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Title: A Budget of Paradoxes, Volume II (of II)
Author: De Morgan, Augustus, 1806-1871
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 "A Budget of Paradoxes, Volume II (of II)" ***

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(OF II)***

Transcriber's note:

      A few typographical errors have been corrected: they are
      listed at the end of the text.

      In mathematical formulae the carat (^) and underscore (_)
      introduce superscripts or subscripts respectively, of one
      character or a group enclosed in curly braces ({xyz}).
      Elsewhere underscores delimit italics in the text, and
      braces enclose the original page numbers thus {123}.








Volume II


       *       *       *       *       *

_This new Dover Edition, published in 1954,
is an unabridged republication of the Second Edition
of 1915, with a new introduction by Professor Ernest Nagel._

_Copyright 1954 by Dover Publications, Inc.
Manufactured in the United States of America_

       *       *       *       *       *





With the general run of the philosophical atheists of the last century the
notion of a God was an hypothesis. There was left an admitted possibility
that the vague somewhat which went by more names than one, might be
personal, intelligent, and superintendent. In the works of Laplace,[1] who
is sometimes called an atheist from his writings, there is nothing from
which such an inference can be drawn: unless indeed a Reverend Fellow of
the Royal Society may be held to be the fool who said in his heart, etc.,
etc., if his contributions to the _Philosophical Transactions_ go no higher
than _nature_. The following anecdote is well known in Paris, but has never
been printed entire. Laplace once went in form to present some edition of
his "Système du Monde" to the First Consul, or Emperor. Napoleon, whom some
wags had told that this book contained no mention of the name of God, and
who was fond of putting embarrassing questions, received it with--"M.
Laplace, they tell me you have written this large book on the system of the
universe, and have never even mentioned its Creator." Laplace, who, though
the most supple of politicians, was as stiff as a martyr on every point of
his philosophy or religion (e. g., even under Charles X he never concealed
his dislike of the priests), drew himself up and answered {2} bluntly, "Je
n'avais pas besoin de cette hypothèse-là."[2] Napoleon, greatly amused,
told this reply to Lagrange, who exclaimed, "Ah! c'est une belle hypothèse;
ça explique beaucoup de choses."[3]

It is commonly said that the last words of Laplace were, "Ce que nous
connaissons est peu de chose; ce que nous ignorons est immense."[4] This
looks like a parody on Newton's pebbles:[5] the following is the true
account; it comes to me through one remove from Poisson.[6] After the
publication (in 1825) of the fifth volume of the _Mécanique Céleste_,
Laplace became gradually weaker, and with it musing and abstracted. He
thought much on the great problems of existence, and often muttered to
himself, _Qu'est ce que c'est que tout cela!_[7] After many alternations,
he appeared at last so permanently prostrated that his family applied to
his favorite pupil, M. Poisson, to try to get a word from him. Poisson paid
a visit, and after a few words of salutation, said, "J'ai une bonne
nouvelle à vous annoncer: on a reçu au Bureau des Longitudes une lettre
d'Allemagne annonçant que M. Bessel a vérifié par l'observation vos
découvertes théoriques sur les satellites de Jupiter."[8] Laplace opened
his eyes and answered with deep {3} gravity, "_L'homme ne poursuit que des
chimères_."[9] He never spoke again. His death took place March 5, 1827.

The language used by the two great geometers illustrates what I have said:
a supreme and guiding intelligence--apart from a blind rule called _nature
of things_--was an _hypothesis_. The absolute denial of such a ruling power
was not in the plan of the higher philosophers: it was left for the smaller
fry. A round assertion of the non-existence of anything which stands in the
way is the refuge of a certain class of minds: but it succeeds only with
things subjective; the objective offers resistance. A philosopher of the
appropriative class tried it upon the constable who appropriated _him_: I
deny your existence, said he; Come along all the same, said the
unpsychological policeman.

Euler[10] was a believer in God, downright and straightforward. The
following story is told by Thiébault,[11] in his _Souvenirs de vingt ans de
séjour à Berlin_,[12] published in his old age, about 1804. This volume was
fully received as trustworthy; and Marshall Mollendorff[13] told the Duc de
Bassano[14] in 1807 that it was the most veracious of books written by the
most honest of men. Thiébault says that he has no personal knowledge of the
truth of the story, but {4} that it was believed throughout the whole of
the north of Europe. Diderot[15] paid a visit to the Russian Court at the
invitation of the Empress. He conversed very freely, and gave the younger
members of the Court circle a good deal of lively atheism. The Empress was
much amused, but some of her councillors suggested that it might be
desirable to check these expositions of doctrine. The Empress did not like
to put a direct muzzle on her guest's tongue, so the following plot was
contrived. Diderot was informed that a learned mathematician was in
possession of an algebraical demonstration of the existence of God, and
would give it him before all the Court, if he desired to hear it. Diderot
gladly consented: though the name of the mathematician is not given, it was
Euler. He advanced towards Diderot, and said gravely, and in a tone of
perfect conviction: _Monsieur,_ (a + b^{n}) / n = x, _donc Dieu existe;
répondez!_[16] Diderot, to whom algebra was Hebrew, was embarrassed and
disconcerted; while peals of laughter rose on all sides. He asked
permission to return to France at once, which was granted.


    An examination of the Astronomical doctrine of the Moon's rotation. By
    J. L.[17] Edinburgh, 1847, 8vo.

A systematic attack of the character afterwards made with less skill and
more notice by Mr. Jellinger Symons.

July 1866, J. L. appears as Mr. James Laurie, with a new pamphlet "The
Astronomical doctrines of the Moon's rotation ..." Edinburgh. Of all the
works I have seen on the question, this is the most confident, and the
sorest. {5} A writer on astronomy said of Mr. Jellinger Symons,[18] "Of
course he convinced no one who knew anything of the subject." This
"ungenerous slur" on the speculator's memory appears to have been keenly
felt; but its truth is admitted. Those who knew anything of the subject are
"the so-called men of science," whose three P's were assailed; prestige,
pride, and prejudice: this the author tries to effect for himself with
three Q's; quibble, quirk, and quiddity. He explains that the Scribes and
Pharisees would not hear Jesus, and that the lordly bishop of Rome will not
cast his tiara and keys at the feet of the "humble presbyter" who now plays
the part of pope in Scotland. I do not know whom he means: but perhaps the
friends of the presbyter-pope may consider this an ungenerous slur. The
best proof of the astronomer is just such "as might have been expected from
the merest of blockheads"; but as the giver is of course not a blockhead,
this circumstance shows how deeply blinded by prejudice he must be.

Of course the paradoxers do not persuade any persons who know their
subjects: and so these Scribes and Pharisees reject the Messiah. We must
suppose that the makers of this comparison are Christians: for if they
thought the Messiah an enthusiast or an impostor, they would be absurd in
comparing those who reject what they take for truth with others who once
rejected what they take for falsehood. And if Christians, they are both
irreverent and blind to all analogy. The Messiah, with His Divine mission
proved by miracles which all might see who chose to look, is degraded into
a prototype of James Laurie, ingeniously astronomizing upon ignorant
geometry and false logic, and comparing to blockheads those who expose his
nonsense. Their comparison is as foolish as--supposing {6} them
Christians--it is profane: but, like errors in general, its other end
points to truth. There were Pseudochrists and Antichrists; and a
Concordance would find the real forerunners of all the paradoxers. But they
are not so clever as the old false prophets: there are none of whom we
should be inclined to say that, if it were possible, they would deceive the
very educated. Not an Egyptian among them all can make uproar enough to
collect four thousand men that are murderers--of common sense--to lead out
into the wilderness. Nothing, says the motto of this work, is so difficult
to destroy as the errors and false facts propagated by illustrious men
whose words have authority. I deny it altogether. There are things much
more difficult to destroy: it is much more difficult to destroy the truths
and real facts supported by such men. And again, it is much more difficult
to prevent men of no authority from setting up false pretensions; and it is
much more difficult to destroy assertions of fancy speculation. Many an
error of thought and learning has fallen before a gradual growth of
thoughtful and learned opposition. But such things as the quadrature of the
circle, etc., are never put down. And why? Because thought can influence
thought, but thought cannot influence self-conceit: learning can annihilate
learning: but learning cannot annihilate ignorance. A sword may cut through
an iron bar; and the severed ends will not reunite: let it go through the
air, and the yielding substance is whole again in a moment.

    Miracles _versus_ Nature: being an application of certain propositions
    in the theory of chances to the Christian miracles. By
    Protimalethes.[19] Cambridge, 1847, 8vo.

The theory, as may be supposed, is carried further than most students of
the subject would hold defensible.


    An astronomical Lecture. By the Rev. R. Wilson.[20] Greenock, 1847,

Against the moon's rotation on her axis.

    [Handed about in the streets in 1847: I quote the whole:] Important
    discovery in astronomy, communicated to the Astronomer Royal, December
    21st, 1846. That the Sun revolve round the Planets in 25748-2/5 years,
    in consequence of the combined attraction of the planets and their
    satellites, and that the Earth revolve round the Moon in 18 years and
    228 days. D. T. GLAZIER [altered with a pen into GLAZION.] Price one

1847. In the _United Service Magazine_ for September, 1847, Mrs.
Borron,[21] of Shrewsbury, published some remarks tending to impeach the
fact that Neptune, the planet found by Galle,[22] really was the planet
which Le Verrier and Adams[23] had a right to claim. This was followed
(September 14) by two pages, separately circulated, of "Further
Observations upon the Planets Neptune and Uranus, with a Theory of
Perturbations"; and (October 19, 1848) by three pages of "A Review of M.
Leverrier's Exposition." Several persons, when the remarkable discovery was
made, contended that the planet actually discovered was an intruder; and
the future histories of the discovery must contain some account of this
little afterpiece. Tim Linkinwater's theory that there is no place like
London for coincidences, would have been utterly overthrown in favor of
what they used to call the celestial spaces, if there had been a planet
which by chance was put {8} near the place assigned to Neptune at the time
when the discovery was made.


    Aerial Navigation; containing a description of a proposed flying
    machine, on a new principle. By Dædalus Britannicus. London, 1847, 8vo.

In 1842-43 a Mr. Henson[24] had proposed what he called an aeronaut
steam-engine, and a Bill was brought in to incorporate an "Aerial Transit
Company." The present plan is altogether different, the moving power being
the explosion of mixed hydrogen and air. Nothing came of it--not even a
Bill. What the final destiny of the balloon may be no one knows: it may
reasonably be suspected that difficulties will at last be overcome.
Darwin,[25] in his "Botanic Garden" (1781), has the following prophecy:

 "Soon shall thy arm, unconquered Steam! afar
  Drag the slow barge, or drive the rapid car;
  Or, on wide-waving wings expanded, bear
  The flying chariot through the fields of air."

Darwin's contemporaries, no doubt, smiled pity on the poor man. It is worth
note that the two true prophecies have been fulfilled in a sense different
from that of the predictions. Darwin was thinking of the suggestion of
Jonathan Hulls,[26] when he spoke of dragging the slow barge: it is only
very recently that the steam-tug has been employed on the canals. The car
was to be driven, not drawn, and on the common roads. Perhaps, the flying
chariot will {9} be something of a character which we cannot imagine, even
with the two prophecies and their fulfilments to help us.[27]


    A book for the public. New Discovery. The causes of the circulation of
    the blood; and the true nature of the planetary system. London, 1848,

Light is the sustainer of motion both in the earth and in the blood. The
natural standard, the pulse of a person in health, four beats to one
respiration, gives the natural second, which is the measure of the earth's
progress in its daily revolution. The Greek fable of the Titans is an
elaborate exposition of the atomic theory: but any attempt to convince
learned classics would only meet their derision; so much does long-fostered
prejudice stand in the way of truth. The author complains bitterly that men
of science will not attend to him and others like him: he observes, that
"in the time occupied in declining, a man of science might test the
merits." This is, alas! too true; so well do applicants of this kind know
how to stick on. But every rule has its exception: I have heard of one. The
late Lord Spencer[28]--the Lord Althorp of the House of Commons--told me
that a speculator once got access to him at the Home Office, and was
proceeding to unfold his way of serving the public. "I do not understand
these things," said Lord Althorp, "but I happen to have ---- (naming an
eminent engineer) upstairs; suppose you talk to him on the subject." The
discoverer went up, and in half-an-hour returned, and said, "I am very much
obliged to your Lordship for introducing me to Mr. ----; he has convinced
me {10} that I am quite wrong." I supposed, when I heard the story--but it
would not have been seemly to say it--that Lord A. exhaled candor and
sense, which infected those who came within reach: he would have done so,
if anybody.


    A method to trisect a series of angles having relation to each other;
    also another to trisect any given angle. By James Sabben. 1848 (two
    quarto pages).

"The consequence of years of intense thought": very likely, and very sad.

1848. The following was sent to me in manuscript. I give the whole of it:

"_Quadrature of the Circle_.--A quadrant is a curvilinear angle traversing
round and at an equal distance from a given point, called a center, no two
points in the curve being at the same angle, but irreptitiously graduating
from 90 to 60. It is therefore a mean angle of 90 and 60, which is 75,
because it is more than 60, and less than 90, approximately from 60 to 90,
and from 90 to 60, with equal generation in each irreptitious
approximation, therefore meeting in 75, and which is the mean angle of the

"Or suppose a line drawn from a given point at 90, and from the same point
at 60. Let each of these lines revolve on this point toward each other at
an equal ratio. They will become one line at 75, and bisect the curve,
which is one-sixth of the entire circle. The result, taking 16 as a
diameter, gives an area of 201.072400, and a circumference of 50.2681.

"The original conception, its natural harmony, and the result, to my own
mind is a demonstrative truth, which I presume it right to make known,
though perhaps at the hazard of unpleasant if not uncourteous remarks."

I have added punctuation: the handwriting and spelling {11} are those of an
educated person; the word _irreptitious_ is indubitable. The whole is a
natural curiosity.

    The quadrature and exact area of the circle demonstrated. By Wm.
    Peters. 8vo. _n. d._ (circa 1848).[29]

    Suggestions as to the necessity for a revolution in philosophy; and
    prospectus for the establishment of a new quarterly, to be called the
    _Physical Philosopher and Heterodox Review_. By Q. E. D. 8vo. 1848.

These works are by one author, who also published, as appears by

"Newton rescued from the precipitancy of his followers through a century
and a half,"[30] and "Dangers along a coast by correcting (as it is called)
a ship's reckoning by bearings of the land at night fall, or in a fog,
nearly out of print. Subscriptions are requested for a new edition."

The area of a circle is made four-fifths of the circumscribed square:
proved on an assumption which it is purposed to explain in a longer
essay.[31] The author, as Q. E. D., was in controversy with the _Athenæum_
journal, and criticised a correspondent, D., who wrote against a certain
class of discoverers. He believed the common theories of hydrostatics to be
wrong, and one of his questions was:

"Have you ever taken into account anent gravity and gravitation the fact
that a five grain cube of cork will of itself half sink in the water,
whilst it will take 20 grains of brass, which will sink of itself, to pull
under the other half? Fit this if you can, friend D., to your notions of
gravity and specific gravity, as applied to the construction of a universal
law of gravitation."

This the _Athenæum_ published--but without some Italics, for which the
editor was sharply reproved, as a sufficient {12} specimen of the _quod
erat_ D. _monstrandum_: on which the author remarks--"D,--Wherefore the e
caret? is it D apostrophe? D', D'M, D'Mo, D'Monstrandum; we cannot find the
_wit_ of it." This I conjecture to contain an illusion to the name of the
supposed author; but whether De Mocritus, De Mosthenes, or De Moivre was
intended, I am not willing to decide.

    The Scriptural Calendar and Chronological Reformer, for the statute
    year 1849. Including a review of recent publications on the Sabbath
    question. London, 1849, 12mo.[32]

This is the almanac of a sect of Christians who keep the Jewish Sabbath,
having a chapel at Mill Yard, Goodman's Fields. They wrote controversial
works, and perhaps do so still; but I never chanced to see one.

    Geometry _versus_ Algebra; or the trisection of an angle geometrically
    solved. By W. Upton, B.A.[33] Bath (circa 1849). 8vo.

The author published two tracts under this title, containing different
alleged proofs: but neither gives any notice of the change. Both contain
the same preface, complaining of the British Association for refusing to
examine the production. I suppose that the author, finding his first proof
wrong, invented the second, of which the Association never had the offer;
and, feeling sure that they would have equally refused to examine the
second, thought it justifiable to {13} present that second as the one which
they had refused. Mr. Upton has discovered that the common way of finding
the circumference is wrong, would set it right if he had leisure, and, in
the mean time, has solved the problem of the duplication of the cube.

_The trisector of an angle, if he demand attention from any mathematician,
is bound to produce, from his construction, an expression for the sine or
cosine of the third part of any angle, in terms of the sine or cosine of
the angle itself, obtained by help of no higher than the square root._ The
mathematician knows that such a thing cannot be; but the trisector
virtually says it can be, and is bound to produce it, to save time. This is
the misfortune of most of the solvers of the celebrated problems, that they
have not knowledge enough to present those consequences of their results by
which they can be easily judged. Sometimes they have the knowledge and
quibble out of the use of it. In many cases a person makes an honest
beginning and presents what he is sure is a solution. By conference with
others he at last feels uneasy, fears the light, and puts self-love in the
way of it. Dishonesty sometimes follows. The speculators are, as a class,
very apt to imagine that the mathematicians are in fraudulent confederacy
against them: I ought rather to say that each one of them consents to the
mode in which the rest are treated, and fancies conspiracy against himself.
The mania of conspiracy is a very curious subject. I do not mean these
remarks to apply to the author before me.

One of Mr. Upton's trisections, if true, would prove the truth of the
following equation:

  3 cos ([theta] / 3) = 1 + [root](4 - sin^2[theta])

which is certainly false.[34]


In 1852 I examined a terrific construction, at the request of the late Dr.
Wallich,[35] who was anxious to persuade a poor countryman of his, that
trisection of the angle was waste of time. One of the principles was, that
"magnitude and direction determine each other." The construction was
equivalent to the assertion that, [theta] being any angle, the cosine of
its third part is

  sin 3[theta] . cos(5[theta]/2) + sin^2 [theta] sin (5[theta]/2)

divided by the square root of

  sin^2 3[theta] . cos^2 (5[theta]/2) + sin^4 [theta] + sin 3[theta] . sin
      5[theta] . sin^2 [theta].

This is from my rough notes, and I believe it is correct.[36] It is so
nearly true, unless the angle be very obtuse, that common drawing, applied
to the construction, will not detect the error. There are many formulae of
this kind: and I have several times found a speculator who has discovered
the corresponding construction, has seen the approximate success of his
drawing--often as great as absolute truth could give in graphical
practice,--and has then set about his demonstration, in which he always
succeeds to his own content.

There is a trisection of which I have lost both cutting and reference: I
think it is in the _United Service Journal_. I could not detect any error
in it, though certain there must {15} be one. At least I discovered that
two parts of the diagram were incompatible unless a certain point lay in
line with two others, by which the angle to be trisected--and which was
trisected--was bound to be either 0° or 180°.

Aug. 22, 1866. Mr. Upton sticks to his subject. He has just published "The
Uptonian Trisection. Respectfully dedicated to the schoolmasters of the
United Kingdom." It seems to be a new attempt. He takes no notice of the
sentence I have put in italics: nor does he mention my notice of him,
unless he means to include me among those by whom he has been "ridiculed
and sneered at" or "branded as a brainless heretic." I did neither one nor
the other: I thought Mr. Upton a paradoxer to whom it was likely to be
worth while to propound the definite assertion now in italics; and Mr.
Upton does not find it convenient to take issue on the point. He prefers
general assertions about algebra. So long as he cannot meet algebra on the
above question, he may issue as many "respectful challenges" to the
mathematicians as he can find paper to write: he will meet with no

There is one trisection which is of more importance than that of the angle.
It is easy to get half the paper on which you write for margin; or a
quarter; but very troublesome to get a third. Show us how, easily and
certainly, to fold the paper into three, and you will be a real benefactor
to society.

Early in the century there was a Turkish trisector of the angle, Hussein
Effendi, who published two methods. He was the father of Ameen Bey, who was
well known in England thirty years ago as a most amiable and cultivated
gentleman and an excellent mathematician. He was then a student at
Cambridge; and he died, years ago, in command of the army in Syria. Hussein
Effendi was instructed in mathematics by Ingliz Selim Effendi, who
translated a work {16} of Bonnycastle[37] into Turkish.[38] This Englishman
was Richard Baily, brother of Francis Baily[39] the astronomer, who
emigrated to Turkey in his youth, and adopted the manners of the Turks, but
whether their religion also I never heard, though I should suppose he did.

I now give the letters from the agricultural laborer and his friend,
described on page 12, Vol. I. They are curiosities; and the history of the
quadrature can never be well written without some specimens of this kind:

"Doctor Morgan, Sir. Permit me to address you

"Brute Creation may perhaps enjoy the faculty of beholding visible things
with a more penitrating eye than ourselves. But Spiritual objects are as
far out of their reach as though they had no being

"Nearest therefore to the brute Creation are those men who Suppose
themselves to be so far governed by external objects as to believe nothing
but what they See and feel And Can accomedate to their Shallow
understanding and Imaginations

"My Dear Sir Let us all Consult ourselves by the wise proverb.

"I believe that evry man^s merit & ability aught to be appreciated and
valued In proportion to its worth & utility

"In whatever State or Circumstances they may fortunately or unfortunately
be placed

"And happy it is for evry man to know his worth and place

"When a Gentleman of your Standing in Society Clad with those honors Can
not understand or Solve a problem That is explicitly explained by words and
Letters and {17} mathematically operated by figuers He had best consult the
wise proverd

"Do that which thou Canst understand and Comprehend for thy good.

"I would recommend that Such Gentleman Change his business

"And appropriate his time and attention to a Sunday School to Learn what he
Could and keep the Litle Children form durting their Close

"With Sincere feelings of Gratitude for your weakness and Inability I am

 "Sir your Superior in Mathematics ----"

"1849 June th29."

"Dor Morgin Sir

"I wrote and Sent my work to Professor ---- of ---- State of ---- United

"I am now in the possession of the facts that he highly approves of my
work. And Says he will Insure me Reward in the States

"I write this that you may understand that I have knowledge of the unfair
way that I am treated In my own nati County

"I am told and have reasons to believe that it is the Clergy that treat me
so unjust.

"I am not Desirous of heaping Disonors upon my own nation. But if I have to
Leave this kingdom without my Just dues. The world Shall know how I am and
have been treated.

 "I am Sir Desirous of my
             "Just dues ----"

"1849 July 3."

"July 7th, 1849.

"Sir, I have been given to understand that a friend of mine one whom I
shall never be ashamed to acknowledge as {18} such tho' lowly his origine;
nay not only not ashamed but proud of doing so for I am one of those who
esteem and respect a man according to his ability and probity, deeming with
Dr. Watts 'that the mind is the standard of the man,'[40] has laid before
you and asked your opinion of his extraordinary performance, viz. the
quadrature of the circle, he did this with the firmest belief that you
would not only treat the matter in a straightforward manner but with the
conviction that from your known or supposed knowledge of mathematicks would
have given an upright and honorable decision upon the subject; but the
question is have you done so? Could I say yes I would with the greatest of
pleasure and have congratulated you upon your decision whatever it might
have been but I am sorry to say that I cannot your letter is a paltry
evasion, you say 'that it is a great pity that you (Mr. ----) should have
attempted this (the quadrature of the circle) for your mathematical
knowledge is not sufficient to make you know in what the problem consists,'
you don't say in what it does consist _according to your ideas_, oh! no
nothing of the sort, you enter into no disquisition upon the subject in
order to show where you think Mr. ---- is wrong and why you have not is
simply--_because you cannot_--you know that he has done it and what is if I
am not wrongly informed _you have been heard to say so_. He has done what
you nor any other mathematician as those who call themselves such have
done. And what is the reason that you will not candidly acknowledge to him
as you have to others that he has squared the circle shall I tell you? it
is because he has performed the feat to obtain the glory of which
mathematicians have battled from time immemorial that they might encircle
their brows with a wreath of laurels far more glorious than ever conqueror
won it is simply this that it is a poor man a {19} humble artisan who has
gained that victory that you don't like to acknowledge it you don't like to
be beaten and worse to acknowledge that you have miscalculated, you have in
short too small a soul to acknowledge that he is right.

"I was asked my opinion and _I_ gave it unhesitatingly in the affirmative
and I am backed in my opinion not only by Mr. ---- a mathematician and
watchmaker residing in the boro of Southwark but by no less an authority
than the Professor of mathematics of ---- College ---- ---- United States
Mr. ---- and I presume that he at least is your equal as an authority and
Mr. ---- says that the government of the U.S. will recompense M. D. for the
discovery he has made if so what a reflection upon Old england the boasted
land of freedom the nursery of arts and sciences that her sons are obliged
to go to a foreign country to obtain that recompense to which they are
justly entitled

"In conclusion I had to contradict an assertion you made to the effect that
'there is not nor ever was any reward offered by the government of this
country for the discovery of the quadrature of the circle.' I beg to inform
you that there _was_ but that it having been deemed an impossibility the
government has withdrawn it. I do this upon no less an authority than the
Marquis of Northampton.[41]

 "I am, sir, yours ----"

"Dr. Morgan."


    Notes on the Kinematic Effects of Revolution and Rotation, with
    reference to the Motions of the Moon and of the earth. By Henry
    Perigal, Jun. Esq. London, 1846-1849, 8vo.

    On the misuse of technical terms. Ambiguity of the terms _Rotation_ and
    _Revolution_, owing to the double meaning improperly {20} attributed to
    each of the words. (No date nor place, but by Mr. Perigal,[42] I have
    no doubt, and containing letters of 1849 and 1850.)

    The moon controversy. Facts _v._ Definitions. By H. P., Jun. London,
    1856, 8vo. (pp. 4.)

Mr. Henry Perigal helped me twenty years ago with the diagrams, direct from
the lathe to the wood, for the article "Trochoidal Curves," in the _Penny
Cyclopædia_: these cuts add very greatly to the value of the article,
which, indeed, could not have been made intelligible without them. He has
had many years' experience, as an amateur turner, in combination of double
and triple circular motions, and has published valuable diagrams in
profusion. A person to whom the double circular motion is familiar in the
lathe naturally looks upon one circle moving upon another as in _simple_
motion, if the second circle be fixed to the revolving radius, so that one
and the same point of the moving circle travels upon the fixed circle. Mr.
Perigal commenced his attack upon the moon for moving about her axis, in
the first of the tracts above, ten years before Mr. Jellinger Symons;[43]
but he did not think it necessary to make it a subject for the _Times_
newspaper. His familiarity with combined motions enabled him to handle his
arguments much better than Mr. J. Symons could do: in fact, he is the
clearest assailant of the lot which turned out with Mr. J. Symons. But he
is as wrong as the rest. The assault is now, I suppose, abandoned, until it
becomes epidemic again. This it will do: it is one of those fallacies which
are very tempting. There was a dispute on the subject in 1748, between
James Ferguson[44] and an anonymous opponent; and I think there have been


A poet appears in the field (July 19, 1863) who calls himself Cyclops, and
writes four octavo pages. He makes a distinction between _rotation_ and
_revolution_; and his doctrines and phrases are so like those of Mr.
Perigal that he is a follower at least. One of his arguments has so often
been used that it is worth while to cite it:

 "Would Mathematicals--forsooth--
  If true, have failed to prove its truth?
  Would not they--if they could--submit
  Some overwhelming proofs of it?
  But still it totters _proofless_! Hence
  There's strong presumptive evidence
  None do--or can--such proof profound
  Because _the dogma is unsound_.
  For, were there means of doing so,
  They would have proved it long ago."

This is only one of the alternatives. Proof requires a person who can give
and a person who can receive. I feel inspired to add the following:

 "A blind man said, As to the Sun,
  I'll take my Bible oath there's none;
  For if there had been one to show
  They would have shown it long ago.
  How came he such a goose to be?
  Did he not know he couldn't see?
                  Not he!"

The absurdity of the verses is in the argument. The writer was not so
ignorant or so dishonest as to affirm that nothing had been offered by the
other side as proof; accordingly, his syllogism amounts to this: If your
proposition were true, you could have given proof satisfactory to _me_; but
this you have not done, therefore, your proposition is not true.

The echoes of the moon-controversy reached Benares in 1857, in which year
was there published a pamphlet "Does the Moon Rotate?" in Sanskrit and
English. The {22} arguments are much the same as those of the discussion at


We see that there are paradoxers in argument as well as in assertion of
fact: my plan does not bring me much into contact with these; but another
instance may be useful. Sects, whether religious or political, give
themselves names which, in meaning, are claimed also by their opponents;
loyal, liberal, conservative (of good), etc. have been severally
appropriated by parties. _Whig_ and _Tory_ are unobjectionable names: the
first--which occurs in English ballad as well as in Scotland--is sour
milk;[45] the second is a robber. In theology, the Greek Church is
_Orthodox_, the Roman is _Catholic_, the modern Puritan is _Evangelical_,

The word _Christian_ (Vol. I, p. 248[46]) is an instance. When words begin,
they carry their meanings. The Jews, who had their Messiah to come, and the
followers of Jesus of Nazareth, who took _Him_ for their Messiah, were both
_Christians_ (which means _Messianites_): the Jews would never have
invented the term to signify _Jesuans_, nor would the disciples have
invented such an ambiguous term for themselves; had they done so, the Jews
would have disputed it, as they would have done in later times if they had
had fair play. The Jews of our day, I see by their newspapers, speak of
Jesus Christ as the _Rabbi Joshua_. But the {23} heathens, who knew little
or nothing about the Jewish hope, would naturally apply the term
_Christians_ to the only followers of a _Messiah_ of whom they had heard.
For the _Jesuans_ invaded them in a missionary way; while the Jews did not
attempt, at least openly, to make proselytes.

All such words as Catholic, etc., are well enough as mere nomenclature; and
the world falls for the most part, into any names which parties choose to
give themselves. Silly people found inferences on this concession; and, as
usually happens, they can cite some of their betters. St. Augustine,[47] a
freakish arguer, or, to put it in the way of an old writer, _lectorem ne
multiloquii tædio fastidiat, Punicis quibusdam argutiis recreare
solet_,[48] asks, with triumph, to what chapel a stranger would be
directed, if he inquired the way to the _Catholic assembly_. But the best
exhibition of this kind in our own century is that made by the excellent
Dr. John Milner,[49] in a work (first published in 1801 or 1802) which I
suppose still circulates, "The End of Religious Controversy": a startling
title which, so far as its truth is concerned, might as well have been "The
floor of the bottomless pit." This writer, whom every one of his readers
will swear to have been a worthy soul, though many, even of his own sect,
will not admire some of his logic, speaks as follows:

"Letter xxv. _On the true Church being Catholic._ In treating of this third
mark of the true Church, as expressed in our common creed, I feel my
spirits sink within me, and I am almost tempted to throw away my pen in
despair. For what chance is there of opening the eyes of candid Protestants
to the other marks of the Church, if they are capable of keeping them shut
to this? Every time they address the {24} God of Truth, either in solemn
worship or in private devotion [stretch of rhetoric], they are forced, each
of them, to repeat: _I believe in_ THE CATHOLIC _Church_, and yet if I ask
any of them the question: _Are you a_ CATHOLIC? he is sure to answer me,
_No, I am a_ PROTESTANT! Was there ever a more glaring instance of
inconsistency and self-condemnation among rational beings!"

 "John Milner, honest and true,
  Did what honest people still may do,
  If they write for the many and not for the few,
  But what by and bye they must eschew."

He _shortened his clause_; and for a reason. If he had used the whole
epithet which he knew so well, any one might have given his argument a
half-turn. Had he written, as he ought, "the _Holy_ Catholic Church" and
then argued as above, some sly Protestant would have parodied him with "and
yet if I ask any of them the question: _Are you_ HOLY? he is sure to answer
me _No, I am a_ SINNER." To take the adjective from the Church, and apply
it to the individual partisan, is recognized slipslop, but not ground of
argument. If Dr. M. had asked his Protestant whether he belonged to the
Catholic _Church_, the answer would have been Yes, but not to the Roman
branch. When he put his question as he did, he was rightly answered and in
his own division. This leaving out words is a common practice, especially
when the omitter is in authority, and cannot be exposed. A year or two ago
a bishop wrote a snubbing letter to a poor parson, who had complained that
he was obliged, in burial, to send the worst of sinners to everlasting
happiness. The bishop sternly said, "_hope_[50] is not _assurance_." {25}
Could the clergyman have dared to answer, he would have said, "No, my Lord!
but '_sure and certain_ hope' is as like assurance as a _minikin_ man is
like a dwarf." Sad to say, a theologian must be illogical: I feel sure that
if you took the clearest headed writer on logic that ever lived, and made a
bishop of him, he would be shamed by his own books in a twelvemonth.

Milner's sophism is glaring: but why should Dr. Milner be wiser than St.
Augustine, one of his teachers? I am tempted to let out the true derivation
of the word _Catholic, as exclusively applied to the Church of Rome_. All
can find it who have access to the _Rituale_ of Bonaventura Piscator[51]
(lib. i. c. 12, _de nomine Sacræ Ecclesiæ_, p. 87 of the Venice {26} folio
of 1537). I am told that there is a _Rituale_ in the Index Expurgatorius,
but I have not thought it worth while to examine whether this be the one: I
am rather inclined to think, as I have heard elsewhere, that the book was
held too dangerous for the faithful to know of it, even by a prohibition:
it would not surprise me at all if Roman Christians should deny its

It amuses me to give, at a great distance of time, a small Rowland for a
small Oliver,[53] which I received, _de par l'Eglise_,[54] so far as lay in
the Oliver-carrier more than twenty years ago. The following contribution
of mine to _Notes and Queries_ (3d Ser. vi. p. 175, Aug. 27, 1864) will
explain what I say. There had been a complaint that a contributor had used
the term _Papist_, which a very excellent dignitary of the Papal system
pronounced an offensive term:


The term _papist_ should be stripped of all except its etymological
meaning, and applied to those who give the higher and final authority to
the declaration _ex cathedrâ_[55] of the Pope. See Dr. Wiseman's[56]
article, _Catholic Church_, in the _Penny Cyclopædia_.

What is one to do about these names? First, it is clear that offence
should, when possible, be avoided: secondly, no one must be required to
give a name which favors _any_ assumption made by those to whom it is
given, and not {27} granted by those who give it. Thus the subdivision
which calls itself distinctly _Evangelical_ has no right to expect others
to concede the title. Now the word _Catholic_, of course, falls under this
rule; and even _Roman Catholic_ may be refused to those who would restrict
the word _Catholic_ to themselves. _Roman Christian_ is unobjectionable,
since the Roman Church does not deny the name of Christian to those whom
she calls heretics. No one is bound in this matter by Acts of Parliament.
In many cases, no doubt, names which have offensive association are used
merely by habit, sometimes by hereditary transmission. Boswell records of
Johnson that he always used the words "dissenting teacher," refusing
_minister_ and _clergyman_ to all but the recipients of episcopal

This distinctive phrase has been widely adopted: it occurs in the Index of
3d S. iv. [_Notes and Queries_]. Here we find "Platts (Rev. John),
Unitarian teacher, 412;" the article indexed has "Unitarian minister."

This, of course is habit: an intentional refusal of the word _minister_
would never occur in an index. I remember that, when I first read about Sam
Johnson's little bit of exclusiveness, I said to myself: "Teacher? Teacher?
surely I remember One who is often called _teacher_, but never _minister_
or _clergyman_: have not the dissenters got the best of it?"

When I said that the Roman Church concedes the epithet Christians to
Protestants, I did not mean that all its adherents do the same. There is,
or was, a Roman newspaper, the _Tablet_, which, seven or eight years ago,
was one of the most virulent of the party journals. In it I read, referring
to some complaint of grievance about mixed marriages, that if _Christians_
would marry _Protestants_ they must take the consequences. My memory notes
this well; because I recollected, when I saw it, that there was in the
stable a horse fit to run in the curricle with this one. About seventeen
years ago an Oxford M. A., who hated {28} mathematics like a genuine
Oxonian of the last century, was writing on education, and was compelled to
give some countenance to the nasty subject. He got out cleverly; for he
gave as his reason for the permission, that man is an arithmetical,
geometrical, and mechanical _animal_, as well as a rational _soul_.

The _Tablet_ was founded by an old pupil of mine, Mr. Frederic Lucas,[57]
who availed himself of his knowledge of me to write some severe
articles--even abusive, I was told, but I never saw them--against me, for
contributing to the _Dublin Review_, and poking my heretic nose into
orthodox places. Dr. Wiseman, the editor, came in for his share, and ought
to have got all. Who ever blamed the pig for intruding himself into the
cabin when the door was left open? When Mr. Lucas was my pupil, he was of
the Society of Friends--in any article but this I should say _Quaker_--and
was quiet and gentlemanly, as members of that Church--in any article but
this I should, from mere habit, say _sect_--usually are. This is due to his
memory; for, by all I heard, when he changed his religion he ceased to be
Lucas couchant, and became Lucas rampant, fanged and langued gules. (I
looked into Guillim[58] to see if my terms were right: I could not find
them; but to prove I have been there, I notice that he calls a violin a
_violent_. How comes the word to take this form?) I met with several Roman
Christians, born and bred, who were very much annoyed at Mr. Lucas and his
doings; and said some severe things about new converts needing


The mention of Dr. Wiseman reminds me of another word, appropriated by
Christians to themselves: _fides_;[59] the Roman faith is _fides_, and
nothing else; and the adherents are _fideles_.[60] Hereby hangs a retort.
When Dr. Wiseman was first in England, he gave a course of lectures in
defence of his creed, which were thought very convincing by those who were
already convinced. They determined to give him a medal, and there was a
very serious discussion about the legend. Dr. Wiseman told me himself that
he had answered to his subscribers that he would not have the medal at all
unless--(naming some Italian authority, whom I forget) approved of the
legend. At last _pro fide vindicata_[61] was chosen: this may be read
either in a Popish or heretical sense. The feminine substantive _fides_
means confidence, trust, (it is made to mean _belief_), but _fidis_, with
the same ablative, _fide_, and also feminine, is a _fiddle-string_.[62] If
a Latin writer had had to make a legend signifying "For the defence of the
fiddle-string," he could not have done it otherwise, in the terseness of a
legend, than by writing _pro fide vindicata_. Accordingly, when a Roman
Christian talks to you of the _faith_, as a thing which is his and not
yours, you may say _fiddle_. I have searched Bonaventura Piscator in vain
for notice of this ambiguity. But the Greeks said fiddle; according to
Suidas,[63] [Greek: skindapsos][64]--a word meaning a four stringed
instrument played with a quill--was an exclamation of contemptuous dissent.
How the wits of different races jump!


I am reminded of a case of _fides vindicata_, which, being in a public
letter, responding to a public invitation, was not meant to be
confidential. Some of the pupils of University College, in which all
subdivisions of religion are (1866; _were_, 1867) on a level, have of
course changed their views in after life, and become adherents of various
high churches. On the occasion of a dinner of old students of the College,
convened by circular, one of these students, whether then Roman or
Tractarian Christian I do not remember, not content with simply giving
negative answer, or none at all, concocted a jorum of theological rebuke,
and sent it to the Dinner Committee. Heyday! said one of them, this man got
out of bed backwards! How is that? said the rest. Why, read his name
backwards, and you will see. As thus read it was--_No grub_![65]


To return to _Notes and Queries_. The substitution in the (editorial) index
of "Unitarian teacher," for the contributor's "Unitarian minister," struck
me very much. I have seldom found such things unmeaning. But as the journal
had always been free from editorial sectarianisms,--and very apt to check
the contributorial,--I could not be sure in this case. True it was, that
the editor and publisher had been changed more than a year before; but this
was not of much force. Though one swallow does not make a summer, I have
generally found it show that summer is coming. However, thought I to
myself, if this be Little Shibboleth, we shall have Big Shibboleth
by-and-bye. At last it came. About a twelvemonth afterwards, (3d S. vii. p.
36) the following was the _editorial_ answer to the question when the
establishment was first called the "Church of England and Ireland":


"That unmeaning clause, 'The United Church of England and Ireland,' which
occurs on the title-page of _The Book of Common Prayer_, was first used at
the commencement of the present century. The authority for this phrase is
the fifth article of the Union of 1800: 'That the Churches of England and
Ireland be united into one _Protestant_ (!) episcopal Church, to be called
"The United Church of England and Ireland."' Of course, churchmen are not
responsible for the theology of Acts of Parliament, especially those passed
during the dark ages of the Georgian era."

That is to say, the journal gives its adhesion to the party which--under
the assumed title of _the_ Church of England--claims for the endowed
corporation for the support of religion rights which Parliament cannot
control, and makes it, in fact, a power above the State. The State has
given an inch: it calls this corporation by the name of the "United
_Church_ of England and Ireland," as if neither England nor Ireland had any
other Church. The corporation, accordingly aspires to an ell. But this the
nation will only give with the aspiration prefixed. To illustrate my
allusion in a delicate way to polite ears, I will relate what happened in a
Johnian lecture-room at Cambridge, some fifty years ago, my informant being
present. A youth of undue aspirations was giving a proposition, and at last
said, "Let E F be produced to 'L':" "Not quite so far, Mr. ----," said the
lecturer, quietly, to the great amusement of the class, and the utter
astonishment of the aspirant, who knew no more than a Tractarian the
tendency of his construction.

This word _Church_ is made to have a very mystical meaning. The following
dialogue between Ecclesiastes and Hæreticus, which I cannot vouch for, has
often taken place in spirit, if not in letter: E. The word _Church_
([Greek: ekklêsia])[66] is never used in the New Testament except generally
or locally for that holy and mystical body to which the sacraments and the
ordinances of Christianity are entrusted. {32} H. Indeed! E. It is beyond a
doubt (here he quoted half a dozen texts in support). H. Do you mean that
any doctrine or ordinance which was solemnly practised by the [Greek:
ekklêsia] is binding upon you and me? E. Certainly, unless we should be cut
off from the congregation of the faithful. H. Have you a couple of hours to
spare? E. What for? H. If you have, I propose we spend them in crying,
Great is Diana of the Ephesians! E. What do you mean? H. You ought to know
the solemn service of the [Greek: ekklêsia] (Acts xix. 32, 41), at Ephesus;
which any one might take to be true Church, by the more part not knowing
wherefore they were come together, and which was dismissed, after one of
the most sensible sermons ever preached, by the Recorder. E. I see your
meaning: it is true, there is that one exception! H. Why, the Recorder's
sermon itself contains another, the [Greek: ennomos ekklêsia],[67]
legislative assembly. E. Ah! the New Testament can only be interpreted by
the Church! H. I see! the Church interprets itself into existence out of
the New Testament, and then interprets the New Testament out of existence
into itself!

I look upon all the Churches as fair game which declare of me that _absque
dubio in æternum peribo_;[68] not for their presumption towards God, but
for their personal insolence towards myself. I find that their sectaries
stare when I say this. Why! they do not speak of you in _particular_! These
poor reasoners seem to think that there could be no meaning, as against me,
unless it should be propounded that "without doubt he shall perish
everlastingly, especially A. De Morgan." But I hold, with the schoolmen,
that "_Omnis_ homo est animal" in conjunction with "Sortes est homo"
amounts to "_Sortes_ est animal."[69] But they do not mean it _personally_!
Every universal proposition is {33} personal to every instance of the
subject. If this be not conceded, then I retort, in their own sense and
manner, "Whosoever would serve God, before all things he must not pronounce
God's decision upon his neighbor. Which decision, except every one leave to
God himself, without doubt he is a bigoted noodle."

The reasoning habit of the educated community, in four cases out of five,
permits universal propositions to be stated at one time, and denied, _pro
re nata_,[70] at another. "Before we proceed to consider any question
involving physical principles, we should set out with _clear ideas_ of the
naturally possible and impossible." The eminent man who said this, when
wanting it for his views of mental education (!) never meant it for more
than what was in hand, never assumed it in the researches which will give
him to posterity! I have heard half-a-dozen defences of his having said
this, not one of which affirmed the truth of what was said. A worthy
clergyman wrote that if A. B. had said a certain thing the point in
question would have been established. It was shown to him that A. B. _had_
said it, to which the reply was a refusal to admit the point because A. B.
said it in a second pamphlet and in answer to objections. And I might give
fifty such instances with very little search. Always assume more than you
want; because you cannot tell how much you may want: put what is over into
the didn't-mean-that basket, or the extreme case what-not.


Something near forty years of examination of the theologies on and
off--more years very much on than quite off--have given me a good title--to
myself, I ask no one else for leave--to make the following remarks: A
conclusion has _premises_, facts or doctrines from proof or authority, and
_mode of inference_. There may be invention or {34} falsehood of premise,
with good logic; and there may be tenable premise, followed by bad logic;
and there may be both false premise _and_ bad logic. The Roman system has
such a powerful manufactory of premises, that bad logic is little wanted;
there is comparatively little of it. The doctrine-forge of the Roman Church
is one glorious compound of everything that could make Heraclitus[71] sob
and Democritus[72] snigger. But not the only one. The Protestants, in
tearing away from the Church of Rome, took with them a fair quantity of the
results of the Roman forge, which they could not bring themselves to give
up. They had more in them of Martin than of Jack. But they would have no
premises, except from the New Testament; though some eked out with a few
general Councils. The consequence is that they have been obliged to find
such a logic as would bring the conclusions they require out of the
canonical books. And a queer logic it is; nothing but the Roman forge can
be compared with the Protestant loom. The picking, the patching, the
piecing, which goes to the Protestant _termini ad quem_,[73] would be as
remarkable to the general eye, as the Roman manufacture of _termini a
quo_,[74] if it were not that the world at large seizes the character of an
asserted fact better than that of a mode of inference. A grand step towards
the deification of a lady, made by alleged revelation 1800 years after her
death, is of glaring evidence: two or three additional shiffle-shuffles
towards defence of saying the Athanasian curse in church and unsaying it
out of church, are hardly noticed. Swift has bungled his satire where he
makes Peter a party to finding out what he wants, _totidem syllabis_ and
_totidem literis_, {35} when he cannot find it _totidem verbis_[75] This is
Protestant method: the Roman plan is _viam faciam_; the Protestant plan is
_viam inveniam_.[76] The public at large begins to be conversant with the
ways of _wriggling out_, as shown in the interpretations of the damnatory
parts of the Athanasian Creed, the phrases of the Burial Service, etc. The
time will come when the same public will begin to see the ways of
_wriggling in_. But one thing at a time: neither Papal Rome nor Protestant
Rome was built--nor will be pulled down--in a day.

The distinction above drawn between the two great antitheses of Christendom
may be illustrated as follows. Two sets of little general dealers lived
opposite to one another: all sold milk. Each vaunted its own produce: one
set said that the stuff on the other side the way was only chalk and water;
the other said that the opposites sold all sorts of filth, of which calves'
brain was the least nasty. Now the fact was that both sets sold milk, and
from the same dairy: but adulterated with different sorts of dirty water:
and both honestly believed that the mixture was what they were meant to
sell and ought to sell. The great difference between them, about which the
apprentices fought each other like Trojans, was that the calves' brain men
poured milk into the water, and the chalk men poured water into the milk.
The Greek and Roman sects on one side, the Protestant sects on the other,
must all have _churches_: the Greek and Roman sects pour the New Testament
into their churches; the Protestant sects pour their churches into the New
Testament. The Greek and Roman insist upon the New Testament being no more
than part and parcel of their churches: the Protestant insist upon their
churches being as much part and parcel of the New Testament. All dwell
vehemently upon the doctrine that there must be milk {36} somewhere; and
each says--I have it. The doctrine is true: and can be verified by any one
who can and will go to the dairy for himself. Him will the several traders
declare to have no milk at all. They will bring their own wares, and
challenge a trial: they want nothing but to name the judges. To vary the
metaphor, those who have looked at Christianity in open day, know that all
who see it through painted windows shut out much of the light of heaven and
color the rest; it matters nothing that the stains are shaped into what are
meant for saints and angels.

But there is another side to the question. To decompose any substance, it
must be placed between the poles of the battery. Now theology is but one
pole; philosophy is the other. No one can make out the combinations of our
day unless he read the writings both of the priest and the philosopher: and
if any one should hold the first word offensive, I tell him that I mean
_both_ words to be _significant_. In reading these writings, he will need
to bring both wires together to find out what it is all about. Time was
when most priests were very explicit about the fate of philosophers, and
most philosophers were very candid about their opinion of priests. But
though some extremes of the old sorts still remain, there is now, in the
middle, such a fusion of the two pursuits that a plain man is wofully
puzzled. The theologian writes a philosophy which seems to tell us that the
New Testament is a system of psychology; and the philosopher writes a
Christianity which is utterly unintelligible as to the question whether the
Resurrection be a fact or a transcendental allegory. What between the
theologian who assents to the Athanasian denunciation in what seems the
sense of no denunciation, and the philosopher who parades a Christianity
which looks like no revelation, there is a maze which threatens to have the
only possible clue in the theory that everything is something else, and
nothing is anything at all. But this is a paradox far beyond my handling:
it is a Budget of itself. {37}


Religion and Philosophy, the two best gifts of Heaven, set up in opposition
to each other at the revival of letters; and never did competing tradesmen
more grossly misbehave. Bad wishes and bad names flew about like swarms of
wasps. The Athanasian curses were intended against philosophers; who, had
they been a corporation, with state powers to protect them, would have
formulized a _per contra_. But the tradesmen are beginning to combine: they
are civil to each other; too civil by half. I speak especially of Great
Britain. Old theology has run off to ritualism, much lamenting, with no
comfort except the discovery that the cloak Paul left at Troas was a
chasuble. Philosophy, which always had a little sense sewed up in its
garments--to pay for its funeral?--has expended a trifle in accommodating
itself to the new system. But the two are poles of a battery; and a
question arises.

  If Peter Piper picked a peck of pepper,
  Where is the peck of pepper Peter Piper picked?

If Religion and Philosophy be the two poles of a battery, whose is the
battery Religion and Philosophy have been made the poles of? Is the change
in the relation of the wires any presumption of a removal of the managers?
We know pretty well who handled the instrument: has he resigned or been[77]
turned out? Has he been put under {38} restriction? A fool may ask more
questions than twenty sages can answer: but there is hope; for twenty sages
cannot ask more questions than one reviewer can answer. I should like to
see the opposite sides employed upon the question, What are the _commoda_,
and what the _pericula_,[78] of the current approximation of Religion and

All this is very profane and irreverent! It has always been so held by
those whose position demands such holding. To describe the Church as it is
passes for assailing the Church as it ought to be with all who cannot do
without it. In Bedlam[79] a poor creature who fancied he was St. Paul, was
told by another patient that he was an impostor; the first maniac lodged a
complaint against the second for calling St. Paul an impostor, which, he
argued, with much appearance of sanity, ought not to be permitted in a well
regulated madhouse. Nothing could persuade him that he had missed the
question, which was whether _he_ was St. Paul. The same thing takes place
in the world _at large_. And especially must be noted the refusal to permit
to the _profane_ the millionth part of the licence assumed by the _sacred_.
I give a sound churchman the epitaph of St. John Long; the usual
pronunciation of whose name must be noted--

 "Behold! ye quacks, the vengeance strong
    On deeds like yours impinging:
  For here below lies St. John Long[80]
    Who now must be _long singeing_."

How shameful to pronounce this of the poor man! What, Mr. Orthodox! may I
not do in joke to one pretender what {39} you do in earnest--unless you
quibble--to all the millions of the Greek Church, and a great many others.
Enough of you and your reasoning! Go and square the circle!

The few years which end with 1867 have shown, not merely the intermediate
fusion of Theology and Philosophy of which I have spoken, but much
concentration of the two extremes, which looks like a gathering of forces
for some very hard fought Armageddon. Extreme theology has been aiming at a
high Church in England, which is to show a new front to all heresy: and
extreme philosophy is contriving a physical organization which is to
_think_, and to show that mind is a consequence of matter, or thought a
recreation of brain. The physical speculators begin with a possible
hypothesis, in which they aim at explanation: and so the bold aspirations
of the author of the _Vestiges_ find standing-ground in the variation of
species by "natural selection." Some relics--so supposed--of extremely
ancient men are brought to help the general cause. Only distant hints are
given that by possibility it may end in the formation of all living
organisms from a very few, if not from one. The better heads above
mentioned know that their theory, if true, does not bear upon morals. The
formation of solar systems from a nebular hypothesis, followed by
organizations gradually emerging from some curious play of particles, nay,
the very evolution of mind and thought from such an apparatus, are all as
consistent with a Personal creative power to whom homage and obedience are
due, and who has declared himself, as with a blind Nature of Things. A pure
materialist, as to all things visible, may be even a bigotted Christian:
this is not frequent, but it is possible. There is a proverb which says, A
pig may fly, but it isn't a likely bird. But when the psychological
speculator comes in, he often undertakes to draw inferences from the
physical conclusions, by joining on his tremendous apparatus of _a priori_
knowledge. He deduces that he can _do without_ a God: he can deduce all
things {40} without any such necessity. With Occam[81] and Newton he will
have no more causes than are necessary to explain phenomena _to him_: and
if by pure head-work combined with results of physical observation he can
construct his universe, he must be a very _unphilosophical_ man who would
encumber himself with a useless Creator! There is something tangible about
my method, says he; yours is vague. He requires it to be granted that his
system is _positive_ and that yours is _impositive_. So reasoned the stage
coachman when the railroads began to depose him--"If you're upset in a
stage-coach, why, there you are! but if you're upset on the railroad, where
are you?" The answer lies in another question, Which is most positive
knowledge, God deduced from man and his history, or the postulates of the
few who think they can reason _a priori_ on the tacit assumption of
unlimited command of data?

We are not yet come to the existence of a school of philosophers who
explicitly deny a Creator: but we are on the way, though common sense may
interpose. There are always straws which show the direction of the wind. I
have before me the printed letter of a medical man--to whose professional
ability I have good testimony--who finds the vital principle in highly
rarefied oxygen. With the usual logic of such thinkers, he dismisses the
"eternal personal identity" because "If soul, spirit, mind, which are
merely modes of sensation, be the attribute or function of nerve-tissue, it
cannot possibly have any existence apart from its material organism!" How
does he know this _impossibility_? If all the mind _we_ know be from
nerve-tissue, how does it appear that mind in other planets may not be
another thing? Nay, when we come to _possibilities_, does not his own
system give a queer one? If highly rarefied oxygen be vital power, more
highly rarefied oxygen {41} may be more vital and more powerful. Where is
this to stop? Is it _impossible_ that a finite quantity, rarefied _ad
infinitum_, may be an Omnipotent? Perhaps the true Genesis, when written,
will open with "In the beginning was an imperial quart of oxygen at 60° of
Fahrenheit, and the pressure of the atmosphere; and this oxygen was
infinitely rarefied; and this oxygen became God." For myself, my
aspirations as to this system are Manichæan. The quart of oxygen is the
Ormuzd, or good principle: another quart, of hydrogen, is the Ahriman, or
evil principle! My author says that his system explains Freewill and
Immortality so obviously that it is difficult to read previous speculations
with becoming gravity. My deduction explains the conflict of good and evil
with such clearness that no one can henceforward read the New Testament
with becoming reverence. The surgeon whom I have described is an early bud
which will probably be nipped by the frost and wither on the ground: but
there is a good crop coming. Material pneuma is destined to high functions;
and man is to read by gas-light.


    The solar system truly solved; demonstrating by the perfect harmony of
    the planets, founded on the four universal laws, the Sun to be an
    electric space; and a source of every natural production displayed
    throughout the solar system. By James Hopkins.[82] London, 1849, 8vo.

The author says:

"I am satisfied that I have given the true _laws_ constituting the _Sun_ to
be _space_; and I call upon those disposed to maintain the contrary, to
give true _laws_ showing him to be a body: until such can be satisfactorily
established, I have an undoubted claim to the credit of my theory,--That
the Sun is an _Electric Space_, fed and governed by the {42} planets, which
have the property of attracting heat from it; and the means of supplying
the necessary _pabulum_ by their degenerated air driven off towards the
central space--the wonderful alembic in which it becomes transmuted to the
revivifying necessities of continuous action; and the central space or Sun
being perfectly electric, has the counter property of repulsing the bodies
that attract it. How wonderful a conception! How beautiful, how magnificent
an arrangement!

"O Centre! O Space! O Electric Space!"


1849. _Joseph Ady_[83] is entitled to a place in this list of discoverers:
his great fault, like that of some others, lay in pushing his method too
far. He began by detecting unclaimed dividends, and disclosing them to
their right owners, exacting his fee before he made his communication. He
then generalized into trying to get fees from all of the _name_ belonging
to a dividend; and he gave mysterious hints of danger impending. For
instance, he would write to a clergyman that a legal penalty was hanging
over him; and when the alarmed divine forwarded the sum required for
disclosure, he was favored with an extract from some old statute or canon,
never repealed, forbidding a clergyman to be a member of a corporation, and
was reminded that he had insured his life in the ---- Office, which had a
royal charter. He was facetious, was Joseph: he described himself in his
circulars as "personally known to Sir Peter Laurie[84] and all other
aldermen"; which was nearly true, {43} as he had been before most of them
on charges of false pretence; but I believe he was nearly always within the
law. Sir James Duke, when Lord Mayor, having particularly displeased him by
a decision, his circulars of 1849 contain the following:

"Should you have cause to complain of any party, Sir J. Duke has contrived
a new law of evidence, viz., write to him, he will consider your letter
sufficient proof, and make the parties complained of pay without judge or
jury, and will frank you from every expense."

I strongly suspect that Joseph Ady believed in himself.

He sometimes issued a second warning, of a Sibylline character:

"Should you find cause to complain of anybody, my voluntary referee, the
Rt. Hon. Sir Peter Laurie, Kt., perpetual Deputy Lord Mayor, will see
justice done you without any charge whatever: he and his toady, -- ----
----. The accursed of Moses can hang any man: thus, by catching him alone
and swearing Naboth spake evil against God and the King. Therefore (!) I
admit no strangers to a personal conference without a prepayment of 20s.
each. Had you attended to my former notice you would have received twice as
much: neglect this and you will lose all."


    Zadkiel's Almanac for 1849. Nineteenth number.

    Raphael's Prophetic Almanac for 1849. Twenty-ninth number.

    Reasons for belief in judicial astrology, and remarks on the dangerous
    character of popish priestcraft. London, 1849, 12mo.

    Astronomy in a nutshell: or the leading problems of the solar system
    solved by simple proportion only, on the theory of magnetic attraction.
    By Lieut. Morrison,[85] N. N. London (_s. a._) 12mo.


Lieut. Morrison is Zadkiel Tao Sze, and declares himself in real earnest an
astrologer. There are a great many books on astrology, but I have not felt
interest enough to preserve many of them which have come in my way. If
anything ever had a fair trial, it was astrology. The idea itself is
natural enough. A human being, set down on this earth, without any
tradition, would probably suspect that the heavenly bodies had something to
do with the guidance of affairs. I think that any one who tries will
ascertain that the planets do not prophesy: but if he should find to the
contrary, he will of course go on asking. A great many persons class
together belief in astrology and belief in apparitions: the two things
differ in precisely the way in which a science of observation differs from
a science of experiment. Many make the mistake which M. le Marquis made
when he came too late, and hoped M. Cassini[86] would do the eclipse over
again for his ladies. The apparition chooses its own time, and comes as
seldom or as often as it pleases, be it departed spirit, nervous
derangement, or imposition. Consequently it can only be observed, and not
experimented upon. But the heavens, if astrology be true, are prophesying
away day and night all the year round, and about every body. Experiments
can be made, then, except only on rare phenomena, such as eclipses: anybody
may choose his time and his question. This is the great difference: and
experiments were made, century after century. If astrology had been true,
it must have lasted in an ever-improving state. If it be true, it is a
truth, and a useful truth, which had experience and prejudice both in its
favor, and yet lost ground as soon as astronomy, its working tool, began to

1850. A letter in the handwriting of an educated man, dated from a street
in which it must be taken that educated persons live, is addressed to the
Secretary of the {45} Astronomical Society about a matter on which the
writer says "his professional pursuit will enable him to give a
satisfactory reply." In a question before a court of law it is sworn on one
side that the moon was shining at a certain hour of a certain night on a
certain spot in London; on the other side it is affirmed that she was
clouded. The Secretary is requested to decide. This is curious, as the
question is not astrological. Persons still send to Greenwich, now and
then, to have their fortunes told. In one case, not very many years ago, a
young gentleman begged to know who his wife was to be, and what fee he was
to remit.

Sometimes the astronomer turns conjurer for fun, and his prophesies are
fulfilled. It is related of Flamsteed[87] that an old woman came to know
the whereabouts of a bundle of linen which had strayed. Flamsteed drew a
circle, put a square into it, and gravely pointed out a ditch, near her
cottage, in which he said it would be found. He meant to have given the
woman a little good advice when she came back: but she came back in great
delight, with the bundle in her hand, found in the very place. The late
Baron Zach[88] received a letter from Pons,[89] a successful finder of
comets, complaining that for a certain period he had found no comets,
though he had searched diligently. Zach, a man of much sly humor, told him
that no spots had been seen on the sun for about the same time--which was
true,--and assured him that when the spots came back, the comets would come
with them. Some time after he got a letter {46} from Pons, who informed him
with great satisfaction that he was quite right, that very large spots had
appeared on the sun, and that he had found a fine comet shortly after. I do
not vouch for the first story, but I have the second in Zach's handwriting.
It would mend the joke exceedingly if some day a real relation should be
established between comets and solar spots: of late years good reason has
been shown for advancing a connection between these spots and the earth's
magnetism.[90] If the two things had been put to Zach, he would probably
have chosen the comets. Here is a hint for a paradox: the solar spots are
the dead comets, which have parted with their light and heat to feed the
sun, as was once suggested. I should not wonder if I were too late, and the
thing had been actually maintained. My list does not contain the twentieth
part of the possible whole.

The mention of coincidences suggests an everlasting source of explanations,
applicable to all that is extraordinary. The great paradox of coincidence
is that of Leibnitz, known as the _pre-established harmony_, or _law of
coincidences_, by which, separately and independently, the body receives
impressions, and the mind proceeds as if it had perceived them from
without. Every sensation, and the consequent state of the soul, are
independent things coincident in time by the pre-established law. The
philosopher could not otherwise _account for_ the connection of mind and
matter; and he never goes by so vulgar a rule as _Whatever is, is_; to him
that which is not clear as to how, is not at all. Philosophers in general,
who tolerate each other's theories much better than Christians do each
other's failings, seldom revive Leibnitz's fantasy: they seem to act upon
the maxim quoted by Father Eustace[91] from the {47} Decretals, _Facinora
ostendi dum puniuntur, flagitia autem abscondi debent_.[92]

The great _ghost-paradox_, and its theory of _coincidences_, will rise to
the surface in the mind of every one. But the use of the word _coincidence_
is here at variance with its common meaning. When A is constantly
happening, and also B, the occurrence of A and B at the same moment is the
mere coincidence which may be casualty. But the case before us is that A is
constantly happening, while B, when it does happen, almost always happens
with A, and very rarely without it. That is to say, such is the phenomenon
asserted: and all who rationally refer it to casualty, affirm that B is
happening very often as well as A, but that it is not thought worthy of
being recorded except when A is simultaneous. Of course A is here a death,
and B the spectral appearance of the person who dies. In talking of this
subject it is necessary to put out of the question all who play fast and
loose with their secret convictions: these had better give us a reason,
when they feel internal pressure for explanation, that there is no
weathercock at Kilve; this would do for all cases. But persons of real
inquiry will see that first, experience does not bear out the asserted
frequency of the spectre, without the alleged coincidence of death: and
secondly, that if the crowd of purely casual spectres were so great that it
is no wonder that, now and then the person should have died at or near the
moment, we ought to expect a much larger proportion of cases in which the
spectre should come at the moment of the death of one or another of all the
cluster who are closely connected with the original of the spectre. But
this, we know, is almost without example. It remains then, for all, who
speculate at all, to look upon the asserted phenomenon, think what they may
of it, the thing which is to be explained, as a _connection_ in time of the
death, and the {48} simultaneous appearance of the dead. Any person the
least used to the theory of probabilities will see that purely casual
coincidence, the _wrong spectre_ being comparatively so rare that it may be
said never to occur, is not within the rational field of possibility.

The purely casual coincidence, from which there is no escape except the
actual doctrine of special providences, carried down to a very low point of
special intention, requires a junction of the things the like of each of
which is always happening. I will give three instances which have occurred
to myself within the last few years: I solemnly vouch for the literal truth
of every part of all three:

In August 1861, M. Senarmont,[93] of the French Institute, wrote to me to
the effect that Fresnel[94] had sent to England, in or shortly after 1824,
a paper for translation and insertion in the _European Review_, which
shortly afterwards expired. The question was what had become of that paper.
I examined the _Review_ at the Museum, found no trace of the paper, and
wrote back to that effect at the Museum, adding that everything now
depended on ascertaining the name of the editor, and tracing his papers: of
this I thought there was no chance. I posted this letter on my way home, at
a Post Office in the Hampstead Road at the junction with Edward Street, on
the opposite side of which is a bookstall. Lounging for a moment over the
exposed books, _sicut meus est mos_,[95] I saw, within a few minutes of the
posting of the letter, a little catch-penny book of anecdotes of Macaulay,
which I bought, and ran over for a minute. My eye was soon caught by this
sentence: "One of the young fellows immediately wrote to the editor (Mr.
Walker) {49} of the _European Review_." I thus got the clue by which I
ascertained that there was no chance of recovering Fresnel's paper. Of the
mention of current reviews, not one in a thousand names the editor.

In the summer of 1865 I made my first acquaintance with the tales of
Nathaniel Hawthorne, and the first I read was about the siege of Boston in
the War of Independence. I could not make it out: everybody seemed to have
got into somebody else's place. I was beginning the second tale, when a
parcel arrived: it was a lot of old pamphlets and other rubbish, as he
called it, sent by a friend who had lately sold his books, had not thought
it worth while to send these things for sale, but thought I might like to
look at them and possibly keep some. The first thing I looked at was a
sheet which, being opened, displayed "A plan of Boston and its environs,
shewing the true situation of his Majesty's army and also that of the
rebels, drawn by an engineer, at Boston Oct. 1775." Such detailed plans of
current sieges being then uncommon, it is explained that "The principal
part of this plan was surveyed by Richard Williams, Lieutenant at Boston;
and sent over by the son of a nobleman to his father in town, by whose
permission it was published." I immediately saw that my confusion arose
from my supposing that the king's troops were besieging the rebels, when it
was just the other way.

April 1, 1853, while engaged in making some notes on a logical point, an
idea occurred which was perfectly new to me, on the mode of conciliating
the notions _omnipresence_ and _indivisibility into parts_. What it was is
no matter here: suffice it that, since it was published elsewhere (in a
paper on _Infinity_, _Camb. Phil. Trans._ vol. xi. p. 1) I have not had it
produced to me. I had just finished a paragraph on the subject, when a
parcel came in from a bookseller containing Heywood's[96] _Analysis of
Kant's Critick_, 1844.

{50} On turning over the leaves I found (p. 109) the identical thought
which up to this day, I only know as in my own paper, or in Kant. I feel
sure I had not seen it before, for it is in Kant's first edition, which was
never translated to my knowledge; and it does not appear in the later
editions. Mr. Heywood gives some account of the first edition.

In the broadsheet which gave account of the dying scene of Charles II, it
is said that the Roman Catholic priest was introduced by P. M. A. C. F. The
chain was this: the Duchess of Portsmouth[97] applied to the Duke of York,
who may have consulted his Cordelier confessor, Mansuete, about procuring a
priest, and the priest was smuggled into the king's room by the Duchess and
Chiffinch.[98] Now the letters are a verbal acrostic of _Père Mansuete a
Cordelier Friar_, and a syllabic acrostic of _PortsMouth and ChifFinch_.
This is a singular coincidence. Macaulay adopted the first interpretation,
preferring it to the second, which I brought before him as the conjecture
of a near relative of my own. But Mansuete is not mentioned in his
narrative: it may well be doubted whether the writer of a broadside for
English readers would use _Père_ instead of _Father_. And the person who
really "reminded" the Duke of "the duty he owed to his brother," was the
Duchess and not Mansuete. But my affair is only with the coincidence.

But there are coincidences which are really connected without the
connection being known to those who find in them matter of astonishment.
Presentiments furnish marked cases: sometimes there is no mystery to those
who have the clue. In the _Gentleman's Magazine_ (vol. 80, part 2, p. 33)
we read, the subject being presentiment of death, as follows: "In 1778, to
come nearer the recollection of {51} survivors, at the taking of
Pondicherry, Captain John Fletcher, Captain De Morgan, and Lieutenant
Bosanquet, each distinctly foretold his own death on the morning of his
fate." I have no doubt of all three; and I knew it of my grandfather long
before I read the above passage. He saw that the battery he commanded was
unduly exposed: I think by the sap running through the fort when produced.
He represented this to the engineer officers, and to the
commander-in-chief; the engineers denied the truth of the statement, the
commander believed them, my grandfather quietly observed that he must make
his will, and the French fulfilled his prediction. His will bore date the
day of his death; and I always thought it more remarkable than the
fulfilment of the prophecy that a soldier should not consider any danger
short of one like the above, sufficient reason to make his will. I suppose
the other officers were similarly posted. I am told that military men very
often defer making their wills until just before an action: but to face the
ordinary risks intestate, and to wait until speedy death must be the all
but certain consequence of a stupid mistake, is carrying the principle very
far. In the matter of coincidences there are, as in other cases, two
wonderful extremes with every intermediate degree. At one end we have the
confident people who can attribute anything to casual coincidence; who
allow Zadok Imposture and Nathan Coincidence to anoint Solomon Selfconceit
king. At the other end we have those who see something _very curious_ in
any coincidence you please, and whose minds yearn for a deep reason. A
speculator of this class happened to find that Matthew viii. 28-33 and Luke
viii. 26-33 contain the same account, that of the demons entering into the
swine. Very odd! chapters tallying, and verses so nearly: is the
versification rightly managed? Examination is sure to show that there are
monstrous inconsistencies in the mode of division, which being corrected,
the verses tally as well as the chapters. And then how comes it? I cannot
go on, {52} for I have no gift at torturing a coincidence, but I would lay
twopence, if I could make a bet--which I never did in all my life--that
some one or more of my readers will try it. Some people say that the study
of chances tends to awaken a spirit of gambling: I suspect the contrary. At
any rate, I myself, the writer of a mathematical book and a comparatively
popular book, have never laid a bet nor played for a stake, however small:
not one single time.

It is useful to record such instances as I have given, with precision and
on the solemn word of the recorder. When such a story as that of Flamsteed
is told, _a priori_ assures us that it could not have been: the story may
have been a _ben trovato_,[99] but not the bundle. It is also useful to
establish some of the good jokes which all take for inventions. My friend
Mr. J. Bellingham Inglis,[100] before 1800, saw the tobacconist's carriage
with a sample of tobacco in a shield, and the motto _Quid rides_[101] (_N._
& _Q._, 3d S. i. 245). His father was able to tell him all about it. The
tobacconist was Jacob Brandon, well known to the elder Mr. Inglis, and the
person who started the motto, the instant he was asked for such a thing,
was Harry Calender of Lloyd's, a scholar and a wit. My friend Mr. H. Crabb
Robinson[102] remembers the King's Counsel (Samuel Marryat) who took the
motto _Causes produce effects_, when his success enabled him to start a

The coincidences of errata are sometimes very remarkable: it may be that
the misprint has a sting. The death of Sir W. Hamilton[103] of Edinburgh
was known in London on a Thursday, and the editor of the _Athenæum_ wrote
to {53} me in the afternoon for a short obituary notice to appear on
Saturday. I dashed off the few lines which appeared without a moment to
think: and those of my readers who might perhaps think me capable of
contriving errata with meaning will, I am sure, allow the hurry, the
occasion, and my own peculiar relation to the departed, as sufficient
reasons for believing in my entire innocence. Of course I could not see a
proof: and two errata occurred. The words "addition to Stewart"[104]
require "_for_ addition to _read_ edition of." This represents what had
been insisted on by the Edinburgh publisher, who, frightened by the edition
of Reid,[105] had stipulated for a simple reprint without notes. Again
"principles of logic and mathematics" required "_for_ mathematics _read_
metaphysics." No four words could be put together which would have so good
a title to be Hamilton's motto.

April 1850, found in the letter-box, three loose leaves, well printed and
over punctuated, being

    Chapter VI. Brethren, lo I come, holding forth the word of life, for so
    I am commanded.... Chapter VII. Hear my prayer, O generations! and walk
    by the way, to drink the waters of the river.... Chapter VIII. Hearken
    o earth, earth, earth, and the kings of the earth, and their armies....

A very large collection might be made of such apostolic writings. They go
on well enough in a misty--meant for mystical--imitation of St. Paul or the
prophets, until at last some prodigious want of keeping shows the education
of the writer. For example, after half a page which might {54} pass for
Irving's[106] preaching--though a person to whom it was presented as such
would say that most likely the head and tail would make something more like
head and tail of it--we are astounded by a declaration from the _Holy
Spirit_, speaking of himself, that he is "not ashamed of the Gospel of
Christ." It would be long before we should find in _educated_ rhapsody--of
which there are specimens enough--such a thing as a person of the Trinity
taking merit for moral courage enough to stand where St. Peter fell. The
following declaration comes next--"I will judge between cattle and cattle,
that use their tongues."


    The figure of the earth. By J. L. Murphy,[107] of Birmingham. (London
    and Birmingham, 4 pages, 12mo.) (1850?)

Mr. Murphy invites attention and objection to some assertions, as that the
earth is prolate, not oblate. "If the philosopher's conclusion be right,
then the pole is the center of a valley (!) thirteen miles deep." Hence it
would be very warm. It is answer enough to ask--Who knows that it is not?

    *** A paragraph in the MS. appears to have been inserted in this place
    by mistake. It will be found in the Appendix at the end of this
    volume.--S. E. De M.


1851. The following letter was written by one of a class of persons whom,
after much experience of them, I {55} do _not_ pronounce insane. But in
this case the second sentence gives a suspicion of actual delusion of the
senses; the third looks like that eye for the main chance which passes for
sanity on the Stock Exchange and elsewhere:

15th Sept. 1851.

"Gentlemen,--I pray you take steps to make known that yesterday I completed
my invention which will give motion to every country on the Earth;--to move
Machinery!--the long sought in vain 'Perpetual Motion'!!--I was supported
at the time by the Queen and H.R.H. Prince Albert. If, Gentlemen, you can
advise me how to proceed to claim the reward, if any is offered by the
Government, or how to secure the PATENT for the machine, or in any way
assist me by advice in this great work, I shall most graciously acknowledge
your consideration.

These are my convictions that my SEVERAL discoveries will be realized: and
this great one can be at once acted upon: although at this moment it only
exists in my mind, from my knowledge of certain fixed principles in
nature:--the Machine I have not made, as I only completed the discovery

          I have, etc. ---- ----"

  To the Directors of the
  London University, Gower Street.


    The Divine Drama of History and Civilisation. By the Rev. James Smith,
    M.A.[108] London, 1854, 8vo.

I have several books on that great paradox of our day, _Spiritualism_, but
I shall exclude all but three. The bibliography of this subject is now very
large. The question is one both of evidence and speculation;--Are the facts
{56} true? Are they caused by spirits? These I shall not enter upon: I
shall merely recommend this work as that of a spiritualist who does not
enter on the subject, which he takes for granted, but applies his derived
views to the history of mankind with learning and thought. Mr. Smith was a
man of a very peculiar turn of thinking. He was, when alive, the editor, or
_an_ editor, of the _Family Herald_: I say when alive, to speak according
to knowledge; for, if his own views be true, he may have a hand in it
still. The answers to correspondents, in his time, were piquant and
original above any I ever saw. I think a very readable book might be made
out of them, resembling "Guesses at Truth:" the turn given to an inquiry
about morals, religion, or socials, is often of the highest degree of
_unexpectedness_; the poor querist would find himself right in a most
unpalatable way.

Answers to correspondents, in newspapers, are very often the fag ends of
literature. I shall never forget the following. A person was invited to
name a rule without exception, if he could: he answered "A man _must_ be
present when he is shaved." A lady--what right have ladies to decide
questions about shaving?--said this was not properly a rule; and the oracle
was consulted. The editor agreed with the lady; he said that "a man _must_
be present when he is shaved" is not a _rule_, but a _fact_.

[Among my anonymous communicants is one who states that I have done
injustice to the Rev. James Smith in "referring to him as a spiritualist,"
and placing his "Divine Drama" among paradoxes: "it is no paradox, nor do
_spiritualistic_ views mar or weaken the execution of the design." Quite
true: for the design is to produce and enforce "spiritualistic views"; and
leather does not mar nor weaken a shoemaker's plan. I knew Mr. Smith well,
and have often talked to him on the subject: but more testimony from me is
unnecessary; his book will speak for itself. {57} His peculiar style will
justify a little more quotation than is just necessary to prove the point.
Looking at the "battle of opinion" now in progress, we see that Mr. Smith
was a prescient:

(P. 588.) "From the general review of parties in England, it is evident
that no country in the world is better prepared for the great Battle of
Opinion. Where else can the battle be fought but where the armies are
arrayed? And here they all are, Greek, Roman, Anglican, Scotch, Lutheran,
Calvinist, Established and Territorial, with Baronial Bishops, and
Nonestablished of every grade--churches with living prophets and apostles,
and churches with dead prophets and apostles, and apostolical churches
without apostles, and philosophies without either prophets or apostles, and
only wanting one more, 'the Christian Church,' like Aaron's rod, to swallow
up and digest them all, and then bud and flourish. As if to prepare our
minds for this desirable and inevitable consummation, different parties
have been favored with a revival of that very spirit of revelation by which
the Church itself was originally founded. There is a complete series of
spiritual revelations in England and the United States, besides mesmeric
phenomena that bear a resemblance to revelation, and thus gradually open
the mind of the philosophical and infidel classes, as well as the professed
believers of that old revelation which they never witnessed in living
action, to a better understanding of that Law of Nature (for it is a Law of
Nature) in which all revelation originates and by which its spiritual
communications are regulated."

Mr. Smith proceeds to say that there are _only_ thirty-five incorporated
churches in England, all formed from the New Testament except five, to each
of which five he concedes a revelation of its own. The five are the
Quakers, the Swedenborgians, the Southcottians, the Irvingites, and the
Mormonites. Of Joanna Southcott he speaks as follows: {58}

(P. 592.) "Joanna Southcott[109] is not very gallantly treated by the
gentlemen of the Press, who, we believe, without knowing anything about
her, merely pick up their idea of her character from the rabble. We once
entertained the same rabble idea of her; but having read her works--for we
really have read them--we now regard her with great respect. However, there
is a great abundance of chaff and straw to her grain; but the grain is
good, and as we do not eat either the chaff or straw if we can avoid it,
nor even the raw grain, but thrash it and winnow it, and grind it and bake
it, we find it, after undergoing this process, not only very palatable, but
a special dainty of its kind. But the husk is an insurmountable obstacle to
those learned and educated gentlemen who judge of books entirely by the
style and the grammar, or those who eat grain as it grows, like the cattle.
Such men would reject all prological revelation; for there never was and
probably never will be a revelation by voice and vision communicated in
classical manner. It would be an invasion of the rights and prerogatives of
Humanity, and as contrary to the Divine and Established order of mundane
government, as a field of quartern loaves or hot French rolls."

Mr. Smith's book is spiritualism from beginning to end; and my anonymous
gainsayer, honest of course, is either ignorant of the work he thinks he
has read, or has a most remarkable development of the organ of


I cut the following from a Sunday paper in 1849:

"X. Y.--The Chaldeans began the mathematics, in which the Egyptians
excelled. Then crossing the sea, by means {59} of Thales,[110] the
Milesian, they came into Greece, where they were improved very much by
Pythagoras,[111] Anaxagoras,[112] and Anopides[113] of Chios. These were
followed by Briso,[114] Antipho, [two circle-squarers; where is Euclid?]
and Hippocrates,[115] but the excellence of the algebraic art was begun by
Geber,[116] an Arabian astronomer, and was carried on by Cardanus,[117]
Tartaglia,[118] Clavius,[119], Stevinus,[120] Ghetaldus,[121]
Herig_e_nius,[122] Fran. Van Schooten [meaning Francis Van Schooten[123]],
Florida de Beau_m_e,[124] etc."

Bryso was a mistaken man. Antipho had the disadvantage of being in advance
of his age. He had the notion of which the modern geometry has made so
much, that of {60} a circle being the polygon of an infinitely great number
of sides. He could make no use of it, but the notion itself made him a
sophist in the eyes of Aristotle, Eutocius,[125] etc. Geber, an Arab
astronomer, and a reputed conjurer in Europe, seems to have given his name
to unintelligible language in the word _gibberish_. At one time _algebra_
was traced to him; but very absurdly, though I have heard it suggested that
_algebra_ and _gibberish_ must have had one inventor.

Any person who meddles with the circle may find himself the crane who was
netted among the geese: as Antipho for one, and Olivier de Serres[126] for
another. This last gentleman ascertained, by weighing, that the area of the
circle is very nearly that of the square on the side of the inscribed
equilateral triangle: which it is, as near as 3.162 ... to 3.141.... He did
not pretend to more than approximation; but Montucla and others
misunderstood him, and, still worse, misunderstood their own
misunderstanding, and made him say the circle was exactly double of the
equilateral triangle. He was let out of limbo by Lacroix, in a note to his
edition of Montucla's _History of Quadrature_.


    Quadratura del cerchio, trisezione dell' angulo, et duplicazione del
    cubo, problemi geometricamente risolute e dimostrate dal Reverendo
    Arciprete di San Vito D. Domenico Angherà,[127] Malta, 1854, 8vo.


    Equazioni geometriche, estratte dalla lettera del Rev. Arciprete ... al
    Professore Pullicino[128] sulla quadratura del cerchio. Milan, 1855 or
    1856, 8vo.

    Il Mediterraneo gazetta di Malta, 26 Decembre 1855, No. 909: also 911,
    912, 913, 914, 936, 939.

    The Malta Times, Tuesday, 9th June 1857.

    Misura esatta del cerchio, dal Rev. D. Angherà. Malta, 1857, 12mo.

    Quadrature of the circle ... by the Rev. D. Angherà, Archpriest of St.
    Vito. Malta, 1858, 12mo.

I have looked for St. Vitus in catalogues of saints, but never found his
legend, though he figures as a day-mark in the oldest almanacs. He must be
properly accredited, since he was an archpriest. And I pronounce and
ordain, by right accruing from the trouble I have taken in this subject,
that he, St. Vitus, who leads his votaries a never-ending and unmeaning
dance, shall henceforth be held and taken to be the patron saint of the
circle-squarer. His day is the 15th of June, which is also that of St.
Modestus,[129] with whom the said circle-squarer often has nothing to do.
And he must not put himself under the first saint with a slantendicular
reference to the other, as is much to be feared was done by the Cardinal
who came to govern England with a title containing St. Pudentiana,[130] who
shares a day with _St. Dunstan_. The Archpriest of St. Vitus will have it
that the square inscribed in a semicircle is half of the semicircle, or the
circumference 3-1/5 diameters. He is active and able, with {62} nothing
wrong about him except his paradoxes. In the second tract named he has
given the testimonials of crowned heads and ministers, etc. as follows.
Louis-Napoleon gives thanks. The minister at Turin refers it to the Academy
of Sciences, and hopes so much labor will be judged _degna di pregio_.[131]
The Vice-Chancellor of Oxford--a blunt Englishman--begs to say that the
University has never proposed the problem, as some affirm. The Prince
Regent of Baden has received the work with lively interest. The Academy of
Vienna is not in a position to enter into the question. The Academy of
Turin offers the most _distinct_ thanks. The Academy della Crusca attends
only to literature, but gives thanks. The Queen of Spain has received the
work with the highest appreciation. The University of Salamanca gives
infinite thanks, and feels true satisfaction in having the book. Lord
Palmerston gives thanks, by the hand of "William San." The Viceroy of
Egypt, not being yet up in Italian, will spend his first moments of leisure
in studying the book, when it shall have been translated into French: in
the mean time he congratulates the author upon his victory over a problem
so long held insoluble. All this is seriously published as a rate in aid of
demonstration. If these royal compliments cannot make the circumference of
a circle about 2 per cent. larger than geometry will have it --which is all
that is wanted--no wonder that thrones are shaky.

I am informed that the legend of St. Vitus is given by Ribadeneira[132] in
his lives of Saints, and that Baronius,[133] in {63} his _Martyrologium
Romanum_, refers to several authors who have written concerning him. There
is an account in Mrs. Jameson's[134] _History of Sacred and Legendary Art_
(ed. of 1863, p. 544). But it seems that St. Vitus is the patron saint of
_all_ dances; so that I was not so far wrong in making him the protector of
the cyclometers. Why he is represented with a cock is a disputed point,
which is now made clear: next after _gallus gallinaceus_[135] himself,
there is no crower like the circle-squarer.


