diamonds found in Wisconsin, Michigan and Ohio near
  the Great Lakes, for they are here found in the terminal moraines of
  the great glacial sheet which is supposed to have spread southwards
  from the region of Hudson Bay; several of the drift minerals of the
  diamantiferous region of Indiana have been identified as probably of
  Canadian origin; no diamonds have however yet been found in the
  intervening country of Ontario. A rock similar to the blue ground of
  Kimberley has been found in the states of Kentucky and New York. The
  occurrence of diamond in meteorites is described below.

  _Origin of the Diamond in Nature._--It appears from the foregoing
  account that at most localities the diamond is found in alluvial
  deposits probably far from the place where it originated. The minerals
  associated with it do not afford much clue to the original conditions;
  they are mostly heavy minerals derived from the neighbouring rocks, in
  which the diamond itself has not been observed. Among the commonest
  associates of the diamond are quartz, topaz, tourmaline, rutile,
  zircon, magnetite, garnet, spinel and other minerals which are common
  accessory constituents of granite, gneiss and the crystalline schists.
  Gold (also platinum) is a not infrequent associate, but this may only
  mean that the sands in which the diamond is found have been searched
  because they were known to be auriferous; also that both gold and
  diamond are among the most durable of minerals and may have survived
  from ancient rocks of which other traces have been lost.

  The localities at which the diamond has been supposed to occur in its
  original matrix are the following:--at Wajra Karur, in the Cuddapah
  district, India, M. Chaper found diamond with corundum in a decomposed
  red pegmatite vein in gneiss. At S[=a]o João da Chapada, in Minas
  Geraes, diamonds occur in a clay interstratified with the itacolumite,
  and are accompanied by sharp crystals of rutile and haematite in the
  neighbourhood of decomposed quartz veins which intersect the
  itacolumite. It has been suggested that these three minerals were
  originally formed in the quartz veins. In both these occurrences the
  evidence is certainly not sufficient to establish the presence of an
  original matrix. At Inverell in New South Wales a diamond (1906) has
  been found embedded in a hornblende diabase which is described as a
  dyke intersecting the granite. Finally there is the remarkable
  occurrence in the blue ground of the African pipes.

  There has been much controversy concerning the nature and origin of
  the blue ground itself; and even granted that (as is generally
  believed) the blue ground is a much serpentinized volcanic breccia
  consisting originally of an olivine-bronzite-biotite rock (the
  so-called kimberlite), it contains so many rounded and angular
  fragments of various rocks and minerals that it is difficult to say
  which of them may have belonged to the original rock, and whether any
  were formed _in situ_, or were brought up from below as inclusions.
  Carvill Lewis believed the blue ground to be true eruptive rock, and
  the carbon to have been derived from the bituminous shales of which it
  contains fragments. The Kimberley shales, which are penetrated by the
  De Beers group of pipes, were, however, certainly not the source of
  the carbon at the Premier (Transvaal) mine, for at this locality the
  shales do not exist. The view that the diamond may have crystallized
  out from solution in its present matrix receives some support from the
  experiments of W. Luzi, who found that it can be corroded by the
  solvent action of fused blue ground; from the experiments of J.
  Friedländer, who obtained diamond by dissolving graphite in fused
  olivine; and still more from the experiments of R. von Hasslinger and
  J. Wolff, who have obtained it by dissolving graphite in a fused
  mixture of silicates having approximately the composition of the blue
  ground. E. Cohen, who regarded the pipes as of the nature of a mud
  volcano, and the blue ground as a kimberlite breccia altered by
  hydrothermal action, thought that the diamond and accompanying
  minerals had been brought up from deep-seated crystalline schists.
  Other authors have sought the origin of the diamond in the action of
  the hydrated magnesian silicates on hydrocarbons derived from
  bituminous schists, or in the decomposition of metallic carbides.

  Of great scientific interest in this connexion is the discovery of
  small diamonds in certain meteorites, both stones and irons; for
  example, in the stone which fell at Novo-Urei in Penza, Russia, in
  1886, in a stone found at Carcote in Chile, and in the iron found at
  Cañon Diablo in Arizona. Graphitic carbon in cubic form (cliftonite)
  has also been found in certain meteoric "irons," for example in those
  from Magura in Szepes county, Hungary, and Youndegin near York in
  Western Australia. The latter is now generally believed to be altered
  diamond. The fact that H. Moissan has produced the diamond
  artificially, by allowing dissolved carbon to crystallize out at a
  high temperature and pressure from molten iron, coupled with the
  occurrence in meteoric iron, has led Sir William Crookes and others to
  conclude that the mineral may have been derived from deep-seated iron
  containing carbon in solution (see the article GEM, ARTIFICIAL). Adolf
  Knop suggested that this may have first yielded hydrocarbons by
  contact with water, and that from these the crystalline diamond has
  been formed. The meteoric occurrence has even suggested the fanciful
  notion that all diamonds were originally derived from meteorites. The
  meteoric iron of Arizona, some of which contains diamond, is actually
  found in and about a huge crater which is supposed by some to have
  been formed by an immense meteorite penetrating the earth's crust.

  It is, at any rate, established that carbon can crystallize as diamond
  from solution in iron, and other metals; and it seems that high
  temperature and pressure and the absence of oxidizing agents are
  necessary conditions. The presence of sulphur, nickel, &c., in the
  iron appears to favour the production of the diamond. On the other
  hand, the occurrence in meteoric stones, and the experiments mentioned
  above, show that the diamond may also crystallize from a basic magma,
  capable of yielding some of the metallic oxides and ferro-magnesian
  silicates; a magma, therefore, which is not devoid of oxygen. This is
  still more forcibly suggested by the remarkable eclogite boulder found
  in the blue ground of the Newlands mine, not far from the Vaal river,
  and described by T. G. Bonney. The boulder is a crystalline rock
  consisting of pyroxene (chrome-diopside), garnet, and a little
  olivine, and is studded with diamond crystals; a portion of it is
  preserved in the British Museum (Natural History). In another eclogite
  boulder, diamond was found partly embedded in pyrope. Similar boulders
  have also been found in the blue ground elsewhere. Specimens of pyrope
  with attached or embedded diamond had previously been found in the
  blue ground of the De Beers mines. In the Newlands boulder the
  diamonds have the appearance of being an original constituent of the
  eclogite. It seems therefore that a holocrystalline pyroxene-garnet
  rock may be one source of the diamond found in blue ground. On the
  other hand many tons of the somewhat similar eclogite in the De Beers
  mine have been crushed and have not yielded diamond. Further, the
  ilmenite, which is the most characteristic associate of the diamond in
  blue ground, and other of the accompanying minerals, may have come
  from basic rocks of a different nature.

  The Inverell occurrence may prove to be another example of diamond
  crystallized from a basic rock.

  In both occurrences, however, there is still the possibility that the
  eclogite or the basalt is not the original matrix, but may have caught
  up the already formed diamond from some other matrix. Some regard the
  eclogite boulders as derived from deep-seated crystalline rocks,
  others as concretions in the blue ground.

  None of the inclusions in the diamond gives any clue to its origin;
  diamond itself has been found as an inclusion, as have also black
  specks of some carbonaceous materials. Other black specks have been
  identified as haematite and ilmenite; gold has also been found; other
  included minerals recorded are rutile, topaz, quartz, pyrites,
  apophyllite, and green scales of chlorite (?). Some of these are of
  very doubtful identification; others (e.g. apophyllite and chlorite)
  may have been introduced along cracks. Some of the fibrous inclusions
  were identified by H. R. Göppert as vegetable structures and were
  supposed to point to an organic origin, but this view is no longer
  held. Liquid inclusions, some of which are certainly carbon dioxide,
  have also been observed.

  Finally, then, both experiment and the natural occurrence in rocks and
  meteorites suggest that diamond may crystallize not only from iron but
  also from a basic silicate magma, possibly from various rocks
  consisting of basic silicates. The blue ground of S. Africa may be
  the result of the serpentinization of several such rocks, and
  although now both brecciated and serpentinized some of these may have
  been the original matrix. A circumstance often mentioned in support of
  this view is the fact that the diamonds in one pipe generally differ
  somewhat in character from those of another, even though they be near
  neighbours.

_History._--All the famous diamonds of antiquity must have been Indian
stones. The first author who described the Indian mines at all fully was
the Portuguese, Garcia de Orta (1565), who was physician to the viceroy
of Goa. Before that time there were only legendary accounts like that of
Sindbad's "Valley of the Diamonds," or the tale of the stones found in
the brains of serpents. V. Ball thinks that the former legend originated
in the Indian practice of sacrificing cattle to the evil spirits when a
new mine is opened; birds of prey would naturally carry off the flesh,
and might give rise to the tale of the eagles carrying diamonds adhering
to the meat.

The following are some of the most famous diamonds of the world:--

A large stone found in the Golconda mines and said to have weighed 787
carats in the rough, before being cut by a Venetian lapidary, was seen
in the treasury of Aurangzeb in 1665 by Tavernier, who estimated its
weight after cutting as 280 (?) carats, and described it as a rounded
rose-cut-stone, tall on one side. The name _Great Mogul_ has been
frequently applied to this stone. Tavernier states that it was the
famous stone given to Shah Jahan by the emir Jumla. The _Orloff_, stolen
by a French soldier from the eye of an idol in a Brahmin temple, stolen
again from him by a ship's captain, was bought by Prince Orloff for
£90,000, and given to the empress Catharine II. It weighs 194¾ carats,
is of a somewhat yellow tinge, and is among the Russian crown jewels.
The _Koh-i-nor_, which was in 1739 in the possession of Nadir Shah, the
Persian conqueror, and in 1813 in that of the raja of Lahore, passed
into the hands of the East India Company and was by them presented to
Queen Victoria in 1850. It then weighed 186-1/16 carats, but was recut in
London by Amsterdam workmen, and now weighs 106-1/16 carats. There has
been much discussion concerning the possibility of this stone and the
Orloff being both fragments of the Great Mogul. The Mogul Baber in his
memoirs (1526) relates how in his conquest of India he captured at Agra
the great stone weighing 8 mishkals, or 320 ratis, which may be
equivalent to about 187 carats. The Koh-i-nor has been identified by
some authors with this stone and by others with the stone seen by
Tavernier. Tavernier, however, subsequently described and sketched the
diamond which he saw as shaped like a bisected egg, quite different
therefore from the Koh-i-nor. Nevil Story Maskelyne has shown reason for
believing that the stone which Tavernier saw was really the Koh-i-nor
and that it is identical with the great diamond of Baber; and that the
280 carats of Tavernier is a misinterpretation on his part of the Indian
weights. He suggests that the other and larger diamond of antiquity
which was given to Shah Jahan may be one which is now in the treasury of
Teheran, and that this is the true Great Mogul which was confused by
Tavernier with the one he saw. (See Ball, Appendix I. to Tavernier's
_Travels_ (1889); and Maskelyne, _Nature_, 1891, 44, p. 555.).

The _Regent_ or _Pitt_ diamond is a magnificent stone found in either
India or Borneo; it weighed 410 carats and was bought for £20,400 by
Pitt, the governor of Madras; it was subsequently, in 1717, bought for
£80,000 (or, according to some authorities, £135,000) by the duke of
Orleans, regent of France; it was reduced by cutting to 1361-4/16 carats;
was stolen with the other crown jewels during the Revolution, but was
recovered and is still in France. The _Akbar Shah_ was originally a
stone of 116 carats with Arabic inscriptions engraved upon it; after
being cut down to 71 carats it was bought by the gaikwar of Baroda for
£35,000. The _Nizam_, now in the possession of the nizam of Hyderabad,
is supposed to weigh 277 carats; but it is only a portion of a stone
which is said to have weighed 440 carats before it was broken. The
_Great Table_, a rectangular stone seen by Tavernier in 1642 at
Golconda, was found by him to weigh 242-3/16 carats; Maskelyne regards it
as identical with the _Darya-i-nur_, which is also a rectangular stone
weighing about 186 carats in the possession of the shah of Persia.
Another stone, the _Taj-e-mah_, belonging to the shah, is a pale rose
pear-shaped stone and is said to weigh 146 carats.

Other famous Indian diamonds are the following:--The _Sancy_, weighing
531-2/16 carats, which is said to have been successively the property of
Charles the Bold, de Sancy, Queen Elizabeth, Henrietta Maria, Cardinal
Mazarin, Louis XIV.; to have been stolen with the Pitt during the French
Revolution; and subsequently to have been the property of the king of
Spain, Prince Demidoff and an Indian prince. The _Nassak_, 78-5/8 carats,
the property of the duke of Westminster. The _Empress Eugénie_, 51
carats, the property of the gaikwar of Baroda. The _Pigott_, 49
carats(?), which cannot now be traced. The _Pasha_, 40 carats. The
_White Saxon_, 48¾ carats. The _Star of Este_, 251-3/32 carats.

Coloured Indian diamonds of large size are rare; the most famous are:--a
beautiful blue brilliant, 67-2/16 carats, cut from a stone weighing
112-3/16 carats brought to Europe by Tavernier. It was stolen from the
French crown jewels with the Regent and was never recovered. The _Hope_,
44¼ carats, has the same colour and is probably a portion of the missing
stone: it was so-called as forming part of the collection of H. T. Hope
(bought for £18,000), and was sold again in 1906 (resold 1909). Two
other blue diamonds are known, weighing 13¾ and 1¾ carats, which may
also be portions of the French diamond. The _Dresden Green_, one of the
Saxon crown jewels, 40 carats, has a fine apple-green colour. The
_Florentine_, 133-1/5 carats, one of the Austrian crown jewels, is a very
pale yellow.

The most famous Brazilian stones are:--The _Star of the South_, found in
1853, when it weighed 254½ carats and was sold for £40,000; when cut it
weighed 125 carats and was bought by the gaikwar of Baroda for £80,000.
Also a diamond belonging to Mr Dresden, 119 carats before, and 76½
carats after cutting.

Many large stones have been found in South Africa; some are yellow but
some are as colourless as the best Indian or Brazilian stones. The most
famous are the following:--the _Star of South Africa_, or _Dudley_,
mentioned above, 83½ carats rough, 46½ carats cut. The _Stewart_, 288-3/8
carats rough, 120 carats cut. Both these were found in the river
diggings. The _Porter Rhodes_ from Kimberley, of the finest water,
weighed about 150 carats. The _Victoria_, 180 carats, was cut from an
octahedron weighing 457½ carats, and was sold to the nizam of Hyderabad
for £400,000. The _Tiffany_, a magnificent orange-yellow stone, weighs
125½ carats cut. A yellowish octahedron found at De Beers weighed 428½
carats, and yielded a brilliant of 288½ carats. Some of the finest and
largest stones have come from the Jagersfontein mine; one, the
_Jubilee_, found in 1895, weighed 640 carats in the rough and 239 carats
when cut. Until 1905 the largest known diamond in the world was the
_Excelsior_, found in 1893 at Jagersfontein by a native while loading a
truck. It weighed 971 carats, and was ultimately cut into ten stones
weighing from 68 to 13 carats. But all previous records were surpassed
in 1905 by a magnificent stone more than three times the size of any
known diamond, which was found in the yellow ground at the newly
discovered Premier mine in the Transvaal. This extraordinary diamond
weighed 3025¾ carats (11/3 lb.) and was clear and water white; the
largest of its surfaces appeared to be a cleavage plane, so that it
might be only a portion of a much larger stone. It was known as the
_Cullinan Diamond_. This stone was purchased by the Transvaal government
in 1907 and presented to King Edward VII. It was sent to Amsterdam to be
cut, and in 1908 was divided into nine large stones and a number of
small brilliants. The four largest stones weigh 516½ carats, 309-3/16
carats, 92 carats and 62 carats respectively. Of these the first and
second are the largest brilliants in existence. All the stones are
flawless and of the finest quality.

  BIBLIOGRAPHY.--Boetius de Boot, _Gemmarum et lapidum historia_ (1609);
  D. Jeffries, _A Treatise on Diamonds and Pearls_ (1757); J. Mawe,
  _Travels in the Interior of Brazil_ (1812); _Treatise on Diamonds and
  Precious Stones_ (1813): Pinder, _De adamante_ (1829); Murray, _Memoir
  on the Nature of the Diamond_ (1831); C. Zerenner, _De adamante
  dissertatio_ (1850); H. Emanuel, _Diamonds and Precious Stones_
  (1865); A. Schrauf, _Edelsteinkunde_ (1869); N. Jacobs and N.
  Chatrian, _Monographie du diamant_ (1880); V. Ball, _Geology of India_
  (1881); C. W. King, _The Natural History of Precious Stones_ _and
  Precious Metals_ (1883); M. E. Boutan, _Le Diamant_ (1886); S. M.
  Burnham, _Precious Stones in Nature, Art and Literature_ (1887); P.
  Groth, _Grundriss der Edelsteinkunde_ (1887); A. Liversidge, _The
  Minerals of New South Wales_ (1888); _Tavernier's Travels in India_,
  translated by V. Ball (1889); E. W. Streeter, _The Great Diamonds of
  the World_ (1896); H. C. Lewis, _The Genesis and Matrix of the
  Diamond_ (1897); L. de Launay, _Les Diamants du Cap_ (1897); C.
  Hintze, _Handbuch der Mineralogie_ (1898); E. W. Streeter, _Precious
  Stones and Gems_ (6th ed., 1898); Dana, _System of Mineralogy_ (1899);
  Kunz and others, _The Production of Precious Stones_ (in annual,
  _Mineral Resources of the United States_); M. Bauer, _Precious Stones_
  (trans. L. J. Spencer, 1904); A. W. Rogers, _An Introduction to the
  Geology of Cape Colony_ (1905); Gardner F. Williams, _The Diamond
  Mines of South Africa_ (revised edition, 1906); George F. Kunz,
  "Diamonds, a study of their occurrence in the United States, with
  descriptions and comparisons of those from all known localities" (U.S.
  Geol. Survey, 1909); P. A. Wagner, _Die Diamantführenden Gesteine
  Südafrikas_ (1909).

  Among papers in scientific periodicals may be mentioned articles by
  Adler, Ball, Baumhauer, Beck, Bonney, Brewster, Chaper, Cohen,
  Crookes, Daubrée, Derby, Des Cloizeaux, Doelter, Dunn, Flight,
  Friedel, Gorceix, Gürich, Goeppert, Harger, Hudleston, Hussak,
  Jannettaz, Jeremejew, de Launay, Lewis, Maskelyne, Meunier, Moissan,
  Molengraaff, Moulle, Rose, Sadebeck, Scheibe, Stelzner, Stow. See
  generally Hintze's _Handbuch der Mineralogie_. (H. A. MI.)


FOOTNOTE:

  [1] Diamonds are invariably weighed in carats and in ½, ¼, 1/8, 1/16,
    1/32, 1/64 of a carat. One (English) carat = 3.17 grains = .2054
    gram. One ounce = 151½ carats. (See CARAT.)



DIAMOND NECKLACE, THE AFFAIR OF THE, a mysterious incident at the court
of Louis XVI. of France, which involved the queen Marie Antoinette. The
Parisian jewellers Boehmer and Bassenge had spent some years collecting
stones for a necklace which they hoped to sell to Madame Du Barry, the
favourite of Louis XV., and after his death to Marie Antoinette. In 1778
Louis XVI. proposed to the queen to make her a present of the necklace,
which cost 1,600,000 livres. But the queen is said to have refused it,
saying that the money would be better spent equipping a man-of-war.
According to others, Louis XVI. himself changed his mind. After having
vainly tried to place the necklace outside of France, the jewellers
attempted again in 1781 to sell it to Marie Antoinette after the birth
of the dauphin. It was again refused, but it was evident that the queen
regretted not being able to acquire it.

At that time there was a personage at the court whom Marie Antoinette
particularly detested. It was the cardinal Louis de Rohan, formerly
ambassador at Vienna, whence he had been recalled in 1774, having
incurred the queen's displeasure by revealing to the empress Maria
Theresa the frivolous actions of her daughter, a disclosure which
brought a maternal reprimand, and for having spoken lightly of Maria
Theresa in a letter of which Marie Antoinette learned the contents.
After his return to France the cardinal was anxious to regain the favour
of the queen in order to obtain the position of prime minister. In March
1784 he entered into relations with a certain Jeanne de St Remy de
Valois, a descendant of a bastard of Henry II., who after many
adventures had married a _soi-disant_ comte de Lamotte, and lived on a
small pension which the king granted her. This adventuress soon gained
the greatest ascendancy over the cardinal, with whom she had intimate
relations. She persuaded him that she had been received by the queen and
enjoyed her favour; and Rohan resolved to use her to regain the queen's
good will. The comtesse de Lamotte assured the cardinal that she was
making efforts on his behalf, and soon announced to him that he might
send his justification to Marie Antoinette. This was the beginning of a
pretended correspondence between Rohan and the queen, the adventuress
duly returning replies to Rohan's notes, which she affirmed to come from
the queen. The tone of the letters became very warm, and the cardinal,
convinced that Marie Antoinette was in love with him, became ardently
enamoured of her. He begged the countess to obtain a secret interview
for him with the queen, and a meeting took place in August 1784 in a
grove in the garden at Versailles between him and a lady whom the
cardinal believed to be the queen herself. Rohan offered her a rose, and
she promised him that she would forget the past. Later a certain Marie
Lejay (renamed by the comtesse "Baronne Gay d'Oliva," the last word
being apparently an anagram of Valoi), who resembled Marie Antoinette,
stated that she had been engaged to play the role of queen in this
comedy. In any case the countess profited by the cardinal's conviction
to borrow from him sums of money destined ostensibly for the queen's
works of charity. Enriched by these, the countess was able to take an
honourable place in society, and many persons believed her relations
with Marie Antoinette, of which she boasted openly and unreservedly, to
be genuine. It is still an unsettled question whether she simply
mystified people, or whether she was really employed by the queen for
some unknown purpose, perhaps to ruin the cardinal. In any case the
jewellers believed in the relations of the countess with the queen, and
they resolved to use her to sell their necklace. She at first refused
their commission, then accepted it. On the 21st of January 1785 she
announced that the queen would buy the necklace, but that not wishing to
treat directly, she left the affair to a high personage. A little while
later Rohan came to negotiate the purchase of the famous necklace for
the 1,600,000 livres, payable in instalments. He said that he was
authorized by the queen, and showed the jewellers the conditions of the
bargain approved in the handwriting of Marie Antoinette. The necklace
was given up. Rohan took it to the countess's house, where a man, in
whom Rohan believed he recognized a valet of the queen, came to fetch
it. Madame de Lamotte had told the cardinal that Marie Antoinette would
make him a sign to indicate her thanks, and Rohan believed that she did
make him a sign. Whether it was so, or merely chance or illusion, no one
knows. But it is certain that the cardinal, convinced that he was acting
for the queen, had engaged the jewellers to thank her; that Boehmer and
Bassenge, before the sale, in order to be doubly sure, had sent word to
the queen of the negotiations in her name; that Marie Antoinette had
allowed the bargain to be concluded, and that after she had received a
letter of thanks from Boehmer, she had burned it. Meanwhile the "comte
de Lamotte" appears to have started at once for London, it is said with
the necklace, which he broke up in order to sell the stones.

When the time came to pay, the comtesse de Lamotte presented the
cardinal's notes; but these were insufficient, and Boehmer complained to
the queen, who told him that she had received no necklace and had never
ordered it. She had the story of the negotiations repeated for her. Then
followed a _coup de théâtre_. On the 15th of August 1785, Assumption
day, when the whole court was awaiting the king and queen in order to go
to the chapel, the cardinal de Rohan, who was preparing to officiate,
was arrested and taken to the Bastille. He was able, however, to destroy
the correspondence exchanged, as he thought, with the queen, and it is
not known whether there was any connivance of the officials, who did not
prevent this, or not. The comtesse de Lamotte was not arrested until the
18th of August, after having destroyed her papers. The police set to
work to find all her accomplices, and arrested the girl Oliva and a
certain Reteaux de Villette, a friend of the countess, who confessed
that he had written the letters given to Rohan in the queen's name, and
had imitated her signature on the conditions of the bargain. The famous
charlatan Cagliostro was also arrested, but it was recognized that he
had taken no part in the affair. The cardinal de Rohan accepted the
parlement of Paris as judges. A sensational trial resulted (May 31,
1786) in the acquittal of the cardinal, of the girl Oliva and of
Cagliostro. The comtesse de Lamotte was condemned to be whipped, branded
and shut up in the Salpetrière. Her husband was condemned, in his
absence, to the galleys for life. Villette was banished.

Public opinion was much excited by this trial. It is generally believed
that Marie Antoinette was stainless in the matter, that Rohan was an
innocent dupe, and that the Lamottes deceived both for their own ends.
People, however, persisted in the belief that the queen had used the
countess as an instrument to satisfy her hatred of the cardinal de
Rohan. Various circumstances fortified this belief, which contributed to
render Marie Antoinette very unpopular--her disappointment at Rohan's
acquittal, the fact that he was deprived of his charges and exiled to
the abbey of la Chaise-Dieu, and finally the escape of the comtesse de
Lamotte from the Salpetrière, with the connivance, as people believed,
of the court. The adventuress, having taken refuge abroad, published
_Mémoires_ in which she accused the queen. Her husband also wrote
_Mémoires_, and lived until 1831, after having, it is said, received
subsidies from Louis XVIII.

  See M. Tourneux, _Marie Antoinette devant l'histoire: Essai
  bibliographique_ (2nd ed., Paris, 1901); Émile Campardon, _Marie
  Antoinette et le procès du collier_ (Paris, 1863); P. Audebert,
  _L'Affaire du collier de la reine, d'après la correspondance inédite
  du chevalier de Pujol_ (Rouen, 1901); F. d'Albini, _Marie Antoinette
  and the Diamond Necklace from another Point of View_ (London, 1900);
  Funck-Brentano, _L'Affaire du collier_ (1903); A. Lang, _Historical
  Mysteries_ (1904). Carlyle's essay on _The Diamond Necklace_ (first
  published in 1837 in _Fraser's Magazine_) is of historical literary
  interest.



DIANA, in Roman mythology, an old Italian goddess, in later times
identified with the Greek Artemis (q.v.). That she was originally an
independent Italian deity is shown by her name, which is the feminine
form of Janus (= Dianus). She is essentially the goddess of the moon and
light generally, and presides over wood, plain and water, the chase and
war. As the goddess of childbirth, she was known, like Juno, by the name
of Lucina, the "bringer to light." As the moon-goddess she was also
identified with Hecate, and invoked as "three-formed" in reference to
the phases of the moon. Her most celebrated shrine was in a grove at
Aricia (whence her title of Nemorensis) near the modern lake of Nemi.
Here she was worshipped side by side with a male deity Virbius, a god of
the forest and the chase. This Virbius was subsequently identified with
Hippolytus, the favourite of Artemis, who was said to have been brought
to life by Aesculapius and conducted by Diana to Aricia (Ovid, _Fasti_,
iii. 263, vi. 731, _Metam._ xv. 497; Virgil, _Aeneid_, vii. 761). A
barbarous custom, perhaps reminiscent of human sacrifice once offered to
her, prevailed in connexion with her ritual here; her priest, called
_Rex Nemorensis_, who was a runaway slave, was obliged to qualify for
office by slaying his predecessor in single combat (Strabo v. p. 239;
Suetonius, _Caligula_, 35). This led to the identification of Diana with
the Tauric Artemis, whose image was said to have been removed by Orestes
to the grove of Aricia (see ARICINI).

After the destruction of Alba Longa this grove was for a long time the
united sanctuary of the neighbouring Latin and Rutulian cities, until at
last it was extinguished beneath the supremacy of Rome. The festival of
the goddess was on the ides (13th) of August, the full moon of the hot
season. She was worshipped with torches, her aid was sought by women
seeking a happy deliverance in childbirth, and many votive offerings
have been found on the site. The worship of Diana was brought to Rome by
Latin plebeians, and hence she was regarded as the protectress of the
lower classes, and especially of slaves. In accordance with this, her
most important temple was that on the Aventine, the chief seat of the
plebeians, founded by Servius Tullius, originally as a sanctuary of the
Latin league (Dion. Halic. iv. 26). No man was allowed to enter the
temple, and on the day of its dedication (August 13) the slaves kept
holiday (Plutarch, _Quaest. Rom._ 100). This Diana was identified with
the sister of Apollo, and at the secular games she was worshipped simply
as Artemis. Another celebrated sanctuary of Diana was that on the slopes
of Mount Tifata near Capua (where she was worshipped under the name of
Tifatina), a sanctuary specially favoured by Sulla and Vespasian. As
Noctiluca ("giving light by night") she had a sanctuary on the Palatine
which was kept illuminated throughout the night (Varro, _L.L._ v. 68).
On the Nemi priesthood see J. G. Frazer, _Golden Bough_.



DIANA MONKEY, a West African representative of the guenon monkeys taking
its name, _Cercopithecus diana_, from the presence of a white crescent
on the forehead; another characteristic feature being the pointed white
beard. The general colour of the fur is greyish, with a deep tinge of
chestnut from the middle of the back to the root of the tail. Together
with _C. neglectus_ of East and Central Africa, _C. ignitus_ of Liberia,
and _C. roloway_ of the Gold Coast, the diana represents the special
subgenus of guenons known as _Pogonocebus_. Although the diana monkey is
commonly seen in menageries, little is known of its habits in the wild
state.



DIANE DE FRANCE (1538-1619), duchess of Montmorency and Angoulême, was
the natural daughter of Henry II. of France and a young Piedmontese,
Filippe Duc. The constable de Montmorency went so far as to assert that
of all the children of Henry II. Diane was the only one who resembled
him. Catherine de' Medici was greatly incensed at this affront, and took
her revenge by having the constable disgraced on the death of Henry II.
Brantôme is loud in praise of Diane. She was a perfect horsewoman and
dancer, played several musical instruments, knew Spanish and Italian,
and "estoit très belle de visage et de taille." Legitimated in 1547, she
was married in 1553 to Horace Farnese, second son of the duke of Parma,
but her husband was killed soon afterwards at the siege of Hesdin. In
order to assure his position, the constable de Montmorency wished to
marry her to his eldest son, Francis. This was a romantic adventure, for
Francis had clandestinely married Mademoiselle de Piennes. The constable
dissolved this union, and after lengthy negotiations obtained the
dispensation of the pope. On the 3rd of May 1559 Francis married Diane.
A wise and moderate woman, Diane undoubtedly helped to make Francis de
Montmorency one of the leaders of the party of the _politiques_. Again a
widow in 1579, she had some influence at the court of Henry III., and
negotiated his reconciliation with Henry of Navarre (1588). She retained
her influence in the reign of Henry IV., conveyed the bodies of
Catherine de' Medici and Henry III. to St Denis, and died in 1619 at her
hôtel of Angoulême.

  See Brantôme, ed. by Lalanne, in the _Coll de la société d'histoire de
  France_, vol. viii. (1875); J. de Thou, _Historia sui temporis..._
  (1733); Matthieu de Morgues, _Oraison funèbre de Diane de France_
  (Paris, 1619).



DIANE DE POITIERS (1499-1566), duchess of Valentinois, and mistress of
Henry II. of France, was the daughter of Jean de Poitiers, seigneur de
St Vallier, who came of an old family of Dauphiné. In 1515 she married
Louis de Brézé, grand seneschal of Normandy, by whom she had two
daughters. She became a widow in 1533, but soon replaced her husband by
a more illustrious lover, the king's second son, Henry, who became
dauphin in 1536. Although he was ten years younger than Diane, she
inspired the young prince with a profound passion, which lasted until
his death. The accession of Henry II. in 1547 was also the accession of
Diane: she was virtual queen, while Henry's lawful wife, Catherine de'
Medici, lived in comparative obscurity. The part Diane played, however,
must not be exaggerated. More rapacious than ambitious, she concerned
herself little with government, but devoted her energies chiefly to
augmenting her income, and providing for her family and friends. Henry
was the most prodigal of lovers, and gave her all rights over the duchy
of Valentinois. Although she showed great tact in her dealings with the
queen, Catherine drove her from the court after Henry's death, and
forced her to restore the crown jewels and to accept Chaumont in
exchange for Chenonceaux. Diane retired to her château at Anet, where
she died in 1566.

Several historians relate that she had been the mistress of Francis I.
before she became the dauphin's mistress, and that she gave herself to
the king in order to obtain the pardon of her father, who had been
condemned to death as an accomplice of the constable de Bourbon. This
rumour, however, has no serious foundation. Men vied with each other in
celebrating Diane's beauty, which, if we may judge from her portraits,
has been slightly exaggerated. She was a healthy, vigorous woman, and,
by dint of great pains, succeeded in retaining her beauty late into
life. It is said that even on the coldest mornings she would wash her
face with well water. Diane was a patroness of the arts. She entrusted
to Philibert de l'Orme the building of her château at Anet, and it was
for her that Jean Goujon executed his masterpiece, the statue of Diana,
now in the Louvre.

  See G. Guiffrey, _Lettres inédites de Diane de Poytiers_ (Paris, 1866)
  and _Procès criminel de Jehan de Poytiers_ (Paris, 1867); Capefigue,
  _Diane de Poitiers_ (Paris, 1860); Hay, _Madame Dianne de Poytiers_
  (London, 1900).



DIAPASON (Gr. [Greek: dia pasôn], through all), a term in music,
originally for an interval of an octave. The Greek is an abbreviation of
[Greek: hê dia pasôn chordôn symphônia], a consonance through all the
tones of the scale. In this sense it is only used now, loosely, for the
compass of an instrument or voice, or for a harmonious melody. The name
is given to the two foundation stops of an organ, the open and the
stopped diapason (see ORGAN), and to a standard of musical pitch, as in
the French _diapason normal_ (see PITCH, MUSICAL).



DIAPER (derived through the Fr, from the Gr. [Greek: dia], through, and
[Greek: aspros], white; the derivation from the town of Ypres,
"d'Ypres," in Belgium is unhistorical, as diapers were known for
centuries before its existence), the name given to a textile fabric,
formerly of a rich and costly nature with embroidered ornament, but now
of linen or cotton, with a simple woven pattern; and particularly
restricted to small napkins. In architecture, the term "diaper" is given
to any small pattern of a conventional nature repeated continuously and
uniformly over a surface; the designs may be purely geometrical, or
based on floral forms, and in early examples were regulated by the
process of their textile origin. Subsequently, similar patterns were
employed in the middle ages for the surface decoration of stone, as in
Westminster Abbey and Bayeux cathedral in the spandrils of the arcades
of the choir and nave; also in mural painting, stained glass, incised
brasses, encaustic tiles, &c. Probably in most cases the pattern was
copied, so far as the general design is concerned, from the tissues and
stuffs of Byzantine manufacture, which came over to Europe and were
highly prized as ecclesiastical vestments.

[Illustration: A B C]

  In its textile use, the term diaper was originally applied to silk
  patterns of a geometrical pattern; it is now almost exclusively used
  for diamond patterns made from linen or cotton yarns. An illustration
  of two patterns of this nature is shown in the figure. The floats of
  the warp and the weft are mostly in three; indeed the patterns are
  made from a base weave which is composed entirely of floats of this
  number. It will be seen that both designs are formed of what may be
  termed concentric figures--alternately black and white. Pattern B
  differs from pattern A only in that more of these concentric figures
  are used for the complete figure. If pattern B, which shows only one
  unit, were extended, the effect would be similar to A, except for the
  size of the unit. In A there are four complete units, and hence the
  pattern appears more striking. Again, the repeating of B would cause
  the four corner pieces to join and to form a diamond similar to the
  one in the centre. The two diamonds in B would then alternate
  diagonally to left and right. Special names are given to certain kinds
  of diapers, e.g. "bird's-eye," "pheasant's-eye"; these terms indicate,
  to a certain extent, the size of the complete diamond in the
  cloth--the smaller kind taking the name "bird's-eye." The size of the
  pattern on paper has little connexion with the size of the pattern in
  the cloth, for it is clearly the number of threads and picks per inch
  which determine the size of the pattern in the cloth from any given
  design. Although A is larger than what is usually termed the
  "bird's-eye" pattern, it is evident that it may be made to appear as
  such, provided that the cloth is fine enough. These designs, although
  adapted mostly for cloths such as nursery-diapers, for pinafores, &c.,
  are sometimes used in the production of towels and table-cloths. In
  the figure, the first pick in A is identical with the first pick in B,
  and the part C shows how each interweaves with the twenty-four
  threads.



DIAPHORETICS (from Gr. [Greek: diaphorein], to carry through), the name
given to those remedies which promote perspiration. In health there is
constantly taking place an exhalation of watery vapour from the skin, by
which not only are many of the effete products of nutrition eliminated,
but the body is kept cool. Under exertion or in a heated atmosphere this
natural function of the skin is increased, sweating more or less profuse
follows, and, evaporation going on rapidly over the whole surface,
little or no rise in the temperature of the body takes place. In many
forms of disease, such as fevers and inflammatory affections, the
action of the skin is arrested, and the surface of the body feels harsh
and dry, while the temperature is greatly elevated. The occurrence of
perspiration not unfrequently marks a crisis in such diseases, and is in
general regarded as a favourable event. In some chronic diseases, such
as diabetes and some cases of Bright's disease, the absence of
perspiration is a marked feature; while, on the other hand, in many
wasting diseases, such as phthisis, the action of the skin is increased,
and copious exhausting sweating occurs. Many means can be used to induce
perspiration, among the best known being baths, either in the form of
hot vapour or hot water baths, or in that part of the process of the
Turkish bath which consists in exposing the body to a dry and hot
atmosphere. Such measures, particularly if followed by the drinking of
hot liquids and the wrapping of the body in warm clothing, seldom fail
to excite copious perspiration. Numerous medicinal substances have the
same effect.



DIAPHRAGM (Gr. [Greek: diaphragma], a partition). The diaphragm or
midriff (Anglo-Saxon, _mid_, middle, _hrif_, belly) in human anatomy is
a large fibro-muscular partition between the cavities of the thorax and
abdomen; it is convex toward the thorax, concave toward the abdomen, and
consists of a central tendon and a muscular margin. The _central tendon_
(q, fig. 1) is trefoil in shape, its leaflets being right, left and
anterior; of these the right is the largest and the left the smallest.
The fleshy fibres rise, in front from the back of the xiphoid cartilage
of the sternum (d), laterally by six serrations, from the inner surfaces
of the lower six ribs, interdigitating with the transversalis,
posteriorly from the arcuate ligaments, of which there are five, a pair
of external, a pair of internal, and a single median one. The _external
arcuate ligament_ (h) stretches from the tip of the twelfth rib (b) to
the costal process of the first lumbar vertebra in front of the
quadratus lumborum muscle (o), the _internal_ and _middle_ are
continuations of the _crura_ which rise from the ventro-lateral aspects
of the bodies of the lumbar vertebrae, the right (e) coming from three,
the left (f) from two. On reaching the level of the twelfth thoracic
vertebra each crus spreads out into a fan-shaped mass of fibres, of
which the innermost join their fellows from the opposite crus, in front
of the aortic opening (k), to form the _middle arcuate ligament_; the
outer ones (g) arch in front of the psoas muscle (n) to the tip of the
costal process of the first lumbar vertebra to form the _internal
arcuate ligament_, while the intermediate ones pass to the central
tendon. There are three large openings in the diaphragm; the _aortic_
(k) is behind the middle arcuate ligament and transmits the aorta, the
vena azygos major, and the thoracic duct. In the right leaflet is an
opening (sometimes called the _hiatus quadratus_) for the inferior vena
cava and a branch of the right phrenic nerve (m), while in front and a
little to the left of the aortic opening is one for the oesophagus and
the two pneumogastric nerves (l), the left being in front and the right
behind. The fleshy fibres on each side of this opening act as a
sphincter. Passing between the xiphoid and costal origins in front are
the superior epigastric arteries, while the other terminal branches of
the internal mammaries, the musculo-phrenics, pass through between two
costal origins.

[Illustration: FIG. 1.--Abdominal Surface of the Diaphragm.]

Through the crura pass the splanchnic nerves, and in addition to these
the left crus is pierced by the vena azygos minor. The sympathetic
nerves usually enter the abdomen behind the internal arcuate ligaments.
The phrenic nerves, which are the main supply of the diaphragm, divide
before reaching the muscle and pierce it in a number of places to enter
its abdominal surface, but some of the lower intercostal nerves assist
in the supply. The last thoracic or subcostal nerves pass behind the
external arcuate ligament.

For the action of the diaphragm see RESPIRATORY SYSTEM.

  _Embryology._--The diaphragm is at first developed in the neck region
  of the embryo, and this accounts for the phrenic nerves, which supply
  it, rising from the fourth and fifth cervical. From the mesoderm on
  the caudal side of the pericardium is developed the _septum
  transversum_, and in this the central tendon is formed. The fleshy
  portion is developed on each side in two parts, an anterior or
  sterno-costal which is derived from the longitudinal neck musculature,
  probably the same layer from which the sternothyroid comes, and a
  spinal part which is a derivative of the transversalis sheet of the
  trunk. Between these two parts is at one time a gap, the _spino-costal
  hiatus_, and this is obliterated by the growth of the
  pleuro-peritoneal membrane, which may occasionally fail to close and
  so may form the site of a phrenic hernia. With the growth of the body
  and the development of the lungs the diaphragm shifts its position
  until it becomes the septum between the thoracic and abdominal
  cavities. (See A. Keith, "On the Development of the Diaphragm," _Jour.
  of Anat. and Phys._ vol. 39.) A. Paterson has recorded cases in which
  the left half of the diaphragm is wanting (_Proceedings_ of the
  Anatomical Society of Gt. Britain, June 1900; _Jour. of Anat. and
  Phys._ vol. 34), and occasionally deficiencies are found elsewhere,
  especially in the sternal portion. For further details see Quain's
  _Anatomy_, vol. i. (London, 1908).

  _Comparative Anatomy._--A complete diaphragm, separating the thoracic
  from the abdominal parts of the coelom, is characteristic of the
  Mammalia; it usually has the human structure and relations except that
  below the Anthropoids it is separated from the pericardium by the
  azygous lobe of the lung. In some Mammals, e.g. Echidna and Phocoena,
  it is entirely muscular. In the Cetacea it is remarkable for its
  obliquity; its vertebral attachment is much nearer the tail than its
  sternal or ventral one; this allows a much larger lung space in the
  dorsal than in the ventral part of the thorax, and may be concerned
  with the equipoise of the animal. (Otto Müller, "Untersuchungen über
  die Veränderung, welche die Respirationsorgane der Säugetiere durch
  die Anpassung an das Leben im Wasser erlitten haben," _Jen. Zeitschr.
  f. Naturwiss._, 1898, p. 93.) In the Ungulata only one crus is found
  (Windle and Parsons, "Muscles of the Ungulata," _Proc. Zool. Soc._,
  1903, p. 287). Below the Mammals incomplete partitions between the
  pleural and peritoneal cavities are found in Chelonians, Crocodiles
  and Birds, and also in Amphibians (Xenopus and Pipa).     (F. G. P.)



DIARBEKR[1] (_Kara Amid_ or Black Amid; the Roman _Amida_), the chief
town of a vilayet of Asiatic Turkey, situated on a basaltic plateau on
the right bank of the Tigris, which here flows in a deep open valley.
The town is still surrounded by the masonry walls of black basalt which
give it the name of _Kara_ or Black Amid; they are well built and
imposing on the west facing the open country, but almost in ruins where
they overlook the river. A mass of gardens and orchards cover the slope
down to the river on the S.W., but there are no suburbs outside the
walls. The houses are rather crowded but only partially fill the walled
area. The population numbers 38,000, nearly half being Christian,
comprising Turks, Kurds, Arabs, Turkomans, Armenians, Chaldeans,
Jacobites and a few Greeks. The streets are 10 ft. to 15 ft. wide, badly
paved and dirty; the houses and shops are low, mostly of stone, and some
of stone and mud. The bazaar is a good one, and gold and silver filigree
work is made, peculiar in character and design. The cotton industry is
declining, but manufacture of silk is increasing. Fruit is good and
abundant as the rich volcanic soil is well watered from the town
springs. The size of the melons is specially famous. To the south, the
walls are some 40 ft. high, faced with large cut stone blocks of very
solid construction, with towers and square bastions rising to 500 ft.
There are four gates: on the north the Kharput gate, on the west the
Rum, on the south the Mardin, and on the east the Yeni Kapu or new
gate. A citadel enclosure stands at the N. E. corner and is now partly
in ruins, but the interior space is occupied by the government konak.
The summer climate in the confined space within the town is excessively
hot and unhealthy. Epidemics of typhus are not unknown, as well as
ophthalmia. The Diarbekr boil is like the "Aleppo button," lasting a
long time and leaving a deep scar. Winters are frequently severe but do
not last long. Snow sometimes lies, and ice is stored for summer use.
Scorpions noted for the virulence of their poison abound as well as
horse leeches in the tanks. The town is supplied with water both by
springs inside the town and by aqueducts from fountains at Ali Punar and
Hamervat. The principal exports are wool, mohair and copper ore, and
imports are cotton and woollen goods, indigo, coffee, sugar, petroleum,
&c.

The Great Mosque, Ulu Jami, formerly a Christian church, occupies the
site of a Sassanian palace and was built with materials from an older
palace, probably that of Tigranes II. The remains consist of the façades
of two palaces 400 ft. apart, each formed by a row of Corinthian columns
surmounted by an equal number of a Byzantine type. Kufic inscriptions
run across the fronts under the entablature. The court of the mosque is
entered by a gateway on which lions and other animals are sculptured.
The churches of greatest interest are those of SS. Cosmas and Damian
(Jacobite) and the church of St James (Greek). In the 19th century
Diarbekr was one of the largest and most flourishing cities of Asia, and
as a commercial centre it now stands at the meeting-point of several
important routes. It is at the head of the navigation of the Tigris,
which is traversed down stream by _keleks_ or rafts supported by
inflated skins. There is a good road to Aleppo and Alexandretta on the
Mediterranean, and to Samsun on the Black Sea by Kharput, Malatia and
Sivas. There are also routes to Mosul and Bitlis.

Diarbekr became a Roman colony in A.D. 230 under the name of Amida, and
received a Christian bishop in A.D. 325. It was enlarged and
strengthened by Constantius II., in whose reign it was taken after a
long siege by Shapur (Sapor) II., king of Persia. The historian Ammianus
Marcellinus, who took part in the defence, gives a detailed account of
it. In the later wars between the Persians and Romans it more than once
changed hands. Though ceded by Jovian to the Persians it again became
annexed to the Roman empire, and in the reign of Anastasius (A.D. 502)
was once more taken by the Persians, when 80,000 of its inhabitants were
slain. It was taken c. 638 by the Arabs, and afterwards passed into the
hands of the Seljuks and Persians, from whom it was finally captured by
Selim I. in 1515; and since that date it has remained under Ottoman
rule. About 2 m. below the town is a masonry bridge over the Tigris; the
older portion being probably Roman, and the western part, which bears a
Kufic inscription, being Arab.

The vilayet of Diarbekr extends south from Palu on the Euphrates to
Mardin and Nisibin on the edge of the Mesopotamian plain, and is divided
into three sanjaks--Arghana, Diarbekr and Mardin. The headwaters of the
main arm of the Tigris have their source in the vilayet.

Cereals, cotton, tobacco, rice and silk are produced, but most of the
fertile lands have been abandoned to semi-nomads, who raise large
quantities of live stock. The richest portion of the vilayet lies east
of the capital in the rolling plains watered by tributaries of the
Tigris. An exceptionally rich copper mine exists at Arghana Maden, but
it is very imperfectly worked; galena mineral oil and silicious sand are
also found.     (C. W. W.; F. R. M.)


FOOTNOTE:

  [1] From _Diar_, land, and Bekr (i.e. Abu Bekr, the caliph).



DIARRHOEA (from Gr. [Greek: dia], through, [Greek: rheô], flow), an
excessive looseness of the bowels, a symptom of irritation which may be
due to various causes, or may be associated with some specific disease.
The treatment in such latter cases necessarily varies, since the symptom
itself may be remedial, but in ordinary cases depends on the removal of
the cause of irritation by the use of aperients, various sedatives being
also prescribed. In chronic diarrhoea careful attention to the diet is
necessary.



DIARY, the Lat. _diarium_ (from _dies_, a day), the book in which are
preserved the daily memoranda regarding events and actions which come
under the writer's personal observation, or are related to him by
others. The person who keeps this record is called a diarist. It is not
necessary that the entries in a diary should be made each day, since
every life, however full, must contain absolutely empty intervals. But
it is essential that the entry should be made during the course of the
day to which it refers. When this has evidently not been done, as in the
case of Evelyn's diary, there is nevertheless an effort made to give the
memoranda the effect of being so recorded, and in point of fact, even in
a case like that of Evelyn, it is probable that what we now read is an
enlargement of brief notes jotted down on the day cited. When this is
not approximately the case, the diary is a fraud, for its whole value
depends on its instantaneous transcript of impressions.

In its primitive form, the diary must always have existed; as soon as
writing was invented, men and women must have wished to note down, in
some almanac or journal, memoranda respecting their business, their
engagements or their adventures. But the literary value of these would
be extremely insignificant until the spirit of individualism had crept
in, and human beings began to be interesting to other human beings for
their own sake. It is not, therefore, until the close of the Renaissance
that we find diaries beginning to have literary value, although, as the
study of sociology extends, every scrap of genuine and unaffected record
of early history possesses an ethical interest. In the 17th century,
diaries began to be largely written in England, although in most cases
without any idea of even eventual publication. Sir William Dugdale
(1605-1686) had certainly no expectation that his slight diary would
ever see the light. There is no surviving record of a journal kept by
Clarendon, Richard Baxter, Lucy Hutchinson and other autobiographical
writers of the middle of the century, but we may take it for granted
that they possessed some such record, kept from day to day. Bulstrode
Whitelocke (1605-1675), whose _Memorials of the English Affairs_ covers
the ground from 1625 to 1660, was a genuine diarist. So was the elder
George Fox (1624-1690), who kept not merely "a great journal," but "the
little journal books," and whose work was published in 1694. The famous
diary of John Evelyn (1620-1706) professes to be the record of seventy
years, and, although large tracts of it are covered in a very
perfunctory manner, while in others many of the entries have the air of
having been written in long after the event, this is a very interesting
and amusing work; it was not published until 1818. In spite of all its
imperfections there is a great charm about the diary of Evelyn, and it
would hold a still higher position in the history of literature than it
does if it were not overshadowed by what is unquestionably the most
illustrious of the diaries of the world, that of Samuel Pepys
(1633-1703). This was begun on the 1st of January 1660 and was carried
on until the 29th of May 1669. The extraordinary value of Pepys' diary
consists in its fidelity to the portraiture of its author's character.
He feigns nothing, conceals nothing, sets nothing down in malice or
insincerity. He wrote in a form of shorthand intelligible to no one but
himself, and not a phrase betrays the smallest expectation that any eye
but his own would ever investigate the pages of his confession. The
importance of this wonderful document, in fact, lay unsuspected until
1819, when the Rev. John Smith of Baldock began to decipher the MS. in
Magdalene College, Cambridge. It was not until 1825 that Lord Braybrooke
published part of what was only fully edited, under the care of Mr
Wheatley, in 1893-1896. In the age which succeeded that of Pepys, a
diary of extraordinary emotional interest was kept by Swift from 1710 to
1713, and was sent to Ireland in the form of a "Journal to Stella"; it
is a surprising amalgam of ambition, affection, wit and freakishness.
John Byrom (1692-1763), the Manchester poet, kept a journal, which was
published in 1854. The diary of the celebrated dissenting divine, Philip
Doddridge (1702-1751), was printed in 1829. Of far greater interest are
the admirably composed and vigorously written journals of John Wesley
(1703-1791). But the most celebrated work of this kind produced in the
latter half of the 18th century was the diary of Fanny Burney (Madame
D'Arblay), published in 1842-1846. It will be perceived that, without
exception, these works were posthumously published, and the whole
conception of the diary has been that it should be written for the
writer alone, or, if for the public, for the public when all prejudice
shall have passed away and all passion cooled down. Thus, and thus only,
can the diary be written so as to impress upon its eventual readers a
sense of its author's perfect sincerity and courage.

Many of the diaries described above were first published in the opening
years of the 19th century, and it is unquestionable that the interest
which they awakened in the public led to their imitation. Diaries ceased
to be rare, but as a rule the specimens which have hitherto appeared
have not presented much literary interest. Exception must be made in
favour of the journals of two minor politicians, Charles Greville
(1794-1865) and Thomas Creevey (1768-1838), whose indiscretions have
added much to the gaiety of nations; the papers of the former appeared
in 1874-1887, those of the latter in 1903. The diary of Henry Crabb
Robinson (1775-1867), printed in 1869, contains excellent biographical
material. Tom Moore's journal, published in 1856 by Lord John Russell,
disappointed its readers. But it is probable, if we reason by the
analogy of the past, that the most curious and original diaries of the
19th century are still unknown to us, and lie jealously guarded under
lock and key by the descendants of those who compiled them.

It was natural that the form of the diary should appeal to a people so
sensitive to social peculiarities and so keen in the observation of them
as the French. A medieval document of immense value is the diary kept by
an anonymous _curé_ during the reigns of Charles VI. and Charles VII.
This _Journal d'un bourgeois de Paris_ was kept from 1409 to 1431, and
was continued by another hand down to 1449. The marquis de Dangeau
(1638-1720) kept a diary from 1684 till the year of his death; this
although dull, and as Saint-Simon said "of an insipidity to make you
sick," is an inexhaustible storehouse of facts about the reign of Louis
XIV. Saint-Simon's own brilliant memoirs, written from 1691 to 1723, may
be considered as a sort of diary. The lawyer, Edmond Barbier
(1689-1771), wrote a journal of the anecdotes and little facts which
came to his knowledge from 1718 to 1762. The studious care which he took
to be correct, and his manifest candour, give a singular value to
Barbier's record; his diary was not printed at all until 1847, nor, in
its entirety, until 1857. The song-writer, Charles Collé (1709-1783),
kept a _journal historique_ from 1758 to 1782; it is full of vivacity,
but very scandalous and spiteful. It saw the light in 1805, and
surprised those to whom Collé, in his lifetime, had seemed the most
placid and good-natured of men. Petit de Bachaumont (1690-1770) had
access to remarkable sources of information, and his _Mémoires secrets_
(a diary the publication of which began in 1762 and was continued after
Bachaumont's death, until 1787, by other persons) contains a valuable
mass of documents. The marquis d'Argenson (1694-1757) kept a diary, of
which a comparatively full text was first published in 1859. In recent
times the posthumous publication of the diaries of the Russian artist,
Marie Bashkirtseff (1860-1884), produced a great sensation in 1887, and
revealed a most remarkable temperament. The brothers Jules and Edmond de
Goncourt kept a very minute diary of all that occurred around them in
artistic and literary Paris; after the death of Jules, in 1870, this was
continued by Edmond, who published the three first volumes in 1888. The
publication of this work was continued, and it produced no little
scandal. It is excessively ill-natured in parts, but of its vivid
picturesqueness, and of its general accuracy as a transcript of
conversation, there can be no two opinions.     (E. G.)



DIASPORE, a native aluminium hydroxide, AlO(OH), crystallizing in the
orthorhombic system and isomorphous with göthite and manganite. It
occurs sometimes as flattened crystals, but usually as lamellar or scaly
masses, the flattened surface being a direction of perfect cleavage on
which the lustre is markedly pearly in character. It is colourless or
greyish-white, yellowish, sometimes violet in colour, and varies from
translucent to transparent. It may be readily distinguished from other
colourless transparent minerals, with a perfect cleavage and pearly
lustre--mica, talc, brucite, gypsum--by its greater hardness of 6½-7.
The specific gravity is 3.4. When heated before the blowpipe it
decrepitates violently, breaking up into white pearly scales; it was
because of this property that the mineral was named diaspore by R. J.
Hauy in 1801, from [Greek: diaspeirein], "to scatter." The mineral
occurs as an alteration product of corundum or emery, and is found in
granular limestone and other crystalline rocks. Well-developed crystals
are found in the emery deposits of the Urals and at Chester,
Massachusetts, and in kaolin at Schemnitz in Hungary. If obtainable in
large quantity it would be of economic importance as a source of
alumina.     (L. J. S)



DIASTYLE (from Gr. [Greek: dia], through, and [Greek: stylos], column),
in architecture, a term used to designate an intercolumniation of three
or four diameters.



DIATOMACEAE. For the knowledge we possess of these beautiful plants, so
minute as to be undiscernible by our unaided vision, we are indebted to
the assistance of the microscope. It was not till towards the close of
the 18th century that the first known forms of this group were
discovered by O. F. Muller. And so slow was the process of discovery in
this field of scientific research that in the course of half a century,
when Agardh published his _Systema algarum_ in 1824, only forty-nine
species included under eight genera had been described. Since that time,
however, with modern microscopes and microscopic methods, eminent
botanists in all parts of the civilized world have studied these minute
plants, with the result that the number of known genera and species has
been greatly increased. Over 10,000 species of diatoms have been
described, and about 1200 species and numerous varieties occur in the
fresh waters and on the coasts of Great Britain and Ireland. Rabenhorst,
in the index to his _Flora Europaea algarum_ (1864) enumerated about
4000 forms which had up to that time been discovered throughout the
continent of Europe.

[Illustration: FIG. 1. A and B, _Melosira arenaria._ C-E, _Melosira
varians._ E, showing formation of auxospore.]

[Illustration: FIG. 2.--_Synedra Ulna._]

The diatoms are more commonly known among systematic botanists as the
Bacillarieae, particularly on the continent of Europe, and although such
an immense number of very diverse forms are included in it, the group as
a whole exhibits a remarkable uniformity of structure. The Bacillarieae
is one of the large groups of Algae, placed by some in close proximity
to the Conjugatae and by others as an order of the Brown Algae (or
Phaeophyceae), but their characters are so distinctive and their
structure is so uniform as to warrant the separation of the diatoms as a
distinct class. The affinities of the group are doubtful.

The diatoms exhibit great variety of form. While some species are
circular and more or less disk-shaped, others are oval in outline. Some
are linear, as _Synedra Ulna_ (fig. 2), others more or less crescentic;
others again are cuneate, as _Podosphenia Lyngbyii_ (fig. 3); some few
have a sigmoid outline, as _Pleurosigma balticum_ (fig. 4); but the
prevailing forms are naviculoid, as in the large family Naviculaceae, of
which the genus _Navicula_ embraces upwards of 1000 species. They vary
also in their modes of growth,--some being free-floating, others
attached to foreign bodies by simple or branched gelatinous stalks,
which in some species are short and thick, while in others they are long
and slender. In some genera the forms are simple, while in others the
frustules are connected together in ribbon-like filaments, or form, as
in other cases, zigzag chains. In some genera the individuals are naked,
while in many others they are enclosed in a more or less definite
gelatinous investment. The conditions necessary to their growth are
moisture and light. Wherever these circumstances coexist, diatomaceous
forms will almost invariably be found. They occur mixed with other
organisms on the surface of moist rocks; in streamlets and pools, they
form a brownish stratum on the surface of the mud, or cover the stems
and leaves of water plants or floating twigs with a furry investment.
Marine forms are usually attached to various sea-weeds, and many are
found in the stomachs of molluscs, holothurians, ascidians and other
denizens of the ocean. The fresh-water forms are specifically distinct
from those incidental to salt or brackish water,--fresh-water species,
however, are sometimes carried some distance into the sea by the force
of the current, and in tidal rivers marine forms are carried up by the
force of the tide. Some notion may be formed of the extreme minuteness
of these forms from the fact that one the length of which is 1/200th of
an inch may be considered as beyond the medium size. Some few, indeed,
are much larger, but by far the greater proportion are of very much
smaller dimensions.

[Illustration: FIG. 3.--_Podosphenia Lyngbyii._]

[Illustration: FIG. 4.--_Pleurosigma balticum._]

[Illustration: FIG. 5. A-C, _Tetracyclus lacustris._ D and E,
_Tabellaria fenestrata._ F and G, _Tabellaria flocculosa._]

Diatoms are unicellular plants distinguished from kindred forms by the
fact of having their soft vegetative part covered by a siliceous case.
Each individual is known as a frustule, and the cell-wall consists of
two similar valves nearly parallel to each other, each valve being
furnished with a rim (or connecting-band) projecting from it at a right
angle.

One of these valves with its rim is slightly smaller than the other,
the smaller fitting into the larger pretty much as a pill-box fits into
its cover. This peculiarity of structure affords ample scope for the
growth of the protoplasmic cell-contents, for as the latter increase in
volume the siliceous valves are pushed out, and their corresponding
siliceous rims become broader. The connecting-bands although closely
fitting their respective valves are distinct from them, and together the
two bands form the girdle.

An individual diatom is usually described from two aspects, one in which
the surface of the valve is exposed to view--the valve view, and one in
which the girdle side is exposed--the girdle view. The valves are thin
and transparent, convex on the outside, and generally ornamented with a
variety of sculptured markings. These sculptures often present the
aspect of striae across the face of the valve, and the best lenses have
shown them to consist of a series of small cavities within the siliceous
wall of the cell. The valves of some of the marine genera exhibit a
beautiful areolated structure due to the presence of larger chambers
within the siliceous cell-wall. Many diatoms possess thickenings of the
cell-wall, visible in the valve view, in the centre of the valve and at
each extremity. These thickenings are known as the nodules, and they are
generally connected by a long median line, the raphe, which is a cleft
in the siliceous valve, extending at least some part of its length.

The protoplasmic contents of this siliceous box-like unicell are very
similar to the contents of many other algal cells. There is a living
protoplasmic layer or primordial utricle, connected either by two broad
bands or by a number of anastomosing threads with a central mass of
protoplasm in which the nucleus is embedded. The greater part of the
cavity of the cell is occupied by one or several fluid vacuoles. The
characteristic brown colour of diatoms is due to the presence of
chromatophores embedded in the lining layer of protoplasm. In number and
form these chromatophores are variable. They contain chlorophyll, but
the green colour is masked by the presence of diatomin, a brown pigment
which resembles that which occurs in the Brown Algae or Phaeophyceae.
The chromatophores contain a variable number of pyrenoids, colourless
proteid bodies of a crystalloidal character.

One of the first phenomena which comes under the notice of the observer
is the extraordinary power of motion with which the frustules are
endowed. Some species move slowly backwards and forwards in pretty much
the same line, but in the case of _Bacillaria paradoxa_ the motion is
very rapid, the frustules darting through the water in a zigzag course.
To account for this motion various theories have been suggested, none of
which appear to be altogether satisfactory. There is little doubt that
the movements are connected with the raphe, and in some diatoms there is
much evidence to prove that they are due to an exudation of mucilage.

_Classification._--The most natural system of classification of the
Bacillarieae is the one put forward by Schütt (1896), and since
generally followed by systematists. He separates them into two primary
divisions, the 'Centricae' and the 'Pennatae.' The former includes all
those diatoms which in the valve view possess a radial symmetry around a
central point, and which are destitute of a raphe (or a pseudoraphe).
The latter includes those which are zygomorphic or otherwise irregular,
and in which the valve view is generally boat-shaped or needle-shaped,
with the markings arranged in a sagittal manner on each side of a raphe
or pseudoraphe.

_Reproduction._--In the Diatomaceae, as well as in the Desmidieae, the
ordinary mode of increase is by simple cell-division. The cell-contents
within the enclosure of the siliceous case separate into two distinct
masses. As these two daughter-masses become more and more developed, the
valves of the mother-cell are pushed more and more widely apart. A new
siliceous valve is secreted by each of the two masses on the side
opposite to the original valve, the new valves being situated within the
girdle of the original frustule. When this process has been completed
the girdle of the mother frustule gives way, and two distinct frustules
are formed, the siliceous valves in each of these new frustules being
one of the valves of the mother-cell, and a newly formed valve similar
and more or less parallel to it.

During the life of the plant this process of self-division is continued
with an almost incredible rapidity. On this subject the observation of
Professor William Smith, writing in 1853, is worthy of special
notice:--"I have been unable to ascertain the time occupied in a single
act of self-division, but supposing it to be completed in twenty-four
hours we should have, as the progeny of a single frustule, the amazing
number of 1,000,000,000 in a single month, a circumstance which will in
some degree explain the sudden, or at least rapid, appearance of these
organisms in localities where they were a short time previously either
unrecognized or sparingly diffused" (_British Diatomaceae_, vol. i. p.
25).

[Illustration: FIG. 6.--Formation of Auxospores.

  A. _Navicula limosa._
  B. _Achnanthes flexella._
  C. _Navicula Amphisbaena._
  D. _Navicula viridis._]

Individual diatoms when once produced by cell-division are incapable of
any increase in size owing to the rigidity of their siliceous
cell-walls, and since the new valves are always formed _within_ the
girdle of the old ones, it would follow that every succeeding generation
is reduced in size by the thickness of the girdle. In some diatoms,
however, this is not strictly true as daughter-cells are sometimes
produced of larger size than the parent-cells. Thus, the reduction in
size of the individuals is not always proportionate to the number of
cell-divisions.

On the diminution in size having reached a limit in any species, the
maximum size is regained by the formation of an auxospore. There are
five known methods of reproduction by auxospores, but it is unnecessary
here to enter into details of these methods. Suffice it to say that a
normal auxospore is produced by the conjugation of two parent-cells, its
distinguishing feature being a rejuvenescence accompanied by a marked
increase in size. These auxospores formed without conjugation are
parthenogenetic.

_Mode of Preparation._--The Diatomaceae are usually gathered in small
bottles, and special care should be taken to collect them as free as
possible from extraneous matter. A small portion having been examined
under the microscope, should the gathering be thought worthy of
preservation, some of the material is boiled in acid for the purpose of
cleaning it. The acids usually employed are hydrochloric, nitric or
sulphuric, according as circumstances require. When the operator
considers that by this process all foreign matter has been eliminated,
the residuum is put into a precipitating jar of a conical shape, broader
at the bottom than at the top, and covered to the brim with filtered or
distilled water. When the diatoms have settled in the bottom of the jar,
the supernatant fluid is carefully removed by a syringe or some similar
instrument, so that the sediment be not disturbed. The jar is again
filled with water, and the process repeated till the acid has been
completely removed. It is desirable afterwards to boil the sediment for
a short time with supercarbonate of soda, the alkali being removed in
the same manner as the acid. A small portion may then be placed with a
pipette upon a slip of glass, and, when the moisture has been thoroughly
evaporated, the film that remains should be covered with dilute Canada
balsam, and, a thin glass cover having been gently laid over the balsam,
the preparation should be laid aside for a short time to harden, and
then is ready for observation.

_General Remarks._--Diatoms are most abundant in cold latitudes, having
a general preference for cold water. In the pelagic waters of lakes and
of the oceans they are often very abundant, and in the cold waters of
the Arctic and Antarctic Oceans they exist in prodigious numbers. They
thus form a large proportion of both the marine and the fresh-water
plankton.

Large numbers of fossil diatoms are known. Not only are these minute
plants assisting at the present time in the accumulation of oceanic and
lake deposits, but in former ages they have been sufficiently active to
give rise to considerable deposits of diatomaceous earths. When the
plant has fulfilled its natural course the siliceous covering sinks to
the bottom of the water in which it had lived, and there forms part of
the sediment. When in the process of ages, as it has often happened, the
accumulated sediment has been hardened into solid rock, the siliceous
frustules of the diatoms remain unaltered, and, if the rock be
disintegrated by natural or artificial means, may be removed from the
enveloping matrix and subjected to examination under the microscope. The
forms found may from their character help in some degree to illustrate
the conditions under which the stratum of rock had been originally
deposited. These earths are generally of a white or grey colour. Some of
them are hard, but most are soft and friable. Many of them are of
economic importance, being used as polishing powders ("Tripoli"), as
absorbents for nitroglycerin in the manufacture of dynamite
("Kieselguhr"), as a dentifrice, and more recently they have been used
to a large extent in the manufacture of non-conducting and sound-proof
materials. Most of these diatomaceous earths are associated with rocks
of Tertiary formations, although it is generally regarded that the
earliest appearance of diatoms is in the Upper Cretaceous (chalk).

Vast deposits of Diatomaceous earths have been discovered in various
parts of the world,--some the deposit of fresh, others of salt water. Of
these deposits the most remarkable for extent, as well as for the number
and beauty of the species contained in it, is that of Richmond, in
Virginia, one of the United States of America. It extends for many
miles, and is in some places at least 40 ft. deep. It is a remarkable
fact that though the generations of a diatom in the space of a few
months far exceed in number the generation of man during the period
usually assigned to the existence of the race, the fossil genera and
species are in most respects to the most minute details identical with
the numerous living representatives of their class.
     (E. O'M.; G. S. W.*)



DIAULOS (from Gr. [Greek: di-], double, and [Greek: aulos], pipe), in
architecture, the peristyle round the great court of the palaestra,
described by Vitruvius (v. II), which measured two stadia (1200 ft.) in
length; on the south side this peristyle had two rows of columns, so
that in stormy weather the rain might not be driven into the inner part.
The word was also used in ancient Greece for a foot-race of twice the
usual length.



DIAVOLO, FRA (1771-1806), the popular name given to a famous Italian
brigand associated with the political revolutions of southern Italy at
the time of the French invasion. His real name was Michele Pezza, and he
was born of low parentage at Itri; he had committed many murders and
robberies in the Terra di Lavoro, but by good luck combined with
audacity he always escaped capture, whence his name of Fra Diavolo,
popular superstition having invested him with the characters of a monk
and a demon, and it seems that at one time he actually was a monk. When
the kingdom of Naples was overrun by the French and the Parthenopaean
Republic established (1799), Cardinal Ruffo, acting on behalf of the
Bourbon king Ferdinand IV., who had fled to Sicily, undertook the
reconquest of the country, and for this purpose he raised bands of
peasants, gaol-birds, brigands, &c., under the name of Sanfedisti or
_bande della Santa Fede_ ("bands of the Holy Faith"). Fra Diavolo was
made leader of one of them, and waged untiring war against the French
troops, cutting off isolated detachments and murdering stragglers and
couriers. Owing to his unrivalled knowledge of the country, he succeeded
in interrupting the enemy's communications between Rome and Naples. But
although, like his fellow-brigands under Ruffo, he styled himself "the
faithful servant and subject of His Sicilian Majesty," wore a military
uniform and held military rank, and was even created duke of Cassano,
his atrocities were worthy of a bandit chief. On one occasion he threw
some of his prisoners, men, women and children, over a precipice, and
on another he had a party of seventy shot. His excesses while at Albano
were such that the Neapolitan general Naselli had him arrested and
imprisoned in the castle of St Angelo, but he was liberated soon after.
When Joseph Bonaparte was made king of Naples, extraordinary tribunals
were established to suppress brigandage, and a price was put on Fra
Diavolo's head. After spreading terror through Calabria, he crossed over
to Sicily, where he concerted further attacks on the French. He returned
to the mainland at the head of 200 convicts, and committed further
excesses in the Terra di Lavoro; but the French troops were everywhere
on the alert to capture him and he had to take refuge in the woods of
Lenola. For two months he evaded his pursuers, but at length, hungry and
ill, he went in disguise to the village of Baronissi, where he was
recognized and arrested, tried by an extraordinary tribunal, condemned
to death and shot. In his last moments he cursed both the Bourbons and
Admiral Sir Sidney Smith for having induced him to engage in this
reckless adventure (1806). Although his cruelty was abominable, he was
not altogether without generosity, and by his courage and audacity he
acquired a certain romantic popularity. His name has gained a world-wide
celebrity as the title of a famous opera by Auber.

  The best known account of Fra Diavolo is in Pietro Colletta's _Storia
  del reame di Napoli_ (2nd ed., Florence, 1848); B. Amante's _Fra
  Diavolo e il suo tempo_ (Florence, 1904) is an attempted
  rehabilitation; but A. Luzio, whose account in _Profili e bozzetti
  storici_ (Milan, 1906) gives the latest information on the subject,
  has demolished Amante's arguments.     (L. V.*)



DIAZ, NARCISSE VIRGILIO (1808-1876), French painter, was born in
Bordeaux of Spanish parents, on the 25th of August 1808. At first a
figure-painter who indulged in strong colour, in his later life Diaz
became a painter of the forest and a "tone artist" of the first order.
He spent much time at Barbizon; and although he is the least exalted of
the half-dozen great artists who are usually grouped round that name, he
sometimes produced works of the highest quality. At the age of ten Diaz
became an orphan, and misfortune dogged his earlier years. His foot was
bitten by a reptile in Meudon wood, near Sèvres, where he had been taken
to live with some friends of his mother. The bite was badly dressed, and
ultimately it cost him his leg. Afterwards his wooden stump became
famous. At fifteen he entered the studios at Sèvres, where the
decoration of porcelain occupied him; but tiring of the restraint of
fixed hours, he took to painting Eastern figures dressed in richly
coloured garments. Turks and Oriental scenes attracted him, and many
brilliant gems remain of this period. About 1831 Diaz encountered
Théodore Rousseau, for whom he entertained a great veneration, although
Rousseau was four years his junior; but it was not until ten years later
that the remarkable incident took place of Rousseau teaching Diaz to
paint trees. At Fontainebleau Diaz found Rousseau painting his wonderful
forest pictures, and determined to paint in the same way if possible.
Rousseau, then in poor health, worried at home, and embittered against
the world, was difficult to approach. Diaz followed him surreptitiously
to the forest,--wooden leg not hindering,--and he dodged round after the
painter, trying to observe his method of work. After a time Diaz found a
way to become friendly with Rousseau, and revealed his anxiety to
understand his painting. Rousseau was touched with the passionate words
of admiration, and finally taught Diaz all he knew. Diaz exhibited many
pictures at the Paris Salon, and was decorated in 1851. During the
Franco-German War he went to Brussels. After 1871 he became fashionable,
his works gradually rose in the estimation of collectors, and he worked
constantly and successfully. In 1876 he caught cold at his son's grave,
and on the 18th of November of that year he died at Mentone, whither he
had gone to recruit his health. Diaz's finest pictures are his forest
scenes and storms, and it is on these, and not on his pretty figures,
that his fame is likely to rest. There are several fairly good examples
of the master in the Louvre, and three small figure pictures in the
Wallace collection, Hertford House. Perhaps the most notable of Diaz's
works are "La Fée aux Perles" (1857), in the Louvre; "Sunset in the
Forest" (1868); "The Storm," and "The Forest of Fontainebleau" (1870)
at Leeds. Diaz had no well-known pupils, but Léon Richet followed
markedly his methods of tree-painting, and J. F. Millet at one period
painted small figures in avowed imitation of Diaz's then popular
subjects.

  See A. Hustin, _Les Artistes célèbres: Diaz_ (Paris); D. Croal
  Thomson, _The Barbizon School of Painters_ (London, 1890); J. W.
  Mollett, _Diaz_ (London, 1890); J. Claretie, _Peintres et sculpteurs
  contemporains: Diaz_ (Paris, 1882); Albert Wolff, _La Capitale de
  l'art: Narcisse Diaz_ (Paris, 1886); Ph. Burty, _Maîtres et
  petit-maîtres: N. Diaz_ (Paris, 1877). (D.C.T.)



DIAZ, PORFIRIO (1830-   ), president of the republic of Mexico (q.v.),
was born in the southern state of Oaxaca, on the 15th of September 1830.
His father was an innkeeper in the little capital of that province, and
died three years after the birth of Porfirio, leaving a family of seven
children. The boy, who had Indian blood in his veins, was educated for
the Catholic Church, a body having immense influence in the country at
that time and ordering and controlling revolutions by the strength of
their filled coffers. Arrived at the age of sixteen Porfirio Diaz threw
off the authority of the priests. Fired with enthusiasm by stories told
by the revolutionary soldiers continually passing through Oaxaca, and
hearing about the war with the United States, a year later he determined
to set out for Mexico city and join the National Guard. There being no
trains, and he being too poor to ride, he walked the greater part of the
250 m., but arrived there too late, as the treaty of Guadalupe-Hidalgo
(1848) had been already signed, and Texas finally ceded to the United
States. Thus his entering the army was for the time defeated. Thereupon
he returned to his native town and began studying law. He took pupils in
order to pay his own fees at the Law Institute, and help his mother. At
this time he came under the notice and influence of Don Marcos Pérez and
Benito Juárez, the first a judge, the second a governor of the state of
Oaxaca, and soon to become famous as the deliverer of Mexico from the
priesthood (War of Reform). Diaz continued in his native town until
1854, when, refusing to vote for the dictator, Santa Anna, he was stung
by a taunt of cowardice, and hastily pushing his way to the voting
place, he recorded his vote in favour of Alvarez and the revolutionists.
Orders were given for his arrest, but seizing a rifle and mounting a
horse he placed himself at the head of a few revolting peasants, and
from that moment became one of the leading spirits in that long struggle
for reform, known as the War of Reform, which, under the leadership of
Juárez, followed the overthrow of Santa Anna. Promotion succeeded
promotion, as Diaz led his troops from victory to victory, amid great
privations and difficulties. He was made captain (1856),
lieutenant-colonel and colonel (1859), brigadier-general (1861), and
general of division for the army (1863). Closely following on civil war,
political strife, open rebellion and the great War of Reform, came the
French invasion of 1862, and the landing of the emperor Maximilian in
1864. From the moment the French disclosed their intentions of settling
in Mexico in 1862, Diaz took a prominent part against the foreign
invasion. He was twice seriously wounded, imprisoned on three different
occasions, had two hairbreadth escapes, and took part in many daring
engagements. So important a personage did he become that both Marshal
Bazaine and the emperor Maximilian made overtures to him. At the time of
Maximilian's death (with which Diaz personally had nothing to do) he was
carrying on the siege of Mexico city, which ended in the surrender of
the town two days after the emperor was shot at Quérétaro between his
two leading generals. Diaz at once set to work to pay up arrears due to
his soldiers, proclaimed death as the penalty of plunder and theft, and
in the few weeks that followed showed his great administrative powers,
the officers as well as the rank and file receiving arrears of pay. On
the very day that he occupied Mexico city, the great commander of the
army of the east, to everyone's surprise, sent in his resignation. He
was, indeed, appointed to the command of the second division of the army
by President Juárez in his military reorganization, but Diaz, seeing men
who had given great and loyal service to the state dismissed from their
positions in the government, and disgusted at this course, retired to
the little city of Oaxaca; there he lived, helping in the
reorganization of the army but taking no active part in the government
until 1871.

On Juárez' death Lerdo succeeded as president, in 1872. His term of
office again brought discord, and when it was known that he was
attempting to be re-elected in 1876, the storm broke. Diaz came from
retirement, took up the leadership against Lerdo, and after desperate
struggles and a daring escape finally made a triumphal entry into Mexico
city on the 24th of November 1876, as provisional president, quickly
followed by the full presidentship. His term of office marks a prominent
change in the history of Mexico; from that date he at once forged ahead
with financial and political reform, the scrupulous settlement of all
national debts, the welding together of the peoples and tribes (there
are 150 different Indian tribes) of his country, the establishment of
railroads and telegraphs, and all this in a land which had been upheaved
for a century with revolutions and bloodshed, and which had had
fifty-two dictators, presidents and rulers in fifty-nine years. In 1880
Diaz was succeeded by Gonzalez, the former minister of war, for four
years (owing to the limit of the presidential office), but in 1884 he
was unanimously re-elected. The government having set aside the
above-mentioned limitation, Diaz was continually re-elected to the
presidency. He married twice and had a son and two daughters. His gifted
second wife (Carmelita), very popular in Mexico, was many years younger
than himself. King Edward VII. made him an honorary grand commander of
the Bath in June 1906, in recognition of his wonderful administration as
perpetual president for over a quarter of a century.

  See also Mrs Alec Tweedie, _Porfirio Diaz, Seven Times President of
  Mexico_ (1906), and _Mexico as I saw it_ (1901); Dr Noll, _From Empire
  to Republic_ (1890); Lieut. Seaton Schroeder, _Fall of Maximilian's
  Empire_ (New York, 1887); R. de Z. Enriquez, _P. Diaz_ (1908); and an
  article by Percy Martin in _Quarterly Review_ for October 1909.
       (E. A. T.)



DIAZ DE NOVAES, BARTHOLOMEU (fl. 1481-1500), Portuguese explorer,
discoverer of the Cape of Good Hope, was probably a kinsman of João
Diaz, one of the first Portuguese to round Cape Bojador (1434), and of
Diniz Diaz, the discoverer of Cape Verde (1445). In 1478 a Bartholomeu
Diaz, probably identical with the discoverer, was exempted from certain
customary payments on ivory brought from the Guinea coast. In 1481 he
commanded one of the vessels sent by King John II. under Diogo
d'Azambuja to the Gold Coast. In 1486 he seems to have been a cavalier
of the king's household, and superintendent of the royal warehouses; on
the 10th of October in this year he received an annuity of 6000 reis
from King John for "services to come"; and some time after this
(probably about July or August 1487, rather than July 1486, the
traditional date) he left Lisbon with three ships to carry on the work
of African exploration so greatly advanced by Diogo Cão (1482-1486).
Passing Cão's farthest point near Cape Cross (in the modern German
South-west Africa and) in 21° 50´ S., he erected a pillar on what is now
known as Diaz Point, south of Angra Pequena or Lüderitz Bay, in 26° 38´
S.; of this fragments still exist. From this point (according to De
Barros) Diaz ran thirteen days southwards before strong winds, which
freshened to dangerous stormy weather, in a comparatively high southern
latitude, considerably south of the Cape. When the storm subsided the
Portuguese stood east; and failing, after several days' search, to find
land, turned north, and so struck the south coast of Cape Colony at
Mossel Bay (Diaz' Bahia dos Vaqueiros), half way between the Cape of
Good Hope and Port Elizabeth (February 3, 1488). Thence they coasted
eastward, passing Algoa Bay (Diaz' Bahia da Roca), erecting pillars (or
perhaps wooden crosses), it is said, on one of the islands in this bay
and at or near Cape Padrone farther east; of these no traces remain. The
officers and men now began to insist on return, and Diaz could only
persuade them to go as far as the estuary of the Great Fish River (Diaz'
Rio do Iffante, so named from his colleague, Captain João Iffante).
Here, however, half way between Port Elizabeth and East London (and
indeed from Cape Padrone), the north-easterly trend of the coast became
unmistakable; the way round Africa had been laid open. On his return
Diaz perhaps named Cape Agulhas after St Brandan; while on the
southernmost projection of the modern Cape peninsula, whose remarkable
highlands (Table Mountain, &c.) doubtless impressed him as the practical
termination of the continent, he bestowed, says De Barros, the name of
Cape of Storms (_Cabo Tormentoso_) in memory of the storms he had
experienced in these far southern waters; this name (in the ordinary
tradition) was changed by King John to that of Good Hope (_Cabo da Boa
Esperança_). Some excellent authorities, however, make Diaz himself give
the Cape its present name. Hard by this "so many ages unknown
promontory" the explorer probably erected his last pillar. After
touching at the Ilha do Principe (Prince's Island, south-west of the
Cameroons) as well as at the Gold Coast, he appeared at Lisbon in
December 1488. He had discovered 1260 m. of hitherto unknown coast; and
his voyage, taken with the letters soon afterwards received from Pero de
Covilhão (who by way of Cairo and Aden had reached Malabar on one side
and the "Zanzibar coast" on the other as far south as Sofala, in
1487-1488) was rightly considered to have solved the question of an
ocean route round Africa to the Indies and other lands of South and East
Asia.

No record has yet been found of any adequate reward for Diaz: on the
contrary, when the great Indian expedition was being prepared (for Vasco
da Gama's future leadership) Bartolomeu only superintended the building
and outfit of the ships; when the fleet sailed in 1497, he only
accompanied da Gama to the Cape Verde Islands, and after this was
ordered to El Mina on the Gold Coast. On Cabral's voyage of 1500 he was
indeed permitted to take part in the discovery of Brazil (April 22), and
thence should have helped to guide the fleet to India; but he perished
in a great storm off his own Cabo Tormentoso. Like Moses, as Galvano
says, he was allowed to see the Promised Land, but not to enter in.

  See João de Barros, _Asia_, Dec. I. bk. iii. ch. 4; Duarte Pacheco
  Pereira, _Esmeraldo de situ orbis_, esp. pp. 15, 90, 92, 94 and
  Raphael Bastos's introduction to the edition of 1892 (Pacheco met
  Diaz, returning from his great voyage, at the Ilha do Principe); a
  marginal note, probably by Christopher Columbus himself, on fol. 13 of
  a copy of Pierre d'Ailly's _Imago mundi_, now in the Colombina at
  Seville (the writer of this note fixes Diaz's return to Lisbon,
  December 1488, and says he was present at Diaz's interview with the
  king of Portugal, when the explorer described his voyage and showed
  his route upon the chart he had kept); a similar but briefer note in a
  copy of Pope Pius II.'s _Historia rerum ubique gestarum_, from the
  same hand; the _Roteiro_ of Vasco da Gama's First Voyage (_Journal of
  the First Voyage of ... Da Gama_, Hakluyt Soc., ed. E. G. Ravenstein
  (1898), pp. 9, 14); Ramusio, _Navigationi_ (3rd ed.), vol. i. fol.
  144; Castanheda, _Historia_, bk. i. ch. 1; Galvano, _Descobrimentos
  (Discoveries of the World)_, Hakluyt Soc. (1862), p. 77; E. G.
  Ravenstein, "Voyages of ... Cão and ... Dias," in _Geog. Journ._
  (London, December 1900), vol. xvi. pp. 638-655), an excellent critical
  summary in the light of the most recent investigations of all the
  material. The fragments of Diaz's only remaining pillar (from Diaz
  Point) are now partly at the Cape Museum, partly at Lisbon: the latter
  are photographed in Ravenstein's paper in _Geog. Journ._ (December
  1900, p. 642).     (C. R. B.)



DIAZO COMPOUNDS, in organic chemistry, compounds of the type R·N·2·X
(where R = a hydrocarbon radical, and X = an acid radical or a hydroxyl
group). These compounds may be divided into two classes, namely, the
true diazo compounds, characterized by the grouping - N = N -, and the
diazonium compounds, characterized by the grouping N:·N<.

The diazonium compounds were first discovered by P. Griess (_Ann._,
1858, 106, pp. 123 et seq.), and may be prepared by the action of
nitrous fumes on a well-cooled solution of a salt of a primary amine,

  C6H5NH2·HNO3 + HNO2 = C6H5N2·NO3 + 2H2O,

or, as is more usually the case (since the diazonium salts themselves
are generally used only in aqueous solution) by the addition of a
well-cooled solution of potassium or sodium nitrite to a well-cooled
dilute acid solution of the primary amine. In order to isolate the
anhydrous diazonium salts, the method of E. Knoevenagel (_Ber._, 1890,
23, p. 2094) may be employed. In this process the amine salt is
dissolved in absolute alcohol and diazotized by the addition of amyl
nitrite; a crystalline precipitate of the diazonium salt is formed on
standing, or on the addition of a small quantity of ether. The diazonium
salts are also formed by the action of zinc-dust and acids on the
nitrates of primary amines (R. Mohlau, _Ber._, 1883, 16, p. 3080), and
by the action of hydroxylamine on nitrosobenzenes. They are colourless
crystalline solids which turn brown on exposure. They dissolve easily in
water, but only to a slight extent in alcohol and ether. They are very
unstable, exploding violently when heated or rubbed. _Benzene diazonium
nitrate_, C6H5N(NO3):·N, crystallizes in long silky needles. The
sulphate and chloride are similar, but they are not quite so unstable as
the nitrate. The bromide may be prepared by the addition of bromine to
an ethereal solution of diazo-amino-benzene (tribromaniline remaining in
solution). By the addition of potassium bromide and bromine water to
diazonium salts they are converted into a _perbromide_, e.g. C6H5N2Br3,
which crystallizes in yellow plates.

  The diazonium salts are characterized by their great reactivity and
  consequently are important reagents in synthetical processes, since by
  their agency the amino group in a primary amine may be exchanged for
  other elements or radicals. The chief reactions are as follows:--

  1. _Replacement of -NH2 by -OH_:--The amine is diazotized and the
  aqueous solution of the diazonium salt is heated, nitrogen being
  eliminated and a phenol formed.

  2. _Replacement of -NH2 by halogens and by the -CN and -CNO
  groups_:--The diazonium salt is warmed with an acid solution of the
  corresponding cuprous salt (T. Sandmeyer, _Ber._, 1884, 17, p. 2650),
  or with copper powder (L. Gattermann, Ber., 1890, 23, p. 1218; 1892,
  25, p. 1074). In the case of iodine, the substitution is effected by
  adding a warm solution of potassium iodide to the diazonium solution,
  no copper or cuprous salt being necessary; whilst for the production
  of nitriles a solution of potassium cuprous cyanide is used. This
  reaction (the so-called "Sandmeyer" reaction) has been investigated by
  A. Hantzsch and J. W. Blagden (_Ber._, 1900, 33, p. 2544), who
  consider that three simultaneous reactions occur, namely, the
  formation of labile double salts which decompose in such a fashion
  that the radical attached to the copper atom wanders to the aromatic
  nucleus; a catalytic action, in which nitrogen is eliminated and the
  acid radical attaches itself to the aromatic nucleus; and finally, the
  formation of azo compounds.

  3. _Replacement of -NH2 by -NO2_:--A well-cooled concentrated solution
  of potassium mercuric nitrate is added to a cooled solution of benzene
  diazonium nitrate, when the crystalline salt 2C6H5N2·NO3, Hg(NO2)2 is
  precipitated. On warming this with copper powder, it gives a
  quantitative yield of nitrobenzene (A. Hantzsch, _Ber._, 1900, 33, p.
  2551).

  4. _Replacement of -NH2 by hydrogen_:--This exchange is brought about,
  in some cases, by boiling the diazonium salt with alcohol; but I.
  Remsen and his pupils (_Amer. Chem. Journ._, 1888, 9, pp. 389 et seq.)
  have shown that the main product of this reaction is usually a
  phenolic ether. This reaction has also been investigated by A.
  Hantzsch and E. Jochem (_Ber._, 1901, 34, p. 3337), who arrived at the
  conclusion that the normal decomposition of diazonium salts by
  alcohols results in the formation of phenolic ethers, but that an
  increase in the molecular weight of the alcohol, or the accumulation
  of negative groups in the aromatic nucleus, diminishes the yield of
  the ether and increases the amount of the hydrocarbon formed. The
  replacement is more readily brought about by the use of sodium
  stannite (P. Friedlander, _Ber._, 1889, 22, p. 587), or by the use of
  a concentrated solution of hypophosphorous acid (J. Mai, _Ber._, 1902,
  35, p. 162). A. Hantzsch (_Ber._, 1896, 29, p. 947; 1898, 31, p. 1253)
  has shown that the chlor- and brom- diazoniumthiocyanates, when
  dissolved in alcohol containing a trace of hydrochloric acid, become
  converted into the isomeric thiocyanbenzene diazonium chlorides and
  bromides. This change only occurs when the halogen atom is in the
  ortho- or para- position to the -N2- group.

  _Metallic Diazo Derivatives._--Benzene diazonium chloride is
  decomposed by silver oxide in aqueous solution, with the formation of
  _benzene diazonium hydroxide_, C6H5·N(OH):·N. This hydroxide, although
  possessing powerful basic properties, is unstable in the presence of
  alkalis and neutralizes them, being converted first into the isomeric
  benzene-diazotic acid, the potassium salt of which is obtained when
  the diazonium chloride is added to an excess of cold concentrated
  potash (A. Hantzsch and W. B. Davidson, _Ber._, 1898, 31, p. 1612).
  _Potassium benzene diazotate_, C6H5N2·OK, crystallizes in colourless
  silky needles. The free acid is not known; by the addition of the
  potassium salt to 50% acetic acid at -20° C., the acid anhydride,
  _benzene diazo oxide_, (C6H5N2)2O, is obtained as a very unstable,
  yellow, insoluble compound, exploding spontaneously at 0° C. Strong
  acids convert it into a diazonium salt, and potash converts it into
  the diazotate. On the constitution, of these anhydrides see E.
  Bamberger, _Ber._, 1896, 29, p. 446, and A. Hantzsch, _Ber._, 1896,
  29, p. 1067; 1898, 31, p. 636. By the addition of the diazonium salts
  to a hot concentrated solution of a caustic alkali, C. Schraube and C.
  Schmidt (_Ber._, 1894, 27, p. 520) obtained an isomer of potassium
  benzene diazotate. These _iso-_diazotates are formed much more readily
  when the aromatic nucleus in the diazonium salt contains negative
  radicals. _Potassium benzene iso-diazotate_ resembles the normal salt,
  but is more stable, and is more highly ionized. Carbon dioxide
  converts it into _phenyl nitrosamine_, C6H5NH·NO (A. Hantzsch). The
  potassium salt of the iso-diazo hydroxide yields on methylation a
  nitrogen ether, R·N(CH3)·NO, whilst the silver salt yields an oxygen
  ether, R·N:N·OCH3. These results point to the conclusion that the
  iso-diazo hydroxide is a tautomeric substance. The same oxygen ether
  is formed by the methylation of the silver salt of the normal diazo
  hydroxide; this points to the conclusion that the isomeric hydroxides,
  corresponding with the silver derivatives, have the same structural
  formulae, namely, R·N:N·OH. These oxygen ethers contain the grouping
  -N:N-, since they couple very readily with the phenols in alkaline
  solution to form azo compounds (q.v.) (E. Bamberger, _Ber._, 1895, 28,
  p. 225); they are also explosive.

  By oxidizing potassium benzene iso-diazotate with alkaline potassium
  ferricyanide, E. Bamberger (_Ber._, 1894, 27, p. 914) obtained the
  _diazoic acids_, R·NH·NO2, substances which he had previously prepared
  by similarly oxidizing the diazonium salts, by dehydrating the
  nitrates of primary amines with acetic anhydride, and by the action of
  nitric anhydride on the primary amines. Concentrated acids convert
  them into the isomeric nitro-amines, the -NO2 group going into the
  nucleus in the ortho- or para- position to the amine nitrogen; this
  appears to indicate that the compounds are nitramines. They behave,
  however, as tautomeric substances, since their alkali salts on
  methylation give nitrogen ethers, whilst their silver salts yield
  oxygen ethers:

            /-->  potassium salt --> R·N(CH3)·NO2 nitramine.
  R·NH·NO2<
            \-->  silver salt    --> R·N:N·O·OCH3 diazoate.

  _Phenyl nitramine_, C6H5NH·NO2, is a colourless crystalline solid,
  which melts at 46° C. Sodium amalgam in alkaline solution reduces it
  to phenylhydrazine.

  _Constitution of the Diazo Compounds._--P. Griess (_Ann._, 1866, 137,
  p. 39) considered that the diazo compounds were formed by the addition
  of complex groupings of the type C6H4N2- to the inorganic acids;
  whilst A. Kekulé (_Zeit. f. Chemie_, 1866, 2, p. 308), on account of
  their ready condensation to form azo compounds and their easy
  reduction to hydrazines, assumed that they were substances of the type
  R·N:N·Cl. The constitution of the diazonium group -N2·X, may be
  inferred from the following facts:--The group C6H5N2- behaves in many
  respects similarly to an alkali metal, and even more so to the
  ammonium group, since it is capable of forming colourless neutral
  salts with mineral acids, which in dilute aqueous solution are
  strongly ionized, but do not show any trace of hydrolytic dissociation
  (A. Hantzsch, _Ber._, 1895, 28, p. 1734). Again, the diazonium
  chlorides combine with platinic chloride to form difficultly soluble
  double platinum salts, such as (C6H5N2Cl)2·PtCl4; similar gold salts,
  C6H5N2Cl·AuCl3, are known. Determinations of the electrical
  conductivity of the diazonium chloride and nitrate also show that the
  diazonium radical is strictly comparable with other quaternary
  ammonium ions. For these reasons, one must assume the existence of
  pentavalent nitrogen in the diazonium salts, in order to account for
  their basic properties.

  The constitution of the isomeric diazo hydroxides has given rise to
  much discussion. E. Bamberger (_Ber._, 1895, 28, pp. 444 et seq.) and
  C. W. Blomstrand (_Journ. prakt. Chem._, 1896, 53, pp. 169 et seq.)
  hold that the compounds are structurally different, the normal
  diazo-hydroxide being a diazonium derivative of the type
  R·N([3:]N)·OH. The recent work of A. Hantzsch and his pupils seems to
  invalidate this view (_Ber._, 1894, 27, pp. 1702 et seq.; see also A.
  Hantzsch, _Die Diazoverbindungen_). According to Hantzsch the isomeric
  diazo hydroxides are structurally identical, and the differences in
  behaviour are due to stereo-chemical relations, the isomerism being
  comparable with that of the oximes (q.v.). On such a hypothesis, the
  relatively unstable normal diazo hydroxides would be the
  _syn-_compounds, since here the nitrogen atoms would be more easily
  eliminated, whilst the stable iso-diazo derivatives would be the
  _anti-_compounds, thus:

         R · N             R · N
             ..                ..
        HO · N                 N · OH
    Normal hydroxide      Iso hydroxide
    (Syn-compound)       (Anti-compound)

  In support of this theory, Hantzsch has succeeded in isolating a
  series of syn- and anti-diazo-cyanides and -sulphonates (_Ber._, 1895,
  28, p. 666; 1900, 33, p. 2161; 1901, 34, p. 4166). By diazotizing
  para-chloraniline and adding a cold solution of potassium cyanide, a
  salt (melting at 29° C.) is obtained, which readily loses nitrogen,
  and forms para-chlorbenzonitrile on the addition of copper powder. By
  dissolving this diazocyanide in alcohol and reprecipitating it by
  water, it is converted into the isomeric diazocyanide (melting at
  105-106° C.), which does not yield para-chlorbenzonitrile when treated
  with copper powder. Similar results have been obtained by using
  diazotized para-anisidine, a syn- and an anti- compound being formed,
  as well as a third isomeric cyanide, obtained by evaporating
  para-methoxy-benzenediazonium hydroxide in the presence of an excess
  of hydrocyanic acid at ordinary temperatures. This salt is a
  colourless crystalline substance of composition
  CH3O·C6H4·N2·CN·HCN·2H2O, and has the properties of a metallic salt;
  it is very soluble in water and its solution is an electrolyte,
  whereas the solutions of the syn-and anti- compounds are not
  electrolytes. The isolation of these compounds is a powerful argument
  in favour of the Hantzsch hypothesis which requires the existence of
  these three different types, whilst the Bamberger-Blomstrand view only
  accounts for the formation of two isomeric cyanides, namely, one of
  the normal diazonium type and one of the iso-diazocyanide type.

  Benzene diazonium hydroxide, although a strong base, reacts with the
  alkaline hydroxides to form salts with the evolution of heat, and
  generally behaves as a weak acid. On mixing dilute solutions of the
  diazonium hydroxide and the alkali together, it is found that the
  molecular conductivity of the mixture is much less than the sum of the
  two electrical conductivities of the solutions separately, from which
  it follows that a portion of the ions present have changed to the
  non-ionized condition. This behaviour is explained by considering the
  non-ionized part of the diazonium hydroxide to exist in solution in a
  hydrated form, the equation of equilibrium being:

          C6H5·N·       --> C6H5·N·OH
    H2O +     ... + OH'          |
               N        <--   HO·N·H

  On adding the alkaline hydroxide to the solution, this hydrate is
  supposed to lose water, yielding the syn-diazo hydroxide, which then
  gives rise to a certain amount of the sodium salt (A. Hantzsch,
  _Ber._, 1898, 31, p. 1612),

           ....
    C6H5·N·:OH:  -->  C6H5·N  -->  C6H5·N
         | :  :            ||           ||
      HO·N·:H :  <--    HO·N  <--   NaO·N
           ....

  This assumption also shows the relationship of the diazonium
  hydroxides to other quaternary ammonium compounds, for most of the
  quaternary ammonium hydroxides (except such as have the nitrogen atom
  attached to four saturated hydrocarbon radicals) are unstable, and
  readily pass over into compounds in which the hydroxyl group is no
  longer attached to the amine nitrogen; thus the syn-diazo hydroxides
  are to be regarded as pseudo-diazonium derivatives. (A. Hantzsch,
  _Ber._, 1899, 32, p. 3109; 1900, 33, p. 278.) It is generally accepted
  that the iso-diazo hydroxides possess the oxime structure R·N:N·OH.

  Hantzsch explains the characteristic reactions of the diazonium
  compounds by the assumption that an addition compound is first formed,
  which breaks down with the elimination of the hydride of the acid
  radical, and the formation of an unstable syn-diazo compound, which,
  in its turn, decomposes with evolution of nitrogen (_Ber._, 1897, 30,
  p. 2548; 1898, 31, p. 2053).

     R         X     R      X     R   X
      \        |      \    /      |   |
        N·:N + | -->   N·:N   --> |   | + HCl --> R·X + N2.
      /        |      /    \      |   |
    Cl         H    Cl      H     N = N

  J. Cain (_Jour. Chem. Soc._, 1907, 91, p. 1049) suggested a quinonoid
  formula for diazonium salts, which has been combated by Hantzsch
  (_Ber._, 1908, 41, pp. 3532 et seq.). G. T. Morgan and F. M. G.
  Micklethwaite (_Jour. Chem. Soc._, 1908, 93, p. 617; 1909, 95, p.
  1319) have pointed out that the salts may possess a dynamic formula,
  Cain's representing the middle stage, thus:

        /||\                /||\                /||\
       / || \              / || \              / || \
    H / N·Cl \ H        H / N·Cl \ H        H / N·Cl \ H
     |\  ||  ||  --->    ||  ||   |  --->    ||  ||  /|
     | \ ||  ||  <---    ||  ||   |  <---    ||  || / |
     |  \N   ||          ||  N    |          ||   N   |
    H\\      / H        H\\  |   / H        H \      //H
      \\    /             \\ |  /              \    //
       \\  /               \\| /                \  //
         H                   H                   H

  _Diazoamines._--The diazoamines, R·N2·NHR, may be prepared by the
  action of the primary and secondary amines on the diazonium salts, or
  by the action of nitrous acid on the free primary amine. In the latter
  reaction it is assumed that the isodiazohydroxide first formed is
  immediately attacked by a second molecule of the amine. They are
  yellow crystalline solids, which do not unite with acids. Nitrous acid
  converts them, in acid solution, into diazonium salts.

    C6H5N2·NHC6H5 + 2HCl + HNO2 = 2C6H5N2Cl + 2H2O.

  They are readily converted into the isomeric aminoazo compounds,
  either by standing in alcoholic solution, or by warming with a mixture
  of the parent base and its hydrochloride; the diazo group preferably
  going into the para-position to the amino group. When the
  para-position is occupied, the diazo group takes the ortho-position.
  H. Goldschmidt and R. U. Reinders (_Ber._, 1896, 29, p. 1369, 1899)
  have shown that the transformation is a monomolecular reaction, the
  velocity of transformation in moderately dilute solution being
  independent of the concentration, but proportional to the amount of
  the catalyst present (amine hydrochloride) and to the temperature. It
  has also been shown that when different salts of the amine are used,
  their catalytic influence varies in amount and is almost proportional
  to their degree of ionization in aqueous solution. Diazoaminobenzene,
  C6H5N2·NHC6H5, crystallizes in golden yellow laminae, which melt at
  96° C. and explode at a slightly higher temperature. It is readily
  soluble in alcohol, ether and benzene. Concentrated hydrochloric acid
  converts it into chlorbenzene, aniline and nitrogen. Zinc dust and
  alcoholic acetic acid reduce it to aniline and phenylhydrazine.

  _Diazoimino compounds_, R·N3, may be regarded as derivatives of
  azoimide (q.v.); they are formed by the action of ammonia on the
  diazoperbromides, or by the action of hydroxylamine on the diazonium
  sulphates (J. Mai, _Ber._, 1892, 25, p. 372; T. Curtius, _Ber._, 1893,
  26, p. 1271). Diazobenzeneimide, C6H5N3, is a yellowish oil of
  stupefying odour. It boils at 59° C. (12 mm.), and explodes when
  heated. Concentrated hydrochloric acid decomposes it with formation of
  chloranilines and elimination of nitrogen, whilst on boiling with
  sulphuric acid it is converted into aminophenols.

  _Aliphatic Diazo Compounds._--The esters of the aliphatic amino acids
  may be diazotized in a manner similar to the primary aromatic amines,
  a fact discovered by T. Curtius (_Ber._, 1833, 16, p. 2230). The first
  aliphatic diazo compound to be isolated was _diazoacetic ester_,
  CH·N2·CO2C2H5, which is prepared by the action of potassium nitrite on
  the ethyl ester of glycocoll hydrochloride, HCl·NH2·CH2·CO2C2H5 + KNO2
  = CHN2·CO2C2H5 + KCl + 2H2O. It is a yellowish oil which melts at -24°
  C.; it boils at 143-144° C., but cannot be distilled safely as it
  decomposes violently, giving nitrogen and ethyl fumarate. It explodes
  in contact with concentrated sulphuric acid. On reduction it yields
  ammonia and glycocoll (aminoacetic acid). When heated with water it
  forms ethyl hydroxy-acetate; with alcohol it yields ethyl
  ethoxyacetate. Halogen acids convert it into monohalogen fatty acids,
  and the halogens themselves convert it into dihalogen fatty acids. It
  unites with aldehydes to form esters of ketonic acids, and with
  aniline yields anilido-acetic acid. It forms an addition product with
  acrylic ester, which on heating loses nitrogen and leaves trimethylene
  dicarboxylic ester. Concentrated ammonia converts it into
  _diazoacetamide_, CHN2·CONH2, which crystallizes in golden yellow
  plates which melt at 114° C. For other reactions see HYDRAZINE. The
  constitution of the diazo fatty esters is inferred from the fact that
  the two nitrogen atoms, when split off, are replaced by two monovalent
  elements or groups, thus leading to the formula

     N \
    ..  > CH·CO2C2H5, for diazoacetic ester.
     N /

  _Diazosuccinic ester_, N2·C(CO2C2H5)2, is similarly prepared by the
  action of nitrous acid on the hydrochloride of aspartic ester. It is
  decomposed by boiling water and yields fumaric ester.

  _Diazomethane_, CH2N2, was first obtained in 1894 by H. v. Pechmann
  (_Ber._, 1894, 27, p. 1888; 1895, 28, p. 855). It is prepared by the
  action of aqueous or alcoholic solutions of the caustic alkalis on the
  nitroso-acidyl derivatives of methylamine (such, for example, as
  _nitrosomethyl urethane_, NO·N(CH3)·CO2C2H5, which is formed on
  passing nitrous fumes into an ethereal solution of methyl urethane).
  E. Bamberger (_Ber._, 1895, 28, p. 1682) regards it as the anhydride
  of iso-diazomethane, CH3·N:N·OH, and has prepared it by a method
  similar to that used for the preparation of iso-diazobenzene. By the
  action of bleaching powder on methylamine hydrochloride, there is
  obtained a volatile liquid (_methyldichloramine_, CH3·N·Cl2), boiling
  at 58-60° C., which explodes violently when heated with water,
  yielding hydrocyanic acid (CH3NCl2 = HCN + 2HCl). Well-dried
  hydroxylamine hydrochloride is dissolved in methyl alcohol and mixed
  with sodium methylate; a solution of methyldichloramine in absolute
  ether is then added and an ethereal solution of diazomethane distils
  over. Diazomethane is a yellow inodorous gas, very poisonous and
  corrosive. It may be condensed to a liquid, which boils at about 0° C.
  It is a powerful methylating agent, reacting with water to form methyl
  alcohol, and converting acetic acid into methylacetate, hydrochloric
  acid into methyl chloride, hydrocyanic acid into acetonitrile, and
  phenol into anisol, nitrogen being eliminated in each case. It is
  reduced by sodium amalgam (in alcoholic solution) to
  _methylhydrazine_, CH3·NH·NH2. It unites directly with acetylene to
  form pyrazole (H. v. Pechmann, _Ber._, 1898, 31, p. 2950) and with
  fumaric methyl ester it forms pyrazolin dicarboxylic ester.
       (F. G. P.*)

    See G. T. Morgan, _B.A. Rep._, 1902; J. Cain, _Diazo Compounds_,
    1908.



DIAZOMATA (Gr. [Greek: diazôma], a girdle), in architecture, the landing
places and passages which were carried round the semicircle and
separated the upper and lower tiers in a Greek theatre.



DIBDIN, CHARLES (1745-1814), British musician, dramatist, novelist,
actor and song-writer, the son of a parish clerk, was born at
Southampton on or before the 4th of March 1745, and was the youngest of
a family of eighteen. His parents designing him for the church, he was
sent to Winchester; but his love of music early diverted his thoughts
from the clerical profession. After receiving some instruction from the
organist of Winchester cathedral, where he was a chorister from 1756 to
1759, he went to London at the age of fifteen. Here he was placed in a
music warehouse in Cheapside, but he soon abandoned this employment to
become a singing actor at Covent Garden. On the 21st of May 1762 his
first work, an operetta entitled _The Shepherd's Artifice_, with words
and music by himself, was produced at this theatre. Other works
followed, his reputation being firmly established by the music to the
play of _The Padlock_, produced at Drury Lane under Garrick's management
in 1768, the composer himself taking the part of Mungo with conspicuous
success. He continued for some years to be connected with Drury Lane,
both as composer and as actor, and produced during this period two of
his best known works, _The Waterman_ (1774) and _The Quaker_ (1775). A
quarrel with Garrick led to the termination of his engagement. In _The
Comic Mirror_ he ridiculed prominent contemporary figures through the
medium of a puppet show. In 1782 he became joint manager of the Royal
circus, afterwards known as the Surrey theatre. In three years he lost
this position owing to a quarrel with his partner. His opera _Liberty
Hall_, containing the successful songs "Jock Ratlin," "The Highmettled
Racer," and "The Bells of Aberdovey," was produced at Drury Lane theatre
on the 8th of February 1785. In 1788 he sailed for the East Indies, but
the vessel having put in to Torbay in stress of weather, he changed his
mind and returned to London. In a musical variety entertainment called
_The Oddities_, he succeeded in winning marked popularity with a number
of songs that included "'Twas in the good ship 'Rover'," "Saturday Night
at Sea," "I sailed from the Downs in the 'Nancy,'" and the immortal "Tom
Bowling," written on the death of his eldest brother, Captain Thomas
Dibdin, at whose invitation he had planned his visit to India. A series
of monodramatic entertainments which he gave at his theatre, Sans Souci,
in Leicester Square, brought his songs, music and recitations more
prominently into notice, and permanently established his fame as a lyric
poet. It was at these entertainments that he first introduced many of
those sea-songs which so powerfully influenced the national spirit. The
words breathe the simple loyalty and dauntless courage that are the
cardinal virtues of the British sailor, and the music was appropriate
and naturally melodious. Their effect in stimulating and ennobling the
spirit of the navy during the war with France was so marked as to call
for special acknowledgment. In 1803 Dibdin was rewarded by government
with a pension of £200 a year, of which he was only for a time deprived
under the administration of Lord Grenville. During this period he opened
a music shop in the Strand, but the venture was a failure. Dibdin died
of paralysis in London on the 25th of July 1814. Besides his _Musical
Tour through England_ (1788), his _Professional Life_, an autobiography
published in 1803, a _History of the Stage_ (1795), and several smaller
works, he wrote upwards of 1400 songs and about thirty dramatic pieces.
He also wrote the following novels:--_The Devil_ (1785); _Hannah Hewitt_
(1792); _The Younger Brother_ (1793). An edition of his songs by G.
Hogarth (1843) contains a memoir of his life. His two sons, Charles and
Thomas John Dibdin (q.v.), whose works are often confused with those of
their father, were also popular dramatists in their day.



DIBDIN, THOMAS FROGNALL (1776-1847), English bibliographer, born at
Calcutta in 1776, was the son of Thomas Dibdin, the sailor brother of
Charles Dibdin. His father and mother both died on the way home to
England in 1780, and Thomas was brought up by a maternal uncle. He was
educated at St John's College, Oxford, and studied for a time at
Lincoln's Inn. After an unsuccessful attempt to obtain practice as a
provincial counsel at Worcester, he was ordained a clergyman at the
close of 1804, being appointed to a curacy at Kensington. It was not
until 1823 that he received the living of Exning in Sussex. Soon
afterwards he was appointed by Lord Liverpool to the rectory of St
Mary's, Bryanston Square, which he held until his death on the 18th of
November 1847. The first of his numerous bibliographical works was his
_Introduction to the Knowledge of Editions of the Classics_ (1802),
which brought him under the notice of the third Earl Spencer, to whom he
owed much important aid in his bibliographical pursuits. The rich
library at Althorp was thrown open to him; he spent much of his time in
it, and in 1814-1815 published his _Bibliotheca Spenceriana_. As the
library was not open to the general public, the information given in the
_Bibliotheca_ was found very useful, but since its author was unable
even to read the characters in which the books he described were
written, the work was marred by the errors which more or less
characterize all his productions. This fault of inaccuracy however was
less obtrusive in his series of playful, discursive works in the form of
dialogues on his favourite subject, the first of which, _Bibliomania_
(1809), was republished with large additions in 1811, and was very
popular, passing through numerous editions. To the same class belonged
the _Bibliographical Decameron_, a larger work, which appeared in 1817.
In 1810 he began the publication of a new and much extended edition of
Ames's _Typographical Antiquities_. The first volume was a great
success, but the publication was checked by the failure of the fourth
volume, and was never completed. In 1818 Dibdin was commissioned by
Earl Spencer to purchase books for him on the continent, an expedition
described in his sumptuous _Bibliographical, Antiquarian and Picturesque
Tour in France and Germany_ (1821). In 1824 he made an ambitious venture
in his _Library Companion, or the Young Man's Guide and Old Man's
Comfort in the Choice of a Library_, intended to point out the best
works in all departments of literature. His culture was not broad
enough, however, to render him competent for the task, and the work was
severely criticized. For some years Dibdin gave himself up chiefly to
religious literature. He returned to bibliography in his _Bibliophobia,
or Remarks on the Present Depression in the State of Literature and the
Book Trade_ (1832), and the same subject furnishes the main interest of
his _Reminiscences of a Literary Life_ (1836), and his _Bibliographical,
Antiquarian and Picturesque Tour in the Northern Counties of England and
Scotland_ (1838). Dibdin was the originator and vice-president, Lord
Spencer being the president, of the Roxburghe Club, founded in
1812,--the first of the numerous book clubs which have done such service
to literature.



