Title: Popular scientific lectures
Author: Mach, Ernst
Language: English
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POPULAR SCIENTIFIC LECTURES.



BY THE SAME AUTHOR.


THE SCIENCE OF MECHANICS. Translated from the Second German Edition
    by T. J. McCormack. 250 Cuts and Illustrations. 534 Pages. Half
    Morocco, Gilt Top. Price, $2.50.

CONTRIBUTIONS TO THE ANALYSIS OF THE SENSATIONS. Translated by C. M.
    Williams. With Notes and New Additions by the Author. 200 Pages.
    36 Cuts. Price, $1.00.

POPULAR SCIENTIFIC LECTURES. Translated by T. J. McCormack. Third
    Revised and Enlarged Edition. 411 Pages. 59 Cuts. Cloth, $1.50;
    Paper, 50 cents.

THE OPEN COURT PUBLISHING CO.,
324 DEARBORN ST., CHICAGO.



  POPULAR SCIENTIFIC LECTURES

  BY ERNST MACH

  FORMERLY PROFESSOR OF PHYSICS IN THE UNIVERSITY OF PRAGUE, NOW
  PROFESSOR OF THE HISTORY AND THEORY OF INDUCTIVE SCIENCE IN THE
  UNIVERSITY OF VIENNA

  TRANSLATED BY THOMAS J. McCORMACK

  THIRD EDITION, REVISED AND ENLARGED

  WITH FIFTY-NINE CUTS AND DIAGRAMS

  CHICAGO THE OPEN COURT PUBLISHING COMPANY

  FOR SALE BY

  KEGAN PAUL, TRENCH, TRUEBNER & CO., LONDON

  1898



  COPYRIGHT

  BY THE OPEN COURT PUBLISHING CO.

  Pages   1-258 }
                } in 1894.
  Pages 338-374 }
  Pages 259-281 in 1896.
  Pages 282-308 in 1897.
  Pages 309-337 in 1898.



AUTHOR'S PREFACE TO THE FIRST EDITION.


Popular lectures, owing to the knowledge they presuppose, and the time
they occupy, can afford only a _modicum_ of instruction. They must
select for this purpose easy subjects, and restrict themselves to the
exposition of the simplest and the most essential points. Nevertheless,
by an appropriate choice of the matter, the _charm_ and the _poetry_ of
research can be conveyed by them. It is only necessary to set forth the
attractive and the alluring features of a problem, and to show what
broad domains of fact can be illuminated by the light radiating from the
solution of a single and ofttimes unobtrusive point.

Furthermore, such lectures can exercise a favorable influence by showing
the substantial sameness of scientific and every-day thought. The
public, in this way, loses its shyness towards scientific questions, and
acquires an interest in scientific work which is a great help to the
inquirer. The latter, in his turn, is brought to understand that his
work is a small part only of the universal process of life, and that the
results of his labors must redound to the benefit not only of himself
and a few of his associates, but to that of the collective whole.

I sincerely hope that these lectures, in the present excellent
translation, will be productive of good in the direction indicated.

    E. MACH.

PRAGUE, December, 1894.



TRANSLATOR'S NOTE TO THE THIRD EDITION.


The present third edition of this work has been enlarged by the addition
of a new lecture, "On Some Phenomena Attending the Flight of
Projectiles." The additions to the second consisted of the following
four lectures and articles: Professor Mach's Vienna Inaugural Lecture,
"The Part Played by Accident in Invention and Discovery," the lecture on
"Sensations of Orientation," recently delivered and summing up the
results of an important psychological investigation, and two historical
articles (see Appendix) on Acoustics and Sight.

The lectures extend over a long period, from 1864 to 1898, and differ
greatly in style, contents, and purpose. They were first published in
collected form in English; afterwards two German editions were called
for.

As the dates of the first five lectures are not given in the footnotes
they are here appended. The first lecture, "On the Forms of Liquids,"
was delivered in 1868 and published with that "On Symmetry" in 1872
(Prague). The second and third lectures, on acoustics, were first
published in 1865 (Graz); the fourth and fifth, on optics, in 1867
(Graz). They belong to the earliest period of Professor Mach's
scientific activity, and with the lectures on electrostatics and
education will more than realise the hope expressed in the author's
Preface.

The eighth, ninth, tenth, eleventh, and twelfth lectures are of a more
philosophical character and deal principally with the methods and nature
of scientific inquiry. In the ideas summarised in them will be found one
of the most important contributions to the theory of knowledge made in
the last quarter of a century. Significant hints in psychological
method, and exemplary specimen-researches in psychology and physics, are
also presented; while in physics many ideas find their first discussion
that afterwards, under other names and other authorship, became
rallying-cries in this department of inquiry.

All the proofs of this translation have been read by Professor Mach
himself.

    T. J. MCCORMACK.

LA SALLE, ILL., May, 1898.



TABLE OF CONTENTS.


  The Forms of Liquids                                             1
  The Fibres of Corti                                             17
  On the Causes of Harmony                                        32
  The Velocity of Light                                           48
  Why Has Man Two Eyes?                                           66
  On Symmetry                                                     89
  On the Fundamental Concepts of Electrostatics                  107
  On the Principle of the Conservation of Energy                 137
  On the Economical Nature of Physical Inquiry                   186
  On Transformation and Adaptation in Scientific Thought         214
  On the Principle of Comparison in Physics                      236
  On the Part Played by Accident in Invention and Discovery      259
  On Sensations of Orientation                                   282
  On Some Phenomena Attending the Flight of Projectiles          309
  On Instruction in the Classics and the Mathematico-Physical
      Sciences                                                   338
  Appendixes.
      I. A Contribution to the History of Acoustics              375
      II. Remarks on the Theory of Spatial Vision                386
  Index                                                          393



THE FORMS OF LIQUIDS.


What thinkest thou, dear Euthyphron, that the holy is, and the just, and
the good? Is the holy holy because the gods love it, or are the gods
holy because they love the holy? By such easy questions did the wise
Socrates make the market-place of Athens unsafe and relieve presumptuous
young statesmen of the burden of imaginary knowledge, by showing them
how confused, unclear, and self-contradictory their ideas were.

You know the fate of the importunate questioner. So called good society
avoided him on the promenade. Only the ignorant accompanied him. And
finally he drank the cup of hemlock--a lot which we ofttimes wish would
fall to modern critics of his stamp.

What we have learned from Socrates, however,--our inheritance from
him,--is scientific criticism. Every one who busies himself with science
recognises how unsettled and indefinite the notions are which he has
brought with him from common life, and how, on a minute examination of
things, old differences are effaced and new ones introduced. The
history of science is full of examples of this constant change,
development, and clarification of ideas.

But we will not linger by this general consideration of the fluctuating
character of ideas, which becomes a source of real uncomfortableness,
when we reflect that it applies to almost every notion of life. Rather
shall we observe by the study of a physical example how much a thing
changes when it is closely examined, and how it assumes, when thus
considered, increasing definiteness of form.

The majority of you think, perhaps, you know quite well the distinction
between a liquid and a solid. And precisely persons who have never
busied themselves with physics will consider this question one of the
easiest that can be put. But the physicist knows that it is one of the
most difficult. I shall mention here only the experiments of Tresca,
which show that solids subjected to high pressures behave exactly as
liquids do; for example, may be made to flow out in the form of jets
from orifices in the bottoms of vessels. The supposed difference of kind
between liquids and solids is thus shown to be a mere difference of
degree.

The common inference that because the earth is oblate in form, it was
originally fluid, is an error, in the light of these facts. True, a
rotating sphere, a few inches in diameter will assume an oblate form
only if it is very soft, for example, is composed of freshly kneaded
clay or some viscous stuff. But the earth, even if it consisted of the
rigidest stone, could not help being crushed by its tremendous weight,
and must perforce behave as a fluid. Even our mountains could not extend
beyond a certain height without crumbling. The earth _may_ once have
been fluid, but this by no means follows from its oblateness.

The particles of a liquid are displaced on the application of the
slightest pressure; a liquid conforms exactly to the shapes of the
vessels in which it is contained; it possesses no form of its own, as
you have all learned in the schools. Accommodating itself in the most
trifling respects to the conditions of the vessel in which it is placed,
and showing, even on its surface, where one would suppose it had the
freest play, nothing but a polished, smiling, expressionless
countenance, it is the courtier _par excellence_ of the natural bodies.

Liquids have no form of their own! No, not for the superficial observer.
But persons who have observed that a raindrop is round and never
angular, will not be disposed to accept this dogma so unconditionally.

It is fair to suppose that every man, even the weakest, would possess a
character, if it were not too difficult in this world to keep it. So,
too, we must suppose that liquids would possess forms of their own, if
the pressure of the circumstances permitted it,--if they were not
crushed by their own weights.

An astronomer once calculated that human beings could not exist on the
sun, apart from its great heat, because they would be crushed to pieces
there by their own weight. The greater mass of this body would also
make the weight of the human body there much greater. But on the moon,
because here we should be much lighter, we could jump as high as the
church-steeples without any difficulty, with the same muscular power
which we now possess. Statues and "plaster" casts of syrup are
undoubtedly things of fancy, even on the moon, but maple-syrup would
flow so slowly there that we could easily build a maple-syrup man on the
moon, for the fun of the thing, just as our children here build
snow-men.

Accordingly, if liquids have no form of their own with us on earth, they
have, perhaps, a form of their own on the moon, or on some smaller and
lighter heavenly body. The problem, then, simply is to get rid of the
effects of gravity; and, this done, we shall be able to find out what
the peculiar forms of liquids are.

The problem was solved by Plateau of Ghent, whose method was to immerse
the liquid in another of the same specific gravity.[1] He employed for
his experiments oil and a mixture of alcohol and water. By Archimedes's
well-known principle, the oil in this mixture loses its entire weight.
It no longer sinks beneath its weight; its formative forces, be they
ever so weak, are now in full play.

As a fact, we now see, to our surprise, that the oil, instead of
spreading out into a layer, or lying in a formless mass, assumes the
shape of a beautiful and perfect sphere, freely suspended in the
mixture, as the moon is in space. We can construct in this way a sphere
of oil several inches in diameter.

If, now, we affix a thin plate to a wire and insert the plate in the oil
sphere, we can, by twisting the wire between our fingers, set the whole
ball in rotation. Doing this, the ball assumes an oblate shape, and we
can, if we are skilful enough, separate by such rotation a ring from the
ball, like that which surrounds Saturn. This ring is finally rent
asunder, and, breaking up into a number of smaller balls, exhibits to us
a kind of model of the origin of the planetary system according to the
hypothesis of Kant and Laplace.

[Illustration: Fig. 1.]

Still more curious are the phenomena exhibited when the formative forces
of the liquid are partly disturbed by putting in contact with the
liquid's surface some rigid body. If we immerse, for example, the wire
framework of a cube in our mass of oil, the oil will everywhere stick to
the wire framework. If the quantity of oil is exactly sufficient we
shall obtain an oil cube with perfectly smooth walls. If there is too
much or too little oil, the walls of the cube will bulge out or cave in.
In this manner we can produce all kinds of geometrical figures of oil,
for example, a three-sided pyramid, a cylinder (by bringing the oil
between two wire rings), and so on. Interesting is the change of form
that occurs when we gradually suck out the oil by means of a glass tube
from the cube or pyramid. The wire holds the oil fast. The figure grows
smaller and smaller, until it is at last quite thin. Ultimately it
consists simply of a number of thin, smooth plates of oil, which extend
from the edges of the cube to the centre, where they meet in a small
drop. The same is true of the pyramid.

[Illustration: Fig. 2.]

The idea now suggests itself that liquid figures as thin as this, and
possessing, therefore, so slight a weight, cannot be crushed or deformed
by their weight; just as a small, soft ball of clay is not affected in
this respect by its weight. This being the case, we no longer need our
mixture of alcohol and water for the production of figures, but can
construct them in the open air. And Plateau, in fact, found that these
thin figures, or at least very similar ones, could be produced in the
air, by dipping the wire nets described in a solution of soap and water
and quickly drawing them out again. The experiment is not difficult. The
figure is formed of itself. The preceding drawing represents to the eye
the forms obtained with cubical and pyramidal nets. In the cube, thin,
smooth films of soap-suds proceed from the edges to a small, quadratic
film in the centre. In the pyramid, a film proceeds from each edge to
the centre.

These figures are so beautiful that they hardly admit of appropriate
description. Their great regularity and geometrical exactness evokes
surprise from all who see them for the first time. Unfortunately, they
are of only short duration. They burst, on the drying of the solution in
the air, but only after exhibiting to us the most brilliant play of
colors, such as is often seen in soap-bubbles. Partly their beauty of
form and partly our desire to examine them more minutely induces us to
conceive of methods of endowing them with permanent form. This is very
simply done.[2] Instead of dipping the wire nets in solutions of soap,
we dip them in pure melted colophonium (resin). When drawn out the
figure at once forms and solidifies by contact with the air.

It is to be remarked that also solid fluid-figures can be constructed
in the open air, if their weight be light enough, or the wire nets of
very small dimensions. If we make, for example, of very fine wire a
cubical net whose sides measure about one-eighth of an inch in length,
we need simply to dip this net in water to obtain a small solid cube of
water. With a piece of blotting paper the superfluous water may be
easily removed and the sides of the cube made smooth.

Yet another simple method may be devised for observing these figures. A
drop of water on a greased glass plate will not run if it is small
enough, but will be flattened by its weight, which presses it against
its support. The smaller the drop the less the flattening. The smaller
the drop the nearer it approaches the form of a sphere. On the other
hand, a drop suspended from a stick is elongated by its weight. The
undermost parts of a drop of water on a support are pressed against the
support, and the upper parts are pressed against the lower parts because
the latter cannot yield. But when a drop falls freely downward all its
parts move equally fast; no part is impeded by another; no part presses
against another. A freely falling drop, accordingly, is not affected by
its weight; it acts as if it were weightless; it assumes a spherical
form.

A moment's glance at the soap-film figures produced by our various wire
models, reveals to us a great multiplicity of form. But great as this
multiplicity is, the common features of the figures also are easily
discernible.

    "All forms of Nature are allied, though none is the same as the other;
    Thus, their common chorus points to a hidden law."

This hidden law Plateau discovered. It may be expressed, somewhat
prosily, as follows:

1) If several plane liquid films meet in a figure they are always three
in number, and, taken in pairs, form, each with another, nearly equal
angles.

2) If several liquid edges meet in a figure they are always four in
number, and, taken in pairs, form, each with another, nearly equal
angles.

This is a strange law, and its reason is not evident. But we might apply
this criticism to almost all laws. It is not always that the motives of
a law-maker are discernible in the form of the law he constructs. But
our law admits of analysis into very simple elements or reasons. If we
closely examine the paragraphs which state it, we shall find that their
meaning is simply this, that the surface of the liquid assumes the shape
of smallest area that is possible under the circumstances.

If, therefore, some extraordinarily intelligent tailor, possessing a
knowledge of all the artifices of the higher mathematics, should set
himself the task of so covering the wire frame of a cube with cloth that
every piece of cloth should be connected with the wire and joined with
the remaining cloth, and should seek to accomplish this feat with the
greatest saving of material, he would construct no other figure than
that which is here formed on the wire frame in our solution of soap and
water. Nature acts in the construction of liquid figures on the
principle of a covetous tailor, and gives no thought in her work to the
fashions. But, strange to say, in this work, the most beautiful fashions
are of themselves produced.

The two paragraphs which state our law apply primarily only to soap-film
figures, and are not applicable, of course, to solid oil-figures. But
the principle that the superficial area of the liquid shall be the least
possible under the circumstances, is applicable to all fluid figures. He
who understands not only the letter but also the reason of the law will
not be at a loss when confronted with cases to which the letter does not
accurately apply. And this is the case with the principle of least
superficial area. It is a sure guide for us even in cases in which the
above-stated paragraphs are not applicable.

Our first task will now be, to show by a palpable illustration the mode
of formation of liquid figures by the principle of least superficial
area. The oil on the wire pyramid in our mixture of alcohol and water,
being unable to leave the wire edges, clings to them, and the given mass
of oil strives so to shape itself that its surface shall have the least
possible area. Suppose we attempt to imitate this phenomenon. We take a
wire pyramid, draw over it a stout film of rubber, and in place of the
wire handle insert a small tube leading into the interior of the space
enclosed by the rubber (Fig. 3). Through this tube we can blow in or
suck out air. The quantity of air in the enclosure represents the
quantity of oil. The stretched rubber film, which, clinging to the wire
edges, does its utmost to contract, represents the surface of the oil
endeavoring to decrease its area. By blowing in, and drawing out the
air, now, we actually obtain all the oil pyramidal figures, from those
bulged out to those hollowed in. Finally, when all the air is pumped or
sucked out, the soap-film figure is exhibited. The rubber films strike
together, assume the form of planes, and meet at four sharp edges in the
centre of the pyramid.

[Illustration: Fig. 3.]

[Illustration: Fig. 4.]

The tendency of soap-films to assume smaller forms may be directly
demonstrated by a method of Van der Mensbrugghe. If we dip a square wire
frame to which a handle is attached into a solution of soap and water,
we shall obtain on the frame a beautiful, plane film of soap-suds. (Fig.
4.) On this we lay a thread having its two ends tied together. If, now,
we puncture the part enclosed by the thread, we shall obtain a soap-film
having a circular hole in it, whose circumference is the thread. The
remainder of the film decreasing in area as much as it can, the hole
assumes the largest area that it can. But the figure of largest area,
with a given periphery, is the circle.

[Illustration: Fig. 5.]

Similarly, by the principle of least superficial area, a freely
suspended mass of oil assumes the shape of a sphere. The sphere is the
form of least surface for a given content. This is evident. The more we
put into a travelling-bag, the nearer its shape approaches the spherical
form.

The connexion of the two above-mentioned paragraphs with the principle
of least superficial area may be shown by a yet simpler example. Picture
to yourselves four fixed pulleys, _a_, _b_, _c_, _d_, and two movable
rings _f_, _g_ (Fig. 5); about the pulleys and through the rings imagine
a smooth cord passed, fastened at one extremity to a nail _e_, and
loaded at the other with a weight _h_. Now this weight always tends to
sink, or, what is the same thing, always tends to make the portion of
the string _e h_ as long as possible, and consequently the remainder of
the string, wound round the pulleys, as short as possible. The strings
must remain connected with the pulleys, and on account of the rings also
with each other. The conditions of the case, accordingly, are similar to
those of the liquid figures discussed. The result also is a similar one.
When, as in the right hand figure of the cut, four pairs of strings
meet, a different configuration must be established. The consequence of
the endeavor of the string to shorten itself is that the rings separate
from each other, and that now at all points only three pairs of strings
meet, every two at equal angles of one hundred and twenty degrees. As a
fact, by this arrangement the greatest possible shortening of the string
is attained; as can be easily proved by geometry.

This will help us to some extent to understand the creation of beautiful
and complicated figures by the simple tendency of liquids to assume
surfaces of least superficial area. But the question arises, _Why_ do
liquids seek surfaces of least superficial area?

The particles of a liquid cling together. Drops brought into contact
coalesce. We can say, liquid particles attract each other. If so, they
seek to come as close as they can to each other. The particles at the
surface will endeavor to penetrate as far as they can into the interior.
This process will not stop, cannot stop, until the surface has become as
small as under the circumstances it possibly can become, until as few
particles as possible remain at the surface, until as many particles as
possible have penetrated into the interior, until the forces of
attraction have no more work to perform.[3]

The root of the principle of least surface is to be sought, accordingly,
in another and much simpler principle, which may be illustrated by some
such analogy as this. We can _conceive_ of the natural forces of
attraction and repulsion as purposes or intentions of nature. As a
matter of fact, that interior pressure which we feel before an act and
which we call an intention or purpose, is not, in a final analysis, so
essentially different from the pressure of a stone on its support, or
the pressure of a magnet on another, that it is necessarily unallowable
to use for both the same term--at least for well-defined purposes.[4] It
is the purpose of nature, accordingly, to bring the iron nearer the
magnet, the stone nearer the centre of the earth, and so forth. If such
a purpose can be realised, it is carried out. But where she cannot
realise her purposes, nature does nothing. In this respect she acts
exactly as a good man of business does.

It is a constant purpose of nature to bring weights lower. We can raise
a weight by causing another, larger weight to sink; that is, by
satisfying another, more powerful, purpose of nature. If we fancy we are
making nature serve our purposes in this, it will be found, upon closer
examination, that the contrary is true, and that nature has employed us
to attain her purposes.

Equilibrium, rest, exists only, but then always, when nature is brought
to a halt in her purposes, when the forces of nature are as fully
satisfied as, under the circumstances, they can be. Thus, for example,
heavy bodies are in equilibrium, when their so-called centre of gravity
lies as low as it possibly can, or when as much weight as the
circumstances admit of has sunk as low as it can.

The idea forcibly suggests itself that perhaps this principle also holds
good in other realms. Equilibrium exists also in the state when the
purposes of the parties are as fully satisfied as for the time being
they can be, or, as we may say, jestingly, in the language of physics,
when the social potential is a maximum.[5]

You see, our miserly mercantile principle is replete with
consequences.[6] The result of sober research, it has become as fruitful
for physics as the dry questions of Socrates for science generally. If
the principle seems to lack in ideality, the more ideal are the fruits
which it bears.

But why, tell me, should science be ashamed of such a principle? Is
science[7] itself anything more than--a business? Is not its task to
acquire with the least possible work, in the least possible time, with
the least possible thought, the greatest possible part of eternal truth?


  FOOTNOTES:

  [Footnote 1: _Statique expérimentale et théorique des liquids_,
  1873. See also _The Science of Mechanics_, p. 384 et seqq., The Open
  Court Publishing Co., Chicago, 1893.]

  [Footnote 2: Compare Mach, _Ueber die Molecularwirkung der
  Flüssigkeiten_, Reports of the Vienna Academy, 1862.]

  [Footnote 3: In almost all branches of physics that are well worked
  out such maximal and minimal problems play an important part.]

  [Footnote 4: Compare Mach, _Vorträge über Psychophysik_, Vienna,
  1863, page 41; _Compendium der Physik für Mediciner_, Vienna, 1863,
  page 234; and also _The Science of Mechanics_, Chicago, 1893, pp. 84
  and 464.]

  [Footnote 5: Like reflexions are found in Quételet, _Du système
  sociale_.]

  [Footnote 6: For the full development of this idea see the essay "On
  the Economical Nature of Physical Inquiry," p. 186, and the chapter
  on "The Economy of Science," in my _Mechanics_ (Chicago: The Open
  Court Publishing Company, 1893), p. 481.]

  [Footnote 7: Science may be regarded as a maximum or minimum
  problem, exactly as the business of the merchant. In fact, the
  intellectual activity of natural inquiry is not so greatly different
  from that exercised in ordinary life as is usually supposed.]



THE FIBRES OF CORTI.


Whoever has roamed through a beautiful country knows that the tourist's
delights increase with his progress. How pretty that wooded dell must
look from yonder hill! Whither does that clear brook flow, that hides
itself in yonder sedge? If I only knew how the landscape looked behind
that mountain! Thus even the child thinks in his first rambles. It is
also true of the natural philosopher.

The first questions are forced upon the attention of the inquirer by
practical considerations; the subsequent ones are not. An irresistible
attraction draws him to these; a nobler interest which far transcends
the mere needs of life. Let us look at a special case.

For a long time the structure of the organ of hearing has actively
engaged the attention of anatomists. A considerable number of brilliant
discoveries has been brought to light by their labors, and a splendid
array of facts and truths established. But with these facts a host of
new enigmas has been presented.