The following is an extract from the _English Cyclopædia_, Art. TABLES:

"1853. William Shanks,[136] _Contributions to Mathematics, comprising
chiefly the Rectification of the Circle to 607 Places of Tables_, London,
1853. (QUADRATURE OF THE CIRCLE.) Here is a _table_, because it tabulates
the results of the subordinate steps of this enormous calculation as far as
527 decimals: the remainder being added as results only during the
printing. For instance, one step is the calculation of the reciprocal of
601.5^{601}; and the result is given. The number of pages required to
describe these results is 87. Mr. Shanks has also thrown off, as chips or
splinters, the values of the base of Napier's logarithms, and of its
logarithms of 2, 3, 5, 10, to 137 decimals; and the value of the modulus
.4342 ... to 136 decimals: with the 13th, 25th, 37th ... up to the 721st
powers of 2. These tremendous stretches of calculation--at least we so call
them in our day--are useful in several respects; they prove more than {64}
the capacity of this or that computer for labor and accuracy; they show
that there is in the community an increase of skill and courage. We say in
the community: we fully believe that the unequalled turnip which every now
and then appears in the newspapers is a sufficient presumption that the
average turnip is growing bigger, and the whole crop heavier. All who know
the history of the quadrature are aware that the several increases of
numbers of decimals to which [pi] has been carried have been indications of
a general increase in the power to calculate, and in courage to face the
labor. Here is a comparison of two different times. In the day of
Cocker,[137] the pupil was directed to perform a common subtraction with a
voice-accompaniment of this kind: '7 from 4 I cannot, but add 10, 7 from 14
remains 7, set down 7 and carry 1; 8 and 1 which I carry is 9, 9 from 2 I
cannot, etc.' We have before us the announcement of the following _table_,
undated, as open to inspection at the Crystal Palace, Sydenham, in two
diagrams of 7 ft. 2 in, by 6 ft. 6 in.: 'The figure 9 involved into the
912th power, and antecedent powers or involutions, containing upwards of
73,000 figures. Also, the proofs of the above, containing upwards of
146,000 figures. By Samuel Fancourt, of Mincing Lane, London, and completed
by him in the year 1837, at the age of sixteen. N.B. The whole operation
performed by simple arithmetic.' The young operator calculated by
successive squaring the 2d, 4th, 8th, etc., powers up to the 512th, with
proof by division. But 511 multiplications by 9, in the short (or 10-1)
way, would have been much easier. The 2d, 32d, 64th, 128th, 256th, and
512th powers are given at the back of the announcement. The powers of 2
have been calculated for many purposes. In Vol. II of his _Magia
Universalis Naturæ et Artis_, Herbipoli, 1658, 4to, the Jesuit Gaspar
Schott[138] having discovered, on some grounds of theological {65} magic,
that the degrees of grace of the Virgin Mary were in number the 256th power
of 2, calculated that number. Whether or no his number correctly
represented the result he announced, he certainly calculated it rightly, as
we find by comparison with Mr. Shanks."

There is a point about Mr. Shanks's 608 figures of the value of [pi] which
attracts attention, perhaps without deserving it. It might be expected
that, in so many figures, the nine digits and the cipher would occur each
about the same number of times; that is, each about 61 times. But the fact
stands thus: 3 occurs 68 times; 9 and 2 occur 67 times each; 4 occurs 64
times; 1 and 6 occur 62 times each; 0 occurs 60 times; 8 occurs 58 times; 5
occurs 56 times; and 7 occurs only 44 times. Now, if all the digits were
equally likely, and 608 drawings were made, it is 45 to 1 against the
number of sevens being as distant from the probable average (say 61) as 44
on one side or 78 on the other. There must be some reason why the number 7
is thus deprived of its fair share in the structure. Here is a field of
speculation in which two branches of inquirers might unite. There is but
one number which is treated with an unfairness which is incredible as an
accident; and that number is the mystic number _seven_! If the cyclometers
and the apocalyptics would lay their heads together until they come to a
unanimous verdict on this phenomenon, and would publish nothing until they
are of one mind, they would earn the gratitude of their race.--I was wrong:
it is the Pyramid-speculator who should have been appealed to. A
correspondent of my friend Prof. Piazzi Smyth[139] notices that 3 is the
number of most frequency, and that 3-1/7 is the nearest approximation to it
in simple digits. Professor Smyth himself, whose word on Egypt is paradox
of a very high order, backed by a great quantity of useful labor, the
results which will be made available by those who do not receive {66} the
paradoxes, is inclined to see confirmation for some of his theory in these


These paradoxes of calculation sometimes appear as illustrations of the
value of a new method. In 1863, Mr. G. Suffield,[140] M.A., and Mr. J. R.
Lunn,[141] M.A., of Clare College and of St. John's College, Cambridge,
published the whole quotient of 10000 ... divided by 7699, throughout the
whole of one of the recurring periods, having 7698 digits. This was done in
illustration of Mr. Suffield's method of _Synthetic division_.

Another instance of computation carried to paradoxical length, in order to
illustrate a method, is the solution of x^3 - 2x = 5, the example given of
Newton's method, on which all improvements have been tested. In 1831,
Fourier's[142] posthumous work on equations showed 33 figures of solution,
got with enormous labor. Thinking this a good opportunity to illustrate the
superiority of the method of W. G. Horner,[143] not yet known in France,
and not much known in {67} England, I proposed to one of my classes, in
1841, to beat Fourier on this point, as a Christmas exercise. I received
several answers, agreeing with each other, to 50 places of decimals. In
1848, I repeated the proposal, requesting that 50 places might be exceeded:
I obtained answers of 75, 65, 63, 58, 57, and 52 places. But one answer, by
Mr. W. Harris Johnston,[144] of Dundalk, and of the Excise Office, went to
101 decimal places. To test the accuracy of this, I requested Mr. Johnston
to undertake another equation, connected with the former one in a way which
I did not explain. His solution verified the former one, but he was unable
to see the connection, even when his result was obtained. My reader may be
as much at a loss: the two solutions are:


The results are published in the _Mathematician_, Vol. III, p. 290. In
1851, another pupil of mine, Mr. J. Power Hicks,[145] carried the result to
152 decimal places, without knowing what Mr. Johnston had done. The result
is in the _English Cyclopædia_, article INVOLUTION AND EVOLUTION.

I remark that when I write the initial of a Christian name, the most usual
name of that initial is understood. I never saw the name of W. G. Horner
written at length, until I applied to a relative of his, who told me that
he was, as I supposed, Wm. _George_, but that he was named after a relative
of that _surname_.

The square root of 2, to 110 decimal places, was given {68} me in 1852 by
my pupil, Mr. William Henry Colvill, now (1867) Civil Surgeon at Baghdad.
It was


Mr. James Steel[146] of Birkenhead verified this by actual multiplication,
and produced

  2 - 2580413 / 10^{117}

as the square.

    Calcolo decidozzinale del Barone Silvio Ferrari. Turin, 1854, 4to.

This is a serious proposal to alter our numeral system and to count by
twelves. Thus 10 would be twelve, 11 thirteen, etc., two new symbols being
invented for ten and eleven. The names of numbers must of course be
changed. There are persons who think such changes practicable. I thought
this proposal absurd when I first saw it, and I think so still:[147] but
the one I shall presently describe beats it so completely in that point,
that I have not a smile left for this one.


    The successful and therefore probably true theory of Comets. London,
    1854. (4pp. duodecimo.)

The author is the late Mr. Peter Legh,[148] of Norbury Booths Hall,
Knutsford, who published for eight or ten {69} years the _Ombrological
Almanac_, a work of asserted discovery in meteorology. The theory of comets
is that the joint attraction of the new moon and several planets in the
direction of the sun, draws off the gases from the earth, and forms these
cometic meteors. But how these meteors come to describe orbits round the
sun, and to become capable of having their returns predicted, is not


    The Mormon, New York, Saturday, Oct. 27, 1855.

A newspaper headed by a grand picture of starred and striped banners,
beehive, and eagle surmounting it. A scroll on each side: on the left,
"Mormon creed. Mind your own business. Brigham Young;"[149] on the right,
"Given by inspiration of God. Joseph Smith."[150] A leading article on the
discoveries of Prof. Orson Pratt[151] says, "Mormonism has long taken the
lead in religion: it will soon be in the van both in science and politics."
At the beginning of the paper is Professor Pratt's "Law of Planetary
Rotation." The cube roots of the densities of the planets are as the square
roots of their periods of rotation. The squares of the cube roots of the
masses divided by the squares of the diameters are as the periods of
rotation. Arithmetical verification attempted, and the whole very modestly
stated {70} and commented on. Dated G. S. L. City, Utah Ter., Aug. 1, 1855.
If the creed, as above, be correctly given, no wonder the Mormonites are in
such bad odor.


    The two estates; or both worlds mathematically considered. London,
    1855, small (pp. 16).

The author has published mathematical works with his name. The present
tract is intended to illustrate mathematically a point which may be guessed
from the title. But the symbols do very little in the way of illustration:
thus, x being the _present value_ of the future estate (eternal happiness),
and a of all that this world can give, the author impresses it on the
mathematician that, x being infinitely greater than a, x + a = x, so that a
need not be considered. This will not act much more powerfully on a
mathematician by virtue of the symbols than if those same symbols had been
dispensed with: even though, as the author adds, "It was this method of
neglecting infinitely small quantities that Sir Isaac Newton was indebted
to for his greatest discoveries."

There has been a moderate quantity of well-meant attempt to enforce,
sometimes motive, sometimes doctrine, by arguments drawn from mathematics,
the proponents being persons unskilled in that science for the most part.
The ground is very dangerous: for the illustration often turns the other
way with greater power, in a manner which requires only a little more
knowledge to see. I have, in my life, heard from the pulpit or read, at
least a dozen times, that all sin is infinitely great, proved as follows.
The greater the being, the greater the sin of any offence against him:
therefore the offence committed against an infinite being is infinitely
great. Now the mathematician, of which the proposers of this argument are
not aware, is perfectly familiar with quantities which increase together,
and never cease increasing, but so that one of them remains finite when
{71} the other becomes infinite. In fact, the argument is a perfect _non
sequitur_.[152] Those who propose it have in their minds, though in a
cloudy and indefinite form, the idea of the increase of guilt being
_proportionate_ to the increase of greatness in the being offended. But
this it would never do to state: for by such statement not only would the
argument lose all that it has of the picturesque, but the asserted premise
would have no strong air of exact truth. How could any one undertake to
appeal to conscience to declare that an offence against a being 4-7/10
times as great as another is exactly, no more and no less, 4-7/10 times as
great an offence against the other?

The infinite character of the offence against an infinite being is laid
down in Dryden's _Religio Laici_,[153] and is, no doubt, an old argument:

 "For, granting we have sinned, and that th' offence
  Of man is made against Omnipotence,
  Some price that bears proportion must be paid,
  And infinite with infinite be weighed.
  See then the Deist lost; remorse for vice
  Not paid; or, paid, inadequate in price."

Dryden, in the words "bears proportion" is in verse more accurate than most
of the recent repeaters in prose. And this is not the only case of the kind
in his argumentative poetry.

My old friend, the late Dr. Olinthus Gregory,[154] who was a sound and
learned mathematician, adopted this dangerous kind of illustration in his
_Letters on the Christian Religion_. {72} He argued, by parallel, from what
he supposed to be the necessarily mysterious nature of the _impossible_
quantity of algebra to the necessarily mysterious nature of certain
doctrines of his system of Christianity. But all the difficulty and mystery
of the impossible quantity is now cleared away by the advance of
algebraical thought: and yet Dr. Gregory's book continues to be sold, and
no doubt the illustration is still accepted as appropriate.

The mode of argument used by the author of the tract above named has a
striking defect. He talks of reducing this world and the next to "present
value," as an actuary does with successive lives or next presentations.
Does value make interest? and if not, why? And if it do, then the present
value of an eternity is _not_ infinitely great. Who is ignorant that a
perpetual annuity at five per cent is worth only twenty years' purchase?
This point ought to be discussed by a person who treats heaven as a
deferred perpetual annuity. I do not ask him to do so, and would rather he
did not; but if he _will_ do it, he must either deal with the question of
discount, or be asked the reason why.

When a very young man, I was frequently exhorted to one or another view of
religion by pastors and others who thought that a mathematical argument
would be irresistible. And I heard the following more than once, and have
since seen it in print, I forget where. Since eternal happiness belonged to
the particular views in question, a benefit infinitely great, then, even if
the probability of their arguments were small, or even infinitely small,
yet the product of the chance and benefit, according to the usual rule,
might give a result which no one ought in prudence to pass over. They did
not see that this applied to all systems as well as their own. I take this
argument to be the most perverse of all the perversions I have heard or
read on the subject: there is some high authority for it, whom I forget.

The moral of all this is, that such things as the preceding should be kept
out of the way of those who are not {73} mathematicians, because they do
not understand the argument; and of those who are, because they do.

[The high authority referred to above is Pascal, an early cultivator of
mathematical probability, and obviously too much enamoured of his new
pursuit. But he conceives himself bound to wager on one side or the other.
To the argument (_Pensées_, ch. 7)[155] that "le juste est de ne point
parier," he answers, "Oui: mais il faut parier: vous êtes embarqué; et ne
parier point que Dieu est, c'est parier qu'il n'est pas."[156] Leaving
Pascal's argument to make its way with a person who, _being a sceptic_, is
yet positive that the issue is salvation or perdition, if a God there
be,--for the case as put by Pascal requires this,--I shall merely observe
that a person who elects to believe in God, as the best chance of gain, is
not one who, according to Pascal's creed, or any other worth naming, will
really secure that gain. I wonder whether Pascal's curious imagination ever
presented to him in sleep his convert, in the future state, shaken out of a
red-hot dice-box upon a red-hot hazard-table, as perhaps he might have
been, if Dante had been the later of the two. The original idea is due to
the elder Arnobius,[157] who, as cited by Bayle,[158] speaks thus:

"Sed et ipse [Christus] quæ pollicetur, non probat. Ita est. Nulla enim, ut
dixi, futurorum potest existere comprobatio. Cum ergo hæc sit conditio
futurorum, ut teneri et comprehendi nullius possint anticipationis attactu;
nonne {74} purior ratio est, ex duobus incertis, et in ambigua expectatione
pendentibus, id potius credere, quod aliquas spes ferat, quam omnino quod
nullas? In illo enim periculi nihil est, si quod dicitur imminere, cassum
fiat et vacuum: in hoc damnum est maximum, id est salutis amissio, si cum
tempus advenerit aperiatur non fuisse mendacium."[159]

Really Arnobius seems to have got as much out of the notion, in the third
century, as if he had been fourteen centuries later, with the arithmetic of
chances to help him.]


    The Sentinel, vol. ix. no. 27. London, Saturday, May 26, 1855.

This is the first London number of an Irish paper, Protestant in politics.
It opens with "Suggestions on the subject of a _Novum Organum Moralium_,"
which is the application of algebra and the differential calculus to
morals, socials, and politics. There is also a leading article on the
subject, and some applications in notes to other articles. A separate
publication was afterwards made, with the addition of a long Preface; the
author being a clergyman who I presume must have been the editor of the

    Suggestions as to the employment of a _Novum Organum Moralium_. Or,
    thoughts on the nature of the Differential Calculus, and on the
    application of its principles to metaphysics, with a view to the
    attainment of demonstration and certainty in moral, {75} political and
    ecclesiastical affairs. By Tresham Dames Gregg,[160] Chaplain of St.
    Mary's, within the church of St. Nicholas intra muros, Dublin. London,
    1859, 8vo. (pp. xl + 32).

I have a personal interest in this system, as will appear from the
following extract from the newspaper:

    "We were subsequently referred to De Morgan's _Formal Logic_ and
    Boole's _Laws of Thought_[161] both very elaborate works, and greatly
    in the direction taken by ourselves. That the writers amazingly surpass
    us in learning we most willingly admit, but we venture to pronounce of
    both their learned treatises, that they deal with the subject in a mode
    that is scholastic to an excess.... That their works have been for a
    considerable space of time before the world and effected nothing, would
    argue that they have overlooked the vital nature of the theme.... On
    the whole, the writings of De Morgan and Boole go to the full
    justification of our principle without in any wise so trenching upon
    our ground as to render us open to reproach in claiming our Calculus as
    a great discovery.... But we renounce any paltry jealousy as to a
    matter so vast. If De Morgan and Boole have had a priority in the case,
    to them we cheerfully shall resign the glory and honor. If such be the
    truth, they have neither done justice to the discovery, nor to
    themselves [quite true]. They have, under the circumstances, acted like
    'the foolish man, who roasteth not that which he taketh {76} in
    hunting.... It will be sufficient for us, however, to be the Columbus
    of these great Americi, and popularize what they found, _if_ they found
    it. We, as from the mountain top, will then become _their_ trumpeters,
    and cry glory to De Morgan and glory to Boole, under Him who is the
    source of all glory, the only good and wise, to Whom be glory for ever!
    _If_ they be our predecessors in this matter, they have, under Him,
    taken moral questions out of the category of probabilities, and
    rendered them perfectly certain. In that case, let their books be read
    by those who may doubt the principles this day laid before the world as
    a great discovery, by our newspaper. Our cry shall be [Greek:
    eurêkasi]![162] Let us hope that they will join us, and henceforth keep
    their light [_sic_] from under their bushel."

For myself, and for my old friend Mr. Boole, who I am sure would join me, I
disclaim both priority, simultaneity, and posteriority, and request that
nothing may be trumpeted from the mountain top except our abjuration of all
community of thought or operation with this _Novum Organum_.

To such community we can make no more claim than Americus could make to
being the forerunner of Columbus who popularized his discoveries. We do not
wish for any [Greek: eurêkasi] and not even for [Greek: heurêkasi]. For
self and Boole, I point out what would have convinced either of us that
this house is divided against itself.

[Alpha] being an apostolic element, [delta] the doctrinal element, and
[Chi] the body of the faithful, the church is [Alpha] [delta] [Chi], we are
told. Also, that if [Alpha] become negative, or the Apostolicity become
Diabolicity [my words]; or if [delta] become negative, and doctrine become
heresy; or if [Chi] become negative, that is, if the faithful become
unfaithful; the church becomes negative, "the very opposite to what it
ought to be." For self and Boole, I admit this. But--which is not
noticed--if [Alpha] and [delta] should _both_ become negative, diabolical
origin {77} and heretical doctrine, then the church, [Alpha] [delta] [Chi],
is still positive, what it ought to be, unless [Chi] be also negative, or
the people unfaithful to it, in which case it is a bad church. Now, self
and Boole--though I admit I have not asked my partner--are of opinion that
a diabolical church with false doctrine does harm when the people are
faithful, and can do good only when the people are unfaithful. We may be
wrong, but this is what we _do_ think. Accordingly, we have caught nothing,
and can therefore roast nothing of our own: I content myself with roasting
a joint of Mr. Gregg's larder.

These mathematical vagaries have uses which will justify a large amount of
quotation: and in a score of years this may perhaps be the only attainable
record. I therefore proceed.

After observing that by this calculus juries (heaven help them! say I) can
calculate damages "almost to a nicety," and further that it is made
abundantly evident that c e x is "the general expression for an
individual," it is noted that the number of the Beast is not given in the
Revelation in words at length, but as [Greek: chxw'].[163] On this the
following remark is made:

"Can it be possible that we have in this case a specimen given to us of the
arithmetic of heaven, and an expression revealed, which indicates by its
function of addibility, the name of the church in question, and of each
member of it; and by its function of multiplicability the doctrine, the
mission, and the members of the great Synagogue of Apostacy? We merely
propound these questions;--we do not pretend to solve them."

After a translation in blank verse--a very pretty one--of the 18th Psalm,
the author proceeds as follows, to render it into differential calculus:


"And the whole tells us just this, that David did what he could. He
augmented those elements of his constitution which were (_exceptis
excipiendis_)[164] subject to himself, and the Almighty then augmented his
personal qualities, and his vocational _status_. Otherwise, to throw the
matter into the expression of our notation, the variable e was augmented,
and c x rose proportionally. The law of the variation, according to our
theory, would be thus expressed. The resultant was David the king c e x [c
= r?] (who had been David the shepherd boy), and from the conditions of the
theorem we have

  du/de = ce(dx/de) + ex(dc/de)x + cx

which, in the terms of ordinary language, just means, the increase of
David's educational excellence or qualities--his piety, his prayerfulness,
his humility, obedience, etc.--was so great, that when multiplied by his
original talent and position, it produced a product so great as to be equal
in its amount to royalty, honor, wealth, and power, etc.: in short, to all
the attributes of majesty."[165]

The "solution of the family problem" is of high interest. It is to
determine the effect on the family in general from a change [of conduct] in
one of them. The person chosen is one of the maid-servants.

"Let c e x be the father; c_1e_1x_1 the mother, etc. The family then
consists of the maid's master, her mistress, her young master, her young
mistress, and fellow servant. Now the master's calling (or c) is to
exercise his share of control over this servant, and mind the rest of his
business: call this remainder a, and let his calling generally, or all his
affairs, be to his maid-servant as m : y, i.e., y = (mz/c); ... {79} and
this expression will represent his relation to the servant. Consequently,

  c e x = (a + mz/c)e x; otherwise (a + mz/c)e x

is the expression for the father when viewed as the girl's master."

I have no objection to repeat so far; but I will not give the formula for
the maid's relation to her young master; for I am not quite sure that all
young masters are to be trusted with it. Suffice it that the son will be
affected directly as his influence over her, and inversely as his
vocational power: if then he should have some influence and no vocational
power, the effect on him would be infinite. This is dismal to think of.
Further, the formula brings out that if one servant improve, the other must
deteriorate, and _vice versa_. This is not the experience of most families:
and the author remarks as follows:

"That is, we should venture to say, a very beautiful result, and we may say
it yielded us no little astonishment. What our calculation might lead to we
never dreamt of; that it should educe a conclusion so recondite that our
unassisted power never could have attained to, and which, if we could have
conjectured it, would have been at best the most distant probability, that
conclusion being itself, as it would appear, the quintessence of truth,
afforded us a measure of satisfaction that was not slight."

That the writings of Mr. Boole and myself "go to the full justification of"
this "principle," is only true in the sense in which the Scotch use, or did
use, the word _justification_.


[The last number of this Budget had stood in type for months, waiting until
there should be a little cessation of correspondence more connected with
the things of the day. {80} I had quite forgotten what it was to contain;
and little thought, when I read the proof, that my allusions to my friend
Mr. Boole, then in life and health, would not be printed till many weeks
after his death. Had I remembered what my last number contained, I should
have added my expression of regret and admiration to the numerous obituary
testimonials, which this great loss to science has called forth.

The system of logic alluded to in the last number of this series is but one
of many proofs of genius and patience combined. I might legitimately have
entered it among my _paradoxes_, or things counter to general opinion: but
it is a paradox which, like that of Copernicus, excited admiration from its
first appearance. That the symbolic processes of algebra, invented as tools
of numerical calculation, should be competent to express every act of
thought, and to furnish the grammar and dictionary of an all-containing
system of logic, would not have been believed until it was proved. When
Hobbes,[166] in the time of the Commonwealth, published his _Computation or
Logique_, he had a remote glimpse of some of the points which are placed in
the light of day by Mr. Boole. The unity of the forms of thought in all the
applications of reason, however remotely separated, will one day be matter
of notoriety and common wonder: and Boole's name will be remembered in
connection with one of the most important steps towards the attainment of
this knowledge.]


    The Decimal System as a whole. By Dover Statter.[167] London and
    Liverpool, 1856, 8vo.


The proposition is to make everything decimal. The day, now 24 hours, is to
be made 10 hours. The year is to have ten months, Unusber, Duober, etc.
Fortunately there are ten commandments, so there will be neither addition
to, nor deduction from, the moral law. But the twelve apostles! Even
rejecting Judas, there is a whole apostle of difficulty. These points the
author does not touch.


    The first book of Phonetic Reading. London, Fred. Pitman,[168] Phonetic
    Depot, 20, Paternoster Row, 1856, 12mo.

    The Phonetic Journal. Devoted to the propagation of phonetic reading,
    phonetic longhand, phonetic shorthand, and phonetic printing. No. 46.
    Saturday, 15 November 1856. Vol. 15.

I write the titles of a couple out of several tracts which I have by me.
But the number of publications issued by the promoters of this spirited
attempt is very large indeed.[169] The attempt itself has had no success
with the mass of the public. This I do not regret. Had the world found that
the change was useful, I should have gone contentedly with the stream; but
not without regretting our old language. I admit the difficulties which our
unpronounceable spelling puts in the way of learning to read: and I have no
doubt that, as affirmed, it is easier to teach children phonetically, and
afterwards to introduce them to our common system, than to proceed in the
usual way. But by the usual way I mean proceeding by letters from the very
beginning. If, which I am sure is a better plan, children be taught at the
commencement very much by _complete words_, as if they were learning
Chinese, and be gradually accustomed to {82} resolve the known words into
letters, a fraction, perhaps a considerable one, of the advantage of the
phonetic system is destroyed. It must be remembered that a phonetic system
can only be an approximation. The differences of pronunciation existing
among educated persons are so great, that, on the phonetic system,
different persons ought to spell differently.

But the phonetic party have produced something which will immortalize their
plan: I mean their _shorthand_, which has had a fraction of the success it
deserves. All who know anything of shorthand must see that nothing but a
phonetic system can be worthy of the name: and the system promulgated is
skilfully done. Were I a young man I should apply myself to it
systematically. I believe this is the only system in which books were ever
published. I wish some one would contribute to a public journal a brief
account of the dates and circumstances of the phonetic movement, not
forgetting a list of the books published in shorthand.

A child beginning to read by himself may owe terrible dreams and waking
images of horror to our spelling, as I did when six years old. In one of
the common poetry-books there is an admonition against confining little
birds in cages, and the child is asked what if a great giant, amazingly
strong, were to take you away, shut you up,

  And feed you with vic-tu-als you ne-ver could bear.

The book was hyphened for the beginner's use; and I had not the least idea
that _vic-tu-als_ were _vittles_: by the sound of the word I judged they
must be of iron; and it entered into my soul.

The worst of the phonetic shorthand book is that they nowhere, so far as I
have seen, give _all_ the symbols, in every stage of advancement, together,
in one or following pages. It is symbols and talk, more symbols and more
talk, etc. A universal view of the signs ought to begin the works. {83}


    Ombrological Almanac. Seventeenth year. An essay on Anemology and
    Ombrology. By Peter Legh,[170] Esq. London, 1856, 12mo.

Mr. Legh, already mentioned, was an intelligent country gentleman, and a
legitimate speculator. But the clue was not reserved for him.

    The proof that the three angles of a triangle are equal to two right
    angles looked for in the inflation of the circle. By Gen. Perronet
    Thompson. London, 1856, 8vo. (pp. 4.)

Another attempt, the third, at this old difficulty, which cannot be put
into few words of explanation.[171]

    Comets considered as volcanoes, and the cause of their velocity and
    other phenomena thereby explained. London (_circa_ 1856), 8vo.

The title explains the book better than the book explains the title.

1856. A stranger applied to me to know what the ideas of a friend of his
were worth upon the magnitude of the earth. The matter being one involving
points of antiquity, I mentioned various persons whose speculations he
seemed to have ignored; among others, Thales. The reply was, "I am
instructed by the author to inform you that he is perfectly acquainted with
the works of Thales, Euclid, Archimedes, ..." I had some thought of asking
whether he had used the Elzevir edition of Thales,[172] which is known to
be very incomplete, or that of Professor Niemand with the lections,
Nirgend, 1824, 2 vols. folio; just to see whether the {84} last would not
have been the very edition he had read. But I refrained, in mercy.

    The moon is the image of the Earth, and is not a solid body. By T^{he}
    Longitude.[173] (Private Circulation.) In five parts. London, 1856,
    1857, 1857; Calcutta, 1858, 1858, 8vo.

The earth is "brought to a focus"; it describes a "looped orbit round the
sun." The eclipse of the sun is thus explained: "At the time of eclipses,
the image is more or less so directly before or behind the earth that, in
the case of new moon, bright rays of the sun fall and bear upon the spot
where the figure of the earth is brought to a focus, that is, bear upon the
image of the earth, when a darkness beyond is produced reaching to the
earth, and the sun becomes more or less eclipsed." How the earth is
"brought to a focus" we do not find stated. Writers of this kind always
have the argument that some things which have been ridiculed at first have
been finally established. Those who put into the lottery had the same kind
of argument; but were always answered by being reminded how many blanks
there were to one prize. I am loath to pronounce against anything: but it
does force itself upon me that the author of these tracts has drawn a


    _Times_, April 6 or 7, 1856. The moon has no rotary motion.

A letter from Mr. Jellinger Symons,[174] inspector of schools, which
commenced a controversy of many letters and pamphlets. This dispute comes
on at intervals, and will continue to do so. It sometimes arises from
inability to understand the character of simple rotation, geometrically;
sometimes from not understanding the mechanical doctrine of rotation.


    Lunar Motion. The whole argument stated, and illustrated by diagrams;
    with letters from the Astronomer Royal. By Jellinger C. Symons. London,
    1856, 8vo.

The Astronomer Royal endeavored to disentangle Mr. J. C. Symons, but
failed. Mr. Airy[175] can correct the error of a ship's compasses, because
he can put her head which way he pleases: but this he cannot do with a

Mr. Symons, in this tract, insinuated that the rotation of the moon is one
of the silver shrines of the craftsmen. To see a thing so clearly as to be
satisfied that all who say they do not see it are telling wilful falsehood,
is the nature of man. Many of all sects find much comfort in it, when they
think of the others; many unbelievers solace themselves with it against
believers; priests of old time founded the right of persecution upon it,
and of our time, in some cases, the right of slander: many of the
paradoxers make it an argument against students of science. But I must say
for men of science, for the whole body, that they are fully persuaded of
the honesty of the paradoxers. The simple truth is, that all those I have
mentioned, believers, unbelievers, priests, paradoxers, are not so sure
they are right in their points of difference that they can safely allow
themselves to be persuaded of the honesty of opponents. Those who know
demonstration are differently situated. I suspect a train might be laid for
the formation of a better habit in this way. We know that Suvaroff[176]
taught his Russians at Ismail not to fear the Turks by accustoming them to
charge bundles of faggots dressed in turbans, etc.

  At which your wise men sneered in phrases witty,
  He made no answer--but he took the city!

Would it not be a good thing to exercise boys, in pairs, in the following
dialogue:--Sir, you are quite wrong!--Sir, {86} I am sure you honestly
think so! This was suggested by what used to take place at Cambridge in my
day. By statute, every B.A. was obliged to perform a certain number of
disputations, and the _father_ of the college had to affirm that it had
been done. Some were performed in earnest: the rest were huddled over as
follows. Two candidates occupied the places of the respondent and the
opponent: _Recte statuit Newtonus_, said the respondent: _Recte non statuit
Newtonus_,[177] said the opponent. This was repeated the requisite number
of times, and counted for as many _acts_ and _opponencies_. The parties
then changed places, and each unsaid what he had said on the other side of
the house: I remember thinking that it was capital drill for the House of
Commons, if any of us should ever get there. The process was repeated with
every pair of candidates.

The real disputations were very severe exercises. I was badgered for two
hours with arguments given and answered in Latin,--or what we called
Latin--against Newton's first section, Lagrange's[178] derived functions,
and Locke[179] on innate principles. And though I _took off_ everything,
and was pronounced by the moderator to have disputed _magno honore_,[180] I
never had such a strain of thought in my life. For the inferior opponents
were made as sharp as their betters by their tutors, who kept lists of
queer objections, drawn from all quarters. The opponents used to meet the
day before to compare their arguments, that the same might not come twice
over. But, after I left Cambridge, it became the fashion to invite the
respondent to be present, who therefore learnt all that was to be brought
against him. This made the whole thing a farce: and the disputations were


    The Doctrine of the Moon's Rotation, considered in a letter to the
    Astronomical Censor of the _Athenæum_. By Jones L. MacElshender.[181]
    Edinburgh, 1856, 8vo.

This is an appeal to those cultivated persons who will read it "to overrule
the _dicta_ of judges who would sacrifice truth and justice to professional
rule, or personal pique, pride, or prejudice"; meaning, the great mass of
those who have studied the subject. But how? Suppose the "cultivated
persons" were to side with the author, would those who have conclusions to
draw and applications to make consent to be wrong because the "general body
of intelligent men," who make no special study of the subject, are against
them? They would do no such thing: they would request the general body of
intelligent men to find their own astronomy, and welcome. But the truth is,
that this intelligent body knows better: and no persons know better that
they know better than the speculators themselves.

But suppose the general body were to combine, in opposition to those who
have studied. Of course all my list must be admitted to their trial; and
then arises the question whether both sides are to be heard. If so, the
general body of the intelligent must hear all the established side have to
say: that is, they must become just as much of students as the inculpated
orthodox themselves. And will they not then get into _professional rule_,
pique, pride, and prejudice, as the others did? But if, which I suspect,
they are intended to judge as they are, they will be in a rare difficulty.
All the paradoxers are of like pretensions: they cannot, as a class, be
right, for each one contradicts a great many of the rest. There will be the
puzzle which silenced the crew of the cutter in Marryat's novel of the Dog
Fiend.[182] "A tog is a tog," said Jansen.--"Yes," replied another, "we all
know a dog is a dog; but the question is--Is _this_ dog {88} a dog?" And
this question would arise upon every dog of them all.


    Zetetic Astronomy: Earth not a globe. 1857 (Broadsheet).

Though only a traveling lecturer's advertisement, there are so many
arguments and quotations that it is a little pamphlet. The lecturer gained
great praise from provincial newspapers for his ingenuity in proving that
the earth is a flat, surrounded by ice. Some of the journals rather incline
to the view: but the _Leicester Advertiser_ thinks that the statements
"would seem very seriously to invalidate some of the most important
conclusions of modern astronomy," while the _Norfolk Herald_ is clear that
"there must be a great error on one side or the other." This broadsheet is
printed at Aylesbury in 1857, and the lecturer calls himself _Parallax_:
but at Trowbridge, in 1849, he was S. Goulden.[183] In this last
advertisement is the following announcement: "A paper on the above subjects
was read before the Council and Members of the Royal Astronomical Society,
Somerset House, Strand, London (Sir John F. W. Herschel,[184] President),
Friday, Dec. 8, 1848." No account of such a paper appears in the _Notice_
for that month: I suspect that the above is Mr. S. Goulden's way of
representing the following occurrence: Dec. 8, 1848, the Secretary of the
Astronomical Society (De Morgan by name) said, at the close of the
proceedings,--"Now, gentlemen, if you will promise not to tell the Council,
I will read something for your amusement": and he then read a few of the
arguments which had been transmitted by the lecturer. The fact is worth
noting that from 1849 to 1857, arguments on the roundness or flatness of
the earth did itinerate. I have {89} no doubt they did much good: for very
few persons have any distinct idea of the evidence for the rotundity of the
earth. The _Blackburn Standard_ and _Preston Guardian_ (Dec. 12 and 16,
1849) unite in stating that the lecturer ran away from his second lecture
at Burnley, having been rather too hard pressed at the end of his first
lecture to explain why the large hull of a ship disappeared before the
sails. The persons present and waiting for the second lecture assuaged
their disappointment by concluding that the lecturer had slipped off the
icy edge of his flat disk, and that he would not be seen again till he
peeped up on the opposite side.

But, strange as it may appear, the opposer of the earth's roundness has
more of a case--or less of a want of case--than the arithmetical squarer of
the circle. The evidence that the earth is round is but cumulative and
circumstantial: scores of phenomena ask, separately and independently, what
other explanation can be imagined except the sphericity of the earth. The
evidence for the earth's figure is tremendously powerful of its kind; but
the proof that the circumference is 3.14159265... times the diameter is of
a higher kind, being absolute mathematical demonstration.

The Zetetic system still lives in lectures and books; as it ought to do,
for there is no way of teaching a truth comparable to opposition. The last
I heard of it was in lectures at Plymouth, in October, 1864. Since this
time a prospectus has been issued of a work entitled "The Earth not a
Globe"; but whether it has been published I do not know. The contents are
as follows:

"The Earth a Plane--How circumnavigated.--How time is lost or gained.--Why
a ship's hull disappears (when outward bound) before the mast head.--Why
the Polar Star sets when we proceed Southward, etc.--Why a pendulum
vibrates with less velocity at the Equator than {90} at the Pole.--The
allowance for rotundity _supposed_ to be made by surveyors, not made in
practice.--Measurement of Arcs of the Meridian unsatisfactory.--Degrees of
Longitude North and South of the Equator considered.--Eclipses and Earth's
form considered.--The Earth no motion on axis or in orbit.--How the Sun
moves above the Earth's surface concentric with the North Pole.--Cause of
Day and Night, Winter and Summer; the long alternation of light and
darkness at the Pole.--Cause of the Sun rising and setting.--Distance of
the Sun from London, 4,028 miles--How measured.--_Challenge to
Mathematicians._--Cause of Tides.--Moon self-luminous, NOT a
reflector.--Cause of Solar and Lunar eclipses.--Stars _not worlds_; their
distance.--Earth, the _only material_ world; its true position in the
universe; its condition and ultimate destruction by fire (2 Peter iii.),

I wish there were geoplatylogical lectures in every town; in England
(_platylogical_, in composition, need not mean _babbling_). The late Mr.
Henry Archer[185] would, if alive, be very much obliged to me for recording
his vehement denial of the roundness of the earth: he was excited if he
heard any one call it a globe. I cannot produce his proof from the
Pyramids, and from some caves in Arabia. He had other curious notions, of
course: I should no more believe that a flat earth was a man's only
paradox, than I should that Dutens,[186] the editor of Leibnitz, was
eccentric only in supplying a tooth which he had lost by one which he found
in an Italian tomb, and fully believed that it had once belonged to Scipio
Africanus, whose family vault was discovered, it is supposed, in 1780. Mr.
Archer is of note as {91} the suggester of the perforated border of the
postage-stamps, and, I think, of the way of doing it; for this he got
4000l. reward. He was a civil engineer.

(_August 28, 1865._) The _Zetetic Astronomy_ has come into my hands. When,
in 1851, I went to see the Great Exhibition, I heard an organ played by a
performer who seemed very desirous to exhibit one particular stop. "What do
you think of that stop?" I was asked.--"That depends on the name of it,"
said I.--"Oh! what can the name have to do with the sound? 'that which we
call a rose,' etc."--"The name has everything to do with it: if it be a
flute-stop, I think it very harsh; but if it be a railway-whistle-stop, I
think it very sweet." So as to this book: if it be childish, it is clever;
if it be mannish, it is unusually foolish. The flat earth, floating
tremulously on the sea; the sun moving always over the flat, giving day
when near enough, and night when too far off; the self-luminous moon, with
a semi-transparent invisible moon, created to give her an eclipse now and
then; the new law of perspective, by which the vanishing of the hull before
the masts, usually thought to prove the earth globular, really proves it
flat;--all these and other things are well fitted to form exercises for a
person who is learning the elements of astronomy. The manner in which the
sun dips into the sea, especially in tropical climates, upsets the whole.
Mungo Park,[187] I think, gives an African hypothesis which explains
phenomena better than this. The sun dips into the western ocean, and the
people there cut him in pieces, fry him in a pan, and then join him
together again, take him round the underway, and set him up in the east. I
hope this book will be read, and that many will be puzzled by it: for there
are many whose notions of astronomy deserve no better fate. There is no
subject on which there is so little {92} accurate conception as that of the
motions of the heavenly bodies. The author, though confident in the
extreme, neither impeaches the honesty of those whose opinions he assails,
nor allots them any future inconvenience: in these points he is worthy to
live on a globe, and to revolve in twenty-four hours.

(_October, 1866._) A follower appears, in a work dedicated to the preceding
author: it is _Theoretical Astronomy examined and exposed by Common Sense_.
The author has 128 well-stuffed octavo pages. I hope he will not be the
last. He prints the newspaper accounts of his work: the _Church Times_
says--not seeing how the satire might be retorted--"We never began to
despair of Scripture until we discovered that 'Common Sense' had taken up
the cudgels in its defence." This paper considers our author as the type of
a _Protestant_. The author himself, who gives a summary of his arguments in
verse, has one couplet which is worth quoting:

 "How is't that sailors, bound to sea, with _a 'globe'_ would never start,
  But in its place will always take _Mercator's_[188] LEVEL _chart_!"

To which I answer:

  Why, really Mr. Common Sense, you've never got so far
  As to think Mercator's planisphere shows countries as they are;
  It won't do to measure distances; it points out how to steer,
  But this distortion's not for you; another is, I fear.
  The earth must be a cylinder, if seaman's charts be true,
  Or else the boundaries, right and left, are one as well as two;
  They contradict the notion that we dwell upon a plain,
  For straight away, without a turn, will bring you home again.
  There are various plane projections; and each one has its use:
  I wish a milder word would rhyme--but really you're a goose!

The great wish of persons who expose themselves as above, is to be argued
with, and to be treated as reputable {93} and refutable opponents. "Common
Sense" reminds us that no amount of "blatant ridicule" will turn right into
wrong. He is perfectly correct: but then no amount of bad argument will
turn wrong into right. These two things balance; and we are just where we
were: but you should answer our arguments, for whom, I ask? Would reason
convince this kind of reasoner? The issue is a short and a clear one. If
these parties be what I contend they are, then ridicule is made for them:
if not, for what or for whom? If they be right, they are only passing
through the appointed trial of all good things. Appeal is made to the
future: and my Budget is intended to show samples of the long line of
heroes who have fallen without victory, each of whom had his day of
confidence and his prophecy of success. Let the future decide: they say
roundly that the earth is flat; I say flatly that it is round.

The paradoxers all want reason, and not ridicule: they are all accessible,
and would yield to conviction. Well then, let them reason with one another!
They divide into squads, each with a subject, and as many different
opinions as persons in each squad. If they be really what they say they
are, the true man of each set can put down all the rest, and can come
crowned with glory and girdled with scalps, to the attack on the orthodox
misbelievers. But they know, to a man, that the rest are not fit to be
reasoned with: they pay the regulars the compliment of believing that the
only chance lies with them. They think in their hearts, each one for
himself, that ridicule is of fit appliance to the rest.

    Miranda. A book divided into three parts, entitled Souls, Numbers,
    Stars, on the Neo-Christian Religion ... Vol. i. London, 1858, 1859,
    1860. 8vo.

The name of the author is Filopanti.[189] He announces himself as the 49th
and last Emanuel: his immediate {94} predecessors were Emanuel Washington,
Emanuel Newton, and Emanuel Galileo. He is to collect nations into one
family. He knows the transmigrations of the whole human race. Thus
Descartes became William III of England: Roger Bacon became Boccaccio. But
Charles IX,[190] in retribution for the massacre of St. Bartholomew, was
hanged in London under the name of Barthélemy for the murder of Collard:
and many of the Protestants whom he killed as King of France were shouting
at his death before the Old Bailey.


    A Letter to the members of the Anglo-Biblical Institute, dated Sept. 7,
    1858, and signed 'Herman Heinfetter.'[191] (Broadsheet.)

This gentleman is well known to the readers of the _Athenæum_, in which,
for nearly twenty years, he has inserted, as advertisements, long arguments
in favor of Christians keeping the Jewish Sabbath, beginning on Friday
Evening. The present letter maintains that, by the force of the definite
article, the _days_ of creation may not be consecutive, but may have any
time--millions of years--between them. This ingenious way of reconciling
the author of Genesis and the indications of geology is worthy to be added
to the list, already pretty numerous. Mr. Heinfetter has taken such pains
to make himself a public agitator, that {95} I do not feel it to be any
invasion of private life if I state that I have heard he is a large
corn-dealer. No doubt he is a member of the congregation whose almanac has
already been described.

    The great Pyramid. Why was it built? And who built it? By John Taylor,
    1859,[192] 12mo.

This work is very learned, and may be referred to for the history of
previous speculations. It professes to connect the dimensions of the
Pyramid with a system of metrology which is supposed to have left strong
traces in the systems of modern times; showing the Egyptians to have had
good approximate knowledge of the dimensions of the earth, and of the
quadrature of the circle. These are points on which coincidence is hard to
distinguish from intention. Sir John Herschel[193] noticed this work, and
gave several coincidences, in the _Athenæum_, Nos. 1696 and 1697, April 28
and May 5, 1860: and there are some remarks by Mr. Taylor in No. 1701, June
2, 1860.

Mr. Taylor's most recent publication is--

    The battle of the Standards: the ancient, of four thousand years,
    against the modern, of the last fifty years--the less perfect of the
    two. London, 1864, 12mo.

This is intended as an appendix to the work on the Pyramid. Mr. Taylor
distinctly attributes the original system to revelation, of which he says
the Great Pyramid is the record. We are advancing, he remarks, towards the
end of the Christian dispensation, and he adds that it is satisfactory to
see that we retain the standards which were given by unwritten revelation
700 years before Moses. This is lighting the candle at both ends; for
myself, I shall not undertake to deny or affirm either what is said about
the dark past or what is hinted about the dark future.


My old friend Mr. Taylor is well known as the author of the argument which
has convinced many, even most, that Sir Philip Francis[194] was Junius:
pamphlet, 1813; supplement, 1817; second edition "The Identity of Junius
with a distinguished living character established," London, 1818, 8vo. He
told me that Sir Philip Francis, in a short conversation with him, made
only this remark, "You may depend upon it you are quite mistaken:" the
phrase appears to me remarkable; it has an air of criticism on the book,
free from all personal denial. He also mentioned that a hearer told him
that Sir Philip said, speaking of writers on the question,--"Those fellows,
for half-a-crown, would prove that Jesus Christ was Junius."

Mr. Taylor implies, I think, that he is the first who started the
suggestion that Sir Philip Francis was Junius, which I have no means either
of confirming or refuting. If it be so [and I now know that Mr. Taylor
himself never heard of any predecessor], the circumstance is very
remarkable: it is seldom indeed that the first proposer of any solution of
a great and vexed question is the person who so nearly establishes his
point in general opinion as Mr. Taylor has done.

As to the Junius question in general, there is a little bit of the
philosophy of horse-racing which may be usefully applied. A man who is so
confident of his horse that he places him far above any other, may
nevertheless, and does, refuse to give odds against all in the field: for
many small adverse chances united make a big chance for one or other of the
opponents. I suspect Mr. Taylor has made it at least 20 to 1 for Francis
against any one competitor who has been named: but what the odds may be
against the {97} whole field is more difficult to settle. What if the real
Junius should be some person not yet named?

Mr. Jopling, _Leisure Hour_, May 23, 1863, relies on the porphyry coffer of
the Great Pyramid, in which he finds "the most ancient and accurate
standard of measure in existence."

I am shocked at being obliged to place a thoughtful and learned writer, and
an old friend, before such a successor as he here meets with. But
chronological arrangement defies all other arrangement.

(I had hoped that the preceding account would have met Mr. Taylor's eye in
print: but he died during the last summer. For a man of a very thoughtful
and quiet temperament, he had a curious turn for vexed questions. But he
reflected very long and very patiently before he published: and all his
works are valuable for their accurate learning, whichever side the reader
may take.)


1859. _The Cottle Church._--For more than twenty years printed papers have
been sent about in the name of Elizabeth Cottle.[195] It is not so
remarkable that such papers should be concocted as that they should
circulate for such a length of time without attracting public attention.
Eighty years ago Mrs. Cottle might have rivalled Lieut. Brothers or Joanna
Southcott.[196] Long hence, when the now current volumes of our journals
are well-ransacked works of reference, those who look into them will be
glad to see this {98} feature of our time: I therefore make a few extracts,
faithfully copied as to type. The Italic is from the New Testament; the
Roman is the requisite interpretation:

"Robert Cottle '_was numbered_ (5196) _with the transgressors_' at the back
of the Church in Norwood Cemetery, May 12, 1858--Isa. liii. 12. The Rev.
J. G. Collinson, Minister of St. James's Church, Chapham, the then district
church, before All Saints was built, read the funeral service _over the
Sepulchre wherein never before man was laid_.

"_Hewn on the stone_, 'at the mouth of the Sepulchre,' is his name,--Robert
Cottle, born at Bristol, June 2, 1774; died at Kirkstall Lodge, Clapham
Park, May 6, 1858. _And that day_ (May 12, 1858) _was the preparation_ (day
and year for 'the PREPARED place for you'--Cottleites---by the widowed
mother of the Father's house, at Kirkstall Lodge--John xiv. 2, 3). _And the
Sabbath_ (Christmas Day, Dec. 25, 1859) _drew on_ (for the resurrection of
the Christian body on 'the third [Protestant Sun]-day'--1 Cor. xv. 35).
_Why seek ye the living_ (God of the New Jerusalem--Heb. xii. 22; Rev. iii.
12) _among the dead_ (men): _he_ (the God of Jesus) _is not here_ (in the
grave), _but is risen_ (in the person of the Holy Ghost, from the supper of
'the dead in the second death' of Paganism). _Remember how he spake unto
you_ (in the church of the Rev. George Clayton,[197] April 14, 1839). _I
will not drink henceforth_ (at this last Cottle supper) _of the fruit of
this_ (Trinity) _vine, until that day_ (Christmas Day, 1859), _when I_
(Elizabeth Cottle) _drink it new with you_ (Cottleites) _in my Father's
kingdom_--John xv. _If this_ (Trinitarian) _cup may not pass away from me_
(Elizabeth Cottle, April 14, 1839), _except I drink it_ ('new with you
Cottleites, in my Father's Kingdom'), _thy will be done_--Matt. xxvi. 29,
42, 64. 'Our Father which art (God) in Heaven,' _hallowed be thy name, thy_
(Cottle) _kingdom_ {99} _come, thy will be done in earth, as it is_ (done)
_in_ (the new) _Heaven_ (and new earth of the new name of Cottle--Rev. xxi.
1; iii. 12).

"... Queen Elizabeth, from A.D. 1558 to 1566. _And this_ WORD _yet once
more_ (by a second Elizabeth--the WORD of his oath) _signifieth_ (at John
Scott's baptism of the Holy Ghost) _the removing of those things_ (those
Gods and those doctrines) _that are made_ (according to the Creeds and
Commandments of men) _that those things_ (in the moral law of God) _which
cannot be shaken_ (as a rule of faith and practice) _may remain, wherefore
we receiving_ (from Elizabeth) _a kingdom_ (of God,) _which cannot be
moved_ (by Satan) _let us have grace_ (in his Grace of Canterbury) _whereby
we may serve God acceptably_ (with the acceptable sacrifice of Elizabeth's
body and blood of the communion of the Holy Ghost) _with reverence_ (for
truth) _and godly fear_ (of the unpardonable sin of blasphemy against the
Holy Ghost) _for our God_ (the Holy Ghost) _is a consuming fire_ (to the
nation that will not serve him in the Cottle Church). We cannot defend
ourselves against the Almighty, and if He is our defence, no nation can
invade us.

"In verse 4 the Church of St. Peter is _in prison between four quaternions
of soldiers_--the Holy Alliance of 1815. Rev. vii. i. Elizabeth, _the Angel
of the Lord_ Jesus _appears_ to the Jewish and Christian body with _the
vision_ of prophecy to the Rev. Geo. Clayton and his clerical brethren,
April 8th, 1839. _Rhoda_ was the name of her maid at Putney Terrace who
used _to open the door to her Peter_, the Rev. Robert Ashton,[198] the
Pastor of 'the little flock' 'of 120 names together, assembled in an upper
(school) room' at Putney Chapel, to which little flock she gave the
revelation (Acts. i. 13, 15) _of Jesus the same_ King of the Jews
_yesterday_ at the prayer meeting, Dec. 31, 1841, _and to-day_, {100} Jan.
1, 1842, _and for ever_. See book of Life, page 24. Matt. xviii. 19, xxi.
13-16. In verse 6 the Italian body of St. Peter _is sleeping_ 'in the
second death' _between the two_ Imperial _soldiers_ of France and Austria.
The Emperor of France from Jan. 1, to July 11, 1859, causes the Italian
_chains of St. Peter to fall off from his_ Imperial _hands_.

"_I say unto thee_, Robert Ashton, _thou art Peter_, a stone, _and upon
this rock_, of truth, _will I_ Elizabeth, the angel of Jesus, _build my_
Cottle _Church, and the gates of hell_, the doors of St. Peter, at Rome,
shall not prevail against it--Matt. xvi. 18. Rev. iii. 7-12."

This will be enough for the purpose. When any one who pleases can circulate
new revelations of this kind, uninterrupted and unattended to, new
revelations will cease to be a good investment of excentricity. I take it
for granted that the gentlemen whose names are mentioned have nothing to do
with the circulars or their doctrines. Any lady who may happen to be
intrusted with a revelation may nominate her own pastor, or any other
clergyman, one of her apostles; and it is difficult to say to what court
the nominees can appeal to get the commission abrogated.

_March 16, 1865._ During the last two years the circulars have continued.
It is hinted that funds are low: and two gentlemen who are represented as
gone "to Bethlehem asylum in despair" say that Mrs. Cottle "will spend all
that she hath, while Her Majesty's Ministers are flourishing on the wages
of sin." The following is perhaps one of the most remarkable passages in
the whole:

"_Extol and magnify Him_ (Jehovah, the Everlasting God, see the Magnificat
and Luke i. 45, 46--68--73--79), _that rideth_ (by rail and steam over land
and sea, from his holy habitation at Kirkstall Lodge, Psa. lxxvii. 19, 20),
_upon the_ (Cottle) _heavens, as it were_ (Sept. 9, 1864, see pages 21,
170), _upon an_ (exercising, Psa. cxxxi. 1), _horse_-(chair, bought of Mr.
John Ward, Leicester-square)." {101}

I have pretty good evidence that there is a clergyman who thinks Mrs.
Cottle a very sensible woman.

[_The Cottle Church._ Had I chanced to light upon it at the time of
writing, I should certainly have given the following. A printed letter to
the _Western Times_, by Mr. Robert Cottle, was accompanied by a manuscript
letter from Mrs. Cottle, apparently a circular. The date was Nov^{r}. 1853,
and the subject was the procedure against Mr. Maurice[199] at King's
College for doubting that God would punish human sins by an existence of
torture lasting through years numbered by millions of millions of millions
of millions (repeat the word _millions_ without end,) etc. The memory of
Mr. Cottle has, I think, a right to the quotation: he seems to have been no
participator in the notions of his wife:

"The clergy of the Established Church, taken at the round number of 20,000,
may, in their first estate, be likened to 20,000 gold blanks, destined to
become sovereigns, in succession,--they are placed between the matrix of
the Mint, when, by the pressure of the screw, they receive the impress that
fits them to become part of the current coin of the realm. In a way
somewhat analogous this great body of the clergy have each passed through
the crucibles of Oxford and Cambridge,--have been assayed by the Bishop's
chaplain, touching the health of their souls, and the validity of their
call by the Divine Spirit, and then the gentle pressure of a prelate's hand
upon their heads; and the words--'Receive the Holy Ghost,' have, in a brief
space of time, wrought a {102} change in them, much akin to the miracle of
transubstantiation--the priests are completed, and they become the current
ecclesiastical coin of our country. The whole body of clergy, here spoken
of, have undergone the preliminary induction of baptism and confirmation;
and all have been duly ordained, _professing_ to hold one faith, and to
believe in the selfsame doctrines! In short, to be as identical as the
20,000 sovereigns, if compared one with the other. But mind is not
malleable and ductile, like gold; and all the preparations of tests,
creeds, and catechisms will not insure uniformity of belief. No stamp of
orthodoxy will produce the same impress on the minds of different men.
Variety is manifest, and patent, upon everything mental and material. The
Almighty has not created, nor man fashioned, two things alike! How futile,
then, is the attempt to shape and mould man's apprehension of divine truth
by one fallible standard of man's invention! If proof of this be required,
an appeal might be made to history and the experience of eighteen hundred

This is an argument of force against the reasonableness of expecting tens
of thousands of educated readers of the New Testament to find the doctrine
above described in it. The lady's argument against the doctrine itself is
very striking. Speaking of an outcry on this matter among the Dissenters
against one of their body, who was the son of "the White Stone (Rev. ii.
17), or the Roman cement-maker," she says--

"If the doctrine for which they so wickedly fight were true, what would
become of the black gentlemen for whose redemption I have been sacrificed
from April 8 1839."

There are certainly very curious points about this revelation. There have
been many surmises about the final restoration of the infernal spirits,
from the earliest ages of Christianity until our own day: a collection of
them would be worth making. On reading this in proof, I see a possibility
that by "black gentlemen" may be meant the clergy: {103} I suppose my first
interpretation must have been suggested by context: I leave the point to
the reader's sagacity.]


    The Problem of squaring the circle solved; or, the circumference and
    area of the circle discovered. By James Smith.[200] London, 1859, 8vo.

    On the relations of a square inscribed in a circle. Read at the British
    Association, Sept. 1859, published in the Liverpool Courier, Oct. 8,
    1859, and reprinted in broadsheet.

    The question: Are there any commensurable relations between a circle
    and other Geometrical figures? Answered by a member of the British
    Association ... London, 1860, 8vo.--[This has been translated into
    French by M. Armand Grange, Bordeaux, 1863, 8vo.]

    The Quadrature of the Circle. Correspondence between an eminent
    mathematician and James Smith, Esq. (Member of the Mersey Docks and
    Harbour Board), London, 1861, 8vo. (pp. 200).

    Letter to the ... British Association ... by James Smith, Esq.
    Liverpool, 1861, 8vo.

    Letter to the ... British Association ... by James Smith, Esq.
    Liverpool, 1862, 8vo.--[These letters the author promised to continue.]

    A Nut to crack for the readers of Professor De Morgan's 'Budget of
    Paradoxes.' By James Smith, Esq. Liverpool, 1863, 8vo.

    Paper read at the Liverpool Literary and Philosophical Society,
    reported in the Liverpool Daily Courier, Jan. 26, 1864. Reprinted as a

    The Quadrature of the circle, or the true ratio between the diameter
    and circumference geometrically and mathematically demonstrated. By
    James Smith, Esq. Liverpool, 1865, 8vo.


    [On the relations between the dimensions and distances of the Sun,
    Moon, and Earth; a paper read before the Literary and Philosophical
    Society of Liverpool, Jan. 25, 1864. By James Smith, Esq.

    The British Association in Jeopardy, and Dr. Whewell, the Master of
    Trinity, in the stocks without hope of escape. Printed for the authors
    (J. S. confessed, and also hidden under _Nauticus_). (No date, 1865).

    The British Association in Jeopardy, and Professor De Morgan in the
    Pillory without hope of escape. London, 1866, 8vo.]

When my work appeared in numbers, I had not anything like an adequate idea
of Mr. James Smith's superiority to the rest of the world in the points in
which he is superior. He is beyond a doubt the ablest head at unreasoning,
and the greatest hand at writing it, of all who have tried in our day to
attach their names to an error. Common cyclometers sink into puny orthodoxy
by his side.

The behavior of this singular character induces me to pay him the
compliment which Achilles paid Hector, to drag him round the walls again
and again. He was treated with unusual notice and in the most gentle
manner. The unnamed mathematician, E. M. bestowed a volume of mild
correspondence upon him; Rowan Hamilton[201] quietly proved him wrong in a
way accessible to an ordinary schoolboy; Whewell,[202] as we shall see,
gave him the means of seeing himself wrong, even more easily than by
Hamilton's method. Nothing would do; it was small kick and silly fling at
all; and he exposed his conceit by alleging that he, James Smith, had
placed Whewell in the stocks. He will therefore be universally pronounced a
proper object of the severest literary punishment: but the opinion of all
who can put two propositions together will be that of the many strokes I
have given, the hardest and most telling are my republications of his own
attempts to reason.

He will come out of my hands in the position he ought {105} to hold, the
Supreme Pontiff of cyclometers, the vicegerent of St. Vitus upon earth, the
Mamamouchi of burlesque on inference. I begin with a review of him which
appeared in the _Athenæum_ of May 11, 1861. Mr. Smith says I wrote it: this
I neither affirm nor deny; to do either would be a sin against the
editorial system elsewhere described. Many persons tell me they know me by
my style; let them form a guess: I can only say that many have declared as
above while fastening on me something which I had never seen nor heard of.

    The Quadrature of the Circle: Correspondence between an Eminent
    Mathematician and James Smith, Esq. (Edinburgh, Oliver & Boyd; London,
    Simpkin, Marshall & Co.)

"A few weeks ago we were in perpetual motion. We did not then suppose that
anything would tempt us on a circle-squaring expedition: but the
circumstances of the book above named have a peculiarity which induces us
to give it a few words.

"Mr. James Smith, a gentleman residing near Liverpool, was some years ago
seized with the _morbus cyclometricus_.[203] The symptoms soon took a
defined form: his circumference shrank into exactly 3-1/8 times his
diameter, instead of close to 3-16/113, which the mathematician knows to be
so near to truth that the error is hardly at the rate of a foot in 2,000
miles. This shrinking of the circumference remained until it became
absolutely necessary that it should be examined by the British Association.
This body, which as Mr. James Smith found to his sorrow, has some interest
in 'jealously guarding the mysteries of their profession,' refused at first
to entertain the question. On this Mr. Smith changed his 'tactics' and the
name of his paper, and smuggled in the subject under the form of 'The
Relations of a Circle inscribed in a Square'! The paper was thus forced
upon the Association, for Mr. Smith informs us that he {106} 'gave the
Section to understand that he was not the man that would permit even the
British Association to trifle with him.' In other words, the Association
bore with and were bored with the paper, as the shortest way out of the
matter. Mr. Smith also circulated a pamphlet. Some kind-hearted man, who
did not know the disorder as well as we do, and who appears in Mr. Smith's
handsome octavo as E. M.--the initials of 'eminent mathematician'--wrote to
him and offered to show him in a page that he was all wrong. Mr. Smith
thereupon opened a correspondence, which is the bulk of the volume. When
the correspondence was far advanced, Mr. Smith announced his intention to
publish. His benevolent instructor--we mean in intention--protested against
the publication, saying 'I do not wish to be gibbeted to the world as
having been foolish enough to enter upon what I feel now to have been a
ridiculous enterprise.'

"For this Mr. Smith cared nothing: he persisted in the publication, and the
book is before us. Mr. Smith has had so much grace as to conceal his kind
adviser's name under E. M., that is to say, he has divided the wrong among
all who may be suspected of having attempted so hopeless a task as that of
putting a little sense into his head. He has violated the decencies of
private life. Against the will of the kind-hearted man who undertook his
case, he has published letters which were intended for no other purpose
than to clear his poor head of a hopeless delusion. He deserves the
severest castigation; and he will get it: his abuse of confidence will
stick by him all his days. Not that he has done his benefactor--in
intention, again--any harm. The patience with which E. M. put the blunders
into intelligible form, and the perseverance with which he tried to find a
cranny-hole for common reasoning to get in at, are more than respectable:
they are admirable. It is, we can assure E. M., a good thing that the
nature of the circle-squarer should be so completely exposed as in this
volume. The benefit which he intended Mr. James Smith may be {107}
conferred upon others. And we should very much like to know his name, and
if agreeable to him, to publish it. As to Mr. James Smith, we can only say
this: he is not mad. Madmen reason rightly upon wrong premises: Mr. Smith
reasons wrongly upon no premises at all.

"E. M. very soon found out that, to all appearance, Mr. Smith got a circle
of 3-1/8 times the diameter by making it the supposition to set out with
that there was such a circle; and then finding certain consequences which,
so it happened, were not inconsistent with the supposition on which they
were made. Error is sometimes self-consistent. However, E. M., to be quite
sure of his ground, wrote a short letter, stating what he took to be Mr.
Smith's hypothesis, containing the following: 'On AC as diameter, describe
the circle D, which by hypothesis shall be equal to three and one-eighth
times the length of AC.... I beg, before proceeding further, to ask whether
I have rightly stated your argument.' To which Mr. Smith replied: 'You have
stated my argument with perfect accuracy.' Still E. M. went on, and we
could not help, after the above, taking these letters as the initials of
Everlasting Mercy. At last, however, when Mr. Smith flatly denied that the
area of the circle lies between those of the inscribed and circumscribed
polygons, E. M. was fairly beaten, and gave up the task. Mr. Smith was left
to write his preface, to talk about the certain victory of truth--which,
oddly enough, is the consolation of all hopelessly mistaken men; to compare
himself with Galileo; and to expose to the world the perverse behavior of
the Astronomer Royal, on whom he wanted to fasten a conversation, and who
replied, 'It would be a waste of time, Sir, to listen to anything you could
have to say on such a subject.'

"Having thus disposed of Mr. James Smith, we proceed to a few remarks on
the subject: it is one which a journal would never originate, but which is
rendered necessary from time to time by the attempts of the autopseustic to
become {108} heteropseustic. To the mathematician we have nothing to say:
the question is, what kind of assurance can be given to the world at large
that the wicked mathematicians are not acting in concert to keep down their
superior, Mr. James Smith, the current Galileo of the quadrature of the

"Let us first observe that this question does not stand alone:
independently of the millions of similar problems which exist in higher
mathematics, the finding of the diagonal of a square has just the same
difficulty, namely, the entrance of a pair of lines of which one cannot be
definitely expressed by means of the other. We will show the reader who is
up to the multiplication-table how he may go on, on, on, ever nearer, never
there, in finding the diagonal of a square from the side.

"Write down the following rows of figures, and more, if you like, in the
way described:

  1   2   5   12   29   70   169   408     985
  1   3   7   17   41   99   239   577   1393

After the second, each number is made up of double the last increased by
the last but one: thus, 5 is 1 more than twice 2, 12 is 2 more than twice
5, 239 is 41 more than twice 99. Now, take out two adjacent numbers from
the upper line, and the one below the first from the lower: as

  70   169

Multiply together 99 and 169, giving 16,731. If, then, you will say that 70
diagonals are exactly equal to 99 sides, you are in error about the
diagonal, but an error the amount of which is not so great as the 16,731st
part of the diagonal. Similarly, to say that five diagonals make exactly
seven sides does not involve an error of the 84th part of the diagonal.

"Now, why has not the question of _crossing the square_ been as celebrated
as that of _squaring the circle_? Merely because Euclid demonstrated the
impossibility of the first {109} question, while that of the second was not
demonstrated, completely, until the last century.

"The mathematicians have many methods, totally different from each other,
of arriving at one and the same result, their celebrated approximation to
the circumference of the circle. An intrepid calculator has, in our own
time, carried his approximation to what they call 607 decimal places: this
has been done by Mr. Shanks,[204] of Houghton-le-Spring, and Dr.
Rutherford[205] has verified 441 of these places. But though 607 looks
large, the general public will form but a hazy notion of the extent of
accuracy acquired. We have seen, in Charles Knight's[206] _English
Cyclopædia_, an account of the matter which may illustrate the
unimaginable, though rationally conceivable, extent of accuracy obtained.

"Say that the blood-globule of one of our animalcules is a millionth of an
inch in diameter. Fashion in thought a globe like our own, but so much
larger that our globe is but a blood-globule in one of its animalcules:
never mind the microscope which shows the creature being rather a bulky
instrument. Call this the first globe _above_ us. Let the first globe above
us be but a blood-globule, as to size, in the animalcule of a still larger
globe, which call the second globe above us. Go on in this way to the
twentieth globe above us. Now go down just as far on the other side. Let
the blood-globule with which we started be a globe peopled with animals
like ours, but rather smaller: {110} and call this the first globe below
us. Take a blood-globule out of this globe, people it, and call it the
second globe below us: and so on to the twentieth globe below us. This is a
fine stretch of progression both ways. Now give the giant of the twentieth
globe _above_ us the 607 decimal places, and, when he has measured the
diameter of his globe with accuracy worthy of his size, let him calculate
the circumference of his equator from the 607 places. Bring the little
philosopher from the twentieth globe _below_ us with his very best
microscope, and set him to see the small error which the giant must make.
He will not succeed, unless his microscopes be much better for his size
than ours are for ours.

"Now it must be remembered by any one who would laugh at the closeness of
the approximation, that the mathematician generally goes _nearer_; in fact
his theorems have usually no error at all. The very person who is
bewildered by the preceding description may easily forget that if there
were _no error at all_, the Lilliputian of the millionth globe below us
could not find a flaw in the Brobdingnagian of the millionth globe above.
The three angles of a triangle, of perfect accuracy of form, are
_absolutely_ equal to two right angles; no stretch of progression will
detect _any_ error.

"Now think of Mr. Lacomme's mathematical adviser (_ante_, Vol. I, p. 46)
making a difficulty of advising a stonemason about the quantity of pavement
in a circular floor!

"We will now, for our non-calculating reader, put the matter in another
way. We see that a circle-squarer can advance, with the utmost confidence,
the assertion that when the diameter is 1,000, the circumference is
accurately 3,125: the mathematician declaring that it is a trifle more than
3,141½. If the squarer be right, the mathematician has erred by about a
200th part of the whole: or has not kept his accounts right by about 10s.
in every 100l. Of course, if he set out with such an error he will
accumulate blunder upon blunder. Now, if there be a process in which {111}
close knowledge of the circle is requisite, it is in the prediction of the
moon's place--say, as to the time of passing the meridian at Greenwich--on
a given day. We cannot give the least idea of the complication of details:
but common sense will tell us that if a mathematician cannot find his way
round the circle without a relative error four times as big as a
stockbroker's commission, he must needs be dreadfully out in his attempt to
predict the time of passage of the moon. Now, what is the fact? His error
is less than a second of time, and the moon takes 27 days odd to revolve.
That is to say, setting out with 10s. in 100l. of error in his
circumference, he gets within the fifth part of a farthing in 100l. in
predicting the moon's transit. Now we cannot think that the respect in
which mathematical science is held is great enough--though we find it not
small--to make this go down. That respect is founded upon a notion that
right ends are got by right means: it will hardly be credited that the
truth can be got to farthings out of data which are wrong by shillings.
Even the celebrated Hamilton[207] of Edinburgh, who held that in
mathematics there was no way of going wrong, was fully impressed with the
belief that this was because error was avoided from the beginning. He never
went so far as to say that a mathematician who begins wrong must end right

"There is always a difficulty about the mode in which the thinking man of
common life is to deal with subjects he has not studied to a professional
extent. He must form opinions on matters theological, political, legal,
medical, and social. If he can make up his mind to choose a guide, there
is, of course, no perplexity: but on all the subjects mentioned the
direction-posts point different ways. Now why should he not form his
opinion upon an abstract mathematical question? Why not conclude that, as
to the circle, it is possible Mr. James Smith may be the man, just {112} as
Adam Smith[208] was the man of things then to come, or Luther, or Galileo?
It is true that there is an unanimity among mathematicians which prevails
in no other class: but this makes the chance of their all being wrong only
different in degree. And more than this, is it not generally thought among
us that priests and physicians were never so much wrong as when there was
most appearance of unanimity among them? To the preceding questions we see
no answer except this, that the individual inquirer may as rationally
decide a mathematical question for himself as a theological or a medical
question, so soon as he can put himself into a position in mathematics,
level with that in which he stands in theology or medicine. The every-day
thought and reading of common life have a certain resemblance to the
thought and reading demanded by the learned faculties. The research, the
balance of evidence, the estimation of probabilities, which are used in a
question of medicine, are closely akin in character, however different the
matter of application, to those which serve a merchant to draw his
conclusions about the markets. But the mathematicians have methods of their
own, to which nothing in common life bears close analogy, as to the nature
of the results or the character of the conclusions. The logic of
mathematics is certainly that of common life: but the data are of a
different species; they do not admit of doubt. An expert arithmetician,
such as is Mr. J. Smith, may fancy that calculation, merely as such, is
mathematics: but the value of his book, and in this point of view it is not
small, is the full manner in which it shows that a practised arithmetician,
venturing into the field of mathematical demonstration, may show himself
utterly destitute of all that distinguishes the reasoning geometrical
investigator from the calculator.


"And further, it should be remembered that in mathematics the power of
verifying results far exceeds that which is found in anything else: and
also the variety of distinct methods by which they can be attained. It
follows from all this that a person who desires to be as near the truth as
he can will not judge the results of mathematical demonstration to be open
to his criticism, in the same degree as results of other kinds. Should he
feel compelled to decide, there is no harm done: his circle may be 3-1/8
times its diameter, if it please him. But we must warn him that, in order
to get this circle, he must, as Mr. James Smith has done, _make it at
home_: the laws of space and thought beg leave respectfully to decline the

I will insert now at length, from the _Athenæum_ of June 8, 1861, the easy
refutation given by my deceased friend, with the remarks which precede.

"Mr. James Smith, of whose performance in the way of squaring the circle we
spoke some weeks ago in terms short of entire acquiescence, has advertised
himself in our columns, as our readers will have seen. He has also
forwarded his letter to the Liverpool _Albion_, with an additional
statement, which he did not make in _our_ journal. He denies that he has
violated the decencies of private life, since his correspondent revised the
proofs of his own letters, and his 'protest had respect only to making his
name public.' This statement Mr. James Smith precedes by saying that we
have treated as true what we well knew to be false: and he follows by
saying that we have not read his work, or we should have known the above
facts to be true. Mr. Smith's pretext is as follows. His correspondent
E. M. says, 'My letters were not intended for publication, and I protest
against their being published,' and he subjoins 'Therefore I must desire
that my name may not be used.' The obvious meaning is that E. M. protested
against the publication altogether, but, judging that Mr. Smith was {114}
determined to publish, desired that his name should not be used. That he
afterwards corrected the proofs merely means that he thought it wiser to
let them pass under his own eyes than to leave them entirely to Mr. Smith.

"We have received from Sir W. Rowan Hamilton[209] a proof that the
circumference is more than 3-1/8 diameters, requiring nothing but a
knowledge of four books of Euclid. We give it in brief as an exercise for
our juvenile readers to fill up. It reminds us of the old days when real
geometers used to think it worth while seriously to demolish pretenders.
Mr. Smith's fame is now assured: Sir W. R. Hamilton's brief and easy
exposure will procure him notice in connection with this celebrated

"It is to be shown that the perimeter of a regular polygon of 20 sides is
greater than 3-1/8 diameters of the circle, and still more, of course, is
the circumference of the circle greater than 3-1/8 diameters.

"1. It follows from the 4th Book of Euclid, that the rectangle under the
side of a regular decagon inscribed in a circle, and that side increased by
the radius, is equal to the square of the radius. But the product 791 (791
+ 1280) is less than 1280 × 1280; if then the radius be 1280 the side of
the decagon is greater than 791.

"2. When a diameter bisects a chord, the square of the chord is equal to
the rectangle under the doubles of the segments of the diameter. But the
product 125 (4 × 1280 - 125) is less than 791 × 791. If then the bisected
chord be a side of the decagon, and if the radius be still 1280, the double
of the lesser segment exceeds 125.

"3. The rectangle under this doubled segment and the radius is equal to the
square of the side of an inscribed regular polygon of 20 sides. But the
product 125 × 1280 is equal to 400 × 400; therefore, the side of the
last-mentioned polygon is greater than 400, if the radius be still 1280. In
other words, if the radius be represented by the new {115} member 16, and
therefore the diameter by 32, this side is greater than 5, and the
perimeter exceeds 100. So that, finally, if the diameter be 8, the
perimeter of the inscribed regular polygon of 20 sides, and still more the
circumference of the circle, is greater than 25: that is, the circumference
is more than 3-1/8 diameters."

The last work in the list was thus noticed in the _Athenæum_, May 27, 1865.

"Mr. James Smith appears to be tired of waiting for his place in the Budget
of Paradoxes, and accordingly publishes a long letter to Professor De
Morgan, with various prefaces and postscripts. The letter opens by a hint
that the Budget appears at very long intervals, and 'apparently without any
sufficient reason for it.' As Mr. Smith hints that he should like to see
Mr. De Morgan, whom he calls an 'elephant of mathematics,' 'pumping his
brains' 'behind the scenes'--an odd thing for an elephant to do, and an odd
place to do it in--to get an answer, we think he may mean to hint that the
Budget is delayed until the pump has worked successfully. Mr. Smith is
informed that we have had the whole manuscript of the Budget, excepting
only a final summing-up, in our hands since October, 1863. [This does not
refer to the Supplement.] There has been no delay: we knew from the
beginning that a series of historical articles would be frequently
interrupted by the things of the day. Mr. James Smith lets out that he has
never been able to get a private line from Mr. De Morgan in answer to his
communications: we should have guessed it. He says, 'The Professor is an
old bird and not to be easily caught, and by no efforts of mine have I been
able, up to the present moment, either to induce or twit him into a
discussion....' Mr. Smith curtails the proverb: old birds are not to be
caught with _chaff_, nor with _twit_, which seems to be Mr. Smith's word
for his own chaff, and, so long as the first letter is sounded, a very
proper word. Why does he not try a little grain of sense? Mr. Smith
evidently {116} thinks that, in his character as an elephant, the Professor
has not pumped up brain enough to furnish forth a bird. In serious earnest,
Mr. Smith needs no answer. In one thing he excites our curiosity: what is
meant by demonstrating 'geometrically _and_ mathematically?'"

I now proceed to my original treatment of the case.

Mr. James Smith will, I have no doubt, be the most uneclipsed
circle-squarer of our day. He will not owe this distinction to his being an
influential and respected member of the commercial world of Liverpool, even
though the power of publishing which his means give him should induce him
to issue a whole library upon one paradox. Neither will he owe it to the
pains taken with him by a mathematician who corresponded with him until the
joint letters filled an octavo volume. Neither will he owe it to the notice
taken of him by Sir William Hamilton, of Dublin, who refuted him in a
manner intelligible to an ordinary student of Euclid, which refutation he
calls a remarkable paradox easily explainable, but without explaining it.
What he will owe it to I proceed to show.

Until the publication of the _Nut to Crack_ Mr. James Smith stood among
circle-squarers in general. I might have treated him with ridicule, as I
have done others: and he says that he does not doubt he shall come in for
his share at the tail end of my Budget. But I can make a better job of him
than so, as Locke would have phrased it: he is such a very striking example
of something I have said on the use of logic that I prefer to make an
example of his writings. On one point indeed he well deserves the
_scutica_,[210] if not the _horribile flagellum_.[211] He tells me that he
will bring his solution to me in such a form as shall compel me to admit it
as _un fait accompli_ [_une faute accomplie?_][212] {117} or leave myself
open to the humiliating charge of mathematical ignorance and folly. He has
also honored me with some private letters. In the first of these he gives
me a "piece of information," after which he cannot imagine that I, "as an
honest mathematician," can possibly have the slightest hesitation in
admitting his solution. There is a tolerable reservoir of modest assurance
in a man who writes to a perfect stranger with what he takes for an
argument, and gives an oblique threat of imputation of dishonesty in case
the argument be not admitted without hesitation; not to speak of the minor
charges of ignorance and folly. All this is blind self-confidence, without
mixture of malicious meaning; and I rather like it: it makes me understand
how Sam Johnson came to say of his old friend Mrs. Cobb,[213]--"I love Moll
Cobb for her impudence." I have now done with my friend's _suaviter in
modo_,[214] and proceed to his _fortiter in re_[215]: I shall show that he
_has_ convicted himself of ignorance and folly, with an honesty and candor
worthy of a better value of [pi].

Mr. Smith's method of proving that every circle is 3-1/8 diameters is to
assume that it is so,--"if you dislike the term datum, then, by hypothesis,
let 8 circumferences be exactly equal to 25 diameters,"--and then to show
that every other supposition is thereby made absurd. The right to this
assumption is enforced in the "Nut" by the following analogy:

"I think you (!) will not dare (!) to dispute my right to this hypothesis,
when I can prove by means of it that every other value of [pi] will lead to
the grossest absurdities; unless indeed, you are prepared to dispute the
right of Euclid to adopt a false line hypothetically for the purpose {118}
of a '_reductio ad absurdum_'[216] demonstration, in pure geometry."

Euclid assumes what he wants to _disprove_, and shows that his _assumption_
leads to absurdity, and so _upsets itself_. Mr. Smith assumes what he wants
to _prove_, and shows that _his_ assumption makes _other propositions_ lead
to absurdity. This is enough for all who can reason. Mr. James Smith cannot
be argued with; he has the whip-hand of all the thinkers in the world.
Montucla would have said of Mr. Smith what he said of the gentleman who
squared his circle by giving 50 and 49 the same square root, _Il a perdu le
droit d'être frappé de l'évidence_.[217]

It is Mr. Smith's habit, when he finds a conclusion agreeing with its own
assumption, to regard that agreement as proof of the assumption. The
following is the "piece of information" which will settle me, if I be
honest. Assuming [pi] to be 3-1/8, he finds out by working instance after
instance that the mean proportional between one-fifth of the area and
one-fifth of eight is the radius. That is,

  if [pi] = 25/8, sqrt(([pi]r^2)/5 · 8/5) = r.

This "remarkable general principle" may fail to establish Mr. Smith's
quadrature, even in an honest mind, if that mind should happen to know
that, a and b being any two numbers whatever, we need only assume--

  [pi] = a^2/b, to get at sqrt(([pi]r^2)/a · b/a) = r.

We naturally ask what sort of glimmer can Mr. Smith have of the subject
which he professes to treat? On this point he has given satisfactory
information. I had mentioned the old problem of finding two mean
proportionals, {119} as a preliminary to the duplication of the cube. On
this mention Mr. Smith writes as follows. I put a few words in capitals;
and I write rq[218] for the sign of the square root, which embarrasses
small type:

"This establishes the following _infallible_ rule, for finding two mean
proportionals OF EQUAL VALUE, and is more than a preliminary, to the famous
old problem of 'Squaring the circle.' Let any finite number, say 20, and
its fourth part = ¼(20) = 5, be given numbers. Then rq(20 × 5) = rq 100 =
10, is their mean proportional. Let this be a given mean proportional TO

  20 × [pi]/4 = 20 × 3.125/4 = 20 × .78125 = 15.625

will be the first number; as

  25 : 16 :: rq 20 : rq 8.192: and (rq 8.192)^2 × [pi]/4 = 8.192 × .78125 =

will be the second number; therefore rq(15.625 × 6.4) = rq 100 = 10, is the
required mean proportional.... Now, my good Sir, however competent you may
be to prove every man a fool [not _every_ man, Mr. Smith! only _some_; pray
learn logical quantification] who now thinks, or in times gone by has
thought, the 'Squaring of the Circle' _a possibility_; I doubt, and, on the
evidence afforded by your Budget, I cannot help doubting, whether you were
ever before competent to find two mean proportionals _by my unique
method_."--(_Nut_, pp. 47, 48.) [That I never was, I solemnly declare!]

All readers can be made to see the following exposure. When 5 and 20 are
given, x is a mean proportional when in 5, x, 20, 5 is to x as x to 20. And
x must be 10. But x and y are two mean proportionals when in 5, x, y, 20, x
{120} is a mean proportional between 5 and y, and y is a mean proportional
between x and 20. And these means are x = 5 [cuberoot]4, y = 5
[cuberoot]16. But Mr. Smith finds _one_ mean, finds it _again_ in a
roundabout way, and produces 10 and 10 as the two (equal!) means, in
solution of the "famous old problem." This is enough: if more were wanted,
there is more where this came from. Let it not be forgotten that Mr. Smith
has found a translator abroad, two, perhaps three, followers at home,
and--most surprising of all--a real mathematician to try to set him right.
And this mathematician did not discover the character of the subsoil of the
land he was trying to cultivate until a goodly octavo volume of letters had
passed and repassed. I have noticed, in more quarters than one, an apparent
want of perception of the _full_ amount of Mr. Smith's ignorance: persons
who have not been in contact with the non-geometrical circle-squarers have
a kind of doubt as to whether anybody can carry things so far. But I am an
"old bird" as Mr. Smith himself calls me; a Simorg, an "all-knowing Bird of
Ages" in matters of cyclometry.

The curious phenomena of thought here exhibited illustrate, as above said,
a remark I have long ago made on the effect of proper study of logic. Most
persons reason well enough on matter to which they are accustomed, and in
terms with which they are familiar. But in unaccustomed matter, and with
use of strange terms, few except those who are practised in the
abstractions of pure logic can be tolerably sure to keep their feet. And
one of the reasons is easily stated: terms which are not quite familiar
partake of the vagueness of the X and Y on which the student of logic
learns to see the formal force of a proposition independently of its
material elements.

I make the following quotation from my fourth paper on logic in the
_Cambridge Transactions_:

"The uncultivated reason proceeds by a process almost entirely material.
Though the necessary law of thought {121} must determine the conclusion of
the ploughboy as much as that of Aristotle himself, the ploughboy's
conclusion will only be tolerably sure when the matter of it is such as
comes within his usual cognizance. He knows that geese being all birds does
not make all birds geese, but mainly because there are ducks, chickens,
partridges, etc. A beginner in geometry, when asked what follows from
'Every A is B,' answers 'Every B is A.' That is, the necessary laws of
thought, except in minds which have examined their tools, are not very sure
to work correct conclusions except upon familiar matter.... As the
cultivation of the individual increases, the laws of thought which are of
most usual application are applied to familiar matter with tolerable
safety. But difficulty and risk of error make a new appearance with a new
subject; and this, in most cases, until new subjects are familiar things,
unusual matter common, untried nomenclature habitual; that is, until it is
a habit to be occupied upon a novelty. It is observed that many persons
reason well in some things and badly in others; and this is attributed to
the consequence of employing the mind too much upon one or another subject.
But those who know the truth of the preceding remarks will not have far to
seek for what is often, perhaps most often, the true reason.... I maintain
that logic tends to make the power of reason over the unusual and
unfamiliar more nearly equal to the power over the usual and familiar than
it would otherwise be. The second is increased; but the first is almost

Mr. James Smith, by bringing ignorance, folly, dishonesty into contact with
my name, in the way of conditional insinuation, has done me a good turn: he
has given me right to a freedom of personal remark which I might have
declined to take in the case of a person who is useful and respected in
matters which he understands.

Tit for tat is logic all the world over. By the way, what has become of the
rest of the maxim: we never hear it {122} now. When I was a boy, in some
parts of the country at least, it ran thus:

 "Tit for tat;
  Butter for fat:
  If you kill my dog,
  I'll kill your cat."

He is a glaring instance of the truth of the observations quoted above. I
will answer for it that, at the Mersey Dock Board, he never dreams of
proving that the balance at the banker's is larger than that in the book by
assuming that the larger sum is there, and then proving that the other
supposition--the smaller balance--is upon that assumption, an absurdity. He
never says to another director, How can you dare to refuse me a right to
assume the larger balance, when you yourself, the other day,
said,--Suppose, for argument's sake, we had 80,000l. at the banker's,
though you knew the book only showed 30,000l.? This is the way in which he
has supported his geometrical paradox by Euclid's example: and this is not
the way he reasons at the board; I know it by the character of him as a man
of business which has reached my ears from several quarters. But in
geometry and rational arithmetic he is a smatterer, though expert at
computation; at the board he is a trained man of business. The language of
geometry is so new to him that he does not know what is meant by "two mean
proportionals:" but all the phrases of commerce are rooted in his mind. He
is most unerasably booked in the history of the squaring of the circle, as
the speculator who took a right to assume a proposition for the destruction
of other propositions, on the express ground that Euclid assumes a
proposition to show that it destroys itself: which is as if the curate
should demand permission to throttle the squire because St. Patrick drove
the vermin to suicide to save themselves from slaughter. He is conspicuous
as a speculator who, more visibly than almost any other known to history,
reasoned in a circle by way of reasoning on a circle. But {123} what I have
chiefly to do with is the force of instance which he has lent to my
assertion that men who have not had real training in pure logic are unsafe
reasoners in matter which is not familiar. It is hard to get first-rate
examples of this, because there are few who find the way to the printer
until practice and reflection have given security against the grossest
slips. I cannot but think that his case will lead many to take what I have
said into consideration, among those who are competent to think of the
great mental disciplines. To this end I should desire him to continue his
efforts, to amplify and develop his great principle, that of proving a
proposition by assuming it and taking as confirmation every consequence
that does not contradict the assumption.