DIBDIN, THOMAS JOHN (1771-1841), English dramatist and song-writer, son
of Charles Dibdin, the song-writer, and of Mrs Davenet, an actress whose
real name was Harriet Pitt, was born on the 21st of March 1771. He was
apprenticed to his maternal uncle, a London upholsterer, and later to
William Rawlins, afterwards sheriff of London. He summoned his second
master unsuccessfully for rough treatment; and after a few years of
service he ran away to join a company of country players. From 1789 to
1795 he played in all sorts of parts; he acted as scene painter at
Liverpool in 1791; and during this period he composed more than 1000
songs. He made his first attempt as a dramatic writer in _Something
New_, followed by _The Mad Guardian_ in 1795. He returned to London in
1795, having married two years before; and in the winter of 1798-1799
his _Jew and the Doctor_ was produced at Covent Garden. From this time
he contributed a very large number of comedies, operas, farces, &c., to
the public entertainment. Some of these brought immense popularity to
the writer and immense profits to the theatres. It is stated that the
pantomime of _Mother Goose_ (1807) produced more than £20,000 for the
management at Covent Garden theatre, and the _High-mettled Racer_,
adapted as a pantomime from his father's play, £18,000 at Astley's.
Dibdin was prompter and pantomime writer at Drury Lane until 1816, when
he took the Surrey theatre. This venture proved disastrous and he became
bankrupt. After this he was manager of the Haymarket, but without his
old success, and his last years were passed in comparative poverty. In
1827 he published two volumes of _Reminiscences_; and at the time of his
death he was preparing an edition of his father's sea songs, for which a
small sum was allowed him weekly by the lords of the admiralty. Of his
own songs "The Oak Table" and "The Snug Little Island" are well-known
examples. He died in London on the 16th of September 1841.



DIBRA (Slav. _Debra_), the capital of a sanjak bearing the same name, in
the vilayet of Monastir, eastern Albania, Turkey. Pop. (1900) about
15,000. Dibra occupies a valley enclosed by mountains, and watered by
the Tsrni Drin and Radika rivers, which meet 3 m. S. It is a fortified
city, and the only episcopal see of the Bulgarian exarchate in Albania;
most of the inhabitants are Albanians, but there is a strong Bulgarian
colony. The local trade is almost entirely agricultural.



DIBRUGARH, a town of British India, in the Lakhimpur district of eastern
Bengal and Assam, of which it is the headquarters, situated on the Dibru
river about 4 m. above its confluence with the Brahmaputra. Pop. (1901)
11,227. It is the terminus of steamer navigation on the Brahmaputra, and
also of a railway running to important coal-mines and petroleum wells,
which connects with the Assam-Bengal system. Large quantities of coal
and tea are exported. There are a military cantonment, the headquarters
of the volunteer corps known as the Assam Valley Light Horse; a
government high school, a training school for masters; and an aided
school for girls. In 1900 a medical school for the province was
established, out of a bequest left by Brigade-Surgeon J. Berry-White,
which is maintained by the government, to train hospital assistants for
the tea gardens. The Williamson artisan school is entirely supported by
an endowment.



DICAEARCHUS, of Messene in Sicily, Peripatetic philosopher and pupil of
Aristotle, historian, and geographer, flourished about 320 B.C. He was a
friend of Theophrastus, to whom he dedicated the majority of his works.
Of his writings, which comprised treatises on a great variety of
subjects, only the titles and a few fragments survive. The most
important of them was his [Greek: bios tês Hellados] (_Life in Greece_),
in which the moral, political and social condition of the people was
very fully discussed. In his _Tripoliticos_ he described the best form
of government as a mixture of monarchy, aristocracy and democracy, and
illustrated it by the example of Sparta. Among the philosophical works
of Dicaearchus may be mentioned the _Lesbiaci_, a dialogue in three
books, in which the author endeavours to prove that the soul is mortal,
to which he added a supplement called _Corinthiaci_. He also wrote a
_Description of the World_ illustrated by maps, in which was probably
included his _Measurements of Mountains_. A description of Greece (150
iambics, in C. Müller, _Frag. hist. Graec_. i. 238-243) was formerly
attributed to him, but, as the initial letters of the first twenty-three
lines show, was really the work of Dionysius, son of Calliphon. Three
considerable fragments of a prose description of Greece (Müller, i.
97-110) are now assigned to an unknown author named Heracleides. The _De
re publica_ of Cicero is supposed to be founded on one of Dicaearchus's
works.

  The best edition of the fragments is by M. Fuhr (1841), a work of
  great learning; see also a dissertation by F. G. Osann, _Beiträge zur
  röm. und griech. Litteratur_, ii. pp. 1-117 (1839); Pauly-Wissowa,
  _Realencyclopädie der klass. Altertumswiss_. v. pt. 1 (1905).



DICE (plural of die, O. Fr. _de_, derived from Lat. _dare_, to give),
small cubes of ivory, bone, wood or metal, used in gaming. The six sides
of a die are each marked with a different number of incised dots in such
a manner that the sum of the dots on any two opposite sides shall be 7.
Dice seem always to have been employed, as is the case to-day, for
gambling purposes, and they are also used in such games as backgammon.
There are many methods of playing, from one to five dice being used,
although two or three are the ordinary numbers employed in Great Britain
and America. The dice are thrown upon a table or other smooth surface
either from the hand or from a receptacle called a dice-box, the latter
method having been in common use in Greece, Rome and the Orient in
ancient times. Dice-boxes have been made in many shapes and of various
materials, such as wood, leather, agate, crystal, metal or paper. Many
contain bars within to ensure a proper agitation of the dice, and thus
defeat trickery. Some, formerly used in England, were employed with
unmarked dice, and allowed the cubes to fall through a kind of funnel
upon a board marked off into six equal parts numbered from 1 to 6. It is
a remarkable fact, that, wherever dice have been found, whether in the
tombs of ancient Egypt, of classic Greece, or of the far East, they
differ in no material respect from those in use to-day, the elongated
ones with rounded ends found in Roman graves having been, not dice but
_tali_, or knucklebones. Eight-sided dice have comparatively lately been
introduced in France as aids to children in learning the multiplication
table. The teetotum, or spinning die, used in many modern games, was
known in ancient times in China and Japan. The increased popularity of
the more elaborate forms of gaming has resulted in the decline of
dicing. The usual method is to throw three times with three dice. If one
or more sixes or fives are thrown the first time they may be reserved,
the other throws being made with the dice that are left. The object is
to throw three sixes = 18 or as near that number as possible, the
highest throw winning, or, when drinks are to be paid for, the lowest
throw losing. (For other methods of throwing consult the _Encyclopaedia
of Indoor Games_, by R. F. Foster, 1903.) The most popular form of pure
gambling with dice at the present day, particularly with the lower
classes in America, is _Craps_, or _Crap-Shooting_, a simple form of
_Hazard_, of French origin. Two dice are used. Each player puts up a
stake and the first caster may cover any or all of the bets. He then
_shoots_, i.e. throws the dice from his open hand upon the table. If the
sum of the dice is 7 or 11 the throw is a _nick_, or _natural_, and the
caster wins all stakes. If the throw is either 2, 3 or 12 it is a
_crap_, and the caster loses all. If any other number is thrown it is a
_point_, and the caster continues until he throws the same number again,
in which case he wins, or a 7, in which case he loses. The now
practically obsolete game of Hazard was much more complicated than
_Craps_. (Consult _The Game of Hazard Investigated_, by George Lowbut.)
_Poker dice_ are marked with ace, king, queen, jack and ten-spot. Five
are used and the object is, in three throws, to make pairs, triplets,
full hands or fours and fives of a kind, five aces being the highest
hand. Straights do not count. In throwing to decide the payment of
drinks the usual method is called _horse and horse_, in which the
highest throws retire, leaving the two lowest to decide the loser by the
best two in three throws. Should each player win one throw both are said
to be _horse and horse_, and the next throw determines the loser. The
two last casters may also agree to _sudden death_, i.e. a single throw.
_Loaded dice_, i.e. dice weighted slightly on the side of the lowest
number, have been used by swindlers from the very earliest times to the
present day, a fact proved by countless literary allusions. Modern dice
are often rounded at the corners, which are otherwise apt to wear off
irregularly.

_History._--Dice were probably evolved from knucklebones. The antiquary
Thomas Hyde, in his _Syntagma_, records his opinion that the game of
"odd or even," played with pebbles, is nearly coeval with the creation
of man. It is almost impossible to trace clearly the development of dice
as distinguished from knucklebones, on account of the confusing of the
two games by the ancient writers. It is certain, however, that both were
played in times antecedent to those of which we possess any written
records. Sophocles, in a fragment, ascribed their invention to
Palamedes, a Greek, who taught them to his countrymen during the siege
of Troy, and who, according to Pausanias (on Corinth, xx.), made an
offering of them on the altar of the temple of Fortune. Herodotus
(_Clio_) relates that the Lydians, during a period of famine in the days
of King Atys, invented dice, knucklebones and indeed all other games
except chess. The fact that dice have been used throughout the Orient
from time immemorial, as has been proved by excavations from ancient
tombs, seems to point clearly to an Asiatic origin. Dicing is mentioned
as an Indian game in the _Rig-veda_. In its primitive form knucklebones
was essentially a game of skill, played by women and children, while
dice were used for gambling, and it was doubtless the gambling spirit of
the age which was responsible for the derivative form of knucklebones,
in which four sides of the bones received different values, which were
then counted, like dice. Gambling with three, sometimes two, dice
([Greek: kuboi]) was a very popular form of amusement in Greece,
especially with the upper classes, and was an almost invariable
accompaniment to the symposium, or drinking banquet. The dice were cast
from conical beakers, and the highest throw was three sixes, called
_Aphrodite_, while the lowest, three aces, was called the _dog_. Both in
Greece and Rome different modes of counting were in vogue. Roman dice
were called _tesserae_ from the Greek word for four, indicative of the
four sides. The Romans were passionate gamblers, especially in the
luxurious days of the Empire, and dicing was a favourite form, though it
was forbidden except during the Saturnalia. The emperor Augustus wrote
in a letter to Suetonius concerning a game that he had played with his
friends: "Whoever threw a _dog_ or a six paid a _denarius_ to the bank
for every die, and whoever threw a _Venus_ (the highest) won
everything." In the houses of the rich the dice-beakers were of carved
ivory and the dice of crystal inlaid with gold. Mark Antony wasted his
time at Alexandria with dicing, while, according to Suetonius, the
emperors Augustus, Nero and Claudius were passionately fond of it, the
last named having written a book on the game. Caligula notoriously
cheated at the game; Domitian played it, and Commodus set apart special
rooms in his palace for it. The emperor Verus, adopted son of Antonine,
is known to have thrown dice whole nights together. Fashionable society
followed the lead of its emperors, and, in spite of the severity of the
laws, fortunes were squandered at the dicing table. Horace derided the
youth of the period, who wasted his time amid the dangers of dicing
instead of taming his charger and giving himself up to the hardships of
the chase. Throwing dice for money was the cause of many special laws in
Rome, according to one of which no suit could be brought by a person who
allowed gambling in his house, even if he had been cheated or assaulted.
Professional gamblers were common, and some of their loaded dice are
preserved in museums. The common public-houses were the resorts of
gamblers, and a fresco is extant showing two quarrelling dicers being
ejected by the indignant host. Virgil, in the _Copa_ generally ascribed
to him, characterizes the spirit of that age in verse, which has been
Englished as follows:--

  "What ho! Bring dice and good wine!
     Who cares for the morrow?
   Live--so calls grinning Death--
     Live, for I come to you soon!"

That the barbarians were also given to gaming, whether or not they
learned it from their Roman conquerors, is proved by Tacitus, who states
that the Germans were passionately fond of dicing, so much so, indeed,
that, having lost everything, they would even stake their personal
liberty. Centuries later, during the middle ages, dicing became the
favourite pastime of the knights, and both dicing schools (_scholae
deciorum_) and gilds of dicers existed. After the downfall of feudalism
the famous German mercenaries called _landsknechts_ established a
reputation as the most notorious dicing gamblers of their time. Many of
the dice of the period were curiously carved in the images of men and
beasts. In France both knights and ladies were given to dicing, which
repeated legislation, including interdictions on the part of St Louis in
1254 and 1256, did not abolish. In Japan, China, Korea, India and other
Asiatic countries dice have always been popular and are so still.

  See Foster's _Encyclopaedia of Indoor Games_ (1903); Raymond's
  _Illustriertes Knobelbrevier_ (Oranienburg, 1888); _Les Jeux des
  Anciens_, by L. Becq de Fouquières (Paris, 1869); _Das Knöchelspiel
  der Alten_, by Bolle (Wismar, 1886); _Die Spiele der Griechen und
  Römer_, by W. Richter (Leipzig, 1887); Raymond's _Alte und neue
  Würfelspiele_; _Chinese Games with Dice_, by Stewart Culin
  (Philadelphia, 1889); _Korean Games_, by Stewart Culin (Philadelphia,
  1895).



DICETO, RALPH DE (d. c. 1202), dean of St Paul's, London, and
chronicler, is first mentioned in 1152, when he received the
archdeaconry of Middlesex. He was probably born between 1120 and 1130;
of his parentage and nationality we know nothing. The common statement
that he derived his surname from Diss in Norfolk is a mere conjecture;
Dicetum may equally well be a Latinized form of Dissai, or Dicy, or
Dizy, place names which are found in Maine, Picardy, Burgundy and
Champagne. In 1152 Diceto was already a master of arts; presumably he
had studied at Paris. His reputation for learning and integrity stood
high; he was regarded with respect and favour by Arnulf of Lisieux and
Gilbert Foliot of Hereford (afterwards of London), two of the most
eminent bishops of their time. Quite naturally, the archdeacon took in
the Becket question the same side as his friends. Although his narrative
is colourless, and although he was one of those who showed some sympathy
for Becket at the council of Northampton (1164), the correspondence of
Diceto shows that he regarded the archbishop's conduct as
ill-considered, and that he gave advice to those whom Becket regarded as
his chief enemies. Diceto was selected, in 1166, as the envoy of the
English bishops when they protested against the excommunications
launched by Becket. But, apart from this episode, which he
characteristically omits to record, he remained in the background. The
natural impartiality of his intellect was accentuated by a certain
timidity, which is apparent in his writings no less than in his life.
About 1180 he became dean of St Paul's. In this office he distinguished
himself by careful management of the estates, by restoring the
discipline of the chapter, and by building at his own expense a
deanery-house. A scholar and a man of considerable erudition, he showed
a strong preference for historical studies; and about the time when he
was preferred to the deanery he began to collect materials for the
history of his own times. His friendships with Richard Fitz Nigel, who
succeeded Foliot in the see of London, with William Longchamp, the
chancellor of Richard I., and with Walter of Coutances, the archbishop
of Rouen, gave him excellent opportunities of collecting information.
His two chief works, the _Abbreviationes Chronicorum_ and the _Ymagines
Historiarum_, cover the history of the world from the birth of Christ to
the year 1202. The former, which ends in 1147, is a work of learning and
industry, but almost entirely based upon extant sources. The latter,
beginning as a compilation from Robert de Monte and the letters of
Foliot, becomes an original authority about 1172, and a contemporary
record about 1181. In precision and fulness of detail the _Ymagines_ are
inferior to the chronicles of the so-called Benedict and of Hoveden.
Though an annalist, Diceto is careless in his chronology; and the
documents which he incorporates, while often important, are selected on
no principle. He has little sense of style; but displays considerable
insight when he ventures to discuss a political situation. For this
reason, and on account of the details with which they supplement the
more important chronicles of the period, the _Ymagines_ are a valuable
though a secondary source.

  See W. Stubbs' edition of the _Historical Works_ of Diceto (Rolls ed.
  1876, 2 vols.), and especially the introduction. The second volume
  contains minor works which are the barest compendia of facts taken
  from well-known sources. Diceto's fragmentary Domesday of the
  capitular estates has been edited by Archdeacon Hale in _The Domesday
  of St Paul's_, pp. 109 ff. (Camden Society, 1858).



DICEY, EDWARD (1832- ), English writer, son of T. E. Dicey of Claybrook
Hall, Leicestershire, was born in 1832. Educated at Trinity College,
Cambridge, where he took mathematical and classical honours, he became
an active journalist, contributing largely to the principal reviews. He
was called to the bar in 1875, became a bencher of Gray's Inn in 1896,
and was treasurer in 1903-1904. He was connected with the _Daily
Telegraph_ as leader writer and then as special correspondent, and after
a short spell in 1870 as editor of the _Daily News_ he became editor of
the _Observer_, a position which he held until 1889. Of his many books
on foreign affairs perhaps the most important are his _England and
Egypt_ (1884), _Bulgaria, the Peasant State_ (1895), _The Story of the
Khedivate_ (1902), and _The Egypt of the Future_ (1907). He was created
C.B. in 1886.

His brother ALBERT VENN DICEY (b. 1835), English jurist, was educated at
Balliol College, Oxford, where he took a first class in the classical
schools in 1858. He was called to the bar at the Inner Temple in 1863.
He held fellowships successively at Balliol, Trinity and All Souls', and
from 1882 to 1909 was Vinerian professor of law. He became Q.C. in 1890.
His chief works are the _Introduction to the Study of the Law of the
Constitution_ (1885, 6th ed. 1902), which ranks as a standard work on
the subject; _England's Case against Home Rule_ (1886); _A Digest of the
Law of England with Reference to the Conflict of Laws_ (1896), and
_Lectures on the Relation between Law and Public Opinion in England
during the 19th century_ (1905).



DICHOTOMY (Gr. [Greek: dicha], apart, [Greek: temnein], to cut),
literally a cutting asunder, the technical term for a form of logical
division, consisting in the separation of a genus into two species, one
of which has and the other has not, a certain quality or attribute. Thus
men may be thus divided into white men, and men who are not white; each
of these may be subdivided similarly. On the principle of contradiction
this division is both exhaustive and exclusive; there can be no
overlapping, and no members of the original genus or the lower groups
are omitted. This method of classification, though formally accurate,
has slight value in the exact sciences, partly because at every step one
of the two groups is merely negatively characterized and therefore
incapable of real subdivision; it is useful, however, in setting forth
clearly the gradual descent from the most inclusive genus (_summum
genus_) through species to the lowest class (_infima species_), which is
divisible only into individual persons or things. (See further
DIVISION.) In astronomy the term is used for the aspect of the moon or
of a planet when apparently half illuminated, so that its disk has the
form of a semicircle.



DICK, ROBERT (1811-1866), Scottish geologist and botanist, was born at
Tullibody, in Clackmannanshire, in January 1811. His father was an
officer of excise. At the age of thirteen, after receiving a good
elementary education at the parish school, Robert Dick was apprenticed
to a baker, and served for three years. In these early days he became
interested in wild flowers--he made a collection of plants and gradually
acquired some knowledge of their names from an old encyclopaedia. When
his time was out he left Tullibody and gained employment as a journeyman
baker at Leith, Glasgow and Greenock. Meanwhile his father, who in 1826
had been removed to Thurso, as supervisor of excise, advised his son to
set up a baker's shop in that town. Thither Robert Dick went in 1830, he
started in business as a baker and worked laboriously until he died on
the 24th of December 1866. Throughout this period he zealously devoted
himself to studying and collecting the plants, mollusca and insects of a
wide area of Caithness, and his attention was directed soon after he
settled in Thurso to the rocks and fossils. In 1835 he first found
remains of fossil fishes; but it was not till some years later that his
interest became greatly stirred. Then he obtained a copy of Hugh
Miller's _Old Red Sandstone_ (published in 1841), and he began
systematically to collect with hammer and chisel the fossils from the
Caithness flags. In 1845 he found remains of _Holoptychius_ and
forwarded specimens to Hugh Miller, and he continued to send the best of
his fossil fishes to that geologist, and to others after the death of
Miller. In this way he largely contributed to the progress of geological
knowledge, although he himself published nothing and was ever averse
from publicity. His herbarium, which consisted of about 200 folios of
mosses, ferns and flowering plants "almost unique in its completeness,"
is now stored, with many of his fossils, in the museum at Thurso. Dick
had a hard struggle for existence, especially through competition during
his late years, when he was reduced almost to beggary: but of this few,
if any, of his friends were aware until it was too late. A monument
erected in the new cemetery at Thurso testifies to the respect which his
life-work created, when the merits of this enthusiastic naturalist came
to be appreciated.

  See _Robert Dick, Baker of Thurso, Geologist and Botanist_, by Samuel
  Smiles (1878).



DICK, THOMAS (1774-1857), Scottish writer on astronomy, was born at
Dundee on the 24th of November 1774. The appearance of a brilliant
meteor inspired him, when in his ninth year, with a passion for
astronomy; and at the age of sixteen he forsook the loom, and supported
himself by teaching. In 1794 he entered the university of Edinburgh, and
set up a school on the termination of his course; then, in 1801, took
out a licence to preach, and officiated for some years as probationer in
the United Presbyterian church. From about 1807 to 1817 he taught in the
secession school at Methven in Perthshire, and during the ensuing decade
in that of Perth, where he composed his first substantive book, _The
Christian Philosopher_ (1823, 8th ed. 1842). Its success determined his
vocation as an author; he built himself, in 1827, a cottage at Broughty
Ferry, near Dundee, and devoted himself wholly to literary and
scientific pursuits. They proved, however, owing to his unpractical turn
of mind, but slightly remunerative, and he was in 1847 relieved from
actual poverty by a crown pension of £50 a year, eked out by a local
subscription. He died on the 29th of July 1857. His best-known works
are: _Celestial Scenery_ (1837), _The Sidereal Heavens_ (1840), and _The
Practical Astronomer_ (1845), in which is contained (p. 204) a
remarkable forecast of the powers and uses of celestial photography.
Written with competent knowledge, and in an agreeable style, they
obtained deserved and widespread popularity.

  See R. Chambers's _Eminent Scotsmen_ (ed. 1868); _Monthly Notices Roy.
  Astr. Society_, xviii. 98; _Athenaeum_ (1857), p. 1008. (A. M. C.)



DICKENS, CHARLES JOHN HUFFAM (1812-1870), English novelist, was born on
the 7th of February 1812 at a house in the Mile End Terrace, Commercial
Road, Landport (Portsea)--a house which was opened as a Dickens Museum
on 22nd July 1904. His father John Dickens (d. 1851), a clerk in the
navy-pay office on a salary of £80 a year, and stationed for the time
being at Portsmouth, had married in 1809 Elizabeth, daughter of Thomas
Barrow, and she bore him a family of eight children, Charles being the
second. In the winter of 1814 the family moved from Portsea in the snow,
as he remembered, to London, and lodged for a time near the Middlesex
hospital. The country of the novelist's childhood, however, was the
kingdom of Kent, where the family was established in proximity to the
dockyard at Chatham from 1816 to 1821. He looked upon himself in later
years as a man of Kent, and his capital abode as that in Ordnance
Terrace, or 18 St Mary's Place, Chatham, amid surroundings classified in
Mr Pickwick's notes as "appearing" to be soldiers, sailors, Jews, chalk,
shrimps, officers and dockyard men. He fell into a family the general
tendency of which was to go down in the world, during one of its easier
periods (John Dickens was now fifth clerk on £250 a year), and he always
regarded himself as belonging by right to a comfortable, genteel, lower
middle-class stratum of society. His mother taught him to read; to his
father he appeared very early in the light of a young prodigy, and by
him Charles was made to sit on a tall chair and warble popular ballads,
or even to tell stories and anecdotes for the benefit of fellow-clerks
in the office. John Dickens, however, had a small collection of books
which were kept in a little room upstairs that led out of Charles's own,
and in this attic the boy found his true literary instructors in
_Roderick Random_, _Peregrine Pickle_, _Humphry Clinker_, _Tom Jones_,
_The Vicar of Wakefield_, _Don Quixote_, _Gil Blas_ and _Robinson
Crusoe_. The story of how he played at the characters in these books and
sustained his idea of Roderick Random for a month at a stretch is
picturesquely told in _David Copperfield_. Here as well as in his first
and last books and in what many regard as his best, _Great
Expectations_, Dickens returns with unabated fondness and mastery to the
surroundings of his childhood. From seven to nine years he was at a
school kept in Clover Lane, Chatham, by a Baptist minister named William
Giles, who gave him Goldsmith's _Bee_ as a keepsake when the call to
Somerset House necessitated the removal of the family from Rochester to
a shabby house in Bayham Street, Camden Town. At the very moment when a
consciousness of capacity was beginning to plump his youthful ambitions,
the whole flattering dream vanished and left not a rack behind.
Happiness and Chatham had been left behind together, and Charles was
about to enter a school far sterner and also far more instructive than
that in Clover Lane. The family income had been first decreased and then
mortgaged; the creditors of the "prodigal father" would not give him
time; John Dickens was consigned to the Marshalsea; Mrs Dickens started
an "Educational Establishment" as a forlorn hope in Upper Gower Street;
and Charles, who had helped his mother with the children, blacked the
boots, carried things to the pawnshop and done other menial work, was
now sent out to earn his own living as a young hand in a blacking
warehouse, at Old Hungerford Stairs, on a salary of six shillings a
week. He tied, trimmed and labelled blacking pots for over a year,
dining off a saveloy and a slice of pudding, consorting with two very
rough boys, Bob Fagin and Pol Green, and sleeping in an attic in Little
College Street, Camden Town, in the house of Mrs Roylance (Pipchin),
while on Sunday he spent the day with his parents in their comfortable
prison, where they had the services of a "marchioness" imported from the
Chatham workhouse.

Already consumed by ambition, proud, sensitive and on his dignity to an
extent not uncommon among boys of talent, he felt his position keenly,
and in later years worked himself up into a passion of self-pity in
connexion with the "degradation" and "humiliation" of this episode. The
two years of childish hardship which ate like iron into his soul were
obviously of supreme importance in the growth of the novelist.
Recollections of the streets and the prison and its purlieus supplied
him with a store of literary material upon which he drew through all the
years of his best activity. And the bitterness of such an experience was
not prolonged sufficiently to become sour. From 1824 to 1826, having
been rescued by a family quarrel and by a windfall in the shape of a
legacy to his father, from the warehouse, he spent two years at an
academy known as Wellington House, at the corner of Granby Street and
the Hampstead Road (the lighter traits of which are reproduced in Salem
House), and was there known as a merry and rather mischievous boy.
Fortunately he learned nothing there to compromise the results of
previous instruction. His father had now emerged from the Marshalsea and
was seeking employment as a parliamentary reporter. A Gray's Inn
solicitor with whom he had had dealings was attracted by the bright,
clever look of Charles, and took him into his office as a boy at a
salary of thirteen and sixpence (rising to fifteen shillings) a week. He
remained in Mr Blackmore's office from May 1827 to November 1828, but he
had lost none of his eager thirst for distinction, and spent all his
spare time mastering Gurney's shorthand and reading early and late at
the British Museum. A more industrious apprentice in the lower grades of
the literary profession has never been known, and the consciousness of
opportunities used to the most splendid advantage can hardly have been
absent from the man who was shortly to take his place at the head of it
as if to the manner born. Lowten and Guppy, and Swiveller had been
observed from this office lad's stool; he was now greatly to widen his
area of study as a reporter in Doctors' Commons and various police
courts, including Bow Street, working all day at law and much of the
night at shorthand. Some one asked John Dickens, during the first eager
period of curiosity as to the man behind "Pickwick," where his son
Charles was educated. "Well really," said the prodigal father, "he may
be said--haw--haw--to have educated himself." He was one of the most
rapid and accurate reporters in London when, at nineteen years of age,
in 1831, he realized his immediate ambition and "entered the gallery" as
parliamentary reporter to the _True Sun_. Later he was reporter to the
_Mirror of Parliament_ and then to the _Morning Chronicle_. Several of
his earliest letters are concerned with his exploits as a reporter, and
allude to the experiences he had, travelling fifteen miles an hour and
being upset in almost every description of known vehicle in various
parts of Britain between 1831 and 1836. The family was now living in
Bentwick Street, Manchester Square, but John Dickens was still no
infrequent inmate of the sponging-houses. With all the accessories of
these places of entertainment his son had grown to be excessively
familiar. Writing about 1832 to his school friend Tom Mitton, Dickens
tells him that his father has been arrested at the suit of a wine firm,
and begs him go over to Cursitor Street and see what can be done. On
another occasion of a paternal disappearance he observes: "I own that
his absence does not give me any great uneasiness, knowing how apt he is
to get out of the way when anything goes wrong." In yet another letter
he asks for a loan of four shillings.

In the meanwhile, however, he had commenced author in a more creative
sense by penning some sketches of contemporary London life, such as he
had attempted in his school days in imitation of the sketches published
in the _London_ and other magazines of that day. The first of these
appeared in the December number of the _Old Monthly Magazine_ for 1833.
By the following August, when the signature "Boz" was first given, five
of these sketches had appeared. By the end of 1834 we find him settled
in rooms in Furnival's Inn, and a little later his salary on the
_Morning Chronicle_ was raised, owing to the intervention of one of its
chiefs, George Hogarth, the father of (in addition to six sons) eight
charming daughters, to one of whom, Catherine, Charles was engaged to be
married before the year was out. Clearly as his career now seemed
designated, he was at this time or a little before it coquetting very
seriously with the stage: but circumstances were rapidly to determine
another stage in his career. A year before Queen Victoria's accession
appeared in two volumes _Sketches by Boz_, _Illustrative of Everyday
Life and Everyday People_. The book came from a prentice hand, but like
the little tract on the Puritan abuse of the Sabbath entitled "Sunday
under three Heads" which appeared a few months later, it contains in
germ all, or almost all, the future Dickens. Glance at the headings of
the pages. Here we have the Beadle and all connected with him, London
streets, theatres, shows, the pawnshop, Doctors' Commons, Christmas,
Newgate, coaching, the river. Here comes a satirical picture of
parliament, fun made of cheap snobbery, a rap on the knuckles of
sectarianism. And what could be more prophetic than the title of the
opening chapter--Our Parish? With the Parish--a large one
indeed--Dickens to the end concerned himself; he began with a rapid
survey of his whole field, hinting at all he might accomplish,
indicating the limits he was not to pass. This year was to be still more
momentous to Dickens, for, on the 2nd of April 1836, he was married to
George Hogarth's eldest daughter Catherine. He seems to have fallen in
love with the daughters collectively, and, judging by subsequent events,
it has been suggested that perhaps he married the wrong one. His wife's
sister Mary was the romance of his early married life, and another
sister, Georgina, was the dearest friend of his last ten years.

A few days before the marriage, just two months after the appearance of
the _Sketches_, the first part of _The Posthumous Papers of the Pickwick
Club_ was announced. One of the chief vogues of the day was the issue of
humorous, sporting or anecdotal novels in parts, with plates, and some
of the best talent of the day, represented by Ainsworth, Bulwer,
Marryat, Maxwell, Egan, Hook and Surtees, had been pressed into this
kind of enterprise. The publishers of the day had not been slow to
perceive Dickens's aptitude for this species of "letterpress." A member
of the firm of Chapman & Hall called upon him at Furnival's Inn in
December 1835 with a proposal that he should write about a Nimrod Club
of amateur sportsmen, foredoomed to perpetual ignominies, while the
comic illustrations were to be etched by Seymour, a well-known rival of
Cruikshank (the illustrator of _Boz_). The offer was too tempting for
Dickens to refuse, but he changed the idea from a club of Cockney
sportsmen to that of a club of eccentric peripatetics, on the sensible
grounds, first that sporting sketches were stale, and, secondly, that he
knew nothing worth speaking of about sport. The first seven pictures
appeared with the signature of Seymour and the letterpress of Dickens.
Before the eighth picture appeared Seymour had blown his brains out.
After a brief interval of Buss, Dickens obtained the services of Hablot
K. Browne, known to all as "Phiz." Author and illustrator were as well
suited to one another and to the common creation of a unique thing as
Gilbert and Sullivan. Having early got rid of the sporting element,
Dickens found himself at once. The subject exactly suited his knowledge,
his skill in arranging incidents--nay, his very limitations too. No
modern book is so incalculable. We commence laughing heartily at
Pickwick and his troupe. The laugh becomes kindlier. We are led on
through a tangle of adventure, never dreaming what is before us. The
landscape changes: Pickwick becomes the symbol of kindheartedness,
simplicity and innocent levity. Suddenly in the Fleet Prison a deeper
note is struck. The medley of human relationships, the loneliness, the
mystery and sadness of human destinies are fathomed. The tragedy of
human life is revealed to us amid its most farcical elements. The droll
and laughable figure of the hero is transfigured by the kindliness of
human sympathy into a beneficent and bespectacled angel in shorts and
gaiters. By defying accepted rules, Dickens had transcended the limited
sphere hitherto allotted to his art: he had produced a book to be
enshrined henceforth in the inmost hearts of all sorts and conditions of
his countrymen, and had definitely enlarged the boundaries of English
humour and English fiction. As for Mr Pickwick, he is a fairy like Puck
or Santa Claus, while his creator is "the last of the mythologists and
perhaps the greatest."

When _The Pickwick Papers_ appeared in book form at the close of 1837
Dickens's popular reputation was made. From the appearance of Sam Weller
in part v. the universal hunger for the monthly parts had risen to a
furore. The book was promptly translated into French and German. The
author had received little assistance from press or critics, he had no
influential connexions, his class of subjects was such as to "expose him
at the outset to the fatal objections of vulgarity," yet in less than
six months from the appearance of the first number, as the _Quarterly
Review_ almost ruefully admits, the whole reading world was talking
about the Pickwickians. The names of Winkle, Wardle, Weller, Jingle,
Snodgrass, Dodson & Fogg, were as familiar as household words. Pickwick
chintzes figured in the linendrapers' windows, and Pickwick cigars in
every tobacconist's; Weller corduroys became the stock-in-trade of every
breeches-maker; Boz cabs might be seen rattling through the streets, and
the portrait of the author of _Pelham_ and _Crichton_ was scraped down
to make way for that of the new popular favourite on the omnibuses. A
new and original genius had suddenly sprung up, there was no denying it,
even though, as the _Quarterly_ concluded, "it required no gift of
prophecy to foretell his fate--he has risen like a rocket and he will
come down like the stick." It would have needed a very emphatic gift of
prophecy indeed to foretell that Dickens's reputation would have gone on
rising until at the present day (after one sharp fall, which reached an
extreme about 1887) it stands higher than it has ever stood before.

Dickens's assumption of the literary purple was as amazing as anything
else about him. Accepting the homage of the luminaries of the literary,
artistic and polite worlds as if it had been his natural due, he
arranges for the settlement of his family, decrees, like another Edmund
Kean, that his son is to go to Eton, carries on the most complicated
negotiations with his publishers and editors, presides and orates with
incomparable force at innumerable banquets, public and private, arranges
elaborate villegiatures in the country, at the seaside, in France or in
Italy, arbitrates in public on every topic, political, ethical,
artistic, social or literary, entertains and legislates for an
increasingly large domestic circle, both juvenile and adult, rules
himself and his time-table with a rod of iron. In his letter-writing
alone, Dickens did a life's literary work. Nowadays no one thinks of
writing such letters; that is to say, letters of such length and detail,
for the quality is Dickens's own. He evidently enjoyed this use of the
pen. Page after page of Forster's _Life_ (750 pages in the _Letters_
edited by his daughter and sister-in-law) is occupied with transcription
from private correspondence, and never a line of this but is thoroughly
worthy of print and preservation. If he makes a tour in any part of the
British Isles, he writes a full description of all he sees, of
everything that happens, and writes it with such gusto, such mirth, such
strokes of fine picturing, as appear in no other private letters ever
given to the public. Naturally buoyant in all circumstances, a holiday
gave him the exhilaration of a schoolboy. See how he writes from
Cornwall, when on a trip with two or three friends, in 1843. "Heavens!
if you could have seen the necks of bottles, distracting in their
immense variety of shape, peering out of the carriage pockets! If you
could have witnessed the deep devotion of the post-boys, the maniac glee
of the waiters! If you could have followed us into the earthy old
churches we visited, and into the strange caverns on the gloomy
seashore, and down into the depths of mines, and up to the tops of giddy
heights, where the unspeakably green water was roaring, I don't know how
many hundred feet below.... I never laughed in my life as I did on this
journey. It would have done you good to hear me. I was choking and
gasping and bursting the buckles off the back of my stock, all the way.
And Stanfield"--the painter--"got into such apoplectic entanglements
that we were obliged to beat him on the back with portmanteaus before we
could recover him."

The animation of Dickens's look would attract the attention of any one,
anywhere. His figure was not that of an Adonis, but his brightness made
him the centre and pivot of every society he was in. The keenness and
vivacity of his eye combined with his inordinate appetite for life to
give the unique quality to all that he wrote. His instrument is that of
the direct, sinewy English of Smollett, combined with much of the
humorous grace of Goldsmith (his two favourite authors), but modernized
to a certain extent under the influence of Washington Irving, Sydney
Smith, Jeffrey, Lamb, and other writers of the _London Magazine_. He
taught himself to speak French and Italian, but he could have read
little in any language. His ideas were those of the inchoate and insular
liberalism of the 'thirties. His unique force in literature he was to
owe to no supreme artistic or intellectual quality, but almost entirely
to his inordinate gift of observation, his sympathy with the humble, his
power over the emotions and his incomparable endowment of unalloyed
human fun. To contemporaries he was not so much a man as an
institution, at the very mention of whose name faces were puckered with
grins or wreathed in smiles. To many his work was a revelation, the
revelation of a new world and one far better than their own. And his
influence went further than this in the direction of revolution or
revival. It gave what were then universally referred to as "the lower
orders" a new sense of self-respect, a new feeling of citizenship. Like
the defiance of another Luther, or the Declaration of a new
Independence, it emitted a fresh ray of hope across the firmament. He
did for the whole English-speaking race what Burns had done for
Scotland--he gave it a new conceit of itself. He knew what a people
wanted and he told what he knew. He could do this better than anybody
else because his mind was theirs. He shared many of their "great useless
virtues," among which generosity ranks before justice, and sympathy
before truth, even though, true to his middle-class vein, he exalts
piety, chastity and honesty in a manner somewhat alien to the mind of
the low-bred man. This is what makes Dickens such a demigod and his
public success such a marvel, and this also is why any exclusively
literary criticism of his work is bound to be so inadequate. It should
also help us to make the necessary allowances for the man. Dickens, even
the Dickens of legend that we know, is far from perfect. The Dickens of
reality to which Time may furnish a nearer approximation is far less
perfect. But when we consider the corroding influence of adulation, and
the intoxication of unbridled success, we cannot but wonder at the
relatively high level of moderation and self-control that Dickens almost
invariably observed. Mr G. K. Chesterton remarks suggestively that
Dickens had all his life the faults of the little boy who is kept up too
late at night. He is overwrought by happiness to the verge of
exasperation, and yet as a matter of fact he does keep on the right side
of the breaking point. The specific and curative in his case was the
work in which he took such anxious pride, and such unmitigated delight.
He revelled in punctual and regular work; at his desk he was often in
the highest spirits. Behold how he pictured himself, one day at
Broadstairs, where he was writing _Chuzzlewit_. "In a bay-window in a
one-pair sits, from nine o'clock to one, a gentleman with rather long
hair and no neckcloth, who writes and grins, as if he thought he was
very funny indeed. At one he disappears, presently emerges from a
bathing-machine, and may be seen, a kind of salmon-colour porpoise,
splashing about in the ocean. After that, he may be viewed in another
bay-window on the ground-floor eating a strong lunch; and after that,
walking a dozen miles or so, or lying on his back on the sand reading a
book. Nobody bothers him, unless they know he is disposed to be talked
to, and I am told he is very comfortable indeed. He's as brown as a
berry, and they do say he is as good as a small fortune to the
innkeeper, who sells beer and cold punch." Here is the secret of such
work as that of Dickens; it is done with delight--done (in a sense)
easily, done with the mechanism of mind and body in splendid order. Even
so did Scott write; though more rapidly and with less conscious care:
his chapter finished before the world had got up to breakfast. Later,
Dickens produced novels less excellent with much more of mental strain.
The effects of age could not have shown themselves so soon, but for the
unfortunate loss of energy involved in his non-literary labours.

While the public were still rejoicing in the first sprightly runnings of
the "new humour," the humorist set to work desperately on the grim
scenes of _Oliver Twist_, the story of a parish orphan, the nucleus of
which had already seen the light in his _Sketches_. The early scenes are
of a harrowing reality, despite the germ of forced pathos which the
observant reader may detect in the pitiful parting between Oliver and
little Dick; but what will strike every reader at once in this book is
the directness and power of the English style, so nervous and unadorned:
from its unmistakable clearness and vigour Dickens was to travel far as
time went on. But the full effect of the old simplicity is felt in such
masterpieces of description as the drive of Oliver and Sikes to
Chertsey, the condemned-cell ecstasy of Fagin, or the unforgettable
first encounter between Oliver and the Artful Dodger. Before November
1837 had ended, Charles Dickens entered on an engagement to write a
successor to _Pickwick_ on similar lines of publication. _Oliver Twist_
was then in mid-career; a _Life of Grimaldi_ and _Barnaby Rudge_ were
already covenanted for. Dickens forged ahead with the new tale of
_Nicholas Nickleby_ and was justified by the results, for its sale far
surpassed even that of _Pickwick_. As a conception it is one of his
weakest. An unmistakably 18th-century character pervades it. Some of the
vignettes are among the most piquant and besetting ever written. Large
parts of it are totally unobserved conventional melodrama; but the
Portsmouth Theatre and Dotheboys Hall and Mrs Nickleby (based to some
extent, it is thought, upon Miss Bates in _Emma_, but also upon the
author's Mamma) live for ever as Dickens conceived them in the pages of
_Nicholas Nickleby_.

Having got rid of _Nicholas Nickleby_ and resigned his editorship of
_Bentley's Miscellany_, in which _Oliver Twist_ originally appeared,
Dickens conceived the idea of a weekly periodical to be issued as
_Master Humphrey's Clock_, to comprise short stories, essays and
miscellaneous papers, after the model of Addison's _Spectator_. To make
the weekly numbers "go," he introduced Mr Pickwick, Sam Weller and his
father in friendly intercourse. But the public requisitioned "a story,"
and in No. 4 he had to brace himself up to give them one. Thus was
commenced _The Old Curiosity Shop_, which was continued with slight
interruptions, and followed by _Barnaby Rudge_. For the first time we
find Dickens obsessed by a highly complicated plot. The tonality
achieved in _The Old Curiosity Shop_ surpassed anything he had attempted
in this difficult vein, while the rich humour of Dick Swiveller and the
Marchioness, and the vivid portraiture of the wandering Bohemians,
attain the very highest level of Dickensian drollery; but in the
lamentable tale of Little Nell (though Landor and Jeffrey thought the
character-drawing of this infant comparable with that of Cordelia), it
is generally admitted that he committed an indecent assault upon the
emotions by exhibiting a veritable monster of piety and long-suffering
in a child of tender years. In _Barnaby Rudge_ he was manifestly
affected by the influence of Scott, whose achievements he always
regarded with a touching veneration. The plot, again, is of the utmost
complexity, and Edgar Allan Poe (who predicted the conclusion) must be
one of the few persons who ever really mastered it. But few of Dickens's
books are written in a more admirable style.

_Master Humphrey's Clock_ concluded, Dickens started in 1842 on his
first visit to America--an episode hitherto without parallel in English
literary history, for he was received everywhere with popular
acclamation as the representative of a grand triumph of the English
language and imagination, without regard to distinctions of nationality.
He offended the American public grievously by a few words of frank
description and a few quotations of the advertisement columns of
American papers illustrating the essential barbarity of the old slave
system (_American Notes_). Dickens was soon pining for home--no English
writer is more essentially and insularly English in inspiration and
aspiration than he is. He still brooded over the perverseness of America
on the copyright question, and in his next book he took the opportunity
of uttering a few of his impressions about the objectionable sides of
American democracy, the result being that "all Yankee-doodle-dom blazed
up like one universal soda bottle," as Carlyle said. _Martin Chuzzlewit_
(1843-1844) is important as closing his great character period. His
_sève originale_, as the French would say, was by this time to a
considerable extent exhausted, and he had to depend more upon artistic
elaboration, upon satires, upon _tours de force_ of description, upon
romantic and ingenious contrivances. But all these resources combined
proved unequal to his powers as an original observer of popular types,
until he reinforced himself by autobiographic reminiscence, as in _David
Copperfield_ and _Great Expectations_, the two great books remaining to
his later career.

After these two masterpieces and the three wonderful books with which he
made his début, we are inclined to rank _Chuzzlewit_. Nothing in Dickens
is more admirably seen and presented than Todgers's, a bit of London
particular cut out with a knife. Mr Pecksniff and Mrs Gamp, Betsy Prig
and "Mrs Harris" have passed into the national language and life. The
coach journey, the windy autumn night, the stealthy trail of Jonas, the
undertone of tragedy in the Charity and Mercy and Chuffey episodes
suggest a blending of imaginative vision and physical penetration hardly
seen elsewhere. Two things are specially notable about this novel--the
exceptional care taken over it (as shown by the interlineations in the
MS.) and the caprice or nonchalance of the purchasing public, its sales
being far lower than those of any of its monthly predecessors.

At the close of 1843, to pay outstanding debts of his now lavish
housekeeping, he wrote that pioneer of Christmas numbers, that national
benefit as Thackeray called it, _A Christmas Carol_. It failed to
realize his pecuniary anticipations, and Dickens resolved upon a drastic
policy of retrenchment and reform. He would save expense by living
abroad and would punish his publishers by withdrawing his custom from
them, at least for a time. Like everything else upon which he ever
determined, this resolution was carried out with the greatest possible
precision and despatch. In June 1844 he set out for Marseilles with his
now rapidly increasing family (the journey cost him £200). In a villa on
the outskirts of Genoa he wrote _The Chimes_, which, during a brief
excursion to London before Christmas, he read to a select circle of
friends (the germ of his subsequent lecture-audiences), including
Forster, Carlyle, Stanfield, Dyce, Maclise and Jerrold. He was again in
London in 1845, enjoying his favourite diversion of private theatricals;
and in January 1846 he experimented briefly as the editor of a London
morning paper--the _Daily News_. By early spring he was back at
Lausanne, writing his customary vivid letters to his friends, craving as
usual for London streets, commencing _Dombey and Son_, and walking his
fourteen miles daily. The success of _Dombey and Son_ completely
rehabilitated the master's finances, enabled him to return to England,
send his son to Eton and to begin to save money. Artistically it is less
satisfactory; it contains some of Dickens's prime curios, such as
Cuttle, Bunsby, Toots, Blimber, Pipchin, Mrs MacStinger and young Biler;
it contains also that masterpiece of sentimentality which trembles upon
the borderland of the sublime and the ridiculous, the death of Paul
Dombey ("that sweet Paul," as Jeffrey, the "critic laureate," called
him), and some grievous and unquestionable blemishes. As a narrative,
moreover, it tails off into a highly complicated and exacting plot. It
was followed by a long rest at Broadstairs before Dickens returned to
the native home of his genius, and early in 1849 "began to prepare for
_David Copperfield_."

"Of all my books," Dickens wrote, "I like this the best; like many fond
parents I have my favourite child, and his name is David Copperfield."
In some respects it stands to Dickens in something of the same relation
in which the contemporary _Pendennis_ stands to Thackeray. As in that
book, too, the earlier portions are the best. They gained in intensity
by the autobiographical form into which they are thrown; as Thackeray
observed, there was no writing against such power. The tragedy of Emily
and the character of Rosa Dartle are stagey and unreal; Uriah Heep is
bad art; Agnes, again, is far less convincing as a consolation than
Dickens would have us believe; but these are more than compensated by
the wonderful realization of early boyhood in the book, by the picture
of Mr Creakle's school, the Peggottys, the inimitable Mr Micawber, Betsy
Trotwood and that monument of selfish misery, Mrs Gummidge.

At the end of March 1850 commenced the new twopenny weekly called
_Household Words_, which Dickens planned to form a direct means of
communication between himself and his readers, and as a means of
collecting around him and encouraging the talents of the younger
generation. No one was better qualified than he for this work, whether
we consider his complete freedom from literary jealousy or his magical
gift of inspiring young authors. Following the somewhat dreary and
incoherent _Bleak House_ of 1852, _Hard Times_ (1854)--an
anti-Manchester School tract, which Ruskin regarded as Dickens's best
work--was the first long story written for _Household Words_. About this
time Dickens made his final home at Gad's Hill, near Rochester, and put
the finishing touch to another long novel published upon the old plan,
_Little Dorrit_ (1855-1857). In spite of the exquisite comedy of the
master of the Marshalsea and the final tragedy of the central figure,
_Little Dorrit_ is sadly deficient in the old vitality, the humour is
often a mock reality, and the repetition of comic catch-words and
overstrung similes and metaphors is such as to affect the reader with
nervous irritation. The plot and characters ruin each other in this
amorphous production. The _Tale of Two Cities_, commenced in _All the
Year Round_ (the successor of _Household Words_) in 1859, is much
better: the main characters are powerful, the story genuinely tragic,
and the atmosphere lurid; but enormous labour was everywhere expended
upon the construction of stylistic ornament.

The _Tale of Two Cities_ was followed by two finer efforts at
atmospheric delineation, the best things he ever did of this kind:
_Great Expectations_ (1861), over which there broods the mournful
impression of the foggy marshes of the Lower Thames; and _Our Mutual
Friend_ (1864-1865), in which the ooze and mud and slime of Rotherhithe,
its boatmen and loafers, are made to pervade the whole book with
cumulative effect. The general effect produced by the stories is,
however, very different. In the first case, the foreground was supplied
by autobiographical material of the most vivid interest, and the
lucidity of the creative impulse impelled him to write upon this
occasion with the old simplicity, though with an added power. Nothing
therefore, in the whole range of Dickens surpassed the early chapters of
_Great Expectations_ in perfection of technique or in mastery of all the
resources of the novelist's art. To have created Abel Magwitch alone is
to be a god indeed, says Mr Swinburne, among the creators of deathless
men. Pumblechook is actually better and droller and truer to imaginative
life than Pecksniff; Joe Gargery is worthy to have been praised and
loved at once by Fielding and by Sterne: Mr Jaggers and his clients, Mr
Wemmick and his parent and his bride, are such figures as Shakespeare,
when dropping out of poetry, might have created, if his lot had been
cast in a later century. "Can as much be said," Mr Swinburne boldly
asks, "for the creatures of any other man or god?"

In November 1867 Dickens made a second expedition to America, leaving
all the writing that he was ever to complete behind him. He was to make
a round sum of money, enough to free him from all embarrassments, by a
long series of exhausting readings, commencing at the Tremont Temple,
Boston, on the 2nd of December. The strain of Dickens's ordinary life
was so tense and so continuous that it is, perhaps, rash to assume that
he broke down eventually under this particular stress; for other
reasons, however, his persistence in these readings, subsequent to his
return, was strongly deprecated by his literary friends, led by the
arbitrary and relentless Forster. It is a long testimony to Dickens's
self-restraint, even in his most capricious and despotic moments, that
he never broke the cord of obligation which bound him to his literary
mentor, though sparring matches between them were latterly of frequent
occurrence. His farewell reading was given on the 15th of March 1870, at
St James's Hall. He then vanished from "those garish lights," as he
called them, "for evermore." Of the three brief months that remained to
him, his last book, _The Mystery of Edwin Drood_, was the chief
occupation. It hardly promised to become a masterpiece (Longfellow's
opinion) as did Thackeray's _Denis Duval_, but contained much fine
descriptive technique, grouped round a scene of which Dickens had an
unrivalled sympathetic knowledge.

In March and April 1870 Dickens, as was his wont, was mixing in the best
society; he dined with the prince at Lord Houghton's and was twice at
court, once at a long deferred private interview with the queen, who had
given him a presentation copy of her _Leaves from a Journal of our Life
in the Highlands_ with the inscription "From one of the humblest of
authors to one of the greatest"; and who now begged him on his
persistent refusal of any other title to accept the nominal distinction
of a privy councillor. He took for four months the Milner Gibsons' house
at 5 Hyde Park Place, opposite the Marble Arch, where he gave a
brilliant reception on the 7th of April. His last public appearance was
made at the Royal Academy banquet early in May. He returned to his
regular methodical routine of work at Gad's Hill on the 30th of May, and
one of the last instalments he wrote of _Edwin Drood_ contained an
ominous speculation as to the next two people to die at Cloisterham:
"Curious to make a guess at the two, or say at one of the two." Two
letters bearing the well-known superscription "Gad's Hill Place, Higham
by Rochester, Kent" are dated the 8th of June, and, on the same
Thursday, after a long spell of writing in the Châlet where he
habitually wrote, he collapsed suddenly at dinner. Startled by the
sudden change in the colour and expression of his face, his
sister-in-law (Miss Hogarth) asked him if he was ill; he said "Yes, very
ill," but added that he would finish dinner and go on afterwards to
London. "Come and lie down," she entreated; "Yes, on the ground," he
said, very distinctly; these were the last words he spoke, and he slid
from her arms and fell upon the floor. He died at 6-10 P.M. on Friday,
the 9th of June, and was buried privately in Poets' Corner, Westminster
Abbey, in the early morning of the 14th of June. One of the most
appealing memorials was the drawing by his "new illustrator" Luke Fildes
in the _Graphic_ of "The Empty Chair; Gad's Hill: ninth of June, 1870."
"Statesmen, men of science, philanthropists, the acknowledged
benefactors of their race, might pass away, and yet not leave the void
which will be caused by the death of Charles Dickens" (_The Times_). In
his will he enjoined his friends to erect no monument in his honour, and
directed his name and dates only to be inscribed on his tomb, adding
this proud provision, "I rest my claim to the remembrance of my country
on my published works."

Dickens had no artistic ideals worth speaking about. The sympathy of his
readers was the one thing he cared about and, like Cobbett, he went
straight for it through the avenue of the emotions. In personality,
intensity and range of creative genius he can hardly be said to have any
modern rival. His creations live, move and have their being about us
constantly, like those of Homer, Virgil, Chaucer, Rabelais, Cervantes,
Shakespeare, Bunyan, Molière and Sir Walter Scott. As to the books
themselves, the backgrounds on which these mighty figures are projected,
they are manifestly too vast, too chaotic and too unequal ever to become
classics. Like most of the novels constructed upon the unreformed model
of Smollett and Fielding, those of Dickens are enormous stock-pots into
which the author casts every kind of autobiographical experience,
emotion, pleasantry, anecdote, adage or apophthegm. The fusion is
necessarily very incomplete and the hotch-potch is bound to fall to
pieces with time. Dickens's plots, it must be admitted, are strangely
unintelligible, the repetitions and stylistic decorations of his work
exceed all bounds, the form is unmanageable and insignificant. The
diffuseness of the English novel, in short, and its extravagant
didacticism cannot fail to be most prejudicial to its perpetuation. In
these circumstances there is very little fiction that will stand
concentration and condensation so well as that of Dickens.

For these reasons among others our interest in Dickens's novels as
integers has diminished and is diminishing. But, on the other hand, our
interest and pride in him as a man and as a representative author of his
age and nation has been steadily augmented and is still mounting. Much
of the old criticism of his work, that it was not up to a sufficiently
high level of art, scholarship or gentility, that as an author he is
given to caricature, redundancy and a shameless subservience to popular
caprice, must now be discarded as irrelevant.

As regards formal excellence it is plain that Dickens labours under the
double disadvantage of writing in the least disciplined of all literary
genres in the most lawless literary milieu of the modern world, that of
Victorian England. In spite of these defects, which are those of masters
such as Rabelais, Hugo and Tolstoy, the work of Dickens is more and more
instinctively felt to be true, original and ennobling. It is already
beginning to undergo a process of automatic sifting, segregation and
crystallization, at the conclusion of which it will probably occupy a
larger segment in the literary consciousness of the English-spoken race
than ever before.

Portraits of Dickens, from the gay and alert "Boz" of Samuel Lawrence,
and the self-conscious, rather foppish portrait by Maclise which served
as frontispiece to _Nicholas Nickleby_, to the sketch of him as Bobadil
by C. R. Leslie, the Drummond and Ary Scheffer portraits of middle age
and the haggard and drawn representations of him from photographs after
his shattering experiences as a public entertainer from 1856 (the year
of his separation from his wife) onwards, are reproduced in Kitton, in
Forster and Gissing and in the other biographies. Sketches are also
given in most of the books of his successive dwelling places at Ordnance
Terrace and 18 St Mary's Place, Chatham; Bayham Street, Camden Town; 15
Furnival's Inn; 48 Doughty Street; 1 Devonshire Terrace, Regent's Park;
Tavistock House, Tavistock Square; and Gad's Hill Place. The manuscripts
of all the novels, with the exception of the _Tale of Two Cities_ and
_Edwin Drood_, were given to Forster, and are now preserved in the Dyce
and Forster Museum at South Kensington. The work of Dickens was a prize
for which publishers naturally contended both before and after his
death. The first collective edition of his works was begun in April
1847, and their number is now very great. The most complete is still
that of Messrs Chapman & Hall, the original publishers of _Pickwick_;
others of special interest are the Harrap edition, originally edited by
F. G. Kitton; Macmillan's edition with original illustrations and
introduction by Charles Dickens the younger; and the edition in the
World's Classics with introductions by G.K. Chesterton. Of the
translations the best known is that done into French by Lorain, Pichot
and others, with B.H. Gausseron's excellent _Pages Choisies_ (1903).

  BIBLIOGRAPHY.--During his lifetime Dickens's biographer was clearly
  indicated in his guide, philosopher and friend, John Forster, who had
  known the novelist intimately since the days of his first triumph with
  _Pickwick_, who had constituted himself a veritable encyclopaedia of
  information about Dickens, and had clung to his subject (in spite of
  many rebuffs which his peremptory temper found it hard to digest) as
  tightly as ever Boswell had enveloped Johnson. Two volumes of
  Forster's _Life of Charles Dickens_ appeared in 1872 and a third in
  1874. He relied much on Dickens's letters to himself and produced what
  must always remain the authoritative work. The first two volumes are
  put together with much art, the portrait as a whole has been regarded
  as truthful, and the immediate success was extraordinary. In the
  opinion of Carlyle, Forster's book was not unworthy to be named after
  that of Boswell. A useful abridgment was carried out in 1903 by the
  novelist George Gissing. Gissing also wrote _Charles Dickens: A
  Critical Study_ (1898), which ranks with G.K. Chesterton's _Charles
  Dickens_(1906) as a commentary inspired by deep insight and adorned by
  great literary talent upon the genius of the master-novelist. The
  names of other lives, sketches, articles and estimates of Dickens and
  his works would occupy a large volume in the mere enumeration. See
  R.H. Shepherd, _The Bibliography of Dickens_ (1880); _James Cooke's
  Bibliography of the Writings of Charles Dickens_ (1879);
  _Dickensiana_, by F. G. Kitton (1886); and _Bibliography_ by J.P.
  Anderson, appended to Sir F.T. Marzials's _Life of Charles Dickens_
  (1887). Among the earlier sketches may be specially cited the lives by
  J. C. Hotten and G. A. Sala (1870), the Anecdote-Biography edited by
  the American R. H. Stoddard (1874), Dr A. W. Ward in the English Men
  of Letters Series (1878), that by Sir Leslie Stephen in the
  _Dictionary of National Biography_, and that by Professor Minto in the
  eighth edition of the _Encyclopaedia Britannica_. The _Letters_ were
  first issued in two volumes edited by his daughter and sister-in-law
  in 1880. For Dickens's connexion with Kent the following books are
  specially valuable:--Robert Langton's _Childhood and Youth of Charles
  Dickens_ (1883); Langton's _Dickens and Rochester_ (1880); Thomas
  Frost's _In Kent with Charles Dickens_ (1880); F. G. Kitton's _The
  Dickens Country_ (1905); H. S. Ward's _The Real Dickens Land_ (1904);
  R. Allbut's _Rambles in Dickens Land_ (1899 and 1903). For Dickens's
  reading tours see G. Dolby's _Charles Dickens as I knew him_ (1884);
  J. T. Fields's _In and Out of Doors with Charles Dickens_ (1876);
  Charles Kent's _Dickens as a Reader_ (1872). And for other aspects of
  his life see M. Dickens's _My Father as I recall him_ (1897); P. H.
  Fitzgerald's _Life of C. Dickens as revealed in his Writings_ (1905),
  and _Bozland_ (1895); F. G. Kitton's _Charles Dickens, his Life,
  Writings and Personality_, a useful compendium (1902); T. E.
  Pemberton's _Charles Dickens and the Stage_, and _Dickens's London_
  (1876); F. Miltoun's _Dickens's London_ (1904); Kitton's _Dickens and
  his Illustrators_; W. Teignmouth Shore's _Charles Dickens and his
  Friends_ (1904 and 1909); B. W. Matz, _Story of Dickens's Life and
  Work_ (1904), and review of solutions to _Edwin Drood_ in _The
  Bookman_ for March 1908; the recollections of Edmund Yates, Trollope,
  James Payn, Lehmann, R. H. Horne, Lockwood and many others. _The
  Dickensian_, a magazine devoted to Dickensian subjects, was started in
  1905; it is the organ of the Dickens Fellowship, and in a sense of the
  Boz Club. _A Dickens Dictionary_ (by G. A. Pierce) appeared in 1872
  and 1878; another (by A. J. Philip) in 1909; and a _Dickens
  Concordance_ by Mary Williams in 1907.     (T. SE.)



DICKINSON, ANNA ELIZABETH (1842-   ), American author and lecturer, was
born, of Quaker parentage, at Philadelphia, Pennsylvania, on the 28th of
October 1842. She was educated at the Friends' Free School in
Philadelphia, and was for a time a teacher. In 1861 she obtained a
clerkship in the United States mint, but was removed for criticizing
General McClellan at a public meeting. She had gradually become widely
known as an eloquent and persuasive public speaker, one of the first of
her sex to mount the platform to discuss the burning questions of the
hour. Before the Civil War she lectured on anti-slavery topics, during
the war she toured the country on behalf of the Sanitary Commission, and
also lectured on reconstruction, temperance and woman's rights. She
wrote several plays, including _The Crown of Thorns_ (1876); _Mary
Tudor_ (1878), in which she appeared in the title rôle; _Aurelian_
(1878); and _An American Girl_ (1880), successfully acted by Fanny
Davenport. She also published a novel, _Which Answer?_ (1868); _A Paying
Investment, a Plea for Education_ (1876); and _A Ragged Register of
People, Places and Opinions_ (1879).



DICKINSON, JOHN (1732-1808), American statesman and pamphleteer, was
born in Talbot county, Maryland, on the 8th of November 1732. He removed
with his father to Kent county, Delaware, in 1740, studied under private
tutors, read law, and in 1753 entered the Middle Temple, London.
Returning to America in 1757, he began the practice of law in
Philadelphia, was speaker of the Delaware assembly in 1760, and was a
member of the Pennsylvania assembly in 1762-1765 and again in
1770-1776.[1] He represented Pennsylvania in the Stamp Act Congress
(1765) and in the Continental Congress from 1774 to 1776, when he was
defeated owing to his opposition to the Declaration of Independence. He
then retired to Delaware, served for a time as private and later as
brigader-general in the state militia, and was again a member of the
Continental Congress (from Delaware) in 1779-1780. He was president of
the executive council, or chief executive officer, of Delaware in
1781-1782, and of Pennsylvania in 1782-1785, and was a delegate from
Delaware to the Annapolis convention of 1786 and the Federal
Constitutional convention of 1787. Dickinson has aptly been called the
"Penman of the Revolution." No other writer of the day presented
arguments so numerous, so timely and so popular. He drafted the
"Declaration of Rights" of the Stamp Act Congress, the "Petition to the
King" and the "Address to the Inhabitants of Quebec" of the Congress of
1774, and the second "Petition to the King"[2] and the "Articles of
Confederation" of the second Congress. Most influential of all, however,
were _The Letters of a Farmer in Pennsylvania_, written in 1767-1768 in
condemnation of the Townshend Acts of 1767, in which he rejected
speculative natural rights theories and appealed to the common sense of
the people through simple legal arguments. By opposing the Declaration
of Independence, he lost his popularity and was never able entirely to
regain it. As the representative of a small state, he championed the
principle of state equality in the constitutional convention, but was
one of the first to advocate the compromise, which was finally adopted,
providing for equal representation, in one house and proportional
representation in the other. He was probably influenced by Delaware
prejudice against Pennsylvania when he drafted the clause which forbids
the creation of a new state by the junction of two or more states or
parts of states without the consent of the states concerned as well as
of congress. After the adjournment of the convention he defended its
work in a series of letters signed "Fabius," which will bear comparison
with the best of the Federalist productions. It was largely through his
influence that Delaware and Pennsylvania were the first two states to
ratify the Constitution. Dickinson's interests were not exclusively
political. He helped to found Dickinson College (named in his honour) at
Carlisle, Pennsylvania, in 1783, was the first president of its board of
trustees, and was for many years its chief benefactor. He died on the
14th of February 1808 and was buried in the Friends' burial ground in
Wilmington, Del.

  See C. J. Stillé, _Life and Times of John Dickinson_, and P. L. Ford
  (editor), _The Writings of John Dickinson_, in vols. xiii. and xiv.
  respectively of the _Memoirs of the Historical Society of
  Pennsylvania_ (Philadelphia, 1891 and 1895).


FOOTNOTES:

  [1] Being under the same proprietor and the same governor,
    Pennsylvania and Delaware were so closely connected before the
    Revolution that there was an interchange of public men.

  [2] The "Declaration of the United Colonies of North America ...
    setting forth the Causes and the Necessity of their Taking up Arms"
    (often erroneously attributed to Thomas Jefferson).



DICKSON, SIR ALEXANDER (1777-1840), British artillerist, entered the
Royal Military Academy in 1793, passing out as second lieutenant in the
Royal Artillery in the following year. As a subaltern he saw service in
Minorca in 1798 and at Malta in 1800. As a captain he took part in the
unfortunate Montevideo Expedition of 1806-07, and in 1809 he accompanied
Howorth to the Peninsular War as brigade-major of the artillery. He soon
obtained a command in the Portuguese artillery, and as a
lieutenant-colonel of the Portuguese service took part in the various
battles of 1810-11. At the two sieges of Budazoz, Ciudad Rodrigo, the
Salamanca forts and Burgos, he was entrusted by Wellington (who had the
highest opinion of him) with most of the detailed artillery work, and at
Salamanca battle he commanded the reserve artillery. In the end he
became commander of the whole of the artillery of the allied army, and
though still only a substantive captain in the British service he had
under his orders some 8000 men. At Vitoria, the Pyrenees battles and
Toulouse he directed the movements of the artillery engaged, and at the
end of the war received handsome presents from the officers who had
served under him, many of whom were his seniors in the army list. He was
at the disastrous affair of New Orleans, but returned to Europe in time
for the Waterloo campaign. He was present at Quatre Bras and Waterloo on
the artillery staff of Wellington's army, and subsequently commanded the
British battering train at the sieges of the French fortresses left
behind the advancing allies. For the rest of his life he was on home
service, principally as a staff officer of artillery. He died, a
major-general and G.C.B., in 1840. A memorial was erected at Woolwich in
1847. Dickson was one of the earliest fellows of the Royal Geographical
Society.

  His diaries kept in the Peninsula were the main source of information
  used in Duncan's _History of the Royal Artillery_.



DICKSON, SIR JAMES ROBERT (1832-1901), Australian statesman, was born in
Plymouth on the 30th of November 1832. He was brought up in Glasgow,
receiving his education at the high school, and became a clerk in the
City of Glasgow Bank. In 1854 he emigrated to Victoria, but after some
years spent in that colony and in New South Wales, he settled in 1862 in
Queensland, where he was connected with many important business
enterprises, among them the Royal Bank of Queensland. He entered the
Queensland House of Assembly in 1872, and became minister of works
(1876), treasurer (1876-1879, and 1883-1887), acting premier (1884), but
resigned in 1887 on the question of taxing land. In 1889 he retired from
business, and spent three years in Europe before resuming political
life. He fought for the introduction of Polynesian labour on the
Queensland sugar plantations at the general election of 1892, and was
elected to the House of Assembly in that year and again at the elections
of 1893 and 1896. He became secretary for railways in 1897, minister for
home affairs in 1898, represented Queensland in the federal council of
Australia in 1896 and at the postal conference at Hobart in 1898, and in
1898 became premier. His energies were now devoted to the formation of
an Australian commonwealth. He secured the reference of the question to
a plebiscite, the result of which justified his anticipations. He
resigned the premiership in November 1899, but in the ministry of Robert
Philp, formed in the next month, he was reappointed to the offices of
chief secretary and vice-president of the executive council which he had
combined with the office of premier. He represented Queensland in 1900
at the conference held in London to consider the question of Australian
unity, and on his return was appointed minister of defence in the first
government of the Australian Commonwealth. He did not long survive the
accomplishment of his political aims, dying at Sydney on the 10th of
January 1901, in the midst of the festivities attending the inauguration
of the new state.



DICOTYLEDONS, in botany, the larger of the two great classes of
angiosperms, embracing most of the common flower-bearing plants. The
name expresses the most universal character of the class, the importance
of which was first noticed by John Ray, namely, the presence of a pair
of seed-leaves or cotyledons, in the plantlet or embryo contained in the
seed. The embryo is generally surrounded by a larger or smaller amount
of foodstuff (endosperm) which serves to nourish it in its development
to form a seedling when the seed germinates; frequently, however, as in
pea or bean and their allies, the whole of the nourishment for future
use is stored up in the cotyledons themselves, which then become thick
and fleshy. In germination of the seed the root of the embryo (radicle)
grows out to get a holdfast for the plant; this is generally followed by
the growth of the short stem immediately above the root, the so-called
"hypocotyl," which carries up the cotyledons above the ground, where
they spread to the light and become the first green leaves of the plant.
Protected between the cotyledons and terminating the axis of the plant
is the first stem-bud (the plumule of the embryo), by the further growth
and development of which the aerial portion of the plant, consisting of
stem, leaves and branches, is formed, while the development of the
radicle forms the root-system. The size and manner of growth of the
adult plant show a great variety, from the small herb lasting for one
season only, to the forest tree living for centuries. The arrangement of
the conducting tissue in the stem is characteristic; a transverse
section of the very young stem shows a number of distinct conducting
strands--vascular bundles--arranged in a ring round the pith; these soon
become united to form a closed ring of bast and wood, separated by a
layer of formative tissue (cambium). In perennials the stem shows a
regular increase in thickness each year by the addition of a new ring of
wood outside the old one--for details of structure see PLANTS: Anatomy.
A similar growth occurs in the root. This increase in the diameter of
stem and root is correlated with the increase in leaf-area each season,
due to the continued production of new leaf-bearing branches. A
characteristic of the class is afforded by the complicated network
formed by the leaf-veins,--well seen in a skeleton leaf, from which the
soft parts have been removed by maceration. The parts of the flower are
most frequently arranged in fives, or multiples of fives; for instance,
a common arrangement is as follows,--five sepals, succeeded by five
petals, ten stamens in two sets of five, and five or fewer carpels; an
arrangement in fours is less frequent, while the arrangement in threes,
so common in monocotyledons, is rare in dicotyledons. In some orders the
parts are numerous, chiefly in the case of the stamens and the carpels,
as in the buttercup and other members of the order Ranunculaceae. There
is a very wide range in the general structure and arrangement of the
parts of the flower, associated with the means for ensuring the
transference of pollen; in the simplest cases the flower consists only
of a few stamens or carpels, with no enveloping sepals or petals, as in
the willow, while in the more elaborate type each series is represented,
the whole forming a complicated structure closely correlated with the
size, form and habits of the pollinating agent (see FLOWER). The
characters of the fruit and seed and the means for ensuring the
dispersal of the seeds are also very varied (see FRUIT).



DICTATOR (from the Lat. _dictare_, frequentative of _dicere_, to speak).
In modern usage this term is loosely used for a personal ruler enjoying
extraordinary and extra-constitutional power. The etymological sense of
one who "dictates"--i.e. one whose word (_dictum_) is law (from which
that of one who "dictates," i.e. speaks for some writer to record, is to
be distinguished)--has been assisted by the historical use of the term,
in ancient times, for an extraordinary magistrate in the Roman
commonwealth. It is unknown precisely how the Roman word came into use,
though an explanation of the earlier official title, magister populi,
throws some light on the subject. That designation may mean "head of the
(infantry) host" as opposed to his subordinate, the magister equitum,
who was "head of the cavalry." If this explanation be accepted, emphasis
was thus laid in early times on the military aspect of the dictatorship,
and in fact the office seems to have been instituted for the purpose of
meeting a military crisis such as might have proved too serious for the
annual consuls with their divided command. Later constitutional theory
held that the repression of civil discord was also one of the motives
for the institution of a dictatorship. Such is the view expressed by
Cicero in the _De legibus_ (iii. 3, 9) and by the emperor Claudius in
his extant _Oratio_ (i. 28). This function of the office, although it
may not have been contemplated at first, is attested by the internal
history of Rome. In the crisis of the agitation that gathered round the
Licinian laws (367 B.C.) a dictator was appointed, and in 314 B.C. we
have the notice of a dictator created for purposes of criminal
jurisdiction (_quaestionibus exercendis_). The dictator appointed to
meet the dangers of war, sedition or crime was technically described as
"the administrative dictator" (_rei gerundae causa_). Minor, or merely
formal, needs of the state might lead to the creation of other types of
this office. Thus we find dictators destined to hold the elections, to
make out the list of the senate, to celebrate games, to establish
festivals, and to drive the nail into the temple of Jupiter--an act of
natural magic which was believed to avert pestilence. These dictators
appointed for minor purposes were expected to retire from office as soon
as their function was completed. The "administrative dictator" held
office for at least six months.

The powers of a dictator were a temporary revival of those of the kings;
but there were some limitations to his authority. He was never concerned
with civil jurisdiction, and was dependent on the senate for supplies of
money. His military authority was confined to Italy; and his power of
life and death over the citizens was at an early period limited by law.
It was probably the _lex Valeria_ of 300 B.C. that made him subject to
the right of criminal appeal (_provocatio_) within the limits of the
city. But during his tenure of power all the magistrates of the people
were regarded as his subordinates; and it was even held that the right
of assistance (_auxilium_), furnished by the tribunes of the plebs to
members of the citizen body, should not be effectively exercised when
the state was under this type of martial law. The dictator was nominated
by one of the consuls. But here as elsewhere the senate asserted its
authority over the magistrates, and the view was finally held that the
senate should not only suggest the need of nomination but also the name
of the nominee. After the nomination, the imperium of the dictator was
confirmed by a _lex curiata_ (see COMITIA). To emphasize the superiority
of this imperium over that of the consuls, the dictator might be
preceded by twenty-four lictors, not by the usual twelve; and, at least
in the earlier period of the office, these lictors bore the axes, the
symbols of life and death, within the city walls.

Tradition represents the dictatorship as having a life of three
centuries in the history of the Roman state. The first dictator is said
to have been created in 501 B.C.; the last of the "administrative"
dictators belongs to the year 216 B.C. It was an office that was
incompatible both with the growing spirit of constitutionalism and with
the greater security of the city; and the epoch of the Second Punic War
was marked by experiments with the office, such as the election of Q.
Fabius Maximus by the people, and the co-dictatorship of M. Minucius
with Fabius, which heralded its disuse (see PUNIC WARS). The emergency
office of the early and middle Republic has few points of contact,
except those of the extraordinary position and almost unfettered
authority of its holder, with the dictatorship as revised by Sulla and
by Caesar. Sulla's dictatorship was the form taken by a provisional
government. He was created "for the establishment of the Republic." It
is less certain whether the dictatorships held by Caesar were of a
consciously provisional character. Since the office represented the only
supreme _Imperium_ in Rome, it was the natural resort of the founder of
a monarchy (see SULLA and CAESAR). Ostensibly to prevent its further use
for such a purpose, M. Antonius in 44 B.C. carried a law abolishing the
dictatorship as a part of the constitution.

  BIBLIOGRAPHY.--Mommsen, _Römisches Staatsrecht_, ii. 141 foll. (3rd
  ed., Leipzig, 1887); Herzog, _Geschichte und System der römischen
  Staatsverfassung_, i. 718 foll. (Leipzig, 1884); Pauly-Wissowa,
  _Realencyclopädie_, v. 370 foll. (new edition, Stuttgart. 1893, &c.);
  Lange, _Römische Alterthümer_, i. 542 foll. (Berlin, 1856, &c.);
  Daremberg-Saglio, _Dictionnaire des antiquités grecques et romaines_,
  ii. 161 foll. (1875, &c.); Haverfield, "The Abolition of the
  Dictatorship," in _Classical Review_, iii. 77.     (A. H. J. G.)



DICTIONARY.

  Definition and history.

In its proper and most usual meaning a dictionary is a book containing a
collection of the words of a language, dialect or subject, arranged
alphabetically or in some other definite order, and with explanations in
the same or some other language. When the words are few in number, being
only a small part of those belonging to the subject, or when they are
given without explanation, or some only are explained, or the
explanations are partial, the work is called a _vocabulary_; and when
there is merely a list of explanations of the technical words and
expressions in some particular subject, a _glossary_. An alphabetical
arrangement of the words of some book or author with references to the
places where they occur is called an index (q.v.). When under each word
the phrases containing it are added to the references, the work is
called a _concordance_. Sometimes, however, these names are given to
true dictionaries; thus the great Italian dictionary of the _Accademia
della Crusca_, in six volumes folio, is called _Vocabolario_, and
Ernesti's dictionary to Cicero is called _Index_. When the words are
arranged according to a definite system of classification under heads
and subdivisions, according to their nature or their meaning, the book
is usually called a classed vocabulary; but when sufficient explanations
are given it is often accepted as a dictionary, like the _Onomasticon_
of Julius Pollux, or the native dictionaries of Sanskrit, Manchu and
many other languages.

Dictionaries were originally books of reference explaining the words of
a language or of some part of it. As the names of things, as well as
those of persons and places, are words, and often require explanation
even more than other classes of words, they were necessarily included in
dictionaries, and often to a very great extent. In time, books were
devoted to them alone, and were limited to special subjects, and these
have so multiplied, that dictionaries of things now rival in number and
variety those of words or of languages, while they often far surpass
them in bulk. There are dictionaries of biography and history, real and
fictitious, general and special, relating to men of all countries,
characters and professions; the English _Dictionary of National
Biography_ (see BIOGRAPHY) is a great instance of one form of these;
dictionaries of bibliography, relating to all books, or to those of some
particular kind or country; dictionaries of geography (sometimes called
_gazetteers_) of the whole world, of particular countries, or of small
districts, of towns and of villages, of castles, monasteries and other
buildings. There are dictionaries of philosophy; of the Bible; of
mathematics; of natural history, zoology, botany; of birds, trees,
plants and flowers; of chemistry, geology and mineralogy; of
architecture, painting and music; of medicine, surgery, anatomy,
pathology and physiology; of diplomacy; of law, canon, civil, statutory
and criminal; of political and social sciences; of agriculture, rural
economy and gardening; of commerce, navigation, horsemanship and the
military arts; of mechanics, machines and the manual arts. There are
dictionaries of antiquities, of chronology, of dates, of genealogy, of
heraldry, of diplomatics, of abbreviations, of useful receipts, of
monograms, of adulterations and of very many other subjects. These works
are separately referred to in the bibliographies attached to the
articles on the separate subjects. And lastly, there are dictionaries of
the arts and sciences, and their comprehensive offspring, encyclopaedias
(q.v.), which include in themselves every branch of knowledge. Neither
under the heading of _dictionary_ nor under that of _encyclopaedia_ do
we propose to include a mention of every work of its class, but many of
these will be referred to in the separate articles on the subjects to
which they pertain. And in this article we confine ourselves to an
account of those dictionaries which are primarily word-books. This is
practically the most convenient distinction from the subject-book or
encyclopaedia; though the two characters are often combined in one work.
Thus the _Century Dictionary_ has encyclopaedic features, while the
present edition of the _Encyclopaedia Britannica_, restoring its
earlier tradition but carrying out the idea more systematically, also
embodies dictionary features.

_Dictionarium_ is a word of low or modern Latinity;[1] _dictio_, from
which it was formed, was used in medieval Latin to mean a word.
_Lexicon_ is a corresponding word of Greek origin, meaning a book of or
for words--a dictionary. A _glossary_ is properly a collection of
unusual or foreign words requiring explanation. It is the name
frequently given to English dictionaries of dialects, which the Germans
usually call _idioticon_, and the Italians _vocabolario_. _Wörterbuch_,
a book of words, was first used among the Germans, according to Grimm,
by Kramer (1719), imitated from the Dutch _woordenboek_. From the
Germans the Swedes and Danes adopted _ordbok_, _ordbog_. The Icelandic
_ordabôk_, like the German, contains the genitive plural. The Slavonic
nations use _slovar_, _slovnik_, and the southern Slavs _ryetshnik_,
from _slovo_, _ryetsh_, a word, formed, like dictionary and lexicon,
without composition. Many other names have been given to dictionaries,
as _thesaurus_, _Sprachschatz_, _cornucopia_, _gazophylacium_,
_comprehensorium_, _catholicon_, to indicate their completeness;
_manipulus predicantium_, _promptorium puerorum_, _liber memorialis_,
_hortus vocabulorum_, _ionia_ (a violet bed), _alveary_ (a beehive),
_kamoos_ (the sea), _haft kulzum_ (the seven seas), _tsze tien_ (a
standard of character), _onomasticon_, _nomenclator_, _bibliotheca_,
_elucidario_, _Mundart-sammlung_, _clavis_, _scala_, _pharetra_,[2] _La
Crusca_ from the great Italian dictionary, and _Calepino_ (in Spanish
and Italian) from the Latin dictionary of Calepinus.

The tendency of great dictionaries is to unite in themselves all the
peculiar features of special dictionaries. A large dictionary is most
useful when a word is to be thoroughly studied, or when there is
difficulty in making out the meaning of a word or phrase. Special
dictionaries are more useful for special purposes; for instance,
synonyms are best studied in a dictionary of synonyms. And small
dictionaries are more convenient for frequent use, as in translating
from an unfamiliar language, for words may be found more quickly, and
they present the words and their meanings in a concentrated and compact
form, instead of being scattered over a large space, and separated by
other matter. Dictionaries of several languages, called _polyglots_, are
of different kinds. Some are polyglot in the vocabulary, but not in the
explanation, like Johnson's dictionary of Persian and Arabic explained
in English; some in the interpretation, but not in the vocabulary or
explanation, like _Calepini octoglotton_, a Latin dictionary of Latin,
with the meanings in seven languages. Many great dictionaries are now
polyglot in this sense. Some are polyglot in the vocabulary and
interpretation, but are explained in one language, like Jal's _Glossaire
nautique_, a glossary of sea terms in many languages, giving the
equivalents of each word in the other languages, but the explanation in
French. Pauthier's _Annamese Dictionary_ is polyglot in a peculiar way.
It gives the Chinese characters with their pronunciation in Chinese and
Annamese. Special dictionaries are of many kinds. There are technical
dictionaries of etymology, foreign words, dialects, secret languages,
slang, neology, barbarous words, faults of expression, choice words,
prosody, pronunciation, spelling, orators, poets, law, music, proper
names, particular authors, nouns, verbs, participles, particles, double
forms, difficulties and many others. Fick's dictionary (Göttingen, 1868,
8vo; 1874-1876, 8vo, 4 vols.) is a remarkable attempt to ascertain the
common language of the Indo-European nations before each of their great
separations. In the second edition of his _Etymologische Forschungen_
(Lemgo and Detmoldt, 1859-1873, 8vo, 7217 pages) Pott gives a
comparative lexicon of Indo-European roots, 2226 in number, occupying
5140 pages.


  Methods.

At no time was progress in the making of general dictionaries so rapid
as during the second half of the 19th century. It is to be seen in three
things: in the perfecting of the theory of what a general dictionary
should be; in the elaboration of methods of collecting and editing
lexicographic materials; and in the magnitude and improved quality of
the work which has been accomplished or planned. Each of these can best
be illustrated from English lexicography, in which the process of
development has in all directions been carried farthest. The advance
that has been made in theory began with a radical change of opinion with
regard to the chief end of the general dictionary of a language. The
older view of the matter was that the lexicographer should furnish a
standard of usage--should register only those words which are, or at
some period of the language have been, "good" from a literary point of
view, with their "proper" senses and uses, or should at least furnish
the means of determining what these are. In other words, his chief duty
was conceived to be to sift and refine, to decide authoritatively
questions with regard to good usage, and thus to fix the language as
completely as might be possible within the limits determined by the
literary taste of his time. Thus the Accademia della Crusca, founded
near the close of the 16th century, was established for the purpose of
purifying in this way the Italian tongue, and in 1612 the _Vocabolario
degli Accademici della Crusca_, long the standard of that language, was
published. The Académie Française, the first edition of whose dictionary
appeared in 1694, had a similar origin. In England the idea of
constructing a dictionary upon this principle arose during the second
quarter of the 18th century. It was imagined by men of letters--among
them Alexander Pope--that the English language had then attained such
perfection that further improvement was hardly possible, and it was
feared that if it were not fixed by lexicographic authority
deterioration would soon begin. Since there was no English "Academy," it
was necessary that the task should fall to some one whose judgment would
command respect, and the man who undertook it was Samuel Johnson. His
dictionary, the first edition of which, in two folio volumes, appeared
in 1755, was in many respects admirable, but it was inadequate even as a
standard of the then existing literary usage. Johnson himself did not
long entertain the belief that the natural development of a language can
be arrested in that or in any other way. His work was, however,
generally accepted as a final authority, and the ideas upon which it was
founded dominated English lexicography for more than a century. The
first effective protest in England against the supremacy of this
literary view was made by Dean (later Archbishop) Trench, in a paper on
"Some Deficiencies in Existing English Dictionaries" read before the
Philological Society in 1857. "A dictionary," he said, "according to
that idea of it which seems to me alone capable of being logically
maintained, is an _inventory of the language_; much more, but this
primarily.... It is no task of the maker of it to select the _good_
words of the language.... The business which he has undertaken is to
collect and arrange _all_ words, whether good or bad, whether they
commend themselves to his judgment or otherwise.... _He is an historian
of_ [the language], _not a critic._" That is, for the literary view of
the chief end of the general dictionary should be substituted the
philological or scientific. In Germany this substitution had already
been effected by Jacob and Wilhelm Grimm in their dictionary of the
German language, the first volume of which appeared in 1854. In brief,
then, the modern view is that the general dictionary of a language
should be a record of all the words--current or obsolete--of that
language, with all their meanings and uses, but should not attempt to
be, except secondarily or indirectly, a guide to "good" usage. A
"standard" dictionary has, in fact, been recognized to be an
impossibility, if not an absurdity.

This theoretical requirement must, of course, be modified considerably
in practice. The date at which a modern language is to be regarded by
the lexicographer as "beginning" must, as a rule, be somewhat
arbitrarily chosen; while considerable portions of its earlier
vocabulary cannot be recovered because of the incompleteness of the
literary record. Moreover, not even the most complete dictionary can
include all the words which the records--earlier and later--actually
contain. Many words, that is to say, which are found in the literature
of a language cannot be regarded as, for lexicographic purposes,
belonging to that language; while many more may or may not be held to
belong to it, according to the judgment--almost the whim--of the
individual lexicographer. This is especially true of the English tongue.
"That vast aggregate of words and phrases which constitutes the
vocabulary of English-speaking men presents, to the mind that endeavours
to grasp it as a definite whole, the aspect of one of those nebulous
masses familiar to the astronomer, in which a clear and unmistakable
nucleus shades off on all sides, through zones of decreasing brightness,
to a dim marginal film that seems to end nowhere, but to lose itself
imperceptibly in the surrounding darkness" (Dr J. A. H. Murray, _Oxford
Dict._ General Explanations, p. xvii). This "marginal film" of words
with more or less doubtful claims to recognition includes thousands of
the terms of the natural sciences (the New-Latin classificatory names of
zoology and botany, names of chemical compounds and of minerals, and the
like); half-naturalized foreign words; dialectal words; slang terms;
trade names (many of which have passed or are passing into common use);
proper names and many more. Many of these even the most complete
dictionary should exclude; others it should include; but where the line
shall be drawn will always remain a vexed question.

Another important principle upon which Trench insisted, and which also
expresses a requirement of modern scientific philology, is that the
dictionary shall be not merely a record, but also an _historical_ record
of words and their uses. From the literary point of view the most
important thing is present usage. To that alone the idea of a "standard"
has any application. Dictionaries of the older type, therefore, usually
make the common, or "proper" or "root" meaning of a word the starting
point of its definition, and arrange its other senses in a logical or
accidental order commonly ignoring the historical order in which the
various meanings arose. Still less do they attempt to give data from
which the vocabulary of the language at any previous period may be
determined. The philologist, however, for whom the growth, or
progressive alteration, of a language is a fact of central importance,
regards no record of a language as complete which does not exhibit this
growth in its successive stages. He desires to know when and where each
word, and each form and sense of it, are first found in the language; if
the word or sense is obsolete, when it died; and any other fact that
throws light upon its history. He requires, accordingly, of the
lexicographer that, having ascertained these data, he shall make them
the foundation of his exposition--in particular, of the division and
arrangement of his definitions, that sense being placed first which
appeared first in order of time. In other words, each article in the
dictionary should furnish an orderly biography of the word of which it
treats, each word and sense being so dated that the exact time of its
appearance and the duration of its use may as nearly as possible be
determined. This, in principle, is the method of the new lexicography.
In practice it is subject to limitations similar to those of the
vocabulary mentioned above. Incompleteness of the early record is here
an even greater obstacle; and there are many words whose history is, for
one reason or another, so unimportant that to treat it elaborately would
be a waste of labour and space.

The adoption of the historical principle involves a further noteworthy
modification of older methods, namely, an important extension of the use
of quotations. To Dr Johnson belongs the credit of showing how useful,
when properly chosen, they may be, not only in corroborating the
lexicographer's statements, but also in revealing special shades of
meaning or variations of use which his definitions cannot well express.
No part of Johnson's work is more valuable than this. This idea was more
fully developed and applied by Dr Charles Richardson, whose _New
Dictionary of the English Language ... Illustrated by Quotations from
the Best Authors_ (1835-1836) still remains a most valuable collection
of literary illustrations. Lexicographers, however, have, with few
exceptions, until a recent date, employed quotations chiefly for the
ends just mentioned--as instances of use or as illustrations of correct
usage--with scarcely any recognition of their value as historical
evidence; and they have taken them almost exclusively from the works of
the "best" authors. But since all the data upon which conclusions with
regard to the history of a word can be based must be collected from the
literature of the language, it is evident that, in so far as the
lexicographer is required to furnish evidence for an historical
inference, a quotation is the best form in which he can give it. In
fact, extracts, properly selected and grouped, are generally sufficient
to show the entire meaning and biography of a word without the aid of
elaborate definitions. The latter simply save the reader the trouble of
drawing the proper conclusions for himself. A further rule of the new
lexicography, accordingly, is that quotations should be used, primarily,
as historical evidence, and that the history of words and meanings
should be exhibited by means of them. The earliest instance of use that
can be found, and (if the word or sense is obsolete) the latest, are as
a rule to be given; while in the case of an important word or sense,
instances taken from successive periods of its currency also should be
cited. Moreover, a quotation which contains an important bit of
historical evidence must be used, whether its source is "good," from the
literary point of view, or not--whether it is a classic of the language
or from a daily newspaper; though where choice is possible, preference
should, of course, be given to quotations extracted from the works of
the best writers. This rule does not do away with the illustrative use
of quotations, which is still recognized as highly important, but it
subordinates it to their historical use. It is necessary to add that it
implies that the extracts must be given exactly, and in the original
spelling and capitalization, accurately dated, and furnished with a
precise reference to author, book, volume, page and edition; for
insistence upon these requirements--which are obviously important,
whatever the use of the quotation may be--is one of the most noteworthy
of modern innovations. Johnson usually gave simply the author's name,
and often quoted from memory and inaccurately; and many of his
successors to this day have followed--altogether or to some extent--his
example.

The chief difficulty in the way of this use of quotations--after the
difficulty of collection--is that of finding space for them in a
dictionary of reasonable size. Preference must be given to those which
are essential, the number of those which are cited merely on methodical
grounds being made as small as possible. It is hardly necessary to add
that the negative evidence furnished by quotations is generally of
little value; one can seldom, that is, be certain that the lexicographer
has actually found the earliest or the latest use, or that the word or
sense has not been current during some intermediate period from which he
has no quotations.

Lastly, a much more important place in the scheme of the ideal
dictionary is now assigned to the _etymology_ of words. This may be
attributed, in part, to the recent rapid development of etymology as a
science, and to the greater abundance of trustworthy data; but it is
chiefly due to the fact that from the historical point of view the
connexion between that section of the biography of a word which lies
within the language--subsequent, that is, to the time when the language
may, for lexicographical purposes, be assumed to have begun, or to the
time when the word was adopted or invented--and its antecedent history
has become more vital and interesting. Etymology, in other words, is
essentially the history of the _form_ of a word up to the time when it
became a part of the language, and is, in a measure, an extension of the
history of the development of the word in the language. Moreover, it is
the only means by which the exact relations of allied words can be
ascertained, and the separation of words of the same form but of diverse
origin (homonyms) can be effected, and is thus, for the dictionary, the
foundation of all _family history_ and correct _genealogy_. In fact, the
attention that has been paid to these two points in the best recent
lexicography is one of its distinguishing and most important
characteristics. Related to the etymology of words are the changes in
their form which may have occurred while they have been in use as parts
of the language--modifications of their pronunciation, corruptions by
popular etymology or false associations, and the like. The facts with
regard to these things which the wide research necessitated by the
historical method furnishes abundantly to the modern lexicographer are
often among the most novel and interesting of his acquisitions.

It should be added that even approximate conformity to the theoretical
requirements of modern lexicography as above outlined is possible only
under conditions similar to those under which the Oxford _New English
Dictionary_ was undertaken (see below). The labour demanded is too vast,
and the necessary bulk of the dictionary too great. When, however, a
language is recorded in one such dictionary, those of smaller size and
more modest pretensions can rest upon it as an authority and conform to
it as a model so far as their special limitations permit.

The ideal thus developed is primarily that of the general dictionary of
the purely philological type, but it applies also to the encyclopaedic
dictionary. In so far as the latter is strictly lexicographic--deals
with words as words, and not with the things they denote--it should be
made after the model of the former, and is defective to the extent in
which it deviates from it. The addition of encyclopaedic matter to the
philological in no way affects the general principles involved. It may,
however, for practical reasons, modify their application in various
ways. For example, the number of obsolete and dialectal words included
may be much diminished and the number of scientific terms (for instance,
new Latin botanical and zoological names) be increased; and the relative
amount of space devoted to etymologies and quotations may be lessened.
In general, since books of this kind are designed to serve more or less
as works of general reference, the making of them must be governed by
considerations of practical utility which the compilers of a purely
philological dictionary are not obliged to regard. The encyclopaedic
type itself, although it has often been criticized as hybrid--as a
mixture of two things which should be kept distinct--is entirely
defensible. Between the dictionary and the encyclopaedia the dividing
line cannot sharply be drawn. There are words the meaning of which
cannot be explained fully without some description of things, and, on
the other hand, the description of things and processes often involves
the definition of names. To the combination of the two objection cannot
justly be made, so long as it is effected in a way--with a selection of
material--that leaves the dictionary essentially a dictionary and not an
encyclopaedia. Moreover, the large vocabulary of the general dictionary
makes it possible to present certain kinds of encyclopaedic matter with
a degree of fulness and a convenience of arrangement which are possible
in no single work of any other class. In fact, it may be said that if
the encyclopaedic dictionary did not exist it would have to be invented;
that its justification is its indispensableness. Not the least of its
advantages is that it makes legitimate the use of diagrams and pictorial
illustrations, which, if properly selected and executed, are often
valuable aids to definition.

On its practical side the advance in lexicography has consisted in the
elaboration of methods long in use rather than in the invention of new
ones. The only way to collect the data upon which the vocabulary, the
definitions and the history are to be based is, of course, to search for
them in the written monuments of the language, as all lexicographers who
have not merely borrowed from their predecessors have done. But the
wider scope and special aims of the new lexicography demand that the
investigation shall be vastly more comprehensive, systematic and
precise. It is necessary, in brief, that, as far as may be possible, the
literature (of all kinds) of every period of the language shall be
examined systematically, in order that all the words, and senses and
forms of words, which have existed during any period may be found, and
that enough excerpts (carefully verified, credited and dated) to cover
all the essential facts shall be made. The books, pamphlets, journals,
newspapers, and so on which must thus be searched will be numbered by
thousands, and the quotations selected may (as in the case of the Oxford
_New English Dictionary_) be counted by millions. This task is beyond
the powers of any one man, even though he be a Johnson, or a Littré or a
Grimm, and it is now assigned to a corps of readers whose number is
limited only by the ability of the editor to obtain such assistance. The
modern method of editing the material thus accumulated--the actual work
of compilation--also is characterized by the application of the
principle of the division of labour. Johnson boasted that his dictionary
was written with but little assistance from the learned, and the same
was in large measure true of that of Littré. Such attempts on the part
of one man to write practically the whole of a general dictionary are no
longer possible, not merely because of the vast labour and philological
research necessitated by modern aims, but more especially because the
immense development of the vocabulary of the special sciences renders
indispensable the assistance, in the work of definition, of persons who
are expert in those sciences. The tendency, accordingly, has been to
enlarge greatly the editorial staff of the dictionary, scores of
sub-editors and contributors being now employed where a dozen or fewer
were formerly deemed sufficient. In other words, the making of a
"complete" dictionary has become a co-operative enterprise, to the
success of which workers in all the fields of literature and science
contribute.

The most complete exemplification of these principles and methods is the
_Oxford New English Dictionary, on historical principles, founded mainly
on materials collected by the Philological Society_. This monumental
work originated in the suggestion of Trench that an attempt should be
made, under the direction of the Philological Society, to complete the
vocabulary of existing dictionaries and to supply the historical
information which they lacked. The suggestion was adopted, considerable
material was collected, and Mr Herbert Coleridge was appointed general
editor. He died in 1861, and was succeeded by Dr F. J. Furnivall.
Little, however, was done, beyond the collection of quotations--about
2,000,000 of which were gathered--until in 1878 the expense of printing
and publishing the proposed dictionary was assumed by the Delegates of
the University Press, and the editorship was entrusted to Dr (afterwards
Sir) J. A. H. Murray. As the historical point of beginning, the middle
of the 12th century was selected, all words that were obsolete at that
date being excluded, though the history of words that were current both
before and after that date is given in its entirety; and it was decided
that the search for quotations--which, according to the original design,
was to cover the entire literature down to the beginning of the 16th
century and as much of the subsequent literature (especially the works
of the more important writers and works on special subjects) as might be
possible--should be made more thorough. More than 800 readers, in all
parts of the world, offered their aid; and when the preface to the first
volume appeared in 1888, the editor was able to announce that the
readers had increased to 1300, and that 3,500,000 of quotations, taken
from the writings of more than 5000 authors, had already been amassed.
The whole work was planned to be completed in ten large volumes, each
issued first in smaller parts. The first part was issued in 1884, and by
the beginning of 1910 the first part of the letter S had been reached.

The historical method of exposition, particularly by quotations, is
applied in the _New English Dictionary_, if not in all cases with entire
success, yet, on the whole, with a regularity and a precision which
leave little to be desired. A minor fault is that excerpts from second
or third rate authors have occasionally been used where better ones from
writers of the first class either must have been at hand or could have
been found. As was said above, the literary quality of the question is
highly important even in historical lexicography, and should not be
neglected unnecessarily. Other special features of the book are the
completeness with which variations of pronunciation and orthography
(with dates) are given; the fulness and scientific excellence of the
etymologies, which abound in new information and corrections of old
errors; the phonetic precision with which the present (British)
pronunciation is indicated; and the elaborate subdivision of meanings.
The definitions as a whole are marked by a high degree of accuracy,
though in a certain number of cases (not explicable by the date of the
volumes) the lists of meanings are not so good as one would expect, as
compared (say) with the _Century Dictionary_. Work of such magnitude
and quality is possible, practically, only when the editor of the
dictionary can command not merely the aid of a very large number of
scholars and men of science, but their gratuitous aid. In this the _New
English Dictionary_ has been singularly fortunate. The conditions under
which it originated, and its aim, have interested scholars everywhere,
and led them to contribute to the perfecting of it their knowledge and
time. The long list of names of such helpers in Sir J. A. H. Murray's
preface is in curious contrast with their absence from Dr Johnson's and
the few which are given in that of Littré. The editor's principal
assistants were Dr Henry Bradley and Dr W. A. Craigie. Of the dictionary
as a whole it may be said that it is one of the greatest achievements,
whether in literature or science, of modern English scholarship and
research.

  The _New English Dictionary_ furnishes for the first time data from
  which the extent of the English word-store at any given period, and
  the direction and rapidity of its growth, can fairly be estimated. For
  this purpose the materials furnished by the older dictionaries are
  quite insufficient, on account of their incompleteness and
  unhistorical character. For example 100 pages of the _New English
  Dictionary_ (from the letter H) contain 1002 words, of which, as the
  dated quotations show, 585 were current in 1750 (though some, of
  course, were very rare, some dialectal, and so on), 191 were obsolete
  at that date, and 226 have since come into use. But of the more than
  700 words--current or obsolete--which Johnson might thus have
  recorded, he actually did record only about 300. Later dictionaries
  give more of them, but they in no way show their status at the date in
  question. It is worth noting that the figures given seem to indicate
  that not very many more words have been added to the vocabulary of the
  language during the past 150 years than had been lost by 1750. The
  pages selected, however, contain comparatively few recent scientific
  terms. A broader comparison would probably show that the gain has been
  more than twice as great as the loss.

In the _Deutsches Wörterbuch_ of Jacob and Wilhelm Grimm the scientific
spirit, as was said above, first found expression in general
lexicography. The desirability of a complete inventory and investigation
of German words was recognized by Leibnitz and by various 18th-century
scholars, but the plan and methods of the Grimms were the direct product
of the then new scientific philology. Their design, in brief, was to
give an exhaustive account of the words of the literary language (New
High German) from about the end of the 15th century, including their
earlier etymological and later history, with references to important
dialectal words and forms; and to illustrate their use and history
abundantly by quotations. The first volume appeared in 1854. Jacob Grimm
(died 1863) edited the first, second (with his brother, who died in
1859), third and a part of the fourth volumes; the others have been
edited by various distinguished scholars. The scope and methods of this
dictionary have been broadened somewhat as the work has advanced. In
general it may be said that it differs from the _New English Dictionary_
chiefly in its omission of pronunciations and other pedagogic matter;
its irregular treatment of dates; its much less systematic and less
lucid statement of etymologies; its less systematic and less fruitful
use of quotations; and its less convenient and less intelligible
arrangement of material and typography.

These general principles lie also at the foundation of the scholarly
_Dictionnaire de la langue française_ of E. Littré, though they are
there carried out less systematically and less completely. In the
arrangement of the definitions the first place is given to the most
primitive meaning of the word instead of to the most common one, as in
the dictionary of the Academy; but the other meanings follow in an order
that is often logical rather than historical. Quotations also are
frequently used merely as literary illustrations, or are entirely
omitted; in the special paragraphs on the history of words before the
16th century, however, they are put to a strictly historical use. This
dictionary--perhaps the greatest ever compiled by one man--was published
1863-1872. (Supplement, 1878.)

The _Thesaurus Linguae Latinae_, prepared under the auspices of the
German Academies of Berlin, Göttingen, Leipzig, Munich and Vienna, is a
notable application of the principles and practical co-operative method
of modern lexicography to the classical tongues. The plan of the work is
to collect quotations which shall register, with its full context, every
word (except the most familiar particles) in the text of each Latin
author down to the middle of the 2nd century A.D., and to extract all
important passages from all writers of the following centuries down to
the 7th; and upon these materials to found a complete historical
dictionary of the Latin language. The work of collecting quotations was
begun in 1894, and the first part of the first volume has been
published.

In the making of all these great dictionaries (except, of course, the
last) the needs of the general public as well as those of scholars have
been kept in view. But the type to which the general dictionary designed
for popular use has tended more and more to conform is the
_encyclopaedic_. This combination of lexicon and encyclopaedia is
exhibited in an extreme--and theoretically objectionable--form in the
_Grand dictionnaire universel du XIX^e siècle_ of Pierre Larousse.
Besides common words and their definitions, it contains a great many
proper names, with a correspondingly large number of biographical,
geographical, historical and other articles, the connexion of which with
the strictly lexicographical part is purely mechanical. Its utility,
which--notwithstanding its many defects--is very great, makes it,
however, a model in many respects. Fifteen volumes were published
(1866-1876), and supplements were brought out later (1878-1890). The
_Nouveau Larousse illustré_ started publication in 1901, and was
completed in 1904 (7 vols.). This is not an abridgment or a fresh
edition of the _Grand Dictionnaire_ of Pierre Larousse, but a new and
distinct publication.

The most notable work of this class, in English, is the _Century
Dictionary_, an American product, edited by Professor W. D. Whitney, and
published 1889-1891 in six volumes, containing 7046 pages (large
quarto). It conforms to the philological mode in giving with great
fulness the older as well as the present vocabulary of the language, and
in the completeness of its etymologies; but it does not attempt to give
the full history of every word within the language. Among its other more
noteworthy characteristics are the inclusion of a great number of modern
scientific and technical words, and the abundance of its quotations. The
quotations are for the most part provided with references, but they are
not dated. Even when compared with the much larger _New English
Dictionary_, the _Century's_ great merit is the excellent enumeration of
meanings, and the accuracy of its explanations; in this respect it is
often better and fuller than the _New English_. In the application of
the encyclopaedic method this dictionary is conservative, excluding,
with a few exceptions, proper names, and restricting, for the most part,
the encyclopaedic matter to descriptive and other details which may
legitimately be added to the definitions. Its pictorial illustrations
are very numerous and well executed. In the manner of its compilation it
is a good example of modern cooperative dictionary-making, being the
joint product of a large number of specialists. Next to the _New English
Dictionary_ it is the most complete and scholarly of English lexicons.

_Bibliography._--The following list of dictionaries (from the 9th
edition of this work, with occasional corrections) is given for its
historical interest, but in recent years dictionary-making has been so
abundant that no attempt is made to be completely inclusive of later
works; the various articles on languages may be consulted for these. The
list is arranged geographically by families of languages, or by regions.
In each group the order, when not alphabetical, is usually from north to
south, extinct languages generally coming first, and dialects being
placed under their language. Dictionaries forming parts of other works,
such as travels, histories, transactions, periodicals, reading-books,
&c., are generally excluded. The system here adopted was chosen as on
the whole the one best calculated to keep together dictionaries
naturally associated. The languages to be considered are too many for an
alphabetical arrangement, which ignores all relations both natural and
geographical, and too few to require a strict classification by
affinities, by which the European languages, which for many reasons
should be kept together, would be dispersed. Under either system,
Arabic, Persian and Turkish, whose dictionaries are so closely
connected, would be widely separated. A wholly geographical arrangement
would be inconvenient, especially in Europe. Any system, however, which
attempts to arrange in a consecutive series the great network of
languages by which the whole world is enclosed, must be open to some
objections; and the arrangement adopted in this list has produced some
anomalies and dispersions which might cause inconvenience if not pointed
out. The old Italic languages are placed under Latin, all dialects of
France under French (but Provençal as a distinct language), and
Wallachian among Romanic languages. Low German and its dialects are not
separated from High German. Basque is placed after Celtic; Albanian,
Gipsy and Turkish at the end of Europe, the last being thus separated
from its dialects and congeners in Northern and Central Asia, among
which are placed the Kazan dialect of Tatar, Samoyed and Ostiak.
Accadian is placed after Assyrian among the Semitic languages, and
Maltese as a dialect of Arabic; while the Ethiopic is among African
languages as it seemed undesirable to separate it from the other
Abyssinian languages, or these from their neighbours to the north and
south. Circassian and Ossetic are joined to the first group of Aryan
languages lying to the north-west of Persia, and containing Armenian,
Georgian and Kurd. The following is the order of the groups, some of the
more important languages, that is, of those best provided with
dictionaries, standing alone:--

EUROPE: Greek, Latin, French, Romance, Teutonic (Scandinavian and
German), Celtic, Basque, Baltic, Slavonic, Ugrian, Gipsy, Albanian.

ASIA: Semitic, Armenian, Persian, Sanskrit, Indian, Indo-Chinese, Malay
Archipelago, Philippines, Chinese, Japanese, Northern and Central Asia.

AFRICA: Egypt and Abyssinia, Eastern Africa, Southern, Western, Central,
Berber.

AUSTRALIA AND POLYNESIA.

AMERICA: North, Central (with Mexico), South.