Whilst in the theory of the organisation and functions of the eye
comparative clearness has been attained; whilst, hand in hand with this,
ophthalmology has reached a degree of perfection which the preceding
century could hardly have dreamed of, and by the help of the
ophthalmoscope the observing physician penetrates into the profoundest
recesses of the eye, the theory of the ear is still much shrouded in
mysterious darkness, full of attraction for the investigator.

Look at this model of the ear. Even at that familiar part by whose
extent we measure the quantity of people's intelligence, even at the
external ear, the problems begin. You see here a succession of helixes
or spiral windings, at times very pretty, whose significance we cannot
accurately state, yet for which there must certainly be some reason.

[Illustration: Fig. 6.]

The shell or concha of the ear, _a_ in the annexed diagram, conducts the
sound into the curved auditory passage _b_, which is terminated by a
thin membrane, the so-called tympanic membrane, _e_. This membrane is
set in motion by the sound, and in its turn sets in motion a series of
little bones of very peculiar formation, _c_. At the end of all is the
labyrinth _d_. The labyrinth consists of a group of cavities filled with
a liquid, in which the innumerable fibres of the nerve of hearing are
imbedded. By the vibration of the chain of bones _c_, the liquid of the
labyrinth is shaken, and the auditory nerve excited. Here the process of
hearing begins. So much is certain. But the details of the process are
one and all unanswered questions.

To these old puzzles, the Marchese Corti, as late as 1851, added a new
enigma. And, strange to say, it is this last enigma, which, perhaps, has
first received its correct solution. This will be the subject of our
remarks to-day.

Corti found in the cochlea, or snail-shell of the labyrinth, a large
number of microscopic fibres placed side by side in geometrically
graduated order. According to Kölliker their number is three thousand.
They were also the subject of investigation at the hands of Max Schultze
and Deiters.

A description of the details of this organ would only weary you, besides
not rendering the matter much clearer. I prefer, therefore, to state
briefly what in the opinion of prominent investigators like Helmholtz
and Fechner is the peculiar function of Corti's fibres. The cochlea, it
seems, contains a large number of elastic fibres of graduated lengths
(Fig. 7), to which the branches of the auditory nerve are attached.
These fibres, called the fibres, pillars, or rods of Corti, being of
unequal length, must also be of unequal elasticity, and, consequently,
pitched to different notes. The cochlea, therefore, is a species of
pianoforte.

[Illustration: Fig. 7.]

What, now, may be the office of this structure, which is found in no
other organ of sense? May it not be connected with some special
property of the ear? It is quite probable; for the ear possesses a very
similar power. You know that it is possible to follow the individual
voices of a symphony. Indeed, the feat is possible even in a fugue of
Bach, where it is certainly no inconsiderable achievement. The ear can
pick out the single constituent tonal parts, not only of a harmony, but
of the wildest clash of music imaginable. The musical ear analyses every
agglomeration of tones.

The eye does not possess this ability. Who, for example, could tell from
the mere sight of white, without a previous experimental knowledge of
the fact, that white is composed of a mixture of other colors? Could it
be, now, that these two facts, the property of the ear just mentioned,
and the structure discovered by Corti, are really connected? It is very
probable. The enigma is solved if we assume that every note of definite
pitch has its special string in this pianoforte of Corti, and,
therefore, its special branch of the auditory nerve attached to that
string. But before I can make this point perfectly plain to you, I must
ask you to follow me a few steps into the dry domain of physics.

Look at this pendulum. Forced from its position of equilibrium by an
impulse, it begins to swing with a definite time of oscillation,
dependent upon its length. Longer pendulums swing more slowly, shorter
ones more quickly. We will suppose our pendulum to execute one
to-and-fro movement in a second.

This pendulum, now, can be thrown into violent vibration in two ways;
either by a _single_ heavy impulse, or by a _number_ of properly
communicated slight impulses. For example, we impart to the pendulum,
while at rest in its position of equilibrium, a very slight impulse. It
will execute a very small vibration. As it passes a third time its
position of equilibrium, a second having elapsed, we impart to it again
a slight shock, in the same direction with the first. Again after the
lapse of a second, on its fifth passage through the position of
equilibrium, we strike it again in the same manner; and so continue. You
see, by this process the shocks imparted augment continually the motion
of the pendulum. After each slight impulse, the pendulum reaches out a
little further in its swing, and finally acquires a considerable
motion.[8]

But this is not the case under all circumstances. It is possible only
when the impulses imparted synchronise with the swings of the pendulum.
If we should communicate the second impulse at the end of half a second
and in the same direction with the first impulse, its effects would
counteract the motion of the pendulum. It is easily seen that our little
impulses help the motion of the pendulum more and more, according as
their time accords with the time of the pendulum. If we strike the
pendulum in any other time than in that of its vibration, in some
instances, it is true, we shall augment its vibration, but in others
again, we shall obstruct it. Our impulses will be less effective the
more the motion of our own hand departs from the motion of the pendulum.

What is true of the pendulum holds true of every vibrating body. A
tuning-fork when it sounds, also vibrates. It vibrates more rapidly when
its sound is higher; more slowly when it is deeper. The standard _A_ of
our musical scale is produced by about four hundred and fifty vibrations
in a second.

I place by the side of each other on this table two tuning-forks,
exactly alike, resting on resonant cases. I strike the first one a sharp
blow, so that it emits a loud note, and immediately grasp it again with
my hand to quench its note. Nevertheless, you still hear the note
distinctly sounded, and by feeling it you may convince yourselves that
the other fork which was not struck now vibrates.

I now attach a small bit of wax to one of the forks. It is thrown thus
out of tune; its note is made a little deeper. I now repeat the same
experiment with the two forks, now of unequal pitch, by striking one of
them and again grasping it with my hand; but in the present case the
note ceases the very instant I touch the fork.

What has happened here in these two experiments? Simply this. The
vibrating fork imparts to the air and to the table four hundred and
fifty shocks a second, which are carried over to the other fork. If the
other fork is pitched to the same note, that is to say, if it vibrates
when struck in the same time with the first, then the shocks first
emitted, no matter how slight they may be, are sufficient to throw the
second fork into rapid sympathetic vibration. But when the time of
vibration of the two forks is slightly different, this does not take
place. We may strike as many forks as we will, the fork tuned to _A_ is
perfectly indifferent to their notes; is deaf, in fact, to all except
its own; and if you strike three, or four, or five, or any number
whatsoever, of forks all at the same time, so as to make the shocks
which come from them ever so great, the _A_ fork will not join in with
their vibrations unless another fork _A_ is found in the collection
struck. It picks out, in other words, from all the notes sounded, that
which accords with it.

The same is true of all bodies which can yield notes. Tumblers resound
when a piano is played, on the striking of certain notes, and so do
window panes. Nor is the phenomenon without analogy in other provinces.
Take a dog that answers to the name "Nero." He lies under your table.
You speak of Domitian, Vespasian, and Marcus Aurelius Antoninus, you
call upon all the names of the Roman Emperors that occur to you, but the
dog does not stir, although a slight tremor of his ear tells you of a
faint response of his consciousness. But the moment you call "Nero" he
jumps joyfully towards you. The tuning-fork is like your dog. It answers
to the name _A_.

You smile, ladies. You shake your heads. The simile does not catch your
fancy. But I have another, which is very near to you: and for punishment
you shall hear it. You, too, are like tuning-forks. Many are the hearts
that throb with ardor for you, of which you take no notice, but are
cold. Yet what does it profit you! Soon the heart will come that beats
in just the proper rhythm, and then your knell, too, has struck. Then
your heart, too, will beat in unison, whether you will or no.

The law of sympathetic vibration, here propounded for sounding bodies,
suffers some modification for bodies incompetent to yield notes. Bodies
of this kind vibrate to almost every note. A high silk hat, we know,
will not sound; but if you will hold your hat in your hand when
attending your next concert you will not only hear the pieces played,
but also feel them with your fingers. It is exactly so with men. People
who are themselves able to give tone to their surroundings, bother
little about the prattle of others. But the person without character
tarries everywhere: in the temperance hall, and at the bar of the
public-house--everywhere where a committee is formed. The high silk hat
is among bells what the weakling is among men of conviction.

A sonorous body, therefore, always sounds when its special note, either
alone or in company with others, is struck. We may now go a step
further. What will be the behaviour of a group of sonorous bodies which
in the pitch of their notes form a scale? Let us picture to ourselves,
for example (Fig. 8), a series of rods or strings pitched to the notes
_c d e f g_.... On a musical instrument the accord _c e g_ is struck.
Every one of the rods of Fig. 8 will see if its special note is
contained in the accord, and if it finds it, it will respond. The rod
_c_ will give at once the note _c_, the rod _e_ the note _e_, the rod
_g_ the note _g_. All the other rods will remain at rest, will not
sound.

[Illustration: Fig. 8.]

We need not look about us long for such an instrument. Every piano is an
instrument of this kind, with which the experiment mentioned may be
executed with splendid success. Two pianos stand here by the side of
each other, both tuned alike. We will employ the first for exciting the
notes, while we will allow the second to respond; after having first
pressed upon the loud pedal, so as to render all the strings capable of
motion.

Every harmony struck with vigor on the first piano is distinctly
repeated on the second. To prove that it is the same strings that are
sounded in both pianos, we repeat the experiment in a slightly changed
form. We let go the loud pedal of the second piano and pressing on the
keys _c e g_ of that instrument vigorously strike the harmony _c e g_ on
the first piano. The harmony _c e g_ is now also sounded on the second
piano. But if we press only on one key _g_ of one piano, while we strike
_c e g_ on the other, only _g_ will be sounded on the second. It is
thus always the like strings of the two pianos that excite each other.

The piano can reproduce any sound that is composed of its musical notes.
It will reproduce, for example, very distinctly, a vowel sound that is
sung into it. And in truth physics has proved that the vowels may be
regarded as composed of simple musical notes.

You see that by the exciting of definite tones in the air quite definite
motions are set up with mechanical necessity in the piano. The idea
might be made use of for the performance of some pretty pieces of
wizardry. Imagine a box in which is a stretched string of definite
pitch. This is thrown into motion as often as its note is sung or
whistled. Now it would not be a very difficult task for a skilful
mechanic to so construct the box that the vibrating cord would close a
galvanic circuit and open the lock. And it would not be a much more
difficult task to construct a box which would open at the whistling of a
certain melody. Sesame! and the bolts fall. Truly, we should have here a
veritable puzzle-lock. Still another fragment rescued from that old
kingdom of fables, of which our day has realised so much, that world of
fairy-stories to which the latest contributions are Casselli's
telegraph, by which one can write at a distance in one's own hand, and
Prof. Elisha Gray's telautograph. What would the good old Herodotus have
said to these things who even in Egypt shook his head at much that he
saw? [Greek: emoi men ou pista], just as simple-heartedly as then, when
he heard of the circumnavigation of Africa.

A new puzzle-lock! But why invent one? Are not we human beings ourselves
puzzle-locks? Think of the stupendous groups of thoughts, feelings, and
emotions that can be aroused in us by a word! Are there not moments in
all our lives when a mere name drives the blood to our hearts? Who that
has attended a large mass-meeting has not experienced what tremendous
quantities of energy and motion can be evolved by the innocent words,
"Liberty, Equality, Fraternity."

But let us return to the subject proper of our discourse. Let us look
again at our piano, or what will do just as well, at some other
contrivance of the same character. What does this instrument do?
Plainly, it decomposes, it analyses every agglomeration of sounds set up
in the air into its individual component parts, each tone being taken up
by a different string; it performs a real spectral analysis of sound. A
person completely deaf, with the help of a piano, simply by touching the
strings or examining their vibrations with a microscope, might
investigate the sonorous motion of the air, and pick out the separate
tones excited in it.

The ear has the same capacity as this piano. The ear performs for the
mind what the piano performs for a person who is deaf. The mind without
the ear is deaf. But a deaf person, with the piano, does hear after a
fashion, though much less vividly, and more clumsily, than with the
ear. The ear, thus, also decomposes sound into its component tonal
parts. I shall now not be deceived, I think, if I assume that you
already have a presentiment of what the function of Corti's fibres is.
We can make the matter very plain to ourselves. We will use the one
piano for exciting the sounds, and we shall imagine the second one in
the ear of the observer in the place of Corti's fibres, which is a model
of such an instrument. To every string of the piano in the ear we will
suppose a special fibre of the auditory nerve attached, so that this
fibre and this alone, is irritated when the string is thrown into
vibration. If we strike now an accord on the external piano, for every
tone of that accord a definite string of the internal piano will sound
and as many different nervous fibres will be irritated as there are
notes in the accord. The simultaneous sense-impressions due to different
notes can thus be preserved unmingled and be separated by the attention.
It is the same as with the five fingers of the hand. With each finger I
can touch something different. Now the ear has three thousand such
fingers, and each one is designed for the touching of a different
tone.[9] Our ear is a puzzle-lock of the kind mentioned. It opens at the
magic melody of a sound. But it is a stupendously ingenious lock. Not
only one tone, but every tone makes it open; but each one differently.
To each tone it replies with a different sensation.

More than once it has happened in the history of science that a
phenomenon predicted by theory, has not been brought within the range of
actual observation until long afterwards. Leverrier predicted the
existence and the place of the planet Neptune, but it was not until
sometime later that Galle actually found the planet at the predicted
spot. Hamilton unfolded theoretically the phenomenon of the so-called
conical refraction of light, but it was reserved for Lloyd some time
subsequently to observe the fact. The fortunes of Helmholtz's theory of
Corti's fibres have been somewhat similar. This theory, too, received
its substantial confirmation from the subsequent observations of V.
Hensen. On the free surface of the bodies of Crustacea, connected with
the auditory nerves, rows of little hairy filaments of varying lengths
and thicknesses are found, which to some extent are the analogues of
Corti's fibres. Hensen saw these hairs vibrate when sounds were excited,
and when different notes were struck different hairs were set in
vibration.

I have compared the work of the physical inquirer to the journey of the
tourist. When the tourist ascends a new hill he obtains of the whole
district a different view. When the inquirer has found the solution of
one enigma, the solution of a host of others falls into his hands.

Surely you have often felt the strange impression experienced when in
singing through the scale the octave is reached, and nearly the same
sensation is produced as by the fundamental tone. The phenomenon finds
its explanation in the view here laid down of the ear. And not only this
phenomenon but all the laws of the theory of harmony may be grasped and
verified from this point of view with a clearness before undreamt of.
Unfortunately, I must content myself to-day with the simple indication
of these beautiful prospects. Their consideration would lead us too far
aside into the fields of other sciences.

The searcher of nature, too, must restrain himself in his path. He also
is drawn along from one beauty to another as the tourist from dale to
dale, and as circumstances generally draw men from one condition of life
into others. It is not he so much that makes the quests, as that the
quests are made of him. Yet let him profit by his time, and let not his
glance rove aimlessly hither and thither. For soon the evening sun will
shine, and ere he has caught a full glimpse of the wonders close by, a
mighty hand will seize him and lead him away into a different world of
puzzles.

Respected hearers, science once stood in an entirely different relation
to poetry. The old Hindu mathematicians wrote their theorems in verses,
and lotus-flowers, roses, and lilies, beautiful sceneries, lakes, and
mountains figured in their problems.

"Thou goest forth on this lake in a boat. A lily juts forth, one palm
above the water. A breeze bends it downwards, and it vanishes two palms
from its previous spot beneath the surface. Quick, mathematician, tell
me how deep is the lake!"

Thus spoke an ancient Hindu scholar. This poetry, and rightly, has
disappeared from science, but from its dry leaves another poetry is
wafted aloft which cannot be described to him who has never felt it.
Whoever will fully enjoy this poetry must put his hand to the plough,
must himself investigate. Therefore, enough of this! I shall reckon
myself fortunate if you do not repent of this brief excursion into the
flowered dale of physiology, and if you take with yourselves the belief
that we can say of science what we say of poetry,

    "Who the song would understand,
    Needs must seek the song's own land;
    Who the minstrel understand
    Needs must seek the minstrel's land."

  FOOTNOTES:

  [Footnote 8: This experiment, with its associated reflexions, is due
  to Galileo.]

  [Footnote 9: A development of the theory of musical audition
  differing in many points from the theory of Helmholtz here
  expounded, will be found in my _Contributions to the Analysis of the
  Sensations_ (English translation by C. M. Williams), Chicago, The
  Open Court Publishing Company, 1897.]



ON THE CAUSES OF HARMONY.


We are to speak to-day of a theme which is perhaps of somewhat more
general interest--_the causes of the harmony of musical sounds_. The
first and simplest experiences relative to harmony are very ancient. Not
so the explanation of its laws. These were first supplied by the
investigators of a recent epoch. Allow me an historical retrospect.

Pythagoras (586 B. C.) knew that the note yielded by a string of steady
tension was converted into its octave when the length of the string was
reduced one-half, and into its fifth when reduced two-thirds; and that
then the first fundamental tone was consonant with the two others. He
knew generally that the same string under fixed tension gives consonant
tones when successively divided into lengths that are in the proportions
of the simplest natural numbers; that is, in the proportions of 1:2,
2:3, 3:4, 4:5.

Pythagoras failed to reveal the causes of these laws. What have
consonant tones to do with the simple natural numbers? That is the
question we should ask to-day. But this circumstance must have appeared
less strange than inexplicable to Pythagoras. This philosopher sought
for the causes of harmony in the occult, miraculous powers of numbers.
His procedure was largely the cause of the upgrowth of a numerical
mysticism, of which the traces may still be detected in our
oneirocritical books and among some scientists, to whom marvels are more
attractive than lucidity.

Euclid (300 B. C.) gives a definition of consonance and dissonance that
could hardly be improved upon, in point of verbal accuracy. The
consonance ([Greek: symphônia]) of two tones, he says, is the mixture,
the blending ([Greek: krasis]) of those two tones; dissonance ([Greek:
diaphônia]), on the other hand, is the incapacity of the tones to blend
([Greek: amixia]), whereby they are made harsh for the ear. The person
who knows the correct explanation of the phenomenon hears it, so to
speak, reverberated in these words of Euclid. Still, Euclid did not know
the true cause of harmony. He had unwittingly come very near to the
truth, but without really grasping it.

Leibnitz (1646-1716 A. D.) resumed the question which his predecessors
had left unsolved. He, of course, knew that musical notes were produced
by vibrations, that twice as many vibrations corresponded to the octave
as to the fundamental tone, etc. A passionate lover of mathematics, he
sought for the cause of harmony in the secret computation and comparison
of the simple numbers of vibrations and in the secret satisfaction of
the soul at this occupation. But how, we ask, if one does not know that
musical notes are vibrations? The computation and the satisfaction at
the computation must indeed be pretty secret if it is unknown. What
queer ideas philosophers have! Could anything more wearisome be imagined
than computation as a principle of æsthetics? Yes, you are not utterly
wrong in your conjecture, yet you may be sure that Leibnitz's theory is
not wholly nonsense, although it is difficult to make out precisely what
he meant by his secret computation.

The great Euler (1707-1783) sought the cause of harmony, almost as
Leibnitz did, in the pleasure which the soul derives from the
contemplation of order in the numbers of the vibrations.[10]

Rameau and D'Alembert (1717-1783) approached nearer to the truth. They
knew that in every sound available in music besides the fundamental note
also the twelfth and the next higher third could be heard; and further
that the resemblance between a fundamental tone and its octave was
always strongly marked. Accordingly, the combination of the octave,
fifth, third, etc., with the fundamental tone appeared to them
"natural." They possessed, we must admit, the correct point of view; but
with the simple naturalness of a phenomenon no inquirer can rest
content; for it is precisely this naturalness for which he seeks his
explanations.

Rameau's remark dragged along through the whole modern period, but
without leading to the full discovery of the truth. Marx places it at
the head of his theory of composition, but makes no further application
of it. Also Goethe and Zelter in their correspondence were, so to speak,
on the brink of the truth. Zelter knew of Rameau's view. Finally, you
will be appalled at the difficulty of the problem, when I tell you that
till very recent times even professors of physics were dumb when asked
what were the causes of harmony.

Not till quite recently did Helmholtz find the solution of the question.
But to make this solution clear to you I must first speak of some
experimental principles of physics and psychology.

1) In every process of perception, in every observation, the attention
plays a highly important part. We need not look about us long for proofs
of this. You receive, for example, a letter written in a very poor hand.
Do your best, you cannot make it out. You put together now these, now
those lines, yet you cannot construct from them a single intelligible
character. Not until you direct your attention to groups of lines which
really belong together, is the reading of the letter possible.
Manuscripts, the letters of which are formed of minute figures and
scrolls, can only be read at a considerable distance, where the
attention is no longer diverted from the significant outlines to the
details. A beautiful example of this class is furnished by the famous
iconographs of Giuseppe Arcimboldo in the basement of the Belvedere
gallery at Vienna. These are symbolic representations of water, fire,
etc.: human heads composed of aquatic animals and of combustibles. At a
short distance one sees only the details, at a greater distance only the
whole figure. Yet a point can be easily found at which, by a simple
voluntary movement of the attention, there is no difficulty in seeing
now the whole figure and now the smaller forms of which it is composed.
A picture is often seen representing the tomb of Napoleon. The tomb is
surrounded by dark trees between which the bright heavens are visible as
background. One can look a long time at this picture without noticing
anything except the trees, but suddenly, on the attention being
accidentally directed to the bright background, one sees the figure of
Napoleon between the trees. This case shows us very distinctly the
important part which attention plays. The same sensuous object can,
solely by the interposition of attention, give rise to wholly different
perceptions.

If I strike a harmony, or chord, on this piano, by a mere effort of
attention you can fix every tone of that harmony. You then hear most
distinctly the fixed tone, and all the rest appear as a mere addition,
altering only the quality, or acoustic color, of the primary tone. The
effect of the same harmony is essentially modified if we direct our
attention to different tones.

Strike in succession two harmonies, for example, the two represented in
the annexed diagram, and first fix by the attention the upper note _e_,
afterwards the base _e_-_a_; in the two cases you will hear the same
sequence of harmonies differently. In the first case, you have the
impression as if the fixed tone remained unchanged and simply altered
its _timbre_; in the second case, the whole acoustic agglomeration seems
to fall sensibly in depth. There is an art of composition to guide the
attention of the hearer. But there is also an art of hearing, which is
not the gift of every person.

[Music: Fig. 9.]

The piano-player knows the remarkable effects obtained when one of the
keys of a chord that is struck is let loose. Bar 1 played on the piano
sounds almost like bar 2. The note which lies next to the key let loose
resounds after its release as if it were freshly struck. The attention
no longer occupied with the upper note is by that very fact insensibly
led to the upper note.

[Music: Fig. 10.]

Any tolerably cultivated musical ear can perform the resolution of a
harmony into its component parts. By much practice we can go even
further. Then, every musical sound heretofore regarded as simple can be
resolved into a subordinate succession of musical tones. For example, if
I strike on the piano the note 1, (annexed diagram,) we shall hear, if
we make the requisite effort of attention, besides the loud fundamental
note the feebler, higher overtones, or harmonics, 2 ... 7, that is, the
octave, the twelfth, the double octave, and the third, the fifth, and
the seventh of the double octave.

[Music: Fig. 11.]

The same is true of every musically available sound. Each yields, with
varying degrees of intensity, besides its fundamental note, also the
octave, the twelfth, the double octave, etc. The phenomenon is
observable with special facility on the open and closed flue-pipes of
organs. According, now, as certain overtones are more or less distinctly
emphasised in a sound, the _timbre_ of the sound changes--that peculiar
quality of the sound by which we distinguish the music of the piano from
that of the violin, the clarinet, etc.