Since my Budget commenced, Mr. Smith has written me notes: the portion
which I have preserved--I suppose several have been mislaid--makes a
hundred and seven pages of note-paper, closely written. To all this I have
not answered one word: but I think I cannot have read fewer than forty
pages. In the last letter the writer informs me that he will not write at
greater length until I have given him an answer, according to the "rules of
good society." Did I not know that for every inch I wrote back he would
return an ell? Surely in vain the net is spread in the eyes of anything
that hath a wing. There were several good excuses for not writing to Mr. J.
Smith: I will mention five. First, I distinctly announced at the beginning
of this Budget that I would not communicate with squarers of the circle.
Secondly, any answer I might choose to give might with perfect propriety be
reserved for this article; had the imputation of incivility been made after
the first note, I should immediately have replied to this effect: but I
presumed it was quite understood. Thirdly, Mr. Smith, by his publication of
E. M.'s letters against the wish of the writer, had put himself out of the
pale of correspondence. Fourthly, he had also gone beyond the rules of good
society in sending {124} letter after letter to a person who had shown by
his silence an intention to avoid correspondence. Fifthly, these same rules
of good society are contrived to be flexible or frangible in extreme cases:
otherwise there would be no living under them; and good society would be
bad. Father Aldrovand has laid down the necessary distinction--"I tell
thee, thou foolish Fleming, the text speaketh but of promises made unto
Christians, and there is in the rubric a special exemption of such as are
made to Welchmen." There is also a rubric to the rules of good society; and
squarers of the circle are among those whom there is special permission not
to answer: they are the wild Welchmen of geometry, who are always
assailing, but never taking, the Garde Douloureuse[219] of the circle. "At
this commentary," proceeds the story, "the Fleming grinned so broadly as to
show his whole case of broad strong white teeth." I know not whether the
Welchman would have done the like, but I hope Mr. James Smith will: and I
hope he has as good a case to show as Wilkin Flammock. For I wish him long
life and long health, and should be very glad to see so much energy
employed in a productive way. I hope he wishes me the same: if not, I will
give him what all his judicious friends will think a good reason for doing
so. His pamphlets and letters are all tied up together, and will form a
curious lot when death or cessation of power to forage among book-shelves
shall bring my little library to the hammer. And this time may not be far
off: for I was X years old in A.D. X^2; not 4 in A.D. 16, nor 5 in A.D. 25,
but still in one case under that law. And now I have made my own age a
problem of quadrature, and Mr. J. Smith may solve it. But I protest against
his method of assuming a result, and making itself prove itself: he might
in this way, as sure as eggs is eggs (a corruption of X is X), make me
1,864 years old, which is a great deal too much.


_April 5, 1864._--Mr. Smith continues to write me long letters, to which he
hints that I am to answer. In his last, of 31 closely written sides of
note-paper, he informs me, with reference to my obstinate silence, that
though I think myself and am thought by others to be a mathematical
Goliath, I have resolved to play the mathematical snail, and keep within my
shell. A mathematical _snail_! This cannot be the thing so called which
regulates the striking of a clock; for it would mean that I am to make Mr.
Smith sound the true time of day, which I would by no means undertake upon
a clock that gains 19 seconds odd in every hour by false quadrature. But he
ventures to tell me that pebbles from the sling of simple truth and common
sense will ultimately crack my shell, and put me _hors de combat_.[220] The
confusion of images is amusing: Goliath turning himself into a snail to
avoid [pi] = 3-1/8, and James Smith, Esq., of the Mersey Dock Board: and
put _hors de combat_--which should have been _caché_[221]--by pebbles from
a sling. If Goliath had crept into a snail-shell, David would have cracked
the Philistine with his foot. There is something like modesty in the
implication that the crack-shell pebble has not yet taken effect; it might
have been thought that the slinger would by this time have been singing--

 "And thrice [and one-eighth] I routed all my foes,
  And thrice [and one-eighth] I slew the slain."

But he promises to give the public his nut-cracker if I do not, before the
Budget is concluded, "unravel" the paradox, which is the
mathematico-geometrical nut he has given me to crack. Mr. Smith is a crack
man: he will crack his own nut; he will crack my shell; in the mean time he
cracks himself up. Heaven send he do not crack himself into lateral
contiguity with himself.

On June 27 I received a letter, in the handwriting of Mr. James Smith,
signed Nauticus. I have ascertained {126} that one of the letters to the
_Athenæum_ signed Nauticus is in the same handwriting. I make a few

"... The important question at issue has been treated by a brace of
mathematical birds with too much levity. It may be said, however, that
sarcasm and ridicule sometimes succeed, where reason fails.... Such a
course is not well suited to a discussion.... For this reason I shall for
the future [this implies there has been a past, so that Nauticus is not
before me for the first time] endeavor to confine myself to dry reasoning
from incontrovertible premises.

... It appears to me that so far as his theory is concerned he comes off
unscathed. You might have found "a hole in Smith's circle" (have you seen a
pamphlet bearing this title? [I never heard of it until now]), but after
all it is quite possible the hole may have been left by design, for the
purpose of entrapping the unwary."

[On the publication of the above, the author of the pamphlet obligingly
forwarded a copy to me of _A Hole in Smith's Circle_--by a Cantab: Longman
and Co., 1859, (pp. 15). "It is pity to lose any fun we can get out of the
affair," says my almamaternal brother: to which I add that in such a case
warning without joke is worse than none at all, as giving a false idea of
the nature of the danger. The Cantab takes some absurdities on which I have
not dwelt: but there are enough to afford a Cantab from every college his
own separate hunting ground.]

Does this hint that his mode of proof, namely, assuming the thing to be
proved, was a design to entrap the unwary? if so, it bangs Banagher. Was
his confounding two mean proportionals with one mean proportional found
twice over a trick of the same intent? if so, it beats cockfighting. That
Nauticus is Mr. Smith appears from other internal evidence. In 1819, Mr.
J. C. Hobhouse[222] was sent to Newgate for a {127} libel on the House of
Commons which was only intended for a libel on Lord Erskine.[223] The
ex-Chancellor had taken Mr. Hobhouse to be thinking of him in a certain
sentence; this Mr. Hobhouse denied, adding, "There is but one man in the
country who is always thinking of Lord Erskine." I say that there is but
one man of our day who would couple me and Mr. James Smith as a "brace of
mathematical birds."

Mr. Smith's "theory" is unscathed by me. Not a doubt about it: but how does
he himself come off? I should never think of refuting a theory proved by
assumption of itself. I left Mr. Smith's [pi] untouched: or, if I put in my
thumb and pulled out a plum, it was to give a notion of the cook, not of
the dish. The "important question at issue" was not the circle: it was,
wholly and solely, whether the abbreviation of _James_ might be spelled
_Jimm_.[224] This is personal to the verge of scurrility: but in literary
controversy the challenger names the weapons, and Mr. Smith begins with
charge of ignorance, folly, and dishonesty, by conditional implication. So
that the question is, not the personality of a word, but its applicability
to the person designated: it is enough if, as the Latin grammar has it,
_Verbum personale concordat cum nominativo_.[225]

I may plead precedent for taking a liberty with the orthography of _Jem_.
An instructor of youth was scandalized at the abrupt and irregular--but
very effective--opening of Wordsworth's little piece:


         "A simple child
  That lightly draws its breath,
  And feels its life in every limb,
  What should it know of death?"

So he mended the matter by instructing his pupils to read the first line

 "A simple child, dear brother ----."

The brother, we infer from sound, was to be James, and the blank must
therefore be filled up with _Jimb_.

I will notice one point of the letter, to make a little more distinction
between the two birds. Nauticus lays down--quite correctly--that the sine
of an angle is less than its circular measure. He then takes 3.1416 for
180°, and finds that 36' is .010472. But this is exactly what he finds for
the sine of 36' in tables: he concludes that either 3.1416 or the tables
must be wrong. He does not know that sines, as well as [pi], are
interminable decimals, of which the tables, to save printing, only take in
a finite number. He is a six-figure man: let us go thrice again to make up
nine, and we have as follows:

  Circular measure of 36'         .010471975...
  Sine of 36'                     .010471784...
  Excess of measure over sine     .000000191...

Mr. Smith invites me to say which is wrong, the quadrature, or the tables:
I leave him to guess. He says his assertions "arise naturally and
necessarily out of the arguments of a circle-squarer:" he might just as
well lay down that all the pigs went to market because it is recorded that
"_This_ pig went to market." I must say for circle-squarers that very few
bring their pigs to so poor a market. I answer the above argument because
it is, of all which Mr. James Smith has produced, the only one which rises
to the level of a schoolboy: to meet him halfway I descend to that level.

Mr. Smith asks me to solve a problem in the _Athenæum_: {129} and I will do
it, because the question will illustrate what is _below_ schoolboy level.

"Let x represent the circular measure of an angle of 15°, and y half the
sine of an angle of 30° = area of the square on the radius of a circle of
diameter unity = .25. If x - y = xy, firstly, what is the arithmetical
value of xy? secondly, what is the angle of which xy represents the
circular measure?"

If x represent 15° and y be ¼, xy represents 3° 45', whether x - y be xy or
no. But, y being ¼, x - y is _not_ xy unless x be 1/3, that is, unless 12x
or [pi] be 4, which Mr. Smith would not admit. How could a person who had
just received such a lesson as I had given immediately pray for further
exposure, furnishing the stuff so liberally as this? Is it possible that
Mr. Smith, because he signs himself Nauticus, means to deny his own very
regular, legible, and peculiar hand? It is enough to make the other members
of the Liverpool Dock Board cry, Mersey on the man!

Mr. Smith says that for the future he will give up what he calls sarcasm,
and confine himself, "as far as possible," to what he calls dry reasoning
from incontrovertible premises. If I have fairly taught him that _his_
sarcasm will not succeed, I hope he will find that his wit's end is his
logic's beginning.

I now reply to a question I have been asked again and again since my last
Budget appeared: Why do you take so much trouble to expose such a reasoner
as Mr. Smith? I answer as a deceased friend of mine used to answer on like
occasions--A man's capacity is no measure of his power to do mischief. Mr.
Smith has untiring energy, which does something; self-evident honesty of
conviction, which does more; and a long purse, which does most of all. He
has made at least ten publications, full of figures which few readers can
criticize. A great many people are staggered to this extent, that they
imagine there must be {130} the indefinite _something_ in the mysterious
_all this_. They are brought to the point of suspicion that the
mathematicians ought not to treat "all this" with such undisguised
contempt, at least. Now I have no fear for [pi]: but I do think it possible
that general opinion might in time demand that the crowd of
circle-squarers, etc. should be admitted to the honors of opposition; and
this would be a time-tax of five per cent., one man with another, upon
those who are better employed. Mr. James Smith may be made useful, in hands
which understand how to do it, towards preventing such opinion from
growing. A speculator who expressly assumes what he wants to prove, and
argues that all which contradicts it is absurd, _because_ it cannot stand
side by side with his assumption, is a case which can be exposed to all.
And the best person to expose it is one who has lived in the past as well
as the present, who takes misthinking from points of view which none but a
student of history can occupy, and who has something of a turn for the

Whether I have any motive but public good must be referred to those who can
decide whether a missionary chooses his pursuit solely to convert the
heathen. I shall certainly be thought to have a little of the spirit of
Col. Quagg, who delighted in strapping the Grace-walking Brethren. I must
quote this myself: if I do not, some one else will, and then where am I?
The Colonel's principle is described as follows:

"I licks ye because I kin, and because I like, and because ye'se critters
that licks is good for. Skins ye have on, and skins I'll have off; hard or
soft, wet or dry, spring or fall. Walk in grace if ye like till pumpkins is
peaches; but licked ye must be till your toe-nails drop off and your noses
bleed blue ink. And--licked--they--were--accordingly."

I am reminded of this by the excessive confidence with which Mr. James
Smith predicted that he would treat me as Zephaniah Stockdolloger (Sam
Slick calls it _slockdollager_) treated Goliah Quagg. He has announced his
{131} intention of bringing me, with a contrite heart, and clean
shaved,--4159265... razored down to 25,--to a camp-meeting of
circle-squarers. But there is this difference: Zephaniah only wanted to
pass the Colonel's smithy in peace; Mr. James Smith sought a fight with me.
As soon as this Budget began to appear, he oiled his own strap, and
attempted to treat me as the terrible Colonel would have treated the
inoffensive brother.

He is at liberty to try again.


    The Moon-hoax; or the discovery that the moon has a vast population of
    human beings. By Richard Adams Locke.[226] New York, 1859, 8vo.

This is a reprint of the hoax already mentioned. I suppose R. A. Locke is
the name assumed by M. Nicollet.[227] The publisher informs us that when
the hoax first appeared day by day in a morning paper, the circulation
increased fivefold, and the paper obtained a permanent footing. Besides
this, an edition of 60,000 was sold off in less than one month.

The discovery was also published under the name of A. R. Grant.[228]
Sohncke's[229] _Bibliotheca Mathematica_ confounds this Grant with Prof. R.
Grant[230] of Glasgow, the author of the _History of Physical Astronomy_,
who is accordingly made to guarantee the discoveries in the moon. I hope
Adams Locke will not merge in J. C. Adams,[231] the co-discoverer of
Neptune. Sohncke gives the titles of {132} three French translations of the
Moon hoax at Paris, of one at Bordeaux, and of Italian translations at
Parma, Palermo, and Milan.

A Correspondent, who is evidently fully master of details, which he has
given at length, informs me that the Moon hoax appeared first in the _New
York Sun_, of which R. A. Locke was editor. It so much resembled a story
then recently published by Edgar A. Poe, in a Southern paper, "Adventures
of Hans Pfaal," that some New York journals published the two side by side.
Mr. Locke, when he left the _New York Sun_, started another paper, and
discovered the manuscript of Mungo Park;[232] but this did not deceive. The
_Sun_, however, continued its career, and had a great success in an account
of a balloon voyage from England to America, in seventy-five hours, by Mr.
Monck Mason,[233] Mr. Harrison Ainsworth,[234] and others. I have no doubt
that M. Nicollet was the author of the Moon hoax,[235] written in a way
which marks the practised observatory astronomer beyond all doubt, and by
evidence seen in the most minute details. Nicollet had an eye to Europe. I
suspect that he took Poe's story, and made it a basis for his own. Mr.
Locke, it would seem, when he attempted a fabrication for himself, did not

    The Earth we inhabit, its past, present, and future. By Capt.
    Drayson.[236] London, 1859, 8vo.

The earth is growing; absolutely growing larger: its diameter increases
three-quarters of an inch per mile every year. The foundations of our
buildings will give way in {133} time: the telegraph cables break, and no
cause ever assigned except ships' anchors, and such things. The book is for
those whose common sense is unwarped, who can judge evidence as well as the
ablest philosopher. The prospect is not a bad one, for population increases
so fast that a larger earth will be wanted in time, unless emigration to
the Moon can be managed, a proposal of which it much surprises me that
Bishop Wilkins has a monopoly.


_Athenæum_, August, 19, 1865. _Notice to Correspondents._

"R. W.--If you will consult the opening chapter of the Budget of Paradoxes,
you will see that the author presents only works in his own library at a
given date; and this for a purpose explained. For ourselves we have
carefully avoided allowing any writers to present themselves in our columns
on the ground that the Budget has passed them over. We gather that Mr. De
Morgan contemplates additions at a future time, perhaps in a separate and
augmented work; if so, those who complain that others of no greater claims
than themselves have been ridiculed may find themselves where they wish to
be. We have done what we can for you by forwarding your letter to Mr. De

The author of "An Essay on the Constitution of the Earth," published in
1844, demanded of the _Athenæum_, as an _act of fairness_, that a letter
from him should be published, proving that he had as much right to be
"impaled" as Capt. Drayson. He holds, on speculative grounds, what the
other claims to have proved by measurement, namely, that the earth is
growing; and he believes that in time--a good long time, not _our_
time--the earth and other planets may grow into suns, with systems of their

This gentleman sent me a copy of his work, after the commencement of my
Budget; but I have no recollection of having received it, and I cannot find
it on the (nursery? {134} quarantine?) shelves on which I keep my
unestablished discoveries. Had I known of this work in time, (see the
Introduction) I should of course, have impaled it (heraldically) with the
other work; but the two are very different. Capt. Drayson professes to
prove his point by results of observation; and I think he does not succeed.
The author before me only speculates; and a speculator can get any
conclusion into his premises, if he will only build or hire them of shape
and size to suit. It reminds me of a statement I heard years ago, that a
score of persons, or near it, were to dine inside the skull of one of the
aboriginal animals, dear little creatures! Whereat I wondered vastly,
nothing doubting; facts being stubborn and not easy drove, as Mrs. Gamp
said. But I soon learned that the skull was not a real one, but
artificially constructed by the methods--methods which have had striking
verifications, too--which enable zoologists to go the whole hog by help of
a toe or a bit of tail. This took off the edge of the wonder: a hundred
people can dine inside an inference, if you draw it large enough. The
method might happen to fail for once: for instance, the toe-bone might have
been abnormalized by therian or saurian malady; and the possibility of such
failure, even when of small probability, is of great alleviation. The
author before me is, apparently, the sole fabricator of his own premises.
With vital force in the earth and continual creation on the part of the
original Creator, he expands our bit of a residence as desired. But, as the
Newtoness of Cookery observed, First catch your hare. When this is done,
when you _have_ a growing earth, you shall dress it with all manner of
proximate causes, and serve it up with a growing Moon for sauce, a growing
Sun, if it please you, at the other end, and growing planets for
side-dishes. Hoping this amount of impalement will be satisfactory, I go on
to something else. {135}


    _The Hailesean System of Astronomy._ By John Davey Hailes[237] (two
    pages duodecimo, 1860).

He offers to _take_ 100,000l. to 1,000l. that he shows the sun to be less
than seven millions of miles from the earth. The earth in the center,
revolving eastward, the sun revolving westward, so that they "meet at half
the circle distance in the 24 hours." The diameter of the circle being
9839458303, the circumference is 30911569920.

The following written challenge was forwarded to the Council of the
Astronomical Society: it will show the "general reader"--and help him
towards earning his name--what sort of things come every now and then to
our scientific bodies. I have added punctuation:

                  1,000 to 30,000.
 "Leverrier's[238] name stand placed first. Do the worthy Frenchman
  By awarding him the medal in a trice.
  Give Adams[239] an extra--of which neck and neck the race.
  Now I challenge to meet them and the F.R.S.'s all,
  For good will and _one_ thousand pounds to their _thirty_ thousand
  That I produce a system, which shall measure the time,
  When the Sun was vertical to Gibeon, afterward to Syene.
  To meet any time in London--name your own period,
  To be decided by a majority of twelve persons--a President, _odd_.
  That mean, if the twelve equally divide, the President decide,
  I should prefer the Bishop of London, over the meeting to preside.
                                JOHN DAVY HAILES."
  Feb. 17, 1847."

Mr. Hailes still issues his flying sheets. The last I have met with
(October 7, 1863) informs us that the latitude of {136} England is slowly
increasing, which is the true cause of the alteration in the variation of
the magnet.

[Mr. Hailes continues his researches. Witness his new Hailesean system of
Astronomy, displaying Joshua's miracle-time, origin of time from science,
with Bible and Egyptian history. Rewards offered for astronomical problems.
With magnetism, etc. etc. Astronomical challenge to all the world.
Published at Cambridge, in 1865. The author agrees with Newton in one
marked point. _Errores quam minimi non sunt contemnendi_,[240] says Isaac:
meaning in figures, not in orthography. Mr. Hailes enters into the spirit,
both positive and negative, of this dictum, by giving the distance of
_Sidius_ from the center of the earth at 163,162,008 miles 10 feet 8 inches
17-28ths of an inch. Of course, he is aware that the center of _figure_ of
the earth is 17.1998 inches from the center of _gravity_. Which of the two
is he speaking of?]

    The Divine Mystery of Life. London [1861], 18mo. (pp.32).

The author has added one class to zoology, which is printed in capitals, as
derived from _zoé_, life, not from _zôon_, animal. That class is of
_Incorporealia_, order I., _Infinitum_, of one genus without plurality,
_Deus_: order II., _Finita_, angels good and evil. The rest is all about a
triune system, with a diagram. The author is not aware that [Greek: zôon]
is not _animal_, but _living being_. Aristotle had classed gods under
[Greek: zôa], and has been called to account for it by moderns who have
taken the word to mean _animal_.


    Explication du Zodiaque de Denderah, des Pyramides, et de Genèse. Par
    le Capitaine au longcours Justin Roblin.[241] Caen, 1861. 8vo.


Capt. Roblin, having discovered the sites of gold and diamond mines by help
of the zodiac of Denderah, offered half to the shareholders of a company
which he proposed to form. One of our journals, by help of the zodiac of
Esné, offered, at five francs a head, to tell the shareholders the exact
amount of gold and diamonds which each would get, and to make up the amount
predicted to those who got less. There are moods of the market in England
in which this company could have been formed: so we must not laugh at our


    A million's worth of property, and five hundred lives annually lost at
    sea by the Theory of Gravitation. A letter on the true figure of the
    earth, addressed to the Astronomer Royal, by Johannes von Gumpach.[242]
    London, 1861, 8vo. (pp. 54).

    The true figure and dimensions of the earth, in a letter addressed to
    the Astronomer Royal. By Joh. von Gumpach. 2nd ed. entirely recast.
    London, 1862, 8vo. (pp. 266).

    Two issues of a letter published with two different title-pages, one
    addressed to the Secretary of the Royal Society, the other to the
    Secretary of the Royal Astronomical Society. It would seem that the
    same letter is also issued with two other titles, addressed to the
    British Association and the Royal Geographical Society. By Joh. von
    Gumpach. London, 1862, 8vo.

    Baby-Worlds. An essay on the nascent members of our solar household. By
    Joh. von Gumpach. London, 1863, 8vo.

The earth, it appears, instead of being flattened, is elongated at the
poles: by ignorance of which the loss above mentioned occurs yearly. There
is, or is to be, a substitute for attraction and an "application hitherto
neglected, of a {138} recognized law of optics to the astronomical theory,
showing the true orbits of the heavenly bodies to be perfectly circular,
and their orbital motions to be perfectly uniform." all irregularities
being, I suppose, optical delusions. Mr. Von Gumpach is a learned man; what
else, time must show.


    Perpetuum Mobile: or Search for self-motive Power. By Henry
    Dircks.[243] London, 1861, 8vo.

A useful collection on the history of the attempts at perpetual motion,
that is, at obtaining the consequences of power without any power to
produce them. September 7, 1863, a correspondent of the _Times_ gave an
anecdote of George Stephenson,[244] which he obtained from Robert
Stephenson.[245] A perpetual motionist wanted to explain his method; to
which George replied--"Sir! I shall believe it when I see you take yourself
up by the waistband, and carry yourself about the room." Never was the
problem better stated.

There is a paradox of which I ought to give a specimen, I mean the
_slander-paradox_; the case of a person who takes it into his head, upon
evidence furnished entirely by the workings of his own thoughts, that some
other person has committed a foul act of which the world at large would no
more suppose him guilty than they would suppose that the earth is a flat
bordered by ice. If I were to determine on giving cases in which the
self-deluded person imagines {139} a conspiracy against _himself_, there
would be no end of choices. Many of the grosser cases are found at last to
be accompanied by mental disorder, and it is difficult to avoid referring
the whole class to something different from simple misuse of the reasoning
power. The first instance is one which puts in a strong light the state of
things in which we live, brought about by our glorious freedom of thought,
speech, and writing. The Government treated it with neglect, the press with
silent contempt, and I will answer for it many of my readers now hear of it
for the first time, when it comes to be enrolled among circle-squarers and
earth-stoppers, where, as the old philosopher said, it will not gravitate,
being _in proprio loco_.[246]

1862. On new year's day, 1862, when the nation was in the full tide of
sympathy with the Queen, and regret for its own loss, a paper called the
_Free Press_ published a number devoted to the consideration of the causes
of the death of the Prince Consort. It is so rambling and inconsecutive
that it takes more than one reading to understand it. It is against the
_Times_ newspaper. First, the following insinuation:

"To the legal mind, the part of [the part taken by] the _Times_ will
present a _prima facie_ case of the gravest nature, in the evident
fore-knowledge of the event, and the preparation to turn it to account when
it should have occurred. The article printed on Saturday must have been
written on Friday. That article could not have appeared had the Prince been
intended to live."

Next, it is affirmed that the _Times_ intended to convey the idea that the
Prince had been poisoned.

"Up to this point we are merely dealing with words which the _Times_
publishes, and these can leave not a shadow of doubt that there is an
intention to promulgate the idea that Prince Albert had been poisoned."

The article then goes on with a strange olio of {140} insinuations to the
effect that the Prince was the obstacle to Russian intrigue, and that if he
should have been poisoned,--which the writer strongly hints may have been
the case,--some Minister under the influence of Russia must have done it.
Enough for this record. _Un sot trouve toujours un plus sot qui
l'admire_:[247] who can he be in this case?


1846. At the end of this year arose the celebrated controversy relative to
the discovery of Neptune. Those who know it are well aware that Mr.
Adams's[248] now undoubted right to rank with Le Verrier[249] was made sure
at the very outset by the manner in which Mr. Airy,[250] the Astronomer
Royal, came forward to state what had taken place between himself and Mr.
Adams. Those who know all the story about Mr. Airy being arrested in his
progress by the neglect of Mr. Adams to answer a letter, with all the
imputations which might have been thrown upon himself for laxity in the
matter, know also that Mr. Airy's conduct exhibited moral courage, honest
feeling, and willingness to sacrifice himself, if need were, to the
attainment of the ends of private justice, and the establishment of a
national claim. A writer in a magazine, in a long and elaborate article,
argued the supposition--put in every way except downright assertion, after
the fashion of such things--that Mr. Airy had communicated Mr. Adams's
results to M. Le Verrier, with intention that they should be used. His
presumption as to motive is that, had Mr. Adams been recognized, "then the
discovery must have been indisputably an _Englishman's_, and that
Englishman not the Astronomer Royal." Mr. Adams's conclusions were
"retouched in France, and sent {141} over the year after." The proof given
is that it cannot be "imagined" otherwise.

"Can it then be imagined that the Astronomer Royal received such results
from Mr. Adams, supported as they were by Professor Challis's[251] valuable
testimony as to their probable accuracy, and did not bring the French
astronomer acquainted with them, especially as he was aware that his friend
was engaged in matters bearing directly upon these results?"

The whole argument the author styles "evidence which I consider it
difficult to refute." He ends by calling upon certain persons, of whom I am
one, to "see ample justice done." This is the duty of every one, according
to his opportunities. So when the reputed author--the article being
anonymous--was, in 1849, proposed as a Fellow of the Astronomical Society,
I joined--if I remember right, I originated--an opposition to his election,
until either the authorship should be denied, or a proper retraction made.
The friends of the author neither denied the first, nor produced the
second: and they judged it prudent to withdraw the proposal. Had I heard of
any subsequent repentance, I would have taken some other instance, instead
of this: should I yet hear of such a thing, I will take care to notice it
in the continuation of this list, which I confidently expect, life and
health permitting, to be able to make in a few years. This much may be
said, that the author, in a lecture on the subject, given in 1849, and
published with his name, did _not_ repeat the charge.

[The libel was published in the _Mechanics' Magazine_,[252] (vol. for 1846,
pp. 604-615): and the editor supported it as follows, (vol. for 1847, p.
476). In answer to Mr. Sheepshanks's charitable hope that he had been
hoaxed, {142} he says: "Mr. Sheepshanks cannot certainly have read the
article referred to.... Severe and inculpatory it is--unjust some may deem
it (though we ourselves are out of the number.)... A 'hoax' forsooth! May
we be often the dupes of such hoaxes!" He then goes on to describe the
article as directed against the Astronomer Royal's alleged neglect to give
Mr. Adams that "encouragement and protection" which was his due, and _does
not hint one word_ about the article containing the charge of having
secretly and fraudulently transmitted news of Mr. Adams's researches to
France, that an Englishman might not have the honor of the discovery. Mr.
Sheepshanks having called this a "deliberate calumny," without a particle
of proof or probability to support it, the editor says "what the reverend
gentleman means by this, we are at a loss to understand." He then proceeds
_not_ to remember. I repeat here, what I have said elsewhere, that the
management of the journal has changed hands; but from 1846 to 1856, it had
the collar of S.S. (scientific slander). The prayer for more such things
was answered (See p. 349).]


I have said that those who are possessed with the idea of conspiracy
against themselves are apt to imagine both conspirators and their bad
motives and actions. A person who should take up the idea of combination
against himself without feeling ill-will and originating accusations would
be indeed a paradox. But such a paradox has existed. It is very well known,
both in and beyond the scientific world, that the late James Ivory was
subject to the {143} impression of which I am speaking; and the diaries and
other sources of anecdote of our day will certainly, sooner or later, make
it a part of his biography. The consequence will be that to his memory will
be attached the unfavorable impression which the usual conduct of such
persons creates; unless it should happen that some one who knows the real
state of the case puts the two sides of it properly together. Ivory was of
that note in the scientific world which may be guessed from Laplace's
description of him as the first geometer in Britain and one of the first in
Europe. Being in possession of accurate knowledge of his peculiarity in
more cases than one; and in one case under his own hand: and having been
able to make full inquiry about him, especially from my friend the late
Thomas Galloway[254]--who came after him at Sandhurst--one of the few
persons with whom he was intimate:--I have decided, after full
deliberation, to forestall the future biographies.

That Ivory was haunted by the fear of which I have spoken, to the fullest
extent, came to my own public and official knowledge, as Secretary of the
Astronomical Society. It was the duty of Mr. Epps,[255] the Assistant
Secretary, at the time when Francis Baily[256] first announced his
discovery of the Flamsteed Papers, to report to me that Mr. Ivory had
called at the Society's apartments to inquire into the contents of those
papers, and to express his hope that Mr. Baily was not attacking living
persons under the names of Newton and Flamsteed.[257] Mr. Galloway, to whom
I communicated this, immediately went to Mr. Ivory, and succeeded, after
some explanation, in setting him right. This is but one of many instances
in which a man of thoroughly sound judgment in every other respect seemed
to be under a complete chain of delusions about the conduct of {144} others
to himself. But the paradox is this:--I never could learn that Ivory,
passing his life under the impression that secret and unprovoked enemies
were at work upon his character, ever originated a charge, imputed a bad
motive, or allowed himself an uncourteous expression. Some letters of his,
now in my possession, referring to a private matter, are, except in the
main impression on which they proceed, unobjectionable in every point: they
might have been written by a cautious friend, whose object was, if
possible, to prevent a difference from becoming a duel without compromising
his principal's rights or character. Knowing that in some quarters the
knowledge of Ivory's peculiarity is more or less connected with a notion
that the usual consequences followed, I think the preceding statement due
to his memory.


In such a record as the present, which mixes up the grossest speculative
absurdities with every degree of what is better, an instance of another
kind may find an appropriate place. The faults of journalism, when merely
exposed by other journalism pass by and are no more regarded. A distinct
account of an undeniable meanness, recorded in a work of amusement and
reference both, may have its use: such a thing may act as a warning. An
editor who is going to indulge his private grudge may be prevented from
counting upon oblivion as a matter of certainty.

There are three kinds of journals, with reference to the mode of entrance
of contributors. First, as a thing which has been, but which now hardly
exists, there is the journal in which the editor receives a fixed sum to
_find the matter_. In such a journal, every article which the editor can
get a friend to give him is so much in his own pocket, which has a great
tendency to lower the character of the articles; but I am not concerned
with this point. Secondly, there is the journal which is supported by
voluntary contributions of {145} matter, the editor selecting. Thirdly,
there is the journal in which the contributor is paid by the proprietors in
a manner with which the literary editor has nothing to do.

The third class is the safe class, as its editors know: and, as a usual
rule, they refuse unpaid contributions of the editorial cast. It is said
that when Canning[258] declined a cheque forwarded for an article in the
_Quarterly_, John Murray[259] sent it back with a blunt threat that if he
did not take his money he could never be admitted again. The great
publisher told him that if men like himself in position worked for nothing,
all the men like himself in talent who could not afford it would not work
for the _Quarterly_. If the above did not happen between Canning and
Murray, it _must have happened_ between some other two. Now journals of the
second class--and of the first, if such there be--have a fault to which
they alone are very liable, to say nothing of the editorial function (see
the paper at the beginning, p. 11 et seq.), being very much cramped, a sort
of gratitude towards effective contributors leads the journal to help their
personal likes and dislikes, and to sympathize with them. Moreover, this
sort of journal is more accessible than others to articles conveying
personal imputation: and when these provoke discussion, the journal is apt
to take the part of the assailant to whom it lent itself in the first


Among the journals which went all lengths with contributors whom they
valued, was the _Mechanics' Magazine_[260] in the period 1846-56. I cannot
say that matters have not mended in the last ten years: and I draw some
{146} presumption that they have mended from my not having heard, since
1856, of anything resembling former proceedings. And on actual inquiry,
made since the last sentence was written, I find that the property has
changed hands, the editor is no longer the same, and the management is of a
different stamp. This journal is chiefly supported by voluntary articles:
and it is the journal in which, as above noted, the ridiculous charge
against the Astronomer Royal was made in 1849. The following instance of
attempt at revenge is so amusing that I select it as the instance of the
defect which I intend to illustrate; for its puerility brings out in better
relief the points which are not so easily seen in more adult attempts.

The _Mechanics' Magazine_, which by its connection with engineering, etc.,
had always taken somewhat of a mathematical character, began, a little
before 1846, to have more to do with abstract science. Observing this, I
began to send short communications, which were always thankfully received,
inserted, and well spoken of. Any one who looks for my name in that journal
in 1846-49, will see nothing but the most respectful and even laudatory
mention. In May 1849 occurred the affair at the Astronomical Society, and
my share in forcing the withdrawal of the name of the alleged contributor
to the journal. In February 1850 occurred the opportunity of payment. The
_Companion to the Almanac_[261] had to be noticed, in which, as then usual,
was an article signed with my name. I shall give the review of this article
entire, as a sample of a certain style, as well as an illustration of my
point. The reader will observe that my name is not mentioned. This would
not have done; the readers of the Magazine would have stared to see a name
of not infrequent occurrence in previous years all of a sudden fallen from
the heaven of respect into the pit of contempt, like Lucifer, son of the
morning. But before {147} giving the review, I shall observe that Mr.
Adams, in whose _favor_ the attack on the Astronomer Royal was made, did
not appreciate the favor; and of course did not come forward to shield his
champion. This gave deadly offence, as appear from the following passage,
(February 16, 1850):

"It was our intention to enter into a comparison of the contents of our
Nautical Almanack with those of its rival, the _Connaissance des Temps_;
but we shall defer it for the present. The Nautical Almanack for 1851 will
contain Mr. Adams's paper 'On the Perturbation of Uranus'; and when it
comes, in due course, before the public, we are quite sure that that
gentleman will expect that we shall again enter upon the subject with
peculiar delight. Whilst we have a thorough loathing for mean, cowardly,
crawlers--we have an especial pleasure in maintaining the claims of men who
are truly grateful as well as highly talented: Mr. Adams, therefore, will
find that he cannot be disappointed--and the occasion will afford us an
opportunity for making the comparison to which we have adverted."

This passage illustrates what I have said on the editorial function (Vol.
I, p. 15). What precedes and follows has some criticism on the Government,
the Astronomer Royal, etc., but reserved in allusion, oblique in sarcasm,
and not fiercely uncourteous. The coarseness of the passage I have quoted
shows editorial insertion, which is also shown by its blunder. The inserter
is waiting for the Almanac of 1851 that he may review Mr. Adams's paper,
which is to be contained in it. His own contributor, only two sentences
before the insertion, had said, "The Nautical Almanac, we believe, is
published three or four years in advance." In fact, the Almanac for
1851--with Mr. Adams's paper at the end--was published at the end of 1847
or very beginning of 1848; it had therefore been more than two years before
the public when the passage quoted was written. And probably every person
in the country who was fit to review Mr. Adams's {148} paper--and most of
those who were fit to read it--knew that it had been widely circulated, in
revise, at the end of 1846: my copy has written on it, "2d revise, December
27, 1846, at noon," in the handwriting of the Superintendent of the
Almanac; and I know that there was an extensive issue of these revises,
brought out by the Le-Verrier-and-Adams discussion. I now give the review
of myself, (February 23, 1850):

"_The British Almanack and Companion._

"The Companion to this Almanack, for some years after its first
publication, annually contained scientific articles by Sir J. Lubbock[262]
and others of a high order and great interest; we have now, however, closed
the publication as a scientific one in remembrance of what it was, and not
in consequence of what it is. Its list of contributors on science, has
grown 'small by degrees and beautifully less,' until it has dwindled down
to one--'a last rose of summer left withering alone.' The one contributor
has contributed one paper 'On Ancient and Modern Usage in Reckoning.'

"The learned critic's _chef d'oeuvre_, is considered, by competent judges,
to be an Essay on _Old Almanacks_ printed a few years ago in this annual,
and supposed to be written with the view of surpassing a profound memoir on
the same subject by James O. Halliwell,[263] Esq., F.R. and A.S.S., but the
tremendous effort which the learned writer then made to excel many titled
competitors for honors in the antique line appears to have had a sad effect
upon his mental powers--at any rate, his efforts have since yearly become
duller and duller; happily, at last, we should suppose, 'the ancient {149}
and modern usage in reckoning' indicates the lowest point to which the _vis
inertia_ of the learned writer's peculiar genius can force him.

"We will give a few extracts from the article.

"The learned author says, 'Those who are accustomed to settle the meaning
of ancient phrases by self-examination will find some _strange_ conclusions
arrived at by us.' The writer never wrote a more correct sentence--it
admits of no kind of dispute.

"'Language and counting,' says the learned author, 'both came before the
logical discussion of either. It is not allowable to argue that something
is or was, because it ought to be or ought to have been. That two negatives
make an affirmative, ought to be; if _no_ man have done _nothing_, the man
who has done nothing does not exist, and _every_ man has done _something_.
But in Greek, and in uneducated English, it is unquestionable that 'no man
has done nothing' is only an emphatic way of saying that no man has done
_anything_; and it would be absurd to reason that it could not have been
so, because it should not.'--p. 5.

"'But there _is_ another difference between old and new times, yet more
remarkable, for we have _nothing_ of it now: whereas in things indivisible
we count with our fathers, and should say in buying an acre of land, that
the result has no parts, and that the purchaser, till he owns all the
ground, owns none, the change of possession being instantaneous. This
second difference lies in the habit of considering nothing, nought, zero,
cipher, or whatever it may be called, to be at the beginning of the scale
of numbers. Count four days from Monday: we should now say Tuesday,
Wednesday, Thursday, Friday; formerly, it would have been Monday, Tuesday,
Wednesday, Thursday. Had we asked, what at that rate is the first day from
Monday, all would have stared at a phrase they had never heard. Those who
were capable of extending language would have said, Why it must be Monday
itself: the rest would have said, there can {150} be no first day from
Monday, for the day after is Tuesday, which must be the second day: Monday,
one; Tuesday, two,'--p. 10.

"We assure our readers that the whole article is equally lucid, and its
logic alike formal.

"There are some exceedingly valuable footnotes; we give one of the most
interesting, taken from the learned Mr. Halliwell's profound book on
Nursery Rhymes[264]--a celebrated production, for which it is supposed the
author was made F.R.S.

 "'_One's nine_,
  Two's some,
  Three's a many,
  Four's a penny,
  Five's a little hundred.'

'The last line refers to five score, the so-called hundred being more
usually six score. The first line, looked at etymologically, is _one is not
one_, and the change of thought by which _nine_, the decimal of _one_, aims
to be associated with the decimal of _plurality_ is curious:'--Very.

"This valuable and profound essay will very probably be transferred to the
next edition of the learned Mr. Halliwell's rare work, of kindred worth,
entitled 'RARA MATHEMATICA,' it will then be deservedly handed down to
posterity as a covering for cheap trunks--a most appropriate archive for
such a treasure."

In December, 1846, the _Mechanics' Magazine_ published a libel on Airy in
the matter of the discovery of Neptune. In May, 1849, one * * * was to have
been brought forward for election at the Astronomical Society, and was
opposed by me and others, on the ground that he was the probable author of
this libel, and that he would not, perhaps could {151} not, deny it. [N.B.
I no more doubt that he was the author then I doubt that I am the author of
this sentence.][265]

Accordingly, * * * was withdrawn, and a discussion took place, for which
see the _Athenæum_, No. 1126, May 26, 1849, p. 544. The _Mechanics'
Magazine_ was very sore, but up to this day has never ventured beyond an
attack on Airy, private whisperings against Adams--(see _ante_, p.
147),--and the above against myself. In due time, I doubt not my name will
appear as one of the _âmes damnées_[266] of the _Mechanics' Magazine_.[267]


First, as to Mr. Halliwell. The late Thomas Stephens Davies,[268] excellent
in geometry, and most learned in its history, was also a good hand at
enmity, though not implacable. He and Mr. Halliwell, who had long before
been very much one, were, at this date, very much two. I do not think T. S.
Davies wrote this article; and I think that by giving my reasons I shall do
service to his memory. It must have been written at the beginning of
February; and within three days of that time T. S. Davies was making over
to me, by his own free act, to be kept until claimed by the relatives, what
all who knew even his writings knew that he considered as the most precious
deposit he had ever had in his keeping--Horner's[269] papers. His letter
announcing the transmission is dated February 2, 1850. This is a strong
point; but there is another quite as strong. Euclid and {152} his writings
were matters on which T. S. Davies knew neither fear nor favor: he could
not have written lightly about a man who stood high with him as a judge of
Euclid. Now in this very letter of Feb. 2, there is a sentence which I
highly value, because, as aforesaid, it is on a point on which he would
never have yielded anything, to which he had paid life-long attention, and
on which he had the bias of having long stood alone. In fact, knowing--and
what I shall quote confirms me,--that in the matter of Euclid his hand was
against every man, I expected, when I sent him a copy of my 22-column
article, "Eucleides" in Smith's _Dictionary_,[270] to have received back a
criticism, that would have blown me out of the water: and I thought it not
unlikely that a man so well up in the subject might have made me feel
demolished on some points. Instead of this, I got the following: "Although
on one or two minor points I do not quite accord with your views, yet as a
whole and without regard to any minor points, I think you are the first who
has succeeded in a delineation of Euclid as a geometer." All this duly
considered, it is utterly incredible that T. S. Davies should have written
the review in question. And yet Mr. Halliwell is treated just as T. S.
Davies would have treated him, as to tone and spirit. The inference in my
mind is that we have here a marked instance of the joining of hatreds which
takes place in journals supported by voluntary contributions of matter.
Should anything ever have revived this article--and no one ever knows what
might have been fished up from the forgotten mass of journals--the
treatment of Mr. Halliwell would certainly have thrown a suspicion on T. S.
Davies, a large and regular contributor to the Magazine. It is good service
to his memory to point out what makes it incredible that he should have
written so unworthy an article.

The fault is this. There are four extracts: the first {153} three are
perfectly well printed. The printing of the _Mechanics' Magazine_ was very
good. I was always exceedingly satisfied with the manner in which my
articles appeared, without my seeing proof. Most likely these extracts were
printed from my printed paper; if not the extractor was a good copier. I
know this by a test which has often served me. I use the subjunctive--"if
no man _have_ done nothing," an ordinary transcriber, narrating a quotation
almost always lets his own habit write _has_. The fourth extract has three
alterations, all tending to make me ridiculous. _None_ is altered, in two
places, into _nine_, _denial_ into _decimal_, and _comes_ into _aims_; so
that "none, the denial of one, comes to be associated with the denial of
plurality," reads as "nine, the decimal of one, aims to be associated with
the decimal of plurality." This is intentional; had it been a compositor's
reading of bad handwriting, these would not have been the only mistakes; to
say nothing of the corrector of the press. And both the compositor and
reader would have guessed, from the first line being translated into "one
is not one," that it must have been "one's none," not "one's nine." But it
was not intended that the gem should be recovered from the unfathomed cave,
and set in a Budget of Paradoxes.

We have had plenty of slander-paradox. I now give a halfpennyworth of bread
to all this sack, an instance of the paradox of benevolence, in which an
individual runs counter to all the ideas of his time, and sees his way into
the next century. At Amiens, at the end of the last century, an institution
was endowed by a M. de Morgan, to whom I hope I am of kin, but I cannot
trace it; the name is common at Amiens. It was the first of the kind I ever
heard of. It is a Salle d'Asyle for children, who are taught and washed and
taken care of during the hours in which their parents must be at work. The
founder was a large wholesale grocer and colonial importer, who was made a
Baron by Napoleon I for his commercial success and his charities. {154}


1862. Mr. Smith replies to me, still signing himself Nauticus: I give an

"By hypothesis [what, again!] let 14° 24' be the chord of an arc of 15°
[but I wont, says 14° 24'], and consequently equal to a side of a regular
polygon of 24 sides inscribed in the circle. Then 4 times 14° 24' = 57° 36'
= the radius of the circle ..."

That is, four times the chord of an arc is the chord of four times the arc:
and the sum of four sides of a certain pentagon is equal to the fifth. This
is the capital of the column, the crown of the arch, the apex of the
pyramid, the watershed of the elevation. Oh! J. S.! J. S.! groans
Geometry--_Summum J. S. summa injuria_![271] The other J. S., Joseph
Scaliger,[272] as already mentioned, had his own way of denying that a
straight line is always the shortest distance between two points. A
parallel might be instituted, but not in half a column. And J. S. the
_second_ has been so tightly handled that he may now be dismissed, with an
inscription for his circular shield, obtained by changing _Lexica contexat_
into _Circus quadrandus_ in an epigram of J. S. the _first_:

 "Si quem dura manet sententia judicis, olim
  Damnatum ærumnis suppliciisque caput,
  Hunc neque fabrili lassent ergastula massa,
  Nec rigidas vexent fossa metalla manus.
  Circus quadrandus: nam--cætera quid moror?--omnes
  Poenarum facies hic labor unus habet."[273]


I had written as far as _damnatum_ when in came the letter of Nauticus as a
printed slip, with a request that I would consider the slip as a 'revised
copy.' Not a word of alteration in the part I have quoted! And in the
evening came a letter desiring that I would alter a gross error; but not
the one above: this is revising without revision! If there were cyclometers
enough of this stamp, they would, as cultivation progresses--and really,
with John Stuart Mill in for Westminster, it seems on the move, even
though, as I learn while correcting the proof, Gladstone be out from
Oxford; for Oxford is no worse than in 1829, while Westminster is far above
what she ever has been: election time excuses even such a parenthesis as
this--be engaged to amuse those who can afford it with paralogism at their
meals, after the manner of the other jokers who wore the caps and bells.
The rich would then order their dinners with _panem et Circenses_,--up with
the victuals and the circle-games--as the poor did in the days of old.

Mr. Smith is determined that half a column shall not do. Not a day without
something from him: letter, printed proof, pamphlet. In what is the last at
this moment of writing he tells me that part of the title of a work of his
will be "Professor De Morgan in the pillory without hope of escape." And
where will he be himself? This I detected by an effort of reasoning which I
never could have made except by following in his steps. In all matters
connected with [pi] the letters l and g are closely related: this appears
in the well-known formula for the time of oscillation [pi] [sqrt](l : g).
Hence g may be written for l, but only once: do it twice, and you require
the time to be [pi] [sqrt](l^2 : g^2). This may be reinforced by observing
that if as a datum, or if you dislike that word, by hypothesis, the first l
be a g, it is absurd that it should be an l. Write g for the first l, and
we have _un fait accompli_. I shall be in pillory; and overhead, in a
cloud, will sit Mr. James Smith on one stick laid across two others, under
a nimbus of 3-1/8 diameters to {156} the circumference--in [pi]-glory. Oh
for a drawing of this scene! Mr. De Morgan presents his compliments to Mr.
James Smith, and requests the honor of an exchange of photographs.

_July 26._--Another printed letter.--Mr. James Smith begs for a distinct
answer to the following plain question: "Have I not in this communication
brought under your notice _truths_ that were never before dreamed of in
your geometrical and mathematical philosophy?" To which, he having taken
the precaution to print the word _truths_ in italics, I can conscientiously
answer, Yes, you have. And now I shall take no more notice of these
_truths_, until I receive something which surpasses all that has yet been


    The Circle secerned from the Square; and its area gauged in terms of a
    triangle common to both. By Wm. Houlston,[274] Esq. London and Jersey,
    1862, 4to.

Mr. Houlston squares at about four poetical quotations in a page, and
brings out [pi] = 3.14213.... His frontispiece is a variegated diagram,
having parts designated Inigo and Outigo. All which relieves the subject,
but does not remove the error.

    Considerations respecting the figure of the Earth.... By C. F.
    Bakewell.[275] London, 1862, 8vo.

Newton and others think that in a revolving sphere the {157} loose surface
matter will tend to the equator: Mr. Bakewell thinks it will tend to the

    On eccentric and centric force: a new theory of projection. By H. F. A.
    Pratt, M.D.[276] London, 1862, 8vo.

Dr. Pratt not only upsets Newton, but cuts away the very ground he stands
on: for he destroys the first law of motion, and will not have the natural
tendency of matter in motion to be rectilinear. This, as we have seen, was
John Walsh's[277] notion. In a more recent work "On Orbital Motion,"
London, 1863, 8vo., Dr. Pratt insists on another of Walsh's notions,
namely, that the precession of the equinoxes is caused by the motion of the
solar system round a distant central sun. In this last work the author
refers to a few notes, which completely destroy the theory of gravitation
in terms "perfectly intelligible as well to the unlearned as to the
learned": to me they are quite unintelligible, which rather tends to
confirm a notion I have long had, that I am neither one thing nor the
other. There is an ambiguity of phrase which delights a writer on logic,
always on the look-out for specimens of _homonymia_ or _æquivocatio_. The
author, as a physician, is accustomed to "appeal from mere formulæ":
accordingly, he sets at nought the whole of the mathematics, which he does
not understand. This equivocation between the formula of the physician and
that of the mathematician is as good, though not so perceptible to the
world at large, as that made by Mr. Briggs's friend in _Punch's_ picture,
which I cut out to paste into my Logic. Mr. Briggs wrote for a couple of
_bruisers_, meaning to prepare oats for his horses: his friend sent him the
Whitechapel Chicken and the Bayswater Slasher, with the gloves, all ready.


    On matter and ether, and the secret laws of physical change. By T. R.
    Birks, M.A.[278] Cambridge, 1862, 8vo.

Bold efforts are made at molecular theories, and the one before me is ably
aimed. When the Newton of this subject shall be seated in his place, books
like the present will be sharply looked into, to see what amount of
anticipation they have made.


    The history of the 'thorn tree and bush' from the earliest to the
    present time: in which is clearly and plainly shown the descent of her
    most gracious Majesty and her Anglo-Saxon people from the half tribe of
    Ephraim, and possibly from the half tribe of Manasseh; and consequently
    her right and title to possess, at the present moment, for herself and
    for them, a share or shares of the desolate cities and places in the
    land of their forefathers! By Theta, M.D.[279] (Private circulation.)
    London, 1862, 8vo.

This is much about _Thorn_, and its connected words, Thor, Thoth, Theta,
etc. It is a very mysterious vagary. The author of it is the person whom I
have described elsewhere as having for his device the round man in the
three-cornered hole, the writer of the little heap of satirical anonymous
letters about the Beast and 666. By accident I discovered the writer: so
that if there be any more thorns to crackle under the pot, they need not be

Nor will they be anonymous. Since I wrote the above, I have received
_onymous_ letters, as _ominous_ as the rest. The writer, William Thorn,
M.D., is obliged to reveal {159} himself, since it is his object to prove
that he himself is one 666. By using W for double Vau (or 12) he cooks the
number out of his own name. But he says it is the number not of a beast but
of a man, and adds, "Thereby hangs a tale!" which sounds like
contradiction. He informs me that he will talk the matter over with me: but
I shall certainly have nothing to say to a gentleman of his number; it is
best to keep on the safe side.

In one letter I am informed that not a line should I have had, but for my
"sneer at 666," which, therefore, I am well pleased to have given. I am
also told that my name means the "'garden of death,' that place in which
the tree of knowledge was plucked, and so you are like your name 'dead' to
the fact that you are an Israelite, like those in Ezekiel 37 ch." Some
hints are given that I shall not fare well in the next world, which any one
who reads the chapter in Ezekiel will see is quite against his comparison.
The reader must not imagine that my prognosticator means _Morgan_ to be a
corruption of _Mortjardin_; he proves his point by Hebrew: but any
philologist would tell him the true derivation of the name, and how
_Glamorgan_ came to get it. It will be of much comfort to those young men
who have not got through to know that the tree of knowledge itself was once
in the same case. And so good bye to 666 for the present, and the
assumption that the enigma is to be solved by the united numeral forces of
the letters of a word.

It is worthy of note that, as soon as my Budget commenced, two guardian
spirits started up, fellow men as to the flesh, both totally unknown to me:
they have stuck to me from first to last. James Smith, Esq., finally
Nauticus, watches over my character in this world, and would fain preserve
me from ignorance, folly, and dishonesty, by inclosing me in a magic circle
of 3-1/8 diameters in circumference. The round man in the three-cornered
hole, finally William Thorn, M.D., takes charge of my future destiny, {160}
and tries to bring me to the truth by unfolding a score of meanings--all
right--of 666. He hints that I, and my wife, are servants of Satan: at
least he desires us both to remember that we cannot serve God and Satan;
and he can hardly mean that we are serving the first, and that he would
have us serve the second. As becomes an interpreter of the Apocalypse, he
uses seven different seals; but not more than one to one letter. If his
seals be all signet-rings, he must be what Aristophanes calls a
sphragidonychargocometical fellow. But--and many thanks to him for the
same--though an M.D., he has not sent me a single vial. And so much for my
tree of secular knowledge and my tree of spiritual life: I dismiss them
with thanks from myself and thanks from my reader. The dual of the
Pythagorean system was Isis and Diana; of the Jewish law, Moses and Aaron;
and of the City of London, Gog and Magog; of the Paradoxiad, James Smith,
Esq., and William Thorn, M.D.

_September, 1866._ Mr. James Biden[280] has favored me with some of his
publications. He is a rival of Dr. Thorn; a prophet by name-right and
crest-right. He is of royal descent through the De Biduns. He is the
_watchman_ of Ezekiel: God has told him so. He is the author of _The True
Church_, a phrase which seems to have a book-meaning and a mission-meaning.
He shall speak for himself:

"A crest of the Bidens has significance. It is a lion rampant between
wings--wings in Scripture denote the flight of time. Thus the beasts or
living creatures of the Revelations have each six wings, intimating a
condition of mankind up to and towards the close of six thousand years of
Bible teaching. The two wings of the crest would thus intimate power
towards the expiration of 2000 years, as time is marked in the history of
Great Britain.


"In a recent publication, _The Pestilence, Why Inflicted_, are given many
reasons why the writer thinks himself to be the appointed watchman foretold
by Ezekiel, chapters iii. and xxxiii. Among the reasons are many prophecies
fulfilled in him. Of these it is now needful to note two as bearing
especially on the subject of the reign of Darius.

"1.--In Daniel it is said, 'Darius the Median took the kingdom, being about
threescore and two years old.'--Daniel v. 31.

"When 'Belshazzar' the king of the Chaldeans is found wanting, Darius takes
the kingdom. It is not given him by the popular voice; he asserts his
right, and this is not denied. He takes it when about sixty-two years of
age. The language of Daniel is prophetic, and Darius has in another an
antitype. The writer was born July 18th, 1803; and the claim was asserted
at the close of 1865, when he was about sixty-two years of age.

"The claims which have been asserted demand a settled faith, and which
could only be reached through a long course of divine teaching."

When I was a little boy at school, one of my school-fellows took it into
his head to set up a lottery of marbles: the thing took, and he made a
stony profit. Soon, one after another, every boy had his lottery, and it
was, "I won't put into yours unless you put into mine." This knocked up the
scheme. It will be the same with the prophets. Dr. Thorn, Mr. Biden, Mrs.
Cottle,[281] etc. will grow imitators, until we are all pointed out in the
Bible: but A will not admit B's claim unless B admits his. For myself, as
elsewhere shown, I am the first Beast in the Revelations.

Every contraband prophet gets a few followers: it is a great point to make
these sequacious people into Buridan's asses, which they will become when
prophets are so numerous that there is no choosing.



    An historical survey of the Astronomy of the Ancients. By the Rt. Hon.
    Sir G. C. Lewis.[282] 8vo. 1862.

There are few men of our day whom I admire more than the late Sir G. Lewis:
he was honest, earnest, sagacious, learned, and industrious. He probably
sacrificed his life to his conjunction of literature and politics: and he
stood high as a minister of state in addition to his character as a man of
letters. The work above named is of great value, and will be read for its
intrinsic merit, consulted for its crowd of valuable references, quoted for
its aid to one side of many a discussion, and opposed for its force against
the other. Its author was also a wit and a satirist. I know of three
classical satires of our day which are inimitable imitations: Mr.
Malden's[283] _Pragmatized Legends_, Mr. Mansel's[284] _Phrontisterion_,
and Sir G. Cornewall Lewis's _Inscriptio Antiqua_. In this last,
HEYDIDDLEDIDDLETHECATANDTHEFIDDLE etc. is treated as an Oscan inscription,
and rendered into Latin by approved methods. As few readers have seen it, I
give the result:

    "Hejus dedit libenter, dedit libenter. Deus propitius [est], deus
    [donatori] libenter favet. Deus in viarum {163} juncturâ ovorum dape
    [colitur], deus mundi. Deus in litatione voluit, benigno animo, hædum,
    taurum intra fines [loci sacri] portandos. Deus, bis lustratus, beat
    fossam sacræ libationis."[285]

How then comes the history of astronomy among the paradoxes? Simply because
the author, so admirably when writing about what he knew, did not know what
he did not know, and blundered like a circle-squarer. And why should the
faults of so good a writer be recorded in such a list as the present? For
three reasons: First, and foremost, because if the exposure be not made by
some one, the errors will gradually ooze out, and the work will get the
character of inaccurate. Nothing hurts a book of which few can fathom the
depths so much as a plain blunder or two on the surface. Secondly, because
the reviews either passed over these errors or treated them too gently,
rather implying their existence than exposing them. Thirdly, because they
strongly illustrate the melancholy truth, that no one knows enough to write
about what he does not know. The distinctness of the errors is a merit; it
proceeds from the clear-headedness of the author. The suppression in the
journals may be due partly to admiration of the talent and energy which
lived two difficult lives at once, partly to respect for high position in
public affairs, partly to some of the critics being themselves men of
learning only, unable to detect the errors. But we know that action and
reaction are equal and contrary. If our generation take no notice of
defects, and allow them to go down undetected among merits, the next
generation will discover them, will perhaps believe us incapable of
detecting them, at least will pronounce our judgment good for nothing, and
will form an {164} opinion in which the merits will be underrated: so it
has been, is, and will be. The best thing that can be done for the memory
of the author is to remove the unsound part that the remainder may thrive.
The errors do not affect the work; they occur in passages which might very
well have been omitted: and I consider that, in making them conspicuous, I
am but cutting away a deleterious fungus from a noble tree.

(P. 154). The periodic times of the five planets were stated by
Eudoxus,[286] as we learn from Simplicius;[287] the following is his
statement, to which the true times are subjoined, for the sake of

  Mercury          1 year             --    87d. 23h.
  Venus            1   "              --   224d. 16h.
  Mars             2   "              1y.  321d. 23h.
  Jupiter         12   "             11y.  315d. 14h.
  Saturn          30   "             29y.  174d.  1h.

Upon this determination two remarks may be made. First, the error with
respect to Mercury and Venus is considerable; with respect to Mercury, it
is, in round numbers, 365 instead of 88 days, more than four times too
much. Aristotle remarks that Eudoxus distinguishes Mercury and Venus from
the other three planets by giving them one sphere each, with the poles in
common. The proximity of Mercury to the sun would render its course
difficult to observe and to measure, but the cause of the large error with
respect to Venus (130 days) is not apparent.


Sir G. Lewis takes Eudoxus as making the planets move round the sun; he has
accordingly compared the _geocentric_ periods of Eudoxus with our
_heliocentric_ periods. What greater blunder can be made by a writer on
ancient astronomy than giving Eudoxus the Copernican system? If Mercury
were a black spot in the middle of the sun it would of course move round
the earth in a year, or appear to do so: let it swing a little on one side
and the other of the sun, and the average period is still a year, with
slight departures both ways. The same for Venus, with larger departures.
Say that a person not much accustomed to the distinction might for once
write down the mistake; how are we to explain its remaining in the mind in
a permanent form, and being made a ground for such speculation as that of
the difficulty of observing Mercury leading to a period four times what it
ought to be, corrected in proof and published by an industrious and
thoughtful person? Only in one way: the writer was quite out of his depth.
This one case is conclusive; be it said with all respect for the real
staple of the work and of the author. He knew well the difference of the
systems, but not the effect of the difference: he is another instance of
what I have had to illustrate by help of a very different person, that it
is difficult to reason well upon matter which is not familiar.

(P. 254). Copernicus, in fact, supposed the axis of the earth to be always
turned towards the Sun.^{(169)} [(169). See Delambre, _Hist. Astr. Mod._,
Vol. I, p. 96]. It was reserved to Kepler to propound the hypothesis of the
constant parallelism of the earth's axis to itself.

If there be one thing more prominent than another in the work of Copernicus
himself, in the popular explanations of it, and in the page of
Delambre[288] cited, it is that the _parallelism of the earth's axis_ is a
glaring part of the {166} theory of Copernicus. What Kepler[289] did was to
throw away, as unnecessary, the method by which Copernicus, _per fas et
nefas_,[290] secured it. Copernicus, thinking of the earth's orbital
revolution as those would think who were accustomed to the _solid
orbs_--and much as the stoppers of the moon's rotation do now: why do they
not strengthen themselves with Copernicus?--thought that the earth's axis
would always incline the same end towards the sun, unless measures were
taken to prevent it. He _did_ take measures: he invented a _compensating_
conical motion of the axis to preserve the parallelism; and, which is one
of the most remarkable points of his system, he obtained the precession of
the equinoxes by giving the necessary trifle more than compensation. What
stares us in the face at the beginning of the paragraph to which the author

"C'est donc pour arriver à ce parallelisme, ou pour le conserver, que
Copernic a cru devoir recourir à ce mouvement égal et opposé qui détruit
l'effet qu'il attribue si gratuitement au premier, de déranger le

Parallelism at any price, is the motto of Copernicus: you need not pay so
dear, is the remark of Kepler.

The opinions given by Sir G. Lewis about the effects of modern astronomy,
which he does not understand and singularly undervalues, will now be seen
to be of no authority. He fancies that--to give an instance--for the
determination of a ship's place, the invention of chronometers has been far
more important than any improvement in astronomical theory (p. 254). Not to
speak of latitude,--though the omission is not without importance,--he
ought to have known that longitude is found by the difference between what
o'clock it is at Greenwich and at the ship's place, at {167} one absolute
moment of time. Now if a chronometer were quite perfect--which no
chronometer is, be it said--and would truly tell Greenwich mean time all
over the world, it ought to have been clear that just as good a watch is
wanted for the time at _the place of observation_, before the longitude of
that place with respect to Greenwich can be found. There is no such watch,
except the starry heaven itself: and that watch can only be read by
astronomical observation, aided by the best knowledge of the heavenly

I think I have done Sir G. Lewis's very excellent book more good than all
the reviewers put together.

I will give an old instance in which literature got into confusion about
astronomy. Theophrastus,[292] who is either the culprit or his historian,
attributes to Meton,[293] the contriver of the lunar calendar of nineteen
years, which lasts to this day, that his solstices were determined for him
by a certain Phaeinus of Elis on Mount Lycabettus. Nobody else mentions
this astronomer: though it is pretty certain that Meton himself made more
than one appointment with him for the purpose of observing solstices; and
we may be sure that if either were behind his time, it was Meton. For
_Phaeinus Helius_ is the shining sun himself; and in the astronomical poet
Aratus[294] we read about the nineteen years of the shining sun:

  [Greek: Enneakaideka kukla phaeinou êelioio].[295]

Some man of letters must have turned Apollo into Phaeinus of Elis; and
there he is in the histories of astronomy to {168} this day. Salmasius[296]
will have Aratus to have meant him, and proposes to read [Greek: êleioio]:
he did not observe that Phaeinus is a very common adjective of Aratus, and
that, if his conjecture were right, this Phaeinus would be the only
non-mythical man in the poems of Aratus.

[When I read Sir George Lewis's book, the points which I have criticized
struck me as not to be wondered at, but I did not remember why at the time.
A Chancellor of the Exchequer and a writer on ancient astronomy are birds
of such different trees that the second did not recall the first. In 1855 I
was one of a deputation of about twenty persons who waited on Sir G. Lewis,
as Chancellor of the Exchequer, on the subject of a decimal coinage. The
deputation was one of much force: Mr. Airy, with myself and others,
represented mathematics; William Brown,[297] whose dealings with the United
States were reckoned by yearly millions, counted duodecimally in England
and decimally in America, was the best, but not the only, representative of
commerce. There were bullionists, accountants, retailers, etc. Sir G. L.
walked into the room, took his seat, and without waiting one moment, began
to read the deputation a smart lecture on the evils of a decimal coinage;
it would require alteration of all the tables, it would impede calculation,
etc. etc. Of those arguments against it which weighed with many of better
knowledge than his, he obviously knew nothing. The members of the
deputation began to make their statements, and met with curious denials. He
interrupted me with "Surely there is no doubt that the calculations of our
books of arithmetic are easier {169} than those in the French books." He
was not aware that the _universally admitted_ superiority of decimal
_calculation_ made many of those who prefer our system for the market and
the counter cast a longing and lingering look towards decimals. My answer
and the smiles which he saw around, made him give a queer puzzled look,
which seemed to say, "I may be out of my depth here!" His manner changed,
and he listened. I saw both the slap-dash mode in which he dealt with
subjects on which he had not thought, and the temperament which admitted
suspicion when the means of knowledge came in his way. Having seen his two
phases, I wonder neither at his more than usual exhibition of shallowness
when shallow, nor at the intensity of the contrast when he had greater


Among the paradoxers are the political paradoxers who care not how far they
go in debate, their only object being to carry the House with them for the
current evening. What I have said of editors I repeat of them. The
preservation of a very marked instance, the association of political
recklessness with cyclometrical and Apocalyptic absurdity, may have a
tendency to warn, not indeed any hardened public-man and sinner, but some
young minds which have yearnings towards politics, and are in formation of

In the debate on decimal coinage of July 12, 1855, Mr. Lowe,[298] then
member for Kidderminster, an effective speaker and a smart man, exhibited
himself in a speech on which I wrote a comment for the Decimal Association.
I have seldom seen a more wretched attempt to distort the points of a
public question than the whole of this speech. Looking at the intelligence
shown by the speaker on other occasions, {170} it is clear that if charity,
instead of believing all things, believed only all things but one, he might
tremble for his political character; for the honesty of his intention on
this occasion might be the incredible exception. I give a few paragraphs
with comments:

"In commenting on the humorous, but still argumentative speech of Mr. Lowe,
the member for Kidderminster, we may observe, in general, that it consists
of points which have been several times set forth, and several times
answered. Mr. Lowe has seen these answers, but does not allude to them, far
less attempt to meet them. There are, no doubt, individuals, who show in
their public speaking the outward and visible signs of a greater degree of
acuteness than they can summon to guide their private thinking. If Mr. Lowe
be not one of these, if the power of his mind in the closet be at all
comparable to the power of his tongue in the House, it may be suspected
that his reserve with respect to what has been put forward by the very
parties against whom he was contending, arises from one or both of two
things--a high opinion of the arguments which he ignored--a low opinion of
the generality of the persons whom he addressed. [Both, I doubt not].

"Did they calculate in florins  In the name of common sense,
?"                              how can it be objected to a
                                system that people do not use
                                it before it is introduced?
                                Let the decimal system be
                                completed, and calculation
                                shall be made in florins; that
                                is, florins shall take their
                                proper place. If florins were
                                introduced _now_, there must
                                be a column for the odd
"He was glad that some hon.     If the hon. gentleman make
gentleman had derived benefit   this assertion of himself, it
from the issue of florins. His  is not for us to gainsay it.
only experience of their        It only proves that he is one
convenience was, that when he   of that class of {171} men who
ought to have received          are described in the old song,
half-a-crown, he had generally  of which one couplet runs
received a florin, and when he  thus:
ought to have paid a florin,
he had generally paid               I sold my cow to buy me a
half-a-crown." (Hear, hear,     calf;
and laughter.)                  I never make a bargain but I
                                lose half,
                                With a etc. etc. etc.

But he cannot mean that Englishmen in general are so easily managed. And as
to Jonathan, who is but John lengthened out a little, he would see creation
whittled into chips before he would even split what may henceforth be
called the Kidderminster difference. The House, not unmoved--for it
laughed--with sly humor decided that the introduction of the florin had
been "eminently successful and satisfactory."

The truth is that Mr. Lowe here attacks nothing except the coexistence of
the florin and half-crown. We are endeavoring to abolish the half-crown.
Let Mr. Lowe join us; and he will, if we succeed, be relieved from the
pressure on his pocket which must arise from having the turn of the market
always against him.

"From a florin they get to 2    Note the sophism of expressing
2-5ths of a penny, but who      our coin in terms of the
ever bought anything, who ever  penny, which we abandon,
reckoned or wished to reckon    instead of the florin, which
in such a coin as that?"        we retain. Remember that this
(Hear, hear.)                   2 2-5ths is the hundredth part
                                of the pound, which is called,
                                as yet, a _cent_. Nobody buys
                                anything at a cent, because
                                the cent is not yet
                                introduced. Nobody reckons in
                                cents for the same reason.
                                Everybody wishes to reckon in
                                cents, who wishes to combine
                                the advantage of decimal
                                reckoning with the
                                preservation of the pound as
                                {172} the highest unit of
                                account; amongst others, a
                                majority of the House of
                                Commons, the Bank of England,
                                the majority of London
                                bankers, the Chambers of
                                Commerce in various places,
                                etc. etc. etc.
"Such a coin could never come   Does 2½d. never pass from hand
into general circulation        to hand? And is 2½d. so
because it represents nothing   precisely the modulus of
which corresponds with any of   popular wants, that an
the wants of the people."       alteration of 4 per cent.
                                would make it useless? Of all
                                the values which 2½d.
                                measures, from three pounds of
                                potatoes down to certain
                                arguments used in the House of
                                Commons, there is not one for
                                which a cent would not do just
                                as well. Mr. Lowe has fallen
                                into the misconception of the
                                person who admired the
                                dispensation of Providence by
                                which large rivers are made to
                                run through cities so great
                                and towns so many. If the cent
                                were to be introduced
                                to-morrow, straightway the
                                buns and cakes, the soda-water
                                bottles, the short omnibus
                                fares, the bunches of
                                radishes, etc. etc. etc.,
                                would adapt themselves to the
"If the proposed system were    The confusion of ideas here
adopted, they would all be      exhibited is most instructive.
compelled to live in decimals   The speaker is under the
for ever; if a man dined at a   impression that _we_ are
public house he would have to   introducing fractions: the
pay for his dinner in decimal   truth is, that we only want to
fractions. (Hear, hear.) He     abandon the _more difficult_
objected to that, for he        fractions which we _have got_,
thought that a man ought to be  and to introduce _easier
able to pay for his dinner in   fractions_. Does he deny this?
integers." (Hear, hear, and a   Let us trace his denial to its
laugh.)                         legitimate consequences. A man
                                ought to pay for his dinner in


Now, if Mr. Lowe insists on it that our integer is the pound, he is bound
to admit that the present integer is the pound, of which a shilling, etc.,
are fractions. The next time he has a chop and a pint of stout in the city,
the waiter should say--"A pound, sir, to you," and should add, "Please to
remember the waiter in integers." Mr. Lowe fancies that when he pays one
and sixpence, he pays in integers, and so he does, if his integer be a
penny or a sixpence. Let him bring his mind to contemplate a mil as the
integer, the lowest integer, and the seven cents five mils which he would
pay under the new system would be payment in integers also. But, as it
happens with some others, he looks _up_ the present system, with
Cocker,[299] and Walkingame,[300] and always looks _down_ the proposed
system. The word _decimal_ is obstinately associated with _fractions_, for
which there is no need. Hence it becomes so much of a bugbear, that, to
parody the lines of Pope, which probably suggested one of Mr. Lowe's

 "Dinner he finds too painful an endeavor,
  Condemned to pay in decimals for ever."

"The present system, however,   A pleasant sum even for an
had not yet been changed into   accomplished mathematician.
decimal system. That change     What does divided by the
might appear very easy to       decimal of a pound mean?
accomplished mathematicians     Perhaps it means _reduced_ to
and men of science, but it was  the decimal of a pound! Mr.
one which it would be very      Lowe supposes, as many others
difficult to carry out. (Hear,  do, that, after the change,
hear). What would have to be    all calculations will be
done? Every sum would have to   _proposed in old money_, and
be reduced into a vulgar        then _converted into new_. He
fraction of a pound, and then   cannot hit the {174} idea that
divided by the decimal of a     the new coins will take the
pound--a pleasant sum for an    place of the old. This lack of
old applewoman to work out!"    apprehension will presently
(Hear, hear, and laughter.)     appear further.
"It would not be an agreeable   Let the members be assured
task, even for some members of  that nine half-pence will be,
that House, to reduce 4½d., or  for every practical purpose,
nine half-pence, to mils."      18 mils. But now to the fact
(Hear, hear.)                   asserted. Davies Gilbert[301]
                                used to maintain that during
                                the long period he sat in the
                                House, he never knew more than
                                three men in it, at one time,
                                who had a tolerable notion of
                                fractions. [I heard him give
                                the names of three at the time
                                when he spoke: they were
                                Warburton,[302] Pollock,[303]
                                and Hume.[304] He himself was
                                then out of Parliament.]
                                Joseph Hume affirmed that he
                                had never met with more than
                                ten members who were
                                arithmeticians. But both these
                                gentlemen had a high standard.
                                Mr. Lowe has given a much more
                                damaging opinion. He evidently
                                means that the general run of
                                members could not do his
                                question. It is done as
                                follows: Since farthings gain
                                on mils, at the rate of a
                                whole mil in 24 farthings (24
                                farthings being 25 mils), it
                                is clear that 18 farthings
                                being three-quarters of 24
                                farthings, will gain
                                three-quarters of a mil; that
                                is, 18 farthings are eighteen
                                {175} mils and three-quarters
                                of a mil. Any number of
                                farthings is as many mils and
                                as many twenty-fourths of a
                                mil. To a certain extent, we
                                feel able to protest against
                                the manner in which
                                Kidderminster has treated the
                                other constituencies. We do
                                not hold it impossible to give
                                the Members of the House in
                                general a sufficient knowledge
                                of the meaning and
                                consequences of the _decimal_
                                succession of units, tens,
                                hundreds, thousands, etc.; and
                                we believe that there are in
                                the House itself competent
                                men, in number enough to teach
                                all the rest. All that is
                                wanted is the power of
                                starting from the known to
                                arrive at the unknown. Now
                                there is one kind of decimals
                                with which every member is
                                acquainted--the _Chiltern
                                Hundreds_. If public opinion
                                would enable the competent
                                minority to start from this in
                                their teaching, not as a
                                basis, but as an alternative,
                                in three weeks the
                                fundamentals would be
                                acquired, and members in
                                general would be as fit to
                                turn 4½d. into mils, as any
                                boys on the lower forms of a
                                commercial school.

                                For a long period of years,
                                allusion to the general
                                ignorance of arithmetic, has
                                been a standing mode of
                                argument, and has always been
                                well received: whenever one
                                member describes others as
                                _knownothings_, those others
                                cry _Hear_ to the country in a
                                transport of delight. In the
                                meanwhile the country is
                                gradually arriving at the
                                conclusion that a true joke is
                                no joke.
"The main objection was, if     Fine words, wrongly used. The
they went below 6d., that the   new coins are commensurable
new scale of coins would not    with, and in a finite ratio
be commensurate in any finite   to, the old ones. The farthing
ratio with anything in this     is to the mil as 25 to 24. The
new currency of mils."          speaker has something here in
                                the bud, which we shall
                                presently meet with in the
                                flower; and fallacies are more
                                easily nipped in flower than
                                in bud. {176}
"No less than five of our       This dreadful change of value
present coins must be called    consists in sixpence farthing
in, or else--which would be     going to the half-shilling
worse--new values must be       instead of sixpence. Whether
given to them."                 the new farthings be called
                                mils or not is of no
"If a poor man put a penny in   Mr. Lowe, who cannot pass a
his pocket, it would come out   half-crown for more than a
a coin of different value,      florin, or get in a florin at
which he would not understand.  less than half-a-crown, has
Suppose he owed another man a   such a high faith in the
penny, how was he to pay him ?  sterner stuff of his fellow
Was he to pay him in mils?      countrymen, that he believes
Four mils would be too little,  any two of them would go to
and five mils would be too      fisty cuffs for the 25th part
much. The hon. gentlemen said   of a farthing. He reasons
there would be only a mil       thus: He has often heard in
between them. That was exactly  the streets, "I'd fight you
it. He believed there would be  for the fiftieth part of a
a 'mill' between them." (Much   farden:" and having (that is,
laughter.)                      for a Member) a notion both of
                                fractions and logic, he infers
                                that those who would fight for
                                the 50th of a farthing would,
                                _a fortiori_, fight for a
                                25th. His mistake arises from
                                his not knowing that when a
                                person offers to fight another
                                for 1/200d., he really means
                                to fight for love; and that
                                the stake is merely a matter
                                of form, a feigned issue, a
                                _pro forma_ report of
                                progress. Do the Members of
                                the House think they have all
                                the forms to themselves?
"What would be the present      We should hardly believe all
expression for four-pence?      this to be uttered in earnest,
Why, 0.166 (a laugh); for       if we had not known {177} that
threepence? .0125; for a        several persons who have not
penny? .004166, and so on _ad   Mr. Lowe's humor, nevertheless
infinitum_ (a laugh); for a     have his impressions on this
half-penny? .002083 _ad         point. It must therefore be
infinitum_. (A laugh). What     answered; but how is this to
would be the present            be done seriously?
expression for a farthing?
Why, .0010416 _ad infinitum_.   _Dialogue between a member of
(A laugh). And this was the     Parliament and an orange-boy,
system which was to cause such  three days after the
a saving in figures, and these  introduction of the complete
were the quantities into which  decimal system. The member,
the poor would have to reduce   going down to the House, wants
the current coin of the realm.  oranges to sustain his voice
(Cheers). With every respect    in a two hours' speech on
for decimal fractions, of       moving that 100000l. be placed
which he boasted no profound    at the disposal of Her
knowledge, he doubted whether   Majesty, to supply the poor
the poor were equal to mental   with ready-reckoners._
arithmetic of this kind,
(hear, hear) and he hoped the   _Boy._ Fine oranges! two a
adoption of the system would    penny! two a penny! {178}
be deferred until there were
some proof that they would be   _Member._ Here boy, two! Now,
able to understand it; for,     how am I to pay you?
after all, this was the
question of the poor, and the   _Boy._ Give you change, your
whole weight of the change      honor.
would fall upon them. Let the
rich by all means have          _Member._ Ah! but how? Where's
permission to perplex           your ready-reckoner?
themselves by any division of
a pound they pleased; but do    _Boy._ I sells a better sort
not let them, by any            nor them. Mine's real Cheyny.
experiment like this, impose
difficulties upon the poor and  _Member._ But you see a
compel men to carry             farthing is now .0014166666
ready-reckoners in their        _ad infinitum_, and if we
pocket to give them all these   multiply this by 4----
fractional quantities." (Hear,
hear.)                          _Boy._ Hold hard, Guv'ner; I
                                sees what you're arter. Now
                                what'll you stand if I puts
                                you up to it? which Bill Smith
                                he put me up in two minutes,
                                cause he goes to the Ragged

                                _Member._ You don't mean that
                                you do without a book!

                                _Boy._ Book be blowed. Come
                                now, old un, here's summut for
                                both on us. I got a florin,
                                you gives me a half-a-crown
                                for it, and I larns you the
                                new money, gives you your
                                oranges, and calls you a brick
                                into the bargain.

                                _Member_ (_to himself_). Never
                                had such a chance of getting
                                off half-a-crown for value
                                since that ---- fellow Bowring
                                carried his crochet.
                                (_Aloud._) Well, boy, it's a
                                bargain. Now!

                                _Boy._ Why, look 'e here, my
                                trump, its a farden more to
                                the tizzy--that's what it is.

                                _Member_. What's that?

                                _Boy._ Why, you knows a
                                sixpence when you sees it.
                                (_Aside_). Blest if I think he
                                does! Well, its six browns and
                                a farden now. A lady buys two
                                oranges, and forks {179} out a
                                sixpence; well in coorse, I
                                hands over fippence farden
                                astead of fippence. I always
                                gives a farden more change,
                                and takes according.

                                _Member_ (_in utter surprise,
                                lets his oranges tumble into
                                the gutter_). Never mind! They
                                won't be wanted now. (_Walks
                                off one way. Boy makes a pass
                                of naso-digital mesmerism, and
                                walks off the other way_).

To the poor, who keep no books, the whole secret is "Sixpence farthing to
the half shilling, twelve pence halfpenny to the shilling." The _new
twopence halfpenny_, or cent, will be at once five to the shilling.

In conclusion, we remark that three very common misconceptions run through
the hon. Member's argument; and, combined in different proportions, give
variety to his patterns.

First, he will have it that we design to bring the uneducated into contact
with _decimal fractions_. If it be so, it will only be as M. Jourdain was
brought into contact with prose. In fact, _Quoi! quand je dis, Nicole,
apportez-moi mes pantoufles, c'est de la prose?_[305] may be rendered:
"What! do you mean that _ten to the florin is a cent a piece_ must be
called decimal reckoning?" If we had to comfort a poor man, horror-struck
by the threat of _decimals_, we should tell him what manner of fractions
had been inflicted upon him hitherto; nothing less awful than
_quarto-duodecimo-vicesimals_, we should assure him.

Secondly, he assumes that the penny, such as it now is, will remain, as a
coin of estimation, after it has ceased to be a coin of exchange; and that
the mass of the people will continue to think of prices in old pence, and
to calculate them in new ones, or else in new mils. No answer is required
to this, beyond the mere statement of the nature of the assumption and


Thirdly, he attributes to the uneducated community a want of perception and
of operative power which really does not belong to them. The evidence
offered to the Committee of the House shows that no fear is entertained on
this point by those who come most in contact with farthing purchasers. And
this would seem to be a rule,--that is, fear of the intelligence of the
lower orders in the minds of those who are not in daily communication with
them, no fear at all in the minds of those who are.

A remarkable instance of this distinction happened five-and-twenty years
ago. The Admiralty requested the Astronomical Society to report on the
alterations which should be made in the _Nautical Almanac_, the seaman's
guide-book over the ocean. The greatest alteration proposed was the
description of celestial phenomena in _mean_ (or clock time), instead of
_apparent_ (or sundial) time, till then always employed. This change would
require that in a great many operations the seaman should let alone what he
formerly altered by addition or subtraction, and alter by addition or
subtraction what he formerly let alone; provided always that what he
formerly altered by addition he should, when he altered at all, alter by
subtraction, and _vice versa_. This was a tolerably difficult change for
uneducated skippers, working by rules they had only learned by rote. The
Astronomical Society appointed a Committee of forty, of whom nine were
naval officers or merchant seamen [I was on this Committee]. Some men of
science were much afraid of the change. They could not trust an ignorant
skipper or mate to make those alterations in their routine, on the
correctness of which the ship might depend. Had the Committee consisted of
men of science only, the change might never have been ventured on. But the
naval men laughed, and said there was nothing to fear; and on their
authority the alteration was made. The upshot was, that, after the new
almanacs appeared, not a word of complaint was ever heard on the matter.
Had the House of Commons had to {181} decide this question, with Mr. Lowe
to quote the description given by Basil Hall[306] (who, by the way, was one
of the Committee) of an observation on which the safety of the ship
depended, worked out by the light of a lantern in a gale of wind off a lee
shore, this simple and useful change might at this moment have been in the
hands of its tenth Government Commission.

[_Aug. 14, 1866._ The Committee was appointed in the spring of 1830: it
consisted of forty members. Death, of course, has been busy; there are now
left Lord Shaftesbury,[307] Mr. Babbage,[308] Sir John Herschel,[309] Sir
Thomas Maclear[310] (Astronomer Royal at the Cape of Good Hope), Dr.
Robinson[311] (of Armagh), Sir James South,[312] Lord Wrottesley,[313] and



    Project of a new system of arithmetic, weight, measure, and coins,
    proposed to be called the tonal system, with sixteen to the base. By
    J. W. Mystrom.[314] Philadelphia, 1862, 8vo.

That is to say, sixteen is to take the place of ten, and to be written 10.
The whole language is to be changed; every man of us is to be
sixteen-stringed Jack and every woman sixteen-stringed Jill. Our old _one_,
_two_, _three_, up to sixteen, are to be (_Noll_ going for nothing, which
will please those who dislike the memory of _Old Noll_) replaced by An, De,
Ti, Go, Su, By, Ra, Me, Ni, Ko, Hu, Vy, La, Po, Fy, Ton; and then Ton-an,
Ton-de, etc. for 17, 18, etc. The number which in the system has the symbol


(using our present compounds instead of new types) is to be pronounced


The year is to have sixteen months, and here they are:

  Anuary, Debrian, Timander, Gostus,
  Suvenary, Bylian, Ratamber, Mesudius,
  Nictoary, Kolumbian, Husamber, Vyctorius,
  Lamboary, Polian, Fylander, Tonborius.