EUROPE

 Greek.---Athenaeus quotes 35 writers of works, known or supposed to be
  dictionaries, for, as they are all lost, it is often difficult to
  decide on their nature. Of these, Anticlides, who lived after the
  reign of Alexander the Great, wrote [Greek: Exêgêtikos], which seems
  to have been a sort of dictionary, perhaps explaining the words and
  phrases occurring in ancient stories. Zenodotus, the first
  superintendent of the great library of Alexandria, who lived in the
  reigns of Ptolemy I. and Ptolemy II., wrote [Greek: Glôssai], and also
  [Greek: Lexeis ethnikai], a dictionary of barbarous or foreign
  phrases. Aristophanes of Byzantium, son of Apelles the painter, who
  lived in the reigns of Ptolemy II. and Ptolemy III., and had the
  supreme management of the Alexandrian library, wrote a number of
  works, as [Greek: Attikai Lexeis, Lakônikai Glôssai] which, from the
  titles, should be dictionaries, but a fragment of his [Greek: Lexeis]
  printed by Boissonade, in his edition of Herodian (London, 1869, 8vo,
  pp. 181-189), is not alphabetical. Artemidorus, a pupil of
  Aristophanes, wrote a dictionary of technical terms used in cookery.
  Nicander Colophonius, hereditary priest of Apollo Clarius, born at
  Claros, near Colophon in Ionia, in reputation for 50 years, from 181
  to 135, wrote [Greek: Glôssai] in at least three books. Parthenius, a
  pupil of the Alexandrian grammarian Dionysius (who lived in the 1st
  century before Christ), wrote on choice words used by historians.
  Didymus, called [Greek: chalkenteros], who, according to Athenaeus,
  wrote 3500 books, and, according to Seneca, 4000, wrote lexicons of
  the tragic poets (of which book 28 is quoted), of the comic poets, of
  ambiguous words and of corrupt expressions. Glossaries of Attic words
  were written by Crates, Philemon, Philetas and Theodorus; of Cretan,
  by Hermon or Hermonax; of Phrygian, by Neoptolemus; of Rhodian, by
  Moschus; of Italian, by Diodorus of Tarsus; of foreign words, by
  Silenus; of synonyms, by Simaristus; of cookery, by Heracleon; and of
  drinking vessels, by Apollodorus of Cyrene. According to Suidas, the
  most ancient Greek lexicographer was Apollonius the sophist, son of
  Archibius. According to the common opinion, he lived in the time of
  Augustus at Alexandria. He composed a lexicon of words used by Homer,
  [Greek: Lexeis Homêrikai], a very valuable and useful work, though
  much interpolated, edited by Villoison, from a MS. of the 10th
  century, Paris, 1773, 4to, 2 vols.; and by Tollius, Leiden, 1788, 8vo;
  ed. Bekker, Berlin, 1833, 8vo. Erotian or Herodian, physician to Nero,
  wrote a lexicon on Hippocrates, arranged in alphabetical order,
  probably by some copyist, whom Klein calls "homo sciolus." It was
  first published in Greek in H. Stephani _Dictionarium Medicum_, Paris,
  1564, 8vo; ed. Klein, Lipsiae, 1865, 8vo, with additional fragments.
  Timaeus the sophist, who, according to Ruhnken, lived in the 3rd
  century, wrote a very short lexicon to Plato, which, though much
  interpolated, is of great value, 1st ed. Ruhnken, Leiden, 1754; ed.
  locupletior, Lugd. Bat. 1789, 8vo. Aelius Moeris, called the Atticist,
  lived about 190 A.D., and wrote an Attic lexicon, 1st ed. Hudson,
  Oxf. 1712, Bekker, 1833. Julius Pollux ([Greek: Ioulios Polydeukês])
  of Naucratis, in Egypt, died, aged fifty-eight, in the reign of
  Commodus (180-192), who made him professor of rhetoric at Athens. He
  wrote, besides other lost works, an Onomasticon in ten books, being a
  classed vocabulary, intended to supply all the words required by each
  subject with the usage of the best authors. It is of the greatest
  value for the knowledge both of language and of antiquities. First
  printed by Aldus, Venice, 1500, fol.; often afterwards; ed. Lederlinus
  and Hemsterhuis, Amst. 1706, 2 vols.; Dindorf, 1824, 5 vols., Bethe
  (1900 f.). Harpocration of Alexandria, probably of the 2nd century,
  wrote a lexicon on the ten Attic orators, first printed by Aldus, Ven.
  1503, fol.; ed. Dindorf, Oxford, 1853, 8vo, 2 vols. from 14 MSS.
  Orion, a grammarian of Thebes, in Egypt, who lived between 390 and
  460, wrote an etymological dictionary, printed by Sturz, Leipzig,
  1820, 4to. Helladius a priest of Jupiter at Alexandria, when the
  heathen temples there were destroyed by Theophilus in 389 or 391
  escaped to Constantinople, where he was living in 408. He wrote an
  alphabetical lexicon, now lost, chiefly of prose, called by Photius
  the largest ([Greek: polystichôtaton]) which he knew. Ammonius,
  professor of grammar at Alexandria, and priest of the Egyptian ape,
  fled to Constantinople with Helladius, and wrote a dictionary of words
  similar in sound but different in meaning, which has been often
  printed in Greek lexicons, as Aldus, 1497, Stephanus, and separately
  by Valckenaer, Lugd. Bat. 1739, 4to, 2 vols., and by others. Zenodotus
  wrote on the cries of animals, printed in Valckenaer's _Ammonius_;
  with this may be compared the work of Vincentio Caralucci, _Lexicon
  vocum quae a brutis animalibus emittuntur_, Perusia, 1779, 12mo.
  Hesychius of Alexandria wrote a lexicon, important for the knowledge
  of the language and literature, containing many dialectic and local
  expressions and quotations from other authors, 1st ed. Aldus, Ven.
  1514, fol.; the best is Alberti and Ruhnken, Lugd. Bat. 1746-1766,
  fol. 2 vols.; collated with the MS. in St Mark's library, Venice, the
  only MS. existing, by Niels Iversen Schow, Leipzig, 1792, 8vo; ed.
  Schmidt, Jena, 1867, 8vo. The foundation of this lexicon is supposed
  to have been that of Pamphilus, an Alexandrian grammarian, quoted by
  Athenaeus, which, according to Suidas, was in 95 books from [Epsilon]
  to [Omega]; [Alpha] to [Delta] had been compiled by Zopirion. Photius,
  consecrated patriarch of Constantinople, 25th December 857, living in
  886, left a lexicon, partly extant, and printed with Zonaras, Lips.
  1808, 4to, 3 vols., being vol. iii.; ed. Naber, Leidae, 1864-1865,
  8vo, 2 vols. The most celebrated of the Greek glossaries is that of
  Suidas, of whom nothing is known. He probably lived in the 10th
  century. His lexicon is an alphabetical dictionary of words including
  the names of persons and places--a compilation of extracts from Greek
  writers, grammarians, scholiasts and lexicographers, very carelessly
  and unequally executed. It was first printed by Demetrius
  Chalcondylas, Milan, 1499, fol.; the best edition, Bernhardy, Halle,
  1853, 4to, 2 vols. John Zonaras, a celebrated Byzantine historian and
  theologian, who lived in the 12th century, compiled a lexicon, first
  printed by Tittmann, Lips. 1808. 4to, 2 vols. An anonymous Greek
  glossary, entitled [Greek: Etymologikon mega], _Etymologicum magnum_,
  has been frequently printed. The first edition is by Musurus, Venitia,
  1499, fol.; the best by Gaisford, Oxonii, 1848, fol. It contains many
  grammatical remarks by famous authorities, many passages of authors,
  and mythological and historical notices. The MSS. vary so much that
  they look like the works of different authors. To Eudocia Augusta of
  Makrembolis, wife of the emperors Constantine XI. and Romanus IV.
  (1059 to 1071), was ascribed a dictionary of history and mythology,
  [Greek: Iônia] (bed of violets), first printed by D'Ansse de
  Villoison, _Anecdota Graeca_, Venetiis, 1781, 4to, vol. i. pp. 1-442.
  It was supposed to have been of much value before it was published.
  Thomas, Magister Officiorum under Andronicus Palaeologus, afterward
  called as a monk Theodulus, wrote [Greek: Eklogai onomatôn Attikôn],
  printed by Callierges, Romae, 1517, 8vo: Papias, _Vocabularium_,
  Mediolani, 1476, fol.: Craston, an Italian Carmelite monk of Piacenza,
  compiled a Greek and Latin lexicon, edited by Bonus Accursius, printed
  at Milan, 1478, fol.: Aldus, Venetiis, 1497, fol.: Guarino, born about
  1450 at Favora, near Camarino, who called himself both Phavorinus and
  Camers, published his _Thesaurus_ in 1504. These three lexicons were
  frequently reprinted. Estienne, _Thesaurus_, Genevae, 1572, fol., 4
  vols.; ed. Valpy, Lond. 1816-1826, 6 vols. fol.; Paris, 1831-1865, 9
  vols. fol., 9902 pages: [Greek: Kibôtos], the ark, was intended to
  give the whole language, ancient and modern, but vol. i.,
  Constantinople, 1819, fol., 763 pages, [Alpha] to [Delta], only
  appeared, as the publication was put an end to by the events of 1821.
  ENGLISH.--Jones, London, 1823, 8vo: Dunbar, Edin. 3rd ed. 1850, 4to:
  Liddell and Scott, 8th ed. Oxford, 1897, 4to. FRENCH.--Alexandre, 12th
  ed. Paris, 1863, 8vo; 1869-1871, 2 vols: Chassang, ib. 1872, 8vo.
  ITALIAN.--Camini, Torino, 1865, 8vo, 972 pages: Müller, ib. 1871, 8vo.
  SPANISH.--_Diccionario manual, por les padres Esculapios_, Madrid,
  1859, 8vo. GERMAN.--Passow, 5th ed. Leipzig, 1841-1857, 4to: Jacobitz
  and Seiler, 4th ed. ib. 1856, 8vo: Benseler, ib. 1859, 8vo: Pape,
  Braunschweig, 1870-1874, 8vo, 4 vols. Prellwitz, _Etymologisches
  Wörterbuch der griechischen Sprache_, new edition, 1906: Herwerden,
  _Lexicon Graecum suppletorium et dialecticum_, 1902.
  DIALECTS.--_Attic_: Moeris, ed. Pierson, Lugd. Bat. 1759. 8vo. _Attic
  Orators_: Reiske, Oxon. 1828, 8vo, 2 vols. _Doric_: Portus, Franckof.
  1605, 8vo. _Ionic_: Id. ib. 1603, 8vo; 1817; 1825. PROSODY.--Morell,
  Etonae, 1762, 4to; ed. Maltby, Lond. 1830, 4to: Brasse, Lond. 1850,
  8vo. RHETORIC.--Ernesti, Lips. 1795, 8vo. MUSIC.--Drieberg, Berlin,
  1855. ETYMOLOGY.--Curtius, Leipzig, 1858-1862: Lancelot, Paris, 1863,
  8vo. SYNONYMS.--Peucer, Dresden, 1766, 8vo: Pillon, Paris, 1847, 8vo.
  PROPER NAMES.--Pape, ed. Sengebusch, 1866, 8vo, 969 pages.
  VERBS.--Veitch, 2nd ed. Oxf. 1866. TERMINATIONS.--Hoogeveen, Cantab.
  1810, 4to: Pape, Berlin, 1836, 8vo. PARTICULAR AUTHORS.--_Aeschylus_:
  Wellauer, 2 vols. Lips. 1830-1831, 8vo. _Aristophanes_: Caravella,
  Oxonii, 1822, 8vo. _Demosthenes_: Reiske, Lips. 1775, 8vo.
  _Euripides_: Beck, Cantab. 1829, 8vo. _Herodotus_: Schweighäuser,
  Strassburg, 1824, 8vo, 2 vols. _Hesiod_: Osoruis, Neapol. 1791, 8vo.
  _Homer_: Apollonius Sophista, ed. Tollius, Lugd. Bat., 1788, 8vo:
  Schaufelberger, Zürich, 1761-1768, 8vo, 8 vols.: Crusius, Hanover,
  1836, 8vo: Wittich, London, 1843, 8vo: Döderlein, Erlangen, 8vo, 3
  vols.: Eberling, Lipsiae, 1875, 8vo: Autenrieth, Leipzig, 1873, 8vo;
  London, 1877, 8vo. _Isocrates_: Mitchell, Oxon. 1828, 8vo. _Pindar_:
  Portus, Hanov. 1606, 8vo. _Plato_: Timaeus, ed. Koch, Lips. 1828, 8vo:
  Mitchell, Oxon. 1832, 8vo: Ast, Lips. 1835-1838, 8vo, 3 vols.
  _Plutarch_: Wyttenbach, Lips. 1835, 8vo, 2 vols. _Sophocles_: Ellendt,
  Regiomonti, 1834-1835, 8vo ed.; Genthe, Berlin, 1872, 8vo.
  _Thucydides_: Bétant, Geneva, 1843-1847, 8vo, 2 vols. _Xenophon_:
  Sturtz, Lips. 1801-1804, 8vo, 4 vols.: Cannesin (Anabasis,
  Gr.-Finnish), Helsirgissä, 1868, 8vo: Sauppe, Lipsiae, 1869, 8vo.
  _Septuagint_: Hutter, Noribergae, 1598, 4to: Biel, Hagae, 1779-1780,
  8vo. _New Testament_: Lithocomus, Colon, 1552, 8vo: Parkhurst, ed.
  Major, London, 1845, 8vo: Schleusner (juxta ed. Lips. quartam),
  Glasguae, 1824, 4to.


 Medieval and Modern Greek.--Meursius, Lugd. Bat. 1614, 4to:
  Critopulos, Stendaliae, 1787, 8vo: Portius, Par. 1635, 4to: Du Cange,
  Paris, 1682, fol., 2 vols.; Ludg. 1688, fol. ENGLISH.--Polymera,
  Hermopolis, 1854, 8vo: Sophocles, Cambr. Mass. 1860-1887: Contopoulos,
  Athens, 1867, 8vo; Smyrna, 1868-1870, 8vo, 2 parts, 1042 pages.
  FRENCH.--Skarlatos, Athens, 1852, 4to: Byzantius, ib. 1856, 8vo, 2
  vols.: Varvati, 4th ed. ib., 1860, 8vo. ITALIAN.--Germano, Romae,
  1622, 8vo: Somavera, Parigi, 1709, fol., 2 vols.: Pericles,
  Hermopolis, 1857, 8vo. GERMAN.--Schmidt, Lips. 1825-1827, 12mo, 2
  vols.: Petraris, Leipz. 1897. POLYGLOTS.--Koniaz (Russian and Fr.),
  Moscow, 1811, 4to; Schmidt (Fr.-Germ.), Leipzig, 1837-1840, 12mo, 3
  vols.: Theocharopulas de Patras (Fr.-Eng.), Munich, 1840, 12mo.


 Latin.--Johannes de Janua, _Catholicon_ or _Summa_, finished in 1286,
  printed Moguntiæ 1460, fol.; Venice, 1487; and about 20 editions
  before 1500: Johannes, _Comprehensorium_, Valentia, 1475, fol.: Nestor
  Dionysius, _Onomasticon_, Milan, 1477, fol.: Stephanus, Paris, 1531,
  fol., 2 vols.: Gesner, Lips. 1749, fol., 4 vols.: Forcellini, Patavii,
  1771, fol., 4 vols. POLYGLOT.--Calepinus, Reggio, 1502, fol. (Aldus
  printed 16 editions, with the Greek equivalents of the Latin words;
  Venetiis, 1575, fol., added Italian, French and Spanish; Basileae,
  1590, fol., is in 11 languages; several editions, from 1609, are
  called Octolingue; many of the latter 2 vol. editions were edited by
  John Facciolati): Verantius (Ital., Germ., Dalmatian, Hungarian),
  Venetiis, 1595, 4to: Lodereckerus (Ital., Germ., Dalm., Hungar.,
  Bohem., Polish), Pragae, 1605, 4to. ENGLISH.--_Promptorium
  parvulorum_, compiled in 1440 by Galfridus Grammaticus, a Dominican
  monk of Lynn Episcopi, in Norfolk, was printed by Pynson, 1499; 8
  editions, 1508-1528, ed. Way, Camden Society, 1843-1865, 3 vols. 4to;
  _Medulla grammaticis_, probably by the same author, MS. written 1483;
  printed as _Ortus vocabulorum_, by Wynkyn de Worde, 1500; 13 editions
  1509-1523; Sir Thomas Elyot, London, 1538, fol.; 2nd ed. 1543;
  _Bibliotheca Eliotae_, ed. Cooper, ib. 1545, fol.: Huloet,
  _Abecedarium_, London, 1552, fol.; _Dictionarie_, 1572, fol.: Cooper,
  London, 1565, fol.; 4th edition, 1584, fol.: Baret, _Alvearie_, ib.
  1575, fol.; 1580, fol.: Fleming, ib. 1583, fol.: Ainsworth, London,
  1736, 4to; ed. Morell, London, 1796, 4to, 2 vols.; ed. Beatson and
  Ellis, ib. 1860, 8vo: Scheller, translated by Riddle, Oxford, 1835,
  fol.: Smith, London, 1855, 8vo; 1870: Lewis and Short, Oxford, 1879.
  ENG.-LATIN.--Levins, _Manipulus puerorum_, Lond. 1570, 4to: Riddle,
  ib. 1838, 8vo: Smith, ib. 1855, 8vo. FRENCH.--_Catholicon parvum_,
  Geneva, 1487: Estienne, _Dictionnaire_, Paris, 1539, fol. 675 pages;
  enlarged 1549; ed. Huggins, Lond. 1572: Id. _Dictionarium
  Latino-Gallicum_, Lutetiae, 1546, fol.; Paris, 1552; 1560: Id.,
  _Dictionariolum puerorum_, Paris, 1542, 4to: _Les Mots français_,
  Paris, 1544, 4to; the copy in the British Museum has the autograph of
  Queen Catherine Parr: Thierry (Fr.-Lat.), Paris, 1564, fol.: Danet, Ad
  usum Delphini, Paris, 1700, 4to, 2 vols.; and frequently: Quicherat,
  9th ed. Paris, 1857, 8vo: Theil, 3rd ed. Paris, 1863, 8vo: Freund, ib.
  1835-1865, 4to, 3 vols. GERMAN.--Joh. Melber, of Gerolzhofen,
  _Vocabularius praedicantium_, of which 26 editions are described by
  Hain (_Repertorium_, No. 11,022, &c.), 15 undated, 7 dated 1480-1495,
  4to, and 3 after 1504: _Vocabularius gemma gemmarum_, Antwerp, 1484,
  4to; 1487; 12 editions, 1505-1518: Herman Torentinus, _Elucidarius
  carminum_, Daventri, 1501, 4to; 22 editions, 1504-1536: Binnart, Ant.
  1649, 8vo: Id., _Biglotton_, ib. 1661; 4th ed. 1688: Faber, ed.
  Gesner, Hagae Com. 1735, fol., 2 vols.: Hederick, Lips. 1766, 8vo, 2
  vols.: Ingerslev, Braunschweig, 1835-1855, 8vo, 2 vols.: _Thesaurus
  linguae Latinae_, Leipzig, 1900: Walde, _Lateinisches etymologisches
  Wörterbuch_, 1906. ITALIAN.--Seebar (Sicilian translation of Lebrixa),
  Venet. 1525, 8vo: Venuti, 1589, 8vo: Galesini, Venez. 1605, 8vo:
  Bazzarini and Bellini, Torino, 1864, 4to, 2 vols. 3100 pages.
  SPANISH.--Salmanticae, 1494, fol.; Antonio de Lebrixa, Nebrissenis,
  Compluti, 1520, fol., 2 vols.: Sanchez de la Ballesta, Salamanca,
  1587, 4to: Valbuena, Madrid, 1826, fol. PORTUGUESE.--Bluteau, Lisbon,
  1712-1728, fol., 10 vols: Fonseca, ib. 1771, fol.: Ferreira, Paris,
  1834, 4to; 1852. ROMANSCH.--_Promptuario di voci volgari_, Valgrisii,
  1565, 4to. VLACH.--Divalitu, Bucuresci, 1852, 8vo.
  SWEDISH.--_Vocabula_, Rostock, 1574, 8vo; Stockholm, 1579: Lindblom,
  Upsala, 1790, 4to. DUTCH.--Binnart, Antw. 1649, 8vo: Scheller, Lugd.
  Bat. 1799, 4to, 2 vols. FLEMISH.--Paludanus, Gandavi, 1544, 4to.
  POLISH.--Macinius, Königsberg, 1564, fol.: Garszynski, Breslau, 1823,
  8vo, 2 vols. BOHEMIAN.--Johannes Aquensis, Pilsnae, 1511, 4to:
  Reschel, Olmucii, 1560-1562, 4to, 2 vols.: Cnapius, Cracovia, 1661,
  fol., 3 vols. ILLYRIAN.--Bellosztenecz, Zagrab, 1740, 4to: Jambresich
  (also Germ. and Hungar.), Zagrab, 1742, 4to. SERVIAN.--Swotlik, Budae,
  1721, 8vo. HUNGARIAN.--Molnar, Frankf. a. M. 1645, 8vo: Pariz-Papai,
  Leutschen, 1708, 8vo; 1767. FINNISH.--Rothsen, Helsingissä, 1864, 8vo.
  POETIC.--_Epithetorum et synonymorum thesaurus_, Paris, 1662, 8vo,
  attributed to Chatillon; reprinted by Paul Aler, a German Jesuit, as
  _Gradus ad Parnassum_, Paris, 1687, 8vo; many subsequent editions:
  _Schirach_, Hal. 1768, 8vo: Noel, Paris, 1810, 8vo; 1826: Quicherat,
  Paris, 1852, 8vo: Young, London, 1856, 8vo. EROTIC.--Rambach,
  Stuttgart, 1836, 8vo. RHETORICAL.--Ernesti, Lips. 1797, 8vo. CIVIL
  LAW.--Dirksen, Berolini, 1837, 4to. SYNONYMS.--Hill, Edinb. 1804, 4to:
  Döderlein, Lips. 1826-1828, 8vo, 6 vols. ETYMOLOGY.--Danet, Paris,
  1677, 8vo: Vossius, Neap. 1762, fol., 2 vols.: Salmon, London, 1796,
  8vo, 2 vols.: Nagel, Berlin, 1869, 8vo; Latin roots, with their French
  and English derivatives, explained in German: Zehetmayr, Vindobonae,
  1873, 8vo: Vani[vc]ek, Leipz. 1874, 8vo. BARBAROUS.--Marchellus,
  Mediol. 1753, 4to; Krebs, Frankf. a. M. 1834, 8vo; 1837. PARTICULAR
  AUTHORS.--_Caesar_: Crusius, Hanov. 1838, 8vo. _Cicero_: Nizzoli,
  Brescia, 1535, fol.; ed. Facciolati, Patavii, 1734, fol.; London,
  1820, 8vo, 3 vols.: Ernesti, Lips. 1739, 8vo; Halle, 1831. _Cornelius
  Nepos_: Schmieder, Halle, 1798, 8vo; 1816: Billerbeck, Hanover, 1825,
  8vo. _Curtius Rufus_: Crusius, Hanov. 1844, 8vo. _Horace_: Ernesti,
  Berlin, 1802-1804, 8vo, 3 vols.: Döring, Leipz. 1829, 8vo. _Justin_:
  Meinecke, Lemgo, 1793, 8vo; 2nd ed. 1818. _Livy_: Ernesti, Lips. 1784,
  8vo; ed Schäfer, 1804. _Ovid_: Gierig, Leipz. 1814: (Metamorphoses)
  Meinecke, 2nd ed., Lemgo, 1825, 8vo: Billerbeck (Do.), Hanover, 1831,
  8vo. _Phaedrus_: Oertel, Nürnberg, 1798, 8vo: Hörstel, Leipz. 1803,
  8vo: Billerbeck Hanover, 1828, 8vo. _Plautus_: Paraeus, Frankf. 1614,
  8vo. _Pliny_: Denso, Rostock, 1766, 8vo_. Pliny, jun._: Wensch,
  Wittenberg, 1837-1839, 4to. _Quintilian_: Bonnellus, Leipz. 1834, 8vo.
  _Sallust_: Schneider, Leipz. 1834, 8vo: Crusius, Hanover, 1840, 8vo.
  _Tacitus_: Bötticher, Berlin, 1830, 8vo. _Velleius Paterculus_: Koch,
  Leipz. 1857, 8vo. _Virgil_: _Clavis_, London, 1742, 8vo: Braunhard,
  Coburg, 1834, 8vo. _Vitruvius_: Rode, Leipz. 1679, 4to, 2 vols.:
  Orsini, Perugia, 1801, 8vo.

  OLD ITALIAN LANGUAGES.--Fabretti, Torini, 1858, 4to. _Umbrian_:
  Huschke, Leipz. 1860, 8vo. _Oscan and Sabellian_: Id. Elberfeld, 1856,
  8vo.

  MEDIEVAL LATIN.--Du Cange, _Glossarium_, Paris, 1733-1736, fol., 6
  vols.; Carpentier, Suppl., Paris, 1766, fol., 4 vols.; ed. Adelung,
  Halae, 1772-1784, 8vo, 6 vols.; ed. Henschel, Paris, 1840-1850, 4to, 7
  vols. (vol. vii. contains a glossary of Old French): Brinckmeier,
  Gotha, 1850-1863, 8vo, 2 vols.: Hildebrand (_Glossarium saec. ix._),
  Götting. 1854, 4to: Diefenbach, _Glossarium_, Frankf. 1857, 4to: Id.
  _Gloss. novum_, ib. 1867, 4to. ECCLESIASTICAL.--Magri, Messina, 1644,
  4to; 8th ed. Venezia, 1732; Latin translation, _Magri Hierolexicon_,
  Romae, 1677, fol.; 6th ed. Bologna, 1765, 4to, 2 vols.


_Romance Languages. _

 Romance Languages generally.--Diez, Bonn, 1853, 8vo; 2nd ed. ib.
  1861-1862, 8vo, 2 vols.; 3rd ed. ib. 1869-1870, 8vo, 2 vols.; transl.
  by Donkin, 1864, 8vo.


 French.--Ranconet, _Thresor_, ed. Nicot, Paris, 1606, fol.; ib. 1618,
  4to: Richelet, Genève, 1680, fol., 2 vols.; ed. Gattel, Paris, 1840,
  8vo, 2 vols.

  The French Academy, after five years' consideration, began their
  dictionary, on the 7th of February 1639, by examining the letter A,
  which took them nine months to go through. The word Académie was for
  some time omitted by oversight. They decided, on the 8th of March
  1638, not to cite authorities, and they have since always claimed the
  right of making their own examples. Olivier justifies them by saying
  that for eighty years all the best writers belonged to their body, and
  they could not be expected to cite each other. Their design was to
  raise the language to its last perfection, and to open a road to reach
  the highest eloquence. Antoine Furetière, one of their members,
  compiled a dictionary which he says cost him forty years' labour for
  ten hours a day, and the manuscript filled fifteen chests. He gave
  words of all kinds, especially technical, names of persons and places,
  and phrases. As a specimen, he published his _Essai_, Paris, 1684,
  4to; Amst. 1685, 12mo. The Academy charged him with using the
  materials they had prepared for their dictionary, and expelled him, on
  the 22nd of January 1685, for plagiarism. He died on the 14th of May
  1688, in the midst of the consequent controversy and law suit. His
  complete work was published, with a preface by Bayle, La Haye and
  Rotterdam, 1690, fol., 3 vols.; again edited by Basnage de Beauval,
  1701; La Haye, 1707, fol., 4 vols. From the edition of 1701 the very
  popular so-called _Dictionnaire de Trevoux_, Trevoux, 1704, fol., 2
  vols., was made by the Jesuits, who excluded everything that seemed to
  favour the Calvinism of Basnage. The last of its many editions is
  Paris, 1771, fol., 8 vols. The Academy's dictionary was first printed
  Paris, 1694, fol., 2 vols. They began the revision in 1700; second
  edition 1718, fol., 2 vols.; 3rd, 1740, fol., 2 vols.; 6th, 1835, 2
  vols. 4to, reprinted 1855; Supplément, by F. Raymond, 1836, 4to;
  Complément, 1842, 4to, reprinted 1856; _Dictionnaire historique_,
  Paris, 1858-1865, 4to, 2 parts (A to Actu), 795 pages, published by
  the Institut: Dochez, Paris, 1859, 4to: Bescherelle, ib. 1844, 4to, 2
  vols.; 5th ed. Paris, 1857, 4to, 2 vols.; 1865; 1887: Landais, Paris,
  1835; 12th ed. ib. 1854, 4to, 2 vols.: Littré, Paris, 1863-1873, 4to,
  4 vols. 7118 pages: Supplément, Paris, 1877, 4to: Godefroy (with
  dialects from 9th to 15th cent.), Paris, 1881-1895, and _Complément_:
  Hatzfield, Darmesteter, and Thomas, Paris, 1890-1900: Larive and
  Fleury, (_mots et choses, illustré_), Paris, 1884-1891.
  ENGLISH.--Palsgrave, _Lesclaircissement de la langue francoyse_,
  London, 1530, 4to, 2 parts; 1852: Hollyband, London, 1533, 4to:
  Cotgrave, ib. 1611, fol.: Boyer, La Haye, 1702, 4to, 2 vols.; 37th ed.
  Paris, 1851, 8vo, 2 vols.: Fleming and Tibbins, Paris, 1846-1849, 4to,
  2 vols.; ib. 1854, 4to, 2 vols.; ib. 1870-1872, 4to, 2 vols.: Tarver,
  London, 1853-1854, 8vo, 2 vols.; 1867-1872: Bellows, Gloucester, 1873,
  16mo; ib. 1876. IDEOLOGICAL, OR ANALOGICAL.--Robertson, Paris, 1859,
  8vo: Boissière, Paris, 1862, 8vo. ETYMOLOGY.--Lebon, Paris, 1571, 8vo:
  Ménage, ib. 1650, 4to. Pougens projected a _Trésor des origines_, his
  extracts for which, filling nearly 100 volumes folio, are in the
  library of the Institut. He published a specimen, Paris, 1819, 4to.
  After his death, _Archéologie française_, Paris, 1821, 8vo, 2 vols.,
  was compiled from his MSS., which were much used by Littré: Scheler,
  Bruxelles, 1862, 8vo; 1873: Brachet, 2nd ed. Paris, 1870, 12mo;
  English trans. Kitchin, Oxf. 1866, 8vo. GREEK WORDS.--Trippault,
  Orleans, 1580, 8vo: Morin, Paris, 1809, 8vo. GERMAN WORDS.--Atzler,
  Cöthen, 1867, 8vo. ORIENTAL WORDS.--Pihan, Paris, 1847, 8vo; 1866:
  Devic, ib. 1876, 8vo. NEOLOGY.--Desfontaines, 3rd ed. Amst. 1728,
  12mo: Mercier, Paris, 1801, 8vo, 2 vols.: Richard, ib. 1842, 8vo; 2nd
  ed. 1845. POETIC.--_Dict. des rimes_ (by La Noue), Geneve, 1596, 8vo;
  Cologny, 1624, 8vo: Carpentier, _Le Gradus français_, Paris, 1825,
  8vo, 2 vols. EROTIC.--De Landes, Bruxelles, 1861, 12mo.
  ORATORY.--Demandre and Fontenai, Paris, 1802, 8vo: Planche, ib.
  1819-1820, 8vo, 3 vols. PRONUNCIATION.--Féline, ib. 1857, 8vo. DOUBLE
  FORMS.--Brachet, ib. 1871, 8vo. EPITHETS.--Daire, ib. 1817, 8vo.
  VERBS.--Bescherelle, ib. 1855, 8vo, 2 vols.: 3rd ed. 1858.
  PARTICIPLES.--Id., ib. 1861, 12mo. DIFFICULTIES.--Boiste, London,
  1828, 12mo: Laveaux, Paris, 1872, 8vo, 843 pages.
  SYNONYMS.--Boinvilliers, Paris, 1826, 8vo: Lafaye, ib. 1858, 8vo;
  1861; 1869: Guizot, ib. 1809, 8vo; 6th ed. 1863; 1873.
  HOMONYMS.--Zlatagorski (Germ., Russian, Eng.), Leipzig, 1862, 8vo, 664
  pages. IMITATIVE WORDS.--Nodier, _Onomatopées_, ib. 1828, 8vo.
  TECHNOLOGY.--D'Hautel, ib. 1808, 8vo, 2 vols.: Desgranges, ib. 1821,
  8vo: Tolhausen (Fr., Eng., Germ.), Leipz. 1873, 8vo, 3 vols. FAULTS OF
  EXPRESSION.--Roland, Gap, 1823, 8vo: Blondin, Paris, 1823, 8vo.
  PARTICULAR AUTHORS.--_Corneille_: Godefroy, ib. 1862, 8vo, 2 vols.:
  Marty-Laveaux, ib. 1868, 8vo, 2 vols. _La Fontaine_: Lorin, ib. 1852,
  8vo. _Malherbe_: Regnier, ib. 1869, 8vo. _Molière_: Genin, ib. 1846,
  8vo: Marty-Laveaux, ib. 8vo. _Racine_: Marty-Laveaux, ib. 1873, 8vo, 2
  vols. _M^me de Sévigné_: Sommer, ib. 1867, 8vo, 2 vols. OLD
  FRENCH.--La Curne de St Palaye prepared a dictionary, of which he only
  published _Projet d'un glossaire_, Paris, 1756, 4to. His MSS. in many
  volumes are in the National Library, and were much used by Littré.
  They were printed by L. Favre, and fasciculi 21-30 (tom. iii.), Niort,
  4to, 484 pages, were published in February 1877. Lacombe (vieux
  langage), Paris, 1766, 2 vols. 4to: Kelham (Norman and Old French),
  London, 1779, 8vo: Roquefort (langue romane), Paris, 1808, 8vo;
  Supplément, ib. 1820, 8vo: Pougens, _Archéologie_, ib. 1821, 8vo, 2
  vols.: Burguy, Berlin, 1851-1856, 8vo, 3 vols.: Laborde (_Notice des
  émaux ... du Louvre_, part ii.), Paris, 1853, 8vo, 564 pages:[3]
  Gachet (rhymed chronicles), Bruxelles, 1859, 4to: Le Héricher (Norman,
  English and French), Paris, 1862, 3 vols. 8vo: Hippeau (12th and 13th
  centuries), Paris, 1875, 8vo. DIALECTS.--Jaubert (central), Paris,
  1856-1857, 8vo, 2 vols.: Baumgarten (north and centre), Coblentz,
  1870, 8vo: Azais, _Idiomes romans du midi_, Montpellier, 1877.
  _Austrasian_: François. Metz, 1773, 8vo. _Auvergne_: Mège, Riom, 1861,
  12mo. _Bearn_: Lespi, Pau, 1858, 8vo. _Beaucaire_: Bonnet (Bouguirén),
  Nismes, 1840, 8vo. _Pays de Bray_: Decorde, Neufchâtel, 1852, 8vo.
  _Burgundy_: Mignard, Dijon, 1870, 8vo. _Pays de Castres_: Couzinié,
  Castres, 1850, 4to. _Dauphiné_: Champollion-Figeac, Paris, 1809, 8vo:
  Jules, Valence, 1835, 8vo; Paris, 1840, 4to. _Dep. of Doubs_: Tissot
  (Patois des Fourg, arr. de Pontarlier) Besançon, 1865, 8vo. _Forez_:
  Gras, Paris, 1864, 8vo; Neolas, Lyon, 1865, 8vo. _Franche Comté_:
  Maisonforte, 2nd ed. Besançon, 1753, 8vo. _Gascony_: Desgrouais
  (Gasconismes corrigés), Toulouse, 1766, 8vo; 1769; 1812, 12mo, 2
  vols.; 1825, 8vo, 2 vols. _Dep. of Gers_: Cenac-Montaut, Paris, 1863,
  8vo. _Geneva_: Humbert, Geneve, 1820, 8vo. _Languedoc_: Odde, Tolose,
  1578, 8vo: Doujat, Toulouse, 1638, 8vo: De S.[auvages], Nismes, 1756,
  2 vols.; 1785; Alais, 1820: Azais, Beziers, 1876, &c., 8vo: Hombres,
  Alais, 1872, 4to: Thomas (_Greek words_) Montpellier, 1843, 4to.
  _Liége_: Forir, Liége, 1866, 8vo, vol i. 455 pages. _Lille_: Vermesse,
  Lille, 1861, 12mo: Debuire du Buc ib., 1867, 8vo. _Limousin_: Beronie,
  ed. Vialle (Corrèze), Tulle, 1823, 4to. _Lyonnais, Forez,
  Beaujolais_: Onofrio, Lyon, 1864, 8vo. _Haut Maine_: R[aoul] de
  M.[ontesson], Paris, 1857; 1859, 503 pages. _Mentone_: Andrews, Nice,
  1877, 12mo. _Dep. de la Meuse_: Cordier, Paris, 1853, 8vo. _Norman_:
  Edélestand and Alfred Duméril, Caen, 1849, 8vo: Dubois, ib. 1857, 8vo:
  Le Héricher (_Philologie topographique_), Caen, 1863, 4to: Id.
  (éléments scandinaves), Avranches, 1861, 12mo: Metivier (Guernsey),
  London, 1870, 8vo: Vasnier (arrond de Pont Audemer), Rouen, 1861, 8vo:
  Delboulle (Vallée d'Yères), Le Havre, 1876. _Picardy_: Corblet,
  Amiens, 1851, 8vo. _Poitou, Saintonge, Aunis_: Favre, Niort, 1867,
  8vo. _Poitou_: Beauchet-Filleau, Paris, 1864, 8vo: Levrier, Niort,
  1867, 8vo: Lalanne, Poitiers, 1868, 8vo. _Saintonge_: Boucherie,
  Angoulême, 1865, 8vo: Jonain, Royan, 1867, 8vo. _Savoy_: Pont
  (Terratzu de la Tarantaise), Chambery, 1869, 8vo. _La Suisse Romande_:
  Bridel, Lausanne, 1866, 8vo. _Dep. of Tarn_: Gary, Castre, 1845, 8vo.
  _Dep. of Vaucluse_: Barjavel, Carpentras, 1849, 8vo. _Walloon
  (Rouchi)_: Cambresier, Liége, 1787, 8vo: Grandgagnage, ib. 1845-1850,
  8vo. 2 vols.: Chavée, Paris, 1857, 18mo: Vermesse, Doudi, 1867, 8vo.
  Sigart (_Montois_), Bruxelles, 1870, 8vo. SLANG.--Oudin, _Curiositez
  Françaises_, Paris, 1640, 8vo: Baudeau de Saumaise (Précieuses, Langue
  de Ruelles), Paris, 1660, 12mo; ed. Livet, ib. 1856: Le Roux, _Dict.
  Comique_, Amst. 1788, and 6 other editions: Carême Prenant [i.e.
  Taumaise], (argot réforme), Paris, 1829, 8vo: Larchey (excentricitées
  du langage), Paris, 1860, 12mo; 5th ed. 1865: Delvau (langue verte,
  Parisian), Paris, 1867, 8vo: Larchey, Paris, 1873, 4to, 236 pages.


 Provençal.--Pallas, Avignon, 1723, 4to: Bastero, _La Crusca
  Provenzale_, Roma, 1724, fol. vol. i. only: Raynouard, Paris,
  1836-1844, 8vo, 6 vols.: Garcin, Draguignand, 1841, 8vo, 2 vols.:
  Honnorat, Digne, 1846-1849, 4to, 4 vols. 107,201 words: Id., _Vocab.
  fr. prov._, ib. 1848, 12mo, 1174 pages.


 Spanish.--Covarruvias Orosco, Madrid, 1611, fol.: ib. 1673-1674, fol.
  2 vols.; Academia Española, Madrid, 1726-1739, fol. 6 vols.; 8th ed.
  1837: Caballero, Madrid, 1849, fol.; 8th ed. ib. 1860, 4to, 2 vols.:
  Cuesta, ib. 1872, fol. 2 vols.: Campano, Paris, 1876, 18mo, 1015
  pages. Cuervo, 1886-1894; Monlau, 1881; Zerola, Toro y Gomes, and
  Isaza, 1895; Serrano (encyclopaedic) 1876-1881. ENGLISH.--Percivall,
  London, 1591, 4to: Pineda, London, 1740, fol.: Connelly and Higgins,
  Madrid, 1797-1798, 4to, 4 vols.: Neuman and Baretti, 9th ed. London,
  1831, 8vo, 2 vols.; 1874. FRENCH.--Oudin, Paris, 1607, 4to, 1660;
  Gattel, Lyon, 1803, 4to, 2 vols.: Dominguez, Madrid, 1846, 8vo, 6
  vols.: Blanc, Paris, 1862, 8vo, 2 vols. GERMAN.--Wagener, Hamb.
  1801-1805, 8vo, 4 vols.: Seckendorp, ib. 1823, 8vo, 3 vols.:
  Franceson, 3rd ed. Leipzig, 1862, 8vo, 2 vols. ITALIAN.--Franciosini,
  Venezia, 1735, 8vo, 2 vols.; Cormon y Manni, Leon, 1843, 16mo, 2
  vols.: Romero, Madrid, 1844, 4to. SYNONYMS.--_Diccionario de
  Sinonimos_, Paris, 1853, 4to. ETYMOLOGY.--Aldrete, Madrid, 1682, fol.:
  Monlau y Roca, ib. 1856, 12mo; Barcia, 1881-1883. ARABIC
  WORDS.--Hammer Purgstall, Wien, 1855, 8vo: Dozy and Engelmann, 2d ed.
  Leiden, 1869, 8vo. ANCIENT.--Sanchez, Paris, 1842, 8vo.
  RHYMING.--Garcia de Rengifo (consonancias) Salmantica, 1592, 4to;
  1876. DON QUIXOTE.--Beneke (German), Leipzig, 1800, 16mo; 4th ed.
  Berlin, 1841, 16mo. DIALECTS.--_Aragonese_: Peralta, Zaragoza, 1836,
  8vo: Borao, ib. 1859, 4to. _Catalan_: Rocha de Girona (Latin),
  Barcinone, 1561, fol.: _Dictionari Catala_ (Lat. Fr. Span.),
  Barcelona, 1642, 8vo: Lacavalleria (Cat.-Lat.), ib. 1696, fol.:
  Esteve, ed. Belvitges, &c. (Catal.-Sp. Lat.), Barcelona, 1805-1835,
  fol. 2 vols.: Saura (Cat.-Span.), ib. 1851, 16mo; 2nd ed.(Span.-Cat.),
  ib. 1854; 3rd ed. (id.) ib. 1862, 8vo: Labernia, ib. 1844-1848, 8vo, 2
  vols. 1864. _Gallegan_: Rodriguez, Coruña, 1863, 4to: Cuveira y Piñol,
  Madrid, 1877, 8vo. Majorca: Figuera, Palma, 1840, 4to: Amengual, ib.
  1845, 4to. _Minorca_: _Diccionario_, Madrid, 1848, 8vo. _Valencian_:
  Palmyreno, Valentiae, 1569: Ros, Valencia, 1764, 8vo: Fuster, ib.
  1827, 8vo: Lamarca, 2nd ed. ib. 1842, 16mo. _Cuba_: _Glossary of
  Creole Words_, London, 1840, 8vo: Pichardo, 1836; 2nd ed. Havana,
  1849, 8vo; 3rd ed. ib. 1862, 8vo; Madrid, 1860, 4to.


 Portuguese.--Lima, Lisbon, 1783, 4to: Moraes da Silva, ib. 1789, 4to,
  2 vols.; 6th ed. 1858: Academia real das Sciencas, ib. 1793, tom. i.,
  ccvi. and 544 pages (A to Azurrar); Faria, ib. 1849, fol. 2 vols.; 3rd
  ed. ib. 1850-1857, fol. 2 vols. 2220 pages. ENGLISH.--Vieyra, London,
  1773, 2 vols. 4to: Lacerda, Lisboa, 1866-1871, 4to, 2 vols.
  FRENCH.--Marquez, Lisboa, 1756-1761, fol. 2 vols.: Roquette, Paris,
  1841, 8vo, 2 vols.; 4th ed. 1860: Marques, Lisbonne, 1875, fol. 2
  vols.: Souza Pinto, Paris, 1877, 32mo, 1024 pages. GERMAN.--Wagener,
  Leipzig, 1811-1812, 8vo, 2 vols.: Wollheim, ib. 1844, 12mo, 2 vols.:
  Bösche, Hamburg, 1858, 8vo, 2 vols. 1660 pages. ITALIAN.--Costa e Sá,
  Lisboa, 1773-1774, fol. 2 vols. 1652 pages: Prefumo, Lisboa, 1853,
  8vo, 1162 pages. ANCIENT.--Joaquim de Sancta Rosa de Viterbo, ib.
  1798, fol. 2 vols.; 1824, 8vo. ARABIC WORDS.--Souza, ib. 1789, 4to;
  2nd ed. by S. Antonio Moura, ib. 1830, 224 pages. ORIENTAL AND AFRICAN
  WORDS, NOT ARABIC.--Saõ Luiz, ib. 1837, 4to, 123 pages. FRENCH
  WORDS.--Id., ib. 1827, 4to; 2nd ed. Rio de Janeiro, 1835, 8vo.
  SYNONYMS.--Id., ib. 1821, 4to; 2nd ed. ib. 1824-1828, 8vo. Fonseca,
  Paris, 1833, 8vo; 1859, 18mo, 863 pages. HOMONYMS.--De Couto, Lisboa,
  1842, fol. POETIC.--Luzitano (i.e. Freire), ib. 1765, 8vo, 2 vols.;
  3rd ed. ib. 1820, 4to, 2 vols. RHYMING.--Couto Guerreiro, Lisboa,
  1763, 4to. NAVAL.--Tiberghien, Rio de Janeiro, 1870, 8vo.
  CEYLON-PORTUGUESE.--Fox, Colombo, 1819, 8vo: Callaway, ib. 1823, 8vo.


 Italian.--Accarigi, _Vocabulario_, Cento, 1543, 4to: Alunno, _La_
  _fabrica del mundo_, Vinezia, 1548, fol.: Porccachi, Venetia, 1588,
  fol.: Accademici della Crusca, _Vocabulario_, Venez. 1612, fol.; 4th
  ed. Firenze, 1729-1738, fol. 6 vols.: Costa and Cardinali, Bologna,
  1819-1826, 4to, 7 vols.: Tommaseo and Bellini, Torino, 1861, &c., 4to,
  4 vols.: Petrocchi, 1884-1891. ENGLISH.--Thomas, London, 1598, 4to:
  Florio, London, 1598, 4to, 1611: Baretti, London, 1794, 2 vols.: 1854,
  8vo, 2 vols.: Petronj and Davenport, Londra, 1828, 8vo, 3 vols.:
  Grassi, Leipz. 1854, 12mo: Millhouse, Lond., 1868, 8vo, 2 vols. 1348
  pages. FRENCH.--Alberti, Paris, 1771, 4to, 2 vols.; Milan, 1862:
  Barberi, Paris, 1838, 4to, 2 vols.: Renzi, Paris, 1850, 8vo.
  GERMAN.--_Libro utilissimo_, Venetiis, 1499, 4to: Valentini, Leipzig,
  1834-1836, 4to, 4 vols. ETYMOLOGY.--Menage, Geneva, 1685, fol.: Bolza,
  Vienna, 1852, 4to. PROVENÇAL WORDS.--Nannucci, Firenze, 1840, 8vo.
  SYNONYMS.--Rabbi, Venezia, 1774, 4to; 10th ed. 1817; Tommaseo,
  Firenze, 1839-1840, 4to, 2 vols.: Milano, 1856, 8vo; 1867.
  VERBS.--Mastrofini, Roma, 1814, 4to, 2 vols. SELECT WORDS AND
  PHRASES.--Redi, Brescia, 1769, 8vo. INCORRECT WORDS AND
  PHRASES.--Molassi, Parma, 1830-1841, 8vo, 854 pages. SUPPOSED
  GALLICISMS.--Viani, Firenze, 1858-1860, 8vo, 2 vols. ADDITIONS TO THE
  DICTIONARIES.--Gherardini, Milano, 1819-1821, 8vo, 2 vols.; ib.
  1852-1857, 8vo, 6 vols. RHYMING.--Falco, Napoli, 1535, 4to: Ruscelli,
  Venetia, 1563, 8vo; 1827: Stigliani, Roma, 1658, 8vo: Rosasco, Padova,
  1763, 4to; Palermo, 1840, 8vo. TECHNICAL.--Bonavilla-Aquilino, Mil.
  1819-1821, 8vo, 5 vols.; 2nd ed. 1829-1831, 4to, 2 vols.: Vogtberg
  (Germ.), Wein, 1831, 8vo. PARTICULAR AUTHORS.--_Boccaccio_: Aluno, _Le
  ricchezze della lingua volgare_, Vinegia, 1543. fol. _Dante_: Blanc,
  Leipzig, 1852, 8vo; Firenze, 1859, 8vo. DIALECTS.--_Bergamo_:
  Gasparini, Mediol. 1565: Zappetini, Bergamo, 1859, 8vo: Tiraboschi
  (anc. and mod.), Turin, 1873, 8vo. _Bologna_: Bumaldi, Bologna, 1660,
  12mo: Ferrari, ib. 1820, 8vo; 1838, 4to. _Brescia_: Gagliardi,
  Brescia, 1759, 8vo: Melchiori, ib. 1817-1820, 8vo: _Vocabularietto_,
  ib. 1872, 4to. _Como_: Monti, Milano, 1845, 8vo. _Ferrara_: Manini,
  Ferrara, 1805, 8vo: Azzi, ib. 1857, 8vo. _Friuli_: Scala, Pordenone,
  1870, 8vo. _Genoa_: Casaccia, Gen. 1842-1851, 8vo; 1873, &c.:
  Paganini, ib. 1857, 8vo. _Lombardy_: Margharini, Tuderti, 1870, 8vo.
  _Mantua_: Cherubini, Milano, 1827, 4to. _Milan_: Varon, ib. 1606, 8vo:
  Cherubini, ib. 1814, 8vo, 2 vols.; 1841-1844, 8vo, 4 vols.; 1851-1861,
  8vo, 5 vols.: Banfi, ib. 1857, 8vo: 1870, 8vo. _Modena_: Galvani,
  Modena, 1868, 8vo. _Naples_: Galiani, Napoli, 1789, 12mo, 2 vols.
  _Parma_: Peschieri, Parma, 1828-1831, 8vo, 3 vols. 1840; Malespina,
  ib. 1856, 8vo, 2 vols. _Pavia_: _Dizionario domestico pavese_, Pavia,
  1829, 8vo: Gambini, ib. 1850, 4to, 346 pages. _Piacenza_: Nicolli,
  Piacenza, 1832: Foresti, ib. 1837-1838, 8vo, 2 pts. _Piedmont_: Pino,
  Torino, 1784, 4to: Capello (Fr.), Turin, 1814, 8vo, 2 pts.: Zalli
  (Ital. Lat. Fr.), Carmagnola, 1815, 8vo, 2 vols: Sant' Albino, Torino,
  1860, 4to. _Reggio_: _Vocabulario Reggiano_, 1832. _Romagna_: Morri,
  Fienza, 1840. _Rome_: _Raccolto di voci Romani e Marchiani_, Osimo,
  1769, 8vo. _Roveretano and Trentino_: Azzolini, Venezia, 1856, 8vo.
  _Sardinia_: Porru, Casteddu, 1832, fol.: Spano, Cagliari, 1851-1852,
  fol. 3 vols. _Sicily_: Bono (It. Lat.), Palermo, 1751-1754, 4to, 3
  vols.; 1783-1785, 4to, 5 vols.: Pasqualino, ib. 1785-1795, 4to, 5
  vols.: Mortillaro, ib. 1853, 4to, 956 pages: Biundi, ib. 1857, 12mo,
  578 pages: Traina, ib. 1870, 8vo. _Siena_: Barbagli, Siena, 1602, 4to.
  _Taranto_: Vincentiis, Taranto, 1872, 8vo. _Turin_: Somis di Chavrie,
  Torino, 1843, 8vo. _Tuscany_: Luna, Napoli, 1536, 4to: Politi, Roma,
  1604, 8vo; Venezia, 1615; 1628; 1665; Paulo, ib. 1740, 4to. _Vaudois_:
  Callet, Lausanne, 1862, 12mo. _Venetian_: Patriarchi (_Veneziano e
  padevano_), Padova, 1755, 4to; 1796, 1821: Boerio, Venezia, 1829, 4to;
  1858-1859; 1861. _Verona_: Angeli, Verona, 1821, 8vo. _Vicenza_:
  Conti, Vicenza, 1871, 8vo. LINGUA FRANCA.--_Dictionnaire de la langue
  Franque, ou Petit Mauresque_, Marseille, 1830, 16mo, 107 pages.
  SLANG.--Sabio (lingua Zerga), Venetia, 1556, 8vo; 1575: _Trattato
  degli bianti_, Pisa, 1828, 8vo.


 Romansh.--_Promptuario de voci volgari e Latine_, Valgrisii, 1565,
  4to: _Der, die, das, oder Nomenclatura_ (German nouns explained in
  Rom.), Scoul, 1744, 8vo: Conradi, Zurich, 1820, 8vo; 1826, 12mo, 2
  vols.: Carisch, Chur, 1821, 8vo; 1852, 16mo.


 Vlach.--_Lesicon Rumanese_ (Lat. Hung. Germ.), Budae, 1825, 4to: Bobb
  (Lat. Hung.), Clus, 1822-1823, 4to, 2 vols. FRENCH.--Vaillant,
  Boucoureshti, 1840, 8vo: Poyenar, Aaron and Hill, Boucourest,
  1840-1841, 4to, 2 vols.; Jassi, 1852, 16mo, 2 vols.: De Pontbriant,
  Bucuresci, 1862, 8vo: Cihac, Frankf. 1870, 8vo: Costinescu, Bucuresci,
  1870, 8vo, 724 pages: Antonescu, Bucharest, 1874, 16mo, 2 vols. 919
  pages. GERMAN.--Clemens, Hermanstadt, 1823, 8vo: Isser, Kronstadt,
  1850: Polyzu, ib. 1857, 8vo.


TEUTONIC: (1) _Scandinavian._

 Icelandic.--LATIN.--Andreae, Havniae, 1683, 8vo: Halderson (Lat.
  Danish), ib. 1814, 4to, 2 vols. ENGLISH.--Cleasby-Vigfusson, Oxford,
  1874, 4to. GERMAN.--Dieterich, Stockholm, 1844, 8vo: Möbius, Leipzig,
  1866, 8vo. DANISH.--Jonssen, Kjöbenhavn, 1863, 8vo. NORWEGIAN.--Kraft,
  Christiania, 1863, 8vo: Fritzner, Kristiania, 1867, 8vo.
  POETIC.--Egilsson (Latin), Hafniae, 1860, 8vo; 1864.


 Swedish.--Kindblad, Stockholm, 1840, 4to: Almqvist, Örebro, 1842-1844,
  8vo: Dalin, _Ordbog._ Stockholm, 1850-1853, 8vo, 2 vols. 1668 pages;
  1867, &c. 4to (vol. i. ii., A to Fjermare, 928 pages): Id.,
  _Handordbog_, ib. 1868, 12mo, 804 pages; Svenska Academien. Stockholm,
  1870, 4to (A) pp. 187. LATIN.--Stjernhjelm, Holm, 1643, 4to: Verelius,
  Upsala, 1691, 8vo: Ihre (Sueo-Gothicum), Upsala, 1769, fol. 2 vols.
  ENGLISH.--Serenius, Nyköping, 1757, 4to: Brisnon, Upsala, 1784, 4to:
  Widegren, Stockholm, 1788, 4to; Brisman, Upsala, 1801, 4to; 3rd ed.
  1815, 2 vols.: Deleen Örebro, 1829, 8vo: Granberg, ib. 1832, 12mo:
  Nilssen, Widmark, &c., Stockholm, 1875, 8vo. FRENCH.--Möller,
  Stockholm, 1745, 4to: Björkengren, ib. 1795, 2 vols.: Nordforss, ib.
  1805, 8vo, 2 vols.: 2nd ed. Örebro, 1827, 12mo: West, Stockh. 1807,
  8vo: Dalin, ib. 1842-1843, 4to, 2 vols.; 1872. GERMAN.--Dähnert,
  Holmiae, 1746, 4to: Heinrich, Christiansund, 1814, 4to, 2 vols.; 4th
  ed. Örebro, 1841, 12mo: Helms, Leipzig, 1858, 8vo; 1872.
  DANISH.--Höst, Kjöbenhavn, 1799, 4to: Welander, Stockholm, 1844, 8vo:
  Dalin, ib. 1869, 16mo: Kaper, Kjöbenhavn, 1876, 16mo.
  ETYMOLOGY.--Tamm, Upsala, 1874, &c., 8vo (A and B), 200 pages. FOREIGN
  WORDS.--Sahlstedt, Wästerås, 1769, 8vo: Andersson (20,000), Stockholm,
  1857, 16mo: Tullberg, ib. 1868, 8vo: Ekbohrn, ib. 1870, 12mo: Dalin,
  ib. 1870, &c., 8vo. SYNONYMS.--Id., ib. 1870, 12mo. NAVAL.--Ramsten,
  ib. 1866, 8vo. TECHNICAL.--Jungberg, ib. 1873, 8vo. DIALECTS.--Ihre,
  Upsala, 1766, 4to: Rietz, Lund, 1862-1867, 4to, 859 pages. _Bohuslän_:
  _Idioticon Bohusiense_, Götaborg, 1776, 4to. _Dalecarlia_: Arborelius,
  Upsala, 1813, 4to. _Gothland_: Hof (Sven), Stockholmiae, 1772, 8vo:
  Rääf (Ydre), Örebro, 1859, 8vo. _Halland_: Möller, Lund, 158, 8vo.
  _Helsingland_: Lenström, ib. 1841, 8vo: Fornminnessällskap,
  Hudikswall, 1870, 8vo.


 Norwegian.--Jenssen, Kjöbenhavn, 1646, 8vo: Pontoppidan, Bergen, 1749,
  8vo: Hanson (German), Christiania, 1840, 8vo: Aasen, ib. 1873, 8vo,
  992 pages.


 Danish.--Aphelen, Kopenh, 1764, 4to, 2 vols.; 1775, 4to, 3 vols.:
  Molbech, Kjöbenhavn, 1833, 8vo, 2 vols.: ib. 1859, 2 vols.:
  Videnskabernes Selskab, ib. 1793-1865, Kalkar. ENGLISH.--Berthelson
  (Eng. Dan.), 1754, 4to: Wolff, London, 1779, 4to. Bay, ib. 1807, 8vo,
  2 vols.; 1824, 8vo: Hornbeck, ib. 1863, 8vo: Ferrall and Repp, ib.
  1814, 16mo; 1873, 8vo: Rosing, Copenhagen, 1869, 8vo: Ancker, ib.
  1874, 8vo. FRENCH.--Aphelen, 1754, 8vo: Id., ib. 1759, 4to, 2 vols.;
  2nd ed. 1772-1777, vol. i. ii. GERMAN.--Id., ib. 1764, 4to, 2 vols.:
  Grönberg, 2nd ed. Kopenh. 1836-1839, 12mo, 2 vols.; 1851, Helms,
  Leipzig, 1858, 8vo. SYNONYMS.--Müller, Kjöbenhavn, 1853, 8vo. FOREIGN
  WORDS.--Hansen, Christiania, 1842, 12mo. NAVAL.--Wilsoet, Copenhagen,
  1830, 8vo: Fisker (French), Kjöbenhavn, 1839, 8vo. OLD
  DANISH.--Molbech, ib. 1857-1868, 8vo, 2 vols. DIALECTS.--Id., ib.
  1841, 8vo. _Bornholm_: Adler, _ib._ 1856, 8vo. _South Jutland_: Kok,
  1867, 8vo. SLANG.--Kristiansen (Gadesproget), ib. 1866, 8vo. p. 452.


(2) _Germanic._

 Teutonic.--COMPARATIVE.--Meidinger, Frankf. a. M. 1833, 8vo, 2nd ed.
  1836, 8vo.


 Gothic.--Junius, Dortrecht, 1665, 4to: 1671; 1684, Diefenbach
  (comparative), Franckf. a. M. 1846-1851, 2 vols. 8vo: Schulze,
  Magdeburg, 1848, 4to: 1867, 8vo: Skeat, London, 1868, 4to: Balg
  (_Comparative Glossary_), Magvike, Wisconsin, 1887-1889. ULPHILAS
  (editions with dictionaries).--Castilionaeus, Mediol, 1829, 4to:
  Gabelentz and Löbe, Altenburg, 1836-1843, 4to, 2 vols.: Gaugengigl,
  Passau, 1848, 8vo: Stamm, Paderborn, 1857: Stamm and Heyne, ib. 1866,
  8vo.


 Anglo-Saxon.--LATIN.--Somner (Lat. Eng.), Oxonii, 1659, fol.: Benson,
  ib. 1701, 8vo: Lye (A.-S. and Gothic), London, 1772, fol. 2 vols.:
  Ettmüller, Quedlinburg, 1851, 8vo. 838 pages. ENGLISH.--Bosworth,
  London, 1838, 8vo, 721 pages: Id. (_Compendious_), 1848, 278 pages.
  Corson (A.-S. and Early English), New York, 1871, 8vo, 587 pages;
  Toller (based on Bosworth), Oxford, 1882-1898. GERMAN.--Bouterwek,
  Gütersloh, 1850, 8vo, 418 pages: Grein (Poets), Göttingen, 1861-1863,
  8vo, 2 vols.: Leo, Halle, 1872, 8vo.


 English.--Cockeram, London, 1623, 8vo: 9th ed. 1650: Blount, ib. 1656,
  8vo: Philips, The new World of Words, London, 1658, fol.: Bailey,
  London, 1721, 8vo; 2nd ed. ib. 1736, fol.; 24th ed. ib. 1782, 8vo:
  Johnson, ib. 1755, fol. 2 vols.; ed. Todd, London, 1818, 4to, 4 vols.;
  ib. 1827. 4to, 3 vols.; ed. Latham, ib. 1866-1874, 4to, 4 vols. (2 in
  4 parts): Barclay, London, 1774, 4to; ed. Woodward, ib. 1848:
  Sheridan, ib. 1780, 4to, 2 vols.: Webster, New York, 1828, 4to, 2
  vols.; London, 1832, 4to, 2 vols.; ed. Goodrich and Porter, 1865, 4to:
  Richardson, ib. 1836, 4to, 2 vols.; Supplement, 1856: Ogilvie,
  _Imperial Dictionary_, Glasgow, 1850-1855, 8vo, 3 vols. (the new
  edition of Ogilvie by Charles Annandale, 4 vols., 1882, was an
  encyclopaedic dictionary, which served to some extent as the
  foundation of the _Century Dictionary_); Boag, _Do._, Edinburgh,
  1852-1853, 8vo, 2 vols.: Craik, ib. 1856, 8vo: Worcester, Boston,
  1863, 4to. Stormouth and Bayne, 1885; Murray and Bradley, _The Oxford
  English Dictionary_, 1884- ; Whitney, _The Century Dict._, New York,
  1889-1891; Porter, _Webster's Internat. Dict._, Springfield,
  Massachusetts, 1890; Funk, _Standard Dict._, New York, 1894; Hunter,
  _The Encyclopaedic Dict._, 1879-1888. ETYMOLOGY.--Skinner, Londini,
  1671, fol.: Junius, Oxonii, 1743, fol.: Wedgewood, London, 1859-1865,
  3 vols.; ib. 1872, 8vo. Skeat, Oxford, 1881; Fennell (Anglicized
  words), Camb. 1892. PRONOUNCING.--Walker, London, 1774, 4to: by Smart,
  2nd ed. ib. 1846, 8vo. PRONOUNCING IN GERMAN.--Hausner, Frankf. 1793,
  8vo; 3rd ed. 1807; Winkelmann, Berlin, 1818, 8vo: Voigtmann, Coburg,
  1835, 8vo: Albert, Leipz. 1839, 8vo: Bassler, ib. 1840, 16mo.
  ANALYTICAL.--Booth, Bath, 1836, 4to: Roget, _Thesaurus_, London, 1852,
  8vo; 6th ed. 1857; Boston, 1874. SYNONYMS.--Piozzi, London, 1794, 8vo,
  2 vols.: L. [abarthe], Paris, 1803, 8vo, 2 vols.: Crabb, London,
  1823, 8vo; 11th ed. 1859: C. J. Smith, ib. 1871, 8vo, 610 pages.
  REDUPLICATED WORDS.--Wheatley, ib. 1866, 8vo. SURNAMES.--Arthur, New
  York, 1857, 12mo, about 2600 names: Lower, ib. 1860, 4to.
  PARTICLES.--Le Febure de Villebrune, Paris, 1774, 8vo.
  RHYMING.--Levins, _Manipulus Puerorum_, London, 1570, 4to; ed.
  Wheatley, ib. 1867, 8vo: Walker, London, 1775, 8vo; 1865, 8vo.
  SHAKESPEARE.--Nares, Berlin, 1822, 4to; ed. Halliwell and Wright,
  London, 1859, 8vo: Schmidt, Berlin, 1874. OLD ENGLISH.--Spelman,
  London [1626], fol. (A to I only); 1664 (completed); 1687 (best ed.):
  Coleridge (1250-1300), ib. 1859, 8vo: Stratmann (Early Eng.), Krefeld,
  1867, 8vo; 2nd ed. 1873, 4to: Bradley (new edition of Stratman),
  Oxford, 1891; Matzner and Bieling, Berlin, 1878- . OLD AND
  PROVINCIAL.--Halliwell, London, 1844-1846, 8vo; 2nd ed. ib. 1850, 2
  vols.: 6th ed. 1904: Wright, ib. 1857, 8vo, 2 vols.; 1862.
  DIALECTS.--Ray, ib. 1674, 12mo: Grose, ib. 1787, 8vo; 1790: Holloway,
  Lewes, 1840, 8vo; Wright, _Eng. Dialect Dict._, London, 1898-1905, 28
  vols. _Scotch_: Jamieson, Edin. 1806, 4to, 2 vols.; Supplement, 1826,
  2 vols.; abridged by Johnstone, ib. 1846, 8vo: Brown, Edin, 1845, 8vo:
  Motherby (German), Königsberg, 1826-1828, 8vo: (_Shetland and
  Orkney_), Edmonston, London, 1866, 8vo: (_Banffshire_), Gregor, ib.
  1866, 8vo. _North Country_: Brockett, London, 1839, 8vo, 2 vols.
  _Berkshire_: [Lousley] ib. 1852, 8vo, _Cheshire_: Wilbraham, ib. 1817,
  4to; 1826, 12mo: Leigh, Chester, 1877, 8vo. _Cumberland_: _Glossary_,
  ib. 1851, 12mo: Dickenson, Whitehaven, 1854, 12mo; Supplement, 1867:
  Ferguson (Scandinavian Words), London, 1856, 8vo. _Derbyshire_: Hooson
  (mining), Wrexham, 1747, 8vo: Sleigh, London, 1865, 8vo. _Dorset_:
  Barnes, Berlin, 1863, 8vo. _Durham_: [Dinsdale] (Teesdale), London,
  1849, 12mo. _Gloucestershire_: Huntley (Cotswold), ib. 1868, 8vo.
  _Herefordshire_: [Sir George Cornewall Lewis,] London, 1839, 12mo.
  _Lancashire_: Nodal and Milner, Manchester Literary Club, 1875, 8vo,
  Morris (Furness), London, 1869, 8vo: R. B. Peacock (Lonsdale, North
  and South of the Sands), ib. 1869, 8vo. _Leicestershire_: A. B. Evans,
  ib. 1848, 8vo. _Lincolnshire_: Brogden, ib. 1866, 12mo: Peacock
  (Manley & Corringham), ib. 1877, 8vo. _Norfolk and Suffolk_; Forby,
  London, 1830, 8vo, 2 vols. _Northamptonshire_: Sternberg, ib. 1851,
  8vo: Miss Anne E. Baker, ib. 1866, 8vo, 2 vols. 868 pages.
  _Somersetshire_: Jennings, ib. 1869, 8vo: W. P. Williams and W. A.
  Jones, Taunton, 1873, 8vo. _Suffolk_: Moor, Woodbridge, 1823, 12mo:
  Bowditch (Surnames), Boston, U.S., 1851, 8vo; 1858; 3rd ed. London,
  1861, 8vo, 784 pages. _Sussex_: Cooper, Brighton, 1836, 8vo: Parish,
  Farncombe, 1875, 8vo. _Wiltshire_: Akerman, London, 1842, 12mo.
  _Yorkshire (North and East)_, Toone, ib. 1832, 8vo: (_Craven_), Carr,
  2nd ed. London, 1828, 8vo, 2 vols.: (_Swaledale_), Harland, ib. 1873,
  8vo: (_Cleveland_), Atkinson, ib. 1868, 4to, 653 pages: (_Whitby_) [F.
  K. Robinson], ib. 1876, 8vo: (_Mid-Yorkshire and Lower Niddersdale_),
  C. Clough Robinson, ib. 1876, 8vo: (_Leeds_), Id., ib. 1861, 12mo:
  (_Wakefield_), Banks, ib. 1865, 16mo: (_Hallamshire_), Hunter, London,
  1829, 8vo. _Ireland: (Forth and Bargy, Co. Wexford)_, Poole, London,
  1867, 8vo. _America_: Pickering, Boston, 1816, 8vo: Bartlett, New
  York, 1848, 8vo; 3rd ed. Boston, 1860. 8vo; Dutch transl. by Keijzer,
  Gorinchen, 1854, 12mo; Germ. transl. by Köhler, Leipz. 1868, 8vo.
  Elwyn, Philadelphia, 1859. 8vo. _Negro English_: Kingos, St Croix,
  1770, 8vo: Focke (Dutch), Leiden, 1855, 8vo: Wullschlaegel, Löbau,
  1856, 8vo. 350 pages. SLANG.--Grose, London, 1785, 8vo; 1796: Hotten,
  ib. 1864, 8vo; 1866; Farmer & Henley (7 vols., 1890-1904).


 Frisic.--Wassenbergh, Leeuwarden, 1802, 8vo: Franeker, 1806, 8vo:
  Outzen, Kopenh. 1837, 4to: Hettema (Dutch), Leuwarden, 1832, 8vo;
  1874, 8vo, 607 pages: Winkler (Nederdeutsch en Friesch Dialectikon),
  's Gravenhage, 1874, 8vo, 2 vols. 1025 pages. OLD FRISIC.--Wiarda
  (Germ.), Aurich, 1786, 8vo: Richthofen, Göttingen, 1840, 4to. NORTH
  FRISIC.--Bendson (Germ.), Leiden, 1860, 8vo: Johansen (Föhringer und
  Amrumer Mundart), Kiel, 1862, 8vo. EAST FRISIC.--Stürenburg, Aurich,
  1857, 8vo. HELIGOLAND.--Oelrichs, s. l., 1836, 16mo.


 Dutch.--Kok, 2nd ed. Amst. 1785-1798, 8vo, 38 vols.: Weiland, Amst.
  1790-1811, 8vo, 11 vols.: Harrebomée, Utrecht, 1857, 4to; 1862-1870,
  8vo, 3 vols.: De Vries and Te Winkel, Gravenh. 1864, &c., 4to (new ed.
  1882- ); Dale, ib. 4th ed. 1898; ENGLISH.--Hexham, ed. Manley,
  Rotterdam, 1675-1678, 4to: Holtrop, Dortrecht, 1823-1824, 8vo, 2
  vols.: Bomhoff, Nimeguen, 1859, 8vo, 2 vols. 2323 pages: Jaeger,
  Gouda, 1862, 16mo: Calisch, Tiel, 1871, &c., 8vo. FRENCH.--Halma,
  Amst. 1710, 4to; 4th ed. 1761: Marin, ib. 1793, 4to, 2 vols.:
  Winkelman, ib. 1793, 4to, 2 vols.: Mook, Zutphen, 1824-1825, 8vo, 4
  vols.; Gouda, 1857, 8vo, 2 vols. 2818 pages: Kramers, ib. 1859-1862, 2
  vols. 16mo. GERMAN.--Kramer, Nürnb. 1719, fol.; 1759, 4to, 2 vols.;
  ed. Titius, 1784, Weiland, Haag, 1812, 8vo: Terwen, Amst. 1844, 8vo.
  ETYMOLOGY.--Franck, 1884-1892. ORIENTAL WORDS.--Dozy, 's Gravenhage,
  1867, 8vo. GENDERS OF NOUNS.--Bilderdijk, Amst. 1822, 8vo, 2 vols.
  SPELLING.--Id., 's Gravenhage, 1829, 8vo. FREQUENTATIVES.--De Jager,
  Gouda, 1875, 8vo, vol. i. OLD DUTCH.--Suringer, Leyden, 1865, 8vo.
  MIDDLE DUTCH.--De Vries, 's Gravenhage, 1864, &c., 4to. Verwijs and
  Verdam, ib. 1885- .


 Flemish.--Kilian, Antw. 1511, 8vo; ed. Hasselt, Utrecht, 1777, 4to, 2
  vols. FRENCH.--Berlemont, Anvers, 1511, 4to: Meurier, ib. 1557, 8vo:
  Rouxell and Halma, Amst. 1708, 4to; 6th ed. 1821: Van de Velde and
  Sleeckx, Brux. 1848-1851, 8vo, 2440 pages; ib. 1860, 8vo, 2 vols.
  ANCIENT NAMES OF PLACES.--Grandgagnage (East Belgium), Bruxelles,
  1859, 8vo.


 German.--Josua Pictorius (Maaler), _Die teütsch Spraach_, Tiguri,
  1561, 8vo; Stieler, Nürnb. 1691, 4to: Adelung, Leipz. 1774-1786, 4to,
  5 vols.; 1793-1818, 5 vols.: Campe, Braunschweig, 1807-1811, 4to, 5
  vols.: Grimm, Leipzig, 1854, &c., 4to: Sanders, ib. 1860-1865, 4to, 3
  vols. 1885: Diefenbach and Wülcker (High and Low German, to supplement
  Grimm), Frankf. a. M. 1874, 1885, 8vo.; Kluge, Strassburg, 1883;
  Heine, Leipzig, 1890-1895; Weigand, Giessen, 1873. ENGLISH.--Adelung,
  1783-1796, 8vo, 3 vols.: Hilpert, Karlsruhe, 1828-1829, 8vo, 2 vols.;
  1845-1846, 4to, 2 vols.: Flügel, Leipz. 1830, 8vo, 2 vols.; London,
  1857, 8vo; Leipzig, 1870: Müller, Cöthen, 1867, 8vo, 2 vols.
  FRENCH.--Laveaux, Strassburg, 1812, 4to: Mozin, Stuttgard, 1811-1812,
  4to, 4 vols.; 1842-1846, 8vo, 4 vols., 3rd ed. 1850-1851, 8vo:
  Schuster, Strasb. 1859, 8vo: Daniel, Paris, 1877, 16mo. OLD HIGH
  GERMAN.--Haltaeus, Lipsiae, 1758, fol. 2 vols.: Graff, Berlin,
  1834-1846, 4to, 7 vols.: Brinckmeier, Gotha, 1850-1863, 4to, 2 vols.:
  Kehrein (from Latin records), Nordhausen, 1863, 8vo. Schade, Halle,
  1872-1882. MIDDLE HIGH GERMAN.--Ziemann, Quedlinburg, 1838, 8vo:
  Benecke, Müller and Zarnche, Leipz. 1854-1866, 8vo, 3 vols.: Lexer,
  Leipzig, 1870, 8vo. MIDDLE LOW GERMAN.--Schiller and Lübben, Bremen,
  1872, &c., 8vo, in progress. LOW GERMAN.--Vollbeding, Zerbst, 1806,
  8vo: Kosegarten, Griefswald, 1839, 4to; 1856, &c., 4to.
  ETYMOLOGY.--Helvigius, Hanov. 1620, 8vo: Wachter, Lipsiae, 1737, fol.
  2 vols.: Kaindl, Salzbach, 1815-1830, 8vo, 7 vols.: Heyse, Magdeburg,
  1843-1849, 8vo, 3 vols.: Kehrein, Wiesbaden, 1847-1852, 2 vols.
  SYNONYMS.--Eberhard, Maas, and Grüber, 4th ed. Leipzig, 1852-1863,
  8vo, 4 vols.: Aue (Engl.), Edinb. 1836, 8vo: Eberhard, 11th ed.
  Berlin, 1854, 12mo: Sanders, Hamburg, 1872, 8vo, 743 pages. FOREIGN
  WORDS.--Campe, Braunschweig, 1813, 4to: Heyse, _Fremdwörterbuch_,
  Hannover, 1848, 8vo. NAMES.--Pott. Leipz. 1853, 8vo: Michaelis
  (Taufnamen), Berlin, 1856, 8vo: Förstemann (Old Germ.) Nordhausen,
  1856-1859, 4to, 2 vols. 1573 pages, 12,000 names: Steub
  (Oberdeutschen), München, 1871, 8vo. LUTHER.--Dietz, Leipzig,
  1869-1872, 8vo, 2 vols. DIALECTS.--Popowitsch, Wien, 1780, 8vo: Fulda,
  Berlin, 1788, 8vo: Klein, Frankf. 1792, 8vo, 2 vols.: Kaltschmidt,
  Nordlingen, 1851, 4to; 1854, 5th ed. 1865. _Aix-la-Chapelle_, Müller
  and Weitz, Aachen, 1836, 12mo. _Appenzell_: Tobler, Zürich, 1837, 8vo.
  _Austria_: Höfer, Linz, 1815, 8vo; Castelli, Wien, 1847, 12mo:
  Scheuchenstül (mining), ib. 1856, 8vo. _Bavaria_: Zaupser, München,
  1789, 8vo: Deling, ib. 1820, 2 vols.: Schmeller, Stuttg. 1827-1837,
  8vo, 4 vols.; 2nd ed. München, 1872, 4to, vol. i. 1799 pages.
  _Berlin_: Trachsel. Berlin, 1873, 8vo. _Bremen_: Bremisch Deutsch
  Gesellschaft, Bremen, 1767-1771, 1869, 8vo, 6 vols. Oelrich (anc.
  statutes), Frankf. a. M. 1767, 8vo. _Carinthia_: Ueberfelder,
  Klagenfurt, 1862, 8vo: Lexe, Leipzig, 1862, 8vo. _Cleves_: De
  Schueren, _Teuthonista_, Colon, 1477, fol.; Leiden, 1804, 4to.
  _Göttingen_: Schambach, Hannover, 1838, 8vo. _Hamburg_: Richey, Hamb.
  1873, 4to; 1755, 8vo. _Henneberg_: Reinwold, Berlin and Stettin, 1793,
  1801, 8vo, 2 vols.: Brückner, Meiningen, 1843, 4to. _Hesse_: Vilmar,
  Marburg, 1868, 8vo, 488 pages. _Holstein_: Schütz Hamb. 1800-1806,
  8vo, 4 vols. _Hungary_: Schoer, Wien, 1858. _Livonia_: Bergmann,
  Salisburg, 1785, 8vo: Gutzeit, Riga, 1859-1864, 8vo, 2 parts. _Upper
  Lusatia_: Anton, Görlitz, 1825-1839, 13 parts. _Luxembourg_: Gangler,
  Lux. 1847, 8vo, 406 pages. _Mecklenburg and Western Pomerania_: M.,
  Leipzig, 1876, 8vo, 114 pages. _Nassau_: Kehrein, Weilburg, 1860, 8vo.
  _Osnaburg_: Strodtmann, Leipz. 1756, 8vo. _Pomerania and Rügen_:
  Dähnert, Stralsund, 1781, 4to. _Posen_: Bernd, Bonn, 1820, 8vo.
  _Prussia_: Bock, Königsb. 1759, 8vo: Hennig, ib. 1785, 8vo. _Saxony_:
  Schmeller (from Heliand, &c.), Stuttg. 1840, 4to. _Silesia_: Berndt,
  Stendal, 1787, 8vo. _Swabia_: Schmid, Berlin, 1795, 8vo; Stuttg. 1831,
  8vo. _Switzerland_: Stalder, Aarau, 1807-1813, 8vo, 2 vols.
  _Thuringia_: Keller, Jena, 1819, 8vo. _Transylvania_: Schuller, Prag,
  1865, 8vo. _Tirol_: Schöpf, Innspruck, 1866, 8vo. _Venetian Alps_:
  Schmeller, Wien, 1854, 8vo. _Vienna_: Hugel, ib. 1873, 8vo.
  HUNTING.--_Westerwald_: Schmidt, Hadamar, 1800, 8vo; Kehrein,
  Wiesbaden, 1871, 12mo. SLANG.--_Gauner Sprache_: Schott, Erlangen,
  1821, 8vo: Grolmann, Giessen, 1822, 8vo: Train, Meissen, 1833, 8vo:
  Anton, 2nd ed. Magdeburg, 1843, 8vo; 1859: Avé-Lallemant, _Das
  Deutsche Gaunerthun_, Leipzig, 1858-1862, 8vo, vol. iv. pp. 515-628.
  _Student Slang_: Vollmann (Burschicoses), Ragaz, 1846, 16mo, 562
  pages.


_Celtic._

 Celtic generally.--Lluyd, Archaeologia Britannica, Oxford, 1707,
  folio: Bullet, Besançon, 1754-1860, fol. 2 vols.


 Irish.--Cormac, bishop of Cashel, born 831, slain in battle 903, wrote
  a Glossary, _Sanas Cormaic_, printed by Dr Whitley Stokes, London,
  1862, 8vo, with another, finished in 1569, by O'Davoren, a
  schoolmaster at Burren Castle, Co. Clare: O'Clery, Lovanii, 1643, 8vo:
  MacCuirtin (Eng.-Irish), Paris, 1732, 4to: O'Brien, ib. 1768, 4to;
  Dublin, 1832, 8vo: O'Reilly, 1817, 4to: 1821; ed. O'Donovan, ib. 1864,
  4to, 725 pages: Foley (Eng.-Irish), ib. 1855, 8vo: Connellan (do.),
  1863, 8vo.


 Gaelic.--Macdonald, Edin. 1741, 8vo: Shaw, London, 1780, 4to, 2 vols.:
  Allan, Edin. 1804, 4to: Armstrong, London, 1825, 4to: Highland
  Society, ib. 1828, 4to, 2 vols.: Macleod and Dewar, Glasgow, 1853,
  8vo.


 Manx.--Cregeen, Douglas, 1835, 8vo: Kelly, ib. 1866, 8vo, 2 vols.


 Welsh.--LATIN.--Davies, London, 1632, fol.: Boxhornius, Amstelodami,
  1654, 4to. ENGLISH.--Salesbury, London, 1547, 4to: 1551: Richards,
  Bristol, 1759, 8vo: Owen (W.), London, 1793-1794, 8vo, 2 vols.; 1803,
  4to, 3 vols.: Walters, ib. 1794, 4to: Owen-Pughe, Denbigh, 1832, 8vo;
  3rd ed. Pryse, ib. 1866, 8vo: D. S. Evans (Eng.-Welsh), ib. 1852-1853,
  8vo; 1887.


 Cornish.--Pryce, _Archaeologia_, Sherborne, 1770, 4to: Williams,
  Llandovery, 1862-1865, 4to. NAMES.--Bannister (20,000), Truro,
  1869-1871, 8vo.


 Breton.--Legadeuc, _Le Catholicon breton_, finished 1464, printed at
  Lantrequier, 1499, fol. 210 pages; 1501, 4to; L'Orient, 1868, 8vo:
  Quicquer de Roskoff, Morlaix, 1633, 8vo: Rostrenen, Rennes, 1732, 4to,
  978 pages; ed. Jolivet, Guingamps, 1834, 8vo, 2 vols.: l'A.[rmerie],
  Leyde, 1744, 8vo; La Haye, 1756: Lepelletier, Paris, 1752, fol.:
  Legonidec, Angouleme, 1821, 8vo; St Brieuc, 1847-1850, 4to, 924 pages.
  DIALECT OF LÉON.--Troude (Fr.-Bret.), Brest, 1870, 8vo; Id.
  (Bret.-Fr.), ib. 1876, 8vo, 845 pages. DIOCESE OF VANNES.--Armerie,
  Leyde, 1774, 8vo.


_Basque._

 Basque.--Larramendi, St Sebastian, 1745, fol. 2 vols.; ed. Zuazua, ib.
  1854, fol.; Chaho, Bayonne, 1856, 4to, 1867: Fabre, ib. 1870, 8vo: Van
  Eys, Paris, 1873, 8vo: Egúren, Madrid, 1877.


_Baltic._

 Lithuanian.--Szyrwid, 3rd ed., Vilnae, 1642, 8vo; 5th ed. 1713:
  Schleicher, Prag, 1856-1857, 8vo, 2 vols.: Kurmin, Wilno, 1858, 8vo:
  Kurschat, Halle, 1870, &c., 8vo.


 Lettic.--Mancelius, Riga, 1638, 4to: Elvers, ib. 1748, 8vo: Lange,
  Mitau, 1777, 4to: Sjögren, Petersburg, 1861, 4to: Ulmann, ed.
  Bielenstein, Riga, 1872, &c., 8vo.


 Prussian.--Bock, Königsberg, 1759, 8vo: Hennig, ib. 1785, 8vo:
  Nesselmann, Berlin, 1873, 8vo: Pierson, ib. 1875, 8vo.


_Slavonic_.

 Slavonic generally.--Franta-Sumavski (Russ. Bulg. Old Slav. Boh.
  Polish), Praga, 1857, 8vo, Miklosich, Wien, 1886.


 Old Slavonic.--Beruinda, Kiev, 1627, 8vo; Kuteinsk, 1653, 4to:
  Polycarpi (Slav. Greek, Latin), Mosque, 1704, 4to: Alexyeev, St
  Petersb. 1773, 8vo; 4th ed. ib. 1817-1819, 8vo, 5 vols.: Russian Imp.
  Academy, ib. 1847, 4to, 4 vols.: Miklosich, Vindobonae, 1850: 4to;
  1862-1865, 8vo, Mikhailovski, St Petersb. 1875, 8vo: Charkovski,
  Warschaw, 1873, 8vo.


 Russian.--Russian Academy, St Petersburg, 1789-1794, 4to, 6 vols.;
  1806-1822, ib. 1869, 8vo, 3 vols.: Dahl, Moskva, 1862-1866, fol. 4
  vols.; d., ib. 1873, &c., 4to; a 3rd edition, 1903, &c.
  FRENCH-GERM.-ENG.--Reiff, ib. 1852-1854, 4to. GERMAN,
  LATIN.--Holterhof, Moskva, 1778, 8vo, 2 vols.; 3rd ed. 1853-1855, 8vo,
  2 vols.: Weismann, ib. 1731, 4to; 1782, and frequently. FRENCH,
  GERMAN.--Nordstet, ib. 1780-1782, 4to, 2 vols.: Heym, Moskau,
  1796-1805, 4to, 4 vols.: Booch-Arkossi and Frey, Leipzig, 1871, &c.,
  8vo. ENGLISH.--Nordstet, London, 1780, 4to: Grammatin and Parenogo,
  Moskva, 1808-1817, 4to, 4 vols. FRENCH.--Tatischeff, 2nd ed. St
  Petersb. 1798, 8vo, 2 vols.; Moskau, 1816, 4to, 2 vols.: Reiff, St
  Petersb. 1835-1836, 8vo, 2 vols.: Makaroff, ib. 1872, 8vo, 2 vols,
  1110 pages; 1873-1874, 12mo, 2 vols. GERMAN.--Pawlowski, Riga, 1859,
  8vo: Lenström, Mitau, 1871, 8vo. SWEDISH.--Geitlin, Helsingfors, 1833,
  12mo: Meurmann, ib. 1846, 8vo. POLISH.--Jakubowicz, Warszawa,
  1825-1828, 8vo, 2 vols.: Amszejewicz, ib. 1866, 8vo: Szlezigier, ib.
  1867, 8vo. TECHNICAL.--Grakov (Germ.), St Petersb. 1872, 8vo.
  NAVAL.--Butakov, ib. 1837. DIALECTS.--_North-west Russia_:
  Gorbachevski (old language, in Russian), Vilna, 1874, 8vo, 418 pages.
  _White Russia_: Nosovich (Russian), St Petersburg, 1870, 4to, 760
  pages. _Red Russia_: Patritzkii (German), Lemberg, 1867, 8vo, 2 vols.
  842 pages. _Ukraine_: Piskanov (Russian), Odessa, 1873, 4to, 156
  pages.


 Polish.--Linde (explained in Lat. Germ. and 13 Slav dialects),
  Warszawie, 1807-1814, 4to, 6 vols. 4574 pages.
  ENGLISH.--[Rykaczewski], _Complete Dictionary_, Berlin, 1849-1851,
  8vo, 2 vols.: Rykaczewski, Berlin, 1866, 16mo, 1161 pages. FRENCH AND
  GERMAN.--Troc, Leipz. 1742-1764, 8vo, 4 vols.; 4th ed. ib. 1806-1822,
  4to, 4 vols.: Bandtke, Breslau, 1806, 8vo, 2 vols.; 1833-1839, 8vo.
  FRENCH.--Schmidt, Leipzig, 1870, 16mo. RUSSIAN AND GERMAN.--Schmidt
  (J. A. E.), Breslau, 1834, 8vo. GERMAN.--Mrongovius, Königsberg, 1765;
  1835, 4to; 1837: Troianski, Berlin, 1835-1838, 8vo, 2 vols.:
  Booch-Arkossi, Leipzig, 1864-1868, 8vo, 2 vols.: Jordan, ib. 1866,
  8vo. ITALIAN.--Plazowski, Warszawa, 1860, 8vo. 2 vols. 730 pages.
  RUSSIAN.--Potocki, Lipsk, 1873, &c., 12mo.


 Wendish.--Matthäi, Budissen, 1721, 8vo: Bose, Grimma, 1840, 8vo:
  Pfuhl, w Budzsinje, 1866, 8vo, 1210 pages. UPPER LUSATIAN.--Pfuhl and
  Jordan, Leipz. 1844, 8vo. LOWER LUSATIAN.--Zwahr, Spremberg, 1847,
  8vo.


 Czech.--Rohn (Germ. Lat.), Prag, 1780, 4to, 4 vols.: Dobrowski and
  Hanka, ib. 1802-1821, 4to, 2 vols. LAT. GERM. HUNGAR.--Jungmann,
  Praze, 1835-1839, 6 vols. 4to, 5316 pages. GERMAN.--Thàm, Prag.
  1805-1807, 8vo, 2 vols.: Sumavski, ib. 1844-1846, 8vo, 2 vols.:
  Koneney, ib. 1855, 18mo, 2 vols.: Rank (Germ. Boh.), ib. 1860, 16mo,
  775 pages. TECHNICAL.--Spatny, ib. 1864, 8vo: Kheil (names of goods,
  Germ. Boh.), ib. 1864, 8vo, 432 pages. HUNTING.--Spatny, ib. 1870,
  8vo, 137 pages.


 South Slavic.--Richter and Ballman, Wien, 1839-1840, 8vo, 2 vols.
  SERVIAN.--Karajic (Germ. Lat.), ib. 1818, 8vo; 1852: Lavrovski
  (Russian), St Petersb. 1870, 8vo, 814 pages. BOSNIAN.--Micalia,
  Laureti, 1649, 8vo. SLOVAK.--Bernolak (Lat. Germ. Hung.), Budae,
  1825-1827, 8vo, 6 vols.: Loos (Hung. and Germ.), Pest, 1869, &c., 3
  vols. SLOVENE.--Gutsmann, Klagenfurt, 1789, 4to: Relkovich, Wien,
  1796, 4to, 2 vols.: Murko, Grätz, 1838, 8vo, 2 vols.: Janezic,
  Klagenfurt, 1851, 12mo. DALMATIAN.--Ardelio della Bella, Venezia,
  1728, 8vo; 2nd ed. Ragusae, 1785, 4to: Stulli, ib. 1801-1810, 4to, 2
  vols. CROATIAN.--Habdelich, Grätz, 1670, 8vo: Sulek, Agram, 1854-1860,
  8vo, 2 vols. 1716 pages. CARINTHIAN.--Lexer, Leipzig, 1862, 8vo. OLD
  SERVIAN.--Danitziye (Servian), Belgrad, 1864, 8vo, 3 vols.


 Bulgarian.--Daniel (Romaic, Albanian, Rumanian, and Bulgarian),
  Moschopolis, 1770; Venice, 1802, 4to. ENGLISH.--Morse and Vassiliev,
  Constantinople, 1860, 8vo. RUSSIAN.--Borogoff, Vienna, 1872, &c., 8vo.


_Ugrian._

 Ugrian, Comparative.--Donner, Helsingfors, 1874, 8vo, in progress:
  Budenz (Ugrian-Magyar), Budapest, 1872-1875, 8vo.


 Lappish.--_Manuale_, Holmiae, 1648, 8vo: Fjellström, ib. 1738, 8vo:
  Leem and Sandberg, Havn. 1768-1781, 4to, 2 parts: Lindahl and
  Oehrling, Holm. 1780, 8vo. NORTH LAPPISH.--Stockfleht, Christiania,
  1852, 8vo.


 Finnish.--Juslenius, Holmiae, 1745, 4to, 567 pages: Renvall, Aboae,
  1826, 4to, 2 vols.: Europaeus, Helsingissä, 1852-1853, 16mo, 2 vols.
  742 pages: Lunin, Derpt, 1853, 8vo: Eurén, Tavashuus, 1860, 8vo:
  Ahlman, ib. 1864, 8vo: Wiedemann, St Petersb. 1869, 4to: Godenhjelm
  (Germ.), Helsingfors, 1871: Lönnrot, Helsingissä, 1874.
  NAVAL.--Stjerncreutz, ib. 1863, 8vo.


 Esthonian.--Hupel, Mitau, 1818, 8vo, 832 pages: Körber, Dorpat, 1860,
  8vo: Wiedemann, St Petersb. 1869, 4to, 1002 pages: Aminoff
  (Esth.-Finnish), Helsingissä, 1869, 8vo: Meves (Russian), Riga, 1876,
  12mo.


 Permian.--Rogord (Russian), St Petersb. 1869, 8vo, 420 pages.


 Votiak.--Wiedemann, Reval, 1847, 8vo: Ahlquist, Helsingfors, 1856,
  4to.


 Cheremiss.--Budenz, Pest, 1866, 8vo.


 Ersa-Mordvine.--Wiedemann, St Petersb. 1865, 4to.
  MOKSHA-MORDVINE.--Ahlquist, ib. 1862, 8vo.


 Magyar.--Szabo, Kassan, 1792, 8vo: Guczor and Fogarazi (Hung.
  Academy), Pesth, 1862, 8vo, in progress. ENGLISH.--Dallos, Pesth,
  1860, 8vo. FRENCH.--Kiss, ib. 1844, 12mo, 2 vols.: Karady, Leipz.
  1848, 12mo: Mole, Pesth, 1865, 8vo, 2 vols. GERMAN.--Schuster, Wien,
  1838, 8vo: Bloch, Pesth, 1857, 4to, 2 vols.: Ballagi, ib. 1857, 8vo;
  6th ed. 1905, 8vo, 2 vols.: Loos, ib. 1870, 8vo, 914 pages.
  ETYMOLOGICAL.--Dankovsky (Lat.-Germ.), Pressburg, 1853, 8vo:
  Kresznerics (under roots, in Hung.), Budân, 1831-1832, 4to, 2 vols.:
  Podhorsky (from Chinese roots, in Germ.), Budapest, 1877, 8vo. NEW
  WORDS.--Kunoss, Pesth, 1836, 8vo; 1844.


 Turkish.--ARAB. PERS.--Esaad Effendi, Constantinople, 1802, fol.
  ROMAIC.--Alexandrides, Vienna, 1812, 4to. POLYGLOTTS.--Pianzola (Ital.
  Grec. volgare, e Turca), Padova, 1789, 4to: Ciakciak (Ital. Armeno,
  Turco), Venice, 1804, 4to; 2nd ed. 1829: Azarian (Ellenico, Ital. Arm.
  Turco), Vienna, 1848, 8vo: Mechitarist Congregation (Ital. Francese,
  Arm. Turco), ib. 1846, 8vo. LATIN.--Mesgnien-Meninski, Viennae, 1680,
  fol. 3 vols.; ed. Jenisch and Klezl, ib. 1780-1802, fol. 4 vols.
  ENGLISH.--Sauerwein, London, 1855, 12mo: Redhouse, ib. 1856, 8vo, 1176
  pages: Id., Eng. Turkish, ib. 860, 8vo. FRENCH.--Kieffer and Bianchi
  (Turk.-Fr.), Paris, 1835-1837, 2 vols. 2118 pages: Bianchi (Fr.-Turk.)
  Paris, 1843-1846, 8vo, 2 vols. 2287 pages; 1850, 8vo, 2 vols.:
  Mallouf, ib. 1863-1867, 8vo, 2 vols. FRENCH AND GERMAN.--Zenker (Arab.
  Pers.), Leipz, 1862-1876, 4to, 2 vols, 982 pages. GERMAN.--Korabinsky,
  Pressburg, 1788, 8vo: Vambéry, Constantinople, 1858, 8vo.
  ITALIAN.--Molina, Roma, 1641, 8vo: Masais, Firenze, 1677, 8vo:
  Ciadyrgy, Milano, 1832-1834, 4to, 2 vols. RUSSIAN.--Budagov
  (Comparative lexicon of the Turkish-Tartar dialects), St Petersburg,
  1869, 8vo, 2 vols.


 Gipsy.--Bischoff, Ilmenau, 1827, 8vo: Truxillo, Madrid, 1844, 8vo:
  Jimenes, Sevilla, 1846, 16mo: Baudrimont, Bordeaux, 1862, 8vo:
  Vaillant, Paris, 1868, 8vo: Paspati; Constantinople, 1870, 4to:
  Borrow, _Romany Lavo Lil_, London, 1874, 8vo: Smart and Crofton,
  London, 1875, 8vo.


 Albanian.--Blanchus, Romae, 1635, 8vo: Kaballioti (Romaic, Wallach.
  Alb.), Venice, 1770, 8vo: Xylander, Frankfurt a. M. 1835, 8vo: Hahn,
  Jena, 1854, 4to: Rossi da Montalto, Roma, 1866, 8vo.


ASIA

 Semitic.--POLYGLOTTS.--Thurneissius, Berolini, 1585, fol.: Thorndike,
  London, 1635, fol.: Schindler, Pentaglotton, Frankf, ad M. 1653, fol.:
  Hottinger, Heptaglotton, ib. 1661, fol.: Castellus, London, 1669, fol.
  2 vols. (Hebrew, Chaldaic, Syriac, Samaritan, Aethiopic and Arabic in
  one alphabet; Persian separately. It occupied him for seventeen years,
  during which he worked sixteen to eighteen hours a day): Otho, Frankf.
  a. M. 1702, 4to (the same languages with Rabbinical).


 Hebrew.--About 875, Zemah, head of the school of Pumbeditha, wrote a
  Talmudical dictionary of words and things, arranged in alphabetical
  order, which is lost. About 880, Jehudah ben 'Alan, of Tiberias, and
  Jehudah ibn Koreish, of Tahurt, in Morocco wrote Hebrew dictionaries.
  Saadia ben Joseph (born 892, died 942), of Fayum, in Upper Egypt,
  wrote [Hebrew: Kefer Igaron], probably a Hebrew-Arabic dictionary.
  Menahem ben Jacob Ibn Sar[=u]q (born 910, died about 970), of Tortosa
  and Cordova, wrote a copious Hebrew dictionary, first printed by
  Herschell F. Filipowski, Edinburgh, 1855, 8vo, from five MSS. David
  ben Abraham, of F[=a]s, wrote, in Arabic, a large Hebrew dictionary,
  the MS. of which, a quarto of 313 leaves on cotton paper, was found
  about 1830 by A. Firkowitz, of Eupatoria, in the cellar of a Qaraite
  synagogue in Jerusalem. The age of this work cannot be ascertained.
  About 1050, Ali ben Suleiman wrote a dictionary in Arabic, on the plan
  of that of David ben Abraham. The MS. of 429 leaves belongs to
  Firkowitz. Haja ben Sherira, the famous teacher of the Academy of
  Pumbeditha, wrote a Hebrew dictionary in Arabic, called _al H[=a]vi_
  (The Gathering), arranged alphabetically in the order of the last
  radical letter. This dictionary is lost, as well as that of the
  Spaniard Isaac ben Saul, of Lucena. Iona ibn Ganah, of Cordova, born
  about 985, wrote a Hebrew dictionary in Arabic called _Kit[=a]b al
  Azul_ (Book of Roots). This, as well as a Hebrew translation by Samuel
  ibn Tab[=o]n, is extant in MS., and was used by Gesenius in his
  _Thesaurus_. Rabbi David ben Joseph Kimhi died soon after 1232. His
  lexicon of roots, called [Hebrew: Shorashim], was printed at Naples
  1490, fol.; Constantinople, 1513, fol.; Naples, 1491, 8vo; Venice,
  1552; Berolini, 1838, 4to. _Tishbi_ (The Tishbite), by Elijah ben
  Asher, the Levite, so called because it contained 712 roots, was
  printed at Isny 1541, 8vo and 4to, and often afterwards.
  LATIN.--Münster, Basileae, 1523, 8vo; 5 editions to 1564: Zamora,
  Compluti, 1526, fol.: Pellicanus, Argentorati, 1540, fol.: Reuchlin,
  Basil, 1556, fol.: Avenarius, Wittebergae, 1568, fol.; auctus, 1589:
  Pagnini, Lugd. Bat. 1575, fol.; 1577; Genevae, 1614; Buxtorf, Basil.
  1607, 8vo; 1615, and many other editions: Frey (Lat.-Eng.), 2nd ed.
  London, 1815, 8vo: Gesenius, _Thesaurus_, Leipz. 1829-1858, 4to, 3
  vols. ENGLISH.--Bale, London, 1767, 4to: Parkhurst, ib. 1792, 4to:
  Lee, ib. 1840, 8vo: Gesenius, translated by Robinson, ib. 1844, 8vo;
  by Tregelles, ib. 1846, 4to: Fuerst, 4th ed. transl. by Davidson, ib.
  1866, 8vo: 1871, 8vo, 1547 pages. FRENCH.--Leigh, Amst. 1703, 4to:
  Glaire, Paris, 1830, 8vo; 1843. GERMAN.--Gesenius, Leipzig, 1810-1812,
  8vo, 2 vols.: Fuerst, ib. 1842, 16mo: ib. 1876, 8vo, 2 vols.
  ITALIAN.--Modena, Venetia, 1612, 4to; 1640; Coen, Reggio, 1811, 8vo:
  Fontanella, Venezia, 1824, 8vo. DUTCH.--Waterman, Rotterdam, 1859,
  &c., 8vo. HUNGARIAN.--Ehrentheil (Pentateuch), Pest, 1868, 8vo.
  ROMAIC.--Loundes, Melité. 1845, 8vo, 987 pages.


 Rabbinical and Chaldee.--Nathan ben Yehiel of Rome wrote in the
  beginning of the 12th century a Talmudic dictionary, _Aruch_, printed
  1480 (?), s. l., fol.; Pesaro, 1517, fol.; Venice, 1531; and often:
  Isaiah ben Loeb, Berlin, wrote a supplement to _Aruch_, vol. i.
  Breslau, 1830, 8vo; vol. ii. ([Hebrew: L] to [Hebrew: T]), Wien, 1859,
  8vo: Münster, Basil. 1527, 4to, 1530, fol.: Elijah ben Asher, the
  Levite, transl. by Fagius, Isnae, 1541, fol.; Venet. 1560: David ben
  Isaac de Pomis, _Zamah David_, Venet. 1587, fol.: Buxtorf,
  Basileae, 1639, fol.: ed. Fischer, Leipz. 1866-1875, 4to: Otho,
  Geneva, 1675, 8vo; Altona, 1757, 8vo: Zanolini, Patavii, 1747, 8vo:
  Hornheim, Halle, 1807, 8vo: Landau, Prag, 1819-1824, 8vo, 5 vols.:
  Dessauer, Erlangen, 1838, 8vo: Nork (i.e. Korn), Grimma, 1842, 4to:
  Schönhak, Warschau, 1858, 8vo, 2 vols. TARGUMS.--Levy, Leipzig,
  1866-68 4to, 2 vols.; 1875: Id. (Eng.), London, 1869, 8vo, 2 vols.
  TALMUD.--Löwy (in Heb.), Wien, 1863, 8vo: Levy, Leipzig, 1876, &c.,
  4to. PRAYER-BOOK.--Hecht, Kreuznach, 1860, 8vo: Nathan, Berlin, 1854,
  12mo. SYNONYMS.--Pantavitius, Lodevae, 1640, fol. FOREIGN
  WORDS.--Rabeini, Lemberg, 1857, 8vo, &c. JEWISH-GERMAN.--Callenberg,
  Halle, 1736, 8vo: Vollbeding, Hamburg, 1808, 8vo: Stern, München,
  1833, 8vo, 2 vols.: Theile, Berlin, 1842-1843, 8vo, 2 vols.:
  Avé-Lallemant, _Das deutsche Gaunerthum_, Leipzig, 1858, 8vo, 4 vols.;
  vol. iv. pp. 321-512.


 Ph[oe]nician.--M. A. Levy, Breslau, 1864, 8vo.


 Samaritan.--Crinesius, Altdorphi, 1613, 4to: Morini, Parisiis, 1657,
  12mo: Hilligerus, Wittebergae, 1679, 4to: Cellarius, Cizae, 1682, 4to;
  Frankof. 1705: Uhlemann, Leipsiae, 1837, 8vo: Nicholls, London, 1859,
  8vo.


 Assyrian.--Norris, London, 1868, 8vo, 3 vols. PROPER NAMES.--Menant,
  Paris, 1861, 8vo.


 Accadian.--Lenormant, Paris, 1875, 8vo.


 Syriac.--Joshua ben Ali, a physician, who lived about 885, made a
  Syro-Arabic lexicon, of which there is a MS. in the Vatican. Hoffmann
  printed this lexicon from Alif to Mim, from a Gotha MS., Kiel, 1874,
  4to. Joshua bar Bahlul, living 963, wrote another, great part of which
  Castelli put into his lexicon. His MS. is now at Cambridge, and, with
  those at Florence and Oxford, was used by Bernstein. Elias bar
  Shinaya, born 975, metropolitan of Nisibis, 1009, wrote a Syriac and
  Arabic lexicon, entitled _Kit[=a]b [=u]t Tarjuman fi Taalem Loghat es
  S[=u]ri[=a]n_ (Book called the Interpreter for teaching the Language
  of the Syrians), of which there is a MS. in the British Museum. It was
  translated into Latin by Thomas à Novaria, a Minorite friar, edited by
  Germanus, and published at Rome by Obicinus, 1636, 8vo. It is a
  classified vocabulary, divided in 30 chapters, each containing several
  sections. Crinesius, Wittebergae, 1612, 4to: Buxforf, Basileae, 1622,
  4to: Ferrarius, Romae, 1622, 4to: Trost, Cothenis Anhaltor, 1643, 4to:
  Gutbir, Hamburgi, 1667, 8vo: Schaaf, Lugd. Bat, 1708, 4to: Zanolini,
  Patavii, 1742, 4to: Castellus, ed. Michaelis, Göttingen, 1788, 4to, 2
  vols.: Bernstein, Berlin, 1857, &c. fol.: Smith (Robt. Paine), Dean of
  Canterbury, Oxonii, 1868, &c. fol.: fasc. 1-3 contain 538 pages:
  Zingerle, Romae, 1873, 8vo, 148 pages.


 Arabic.--The native lexicons are very many, voluminous and copious. In
  the preface to his great Arabic-English lexicon, Lane describes 33,
  the most remarkable of which are-the _'Ain_, so called from the letter
  which begins its alphabet, commonly ascribed to al Khalil (who died
  before A.H. 175 [A.D. 791], aged seventy-four): the _Sihah_ of Jauhari
  (died 398 [1003]): the _Mohkam_ of Ibn Sidah the Andalusian, who was
  blind, and died A.H. 458 [A.D. 1066], aged about sixty: the _Asas_ of
  Zamakhshari (born 467 [1075], died 538 [1144]), "a most excellent
  repertory of choice words and phrases": the _Lis[=a]n el 'Arab_ of Ibn
  Mukarram (born 630 [1232], died 711 [1311]); Lane's copy is in 28
  vols. 4to: the _Kamus_ (The Sea) of Fairuzabadi (born 729 [1328], died
  816 [1413]),: the _Taj el Arus_, by Murtada Ez Zebadi (born A.D. 1732,
  died 1791)--the copy made for Lane is in 24 vols. thick 4to. The
  _Sihah_ was printed Hardervici Getorum, 1774, 4to; Bulak, 1865, fol. 2
  vols.: _Kamus_, Calcutta, 1817, fol. 2 vols.; Bombay, 1855, fol. 920
  pages: _Sirr el Lagal_, by Farish esh Shidiac, Tunis, fol. 609 pages:
  _Muh[=i]t al Muh[=i]t_, by Beitrus Al Bustani Beirut, 1867-1870, 2
  vols. 4to, 2358 pages (abridged as _Katr Al Muhit_, ib. 1867-1869, 2
  vols. 8vo, 2352 pages), is excellent for spoken Arabic. PERSIAN.--The
  _Surah_, by Jumal, Calcutta, 1812-1815, 2 vols. 4to: _Samachsharii
  Lexicon_, ed. Wetzstein, Leipz. 1845, 4to; 1850: _Muntakhal al
  Loghat_, Calcutta, 1808; ib. 1836; Lucknow, 1845; Bombay, 1862, 8vo, 2
  vols.: _Muntaha l'Arabi_, 4 vols. fol. 1840: _Shams al Loghat_,
  Bombay, 1860, fol. 2 vols. 509 pages. TURKISH.--_Achteri Kabir_,
  Constantinople. 1827, fol.: _El Kamus_, ib. 1816, fol. 3 vols.;
  translated by Açan Effendi, Bulak, fol. 3 vols.; _El Sihah_,
  translated by Al Vani, Constantinople, 1728, fol. 2 vols.: 1755-1756;
  Scutari, 1802, fol. 2 vols. LATIN.--Raphelengius, Leiden, 1613, fol.:
  Giggeius, Mediolani, 1632, fol. 4 vols.: Golius Lugd. Bat. 1653, fol.
  (the best before Lane's): Jahn, Vindobonae, 1802, 8vo: Freytag, Halle,
  1830-1838, 4 vols. 4to; abridged, ib. 1837, 4to. ENGLISH.--Catafago
  (Arab.-Eng. and Eng.-Arab.), London, 1858, 8vo, 2 vols.; 2nd ed. 1873,
  8vo: Lane, London, 1863-1893 (edited after Lane's death, from 1876, by
  his grandnephew, Stanley Lane-Poole. The Arabic title is _Medd el
  Kamoos_, meaning either the Flow of the Sea, or The Extension of the
  Kamus. It was undertaken in 1842, at the suggestion and at the cost of
  the 6th duke of Northumberland, then Lord Prudhoe, by Mr Lane, who
  returned to Egypt for the purpose, and lived in Cairo for seven years
  to study, and obtain copies of, the great MS. lexicons in the
  libraries of the mosques, few of which had ever been seen by a
  European, and which were so quickly disappearing through decay,
  carelessness and theft, that the means of composing such a work would
  not long have existed). Newman (modern), ib. 1872, 8vo, 2 vols. 856
  pages. FRENCH.--Ruphy (Fr.-Ar.), Paris, 1802, 4to: Bochtor (do.),
  Paris, 1828, 4to, 2 vols.; 2nd ed. ib. 1850: Roland de Bussy (Algiers,
  Fr.-Ar.), Alger, 1835, 16mo: Id., 1836, 8vo; 1839: Berggren (Fr.-vulg.
  Ar., Syria and Egypt.), Upsala, 1844, 4to: Farhat (Germanos), revu par
  Rochaid ed Dahdah, Marseille, 1849, 4to: Biberstein Kasimirski, Paris,
  1846, 8vo, 2 vols.; 1853-1856; 1860, 2 vols. 3032 pages: Marcel
  (vulgar dialects of Africa), Paris, 1830; 1835, 8vo; 1837; enlarged,
  1869, 8vo; Paulmier (Algeria), 2nd ed. Paris, 1860, 8vo, 931 pages;
  1872: Bernard (Egypt), Lyon, 1864, 18mo: Cuche, Beirut, 1862, 8vo;
  1867: Nar Bey (A. Calfa), 2nd ed. Paris, 1872, 12mo, 1042 pages:
  Cherbonneau (written language), Paris, 1876, 2 vols. 8vo: Id.
  (Fr.-Ar.), Paris, 1872, 8vo: Beausier (Algiers, Tunis, legal,
  epistolary), Alger, 1871, 4to, 764 pages; 1873. GERMAN.--Seyfarth
  (Algeria), Grimma, 1849, 16mo: Wolff (Mod. Ar.), Leipzig, 1867, 8vo:
  Wahrmund (do.), Giessen, 1870-1875, 8vo, 4 vols. ITALIAN.--Germano,
  Roma, 1636, 8vo; (Ar. Lat. It.), Romae, 1639, fol.: _Dizionario_,
  Bulak. 1824, 4to: Schiaparelli, Firenze, 1871, 4to, 641 pages.
  SPANISH.--Alcala, Grenada, 1505, 4to: Cañes, Madrid, 1787, fol. 3
  vols. SUFI TECHNICAL TERMS.--Abd Errahin, ed. Sprenger, Calcutta,
  1845, 8vo. TECHNICAL TERMS OF THE MUSSULMAN SCIENCES.--Abd al Hagg and
  Gholam Kadir, Calcutta, 1853-1862, 4to, 1593 pages. MEDICAL
  TERMS.--Pharaon and Bertherand, Paris, 1860, 12mo. MATERIA
  MEDICA.--Muhammed Abd Allah Shirazi, _Ulfaz Udwiyeh_, translated by
  Gladwin (Eng. Pers. Hindi), Calcutta, 1793, 4to, 1441 words. NOMS DES
  VÊTEMENTS.--Dozy, Amst. 1845, 8vo. WÖRTER IN ENTGEGENGESETZTEN
  BEDEUTUNGEN.--Redslob, Göttingen, 1873, 8vo. KORAN.--Willmet (also in
  Haririum et vitam Timuri), Lugd. Bat. 1784, 4to; Amst. 1790: Fluegel,
  _Concordantia_, Leipz. 1842, 4to: Penrice, _Dictionary and Glossary_,
  London, 1873, 4to. EL TABRIZI'S LOGIC.--Mir Abufeth (French), Bulak,
  1842, 8vo. MALTESE.--Vassali, Romae, 1796, 4to: Falzon (Malt. Ital.
  Eng.), Malta, _s.a._ 8vo: Vella, Livorno, 1843, 8vo.


 Armenian.--Mechitar, Venice, 1749-1769, 4to, 2 vols.: Avedichiam,
  Sürmelian and Aucher (Aukerian), ib. 1836-1837, 4to, 2 vols.: Aucher,
  ib. 1846, 4to. POLYGLOT.--Villa (Arm.-vulg., litteralis, Lat. Indicae
  et Gallicae), Romae, 1780. GREEK AND LATIN.--Lazarists, Venice,
  1836-1837, 4to, 2 vols. 2217 pages. LATIN.--Rivola, Mediolani, 1621,
  fol.: Nierszesovicz, Romae, 1695, 4to; Villotte, ib. 1714, fol.:
  Mechitar, Venetiae, 1747-1763, 4to, 2 vols. ENGLISH.--Aucher, Venice,
  1821-1825, 4to, 2 vols. FRENCH.--Aucher, Venise, 1812-1817, 8vo, 2
  vols.; (Fr.-Arm. Turc.), ib. 1840, 4to: Eminian, Vienna, 1853, 4to:
  Calfa, Paris, 1861, 8vo, 1016 pages; 1872. ITALIAN.--Ciakciak,
  Venezia, 1837, 4to. RUSSIAN.--Khudobashev [Khutapashian], Moskva,
  1838, 8vo, 2 vols. RUSS. ARM.--Adamdarov, ib. 1821, 8vo: Popov, ib.
  1841, 8vo, 2 vols. MODERN WORDS.--Riggs, Smyrna, 1847, 8vo.


 Georgian.--Paolini (Ital.), Roma, 1629, 4to: Klaproth (Fr.), Paris,
  1827, 8vo: Tshubinov (Russian, French), St Petersburg, 1840, 4to;
  1846, 8vo, 2 vols. 1187 pages.


 Circassian.--Loewe, London, 1854, 8vo.


 Ossetic.--Sjörgen, St Petersb. 1844, 4to.


 Kurd.--Garzoni, Roma, 1787, 8vo: Lerch (German), St Petersburg, 1857,
  8vo: Id. (Russian), ib. 1856-1858, 8vo.


 Persian.--_Burhani Qatiu_, arranged by J. Roebuck, Calcutta, 1818,
  4to: _Burhan i Kati_, Bulak, 1836, fol.: Muhammed Kazim, Tabriz, 1844,
  fol.: _Haft Kulzum_ (The Seven Seas), by Ghazi ed din Haidar, King of
  Oude, Lucknow, 1822, fol. 7 vols. ARABIC.--_Shums ul Loghat_,
  Calcutta, 1806, 4to, 2 vols. TURKISH.--Ibrahim Effendi, _Farhangi
  Shu'uri_, ib. 1742, fol. 2 vols. 22,530 words, and 22,450 poetical
  quotations: _Burhan Kati_, by Ibn Kalif, translated by Ahmed Asin
  Aintabi, ib. 1799, fol.; Bulak, 1836, fol.: Hayret Effendi, ib. 1826,
  8vo. ARMENIAN.--Douzean, Constantinople, 1826, fol. BENGALI.--Jay
  Gopal, Serampore, 1818, 8vo. LATIN.--Vullers (Zend appendix), Bonnae
  ad Rhen, 1855-1868, 4to, 2 vols. 2544 pages; Supplement of Roots,
  1867, 142 pages. ENGLISH.--Gladwin, Malda in Bengal, 1780, 4to;
  Calcutta, 1797: Kirkpatrick, London, 1785, 4to: Moises, Newcastle,
  1794, 4to: Rousseau, London, 1802, 8vo; 1810: Richardson (Arab, and
  Pers.), ib. 1780-1800, fol. 2 vols.; ed. Wilkins, ib. 1806-1810, 4to,
  2 vols.; ed Johnson, ib. 1829, 4to: Ramdhen Sen, Calcutta, 1829, 8vo;
  1831: Tucker (Eng.-Pers.), London, 1850, 4to: Johnson (Pers. and
  Arab.), ib. 1852, 4to: Palmer, ib. 1876, 8vo, 726 pages.
  FRENCH.--Handjeri (Pers. Arab. and Turkish), Moscou, 1841, 4to, 3
  vols. 2764 pages: Bergé, Leipzig, 1869, 12mo. GERMAN.--Richardson,
  translated by Wahl as _Orientalische Bibliotheque_, Lemg, 1788-1792,
  8vo, 3 vols. ITALIAN.--Angelus a S. Josepho [i.e. Labrosse] (Ital.
  Lat. Fr.), Amst. 1684, fol.


 Old Persian.--(Cuneiform), Benfey (German), Leipzig, 1847, 8vo:
  Spiegel (id.), ib. 1862, 8vo: Kossovich (Latin), Petropoli, 1872, 8vo.


 Zend.--Justi, Leipzig, 1864, 4to: Vullers, Persian Lexicon, Appendix:
  Lagarde, Leipzig, 1868, 8vo.


 Pahlavi.--_An old Pahlavi and Pazend Glossary_, translated by Destur
  Hoshengi Jamaspji, ed. Haug, London, 1867, 8vo; 1870, 8vo: West,
  Bombay, 1874, 8vo.


  INDIAN TERMS.--_The Indian Vocabulary_, London, 1788, 16mo: Gladwin,
  Calcutta, 1797, 4to: Roberts, London, 1800, 8vo: Rousseau, ib. 1802,
  8vo: Roebuck (naval), ib. 1813, 12mo: C. P. Brown, _Zillah Dict._,
  Madras, 1852, 8vo: Robinson (Bengal Courts), Calcutta, 1854, 8vo;
  1860: Wilson, London, 1855, 4to: Fallon, Calcutta, 1858, 8vo.