On the piano these overtones can be very easily rendered audible. If I
strike, for example, sharply note 1 of the foregoing series, whilst I
simply press down upon, one after another, the keys 2, 3, ... 7, the
notes 2, 3, ... 7 will continue to sound after the striking of 1,
because the strings corresponding to these notes, now freed from their
dampers, are thrown into sympathetic vibration.

As you know, this sympathetic vibration of the like-pitched strings with
the overtones is really not to be conceived as sympathy, but rather as
lifeless mechanical necessity. We must not think of this sympathetic
vibration as an ingenious journalist pictured it, who tells a gruesome
story of Beethoven's F minor sonata, Op. 2, that I cannot withhold from
you. "At the last London Industrial Exhibition nineteen virtuosos played
the F minor sonata on the same piano. When the twentieth stepped up to
the instrument to play by way of variation the same production, to the
terror of all present the piano began to render the sonata of its own
accord. The Archbishop of Canterbury, who happened to be present, was
set to work and forthwith expelled the F minor devil."

Although, now, the overtones or harmonics which we have discussed are
heard only upon a special effort of the attention, nevertheless they
play a highly important part in the formation of musical _timbre_, as
also in the production of the consonance and dissonance of sounds. This
may strike you as singular. How can a thing which is heard only under
exceptional circumstances be of importance generally for audition?

But consider some familiar incidents of your every-day life. Think of
how many things you see which you do not notice, which never strike your
attention until they are missing. A friend calls upon you; you cannot
understand why he looks so changed. Not until you make a close
examination do you discover that his hair has been cut. It is not
difficult to tell the publisher of a work from its letter-press, and yet
no one can state precisely the points by which this style of type is so
strikingly different from that style. I have often recognised a book
which I was in search of from a simple piece of unprinted white paper
that peeped out from underneath the heap of books covering it, and yet I
had never carefully examined the paper, nor could I have stated its
difference from other papers.

What we must remember, therefore, is that every sound that is musically
available yields, besides its fundamental note, its octave, its twelfth,
its double octave, etc., as overtones or harmonics, and that these are
important for the agreeable combination of several musical sounds.

2) One other fact still remains to be dealt with. Look at this
tuning-fork. It yields, when struck, a perfectly smooth tone. But if you
strike in company with it a second fork which is of slightly different
pitch, and which alone also gives a perfectly smooth tone, you will
hear, if you set both forks on the table, or hold both before your ear,
a uniform tone no longer, but a number of shocks of tones. The rapidity
of the shocks increases with the difference of the pitch of the forks.
These shocks, which become very disagreeable for the ear when they
amount to thirty-three in a second, are called "beats."

Always, when one of two like musical sounds is thrown out of unison with
the other, beats arise. Their number increases with the divergence from
unison, and simultaneously they grow more unpleasant. Their roughness
reaches its maximum at about thirty-three beats in a second. On a still
further departure from unison, and a consequent increase of the number
of beats, the unpleasant effect is diminished, so that tones which are
widely apart in pitch no longer produce offensive beats.

To give yourselves a clear idea of the production of beats, take two
metronomes and set them almost alike. You can, for that matter, set the
two exactly alike. You need not fear that they will strike alike. The
metronomes usually for sale in the shops are poor enough to yield, when
set alike, appreciably unequal strokes. Set, now, these two metronomes,
which strike at unequal intervals, in motion; you will readily see that
their strokes alternately coincide and conflict with each other. The
alternation is quicker the greater the difference of time of the two
metronomes.

If metronomes are not to be had, the experiment may be performed with
two watches.

Beats arise in the same way. The rhythmical shocks of two sounding
bodies, of unequal pitch, sometimes coincide, sometimes interfere,
whereby they alternately augment and enfeeble each other's effects.
Hence the shock-like, unpleasant swelling of the tone.

Now that we have made ourselves acquainted with overtones and beats, we
may proceed to the answer of our main question, Why do certain relations
of pitch produce pleasant sounds, consonances, others unpleasant sounds,
dissonances? It will be readily seen that all the unpleasant effects of
simultaneous sound-combinations are the result of beats produced by
those combinations. Beats are the only sin, the sole evil of music.
Consonance is the coalescence of sounds without appreciable beats.

[Illustration: Fig. 12.]

To make this perfectly clear to you I have constructed the model which
you see in Fig. 12. It represents a claviatur. At its top a movable
strip of wood _aa_ with the marks 1, 2 ... 6 is placed. By setting this
strip in any position, for example, in that where the mark 1 is over the
note _c_ of the claviatur, the marks 2, 3 ... 6, as you see, stand over
the overtones of _c_. The same happens when the strip is placed in any
other position. A second, exactly similar strip, _bb_, possesses the
same properties. Thus, together, the two strips, in any two positions,
point out by their marks all the tones brought into play upon the
simultaneous sounding of the notes indicated by the marks 1.

The two strips, placed over the same fundamental note, show that also
all the overtones of those notes coincide. The first note is simply
intensified by the other. The single overtones of a sound lie too far
apart to permit appreciable beats. The second sound supplies nothing
new, consequently, also, no new beats. Unison is the most perfect
consonance.

Moving one of the two strips along the other is equivalent to a
departure from unison. All the overtones of the one sound now fall
alongside those of the other; beats are at once produced; the
combination of the tones becomes unpleasant: we obtain a dissonance. If
we move the strip further and further along, we shall find that as a
general rule the overtones always fall alongside each other, that is,
always produce beats and dissonances. Only in a few quite definite
positions do the overtones partially coincide. Such positions,
therefore, signify higher degrees of euphony--they point out _the
consonant intervals_.

These consonant intervals can be readily found experimentally by cutting
Fig. 12 out of paper and moving _bb_ lengthwise along _aa_. The most
perfect consonances are the octave and the twelfth, since in these two
cases the overtones of the one sound coincide absolutely with those of
the other. In the octave, for example, 1_b_ falls on 2_a_, 2_b_ on 4_a_,
3_b_ on 6_a_. Consonances, therefore, are simultaneous
sound-combinations not accompanied by disagreeable beats. This, by the
way, is, expressed in English, what Euclid said in Greek.

Only such sounds are consonant as possess in common some portion of
their partial tones. Plainly we must recognise between such sounds, also
when struck one after another, a certain affinity. For the second sound,
by virtue of the common overtones, will produce partly the same
sensation as the first. The octave is the most striking exemplification
of this. When we reach the octave in the ascent of the scale we actually
fancy we hear the fundamental tone repeated. The foundations of harmony,
therefore, are the foundations of melody.

Consonance is the coalescence of sounds without appreciable beats! This
principle is competent to introduce wonderful order and logic into the
doctrines of the fundamental bass. The compendiums of the theory of
harmony which (Heaven be witness!) have stood hitherto little behind the
cook-books in subtlety of logic, are rendered extraordinarily clear and
simple. And what is more, all that the great masters, such as
Palestrina, Mozart, Beethoven, unconsciously got right, and of which
heretofore no text-book could render just account, receives from the
preceding principle its perfect verification.

But the beauty of the theory is, that it bears upon its face the stamp
of truth. It is no phantom of the brain. Every musician can hear for
himself the beats which the overtones of his musical sounds produce.
Every musician can satisfy himself that for any given case the number
and the harshness of the beats can be calculated beforehand, and that
they occur in exactly the measure that theory determines.

This is the answer which Helmholtz gave to the question of Pythagoras,
so far as it can be explained with the means now at my command. A long
period of time lies between the raising and the solving of this
question. More than once were eminent inquirers nearer to the answer
than they dreamed of.

The inquirer seeks the truth. I do not know if the truth seeks the
inquirer. But were that so, then the history of science would vividly
remind us of that classical rendezvous, so often immortalised by
painters and poets. A high garden wall. At the right a youth, at the
left a maiden. The youth sighs, the maiden sighs! Both wait. Neither
dreams how near the other is.

I like this simile. Truth suffers herself to be courted, but she has
evidently no desire to be won. She flirts at times disgracefully. Above
all, she is determined to be merited, and has naught but contempt for
the man who will win her too quickly. And if, forsooth, one breaks his
head in his efforts of conquest, what matter is it, another will come,
and truth is always young. At times, indeed, it really seems as if she
were well disposed towards her admirer, but that admitted--never! Only
when Truth is in exceptionally good spirits does she bestow upon her
wooer a glance of encouragement. For, thinks Truth, if I do not do
something, in the end the fellow will not seek me at all.

This one fragment of truth, then, we have, and it shall never escape us.
But when I reflect what it has cost in labor and in the lives of
thinking men, how it painfully groped its way through centuries, a
half-matured thought, before it became complete; when I reflect that it
is the toil of more than two thousand years that speaks out of this
unobtrusive model of mine, then, without dissimulation, I almost repent
me of the jest I have made.

And think of how much we still lack! When, several thousand years hence,
boots, top-hats, hoops, pianos, and bass-viols are dug out of the earth,
out of the newest alluvium as fossils of the nineteenth century; when
the scientists of that time shall pursue their studies both upon these
wonderful structures and upon our modern Broadways, as we to-day make
studies of the implements of the stone age and of the prehistoric
lake-dwellings--then, too, perhaps, people will be unable to comprehend
how we could come so near to many great truths without grasping them.
And thus it is for all time the unsolved dissonance, for all time the
troublesome seventh, that everywhere resounds in our ears; we feel,
perhaps, that it will find its solution, but we shall never live to see
the day of the pure triple accord, nor shall our remotest descendants.

Ladies, if it is the sweet purpose of your life to sow confusion, it is
the purpose of mine to be clear; and so I must confess to you a slight
transgression that I have been guilty of. On one point I have told you
an untruth. But you will pardon me this falsehood, if in full repentance
I make it good. The model represented in Fig. 12 does not tell the whole
truth, for it is based upon the so-called "even temperament" system of
tuning. The overtones, however, of musical sounds are not tempered, but
purely tuned. By means of this slight inexactness the model is made
considerably simpler. In this form it is fully adequate for ordinary
purposes, and no one who makes use of it in his studies need be in fear
of appreciable error.

If you should demand of me, however, the full truth, I could give you
that only by the help of a mathematical formula. I should have to take
the chalk into my hands and--think of it!--reckon in your presence. This
you might take amiss. Nor shall it happen. I have resolved to do no more
reckoning for to-day. I shall reckon now only upon your forbearance, and
this you will surely not gainsay me when you reflect that I have made
only a limited use of my privilege to weary you. I could have taken up
much more of your time, and may, therefore, justly close with Lessing's
epigram:

    "If thou hast found in all these pages naught that's worth the thanks,
    At least have gratitude for what I've spared thee."

  FOOTNOTES:

  [Footnote 10: Sauveur also set out from Leibnitz's idea, but arrived
  by independent researches at a different theory, which was very near
  to that of Helmholtz. Compare on this point Sauveur, _Mémoires de
  l'Académie des Sciences_, Paris, 1700-1705, and R. Smith,
  _Harmonics_, Cambridge, 1749. (See _Appendix_, p. 346.)]



THE VELOCITY OF LIGHT.


When a criminal judge has a right crafty knave before him, one well
versed in the arts of prevarication, his main object is to wring a
confession from the culprit by a few skilful questions. In almost a
similar position the natural philosopher seems to be placed with respect
to nature. True, his functions here are more those of the spy than the
judge; but his object remains pretty much the same. Her hidden motives
and laws of action is what nature must be made to confess. Whether a
confession will be extracted depends upon the shrewdness of the
inquirer. Not without reason, therefore, did Lord Bacon call the
experimental method a questioning of nature. The art consists in so
putting our questions that they may not remain unanswered without a
breach of etiquette.

Look, too, at the countless tools, engines, and instruments of torture
with which man conducts his inquisitions of nature, and which mock the
poet's words:

    "Mysterious even in open day,
    Nature retains her veil, despite our clamors;
    That which she doth not willingly display
    Cannot be wrenched from her with levers, screws, and hammers."

Look at these instruments and you will see that the comparison with
torture also is admissible.[11]

This view of nature, as of something designedly concealed from man, that
can be unveiled only by force or dishonesty, chimed in better with the
conceptions of the ancients than with modern notions. A Grecian
philosopher once said, in offering his opinion of the natural science of
his time, that it could only be displeasing to the gods to see men
endeavoring to spy out what the gods were not minded to reveal to
them.[12] Of course all the contemporaries of the speaker were not of
his opinion.

Traces of this view may still be found to-day, but upon the whole we are
now not so narrow-minded. We believe no longer that nature designedly
hides herself. We know now from the history of science that our
questions are sometimes meaningless, and that, therefore, no answer can
be forthcoming. Soon we shall see how man, with all his thoughts and
quests, is only a fragment of nature's life.

Picture, then, as your fancy dictates, the tools of the physicist as
instruments of torture or as engines of endearment, at all events a
chapter from the history of those implements will be of interest to you,
and it will not be unpleasant to learn what were the peculiar
difficulties that led to the invention of such strange apparatus.

Galileo (born at Pisa in 1564, died at Arcetri in 1642) was the first
who asked what was the velocity of light, that is, what time it would
take for a light struck at one place to become visible at another, a
certain distance away.[13]

The method which Galileo devised was as simple as it was natural. Two
practised observers, with muffled lanterns, were to take up positions in
a dark night at a considerable distance from each other, one at _A_ and
one at _B_. At a moment previously fixed upon, _A_ was instructed to
unmask his lantern; while as soon as _B_ saw the light of _A_'s lantern
he was to unmask his. Now it is clear that the time which _A_ counted
from the uncovering of his lantern until he caught sight of the light of
_B_'s would be the time which it would take light to travel from _A_ to
_B_ and from _B_ back to _A_.

[Illustration: Fig. 13.]

The experiment was not executed, nor could it, in the nature of the
case, have been a success. As we now know, light travels too rapidly to
be thus noted. The time elapsing between the arrival of the light at _B_
and its perception by the observer, with that between the decision to
uncover and the uncovering of the lantern, is, as we now know,
incomparably greater than the time which it takes light to travel the
greatest earthly distances. The great velocity of light will be made
apparent, if we reflect that a flash of lightning in the night
illuminates instantaneously a very extensive region, whilst the single
reflected claps of thunder arrive at the observer's ear very gradually
and in appreciable succession.

During his life, then, the efforts of Galileo to determine the velocity
of light remained uncrowned with success. But the subsequent history of
the measurement of the velocity of light is intimately associated with
his name, for with the telescope which he constructed he discovered the
four satellites of Jupiter, and these furnished the next occasion for
the determination of the velocity of light.

The terrestrial spaces were too small for Galileo's experiment. The
measurement was first executed when the spaces of the planetary system
were employed. Olaf Römer, (born at Aarhuus in 1644, died at Copenhagen
in 1710) accomplished the feat (1675-1676), while watching with Cassini
at the observatory of Paris the revolutions of Jupiter's moons.

[Illustration: Fig. 14.]

Let _AB_ (Fig. 14) be Jupiter's orbit. Let _S_ stand for the sun, _E_
for the earth, _J_ for Jupiter, and _T_ for Jupiter's first satellite.
When the earth is at _E₁_ we see the satellite enter regularly into
Jupiter's shadow, and by watching the time between two successive
eclipses, can calculate its time of revolution. The time which Römer
noted was forty-two hours, twenty-eight minutes, and thirty-five
seconds. Now, as the earth passes along in its orbit towards E₂, the
revolutions of the satellite grow apparently longer and longer: the
eclipses take place later and later. The greatest retardation of the
eclipse, which occurs when the earth is at _E₂_, amounts to sixteen
minutes and twenty-six seconds. As the earth passes back again to _E₁_,
the revolutions grow apparently shorter, and they occur in exactly the
time that they first did when the earth arrives at _E₁_. It is to be
remarked that Jupiter changes only very slightly its position during one
revolution of the earth. Römer guessed at once that these periodical
changes of the time of revolution of Jupiter's satellite were not
actual, but apparent changes, which were in some way connected with the
velocity of light.

Let us make this matter clear to ourselves by a simile. We receive
regularly by the post, news of the political status at our capital.
However far away we may be from the capital, we hear the news of every
event, later it is true, but of all equally late. The events reach us in
the same succession of time as that in which they took place. But if we
are travelling away from the capital, every successive post will have a
greater distance to pass over, and the events will reach us more slowly
than they took place. The reverse will be the case if we are approaching
the capital.

At rest, we hear a piece of music played in the same _tempo_ at all
distances. But the _tempo_ will be seemingly accelerated if we are
carried rapidly towards the band, or retarded if we are carried rapidly
away from it.[14]

[Illustration: Fig. 15.]

Picture to yourself a cross, say the sails of a wind-mill (Fig. 15), in
uniform rotation about its centre. Clearly, the rotation of the cross
will appear to you more slowly executed if you are carried very rapidly
away from it. For the post which in this case conveys to you the light
and brings to you the news of the successive positions of the cross will
have to travel in each successive instant over a longer path.

Now this must also be the case with the rotation (the revolution) of the
satellite of Jupiter. The greatest retardation of the eclipse (16½
minutes), due to the passage of the earth from _E₁_ to _E₂_, or to its
removal from Jupiter by a distance equal to the diameter of the orbit of
the earth, plainly corresponds to the time which it takes light to
traverse a distance equal to the diameter of the earth's orbit. The
velocity of light, that is, the distance described by light in a second,
as determined by this calculation, is 311,000 kilometres,[15] or 193,000
miles. A subsequent correction of the diameter of the earth's orbit,
gives, by the same method, the velocity of light as approximately
186,000 miles a second.

The method is exactly that of Galileo; only better conditions are
selected. Instead of a short terrestrial distance we have the diameter
of the earth's orbit, three hundred and seven million kilometres; in
place of the uncovered and covered lanterns we have the satellite of
Jupiter, which alternately appears and disappears. Galileo, therefore,
although he could not carry out himself the proposed measurement, found
the lantern by which it was ultimately executed.

Physicists did not long remain satisfied with this beautiful discovery.
They sought after easier methods of measuring the velocity of light,
such as might be performed on the earth. This was possible after the
difficulties of the problem were clearly exposed. A measurement of the
kind referred to was executed in 1849 by Fizeau (born at Paris in 1819).

I shall endeavor to make the principle of Fizeau's apparatus clear to
you. Let _s_ (Fig. 16) be a disk free to rotate about its centre, and
perforated at its rim with a series of holes. Let _l_ be a luminous
point casting its light on an unsilvered glass, _a_, inclined at an
angle of forty-five degrees to the axis of the disk. The ray of light,
reflected at this point, passes through one of the holes of the disk and
falls at right angles upon a mirror _b_, erected at a point about five
miles distant. From the mirror _b_ the light is again reflected, passes
once more through the hole in _s_, and, penetrating the glass plate,
finally strikes the eye, _o_, of the observer. The eye, _o_, thus, sees
the image of the luminous point _l_ through the glass plate and the hole
of the disk in the mirror _b_.

[Illustration: Fig. 16.]

If, now, the disk be set in rotation, the unpierced spaces between the
apertures will alternately take the place of the apertures, and the eye
o will now see the image of the luminous point in _b_ only at
interrupted intervals. On increasing the rapidity of the rotation,
however, the interruptions for the eye become again unnoticeable, and
the eye sees the mirror _b_ uniformly illuminated.

But all this holds true only for relatively small velocities of the
disk, when the light sent through an aperture in _s_ to _b_ on its
return strikes the aperture at almost the same place and passes through
it a second time. Conceive, now, the speed of the disk to be so
increased that the light on its return finds before it an unpierced
space instead of an aperture, it will then no longer be able to reach
the eye. We then see the mirror _b_ only when no light is emitted from
it, but only when light is sent to it; it is covered when light comes
from it. In this case, accordingly, the mirror will always appear dark.

If the velocity of rotation at this point were still further increased,
the light sent through one aperture could not, of course, on its return
pass through the same aperture but might strike the next and reach the
eye by that. Hence, by constantly increasing the velocity of the
rotation, the mirror _b_ may be made to appear alternately bright and
dark. Plainly, now, if we know the number of apertures of the disk, the
number of rotations per second, and the distance _sb_, we can calculate
the velocity of light. The result agrees with that obtained by Römer.

The experiment is not quite as simple as my exposition might lead you to
believe. Care must be taken that the light shall travel back and forth
over the miles of distance _sb_ and _bs_ undispersed. This difficulty
is obviated by means of telescopes.

If we examine Fizeau's apparatus closely, we shall recognise in it an
old acquaintance: the arrangement of Galileo's experiment. The luminous
point _l_ is the lantern _A_, while the rotation of the perforated disk
performs mechanically the uncovering and covering of the lantern.
Instead of the unskilful observer _B_ we have the mirror _b_, which is
unfailingly illuminated the instant the light arrives from _s_. The disk
_s_, by alternately transmitting and intercepting the reflected light,
assists the observer _o_. Galileo's experiment is here executed, so to
speak, countless times in a second, yet the total result admits of
actual observation. If I might be pardoned the use of a phrase of
Darwin's in this field, I should say that Fizeau's apparatus was the
descendant of Galileo's lantern.

A still more refined and delicate method for the measurement of the
velocity of light was employed by Foucault, but a description of it here
would lead us too far from our subject.

The measurement of the velocity of sound is easily executed by the
method of Galileo. It was unnecessary, therefore, for physicists to rack
their brains further about the matter; but the idea which with light
grew out of necessity was applied also in this field. Koenig of Paris
constructs an apparatus for the measurement of the velocity of sound
which is closely allied to the method of Fizeau.

The apparatus is very simple. It consists of two electrical clock-works
which strike simultaneously, with perfect precision, tenths of seconds.
If we place the two clock-works directly side by side, we hear their
strokes simultaneously, wherever we stand. But if we take our stand by
the side of one of the works and place the other at some distance from
us, in general a coincidence of the strokes will now not be heard. The
companion strokes of the remote clock-work arrive, as sound, later. The
first stroke of the remote work is heard, for example, immediately after
the first of the adjacent work, and so on. But by increasing the
distance we may produce again a coincidence of the strokes. For example,
the first stroke of the remote work coincides with the second of the
near work, the second of the remote work with the third of the near
work, and so on. If, now, the works strike tenths of seconds and the
distance between them is increased until the first coincidence is noted,
plainly that distance is travelled over by the sound in a tenth of a
second.

We meet frequently the phenomenon here presented, that a thought which
centuries of slow and painful endeavor are necessary to produce, when
once developed, fairly thrives. It spreads and runs everywhere, even
entering minds in which it could never have arisen. It simply cannot be
eradicated.

The determination of the velocity of light is not the only case in which
the direct perception of the senses is too slow and clumsy for use. The
usual method of studying events too fleet for direct observation
consists in putting into reciprocal action with them other events
already known, the velocities of all of which are capable of comparison.
The result is usually unmistakable, and susceptible of direct inference
respecting the character of the event which is unknown. The velocity of
electricity cannot be determined by direct observation. But it was
ascertained by Wheatstone, simply by the expedient of watching an
electric spark in a mirror rotating with tremendous known velocity.

[Illustration: Fig. 17.]

[Illustration: Fig. 18.]

If we wave a staff irregularly hither and thither, simple observation
cannot determine how quickly it moves at each point of its course. But
let us look at the staff through holes in the rim of a rapidly rotating
disk (Fig. 17). We shall then see the moving staff only in certain
positions, namely, when a hole passes in front of the eye. The single
pictures of the staff remain for a time impressed upon the eye; we think
we see several staffs, having some such disposition as that represented
in Fig. 18. If, now, the holes of the disk are equally far apart, and
the disk is rotated with uniform velocity, we see clearly that the staff
has moved slowly from _a_ to _b_, more quickly from _b_ to _c_, still
more quickly from _c_ to _d_, and with its greatest velocity from _d_ to
_e_.