Surely An-month, De-month, etc. would do as well. Probably the wants of
poetry were considered. But what are we to do with our old poets? For

 "It was a night of lovely June,
  High rose in cloudless blue the moon."

Let us translate--

 "It was a night of lovely Nictoary,
  High rose in cloudless blue the (what, in the name of all that is

And again, _Fylander_ thrown into our December! What is {183} to become of
those lines of Praed, which I remember coming out when I was at

 "Oh! now's the time of all the year for flowers and fun, the Maydays;
  To trim your whiskers, curl your hair, and sinivate the ladies."

If I were asked which I preferred, this system or that of Baron
Ferrari[315] already mentioned, proceeding by _twelves_, I should reply,
with Candide, when he had the option given of running the gauntlet or being
shot: Les volontés sont libres, et je ne veux ni l'un ni l'autre.[316] We
can imagine a speculator providing such a system for Utopia as it would be
in the mind of a Laputan: but to explain how an engineer who has surveyed
mankind from Philadelphia to Rostof on the Don should for a moment
entertain the idea of such a system being actually adopted, would beat a
jury of solar-system-makers, though they were shut up from the beginning of
Anuary to the end of Tonborius. When I see such a scheme as this imagined
to be practicable, I admire the wisdom of Providence in providing the
quadrature of the circle, etc., to open a harmless sphere of action to the
possessors of the kind of ingenuity which it displays. Those who cultivate
mathematics have a right to speak strongly on such efforts of arithmetic as
this: for, to my knowledge, persons who have no knowledge are frequently
disposed to imagine that their makers are true brothers of the craft, a
little more intelligible than the rest.


    Vis inertiae victa,[317] or Fallacies affecting science. By James
    Reddie.[318] London, 1862, 8vo.


An attack on the Newtonian mechanics; revolution by gravitation
demonstrably impossible; much to be said for the earth being the immovable
center. A good analysis of contents at the beginning, a thing seldom found.
The author has followed up his attack in a paper submitted to the British
Association, but which it appears the Association declined to consider. It
is entitled--

    _Victoria Toto Coelo_; or, Modern Astronomy recast. London, 1863, 8vo.

At the end is a criticism of Sir G. Lewis's _History of Ancient Astronomy_.

    On the definition and nature of the Science of Political Economy. By H.
    Dunning Macleod,[319] Esq. Cambridge, 1862, 8vo.

A paper read--but, according to the report, not understood--at the British
Association. There is a notion that political economy is entirely
mathematical; and its negative quantity is strongly recommended for study:
it contains "the whole of the Funds, Credit, 32 parts out of 33 of the
value of Land...." The mathematics are described as consisting of--first,
number, or Arithmetic; secondly, the theory of dependent quantities,
subdivided into dependence by cause and effect, and dependence by
simultaneous variations; thirdly, "independent quantities or unconnected
events, which is the theory of probabilities." I am not ashamed, having the
British Association as a co-non-intelligent, to say I do not understand
this: there is a paradox in it, and the author should give further
explanation, especially of his negative quantity. Mr. Macleod has gained
{185} praise from great names for his political economy; but this, I
suspect, must have been for other parts of his system.

    On the principles and practice of just Intonation, with a view to the
    abolition of temperament.... By General Perronet Thompson.[320] Sixth
    Edition. London, 1862, 8vo.

Here is General Thompson again, with another paradox: but always master of
the subject, always well up in what his predecessors have done, and always
aiming at a useful end. He desires to abolish temperament by additional
keys, and has constructed an enharmonic organ with forty sounds in the
octave. If this can be introduced, I, for one, shall delight to hear it:
but there are very great difficulties in the way, greater than stood even
in the way of the repeal of the bread-tax.

In a paper on the beats of organ-pipes and on temperament published some
years ago, I said that equal temperament appeared to me insipid, and not so
agreeable as the effect of the instrument when in progress towards being
what is called out of tune, before it becomes offensively wrong. There is
throughout that period unequal temperament, determined by accident. General
Thompson, taking me one way, says I have launched a declaration which is
likely to make an epoch in musical practice; a public musical critic,
taking me another way, quizzes me for preferring music _out of tune_. I do
not think I deserve either one remark or the other. My opponent critic, I
suspect, takes _equally tempered_ and _in tune_ to be phrases of one
meaning. But by equal temperament is meant equal distribution among all the
keys of the error which an instrument _must_ have, which, with twelve
sounds only in the octave, professes to be fit for all the keys. I am
reminded of the equal temperament which was once applied to the postmen's
jackets. The coats were all made for the average man: the {186} consequence
was that all the tall men had their tails too short; all the short men had
them too long. Some one innocently asked why the tall men did not change
coats with the short ones.

    A diagram illustrating a discovery in the relation of circles to
    right-lined geometrical figures. London, 1863, 12mo.

The circle is divided into equal sectors, which are joined head and tail:
but a property is supposed which is not true.

    An attempt to assign the square roots of negative powers; or what is
    [sqrt] -1? By F.H. Laing.[321] London, 1863, 8vo.

If I understand the author, -a and +a are the square roots of -a^2, as
proved by multiplying them together. The author seems quite unaware of what
has been done in the last fifty years.


    Dual Arithmetic. A new art. By Oliver Byrne.[322] London, 1863, 8vo.

The plan is to throw numbers into the form a(1.1)^{b} (1.01)^{c}
(1.001)^{d}... and to operate with this form. This is an ingenious and
elaborate speculation; and I have no doubt the author has practised his
method until he could surprise any one else by his use of it. But I doubt
if he will persuade others to use it. As asked of Wilkins's universal
language, Where is the second man to come from?

An effective predecessor in the same line of invention {187} was the late
Mr. Thomas Weddle,[323] in his "New, simple, and general method of solving
numeric equations of all orders," 4to, 1842. The Royal Society, to which
this paper was offered, declined to print it: they ought to have printed an
organized method, which, without subsidiary tables, showed them, in six
quarto pages, the solution (x=8.367975431) of the equation

  1379.664 x^{622} + 2686034 × 10^{432} x^{152} - 17290224 × 10^{518}
      x^{60} + 2524156 × 10^{574} = 0.

The method proceeds by successive factors of the form, a being the first
approximation, a × 1.b × 1.0c × 1.00d.... In my copy I find a few
corrections made by me at the time in Mr. Weddle's announcement. "It was
read before that learned body [the R. S.] and they were pleased [but] to
transmit their thanks to the author. The en[dis]couragement which he
received induces [obliges] him to lay the result of his enquiries in this
important branch of mathematics before the public [, at his own expense; he
being an usher in a school at Newcastle]." Which is most satirical, Mr.
Weddle or myself? The Society, in the account which it gave of this paper,
described it as a "new and remarkably simple method" possessing "several
important advantages." Mr. Rutherford's[324] extended value of [pi] was
read at the very next meeting, and was printed in the _Transactions_; and
very properly: Mr. Weddle's paper was excluded, and very very improperly.


I think it may be admited that the indisposition to look at and encourage
improvements of calculation which once {188} marked the Royal Society is no
longer in existence. But not without severe lessons. They had the luck to
accept Horner's[325] now celebrated paper, containing the method which is
far on the way to become universal: but they refused the paper in which
Horner developed his views of this and other subjects: it was printed by
T. S. Davies[326] after Horner's death. I make myself responsible for the
statement that the Society could not reject this paper, yet felt unwilling
to print it, and suggested that it should be withdrawn; which was done.

But the severest lesson was the loss of _Barrett's Method_,[327] now the
universal instrument of the actuary in his highest calculations. It was
presented to the Royal Society, and refused admission into the
_Transactions_: Francis Baily[328] printed it. The Society is now better
informed: "_live and learn_," meaning "_must live, so better learn_," ought
to be the especial motto of a corporation, and is generally acted on, more
or less.

Horner's method begins to be introduced at Cambridge: it was published in
1820. I remember that when I first went to Cambridge (in 1823) I heard my
tutor say, in conversation, there is no doubt that the true method of
solving equations is the one which was published a few years ago in the
_Philosophical Transactions_. I wondered it was not taught, but presumed
that it belonged to the higher mathematics. This Horner himself had in his
head: and in a sense it is true; for all lower branches belong to the
higher: but he would have stared to have been told that he, Horner, {189}
was without a European predecessor, and in the distinctive part of his
discovery was heir-at-law to the nameless
Brahmin--Tartar--Antenoachian--what you please--who concocted the
extraction of the square root.

It was somewhat more than twenty years after I had thus heard a Cambridge
tutor show sense of the true place of Horner's method, that a pupil of mine
who had passed on to Cambridge was desired by his college tutor to solve a
certain cubic equation--one of an integer root of two figures. In a minute
the work and answer were presented, by Horner's method. "How!" said the
tutor, "this can't be, you know." "There is the answer, Sir!" said my
pupil, greatly amused, for my pupils learnt, not only Horner's method, but
the estimation it held at Cambridge. "Yes!" said the tutor, "there is the
answer certainly; but it _stands to reason_ that a cubic equation cannot be
solved in this space." He then sat down, went through a process about ten
times as long, and then said with triumph: "There! that is the way to solve
a cubic equation!"

I think the tutor in this case was never matched, except by the country
organist. A master of the instrument went into the organ-loft during
service, and asked the organist to let him _play the congregation out_;
consent was given. The stranger, when the time came, began a voluntary
which made the people open their ears, and wonder who had got into the
loft: they kept their places to enjoy the treat. When the organist saw
this, he pushed the interloper off the stool, with "You'll never play 'em
out this side Christmas." He then began his own drone, and the congregation
began to move quietly away. "There," said he, "that's the way to play 'em

I have not scrupled to bear hard on my own university, on the Royal
Society, and on other respectable existences: being very much the friend of
all. I will now clear the Royal Society from a very small and obscure
slander, simply because I know how. This dissertation began with {190} the
work of Mr. Oliver Byrne, the dual arithmetician, etc. This writer
published, in 1849, a method of calculating logarithms.[329] First, a long
list of instances in which, as he alleges, foreign discoverers have been
pillaged by Englishmen, or turned into Englishmen: for example,
O'Neill,[330] so called by Mr. Byrne, the rectifier of the semi-cubical
parabola claimed by the Saxons under the name of _Neal_: the grandfather of
this mathematician was conspicuous enough as _Neal_; he was archbishop of
York. This list, says the writer, might be continued without end; but he
has mercy, and finishes with his own case, as follows:--"About twenty years
ago, I discovered this method of directly calculating logarithms. I could
generally find the logarithm of any number in a minute or two without the
use of books or tables. The importance of the discovery subjected me to all
sorts of prying. Some asserted that I committed a table of logarithms to
memory; others attributed it to a peculiar mental property; and when
Societies and individuals failed to extract my secret, they never failed to
traduce the inventor and the invention. Among the learned Societies, the
Royal Society of London played a very base part. When I have more space and
time at my disposal, I will revert to this subject again."

Such a trumpery story as this remains unnoticed at the time; but when all
are gone, a stray copy from a stall falls into hands which, not knowing
what to make of it, make history of it. It is a very curious distortion.
The reader may take it on my authority, that the Royal Society played no
part, good or bad, nor had the option of playing a part. {191} But I myself
_pars magna fui_:[331] and when the author has "space and time" at his
disposal, he must not take all of them; I shall want a little of both.


    The mystery of being; or are ultimate atoms inhabited worlds? By
    Nicholas Odgers.[332] Redruth and London, 1863, 8vo.

This book, as a paradox, beats quadrature, duplication, trisection,
philosopher's stone, perpetual motion, magic, astrology, mesmerism,
clairvoyance, spiritualism, homoeopathy, hydropathy, kinesipathy, Essays
and Reviews, and Bishop Colenso,[333] all put together. Of all the
suppositions I have given as actually argued, this is the one which is
hardest to deny, and hardest to admit. Reserving the question--as beyond
human discussion--whether our particles of carbon, etc. are _clusters_ of
worlds, the author produces his reasons for thinking that they are at least
single worlds. Of course--though not mentioned--the possibility is to be
added of the same thing being true of the particles which make up our
particles, and so down, for ever: and, on the other hand, of our planets
and stars as being particles in some larger universe, and so up, for ever.

 "Great fleas have little fleas upon their backs to bite 'em,
  And little fleas have lesser fleas, and so _ad infinitum._
  And the great fleas themselves, in turn, have greater fleas to go on;
  While these again have greater still, and greater still, and so on."[334]

I have often had the notion that all the nebulæ we see, including our own,
which we call the Milky Way, may be particles of snuff in the box of a
giant of a proportionately {192} larger universe. Of course the minim of
time--a million of years or whatever the geologists make it[335]--which our
little affair has lasted, is but a very small fraction of a second to the
great creature in whose nose we shall all be in a few tens of thousands of
millions of millions of millions of years.

All this is quite possible, and the probabilities for and against are quite
out of reach. Perhaps also all the worlds, both above and below us, are
fac-similes of our own. If so, away goes free will for good and all;
unless, indeed, we underpin our system with the hypothesis that all the
fac-simile bodies of different sizes are actuated by a common soul. These
acute supplementary notions of mine go far to get rid of the difficulty
which some have found in the common theory that the soul inhabits the body:
it has been stated that there is, somewhere or another, a world of souls
which communicate with their bodies by wondrous filaments of a nature
neither mental nor material, but of a _tertium quid_ fit to be a
go-between; as it were a corporispiritual copper encased in a
spiritucorporeal gutta-percha. My theory is that every soul is everywhere
_in posse_, as the schoolmen said, but not anywhere _in actu_, except where
it finds one of its bodies. These _a priori_ difficulties being thus
removed, the system of particle-worlds is reduced to a dry question of
fact, and remitted to the decision of the microscope. And a grand field may
thus be opened, as optical science progresses! For the worlds are not
fac-similes of ours in time: there is not a moment of _our_ past, and not a
moment of _our_ future, but is the _present_ of one or more of the
particles. A will write the death of Cæsar, and B the building of the
Pyramids, by actual observation of the processes with a power of a thousand
millions; C will discover the commencement of the Millennium, and D the
{193} termination of Ersch and Gruber's Lexicon,[336] as mere physical
phenomena. Against this glorious future there is a sad omen: the initials
of the forerunner of this discovery are--NO!


    The History of the Supernatural in all ages and nations, and in all
    Churches, Christian and Pagan: demonstrating a universal faith. By Wm.
    Howitt.[337] London, 2 vols. 8vo. 1863.

Mr. Howitt is a preacher of spiritualism. He cements an enormous collection
of alleged facts with a vivid outpouring of exhortation, and an unsparing
flow of sarcasm against the scorners of all classes. He and the Rev. J.
Smith[338] (_ante_, 1854) are the most thoroughgoing universalists of all
the writers I know on spiritualism. If either can insert the small end of
the wedge, he will not let you off one fraction of the conclusion that all
countries, in all ages, have been the theaters of one vast spiritual
display. And I suspect that this consequence cannot be avoided, if any part
of the system be of truly spiritual origin. Mr. Howitt treats the
philosophers either as ignorant babies, or as conscious spirit-fearers: and
seems much inclined to accuse the world at large of dreading, lest by the
actual presence of the other world their Christianity should imbibe a
spiritual element which would unfit it for the purposes of their lives.



    From Matter to Spirit. By C. D. With a preface by A. B.[339] London,
    1863, 8vo.

This is a work on Spiritual Manifestations. The author upholds the facts
for spiritual phenomena: the prefator suspends his opinion as to the cause,
though he upholds the facts. The work begins systematically with the lower
class of phenomena, proceeds to the higher class, and offers a theory,
suggested by the facts, of the connection of the present and future life. I
agree in the main with A. B.; but can, of course, make none but horrescent
reference to his treatment of the smaller philosophers. This is always the
way with your paradoxers: they behave towards orthodoxy as the thresher
fish behaves towards the whale. But if true, as is said, that the drubbing
clears the great fish of parasites which he could not otherwise get rid of,
he ought to bear no malice. This preface retorts a little of that contempt
which the "philosophical world" has bestowed with heaped measure upon those
who have believed their senses, and have drawn natural, even if hasty,
inferences. There is philosophercraft as well as priestcraft, both from one
source, both of one spirit. In English cities and towns, the minister of
religion has been tamed: so many weapons are bared against him when he
obtrudes his office in a dictatory manner, that, as a rule, there is no
more quiet and modest member of society than the urban clergyman.
Domination over religious belief is reserved for the exclusive use of those
who admit the right: the rare exception to this mode of behavior is laughed
at as a bigot, or shunned as a nuisance. But the overbearing minister of
nature, who snaps you with _unphilosophical_ as the clergyman once
frightened you with _infidel_, is still a recognized member of society,
wants taming, and will get it. He wears the priest's cast-off {195}
clothes, dyed to escape detection: the better sort of philosophers would
gladly set him to square the circle.

The book just named appeared about the same time as this Budget began in
the _Athenæum_. It was commonly attributed, the book to my wife, the
preface to myself. Some time after, our names were actually announced by
the publisher, who ought to know. It will be held to confirm this statement
that I announce our having in our possession some twenty reviews of
different lengths, and of all characters: who ever collects a number of
reviews of a book, except the author?

A great many of these reviews settle the matter _a priori_. If there had
been spirits in the matter, they would have done this, and they would not
have done that. Jean Meslier[340] said there could be no God over all, for,
_if_ there had been one, He would have established a universal religion. If
J. M. _knew_ that, J. M. was right: but if J. M. did not know that, then
J. M. was on the "high priori road," and may be left to his course. The
same to all who know what spirits would do and would not do.

A. B. very distinctly said that he knew some of the asserted facts,
believed others on testimony, but did not pretend to know whether they were
caused by spirits, or had some unknown and unimagined origin. This he said
as clearly as I could have said it myself. But a great many persons cannot
understand such a frame of mind: their own apparatus is a kind of
spirit-level, and their conclusion on any subject is the little bubble,
which is always at one end or the other. Many of the reviewers declare that
A. B. is a secret believer in the spirit-hypothesis: and one of them wishes
that he had "endorsed his opinion more boldly." According to this reviewer,
any one who writes "I boldly {196} say I am unable to choose," contradicts
himself. In truth, a person who does say it has a good deal of courage, for
each side believes that he secretly favors the other; and both look upon
him as a coward. In spite of all this, A. B. boldly repeats that he feels
assured of many of the facts of _spiritualism_, and that he cannot pretend
to affirm or deny anything about their cause.

The great bulk of the illogical part of the educated community--whether
majority or minority I know not; perhaps six of one and half-a-dozen of the
other--have not power to make a distinction, cannot be made to take a
distinction, and of course, never attempt to shake a distinction. With them
all such things are evasions, subterfuges, come-offs, loopholes, etc. They
would hang a man for horse-stealing under a statute against sheep-stealing;
and would laugh at you if you quibbled about the distinction between a
horse and a sheep. I divide the illogical--I mean people who have not that
amount of natural use of sound inference which is really not uncommon--into
three classes:--First class, three varieties: the Niddy, the Noddy, and the
Noodle. Second class, three varieties: the Niddy-Noddy, the Niddy-Noodle,
and the Noddy-Noodle. Third class, undivided: the Niddy-Noddy-Noodle. No
person has a right to be angry with me for more than one of these

The want of distinction was illustrated to me, when a boy, about 1820, by
the report of a trial which I shall never forget: boys read newspapers more
keenly than men. Every now and then a bench of country magistrates rather
astonishes the town populations, accustomed to rub their brains[341]
against one another. Such a story as the following would, {197} in our day,
bring down grave remarks from above: but I write of the olden (or
Eldon[342]) time, when nothing but conviction in a court of record would
displace a magistrate. In that day the third-class amalgamator of distinct
things was often on the bench of quarter-sessions.

An attorney was charged with having been out at night poaching. A clear
_alibi_ was established; and perjury had certainly been committed. The
whole gave reason to suspect that some ill-willers thought the bench
disliked the attorney so much that any conviction was certain on any
evidence. The bench did dislike the attorney: but not to the extent of
thinking he could snare any partridges in the fields while he was asleep in
bed, except the dream-partridges which are not always protected by the
dream-laws. So the chairman said, "Mr. ----, you are discharged; but you
should consider this one of the most fortunate days of your life." The
attorney indignantly remonstrated, but the magistrate was right; for he
said, "Mr. ----, you have frequently been employed to defend poachers: have
you been careful to impress upon them the enormity of their practices?" It
appeared in a wrangling conversation that the magistrates saw little moral
difference between poaching and being a poacher's professional defender
without lecturing him on his wickedness: but they admitted with reluctance,
that there was a legal distinction; and the brain of N^3 could no further
go. This is nearly fifty years ago; and Westernism was not quite extinct.
If the present lords of the hills and the valleys want to shine, let them
publish a true history of their own order. I am just old enough to remember
some of the last of the squires and parsons who protested against teaching
the poor to read and write. They now write books for the working classes,
give them lectures, and the like. There is now no class, as a class, more
highly educated, broadly educated, and deeply educated, {198} than those
who were, in old times, best described as partridge-popping squireens. I
have myself, when a boy, heard Old Booby speaking with pride of Young Booby
as having too high a spirit to be confined to books: and I suspected that
his dislike to teaching the poor arose in fact from a feeling that they
would, if taught a little, pass his heir.

A. B. recommended the spirit-theory as an hypothesis on which to ground
inquiry; that is, as the means of suggestion for the direction of inquiry.
Every person who knows anything of the progress of physics understands what
is meant; but not the reviewers I speak of. Many of them consider A. B. as
_adopting_ the spirit-hypothesis. The whole book was written, as both the
authors point out, to suggest inquiry to those who are curious; C. D.
firmly believing, A. B. as above. Neither C. D. nor A. B. make any other
pretence. Both dwell upon the absence of authentications and the
suppression of names as utterly preventive of anything like proof. And
A. B. says that his reader "will give him credit, if not himself a goose,
for seeing that the tender of an anonymous cheque would be of equal effect,
whether drawn on the Bank of England or on Aldgate Pump." By this test a
number of the reviewers are found to be geese: for they take the authors as
offering proof, and insist, against the authors, on the very point on which
the authors had themselves insisted beforehand.

Leaving aside imperceptions of this kind, I proceed to notice a clerical
and medical review. I have lived much in the middle ages, especially since
the invention of printing; and from thence I have brought away a high
respect for and grateful recollection of--the priest in everything but
theology, and the physician in everything but medicine. The professional
harness was unfavorable to all progress, except on a beaten road; the
professional blinkers prevented all but the beaten road from being seen:
the professional reins were pulled at the slightest attempt to quicken
pace, even on the permitted path; and the {199} professional whip was
heavily laid on at the slightest attempt to diverge. But when the
intelligent man of either class turned his attention out of his ordinary
work, he had, in most cases, the freshness and vigor of a boy at play, and
like the boy, he felt his freedom all the more from the contrast of

In the case of medicine, and physics generally, the learned were, in some
essential points, more rational than many of their present impugners. They
pass for having put _a priori_ obstacles in the way of progress: they might
rather be reproved for too much belief in progress obtained by _a priori_
means. They would have shouted with laughter at a dunce who--in a review I
read, but without making a note--declared that he would not believe his
senses except when what they showed him was capable of explanation upon
some known principle. I have seen such stuff as this attributed to the
schoolmen; but only by those who knew nothing about them. The following,
which I wrote some years ago, will give a notion of a distinction worth
remembering. It is addressed to the authorities of the College of

"The ignominy of the word _empiric_ dates from the ages in which scholastic
philosophy deduced physical consequences _a priori_;--the ages in which,
because a lion is strong, rubbing with lion's fat would have been held an
infallible tonic. In those happy days, if a physician had given decoction
of a certain bark, only because in numberless instances that decoction had
been found to strengthen the patient, he would have been a miserable
empiric. Not that the colleges would have passed over his returns because
they were empirical: they knew better. They were as skilful in finding
causes for facts, as facts for causes. The president and the elects of that
day would have walked out into the forest with a rope, and would have
pulled heartily at the tree which yielded the bark: nor would they ever
have left it until they had pulled out a legitimate {200} reason. If the
tree had resisted all their efforts, they would have said, 'Ah! no wonder
now; the bark of a strong tree makes a strong man.' But if they had managed
to serve the tree as you would like to serve homoeopathy, then it would
have been 'We might have guessed it; all the _virtus roborativa_ has
settled in the bark.' They admitted, as we know from Molière, the _virtus
dormitiva_[343] of opium, for no other reason than that opium _facit
dormire_.[344] Had the medicine not been previously _known_, they would,
strange as it may seem to modern pharmacopoeists, have accorded a _virtus
dormitiva_ to the new _facit dormire_. On this point they have been
misapprehended. They were prone to infer _facit_ from a _virtus_ imagined
_a priori_; and they were ready in supplying _facit_ in favor of an
orthodox _virtus_. They might have gone so far, for example, under
pre-notional impressions, as the alliterative allopath, who, when
maintenance of truth was busy opposing the progress of science called
_vaccination_, declared that some of its patients coughed like cows, and
bellowed like bulls; but they never refused to find _virtus_ when _facit_
came upon them, no matter whence. They would rather have accepted Tenterden
steeple than have rejected the Goodwin Sands. They would have laughed their
modern imitators to scorn: but as they are not here, we do it for them.

"The man of our day--the _a priori_ philosopher--tries the question whether
opium can cause sleep by finding out in the recesses of his own noddle
whether the drug can have a dormitive power: Well! but did not the
schoolman do the same? He did; but mark the distinction. The schoolman had
recourse to first principles, when there was no opium to try it by: our man
settles the point in the same way _with a lump of opium before him_. The
schoolman shifted his principles with his facts: the man of our
drawing-rooms will fight facts with his principles, just as an old {201}
physician would have done in actual practice, with the rod of his _Church_
at his back.

"The story about Galileo--which seems to have been either a joke made
against him, or by him--illustrates this. _Nature abhors a vacuum_ was the
explanation of the water rising in a pump: but they found that the water
would not rise more than 32 feet. They asked for explanation: what does the
satirist make the schoolmen say? That the stoppage is _not_ a fact, because
nature abhors a vacuum? No! but that the principle should be that nature
abhors a vacuum as far as 32 feet. And this is what would have been done.

"There are still among us both priests and physicians who would have
belonged, had they lived three or four centuries ago, to the glorious band
of whom I have spoken, the majority of the intelligent, working well for
mankind out of the professional pursuit. But we have a great many who have
helped to abase their classes. Go where we may, we find specimens of the
lower orders of the ministry of religion and the ministry of health showing
themselves smaller than the small of other pursuits. And how is this?
First, because each profession is entered upon a mere working smack of its
knowledge, without any depth of education, general or professional. Not
that this is the whole explanation, nor in itself objectionable: the great
mass of the world must be tended, soul and body, by those who are neither
Hookers[345] nor Harveys[346]: let such persons not venture _ultra
crepidam_, and they are useful and respectable. But, secondly, there is a
vast upheaving of thought from the depths of commonplace learning. I am a
clergyman! Sir! I am a medical man! Sir! and forthwith the nature of things
is picked to pieces, and there is a race, with the last the winner, between
Philosophy mounted on Folly's donkey, and Folly mounted on Philosophy's
donkey. How fortunate {202} it is for Law that her battles are fought by
politicians in the Houses of Parliament. Not that it is better done: but
then _politics_ bears the blame."

I now come to the medical review. After a quantity of remark which has been
already disposed of, the writer shows Greek learning, a field in which the
old physician would have had a little knowledge. A. B., for the joke's
sake, had left untranslated, as being too deep, a remarkably easy sentence
of Aristotle, to the effect that what has happened was possible, for if
impossible it would not have happened. The reviewer, in "simple
astonishment,"--it was simple--at the pretended incapacity--I was told by
A. B. that the joke was intended to draw out a reviewer--translates:--He
says that this sentence is A. B.'s summing up of the evidence of
Spiritualism. Now, being a sort of _alter ego_[347] of A. B., I do declare
that he is not such a fool as to rest the evidence of Spiritualism--the
_spirit explanation_--upon the occurrence of certain facts proving the
possibility of those very facts. In truth, A. B. refuses to receive
spiritualism, while he receives the facts: this is the gist of his whole
preface, which simply admits spiritualism among the qualified candidates,
and does not know what others there may be.

The reviewer speaks of Aristotle as "that clear thinker and concise
writer." I strongly suspect that his knowledge of Aristotle was limited to
the single sentence which he had translated or got translated. Aristotle is
concise in _phrase_, not in book, and is powerful and profound in thought:
but no one who knows that his writing, all we have of him, is the very
opposite of clear, will pretend to decide that he thought clearly. As his
writing, so probably was his thought; and his books are, if not anything
but clear, at least anything good but clear. Nobody thinks them clear
except a person who always clears difficulties: which I have no doubt was
the reviewer's habit; that is, if he ever took the field {203} at all. The
gentleman who read Euclid, all except the As and Bs and the pictures of
scratches and scrawls, is the type of a numerous class.

The reviewer finds that the word _amosgepotically_, used by A. B., is
utterly mysterious and incomprehensible. He hopes his translation of the
bit of Greek will shield him from imputation of ignorance: and thinks the
word may be referred to the "obscure dialect" out of which sprung
_aneroid_, _kalos geusis sauce_, and _Anaxyridian trousers_. To lump the
first two phrases with the third smacks of ignorance in a Greek critic; for
[Greek: anaxuridia], _breeches_, would have turned up in the lexicon; and
_kalos geusis_, though absurd, is not obscure. And [Greek: amôsgepôs],
_somehow or other_, is as easily found as [Greek: anaxuridia]. The word
_aneroid_, I admit, has puzzled better scholars than the critic: but never
one who knows the unscholarlike way in which words ending in [Greek: eidês]
have been rendered. The _aneroid barometer_ does _not_ use a column of air
in the same way as the old instrument. Now [Greek: aeroeidês]--properly
_like_ the atmosphere--is by scientific non-scholarship rendered having to
do with the atmosphere; and [Greek: anaeroeidês]--say _anaëroid_--denies
having to do with the atmosphere; a nice thing to say of an instrument
which is to measure the weight of the atmosphere. One more absurdity, and
we have _aneroid_, and there you are. The critic ends with a declaration
that nothing in the book shakes his faith in a _Quarterly_ reviewer who
said that suspension of opinion, until further evidence arrives, is
justifiable: a strange summing up for an article which insists upon utter
rejection being unavoidable.[348] The expressed aim of both A. B. and C. D.
was to excite inquiry, and get further evidence: until this is done,
neither asks for a verdict.

Oh where! and oh where! is old Medicine's learning gone! There _was_ some
in the days of yore, when Popery {204} was on! And it's oh! for some Greek,
just to find a word upon! The reviewer who, lexicon in hand, can neither
make out _anaxyridical_, _amosgepotical_, _kalos geusis_, nor distinguish
them from _aneroid_, cannot be trusted when he says he has translated a
sentence of Aristotle. He may have done it; but, as he says of
spiritualism, we must suspend our opinion until further evidence shall

We now come to the theological review. I have before alluded to the faults
of logic which are Protestant necessities: but I never said that Protestant
argument had _nothing but_ paralogism. The writer before me attains this
completeness: from beginning to end he is of that confusion and perversion
which, as applied to interpretation of the New Testament, is so common as
to pass unnoticed by sermon-hearers; but which, when applied out of church,
is exposed with laughter in all subjects except theology. I shall take one
instance, putting some words in italics.

_A. B._                         _Theological Critic._

My state of mind, which refers  ... he proceeds to argue that
the whole _either_ to unseen    he himself is outside its
intelligence, _or something     sacred pale because he refers
which man has never had any     all these strange phenomena to
conception of_, proves me to    _unseen spiritual
be out of the pale of the       intelligence_.
Royal Society.

The possibility of a _yet unimagined_ cause is insisted on in several
places. On this ground it is argued by A. B. that spiritualists are
"incautious" for giving in at once to the spirit doctrine. But, it is said,
they may be justified by the philosophers, who make the flint _axes_, as
they call them, to be the works of men, because no one can see _what else
they can be_. This kind of adoption, _condemned_ as a conclusion, is
_approved_ as a provisional theory, suggestive of direction of inquiry:
experience having shown that {205} inquiry directed by a _wrong_ theory has
led to more good than inquiry without any theory at all. All this A. B. has
fully set forth, in several pages. On it the reviewer remarks that "with
infinite satisfaction he tries to justify his view of the case by urging
that there is no other way of accounting for it; after the fashion of the
philosophers of our own day, who conclude that certain flints found in the
drift are the work of men, because the geologist does not see what else
they can be." After this twist of meaning, the reviewer proceeds to say,
and A. B. would certainly join him, "There is no need to combat any such
mode of reasoning as this, because it would apply with equal force and
justice to any theory whatever, however fantastic, profane, or silly." And
so, having shown how the reviewer has hung himself, I leave him

One instance more, and I have done. A reviewer, not theological, speaking
of the common argument that things which are derided are not _therefore_ to
be rejected, writes as follows:--"It might as well be said that they who
laughed at Jenner[349] and vaccination were, in a certain but very
unsatisfactory way, witnesses to the possible excellence of the system of
St. John Long."[350] Of course it _might_: and of course it _is_ said by
all people of common sense. In introducing the word "possible," the
reviewer has hit the point: I suspect that this word was introduced during
revision, to put the sentence into fighting order; hurry preventing it
being seen that the sentence was thus made to fight on the wrong side.
Jenner, who was laughed at, was right; therefore, it is not
impossible--that is, it is _possible_--that a derided system may be right.
Mark the three gradations: _in medio tutissimus ibis_.[351]


_Reviewer._--If a system be derided, it is no ground of suspense that
derided systems have turned out true: if it were, you would suspend your
opinion about St. John Long on account of Jenner.--_Ans._ You ought to do
so, as to _possibility_; and _before examination_; not with the notion that
J. proves St. J. _probable_; only _possible_.

_Common Sense._--The past emergence of truths out of derided systems proves
that there is a practical certainty of like occurrence to come. But,
inasmuch as a hundred speculative fooleries are started for one truth, the
mind is permitted to approach the examination of any one given novelty with
a bias against it of a hundred to one: and this permission is given because
so it will be, leave or no leave. Every one has licence not to jump over
the moon.

_Paradoxer._--Great men have been derided, and I am derided: which proves
that my system ought to be adopted. This is a summary of all the degrees in
which paradoxers contend for the former derision of truths now established,
giving their systems _probability_. I annex a paragraph which D [e &c.]
inserted in the _Athenæum_ of October 23, 1847.

"_Discoverers and Discoveries._

"Aristotle once sent his servant to the cellar to fetch wine:--and the
fellow brought him back small beer. The Stagirite (who knew the difference)
called him a blockhead. 'Sir,' said the man, 'all I can say is, that I
found it in the cellar.' The philosopher muttered to himself that an
affirmative conclusion could not be proved in the second figure,--and Mrs.
Aristotle, who was by, was not less effective in her remark, that small
beer was not wine because it was in the same cellar. Both were right
enough: and our philosophers might take a lesson from either--for they
insinuate an affirmative conclusion in the second figure. Great discoverers
have been little valued by established {207} schools,--and they are little
valued. The results of true science are strange at first,--and so are
their's. Many great men have opposed existing notions,--and so do they. All
great men were obscure at first,--and they are obscure. Thinking men
doubt,--and they doubt. Their small beer, I grant, has come out of the same
cellar as the wine; but this is not enough. If they had let it stand awhile
in the old wine-casks, it might have imbibed a little of the flavor."

There are better reviews than I have noticed; which, though entirely
dissenting, are unassailable on their own principles. What I have given
represents five-sixths of the whole. But it must be confessed that the
fraction of fairness and moderation and suspended opinion which the
doctrine of _Spirit Manifestations_ has met with--even in the lower
reviews--is strikingly large compared to what would have been the case
fifty years ago. It is to be hoped that our popular and periodical
literatures are giving us one thinker created for twenty geese
double-feathered: if this hope be realized, we shall do! Seeing all that I
see, I am not prepared to go the length of a friend of mine who, after
reading a good specimen of the lower reviewing, exclaimed--Oh! if all the
fools in the world could be rolled up into one fool, what a reviewer he
would make!

    Calendrier Universel et Perpétuel; par le Commandeur P. J. Arson.[352]
    Publié par ses Enfans (Oeuvre posthume). Nice, 1863, 4to.

I shall not give any account of this curious calendar, with all its changes
and symbols. But there is one proposal, which, could we alter the general
notions of time--a thing of very dubious possibility--would be convenient.
The week is made to wax and wane, culminating on the Sunday, {208} which
comes in the middle. Thursday, Friday, Saturday, are ascending or waxing
days; Monday, Tuesday, Wednesday, are descending or waning days. Our six
days, lumped together after the great distinguishing day, Sunday, are too
many to be distinctly thought of together: a division of three preceding
and three following the day of most note would be much more easily used.
But all this comes too late. It may be, nevertheless, that some individuals
may be able to adjust their affairs with advantage by referring Thursday,
Friday, Saturday, to the following Sunday, and Monday, Tuesday, Wednesday,
to the preceding Sunday. But M. Arson's proposal to alter the names of the
days is no more necessary than it is practicable.


I am not to enter anything I do not possess. The reader therefore will not
learn from me the feats of many a man-at-arms in these subjects. He must be
content, unless he will bestir himself for himself, not to know how Mr.
Patrick Cody trisects the angle at Mullinavat, or Professor Recalcati
squares the circle at Milan. But this last is to be done by subscription,
at five francs a head: a banker is named who guarantees restitution if the
solution be not perfectly rigorous; the banker himself, I suppose, is the
judge. I have heard of a man of business who settled the circle in this
way: if it can be reduced to a debtor and creditor account, it can
certainly be done; if not, it is not worth doing. Montucla will give the
accounts of the lawsuits which wagers on the problem have produced in

Neither will I enter at length upon the success of the new squarer who
advertises (Nov. 1863) in a country paper that, having read that the
circular ratio was undetermined, "I thought it very strange that so many
great scholars in all ages should have failed in finding the true ratio,
and have been determined to try myself.... I am about to secure the {209}
benefit of the discovery, so until then the public cannot know my new and
true ratio." I have been informed that this trial makes the diameter to the
circumference as 64 to 201, giving [pi] = 3.140625 exactly. The result was
obtained by the discoverer in three weeks after he first heard of the
existence of the difficulty. This quadrator has since published a little
slip, and entered it at Stationers' Hall. He says he has done it by actual
measurement; and I hear from a private source that he uses a disk of 12
inches diameter, which he rolls upon a straight rail. Mr. James Smith did
the same at one time; as did also his partisan at Bordeaux. We have, then,
both 3.125 and 3.140625, by actual measurement. The second result is more
than the first by about one part in 200. The second rolling is a very
creditable one; it is about as much below the mark as Archimedes was above
it. Its performer is a joiner, who evidently knows well what he is about
when he measures; he is not wrong by 1 in 3,000.

The reader will smile at the quiet self-sufficiency with which "I have been
determined to try myself" follows the information that "so many great
scholars in all ages" have failed. It is an admirable spirit, when
accompanied by common sense and uncommon self-knowledge. When I was an
undergraduate there was a little attendant in the library who gave me the
following,--"As to cleaning this library, Sir, if I have spoken to the
Master once about it, I have spoken fifty times: but it is of no use; he
will not employ _littery_ men; and so I am obliged to look after it

I do not think I have mentioned the bright form of quadrature in which a
square is made equal to a circle by making each side equal to a quarter of
the circumference. The last squarer of this kind whom I have seen figures
in the last number of the _Athenæum_ for 1855: he says the thing is no
longer a _problem_, but an _axiom_. He does not know that the area of the
circle is greater than that of any other figure of the same circuit. This
any one might see without {210} mathematics. How is it possible that the
figure of greatest area should have any one length in its circuit unlike in
form to any other part of the same length?

The feeling which tempts persons to this problem is that which, in romance,
made it impossible for a knight to pass a castle which belonged to a giant
or an enchanter. I once gave a lecture on the subject: a gentleman who was
introduced to it by what I said remarked, loud enough to be heard by all
around, "Only prove to me that it is impossible, and I will set about it
this very evening."

This rinderpest of geometry cannot be cured, when once it has seated itself
in the system: all that can be done is to apply what the learned call
prophylactics to those who are yet sound. When once the virus gets into the
brain, the victim goes round the flame like a moth; first one way and then
the other, beginning where he ended, and ending where he begun: thus
verifying the old line

 "In girum imus nocte, ecce! et consumimur igni."[353]

Every mathematician knows that scores of methods, differing altogether from
each other in process, all end in this mysterious 3.14159..., which insists
on calling itself the circumference to a unit of diameter. A reader who is
competent to follow processes of arithmetic may be easily satisfied that
such methods do actually exist. I will give a sketch, carried out to a few
figures, of three: the first two I never met with in my reading; the third
is the old method of Vieta.[354] [I find that both the first and second
methods are contained in a theorem of Euler.]

What Mr. James Smith says of these methods is worth noting. He says I have
given three "_fancy_ proofs" of the value of [pi]: he evidently takes me to
be offering demonstration. He proceeds thus:--

"His first proof is traceable to the diameter of a circle {211} of radius
1. His second, to the side of any inscribed equilateral triangle to a
circle of radius 1. His third, to a radius of a circle of diameter 1. Now,
it may be frankly admitted that we can arrive at the same result by many
other modes of arithmetical calculation, all of which may be shown to have
some sort of relation to a circle; but, after all, these results are mere
exhibitions of the properties of numbers, and have no more to do with the
ratio of diameter to circumference in a circle than the price of sugar with
the mean height of spring tides. (_Corr._ Oct. 21, 1865)."

I quote this because it is one of the few cases--other than absolute
assumption of the conclusion--in which Mr. Smith's conclusions would be
true if his premise were true. Had I given what follows as _proof_, it
would have been properly remarked, that I had only exhibited properties of
numbers. But I took care to tell my reader that I was only going to show
him _methods_ which end in 3.14159.... The proofs that these methods
establish the value of [pi] are for those who will read and can understand.

  200000000      31415       3799
   66666667                  2817
   26666667                  1363
   11428571                   661
    5079365                   321
    2308802                   156
    1065601                    76
     497281                    37
     234014                    18
     110849                     9
      52785                     5
      25245                     2
      12118                     1
   --------     --------  -------
  314153799      31415       9265


1. Take any diameter, double it, take 1-3d of that double, 2-5ths of the
last, 3-7ths of the last, 4-9ths of the last, 5-11ths of the last, and so
on. The sum of all is the circumference of that diameter. The preceding is
the process when the diameter is a hundred millions: the errors arising
from rejection of fractions being lessened by proceeding on a thousand
millions, and striking off one figure. Here 200 etc. is double of the
diameter; 666 etc. is 1-3rd of 200 etc.; 266 etc. is 2-5ths of 666 etc.;
114 etc. is 3-7ths of 266 etc.; 507 etc. is 4-9ths of 114 etc.; and so on.

2. To the square root of 3 add its half. Take _half_ the third part of
this; half 2-5ths of the last; half 3-7ths of the last; and so on. The sum
is the circumference to a unit of diameter.

    Square root of 3....  1.73205081

3. Take the square root of ½; the square root of half of one more than
this; the square root of half of one more {213} than the last; and so on,
until we come as near to unity as the number of figures chosen will permit.
Multiply all the results together, and divide 2 by the product: the
quotient is an approximation to the circumference when the diameter is
unity. Taking aim at four figures, that is, working to five figures to
secure accuracy in the fourth, we have .70712 for the square root of ½;
.92390 for the square root of half one more than .70712; and so on, through
.98080, .99520, .99880, .99970, .99992, .99998. The product of the eight
results is .63667; divide 2 by this, and the quotient is 3.1413..., of
which four figures are correct. Had the product been .636363... instead of
.63667..., the famous result of Archimedes, 22-7ths, would have been
accurately true. It is singular that no cyclometer maintains that
Archimedes hit it exactly.

A literary journal could hardly admit as much as the preceding, if it stood
alone. But in my present undertaking it passes as the halfpennyworth of
bread to many gallons of sack. Many more methods might be given, all ending
in the same result, let that result mean what it may.

Now since dozens of methods, to which dozens more might be added at
pleasure, concur in giving one and the same result; and since these methods
are declared by all who have shown knowledge of mathematics to be
_demonstrated_: it is not asking too much of a person who has just a little
knowledge of the first elements that he should learn more, and put his hand
upon the error, before he intrudes his assertion of the existence of error
upon those who have given more time and attention to it than himself, and
who are in possession, over and above many demonstrations, of many
consequences verifying each other, of which he can know nothing. This is
all that is required. Let any one square the circle, and persuade his
friends, if he and they please: let him print, and let all read who choose.
But let him abstain from intruding himself upon those who have been
satisfied by existing demonstration, until he is prepared {214} to lay his
finger on the point in which existing demonstration is wrong. Let him also
say what this mysterious 3.14159... really is, which comes in at every door
and window, and down every chimney, calling itself the circumference to a
unit of diameter. This most impudent and successful impostor holds false
title-deeds in his hands, and invites examination: surely those who can
find out the rightful owner are equally able to detect the forgery. All the
quadrators are agreed that, be the right what it may, 3.14159... is wrong.
It would be well if they would put their heads together, and say what this
wrong result really means. The mathematicians of all ages have tried all
manner of processes, with one object in view, and by methods which are
admitted to yield demonstration in countless cases. They have all arrived
at one result. A large number of opponents unite in declaring this result
wrong, and all agree in two points: first, in differing among themselves;
secondly, in declining to point out what that curious result really is
which the mathematical methods all agree in giving.

Most of the quadrators are not aware that it has been fully demonstrated
that no two numbers whatsoever can represent the ratio of the diameter to
the circumference with perfect accuracy. When therefore we are told that
either 8 to 25 or 64 to 201 is the true ratio, we know that it is no such
thing, without the necessity of examination. The point that is left open,
as not fully demonstrated to be impossible, is the _geometrical_
quadrature, the determination of the circumference by the straight line and
circle, used as in Euclid. The general run of circle-squarers, hearing that
the quadrature is not pronounced to be _demonstratively_ impossible,
imagine that the _arithmetical_ quadrature is open to their ingenuity.
Before attempting the arithmetical problem, they ought to acquire knowledge
enough to read Lambert's[355] demonstration (last given in Brewster's[356]
translation {215} of Legendre's[357] Geometry) and, if they can, to refute
it. [It will be given in an Appendix.] Probably some have begun this way,
and have caught a Tartar who has refused to let them go: I have never heard
of any one who, in producing his own demonstration, has laid his finger on
the faulty part of Lambert's investigation. This is the answer to those who
think that the mathematicians treat the arithmetical squarers too lightly,
and that as some person may succeed at last, all attempts should be
examined. Those who have so thought, not knowing that there is
demonstration on the point, will probably admit that a person who
contradicts a theorem of which the demonstration has been acknowledged for
a century by all who have alluded to it as read by themselves, may
reasonably be required to point out the error before he demands attention
to his own result.

_Apopempsis of the Tutelaries._--Again and again I am told that I spend too
much time and trouble upon my two tutelaries: but when I come to my
summing-up I shall make it appear that I have a purpose. Some say I am too
hard upon them: but this is quite a mistake. Both of them beat little
Oliver himself in the art and science of asking for more; but without
Oliver's excuse, for I had given good allowance. Both began with me, not I
with them: and both knew what they had to expect when they applied for a
second helping.

On July 31, the Monday after the publication of my remarks on my 666
correspondent, I found _three_ notes in separate envelopes, addressed to me
at "7A, University College." When I saw the three new digits I was taken
rhythmopoetic, as follows--

  Here's the Doctor again with his figs, and by Heavens!
  He was always at sixes, and now he's at sevens.

To understand this fully the reader must know that the greater part of
Apocalyptic interpretation has long been condensed, in my mind, into the
Turkish street-cry--In the {216} name of the Prophet! figs! I make a few
extracts. The reader will observe that Dr. Thorn grumbles at his _private_
letters being _publicly_ ridiculed. A man was summoned for a glutolactic
assault; he complained of the publication of his proceeding: I kicked etc.
_in confidence_, he said.

"After reading your last, which tries in every way to hold me up to public
ridicule for daring to write you privately ['that you would be d----d,'
omitted by accident] one would say, Why have anything to do with such a
testy person? [Wrong word; no testy person can manage cool and consecutive
ridicule. Quære, what is this word? Is it anything but a corruption of the
obsolete word _tetchy_ of the same meaning? Some think _touchy_ is our
modern form of _tetchy_, which I greatly doubt]. My answer is, the poor man
is lamentably ignorant; he is not only so, but 'out of the way' [quite
true; my readers know me by this time for an out-of-the-way person. What
other could tackle my squad of paradoxers? What other would undertake the
job?] Can he be brought back and form one of those who in Ezekiel 37 ch.
have the Spirit breathed into them and live.... Have I any other feeling
towards you except that of peace and goodwill? [Not to your distinct
knowledge; but in all those who send people to 'the other place' for
contempt of their interpretations, there is a lurking wish which is father
to the thought; 'you _will_ be d----d' and 'you _be_ d--d' are Siamese
twins]. Of course your sneer at 666 brought plain words; but when men
meddle with what they do not understand (not having the double _Vahu_) they
must be dealt with faithfully by those who do.... [They must; which
justifies the Budget of Paradoxes: but no occasion to send them anywhere;
no preachee and floggee too, as the negro said]. Many will find the text
Prov. i. 26 fully realized. [All this contains distinct assumption of a
right 'of course' to declare accursed those who do not respect the writer's
vagary].... If I could but get the [Hebrew: A], the Ox-head, which in Old
Hebrew was just the Latin Digamma, F, out {217} of your name, and could
then Thau you with the Thau of Ezekiel ix, 4, the [chi], then you would
bear the number of a man! But this is too hard for me, although not so for
the Lord! Jer. xxxii. 17.... And now a word: is ridicule the right thing in
so solemn a matter as the discussion of Holy Writ? [Is food for ridicule
the right thing? Did I discuss Holy Writ? I did not: I concussed profane
scribble. Even the Doctor did not _discuss_; he only enunciated and
denunciated out of the mass of inferences which a mystical head has found
premises for in the Bible]."

  M    40
  O    70
  R   100
  G     6
  N    50
  [Hebrew: t]=[chi] 400

[That ill opinions are near relations of ill wishes, will be detected by
those who are on the look out. The following was taken down in a Scotch
Church by Mr. Cobden,[358] who handed it to a Roman friend of mine, for his
delectation (in 1855): "Lord, we thank thee that thou hast brought the Pope
into trouble; and we pray that thou wouldst be mercifully pleased to
increase the same."]

Here is a martyr who quarrels with his crown; a missionary who reviles his
persecutor: send him to New Zealand, and he would disagree with the Maoris
who ate him. Man of unilateral reciprocity! have you, who write to a
stranger with hints that that stranger and his wife are children of
perdition, the bad taste to complain of a facer in return? As James
Smith[359]--the Attorney-wit, not the Dock-cyclometer--said, or nearly

   "A pretty thing, forsooth!
  Is he to burn, all scalding hot,
  Me and my wife, and am I not
    To job him out a tooth?"


Those who think parody vulgar will be pleased to substitute for the above a
quotation from Butler[360]:--

 "There's nothing so absurd or vain
  Or barbarous or inhumane,
  But if it lay the least pretence
  To piety and godliness,
  Or tender-hearted conscience,
  And zeal for gospel truths profess,--
  Does sacred instantly commence,
  And all that dare but question it are straight
  Pronounced th' uncircumcised and reprobate,
  As malefactors that escape and fly
  Into a sanctuary for defence,
  Must not be brought to justice thence,
  Although their crimes be ne'er so great and high.
  And he that dares presume to do't
  Is sentenced and delivered up
  To Satan that engaged him to't."


Of all the drolleries of controversy none is more amusing than the manner
in which those who provoke a combat expect to lay down the laws of
retaliation. You must not strike this way! you must not parry that way! If
you don't take care, we shall never meddle with you again! We were not
_prepared_ for such as this! Why did we have anything to do with such a
testy person? M. Jourdain must needs show Nicole, his servant-maid, how
good a thing it was to be sure of fighting without being killed, by care
and tierce.[361] "Et cela n'est il pas beau d'être assuré de son fait quand
on se bat contre quelqu'un? Là, pousse moi un peu, pour voir. NICOLE. Eh
bien! quoi? M. JOURDAIN. Tout beau. Hola! {219} Ho! doucement. Diantre soit
la coquine! NICOLE. Vous me dites de pousser. M. JOURDAIN. Oui; mais tu me
pousses en tierce, avant que de pousser en quarte, et tu n'as pas la
patience que je pare."

His colleague, my secular tutelary, who also made an anachronistic onset,
with his repartees and his retorts, before there was anything to fire at,
takes what I give by way of subsequent provocation with a good humor which
would make a convert of me if he could afford .01659265 ... of a grain of
logic. He instantly sent me his photograph for the asking, and another
letter in proof. The Thor-hammerer does nothing but grumble, except when he
tells a good story, which he says he had from Dr. Abernethy.[362] A Mr.
James Dunlop was popping at the Papists with a 666-rifled gun, when Dr.
Chalmers[363] quietly said, "Why, Dunlop, you bear it yourself," and handed
him a paper on which the numerals in

  I A C O B V S         D V N L O P V S
  1  100    5         500 5  50     5

were added up. This is almost as good as the _Filii Dei Vicarius_, the
numerical letters of which also make 666. No more of these crazy--I first
wrote _puerile_, but why should young cricketers be libelled?--attempts to
extract religious use from numerical vagaries, and to make God over all a
proposer of _salvation conundrums_: and no more of the trumpery hints about
future destiny which is too great a compliment to call blasphemous. If the
Doctor will cipher upon the letter in [Greek: en hôi metrôi metreite
metrêthêsetai humin][364] with _double Vahu_ cubic measure, he will perhaps
learn to leave off trying to frighten me into gathering grapes from thorns.

Mystical hermeneutics may be put to good use by out-of-the-way people. They
may be made to call the attention {220} of the many to a distinction well
known among the learned. The books of the New Testament have been for 1,500
years divided into two classes: the _acknowledged_ ([Greek:
homologoumena]), which it has always been paradox not to receive; and the
_controverted_ ([Greek: antilegomena]), about which there has always been
that difference of opinion which no scholar overlooks, however he may
decide for himself after balance of evidence. Eusebius,[365] who first (l.
3, c. 25) recorded the distinction--which was much insisted on by the early
Protestants--states the books which are questioned as doubtful, but which
yet are approved and acknowledged by _many_--or _the many_, it is not easy
to say which he means--to be the Epistles of James and Jude, the second of
Peter and the second and third of John. In other places he speaks
doubtingly of the Epistle to the Hebrews. The Apocalypse he does not even
admit into this class, for he proceeds as follows--I use the second edition
of the English folio translation (1709), to avert suspicion of bias from

"Among the _spurious_ [[Greek: nothoi]] let there be ranked both the work
entitled the _Acts of Paul_, and the book called _Pastor_, and the
_Revelation of Peter_: and moreover, that which is called the _Epistle of
Barnabas_, and that named the _Doctrines of the Apostles_: and moreover, as
I said, the _Revelation of John_ (if you think good), which some, as I have
said, do reject, but others allow of, and admit among those books which are
received as unquestionable and undoubted."

Eusebius, though he will not admit the Apocalypse even into the
_controverted_ list, but gives permission to call it _spurious_, yet
qualifies his permission in a manner which almost annihilates the
distinctive force of [Greek: nothos], and gives the book a claim to rank
(if you think good, again) in the controverted list. And this is the
impression received by {221} the mind of Lardner, who gives Eusebius fully
and fairly, but when he sums up, considers his author as admitting the
Apocalypse into the second list. A stick may easily be found to beat the
father of ecclesiastical history. There are whole faggots in writers as
opposite as Baronius and Gibbon, who are perhaps his two most celebrated
sons. But we can hardly imagine him totally misrepresenting the state of
opinion of those for whom and among whom he wrote. The usual plan, that of
making an author take the views of his readers, is more easy in his case
than in that of any other writer: for, as the riddle says, he is
You-see-by-us; and to this reading of his name he has often been subjected.
Dr. Nathaniel Lardner,[366] who, though heterodox in doctrine, tries hard
to be orthodox as to the Canon, is "sometimes apt to think" that the list
should be collected and divided as in Eusebius. He would have no one of the
controverted books to be allowed, by itself, to establish any doctrine.
Even without going so far, a due use of early opinion and long continued
discussion would perhaps prevent rational people from being induced by
those who have the _double Vahu_ to place the Apocalypse _above_ the
Gospels, which all the Bivahuites do in effect, and some are said to have
done in express words. But my especial purpose is to point out that an easy
way of getting rid of 665 out of 666 of the mystics is to require them to
establish the Apocalypse before they begin. See if they even know so much
as that there is a crowd of testimonies for and against, running through
the first four centuries, which makes this book the most difficult of the
whole Canon. Try this method, and you will escape beautiful, as the French
say. Dean Alford,[367] in Vol. IV, p. 8, of his New Testament, gives an
elaborate handling of this question. He concludes by saying that he cannot
{222} venture to refuse his consent to the tradition that the Apostle is
the author. This modified adherence, or non-nonadherence, pretty well
represents the feeling of orthodox Protestants, when learning and common
sense come together.

I have often, in former days, had the attempt made to place the Apocalypse
on my neck as containing prophecies yet unfulfilled. The preceding method
prevents success; and so does the following. It may almost be taken for
granted that theological system-fighters do not read the New Testament:
they hunt it for detached texts; they listen to it in church in that state
of quiescent nonentity which is called reverent attention: but they never
read it. When it is brought forward, you must pretend to find it necessary
to turn to the book itself: you must read "The revelation ... to show unto
his servants _things which must shortly come to pass_.... Blessed is he
that readeth ... _for the time is at hand_." You must then ask your mystic
whether things deferred for 1800 years were shortly to come to pass, etc.?
You must tell him that the Greek [Greek: en tachei], rendered "shortly," is
as strong a phrase as the language has to signify _soon_. The interpreter
will probably look as if he had never read this opening: the chances are
that he takes up the book to see whether you have been committing a fraud.
He will then give you some exquisite evasion: I have heard it pleaded that
the above was a _mere preamble_. This word _mere_ is all-sufficient: it
turns anything into nothing. Perhaps he will say that the argument is that
of the Papists: if so, tell him that there is no Christian sect but bears
true witness against some one or more absurdities in other sects.

An anonyme suggests that [Greek: en tachei] may not be "soon," it may be
"quickly, without reference to time when:" he continues thus, "May not time
be 'at hand' when it is ready to come, no matter how long delayed?" I now
understand what *** and *** meant when they borrowed my books and promised
to return them quickly, it was "without {223} reference to time when." As
to time at _hand_--provided you make a long _arm_--I admire the quirk, but
cannot receive it: the word is [Greek: engus], which is a word of
_closeness_ in time, in place, in reckoning, in kindred, etc.

Another gentleman is not surprised that Apocalyptic reading leads to a
doubt of the "canonicity" of the book: it ought not to rest on church
testimony, but on visible miracle. He offers me, or any reader of the
_Athenæum_, the "sight of a miracle to that effect, and within forty-eight
hours' journey (fare paid)." I seldom travel, and my first thought was
whether my carpet-bag would be found without a regular hunt: but, on
reading further, I found that it was only a concordance that would be
wanted. Forty hours' collection and numerical calculation of Greek nouns
would make it--should I happen to agree with the writer--many hundred
millions to one that Revelation xiii is superhuman. There is but one verse
(the fifth) which the writer does not see verified. I looked at this verse,
and was much startled. The Budget began in October 1863: should it last
until March 1867--it is now August 1866--it is clear that I am the first
Beast, and my paradoxers are the saints whom I persecute.

[The Budget _did_ terminate in March 1867: I hope the gentleman will be
satisfied with the resulting interpretation.]

The same opponent is surprised that I should suppose a thing which "comes
to pass" must be completed, and cannot contain what is to happen 1800 years
after. All who have any knowledge of English idiom know that a thing
_comes_ to pass when it happens, and _came_ to pass afterwards. But as the
original is Greek, we must look at the Greek: it is [Greek: dei genesthai]
for "must come to pass," and we know that [Greek: egeneto] is what is
usually translated "came to pass." No word of more finished completion
exists in Greek.

And now for a last round of biter-bit with the Thor-hammerer, of whom, as
in the other case, I shall take no {224} more notice until he can contrive
to surpass himself, which I doubt his being able to do. He informs me that
by changing A into [Hebrew: t] in my name he can make a 666 of _me_;
adding, "This is too hard for me, although not so for the Lord!" Sheer
nonsense! He could just as easily have directed to "Prof. De Morg[Hebrew:
t]n" as have assigned me apartment 7A in University College. It would have
been seen for whom it was intended: and if not, it would still have reached
me, for my colleagues have for many a year handed all out-of-the-way things
over to me. There is no 7A: but 7 is the Museum of Materia Medica. I took
the only hint which the address gave: I inquired for hellebore, but they
told me it was not now recognized, that the old notion of its value was
quite obsolete, and that they had nothing which was considered a specific
in senary or septenary cases. The great platitude is the reference of such
a difficulty as writing [Hebrew: t] for A to the Almighty! Not childish,
but fatuous: real childishness is delightful. I knew an infant to whom,
before he could speak plain, his parents had attempted to give notions of
the Divine attributes: a wise plan, many think. His father had dandled him
up-side-down, ending with, There now! Papa could not dance on his head! The
mannikin made a solemn face, and said, _But Dod tood_! I think the Doctor
has rather mistaken the way of becoming as a little child, intended in
Matt. xviii. 3: let us hope the will may be taken for the deed.

Two poets have given images of transition from infancy to manhood:
Dryden,--for the Hind is Dryden himself on all fours! and Wordsworth, in
his own character of broad-nailed, featherless biped:

 "The priest continues what the nurse began,
  And thus the child imposes on the man."
   "The child's the father of the man,
    And I could wish my days to be
    Bound each to each by natural piety."


In Wordsworth's aspiration it is meant that sense and piety should grow
together: in Dryden's description a combination of Mysticism And Bigotry
(can this be the _double Vahu_?), personified as "the priest,"--who always
catches it on this score, though the same spirit is found in all
associations,--succeeds the boguey-teaching of the nurse. Never was the
contrast of smile and scowl, of light and darkness, better seen than in the
two pictures. But an acrostic distinction may be drawn. When mysticism
predominates over bigotry, we have the grotesque picturesque, and the
natural order of words gives us _Mab_, an appropriate suggestion. But when
bigotry has the upper hand, we see _Bam_, which is just as appropriate; for
bigotry nearly always deals with facts and logic so as to require the
application of at least one of the minor words by which dishonesty is
signified. I think that M is the Doctor's initial, and that Queen Mab
tickles him in his sleep with the sharp end of a 6.

(_Monday, August 21._) Three weeks having elapsed without notice from me of
the Doctor, I receive a reminder of his existence, in which I find that as
I am the Daniel who judges the Magi of Babylon, it is to be pointed out
that Daniel "bore a certain number, that of a man (beloved), Daniel, ch.
10. v. 11, and which you certainly do not." Then, "by Greek power,"
Belteshazzar is made = 666. Here is another awkward imitation of the way of
a baby child. When you have sported with the tiny creature until it runs
away offended, by the time you have got into conversation again you will
find the game is to be renewed: a little head peeps out from a hiding-place
with "I don't love you." The proper rejoinder is, "Very well! then I'll
have pussy." But in the case before me there is a rule of three sums to do;
as baby : Pussy Dr. :: 666 : the answer required. I will work it out, if I

The squaring of the circle and the discovery of the Beast are the two
goals--and goals also--of many unbalanced intellects, and of a few
instances of the better kind. {226} I might have said more of 666, but I am
not deep in its bibliography. A work has come into my hands which contains
a large number of noted cases: to some of my readers it will be a treat to
see the collection; and the sight will perhaps be of some use to those who
have read controversy on the few celebrated cases which are of general
notoriety. It is written by a learned decipherer, a man who really knew the
history of the subject, the Rev. David Thom,[368] of Bold Street Chapel,
Liverpool, who died, I am told, a few years ago.

Anybody who reads his book will be inclined to parody a criticism which was
once made on Paley's[369] Evidences--"Well! if there be anything in
Christianity, this man is no fool." And, if he should chance to remember
it, he will be strongly reminded of a sentence in my opening chapter,--"The
manner in which a paradoxer will show himself, as to sense or nonsense,
will not depend upon what he maintains, but upon whether he has or has not
made a sufficient knowledge of what has been done by others, _especially as
to the mode of doing it_, a preliminary to inventing knowledge for
himself." And this is reinforced by the fact that Mr. Thom, though a
scholar, was not conspicuous for learning, except in this his great
pursuit. He was a paradoxer on other points. He reconciled Calvinism and
eternal reprobation with Universalism and final salvation; showing these
two doctrines to be all one.

This gentleman must not be confounded with the Rev. John Hamilton Thom[370]
(no relation), at or near the same {227} time and until recently, of
Renshaw Street Chapel, Liverpool who was one of the minority in the
Liverpool controversy when, nearly thirty years ago, _three_ heretical
Unitarian schooners exchanged shotted sermons with _thirteen_ Orthodox
ships of the line, and put up their challengers' dander--an American
corruption of _d--d anger_--to such an extent, by quiet and respectful
argument, that those opponents actually addressed a printed intercession to
the Almighty for the Unitarian triad, as for "Jews, Turks, Infidels, and
Heretics." So much for the distinction, which both gentlemen would thank me
for making very clear: I take it quite for granted that a guesser at 666
would feel horrified at being taken for a Unitarian, and that a Unitarian
would feel queerified at being taken for a guesser at 666. Mr. David Thom's
book is _The Number and Names of the Apocalyptic Beasts_, Part I, 1848,
8vo.: I think the second part was never published. I give the Greek and
Latin solutions, omitting the Hebrew: as usual, all the Greek letters are
numeral, but only M D C L X V I of the Latin. I do not give either the
decipherers or their reasons: I have not room for this; nor would I, if I
could, bias my reader for one rather than another.

D. F. Julianus Cæsar Atheus (or Aug.[371]); Diocles Augustus; Ludovicus;
Silvester Secundus; Linus Secundus; {228} Vicarius Filii Dei; Doctor et Rex
Latinus; Paulo V. Vice-Deo; Vicarius Generalis Dei in Terris; Ipse
Catholicæ Ecclesiæ Visibile Caput; Dux Cleri; Una, Vera, Catholica,
Infallibilis Ecclesia; Auctoritas politica ecclesiasticaque Papalis (Latina
will also do); Lutherus Ductor Gregis; Calvinus tristis fidei interpres;
Dic Lux ; Ludvvic; Will. Laud; [Greek: Lateinos];[372] [Greek: hê latinê
basileia]; [Greek: ekklêsia italika]; [Greek: euanthas]; [Greek: teitan];
[Greek: arnoume]; [Greek: lampetis]; [Greek: ho nikêtês]; [Greek: kakos
hodêgos]; [Greek: alêthês blaberos]; [Greek: palai baskanos]; [Greek: amnos
adikos]; [Greek: antemos]; [Greek: gensêrikos]; [Greek: euinas]; [Greek:
Benediktos]; [Greek: Bonibazios g. papa x. ê. e. e. a.], meaning Boniface
III. Pope 68th, bishop of bishops the first! [Greek: oulpios]; [Greek: dios
eimi hê hêras]; [Greek: hê missa hê papikê]; [Greek: loutherana]; [Greek:
saxoneios]; [Greek: Bezza antitheos] (Beza); [Greek: hê alazoneia biou];
[Greek: Maometis]; [Greek: Maometês b.]; [Greek: theos eimi epi gaiês];
[Greek: iapetos]; [Greek: papeiskos]; [Greek: dioklasianos]; [Greek:
cheina]; [Greek: braski]; [Greek: Ion Paune]; [Greek: koupoks]; (cowpox,
[Greek: s] being the _vau_; certainly the {229} vaccinated have the mark of
the Beast); [Greek: Bonnepartê]; [Greek: N. Bonêparte]; [Greek: euporia];
[Greek: paradosis]; [Greek: to megathêrion].

All sects fasten this number on their opponents. It is found in _Martin
Lauter_, affirmed to be the true way of writing the name, by carrying
numbers through the Roman Alphabet. Some Jews, according to Mr. Thorn,
found it in [Hebrew: JSHW NTSRJ] _Jesus of Nazareth_. I find on inquiry
that this satire was actually put forth by some medieval rabbis, but that
it is not idiomatic: it represents quite fairly "Jesus Nazarene," but the
Hebrew wants an article quite as much as the English wants "the."

Mr. David Thom's own solution hits hard at all sides: he finds a 666 for
both beasts; [Greek: hê phrên] (the mind) for the first, and [Greek:
ekklêsiai sarkikai] (fleshly churches) for the second. A solution which
embodies all mental philosophy in one beast and all dogmatic theology in
the other, is very tempting: for in these are the two great supports of
Antichrist. It will not, however, mislead me, who have known the true
explanation a long time. The three sixes indicate that any two of the three
subdivisions, Roman, Greek, and Protestant, are, in corruption of
Christianity, six of one and half a dozen of the other: the distinctions of
units, tens, hundreds, are nothing but the old way (1 Samuel xviii. 7, and
Concordance at _ten_, _hundred_, _thousand_) of symbolizing differences of
number in the subdivisions.

It may be good to know that, even in speculations on 666, there are
different degrees of unreason. All the diviners, when they get a colleague
or an opponent, at once proceed to reckon him up: but some do it in play
and some in earnest. Mr. David Thom found a young gentleman of the name St.
Claire busy at the Beast number: he forthwith added the letters in [Greek:
st klaire] and found 666: this was good fun. But my spiritual tutelary,
when he found that he could not make a beast of me, except by changing
[Hebrew: A] into [Hebrew: T], solemnly referred the difficulty to the
Almighty: this was poor earnest. {230}

I am glad I did not notice, in time to insert it in the _Athenæum_, a very
remarkable paradoxer brought forward by Mr. Thom, his friend Mr.
Wapshare[373]: it is a little too strong for the general public. In the
_Athenæum_ they would have seen and read it: but this book will be avoided
by the weaker brethren. It is as follows:

"God, the Elohim, was six days in creating all things, and having made MAN
he entered into his rest. He is no more seen as a Creator, as Elohim, but
as Jehovah, the _Lord_ of the Sabbath, and the Spirit of life in MAN, which
Spirit worketh _sin in the flesh_; for the Spirit of Love, in all flesh, is
Lust, or the spirit of a beast, So Rom. vii. And which Spirit is
_crucified_ in the flesh. He then, as Jehovah--as the power of the Law,
_in_ and _over_ all flesh, John viii. 44--increases that which he has made
as the Elohim, and his power shall last for 6 days, or 6 periods of time,
computed at a millennium of years; and at the end of which six days, he who
is the Spirit of all flesh shall manifest himself as the Holy Spirit of
Almighty Love, and of all truth; and so shall the Church have her Sabbath
of Rest--all contention being at an end. This is, as well as I may now
express it, my solution of the mystery in Hebrew, and in Greek, and also in
Latin, IHS. For he that was lifted up _is_ King of the Jews, and is the
Lord of all Life, working in us, both to will and to do; as is manifest in
the Jews--they slaying him that his blood might be _good_ for the healing
of the nations, of all people and tongues. As the Father of all _natural_
flesh, he is the Spirit of Lust, as in all _beasts_; as the Father, or King
of the Jews, he is the Devil, as he himself witnesseth in John viii.,
already referred to. As lifted up, he is transformed into the Spirit of
Love, a light to the Gentiles, and the glory of his people Israel.... For
there is but ONE God, ONE Lord, ONE Spirit, ONE body, etc. and he who was
Satan, the Spirit of life in that body, is, in {231} Christ crucified, seen
in the Spirit that is in all, and through all and over all, God blessed for

All this seems well meant, and Mr. Thom prints it as convinced of its
piety, and "pronounces no opinion." Mystics of all sorts! see what you may
come to, or what may come to you! I have inserted the above for your good.

There is nothing in this world so steady as some of the paradoxers. They
are like the spiders who go on spinning after they have web enough to catch
all the flies in the neighborhood, if the flies would but come. They are
like the wild bees who go on making honey which they never can eat, proving
_sic vos non vobis_ to be a physical necessity of their own contriving. But
nobody robs their hives: no, unlike the bees, they go about offering their
ware to any who will take it as a gift. I had just written the last
sentence (Oct. 30, 1866, 8.45 A.M.) when in comes the second note received
this morning from Dr. Thorn: at 1.30 P.M. came in a third. These arise out
of the above account of the Rev. D. Thom, published Oct. 27: three notes
had arrived before.

For curiosity I give one day's allowance, supposing these to be all: more
may arrive before night.

29th Oct. 1866.

"Dear Sir,--

In re [swastika].[374]

"So that 'Zaphnath Paaneah' may be after all the revealer of the 'Northern
Tau' [Greek: Phaneroô]--To make manifest, shew, or explain; and this may
satisfy the House of Joseph in Amos 5^c. While Belteshazzar = 666 may be
also satisfactory to the House of David, and so we may have Zech. 10^c.
6^v. in operation when Ezekiel 37^c. 16^v. has been realised;--but there,
what is the use of writing, it is all Coptic {232} to a man who has not
[swastika], The Thau of the North, the double Vahu [Hebrew: W\qamats\W].
Look at Jeremiah 3^c. 8^v. and then to Psalm 83 for 'hidden ones' [Hebrew:
TS\sheva\PW\dagesh\N\segol\Y Y\sheva\HW\qamats\H]--The Zephoni Jehovah, and
say whether they have any connection with the Zephon _Thau_. The Hammer of
Thor of Jeremiah 23^c. 29^v. as I gave you in No. 3 of my present edition.

Yours truly


_By Greek Power._

  C =  20
  H =   8
  E =   5
  V =   6
  A =   1
  L =  30
  I =  10
  E =   5
  R = 100

  A =   1
  U = 400

  C =  20
  I =  10
  N =  50

There will be thousands of Morgans who will be among the wise and prudent
of Hosea 14^c. 9^v. when the Seventh Angel sounds, let me number _that One_
by Greek, Rev. 17^c. 1^v: {233}

    S = 200
    E  =   5
  × V  =   6
    E  =   5
    N  =  50
    T  = 300
    H  =   8

    A =   1
    N  =  50
  × G  =   6
    E  =   5
    L  =  30

    V and G = 12 ought to be equal to one Gammadion or ^3[swastika]3 × 4 =
    12, what say you?

London, October 29, 1866.

"Dear Sir,--

In re [swastika] versus [maltese cross].

However pretentious the X or [maltese cross] may be, and it is peculiarly
so just now in this land; after all it is only made of two Roman V's--and
so is only = [ one inverted](10)--and therefore is not the perfect number
12 of Revel^n, but is the mark of the goddess _Decima_!

Yours truly


Had the _one_ who sent forth a pastoral (Romish) the other day, remained
amongst the faithful expectants, see how he would have numbered, whereas he
sold himself for the privilege of signing

[maltese cross] HENRY E. MANNING.[375]


_By English Key._

  H =   8
  E =   5
  N =  40
  R =  80
  Y = 140

  E =   5
  D =   4
  W = 120
  A =   1
  R =  80
  D =   4

  M =  30
  A =   1
  N =  40
  N =  40
  I =   9
  N =  40
  G =   7
  [swastika] = 12

    Can you now understand the difference between [swastika] and [maltese
    cross] or X? Look to my challenge.

Cutting from newspaper:--


Rome (_via_ Marseilles), October

Mr. Gladstone has paid a visit to the Pope.

_By Greek Power._

  G =   6
  L =  30
  A =   1
  D =   4
  S = 200
  T = 300
  O =  70
  N =  50
  E =   5

And what then [swastika]?


In other letters _John Stuart Mill_ is 666 if the _a_ be left out;
_Chasuble_ is perfect. _John Brighte_[376] is a _fait accompli_; and I am
asked whether intellect can account for the final e. Very easily: this
Beast is not the M. P., but another person who spells his name differently.
But if John Sturt Mill and John Brighte choose so to write themselves, they

A curious collection; a mystical phantasmagoria! There are those who will
try to find meaning: there are those who will try to find purpose.

 "And some they said--What are you at?
  And some--What are you arter?"

My account of Mr. Thom and his 666 appeared on October 27: and on the 29th
I received from the editor a copy of Mr. Thom's sermons published in 1863
(he died Feb. 27, 1862) with best wishes for my health and happiness. The
editor does not name himself in the book; but he signed his name in my
copy: and may my circumference never be more than 3-1/8 of my diameter if
the signature, name and writing both, were not that of my [circle square]
ing friend Mr. James Smith! And so I have come in contact with him on 666
as well as on [pi]! I should have nothing left to live for, had I not
happened to hear that he has a perpetual motion on hand. I returned thanks
and kind regards: and Miss Miggs's words--"Here's forgivenesses of
injuries! here's amicablenesses!"--rang in my ears. But I was made slightly
uncomfortable: how could the war go on after this armistice? Could I ever
make it understood that the truce only extended to the double Vahu and
things thereunto relating? It was once held by seafaring men that there was
no peace with Spaniards beyond the line: I was determined that there must
be no concord with J. S. inside the circle; that this must be a special
exception, like Father Huddleston {236} and old Grouse in the gun-room. I
was not long in anxiety; twenty-four hours after the book of sermons there
came a copy of the threatened exposure--_The British Association in
Jeopardy, and Professor De Morgan in the Pillory without hope of escape_.
By James Smith, Esq. London and Liverpool, 8vo., 1866 (pp. 94). This
exposure consists of reprints from the _Athenæum_ and _Correspondent_: of
things new there is but one. In a short preface Mr. J. S. particularly
recommends to "_read to the end_." At the end is an appendix of two pages,
in type as large as the work; a very prominent peroration. It is an article
from the _Athenæum_, left out of its place. In the last sentence Mr. J.
Smith, who had asked whether his character as an honest Geometer and
Mathematician was not at stake, is warned against the _fallacia plurium
interrogationum_.[377] He is told that there is not a more honest
what's-his-name in the world: but that as to the counter which he calls his
character as a mathematician, he is assured that it has been staked years
ago, and lost. And thus truth has the last word. There is no occasion to
say much about reprints. One of them is a letter [that given above] of
August 25, 1865, written by Mr. J. S. to the _Correspondent_. It is one of
his quadratures; and the joke is that I am made to be the writer: it
appears as what Mr. J. S. hopes I shall have the sense to write in the
_Athenæum_ and forestall him. When I saw myself thus quoted--yes! quoted!
double commas, first person--I felt as I suppose did Wm. Wilberforce[378]
when he set eyes on the affectionate benediction of the potato which
waggish comrades had imposed on a raw Irish reporter as part of his speech.
I felt as Martin[379] of {237} Galway--kind friend of the poor dumb
creatures!--when he was told that the newspapers had put him in Italics. "I
appeal to you, Mr. Speaker! I appeal to the House! Did I speak in Italics?
Do I ever speak in Italics?" I appeal to editor and readers, whether I ever
squared the circle until a week or two ago, when I gave my charitable mode
of reconciling the discrepant cyclometers.

The absurdity of the imitation of symbolic reasoning is so lusciously rich,
that I shall insert it when I make up my final book. Somebody mastered
Spanish merely to read Don Quixote: it would be worth while to learn a
little algebra merely to enjoy this a b-istical attack on the windmills.
The principle is, Prove something in as roundabout a way as possible,
mention the circle once or twice irrelevantly in the course of your proof,
and then make an act of Q. E. D. in words at length. The following is
hardly caricature:--

To prove that 2 and 2 make 5. Let a = 2, b = 5: let c = 658, the number of
the House: let d = 666, the number of the Beast. Then of necessity d = a +
b + c + 1; so that 1 is a harmonious and logical quantification of the
number of which we are to take care. Now, b, the middle of our digital
system, is, by mathematical and geometrical combination, a mean between 5 +
1 and 2 + 2. Let 1 be removed to be taken care of, a thing no real
mathematician can refuse without serious injury to his mathematical and
geometrical reputation. It follows of necessity that 2 + 2 = 5, _quod erat
demonstrumhorrendum_. If Simpkin & Marshall have not, after my notice, to
account for a gross of copies more than would have gone off without me, the
world is not worthy of its James Smith!

The only fault of the above is, that there is more {238} connection than in
the process of Faber Cyclometricus: so much, in fact, that the blunders are
visible. The utter irrelevance of premises to conclusion cannot be
exhibited with the requisite obscurity by any one who is able to follow
reasoning: it is high art displayed in a certain toning down of the _ægri
somnia_, which brings them to a certain look of reproach to reasoning which
I can only burlesque. Mr. J. S. produces something which resembles argument
much as a chimpanzee in dolor, because balked of his dinner, resembles a
thinking man at his studies. My humble attempt at imitation of him is more
like a monkey hanging by his tail from a tree and trying to crack a
cocoa-nut by his chatter.

I could forgive Mr. J. S. anything, properly headed. I would allow him to
prove--_for himself_--that the Quadrature of the Circle is the child of a
private marriage between the Bull Unigenitus and the Pragmatic Sanction,
claiming tithe of onions for repeal of the Mortmain Act, before the Bishops
in Committee under the kitchen table: his mode of imitating reason would do
this with ease. But when he puts his imitation into my mouth, to make me
what _he_ calls a "real mathematician," my soul rises in epigram against
him. I say with the doll's dressmaker--such a job makes me feel like a
puppet's tailor myself--"He ought to have a little pepper? just a few
grains? I think the young man's tricks and manners make a claim upon his
friends for a little pepper?" De Fauré[380] and Joseph Scaliger[381] come
into my head: my reader may look back for them.

 "Three circlesquarers to the manner born,
  Switzerland, France, and England did adorn,
  De Fauré in equations did surpass,
  Joseph at contradictions was an ass.
  Groaned Folly, I'm used up! What shall I do
  To make James Smith? Grinned Momus, _Join the two_!"


As to my _locus poenitentiæ_,[382] the reader who is fit to enjoy the
letter I have already alluded to will see that I have a soft and easy
position; that the thing is really a _pillowry_; and that I am, like
Perrette's pot of milk,

 "Bien posé sur un coussinet."[383]

Joanna Southcott[384] never had a follower who believed in her with more
humble piety than Mr. James Smith believes in himself. After all that has
happened to him, he asks me with high confidence to "favor the writer with
a proof" that I still continue of opinion that "the best of the argument is
in my jokes, and the best of the joke is in his arguments." I will not so
favor him. At the very outset I told him in plain English that he has the
whiphand of all the reasoners in the world, and in plain French that _il a
perdu le droit d'être frappé de l'évidence_[385]; I might have said
_pendu_.[386] To which I now add, in plain Latin, _Sapienti pauca, indocto
nihil_.[387] The law of Chancery says that he who will have equity must do
equity: the law of reasoning says that he who will have proof must see

The introduction of things quite irrelevant, by way of reproach, is an
argument in universal request: and it often happens that the argument so
produced really tells against the producer. So common is it that we forget
how boyish it is; but we are strikingly reminded when it actually comes
from a boy. In a certain police court, certain small boys were arraigned
for conspiring to hoot an obnoxious individual on his way from one of their
school exhibitions. This proceeding was necessary, because there seemed to
be a permanent conspiracy to annoy the gentleman; and the {240} masters did
not feel able to interfere in what took place outside the school. So the
boys were arraigned; and their friends, as silly in their way as
themselves, allowed one of them to make the defence, instead of employing
counsel; and did not even give them any useful hints. The defence was as
follows; and any one who does not see how richly it sets off the defences
of bigger boys in bigger matters has much to learn. The innocent conviction
that there was answer in the latter part is delightful. Of course fine and
recognizance followed.

A---- said the boys had received great provocation from B----. He was
constantly threatening them with a horsewhip which he carried in his hand
[the boy did not say what had passed to induce him to take such a weapon],
and he had repeatedly insulted the master, which the boys could not stand.
B---- had in his own drawing-room told him (A----) that he had drawn his
sword against the master and thrown away the scabbard. B---- knew well that
if he came to the college he would catch it, and then he went off through a
side door--which was no sign of pluck; and then he brought Mrs. B---- with
him, thinking that her presence would protect him.

My readers may expect a word on Mr. Thom's sermons, after my account of his
queer doings about 666. He is evidently an honest and devout man, much
wanting in discrimination. He has a sermon about private _judgment_, in
which he halts between the logical and legal meanings of the word. He
loathes those who apply their private judgment to the word of God: here he
means those who decide what it _ought to be_. He seems in other places
aware that the theological phrase means taking right to determine what it
_is_. He uses his own private judgment very freely, and is strong in the
conclusion that others ought not to use theirs except as he tells them how;
he leaves all the rest of mankind free to think with him. In this he is not
original: his fame must rest on his senary tripod. {241}


Mr. James Smith's procedures are not caricature of reasoning; they are
caricature of blundering. The old way of proving that 2 = 1 is solemn
earnest compared with his demonstrations. As follows:[388]

  Let x = 1
  Then x^2 = x
  And x^2 - 1 = x - 1
  Divide both sides by x - 1; then x + 1 = 1; but x = 1, whence 2 = 1.

When a man is regularly snubbed, bullied, blown up, walked into, and put
down, there is usually some reaction in his favor, a kind of deostracism,
which cannot bear to hear him always called the blunderer. I hope it will
be so in this case. There is nothing I more desire than to see _sects_ of
paradoxers. There are fully five thousand adults in England who ought to be
the followers of some one false quadrature. And I have most hope of 3-1/8,
because I think Mr. James Smith better fitted to be the leader of an
organized infatuation than any one I know of. He wants no pity, and will
get none. He has energy, means, good humor, strong conviction, character,
and popularity in his own circle. And, most indispensable point of all, he
sticks at nothing;

 "In coelum jusseris, ibit."[389]

When my instructor found I did not print an acceptance of what I have
quoted, he addressed me as follows (_Corr._, Sept 23):--

"In this life, however, we must do our duty, and, when {242} necessary, use
the rod, not in a spirit of revenge, but for the benefit of the culprit and
the good of society. Now, Sir, the opportunity has been thrown in your way
of slipping out of the pillory without risk of serious injury; but, like an
obstinate urchin, you have chosen to quarrel with your opportunity and
remain there, and thus you compel me to deal with you as schoolmasters used
to do with stupid boys in bygone days--that is to say, you force me to the
use of the critic's rod, compel me to put you where little Jack Horner sat,
and, as a warning to other naughty boys, to ornament you with a dunce's
cap. The task I set you was a very simple one, as I shall make manifest at
the proper time."