 Sanskrit.--Amarasimha (lived before A.D. 1000), _Amarakosha_ Calcutta,
  1807, 8vo; ib. 1834, 4to; Bombay, 1860, 4to; Lucknow, 1863, 4to;
  Madras, 1870, 8vo, in Grantha characters; Cottayam, 1873, 8vo, in
  Malaylim characters; Benares, 1867, fol. with _Amaraviveka_, a
  commentary by Mahesvara: Rajah Radhakanta Deva, _Sabdakalpadruma_,
  Calcutta, 1821-1857, 4to, 8 vols. 8730 pages: 2nd ed. 1874, &c.:
  Bhattachdrya, _Sabdastoma Mahanidhi_, Calcutta, 1869-1870, 8vo, parts
  i.-vii. 528 pages: _Abhidhanaratnamala_, by Halayudha, ed. Aufrecht,
  London, 1861, 8vo: VACHASPATYA, by Taranatha Tarkavachaspati,
  Calcutta, 1873, &c., 4to (parts i.-vii., 1680 pages).
  BENGALI.--_Sabdasindhu_, Calcutta, 1808: _Amarakosa_, translated by
  Ramodoyu Bidjalunker, Calcutta, 1831, 4to: Mathurana Tarkaratna,
  _Sabdasandarbhasindhu_, Calcutta, 1863, 4to. MARATHI.--Ananta Sastri
  Talekar, Poona, 1853, 8vo, 495 pages: Madhava Chandora, Bombay, 1870,
  4to, 695 pages. TELUGU.--_Amarakosha_, Madras, 1861, ed. Kala, with
  _Gurubalala prabodhika_, a commentary, ib. 1861, 4to; with the same,
  ib. 1875, 4to, 516 pages; with _Amarapadaparijata_ (Sans. and Tel.),
  by Vavilla Ramasvani Sastri, ib. 1862, 4to; ib. 1863, 8vo; 3rd ed. by
  Jaganmohana Tarkalankara and Khetramohana, 1872, &c., parts i.-iv. 600
  pages: Suria Pracasa Row, _Sarva-Sabda-Sambodhini_, ib. 1875, 4to,
  1064 pages. TIBETAN AND MONGOL.--Schiefner, _Buddhistische Triglotte_,
  St Petersburg, 1859, fol., the _Vyupatti_ or _Mahavyupatti_ from the
  _Tanguir_, vol. 123 of the Sutra. LATIN.--Paulinus a Sancto
  Bartholomeo, Amarasinha, sectio i. de coelo, Romae, 1798, 4to: Bopp.
  Berlin, 1828-1830, 4to; 2nd ed. 1840-1844; 3rd, 1866, 4to.
  ENGLISH.--_Amarakosha_, trans. by Colebrooke, Serampore, 1808, 4to;
  1845, 8vo: Rousseau, London, 1812, 4to: Wilson, Calcutta, 1819, 4to;
  2nd ed. 1832: ed. Goldstücker, Berlin, 1862, &c., folio, to be in 20
  parts: Yates, Calcutta, 1846, 4to: Benfey, London, 1865, 8vo: Ram
  Jasen, Benares, 1871, 8vo, 713 pages: Williams, Oxford, 1872, 4to.
  ENGLISH-SANSKRIT.--Williams, London, 1851, 4to. FRENCH.--Amarakosha,
  transl. by Loiseleur Deslongchamps, Paris, 1839-1845, 8vo, 2 vols. 796
  pages: Burnouf and Leupol, Nancy, 1863-1864, 8vo. GERMAN.--Böhtlingk
  and Roth, St Petersb. 1853, &c., 4to, 7 vols. to 1875.
  ITALIAN.--Gubernatis, Torino, 1856, &c. 8vo, unfinished, 2 parts.
  RUSSIAN.--Kossovich, St Petersburg, 1859, 8vo. ROOTS.--Wilkins,
  London, 1815, 4to: Rosen, Berolini, 1827, 8vo: Westergaard, Bonnae,
  1840-1841, 8vo: Vishnu Parasurama Sastri Pandita (Sans. and Marathi),
  Bombay, 1865, 8vo: Taranatha Tarkavachaspati, _Dhatupadarsa_,
  Calcutta, 1869, 8vo: Leupol, Paris, 1870, 8vo.
  SYNONYMS.--_Abhidhanacintamani_, by Hemachadra, ed. Colebrooke,
  Calcutta, 1807, 8vo; translated by Böhtlingk and Rieu (German), St
  Petersburg, 1847, 8vo. HOMONYMS.--Medinikara, _Medinikosha_, Benares,
  1865, 4to; Calcutta, 1869, 8vo; ib. 1872, 8vo. DERIVATIVES.--Hirochand
  and Rooji Rangit, _Dhatumanjari_, Bombay, 1865, 8vo. TECHNICAL TERMS
  OF THE NYÂYA PHILOSOPHY.--_Nyâyakosa_, by Bhimachârya Jhalakîkar
  (Sanskrit), Bombay, 1875, 8vo, 183 pages. RIG VEDA.--Grassmann,
  Leipzig, 1873-1875, 8vo.


 Bengali.--Manoel, Lisboa, 1743, 8vo: Forster, Calcutta, 1799-1802,
  4to, 2 vols. 893 pages: Carey, Serampore, 1815-1825, 4to, 2 vols.; ed.
  Marshman, ib. 1827-1828, 8vo, 2 vols.; 3rd ed. ib. 1864-1867, 8vo;
  abridged by Marshman, ib. 1865, 8vo; ib. 1871, 8vo, 2 vols. 936 pages:
  Morton, Calcutta, 1828, 8vo: Houghton, London, 1833, 4to: Adea,
  _Shabdabudhi_, Calcutta, 1854, 604 pages. ENGLISH.--Ram Comul Sen, ib.
  1834, 4to, 2 vols.; London, 1835, 4to: D'Rozario, Calcutta, 1837, 8vo:
  Adea, _Abhidan_, Calcutta, 1854, 761 pages. ENGLISH LAT.--Ramkissen
  Sen, ib. 1821, 4to. ENG.-BENG. AND MANIPURI.--[Gordon], Calcutta,
  1837, 8vo.


 Canarese.--Reeve, Madras, 1824-1832, 4to, 2 vols.; ed. Sanderson,
  Bangalore, 1858, 8vo, 1040 pages; abridged by the same, 1858, 8vo, 276
  pages: _Dictionarium Canarense_, Bengalori, 1855, 8vo: _School
  Dictionary_, Mangalore, 1876, 8vo, 575 pages.


 Dardic Languages.--Leitner (Astori, Ghilghiti, Chilasi, and dialects
  of Shina, viz. Arnyia, Khajuna and Kalasha), Lahore, 1868, 4to.


 Guzarati.--(English) Mirza Mohammed Cauzim, Bombay, 1846, 4to;
  Shapurji Edalji, ib. 1868, 8vo, 896 pages: Karsandas Mulji, ib. 1868,
  8vo, 643 pages.


 Hindi.--Rousseau, London, 1812, 4to: Adam, Calcutta, 1829, 8vo:
  Thompson, ib. 1846, 8vo: J. D. Bate, London, 1876, 8vo, 809 pages.
  ENGLISH.--Adam, Calcutta, 1833, 8vo. ENGLISH, URDU AND
  HINDI.--Mathuraprasada Mirsa, Benares, 1865, 8vo, 1345 pages.


 Hindustani.--Ferguson, London, 1773, 4to: Gilchrist, Calcutta, 1800,
  8vo; ed. Hunter, Edinb. 1810; Lond. 1825: Taylor, Calcutta, 1808, 4to,
  2 vols.: Gladwin (Persian and Hind.), Calcutta, 1809, 8vo, 2 vols.:
  Shakespeare, London, 1817, 4to; 1820; 1834; 1849: Forbes, London,
  1847, 8vo; 1857: Bertrand (French), Paris, 1858, 8vo: Brice, London,
  1864, 12mo: Fallon, Banaras, 1876, &c., to be in about 25 parts and
  1200 pages. ENGLISH.--Gilchrist, 1787-1780, 4to, 2 parts: Thompson,
  Serampore, 1838, 8vo.


 Kashmiri.--Elmslie, London, 1872, 12mo.


 Khassia.--Roberts, Calcutta, 1875, 12mo.


 Malayalim.--Fabricius and Breithaupt, Weperg, 1779, 4to: Bailey,
  Cottayam, 1846, 8vo: Gundert, Mangalore, 1871, 8vo, 1171 pages.


 Marathi.--Carey, Serampore, 1810, 8vo: Kennedy, Bombay, 1824, fol.:
  Jugunnauth Shastri Kramavant, Bombay, 1829-1831, 4to, 3 vols.:
  Molesworth, ib. 1831, 4to; 2nd ed. 1847, 4to; ed. Candy, Bombay, 1857,
  4to, 957 pages; abridged by Baba Padmanji, ib. 1863, 8vo; 2nd ed.
  (abridged), London, 1876, 8vo, 644 pages. ENGLISH.--Molesworth,
  Bombay, 1847, 4to.


 Oriya.--Mohunpersaud Takoor, Serampore, 1811, 8vo: Sutton, Cuttack,
  1841-1848, 8vo, 3 vols. 856 pages.


 Pali.--Clough, Colombo, 1824, 8vo: Moggallana Thero (a Sinhalese
  priest of the 12th century), _Abhidhanappika_ (Pali, Eng. Sinhalese),
  ed. Waskeduwe Subheti, Colombo, 1865, 8vo: Childers, London,
  1872-1875, 8vo, 658 pages. ROOTS.--Silavansa, _Dhatumanjusa_ (Pali
  Sing. and Eng.), Colombo, 1872, 8vo.


 Prakrit.--Delius, _Radices_, Bonnae ad Rh., 1839, 8vo.


 Punjabi.--Starkey, 1850, 8vo; Lodiana Mission, Lodiana, 1854-1860, 444
  pages.


 Pushtu or Afghan.--Dorn, St Petersb. 1845, 4to: Raverty, London, 1860,
  4to; 2nd ed. ib. 1867, 4to: Bellew, 1867, 8vo.


 Sindhi.--Eastwick, Bombay, 1843, fol. 73 pages: Stack, ib. 1855, 8vo,
  2 vols.


 Sinhalese.--Clough, Colombo, 1821-1830, 8vo, 2 vols.: Callaway (Eng.,
  Portuguese and Sinhalese), ib. 1818, 8vo: Id., _School Dictionary_,
  ib. 1821, 8vo: Bridgenell (Sinh.-Eng.), ib. 1847, 18mo: Nicholson
  (Eng.-Sinh.), 1864, 32mo, 646 pages.


 Tamil.--Provenza (Portug.), Ambalacotae, 1679, 8vo: _Sadur Agurardi_,
  written by Beschi in 1732, Madras, 1827, fol.; Pondicherry, 1875, 8vo:
  Blin (French), Paris, 1834, 8vo: Rottler, Madras, 1834-1841, 4to, 4
  vols.: Jaffna Book Society (Tamil), Jaffna, 1842, 8vo, about 58,500
  words: Knight and Spaulding (Eng. Tam.), ib. 1844, 8vo; _Dictionary_,
  ib. 1852, 4to: Pope, 2nd ed. ib. 1859, 8vo: Winslow, Madras, 1862,
  4to, 992 pages, 67,452 words.


 Telugu.--Campbell, Madras, 1821, 4to: C. P. Brown, Madras (Eng.-Tel.),
  1852, 8vo, 1429 pages: Id. (Tel.-Eng.), ib. 1852, 8vo, 1319 pages.
  MIXED TELUGU.--Id., ib. 1854, 8vo.


 Thuggee.--Sleeman, Calcutta, 1830, 8vo, 680 Ramasi words.


 Indo-Chinese Languages.--Leyden, _Comparative Vocabulary of Barma,
  Malaya and Thai_, Serampore, 1810, 8vo. _Annamese_: Rhodes (Portug.
  and Lat.), Romae, 1651, 4to: Pigneaux and Taberd, Fredericinagori,
  1838, 4to; Legrand de la Liraye, Paris, 1874, 8vo: Pauthier (Chin.
  Ann.-Fr. Lat.), Paris, 1867, &c., 8vo. _Assamese_: Mrs Cutter, Saipur,
  1840, 12 mo; Bronson, London, 1876, 8vo, 617 pages. _Burmese_: Hough
  (Eng.-Burm.), Serampore, 1825, Moulmain, 1845, 8vo, 2 vols. 955 pages:
  Judson, Calcutta, 1826, 8vo; (Eng. Burm.), Moulmain, 1849, 4to; (Burm.
  Eng.), ib. 1852, 8vo; 2nd ed., Rangoon, 1866, 8vo, 2 vols. 968 pages:
  Lane, Calcutta, 1841, 4to. _Cambodian_: Aymonier (Fr.-Camb.), Saigon,
  1874, 4to; Id. (Camb.-Fr.), ib. 1875, fol. _Karen_: Sau-kau Too
  (Karen), Tavoy, 1847, 12mo, 4 vols.: Mason, Tavoy, 1840, 4to.
  _Sgau-Karen_: Wade, ib. 1849, 8vo. _Siamese (Thai)_: Pallegoix (Lat.
  French, Eng.), Paris, 1854, 4to: _Dictionarium Latinum Thai_, Bangkok,
  1850, 4to, 498 pages.


 Malay.--LATIN.--Haex, Romae, 1631, 4to; Batavia, 1707.
  DUTCH.--Houtmann (Malay and Malagasy), Amst. 1603, 4to; 1673; 1680;
  1687; 1703; Batavia, 1707: Wiltens and Dankaarts, Gravenhage, 1623,
  4to; Amst. 1650; 1677; Batavia, 1708, 4to: Heurnius, Amst. 1640, 4to:
  Gueynier, Batavia, 1677, 4to; 1708: Loder, ib. 1707-1708, 4to: Van der
  Worm, ib. 1708, 4to: Roorda van Eysinga (Low), ib. 1824-1825, 8vo, 2
  vols.; 12th ed. 's Gravenhage, 1863, 8vo; Id. (Hof, Volks en Lagen
  Taal), ib. 1855, 8vo: Dissel and Lucardie (High Malay), Leiden, 1860,
  12mo: Pijnappel, Amst. 1863, 8vo: Badings, Schoonhoven, 1873, 8vo.
  ENGLISH.--Houtmann (Malay and Malagasy), translated by A. Spaulding,
  London, 1614, 4to: Bowrey, ib. 1701, 4to: Howison, ib. 1801, 4to:
  Marsden, ib. 1812, 4to: Thomsen, Malacca, 1820, 8vo; 1827: Crawford,
  London, 1851, 8vo, 2 vols. FRENCH.--Boze, Paris, 1825, 16mo: Elout
  (Dutch-Malay and French-Malay), Harlem, 1826, 4to: Bougourd, Le Havre,
  1856, 8vo: Richard, Paris, 1873, 8vo, 2 vols.: Favre, Vienna, 1875,
  8vo, 2 vols.


 Malay Archipelago.--_Batak_: Van der Tuuk, Amsterdam, 1861, 8vo, 564
  pages. _Bugis_: Mathes, Gravenh. 1874, 8vo, 1188 pages: Thomsen
  (Eng.-Bugis and Malay), Singapore, 1833, 8vo. _Dyak_: Hardeland
  (German), Amst. 1859, 8vo, 646 pages. _Javanese_: Senerpont Domis,
  Samarang, 1827, 4to, 2 vols.: Roorda van Eysinga, Kampen, 1834-1835,
  8vo, 2 vols.: Gericke, Amst. 1847, 8vo; ed. Taco Roorda, ib. 1871, &c.
  parts i.-v., 880 pages: Jansz and Klinkert, Samarang, 1851, 8vo; 1865:
  Favre (French), Vienne, 1870, 8vo. _Macassar_: Matthes, Amst. 1859,
  8vo, 951 pages. Sunda: De Wilde (Dutch, Malay and Sunda), Amsterdam,
  1841, 8vo: Rigg (Eng.), Batavia, 1862, 4to, 573 pages. _Formosa_:
  Happart (Favorlang dialect, written about 1650), Parrapattan, 1840,
  12mo.


 Philippines.--_Bicol_: Marcos, Sampaloc, 1754, fol. _Bisaya_: Sanchez,
  Manila, 1711, fol.: Bergaño, ib. 1735, fol.: Noceda, ib. 1841:
  Mentrida (also Hiliguena and Haraya) ib. 1637, 4to; 1841, fol. 827
  pages: Felis de la Encarnacion, ib. 1851, 4to, 2 vols. 1217 pages.
  _Ibanac_: Bugarin, ib. 1854, 4to. _Ilocana_, Carro, ib. 1849, fol.
  _Pampanga_: Bergaño, ib. 1732, fol. _Tagala_: Santos, Toyabas, 1703,
  fol.; ib. 1835, 4to, 857 pages: Noceda and San Lucar, Manila, 1754,
  fol.; 1832.


 Chinese.--Native Dictionaries are very numerous. Many are very copious
  and voluminous, and have passed through many editions. _Shwo wan_, by
  Hü Shin, is a collection of the ancient characters, about 10,000 in
  number, arranged under 540 radicals, published 150 B.C., usually in 12
  vols.: _Yu pien_, by Ku Ye Wang, published A.D. 530, arranged under
  542 radicals, is the basis of the Chinese Japanese Dictionaries used
  in Japan: _Ping tseu loui pien_, Peking, 1726, 8vo, 130 vols.: _Pei
  wan yün fu_ (Thesaurus of Literary Phrases), 1711, 131 vols. 8vo,
  prepared by 66 doctors of the Han lin Academy in seven years. It
  contains 10,362 characters, and countless combinations of two, three
  or four characters, forming compound words and idioms, with numerous
  and copious quotations. According to Williams (_On the word Shin_, p.
  79), an English translation would fill 140 volumes octavo of 1000
  pages each. _Kanghi tsze tien_ (Kanghi's Standard or Canon of the
  Character), the dictionary of Kanghi, the first emperor of the present
  dynasty, was composed by 30 members of the Han lin, and published in
  1716, 40 vols. 4to, with a preface by the emperor. It contains 49,030
  characters, arranged under the 214 radicals. It is generally in 12
  vols., and is universally used in China, being the standard authority
  among native scholars for the readings as well as the meanings of
  characters. LATIN.--De Guignes (French, Lat.), Paris, 1813, fol.;
  Klaproth, Supplément, 1819; ed. Bazil (Latin), Hong-Kong, 1853, 4to:
  Gonçalves (Lat.-Chin.), Macao, 1841, fol.: Callery, _Systema
  phoneticum_, Macao, 1841, 8vo: Schott, _Vocabularium_, Berlin, 1844,
  4to. ENGLISH.--Raper, London, 1807, fol. 4 vols.: Morrison, Macao,
  1815-1823, 4to, 3 parts in 6 vols.: Medhurst, Batavia, 1842-1843, 8vo,
  2 vols.: Thom, Canton, 1843, 8vo: Lobscheid, Hong-Kong, 1871, 4to:
  Williams, Shanghai, 1874, 4to. ENG. CHINESE.--Morrison, part iii.:
  Williams, Macao, 1844, 8vo: Medhurst, Shanghai, 1847-1848, 8vo, 2
  vols.: Hung Maou, _Tung yung fan hwa_ (Common words of the Red-haired
  Foreigners), 1850, 8vo. Doolittle, Foochow, 1872, 4to, vol. i. 550
  pages. FRENCH,--Callery, _Dict. encyclopédique_, Macao and Paris, 1845
  (radicals 1-20 only): M. A. H., 1876, 8vo, autographié, 1730 pages.
  FRENCH-CHIN.--Perny (Fr.-Latin, Spoken Mandarin), Paris, 1869, 4to;
  Appendice, 1770; Lemaire and Giguel, Shanghai, 1874, 16mo.
  PORTUGUESE.--Gonçalves (Port.-Chin.), Macao, 1830, 8vo, 2 vols.: Id.
  (Chin.-Port.), ib. 1833, 8vo. IDIOMS.--Giles, Shanghai, 1873, 4to.
  PHRASES.--Yaou Pei-keen, _Luy yih_, 1742-1765, 8vo, 55 vols.: Tseen
  Ta-hin, _Shing luy_, 1853, 8vo, 4 vols. CLASSICAL EXPRESSIONS.--Keang
  Yang and 30 others, _Sze Shoo teen Lin_, 1795, 8vo, 30 vols. ELEGANT
  EXPRESSIONS.--Chang ting yuh, _Fun luy tsze kin_, 1722, 8vo, 64 vols.
  PHRASES OF THREE WORDS.--Julien (Latin), Paris, 1864, 8vo.
  POETICAL.--_Pei wan she yun_, 1800, 8vo, 5 vols. PROPER NAMES.--F.
  Porter Smith (China, Japan, Corea, Annam, &c., Chinese-Eng.),
  Shanghai, 1870, 8vo. TOPOGRAPHY.--Williams, Canton, 1841, 8vo. NAMES
  OF TOWNS.--Biot, Paris, 1842, 8vo. ANCIENT CHARACTERS.--Foo
  Lwantseang, _Luh shoo fun luy_, 1800, 8vo, 12 vols. SEAL
  CHARACTER.--Heu Shin, _Shwo wan_, ed. Seu Heuen, 1527, 8vo, 12 vols.
  RUNNING HAND.--St Aulaire and Groeneveld (Square Characters, Running
  Hand; Running, Square), Amst. 1861, 4to, 117 pages. TECHNICAL TERMS
  (in Buddhist translations from Sanskrit)--Yuen Ying, _Yih 'see king
  pin e_, 1848, 8vo. DIALECTS.--_Amoy_: Douglas, London, 1873, 4to, 632
  pages: Macgowan, Hong-Kong, 1869, 8vo. _Canton_: Yu Heo-poo and Wan
  ke-shih, _Keang hoo chih tuh fun yun tso yaou ho tseih_, Canton, 1772,
  8vo, 4 vols.; 1803, 8vo, 4 vols.; Fuh-shan, 1833, 8vo, 4 vols.:
  Morrison, Macao, 1828, 8vo: Wan ke shih, Canton, 1856, 8vo: Williams
  (tonic, Eng.-Chinese), Canton, 1856, 8vo: Chalmers, Hong-Kong, 1859,
  12mo; 3rd ed. 1873, 8vo. _Changchow in Fuhkeen_: Seay Sew-lin, _Ya suh
  tung shih woo yin_, 1818, 8vo, 8 vols.; 1820. _Foo-chow_: Tseih (a
  Japanese general) and Lin Peih shan, _Pa yin ho ting_, ed. Tsin Gan,
  1841, 8vo: Maclay and Baldwin, Foochow, 1870, 8vo, 1123 pages.
  _Hok-keen_: Medhurst, Macao, 1832, 4to: _Peking_, Stent, Shanghai,
  1871, 8vo.


 Corean.--CHINESE, COREAN AND JAPANESE.--_Cham Seen Wo Kwo tsze mei_,
  translated by Medhurst, Batavia, 1835, 8vo. RUSSIAN.--Putzillo, St
  Petersburg, 1874, 12mo, 746 pages.


 Japanese.--_Sio Ken Zi Ko_ (Examination of Words and Characters),
  1608, 8vo, 10 vols.: _Wa Kan Won Se Ki Sio Gen Zi Ko_, lithographed by
  Siebold, Lugd. Bat., 1835, fol. JAP.-CHINESE.--_Faga biki set yo siu_.
  CHINESE-JAP.--_Kanghi Tse Tein_, 30 vols. 12mo: _Zi rin gioku ben_.
  DUTCH DICTIONARIES PRINTED BY JAPANESE.--_Nieeu verzameld Japansch en
  Hollandsch Woordenbock_, by the interpreter, B. Sadayok, 1810:
  Minamoto Masataka, Prince of Nakats (Jap. Chinese-Dutch), 5 vols. 4to,
  printed at Kakats by his servants: _Jedo-Halma_ (Dutch-Jap.), Jedo,
  4to, 20 vols.: _Nederduitsche taal_, Dutch Chinese, for the use of
  interpreters. LATIN AND PORTUGUESE.--Calepinus, _Dictionarium_,
  Amacusa, 1595, 4to. LATIN.--Collado, _Compendium_, Romae, 1632, 4to:
  _Lexicon_, Romae, 1870, 4to, from Calepinus. ENGLISH.--Medhurst,
  Batavia, 1830, 8vo: Hepburn, Shanghai, 1867, 8vo; 1872.
  ENG.-JAP.--Hori Tatnoskoy, Yedo, 1862, 8vo; 2nd ed. Yeddo, 1866, 8vo:
  Satow and Ishibashi Masakata (spoken language), London, 1876, 8vo.
  FRENCH.--Rosny (Jap. Fr. Eng.), Paris, 1857, 4to, vol. i.: Pagés,
  Paris, 1869, 4to, translated from Calepinus. FR.-JAP.--Soutcovey,
  Paris, 1864, 8vo. FR. ENG. JAP.--Mermet de Cachon, Paris, 1866, 8vo,
  unfinished. GERMAN.--Pfizmaier (Jap.-Ger., Eng.), Wien, 1851, 4to,
  unfinished. SPANISH.--_Vocabulario del Japon_, Manila, 1630, 4to,
  translated from the next. PORTUGUESE.--_Vocabulario da Lingua de
  Japam_, Nagasaki, 1603, 4to. RUSSIAN.--Goshkevich, St Petersburg,
  1857, 8vo, 487 pages. CHINESE CHARACTERS WITH JAPANESE
  PRONUNCIATION.--Rosny, Paris, 1867, 8vo. CHINESE AND JAPANESE NAMES OF
  PLANTS.--Hoffmann, Leyde, 1864, 8vo.


 Aino.--Pfizmaier, Wien, 1854, 4to.


 Northern and Central Asia.--_Buriat_: Castrén, St Petersburg, 1857,
  8vo. _Calmuck_: Zwick, Villingen, 1853, 4to: Smirnov, Kazan, 1857,
  12mo: Jügl, _Siddhi Kur_, Leipzig, 1866, 8vo. _Chuvash_: Clergy of the
  school of the Kazan Eparchia, Kazan, 1836, 8vo, 2481 words: Lyulé
  (Russ.-Chuv. French), Odessa, 1846, 8vo, 244 pages: Zolotnitski,
  Kazan, 1875, 8vo, 287 pages. _Jagatai_: Mir Ali Shir, _Abuska_, ed.
  Vámbéry, with Hungarian translation, Pesth, 1862, 8vo: Vámbéry,
  Leipzig, 1867, 8vo: Pavet de Courteille, Paris, 1870, 8vo. _Koibal and
  Karagas_: Castrén, St Petersburg, 1857, 8vo. _Manchu_: _Yutchi tseng
  ting tsing wen kian_ (Manchu Chinese), 1771, 4to, 6 vols.: _Sze li hoh
  pik wen kian_ (Manchu-Mongol, Tibetan, Chinese) 10 vols. 4to, the
  Chinese pronunciation represented in Manchu: _San hoh pien lan_
  (Manchu-Chinese, Mongol), 1792, 8vo, 12 vols.;--all three classed
  vocabularies: Langlès (French), Paris, 1789-1790, 4to, 3 vols.:
  Gabelentz (German), Leipzig, 1864, 8vo: Zakharov (Russian), St
  Petersburg, 1875, 8vo, 1235 pages: _Mongol_: I. J. Schmidt (German,
  Russian), St Petersburg, 1835, 4to: Schergin, Kazan, 1841, 8vo:
  Kovalevski, Kasan, 1844-1849, 4to, 3 vols. 2703 pages. _Ostiak_:
  Castrén, St Petersb. 1858, 8vo. _Samoyed_: Castrén, St Petersb. 1855,
  8vo, 308 pages. Tartar: Giganov (Tobolsk), St Petersburg, 1804, 4to;
  (Russ.-Tartar), ib. 1840, 4to: Troyanski (Karan), Kasan, 1835-1855,
  4to. _Tibetan_: _Minggi djamtoo_ (Tibet-Mongol): _Bodschi dajig togpar
  lama_: _Kad shi schand scharwi melonggi jige_
  (Manchu-Mongol-Tibetan-Chinese), Kanghi's Dictionary with the Tibetan
  added in the reign of Khian lung (1736-1795); Csoma de Körös (Eng.),
  Calcutta, 1834, 4to: I. J. Schmidt (German), St Petersburg, 1841, 4to:
  Id. (Russian), ib. 1843, 4to: Jaeschke (Eng.), London, 1870, 8vo, 160
  pages: Id. (Germ.), Gnadau, 1871, 658 pages: (Bhotanta), Schroeter,
  Serampore, 1826, 4to. _Tungusian_: Castrén, St Petersburg, 1856, 8vo,
  632 pages. _Uigur_: Vámbéry, Innspruck, 1870, 4to. _Yakut_: Böhtlingk,
  ib. 1854, 4to, 2 vols. _Yenissei Ostiak_: Castrén, ib. 1849, 8vo.


AFRICA

 Egyptian.--Young (enchorial), London, 1830-1831, 8vo: Sharpe, London,
  1837, 4to: Birch, London. 1838, 4to: Champollion (died March 4, 1832),
  _Dictionnaire égyptien_, Paris, 1841, 4to: Brugsch,
  _Hieroglyphisch-Demotisches Wörterbuch_, Leipzig, 1867-1868, 4to, 4
  vols. 1775 pages, nearly 4700 words, arranged according to the
  hieroglyphic alphabet of 28 letters: Pierret, _Vocabulaire hiérog._,
  Paris, 1875, 8vo, containing also names of persons and places: Birch,
  in vol. v. pp. 337-580 of Bunsen's _Egypt's Place_, 2nd ed. London,
  1867, &c. 8vo, 5010 words. PROPER NAMES.--Brugsch, Berlin, 1851, 8vo,
  726 names: Parthey, ib. 1864, 8vo, about 1500 names: Lieblein,
  Christiania, 1871, 8vo, about 3200 from hieroglyphic texts. BOOK OF
  THE DEAD.--Id., Paris, 1875, 12mo.


 Coptic.--Veyssière de la Croze, Oxon. 1775, 8vo: Rossi, Romae, 1807,
  4to: Tattam, Oxon. 1855, 8vo: Peyron, 1835, 4to (the standard):
  Parthey, Berolini, 1844, 8vo.


 Ethiopic.--Wemmer, Romae, 1638, 4to: Ludolf, London, 1661, 4to:
  Francof. ad M., 1699, fol.: Dillmann (Tigré appendix), Leipzig,
  1863-1865, 4to, 828 pages.


 Amharic.--Ludolphus, Franc. ad Maenum, 1698, fol.: Isenberg, London,
  1841, 4to, 442 pages. _Tigré_: Munzinger, Leipzig, 1865, 8vo:
  Beurmann, ib. 1868, 8vo.


 East Coast.--_Dankali_: Isenberg, London, 1840, 12mo. _Galla_: Krapf,
  London, 1842, 8vo: Tutschek, München, 1844, 8vo. _Engutuk Iloigob_:
  Erhardt, Ludwigsberg, 1857, 8vo. _Kisuaheli_: _Vocabulary of the
  Soahili_, Cambridge, U.S. 1845, 8vo: Steere, London, 1870, 8vo, about
  5800 words. _Kisuaheli, Kinika, Kikamba, Kipokono, Kikian, Kigalla_:
  Krapf, Tübingen, 1850, 8vo.


 Malagasy.--Houtmann (Malaysche en Madagask Talen), Amst. 1603, 2nd ed.
  Matthysz, ib. 1680, 8vo: Huet de Froberville, Isle de France, fol. 2
  vols.: Flacourt, Paris, 1658, 8vo: Challand (Southern), Isle de
  France, 1773, 4to: Freeman and Johns, London, 1835, 8vo, 2 vols.:
  Dalmont (Malgache, Salalave, et Betsimara), 1842, 8vo: Kessler,
  London, 1870, 8vo.


 Southern Africa.--Bleek, _The Languages of Mozambique_, London, 1856,
  8vo. _Kaffre_: Bennie, Lovedale, 1826, 16mo: Ayliffe, Graham's Town,
  1846, 12mo: Appleyard, 1850, 8vo: Bleek, Bonn, 1853, 4to, 646 pages.
  _Zulu-Kaffre_: Perrin (Kaffre-Eng.), London, 1855, 24mo, 172 pages:
  Id. (Eng.-Kaffre), Pietermaritzburg, 1855, 24mo, 227 pages: Id.
  (Eng.-Zulu), ib. 1865, 12mo, 226 pages: Dohne, Cape Town, 1857, 8vo,
  428 pages: Colenso, Pietermaritzburg, 1861, 8vo, 560 pages, about 8000
  words. _Hottentot_: Bleek, Cape Town, 1857, 4to, 261 pages. _Namaqua_:
  Tindall, ib. 1852, 8vo: _Vocabulary_, Barmen, 1854, 8vo: Hahn,
  Leipzig, 1870, 12mo. Sechuana: Casalis, Paris, 1841, 8vo. _Herero_:
  Hahn, Berlin, 1857, 8vo, 207 pages, 4300 words.


 Western Africa.--_Akra_ or _Ga_: Zimmermann, Stuttgart, 1858, 8vo, 690
  pages. _Ashantee_: Christaller (also Akra), Basel, 1874, 8vo, 299
  pages. _Bullom_: Nylander, London, 1814, 12mo. _Bunda or Angola_:
  Cannecatim, Lisboa, 1804, 4to, 722 pages. _Dualla Grammatical
  Elements_, &c., Cameroons, 1855, 8vo. _Efik_ or _Old Calabar_:
  Waddell, Old Calabar, 1846, 16mo, 126 pages; Edinb, 1849, 8vo, 95
  pages. _Eyo_: Raban, London, 1830-1831, 12mo, 2 parts. _Grebo_:
  _Vocabulary_, Cape Palmas, 1837, 8vo; _Dictionary_, ib. 1839, 8vo, 119
  pages. _Ifa_: Schlegel, Stuttgart, 1857, 8vo. _Mpongwe_: De Lorme
  (Franç.-Pongoué), Paris, 1876, 12mo, 354 pages. _Oji_: Riis, Basel,
  1854, 8vo, 284 pages. _Sherbro'_: Schön, _s. a. et l._ 8vo, written in
  1839, 42 pages. _Susu_: Brunton, Edinburgh, 1802, 8vo, 145 pages.
  _Vei_: Koelle, London, 1854, 8vo, 266 pages. _Wolof and Bambarra_:
  Dard, Paris, 1825, 8vo. _Wolof_: Roger, ib. 1829, 8vo: Missionnaires
  de S. Esprit, Dakar, 1855, &c. 16mo. Faidherbe (French-Wolof, Poula
  and Soninke), St Louis, Senegambia, 1860, 12mo. _Yoruba_: Crowther,
  London, 1843, 8vo; 1852, 298 pages: Vidal, ib. 1852, 8vo: Bowen,
  Washington, 1858, 4to.


 Central Africa.--Barth, _Vocabularies_. Gotha, 1862-1866, 4to. _Bari_:
  Mitterreutzner, Brixen, 1867, 8vo: Reinisch, Vienna, 1874, 8vo.
  _Dinka_: Mitterreutzner, Brixen, 1866, 8vo. _Haussa_: Schön (Eng.),
  London, 1843, 8vo.


 Berber.--Venture de Paradis, Paris, 1844, 8vo: Brosselard, ib. 1844,
  8vo: Delaporte, ib. 1844, 4to, by order of the Minister of War:
  Creusat, Franç.-Kabyle (Zouaoua), Alger, 1873, 8vo. _Siwah_: Minutoli,
  Berlin, 1827, 4to.


AUSTRALIA AND POLYNESIA

 Australia.--_New South Wales_: Threlkeld (Lake Macquarie Language),
  Sydney, 1834, 8vo. _Victoria_: Bunce, Melbourne, 1856, 12mo, about
  2200 words. _South Australia_: Williams, South Australia, 1839, 8vo:
  Teichelmann and Schürmann, Adelaide, 1840, 8vo: Meyer, ib. 1843, 8vo.
  _Murray River_: Moorhouse, ib. 1846, 8vo. _Parnkalla_: Schürmann,
  Adelaide, 1844, 8vo. _Woolner District_: _Vocabulary_, ib. 1869, 12mo.
  _Western Australia_: Sir George Grey, Perth, 1839, 4to; London, 1840,
  8vo: Moore, ib. 1843: Brady, Roma, 1845, 24mo, 8vo, 187 pages.
  _Tasmania_: Millegan, Tasmania, 1857.


 Polynesia.--Hale, _Grammars and Vocabularies of all the Polynesian
  Languages_, Philadelphia, 1846, 4to. _Marquesas, Sandwich Gambier_:
  Mosblech, Paris, 1843, 8vo. _Hawaiian_: Andrews, _Vocabulary_,
  Lahainaluna, 1636, 8vo: Id., _Dictionary_, Honolulu, 1865, 8vo, 575
  pages, about 15,500 words. _Marquesas_: Pierquin, de Gembloux,
  Bourges, 1843, 8vo: Buschmann, Berlin, 1843, 8vo. _Samoan_:
  _Dictionary_, Samoa, 1862, 8vo. _Tahitian_: _A Tahitian and English
  Dictionary_, Tahiti, 1851, 8vo, 314 pages. _Tonga_: Rabone, Vavau,
  1845, 8vo. _Fijian_: Hazlewood (Fiji-Eng.), Vewa. 1850, 12mo: Id.
  (Eng.-Fiji), ib. 1852, 12mo: Id., London, 1872, 8vo. _Maori_: Kendall,
  1820, 12mo: Williams, Paihia, 1844, 8vo; 3rd ed. London, 1871, 8vo:
  Taylor, Auckland, 1870, 12mo.


AMERICA

 North America.--_Eskimo_: Washington, London, 1850, 8vo: Petitot
  (Mackenzie and Anderson Rivers), Paris, 1876, 4to. _Kinai_: Radloff,
  St Petersburg, 1874, 4to. _Greenland_: Egede (Gr. Dan. Lat., 3 parts),
  Hafn, 1750, 8vo; 1760, Fabricius, Kjöbenhavn, 1804, 4to. _Hudson's Bay
  Indians_: Bowrey, London, 1701, fol. _Abnaki_: Rasles, Cambridge,
  U.S., 1833, 4to. _Chippewa_: Baraga, Cincinnati, 1853, 12mo, 622
  pages: Petitot, Paris, 1876, 4to, 455 pages. _Massachusetts_ or
  _Natick_: Cotton, Cambridge, U.S. 1829, 8vo. _Onondaga_: Shea
  (French-Onon.), from a MS. (of 17th century), London, 1860, 4to, 109
  pages. _Dacota_: Riggs, New York, 1851, 4to, 424 pages: Williamson
  (Eng. Dac.), Santos Agency, Nebraska, 12mo, 139 pages. _Mohawk_:
  Bruyas, New York, 1863, 8vo. _Hidatsa (Minnetarees, Gros Ventres of
  the Missouri)_: Matthews, ib. 1874, 8vo. _Choctaw_: Byington, ib.
  1852, 16mo. _Clallam and Lummi_: Gibbs, ib. 1863, 8vo. _Yakama_:
  Pandosy, translated by Gibbs and Shea, ib. 1862, 8vo. _Chinook_:
  Gibbs, New York, 1863, 4to. _Chinook Jargon, the trade language of
  Oregon_: Id., ib. 1863, 8vo. _Tatche_ or _Telamé_: Sitjar, ib. 1841,
  8vo.


 Mexico and Central America.--_Tepehuan_: Rinaldini, Mexico, 1743, 4to.
  _Cora_: Ortega, Mexico, 1732, 4to. _Tarahumara_: Steffel, Brünn, 1791,
  8vo. _Otomi_: Carochi, Mexico, 1645, 4to: Neve y Molina, ib. 1767,
  8vo: Yepes, ib. 1826, 4to: Piccolomini, Roma, 1841, 8vo. _Mexican_ or
  _Aztec_: Molina, Mexico, 1555, 4to; 1571, fol. 2 vols.: Arenas, ib.
  1583; 1611, 8vo; 1683; 1725; 1793, 12mo: Biondelli, Milan, 1869, fol.
  _Mexican, Tontonacan, and Huastecan_: Olmos, Mexico, 1555-1560, 4to, 2
  vols. _Huastecan_: Tapia Zenteno, ib. 1767, 4to, 128 pages. _Opata_ or
  _Tequima_: Lombardo, ib. 1702, 4to. _Tarasca_: Gilberti, ib. 1559,
  4to: Lagunas, ib. 1574, 8vo. _Mixtecan_: Alvarado, Mexico, 1593, 4to.
  _Zapoteca_: Cordova, ib. 1578, 4to. _Maya_: Beltran de Santa Rosa
  Maria, ib. 1746, 4to; Merida de Yucatan, 1859, 4to, 250 pages:
  Brasseur de Bourbourg, Paris, 1874, 8vo, 745 pages. _Quiché_: Id.
  (also Cakchiquel and Trutuhil dialects), ib. 1862, 8vo.


 South America.--_Chibcha_: Uricoechea, Paris, 1871, 8vo. _Chayma_:
  Tauste, Madrid, 1680, 4to: Yanguas, Burgos, 1683, 4to. _Carib_:
  Raymond, Auxerre, 1665-1666, 8vo. _Galibi_: D.[e]. L.[a] S.[auvage],
  Paris, 1763, 8vo. _Tupi_: Costa Rubim, Rio de Janeiro, 1853, 8vo:
  Silva Guimaräes, Bahia, 1854, 8vo: Diaz, Lipsia, 1858, 16mo.
  _Guarani_: Ruiz de Montoyo, Madrid, 1639, 4to; 1640; 1722, 4to; ed.
  Platzmann, Leipzig, 1876, &c., 8vo, to be in 4 vols. 1850 pages.
  _Moxa_: Marban, Lima, 1701, 8vo. _Lule_: Machoni de Corderia, Madrid,
  1732, 12mo. _Quichua_: Santo Thomas, Ciudad de los Reyes, 1586, 8vo:
  Torres Rubio, Sevilla, 1603, 8vo; Lima, 1609, 8vo; ed. Figueredo,
  Lima, 1754, 8vo; Holguin, Ciudad de los Reyes, 1608, 8vo: Tschudi,
  Wien, 1853, 8vo, 2 vols.: Markham, London, 1864, 8vo: Lopez, _Les
  Races Aryennes de Perou_, Paris, 1871, 8vo, comparative vocabulary,
  pp. 345-421. _Aymara_: Bertonio, Chicuyto, 1612, 4to, 2 vols.
  _Chileno_: Valdivia (also Allentiac and Milcocayac), Lima, 1607, 8vo:
  Febres, ib. 1765, 12mo; ed. Hernandez y Caluza, Santiago, 1846, 8vo, 2
  vols. _Tsonecan_ (Patagonian): Schmid, Bristol, 1860, 12mo.

The above article incorporates the salient features of the 9th-edition
article by the Rev. Ponsonby A. Lyons, and the 10th-edition article by
Benjamin E. Smith.


FOOTNOTES:

  [1] Joannes de Garlandia (John Garland; fl. 1202-1252) gives the
    following explanation in his _Dictionarius_, which is a classed
    vocabulary:--"Dictionarius dicitur libellus iste a dictionibus magis
    necessariis, quas tenetur quilibet scolaris, non tantum in scrinio de
    lignis facto, sed in cordis armariolo firmiter retinere." This has
    been supposed to be the first use of the word.

  [2] An excellent dictionary of quotations, perhaps the first of the
    kind; a large folio volume printed in Strassburg about 1475 is
    entitled "Pharetra auctoritates et dicta doctorum, philosophorum, et
    poetarum continens."

  [3] This volume was issued with a new title-page as _Glossaire du
    moyen âge_, Paris, 1872.



DICTYOGENS (Gr. [Greek: diktyon], a net, and the termination [Greek:
-genês], produced), a botanical name proposed by John Lindley for a
class including certain families of Monocotyledons which have net-veined
leaves. The class was not generally recognized.



DICTYS CRETENSIS, of Cnossus in Crete, the supposed companion of
Idomeneus during the Trojan War, and author of a diary of its events.
The MS. of this work, written in Phoenician characters, was said to have
been found in his tomb (enclosed in a leaden box) at the time of an
earthquake during the reign of Nero, by whose order it was translated
into Greek. In the 4th century A.D. a certain Lucius Septimius brought
out _Dictys Cretensis Ephemeris belli Trojani_, which professed to be a
Latin translation of the Greek version. Scholars were not agreed whether
any Greek original really existed; but all doubt on the point was
removed by the discovery of a fragment in Greek amongst the papyri found
by B. P. Grenfell and A. S. Hunt in 1905-1906. Possibly the Latin
Ephemeris was the work of Septimius himself. Its chief interest lies in
the fact that (together with Dares Phrygius's _De excidio Trojae_) it
was the source from which the Homeric legends were introduced into the
romantic literature of the middle ages.

  Best edition by F. Meister (1873), with short but useful introduction
  and index of Latinity; see also G. Körting, _Diktys und Dares_
  (1874), with concise bibliography; H. Dunger, _Die Sage vom
  trojanischen Kriege in den Bearbeitungen des Mittelalters und ihren
  antiken Quellen_ (1869, with a literary genealogical table); E.
  Collilieux, _Étude sur Dictys de Crète et Darès de Phrygie_ (1887),
  with bibliography; W. Greif, "Die mittelalterlichen Bearbeitungen der
  Trojanersage," in E. M. Stengel's _Ausgaben und Abhandlungen aus dem
  Gebiete der romanischen Philologie_, No. 61 (1886, esp. sections 82,
  83, 168-172); F. Colagrosso, "Ditte Cretese" in _Atti della r.
  Accademia di Archeologia_ (Naples, 1897, vol. 18, pt. ii. 2); F.
  Noack, "Der griechische Dictys," in _Philologus_, supp. vi. 403 ff.;
  N. E. Griffin, _Dares and Dictys, Introduction to the Study of the
  Medieval Versions of the Story of Troy_ (1907).



DICUIL (fl. 825), Irish monastic scholar, grammarian and geographer. He
was the author of the _De mensura orbis terrae_, finished in 825, which
contains the earliest clear notice of a European discovery of and
settlement in Iceland and the most definite Western reference to the old
freshwater canal between the Nile and the Red Sea, finally blocked up in
767. In 795 (February 1-August 1) Irish hermits had visited Iceland; on
their return they reported the marvel of the perpetual day at midsummer
in "Thule," where there was then "no darkness to hinder one from doing
what one would." These eremites also navigated the sea north of Iceland
on their first arrival, and found it ice-free for one day's sail, after
which they came to the ice-wall. Relics of this, and perhaps of other
Irish religious settlements, were found by the permanent Scandinavian
colonists of Iceland in the 9th century. Of the old Egyptian freshwater
canal Dicuil learnt from one "brother Fidelis," probably another Irish
monk, who, on his way to Jerusalem, sailed along the "Nile" into the Red
Sea--passing on his way the "Barns of Joseph" or Pyramids of Giza, which
are well described. Dicuil's knowledge of the islands north and west of
Britain is evidently intimate; his references to Irish exploration and
colonization, and to (more recent) Scandinavian devastation of the same,
as far as the Faeroes, are noteworthy, like his notice of the elephant
sent by Harun al-Rashid (in 801) to Charles the Great, the most curious
item in a political and diplomatic intercourse of high importance.
Dicuil's reading was wide; he quotes from, or refers to, thirty Greek
and Latin writers, including the classical Homer, Hecataeus, Herodotus,
Thucydides, Virgil, Pliny and King Juba, the sub-classical Solinus, the
patristic St Isidore and Orosius, and his contemporary the Irish poet
Sedulius;--in particular, he professes to utilize the alleged surveys of
the Roman world executed by order of Julius Caesar, Augustus and
Theodosius (whether Theodosius the Great or Theodosius II. is
uncertain). He probably did not know Greek; his references to Greek
authors do not imply this. Though certainly Irish by birth, it has been
conjectured (from his references to Sedulius and the caliph's elephant)
that he was in later life in an Irish monastery in the Frankish empire.
Letronne inclines to identify him with Dicuil or Dichull, abbot of
Pahlacht, born about 760.

  There are seven chief MSS. of the _De mensura_ (Dicuil's tract on
  grammar is lost); of these the earliest and best are (1) Paris,
  National Library, Lat. 4806; (2) Dresden, Regius D. 182; both are of
  the 10th century. Three editions exist: (1) C. A. Walckenaer's, Paris,
  1807; (2) A. Letronne's, Paris, 1814, best as to commentary; (3) G.
  Parthey's, Berlin, 1870, best as to text. See also C. R. Beazley,
  _Dawn of Modern Geography_ (London, 1897), i. 317-327, 522-523, 529;
  T. Wright, _Biographia Britannica literaria, Anglo-Saxon Period_
  (London, 1842), pp. 372-376.     (C. R. B.)



DIDACHE, THE, or _Teaching of the (twelve) Apostles_,--the most
important of the recent recoveries in the region of early Christian
literature (see APOCRYPHAL LITERATURE). It was previously known by name
from lists of canonical and extra-canonical books compiled by Eusebius
and other writers. Moreover, it had come to be suspected by several
scholars that a lost book, variously entitled _The Two Ways_ or _The
Judgment of Peter_, had been freely used in a number of works, of which
mention must presently be made. In 1882 a critical reconstruction of
this book was made by Adam Krawutzcky with marvellous accuracy, as was
shown when in the very next year the Greek bishop and metropolitan,
Philotheus Bryennius, published _The Teaching of the Twelve Apostles_
from the same manuscript from which he had previously published the
complete form of the Epistle of Clement.[1]

_The Didach[=e]_, as we now have it in the Greek, falls into two marked
divisions: (a) a book of moral precepts, opening with the words, "There
are two ways"; (b) a manual of church ordinances, linked on to the
foregoing by the words, "Having first said all these things, baptize,
&c." Each of these must be considered separately before we approach the
question of the locality and date of the whole book in its present form.

1. _The Two Ways._--The author of the complete work, as we now have it,
has modified the original _Two Ways_ by inserting near the beginning a
considerable section containing, among other matter, passages from the
Sermon on the Mount, in which the language of St Matthew's Gospel is
blended with that of St Luke's. He has also added at the close a few
sentences, beginning, "If thou canst not bear (the whole yoke of the
Lord), bear what thou canst" (vi. 2); and among minor changes he has
introduced, in dealing with confession, reference to "the church" (iv.
14). No part of this matter is to be found in the following documents,
which present us in varying degrees of accuracy with _The Two Ways_:
(i.) the Epistle of Barnabas, chaps. xix., xx. (in which the order of
the book has been much broken up, and a good deal has been omitted);
(ii.) the _Ecclesiastical Canons of the Holy Apostles_, usually called
the _Apostolic Church Order_, a book which presents a parallel to the
_Teaching_, in so far as it consists first of a form of _The Two Ways_,
and secondly of a number of church ordinances (here, however, as in the
Syriac _Didascalia_, which gives about the same amount of _The Two
Ways_, various sections are ascribed to individual apostles, e.g. "John
said, There are two ways," &c.); (iii.) a discourse of the Egyptian monk
Schnudi (d. 451), preserved in Arabic (see Iselin, _Texte u. Unters._,
1895); (iv.) a Latin version, of which a fragment was published by O.
von Gebhardt in 1884, and the whole by J. Schlecht in 1900. When by the
aid of this evidence _The Two Ways_ is restored to us free of glosses,
it has the appearance of being a Jewish manual which has been carried
over into the use of the Christian church. This is of course only a
probable inference; there is no prototype extant in Jewish literature,
and, comparing the moral (non-doctrinal) instruction for Christian
catechumens in Hermas, _Shepherd_ (_Mand._ i.-ix.), no real need to
assume one. There was a danger of admitting Gentile converts to the
church on too easy moral terms; hence the need of such insistence on the
ideal as in The Two Ways and the _Mandates_. The recent recovery of the
Latin version is of singular interest, as showing that, even without the
distinctively Christian additions and interpolations which our full form
of the _Teaching_ presents, it was circulating under the title _Doctrina
apostolorum_.[2]

2. The second part of our _Teaching_ might be called a church directory.
It consists of precepts relating to church life, which are couched in
the second person plural; whereas _The Two Ways_ uses throughout the
second person singular. It appears to be a composite work. First (vii.
1-xi. 2) is a short sacramental manual intended for the use of local
elders or presbyters, though such are not named, for they were not yet a
distinctive order or clergy. This section was probably added to _The Two
Ways_ before the addition of the remainder. It orders baptism in the
threefold name, making a distinction as to waters which has Jewish
parallels, and permitting a threefold pouring on the head, if sufficient
water for immersion cannot be had. It prescribes a fast before baptism
for the baptizer as well as the candidate. Fasts are to be kept on
Wednesday and Friday, not Monday and Thursday, which are the fast days
of "the hypocrites," i.e. by a perversion of the Lord's words, the Jews.
"Neither pray ye as the hypocrites; but as the Lord commanded in His
Gospel." Then follows the Lord's Prayer, almost exactly as in St
Matthew, with a brief doxology--"for Thine is the power and the glory
forever." This is to be said three times a day. Next come three
eucharistic prayers, the language of which is clearly marked off from
that of the rest of the book, and shows parallels with the diction of St
John's Gospel. They are probably founded on Jewish thanksgivings, and it
is of interest to note that a portion of them is prescribed as a grace
before meat in (pseudo-) Athanasius' _De virginitate_. A trace of them
is found in one of the liturgical prayers of Serapion, bishop of Thmui,
in Egypt, but they have left little mark on the liturgies of the church.
As in Ignatius and other early writers, the eucharist, a real meal (x.
1) of a family character, is regarded as producing immortality (cf.
"spiritual food and drink and eternal life"). None are to partake of it
save those who have been "baptized in the name of the Lord" (an
expression which is of interest in a document which prescribes the
threefold formula). The prophets are not to be confined to these forms,
but may "give thanks as much as they will." This appears to show that a
prophet, if present, would naturally preside over the eucharist. The
next section (xi. 3-xiii.) deals with the ministry of spiritual gifts as
exercised by apostles, prophets and teachers. An apostle is to be
"received as the Lord"; but he must follow the Gospel precepts, stay but
one or two days, and take no money, but only bread enough for a day's
journey. Here we have that wider use of the term "apostle" to which
Lightfoot had already drawn attention. A prophet, on the contrary, may
settle if he chooses, and in that case he is to receive tithes and
first-fruits; "for they are your high priests." If he be once approved
as a true prophet, his words and acts are not to be criticized; for this
is the sin that shall not be forgiven. Next comes a section (xiv., xv.)
reflecting a somewhat later development concerning fixed services and
ministry; the desire for a stated service, and the need of regular
provision for it, is leading to a new order of things. The eucharist is
to be celebrated every Lord's Day, and preceded by confession of sins,
"that your sacrifice may be pure ... for this is that sacrifice which
was spoken of by the Lord, In every place and time to offer unto Me a
pure sacrifice. Appoint therefore unto yourselves bishops and deacons,
worthy of the Lord, men meek and uncovetous, and true and approved; for
they also minister unto you the ministration of the prophets and
teachers. Therefore despise them not; for they are your honoured ones,
together with the prophets and teachers." This is an arrangement
recommended by one who has tried it, and he reassures the old-fashioned
believer who clings to the less formal régime (and whose protest was
voiced in the Montanist movement), that there will be no spiritual loss
under the new system. The book closes (chap. xvi.) with exhortations to
steadfastness in the last days, and to the coming of the
"world-deceiver" or Antichrist, which will precede the coming of the
Lord. This section is perhaps the actual utterance of a Christian
prophet, and may be of earlier origin than the two preceding sections.

3. It will now be clear that indications of the locality and date of our
present _Teaching_ must be sought for only in the second part, and in
the Christian interpolations in the first part. We have no ground for
thinking that the second part ever existed independently as a separate
book. The whole work was in the hands of the writer of the seventh book
of the _Apostolic Constitutions_, who embodies almost every sentence of
it, interspersing it with passages of Scripture, and modifying the
precepts of the second part to suit a later (4th-century) stage of
church development; this writer was also the interpolator of the
Epistles of Ignatius, and belonged to the Syrian Church. Whether the
second part was known to the writer of the _Apostolic Church Order_ is
not clear, as his only quotation of it comes from one of the eucharistic
prayers. The allusions of early writers seem to point to Egypt, but
their references are mostly to the first part, so that we must be
careful how we argue from them as to the provenance of the book as a
whole. Against Egypt has been urged the allusion in one of the
eucharistic prayers to "corn upon the mountains." This is found in the
Prayer-book of Serapion (c. 350) but omitted in a later Egyptian
prayer; the form as we have it in _The Didach[=e]_ may have passed into
Egypt with the authority of tradition which was afterwards weakened. The
anti-Jewish tone of the second part suggests the neighbourhood of Jews,
from whom the Christians were to be sharply distinguished. Either Egypt
or Syria would satisfy this condition, and in favour of Syria is the
fact that the presbyterate there was to a late date regarded as a rank
rather than an office. If we can connect the injunctions (vi. 3)
concerning (abstinence from certain) food and that which is offered to
idols with the old trouble that arose at Antioch (Acts xv. 1) and was
legislated for by the Jerusalem council, we have additional support for
the Syrian claim. But all that we can safely say as to locality is that
the community here represented seems to have been isolated, and out of
touch with the larger centres of Christian life.

This last consideration helps us in discussing the question of date. For
such an isolated community may have preserved primitive customs for some
time after they had generally disappeared. Certainly the stage of
development is an early one, as is shown, e.g., by the prominence of
prophets, and the need that was felt for the vindication of the position
of the bishops and deacons (there is no mention at all of presbyters);
moreover, there is no reference to a canon of Scripture (though the
written Gospel is expressly mentioned) or to a creed. On the other hand
the "apostles" of the second part are obviously not "the twelve
apostles" of the title; and the prophets seem in some instances to have
proved unworthy of their high position. The ministry of enthusiasm which
they represent is about to give way to the ministry of office, a
transition which is reflected in the New Testament in the 3rd Epistle of
John. Three of the Gospels have clearly been for some time in
circulation; St Matthew's is used several times, and there are phrases
which occur only in St Luke's, while St John's Gospel lies behind the
eucharistic prayers which the writer has embodied in his work. There are
no indications of any form of doctrinal heresy as needing rebuke; the
warnings against false teaching are quite general. While the first part
must be dated before the Epistle of Barnabas, i.e. before A.D. 90, it
seems wisest not to place the complete work much earlier than A.D. 120,
and there are passages which may well be later.

  A large literature has sprung up round The _Didach[=e]_ since 1884.
  Harnack's edition in _Texte u. Unters._ vol. ii. (1884) is
  indispensable to the student; and his discussions in _Altchristl.
  Litteratur_ and _Chronologie_ give clear summaries of his work. Other
  editions of the text are those of F. X. Funk, _Patres Apostolici_,
  vol. i. (Tübingen, 1901); H. Lietzmann (Bonn, 1903; with Latin
  version). Dr J. E. Odgers has published an English translation with
  introduction and notes (London, 1906). Dr C. Taylor in 1886 drew
  attention to some important parallels in Jewish literature; his
  edition contains an English translation. Dr Rendel Harris published in
  1887 a complete facsimile, and gathered a great store of patristic
  illustration. Text and translation will also be found in Lightfoot's
  _Apostolic Fathers_ (ed. min.) The fullest critical treatment in
  English is by Dr Vernon Bartlet in the extra volume of Hastings's
  _Dictionary of the Bible_; the most complete commentary on the text is
  by P. Drews in Hennecke's _Handbuch zu den N.T. Apocryphen_ (1904).
  Other references to the literature may be found by consulting
  Harnack's _Altchristl. Litteratur_.


FOOTNOTES:

  [1] The MS. was found in the Library of the Jerusalem Monastery of
    the Most Holy Sepulchre, in Phanar, the Greek quarter of
    Constantinople. It is a small octavo volume of 120 parchment leaves,
    written throughout by Leo, "notary and sinner," who finished his task
    on the 11th of June 1156. Besides The _Didach[=e]_ and the Epistles
    of Clement it contains several spurious Ignatian epistles.

  [2] The word _twelve_ had no place in the original title and was
    inserted when the original _Didach[=e]_ or _Teaching_ (e.g. _The Two
    Ways_) was combined with the church manual which mentions apostles
    outside of the twelve. It may be noted that the division of the
    _Didach[=e]_ into chapters is due to Bryennius, that into verses to
    A. Harnack.



DIDACTIC POETRY, that form of verse the aim of which is, less to excite
the hearer by passion or move him by pathos, than to instruct his mind
and improve his morals. The Greek word [Greek: didaktikos] signifies a
teacher, from the verb [Greek: didaskein], and poetry of the class under
discussion approaches us with the arts and graces of a schoolmaster. At
no time was it found convenient to combine lyrical verse with
instruction, and therefore from the beginning of literature the didactic
poets have chosen a form approaching the epical. Modern criticism, which
discourages the epic, and is increasingly anxious to limit the word
"poetry" to lyric, is inclined to exclude the term "didactic poetry"
from our nomenclature, as a phrase absurd in itself. It is indeed more
than probable that didactic verse is hopelessly obsolete. Definite
information is now to be found in a thousand shapes, directly and boldly
presented in clear and technical prose. No farmer, however elegant,
will, any longer choose to study agriculture in hexameters, or even in
Tusser's shambling metre. The sciences and the professions will not
waste their time on methods of instruction which must, from their very
nature, be artless, inexact and vague. But in the morning of the world,
those who taught with authority might well believe that verse was the
proper, nay, the only serious vehicle of their instruction. What they
knew was extremely limited, and in its nature it was simple and
straightforward; it had little technical subtlety; it constantly lapsed
into the fabulous and the conjectural. Not only could what early sages
knew, or guessed, about astronomy and medicine and geography be
conveniently put into rolling verse, but, in the absence of all written
books, this was the easiest way in which information could be made
attractive to the ear and be retained by the memory.

In the prehistoric dawn of Greek civilization there appear to have been
three classes of poetry, to which the literature of Europe looks back as
to its triple fountain-head. There were romantic epics, dealing with the
adventures of gods and heroes; these Homer represents. There were mystic
chants and religious odes, purely lyrical in character, of which the
best Orphic Hymns must have been the type. And lastly there was a great
body of verse occupied entirely with increasing the knowledge of
citizens in useful branches of art and observation; these were the
beginnings of didactic poetry, and we class them together under the dim
name of Hesiod. It is impossible to date these earliest didactic poems,
which nevertheless set the fashion of form which has been preserved ever
since. The _Works and Days_, which passes as the direct masterpiece of
Hesiod (q.v.), is the type of all the poetry which has had education as
its aim. Hesiod is supposed to have been a tiller of the ground in a
Boeotian village, who determined to enrich his neighbours' minds by
putting his own ripe stores of useful information into sonorous metre.
Historically examined, the legend of Hesiod becomes a shadow, but the
substance of the poems attributed to him remains. The genuine parts of
the _Works and Days_, which Professor Gilbert Murray has called "a slow,
lowly, simple poem," deal with rules for agriculture. The _Theogony_ is
an annotated catalogue of the gods. Other poems attributed to Hesiod,
but now lost, were on astronomy, on auguries by birds, on the character
of the physical world; still others seem to have been genealogies of
famous women. All this mass of Boeotian verse was composed for
educational purposes, in an age when even preposterous information was
better than no knowledge at all. In slightly later times, as the Greek
nation became better supplied with intellectual appliances, the stream
of didactic poetry flowed more and more closely in one, and that a
theological, channel. The great poem of Parmenides _On Nature_ and those
of Empedocles exist only in fragments, but enough remains to show that
these poets carried on the didactic method in mythology. Cleostratus of
Tenedos wrote an astronomical poem in the 6th century, and Periander a
medical one in the 4th, but didactic poetry did not flourish again in
Greece until the 3rd century, when Aratus, in the Alexandrian age, wrote
his famous _Phenomena_, a poem about things seen in the heavens. Other
later Greek didactic poets were Nicander, and perhaps Euphorion.

It was from the hands of these Alexandrian writers that the genius of
didactic poetry passed over to Rome, since, although it is possible that
some of the lost works of the early republic, and in particular those of
Ennius, may have possessed an educational character, the first and by
far the greatest didactic Latin poet known to us is Lucretius. A highly
finished translation by Cicero into Latin hexameters of the principal
works of Aratus is believed to have drawn the attention of Lucretius to
this school of Greek poetry, and it was not without reference to the
Greeks, although in a more archaic and far purer taste, that he
composed, in the 1st century before Christ, his magnificent _De rerum
natura_. By universal consent, this is the noblest didactic poem in the
literature of the world. It was intended to instruct mankind in the
interpretation and in the working of the system of philosophy revealed
by Epicurus, which at that time was exciting the sympathetic attention
of all classes of Roman society. What gave the poem of Lucretius its
extraordinary interest, and what has prolonged and even increased its
vitality, was the imaginative and illustrative insight of the author,
piercing and lighting up the recesses of human experience. On a lower
intellectual level, but of a still greater technical excellence, was the
_Georgics_ of Virgil, a poem on the processes of agriculture, published
about 30 B.C. The brilliant execution of this famous work has justly
made it the type and unapproachable standard of all poetry which desires
to impart useful information in the guise of exquisite literature.
Himself once a farmer on the banks of the Mincio, Virgil, at the apex of
his genius, set himself in his Campanian villa to recall whatever had
been essential in the agricultural life of his boyish home, and the
result, in spite of the ardours of the subject, was what J. W. Mackail
has called "the most splendid literary production of the Empire." In the
rest of surviving Latin didactic poetry, the influence and the imitation
of Virgil and Lucretius are manifest. Manilius, turning again to
Alexandria, produced a fine _Astronomica_ towards the close of the reign
of Augustus. Columella, regretting that Virgil had omitted to sing of
gardens, composed a smooth poem on horticulture. Natural philosophy
inspired Lucilius junior, of whom a didactic poem on Etna survives. Long
afterwards, under Diocletian, a poet of Carthage, Nemesianus, wrote in
the manner of Virgil the _Cynegetica_, a poem on hunting with dogs,
which has had numerous imitations in later European literatures. These
are the most important specimens of didactic poetry which ancient Rome
has handed down to us.

In Anglo-Saxon and early English poetic literature, and especially in
the religious part of it, an element of didacticism is not to be
overlooked. But it would be difficult to say that anything of importance
was written in verse with the sole purpose of imparting information,
until we reach the 16th century. Some of the later medieval allegories
are didactic or nothing. The first poem, however, which we can in any
reasonable way compare with the classic works of which we have been
speaking is the _Hundreth Pointes of Good Husbandrie_, published in 1557
by Thomas Tusser; these humble Georgics aimed at a practical description
of the whole art of English farming. Throughout the early part of the
17th century, when our national poetry was in its most vivid and
brilliant condition, the last thing a poet thought of doing was the
setting down of scientific facts in rhyme. We come across, however, one
or two writers who were as didactic as the age would permit them to be,
Samuel Daniel with his philosophy, Fulke Greville, Lord Brooke with his
"treatises" of war and monarchy. After the Restoration, as the lyrical
element rapidly died out of English poetry, there was more and more room
left for educational rhetoric in verse. The poems about prosody, founded
upon Horace, and signed by John Sheffield, 3rd earl of Mulgrave
(1648-1721), and Lord Roscommon, were among the earliest purely didactic
verse-studies in English. John Philips deserves a certain pre-eminence,
as his poem called Cyder, in 1706, set the fashion which lasted all down
the 18th century, of writing precisely in verse about definite branches
of industry or employment. None of the greater poets of the age of Anne
quite succumbed to the practice, but there is a very distinct flavour of
the purely didactic about a great deal of the verse of Pope and Gay. In
such productions as Gilbert West's (1703-1756) _Education_, Dyer's
_Fleece_, and Somerville's _Chase_, we see technical information put
forward as the central aim of the poet. Instead of a passionate
pleasure, or at least an uplifted enthusiasm, being the poet's object,
he frankly admits that, first and foremost, he has some facts about wool
or dogs or schoolmasters which he wishes to bring home to his readers,
and that, secondly, he consents to use verse, as brilliantly as he can,
for the purpose of gilding the pill and attracting an unwilling
attention. As we descend the 18th century, these works become more and
more numerous, and more dry, especially when opposed by the descriptive
and rural poets of the school of Thomson, the poet of _The Seasons_. But
Thomson himself wrote a huge poem of _Liberty_ (1732), for which we have
no name if we must not call it didactic. Even Gray began, though he
failed to finish, a work of this class, on _The Alliance of Education
and Government_. These poems were discredited by the publication of _The
Sugar-Cane_ (1764), a long verse-treatise about the cultivation of sugar
by negroes in the West Indies, by James Grainger (1721-1766), but,
though liable to ridicule, such versified treatises continued to
appear. Whether so great a writer as Cowper is to be counted among the
didactic poets is a question on which readers of _The Task_ may be
divided; this poem belongs rather to the class of descriptive poetry,
but a strong didactic tendency is visible in parts of it. Perhaps the
latest frankly educational poem which enjoyed a great popularity was
_The Course of Time_ by Robert Pollok (1798-1827), in which a system of
Calvinistic divinity is laid down with severity and in the pomp of blank
verse. This kind of literature had already been exposed, and
discouraged, by the teaching of Wordsworth, who had insisted on the
imperative necessity of charging all poetry with imagination and
passion. Oddly enough, _The Excursion_ of Wordsworth himself is perhaps
the most didactic poem of the 19th century, but it must be acknowledged
that his influence, in this direction, was saner than his practice.
Since the days of Coleridge and Shelley it has been almost impossible to
conceive a poet of any value composing in verse a work written with the
purpose of inculcating useful information.

The history of didactic poetry in France repeats, in great measure, but
in drearier language, that of England. Boileau, like Pope, but with a
more definite purpose as a teacher, offered instruction in his _Art
poétique_ and in his _Epistles_. But his doctrine was always literary,
not purely educational. At the beginning of the 18th century, the
younger Racine (1692-1763) wrote sermons in verse, and at the close of
it the Abbé Delille (1738-1813) tried to imitate Virgil in poems about
horticulture. Between these two there lies a vast mass of verse written
for the indulgence of intellect rather than at the dictates of the
heart; wherever this aims at increasing knowledge, it at once becomes
basely and flatly didactic. There is nothing in French literature of the
transitional class that deserves mention beside _The Task_ or _The
Excursion_.

During the century which preceded the Romantic revival of poetry in
Germany, didactic verse was cultivated in that country on the lines of
imitation of the French, but with a greater dryness and on a lower level
of utility. Modern German literature began with Martin Opitz (1597-1639)
and the Silesian School, who were in their essence rhetorical and
educational, and who gave their tone to German verse. Albrecht von
Haller (1708-1777) brought a very considerable intellectual force to
bear on his huge poems, _The Origin of Evil_, which was theological, and
_The Alps_ (1729), botanical and topographical. Johann Peter Uz
(1720-1796) wrote a _Theodicée_, which was very popular, and not without
dignity. Johann Jacob Dusch (1725-1787) undertook to put _The Sciences_
into the eight books of a great didactic poem. Tiedge (1752-1840) was
the last of the school; in a once-famous _Urania_, he sang of God and
Immortality and Liberty. These German pieces were the most unswervingly
didactic that any modern European literature has produced. There was
hardly the pretence of introducing into them descriptions of natural
beauty, as the English poets did, or of grace and wit like the French.
The German poets simply poured into a lumbering mould of verse as much
solid information and direct instruction as the form would hold.

Didactic poetry has, in modern times, been antipathetic to the spirit of
the Latin peoples, and neither Italian nor Spanish literature has
produced a really notable work in this class. An examination of the
poems, ancient and modern, which have been mentioned above, will show
that from primitive times there have been two classes of poetic work to
which the epithet didactic has been given. It is desirable to
distinguish these a little more exactly. One is the pure instrument of
teaching, the poetry which desires to impart all that it knows about the
growing of cabbages or the prevention of disasters at sea, the
revolution of the planets or the blessings of inoculation. This is
didactic poetry proper, and this, it is almost certain, became
irrevocably obsolete at the close of the 18th century. No future Virgil
will give the world a second _Georgics_. But there is another species
which it is very improbable that criticism has entirely dislodged; that
is the poetry which combines, with philosophical instruction, an impetus
of imaginative movement, and a certain definite cultivation of fire and
beauty. In hands so noble as those of Lucretius and Goethe this species
of didactic poetry has enriched the world with durable masterpieces,
and, although the circle of readers which will endure scientific
disquisition in the bonds of verse grows narrower and narrower, it is
probable that the great poet who is also a great thinker will now and
again insist on being heard. In Sully-Prudhomme France has possessed an
eminent writer whose methods are directly instructive, and both _La
Justice_ (1878) and _Le Bonheur_ (1888) are typically didactic poems.
Perhaps future historians may name these as the latest of their class.
     (E. G.)



DIDEROT, DENIS (1713-1784), French man of letters and encyclopaedist,
was born at Langres on the 5th of October 1713. He was educated by the
Jesuits, like most of those who afterwards became the bitterest enemies
of Catholicism; and, when his education was at an end, he vexed his
brave and worthy father's heart by turning away from respectable
callings, like law or medicine, and throwing himself into the vagabond
life of a bookseller's hack in Paris. An imprudent marriage (1743) did
not better his position. His wife, Anne Toinette Champion, was a devout
Catholic, but her piety did not restrain a narrow and fretful temper,
and Diderot's domestic life was irregular and unhappy. He sought
consolation for chagrins at home in attachments abroad, first with a
Madame Puisieux, a fifth-rate female scribbler, and then with Sophie
Voland, to whom he was constant for the rest of her life. His letters to
her are among the most graphic of all the pictures that we have of the
daily life of the philosophic circle in Paris. An interesting contrast
may be made between the Bohemianism of the famous English literary set
who supped at the Turk's Head with the Tory Johnson and the Conservative
Burke for their oracles, and the Bohemianism of the French set who about
the same time dined once a week at the baron D'Holbach's, to listen to
the wild sallies and the inspiring declamations of Diderot. For Diderot
was not a great writer; he stands out as a fertile, suggestive and
daring thinker, and a prodigious and most eloquent talker.

Diderot's earliest writings were of as little importance as Goldsmith's
_Enquiry into the State of Polite Learning_ or Burke's _Abridgement of
English History_. He earned 100 crowns by translating Stanyan's _History
of Greece_ (1743); with two colleagues he produced a translation of
James's _Dictionary of Medicine_ (1746-1748) and about the same date he
published a free rendering of Shaftesbury's _Inquiry Concerning Virtue
and Merit_ (1745), with some original notes of his own. With strange and
characteristic versatility, he turned from ethical speculation to the
composition of a volume of stories, the _Bijoux indiscrets_ (1748),
gross without liveliness, and impure without wit. In later years he
repented of this shameless work, just as Boccaccio is said in the day of
his grey hairs to have thought of the sprightliness of the _Decameron_
with strong remorse. From tales Diderot went back to the more congenial
region of philosophy. Between the morning of Good Friday and the evening
of Easter Monday he wrote the _Pensées philosophiques_ (1746), and he
presently added to this a short complementary essay on the sufficiency
of natural religion. The gist of these performances is to press the
ordinary rationalistic objections to a supernatural revelation; but
though Diderot did not at this time pass out into the wilderness beyond
natural religion, yet there are signs that he accepted that less as a
positive doctrine, resting on grounds of its own, than as a convenient
point of attack against Christianity. In 1747 he wrote the _Promenade du
sceptique_, a rather poor allegory--pointing first to the extravagances
of Catholicism; second, to the vanity of the pleasures of that world
which is the rival of the church; and third, to the desperate and
unfathomable uncertainty of the philosophy which professes to be so high
above both church and world.

Diderot's next piece was what first introduced him to the world as an
original thinker, his famous _Lettre sur les aveugles_ (1749). The
immediate object of this short but pithy writing was to show the
dependence of men's ideas on their five senses. It considers the case of
the intellect deprived of the aid of one of the senses; and in a second
piece, published afterwards, Diderot considered the case of a similar
deprivation in the deaf and dumb. The _Lettre sur les sourds et muets_,
however, is substantially a digressive examination of some points in
aesthetics. The philosophic significance of the two essays is in the
advance they make towards the principle of Relativity. But what
interested the militant philosophers of that day was an episodic
application of the principle of relativity to the master-conception of
God. What makes the _Lettre sur les aveugles_ interesting is its
presentation, in a distinct though undigested form, of the modern theory
of variability, and of survival by superior adaptation. It is worth
noticing, too, as an illustration of the comprehensive freedom with
which Diderot felt his way round any subject that he approached, that in
this theoretic essay he suggests the possibility of teaching the blind
to read through the sense of touch. If the _Lettre sur les aveugles_
introduced Diderot into the worshipful company of the philosophers, it
also introduced him to the penalties of philosophy. His speculation was
too hardy for the authorities, and he was thrown into the prison of
Vincennes. Here he remained for three months; then he was released, to
enter upon the gigantic undertaking of his life.

The bookseller Lebreton had applied to him with a project for the
publication of a translation into French of Ephraim Chambers's
_Cyclopaedia_, undertaken in the first instance by an Englishman, John
Mills, and a German, Gottfried Sellius (for particulars see
ENCYCLOPAEDIA). Diderot accepted the proposal, but in his busy and
pregnant intelligence the scheme became transformed. Instead of a mere
reproduction of Chambers, he persuaded the bookseller to enter upon a
new work, which should collect under one roof all the active writers,
all the new ideas, all the new knowledge, that were then moving the
cultivated class to its depths, but still were comparatively ineffectual
by reason of their dispersion. His enthusiasm infected the publishers;
they collected a sufficient capital for a vaster enterprise than they
had at first planned; D'Alembert was persuaded to become Diderot's
colleague; the requisite permission was procured from the government; in
1750 an elaborate prospectus announced the project to a delighted
public; and in 1751 the first volume was given to the world. The last of
the letterpress was issued in 1765, but it was 1772 before the
subscribers received the final volumes of the plates. These twenty years
were to Diderot years not merely of incessant drudgery, but of harassing
persecution, of sufferings from the cabals of enemies, and of injury
from the desertion of friends. The ecclesiastical party detested the
_Encyclopaedia_, in which they saw a rising stronghold for their
philosophic enemies. By 1757 they could endure the sight no longer. The
subscribers had grown from 2000 to 4000, and this was a right measure of
the growth of the work in popular influence and power. To any one who
turns over the pages of these redoubtable volumes now, it seems
surprising that their doctrines should have stirred such portentous
alarm. There is no atheism, no overt attack on any of the cardinal
mysteries of the faith, no direct denunciation even of the notorious
abuses of the church. Yet we feel that the atmosphere of the book may
well have been displeasing to authorities who had not yet learnt to
encounter the modern spirit on equal terms. The _Encyclopaedia_ takes
for granted the justice of religious tolerance and speculative freedom.
It asserts in distinct tones the democratic doctrine that it is the
common people in a nation whose lot ought to be the main concern of the
nation's government. From beginning to end it is one unbroken process of
exaltation of scientific knowledge on the one hand, and pacific industry
on the other. All these things were odious to the old governing classes
of France; their spirit was absolutist, ecclesiastical and military.
Perhaps the most alarming thought of all was the current belief that the
_Encyclopaedia_ was the work of an organized band of conspirators
against society, and that a pestilent doctrine was now made truly
formidable by the confederation of its preachers into an open league.
When the seventh volume appeared, it contained an article on "Geneva,"
written by D'Alembert. The writer contrived a panegyric on the pastors
of Geneva, of which every word was a stinging reproach to the abbés and
prelates of Versailles. At the same moment Helvétius's book, _L'Esprit_,
appeared, and gave a still more profound and, let us add, a more
reasonable shock to the ecclesiastical party. Authority could brook no
more, and in 1759 the _Encyclopaedia_ was formally suppressed.

The decree, however, did not arrest the continuance of the work. The
connivance of the authorities at the breach of their own official orders
was common in those times of distracted government. The work went on,
but with its difficulties increased by the necessity of being
clandestine. And a worse thing than troublesome interference by the
police now befell Diderot. D'Alembert, wearied of shifts and
indignities, withdrew from the enterprise. Other powerful colleagues,
Turgot among them, declined to contribute further to a book which had
acquired an evil fame. Diderot was left to bring the task to an end as
he best could. For seven years he laboured like a slave at the oar. He
wrote several hundred articles, some of them very slight, but many of
them most laborious, comprehensive and ample. He wore out his eyesight
in correcting proofs, and he wearied his soul in bringing the manuscript
of less competent contributors into decent shape. He spent his days in
the workshops, mastering the processes of manufactures, and his nights
in reproducing on paper what he had learnt during the day. And he was
incessantly harassed all the time by alarms of a descent from the
police. At the last moment, when his immense work was just drawing to an
end, he encountered one last and crowning mortification: he discovered
that the bookseller, fearing the displeasure of the government, had
struck out from the proof sheets, after they had left Diderot's hands,
all passages that he chose to think too hardy. The monument to which
Diderot had given the labour of twenty long and oppressive years was
irreparably mutilated and defaced. It is calculated that the average
annual salary received by Diderot for his share in the _Encyclopaedia_
was about £120 sterling. "And then to think," said Voltaire, "that an
army contractor makes £800 in a day!"

Although the _Encyclopaedia_ was Diderot's monumental work, he is the
author of a shower of dispersed pieces that sowed nearly every field of
intellectual interest with new and fruitful ideas. We find no
masterpiece, but only thoughts for masterpieces; no creation, but a
criticism with the quality to inspire and direct creation. He wrote
plays--_Le Fils naturel_ (1757) and _Le Père de famille_ (1758)--and
they are very insipid performances in the sentimental vein. But he
accompanied them by essays on dramatic poetry, including especially the
_Paradoxe sur le comédien_, in which he announced the principles of a
new drama,--the serious, domestic, bourgeois drama of real life, in
opposition to the stilted conventions of the classic French stage. It
was Diderot's lessons and example that gave a decisive bias to the
dramatic taste of Lessing, whose plays, and his _Hamburgische
Dramaturgie_ (1768), mark so important an epoch in the history of the
modern theatre. In the pictorial art, Diderot's criticisms are no less
rich, fertile and wide in their ideas. His article on "Beauty" in the
_Encyclopaedia_ shows that he had mastered and passed beyond the
metaphysical theories on the subject, and the _Essai sur la peinture_
was justly described by Goethe, who thought it worth translating, as "a
magnificent work, which speaks even more helpfully to the poet than to
the painter, though to the painter too it is as a blazing torch."
Diderot's most intimate friend was Grimm, one of the conspicuous figures
of the philosophic body. Grimm wrote news-letters to various high
personages in Germany, reporting what was going on in the world of art
and literature in Paris, then without a rival as the capital of the
intellectual activity of Europe. Diderot helped his friend at one time
and another between 1759 and 1779, by writing for him an account of the
annual exhibitions of paintings. These _Salons_ are among the most
readable of all pieces of art criticism. They have a freshness, a
reality, a life, which take their readers into a different world from
the dry and conceited pedantries of the ordinary virtuoso. As has been
said by Sainte-Beuve, they initiated the French into a new sentiment,
and introduced people to the mystery and purport of colour by ideas.
"Before Diderot," Madame Necker said, "I had never seen anything in
pictures except dull and lifeless colours; it was his imagination that
gave them relief and life, and it is almost a new sense for which I am
indebted to his genius."

Greuze was Diderot's favourite among contemporary artists, and it is
easy to see why. Greuze's most characteristic pictures were the
rendering in colour of the same sentiment of domestic virtue and the
pathos of common life, which Diderot attempted with inferior success to
represent upon the stage. For Diderot was above all things interested in
the life of men,--not the abstract life of the race, but the incidents
of individual character, the fortunes of a particular family, the
relations of real and concrete motives in this or that special case. He
delighted with the enthusiasm of a born casuist in curious puzzles of
right and wrong, and in devising a conflict between the generalities of
ethics and the conditions of an ingeniously contrived practical dilemma.
Mostly his interest expressed itself in didactic and sympathetic form;
in two, however, of the most remarkable of all his pieces, it is not
sympathetic, but ironical. _Jacques le fataliste_ (written in 1773, but
not published until 1796) is in manner an imitation of _Tristram Shandy_
and _The Sentimental Journey_. Few modern readers will find in it any
true diversion. In spite of some excellent criticisms dispersed here and
there, and in spite of one or two stories that are not without a certain
effective realism, it must as a whole be pronounced savourless, forced,
and as leaving unmoved those springs of laughter and of tears which are
the common fountain of humour. _Le Neveu de Rameau_ is a far superior
performance. If there were any inevitable compulsion to name a
masterpiece for Diderot, one must select this singular "farce-tragedy."
Its intention has been matter of dispute; whether it was designed to be
merely a satire on contemporary manners, or a reduction of the theory of
self-interest to an absurdity, or the application of an ironical
clincher to the ethics of ordinary convention, or a mere setting for a
discussion about music, or a vigorous dramatic sketch of a parasite and
a human original. There is no dispute as to its curious literary
flavour, its mixed qualities of pungency, bitterness, pity and, in
places, unflinching shamelessness. Goethe's translation (1805) was the
first introduction of _Le Neveu de Rameau_ to the European public. After
executing it, he gave back the original French manuscript to Schiller,
from whom he had it. No authentic French copy of it appeared until the
writer had been nearly forty years in his grave (1823).

It would take several pages merely to contain the list of Diderot's
miscellaneous pieces, from an infinitely graceful trifle like the
_Regrets sur ma vieille robe de chambre_ up to _Le Rêve de D'Alembert_,
where he plunges into the depths of the controversy as to the ultimate
constitution of matter and the meaning of life. It is a mistake to set
down Diderot for a coherent and systematic materialist. We ought to look
upon him "as a philosopher in whom all the contradictions of the time
struggle with one another" (Rosenkranz). That is to say, he is critical
and not dogmatic. There is no unity in Diderot, as there was in Voltaire
or in Rousseau. Just as in cases of conduct he loves to make new ethical
assumptions and argue them out as a professional sophist might have
done, so in the speculative problems as to the organization of matter,
the origin of life, the compatibility between physiological machinery
and free will, he takes a certain standpoint, and follows it out more or
less digressively to its consequences. He seizes a hypothesis and works
it to its end, and this made him the inspirer in others of materialist
doctrines which they held more definitely than he did. Just as Diderot
could not attain to the concentration, the positiveness, the finality of
aim needed for a masterpiece of literature, so he could not attain to
those qualities in the way of dogma and system. Yet he drew at last to
the conclusions of materialism, and contributed many of its most
declamatory pages to the _Système de la nature_ of his friend
D'Holbach,--the very Bible of atheism, as some one styled it. All that
he saw, if we reduce his opinions to formulae, was motion in space:
"attraction and repulsion, the only truth." If matter produces life by
spontaneous generation, and if man has no alternative but to obey the
compulsion of nature, what remains for God to do?

In proportion as these conclusions deepened in him, the more did
Diderot turn for the hope of the race to virtue; in other words, to such
a regulation of conduct and motive as shall make us tender, pitiful,
simple, contented. Hence his one great literary passion, his enthusiasm
for Richardson, the English novelist. Hence, also, his deepening
aversion for the political system of France, which makes the realization
of a natural and contented domestic life so hard. Diderot had almost as
much to say against society as even Rousseau himself. The difference
between them was that Rousseau was a fervent theist. The atheism of the
Holbachians, as he called Diderot's group, was intolerable to him; and
this feeling, aided by certain private perversities of humour, led to a
breach of what had once been an intimate friendship between Rousseau and
Diderot (1757). Diderot was still alive when Rousseau's _Confessions_
appeared, and he was so exasperated by Rousseau's stories about Grimm,
then and always Diderot's intimate, that in 1782 he transformed a life
of Seneca, that he had written four years earlier, into an _Essai sur
les règnes de Claude et de Néron_ (1778-1782), which is much less an
account of Seneca than a vindication of Diderot and Grimm, and is one of
the most rambling and inept productions in literature. As for the merits
of the old quarrel between Rousseau and Diderot, we may agree with the
latter, that too many sensible people would be in the wrong if Jean
Jacques was in the right.

Varied and incessant as was Diderot's mental activity, it was not of a
kind to bring him riches. He secured none of the posts that were
occasionally given to needy men of letters; he could not even obtain
that bare official recognition of merit which was implied by being
chosen a member of the Academy. The time came for him to provide a dower
for his daughter, and he saw no other alternative than to sell his
library. When the empress Catherine of Russia heard of his straits, she
commissioned an agent in Paris to buy the library at a price equal to
about £1000 of English money, and then handsomely requested the
philosopher to retain the books in Paris until she required them, and to
constitute himself her librarian, with a yearly salary. In 1773 Diderot
started on an expedition to thank his imperial benefactress in person,
and he passed some months at St Petersburg. The empress received him
cordially. The strange pair passed their afternoons in disputes on a
thousand points of high philosophy, and they debated with a vivacity and
freedom not usual in courts. "_Fi, donc,_" said Catherine one day, when
Diderot hinted that he argued with her at a disadvantage, "_is there any
difference among men?_" Diderot returned home in 1774. Ten years
remained to him, and he spent them in the industrious acquisition of new
knowledge, in the composition of a host of fragmentary pieces, some of
them mentioned above, and in luminous declamations with his friends. All
accounts agree that Diderot was seen at his best in conversation. "He
who only knows Diderot in his writings," says Marmontel, "does not know
him at all. When he grew animated in talk, and allowed his thoughts to
flow in all their abundance, then he became truly ravishing. In his
writings he had not the art of ensemble; the first operation which
orders and places everything was too slow and too painful to him."
Diderot himself was conscious of the want of literary merit in his
pieces. In truth he set no high value on what he had done. It is
doubtful whether he was ever alive to the waste that circumstance and
temperament together made of an intelligence from which, if it had been
free to work systematically, the world of thought had so much to hope.
He was one of those simple, disinterested and intellectually sterling
workers to whom their own personality is as nothing in presence of the
vast subjects that engage the thoughts of their lives. He wrote what he
found to write, and left the piece, as Carlyle has said, "on the waste
of accident, with an ostrich-like indifference." When he heard one day
that a collected edition of his works was in the press at Amsterdam, he
greeted the news with "peals of laughter," so well did he know the haste
and the little heed with which those works had been dashed off.

Diderot died on the 30th of July 1784, six years after Voltaire and
Rousseau, one year after his old colleague D'Alembert, and five years
before D'Holbach, his host and intimate for a lifetime. Notwithstanding
Diderot's peals of laughter at the thought, an elaborate and exhaustive
collection of his writings in twenty stout volumes, edited by MM.
Assézat and Tourneux, was completed in 1875-1877.

  AUTHORITIES.--Studies on Diderot by Scherer (1880); by E. Faguet
  (1890); by Sainte-Beuve in the _Causeries du lundi_; by F. Brunetière
  in the _Études critiques_, 2nd series, may be consulted. In English,
  Diderot has been the subject of a biography by John Morley [Viscount
  Morley of Blackburn] (1878). See also Karl Rosenkranz, _Diderots Leben
  und Werke_ (1866). For a discussion of the authenticity of the
  posthumous works of Diderot see R. Dominic in the _Revue des deux
  mondes_ (October 15, 1902).     (J. Mo.)



DIDIUS SALVIUS JULIANUS, MARCUS, Roman emperor for two months (March
28-June 2) during the year A.D. 193. He was the grandson of the famous
jurist Salvius Julianus (under Hadrian and the Antonines), and the son
of a distinguished general, who might have ascended the throne after the
death of Antoninus Pius, had not his loyalty to the ruling house
prevented him. Didius filled several civil and military offices with
distinguished success, but subsequently abandoned himself to
dissipation. On the death of Pertinax, the praetorian guards offered the
throne to the highest bidder. Flavius Sulpicianus, the father-in-law of
Pertinax and praefect of the city, had already made an offer; Didius,
urged on by the members of his family, his freedmen and parasites,
hurried to the praetorian camp to contend for the prize. He and
Sulpicianus bid against each other, and finally the throne was knocked
down to Didius. The senate and nobles professed their loyalty; but the
people made no attempt to conceal their indignation at this insult to
the state, and the armies of Britain, Syria and Illyricum broke out into
open revolt. Septimius Severus, the commander of the Pannonian legions,
was declared emperor and hastened by forced marches to Italy. Didius,
abandoned by the praetorians, was condemned and executed by order of the
senate, which at once acknowledged Severus.

  AUTHORITIES.--Dio Cassius lxxiii. 11-17, who was actually in Rome at
  the time; Aelius Spartianus, _Didius Julianus_; Julius Capitolinus,
  _Pertinax_; Herodian ii.; Aurelius Victor, _De Caesaribus_, 19;
  Zosimus i. 7; Gibbon, _Decline and Fall_, chap. 5.



DIDO, or ELISSA, the reputed founder of Carthage (q.v.), in Africa,
daughter of the Tyrian king Metten (Mutto, Methres, Belus), wife of
Acerbas (more correctly Sicharbas; Sychaeus in Virgil), a priest of
Hercules. Her husband having been slain by her brother Pygmalion, Dido
fled to Cyprus, and thence to the coast of Africa, where she purchased
from a local chieftain Iarbas a piece of land on which she built
Carthage. The city soon began to prosper and Iarbas sought Dido's hand
in marriage, threatening her with war in case of refusal. To escape from
him, Dido constructed a funeral pile, on which she stabbed herself
before the people (Justin xviii. 4-7). Virgil, in defiance of the
usually accepted chronology, makes Dido a contemporary of Aeneas, with
whom she fell in love after his landing in Africa, and attributes her
suicide to her abandonment by him at the command of Jupiter (_Aeneid_,
iv.). Dido was worshipped at Carthage as a divinity under the name of
Caelestis, the Roman counterpart of Tanit, the tutelary goddess of
Carthage. According to Timaeus, the oldest authority for the story, her
name was Theiosso, in Phoenician Helissa, and she was called Dido from
her wanderings, Dido being the Phoenician equivalent of [Greek:
planêtis] (_Etymologicum Magnum_, s.v.); some modern scholars, however,
translate the name by "beloved." Timaeus makes no mention of Aeneas, who
seems to have been introduced by Naevius in his _Bellum Poenicum_,
followed by Ennius in his _Annales_.

  For the variations of the legend in earlier and later Latin authors,
  see O. Rossbach in Pauly-Wissowa's _Realencyclopädie_, v. pt. 1
  (1905); O. Meltzer's _Geschichte der Karthager_, i. (1879), and his
  article in Roscher's _Lexikon der Mythologie_.



DIDON, HENRI (1840-1900), French Dominican, was born at Trouvet, Isère,
on the 17th of March 1840. He joined the Dominicans, under the influence
of Lacordaire, in 1858, and completed his theological studies at the
Minerva convent at Rome. The influence of Lacordaire was shown in the
zeal displayed by Didon in favour of a reconciliation between philosophy
and science. In 1871 his fame had so much grown that he was chosen to
deliver the funeral oration over the murdered archbishop of Paris,
Monseigneur G. Darboy. He also delivered some discourses at the church
of St Jean de Beauvais in Paris on the relations between science and
religion; but his utterances, especially on the question of divorce,
were deemed suspicious by his superiors, and his intimacy with Claude
Bernard the physiologist was disapproved. He was interdicted from
preaching and sent into retirement at the convent of Corbara in Corsica.
After eighteen months he emerged, and travelled in Germany, publishing
an interesting work upon that country, entitled _Les Allemands_ (English
translation by R. Ledos de Beaufort, London, 1884). On his return to
France in 1890 he produced his best known work, _Jésus-Christ_ (2 vols.,
Paris), for which he had qualified himself by travel in the Holy Land.
In the same year he became director of the Collège Albert-le-Grand at
Arcueil, and founded three auxiliary institutions, École Lacordaire,
École Laplace and École St Dominique. He wrote, in addition, several
works on educational questions, and augmented his fame as an eloquent
preacher by discourses preached during Lent and Advent. He died at
Toulouse on the 13th of March 1900.

  See the biographies by J. de Romano (1891), and A. de Coulanges
  (Paris, 1900); and especially the work of Stanislas Reynaud, entitled
  _Le Père Didon, sa vie et son oeuvre_ (Paris, 1904).



DIDOT, the name of a family of learned French printers and publishers.
FRANÇOIS DIDOT (1689-1757), founder of the family, was born at Paris. He
began business as a bookseller and printer in 1713, and among his
undertakings was a collection of the travels of his friend the Abbé
Prévost, in twenty volumes (1747). It was remarkable for its
typographical perfection, and was adorned with many engravings and maps.
FRANÇOIS AMBROISE DIDOT (1730-1804), son of François, made important
improvements in type-founding, and was the first to attempt printing on
vellum paper. Among the works which he published was the famous
collection of French classics prepared by order of Louis XVI. for the
education of the Dauphin, and the folio edition of _L'Art de vérifier
les dates_. PIERRE FRANÇOIS DIDOT (1732-1795), his brother, devoted much
attention to the art of type-founding and to paper-making. Among the
works which issued from his press was an edition in folio of the
_Imitatio Christi_ (1788). HENRI DIDOT (1765-1852), son of Pierre
François, is celebrated for his "microscopic" editions of various
standard works, for which he engraved the type when nearly seventy years
of age. He was also the engraver of the _assignats_ issued by the
Constituent and Legislative Assemblies and the Convention. DIDOT
SAINT-LÉGER, second son of Pierre François, was the inventor of the
paper-making machine known in England as the Didot machine. PIERRE DIDOT
(1760-1853), eldest son of François Ambroise, is celebrated as the
publisher of the beautiful "Louvre" editions of Virgil, Horace and
Racine. The Racine, in three volumes folio, was pronounced in 1801 to be
"the most perfect typographical production of all ages." FIRMIN DIDOT
(1764-1836), his brother, second son of François Ambroise, sustained the
reputation of the family both as printer and type-founder. He revived
(if he did not invent--a distinction which in order of time belongs to
William Ged) the process of stereotyping, and coined its name, and he
first used the process in his edition of Callet's _Tables of Logarithms_
(1795), in which he secured an accuracy till then unattainable. He
published stereotyped editions of French, English and Italian classics
at a very low price. He was the author of two tragedies--_La Reine de
Portugal_ and _La Mort d'Annibal_; and he wrote metrical translations
from Virgil, Tyrtaeus and Theocritus. AMBROISE FIRMIN DIDOT (1790-1876)
was his eldest son. After receiving a classical education, he spent
three years in Greece and in the East; and on the retirement of his
father in 1827 he undertook, in conjunction with his brother Hyacinthe,
the direction of the publishing business. Their greatest undertaking was
a new edition of the _Thesaurus Graecae linguae_ of Henri Estienne,
under the editorial care of the brothers Dindorf and M. Hase (9 vols.,
1855-1859). Among the numerous important works published by the
brothers, the 200 volumes forming the _Bibliothèque des auteurs grecs_,
_Bibliothèque latine_, and _Bibliothèque française_ deserve special
mention. Ambroise Firmin Didot was the first to propose (1823) a
subscription in favour of the Greeks, then in insurrection against
Turkish tyranny. Besides a translation of Thucydides (1833), he wrote
the articles "Estienne" in the _Nouvelle Biographie générale_, and
"Typographie" in the _Ency. mod._, as well as _Observations sur
l'orthographie française_ (1867), &c. In 1875 he published a very
learned and elaborate monograph on Aldus Manutius. His collection of
MSS., the richest in France, was said to have been worth, at the time of
his death, not less than 2,000,000 francs.



DIDRON, ADOLPHE NAPOLÉON (1806-1867), French archaeologist, was born at
Hautvillers, in the department of Marne, on the 13th of March 1806. At
first a student of law, he began in 1830, by the advice of Victor Hugo,
a study of the Christian archaeology of the middle ages. After visiting
and examining the principal churches, first of Normandy, then of central
and southern France, he was on his return appointed by Guizot secretary
to the Historical Committee of Arts and Monuments (1835); and in the
following years he delivered several courses of lectures on Christian
iconography at the Bibliothèque Royale. In 1839 he visited Greece for
the purpose of examining the art of the Eastern Church, both in its
buildings and its manuscripts. In 1844 he originated the _Annales
archéologiques_, a periodical devoted to his favourite subject, which he
edited until his death. In 1845 he established at Paris a special
archaeological library, and at the same time a manufactory of painted
glass. In the same year he was admitted to the Legion of Honour. His
most important work is the _Iconographie chrétienne_, of which, however,
the first portion only, _Histoire de Dieu_ (1843), was published. It was
translated into English by E. J. Millington. Among his other works may
be mentioned the _Manuel d'iconographie chrétienne grecque et latine_
(1845), the _Iconographie des chapiteaux du palais ducal de Venise_
(1857), and the _Manuel des objets de bronze et d'orfèvrerie_ (1859). He
died on the 13th of November 1867.



DIDYMI, or DIDYMA (mod. _Hieronta_), an ancient sanctuary of Apollo in
Asia Minor situated in the territory of Miletus, from which it was
distant about 10 m. S. and on the promontory Poseideion. It was
sometimes called _Branchidae_ from the name of its priestly caste which
claimed descent from Branchus, a youth beloved by Apollo. As the seat of
a famous oracle, the original temple attracted offerings from Pharaoh
Necho (in whose army there was a contingent of Milesian mercenaries),
and the Lydian Croesus, and was plundered by Darius of Persia. Xerxes
finally sacked and burnt it (481 B.C.) and exiled the Branchidae to the
far north-east of his empire. This exile was believed to be voluntary,
the priests having betrayed their treasures to the Persian; and on this
belief Alexander the Great acted 150 years later, when, finding the
descendants of the Branchidae established in a city beyond the Oxus, he
ordered them to be exterminated for the sin of their fathers (328). The
celebrated cult-statue of Apollo by Canachus, familiar to us from
reproductions on Milesian coins, was also carried to Persia, there to
remain till restored by Seleucus I. in 295, and the oracle ceased to
speak for a century and a half. The Milesians were not able to undertake
the rebuilding till about 332 B.C., when the oracle revived at the
bidding of Alexander. The work proved too costly, and despite a special
effort made by the Asian province nearly 400 years later, at the bidding
of the emperor Caligula, the structure was never quite finished: but
even as it was, Strabo ranked the Didymeum the greatest of Greek temples
and Pliny placed it among the four most splendid and second only to the
Artemisium at Ephesus. In point of fact it was a little smaller than the
Samian Heraeum and the temple of Cybele at Sardis, and almost exactly
the same size as the Artemisium. The area covered by the platform
measures roughly 360 × 160 ft.

When Cyriac of Ancona visited the spot in 1446, it seems that the temple
was still standing in great part, although the _cella_ had been
converted into a fortress by the Byzantines: but when the next European
visitor, the Englishman Dr Pickering, arrived in 1673, it had collapsed.
It is conjectured that the cause was the great earthquake of 1493. The
Society of Dilettanti sent two expeditions to explore the ruins, the
first in 1764 under Richard Chandler, the second in 1812 under Sir Wm.
Gell; and the French "Rothschild Expedition" of 1873 under MM. O. Rayet
and A. Thomas sent a certain amount of architectural sculpture to the
Louvre. But no excavation was attempted till MM. E. Pontremoli and B.
Haussoullier were sent out by the French Schools of Rome and Athens in
1895. They cleared the western façade and the _prodomos_, and discovered
inscriptions giving information about other parts which they left still
buried. Finally the site was purchased by, and the French rights were
ceded to, Dr Th. Wiegand, the German explorer of Miletus, who in 1905
began a thorough clearance of what is incomparably the finest temple
ruin in Asia Minor.

The temple was a decastyle peripteral structure of the Ionic order,
standing on seven steps and possessing double rows of outer columns 60
ft. high, twenty-one in each row on the flanks. It is remarkable not
only for its great size, but (_inter alia_) for (1) the rich ornament of
its column bases, which show great variety of design; (2) its various
developments of the Ionic capital, e.g. heads of gods, probably of
Pergamene art, spring from the "eyes" of the volutes with bulls' heads
between them; (3) the massive building two storeys high at least, which
served below for _prodomos_, and above for a dispensary of oracles
([Greek: chrêsmographia] mentioned in the inscriptions) and a treasury;
two flights of stairs called "labyrinths" in the inscriptions, led up to
these chambers; (4) the pylon and staircase at the west; (5) the frieze
of Medusa heads and foliage. Two outer columns are still erect on the
north-east flank, carrying their entablature, and one of the inner order
stands on the south-west. The fact that the temple was never finished is
evident from the state in which some bases still remain at the west.
There were probably no pedimental sculptures. A sacred way led from the
temple to the sea at Panormus, which was flanked with rows of archaic
statues, ten of which were excavated and sent to the British Museum in
1858 by C. T. Newton. Fragments of architectural monuments, which once
adorned this road, have also been found. Modern Hieronta is a large and
growing Greek village, the only settlement within a radius of several
miles. Its harbour is Kovella, distant about 2½ m., and on the N. of the
promontory.

  See Dilettanti Society, _Ionian Antiquities_, ii. (1821); C. T.
  Newton, _Hist. of Discoveries_, &c. (1862) and _Travels in the
  Levant_, ii. (1865); O. Rayet and A. Thomas, _Milet et le Golfe
  Latmique_ (1877); E. Pontremoli and B. Haussoullier, _Didymes_ (1904).
       (D. G. H.)



DIDYMIUM (from the Gr. [Greek: didymos], twin), the name given to the
supposed element isolated by C. G. Mosander from cerite (1839-1841). In
1879, however, Lecoq de Boisbaudran showed that Mosander's "didymium"
contained samarium; while the residual "didymium," after removal of
samarium, was split by Auer v. Welsbach (_Monats. f. Chemie_, 1885, 6,
477) into two components (known respectively as neodymium and
praseodymium) by repeated fractional crystallization of the double
nitrate of ammonium and didymium in nitric acid. _Neodymium_ (Nd) forms
the chief portion of the old "didymium." Its salts are reddish violet in
colour, and give a characteristic absorption spectrum. It forms oxides
of composition Nd2O3 and Nd2O5, the latter being obtained by ignition of
the nitrate (B. Brauner). The atomic weight of neodymium is 143.6 (B.
Brauner, _Proc. Chem. Soc._, 1897-1898, p. 70). _Praseodymium_ (Pr)
forms oxides of composition Pr2O3, Pr2O5, xH2O (B. Brauner), and Pr4O7.
The peroxide, Pr4O7, forms a dark brown powder, and is obtained by
ignition of the oxalate or nitrate. The sesquioxide, Pr2O3, is obtained
as a greenish white mass by the reduction of the peroxide. The salts of
praseodymium are green in colour, and give a characteristic spark
spectrum. The atomic weight of praseodymium is 140.5.



DIDYMUS (?309-?394), surnamed "the Blind," ecclesiastical writer of
Alexandria, was born about the year 309. Although he became blind at the
age of four, before he had learned to read; he succeeded in mastering
the whole circle of the sciences then known; and on entering the service
of the Church he was placed at the head of the Catechetical school in
Alexandria, where he lived and worked till almost the close of the
century. Among his pupils were Jerome and Rufinus. He was a loyal
follower of Origen, though stoutly opposed to Arian and Macedonian
teaching. Such of his writings as survive show a remarkable knowledge of
scripture, and have distinct value as theological literature. Among them
are the _De Trinitate_, _De Spiritu Sancto_ (Jerome's Latin
translation), _Adversus Manichaeos_, and notes and expositions of
various books, especially the Psalms and the Catholic Epistles.

  See Migne, _Patrol. Graec._ xxxix.; O. Bardenhewer, _Patrologie_, pp.
  290-293 (Freiburg, 1894).



DIDYMUS CHALCENTERUS (c. 63 B.C.-A.D. 10), Greek scholar and grammarian,
flourished in the time of Cicero and Augustus. His surname (Gr. [Greek:
Chalkenteros], brazen-bowelled) came from his indefatigable industry; he
was said to have written so many books (more than 3500) that he was
unable to recollect their names ([Greek: bibliolathas]). He lived and
taught in Alexandria and Rome, where he became the friend of Varro. He
is chiefly important as having introduced Alexandrian learning to the
Romans. He was a follower of the school of Aristarchus, upon whose
recension of Homer he wrote a treatise, fragments of which have been
preserved in the Venetian Scholia. He also wrote commentaries on many
other Greek poets and prose authors. In his work on the lyric poets he
treated of the various classes of poetry and their chief
representatives, and his lists of words and phrases (used in tragedy and
comedy and by orators and historians), of words of doubtful meaning, and
of corrupt expressions, furnished the later grammarians with valuable
material. His activity extended to all kinds of subjects: grammar
(orthography, inflexions), proverbs, wonderful stories, the law-tablets
([Greek: axones]) of Solon, stones, and different kinds of wood. His
polemic against Cicero's _De republica_ (Ammianus Marcellinus xxii. 16)
provoked a reply from Suetonius. In spite of his stupendous industry,
Didymus was little more than a compiler, of little critical judgment and
doubtful accuracy, but he deserves recognition for having incorporated
in his numerous writings the works of earlier critics and commentators.

  See M. W. Schmidt, _De Didymo Chalcentero_ (1853) and _Didymi
  Chalcenteri fragmenta_ (1854); also F. Susemihl, _Geschichte der
  griech. Literatur in der Alexandrinerzeit_, ii. (1891); J. E. Sandys,
  _History of Classical Scholarship_, i. (1906).



DIE, a town of south-eastern France, capital of an arrondissement in the
department of Drôme, 43 m. E.S.E. of Valence on the Paris-Lyon railway.
Pop. (1906) 3090. The town is situated in a plain enclosed by mountains
on the right bank of the Drôme below its confluence with the Meyrosse,
which supplies power to some of the industries. The most interesting
structures of Die are the old cathedral, with a porch of the 11th
century supported on granite columns from an ancient temple of Cybele;
and the Porte St Marcel, a Roman gateway flanked by massive towers. The
Roman remains also include the ruins of aqueducts and altars. Die is the
seat of a sub-prefect, and of a tribunal of first instance. The
manufactures are silk, furniture, cloth, lime and cement, and there are
flour and saw mills. Trade is in timber, especially walnut, and in white
wine known as _clairette de Die_. The mulberry is largely grown for the
rearing of silkworms. Under the Romans, Die (_Dea Augusta Vocontiorum_)
was an important colony. It was formerly the seat of a bishopric, united
to that of Valence from 1276 to 1687 and suppressed in 1790. Previous to
the revocation of the edict of Nantes in 1685 it had a Calvinistic
university.



DIE (Fr. _dé_, from Lat. _datum_, given), a word used in various senses,
for a small cube of ivory, &c. (see DICE), for the engraved stamps used
in coining money, &c., and various mechanical appliances in engineering.
In architecture a "die" is the term used for the square base of a
column, and it is applied also to the vertical face of a pedestal or
podium.

The fabrics known as "dice" take their name from the rectangular form of
the figure. The original figures would probably be perfectly square, but
to-day the same principle of weaving is applied, and the name dice is
given to all figures of rectangular form. The different effects in the
adjacent squares or rectangles are due to precisely the same reasons as
those explained in connexion with the ground and the figure of damasks.
The same weaves are used in both damasks and dices, but simpler weaves
are generally employed for the commoner classes of the latter. The
effect is, in every case, obtained by what are technically called warp
and weft float weaves. The illustration B shows the two double damask
weaves arranged to form a dice pattern, while A shows a similar pattern
made from two four-thread twill weaves. C and D represent respectively
the disposition of the threads in A and B with the first pick, and the
solid marks represent the floats of warp. The four squares, which are
almost as pronounced in the cloth as those of a chess-board, may be made
of any size by repeating each weave for the amount of surface required.
It is only in the finest cloths that the double damask weaves B are used
for dice patterns, the single damask weaves and the twill weaves being
employed to a greater extent. This class of pattern is largely employed
for the production of table-cloths of lower and medium qualities. The
term damask is also often applied to cloths of this character, and
especially so when the figure is formed by rectangles of different
sizes.

[Illustration: A B C D]



DIEBITSCH, HANS KARL FRIEDRICH ANTON, count von Diebitsch and Narden,
called by the Russians Ivan Ivanovich, Count Diebich-Zabalkansky
(1785-1831), Russian field-marshal, was born in Silesia on the 13th of
May 1785. He was educated at the Berlin cadet school, but by the desire
of his father, a Prussian officer who had passed into the service of
Russia, he also did the same in 1801. He served in the campaign of 1805,
and was wounded at Austerlitz, fought at Eylau and Friedland, and after
Friedland was promoted captain. During the next five years of peace he
devoted himself to the study of military science, engaging once more in
active service in the War of 1812. He distinguished himself very greatly
in Wittgenstein's campaign, and in particular at Polotzk (October 18 and
19), after which combat he was raised to the rank of major-general. In
the latter part of the campaign he served against the Prussian
contingent of General Yorck (von Wartenburg), with whom, through
Clausewitz, he negotiated the celebrated convention of Tauroggen,
serving thereafter with Yorck in the early part of the War of
Liberation. After the battle of Lützen he served in Silesia and took
part in negotiating the secret treaty of Reichenbach. Having
distinguished himself at the battles of Dresden and Leipzig he was
promoted lieutenant-general. At the crisis of the campaign of 1814 he
strongly urged the march of the allies on Paris; and after their entry
the emperor Alexander conferred on him the order of St Alexander Nevsky.
In 1815 he attended the congress of Vienna, and was afterwards made
adjutant-general to the emperor, with whom, as also with his successor
Nicholas, he had great influence. By Nicholas he was created baron, and
later count. In 1820 he had become chief of the general staff, and in
1825 he assisted in suppressing the St Petersburg _émeute_. His greatest
exploits were in the Russo-Turkish War of 1828-1829, which, after a
period of doubtful contest, was decided by Diebitsch's brilliant
campaign of Adrianople; this won him the rank of field-marshal and the
honorary title of Zabalkanski to commemorate his crossing of the
Balkans. In 1830 he was appointed to command the great army destined to
suppress the insurrection in Poland. He won the terrible battle of
Gróchow on the 25th of February, and was again victorious at Ostrolenka
on the 26th of May, but soon afterwards he died of cholera (or by his
own hand) at Klecksewo near Pultusk, on the 10th of June 1831.

  See Belmont (Schümberg), _Graf Diebitsch_ (Dresden, 1830); Stürmer,
  _Der Tod des Grafen Diebitsch_ (Berlin, 1832); Bantych-Kamenski,
  _Biographies of Russian Field-Marshals_ (in Russian, St Petersburg,
  1841).



DIEDENHOFEN (Fr. _Thionville_), a fortified town of Germany, in
Alsace-Lorraine, dist. Lorraine, on the Mosel, 22 m. N. from Metz by
rail. Pop. (1905) 6047. It is a railway junction of some consequence,
with cultivation of vines, fruit and vegetables, brewing, tanning, &c.
Diedenhofen is an ancient Frank town (Theudonevilla, Totonisvilla), in
which imperial diets were held in the 8th century; was captured by Condé
in 1643 and fortified by Vauban; capitulated to the Prussians, after a
severe bombardment, on the 25th of November 1870.



DIEKIRCH, a small town in the grand duchy of Luxemburg, charmingly
situated on the banks of the Sûre. Pop. (1905) 3705. Its name is said to
be derived from Dide or Dido, granddaughter of Odin and niece of Thor.
The mountain at the foot of which the town lies, now called Herrenberg,
was formerly known as Thorenberg, or Thor's mountain. On the summit of
this rock rises a perennial stream which flows down into the town under
the name of Bellenflesschen. Diekirch was an important Roman station,
and in the 14th century John of Luxemburg, the blind king of Bohemia,
fortified it, surrounding the place with a castellated wall and a ditch
supplied by the stream mentioned. It remained more or less fortified
until the beginning of the 19th century when the French during their
occupation levelled the old walls, and substituted the avenues of trees
that now encircle the town. Diekirch is the administrative centre of one
of the three provincial divisions of the grand duchy. It is visited
during the summer by many thousand tourists and travellers from Holland,
Belgium and Germany.



DIELECTRIC, in electricity, a non-conductor of electricity; it is the
same as insulator. The "dielectric constant" of a medium is its specific
inductive capacity, and on the electromagnetic theory of light it equals
the square of its refractive index for light of infinite wave length
(see ELECTROSTATICS; MAGNETO-OPTICS).