A jet of water flowing from an orifice in the bottom of a vessel has the
appearance of perfect quiet and uniformity, but if we illuminate it for
a second, in a dark room, by means of an electric flash we shall see
that the jet is composed of separate drops. By their quick descent the
images of the drops are obliterated and the jet appears uniform. Let us
look at the jet through the rotating disk. The disk is supposed to be
rotated so rapidly that while the second aperture passes into the place
of the first, drop 1 falls into the place of 2, 2 into the place of 3,
and so on. We see drops then always in the same places. The jet appears
to be at rest. If we turn the disk a trifle more slowly, then while the
second aperture passes into the place of the first, drop 1 will have
fallen somewhat lower than 2, 2 somewhat lower than 3, etc. Through
every successive aperture we shall see drops in successively lower
positions. The jet will appear to be flowing slowly downwards.

[Illustration: Fig. 19.]

Now let us turn the disk more rapidly. Then while the second aperture is
passing into the place of the first, drop 1 will not quite have reached
the place of 2, but will be found slightly above 2, 2 slightly above 3,
etc. Through the successive apertures we shall see the drops at
successively higher places. It will now look as if the jet were flowing
upwards, as if the drops were rising from the lower vessel into the
higher.

You see, physics grows gradually more and more terrible. The physicist
will soon have it in his power to play the part of the famous lobster
chained to the bottom of the Lake of Mohrin, whose direful mission, if
ever liberated, the poet Kopisch humorously describes as that of a
reversal of all the events of the world; the rafters of houses become
trees again, cows calves, honey flowers, chickens eggs, and the poet's
own poem flows back into his inkstand.

       *       *       *       *       *

You will now allow me the privilege of a few general remarks. You have
seen that the same principle often lies at the basis of large classes of
apparatus designed for different purposes. Frequently it is some very
unobtrusive idea which is productive of so much fruit and of such
extensive transformations in physical technics. It is not otherwise here
than in practical life.

The wheel of a waggon appears to us a very simple and insignificant
creation. But its inventor was certainly a man of genius. The round
trunk of a tree perhaps first accidentally led to the observation of the
ease with which a load can be moved on a roller. Now, the step from a
simple supporting roller to a fixed roller, or wheel, appears a very
easy one. At least it appears very easy to us who are accustomed from
childhood up to the action of the wheel. But if we put ourselves vividly
into the position of a man who never saw a wheel, but had to invent one,
we shall begin to have some idea of its difficulties. Indeed, it is
even doubtful whether a single man could have accomplished this feat,
whether perhaps centuries were not necessary to form the first wheel
from the primitive roller.[16]

History does not name the progressive minds who constructed the first
wheel; their time lies far back of the historic period. No scientific
academy crowned their efforts, no society of engineers elected them
honorary members. They still live only in the stupendous results which
they called forth. Take from us the wheel, and little will remain of the
arts and industries of modern life. All disappears. From the
spinning-wheel to the spinning-mill, from the turning-lathe to the
rolling-mill, from the wheelbarrow to the railway train, all vanishes.

In science the wheel is equally important. Whirling machines, as the
simplest means of obtaining quick motions with inconsiderable changes of
place, play a part in all branches of physics. You know Wheatstone's
rotating mirror, Fizeau's wheel, Plateau's perforated rotating disks,
etc. Almost the same principle lies at the basis of all these apparatus.
They differ from one another no more than the pen-knife differs, in the
purposes it serves, from the knife of the anatomist or the knife of the
vine-dresser. Almost the same might be said of the screw.

It will now perhaps be clear to you that new thoughts do not spring up
suddenly. Thoughts need their time to ripen, grow, and develop in, like
every natural product; for man, with his thoughts, is also a part of
nature.

Slowly, gradually, and laboriously one thought is transformed into a
different thought, as in all likelihood one animal species is gradually
transformed into new species. Many ideas arise simultaneously. They
fight the battle for existence not otherwise than do the Ichthyosaurus,
the Brahman, and the horse.

A few remain to spread rapidly over all fields of knowledge, to be
redeveloped, to be again split up, to begin again the struggle from the
start. As many animal species long since conquered, the relicts of ages
past, still live in remote regions where their enemies cannot reach
them, so also we find conquered ideas still living on in the minds of
many men. Whoever will look carefully into his own soul will acknowledge
that thoughts battle as obstinately for existence as animals. Who will
gainsay that many vanquished modes of thought still haunt obscure
crannies of his brain, too faint-hearted to step out into the clear
light of reason? What inquirer does not know that the hardest battle, in
the transformation of his ideas, is fought with himself.

Similar phenomena meet the natural inquirer in all paths and in the most
trifling matters. The true inquirer seeks the truth everywhere, in his
country-walks and on the streets of the great city. If he is not too
learned, he will observe that certain things, like ladies' hats, are
constantly subject to change. I have not pursued special studies on this
subject, but as long as I can remember, one form has always gradually
changed into another. First, they wore hats with long projecting rims,
within which, scarcely accessible with a telescope, lay concealed the
face of the beautiful wearer. The rim grew smaller and smaller; the
bonnet shrank to the irony of a hat. Now a tremendous superstructure is
beginning to grow up in its place, and the gods only know what its
limits will be. It is not otherwise with ladies' hats than with
butterflies, whose multiplicity of form often simply comes from a slight
excrescence on the wing of one species developing in a cognate species
to a tremendous fold. Nature, too, has its fashions, but they last
thousands of years. I could elucidate this idea by many additional
examples; for instance, by the history of the evolution of the coat, if
I were not fearful that my gossip might prove irksome to you.

       *       *       *       *       *

We have now wandered through an odd corner of the history of science.
What have we learned? The solution of a small, I might almost say
insignificant, problem--the measurement of the velocity of light. And
more than two centuries have worked at its solution! Three of the most
eminent natural philosophers, Galileo, an Italian, Römer, a Dane, and
Fizeau, a Frenchman, have fairly shared its labors. And so it is with
countless other questions. When we contemplate thus the many blossoms of
thought that must wither and fall before one shall bloom, then shall we
first truly appreciate Christ's weighty but little consolatory words:
"Many be called but few are chosen."

Such is the testimony of every page of history. But is history right?
Are really only those chosen whom she names? Have those lived and
battled in vain, who have won no prize?

I doubt it. And so will every one who has felt the pangs of sleepless
nights spent in thought, at first fruitless, but in the end successful.
No thought in such struggles was thought in vain; each one, even the
most insignificant, nay, even the erroneous thought, that which
apparently was the least productive, served to prepare the way for those
that afterwards bore fruit. And as in the thought of the individual
naught is in vain, so, also, it is in that of humanity.

Galileo wished to measure the velocity of light. He had to close his
eyes before his wish was realised. But he at least found the lantern by
which his successor could accomplish the task.

And so I may maintain that we all, so far as inclination goes, are
working at the civilisation of the future. If only we all strive for the
right, then are we _all_ called and _all_ chosen!

  FOOTNOTES:

  [Footnote 11: According to Mr. Jules Andrieu, the idea that nature
  must be tortured to reveal her secrets is preserved in the name
  _crucible_--from the Latin _crux_, a cross. But, more probably,
  _crucible_ is derived from some Old French or Teutonic form, as
  _cruche_, _kroes_, _krus_, etc., a pot or jug (cf. Modern English
  _crock_, _cruse_, and German _Krug_).--_Trans._]

  [Footnote 12: Xenophon, Memorabilia iv, 7, puts into the mouth of
  Socrates these words: [Greek: oute gar heureta anthrôpois auta
  enomizen einai, oute chaoizesthai theois an hêgeito ton zêtounta ha
  ekeinoi saphênisai ouk eboulêthêsan].]

  [Footnote 13: Galilei, _Discorsi e dimostrazione matematiche_.
  Leyden, 1638. _Dialogo Primo._]

  [Footnote 14: In the same way, the pitch of a locomotive-whistle is
  higher as the locomotive rapidly approaches an observer, and lower
  when rapidly leaving him than if the locomotive were at
  rest.--_Trans._]

  [Footnote 15: A kilometre is 0.621 or nearly five-eighths of a
  statute mile.]

  [Footnote 16: Observe, also, the respect in which the wheel is held
  in India, Japan and other Buddhistic countries, as the emblem of
  power, order, and law, and of the superiority of mind over matter.
  The consciousness of the importance of this invention seems to have
  lingered long in the minds of these nations.--_Tr._]



WHY HAS MAN TWO EYES?


Why has man two eyes? That the pretty symmetry of his face may not be
disturbed, the artist answers. That his second eye may furnish a
substitute for his first if that be lost, says the far-sighted
economist. That we may weep with two eyes at the sins of the world,
replies the religious enthusiast.

Odd opinions! Yet if you should approach a modern scientist with this
question you might consider yourself fortunate if you escaped with less
than a rebuff. "Pardon me, madam, or my dear sir," he would say, with
stern expression, "man fulfils no purpose in the possession of his eyes;
nature is not a person, and consequently not so vulgar as to pursue
purposes of any kind."

Still an unsatisfactory answer! I once knew a professor who would shut
with horror the mouths of his pupils if they put to him such an
unscientific question.

But ask a more tolerant person, ask me. I, I candidly confess, do not
know exactly why man has two eyes, but the reason partly is, I think,
that I may see you here before me to-night and talk with you upon this
delightful subject.

Again you smile incredulously. Now this is one of those questions that a
hundred wise men together could not answer. You have heard, so far, only
five of these wise men. You will certainly want to be spared the
opinions of the other ninety-five. To the first you will reply that we
should look just as pretty if we were born with only one eye, like the
Cyclops; to the second we should be much better off, according to his
principle, if we had four or eight eyes, and that in this respect we are
vastly inferior to spiders; to the third, that you are not just in the
mood to weep; to the fourth, that the unqualified interdiction of the
question excites rather than satisfies your curiosity; while of me you
will dispose by saying that my pleasure is not as intense as I think,
and certainly not great enough to justify the existence of a double eye
in man since the fall of Adam.

But since you are not satisfied with my brief and obvious answer, you
have only yourselves to blame for the consequences. You must now listen
to a longer and more learned explanation, such as it is in my power to
give.

As the church of science, however, debars the question "Why?" let us put
the matter in a purely orthodox way: Man has two eyes, what _more_ can
he see with two than with one?

I will invite you to take a walk with me? We see before us a wood. What
is it that makes this real wood contrast so favorably with a painted
wood, no matter how perfect the painting may be? What makes the one so
much more lovely than the other? Is it the vividness of the coloring,
the distribution of the lights and the shadows? I think not. On the
contrary, it seems to me that in this respect painting can accomplish
very much.

The cunning hand of the painter can conjure up with a few strokes of his
brush forms of wonderful plasticity. By the help of other means even
more can be attained. Photographs of reliefs are so plastic that we
often imagine we can actually lay hold of the elevations and
depressions.

[Illustration: Fig. 20.]

But one thing the painter never can give with the vividness that nature
does--the difference of near and far. In the real woods you see plainly
that you can lay hold of some trees, but that others are inaccessibly
far. The picture of the painter is rigid. The picture of the real woods
changes on the slightest movement. Now this branch is hidden behind
that; now that behind this. The trees are alternately visible and
invisible.

Let us look at this matter a little more closely. For convenience sake
we shall remain upon the highway, I, II. (Fig. 20.) To the right and the
left lies the forest. Standing at I, we see, let us say, three trees (1,
2, 3) in a line, so that the two remote ones are covered by the nearest.
Moving further along, this changes. At II we shall not have to look
round so far to see the remotest tree 3 as to see the nearer tree 2, nor
so far to see this as to see 1. _Hence, as we move onward, objects that
are near to us seem to lag behind as compared with objects that are
remote from us, the lagging increasing with the proximity of the
objects._ Very remote objects, towards which we must always look in the
same direction as we proceed, appear to travel along with us.

If we should see, therefore, jutting above the brow of yonder hill the
tops of two trees whose distance from us we were in doubt about, we
should have in our hands a very easy means of deciding the question. We
should take a few steps forward, say to the right, and the tree-top
which receded most to the left would be the one nearer to us. In truth,
from the amount of the recession a geometer could actually determine the
distance of the trees from us without ever going near them. It is simply
the scientific development of this perception that enables us to
measure the distances of the stars.

_Hence, from change of view in forward motion the distances of objects
in our field of vision can be measured._

Rigorously, however, even forward motion is not necessary. For every
observer is composed really of _two_ observers. Man has _two_ eyes. The
right eye is a short step ahead of the left eye in the right-hand
direction. Hence, the two eyes receive _different_ pictures of the same
woods. The right eye will see the near trees displaced to the left, and
the left eye will see them displaced to the right, the displacement
being greater, the greater the proximity. This difference is sufficient
for forming ideas of distance.

We may now readily convince ourselves of the following facts:

1. With one eye, the other being shut, you have a very uncertain
judgment of distances. You will find it, for example, no easy task, with
one eye shut, to thrust a stick through a ring hung up before you; you
will miss the ring in almost every instance.

2. You see the same object differently with the right eye from what you
do with the left.

Place a lamp-shade on the table in front of you with its broad opening
turned downwards, and look at it from above. (Fig. 21.) You will see
with your right eye the image 2, with your left eye the image 1. Again,
place the shade with its wide opening turned upwards; you will receive
with your right eye the image 4, with your left eye the image 3. Euclid
mentions phenomena of this character.

3. Finally, you know that it is easy to judge of distances with both
eyes. Accordingly your judgment must spring in some way from a
co-operation of the two eyes. In the preceding example the openings in
the different images received by the two eyes seem displaced with
respect to one another, and this displacement is sufficient for the
inference that the one opening is nearer than the other.

[Illustration: Fig. 21.]

I have no doubt that you, ladies, have frequently received delicate
compliments upon your eyes, but I feel sure that no one has ever told
you, and I know not whether it will flatter you, that you have in your
eyes, be they blue or black, little geometricians. You say you know
nothing of them? Well, for that matter, neither do I. But the facts are
as I tell you.

You understand little of geometry? I shall accept that confession. Yet
with the help of your two eyes you judge of distances? Surely that is a
geometrical problem. And what is more, you know the solution of this
problem: for you estimate distances correctly. If, then, _you_ do not
solve the problem, the little geometricians in your eyes must do it
clandestinely and whisper the solution to you. I doubt not they are
fleet little fellows.

What amazes me most here is, that you know nothing about these little
geometricians. But perhaps they also know nothing about you. Perhaps
they are models of punctuality, routine clerks who bother about nothing
but their fixed work. In that case we may be able to deceive the
gentlemen.

If we present to our right eye an image which looks exactly like the
lamp-shade for the right eye, and to our left eye an image which looks
exactly like a lamp-shade for the left eye, we shall imagine that we see
the whole lamp-shade bodily before us.

You know the experiment. If you are practised in squinting, you can
perform it directly with the figure, looking with your right eye at the
right image, and with your left eye at the left image. In this way the
experiment was first performed by Elliott. Improved and perfected, its
form is Wheatstone's stereoscope, made so popular and useful by
Brewster.

By taking two photographs of the same object from two different points,
corresponding to the two eyes, a very clear three-dimensional picture of
distant places or buildings can be produced by the stereoscope.

But the stereoscope accomplishes still more than this. It can visualise
things for us which we never see with equal clearness in real objects.
You know that if you move much while your photograph is being taken,
your picture will come out like that of a Hindu deity, with several
heads or several arms, which, at the spaces where they overlap, show
forth with equal distinctness, so that we seem to see the one picture
_through_ the other. If a person moves quickly away from the camera
before the impression is completed, the objects behind him will also be
imprinted upon the photograph; the person will look transparent.
Photographic ghosts are made in this way.

Some very useful applications may be made of this discovery. For
example, if we photograph a machine stereoscopically, successively
removing during the operation the single parts (where of course the
impression suffers interruptions), we obtain a transparent view, endowed
with all the marks of spatial solidity, in which is distinctly
visualised the interaction of parts normally concealed. I have employed
this method for obtaining transparent stereoscopic views of anatomical
structures.

You see, photography is making stupendous advances, and there is great
danger that in time some malicious artist will photograph his innocent
patrons with solid views of their most secret thoughts and emotions. How
tranquil politics will then be! What rich harvests our detective force
will reap!

       *       *       *       *       *

By the joint action of the two eyes, therefore, we arrive at our
judgments of distances, as also of the forms of bodies.

Permit me to mention here a few additional facts connected with this
subject, which will assist us in the comprehension of certain phenomena
in the history of civilisation.

You have often heard, and know from personal experience, that remote
objects appear perspectively dwarfed. In fact, it is easy to satisfy
yourself that you can cover the image of a man a few feet away from you
simply by holding up your finger a short distance in front of your eye.
Still, as a general rule, you do not notice this shrinkage of objects.
On the contrary, you imagine you see a man at the end of a large hall,
as large as you see him near by you. For your eye, in its measurement of
the distances, makes remote objects correspondingly larger. The eye, so
to speak, is aware of this perspective contraction and is not deceived
by it, although its possessor is unconscious of the fact. All persons
who have attempted to draw from nature have vividly felt the difficulty
which this superior dexterity of the eye causes the perspective
conception. Not until one's judgment of distances is made uncertain, by
their size, or from lack of points of reference, or from being too
quickly changed, is the perspective rendered very prominent.

On sweeping round a curve on a rapidly moving railway train, where a
wide prospect is suddenly opened up, the men upon distant hills appear
like dolls.[17] You have at the moment, here, no known references for
the measurement of distances. The stones at the entrance of a tunnel
grow visibly larger as we ride towards it; they shrink visibly in size
as we ride from it.

Usually both eyes work together. As certain views are frequently
repeated, and lead always to substantially the same judgments of
distances, the eyes in time must acquire a special skill in geometrical
constructions. In the end, undoubtedly, this skill is so increased that
a single eye alone is often tempted to exercise that office.

Permit me to elucidate this point by an example. Is any sight more
familiar to you than that of a vista down a long street? Who has not
looked with hopeful eyes time and again into a street and measured its
depth. I will take you now into an art-gallery where I will suppose you
to see a picture representing a vista into a street. The artist has not
spared his rulers to get his perspective perfect. The geometrician in
your left eye thinks, "Ah ha! I have computed that case a hundred times
or more. I know it by heart. It is a vista into a street," he continues;
"where the houses are lower is the remote end." The geometrician in the
right eye, too much at his ease to question his possibly peevish comrade
in the matter, answers the same. But the sense of duty of these punctual
little fellows is at once rearoused. They set to work at their
calculations and immediately find that all the points of the picture are
equally distant from them, that is, lie all upon a plane surface.

What opinion will you now accept, the first or the second? If you accept
the first you will see distinctly the vista. If you accept the second
you will see nothing but a painted sheet of distorted images.

It seems to you a trifling matter to look at a picture and understand
its perspective. Yet centuries elapsed before humanity came fully to
appreciate this trifle, and even the majority of you first learned it
from education.

I can remember very distinctly that at three years of age all
perspective drawings appeared to me as gross caricatures of objects. I
could not understand why artists made tables so broad at one end and so
narrow at the other. Real tables seemed to me just as broad at one end
as at the other, because my eye made and interpreted its calculations
without my intervention. But that the picture of the table on the plane
surface was not to be conceived as a plane painted surface but stood for
a table and so was to be imaged with all the attributes of extension was
a joke that I did not understand. But I have the consolation that whole
nations have not understood it.

Ingenuous people there are who take the mock murders of the stage for
real murders, the dissembled actions of the players for real actions,
and who can scarcely restrain themselves, when the characters of the
play are sorely pressed, from running in deep indignation to their
assistance. Others, again, can never forget that the beautiful
landscapes of the stage are painted, that Richard III. is only the
actor, Mr. Booth, whom they have met time and again at the clubs.

Both points of view are equally mistaken. To look at a drama or a
picture properly one must understand that both are _shows_, simply
_denoting_ something real. A certain preponderance of the intellectual
life over the sensuous life is requisite for such an achievement, where
the intellectual elements are safe from destruction by the direct
sensuous impressions. A certain liberty in choosing one's point of view
is necessary, a sort of humor, I might say, which is strongly wanting in
children and in childlike peoples.

Let us look at a few historical facts. I shall not take you as far back
as the stone age, although we possess sketches from this epoch which
show very original ideas of perspective. But let us begin our
sight-seeing in the tombs and ruined temples of ancient Egypt, where the
numberless reliefs and gorgeous colorings have defied the ravages of
thousands of years.

A rich and motley life is here opened to us. We find the Egyptians
represented in all conditions of life. What at once strikes our
attention in these pictures is the delicacy of their technical
execution. The contours are extremely exact and distinct. But on the
other hand only a few bright colors are found, unblended and without
trace of transition. Shadows are totally wanting. The paint is laid on
the surfaces in equal thicknesses.

Shocking for the modern eye is the perspective. All the figures are
equally large, with the exception of the king, whose form is unduly
exaggerated. Near and far appear equally large. Perspective contraction
is nowhere employed. A pond with water-fowl is represented flat, as if
its surface were vertical.

Human figures are portrayed as they are never seen, the legs from the
side, the face in profile. The breast lies in its full breadth across
the plane of representation. The heads of cattle appear in profile,
while the horns lie in the plane of the drawing. The principle which the
Egyptians followed might be best expressed by saying that their figures
are pressed in the plane of the drawing as plants are pressed in a
herbarium.

The matter is simply explained. If the Egyptians were accustomed to
looking at things ingenuously with both eyes at once, the construction
of perspective pictures in space could not be familiar to them. They saw
all arms, all legs on real men in their natural lengths. The figures
pressed into the planes resembled more closely, of course, in their eyes
the originals than perspective pictures could.

This will be better understood if we reflect that painting was developed
from relief. The minor dissimilarities between the pressed figures and
the originals must gradually have compelled men to the adoption of
perspective drawing. But physiologically the painting of the Egyptians
is just as much justified as the drawings of our children are.

A slight advance beyond the Egyptians is shown by the Assyrians. The
reliefs rescued from the ruined mounds of Nimrod at Mossul are, upon the
whole, similar to the Egyptian reliefs. They were made known to us
principally by Layard.

Painting enters on a new phase among the Chinese. This people have a
marked feeling for perspective and correct shading, yet without being
very logical in the application of their principles. Here, too, it
seems, they took the first step but did not go far. In harmony with this
immobility is their constitution, in which the muzzle and the bamboo-rod
play significant functions. In accord with it, too, is their language,
which like the language of children has not yet developed into a
grammar, or, rather, according to the modern conception, has not yet
degenerated into a grammar. It is the same also with their music which
is satisfied with the five-toned scale.

The mural paintings at Herculaneum and Pompeii are distinguished by
grace of representation, as also by a pronounced sense for perspective
and correct illumination, yet they are not at all scrupulous in
construction. Here still we find abbreviations avoided. But to offset
this defect, the members of the body are brought into unnatural
positions, in which they appear in their full lengths. Abridgements are
more frequently observed in clothed than in unclothed figures.

A satisfactory explanation of these phenomena first occurred to me on
the making of a few simple experiments which show how differently one
may see the same object, after some mastery of one's senses has been
attained, simply by the arbitrary movement of the attention.

[Illustration: Fig. 22.]

Look at the annexed drawing (Fig. 22). It represents a folded sheet of
paper with either its depressed or its elevated side turned towards you,
as you wish. You can conceive the drawing in either sense, and in either
case it will appear to you differently.

If, now, you have a real folded sheet of paper on the table before you,
with its sharp edges turned towards you, you can, on looking at it with
one eye, see the sheet alternately elevated, as it really is, or
depressed. Here, however, a remarkable phenomenon is presented. When you
see the sheet properly, neither illumination nor form presents anything
conspicuous. When you see it bent back you see it perspectively
distorted. Light and shadow appear much brighter or darker, or as if
overlaid thickly with bright colors. Light and shadow now appear devoid
of all cause. They no longer harmonise with the body's form, and are
thus rendered much more prominent.