In one or more places, as well as this, Mr. Smith shows that he does not
know the legend of little Jack Horner, whom he imagines to be put in the
corner as a bad boy. This is curious; for there had been many allusions to
the story in the journal he was writing in, and the Christmas pie had
become altered into the Seaforth [pi].

Mr. Smith is satisfied at last that--what between argument and punishment
he has convinced me. He says (_Corr._, Jan. 27, 1866): "I tell him without
hesitation that he knows the true ratio of diameter to circumference as
well as I do, and if he be wise he will admit it." I should hope I do, and
better; but there is no occasion to admit what everybody knows.

I have often wished that we could have a slight glimpse of the reception
which was given to some of the old cyclometers: but we have nothing, except
the grave disapprobation of historians. I am resolved to give the New
Zealander a chance of knowing a little more than this about one of them at
least; and, by the fortunate entrance into life of the _Correspondent_, I
am able to do it. I omit sober mathematical answers, of which there were
several. The following letter is grave earnest:

"Sir,--I have watched Mr. James Smith's writings on this subject from the
first, and I did hope that, as the more {243} he departs from truth the
more easy it must be to refute him, [this by no means always true] some of
your correspondents would by this time have done so. I own that I am unable
to detect the fallacy of his argument; and I am quite certain that '[Pi]'
is wrong, in No. 23, where he declares that Mr. Smith is 'ignorant of the
very elements of mathematical truth.' I have observed an immense amount of
geometrical reasoning on his part, and I cannot see that it is either fair
or honest to deny this, which may be regarded as the 'elements' of
mathematical truth. Would it not be better for '[Pi]' to answer Mr. Smith,
to refute his arguments, to point out their fallacies, and to save learners
from error, than to plunge into gross insult and unmanly abuse? Would it
not be well, also, that Professor De Morgan should favour us with a little

"I have hitherto seen no attempt to overthrow Mr. Smith's arguments; I
trust that this will not continue, since the subject is one of immense
importance to science in general, especially to nautical science, and all
that thereto belongs.

Yours, etc.,


On looking at this homoeopathic treatment of the 3-1/8
quadrature--remember, homoeopathic, _similia similibus_,[390] not
infinitesimal--and at the imputation thrown upon it, I asked myself, what
_is_ vulgarity? No two agree, except in this, that every one sees vulgarity
in what is directed against himself. Mark the world, and see if anything be
so common as the description of the other side's remarks as "vulgar attempt
at wit." "I suppose you think that very witty:" the answer is "No my
friend! your remark shows that you feel it as wit, so that the purpose is
answered; I keep my razor for something else than cutting blocks;" I am
inclined to think that "out of place" is a necessary attribute of true
vulgarity. And further, it is to be noticed that nothing is {244}
unproducible--_salvo pudore_[391]--which has classical authority, modern or
ancient, in its favor. "He is a vulgar fellow; I asked him what he was
upon, and what do you think he answered, My legs!"--"Well, and has he not
justification? what do you find in Terence? _Quid agitur? Statur._"[392] I
do not even blench from my principle where I find that it brings what is
called "taking a sight" within permissible forms of expression: Rabelais
not only establishes its antiquity, but makes it English. Our old
translation[393] has it thus (book 2. ch. 19):

"Then made the Englishman this sign. His left hand, all open, he lifted up
into the air, then instantly shut into his fist the four fingers thereof;
and his thumb extended at length he placed upon the tip of his nose.
Presently after he lifted up his right hand all open and abased and bent it
downwards, putting the thumb thereof in the very place where the little
finger of the left hand did close in the fist, and the four right hand
fingers he softly moved in the air. Then contrarily he did with the right
hand what he had done with the left, and with the left what he had done
with the right."

An impressive sight! The making of a fist of the left hand is a great
addition of power, and should be followed in modern practice. The gentle
sullation of the front fingers, with the clenched fist behind them, says as
plainly as possible, Put _suaviter in modo_ in the van, but don't forget to
have _fortiter in re_[394] in the rear.


My Budget was announced (March 23, 1867) for completion on the 30th. Mr.
James Smith wrote five letters, one before the completion, four after it;
the five contained 68 pages of quarto letter paper. Mr. J. S. had picked up
a clerical correspondent, with whom he was in the heat of battle.

"_March 27._--Dear Sir. Very truly yours. Duty; for my own sake; just time
left to retrieve my errors; sends copy of letter to clergyman; new proof
never before thought of; merest tyro would laugh if I were to stifle it,
whether by rhodomontade or silent contempt; keep your temper. I shall be
convinced; and if world be right in supposing me incapable of a foul act, I
shall proclaim glorious discovery in the _Athenæum_.

"_April 15._--Sir,... My dear Sir, Your sincere tutelary. Copy of another
letter to clergyman; discovery tested by logarithms; reasons such as none
but a knave or a sinner can resist. Let me advise you to take counsel
before it is too late! Keep your temper. Let not your _pride_ get the
better of your discretion! Screw up your courage, my good friend, and
_resolve_ to show the world that you are an _honest_ man....

"_April 20._--Sir ... Your very sincere and favorite tutelary. I have long
played the _cur_, snapping and snarling...; suddenly lost my power, and
became _half-starved_ dog without _spirit_ to bark; try if air cannot
restore me; calls himself the _thistle_ in allusion to my other tutelary,
the _thorn_; Would I prefer his next work to be, 'A whip for the
Mathematical Cur, Prof. De M.' In some previous letter which I have
mislaid, he told me his next would be 'a muzzle for the Mathematical Bull
dog, Prof. De M.'

"_April 23._--Sir. Very sincerely yours. More letters to clergyman; you may
as well knock your head against a stone wall to improve your intellect as
attempt to controvert my proofs. [I thought so too; and tried neither].

"_May 6._--My dear Sir. Very sincerely yours. All to myself, and nothing to

"_July 2._--No more in this interval. All that precedes is a desperate
attempt to induce me to continue my descriptions: notoriety at any price."

I dare say the matter is finished: the record of so marked an instance of
self-delusion will be useful.

I append to the foregoing a letter from Dr. Whewell[395] to Mr. James
Smith. The Master of Trinity was conspicuous as a rough customer, an
intellectual bully, an overbearing disputant: the character was as well
established as that of Sam Johnson. But there was a marked difference. It
was said of Johnson that if his pistol missed fire, he would knock you down
with the butt end of it: but Whewell, in like case, always acknowledged the
miss, and loaded again or not, as the case might be. He reminded me of
Dennis Brulgruddery, who says to Dan, Pacify me with a good reason, and
you'll find me a dutiful master. I knew him from the time when he was my
teacher at Cambridge, more than forty years. As a teacher, he was anything
but dictatorial, and he was perfectly accessible to proposal of objections.
He came in contact with me in his slashing way twice in our after joint
lives, and on both occasions he acknowledged himself overcome, by that
change of manner, and apologetic mode of continuance, which I had seen him
employ towards others under like circumstances.

I had expressed my wish to have a _thermometer of probability_, with
impossibility at one end, as 2 and 2 make 5, and necessity at the other, as
2 and 2 make 4, and a graduated rise of examples between them. Down came a
blow: "What! put necessary and contingent propositions together! It's
absurd!" I pointed out that the two kinds of necessity are but such
extremes of probability as 0 and [infinity] are of number, and illustrated
by an urn with 1 white and _n_ black {247} balls, _n_ increasing without
limit. It was frankly seen, and the point yielded; a large company was

Again, in a large party, after dinner, and politics being the subject, I
was proceeding, in discussion with Mr. Whewell, with "I think"...--"Ugh!
_you_ think!" was the answer. I repeated my phrase, and gave as a reason
the words which Lord Grey[396] had used in the House of Lords the night
before (the celebrated advice to the Bishops to set their houses in order).
He had not heard of this, and his manner changed in an instant: he was the
rational discutient all the rest of the evening, having previously been
nothing but a disputant with all the distinctions strongly marked.

I have said that Whewell was gentle with his pupils; it was the same with
all who wanted teaching: it was only on an armed enemy that he drew his
weapon. The letter which he wrote to Mr. J. Smith is an instance: and as it
applies with perfect fidelity to the efforts of unreasoning above
described, I give it here. Mr. James Smith is skilfully exposed, and felt
it; as is proved by "putting the writer in the stocks."

"The Lodge, Cambridge, September 14th, 1862.

"Sir,--I have received your explanation of your proposition that the
circumference of the circle is to its diameter as 25 to 8. I am afraid I
shall disappoint you by saying that I see no force in your proof: and I
should hope that you will see that there is no force in it if you consider
this: In the whole course of the proof, though the word cycle occurs, there
is no property of the circle employed. You may do this: you may put the
word _hexagon_ or _dodecagon_, or any other word describing a polygon in
the place of _Circle_ in your proof, and the proof would be just as good as
before. Does not this satisfy you that you cannot have proved a property of
that special figure--a circle? {248}

"Or you may do this: calculate the side of a polygon of 24 sides inscribed
in a circle. I think you are a Mathematician enough to do this. You will
find that if the radius of the circle be one, the side of this polygon is
.264 etc. Now, the arc which this side subtends is according to your
proposition 3.125/12 = .2604, and therefore the chord is greater than its
arc, which you will allow is impossible.

"I shall be glad if these arguments satisfy you, and

"I am, Sir, your obedient Servant,



In the debate of May, 1866, on Electoral Qualifications, a question arose
about arithmetical capability. Mr. Gladstone asked how many members of the
House could divide 1330l. 7s. 6d. by 2l. 13s. 8d. Six hundred and
fifty-eight, answered one member; the thing cannot be done, answered
another. There is an old paradox to which this relates: it arises out of
the ignorance of the distinction between abstract and concrete arithmetic.
_Magnitude_ may be divided by _magnitude_; and the answer is number: how
often does 12d. contain 4d.; answer three times. _Magnitude_ may be divided
by _number_, and the answer is _magnitude_: 12d. is divided in four equal
parts, what is each part? Answer three _pence_. The honorable objector,
whose name I suppress, trusting that he has mended his ways, gave the
following utterance:

"With regard to the division sum, it was quite possible to divide by a sum,
but not by money. How could any one divide money by 2l. 16s. 8d.?
(Laughter.) The question might be asked, 'How many times 2s. will go into
1l.?' but that was not dividing by money; it was simply dividing 20 by 2.
He might be asked, 'How many times will 6s. 8d. go into a pound?' but it
was only required to divide 240 by 80. If the right hon. gentleman were to
ask the hon. {249} member for Brighton (Professor Fawcett),[397] or any
other authority, he would receive the same answer--viz., that it was
possible to divide by a sum, but not by money. (Hear.)"

I shall leave all comment for the second edition, if I publish one.[398] I
shall be sure to have something to laugh at. Anything said from a
respectable quarter, or supposed to be said, is sure to find defenders. Sam
Johnson, a sound arithmetician, comparing himself, and what he alone had
done in three years, with forty French Academicians and their forty years,
said it proved that an Englishman is to a Frenchman as 40 × 40 to 3, or as
1600 to 3. Boswell, who was no great hand at arithmetic, made him say that
an Englishman is to a Frenchman as 3 to 1600. When I pointed this out, the
supposed Johnson was defended through thick and thin in _Notes and

I am now curious to see whether the following will find a palliator. It is
from "Tristram Shandy," book V. chapter 3. There are two curious idioms,
"for for" and "half in half"; but these have nothing to do with my point:

"A blessing which tied up my father's tongue, and a misfortune which set it
loose with a good grace, were pretty equal: sometimes, indeed, the
misfortune was the better of the two; for, for instance, where the pleasure
of harangue was as _ten_, and the pain of the misfortune but as _five_, my
father gained half in half; and consequently was as well again off as if it
had never befallen him."

This is a jolly confusion of ideas; and wants nothing but a defender to
make it perfect. A person who invests five {250} with a return of ten, and
one who loses five with one hand and gains ten with the other, both leave
off five richer than they began, no doubt. The first gains "half in half,"
more properly "half _on_ half," that is, of the return, 10, the second 5 is
gain upon the first 5 invested. "Half _in_ half" is a queer way of saying
cent. per cent. If the 5l. invested be all the man had in the world, he
comes out, after the gain, twice as well off as he began, with reference to
his whole fortune. But it is very odd to say that balance of 5l. gain is
_twice_ as good as if nothing had befallen, either loss or gain. A
mathematician thinks 5 an infinite number of times as great as 0. The whole
confusion is not so apparent when money is in question: for money is money
whether gained or lost. But though pleasure and pain stand to one another
in the same algebraical relation as money gained and lost, yet there is
more than algebra can take account of in the difference.

Next, Ri. Milward[399] (Richard, no doubt, but it cannot be proved) who
published Selden's[400] Table Talk, which he had collected while serving as
amanuensis, makes Selden say, "A subsidy was counted the fifth part of a
man's estate; and so fifty subsidies is five and forty times more than a
man is worth." For _times_ read _subsidies_, which seems part of the
confusion, and there remains the making all the subsidies equal to the
first, though the whole of which they are to be the fifths is perpetually

Thirdly, there is the confusion of the great misomath {251} of our own day,
who discovered two quantities which he avers to be identically the same,
but the greater the one the less the other. He had a truth in his mind,
which his notions of quantity were inadequate to clothe in language. This
erroneous phraseology has not found a defender; and I am almost inclined to
say, with Falstaff, The poor abuses of the time want countenance.


"Shallow numerists," as Cocker[401] is made to call them, have long been at
work upon the question how to _multiply_ money by money. It is, I have
observed, a very common way of amusing the tedium of a sea voyage: I have
had more than one bet referred to me. Because an oblong of five inches by
four inches contains 5 × 4 or 20 _square_ inches, people say that five
inches multiplied by four inches _is_ twenty _square_ inches: and, thinking
that they have multiplied length by length, they stare when they are told
that money cannot be multiplied by money. One of my betters made it an
argument for the thing being impossible, that there is no _square money_:
what could I do but suggest that postage-stamps should be made legal
tender. Multiplication must be _repetition_: the repeating process must be
indicated by _number_ of times. I once had difficulty in persuading another
of my betters that if you repeat five shillings as often as there are hairs
in a horse's tail, you do not _multiply five shillings by a

I am very sorry to say that these wrong notions have found support--I think
they do so no longer--in the University of Cambridge. In 1856 or 1857, an
examiner was displaced by a vote of the Senate. The pretext was that he was
too severe an examiner: but it was well known that {252} great
dissatisfaction had been expressed, far and wide through the Colleges, at
an absurd question which he had given. He actually proposed such a fraction

   6s. 3d.
  17s. 4d.

As common sense gained a hearing very soon, there is no occasion to say
more. In 1858, it was proposed at a college examination, to divide 22557
days, 20 hours, 20 minutes, 48 seconds, by 57 minutes, 12 seconds, and also
to explain the fraction

  32l. 18s. 8d.
  62l. 12s. 9d.

All paradoxy, in matters of demonstration, arises out of muddle about first
principles. Who can say how much of it is to be laid at the door of the
University of Cambridge, for not taking care of the elements of
arithmetical thought?


The phenomena of the two ends of society, when brought together, give
interesting comparisons: I mean the early beginnings of thought and
literature, and our own high and finished state, as we think it. There is
one very remarkable point. In the early day, the letter was matter of the
closest adherence, and implied meanings were not admitted.

The blessing of Isaac meant for Esau, went to false Jacob, in spite of the
imposition; and the writer of Genesis seems to intend to give the notion
that Isaac had no power to pronounce it null and void. And "Jacob's policy,
whereby he became rich"--as the chapter-heading puts it--in speckled and
spotted stock, is not considered as a violation of the agreement, which
contemplated natural proportions. In {253} the story of Lycurgus the
lawgiver is held to have behaved fairly when he bound the Spartans to obey
his laws until he returned--intimating a short absence--he intending never
to return. And Vishnoo, when he asked the usurper for three steps of
territory as a dwarf, and then enlarged himself until he could bring heaven
and earth under the bargain, was thought clever, certainly, but quite fair.

There is nothing of this kind recognized in our day: so far good. But there
is a bad contrary: the age is apt, in interpretation, to upset the letter
in favor of the view--very often the after thought--of one side only. The
case of John Palmer,[403] the improver of the mail coach system, is
smothered. He was to have an office and a salary, and 2½ per cent for life
on the increased _revenue_ of the Post-Office. His rights turned out so
large, that Government would not pay them. For misconduct, real or
pretended, they turned him out of his _office_: but his bargain as to the
percentage had nothing to do with his future conduct; it was payment for
his _plan_. I know nothing, except from the debates of 1808 in the two
Houses: if any one can redeem the credit of the nation, the field is open.
When I was young, the old stagers spoke of this transaction sparingly, and
dismissed it speedily.

The government did not choose to remember what private persons must
remember, and are made to remember, if needful. When Dr. Lardner[404] made
his bargain with the {254} publishers for the _Cabinet Cyclopædia_ he
proposed that he, as editor, should have a certain sum for every hundred
sold above a certain number: the publishers, who did not think there was
any chance of reaching the turning sale of this stipulation, readily
consented. But it turned out that Dr. Lardner saw further than they: the
returns under this stipulation gave him a very handsome addition to his
other receipts. The publishers stared; but they paid. They had no idea of
standing out that the amount was too much for an editor; they knew that,
though the editor had a percentage, they had all the rest; and they would
not have felt aggrieved if he had received ten times as much. But
governments, which cannot be brought to book before a sworn jury, are ruled
only by public opinion. John Palmer's day was also the day of Thomas Fyshe
Palmer,[405] and the governments, in their prosecutions for sedition, knew
that these would have a reflex action upon the minds of all who wrote about
public affairs.


1864-65.--It often happens that persons combine to maintain and enforce an
opinion; but it is, in our state of society, a paradox to unite for the
sole purpose of blaming the opposite side. To invite educated men to do
this, and above all, men of learning or science, is the next paradoxical
thing of all. But this was done by a small combination in 1864. They got
together and drew up a _declaration_, to be signed by "students of the
natural sciences," who were to express their "sincere regret that
researches into {255} scientific truth are perverted by some in our own
times into occasion for casting doubt upon the truth and authenticity of
the Holy Scriptures." In words of ambiguous sophistry, they proceeded to
request, in effect, that people would be pleased to adopt the views of
churches as to the _complete_ inspiration of all the canonical books. The
great question whether the Word of God is _in_ the Bible, or whether the
Word of God is _all_ the Bible, was quietly taken for granted in favor of
the second view; to the end that men of science might be induced to blame
those who took the first view. The first public attention was drawn to the
subject by Sir John Herschel,[406] who in refusing to sign the writ sent to
him, administered a rebuke in the _Athenæum_, which would have opened most
eyes to see that the case was hopeless. The words of a man whose _suaviter
in modo_ makes his _fortiter in re_[407] cut blocks with a razor are worth

"I consider the act of calling upon me publicly to avow or disavow, to
approve or disapprove, in writing, any religious doctrine or statement,
however carefully or cautiously drawn up (in other words, to append my name
to a religious manifesto) to be an infringement of that social forbearance
which guards the freedom of religious opinion in this country with especial
sanctity.... I consider this movement simply mischievous, having a direct
tendency (by putting forward a new Shibboleth, a new verbal test of
religious partisanship) to add a fresh element of discord to the already
too discordant relations of the Christian world.... But no nicety of
wording, no artifice of human language, will suffice to discriminate the
hundredth part of the shades of meaning in which the most world-wide
differences of thought on such subjects may be involved; or prevent the
most gentle worded and apparently justifiable expression of regret, so
embodied, from grating on the {256} feelings of thousands of estimable and
well-intentioned men with all the harshness of controversial hostility."

Other doses were administered by Sir J. Bowring,[408] Sir W. Rowan
Hamilton,[409] and myself. The signed declaration was promised for
Christmas, 1864: but nothing presentable was then ready; and it was near
Midsummer, 1865, before it was published. Persons often incautiously put
their names without seeing the _character_ of a document, because they
coincide in its _opinions_. In this way, probably, fifteen respectable
names were procured before printing; and these, when committed, were hawked
as part of an application to "solicit the favor" of other signatures. It is
likely enough no one of the fifteen saw that the declaration was, not
_maintenance_ of their own opinion, but _regret_ (a civil word for _blame_)
that others should _think differently_.

When the list appeared, there were no fewer than 716 names! But analysis
showed that this roll was not a specimen of the mature science of the
country. The collection was very miscellaneous: 38 were designated as
"students of the College of Chemistry," meaning young men who attended
lectures in that college. But as all the Royal Society had been applied to,
a test results as follows. Of Fellows of the Royal Society, 600 in number,
62 gave their signatures; of writers in the _Philosophical Transactions_,
166 in number, 19 gave their signatures. Roughly speaking, then, only one
out of ten could be got to express disapprobation of the free comparison of
the results of science with the statements of the canonical books. And I am
satisfied that many of these thought they were signing only a declaration
of difference of opinion, not of blame for that difference. The number of
persons is not small who, when it comes to signing printed documents, would
put their names to a declaration that the coffee-pot ought to be taken
down-stairs, meaning that the teapot ought to be brought {257} up-stairs.
And many of them would defend it. Some would say that the two things are
not contradictory; which, with a snort or two of contempt, would be very
effective. Others would, in the candid and quiet tone, point out that it is
all one, because coffee is usually taken before tea, and it keeps the table
clear to send away the coffee-pot before the teapot is brought up.

The original signatures were decently interred in the Bodleian Library: and
the advocates of scattering indefinite blame for indefinite sins of opinion
among indefinite persons are, I understand, divided in opinion about the
time at which the next attempt shall be made upon men of scientific
studies: some are for the Greek Calends, and others for the Roman
Olympiads. But, with their usual love of indefiniteness, they have
determined that the choice shall be argued upon the basis that which comes
first cannot be settled, and is of no consequence.

I give the declaration entire, as a curiosity: and parallel with it I give
a substitute which was proposed in the _Athenæum_, as worthy to be signed
both by students of theology, and by students of science, especially in
past time. When a new attempt is made, it will be worth while to look at

_Declaration._                  _Proposed Substitute._

We, the undersigned Students    We, the undersigned Students
of the Natural Sciences,        of Theology and of Nature,
desire to express our sincere   desire to express our sincere
regret, that researches into    regret, that common notions of
scientific truth are perverted  religious truth are perverted
by some in our own times into   by some in our own times into
occasion for casting doubt      occasion for casting reproach
upon the Truth and              upon the advocates of
Authenticity of the Holy        demonstrated or highly
Scriptures.                     probable scientific theories.

We conceive that it is          We conceive that it is
impossible for the Word of      impossible for the Word of
God, as written in the book of  God, as correctly read in the
nature, and God's Word written  Book of Nature, and the Word
in Holy Scripture, to           of God, as truly interpreted
contradict one another,         out of the Holy Scripture, to
however much they may appear    contradict one another,
to differ.                      however much they may appear
                                to differ.
We are not forgetful that       We are not forgetful that
Physical Science is not         neither theological
complete, but is only in a      interpretation nor physical
condition of progress, and      knowledge is yet complete, but
that at present our finite      that both are in a condition
reason enables us only to see   of progress; and that at
as through a glass darkly,      present our finite reason
                                enables us only to see both
                                one and the other as through a
                                glass darkly [the writers of
                                the original declaration have
                                distinctively applied to
                                physical science the phrase by
                                which St. Paul denotes the
                                imperfections of theological
                                vision, which they tacitly
                                assume to be quite perfect],
and we confidently believe,     and we confidently believe,
that a time will come when the  that a time will come when the
two records will be seen to     two records will be seen to
agree in every particular. We   agree in every particular. We
cannot but deplore that         cannot but deplore that
Natural Science should be       Religion should be looked upon
looked upon with suspicion by   with suspicion by some and
many who do not make a study    Science by others, of the
of it, merely on account of     students of either who do not
the unadvised manner in which   make a study of the {259}
some are placing it in          other, merely on account of
opposition to Holy Writ.        the unadvised manner in which
                                some are placing Religion in
                                opposition to Science, and
                                some are placing Science in
                                opposition to Religion.
We believe that it is the duty  We believe that it is the duty
of every Scientific Student to  of every theological student
investigate nature simply for   to investigate the Scripture,
the purpose of elucidating      and of every scientific
truth,                          student to investigate Nature,
                                simply for the purpose of
                                elucidating truth.
and that if he finds that some  And if either should find that
of his results appear to be in  some of his results appear to
contradiction to the Written    be in contradiction, whether
Word, or rather to his own      to Scripture or to Nature, or
_interpretations_ of it, which  rather to his own
may be erroneous, he should     _interpretation_ of one or the
not presumptuously affirm that  other, which may be erroneous,
his own conclusions must be     he should not affirm as with
right, and the statements of    certainty that his own
Scripture wrong;                conclusion must be right, and
                                the other interpretation
rather, leave the two side by   but should leave the two side
side till it shall please God   by side for further inquiry
to allow us to see the manner   into both, until it shall
in which they may be            please God to allow us to
reconciled;                     arrive at the manner in which
                                they may be reconciled.
and, instead of insisting upon  In the mean while, instead of
the seeming differences         insisting, and least of all
between Science and the         with acrimony or injurious
Scriptures, it would be as      {260} statements about others,
well to rest in faith upon the  upon the seeming differences
points in which they agree.     between Science and the
                                Scriptures, it would be a
                                thousand times better to rest
                                in faith as to our future
                                state, in hope as to our
                                coming knowledge, and in
                                charity as to our present

The distinctness of the fallacies is creditable to the composers, and shows
that scientific habits tend to clearness, even to sophistry. Nowhere does
it so plainly stand out that the _Written Word_ means the sense in which
the accuser takes it, while the sense of the other side is _their
interpretation_. The infallible church on one side, arrayed against
heretical pravity on the other, is seen in all subjects in which men
differ. At school there were various games in which one or another
advantage was the right of those who first called for it. In adult argument
the same thing is often attempted: we often hear--I cried _Church_ first!

I end with the answer which I myself gave to the application: its revival
may possibly save me from a repetition of the like. If there be anything I
hate more than another it is the proposal to place any persons, especially
those who allow freedom to me, under any abridgment of their liberty to
think, to infer, and to publish. If they break the law, take the law; but
do not make the law: [Greek: agoraioi agontai enkaleitôsan allêlois.][410]
I would rather be asked to take shares in an argyrosteretic company (with
limited liability) for breaking into houses by night on fork and spoon
errands. I should put aside this proposal with _nothing but laughter_. It
was a joke against Sam Rogers[411] that his appearance was very like that
of a corpse. The _John Bull_ {261} newspaper--suppose we now say Theodore
Hook[412]--averred that when he hailed a coach one night in St. Paul's
Churchyard, the jarvey said, "Ho! ho! my man; I'm not going to be taken in
that way: go back to your grave!" This is the answer I shall make for the
future to any relics of a former time who shall want to call me off the
stand for their own purposes. What obligation have I to admit that they
belong to our world?


"_The Writ De Hæretico Commiserando._[413]

Nov. 14, 1864.

"This document was sent to me four days ago. It 'solicits the favor'--I
thought at first it was a grocer's supplication for tea and sugar
patronage--of my signature to expression of 'sincere regret' that some
persons unnamed--general warrants are illegal--differ from what I am
supposed--by persons whom it does not concern--to hold about Scripture and
Science in their real or alleged discrepancies.

"No such favor from me: for three reasons. First, I agree with Sir. J.
Herschel that the solicitation is an intrusion to be publicly repelled.
Secondly, I do _not_ regret that others should differ from me, think what I
may: those others are as good as I, and as well able to think, and as much
entitled to their conclusions. Thirdly, even if I did regret, I should be
ashamed to put my name to bad chemistry made to do duty for good reasoning.
The declaration is an awkward attempt to saturate sophism with truism; but
the sophism is left largely in excess.


"I owe the inquisitors a grudge for taking down my conceit of myself. For
two months I have crowed in my own mind over my friend Sir J. Herschel,
fancying that the promoters instinctively knew better than to bring their
fallacies before a writer on logic. Ah! my dear Sir John! thought I, if you
had shown yourself to be well up in _Barbara Celarent_,[414] and had ever
and anon astonished the natives with the distinction between _simpliciter_
and _secundum quid_, no autograph-hunters would have baited a trap with
_non sequitur_[415] to catch your signature. What can I say now? I hide my
diminished head, diminished by the horns which I have been compelled to
draw in.

"Those who make personal solicitation for support to an opinion about
religion are bound to know their men. The king had a right to Brother
Neale's money, because Brother Neale offered it. Had he put his hand into
purse after purse by way of finding out all who were of Brother Neale's
mind, he would have been justly met by a rap on the knuckles whenever he
missed his mark.

"The kind of test before me is the utmost our time will allow of that
inquisition into opinion which has been the curse of Christianity ever
since the State took Providence under its protection. The writ _de hæretico
commiserando_ is little more than the smell of the empty cask: and those
who issue it may represent the old woman with her

 "O suavis anima, quale in te dicam bonum
  Antehac fuisse; tales cum sint reliquiæ."[416]

It is no excuse that the illegitimate bantling is a very little one. Its
parents may think themselves hardly treated when they are called lineal
successors of Tony Fire-the-faggot: {263} but, degenerate though they be,
such is their ancestry. Let every allowance be made for them: but their
unholy fire must be trodden out; so long as a spark is left, nothing but
fuel is wanted to make a blaze. If this cannot be done, let the flame be
confined to theology, though even there it burns with diminished vigor: and
let charity, candor, sense, and ridicule, be ready to play upon it whenever
there is any chance of its extending to literature and science.

"What would be the consequence if this test-signing absurdity were to grow?
Deep would call unto deep; counter-declaration would answer declaration,
each stronger than the one before. The moves would go on like the dispute
of two German students, of whom each is bound to a sharper retort on a
graduated scale, until at last comes _dummer Junge_![417]--and then they
must fight. There is a gentleman in the upper fifteen of the signers of the
writ--the hawking of whose names appears to me very bad taste--whom I met
in cordial cooperation for many a year at a scientific board. All I knew
about his religion was that he, as a clergyman, must in some sense or other
receive the 39 Articles:--all that he could know about mine was that I was
some kind of heretic, or so reputed. If we had come to signing opposite
manifestoes, turn-about, we might have found ourselves in the lowest depths
of party discussion at our very council-table. I trust the list of
subscribers to the declaration, when it comes to be published, will show
that the bulk of those who have really added to our knowledge have seen the
thing in its true light.

"The promoters--I say nothing about the subscribers--of the movement will,
I trust, not feel aggrieved at the course I have taken or the remarks I
have made. Walter Scott says that before we judge Napoleon by the
temptation to which he yielded, we ought to remember how much he may have
resisted: I invite them to apply this rule to myself; they can have no idea
of the feeling with which I {264} contemplate all attempts to repress
freedom of inquiry, nor of the loathing with which I recoil from the
proposal to be art and part. They have asked me to give a public opinion
upon a certain point. It is true that they have had the kindness to tender
both the opinion they wish me to form, and the shape in which they would
have it appear: I will let them draw me out, but I will not let them take
me in. If they will put an asterisk to my name, and this letter to the
asterisk, they are welcome to my signature. As I do not expect them to
relish this proposal, I will not solicit the favor of its adoption. But
they have given a right to think, for they have asked me to think; to
publish, for they have asked me to allow them to publish; to blame them,
for they have asked me to blame their betters. Should they venture to find
fault because my direction of disapproval, publicly given, is half a
revolution different from theirs, they will be known as having presented a
loaded document at the head of a traveler in the highway of discussion,
with--Your signature or your silence!"


The paradox being the proposition of something which runs counter to what
would generally be thought likely, may present itself in many ways. There
is a _fly-leaf paradox_, which puzzled me for many years, until I found a
probable solution. I frequently saw, in the blank leaves of old books,
learned books, Bibles of a time when a Bible was very costly, etc., the
name of an owner who, by the handwriting and spelling, must have been an
illiterate person or a child, followed by the date of the book itself.
Accordingly, this uneducated person or young child seemed to be the first
owner, which in many cases was not credible. Looking one day at a
Barker's[418] Bible of 1599, I saw an {265} inscription in a child's
writing, which certainly belonged to a much later date. It was "Martha
Taylor, her book, giuen me by Granny Scott to keep for her sake." With this
the usual verses, followed by 1599, the date of the book. But it so chanced
that the blank page opposite the title, on which the above was written, was
a verso of the last leaf of a prayer book, which had been bound before the
Bible; and on the recto of this leaf was a colophon, with the date 1632. It
struck me immediately that uneducated persons and children, having seen
dates written under names, and not being quite up in chronology, did
frequently finish off with the date of the book, which stared them in the

Always write in your books. You may be a silly person--for though your
reading my book is rather a contrary presumption, yet it is not
conclusive--and your observations may be silly or irrelevant, but you
cannot tell what use they may be of long after you are gone where
Budgeteers cease from troubling.

I picked up the following book, printed by J. Franklin[419] at Boston,
during the period in which his younger brother Benjamin was his apprentice.
And as Benjamin was apprenticed very early, and is recorded as having
learned the mechanical art very rapidly, there is some presumption that
part of it may be his work, though he was but thirteen at the time. As this
set of editions of Hodder[420] (by {266} Mose[421]) is not mentioned, to my
knowledge, I give the title in full:

    "Hodder's Arithmetick: or that necessary art made most easy: Being
    explained in a way familiar to the capacity of any that desire to learn
    it in a little time. By James Hodder, Writing-master. The Five and
    twentieth edition, revised, augmented, and above a thousand faults
    amended, by Henry Mose, late servant and successor to the author.
    Boston: printed by J. Franklin, for S. Phillips, N. Buttolph, B.
    Elliot, D. Henchman, G. Phillips, J. Elliot, and E. Negus, booksellers
    in Boston, and sold at their shops. 1719."

The book is a very small octavo, the type and execution are creditable, the
woodcut at the beginning is clumsy. It is a perfect copy, page for page, of
the English editions of Mose's Hodder, of which the one called seventeenth
is of London, 1690. There is not a syllable to show that the edition above
described might not be of Boston in England. Presumptions, but not very
strong ones, might be derived from the name of _Franklin_, and from the
large number of booksellers who combined in the undertaking. It chanced,
however, that a former owner had made the following note in my copy:

    "Wednessday, July y^e 14, 1796, att ten in y^e forenoon we sail^d from
    Boston, came too twice, once in King Rode, and once in y^e Narrows.
    Sail^d by y^e lighthouse in y^e even^g."


No ordinary map would decide these points: so I had to apply to my friend
Sir Francis Beaufort,[422] and the charts at the Admiralty decided
immediately for Massachusetts.


The French are able paradoxers in their spelling of foreign names. The Abbé
Sabatier de Castres,[423] in 1772, gives an account of an imaginary
dialogue between Swif, Adisson, Otwai, and Bolingbrocke. I had hoped that
this was a thing of former days, like the literal roasting of heretics; but
the charity which hopeth all things must hope for disappointments. Looking
at a recent work on the history of the popes, I found referred to, in the
matter of Urban VIII[424] and Galileo, references to the works of two
Englishmen, the Rev. Win Worewel and the Rev. Raden Powen. [Wm. Whewell and
Baden Powell].[425]

I must not forget the "moderate computation" paradox. This is the way by
which large figures are usually obtained. Anything surprisingly great is
got by the "lowest computation," anything as surprisingly small by the
"utmost computation"; and these are the two great subdivisions of "moderate
computation." In this way we learn that 70,000 persons were executed in one
reign, and 150,000 persons {268} burned for witchcraft in one century.
Sometimes this computation is very close. By a card before me it appears
that all the Christians, including those dispersed in heathen countries,
those of Great Britain and Ireland excepted, are 198,728,000 people, and
pay their clergy 8,852,000l. But 6,400,000 people pay the clergy of the
Anglo-Irish Establishment 8,896,000l.; and 14,600,000 of other
denominations pay 1,024,000l. When I read moderate computations, I always
think of Voltaire and the "mémoires du fameux évêque de Chiapa, par
lesquels il paraît qu'il avait égorgé, ou brulé, ou noyé dix millions
d'infidèles en Amérique pour les convertir. Je crus que cet évêque
exaggérait; mais quand on réduisait ces sacrifices à cinq millions de
victimes, cela serait encore admirable."[426]


My Budget has been arranged by authors. This is the only plan, for much of
the remark is personal: the peculiarities of the paradoxer are a large part
of the interest of the paradox. As to subject-matter, there are points
which stand strongly out; the quadrature of the circle, for instance. But
there are others which cannot be drawn out so as to be conspicuous in a
review of writers: as one instance, I may take the _centrifugal force_.

When I was about nine years old I was taken to hear a course of lectures,
given by an itinerant lecturer in a country town, to get as much as I could
of the second half of a good, sound, philosophical omniscience. The first
half (and sometimes more) comes by nature. To this end I smelt chemicals,
learned that they were different kinds of _gin_, saw young wags try to kiss
the girls under the excuse of what was called _laughing gas_--which I was
sure {269} was not to blame for more than five per cent of the requisite
assurance--and so forth. This was all well so far as it went; but there was
also the excessive notion of creative power exhibited in the millions of
miles of the solar system, of which power I wondered they did not give a
still grander idea by expressing the distances in inches. But even this was
nothing to the ingenious contrivance of the centrifugal force. "You have
heard what I have said of the wonderful centripetal force, by which Divine
Wisdom has retained the planets in their orbits round the Sun. But, ladies
and gentlemen, it must be clear to you that if there were no other force in
action, this centripetal force would draw our earth and the other planets
into the Sun, and universal ruin would ensue. To prevent such a
catastrophe, the same wisdom has implanted a centrifugal force of the same
amount, and directly opposite," etc. I had never heard of Alfonso X of
Castile,[427] but I ventured to think that if Divine Wisdom had just let
the planets alone it would come to the same thing, with equal and opposite
troubles saved. The paradoxers deal largely in speculation conducted upon
the above explanation. They provide external agents for what they call the
centrifugal force. Some make the sun's rays keep the planets off, without a
thought about what would become of our poor eyes if the _push_ of the light
which falls on the earth were a counterpoise to all its gravitation. The
true explanation cannot be given here, for want of room.


Sometimes a person who has a point to carry will assert a singular fact or
prediction for the sake of his point; and {270} this paradox has almost
obtained the sole use of the name. Persons who have reputation to care for
should beware how they adopt this plan, which now and then eventuates a
spanker, as the American editor said. Lord Byron, in "English Bards, etc."
(1809), ridiculing Cambridge poetry, wrote as follows:

 "But where fair Isis rolls her purer wave,
  The partial muse delighted loves to lave;
  On her green banks a greener wreath she wove,
  To crown the bards that haunt her classic grove;
  Where Richards[428] wakes a genuine poet's fires,
  And modern Britons glory in their sires."[429]

There is some account of the Rev. Geo. Richards, Fellow of Oriel and Vicar
of Bampton, (M.A. in 1791) in the _Living Authors_ by Watkins[430] and
Shoberl[431] (1816). In Rivers's _Living Authors_, of 1798, which is best
fitted for citation, as being published before Lord Byron wrote, he is
spoken of in high terms. The _Aboriginal Britons_ was an Oxford (special)
prize poem, of 1791. Charles Lamb mentions Richards as his school-fellow at
Christ's Hospital, "author of the _Aboriginal Britons_, the most spirited
of the Oxford Prize Poems: a pale, studious Grecian."

As I never heard of Richards as a poet,[432] I conclude that his fame is
defunct, except in what may prove to be a very ambiguous kind of
immortality, conferred by Lord Byron. The awkwardness of a case which time
has broken down {271} is increased by the eulogist himself adding so
powerful a name to the list of Cambridge poets, that his college has placed
his statue in the library, more conspicuously than that of Newton in the
chapel; and this although the greatness of poetic fame had some serious
drawbacks in the moral character of some of his writings. And it will be
found on inquiry that Byron, to get his instance against Cambridge, had to
go back eighteen years, passing over seven intermediate productions, of
which he had either never heard, or which he would not cite as waking a
genuine poet's fires.

The conclusion seems to be that the _Aboriginal Britons_ is a remarkable
youthful production, not equalled by subsequent efforts.

To enhance the position in which the satirist placed himself, two things
should be remembered. First, the glowing and justifiable terms in which
Byron had spoken,--a hundred and odd lines before he found it convenient to
say no Cambridge poet could compare with Richards,--of a Cambridge poet who
died only three years before Byron wrote, and produced greatly admired
works while actually studying in the University. The fame of Kirke
White[433] still lives; and future literary critics may perhaps compare his
writings and those of Richards, simply by reason of the curious relation in
which they are here placed alongside of each other. And it is much to
Byron's credit that, in speaking of the deceased Cambridge poet, he forgot
his own argument and its exigencies, and proved himself only a paradoxer
_pro re nata_.

Secondly, Byron was very unfortunate in another passage of the same poem:


 "What varied wonders tempt us as they pass!
  The cow-pox, tractors, galvanism, and gas.
  In turns appear, to make the vulgar stare,
  Till the swoln bubble bursts--and all is air!"

Three of the bubbles have burst to mighty ends. The metallic tractors are
disused; but the force which, if anything, they put in action, is at this
day, under the name of mesmerism, used, prohibited, respected, scorned,
assailed, defended, asserted, denied, declared utterly obscure, and
universally known. It was hard lines to select for candidates for oblivion
not one of whom got in. I shall myself, I am assured, be some day cited for
laughing at the great discovery of ----: the blank is left for my reader to
fill up in his own way; but I think I shall not be so unlucky in four
different ways.


The narration before the fact, as prophecy has been called, sometimes quite
as true as the narration after the fact, is very ridiculous when it is
wrong. Why, the pre-narrator could not know; the post-narrator might have
known. A good collection of unlucky predictions might be made: I hardly
know one so fit to go with Byron's as that of the Rev. Daniel Rivers,
already quoted, about Johnson's biographers. Peter Pindar[434] may be
excused, as personal satire was his object, for addressing Boswell and Mrs.
Piozzi[435] as follows:

 "Instead of adding splendor to his name,
  Your books are downright gibbets to his fame;
  You never with posterity can thrive,
 'Tis by the Rambler's death alone you live."

But Rivers, in prose narrative, was not so excusable. He says:


"As admirers of the learning and moral excellence of their hero, we glow at
almost every page with indignation that his weaknesses and his failings
should be disclosed to public view.... Johnson, after the luster he had
reflected on the name of Thrale ... was to have his memory tortured and
abused by her detested itch for scribbling. More injury, we will venture to
affirm, has been done to the fame of Johnson by this Lady and her late
biographical helpmate, than his most avowed enemies have been able to
effect: and if his character becomes unpopular with some of his successors,
it is to those gossiping friends he is indebted for the favor."

Poor dear old Sam! the best known dead man alive! clever, good-hearted,
logical, ugly bear! Where would he have been if it had not been for Boswell
and Thrale, and their imitators? What would biography have been if Boswell
had not shown how to write a life?

Rivers is to be commended for not throwing a single Stone at Mrs. Thrale's
second marriage. This poor lady begins to receive a little justice. The
literary world seems to have found out that a blue-stocking dame who keeps
open house for a set among them has a right, if it so please her, to marry
again without taking measures to carry on the cake-shop. I was before my
age in this respect: as a boy-reader of Boswell, and a few other things
that fell in my way, I came to a clearness that the conduct of society
towards Mrs. Piozzi was _blackguard_. She wanted nothing but what was in
that day a woman's only efficient protection, a male relation with a brace
of pistols, and a competent notion of using them.


Byron's mistake about Hallam in the Pindar story may be worth placing among
absurdities. For elucidation, suppose that some poet were now to speak--

 "Of man's first disobedience, and the fruit
  Eve gave to Adam in his birthday suit--"

and some critic were to call it nonsense, would that critic be laughing at
Milton? Payne Knight,[436] in his _Taste_, translated part of Gray's _Bard_
into Greek. Some of his lines are

  [Greek: therma d' ho tengôn dakrua stonachais]
  [Greek: oulon melos phoberai]
  [Greek: êeide phônai.]

Literally thus:

 "Wetting warm tears with groans,
  Continuous chant with fearful
  Voice he sang."

On which Hallam remarks: "The twelfth line [our first] is nonsense." And so
it is, a poet can no more wet his tears with his groans than wet his ale
with his whistle. Now this first line is from Pindar, but is only part of
the sense; in full it is:

  [Greek: therma de tengôn dakrua stonachais]
      [Greek: horthion phônase.]

Pindar's [Greek: tengôn] must be Englished by _shedding_, and he stands
alone in this use. He says, "shedding warm tears, he cried out loud, with
groans." Byron speaks of

 "Classic Hallam, much renowned for Greek:"

and represents him as criticising _the Greek_ of all Payne's lines, and not
discovering that "the lines" were Pindar's {275} until after publication.
Byron was too much of a scholar to make this blunder himself: he either
accepted the facts from report, or else took satirical licence. And why
not? If you want to laugh at a person, and he will not give occasion, whose
fault is it that you are obliged to make it? Hallam did criticise some of
Payne Knight's Greek; but with the caution of his character, he remarked
that possibly some of these queer phrases might be "critic-traps" justified
by some one use of some one author. I remember well having a Latin essay to
write at Cambridge, in which I took care to insert a few monstrous and
unusual idioms from Cicero: a person with a Nizolius,[437] and without
scruples may get scores of them. So when my tutor raised his voice against
these oddities, I was up to him, for I came down upon him with Cicero,
chapter and verse, and got round him. And so my own solecisms, many of
them, passed unchallenged.

Byron had more good in his nature than he was fond of letting out: whether
he was a soured misanthrope, or whether his _vein_ lay that way in poetry,
and he felt it necessary to fit his demeanor to it, are matters far beyond
me. Mr. Crabb Robinson[438] told me the following story more than once. He
was at Charles Lamb's chambers in the Temple when Wordsworth came in, with
the new _Edinburgh Review_ in his hand, and fume on his countenance. "These
reviewers," said he, "put me out of patience! Here is a young man--they say
he is a lord--who has written a volume of poetry; and these fellows, just
because he is a lord, set upon him, laugh at him, and sneer at his writing.
The young man will do something, if he goes on as he has begun. But these
reviewers seem to think {276} that nobody may write poetry, unless he lives
in a garret." Crabb Robinson told this long after to Lady Byron, who said,
"Ah! if Byron had known that, he would never have attacked Wordsworth. He
went one day to meet Wordsworth at dinner; when he came home I said, 'Well,
how did the young poet get on with the old one?' 'Why, to tell you the
truth,' said he, 'I had but one feeling from the beginning of the visit to
the end, and that was--_reverence_!'" Lady Byron told my wife that her
husband had a very great respect for Wordsworth. I suppose he would have
said--as the Archangel said to his Satan--"Our difference is po[li =

I suspect that Fielding would, if all were known, be ranked among the
unlucky railers at supposed paradox. In his _Miscellanies_ (1742, 8vo) he
wrote a satire on the Chrysippus or Guinea, an animal which multiplies
itself by division, like the polypus. This he supposes to have been drawn
up by Petrus Gualterus, meaning the famous usurer, Peter Walter. He calls
it a paper "proper to be read before the R----l Society": and next year,
1743, a quarto reprint was made to resemble a paper in the _Philosophical
Transactions_. So far as I can make out, one object is ridicule of what the
zoologists said about the polypus: a reprint in the form of the
_Transactions_ was certainly satire on the Society, not on Peter Walter and
his knack of multiplying guineas.

Old poets have recognized the quadrature of the circle as a well-known
difficulty. Dante compares himself, when bewildered, to a geometer who
cannot find the principle on which the circle is to be measured:

 "Quale è 'l geometra che tutto s' affige
  Per misurar lo cerchio, e non ritruova,
  Pensando qual principio ond' egli indige."[439]

{277} And Quarles[440] speaks as follows of the _summum bonum_:

 "Or is't a tart idea, to procure
  An edge, and keep the practic soul in ure,
  Like that dear chymic dust, or puzzling quadrature?"

The poetic notion of the quadrature must not be forgotten. Aristophanes, in
the _Birds_, introduces a geometer who announces his intention to _make a
square circle_. Pope, in the _Dunciad_, delivers himself as follows, with a
Greek pronunciation rather strange in a translator of Homer. Probably Pope
recognized, as a general rule, the very common practice of throwing back
the accent in defiance of quantity, seen in o´rator, au´ditor, se´nator,
ca´tenary, etc.

 "Mad _Mathesis_ alone was unconfined,
  Too mad for mere material chains to bind,--
  Now to pure space lifts her ecstatic stare,
  Now, running round the circle, finds it square."

The author's note explains that this "regards the wild and fruitless
attempts of squaring the circle." The poetic idea seems to be that the
geometers try to make a square circle. Disraeli quotes it as "finds _its_
square," but the originals do not support this reading.


I have come in the way of a work, entitled _The Grave of Human
Philosophies_ (1827), translated from the French of R. de Bécourt[441] by
A. Dalmas. It supports, but I suspect not very accurately, the views of the
old Hindu books. {278} That the sun is only 450 miles from us, and only 40
miles in diameter, may be passed over; my affair is with the state of mind
into which persons of M. Bécourt's temperament are brought by a fancy. He
fully grants, as certain, four millions of years as the duration of the
Hindu race, and 1956 as that of the universe. It must be admitted he is not
wholly wrong in saying that our errors about the universe proceed from our
ignorance of its origin, antiquity, organization, laws, and final
destination. Living in an age of light, he "avails himself of that
opportunity" to remove this veil of darkness, etc. The system of the
Brahmins is the only true one: he adds that it has never before been
attempted, as it could not be obtained except by him. The author requests
us first, to lay aside prejudice; next, to read all he says in the order in
which he says it: we may then pronounce judgment upon a work which begins
by taking the Brahmins for granted. All the paradoxers make the same
requests. They do not see that compliance would bring thousands of systems
before the world every year: we have scores as it is. How is a poor candid
inquirer to choose. Fortunately, the mind has its grand jury as well as its
little one: and it will not put a book upon its trial without a _prima
facie_ case in its favor. And with most of those who really search for
themselves, that case is never made out without evidence of knowledge,
standing out clear and strong, in the book to be examined.


There is much private history which will never come to light, _caret quia
vate sacro_,[442] because no Budgeteer comes across it. Many years ago a
man of business, whose life was passed in banking, amused his leisure with
quadrature, was successful of course, and bequeathed the result in a sealed
book, which the legatee was enjoined not to sell {279} under a thousand
pounds. The true ratio was 3.1416: I have the anecdote from the legatee's
executor, who opened the book. That a banker should square the circle is
very credible: but how could a City man come by the notion that a thousand
pounds could be got for it? A friend of mine, one of the twins of my
zodiac, will spend a thousand pounds, if he have not done it already, in
black and white cyclometry: but I will answer for it that he, a man of
sound business notions, never entertained the idea of [pi] recouping him,
as they now say. I speak of individual success: of course if a company were
formed, especially if it were of unlimited lie-ability, the shares would be
taken. No offence; there is nothing but what a pun will either sanctify,
justify, or nullify:

 "It comes o'er the soul like the sweet South
  That breathes upon a bank of _vile hits_."

The shares would be at a premium of 3-1/8 on the day after issue. If they
presented me with the number of shares I deserve, for suggestion and
advertisement, I should stand up for the Archpriest of St. Vitus[443] and
3-1/5, with a view to a little more gold on the bridge.

I now insert a couple of reviews, one about Cyclopædias, one about
epistolary collections. Should any reader wish for explanation of this
insertion, I ask him to reflect a moment, and imagine me set to justify all
the additions now before him! In truth these reviews are the repositories
of many odds and ends: they were not made to the books; the materials were
in my notes, and the books came as to a ready-made clothes shop, and found
what would fit them. Many remember Curll's[444] bequest of some very good
titles {280} which only wanted treatises written to them. Well! here were
some tolerable reviews--as times go--which only wanted books fitted to
them. Accordingly, some tags were made to join on the books; and then as
the reader sees.

I should find it hard to explain why the insertion is made in this place
rather than another. But again, suppose I were put to make such an
explanation throughout the volume. The improver who laid out grounds and
always studied what he called _unexpectedness_, was asked what name he gave
it for those who walked over his grounds a second time. He was silenced;
but I have an answer: It is that which is given by the very procedure of
taking up my book a second time.


    October 19, 1861. _The English Cyclopædia._ Conducted by Charles
    Knight.[445] 22 vols.: viz., _Geography_, 4 vols.; _Biography_, 6
    vols.; _Natural History_, 4 vols.; _Arts and Sciences_, 8 vols.
    (Bradbury & Evans.)

    _The Encyclopædia Britannica: a Dictionary of Arts, Sciences, and
    General Literature._ Eighth Edition. 21 vols. and Index. (Black.)

The two editions above described are completed at the same time: and they
stand at the head of the two great branches into which pantological
undertakings are divided, as at once the largest and the best of their

When the works are brought together, the first thing that strikes the eye
is the syllable of difference in the names. The word _Cyclopædia_ is a bit
of modern purism. Though [Greek: enkuklopaideia][446] is not absolutely
Greek of Greece, we learn from both Pliny[447] and Quintilian[448] that the
circle {281} of the sciences was so called by the Greeks, and
Vitruvius[449] has thence naturalized _encyclium_ in Latin. Nevertheless we
admit that the initial _en_ would have euphonized but badly with the word
_Penny_: and the _English Cyclopædia_ is the augmented, revised, and
distributed edition of the _Penny Cyclopædia_. It has indeed been said that
Cyclopædia should mean the education _of_ a circle, just as Cyropædia is
the education _of_ Cyrus. But this is easily upset by Aristotle's word
[Greek: kuklophoria],[450] motion _in_ a circle, and by many other cases,
for which see the lexicon.

The earliest printed Encyclopædia of this kind was perhaps the famous
"myrrour of the worlde," which Caxton[451] translated from the French and
printed in 1480. The original Latin is of the thirteenth century, or
earlier. This is a collection of very short treatises. In or shortly after
1496 appeared the _Margarita Philosophica_ of Gregory Reisch,[452] the same
we must suppose, who was confessor to the Emperor Maximilian.[453] This is
again a collection of treatises, of much more pretension: and the
estimation formed of it is proved by the number of editions it went
through. In 1531 appeared the little collection of _works_ of
Ringelberg,[454] which is truly called an Encyclopædia by {282} Morhof,
though the thumbs and fingers of the two hands will meet over the length of
its one volume. There are more small collections; but we pass on to the
first work to which the name of _Encyclopædia_ is given. This is a
ponderous _Scientiarum Omnium Encyclopædia_ of Alsted,[455] in four folio
volumes, commonly bound in two: published in 1629 and again in 1649; the
true parent of all the Encyclopædias, or collections of treatises, or works
in which that character predominates. The first great _dictionary_ may
perhaps be taken to be Hofman's _Lexicon Universale_[456] (1677); but
Chambers's[457] (so called) _Dictionary_ (1728) has a better claim. And we
support our proposed nomenclature by observing that Alsted accidentally
called his work _En_cyclopædia, and Chambers simply Cyclopædia.

We shall make one little extract from the _myrrour_, and one from
Ringelberg. Caxton's author makes a singular remark for his time; and one
well worthy of attention. The grammar rules of a language, he says, must
have been invented by foreigners: "And whan any suche tonge was perfytely
had and usyd amonge any people, than other people not used to the same
tonge caused rulys to be made wherby they myght lerne the same tonge ...
and suche rulys be called the gramer of that tonge." Ringelberg says that
if the right nostril bleed, the little finger of the right hand should be
crooked, and squeezed with great force; and the same for the left.


We pass on to _the_ Encyclopédie,[458] commenced in 1751; the work which
has, in many minds, connected the word _encyclopædist_ with that of
_infidel_. Readers of our day are surprised when they look into this work,
and wonder what has become of all the irreligion. The truth is, that the
work--though denounced _ab ovo_[459] on account of the character of its
supporters--was neither adapted, nor intended, to excite any particular
remark on the subject: no work of which D'Alembert[460] was co-editor would
have been started on any such plan. For, first, he was a real _sceptic_:
that is, doubtful, with a mind not made up. Next, he valued his quiet more
than anything; and would as soon have gone to sleep over an hornet's nest
as have contemplated a systematic attack upon either religion or
government. As to Diderot[461]--of whose varied career of thought it is
difficult to fix the character of any one moment, but who is very
frequently taken among us for a pure atheist--we will quote one sentence
from the article "_Encyclopédie_," which he wrote himself:--"Dans le moral,
il n'y a que Dieu qui doit servir de modèle a 1'homme; dans les art, que la

A great many readers in our country have but a very hazy idea of the
difference between the political Encyclopædia, as we may call it, and the
_Encyclopédie Méthodique_,[463] which we always take to be meant--whether
rightly or not we cannot tell--when we hear of the "great French
Encyclopædia." This work, which takes much from its {284} predecessor,
professing to correct it, was begun in 1792, and finished in 1832. There
are 166 volumes of text, and 6439 plates, which are sometimes incorporated
with the text, sometimes make about 40 more volumes. This is still the
monster production of the kind; though probably the German Cyclopædia of
Ersch and Gruber,[464] which was begun in 1818, and is still in progress,
will beat it in size. The great French work is a collection of
dictionaries; it consists of Cyclopædias of all the separate branches of
knowledge. It is not a work, but a collection of works, one or another
department is to be bought from time to time; but we never heard of a
complete set for sale in one lot. As ships grow longer and longer, the
question arises what limit there is to the length. One answer is, that it
will never do to try such a length that the stern will be rotten before the
prow is finished. This wholesome rule has not been attended to in the
matter before us; the earlier parts of the great French work were
antiquated before the whole were completed: something of the kind will
happen to that of Ersch and Gruber.

The production of a great dictionary of either of the kinds is far from an
easy task. There is one way of managing the _En_cyclopædia which has been
largely resorted to; indeed, we may say that no such work has been free
from it. This plan is to throw all the attention upon the great treatises,
and to resort to paste and scissors, or some process of equally easy
character, for the smaller articles. However it may be done, it has been
the rule that the Encyclopædia of treatises should have its supplemental
Dictionary of a very incomplete character. It is true that the treatises
are intended to do a good deal; and that the Index, if it be good, knits
the treatises and the dictionary into one whole of reference. Still there
are two stools, and between them a great deal will fall to the ground. The
dictionary portion of the _Britannica_ is not to be compared with its {285}
treatises; the part called Miscellaneous and Lexicographical in the
_Metropolitana_[465] is a great failure. The defect is incompleteness. The
biographical portion, for example, of the Britannica is very defective: of
many names of note in literature and science, which become known to the
reader from the treatises, there is no account whatever in the dictionary.
So that the reader who has learnt the results of a life in astronomy, for
example, must go to some other work to know when that life began and ended.
This defect has run through all the editions; it is in the casting of the
work. The reader must learn to take the results at their true value, which
is not small. He must accustom himself to regard the Britannica as a
splendid body of treatises on all that can be called heads of knowledge,
both greater and smaller; with help from the accompanying dictionary, but
not of the most complete character. Practically, we believe, this defect
cannot be avoided: two plans of essentially different structure cannot be
associated on the condition of each or either being allowed to abbreviate
the other.

The defect of all others which it is most difficult to avoid is inequality
of performance. Take any dictionary you please, of any kind which requires
the association of a number of contributors, and this defect must result.
We do not merely mean that some will do their work better than others; this
of course: we mean that there will be structural differences of execution,
affecting the relative extent of the different parts of the whole, as well
as every other point by which a work can be judged. A wise editor will not
attempt any strong measures of correction: he will remember that if some
portions be below the rest, which is a disadvantage, it follows that some
portions must be above the rest, which is an advantage. The only practical
level, if {286} level there must be, is that of mediocrity, if not of
absolute worthlessness: any attempt to secure equality of strength will
result in equality of weakness. Efficient development may be cut down into
meager brevity, and in this way only can apparent equality of plan be
secured throughout. It is far preferable to count upon differences of
execution, and to proceed upon the acknowledged expectation that the
prominent merits of the work will be settled by the accidental character of
the contributors; it being held impossible that any editorial efforts can
secure a uniform standard of goodness. Wherever the greatest power is
found, it should be suffered to produce its natural effect. There are,
indeed, critics who think that the merit of a book, like the strength of a
chain, is that of its weakest part: but there are others who know that the
parallel does not hold, and who will remember that the union of many
writers must show exaggeration of the inequalities which almost always
exist in the production of one person. The true plan is to foster all the
good that can be got, and to give development in the directions in which
most resources are found: a Cyclopædia, like a plant, should grow towards
the light.

The _Penny Cyclopædia_ had its share of this kind of defect or excellence,
according to the way in which the measure is taken. The circumstance is not
so much noticed as might be expected, and this because many a person is in
the habit of using such a dictionary chiefly with relation to one subject,
his own; and more still want it for the pure dictionary purpose, which does
not go much beyond the meaning of the word. But the person of full and
varied reference feels the differences; and criticism makes capital of
them. The Useful Knowledge Society was always odious to the organs of
religious bigotry; and one of them, adverting to the fact that geography
was treated with great ability, and most unusual fullness, in the _Penny
Cyclopædia_, announced it by making it the sole merit of {287} the work
that, with sufficient addition, it would make a tolerably good gazetteer.

Some of our readers may still have hanging about them the feelings derived
from this old repugnance of a class to all that did not associate direct
doctrinal teaching of religion with every attempt to communicate knowledge.
I will take one more instance, by way of pointing out the extent to which
stupidity can go. If there be an astronomical fact of the telescopic
character which, next after Saturn's ring and Jupiter's satellites, was
known to all the world, it was the existence of multitudes of double stars,
treble stars, etc. A respectable quarterly of the theological cast, which
in mercy we refrain from naming, was ignorant of this common
knowledge,--imagined that the mention of such systems was a blunder of one
of the writers in the _Penny Cyclopædia_, and lashed the presumed ignorance
of the statement in the following words, delivered in April, 1837:

    "We have forgotten the name of that Sidrophel who lately discovered
    that the fixed stars were not single stars, but appear in the heavens
    like soles at Billingsgate, in pairs; while a second astronomer, under
    the influence of that competition in trade which the political
    economists tell us is so advantageous to the public, professes to show
    us, through his superior telescope, that the apparently single stars
    are really three. Before such wondrous mandarins of science, how
    continually must _homunculi_ like ourselves keep in the background,
    lest we come between the wind and their nobility."

Certainly these little men ought to have kept in the background; but they
did not: and the growing reputation of the work which they assailed has
chronicled them in literary history; grubs in amber.

This important matter of inequality, which has led us so far, is one to
which the _Encyclopædia_ is as subject as the _Cyclopædia_; but it is not
so easily recognized as a fault. {288} We receive the first book as mainly
a collection of treatises: we know their authors, and we treat them as
individuals. We see, for instance, the names of two leading writers on
Optics, Brewster[466] and Herschel.[467] It would not at all surprise us if
either of these writers should be found criticising the other by name, even
though the very view opposed should be contained in the same _Encyclopædia_
with the criticism. And in like manner, we should hold it no wonder if we
found some third writer not comparable to either of those we have named. It
is not so in the _Cyclopædia_: here we do not know the author, except by
inference from a list of which we never think while consulting the work. We
do not dissent from this or that author: we blame the book.

The _Encyclopædia Britannica_ is an old friend. Though it holds a proud
place in our present literature, yet the time was when it stood by itself,
more complete and more clear than anything which was to be found elsewhere.
There must be studious men alive in plenty who remember when they were
studious boys, what a literary luxury it was to pass a few days in the
house of a friend who had a copy of this work. The present edition is a
worthy successor of those which went before. The last three editions,
terminating in 1824, 1842, and 1861, seem to show that a lunar cycle cannot
pass without an amended and augmented edition. Detailed criticism is out of
the question; but we may notice the effective continuance of the plan of
giving general historical dissertations on the progress of knowledge. Of
some of these dissertations we have had to take separate notice; and all
will be referred to in our ordinary treatment of current literature.[468]

The literary excellence of these two extensive undertakings is of the same
high character. To many this will {289} need justification: they will not
easily concede to the cheap and recent work a right to stand on the same
shelf with the old and tried magazine, newly replenished with the best of
everything. Those who are cognizant by use of the kind of material which
fills the _Penny Cyclopædia_ will need no further evidence: to others we
shall quote a very remarkable and certainly very complete testimony. The
_Cyclopædia of the Physical Sciences_, published by Dr. Nichol[469] in 1857
(noticed by us, April 4), is one of the most original of our special
dictionaries. The following is an extract from the editor's preface:

    "When I assented to Mr. Griffin's proposal that I should edit such a
    Cyclopædia, I had it in my mind that I might make the _scissors_
    eminently effective. Alas! on narrowly examining our best Cyclopædias,
    I found that the scissors had become blunted through too frequent and
    vigorous use. One great exception exists: viz., the _Penny Cyclopædia_
    of Charles Knight.[470] The cheapest and the least pretending, it is
    really the most philosophical of our _scientific_ dictionaries. It is
    not made up of a series of treatises, some good and many indifferent,
    but is a thorough _Dictionary_, well proportioned and generally written
    by the best men of the time. The more closely it is examined, the more
    deeply will our obligation be felt to the intelligence and
    conscientiousness of its projector and editor."

After Dr. Nichol's candid and amusing announcement of his scissorial
purpose, it is but fair to state that nothing of the kind was ultimately
carried into effect, even upon the work in which he found so much to
praise. I quote this testimony because it is of a peculiar kind.


The success of the _Penny Magazine_ led Mr. Charles Knight in 1832 to
propose to the Useful Knowledge Society a Cyclopædia in weekly penny
numbers. These two works stamp the name of the projector on the literature
of our day in very legible characters. Eight volumes of 480 pages each were
contemplated; and Mr. Long[471] and Mr. Knight were to take the joint
management. The plan embraced a popular account of Art and Science, with
very brief biographical and geographical information. The early numbers of
the work had some of the _Penny Magazine_ character: no one can look at the
pictures of the Abbot and Abbess in their robes without seeing this. By the
time the second volume was completed, it was clearly seen that the plan was
working out its own extension: a great development of design was submitted
to, and Mr. Long became sole editor. Contributors could not be found to
make articles of the requisite power in the assigned space. One of them
told us that when he heard of the eight volumes, happening to want a shelf
to be near at hand for containing the work as it went on, he ordered it to
be made to hold twenty-five volumes easily. But the inexorable logic of
facts beat him after all: for the complete work contained twenty-six
volumes and two thick volumes of Supplement.

The penny issue was brought to an end by the state of the law, which
required, in 1833, that the first and last page of everything sold
separately should contain the name and address of the printer. The penny
numbers contained this imprint on the fold of the outer leaf: and _qui
tam_[472] informations were laid against the agents in various towns. {291}
It became necessary to call in the stock; and the penny issue was
abandoned. Monthly parts were substituted, which varied in bulk, as the
demands of the plan became more urgent, and in price from one sixpence to
three. The second volume of Supplement appeared in 1846, and during the
fourteen years of issue no one monthly part was ever behind its time. This
result is mainly due to the peculiar qualities of Mr. Long, who unites the
talents of the scholar and the editor in a degree which is altogether
unusual. If any one should imagine that a mixed mass of contributors is a
punctual piece of machinery, let him take to editing upon that hypothesis,
and he shall see what he shall see and learn what he shall learn.

The _English_ contains about ten per cent more matter than the _Penny
Cyclopædia_ and its supplements; including the third supplementary volume
of 1848, which we now mention for the first time. The literary work of the
two editions cost within 500l. and 50,000l.: that of the two editions of
the _Britannica_ cost 41,000l. But then it is to be remembered that the
_Britannica_ had matter to begin upon, which had been paid for in the
former editions. Roughly speaking, it is probable that the authorship of a
page of the same size would have cost nearly the same in one as in the

The longest articles in the _Penny Cyclopædia_ were "Rome" in 98 columns
and "Yorkshire" in 86 columns. The only article which can be called a
treatise is the Astronomer Royal's "Gravitation," founded on the method of
Newton in the eleventh section, but carried to a much greater extent. In
the _English Cyclopædia_, the longest article of geography is "Asia," in 45
columns. In natural history the antelopes demand 36 columns. In biography,
"Wellington" uses up 42 columns, and his great military opponent 41
columns. In the division of Arts and Sciences, which includes much of a
social and commercial character, the length of articles often depends upon
the state of the {292} times with regard to the subject. Our readers would
not hit the longest article of this department in twenty guesses: it is
"Deaf and Dumb" in 60 columns. As other specimens, we may cite Astronomy,
19; Banking, 36; Blind, 24; British Museum, 35; Cotton, 27; Drama, 26;
Gravitation, 50; Libraries, 50; Painting, 34; Railways, 18; Sculpture, 36;
Steam, etc., 37; Table, 40; Telegraph, 30; Welsh language and literature,
39; Wool, 21. These are the long articles of special subdivisions: the
words under which the _En_cyclopædia gives treatises are not so prominent.
As in Algebra, 10; Chemistry, 12; Geometry, 8; Logic, 14; Mathematics, 5;
Music, 9. But the difference between the collection of treatises and the
dictionary may be illustrated thus: though "Mathematics" have only five
columns, "Mathematics, recent terminology of," has eight: and this article
we believe to be by Mr. Cayley,[473] who certainly ought to know his
subject, being himself a large manufacturer of the new terms which he
explains. Again, though "Music" _in genere_, as the schoolmen said, has
only nine columns, "Temperament and Tuning," has eight, and "Chord" alone
has two. And so on.

In a dictionary of this kind it is difficult to make a total clearance of
_personality_: by which we mean that exhibition of peculiar opinion which
is offensive to taste when it is shifted from the individual on the
corporate book. The treatise of the known author may, as we have said,
carry that author's controversies on its own shoulders: and even his
crotchets, if we may use such a word. But {293} the dictionary should not
put itself into antagonism with general feeling, nor even with the feelings
of classes. We refer particularly to the ordinary and editorial teaching of
the article. If, indeed, the writer, being at issue with mankind, should
confess the difference, and give abstract of his full grounds, the case is
altered: the editor then, as it were, admits a correspondent to a statement
of his own individual views. The dictionary portion of the Britannica is
quite clear of any lapses on this point, so far as we know: the treatises
and dissertations rest upon their authors. The Penny Cyclopædia was all but
clear: and great need was there that it should have been so. The Useful
Knowledge Society, starting on the principle of perfect neutrality in
politics and religion, was obliged to keep strict watch against the
entrance of all attempt even to look over the hedge. There were two--we
believe only two--instances of what we have called personality. The first
was in the article "Bunyan." It is worth while to extract all that is
said--in an article of thirty lines--about a writer who is all but
universally held to be the greatest master of allegory that ever wrote:

    "His works were collected in two volumes, folio, 1736-7: among them
    'The Pilgrim's Progress' has attained the greatest notoriety. If a
    judgment is to be formed of the merits of a book by the number of times
    it has been reprinted, and the many languages into which it has been
    translated, no production in English literature is superior to this
    coarse allegory. On a composition which has been extolled by Dr.
    Johnson, and which in our own times has received a very high critical
    opinion in its favor [probably Southey], it is hazardous to venture a
    disapproval, and we, perhaps, speak the opinion of a small minority
    when we confess that to us it appears to be mean, jejune and

--If the unfortunate critic who thus individualized himself had been a
sedulous reader of Bunyan, his power over {294} English would not have been
so _jejune_ as to have needed that fearful word. This little bit of
criticism excited much amusement at the time of its publication: but it was
so thoroughly exceptional and individual that it was seldom or never
charged on the book. The second instance occurred in the article
"Socinians." It had been arranged that the head-words of Christian sects
should be intrusted to members of the sects themselves, on the
understanding that the articles should simply set forth the accounts which
the sects themselves give of their own doctrines. Thus the article on the
Roman Church was written by Dr. Wiseman.[474] But the Unitarians were not
allowed to come within the rule: as in other quarters, they were treated as
the gypsies of Christianity. Under the head "Socinians"--a name repudiated
by themselves--an opponent was allowed not merely to state their alleged
doctrines in his own way, but to apply strong terms, such as "audacious
unfairness," to some of their doings. The protests which were made against
this invasion of the understanding produced, in due time, the article
"Unitarians," written by one of that persuasion. We need not say that these
errors have been amended in the English Cyclopædia: and our chief purpose
in mentioning them is to remark, that this is all we can find on the points
in question against twenty-eight large volumes produced by an editor whose
task was monthly, and whose issue was never delayed a single hour. How much
was arrested before publication none but himself can say. We have not
alluded to one or two remonstrances on questions of absolute fact, which
are beside the present purpose.

Both kinds of encyclopædic works have been fashioned upon predecessors,
from the very earliest which had a predecessor to be founded upon; and the
undertakings before us will be themselves the ancestors of a line of
successors. Those who write in such collections should be {295} careful
what they say, for no one can tell how long a mis-statement may live. On
this point we will give the history of a pair of epithets. When the
historian De Thou[475] died, and left the splendid library which was
catalogued by Bouillaud[476] and the brothers Dupuis[477] (Bullialdus and
Puteanus), there was a manuscript of De Thou's friend Vieta,[478] the
_Harmonicon Coeleste_, of which it is on record, under Bouillaud's hand,
that he himself lent it to Cosmo de' Medici,[479] to which must be added
that M. Libri[480] found it in the Magliabecchi Library at Florence in our
own day. Bouillaud, it seems, entirely forgot what he had done. Something,
probably, that Peter Dupuis said to Bouillaud, while they were at work on
the catalogue, remained on his memory, and was published by him in 1645,
long after; to the effect that Dupuis lent the manuscript to Mersenne,[481]
from whom it was procured by some intending plagiarist, who would not give
it back. This was repeated by Sherburne,[482] in 1675, who speaks of the
work, which "being communicated to Mersennus was, by some perfidious
acquaintance of that honest-minded person, surreptitiously taken from him,
and irrecoverably lost or suppressed, to the unspeakable detriment of the
lettered world." Now let the {296} reader look through the dictionaries of
the last century and the present, scientific or general, at the article,
"Vieta," and he will be amused with the constant recurrence of
"honest-minded" Mersenne, and his "surreptitious" acquaintance. We cannot
have seen less than thirty copies of these epithets.


    October 18, 1862. _Correspondence of Scientific Men of the Seventeenth
    Century, in the Collection of the Earl of Macclesfield._[483] 2 vols.
    (Oxford, University Press.)

Though the title-page of this collection bears the date 1841, it is only
just completed by the publication of its Table of Contents and Index.
Without these, a work of the kind is useless for consultation, and cannot
make its way. The reason of the delay will appear: its effect is well known
to us. We have found inquirers into the history of science singularly
ignorant of things which this collection might have taught them.

In the same year, 1841, the Historical Society of Science, which had but a
brief existence, published a collection of letters, eighty-three in number,
edited by Mr. Halliwell,[484] of English men of science, which dovetails
with the one before us, and is for the most part of a prior date. The two
should be bound up together. The smaller collection runs from 1562 to 1682;
the larger, from 1606 to past 1700. We shall speak of the two as the Museum
collection and the Macclesfield collection. And near them should be placed,
in every scientific library, the valuable collection published, by Mr.
Edleston,[485] for Trinity College, in 1850.


The history of these letters runs back to famous John Collins, the
attorney-general of the mathematics, as he has been called, who wrote to
everybody, heard from everybody, and sent copies of everybody's letter to
everybody else. He was in England what Mersenne[486] was in France: as
early as 1671, E. Bernard[487] addresses him as "the very Mersennus and
intelligence of this age." John Collins[488] was never more than accountant
to the Excise Office, to which he was promoted from teaching writing and
ciphering, at the Restoration: he died in 1682. We have had a man of the
same office in our own day, the late Prof. Schumacher,[489] who made the
little Danish Observatory of Altona the junction of all the lines by which
astronomical information was conveyed from one country to another. When the
collision took place between Denmark and the Duchies, the English
Government, moved by the Astronomical Society, instructed its diplomatic
agents to represent strongly to the Danish Government, when occasion should
arise, the great importance of the Observatory of Altona to the
astronomical communications of the whole world. But Schumacher had his own
celebrated journal, the _Astronomische Nachrichten_, by which to work out
part of his plan; private correspondence was his supplementary assistant.
Collins had only correspondence to rely on. Nothing is better known than
that it was Collins's collection which furnished the materials put forward
by the Committee of the Royal Society in 1712, as a defence of Newton
against the partisans of Leibnitz. The noted _Commercium Epistolicum_ is
but the abbreviation of a title which runs on with "D. Johannis Collins et
aliorum ..."

The whole of this collection passed into the hands of {298} William
Jones,[490] the father of the Indian Judge of the same name, who died in
1749. Jones was originally a teacher, but was presented with a valuable
sinecure by the interest of George, second Earl of Macclesfield, the mover
of the bill for the change of style in Britain, who died President of the
Royal Society. This change of style may perhaps be traced to the union of
energies which were brought into concert by the accident of a common
teacher: Lord Macclesfield and Lord Chesterfield,[491] the mover and the
seconder, and Daval,[492] who drew the bill, were pupils of De Moivre.[493]
Jones, who was a respectable mathematician though not an inventor,
collected the largest mathematical library of his day, and became possessor
of the papers of Collins, which contained those of Oughtred[494] and
others. Some of these papers passed into the custody of the Royal Society:
but the bulk was either bequeathed to, or purchased by, Lord Macclesfield;
and thus they found their way to Shirburn Castle, where they still remain.

A little before 1836, this collection attracted the attention of a
searching inquirer into points of mathematical history, the late Professor
Rigaud,[495] who died in 1839. He examined the whole collection of letters,
obtained Lord Macclesfield's consent to their publication, and induced the
Oxford Press to bear the expense. It must be particularly remembered that
there still remains at Shirburn Castle a {299} valuable mass of
non-epistolary manuscripts. So far as we can see, the best chance of a
further examination and publication lies in public encouragement of the
collection now before us: the Oxford Press might be induced to extend its
operations if it were found that the results were really of interest to the
literary and scientific world. Rigaud died before the work was completed,
and the publication was actually made by one of his sons, S. Jordan
Rigaud,[496] who died Bishop of Antigua. But this publication was little
noticed, for the reasons given. The completion now published consists of a
sufficient table of contents, of the briefest kind, by Professor De Morgan,
and an excellent index by the Rev. John Rigaud.[497] The work is now fairly
started on its career.

If we were charged to write a volume with the title "Small things in their
connection with great," we could not do better than choose the small part
of this collection of letters as our basis. The names, as well as the
contents, are both great and small: the great names, those which are known
to every mathematician who has any infusion of the history of his pursuit,
are Briggs,[498] Oughtred, Charles Cavendish,[499] Gascoigne,[500] Seth
Ward,[501] Wallis,[502] {300} Hu[y]gens,[503] Collins,[504] William
Petty,[505] Hooke,[506] Boyle,[507] Pell,[508] Oldenburg,[509]
Brancker,[510] Slusius,[511] Bertit,[512] Bernard,[513] Borelli,[514]
Mouton,[515] Pardies,[516] Fermat,[517] Towneley,[518] Auzout,[519] {301}
D. Gregory,[520] Halley,[521] Machin,[522] Montmort,[523] Cotes,[524]
Jones,[525] Saunderson,[526] Reyneau,[527] Brook Taylor,[528]
Maupertuis,[529] Bouguer,[530] La Condamine,[531] Folkes,[532]
Macclesfield,[533] {302} Baker,[534] Barrow,[535] Flamsteed,[536] Lord
Brounker,[537] J. Gregory,[538] Newton[539] and Keill.[540] To these the
Museum collection adds the names of Thomas Digges,[541] Dee,[542] Tycho
Brahe,[543] Harriot,[544] Lydyat,[545] Briggs,[546] Warner,[547] Tarporley,
Pell,[548] Lilly,[549] Oldenburg,[550] Collins,[551] Morland.[552]


The first who appears on the scene is the celebrated Oughtred, who is
related to have died of joy at the Restoration: but it should be added, by
way of excuse, that he was eighty-six years old. He is an animal of extinct
race, an Eton mathematician. Few Eton men, even of the minority which knows
what a sliding rule is, are aware that the inventor was of their own school
and college: but they may be excused, for Dr. Hutton,[553] so far as his
Dictionary bears witness, seems not to have known it any more than they. A
glance at one of his letters reminds us of a letter from the Astronomer
Royal on the discovery of Neptune, which we printed March 20, 1847. Mr.
Airy[554] there contends, and proves it both by Leverrier[555] and by
Adams,[556] that the limited publication of a private letter is more
efficient than the more general publication of a printed memoir. The same
may be true of a dead letter, as opposed to a dead book. Our eye was caught
by a letter of Oughtred (1629), containing systematic use of contractions
for the words _sine_, _cosine_, etc., prefixed to the symbol of the angle.
This is so very important a step, simple as it is, that Euler[557] is
justly held to have greatly advanced trigonometry by its introduction.
Nobody that we know of has noticed that Oughtred was master of the
improvement, and willing to have taught it, if people would have learnt.
After looking at his dead letter, we naturally turned to his dead book on
trigonometry, and there we found the abbreviations _s_, _sco_, _t_, _tco_,
_se_, _seco_, regularly established as part of the system of the work. But
not one of those who have investigated the contending claims of Euler and
Thomas {304} Simpson[558] has chanced to know of Oughtred's
"Trigonometrie": and the present revival is due to his letter, not to his

A casual reader, turning over the pages, would imagine that almost all the
letters had been printed, either in the General Dictionary, or in
Birch,[559] etc.: so often does the supplementary remark begin with "this
letter has been printed in ----." For ourselves we thought, until we
counted, that a large majority of the letters had been given, either in
whole or in part. But the positive strikes the mind more forcibly than the
negative: we find that all of which any portion has been in type makes up
very little more than a quarter; the cases in which the whole letter is
given being a minority of this quarter. The person who has been best
ransacked is Flamsteed: of 36 letters from him, 34 had been previously
given in whole or in part. Of 59 letters to and from Newton, only 17 have
been culled.

The letters have been modernized in spelling, and, to some extent, in
algebraical notation; it also seems that conjectural methods of introducing
interpolations into the text have been necessary. For all this we are
sorry: the scientific value of the collection is little altered, but its
literary value is somewhat lowered. But it could not be helped: the
printers could not work from the originals, and Professor Rigaud had to
copy everything himself. A fac-simile must have been the work of more time
than he had to give: had he attempted it, his death would have cut short
the whole undertaking, instead of allowing him to prepare everything but a
preface, and to superintend the printing of one of the volumes. We may also
add, that we believe we have notices of _all_ the letters in the
Macclesfield collection. We judge this because several which are too
trivial to print are numbered and described; and those would certainly not
have been noticed if _any_ omissions had {305} been made. And we know that
every letter was removed from Shirburn Castle to Oxford.

Two persons emerge from oblivion in this series of letters. The first is
Michael Dary,[560] an obscure mathematician, who was in correspondence with
Newton and other stars. He was a gauger at Bristol, by the interest of
Collins; afterwards a candidate for the mathematical school at Christ's
Hospital, with a certificate from Newton: he was then a gunner in the
Tower, and is lastly described by Wallis as "Mr. Dary, the tobacco-cutter,
a knowing man in algebra." In 1674, Dary writes to Newton at Cambridge, as
follows:--"Although I sent you three papers yesterday, I cannot refrain
from sending you this. I have had fresh thoughts this morning." Two months
afterwards poor Newton writes to Collins, "Mr. Dary is very solicitous
about mathematics": but in spite of the persecution, he subscribes himself
to Dary "your loving friend." Dary's _problem_ is that of finding the rate
of interest of an annuity of which the value and term are given. Dary's
_theorem_, which he seems to have invented specially for the solution of
his problem, though it is of wide range, can be exhibited to mathematical
readers even in our columns. In modern language, it is that the limit of
[phi]^{_n_}_x_, when _n_ increases without limit, is a solution of [phi]_x_
= _x_. We have mentioned the I. Newton to whom Dary looked up; we add a
word about the one on whom he looked down. Dr. John Newton,[561] a sedulous
publisher of logarithms, tables of interest, etc., who began his career
before Isaac Newton, sometimes puzzles those who do not know him, when
described as I. Newton. The scientific world was of opinion that all that
was valuable in one of his works was taken from Dary's private


The second character above alluded to is one who carried mathematical
researches a far greater length than Newton himself: the assistance which
he rendered in this respect, even to Newton, has never been acknowledged in
modern times: though the work before us shows that his contemporaries were
fully aware of it, and never thought of concealing it. In his theory of
gravitation, in which, so far as he went, we have every reason to believe
he was prior to Newton, he did not extend his calculations to the distance
of the moon; his views in this matter were purely terrestrial, and led him
to charge according to weight. He was John Stiles, the London and Cambridge
carrier: his name is a household word in the Macclesfield Letters, and is
even enshrined in the depths of Birch's quartos. Dary informs Newton--let
us do his memory this justice--that he had paid John Stiles for the
carriage. At the time when the railroad to Cambridge was opened, a
correspondent recommended the directors, in our columns, to call an engine
by the name of John Stiles, and never to let that name go off the road. We
do not know whether the advice was followed: if not, we repeat it.