DIELMANN, FREDERICK (1847- ), American portrait and figure painter, was
born at Hanover, Germany, on the 25th of December 1847. He was taken to
the United States in early childhood; studied under Diez at the Royal
Academy at Munich; was first an illustrator, and became a distinguished
draughtsman and painter of genre pictures. His mural decorations and
mosaic panels for the Congressional library, Washington, are notable. He
was elected in 1899 president of the National Academy of Design.



DIEMEN, ANTHONY VAN (1593-1645), Dutch admiral and governor-general of
the East Indian settlements, was born at Kuilenburg in 1593. He was
educated in commerce, and on entering the service of the East India
Company speedily attained high rank. In 1631 he led a Dutch fleet from
the Indies to Holland, and in 1636 he was raised to the
governor-generalship. He came into conflict with the Portuguese, and
took their possessions in Ceylon and Malacca from them. He greatly
extended the commercial relationships of the Dutch, opening up trade
with Tong-king, China and Japan. As an administrator also he showed
ability, and the foundation of a Latin school and several churches in
Batavia is to be ascribed to him. Exploring expeditions were sent to
Australia under his auspices in 1636 and 1642, and Abel Tasman named
after him (Van Diemen's Land) the island now called Tasmania. Van Diemen
died at Batavia on the 19th of April 1645.



DIEPENBECK, ABRAHAM VAN (1599-1675), Flemish painter, was born at
Herzogenbusch, and studied painting at Antwerp, where he became one of
Rubens's "hundred pupils." But he was not one of the cleverest of
Rubens's followers, and he succeeded, at the best, in imitating the
style and aping the peculiarities of his master. We see this in his
earliest pictures--a portrait dated 1629 in the Munich Pinakothek, and a
"Distribution of Alms" of the same period in the same collection. Yet
even at this time there were moments when Diepenbeck probably fancied
that he might take another path. A solitary copperplate executed with
his own hand in 1630 represents a peasant sitting under a tree holding
the bridle of an ass, and this is a minute and finished specimen of the
engraver's art which shows that the master might at one time have hoped
to rival the animal draughtsmen who flourished in the schools of
Holland. However, large commissions now poured in upon him; he was asked
for altarpieces, subject-pieces and pagan allegories. He was tempted to
try the profession of a glass-painter, and at last he gave up every
other occupation for the lucrative business of a draughtsman and
designer for engravings. Most of Diepenbeck's important canvases are in
continental galleries. The best are the "Marriage of St Catherine" at
Berlin and "Mary with Angels Wailing over the Dead Body of Christ" in
the Belvedere at Vienna, the first a very fair specimen of the artist's
skill, the second a picture of more energy and feeling than might be
expected from one who knew more of the outer form than of the spirit of
Rubens. Then we have the fine "Entombment" at Brunswick, and "St Francis
Adoring the Sacrament" at the museum at Brussels, "Clelia and her Nymphs
Flying from the Presence and Pursuit of Porsenna" in two examples at
Berlin and Paris, and "Neptune and Amphitrite" at Dresden. In all these
compositions the drawing and execution are after the fashion of Rubens,
though inferior to Rubens in harmony of tone and force of contrasted
light and shade. Occasionally a tendency may be observed to imitate the
style of Vandyck, for whom, in respect of pictures, Diepenbeck in his
lifetime was frequently taken. But Diepenbeck spent much less of his
leisure on canvases than on glass-painting. Though he failed to master
the secrets of gorgeous tinting, which were lost, apparently for ever in
the 16th century, he was constantly employed during the best years of
his life in that branch of his profession. In 1635 he finished forty
scenes from the life of St Francis of Paula in the church of the Minimes
at Antwerp. In 1644 he received payment for four windows in St Jacques
of Antwerp, two of which are still preserved, and represent Virgins to
whom Christ appears after the Resurrection. The windows ascribed to him
at St Gudule of Brussels were executed from the cartoons of Theodore van
Thulden. On the occasion of his matriculation at Antwerp in 1638-1639,
Diepenbeck was registered in the guild of St Luke as a glass-painter. He
resigned his membership in the Artist Club of the Violette in 1542,
apparently because he felt hurt by a valuation then made of drawings
furnished for copperplates to the engraver Pieter de Jode. The earliest
record of his residence at Antwerp is that of his election to the
brotherhood (Sodalität) "of the Bachelors" in 1634. It is probable that
before this time he had visited Rome and London, as noted in the work of
Houbraken. In 1636 he was made a burgess of Antwerp. He married twice,
in 1637 and 1652. He died in December 1675, and was buried at St Jacques
of Antwerp.



DIEPPE, a seaport of northern France, capital of an arrondissement in
the department of Seine-Inférieure, on the English Channel, 38 m. N. of
Rouen, and 105 m. N.W. of Paris by the Western railway. Pop. (1906)
22,120. It is situated at the mouth of the river Arques in a valley
bordered on each side by steep white cliffs. The main part of the town
lies to the west, and the fishing suburb of Le Pollet to the east of the
river and harbour. The sea-front of Dieppe, which in summer attracts
large numbers of visitors, consists of a pebbly beach backed by a
handsome marine promenade. Dieppe has a modern aspect; its streets are
wide and its houses, in most cases, are built of brick. Two squares side
by side and immediately to the west of the outer harbour form the
nucleus of the town, the Place Nationale, overlooked by the statue of
Admiral A. Duquesne, and the Place St Jacques, named after the beautiful
Gothic church which stands in its centre. The Grande Rue, the busiest
and handsomest street, leads westward from the Place Nationale. The
church of St Jacques was founded in the 13th century, but consists in
large measure of later workmanship and was in some portions restored in
the 19th century. The castle, overlooking the beach from the summit of
the western cliff, was erected in 1435. The church of Notre-Dame de Bon
Secours on the opposite cliff, and the church of St Remy, of the 16th
and 17th centuries, are other noteworthy buildings. A well-equipped
casino stands at the west end of the sea-front. The public institutions
include the subprefecture, tribunals of first instance and commerce, a
chamber of commerce, a communal college and a school of navigation.

Dieppe has one of the safest and deepest harbours on the English
Channel. A curved passage cut in the bed of the Arques and protected by
an eastern and a western jetty gives access to the outer harbour, which
communicates at the east end by a lockgate with the Bassin Duquesne and
the Bassin Bérigny, and at the west end by the New Channel, with an
inner tidal harbour and two other basins. Vessels drawing 20 ft. can
enter the new docks at neap tide. A dry-dock and a gridiron are included
among the repairing facilities of the port. The harbour railway station
is on the north-west quay of the outer harbour alongside which the
steamers from Newhaven lie. The distance of Dieppe from Newhaven, with
which there has long been daily communication, is 64 m. The imports
include silk and cotton goods, thread, oil-seeds, timber, coal and
mineral oil; leading exports are wine, silk, woollen and cotton fabrics,
vegetables and fruit and flint-pebbles. The average annual value of
imports for the five years 1901-1905 was £4,916,000 (£4,301,000 for the
years 1896-1900); the exports were valued at £9,206,000 (£7,023,000 for
years 1896-1900). The industries comprise shipbuilding, cotton-spinning,
steam-sawing, the manufacture of machinery, porcelain, briquettes, lace,
and articles in ivory and bone, the production of which dates from the
15th century. There is also a tobacco factory of some importance. The
fishermen of Le Pollet, to whom tradition ascribes a Venetian origin,
are among the main providers of the Parisian market. The sea-bathing
attracts many visitors in the summer. Two miles to the north-east of the
town is the ancient camp known as the Cité de Limes, which perhaps
furnished the nucleus of the population of Dieppe.

It is suggested on the authority of its name, that Dieppe owed its
origin to a band of Norman adventurers, who found its "diep" or inlet
suitable for their ships, but it was unimportant till the latter half of
the 12th century. Its first castle was probably built in 1188 by Henry
II. of England, and it was counted a place of some consideration when
Philip Augustus attacked it in 1195. By Richard I. of England it was
bestowed in 1197 on the archbishop of Rouen in return for certain
territory in the neighbourhood of the episcopal city. In 1339 it was
plundered by the English, but it soon recovered from the blow, and in
spite of the opposition of the lords of Hantot managed to surround
itself with fortifications. Its commercial activity was already great,
and it is believed that its seamen visited the coast of Guinea in 1339,
and founded there a Petit Dieppe in 1365. The town was occupied by the
English from 1420 to 1435. A siege undertaken in 1442 by John Talbot,
first earl of Shrewsbury, was raised by the dauphin, afterwards Louis
XI., and the day of the deliverance continued for centuries to be
celebrated by a great procession and miracle plays. In the beginning of
the 16th century Jean Parmentier, a native of the town, made voyages to
Brazil and Sumatra; and a little later its merchant prince, Jacques
Ango, was able to blockade the Portuguese fleet in the Tagus. Francis I.
began improvements which were continued under his successor. Its
inhabitants in great number embraced the reformed religion; and they
were among the first to acknowledge Henry IV., who fought one of his
great battles at the neighbouring village of Arques. Few of the cities
of France suffered more from the revocation of the edict of Nantes in
1685; and this blow was followed in 1694 by a terrible bombardment on
the part of the English and Dutch. The town was rebuilt after the peace
of Ryswick, but the decrease of its population and the deterioration of
its port prevented the restoration of its commercial prosperity. During
the 19th century it made rapid advances, partly owing to Marie Caroline,
duchess of Berry, who brought it into fashion as a watering-place; and
also because the establishment of railway communication with Paris gave
an impetus to its trade. During the Franco-German War the town was
occupied by the Germans from December 1870 till July 1871.

  See L. Vitet, _Histoire de Dieppe_ (Paris, 1844); D. Asseline, _Les
  Antiquités et chroniques de la ville de Dieppe_, a 17th-century
  account published at Paris in 1874.



DIERX, LÉON (1838-   ), French poet, was born in the island of Réunion
in 1838. He came to Paris to study at the Central School of Arts and
Manufactures, and subsequently settled there, taking up a post in the
education office. He became a disciple of Leconte de Lisle and one of
the most distinguished of the Parnassians. In the death of Stéphane
Mallarmé in 1898 he was acclaimed "prince of poets" by "les jeunes." His
works include: _Poèmes et poésies_ (1864); _Lèvres closes_ (1867);
_Paroles d'un vaincu_ (1871); _La Rencontre_, a dramatic scene (1875)
and _Les Amants_ (1879). His _Poésies complètes_ (1872) were crowned by
the French Academy. A complete edition of his works was published in 2
vols., 1894-1896.



DIES, CHRISTOPH ALBERT (1755-1822), German painter, was born at Hanover,
and learned the rudiments of art in his native place. For one year he
studied in the academy of Dusseldorf, and then he started at the age of
twenty with thirty ducats in his pocket for Rome. There he lived a
frugal life till 1796. Copying pictures, chiefly by Salvator Rosa, for a
livelihood, his taste led him to draw and paint from nature in Tivoli,
Albano and other picturesque places in the vicinity of Rome. Naples, the
birthplace of his favourite master, he visited more than once for the
same reasons. In this way he became a bold executant in water-colours
and in oil, though he failed to acquire any originality of his own. Lord
Bristol, who encouraged him as a copyist, predicted that he would be a
second Salvator Rosa. But Dies was not of the wood which makes original
artists. Besides other disqualifications, he had necessities which
forced him to give up the great career of an independent painter. David,
then composing his Horatii at Rome, wished to take him to Paris. But
Dies had reasons for not accepting the offer. He was courting a young
Roman whom he subsequently married. Meanwhile he had made the
acquaintance of Volpato, for whom he executed numerous drawings, and
this no doubt suggested the plan, which he afterwards carried out, of
publishing, in partnership with Méchan, Reinhardt and Frauenholz, the
series of plates known as the _Collection de vues pittoresques de
l'Italie_, published in seventy-two sheets at Nuremberg in 1799. With so
many irons in the fire Dies naturally lost the power of concentration.
Other causes combined to affect his talent. In 1787 he swallowed by
mistake three-quarters of an ounce of sugar of lead. His recovery from
this poison was slow and incomplete. He settled at Vienna, and lived
there on the produce of his brush as a landscape painter, and on that of
his pencil or graver as a draughtsman and etcher. But instead of getting
better, his condition became worse, and he even lost the use of one of
his hands. In this condition he turned from painting to music, and spent
his leisure hours in the pleasures of authorship. He did not long
survive, dying at Vienna in 1822, after long years of chronic suffering.
From two pictures now in the Belvedere gallery, and from numerous
engraved drawings from the neighbourhood of Tivoli, we gather that Dies
was never destined to rise above a respectable mediocrity. He followed
Salvator Rosa's example in imitating the manner of Claude Lorraine. But
Salvator adapted the style of Claude, whilst Dies did no more than copy
it.



DIEST, a small town in the province of Brabant, Belgium, situated on the
Demer at its junction with the Bever. Pop. (1904) 8383. It lies about
half-way between Hasselt and Louvain, and is still one of the five
fortified places in Belgium. It contains many breweries, and is famous
for the excellence of its beer.



DIESTERWEG, FRIEDRICH ADOLF WILHELM (1790-1866), German educationist,
was born at Siegen on the 29th of October 1790. Educated at Herborn and
Tübingen universities, he took to the profession of teaching in 1811. In
1820 he was appointed director of the new school at Mörs, where he put
in practice the methods of Pestalozzi. In 1832 he was summoned to Berlin
to direct the new state-schools seminary in that city. Here he proved
himself a strong supporter of unsectarian religious teaching. In 1846 he
established the Pestalozzi institution at Pankow, and the Pestalozzi
societies for the support of teachers' widows and orphans. In 1850 he
retired on a pension, but continued vigorously to advocate his
educational views. In 1858 he was elected to the chamber of deputies as
member for the city of Berlin, and voted with the Liberal opposition. He
died in Berlin on the 7th of July 1866. Diesterweg was a voluminous
writer on educational subjects, and was the author of various school
text-books.



DIET, a term used in two senses, (1) food or the regulation of feeding
(see DIETARY and DIETETICS), (2) an assembly or council (Fr. _diète_;
It. _dieta_; Low Lat. _diaeta_; Ger. _Tag_). We are here concerned only
with this second sense. In modern usage, though in Scotland the term is
still sometimes applied to any assembly or session, it is practically
confined to the sense of an assembly of estates or of national or
federal representatives. The origin of the word in this connotation is
somewhat complicated. It is undoubtedly ultimately derived from the
Greek [Greek: diaita] (Lat. _diaeta_), which meant "mode of life" and
thence "prescribed mode of life," the English "diet" or "regimen." This
was connected with the verb [Greek: diaitan], in the sense of "to rule,"
"to regulate"; compare the office of [Greek: diaitêtês] at Athens, and
_dieteta_, "umpire," in Late Latin. In both Greek and Latin, too, the
word meant "a room," from which the transition to "a place of assembly"
and so to "an assembly" would be easy. In the latter sense the word,
however, actually occurs only in Low Latin, Du Cange (_Glossarium_,
s.v.) deriving it from the late sense of "meal" or "feast," the Germans
being accustomed to combine their political assemblies with feasting. It
is clear, too, that the word _diaeta_ early became confused with Lat.
_dies_, "day" (Ger. _Tag_), "especially a set day, a day appointed for
public business; whence, by extension, meeting for business, an
assembly" (Skeat). Instances of this confusion are given by Du Cange,
e.g. _diaeta_ for _dieta_, "a day's journey" (also an obsolete sense of
"diet" in English), and _dieta_ for "the ordinary course of the church,"
i.e. "the daily office," which suggests the original sense of _diaeta_
as "a prescribed mode of life."

The word "diet" is now used in English for the _Reichstag_, "imperial
diet" of the old Holy Roman Empire; for the _Bundestag_, "federal diet,"
of the former Germanic confederation; sometimes for the _Reichstag_ of
the modern German empire; for the _Landtage_, "territorial diets" of the
constituent states of the German and Austrian empires; as well as for
the former or existing federal or national assemblies of Switzerland,
Hungary, Poland, &c. Although, however, the word is still sometimes used
of all the above, the tendency is to confine it, so far as contemporary
assemblies are concerned, to those of subordinate importance. Thus
"parliament" is often used of the German _Reichstag_ or of the Russian
Landtag, while the _Landtag_, e.g. of Styria, would always be rendered
"diet." In what follows we confine ourselves to the diet of the Holy
Roman Empire and its relation to its successors in modern Germany.

The origin of the diet, or deliberative assembly, of the Holy Roman
Empire must be sought in the _placitum_ of the Frankish empire. This
represented the tribal assembly of the Franks, meeting (originally in
March, but after 755 in May, whence it is called the Campus Maii) partly
for a military review on the eve of the summer campaign, partly for
deliberation on important matters of politics and justice. By the side
of this larger assembly, however, which contained in theory, if not in
practice, the whole body of Franks available for war, there had
developed, even before Carolingian times, a smaller body composed of the
magnates of the Empire, both lay and ecclesiastical. The germ of this
smaller body is to be found in the episcopal synods, which, afforced by
the attendance of lay magnates, came to be used by the king for the
settlement of national affairs. Under the Carolingians it was usual to
combine the assembly of magnates with the _generalis conventus_ of the
"field of May," and it was in this inner assembly, rather than in the
general body (whose approval was merely formal, and confined to matters
momentous enough to be referred to a general vote), that the centre of
power really lay. It is from the assembly of magnates that the diet of
medieval Germany springs. The general assembly became meaningless and
unnecessary, as the feudal array gradually superseded the old levy _en
masse_, in which each freeman had been liable to service; and after the
close of the 10th century it no longer existed.

The imperial diet (_Reichstag_) of the middle ages might sometimes
contain representatives of Italy, the _regnum Italicum_; but it was
practically always confined to the magnates of Germany, the _regnum
Teutonicum_. Upon occasion a summons to the diet might be sent even to
the knights, but the regular members were the princes (_Fürsten_), both
lay and ecclesiastical. In the 13th century the seven electors began to
disengage themselves from the prince as a separate element, and the
Golden Bull (1356) made their separation complete; from the 14th century
onwards the nobles (both counts and other lords) are regarded as regular
members; while after 1250 the imperial and episcopal towns often appear
through their representatives. By the 14th century, therefore, the
originally homogeneous diet of princes is already, at any rate
practically if not yet in legal form, divided into three colleges--the
electors, the princes and nobles, and the representatives of the towns
(though, as we shall see, the latter can hardly be reckoned as regular
members until the century of the Reformation). Under the Hohenstaufen it
is still the rule that every member of the diet must attend personally,
or lose his vote; at a later date the principle of representation by
proxy, which eventually made the diet into a mere congress of envoys,
was introduced. By the end of the 13th century the vote of the majority
had come to be regarded as decisive; but in accordance with the strong
sense of social distinctions which marks German history, the quality as
well as the quantity of votes was weighed, and if the most powerful of
the princes were agreed, the opinion of the lesser magnates was not
consulted. The powers of the medieval diet extended to matters like
legislation, the decision upon expeditions (especially the _expeditio
Romana_), taxation and changes in the constitution of the principalities
or the Empire. The election of the king, which was originally regarded
as one of the powers of the diet, had passed to the electors by the
middle of the 13th century.

A new era in the history of the diet begins with the Reformation. The
division of the diet into three colleges becomes definite and precise;
the right of the electors, for instance, to constitute a separate
college is explicitly recognized as a matter of established custom in
1544. The representatives of the towns now become regular members. In
the 15th century they had only attended when special business, such as
imperial reform or taxation, fell under discussion; in 1500, however,
they were recognized as a separate and regular estate, though it was not
until 1648 that they were recognized as equal to the other estates of
the diet. The estate of the towns, or college of municipal
representatives, was divided into two benches, the Rhenish and the
Swabian. The estate of the princes and counts, which stood midway
between the electors and the towns, also attained, in the years that
followed the Reformation, its final organization. The vote of the great
princes ceased to be personal, and began to be territorial. This had two
results. The division of a single territory among the different sons of
a family no longer, as of old, multiplied the voting power of the
family; while in the opposite case, the union of various territories in
the hands of a single person no longer meant the extinction of several
votes, since the new owner was now allowed to give a vote for each of
his territories. The position of the counts and other lords, who joined
with the princes in forming the middle estate, was finally fixed by the
middle of the 17th century. While each of the princes enjoyed an
individual vote, the counts and other lords were arranged in groups,
each of which voted as a whole, though the whole of its vote
(_Kuriatstimme_) only counted as equal to the vote of a single prince
(_Virilstimme_). There were six of these groups; but as the votes of the
whole college of princes and counts (at any rate in the 18th century)
numbered 100, they could exercise but little weight.

The last era in the history of the diet may be said to open with the
treaty of Westphalia (1648). The treaty acknowledged that Germany was no
longer a unitary state, but a loose confederation of sovereign princes;
and the diet accordingly ceased to bear the character of a national
assembly, and became a mere congress of envoys. The "last diet" which
issued a regular recess (_Reichsabschied_--the term applied to the
_acta_ of the diet, as formally compiled and enunciated at its
dissolution) was that of Regensburg in 1654. The next diet, which met at
Regensburg in 1663, never issued a recess, and was never dissolved; it
continued in permanent session, as it were, till the dissolution of the
Empire in 1806. This result was achieved by the process of turning the
diet from an assembly of principals into a congress of envoys. The
emperor was represented by two _commissarii_; the electors, princes and
towns were similarly represented by their accredited agents. Some
legislation was occasionally done by this body; a _conclusum imperii_
(so called in distinction from the old _recessus imperii_ of the period
before 1663) might slowly (very slowly--for the agents, imperfectly
instructed, had constantly to refer matters back to their principals) be
achieved; but it rested with the various princes to promulgate and
enforce the _conclusum_ in their territories, and they were sufficiently
occupied in issuing and enforcing their own decrees. In practice the
diet had nothing to do; and its members occupied themselves in
"wrangling about chairs"--that is to say, in unending disputes about
degrees and precedences.

In the Germanic Confederation, which occupies the interval between the
death of the Holy Roman Empire and the formation of the North German
Confederation (1815-1866), a diet (_Bundestag_) existed, which was
modelled on the old diet of the 18th century. It was a standing congress
of envoys at Frankfort-on-Main. Austria presided in the diet, which, in
the earlier years of its history, served, under the influence of
Metternich, as an organ for the suppression of Liberal opinion. In the
North German Confederation (1867-1870) a new departure was made, which
has been followed in the constitution of the present German empire. Two
bodies were instituted--a _Bundesrat_, which resembles the old diet in
being a congress of envoys sent by the sovereigns of the different
states of the confederation, and a _Reichstag_, which bears the name of
the old diet, but differs entirely in composition. The new Reichstag is
a popular representative assembly, based on wide suffrage and elected by
ballot; and, above all, it is an assembly representing, not the several
states, but the whole Empire, which is divided for this purpose into
electoral districts. Both as a popular assembly, and as an assembly
which represents the whole of a united Germany, the new Reichstag goes
back, one may almost say, beyond the diet even of the middle ages, to
the days of the old Teutonic folk-moot.

  See R. Schröder, _Lehrbuch der deutschen Rechtsgeschichte_ (1902), pp.
  149, 508, 820, 880. Schröder gives a bibliography of monographs
  bearing on the history of the medieval diet.     (E. BR.)



DIETARY, in a general sense, a system or course of diet, in the sense of
food; more particularly, such an allowance and regulation of food as
that supplied to workhouses, the army and navy, prisons, &c. Lowest in
the scale of such dietaries comes what is termed "bare existence" diet,
administered to certain classes of the community who have a claim on
their fellow-countrymen that their lives and health shall be preserved
_in statu quo_, but nothing further. This applies particularly to the
members of a temporarily famine-stricken community. Before the days of
prison reform, too, the dietary scale of many prisons was to a certain
extent penal, in that the food supplied to prisoners was barely
sufficient for existence. Nowadays more humane principles apply; there
is no longer the obvious injustice of applying the same scale of
quantity and quality to all prisoners under varying circumstances of
constitution and surroundings, and whether serving long or short periods
of imprisonment.

  The system of dietary in force in the local and convict prisons of
  England and Wales is that recommended by the Home Office on the advice
  of a departmental committee. As to the local prison dietary, its
  application is based on (1) the principle of variation of diet with
  length of sentence; (2) the system of progressive dietary; (3) the
  distinction between hard labour diets and non-hard labour diets; (4)
  the differentiation of diet according to age and sex. There are three
  classes of diet, classes A, B and C. Class A diet is given to
  prisoners undergoing not more than seven days' imprisonment. The food
  is good and wholesome, but sufficiently plain and unattractive, so as
  not to offer temptation to the loafer or mendicant. It is given in
  quantity sufficient to maintain health and strength during the single
  week. Prisoners sentenced to more than seven days and not more than
  fourteen days are given class A diet for the first seven days and
  class B for the remainder of the sentence. In most of the local
  prisons in England and Wales prisoners sentenced to hard labour
  received hard labour diet, although quite 60% were unable to perform
  the hardest forms of prison labour either through physical defect, age
  or infirmity. The departmental committee of 1899 in their report
  recommended that no distinction should be made between hard labour and
  non-hard labour diets. Class A diet is as follows:--_Breakfast_,
  Bread, 8 oz. daily (6 oz. for women and juveniles) with 1 pint of
  gruel. Juveniles (males and females under sixteen years of age) get,
  in addition, ½ pint of milk. _Dinner_, 8 oz. of bread daily, with 1
  pint of porridge on three days of the week, 8 oz. of potatoes
  (representing the vegetable element) on two other days, and 8 oz. of
  suet pudding (representing the fatty element) on the other two days.
  _Supper_, the breakfast fare repeated.

  Class B diet, which is also given to (1) prisoners on remand or
  awaiting trial, (2) offenders of the 1st division who do not maintain
  themselves, (3) offenders of the 2nd division and (4) debtors, is as
  shown in Table I.

  Class C diet is class B amplified, and is given to those prisoners
  serving sentences of three months and over.


      TABLE I.

    +------------+------------------+--------+--------+-----------+
    |            |                  |  Men.  | Women. | Juveniles.|
    +------------+------------------+--------+--------+-----------+
    |            |                  |        |        |           |
    | Breakfast. | Daily:--         |        |        |           |
    |            |    Bread         |  8 oz. |  6 oz. |   6 oz.   |
    |            |    Gruel         |  1 pt. |  1 pt. |   1 pt.   |
    |            |    Milk          |    ..  |    ..  |   ½ pt.   |
    |            |                  |        |        |           |
    | Dinner.    | Sunday:--        +--------+--------+-----------+
    |            |    Bread         |  6 oz. |       6 oz.        |
    |            |    Potatoes      |  8  "  |       8  "         |
    |            |    Cooked meat,  |        |                    |
    |            |      preserved   |  4  "  |       3  "         |
    |            |      by heat     |        |                    |
    |            | Monday:--        |        |                    |
    |            |    Bread         |  6 oz. |       6 oz.        |
    |            |    Potatoes      |  8  "  |       8  "         |
    |            |    Beans         | 10  "  |       8  "         |
    |            |    Fat bacon     |  2  "  |       1  "         |
    |            |                  |        |                    |
    |            | Tuesday:--       |        |                    |
    |            |    Bread         |  6 oz  |       6 oz.        |
    |            |    Potatoes      |  8  "  |       8  "         |
    |            |    Soup          |  1 pt. |       1 pt.        |
    |            |                  |        |                    |
    |            | Wednesday:--     |        |                    |
    |            |    Bread         |  6 oz. |       6 oz.        |
    |            |    Potatoes      |  8  "  |       8  "         |
    |            |    Suet pudding  | 10  "  |       8  "         |
    |            |                  |        |                    |
    |            | Thursday:--      |        |                    |
    |            |    Bread         |  6 oz. |       6 oz.        |
    |            |    Potatoes      |  8  "  |       8  "         |
    |            |    Cooked beef,  |        |                    |
    |            |      without bone|  4  "  |       3  "         |
    |            |                  |        |                    |
    |            | Friday:--        |        |                    |
    |            |    Bread         |  6 oz. |       6 oz.        |
    |            |    Potatoes      |  8  "  |       8  "         |
    |            |    Soup          |  1 pt. |       1 pt.        |
    |            |                  |        |                    |
    |            | Saturday:--      |        |                    |
    |            |    Bread         |  6 oz. |       6 oz.        |
    |            |    Potatoes      |  8  "  |       8  "         |
    |            |    Suet pudding  | 10  "  |       8  "         |
    |            |                  |        |                    |
    | Supper.    | Daily:--         |        +---------+----------+
    |            |    Bread         |  8 oz. |   6 oz. |   6 oz.  |
    |            |    Porridge      |  1 pt. |         |          |
    |            |    Gruel         |        |   1 pt. |          |
    |            |    Cocoa         |        |         |   1 pt.  |
    +------------+------------------+--------+---------+----------+


      TABLE II.
    +------------------+-----+------+------+------+------+------+------+
    |                  |Sun. | Mon. | Tue. | Wed. | Thu. | Fri. | Sat. |
    +------------------+-----+------+------+------+------+------+------+
    | Breakfast.       |     |      |      |      |      |      |      |
    |  Bread.       oz.|  8  |   4  |   4  |   4  |   4  |   4  |   4  |
    |  Porridge.    pt.|  *  |   1½ |   1½ |   1½ |   1½ |   1½ |   1½ |
    +------------------+-----+------+------+------+------+------+------+
    | Dinner.          |     |      |      |      |      |      |      |
    |  Bread.       oz.|  4  |   6  |  ..  |   4  |   4  |   8  |   6  |
    |  Beef.        oz.|  4½ |  ..  |  ..  |  ..  |   4½ |  ..  |  ..  |
    |  Vegetables.  oz.| 12  |  ..  |  ..  |  12  |  12  |  ..  |  ..  |
    |  Barley Soup. pt.| ..  |   1½ |  ..  |  ..  |  ..  |  ..  |  ..  |
    |  Pork.        oz.| ..  |  ..  |   4½ |  ..  |  ..  |  ..  |  ..  |
    |  Beans.       oz.| ..  |  ..  |  12  |  ..  |  ..  |  ..  |  ..  |
    |  Fish.        oz.| ..  |  ..  |  ..  |  10  |  ..  |  ..  |  ..  |
    |  Cheese.      oz.| ..  |  ..  |  ..  |  ..  |  ..  |   3  |  ..  |
    |  Broth.       pt.| ..  |  ..  |  ..  |  ..  |  ..  |   1  |  ..  |
    |  Irish Stew.  pt.| ..  |  ..  |  ..  |  ..  |  ..  |  ..  |   1  |
    +------------------+-----+------+------+------+------+------+------+
    | Supper.          |     |      |      |      |      |      |      |
    |  Bread.       oz.|  8  |   6  |   6  |   6  |   8  |   6  |   6  |
    |  Butter.      oz.|   ½ |  ..  |  ..  |  ..  |  ..  |  ..  |  ..  |
    |  Tea.         pt.|  1  |  ..  |  ..  |  ..  |  ..  |  ..  |  ..  |
    |  Gruel.       pt.| ..  |   1½ |   1½ |   1½ |  ..  |   1½ |   1½ |
    |  Broth.       pt.| ..  |  ..  |  ..  |  ..  |   1  |  ..  |  ..  |
    |  Cheese.      oz.| ..  |  ..  |  ..  |  ..  |   2  |  ..  |  ..  |
    +------------------+-----+------+------+------+------+------+------+
        * On Sundays 1 pint of tea and 2½ oz. of butter are given instead
            of porridge.

  The dietary of convict prisons, in which prisoners are all under long
  sentence, is divided into a diet for convicts employed at hard labour
  and a diet for convicts employed at sedentary, indoor and light
  labour. It will be found set forth in the Blue-book mentioned above.
  The sparest of all prison diets is called "punishment diet," and is
  administered for offences against the internal discipline of the
  prison. It is limited to a period of three days. It consists of 1 lb.
  of bread and as much water as the prisoner chooses to drink.

  In French prisons the dietary is nearly two pounds weight of bread,
  with two meals of thin soup (breakfast and dinner) made from potatoes,
  beans or other vegetables, and on two days a week made from meat. In
  France the canteen system is in vogue, additional food, such as
  sausages, cheese, fruit, &c., may be obtained by the prisoner,
  according to the wages he receives for his labours. The dietary of
  Austrian prisons is 1½ lb. of bread daily, a dinner of soup on four
  days of the week, and of meat on the other three days, with a supper
  of soup or vegetable stew. Additional food can be purchased by the
  prisoner out of his earnings.

  These dietaries may be taken as more or less typical of the ordinary
  prison fare in most civilized countries, though in some countries it
  may err on the side of severity, as in Sweden, prisoners being given
  only two meals a day, one at mid-day and one at seven p.m., porridge
  or gruel being the principal element in both meals. On the other hand,
  the prison dietaries of many of the United States prisons go to the
  other extreme, fresh fish, green vegetables, even coffee and fruit,
  figuring in the dietary.

  Another class of dietary is that given to paupers. In England, until
  1900, almost every individual workhouse had its own special dietary,
  with the consequence that many erred on the side of scantiness and
  unsuitability, while others were too lavish. By an order of the Local
  Government Board of that year, acting on a report of a committee, all
  inmates of workhouses, with the exception of the sick, children under
  three years of age, and certain other special cases, are dieted in
  accordance with certain dietary tables as framed and settled by the
  board. The order contained a great number of different rations, it
  being left to the discretion of the guardians as to the final
  settlement of the tables. For adult inmates the dietary tables are for
  each sex respectively, two in number, one termed "plain diet" and the
  other "infirm diet." All male inmates certified as healthy able-bodied
  persons receive plain diet only. All inmates, however, in workhouses
  are kept employed according to their capacity and ability, and this is
  taken into consideration in giving allowances of food. For instance,
  for work with sustained exertion, such as stone-breaking, digging,
  &c., more food is given than for work without sustained exertion, such
  as wood-chopping, weeding or sewing. Table II. shows an example of a
  workhouse dietary.

  In the casual wards of workhouses the dietary is plainer, consisting
  of 8 oz. of bread, or 6 oz. of bread and one pint of gruel or broth
  for breakfast; the same for supper; for dinner 8 oz. of bread and 1½
  oz. of cheese or 6 oz. of bread and one pint of soup. The American
  poor law system is based broadly on that of England, and the methods
  of relief are much the same. Each state, however, makes its own
  regulations, and there is considerable diversity in workhouse
  dietaries in consequence. The German system of poor relief is more
  methodical than those of England and America. The really deserving are
  treated with more commiseration, and a larger amount of outdoor
  relief is given than in England. There is no casual ward, tramps and
  beggars being liable to penal treatment, but there are "relief
  stations," somewhat corresponding to casual wards, where destitute
  persons tramping from one place to another can obtain food and lodging
  in return for work done.

  In the British navy certain staple articles of diet are supplied to
  the men to the value approximately of 6d. per diem--the standard
  government ration--and, in addition, a messing allowance of 4d. per
  diem, which may either be expended on luxuries in the canteen, or in
  taking up government provisions on board ship, in addition to the
  standard ration. The standard ration as recommended in 1907 by a
  committee appointed to inquire into the question of victualling in the
  navy is as follows:--

  _Service Afloat._

    1 lb. bread (or ¾ lb. bread and ¼ lb. trade flour).
    ½ lb. fresh meat.
    1 lb. fresh vegetables.
    1/8 pint spirit.
    4 oz. sugar.
    ½ oz. tea (or 1 oz. coffee for every ¼ oz. tea).
    ½ oz. ordinary or soluble chocolate (or 1 oz. coffee).
    ¾ oz. condensed milk.
    1 oz. jam or marmalade.
    4 oz. preserved meat on _one_ day of the week in harbour, or on
       _two_ days at sea.

  Mustard, pepper, vinegar, and salt as required.

  Substitute for soft bread when the latter is not available--
      ½ lb. biscuit (new type) or 1 lb. flour.

  Substitutes for fresh meat when the latter is not available:--

              /(1) Salt pork day:--
             |     ½ lb. salt pork.
             |     ¼ lb. split peas.
             |     Celery seed, ½ oz. to every 8 lb. of split peas put
             |       into the coppers.
      On     |     ½ lb. potatoes (or 1 oz. compressed vegetables).
   alternate<
     days    | (2) Preserved meat day:--
             |     6 oz. preserved meat.
             |     8 oz. trade flour   \
             |     ¾ oz. refined suet   > or 4 oz. rice.
             |     2 oz. raisins       /
              \    ½ lb. potatoes (or 1 oz. compressed vegetables).

  On shore establishments and depot ships ¼ pt. fresh milk is issued in
  lieu of the ¾ oz. of condensed milk.

  In the United States navy there is more liberality and variety of
  diet, the approximate daily cost of the rations supplied being 1s. 3d.
  per head. In the American mercantile marine, too, according to the
  scale sanctioned by act of Congress (December 21, 1898) for American
  ships, the seaman is better off than in the British merchant service.
  The scale is shown in Table III.


      TABLE III.

    +---------+-----------------------++-----------+----------------+
    | Weekly  |       Articles.       ||  Weekly   |    Articles.   |
    | Scale.  |                       ||  Scale.   |                |
    +---------+-----------------------++-----------+----------------+
    | 3½  lb. | Biscuits.             || 7/8  oz.  | Tea.           |
    | 3¾   "  | Salt beef.            || 21    "   | Sugar.         |
    | 3    "  |  "   pork.            ||  1½  lb.  | Molasses.      |
    | 1½   "  | Flour.                ||  9   oz.  | Fruits, dried. |
    | 2    "  | Meats, preserved.     ||   ¾  pt.  | Pickles.       |
    | 10½  "  | Bread, fresh (8 lb.   ||   1   "   | Vinegar.       |
    |         |   flour in lieu).     ||  8   oz.  | Corn Meal.     |
    | 1    "  | Fish, dried.          || 12    "   | Onions.        |
    | 7    "  | Potatoes or yams.     ||  7    "   | Lard.          |
    | 1    "  | Tomatoes, preserved.  ||  7    "   | Butter.        |
    |2/3   "  | Peas.                 ||   ¼   "   | Mustard.       |
    |2/3   "  | Calavances.           ||   ¼   "   | Pepper.        |
    |2/3   "  | Rice.                 ||   ¼   "   | Salt.          |
    | 5¼  oz. | Coffee, green.        ||           |                |
    +---------+-----------------------++-----------+----------------+

  In the British mercantile marine there is no scale of provisions
  prescribed by the Board of Trade; there is, however, a traditional
  scale very generally adopted, having the sanction of custom only and
  seldom adhered to. The following dietary scale for steerage
  passengers, laid down in the 12th schedule of the Merchant Shipping
  Act 1894, is of interest. See Table IV.

  Certain substitutions may be made in this scale at the option of the
  master of any emigrant ship, provided that the substituted articles
  are set forth in the contract tickets of the steerage passengers.

  In the British army the soldier is fed partly by a system of
  co-operation. He gets a free ration from government of 1 lb. of bread
  and ¾ lb. of meat; in addition there is a messing allowance of 3½d.
  per man per day. He is able to supplement his food by purchases from
  the canteen. Much depends on the individual management in each
  regiment as to the satisfactory expenditure of the messing allowance.
  In some regiments an allowance is made from the canteen funds towards
  messing in addition to that granted by the government. The ordinary
  _field_ ration of the British soldier is 1½ lb. of bread or 1 lb. of
  biscuit; 1 lb. of fresh, salt or preserved meat; ½ oz. of coffee; 1/6
  oz. of tea; 2 oz. of sugar; ½ oz. of salt, 1/36 oz. of pepper, the
  whole weighing something over 2 lb. 3 oz. This cannot be looked on as
  a fixed ration, as it varies in different campaigns, according to the
  country into which the troops may be sent. The Prussian soldier during
  peace gets weekly from his canteen 11 lb. 1 oz. of rye bread and not
  quite 2½ lb. of meat. This is obviously insufficient, but under


      TABLE IV.--_Weekly, per Statute Adult._

    +---------------------------+-------------------+-------------------+
    |                           |      Scale A.     |      Scale B.     |
    |                           |For voyages not    |For voyages        |
    |                           | exceeding 84 days | exceeding 84 days |
    |                           | for sailing ships | for sailing ships |
    |                           | or 50 days        | or 50 days        |
    |                           | for steamships.   | for steamships.   |
    +---------------------------+-------------------+-------------------+
    |                           |     lb.   oz.     |     lb.   oz.     |
    | Bread or biscuit, not     |                   |                   |
    |  inferior to navy biscuit |      3    8       |      3    8       |
    | Wheaten flour             |      1    0       |      2    0       |
    | Oatmeal                   |      1    8       |      1    0       |
    | Rice                      |      1    8       |      0    8       |
    | Peas                      |      1    8       |      1    8       |
    | Beef                      |      1    4       |      1    4       |
    | Pork                      |      1    0       |      1    0       |
    | Butter                    |        ..         |      0    4       |
    | Potatoes                  |      2    0       |      2    0       |
    | Sugar                     |      1    0       |      1    0       |
    | Tea                       |      0    2       |      0    2       |
    | Salt                      |      0    2       |      0    2       |
    | Pepper (white or          |                   |                   |
    |  black), ground           |      0    0½      |      0    0½      |
    | Vinegar                   |      1 gill       |      1 gill       |
    | Preserved meat            |        ..         |      1    0       |
    | Suet                      |                   |      0    6       |
    | Raisins                   |                   |      0    8       |
    | Lime juice                |                   |      0    6       |
    +---------------------------+-------------------+-------------------+

  the conscription system it is reckoned that he will be able to make up
  the deficiency out of his own private means, or obtain charitable
  contributions from his friends. In the French infantry of the line
  each man during peace gets weekly 15 lb. of bread, 3-3/10 lb. of meat,
  2½ lb. of haricot beans or other vegetables, with salt and pepper, and
  1¾ oz. of brandy.

  An Austrian under the same circumstances receives 13.9 lb. of bread, ½
  lb. of flour and 3.3 lb. of meat.

  The Russian conscript is allowed weekly:--

    Black bread              7 lb.
    Meat                     7 lb.
    Kvass (beer)             7.7 quarts.
    Sour cabbage             24½ gills = 122½ oz.
    Barley                   24½ gills = 122½ oz.
    Salts                    10½ oz.
    Horse-radish             28 grains.
    Pepper                   28 grains.
    Vinegar                  5½ gills = 26½ oz.



DIETETICS, the science of diet, i.e. the food and nutrition of man in
health and disease (see NUTRITION). This article deals mainly with that
part of the subject which has to do with the composition and nutritive
values of foods and their adaptation to the use of people in health. The
principal topics considered are: (1) Food and its functions; (2)
Metabolism of matter and energy; (3) Composition of food materials; (4)
Digestibility of food; (5) Fuel value of food; (6) Food consumption; (7)
Quantities of nutrients needed; (8) Hygienic economy of food; (9)
Pecuniary economy of food.

1. _Food and its Functions._--For practical purposes, food may be
defined as that which, when taken into the body, may be utilized for the
formation and repair of body tissue, and the production of energy. More
specifically, food meets the requirements of the body in several ways.
It is used for the formation of the tissues and fluids of the body, and
for the restoration of losses of substance due to bodily activity. The
potential energy of the food is converted into heat or muscular work or
other forms of energy. In being thus utilized, food protects body
substance or previously acquired nutritive material from consumption.
When the amount of food taken into the body is in excess of immediate
needs, the surplus may be stored for future consumption.

Ordinary food materials, such as meat, fish, eggs, vegetables, &c.,
consist of inedible materials, or _refuse_, e.g. bone of meat and fish,
shell of eggs, rind and seed of vegetables; and _edible material_, as
flesh of meat and fish, white and yolk of eggs, wheat flour, &c. The
edible material is by no means a simple substance, but consists of
_water_, and some or all of the compounds variously designated as food
stuffs, proximate principles, nutritive ingredients or nutrients, which
are classified as _protein_, _fats_, _carbohydrates_ and _mineral
matters_. These have various functions in the nourishment of the body.

The _refuse_ commonly contains compounds similar to those in the food
from which it is derived, but since it cannot be eaten, it is usually
considered as a non-nutrient. It is of importance chiefly in a
consideration of the pecuniary economy of food. _Water_ is also
considered as a non-nutrient, because although it is a constituent of
all the tissues and fluids of the body, the body may obtain the water it
needs from that drunk; hence, that contained in the food materials is of
no special significance as a nutrient.

_Mineral matters_, such as sulphates, chlorides, phosphates and
carbonates of sodium, potassium, calcium, &c., are found in different
combinations and quantities in most food materials. These are used by
the body in the formation of the various tissues, especially the
skeletal and protective tissues, in digestion, and in metabolic
processes within the body. They yield little or no energy, unless
perhaps the very small amount involved in their chemical transformation.

Protein[1] is a term used to designate the whole group of nitrogenous
compounds of food except the nitrogenous fats. It includes the
albuminoids, as albumin of egg-white, and of blood serum, myosin of meat
(muscle), casein of milk, globulin of blood and of egg yolk, fibrin of
blood, gluten of flour; the gelatinoids, as gelatin and allied
substances of connective tissue, collagen of tendon, ossein of bone and
the so-called extractives (e.g. creatin) of meats; and the amids (e.g.
asparagin) and allied compounds of vegetables and fruits.

The albuminoids and gelatinoids, classed together as proteids, are the
most important constituents of food, because they alone can supply the
nitrogenous material necessary for the formation of the body tissues.
For this purpose, the albuminoids are most valuable. Both groups of
compounds, however, supply the body with energy, and the gelatinoids in
being thus utilized protect the albuminoids from consumption for this
purpose. When their supply in the food is in excess of the needs of the
body, the surplus proteids may be converted into body fat and stored.

The so-called extractives, which are the principal constituents of meat
extract, beef tea and the like, act principally as stimulants and
appetizers. It has been believed that they serve neither to build tissue
nor to yield energy, but recent investigations[2] indicate that creatin
may be metabolized in the body.

The _fats_ of food include both the animal fats and the vegetable oils.
The _carbohydrates_ include such compounds as starches, sugars and the
fibre of plants or cellulose, though the latter has but little value as
food for man. The more important function of both these classes of
nutrients is to supply energy to the body to meet its requirements above
that which it may obtain from the proteids. It is not improbable that
the atoms of their molecules as well as those from the proteids are
built up into the protoplasmic substance of the tissues. In this sense,
these nutrients may be considered as being utilized also for the
formation of tissue; but they are rather the accessory ingredients,
whereas the proteids are the essential ingredients for this purpose. The
fats in the food in excess of the body requirements may be stored as
body fat, and the surplus carbohydrates may also be converted into fat
and stored.

To a certain extent, then, the nutrients of the food may substitute each
other. All may be incorporated into the protoplasmic structure of body
tissue, though only the proteids can supply the essential nitrogenous
ingredients; and apart from the portion of the proteid material that is
indispensable for this purpose, all the nutrients are used as a source
of energy. If the supply of energy in the food is not sufficient, the
body will use its own proteid and fat for this purpose. The gelatinoids,
fats and carbohydrates in being utilized for energy protect the body
proteids from consumption. The fat stored in the body from the excess of
food is a reserve of energy material, on which the body may draw when
the quantity of energy in the food is insufficient for its immediate
needs.

What compounds are especially concerned in intellectual activity is not
known. The belief that fish is especially rich in phosphorus and
valuable as a brain food has no foundation in observed fact.

2. _Metabolism of Matter and Energy._--The processes of nutrition thus
consist largely of the transformation of food into body material and the
conversion of the potential energy of both food and body material into
the kinetic energy of heat and muscular work and other forms of energy.
These various processes are generally designated by the term metabolism.
The metabolism of matter in the body is governed largely by the needs of
the body for energy. The science of nutrition, of which the present
subject forms a part, is based on the principle that the transformations
of matter and energy in the body occur in accordance with the laws of
the conservation of matter and of energy. That the body can neither
create nor destroy matter has long been universally accepted. It would
seem that the transformation of energy must likewise be governed by the
law of the conservation of energy; indeed there is every reason a priori
to believe that it must; but the experimental difficulties in the way of
absolute demonstration of the principle are considerable. For such
demonstration it is necessary to prove that the income and expenditure
of energy are equal. Apparatus and methods of inquiry devised in recent
years, however, afford means for a comparison of the amounts of both
matter and energy received and expended by the body, and from the
results obtained in a large amount of such research, it seems probable
that the law obtains in the living organism in general.

The first attempt at such demonstration was made by M. Rubner[3] in
1894, experimenting with dogs doing no external muscular work. The
income of energy (as heat) was computed, but the heat eliminated was
measured. In the average of eight experiments continuing forty-five
days, the two quantities agreed within 0.47%, thus demonstrating what it
was desired to prove--that the heat given off by the body came solely
from the oxidation of food within it. Results in accordance with these
were reported by Studenski[4] in 1897, and by Laulanie[5] in 1898.

The most extensive and complete data yet available on the subject have
been obtained by W. O. Atwater, F. G. Benedict and associates[6] in
experiments with men in the respiration calorimeter, in which a subject
may remain for several consecutive days and nights. These experiments
involve actual weighing and analyses of the food and drink, and of the
gaseous, liquid and solid excretory products; determinations of
potential energy (heat of oxidation) of the oxidizable material received
and given off by the body (including estimation of the energy of the
material gained or lost by the body); and measurements of the amounts of
energy expended as heat and as external muscular work. By October 1906
eighty-eight experiments with fifteen different subjects had been
completed. The separate experiments continued from two to thirteen days,
making a total of over 270 days.


    TABLE I.--_Percentage Composition of some Common Food Materials._

  +--------------------------+--------+--------+---------+------+---------+--------+-----------+
  |     Food Material.       | Refuse.|  Water.| Protein.| Fat. | Carbo-  | Mineral| Fuel Value|
  |                          |        |        |         |      |hydrates.| Matter.| per lb.   |
  +--------------------------+--------+--------+---------+------+---------+--------+-----------+
  |                          |        |        |         |      |         |        |           |
  |                          |    %   |    %   |    %    |   %  |    %    |    %   | Calories. |
  |Beef, fresh (medium fat)--|        |        |         |      |         |        |           |
  |  Chuck                   |  16.3  |  52.6  |  15.5   | 15.0 |   . .   |   0.8  |    910    |
  |  Loin                    |  13.3  |  52.5  |  16.1   | 17.5 |   . .   |   0.9  |   1025    |
  |  Ribs                    |  20.8  |  43.8  |  13.9   | 21.2 |   . .   |   0.7  |   1135    |
  |  Round                   |   7.2  |  60.7  |  19.0   | 12.8 |   . .   |   1.0  |    890    |
  |  Shoulder                |  16.4  |  56.8  |  16.4   |  9.8 |   . .   |   0.9  |    715    |
  |Beef, dried and smoked    |   4.7  |  53.7  |  26.4   |  6.9 |   . .   |   8.9  |    790    |
  |Veal--                    |        |        |         |      |         |        |           |
  |  Leg                     |  14.2  |  60.1  |  15.5   |  7.9 |   . .   |   0.9  |    625    |
  |  Loin                    |  16.5  |  57.6  |  16.6   |  9.0 |   . .   |   0.9  |    685    |
  |  Breast                  |  21.3  |  52.0  |  15.4   | 11.0 |   . .   |   0.8  |    745    |
  |Mutton--                  |        |        |         |      |         |        |           |
  |  Leg                     |  18.4  |  51.2  |  15.1   | 14.7 |   . .   |   0.8  |    890    |
  |  Loin                    |  16.0  |  42.0  |  13.5   | 28.3 |   . .   |   0.7  |   1415    |
  |  Flank                   |   9.9  |  39.0  |  13.8   | 36.9 |   . .   |   0.6  |   1770    |
  |Pork--                    |        |        |         |      |         |        |           |
  |  Loin                    |  19.7  |  41.8  |  13.4   | 24.2 |   . .   |   0.8  |   1245    |
  |  Ham, fresh              |  10.7  |  48.0  |  13.5   | 25.9 |   . .   |   0.8  |   1320    |
  |  Ham, smoked and salted  |  13.6  |  34.8  |  14.2   | 33.4 |   . .   |   4.2  |   1635    |
  |  Fat, salt               |  . .   |   7.9  |   1.9   | 86.2 |   . .   |   3.9  |   3555    |
  |  Bacon                   |   7.7  |  17.4  |   9.1   | 62.2 |   . .   |   4.1  |   2715    |
  |  Lard, refined           |  . .   |  . .   |  . .    |100.0 |   . .   |  . .   |   4100    |
  |Chicken                   |  25.9  |  47.1  |  13.7   | 12.3 |   . .   |   0.7  |    765    |
  |Turkey                    |  22.7  |  42.4  |  16.1   | 18.4 |   . .   |   0.8  |   1060    |
  |Goose                     |  17.6  |  38.5  |  13.4   | 29.8 |   . .   |   0.7  |   1475    |
  |Eggs                      |  11.2  |  65.5  |  13.1   |  9.3 |   . .   |   0.9  |    635    |
  |Cod, fresh                |  29.9  |  58.5  |  11.1   |  0.2 |   . .   |   0.8  |    220    |
  |Cod, salted               |  24.9  |  40.2  |  16.0   |  0.4 |   . .   |  18.5  |    325    |
  |Mackerel, fresh           |  44.7  |  40.4  |  10.2   |  4.2 |   . .   |   0.7  |    370    |
  |Herring, smoked           |  44.4  |  19.2  |  20.5   |  8.8 |   . .   |   7.4  |    755    |
  |Salmon, tinned            |  . .   |  63.5  |  21.8   | 12.1 |   . .   |   2.6  |    915    |
  |Oysters, shelled          |  . .   |  88.3  |   6.0   |  1.3 |    3.3  |   1.1  |    225    |
  |Butter                    |  . .   |  11.0  |   1.0   | 85.0 |   . .   |   3.0  |   3410    |
  |Cheese                    |  . .   |  34.2  |  25.9   | 33.7 |    2.4  |   3.8  |   1885    |
  |Milk, whole               |  . .   |  87.0  |   3.3   |  4.0 |    5.0  |   0.7  |    310    |
  |Milk, skimmed             |  . .   |  90.5  |   3.4   |  0.3 |    5.1  |   0.7  |    165    |
  |Oatmeal                   |  . .   |   7.7  |  16.7   |  7.3 |   66.2  |   2.1  |   1800    |
  |Corn (maize) meal         |  . .   |  12.5  |   9.2   |  1.9 |   75.4  |   1.0  |   1635    |
  |Rye flour                 |  . .   |  12.9  |   6.8   |  0.9 |   78.7  |   0.7  |   1620    |
  |Buckwheat flour           |  . .   |  13.6  |   6.4   |  1.2 |   77.9  |   0.9  |   1605    |
  |Rice                      |  . .   |  12.3  |   8.0   |  0.3 |   79.0  |   0.4  |   1620    |
  |Wheat flour, white        |  . .   |  12.0  |  11.4   |  1.0 |   75.1  |   0.5  |   1635    |
  |Wheat flour, graham       |  . .   |  11.3  |  13.3   |  2.2 |   71.4  |   1.8  |   1645    |
  |Wheat, breakfast food     |  . .   |   9.6  |  12.1   |  1.8 |   75.2  |   1.3  |   1680    |
  |Wheat bread, white        |  . .   |  35.3  |   9.2   |  1.3 |   53.1  |   1.1  |   1200    |
  |Wheat bread, graham       |  . .   |  35.7  |   8.9   |  1.8 |   52.1  |   1.5  |   1195    |
  |Rye bread                 |  . .   |  35.7  |   9.0   |  0.6 |   53.2  |   1.5  |   1170    |
  |Biscuit (crackers)        |  . .   |   6.8  |   9.7   | 12.1 |   69.7  |   1.7  |   1925    |
  |Macaroni                  |  . .   |  10.3  |  13.4   |  0.9 |   74.1  |   1.3  |   1645    |
  |Sugar                     |  . .   |  . .   |  . .    | . .  |  100.0  |  . .   |   1750    |
  |Starch (corn starch)      |  . .   |  . .   |  . .    | . .  |   90.0  |  . .   |   1680    |
  |Beans, dried              |  . .   |  12.6  |  22.5   |  1.8 |   59.6  |   3.5  |   1520    |
  |Peas, dried               |  . .   |  9.5   |  24.6   |  1.0 |   62.0  |   2.9  |   1565    |
  |Beets                     |  20.0  |  70.0  |   1.3   |  0.1 |    7.7  |   0.9  |    160    |
  |Cabbage                   |  50.0  |  44.2  |   0.7   |  0.2 |    4.5  |   0.4  |    100    |
  |Potatoes                  |  20.0  |  62.6  |   1.8   |  0.1 |   14.7  |   0.8  |    295    |
  |Sweet potatoes            |  20.0  |  55.2  |   1.4   |  0.6 |   21.9  |   0.9  |    440    |
  |Tomatoes                  |  . .   |  94.3  |   0.9   |  0.4 |    3.9  |   0.5  |    100    |
  |Apples                    |  25.0  |  63.3  |   0.3   |  0.3 |   10.8  |   0.3  |    190    |
  |Bananas                   |  35.0  |  48.9  |   0.8   |  0.4 |   14.3  |   0.6  |    260    |
  |Grapes                    |  25.0  |  58.0  |   1.0   |  1.2 |   14.4  |   0.4  |    295    |
  |Strawberries              |   5.0  |  85.9  |   0.9   |  0.6 |    7.0  |   0.6  |    150    |
  |Almonds                   |  45.0  |   2.7  |  11.5   | 30.2 |    9.5  |   1.1  |   1515    |
  |Brazil nuts               |  49.6  |   2.6  |   8.6   | 33.7 |    3.5  |   2.0  |   1485    |
  |Chestnuts                 |  16.0  |  37.8  |   5.2   |  4.5 |   35.4  |   1.1  |    915    |
  |Walnuts                   |  58.1  |   1.0  |   6.9   | 26.6 |    6.8  |   0.6  |   1250    |
  +--------------------------+--------+--------+---------+------+---------+--------+-----------+

In some cases the subjects were at rest; in others they performed
varying amounts of external muscular work on an apparatus by means of
which the amount of work done was measured. In some cases they fasted,
and in others they received diets generally not far from sufficient to
maintain nitrogen, and usually carbon, equilibrium in the body. In these
experiments the amount of energy expended by the body as heat and as
external muscular work measured in terms of heat agreed on the average
very closely with the amount of heat that would be produced by the
oxidation of all the matter metabolized in the body. The variations for
individual days, and in the average for individual experiments as well,
were in some cases appreciable, amounting to as much as 6%, which is not
strange in view of the uncertainties in physiological experimenting; but
in the average of all the experiments the energy of the expenditure was
above 99.9% of the energy of the income,--an agreement within one part
in 1000. While these results do not absolutely prove the application of
the law of the conservation of energy in the human body, they certainly
approximate very closely to such demonstration. It is of course possible
that energy may have given off from the body in other forms than heat
and external muscular work. It is conceivable, for example, that
intellectual activity may involve the transformation of physical energy,
and that the energy involved may be eliminated in some form now unknown.
But if the body did give off energy which was not measured in these
experiments, the quantity must have been extremely small. It seems fair
to infer from the results obtained that the metabolism of energy in the
body occurred in conformity with the law of the conservation of energy.

3. _Composition of Food Materials._--The composition of food is
determined by chemical analyses, the results of which are conventionally
expressed in terms of the nutritive ingredients previously described. As
a result of an enormous amount of such investigation in recent years,
the kinds and proportions of nutrients in our common sorts of food are
well known. Average values for percentage composition of some ordinary
food materials are shown in Table I. (Table I. also includes figures for
fuel value.)

It will be observed that different kinds of food materials vary widely
in their proportions of nutrients. In general the animal foods contain
the most protein and fats, and vegetable foods are rich in
carbohydrates. The chief nutrient of lean meat and fish is protein; but
in medium fat meats the proportion of fat is as large as that of
protein, and in the fatter meats it is larger. Cheese is rich in both
protein and fat. Among the vegetable foods, dried beans and peas are
especially rich in protein. The proportion in oatmeal is also fairly
large, in wheat it is moderate, and in maize meal and rice it is rather
small. Oats contain more oil than any of the common cereals, but in none
of them is the proportion especially large. The most abundant nutrient
in all the cereals is starch, which comprises from two-thirds to
three-fourths or more of their total nutritive substance. Cotton-seed is
rich in edible oil, and so are olives. Some of the nuts contain fairly
large proportions of both protein and fat. The nutrient of potatoes is
starch, present in fair proportion. Fruits contain considerable
carbohydrates, chiefly sugar. Green vegetables are not of much account
as sources of any of the nutrients or energy.

Similar food materials from different sources may also differ
considerably in composition. This is especially true of meats. Thus, the
leaner portions from a fat animal may contain nearly as much fat as the
fatter portions from a lean animal. The data here presented are largely
those for American food products, but the available analyses of English
food materials indicate that the latter differ but little from the
former in composition. The analyses of meats produced in Europe imply
that they commonly contain somewhat less fat and more water, and often
more protein, than American meats. The meats of English production
compare with the American more than with the European meats. Similar
vegetable foods from the different countries do not differ so much in
composition.

4. _Digestibility or Availability of Food Materials._--The value of any
food material for nutriment depends not merely upon the kinds and
amounts of nutrients it contains, but also upon the ease and convenience
with which the nutrients may be digested, and especially upon the
proportion of the nutrients that will be actually digested and
absorbed. Thus, two foods may contain equal amounts of the same
nutrient, but the one most easily digested will really be of most value
to the body, because less effort is necessary to utilize it.
Considerable study of this factor is being made, and much valuable
information is accumulating, but it is of more especial importance in
cases of disordered digestion.

    TABLE II.--_Coefficients of Digestibility (or Availability) of
    Nutrients in Different Classes of Food Materials._

  +--------------------------+----------+----------+----------------+
  |       Kind of Food.      | Protein. |   Fat.   | Carbohydrates. |
  +--------------------------+----------+----------+----------------+
  |                          |          |          |                |
  |                          |     %    |     %    |        %       |
  | Meats                    |    98    |    98    |       ..       |
  | Fish                     |    96    |    97    |       ..       |
  | Poultry                  |    96    |    97    |       ..       |
  | Eggs                     |    97    |    98    |       ..       |
  | Dairy products           |    97    |    96    |       98       |
  | Total animal food of     |          |          |                |
  |   mixed diet             |    97    |    97    |       98       |
  | Potatoes                 |    73    |    ..    |       98       |
  | Beets, carrots, &c.      |    72    |    ..    |       97       |
  | Cabbage, lettuce, &c.    |    ..    |    ..    |       83       |
  | Legumes                  |    78    |    90    |       95       |
  | Oatmeal                  |    78    |    90    |       97       |
  | Corn meal                |    80    |    ..    |       99       |
  | Wheat meals without bran |    83    |    ..    |       93       |
  | Wheat meals with bran    |    75    |    ..    |       92       |
  | White bread              |    88    |    ..    |       98       |
  | Entire wheat bread       |    82    |    ..    |       94       |
  | Graham bread             |    76    |    ..    |       90       |
  | Rice                     |    76    |    ..    |       91       |
  | Fruits and nuts          |    80    |    86    |       96       |
  | Sugars and starches      |    ..    |    ..    |       98       |
  | Total vegetable food of  |          |          |                |
  |   mixed diet             |    85    |    90    |       97       |
  | Total food of mixed diet |    92    |    95    |       97       |
  +--------------------------+----------+----------+----------------+

The digestibility of food in the sense of thoroughness of digestion,
however, is of particular importance in the present discussion. Only
that portion of the food that is digested and absorbed is available to
the body for the building of tissue and the production of energy. Not
all the food eaten is thus actually digested; undigested material is
excreted in the faeces. The thoroughness of digestion is determined
experimentally by weighing and analysing the food eaten and the faeces
pertaining to it. The difference between the corresponding ingredients
of the two is commonly considered to represent the amounts of the
ingredients digested. Expressed in percentages, these are called
coefficients of digestibility. See Table II.

Such a method is not strictly accurate, because the faeces do not
consist entirely of undigested food but contain in addition to this the
so-called metabolic products, which include the residuum of digestive
juices not resorbed, fragments of intestinal epithelium, &c. Since there
is as yet no satisfactory method of separating these constituents of the
excreta, the actual digestibility of the food is not determined. It has
been suggested that since these materials must originally come from
food, they represent, when expressed in terms of food ingredients, the
cost of digestion; hence that the values determined as above explained
represent the portion of food available to the body for the building of
tissue and the yielding of energy, and what is commonly designated as
digestibility should be called availability. Other writers retain the
term "digestibility," but express the results as "apparent
digestibility," until more knowledge regarding the metabolic products of
the excreta is available and the actual digestibility may be
ascertained.

Experimental inquiry of this nature has been very active in recent
years, especially in Europe, the United States and Japan; and the
results of considerably over 1000 digestion experiments with single
foods or combinations of food materials are available. These were mostly
with men, but some were with women and with children. The larger part of
these have been taken into account in the following estimations of the
digestibility of the nutrients in different classes of food materials.
The figures here shown are subject to revision as experimental data
accumulate. They are not to be taken as exact measures of the
digestibility (or availability) of every kind of food in each given
class, but they probably represent fairly well the average digestibility
of the classes of food materials as ordinarily utilized in the mixed
diet.

5. _Fuel Value of Food._--The potential energy of food is commonly
measured as the amount of heat evolved when the food is completely
oxidized. In the laboratory this is determined by burning the food in
oxygen in a calorimeter. The results, which are known as the heat of
combustion of the food, are expressed in calories, one calory being the
amount of heat necessary to raise the temperature of one kilogram of
water one degree centigrade. But it is to be observed that this unit is
employed simply from convenience, and without implication as to what
extent the energy of food is converted into heat in the body. The unit
employed in the measurement of some other form of energy might be used
instead, as, for example, the foot-ton, which represents the amount of
energy necessary to raise one ton through one foot.

  TABLE III.--_Estimates of Heats of Combustion and of Fuel Value of
  Nutrients in Ordinary Mixed Diet._

  +---------------------------+-------------+-------------+
  |         Nutrients.        |   Heat of   | Fuel Value. |
  |                           | Combustion. |             |
  +---------------------------+-------------+-------------+
  |                           |             |             |
  |                           |  Calories.  |  Calories.  |
  |                           |             |             |
  | One gram of protein       |    5.65     |    4.05     |
  | One gram of fats          |    9.40     |    8.93     |
  | One gram of carbohydrates |    4.15     |    4.03     |
  |                           |             |             |
  +---------------------------+-------------+-------------+

  The amount of energy which a given quantity of food will produce on
  complete oxidation outside the body, however, is greater than that
  which the body will actually derive from it. In the first place, as
  previously shown, part of the food will not be digested and absorbed.
  In the second place, the nitrogenous compounds absorbed are not
  completely oxidized in the body, the residuum being excreted in the
  urine as urea and other bodies that are capable of further oxidation
  in the calorimeter. The total heat of combustion of the food eaten
  must therefore be diminished by the heat of combustion of the
  oxidizable material rejected by the body, to find what amount of
  energy is actually available to the organism for the production of
  work and heat. The amount thus determined is commonly known as the
  fuel value of food.

  Rubner's[7] commonly quoted estimates for the fuel value of the
  nutrients of mixed diet are,--for protein and carbohydrates 4.1, and
  for fats 9.3 calories per gram. According to the method of deduction,
  however, these factors were more applicable to digested than to total
  nutrients. Atwater[8] and associates have deduced, from data much more
  extensive than those available to Rubner, factors for total nutrients
  somewhat lower than these, as shown in Table III. These estimates seem
  to represent the best average factors at present available, but are
  subject to revision as knowledge is extended.


      TABLE IV.--_Quantities of Available Nutrients and Energy in Daily
      Food Consumption of Persons in Different Circumstances._

    +------------------------------------------+--------+--------------------------------+
    |                                          |        |     Nutrients and Energy       |
    |                                          | Number |        per Man per Day.        |                                |
    |                                          |   of   +------+------+--------+---------+
    |                                          |Studies.| Pro- | Fat. |Carbohy-|  Fuel   |
    |                                          |        | tein.|      | drates.| Value.  |
    +------------------------------------------+--------+------+------+--------+---------+
    |                                          |        |      |      |        |         |
    |     _Persons with Active Work._          |        |Grams.|Grams.|  Grams.|Calories.|
    | English royal engineers                  |    1   |  132 |   79 |   612  |   3835  |
    | Prussian machinists                      |    1   |  129 |  107 |   657  |   4265  |
    | Swedish mechanics                        |    5   |  174 |  105 |   693  |   4590  |
    | Bavarian lumbermen                       |    3   |  120 |  277 |   702  |   6015  |
    | American lumbermen                       |    5   |  155 |  327 |   804  |   6745  |
    | Japanese rice cleaner                    |    1   |  103 |   11 |   917  |   4415  |
    | Japanese jinrikshaw runner               |    1   |  137 |   22 |  1010  |   5050  |
    | Chinese farm labourers in California     |    1   |  132 |   90 |   621  |   3980  |
    | American athletes                        |   19   |  178 |  192 |   525  |   4740  |
    | American working-men's families          |   13   |  156 |  226 |   694  |   5650  |
    |                                          |        |      |      |        |         |
    |                                          |        |      |      |        |         |
    |     _Persons with Ordinary Work._        |        |      |      |        |         |
    | Bavarian mechanics.                      |   11   |  112 |   32 |   553  |   3060  |
    | Bavarian farm labourers                  |    5   |  126 |   52 |   526  |   3200  |
    | Russian peasants                         |   ..   |  119 |   31 |   571  |   3155  |
    | Prussian prisoners                       |    1   |  117 |   28 |   620  |   3320  |
    | Swedish mechanics.                       |    6   |  123 |   75 |   507  |   3325  |
    | American working-men's families          |   69   |  105 |  135 |   426  |   3480  |
    |                                          |        |      |      |        |         |
    |     _Persons with Light Work._           |        |      |      |        |         |
    | American artisans' families              |   21   |   93 |  107 |   358  |   2880  |
    | English tailors (prisoners)              |    1   |  121 |   37 |   509  |   2970  |
    | German shoemakers                        |    1   |   99 |   73 |   367  |   2629  |
    | Japanese prisoners                       |    1   |   43 |    6 |   444  |   2110  |
    |                                          |        |      |      |        |         |
    |     _Professional and Business Men._     |        |      |      |        |         |
    | Japanese professional men.               |   13   |   75 |   15 |   408  |   2190  |
    | Japanese students                        |    8   |   85 |   18 |   537  |   2800  |
    | Japanese military cadets                 |   11   |   98 |   20 |   611  |   3185  |
    | German physicians                        |    2   |  121 |   90 |   317  |   2685  |
    | Swedish medical students                 |    5   |  117 |  108 |   291  |   2725  |
    | Danish physicians                        |    1   |  124 |  133 |   242  |   2790  |
    | American professional and business       |        |      |      |        |         |
    |   men and students                       |   51   |   98 |  125 |   411  |   3285  |
    |                                          |        |      |      |        |         |
    |     _Persons with Little or no Exercise._|        |      |      |        |         |
    | Prussian prisoners                       |    2   |   90 |   27 |   427  |   2400  |
    | Japanese prisoners                       |    1   |   36 |    6 |   360  |   1725  |
    | Inmates of home for aged--Germany        |    1   |   85 |   43 |   322  |   2097  |
    | Inmates of hospitals for insane--America |   49   |   80 |   86 |   353  |   2590  |
    |                                          |        |      |      |        |         |
    |     _Persons in Destitute Circumstances._|        |      |      |        |         |
    | Prussian working people                  |   13   |   63 |   43 |   372  |   2215  |
    | Italian mechanics                        |    5   |   70 |   36 |   384  |   2225  |
    | American working-men's families          |   11   |   69 |   75 |   263  |   2085  |
    +------------------------------------------+--------+------+------+--------+---------+

The heats of combustion of all the fats in an ordinary mixed diet would
average about 9.40 calories per gram, but as only 95% of the fat would
be available to the body, the fuel value per gram would be (9.40 × 0.95
=) 8.93 calories. Similarly, the average heat of combustion of
carbohydrates of the diet would be about 4.15 calories per gram, and as
97% of the total quantity is available to the body, the fuel value per
gram would be 4.03. (It is commonly assumed that the resorbed fats and
carbohydrates are completely oxidized in the body.) The heats of
combustion of all the kinds of protein in the diet would average about
5.65 calories per gram. Since about 92% of the total protein would be
available to the body, the potential energy of the available protein
would be equivalent to (5.65 × 0.92 =) 5.20 calories; but as the
available protein is not completely oxidized allowance must be made for
the potential energy of the incompletely oxidized residue. This is
estimated as equivalent to 1.15 calories for the 0.92 gram of available
protein; hence, the fuel value of the total protein is (5.20 - 1.15 =)
4.05 calories per gram. Nutrients of the same class, but from different
food materials, vary both in digestibility and in heat of combustion,
and hence in fuel value. These factors are therefore not so applicable
to the nutrients of the separate articles in a diet as to those of the
diet as a whole.

6. _Food Consumption._--Much information regarding the food consumption
of people in various circumstances in different parts of the world has
accumulated during the past twenty years, as a result of studies of
actual dietaries in England, Germany, Italy, Russia, Sweden and
elsewhere in Europe, in Japan and other oriental countries, and
especially in the United States. These studies commonly consist in
ascertaining the kinds, amounts and composition of the different food
materials consumed by a group of persons during a given period and the
number of meals taken by each member of the group, and computing the
quantities of the different nutrients in the food on the basis of one
man for one day. When the members of the group are of different age,
sex, occupation, &c., account must be taken of the effect of these
factors on consumption in estimating the value "per man." Men as a rule
eat more than women under similar conditions, women more than children,
and persons at active work more than those at sedentary occupation. The
navvy, for example, who is constantly using up more nutritive material
or body tissue to supply the energy required for his muscular work needs
more protein and energy in his food than a bookkeeper who sits at his
desk all day.

In making allowance for these differences, the various individuals are
commonly compared with a man at moderately active muscular work, who is
taken as unity. A man at hard muscular work is reckoned at 1.2 times
such an individual; a man with light muscular work or a boy 15-16 years
old, .9; a man at sedentary occupation, woman at moderately active
muscular work, boy 13-14 or girl 15-16 years old, .8; woman at light
work, boy 12 or girl 13-14 years old, .7; boy 10-11 or girl 10-12 years
old, .6; child 6-9 years old, .5; child 2-5 years old, .4; child under 2
years, .3. These factors are by no means absolute or final, but are
based in part upon experimental data and in part upon arbitrary
assumption.

The total number of dietary studies on record is very large, but not all
of them are complete enough to furnish reliable data. Upwards of 1000
are sufficiently accurate to be included in statistical averages of food
consumed by people in different circumstances, nearly half of which have
been made in the United States in the past decade. The number of persons
in the individual studies has ranged from one to several hundred. Some
typical results are shown in Table IV.

7. _Quantities of Nutrients needed._--For the proper nourishment of the
body, the important problem is how much protein, fats and carbohydrates,
or more simply, what amounts of protein and potential energy are needed
under varying circumstances, to build and repair muscular and other
tissues and to supply energy for muscular work, heat and other forms of
energy. The answer to the problem is sought in the data obtained in
dietary studies with considerable numbers of people, and in metabolism
experiments with individuals in which the income and expenditure of the
body are measured. From the information thus derived, different
investigators have proposed so-called dietary standards, such as are
shown in the table below, but unfortunately the experimental data are
still insufficient for entirely trustworthy figures of this sort; hence
the term "standard" as here used is misleading. The figures given are
not to be considered as exact and final as that would suggest; they are
merely tentative estimates of the average daily amounts of nutrients and
energy required. (It is to be especially noted that these are available
nutrients and fuel value rather than total nutrients and energy.) Some
of the values proposed by other investigators are slightly larger than
these, and others are decidedly smaller, but these are the ones that
have hitherto been most commonly accepted in Europe and America.


    TABLE V.--_Standards for Dietaries. Available Nutrients and Energy
    per Man per Day._

  +---------------------------+---------+--------+---------+---------+
  |                           | Protein.|  Fat.  | Carbo-  |  Fuel   |
  |                           |         |        |hydrates.|  Value. |
  +---------------------------+---------+--------+---------+---------+
  |                           |         |        |         |         |
  |   _Voit's Standards._     |Grams.[9]| Grams. |  Grams. |Calories.|
  | Man at hard work          |   133   |   95   |   437   |  3270   |
  | Man at moderate work      |   109   |   53   |   485   |  2965   |
  |    _Atwater's Standards._ |         |        |         |         |
  | Man at very hard          |         |        |         |         |
  |   muscular work           |   161   |  ..[10]|   ..[10]|  5500   |
  | Man at hard muscular work |   138   |   ..   |   ..    |  4150   |
  | Man at moderately         |         |        |         |         |
  |   active muscular work    |   115   |   ..   |   ..    |  3400   |
  | Man at light to moderate  |         |        |         |         |
  |    muscular work          |   103   |   ..   |   ..    |  3050   |
  | Man at "sedentary"        |         |        |         |         |
  |    or woman at moderately |         |        |         |         |
  |    active work            |    92   |   ..   |   ..    |  2700   |
  | Woman at light muscular   |         |        |         |         |
  |    work, or man without   |         |        |         |         |
  |    muscular exercise      |    83   |   ..   |   ..    |  2450   |
  |                           |         |        |         |         |
  +---------------------------+---------+--------+---------+---------+

8. _Hygienic Economy of Food._--For people in good health, there are two
important rules to be observed in the regulation of the diet. One is to
choose the foods that "agree" with them, and to avoid those which they
cannot digest and assimilate without harm; and the other is to use such
sorts and quantities of foods as will supply the kinds and amounts of
nutrients needed by the body and yet to avoid burdening it with
superfluous material to be disposed of at the cost of health and
strength.

As for the first-mentioned rule, it is practically impossible to give
information that may be of more than general application. There are
people who, because of some individual peculiarity, cannot use foods
which for people in general are wholesome and nutritious. Some persons
cannot endure milk, others suffer if they eat eggs, others have to
eschew certain kinds of meat, or are made uncomfortable by fruit; but
such cases are exceptions. Very little is known regarding the cause of
these conditions. It is possible that in the metabolic processes to
which the ingredients of the food are subjected in the body, or even
during digestion before the substances are actually taken into the body,
compounds may be formed that are in one way or another injurious.
Whatever the cause may be, it is literally true in this sense that "what
is one man's meat is another man's poison," and each must learn for
himself what foods "agree" with him and what ones do not. But for the
great majority of people in health, suitable combinations of the
ordinary sorts of wholesome food materials make a healthful diet. On the
other hand, some foods are of particular value at times, aside from
their use for nourishment. Fruits and green vegetables often benefit
people greatly, not as nutriment merely, for they may have very little
actual nutritive material, but because of fruit or vegetable acids or
other substances which they contain, and which sometimes serve a most
useful purpose.


    TABLE VI.--_Amounts of Nutrients and Energy Furnished for One
    Shilling in Food Materials at Ordinary Prices._

  +----------------------+-------+----------------------------------------------+
  |                      |       |            One Shilling will buy             |
  |                      |       +----------+-------------------------+---------+
  |   Food Materials     |Prices |          |   Available Nutrients.  |         |
  |    as Purchased.     |  per  |Total Food+-------------------------+   Fuel  |
  |                      |  lb.  |materials.|        |      | Carbo-  |  Value. |
  |                      |       |          |Protein.| Fat. |hydrates.|         |
  +----------------------+-------+----------+--------+------+---------+---------+
  |                      | s. d. |    lb.   |    lb. |  lb. |    lb.  |Calories.|
  | Beef, round          | 0 10  |   1.20   |   .22  |  .14 |    ..   |  1,155  |
  |                      | 0  8½ |   1.41   |   .26  |  .17 |    ..   |  1,235  |
  |                      | 0  5  |   2.40   |   .44  |  .29 |    ..   |  2,105  |
  |                      |       |          |        |      |         |         |
  | Beef, sirloin        | 0 10  |   1.20   |   .19  |  .20 |    ..   |  1,225  |
  |                      | 0  9  |   1.33   |   .21  |  .22 |    ..   |  1,360  |
  |                      | 0  8  |   1.50   |    ..  |   .. |    ..   |    ..   |
  |                      | 0  5  |   2.40   |    ..  |   .. |    ..   |    ..   |
  |                      |       |          |        |      |         |         |
  | Beef, rib            | 0  9  |   1.33   |   .19  |  .19 |    ..   |  1,200  |
  |                      | 0  7½ |   1.60   |    ..  |   .. |    ..   |    ..   |
  |                      | 0  4½ |   2.67   |    ..  |   .. |    ..   |    ..   |
  |                      |       |          |        |      |         |         |
  | Mutton, leg          | 0  9  |   1.33   |   .20  |  .20 |    ..   |  1,245  |
  |                      | 0  5  |   2.40   |   .37  |  .35 |    ..   |  2,245  |
  |                      |       |          |        |      |         |         |
  | Pork, spare-rib      | 0  9  |   1.33   |   .17  |  .31 |    ..   |  1,645  |
  |                      | 0  7  |   1.71   |   .22  |  .39 |    ..   |  2,110  |
  |                      |       |          |        |      |         |         |
  | Pork, salt, fat      | 0  7  |   1.71   |   .03  | 1.40 |    ..   |  6,025  |
  |                      | 0  5  |   2.40   |   .04  | 1.97 |    ..   |  8,460  |
  |                      |       |          |        |      |         |         |
  | Pork, smoked ham     | 0  8  |   1.50   |   .20  |  .48 |    ..   |  2,435  |
  |                      | 0  4½ |   2.67   |   .36  |  .85 |    ..   |  4,330  |
  |                      |       |          |        |      |         |         |
  | Fresh cod            | 0  4  |   3.00   |   .34  |  .01 |    ..   |    710  |
  |                      | 0  3  |   4.00   |   .45  |  .01 |    ..   |    945  |
  |                      |       |          |        |      |         |         |
  | Salt cod             | 0  3½ |   3.43   |   .54  |  .07 |    ..   |  1,370  |
  |                      | 0 10  |   1.20   |   .07  |  .01 |   .04   |    275  |
  |                      |       |          |        |      |         |         |
  |Milk, whole, 4d. a qt.| 0  2  |   6.00   |   .19  |  .23 |   .30   |  1,915  |
  |    "        3d. a qt.| 0  1½ |   8.00   |   .26  |  .30 |   .40   |  2,550  |
  |    "        2d. a qt.| 0  1  |  12.00   |   .38  |  .46 |   .60   |  3,825  |
  |                      |       |          |        |      |         |         |
  | Milk, skimmed, 2d. a | 0  1  |  12.00   |   .40  |  .03 |   .61   |  2,085  |
  |  qt.                 |       |          |        |      |         |         |
  | Butter               | 1  6  |    .67   |   .01  |  .54 |    ..   |  2,320  |
  |                      | 1  3  |    .80   |   .01  |  .64 |    ..   |  2,770  |
  |                      | 1  0  |   1.00   |   .01  |  .81 |    ..   |  3,460  |
  |                      |       |          |        |      |         |         |
  | Margarine            | 0  4  |   3.00   |    ..  | 2.37 |    ..   | 10,080  |
  |                      |       |          |        |      |         |         |
  | Eggs, 2s. a dozen    | 1  4  |    .75   |   .10  |  .07 |    ..   |    475  |
  |   "   1½s. a dozen   | 1  0  |   1.00   |   .13  |  .09 |    ..   |    635  |
  |   "    1s. a dozen   | 0  8  |   1.50   |   .19  |  .13 |    ..   |    950  |
  |                      |       |          |        |      |         |         |
  | Cheese               | 0  8  |   1.50   |   .38  |  .48 |   .04   |  2,865  |
  |                      | 0  7  |   1.71   |   .43  |  .55 |   .04   |  3,265  |
  |                      | 0  5  |   2.40   |   .60  |  .77 |   .06   |  4,585  |
  |                      |       |          |        |      |         |         |
  | Wheat bread          |0 1-1/8|  10.67   |   .76  |  .13 |  5.57   | 12,421  |
  |                      |       |          |        |      |         |         |
  | Wheat flour          |0 1-3/5|   7.64   |   .67  |  .07 |  5.63   | 12,110  |
  |                      | 0 1½  |   8.16   |   .72  |  .07 |  6.01   | 12,935  |
  |                      |       |          |        |      |         |         |
  | Oatmeal              |0 1-2/5|   8.39   |  1.11  |  .54 |  5.54   | 14,835  |
  |                      | 0 1½  |   8.16   |  1.08  |  .53 |  5.39   | 14,430  |
  |                      |       |          |        |      |         |         |
  | Rice                 | 0 1¾  |   6.86   |   .45  |  .02 |  5.27   | 10,795  |
  |                      |       |          |        |      |         |         |
  | Potatoes             |0 0-2/3|  18.00   |   .25  |  .02 |  2.70   |  5,605  |
  |                      | 0 0½  |  24.00   |   .34  |  .02 |  3.60   |  7,470  |
  |                      |       |          |        |      |         |         |
  | Beans                | 0 2   |   6.00   |  1.05  |  .10 |  3.47   |  8,960  |
  |                      |       |          |        |      |         |         |
  | Sugar                | 1 ¾   |   6.86   |   ..   |  ..  |  6.86   | 12,760  |
  +----------------------+-------+----------+--------+------+---------+---------+

The proper observance of the second rule mentioned requires information
regarding the demands of the body for food under different
circumstances. To supply this information is one purpose of the effort
to determine the so-called dietary standards mentioned above. It should
be observed, however, that these are generally more applicable to the
proper feeding of a group or class of people as a whole than for
particular individuals in this class. The needs of individuals will vary
largely from the average in accordance with the activity and
individuality. Moreover, it is neither necessary nor desirable for the
individual to follow any standard exactly from day to day. It is
requisite only that the average supply shall be sufficient to meet the
demands of the body during a given period.

The cooking of food and other modes of preparing it for consumption have
much to do with its nutritive value. Many materials which, owing to
their mechanical condition or to some other cause, are not particularly
desirable food materials in their natural state, are quite nutritious
when cooked or otherwise prepared for consumption. It is also a matter
of common experience that well-cooked food is wholesome and appetizing,
whereas the same material poorly prepared is unpalatable. There are
three chief purposes of cooking; the first is to change the mechanical
condition of the food. Heating changes the structure of many food
materials very materially, so that they may be more easily chewed and
brought into a condition in which the digestive juices can act upon them
more freely, and in this way probably influencing the ease and
thoroughness of digestion. The second is to make the food more
appetizing by improving the appearance or flavour or both. Food which is
attractive to the eye and pleasing to the palate quickens the flow of
saliva and other digestive juices and thus aids digestion. The third is
to kill, by heat, disease germs, parasites or other dangerous organisms
that may be contained in food. This is often a very important matter and
applies to both animal and vegetable foods. Scrupulous neatness should
always be observed in storing, handling and serving food. If ever
cleanliness is desirable it must be in the things we eat, and every care
should be taken to ensure it for the sake of health as well as of
decency. Cleanliness in this connexion means not only absence of visible
dirt, but freedom from undesirable bacteria and other minute organisms
and from worms and other parasites. If food, raw or cooked, is kept in
dirty places, peddled from dirty carts, prepared in dirty rooms and in
dirty dishes, or exposed to foul air, disease germs and other offensive
and dangerous substances may easily enter it.

9. _Pecuniary Economy of Food._--Statistics of economy and of cost of
living in Great Britain, Germany and the United States show that at
least half, and commonly more, of the income of wage-earners and other
people in moderate circumstances is expended for subsistence. The
relatively large cost of food, and the important influence of diet upon
health and strength, make a more widespread understanding of the subject
of dietetics very desirable. The maxim that "the best is the cheapest"
does not apply to food. The "best" food, in the sense of that which is
the finest in appearance and flavour and which is sold at the highest
price, is not generally the most economical.

The price of food is not regulated largely by its value for nutriment.
Its agreeableness to the palate or to the buyer's fancy is a large
factor in determining the current demand and market price. There is no
more nutriment in an ounce of protein or fat from the tender-loin of
beef than from the round or shoulder. The protein of animal food has,
however, some advantage over that of vegetable foods in that it is more
thoroughly, and perhaps more easily, digested, for which reason it would
be economical to pay somewhat more for the same quantity of nutritive
material in the animal food. Furthermore, animal foods such as meats,
fish and the like, gratify the palate as most vegetable foods do not.
For persons in good health, foods in which the nutrients are the most
expensive are like costly articles of adornment. People who can well
afford them may be justified in buying them, but they are not
economical. The most economical food is that which is at the same time
most healthful and cheapest.

The variations in the cost of the actual nutriment in different food
materials may be illustrated by comparison of the amounts of nutrients
obtained for a given sum in the materials as bought at ordinary market
prices. This is done in Table VI., which shows the amounts of available
nutrients contained in the quantities of different food materials that
may be purchased for one shilling at prices common in England.

When proper attention is given to the needs of the body for food and the
relation between cost and nutritive value of food materials, it will be
found that with care in the purchase and skill in the preparation of
food, considerable control may be had over the expensiveness of a
palatable, nutritious and healthful diet.

  AUTHORITIES.--COMPOSITION OF FOODS:--König, _Chemie der menschlichen
  Nahrungs- und Genussmittel_; Atwater and Bryant, "Composition of
  American Food Materials," Bul. 28, Office of Experiment Stations, U.S.
  Department of Agriculture. NUTRITION AND DIETETICS:--Armsby,
  _Principles of Animal Nutrition_; Lusk, _The Science of Nutrition_;
  Burney Yeo, _Food in Health and Disease_; Munk and Uffelmann, _Die
  Ernährung des gesunden und kranken Menschen_; Von Leyden,
  _Ernährungstherapie und Diätetik_; Dujardin-Beaumetz, Hygiène
  alimentaire; Hutchison, _Food and Dietetics_; R. H. Chittenden,
  _Physiological Economy in Nutrition_ (1904), _Nutrition of Man_
  (1907); Atwater, "Chemistry and Economy of Food," Bul. 21, Office of
  Experiment Stations, U.S. Department of Agriculture. See also other
  Bulletins of the same office on composition of food, results of
  dietary studies, metabolism experiments, &c., in the United States.
  GENERAL METABOLISM:--Voit, _Physiologie des allgemeinen Stoffwechsels
  und der Ernährung_; Hermann, _Handbuch der Physiologie_, Bd. vi.; Von
  Noorden, _Pathologie des Stoffwechsels_; Schäfer, _Text-Book of
  Physiology_, vol. i.; Atwater and Langworthy, "Digest of Metabolism
  Experiments," Bull. 45, Office of Experiment Stations, U.S. Department
  of Agriculture.     (W. O. A.; R. D. M.)


FOOTNOTES:

  [1] The terms applied by different writers to these nitrogenous
    compounds are conflicting. For instance, the term "proteid" is
    sometimes used as protein is here used, and sometimes to designate
    the group here called albuminoids. The classification and terminology
    here followed are those tentatively recommended by the Association of
    American Agricultural Colleges and Experiment Stations.

  [2] Folin, _Festschrift für Olaf Hammarsten_, iii. (Upsala, 1906).

  [3] _Ztschr. Biol._ 30, 73.

  [4] In Russian. Cited in United States Department of Agriculture,
    Office of Experiment Stations, Bul. No. 45, _A Digest of Metabolism
    Experiments_, by W. O. Atwater and C. F. Langworthy.

  [5] _Arch. physiol. norm. et path._ (1894) 4.

  [6] U.S. Department of Agriculture, Office of Experiment Stations,
    Bulletins Nos. 63, 69, 109, 136, 175. For a description of the
    respiration calorimeter here mentioned see also publication No. 42 of
    the Carnegie Institution of Washington.

  [7] _Ztschr. Biol._ 21 (1885), p. 377.

  [8] _Connecticut_ (Storrs) _Agricultural Experiment Station Report_
    (1899), 73.

  [9] One ounce equals 28.35 grams.

  [10] As the chief function of both fats and carbohydrates is to
    furnish energy, their exact proportion in the diet is of small
    account. The amount of either may vary largely according to taste,
    available supply, or other condition, as long as the total amount of
    both is sufficient, together with the protein to furnish the required
    energy.



DIETRICH, CHRISTIAN WILHELM ERNST (1712-1774), German painter, was born
at Weimar, where he was brought up early to the profession of art by his
father Johann George, then painter of miniatures to the court of the
duke. Having been sent to Dresden to perfect himself under the care of
Alexander Thiele, he had the good fortune to finish in two hours, at the
age of eighteen, a picture which attracted the attention of the king of
Saxony. Augustus II. was so pleased with Dietrich's readiness of hand
that he gave him means to study abroad, and visit in succession the
chief cities of Italy and the Netherlands. There he learnt to copy and
to imitate masters of the previous century with a versatility truly
surprising. Winckelmann, to whom he had been recommended, did not
hesitate to call him the Raphael of landscape. Yet in this branch of his
practice he merely imitated Salvator Rosa and Everdingen. He was more
successful in aping the style of Rembrandt, and numerous examples of
this habit may be found in the galleries of St Petersburg, Vienna and
Dresden. At Dresden, indeed, there are pictures acknowledged to be his,
bearing the fictitious dates of 1636 and 1638, and the name of
Rembrandt. Among Dietrich's cleverest reproductions we may account that
of Ostade's manner in the "Itinerant Singers" at the National Gallery.
His skill in catching the character of the later masters of Holland is
shown in candlelight scenes, such as the "Squirrel and the Peep-Show" at
St Petersburg, where we are easily reminded of Godfried Schalcken.
Dietrich tried every branch of art except portraits, painting Italian
and Dutch views alternately with Scripture scenes and still life. In
1741 he was appointed court painter to Augustus III. at Dresden, with an
annual salary of 400 thalers (£60), conditional on the production of
four cabinet pictures a year. This condition, no doubt, accounts for the
presence of fifty-two of the master's panels and canvases in one of the
rooms at the Dresden museum. Dietrich, though popular and probably the
busiest artist of his time, never produced anything of his own; and his
imitations are necessarily inferior to the originals which he affected
to copy. His best work is certainly that which he gave to engravings. A
collection of these at the British Museum, produced on the general lines
of earlier men, such as Ostade and Rembrandt, reveal both spirit and
skill. Dietrich, after his return from the Peninsula, generally signed
himself "Dietericij," and with this signature most of his extant
pictures are inscribed. He died at Dresden, after he had successively
filled the important appointments of director of the school of painting
at the Meissen porcelain factory and professor of the Dresden academy of
arts.



DIETRICH OF BERN, the name given in German popular poetry to Theodoric
the Great. The legendary history of Dietrich differs so widely from the
life of Theodoric that it has been suggested that the two were
originally unconnected. Medieval chroniclers, however, repeatedly
asserted the identity of Dietrich and Theodoric, although the more
critical noted the anachronisms involved in making Ermanaric (d. 376)
and Attila (d. 453) contemporary with Theodoric (b. 455). That the
legend is based on vague historical reminiscences is proved by the
retention of the names of Theodoric (Thiuda-reiks, Dietrich) and his
father Theudemir (Dietmar), by Dietrich's connexion with Bern (Verona)
and Raben (Ravenna). Something of the Gothic king's character descended
to Dietrich, familiarly called the Berner, the favourite of German
medieval saga heroes, although his story did not leave the same mark on
later German literature as did that of the Nibelungs. The cycle of songs
connected with his name in South Germany is partially preserved in the
Heldenbuch (q.v.) in _Dietrich's Flucht_, the _Rabenschlacht_ and
_Alpharts Tod_; but it was reserved for an Icelandic author, writing in
Norway in the 13th century, to compile, with many romantic additions, a
consecutive account of Dietrich. In this Norse prose redaction, known as
the _Vilkina Saga_, or more correctly the _Thidrekssaga_, is
incorporated much extraneous matter from the Nibelungen and Wayland
legends, in fact practically the whole of south German heroic tradition.

There are traces of a form of the Dietrich legend in which he was
represented as starting out from Byzantium, in accordance with
historical tradition, for his conquest of Italy. But this early
disappeared, and was superseded by the existing legend, in which,
perhaps by an "epic fusion" with his father Theudemir, he was associated
with Attila, and then by an easy transition with Ermanaric. Dietrich was
driven from his kingdom of Bern by his uncle Ermanaric. After years of
exile at the court of Attila he returned with a Hunnish army to Italy,
and defeated Ermanaric in the Rabenschlacht, or battle of Ravenna.
Attila's two sons, with Dietrich's brother, fell in the fight, and
Dietrich returned to Attila's court to answer for the death of the young
princes. This very improbable renunciation of the advantages of his
victory suggests that in the original version of the story the
Rabenschlacht was a defeat. In the poem of _Ermenrichs Tod_ he is
represented as slaying Ermanaric, as in fact Theodoric slew Odoacer.
"Otacher" replaces Ermanaric as his adversary in the _Hildebrandslied_,
which relates how thirty years after the earlier attempt he reconquered
his Lombard kingdom. Dietrich's long residence at Attila's court
represents the youth and early manhood of Theodoric spent at the
imperial court and fighting in the Balkan peninsula, and, in accordance
with epic custom, the period of exile was adorned with war-like
exploits, with fights with dragons and giants, most of which had no
essential connexion with the cycle. The romantic poems of _König
Laurin_, _Sigenot_, _Eckenlied_ and _Virginal_ are based largely on
local traditions originally independent of Dietrich. The court of Attila
(Etzel) was a ready bridge to the Nibelungen legend. In the final
catastrophe he was at length compelled, after steadily holding aloof
from the combat, to avenge the slaughter of his Amelungs by the
Burgundians, and delivered Hagen bound into the hands of Kriemhild. The
flame breath which anger induced from him shows the influence of pure
myth, but the tales of his demonic origin and of his being carried off
by the devil in the shape of a black horse may safely be put down to the
clerical hostility to Theodoric's Arianism.

Generally speaking, Dietrich of Bern was the wise and just monarch as
opposed to Ermanaric, the typical tyrant of Germanic legend. He was
invariably represented as slow of provocation and a friend of peace, but
once roused to battle not even Siegfried could withstand his onslaught.
But probably Dietrich's fight with Siegfried in Kriemhild's rose garden
at Worms is a late addition to the Rosengarten myth. The chief heroes of
the Dietrich cycle are his tutor and companion in arms, Hildebrand (see
HILDEBRAND, lay of), with his nephews the Wolfings Alphart and Wolfhart;
Wittich, who renounced his allegiance to Dietrich and slew the sons of
Attila; Heime and Biterolf.

  The contents of the poems dealing with the Dietrich cycle are
  summarized by Uhland in _Schriften zur Geschichte der Dichtung und
  Sage_ (Stuttgart, 1873). The _Thidrekssaga_ (ed. C. Unger,
  Christiania, 1853) is translated into German by F. H. v. der Hagen in
  _Altdeutsche und altnordische Heldensagen_ (vols. i. and ii. 3rd ed.,
  Breslau, 1872). A summary of it forms the concluding chapter of T.
  Hodgkin's _Theodoric the Goth_ (1891). The variations in the Dietrich
  legend in the Latin historians, in Old and Middle High German
  literature, and in the northern saga, can be studied in W. Grimm's
  _Deutsche Heldensage_ (2nd ed., Berlin, 1867). There is a good account
  in English in F. E. Sandbach's _Heroic Saga-cycle of Dietrich of Bern_
  (1906), forming No. 15 of Alfred Nutt's _Popular Studies in
  Mythology_, and another in M. Bentinck Smith's translation of Dr O. L.
  Jiriczek's _Deutsche Heldensage_ (_Northern Legends_, London, 1902).
  For modern German authorities and commentators see B. Symons,
  "Deutsche Heldensage" in H. Paul's _Grd. d. german. Phil._
  (Strassburg, new ed., 1905); also Goedeke, _Geschichte der deutschen
  Dichtung_ (i. 241-246).



DIEZ, FRIEDRICH CHRISTIAN (1794-1876), German philologist, was born at
Giessen, in Hesse-Darmstadt, on the 15th of March 1794. He was educated
first at the gymnasium and then at the university of his native town.
There he studied classics under Friedrich Gottlieb Welcker (1784-1868)
who had just returned from a two years' residence in Italy to fill the
chair of archaeology and Greek literature. It was Welcker who kindled in
him a love of Italian poetry, and thus gave the first bent to his
genius. In 1813 he joined the Hesse corps as a volunteer and served in
the French campaign. Next year he returned to his books, and this short
taste of military service was the only break in a long and uneventful
life of literary labours. By his parents' desire he applied himself for
a short time to law, but a visit to Goethe in 1818 gave a new direction
to his studies, and determined his future career. Goethe had been
reading Raynouard's _Selections from the Romance Poets_, and advised the
young scholar to explore the rich mine of Provençal literature which the
French savant had opened up. This advice was eagerly followed, and
henceforth Diez devoted himself to Romance literature. He thus became
the founder of Romance philology. After supporting himself for some
years by private teaching, he removed in 1822 to Bonn, where he held the
position of privatdocent. In 1823 he published his first work, _An
Introduction to Romance Poetry_; in the following year appeared _The
Poetry of the Troubadours_, and in 1829 _The Lives and Works of the
Troubadours_. In 1830 he was called to the chair of modern literature.
The rest of his life was mainly occupied with the composition of the two
great works on which his fame rests, the _Grammar of the Romance
Languages_ (1836-1844), and the _Lexicon of the Romance
Languages--Italian, Spanish and French_ (1853); in these two works Diez
did for the Romance group of languages what Jacob Grimm did for the
Teutonic family. He died at Bonn on the 29th of May 1876.

  The earliest French philologists, such as Perion and Henri Estienne,
  had sought to discover the origin of French in Greek and even in
  Hebrew. For more than a century Ménage's _Etymological Dictionary_
  held the field without a rival. Considering the time at which it was
  written (1650), it was a meritorious work, but philology was then in
  the empirical stage, and many of Ménage's derivations (such as that of
  "rat" from the Latin "mus," or of "haricot" from "faba") have since
  become bywords among philologists. A great advance was made by
  Raynouard, who by his critical editions of the works of the
  Troubadours, published in the first years of the 19th century, laid
  the foundations on which Diez afterwards built. The difference between
  Diez's method and that of his predecessors is well stated by him in
  the preface to his dictionary. In sum it is the difference between
  science and guess-work. The scientific method is to follow implicitly
  the discovered principles and rules of phonology, and not to swerve a
  foot's breadth from them unless plain, actual exceptions shall justify
  it; to follow the genius of the language, and by cross-questioning to
  elicit its secrets; to gauge each letter and estimate the value which
  attaches to it in each position; and lastly to possess the true
  philosophic spirit which is prepared to welcome any new fact, though
  it may modify or upset the most cherished theory. Such is the
  historical method which Diez pursues in his grammar and dictionary. To
  collect and arrange facts is, as he tells us, the sole secret of his
  success, and he adds in other words the famous apophthegm of Newton,
  "hypotheses non fingo." The introduction to the grammar consists of
  two parts:--the first discusses the Latin, Greek and Teutonic elements
  common to the Romance languages; the second treats of the six dialects
  separately, their origin and the elements peculiar to each. The
  grammar itself is divided into four books, on phonology, on flexion,
  on the formation of words by composition and derivation, and on
  syntax.

  His dictionary is divided into two parts. The first contains words
  common to two at least of the three principal groups of
  Romance:--Italian, Spanish and Portuguese, and Provençal and French.
  The Italian, as nearest the original, is placed at the head of each
  article. The second part treats of words peculiar to one group. There
  is no separate glossary of Wallachian.

  Of the introduction to the grammar there is an English translation by
  C. B. Cayley. The dictionary has been published in a remodelled form
  for English readers by T. C. Donkin.



DIEZ, a town of Germany, in the Prussian province of Hesse-Nassau,
romantically situated in the deep valley of the Lahn, here crossed by an
old bridge, 30 m. E. from Coblenz on the railway to Wetzlar. Pop. 4500.
It is overlooked by a former castle of the counts of Nassau-Dillenburg,
now a prison. Close by, on an eminence above the river, lies the castle
of Oranienstein, formerly a Benedictine nunnery and now a cadet school,
with beautiful gardens. There are a Roman Catholic and two Evangelical
churches. The new part of the town is well built and contains numerous
pretty villa residences. In addition to extensive iron-works there are
sawmills and tanneries. In the vicinity are Fachingen, celebrated for
its mineral waters, and the majestic castle of Schaumburg belonging to
the prince of Waldeck-Pyrmont.



DIFFERENCES, CALCULUS OF (_Theory of Finite Differences_), that branch
of mathematics which deals with the successive differences of the terms
of a series.

1. The most important of the cases to which mathematical methods can be
applied are those in which the terms of the series are the values, taken
at stated intervals (regular or irregular), of a continuously varying
quantity. In these cases the formulae of finite differences enable
certain quantities, whose exact value depends on the law of variation
(i.e. the law which governs the relative magnitude of these terms) to be
calculated, often with great accuracy, from the given terms of the
series, without explicit reference to the law of variation itself. The
methods used may be extended to cases where the series is a double
series (series of double entry), i.e. where the value of each term
depends on the values of a pair of other quantities.

2. The _first differences_ of a series are obtained by subtracting from
each term the term immediately preceding it. If these are treated as
terms of a new series, the first differences of this series are the
_second differences_ of the original series; and so on. The successive
differences are also called _differences of the first, second, ...
order_. The differences of successive orders are most conveniently
arranged in successive columns of a table thus:--

  +-----+----------+-----------+-----------------+----------------------+
  |Term.| 1st Diff.| 2nd Diff. |    3rd Diff.    |      4th Diff.       |
  +-----+----------+-----------+-----------------+----------------------+
  |     |          |           |                 |                      |
  |  a  |          |           |                 |                      |
  |     |   b - a  |           |                 |                      |
  |  b  |          | c - 2b +a |                 |                      |
  |     |   c - b  |           | d - 3c + 3b - a |                      |
  |  c  |          | d - 2c +b |                 | e - 4d + 6c - 4b + a |
  |     |   d - c  |           | e - 3d + 3c - b |                      |
  |  d  |          | e - 2d +c |                 |                      |
  |     |   e - d  |           |                 |                      |
  |  e  |          |           |                 |                      |
  +-----+----------+-----------+-----------------+----------------------+

    _Algebra of Differences and Sums._

  [Illustration: FIG. 1.]

  3. The formal relations between the terms of the series and the
  differences may be seen by comparing the arrangements (A) and (B) in
  fig. 1. In (A) the various terms and differences are the same as in §
  2, but placed differently. In (B) we take a new series of terms
  [alpha], [beta], [gamma], [delta], commencing with the same term
  [alpha], and take the successive sums of pairs of terms, instead of
  the successive differences, but place them to the left instead of to
  the right. It will be seen, in the first place, that the successive
  terms in (A), reading downwards to the right, and the successive terms
  in (B), reading downwards to the left, consist each of a series of
  terms whose coefficients follow the binomial law; i.e. the
  coefficients in b - a, c - 2b + a, d - 3c + 3b - a, ... and in [alpha]
  + [beta], [alpha] + 2[beta] + [gamma], [alpha] + 3[beta] + 3[gamma] +
  [delta], ... are respectively the same as in y - x, (y - x)², (y -
  x)³, ... and in x + y, (x + y)², (x + y)³,.... In the second place, it
  will be seen that the relations between the various terms in (A) are
  identical with the relations between the similarly placed terms in
  (B); e.g. [beta] + [gamma] is the difference of [alpha] + 2[beta] +
  [gamma] and [alpha] + [beta], just as c - b is the difference of c and
  b: and d - c is the sum of c - b and d - 2c + b, just as [beta] +
  2[gamma] + [delta] is the sum of [beta] + [gamma] and [gamma] +
  [delta]. Hence if we take [beta], [gamma], [delta], ... of (B) as
  being the same as b - a, c - 2b + a, d -3c + 3b - a, ... of (A), all
  corresponding terms in the two diagrams will be the same.

  Thus we obtain the two principal formulae connecting terms and
  differences. If we provisionally describe b - a, c - 2b + a, ... as
  the first, second, ... differences of the particular term a (§ 7),
  then (i.) the nth difference of a is

                              n·n - 1
    l - nk + ... + (-1)^(n-2) ------- c + (-1)^(n-1) nb + (-1)^n a,
                                1·2

  where l, k ... are the (n + 1)th, nth, ... terms of the series a, b,
  c, ...; the coefficients being those of the terms in the expansion of
  (y -x)^n: and (ii.) the (n + 1)th term of the series, i.e. the nth
  term after a, is

                  n·n - 1
    a + n[beta] + ------- [gamma] + ...
                    1·2

  where [beta], [gamma], ... are the first, second, ... differences of
  a; the coefficients being those of the terms in the expansion of (x +
  y)^n.

  4. Now suppose we treat the terms a, b, c, ... as being themselves the
  first differences of another series. Then, if the first term of this
  series is N, the subsequent terms are N + a, N + a + b, N + a + b + c,
  ...; i.e. the difference between the (n + 1)th term and the first term
  is the sum of the first n terms of the original series. The term N, in
  the diagram (A), will come above and to the left of a; and we see, by
  (ii.) of § 3, that the sum of the first n terms of the original series
  is

     /         n·n - 1            \             n·n - 1          n·n - 1·n - 2
    ( N + na + ------- [beta] + ...) - N = na + ------- [beta] + ------------- [gamma] + ...
     \           1·2              /               1·2              1 · 2 · 3

  5. As an example, take the arithmetical series

    a, a + p, a + 2p, ...

  The first differences are p, p, p, ... and the differences of any
  higher order are zero. Hence, by (ii.) of § 3, the (n + 1)th term is a
  + np, and, by § 4, the sum of the first n terms is na + ½n(n - 1)p =
  ½n{2a + (n - 1)p}.

  6 As another example, take the series 1, 8, 27, ... the terms of which
  are the cubes of 1, 2, 3, ... The first, second and third differences
  of the first term are 7, 12 and 6, and it may be shown (§ 14 (i.))
  that all differences of a higher order are zero. Hence the sum of the
  first n terms is

          n·n - 1      n·n - 1·n - 2     n·n - 1·n - 2·n - 3
    n + 7 ------- + 12 ------------- + 6 ------------------- =
            1·2            1·2·3              1·2·3·4

      ¼n^4 + ½n³ + ¼n² = {½n(n + 1)}².

  7. In § 3 we have described b - a, c - 2b + a, ... as the first,
  second, ... differences of a. This ascription of the differences to
  particular terms of the series is quite arbitrary. If we read the
  differences in the table of § 2 upwards to the right instead of
  downwards to the right, we might describe e - d, e - 2d + c, ... as
  the first, second, ... differences of e. On the other hand, the term
  of greatest weight in c -2b + a, i.e. the term which has the
  numerically greatest coefficient, is b, and therefore c - 2b + a might
  properly be regarded as the second difference of b, and similarly e -
  4d + 6c - 4b + a might be regarded as the fourth difference of c.
  These three methods of regarding the differences lead to three
  different systems of notation, which are described in §§ 9, 10 and 11.


  _Notation of Differences and Sums._

  8. It is convenient to denote the terms a, b, c, ... of the series by
  u0, u1, u2, u3, ... If we merely have the terms of the series, un may
  be regarded as meaning the (n + 1)th term. Usually, however, the terms
  are the values of a quantity u, which is a function of another
  quantity x, and the values of x, to which a, b, c, ... correspond,
  proceed by a constant difference h. If x0 and u0 are a pair of
  corresponding values of x and u, and if any other value x0 + mh of x
  and the corresponding value of u are denoted by xm and um, then the
  terms of the series will be ... u_(n-2), u_(n-1), u_n, u_(n+1),
  u_(n+2) ..., corresponding to values Of x denoted by ... x_(n-2),
  x_(n-1), x_n, x_(n+1), x_(n+2)....

  9. In the _advancing-difference notation_ u_(n+1) - u_n is denoted by
  [Delta]un. The differences [Delta]u0, [Delta]u1, [Delta]u2 ... may
  then be regarded as values of a function [Delta]u corresponding to
  values of x proceeding by constant difference h; and therefore
  [Delta]u_(n+1) -[Delta]u_n denoted by [Delta][Delta]u_n, or, more
  briefly, [Delta]²u_n; and so on. Hence the table of differences in §
  2, with the corresponding values of x and of u placed opposite each
  other in the ordinary manner of mathematical tables, becomes

    +---------+---------+----------------+-----------------+-----------------+----------------------+
    |    x    |    u    |    1st Diff.   |    2nd Diff.    |     3rd Diff.   |       4th Diff.      |
    +---------+---------+----------------+-----------------+-----------------+----------------------+
    |    .    |    .    |        .       |        .        |        .        |          .           |
    |    .    |    .    |        .       |        .        |        .        |          .           |
    |    .    |    .    |        .       |        .        |        .        |          .           |
    |         |         |                |                 |                 |                      |
    | x_(n-2) | u_(n-2) |                | [Delta]²u_(n-3) |                 | [Delta]^4u_(n-4) ... |
    |         |         | [Delta]u_(n-2) |                 | [Delta]³u_(n-3) |                      |
    | x_(n-1) | u_(n-1) |                | [Delta]²u_(n-2) |                 | [Delta]^4u_(n-3) ... |
    |         |         | [Delta]u_(n-1) |                 | [Delta]³u_(n-2) |                      |
    | xn      | u_n     |                | [Delta]²u_(n-1) |                 | [Delta]^4u_(n-2) ... |
    |         |         | [Delta]u_n     |                 | [Delta]³u_(n-1) |                      |
    | x_(n+1) | u_(n+1) |                | [Delta]²u_n     |                 | [Delta]^4u_(n-1) ... |
    |         |         | [Delta]u_(n+1) |                 | [Delta]³u_n     |                      |
    | x_(n+2) | u_(n+2) |                | [Delta]²u_(n+1) |                 | [Delta]^4u_n     ... |
    |    .    |    .    |        .       |        .        |        .        |          .           |
    |    .    |    .    |        .       |        .        |        .        |          .           |
    |    .    |    .    |        .       |        .        |        .        |          .           |
    +---------+---------+----------------+-----------------+-----------------+----------------------+

  The terms of the series of which ... u_(n-1), u_n, u_(n+1), ... are
  the first differences are denoted by [Sigma]u, with proper suffixes,
  so that this series is ... [Sigma]u_(n-1), [Sigma]u_n,
  [Sigma]u_(n+1).... The suffixes are chosen so that we may have
  [Delta][Sigma]un = un, whatever n may be; and therefore (§ 4)
  [Sigma]un may be regarded as being the sum of the terms of the series
  up to and including un-1. Thus if we write [Sigma]u_(n-1) = C + un-2,
  where C is any constant, we shall have

    [Sigma]u_n = [Sigma]u_(n-1) + [Delta][Sigma]u_(n-1) = C + u_(n-2) + u_(n-1),
    [Sigma]u_(n+1) = C + u_(n-2) + u_(n-1) + u_n,

  and so on. This is true whatever C may be, so that the knowledge of
  ... u_n-1, u_n, ... gives us no knowledge of the exact value of
  [Sigma]u_n; in other words, C is an arbitrary constant, the value of
  which must be supposed to be the same throughout any operations in
  which we are concerned with values of [Sigma]_u corresponding to
  different suffixes.

  There is another symbol E, used in conjunction with u to denote the
  next term in the series. Thus Eun means u_(n+1), so that Eun = u_n +
  [Delta]u_n.