In common life we employ the perspective and illumination of objects to
determine their forms and position. Hence we do not notice the lights,
the shadows, and the distortions. They first powerfully enter
consciousness when we employ a different construction from the usual
spatial one. In looking at the planar image of a camera obscura we are
amazed at the plenitude of the light and the profundity of the shadows,
both of which we do not notice in real objects.

In my earliest youth the shadows and lights on pictures appeared to me
as spots void of meaning. When I began to draw I regarded shading as a
mere custom of artists. I once drew the portrait of our pastor, a friend
of the family, and shaded, from no necessity, but simply from having
seen something similar in other pictures, the whole half of his face
black. I was subjected for this to a severe criticism on the part of my
mother, and my deeply offended artist's pride is probably the reason
that these facts remained so strongly impressed upon my memory.

You see, then, that many strange things, not only in the life of
individuals, but also in that of humanity, and in the history of general
civilisation, may be explained from the simple fact that man has two
eyes.

Change man's eye and you change his conception of the world. We have
observed the truth of this fact among our nearest kin, the Egyptians,
the Chinese, and the lake-dwellers; how must it be among some of our
remoter relatives,--with monkeys and other animals? Nature must appear
totally different to animals equipped with substantially different eyes
from those of men, as, for example, to insects. But for the present
science must forego the pleasure of portraying this appearance, as we
know very little as yet of the mode of operation of these organs.

It is an enigma even how nature appears to animals closely related to
man; as to birds, who see scarcely anything with two eyes at once, but
since their eyes are placed on opposite sides of their heads, have a
separate field of vision for each.[18]

The soul of man is pent up in the prison-house of his head; it looks at
nature through its two windows, the eyes. It would also fain know how
nature looks through other windows. A desire apparently never to be
fulfilled. But our love for nature is inventive, and here, too, much has
been accomplished.

Placing before me an angular mirror, consisting of two plane mirrors
slightly inclined to each other, I see my face twice reflected. In the
right-hand mirror I obtain a view of the right side, and in the
left-hand mirror a view of the left side, of my face. Also I shall see
the face of a person standing in front of me, more to the right with my
right eye, more to the left with my left. But in order to obtain such
widely different views of a face as those shown in the angular mirror,
my two eyes would have to be set much further apart from each other than
they actually are.

[Illustration: Fig. 23.]

Squinting with my right eye at the image in the right-hand mirror, with
my left eye at the image in the left-hand mirror, my vision will be the
vision of a giant having an enormous head with his two eyes set far
apart. This, also, is the impression which my own face makes upon me. I
see it now, single and solid. Fixing my gaze, the relief from second to
second is magnified, the eyebrows start forth prominently from above the
eyes, the nose seems to grow a foot in length, my mustache shoots forth
like a fountain from my lip, the teeth seem to retreat immeasurably. But
by far the most horrible aspect of the phenomenon is the nose.

Interesting in this connexion is the telestereoscope of Helmholtz. In
the telestereoscope we view a landscape by looking with our right eye
(Fig. 24) through the mirror _a_ into the mirror _A_, and with our left
eye through the mirror _b_ into the mirror _B_. The mirrors _A_ and _B_
stand far apart. Again we see with the widely separated eyes of a giant.
Everything appears dwarfed and near us. The distant mountains look like
moss-covered stones at our feet. Between, you see the reduced model of a
city, a veritable Liliput. You are tempted almost to stroke with your
hand the soft forest and city, did you not fear that you might prick
your fingers on the sharp, needle-shaped steeples, or that they might
crackle and break off.

[Illustration: Fig. 24.]

Liliput is no fable. We need only Swift's eyes, the telestereoscope, to
see it.

Picture to yourself the reverse case. Let us suppose ourselves so small
that we could take long walks in a forest of moss, and that our eyes
were correspondingly near each other. The moss-fibres would appear like
trees. On them we should see strange, unshapely monsters creeping about.
Branches of the oak-tree, at whose base our moss-forest lay, would seem
to us dark, immovable, myriad-branched clouds, painted high on the vault
of heaven; just as the inhabitants of Saturn, forsooth, might see their
enormous ring. On the tree-trunks of our mossy woodland we should find
colossal globes several feet in diameter, brilliantly transparent,
swayed by the winds with slow, peculiar motions. We should approach
inquisitively and should find that these globes, in which here and there
animals were gaily sporting, were liquid globes, in fact that they were
water. A short, incautious step, the slightest contact, and woe betide
us, our arm is irresistibly drawn by an invisible power into the
interior of the sphere and held there unrelentingly fast! A drop of dew
has engulfed in its capillary maw a manikin, in revenge for the
thousands of drops that its big human counterparts have quaffed at
breakfast. Thou shouldst have known, thou pygmy natural scientist, that
with thy present puny bulk thou shouldst not joke with capillarity!

My terror at the accident brings me back to my senses. I see I have
turned idyllic. You must pardon me. A patch of greensward, a moss or
heather forest with its tiny inhabitants have incomparably more charms
for me than many a bit of literature with its apotheosis of human
character. If I had the gift of writing novels I should certainly not
make John and Mary my characters. Nor should I transfer my loving pair
to the Nile, nor to the age of the old Egyptian Pharaohs, although
perhaps I should choose that time in preference to the present. For I
must candidly confess that I hate the rubbish of history, interesting
though it may be as a mere phenomenon, because we cannot simply observe
it but must also _feel_ it, because it comes to us mostly with
supercilious arrogance, mostly unvanquished. The hero of my novel would
be a cockchafer, venturing forth in his fifth year for the first time
with his newly grown wings into the light, free air. Truly it could do
no harm if man would thus throw off his inherited and acquired
narrowness of mind by making himself acquainted with the world-view of
allied creatures. He could not help gaining incomparably more in this
way than the inhabitant of a small town would in circumnavigating the
globe and getting acquainted with the views of strange peoples.

       *       *       *       *       *

I have now conducted you, by many paths and by-ways, rapidly over hedge
and ditch, to show you what wide vistas we may reach in every field by
the rigorous pursuit of a single scientific fact. A close examination of
the two eyes of man has conducted us not only into the dim recesses of
humanity's childhood, but has also carried us far beyond the bourne of
human life.

It has surely often struck you as strange that the sciences are divided
into two great groups; that the so-called humanistic sciences, belonging
to the so-called "higher education," are placed in almost a hostile
attitude to the natural sciences.

I must confess I do not overmuch believe in this partition of the
sciences. I believe that this view will appear as childlike and
ingenuous to a matured age as the want of perspective in the old
paintings of Egypt does to us. Can it really be that "higher culture" is
to be gotten only from a few old pots and palimpsests, which are at best
mere scraps of nature, or that more is to be learned from them alone
than from all the rest of nature? I believe that both these sciences are
simply parts of the same science, which have begun at different ends. If
these two ends still act towards each other as the Montagues and
Capulets, if their retainers still indulge in lively tilts, I believe
that after all they are not in earnest. On the one side there is surely
a Romeo, and on the other a Juliet, who, some day, it is hoped, will
unite the two houses with a less tragic sequel than that of the play.

Philology began with the unqualified reverence and apotheosis of the
Greeks. Now it has begun to draw other languages, other peoples and
their histories, into its sphere; it has, through the mediation of
comparative linguistics, already struck up, though as yet somewhat
cautiously, a friendship with physiology.

Physical science began in the witch's kitchen. It now embraces the
organic and inorganic worlds, and with the physiology of articulation
and the theory of the senses, has even pushed its researches, at times
impertinently, into the province of mental phenomena.

In short, we come to the understanding of much within us solely by
directing our glance without, and _vice versa_. Every object belongs to
both sciences. You, ladies, are very interesting and difficult problems
for the psychologist, but you are also extremely pretty phenomena of
nature. Church and State are objects of the historian's research, but
not less phenomena of nature, and in part, indeed, very curious
phenomena. If the historical sciences have inaugurated wide extensions
of view by presenting to us the thoughts of new and strange peoples, the
physical sciences in a certain sense do this in a still greater degree.
In making man disappear in the All, in annihilating him, so to speak,
they force him to take an unprejudiced position without himself, and to
form his judgments by a different standard from that of the petty human.

But if you should ask me now why man has two eyes, I should answer:

That he may look at nature justly and accurately; that he may come to
understand that he himself, with all his views, correct and incorrect,
with all his _haute politique_, is simply an evanescent shred of nature;
that, to speak with Mephistopheles, he is a part of the part, and that
it is absolutely unjustified,

    "For man, the microcosmic fool, to see
    Himself a whole so frequently."

  FOOTNOTES:

  [Footnote 17: This effect is particularly noticeable in the size of
  workmen on high chimneys and church-steeples--"steeple Jacks." When
  the cables were slung from the towers of the Brooklyn bridge (277
  feet high), the men sent out in baskets to paint them, appeared,
  against the broad background of heaven and water, like
  flies.--_Trans._]

  [Footnote 18: See Joh. Müller, _Vergleichende Physiologie des
  Gesichtssinnes_, Leipsic, 1826.]



ON SYMMETRY.[19]


An ancient philosopher once remarked that people who cudgelled their
brains about the nature of the moon reminded him of men who discussed
the laws and institutions of a distant city of which they had heard no
more than the name. The true philosopher, he said, should turn his
glance within, should study himself and his notions of right and wrong;
only thence could he derive real profit.

This ancient formula for happiness might be restated in the familiar
words of the Psalm:

    "Dwell in the land, and verily thou shalt be fed."

To-day, if he could rise from the dead and walk about among us, this
philosopher would marvel much at the different turn which matters have
taken.

The motions of the moon and the other heavenly bodies are accurately
known. Our knowledge of the motions of our own body is by far not so
complete. The mountains and natural divisions of the moon have been
accurately outlined on maps, but physiologists are just beginning to
find their way in the geography of the brain. The chemical constitution
of many fixed stars has already been investigated. The chemical
processes of the animal body are questions of much greater difficulty
and complexity. We have our _Mécanique céleste_. But a _Mécanique
sociale_ or a _Mécanique morale_ of equal trustworthiness remains to be
written.

Our philosopher would indeed admit that we have made great progress. But
we have not followed his advice. The patient has recovered, but he took
for his recovery exactly the opposite of what the doctor prescribed.

Humanity is now returned, much wiser, from its journey in celestial
space, against which it was so solemnly warned. Men, after having become
acquainted with the great and simple facts of the world without, are now
beginning to examine critically the world within. It sounds absurd, but
it is true, that only after we have thought about the moon are we able
to take up ourselves. It was necessary that we should acquire simple and
clear ideas in a less complicated domain, before we entered the more
intricate one of psychology, and with these ideas astronomy principally
furnished us.

To attempt any description of that stupendous movement, which,
originally springing out of the physical sciences, went beyond the
domain of physics and is now occupied with the problems of psychology,
would be presumptuous in this place. I shall only attempt here, to
illustrate to you by a few simple examples the methods by which the
province of psychology can be reached from the facts of the physical
world--especially the adjacent province of sense-perception. And I wish
it to be remembered that my brief attempt is not to be taken as a
measure of the present state of such scientific questions.

       *       *       *       *       *

It is a well-known fact that some objects please us, while others do
not. Generally speaking, anything that is constructed according to fixed
and logically followed rules, is a product of tolerable beauty. We see
thus nature herself, who always acts according to fixed rules,
constantly producing such pretty things. Every day the physicist is
confronted in his workshop with the most beautiful vibration-figures,
tone-figures, phenomena of polarisation, and forms of diffraction.

A rule always presupposes a repetition. Repetitions, therefore, will
probably be found to play some important part in the production of
agreeable effects. Of course, the nature of agreeable effects is not
exhausted by this. Furthermore, the repetition of a physical event
becomes the source of agreeable effects only when it is connected with
a repetition of sensations.

An excellent example that repetition of sensations is a source of
agreeable effects is furnished by the copy-book of every schoolboy,
which is usually a treasure-house of such things, and only in need of an
Abbé Domenech to become celebrated. Any figure, no matter how crude or
poor, if several times repeated, with the repetitions placed in line,
will produce a tolerable frieze.

[Illustration: Fig. 25.]

Also the pleasant effect of symmetry is due to the repetition of
sensations. Let us abandon ourselves a moment to this thought, yet not
imagine when we have developed it, that we have fully exhausted the
nature of the agreeable, much less of the beautiful.

First, let us get a clear conception of what symmetry is. And in
preference to a definition let us take a living picture. You know that
the reflexion of an object in a mirror has a great likeness to the
object itself. All its proportions and outlines are the same. Yet there
is a difference between the object and its reflexion in the mirror,
which you will readily observe.

Hold your right hand before a mirror, and you will see in the mirror a
left hand. Your right glove will produce its mate in the glass. For you
could never use the reflexion of your right glove, if it were present to
you as a real thing, for covering your right hand, but only for covering
your left. Similarly, your right ear will give as its reflexion a left
ear; and you will at once perceive that the left half of your body could
very easily be substituted for the reflexion of your right half. Now
just as in the place of a missing right ear a left ear cannot be put,
unless the lobule of the ear be turned upwards, or the opening into the
concha backwards, so, despite all similarity of form, the reflexion of
an object can never take the place of the object itself.[20]

The reason of this difference between the object and its reflexion is
simple. The reflexion appears as far behind the mirror as the object is
in front of it. The parts of the object, accordingly, which are nearest
the mirror will also be nearest the mirror in the reflexion.
Consequently, the succession of the parts in the reflexion will be
reversed, as may best be seen in the reflexion of the face of a watch or
of a manuscript.

It will also be readily seen, that if a point of the object be joined
with its reflexion in the image, the line of junction will cut the
mirror at right angles and be bisected by it. This holds true of all
corresponding points of object and image.

If, now, we can divide an object by a plane into two halves so that each
half, as seen in the reflecting plane of division, is a reproduction of
the other half, such an object is termed symmetrical, and the plane of
division is called the plane of symmetry.

If the plane of symmetry is vertical, we can say that the body is
vertically symmetrical. An example of vertical symmetry is a Gothic
cathedral.

If the plane of symmetry is horizontal, we can say that the object is
horizontally symmetrical. A landscape on the shores of a lake with its
reflexion in the water, is a system of horizontal symmetry.

Exactly here is a noticeable difference. The vertical symmetry of a
Gothic cathedral strikes us at once, whereas we can travel up and down
the whole length of the Rhine or the Hudson without becoming aware of
the symmetry between objects and their reflexions in the water. Vertical
symmetry pleases us, whilst horizontal symmetry is indifferent, and is
noticed only by the experienced eye.

Whence arises this difference? I say from the fact that vertical
symmetry produces a repetition of the same sensation, while horizontal
symmetry does not. I shall now show that this is so.

Let us look at the following letters:

  d b
  q p

It is a fact known to all mothers and teachers, that children in their
first attempts to read and write, constantly confound d and b, and q and
p, but never d and q, or b and p. Now d and b and q and p are the two
halves of a _vertically_ symmetrical figure, while d and q, and b and p
are two halves of a _horizontally_ symmetrical figure. The first two are
confounded; but confusion is only possible of things that excite in us
the same or similar sensations.

Figures of two flower-girls are frequently seen on the decorations of
gardens and of drawing-rooms, one of whom carries a flower-basket in her
right hand and the other a flower-basket in her left. All know how apt
we are, unless we are very careful, to confound these figures with one
another.

While turning a thing round from right to left is scarcely noticed, the
eye is not at all indifferent to the turning of a thing upside down. A
human face which has been turned upside down is scarcely recognisable as
a face, and makes an impression which is altogether strange. The reason
of this is not to be sought in the unwontedness of the sight, for it is
just as difficult to recognise an arabesque that has been inverted,
where there can be no question of a habit. This curious fact is the
foundation of the familiar jokes played with the portraits of unpopular
personages, which are so drawn that in the upright position of the page
an exact picture of the person is presented, but on being inverted some
popular animal is shown.

It is a fact, then, that the two halves of a vertically symmetrical
figure are easily confounded and that they therefore probably produce
very nearly the same sensations. The question, accordingly, arises,
_why_ do the two halves of a vertically symmetrical figure produce the
same or similar sensations? The answer is: Because our apparatus of
vision, which consists of our eyes and of the accompanying muscular
apparatus is itself vertically symmetrical.[21]

Whatever external resemblances one eye may have with another they are
still not alike. The right eye of a man cannot take the place of a left
eye any more than a left ear or left hand can take the place of a right
one. By artificial means, we can change the part which each of our eyes
plays. (Wheatstone's pseudoscope.) But we then find ourselves in an
entirely new and strange world. What is convex appears concave; what is
concave, convex. What is distant appears near, and what is near appears
far.

The left eye is the reflexion of the right. And the light-feeling retina
of the left eye is a reflexion of the light-feeling retina of the right,
in all its functions.

The lense of the eye, like a magic lantern, casts images of objects on
the retina. And you may picture to yourself the light-feeling retina of
the eye, with its countless nerves, as a hand with innumerable fingers,
adapted to feeling light. The ends of the visual nerves, like our
fingers, are endowed with varying degrees of sensitiveness. The two
retinæ act like a right and a left hand; the sensation of touch and the
sensation of light in the two instances are similar.

Examine the right-hand portion of this letter T: namely, T. Instead of
the two retinæ on which this image falls, imagine feeling the object, my
two hands. The T, grasped with the right hand, gives a different
sensation from that which it gives when grasped with the left. But if we
turn our character about from right to left, thus: T, it will give the
same sensation in the left hand that it gave before in the right. The
sensation is repeated.

If we take a whole T, the right half will produce in the right hand the
same sensation that the left half produces in the left, and _vice
versa_.

The symmetrical figure gives the same sensation twice.

If we turn the T over thus: T, or invert the half T thus: L, so long as
we do not change the position of our hands we can make no use of the
foregoing reasoning.

The retinæ, in fact, are exactly like our two hands. They, too, have
their thumbs and index fingers, though they are thousands in number; and
we may say the thumbs are on the side of the eye near the nose, and the
remaining fingers on the side away from the nose.

With this I hope to have made perfectly clear that the pleasing effect
of symmetry is chiefly due to the repetition of sensations, and that
the effect in question takes place in symmetrical figures, only where
there is a repetition of sensation. The pleasing effect of regular
figures, the preference which straight lines, especially vertical and
horizontal straight lines, enjoy, is founded on a similar reason. A
straight line, both in a horizontal and in a vertical position, can cast
on the two retinæ the same image, which falls moreover on symmetrically
corresponding spots. This also, it would appear, is the reason of our
psychological preference of straight to curved lines, and not their
property of being the shortest distance between two points. The straight
line is felt, to put the matter briefly, as symmetrical to itself, which
is the case also with the plane. Curved lines are felt as deviations
from straight lines, that is, as deviations from symmetry.[22] The
presence of a sense for symmetry in people possessing only one eye from
birth, is indeed a riddle. Of course, the sense of symmetry, although
primarily acquired by means of the eyes, cannot be wholly limited to the
visual organs. It must also be deeply rooted in other parts of the
organism by ages of practice and can thus not be eliminated forthwith by
the loss of one eye. Also, when an eye is lost, the symmetrical muscular
apparatus is left, as is also the symmetrical apparatus of innervation.


It appears, however, unquestionable that the phenomena mentioned have,
in the main, their origin in the peculiar structure of our eyes. It will
therefore be seen at once that our notions of what is beautiful and ugly
would undergo a change if our eyes were different. Also, if this view is
correct, the theory of the so-called eternally beautiful is somewhat
mistaken. It can scarcely be doubted that our culture, or form of
civilisation, which stamps upon the human body its unmistakable traces,
should not also modify our conceptions of the beautiful. Was not
formerly the development of all musical beauty restricted to the narrow
limits of a five-toned scale?

The fact that a repetition of sensations is productive of pleasant
effects is not restricted to the realm of the visible. To-day, both the
musician and the physicist know that the harmonic or the melodic
addition of one tone to another affects us agreeably only when the added
tone reproduces a part of the sensation which the first one excited.
When I add an octave to a fundamental tone, I hear in the octave a part
of what was heard in the fundamental tone. (Helmholtz.) But it is not my
purpose to develop this idea fully here.[23] We shall only ask to-day,
whether there is anything similar to the symmetry of figures in the
province of sounds.

Look at the reflexion of your piano in the mirror.

You will at once remark that you have never seen such a piano in the
actual world, for it has its high keys to the left and its low ones to
the right. Such pianos are not manufactured.

If you could sit down at such a piano and play in your usual manner,
plainly every step which you imagined you were performing in the upward
scale would be executed as a corresponding step in the downward scale.
The effect would be not a little surprising.

For the practised musician who is always accustomed to hearing certain
sounds produced when certain keys are struck, it is quite an anomalous
spectacle to watch a player in the glass and to observe that he always
does the opposite of what we hear.

But still more remarkable would be the effect of attempting to strike a
harmony on such a piano. For a melody it is not indifferent whether we
execute a step in an upward or a downward scale. But for a harmony, so
great a difference is not produced by reversal. I always retain the same
consonance whether I add to a fundamental note an upper or a lower
third. Only the order of the intervals of the harmony is reversed. In
point of fact, when we execute a movement in a major key on our
reflected piano, we hear a sound in a minor key, and _vice versa_.

It now remains to execute the experiments indicated. Instead of playing
upon the piano in the mirror, which is impossible, or of having a piano
of this kind built, which would be somewhat expensive, we may perform
our experiments in a simpler manner, as follows:

1) We play on our own piano in our usual manner, look into the mirror,
and then repeat on our real piano what we see in the mirror. In this way
we transform all steps upwards into corresponding steps downwards. We
play a movement, and then another movement, which, with respect to the
key-board, is symmetrical to the first.

2) We place a mirror beneath the music in which the notes are reflected
as in a body of water, and play according to the notes in the mirror. In
this way also, all steps upwards are changed into corresponding, equal
steps downwards.

3) We turn the music upside down and read the notes from right to left
and from below upwards. In doing this, we must regard all sharps as
flats and all flats as sharps, because they correspond to half lines and
spaces. Besides, in this use of the music we can only employ the bass
clef, as only in this clef are the notes not changed by symmetrical
reversal.

You can judge of the effect of these experiments from the examples which
appear in the annexed musical cut. (Page 102.) The movement which
appears in the upper lines is symmetrically reversed in the lower.

The effect of the experiments may be briefly formulated. The melody is
rendered unrecognisable. The harmony suffers a transposition from a
major into a minor key and _vice versa_. The study of these pretty
effects, which have long been familiar to physicists and musicians, was
revived some years ago by Von Oettingen.[24]

[Music: Fig. 26.

(See pages 101 and 103.)]

Now, although in all the preceding examples I have transposed steps
upward into equal and similar steps downward, that is, as we may justly
say, have played for every movement the movement which is symmetrical to
it, yet the ear notices either little or nothing of symmetry. The
transposition from a major to a minor key is the sole indication of
symmetry remaining. The symmetry is there for the mind, but is wanting
for sensation. No symmetry exists for the ear, because a reversal of
musical sounds conditions no repetition of sensations. If we had an ear
for height and an ear for depth, just as we have an eye for the right
and an eye for the left, we should also find that symmetrical
sound-structures existed for our auditory organs. The contrast of major
and minor for the ear corresponds to inversion for the eye, which is
also only symmetry for the mind, but not for sensation.

By way of supplement to what I have said, I will add a brief remark for
my mathematical readers.

Our musical notation is essentially a graphical representation of a
piece of music in the form of curves, where the time is the abscissæ,
and the logarithms of the number of vibrations the ordinates. The
deviations of musical notation from this principle are only such as
facilitate interpretation, or are due to historical accidents.