Little points of life and manners come out occasionally. Baker, the author
of a work on algebra much esteemed at the time, wrote to Collins that their
circumstances are alike, "having a just and equal number of chargeable
olive-branches, and being in the same predicament and blessed condemnation
with you, not more preaching than unpaid, and preaching the art of
contentment to others, am forced to practise it." But the last sentence of
his letter runs as follows: "I have sent by the bearer ... twenty
shillings, as a token to you; desiring you to accept of it, as a small
taste from Yours, Thos. Baker." In our day, men of a station to pay parish
taxes do not offer their friends hard money to buy liquor. But
Flamsteed[562] writes to Collins as follows: "Last week he sent us down the
counterpart, which {307} my father has scaled, and I return up to you by
the carrier, with 5l. to be paid to Mr. Leneve for the writing, I have
added 2s. 6d. over, which will pay the expenses and serve to drink, with
him." This would seem as odd to us as it would have seemed thirty years ago
that half-a-crown should pay carriage for a deed from Derby to London, and
leave margin for a bottle of wine: in our day, the Post-office and the
French treaty would just manage it between them. But Flamsteed does not
limit his friend to one bottle; he adds, "If you expend more than the
half-crown, I will make it good after Whitsuntide." Collins does not
remember exactly where he had met James Gregory, and mentions two equally
likely places thus: "Sir, it was once my good hap to meet with you in an
alehouse or in Sion College." There is a little proof how universally the
dinner-hour was twelve o'clock. Astronomers well know the method of finding
time by equal altitudes of the sun before and after noon: Huyghens calls it
"le moyen de deux égales hauteurs du soleil devant et après _dîner_."[563]

There is one mention of "Mr. Cocker,[564] our famous English graver and
writer, now a schoolmaster at Northampton." This is the true Cocker: his
genuine works are specimens of writing, such as engraved copy-books,
including some on arithmetic, with copper-plate questions and space for the
working; also a book of forms for law-stationers, with specimens of legal
handwriting. It is recorded somewhere that Cocker and another, whose name
we forget, competed with the Italians in the beauty of their flourishes.
This was his real fame: and in these matters he was great. The eighth
edition of his book of law forms (1675), published shortly after Cocker's
death, has a preface signed "J. H." This was John Hawkins, who became
possessed of Cocker's papers--at least he said so--and {308} subsequently
forged the famous Arithmetic,[565] a second work on Decimal Arithmetic, and
an English dictionary, all attributed to Cocker. The proofs of this are set
out in De Morgan's _Arithmetical Books_. Among many other corroborative
circumstances, the clumsy forger, after declaring that Cocker to his dying
day resisted strong solicitation to publish his Arithmetic, makes him write
in the preface _Ille ego qui quondam_[566] of this kind: "I have been
instrumental to the benefit of many, by virtue of those useful arts,
writing and engraving; and do _now_, with the same _wonted alacrity_, cast
this my arithmetical mite into the public treasury." The book itself is not
comparable in merit to at least half-a-dozen others. How then comes Cocker
to be the impersonation of Arithmetic? Unless some one can show proof,
which we have never found, that he was so before 1756, the matter is to be
accounted for thus.

Arthur Murphy,[567] the dramatist, was by taste a man of letters, and ended
by being the translator of Tacitus; though many do not know that the two
are one. His friends had tried to make him a man of business; and no doubt
he had been well plied with commercial arithmetic. His first dramatic
performance, the farce of "The Apprentice," produced in 1756, is about an
idle young man who must needs turn actor. Two of the best known books of
the day in arithmetic were those of Cocker and Wingate.[568] Murphy chooses
_Wingate_ to be the name of an old merchant who {309} delights in vulgar
fractions, and _Cocker_ to be his arithmetical catchword--"You read
Shakespeare! get Cocker's Arithmetic! you may buy it for a shilling on any
stall; best book that ever was wrote!" and so on. The farce became very
popular, and, as we believe, was the means of elevating Cocker to his
present pedestal, where Wingate would have been, if his name had had the
droller sound of the two to English ears.

A notoriety of an older day turns up, Major-General Lambert.[569] The
common story is that he was banished to Guernsey, where he passed thirty
years in confinement, rearing and painting flowers. But Baker, in 1678,
represents him as a prisoner at Plymouth, sending equations for solution as
a challenge: probably his place of confinement was varied, and his
occupation also.

[General Lambert was removed to Plymouth, probably about 1668. His daughter
captured the son of the Governor of Guernsey, who therefore probably was
reckoned an unsafe custodier thenceforward; though he assured the king that
he had turned the young couple out of doors, and had never given them a
penny. Great importance was attached to Lambert's safe detention: probably
the remaining republicans looked upon him as to be their next Cromwell, if
such a thing were to be. There were standing orders to shoot him at once on
the first appearance of any enemy before the island. See _Notes and
Queries_, 3d S. iv. 89.]

Collins informs James Gregory that "some of the Royal Academy wrote over to
Mr. Oldenburg, who was desired to impart the same to the Council of the
Royal Society, that the French King was willing to allow pensions to one or
two learned Englishmen, but they never made any answer {310} to such a
proposal." This was written in 1671, and the thing probably happened
several years before. Mr. De Morgan communicated the account of the
proposal to Lord Macaulay, who replied that he did not think that any
Englishman _received_ a literary pension from Louis; but that there is a
curious letter, about 1664, from the French Ambassador, in which he says
that he has, by his master's orders, been making inquiries as to the state
of learning in England, and that he is sorry to find that the best writer
is _the infamous Miltonus_. On two such independent testimonies it may be
held proved that the French King had attempted to buy a little adherence
from English literature and science; and the silent contempt of the Royal
Society is an honorable fact in their history.

Another little bit of politics is as follows. Oughtred is informed that
"Mr. Foster,[570] our Lecturer on Astronomy at Gresham College, is put out
because he will not kneel down at the communion-table. A Scotsman [Mungo
Murray], one that is _verbi bis minister_,[571] is now lecturer in Mr.
Foster's place." Ward in his work on the Gresham Professors,[572]
suppresses the reason, and the suppression lowers the character of his
book. Foster was expelled in 1636, and re-elected on a vacancy in 1641,
when Puritanism had gained strength.

The correspondence of Newton would require deeper sifting than could be
given in such an article as the present. The first of the letters (1669) is
curious, as presenting the {311} appearance of forms belonging to the great
calculus which, in this paragraph, we ought to call that of fluxions. We
find, of the date February 18, 1669-70, what we believe is the earliest
manifestation of that morbid part of Newton's temperament which has been so
variously represented. He had solved a problem--being that which we have
called Dary's--on which he writes as follows: "The solution of the annuity
problem, if it will be of any use, you have my leave to insert into the
_Philosophical Transactions_, so it be without my name to it. For I see not
what there is desirable in public esteem, were I able to acquire and
maintain it. It would perhaps increase my acquaintance, the thing which I
chiefly study to decline."

Three letters touch upon "the experiment of glass rubbed to cause various
motions in bits of paper underneath": they are supplements to the account
given by Newton to the Royal Society, and printed by Birch. It was Newton,
so far as appears, who added _glass_ to the substances known to be
electric. Soon afterwards we come to a little bit of the history of the
appointment to the Mint. It has appeared from the researches of late years
that Newton was long an aspirant for public employment: the only coolness
which is known to have taken place between him and Charles Montague[573]
[Halifax] arose out of his imagining that his friend was not in earnest
about getting him into the public service. March 14, 1696, Newton writes
thus to Halley: "And if the rumour of preferment for me in the Mint should
hereafter, upon the death of Mr. Hoar [the comptroller], or any other
occasion, be revived, I pray that you would {312} endeavour to obviate it
by acquainting your friends that I neither _put in_ for _any_ place in the
Mint, nor would meddle with _Mr. Hoar's place_, were it offered to me."
This means that Mr. Hoar's place had been suggested, which Newton seems to
have declined. Five days afterwards, Montague writes to Newton that he is
to have the _Wardenship_. It is fair to Newton to say that in all
probability this was not--or only in a smaller degree--a question of
personal dignity, or of salary. It must by this time have been clear to him
that the minister, though long bound to make him an object of patronage,
was actually seeking him for the Mint, because he wanted both Newton's name
and his talents for business--which he knew to be great--in the weighty and
dangerous operation of restoring the coinage. It may have been, and
probably was, the case that Newton had a tolerably accurate notion of what
he would have to do, and of what degree of power would be necessary to
enable him to do it in his own way.

We have said that the non-epistolary manuscripts are still unexamined.
There is a chance that one of them may answer a question of two centuries'
standing, which is worth answering, because it has been so often asked.
About 1640, Warner,[574] afterwards assisted by Pell,[575] commenced a
table of _antilogarithms_, of the kind which Dodson[576] afterwards
constructed anew and published. In the Museum collection there is inquiry
after inquiry from Charles Cavendish,[577] first, as to when the
_Analogics_, as he called them, would be finished; next, when they would be
printed. Pell answers, in 1644, that Warner left his papers to a kinsman,
who had become bankrupt, and proceeds thus:

"I am not a little afraid that all Mr. Warner's papers, {313} and no small
share of my labours therein, are seazed upon, and most unmathematically
divided between the sequestrators and creditors, who (not being able to
ballance the account where there appeare so many numbers, and much troubled
at the sight of so many crosses and circles in the superstitious Algebra
and that black art of Geometry) will, no doubt, determine once in their
lives to become figure-casters, and so vote them all to be throwen into the
fire, if some good body doe not reprieve them for pye-bottoms, for which
purposes you know analogicall numbers are incomparably apt, if they be
accurately calculated."

Pell afterwards told Wallis[578] that the papers had fallen into the hands
of Dr. Busby,[579] and Collins[580] writes that they were left in the hands
of Dr. Thorndike,[581] a prebendary of Westminster; whence Rigaud[582]
seems to say that Thorndike had left them to Dr. Busby. Birch[583] says
that he procured for the Royal Society four boxes from Busby's trustees,
containing papers of Warner and Pell: but there is no other tradition of
such things in the Society. But in the Birch manuscripts at the British
Museum, there turns up, as printed in what we call the Museum collection, a
list of Warner's papers, with _Collins's_ receipt to Dr. Thorndike at the
bottom, and engagement to restore them on demand. The date is December 14,
1667; Wallis's statement being in 1693. It is possible that Busby may be a
mistake altogether: he was very unlikely to have had charge of any
mathematical papers: there may have been a confusion between the Prebendary
of Westminster and the Head Master of Westminster School. If so, in all
probability Thorndike handed {314} the cumbrous lot over to the notorious
collector of mathematical papers, blessing himself that he got rid of them
in a manner which would insure their return if he were called upon by the
owners to restore them. It is much against this hypothesis that Dodson, who
certainly recalculated, can say nothing more about Warner than a repetition
of Wallis's story: though, had Collins kept the papers, they would probably
have been in Jones's possession at the very time when Dodson, who was a
friend of Jones and a user of his library, was engaged on his own
computations. But even books, and still more manuscripts, are often
singularly overlooked; and it remains not very improbable that Warner's
table is now at Shirburn Castle, among the unexamined manuscripts.


_Redit labor actus in orbem._[584] Among the matters which have come to me
since the Budget opened, there is a pamphlet of quadrature of two pages and
a half from Professor Recalcati,[585] already mentioned. It ends with
"Quelque objection qu'on fasse touchant les raisonnements ci-dessus on
tombera toujours dans l'absurde."[586] A civil engineer--so he says--has
made the quadrature "no longer a problem, but an axiom." As follows: "Take
the quadrant of a circle whose circumference is given, square the quadrant
which gives the true square of the circle. Because 30 ÷ 4 = 7.5 × 7.5 =
56.25 = the positive square of a circle whose circumference is 30."
Brevity, the soul of wit, is the "wings of mighty-winds" to quadrature, and
sends it "flying all abroad." A _surbodhicary_--something like M.A. or
LL.D., I understand--at Calcutta, published in 1863 the division of an
{315} angle into any odd number of parts, demonstration and all in--when
the diagram is omitted--one page, good-sized, well-leaded type, small
duodecimo. But in the Preface he acknowledges "sheer inability" to execute
his task. Mr. William Dean, of Todmorden, in 1863, announced 3-9/64 as
proved both practically and geometrically: he has been already mentioned
anonymously. Next I have the tract of Don Juan Larriva, published at Leiria
in 1856, and dedicated to Queen Victoria. Mr. W. Peters,[587] already
mentioned, who has for some months been circulating diagrams on a card,
publishes (August, 1865) _The Circle Squared_. He agrees with the
Archpriest of St. Vitus. He hints that a larger publication will depend
partly on the support he receives, and partly on the castigation, for which
last, of course, he looks to me. Cyclometers have their several styles of
wit; so have anticyclometers too, for that matter. Mr. Peters will not
allow me any extra-journal being: I am essentially a quotation from the
_Athenæum_; "A. De Morgan" _et præterea nihil_.[588] If he had to pay for
keeping me set up, he would find out his mistake, and would be glad to
compound handsomely for a stereotype. Next comes a magnificent sheet of
pasteboard, printed on both sides. Having glanced at it and detected
quadrature, I began methodically at the beginning--"By Royal Command," with
the lion and unicorn, and all that comes between. Mercy on us! thought I to
myself: has Her Majesty referred the question to the Judicial Committee of
the Privy Council, where all the great difficulties go now-a-days, and is
this proclamation the result? On reading further I was relieved by finding
that the first side is entirely an advertisement of Joseph Gillott's[589]
steel pens, with engraving of his {316} premises, and notice of novel
application of his unrivalled machinery. The second side begins with "the
circle rectified" by W. E. Walker,[590] who finds [pi] =
3.141594789624155.... This is an off-shoot from an accurate geometrical
rectification, on which is to be presumed Mr. Gillott's new machinery is
founded. I have no doubt that Mr. Walker's error, which is only in the
sixth place of decimals, will not hurt the pens, unless it be by the
slightest possible increase of the tendency to open at the points. This
arises from Mr. Walker having rectified above proof by .000002136034362....

Lastly, I, even I myself, who have long felt that I was a quadrature below
par, have solved the problem by means which, in the present state of the
law of libel, I dare not divulge. But the result is permitted; and it goes
far to explain all the discordances. The ratio of the circumference to the
diameter is not always the same! Not that it varies with the radius; the
geometers are right enough on that point: but it varies with the time, in a
manner depending upon the difference of the true longitudes of the Sun and
Moon. A friend of mine--at least until he misbehaved--insisted on the mean
right ascensions: but I served him as Abraham served his guest in
Franklin's parable. The true formula is, A and a being the Sun's and Moon's

  [pi] = 3-13/80 + 3/80 cos(A - a).

Mr. James Smith obtained his quadrature at full moon; the Archpriest of St.
Vitus and some others at new moon. Until I can venture to publish the
demonstration, I recommend the reader to do as I do, which is to adopt
3.14159..., and to think of the matter only at the two points of the lunar
month at which it is correct. The _Nautical Almanac_ will no doubt give
these points in a short time: I am in correspondence with the Admiralty,
with nothing {317} to get over except what I must call a perverse notion on
the part of the Superintendent of the _Almanac_, who suspects one
correction depending on the Moon's latitude; and the Astronomer Royal leans
towards another depending on the date of the Queen's accession. I have no
patience with these men: what can the Moon's node of the Queen's reign
possibly have to do with the ratio in question? But this is the way with
all the regular men of science; Newton is to them etc. etc. etc. etc.

The following method of finding the circumference of a circle (taken from a
paper by Mr. S. Drach[591] in the _Phil. Mag._, Jan. 1863, Suppl.) is as
accurate as the use of 3.14159265. From three diameters deduct
8-thousandths and 7-millionths of a diameter; to the result add five per
cent. We have then not quite enough; but the shortcoming is at the rate of
about an inch and a sixtieth of an inch in 14,000 miles.


Though I have met with nothing but a little tract from the school of Jacob
Behmen[592] (or Böhme; I keep to the old English version of his name), yet
there has been more, and of a more recent date. I am told of an
"Introduction to Theosophy [_Theo_ private, I suppose, as in theological];
or, the Science of the Mystery of Christ," published in 1854, mostly from
the writings of William Law[593]: and also of a volume of 688 pages, of the
same year, printed for private circulation, containing notes for a
biography of William Law. The editor of the first work wishes to grow "a
{318} generation of perfect Christians" by founding a Theosophic College,
for which he requests the public to raise a hundred thousand pounds. There
is a good account of Jacob Behmen in the _Penny Cyclopædia_. The author
mentions inaccurate accounts, one of which he quotes, as follows: "He
derived all his mystical and rapturous doctrine from Wood's[594] _Athenæ
Oxonienses_, Vol. I, p. 610, and _Hist. et Antiq. Acad. Oxon._, Vol. II, p.
308." On which the author remarks that Wood was born after Behmen's death.
There must have been a few words which slipped out: what is meant is that
Behmen "derived his doctrine from _Robert Fludd_,[595] _for whom see_
Wood's etc. etc." Even this is absurd enough: for Behmen began to publish
in 1610, and Fludd in 1616. Fludd was a Rosicrucian, and a mystic of a
different type from Behmen. I have some of his works, and could produce out
of them paradoxes enough, according to our ways of thinking, to fit out a
host. But the Rosicrucian system was a recognized school of its day, and
Fludd, a man of great learning, had abettors enough in all which he
advanced, and predecessors in most of it.

[A Correspondent has recently sent a short summary of the claims of Jacob
Behmen to rank higher than I have placed him. I shall gladly insert this
summary in the book I contemplate, as a statement of what is said of Behmen
far less liable to suspicion of exaggeration than anything I could write. I
shall add a few extracts from Behmen himself, in support of his right to be
in my list.]

"_Jacob Behmen._--That Prof. De Morgan classes Jacob Behmen among
paradoxers can only be attributed to the fact of his being avowedly
unacquainted with the writings {319} of that author. Perhaps you may think
a few words from one who knows them well of sufficient interest to the
learned Professor, and your readers in general, to be worthy of space in
your columns. The metaphysical system of Behmen--the most perfect and only
true one--still awaits a qualified commentator. Behmen's countryman,
Dionysius Andreas Freher,[596] who spent the greater part of his life in
this country, and whose exposition of Behmen exists only in MS., filling
many volumes, written in English, with the exception of two, written in
German, with numerous beautiful, highly ingenious, and elaborate
illustrations,--copies of some of which are in the British Museum, but all
the originals of which are in the possession of the gentleman who is the
editor of the two works alluded to by Professor De Morgan,--this Freher was
the first to philosophically expound Behmen's system, which was afterwards,
with the help of these MSS., as it were, popularized by William Law; but
both Freher and Law confined themselves chiefly to its theological aspect.
In Behmen, however, is to be found, not only the true ground of all
theology, but also that of all physical science. He demonstrated with a
fullness, accuracy, completeness and certainty that leave nothing to be
desired, the innermost ground of Deity and Nature; and, confining myself to
the latter, I can from my own knowledge assert, that in Behmen's writings
is to be found the true and clear demonstration of every physical fact that
has been discovered since his day. Thus, the science of electricity, which
was not yet in existence when he wrote, is there anticipated; and not only
does Behmen describe all the now known phenomena of that force, but he even
gives us the origin, generation and birth of electricity itself. Again,
positive evidence can be adduced that Newton derived all his knowledge of
gravitation and its {320} laws from Behmen, with whom gravitation or
attraction is, and very properly so, as he shows us, the first of the seven
properties of Nature. The theory defended by Mr. Grove,[597] at the
Nottingham meeting of last year, that all the apparently distinct causes of
moral and physical phenomena are but so many manifestations of one central
force, and that Continuity is the law of nature, is clearly laid down, and
its truth demonstrated, by Behmen, as well as the distinction between
spirit and matter, and that the moral and material world is pervaded by a
sublime unity. And though all this was not admitted in Behmen's days,
because science was not then sufficiently advanced to understand the deep
sense of our author, many of his passages, then unintelligible, or
apparently absurd, read by the light of the present age, are found to
contain the positive enunciation of principles at whose discovery and
establishment science has only just arrived by wearisome and painful
investigations. Every new scientific discovery goes to prove his profound
and intuitive insight into the most secret workings of nature; and if
scientific men, instead of sharing the prejudice arising from ignorance of
Behmen's system, would place themselves on the vantage ground it affords,
they would at once find themselves on an eminence whence they could behold
all the arcana of nature. Behmen's system, in fact, shows us the _inside_
of things, while modern physical science is content with looking at the
_outside_. Behmen traces back every outward manifestation or development to
its one central root,--to that one central energy which, as yet, is only
suspected; every link in the chain of his demonstration is perfect, and
there is not one link wanting. He carries us from the out-births of the
circumference, along the radius to the center, {321} or point, and beyond
that even to the zero, demonstrating the constitution of the zero, or
nothing, with mathematical precision. C. W. H."

And so Behmen is no subject for the Budget! I waited until I should chance
to light on one of his volumes, knowing that any volume would do, and
almost any page. My first hap was on the second volume of the edition of
1664 (4to, published by M. Richardson) and opening near the beginning, a
turn or two brought me to page 13, where I saw about _sulphur_ and
_mercurius_ as follows:

"Thus SUL is the soul, in an herb it is the oil, and in man also, according
to the spirit of _this_ world in the third principle, which is continually
generated out of the anguish of the will in the mind, and the
Brimstone-worm is the Spirit, which hath the fire and _burneth_: PHUR is
the sour wheel in itself which causeth that.

"_Mercurius_ comprehendeth all the four forms, even as the life springeth
up, and yet hath not its dark beginning in the Center as the PHUR hath, but
after the flash of fire, when the sour dark form is terrified, where the
hardness is turned into pliant sharpness, and where the second will (_viz._
the will of nature, which is called the Anguish) ariseth, there Mercurius
hath its original. For MER is the shivering wheel, very horrible, sharp,
venomous, and hostile; which assimulateth it thus in the sourness in the
flash of fire, where the sour wrathful life _ariseth_. The syllable CU is
the pressing out, of the _Anxious_ will of the mind, from Nature: which is
climbing up, and _willeth_ to be out aloft. RI is the comprehension of the
flash of fire, which in MER giveth a clear sound and tune. For the flash
maketh the tune, and it is the Salt-Spirit which _soundeth_, and its form
(or quality) is gritty like sand, and herein arise noises, sounds and
voices, and thus CU comprehendeth the flash, and so the pressure is as a
_wind_ which thrusteth, and giveth a spirit to the flash, so that it liveth
and burneth. Thus the {322} syllable US is called the burning fire, which
with the spirit continually driveth itself forth: and the syllable CU
presseth continually upon the flash."

Shades of Tauler[598] and Paracelsus,[599] how strangely you do mix! Well
may Hallam call Germany the native soil of Mysticism. Had Behmen been the
least of a scholar, he would not have divided _sulph-ur_ and _merc-ur-i-us_
as he has done: and the inflexion _us_, that boy of all work, would have
been rejected. I think it will be held that a writer from whom hundreds of
pages like the above could be brought together, is fit for the Budget. If
Sampson Arnold Mackay[600] had tied his etymologies to a mystical
Christology, instead of a mystical infidelity, he might have had a school
of followers. The nonsense about Newton borrowing gravitation from Behmen
passes only with those who know neither what Newton did, nor what was done
before him.

The above reminds me of a class of paradoxers whom I wonder that I forgot;
they are without exception the greatest bores of all, because they can put
the small end of their paradox into any literary conversation whatever. I
mean the people who have heard the local pronunciation of celebrated names,
and attempt not only to imitate it, but to impose on others their broken
German or Arabic, or what not. They also learn the vernacular names of
those who are generally spoken of in their Latin forms; at least, they
learn a few cases, and hawk them as evidences of erudition. They are
miserably mistaken: scholarship, as a rule, {323} always accepts the
vernacular form of a name which has vernacular celebrity. Hallam writes
Behmen: his index-maker, rather superfluously, gives "_Behmen_ or Boehm."
And he retains Melanchthon,[601] the name given by Reuchlin[602] to his
little kinsman Schwartzerd, because the world has adopted it: but he will
none of Capnio, the name which Reuchlin fitted on to himself, because the
world has not adopted it. He calls the old forms pedantry: but he sees that
the rejection of well-established results of pedantry would be greater
pedantry still. The paradoxers assume the question that it is more
_correct_ to sound a man by lame imitation of his own countrymen than as
usual in the country in which the sound is to be made. Against them are,
first, the world at large; next, an overpowering majority of those who know
something about surnames and their history. Some thirty years ago--a
fact--there appeared at the police-office a complainant who found his own
law. In the course of his argument, he asked, "What does Kitty
say?"--"Who's Kitty?" said the magistrate, "your wife, or your
nurse?"--"Sir! I mean Kitty, the celebrated lawyer."--"Oh!" said the
magistrate, "I suspect you mean Mr. Chitty,[603] the author of the great
work on pleading."--"I do sir! But Chitty is an Italian name, and ought to
be pronounced _Kitty_." This man was a full-blown flower: but there is many
a modest bud; and all ought either to blush when seen or to waste their
pronunciation on the desert air.



I stand up for King Custom, or _Usus_, as Horace called him, with whom is
_arbitrium_ the decision, and _jus_ the right, and _norma_ the way of
deciding, simply because he has _potestas_ the power. He may admit one and
another principle to advise: but Custom is not a constitutional king; he
may listen to his cabinet, but he decides for himself: and if the ministry
should resign, he blesses his stars and does without them. We have a
glorious liberty in England of owning neither dictionary, grammar, nor
spelling-book: as many as choose write by either of the three, and decide
all disputed points their own way, those following them who please.

Throughout this book I have called people by the names which denote them in
their books, or by our vernacular names. This is the intelligible way of
proceeding. I might, for instance (Vol. I, p. 44), have spoken of Charles
de Bovelles,[604] of Lefèvre d'Étaples,[605] of Pèlerin,[606] and of
Etienne.[607] But I prefer the old plan. Those who like another plan
better, are welcome to substitute with a pen, when they know what to write;
when they do not, it is clear that they would not have understood me if I
had given modern names.

The principal advisers of King Custom are as follows. First, there is
Etymology, the _chiffonnier_, or general rag-merchant, who has made such a
fortune of late years in his own business that he begins to be considered
highly respectable. He gives advice which is more thought of than followed,
partly on account of the fearful extremes into which he runs. He lately
asked some boys of sixteen, at a matriculation examination in _English_, to
what branch of {325} the Indo-Germanic family they felt inclined to refer
the Pushto language, and what changes in the force of the letters took
place in passing from Greek into Moeso-Gothic. Because all syllables were
once words, he is a little inclined to insist that they shall be so still.
He would gladly rule English with a Saxon rod, which might be permitted
with a certain discretion which he has never attained: and when opposed, he
defends himself with analogies of the Aryan family until those who hear him
long for the discovery of an Athanasyus. He will transport a word beyond
seas--he is recorder of Rhematopolis--on circumstantial evidence which
looks like mystery gone mad; but, strange to say, something very often
comes to light after sentence is passed which proves the soundness of the

The next adviser is Logic, a swearing old justice of peace, quorum, and
rotulorum, whose excesses brought on such a fit of the gout that for many
years he was unable to move. He is now mending, and his friends say he has
sown his wild oats. He has some influence with the educated subjects of
Custom, and will have more, if he can learn the line at which interference
ought to stop: with them he has succeeded in making an affirmative of two
negatives; but the vulgar won't never have nothing to say to him. He has
always railed at Milton for writing that Eve was the fairest of her
daughters; but has never satisfactorily shown what Milton ought to have
said instead.

The third adviser has more influence with the mass of the subjects of King
Custom than the other two put together; his name is Fiddlefaddle, the
toy-shop keeper; and the other two put him forward to do their worst work.
In return, he often uses their names without authority. He took Etymology
to witness that _means_ to an end must be plural: and he would have any one
method to be a _mean_. But Etymology proved him wrong, King Custom referred
him to his Catechism, in which is "a means whereby we receive the same,"
and Analogy--a subordinate of {326} Etymology--asked whether he thought it
a great _new_ to hear that he was wrong. It was either this Fiddlefaddle,
or Lindley Murray[608] his traveler, who persuaded the Miss Slipslops, of
the Ladies Seminary, to put "The Misses Slipslop" over the gate. Sixty
years ago, this bagman called at all the girls' schools, and got many of
the teachers to insist on the pupils saying "Is it not" and "Can I not" for
"Isn't it" and "Can't I": of which it came that the poor girls were
dreadfully laughed at by their irreverent brothers when they went home for
the holidays. Had this bad adviser not been severely checked, he might by
this time have proposed our saying "The Queen's of England son," declaring,
in the name of Logic, that the prince was the Queen's son, not England's.

Lastly, there is Typography the metallurgist, an executive officer who is
always at work in secret, and whose lawless mode of advising is often done
by carrying his notions into effect without leave given. He it is who never
ceases suggesting that the same word is not to occur in a second place
within sight of the first. When the Authorized Version was first printed,
he began this trick at the passage, "Let there be light, and there was
light;" he drew a line on the proof under the second _light_, and wrote
"_luminosity?_" opposite. He is strongest in the punctuations and other
signs; he has a pepper-box full of commas always by his side. He puts
everything under marks of quotation which he has ever heard before. An
earnest preacher, in a very moving sermon, used the phrase Alas! and alack
a day! Typography stuck up the inverted commas because he had read the old
Anglo-Indian toast, "A lass and a lac a day!" If any one should have the
sense to leave out of his Greek {327} the unmeaning scratches which they
call accents, he goes to a lexicon and puts them in. He is powerful in
routine; but when two routines interlace or overlap, he frequently takes
the wrong one.

Subject to bad advice, and sometimes misled for a season, King Custom goes
on his quiet way and is sure to be right at last.

 "Treason does never prosper: what's the reason?
  Why, when it prospers, none dare call it treason."

Language is in constant fermentation, and all that is thrown in, so far as
it is not fit to assimilate, is thrown off; and this without any obvious
struggle. In the meanwhile every one who has read good authors, from
Shakspeare downward, knows what is and what is not English; and knows,
also, that our language is not one and indivisible. Two very different
turns of phrase may both be equally good, and as good as can be: we may be
relieved of the consequences of contempt of one court by _habeas corpus_
issuing out of another.


Hallam remarks that the Authorized Version of the Bible is not in the
language of the time of James the First: that it is not the English of
Raleigh or of Bacon. Here arises the question whether Raleigh and Bacon are
the true expositors of the language of their time; and whether they were
not rather the incipient promoters of a change which was successfully
resisted by--among other things--the Authorized Version of the Testaments.
I am not prepared to concede that I should have given to the English which
would have been fashioned upon that of Bacon by imitators, such as they
usually are, the admiration which is forced from me by Bacon's English from
Bacon's pen. On this point we have a notable parallel. Samuel Johnson {328}
commands our admiration, at least in his matured style: but we nauseate his
followers. It is an opinion of mine that the works of the leading writers
of an age are seldom the proper specimens of the language of their day,
when that language is in its state of progression. I judge of a language by
the colloquial idiom of educated men: that is, I take this to be the best
medium between the extreme cases of one who is ignorant of grammar and one
who is perched upon a style. Dialogue is what I want to judge by, and plain
dialogue: so I choose Robert Recorde[609] and his pupil in the _Castle of
Knowledge_, written before 1556. When Dr. Robert gets into his altitudes of
instruction, he differs from his own common phraseology as much as probably
did Bacon when he wrote morals and philosophy. But every now and then I
come to a little plain talk about a common thing, of which I propose to
show a specimen. Anything can be made to look old by such changes as
_makes_ into _maketh_, with a little old spelling. I shall invert these
changes, using the newer form of inflexion, and the modern spelling: with
no other variation whatever.

"_Scholar._ Yet the reason of that is easy enough to be conceived, for when
the day is at the longest the Sun must needs shine the more time, and so
must it needs shine the less time when the day is at the shortest: this
reason I have heard many men declare.

_Master._ That may be called a crabbed reason, for it {329} goes backward
like a crab. The day makes not the Sun to shine, but the Sun shining makes
the day. And so the length of the day makes not the Sun to shine long,
neither the shortness of the day causes not [_sic_] the Sun to shine the
lesser time, but contrariwise the long shining of the Sun makes the long
day, and the short shining of the Sun makes the lesser day: else answer me
what makes the days long or short?

_Scholar._ I have heard wise men say that Summer makes the long days, and
Winter makes the long nights.

_Master._ They might have said more wisely, that long days make summer and
short days make winter.

_Scholar._ Why, all that seems one thing to me.

_Master._ Is it all one to say, God made the earth, and the earth made God?
Covetousness overcomes all men, and all men overcome covetousness?

_Scholar._ No, not so; for here the effect is turned to be the cause, and
the agent is made the patient.

_Master._ So is it to say Summer makes long days, when you should say: Long
days make summer.

_Scholar._ I perceive it now: but I was so blinded with the vulgar error,
that if you had demanded of me further what did make the summer, I had been
like to have answered that green leaves do make summer; and the sooner by
remembrance of an old saying that a year should come in which the summer
should not be known but by the green leaves.

_Master._ Yet this saying does not import that green leaves do make summer,
but that they betoken summer; so are they the sign and not the cause of

I have taken a whole page of our author, without omission, that the reader
may see that I do not pick out sentences convenient for my purpose. I have
done nothing but alter the third person of the verb and the spelling: but
great is the effect thereof. We say "the Sun shining makes the day";
Recorde, "the Sonne shynynge maketh the daye." {330} These points apart, we
see a resemblance between our English and that of three hundred years ago,
in the common talk of educated persons, which will allow us to affirm that
the language of the authorized Bible must have been very close to that of
its time. For I cannot admit that much change can have taken place in fifty
years: and the language of the version represents both our common English
and that of Recorde with very close approximation. Take sentences from
Bacon and Raleigh, and it will be apparent that these writers will be held
to differ from all three, Recorde, the version, and ourselves, by
differences of the same character. But we speak of Recorde's conversation,
and of our own. We conclude that it is the plain and almost colloquial
character of the Authorized Version which distinguishes it from the English
of Bacon and Raleigh, by approximating it to the common idiom of the time.
If any one will cast an eye upon the letters of instruction written by
Cecil[610] and the Bishop of London to the translators themselves, or to
the general directions sent to them in the King's name, he will find that
these plain business compositions differ from the English of Bacon and
Raleigh by the same sort of differences which distinguish the version


The foreign word, or the word of a district, or class of people, passes
into the general vernacular; but it is long before the specially learned
will acknowledge the right of those with whom they come in contact to
follow general usage. The rule is simple: so long as a word is technical or
local, those who know its technical or local pronunciation may reasonably
employ it. But when the word has become general, the specialist is not very
wise if he refuse to follow {331} the mass, and perfectly foolish if he
insist on others following him. There have been a few who demanded that
Euler should be pronounced in the German fashion:[611] Euler has long been
the property of the world at large; what does it matter how his own
countrymen pronounce the letters? Shall we insist on the French pronouncing
_Newton_ without that final _tong_ which they never fail to give him? They
would be wise enough to laugh at us if we did. We remember that a pedant
who was insisting on all the pronunciations being retained, was met by a
maxim in contradiction, invented at the moment, and fathered upon
Kaen-foo-tzee,[612] an authority which he was challenged to dispute. Whom
did you speak of? said the bewildered man of accuracy. Learn your own
system, was the answer, before you impose it on others; Confucius says that

The old English has _fote_, _fode_, _loke_, _coke_, _roke_, etc., for
_foot_, etc. And _above_ rhymes in Chaucer to _remove_. Suspecting that the
broader sounds are the older, we may surmise that _remove_ and _food_ have
retained their old sounds, and that _cook_, once _coke_, would have rhymed
to our _Luke_, the vowel being brought a little nearer, perhaps, to the _o_
in our present _coke_, the fuel, probably so called as used by cooks. If
this be so, the Chief Justice _Cook_[614] of our lawyers, and the _Coke_
(pronounced like the fuel) of the greater part of the world, are equally
wrong. The lawyer has no right whatever to fasten his pronunciation upon
us: even leaving aside the general custom, he cannot prove himself right,
and is probably wrong. Those who {332} know the village of Rokeby
(pronounced Rookby) despise the world for not knowing how to name Walter
Scott's poem: that same world never asked a question about the matter, and
the reception of the parody of _Jokeby_, which soon appeared, was a
sufficient indication of their notion. Those who would fasten the hodiernal
sound upon us may be reminded that the question is, not what they call it
now, but what it was called in Cromwell's time. Throw away general usage as
a lawgiver, and this is the point which emerges. Probably _R[=u]ke-by_
would be right, with a little turning of the Italian [=u] towards [=o] of
modern English.

[Some of the above is from an old review. I do not always notice such
insertions: I take nothing but my own writings. A friend once said to me,
"Ah! you got that out of the _Athenæum_!" "Excuse me," said I. "the
_Athenæum_ got that out of me!"]


It is part of my function to do justice to any cyclometers whose methods
have been wrongly described by any orthodox sneerers (myself included). In
this character I must notice _Dethlevus Cluverius_,[615] as the Leipzig
Acts call him (probably Dethleu Cluvier), grandson of the celebrated
geographer, Philip Cluvier. The grandson was a Fellow of the Royal Society,
elected on the same day as Halley,[616] November 30, 1678: I suppose he
lived in England. This {333} man is quizzed in the Leipzig Acts for 1686;
and, if Montucla insinuate rightly, by Leibnitz, who is further suspected
of wanting to embroil Cluvier with his own opponent Nieuwentiit,[617] on
the matter of infinitesimals. So far good: I have nothing against Leibnitz,
who though he was ironical, told us what he laughed at. But Montucla has
behaved very unfairly: he represents Cluvier as placing the essence of his
method in the solution of the problem _construere mundum divinæ menti
analogum_, to construct a world corresponding to the divine mind. Nothing
to begin with: no way of proceeding. Now, it ought to have been _ex data
linea construere_,[618] etc.: there is a given line, which is something to
go on. Further, there is a way of proceeding: it is to find the product of
1, 2, 3, 4, etc. for ever. Moreover, Montucla charges Cluvier with
_unsquaring_ the parabola, which Archimedes had squared as tight as a
glove. But he never mentions how very nearly Cluvier agrees with the Greek:
they only differ by 1 divided by 3n^2, where n is the infinite number of
parts of which a parabola is composed. This must have been the conceit that
tickled Leibnitz, and made him wish that Cluvier and Nieuwentiit should
fight it out. Cluvier, was admitted, on terms of irony, into the Leipzig
Acts: he appeared on a more serious footing in London. It is very rare for
one cyclometer to refute another: _les corsaires ne se battent pas_.[619]
The only instance I recall is that of M. Cluvier, who (_Phil. Trans._,
1686, No. 185) refuted M. Mallemont de Messange,[620] who {334} published
at Paris in 1686. He does it in a very serious style, and shows himself a
mathematician. And yet in the year in which, in the _Phil. Trans._, he was
a geometer, and one who rebukes his squarer for quoting Matthew xi. 25, in
that very year he was the visionary who, in the Leipzig Acts, professed to
build a world resembling the divine mind by multiplying together 1, 2, 3,
4, etc. up to infinity.


There is a very pretty opening for a paradox which has never found its
paradoxer in print. The philosophers teach that the rainbow is not
material: it comes from rain-drops, but those rain-drops do not _take_
color. They only _give_ it, as lenses and mirrors; and each one drop gives
_all_ the colors, but throws them in different directions. Accordingly, the
same drop which furnishes red light to one spectator will furnish violet to
another, properly placed. Enter the paradoxer whom I have to invent. The
philosopher has gulled you nicely. Look into the water, and you will see
the reflected rainbow: take a looking-glass held sideways, and you see
another reflection. How could this be, if there were nothing colored to
reflect? The paradoxer's facts are true: and what are called the reflected
rainbows are _other_ rainbows, caused by those _other_ drops which are
placed so as to give the colors to the eye after reflection, at the water
or the looking-glass. A few years ago an artist exhibited a picture with a
rainbow and its apparent reflection: he simply copied what he had seen.
When his picture was examined, some started the idea that there could be no
reflection of a rainbow; they were right: they inferred that the artist had
made a mistake; they were wrong. When it was explained, some agreed and
some dissented. Wanted, {335} immediately, an able paradoxer: testimonials
to be forwarded to either end of the rainbow, No. 1. No circle-squarer need
apply, His Variegatedness having been pleased to adopt 3.14159... from Noah


The system of Tycho Brahé,[621] with some alteration and addition, has been
revived and contended for in our own day by a Dane, W. Zytphen,[622] who
has published _The Motion of the Sun in the Universe_, (second edition)
Copenhagen, 1865, 8vo, and _Le Mouvement Sidéral_, 1865, 8vo. I make an

"How can one explain Copernically that the velocity of the Moon must be
added to the velocity of the Earth on the one place in the Earth's orbit,
to learn how far the Moon has advanced from one fixed star to another; but
in another place in the orbit these velocities must be subtracted (the
movements taking place in opposite directions) to attain the same result?
In the Copernican and other systems, it is well known that the Moon,
abstracting from the insignificant excentricity of the orbit, always in
twenty-four hours performs an equally long distance. Why has Copernicus
never been denominated Fundamentus or Fundator? Because he has never
convinced anybody so thoroughly that this otherwise so natural epithet has
occurred to the mind."

Really the second question is more effective against Newton than against
Copernicus; for it upsets gravity: the first is of great depth.



The _Correspondent_ journal makes a little episode in the history of my
Budget (born May, 1865, died April, 1866). It consisted entirely of letters
written by correspondents. In August, a correspondent who signed "Fair
Play"--and who I was afterwards told was a lady--thought it would be a good
joke to bring in the Cyclometers. Accordingly a letter was written,
complaining that though Mr. Sylvester's[623] demonstration of Newton's
theorem--then attracting public attention--was duly lauded, the possibly
greater discovery of the quadrature seemed to be blushing unseen, and
wasting etc. It went on as follows:

"Prof. De Morgan, who, from his position in the scientific world, might
fairly afford to look favourably on less practised efforts than his own,
seems to delight in ridiculing the discoverer. Science is, of course, a
very respectable person when he comes out and makes himself useful in the
world [it must have been a lady; each sex gives science to the other]: but
when, like a monk of the Middle Ages, he shuts himself up [it must have
been a lady; they always snub the bachelors] in his cloistered cell,
repeating his mumpsimus from day to day, and despising the labourers on the
outside, we begin to think of Galileo,[624] Jenner,[625] Harvey,[626] and
other glorious trios, who have been contemned ..."

The writer then called upon Mr. James Smith[627] to come {337} forward. The
irony was not seen; and that day fortnight appeared the first of more than
thirty letters from his pen. Mr. Smith was followed by Mr. Reddie,[628]
Zadkiel,[629] and others, on their several subjects. To some of the letters
I have referred; to others I shall come. The _Correspondent_ was to become
a first-class scientific journal; the time had arrived at which truth had
an organ: and I received formal notice that I could not stifle it by
silence, nor convert it into falsehood by ridicule. When my reader sees my
extracts, he will readily believe my declaration that I should have been
the last to stifle a publication which was every week what James Mill[630]
would call a dose of capital for my Budget. A few anti-paradoxers brought
in common sense: but to the mass of the readers of the journal it all
seemed to be the difference between Tweedledum and Tweedledee. Some said
that the influx of scientific paradoxes killed the journal: but my belief
is that they made it last longer than it otherwise would have done. Twenty
years ago I recommended the paradoxers to combine and publish their views
in a common journal: with a catholic editor, who had no pet theory, but a
stern determination not to exclude anything merely for absurdity. I suspect
it would answer very well. A strong title, or motto, would be wanted: not
so coarse as was roared out in a Cambridge mob when I was an
undergraduate--"No King! No Church! No House of Lords! No nothing, blast
me!"--but something on that _principle_.

At the end of 1867 I addressed the following letter to the _Athenæum_:


_December 31, 1867_

Many thanks for the present of Mr. James Smith's letters {338} of Sept. 28
and of Oct. 10 and 12. He asks where you will be if you read and digest his
letters: you probably will be somewhere first. He afterwards asks what the
WE of the _Athenæum_ will be if, finding it impossible to controvert, it
should refuse to print. I answer for you, that We-We of the _Athenæum_, not
being Wa-Wa the wild goose, so conspicuous in "Hiawatha," will leave what
controverts itself to print itself, if it please.

_Philomath_ is a good old word, easier to write and speak than
_mathematician_. It wants the words between which I have placed it. They
are not well formed: _pseudomathete_ and _graphomathete_ would be better:
but they will do. I give an instance of each.

The _pseudomath_ is a person who handles mathematics as the monkey handled
the razor. The creature tried to shave himself as he had seen his master
do; but, not having any notion of the angle at which the razor was to be
held, he cut his own throat. He never tried a second time, poor animal! but
the pseudomath keeps on at his work, proclaims himself clean-shaved, and
all the rest of the world hairy. So great is the difference between moral
and physical phenomena! Mr. James Smith is, beyond doubt, the great
pseudomath of our time. His 3-1/8 is the least of a wonderful chain of
discoveries. His books, like Whitbread's barrels, will one day reach from
Simpkin & Marshall's to Kew, placed upright, or to Windsor laid
length-ways. The Queen will run away on their near approach, as Bishop
Hatto did from the rats: but Mr. James Smith will follow her were it to
John o' Groats.

The _philomath_, for my present purpose, must be exhibited as giving a
lesson to presumption. The following anecdote is found in Thiébault's[631]
_Souvenirs de vingt ans de séjours à Berlin_, published in 1804. The book
itself got a high character for truth. In 1807 Marshal Mollendorff[632]
{339} answered an inquiry of the Duc de Bassano,[633] by saying that it was
the most veracious of books, written by the most honest of men. Thiébault
does not claim personal knowledge of the anecdote, but he vouches for its
being received as true all over the north of Europe.[634]

Diderot[635] paid a visit to Russia at the invitation of Catherine the
Second. At that time he was an atheist, or at least talked atheism: it
would be easy to prove him either one thing or the other from his writings.
His lively sallies on this subject much amused the Empress, and all the
younger part of her Court. But some of the older courtiers suggested that
it was hardly prudent to allow such unreserved exhibitions. The Empress
thought so too, but did not like to muzzle her guest by an express
prohibition: so a plot was contrived. The scorner was informed that an
eminent mathematician had an algebraical proof of the existence of God,
which he would communicate before the whole Court, if agreeable. Diderot
gladly consented. The mathematician, who is not named, was Euler.[636] He
came to Diderot with the gravest air, and in a tone of perfect conviction
said, "_Monsieur!_

  (a + b^n)/n = x

_donc Dieu existe; répondez!_"[637] Diderot, to whom algebra was Hebrew,
though this is expressed in a very roundabout way by Thiébault--and whom we
may suppose to have expected some verbal argument of alleged algebraical
closeness, was disconcerted; while peals of laughter sounded on all sides.
Next day he asked permission to return to France, which was granted. An
algebraist would have {340} turned the tables completely, by saying,
"Monsieur! vous savez bien que votre raisonnement demande le développement
de x suivant les puissances entières de n".[638] Goldsmith could not have
seen the anecdote, or he might have been supposed to have drawn from it a
hint as to the way in which the Squire demolished poor Moses.

The _graphomath_ is a person who, having no mathematics, attempts to
describe a mathematician. Novelists perform in this way: even Walter Scott
now and then burns his fingers. His dreaming calculator, Davy Ramsay,
swears "by the bones of the immortal Napier." Scott thought that the the
philomaths worshiped relics: so they do, in one sense. Look into
Hutton's[639] Dictionary for _Napier's Bones_, and you shall learn all
about the little knick-knacks by which he did multiplication and division.
But never a bone of his own did he contribute; he preferred elephants'
tusks. The author of _Headlong Hall_[640] makes a grand error, which is
quite high science: he says that Laplace proved the precession of the
equinoxes to be a periodical inequality. He should have said the variation
of the obliquity. But the finest instance is the following: Mr.
Warren,[641] in his well-wrought tale of the martyr-philosopher, was
incautious enough to invent the symbols by which his _savant_ satisfied
himself Laplace[642] was right on a doubtful point. And this is what he put

  [sqrt]-3a^2, [rectangle]y^2 / z^2 + 9 - n = 9, n × log e.

Now, to Diderot and the mass of mankind this might be Laplace all over:
and, in a forged note of Pascal, would {341} prove him quite up to
gravitation. But I know of nothing like it, except in the lately received
story of the American orator, who was called on for some Latin, and
perorated thus: "Committing the destiny of the country to your hands,
Gentlemen, I may without fear declare, in the language of the noble Roman

  E pluribus unum,
  Multum in parvo,
  Ultima Thule,
  Sine qua non."[643]

But the American got nearer to Horace than the martyr-philosopher to
Laplace. For all the words are in Horace, except _Thule_, which might have
been there. But [rectangle] is not a symbol wanted by Laplace; nor can we
see how it could have been; in fact, it is not recognized in algebra. As to
the junctions, etc., Laplace and Horace are about equally well imitated.

Further thanks for Mr. Smith's letters to you of Oct. 15, 18, 19, 28, and
Nov. 4, 15. The last of these letters has two curious discoveries. First,
Mr. Smith declares that he has _seen_ the editor of the _Athenæum_: in
several previous letters he mentions a name. If he knew a little of
journalism he would be aware that editors are a peculiar race, obtained by
natural selection. They are never seen, even by their officials; only heard
down a pipe. Secondly, "an ellipse or oval" is composed of four arcs of
circles. Mr. Smith has got hold of the construction I was taught, when a
boy, for a pretty four-arc oval. But my teachers knew better than to call
it an ellipse: Mr. Smith does not; but he produces from it such
confirmation of 3-1/8 as would convince any _honest_ editor.

Surely the cyclometer is a Darwinite development of a spider, who is always
at circles, and always begins again when his web is brushed away. He
informs you that he {342} has been privileged to discover truths unknown to
the scientific world. This we know; but he proceeds to show that he is
equally fortunate in art. He goes on to say that he will make use of you to
bring those truths to light, "just as an artist makes use of a dummy for
the purpose of arranging his drapery." The painter's lay-figure is for
flowing robes; the hairdresser's dummy is for curly locks. Mr. James Smith
should read Sam Weller's pathetic story of the "four wax dummies." As to
_his_ use of a dummy, it is quite correct. When I was at University
College, I walked one day into a room in which my Latin colleague was
examining. One of the questions was, "Give the lives and fates of Sp.
Mælius,[644] and Sp. Cassius."[645] Umph! said I, surely all know that
Spurius Mælius was whipped for adulterating flour, and that Spurius Cassius
was hanged for passing bad money. Now, a robe arranged on a dummy would
look just like the toga of Cassius on the gallows. Accordingly, Mr. Smith
is right in the drapery-hanger which he has chosen: he has been detected in
the attempt to pass bad circles. He complains bitterly that his geometry,
instead of being read and understood by you, is handed over to me to be
treated after my scurrilous fashion. It is clear enough that he would
rather be handled in this way than not handled at all, or why does he go on
writing? He must know by this time that it is a part of the institution
that his "untruthful and absurd trash" shall be distilled into mine at the
rate of about 3-1/8 pages of the first to one column of the second. Your
readers will never know how much they gain by the process, until Mr. James
Smith publishes it all in a big book, or until they get hold of what he has
already published. I have six pounds avoirdupois of pamphlets and letters;
and there is more than half a pound of letters {343} written to you in the
last two months. Your compositor must feel aggrieved by the rejection of
these clearly written documents, without erasures, and on one side only.
Your correspondent has all the makings of a good contributor, except the
knowledge of his subject and the sense to get it. He is, in fact, only a
mask: of whom the fox

 "O quanta species, inquit, cerebrum non habet."[646]

I do not despair of Mr. Smith on any question which does not involve that
unfortunate two-stick wicket at which he persists in bowling. He has
published many papers; he has forwarded them to mathematicians: and he
cannot get answers; perhaps not even readers. Does he think that he would
get more notice if you were to print him in your journal? Who would study
his columns? Not the mathematician, we know; and he knows. Would others?
His balls are aimed too wide to be blocked by any one who is near the
wicket. He has long ceased to be worth the answer which a new invader may
get. Rowan Hamilton,[647] years ago, completely knocked him over; and he
has never attempted to point out any error in the short and easy method by
which that powerful investigator condescended to show that, be right who
may, he must be wrong. There are some persons who feel inclined to think
that Mr. Smith should be argued with: let those persons understand that he
has been argued with, refuted, and has never attempted to stick a pen into
the refutation. He stated that it was a remarkable paradox, easily
explicable; and that is all. After this evasion, Mr. James Smith is below
the necessity of being told that he is unworthy of answer. His friends
complain that I do nothing but _chaff_ him. Absurd! I winnow him; and if
nothing but chaff results, whose fault is that? I am usefully employed: for
he is the type of a class which ought to be known, and which I have done
much to make known.


Nothing came of this until July 1869, when I received a reprint of the
above letter, with a comment, described as Appendix D of a work in course
of publication on the geometry of the circle. The _Athenæum_ journal
received the same: but the Editor, in his private capacity, received the
whole work, being _The Geometry of the Circle and Mathematics as applied to
Geometry by Mathematicians, shown to be a mockery, delusion, and a snare_,
Liverpool, 8vo, 1869. Mr. J. S. here appears in deep fight with Professor
Whitworth,[648] and Mr. Wilson,[649] the author of the alleged amendment of
Euclid. How these accomplished mathematicians could be inveigled into
continued discussion is inexplicable. Mr. Whitworth began by complaining of
Mr. Smith's attacks upon mathematicians, continued to correspond after he
was convinced that J. S. proved an arc and its chord to be equal, and only
retreated when J. S. charged him with believing in 3-1/8, and refusing
acknowledgment. Mr. Wilson was introduced to J. S. by a volunteer defense
of his geometry from the assaults of the _Athenæum_. This the editor would
not publish; so J. S. sent a copy to Mr. Wilson himself. Some
correspondence ensued, but Mr. Wilson soon found out his man, and withdrew.

There is a little derision of the _Athenæum_ and a merited punishment for
"that unscrupulous critic and contemptible mathematical twaddler, De


At p. 183 I mentioned Mr. Reddie,[650] the author of _Vis Inertiæ Victa_
and of _Victoria toto coelo_,[651] which last is not {345} an address to
the whole heaven, either from a Roman Goddess or a British Queen, whatever
a scholar may suppose. Between these Mr. Reddie has published _The
Mechanics of the Heavens_, 8vo, 1862: this I never saw until he sent it to
me, with an invitation to notice it, he very well knowing that it would
catch. His speculations do battle with common notions of mathematics and of
mechanics, which, to use a feminine idiom, he blasphemes so you can't
think! and I suspect that if you do not blaspheme them too, _you_ can't
think. He appeals to the "truly scientific," and would be glad to have
readers who have read what he controverts, i.e., Newton's _Principia_: I
wish he may get them; I mean I hope he may obtain them. To none but these
would an account of his speculations be intelligible: I accordingly
disposed of him in a very short paragraph of description. Now many
paradoxers desire notice, even though it be disparaging. I have letters
from more than one--besides what have been sent to the Editor of the
_Athenæum_--complaining that they are not laughed at; although they deserve
it, they tell me, as much as some whom I have inserted. Mr. Reddie informs
me that I have not said a single word against his books, though I have
given nearly a column to sixteen-string arithmetic, and as much to
animalcule universes. What need to say anything to readers of Newton
against a book from which I quoted that revolution by gravitation is
_demonstrably_ impossible? It would be as useless as evidence against a man
who has pleaded guilty. Mr. Reddie derisively thanks me for "small
mercies"; he wrote me private letters; he published them, and more, in the
_Correspondent_. He gave me, _pro viribus suis_,[652] such a dressing you
can't think, both for my Budget non-notice, and for reviews which he
assumed me to have written. He outlawed himself by declaring
(_Correspondent_, Nov. 11, 1856) that I--in a review--had made a quotation
which was "garbled, evidently on purpose {346} to make it appear that" he
"was advocating solely a geocentric hypothesis, which is not true." In
fact, he did his best to get larger "mercy." And he shall have it; and at a
length which shall content him, unless his mecometer be an insatiable
apparatus. But I fear that in other respects I shall no more satisfy him
than the Irish drummer satisfied the poor culprit when, after several times
changing the direction of the stroke at earnest entreaty, he was at last
provoked to call out, "Bad cess to ye, ye spalpeen! strike where one will,
there's no _plasing_ ye!"

Mr. Reddie attaches much force to Berkeley's[653] old arguments against the
doctrine of fluxions, and advances objections to Newton's second section,
which he takes to be new. To me they appear "such as have been often made,"
to copy a description given in a review: though I have no doubt Mr. Reddie
got them out of himself. But the whole matter comes to this: Mr. Reddie
challenged answer, especially from the British Association, and got none.
He presumes that this is because he is right, and cannot be answered: the
Association is willing to risk itself upon the counter-notion that he is
wrong, and need not be answered; because so wrong that none who could
understand an answer would be likely to want one.

Mr. Reddie demands my attention to a point which had already particularly
struck me, as giving the means of showing to _all_ readers the kind of
confusion into which paradoxers are apt to fall, in spite of the clearest
instruction. It is a very honest blunder, and requires notice: it may
otherwise mislead some, who may suppose that no one able to read could be
mistaken about so simple a matter, {347} let him be ever so wrong about
Newton. According to his own mis-statement, in less than five months he
made the Astronomer Royal abandon the theory of the solar motion in space.
The announcement is made in August, 1865, as follows: the italics are not

    "The third (_Victoria ..._), although only published in September,
    1863, has already had its triumph. _It is the book that forced the
    Astronomer Royal of England, after publicly teaching the contrary for
    years, to come to the conclusion, "strange as it may appear," that "the
    whole question of solar motion in space is at the present time in doubt
    and abeyance."_ This admission is made in the Annual Report of the
    Council of the Royal Astronomical Society, published in the Society's
    _Monthly Notices_ for February, 1864."

It is added that solar motion is "full of self-contradiction, which "the
astronomers" simply overlooked, but which they dare not now deny after
being once pointed out."

The following is another of his accounts of the matter, given in the
_Correspondent_, No. 18, 1865:

    "... You ought, when you came to put me in the 'Budget,' to have been
    aware of the Report of the Council of the Royal Astronomical Society,
    where it appears that Professor Airy,[654] with a better appreciation
    of my demonstrations, had admitted--'strange,' say the Council, 'as it
    may appear,'--that 'the whole question of solar motion in space [and
    here Mr. Reddie omits some words] is now in _doubt and abeyance_.' You
    were culpable as a public teacher of no little pretensions, if you were
    'unaware' of this. If aware of it, you ought not to have suppressed
    such an important testimony to my really having been 'very successful'
    in drawing the teeth of the pegtops, though you thought them so firmly
    fixed. And if you still suppress {348} it, in your Appendix, or when
    you reprint your 'Budget,' you will then be guilty of a _suppressio
    veri_, also of further injury to me, who have never injured you...."

Mr. Reddie must have been very well satisfied in his own mind before he
ventured such a challenge, with an answer from me looming in the distance.
The following is the passage of the Report of the Council, etc., from which
he quotes:

    "And yet, strange to say, notwithstanding the near coincidence of all
    the results of the before-mentioned independent methods of
    investigation, the inevitable logical inference deduced by Mr. Airy is,
    that the whole question of solar motion in space, _so far at least as
    accounting for the proper motion of the stars is concerned_, [I have
    put in italics the words omitted by Mr. Reddie] appears to remain at
    this moment in doubt and abeyance."

Mr. Reddie has forked me, as he thinks, on a dilemma: if unaware, culpable
ignorance; if aware, suppressive intention. But the thing is a _trilemma_,
and the third horn, on which I elect to be placed, is surmounted by a
doubly-stuffed seat. First, Mr. Airy has not changed his opinion about the
_fact_ of solar motion in space, but only suspends it as to the sufficiency
of present means to give the amount and direction of the motion. Secondly,
all that is alluded to in the Astronomical Report was said and printed
before the Victoria proclamation appeared. So that the author, instead of
drawing the tooth of the Astronomer Royal's pegtop, has burnt his own
doll's nose.

William Herschel,[655] and after him about six other astronomers, had aimed
at determining, by the proper motions of the stars, the point of the
heavens towards which the solar system is moving: their results were
tolerably accordant. Mr. Airy, in 1859, proposed an improved method, and,
applying it to stars of large proper motion, produced {349} much the same
result as Herschel. Mr. E. Dunkin,[656] one of Mr. Airy's staff at
Greenwich, applied Mr. Airy's method to a very large number of stars, and
produced, again, nearly the same result as before. This paper was read to
the Astronomical Society in _March_, 1863, was printed in abstract in the
_Notice_ of that month, was printed in full in the volume then current, and
was referred to in the Annual Report of the Council in _February_, 1864,
under the name of "the Astronomer Royal's elaborate investigation, as
exhibited by Mr. Dunkin." Both Mr. Airy and Mr. Dunkin express grave doubts
as to the sufficiency of the data: and, regarding the coincidence of all
the results as highly curious, feel it necessary to wait for calculations
made on better data. The report of the Council states these doubts. Mr.
Reddie, who only published in _September_, 1863, happened to see the Report
of February, 1864, assumes that the doubts were then first expressed, and
declares that his book of September had the triumph of forcing the
Astronomer Royal to abandon the _fact_ of motion of the solar system by the
February following. Had Mr. Reddie, when he saw that the Council were
avowedly describing a memoir presented some time before, taken the
precaution to find out _when_ that memoir was presented, he would perhaps
have seen that doubts of the results obtained, expressed by one astronomer
in March, 1863, and by another in 1859, could not have been due to his
publication of September, 1863. And any one else would have learnt that
neither astronomer doubts the _solar motion_, though both doubt the
sufficiency of present means to determine its _amount_ and _direction_.
This is implied in the omitted words, which Mr. Reddie--whose omission
would have been dishonest if he had seen their meaning--no doubt took for
pleonasm, superfluity, overmuchness. The rashness which pushed him headlong
{350} into the quillet that _his_ thunderbolt had stopped the chariot of
the Sun and knocked the Greenwich Phaeton off the box, is the same which
betrayed him into yet grander error--which deserves the full word,
_quidlibet_--about the _Principia_ of Newton. There has been no change of
opinion at all. When a person undertakes a long investigation, his opinion
is that, at a certain date, there is _prima facie_ ground for thinking a
sound result may be obtained. Should it happen that the investigation ends
in doubt upon the sufficiency of the grounds, the investigator is not put
in the wrong. He knew beforehand that there was an alternative: and he
takes the horn of the alternative indicated by his calculations. The two
sides of this case present an instructive contrast. Eight astronomers
produce nearly the same result, and yet the last two doubt the sufficiency
of their means: compare them with the what's-his-name who rushes in where
thing-em-bobs fear to tread.

I was not aware, until I had written what precedes, that Mr. Airy had given
a sufficient answer on the point. Mr. Reddie says (_Correspondent_, Jan.
20, 1866):

    "I claim to have forced Professor Airy to give up the notion of 'solar
    motion in space' altogether, for he admits it to be 'at present in
    doubt and abeyance.' I first made that claim in a letter addressed to
    the Astronomer Royal himself in June, 1864, and in replying, very
    courteously, to other portions of my letter, he did not gainsay that
    part of it."

Mr. Reddie is not ready at reading satire, or he never would have so missed
the meaning of the courteous reply on one point, and the total silence upon
another. Mr. Airy must be one of those peculiar persons who, when they do
not think an assertion worth notice, let it alone, without noticing it by a
notification of non-notice. He would never commit the bull of "Sir! I will
not say a word on that subject." He would put it thus, "Sir! I will only
say ten words on that subject,"--and, having thus said them, would {351}
proceed to something else. He assumed, as a matter of form, that Mr. Reddie
would draw the proper inference from his silence: and this because he did
not care whether or no the assumption was correct.

The _Mechanics of the Heavens_, which Mr. Reddie sends to be noticed, shall
be noticed, so far as an extract goes:

"My connection with this subject is, indeed, very simply explained. In
endeavoring to understand the laws of physical astronomy as generally
taught, I happened to entertain some doubt whether gravitating bodies could
revolve, and having afterwards imbibed some vague idea that the laws of the
universe were chemical and physical rather than mechanical, and somehow
connected with electricity and magnetism as opposing correlative
forces--most probably suggested to my mind, as to many others, by the
transcendent discoveries made in electro-magnetism by Professor
Faraday[657]--my former doubts about gravitation were revived, and I was
led very naturally to try and discover whether a gravitating body really
could revolve; and I became convinced it could not, before I had ever
presumed to look into the demonstrations of the _Principia_."

This is enough against the book, without a word from me: I insert it only
to show those who know the subject what manner of writer Mr. Reddie is. It
is clear that "presumed" is a slip of the pen; it should have been

Mr. Reddie represents me as dreaming over paltry paradoxes. He is right;
many of my paradoxes are paltry: he is wrong; I am wide awake to them. A
single moth, beetle, or butterfly, may be a paltry thing; but when a
cabinet is arranged by genus and species, we then begin to admire the {352}
infinite variety of a system constructed on a wonderful sameness of leading
characteristics. And why should paradoxes be denied that collective
importance, paltry as many of them may individually be, which is accorded
to moths, beetles, or butterflies? Mr. Reddie himself sees that "there is a
method in" my "mode of dealing with paradoxes." I hope I have atoned for
the scantiness of my former article, and put the demonstrated impossibility
of gravitation on that level with Hubongramillposanfy arithmetic and
inhabited atoms which the demonstrator--not quite without reason--claims
for it.

In the Introduction to a collected edition of the three works, Mr. Reddie
describes his _Mechanism of the Heavens_, from which I have just quoted,

"a public challenge offered to the British Association and the
mathematicians at Cambridge, in August, 1862, calling upon them to point to
a single demonstration in the _Principia_ or elsewhere, which even attempts
to prove that Universal Gravitation is possible, or to show that a
gravitating body could possibly revolve about a center of attraction. The
challenge was not accepted, and never will be. No such demonstration
exists. And the public must judge for themselves as to the character of a
so-called "certain science," which thus shrinks from rigid examination, and
dares not defend itself when publicly attacked: also of the character of
its teachers, who can be content to remain dumb under such circumstances."


The above is the commonplace talk of the class, of which I proceed to speak
without more application to this paradoxer than to that. It reminds one of
the funny young rascals who used, in times not yet quite forgotten, to
abuse the passengers, as long as they could keep up with the {353} stage
coach; dropping off at last with "Why don't you get down and thrash us?
You're afraid, you're afraid!" They will allow the public to judge for
themselves, but with somewhat of the feeling of the worthy uncle in _Tom
Jones_, who, though he would let young people choose for themselves, would
_have them_ choose wisely. They try to be so awfully moral and so ghastly
satirical that they must be answered: and they are best answered in their
own division. We have all heard of the way in which sailors cat's-pawed the
monkeys: they taunted the dwellers in the trees with stones, and the
monkeys taunted them with cocoa-nuts in return. But these were silly
dendrobats: had they belonged to the British Association they would have
said--No! No! dear friends; it is not in the itinerary: if you want nuts,
you must climb, as we do. The public has referred the question to Time: the
procedure of this great king I venture to describe, from precedents, by an
adaptation of some smart anapæstic tetrameters--your anapæst is the foot
for satire to halt on, both in Greek and English--which I read about twenty
years ago, and with the point of which I was much tickled. Poetasters were
laughed at; but Mr. Slum, whom I employed--Mr. Charles Dickens obliged me
with his address--converted the idea into that of a hit at
mathematicasters, as easily as he turned the Warren acrostic into Jarley.
As he observed, when I settled his little account, it is cheaper than any
prose, though the broom was not stolen quite ready made:

  _Forty stripes save one for the smaller Paradoxers._

  Hark to the wisdom the sages preach
  Who never have learnt what they try to teach.
  We are the lights of the age, they say!
  We are the men, and the thinkers we!
  So we build up guess-work the livelong day,
  In a topsy-turvy sort of way,
  Some with and some wanting _a_ plus b.
  Let the British Association fuss;
  What are theirs to the feats to be wrought by us?
  Shall the earth stand still? Will the round come square?
  Must Isaac's book be the nest of a mare?
  Ought the moon to be taught by the laws of space
  To turn half round without right-about-face?
  Our whimsey crotchets will manage it all;
  Deep! Deep! posterity will them call!
  Though the world, for the present, lets them fall
  Down! Down! to the twopenny box of the stall!

  Thus they--But the marplot Time stands by,
  With a knowing wink in his funny old eye.
  He grasps by the top an immense fool's cap,
  Which he calls a philosophaster-trap:
  And rightly enough, for while these little men
  Croak loud as a concert of frogs in a fen,
  He first singles out one, and then another,
  Down goes the cap--lo! a moment's pother,
  A spirit like that which a rushlight utters
  As just at the last it kicks and gutters:
  When the cruel smotherer is raised again
  Only snuff, and but little of that, will remain.

  But though _uno avulso_ thus comes every day
  _Non deficit alter_ is also in play:
  For the vacant parts are, one and all,
  Soon taken by puppets just as small;
  Who chirp, chirp, chirp, with a grasshopper's glee,
  We're the lamps of the Universe, We! We! We!
  But Time, whose speech is never long,--
  He hasn't time for it--stops the song
  And says--Lilliput lamps! leave the twopenny boxes,
  And shine in the Budget of Paradoxes!

When a paradoxer parades capital letters and diagrams which are as good as
Newton's to all who know nothing about it, some persons wonder why science
does not rise and triturate the whole thing. This is why: all who are fit
to read the refutation are satisfied already, and can, if they please,
detect the paradoxer for themselves. Those who are not fit to do this would
not know the difference between the true answer and the new capitals and
diagrams on which the delighted paradoxer would declare {355} that he had
crumbled the philosophers, and not they him. Trust him for having the last
word: and what matters it whether he crow the unanswerable sooner or later?
There are but two courses to take. One is to wait until he has committed
himself in something which all can understand, as Mr. Reddie has done in
his fancy about the Astronomer Royal's change of opinion: he can then be
put in his true place. The other is to construct a Budget of Paradoxes,
that the world may see how the thing is always going on, and that the
picture I have concocted by cribbing and spoiling a bit of poetry is drawn
from life. He who wonders at there being no answer has seen one or two: he
does not know that there are always fifty with equal claims, each of whom
regards his being ranked with the rest as forty-nine distinct and several
slanders upon himself, the great Mully Ully Gue. And the fifty would soon
be five hundred if any notice were taken of them. They call mankind to
witness that science _will not_ defend itself, though publicly attacked in
terms which might sting a pickpocket into standing up for his character:
science, in return, allows mankind to witness or not, at pleasure, that it
_does not_ defend itself, and yet receives no injury from centuries of
assault. Demonstrative reason never raises the cry of _Church in Danger_!
and it cannot have any Dictionary of Heresies except a Budget of Paradoxes.
Mistaken claimants are left to Time and his extinguisher, with the
approbation of all thinking non-claimants: there is no need of a succession
of exposures. Time gets through the job in his own workmanlike manner as
already described.

On looking back more than twenty years, I find among my cuttings the
following passage, relating to a person who had signalized himself by an
effort to teach comets to the conductor of the _Nautical Almanac_:

"Our brethren of the literary class have not the least idea of the small
amount of appearance of knowledge {356} which sets up the scientific
charlatan. Their world is large, and there are many who have that moderate
knowledge, and perception of what is knowledge, before which extreme
ignorance is detected in its first prank. There is a public of moderate
cultivation, for the most part sound in its judgment, always ready in its
decisions. Accordingly, all their successful pretenders have _some
pretension_. It is not so in science. Those who have a right to judge are
fewer and farther between. The consequence is, that many scientific
pretenders have _nothing but pretension_."

This is nearly as applicable now as then. It is impossible to make those
who have not studied for themselves fully aware of the truth of what I have
quoted. The best chance is collection of cases; in fact, a Budget of
Paradoxes. Those who have no knowledge of the subject can thus argue from
the seen to the unseen. All can feel the impracticability of the
Hubongramillposanfy numeration, and the absurdity of the equality of
contour of a regular pentagon and hexagon in one and the same circle. Many
may accordingly be satisfied, on the assurance of those who have studied,
that there is as much of impracticability, or as much of absurdity, in
things which are hidden under

 "Sines, tangents, secants, radius, cosines
  Subtangents, segments and all those signs;
  Enough to prove that he who read 'em
  Was just as mad as he who made 'em."

Not that I mean to be disrespectful to mathematical terms: they are short
and easily explained, and compete favorably with those of most other
subjects: for instance, with

 "Horse-pleas, traverses, demurrers,
  Jeofails, imparlances, and errors,
  Averments, bars, and protestandos,
  And puis d'arreign continuandos."


From which it appears that, taking the selections made by satirists for our
samples, there are, one with another, four letters more in a law term than
in one of mathematics. But pleading has been simplified of late years.

All paradoxers can publish; and any one who likes may read. But this is not
enough; they find that they cannot publish, or those who can find they are
_not_ read, and they lay their plans athwart the noses of those who, they
think, ought to read. To recommend them to be content with publication,
like other authors, is an affront: of this I will give the reader an
amusing instance. My good nature, of which I keep a stock, though I do not
use it all up in this Budget, prompts me to conceal the name.

I received the following letter, accompanied by a prospectus of a work on
metaphysics, physics, astronomy, etc. The author is evidently one whom I
should delight to honor:

"Sir,--A friend of mine has mentioned your name in terms of panigeric
[_sic_], as being of high standing in mathematics, and of greatly original
thought. I send you the enclosed without comment; and, assuming that the
bent of your mind is in free inquiry, shall feel a pleasure in showing you
my portfolio, which, as a mathematician, you will acknowledge to be deeply
interesting, even in an educational point of view. The work is complete,
and the system so far perfected as to place it above criticism; and, so far
as regards astronomy, as will Ptolemy beyond rivalry [_sic_: no doubt some
words omitted]. Believe me to be, Sir, with the profoundest respect, etc.
The work is the result of thirty-five years' travel and observation, labor,
expense, and self-abnegation."

I replied to the effect that my time was fully occupied, and that I was
obliged to decline discussion with many persons who have views of their
own; that the proper way is to publish, so that those who choose may read
when they can find leisure. I added that I should advise a precursor in the
shape of a small pamphlet, as two octavo volumes {358} would be too much
for most persons. This was sound advice; but it is not the first, second,
or third time that it has proved very unpalatable. I received the following
answer, to which I take the liberty of prefixing a bit of leonine wisdom:

 "Si doceas stultum, lætum non dat tibi vultum;
  Odit te multum; vellet te scire sepultum.[658]"

"Sir,--I pray you pardon the error I unintentionally have fallen into;
deceived by the F.R.S. [I am not F.R.S.] I took you to be a man of science
[_omnis homo est animal, Sortes est homo, ergo Sortes est animal_][659]
instead of the mere mathematician, or human calculating-machine. Believe
me, Sir, you also have mistaken your mission, as I have mine. I wrote to
you as I would to any other man well up in mathematics, with the intent to
call your attention to a singular fact of omission by Euclid, and other
great mathematicians: and, in selecting you, I did you an honor which, from
what I have just now heard, was entirely out of place. I think, considering
the nature of the work set forth in the prospectus, you are guilty of both
folly and presumption, in assuming the character of a patron; for your own
sense ought to have assured you that was such my object I should not have
sought him in a De Morgan, who exists only by patronage of others. On the
other hand, I deem it to be an unpardonable piece of presumption in
offering your advice upon a subject the magnitude, importance, and real
utility of which you know nothing about: by doing so you have offered me a
direct insult. The system is a manual of Philosophy, a one inseparable
whole of metaphysics and physic; embracing points the most interesting,
laws the most important, {359} doctrines the most essential to advance man
in accordance with the spirit of the times. I may not live to see it in
print; for, at ----, life at best is uncertain: but, live or die, be
assured Sir, it is not my intention to debase the work by seeking
patronage, or pandering to the public taste. Your advice was the less
needed, seeing I am an old-established ----. I remain, etc.--P.S. You will
oblige me by returning the prospectus of my work."

My reader will, I am sure, not take this transition from the "profoundest
respect" to the loftiest insolence for an _apocraphical_ correspondence, to
use a word I find in the Prospectus: on my honor it is genuine. He will be
better employed in discovering whether I exist by patronizing others, or by
being patronized by them. I make any one who can find it out a fair offer:
I will give him my patronage if I turn out to be Bufo, on condition he
gives me his, if I turn out to be Bavius.[660] I need hardly say that I
considered the last letter to be one of those to which no answer is so good
as no answer.

These letters remind me in one respect of the correspondents of the
newspapers. My other party wrote because a friend had pointed me out: but
he would not have written if he had known what another friend told him just
in time for the second letter. The man who sends his complaint to the
newspaper very often says, in effect, "Don't imagine, Sir, that I read your
columns; but a friend who sometimes does has told me ..." It is worded
thus: "My attention {360} has been directed to an article in your paper of
..." Many thanks to my friend's friends for not mentioning the Budget: had
my friend's attention been directed to it I might have lost a striking
example of the paradoxer in search of a patron. That my Friend was on this
scent in the first letter is revealed in the second. Language was given to
man to conceal his thoughts; but it is not every one who can do it.

Among the most valuable information which my readers will get from me is
comparison of the reactions of paradoxers, when not admitted to argument,
or when laughed at. Of course, they are misrepresented; and at this they
are angry, or which is the same thing, take great pains to assure the
reader that they are not. So far natural, and so far good; anything short
of concession of a case which must be seriously met by counter-reasons is
sure to be misrepresentation. My friend Mr. James Smith and my friend Mr.
Reddie are both terribly misrepresented: they resent it by some
insinuations in which it is not easy to detect whether I am a conscious
smotherer of truth, or only muddle-headed and ignorant. [This was written
before I received my last communication from Mr. James Smith. He tells me
that I am wrong in saying that his work in which I stand in the pillory is
all reprint: I have no doubt I confounded some of it with some of the
manuscript or slips which I had received from my much not-agreed-with
correspondent. He adds that my mistake was intentional, and that my reason
is obvious to the reader. This _is_ information, as the sea-serpent said
when he read in the newspaper that he had a mane and tusks.]


My friend Dr. Thorn[661] sees deeper into my mystery. By the way, he still
sends an occasional touch at the old {361} subject; and he wants me
particularly to tell my readers that the Latin numeral letters, if M be
left out, give 666. And so they do: witness DCLXVI. A person who thinks of
the origin of symbols will soon see that 666 is our number because we have
five fingers on each hand: had we had but four, our mystic number would
have been expressed by 555, and would have stood for our present 365. Had n
been the number on each hand, the great number would have been

  (n + 1) (4n^2 + 2n + 1)

With no finger on each hand, the number would have been 1: with one finger
less than none at all on each hand, it would have been 0. But what does
this mean? Here is a question for an algebraical paradoxer! So soon as we
have found out how many fingers the inhabitants of any one planet have on
each hand, we have the means of knowing their number of the Beast, and
thence all about them. Very much struck with this hint of discovery, I
turned my attention to the means of developing it. The first point was to
clear my vision of all the old cataracts. I propose the following
experiment, subject of course to the consent of parties. Let Dr. Thorn
Double-Vahu Mr. James Smith, and Thau Mr. Reddie: if either be deparadoxed
by the treatment, I will consent to undergo it myself. Provided always that
the temperature required be not so high as the Doctor hints at: if the
Turkish Baths will do for this world, I am content.

The three paradoxers last named and myself have a pentasyllable convention,
under which, though we go far beyond civility, we keep within civilization.
Though Mr. James Smith pronounced that I must be dishonest if I did not see
his argument, which he knew I should not do [to say nothing of recent
accusation]; though Dr. Thorn declared me a competitor for fire and
brimstone--and my wife, too, which doubles the joke: though Mr. Reddie
{362} was certain I had garbled him, evidently on purpose to make falsehood
appear truth; yet all three profess respect for me as to everything but
power to see truth, or candor to admit it. And on the other hand, though
these were the modes of opening communication with me, and though I have no
doubt that all three are proper persons of whom to inquire whether I should
go up-stairs or down-stairs, etc., yet I am satisfied they are thoroughly
respectable men, as to everything but reasoning. And I dare say our several
professions are far more true in extent than in many which are made under
more parliamentary form. We find excuses for each other: they make
allowances for my being hoodwinked by Aristotle, by Newton, by the Devil;
and I permit them to feel, for I know they cannot get on without it, that
their reasons are such as none but a knave or a sinner can resist. But
_they_ are content with cutting a slice each out of my character: neither
of them is more than an uncle, a Bone-a-part; I now come to a dreadful
nephew, Bone-the-whole.

I will not give the name of the poor fellow who has fallen so far below
both the _honestum_ and the _utile_, to say nothing of the _decorum_ or the
_dulce_.[662] He is the fourth who has taken elaborate notice of me; and my
advice to him would be, _Nec quarta loqui persona laboret_.[663] According
to him, I scorn humanity, scandalize learning, and disgrace the press; it
admits of no manner of doubt that my object is to mislead the public and
silence truth, at the expense of the interests of science, the wealth of
the nation, and the lives of my fellow men. The only thing left to be
settled is, whether this is due to ignorance, natural distaste for truth,
personal malice, a wish to curry favor with the Astronomer Royal, or mere
toadyism. The only accusation which has truth in it is, that I have made
myself a "public scavenger of science": the assertion, which is the {363}
most false of all is, that the results of my broom and spade are "shot
right in between the columns of" the _Athenæum_. I declare I never in my
life inserted a word between the columns of the _Athenæum_: I feel huffed
and miffed at the very supposition. I _have_ made myself a public
scavenger; and why not? Is the mud never to be collected into a heap? I
look down upon the other scavengers, of whom there have been a few--mere
historical drudges; Montucla, Hutton, etc.--as not fit to compete with me.
I say of them what one crossing-sweeper said of the rest: "They are well
enough for the common thing; but put them to a bit of fancy-work, such as
sweeping round a post, and see what a mess they make of it!" Who can touch
me at sweeping round a paradoxer? If I complete my design of publishing a
separate work, an old copy will be fished up from a stall two hundred years
hence by the coming man, and will be described in an article which will end
by his comparing our century with his own, and sighing out in the best New
Zealand pronunciation--

 "Dans ces tems-là
  C'était déjà comme ça!"[664]


And pray, Sir! I have been asked by more than one--do your orthodox never
fall into mistake, nor rise into absurdity? They not only do both, but they
admit it of each other very freely; individually, they are convinced of
sin, but not of any particular sin. There is not a syndoxer among them all
but draws his line in such a way as to include among paradoxers a great
many whom I should exclude altogether from this work. My worst specimens
are but exaggerations of what may be found, occasionally, in the thoughts
of sagacious investigators. At the end of the {364} glorious dream, we
learn that there is a way to Hell from the gates of Heaven, as well as from
the City of Destruction: and that this is true of other things besides
Christian pilgrimage is affirmed at the end of the Budget of Paradoxes. If
D'Alembert[665] had produced _enough_ of a quality to match his celebrated
mistake on the chance of throwing head in two throws, he would have been in
my list. If Newton had produced _enough_ to match his reception of the
story that Nausicaa, Homer's Phæacian princess, invented the celestial
sphere, followed by his serious surmise that she got it from the
Argonauts,--then Newton himself would have had an appearance entered for
him, in spite of the _Principia_. In illustration, I may cite a few words
from _Tristram Shandy_:

"'A soldier,' cried my uncle Toby, interrupting the Corporal, 'is no more
exempt from saying a foolish thing, Trim, than a man of letters.'--'But not
so often, an' please your honor,' replied the Corporal. My uncle Toby gave
a nod."

I now proceed to die out. Some prefatory remarks will follow in time.[666]
I shall have occasion to insist that all is not barren: I think I shall
find, on casting up, that two out of five of my paradoxers are not to be
utterly condemned. Among the better lot will be found all gradations of
merit; at the same time, as was remarked on quite a different subject,
there may be little to choose between the last of the saved and the first
of the lost. The higher and better class is worthy of blame; the lower and
worse class is worthy of praise. The higher men are to be reproved for not
taking up things in which they could do some good: the lower men are to be
commended for taking up things in which they can do no great harm. The
circle problem is like Peter Peebles's lawsuit:


"'But, Sir, I should really spoil any cause thrust on me so hastily.'--'Ye
cannot spoil it, Alan,' said my father, 'that is the very cream of the
business, man,--... the case is come to that pass that Stair or Arniston
could not mend it, and I don't think even you, Alan, can do it much harm.'"

I am strongly reminded of the monks in the darker part of the Middle Ages.
To a certain proportion of them, perhaps two out of five, we are indebted
for the preservation of literature, and their contemporaries for good
teaching and mitigation of socials evils. But the remaining three were the
fleas and flies and thistles and briars with whom the satirist lumps them,
about a century before the Reformation:

 "Flen, flyys, and freris, populum domini male cædunt;
  Thystlis and breris crescentia gramina lædunt.
  Christe nolens guerras qui cuncta pace tueris,
  Destrue per terras breris, flen, flyys, and freris.
  Flen, flyys, and freris, foul falle hem thys fyften yeris,
  For non that her is lovit flen, flyys ne freris."[667]

I should not be quite so savage with my second class. Taken together, they
may be made to give useful warning to those who are engaged in learning
under better auspices: aye, even useful hints; for bad things are very
often only good things spoiled or misused. My plan is that of a predecessor
in the time of Edward the Second:

 "Meum est propositum genti imperitæ
  Artes frugi reddere melioris vitæ."[668]

To this end I have spoken with freedom of books as books, of opinions as
opinions, of ignorance as ignorance, of {366} presumption as presumption;
and of writers as I judge may be fairly inferred from what they have
written. Some--to whom I am therefore under great obligation--have
permitted me to enlarge my plan by assaults to which I have alluded;
assaults which allow a privilege of retort, of which I have often availed
myself; assaults which give my readers a right of partnership in the
amusement which I myself have received.

For the present I cut and run: a Catiline, pursued by a chorus of Ciceros,
with _Quousque tandem? Quamdiu nos? Nihil ne te?_[669] ending with, _In te
conferri pestem istam jam pridem oportebat, quam tu in nos omnes jamdiu
machinaris!_ I carry with me the reflection that I have furnished to those
who need it such a magazine of warnings as they will not find elsewhere; _a
signatis cavetote_:[670] and I throw back at my pursuers--_Valete, doctores
sine doctrina; facite ut proxima congressu vos salvos corporibus et sanos
mentibus videamus._[671] Here ends the Budget of Paradoxes.


       *       *       *       *       *


I think it right to give the proof that the ratio of the circumference to
the diameter is incommensurable. This method of proof was given by
Lambert,[672] in the _Berlin Memoirs_ for 1761, and has been also given in
the notes to Legendre's[673] Geometry, and to the English translation of
the same. Though not elementary algebra, it is within the reach of a
student of ordinary books.[674]

Let a continued fraction, such as

  b + c
      d + e
          f + etc.,

be abbreviated into a/b+ c/d+ e/f+ etc.: each fraction being understood as
falling down to the side of the preceding sign +. In every such fraction we
may suppose b, d, f, etc. {368} positive; a, c, e, &c. being as required:
and all are supposed integers. If this succession be continued ad
infinitum, and if a/b, c/d, e/f, etc. all lie between -1 and +1, exclusive,
the limit of the fraction must be incommensurable with unity; that is,
cannot be A/B, where A and B are integers.

First, whatever this limit may be, it lies between -1 and +1. This is
obviously the case with any fraction p/(q + [omega]), where [omega] is
between ±1: for, p/q, being < 1, and p and q integer, cannot be brought up
to 1, by the value of [omega]. Hence, if we take any of the fractions

  a/b, a/b+ c/d, a/b+ c/d+ e/f, etc.

say a/b+ c/d+ e/f+ g/h we have, g/h being between ±1, so is e/f+ g/h, so
therefore is c/d+ e/f+ g/h; and so therefore is a/b+ c/d+ e/f+ g/h.