  10. Corresponding to the advancing-difference notation there is a
  _receding-difference_ notation, in which u_(n+1) - u_n is regarded as
  a difference of u_(n+1), and may be denoted by [Delta]'u_(n+1), and
  similarly u_(n+1) - 2u_n + u_(n-1) may be denoted by [Delta]'²u_(n+1).
  This notation is only required for certain special purposes, and the
  usage is not settled (§ 19 (ii.)).

  11. The _central-difference_ notation depends on treating u_(n+1) -
  2u_n -u_(n-1) as the second difference of un, and therefore as
  corresponding to the value x_n; but there is no settled system of
  notation. The following seems to be the most convenient. Since un is a
  function of x_n, and the second difference u_(n+2) - 2u_(n+1) + u_n is
  a function of x_(n+1), the first difference u_(n+1) - u_n must be
  regarded as a function of x_(n+½), i.e. of ½{x_n + x_(n+1)}. We
  therefore write u_(n+1) - u_n = [delta]u_(n+½), and each difference in
  the table in § 9 will have the same suffix as the value of x in the
  same horizontal line; or, if the difference is of an odd order, its
  suffix will be the means of those of the two nearest values of x. This
  is shown in the table below.

  In this notation, instead of using the symbol E, we use a symbol [mu]
  to denote the mean of two consecutive values of u, or of two
  consecutive differences of the same order, the suffixes being assigned
  on the same principle as in the case of the differences. Thus

    [mu]u_(n+½) = ½{u_n + u_(n+1)}, [mu][delta]u_n = ½{[delta]u_(n-½)} + [delta]u_(n+½), &c.

  If we take the means of the differences of odd order immediately above
  and below the horizontal line through any value of x, these means,
  with the differences of even order in that line, constitute the
  _central differences_ of the corresponding value of u. Thus the table
  of central differences is as follows, the values obtained as means
  being placed in brackets to distinguish them from the actual
  differences:--

    +-------+-------+---------------------+----------------+----------------------+----------------------+
    |   x   |   u   |      1st Diff.      |    2nd Diff.   |       3rd Diff.      |      4th Diff.       |
    +-------+-------+---------------------+----------------+----------------------+----------------------+
    |   .   |   .   |          .          |        .       |           .          |           .          |
    |   .   |   .   |          .          |        .       |           .          |           .          |
    |   .   |   .   |          .          |        .       |           .          |           .          |
    |x_(n-2)|u_(n-2)| {[mu][delta]u_(n-2)}| [delta]²u_(n-2)| {[mu][delta]³u_(n-2)}| [delta]^4u_(n-2) ... |
    |       |       |   [delta]u_(n-3/2)  |                |  [delta]³u_(n-3/2)   |                      |
    |x_(n-1)|u_(n-1)| {[mu][delta]u_(n-1)}| [delta]²u_(n-1)| {[mu][delta]³u_(n-1)}| [delta]^4u_(n-1) ... |
    |       |       |   [delta]u_(n-½)    |                |    [delta]³u_(n-2    |                      |
    |x_n    |u_n    |  ([mu][delta]u_n)   | [delta]²u_n    |  ([mu][delta]³u_n)   | [delta]^4u_n     ... |
    |       |       |   [delta]u_(n+½)    |                |     [delta]³u_(n+½)  |                      |
    |x_(n+1)|u_(n+1)| {[mu][delta]u_(n+1)}| [delta]²u_(n+1)| {[mu][delta]³u_(n+1)}| [delta]^4u_(n+1) ... |
    |       |       |   [delta]u_(n+3/2)  |                |  [delta]³u_(n+3/2)   |                      |
    |x_(n+2)|u_(n+2)| {[mu][delta]u_(n+2)}| [delta]²u_(n+2)| {[mu][delta]³u_(n+2)}| [delta]^4u_(n+2) ... |
    |   .   |   .   |          .          |        .       |           .          |           .          |
    |   .   |   .   |          .          |        .       |           .          |           .          |
    |   .   |   .   |          .          |        .       |           .          |           .          |
    +-------+-------+---------------------+----------------+----------------------+----------------------+

  Similarly, by taking the means of consecutive values of u and also of
  consecutive differences of even order, we should get a series of terms
  and differences central to the intervals x_(n-2) to x_(n-1), x_(n-1)
  to x_n, ....

  The terms of the series of which the values of u are the first
  differences are denoted by [sigma]u, with suffixes on the same
  principle; the suffixes being chosen so that [delta][sigma]un shall be
  equal to un. Thus, if

    [sigma]u_(n-3/2) = C + u_(n-2),

  then

    [sigma]u_(n-½) = C + u_(n-2) + u_(n-1), [sigma]_(n+½)
       = C + u_(n-2) + u_(n-1) + u_n, &c.,

  and also

    [mu][sigma]u_(n-1) = C + u_(n-2) + ½u_(n-1), [mu][sigma]u_n
       = C + u_(n-2) + u_(n-1) + ½u_n, &c.,

  C being an arbitrary constant which must remain the same throughout
  any series of operations.


  _Operators and Symbolic Methods._

  12. There are two further stages in the use of the symbols [Delta],
  [Sigma], [delta], [sigma], &c., which are not essential for elementary
  treatment but lead to powerful methods of deduction.

  (i.) Instead of treating [Delta]u as a function of x, so that
  [Delta]u_n means ([Delta]u)_n, we may regard [Delta] as denoting an
  _operation_ performed on u, and take [Delta]un as meaning [Delta].u_n.
  This applies to the other symbols E, [delta], &c., whether taken
  simply or in combination. Thus [Delta]Eu_n means that we first replace
  un by un+1, and then replace this by u_(n+2) - u_(n+1).

  (ii.) The operations [Delta], E, [delta], and [mu], whether performed
  separately or in combination, or in combination also with numerical
  multipliers and with the operation of differentiation denoted by D (:=
  d/dx), follow the ordinary rules of algebra: e.g. [Delta](u_n + v_n) =
  [Delta]u_n + [Delta]v_n, [Delta]Du_n = D[Delta]u_n, &c. Hence the
  symbols can be separated from the functions on which the operations
  are performed, and treated as if they were algebraical quantities. For
  instance, we have

    E·u_n = u_(n+1) = u_n + [Delta]u_n = 1·u_n + [Delta]·u_n,

  so that we may write E = 1 + [Delta], or [Delta] = E - 1. The first of
  these is nothing more than a statement, in concise form, that if we
  take two quantities, subtract the first from the second, and add the
  result to the first, we get the second. This seems almost a truism.
  But, if we deduce E^n = (1 + [Delta])^n, [Delta]^n = (E-1)^n, and
  expand by the binomial theorem and then operate on u0, we get the
  general formulae

                           n·n - 1
    un = u0 + n[Delta]u0 + ------- [Delta]^2u0 + ... + [Delta]^nu0,
                             1·2
                                   n·n - 1
    [Delta]^nu0 = u_n - nu_(n-1) + ------- u_(n-2) + ... + (-1)^nu0,
                                     1·2

  which are identical with the formulae in (ii.) and (i.) of § 3.

  (iii.) What has been said under (ii.) applies, with certain
  reservations, to the operations [Sigma] and [sigma], and to the
  operation which represents integration. The latter is sometimes
  denoted by D^-1; and, since [Delta][Sigma]un = un, and
  [delta][sigma]u_n = u_n, we might similarly replace [Sigma] and
  [sigma] by [Delta]^-1 and [delta]^-1. These symbols can be combined
  with [Delta], E, &c. according to the ordinary laws of algebra,
  provided that proper account is taken of the arbitrary constants
  introduced by the operations D^-1, [Delta]^-1, [delta]^-1.


  _Applications to Algebraical Series._

  13. _Summation of Series._--If ur, denotes the (r+1)th term of a
  series, and if vr is a function of r such that [Delta]v_r = u_r for
  all integral values of r, then the sum of the terms u_m, u_(m+1), ...
  un is v_(n+1) -v_m. Thus the sum of a number of terms of a series may
  often be found by inspection, in the same kind of way that an integral
  is found.

  14. _Rational Integral Functions._--(i.) If u_r is a rational integral
  function of r of degree p, then [Delta]ur, is a rational integral
  function of r of degree p-1.

  (ii.) A particular case is that of a _factorial_, i.e. a product of
  the form (r+a+1) (r+a+2) ... (r+b), each factor exceeding the
  preceding factor by 1. We have

    [Delta]·(r+a+1) (r+a+2) ... (r+b) = (b-a)·(r+a+2) ... (r+b),

  whence, changing a into a-1,

    [Sigma](r+a+1)(r+a+2) ... (r+b) = _const._ + (r+a)(r+a+1) ...
      (r+b)/(b-a+1).

  A similar method can be applied to the series whose (r+1)th term is of
  the form 1/(r+a+1) (r+a+2) ... (r+b).

  (iii.) Any rational integral function can be converted into the sum of
  a number of factorials; and thus the sum of a series of which such a
  function is the general term can be found. For example, it may be
  shown in this way that the sum of the pth powers of the first n
  natural numbers is a rational integral function of n of degree p+1,
  the coefficient of n^p+1 being 1/(p+1).

  15. _Difference-equations._--The summation of the series ... + u_(n+2)
  + u_(n-1) + u_n is a solution of the _difference-equation_ [Delta]v_n
  = u_(n+1), which may also be written (E-1)v_n = u_(n+1). This is a
  simple form of difference-equation. There are several forms which have
  been investigated; a simple form, more general than the above, is the
  _linear equation_ with _constant coefficients_--

    v_(n+m) + a1v_(n+m-1) + a2v_(n+m-2) + ... + a_mv_n = N,

  where a1, a2, ... am are constants, and N is a given function of n.
  This may be written

    (E^m + a1E^(m-1) + ... + a_m)v_n = N

  or

    (E-p1)(E-p2) ... (E-p_m)v_n = N.

  The solution, if p1, p2, ... pm are all different, is vn = C1p1^n +
  C2p2^n + ... + C_mp_m^n + V_n, where C1, C2 ... are constants, and v_n
  = V_n is any one solution of the equation. The method of finding a
  value for Vn depends on the form of N. Certain modifications are
  required when two or more of the p's are equal.

  It should be observed, in all cases of this kind, that, in describing
  C1, C2 as "constants," it is meant that the value of any one, as C1,
  is the same for all values of n occurring in the series. A "constant"
  may, however, be a periodic function of n.


  _Applications to Continuous Functions._

  16. The cases of greatest practical importance are those in which u is
  a continuous function of x. The terms u1, u2 ... of the series then
  represent the successive values of u corresponding to x = x1, x2....
  The important applications of the theory in these cases are to (i.)
  relations between differences and differential coefficients, (ii.)
  interpolation, or the determination of intermediate values of u, and
  (iii.) relations between sums and integrals.

  17. Starting from any pair of values x0 and u0, we may suppose the
  interval h from x0 to x1 to be divided into q equal portions. If we
  suppose the corresponding values of u to be obtained, and their
  differences taken, the successive advancing differences of u0 being
  denoted by dPu0, dP²u0 ..., we have (§ 3 (ii.))

                         q·q - 1
    u1  = u0  + qdPu0  + ------- dP²u0  + ....
                           1·2

  When q is made indefinitely great, this (writing f(x) for u) becomes
  Taylor's Theorem (INFINITESIMAL CALCULUS)

                                h²
    f(x + h) = f(x) + hf'(x) + --- f"(x) + ...,
                               1·2

  which, expressed in terms of operators, is

                  h²       h³
    E = 1 + hD + ---D² + ----- D³ + ... = e^(hD).
                 1·2     1·2·3

  This gives the relation between [Delta] and D. Also we have

                         2q·2q - 1
    u2 = u0  + 2qdPu0  + --------- dP²u0  + ...
                            1·2

                         3q·3q - 1
    u3 = u0  + 3qdPu0  + --------- dP²u0  + ...
                            1·2
    .            .
    .            .
    .            .

  and, if p is any integer,

                           p·p - 1
    u_(p/q) = u0 + pdPu0 + ------- dP²u0 + ....
                             1·2

  From these equations up/q could be expressed in terms of u0, u1, u2,
  ...; this is a particular case of interpolation (q.v.).

  18. _Differences and Differential Coefficients._--The various formulae
  are most quickly obtained by symbolical methods; i.e. by dealing with
  the operators [Delta], E, D, ... as if they were algebraical
  quantities. Thus the relation E = e^(hD) (§ 17) gives

    hD = log_e  (1 + [Delta]) = [Delta] - ½[Delta]² + 1/3 [Delta]³ ...

         /du\
    or h( -- ) = [Delta]u0 - ½[Delta]²u0 + 1/3 [Delta]³u0  ....
         \dx/0

  The formulae connecting central differences with differential
  coefficients are based on the relations [mu] = cosh ½hD = ½(e^ ½hD +
  e^ -½hD), [delta] = 2 sinh ½hD - e^ ½hD - e^ -½hD, and may be grouped
  as follows:--

              u0  = u0                                              \
                                                                    |
    [mu][delta]u0 = (hD + 1/6 h³D³ + 1/120 h^5 D^5 + ...)u0         |
                                                                    |
       [delta]²u0 = (h²D² + 1/12 h^4 D^4 + 1/360 h^6 D^6 + ...)u0    >
                                                                    |
   [mu][delta]³u0 = (h³D³ + 1/4 h^5 D^5 + ...)u0                    |
                                                                    |
     [delta]^4 u0 = (h^4 D^4 + 1/6 h^6 D^6 + ...)u0                 /

                  .                .             .
                  .                .             .
                  .                .             .

          [mu]u_½ = (1 + 1/8 h²D² + 1/384 h^4 D^4 + 1/46080 h^6 D^6 + ...)u_½ \
                                                                              |
       [delta]u_½ = (hD + 1/24 h³D³ + 1/1920 h^5 D^5 + ...)u_½                |
                                                                              |
   [mu][delta]²u_½ = (h²D² + 5/24 h^4 D^4 + 91/5760 h^6 D^6 + ...)u_½          >
                                                                              |
      [delta]³u_½ = (h³D³ + 1/8 h^5 D^5 + ...)u_½                             |
                                                                              |
   [mu][delta]^4 u_½ = (h^4 D^4 + 7/24 h^6 D^6 + ...)u_½                      /

                  .                .             .
                  .                .             .
                  .                .             .

               u0 = u0                                                             \
                                                                                   |
            hDu0  = ([mu][delta] - 1/6 [mu][delta]³ + 1/30 [mu][delta]^5  - ...)u0 |
                                                                                   |
           h²D²u0 = ([delta]² - 1/12 [delta]^4 + 1/90 [delta]^6 - ...)u0            >
                                                                                   |
          h³D³u0  = ([mu][delta]³ 1/4 [mu][delta]^5  + ...)u0                      |
                                                                                   |
      h^4 D^4 u_0 = ([delta]^4 - 1/6 [delta]^6 + ...)u0                            /

                  .                .             .
                  .                .             .
                  .                .             .

              u_½ = ([mu] - 1/8 [mu][delta]² + 3/128 [mu][delta]^4 - 5/1024 [mu][delta]^6 + ...)u_½ \
                                                                                                    |
           hDu_½  = ([delta] - 1/24 [delta]³ + 3/640 [delta]^5 - ...)u_½                            |
                                                                                                    |
          h²D²u_½ = ([mu][delta]² - 5/24 [mu][delta]^ + 259/5760 [mu][delta]^6 - ...)u_½             >
                                                                                                    |
         h³D³u_½  = ([delta]³ - 1/8 [delta]^5 + ...)u_½                                             |
                                                                                                    |
      h^4 D^4 u_½ = ([mu][delta]^4 - 7/24 [mu][delta]^6 + ...)u_½                                   /

                  .                .             .
                  .                .             .
                  .                .             .

  When u is a rational integral function of x, each of the above series
  is a terminating series. In other cases the series will be an infinite
  one, and may be divergent; but it may be used for purposes of
  approximation up to a certain point, and there will be a "remainder,"
  the limits of whose magnitude will be determinate.

  19. _Sums and Integrals._--The relation between a sum and an integral
  is usually expressed by the _Euler-Maclaurin formula_. The principle
  of this formula is that, if um and um+1, are ordinates of a curve,
  distant h from one another, then for a first approximation to the area
  of the curve between um and um+1 we have ½h(u_m + u_m+1), and the
  difference between this and the true value of the area can be
  expressed as the difference of two expressions, one of which is a
  function of x_m, and the other is the same function of x_m+1. Denoting
  these by [phi](x_m) and [phi](xm+1), we have

      _ x_m+1
     /
     |   udx = ½h(u_m  + u_m+1) + [phi](x_m+1 ) - [phi](x_m).
    _/x_m

  Adding a series of similar expressions, we find

      _ x_n
     /
     |  udx = h{½u_m + u_m+1 + u_m+2 + ... + u_n-1 + ½u_n} + [phi](x_n) - [phi](x_m).
    _/x_m

  The function [phi](x) can be expressed in terms either of differential
  coefficients of u or of advancing or central differences; thus there
  are three formulae.

  (i.) The Euler-Maclaurin formula, properly so called, (due
  independently to Euler and Maclaurin) is

      _ x_n
     /                             1    du_n    1       d³u_n     1       d^5 u_n
     |   udx = h·[mu][sigma]u_n - -- h² ---- + --- h^4  ----- - ----- h^6 ------- + ...
    _/x_m                         12     dx    720       dx³    30240       dx^5

                                  B1    du_n   B2     d³u_n   B3     d^5u_n
             = h·[mu][sigma]u_n - -- h2 ---- + -- h^4 ----- - -- h^6 ------ + ...,
                                  2!     dx    4!      dx³    6!      dx^5

  where B1, B2, B3 ... are _Bernoulli's numbers_.

  (ii.) If we express differential coefficients in terms of advancing
  differences, we get a theorem which is due to Laplace:--

        _ x_n
    1  /
    -  |  udx = [mu][sigma](u_n - u0) - 1/12 ([Delta]u_n - [Delta]u0) + 1/24 ( [Delta]²u_n - [Delta]²u0)
    h _/x0

           - 19/720 ([Delta]³u_n - [Delta]³u_0) + 3/160 ([Delta]^4 u_n - [Delta]^4 u0) - ...


  For practical calculations this may more conveniently be written

        _ x_n
    1  /
    -  |  udx = [mu][sigma](u_n - u0) + 1/12 ([Delta]u0 - ½[Delta]²u0 + 19/60 [Delta]³u0 - ...)
    h _/x0

                 + 1/12 ([Delta]'u_n - ½[Delta]'²u_n + 19/60 [Delta]'³u_n - ...),

  where accented differences denote that the values of u are read
  backwards from un; i.e. [Delta]'un denotes u_n-1 - u_n, not (as in §
  10) u_n - u_n-1.

  (iii.) Expressed in terms of central differences this becomes

        _ x_n
    1  /
    -  |   udx = [mu][sigma](u_n - u0) - 1/12 [mu][delta]u_n + 11/720 [mu][delta]³u_n - ...
    h _/x0
                            + 1/12 [mu][delta]u0  - 11/720 [mu][delta]³u0 + ...

           /          1            11             191               2497                 \  /      \
    = [mu]([sigma] - -- [delta] + --- [delta]³ - ----- [delta]^5 + ------- [delta]^7 - ...)(u_n - u0).
           \         12           720            60480             3628800               /  \      /

  (iv.) There are variants of these formulae, due to taking hum+½ as the
  first approximation to the area of the curve between um and um+1; the
  formulae involve the sum u_½ + u_3/2 + ... + u_n-½ := [sigma](u_n -
  u0) (see MENSURATION).

  20. The formulae in the last section can be obtained by symbolical
  methods from the relation

        _
    1  /       1         1
    -  | udx = - D^1 u = --·u.
    h _/       h         hD

  Thus for central differences, if we write [theta] := ½hD, we have [mu]
  = cosh [theta], [delta] = 2 sinh [theta], [sigma] = [delta]^-1, and
  the result in (iii.) corresponds to the formula

                                        / /   1                 2                   2·4                    \
    sinh [theta] = [theta] cosh [theta]/ (1 + - sinh²[theta] - --- sinh^4[theta] + ----- sinh^6[theta] - ...).
                                      /   \   3                3·5                 3·5·7                   /

  REFERENCES.--There is no recent English work on the theory of finite
  differences as a whole. G. Boole's _Finite Differences_ (1st ed.,
  1860, 2nd ed., edited by J. F. Moulton, 1872) is a comprehensive
  treatise, in which symbolical methods are employed very early. A. A.
  Markoff's _Differenzenrechnung_ (German trans., 1896) contains general
  formulae. (Both these works ignore central differences.) _Encycl. der
  math. Wiss._ vol. i. pt. 2, pp. 919-935, may also be consulted. An
  elementary treatment of the subject will be found in many text-books,
  e.g. G. Chrystal's _Algebra_ (pt. 2, ch. xxxi.). A. W. Sunderland,
  _Notes on Finite Differences_ (1885), is intended for actuarial
  students. Various central-difference formulae with references are
  given in _Proc. Lond. Math. Soc._ xxxi. pp. 449-488. For other
  references see INTERPOLATION.     (W. F. SH.)



DIFFERENTIAL EQUATION, in mathematics, a relation between one or more
functions and their differential coefficients. The subject is treated
here in two parts: (1) an elementary introduction dealing with the more
commonly recognized types of differential equations which can be solved
by rule; and (2) the general theory.


  _Part I.--Elementary Introduction._

  Of equations involving only one independent variable, x (known as
  _ordinary_ differential equations), and one dependent variable, y, and
  containing only the first differential coefficient dy/dx (and
  therefore said to be of the first _order_), the simplest form is that
  reducible to the type

    dy/dx = f(x)/F(y),

  leading to the result fF(y)dy - ff(x)dx = A, where A is an arbitrary
  constant; this result is said to solve the differential equation, the
  problem of evaluating the integrals belonging to the integral
  calculus.


  Another simple form is

    dy/dx + yP = Q,

  where P, Q are functions of x only; this is known as the linear
  equation, since it contains y and dy/dx only to the first degree. If

    fPdx = u, we clearly have

     d             /dy    \
    --(ye^u) =e^u ( -- + Py) = e^u Q,
    dx             \dx    /

  so that y = e^-u(fe^u Qdx + A) solves the equation, and is the only
  possible solution, A being an arbitrary constant. The rule for the
  solution of the linear equation is thus to multiply the equation by
  e^u, where u = fPdx.

  A third simple and important form is that denoted by

    y = px + f(p),

  where p is an abbreviation for dy/dx; this is known as Clairaut's
  form. By differentiation in regard to x it gives

             dp        dp
    p = p + x-- + f'(p)--,
             dx        dx

  where

             d
    f'(p) = -- f(p);
            dp

  thus, either (i.) dp/dx = 0, that is, p is constant on the curve
  satisfying the differential equation, which curve is thus any one of
  the straight lines y = cx = f(c), where c is an arbitrary constant, or
  else, (ii.) x + [f]'(p) = 0; if this latter hypothesis be taken, and p
  be eliminated between x + f'(p) = 0 and y = px + f(p), a relation
  connecting x and y, not containing an arbitrary constant, will be
  found, which obviously represents the envelope of the straight lines y
  = cx + f(c).

  In general if a differential equation [phi](x, y, dy/dx) = 0 be
  satisfied by any one of the curves F(x, y, c) = 0, where c is an
  arbitrary constant, it is clear that the envelope of these curves,
  when existent, must also satisfy the differential equation; for this
  equation prescribes a relation connecting only the co-ordinates x, y
  and the differential coefficient dy/dx, and these three quantities are
  the same at any point of the envelope for the envelope and for the
  particular curve of the family which there touches the envelope. The
  relation expressing the equation of the envelope is called a
  _singular_ solution of the differential equation, meaning an
  _isolated_ solution, as not being one of a family of curves depending
  upon an arbitrary parameter.

  An extended form of Clairaut's equation expressed by

    y = xF(p) + f(p)

  may be similarly solved by first differentiating in regard to p, when
  it reduces to a linear equation of which x is the dependent and p the
  independent variable; from the integral of this linear equation, and
  the original differential equation, the quantity p is then to be
  eliminated.

  Other types of solvable differential equations of the first order are
  (1)

    M dy/dx = N,

  where M, N are homogeneous polynomials in x and y, of the same order;
  by putting v = y/x and eliminating y, the equation becomes of the
  first type considered above, in v and x. An equation (aB <> bA)

    (ax + by + c)dy/dx = Ax + By + C

  may be reduced to this rule by first putting x + h, y + k for x and y,
  and determining h, k so that ah + bk + c = 0, Ah + Bk + C = 0.

  (2) An equation in which y does not explicitly occur,

    f(x, dy/dx) = 0,

  may, theoretically, be reduced to the type dy/dx = F(x); similarly an
  equation F(y, dy/dx) = 0.

  (3) An equation

    f(dy/dx, x, y) = 0,

  which is an integral polynomial in dy/dx, may, theoretically, be
  solved for dy/dx, as an algebraic equation; to any root dy/dx = F1(x,
  y) corresponds, suppose, a solution [phi]1(x, y, c) = 0, where c is an
  arbitrary constant; the product equation [phi]1(x, y, c)[phi]2(x, y,
  c) ... = 0, consisting of as many factors as there were values of
  dy/dx, is effectively as general as if we wrote [phi]1(x, y, c1)
  [phi]2(x, y, c2) ... = 0; for, to evaluate the first form, we must
  necessarily consider the factors separately, and nothing is then
  gained by the multiple notation for the various arbitrary constants.
  The equation [phi]1(x, y, c)[phi]2(x, y, c) ... = 0 is thus the
  solution of the given differential equation.

  In all these cases there is, except for cases of singular solutions,
  one and only one arbitrary constant in the most general solution of
  the differential equation; that this must necessarily be so we may
  take as obvious, the differential equation being supposed to arise by
  elimination of this constant from the equation expressing its solution
  and the equation obtainable from this by differentiation in regard to
  x.

  A further type of differential equation of the first order, of the
  form

    dy/dx = A + By + Cy²

  in which A, B, C are functions of x, will be briefly considered below
  under differential equations of the second order.

  When we pass to ordinary differential equations of the second order,
  that is, those expressing a relation between x, y, dy/dx and d²y/dx²,
  the number of types for which the solution can be found by a known
  procedure is very considerably reduced. Consider the general linear
  equation

    d²y    dy
    --- + P-- + Qy = R,
    dx²    dx

  where P, Q, R are functions of x only. There is no method always
  effective; the main general result for such a linear equation is that
  if any particular function of x, say y1, can be discovered, for which

    d²y1    dy1
    ---- + P--- + Qy1 = 0,
    dx²     dx

  then the substitution y = y1[eta] in the original equation, with R on
  the right side, reduces this to a linear equation of the first order
  with the dependent variable d[eta]/dx. In fact, if y = y1[eta] we have

    dy     d[eta]        dy1     d²y     d²[eta]    dy1 d[eta]        d²y1
    -- = y1------ + [eta]--- and --- = y1------- + 2--- ------ + [eta]----,
    dx       dx          dx      dx²       dx²      dx    dx           dx²

  and thus

    d²y     dy          d²[eta]    / dy1     \  d[eta]    /d²y1    dy1     \
    --- + P -- + Qy = y1------- + ( 2--- + Py1) ------ + ( ---- + P--- + Qy1)[eta];
    dx²     dx            dx²      \ dx      /    dx      \ dx²    dx      /

  if then

    d²y1     dy1
    ---- + P --- + Qy1 = 0,
    dx²      dx

  and z denote d[eta]/dx, the original differential equation becomes

      dz    / dy1     \
    y1-- + ( 2--- + Py1)z = R.
      dx    \ dx      /

  From this equation z can be found by the rule given above for the
  linear equation of the first order, and will involve one arbitrary
  constant; thence y = y1 [eta] = y1 [int] zdx + Ay1, where A is another
  arbitrary constant, will be the general solution of the original
  equation, and, as was to be expected, involves two arbitrary
  constants.

  The case of most frequent occurrence is that in which the coefficients
  P, Q are constants; we consider this case in some detail. If [t]*
  be a root of the quadratic equation [t]² + [t]P + Q = 0, it
  can be at once seen that a particular integral of the differential
  equation with zero on the right side is y1 = e^[theta]x. Supposing
  first the roots of the quadratic equation to be different, and [phi]
  to be the other root, so that [p] + [t] = -P, the auxiliary
  differential equation for z, referred to above, becomes dz/dx +
  ([t] - [p])z = Re^(-[t]^x), which leads to
  ze^{([t]-[p])^x} = B + [int] Re^(-[p]^x)dx, where B is an
  arbitrary constant, and hence to

(*) [t] = [theta]; [p] = [phi].
                               _                             _               _
                              /                             /               /
    y = Ae^([t]^x) + e^([t]^x)| Be^([p]-[t])^x dx + e^[t]^x | e^([p]-[t])^x | Re^-[p]^x dxdx,
                             _/                            _/              _/

  or say to y = Ae^[t]^x + Ce^[p]^x + U, where A, C are arbitrary
  constants and U is a function of x, not present at all when R = 0. If
  the quadratic equation [t]² + P[t] + Q = 0 has equal roots, so that
  2[t] = -P, the auxiliary equation in z becomes dz/dx = Re^-[t]^x,
  giving z = B + [int] Re^-[t]^x dx, where B is an arbitrary constant,
  and hence
                                   _  _
                                  /  /
    y = (A + Bx)e^[t]^x + e^[t]^x |  | Re^-[t]^x dxdx,
                                 _/ _/

  or, say, y = (A + Bx)e^[t]^x + U, where A, B are arbitrary constants,
  and U is a function of x not present at all when R = 0. The portion
  Ae^[t]^x + Be^[p]^x or (A + Bx)e^[t]^x of the solution, which is known
  as the _complementary function_, can clearly be written down at once
  by inspection of the given differential equation. The remaining
  portion U may, by taking the constants in the complementary function
  properly, be replaced by any particular solution whatever of the
  differential equation

    d²v     dy
    --- + P -- + Qy = R;
    dx²     dx

  for if u be any particular solution, this has a form

    u = A0  e^[t]^x + B0 e^[p]^x + U,

  or a form

    u = (A0 + B0x)e^[t]^x + U;

  thus the general solution can be written

    (A - A0)e^[t]^x + (B - B0)e^[p]^x + u,

  or

    {A - A0  + (B - B0)x}e^[t]^x + u,

  where A - A0, B - B0, like A, B, are arbitrary constants.

  A similar result holds for a linear differential equation of any
  order, say

    d^n y      d^n-1 y
    ----- + P1 ------- + ... + P_n y = R,
    dx_n       dx^n-1

  where P1, P2, ... Pn are constants, and R is a function of x. If we
  form the algebraic equation [t]^n + P1[t]^n-1 + ... + P_n = 0, and all
  the roots of this equation be different, say they are [t]1, [t]2, ...
  [t]n, the general solution of the differential equation is

    y = A1 e^[t]1^x + A2 e^[t]2^x + ... + A_n e^[t]_n^x + u,

  where A1, A2, ... An are arbitrary constants, and u is any particular
  solution whatever; but if there be one root [t]1 repeated r times, the
  terms A1 e^[t]1^x + ... + A_r e^[t]_r^x must be replaced by (A1 + A2x
  + ... + A_r x^r-1)e^[t]1x where A1, ... An are arbitrary constants;
  the remaining terms in the complementary function will similarly need
  alteration of form if there be other repeated roots.

  To complete the solution of the differential equation we need some
  method of determining a particular integral u; we explain a procedure
  which is effective for this purpose in the cases in which R is a sum
  of terms of the form e^ax[p](x), where [p](x) is an integral
  polynomial in x; this includes cases in which R contains terms of the
  form cos bx·[p](x) or sin bx·[p](x). Denote d/dx by D; it is clear
  that if u be any function of x, D(e^ax u) = e^ax Du + ae^ax u, or say,
  D(e^ax u) = e^ax (D + a)u; hence D²(e^ax u), i.e. d²/dx² (e^ax u),
  being equal to D(e^ax v), where v=(D + a)u, is equal to e^ax(D + a)v,
  that is to e^ax(D + a)²u. In this way we find D^n(e^ax u) = e^ax(D +
  a)^n u, where n is any positive integer. Hence if [psi](D) be any
  polynomial in D with constant coefficients, [psi](D)(e^ax u) = e^ax
  [psi](D + a)u. Next, denoting [int] udx by D^-1 u, and any solution of
  the differential equation dz/dx + az = u by z = (d + a)^-1 u, we have
  D[e^ax(D + a)^-1 u] = D(e^ax z) = e^ax(D + a)z = e^ax u, so that we
  may write D^-1(e^ax u) = e^ax(D+a)^-1 u, where the meaning is that one
  value of the left side is equal to one value of the right side; from
  this, the expression D^-2(e^axu), which means D^-1[D^-1(e^ax u)], is
  equal to D^-1(e^ax z) and hence to e^ax(D + a)^-1 z, which we write
  e^ax(D + a)^-2 u; proceeding thus we obtain

    D^-n(e^ax u) = e^ax(D + a)^-n u,

  where n is any positive integer, and the meaning, as before, is that
  one value of the first expression is equal to one value of the second.
  More generally, if [psi](D) be any polynomial in D with constant
  coefficients, and we agree to denote by 1/[psi](D) u any solution z of
  the differential equation [psi](D)z = u, we have, if v = 1/[psi](D +
  a) u, the identity [psi](D)(e^ax v) = e^ax [psi](D + a)v = e^ax u,
  which we write in the form

        1                        1
    --------(e^ax u) = e^ax ------------ u.
    [psi](D)                [psi](D + a)

  This gives us the first step in the method we are explaining, namely
  that a solution of the differential equation [psi](D)y = e^ax u + e^bx
  v + ... where u, v, ... are any functions of x, is any function
  denoted by the expression

              1                     1
    e^ax ------------ u + e^ax ------------ v + ....
         [psi](D + a)          [psi](D + b)

  It is now to be shown how to obtain one value of 1/[psi](D + a) u,
  when u is a polynomial in x, namely one solution of the differential
  equation [psi](D + a)z = u. Let the highest power of x entering in u
  be x^m; if t were a variable quantity, the rational fraction in t,
  1/[psi](t + a), by first writing it as a sum of partial fractions, or
  otherwise, could be identically written in the form

    K_r t^-r + K_r-1 t^-r+1 + ... + K1 t^-1 + H + H1t + ... + H_m t^m + t^m+1 [p](t)/[psi](t + a),

  where [p](t) is a polynomial in t; this shows that there exists an
  identity of the form

    1 = [psi](t + a)(K_r t^-r + ... + K1t^-1 + H + H1t + ... + H_m t^m) + [p](t)t^m+1,

  and hence an identity

    u = [psi](D + a)[K_r D^-r + ... + K1D^-1 + H + H1D + ... + H_m D^m]u + [p](D)D^m+1 u;

  in this, since u contains no power of x higher than x^m, the second
  term on the right may be omitted. We thus reach the conclusion that a
  solution of the differential equation [psi](D + a)z = u is given by

    z = (K_r D^-r + ... + K1D^-1 + H + H1D + ... + H_m D^m)u,

  of which the operator on the right is obtained simply by expanding
  1/[psi](D + a) in ascending powers of D, as if D were a numerical
  quantity, the expansion being carried as far as the highest power of D
  which, operating upon u, does not give zero. In this form every term
  in z is capable of immediate calculation.

  _Example._--For the equation

    d^4v    d²y
    ---- + 2--- + y = x³ cos x or (D² + 1)²y = x³ cos x,
    dx^4    dx³

  the roots of the associated algebraic equation ([t]²+1)² = 0 are [t] =
  ±i, each repeated; the complementary function is thus

    (A + Bx)e^ix + (C + Dx)e^ix,

  where A, B, C, D are arbitrary constants; this is the same as

    (H + Kx) cos x + (M + Nx) sin x,

  where H, K, M, N are arbitrary constants. To obtain a particular
  integral we must find a value of (1 + D²)^-2 x³ cos x; this is the
  real part of (1+D²)^-2 e^ix x³ and hence of e^ix [1 + (D + i)²]^-2 x³

  or    e^ix [2iD(1 + ½iD)]^-2 x³,

  or    -¼e^ix D^-2 (1 + iD - ¾D² - ½iD³ + 5/16 D^4 + 3/16 iD^5 ...)x³,

  or    -¼e^ix(1/20 x^5 + ¼ix^4 - ¾x³ - 3/2 ix² + 15/8 x + 9/8 i);

  the real part of this is

    -¼(1/20 x^5 - ¾x² + 15/8 x) cos x + ¼(¼x^4 - 3/2 x² + 9/8) sin x.

  This expression added to the complementary function found above gives
  the complete integral; and no generality is lost by omitting from the
  particular integral the terms -15/32 x cos x + 9/32 sin x, which are
  of the types of terms already occurring in the complementary function.

  The symbolical method which has been explained has wider applications
  than that to which we have, for simplicity of explanation, restricted
  it. For example, if [psi](x) be any function of x, and a1, a2, ... an
  be different constants, and [(t + a1) (t + a2) ... (t + an)]^-1 when
  expressed in partial fractions be written [Sigma]c_m(t + a_m)^-1, a
  particular integral of the differential equation (D + a1)(D + a2) ...
  (D + a_n)y = [psi](x) is given by

    y = [Sigma]c_m(D + a_m)^-1 [psi](x) = [Sigma]c_m(D + a_m)^-1 e^-a m^x e^a m^x [psi](x) =

    [Sigma]c_m e^-a m^x D^-1 (e^a m^x [psi](x)) = [Sigma]c_m e^-a m^x [int] e^a m^x [psi](x)dx.

  The particular integral is thus expressed as a sum of n integrals.

  A linear differential equation of which the left side has the form

        d^ny           d^n-1 y                dy
    x^n ---- + P1x^n-1 ------- + ... + P_n-1 x-- + P_n y,
        dx^n           dx^n-1                 dx

  where P1, ... Pn are constants, can be reduced to the case considered
  above. Writing x = e^t we have the identity

        d^mu
    x^m ----  = [t]([t] - 1)([t] - 2) ... ([t] - m + 1)u, where [t] = d/dt.
        dx^m

  When the linear differential equation, which we take to be of the
  second order, has variable coefficients, though there is no general
  rule for obtaining a solution in finite terms, there are some results
  which it is of advantage to have in mind. We have seen that if one
  solution of the equation obtained by putting the right side zero, say
  y1, be known, the equation can be solved. If y2 be another solution of

    d²y    dy
    --- + P-- + Qy = 0,
    dx²    dx

  there being no relation of the form my1 + ny2 = k, where m, n, k are
  constants, it is easy to see that

    d/dx(y1'y2 - y1y2') = P(y1'y2 - y1y2'),

  so that we have

    y1'y2 - y1y2' = A exp.([int] Pdx),

  where A is a suitably chosen constant, and exp. z denotes e^z. In
  terms of the two solutions y1, y2 of the differential equation having
  zero on the right side, the general solution of the equation with R =
  [phi](x) on the right side can at once be verified to be Ay1 + By2 +
  y1u - y2v, where u, v respectively denote the integrals
         _                                     _
        /                                     /
    u = |y2[phi](x)(y1'y2 - y2'y1)^-1 dx, v = |y1[phi](x)(y1'y2 - y2'y1)^-1 dx.
       _/                                    _/

  The equation

    d²y    dy
    --- + P-- + Qy = 0,
    dx²    dx

  by writing y = v exp. (-½ [int] Pdx), is at once seen to be reduced to
  d²v/dx² + 1v = 0, where 1 = Q - ½dP/dx - ¼P². If [eta] = - 1/v dv/dx,
  the equation d²v/dx² + 1v = 0 becomes d[eta]/dx = 1 + [eta]², a
  non-linear equation of the first order.

  More generally the equation

    d[eta]
    ------ = A + B[eta] + C[eta]²,
      dx

  where A, B, C are functions of x, is, by the substitution

               1 dy
    [eta] = - -- --,
              Cy dx

  reduced to the linear equation

  d²y    /    1 dC\ dy
  --- - ( B + - -- )-- + ACy = 0.
  dx²    \    C dx/ dx

  The equation

    d[eta]
    ------ = A + B[eta] + C[eta]²,
      dx

  known as Riccati's equation, is transformed into an equation of the
  same form by a substitution of the form [eta] = (aY + b)/(cY + d),
  where a, b, c, d are any functions of x, and this fact may be utilized
  to obtain a solution when A, B, C have special forms; in particular if
  any particular solution of the equation be known, say [eta]0, the
  substitution [eta] = [eta]0 - 1/Y enables us at once to obtain the
  general solution; for instance, when

         d      /A\
    2B = -- log( - ),
         dx     \C/

  a particular solution is [eta]0 = [root](-A/C). This is a case of the
  remark, often useful in practice, that the linear equation

            d²y    d[phi] dy
    [phi](x)--- + ½------ -- + [mu]y = 0,
            dx²      dx   dx

  where [mu] is a constant, is reducible to a standard form by taking a
  new independent variable
         _
        /
    z = | dx[[p](x)]^-½.
       _/

  We pass to other types of equations of which the solution can be
  obtained by rule. We may have cases in which there are two dependent
  variables, x and y, and one independent variable t, the differential
  coefficients dx/dt, dy/dt being given as functions of x, y and t. Of
  such equations a simple case is expressed by the pair

    dx                dy
    -- = ax + by + c, -- = a'x + b'y + c',
    dt                dt

  wherein the coefficients a, b, c, a', b', c', are constants. To
  integrate these, form with the constant [lambda] the differential
  coefficient of z = x + [lambda]y, that is dz/dt = (a + [lambda]a')x +
  (b + [lambda]b')y + c + [lambda]c', the quantity [lambda] being so
  chosen that b + [lambda]b' = [lambda](a + [lambda]a'), so that we have
  dz/dt = (a + [lambda]a')z + c + [lambda]c'; this last equation is at
  once integrable in the form z(a + [lambda]a') + c + [lambda]c' = Ae^(a
  + [lambda]a')t, where A is an arbitrary constant. In general, the
  condition b + [lambda]b' = [lambda](a + [lambda]a') is satisfied by
  two different values of [lambda], say [lambda]1, [lambda]2; the
  solutions corresponding to these give the values of x +[lambda]1y and
  x + [lambda]2y, from which x and y can be found as functions of t,
  involving two arbitrary constants. If, however, the two roots of the
  quadratic equation for [lambda] are equal, that is, if (a - b')² +
  4a'b = 0, the method described gives only one equation, expressing x +
  [lambda]y in terms of t; by means of this equation y can be eliminated
  from dx/dt = ax + by + c, leading to an equation of the form dx/dt =
  Px + Q + Re^(a + [lambda]a')t, where P, Q, R are constants. The
  integration of this gives x, and thence y can be found.

  A similar process is applicable when we have three or more dependent
  variables whose differential coefficients in regard to the single
  independent variables are given as linear functions of the dependent
  variables with constant coefficients.

  Another method of solution of the equations

    dx/dt = ax + by + c, dy/dt = a'x + b'y + c',

  consists in differentiating the first equation, thereby obtaining

    d²x    dx    dy
    --- = a-- + b--;
    dt²    dt    dx

  from the two given equations, by elimination of y, we can express
  dy/dt as a linear function of x and dx/dt; we can thus form an
  equation of the shape d²x/dt² = P + Qx + Rdx/dt, where P, Q, R are
  constants; this can be integrated by methods previously explained, and
  the integral, involving two arbitrary constants, gives, by the
  equation dx/dt = ax + by + c, the corresponding value of y. Conversely
  it should be noticed that any single linear differential equation

    d²x             dx
    --- = u + vx + w--,
    dt²             dt

  where u, v, w are functions of t, by writing y for dx/dt, is
  equivalent with the two equations dx/dt = y, dy/dt = u + vx + wy. In
  fact a similar reduction is possible for any system of differential
  equations with one independent variable.

  Equations occur to be integrated of the form

    Xdx + Ydy + Zdz = 0,

  where X, Y, Z are functions of x, y, z. We consider only the case in
  which there exists an equation [phi](x, y, z) = C whose differential

    dP[phi]     dP[phi]     dP[phi]
    -------dx + -------dy + -------dz = 0
      dPx         dPy         dPz

  is equivalent with the given differential equation; that is, [mu]
  being a proper function of x, y, z, we assume that there exist
  equations

    dP[phi]          dP[phi]          v[phi]
    ------- = [mu]X, ------- = [mu]Y, ------ = [mu]Z;
      dPx              vy               vz

  these equations require

    dP           dP
    ---([mu]Y) = ---([mu]Z), &c.,
    dPz          dPy

  and hence

      /dPZ   dPY\      /dPX   dPZ\      /dPY   dPX\
    X( --- - --- ) + Y( --- - --- ) + Z( --- - --- ) = 0;
      \dPy   dPz/      \dPz   dPx/      \dPx   dPy/

  conversely it can be proved that this is sufficient in order that [mu]
  may exist to render [mu](Xdx + Ydy + Zdz) a perfect differential; in
  particular it may be satisfied in virtue of the three equations such
  as

    dPZ   dPY
    --- - --- = 0;
    dPy   dPz

  in which case we may take [mu] = 1. Assuming the condition in its
  general form, take in the given differential equation a plane section
  of the surface [phi] = C parallel to the plane z, viz. put z constant,
  and consider the resulting differential equation in the two variables
  x, y, namely Xdx + Ydy = 0; let [psi](x, y, z) = constant, be its
  integral, the constant z entering, as a rule, in [psi] because it
  enters in X and Y. Now differentiate the relation [psi](x, y, z) =
  [f](z), where [f] is a function to be determined, so obtaining

    dP[psi]     dP[psi]      /dP[psi]   df\
    -------dx + -------dy + ( ------- - -- )dz = 0;
      dPx         dPy        \  dPz     dz/

  there exists a function [sigma] of x, y, z such that

    dP[psi]              dP[psi]
    -------- = [sigma]X, ------- = [sigma]Y,
      dPx                  dPy

  because [psi] = constant, is the integral of Xdx + Ydy = 0; we desire
  to prove that [f] can be chosen so that also, in virtue of [psi](x, y,
  z) = f(z), we have

    dP[psi]   df                    df   dP[psi]
    ------- - -- = [sigma]Z, namely -- = ------- - [sigma]Z;
      dPz     dz                    dz     dPz

  if this can be proved the relation [psi](x, y, z) - f(z) = constant,
  will be the integral of the given differential equation. To prove this
  it is enough to show that, in virtue of [psi](x, y, z) = [f](z), the
  function dP[psi]/dPx - [sigma]Z can be expressed in terms of z only.
  Now in consequence of the originally assumed relations,

    dP[psi]          dP[phi]          dP[phi]
    ------- = [mu]X, ------- = [mu]Y, ------- = [mu]Z,
      dPx              dPy              dPz

  we have

    dP[psi]  /dP[phi]   [sigma]   dP[psi]  /dP[phi]
    ------- / ------- = ------- = ------- / -------,
      dPx  /    dPx      [mu]       dPy  /    dPy

  and hence

    dP[psi] dP[phi]   dP[psi] dP[phi]
    ------- ------- - ------- ------- = 0;
      dPx     dPy       dPy     dPx

  this shows that, as functions of x and y, [psi] is a function of [phi]
  (see the note at the end of part i. of this article, on Jacobian
  determinants), so that we may write [psi] = F(z, [phi]), from which

    [sigma]     dPF         dP[psi]   dPF     dPF   dP[phi]   dPF   [sigma]           dPF
    ------- = -------; then ------- = --- + ------- ------- = --- + ------- · [mu]Z = --- + [sigma]Z
     [mu]     dP[phi]         dPz     dPz   dP[phi]   dPz     dPz    [mu]             dPz

         dP[psi]              dPF
      or ------- - [sigma]Z = ---;
           dPz                dPz

  in virtue of [psi](x, y, z) = f(z), and [psi] = F(z, [phi]), the
  function [phi] can be written in terms of z only, thus dPF/dPz can be
  written in terms of z only, and what we required to prove is proved.

  Consider lastly a simple type of differential equation containing
  _two_ independent variables, say x and y, and one dependent variable
  z, namely the equation

     dPz    dPz
    P--- + Q--- = R,
     dPx    dPy

  where P, Q, R are functions of x, y, z. This is known as Lagrange's
  linear partial differential equation of the first order. To integrate
  this, consider first the ordinary differential equations dx/dz = P/R,
  dy/dz = Q/R, and suppose that two functions u, v, of x, y, z can be
  determined, independent of one another, such that the equations u = a,
  v = b, where a, b are arbitrary constants, lead to these ordinary
  differential equations, namely such that

     dPu    dPu    dPu          dPv    dPv    dPv
    P--- + Q--- = R--- = 0 and P--- + Q--- = R--- = 0.
     dPx    dPy    dPz          dPx    dPy    dPz

  Then if F(x, y, z) = 0 be a relation satisfying the original
  differential equations, this relation giving rise to

    dPF   dPF dPz         dPF   dPF dPz               dPF    dPF    dPF
    --- + --- --- = 0 and --- + --- --- = 0, we have P--- + Q--- = R--- = 0.
    dPx   dPz dPx         dPy   dPz dPy               dPx    dPy    dPz

  It follows that the determinant of three rows and columns vanishes
  whose first row consists of the three quantities dPF/dPx, dPF/dPy,
  dPF/dPz, whose second row consists of the three quantities dPu/dPx,
  dPu/dPy, dPu/dPz, whose third row consists similarly of the partial
  derivatives of v. The vanishing of this so-called Jacobian determinant
  is known to imply that F is expressible as a function of u and v,
  unless these are themselves functionally related, which is contrary to
  hypothesis (see the note below on Jacobian determinants). Conversely,
  any relation [phi](u, v) = 0 can easily be proved, in virtue of the
  equations satisfied by u and v, to lead to

     dz    dz
    P-- + Q-- = R.
     dx    dx

  The solution of this partial equation is thus reduced to the solution
  of the two ordinary differential equations expressed by dx/P = dy/Q =
  dz/R. In regard to this problem one remark may be made which is often
  of use in practice: when one equation u = a has been found to satisfy
  the differential equations, we may utilize this to obtain the second
  equation v = b; for instance, we may, by means of u = a, eliminate
  z--when then from the resulting equations in x and y a relation v = b
  has been found containing x and y and a, the substitution a = u will
  give a relation involving x, y, z.

  _Note on Jacobian Determinants._--The fact assumed above that the
  vanishing of the Jacobian determinant whose elements are the partial
  derivatives of three functions F, u, v, of three variables x, y, z,
  involves that there exists a functional relation connecting the three
  functions F, u, v, may be proved somewhat roughly as follows:--

  The corresponding theorem is true for any number of variables.
  Consider first the case of two functions p, q, of two variables x, y.
  The function p, not being constant, must contain one of the variables,
  say x; we can then suppose x expressed in terms of y and the function
  p; thus the function q can be expressed in terms of y and the function
  p, say q = Q(p, y). This is clear enough in the simplest cases which
  arise, when the functions are rational. Hence we have

    dPq   dPQ dPp     dPq   dPQ dPp   dPQ
    --- = --- --- and --- = --- --- + ---;
    dPx   dPp dPx     dPy   dPp dPy   dPy

  these give

    dPp dPq   dPp dPq   dPp dPQ
    --- --- - --- --- = --- ---;
    dPx dPy   dPy dPx   dPx dPy

  by hypothesis dPp/dPx is not identically zero; therefore if the
  Jacobian determinant of p and q in regard to x and y is zero
  identically, so is dPQ/dPy, or Q does not contain y, so that q is
  expressible as a function of p only. Conversely, such an expression
  can be seen at once to make the Jacobian of p and q vanish
  identically.

  Passing now to the case of three variables, suppose that the Jacobian
  determinant of the three functions F, u, v in regard to x, y, z is
  identically zero. We prove that if u, v are not themselves
  functionally connected, F is expressible as a function of u and v.
  Suppose first that the minors of the elements of dPF/dPx, dPF/dPy,
  dPF/dPz in the determinant are all identically zero, namely the three
  determinants such as

    dPu dPv   dPu dPv
    --- --- - --- ---;
    dPy dPz   dPz dPy

  then by the case of two variables considered above there exist three
  functional relations. [psi]1(u, v, x) = 0, [psi]2(u, v, y) = 0,
  [psi]3(u, v, z) = 0, of which the first, for example, follows from the
  vanishing of

    dPu dPv   dPu dPv
    --- --- - --- ---.
    dPy dPz   dPz dPy

  We cannot assume that x is absent from [psi]1, or y from [psi]2, or z
  from [psi]3; but conversely we cannot simultaneously have x entering
  in [psi]1, and y in [psi]2, and z in [psi]3, or else by elimination of
  u and v from the three equations [psi]1 = 0, [psi]2 = 0, [psi]3 = 0,
  we should find a necessary relation connecting the three independent
  quantities x, y, z; which is absurd. Thus when the three minors of
  dPF/dPx, dPF/dPy, dPF/dPz in the Jacobian determinant are all zero,
  there exists a functional relation connecting u and v only. Suppose no
  such relation to exist; we can then suppose, for example, that

    dPu dPv   dPu dPv
    --- --- - --- ---
    dPy dPz   dPz dPy

  is not zero. Then from the equations u(x, y, z) = u, v(x, y, z) = v we
  can express y and z in terms of u, v, and x (the attempt to do this
  could only fail by leading to a relation connecting u, v and x, and
  the existence of such a relation would involve that the determinant

    dPu dPv   dPu dPv
    --- --- - --- ---
    dPy dPz   dPz dPy

  was zero), and so write F in the form F(x, y, z) = [Phi](u, v, x). We
  then have

    dPF   dP[Phi] dPu   dP[Phi] dPv   dP[Phi]  dPF   dP[Phi] dPu   dP[Phi] dPv  dPF   dP[Phi] dPu   dP[Phi] dPv
    --- = ------- --- + ------- --- + -------, --- = ------- --- + ------- ---, --- = ------- --- + ------- ---;
    dPx     dPu   dPx     dPv   dPx     dPx    dPy     dPu   dPy     dPv   dPy  dPz     dPu   dPz     dPv   dPz

  thereby the Jacobian determinant of F, u, v is reduced to

    dP[Phi] /dPu dPv   dPu dPv\
    -------( --- --- - --- --- );
      dPx   \dPy dPz   dPz dPy/

  by hypothesis the second factor of this does not vanish identically;
  hence dP[Phi]/dPx = 0 identically, and [Phi] does not contain x; so
  that F is expressible in terms of u, v only; as was to be proved.


_Part II.--General Theory._

Differential equations arise in the expression of the relations between
quantities by the elimination of details, either unknown or regarded as
unessential to the formulation of the relations in question. They give
rise, therefore, to the two closely connected problems of determining
what arrangement of details is consistent with them, and of developing,
apart from these details, the general properties expressed by them. Very
roughly, two methods of study can be distinguished, with the names
Transformation-theories, Function-theories; the former is concerned with
the reduction of the algebraical relations to the fewest and simplest
forms, eventually with the hope of obtaining explicit expressions of the
dependent variables in terms of the independent variables; the latter is
concerned with the determination of the general descriptive relations
among the quantities which are involved by the differential equations,
with as little use of algebraical calculations as may be possible. Under
the former heading we may, with the assumption of a few theorems
belonging to the latter, arrange the theory of partial differential
equations and Pfaff's problem, with their geometrical interpretations,
as at present developed, and the applications of Lie's theory of
transformation-groups to partial and to ordinary equations; under the
latter, the study of linear differential equations in the manner
initiated by Riemann, the applications of discontinuous groups, the
theory of the singularities of integrals, and the study of potential
equations with existence-theorems arising therefrom. In order to be
clear we shall enter into some detail in regard to partial differential
equations of the first order, both those which are linear in any number
of variables and those not linear in two independent variables, and also
in regard to the function-theory of linear differential equations of the
second order. Space renders impossible anything further than the
briefest account of many other matters; in particular, the theories of
partial equations of higher than the first order, the function-theory of
the singularities of ordinary equations not linear and the applications
to differential geometry, are taken account of only in the bibliography.
It is believed that on the whole the article will be more useful to the
reader than if explanations of method had been further curtailed to
include more facts.

When we speak of a function without qualification, it is to be
understood that in the immediate neighbourhood of a particular set x0,
y0, ... of values of the independent variables x, y, ... of the
function, at whatever point of the range of values for x, y, ... under
consideration x0, y0, ... may be chosen, the function can be expressed
as a series of positive integral powers of the differences x - x0, y
-y0, ..., convergent when these are sufficiently small (see FUNCTION:
Functions of Complex Variables). Without this condition, which we
express by saying that the function is developable about x0, y0, ...,
many results provisionally stated in the transformation theories would
be unmeaning or incorrect. If, then, we have a set of k functions, f1
... fk of n independent variables x1 ... xn, we say that they are
independent when n >= k and not every determinant of k rows and columns
vanishes of the matrix of k rows and n columns whose r-th row has the
constituents dfr/dx1, ... dfr/dxn; the justification being in the
theorem, which we assume, that if the determinant involving, for
instance, the first k columns be not zero for x1 = x1^0 ... xn = xn^0,
and the functions be developable about this point, then from the
equations f1 = c1, ... fk = ck we can express x1, ... xk by convergent
power series in the differences x_k+1 - x_k+1^0, ... x_n - x_n^0, and so
regard x1, ... xk as functions of the remaining variables. This we often
express by saying that the equations f1 = c1, ... fk = ck can be solved
for x1, ... xk. The explanation is given as a type of explanation often
understood in what follows.


    Ordinary equations of the first order.

    Single homogeneous partial equation of the first order.

    Proof of the existence of integrals.

  We may conveniently begin by stating the theorem: If each of the n
  functions [phi]1, ... [phi]n of the (n + 1) variables x1, ... x_nt be
  developable about the values x1^0, ... x_n^0t^0, the n differential
  equations of the form dx1/dt = [phi]1(tx1, ... xn) are satisfied by
  convergent power series

    x_r = x_r^0 + (t - t^0 ) A_r1 + (t - t0 )²A_r2 + ...

  reducing respectively to x1^0, ... xn^0 when t = t^0; and the only
  functions satisfying the equations and reducing respectively to x1^0,
  ... xn^0 when t = t^0, are those determined by continuation of these
  series. If the result of solving these n equations for x1^0, ... xn^0
  be written in the form [omega]1(x1, ... xnt) = x1^0, ... [omega]n(x1,
  ... xnt) = xn^0, it is at once evident that the differential equation

    df/dt + [phi]1 df/dx1 + ... + [phi]n df/dxn = 0

  possesses n integrals, namely, the functions [omega]1, ... [omega]n,
  which are developable about the values (x1^0 ... xn^0t^0) and reduce
  respectively to x1, ... xn when t = t^0. And in fact it has no other
  integrals so reducing. Thus this equation also possesses a unique
  integral reducing when t = t^0 to an arbitrary function [psi](x1, ...
  xn), this integral being. [psi]([omega]1, ... [omega]n). Conversely
  the existence of these _principal_ integrals [omega]1, ... [omega]n of
  the partial equation establishes the existence of the specified
  solutions of the ordinary equations dxi/dt = [phi]i. The following
  sketch of the proof of the existence of these principal integrals for
  the case n = 2 will show the character of more general investigations.
  Put x for x - x^0, &c., and consider the equation a(xyt) df/dx +
  b(xyt) df/dy = df/dt, wherein the functions a, b are developable about
  x = 0, y = 0, t = 0; say

    a(xyt) = a0 + ta1 + t²a2/2! + ..., b(xyt) = b0 + tb1 + t²b2/2! + ...,

  so that

    ad/dx + bd/dy = [delta]0 + t[delta]1 + ½t²[delta]2 + ...,

  where [delta] = a_r d/dx + b_r d/dy. In order that

    f = p0 + tp1 + t²p2/2! + ...

  wherein p0, p1 ... are power series in x, y, should satisfy the
  equation, it is necessary, as we find by equating like terms, that

    p1 = [delta]0 p0, p2 = [delta]0 p1 + [delta]1 p0, &c.

  and in general

    p_s+1 = [delta]0 p_s + s1 [delta]1 p_s-1 + ... + [delta]_s p0,

  where s_r = (s!)/(r!) (s - r)!

  Now compare with the given equation another equation

    A(xyt)dF/dx + B(xyt)dF/dy = dF/dt,

  wherein each coefficient in the expansion of either A or B is real and
  positive, and not less than the absolute value of the corresponding
  coefficient in the expansion of a or b. In the second equation let us
  substitute a series

    F = P0 + tP1 + t²P2/2! + ...,

  wherein the coefficients in P0 are real and positive, and each not
  less than the absolute value of the corresponding coefficient in p0;
  then putting [Delta]r = A_r d/dx + B_r d/dy we obtain necessary
  equations of the same form as before, namely,

    P1 = [Delta]0 P0, P2= [Delta]0 P1 + [Delta]1 P0, ...

  and in general P_s+1 = [Delta]0 P_s, + s1[Delta]1 P_s-1 + ... +
  [Delta]_s P0. These give for every coefficient in Ps+1 an integral
  aggregate with real positive coefficients of the coefficients in P_s,
  P_s-1, ..., P0 and the coefficients in A and B; and they are the same
  aggregates as would be given by the previously obtained equations for
  the corresponding coefficients in p_s+1 in terms of the coefficients
  in ps, p_s-1, ..., p0 and the coefficients in a and b. Hence as the
  coefficients in P0 and also in A, B are real and positive, it follows
  that the values obtained in succession for the coefficients in P1, P2,
  ... are real and positive; and further, taking account of the fact
  that the absolute value of a sum of terms is not greater than the sum
  of the absolute values of the terms, it follows, for each value of s,
  that every coefficient in p_s+1 is, in absolute value, not greater
  than the corresponding coefficient in P_s+1. Thus if the series for F
  be convergent, the series for f will also be; and we are thus reduced
  to (1), specifying functions A, B with real positive coefficients,
  each in absolute value not less than the corresponding coefficient in
  a, b; (2) proving that the equation

    AdF/dx + BdF/dy = dF/dt

  possesses an integral P0 + tP1 + t²P2/2! + ... in which the
  coefficients in P0 are real and positive, and each not less than the
  absolute value of the corresponding coefficient in p0. If a, b be
  developable for x, y both in absolute value less than r and for t less
  in absolute value than R, and for such values a, b be both less in
  absolute value than the real positive constant M, it is not difficult
  to verify that we may take

              /    x + y\-1 /    t\-1
    A = B = M( 1 - ----- ) ( 1 - - ),
              \      r  /   \    R/

  and obtain
                         _                                 _
                        |     4MR /   x + y\-2     /   t\-1 |½
    F = r - (r - x - y) | 1 - ---(1 - ------) log (1 - - )  |,
                        |_     r  \     r  /       \   R/  _|

  and that this solves the problem when x, y, t are sufficiently small
  for the two cases p0 = x, p0 = y. One obvious application of the
  general theorem is to the proof of the existence of an integral of an
  ordinary linear differential equation given by the n equations dy/dx =
  y1, dy1/dx = y2, ...,

    dy_n-1/dx = p - p1 y_n-1 - ... - p_n y;

  but in fact any simultaneous system of ordinary equations is reducible
  to a system of the form

    dx1/dt = [phi](tx1, ... x_n).


    Simultaneous linear partial equations.

    Complete systems of linear partial equations.

    Jacobian systems.

  Suppose we have k homogeneous linear partial equations of the first
  order in n independent variables, the general equation being
  a_[sigma]1 df/dx1 + ... + a_[sigma]n df/dx_n = 0, where [sigma] = 1,
  ... k, and that we desire to know whether the equations have common
  solutions, and if so, how many. It is to be understood that the
  equations are linearly independent, which implies that k <= n and not
  every determinant of k rows and columns is identically zero in the
  matrix in which the i-th element of the [sigma]-th row is a[sigma]_i(i
  = 1, ... n, [sigma] = 1, ... k). Denoting the left side of the
  [sigma]-th equation by P[sigma]f, it is clear that every common
  solution of the two equations P_[sigma]f = 0, P_[rho]f = 0, is also a
  solution of the equation P_[rho](P_[sigma]f), P_[sigma](P_[rho]f), We
  immediately find, however, that this is also a linear equation,
  namely, [Sigma]H_i df/dx_i = 0 where H_i = P[rho]a[sigma]_i -
  P[sigma]a[rho]_i, and if it be not already contained among the given
  equations, or be linearly deducible from them, it may be added to
  them, as not introducing any additional limitation of the possibility
  of their having common solutions. Proceeding thus with every pair of
  the original equations, and then with every pair of the possibly
  augmented system so obtained, and so on continually, we shall arrive
  at a system of equations, linearly independent of each other and
  therefore not more than n in number, such that the combination, in the
  way described, of every pair of them, leads to an equation which is
  linearly deducible from them. If the number of this so-called
  _complete system_ is n, the equations give df/dx1 = 0 ... df/dxn = 0,
  leading to the nugatory result f = a constant. Suppose, then, the
  number of this system to be r < n; suppose, further, that from the
  matrix of the coefficients a determinant of r rows and columns not
  vanishing identically is that formed by the coefficients of the
  differential coefficients of f in regard to x1 ... x_r; also that the
  coefficients are all developable about the values x1 = x1^0, ... xn=
  xn^0, and that for these values the determinant just spoken of is not
  zero. Then the main theorem is that the complete system of r
  equations, and therefore the originally given set of k equations,
  have in common n - r solutions, say [omega]r+1, ... [omega]n, which
  reduce respectively to x_r+1, ... x_n when in them for x1, ... x_r are
  respectively put x1^0, ... x_r^0; so that also the equations have in
  common a solution reducing when x1 = x1^0, ... x_r = x_r^0 to an
  arbitrary function [psi](x_r+1, ... x_n) which is developable about
  x_r+1^0, ... x_n^0, namely, this common solution is [psi]([omega]_r+1,
  ... [omega]_n). It is seen at once that this result is a
  generalization of the theorem for r = 1, and its proof is conveniently
  given by induction from that case. It can be verified without
  difficulty (1) that if from the r equations of the complete system we
  form r independent linear aggregates, with coefficients not
  necessarily constants, the new system is also a complete system; (2)
  that if in place of the independent variables x1, ... xn we introduce
  any other variables which are independent functions of the former, the
  new equations also form a complete system. It is convenient, then,
  from the complete system of r equations to form r new equations by
  solving separately for df/dx1, ..., df/dx_r; suppose the general
  equation of the new system to be

    Q_[sigma]f = df/dx_[sigma] + c_[sigma],r+1 df/dx_r+1 + ... + c_[sigma]n df/dx_n = 0 ([sigma] = 1, ... r).

  Then it is easily obvious that the equation Q_[rho]Q_[sigma]f -
  Q_[sigma]Q_[rho]f = 0 contains only the differential coefficients of f
  in regard to x_r+1 ... xn; as it is at most a linear function of Q1f,
  ... Qrf, it must be identically zero. So reduced the system is called
  a Jacobian system. Of this system Q1f=0 has n - 1 principal solutions
  reducing respectively to x2, ... xn when

    x1 = x1^0,

  and its form shows that of these the first r - 1 are exactly x2 ...
  xr. Let these n - 1 functions together with x1 be introduced as n new
  independent variables in all the r equations. Since the first equation
  is satisfied by n - 1 of the new independent variables, it will
  contain no differential coefficients in regard to them, and will
  reduce therefore simply to df/dx1 = 0, expressing that any common
  solution of the r equations is a function only of the n - 1 remaining
  variables. Thereby the investigation of the common solutions is
  reduced to the same problem for r - 1 equations in n - 1 variables.
  Proceeding thus, we reach at length one equation in n - r + 1
  variables, from which, by retracing the analysis, the proposition
  stated is seen to follow.


    System of total differential equations.

  The analogy with the case of one equation is, however, still closer.
  With the coefficients c_[sigma]j, of the equations Q_[sigma]f = 0 in
  transposed array ([sigma] = 1, ... r, j = r + 1, ... n) we can put
  down the (n - r) equations, dx_j = c1_j dx1 + ... + c_rj dx_r,
  equivalent to the r(n - r) equations dx_j/dx_[sigma] = c_[sigma]r.
  That consistent with them we may be able to regard x_r+1, ... x_n as
  functions of x1, ... x_r, these being regarded as independent
  variables, it is clearly necessary that when we differentiate
  c_[sigma]j in regard to x_[rho] on this hypothesis the result should
  be the same as when we differentiate c[rho]j, in regard to x[sigma] on
  this hypothesis. The differential coefficient of a function f of x1,
  ... xn on this hypothesis, in regard to x_[rho]j is, however,

    df/dx_[rho] + c_[rho],r+1 df/dx_r+1 + ... + c_[rho]n df/dx_n,

  namely, is Q_[rho]f. Thus the consistence of the n - r total equations
  requires the conditions Q_[rho]c_[sigma]j - Q_[sigma]c_[rho]j = 0,
  which are, however, verified in virtue of Q[rho](Q[sigma][f]) -
  Q_[sigma](Q_[rho]f) = 0. And it can in fact be easily verified that if
  [omega]_r+1, ... [omega]_n be the principal solutions of the Jacobian
  system, Q_[sigma]f = 0, reducing respectively to x_r+1, ... xn when x1
  = x1^0, ... x_r = x_r^0, and the equations [omega]_r+1 = x_r+1^0, ...
  [omega]_n = x_n^0 be solved for x_r+1, ... x_n to give x_j =
  [psi]_j(x1, ... x_r, x_r+1^0, ... x_n^0), these values solve the total
  equations and reduce respectively to x_r+1^0, ... x_n^0 when x1 = x1^0
  ... x_r = x_r^0. And the total equations have no other solutions with
  these initial values. Conversely, the existence of these solutions of
  the total equations can be deduced a priori and the theory of the
  Jacobian system based upon them. The theory of such total equations,
  in general, finds its natural place under the heading _Pfaffian
  Expressions_, below.


    Geometrical interpretation and solution.

    Mayer's method of integration.

  A practical method of reducing the solution of the r equations of a
  Jacobian system to that of a single equation in n - r + 1 variables
  may be explained in connexion with a geometrical interpretation which
  will perhaps be clearer in a particular case, say n = 3, r = 2. There
  is then only one total equation, say dz = adz + bdy; if we do not take
  account of the condition of integrability, which is in this case da/dy
  + bda/dz = db/dx + adb/dz, this equation may be regarded as defining
  through an arbitrary point (x0, y0, z0) of three-dimensioned space
  (about which a, b are developable) a plane, namely, z - z0 = a0(x -
  x0) + b0(y - y0), and therefore, through this arbitrary point [oo]²
  directions, namely, all those in the plane. If now there be a surface
  z = [psi](x, y), satisfying dz = adz + bdy and passing through (x0,
  y0, z0), this plane will touch the surface, and the operations of
  passing along the surface from (x0, y0, z0) to

    (x0 + dx0, y0, z0 + dz0)

  and then to (x0 + dx0, y0 + dy0, Z0 + d¹z0), ought to lead to the same
  value of d^1z0 as do the operations of passing along the surface from
  (x0, y0, z0) to (x0, y0 + dy0, z0 + [delta]z0), and then to

  (x_  + dx_  , y_  + dy_  , Z_  + [delta]¹z_  ),
    0    0   0    0   0           0

  namely, [delta]¹z0 ought to be equal to d¹z0. But we find

    d¹z0 = a0dx0 + b(x0 + dx0 , y0, z0 + a0dx0)dy0 =

                           /db      db \
    a0dx0 + b0dy0 + dx0dy0( --- + a0--- ),
                           \dx0     dz0/

  and so at once reach the condition of integrability. If now we put x
  = x0 + t, y = y0 + mt, and regard m as constant, we shall in fact be
  considering the section of the surface by a fixed plane y - y0 = m(x -
  x0); along this section dz = dt(a + bm); if we then integrate the
  equation dx/dt = a + bm, where a, b are expressed as functions of m
  and t, with m kept constant, finding the solution which reduces to z0
  for t = 0, and in the result again replace m by (y - y0)/(x - x0), we
  shall have the surface in question. In the general case the equations

    dx_j - c_1j dx1 + ... c_rj dx_r

  similarly determine through an arbitrary point x1^0, ... xn^0 a planar
  manifold of r dimensions in space of n dimensions, and when the
  conditions of integrability are satisfied, every direction in this
  manifold through this point is tangent to the manifold of r
  dimensions, expressed by [omega]_r+1 = x_r+1^0, ... [omega]_n = x_n^0,
  which satisfies the equations and passes through this point. If we put
  x1 = x1^0 = t, x2 = x2^0 = m2t, ... xr = xr^0 = mrt, and regard m2,
  ... mr as fixed, the (n-r) total equations take the form dx_j/dt =
  c_1j + m2c_2j + ... + m_rc_rj, and their integration is equivalent to
  that of the single partial equation

                n
    df/dt + [Sigma](c_1j + m2c_2j + ... + m_rc_rj)df/dx_j = 0
              j=r+1

  in the n - r + 1 variables t, xr+1, ... xn. Determining the solutions
  [Omega]_r+1, ... [Omega]_n which reduce to respectively x_r+1, ... x_n
  when t = 0, and substituting t = x1 - x1^0, m2 = (x2 - x2^0)/(x1 -
  x1^0), ... mr = (xr - xr^0)/(x1 - x1^0), we obtain the solutions of
  the original system of partial equations previously denoted by
  [omega]_r+1, ... [omega]_n. It is to be remarked, however, that the
  presence of the fixed parameters m2, ... mr in the single integration
  may frequently render it more difficult than if they were assigned
  numerical quantities.


    Pfaffian Expressions.

  We have above considered the integration of an equation

    dz = adz + bdy

  on the hypothesis that the condition

    da/dy + bda/dz = db/dz + adb/dz.

  It is natural to inquire what relations among x, y, z, if any, are
  implied by, or are consistent with, a differential relation adx + bdy
  + cdx = 0, when a, b, c are unrestricted functions of x, y, z. This
  problem leads to the consideration of the so-called _Pfaffian
  Expression_ adx + bdy + cdz. It can be shown (1) if each of the
  quantities db/dz - dc/dy, dc/dx - da/dz, da/dy - db/dz, which we shall
  denote respectively by u23, u31, u12, be identically zero, the
  expression is the differential of a function of x, y, z, equal to dt
  say; (2) that if the quantity au23 + bu31 + cu12 is identically zero,
  the expression is of the form udt, i.e. it can be made a perfect
  differential by multiplication by the factor 1/u; (3) that in general
  the expression is of the form dt + u1dt1. Consider the matrix of four
  rows and three columns, in which the elements of the first row are a,
  b, c, and the elements of the (r+1)-th row, for r = 1, 2, 3, are the
  quantities u_r1, u_r2, u_r3, where u11 = u22 = u33 = 0. Then it is
  easily seen that the cases (1), (2), (3) above correspond respectively
  to the cases when (1) every determinant of this matrix of two rows and
  columns is zero, (2) every determinant of three rows and columns is
  zero, (3) when no condition is assumed. This result can be generalized
  as follows: if a1, ... an be any functions of x1, ... xn, the
  so-called Pfaffian expression a1dx1 + ... + a_ndx_n can be reduced to
  one or other of the two forms

    u1dt1 + ... + u_kdt_k, dt + u1dt1 +  ... + u_k-1 dt_k-1,

  wherein t, u1 ..., t1, ... are independent functions of x1, ... xn,
  and k is such that in these two cases respectively 2k or 2k - 1 is the
  rank of a certain matrix of n + 1 rows and n columns, that is, the
  greatest number of rows and columns in a non-vanishing determinant of
  the matrix; the matrix is that whose first row is constituted by the
  quantities a1, ... an, whose s-th element in the (r+1)-th row is the
  quantity da_r/dx_s - da_s/dx_r. The proof of such a reduced form can
  be obtained from the two results: (1) If t be any given function of
  the 2m independent variables u1, ... um, t1, ... tm, the expression dt
  + u1 dt1 + ... + u_m dt_m can be put into the form u'1 dt'1 + ... +
  u'_mdt'_m. (2) If the quantities u1, ..., u1, t1, ... tm be connected
  by a relation, the expression n1dt1 + ... + umdtm can be put into the
  format dt' + u'1 dt'1 + ... + u'_m-1 dt'_m-1; and if the relation
  connecting u1, um, t1, ... tm be homogeneous in u1, ... um, then t'
  can be taken to be zero. These two results are deductions from the
  theory of _contact transformations_ (see below), and their
  demonstration requires, beside elementary algebraical considerations,
  only the theory of complete systems of linear homogeneous partial
  differential equations of the first order. When the existence of the
  reduced form of the Pfaffian expression containing only independent
  quantities is thus once assured, the identification of the number k
  with that defined by the specified matrix may, with some difficulty,
  be made _a posteriori_.


    Single linear Pfaffian equation.

  In all cases of a single Pfaffian equation we are thus led to consider
  what is implied by a relation dt - u1dt1 - ... - umdtm = 0, in which
  t, u1, ... um, t1 ..., tm are, except for this equation, independent
  variables. This is to be satisfied in virtue of one or several
  relations connecting the variables; these must involve relations
  connecting t, t1, ... tm only, and in one of these at least t must
  actually enter. We can then suppose that in one actual system of
  relations in virtue of which the Pfaffian equation is satisfied, all
  the relations connecting t, t1 ... tm only are given by

    t = [psi](t_s+1 ... t_m), t1 = [psi]1(t_s+1 ... t_m), ... t_s = [psi]_s(t_s+1 ... t_m);

  so that the equation

    d[psi] - u1d[psi]1 - ... - u_s d[psi]_s - u_s+1 dt_s+1 - ... - u_m dt_m = 0

  is identically true in regard to u1, ... um, t_s+1 ..., t_m; equating
  to zero the coefficients of the differentials of these variables, we
  thus obtain m - s relations of the form

    d[psi]/dt_j - u1 d[psi]1/dt_j - ... - u_s d[psi]_s/dt_j - u_j = 0;

  these m - s relations, with the previous s + 1 relations, constitute a
  set of m + 1 relations connecting the 2m + 1 variables in virtue of
  which the Pfaffian equation is satisfied independently of the form of
  the functions [psi],[psi]1, ... [psi]s. There is clearly such a set
  for each of the values s = 0, s = 1, ..., s = m - 1, s = m. And for
  any value of s there may exist relations additional to the specified m
  + 1 relations, provided they do not involve any relation connecting t,
  t1, ... tm only, and are consistent with the m - s relations
  connecting u1, ... um. It is now evident that, essentially, the
  integration of a Pfaffian equation

    a1dx1 + ... + a_n dx_n = 0,

  wherein a1, ... an are functions of x1, ... xn, is effected by the
  processes necessary to bring it to its reduced form, involving only
  independent variables. And it is easy to see that if we suppose this
  reduction to be carried out in all possible ways, there is no need to
  distinguish the classes of integrals corresponding to the various
  values of s; for it can be verified without difficulty that by putting
  t' = t - u1t1 - ... - u_s t_s, t'1 = u1, ... t'_s = u_s, u'1 = -t1,
  ..., u'_s = -t_s, t'_s+1 = t_s+1, ... t'_m = t_m, u'_s+1 = u_s+1, ...
  u'_m = u_m, the reduced equation becomes changed to dt' - u'1 dt'1 -
  ... - u'_m dt'_m = 0, and the general relations changed to

    t' = [psi](t'_s+l, ... t'_m) - t'1[psi]1(t'_s+1, ... t'_m) -  ... -t'_s[psi]_s(t'_s+1, ... t'_m), = [phi],

  say, together with u'1 = d[phi]/dt'1, ..., u'm = d[phi]/dt'm, which
  contain only one relation connecting the variables t', t'1, ... t'm
  only.


    Simultaneous Pfaffian equations.

  This method for a single Pfaffian equation can, strictly speaking, be
  generalized to a simultaneous system of (n - r) Pfaffian equations dxj
  = c_1j dx1 + ... + c_rj dxr only in the case already treated, when
  this system is satisfied by regarding x_r+1, ... x_n as suitable
  functions of the independent variables x1, ... xr; in that case the
  integral manifolds are of r dimensions. When these are non-existent,
  there may be integral manifolds of higher dimensions; for if

  d[phi] = [phi]1 dx_r +  ... + [phi]_r dx_r + [phi]_r+1(c_1,r+1 dx1 +  ...  + c_r,r+1 dx_r) + [phi]_r+2 ( ) +  ...

  be identically zero, then [phi][sigma] + c[sigma]_,r+1 [phi]_r+1 + ...
  + c[sigma]_,n [phi]_n = 0, or [phi] satisfies the r partial
  differential equations previously associated with the total equations;
  when these are not a complete system, but included in a complete
  system of r - [mu] equations, having therefore n - r - [mu]
  independent integrals, the total equations are satisfied over a
  manifold of r + [mu] dimensions (see E. v. Weber, _Math. Annal._ 1v.
  (1901), p. 386).


    Contact transformations.

  It seems desirable to add here certain results, largely of algebraic
  character, which naturally arise in connexion with the theory of
  contact transformations. For any two functions of the 2n independent
  variables x1, ... xn, p1, ... pn we denote by ([phi][psi]) the sum of
  the n terms such as d[phi]d[psi]/dp_idx_i - d[psi]d[phi]/dp_idx_i. For
  two functions of the (2n + 1) independent variables z, x1, ... xn, p1,
  ... pn we denote by [phi][psi] the sum of the n terms such as

    d[phi] /d[psi]      d[psi]\    d[psi] /d[phi]      d[phi]\
    ------( ------ + p_i------ ) - ------( ------ + p_i------ ).
     dpi   \ dxi          dz  /     dpi   \ dxi          dz  /

  It can at once be verified that for any three functions
  [f[[phi][psi]]] + [[phi][psi]f]] + [[psi][f[phi]]] = df/dz
  [[phi][psi]] + d[phi]/dz [[psi]f] + d[psi]/dz [f[phi]], which when f,
  [phi],[psi] do not contain z becomes the identity (f([phi][psi])) +
  (phi([psi]f)) + ([psi](f[phi])) = 0. Then, if X1, ... Xn, P1, ... Pn
  be such functions Of x1, ... xn, p1 ... pn that P1 dX1 + ... + Pn dXn
  is identically equal to p1dx1 + ... + pn dxn, it can be shown by
  elementary algebra, after equating coefficients of independent
  differentials, (1) that the functions X1, ... Pn are independent
  functions of the 2n variables x1, ... pn, so that the equations x'i =
  Xi, p'i = Pi can be solved for x1, ... xn, p1, ... pn, and represent
  therefore a transformation, which we call a homogeneous contact
  transformation; (2) that the X1, ... Xn are homogeneous functions of
  p1, ... pn of zero dimensions, the P1, ... Pn are homogeneous
  functions of p1, ... pn of dimension one, and the ½n(n - 1) relations
  (Xi Xj) = 0 are verified. So also are the n² relations (Pi Xi) = 1,
  (Pi Xj) = 0, (Pi Pj) = 0. Conversely, if X1, ... Xn be independent
  functions, each homogeneous of zero dimension in p1, ... pn satisfying
  the ½n(n - 1) relations (Xi Xj) = 0, then P1, ... Pn can be uniquely
  determined, by solving linear algebraic equations, such that P1 dX1 +
  ... + Pn dXn = p1 dx1 + ... + pn dxn. If now we put n + 1 for n, put z
  for x_n+1, Z for X_n+1, Qi for -Pi/P_n+1, for i = 1, ... n, put qi for
  -p_i/p_n+1 and [sigma] for q_n+1/Q_n+1, and then finally write P1, ...
  Pn, p1, ... pn for Q1, ... Qn, q1, ... qn, we obtain the following
  results: If ZX1 ... Xn, P1, ... Pn be functions of z, x1, ... xn, p1,
  ... pn, such that the expression dZ - P1 dX1 - ... - Pn dXn is
  identically equal to [sigma](dz - p1 dx1 - ... - pn dxn), and [sigma]
  not zero, then (1) the functions Z, X1, ... Xn, P1, ... Pn are
  independent functions of z, x1, ... xn, p1, ... pn, so that the
  equations z' = Z, x'i = Xi, p'i = Pi can be solved for z, x1, ... xn,
  p1, ... pn and determine a transformation which we call a
  (non-homogeneous) contact transformation; (2) the Z, X1, ... Xn verify
  the ½n(n + 1) identities [Z Xi] = 0, [Xi Xj] = 0. And the further
  identities

    [Pi Xi] = [sigma], [Pi Xj] = 0, [Pi Z] = [sigma]Pi, [Pi Pj] = 0,

                        dZ                                   dXi                        dPi
    [Z[sigma]] = [sigma]-- - [sigma]², [Xi [sigma]] = [sigma]---, [Pi [sigma]] = [sigma]---
                        dz                                   dz                         dz

  are also verified. Conversely, if Z, x1, ... Xn be independent
  functions satisfying the identities [Z Xi] = 0, [Xi Xj] = 0, then
  [sigma], other than zero, and P1, ... Pn can be uniquely determined,
  by solution of algebraic equations, such that

    dZ - P1 dX1 - ... - Pn dXn = [sigma](dz - p1 dx1 - ... - p_n dx_n).

  Finally, there is a particular case of great importance arising when
  [sigma] = 1, which gives the results: (1) If U, X1, ... Xn, P1, ... Pn
  be 2n + 1 functions of the 2n independent variables x1, ... xn, p1,
  ... pn, satisfying the identity

    dU + P1 dx1 + ... + Pn dXn = p1 dx1 + ... + p_n dx_n,

  then the 2n functions P1, ... Pn, X1, ... Xn are independent, and we
  have

    (Xi Xj) = 0, (Xi U) = [delta]Xi, (Pi Xi) = 1, (Pi Xj) = 0, (Pi Pj ) = 0, (Pi U) + Pi = [delta]Pi,

  where [delta] denotes the operator p1d/dp1 + ... + pnd/dpn; (2) If X1,
  ... Xn be independent functions of x1, ... xn, p1, ... pn, such that
  (Xi Xj) = 0, then U can be found by a quadrature, such that

    (Xi U) = [delta]Xi;

  and when Xi, ... Xn, U satisfy these ½n(n + 1) conditions, then P1,
  ... Pn can be found, by solution of linear algebraic equations, to
  render true the identity dU + P1 dX1 + ... + Pn dXn = p1 dx1 + ... +
  pn dxn; (3) Functions X1, ... Xn, P1, ... Pn can be found to satisfy
  this differential identity when U is an arbitrary given function of
  x1, ... xn, p1, ... pn; but this requires integrations. In order to
  see what integrations, it is only necessary to verify the statement
  that if U be an arbitrary given function of x1, ... xn, p1, ... pn,
  and, for r < n, X1, ... Xr be independent functions of these
  variables, such that (X_[sigma] U) = [delta]X_[sigma], (X_[rho]
  X_[sigma]) = 0, for [rho], [sigma] = 1 ... r, then the r + 1
  homogeneous linear partial differential equations of the first order
  (Uf) + [delta]f = 0, (X[rho]f) = 0, form a complete system. It will be
  seen that the assumptions above made for the reduction of Pfaffian
  expressions follow from the results here enunciated for contact
  transformations.


  Partial differential equation of the first order.

  Meaning of a solution of the equation.

We pass on now to consider the solution of any partial differential
equation of the first order; we attempt to explain certain ideas
relatively to a single equation with any number of independent variables
(in particular, an ordinary equation of the first order with one
independent variable) by speaking of a single equation with two
independent variables x, y, and one dependent variable z. It will be
seen that we are naturally led to consider systems of such simultaneous
equations, which we consider below. The central discovery of the
transformation theory of the solution of an equation F(x, y, z, dz/dx,
dz/dy) = 0 is that its solution can always be reduced to the solution of
partial equations which are _linear_. For this, however, we must regard
dz/dx, dz/dy, during the process of integration, not as the differential
coefficients of a function z in regard to x and y, but as variables
independent of x, y, z, the too great indefiniteness that might thus
appear to be introduced being provided for in another way. We notice
that if z = [psi](x, y) be a solution of the differential equation, then
dz = dxd[psi]/dx + dyd[psi]/dy; thus if we denote the equation by F(x,
y, z, p, q,) = 0, and prescribe the condition dz = pdx + qdy for every
solution, any solution such as z = [psi](x, y) will necessarily be
associated with the equations p = dz/dx, q = dz/dy, and z will satisfy
the equation in its original form. We have previously seen (under
_Pfaffian Expressions_) that if five variables x, y, z, p, q, otherwise
independent, be subject to dz - pdx - qdy = 0, they must in fact be
subject to at least three mutual relations. If we associate with a point
(x, y, z) the plane

  Z - z = p(X - x) + q(Y - y)

passing through it, where X, Y, Z are current co-ordinates, and call
this association a surface-element; and if two consecutive elements of
which the point(x + dx, y + dy, z + dz) of one lies on the plane of the
other, for which, that is, the condition dz = pdx + qdy is satisfied, be
said to be _connected,_ and an infinity of connected elements following
one another continuously be called a _connectivity_, then our statement
is that a connectivity consists of not more than [oo]² elements, the
whole number of elements (x, y, z, p, q) that are possible being called
[oo]^5. The solution of an equation F(x, y, z, dz/dx, dz/dy) = 0 is then
to be understood to mean finding in all possible ways, from the [oo]^4
elements (x, y, z, p, q) which satisfy F(x, y, z, p, q) = 0 a set of
[oo]² elements forming a connectivity; or, more analytically, finding in
all possible ways two relations G = 0, H = 0 connecting x, y, z, p, q
and independent of F = 0, so that the three relations together may
involve

  dz = pdx + qdy.

Such a set of three relations may, for example, be of the form z =
[psi](x, y), p = d[psi]/dx, q = d[psi]/dy; but it may also, as another
case, involve two relations z = [psi](y), x = [psi]1(y) connecting x, y,
z, the third relation being

  [psi]'(y) = p[psi]'1(y) + q,

the connectivity consisting in that case, geometrically, of a curve in
space taken with [oo]¹ of its tangent planes; or, finally, a
connectivity is constituted by a fixed point and all the planes passing
through that point. This generalized view of the meaning of a solution
of F = 0 is of advantage, moreover, in view of anomalies otherwise
arising from special forms of the equation itself. For instance, we may
include the case, sometimes arising when the equation to be solved is
obtained by transformation from another equation, in which F does not
contain either p or q. Then the equation has [oo]² solutions, each
consisting of an arbitrary point of the surface F = 0 and all the [oo]²
planes passing through this point; it also has [oo]² solutions, each
consisting of a curve drawn on the surface F = 0 and all the tangent
planes of this curve, the whole consisting of [oo]² elements; finally,
it has also an isolated (or singular) solution consisting of the points
of the surface, each associated with the tangent plane of the surface
thereat, also [oo]² elements in all. Or again, a linear equation F = Pp
+ Qq - R = 0, wherein P, Q, R are functions of x, y, z only, has [oo]²
solutions, each consisting of one of the curves defined by

  dx/P = dy/Q = dz/R

taken with all the tangent planes of this curve; and the same equation
has [oo]² solutions, each consisting of the points of a surface
containing [oo]¹ of these curves and the tangent planes of this surface.
And for the case of n variables there is similarly the possibility of n
+ 1 kinds of solution of an equation F(x1, ... xn, z, p1, ... pn) = 0;
these can, however, by a simple contact transformation be reduced to one
kind, in which there is only one relation z' = [psi](x'1, ... x'n)
connecting the new variables x'1, ... x'n, z' (see under PFAFFIAN
EXPRESSIONS); just as in the case of the solution

  z = [psi](y), x = [psi]1(y), [psi]'(y) = p[psi]'1(y) + q

of the equation Pp + Qq = R the transformation z' = z - px, x' = p, p' =
-x, y' = y, q' = q gives the solution

  z' = [psi](y') + x'[psi]1(y'), p' = dz'/dx', q' = dz'/dy'

of the transformed equation. These explanations take no account of the
possibility of p and q being infinite; this can be dealt with by writing
p = -u/w, q = -v/w, and considering homogeneous equations in u, v, w,
with udx + vdy + wdz = 0 as the differential relation necessary for a
connectivity; in practice we use the ideas associated with such a
procedure more often without the appropriate notation.


  Order of the ideas.

In utilizing these general notions we shall first consider the theory of
characteristic chains, initiated by Cauchy, which shows well the nature
of the relations implied by the given differential equation; the
alternative ways of carrying out the necessary integrations are
suggested by considering the method of Jacobi and Mayer, while a good
summary is obtained by the formulation in terms of a Pfaffian
expression.


    Characteristic chains.

  Consider a solution of F = 0 expressed by the three independent
  equations F = 0, G = 0, H = 0. If it be a solution in which there is
  more than one relation connecting x, y, z, let new variables x', y',
  z', p', q' be introduced, as before explained under PFAFFIAN
  EXPRESSIONS, in which z' is of the form

    z' = z - p1x1 - ... - p_s x_s (s = 1 or 2),

  so that the solution becomes of a form z' = [psi](x'y'), p' =
  d[psi]/dx', q' = d[psi]/dy', which then will identically satisfy the
  transformed equations F' = 0, G' = 0, H' = 0. The equation F' = 0, if
  x', y', z' be regarded as fixed, states that the plane Z - z' = p'(X -
  x') + q'(Y - y') is tangent to a certain cone whose vertex is (x', y',
  z'), the consecutive point (x' + dx', y' + dy', z' + dz') of the
  generator of contact being such that

        /dF'       /dF'       / /  dF'      dF'\
    dx'/ --  = dy'/ --  = dz'/ ( p'--- + q' --- ).
      /  dp'     /  dq'     /   \  dp'      dq'/


  Passing in this direction on the surface z' = [psi](x', y') the
  tangent plane of the surface at this consecutive point is (p' + dp',
  q' + dq'), where, since F'(x', y', [psi], d[psi]/dx', d[psi]/dy') = 0
  is identical, we have dx' (dF'/dx' + p'dF'/dz') + dp'dF'/dp' = 0. Thus
  the equations, which we shall call the characteristic equations,

        /dF'       /dF'       //   dF'     dF'\        //  dF'     dF'\
    dx'/ --- = dy'/ --- = dz'/( p' --- + q'--- ) = dp'/( - --- - p'--- )
      /  dp'     /  dq'     /  \   dp'     dq'/      /  \  dx'     dz'/

                                                       //  dF'     dF'\
                                                 = dq'/( - --- - q'--- )
                                                     /  \  dy'     dz'/

  are satisfied along a connectivity of [oo]¹ elements consisting of a
  curve on z' = [psi](x', y') and the tangent planes of the surface
  along this curve. The equation F' = 0, when p', q' are fixed,
  represents a curve in the plane Z - z' = p'(X - x') + q'(Y - y')
  passing through (x', y', z'); if (x' + [delta]x', y' + [delta]y', z' +
  [delta]z') be a consecutive point of this curve, we find at once

              /dF'     dF'\              /dF'     dF'\
    [delta]x'( --- + p'--- ) + [delta]y'( --- + q'--- ) = 0;
              \dx'     dz'/              \dy'     dz'/

  thus the equations above give [delta]x'dp' + [delta]y'dq' = 0, or the
  tangent line of the plane curve, is, on the surface z' = [psi](x',
  y'), in a direction conjugate to that of the generator of the cone.
  Putting each of the fractions in the characteristic equations equal to
  dt, the equations enable us, starting from an arbitrary element x'0,
  y'0, z'0, p'0, q'0, about which all the quantities F', dF'/dp', &c.,
  occurring in the denominators, are developable, to define, from the
  differential equation F' = 0 alone, a connectivity of [oo]¹ elements,
  which we call a _characteristic chain_; and it is remarkable that when
  we transform again to the original variables (x, y, z, p, q), the form
  of the differential equations for the chain is unaltered, so that they
  can be written down at once from the equation F = 0. Thus we have
  proved that the characteristic chain starting from any ordinary
  element of any integral of this equation F = 0 consists only of
  elements belonging to this integral. For instance, if the equation do
  not contain p, q, the characteristic chain, starting from an arbitrary
  plane through an arbitrary point of the surface F = 0, consists of a
  pencil of planes whose axis is a tangent line of the surface F = 0. Or
  if F = 0 be of the form Pp + Qq = R, the chain consists of a curve
  satisfying dx/P = dy/Q = dz/R and a single infinity of tangent planes
  of this curve, determined by the tangent plane chosen at the initial
  point. In all cases there are [oo]³ characteristic chains, whose
  aggregate may therefore be expected to exhaust the [oo]^4 elements
  satisfying F = 0.


    Complete integral constructed with characteristic chains.

  Consider, in fact, a single infinity of connected elements each
  satisfying F = 0, say a chain connectivity T, consisting of elements
  specified by x0, y0, z0, p0, q0, which we suppose expressed as
  functions of a parameter u, so that

    U0 = dz0/du - p0dx0/du - q0dy0/du

  is everywhere zero on this chain; further, suppose that each of F,
  dF/dp, ... , dF/dx + pdF/dz is developable about each element of this
  chain T, and that T is _not_ a characteristic chain. Then consider the
  aggregate of the characteristic chains issuing from all the elements
  of T. The [oo]² elements, consisting of the aggregate of these
  characteristic chains, satisfy F = 0, provided the chain connectivity
  T consists of elements satisfying F = 0; for each characteristic chain
  satisfies dF = 0. It can be shown that these chains are connected; in
  other words, that if x, y, z, p, q, be any element of one of these
  characteristic chains, not only is

    dz/dt - pdx/dt - qdy/dt = 0,

  as we know, but also U = dz/du - pdx/du - qdy/du is also zero. For we
  have

    dU   d  /dz    dx    dy\    d  /dz    dx    dy\
    -- = --( -- - p-- - q-- ) - --( -- - p-- - q-- )
    dt   dt \du    du    du/    du \dt    dt    dt/

         dp dx   dp dx   dq dy   dq dy
       = -- -- - -- -- + -- -- - -- -- ,
         du dt   dt du   du dt   dt du

  which is equal to

    dp dF   dx /dF    dF\    dq dF   dy /dF    dF\      dF
    -- -- + --( -- + p-- ) + -- -- + --( -- + q-- ) = - -- U.
    du dp   du \dx    dz/    du dq   du \dy    dz/      dz

     dF
  As -- is a developable function of t, this, giving
     dz
                       _
                  /   / t  dF  \
    U = U_{0} exp( -  |    --dt ),
                  \  _/t0  dz  /

  shows that U is everywhere zero. Thus integrals of F = 0 are
  obtainable by considering the aggregate of characteristic chains
  issuing from arbitrary chain connectivities T satisfying F = 0; and
  such connectivities T are, it is seen at once, determinable without
  integration. Conversely, as such a chain connectivity T can be taken
  out from the elements of any given integral all possible integrals are
  obtainable in this way. For instance, an arbitrary curve in space,
  given by x0 = [theta](u), y0 = [phi](u), z0 = [psi](u), determines by
  the two equations F(x0, y0, z0, p0, q0) = 0, [psi]'(u) = p0[theta]'(u)
  + q0[phi]'(u), such a chain connectivity T, through which there passes
  a perfectly definite integral of the equation F = 0. By taking [oo]²
  initial chain connectivities T, as for instance by taking the curves
  x0 = [theta], y0 = [phi], z0 = [psi] to be the [oo]² curves upon an
  arbitrary surface, we thus obtain [oo]² integrals, and so [oo]^4
  elements satisfying F = 0. In general, if functions G, H, independent
  of F, be obtained, such that the equations F = 0, G = b, H = c
  represent an integral for all values of the constants b, c, these
  equations are said to constitute a _complete integral_. Then [oo]^4
  elements satisfying F = 0 are known, and in fact every other form of
  integral can be obtained without further integrations.


    Operations necessary for integration of F = a.

  In the foregoing discussion of the differential equations of a
  characteristic chain, the denominators dF/dp, ... may be supposed to
  be modified in form by means of F = 0 in any way conducive to a simple
  integration. In the immediately following explanation of ideas,
  however, we consider indifferently all equations F = constant; when a
  function of x, y, z, p, q is said to be zero, it is meant that this is
  so identically, not in virtue of F = 0; in other words, we consider
  the integration of F = a, where a is an arbitrary constant. In the
  theory of linear partial equations we have seen that the integration
  of the equations of the characteristic chains, from which, as has just
  been seen, that of the equation F = a follows at once, would be
  involved in completely integrating the single linear homogeneous
  partial differential equation of the first order [Ff] = 0 where the
  notation is that explained above under CONTACT TRANSFORMATIONS. One
  obvious integral is f = F. Putting F = a, where a is arbitrary, and
  eliminating one of the independent variables, we can reduce this
  equation [Ff] = 0 to one in four variables; and so on. Calling, then,
  the determination of a single integral of a single homogeneous partial
  differential equation of the first order in n independent variables,
  _an operation of order_ n - 1, the characteristic chains, and
  therefore the most general integral of F = a, can be obtained by
  successive operations of orders 3, 2, 1. If, however, an integral of F
  = a be represented by F = a, G = b, H = c, where b and c are arbitrary
  constants, the expression of the fact that a characteristic chain of F
  = a satisfies dG = 0, gives [FG] = 0; similarly, [FH] = 0 and [GH] =
  0, these three relations being identically true. Conversely, suppose
  that an integral G, independent of F, has been obtained of the
  equation [Ff] = 0, which is an operation of order three. Then it
  follows from the identity [f[[phi][psi]]] + [[phi][[psi]f]] +
  [[psi][f[phi]]] = df/dz [[psi][phi]] + d[phi]/dz [psif] + d[psi]/dz
  [f[phi]] before remarked, by putting [phi] = F, [psi] = G, and then
  [Ff] = A(f), [Gf] = B(f), that AB(f) - BA(f) = dF/dz B(f) - dG/dz
  A(f), so that the two linear equations [Ff] = 0, [Gf] = 0 form a
  complete system; as two integrals F, G are known, they have a common
  integral H, independent of F, G, determinable by an operation of order
  one only. The three functions F, G, H thus identically satisfy the
  relations [FG] = [GH] = [FH] = 0. The [oo]² elements satisfying F = a,
  G = b, H = c, wherein a, b, c are assigned constants, can then be seen
  to constitute an integral of F = a. For the conditions that a
  characteristic chain of G = b issuing from an element satisfying F =
  a, G = b, H = c should consist only of elements satisfying these three
  equations are simply [FG] = 0, [GH] = 0. Thus, starting from an
  arbitrary element of (F = a, G = b, H = c), we can single out a
  connectivity of elements of (F = a, G = b, H = c) forming a
  characteristic chain of G = b; then the aggregate of the
  characteristic chains of F = a issuing from the elements of this
  characteristic chain of G = b will be a connectivity consisting only
  of elements of

    (F = a, G = b, H = c),

  and will therefore constitute an integral of F = a; further, it will
  include all elements of (F = a, G = b, H = c). This result follows
  also from a theorem given under CONTACT TRANSFORMATIONS, which shows,
  moreover, that though the characteristic chains of F = a are not
  determined by the three equations F = a, G = b, H = c, no further
  integration is now necessary to find them. By this theorem, since
  identically [FG] = [GH] = [FH] = 0, we can find, by the solution of
  linear algebraic equations only, a non-vanishing function [sigma] and
  two functions A, C, such that

    dG - AdF - CdH = [sigma](dz - pdz - qdy);

  thus all the elements satisfying F = a, G = b, H = c, satisfy dz = pdx
  + qdy and constitute a connectivity, which is therefore an integral of
  F = a. While, further, from the associated theorems, F, G, H, A, C are
  independent functions and [FC] = 0. Thus C may be taken to be the
  remaining integral independent of G, H, of the equation [Ff] = 0,
  whereby the characteristic chains are entirely determined.


    The single equation F = 0 and Pfaffian formulations.

  When we consider the particular equation F = 0, neglecting the case
  when neither p nor q enters, and supposing p to enter, we may express
  p from F = 0 in terms of x, y, z, q, and then eliminate it from all
  other equations. Then instead of the equation [Ff] = 0, we have, if F
  = 0 give p = [psi](x, y, z, q), the equation

                  /df         df\    d[psi]  /df     df\     /d[psi]     d[psi]\  df
    [Sigma]f = - ( -- + [psi] -- ) + ------ ( -- + q -- ) - ( ------ + q ------ ) -- = 0,
                  \dx         dz/      dq    \dy     dz/     \  dy         dz  /  dq

  moreover obtainable by omitting the term in df/dp in [p-[psi], f] = 0.
  Let x0, y0, z0, q0, be values about which the coefficients in this
  equation are developable, and let [zeta], [eta], [omega] be the
  principal solutions reducing respectively to z, y and q when x = x0.
  Then the equations p = [psi], [zeta] = z0, [eta] = y0, [omega] = q0
  represent a characteristic chain issuing from the element x0, y0, z0,
  [psi]0, q0; we have seen that the aggregate of such chains issuing
  from the elements of an arbitrary chain satisfying

    dz0 = p0dx0 - q0dy0 = 0

  constitute an integral of the equation p = [psi]. Let this arbitrary
  chain be taken so that x0 is constant; then the condition for initial
  values is only

    dz0 - q0dy0 = 0,

  and the elements of the integral constituted by the characteristic
  chains issuing therefrom satisfy

    d[zeta] - [omega]d[eta] = 0.

  Hence this equation involves dz - [psi]dx - qdy = 0, or we have

    dz - [psi]dx - qdy = [sigma](d[zeta] - [omega]d[eta]),

  where [sigma] is not zero. Conversely, the integration of p = [psi]
  is, essentially, the problem of writing the expression dz - [psi]dx -
  qdy in the form [sigma](d[zeta] - [omega]d[eta]), as must be possible
  (from what was said under _Pfaffian Expressions_).


    System of equations of the first order.

  To integrate a system of simultaneous equations of the first order X1
  = a1, ... Xr = ar in n independent variables x1, ... xn and one
  dependent variable z, we write p1 for dz/dx1, &c., and attempt to find
  n + 1 - r further functions Z, X_r+1 ... Xn, such that the equations Z
  = a, Xi = ai,(i = 1, ... n) involve dz - p1dx1 - ... - pndxn = 0. By
  an argument already given, the common integral, if existent, must be
  satisfied by the equations of the characteristic chains of any one
  equation Xi = ai; thus each of the expressions [Xi Xj] must vanish in
  virtue of the equations expressing the integral, and we may without
  loss of generality assume that each of the corresponding ½r(r - 1)
  expressions formed from the r given differential equations vanishes in
  virtue of these equations. The determination of the remaining n + 1 -
  r functions may, as before, be made to depend on characteristic
  chains, which in this case, however, are manifolds of r dimensions
  obtained by integrating the equations [X1f] = 0, ... [Xrf] = 0; or
  having obtained one integral of this system other than X1, ... Xr, say
  Xr+1, we may consider the system [X1f] = 0, ... [X_r+1 f] = 0, for
  which, again, we have a choice; and at any stage we may use Mayer's
  method and reduce the simultaneous linear equations to one equation
  involving parameters; while if at any stage of the process we find
  some but not all of the integrals of the simultaneous system, they can
  be used to simplify the remaining work; this can only be clearly
  explained in connexion with the theory of so-called function groups
  for which we have no space. One result arising is that the
  simultaneous system p1 = [phi]1, ... pr = [phi]r, wherein p1, ... pr
  are not involved in [phi]1, ... [phi]r, if it satisfies the ½r(r - 1)
  relations [pi - [phi]i, pj - [phi]j] = 0, has a solution z = [psi](x1,
  ... xn), p1 = d[psi]/dx1, ... pn = d[psi]/dxn, reducing to an
  arbitrary function of x_r+1, ... xn only, when x1 = x1^0, ... xr =
  xr^0 under certain conditions as to developability; a generalization
  of the theorem for linear equations. The problem of integration of
  this system is, as before, to put

    dz - [phi]1dx1 - ... - [phi]_r dx_r - p_r+1 dx_r+1 - ... - p_n dx_n

  into the form [sigma](d[zeta] - [omega]_r+1 + d[xi]_r+1 - ... -
  [omega]_n d[xi]_n); and here [zeta], [xi]_r+1, ... [xi]_n,
  [omega]_r+1, ... [omega]_n may be taken, as before, to be principal
  integrals of a certain complete system of linear equations; those,
  namely, determining the characteristic chains.


    Equations of dynamics.

  If L be a function of t and of the 2n quantities x1, ... xn, [.x]1,
  ... [.x]n, where [.x]i, denotes dxi/dt, &c., and if in the n equations

     d   /  dL  \     dL
    --- (--------) = ----
    dt   \ dx_i /    dx_i

  we put p_i = dL/d[.x]_i, and so express [.x]1 , ... [.x]_n in terms of
  t, x_i, ... x_n, p1, ... p_n, assuming that the determinant of the
  quantities d²L/dx_i d[.x]_j is not zero; if, further, H denote the
  function of t, x1, ... xn, p1, ... pn, numerically equal to p1[.x]1 +
  ... + pn[.x]n - L, it is easy to prove that dpi/dt = -dH/dxi, dxi/dt =
  dH/dp_i. These so-called _canonical_ equations form part of those for
  the characteristic chains of the single partial equation dz/dt + H(t,
  x1, ... xn, dz/dx1, ..., dz/dx_n) = 0, to which then the solution of
  the original equations for x1 ... xn can be reduced. It may be shown
  (1) that if z = [psi](t, x1, ... xn, c1, .. cn) + c be a complete
  integral of this equation, then pi = d[psi]/dx_i, d[psi]/dc_i = e_i are
  2n equations giving the solution of the canonical equations referred
  to, where c1 ... cn and e1, ... en are arbitrary constants; (2) that
  if xi = Xi(t, x^01, ... pn^0), pi=Pi(t, x1^0, ... p^0n) be the
  principal solutions of the canonical equations for t = t^0, and
  [omega] denote the result of substituting these values in p1dH/dp1 +
  ... + pndH/dpn - H, and [Omega] = [int] [t0 to t] [omega]dt, where,
  after integration, [Omega] is to be expressed as a function of t, x1,
  ... xn, x1^0, ... xn^0, then z = [Omega] + z^0 is a complete integral
  of the partial equation.


  Application of theory of continuous groups to formal theories.

A system of differential equations is said to allow a certain continuous
group of transformations (see GROUPS, THEORY OF) when the introduction
for the variables in the differential equations of the new variables
given by the equations of the group leads, for all values of the
parameters of the group, to the same differential equations in the new
variables. It would be interesting to verify in examples that this is
the case in at least the majority of the differential equations which
are known to be integrable in finite terms. We give a theorem of very
general application for the case of a simultaneous complete system of
linear partial homogeneous differential equations of the first order, to
the solution of which the various differential equations discussed have
been reduced. It will be enough to consider whether the given
differential equations allow the infinitesimal transformations of the
group.

  It can be shown easily that sufficient conditions in order that a
  complete system [Pi]1f = 0 ... [Pi]kf = 0, in n independent variables,
  should allow the infinitesimal transformation Pf = 0 are expressed by
  k equations [Pi]_i Pf - P[Pi]_i f = [lambda]_i1 [Pi]1f + ... +
  [lambda]_ik [Pi]_kf. Suppose now a complete system of n - r equations
  in n variables to allow a group of r infinitesimal transformations
  (P1f, ..., Prf) which has an invariant subgroup of r - 1 parameters
  (P1f, ..., Pr-1f), it being supposed that the n quantities [Pi]1f,
  ..., [Pi]_n-r f, P1 f, ..., P_r f are not connected by an identical
  linear equation (with coefficients even depending on the independent
  variables). Then it can be shown that one solution of the complete
  system is determinable by a quadrature. For each of [Pi]_i P_[sigma] f
  - P_[sigma] [Pi]_i f is a linear function of [Pi]1f, ..., [Pi]_n-r f
  and the simultaneous system of independent equations [Pi]1f = 0, ...
  [Pi]_n-r f = 0, P1f = 0, ... P_r-1 f = 0 is therefore a complete
  system, allowing the infinitesimal transformation Prf. This complete
  system of n - 1 equations has therefore one common solution [omega],
  and P_r([omega]) is a function of [omega]. By choosing [omega]
  suitably, we can then make Pr([omega]) = 1. From this equation and the
  n - 1 equations [Pi]_i[omega] = 0, P_[sigma][omega] = 0, we can
  determine [omega] by a quadrature only. Hence can be deduced a much
  more general result, _that if the group of r parameters be integrable,
  the complete system can be entirety solved by quadratures_; it is only
  necessary to introduce the solution found by the first quadrature as
  an independent variable, whereby we obtain a complete system of n - r
  equations in n - 1 variables, subject to an integrable group of r - 1
  parameters, and to continue this process. We give some examples of the
  application of the theorem. (1) If an equation of the first order y' =
  [psi](x, y) allow the infinitesimal transformation [xi]df/dx +
  [eta]df/dy, the integral curves [omega](x, y) = y°, wherein [omega](x,
  y) is the solution of df/dx + [psi](x, y) df/dy = 0 reducing to y for
  x = x°, are interchanged among themselves by the infinitesimal
  transformation, or [omega](x, y) can be chosen to make [xi]d[omega]/dx
  + [eta]d[omega]/dy = 1; this, with d[omega]/dx + [psi]d[omega]/dy = 0,
  determines [omega] as the integral of the complete differential (dy -
  [psi]dx)/([eta] - [psi][xi]). This result itself shows that every
  ordinary differential equation of the first order is subject to an
  infinite number of infinitesimal transformations. But every
  infinitesimal transformation [xi]df/dx + [eta]df/dy can by change of
  variables (after integration) be brought to the form df/dy, and all
  differential equations of the first order allowing this group can then
  be reduced to the form F(x, dy/dx) = 0. (2) In an ordinary equation of
  the second order y" = [psi](x, y, y'), equivalent to dy/dx = y1,
  dy1/dx = [psi](x, y, y1), if H, H1 be the solutions for y and y1
  chosen to reduce to y^0 and y1° when x = x°, and the equations H = y,
  H1= y1 be equivalent to [omega] = y°, [omega]1 = y1°, then [omega],
  [omega]1 are the principal solutions of [Pi]f = df/dx + y1df/dy +
  [psi]df/dy1 = 0. If the original equation allow an infinitesimal
  transformation whose first _extended_ form (see GROUPS) is Pf =
  [xi]df/dx + [eta]df/dy + [eta]1df/dy1, where [eta]1[delta]t is the
  increment of dy/dx when [xi][delta]t, [eta][delta]t are the increments
  of x, y, and is to be expressed in terms of x, y, y1, then each of
  P[omega] and P[omega]1 must be functions of [omega] and [omega]1, or
  the partial differential equation [Pi]f must allow the group Pf. Thus
  by our general theorem, if the differential equation allow a group of
  two parameters (and such a group is always integrable), it can be
  solved by quadratures, our explanation sufficing, however, only
  provided the form [Pi]f and the two infinitesimal transformations are
  not linearly connected. It can be shown, from the fact that [eta]1 is
  a quadratic polynomial in y1, that no differential equation of the
  second order can allow more than 8 really independent infinitesimal
  transformations, and that every homogeneous linear differential
  equation of the second order allows just 8, being in fact reducible to
  d²y/dx² = 0. Since every group of more than two parameters has
  subgroups of two parameters, a differential equation of the second
  order allowing a group of more than two parameters can, as a rule, be
  solved by quadratures. By transforming the group we see that if a
  differential equation of the second order allows a single
  infinitesimal transformation, it can be transformed to the form F(x,
  d[gamma]/dx, d²[gamma]/dx²); this is not the case for every
  differential equation of the second order. (3) For an ordinary
  differential equation of the third order, allowing an integrable group
  of three parameters whose infinitesimal transformations are not
  linearly connected with the partial equation to which the solution of
  the given ordinary equation is reducible, the similar result follows
  that it can be integrated by quadratures. But if the group of three
  parameters be simple, this result must be replaced by the statement
  that the integration is reducible to quadratures and that of a
  so-called Riccati equation of the first order, of the form dy/dx = A +
  By + Cy², where A, B, C are functions of x. (4) Similarly for the
  integration by quadratures of an ordinary equation yn = [psi](x, y,
  y1, ... yn-1) of any order. Moreover, the group allowed by the
  equation may quite well consist of extended contact transformations.
  An important application is to the case where the differential
  equation is the resolvent equation defining the group of
  transformations or rationality group of another differential equation
  (see below); in particular, when the rationality group of an ordinary
  linear differential equation is integrable, the equation can be solved
  by quadratures.