If, now, it be further observed that the sensation of pitch also is
proportional to the logarithm of the number of vibrations, and that the
intervals between the notes correspond to the differences of the
logarithms of the numbers of vibrations, the justification will be found
in these facts of calling the harmonies and melodies which appear in the
mirror, symmetrical to the original ones.

       *       *       *       *       *

I simply wish to bring home to your minds by these fragmentary remarks
that the progress of the physical sciences has been of great help to
those branches of psychology that have not scorned to consider the
results of physical research. On the other hand, psychology is beginning
to return, as it were, in a spirit of thankfulness, the powerful
stimulus which it received from physics.

The theories of physics which reduce all phenomena to the motion and
equilibrium of smallest particles, the so-called molecular theories,
have been gravely threatened by the progress of the theory of the senses
and of space, and we may say that their days are numbered.

I have shown elsewhere[25] that the musical scale is simply a species of
space--a space, however, of only one dimension, and that, a one-sided
one. If, now, a person who could only hear, should attempt to develop a
conception of the world in this, his linear space, he would become
involved in many difficulties, as his space would be incompetent to
comprehend the many sides of the relations of reality. But is it any
more justifiable for us, to attempt to force the whole world into the
space of our eye, in aspects in which it is not accessible to the eye?
Yet this is the dilemma of all molecular theories.

We possess, however, a sense, which, with respect to the scope of the
relations which it can comprehend, is richer than any other. It is our
reason. This stands above the senses. It alone is competent to found a
permanent and sufficient view of the world. The mechanical conception of
the world has performed wonders since Galileo's time. But it must now
yield to a broader view of things. A further development of this idea is
beyond the limits of my present purpose.

One more point and I have done. The advice of our philosopher to
restrict ourselves to what is near at hand and useful in our researches,
which finds a kind of exemplification in the present cry of inquirers
for limitation and division of labor, must not be too slavishly
followed. In the seclusion of our closets, we often rack our brains in
vain to fulfil a work, the means of accomplishing which lies before our
very doors. If the inquirer must be perforce a shoemaker, tapping
constantly at his last, it may perhaps be permitted him to be a
shoemaker of the type of Hans Sachs, who did not deem it beneath him to
take a look now and then at his neighbor's work and to comment on the
latter's doings.

Let this be my apology, therefore, if I have forsaken for a moment
to-day the last of my specialty.

  FOOTNOTES:

  [Footnote 19: Delivered before the German Casino of Prague, in the
  winter of 1871.

  A fuller treatment of the problems of this lecture will be found in
  my _Contributions to the Analysis of the Sensations_ (Jena, 1886),
  English Translation, Chicago, 1895. J. P. Soret, _Sur la perception
  du beau_ (Geneva, 1892), also regards repetition as a principle of
  æsthetics. His discussions of the _æsthetical_ side of the subject
  are much more detailed than mine. But with respect to the
  psychological and physiological foundation of the principle, I am
  convinced that the _Contributions to the Analysis of the Sensations_
  go deeper.--MACH (1894).]

  [Footnote 20: Kant, in his _Prolegomena zu jeder künftigen
  Metaphysik_, also refers to this fact, but for a different purpose.]

  [Footnote 21: Compare Mach, _Fichte's Zeitschrift für Philosophie_,
  1864, p. 1.]

  [Footnote 22: The fact that the first and second differential
  coefficients of a curve are directly seen, but the higher
  coefficients not, is very simply explained. The first gives the
  position of the tangent, the declination of the straight line from
  the position of symmetry, the second the declination of the curve
  from the straight line. It is, perhaps, not unprofitable to remark
  here that the ordinary method of testing rulers and plane surfaces
  (by reversed applications) ascertains the deviation of the object
  from symmetry to itself.]

  [Footnote 23: See the lecture _On the Causes of Harmony_.]

  [Footnote 24: A. von Oettingen, _Harmoniesystem in dualer
  Entwicklung_. Leipsic and Dorpat, 1866.]

  [Footnote 25: Compare Mach's _Zur Theorie des Gehörorgans_, Vienna
  Academy, 1863.]



ON THE FUNDAMENTAL CONCEPTS OF ELECTROSTATICS.[26]


The task has been assigned me to develop before you in a
popular manner the fundamental quantitative concepts of
electrostatics--"quantity of electricity," "potential," "capacity,"
and so forth. It would not be difficult, even within the brief
limits of an hour, to delight the eye with hosts of beautiful
experiments and to fill the imagination with numerous and varied
conceptions. But we should, in such a case, be still far from a
lucid and easy grasp of the phenomena. The means would still fail us
for reproducing the facts accurately in thought--a procedure which
for the theoretical and practical man is of equal importance. These
means are the _metrical concepts_ of electricity.

As long as the pursuit of the facts of a given province of phenomena
is in the hands of a few isolated investigators, as long as every
experiment can be easily repeated, the fixing of the collected facts
by provisional description is ordinarily sufficient. But the case
is different when the whole world must make use of the results
reached by many, as happens when the science acquires broader
foundations and scope, and particularly so when it begins to supply
intellectual nourishment to an important branch of the practical
arts, and to draw from that province in return stupendous empirical
results. Then the facts must be so described that individuals in all
places and at all times can, from a few easily obtained elements,
put the facts accurately together in thought, and reproduce them
from the description. This is done with the help of the metrical
concepts and the international measures.

The work which was begun in this direction in the period of the
purely scientific development of the science, especially by Coulomb
(1784), Gauss (1833), and Weber (1846), was powerfully stimulated by
the requirements of the great technical undertakings manifested
since the laying of the first transatlantic cable, and brought to a
brilliant conclusion by the labors of the British Association, 1861,
and of the Paris Congress, 1881, chiefly through the exertions of
Sir William Thomson.

It is plain, that in the time allotted to me I cannot conduct you
over all the long and tortuous paths which the science has actually
pursued, that it will not be possible at every step to remind you of
all the little precautions for the avoidance of error which the
early steps have taught us. On the contrary, I must make shift with
the simplest and rudest tools. I shall conduct you by the shortest
paths from the facts to the ideas, in doing which, of course, it
will not be possible to anticipate all the stray and chance ideas
which may and must arise from prospects into the by-paths which we
leave untrodden.

     *       *       *       *       *

Here are two small, light bodies (Fig. 27) of equal size, freely
suspended, which we "electrify" either by friction with a third body
or by contact with a body already electrified. At once a repulsive
force is set up which drives the two bodies away from each other in
opposition to the action of gravity. This force could accomplish
anew the same mechanical work which was expended to produce it.[27]

[Illustration: Fig. 27.]

[Illustration: Fig. 28.]

Coulomb, now, by means of delicate experiments with the
torsion-balance, satisfied himself that if the bodies in question,
say at a distance of two centimetres, repelled each other with the
same force with which a milligramme-weight strives to fall to the
ground, at half that distance, or at one centimetre, they would
repel each other with the force of four milligrammes, and at double
that distance, or at four centimetres, they would repel each other
with the force of only one-fourth of a milligramme. He found that
the electrical force acts inversely as the square of the distance.

Let us imagine, now, that we possessed some means of measuring
electrical repulsion by weights, a means which would be supplied,
for example, by our electrical pendulums; then we could make the
following observation.

The body _A_ (Fig. 28) is repelled by the body _K_ at a distance of
two centimetres with a force of one milligramme. If we touch _A_,
now, with an equal body _B_, the half of this force of repulsion
will pass to the body _B_; both _A_ and _B_, now, at a distance of
two centimetres from _K_, are repelled only with the force of
one-half a milligramme. But both together are repelled still with
the force of one milligramme. Hence, _the divisibility of electrical
force_ among bodies in contact _is a fact_. It is a useful, but by
no means a necessary supplement to this fact, to imagine an
electrical fluid present in the body _A_, with the quantity of which
the electrical force varies, and half of which flows over to _B_.
For, in the place of the new physical picture, thus, an old,
familiar one is substituted, which moves spontaneously in its wonted
courses.

Adhering to this idea, we define the _unit_ of electrical
quantity, according to the now almost universally adopted
centimetre-gramme-second (C. G. S.) system, as that quantity which
at a distance of one centimetre repels an equal quantity with unit
of force, that is, with a force which in one second would impart to
a mass of one gramme a velocity-increment of a centimetre.
As a gramme mass acquires through the action of gravity a
velocity-increment of about 981 centimetres in a second,
accordingly, a gramme is attracted to the earth with 981, or, in
round numbers, 1000 units of force of the centimetre-gramme-second
system, while a milligramme-weight would strive to fall to the earth
with approximately the unit force of this system.

We may easily obtain by this means a clear idea of what the unit
quantity of electricity is. Two small bodies, _K_, weighing each a
gramme, are hung up by vertical threads, five metres in length and
almost weightless, so as to touch each other. If the two bodies be
equally electrified and move apart upon electrification to a
distance of one centimetre, their charge is approximately equivalent
to the electrostatic unit of electric quantity, for the repulsion
then holds in equilibrium a gravitational force-component of
approximately one milligramme, which strives to bring the bodies
together.

Vertically beneath a small sphere suspended from the equilibrated
beam of a balance a second sphere is placed at a distance of a
centimetre. If both be equally electrified the sphere suspended
from the balance will be rendered apparently lighter by the
repulsion. If by adding a weight of one milligramme equilibrium is
restored, each of the spheres contains in round numbers the
electrostatic unit of electrical quantity.

In view of the fact that the same electrical bodies exert at
different distances different forces upon one another, exception
might be taken to the measure of quantity here developed. What kind
of a quantity is that which now weighs more, and now weighs less, so
to speak? But this apparent deviation from the method of
determination commonly used in practical life, that by weight, is,
closely considered, an agreement. On a high mountain a heavy mass
also is less powerfully attracted to the earth than at the level of
the sea, and if it is permitted us in our determinations to neglect
the consideration of level, it is only because the comparison of a
body with fixed conventional weights is invariably effected at the
same level. In fact, if we were to make one of the two weights
equilibrated on our balance approach sensibly to the centre of the
earth, by suspending it from a very long thread, as Prof. von Jolly
of Munich suggested, we should make the gravity of that weight, its
heaviness, proportionately greater.

Let us picture to ourselves, now, two different electrical fluids, a
positive and a negative fluid, of such nature that the particles of
the one attract the particles of the other according to the law of
the inverse squares, but the particles of the same fluid repel each
other by the same law; in non-electrical bodies let us imagine the
two fluids uniformly distributed in equal quantities, in electric
bodies one of the two in excess; in conductors, further, let us
imagine the fluids mobile, in non-conductors immobile; having formed
such pictures, we possess the conception which Coulomb developed and
to which he gave mathematical precision. We have only to give this
conception free play in our minds and we shall see as in a clear
picture the fluid particles, say of a positively charged conductor,
receding from one another as far as they can, all making for the
surface of the conductor and there seeking out the prominent parts
and points until the greatest possible amount of work has been
performed. On increasing the size of the surface, we see a
dispersion, on decreasing its size we see a condensation of the
particles. In a second, non-electrified conductor brought into the
vicinity of the first, we see the two fluids immediately separate,
the positive collecting itself on the remote and the negative on the
adjacent side of its surface. In the fact that this conception
reproduces, lucidly and spontaneously, all the data which arduous
research only slowly and gradually discovered, is contained its
advantage and scientific value. With this, too, its value is
exhausted. We must not seek in nature for the two hypothetical
fluids which we have added as simple mental adjuncts, if we would
not go astray. Coulomb's view may be replaced by a totally
different one, for example, by that of Faraday, and the most proper
course is always, after the general survey is obtained, to go back
to the actual facts, to the electrical forces.

[Illustration: Fig. 29.]

[Illustration: Fig. 30.]

We will now make ourselves familiar with the concept of electrical
quantity, and with the method of measuring or estimating it. Imagine
a common Leyden jar (Fig. 29), the inner and outer coatings of which
are connected together by means of two common metallic knobs placed
about a centimetre apart. If the inside coating be charged with the
quantity of electricity +_q_, on the outer coating a distribution of
the electricities will take place. A positive quantity almost
equal[28] to the quantity +_q_ flows off to the earth, while a
corresponding quantity-_q_ is still left on the outer coating. The
knobs of the jar receive their portion of these quantities and when
the quantity _q_ is sufficiently great a rupture of the insulating
air between the knobs, accompanied by the self-discharge of the
jar, takes place. For any given distance and size of the knobs, a
charge of a definite electric quantity _q_ is always necessary for
the spontaneous discharge of the jar.

Let us insulate, now, the outer coating of a Lane's unit jar _L_,
the jar just described, and put in connexion with it the inner
coating of a jar _F_ exteriorly connected with the earth (Fig. 30).
Every time that _L_ is charged with +_q_, a like quantity +_q_ is
collected on the inner coating of _F_, and the spontaneous discharge
of the jar _L_, which is now again empty, takes place. The number of
the discharges of the jar _L_ furnishes us, thus, with a measure of
the quantity collected in the jar _F_, and if after 1, 2, 3, ...
spontaneous discharges of _L_ the jar _F_ is discharged, it is
evident that the charge of _F_ has been proportionately augmented.

[Illustration: Fig. 31.]

Let us supply now, to effect the spontaneous discharge, the jar _F_
with knobs of the same size and at the same distance apart as those
of the jar _L_ (Fig. 31). If we find, then, that five discharges of
the unit jar take place before one spontaneous discharge of the jar
_F_ occurs, plainly the jar _F_, for equal distances between the
knobs of the two jars, equal striking distances, is able to hold
five times the quantity of electricity that _L_ can, that is, has
five times the _capacity_ of _L_.[29]

[Illustration: Fig. 32.]

We will now replace the unit jar _L_, with which we measure
electricity, so to speak, _into_ the jar _F_, by a Franklin's pane,
consisting of two parallel flat metal plates (Fig. 32), separated
only by air. If here, for example, thirty spontaneous discharges of
the pane are sufficient to fill the jar, ten discharges will be
found sufficient if the air-space between the two plates be filled
with a cake of sulphur. Hence, the capacity of a Franklin's pane of
sulphur is about three times greater than that of one of the same
shape and size made of air, or, as it is the custom to say, the
specific inductive capacity of sulphur (that of air being taken as
the unit) is about 3.[30] We are here arrived at a very simple fact,
which clearly shows us the significance of the number called
_dielectric constant_, or _specific inductive capacity_, the
knowledge of which is so important for the theory of submarine
cables.

Let us consider a jar _A_, which is charged with a certain quantity
of electricity. We can discharge the jar directly. But we can also
discharge the jar _A_ (Fig. 33) partly into a jar _B_, by connecting
the two outer coatings with each other. In this operation a portion
of the quantity of electricity passes, accompanied by sparks, into
the jar _B_, and we now find both jars charged.

[Illustration: Fig. 33.]

[Illustration: Fig. 34.]

It may be shown as follows that the conception of a constant
quantity of electricity can be regarded as the expression of a pure
fact. Picture to yourself any sort of electrical conductor (Fig.
34); cut it up into a large number of small pieces, and place these
pieces by means of an insulated rod at a distance of one centimetre
from an electrical body which acts with unit of force on an equal
and like-constituted body at the same distance. Take the sum of the
forces which this last body exerts on the single pieces of the
conductor. The sum of these forces will be the quantity of
electricity on the whole conductor. It remains the same, whether we
change the form and the size of the conductor, or whether we bring
it near or move it away from a second electrical conductor, so long
as we keep it insulated, that is, do not discharge it.

A basis of reality for the notion of electric quantity seems also to
present itself from another quarter. If a current, that is, in the
usual view, a definite quantity of electricity per second, is sent
through a column of acidulated water; in the direction of the
positive stream, hydrogen, but in the opposite direction, oxygen is
liberated at the extremities of the column. For a given quantity of
electricity a given quantity of oxygen appears. You may picture the
column of water as a column of hydrogen and a column of oxygen,
fitted into each other, and may say the electric current is a
chemical current and _vice versa_. Although this notion is more
difficult to adhere to in the field of statical electricity and with
non-decomposable conductors, its further development is by no means
hopeless.

The concept quantity of electricity, thus, is not so aerial as might
appear, but is able to conduct us with certainty through a multitude
of varied phenomena, and is suggested to us by the facts in almost
palpable form. We can collect electrical force in a body, measure it
out with one body into another, carry it over from one body into
another, just as we can collect a liquid in a vessel, measure it out
with one vessel into another, or pour it from one into another.

For the analysis of mechanical phenomena, a metrical notion, derived
from experience, and bearing the designation _work_, has proved
itself useful. A machine can be set in motion only when the forces
acting on it can perform work.

[Illustration: Fig. 35.]

Let us consider, for example, a wheel and axle (Fig. 35) having the
radii 1 and 2 metres, loaded respectively with the weights 2 and 1
kilogrammes. On turning the wheel and axle, the 1 kilogramme-weight,
let us say, sinks two metres, while the 2 kilogramme-weight rises
one metre. On both sides the product

KGR. M. KGR. M.

1 × 2 = 2 × 1.

is equal. So long as this is so, the wheel and axle will not move of
itself. But if we take such loads, or so change the radii of the
wheels, that this product (kgr. × metre) on displacement is in
excess on one side, that side will sink. As we see, this product is
characteristic for mechanical events, and for this reason has been
invested with a special name, _work_.

In all mechanical processes, and as all physical processes present a
mechanical side, in all physical processes, work plays a
determinative part. Electrical forces, also, produce only changes in
which work is performed. To the extent that forces come into play in
electrical phenomena, electrical phenomena, be they what they may,
extend into the domain of mechanics and are subject to the laws
which hold in this domain. The universally adopted measure of work,
now, is the product of the force into the distance through which it
acts, and in the C. G. S. system, the unit of work is the action
through one centimetre of a force which would impart in one second
to a gramme-mass a velocity-increment of one centimetre, that is, in
round numbers, the action through a centimetre of a pressure equal
to the weight of a milligramme. From a positively charged body,
electricity, yielding to the force of repulsion and performing work,
flows off to the earth, providing conducting connexions exist. To a
negatively charged body, on the other hand, the earth under the
same circumstances gives off positive electricity. The electrical
work possible in the interaction of a body with the earth,
characterises the electrical condition of that body. We will call
the work which must be expended on the unit quantity of positive
electricity to raise it from the earth to the body _K_ the
_potential_ of the body _K_.[31]

We ascribe to the body _K_ in the C. G. S. system the potential +1,
if we must expend the unit of work to raise the positive
electrostatic unit of electric quantity from the earth to that body;
the potential -1, if we gain in this procedure the unit of work; the
potential 0, if no work at all is performed in the operation.

The different parts of one and the same electrical conductor in
electrical equilibrium have the same potential, for otherwise the
electricity would perform work and move about upon the conductor,
and equilibrium would not have existed. Different conductors of
equal potential, put in connexion with one another, do not exchange
electricity any more than bodies of equal temperature in contact
exchange heat, or in connected vessels, in which the same pressures
exist, liquids flow from one vessel to the other. Exchange of
electricity takes place only between conductors of different
potentials, but in conductors of given form and position a definite
difference of potential is necessary for a spark, which pierces the
insulating air, to pass between them.

On being connected, every two conductors assume at once the same
potential. With this the means is given of determining the potential
of a conductor through the agency of a second conductor expressly
adapted to the purpose called an electrometer, just as we determine
the temperature of a body with a thermometer. The values of the
potentials of bodies obtained in this way simplify vastly our
analysis of their electrical behavior, as will be evident from what
has been said.

Think of a positively charged conductor. Double all the electrical
forces exerted by this conductor on a point charged with unit
quantity, that is, double the quantity at each point, or what is the
same thing, double the total charge. Plainly, equilibrium still
subsists. But carry, now, the positive electrostatic unit towards
the conductor. Everywhere we shall have to overcome double the force
of repulsion we did before, everywhere we shall have to expend
double the work. By doubling the charge of the conductor a double
potential has been produced. Charge and potential go hand in hand,
are proportional. Consequently, calling the total quantity of
electricity of a conductor _Q_ and its potential _V_, we can write:
_Q = CV_, where _C_ stands for a constant, the import of which will
be understood simply from noting that _C = Q/V_.[32] But the
division of a number representing the units of quantity of a
conductor by the number representing its units of potential tells us
the quantity which falls to the share of the unit of potential. Now
the number _C_ here we call the capacity of a conductor, and have
substituted, thus, in the place of the old relative determination of
capacity, an absolute determination.[33]

In simple cases the connexion between charge, potential, and
capacity is easily ascertained. Our conductor, let us say, is a
sphere of radius _r_, suspended free in a large body of air. There
being no other conductors in the vicinity, the charge _q_ will then
distribute itself uniformly upon the surface of the sphere, and
simple geometrical considerations yield for its potential the
expression _V = q/r_. Hence, _q/V = r_; that is, the capacity of a
sphere is measured by its radius, and in the C. G. S. system in
centimetres.[34] It is clear also, since a potential is a quantity
divided by a length, that a quantity divided by a potential must be
a length.

Imagine (Fig. 36) a jar composed of two concentric conductive
spherical shells of the radii _r_ and _r₁_, having only air between
them. Connecting the outside sphere with the earth, and charging the
inside sphere by means of a thin, insulated wire passing through the
first, with the quantity _Q_, we shall have _V = (r₁-r)/(r₁r)Q_, and
for the capacity in this case _(r₁r)/(r₁-r)_, or, to take a specific
example, if _r = 16_ and _r₁ = 19_, a capacity of about 100
centimetres.

[Illustration: Fig. 36.]

We shall now use these simple cases for illustrating the principle
by which capacity and potential are determined. First, it is clear
that we can use the jar composed of concentric spheres with its
known capacity as our unit jar and by means of this ascertain, in
the manner above laid down, the capacity of any given jar _F_. We
find, for example, that 37 discharges of this unit jar of the
capacity 100, just charges the jar investigated at the same
striking distance, that is, at the same potential. Hence, the
capacity of the jar investigated is 3700 centimetres. The large
battery of the Prague physical laboratory, which consists of sixteen
such jars, all of nearly equal size, has a capacity, therefore, of
something like 50,000 centimetres, or the capacity of a sphere, a
kilometre in diameter, freely suspended in atmospheric space. This
remark distinctly shows us the great superiority which Leyden jars
possess for the storage of electricity as compared with common
conductors. In fact, as Faraday pointed out, jars differ from simple
conductors mainly by their great capacity.

[Illustration: Fig. 37.]

For determining potential, imagine the inner coating of a jar _F_,
the outer coating of which communicates with the ground, connected
by a long, thin wire with a conductive sphere _K_ placed free in a
large atmospheric space, compared with whose dimensions the radius
of the sphere vanishes. (Fig. 37.) The jar and the sphere assume at
once the same potential. But on the surface of the sphere, if that
be sufficiently far removed from all other conductors, a uniform
layer of electricity will be found. If the sphere, having the radius
_r_, contains the charge _q_, its potential is _V = q/r_. If the
upper half of the sphere be severed from the lower half and
equilibrated on a balance with one of whose beams it is connected by
silk threads, the upper half will be repelled from the lower half
with the force _P = q²/8r² = 1/8V²_. This repulsion _P_ may be
counter-balanced by additional weights placed on the beam-end, and
so ascertained. The potential is then _V = [sqrt](8P)_.[35]

That the potential is proportional to the square root of the force
is not difficult to see. A doubling or trebling of the potential
means that the charge of all the parts is doubled or trebled; hence
their combined power of repulsion quadrupled or nonupled.