Now, if possible, let a/b+ c/d+ etc. be A/B at the limit; A and B being
integers. Let

  P = A c/d+ e/f+ etc., Q = P e/f+ g/h+ etc., R = Q g/h + i/k + etc.

P, Q, R, etc. being integer or fractional, as may be. It is easily shown
that all must be integer: for


  A/B = a/b+ P/A, or, P = aB - bA

  P/A = c/d+ Q/P, or, Q = cA - dP

  Q/P = e/f+ R/Q, or, R = eP - fQ

etc., etc. Now, since a, B, b, A, are integers, so also is P; and thence Q;
and thence R, etc. But since A/B, P/A, Q/P, R/Q, etc. are all between -1
and +1, it follows that the unlimited succession of integers P, Q, R, are
each less in numerical value than the preceding. Now there can be no such
_unlimited_ succession of _descending_ integers: consequently, it is
impossible that a/b+ c/d+, etc. can have a commensurable limit.

It easily follows that the continued fraction is incommensurable if a/b,
c/d, etc., being at first greater than unity, become and continue less than
unity after some one point. Say that i/k, l/m,... are all less than unity.
Then the fraction i/k+ l/m+ ... is incommensurable, as proved: let it be
[kappa]. Then g/(h + [kappa]) is incommensurable, say [lambda]; e/(f +
[lambda]) is the same, say [mu]; also c/(d + [mu]), say [nu], and a/(b +
[nu]), say [rho]. But [rho] is the fraction a/b+ c/d+ ... itself; which is
therefore incommensurable.

Let [phi]z represent

      a     a^2          a^3
  1 + - + ------- + -------------- + ....
      z   2z(z+1)   2·3·z(z+1)(z+2)

{370} Let z be positive: this series is convergent for all values of a, and
approaches without limit to unity as z increases without limit. Change z
into z + 1, and form [phi]z - [phi](z+1): the following equation will

  [phi]z-[phi](z+1) = ------([phi](z+2))

         a [phi](z+1)       a [phi](z+1)    a  [phi](z+2)
  or a = - ---------- · z + - ---------- · --- ----------
         z   [phi]z         z   [phi]z     z+1 [phi](z+1)

  a = [psi]z(z+[psi](z+1))

[psi]z being (a/z)([phi](z+1)/[phi]z); of which observe that it diminishes
without limit as z increases without limit. Accordingly, we have

  [psi]z = a/z+ [psi](z+1) = a/z+ a/(z+1)+ [psi](z+2)
                           = a/z+ a/(z+1)+ a/(z+2)+ [psi](z+3), etc.

And, [psi](z + n) diminishing without limit, we have

  a/z · [phi](z+1)/[phi]z = (a/z+) (a/(z+1)+) (a/(z+2)+) (a/((z+3)+ ...))

Let z = ½; and let 4a = -x^2. Then (a/z)[phi](z+1) is -(x^2/2) ( 1 -
x^2/(2·3) + x^4/(2·3·4·5...)) or -(x/2) sin x. Again [phi]z is 1 - x^2/2 +
x^4/(2·3·4) or cos x: and the continued fraction is

  (¼)x^2/(½)+ (¼)x^2/(3/2)+ (¼)x^2/(5/2)+ ... or -x/2 x/1+ -x^2/3+ -x^2/5+

{371} whence tan x = x/1+ -x^2/3+ -x^2/5+ -x^2/7+ ...

Or, as written in the usual way,

  tan x = x
          1 - x^2
              3 - x^2
                  5 - x^2
                      7 - ...

This result may be proved in various ways: it may also be verified by
calculation. To do this, remember that if

  a_1/b_1+ a_2/b_2+ a_3/b_3+ ...  a_n/b_n = P_n/Q_n; then

  P_1=a_1, P_2=b_2 P_1,     P_3=b_3 P_2+a_3 P_1, P_4=b_4 P_3+a_4 P_2, etc.
  Q_1=b_1, Q_2=b_2 Q_1+a_2, Q_3=b_3 Q_2+a_3 Q_1, Q_4=b_4 Q_3+a_4 Q_2, etc.

in the case before us we have

  a_1=x, a_2=-x^2, a_3=-x^2, a_4=-x^2, a_5=-x^2, etc.
  b_1=1, b_2=3,    b_3=5,    b_4=7,    b_5=9, etc.

  P_1=x                     Q_1=1
  P_2=3x                    Q_2=3-x^2
  P_3=15x-x^3               Q_3=15-6x^2
  P_4=105x-10x^3            Q_4=105-45x^2+x^4
  P_5=945x-105x^3+x^5       Q_5=945-420x^2+15x^4
  P_6=10395x-1260x^3+21x^5  Q_6=10395-4725x^2+210x^4-x^6

We can use this algebraically, or arithmetically. If we divide P_n by Q_n,
we shall find a series agreeing with the known series for tan x, _as far
as_ n _terms_. That series is

  x + x^3/3 + 2x^5/15 + 17x^7/315 + 62x^9/2835 + ...

{372} Take P_5, and divide it by Q_5 in the common way, and the first five
terms will be as here written. Now take _x_ = .1, which means that the
angle is to be one tenth of the actual unit, or, in degrees 5°.729578. We
find that when x = .1, P_6 = 1038.24021, Q_6 = 10347.770999; whence P_6
divided by Q_6 gives .1003346711. Now 5°.729578 is 5°43'46½"; and from the
old tables of Rheticus[675]--no modern tables carry the tangents so
far--the tangent of this angle is .1003347670.

Now let x = ¼[pi]; in which case tan x = 1. If ¼[pi] be commensurable with
the unit, let it be (m/n), m and n being integers: we know that ¼[pi] < 1.
We have then

  1=(m/n)/1- (m^2/n^2)/3- (m^2/n^2)/5- ...  = m/n- m^2/3n- m^2/5n- m^2/7n-

Now it is clear that m^2/3n, m^2/5n, m^2/7n, etc. must at last become and
continue severally less than unity. The continued fraction is therefore
incommensurable, and cannot be unity. Consequently [pi]^2 cannot be
commensurable: that is, [pi] is an incommensurable quantity, and so also is

I thought I should end with a grave bit of appendix, deeply mathematical:
but paradox follows me wherever I go. The foregoing is--in my own
language--from Dr. (now Sir David) Brewster's[676] English edition of
Legendre's Geometry, (Edinburgh, 1824, 8vo.) translated by some one who is
not named. I picked up a notion, which others had at Cambridge in 1825,
that the translator was the late Mr. Galbraith,[677] then known at
Edinburgh as a writer and teacher.

{373} But it turns out that it was by a very different person, and one
destined to shine in quite another walk; it was a young man named Thomas
Carlyle.[678] He prefixed, from his own pen, a thoughtful and ingenious
essay on Proportion, as good a substitute for the fifth Book of Euclid as
could have been given in the space; and quite enough to show that he would
have been a distinguished teacher and thinker on first principles. But he
left the field immediately.

       *       *       *       *       *

(The following is the passage referred to at Vol. II, page 54.)

Michael Stifelius[679] edited, in 1554, a second edition of the Algebra
(_Die Coss._), of Christopher Rudolff.[680] This is one of the earliest
works in which + and - are used.

Stifelius was a queer man. He has introduced into this very work of Rudolff
his own interpretation of the number of the Beast. He determined to fix the
character of Pope Leo: so he picked the numeral letters from LEODECIMVS,
and by taking in X from LEO X. and striking out M as standing for
_mysterium_, he hit the number exactly. This discovery completed his
conversion to Luther, and his determination to throw off his monastic vows.
Luther dealt with him as straight-forwardly as with Melanchthon about his
astrology: he accepted the conclusions, but told him to clear his mind of
all the premises about the Beast. Stifelius {374} did not take the advice,
and proceeded to settle the end of the world out of the prophet Daniel: he
fixed on October, 1533. The parishioners of some cure which he held, having
full faith, began to spend their savings in all kinds of good eating and
drinking; we may charitably hope this was not the way of preparing for the
event which their pastor pointed out. They succeeded in making themselves
as fit for Heaven as Lazarus, so far as beggary went: but when the time
came, and the world lasted on, they wanted to kill their deceiver, and
would have done so but for the interference of Luther. {375}

       *       *       *       *       *


Pages denoted by numerals of this kind (_287_) refer to biographical notes,
chiefly by the editor. Numerals like 426 refer to books discussed by De
Morgan, or to leading topics in the text. Numerals like 126 indicate minor

  Abbott, Justice, I, _181_.
  Abernethy, J., II, _219_.
  Aboriginal Britons, a poem, II, 270.
  Academy of Sciences, French, I, 163.
  Adair, J., I, _223_.
  Adam, M., I, _65_.
  Adams, J. C., I, _43_, 82, 385, 388; II, 131, 135, 140, 303.
  Ady, Joseph, II, 42, _42_.
  Agnew, H. C., I, 328.
  Agricola, J., I, 394.
  Agricultural Laborer's letter, II, 16.
  Agrippa, H. C., I, _48_, 48.
  Ainsworth, W. H., II, _132_.
  Airy, I, _85_, 88, 152, 242; II, 85, 140, 150, 303, 347.
  Alchemy, I, 125.
  Alfonso X (El Sabio), II, _269_.
  Alford, H., II, _221_.
  Alfred, King, Ballad of, II, 22.
  Algebra, I, 121.
  Algebraic symbols, I, 121.
  Almanac, I, 300; II, 147, 148, 207.
    (_See Easter._)
  Aloysius Lilius, I, 362.
  Alsted, J. H., II, _282_.
  Ameen Bey, II, 15.
  Amicable Society, I, 347.
  Ampère, I, 86.
  Amphisbæna serpent, I, 31.
  Anagrams, De Morgan, I, 138.
  Anaxagoras, II, _59_.
  Angherà, II, 60, _60_, 61, 279.
  Annuities, Fallacies of, I, 157.
  Antichrist, I, 130.
  Antimony, I, 125.
  Antinewtonism, I, 162.
  Antinomians, I, 394.
  Antiphon, II, _59_.
  Antonie, F., I, _126_, 126.
  Apollonius, I, 41, 107.
  Apparitions, II, 47.
  Arago, I, _243_, 390.
  Aratus, II, _167_.
  Arbuthnot, I, _133_, 134.
  Archer, H., II, 90.
  Archimedes,  I, 5, 11, 42, 83, 107.
  Archytas, I, _53_.
  Argoli, I, _104_.
  Aristocrat, as a scientist, I, 131.
  Aristotle, I, 5, 331.
  Arnobius, II, _73_.
  Arson, P. J., II, _207_.
  Ashton, R., II, _99_.
  Astrology, I, 118, 127, 128, 350; II, 43.
  Astronomer's Drinking Song, I, 380.
  Astronomical Aphorisms, I, 398.
    Paradox, I, 394.
    Police Report, I, 390.
    Society. (_See Royal Astronomical Society._)
  Astronomy, Bailly's exaggerated view of, I, 166.
  Astunica, Didacas, I, 90.
  Athanasian Creed, I, 371.
  Atheists, Philosophical, I, 1.
  Atoms, II, 191.
  Attraction, I, 136, 151, 155.
  Augustine, St., II, _23_.
  Aurora borealis, I, 134.
  Austen, Jane, I, 191.
  Auzout, A., II, _300_.
  Aviation, Early ideas of, II, 8.

  Babbage, C., I, _207_, 290, 291; II, 181.
  Bachet, de Méziriac, I, _161_.
  Bacon, F., I, 5, _75_, 75, 76, 79, 89, 145, 331.
  Bacon, R., I, 5, _126_, 126, 360; II, 94.
  Baconian controversy, I, 141.
  Baden Powell, II, _267_.
  Bailly, J. S., I, 166, _166_, 308.
  Baily, F., I, 308, _309_; II, 16, 143, 188.
  Baily, R., II, 16.
  Baker, T., II, _302_.
  Bakewell, F. C., II, 156, _156_.
  Banks, J., I, 28.
  Barberini, M., II, _267_.
  Barker, C., II, _262_.
  Baronius, I, 33, 35; II, _62_.
  Barrême, I, _42_.
  Barrett, G., II, _188_.
  Barrow, I., I, _160_; II, 302.
  Baruel, de, I, 165.
  Bassano, Duc de, II, 3, 339.
  Baxter, T., I, 146.
  Bayle, P., II, _73_.
  Beaufort, F., II, _267_.
  Beaugrand, I, 119, _121_.
  Beaulieu, I, _119_, 119, 121.
  Beaune, de, II, _59_.
  Bécourt, R., II, 277.
  Bedford, Duke of, (6th), I, _182_.
  Behmen, I, _168_, 254; II, 317.
  Bellenden, W., I, _175_.
  Bentley, I, _110_.
  Berkeley, G., II, 346.
  Bernard, E., II, _297_, 300.
  Bernardus Trevisanus, I, _126_, 126.
  Bernoullis, I, 130, _150_, _335_, 336.
  Bertius, P., II, _300_.
  Bèse, I, _66_.
  Bessel, I, _384_; II, 2.
  Bethune, I, _99_, 279, 291.
  Bettesworth, I, 19.
  Beza. (_See Bèse._)
  Bickersteth, E. H., I, _238_.
  Bidder, I, _86_.
  Biden, J., II, 158, _160_.
  Bidle, (Biddle), I, 239.
  Biot, I, _85_.
  Birch, T., I, _108_; II, 304, 313.
  Birks, T. R., II, 158, _158_.
  Bishop, G., I, _386_.
  Bishops as Paradoxers, I, 226.
  Boccaccio, I, 38.
  Boethius, I, _42_, 45.
  Böhme. (_See Behmen._)
  Boncompagni, I, _298_.
  Boniface, St., I, 32.
  Bonnycastle, J., II, _16_.
  Booker, I, 115.
  Boole, G., I, _261_, 332; II, 75, 79.
    --A tribute to, II, 79.
  Borelli, G. A., II, _300_.
  Borello, I, _69_.
  Boreman,  I, 113.
  Borron, Mrs., II, 7.
  Boscovich, I, _156_, 164.
  Bouguer, II, _301_.
  Bouillaud, I, _87_; II, 295.
  Bouvard, A., I, _327_.
  Bovillus, I, _44_; II, 324.
    --Epitome of, I, 44.
  Bowdler, H. M., I, _194_.
  Bowring, J., I, _352_; II, 256.
  Boyle, R., I, 24, _125_; II, 300.
  Bradley, I, 24.
  Bradwardine, I, _227_, 228, 229.
  Brahe. (_See Tycho B._)
  Brancker, I, 107; II, _300_.
  Brenan, J., I, 330, _330_.
  Brewster, D., I, 39, _137_, 140; II, 214, 288, 372.
  Briggs, I, _69_; II, 299, 302.
  Bright, J., II, _235_.
  Brinkley, J., I, _311_.
  Britannicus, D., II, 8.
  British Museum library, I, 151.
  Brothers, R., I, _315_; II, 97.
  Brougham, Henry, Lord, I, _191_.
  Brouncker (Brounker), I, _132_; II, 302.
  Brown, W., II, _168_.
  Browne, T., I, 31.
  Brucker, I, _61_.
  Brunet, I, _402_.
  Brünnow, I, _386_.
  Bruno, I, _59_, 59.
  Bryson, II, _59_.
  Bürgi, I, 52.
  Buffon, I, _282_.
  Bulstrode, II, 84.
  Bungus, I, _55_, 55, 57.
  Buoncompagno, U., I, _362_.
  Burgon, J. W., II, _30_.
  Buridan, I, _37_.
    --Questiones morales, I, 37.
  Buridan's Ass, I, 37.
  Burke, E., I, _173_.
  Burlesque, Frend's, I, 208.
  Burnet, G., I, _107_.
  Burney, Frances, I, _190_.
  Burton, Frances B., I, 374.
  Busby, R., II, _313_.
  Buteo, I, _51_.
  Butler, G., I, _199_.
  Butler, S., II, _218_.
  Buxton, J., I, _86_.
  Byrgius. (_See Bürgi._)
  Byrne, O., I, _329_; II, 186, 190.
  Byron, I, 186; II, 270, 273.

  Cabbala, I, 272.
  Calculating Boys, I, 86.
  Calculus, I, 129.
  Calendar. (_See Easter._)
  Cambridge Poets, II, 269.
  Campanus, I, 42, _43_.
  Canning, Geo., II, _145_.
  Carcavi, I, _106_.
  Cardanus, II, _59_.
  Carlile, R., I, _271_.
  Carlyle, T., II, _373_.
  Carnot, I, 107.
  Caroline tables, I, 124.
  Casaubon, I, _111_.
  Case, J., I, _128_, 128.
  Cassini, J., I, _172_.
  Castel, I, _148_, 148.
  Castiglioni, I, _139_.
  Castlereagh, I, 185, _186_.
  Cataldi, I, _69_, 69.
  Catcott, A., I, _237_.
  Causans, de, I, _298_.
  Cavalieri, I, _106_.
  Cavendish, C., I, _106_; II, 299, 312.
  Cavendish, W., I, _290_.
  Caxton, W., II, _281_.
  Cayley, A., II, _292_.
  Cecil, R. (1st Earl of Salisbury), II, _330_.
  Centrifugal force, II, 268.
  Ceulen. (_See Van Ceulen._)
  Challis, J., I, _390_; II, 141.
  Chalmers, I, _102_; II, 219.
  Chambers, E., II, _282_.
  Chambers, R., I, _344_, 344.
  Charles IX, II, 94.
  Charles X, II, 1.
  Chasles, I, _39_, 229.
  Chesterfield, Earl of (4th), II, _298_.
  Chiffinch, W., II, _50_.
  Ch'in Chiu-shang, II, 66.
  Chitty, J., II, _323_.
  Chiu-chang, Suan-shu, II, 67.
  Christian Evidence Society, I, 270.
  Christie, I, 27.
  Christmann, I, 272, _272_.
  Church question, I, 62.
  Church, The word, II, 30.
  Circle squarers. (_See Squaring the Circle._)
  Circulating media of mathematics, I, 107.
  Ciruelo. (_See Sanchez._)
  Clairaut, I, _219_, 382.
  Clarence, Duke of, I, 179.
  Clarke, R., I, _255_.
  Clavius, I, 11, _69_, 111, 112, 335, 362, 363, 372; II,
  Clayton., Geo., II, _98_.
  Cluvier, D., II, 332, _332_.
  Cobb, Mary, II, 117.
  Cobbett, W., I, _177_, 200, 399.
  Cobden, R., II, 217.
  Cocker, I, _42_; II, 64, 173, 251, 307.
  Cody, P., II, 208.
  Coke, E., II, _331_.
  Colburn, Z., I, _86_.
  Colenso, I, _63_, 247; II,  191.
  Collins, J., I, _107_; II, 297, 300,  302, 313.
  Colvill, W. H., II, 68.
  Cometic astrology, I, 128.
  Comets, I, 128; II, 68, 83.
  Cominale, C., I, _162_, 162.
  Compton, S. J. A., II, _19_.
  Computation, Paradoxes of, II, 251, 267.
  Condamine, C. M. de la, II, _301_.
  Conduitt, John, I, _397_.
  Conduitt, Mrs., I, _136_.
  Congregation of the Index, I, 90.
  Converse propositions, I, 295.
  Convocation at Oxford, I, 96.
  Cooke, Margaret, I, _310_.
  Cooper, A. A. (Shaftsbury), II, _181_.
  Copernicus, I, 5, 6, _76_, 90, 121, 172, 380; II, 165, 335.
  Copley, J. S., I, _198_.
  Cormouls, I, 225.
  Cosmology, I, 172.
  Cotes, R., II, _301_.
  Cottle, Mrs., II, 97, _97_, 161.
  Craig, J., I, _129_, 129.
  Creed, Mathematics of a, I, 329.
  Cribb, T., I, _314_.
  Crotus, J., I, 318.
  Cruickshank, G., I, _186_.
  Cube, Duplication of, I, 349.
  Cumyns, Eliza, I, 299.
  Cunningham, I, 172, _172_.
  Curabelle, I, _221_.
  Curious Calculations, II, 66.
  Curll, E., II, _279_.
  Cusa, I, _44_, 47, 360.
  Custom, II, 324.
  Cyclometry, II, 208. (_See  Squaring of the Circle._)
  Cyclopædias, Review of, II, 280.

  D'Alembert, I, _382_; II, 283, 364.
  Dalgarno, I, 116, _117_.
  Dalton, J., I, _255_.
  D'Arblay, Mme., I, _190_.
  Darwin, E., II, _8_.
  Darwinism, Primitive, I, 344.
  Dary, M., II, _305_.
  Daval, P., II, _298_.
  Davies, T. S., II, 151, _151_, 188.
  Day, A., I, 295, _295_.
  De Baruel, I, 165.
  De Beaune. (_See Beaune._)
  De Becourt, II, 277, _277_.
  Debenham, J., I, _393_.
  De Causans. (_See Causans._)
  Dechales. (_See de Challes._)
  De Challes, I, _45_.
  Decimal coinage, II, 80, 168, 169.
  Decimals run riot, II, 80.
  Dee, J., II, _302_.
  De Fauré, I, 149.
  De la Leu, I, _297_.
  Delambre, I, _160_, 167, 354; II, 165.
  Democritus, II, _34_.
  De Moivre, I, 24, _376_; II, 298.
  De Molières, I, _220_.
  De Molina, I, _297_.
  Demonville, I, 291, 293.
  De Morgan, A., I, 191, 383; II, 194.
    --Refusal of LL. D., I, 191.
  De Morgan, G. C., I, 383.
  De Morgan, Mrs., I, _196_; II, 194.
  Denison, J., I, 348, 353.
  Desaguliers, I, _153_, 156, 157.
  Desargues, I, _119_, 221.
  Descartes, I, 5, 59, 105, 132, 165, 204, 220; II, 94.
  De Serres, II, _60_.
  De Sluse. (_See Sluse._)
  De Thou, I, 51, _111_, 113; II, 295.
  De Vausenville, I, 12.
  Devonshire, Duke of (7th), I, _290_.
  Diamandi, I, 86.
  Didacus Astunica, I, 90.
  Diderot, II, _4_, 283, 339.
  Digby, K., I, _108_.
  Digges, T., and L., II, _302_.
  Dionysius Exiguus, I, _360_.
  Dircks, H., II, 138, _138_.
  Discoverers and discoveries, II, 206.
  Discovery, Basis of, I, 85.
  D'Israeli, I., I, _115_, 118, 136, 188, 227.
  Ditton, I, _133_, 133.
  Division, Nature of, II, 248.
  Dobson, J., I, _234_, 234.
  Dodson, J., II, _312_.
  Dodt, I, _52_.
  Doggerel verse, I, 341.
  Dolland, I, _377_.
  Double Vahu Process, II, 360.
  Douglas, G., I, _232_.
  Drach, S. M., II, _317_.
  Drayson, G. A. W., II, _132_, 132.
  Dryden, II, _71_.
  Dual arithmetic, II, 186.
  Duchesne, I, _52_.
  Dumortier, I, 313.
  Duncan, A., I, _179_.
  Dunkin, E., II, _349_.
  Duodecimal scale, II, 68.
  Duplication Problem, I, 349.
  Dupuy, J. and P., II, _295_.
  Dutens, L., II, _90_.
  Dyer, G., I, _178_.

  Earth, Figure of, II, 54.
  Easter, I, 359.
  Easter Day Paradoxes, I, 353.
  Ebrington, Thos., I, _247_.
  Edgeworth, Maria, I,  _191_.
  Editorial System, I, 15.
  Edleston, I, _140_; II, 296.
  Edwards, J., I, _144_.
  Edwards, T., I, _112_.
  Eirenæus Philalethes, I, _125_, 125, 126.
  Eldon, Lord (1st), II, _197_.
  Elephant story, I, 58.
  Elizabeth, Queen, I, 128.
  Ellenborough, Baron, I, _181_.
  Ellicot, I, 24.
  Ellis, I, _76_, 82.
  Engel, I, 230.
  English language, Origin of, I, 215.
  Enriques, F., II, _367_.
  Epps, J., I, _153_; II, 143.
  Equation of fifth degree, I, 250, 373.
  Erasmus, I, 110.
  Erastus, I, _65_.
  Erichsen, I, 163.
  Ersch, II, _193_, 282.
  Erskine, T., II, _127_.
  Esperanto, Forerunner of, I, 116.
  Euclid, I, 5, 43; II, 118, 151.
    --Without Axioms, I, 287.
  Eudoxus, II, _164_.
  Euler, I, 221, _382_; II, 3, 4, 303, 331, 339.
  Eusebius, II, _220_.
  Eustace, J. C., II, _46_.
  Eutocius, I, _41_; II, 60.
  Evelyn, J., I, _108_.
  Everett, J., I, _346_.
  Evidence, I, 57, 58.

  Faber. (_See Stapulensis._)
  Fairfax, Mary, I, _242_.
  Falco, I, 53.
  Faraday, M., II, _351_.
  Fauré, de, I, 149; II, 238.
  Fawcett, H., II, _249_.
  Ferguson, J., II, _20_.
  Fermat, I, 122, 221; II, _300_.
  Ferrari, S., II, 68.
  Fiction, New era in, I, 189.
  Fienus, I, _74_, 74.
  Filopanti, Q. B., II, _93_.
  Finæus, I, _50_, 50, 113.
  Finleyson, J., I, 314, _314_.
  Flamsteed, I, _87_, 309; II, 45, 143, 302, 306.
  Fletcher, I, 378.
  Fludd, II, _318_.
  Fly-leaf Paradox, II, 264.
  Folkes, M., I, _136_; II, 301.
  Fontenelle, I, _103_.
  Forbes, D., I, _237_.
  Forman, W., I, 296, _296_, 306.
  Forster, T. I. M., I, 320, _320_.
  Foscarini, I, _90_.
  Foster, S., II, _310_.
  Fourier, II, _66_.
  Fox, G., I, _397_.
  Francis, P., II, _96_.
  Francoeur, I, _365_.
  Frankland, W. B., I, 230, 287.
  Franklin, J., II, _265_.
  Freedom of opinion, Growth of, I, 265.
  Freher, A., II, 319.
  French academy on circle squaring, I, 163.
  Frend, W., I, _196_, 196, 206, 208, 252.
  Fresnel, II, _48_.
  Fromondus, I, _74_, 74, 99.
  Frost, I. and J., I, 394.
  Fry, Elizabeth, I, 224.
  Fuller, T., I, _86_.
  Fulton, R., I, _148_.

  Gadbury, J., I, _115_, 115.
  Galbraith, J. A., II, 372.
  Galileo, I, 5, 6, 32, _76_, 82, 83, 96, 122, 381.
  Galle, J. G., I, _386_; II, 7.
  Galloway, I, _56_, 57; II, 143.
  Gamblers, I, 280.
  Garrick, I, 21.
  Gascoigne, W., II, _299_.
  Gassendi, I, _107_.
  Gauss, I, 86, 107, 310.
  Gemistus, G., I, _188_.
  Gentleman's Monthly, Miscellany, I, 208.
  Gephryander. (_See Salicetus._)
  Gergonne, I, _336_.
  Ghetaldi, I, _83_; II, 59.
  Ghost paradox, II, 47.
  Giddy (Gilbert), II, _174_.
  Gilbert, Davies, II, 66, _174_.
  Gilbert, William, I, 6, _68_, 68, 76.
  Gillot, II, _315_.
  Glazier (Glazion), II, 7.
  Godwin, F., I, _103_.
  Godwin, W., I, _174_.
  Golius, I, _106_.
  Gompertz, B., I, _378_.
  Goulburn, I, _288_.
  Goulden, S., II, 88.
  Graham, I, 24.
  Grandamicus, I, _104_, 104.
  Granger, J., I, _156_.
  Grant, A. R., II, 131.
  Grant, R., I, _392_; II, 131.
  Grassi, O., I, _262_.
  Grassini, I, _231_.
  Graunt, J., I, 113, _114_, 154.
  Gravity, I, 151, 244, 348, 353.
    --Newton's apple, I, 136.
  Greek numerals, II, _77_.
  Greene, R., I, _135_, 135.
  Greenhill, Sir G., I, 136.
  Greenwich observatory,  I, 87.
  Gregg, T. D., II, 75, _75_.
  Gregorian Calendar, I, 363.
  Gregory, D., I, _66_; II, 301.
  Gregory, J., I, _118_, 118, 207; II, 302.
  Gregory O., II, _71_.
  Gregory, Pope, I, 362.
  Grevil, I, _202_.
  Grey, C., (2d Earl), I, _315_; II, 247.
  Grosart, I, _141_, 141, 145.
  Grove, W. R., II, _320_.
  Gruber, II, _193_, 282.
  Gruenberger, I, _70_.
  Grynaeus, I, _66_.
  Guaricus, I, _43_.
  Guillim, J., II, _28_.
  Guldin, I, _83_.
  Gumpach, Von, II, 137, _137_.
  Gunning, H., I, _198_.
  Gurney. (_See Fry, E._)
  Guthrie, W., I, _395_.

  Hailes, J. D., II, 135, _135_.
  Hailesean system of astronomy, II, 135.
  Hale, M., I, _123_, 123.
  Hales, S., I, _123_.
  Hall, B., II, _181_.
  Hallam, I, 159.
  Halley, I, 24, _124_, 311; II, 301, 332.
  Halliwell-Phillips, II, _148_, 296.
  Hamilton, W., I, _112_, 117, 331, 335, 339, 341, 342; II,
      52, 53, 111.
  Hamilton, W. Rowan, I, _332_; II, 104, 256, 343.
  Hanover, King of, I, 201.
  Hardy, C., I, _298_.
  Hardy, T., I, _178_.
  Harriot, T., II, _302_.
  Harvey, I, _76_, 78; II, 201.
  Hauff, I, _230_.
  Haughton, S., II, 372.
  Hauksbee, F., I, _156_.
  Hayes, C., I, _132_, 132.
  Heath, D. D., I, _76_.
  Heinfetter, H., II, 94, _94_.
  Helmont, J. B. van, I, _125_.
  Henson, II, 8.
  Heraclitus, II, _34_.
  Herbart, J. F., I, 253, _253_; II, 78.
  Hérigone, II, _59_.
  Herschel, J., I, _80_, 299, 326, 383, 386; II, 88, 95, 181,
      255, 261, 262.
  Herschel, W., I, _81_, 151, 192, 225, 233, 299; II, 288, 348.
  Heywood, F., II, _49_.
  Hicks, J. P., II, _67_.
  Higgins, G., I, _257_, 274.
  Hilarius, Pope, I, _359_.
  Hill, J., I, 21, 22, 23, 24.
  Hill, Rev. R., I, _192_.
  Hill, Sir R., I, _165_, 232.
  Hind, J. R., I, _384_.
  Hippocrates, II, _59_.
  Hoax, An interesting, I, 163.
    --Lunar Caustic, I, 288.
    --Moon (Herschel), I, 326; II, 131.
  Hobbes, I, _105_, 109, 143, 144; II, 80.
  Hobhouse, J. C., II, _126_.
  Hodder, J., II, _265_.
  Hodge, C. B., I, 114.
  Hodges, W., I, 237.
  Hoffmann, J. J., II, _282_.
  Hoffmann, J. J. I. von, I, _230_.
  Holloway, B., I, _237_.
  Holmes, O. W., I, 109.
  Holyoake, G. J., I, _399_, 399.
  Hone, W., I, 124, 180, 184, 185.
  Hook, T. E., II, _261_.
  Hooke, I, _77_; II, 300.
  Hooker, R., II, _201_.
  Hopkins, J., II, 41.
  Horace, I, 40.
  Horne, G., I, _152_, 152, 154, 155, 236.
  Horne, J., I, _178_.
  Horner, L., I, _176_.
  Horner, W. G., II, _66_, 151, 187.
  Houlston, W., II, 156, _156_.
  Howard, E., I, 131.
  Howison, W., I, 256, _256_.
  Howitt, W., II, 193, _193_.
  Howley, I, 63.
  Hulls, I, _147_, 147; II, 8.
  Hume, J., I, _352_; II, 174.
  Husaín Rifki, II, 16.
  Hussein Effendi, II, 15.
  Hutchinson, J., I, 154, _154_.
  Hutton, C., I, _153_, 161; II, 303, 340.
  Huyghens, I, _100_, 133; II, 300.

  Imaginary numbers, II, 186.
  Impalement by request, II, 133.
  Inaudi, I, 86.
  Index Expurgatorius, I, 90.
  Infant prodigies, I, 86.
  Inglis, J. B., II, _52_.
  Inglis, R. H., I, _352_.
  Ingliz Selim Effendi, II, 15.
  Innocent I., I, _359_.
  Irving, E., II, _54_.
  Ivory, J., II, 142, _142_.

  Jabir ben Aflah, II, _59_.
  Jack, R., I, 149.
  Jacotot, J., I, 278, _278_.
  Jameson, Mrs., II, _63_.
  Jeffreys, G., I, _183_.
  Jenner, E., II, _205_.
  Jesuit contributions, I, 164.
  Johnson, H. C., I, 350.
  Johnson, S., I, 20, _190_, 259; II, 117.
  Johnston, W. H., II, 67.
  Jombert, I, _161_.
  Jonchère, I, _146_, 146.
  Jones, W., I, _135_; II, 298, 301.
  Jones, Rev. W., I, _237_.
  Jonson, B., I, 13.
  Journals, Three classes of, II,  144.

  Kantesian Jeweler,  I, 258.
  Karsten, I, _230_.
  Kästner, I, _43_, 110, 112.
  Kater, I, 11.
  Keckermann, I, 3.
  Keill, J., II, _302_.
  Kepler, I, 52, _76_, 82, 381; II, 166.
  Kerigan, T., I, _308_, 353.
  Keroualle, De, II, 50.
  Kersey, I, 107.
  King, Wm., I, _246_.
  Kircher, Adolphe, I, 229.
  Kircher, Athanasius, I, _103_.
  Kirkringius, T., I, 125, _125_.
  Kittle, I, 236.
  Klein, F., II, 367.
  Knight, C., II, _109_, 280, 289.
  Knight, G., I, 151, _151_.
  Knight, R. P., II, _274_.
  Knight, Wm., I, _97_.
  Koenig, S., I, _150_.

  Lacomme, I, 46.
  La Condamine, II, _301_.
  Lacroix, I, 41, 159, 207.
  Lactantius, I, 33, _96_.
  Lagrange, I, 221, _288_, 313, 382; II, 86.
  Laing, F. H., II, _186_, 186.
  Lalande, I, _159_.
  Lamb, C., I, 178; II, 270.
  Lambert, J. H., I, _336_; II, 214, 367.
  Lambert, John, II, _309_.
  Language, Test of, II, 327.
  Lansbergius, I, _70_, 70.
  Laplace, I, 24, _255_, 382; II, 1, 340.
  Lardner, D., II, _253_.
  Lardner, N., I, 14; II, _221_.
  Laud, I, _145_.
  Lauder, W., I, 297.
  Laurent, P., I, _309_, 309.
  Laurie, J., II, 4.
  Laurie, P., II, _42_.
  Laurus, I, 381.
  Law, Edmund, I, _181_.
  Law, Edward, I, _181_.
  Law, W., I, 168, _254_; II, 317.
  Le Coq, I, _86_.
  Lee, R., I, _66_.
  Lee, S., I, _131_.
  Lee, W., I, 157.
  Legate, I, _59_.
  Legendre, I, _229_; II, 215, 367.
  Legh, P., II, _68_, 83.
  Leibnitz, I, 5, 7; II, 46.
  Leo, St., I, _359_.
  Leverrier, I, _43_, 82, 383, 386, 388, 390; II, 7, 135, 140,
  Lewis, G. C., II, 162, _162_.
  Libri, I, _40_, 62; II, 295.
  Lilius, Aloysius, I, 362.
  Lilly, I, _115_; II, 302.
  Lipen, M., I, _298_.
  Little, J., I, 206.
  Livingston, R., I, _148_.
  Locke, J., I, _142_, 142, 144;
    --and Socinianism, I, 142.
  Locke, R., I, 146.
  Locke, R. A., I, _326_; II, 86, 131.
  Logan, W. E., I, _337_.
  Logic, Formal, I, 158; II, 75.
    --Has no paradoxes, I, 330.
  London Mathematical Society, I, 383.
  London, University of, I, 259; II, 71.
  Long, G., II, _290_.
  Long, J. St. J., II. _38_, 205.
  Longitude problems, I, 132, 146, 249.
  Longley, C. T., I, _325_.
  Longomontanus, I, _105_, 105.
  Lottery, I. 281.
  Lovett, R., I, 165, _166_.
  Lowe, R., II, _169_.
  Lowndes, W. T., I. _402_.
  Lubbock, J., I, _279_; II, 148.
  Lucas, F., II, _28_.
  Lucian, I, _102_.
  Lunar Caustic Joke, I, 288.
  Lunn, J. R., II, _66_.
  Lydiat, T., II, _302_.
  Lyndhurst, I, _198_.

  Macclesfield, Earls of, I, 7; II, _296_, 301.
  Macclesfield, Letters, II, 296.
  MacElshender, II, 87.
  Machin, J., II, _301_.
  Mackey, John,  I, 349.
  Mackey, S. A., I, _256_.
  Maclear, T., II, _181_.
  Macleod, H. D., II, 184, _184_.
  Magic, I, 118.
  Magna Charta, I, 25.
  Magnus, I, 42.
  Maitland, S., I, _63_, 163.
  Malacarne, I, 119.
  Malden, H., II, _162_.
  Malius, II, _342_.
  Mallemens, II, _333_.
  Mankind gullible, I, 115.
  Manning, H. E., (Card.), II, _233_.
  Mansel, H. L., II, _162_.
  Marcelis, J., I, _129_, 129.
  Maret, II, _3_.
  Margarita Philosophica. (_See Reisch._)
  Marryat, Capt., II, _87_.
  Marsh, H., I, _199_, 271.
  Martin, B., I, _152_, 153.
  Martin, R., II, _236_.
  Maseres, F., I, _197_, 203.
  Mason, M., II, _132_.
  Mathematical Illustrations of Doctrine, II, 70;
    --Psychology, I, 253;
    --Society, I, 374, 376, 382;
    --Theology, I, 149.
  Mathematics, Condensed history of, II, 58.
  Matter to Spirit, II, 194.
  Maty, I, 23.
  Maupertuis, II, _301_.
  Maurice, F. D., II, _101_.
  Maurolycus, I, _121_.
  Maxwell, A., I, _102_.
  Meadley, G. W., I, _223_.
  Mechanics Magazine, II, 141, 145.
  Medici, Cosmo de, II, _295_.
  Medicine, Status of, I, 266.
  Melanchthon, II, _323_.
  Menestrier, I, _127_, 127.
  Mercator, G., II, _92_.
  Mercator's projection, I, 84.
  Mersenne, I, _106_, 107; II, 295, 297.
  Meslier, J., II, _195_.
  Meteorologist, An early, I, 320.
  Meteorology, I, 327.
  Metius, A. and P., I, _52_, _99_, 99.
  Meton, II, 167.
  Metric System, Forerunner of, I, 171.
  Méziriac, I, 161.
  Milbanke, A. I., I, 225.
  Mill, Jas., I, 260.
  Miller, Joe, I, _182_.
  Miller, S., I, 167.
  Mills, Elizabeth, W., II, 7.
  Milne, J., I, _286_.
  Milner, I., I, _251_, 251.
  Milner, J., II, _23_.
  Milner's lamp, I, 252.
  Milward, II, _250_.
  Miracles _vs_. Nature, II, 6.
  Mitchell, J., I, _242_.
  Molière, I, _232_.
  Molina, A. C. de, I, _297_.
  Mollendorff, von, II, _3_, 338.
  Mondeux, I, 86.
  Montague, C., II, _311_.
  Montmort, P. R. de, II, _301_.
  Montucla, I, 40, 45, 54, 65, 117, 120, 159, 163; II, 60.
  Moon Hoax, I, 326; II, 131.
  Moon, Nature of, II, 84;
    --Rotation of, II, 4, 19, 84, 87.
  More, Hannah, I, _189_, 192.
  More, Henry, I, 123.
  Moore, Dr. John, I, _190_.
  Moore, Sir John, I, _190_.
  Morgan, S., I, 6.
  Morgan, T., I, 191.
  Morgan, W., I, _223_, 224.
  Morhof, I, _61_.
  Morin, I, _99_, 99.
  Morinus, J. B., I, 149.
  Morland, S., II, _302_.
  Mormonism, II, 69.
  Morrison, R. J., I, _321_; II, 43.
  Mose, H., II, _266_.
  Mottelay, I, 68.
  Motti, II, 60.
  Mouton, I, _172_; II, 300.
  Muggleton, I, 394, _395_.
  Multiplication, Nature of, II, 251.
  Murchison, R. I., I, _384_.
  Murhard, I, _43_, 298.
  Murphy, A., II, _308_.
  Murphy, J. L., II, 54, _54_.
  Murphy, P., I, _327_, 398.
  Murphy, R., I, 349, _349_.
  Murray, J., I, _186_; II, 145.
  Murray, L., II, _326_.
  Murray, Mungo, II, 310.
  Musgrave, T., I, _324_.
  Mydorge, I, _298_.
  Mystrom, J. W., II, 182.
  Mythological paradoxes, I, 256.

  Names of Religious Bodies, II, 22.
  Napier, J., I, 5, 66, _67_, 82.
  Napoleon, Doubts as to, I, 246.
  Nautical Almanac, I, 300; II, 147.
  Neal, I, _98_.
  Negative numbers, I, 196, 203.
  Neile, W., II, _190_.
  Neptune, Discovery of, I, 387; II, 140. (_See Adams, Leverrier._)
  Nesse, C, I, _128_, 128.
  Newcomb, S., I, 162.
  Newcomen, T., I, _147_.
  Newton, Sir Isaac, I, 5, 6, 8, 24, 39, 84, 88, 130,
      136, 139, 144, 145, 148, 152, 154, 155, 162, 165,
      167, 197, 225, 237, 242, 257, 282, 296, 297, 309,
      311, 382, 394, 395, 396, 397; II, 2, 70, 184, 297, _302_,
  Newton, John, II, _305_.
  Nicene Creed, I, 371.
  Nichol, J. P., II, _289_.
  Nicholas. (_See Cusa._)
  Nichols, J., I, _175_.
  Nicolas, N. H., I, _354_.
  Nicollet, I, _326_; II, 131.
  Nicolson, W., I, _201_.
  Nieuwentijt, II, _333_.
  Nizzoli, M., II, 275.
  Non-Euclidean geometry, II, 83.
  Northampton, Marquis of (2d), II, _19_.
  Novum Organum Moralium, II, 74.
  Number, Mystery of, I, 55, 56, 169.
  Number of the Beast (666), I, 55, 130, 272, 298, 352; II, 77,
      159, 217, 218, 361, 373.
  Numeral system, II, 68.
  Nursery rhymes, II, 150.

  Occam, Wm. of, II, _40_.
  Odgers, N., II, 191, _191_.
  Oinopides of Chios, II, _59_.
  Oldenburgh, H., II, _300_, 302.
  Orthodox Paradoxes, II, 363.
  Orthography, Paradoxes of, II, 267.
  Ortwinus, I, 319.
  Oughtred, W., II, _298_, 303.
  Owenson, I, _191_.
  Ozanam, I, _161_, 312.

  Pagi, I, 32.
  Paine, T., I, 173, _173_, 271.
  Paley, W., I, _222_; II, 226.
  Palmer, C., I, 225.
  Palmer, H., I, _141_, 141, 145.
  Palmer, J., II, _253_.
  Palmer, T. F., II, _254_.
  Palmer, W., II, _37_.
  Palmerston, Viscount (3d), I, _290_, 352.
  Palmézeaux, I, 167.
  Panizzi, A., I, _151_.
  Papist, II, 26.
  Paracelsus, II, 322.
  Paradox defined, I, 2, 31.
  Paradox, religious, I, 236.
  Paradoxers in general, II, 352.
  Parallels, Theory of, I, 229, 344.
  Pardies, I. G., II, _300_.
  Park, Mungo, II, _91_, 132.
  Parker, F., II, _94_.
  Parker, G. (Earl of Macclesfield), II, _296_.
  Parr, S., I, 173, _173_, 175, 176, 184.
  Parsey, I, 293, _293_.
  Partridge, J., I, _305_.
  Pasbergius, I, 381.
  Pascal, I, 39, 119, 122, _220_, _221_; II, 73.
  Pascal's Hexagram, I, 221.
  Passot, I, 279, _279_.
  Passover, I, 358, 372.
  Patriotic paradox, I, 231.
  Paucton, I, _172_.
  Paulian, I, 165, _165_.
  Peacock, Geo., I, _196_, 350.
  Peacock, T. L., I, _190_, 340.
  Pearce, A. J., II, 43.
  Pearne, T., I, _239_.
  Peel, Sir R., I, 290, _352_.
  Peel, W. Y., I, _290_.
  Pèlerin, J., II, _324_.
  Pell, J., I, _105_, 105, 107; II, 300, 302, 312.
  Pepys, I, _113_, 114.
  Perigal, H., II, 19, _20_.
  Perpetual motion, I, 118, 348; II, 55, 138.
  Perspective, New theory of, I, 293.
  Peters, W., II, 11, 315.
  Petitioning Comet, I, 127.
  Petrie, W. M. F., I, _328_.
  Petty, I, _114_; II, 300.
  Philalethes, Eirenaeus, I, _125_, 125, 126.
  Philalethes, Eugenius, I, _255_.
  Phillips, R., I, 242, _242_, 245.
  Philo of Gadara, I, 40, _40_.
  Philosopher's stone, I, 118, 125.
  Philosophical atheists, II, 1.
  Philosophy and Religion, II, 37.
  Phonetic spelling, II, 81.
  [pi], values of, I, 11, 43, 45, 46, 52, 69, 100, 110,
      129, 135, 146, 245, 283, 284, 294, 347, 348, 349,
      350; II, 60, 63, 105, 110, 118, 135, 156, 209, 279, 315, 316.
  Pighius, I, _372_.
  Pike, S., I, 236, _236_.
  Pindar, P., II, _272_.
  Piozzi, Mrs., I, _235_; II, 272.
  Piscator, B., II, 25.
  Pitman, F., II, 81, _81_.
  Place, F., I, _199_.
  Planets inhabitable, I, 100, 102.
  Plato, I, 5.
  Platt, H., I, _126_, 126.
  Playfair, J., I, _233_.
  Pletho, G., I, _188_.
  Pliny, II, 280.
  Ploucquet, I, _336_, 337.
  Poe, E. A., II, 132.
  Poincaré, I, 136.
  Poisson, I, _292_; II, 2.
  Pollock, J. F., II, _174_.
  Pons, II, _45_.
  Pope, Wm., I, 277, _277_.
  Porta, I, _68_, 68, 83.
  Porteus, B., I, _193_, 203.
  Porteus, H. F. A., II, 157, _157_.
  Porus, I, 44.
  Powell, Baden, II, _267_.
  Powell, W. S., I, _222_.
  Pratt, H. F. A., II, 157, _157_.
  Pratt, O., II, _69_.
  Predaval, Count de, I, _348_.
  Prescot, B., I, 270, _270_, 278.
  Prester John, I, 70, _71_, 152.
  Price, R., I, _223_.
  Probability, Discourse on, I, 279.
  Proclus, I, 188, _188_.
  Prodigies, Youthful, I, 219, 332.
  Pronunciation, II, 330.
  Protestant and Papal Christendom, II, 33.
  Protimalethes, II, 6.
  Ptolemy, I, 5, 33, _380_.
  Pullicino, II, 61, _61_.
  Pusey, I, _64_.
  Pyramids, The, I, 328; II, 95, 136.
  Pythagoras, II, _59_.

  Quadrature problem. (_See Squaring the circle._)
  Quarles, F., II, _277_.
  Quintilian, II, 280.
  Quotem, C., I, 399.

  Rabelais, I, _102_.
  Rainbow Paradox, II, 334.
  Ramachandra, Y., I, _374_.
  Ramchundra, I, 374.
  Ramus, I, 5.
  Recalcati, II, 208, 314.
  Recorde, R., II, _328_.
  Reddie, Jas., II, 183, _183_, 344, 360.
  Reeve, J., I, _395_.
  Regiomontanus, I, _48_, 360.
  Reisch, I, _45_; II, 281.
  Religion and Philosophy, II, 37.
  Religious bodies, Names of, II, 22;
    --customs, Attacks on, I, 177;
    --Insurance, I, 345;
    --Paradox, I, 236;
    --Tract society, I, 192.
  Remigius, I, _50_.
  Reuchlin, J., II, _323_.
  Revelations, Napier on, I, 66.
  Revilo, (O. Byrne), I, 241, 329, _329_.
  Reyneau, C. R., II, _301_.
  Rheticus, I, _69_; II, 372.
  Rhonius, II, 300.
  Ribadeneira, P. de, II, _62_.
  Riccioli, I, _96_.
  Richards, G., II, _270_.
  Rigaud, J., II, _299_.
  Rigaud, S. J., II, _299_.
  Rigaud, S. P., I, _140_; II, 298, 313.
  Ringelbergh, J. S., II, _281_.
  Ripley, G., I, _126_, 126.
  Ritchie, W., I, 295, _295_.
  Ritterhusius, I, _60_.
  Rive, J.-J., I, _160_.
  Robertson, Jas., I, _237_.
  Roberval, I, _105_, 122.
  Robinson, B., I, _148_, 148.
  Robinson, H. C., I, _314_; II, 52, 275.
  Robinson, R., I, _177_.
  Robinson, T. R., II, _181_.
  Roblin, J., II, 136.
  Rogers, S., II, _260_.
  Roget, P. M., I, _398_.
  Roomen, A. van, I, _110_.
  Ross, J. C., I, _303_.
  Rosse, I, 26.
  Rossi, G., I, 231, _231_.
  Rotation of the Moon, II, 4, 19.
  Rough, W., I, 198.
  Rowning, J., I, _155_.
  Royal Astronomical Society, I, 27;
    --Forerunner of, I, 374.
  Royal Society, I, 21, 22, 24-30, 56, 57, 136, 153, 163,
      164, 165.
  Rudio, I, 159; II, 367.
  Rudolff, C., II, 373.
  Russell, Earl (1st), I, _296_.
  Rutherford, W., II, _109_.

  Sabatíer, A., II, _267_.
  Sabellius, I, _241_.
  Sacrobosco, I, _360_.
  Sadler, T., I, _238_, 241.
  Saint-Martin, I, 167, _168_, 206.
  St.-Mesmin, M. de., I, 280.
  St. Vincent, G. de., I, _110_, 117.
  St. Vitus, Patron of Cyclometers, II, 60.
  Sales, de, I, 167.
  Salicetus, I, 64.
  Salisbury, Earl of (1st), II, _330_.
  Salmasius, Claudius, II, _168_.
  Salusbury, Hester, I, _235_.
  Sanchez, Petro, I, 229, _229_.
  Sanders, W., I, 207.
  Sanderson, R., I, _135_.
  Sara, R., I, _297_.
  Saunderson, N., I, _377_; II, 301.
  Scaliger, I, _44_, 110, 111, 112, 113; II, 238.
  Scévole de St. Marthe, I, 113.
  Schooten, Van, II, _59_.
  Schopp, I, _60_.
  Schott, I, _64_; II, 64.
  Schumacher, H. C., I, _107_; II, 297.
  Schwab, I, _230_.
  Scientific paradoxes, I, 232.
  Scott, Michael, I, _38_.
  Scott's Devils, I, 38.
  Scott, W., I, 20, 27, 38, 39, 155, _191_.
  Scripture and Science, II, 261.
  Search, John, I, 247.
  Selden, J., II, _250_.
  Sénarmont, II, _48_.
  Serres, De, II, _60_.
  Shaftesbury, Earl of, II, _181_.
  Shakespeare, I, 13.
  Shanks, II, _63_, 65, 109.
  Shaw, P., I, _142_.
  Sheepshanks, J., I, _147_.
  Sheepshanks, R., I, _290_.
  Shelley, I, 174.
  Shepherd, S., I, _124_.
  Sherburne, E., II, _295_.
  Sheridan, R. B., I, _175_.
  Sheridan, T., I, _175_.
  Shoberl, F., II, _270_.
  Shrewsbury, I, _108_.
  Siddons, Mrs., I, 189.
  Simms, W., I, _152_.
  Simplicius, II, _164_.
  Simpson, T., I, 377; II, 304.
  Simson, R., I, _197_, 202, 233.
  Sinclair, G., I, _207_.
  Slander Paradoxes, II, 138.
  Sloane, I, 24.
  Sluse, R. de, I, _118_, 118; II, 300.
  Smith, Adam, II, _112_.
  Smith, Jas., I, _46_; II, 103, _103_, 154, 236, 237, 238, 241, 336,
  Smith, Jas., II, _217_.
  Smith, Jas. (Shepherd), II, _55_, 193.
  Smith, Joseph, II, _69_.
  Smith, Richarda, I, _242_.
  Smith, Thomas, I, 346, _346_.
  Smith, Wm., II, _152_.
  Smyth, C. P., I, _328_; II, 65.
  Snell, I, _75_, 75.
  Socinianism, I, 142, 143.
  Socinus, I, 3, _143_.
  Socrates Scholasticus, I, _358_.
  Sohncke, L. A., II, _131_.
  Somerville, Mrs., I, _242_.
  South, J., II, _181_.
  Southcott, Joanna, II, _58_, 97, 239.
  Spearman, R., I, _237_.
  Speculative thought in England, I, 374.
  Spedding, I, _76_, 82, 142.
  Speed, J., I, _201_.
  Speke, I, _70_.
  Spelling, phonetic, II, 81.
  Spence, W., I, _231_, 231.
  Spencer, Earl (3d), II, _9_.
  Spinoza, I, 3, _37_.
  Spiritualism, II, 47, 55, 207.
  Spurius Cassius Viscellinus, II, _342_.
  Spurius Maelius, II, _342_.
  Squaring the circle, I, 8, 42, 44, 46, 47, 50, 52, 53,
      69, 70, 75, 109, 117, 119, 129, 135, 146, 149,
      159, 163, 164, 347, 348, 374; II, 10, 11, 60, 105, 154,
      156, 208, 278, 314.
  Stäckel, I, 230.
  Stanhope, P. D., (Earl of Chesterfield), II, _298_.
  Stapulensis, I, _44_; II, 324.
  Star polygons, I, 229.
  Starkie, G., I, _126_, 126.
  Statter, D., II, 80.
  Steamship suggested, I, 147.
  Steel, Jas., II, 68.
  Stenography, II, 81.
  Stephens, I, _44_; II, 324.
  Stephenson, G., II, _138_.
  Stephenson, R., II, _138_.
  Stevin, I, _83_, 313; II, 59.
  Stewart, D., II, _53_.
  Stewart, R., I, _186_.
  Stifel, M., II, _373_.
  Strafford, Earl of, I, _240_.
  Stratford, W. S., I, _300_.
  Street, T., I, _124_.
  Stukely, W., I, _236_.
  Suffield, G., II, _66_.
  Suidas, II, _29_.
  Sumner, C. R., I, _324_.
  Sumner, J. B., I, _324_.
  Sun as an electric space, II, 41.
  Supernatural, The, II, 193.
  Suvaroff, II, _85_.
  Swastika, II, 231.
  Swedenborg, E., I, _255_.
  Swift, I, 19, 133.
  Sylvester, J. J., II, _336_.
  Symington, W., I, _148_.
  Symons, II, 4, 5, 20, 84, 85.
  Sympathetic powder, I, 108.
  Synesius, I, 125.

  Talbot, G., I, _22_, _108_.
  Talbot's powder, I, 108.
  Tartaglia, II, 59.
  Tasse, I, _106_.
  Tate, J., I, _199_.
  Tauler, J., II, _322_.
  Taylor, Brook, II, _301_.
  Taylor, John, I, _352_; II, 95.
  Taylor, Robt., I, _270_.
  Taylor, T., I, 188, _188_.
  Teissier, I, _113_.
  Temple, H. J., I, 290.
  Tenterden, Chief Justice, I, _181_.
  Thales, II, _59_, 83.
  Theism independent of Revelation, I, 399.
  Thelwall, J., I, _178_.
  Theodoretus, I, _358_.
  Theological Paradoxes, I, 316.
  Theology, Mathematical, I, 129, 149.
  Theophrastus, II, _167_.
  Thiébault, II, _3_, 338.
  Thom, D., II, _226_, 240.
  Thom, J. H., II, _226_.
  Thompson, P., I, _7_.
  Thompson, T. P., I, _252_, 287, 344; II, 83, 185.
  Thomson, Dr., I, 21.
  Thomson, W., I, _325_.
  Thorn, W., II, 158, _158_, 360.
  Thorndike, H., II, _313_.
  Thrale, Mrs., I, _235_.
  Thurlow, Baron, I, _222_.
  Thyræus, I, 50.
  Tides, New theory of, I, 393.
  Tombstones of mathematicians, I, 106.
  Tonal System, II, 182.
  Tooke, H., I, _178_.
  Torriano, E., I, 250.
  Towneley, II, 300.
  Townley, C., II, _300_.
  Trisection problem, I, 118; II, 10, 12, 13, 15.
  Troughton, I, _152_.
  Turnor, E., I, _137_.
  Tycho Brahe, I, 5, _76_, 381; II, 302, 335.

  Upton, W., II, 12, _12_, 15.
  Ursus, I, _52_.

  Valentine, B., I, _125_, 125.
  Van Ceulen, I, 52, 70, 100.
  Van de Weyer, I, _313_.
  Van Etten, I, _161_.
  Van Helmont, I, _125_, 125.
  Van Roomen, I, _110_.
  Van Schooten, II, _59_.
  Vaughan, T., I, _255_.
  Victorinus, I, _359_.
  Viète, I, _51_; II, 210, 295.
  Virgil, St., I, 32, _33_, 34, 99.
  Virginia, University of, I, 233.
  Viscellinus, II, _342_.
  Vitruvius, II, _281_.
  Vivian, T., I, 172, _172_.
  Vogel, A. F., I, 373.
  Voltaire, I, 103, 165, 166, 167, 168, 248; II, 268.
  Von Gumpach, II, 137, _137_.
  Von Hutten, I, 318.
  Von Wolzogen. (_See Wolzogen._)
  Vyse, R. W. H., I, _328_.

  Walker, W. E., II, _316_.
  Walkingame, F., II, _173_.
  Wallich, N., II, _14_.
  Wallis, J., I, 107, 109, 110; II, 299, 313.
  Walpole, I, _23_, 131.
  Walsh, John, I, 260, _260_; II, 157.
  Wapshare, J., II, _230_.
  Warburton, H., I, _349_.
  Warburton, Wm., I, _55_, 112; II, 174.
  Ward, S., II, _299_.
  Waring, E., I, _203_, 222.
  Warner, W., II, _302_, 312.
  Warren, S., II, _340_.
  Watkins, J., II, _270_.
  Watson, Bp., I, _223_.
  Watt, R., I, _102_, 402.
  Watts, I., II, 18.
  Weddle, T., II, _187_.
  Wentworth, Thos., I, _240_.
  Wharton, I, 115.
  Whately, R., I, 246, _246_, 324.
  Whately's Paradox, I, 246.
  Whewell, I, _101_, 101, 273, 314, 380; II, 104, 246, 247.
  Whigs, II, 22.
  Whiston, J., I, _147_.
  Whiston, W., I, 133, _133_, 146, 156, 311.
  White, H. K., II, _271_.
  White, J. B., I, 248.
  White, R., I, 11.
  Whitford, I, 105.
  Whitworth, W. A., II, _344_.
  Whizgig, On the, I, 254.
  Wightman, I, 59.
  Wilberforce, W., II, _236_.
  Wilkins, J., I, 96, _100_, 116, 226.
  Williams, J. B., I, _378_.
  Williams, T., I, 171, _171_.
  Wilson, Sir J., I, _221_.
  Wilson, J. M., II, _344_.
  Wilson, R., II, 7, _7_.
  Wilson's Theorem, I, _222_.
  Wingate, E., II, _308_.
  Winter, I, 46.
  Wirgman, T., I, 258, _258_.
  Wiseman, N. P. S., II, _26_, 61, 294.
  Wolcot, J. (Peter Pindar), II, _272_.
  Wollstonecraft, I, 173, _173_.
  Wolzogen, I, _106_.
  Wood, A., I, _98_.
  Wood, John, I, _233_.
  Wood, Wm., I, 246, _246_.
  Woodley, W., I, 307, _307_.
  Wordsworth, II, 273.
  Wright, E., I, 84.
  Wright, T., I, 151, _151_, 152, 153.
  Wright, W., II, 9.
  Wronski, I, 249, _250_.
  Wrottesley, J. (Baron), II, _181_.

  Young, B., II, _69_.
  Young, J. W. A., II, 367.
  Young, T., I, 24, 30, _250_.
  Youthful Prodigies, I, 219.
  Yvon, I, _297_.

  Zach, von, II, _45_, 196.
  Zachary, Pope, I, 32, 34.
  Zadkiel, I, _321_; II, 43.
  Zetetic Astronomy, II, 88.
  Zodiac, II, 136.
  Zytphen, II, _335_.

       *       *       *       *       *


Transcriber's note: References to Notes in Volume I are shown as in the
printed book, with the resequenced footnote numbers in the Project
Gutenberg Edition (EText-No. 23100) added thus {123}.

[1] See Vol. I, page 255, note 6 {584}.

[2] "I have no need for this hypothesis."

[3] "Ah, it is a beautiful hypothesis; it explains many things."

[4] "What we know is very slight; what we don't know is immense."

[5] Brewster relates (_Life of Sir Isaac Newton_, Vol. II, p. 407) that, a
short time before his death, Newton remarked: "I do not know what I may
appear to the world, but to myself I seem to have been only like a boy
playing on the seashore, and diverting myself in now and then finding a
smoother pebble or a prettier shell than ordinary, whilst the great ocean
of truth lay all undiscovered before me."

[6] See Vol. I, p. 292, note 1 {632}.

[7] "What is all that!"

[8] "I have some good news to tell you: at the Bureau of Longitudes they
have just received a letter from Germany announcing that M. Bessel has
verified by observation your theoretical discoveries on the satellites of

[9] "Man follows only phantoms."

[10] See Vol. I, page 382, note 13 {786}.

[11] Dieudonné Thiébault (1733-1807) was a Jesuit in his early life, but he
left the order and took up the study of law. In 1765 he went to Prussia and
became a favorite of Frederick the Great. He returned to France in 1785 and
became head of the Lycée at Versailles.

[12] _Memories of Twenty Years of Residence in Berlin._ There was a second
French and an English edition in 1805.

[13] Richard Joachim Heinrich von Mollendorff (1724-1816) began his career
as a page of Frederick the Great (1740) and became field marshal (1793) and
commander of the Prussian army on the Rhine (1794).

[14] Hugues Bernard Maret (1763-1839) was not Duc de Bassano in 1807, this
title not being conferred upon him until 1809. He was ambassador to England
in 1792 and to Naples in 1793. Napoleon made him head of the cabinet and
his special confidant. The Bourbons exiled him in 1816.

[15] Denis Diderot (1713-1784), whose _Lettre sur les aveugles_ (1749)
introduced him to the world as a philosopher, and whose work on the
_Encyclopédie_ is so well known.

[16] "Sir, (a + b^{n}) / n = x, whence God exists; answer!"

[17] This was one James Laurie of Musselburgh.

[18] Jelinger Cookson Symons (1809-1860) was an office-holder with a
decided leaning towards the improvement of education and social conditions.
He wrote _A Plea for Schools_ (1847), _The Industrial Capacities of South
Wales_ (1855), and _Lunar Motion_ (1856), to which last work the critic
probably refers.

[19] "Protimalethes" followed this by another work along the same line the
following year, _The Independence of the Testimony of St. Matthew and St.
John tested and vindicated by the theory of chances_.

[20] Wilson had already taken up the lance against science in his
_Strictures on Geology and Astronomy, in reference to a supposed want of
harmony between these sciences and some parts of Divine Revelation_,
Glasgow, 1843. He had also ventured upon poetry in his _Pleasures of
Piety_, Glasgow, 1837.

[21] Mrs. Borron was Elizabeth Willesford Mills before her marriage. She
made an attempt at literature in her _Sibyl's Leaves_, London (printed at
Devonport), 1826.

[22] See Vol. I, page 386, note 10 {801}.

[23] See Vol. I, page 43, notes 7 {32} and 8 {33}.

[24] His flying machine, designed in 1843, was one of the earliest attempts
at aviation on any extensive scale.

[25] Erasmus Darwin (1731-1802) was the grandfather of Charles Darwin. The
work here mentioned had great influence, being translated into French,
Portuguese, and Italian. Canning parodied it in his _Loves of the

[26] See Vol. I, page 147, note 1 {312}.

[27] The notes on this page were written on the day of the funeral of
Wilbur Wright, June 1, 1912, the man who realized all of these prophecies,
and then died a victim of municipal crime,--of typhoid fever.

[28] John Charles, third Earl Spencer (1782-1845), to whose efforts the
Reform Bill was greatly indebted for its final success.

[29] This was published in London in 1851 instead of 1848.

[30] This appeared in 1846.

[31] This was done in _The Circle Squared_, published at Brighton in 1865.

[32] It first appeared in 1847, under the title, _The Scriptural Calendar
and Chronological Reformer, 1848. Including a review of tracts by Dr.
Wardlaw and others on the Sabbath question. By W. H. Black._ The one above
mentioned, for 1849, was printed in 1848, and was also by Black
(1808-1872). He was pastor of the Seventh Day Baptists and was interested
in archeology and in books. He catalogued the manuscripts of the Ashmolean
Museum at Oxford.

[33] William Upton, a Trinity College man, Dublin. He also wrote _Upton's
Physioglyphics_, London, 1844; _Pars prima. Geometria vindicata;
antiquorumque Problematum, ad hoc tempus desperatorum, Trisectionis Anguli,
Circulique Quadraturae, Solutio, per Eucliden effecta, London_ (printed at
Southampton), 1847; _The Uptonian Trisection_, London, 1866; and _The
Circle Squared_, London, 1872.

[34] For example, if [theta] = 90° we should have 3 cos 30° = 1 + [root](4
- sin^2 90°), or 3.½ [root]3 = 1 + [root]3, or ½ [root]3 = 1.

[35] Nathaniel Wallich (1786-1854) was surgeon at the Danish settlement at
Serampore when the East India Company took over the control in 1807. He
entered the British medical service and was invalided to England in 1828.
His _Plantae Asiaticae Rariores_ (3 vols., London, 1830-1832) was
recognized as a standard. He became vice-president of the Linnean Society,
F. R. S., and fellow of the Royal Asiatic Society.

[36] But if [theta] = 90° this asserts that

  cos 30° = (sin 270° . cos 225° + sin^2 90° . sin 225°) / [root](sin^2
      270° . cos^2 225° + sin^{4} 90° + sin 270° . sin 450° . sin^2 90°),

or that

  ½ [root]3 = (-1 . (-1 /[root]2) + 1 . (-1/[root]2) / [root]1 . ½ + 1 - 1
      . 1 . 1) = 0 / [root]½,

so that De Morgan must have made some error in copying.

[37] John Bonnycastle (died in 1821) was professor of mathematics at
Woolwich. His edition of Bossut's _History of Mathematics_ (1803), and his
works on elementary mathematics were well known.

[38] The bibliographies give Husaín Rifki as the translator, a practical
geometry as the work, and 1802 as the date.

[39] See Vol. I, page 309, note 2 {670}.

[40] Probably in _The Improvement of the Mind_ which Isaac Watts
(1674-1748) published in 1741. His _Horae Lyricae_ appeared in 1706, and
the _Hymns_, by which he is still well known, in 1707.

[41] Spencer Joshua Alwyne Compton, second Marquis of Northampton
(1790-1851), was a poet, a scientist, and a statesman. He was president of
the Royal Society from 1838 to 1849.

[42] Besides the writings here mentioned Perigal published a work on
_Geometric Maps_ (London, 1853), and _Graphic Demonstrations of Geometric
Problems_ (1891).

[43] See Vol. II, page 5, note 18.

[44] James Ferguson (1710-1776) was a portrait painter, an astronomer, and
a popular writer and lecturer on various subjects.

[45] In the old ballad of King Alfred and the Shepherd, when the latter is
tempting the disguised king into his service, he says:

 "Of whig and whey we have good store,
  And keep good pease-straw fire."

_Whig_ is then a preparation of milk. But another commonly cited derivation
may be suspected from the word _whiggamor_ being used before _whig_, as
applied to the political party; _whig_ may be a contraction. Perhaps both
derivations conspired: the word _whiggamor_, said to be a word of command
to the horses, might contract into _whig_, and the contraction might be
welcomed for its own native meaning.--A. De M.

[46] This was p. 147 in the first edition.

[47] St. Augustine (354-430) was bishop of Hippo. His _Confessiones_, in 13
books, was written in 397, and his _De Civitate Dei_ in 426.

[48] "He was wont to indulge in certain Punic subtleties lest he should
weary the reader by much speaking."

[49] John Milner (1751-1826), bishop of Castabala, a well-known

[50] It will be said that when the final happiness is spoken of in "sure
and certain hope," it is _the_ Resurrection, generally; but when afterwards
application is made to the individual, simple "hope" is all that is
predicated which merely means "wish?" I know it: but just before the
general declaration, it is declared that it _has_ pleased God of his great
mercy to _take unto Himself_, the soul of our dear brother: and between the
"hopes" hearty thanks are given that it _has_ pleased God to deliver our
dear brother out of the miseries of this wicked world, with an additional
prayer that the number of the elect may shortly be accomplished. All which
means, that our dear brother is declared to be taken to God, to be in a
place not so miserable as this world--a description which excludes the
"wicked place"--and to be of the elect. Yes, but it will be said again! do
you not know that when this Liturgy was framed, all who were not in the
road to Heaven were excommunicated burial service read over them. Supposing
the fact to have been true in old time, which is a very spicy supposition,
how does that excuse the present practice? Have you a right _always_ to say
what you believe _cannot always_ be true, because you think it was once
_always_ true? Yes, but, choose whom you please, you cannot be _certain_ He
is _not_ gone to Heaven. True, and choose which Bishop you please, you
cannot be demonstratively _certain_, he is _not_ a concealed unbeliever:
may I therefore say of the whole bench, _singulatim et seriatim_, that they
_are_ unbelievers? No! No! The voice of common sense, of which common logic
is a part, is slowly opening the eyes of the multitude to the unprincipled
reasoning of theologians. Remember 1819. What chance had Parliamentary
Reform when the House of Commons thanked the Manchester sabre-men? If you
do not reform your Liturgy, it will be reformed for you, and sooner than
you think! The dishonest interpretations, by defence of which even the
minds of children are corrupted, and which throw their shoots into
literature and commerce, will be sent to the place whence they came: and
over the door of the established organization for teaching religion will be
posted the following notice:

 "Shift and Subterfuge, Shuffle and Dodge,
  No longer here allowed to lodge!"

All this ought to be written by some one who belongs to the Establishment:
in him, it would be quite prudent and proper; in me, it is kind and
charitable.--A. De M.

[51] But few do have access to it, for the work is not at all common, and
this Piscator is rarely mentioned.

[52] This derivation has been omitted.--S. E. De M.

[53] A blow for a blow. Roland and Oliver were two of the paladins of
Charlemagne whose exploits were so alike that each was constantly receiving
credit for what the other did. Finally they met and fought for five days on
an island in the Rhine, but even at the end of that period it was merely a
drawn battle.

[54] "In the name of the church."

[55] "From the chair," officially.

[56] Nicholas Patrick Stephen Wiseman (1802-1865), whose elevation to the
archbishopric of Westminster and the cardinalate (1850) led to the act
prohibiting Roman Catholics from assuming episcopal titles in England, a
law that was never enforced.

[57] He was born in 1812 and was converted to Catholicism in 1839. He
founded the _Tablet_ in London in 1840, removing its office to Dublin in
1849. He became M. P. in 1852, and at the time of his death (1855) he was
preparing a memorial to the Pope asking him to annul the proclamation of an
Irish bishop prohibiting his priests from taking part in politics.

[58] John Guillim (1565-1621) was the first to systematize and illustrate
the whole science of heraldry. He published _A display of Heraldrie:
manifesting a more easie accesse to the knowledge thereof_ in 1610.

[59] "Faith."

[60] "Faithful."

[61] "For the faith vindicated."

[62] The words are of the same root, and hence our word _fiddle_. Some
suppose this root means a _rope_, which, as that to which you trust,
becomes, in one divergence, confidence itself--just as a _rock_, and other
words, come to mean reliance--and in another, a little string.--A. De M.

[63] The Greek lexicographer, a Christian, living after 1000 A. D. His
lexicon was first printed at Milan in 1499.

[64] _Skindapsos._

[65] This was John William Burgon (1813-1888), Gresham professor of
theology (1867) and dean of Chichester. He was an ultra-conservative,
opposing the revised version of the New Testament, and saying of the
admission of women to the university examinations that it was "a thing
inexpedient and immodest."

[66] _Ekklesia_, or _ecclesia_.

[67] _Ennomos ekklesia._

[68] "Without doubt I shall perish forever."

[69] "Every man is an animal." "Sortes is a man." "Sortes is an animal."

[70] "For a special purpose."

[71] Heraclitus of Ephesus, the weeping philosopher, 6th century B. C.

[72] Democritus, the laughing philosopher, founder of the atomistic theory,
5th century B. C.

[73] "Ends to which."

[74] "Ends from which."

[75] "In just as many syllables," "With just as many letters," "In just as
many words."

[76] "I shall make a way," "I shall find a way."

[77] The notion that the Evil Spirit is a functionary liable to be
dismissed for not attending to his duty, is, so far as my reading goes,
utterly unknown in theology. My first wrinkle on the subject was the remark
of the Somersetshire farmer upon Palmer the poisoner-- "Well! if the Devil
don't take he, he didn't ought to be allowed to be devil no longer."--A. De

William Palmer (1824-1856) was a member of the Royal College of Surgeons
and practised medicine at London. He was hanged in 1856 for having poisoned
a friend and was also suspected of having poisoned his wife and brother for
their insurance money, besides being guilty of numerous other murders. His
trial was very much in the public attention at the time.

[78] Advantages and dangers.

[79] The old priory of St. Mary of Bethlehem in London, was used as an
asylum for the insane. The name was corrupted to Bedlam.

[80] Referring to the common English pronunciation of St. John, almost
Sinjin. John St. John Long (1798-1834), an Irishman by birth, practised
medicine in London. He claimed to have found a specific for rheumatism and
tuberculosis, but upon the death of one of his patients in 1830 he was
tried for manslaughter. He died of tuberculosis four years later, refusing
to take his own treatment.

[81] William of Occam (d. 1349), so called from his birthplace, Ockham, in
Surrey. He was a Franciscan, and lectured on philosophy in the Sorbonne.

[82] He signs himself "James Hopkins, schoolmaster," and this seems to have
been his only published effort.

[83] Joseph Ady (1770-1852) was a famous swindler. One of his best-known
schemes was to send out letters informing the recipients that they would
learn something to their advantage on payment of a certain sum. He spent
some time in prison.

[84] Sir Peter Laurie (c. 1779-1861) was worth referring to, for he was
prominent as a magistrate and was honored because of his interest in all
social reforms. He made a fortune as a contractor, became sheriff of London
in 1823, and was knighted in the following year. He became Lord Mayor of
London in 1832.

[85] See Vol. I, page 321, note 2 {691}. The _Astronomy in a nutshell_
appeared in 1860. _The Herald of Astrology_ was first published in London
in 1831, "by Zadkiel the Seer." It was continued as _The Astrological
Almanac_ (London, 1834), as _Zadkiel's Almanac and Herald of Astrology_
(_ibid._, 1835, edited by R. J. Morrison, and subsequently by A. J.
Pearce), and as _Raphael's Prophetic Almanac_ (1840-1855).

[86] See Vol. I, page 172, note 3 {382}.

[87] See Vol. I, page 87, note 4 {133}.

[88] Franz Xaver, Freiherr von Zach (1754-1832) was director of the
observatory at Seeberge near Gotha. He wrote the _Tabulae speciales
aberrationis et mutationis_ (1806-7), _Novae et correctae tabulae solis_
(1792), and _L'attraction des montagnes et ses effets sur le fil à plomb_

[89] Jean Louis Pons (1761-1831) was connected with the observatory at
Marseilles for thirty years (1789-1819). He later became director of the
observatory at Marlia, near Lucca, and subsequently filled the same office
at Florence. He was an indefatigable searcher for comets, discovering 37
between 1801 and 1827, among them being the one that bears Encke's name.

[90] This hypothesis has now become an established fact.

[91] John Chetwode Eustace (c. 1762-1815) was born in Ireland. Although a
Roman Catholic priest he lived for a time at Cambridge where he did some
tutoring. His _Classical Tour_ appeared in 1813 and went through several

[92] "Crimes should be exposed when they are punished, but disgraceful acts
should be hidden."

[93] Henri Hureau de Sénarmont (1808-1862) was professor of mineralogy at
the _Ecole des mines_ and examiner at the _Ecole polytechnique_ at Paris.

[94] Augustin Jean Fresnel (1788-1827), "Ingenieur des ponts et chaussées,"
gave the first experimental proofs of the wave theory of light. He studied
the questions of interference and polarization, and determined the
approximate velocity of light.

[95] "As is my custom."

[96] Francis Heywood (1796-1858) made the first English translation of
Kant's _Critick of Pure Reason_ (1838, reprinted in 1848). The _Analysis_
came out, as here stated, in 1844.

[97] Louise Renée de Keroualle, Duchess of Portsmouth and Aubigny
(1649-1734), was a favorite of Charles II. She used her influence to keep
him under the control of Louis XIV.

[98] William Chiffinch (c. 1602-1688) was page of the king's bed-chamber
and keeper of the private closet to Charles II. He was one of the king's
intimates and was an unscrupulous henchman.

[99] "Well devised."

[100] "John Bellingham Inglis. His _Philobiblion_ "translated from the
first edition (of Ricardus d'Aungervile, Bishop of Durham), 1473," appeared
at London in 1832. It was republished in America (Albany, N. Y.) in 1864.

[101] "What are you laughing at?"

[102] See Vol. I, page 314, note 4 {681}.

[103] See Vol. I, page 112, note 7 {211}.

[104] Referring to Hamilton's edition of the _Collected Works of Dugald
Stewart_, 10 volumes, Edinburgh, 1854-58. It is not commonly remembered
that Stewart (1753-1828) taught mathematics at the University of Edinburgh
before he took up philosophy.

[105] This was Hamilton's edition of the _Works of Thomas Reid_ (2 vols.,
Edinburgh, 1846-1863). Reid (1710-1796) included mathematics in his work in
philosophy at Aberdeen. In 1764 he succeeded Adam Smith at Glasgow.

[106] Edward Irving (1792-1834), the famous preacher. At first he assisted
Dr. Chalmers at Glasgow, but in 1822 he went to London where he met with
great success. A few years later he became mentally unbalanced and was
finally expelled from his church (1832) for heresy. He was a great friend
of Carlyle.

[107] He also wrote a number of other paradoxes, including _An Essay
towards a Science of Consciousness_ (1838), _Instinctive Natural Religion_
(1858), _Popular Treatise on the structure, diseases, and treatment of the
human teeth_ (1837), and _On Headache_ (1859).

[108] James Smith (1801-1857), known as Shepherd Smith, was a socialist and
a mystic, with a philosophy that was wittily described as "Oriental
pantheism translated into Scotch." He was editor of several journals.

[109] Joanna Southcott (1750-1814) was known for her rhyming prophecies in
which she announced herself as the woman spoken of in Revelations xii. She
had at one time as many as 100,000 disciples, and she established a sect
that long survived her.

[110] Thales, c. 640-548 B. C.

[111] Pythagoras, 580-501 B. C.

[112] Anaxagoras, 499-428 B. C., the last of the Ionian school, teacher of
Euripides and Pericles. Plutarch speaks of him as having squared the

[113] Oinopides of Chios, contemporary of Anaxagoras. Proclus tells us that
Oinopides was the first to show how to let fall a perpendicular to a line
from an external point.

[114] Bryson and Antiphon, contemporaries of Socrates, invented the
so-called method of exhaustions, one of the forerunners of the calculus.

[115] He wrote, c. 440 B. C., the first elementary textbook on mathematics
in the Greek language. The "lunes of Hippocrates" are well known in

[116] Jabir ben Aflah. He lived c. 1085, at Seville, and wrote on astronomy
and spherical trigonometry. The _Gebri filii Affla Hispalensis de
astronomia libri_ IX was published at Nuremberg in 1533.

[117] Hieronymus Cardanus, or Girolamo Cardano (1501-1576), the great
algebraist. His _Artis magnae sive de regulis Algebrae_ was published at
Nuremberg in 1545.

[118] Nicolo Tartaglia (c. 1500-1557), the great rival of Cardan.

[119] See note 5 {98}, Vol. I, page 69.

[120] See note 10 {124}, Vol. I., page 83.

[121] See note 9 {123}, Vol. I, page 83.

[122] Pierre Hérigone lived in Paris the first half of the 17th century.
His _Cours mathématique_ (6 vols., 1634-1644) had some standing but was not
at all original.

[123] Franciscus van Schooten (died in 1661) was professor of mathematics
at Leyden. He edited Descartes's _La Géométrie_.

[124] Florimond de Beaune (1601-1652) was the first Frenchman to write a
commentary on Descartes's _La Géométrie_. He did some noteworthy work in
the theory of curves.

[125] See note 3 {23}, Vol. I, page 41.

[126] Olivier de Serres (b. in 1539) was a writer on agriculture. Montucla
speaks of him in his _Quadrature du cercle_ (page 227) as having asserted
that the circle is twice the inscribed equilateral triangle, although, as
De Morgan points out, this did not fairly interpret his position.

[127] Angherà wrote not only the three works here mentioned, but also the
_Problemi del più alto interesse scientifico, geometricamente risoluti e
dimostrati_, Naples, 1861. His quadrature was defended by Giovanni Motti in
a work entitled _Matematica Vera. Falsità del sistema ciclometrico
d'Archimede, quadratura del cerchio d'Angherà, ricerca algebraica dei lati
di qualunque poligono regolare inscritto in un circolo_, Voghera, 1877. The
_Problemi_ of 1861 contains Angherà's portrait, and states that he lived at
Malta from 1849 to 1861. It further states that the Malta publications are
in part reproduced in this work.

[128] This was his friend Paolo Pullicino whose _Elogio_ was pronounced by
L. Farrugia at Malta in 1890. He wrote a work _La Santa Effegie della Blata
Vergine Maria_, published at Valetta in 1868.

[129] St. Vitus, St. Modestus, and St. Crescentia were all martyred the
same day, being torn limb from limb after lions and molten lead had proved
of no avail. At least so the story runs.

[130] The reference is to Cardinal Wiseman. See Vol. II, page 26, note 56.

[131] "Worthy of esteem."

[132] Pedro de Ribadeneira (Ribadeneyra, Rivadeneira), was born at Toledo
in 1526 and died in 1611. He held high position in the Jesuit order. The
work referred to is the _Flos Sanctorum o libro de las vidas de los
santos_, of which there was an edition at Barcelona in 1643. His life of
Loyola (1572) and _Historia ecclesiástica del Cisma del reino de
Inglaterra_ were well known.

[133] Cæsar Baronius (1538-1607) was made a cardinal in 1595 and became
librarian at the Vatican in 1597. The work referred to appeared at Rome in

[134] Mrs. Jameson's (1794-1860) works were very popular half a century
ago, and still have some circulation among art lovers. The first edition of
the work mentioned appeared in 1848.

[135] The barnyard cock.

[136] Shanks did nothing but computing. The title should, of course, read
"to 607 Places of _Decimals_." He later carried the computation to 707
decimal places. (_Proc. Roy. Society_, XXI, p. 319.) He also prepared a
table of prime numbers up to 60,000. (_Proc. Roy. Society_, XXII, p. 200.)

[137] See Vol. I, page 42, note 4 {24}.

[138] See Vol. I, page 64, note 1 {78}.

[139] See Vol. I, page 328, note 1 {704}.

[140] George Suffield published _Synthetic Division in Arithmetic_, to
which reference is made, in 1863.

[141] John Robert Lunn wrote chiefly on Church matters, although he
published a work on motion in 1859.

[142] Jean Baptiste Joseph, Baron Fourier (1768-1830), sometime professor
in the Military School at Paris, and later at the _Ecole polytechnique_. He
is best known by his _Théorie analytique de la chaleur_ (Paris, 1822), in
which the Fourier series is used. The work here referred to is the _Analyse
des équations déterminées_ (Paris, 1831).

[143] William George Horner (1786-1837) acquired a name for himself in
mathematics in a curious manner. He was not a university man nor was he a
mathematician of any standing. He taught school near Bristol and at Bath,
and seems to have stumbled upon his ingenious method for finding the
approximate roots of numerical higher equations, including as a special
case the extracting of the various roots of numbers. Davies Gilbert
presented the method to the Royal Society in 1819, and it was reprinted in
the _Ladies' Diary_ for 1838, and in the _Mathematician_ in 1843. The
method was original as far as Horner was concerned, but it is practically
identical with the one used by the Chinese algebraist Ch'in Chiu-shang, in
his _Su-shu Chiu-chang_ of 1247. But even Ch'in Chiu-shang can hardly be
called the discoverer of the method since it is merely the extension of a
process for root extracting that appeared in the _Chiu-chang Suan-shu_ of
the second century B. C.

[144] He afterwards edited Loftus's _Inland Revenue Officers' Manual_
(London, 1865). The two equations mentioned were x^3 - 2x = 5 and y^3 -
90y^2 + 2500y - 16,000 = 0, in which y = 30 - 10x. Hence each place of y is
the complement of the following place of x with respect to 9.

[145] Probably the John Power Hicks who wrote a memoir on T. H. Key,
London, 1893.

[146] Possibly the one who wrote on the quadrature of the circle in 1881.