  Consideration of function theories of differential equations.

Following the practical and provisional division of theories of
differential equations, to which we alluded at starting, into
transformation theories and function theories, we pass now to give some
account of the latter. These are both a necessary logical complement of
the former, and the only remaining resource when the expedients of the
former have been exhausted. While in the former investigations we have
dealt only with values of the independent variables about which the
functions are developable, the leading idea now becomes, as was long ago
remarked by G. Green, the consideration of the neighbourhood of the
values of the variables for which this developable character ceases.
Beginning, as before, with existence theorems applicable for ordinary
values of the variables, we are to consider the cases of failure of such
theorems.


  A general existence theorem.

When in a given set of differential equations the number of equations is
greater than the number of dependent variables, the equations cannot be
expected to have common solutions unless certain conditions of
compatibility, obtainable by equating different forms of the same
differential coefficients deducible from the equations, are satisfied.
We have had examples in systems of linear equations, and in the case of
a set of equations p1 = [phi]1, ..., pr = [phi]r. For the case when the
number of equations is the same as that of dependent variables, the
following is a general theorem which should be referred to: Let there be
r equations in r dependent variables z1, ... zr and n independent
variables x1, ... xn; let the differential coefficient of z[sigma] of
highest order which enters be of order h[sigma], and suppose d^h_[sigma]
z_[sigma]/dx1^h_[sigma] to enter, so that the equations can be written
d^h_[sigma] z_[sigma]/dx1^h_[sigma] = [Phi]_[sigma], where in the
general differential coefficient of z_[rho] which enters in
[Phi]_[sigma], say

  d^(k1 + ... + kn) z_[rho]/dx1^k1 ... dx_n^k_n,

we have k1 < h_[rho] and k1 + ... + k_n <= h_[rho]. Let a1, ... an, b1,
... br, and b[rho]_(k1 ... kn) be a set of values of

  x1, ... x_n, z1, ... z_r

and of the differential coefficients entering in [Phi]_[sigma] about
which all the functions [Phi]1, ... [Phi]_r, are developable.
Corresponding to each dependent variable z_[sigma], we take now a set of
h_[sigma] functions of x2, ... xn, say [phi][sigma], [phi][sigma]^(1),
..., [phi][sigma]^(h-1) arbitrary save that they must be developable
about a2, a3, ... an, and such that for these values of x2, ... xn, the
function [phi]_[rho] reduces to b_[rho], and the differential
coefficient

  d^(k2 + ... + kn) [phi]_[rho]^(k1)/dx2^k2 ... dx_n^kn

reduces to b^kn_(k1 ... kn). Then the theorem is that there exists one,
and only one, set of functions z1, ... z_r, of x2, ... x_n developable
about a1, ... an satisfying the given differential equations, and such
that for x1 = a1 we have

  z_[sigma] = [phi]_[sigma], dz_[sigma]/dx1 = [phi]_[sigma]^(1), ...
         d^(h_[sigma]-1) z_[sigma]/d^(h_[sigma]-1) x1 = [phi][sigma]^(h_[sigma]-1).

And, moreover, if the arbitrary functions [phi]_[sigma],
[phi]_[sigma]^(1) ... contain a certain number of arbitrary variables
t1, ... tm, and be developable about the values t1°, ... tm° of these
variables, the solutions z1, ... zr will contain t1, ... tm, and be
developable about t1°, ... tm°.


    Singular points of solutions.

  The proof of this theorem may be given by showing that if ordinary
  power series in x1 - -a1, ... xn - an, t1 - t1°, ... tm - tm° be
  substituted in the equations wherein in z[sigma] the coefficients of
  (x1 - a1)°, x1 - a1, ..., (x1 - a1)^(h_[sigma]-1) are the arbitrary
  functions [phi]_[sigma], [phi]_[sigma]^(1), ..., [phi]_[sigma]^h-1,
  divided respectively by 1, 1!, 2!, &c., then the differential
  equations determine uniquely all the other coefficients, and that the
  resulting series are convergent. We rely, in fact, upon the theory of
  monogenic analytical functions (see FUNCTION), a function being
  determined entirely by its development in the neighbourhood of one set
  of values of the independent variables, from which all its other
  values arise by _continuation_; it being of course understood that the
  coefficients in the differential equations are to be continued at the
  same time. But it is to be remarked that there is no ground for
  believing, if this method of continuation be utilized, that the
  function is single-valued; we may quite well return to the same values
  of the independent variables with a different value of the function;
  belonging, as we say, to a different branch of the function; and there
  is even no reason for assuming that the number of branches is finite,
  or that different branches have the same singular points and regions
  of existence. Moreover, and this is the most difficult consideration
  of all, all these circumstances may be dependent upon the values
  supposed given to the arbitrary constants of the integral; in other
  words, the singular points may be either _fixed_, being determined by
  the differential equations themselves, or they may be _movable_ with
  the variation of the arbitrary constants of integration. Such
  difficulties arise even in establishing the reversion of an elliptic
  integral, in solving the equation

     /dx\²
    ( -- ) = (x-a1)(x - a2)(x - a3)(x - a4);
     \ds/

  about an ordinary value the right side is developable; if we put x -
  a1 = t1², the right side becomes developable about t1 = 0; if we put x
  = 1/t, the right side of the changed equation is developable about t =
  0; it is quite easy to show that the integral reducing to a definite
  value x0 for a value s0 is obtainable by a series in integral powers;
  this, however, must be supplemented by showing that for no value of s
  does the value of x become entirely undetermined.


    Linear differential equations with rational coefficients.

  These remarks will show the place of the theory now to be sketched of
  a particular class of ordinary linear homogeneous differential
  equations whose importance arises from the completeness and generality
  with which they can be discussed. We have seen that if in the
  equations dy/dx = y1, dy1/dx = y2, ..., dy_n-2/dx = y_n-1,

    dy_n-1/dx = a_n y + a_n-1 y1 + ... + a1 y_n-1,

  where a1, a2, ..., an are now to be taken to be rational functions of
  x, the value x = xº be one for which no one of these rational
  functions is infinite, and yº, yº1, ..., yº_n-1 be quite arbitrary
  finite values, then the equations are satisfied by

    y = yºu + yº1u1 + ... + yº_n-1 u_n-1,

  where u, u1, ..., un-1 are functions of x, independent of yº, ...
  yº_n-1, developable about x = xº; this value of y is such that for x =
  xº the functions y, y1 ... y_n-1 reduce respectively to yº, yº1, ...
  yº_n-1; it can be proved that the region of existence of these series
  extends within a circle centre xº and radius equal to the distance
  from xº of the nearest point at which one of a1, ... an becomes
  infinite. Now consider a region enclosing xº and only one of the
  places, say [Sigma], at which one of a1, ... an becomes infinite. When
  x is made to describe a closed curve in this region, including this
  point [Sigma] in its interior, it may well happen that the
  continuations of the functions u, u1, ..., u_n-1 give, when we have
  returned to the point x, values v, v1, ..., v_n-1, so that the
  integral under consideration becomes changed to yº + yº1v1 + ... +
  yº_n-1 v_n-1. At xº let this branch and the corresponding values of
  y1, ... y_n-1 be [eta]º, [eta]º1, ... [eta]º_n-1; then, as there is
  only one series satisfying the equation and reducing to ([eta]º,
  [eta]º1, ... [eta]º_n-1) for x = xº and the coefficients in the
  differential equation are single-valued functions, we must have
  [eta]ºu + [eta]º1u1 + ... + [eta]º_n-1 u_n-1 = yºv + yº1v1 + ... +
  yº_n-1 v_n-1; as this holds for arbitrary values of yº ... yº_n-1,
  upon which u, ... u_n-1 and v, ... v_n-1 do not depend, it follows
  that each of v, ... v_n-1 is a linear function of u, ... u_n-1 with
  constant coefficients, say v_i = A_i1 u + ... + A_in u_n-1. Then

    yºv + ... + yº_n-1 v_n-1 = ([Sigma]_i A_i1 y_iº)u + ... + ([Sigma]_i A_in yº_i)u_n-1;

  this is equal to [mu](yºu + ... + yº_n-1 u_n-1) if [Sigma]_i A_ir yº_i
  = [mu]yº_r-1; eliminating yº ... yº_n-1 from these linear equations,
  we have a determinantal equation of order n for [mu]; let [mu]1 be one
  of its roots; determining the ratios of yº, y1º, ... yº_n-1 to satisfy
  the linear equations, we have thus proved that there exists an
  integral, H, of the equation, which when continued round the point
  [Sigma] and back to the starting-point, becomes changed to H1 =
  [mu]1H. Let now [xi] be the value of x at [Sigma] and r1 one of the
  values of (1/2[pi]i) log [mu]1; consider the function (x - [xi])^r1 H;
  when x makes a circuit round x = [xi], this becomes changed to

    exp(-2[pi]ir1) (x - [xi])^-r1 [mu]H,

  that is, is unchanged; thus we may put H = (x - [xi])^r1 [phi]1,
  [phi]1 being a function single-valued for paths in the region
  considered described about [Sigma], and therefore, by Laurent's
  Theorem (see FUNCTION), capable of expression in the annular region
  about this point by a series of positive and negative integral powers
  of x - [xi], which in general may contain an infinite number of
  negative powers; there is, however, no reason to suppose r1 to be an
  integer, or even real. Thus, if all the roots of the determinantal
  equation in [mu] are different, we obtain n integrals of the forms (x
  -[xi])^r1 phi1, ..., (x - [xi])^rn [phi]_n. In general we obtain as
  many integrals of this form as there are really different roots; and
  the problem arises to discover, in case a root be k times repeated, k
  - 1 equations of as simple a form as possible to replace the k - 1
  equations of the form yº + ... + yº_n-1 v_n-1 = [mu](yº + ... + yº_n-1
  u_n-1) which would have existed had the roots been different. The most
  natural method of obtaining a suggestion lies probably in remarking
  that if r2 = r1 + h, there is an integral [(x - [xi])^(r1 + h) [phi]2
  - (x -[xi])^r1 [phi]1]/h, where the coefficients in [phi]2 are the
  same functions of r1 + h as are the coefficients in [phi]1 of r1; when
  h vanishes, this integral takes the form
                    _                              _
                   | d[phi]1                        |
    (x - [xi])^r1  | ------- + [phi]1 log (x - [xi])|,
                   |_  dr1                         _|

  or say       (x-[xi])^r1 [[phi]1 + [psi]1 log (x - [xi])];

  denoting this by 2[pi]i[mu]1K, and (x-[xi])^r1 [phi]1 by H, a circuit
  of the point [xi] changes K into

              1
    K' = ----------- [e^(2[pi]ir1) (x - [xi])^r1 [psi]1 + e^(2[pi]ir1) (x - [xi])^r1 [phi]1 (2[pi]i + log(x - [xi]))]
         2[pi]i[mu]1

       = [mu]1K + H.

  A similar artifice suggests itself when three of the roots of the
  determinantal equation are the same, and so on. We are thus led to the
  result, which is justified by an examination of the algebraic
  conditions, that whatever may be the circumstances as to the roots of
  the determinantal equation, n integrals exist, breaking up into
  batches, the values of the constituents H1, H2, ... of a batch after
  circuit about x = [xi] being H1' = [mu]1H1, H2' = [mu]1H2 + H1, H3' =
  [mu]1H3 + H2, and so on. And this is found to lead to the forms (x -
  [xi])^r1 [phi]1, (x - [xi])^r1 [[psi]1 + [phi]1 log (x - [xi])], (x -
  [xi])^r1 [[chi]1 + [chi]2 log (x - [xi]) + [phi]1(log(x - [xi]))²],
  and so on. Here each of [phi]1, [psi]1, [chi]1, [chi]2, ... is a
  series of positive and negative integral powers of x - [xi] in which
  the number of negative powers may be infinite.


    Regular equations.

  It appears natural enough now to inquire whether, under proper
  conditions for the forms of the rational functions a1, ... an, it may
  be possible to ensure that in each of the series [phi]1, [psi]1,
  [chi]1, ... the number of negative powers shall be finite. Herein
  lies, in fact, the limitation which experience has shown to be
  justified by the completeness of the results obtained. Assuming n
  integrals in which in each of [phi]1, [psi]1, [chi]1 ... the number of
  negative powers is finite, there is a definite homogeneous linear
  differential equation having these integrals; this is found by forming
  it to have the form

    y'^n = (x - [xi])^-1 b1y'^(n-1) + (x - [xi])^-2 b2y'^(n-2) + ... +(x - [xi])^-n b_n y,

  where b1, ... bn are finite for x = [xi]. Conversely, assume the
  equation to have this form. Then on substituting a series of the form
  (x - [xi])^r [1 + A1(x - [xi]) + A2(x - [xi])² + ... ] and equating
  the coefficients of like powers of x-[xi], it is found that r must be
  a root of an algebraic equation of order n; this equation, which we
  shall call the index equation, can be obtained at once by substituting
  for y only (x - [xi])^r and replacing each of b1, ... bn by their
  values at x = [xi]; arrange the roots r1, r2, ... of this equation so
  that the real part of ri is equal to, or greater than, the real part
  of r_i+1, and take r equal to r1; it is found that the coefficients
  A1, A2 ... are uniquely determinate, and that the series converges
  within a circle about x = [xi] which includes no other of the points
  at which the rational functions a1 ... an become infinite. We have
  thus a solution H1 = (x -[xi])^r1 [phi]1 of the differential equation.
  If we now substitute in the equation y = H1 f[eta]dx, it is found to
  reduce to an equation of order n - 1 for [eta] of the form

    [eta]'^(n-1) = (x - [xi])^-1 c1[eta]'^(n-2) + ... + (x-[xi])^(n-1) c_n-1 [eta],

  where c1, ... c_n-1 are not infinite at x = [xi]. To this equation
  precisely similar reasoning can then be applied; its index equation
  has in fact the roots r2 - r1 - 1, ... , rn - r1 - 1; if r2 - r1 be
  zero, the integral (x - [xi])^-1 [psi]1 of the [eta] equation will
  give an integral of the original equation containing log (x - [xi]);
  if r2 - r1 be an integer, and therefore a negative integer, the same
  will be true, unless in [psi]1 the term in (x - [xi])^(r1 - r2) be
  absent; if neither of these arise, the original equation will have an
  integral (x -[xi])^r2 [phi]2. The [eta] equation can now, by means of
  the one integral of it belonging to the index r2 - r1 - 1, be
  similarly reduced to one of order n - 2, and so on. The result will be
  that stated above. We shall say that an equation of the form in
  question is _regular_ about x = [xi].


    Fuchsian equations.

    Equation of the second order.

  We may examine in this way the behaviour of the integrals at all the
  points at which any one of the rational functions a1 ... an becomes
  infinite; in general we must expect that beside these the value x =
  [oo] will be a singular point for the solutions of the differential
  equation. To test this we put x = 1/t throughout, and examine as
  before at t = 0. For instance, the ordinary linear equation with
  constant coefficients has no singular point for finite values of x; at
  x = [oo] it has a singular point and is not regular; or again,
  Bessel's equation x² + xy' + (x² - n²)y = 0 is regular about x = 0,
  but not about x = [oo]. An equation regular at all the finite
  singularities and also at x = [oo] is called a Fuchsian equation. We
  proceed to examine particularly the case of an equation of the second
  order

    y" + ay' + by = 0.

  Putting x = 1/t, it becomes

    d²y/dt² + (2t^-1 - at^-2)dy/dt + bt^-4 y = 0,

  which is not regular about t = 0 unless 2 - at^-1 and bt^-2, that is,
  unless ax and bx² are finite at x =[oo]; which we thus assume; putting
  y = t^r(1 + A1t + ... ), we find for the index equation at x =
  [inifinity] the equation r(r - 1) + r(2 - ax)_0 + (bx²)_0 = 0. If
  there be finite singular points at [xi]1, ... [xi]m, where we assume
  m>1, the cases m = 0, m = 1 being easily dealt with, and if [phi](x) =
  (x - [xi]1) ... (x -[xi]m), we must have a.[phi](x) and b·[[phi](x)]²
  finite for all finite values of x, equal say to the respective
  polynomials [psi](x) and [theta](x), of which by the conditions at x =
  [oo] the highest respective orders possible are m - 1 and 2(m - 1).
  The index equation at x = [xi]1 is r(r - 1) +
  r[psi]([xi]1)/[phi]'([xi]1) + [theta]([xi])1/[[phi]'([xi]1)]² = 0, and
  if [alpha]1, [beta]1 be its roots, we have [alpha]1 + [beta]1 = 1 -
  [psi]([xi]1)/[phi]'([xi]1) and [alpha]1[beta]1 =
  [theta]([xi])1/[[phi]'([xi]1)]². Thus by an elementary theorem of
  algebra, the sum [Sigma](1 - [alpha]i - [beta]i)/(x - [xi]i), extended
  to the m finite singular points, is equal to [psi](x)/[phi](x), and
  the sum [Sigma](1 - [alpha]i - [beta]i) is equal to the ratio of the
  coefficients of the highest powers of x in [psi](x) and [phi](x), and
  therefore equal to 1 + [alpha] + [beta], where [alpha], [beta] are the
  indices at x = [oo]. Further, if (x, 1)m-2 denote the integral part of
  the quotient [theta](x)/[phi](x), we have
  [Sigma][alpha]_i[beta]_i[phi]'([xi]_i)/(x - [xi]_i) equal to -(x,
  1)_m-2 + [theta](x)/[phi](x), and the coefficient of x^m-2 in (x,
  1)_m-2 is [alpha][beta]. Thus the differential equation has the form

    y" + y'[Sigma](1 - [alpha]_i - [beta]_i)/(x - [xi]_i) + y[(x, 1)_m-2 +
         [Sigma][alpha]_i[beta]_i[phi]'([xi]_i)/(x - [xi]_i)]/[phi](x) = 0.

  If, however, we make a change in the dependent variable, putting y =
  (x - [xi]1)^[alpha]1 ... (x - [xi]_m)^[alpha] m[eta], it is easy to
  see that the equation changes into one having the same singular points
  about each of which it is regular, and that the indices at x = [xi]_i
  become 0 and [beta]_i - [alpha]_i, which we shall denote by [lambda]i,
  for (x -[xi]_i)^[alpha]j can be developed in positive integral powers
  of x -[xi]_i about x = [xi]_i; by this transformation the indices at x
  = [oo] are changed to

    [alpha] + [alpha]1 + ... + [alpha]m, [beta] + [beta]1 + ... + [beta]m

  which we shall denote by [lambda], [mu]. If we suppose this change to
  have been introduced, and still denote the independent variable by y,
  the equation has the form

    y" + y'[Sigma](1 - [lambda]_i)/(x - [xi]_i) + y(x, 1)_m-2/[phi](x) = 0,

  while [lambda] + [mu] + [lambda]1 + ... + [lambda]_m = m - 1.
  Conversely, it is easy to verify that if [lambda][mu] be the
  coefficient of x^m-2 in (x, 1)_m-2, this equation has the specified
  singular points and indices whatever be the other coefficients in (x,
  1)_m-2.


    Hypergeometric equation.

  Thus we see that (beside the cases m = 0, m = 1) the "Fuchsian
  equation" of the second order with _two_ finite singular points is
  distinguished by the fact that it has a definite form when the
  singular points and the indices are assigned. In that case, putting (x
  - [xi]1)/(x - [xi]2) = t/(t - 1), the singular points are transformed
  to 0, 1, [oo], and, as is clear, without change of indices. Still
  denoting the independent variable by x, the equation then has the form

    x(1 - x)y" + y'[1 - [lambda]1 - x(1  + [lambda] + [mu])] - [lambda][mu]y = 0,

  which is the ordinary hypergeometric equation. Provided none of
  [lambda]1, [lambda]2, [lambda] - [mu] be zero or integral about x = 0,
  it has the solutions

    F([lambda], [mu], 1 - [lambda]1, x), x^[lambda]1 F([lambda] + [lambda]1, [mu] + [lambda]1, 1 + [lambda]1, x);

  about x = 1 it has the solutions

    F([lambda], [mu], 1 - [lambda]2, 1 - x), (1 - x)^[lambda]1 F([lambda] + [lambda]2, [mu] + [lambda]2, 1 + [lambda]2, 1 - x),

  where [lambda] + [mu] + [lambda]1 + [lambda]2 = 1; about x = [oo] it
  has the solutions

    x^-[lambda] F([lambda], [lambda] + [lambda]1, [lambda] - [mu] + 1, x^-1),
      x^-[mu] F([mu], [mu] + [lambda]1, [mu] - [lambda] + 1, x^-1),

  where F([alpha], [beta], [gamma], x) is the series

        [alpha][beta]x   [alpha]([alpha] + 1)[beta]([beta] + 1)x²
    1 + -------------- + ---------------------------------------- ...,
           [gamma]               1·2·[gamma]([gamma] + 1)

  which converges when |x| < 1, whatever [alpha], [beta], [gamma] may
  be, converges for all values of x for which |x| = 1 provided the real
  part of [gamma] - [alpha] - [beta] < 0 algebraically, and converges
  for all these values except x = 1 provided the real part of [gamma] -
  [alpha] -[beta] > -1 algebraically.

  In accordance with our general theory, logarithms are to be expected
  in the solution when one of [lambda]1, [lambda]2, [lambda] - [mu] is
  zero or integral. Indeed when [lambda]1 is a negative integer, not
  zero, the second solution about x = 0 would contain vanishing factors
  in the denominators of its coefficients; in case [lambda] or [mu] be
  one of the positive integers 1, 2, ... (-[lambda]1), vanishing factors
  occur also in the numerators; and then, in fact, the second solution
  about x = 0 becomes x^[lambda]1 times an integral polynomial of degree
  (-[lambda]1) - [lambda] or of degree (-[lambda]1) - [mu]. But when
  [lambda]1 is a negative integer including zero, and neither [lambda]
  nor [mu] is one of the positive integers 1, 2 ... (-[lambda]1), the
  second solution about x = 0 involves a term having the factor log x.
  When [lambda]1 is a positive integer, not zero, the second solution
  about x = 0 persists as a solution, in accordance with the order of
  arrangement of the roots of the index equation in our theory; the
  first solution is then replaced by an integral polynomial of degree
  -[lambda] or -[mu]1, when [lambda] or [mu] is one of the negative
  integers 0, -1, -2, ..., 1 - [lambda]1, but otherwise contains a
  logarithm. Similarly for the solutions about x = 1 or x = [oo]; it
  will be seen below how the results are deducible from those for x = 0.


    March of the Integral.

  Denote now the solutions about x = 0 by u1, u2; those about x = 1 by
  v1, v2; and those about x = [oo] by w1, w2; in the region (S0S1)
  common to the circles S0, S1 of radius 1 whose centres are the points
  x = 0, x = 1, all the first four are valid, and there exist equations
  u1 =Av1 + Bv2, u2 = Cv1 + Dv2 where A, B, C, D are constants; in the
  region (S1S) lying inside the circle S1 and outside the circle S0,
  those that are valid are v1, v2, w1, w2, and there exist equations v1
  = Pw1 + Qw2, v2 = Rw1 + Tw2, where P, Q, R, T are constants; thus
  considering any integral whose expression within the circle S0 is au1
  + bu2, where a, b are constants, the same integral will be represented
  within the circle S1 by (aA + bC)v1 + (aB + bD)v2, and outside these
  circles will be represented by

    [(aA + bC)P + (aB + bD)R]w1 + [(aA + bC)Q + (aB + bD)T]w2.

  A single-valued branch of such integral can be obtained by making a
  barrier in the plane joining [oo] to 0 and 1 to [oo]; for instance, by
  excluding the consideration of real negative values of x and of real
  positive values greater than 1, and defining the phase of x and x - 1
  for real values between 0 and 1 as respectively 0 and [pi].


    Transformation of the equation into itself.

  We can form the Fuchsian equation of the second order with three
  arbitrary singular points [xi]1, [xi]2, [xi]3, and no singular point
  at x = [oo], and with respective indices [alpha]1, [beta]1, [alpha]2,
  [beta]2, [alpha]3, [beta]3 such that [alpha]1 + [beta]1 + [alpha]2 +
  [beta]2 + [alpha]3 + [beta]3 = 1. This equation can then be
  transformed into the hypergeometric equation in 24 ways; for out of
  [xi]1, [xi]2, [xi]3 we can in six ways choose two, say [xi]1, [xi]2,
  which are to be transformed respectively into 0 and 1, by (x -
  [xi]1)/(x - [xi]2) = t(t - 1); and then there are four possible
  transformations of the dependent variable which will reduce one of the
  indices at t = 0 to zero and one of the indices at t = 1 also to zero,
  namely, we may reduce either [alpha]1 or [beta]1 at t = 0, and
  simultaneously either [alpha]2 or [beta]2 at t = 1. Thus the
  hypergeometric equation itself can be transformed into itself in 24
  ways, and from the expression F([lambda], [mu], 1 - [lambda]1, x)
  which satisfies it follow 23 other forms of solution; they involve
  four series in each of the arguments, x, x-1, 1/x, 1/(1-x), (x-1)/x,
  x/(x-1). Five of the 23 solutions agree with the fundamental solutions
  already described about x = 0, x = 1, x = [oo]; and from the
  principles by which these were obtained it is immediately clear that
  the 24 forms are, in value, equal in fours.


    Inversion. Modular functions.

  The quarter periods K, K' of Jacobi's theory of elliptic functions, of
  which K = [int] [0 to [pi]/2] (1 - h sin²[theta])^-½ d[theta], and K'
  is the same function of 1-h, can easily be proved to be the solutions
  of a hypergeometric equation of which h is the independent variable.
  When K, K' are regarded as defined in terms of h by the differential
  equation, the ratio K'/K is an infinitely many valued function of h.
  But it is remarkable that Jacobi's own theory of theta functions leads
  to an expression for h in terms of K'/K (see FUNCTION) in terms of
  single-valued functions. We may then attempt to investigate, in
  general, in what cases the independent variable x of a hypergeometric
  equation is a single-valued function of the ratio s of two independent
  integrals of the equation. The same inquiry is suggested by the
  problem of ascertaining in what cases the hypergeometric series
  F([alpha], [beta], [gamma], x) is the expansion of an algebraic
  (irrational) function of x. In order to explain the meaning of the
  question, suppose that the plane of x is divided along the real axis
  from -[oo] to 0 and from 1 to +[oo], and, supposing logarithms not to
  enter about x = 0, choose two quite definite integrals y1, y2 of the
  equation, say

    y1 = F([lambda], [mu], 1-[lambda]1, x),
    y2 = x^[lambda]1 F([lambda] + [lambda]1, [mu] + [lambda]1, 1 + [lambda]1, x),

  with the condition that the phase of x is zero when x is real and
  between 0 and 1. Then the value of [sigma] = y2/y1 is definite for all
  values of x in the divided plane, [sigma] being a single-valued
  monogenic branch of an analytical function existing and without
  singularities all over this region. If, now, the values of [sigma]
  that so arise be plotted on to another plane, a value p + iq of
  [sigma] being represented by a point (p, q) of this [stigma]-plane, and
  the value of x from which it arose being mentally associated with this
  point of the [sigma]-plane, these points will fill a connected region
  therein, with a continuous boundary formed of four portions
  corresponding to the two sides of the two barriers of the x-plane. The
  question is then, firstly, whether the same value of s can arise for
  two different values of x, that is, whether the same point (p, q) of
  the [sigma]-plane can arise twice, or in other words, whether the
  region of the [sigma]-plane overlaps itself or not. Supposing this is
  not so, a second part of the question presents itself. If in the
  x-plane the barrier joining -[oo] to 0 be momentarily removed, and x
  describe a small circle with centre at x = 0 starting from a point x =
  -h - ik, where h, k are small, real, and positive and coming back to
  this point, the original value s at this point will be changed to a
  value [sigma], which in the original case did not arise for this value
  of x, and possibly not at all. If, now, after restoring the barrier
  the values arising by continuation from [sigma] be similarly plotted
  on the s-plane, we shall again obtain a region which, while not
  overlapping itself, may quite possibly overlap the former region. In
  that case two values of x would arise for the same value or values of
  the quotient y2/y1, arising from two different branches of this
  quotient. We shall understand then, by the condition that x is to be a
  single-valued function of x, that the region in the [stimga]-plane
  corresponding to any branch is not to overlap itself, and that no two
  of the regions corresponding to the different branches are to overlap.
  Now in describing the circle about x = 0 from x = -h - ik to -h + ik,
  where h is small and k evanescent,

    [stigma] = x^[lambda]1 F([lambda] + [lambda]1, [mu] + [lambda]1, 1 + [lambda]1, x)/F([lambda], [mu], 1 - [lambda]1, x)

  is changed to [sigma] = [stigma]e^(2[pi]i[lambda])1. Thus the two
  portions of boundary of the s-region corresponding to the two sides of
  the barrier (-[oo], 0) meet (at [sigmaf] = 0 if the real part of
  [lambda]1 be positive) at an angle 2[pi]L1, where L1 is the absolute
  value of the real part of [lambda]1; the same is true for the
  [sigma]-region representing the branch [sigma]. The condition that the
  s-region shall not overlap itself requires, then, L1 = 1. But,
  further, we may form an infinite number of branches [sigma] =
  [stigma]e^(2[pi]i[lambda])1, [sigma]1 = e^(2[pi]i[lambda])1, ... in
  the same way, and the corresponding regions in the plane upon which
  y2/y1 is represented will have a common point and each have an angle
  2[pi]L1; if neither overlaps the preceding, it will happen, if L1 is
  not zero, that at length one is reached overlapping the first, unless
  for some positive integer [alpha] we have 2[pi][alpha]L1 = 2[pi], in
  other words L1 = 1/a. If this be so, the branch [sigma]_a-1 =
  [stigma]e^(2[pi]ia[lambda])1 will be represented by a region having
  the angle at the common point common with the region for the branch
  [stigma]; but not altogether coinciding with this last region unless
  [lambda]1 be real, and therefore = ±1/a; then there is only a finite
  number, a, of branches obtainable in this way by crossing the barrier
  (-[oo], 0). In precisely the same way, if we had begun by taking the
  quotient

    [stigma]' = (x - 1)^[lambda]2 F([lambda] + [lambda]2, [mu] + [lambda]2, 1 + [lambda]2, 1 - x)/F([lambda], [mu], 1 - [lambda]2, 1 - x)

  of the two solutions about x = 1, we should have found that x is not a
  single-valued function of [stigma]' unless [lambda]2 is the inverse of
  an integer, or is zero; as [stigma]' is of the form (A[stigma] +
  B)/(C[stigma] + D), A, B, C, D constants, the same is true in our
  case; equally, by considering the integrals about x = [oo] we find, as
  a third condition necessary in order that x may be a single-valued
  function of [stigma], that [lambda] - [mu] must be the inverse of an
  integer or be zero. These three differences of the indices, namely,
  [lambda]1, [lambda]2, [lambda] - [mu], are the quantities which enter
  in the differential equation satisfied by x as a function of [stigma],
  which is easily found to be

       x111   3²x²11
    -  ---- + ------ = ½(h - h1  - h2)x^-1 (x - 1)^-1 + ½h1 x^-2 + ½h2(x - 1)^-2,
       x1³    2x1^4

  where x1 = dx/d[stigma], &c.; and h1 = 1 - y1², h2 = 1 - [lambda]2²,
  h3 = 1 - ([lambda] - [mu])². Into the converse question whether the
  three conditions are sufficient to ensure (1) that the [stigma] region
  corresponding to any branch does not overlap itself, (2) that no two
  such regions overlap, we have no space to enter. The second question
  clearly requires the inquiry whether the group (that is, the monodromy
  group) of the differential equation is properly discontinuous. (See
  GROUPS, THEORY OF.)

The foregoing account will give an idea of the nature of the function
theories of differential equations; it appears essential not to exclude
some explanation of a theory intimately related both to such theories
and to transformation theories, which is a generalization of Galois's
theory of algebraic equations. We deal only with the application to
homogeneous linear differential equations.


    Rationality group of a linear equation.

    Irreducibility of a rational equation.

  In general a function of variables x1, x2 ... is said to be rational
  when it can be formed from them and the integers 1, 2, 3, ... by a
  finite number of additions, subtractions, multiplications and
  divisions. We generalize this definition. Assume that we have assigned
  a fundamental series of quantities and functions of x, in which x
  itself is included, such that all quantities formed by a finite number
  of additions, subtractions, multiplications, divisions _and
  differentiations in regard to x_, of the terms of this series, are
  themselves members of this series. Then the quantities of this series,
  and only these, are called _rational_. By a rational function of
  quantities p, q, r, ... is meant a function formed from them and any
  of the fundamental rational quantities by a finite number of the five
  fundamental operations. Thus it is a function which would be called,
  simply, rational if the fundamental series were widened by the
  addition to it of the quantities p, q, r, ... and those derivable from
  them by the five fundamental operations. A rational ordinary
  differential equation, with x as independent and y as dependent
  variable, is then one which equates to zero a rational function of y,
  the order k of the differential equation being that of the highest
  differential coefficient y^(k) which enters; only such equations are
  here discussed. Such an equation P = 0 is called _irreducible_ when,
  firstly, being arranged as an integral polynomial in y^(k), this
  polynomial is not the product of other polynomials in y^(k) also of
  rational form; and, secondly, the equation has no solution satisfying
  also a rational equation of lower order. From this it follows that if
  an irreducible equation P = 0 have one solution satisfying another
  rational equation Q = 0 of the same or higher order, then all the
  solutions of P = 0 also satisfy Q = 0. For from the equation P = 0 we
  can by differentiation express y^(k+1), y^(k+2), ... in terms of x, y,
  y^(1), ... , y^(k), and so put the function Q rationally in terms of
  these quantities only. It is sufficient, then, to prove the result
  when the equation Q = 0 is of the same order as P = 0. Let both the
  equations be arranged as integral polynomials in y^(k); their
  algebraic eliminant in regard to y^(k) must then vanish identically,
  for they are known to have one common solution not satisfying an
  equation of lower order; thus the equation P = 0 involves Q = 0 for
  all solutions of P = 0.


    The variant function for a linear equation.

  Now let y^(n) = [alpha]1y^(n-1) + ... + [alpha]_n y be a given
  rational homogeneous linear differential equation; let y1, ... yn be n
  particular functions of x, unconnected by any equation with constant
  coefficients of the form c1y1 + ... + cnyn = 0, all satisfying the
  differential equation; let [eta]1, ... [eta]n be linear functions of
  y1, ... yn, say [eta]i = A_i1 y1 + ... + A_in yn, where the constant
  coefficients Aij have a non-vanishing determinant; write ([eta]) =
  A(y), these being the equations of a general linear homogeneous group
  whose transformations may be denoted by A, B, .... We desire to form a
  rational function [phi]([eta]), or say [phi](A(y)), of [eta]1, ...
  [eta], in which the [eta]² constants Aij shall all be essential, and
  not reduce effectively to a fewer number, as they would, for instance,
  if the y1, ... yn were connected by a linear equation with constant
  coefficients. Such a function is in fact given, if the solutions y1,
  ... yn be developable in positive integral powers about x = a, by
  [phi]([eta]) = [eta]1 + (x - a)^n[eta]2 + ... + (x - a)^(n-1)n[eta]n.
  Such a function, V, we call a _variant_.


    The resolvent eqution.

  Then differentiating V in regard to x, and replacing [eta]i^(n) by
  its value a1[eta]^(n-1) + ... + an[eta], we can arrange dV/dx, and
  similarly each of d²/dx² ... d^NV/dx^N, where N = n², as a linear
  function of the N quantities [eta]1, ... [eta]n, ... [eta]1^(n-1), ...
  [eta]n^(n-1), and thence by elimination obtain a linear differential
  equation for V of order N with rational coefficients. This we denote
  by F = 0. Further, each of [eta]1 ... [eta]n is expressible as a
  linear function of V, dV/dx, ... d^(N-1)V/dx^(N-1), with rational
  coefficients not involving any of the n² coefficients A_ij, since
  otherwise V would satisfy a linear equation of order less than N,
  which is impossible, as it involves (linearly) the n² arbitrary
  coefficients Aij, which would not enter into the coefficients of the
  supposed equation. In particular, y1 ,.. yn are expressible rationally
  as linear functions of [omega], d[omega]/dx, ...
  d^(N-1)[omega]/dx^(N-1), where [omega] is the particular function
  [phi](y). Any solution W of the equation F = 0 is derivable from
  functions [zeta]1, ... [zeta]n, which are linear functions of y1, ...
  yn, just as V was derived from [eta]1, ... [eta]n; but it does not
  follow that these functions [zeta]i, ... [zeta]n are obtained from y1,
  ... yn by a transformation of the linear group A, B, ... ; for it may
  happen that the determinant d([zeta]1, ... [zeta]n)/(dy1, ... yn) is
  zero. In that case [zeta]1, ... [zeta]n may be called a singular set,
  and W a singular solution; it satisfies an equation of lower than the
  N-th order. But every solution V, W, ordinary or singular, of the
  equation F = 0, is expressible rationally in terms of [omega],
  d[omega]/dx, ... d^(N-1)[omega]/dx^(N-1); we shall write, simply, V =
  r([omega]). Consider now the rational irreducible equation of lowest
  order, not necessarily a linear equation, which is satisfied by
  [omega]; as y1, ... yn are particular functions, it may quite well be
  of order less than N; we call it the _resolvent equation_, suppose it
  of order p, and denote it by [gamma](v). Upon it the whole theory
  turns. In the first place, as [gamma](v) = 0 is satisfied by the
  solution [omega] of F = 0, all the solutions of [gamma](v) are
  solutions F = 0, and are therefore rationally expressible by [omega];
  any one may then be denoted by r([omega]). If this solution of F = 0
  be not singular, it corresponds to a transformation A of the linear
  group (A, B, ...), effected upon y1, ... yn. The coefficients Aij of
  this transformation follow from the expressions before mentioned for
  [eta]1 ... [eta]n in terms of V, dV/dx, d²V/dx², ... by substituting V
  = r([omega]); thus they depend on the p arbitrary parameters which
  enter into the general expression for the integral of the equation
  [gamma](v) = 0. Without going into further details, it is then clear
  enough that the resolvent equation, being irreducible and such that
  any solution is expressible rationally, with p parameters, in terms of
  the solution [omega], enables us to define a linear homogeneous group
  of transformations of y1 ... yn depending on p parameters; and every
  operation of this (continuous) group corresponds to a rational
  transformation of the solution of the resolvent equation. This is the
  group called the _rationality group_, or the _group of
  transformations_ of the original homogeneous linear differential
  equation.

  The group must not be confounded with a subgroup of itself, the
  _monodromy group_ of the equation, often called simply the group of
  the equation, which is a set of transformations, not depending on
  arbitrary variable parameters, arising for one particular fundamental
  set of solutions of the linear equation (see GROUPS, THEORY OF).


    The fundamental theorem in regard to the rationality group.

  The importance of the rationality group consists in three
  propositions. (1) Any rational function of y1, ... yn which is
  unaltered in value by the transformations of the group can be written
  in rational form. (2) If any rational function be changed in form,
  becoming a rational function of y1, ... yn, a transformation of the
  group applied to its new form will leave its value unaltered. (3) Any
  homogeneous linear transformation leaving unaltered the value of every
  rational function of y1, ... yn which has a rational value, belongs to
  the group. It follows from these that any group of linear homogeneous
  transformations having the properties (1) (2) is identical with the
  group in question. It is clear that with these properties the group
  must be of the greatest importance in attempting to discover what
  functions of x must be regarded as rational in order that the values
  of y1 ... yn may be expressed. And this is the problem of solving the
  equation from another point of view.

  LITERATURE.--([alpha]) _Formal or Transformation Theories for
  Equations of the First Order_:--E. Goursat, _Leçons sur l'intégration
  des équations aux dérivées partielles du premier ordre_ (Paris, 1891);
  E. v. Weber, _Vorlesungen über das Pfaff'sche Problem und die Theorie
  der partiellen Differentialgleichungen erster Ordnung_ (Leipzig,
  1900); S. Lie und G. Scheffers, _Geometrie der
  Berührungstransformationen_, Bd. i. (Leipzig, 1896); Forsyth, _Theory
  of Differential Equations, Part i., Exact Equations and Pfaff's
  Problem_ (Cambridge, 1890); S. Lie, "Allgemeine Untersuchungen über
  Differentialgleichungen, die eine continuirliche endliche Gruppe
  gestatten" (Memoir), _Mathem. Annal._xxv. (1885), pp. 71-151; S. Lie
  und G. Scheffers, _Vorlesungen über Differentialgleichungen mit
  bekannten infinitesimalen Transformationen_ (Leipzig, 1891). A very
  full bibliography is given in the book of E. v. Weber referred to;
  those here named are perhaps sufficiently representative of modern
  works. Of classical works may be named: Jacobi, _Vorlesungen über
  Dynamik_ (von A. Clebsch, Berlin, 1866); _Werke, Supplementband_; G
  Monge, _Application de l'analyse à la géométrie_ (par M. Liouville,
  Paris, 1850); J. L. Lagrange, _Leçons sur le calcul des fonctions_
  (Paris, 1806), and _Théorie des fonctions analytiques_ (Paris,
  Prairial, an V); G. Boole, _A Treatise on Differential Equations_
  (London, 1859); and _Supplementary Volume_ (London, 1865); Darboux,
  _Leçons sur la théorie générale des surfaces_, tt. i.-iv. (Paris,
  1887-1896); S. Lie, _Théorie der transformationsgruppen_ ii. (on
  Contact Transformations) (Leipzig, 1890).

  ([beta]) _Quantitative or Function Theories for Linear Equations_:--C.
  Jordan, _Cours d'analyse_, t. iii. (Paris, 1896); E. Picard, _Traité
  d'analyse_, tt. ii. and iii. (Paris, 1893, 1896); Fuchs, _Various
  Memoirs, beginning with that in Crelle's Journal_, Bd. lxvi. p. 121;
  Riemann, _Werke_, 2^r Aufl. (1892); Schlesinger, _Handbuch der Theorie
  der linearen Differentialgleichungen_, Bde. i.-ii. (Leipzig,
  1895-1898); Heffter, _Einleitung in die Theorie der linearen
  Differentialgleichungen mit einer unabhängigen Variablen_ (Leipzig,
  1894); Klein, _Vorlesungen über lineare Differentialgleichungen der
  zweiten Ordnung_ (Autographed, Göttingen, 1894); and _Vorlesungen über
  die hypergeometrische Function_ (Autographed, Göttingen, 1894);
  Forsyth, _Theory of Differential Equations, Linear Equations_.

  ([gamma]) _Rationality Group (of Linear Differential
  Equations)_:--Picard, _Traité d'Analyse_, as above, t. iii.; Vessiot,
  _Annales de l'École Normale_, série III. t. ix. p. 199 (Memoir); S.
  Lie, _Transformationsgruppen_, as above, iii. A connected account is
  given in Schlesinger, as above, Bd. ii., erstes Theil.

  ([delta]) _Function Theories of Non-Linear Ordinary
  Equations_:--Painlevé, _Leçons sur la théorie analytique des équations
  différentielles_ (Paris, 1897, Autographed); Forsyth, _Theory of
  Differential Equations, Part ii., Ordinary Equations not Linear_ (two
  volumes, ii. and iii.) (Cambridge, 1900); Königsberger, _Lehrbuch der
  Theorie der Differentialgleichungen_ (Leipzig, 1889); Painlevé,
  _Leçons sur l'intégration des équations differentielles de la
  mécanique et applications_ (Paris, 1895).

  ([epsilon]) _Formal Theories of Partial Equations of the Second and
  Higher Orders_:--E. Goursat, _Leçons sur l'intégration des équations
  aux dérivées partielles du second ordre_, tt. i. and ii. (Paris, 1896,
  1898); Forsyth, _Treatise on Differential Equations_ (London, 1889);
  and _Phil. Trans. Roy. Soc._ (A.), vol. cxci. (1898), pp. 1-86.

  ([zeta]) See also the six extensive articles in the second volume of
  the German _Encyclopaedia of Mathematics_.     (H. F. BA.)



DIFFLUGIA (L. Leclerc), a genus of lobose Rhizopoda, characterized by a
shell formed of sand granules cemented together; these are swallowed by
the animal, and during the process of bud-fission they pass to the
surface of the daughter-bud and are cemented there. _Centropyxis_
(Steia) and _Lecqueureuxia_ (Schlumberg) differ only in minor points.



DIFFRACTION OF LIGHT.--1. When light proceeding from a small source
falls upon an opaque object, a shadow is cast upon a screen situated
behind the obstacle, and this shadow is found to be bordered by
alternations of brightness and darkness, known as "diffraction bands."
The phenomena thus presented were described by Grimaldi and by Newton.
Subsequently T. Young showed that in their formation interference plays
an important part, but the complete explanation was reserved for A. J.
Fresnel. Later investigations by Fraunhofer, Airy and others have
greatly widened the field, and under the head of "diffraction" are now
usually treated all the effects dependent upon the limitation of a beam
of light, as well as those which arise from irregularities of any kind
at surfaces through which it is transmitted, or at which it is
reflected.

2. _Shadows._--In the infancy of the undulatory theory the objection
most frequently urged against it was the difficulty of explaining the
very existence of shadows. Thanks to Fresnel and his followers, this
department of optics is now precisely the one in which the theory has
gained its greatest triumphs. The principle employed in these
investigations is due to C. Huygens, and may be thus formulated. If
round the origin of waves an ideal closed surface be drawn, the whole
action of the waves in the region beyond may be regarded as due to the
motion continually propagated across the various elements of this
surface. The wave motion due to any element of the surface is called a
_secondary_ wave, and in estimating the total effect regard must be paid
to the phases as well as the amplitudes of the components. It is usually
convenient to choose as the surface of resolution a _wave-front_, i.e. a
surface at which the primary vibrations are in one phase. Any obscurity
that may hang over Huygens's principle is due mainly to the
indefiniteness of thought and expression which we must be content to put
up with if we wish to avoid pledging ourselves as to the character of
the vibrations. In the application to sound, where we know what we are
dealing with, the matter is simple enough in principle, although
mathematical difficulties would often stand in the way of the
calculations we might wish to make. The ideal surface of resolution may
be there regarded as a flexible lamina; and we know that, if by forces
locally applied every element of the lamina be made to move normally to
itself exactly as the air at that place does, the external aerial motion
is fully determined. By the principle of superposition the whole effect
may be found by integration of the partial effects due to each element
of the surface, the other elements remaining at rest.

  We will now consider in detail the important case in which uniform
  plane waves are resolved at a surface coincident with a wave-front
  (OQ). We imagine a wave-front divided into elementary rings or
  zones--often named after Huygens, but better after Fresnel--by spheres
  described round P (the point at which the aggregate effect is to be
  estimated), the first sphere, touching the plane at O, with a radius
  equal to PO, and the succeeding spheres with radii increasing at each
  step by ½[lambda]. There are thus marked out a series of circles,
  whose radii x are given by x² + r² = (r + ½n[lambda])², or x² =
  n[lambda]r nearly; so that the rings are at first of nearly equal
  area. Now the effect upon P of each element of the plane is
  proportional to its area; but it depends also upon the distance from
  P, and possibly upon the inclination of the secondary ray to the
  direction of vibration and to the wave-front.

        O     x          Q
    ---------------------------
        |                /
        |               /
        |              /
        |             /
        |            /
        |           /
        |          /
       r|         /
        |        /
        |       /
        |      /
        |     /
        |    /
        |   /
        |  /
        | /
       P|/

    FIG. 1.

  The latter question can only be treated in connexion with the
  dynamical theory (see below, § 11); but under all ordinary
  circumstances the result is independent of the precise answer that may
  be given. All that it is necessary to assume is that the effects of
  the successive zones gradually diminish, whether from the increasing
  obliquity of the secondary ray or because (on account of the
  limitation of the region of integration) the zones become at last more
  and more incomplete. The component vibrations at P due to the
  successive zones are thus nearly equal in amplitude and opposite in
  phase (the phase of each corresponding to that of the infinitesimal
  circle midway between the boundaries), and the series which we have to
  sum is one in which the terms are alternately opposite in sign and,
  while at first nearly constant in numerical magnitude, gradually
  diminish to zero. In such a series each term may be regarded as very
  nearly indeed destroyed by the halves of its immediate neighbours, and
  thus the sum of the whole series is represented by half the first
  term, which stands over uncompensated. The question is thus reduced to
  that of finding the effect of the first zone, or central circle, of
  which the area is [pi][lambda]r.

  We have seen that the problem before us is independent of the law of
  the secondary wave as regards obliquity; but the result of the
  integration necessarily involves the law of the intensity and phase of
  a secondary wave as a function of r, the distance from the origin. And
  we may in fact, as was done by A. Smith (_Camb. Math. Journ._, 1843,
  3, p. 46), determine the law of the secondary wave, by comparing the
  result of the integration with that obtained by supposing the primary
  wave to pass on to P without resolution.

  Now as to the phase of the secondary wave, it might appear natural to
  suppose that it starts from any point Q with the phase of the primary
  wave, so that on arrival at P, it is retarded by the amount
  corresponding to QP. But a little consideration will prove that in
  that case the series of secondary waves could not reconstitute the
  primary wave. For the aggregate effect of the secondary waves is the
  half of that of the first Fresnel zone, and it is the central element
  only of that zone for which the distance to be travelled is equal to
  r. Let us conceive the zone in question to be divided into
  infinitesimal rings of equal area. The effects due to each of these
  rings are equal in amplitude and of phase ranging uniformly over half
  a complete period. The phase of the resultant is midway between those
  of the extreme elements, that is to say, a quarter of a period behind
  that due to the element at the centre of the circle. It is accordingly
  necessary to suppose that the secondary waves start with a phase
  one-quarter of a period in advance of that of the primary wave at the
  surface of resolution.

  Further, it is evident that account must be taken of the variation of
  phase in estimating the magnitude of the effect at P of the first
  zone. The middle element alone contributes without deduction; the
  effect of every other must be found by introduction of a resolving
  factor, equal to cos [theta], if [theta] represent the difference of
  phase between this element and the resultant. Accordingly, the
  amplitude of the resultant will be less than if all its components had
  the same phase, in the ratio

      _ +½[pi]
     /
     |      cos [theta]d[theta] : [pi],
    _/-½[pi]

  or 2 : [pi]. Now 2 area /[pi] = 2[lambda]r; so that, in order to
  reconcile the amplitude of the primary wave (taken as unity) with the
  half effect of the first zone, the amplitude, at distance r, of the
  secondary wave emitted from the element of area dS must be taken to be

    dS/[lambda]r                   (1).


  By this expression, in conjunction with the quarter-period
  acceleration of phase, the law of the secondary wave is determined.

  That the amplitude of the secondary wave should vary as r^-1 was to be
  expected from considerations respecting energy; but the occurrence of
  the factor [lambda]^-1, and the acceleration of phase, have sometimes
  been regarded as mysterious. It may be well therefore to remember that
  precisely these laws apply to a secondary wave of sound, which can be
  investigated upon the strictest mechanical principles.

  The recomposition of the secondary waves may also be treated
  analytically. If the primary wave at O be cos kat, the effect of the
  secondary wave proceeding from the element dS at Q is

         dS                                           dS
    ------------- cos k(at - [rho] + ¼[lambda]) = ------------- sin k(at - [rho]).
    [lambda][rho]                                 [lambda][rho]


  If dS = 2[pi]xdx, we have for the whole effect

                  _[oo]
       2[pi]     /   sin k(at - [rho])x dx
    - --------   |   ---------------------,
      [lambda]  _/ 0         [rho]

  or, since xdx = [rho]d[rho], k = 2[pi]/[lambda],

        _[oo]                       _                 _
       /                           |                   |[oo]
    -k | sin k(at - [rho])d[rho] = | -cos k(at - [rho])|    .
      _/r                          |_                 _|r

  In order to obtain the effect of the primary wave, as retarded by
  traversing the distance r, viz. cos k(at - r), it is necessary to
  suppose that the integrated term vanishes at the upper limit. And it
  is important to notice that without some further understanding the
  integral is really ambiguous. According to the assumed law of the
  secondary wave, the result must actually depend upon the precise
  radius of the outer boundary of the region of integration, supposed to
  be exactly circular. This case is, however, at most very special and
  exceptional. We may usually suppose that a large number of the outer
  rings are incomplete, so that the integrated term at the upper limit
  may properly be taken to vanish. If a formal proof be desired, it may
  be obtained by introducing into the integral a factor such as
  e^-h[rho], in which h is ultimately made to diminish without limit.

  When the primary wave is plane, the area of the first Fresnel zone is
  [pi][lambda]r, and, since the secondary waves vary as r^-1, the
  intensity is independent of r, as of course it should be. If, however,
  the primary wave be spherical, and of radius a at the wave-front of
  resolution, then we know that at a distance r further on the amplitude
  of the primary wave will be diminished in the ratio a:(r + a). This
  may be regarded as a consequence of the altered area of the first
  Fresnel zone. For, if x be its radius, we have

                                /
    {(r + ½[lambda])² - x²} + \/ {a² - x²} = r + a,

  so that

    x² = [lambda]ar/(a + r) nearly.

  Since the distance to be travelled by the secondary waves is still r,
  we see how the effect of the first zone, and therefore of the whole
  series is proportional to a/(a + r). In like manner may be treated
  other cases, such as that of a primary wave-front of unequal principal
  curvatures.

  The general explanation of the formation of shadows may also be
  conveniently based upon Fresnel's zones. If the point under
  consideration be so far away from the geometrical shadow that a large
  number of the earlier zones are complete, then the illumination,
  determined sensibly by the first zone, is the same as if there were no
  obstruction at all. If, on the other hand, the point be well immersed
  in the geometrical shadow, the earlier zones are altogether missing,
  and, instead of a series of terms beginning with finite numerical
  magnitude and gradually diminishing to zero, we have now to deal with
  one of which the terms diminish to zero _at both ends_. The sum of
  such a series is very approximately zero, each term being neutralized
  by the halves of its immediate neighbours, which are of the opposite
  sign. The question of light or darkness then depends upon whether the
  series begins or ends abruptly. With few exceptions, abruptness can
  occur only in the presence of the first term, viz. when the secondary
  wave of least retardation is unobstructed, or when a _ray_ passes
  through the point under consideration. According to the undulatory
  theory the light cannot be regarded strictly as travelling along a
  ray; but the existence of an unobstructed ray implies that the system
  of Fresnel's zones can be commenced, and, if a large number of these
  zones are fully developed and do not terminate abruptly, the
  illumination is unaffected by the neighbourhood of obstacles.
  Intermediate cases in which a few zones only are formed belong
  especially to the province of diffraction.

  An interesting exception to the general rule that full brightness
  requires the existence of the first zone occurs when the obstacle
  assumes the form of a small circular disk parallel to the plane of the
  incident waves. In the earlier half of the 18th century R. Delisle
  found that the centre of the circular shadow was occupied by a bright
  point of light, but the observation passed into oblivion until S. D.
  Poisson brought forward as an objection to Fresnel's theory that it
  required at the centre of a circular shadow a point as bright as if no
  obstacle were intervening. If we conceive the primary wave to be
  broken up at the plane of the disk, a system of Fresnel's zones can be
  constructed which begin from the circumference; and the first zone
  external to the disk plays the part ordinarily taken by the centre of
  the entire system. The whole effect is the half of that of the first
  existing zone, and this is sensibly the same as if there were no
  obstruction.

  When light passes through a small circular or annular aperture, the
  illumination at any point along the axis depends upon the precise
  relation between the aperture and the distance from it at which the
  point is taken. If, as in the last paragraph, we imagine a system of
  zones to be drawn commencing from the inner circular boundary of the
  aperture, the question turns upon the manner in which the series
  terminates at the outer boundary. If the aperture be such as to fit
  exactly an integral number of zones, the aggregate effect may be
  regarded as the half of those due to the first and last zones. If the
  number of zones be even, the action of the first and last zones are
  antagonistic, and there is complete darkness at the point. If on the
  other hand the number of zones be odd, the effects conspire; and the
  illumination (proportional to the square of the amplitude) is four
  times as great as if there were no obstruction at all.

  The process of augmenting the resultant illumination at a particular
  point by stopping some of the secondary rays may be carried much
  further (Soret, _Pogg. Ann._, 1875, 156, p. 99). By the aid of
  photography it is easy to prepare a plate, transparent where the zones
  of odd order fall, and opaque where those of even order fall. Such a
  plate has the power of a condensing lens, and gives an illumination
  out of all proportion to what could be obtained without it. An even
  greater effect (fourfold) can be attained by providing that the
  stoppage of the light from the alternate zones is replaced by a
  phase-reversal without loss of amplitude. R. W. Wood (_Phil. Mag._,
  1898, 45, p 513) has succeeded in constructing zone plates upon this
  principle.

  In such experiments the narrowness of the zones renders necessary a
  pretty close approximation to the geometrical conditions. Thus in the
  case of the circular disk, equidistant (r) from the source of light
  and from the screen upon which the shadow is observed, the width of
  the first exterior zone is given by

    dx = [lambda](2r)/4(2x),

  2x being the diameter of the disk. If 2r = 1000 cm., 2x = 1 cm.,
  [lambda] = 6 × 10^-5 cm., then dx = .0015 cm. Hence, in order that
  this zone may be perfectly formed, there should be no error in the
  circumference of the order of .001 cm. (It is easy to see that the
  radius of the bright spot is of the same order of magnitude.) The
  experiment succeeds in a dark room of the length above mentioned, with
  a threepenny bit (supported by three threads) as obstacle, the origin
  of light being a small needle hole in a plate of tin, through which
  the sun's rays shine horizontally after reflection from an external
  mirror. In the absence of a heliostat it is more convenient to obtain
  a point of light with the aid of a lens of short focus.

  The amplitude of the light at any point in the axis, when plane waves
  are incident perpendicularly upon an annular aperture, is, as above,

    cos k(at - r1) - cos k(at - r2) = 2 sin kat sin k(r1 - r2),

  r2, r1 being the distances of the outer and inner boundaries from the
  point in question. It is scarcely necessary to remark that in all such
  cases the calculation applies in the first instance to homogeneous
  light, and that, in accordance with Fourier's theorem, each
  homogeneous component of a mixture may be treated separately. When the
  original light is white, the presence of some components and the
  absence of others will usually give rise to coloured effects, variable
  with the precise circumstances of the case.

  [Illustration: FIG. 2.]

  Although the matter can be fully treated only upon the basis of a
  dynamical theory, it is proper to point out at once that there is an
  element of assumption in the application of Huygens's principle to the
  calculation of the effects produced by opaque screens of limited
  extent. Properly applied, the principle could not fail; but, as may
  readily be proved in the case of sonorous waves, it is not in
  strictness sufficient to assume the expression for a secondary wave
  suitable when the primary wave is undisturbed, with mere limitation of
  the integration to the transparent parts of the screen. But, except
  perhaps in the case of very fine gratings, it is probable that the
  error thus caused is insignificant; for the incorrect estimation of
  the secondary waves will be limited to distances of a few wave-lengths
  only from the boundary of opaque and transparent parts.

3. _Fraunhofer's Diffraction Phenomena._--A very general problem in
diffraction is the investigation of the distribution of light over a
screen upon which impinge divergent or convergent spherical waves after
passage through various diffracting apertures. When the waves are
convergent and the recipient screen is placed so as to contain the
centre of convergency--the image of the original radiant point, the
calculation assumes a less complicated form. This class of phenomena was
investigated by J. von Fraunhofer (upon principles laid down by
Fresnel), and are sometimes called after his name. We may conveniently
commence with them on account of their simplicity and great importance
in respect to the theory of optical instruments.

  If f be the radius of the spherical wave at the place of resolution,
  where the vibration is represented by cos kat, then at any point M
  (fig. 2) in the recipient screen the vibration due to an element dS of
  the wave-front is (§ 2)

           dS
    - ------------- sin k(at - [rho]),
      [lambda][rho]

  [rho] being the distance between M and the element dS.

  Taking co-ordinates in the plane of the screen with the centre of the
  wave as origin, let us represent M by [xi], [eta], and P (where dS is
  situated) by x, y, z. Then

    [rho]² = (x - [xi])² + (y - [eta])² + z², f² = x² + y² + z²;

  so that

    [rho]² = f² - 2x[xi] - 2y[eta] + [xi]² + [eta]².

  In the applications with which we are concerned, [xi], [eta] are very
  small quantities; and we may take

               /    x[xi] + y[eta]\
    [rho] = f ( 1 - -------------- ).
               \          f²      /

  At the same time dS may be identified with dxdy, and in the
  denominator [rho] may be treated as constant and equal to f. Thus the
  expression for the vibration at M becomes

                      _ _
            1        / /        /         x[xi] + y[eta]\
    - -------------  | | sin k ( at - f + -------------- ) dxdy    (1);
      [lambda]²[f]² _/_/        \               f       /

  and for the intensity, represented by the square of the amplitude,

                       _   _ _                         _
               1      |   / /       x[xi] + y[eta]      |²
    I² = ------------ |   | | sin k -------------- dxdy |
          [lambda]²f² |_ _/_/             f            _|
                      _   _ _                          _
               1     |   / /        x[xi] + y[eta]      |²
       + ----------- |   | |  cos k -------------- dxdy |         (2).
         [lambda]²f² |_ _/_/              f            _|

  This expression for the intensity becomes rigorously applicable when f
  is indefinitely great, so that ordinary optical aberration disappears.
  The incident waves are thus plane, and are limited to a plane aperture
  coincident with a wave-front. The integrals are then properly
  functions of the _direction_ in which the light is to be estimated.

  In experiment under ordinary circumstances it makes no difference
  whether the collecting lens is in front of or behind the diffracting
  aperture. It is usually most convenient to employ a telescope focused
  upon the radiant point, and to place the diffracting apertures
  immediately in front of the object-glass. What is seen through the
  eye-piece in any case is the same as would be depicted upon a screen
  in the focal plane.

  Before proceeding to special cases it may be well to call attention to
  some general properties of the solution expressed by (2) (see Bridge,
  _Phil. Mag._, 1858).

  If when the aperture is given, the wave-length (proportional to k^-1)
  varies, the composition of the integrals is unaltered, provided [xi]
  and [eta] are taken universely proportional to [lambda]. A diminution
  of [lambda] thus leads to a simple proportional shrinkage of the
  diffraction pattern, attended by an augmentation of brilliancy in
  proportion to [lambda]^-2.

  If the wave-length remains unchanged, similar effects are produced by
  an increase in the scale of the aperture. The linear dimension of the
  diffraction pattern is inversely as that of the aperture, and the
  brightness at corresponding points is as the _square_ of the area of
  aperture.

  If the aperture and wave-length increase in the same proportion, the
  size and shape of the diffraction pattern undergo no change.

  We will now apply the integrals (2) to the case of a rectangular
  aperture of width a parallel to x and of width b parallel to y. The
  limits of integration for x may thus be taken to be -½a and +½a, and
  for y to be -½b, +½b. We readily find (with substitution for k of
  2[pi]/[lambda])

                            [pi]a[xi]           [pi]b[eta]
                       sin² ---------      sin² ----------
              a²b²          f[lambda]           f[lambda]
    I² = ----------- · ----------------- · ---------------     (3),
         f²[lambda]²     [pi]²a²[xi]²       [pi]²b²[eta]²
                         ------------       -------------
                         f²[lambda]²         f²[lambda]²

  as representing the distribution of light in the image of a
  mathematical point when the aperture is rectangular, as is often the
  case in spectroscopes.

  The second and third factors of (3) being each of the form sin²u/u²,
  we have to examine the character of this function. It vanishes when u
  = m[pi], m being any whole number other than zero. When u = 0, it
  takes the value unity. The maxima occur when

    u = tan u,                                                 (4),

  and then

    sin²u/u² = cos²u                                           (5).

  To calculate the roots of (5) we may assume

    u = (m + ½)[pi] - y = U - y,



  where y is a positive quantity which is small when u is large.
  Substituting this, we find cot y = U - y, whence

        1  /    y   y-      \    y³ 2y^5   17y^7
    y = - ( 1 + - + -- + ... ) - -- ---- - -----.
        U  \    U   U²      /    3   15     315

  This equation is to be solved by successive approximation. It will
  readily be found that

                           2         13        146
    u = U - y = U - U^-1 - -- U^-3 - -- U^-5 - --- U^-7 - ...   (6).
                           3         15        105

  In the first quadrant there is no root after zero, since tan u > u,
  and in the second quadrant there is none because the signs of u and
  tan u are opposite. The first root after zero is thus in the third
  quadrant, corresponding to m = 1. Even in this case the series
  converges sufficiently to give the value of the root with considerable
  accuracy, while for higher values of m it is all that could be
  desired. The actual values of u/[pi] (calculated in another manner by
  F. M. Schwerd) are 1.4303, 2.4590, 3.4709, 4.4747, 5.4818, 6.4844, &c.

  Since the maxima occur when u = (m + ½)[pi] nearly, the successive
  values are not very different from

      4        4       4
    ------, ------, -------, &c.
    9[pi]²  25[pi]  49[pi]²

  The application of these results to (3) shows that the field is
  brightest at the centre [xi] = 0, [eta] = 0, viz. at the geometrical
  image of the radiant point. It is traversed by dark lines whose
  equations are

    [xi] = mf[lambda]/a, [eta] = mf[lambda]/b.

  Within the rectangle formed by pairs of consecutive dark lines, and
  not far from its centre, the brightness rises to a maximum; but these
  subsequent maxima are in all cases much inferior to the brightness at
  the centre of the entire pattern ([xi] = 0, [eta] = 0).

  By the principle of energy the illumination over the entire focal
  plane must be equal to that over the diffracting area; and thus, in
  accordance with the suppositions by which (3) was obtained, its value
  when integrated from [xi] = [oo] to [xi] = +[oo], and from [eta] =
  -[oo] to [eta] = +[oo] should be equal to ab. This integration,
  employed originally by P. Kelland (_Edin. Trans._ 15, p. 315) to
  determine the absolute intensity of a secondary wave, may be at once
  effected by means of the known formula

      _+[oo]        _+[oo]
     / sin²u       / sin u
     | ----- du =  | ----- du = [pi].
    _/   u²       _/   u
     -[oo]         -[oo]

  It will be observed that, while the total intensity is proportional to
  ab, the intensity at the focal point is proportional to a²b². If the
  aperture be increased, not only is the total brightness over the focal
  plane increased with it, but there is also a concentration of the
  diffraction pattern. The form of (3) shows immediately that, if a and
  b be altered, the co-ordinates of any characteristic point in the
  pattern vary as a^-1 and b^-1.

  The contraction of the diffraction pattern with increase of aperture
  is of fundamental importance in connexion with the resolving power of
  optical instruments. According to common optics, where images are
  absolute, the diffraction pattern is supposed to be infinitely small,
  and two radiant points, however near together, form separated images.
  This is tantamount to an assumption that [lambda] is infinitely small.
  The actual finiteness of [lambda] imposes a limit upon the separating
  or resolving power of an optical instrument.

  This indefiniteness of images is sometimes said to be due to
  diffraction by the edge of the aperture, and proposals have even been
  made for curing it by causing the transition between the interrupted
  and transmitted parts of the primary wave to be less abrupt. Such a
  view of the matter is altogether misleading. What requires explanation
  is not the imperfection of actual images so much as the possibility of
  their being as good as we find them.

  At the focal point ([xi] = 0, [eta] = 0) all the secondary waves agree
  in phase, and the intensity is easily expressed, whatever be the form
  of the aperture. From the general formula (2), if A be the _area_ of
  aperture,

    I0² = A²/[lambda]²f²           (7).

  The formation of a sharp image of the radiant point requires that the
  illumination become insignificant when [xi], [eta] attain small
  values, and this insignificance can only arise as a consequence of
  discrepancies of phase among the secondary waves from various parts of
  the aperture. So long as there is no sensible discrepancy of phase
  there can be no sensible diminution of brightness as compared with
  that to be found at the focal point itself. We may go further, and lay
  it down that there can be no considerable loss of brightness until the
  difference of phase of the waves proceeding from the nearest and
  farthest parts of the aperture amounts to ¼[lambda].

  When the difference of phase amounts to [lambda], we may expect the
  resultant illumination to be very much reduced. In the particular case
  of a rectangular aperture the course of things can be readily
  followed, especially if we conceive f to be infinite. In the direction
  (suppose horizontal) for which [eta] = 0, [xi]/f = sin [theta], the
  phases of the secondary waves range over a complete period when sin
  [theta] = [lambda]/a, and, since all parts of the horizontal aperture
  are equally effective, there is in this direction a complete
  compensation and consequent absence of illumination. When sin [theta]
  = 3/2[lambda]/a, the phases range one and a half periods, and there
  is revival of illumination. We may compare the brightness with that in
  the direction [theta] = 0. The phase of the resultant amplitude is the
  same as that due to the central secondary wave, and the discrepancies
  of phase among the components reduce the amplitude in the proportion

            _+3/2[pi]
      1    /
    -----  | cos [phi] d[phi]: 1,
    3[pi] _/-3/2[pi]

  or -2/3[pi]:1; so that the brightness in this direction is 4/9[pi]² of
  the maximum at [theta] = 0. In like manner we may find the
  illumination in any other direction, and it is obvious that it
  vanishes when sin [theta] is any multiple of [lamba]/a.

  The reason of the augmentation of resolving power with aperture will
  now be evident. The larger the aperture the smaller are the angles
  through which it is necessary to deviate from the principal direction
  in order to bring in specified discrepancies of phase--the more
  concentrated is the image.

  In many cases the subject of examination is a luminous line of uniform
  intensity, the various points of which are to be treated as
  independent sources of light. If the image of the line be [xi] = 0,
  the intensity at any point [xi], [eta] of the diffraction pattern may
  be represented by

                                 [pi]a[xi]
      _+[oo]                 sin²---------
     /               a²b         [lambda]f
     | I²d[eta] = ---------  -------------     (8),
    _/            [lambda]f  [pi]²a²[xi]²
     -[oo]                   ------------
                              [lambda]²f²

  the same law as obtains for a luminous point when horizontal
  directions are alone considered. The definition of a fine vertical
  line, and consequently the resolving power for contiguous vertical
  lines, is thus _independent of the vertical aperture of the
  instrument_, a law of great importance in the theory of the
  spectroscope.

  The distribution of illumination in the image of a luminous line is
  shown by the curve ABC (fig. 3), representing the value of the
  function sin²u/u² from u = 0 to u = 2[pi]. The part corresponding to
  negative values of u is similar, OA being a line of symmetry.

  [Illustration: Fig. 3.]

  Let us now consider the distribution of brightness in the image of a
  double line whose components are of equal strength, and at such an
  angular interval that the central line in the image of one coincides
  with the first zero of brightness in the image of the other. In fig. 3
  the curve of brightness for one component is ABC, and for the other
  OA'C'; and the curve representing half the combined brightnesses is
  E'BE. The brightness (corresponding to B) midway between the two
  central points AA' is .8106 of the brightness at the central points
  themselves. We may consider this to be about the limit of closeness at
  which there could be any decided appearance of resolution, though
  doubtless an observer accustomed to his instrument would recognize the
  duplicity with certainty. The obliquity, corresponding to u = [pi], is
  such that the phases of the secondary waves range over a complete
  period, i.e. such that the projection of the horizontal aperture upon
  this direction is one wave-length. We conclude that a _double line
  cannot be fairly resolved unless its components subtend an angle
  exceeding that subtended by the wave-length of light at a distance
  equal to the horizontal aperture_. This rule is convenient on account
  of its simplicity; and it is sufficiently accurate in view of the
  necessary uncertainty as to what exactly is meant by resolution.

  If the angular interval between the components of a double line be
  half as great again as that supposed in the figure, the brightness
  midway between is .1802 as against 1.0450 at the central lines of each
  image. Such a falling off in the middle must be more than sufficient
  for resolution. If the angle subtended by the components of a double
  line be twice that subtended by the wave-length at a distance equal to
  the horizontal aperture, the central bands are just clear of one
  another, and there is a line of absolute blackness in the middle of
  the combined images.

  The resolving power of a telescope with circular or rectangular
  aperture is easily investigated experimentally. The best object for
  examination is a grating of fine wires, about fifty to the inch,
  backed by a sodium flame. The object-glass is provided with diaphragms
  pierced with round holes or slits. One of these, of width equal, say,
  to one-tenth of an inch, is inserted in front of the object-glass, and
  the telescope, carefully focused all the while, is drawn gradually
  back from the grating until the lines are no longer seen. From a
  measurement of the maximum distance the least angle between
  consecutive lines consistent with resolution may be deduced, and a
  comparison made with the rule stated above.

  Merely to show the dependence of resolving power on aperture it is not
  necessary to use a telescope at all. It is sufficient to look at wire
  gauze backed by the sky or by a flame, through a piece of blackened
  cardboard, pierced by a needle and held close to the eye. By varying
  the distance the point is easily found at which resolution ceases; and
  the observation is as sharp as with a telescope. The function of the
  telescope is in fact to allow the use of a wider, and therefore more
  easily measurable, aperture. An interesting modification of the
  experiment may be made by using light of various wave-lengths.

  Since the limitation of the width of the central band in the image of
  a luminous line depends upon discrepancies of phase among the
  secondary waves, and since the discrepancy is greatest for the waves
  which come from the edges of the aperture, the question arises how far
  the operation of the central parts of the aperture is advantageous. If
  we imagine the aperture reduced to two equal narrow slits bordering
  its edges, compensation will evidently be complete when the projection
  on an oblique direction is equal to ½[lambda], instead of [lambda] as
  for the complete aperture. By this procedure the width of the central
  band in the diffraction pattern is halved, and so far an advantage is
  attained. But, as will be evident, the bright bands bordering the
  central band are now not inferior to it in brightness; in fact, a band
  similar to the central band is reproduced an indefinite number of
  times, so long as there is no sensible discrepancy of phase in the
  secondary waves proceeding from the various parts of the _same_ slit.
  Under these circumstances the narrowing of the band is paid for at a
  ruinous price, and the arrangement must be condemned altogether.

  A more moderate suppression of the central parts is, however,
  sometimes advantageous. Theory and experiment alike prove that a
  double line, of which the components are equally strong, is better
  resolved when, for example, one-sixth of the horizontal aperture is
  blocked off by a central screen; or the rays quite at the centre may
  be allowed to pass, while others a little farther removed are blocked
  off. Stops, each occupying one-eighth of the width, and with centres
  situated at the points of trisection, answer well the required
  purpose.

  It has already been suggested that the principle of energy requires
  that the general expression for I² in (2) when integrated over the
  whole of the plane [xi], [eta] should be equal to A, where A is the
  area of the aperture. A general analytical verification has been given
  by Sir G. G. Stokes (_Edin. Trans._, 1853, 20, p. 317). Analytically
  expressed--

      _ _+[oo]             _ _
     / /                  / /
     | | I² d[xi]d[eta] = | | dxdy  = A        (9).
    _/_/-[oo]            _/_/

  We have seen that I0² (the intensity at the focal point) was equal to
  A²/[lambda]²f². If A' be the area over which the intensity must be I0²
  in order to give the actual total intensity in accordance with

             _ _+[oo]
            / /
    A'I0² = | | I² d[xi]d[eta],
           _/_/-[oo]

  the relation between A and A' is AA' = [lambda]²f². Since A' is in
  some sense the area of the diffraction pattern, it may be considered
  to be a rough criterion of the definition, and we infer that the
  definition of a point depends principally upon the area of the
  aperture, and only in a very secondary degree upon the shape when the
  area is maintained constant.

4. _Theory of Circular Aperture._--We will now consider the important
case where the form of the aperture is circular.

  Writing for brevity

    k[xi]/f = p, k[eta]/f = q,        (1),

  we have for the general expression (§ 11) of the intensity

    [lambda]²f²I² = S² + C²           (2),

  where
          _ _
         / /
    S =  | | sin(px + qy)dx dy,       (3),
        _/_/
          _ _
         / /
    C =  | | cos(px + qy)dx dy,       (4).
        _/_/

  When, as in the application to rectangular or circular apertures, the
  form is symmetrical with respect to the axes both of x and y, S = 0,
  and C reduces to
          _ _
         / /
    C =  | | cos px cos qy dx dy,     (5).
        _/_/

  In the case of the circular aperture the distribution of light is of
  course symmetrical with respect to the focal point p = 0, q = 0; and C
  is a function of p and q only through [sqrt](p² + q²). It is thus
  sufficient to determine the intensity along the axis of p. Putting q =
  0, we get
          _ _                  _+R
         / /                  /          /
    C =  | | cos px dx dy = 2 | cos px \/(R² - x²) dx,
        _/_/                 _/-R

  R being the radius of the aperture. This integral is the Bessel's
  function of order unity, defined by

                   _[pi]
             z    /
    J1(z) = ----  | cos(z cos [phi]) sin² [phi] d[phi]    (6).
            [pi] _/0

  Thus, if x = R cos [phi],

               2J1(pR)
    C = [pi]²R -------                                    (7);
                 pR

  and the illumination at distance r from the focal point is

                            / 2[pi]Rr \
                       4J1²( --------- )
          [pi]²R^4          \f[lambda]/
    I² = ----------- · -----------------                  (8).
         [lambda]²f²     / 2[pi]Rr \²
                        ( --------- )
                         \f[lambda]/

  The ascending series for J1(z), used by Sir G. B. Airy (_Camb.
  Trans._, 1834) in his original investigation of the diffraction of a
  circular object-glass, and readily obtained from (6), is

            z    z³      z^5         z^7
    J1(z) = - - ---- + ------- - ---------- + ...         (9).
            2   2²·4   2²·4²·6   2²·4²·6²·8

  When z is great, we may employ the semi-convergent series
                                          _
              / /  2  \                  |    3·5·1   /1\²
    J1(z) =  / ( ----- ) sin (z - ¼[pi]) |1 + ------ ( - )
           \/   \[pi]z/                  |_    8·16   \z/
                                          _
               3·5·7·9·1·3·5  /1\^4        |
             - ------------- ( - )   + ... |
                 8·16·24·32   \z/         _|
                                       _
           / /  2  \                  | 3   1   3·5·7·1·3  /1\ ³
       +  / ( ----- ) cos (z - ¼[pi]) | - · - - --------- ( - )
        \/   \[pi]z/                  |_8   z    8·16·24   \z/
                                         _
         3·5·7·9·11·1·3·5·7  /1\^5        |
       + ------------------ ( - )   - ... | ...          (10).
           8·16·24·32·40     \z/         _|

  A table of the values of 2z^-1J1(z) has been given by E. C. J. Lommel
  (_Schlömilch_, 1870, 15, p. 166), to whom is due the first systematic
  application of Bessel's functions to the diffraction integrals.

  The illumination vanishes in correspondence with the roots of the
  equation J1(z) = 0. If these be called z1 z2, z3, ... the radii of the
  dark rings in the diffraction pattern are

    f[lambda]z1  f[lambda]z2
    -----------, -----------, ...
      2[pi]R       2[pi]R

  being thus _inversely_ proportional to R.

  The integrations may also be effected by means of polar co-ordinates,
  taking first the integration with respect to [phi] so as to obtain the
  result for an infinitely thin annular aperture. Thus, if

    x = [rho] cos [phi], y = [rho] sin [phi],

         _ _                _R  _2[pi]
        / /                /   /
    C = | | cos px dx dy = |   |   cos (p[rho] cos [theta]) [rho]d[rho] d[theta].
       _/_/               _/0 _/0

  Now by definition

                   _½[pi]
              2   /                              z²    z^4      z^6
    J0(z) = ----  |  cos(z cos[theta])d[theta] = -- + ----- - -------- + ...  (11).
            [pi] _/0                             2²   2²·4²   2²·4²·6²

  The value of C for an annular aperture of radius r and width dr is thus

    dC = 2 [pi]J0 (p[rho]) [rho] d[rho],                                       (12).

  For the complete circle,

               _ pR
        2[pi] /
    C = ----- | J0(z) zdz
         p²  _/0

         2[pi]  /p²R²   p^4 R^4   p^6 R^6       \
      = ------ ( ---- - ------- + -------- - ... )
          p²    \ 2      2²·4²    2²·4²·6²      /

                 2J1(pR)
      = [pi]R² · ------- as before.
                   pR

  In these expressions we are to replace p by k[xi]/f, or rather, since
  the diffraction pattern is symmetrical, by kr/f, where r is the
  distance of any point in the focal plane from the centre of the
  system.

  The roots of J0(z) after the first may be found from

     z               .050561    .053041     .262051
    ---- = i - .25 + ------- - --------- + ---------- ...   (13),
    [pi]             4i - 1    (4i - 1)³   (4i - 1)^5

  and those of J1(z) from

     z               .151982    .015399     .245835
    ---- = i + .25 - ------- + --------- + ---------- ...   (14),
    [pi]             4i + 1    (4i + 1)³   (4i + 1)^5

  formulae derived by Stokes (_Camb. Trans._, 1850, vol. ix.) from the
  descending series.[1] The following table gives the actual values:--

    +---+--------------------+--------------------+
    |   |   z                |  z                 |
    | i | ---- for J0(z) = 0 | ---- for J1(z) = 0 |
    |   | [pi]               | [pi]               |
    +---+--------------------+--------------------+
    | 1 |        7655        |       1 2197       |
    | 2 |      1 7571        |       2 2330       |
    | 3 |      2 7546        |       3 2383       |
    | 4 |      3 7534        |       4 2411       |
    | 5 |      4 7527        |       5 2428       |
    | 6 |      5 7522        |       6 2439       |
    | 7 |      6 7519        |       7 2448       |
    | 8 |      7 7516        |       8 2454       |
    | 9 |      8 7514        |       9 2459       |
    |10 |      9 7513        |      10 2463       |
    +---+--------------------+--------------------+

  In both cases the image of a mathematical point is thus a symmetrical
  ring system. The greatest brightness is at the centre, where

    dC = 2[pi][rho] d[rho], C = [pi]R².

  For a certain distance outwards this remains sensibly unimpaired and
  then gradually diminishes to zero, as the secondary waves become
  discrepant in phase. The subsequent revivals of brightness forming the
  bright rings are necessarily of inferior brilliancy as compared with
  the central disk.

  The first dark ring in the diffraction pattern of the complete
  circular aperture occurs when

    r/f = 1.2197 × [lambda]/2R                  (15).

  We may compare this with the corresponding result for a rectangular
  aperture of width a,

    [xi]/f =[lambda]/a;

  and it appears that in consequence of the preponderance of the central
  parts, the compensation in the case of the circle does not set in at
  so small an obliquity as when the circle is replaced by a rectangular
  aperture, whose side is equal to the diameter of the circle.

  Again, if we compare the complete circle with a narrow annular
  aperture of the same radius, we see that in the latter case the first
  dark ring occurs at a much smaller obliquity, viz.

    r/f = .7655 × [lambda]/2R.

  It has been found by Sir William Herschel and others that the
  definition of a telescope is often improved by stopping off a part of
  the central area of the object-glass; but the advantage to be obtained
  in this way is in no case great, and anything like a reduction of the
  aperture to a narrow annulus is attended by a development of the
  external luminous rings sufficient to outweigh any improvement due to
  the diminished diameter of the central area.[2]

  The maximum brightnesses and the places at which they occur are easily
  determined with the aid of certain properties of the Bessel's
  functions. It is known (see SPHERICAL HARMONICS) that

    J0'(z) = -J1(z),                    (16);

            1
    J2(z) = - J1(z) - J1'(z)            (17);
            z

                    2
    J0(z) + J2(z) = - J1(z)             (18).
                    z

  The maxima of C occur when

     d  /J1(z)\    J1'(z)   J1(z)
    -- (-------) = ------ - ----- = 0;
    dz  \  z  /      z        z²

  or by 17 when J2(z) = 0. When z has one of the values thus determined,

    2
    - J1(z) = J0(z).
    z

  The accompanying table is given by Lommel, in which the first column
  gives the roots of J2(z) = 0, and the second and third columns the
  corresponding values of the functions specified. If appears that the
  maximum brightness in the first ring is only about 1/57 of the
  brightness at the centre.

    +-------------------------------------------+
    |       z        2z^-1 J1(z)  4z^-2 J1²(z)  |
    +-------------------------------------------+
    |                                           |
    |    .000000     +1.000000      1.000000    |
    |   5.135630     - .132279       .017498    |
    |   8.417236     + .064482       .004158    |
    |  11.619857     - .040008       .001601    |
    |  14.795938     + .027919       .000779    |
    |  17.959820     - .020905       .000437    |
    +-------------------------------------------+

  We will now investigate the total illumination distributed over the
  area of the circle of radius r. We have

           [pi]²R^4     4J1²(z)
    I^2 = ----------- · -------                (19),
          [lambda]²f²     z²

  where

    z = 2[pi]Rr/[lambda]f                      (20).

  Thus
           _                     _                  _
          /         [lambda]²f² /                  /
    2[pi] | I²rdr = ----------- | I²zdz = [pi]R²·2 | z^-1 J1²(z)dz.
         _/           2[pi]R²  _/                 _/

  Now by (17), (18)

    z^-1 J1(z) = J0(z) - J1'(z);

  so that

                   d          d
    z^-1J1²(z) = ½ -- J0² - ½ -- J1²(z),
                   dz         dz

  and

       _z
      /
    2 | z^-1 J1²(z)dz = 1 - J0²(z) - J1²(z)    (21).
     _/0

  If r, or z, be infinite, J0(z), J1(z) vanish, and the whole
  illumination is expressed by [pi]R², in accordance with the general
  principle. In any case the proportion of the whole illumination to be
  found outside the circle of radius r is given by

    J0²(z) + J1²(z).

  For the dark rings J1(z) = 0; so that the fraction of illumination
  outside any dark ring is simply J0²(z). Thus for the first, second,
  third and fourth dark rings we get respectively .161, .090, .062,
  .047, showing that more than 9/10ths of the whole light is
  concentrated within the area of the second dark ring (_Phil. Mag._,
  1881).

  When z is great, the descending series (10) gives

    2J1(z)   2    / /  2  \
    ------ = -   / ( ----- ) sin(z - ¼[pi])    (22);
      z      z \/   \[pi]z/

  so that the places of maxima and minima occur at equal intervals.

  The mean brightness varies as z^-3 (or as r^-3), and the integral
  found by multiplying it by zdz and integrating between 0 and [oo]
  converges.

  It may be instructive to contrast this with the case of an infinitely
  narrow annular aperture, where the brightness is proportional to
  J0²(z). When z is great,

                /  2
    J0(z) = \  / ----- cos(z^-¼ [pi]).
             \/  [pi]z

  The mean brightness varies as z^-1; and the integral
      _
     / [oo]
     |   J0²(z)z dz  is not convergent.
    _/ 0

5. _Resolving Power of Telescopes._--The efficiency of a telescope is of
course intimately connected with the size of the disk by which it
represents a mathematical point. In estimating theoretically the
resolving power on a double star we have to consider the illumination of
the field due to the superposition of the two independent images. If the
angular interval between the components of a double star were equal to
twice that expressed in equation (15) above, the central disks of the
diffraction patterns would be just in contact. Under these conditions
there is no doubt that the star would appear to be fairly resolved,
since the brightness of its external ring system is too small to produce
any material confusion, unless indeed the components are of very unequal
magnitude. The diminution of the star disks with increasing aperture was
observed by Sir William Herschel, and in 1823 Fraunhofer formulated the
law of inverse proportionality. In investigations extending over a long
series of years, the advantage of a large aperture in separating the
components of close double stars was fully examined by W. R. Dawes.

The resolving power of telescopes was investigated also by J. B. L.
Foucault, who employed a scale of equal bright and dark alternate parts;
it was found to be proportional to the aperture and independent of the
focal length. In telescopes of the best construction and of moderate
aperture the performance is not sensibly prejudiced by outstanding
aberration, and the limit imposed by the finiteness of the waves of
light is practically reached. M. E. Verdet has compared Foucault's
results with theory, and has drawn the conclusion that the radius of the
visible part of the image of a luminous point was equal to half the
radius of the first dark ring.

The application, unaccountably long delayed, of this principle to the
microscope by H. L. F. Helmholtz in 1871 is the foundation of the
important doctrine of the _microscopic limit_. It is true that in 1823
Fraunhofer, inspired by his observations upon gratings, had very nearly
hit the mark.[3] And a little before Helmholtz, E. Abbe published a
somewhat more complete investigation, also founded upon the phenomena
presented by gratings. But although the argument from gratings is
instructive and convenient in some respects, its use has tended to
obscure the essential unity of the principle of the limit of resolution
whether applied to telescopes or microscopes.

  [Illustration: Fig. 4.]

  In fig. 4, AB represents the axis of an optical instrument (telescope
  or microscope), A being a point of the object and B a point of the
  image. By the operation of the object-glass LL' all the rays issuing
  from A arrive in the same phase at B. Thus if A be self-luminous, the
  illumination is a maximum at B, where all the secondary waves agree in
  phase. B is in fact the centre of the diffraction disk which
  constitutes the image of A. At neighbouring points the illumination is
  less, in consequence of the discrepancies of phase which there enter.
  In like manner if we take a neighbouring point P, also self-luminous,
  in the plane of the object, the waves which issue from it will arrive
  at B with phases no longer absolutely concordant, and the discrepancy
  of phase will increase as the interval AP increases. When the
  interval is very small the discrepancy, though mathematically
  existent, produces no practical effect; and the illumination at B due
  to P is as important as that due to A, the intensities of the two
  luminous sources being supposed equal. Under these conditions it is
  clear that A and P are not separated in the image. The question is to
  what amount must the distance AP be increased in order that the
  difference of situation may make itself felt in the image. This is
  necessarily a question of degree; but it does not require detailed
  calculations in order to show that the discrepancy first becomes
  conspicuous when the phases corresponding to the various secondary
  waves which travel from P to B range over a complete period. The
  illumination at B due to P then becomes comparatively small, indeed
  for some forms of aperture evanescent. The extreme discrepancy is that
  between the waves which travel through the outermost parts of the
  object-glass at L and L'; so that if we adopt the above standard of
  resolution, the question is where must P be situated in order that the
  relative retardation of the rays PL and PL' may on their arrival at B
  amount to a wave-length ([lambda]). In virtue of the general law that
  the reduced optical path is stationary in value, this retardation may
  be calculated without allowance for the different paths pursued on the
  farther side of L, L', so that the value is simply PL - PL'. Now since
  AP is very small, AL' - PL' = AP sin [alpha], where [alpha] is the
  angular semi-aperture L'AB. In like manner PL - AL has the same value,
  so that

    PL - PL' = 2AP sin [alpha].

  According to the standard adopted, the condition of resolution is
  therefore that AP, or [epsilon], should exceed ½[lambda]/sin [alpha].
  If [epsilon] be less than this, the images overlap too much; while if
  [epsilon] greatly exceed the above value the images become
  unnecessarily separated.

  In the above argument the whole space between the object and the lens
  is supposed to be occupied by matter of one refractive index, and
  [lambda] represents the wave-length _in this medium_ of the kind of
  light employed. If the restriction as to uniformity be violated, what
  we have ultimately to deal with is the wave-length in the medium
  immediately surrounding the object.

  Calling the refractive index [mu], we have as the critical value of
  [epsilon],

    [epsilon] = ½[lambda]0/[mu] sin[alpha],      (1),

  [lambda]0 being the wave-length _in vacuo_. The denominator [mu] sin
  [alpha] is the quantity well known (after Abbe) as the "numerical
  aperture."

  The extreme value possible for [alpha] is a right angle, so that for
  the microscopic limit we have

    [epsilon] = ½[lambda]0/[mu]                  (2).

  The limit can be depressed only by a diminution in [lambda]0, such as
  photography makes possible, or by an increase in [mu], the refractive
  index of the medium in which the object is situated.

  The statement of the law of resolving power has been made in a form
  appropriate to the microscope, but it admits also of immediate
  application to the telescope. If 2R be the diameter of the
  object-glass and D the distance of the object, the angle subtended by
  AP is [epsilon]/D, and the angular resolving power is given by

    [lambda]/2D sin[alpha] = [lambda]/2R         (3).

  This method of derivation (substantially due to Helmholtz) makes it
  obvious that there is no essential difference of principle between the
  two cases, although the results are conveniently stated in different
  forms. In the case of the telescope we have to deal with a linear
  measure of aperture and an angular limit of resolution, whereas in the
  case of the microscope the limit of resolution is linear, and it is
  expressed in terms of angular aperture.

  It must be understood that the above argument distinctly assumes that
  the different parts of the object are self-luminous, or at least that
  the light proceeding from the various points is without phase
  relations. As has been emphasized by G. J. Stoney, the restriction is
  often, perhaps usually, violated in the microscope. A different
  treatment is then necessary, and for some of the problems which arise
  under this head the method of Abbe is convenient.

  The importance of the general conclusions above formulated, as
  imposing a limit upon our powers of direct observation, can hardly be
  overestimated; but there has been in some quarters a tendency to
  ascribe to it a more precise character than it can bear, or even to
  mistake its meaning altogether. A few words of further explanation may
  therefore be desirable. The first point to be emphasized is that
  nothing whatever is said as to the smallness of a single object that
  may be made visible. The eye, unaided or armed with a telescope, is
  able to see, as points of light, stars subtending no sensible angle.
  The visibility of a star is a question of brightness simply, and has
  nothing to do with resolving power. The latter element enters only
  when it is a question of recognizing the duplicity of a double star,
  or of distinguishing detail upon the surface of a planet. So in the
  microscope there is nothing except lack of light to hinder the
  visibility of an object however small. But if its dimensions be much
  less than the half wave-length, it can only be seen as a whole, and
  its parts cannot be distinctly separated, although in cases near the
  border line some inference may be possible, founded upon experience of
  what appearances are presented in various cases. Interesting
  observations upon particles, _ultra-microscopic_ in the above sense,
  have been recorded by H. F. W. Siedentopf and R. A. Zsigmondy
  (_Drude's Ann._, 1903, 10, p. 1).

  In a somewhat similar way a dark linear interruption in a bright
  ground may be visible, although its actual width is much inferior to
  the half wave-length. In illustration of this fact a simple experiment
  may be mentioned. In front of the naked eye was held a piece of copper
  foil perforated by a fine needle hole. Observed through this the
  structure of some wire gauze just disappeared at a distance from the
  eye equal to 17 in., the gauze containing 46 meshes to the inch. On
  the other hand, a single wire 0.034 in. in diameter remained fairly
  visible up to a distance of 20 ft. The ratio between the limiting
  angles subtended by the periodic structure of the gauze and the
  diameter of the wire was (.022/.034) × (240/17) = 9.1. For further
  information upon this subject reference may be made to _Phil. Mag._,
  1896, 42, p. 167; _Journ. R. Micr. Soc._, 1903, p. 447.

6. _Coronas or Glories._--The results of the theory of the diffraction
patterns due to circular apertures admit of an interesting application
to _coronas_, such as are often seen encircling the sun and moon. They
are due to the interposition of small spherules of water, which act the
part of diffracting obstacles. In order to the formation of a
well-defined corona it is essential that the particles be exclusively,
or preponderatingly, of one size.

  If the origin of light be treated as infinitely small, and be seen in
  focus, whether with the naked eye or with the aid of a telescope, the
  whole of the light in the absence of obstacles would be concentrated
  in the immediate neighbourhood of the focus. At other parts of the
  field the effect is the same, in accordance with the principle known
  as Babinet's, whether the imaginary screen in front of the
  object-glass is generally transparent but studded with a number of
  opaque circular disks, or is generally opaque but perforated with
  corresponding apertures. Since at these points the resultant due to
  the whole aperture is zero, any two portions into which the whole may
  be divided must give equal and opposite resultants. Consider now the
  light diffracted in a direction many times more oblique than any with
  which we should be concerned, were the whole aperture uninterrupted,
  and take first the effect of a single small aperture. The light in the
  proposed direction is that determined by the size of the small
  aperture in accordance with the laws already investigated, and its
  phase depends upon the position of the aperture. If we take a
  direction such that the light (of given wave-length) from a single
  aperture vanishes, the evanescence continues even when the whole
  series of apertures is brought into contemplation. Hence, whatever
  else may happen, there must be a system of dark rings formed, the same
  as from a single small aperture. In directions other than these it is
  a more delicate question how the partial effects should be compounded.
  If we make the extreme suppositions of an infinitely small source and
  absolutely homogeneous light, there is no escape from the conclusion
  that the light in a definite direction is arbitrary, that is,
  dependent upon the chance distribution of apertures. If, however, as
  in practice, the light be heterogeneous, the source of finite area,
  the obstacles in motion, and the discrimination of different
  directions imperfect, we are concerned merely with the mean brightness
  found by varying the arbitrary phase-relations, and this is obtained
  by simply multiplying the brightness due to a single aperture by the
  number of apertures (n) (see INTERFERENCE OF LIGHT, § 4). The
  diffraction pattern is therefore that due to a single aperture, merely
  brightened n times.

  In his experiments upon this subject Fraunhofer employed plates of
  glass dusted over with lycopodium, or studded with small metallic
  disks of uniform size; and he found that the diameters of the rings
  were proportional to the length of the waves and inversely as the
  diameter of the disks.

  In another respect the observations of Fraunhofer appear at first
  sight to be in disaccord with theory; for his measures of the
  diameters of the red rings, visible when white light was employed,
  correspond with the law applicable to dark rings, and not to the
  different law applicable to the luminous maxima. Verdet has, however,
  pointed out that the observation in this form is essentially different
  from that in which homogeneous red light is employed, and that the
  position of the red rings would correspond to the _absence_ of
  blue-green light rather than to the greatest abundance of red light.
  Verdet's own observations, conducted with great care, fully confirm
  this view, and exhibit a complete agreement with theory.

  By measurements of coronas it is possible to infer the size of the
  particles to which they are due, an application of considerable
  interest in the case of natural coronas--the general rule being the
  larger the corona the smaller the water spherules. Young employed this
  method not only to determine the diameters of cloud particles (e.g.
  1/1000 in.), but also those of fibrous material, for which the theory
  is analogous. His instrument was called the _eriometer_ (see
  "Chromatics," vol. iii. of supp. to _Ency. Brit._, 1817).

7. _Influence of Aberration. Optical Power of Instruments._--Our
investigations and estimates of resolving power have thus far proceeded
upon the supposition that there are no optical imperfections, whether of
the nature of a regular aberration or dependent upon irregularities of
material and workmanship. In practice there will always be a certain
aberration or error of phase, which we may also regard as the deviation
of the actual wave-surface from its intended position. In general, we
may say that aberration is unimportant when it nowhere (or at any rate
over a relatively small area only) exceeds a small fraction of the
wave-length ([lamda]). Thus in estimating the intensity at a focal point,
where, in the absence of aberration, all the secondary waves would have
exactly the same phase, we see that an aberration nowhere exceeding
¼[lambda] can have but little effect.

  The only case in which the influence of small aberration upon the
  entire image has been calculated (_Phil. Mag._, 1879) is that of a
  rectangular aperture, traversed by a cylindrical wave with aberration
  equal to cx³. The aberration is here unsymmetrical, the wave being in
  advance of its proper place in one half of the aperture, but behind in
  the other half. No terms in x or x² need be considered. The first
  would correspond to a general turning of the beam; and the second
  would imply imperfect focusing of the central parts. The effect of
  aberration may be considered in two ways. We may suppose the aperture
  (a) constant, and inquire into the operation of an increasing
  aberration; or we may take a given value of c (i.e. a given
  wave-surface) and examine the effect of a varying aperture. The
  results in the second case show that an increase of aperture up to
  that corresponding to an extreme aberration of half a period has no
  ill effect upon the central band (§ 3), but it increases unduly the
  intensity of one of the neighbouring lateral bands; and the practical
  conclusion is that the best results will be obtained from an aperture
  giving an extreme aberration of from a quarter to half a period, and
  that with an increased aperture aberration is not so much a direct
  cause of deterioration as an obstacle to the attainment of that
  improved definition which should accompany the increase of aperture.

  If, on the other hand, we suppose the aperture given, we find that
  aberration begins to be distinctly mischievous when it amounts to
  about a quarter period, i.e. when the wave-surface deviates at each
  end by a quarter wave-length from the true plane.

  As an application of this result, let us investigate what amount of
  temperature disturbance in the tube of a telescope may be expected to
  impair definition. According to J. B. Biot and F. J. D. Arago, the
  index [mu] for air at t° C. and at atmospheric pressure is given by

                .00029
    [mu] - 1 = -----------.
               1 + .0037 t

  If we take 0° C. as standard temperature,

    [delta][mu] = -1.1 × 10^-6.

  Thus, on the supposition that the irregularity of temperature t
  extends through a length l, and produces an acceleration of a quarter
  of a wave-length,

    ¼[lambda] = 1.1 lt × 10^-6;

  or, if we take [lambda] = 5.3 × 10^-5,

    lt = 12,

  the unit of length being the centimetre.

  We may infer that, in the case of a telescope tube 12 cm. long, a
  stratum of air heated 1° C. lying along the top of the tube, and
  occupying a moderate fraction of the whole volume, would produce a not
  insensible effect. If the change of temperature progressed uniformly
  from one side to the other, the result would be a lateral displacement
  of the image without loss of definition; but in general both effects
  would be observable. In longer tubes a similar disturbance would be
  caused by a proportionally less difference of temperature. S. P.
  Langley has proposed to obviate such ill-effects by stirring the air
  included within a telescope tube. It has long been known that the
  definition of a carbon bisulphide prism may be much improved by a
  vigorous shaking.

  We will now consider the application of the principle to the formation
  of images, unassisted by reflection or refraction (_Phil. Mag._,
  1881). The function of a lens in forming an image is to compensate by
  its variable thickness the differences of phase which would otherwise
  exist between secondary waves arriving at the focal point from various
  parts of the aperture. If we suppose the diameter of the lens to be
  given (2R), and its focal length f gradually to increase, the original
  differences of phase at the image of an infinitely distant luminous
  point diminish without limit. When f attains a certain value, say f1,
  the extreme error of phase to be compensated falls to ¼[lambda]. But,
  as we have seen, such an error of phase causes no sensible
  deterioration in the definition; so that from this point onwards the
  lens is useless, as only improving an image already sensibly as
  perfect as the aperture admits of. Throughout the operation of
  increasing the focal length, the resolving power of the instrument,
  which depends only upon the aperture, remains unchanged; and we thus
  arrive at the rather startling conclusion that a telescope of any
  degree of resolving power might be constructed without an
  object-glass, if only there were no limit to the admissible focal
  length. This last proviso, however, as we shall see, takes away almost
  all practical importance from the proposition.

  To get an idea of the magnitudes of the quantities involved, let us
  take the case of an aperture of 1/5 in., about that of the pupil of
  the eye. The distance f1, which the actual focal length must exceed,
  is given by

      /
    \/ (f1² + R²) - f1  = ¼[lambda];

  so that

    f1  = 2R²/[lambda]                        (1).

  Thus, if [lambda] = 1/4000, R = 1/10, we find

    f1 = 800 inches.

  The image of the sun thrown upon a screen at a distance exceeding 66
  ft., through a hole 1/5 in. in diameter, is therefore at least as well
  defined as that seen direct.

  As the minimum focal length increases with the square of the aperture,
  a quite impracticable distance would be required to rival the
  resolving power of a modern telescope. Even for an aperture of 4 in.,
  f1 would have to be 5 miles.

  A similar argument may be applied to find at what point an achromatic
  lens becomes sensibly superior to a single one. The question is
  whether, when the adjustment of focus is correct for the central rays
  of the spectrum, the error of phase for the most extreme rays (which
  it is necessary to consider) amounts to a quarter of a wave-length. If
  not, the substitution of an achromatic lens will be of no advantage.
  Calculation shows that, if the aperture be 1/5 in., an achromatic lens
  has no sensible advantage if the focal length be greater than about 11
  in. If we suppose the focal length to be 66 ft., a single lens is
  practically perfect up to an aperture of 1.7 in.

  Another obvious inference from the necessary imperfection of optical
  images is the uselessness of attempting anything like an absolute
  destruction of spherical aberration. An admissible error of phase of
  ¼[lambda] will correspond to an error of 1/8[lambda] in a reflecting
  and ½[lambda] in a (glass) refracting surface, the incidence in both
  cases being perpendicular. If we inquire what is the greatest
  admissible longitudinal aberration ([delta]f) in an object-glass
  according to the above rule, we find

    [delta]f = [lambda][alpha]^-2               (2),

  [alpha] being the angular semi-aperture.

  In the case of a single lens of glass with the most favourable
  curvatures, [delta]f is about equal to [alpha]²f, so that [alpha]^4
  must not exceed [lambda]/f. For a lens of 3 ft. focus this condition
  is satisfied if the aperture does not exceed 2 in.

  When parallel rays fall directly upon a spherical mirror the
  longitudinal aberration is only about one-eighth as great as for the
  most favourably shaped single lens of equal focal length and aperture.
  Hence a spherical mirror of 3 ft. focus might have an aperture of 2½
  in., and the image would not suffer materially from aberration.

  On the same principle we may estimate the least visible displacement
  of the eye-piece of a telescope focused upon a distant object, a
  question of interest in connexion with range-finders. It appears
  (_Phil. Mag._, 1885, 20, p. 354) that a displacement [delta]f from the
  true focus will not sensibly impair definition, provided

    [delta]f < f²[lambda]/R²                    (3),

  2R being the diameter of aperture. The linear accuracy required is
  thus a function of the _ratio_ of aperture to focal length. The
  formula agrees well with experiment.

  The principle gives an instantaneous solution of the question of the
  ultimate optical efficiency in the method of "mirror-reading," as
  commonly practised in various physical observations. A rotation by
  which one edge of the mirror advances ¼[lambda] (while the other edge
  retreats to a like amount) introduces a phase-discrepancy of a whole
  period where before the rotation there was complete agreement. A
  rotation of this amount should therefore be easily visible, but the
  limits of resolving power are being approached; and the conclusion is
  independent of the focal length of the mirror, and of the employment
  of a telescope, provided of course that the reflected image is seen in
  focus, and that the full width of the mirror is utilized.

  A comparison with the method of a material pointer, attached to the
  parts whose rotation is under observation, and viewed through a
  microscope, is of interest. The limiting efficiency of the microscope
  is attained when the angular aperture amounts to 180°; and it is
  evident that a lateral displacement of the point under observation
  through ½[lambda] entails (at the old image) a phase-discrepancy of a
  whole period, one extreme ray being accelerated and the other retarded
  by half that amount. We may infer that the limits of efficiency in the
  two methods are the same when the length of the pointer is equal to
  the width of the mirror.

  [Illustraton: FIG. 5.]

  We have seen that in perpendicular reflection a surface error not
  exceeding 1/8[lambda] may be admissible. In the case of oblique
  reflection at an angle [phi], the error of retardation due to an
  elevation BD (fig. 5) is

    QQ' - QS = BD sec [phi](1 - cos SQQ') = BD sec [phi] (1 + cos 2[phi]) = 2BD cos [phi];

  from which it follows that an error of given magnitude in the figure
  of a surface is less important in oblique than in perpendicular
  reflection. It must, however, be borne in mind that errors can
  sometimes be compensated by altering adjustments. If a surface
  intended to be flat is affected with a slight general curvature, a
  remedy may be found in an alteration of focus, and the remedy is the
  less complete as the reflection is more oblique.

  The formula expressing the optical power of prismatic spectroscopes
  may readily be investigated upon the principles of the wave theory.
  Let A0B0 be a plane wave-surface of the light before it falls upon the
  prisms, AB the corresponding wave-surface for a particular part of the
  spectrum after the light has passed the prisms, or after it has passed
  the eye-piece of the observing telescope. The path of a ray from the
  wave-surface A0B0 to A or B is determined by the condition that the
  optical distance, [int] [mu]ds, is a minimum; and, as AB is by
  supposition a wave-surface, this optical distance is the same for both
  points. Thus
      _                  _
     /                  /
     | [mu]ds (for A) = | [mu]ds (for B)          (4).
    _/                 _/

  We have now to consider the behaviour of light belonging to a
  neighbouring part of the spectrum. The path of a ray from the
  wave-surface A0B0 to the point A is changed; but in virtue of the
  minimum property the change may be neglected in calculating the
  optical distance, as it influences the result by quantities of the
  second order only in the changes of refrangibility. Accordingly, the
  optical distance from A0B0 to A is represented by [int]([mu] +
  [delta][mu])ds, the integration being along the original path A0 ...
  A; and similarly the optical distance between A0B0 and B is
  represented by [int] ([mu] + [delta][mu])ds, the integration being
  along B0 ... B. In virtue of (4) the difference of the optical
  distances to A and B is
      _                                  _
     /                                  /
     | [delta][mu]ds (along B0 ... B) - | [delta][mu]ds (along A0 ... A)  (5).
    _/                                 _/

  The new wave-surface is formed in such a position that the optical
  distance is constant; and therefore the _dispersion_, or the angle
  through which the wave-surface is turned by the change of
  refrangibility, is found simply by dividing (5) by the distance AB.
  If, as in common flint-glass spectroscopes, there is only one
  dispersing substance, [int] [delta][mu] ds = [delta][mu]·s, where s is
  simply the thickness traversed by the ray. If t2 and t1 be the
  thicknesses traversed by the extreme rays, and a denote the width of
  the emergent beam, the dispersion [theta] is given by

    [theta] = [delta][mu](t2 - t1)/a,

  or, if t1 be negligible,

    [theta] = [delta][mu]t/a                 (6).

  The condition of resolution of a double line whose components subtend
  an angle [theta] is that [theta] must exceed [lambda]/a. Hence, in
  order that a double line may be resolved whose components have indices
  [mu] and [mu] + [delta][mu], it is necessary that t should exceed the
  value given by the following equation:--

    t = [lambda]/[delta][mu]                 (7).

8. _Diffraction Gratings._--Under the heading "Colours of Striated
Surfaces," Thomas Young (_Phil. Trans._, 1802) in his usual summary
fashion gave a general explanation of these colours, including the law
of sines, the striations being supposed to be straight, parallel and
equidistant. Later, in his article "Chromatics" in the supplement to the
5th edition of this encyclopaedia, he shows that the colours "lose the
mixed character of periodical colours, and resemble much more the
ordinary prismatic spectrum, with intervals completely dark interposed,"
and explains it by the consideration that any phase-difference which may
arise at neighbouring striae is multiplied in proportion to the total
number of striae.

The theory was further developed by A. J. Fresnel (1815), who gave a
formula equivalent to (5) below. But it is to J. von Fraunhofer that we
owe most of our knowledge upon this subject. His recent discovery of the
"fixed lines" allowed a precision of observation previously impossible.
He constructed gratings up to 340 periods to the inch by straining fine
wire over screws. Subsequently he ruled gratings on a layer of gold-leaf
attached to glass, or on a layer of grease similarly supported, and
again by attacking the glass itself with a diamond point. The best
gratings were obtained by the last method, but a suitable diamond point
was hard to find, and to preserve. Observing through a telescope with
light perpendicularly incident, he showed that the position of any ray
was dependent only upon the grating interval, viz. the distance from the
centre of one wire or line to the centre of the next, and not otherwise
upon the thickness of the wire and the magnitude of the interspace. In
different gratings the lengths of the spectra and their distances from
the axis were inversely proportional to the grating interval, while with
a given grating the distances of the various spectra from the axis were
as 1, 2, 3, &c. To Fraunhofer we owe the first accurate measurements of
wave-lengths, and the method of separating the overlapping spectra by a
prism dispersing in the perpendicular direction. He described also the
complicated patterns seen when a point of light is viewed through two
superposed gratings, whose lines cross one another perpendicularly or
obliquely. The above observations relate to transmitted light, but
Fraunhofer extended his inquiry to the light _reflected_. To eliminate
the light returned from the hinder surface of an engraved grating, he
covered it with a black varnish. It then appeared that under certain
angles of incidence parts of the resulting spectra were _completely
polarized_. These remarkable researches of Fraunhofer, carried out in
the years 1817-1823, are republished in his _Collected Writings_
(Munich, 1888).

  The principle underlying the action of gratings is identical with that
  discussed in § 2, and exemplified in J. L. Soret's "zone plates." The
  alternate Fresnel's zones are blocked out or otherwise modified; in
  this way the original compensation is upset and a revival of light
  occurs in unusual directions. If the source be a point or a line, and
  a collimating lens be used, the incident waves may be regarded as
  plane. If, further, on leaving the grating the light be received by a
  focusing lens, e.g. the object-glass of a telescope, the Fresnel's
  zones are reduced to parallel and equidistant straight strips, which
  at certain angles coincide with the ruling. The directions of the
  lateral spectra are such that the passage from one element of the
  grating to the corresponding point of the next implies a retardation
  of an integral number of wave-lengths. If the grating be composed of
  alternate transparent and opaque parts, the question may be treated by
  means of the general integrals (§ 3) by merely limiting the
  integration to the transparent parts of the aperture. For an
  investigation upon these lines the reader is referred to Airy's
  _Tracts_, to Verdet's _Leçons_, or to R. W. Wood's _Physical Optics_.
  If, however, we assume the theory of a simple rectangular aperture (§
  3); the results of the ruling can be inferred by elementary methods,
  which are perhaps more instructive.

  Apart from the ruling, we know that the image of a mathematical line
  will be a series of narrow bands, of which the central one is by far
  the brightest. At the middle of this band there is complete agreement
  of phase among the secondary waves. The dark lines which separate the
  bands are the places at which the phases of the secondary wave range
  over an integral number of periods. If now we suppose the aperture AB
  to be covered by a great number of opaque strips or bars of width d,
  separated by transparent intervals of width a, the condition of things
  in the directions just spoken of is not materially changed. At the
  central point there is still complete agreement of phase; but the
  amplitude is diminished in the ratio of a : a + d. In another
  direction, making a small angle with the last, such that the
  projection of AB upon it amounts to a few wave-lengths, it is easy to
  see that the mode of interference is the same as if there were no
  ruling. For example, when the direction is such that the projection of
  AB upon it amounts to one wave-length, the elementary components
  neutralize one another, because their phases are distributed
  symmetrically, though discontinuously, round the entire period. The
  only effect of the ruling is to diminish the amplitude in the ratio a
  : a + d; and, except for the difference in illumination, the
  appearance of a line of light is the same as if the aperture were
  perfectly free.