Let us consider a special case. I wish to produce the potential 40
on the sphere. What additional weight must I give to the half sphere
in grammes that the force of repulsion shall maintain the balance in
exact equilibrium? As a gramme weight is approximately equivalent
to 1000 units of force, we have only the following simple example to
work out: _40×40 = 8× 1000.x_, where _x_ stands for the number of
grammes. In round numbers we get _x_ = 0.2 gramme. I charge the jar.
The balance is deflected; I have reached, or rather passed, the
potential 40, and you see when I discharge the jar the associated
spark.[36]

The striking distance between the knobs of a machine increases with
the difference of the potential, although not proportionately to
that difference. The striking distance increases faster than the
potential difference. For a distance between the knobs of one
centimetre on this machine the difference of potential is 110. It
can easily be increased tenfold. Of the tremendous differences of
potential which occur in nature some idea may be obtained from the
fact that the striking distances of lightning in thunder-storms is
counted by miles. The differences of potential in galvanic batteries
are considerably smaller than those of our machine, for it takes
fully one hundred elements to give a spark of microscopic striking
distance.

     *       *       *       *       *

We shall now employ the ideas reached to shed some light upon
another important relation between electrical and mechanical
phenomena. We shall investigate what is the potential _energy_, or
the _store of work_, contained in a charged conductor, for example,
in a jar.

If we bring a quantity of electricity up to a conductor, or, to
speak less pictorially, if we generate by work electrical force in a
conductor, this force is able to produce anew the work by which it
was generated. How great, now, is the energy or capacity for work of
a conductor of known charge _Q_ and known potential _V_?

Imagine the given charge _Q_ divided into very small parts _q_,
_q₁_, _q₂_ ..., and these little parts successively carried up to
the conductor. The first very small quantity _q_ is brought up
without any appreciable work and produces by its presence a small
potential _V__{'}. To bring up the second quantity, accordingly, we
must do the work _q__{'}_V__{'}, and similarly for the quantities
which follow the work _q__{''}_V__{''}, _q__{'''}_V__{'''}, and so
forth. Now, as the potential rises proportionately to the quantities
added until the value _V_ is reached, we have, agreeably to the
graphical representation of Fig. 38, for the total work performed,

_W = 1/2QV_,

which corresponds to the total energy of the charged conductor.
Using the equation _Q_ = _CV_, where _C_ stands for capacity, we
also have,

_W = 1/2CV²_, or _W = Q²/2C_.

It will be helpful, perhaps, to elucidate this idea by an analogy
from the province of mechanics. If we pump a quantity of liquid,
_Q_, gradually into a cylindrical vessel (Fig. 39), the level of the
liquid in the vessel will gradually rise. The more we have pumped
in, the greater the pressure we must overcome, or the higher the
level to which we must lift the liquid. The stored-up work is
rendered again available when the heavy liquid _Q_, which reaches up
to the level _h_, flows out. This work _W_ corresponds to the fall
of the whole liquid weight _Q_, through the distance _h_/2 or
through the altitude of its centre of gravity. We have

_W = 1/2Qh_.

Further, since _Q_ = _Kh_, or since the weight of the liquid and the
height _h_ are proportional, we get also

_W = 1/2Kh²_ and _W = Q²/2K_.

[Illustration: Fig. 38.]

[Illustration: Fig. 39.]

As a special case let us consider our jar. Its capacity is _C_ =
3700, its potential _V_ = 110; accordingly, its quantity _Q = CV_ =
407,000 electrostatic units and its energy _W = 1/2QV_ = 22,385,000
C. G. S. units of work.

The unit of work of the C. G. S. system is not readily appreciable
by the senses, nor does it well admit of representation, as we are
accustomed to work with weights. Let us adopt, therefore, as our
unit of work the gramme-centimetre, or the gravitational pressure of
a gramme-weight through the distance of a centimetre, which in round
numbers is 1000 times greater than the unit assumed above; in this
case, our numerical result will be approximately 1000 times smaller.
Again, if we pass, as more familiar in practice, to the
kilogramme-metre as our unit of work, our unit, the distance being
increased a hundred fold, and the weight a thousand fold, will be
100,000 times larger. The numerical result expressing the work done
is in this case 100,000 times less, being in round numbers 0.22
kilogramme-metre. We can obtain a clear idea of the work done here
by letting a kilogramme-weight fall 22 centimetres.

This amount of work, accordingly, is performed on the charging of
the jar, and on its discharge appears again, according to the
circumstances, partly as sound, partly as a mechanical disruption of
insulators, partly as light and heat, and so forth.

The large battery of the Prague physical laboratory, with its
sixteen jars charged to equal potentials, furnishes, although the
effect of the discharge is imposing, a total amount of work of only
three kilogramme-metres.

In the development of the ideas above laid down we are not
restricted to the method there pursued; in fact, that method was
selected only as one especially fitted to familiarise us with the
phenomena. On the contrary, the connexion of the physical processes
is so multifarious that we can come at the same event from very
different directions. Particularly are electrical phenomena
connected with all other physical events; and so intimate is this
connexion that we might justly call the study of electricity the
theory of the general connexion of physical processes.

With respect to the principle of the conservation of energy which
unites electrical with mechanical phenomena, I should like to point
out briefly two ways of following up the study of this connexion.

A few years ago Professor Rosetti, taking an influence-machine,
which he set in motion by means of weights alternately in the
electrical and non-electrical condition with the same velocities,
determined the mechanical work expended in the two cases and was
thus enabled, after deducting the work of friction, to ascertain the
mechanical work consumed in the development of the electricity.

I myself have made this experiment in a modified, and, as I think,
more advantageous form. Instead of determining the work of friction
by special trial, I arranged my apparatus so that it was eliminated
of itself in the measurement and could consequently be neglected.
The so-called fixed disk of the machine, the axis of which is
placed vertically, is suspended somewhat like a chandelier by three
vertical threads of equal lengths _l_ at a distance _r_ from the
axis. Only when the machine is excited does this fixed disk, which
represents a Prony's brake, receive, through its reciprocal action
with the rotating disk, a deflexion _[alpha]_ and a moment of
torsion which is expressed by _D = (Pr²/l)[alpha]_, where _P_ is the
weight of the disk.[37] The angle _[alpha]_ is determined by a
mirror set in the disk. The work expended in _n_ rotations is given
by _2n[pi]D_.

If we close the machine, as Rosetti did, we obtain a continuous
current which has all the properties of a very weak galvanic
current; for example, it produces a deflexion in a multiplier which
we interpose, and so forth. We can directly ascertain, now, the
mechanical work expended in the maintenance of this current.

If we charge a jar by means of a machine, the energy of the jar
employed in the production of sparks, in the disruption of the
insulators, etc., corresponds to a part only of the mechanical work
expended, a second part of it being consumed in the arc which forms
the circuit.[38] This machine, with the interposed jar, affords in
miniature a picture of the transference of force, or more properly
of work. And in fact nearly the same laws hold here for the
economical coefficient as obtain for large dynamo-machines.

Another means of investigating electrical energy is by its
transformation into heat. A long time ago (1838), before the
mechanical theory of heat had attained its present popularity, Riess
performed experiments in this field with the help of his electrical
air-thermometer or thermo-electrometer.

[Illustration: Fig. 40.]

If the discharge be conducted through a fine wire passing through
the globe of the air-thermometer, a development of heat is observed
proportional to the expression above-discussed _W = 1/2QV_. Although
the total energy has not yet been transformed into measurable heat
by this means, in as much as a portion is left behind in the spark
in the air outside the thermometer, still everything tends to show
that the total heat developed in all parts of the conductor and
along all the paths of discharge is the equivalent of the work
1/2_QV_.

It is not important here whether the electrical energy is
transformed all at once or partly, by degrees. For example, if of
two equal jars one is charged with the quantity _Q_ at the potential
_V_ the energy present is 1/2_QV_. If the first jar be discharged
into the second, _V_, since the capacity is now doubled, falls to
_V_/2. Accordingly, the energy 1/4_QV_ remains, while 1/4_QV_ is
transformed in the spark of discharge into heat. The remainder,
however, is equally distributed between the two jars so that each on
discharge is still able to transform 1/8_QV_ into heat.

     *       *       *       *       *

We have here discussed electricity in the limited phenomenal form in
which it was known to the inquirers before Volta, and which has been
called, perhaps not very felicitously, "statical electricity." It is
evident, however, that the nature of electricity is everywhere one
and the same; that a substantial difference between statical and
galvanic electricity does not exist. Only the quantitative
circumstances in the two provinces are so widely different that
totally new aspects of phenomena may appear in the second, for
example, magnetic effects, which in the first remained unnoticed,
whilst, _vice versa_, in the second field statical attractions and
repulsions are scarcely appreciable. As a fact, we can easily
show the magnetic effect of the current of discharge of an
influence-machine on the galvanoscope although we could hardly have
made the original discovery of the magnetic effects with this
current. The statical distant action of the wire poles of a galvanic
element also would hardly have been noticed had not the phenomenon
been known from a different quarter in a striking form.

If we wished to characterise the two fields in their chief and most
general features, we should say that in the first, high potentials
and small quantities come into play, in the second small potentials
and large quantities. A jar which is discharging and a galvanic
element deport themselves somewhat like an air-gun and the bellows
of an organ. The first gives forth suddenly under a very high
pressure a small quantity of air; the latter liberates gradually
under a very slight pressure a large quantity of air.

In point of principle, too, nothing prevents our retaining the
electrostatical units in the domain of galvanic electricity and in
measuring, for example, the strength of a current by the number of
electrostatic units which flow per second through its cross-section.
But this would be in a double aspect impractical. In the first
place, we should totally neglect the magnetic facilities for
measurement so conveniently offered by the current, and substitute
for this easy means a method which can be applied only with
difficulty and is not capable of great exactness. In the second
place our units would be much too small, and we should find
ourselves in the predicament of the astronomer who attempted to
measure celestial distances in metres instead of in radii of the
earth and the earth's orbit; for the current which by the magnetic
C. G. S. standard represents the unit, would require a flow of some
30,000,000,000 electrostatic units per second through its
cross-section. Accordingly, different units must be adopted here.
The development of this point, however, lies beyond my present
task.

  FOOTNOTES:

  [Footnote 26: A lecture delivered at the International Electrical
  Exhibition, in Vienna, on September 4, 1883.]

  [Footnote 27: If the two bodies were oppositely electrified they
  would exert attractions upon each other.]

  [Footnote 28: The quantity which flows off is in point of fact less
  than _q_. It would be equal to the quantity _q_ only if the inner
  coating of the jar were wholly encompassed by the outer coating.]

  [Footnote 29: Rigorously, of course, this is not correct. First, it
  is to be noted that the jar _L_ is discharged simultaneously with
  the electrode of the machine. The jar _F_, on the other hand, is
  always discharged simultaneously with the outer coating of the jar
  _L_. Hence, if we call the capacity of the electrode of the machine
  _E_, that of the unit jar _L_, that of the outer coating of _L_,
  _A_, and that of the principal jar _F_, then this equation would
  exist for the example in the text: _(F + A)/(L + E) = 5_. A cause of
  further departure from absolute exactness is the residual charge.]

  [Footnote 30: Making allowance for the corrections indicated in the
  preceding footnote, I have obtained for the dielectric constant of
  sulphur the number 3.2, which agrees practically with the results
  obtained by more delicate methods. For the highest attainable
  precision one should by rights immerse the two plates of the
  condenser first wholly in air and then wholly in sulphur, if the
  ratio of the capacities is to correspond to the dielectric constant.
  In point of fact, however, the error which arises from inserting
  simply a plate of sulphur that exactly fills the space between the
  two plates, is of no consequence.]

  [Footnote 31: As this definition in its simple form is apt to give
  rise to misunderstandings, elucidations are usually added to it. It
  is clear that we cannot lift a quantity of electricity to _K_,
  without changing the distribution on _K_ and the potential on _K_.
  Hence, the charges on _K_ must be conceived as fixed, and so small a
  quantity raised that no appreciable change is produced by it. Taking
  the work thus expended as many times as the small quantity in
  question is contained in the unit of quantity, we shall obtain the
  potential. The potential of a body _K_ may be briefly and precisely
  defined as follows: If we expend the element of work _dW_ to raise
  the element of positive quantity _dQ_ from the earth to the
  conductor, the potential of a conductor _K_ will be given by _V =
  dW/dQ_.]

  [Footnote 32: In this article the solidus or slant stroke is used
  for the usual fractional sign of division. Where plus or minus signs
  occur in the numerator or denominator, brackets or a vinculum is
  used.--_Tr._]

  [Footnote 33: A sort of agreement exists between the notions of
  thermal and electrical capacity, but the difference between the two
  ideas also should be carefully borne in mind. The thermal capacity
  of a body depends solely upon that body itself. The electrical
  capacity of a body _K_ is influenced by all bodies in its vicinity,
  inasmuch as the charge of these bodies is able to alter the
  potential of _K_. To give, therefore, an unequivocal significance to
  the notion of the capacity (_C_) of a body _K_, _C_ is defined as
  the relation _Q_/_V_ for the body _K_ in a certain given position of
  all neighboring bodies, and during connexion of all neighboring
  conductors with the earth. In practice the situation is much
  simpler. The capacity, for example, of a jar, the inner coating of
  which is almost enveloped by its outer coating, communicating with
  the ground, is not sensibly affected by charged or uncharged
  adjacent conductors.]

  [Footnote 34: These formulæ easily follow from Newton's theorem that
  a homogeneous spherical shell, whose elements obey the law of the
  inverse squares, exerts no force whatever on points within it but
  acts on points without as if the whole mass were concentrated at its
  centre. The formulæ next adduced also flow from this proposition.]

  [Footnote 35: The energy of a sphere of radius _r_ charged with the
  quantity _q_ is 1/2(_q_²/_r_). If the radius increase by the space
  _dr_ a loss of energy occurs, and the work done is
  1/2(_q_²/_r_²)_dr_. Letting _p_ denote the uniform electrical
  pressure on unit of surface of the sphere, the work done is also
  4_r_²[pi]_pdr_. Hence _p = (1/8r²[pi])(q²/r²)_. Subjected to the
  same superficial pressure on all sides, say in a fluid, our half
  sphere would be an equilibrium. Hence we must make the pressure _p_
  act on the surface of the great circle to obtain the effect on the
  balance, which is _r²[pi]p = 1/8(q²/r²) = 1/8V²_.]

  [Footnote 36: The arrangement described is for several reasons not
  fitted for the actual measurement of potential. Thomson's absolute
  electrometer is based upon an ingenious modification of the
  electrical balance of Harris and Volta. Of two large plane parallel
  plates, one communicates with the earth, while the other is brought
  to the potential to be measured. A small movable superficial portion
  _f_ of this last hangs from the balance for the determination of the
  attraction _P_. The distance of the plates from each other being _D_
  we get _V = D[sqrt](8[pi]P/f)_.]

  [Footnote 37: This moment of torsion needs a supplementary
  correction, on account of the vertical electric attraction of the
  excited disks. This is done by changing the weight of the disk by
  means of additional weights and by making a second reading of the
  angles of deflexion.]

  [Footnote 38: The jar in our experiment acts like an accumulator,
  being charged by a dynamo machine. The relation which obtains
  between the expended and the available work may be gathered from the
  following simple exposition. A Holtz machine _H_ (Fig. 40) is
  charging a unit jar _L_, which after _n_ discharges of quantity _q_
  and potential _v_, charges the jar _F_ with the quantity _Q_ at the
  potential _V_. The energy of the unit-jar discharges is lost and
  that of the jar _F_ alone is left. Hence the ratio of the available
  work to the total work expended is

_½QV/[½QV + (n/2)qv]_ and as _Q = nq_, also _V/(V + v)_.

  If, now, we interpose no unit jar, still the parts of the machine
  and the wires of conduction are themselves virtually such unit jars
  and the formula still subsists _V/(V + [sum]v)_, in which [sum]_v_
  represents the sum of all the successively introduced differences of
  potential in the circuit of connexion.]



ON THE PRINCIPLE OF THE CONSERVATION OF ENERGY.[39]


In a popular lecture, distinguished for its charming simplicity and
clearness, which Joule delivered in the year 1847,[40] that famous
physicist declares that the living force which a heavy body has acquired
by its descent through a certain height and which it carries with it in
the form of the velocity with which it is impressed, is the _equivalent_
of the attraction of gravity through the space fallen through, and that
it would be "absurd" to assume that this living force could be destroyed
without some restitution of that equivalent. He then adds: "You will
therefore be surprised to hear that until very _recently_ the universal
opinion has been that living force could be absolutely and irrevocably
destroyed at any one's option." Let us add that to-day, after
forty-seven years, the _law of the conservation of energy_, wherever
civilisation exists, is accepted as a fully established truth and
receives the widest applications in all domains of natural science.

The fate of all momentous discoveries is similar. On their first
appearance they are regarded by the majority of men as errors. J. R.
Mayer's work on the principle of energy (1842) was rejected by the first
physical journal of Germany; Helmholtz's treatise (1847) met with no
better success; and even Joule, to judge from an intimation of Playfair,
seems to have encountered difficulties with his first publication
(1843). Gradually, however, people are led to see that the new view was
long prepared for and ready for enunciation, only that a few favored
minds had perceived it much earlier than the rest, and in this way the
opposition of the majority is overcome. With proofs of the fruitfulness
of the new view, with its success, confidence in it increases. The
majority of the men who employ it cannot enter into a deep-going
analysis of it; for them, its success is its proof. It can thus happen
that a view which has led to the greatest discoveries, like Black's
theory of caloric, in a subsequent period in a province where it does
not apply may actually become an obstacle to progress by its blinding
our eyes to facts which do not fit in with our favorite conceptions. If
a theory is to be protected from this dubious rôle, the grounds and
motives of its evolution and existence must be examined from time to
time with the utmost care.

The most multifarious physical changes, thermal, electrical, chemical,
and so forth, can be brought about by mechanical work. When such
alterations are reversed they yield anew the mechanical work in exactly
the quantity which was required for the production of the part reversed.
This is the _principle of the conservation of energy_; "energy" being
the term which has gradually come into use for that "indestructible
something" of which the measure is mechanical _work_.

How did we acquire this idea? What are the sources from which we have
drawn it? This question is not only of interest in itself, but also for
the important reason above touched upon. The opinions which are held
concerning the foundations of the law of energy still diverge very
widely from one another. Many trace the principle to the impossibility
of a perpetual motion, which they regard either as sufficiently proved
by experience, or as self-evident. In the province of pure mechanics the
impossibility of a perpetual motion, or the continuous production of
_work_ without some _permanent_ alteration, is easily demonstrated.
Accordingly, if we start from the theory that all physical processes are
purely _mechanical_ processes, motions of molecules and atoms, we
embrace also, by this _mechanical_ conception of physics, the
impossibility of a perpetual motion in the _whole_ physical domain. At
present this view probably counts the most adherents. Other inquirers,
however, are for accepting only a purely _experimental_ establishment of
the law of energy.

It will appear, from the discussion to follow, that _all_ the factors
mentioned have co-operated in the development of the view in question;
but that in addition to them a logical and purely formal factor,
hitherto little considered, has also played a very important part.


I. THE PRINCIPLE OF THE EXCLUDED PERPETUAL MOTION.

The law of energy in its modern form is not identical with the principle
of the excluded perpetual motion, but it is very closely related to it.
The latter principle, however, is by no means new, for in the province
of mechanics it has controlled for centuries the thoughts and
investigations of the greatest thinkers. Let us convince ourselves of
this by the study of a few historical examples.

[Illustration: Fig. 41.]

S. Stevinus, in his famous work _Hypomnemata mathematica_, Tom. IV, _De
statica_, (Leyden, 1605, p. 34), treats of the equilibrium of bodies on
inclined planes.

Over a triangular prism _ABC_, one side of which, _AC_, is horizontal,
an endless cord or chain is slung, to which at equal distances apart
fourteen balls of equal weight are attached, as represented in
cross-section in Figure 41. Since we can imagine the lower symmetrical
part of the cord _ABC_ taken away, Stevinus concludes that the four
balls on _AB_ hold in equilibrium the two balls on _BC_. For if the
equilibrium were for a moment disturbed, it could never subsist: the
cord would keep moving round forever in the same direction,--we should
have a perpetual motion. He says:

     "But if this took place, our row or ring of balls would come once
     more into their original position, and from the same cause the
     eight globes to the left would again be heavier than the six to the
     right, and therefore those eight would sink a second time and these
     six rise, and all the globes would keep up, of themselves, _a
     continuous and unending motion, which is false_."[41]

Stevinus, now, easily derives from this principle the laws of
equilibrium on the inclined plane and numerous other fruitful
consequences.

In the chapter "Hydrostatics" of the same work, page 114, Stevinus sets
up the following principle: "Aquam datam, datum sibi intra aquam locum
servare,"--a given mass of water preserves within water its given place.

[Illustration: Fig. 42.]

This principle is demonstrated as follows (see Fig. 42):

     "For, assuming it to be possible by natural means, let us suppose
     that A does not preserve the place assigned to it, but sinks down
     to D. This being posited, the water which succeeds A will, for the
     same reason, also flow down to _D_; _A_ will be forced out of its
     place in _D_; and thus this body of water, for the conditions in it
     are everywhere the same, _will set up a perpetual motion, which is
     absurd_."[42]

From this all the principles of hydrostatics are deduced. On this
occasion Stevinus also first develops the thought so fruitful for modern
analytical mechanics that the equilibrium of a system is not destroyed
by the addition of rigid connexions. As we know, the principle of the
conservation of the centre of gravity is now sometimes deduced from
D'Alembert's principle with the help of that remark. If we were to
reproduce Stevinus's demonstration to-day, we should have to change it
slightly. We find no difficulty in imagining the cord on the prism
possessed of unending uniform motion if all hindrances are thought away,
but we should protest against the assumption of an accelerated motion or
even against that of a uniform motion, if the resistances were not
removed. Moreover, for greater precision of proof, the string of balls
might be replaced by a heavy homogeneous cord of infinite flexibility.
But all this does not affect in the least the historical value of
Stevinus's thoughts. It is a fact, Stevinus deduces apparently much
simpler truths from the principle of an impossible perpetual motion.

In the process of thought which conducted Galileo to his discoveries at
the end of the sixteenth century, the following principle plays an
important part, that a body in virtue of the velocity acquired in its
descent can rise exactly as high as it fell. This principle, which
appears frequently and with much clearness in Galileo's thought, is
simply another form of the principle of excluded perpetual motion, as we
shall see it is also in Huygens.