[147] As it is. But what a pity that we have not 12 fingers, with
duodecimal fractions instead of decimals! We should then have 0.6 for ½,
0.4 for 1/3, 0.8 for 2/3, 0.3 for ¼, 0.9 for ¾, and 0.16 for 1/8, instead
of 0.5, 0.333+, 0.666+, 0.25, 0.75, and 0.125 as we now have with our
decimal system. In other words, the most frequently used fractions in
business would be much more easily represented on the duodecimal scale than
on the decimal scale that we now use.

[148] He wrote Hints for an _Essay on Anemology and Ombrology_ (London,
1839-40) and _The Music of the Eye_ (London, 1831).

[149] Brigham Young (1801-1877) was born at Whitingham, Vermont, and
entered the Mormon church in 1832. In 1840 he was sent as a missionary to
England. After the death of Joseph Smith he became president of the Mormons
(1847), leading the church to Salt Lake City (1848).

[150] Joseph Smith (1805-1844) was also born in Vermont, and was four years
the junior of Brigham Young. The _Book of Mormon_ appeared in 1827, and the
church was founded in 1830. He was murdered in 1844.

[151] Orson Pratt (1811-1881) was one of the twelve apostles of the Mormon
Church (1835), and made several missionary journeys to England. He was
professor of mathematics in the University of Deseret (the Mormon name for
Utah). Besides the paper mentioned Pratt wrote the _Divine Authenticity of
the Book of Mormon_ (1849), _Cubic and Biquadratic Equations_ (1866), and a
_Key to the Universe_ (1866).

[152] "It does not follow."

[153] Dryden (1631-1700) published his _Religio Laici_ in 1682. The use of
the word "proportion" in the sense of ratio was common before his time, but
he uses it in the sense of having four terms; that is, that price is to
price as offence is to offence.

[154] Olinthus Gilbert Gregory (1774-1841) succeeded Hutton as professor of
mathematics at Woolwich. He was, with De Morgan, much interested in
founding the University of London. He wrote on astronomy (1793), mechanics
(1806), practical mathematics (1825), and Christian evidences (1811).

[155] See Vol. I, page 220, note 6 {482}. The _Pensées_ appeared
posthumously in 1670.

[156] "The right thing to do is not to wager at all." "Yes, but you ought
to wager; you have started out; and not to wager at all that God exists is
to wager that he does not exist."

[157] He lived about 300 A.D., in Africa, and wrote _Libri septem adversus
Gentes_. This was printed at Rome in 1542-3.

[158] Pierre Bayle (1647-1706) was professor of philosophy at the
Prostestant University at Sedan from 1675 until its dissolution in 1681. He
then became professor at Rotterdam (1681-1693). In 1684 he began the
publication of his journal of literary criticism _Nouvelles de la
République des Lettres_. He is best known for his erudite _Dictionnaire
historique et critique_ (1697).

[159] "But Christ himself does not prove what he promises. It is true. For,
as I have said, there cannot be any absolute proof of future events.
Therefore since it is a condition of future events that they cannot be
grasped or comprehended by any efforts of anticipation, is it not more
reasonable, out of two alternatives that are uncertain and that are hanging
in doubtful expectation, to give credence to the one that gives some hope
rather than to the one that offers none at all? For in the former case
there is no danger if, as is said to threaten, it becomes empty and void;
while in the latter case the danger is greatest, that is, the loss of
salvation, if when the time comes it is found that it was not a falsehood."

[160] Gregg wrote several other paradoxes, including the following: _The
Authentic Report of the extraordinary case of Tresham Dames Gregg ... his
committal to Bridewell for refusing to give his recognizance_ (Dublin,
1841), _An Appeal to Public Opinion upon a Case of Injury and Wrong ... in
the case of a question of prerogative that arose between_ [R. Whately] _...
Archbishop of Dublin and the author_ (London, 1861), _The Cosmology of Sir
Isaac Newton proved to be in accordance with the Bible_ (London, 1871),
_The Steam Locomotive as revealed in the Bible_ (London 1863) and _On the
Sacred Law of 1866, conferring perpetual life with immunity from decay and
disease. A cento of decisive scriptural oracles strangely discovered_
(London and Dublin, 1875). These titles will help the reader to understand
the man whom De Morgan so pleasantly satirizes.

[161] See Vol. I, page 261, note 2 {592}.

[162] "They have found it."

[163] The late Greeks used the letters of their alphabet as numerals,
adding three early alphabetic characters. The letter [chi] represented 600,
[xi] represented 60, and [digamma] stood for 6. This gives 666, the number
of the Beast given in the Revelation.

[164] "Allowing for necessary exceptions."

[165] Mr. Gregg is not alone in his efforts to use the calculus in original
lines, as any one who has read Herbart's application of the subject to
psychology will recall.

[166] See Vol. I, page 105, note 4 {188}; page 109, note 1 {197}.

[167] The full title shows the plan,--_The Decimal System as a whole, in
its relation to time, measure, weight, capacity, and money, in unison with
each other._ But why is this so much worse than the French plan of which we
have only the metric system and the decimal division of the angle left?

[168] One of the brothers of Sir Isaac Pitman (1813-1897), the inventor of
modern stenography. Of these brothers, Benjamin taught the art in America,
Jacob in Australia, and Joseph, Henry, and Frederick in England.

[169] For example, _The Phonographic Lecturer_ (London, 1871 etc.), _The
Phonographic Student_ (1867, etc.), and _The Shorthand Magazine_ (1866,

[170] See Vol. II, page 68, note 148.

[171] It involves the theory of non-Euclidean geometry, Euclid's postulate
of parallels being used in proving this theorem.

[172] Referring to the fact that none of the works of Thales is extant.

[173] The author was one B. Bulstrode. Parts 4 and 5 were printed at

[174] See Vol. II, page 5, note 18.

[175] See Vol. I, page 85, note 2 {129}.

[176] Alexander Vasilievich Suvaroff (1729-1800), a Russian general who
fought against the Turks, in the Polish wars, and in the early Napoleonic
campaigns. When he took Ismail in 1790 he sent this couplet to Empress

[177] "Newton hath determined rightly," "Newton hath not determined

[178] See Vol. I, page 288, note 3 {621}.

[179] See Vol. I, page 326, note 1 {700}.

[180] "With great honor."

[181] Apparently unknown to biographers. He seems to have written nothing

[182] Captain Marryat (1792-1848) published _Snarley-yow, or the Dog Fiend_
in 1837.

[183] He is not known to biographers, and published nothing else under this

[184] See Vol. I, page 80, note 5 {119}.

[185] He published a _Family and Commercial Illustrated Almanack and Year
Book ... for 1861_ (Bath, 1860).

[186] Louis Dutens (1730-1812) was born at Tours, but went to England as a
young man. He made the first collection of the works of Leibnitz, against
the advice of Voltaire, who wrote to him: "Les écrits de Leibnitz sont
épars comme les feuilles de la Sybille, et aussi obscurs que les écrits de
cette vieille." The work appeared at Geneva, in six volumes, in 1769.

[187] Mungo Park (1771-1806), the first European to explore the Niger
(1795-6). His _Travels in the Interior of Africa_ appeared in 1799. He died
in Africa.

[188] Gerhard Mercator (1512-1594) the well-known map maker of Louvain. The
"Mercator's Projection" was probably made as early as 1550, but the
principle of its construction was first set forth by Edward Wright (London,

[189] Quirico Barilli Filopanti wrote a number of works and monographs. He
succeeded in getting his _Cesare al Rubicone_ and _Degli_ _usi idraulici
della Tela_ in the _Memoria letta ... all' Accademia delle Scienze in
Bologna_ (1847, 1866). He also wrote _Dio esiste_ (1881), _Dio Liberale_
(1880), and _Sunto della memoria sulle geuranie ossia di alcune singolari
relazioni cosmiche della terra e del cielo_ (1862).

[190] The periods of disembodiment may be interesting. They will be seen
from the following dates: Descartes (1596-1650), William III (1650-1702);
Roger Bacon (1214 to c. 1294), Boccaccio (1313-1375). Charles IX was born
in 1550 and died in 1574.

[191] His real name was Frederick Parker, and he wrote several works on the
Greek language and on religion. Among these were a translation of the New
Testament from the Vatican MS. (1864), _The Revealed History of Man_
(1854), _An Enquiry respecting the Punctuation of Ancient Greek_ (1841),
and _Rules for Ascertaining the sense conveyed in Ancient Greek
Manuscripts_ (1848, the seventh edition appearing in 1862).

[192] See Vol. I, page 352, second note 1 {736}.

The literature on the subject of the Great Pyramid, considered from the
standpoint of metrology, is extensive.

[193] See Vol. I, page 80, note 5 {119}.

[194] Sir Philip Francis (1740-1818) was a Whig politician. The evidence
that he was the author of the _Letters of Junius_ (1769-1772) is purely
circumstantial. He was clerk in the war office at the time of their
publication. In 1774 he was made a member of the Supreme Council of Bengal,
and was a vigorous opponent of Warren Hastings, the two fighting a duel in
1780. He entered parliament in 1784 and was among the leaders in the
agitation for parliamentary reform.

[195] Mrs. Cottle published a number of letters that attracted attention at
the time. Among these were letters to the emperor of France and king of
Sardinia (1859) relating to the prophecies of the war between France and
Austria; to G. C. Lavis and Her Majesty's Ministers (1859) relating to her
claims as a prophetess; and to the "Crowned Heads" at St. James, the King
of Prussia, and others (1860), relating to certain passages of Scripture.
She also wrote _The Lamb's Book of Life for the New Jerusalem Church and
Kingdom, interpreted for all nations_ (1861).

[196] See Vol. I, page 315, note 2 {685}, and Vol. II, page 58, note 109.

[197] A Congregational minister, who published a number of sermons, chiefly
obituaries, between 1804 and 1851. His _Frailty of Human Life_, two sermons
delivered on the occasion of the death of Princess Charlotte, went through
at least three editions.

[198] He was secretary of the Congregational Board and editor of the
_Congregational Year Book_ (from 1846) and the _Congregational Manual_.

[199] Frederick Denison Maurice (1805-1872) began his preaching as a
Unitarian but entered the Established Church in 1831, being ordained in
1834. He was professor of English and History at King's College, London,
from 1840 to 1853. He was one of the founders of Queen's College for women,
and was the first principal of the Working Men's College, London. The
subject referred to by De Morgan is his expression of opinion in his
_Theological Essays_ (1853) that future punishment is not eternal. As a
result of this expression he lost his professorship at King's College. In
1866 he was made Knightbridge Professor of Casuistry, Moral Theology, and
Moral Philosophy at Cambridge.

[200] See Vol. I, page 46, note 1 {42}. Besides the books mentioned in
this list he wrote _The Ratio between Diameter and Circumference
demonstrated by angles, and Euclid's Theorem, Proposition 32, Book I,
proved to be fallacious_ (Liverpool, 1870). This is the theorem which
asserts that the exterior angle of a triangle is equal to the sum of the
two opposite interior angles, and that the sum of the interior angles
equals two right angles. He also published his _Curiosities of Mathematics_
in 1870, a work containing an extensive correspondence with every one who
would pay any attention to him. De Morgan was then too feeble to show any
interest in the final effort of the subject of some of his keenest satire.

[201] See Vol. I, page 332, note 4 {709}.

[202] See Vol. I, page 101, note 4 {174}.

[203] "The circle-squaring disease"; literally, "the circle-measuring

[204] See Vol. II, page 63, note 136.

[205] William Rutherford (c. 1798-1871), teacher of mathematics at
Woolwich, secretary of the Royal Astronomical Society, editor of _The
Mathematician_, and author of various textbooks. _The Extension of [pi] to
440 places_, appeared in the _Proceedings_ of the Royal Society in 1853 (p.

[206] Charles Knight (1791-1873) was associated with De Morgan for many
years. After 1828 he superintended the publications of the Society for the
Diffusion of Useful Knowledge, to which De Morgan contributed, and he
edited the _Penny Cyclopedia_ (1833-1844) for which De Morgan wrote the
articles on mathematics.

[207] Sir William Hamilton. See Vol. I, page 112, note 7 {211}.

[208] Adam Smith (1723-1790) was not only known for his _Wealth of Nations_
(1776), but for his _Theory of Moral Sentiments_ (1759), published while he
was professor of moral philosophy at Glasgow (1752-1764). He was Lord
Rector of the university in 1787.

[209] See Vol. I, page 332, note 4 {709}.

[210] "Whip."

[211] "Terrible lash."

[212] "An accomplished fact [an accomplished fault]."

[213] See _Extracts from the Diary and Letters of Mrs. Mary Cobb_, London,

[214] "Gentle in manner."

[215] "Brave in action." The motto of Earl Newborough was "Suaviter in
modo, fortiter in re."

[216] "Reduction to an absurdity," a method of proof occasionally used in
geometry and in logic.

[217] "He has lost the right of being moved (struck) by evidence."

[218] For _radix quadratus_. The usual root sign is supposed to be derived
from _r_ (for radix), and at one time _q_ was commonly used for square, as
in Viète's style of writing Aq for A^2.

[219] The Garde Douloureuse was a castle in the marches of Wales and
received its name because of its exposure to attacks by the Welsh.

[220] "Out of the fight."

[221] "Hidden."

[222] John Cam Hobhouse (1786-1869), Baron Broughton, was committed to
Newgate for two months in 1819 for his anonymous pamphlet, _A Trifling
Mistake_. This was a great advertisement for him, and upon his release he
was at once elected to parliament for Westminster. He was a strong
supporter of all reform measures, and was Secretary for War in 1832. He was
created Baron Broughton de Gyfford in 1851.

[223] Thomas Erskine (1750-1823), the famous orator. He became Lord
Chancellor in 1806, but sat in the House of Commons most of his life.

[224] The above is explained in the MS. by a paragraph referring to some
anagrams, in one of which, by help of the orthography suggested, a
designation for this cyclometer was obtained from the letters of his
name.--S. E. De M.

[225] "A personal verb agrees with its subject."

[226] See Vol. I, page 326, note 1 {700}.

[227] See Vol. I, page 326, note 2 {701}.

[228] Apparently unknown to biographers.

[229] The _Bibliotheca Mathematica_ of Ludwig Adolph Sohncke (1807-1853),
professor of mathematics at Königsberg and Halle, covered the period from
1830 to 1854, being completed by W. Engelmann. It appeared in 1854.

[230] See Vol. I, page 392, note 2 {805}.

[231] See Vol. I, page 43, note 7 {32}.

[232] See Vol. II, page 91, note 187.

[233] Mason made a notable balloon trip from London to Weilburg, in the
Duchy of Nassau, in November, 1836, covering 500 miles in 18 hours. He
published an account of this trip in 1837, and a work entitled
_Aeronautica_ in 1838.

[234] William Harrison Ainsworth (1805-1885) the novelist.

[235] On this question see Vol. I, page 326, note 2 {701}.

[236] Major General Alfred Wilks Drayson, author of various works on
geology, astronomy, military surveying, and adventure.

[237] Hailes also wrote several other paradoxes on astronomy and circle
squaring during the period 1843-1872.

[238] See Vol. I, page 43, note 8 {33}.

[239] See Vol. I, page 43, note 7 {32}.

[240] "Very small errors are not to be condemned."

[241] He seems to have written nothing else.

[242] Besides the paradoxes here mentioned by De Morgan he wrote several
other works, including the following: _Abriss der Babylonisch-Assyrischen
Geschichte_ (Mannheim, 1854), _A Popular Inquiry into the Moon's rotation
on her axis_ (London, 1856), _Practical Tables for the reduction of the
Mahometan dates to the Christian kalendar_ (London, 1856), _Grundzüge einer
neuen Weltlehre_ (Munich, 1860), and _On the historical Antiquity of the
People of Egypt_ (London, 1863).

[243] Dircks (1806-1873) was a civil engineer of prominence, and a member
of the British Association and the Royal Society of Edinburgh. He wrote
(1863) on "Pepper's Ghost," an ingenious optical illusion invented by him.
There was a second edition of the _Perpetuum Mobile_ in 1870.

[244] George Stephenson (1781-1848), the inventor of the first successful
steam locomotive. His first engine was tried in 1814.

[245] Robert Stephenson (1803-1859), the only son of George. Most of the
early improvements in locomotive manufacture were due to him. He was also
well known for his construction of great bridges.

[246] "In its proper place."

[247] "A fool always finds a bigger fool to admire him."

[248] See Vol. I, page 43, note 7 {32}.

[249] See Vol. I, page 43, note 8 {33}.

[250] See Vol. I, page 85, note 2 {129}.

[251] See Vol. I, page 390, note 1 {390}.

[252] From 1823 to 1852 it was edited by I. C. Robertson; from 1852 to 1857
by R. A. Brooman; and from 1857 to 1863 by Brooman and E. J. Reed.

[253] Sir James Ivory (1765-1842) was, as a young man, manager of a flax
mill in Scotland. In 1804 he was made professor of mathematics at the Royal
Military College, then at Marlow and later at Sandhurst. He was deeply
interested in mathematical physics, and there is a theorem on the
attraction of ellipsoids that bears his name. He was awarded three medals
of the Royal Society, and was knighted together with Herschel and Brewster,
in 1831.

[254] See Vol. I, page 56, note 1 {64}.

[255] See Vol. I, page 153, note 5 {338}.

[256] See Vol. I, page 309, note 2 {670}.

[257] See Vol. I, page 87, note 4 {133}.

[258] George Canning (1770-1857), the Tory statesman and friend of Scott,
was much interested in founding the _Quarterly Review_ (1808) and was a
contributor to its pages.

[259] See Vol. I, page 186, note 14 {418}.

[260] See Vol. II, page 141, note 252.

[261] De Morgan had a number of excellent articles in this publication.

[262] See Vol. I, page 279, note 1 {611}.

[263] James Orchard Halliwell (1820-1889), afterwards Halliwell-Phillips,
came into prominence as a writer at an early age. When he was seventeen he
wrote a series of lives of mathematicians for the _Parthenon_. His _Rara
Mathematica_ appeared when he was but nineteen. He was a great bibliophile
and an enthusiastic student of Shakespeare.

[264] This was written at the age of twenty-two.

[265] The subject of this criticism is of long past date, and as it has
only been introduced by the author as an instance of faulty editorship, I
have omitted the name of the writer of the libel, and a few lines of
further detail.--S. E. De M.

[266] "Condemned souls."

[267] The editor of the _Mechanics' Magazine_ died soon after the above was
written.--S. E. De M.

[268] Thomas Stephens Davies (1795-1851) was mathematical master at
Woolwich and F. R. S. He contributed a series of "Geometrical Notes" to the
_Mechanics' Magazine_ and edited the _Mathematician_. He also published a
number of text-books.

[269] See Vol. II, page 66, note 143.

[270] The _Dictionary of Greek and Roman Biography_ (1849), edited by Sir
William Smith (1813-1893), whose other dictionaries on classical and
biblical matters are well known.

[271] "O J. S.! This is the worst! the greatest possible injury!"

[272] See Vol. I, page 44, note 9 {34} and page 110, note 5 {201}.


 "If there's a man whom the judge's pitiless sentence awaiteth,
  His head condemned to penalties and tribulations,
  Let neither penitentiaries tire him with laborer's burdens
  Nor let his stiffened hands be harrassed by work in the mines.
  He must square the circle! For what else do I care?--all
  Known punishments this one task hath surely included."

[274] Houlston was in the customs service. He also published _Inklings of
Areal Autometry_, London, 1874.

[275] This is Frederick C. Bakewell. He had already published _Natural
Evidence of a Future Life_ (London, 1835), _Philosophical Conversations_
(London, 1833, with other editions), and _Electric Science_ (London, 1853,
with other editions).

[276] Henry F. A. Pratt had already published _A Dissertation on the power
of the intercepted pressure of the Atmosphere_ (London, 1844) and _The
Genealogy of Creation_ (1861). Later he published a work _On Orbital
Motion_ (1863), and _Astronomical Investigations_ (1865).

[277] See Vol. I, page 260, note 1 {591}.

[278] Thomas Rawson Birks (1810-1883), a theologian and controversialist,
fellow of Trinity College, Cambridge, and (1872) professor of moral
philosophy in that university. He wrote _Modern Rationalism_ (1853), _The
Bible and Modern Thought_ (1861), _The First Principles of Moral Science_
(1873), and _Modern Physical Fatalism and the Doctrine of Evolution_
(1876), the last being an attack on Herbert Spencer's _First Principles_.

[279] Pseudonym for William Thorn. In the following year (1863) he
published a second work, _The Thorn-Tree: being a History of Thorn
Worship_, a reply to Bishop Colenso's work entitled _The Pentateuch and the
Book of Joshua critically examined_.

[280] Besides _The Pestilence_ (1866) he published _The True Church_
(1851), _The Church and her destinies_ (1855), _Religious reformation
imperatively demanded_ (1864), and _The Bible plan unfolded_ (second
edition, 1872).

[281] See Vol. II, page 97, note 195.

[282] Sir George Cornewall Lewis (1806-1863) also wrote an _Essay on the
Origin and Formation of the Romance Languages_ (1835), an _Essay on the
Government of Dependencies_ (1841), and an _Essay on Foreign Jurisdiction
and the Extradition of Criminals_ (1859). He was Chancellor of the
Exchequer in 1855 and Home Secretary in 1859.

[283] Henry Malden (1800-1876), a classical scholar, fellow of Trinity
College, Cambridge, and professor of Greek at University College
(1831-1876), then (1831) the University of London. He wrote a _History of
Rome to 390 B. C._ (1830), and _On the Origin of Universities and
Academical Degrees_ (1835).

[284] Henry Longueville Mansel (1820-1871), theologian and metaphysician,
reader in theology at Magdalen College, Oxford (1855), and professor of
ecclesiastical history and Dean of St. Paul's (1866). He wrote on
metaphysics, and his Bampton Lectures (1858) were reprinted several times.

[285] "Hejus gave freely, gave freely. God is propitious, God is favorable
to him who gives freely. God is honored with a banquet of eggs at the cross
roads, the god of the world. God, with benignant spirit, desired in
sacrifice a goat, a bull to be carried within the precincts of the holy
place. God, twice propitiated, blesses the pit of the sacred libation."

[286] Eudoxus of Cnidus (408-355 B. C.) had much to do with the early
scientific astronomy of the Greeks. The fifth book of Euclid is generally
attributed to him. His astronomical works are known chiefly through the
poetical version of Aratus mentioned in note 13, page 167.

[287] Simplicius, a native of Cilicia, lived in the 6th century of our era.
He was driven from Athens by Justinian and went to Persia (531), but he
returned later and had some fame as a teacher.

[288] See Vol. I, page 160, note 3 {348}.

[289] See Vol. I, page 76, note 3 {112}.

[290] "Through right and wrong."

[291] "It is therefore to arrive at this parallelism, or to preserve it,
that Copernicus feared to be obliged to have recourse to this equal and
opposite movement which destroys the effect which he attributed so freely
to the first, of deranging the parallelism."

[292] A contemporary of Plato and a disciple of Aristotle.

[293] Meton's solstice, the beginning of the Metonic cycles, has been
placed at 432 B. C. Ptolemy states that he made the length of the year 365¼
+ 1/72 days.

[294] Aratus lived about 270 B. C., at the court of Antigonus of Macedonia,
and probably practiced medicine there. He was the author of two
astronomical poems, the [Greek: Phainomena], apparently based on the lost
work of Eudoxus, and the [Greek: Diosêeia] based on Aristotle's
_Meteorologica_ and _De Signis Ventorum_ of Theophrastus.

[295] "The nineteen (-year) cycle of the shining sun."

[296] Claudius Salmasius (1588-1653), or Claude Saumaise, was a
distinguished classicist, and professor at the University of Leyden. The
word [Greek: êleioio] means Elian, thus making the phrase refer to the
brilliant one of Elis.

[297] Sir William Brown (1784-1864). In 1800 the family moved to Baltimore,
and there the father, Alexander Brown, became prominent in the linen trade.
William went to Liverpool where he acquired great wealth as a merchant and
banker. He was made a baronet in 1863.

[298] Robert Lowe (1811-1892), viscount Sherbrooke, was a fellow of
Magdalen College, Oxford (1835). He went to Australia in 1842 and was very
successful at the bar. He returned to England in 1850 and became leader
writer on the _Times_. He was many years in parliament, and in 1880 was
raised to the peerage.

[299] See Vol. I, page 42, note 4 {24}.

[300] Francis Walkingame (fl. about 1751-1785), whose _Tutor's Assistant_
went through many editions from 1751-1854.

[301] Davies Gilbert (1767-1839). His family name was Giddy, but he assumed
his wife's name. He sat in parliament from 1806 to 1832. In 1819 he secured
the establishment of the Cape of Good Hope observatory. He was Treasurer
(1820-1827) and President (1827-1830) of the Royal Society.

[302] See Vol. I, page 55, note 2 {63}.

[303] Sir Jonathan Frederick Pollock (1783-1870) entered parliament in 1831
and was knighted in 1834.

[304] Joseph Hume (1777-1855) entered parliament in 1812 and for thirty
years was leader of the Radical party.

[305] "What! when I say, 'Nicole, bring me my slippers,' is that prose?"

[306] Captain Basil Hall (1788-1844), a naval officer, carried on a series
of pendulum observations in 1820-1822, while on a cruise of the west coast
of North America. The results were published in 1823 in the _Philosophical
Transactions_. He also wrote two popular works on travel that went through
numerous editions.

[307] Anthony Ashley Cooper (1801-1885), Earl of Shaftesbury. His name is
connected with philanthropic work and factory legislation.

[308] See Vol. I, page 207, note 12 {469}.

[309] See Vol. I, page 80, note 5 {119}.

[310] Sir Thomas Maclear (1794-1879), an Irishman by birth, became
Astronomer Royal at the Cape of Good Hope in 1833. He was an indefatigable
observer. He was knighted in 1860.

[311] Thomas Romney Robinson (1792-1882), another Irish astronomer of
prominence. He was a deputy professor at Trinity College, Dublin, but took
charge of the Armagh observatory in 1823 and remained there until his

[312] Sir James South (1785-1867) was in early life a surgeon, but gave up
his practice in 1816 and fitted up a private observatory. He contributed to
the science of astronomy, particularly with respect to the study of double

[313] Sir John Wrottesley (1798-1867), second Baron Wrottesley. Like Sir
James South, he took up the study of astronomy after a professional
career,--in his case in law. He built a private observatory in 1829 and
made a long series of observations, publishing three star catalogues. He
was president of the Astronomical Society from 1841 to 1843, and of the
Royal Society from 1854 to 1857.

[314] He seems to have written nothing else.

[315] See Vol. II, page 68, note 147.

[316] "The wills are free, and I wish neither the one nor the other."

[317] "The force of inertia conquered."

[318] Reddie also wrote _The Mechanics of the Heavens_, referred to later
in this work. He must not be confused with Judge James Reddie (1773-1852),
of Glasgow, who wrote on international law, although this is done in the
printed edition of the British Museum catalogue, for he is mentioned by De
Morgan somewhat later as alive in 1862.

[319] Henry Dunning Macleod (1821-1902), a lawyer and writer on political
economy, was a Scotchman by birth. He wrote on economical questions, and
lectured on banking at Cambridge (1877) and at King's College, London
(1878). He was a free lance in his field, and was not considered orthodox
by the majority of economists of his time. He was an unsuccessful candidate
for the chairs of political economy at Cambridge (1863), Edinburgh (1871),
and Oxford (1888).

[320] See Vol. I, page 252, note 2 {576}.

[321] Francis Henry Laing (1816-1889) was a graduate of Queen's College,
Cambridge, and a clergyman in the Church of England until 1846, when he
entered the Church of Rome. He taught in various Jesuit colleges until
1862, when his eccentricity was too marked to warrant the Church in
allowing him to continue. He published various controversial writings
during his later years. Of course if he had known the works of Wessel,
Gaus, Buée, Argand, and others, he would not have made such a sorry
exhibition of his ignorance of mathematics.

[322] See Vol. I, page 329, note 1 {705}. The book went into a second
edition in 1864.

[323] Thomas Weddle (1817-1853) was, at the time of publishing this paper,
a teacher in a private school. In 1851 he became professor of mathematics
at Sandhurst. He contributed several papers to the _Cambridge and Dublin
Mathematical Journal_, chiefly on geometry.

[324] See Vol. II, page 109, note 205.

[325] See Vol. II, page 66, note 143.

[326] See Vol. II, page 151, note 268.

[327] George Barrett (1752-1821) worked from 1786 to 1811 on a set of life
insurance and annuity tables. He invented a plan known as the "columnar
method" for the construction of such tables, and as De Morgan states, this
was published by Francis Baily, appearing in the appendix to his work on
annuities, in the edition of 1813. Some of his tables were used in
Babbage's _Comparative View of the various Institutions for the Assurance
of Lives_ (1826).

[328] See Vol. I, page 309, note 2 {670}.

[329] This was his _Practical short and direct Method of Calculating the
Logarithm of any given Number, and the Number corresponding to any given
Logarithm_ (1849).

[330] This is William Neile (1637-1670), grandson of Richard Neile (not
Neal), Archbishop of York. At the age of 19, in 1657, he gave the first
rectification of the semicubical parabola. Although he communicated it to
Brouncker, Wren, and others, it was not published until 1639, when it
appeared in John Wallis's _De Cycloide_.

[331] I myself "was a considerable part."

[332] He also wrote _A Glance at the Universe_ ("2d thousand" in 1862), and
_The Resurrection Body_ (1869).

[333] See Vol. I, page 63, note 1 {74}.

[334] As Swift gave it in his _Poetry. A Rhapsody_, it is as follows:

 "So, naturalists observe, a flea
  Has smaller fleas that on him prey;
  And these have smaller still to bite 'em.
  And so proceed _ad infinitum_."

[335] Perhaps 1,600,000,000 years, if Boltwood's recent computations based
on radium disintegration stand the test. This would mean, according to
MacCurdy's estimate, 60,000,000 years since life first appeared on the

[336] De Morgan wrote better than he knew, for this work, the _Allgemeine
Encyclopädie der Wissenschaften und Künste_, begun at Leipsic in 1818, is
still (1913) unfinished. Section I, A-G, consists of 99 parts in 56
volumes; Section II, H-N, consists of 43 volumes and is not yet completed;
and Section III, O-Z, consists of 25 volumes thus far, with most of the
work still to be done. Johann Samuel Ersch (1766-1828), the founder, was
head librarian at Halle. Johann Gottfried Gruber (1774-1851), his
associate, was professor of philosophy at the same university.

[337] William Howitt (1792-1879) was a poet, a spiritualist, and a
miscellaneous writer. He and his wife became spiritualists about 1850. He
wrote numerous popular works on travel, nature and history.

[338] See Vol. II, page 55, note 108.

[339] As will be inferred from the text, C. D. was Mrs. De Morgan, and
A. B. was De Morgan.

[340] Jean Meslier (1678-1733), curé of Estrepigny, in Champagne, was a
skeptic, but preached only strict orthodoxy to his people. It was only in
his manuscript, _Mon Testament_, that was published after his death, and
that caused a great sensation in France, that his antagonism to
Christianity became known.

[341] Baron Zach relates that a friend of his, in a writing intended for
publication, said _Un esprit doit se frotter contre un autre_. The censors
struck it out. The Austrian police have a keen eye for consequences.--A. De

"One mind must rub against another." On Baron Zach, see Vol. II, page 45,
note 4.

[342] Referring to the first Lord Eldon (1751-1838), who was Lord
Chancellor from 1799 to 1827, with the exception of one year.

[343] "Sleeping power."

[344] "Causes sleep."

[345] Richard Hooker (c. 1554-1600), a theologian, "the ablest living
advocate of the Church of England as by law established."

[346] See Vol. I, page 76, note 3 {112}.

[347] "Other I,"--other self.

[348] This "utter rejection" has been repeated (1872) by the same
writer.--S. E. De M.

[349] Edward Jenner (1749-1823) was a physician and biologist. His first
experiments in vaccination were made in 1796, and his discovery was
published in 1798.

[350] See Vol. II, page 38, note 80.

[351] "You will go most safely in the middle (way)."

[352] Pierre Joseph Arson was known early in the 19th century for his
controversy with Hoëné Wronski the mathematician, whom he attacked in his
_Document pour l'histoire des grands fourbes qui ont figuré sur la terre_

[353] "We enter the course by night and are consumed by fire."

[354] See Vol. I, page 51, note 3 {51}.

[355] See Vol. I, page 336, note 8 {713}.

[356] See Vol. I, page 137, note 8 {286}.

[357] See Vol. I, page 229, note 2 {515}.

[358] Richard Cobden (1804-1865), the cotton manufacturer and statesman who
was prominent in his advocacy of the repeal of the Corn Laws.

[359] James Smith (1775-1839), solicitor to the Board of Ordnance. With his
brother Horatio he wrote numerous satires. His _Horace in London_ (1813)
imitated the Roman poet. His works were collected and published in 1840.

[360] Samuel Butler (1612-1680), the poet and satirist, author of
_Hudibras_ (1663-1678).

[361] "Is it not fine to be sure of one's action when entering in a combat
with another? There, push me a little in order to see. NICOLE. Well! what's
the matter? M. JOURDAIN. Slowly. Ho there! Ho! gently. Deuce take the
rascal! NICOLE. You told me to push. M. JOURDAIN. Yes, but you pushed me
_en tierce_, before you pushed _en quarte_, and you did not give me time to

[362] John Abernethy (1764-1831), the famous physician and surgeon.

[363] See Vol. I, page 102, note 5 {175}.

[364] "With what measure ye mete, it shall be measured to you again."

[365] Eusebius of Cæsarea (c. 260-340), leader of the moderate party at the
Council of Nicæa, and author of a _History of the Christian Church_ in ten
books (c. 324 A. D.).

[366] Nathaniel Lardner (1684-1768), a non-conformist minister and one of
the first to advocate the scientific study of early Christian literature.

[367] Henry Alford (1810-1871) Dean of Canterbury (1857-1871) and editor of
the Greek Testament (1849-1861).

[368] The work was _The Number and Names of the Apocalyptic Beasts: with an
explanation and application. Part I._ London, 1848, as mentioned below.
Thom also wrote _The Assurance of Faith, or Calvinism identified with
Universalism_ (London, 1833), and various other religious works.

[369] See Vol. I, page 222, note 14 {490}.

[370] John Hamilton Thom (1808-1894) was converted to Unitarianism and was
long a minister in that church, preaching in the Renshaw Street Chapel from
1831 to 1866. De Morgan refers to the Liverpool Unitarian controversy
conducted by James Martineau and Henry Giles in response to a challenge by
thirteen Anglican Clergy. In 1839 Thom contributed four lectures and a
letter to this controversy. Among his religious works were a _Life of
Blanco White_ (1845) and _Hymns, Chants, and Anthems_ (1854).

[371] The spelling of these names is occasionally changed to meet the
condition that the numerical value of the letters shall be 666, "the number
of the beast" of Revelations. The names include Julius Cæsar; Valerius
Jovius Diocletianus (249-313), emperor from 287 to 305, persecutor of the
Christians; Louis, presumably Louis XIV; Gerbert (940-1003), who reigned as
Pope Sylvester II from 999 to 1003, known to mathematicians for his abacus
and his interest in geometry, and accused by his opponents as being in
league with the devil; Linus, the second Bishop of Rome, the successor of
Peter; Camillo Borghese (1552-1621), who reigned as Pope Paul V from 1605
to 1621, and who excommunicated all Venice in 1606 for its claim to try
ecclesiastics before lay tribunals, thus taking a position which he was
forced to abandon; Luther, Calvin; Laud (see Vol. I, page 145, note 7
{307}); Genseric (c. 406-477), king of the Vandals, who sacked Rome in 455
and persecuted the orthodox Christians in Africa; Boniface III, who was
pope for nine months in 606; Beza (see Vol. I, page 66, note 6 {83});
Mohammed; [Greek: braski], who was Giovanni Angelo Braschi (1717-1799), and
who reigned as Pope Pius VI from 1775 to 1799, dying in captivity because
he declined to resign his temporal power to Napoleon; Bonaparte; and, under
[Greek: Ion Paune], possibly Pope John XIV, who reigned in 983 and 984
during the absence of Boniface VII in Constantinople.

[372] The Greek words and names are also occasionally misspelled so as to
fit them to the number 666. They are [Greek: Lateinos] (Latin), [Greek: hê
latinê basileia] (the Latin kingdom), [Greek: ekklêsia italika] (the
Italian Church), [Greek: euanthas] (blooming), [Greek: teitan] (Titan),
[Greek: arnoume] (renounce), [Greek: lampetis] (the lustrous), [Greek: ho
nikêtês] (conqueror), [Greek: kakos hodêgos] (bad guide), [Greek: alêthês
blaberos] (truthful harmful one), [Greek: palai baskanos] (a slanderer of
old), [Greek: amnos adikos] (unmanageable lamb), [Greek: antemos]
(Antemos), [Greek: gensêrikos] (Genseric), [Greek: euinas] (with stout
fibers), [Greek: Benediktos] (Benedict), [Greek: Bonibazios g. papa x. ê.
e. e. a.] (Boniface III, pope 68, bishop of bishops I), [Greek: oulpios]
(baneful), [Greek: dios eimi hê hêras] (I, a god, am the), [Greek: hê missa
hê papikê] (the papal brief), [Greek: loutherana] (Lutheran), [Greek:
saxoneios] (Saxon), [Greek: Bezza antitheos] (Beza antigod), [Greek: hê
alazoneia biou] (the illusion of life), [Greek: Maometis] (Mahomet);
[Greek: Maometês b.] (Mahomet II), [Greek: theos eimi epi gaiês] (I am lord
over the earth), [Greek: iapetos] (Iapetos, father of Atlas), [Greek:
papeiskos] (Papeiskos), [Greek: dioklasianos] (Diocletian), [Greek: cheina]
(Cheina = Cain? China?), [Greek: braski] (Braschi, as explained in note
10), [Greek: Ion Paune] (Paunian violet, but see note 10), [Greek: koupoks]
(cowpox), [Greek: Bonnepartê] (Bonneparte), [Greek: N. Bonêparte] (N.
Boneparte), [Greek: euporia] (facility), [Greek: paradosis] (surrender),
[Greek: to megathêrion] (the megathereum, the beast).

[373] James Wapshare, whose _Harmony of the Word of God in Spirit and in
Truth_ appeared in 1849.

[374] The literature relating to the _Swastika_ is too extended to permit
of any adequate summary in these notes.

[375] Henry Edward Manning (1808-1892), at first an Anglican clergyman, he
became a Roman Catholic priest in 1851, and became Cardinal in 1875. He
succeeded Cardinal Wiseman as Archbishop of Westminster in 1865. He wrote a
number of religious works.

[376] John Bright (1811-1889), Quaker, cotton manufacturer, and statesman.
He worked with Cobden for free trade, peace, and reform of the electorate.

[377] "The fallacy of many questions."

[378] William Wilberforce (1759-1833), best known for his long fight for
the abolition of the slave trade.

[379] Richard Martin (1754-1834), high sheriff of County Galway and owner
of a large estate in Connemara. Curiously enough, he was known both for his
readiness in duelling and for his love for animals. He was known as
"Humanity Martin," and in 1822 secured the passage of an act "to prevent
the cruel and improper treatment of cattle." He was one of the founders
(1824) of the Royal Society for the Prevention of Cruelty to Animals. He is
usually considered the original of Godfrey O'Malley in Lever's novel,
_Charles O'Malley_.

[380] See Vol. I, page 149, note 1 {323}, also text on same page.

[381] See Vol. I, page 44, note 9 {34}, also text, Vol. I, page 110.

[382] "Penitential seat."

[383] "Well placed upon the cushion."

[384] See Vol. II, page 58, note 109.

[385] "He has lost the right of being influenced by evidence."

[386] "Hung up."

[387] "A few things to the wise, nothing to the unlettered."

[388] The fallacy results from dividing both members of an equation by 0, x
- 1 being the same as 1 - 1, and calling the quotients finite.

[389] "If you order him to the sky he will go."

[390] _Similia similibus curanter_, "Like cures like," the homeopathic

[391] "Without harm to the proprieties."

[392] "What are you doing? I am standing here."

[393] Lors feist l'Anglois tel signe. La main gausche toute ouverte il leva
hault en l'aer, puis ferma au poing les quatres doigtz d'icelle et le
poulce estendu assit sus la pinne du nez. Soubdain après leva la dextre
toute ouverte, et toute ouverte la baissa, joignant la poulce au lieu que
fermait le petit doigt de la gausche, et les quatre doigtz d'icelle mouvoit
lentement en l'aer. Puis au rebours feit de la dextre ce qu'il avoit faict
de la gausche, et de la gausche ce que avoit faict de la dextre.--A. De M.

[394] _Suaviter in modo, fortiter in re_, "Gentle in manners, firm in

[395] See Vol. I, page 101, note 4 {174}.

[396] See Vol. I, page 315, note 3 {686}.

[397] Henry Fawcett (1833-1884) became totally blind in 1858, but in spite
of this handicap he became professor of political economy at Cambridge and
sat in parliament for a number of years. He championed the cause of reform
and in particular he was prominent in the protection of the native
interests of India. The establishing of the parcels post (1882) took place
while he was postmaster general (1880-1884).

[398] Of course the whole thing depends upon what definition of division is
taken. We can multiply 2 ft. by 3 ft. if we define multiplication so as to
allow it, or 2 ft. by 3 lb, getting foot-pounds, as is done in physics.

[399] Richard Milward (1609-1680), for so the name is usually given, was
rector of Great Braxted (Essex) and canon of Windsor. He was long the
amanuensis of John Selden, and the _Table Talk_ was published nine years
after Milward's death, from notes that he left. Some doubt has been cast
upon the authenticity of the work owing to many of the opinions that it
ascribes to Selden.

[400] John Selden (1584-1654) was a jurist, legal antiquary, and Oriental
scholar. He sat in the Long Parliament, and while an advocate of reform he
was not an extremist. He was sent to the Tower for his support of the
resolution against "tonnage and poundage," in 1629. His _History of Tythes_
(1618) was suppressed at the demand of the bishops. His _De Diis Syriis_
(1617) is still esteemed a classic on Semitic mythology.

[401] See Vol. I, page 42, note 4 {24}.

[402] See Vol. II, page 249, note 398.

[403] John Palmer (1742-1818) was a theatrical manager. In 1782 he set
forth a plan for forwarding the mails by stage coaches instead of by
postmen. Pitt adopted the plan in 1784. Palmer was made comptroller-general
of the post office in 1786 and was dismissed six years later for
arbitrarily suspending a deputy. He had been verbally promised 2½% on the
increased revenue, but Pitt gave him only a pension of £3000. In 1813 he
was awarded £50,000 in addition to his pension.

[404] Dionysius Lardner (1793-1859), professor of natural philosophy in
London University (now University College). His _Cabinet Cyclopædia_
(1829-1849) contained 133 volumes. De Morgan wrote on probabilities, and
Lardner on various branches of mathematics, and there were many other
well-known contributors. Lardner is said to have made $200,000 on a lecture
tour in America.

[405] Thomas Fysche Palmer (1747-1802) joined the Unitarians in 1783, and
in 1785 took a charge in Dundee. He was arrested for sedition because of an
address that it was falsely alleged that he gave before a society known as
the "Friends of Liberty." As a matter of fact the address was given by an
uneducated weaver, and Palmer was merely asked to revise it, declining to
do even this. Nevertheless he was sentenced to Botany Bay (1794) for seven
years. The trial aroused great indignation.

[406] See Vol. I, page 80, note 5 {119}.

[407] See Vol. II, page 244, note 394.

[408] See Vol. I, page 352, note 1 {731}.

[409] See Vol. I, page 332, note 4 {709}.

[410] "The lawyers are brought into court; let them accuse each other."

[411] Samuel Rogers (1763-1855), the poet and art connoisseur. He declined
the laureateship on the death of Wordsworth (1850). Byron, his pretended
friend, wrote a lampoon (1818) ridiculing his cadaverous appearance.

[412] Theodore Edward Hook (1788-1841), the well-known wit. He is satirized
as Mr. Wagg in _Vanity Fair_. The _John Bull_ was founded in 1820 and Hook
was made editor.

[413] "On pitying the heretic."

[414] A term of medieval logic. Barbara: All M is P, all S is M, hence all
S is P. Celarent: No M is P, all S is M, hence no S is P.

[415] "Simply," "According to which," "It does not follow."


 "O sweet soul, what good shall I declare
  That heretofore was thine, since such are thy remains!"

[417] "Stupid fellow!"

[418] Christopher Barker (c. 1529-1599), also called Barkar, was the
Queen's printer. He began to publish books in 1569, but did no actual
printing until 1576. In 1575 the Geneva Bible was first printed in England,
the work being done for Barker. He published 38 partial or complete
editions of the Bible from 1575 to 1588, and 34 were published by his
deputies (1588-1599).

[419] James Franklin (1697-1735) was born in Boston, Mass., and was sent to
London to learn the printer's trade. He returned in 1717 and started a
printing house. Benjamin, his brother, was apprenticed to him but ran away
(1723). James published the _New England Courant_ (1721-1727), and Benjamin
is said to have begun his literary career by writing for it.

[420] James Hodder was a writing master in Tokenhouse Yard, Lothbury, in
1661, and later kept a boarding school in Bromley-by-Bow. His famous
arithmetic appeared at London in 1661 and went through many editions. It
was the basis of Cocker's work. (See Vol. I, page 42, note 4 {24}.) It was
long thought to have been the first arithmetic published in America, and it
was the first English one. There was, however, an arithmetic published much
earlier than this, in Mexico, the _Sumario compendioso ... con algunas
reglas tocantes al Aritmética_, by "Juan Diaz Freyle," in 1556.

[421] Henry Mose, Hodder's successor, kept a school in Sherborne Lane,

[422] Rear Admiral Sir Francis Beaufort (1774-1857), F.R.S., was
hydrographer to the Navy from 1829 to 1855. He prepared an atlas that was
printed by the Society for the Diffusion of Useful Knowledge.

[423] Antoine Sabatier (1742-1817), born at Castres, was known as the Abbé
but was really nothing more than a "clerc tonsuré." He lived at Court and
was pensioned to write against the philosophers of the Voltaire group. He
posed as the defender of morality, a commodity of which he seems to have
possessed not the slightest trace.

[424] Maffeo Barberini was pope, as Urban VIII, from 1623 to 1644. It was
during his ambitious reign that Galileo was summoned to Rome to make his
recantation (1633), the exact nature of which is still a matter of dispute.

[425] This Baden Powell (1796-1860) was the Savilian professor of geometry
(1827-1860) at Oxford.

[426] "Memoirs of the famous bishop of Chiapa, by which it appears that he
had butchered or burned or drowned ten million infidels in America in order
to convert them. I believe that this bishop exaggerated; but if we should
reduce these sacrifices to five million victims, this would still be

[427] Alfonso X (1221-1284), known as El Sabio (the Wise), was interested
in astronomy and caused the Alphonsine Tables to be prepared. These table
were used by astronomers for a long time. It is said that when the
Ptolemaic system of the universe was explained to him he remarked that if
he had been present at the Creation he could have shown how to arrange
things in a much simpler fashion.

[428] George Richards (c. 1767-1837), fellow of Oriel (1790-1796), Bampton
lecturer (1800), Vicar of St. Martin's-in-the-Fields, Westminster (1824),
and a poet of no mean ability.

[429] The "Aboriginal Britons," an excellent poem, by Richards. (Note by
Byron.)--A. De M.

[430] John Watkins (d. after 1831), a teacher and miscellaneous writer.

[431] Frederic Shoberl (1775-1853), a miscellaneous writer.

[432] He wrote, besides the _Aboriginal Britons_, _Songs of the Aboriginal
Bards_ (1792), _Modern France: a Poem_ (1793), _Odin, a drama_ (1804),
_Emma, a drama on the model of the Greek theatre_ (1804), _Poems_ (2
volumes, 1804), and a _Monody on the Death of Lord Nelson_ (1806).

[433] Henry Kirke White (1785-1806), published his first volume of poems at
the age of 18. Southey and William Wilberforce became interested in him and
procured for him a sizarship at St. John's College, Cambridge. He at once
showed great brilliancy, but he died of tuberculosis at the age of 21.

[434] John Wolcot, known as Peter Pindar (1738-1819), was a London
physician. He wrote numerous satirical poems. His _Bozzy and Piozzi, or the
British Biographers_, appeared in 1786, and reached the 9th edition in

[435] See Vol. I, page 235, note 8 {532}.

[436] Richard Payne Knight (1750-1824) was a collector of bronzes, gems,
and coins, many of his pieces being now in the British Museum. He sat in
parliament for twenty-six years (1780-1806), but took no active part in
legislation. He opposed the acquisition of the Elgin Marbles, holding them
to be of little importance. His _Analytical Inquiry into the Principles of
Taste_ appeared in 1808.

[437] Mario Nizzoli (1498-1566), a well-known student of Cicero, was for a
time professor at the University of Parma. His _Observationes in M. Tullium
Ciceronem_ appeared at Pratalboino in 1535. It was revised by his nephew
under the title _Thesaurus Ciceronianus_ (Venice, 1570).

[438] See Vol. I, page 314, note 4 {681}.


 "Like the geometer, who bends all his powers
  To measure the circle, and does not succeed,
  Thinking what principle he needs."

[440] Francis Quarles (1592-1644), a religious poet. He wrote paraphrases
of the Bible and numerous elegies. In the early days of the revolutionary
struggle he sided with the Royalists. One of his most popular works was the
_Emblems_ (1635), with illustrations by William Marshall.

[441] Regnault de Bécourt wrote _La Création du monde, ou Système
d'organisation primitive suivi de l'interprétation des principaux
phénomènes et accidents que se sont opérés dans la nature depuis l'origine
de univers jusqu'à nos jours_ (1816). This may be the work translated by

[442] "Because it lacks a holy prophet."

[443] Angherà. See Vol. II, page 60, note 127.

[444] Edmund Curll (1675-1747), a well-known bookseller, publisher, and
pamphleteer. He was for a time at "The Peacock without Temple Bar," and
later at "The Dial and Bible against St. Dunstan's Church." He was fined
repeatedly for publishing immoral works, and once stood in the pillory for
it. He is ridiculed in the _Dunciad_ for having been tossed in a blanket by
the boys of Westminster School because of an oration that displeased them.

[445] See Vol. II, page 109, note 206.

[446] Encyclopædia.

[447] Author of the _Historia Naturalis_ (77 A.D.)

[448] Author of the _De Institutione Oratorio Libri_ XII (c. 91 A.D.)

[449] His _De Architectures Libri_ X was not merely a work on architecture
and building, but on the education of the architect.

[450] Cyclophoria.

[451] William Caxton (c. 1422-c.1492), sometime Governor of the Company of
Merchant Adventurers in Bruges (between 1449 and 1470). He learned the art
of printing either at Bruges or Cologne, and between 1471 and 1477 set up a
press at Westminster. Tradition says that the first book printed in England
was his _Game and Playe of Chesse_ (1474). The _Myrrour of the Worlde and
th'ymage of the same_ appeared in 1480. It contains a brief statement on
arithmetic, the first mathematics to appear in print in England.

[452] See Vol. I, page 45, note 6 {40}. De Morgan is wrong as to the date
of the _Margarita Philosophica_. The first edition appeared at Freiburg in

[453] Reisch was confessor to Maximilian I (1459-1519), King of the Romans
(1486) and Emperor (1493-1519).

[454] Joachim Sterck Ringelbergh (c. 1499-c. 1536), teacher of philosophy
and mathematics in various cities of France and Germany. His _Institutionum
astronomicarum libri III_ appeared at Basel in 1528, his _Cosmographia_ at
Paris in 1529, and his _Opera_ at Leyden in 1531.

[455] Johannes Heinrich Alsted (1588-1638) was professor of philosophy and
theology at his birthplace, Herborn, in Nassau, and later at Weissenberg.
He published several works, including the _Elementale mathematicum_ (1611),
_Systema physicae harmonicae_ (1612), _Methodus admirandorum
mathematicorum_ (1613), _Encyclopædia septem tomis distincta_ (1630), and
the work mentioned above.

[456] Johann Jakob Hoffmann (1635-1706), professor of Greek and history at
his birthplace, Basel. He also wrote the _Epitome metrica historiæ
universalis civilis et sacræ ab orbe condito_ (1686).

[457] Ephraim Chambers (c. 1680-1740), a crotchety, penurious, but
kind-hearted freethinker. His _Cyclopædia, or an Universal Dictionary_ was
translated into French and is said to have suggested the great

[458] _Encyclopédie, ou Dictionnaire raisonné des sciences, des arts et des
métiers, par un société de gens de lettres. Mis en ordre et publié par M.
Diderot, et quant à la partie mathématique, par M. d'Alembert._ Paris,
1751-1780, 35 volumes.

[459] "From the egg" (state).

[460] See Vol. I, page 382, note 12 {785}.

[461] See Vol. II, page 4, note 15.

[462] "In morals nothing should serve man as a model but God; in the arts,
nothing but nature."

[463] _Encyclopédie Méthodique, ou par ordre de matières._ Paris,
1782-1832, 166½ volumes.

[464] See Vol. II, page 193, note 336.

[465] _Encyclopædia Metropolitana; or, Universal Dictionary of Knowledge._
London, 1845, 29 volumes. A second edition came out in 1848-1858 in 40

[466] See Vol. I, page 137, note 8 {286}.

[467] See Vol. I, page 80, note 5 {119}.

[468] De Morgan should be alive to satirize some of the statements on the
history of mathematics in the eleventh edition.

[469] John Pringle Nichol (1804-1859), Regius professor of astronomy at
Glasgow and a popular lecturer on the subject. He lectured in the United
States in 1848-1849. His _Views of the Architecture of the Heavens_ (1838)
was a very popular work, and his _Planetary System_ (1848, 1850) contains
the first suggestion for the study of sun spots by the aid of photography.

[470] See Vol. II, page 109, note 206.

[471] George Long (1800-1879), a native of Poulton, in Lancashire, was
called to the University of Virginia when he was only twenty-four years old
as professor of ancient languages. He returned to England in 1828 to become
professor of Greek at London University. From 1833 to 1849 he edited the
twenty-nine volumes of the _Penny Cyclopædia_. He was an authority on Roman

[472] A legal phrase, "Qui tam pro domina regina, quam pro se ipso
sequitur,"--"Who sues as much on the Queen's account as on his own."

[473] Arthur Cayley (1821-1895) was a fellow of Trinity College, Cambridge
(1842-1846) and was afterwards a lawyer (1849-1863). During his fourteen
years at the bar he published some two hundred mathematical papers. In 1863
he became professor of mathematics at Cambridge, and so remained until his
death. His collected papers, nine hundred in number, were published by the
Cambridge Press in 13 volumes (1889-1898). He contributed extensively to
the theory of invariants and covariants. De Morgan's reference to his
coining of new names is justified, although his contemporary, Professor
Sylvester, so far surpassed him in this respect as to have been dubbed "the
mathematical Adam."

[474] See Vol. II, page 26, note 56.

[475] See Vol. I, page 111, note 3 {207}.

[476] See Vol. I, page 87, note 6 {135}.

[477] Pierre Dupuy (1582-1651) was a friend and relative of De Thou. With
the collaboration of his brother and Nicolas Rigault he published the 1620
and 1626 editions of De Thou's History. He also wrote on law and history.
His younger brother, Jacques (died in 1656), edited his works. The two had
a valuable collection of books and manuscripts which they bequeathed to the
Royal Library at Paris.

[478] See Vol. I, page 51, note 3 {51}.

[479] It was Cosmo de' Medici (1590-1621) who was the patron of Galileo.

[480] See Vol. I, page 40, note 4 {20}.

[481] See Vol. I, page 106, note 4 {188}.

[482] Sir Edward Sherburne (1618-1702), a scholar of considerable
reputation. The reference by De Morgan is to _The Sphere of Marcus
Manilius_, in the appendix to which is a _Catalogue of Astronomers, ancient
and modern_.

[483] George Parker, second Earl of Macclesfield (1697-1764). He erected an
observatory at Shirburn Castle, Oxfordshire, in 1739, and fitted it out
with the best equipment then available. He was President of the Royal
Society in 1752.

[484] See Vol. II, page 148, note 263.

[485] See Vol. I, page 140, note 7 {296}.

[486] See Vol. I, page 106, note 4 {188}.

[487] Edward Bernard (1638-1696), although Savilian professor of astronomy
at Oxford, was chiefly interested in archeology.

[488] See Vol. I, page 107, note 1 {190}.

[489] See Vol. I, page 107, note 1 {190}.

[490] See Vol. I, page 135, note 3 {281}.

[491] Philip Dormer Stanhope, fourth Earl of Chesterfield (1694-1773), well
known for the letters written to his son which were published posthumously

[492] Peter Daval (died in 1763), Vice-President of the Royal Society, and
an astronomer of some ability.

[493] See Vol. I, page 376, note 1 {766}.

[494] William Oughtred (c. 1573-1660), a fellow of King's College,
Cambridge, and afterwards vicar of Aldbury, Surrey, wrote the best-known
arithmetic and trigonometry of his time. His _Arithmeticæ in Numero &
Speciebus Institutio ... quasi Clavis Mathematicæ est_ (1631) went through
many editions and appeared in English as _The Key to the Mathematicks new
forged and filed_ in 1647.

[495] See Vol. I, page 140, note 5 {294}.

[496] Stephen Jordan Rigaud (1816-1859) was senior assistant master of
Westminster School (1846) and head master of Queen Elizabeth's School at
Ipswich (1850). He was made Bishop of Antigua in 1858 and died of yellow
fever the following year.

[497] He also wrote a memoir of his father, privately printed at Oxford in

[498] See Vol. I, page 69, note 3 {96}.

[499] See Vol. I, page 106, note 4 {188}.

[500] William Gascoigne was born at Middleton before 1612 and was killed in
the battle of Marston Moor in 1644. He was an astronomer and invented the
micrometer with movable threads (before 1639).

[501] Seth Ward (1617-1689) was deprived of his fellowship at Cambridge for
refusing to sign the covenant. He became professor of astronomy at Oxford
(1649), Bishop of Exeter (1662), Bishop of Salisbury (1667), and Chancellor
of the Garter (1671). He is best known for his solution of Kepler's problem
to approximate a planet's orbit, which appeared in his _Astronomia
geometrica_ in 1656.

[502] See Vol. I, page 110, note 2 {198}.

[503] See Vol. I, page 100, note 2 {172}.

[504] See Vol. I, page 107, note 1 {190}.

[505] See Vol. I page 114, note 6 {220}.

[506] See Vol. I, page 77, note 4 {118}.

[507] See Vol. I, page 125, note 3 {253}.

[508] See Vol. I, page 105, note 2 {186}.

[509] Heinrich Oldenburgh (1626-1678) was consul in England for the City of
Bremen, his birthplace, and afterwards became a private teacher in London.
He became secretary of the Royal Society and contributed on physics and
astronomy to the _Philosophical Transactions_.

[510] Thomas Brancker, or Branker (1636-1676) wrote the _Doctrinæ sphæricæ
adumbratio et usus globorum artificialium_ (1662) and translated the
algebra of Rhonius with the help of Pell. The latter work appeared under
the title of _An Introduction to Algebra_ (1668), and is noteworthy as
having brought before English mathematicians the symbol ÷ for division. The
symbol never had any standing on the Continent for this purpose, but
thereafter became so popular in England that it is still used in all the
English-speaking world.

[511] See Vol. I, page 118, note 1 {230}.

[512] Pierre Bertius (1565-1629) was a native of Flanders and was educated
at London and Leyden. He became a professor at Leyden, and later held the
chair of mathematics at the Collège de France. He wrote chiefly on

[513] See Vol. II, page 297, note 487.

[514] Giovanni Alphonso Borelli (1608-1679) was professor of mathematics at
Messina (1646-1656) and at Pisa (1656-1657), after which he taught in Rome
at the Convent of St. Panteleon. He wrote several works on geometry,
astronomy, and physics.

[515] See Vol. I, page 172, note 2 {381}.

[516] Ignace Gaston Pardies (c. 1636-1673), a Jesuit, professor of ancient
languages and later of mathematics and physics at the Collège of Pau, and
afterwards professor of rhetoric at the Collège Louis-le-Grand at Paris. He
wrote on geometry, astronomy and physics.

[517] Pierre Fermat was born in 1608 (or possibly in 1595) near Toulouse,
and died in 1665. Although connected with the parliament of Toulouse, his
significant work was in mathematics. He was one of the world's geniuses in
the theory of numbers, and was one of the founders of the theory of
probabilities and of analytic geometry. After his death his son published
his edition of Diophantus (1670) and his _Varia opera mathematica_ (1679).

[518] This may be Christopher Townley (1603-1674) the antiquary, or his
nephew, Richard, who improved the micrometer already invented by Gascoigne.

[519] Adrien Auzout a native of Rouen, who died at Rome in 1691. He
invented a screw micrometer with movable threads (1666) and made many
improvements in astronomical instruments.

[520] See Vol. I, page 66, note 9 {86}.

[521] See Vol. I, page 124, note 7 {248}.

[522] John Machin (d. 1751) was professor of astronomy at Gresham College
(1713-1751) and secretary of the Royal Society. He translated Newton's
_Principia_ into English. His computation of [pi] to 100 places is given in
William Jones's _Synopsis palmariorum matheseos_ (1706).

[523] Pierre Rémond de Montmort (1678-1719) was canon of Notre Dame until
his marriage. He was a gentleman of leisure and devoted himself to the
study of mathematics, especially of probabilities.

[524] Roger Cotes (1682-1716), first Plumian professor of astronomy and
physics at Cambridge, and editor of the second edition of Newton's
_Principia_. His posthumous _Harmonia Mensurarum_ (1722) contains "Cotes's
Theorem" on the binomial equation. Newton said of him, "If Mr. Cotes had
lived we had known something."

[525] See Vol. I, page 135, note 3 {281}.

[526] See Vol. I, page 377, note 4 {769}.

[527] Charles Réné Reyneau (1656-1728) was professor of mathematics at
Angers. His _Analyse démontrée, ou Manière de resoudre les problèmes de
mathématiques_ (1708) was a successful attempt to popularize the theories
of men like Descartes, Newton, Leibnitz, and the Bernoullis.

[528] Brook Taylor (1685-1731), secretary of the Royal Society, and student
of mathematics and physics. His _Methodus incrementorum directa et inversa_
(1715) was the first treatise on the calculus of finite differences. It
contained the well-known theorem that bears his name.

[529] Pierre Louis Moreau de Maupertuis (1698-1759) was sent with Clairaut
(1735) to measure an arc of a meridian in Lapland. He was head of the
physics department in the Berlin Academy from 1745 until 1753. He wrote
_Sur la figure de la terre_ (1738) and on geography and astronomy.

[530] Pierre Bouguer (1698-1758) was professor of hydrography at Paris, and
was one of those sent by the Academy of Sciences to measure an arc of a
meridian in Peru (1735). The object of this and the work of Maupertuis was
to determine the shape of the earth and see if Newton's theory was

[531] Charles Marie de la Condamine (1701-1774) was a member of the Paris
Academy of Sciences and was sent with Bouguer to Peru, for the purpose
mentioned in the preceding note. He wrote on the figure of the earth, but
was not a scientist of high rank.

[532] See Vol. I, page 136, note 5 {283}.

[533] See Vol. II, page 296, note 483.

[534] Thomas Baker (c. 1625-1689) gave a geometric solution of the
biquadratic in his _Geometrical Key, or Gate of Equations unlocked_ (1684).

[535] See Vol. I, page 160, note 5 {350}.

[536] See Vol. I, page 87, note 4 {133}.

[537] See Vol. I, page 132, note 2 {272}.

[538] See Vol. I, page 118, second note 1 {231}.

[539] The name of Newton is so well known that no note seems necessary. He
was born at Woolsthorpe, Lincolnshire, in 1642, and died at Kensington in

[540] John Keill (1671-1721), professor of astronomy at Oxford from 1710,
is said to have been the first to teach the Newtonian physics by direct
experiment, the apparatus being invented by him for the purpose. He wrote
on astronomy and physics. His _Epistola de legibus virium centripetarum_,
in the Philosophical Transactions for 1708, accused Leibnitz of having
obtained his ideas of the calculus from Newton, thus starting the priority

[541] Thomas Digges (d. in 1595) wrote _An Arithmeticall Militare Treatise,
named Stratioticos_ (1579), and completed _A geometrical practise, named
Pantometria_ (1571) that had been begun by his father, Leonard Digges.

[542] John Dee (1527-1608), the most famous astrologer of his day, and
something of a mathematician, wrote a preface to Billingsley's translation
of Euclid into English (1570).

[543] See Vol. I, page 76, note 3 {112}.

[544] Thomas Harriot (1560-1621) was tutor in mathematics to Sir Walter
Raleigh, who sent him to survey Virginia (1585). He was one of the best
English algebraists of his time, but his _Artis Analyticæ Praxis ad
Aequationes Algebraicas resolvendas_ (1631) did not appear until ten years
after his death.

[545] Thomas Lydiat (1572-1626), rector of Alkerton, devoted his life
chiefly to the study of chronology, writing upon the subject and taking
issue with Scaliger (1601).

[546] See Vol. I, page 69, note 3 {96}.

[547] Walter Warner edited Harriot's _Artis Analyticae Praxis_ (1631).
Tarporley is not known in mathematics.

[548] See Vol. I, page 105, note 2 {186}.

[549] See Vol. I, page 115, note 3 {224}.

[550] See Vol. II, page 300, note 509.

[551] See Vol. I, page 107, note 1 {190}.

[552] Sir Samuel Morland (1625-1695) was a diplomat and inventor. For some
years he was assistant to John Pell, then ambassador to Switzerland. He
wrote on arithmetical instruments invented by him (1673), on hydrostatics
(1697) and on church history (1658).

[553] See Vol. I, page 153, note 4 {337}.

[554] See Vol. I, page 85, note 2 {129}.

[555] See Vol. I, page 43, note 8 {33}.

[556] See Vol. I, page 43, note 7 {32}.

[557] See Vol. I, page 382, note 13 {786}. The history of the subject may
be followed in Braunmühl's _Geschichte der Trigonometrie_.

[558] See Vol. I, page 377, note 3 {768}.

[559] See Vol. I, page 108, note 2 {192}.

[560] Michael Dary wrote _Dary's Miscellanies_ (1669), _Gauging epitomised_
(1669), and _The general Doctrine of Equation_ (1664).

[561] John Newton (1622-1678), canon of Hereford (1673), educational
reformer, and writer on elementary mathematics and astronomy.

[562] See Vol. I, page 87, note 4 {133}.

[563] "The average of the two equal altitudes of the sun before and after

[564] See Vol. I, page 42, note 4 {24}.

[565] London, 1678. It went though many editions.

[566] "This I who once ..."

[567] Arthur Murphy (1727-1805) worked in a banking house until 1754. He
then went on the stage and met with some success at Covent Garden. His
first comedy, _The Apprentice_ (1756) was so successful that he left the
stage and took to play writing. His translation of Tacitus appeared in
1793, in four volumes.

[568] Edmund Wingate (1596-1656) went to Paris in 1624 as tutor to Princess
Henrietta Maria and remained there several years. He wrote _L'usage de la
règle de proportion_ (Paris, 1624, with an English translation in 1626),
_Arithmétique Logarithmétique_ (Paris, 1626, with an English translation in
1635), and _Of Natural and Artificial Arithmetick_ (London, 1630, revised
in 1650-1652), part I of which was one of the most popular textbooks ever
produced in England.

[569] John Lambert (1619-1694) was Major-General during the Revolution and
helped to draw up the request for Cromwell to assume the protectorate. He
was imprisoned in the Tower by the Rump Parliament. He was confined in
Guernsey until the clandestine marriage of his daughter Mary to Charles
Hatton, son of the governor, after which he was removed (1667) to St.
Nicholas in Plymouth Sound.

[570] Samuel Foster (d. in 1652) was made professor of astronomy at Gresham
College in March, 1636, but resigned in November of that year, being
succeeded by Mungo Murray. Murray vacated his chair by marriage in 1641 and
Foster succeeded him. He wrote on dialling and made a number of
improvements in geometric instruments.

[571] "Twice of the word a minister," that is, twice a minister of the

[572] This is _The Lives of the Professors of Gresham College to which is
prefixed the Life of the Founder, Sir Thomas Gresham_, London, 1740. It was
written by John Ward (c. 1679-1758), professor of rhetoric (1720) at
Gresham College and vice-president (1752) of the Royal Society.

[573] Charles Montagu (1661-1715), first Earl of Halifax, was Lord of the
Treasury in 1692, and was created Baron Halifax in 1700 and Viscount
Sunbury and Earl of Halifax in 1714. He introduced the bill establishing
the Bank of England, the bill becoming a law in 1694. He had troubles of
his own, without considering Newton, for he was impeached in 1701, and was
the subject of a damaging resolution of censure as auditor of the exchequer
in 1703. Although nothing came of either of these attacks, he was out of
office during much of Queen Anne's reign.

[574] See Vol. II, page 302, note 547.

[575] See Vol. I, page 105, note 2 {186}.

[576] James Dodson (d. 1757) was master of the Royal Mathematical School,
Christ's Hospital. He was De Morgan's great-grandfather. The
_Anti-Logarithmic Canon_ was published in 1742.

[577] See Vol. I, page 106, note 4 {188}.

[578] See Vol. I, page 110, note 2 {198}.

[579] Richard Busby, (1606-1695), master of Westminster School (1640) had
among his pupils Dryden and Locke.

[580] See Vol. I, page 107, note 1 {190}.

[581] Herbert Thorndike (1598-1672), fellow of Trinity College, Cambridge
(1620-1646), and Prebend of Westminster (1661), was a well-known
theological writer of the time.

[582] See Vol. I, page 140, note 5 {294}.

[583] See Vol. I, page 108, note 2 {192}.

[584] "Labor performed returns in a circle."

[585] See Vol. II, page 208.

[586] "Whatever objections one may make to the above arguments, one always
falls into an absurdity."

[587] See Vol. II. page 11, note 29. _The Circle Squared; and the solution
of the problem adapted to explain the difference between square and
superficial measurement_ appeared at Brighton in 1865.

[588] "And beyond that nothing."

[589] Gillott (1759-1873) was the pioneer maker of steel pens by machinery,
reducing the price from 1s. each to 4d. a gross. He was a great collector
of paintings and old violins.

[590] William Edward Walker wrote five works on circle squaring (1853,
1854, 1857, 1862, 1864), mostly and perhaps all published at Birmingham.

[591] Solomon M. Drach wrote _An easy Rule for formulizing all Epicyclical
Curves_ (London, 1849), _On the Circle area and Heptagon-chord_ (London,
1864), _An easy general Rule for filling up all Magic Squares_ (London,
1873), and _Hebrew Almanack-Signs_ (London, 1877), besides numerous
articles in journals.

[592] See Vol. I, page 168, note 3 {371}.

[593] See Vol. I, page 254, note 2 {580}.

[594] See Vol. I, page 98, note 6 {163}.

[595] Robert Fludd or Flud (1574-1637) was a physician with a large London
practice. He denied the diurnal rotation of the earth, and was attacked by
Kepler and Mersenne, and accused of magic by Gassendi. His _Apologia
Compendiania, Fraternitatem de Rosea Cruce suspicionis ... maculis
aspersam, veritatis quasi Fluctibus abluens_ (Leyden, 1616) is one of a
large number of works of the mystic type.

[596] Consult _To the Christianity of the Age. Notes ... comprising an
elucidation of the scope and contents of the writings ... of Dionysius
Andreas Freher_ (1854).

[597] Sir William Robert Grove (1811-1896), although called to the bar
(1835) and to the bench (1853), is best known for his work as a physicist.
He was professor of experimental philosophy (1840-1847) at the London
Institution, and invented a battery (1839) known by his name. His
_Correlation of Physical Forces_ (1846) went through six editions and was
translated into French.

[598] Johann Tauler (c. 1300-1361), a Dominican monk of Strassburg, a
mystic, closely in touch with the Gottesfreunde of Basel. His _Sermons_
first appeared in print at Leipsic in 1498.

[599] Paracelsus (c. 1490-1541). His real name was Theophrastes Bombast von
Hohenheim, and he took the name by which he is generally known because he
held himself superior to Celsus. He was a famous physician and pharmacist,
but was also a mystic and neo-Platonist. He lectured in German on medicine
at Basel, but lost his position through the opposition of the orthodox
physicians and apothecaries.

[600] See Vol. I, page 256, note 2 {588}.

[601] Philip Schwarzerd (1497-1560) was professor of Greek at Wittenberg.
He helped Luther with his translation of the Bible.

[602] Johann Reuchlin (1455-1522), the first great German humanist, was
very influential in establishing the study of Greek and Hebrew in Germany.
His lectures were mostly delivered privately in Heidelberg and Stuttgart.
Unlike Melanchthon, he remained in the Catholic Church.

[603] Joseph Chitty (1776-1841) published his _Precedents of Pleading_ in
1808 and his _Reports of Cases on Practice and Pleading_ in 1820-23 (2

[604] See Vol. I, page 44, note 1 {35}.

[605] See Vol. I, page 44, note 4 {38}.

[606] Jean Pèlerin, also known as Viator, who wrote on perspective. His
work appeared in 1505, with editions in 1509 and 1521.

[607] Henry Stephens. See Vol. I, page 44, note 3 {37}.

[608] The well-known grammarian (1745-1826). He was born at Swatara, in
Pennsylvania, and practised law in New York until 1784, after which he
resided in England. His grammar (1795) went through 50 editions, and the
abridgment (1818) through 120 editions. Murray's friend Dalton, the
chemist, said that "of all the contrivances invented by human ingenuity for
puzzling the brains of the young, Lindley Murray's grammar was the worst."

[609] Robert Recorde (c. 1510-1558) read and probably taught mathematics
and medicine at Cambridge up to 1545. After that he taught mathematics at
Oxford and practised medicine in London. His _Grounde of Artes_, published
about 1540, was the first arithmetic published in English that had any
influence. It went through many editions. The _Castle of Knowledge_
appeared in 1551. It was a textbook on astronomy and the first to set forth
the Copernican theory in England. Like Recorde's other works it was written
on the catechism plan. His _Whetstone of Witte ... containying thextraction
of Rootes: The Cosike practise, with the rule of Equation: and the woorkes
of Surde Nombres_ appeared in 1557, and it is in this work that the modern
sign of equality first appears in print. The word "Cosike" is an adjective
that was used for a long time in Germany as equivalent to algebraic, being
derived from the Italian _cosa_, which stood for the unknown quantity.

[610] Robert Cecil (c. 1563-1612), first Earl of Salisbury, Secretary of
State under Elizabeth (1596-1603) and under James I (1603-1612).

[611] In America the German pronunciation is at present universal among
mathematicians, as in the case of most other German names. This is due, no
doubt, to the great influence that Germany has had on American education in
the last fifty years.

[612] The latest transliteration is substantially K'ung-fu-tz[vu].

[613] The tendency seems to be, however, to adopt the forms used of
individuals or places as rapidly as the mass of people comes to be prepared
for it. Thus the spelling Leipzig, instead of Leipsic, is coming to be very
common in America.

[614] Sir Edward Coke (1552-1634), the celebrated jurist.

[615] Dethlef Cluvier or Clüver (d. 1708 at Hamburg) was a nephew, not a
grandson, of Philippe Cluvier, or Philipp Clüver (1580-c. 1623). Dethlef
traveled in France and Italy and then taught mathematics in London. He
wrote on astronomy and philosophy and also published in the _Acta
Eruditorum_ (1686) his _Schediasma geometricum de nova infinitorum
scientia_. _Quadratura circuli infinitis modis demonstrata_, and his
_Monitum ad geometras_ (1687). Philippe was geographer of the Academy of
Leyden. His _Introductionis in universam geographiam tam veterem quam novam
libri sex_ appeared at Leyden in 1624, about the time of his death.

[616] See Vol. I, page 124, note 7 {248}.

[617] Bernard Nieuwentijt (1654-1718), a physician and burgomaster at
Purmerend. His _Considerationes circa Analyseos ad quantitates infinite
parvas applicatæ Principia et Calculi Differentialis usum_ (Amsterdam,
1694) was attacked by Leibnitz. He replied in his _Considerationes secundæ_
(1694), and also wrote the _Analysis Infinitorum, seu Curvilineorum
Proprietates ex Polygonorum Natura deductæ_ (1695). His most famous work
was on the existence of God, _Het Regt Gebruik der Werelt Beschouwingen_

[618] "From a given line to construct" etc.

[619] "Pirates do not fight one another."

[620] Claude Mallemens (Mallement) de Messanges (1653-1723) was professor
of philosophy at the Collège du Plessis, in Paris, for 34 years. The work
to which De Morgan refers is probably the _Fameux Problème de la quadrature
du cercle, résolu géometriquement par le cercle et a ligne droite_ that
appeared in 1683.

[621] On Tycho Brahe see Vol. I, page 76, note 3 {112}.

[622] Wilhelm Frederik von Zytphen also published the _Tidens Ström_, a
chronological table, in 1840. The work to which De Morgan refers, the
_Solens Bevægelse i Verdensrummet_, appeared first in 1861. De Morgan seems
to have missed his _Nogl Ord om Cirkelens Quadratur_ which appeared in
1865, at Copenhagen.

[623] James Joseph Sylvester (1814-1897), professor of natural philosophy
at University College, London (1837-1841), professor of mathematics at the
University of Virginia (1841-1845), actuary in London (1845-1855),
professor of mathematics at Woolwich (1877-1884) and at Johns Hopkins
University, Baltimore (1877-1884), and Savilian professor of geometry at
Oxford (1884-1894).

[624] See Vol. I, page 76, note 3 {112}.

[625] See Vol. II, page 205, note 349.

[626] See Vol. I, page 76, note 3 {112}.

[627] See Vol. I, page 46, note 1 {42}.

[628] See Vol. II, page 183, note 318.

[629] See Vol. I, page 321, note 2 {691}.

[630] James Mill, born 1773, died 1836.

[631] See Vol. II, page 3, note 11.

[632] See Vol. II, page 3, note 13.

[633] See Vol. II, page 3, note 14.

[634] This anecdote is printed at page 4 (Vol. II); but as it is used in
illustration here, and is given more in detail, I have not omitted
it.--S.E. De M.

[635] See Vol. II, page 4, note 15.

[636] See Vol. I, page 382, note 13 {786}.

[637] "Monsieur, (a + b^{n})/n = x, whence God exists; answer that!"

[638] "Monsieur, you know very well that your argument requires the
development of x according to integral powers of n."

[639] See Vol. I, page 153, note 4 {337}.

[640] Thomas Love Peacock (1785-1866) an English novelist and poet.

[641] Perhaps Dr. Samuel Warren (1807-1877), the author of _Ten Thousand a
Year_ (serially in Blackwood's in 1839; London, 1841).

[642] See Vol. I, page 255, note 6 {584}.

[643] "From many, one; much in little; Ultima Thule (the most remote
region); without which not."

[644] Spurius Mælius (fl. 440 B. C.), who distributed corn freely among the
poor in the famine of 440 B. C. and was assassinated by the patricians.

[645] Spurius Cassius Viscellinus, Roman consul in 502, 493, and 486 B. C.
Put to death in 485.

[646] "O what a fine bearing, he said, that has no brain."

[647] Sir William Rowan Hamilton. See Vol. I, page 332, note 4 {709}.

[648] William Allen Whitworth, the author of the well-known _Choice and
Chance_ (Cambridge, 1867), and other works.

[649] James Maurice Wilson, whose _Elementary Geometry_ appeared in 1868
and went through several editions.

[650] See Vol. II, page 183, note 315.

[651] "Force of inertia conquered," and "Victory in the whole heavens."

[652] "With all his might."

[653] George Berkeley (1685-1753), Bishop of Cloyne, the idealistic
philosopher and author of the _Principles of Human Knowledge_ (1710), _The
Analyst, or a Discourse addressed to an Infidel Mathematician_ (1734), and
_A Defense of Freethinking in Mathematics_ (1735). He asserted that space
involves the idea of movement without the sensation of resistance. Space
sensation less than the "minima sensibilia" is, therefore, impossible. From
this he argues that infinitesimals are impossible concepts.

[654] See Vol. I, page 85, note 2 {129}.

[655] See Vol. I, page 81, note 6 {120}.

[656] Edwin Dunkin revised Lardner's _Handbook of Astronomy_ (1869) and
Milner's _The Heavens and the Earth_ (1873) and wrote _The Midnight Sky_

[657] Michael Faraday (1791-1867) the celebrated physicist and chemist. He
was an assistant to Sir Humphrey Davy (1813) and became professor of
chemistry at the Royal Institution, London, in 1827.

[658] "If you teach a fool he shows no joyous countenance; he cordially
hates you; he wishes you buried."

[659] "Every man is an animal, Sortes is a man, therefore Sortes is an


 "May some choice patron bless each grey goose quill;
  May every Bavius have his Bufo still."--POPE, _Prologue to the Satires._

Bavius has become proverbial as a bad poet from the lines in Vergil's
_Eclogues_ (III, 90):

 "Qui Bavium non odit, amet tua carmina, Maevi,
  Atque idem jungat vulpes, et mulgeat hircos."

"He who does not hate Bavius shall love thy verses, O Maevius; and the same
shall yoke foxes and shall milk he-goats."

Bavius and Maevius were the worst of Latin poets, condemned by Horace as
well as Vergil.

[661] See Vol. II, page 158, note 279.

[662] "Honest," "useful," "handsome," "sweet."

[663] "Let not the fourth man attempt to speak."


 "In those old times,--ah
 'Twas just like this, ah!"

[665] See Vol. I, page 382, note 12 {785}.

[666] These remarks were never written.--S. E. De M.


 "Fleas, flies, and friars, are masters who sadly the people abuse,
  And thistles and briars are sure growing grains to abuse.
  O Christ, who hatest strife and slayst all things in peace,
  Destroy where'er are rife, briars, friars, flies and fleas.
  Fleas, flies, and friars foul fall them these fifteen years
  For none that there is loveth fleas, flies, nor freres."

[668] "It is my plan to restore to an unskilled race the worthy arts of a
better life."

[669] The first sentences of the first oration of Cicero against Catiline:
"Quo usque tandem abutere, Catilina, patientia nostra?" (How long, O
Catiline, will you abuse our patience?) "Quamdiu etiam furor iste tuus nos
eludet?" (How long will this your madness baffle us?) "Nihilne te nocturnum
praesidium Palati, ... nihil horum ora voltusque moverunt?" (Does the night
watch of the Palatium, ... do the faces and expressions of all these men
fail to move you?) "In te conferri ..." (This plague should have been
inflicted upon you long ago, which you have plotted against us so long.)

[670] "Beware of the things that are marked."

[671] "Farewell, ye teachers without learning! See to it that at our next
meeting we may find you strong in body and sound in mind."

[672] See Vol. I, page 336, note 8 {713}.

[673] See Vol. I, page 229, note 2 {515}.

[674] This proof, although capable of improvement, is left as in the
original. Those who may be interested in the mathematics of the question,
may consult F. Enriques, _Fragen der Elementargeometrie_ (German by
Fleischer), Leipsic, 1907, Part II, p. 267; F. Rudio, _Archimedes_,
_Huygens_, _Lambert_, _Legendre_. _Vier Abhandlungen über die
Kreismessung_, Leipsic, 1892; F. Klein, _Famous Problems of Elementary
Geometry_ (English by Beman and Smith), Boston, 1895; J. W. A. Young,
_Monographs on Modern Mathematics_, New York, 1911, Chap. IX (by the editor
of the present edition of De Morgan.)

[675] See Vol. I, page 69, note 2 {95}.

[676] See Vol. I, page 137, note 8 {286}.

[677] Joseph Allen Galbraith who, with Samuel Haughton, wrote the Galbraith
and Haughton's _Scientific Manuals_. (Euclid, 1856; Algebra, 1860;
Trigonometry, 1854; Optics, 1854, and others.)

[678] This note on Carlyle (1795-1881) is interesting. The translation of
Legendre appeared in the same year (1824) as his translation of Goethe's
_Wilhelm Meister_.

[679] Michael Stifel (1487-1567), also known as Stiefel, Styfel, and
Stifelius, was an Augustine monk but became a convert to Lutheranism. He
was professor of mathematics at Jena (1559-1567). His edition of the _Coss_
appeared at Königsberg in 1553, the first edition having been published in
1525. The + and - signs first appeared in print in Widman's arithmetic of
1489, but for purposes of algebra this book was one of the first to make
them known.

[680] Christoff Rudolff was born about 1500 and died between 1540 and 1552.
_Die Coss_ appeared in 1525 and his arithmetic in 1526.

       *       *       *       *       *

Corrections made to printed original.

Page 9, "long-fostered prejudice": 'perjudice' in original.

Page 73, "Pensées, ch. 7": 'Pansées' in original.

Page 127, "and pulled out a plum": 'und' in original.

Page 147, "did not come forward": 'forword' in original.

Page 172, "come into general circulation": 'circulalation' in original.

Ibid., "the more difficult fractions which we have got": 'he have got' in

Page 192, "it has been stated": 'started' in original.

Page 216, "the obsolete word tetch of the same meaning": 'meaing' in

Page 228, "[Greek: dioklasianos]": 'dioklalasianos' in original.

Page 233. After `Henry E. Manning' were printed two paragraphs `Shilling
versus Franc.' and `Teutonic Long Hundred 120 versus 100 or the Decimal
question.' These appear to have been set in error, there is no applicable

Page 316, "in a manner depending upon the difference": 'maner' in original.

Page 322, "neither what Newton did, nor what was done before him.": 'not'
(for 'nor') in original.

Page 344, "Victoria toto coelo": 'tolo' in original.

Page 368, "cannot be brought up to 1": 'up to ±' in original.

Page 371, "Q_2=b_2 Q_1+a_2": 'Q_2=b_2 Q_1-a_2' in original.

Note 50, "all who were not in the road to Heaven were excommunicated":
'excomunicated' in original.

Note 372, "[Greek: hê alazoneia biou]": 'alaxoneia' in original.

Ibid., "Iapetos": 'Ispetos' in original.

Ibid., "Papeiskos": 'Paspeisoks' in original.

Ibid., "[Greek: dioklasianos]": 'dioklalasianos' in original.

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