  The lateral (spectral) images occur in such directions that the
  projection of the element (a + d) of the grating upon them is an exact
  multiple of [lambda]. The effect of each of the n elements of the
  grating is then the same; and, unless this vanishes on account of a
  particular adjustment of the ratio a : d, the resultant amplitude
  becomes comparatively very great. These directions, in which the
  retardation between A and B is exactly mn[lambda], may be called the
  principal directions. On either side of any one of them the
  illumination is distributed according to the same law as for the
  central image (m = 0), vanishing, for example, when the retardation
  amounts to (mn ± 1)[lambda]. In considering the relative brightnesses
  of the different spectra, it is therefore sufficient to attend merely
  to the principal directions, provided that the whole deviation be not
  so great that its cosine differs considerably from unity.

  We have now to consider the amplitude due to a single element, which
  we may conveniently regard as composed of a transparent part a bounded
  by two opaque parts of width ½d. The phase of the resultant effect is
  by symmetry that of the component which comes from the middle of a.
  The fact that the other components have phases differing from this by
  amounts ranging between ± am[pi]/(a + d) causes the resultant
  amplitude to be less than for the central image (where there is
  complete phase agreement). If Bm denote the brightness of the m^th
  lateral image, and B0 that of the central image, we have

                _    _+ am[pi]/(a + d)       _
               |    /                 2am[pi] |²   /a + d \²      am[pi]
    B_m : B0 = |    |       cosx dx ÷ ------- | = ( ------ ) sin² ------ (1).
               |_  _/                  a + d _|    \am[pi]/       a + d
                     -am[pi]/(a + d)

  If B denotes the brightness of the central image when the whole of the
  space occupied by the grating is transparent, we have

    B0 : B = a² : (a + d)²,

  and thus

                1         am[pi]
    Bm : B = ------- sin² ------          (2).
             m²[pi]²      a + d

  The sine of an angle can never be greater than unity; and consequently
  under the most favourable circumstances only 1/m²[pi]² of the original
  light can be obtained in the m^th spectrum. We conclude that, with a
  grating composed of transparent and opaque parts, the utmost light
  obtainable in any one spectrum is in the first, and there amounts to
  1/[pi]², or about 1/10, and that for this purpose a and d must be
  equal. When d = a the general formula becomes

              sin² ½m[pi]
    Bm : B =  -----------                 (3),
                m²[pi]²

  showing that, when m is even, Bm vanishes, and that, when m is odd,

    Bm : B = 1/m²[pi]².

  The third spectrum has thus only 1/9 of the brilliancy of the first.

  Another particular case of interest is obtained by supposing a small
  relatively to (a + d). Unless the spectrum be of very high order, we
  have simply

    Bm : B = a/(a + d)²                   (4);

  so that the brightnesses of all the spectra are the same.

  The light stopped by the opaque parts of the grating, together with
  that distributed in the central image and lateral spectra, ought to
  make up the brightness that would be found in the central image, were
  all the apertures transparent. Thus, if a = d, we should have

        1   1     2    /    1    1      \
    1 = - + - + ----- ( 1 + - + -- + ... ),
        2   4   [pi]²  \    9   25      /

  which is true by a known theorem. In the general case

                              ___m=[oo]
      a      /  a  \²     2   \   1       /m[pi]a\
    ----- = ( ----- ) + -----  >  -- sin²( ------ ),
    a + d    \a + d/    [pi]² /__ m²      \ a + d/
                               m=1

  a formula which may be verified by Fourier's theorem.

  According to a general principle formulated by J. Babinet, the
  brightness of a lateral spectrum is not affected by an interchange of
  the transparent and opaque parts of the grating. The vibrations
  corresponding to the two parts are precisely antagonistic, since if
  both were operative the resultant would be zero. So far as the
  application to gratings is concerned, the same conclusion may be
  derived from (2).

  [Illustration: FIG. 6.]

  From the value of Bm : B0 we see that no lateral spectrum can surpass
  the central image in brightness; but this result depends upon the
  hypothesis that the ruling acts by opacity, which is generally very
  far from being the case in practice. In an engraved glass grating
  there is no opaque material present by which light could be absorbed,
  and the effect depends upon a difference of retardation in passing the
  alternate parts. It is possible to prepare gratings which give a
  lateral spectrum brighter than the central image, and the explanation
  is easy. For if the alternate parts were equal and alike transparent,
  but so constituted as to give a relative retardation of ½[lambda], it
  is evident that the central image would be entirely extinguished,
  while the first spectrum would be four times as bright as if the
  alternate parts were opaque. If it were possible to introduce at every
  part of the aperture of the grating an arbitrary retardation, all the
  light might be concentrated in any desired spectrum. By supposing the
  retardation to vary uniformly and continuously we fall upon the case
  of an ordinary prism: but there is then no diffraction spectrum in the
  usual sense. To obtain such it would be necessary that the retardation
  should gradually alter by a wave-length in passing over any element of
  the grating, and then fall back to its previous value, thus springing
  suddenly over a wave-length (_Phil. Mag._, 1874, 47, p. 193). It is
  not likely that such a result will ever be fully attained in practice;
  but the case is worth stating, in order to show that there is no
  theoretical limit to the concentration of light of assigned
  wave-length in one spectrum, and as illustrating the frequently
  observed unsymmetrical character of the spectra on the two sides of
  the central image.[4]

  We have hitherto supposed that the light is incident perpendicularly
  upon the grating; but the theory is easily extended. If the incident
  rays make an angle [theta] with the normal (fig. 6), and the
  diffracted rays make an angle [phi] (upon the same side), the relative
  retardation from each element of width (a + d) to the next is (a + d)
  (sin[theta] + sin[phi]); and this is the quantity which is to be
  equated to m[lambda]. Thus

    sin[theta] + sin[phi] = 2 sin ½([theta] + [phi]) cos ½([theta] - [phi]) = m[lambda]/(a + d) (5).

  The "deviation" is ([theta] + [phi]), and is therefore a minimum when
  [theta] = [phi], i.e. when the grating is so situated that the angles
  of incidence and diffraction are equal.

  In the case of a reflection grating the same method applies. If
  [theta] and [phi] denote the angles with the normal made by the
  incident and diffracted rays, the formula (5) still holds, and, if the
  deviation be reckoned from the direction of the regularly reflected
  rays, it is expressed as before by ([theta] + [phi]), and is a minimum
  when [theta] = [phi], that is, when the diffracted rays return upon
  the course of the incident rays.

  [Illustration: FIG. 7.]

  In either case (as also with a prism) the position of minimum
  deviation leaves the width of the beam unaltered, i.e. neither
  magnifies nor diminishes the angular width of the object under view.

  From (5) we see that, when the light falls perpendicularly upon a
  grating ([theta] = 0), there is no spectrum formed (the image
  corresponding to m = 0 not being counted as a spectrum), if the
  grating interval [sigma] or (a + d) is less than [lambda]. Under these
  circumstances, if the material of the grating be completely
  transparent, the whole of the light must appear in the direct image,
  and the ruling is not perceptible. From the absence of spectra
  Fraunhofer argued that there must be a microscopic limit represented
  by [lambda]; and the inference is plausible, to say the least (_Phil.
  Mag._, 1886). Fraunhofer should, however, have fixed the microscopic
  limit at ½[lambda], as appears from (5), when we suppose [theta] =
  ½[pi], [phi] = ½[pi].

  [Illustration: FIG. 8.]

  We will now consider the important subject of the resolving power of
  gratings, as dependent upon the number of lines (n) and the order of
  the spectrum observed (m). Let BP (fig. 8) be the direction of the
  principal maximum (middle of central band) for the wave-length
  [lambda] in the m^th spectrum. Then the relative retardation of the
  extreme rays (corresponding to the edges A, B of the grating) is
  mn[lambda]. If BQ be the direction for the first minimum (the darkness
  between the central and first lateral band), the relative retardation
  of the extreme rays is (mn + 1)[lambda]. Suppose now that [lambda] +
  [delta][lambda] is the wave-length for which BQ gives the principal
  maximum, then

    (mn + 1)[lambda] = mn([lambda] + [delta][lambda]);

  whence

    [delta][lambda]/[lambda] = 1/mn            (6).

  According to our former standard, this gives the smallest difference
  of wave-lengths in a double line which can be just resolved; and we
  conclude that the resolving power of a grating depends only upon the
  total number of lines, and upon the order of the spectrum, without
  regard to any other considerations. It is here of course assumed that
  the n lines are really utilized.

  In the case of the D lines the value of [delta][lambda]/[lambda] is
  about 1/1000; so that to resolve this double line in the first
  spectrum requires 1000 lines, in the second spectrum 500, and so on.

  It is especially to be noticed that the resolving power does not
  depend directly upon the closeness of the ruling. Let us take the case
  of a grating 1 in. broad, and containing 1000 lines, and consider the
  effect of interpolating an additional 1000 lines, so as to bisect the
  former intervals. There will be destruction by interference of the
  first, third and odd spectra generally; while the advantage gained in
  the spectra of even order is not in dispersion, nor in resolving
  power, but simply in brilliancy, which is increased four times. If we
  now suppose half the grating cut away, so as to leave 1000 lines in
  half an inch, the dispersion will not be altered, while the brightness
  and resolving power are halved.

  There is clearly no theoretical limit to the resolving power of
  gratings, even in spectra of given order. But it is possible that, as
  suggested by Rowland,[5] the structure of natural spectra may be too
  coarse to give opportunity for resolving powers much higher than those
  now in use. However this may be, it would always be possible, with the
  aid of a grating of given resolving power, to construct artificially
  from white light mixtures of slightly different wave-length whose
  resolution or otherwise would discriminate between powers inferior and
  superior to the given one.[6]

  If we define as the "dispersion" in a particular part of the spectrum
  the ratio of the angular interval d[theta] to the corresponding
  increment of wave-length d[lambda], we may express it by a very simple
  formula. For the alteration of wave-length entails, at the two limits
  of a diffracted wave-front, a relative retardation equal to
  mnd[lambda]. Hence, if a be the width of the diffracted beam, and
  d[theta] the angle through which the wave-front is turned,

    ad[theta] = mn d[lambda],

  or  dispersion = mn/a                       (7).

  The resolving power and the width of the emergent beam fix the optical
  character of the instrument. The latter element must eventually be
  decreased until less than the diameter of the pupil of the eye. Hence
  a wide beam demands treatment with further apparatus (usually a
  telescope) of high magnifying power.

  In the above discussion it has been supposed that the ruling is
  accurate, and we have seen that by increase of m a high resolving
  power is attainable with a moderate number of lines. But this
  procedure (apart from the question of illumination) is open to the
  objection that it makes excessive demands upon accuracy. According to
  the principle already laid down it can make but little difference in
  the principal direction corresponding to the first spectrum, provided
  each line lie within a quarter of an interval (a + d) from its
  theoretical position. But, to obtain an equally good result in the
  m^th spectrum, the error must be less than 1/m of the above amount.[7]

  There are certain errors of a systematic character which demand
  special consideration. The spacing is usually effected by means of a
  screw, to each revolution of which corresponds a large number (e.g.
  one hundred) of lines. In this way it may happen that although there
  is almost perfect periodicity with each revolution of the screw after
  (say) 100 lines, yet the 100 lines themselves are not equally spaced.
  The "ghosts" thus arising were first described by G. H. Quincke
  (_Pogg. Ann._, 1872, 146, p. 1), and have been elaborately
  investigated by C. S. Peirce (_Ann. Journ. Math._, 1879, 2, p. 330),
  both theoretically and experimentally. The general nature of the
  effects to be expected in such a case may be made clear by means of an
  illustration already employed for another purpose. Suppose two similar
  and accurately ruled transparent gratings to be superposed in such a
  manner that the lines are parallel. If the one set of lines exactly
  bisect the intervals between the others, the grating interval is
  practically halved, and the previously existing spectra of odd order
  vanish. But a very slight relative displacement will cause the
  apparition of the odd spectra. In this case there is approximate
  periodicity in the half interval, but complete periodicity only after
  the whole interval. The advantage of approximate bisection lies in the
  superior brilliancy of the surviving spectra; but in any case the
  compound grating may be considered to be perfect in the longer
  interval, and the definition is as good as if the bisection were
  accurate.

  [Illustration:

     | | | | |         ( ( (            | |  |   |         ) | (

    FIG. 9.--x².    FIG. 10.--y².     FIG. 11.--x³.    FIG. 12.--xy².

                                        /    /    /
      \ | | /           |  \  |          |    |    |
                                        /    /    /

    FIG. 13.--xy.   FIG. 14.--x²y.    FIG. 15.--y³.]

  The effect of a gradual increase in the interval (fig. 9) as we pass
  across the grating has been investigated by M. A. Cornu (_C.R._, 1875,
  80, p. 655), who thus explains an anomaly observed by E. E. N.
  Mascart. The latter found that certain gratings exercised a converging
  power upon the spectra formed upon one side, and a corresponding
  diverging power upon the spectra on the other side. Let us suppose
  that the light is incident perpendicularly, and that the grating
  interval increases from the centre towards that edge which lies
  nearest to the spectrum under observation, and decreases towards the
  hinder edge. It is evident that the waves from _both_ halves of the
  grating are accelerated in an increasing degree, as we pass from the
  centre outwards, as compared with the phase they would possess were
  the central value of the grating interval maintained throughout. The
  irregularity of spacing has thus the effect of a convex lens, which
  accelerates the marginal relatively to the central rays. On the other
  side the effect is reversed. This kind of irregularity may clearly be
  present in a degree surpassing the usual limits, without loss of
  definition, when the telescope is focused so as to secure the best
  effect.

  It may be worth while to examine further the other variations from
  correct ruling which correspond to the various terms expressing the
  deviation of the wave-surface from a perfect plane. If x and y be
  co-ordinates in the plane of the wave-surface, the axis of y being
  parallel to the lines of the grating, and the origin corresponding to
  the centre of the beam, we may take as an approximate equation to the
  wave-surface

          x²              y²
    z = ------ + Bxy + ------- + [alpha]x³ + [beta]x²y + [gamma]xy² + [delta]y³ + ... (8);
        2[rho]         2[rho]'

  and, as we have just seen, the term in x² corresponds to a linear
  error in the spacing. In like manner, the term in y² corresponds to a
  general _curvature_ of the lines (fig. 10), and does not influence the
  definition at the (primary) focus, although it may introduce
  astigmatism.[8] If we suppose that everything is symmetrical on the
  two sides of the primary plane y = 0, the coefficients B, [beta],
  [delta] vanish. In spite of any inequality between [rho] and [rho]',
  the definition will be good to this order of approximation, provided
  [alpha] and [gamma] vanish. The former measures the _thickness_ of the
  primary focal line, and the latter measures its _curvature_. The error
  of ruling giving rise to [alpha] is one in which the intervals
  increase or decrease in _both_ directions from the centre outwards
  (fig. 11), and it may often be compensated by a slight rotation in
  azimuth of the object-glass of the observing telescope. The term in
  [gamma] corresponds to a _variation_ of curvature in crossing the
  grating (fig. 12).

  When the plane zx is not a plane of symmetry, we have to consider the
  terms in xy, x²y, and y³. The first of these corresponds to a
  deviation from parallelism, causing the interval to alter gradually as
  we pass _along_ the lines (fig. 13). The error thus arising may be
  compensated by a rotation of the object-glass about one of the
  diameters y = ± x. The term in x²y corresponds to a deviation from
  parallelism in the same direction on both sides of the central line
  (fig. 14); and that in y³ would be caused by a curvature such that
  there is a point of inflection at the middle of each line (fig. 15).

  All the errors, except that depending on [alpha], and especially those
  depending on [gamma] and [delta], can be diminished, without loss of
  resolving power, by contracting the _vertical_ aperture. A linear
  error in the spacing, and a general curvature of the lines, are
  eliminated in the ordinary use of a grating.

  The explanation of the difference of focus upon the two sides as due
  to unequal spacing was verified by Cornu upon gratings purposely
  constructed with an increasing interval. He has also shown how to rule
  a plane surface with lines so disposed that the grating shall of
  itself give well-focused spectra.

  [Illustration: FIG. 16.]

  A similar idea appears to have guided H. A. Rowland to his brilliant
  invention of concave gratings, by which spectra can be photographed
  without any further optical appliance. In these instruments the lines
  are ruled upon a spherical surface of speculum metal, and mark the
  intersections of the surface by a system of parallel and equidistant
  planes, of which the middle member passes through the centre of the
  sphere. If we consider for the present only the primary plane of
  symmetry, the figure is reduced to two dimensions. Let AP (fig. 16)
  represent the surface of the grating, O being the centre of the
  circle. Then, if Q be any radiant point and Q' its image (primary
  focus) in the spherical mirror AP, we have

    1    1   2cos[phi]
    -- + - = ---------,
    v1   u       a

  where v1 = AQ', u = AQ, a = OA, [phi] = angle of incidence QAO, equal
  to the angle of reflection Q'AO. If Q be on the circle described upon
  OA as diameter, so that u = a cos [phi], then Q' lies also upon the
  same circle; and in this case it follows from the symmetry that the
  unsymmetrical aberration (depending upon a) vanishes.

  This disposition is adopted in Rowland's instrument; only, in addition
  to the central image formed at the angle [phi]' = [phi], there are a
  series of spectra with various values of [phi]', but all disposed upon
  the same circle. Rowland's investigation is contained in the paper
  already referred to; but the following account of the theory is in the
  form adopted by R. T. Glazebrook (_Phil. Mag._, 1883).

  In order to find the difference of optical distances between the
  courses QAQ', QPQ', we have to express QP - QA, PQ' - AQ'. To find the
  former, we have, if OAQ = [phi], AOP = [omega],

    QP² = u² + 4a²sin²½[omega] - 4au sin ½[omega] sin (½[omega] - [phi])
   = (u + a sin[phi] sin[omega])² - a² sin²[phi] sin²[omega] + 4a sin² ½[omega](a - u cos[phi]).

  Now as far as [omega]^4

    4 sin² ½[omega] = sin²[omega] + ¼sin^4[omega],

  and thus to the same order

      QP² = (u + a sin [phi] sin [omega])²
    -a cos [phi](u - a cos [phi]) sin²[omega] + ¼ a(a - u cos[phi]) sin^4 [omega].

  But if we now suppose that Q lies on the circle u = a cos [phi], the
  middle term vanishes, and we get, correct as far as [omega]^4,

                                       / /    a² sin²[phi] sin^4[omega]\
    QP = (u + a sin[phi] sin[omega])  / ( 1 + ------------------------- );
                                    \/   \              4u             /
  so that

    QP - u = a sin [phi] sin [omega] + 1/8 a sin[phi] tan[phi] sin^4 [omega] (9),

  in which it is to be noticed that the adjustment necessary to secure
  the disappearance of sin²[omega] is sufficient also to destroy the
  term in sin³[omega].

  A similar expression can be found for Q'P - Q'A; and thus, if Q'A = v,
  Q'AO = [phi]', where v = a cos [phi]', we get

    QP + PQ' - QA -AQ' = a sin[omega] (sin[phi] - sin[phi]')
       + 1/8 a sin^4 [omega] (sin[phi] tan[phi] + sin[phi]' tan[phi]')         (10).

  If [phi]' = [phi], the term of the first order vanishes, and the
  reduction of the difference of path _via_ P and _via_ A to a term of
  the fourth order proves not only that Q and Q' are conjugate foci, but
  also that the foci are exempt from the most important term in the
  aberration. In the present application [phi]' is not necessarily equal
  to [phi]; but if P correspond to a line upon the grating, the
  difference of retardations for consecutive positions of P, so far as
  expressed by the term of the first order, will be equal to [-+]
  m[lambda] (m integral), and therefore without influence, provided

    [sigma] (sin[phi] - sin[phi]') = ± m[lambda]      (11),

  where [sigma] denotes the constant interval between the planes
  containing the lines. This is the ordinary formula for a reflecting
  plane grating, and it shows that the spectra are formed in the usual
  directions. They are here focused (so far as the rays in the primary
  plane are concerned) upon the circle OQ'A, and the outstanding
  aberration is of the fourth order.

  In order that a large part of the field of view may be in focus at
  once, it is desirable that the locus of the focused spectrum should be
  nearly perpendicular to the line of vision. For this purpose Rowland
  places the eye-piece at O, so that [phi] = 0, and then by (11) the
  value of [phi]' in the m^th spectrum is

    [sigma] sin [phi]' = ± m[lambda]                  (12).

  If [omega] now relate to the edge of the grating, on which there are
  altogether n lines,

  n[sigma] = 2a sin [omega],

  and the value of the last term in (10) becomes

    1/16 n[sigma] sin³[omega] sin[phi]' tan[phi]',

  or

    1/16 mn[lambda] sin³[omega] tan [phi]'            (13).

  This expresses the retardation of the extreme relatively to the
  central ray, and is to be reckoned positive, whatever may be the signs
  of [omega], and [phi]'. If the semi-angular aperture ([omega]) be
  1/100, and tan [phi]' = 1, mn might be as great as four millions
  before the error of phase would reach ¼[lambda]. If it were desired to
  use an angular aperture so large that the aberration according to (13)
  would be injurious, Rowland points out that on his machine there would
  be no difficulty in applying a remedy by making [sigma] slightly
  variable towards the edges. Or, retaining [sigma] constant, we might
  attain compensation by so polishing the surface as to bring the
  circumference slightly forward in comparison with the position it
  would occupy upon a true sphere.

  It may be remarked that these calculations apply to the rays in the
  primary plane only. The image is greatly affected with astigmatism;
  but this is of little consequence, if [gamma] in (8) be small enough.
  Curvature of the primary focal line having a very injurious effect
  upon definition, it may be inferred from the excellent performance of
  these gratings that [gamma] is in fact small. Its value does not
  appear to have been calculated. The other coefficients in (8) vanish
  in virtue of the symmetry.

  The mechanical arrangements for maintaining the focus are of great
  simplicity. The grating at A and the eye-piece at O are rigidly
  attached to a bar AO, whose ends rest on carriages, moving on rails
  OQ, AQ at right angles to each other. A tie between the middle point
  of the rod OA and Q can be used if thought desirable.

  The absence of chromatic aberration gives a great advantage in the
  comparison of overlapping spectra, which Rowland has turned to
  excellent account in his determinations of the relative wave-lengths of
  lines in the solar spectrum (_Phil. Mag._, 1887).

  For absolute determinations of wave-lengths plane gratings are used.
  It is found (Bell, _Phil. Mag._, 1887) that the angular measurements
  present less difficulty than the comparison of the grating interval
  with the standard metre. There is also some uncertainty as to the
  actual temperature of the grating when in use. In order to minimize
  the heating action of the light, it might be submitted to a
  preliminary prismatic analysis before it reaches the slit of the
  spectrometer, after the manner of Helmholtz.

In spite of the many improvements introduced by Rowland and of the care
with which his observations were made, recent workers have come to the
conclusion that errors of unexpected amount have crept into his
measurements of wave-lengths, and there is even a disposition to discard
the grating altogether for fundamental work in favour of the so-called
"interference methods," as developed by A. A. Michelson, and by C. Fabry
and J. B. Pérot. The grating would in any case retain its utility for
the reference of new lines to standards otherwise fixed. For such
standards a relative accuracy of at least one part in a million seems
now to be attainable.

Since the time of Fraunhofer many skilled mechanicians have given their
attention to the ruling of gratings. Those of Nobert were employed by A.
J. Ångström in his celebrated researches upon wave-lengths. L. M.
Rutherfurd introduced into common use the reflection grating, finding
that speculum metal was less trying than glass to the diamond point,
upon the permanence of which so much depends. In Rowland's dividing
engine the screws were prepared by a special process devised by him, and
the resulting gratings, plane and concave, have supplied the means for
much of the best modern optical work. It would seem, however, that
further improvements are not excluded.

There are various copying processes by which it is possible to reproduce
an original ruling in more or less perfection. The earliest is that of
Quincke, who coated a glass grating with a chemical silver deposit,
subsequently thickened with copper in an electrolytic bath. The metallic
plate thus produced formed, when stripped from its support, a reflection
grating reproducing many of the characteristics of the original. It is
best to commence the electrolytic thickening in a silver acetate bath.
At the present time excellent reproductions of Rowland's speculum
gratings are on the market (Thorp, Ives, Wallace), prepared, after a
suggestion of Sir David Brewster, by coating the original with a
varnish, e.g. of celluloid. Much skill is required to secure that the
film when stripped shall remain undeformed.

A much easier method, applicable to glass originals, is that of
photographic reproduction by contact printing. In several papers dating
from 1872, Lord Rayleigh (see _Collected Papers_, i. 157, 160, 199, 504;
iv. 226) has shown that success may be attained by a variety of
processes, including bichromated gelatin and the old bitumen process,
and has investigated the effect of imperfect approximation during the
exposure between the prepared plate and the original. For many purposes
the copies, containing lines up to 10,000 to the inch, are not inferior.
It is to be desired that transparent gratings should be obtained from
first-class ruling machines. To save the diamond point it might be
possible to use something softer than ordinary glass as the material of
the plate.

9. _Talbot's Bands._--These very remarkable bands are seen under certain
conditions when a tolerably pure spectrum is regarded with the naked
eye, or with a telescope, _half the aperture being covered by a thin
plate_, e.g. _of glass or mica_. The view of the matter taken by the
discoverer (_Phil. Mag._, 1837, 10, p. 364) was that any ray which
suffered in traversing the plate a retardation of an odd number of half
wave-lengths would be extinguished, and that thus the spectrum would be
seen interrupted by a number of dark bars. But this explanation cannot
be accepted as it stands, being open to the same objection as Arago's
theory of stellar scintillation.[9] It is as far as possible from being
true that a body emitting homogeneous light would disappear on merely
covering half the aperture of vision with a half-wave plate. Such a
conclusion would be in the face of the principle of energy, which
teaches plainly that the retardation in question leaves the aggregate
brightness unaltered. The actual formation of the bands comes about in
a very curious way, as is shown by a circumstance first observed by
Brewster. When the retarding plate is held on the side towards the red
of the spectrum, _the bands are not seen_. Even in the contrary case,
the thickness of the plate must not exceed a certain limit, dependent
upon the purity of the spectrum. A satisfactory explanation of these
bands was first given by Airy (_Phil. Trans._, 1840, 225; 1841, 1), but
we shall here follow the investigation of Sir G. G. Stokes (_Phil.
Trans._, 1848, 227), limiting ourselves, however, to the case where the
retarded and unretarded beams are contiguous and of equal width.

  The aperture of the unretarded beam may thus be taken to be limited by
  x = -h, x = 0, y = -l, y= +l; and that of the beam retarded by R to be
  given by x = 0, x = h, y= -l, y = +l. For the former (1) § 3 gives
                   _     _
          1       / 0   / +l       /        x[xi] + y[eta]\
    - ---------   |     |   sin k (at - f + -------------- )dxdy
      [lambda]f  _/-h  _/-l        \              f       /

           2lh         f        k[eta]l     2f       k[xi]h          /       [xi]h \
    = - --------- · ------- sin ------- · ------ sin ------ · sin k (at - f - ----- ) (1),
        [lambda]f   k[eta]l        f      k[xi]h       2f            \         2f  /

  on integration and reduction.

  For the retarded stream the only difference is that we must subtract R
  from at, and that the limits of x are 0 and +h. We thus get for the
  disturbance at [xi], [eta], due to this stream

         2lh         f        k[eta]l     2f       k[xi]h          /           [xi]h \
    - --------- · ------- sin ------- · ------ sin ------ . sin k (at - f - R + ----- ) (2).
      [lambda]f   k[eta]l        f      k[xi]h       2f            \            2f   /

  If we put for shortness [pi] for the quantity under the last circular
  function in (1), the expressions (1), (2) may be put under the forms u
  sin [tau], v sin ([tau] - [alpha]) respectively; and, if I be the
  intensity, I will be measured by the sum of the squares of the
  coefficients of sin [tau] and cos [tau] in the expression

    u sin[tau] + v sin([tau] - [alpha]),

  so that

    I = u² + v² + 2uv cos[alpha],

  which becomes on putting for u, v, and [alpha] their values, and
  putting

     /   f        k[eta]l \²
    ( ------- sin -------  )  = Q           (3),
     \k[eta]l        f    /
                                       _                                  _
                4l²         [pi][xi]h |           / 2[pi]R    2[pi][xi]h\  |
    I = Q · ---------- sin² --------- |2 + 2 cos ( -------- - ---------- ) | (4).
            [pi]²[xi]²      [lambda]f |_          \[lambda]   [lambda]f / _|

  If the subject of examination be a luminous line parallel to [eta], we
  shall obtain what we require by integrating (4) with respect to [eta]
  from -[oo] to +[oo]. The constant multiplier is of no especial
  interest so that we may take as applicable to the image of a line
                              _                                 _
          2        [pi][xi]h |         / 2[pi]R    2[pi][xi]h \  |
    I = ----- sin² --------- |1 + cos ( -------- - ----------  ) | (5).
        [xi]²      [lambda]f |_        \[lambda]   [lambda]f  / _|

  If R = ½[lambda], I vanishes at [xi]= 0; but the whole illumination,
  represented by
      _
     / +[oo]
     |    I d[xi], is independent of the value of R. If R = 0,
    _/-[oo]

          1        2[pi][xi]h
    I = ----- sin² ----------,
        [xi]²      [lambda]f

  in agreement with § 3, where a has the meaning here attached to 2h.

  The expression (5) gives the illumination at [xi] due to that part of
  the complete image whose geometrical focus is at [xi] = 0, the
  retardation for this component being R. Since we have now to integrate
  for the whole illumination at a particular point O due to all the
  components which have their foci in its neighbourhood, we may
  conveniently regard O as origin. [xi] is then the co-ordinate
  relatively to O of any focal point O' for which the retardation is R;
  and the required result is obtained by simply integrating (5) with
  respect to [xi] from -[oo] to +[oo]. To each value of [xi] corresponds
  a different value of [lambda], and (in consequence of the dispersing
  power of the plate) of R. The variation of [lambda] may, however, be
  neglected in the integration, except in 2[pi]R/[lambda], where a small
  variation of [lambda] entails a comparatively large alteration of
  phase. If we write

    [rho] = 2[pi]R/[lambda]                                                     (6),

  we must regard [rho] as a function of [xi], and we may take with
  sufficient approximation under any ordinary circumstances

    [rho] = [rho]' + [=omega][xi]                                               (7),

  where [rho]' denotes the value of [rho] at O, and [=omega] is a
  constant, which is positive when the retarding plate is held at the
  side on which the lue of the spectrum _is seen_. The possibility of
  dark bands depends upon [=omega] being positive. Only in this case can

    cos {[rho]' + ([=omega] - 2[pi]h/[lambda]f)[xi]}

  retain the constant value -1 throughout the integration, and then only
  when

    [=omega] = 2[pi]h / [lambda]f                                               (8)

  and

    cos [rho]' = -1                                                             (9).

  The first of these equations is the condition for the formation of
  dark bands, and the second marks their situation, which is the same
  as that determined by the imperfect theory.

  The integration can be effected without much difficulty. For the first
  term in (5) the evaluation is effected at once by a known formula. In
  the second term if we observe that

    cos {[rho]' +([=omega] - 2[pi]h/[lambda]f)[xi]} = cos {[rho]'- g1[xi]}
      = cos [rho]' cos g1[xi] + sin [rho]' sin g1[xi],

  we see that the second part vanishes when integrated, and that the
  remaining integral is of the form

          _+[oo]
         /                        d[xi]
    w =  | sin² h1[xi] cos g1[xi] -----,
        _/-[oo]                   [xi]²

  where

    h1 = [pi]h/[lambda]f,  g1 = [omega] - 2[pi]h/[lambda]f    (10).

  By differentiation with respect to g1 it may be proved that

    w = 0                  from g1 = -[oo] to g1 = -2h1,
    w = ½[pi](2h1 + g1)    from g1 = -2h1  to g1 = 0,
    w = ½[pi](2h1 - g1)    from g1 =  0    to g1 = 2h1,
    w = 0                  from g1 = 2h1   to g1 = [oo].

  The integrated intensity, I', or

    2[pi]h1 + 2 cos[rho]w,

  is thus

    I' = 2[pi]h1              (11),

  when g1 numerically exceeds 2h1; and, when g1 lies between ±2h1,

    I = [pi]2h1 + (2h1 - [sqrt] g1²) cos[rho]'     (12).

  It appears therefore that there are no bands at all unless [omega]
  lies between 0 and +4h1, and that within these limits the best bands
  are formed at the middle of the range when [omega] = 2h1. The
  formation of bands thus requires that the retarding plate be held upon
  the side already specified, so that [omega] be positive; and that the
  thickness of the plate (to which [omega] is proportional) do not
  exceed a certain limit, which we may call 2T0. At the best thickness
  T0 the bands are black, and not otherwise.

  The linear width of the band (e) is the increment of [xi] which alters
  [rho] by 2[pi], so that

    e = 2[pi]/[=omega]                             (13).

  With the best thickness

    [=omega] = 2[pi]h/[lambda]f                    (14),

  so that in this case

    e = [lambda]f/h                                (15).

  The bands are thus of the same width as those due to two infinitely
  narrow apertures coincident with the central lines of the retarded and
  unretarded streams, the subject of examination being itself a fine
  luminous line.

  If it be desired to see a given number of bands in the whole or in any
  part of the spectrum, the thickness of the retarding plate is thereby
  determined, independently of all other considerations. But in order
  that the bands may be really visible, and still more in order that
  they may be black, another condition must be satisfied. It is
  necessary that the aperture of the pupil be accommodated to the
  angular extent of the spectrum, or reciprocally. Black bands will be
  too fine to be well seen unless the aperture (2h) of the pupil be
  somewhat contracted. One-twentieth to one-fiftieth of an inch is
  suitable. The aperture and the number of bands being both fixed, the
  condition of blackness determines the angular magnitude of a band and
  of the spectrum. The use of a grating is very convenient, for not only
  are there several spectra in view at the same time, but the dispersion
  can be varied continuously by sloping the grating. The slits may be
  cut out of tin-plate, and half covered by mica or "microscopic glass,"
  held in position by a little cement.

  If a telescope be employed there is a distinction to be observed,
  according as the half-covered aperture is between the eye and the
  ocular, or in front of the object-glass. In the former case the
  function of the telescope is simply to increase the dispersion, and
  the formation of the bands is of course independent of the particular
  manner in which the dispersion arises. If, however, the half-covered
  aperture be in front of the object-glass, the phenomenon is magnified
  as a whole, and the desirable relation between the (unmagnified)
  dispersion and the aperture is the same as without the telescope.
  There appears to be no further advantage in the use of a telescope
  than the increased facility of accommodation, and for this of course a
  very low power suffices.

  The original investigation of Stokes, here briefly sketched, extends
  also to the case where the streams are of unequal width h, k, and are
  separated by an interval 2g. In the case of unequal width the bands
  cannot be black; but if h = k, the finiteness of 2g does not preclude
  the formation of black bands.

  The theory of Talbot's bands with a half-covered _circular_ aperture
  has been considered by H. Struve (_St Peters. Trans._, 1883, 31, No.
  1).

  The subject of "Talbot's bands" has been treated in a very instructive
  manner by A. Schuster (_Phil. Mag._, 1904), whose point of view offers
  the great advantage of affording an instantaneous explanation of the
  peculiarity noticed by Brewster. A plane _pulse_, i.e. a disturbance
  limited to an infinitely thin slice of the medium, is supposed to fall
  upon a parallel grating, which again may be regarded as formed of
  infinitely thin wires, or infinitely narrow lines traced upon glass.
  The secondary pulses diverted by the ruling fall upon an object-glass
  as usual, and on arrival at the focus constitute a procession equally
  spaced in time, the interval between consecutive members depending
  upon the obliquity. If a retarding plate be now inserted so as to
  operate upon the pulses which come from one side of the grating, while
  leaving the remainder unaffected, we have to consider what happens at
  the focal point chosen. A full discussion would call for the formal
  application of Fourier's theorem, but some conclusions of importance
  are almost obvious.

  Previously to the introduction of the plate we have an effect
  corresponding to wave-lengths closely grouped around the principal
  wave-length, viz. [sigma] sin [phi], where [sigma] is the
  grating-interval and [phi] the obliquity, the closeness of the
  grouping increasing with the number of intervals. In addition to these
  wave-lengths there are other groups centred round the wave-lengths
  which are submultiples of the principal one--the overlapping spectra
  of the second and higher orders. Suppose now that the plate is
  introduced so as to cover naif the aperture and that it retards those
  pulses which would otherwise arrive first. The consequences must
  depend upon the amount of the retardation. As this increases from
  zero, the two processions which correspond to the two halves of the
  aperture begin to overlap, and the overlapping gradually increases
  until there is almost complete superposition. The stage upon which we
  will fix our attention is that where the one procession bisects the
  intervals between the other, so that a new simple procession is
  constituted, containing the same number of members as before the
  insertion of the plate, but now spaced at intervals only half as
  great. It is evident that the effect at the focal point is the
  obliteration of the first and other spectra of odd order, so that as
  regards the spectrum of the first order we may consider that the two
  beams _interfere_. The formation of black bands is thus explained, and
  it requires that the plate be introduced upon one particular side, and
  that the amount of the retardation be adjusted to a particular value.
  If the retardation be too little, the overlapping of the processions
  is incomplete, so that besides the procession of half period there are
  residues of the original processions of full period. The same thing
  occurs if the retardation be too great. If it exceed the double of the
  value necessary for black bands, there is again no overlapping and
  consequently no interference. If the plate be introduced upon the
  other side, so as to retard the procession originally in arrear, there
  is no overlapping, whatever may be the amount of retardation. In this
  way the principal features of the phenomenon are accounted for, and
  Schuster has shown further how to extend the results to spectra having
  their origin in prisms instead of gratings.

10. _Diffraction when the Source of Light is not seen in Focus._--The
phenomena to be considered under this head are of less importance than
those investigated by Fraunhofer, and will be treated in less detail;
but in view of their historical interest and of the ease with which many
of the experiments may be tried, some account of their theory cannot be
omitted. One or two examples have already attracted our attention when
considering Fresnel's zones, viz. the shadow of a circular disk and of a
screen circularly perforated.

Fresnel commenced his researches with an examination of the fringes,
external and internal, which accompany the shadow of a narrow opaque
strip, such as a wire. As a source of light he used sunshine passing
through a very small hole perforated in a metal plate, or condensed by a
lens of short focus. In the absence of a heliostat the latter was the
more convenient. Following, unknown to himself, in the footsteps of
Young, he deduced the principle of interference from the circumstance
that the darkness of the interior bands requires the co-operation of
light from both sides of the obstacle. At first, too, he followed Young
in the view that the exterior bands are the result of interference
between the direct light and that reflected from the edge of the
obstacle, but he soon discovered that the character of the edge--e.g.
whether it was the cutting edge or the back of a razor--made no material
difference, and was thus led to the conclusion that the explanation of
these phenomena requires nothing more than the application of Huygens's
principle to the unobstructed parts of the wave. In observing the bands
he received them at first upon a screen of finely ground glass, upon
which a magnifying lens was focused; but it soon appeared that the
ground glass could be dispensed with, the diffraction pattern being
viewed in the same way as the image formed by the object-glass of a
telescope is viewed through the eye-piece. This simplification was
attended by a great saving of light, allowing measures to be taken such
as would otherwise have presented great difficulties.

  In theoretical investigations these problems are usually treated as of
  two dimensions only, everything being referred to the plane passing
  through the luminous point and perpendicular to the diffracting edges,
  supposed to be straight and parallel. In strictness this idea is
  appropriate only when the source is a luminous line, emitting
  cylindrical waves, such as might be obtained from a luminous point
  with the aid of a cylindrical lens. When, in order to apply Huygens's
  principle, the wave is supposed to be broken up, the phase is the same
  at every element of the surface of resolution which lies upon a line
  perpendicular to the plane of reference, and thus the effect of the
  whole line, or rather infinitesimal strip, is related in a constant
  manner to that of the element which lies in the plane of reference,
  and may be considered to be represented thereby. The same method of
  representation is applicable to spherical waves, issuing from a
  _point_, if the radius of curvature be large; for, although there is
  variation of phase along the length of the infinitesimal strip, the
  whole effect depends practically upon that of the central parts where
  the phase is sensibly constant.[10]

  [Illustration: FIG. 17.]

  In fig. 17 APQ is the arc of the circle representative of the
  wave-front of resolution, the centre being at O, and the radius QA
  being equal to a. B is the point at which the effect is required,
  distant a + b from O, so that AB = b, AP = s, PQ = ds.

  Taking as the standard phase that of the secondary wave from A, we may
  represent the effect of PQ by

               /t   [delta] \
    cos 2[pi] ( - - -------- )·ds,
               \r   [lambda]/

  where [delta] = BP - AP is the retardation at B of the wave from P
  relatively to that from A.

  Now

    [delta] = (a + b) s²/2ab                    (1),

  so that, if we write

    2[pi][delta] =  [pi](a + b)s²     [pi]v²
    ------------   ---------------  = ------    (2),
      [lambda]       ab[lambda]         2

  the effect at B is
                                 _                             _
     /ab[lambda]\½  /    2[pi]t /                      2[pi]t /               \
    ( ---------- ) ( cos ------ | cos ½[pi]v²·dv + sin ------ | sin ½[pi]v²·dv ) (3),
     \2(a + b)  /   \    [tau] _/                      [tau] _/               /

  the limits of integration depending upon the disposition of the
  diffracting edges. When a, b, [lambda] are regarded as constant, the
  first factor may be omitted,--as indeed should be done for
  consistency's sake, inasmuch as other factors of the same nature have
  been omitted already.

  The intensity I², the quantity with which we are principally
  concerned, may thus be expressed

             _                        _
         /  /               \²    /  /               \²
    I²= (   | cos ½[pi]v²·dv ) + (   | sin ½[pi]v²·dv ) (4).
         \ _/               /     \ _/               /

  These integrals, taken from v = 0, are known as Fresnel's integrals;
  we will denote them by C and S, so that
         _                       _
        / v                     / v
    C = | cos ½[pi]v²·dv,   S = | cos ½[pi]v²·dv               (5).
       _/0                     _/0

  When the upper limit is infinity, so that the limits correspond to the
  inclusion of half the primary wave, C and S are both equal to ½, by a
  known formula; and on account of the rapid fluctuation of sign the
  parts of the range beyond very moderate values of v contribute but
  little to the result.

  Ascending series for C and S were given by K. W. Knockenhauer, and are
  readily investigated. Integrating by parts, we find

              _v                                        _v
             /   i·½[pi]v²      i·½[pi]v²      1       /   i·½[pi]v²
    C + iS = |  e         dv = e         · v - - i[pi] |  e         dv³;
            _/0                                3      _/0

  and, by continuing this process,

              i.½[pi]v² /    i[pi]      i[pi] i[pi]       i[pi] i[pi] i[pi]          \
    C + iS = e         ( v - ----- v³ + ----- ----- v^5 - ----- ----- ----- v^7 + ... ).
                        \      3          3     5           3     5     7            /

  By separation of real and imaginary parts,

    C = M cos ½[pi]v² - N sin ½[pi]v²  \
    S = M sin ½[pi]v² - N cos ½[pi]v²  /          (6)

  where

        v   [pi]²v^5    [pi]^4v^9
    M = - - --------- + --------- - ...           (7)
        1     3·5        3·5·7·9

        [pi]v³   [pi]^3v^7    [pi]^5v^11
    N = ------ - --------- + ------------ ...     (8)
         1·3      1·3·5·7    1·3·5·7·9·11

  These series are convergent for all values of v, but are practically
  useful only when v is small.

  Expressions suitable for discussion when v is large were obtained by
  L. P. Gilbert (_Mem. cour. de l'Acad. de Bruxelles_, 31, p. 1). Taking

    ½[pi]v² = u                                   (9),

  we may write
                                _
                    1          /u  e^iu du
    C + iS =  -------------    |   --------      (10).
              [sqrt](2[pi])   _/0  [sqrt] u

Again, by a known formula,

                             _[oo]
        1           1       /  e^-ux dx
    --------  = ----------  |  --------         (11).
    [sqrt] u    [sqrt][pi] _/0  [sqrt]x

  Substituting this in (10), and inverting the order of integration, we
  get

                       _[oo]         _u
                1     /     dx      /  e^u(i - x)
    C + iS = -------  |  --------   |  ----------- dx
             [sqrt]2 _/0 [sqrt] x  _/0   [sqrt]x

                   _[oo]
           1     /     dx    e^u(i - x) - 1
      = -------  |  -------- -------------- dx     (12).
        [sqrt]2 _/0 [sqrt] x     i - x

  Thus, if we take

                      _[oo]
             1       /  e^-ux [sqrt](x)·dx
    G = -----------  |  ------------------,
        [pi][sqrt]2 _/0       1 + x²

                      _[oo]
             1       /      e^-ux dx
    H = -----------  | ------------------         (13).
        [pi][sqrt]2 _/ [sqrt]x · (1 + x²)
                     0

    C = ½ - G cos u + H sin u,  S = ½ - G sin u - H cos u   (14).

  The constant parts in (14), viz. ½, may be determined by direct
  integration of (12), or from the observation that by their
  constitution G and H vanish when u = [oo], coupled with the fact that
  C and S then assume the value ½.

  Comparing the expressions for C, S in terms of M, N, and in terms of
  G, H, we find that

    G = ½ (cos u + sin u) - M,  H = ½ (cos u - sin u) + N  (15),

  formulae which may be utilized for the calculation of G, H when u (or
  v) is small. For example, when u = 0, M = 0, N = 0, and consequently G
  = H = ½.

  Descending series of the semi-convergent class, available for
  numerical calculation when u is moderately large, can be obtained from
  (12) by writing x = uy, and expanding the denominator in powers of y.
  The integration of the several terms may then be effected by the
  formula

      _ [oo]
     /  -y   q-½
     | e    y    dy = [Gamma](q + ½) = (q - ½)(q - 3/2) ... ½[sqrt][pi];
    _/0

  and we get in terms of v

           1         1·3·5       1·3·5·9
    G = ------- - ---------- + ----------- -          (16),
        [pi]²v³   [pi]^4 v^7   [pi]^6 v^11

          1        1·3        1·3·5·7
    H = ----- - --------- + ---------- -              (17).
        [pi]v   [pi]³ v^5   [pi]^5 v^9

  The corresponding values of C and S were originally derived by A. L.
  Cauchy, without the use of Gilbert's integrals, by direct integration
  by parts.

  From the series for G and H just obtained it is easy to verify that

    dH             dG
    -- = - [pi]vG, -- = [pi]vH - 1                    (18).
    dv             dv

  We now proceed to consider more particularly the distribution of light
  upon a screen PBQ near the shadow of a straight edge A. At a point P
  within the geometrical shadow of the obstacle, the half of the wave to
  the right of C (fig. 18), the nearest point on the wave-front, is
  wholly intercepted, and on the left the integration is to be taken
  from s = CA to s = [oo]. If V be the value of v corresponding to CA,
  viz.

         / /  2(a + b)  \
    V=  / (  ----------  )·CA,                       (19),
      \/   \ ab[lambda] /

  we may write

              _[oo]                    _[oo]
          /  /               \²    /  /               \²
    I² = (   | cos ½[pi]v²·dv ) + (   | sin ½[pi]v²·dv ) (20),
          \ _/v              /     \ _/v              /

  or, according to our previous notation,

    I² = (½ - Cv)² + (½ - Sv)² = G² + H²             (21).

  Now in the integrals represented by G and H every element diminishes
  as V increases from zero. Hence, as CA increases, viz. as the point P
  is more and more deeply immersed in the shadow, the illumination
  _continuously_ decreases, and that without limit. It has long been
  known from observation that there are no bands on the interior side of
  the shadow of the edge.

  [Illustration: FIG. 18.]

  The law of diminution when V is moderately large is easily expressed
  with the aid of the series (16), (17) for G, H. We have ultimately G =
  0, H = ([pi]V)^-1, so that

    I² = 1/[pi]²V²,

  or the illumination is inversely as the square of the distance from
  the shadow of the edge.

  For a point Q outside the shadow the integration extends over _more_
  than half the primary wave. The intensity may be expressed by

    I² = (½ + Cv)² + (½ + Sv)²                   (22);

  and the maxima and minima occur when

              dC             dS
    (½ + C_v) -- + (½ + S_v) -- = 0,
              dV             dV

  whence

    sin ½[pi]V² + cos ½[pi]V² = G               (23).

  When V = 0, viz. at the edge of the shadow, I² = ½; when V = [oo], I²
  = 2, on the scale adopted. The latter is the intensity due to the
  uninterrupted wave. The quadrupling of the intensity in passing
  outwards from the edge of the shadow is, however, accompanied by
  fluctuations giving rise to bright and dark bands. The position of
  these bands determined by (23) may be very simply expressed when V is
  large, for then sensibly G = 0, and

    ½[pi]V² = ¾[pi] + n[pi]                     (24),

  n being an integer. In terms of [delta], we have from (2)

    [delta] = (3/8 + ½n)[lambda]                (25).

  The first maximum in fact occurs when [delta] = 3/8[lambda]
  -.0046[lambda], and the first minimum when [delta] = 7/8[lambda]
  -.0016[lambda], the corrections being readily obtainable from a table
  of G by substitution of the approximate value of V.

  The position of Q corresponding to a given value of V, that is, to a
  band of given order, is by (19)

         a + b          / / b[lambda](a + b) \
    BQ = ----- AD = V  / (  ----------------- )                               (26).
           a         \/   \        2a        /

  By means of this expression we may trace the locus of a band of given
  order as b varies. With sufficient approximation we may regard BQ and
  b as rectangular co-ordinates of Q. Denoting them by x, y, so that AB
  is axis of y and a perpendicular through A the axis of x, and
  rationalizing (26), we have

    2ax² - V²[lambda]y² - V²a[lambda]y = 0,

  which represents a hyperbola with vertices at O and A.

  From (24), (26) we see that the width of the bands is of the order
  [sqrt] {b[lambda](a + b)/a}. From this we may infer the limitation
  upon the width of the source of light, in order that the bands may be
  properly formed. If [omega] be the apparent magnitude of the source
  seen from A, [omega]b should be much smaller than the above quantity,
  or

    [omega] < [sqrt] {[lambda](a + b)/ab}        (27).

  If a be very great in relation to b, the condition becomes

    [omega] < [sqrt] ([lambda]/b)                (28).

  so that if b is to be moderately great (1 metre), the apparent
  magnitude of the sun must be greatly reduced before it can be used as
  a source. The values of V for the maxima and minima of intensity, and
  the magnitudes of the latter, were calculated by Fresnel. An extract
  from his results is given in the accompanying table.

    +--------------------+----------+------------+
    |                    |     V    |     I²     |
    +--------------------+----------+------------+
    | First maximum      |  1.2172  |   2.7413   |
    | First minimum      |  1.8726  |   1.5570   |
    | Second maximum     |  2.3449  |   2.3990   |
    | Second minimum     |  2.7392  |   1.6867   |
    | Third maximum.     |  3.0820  |   2.3022   |
    | Third minimum      |  3.3913  |   1.7440   |
    +--------------------+----------+------------+

  A very thorough investigation of this and other related questions,
  accompanied by fully worked-out tables of the functions concerned,
  will be found in a paper by E. Lommel (_Abh. bayer. Akad. d. Wiss._
  II. CI., 15, Bd., iii. Abth., 1886).

  When the functions C and S have once been calculated, the discussion
  of various diffraction problems is much facilitated by the idea, due
  to M. A. Cornu (_Journ. de Phys._, 1874, 3, p. 1; a similar suggestion
  was made independently by G. F. Fitzgerald), of exhibiting as a curve
  the relationship between C and S, considered as the rectangular
  co-ordinates (x, y) of a point. Such a curve is shown in fig. 19,
  where, according to the definition (5) of C, S,

          _ v                        _ v
         /                          /
    x =  | cos ½[pi]v²·dv,      y = | sin ½[pi]v²·dv   (29).
        _/0                        _/0

  The origin of co-ordinates O corresponds to v = 0; and the asymptotic
  points J, J', round which the curve revolves in an ever-closing
  spiral, correspond to v = ±[oo].

  The intrinsic equation, expressing the relation between the arc
  [sigma] (measured from O) and the inclination [phi] of the tangent at
  any points to the axis of x, assumes a very simple form. For

    dx = cos ½[pi]v²·dv,   dy = sin ½[pi]v²·dv;

  so that
               _
              /
    [sigma] = | [sqrt] (dx² + dy²) = v,                (30),
             _/

    [phi] = tan^-1 (dy/dx) = ½[pi]v²                   (31).



  Accordingly,

    [phi] = ½[pi][sigma]²                              (32);

  and for the curvature,

    d[phi]/d[sigma] = [pi][sigma]                      (33).

  Cornu remarks that this equation suffices to determine the general
  character of the curve. For the osculating circle at any point
  includes the whole of the curve which lies beyond; and the successive
  convolutions envelop one another without intersection.

  [Illustration: Fig. 19.]

  The utility of the curve depends upon the fact that the elements of
  arc represent, in amplitude and phase, the component vibrations due to
  the corresponding portions of the primary wave-front. For by (30)
  d[sigma] = dv, and by (2) dv is proportional to ds. Moreover by (2)
  and (31) the retardation of phase of the elementary vibration from PQ
  (fig. 17) is 2[pi][delta]/[lambda], or [phi]. Hence, in accordance
  with the rule for compounding vector quantities, the resultant
  vibration at B, due to any finite part of the primary wave, is
  represented in amplitude and phase by the chord joining the
  extremities of the corresponding arc ([sigma]2 - [sigma]1).

  In applying the curve in special cases of diffraction to exhibit the
  effect at any point P (fig. 18) the centre of the curve O is to be
  considered to correspond to that point C of the primary wave-front
  which lies nearest to P. The operative part, or parts, of the curve
  are of course those which represent the unobstructed portions of the
  primary wave.

  Let us reconsider, following Cornu, the diffraction of a screen
  unlimited on one side, and on the other terminated by a straight edge.
  On the illuminated side, at a distance from the shadow, the vibration
  is represented by JJ'. The co-ordinates oí J, J' being (½, ½), (-½,
  -½), I² is 2; and the phase is 1/8 period in arrear of that of the
  element at O. As the point under contemplation is supposed to approach
  the shadow, the vibration is represented by the chord drawn from J to
  a point on the other half of the curve, which travels inwards from J'
  towards O. The amplitude is thus subject to fluctuations, which
  increase as the shadow is approached. At the point O the intensity is
  one-quarter of that of the entire wave, and after this point is
  passed, that is, when we have entered the geometrical shadow, the
  intensity falls off gradually to zero, _without fluctuations_. The
  whole progress of the phenomenon is thus exhibited to the eye in a
  very instructive manner.

  We will next suppose that the light is transmitted by a slit, and
  inquire what is the effect of varying the width of the slit upon the
  illumination at the projection of its centre. Under these
  circumstances the arc to be considered is bisected at O, and its
  length is proportional to the width of the slit. It is easy to see
  that the length of the chord (which passes in all cases through O)
  increases to a maximum near the place where the phase-retardation is
  3/8 of a period, then diminishes to a minimum when the retardation is
  about 7/8 of a period, and so on.

  If the slit is of constant width and we require the illumination at
  various points on the screen behind it, we must regard the arc of the
  curve as of _constant length_. The intensity is then, as always,
  represented by the square of the length of the chord. If the slit be
  narrow, so that the arc is short, the intensity is constant over a
  wide range, and does not fall off to an important extent until the
  discrepancy of the extreme phases reaches about a quarter of a period.

  We have hitherto supposed that the shadow of a diffracting obstacle is
  received upon a diffusing screen, or, which comes to nearly the same
  thing, is observed with an eye-piece. If the eye, provided if
  necessary with a perforated plate in order to reduce the aperture, be
  situated inside the shadow at a place where the illumination is still
  sensible, and be focused upon the diffracting edge, the light which it
  receives will appear to come from the neighbourhood of the edge, and
  will present the effect of a silver lining. This is doubtless the
  explanation of a "pretty optical phenomenon, seen in Switzerland, when
  the sun rises from behind distant trees standing on the summit of a
  mountain."[11]

II. _Dynamical Theory of Diffraction._--The explanation of diffraction
phenomena given by Fresnel and his followers is independent of special
views as to the nature of the aether, at least in its main features; for
in the absence of a more complete foundation it is impossible to treat
rigorously the mode of action of a solid obstacle such as a screen. But,
without entering upon matters of this kind, we may inquire in what
manner a primary wave may be resolved into elementary secondary waves,
and in particular as to the law of intensity and polarization in a
secondary wave as dependent upon its direction of propagation, and upon
the character as regards polarization of the primary wave. This question
was treated by Stokes in his "Dynamical Theory of Diffraction" (_Camb.
Phil. Trans._, 1849) on the basis of the elastic solid theory.

  Let x, y, z be the co-ordinates of any particle of the medium in its
  natural state, and [chi], [eta], [zeta] the displacements of the same
  particle at the end of time t, measured in the directions of the three
  axes respectively. Then the first of the equations of motion may be
  put under the form

    d²[xi]      /d²[xi]   d²[xi]   d²[xi]\             d² /d²[xi]   d²[eta]   d²[zeta]\
    ------ = b²( ------ + ------ + ------ ) + (a² - b²)--( ------ + ------- + -------- ),
     dt²        \ dx²      dy²      dz²  /             dx \ dx²       dy²        dz²  /

  where a2 and b2 denote the two arbitrary constants. Put for shortness

    d²[xi]   d²[eta]   d²[zeta]
    ------ + ------- + -------- = [delta]    (1),
     dx²       dy²        dz²

  and represent by [Delta]²[chi] the quantity multiplied by b².
  According to this notation, the three equations of motion are

    d²[xi]                                  d[delta] \
    ------   = b²[Delta]²[xi]   + (a² - b²) -------- |
      dt²                                      dx    |
                                                     |
    d²[eta]                                 d[delta] |
    -------  = b²[Delta]²[eta]  + (a² - b²) --------  >   (2).
      dt²                                      dy    |
                                                     |
    d²[zeta]                                d[delta] |
    -------- = b²[Delta]²[zeta] + (a² - b²) -------- |
      dt²                                      dz    /

  It is to be observed that S denotes the dilatation of volume of the
  element situated at (x, y, z). In the limiting case in which the
  medium is regarded as absolutely incompressible [delta] vanishes; but,
  in order that equations (2) may preserve their generality, we must
  suppose a at the same time to become infinite, and replace a²[delta]
  by a new function of the co-ordinates.

  These equations simplify very much in their application to plane
  waves. If the ray be parallel to OX, and the direction of vibration
  parallel to OZ, we have [xi] = 0, [eta] = 0, while [zeta] is a
  function of x and t only. Equation (1) and the first pair of equations
  (2) are thus satisfied identically. The third equation gives

    d²[zeta]   d²[zeta]
    -------- = --------                   (3),
       dt²        dx²

  of which the solution is

    [zeta] = f(bt - x)                    (4),

  where f is an arbitrary function.

  The question as to the law of the secondary waves is thus answered by
  Stokes. "Let [xi] = 0, [eta] = 0, [zeta] = f(bt-x) be the
  displacements corresponding to the incident light; let O1 be any point
  in the plane P (of the wave-front), dS an element of that plane
  adjacent to O1, and consider the disturbance due to that portion only
  of the incident disturbance which passes continually across dS. Let O
  be any point in the medium situated at a distance from the point O1
  which is large in comparison with the length of a wave; let O1O = r,
  and let this line make an angle [theta] with the direction of
  propagation of the incident light, or the axis of x, and [phi] with
  the direction of vibration, or axis of z. Then the displacement at O
  will take place in a direction perpendicular to O1O, and lying in the
  plane ZO1O; and, if [zeta]' be the displacement at O, reckoned
  positive in the direction nearest to that in which the incident
  vibrations are reckoned positive,

                dS
    [zeta]' = ------ ( 1 + cos[theta]) sin[phi] f'(bt - r).
              4[pi]r

  In particular, if

                          2[pi]
    f(bt - x) =   c sin -------- (bt - x)        (5),
                        [lambda]

  we shall have

                   cdS                                  2[pi]
    [zeta]' =  ---------- (1 + cos[theta]) sin[phi]cos -------- (bt - r)  (6)."
               2[lambda]r                              [lambda]

  It is then verified that, after integration with respect to dS, (6)
  gives the same disturbance as if the primary wave had been supposed to
  pass on unbroken.

  The occurrence of sin [phi] as a factor in (6) shows that the relative
  intensities of the primary light and of that diffracted in the
  direction [theta] depend upon the condition of the former as regards
  polarization. If the direction of primary vibration be perpendicular
  to the plane of diffraction (containing both primary and secondary
  rays), sin [phi] = 1; but, if the primary vibration be in the plane of
  diffraction, sin [phi] = cos [theta]. This result was employed by
  Stokes as a criterion of the direction of vibration; and his
  experiments, conducted with gratings, led him to the conclusion that
  the vibrations of polarized light are executed in a direction
  _perpendicular_ to the plane of polarization.

  The factor (1 + cos [theta]) shows in what manner the secondary
  disturbance depends upon the direction in which it is propagated with
  respect to the front of the primary wave.

  If, as suffices for all practical purposes, we limit the application
  of the formulae to points in advance of the plane at which the wave is
  supposed to be broken up, we may use simpler methods of resolution
  than that above considered. It appears indeed that the purely
  mathematical question has no definite answer. In illustration of this
  the analogous problem for sound may be referred to. Imagine a flexible
  lamina to be introduced so as to coincide with the plane at which
  resolution is to be effected. The introduction of the lamina (supposed
  to be devoid of inertia) will make no difference to the propagation of
  plane parallel sonorous waves through the position which it occupies.
  At every point the motion of the lamina will be the same as would have
  occurred in its absence, the pressure of the waves impinging from
  behind being just what is required to generate the waves in front. Now
  it is evident that the aerial motion in front of the lamina is
  determined by what happens at the lamina without regard to the cause
  of the motion there existing. Whether the necessary forces are due to
  aerial pressures acting on the rear, or to forces directly impressed
  from without, is a matter of indifference. The conception of the
  lamina leads immediately to two schemes, according to which a primary
  wave may be supposed to be broken up. In the first of these the
  element dS, the effect of which is to be estimated, is supposed to
  execute its actual motion, while every other element of the plane
  lamina is maintained at rest. The resulting aerial motion in front is
  readily calculated (see Rayleigh, _Theory of Sound_, § 278); it is
  symmetrical with respect to the origin, i.e. independent of [theta].
  When the secondary disturbance thus obtained is integrated with
  respect to dS over the entire plane of the lamina, the result is
  necessarily the same as would have been obtained had the primary wave
  been supposed to pass on without resolution, for this is precisely the
  motion generated when every element of the lamina vibrates with a
  common motion, equal to that attributed to dS. The only assumption
  here involved is the evidently legitimate one that, when two systems
  of variously distributed motion at the lamina are superposed, the
  corresponding motions in front are superposed also.

  The method of resolution just described is the simplest, but it is
  only one of an indefinite number that might be proposed, and which are
  all equally legitimate, so long as the question is regarded as a
  merely mathematical one, without reference to the physical properties
  of actual screens. If, instead of supposing the _motion_ at dS to be
  that of the primary wave, and to be zero elsewhere, we suppose the
  _force_ operative over the element dS of the lamina to be that
  corresponding to the primary wave, and to vanish elsewhere, we obtain
  a secondary wave following quite a different law. In this case the
  motion in different directions varies as cos[theta], vanishing at
  right angles to the direction of propagation of the primary wave. Here
  again, on integration over the entire lamina, the aggregate effect of
  the secondary waves is necessarily the same as that of the primary.

  In order to apply these ideas to the investigation of the secondary
  wave of light, we require the solution of a problem, first treated by
  Stokes, viz. the determination of the motion in an infinitely extended
  elastic solid due to a locally applied periodic force. If we suppose
  that the force impressed upon the element of mass D dx dy dz is

    DZ dx dy dz,

  being everywhere parallel to the axis of Z, the only change required
  in our equations (1), (2) is the addition of the term Z to the second
  member of the third equation (2). In the forced vibration, now under
  consideration, Z, and the quantities [xi], [eta], [zeta], [delta]
  expressing the resulting motion, are to be supposed proportional to
  e^int, where i = [sqrt](-1), and n = 2[pi]/[tau], [tau] being the
  periodic time. Under these circumstances the double differentiation
  with respect to t of any quantity is equivalent to multiplication by
  the factor -n², and thus our equations take the form

                                        d[delta]      \
    (b²[Delta]² + n²)[xi]   + (a² - b²) -------- = 0  |
                                           dx         |
                                                      |
                                        d[delta]      |
    (b²[Delta]² + n²)[eta]  + (a² - b²) -------- = 0   >   (7).
                                           dx         |
                                                      |
                                        d[delta]      |
    (b²[Delta]² + n²)[zeta] + (a² - b²) -------- = -Z |
                                           dx         /

  It will now be convenient to introduce the quantities.[=omega]1,
  [=omega]2, [=omega]3 which express the _rotations_ of the elements of
  the medium round axes parallel to those of co-ordinates, in accordance
  with the equations

                d[xi]   d[eta]              d[eta]   d[zeta]
    [=omega]3 = ----- - ------, [=omega]1 = ------ - -------,
                 dy       dx'                 dz       dy

                d[zeta]   d[xi]
    [=omega]2 = ------- - -----                (8).
                  dx       dz

  In terms of these we obtain from (7), by differentiation and
  subtraction,

    (b²[Delta]² + n²) [=omega]3 = 0       \
    (b²[Delta]² + n²) [=omega]1 = dZ/dy    >   (9).
    (b²[Delta]² + n²) [=omega]2 = -dZ/dx  /


  The first of equations (9) gives

    [=omega]3 = 0                             (10).

  For =[omega]1, we have
                          _ _ _    -ikr
                   1     / / / dZ e
    [=omega]1 = -------  | | | -- ----- dx dy dz   (11),
                4[pi]b² _/_/_/ dy   r

  where r is the distance between the element dx dy dz and the point
  where [=omega]1 is estimated, and

    k = n/b = 2[pi]/[lambda]                  (12),

  [lambda] being the wave-length.

  (This solution may be verified in the same manner as Poisson's
  theorem, in which k = 0.)

  We will now introduce the supposition that the force Z acts only
  within a small space of volume T, situated at (x, y, z), and for
  simplicity suppose that it is at the origin of co-ordinates that the
  rotations are to be estimated. Integrating by parts in (11), we get

      _ -ikr           _       _     _
     / e      dZ      | Ze^-ikr |   /   d   / e^-ikr\
     | ------ -- dy = | ------- | - | Z -- ( ------- ) dy,
    _/    r   dy      |_   r   _|  _/   dy  \    r  /

  in which the integrated terms at the limits vanish, Z being finite
  only within the region T. Thus

                          _ _ _         -ikr
                   1     / / /   d   /e^      \
    [=omega]1 = -------  | | | Z -- ( -------- ) dx dy dz.
                4[pi]b² _/_/_/   dy  \    r   /


  Since the dimensions of T are supposed to be very small in comparison
  with [lambda], the factor d/dy (e^-ikr / r) is sensibly constant; so
  that, if Z stand for the mean value of Z over the volume T, we may
  write

                  TZ      y   d   / e^-ikr \
    [=omega]1 = ------- · - · -- (  ------  )        (13).
                4[pi]b²   r   dr  \   r    /

  In like manner we find

                  TZ      x   d   / e^-ikr \
    [=omega]2 = ------  · - · -- ( -------  )        (14).
                4[pi]b²   r   dr  \     r  /


  From (10), (13), (14) we see that, as might have been expected, the
  rotation at any point is about an axis perpendicular both to the
  direction of the force and to the line joining the point to the source
  of disturbance. If the resultant rotation be [omega], we have

                 TZ      [sqrt](x² + y²)   d   /e^-ikr\
    [=omega] = ------- · --------------- · -- ( ------ ) =
               4[pi]b²          r          dr  \   r  /

                TZ sin[phi] d   /e^-ikr\
             =  ----------- -- ( ------ ),
                  4[pi]b²   dr  \   r  /

  [phi] denoting the angle between r and z. In differentiating
  e^(-ikr)/r with respect to r, we may neglect the term divided by r² as
  altogether insensible, kr being an exceedingly great quantity at any
  moderate distance from the origin of disturbance. Thus

               -ik·TZ sin[phi]    /e^-ikr\
    [=omega] = --------------- · ( ------ )         (15),
                   4[pi]b²        \  r   /

  which completely determines the rotation at any point. For a
  disturbing force of given integral magnitude it is seen to be
  everywhere about an axis perpendicular to r and the direction of the
  force, and in magnitude dependent only upon the angle ([phi]) between
  these two directions and upon the distance (r).

  The intensity of light is, however, more usually expressed in terms of
  the actual displacement in the plane of the wave. This displacement,
  which we may denote by [zeta]', is in the plane containing z and r,
  and perpendicular to the latter. Its connexion with [=omega]is
  expressed by [=omega] = d[zeta]'/dr; so that

              TZ sin [phi]   /e^-ikr\
    [zeta]' = ----------- · ( ------ )              (16),
                4[pi]b²      \  r   /

  where the factor e^int is restored.

  Retaining only the real part of (16), we find, as the result of a
  local application of force equal to

    DTZ cos nt                                      (17),

  the disturbance expressed by

              TZ sin [phi]    /cos(nt - kr)\
    [zeta]' = ------------ · ( ------------ )       (18).
                4[pi]b²       \     r      /

  The occurrence of sin [phi] shows that there is no disturbance
  radiated in the direction of the force, a feature which might have
  been anticipated from considerations of symmetry.

  We will now apply (18) to the investigation of a law of secondary
  disturbance, when a primary wave

    [zeta] = sin(nt - kx)                           (19)

  is supposed to be broken up in passing the plane x = 0. The first step
  is to calculate the force which represents the reaction between the
  parts of the medium separated by x = 0. The force operative upon the
  positive half is parallel to OZ, and of amount per unit of area equal
  to

    -b²D d[zeta]/dx = b²kD cos nt;

  and to this force acting over the whole of the plane the actual motion
  on the positive side may be conceived to be due. The secondary
  disturbance corresponding to the element dS of the plane may be
  supposed to be that caused by a force of the above magnitude acting
  over dS and vanishing elsewhere; and it only remains to examine what
  the result of such a force would be.

  Now it is evident that the force in question, supposed to act upon the
  positive half only of the medium, produces just double of the effect
  that would be caused by the same force if the medium were undivided,
  and on the latter supposition (being also localized at a point) it
  comes under the head already considered. According to (18), the effect
  of the force acting at dS parallel to OZ, and of amount equal to

    2b²kD dS cos nt,

  will be a disturbance

              dS sin [phi]
    [zeta]' = ------------ cos(nt - kr)            (20),
               [lambda]r

  regard being had to (12). This therefore expresses the secondary
  disturbance at a distance r and in a direction making an angle [phi]
  with OZ (the direction of primary vibration) due to the element dS of
  the wave-front.

  The proportionality of the secondary disturbance to sin [phi] is
  common to the present law and to that given by Stokes, but here there
  is no dependence upon the angle [theta] between the primary and
  secondary rays. The occurrence of the factor [lambda]r^-1, and the
  necessity of supposing the phase of the secondary wave accelerated by
  a quarter of an undulation, were first established by Archibald Smith,
  as the result of a comparison between the primary wave, supposed to
  pass on without resolution, and the integrated effect of all the
  secondary waves (§ 2). The occurrence of factors such as sin [phi], or
  ½(1 + cos [theta]), in the expression of the secondary wave has no
  influence upon the result of the integration, the effects of all the
  elements for which the factors differ appreciably from unity being
  destroyed by mutual interference.

  The choice between various methods of resolution, all mathematically
  admissible, would be guided by physical considerations respecting the
  mode of action of obstacles. Thus, to refer again to the acoustical
  analogue in which plane waves are incident upon a perforated rigid
  screen, the circumstances of the case are best represented by the
  first method of resolution, leading to symmetrical secondary waves, in
  which the normal motion is supposed to be zero over the unperforated
  parts. Indeed, if the aperture is very small, this method gives the
  correct result, save as to a constant factor. In like manner our
  present law (20) would apply to the kind of obstruction that would be
  caused by an actual physical division of the elastic medium, extending
  over the whole of the area supposed to be occupied by the intercepting
  screen, but of course not extending to the parts supposed to be
  perforated.

  On the electromagnetic theory, the problem of diffraction becomes
  definite when the properties of the obstacle are laid down. The
  simplest supposition is that the material composing the obstacle is
  perfectly conducting, i.e. perfectly reflecting. On this basis A. J.
  W. Sommerfeld (_Math. Ann._, 1895, 47, p. 317), with great
  mathematical skill, has solved the problem of the shadow thrown by a
  semi-infinite plane screen. A simplified exposition has been given by
  Horace Lamb (_Proc. Lond. Math. Soc._, 1906, 4, p. 190). It appears
  that Fresnel's results, although based on an imperfect theory, require
  only insignificant corrections. Problems not limited to two
  dimensions, such for example as the shadow of a circular disk, present
  great difficulties, and have not hitherto been treated by a rigorous
  method; but there is no reason to suppose that Fresnel's results would
  be departed from materially.     (R.)


FOOTNOTES:

  [1] The descending series for J0(z) appears to have been first given
    by Sir W. Hamilton in a memoir on "Fluctuating Functions," _Roy.
    Irish Trans._, 1840.

  [2] Airy, loc. cit. "Thus the magnitude of the central spot is
    diminished, and the brightness of the rings increased, by covering
    the central parts of the object-glass."

  [3] _"Man kann daraus schliessen, was moglicher Weise durch
    Mikroskope noch zu sehen ist. Ein mikroskopischer Gegenstand z. B,
    dessen Durchmesser = ([lambda]) ist, und der aus zwei Theilen
    besteht, kann nicht mehr als aus zwei Theilen bestehend erkannt
    werden. Dieses zeigt uns eine Grenze des Sehvermogens durch
    Mikroskope"_ (_Gilbert's Ann._ 74, 337). Lord Rayleigh has recorded
    that he was himself convinced by Fraunhofer's reasoning at a date
    antecedent to the writings of Helmholtz and Abbe.

  [4] The last sentence is repeated from the writer's article "Wave
    Theory" in the 9th edition of this work, but A. A. Michelson's
    ingenious échelon grating constitutes a realization in an unexpected
    manner of what was thought to be impracticable.--[R.]

  [5] Compare also F. F. Lippich, _Pogg. Ann._ cxxxix. p. 465, 1870;
    Rayleigh, _Nature_ (October 2, 1873).

  [6] The power of a grating to construct light of nearly definite
   wave-length is well illustrated by Young's comparison with the
   production of a musical note by reflection of a sudden sound from a
   row of palings. The objection raised by Herschel (_Light_, § 703) to
   this comparison depends on a misconception.

  [7] It must not be supposed that errors of this order of magnitude
    are unobjectionable in all cases. The position of the middle of the
    bright band representative of a mathematical line can be fixed with a
    spider-line micrometer within a small fraction of the width of the
    band, just as the accuracy of astronomical observations far
    transcends the separating power of the instrument.

  [8] "In the same way we may conclude that in flat gratings any
    departure from a straight line has the effect of causing the dust in
    the slit and the spectrum to have different foci--a fact sometimes
    observed." (Rowland, "On Concave Gratings for Optical Purposes,"
    _Phil. Mag._, September 1883).

  [9] On account of inequalities in the atmosphere giving a variable
    refraction, the light from a star would be irregularly distributed
    over a screen. The experiment is easily made on a laboratory scale,
    with a small source of light, the rays from which, in their course
    towards a rather distant screen, are disturbed by the neighbourhood
    of a heated body. At a moment when the eye, or object-glass of a
    telescope, occupies a dark position, the star vanishes. A fraction of
    a second later the aperture occupies a bright place, and the star
    reappears. According to this view the chromatic effects depend
    entirely upon atmospheric dispersion.

  [10] In experiment a line of light is sometimes substituted for a
    point in order to increase the illumination. The various parts of the
    line are here _independent_ sources, and should be treated
    accordingly. To assume a cylindrical form of primary wave would be
    justifiable only when there is synchronism among the secondary waves
    issuing from the various centres.

  [11] H. Necker (_Phil. Mag._, November 1832); Fox Talbot (_Phil.
    Mag._, June 1833). "When the sun is about to emerge ... every branch
    and leaf is lighted up with a silvery lustre of indescribable
    beauty.... The birds, as Mr Necker very truly describes, appear like
    flying brilliant sparks." Talbot ascribes the appearance to
    diffraction; and he recommends the use of a telescope.



DIFFUSION (from the Lat. _diffundere; dis-_, asunder, and _fundere_, to
pour out), in general, a spreading out, scattering or circulation; in
physics the term is applied to a special phenomenon, treated below.

1. _General Description._--When two different substances are placed in
contact with each other they sometimes remain separate, but in many
cases a gradual mixing takes place. In the case where both the
substances are gases the process of mixing continues until the result is
a uniform mixture. In other cases the proportions in which two different
substances can mix lie between certain fixed limits, but the mixture is
distinguished from a chemical compound by the fact that between these
limits the composition of the mixture is capable of continuous
variation, while in chemical compounds, the proportions of the different
constituents can only have a discrete series of numerical values, each
different ratio representing a different compound. If we take, for
example, air and water in the presence of each other, air will become
dissolved in the water, and water will evaporate into the air, and the
proportions of either constituent absorbed by the other will vary
continuously. But a limit will come when the air will absorb no more
water, and the water will absorb no more air, and throughout the change
a definite surface of separation will exist between the liquid and the
gaseous parts. When no surface of separation ever exists between two
substances they must necessarily be capable of mixing in all
proportions. If they are not capable of mixing in all proportions a
discontinuous change must occur somewhere between the regions where the
substances are still unmixed, thus giving rise to a surface of
separation.

The phenomena of mixing thus involves the following processes:--(1) A
motion of the substances relative to one another throughout a definite
_region_ of space in which mixing is taking place. This relative motion
is called "diffusion." (2) The passage of portions of the mixing
substances across the _surface_ of separation when such a surface
exists. These surface actions are described under various terms such as
solution, evaporation, condensation and so forth. For example, when a
soluble salt is placed in a liquid, the process which occurs at the
surface of the salt is called "solution," but the salt which enters the
liquid by solution is transported from the surface into the interior of
the liquid by "diffusion."

Diffusion may take place in solids, that is, in regions occupied by
matter which continues to exhibit the properties of the solid state.
Thus if two liquids which can mix are separated by a membrane or
partition, the mixing may take place through the membrane. If a solution
of salt is separated from pure water by a sheet of parchment, part of
the salt will pass through the parchment into the water. If water and
glycerin are separated in this way most of the water will pass into the
glycerin and a little glycerin will pass through in the opposite
direction, a property frequently used by microscopists for the purpose
of gradually transferring minute algae from water into glycerin. A still
more interesting series of examples is afforded by the passage of gases
through partitions of metal, notably the passage of hydrogen through
platinum and palladium at high temperatures. When the process is
considered with reference to a membrane or partition taken as a whole,
the passage of a substance from one side to the other is commonly known
as "osmosis" or "transpiration" (see SOLUTION), but what occurs in the
material of the membrane itself is correctly described as diffusion.

Simple cases of diffusion are easily observed qualitatively. If a
solution of a coloured salt is carefully introduced by a funnel into the
bottom of a jar containing water, the two portions will at first be
fairly well defined, but if the mixture can exist in all proportions,
the surface of separation will gradually disappear; and the rise of the
colour into the upper part and its gradual weakening in the lower part,
may be watched for days, weeks or even longer intervals. The diffusion
of a strong aniline colouring matter into the interior of gelatine is
easily observed, and is commonly seen in copying apparatus. Diffusion of
gases may be shown to exist by taking glass jars containing vapours of
hydrochloric acid and ammonia, and placing them in communication with
the heavier gas downmost. The precipitation of ammonium chloride shows
that diffusion exists, though the chemical action prevents this example
from forming a typical case of diffusion. Again, when a film of Canada
balsam is enclosed between glass plates, the disappearance during a few
weeks of small air bubbles enclosed in the balsam can be watched under
the microscope.

In fluid media, whether liquids or gases, the process of mixing is
greatly accelerated by stirring or agitating the fluids, and liquids
which might take years to mix if left to themselves can thus be mixed in
a few seconds. It is necessary to carefully distinguish the effects of
agitation from those of diffusion proper. By shaking up two liquids
which do not mix we split them up into a large number of different
portions, and so greatly increase the area of the surface of separation,
besides decreasing the thicknesses of the various portions. But even
when we produce the appearance of a uniform turbid mixture, the small
portions remain quite distinct. If however the fluids can really mix,
the final process must in every case depend on diffusion, and all we do
by shaking is to increase the sectional area, and decrease the thickness
of the diffusing portions, thus rendering the completion of the
operation more rapid. If a gas is shaken up in a liquid the process of
absorption of the bubbles is also accelerated by capillary action, as
occurs in an ordinary sparklet bottle. To state the matter precisely,
however finely two fluids have been subdivided by agitation, the
molecular constitution of the different portions remains unchanged. The
ultimate process by which the individual molecules of two different
substances become mixed, producing finally a homogeneous mixture, is in
every case diffusion. In other words, diffusion is that relative motion
of the molecules of two different substances by which the proportions of
the molecules in any region containing a finite number of molecules are
changed.

  In order, therefore, to make accurate observations of diffusion in
  fluids it is necessary to guard against any cause which may set up
  currents; and in some cases this is exceedingly difficult. Thus, if
  gas is absorbed at the upper surface of a liquid, and if the gaseous
  solution is heavier than the pure liquid, currents may be set up, and
  a steady state of diffusion may cease to exist. This has been tested
  experimentally by C. G. von Hüfner and W. E. Adney. The same thing may
  happen when a gas is evolved into a liquid at the surface of a solid
  even if no bubbles are formed; thus if pieces of aluminium are placed
  in caustic soda, the currents set up by the evolution of hydrogen are
  sufficient to set the aluminium pieces in motion, and it is probable
  that the motions of the Diatomaceae are similarly caused by the
  evolution of oxygen. In some pairs of substances diffusion may take
  place more rapidly than in others. Of course the progress of events in
  any experiment necessarily depends on various causes, such as the size
  of the containing vessels, but it is easy to see that when experiments
  with different substances are carried out under similar conditions,
  however these "similar conditions" be defined, the rates of diffusion
  must be capable of numerical comparison, and the results must be
  expressible in terms of at least one physical quantity, which for any
  two substances can be called their coefficient of diffusion. How to
  select this quantity we shall see later.

2 _Quantitative Methods of observing Diffusion._--The simplest plan of
determining the progress of diffusion between two liquids would be to
draw off and examine portions from different strata at some stage in the
process; the disturbance produced would, however, interfere with the
subsequent process of diffusion, and the observations could not be
continued. By placing in the liquid column hollow glass beads of
different average densities, and observing at what height they remain
suspended, it is possible to trace the variations of density of the
liquid column at different depths, and different times. In this method,
which was originally introduced by Lord Kelvin, difficulties were caused
by the adherence of small air bubbles to the beads.

In general, optical methods are the most capable of giving exact
results, and the following may be distinguished, (a) _By refraction in a
horizontal plane._ If the containing vessel is in the form of a prism,
the deviation of a horizontal ray of light in passing through the prism
determines the index of refraction, and consequently the density of the
stratum through which the ray passes, (b) _By refraction in a vertical
plane._ Owing to the density varying with the depth, a horizontal ray
entering the liquid also undergoes a small vertical deviation, being
bent downwards towards the layers of greater density. The observation of
this vertical deviation determines not the actual density, but its rate
of variation with the depth, i.e. the "density gradient" at any point,
(c) _By the saccharimeter._ In the cases of solutions of sugar, which
cause rotation of the plane of polarized light, the density of the sugar
at any depth may be determined by observing the corresponding angle of
rotation, this was done originally by W. Voigt.

3. _Elementary Definitions of Coefficient of Diffusion._--The simplest
case of diffusion is that of a substance, say a gas, diffusing in the
interior of a homogeneous solid medium, which remains at rest, when no
external forces act on the system. We may regard it as the result of
experience that: (1) if the density of the diffusing substance is
everywhere the same no diffusion takes place, and (2) if the density of
the diffusing substance is different at different points, diffusion will
take place from places of greater to those of lesser density, and will
not cease until the density is everywhere the same. It follows that the
rate of flow of the diffusing substance at any point in any direction
must depend on the density gradient at that point in that direction,
i.e. on the rate at which the density of the diffusing substance
decreases as we move in that direction. We may define the _coefficient
of diffusion_ as the ratio of the total mass per unit area which flows
across any small section, to the rate of decrease of the density per
unit distance in a direction perpendicular to that section.

  In the case of steady diffusion parallel to the axis of x, if [rho] be
  the density of the diffusing substance, and q the mass which flows
  across a unit of area in a plane perpendicular to the axis of x, then
  the density gradient is -d[rho]/dx and the ratio of q to this is
  called the "coefficient of diffusion." By what has been said this
  ratio remains finite, however small the actual gradient and flow may
  be., and it is natural to assume, at any rate as a first
  approximation, that it is constant as far as the quantities in
  question are concerned. Thus if the coefficient of diffusion be
  denoted by K we have q= -K(d[rho]/dx).

  Further, the rate at which the quantity of substance is increasing in
  an element between the distances x and x+dx is equal to the difference
  of the rates of flow in and out of the two faces, whence as in
  hydrodynamics, we have d[rho]/dt =-dq/dx.

  It follows that the equation of diffusion in this case assumes the
  form

    d[rho]   d   /  d[rho] \
    ------ = -- ( K ------  ),
      dt     dx  \    dx   /

  which is identical with the equations representing conduction of heat,
  flow of electricity and other physical phenomena. For motion in three
  dimensions we have in like manner

    d[rho]   d   /  d[rho]\    d   /  d[rho]\    d   /  d[rho]\
    ------ = -- ( K ------ ) + -- ( K ------ ) + -- ( K ------ );
      dt     dx  \   dx   /    dy  \   dy   /    dz  \   dz   /

  and the corresponding equations in electricity and heat for
  anisotropic substances would be available to account for any parallel
  phenomena, which may arise, or might be conceived, to exist in
  connexion with diffusion through a crystalline solid.

In the case of a very dilute solution, the coefficient of diffusion of
the dissolved substance can be defined in the same way as when the
diffusion takes place in a solid, because the effects of diffusion will
not have any perceptible influence on the solvent, and the latter may
therefore be regarded as remaining practically at rest. But in most
cases of diffusion between two fluids, both of the fluids are in motion,
and hence there is far greater difficulty in determining the motion, and
even in defining the coefficient of diffusion. It is important to notice
in the first instance, that it is only the relative motion of the two
substances which constitutes diffusion. Thus when a current of air is
blowing, under ordinary circumstances the changes which take place are
purely mechanical, and do not depend on the separate diffusions of the
oxygen and nitrogen of which the air is mainly composed. It is only when
two gases are flowing with unequal velocity, that is, when they have a
relative motion, that these changes of relative distribution, which are
called diffusion, take place. The best way out of the difficulty is to
investigate the separate motions of the two fluids, taking account of
the mechanical actions exerted on them, and supposing that the mutual
action of the fluids causes either fluid to resist the relative motion
of the other.

4. _The Coefficient of Resistance._--Let us call the two diffusing
fluids A and B. If B were absent, the motion of the fluid A would be
determined entirely by the variations of pressure of the fluid A, and by
the external forces, such as that due to gravity acting on A. Similarly
if A were absent, the motion of B would be determined entirely by the
variations of pressure due to the fluid B, and by the external forces
acting on B. When both fluids are mixed together, each fluid tends to
resist the relative motion of the other, and by the law of equality of
action and reaction, the resistance which A experiences from B is
everywhere equal and opposite to the resistance which B experiences from
A. If the amount of this resistance per unit volume be divided by the
relative velocity of the two fluids, and also by the product of their
densities, the quotient is called the "coefficient of resistance." If
then [rho]1, [rho]2 are the densities cf the two fluids, u1, u2 their
velocities, C the coefficient of resistance, then the portion of the
fluid A contained in a small element of volume v will experience from
the fluid B a resistance C[rho]1[rho]2v(u1- u2), and the fluid B
contained in the same volume element will experience from the fluid A an
equal and opposite resistance, C[rho]1[rho]2v(u2 - u1).

This definition implies the following laws of resistance to diffusion,
which must be regarded as based on experience, and not as self-evident
truths: (1) each fluid tends to assume, so far as diffusion is
concerned, the same equüibrium distribution that it would assume if its
motion were unresisted by the presence of the other fluid. (Of course,
the mutual attraction of gravitation of the two fluids might affect the
final distribution, but this is practically negligible. Leaving such
actions as this out of account the following statement is correct.) In
a state of equilibrium, the density of each fluid at any point thus
depends only on the partial pressure of that fluid alone, and is the
same as if the other fluids were absent. It does not depend on the
partial pressures of the other fluids. If this were not the case, the
resistance to diffusion would be analogous to friction, and would
contain terms which were independent of the relative velocity u2 - u1.
(2) For slow motions the resistance to diffusion is (approximately at
any rate) proportional to the relative velocity. (3) The coefficient of
resistance C is not necessarily always constant; it may, for example,
and, in general, does, depend on the temperature.

  If we form the equations of hydrodynamics for the different fluids
  occurring in any mixture, taking account of diffusion, but neglecting
  viscosity, and using suffixes 1, 2 to denote the separate fluids,
  these assume the form given by James Clerk Maxwell ("Diffusion," in
  _Ency. Brit._, 9th ed.):--

          Du1   dp1
    [rho] --- + --- - X1[rho]1 + C12[rho]1[rho]2(u1 - u2) + &c. = 0,
          Dt    dx

  where

    Du1   du1      du1      du1      du1
    --- = --- + u1 --- + v1 --- + w1 ---,
    Dt    dt       dx       dy       dz

  and these equations imply that when diffusion and other motions cease,
  the fluids satisfy the separate conditions of equilibrium dp1/dx -
  X1[rho]1 = 0. The assumption made in the following account is that
  terms such as Du1/Dt may be neglected in the cases considered.

A further property based on experience is that the motions set up in a
mixture by diffusion are very slow compared with those set up by
mechanical actions, such as differences of pressure. Thus, if two gases
at equal temperature and pressure be allowed to mix by diffusion, the
heavier gas being below the lighter, the process will take a long time;
on the other hand, if two gases, or parts of the same gas, at different
pressures be connected, equalization of pressure will take place almost
immediately. It follows from this property that the forces required to
overcome the "inertia" of the fluids in the motions due to diffusion are
quite imperceptible. At any stage of the process, therefore, any one of
the diffusing fluids may be regarded as in equilibrium under the action
of its own partial pressure, the external forces to which it is
subjected and the resistance to diffusion of the other fluids.

5. _Slow Diffusion of two Gases. Relation between the Coefficients of
Resistance and of Diffusion._--We now suppose the diffusing substances
to be two gases which obey Boyle's law, and that diffusion takes place
in a closed cylinder or tube of unit sectional area at constant
temperature, the surfaces of equal density being perpendicular to the
axis of the cylinder, so that the direction of diffusion is along the
length of the cylinder, and we suppose no external forces, such as
gravity, to act on the system.

  The densities of the gases are denoted by [rho]1, [rho]2, their
  velocities of diffusion by u1, u2, and if their partial pressures are
  p1, p2, we have by Boyle's law p1 = k1[rho]1, p2 = k2[rho]2, where
  k1, k2 are constants for the two gases, the temperature being constant.
  The axis of the cylinder is taken as the axis of x.

  From the considerations of the preceding section, the effects of
  inertia of the diffusing gases may be neglected, and at any instant of
  the process either of the gases is to be treated as kept in
  equilibrium by its partial pressure and the resistance to diffusion
  produced by the other gas. Calling this resistance per unit volume R,
  and putting R = C[rho]1[rho]2(u1 - u2), where C is the coefficient of
  resistance, the equations of equilibrium give

    dp1                                  dp2
    --- + C[rho]1[rho]2(u1 - u2)= 0, and --- + C[rho]1[rho]2(u2 - u1)= 0 (1).
    dx                                   dx

  These involve

    dp1   dp2
    --- + --- = 0 or p1 + p2 = P                      (2)
    dx    dx

  where P is the total pressure of the mixture, and is everywhere
  constant, consistently with the conditions of mechanical equilibrium.

  Now dp1/dx is the pressure-gradient of the first gas, and is, by
  Boyle's law, equal to k1 times the corresponding density-gradient.
  Again [rho]1u1 is the mass of gas flowing across any section per unit
  time, and k1[rho]1u1 or p1u1 can be regarded as representing the flux
  of partial pressure produced by the motion of the gas. Since the total
  pressure is everywhere constant, and the ends of the cylinder are
  supposed fixed, the fluxes of partial pressure due to the two gases
  are equal and opposite, so that

    p1u1 + p2u2 = 0 or k1[rho]1u1 + k2[rho]2u2 = 0    (3).

  From (2) (3) we find by elementary algebra

    u1/p2 = - u2/p1 = (u1 - u2)/(p1 + p2) = (u1 - u2)/P,

  and therefore

    p2u1 = - p2u2 = p1p2(u1 - u2)/P = k1k2[rho]1[rho]2(u1 - u2)/P

  Hence equations (1) (2) gives

    dp1    CP                   dp2    CP
    --- + ---- (p1u1) = 0, and  --- + ---- (p2u2) = 0;
    dx    k1k2                  dx    k1k2

  whence also substituting p1 = k1[rho]1, p2 = k2[rho]2, and by
  transposing

                 k1k2 d[rho]1                   k1k2 d[rho]2
    [rho]1u1 = - ---- -------, and [rho]2u2 = - ---- -------.
                  CP    dx                       CP     dx

  We may now define the "coefficient of diffusion" of either gas as the
  ratio of the rate of flow of that gas to its density-gradient. With
  this definition, the coefficients of diffusion of both the gases in a
  mixture are equal, each being equal to k1k2/CP. The ratios of the
  fluxes of partial pressure to the corresponding pressure-gradients are
  also equal to the same coefficient. Calling this coefficient K, we
  also observe that the equations of continuity for the two gases are

    d[rho]1   d([rho]1u1)          d[rho]2   d([rho]2u2)
    ------- + ----------- = 0, and ------- + ----------- = 0,
       dt         dx                  dt         dx

  leading to the equations of diffusion

    d[rho]1   d   /  d[rho]1\        d[rho]2   d   /  d[rho]2\
    ------- = -- ( K ------- ) , and ------- = -- ( K ------- ),
      dt      dx  \    dx   /          dt      dx  \    dx   /

  exactly as in the case of diffusion through a solid.

If we attempt to treat diffusion in liquids by a similar method, it is,
in the first place, necessary to define the "partial pressure" of the
components occurring in a liquid mixture. This leads to the conception
of "osmotic pressure," which is dealt with in the article SOLUTION. For
dilute solutions at constant temperature, the assumption that the
osmotic pressure is proportional to the density, leads to results
agreeing fairly closely with experience, and this fact may be
represented by the statement that a substance occurring in a dilute
solution behaves like a perfect gas.

6. _Relation of the Coefficient of Diffusion to the Units of Length and
Time._--We may write the equation defining K in the form

             I   d[rho]
  u = -K × ----- ------.
           [rho]   dx

Here -d[rho]/[rho]dx represents the "percentage rate" at which the
density decreases with the distance x; and we thus see that the
coefficient of diffusion represents the ratio of the velocity of flow to
the percentage rate at which the density decreases with the distance
measured in the direction of flow. This percentage rate being of the
nature of a number divided by a length, and the velocity being of the
nature of a length divided by a time, we may state that K is of two
dimensions in length and - 1 in time, i.e. dimensions L²/T.

  _Example 1._ Taking K = 0.1423 for carbon dioxide and air (at
  temperature 0° C. and pressure 76 cm. of mercury) referred to a
  centimetre and a second as units, we may interpret the result as
  follows:--Supposing in a mixture of carbon dioxide and air, the
  density of the carbon dioxide decreases by, say, 1, 2 or 3% of itself
  in a distance of 1 cm., then the corresponding velocities of the
  diffusing carbon dioxide will be respectively 0.01, 0.02 and 0.03
  times 0.1423, that is, 0.001423, 0.002846 and 0.004269 cm. per second
  in the three cases.

  _Example 2._ If we wished to take a foot and a second as our units, we
  should have to divide the value of the coefficient of diffusion in
  Example 1 by the square of the number of centimetres in 1 ft., that
  is, roughly speaking, by 900, giving the new value of K = 0.00016
  roughly.

7. _Numerical Values of the Coefficient of Diffusion._--The table on p.
258 gives the values of the coefficient of diffusion of several of the
principal pairs of gases at a pressure of 76 cm. of mercury, and also of
a number of other substances. In the gases the centimetre and second are
taken as fundamental units, in other cases the centimetre and day.

8. _Irreversible Changes accompanying Diffusion._--The diffusion of two
gases at constant pressure and temperature is a good example of an
"irreversible process." The gases always tend to mix, never to separate.
In order to separate the gases a change must be effected in the external
conditions to which the mixture is subjected, either by liquefying one
of the gases, or by separating them by diffusion through a membrane, or
by bringing other outside influences to bear on them. In the case of
liquids, electrolysis affords a means of separating the constituents of
a mixture. Every such method involves some change taking place outside
the mixture, and this change may be regarded as a "compensating
transformation." We thus have an instance of the property that every
irreversible change leaves an indelible imprint somewhere or other on
the progress of events in the universe. That the process of diffusion
obeys the laws of irreversible thermodynamics (if these laws are
properly stated) is proved by the fact that the compensating
transformations required to separate mixed gases do not essentially
involve anything but transformation of energy. The process of allowing
gases to mix by diffusion, and then separating them by a compensating
transformation, thus constitutes an irreversible cycle, the outside
effects of which are that energy somewhere or other must be less capable
of transformation than it was before the change. We express this fact by
stating that an irreversible process essentially implies a loss of
availability. To measure this loss we make use of the laws of
thermodynamics, and in particular of Lord Kelvin's statement that "It is
impossible by means of inanimate material agency to derive mechanical
effect from any portion of matter by cooling it below the temperature of
the coldest of the surrounding objects."

  +-------------------------------------------+---------+---------------------+--------------+
  |                Substances.                |  Temp.  |         K.          |   Author.    |
  +-------------------------------------------+---------+---------------------+--------------+
  | Carbon dioxide and air                    |    0°C. | 0.1423 cm²/sec.     | J. Loschmidt.|
  |    "      "        hydrogen               |    0°C. | 0.5558    "         |      "       |
  |    "      "        oxygen                 |    0°C. | 0.1409    "         |      "       |
  |    "      "        carbon monoxide        |    0°C. | 0.1406    "         |      "       |
  |    "      "        marsh gas (methane)    |    0°C. | 0.1586    "         |      "       |
  |    "      "        nitrous oxide          |    0°C. | 0.0983    "         |      "       |
  | Hydrogen and oxygen                       |    0°C. | 0.7214    "         |      "       |
  |    "      "  carbon monoxide              |    0°C. | 0.6422    "         |      "       |
  |    "      "  sulphur dioxide              |    0°C. | 0.4800    "         |      "       |
  | Oxygen and carbon monoxide                |    0°C. | 0.1802    "         |      "       |
  | Water and ammonia                         |   20°C. | 1.250     "         | G. Hüfner.   |
  |    "        "                             |    5°C. | 0.822     "         |      "       |
  |    "      common salt (density 1.0269)    |         | 0.355 cm²/hour.     | J. Graham.   |
  |    "         "          "        "        |14.33°C. | 1.020, 0.996, 0.972,|      "       |
  |                                           |         | 0.932 cm²/day.      | F. Heimbrodt.|
  |    "      zinc sulphate (0.312 gm/cm³)    |         | 0.1162    "         | W. Seitz.    |
  |    "      zinc sulphate (normal)          |         | 0.2355    "         |      "       |
  |    "      zinc acetate (double normal)    |         | 0.1195    "         |      "       |
  |    "      zinc formate (half normal)      |         | 0.4654    "         |      "       |
  |    "      cadmium sulphate (double normal)|         | 0.2456    "         |      "       |
  |    "      glycerin (1/8n, ½n, 7/8n, 7/8n) |10.14°C. | 0.356, 0.350, 0.342,| F. Heimbrodt.|
  |                                           |         | 0.315 cm²/day.      |      "       |
  |    "      urea          "         "       |14.83°C. | 0.973, 0.946, 0.926,|      "       |
  |                                           |         | 0.883 cm²/day.      |      "       |
  |    "      hydrochloric acid               |14.30°C. | 2.208, 2.331,       |      "       |
  |                                           |         | 2.480 cm²/day       |      "       |
  | Gelatin 20% and ammonia                   |   17°C. | 127.1     "         | A. Hagenbach.|
  |    "     "      carbon dioxide            |         | 0.845     "         |      "       |
  |    "     "      nitrous oxide             |         | 0.509     "         |      "       |
  |    "     "      oxygen                    |         | 0.230     "         |      "       |
  |    "     "      hydrogen                  |         | 0.0565    "         |      "       |
  +-------------------------------------------+---------+---------------------+--------------+

  Let us now assume that we have any syste m such as the gases above
  considered, and that it is in the presence of an indefinitely extended
  medium which we shall call the "auxiliary medium." If heat be taken
  from any part of the system, only part of this heat can be converted
  into work by means of thermodynamic engines; and the rest will be
  given to the auxiliary medium, and will constitute unavailable energy
  or waste. To understand what this means, we may consider the case of a
  condensing steam engine. Only part of the energy liberated by the
  combustion of the coal is available for driving the engine, the rest
  takes the form of heat imparted to the condenser. The colder the
  condenser the more efficient is the engine, and the smaller is the
  quantity of waste.

  The amount of unavailable energy associated with any given
  transformation is proportional to the absolute temperature of the
  auxiliary medium. When divided by that temperature the quotient is
  called the change of "entropy" associated with the given change (see
  THERMODYNAMICS). Thus if a body at temperature T receives a quantity
  of heat Q, and if T0 is the temperature of the auxiliary medium, the
  quantity of work which could be obtained from Q by means of ideal
  thermodynamic engines would be Q(1 - T0/T), and the balance, which is
  QT0/T, would take the form of unavailable or waste energy given to the
  medium. The quotient of this, when divided by T0, is Q/T, and this
  represents the quantity of entropy associated with Q units of heat at
  temperature T.

  Any irreversible change for which a compensating transformation of
  energy exists represents, therefore, an increase of unavailable
  energy, which is measurable in terms of entropy. The increase of
  entropy is independent of the temperature of the auxiliary medium. It
  thus affords a measure of the extent to which energy has run to waste
  during the change. Moreover, when a body is heated, the increase of
  entropy is the factor which determines how much of the energy imparted
  to the body is unavailable for conversion into work under given
  conditions. In all cases we have

    increase of unavailable energy
    ------------------------------- = increase of entropy.
    temperature of auxiliary medium

  When diffusion takes place between two gases inside a closed vessel at
  uniform pressure and temperature no energy in the form of heat or work
  is received from without, and hence the entropy gained by the gases
  from without is zero. But the irreversible processes inside the vessel
  may involve a gain of entropy, and this can only be estimated by
  examining by what means mixed gases can be separated, and, in
  particular, under what conditions the process of mixing and separating
  the gases could (theoretically) be made reversible.

9. _Evidence derived from Liquefaction of one or both of the
Gases._--The gases in a mixture can often be separated by liquefying, or
even solidifying, one or both of the components. In connexion with this
property we have the important law according to which "The pressure of a
vapour in equilibrium with its liquid depends only on the temperature
and is independent of the pressures of any other gases or vapours which
may be mixed with it." Thus if two closed vessels be taken containing
some water and one be exhausted, the other containing air, and if the
temperatures be equal, evaporation will go on until the pressure of the
vapour in the exhausted vessel is equal to its _partial_ pressure in the
other vessel, notwithstanding the fact that the _total_ pressure in the
latter vessel is greater by the pressure of the air.

  To separate mixed gases by liquefaction, they must be compressed and
  cooled till one separates in the form of a liquid. If no changes are
  to take place outside the system, the separate components must be
  allowed to expand until the work of expansion is equal to the work of
  compression, and the heat given out in compression is reabsorbed in
  expansion. The process may be made as nearly reversible as we like by
  performing the operations so slowly that the substances are
  practically in a state of equilibrium at every stage. This is a
  consequence of an important axiom in thermodynamics according to which
  "any small change in the neighbourhood of a state of equilibrium is to
  a first approximation reversible."

  Suppose now that at any stage of the compression the partial pressures
  of the two gases are p1 and p2, and that the volume is changed from V
  to V - dV. The work of compression is (p1 + p2)dV, and this work will
  be restored at the corresponding stage if each of the separated gases
  increases in volume from V - dV to V. The ultimate state of the
  separated gases will thus be one in which each gas occupies the volume
  V originally occupied by the mixture.

  We may now obtain an estimate of the amount of energy rendered
  unavailable by diffusion. We suppose two gases occupying volumes V1
  and V2 at equal pressure p to mix by diffusion, so that the final
  volume is V1 + V2. Then if before mixing each gas had been allowed to
  expand till its volume was V1 + V2, work would have been done in the
  expansion, and the gases could still have been mixed by a reversal of
  the process above described. In the actual diffusion this work of
  expansion is lost, and represents energy rendered unavailable at the
  temperature at which diffusion takes place. When divided by that
  temperature the quotient gives the increase of entropy. Thus the
  irreversible processes, and, in particular, the entropy changes
  associated with diffusion of two gases at uniform pressure, are the
  same as would take place if each of the gases in turn were to expand
  by rushing into a vacuum, till it occupied the whole volume of the
  mixture. A more rigorous proof involves considerations of the
  thermodynamic potentials, following the methods of J. Willard Gibbs
  (see ENERGETICS).

  Another way in which two or more mixed gases can be separated is by
  placing them in the presence of a liquid which can freely absorb one
  of the gases, but in which the other gas or gases are insoluble. Here
  again it is found by experience that when equilibrium exists at a
  given temperature between the dissolved and undissolved portions of
  the first gas, the partial pressure of that gas in the mixture depends
  on the temperature alone, and is independent of the partial pressures
  of the insoluble gases with which it is mixed, so that the conclusions
  are the same as before.

10. _Diffusion through a Membrane or Partition. Theory of the
semi-permeable Membrane._--It has been pointed out that diffusion of
gases frequently takes place in the interior of solids; moreover,
different gases behave differently with respect to the same solid at the
same temperature. A membrane or partition formed of such a solid can
therefore be used to effect a more or less complete separation of gases
from a mixture. This method is employed commercially for extracting
oxygen from the atmosphere, in particular for use in projection lanterns
where a high degree of purity is not required. A similar method is often
applied to liquids and solutions and is known as "dialysis."

In such cases as can be tested experimentally it has been found that a
gas always tends to pass through a membrane from the side where its
density, and therefore its partial pressure, is greater to the side
where it is less; so that for equilibrium the partial pressures on the
two sides must be equal. This result is unaffected by the presence of
other gases on one or both sides of the membrane. For example, if
different gases at the same pressure are separated by a partition
through which one gas can pass more rapidly than the other, the
diffusion will give rise to a difference of pressure on the two sides,
which is capable of doing mechanical work in moving the partition. In
evidence of this conclusion Max Planck quotes a test experiment made by
him in the Physical Institute of the university of Munich in 1883,
depending on the fact that platinum foil at white heat is permeable to
hydrogen but impermeable to air, so that if a platinum tube filled with
hydrogen be heated the hydrogen will diffuse out, leaving a vacuum.

  The details of the experiment may be quoted here:--"A glass tube of
  about 5 mm. internal diameter, blown out to a bulb at the middle, was
  provided with a stop-cock at one end. To the other a platinum tube 10
  cm. long was fastened, and closed at the end. The whole tube was
  exhausted by a mercury pump, filled with hydrogen at ordinary
  atmospheric pressure, and then closed. The closed end of the platinum
  portion was then heated in a horizontal position by a Bunsen burner.
  The connexion between the glass and platinum tubes, having been made
  by means of sealing-wax, had to be kept cool by a continuous current
  of water to prevent the softening of the wax. After four hours the
  tube was taken from the flame, cooled to the temperature of the room,
  and the stop-cock opened under mercury. The mercury rose rapidly,
  almost completely filling the tube, proving that the tube had been
  very nearly exhausted."

[Illustration]

In order that diffusion through a membrane may be reversible so far as a
particular gas is concerned, the process must take place so slowly that
equilibrium is set up at every stage (see § 9 above). In order to
separate one gas from another consistently with this condition it is
necessary that no diffusion of the latter gas should accompany the
process. The name "semi-permeable" is applied to an ideal membrane or
partition through which one gas can pass, and which offers an
insuperable barrier to any diffusion whatever of a second gas. By means
of two semi-permeable partitions acting oppositely with respect to two
different gases A and B these gases could be mixed or separated by
reversible methods. The annexed figure shows a diagrammatic
representation of the process.

  We suppose the gases contained in a cylindrical tube; P, Q, R, S are
  four pistons, of which P and R are joined to one connecting rod, Q and
  S to another. P, S are impermeable to both gases; Q is semi-permeable,
  allowing the gas A to pass through but not B, similarly R allows the
  gas B to pass through but not A. The distance PR is equal to the
  distance QS, so that if the rods are pushed towards each other as far
  as they will go, P and Q will be in contact, as also R and S. Imagine
  the space RQ filled with a mixture of the two gases under these
  conditions. Then by slowly drawing the connecting rods apart until R,
  Q touch, the gas A will pass into the space PQ, and B will pass into
  the space RS, and the gases will finally be completely separated;
  similarly, by pushing the connecting rods together, the two gases will
  be remixed in the space RQ. By performing the operations slowly enough
  we may make the processes as nearly reversible as we please, so that
  no available energy is lost in either change. The gas A being at every
  instant in equilibrium on the two sides of the piston Q, its density,
  and therefore its partial pressure, is the same on both sides, and the
  same is true regarding the gas B on the two sides of R. Also _no work
  is done in moving the pistons_, for the partial pressures of B on the
  two sides of R balance each other, consequently, the resultant thrust
  on R is due to the gas A alone, and is equal and opposite to its
  resultant thrust on P, so that the connecting rods are at every
  instant in a state of mechanical equilibrium so far as the pressures
  of the gases A and B are concerned. We conclude that in the reversible
  separation of the gases by this method at constant temperature without
  the production or absorption of mechanical work, the densities and the
  partial pressures of the two separated gases are the same as they were
  in the mixture. These conclusions are in entire agreement with those
  of the preceding section. If this agreement did not exist it would be
  possible, theoretically, to obtain perpetual motion from the gases in
  a way that would be inconsistent with the second law of
  thermodynamics.

Most physicists admit, as Planck does, that it is impossible to obtain
an ideal semi-permeable substance; indeed such a substance would
necessarily have to possess an infinitely great resistance to diffusion
for such gases as could not penetrate it. But in an experiment performed
under actual conditions the losses of available energy arising from this
cause would be attributable to the imperfect efficiency of the
partitions and not to the gases themselves; moreover, these losses are,
in every case, found to be completely in accordance with the laws of
irreversible thermodynamics. The reasoning in this article being
somewhat condensed the reader must necessarily be referred to treatises
on thermodynamics for further information on points of detail connected
with the argument. Even when he consults these treatises he may find
some points omitted which have been examined in full detail at some time
or other, but are not sufficiently often raised to require mention in
print.

II. _Kinetic Models of Diffusion._--Imagine in the first instance that a
very large number of red balls are distributed over one half of a
billiard table, and an equal number of white balls over the other half.
If the balls are set in motion with different velocities in various
directions, diffusion will take place, the red balls finding their way
among the white ones, and vice versa; and the process will be retarded
by collisions between the balls. The simplest model of a perfect gas
studied in the kinetic theory of gases (see MOLECULE) differs from the
above illustration in that the bodies representing the molecules move in
space instead of in a plane, and, unlike billiard balls, their motion is
unresisted, and they are perfectly elastic, so that no kinetic energy is
lost either during their free motions, or at a collision.

  The mathematical analysis connected with the application of the
  kinetic theory to diffusion is very long and cumbersome. We shall
  therefore confine our attention to regarding a medium formed of
  elastic spheres as a mechanical model, by which the most important
  features of diffusion can be illustrated. We shall assume the results
  of the kinetic theory, according to which:--(1) In a dynamical model
  of a perfect gas the mean kinetic energy of translation of the
  molecules represents the absolute temperature of the gas. (2) The
  pressure at any point is proportional to the product of the number of
  molecules in unit volume about that point into the mean square of the
  velocity. (The mean square of the velocity is different from but
  proportional to the square of the mean velocity, and in the subsequent
  arguments either of these two quantities can generally be taken.) (3)
  In a gas mixture represented by a mixture of molecules of unequal
  masses, the mean kinetic energies of the different kinds are equal.

  Consider now the problem of diffusion in a region containing two kinds
  of molecules A and B of unequal mass. The molecules of A in the
  neighbourhood of any point will, by their motion, spread out in every
  direction until they come into collision with other molecules of
  either kind, and this spreading out from every point of the medium
  will give rise to diffusion. If we imagine the velocities of the A
  molecules to be equally distributed in all directions, as they would
  be in a homogeneous mixture, it is obvious that the process of
  diffusion will be greater, _ceteris paribus_, the greater the velocity
  of the molecules, and the greater the length of the free path before a
  collision takes place. If we assume consistently with this, that the
  coefficient of diffusion of the gas A is proportional to the mean
  value of Wala, where wa is the velocity and la is the length of the
  path of a molecule of A, this expression for the coefficient of
  diffusion is of the right dimensions in length and time. If, moreover,
  we observe that when diffusion takes place in a fixed direction, say
  that of the axis of x, it depends only on the resolved part of the
  velocity and length of path in that direction: this hypothesis readily
  leads to our taking the mean value of 1/3w_a l_a as the coefficient of
  diffusion for the gas A. This value was obtained by O. E. Meyer and
  others.

  Unfortunately, however, it makes the coefficients of diffusion unequal
  for the two gases, a result inconsistent with that obtained above from
  considerations of the coefficient of resistance, and leading to the
  consequence that differences of pressure would be set up in different
  parts of the gas. To equalize these differences of pressure, Meyer
  assumed that a counter current is set up, this current being, of
  course, very slow in practice; and J. Stefan assumed that the
  diffusion of one gas was not affected by collisions between molecules
  of the _same gas_. When the molecules are mixed in equal proportions
  both hypotheses lead to the value 1/6([w_a l_a] + [w_b l_b]