Galileo, as we know, arrived at the law of uniformly accelerated motion
by _a priori_ considerations, as that law which was the "simplest and
most natural," after having first assumed a different law which he was
compelled to reject. To verify his law he executed experiments with
falling bodies on inclined planes, measuring the times of descent by the
weights of the water which flowed out of a small orifice in a large
vessel. In this experiment he assumes as a fundamental principle, that
the velocity acquired in descent down an inclined plane always
corresponds to the vertical height descended through, a conclusion which
for him is the immediate outcome of the fact that a body which has
fallen down one inclined plane can, with the velocity it has acquired,
rise on another plane of any inclination only to the same vertical
height. This principle of the height of ascent also led him, as it
seems, to the law of inertia. Let us hear his own masterful words in the
_Dialogo terzo_ (_Opere_, Padova, 1744, Tom. III). On page 96 we read:

     "I take it for granted that the velocities acquired by a body in
     descent down planes of different inclinations are equal if the
     heights of those planes are equal."[43]

Then he makes Salviati say in the dialogue:[44]

     "What you say seems very probable, but I wish to go further and by
     an experiment so to increase the probability of it that it shall
     amount almost to absolute demonstration. Suppose this sheet of
     paper to be a vertical wall, and from a nail driven in it a ball of
     lead weighing two or three ounces to hang by a very fine thread
     _AB_ four or five feet long. (Fig. 43.) On the wall mark a
     horizontal line _DC_ perpendicular to the vertical _AB_, which
     latter ought to hang about two inches from the wall. If now the
     thread _AB_ with the ball attached take the position _AC_ and the
     ball be let go, you will see the ball first descend through the arc
     _CB_ and passing beyond _B_ rise through the arc _BD_ almost to the
     level of the line _CD_, being prevented from reaching it exactly by
     the resistance of the air and of the thread. From this we may truly
     conclude that its impetus at the point _B_, acquired by its descent
     through the arc _CB_, is sufficient to urge it through a similar
     arc _BD_ to the same height. Having performed this experiment and
     repeated it several times, let us drive in the wall, in the
     projection of the vertical _AB_, as at _E_ or at _F_, a nail five
     or six inches long, so that the thread _AC_, carrying as before the
     ball through the arc _CB_, at the moment it reaches the position
     _AB_, shall strike the nail _E_, and the ball be thus compelled to
     move up the arc _BG_ described about _E_ as centre. Then we shall
     see what the same impetus will here accomplish, acquired now as
     before at the same point _B_, which then drove the same moving body
     through the arc _BD_ to the height of the horizontal _CD_. Now
     gentlemen, you will be pleased to see the ball rise to the
     horizontal line at the point _G_, and the same thing also happen if
     the nail be placed lower as at _F_, in which case the ball would
     describe the arc _BJ_, always terminating its ascent precisely at
     the line _CD_. If the nail be placed so low that the length of
     thread below it does not reach to the height of _CD_ (which would
     happen if _F_ were nearer _B_ than to the intersection of _AB_ with
     the horizontal _CD_), then the thread will wind itself about the
     nail. This experiment leaves no room for doubt as to the truth of
     the supposition. For as the two arcs _CB_, _DB_ are equal and
     similarly situated, the momentum acquired in the descent of the arc
     _CB_ is the same as that acquired in the descent of the arc _DB_;
     but the momentum acquired at _B_ by the descent through the arc
     _CB_ is capable of driving up the same moving body through the arc
     _BD_; hence also the momentum acquired in the descent _DB_ is equal
     to that which drives the same moving body through the same arc from
     _B_ to _D_, so that in general every momentum acquired in the
     descent of an arc is equal to that which causes the same moving
     body to ascend through the same arc; but all the momenta which
     cause the ascent of all the arcs _BD_, _BG_, _BJ_, are equal since
     they are made by the same momentum acquired in the descent _CB_, as
     the experiment shows: therefore all the momenta acquired in the
     descent of the arcs _DB_, _GB_, _JB_ are equal."

[Illustration: Fig. 43.]

The remark relative to the pendulum may be applied to the inclined plane
and leads to the law of inertia. We read on page 124:[45]

     "It is plain now that a movable body, starting from rest at _A_ and
     descending down the inclined plane _AB_, acquires a velocity
     proportional to the increment of its time: the velocity possessed
     at _B_ is the greatest of the velocities acquired, and by its
     nature immutably impressed, provided all causes of new acceleration
     or retardation are taken away: I say acceleration, having in view
     its possible further progress along the plane extended;
     retardation, in view of the possibility of its being reversed and
     made to mount the ascending plane _BC_. But in the horizontal plane
     _GH_ its equable motion, according to its velocity as acquired in
     the descent from _A_ to _B_, will be continued _ad infinitum_."
     (Fig. 44.)

[Illustration: Fig. 44.]

Huygens, upon whose shoulders the mantel of Galileo fell, forms a
sharper conception of the law of inertia and generalises the principle
respecting the heights of ascent which was so fruitful in Galileo's
hands. He employs the latter principle in the solution of the problem of
the centre of oscillation and is perfectly clear in the statement that
the principle respecting the heights of ascent is identical with the
principle of the excluded perpetual motion.

The following important passages then occur (Hugenii, _Horologium
oscillatorium, pars secunda_). _Hypotheses_:

     "If gravity did not exist, nor the atmosphere obstruct the motions
     of bodies, a body would keep up forever the motion once impressed
     upon it, with equable velocity, in a straight line."[46]

In part four of the _Horologium de centro oscillationis_ we read:

     "If any number of weights be set in motion by the force of gravity,
     the common centre of gravity of the weights as a whole cannot
     possibly rise higher than the place which it occupied when the
     motion began.

     "That this hypothesis of ours may arouse no scruples, we will state
     that it simply imports, what no one has ever denied, that heavy
     bodies do not move _upwards_.--And truly if the devisers of the new
     machines who make such futile attempts to construct a perpetual
     motion would acquaint themselves with this principle, they could
     easily be brought to see their errors and to understand that the
     thing is utterly impossible by mechanical means."[47]

There is possibly a Jesuitical mental reservation contained in the words
"mechanical means." One might be led to believe from them that Huygens
held a non-mechanical perpetual motion for possible.

The generalisation of Galileo's principle is still more clearly put in
Prop. IV of the same chapter:

     "If a pendulum, composed of several weights, set in motion from
     rest, complete any part of its full oscillation, and from that
     point onwards, the individual weights, with their common connexions
     dissolved, change their acquired velocities upwards and ascend as
     far as they can, the common centre of gravity of all will be
     carried up to the same altitude with that which it occupied before
     the beginning of the oscillation."[48]

On this last principle now, which is a generalisation, applied to a
system of masses, of one of Galileo's ideas respecting a single mass and
which from Huygens's explanation we recognise as the principle of
excluded perpetual motion, Huygens grounds his theory of the centre of
oscillation. Lagrange characterises this principle as precarious and is
rejoiced at James Bernoulli's successful attempt, in 1681, to reduce the
theory of the centre of oscillation to the laws of the lever, which
appeared to him clearer. All the great inquirers of the seventeenth and
eighteenth centuries broke a lance on this problem, and it led
ultimately, in conjunction with the principle of virtual velocities, to
the principle enunciated by D'Alembert in 1743 in his _Traité de
dynamique_, though previously employed in a somewhat different form by
Euler and Hermann.

Furthermore, the Huygenian principle respecting the heights of ascent
became the foundation of the "law of the conservation of living force,"
as that was enunciated by John and Daniel Bernoulli and employed with
such signal success by the latter in his _Hydrodynamics_. The theorems
of the Bernoullis differ in form only from Lagrange's expression in the
_Analytical Mechanics_.

The manner in which Torricelli reached his famous law of efflux for
liquids leads again to our principle. Torricelli assumed that the liquid
which flows out of the basal orifice of a vessel cannot by its velocity
of efflux ascend to a greater height than its level in the vessel.

Let us next consider a point which belongs to pure mechanics, the
history of the principle of _virtual motions_ or _virtual velocities_.
This principle was not first enunciated, as is usually stated, and as
Lagrange also asserts, by Galileo, but earlier, by Stevinus. In his
_Trochleostatica_ of the above-cited work, page 72, he says:

     "Observe that this axiom of statics holds good here:

     "As the space of the body acting is to the space of the body acted
     upon, so is the power of the body acted upon to the power of the
     body acting."[49]

Galileo, as we know, recognised the truth of the principle in the
consideration of the simple machines, and also deduced the laws of the
equilibrium of liquids from it.

Torricelli carries the principle back to the properties of the centre of
gravity. The condition controlling equilibrium in a simple machine, in
which power and load are represented by weights, is that the common
centre of gravity of the weights shall not sink. Conversely, if the
centre of gravity cannot sink equilibrium obtains, because heavy bodies
of themselves do not move upwards. In this form the principle of virtual
velocities is identical with Huygens's principle of the impossibility of
a perpetual motion.

John Bernoulli, in 1717, first perceived the universal import of the
principle of virtual movements for all systems; a discovery stated in a
letter to Varignon. Finally, Lagrange gives a general demonstration of
the principle and founds upon it his whole _Analytical Mechanics_. But
this general demonstration is based after all upon Huygens and
Torricelli's remarks. Lagrange, as is known, conceives simple pulleys
arranged in the directions of the forces of the system, passes a cord
through these pulleys, and appends to its free extremity a weight which
is a common measure of all the forces of the system. With no difficulty,
now, the number of elements of each pulley may be so chosen that the
forces in question shall be replaced by them. It is then clear that if
the weight at the extremity cannot sink, equilibrium subsists, because
heavy bodies cannot of themselves move upwards. If we do not go so far,
but wish to abide by Torricelli's idea, we may conceive every individual
force of the system replaced by a special weight suspended from a cord
passing over a pulley in the direction of the force and attached at its
point of application. Equilibrium subsists then when the common centre
of gravity of all the weights together cannot sink. The fundamental
supposition of this demonstration is plainly the impossibility of a
perpetual motion.

Lagrange tried in every way to supply a proof free from extraneous
elements and fully satisfactory, but without complete success. Nor were
his successors more fortunate.

The whole of mechanics, thus, is based upon an idea, which, though
unequivocal, is yet unwonted and not coequal with the other principles
and axioms of mechanics. Every student of mechanics, at some stage of
his progress, feels the uncomfortableness of this state of affairs;
every one wishes it removed; but seldom is the difficulty stated in
words. Accordingly, the zealous pupil of the science is highly rejoiced
when he reads in a master like Poinsot (_Théorie générale de l'équilibre
et du mouvement des systèmes_) the following passage, in which that
author is giving his opinion of the _Analytical Mechanics_:

     "In the meantime, because our attention in that work was first
     wholly engrossed with the consideration of its beautiful
     development of mechanics, which seemed to spring complete from a
     single formula, we naturally believed that the science was
     completed or that it only remained to seek the demonstration of the
     principle of virtual velocities. But that quest brought back all
     the difficulties that we had overcome by the principle itself. That
     law so general, wherein are mingled the vague and unfamiliar ideas
     of infinitely small movements and of perturbations of equilibrium,
     only grew obscure upon examination; and the work of Lagrange
     supplying nothing clearer than the march of analysis, we saw
     plainly that the clouds had only appeared lifted from the course of
     mechanics because they had, so to speak, been gathered at the very
     origin of that science.

     "At bottom, a general demonstration of the principle of virtual
     velocities would be equivalent to the establishment of the whole of
     mechanics upon a different basis: for the demonstration of a law
     which embraces a whole science is neither more nor less than the
     reduction of that science to another law just as general, but
     evident, or at least more simple than the first, and which,
     consequently, would render that useless."[50]

According to Poinsot, therefore, a proof of the principle of virtual
movements is tantamount to a total rehabilitation of mechanics.

Another circumstance of discomfort to the mathematician is, that in the
historical form in which mechanics at present exists, dynamics is
founded on statics, whereas it is desirable that in a science which
pretends to deductive completeness the more special statical theorems
should be deducible from the more general dynamical principles.

In fact, a great master, Gauss, gave expression to this desire in his
presentment of the principle of least constraint (Crelle's _Journal für
reine und angewandte Mathematik_, Vol. IV, p. 233) in the following
words: "Proper as it is that in the gradual development of a science,
and in the instruction of individuals, the easy should precede the
difficult, the simple the complex, the special the general, yet the
mind, when once it has reached a higher point of view, demands the
contrary course, in which all statics shall appear simply as a special
case of mechanics." Gauss's own principle, now, possesses all the
requisites of universality, but its difficulty is that it is not
immediately intelligible and that Gauss deduced it with the help of
D'Alembert's principle, a procedure which left matters where they were
before.

Whence, now, is derived this strange part which the principle of virtual
motion plays in mechanics? For the present I shall only make this reply.
It would be difficult for me to tell the difference of impression which
Lagrange's proof of the principle made on me when I first took it up as
a student and when I subsequently resumed it after having made
historical researches. It first appeared to me insipid, chiefly on
account of the pulleys and the cords which did not fit in with the
mathematical view, and whose action I would much rather have discovered
from the principle itself than have taken for granted. But now that I
have studied the history of the science I cannot imagine a more
beautiful demonstration.

In fact, through all mechanics it is this self-same principle of
excluded perpetual motion which accomplishes almost all, which
displeased Lagrange, but which he still had to employ, at least tacitly,
in his own demonstration. If we give this principle its proper place and
setting, the paradox is explained.

The principle of excluded perpetual motion is thus no new discovery; it
has been the guiding idea, for three hundred years, of all the great
inquirers. But the principle cannot properly be _based_ upon mechanical
perceptions. For long before the development of mechanics the conviction
of its truth existed and even contributed to that development. Its power
of conviction, therefore, must have more universal and deeper roots. We
shall revert to this point.


II. MECHANICAL PHYSICS.

It cannot be denied that an unmistakable tendency has prevailed, from
Democritus to the present day, to explain _all_ physical events
_mechanically_. Not to mention earlier obscure expressions of that
tendency we read in Huygens the following:[51]

     "There can be no doubt that light consists of the _motion_ of a
     certain substance. For if we examine its production, we find that
     here on earth it is principally fire and flame which engender it,
     both of which contain beyond doubt bodies which are in rapid
     movement, since they dissolve and destroy many other bodies more
     solid than they: while if we regard its effects, we see that when
     light is accumulated, say by concave mirrors, it has the property
     of combustion just as fire has, that is to say, it disunites the
     parts of bodies, which is assuredly a proof of _motion_, at least
     in the _true philosophy_, in which the causes of all natural
     effects are conceived as _mechanical_ causes. Which in my judgment
     must be accomplished or all hope of ever understanding physics
     renounced."[52]

S. Carnot,[53] in introducing the principle of excluded perpetual motion
into the theory of heat, makes the following apology:

     "It will be objected here, perhaps, that a perpetual motion proved
     impossible for _purely mechanical actions_, is perhaps not so when
     the influence of _heat_ or of electricity is employed. But can
     phenomena of heat or electricity be thought of as due to anything
     else than to _certain motions of bodies_, and as such must they not
     be subject to the general laws of mechanics?"[54]

These examples, which might be multiplied by quotations from recent
literature indefinitely, show that a tendency to explain all things
mechanically actually exists. This tendency is also intelligible.
Mechanical events as simple motions in space and time best admit of
observation and pursuit by the help of our highly organised senses. We
reproduce mechanical processes almost without effort in our imagination.
Pressure as a circumstance that produces motion is very familiar to us
from daily experience. All changes which the individual personally
produces in his environment, or humanity brings about by means of the
arts in the world, are effected through the instrumentality of
_motions_. Almost of necessity, therefore, motion appears to us as the
most important physical factor. Moreover, mechanical properties may be
discovered in all physical events. The sounding bell trembles, the
heated body expands, the electrified body attracts other bodies. Why,
therefore, should we not attempt to grasp all events under their
mechanical aspect, since that is so easily apprehended and most
accessible to observation and measurement? In fact, no objection _is_ to
be made to the attempt to elucidate the properties of physical events by
mechanical _analogies_.

But modern physics has proceeded _very far_ in this direction. The point
of view which Wundt represents in his excellent treatise _On the
Physical Axioms_ is probably shared by the majority of physicists. The
axioms of physics which Wundt sets up are as follows:

1. All natural causes are motional causes.

2. Every motional cause lies outside the object moved.

3. All motional causes act in the direction of the straight line of
junction, and so forth.

4. The effect of every cause persists.

5. Every effect involves an equal countereffect.

6. Every effect is equivalent to its cause.

These principles might be studied properly enough as fundamental
principles of mechanics. But when they are set up as axioms of physics,
their enunciation is simply tantamount to a negation of all events
except motion.

According to Wundt, all changes of nature are mere changes of place. All
causes are motional causes (page 26). Any discussion of the
philosophical grounds on which Wundt supports his theory would lead us
deep into the speculations of the Eleatics and the Herbartians. Change
of place, Wundt holds, is the _only_ change of a thing in which a thing
remains identical with itself. If a thing changed _qualitatively_, we
should be obliged to imagine that something was annihilated and
something else created in its place, which is not to be reconciled with
our idea of the identity of the object observed and of the
indestructibility of matter. But we have only to remember that the
Eleatics encountered difficulties of exactly the same sort in motion.
Can we not also imagine that a thing is destroyed in _one_ place and in
_another_ an exactly similar thing created? After all, do we really know
_more_ why a body leaves one place and appears in another, than why a
_cold_ body grows _warm_? Granted that we had a perfect knowledge of the
mechanical processes of nature, could we and should we, for that reason,
_put out of the world_ all other processes that we do not understand? On
this principle it would really be the simplest course to deny the
existence of the whole world. This is the point at which the Eleatics
ultimately arrived, and the school of Herbart stopped little short of
the same goal.

Physics treated in this sense supplies us simply with a diagram of the
world, in which we do not know reality again. It happens, in fact, to
men who give themselves up to this view for many years, that the world
of sense from which they start as a province of the greatest
familiarity, suddenly becomes, in their eyes, the supreme
"world-riddle."

Intelligible as it is, therefore, that the efforts of thinkers have
always been bent upon the "reduction of all physical processes to the
motions of atoms," it must yet be affirmed that this is a chimerical
ideal. This ideal has often played an effective part in popular
lectures, but in the workshop of the serious inquirer it has discharged
scarcely the least function. What has really been achieved in mechanical
physics is either the _elucidation_ of physical processes by more
familiar _mechanical analogies_, (for example, the theories of light and
of electricity,) or the exact _quantitative_ ascertainment of the
connexion of mechanical processes with other physical processes, for
example, the results of thermodynamics.


III. THE PRINCIPLE OF ENERGY IN PHYSICS.

We can know only from _experience_ that mechanical processes produce
other physical transformations, or _vice versa_. The attention was first
directed to the connexion of mechanical processes, especially the
performance of work, with changes of thermal conditions by the invention
of the steam-engine, and by its great technical importance. Technical
interests and the need of scientific lucidity meeting in the mind of S.
Carnot led to the remarkable development from which thermodynamics
flowed. It is simply _an accident of history_ that the development in
question was not connected with the practical applications of
_electricity_.

In the determination of the maximum quantity of _work_ that, generally,
a heat-machine, or, to take a special case, a steam-engine, can perform
with the expenditure of a _given_ amount of heat of combustion, Carnot
is guided by mechanical analogies. A body can do work on being heated,
by expanding under pressure. But to do this the body must receive heat
from a _hotter_ body. Heat, therefore, to do work, must pass from a
hotter body to a colder body, just as water must fall from a higher
level to a lower level to put a mill-wheel in motion. Differences of
temperature, accordingly, represent forces able to do work exactly as do
differences of height in heavy bodies. Carnot pictures to himself an
ideal process in which no heat flows away unused, that is, without doing
work. With a given expenditure of heat, accordingly, this process
furnishes the maximum of work. An analogue of the process would be a
mill-wheel which scooping its water out of a higher level would slowly
carry it to a lower level without the loss of a drop. A peculiar
property of the process is, that with the expenditure of the same work
the water can be raised again exactly to its original level. This
property of _reversibility_ is also shared by the process of Carnot. His
process also can be reversed by the expenditure of the same amount of
work, and the heat again brought back to its original temperature level.

Suppose, now, we had _two_ different reversible processes _A_, _B_, such
that in _A_ a quantity of heat, _Q_, flowing off from the temperature
_t₁_ to the lower temperature _t₂_ should perform the work _W_, but in
_B_ under the same circumstances it should perform a greater quantity of
work _W_ + _W'_; then, we could join _B_ in the sense assigned and _A_
in the reverse sense into a _single_ process. Here _A_ would reverse the
transformation of heat produced by _B_ and would leave a surplus of work
_W'_, produced, so to speak, from nothing. The combination would present
a perpetual motion.

With the feeling, now, that it makes little difference whether the
mechanical laws are broken directly or indirectly (by processes of
heat), and convinced of the existence of a _universal_ law-ruled
connexion of nature, Carnot here excludes for the first time from the
province of _general_ physics the possibility of a perpetual motion.
_But it follows, then, that the quantity of work W, produced by the
passage of a quantity of heat Q from a temperature t₁ to a temperature
t₂, is independent of the nature of the substances as also of the
character of the process, so far as that is unaccompanied by loss, but
is wholly dependent upon the temperature t₁, t₂.

This important principle has been fully confirmed by the special
researches of Carnot himself (1824), of Clapeyron (1834), and of Sir
William Thomson (1849), now Lord Kelvin. The principle was reached
_without any assumption whatever_ concerning the nature of heat, simply
by the exclusion of a perpetual motion. Carnot, it is true, was an
adherent of the theory of Black, according to which the sum-total of the
quantity of heat in the world is constant, but so far as his
investigations have been hitherto considered the decision on this point
is of no consequence. Carnot's principle led to the most remarkable
results. W. Thomson (1848) founded upon it the ingenious idea of an
"absolute" scale of temperature. James Thomson (1849) conceived a Carnot
process to take place with water freezing under pressure and, therefore,
performing work. He discovered, thus, that the freezing point is lowered
0·0075° Celsius by every additional atmosphere of pressure. This is
mentioned merely as an example.

About twenty years after the publication of Carnot's book a further
advance was made by J. R. Mayer and J. P. Joule. Mayer, while engaged as
a physician in the service of the Dutch, observed, during a process of
bleeding in Java, an unusual redness of the venous blood. In agreement
with Liebig's theory of animal heat he connected this fact with the
diminished loss of heat in warmer climates, and with the diminished
expenditure of organic combustibles. The total expenditure of heat of a
man at rest must be equal to the total heat of combustion. But since
_all_ organic actions, even the mechanical actions, must be set down to
the credit of the heat of combustion, some connexion must exist between
mechanical work and expenditure of heat.

Joule started from quite similar convictions concerning the galvanic
battery. A heat of association equivalent to the consumption of the zinc
can be made to appear in the galvanic cell. If a current is set up, a
part of this heat appears in the conductor of the current. The
interposition of an apparatus for the decomposition of water causes a
part of this heat to disappear, which on the burning of the explosive
gas formed, is reproduced. If the current runs an electromotor, a
portion of the heat again disappears, which, on the consumption of the
work by friction, again makes its appearance. Accordingly, both the
heat produced and the work produced, appeared to Joule also as
connected with the consumption of material. The thought was therefore
present, both to Mayer and to Joule, of regarding heat and work as
equivalent quantities, so connected with each other that what is lost in
one form universally appears in another. The result of this was a
_substantial_ conception of heat and of work, and _ultimately a
substantial conception of energy_. Here every physical change of
condition is regarded as energy, the destruction of which generates work
or equivalent heat. An electric charge, for example, is energy.

In 1842 Mayer had calculated from the physical constants then
universally accepted that by the disappearance of one kilogramme-calorie
365 kilogramme-metres of work could be performed, and _vice versa_.
Joule, on the other hand, by a long series of delicate and varied
experiments beginning in 1843 ultimately determined the mechanical
equivalent of the kilogramme-calorie, more exactly, as 425
kilogramme-metres.

If we estimate every change of physical condition by the _mechanical
work_ which can be performed upon the _disappearance_ of that condition,
and call this measure _energy_, then we can measure all physical changes
of condition, no matter how different they may be, with the same common
measure, and say: _the sum-total of all energy remains constant_. This
is the form that the principle of excluded perpetual motion received at
the hands of Mayer, Joule, Helmholtz, and W. Thomson in its extension to
the whole domain of physics.

After it had been proved that heat must _disappear_ if mechanical work
was to be done at its expense, Carnot's principle could no longer be
regarded as a complete expression of the facts. Its improved form was
first given, in 1850, by Clausius, whom Thomson followed in 1851. It
runs thus: "If a quantity of heat _Q'_ is transformed into work in a
reversible process, _another_ quantity of heat _Q_ of the absolute[55]
temperature _T₁_ is lowered to the absolute temperature _T₂_." Here
_Q'_ is dependent only on _Q_, _T₁_, _T₂_, but is independent of the
substances used and of the character of the process, so far as that is
unaccompanied by loss. Owing to this last fact, it is sufficient to find
the relation which obtains for some one well-known physical substance,
say a gas, and some definite simple process. The relation found will be
the one that holds generally. We get, thus,

_Q'/(Q' + Q) = (T₁-T₂)/T₁_ (1)

that is, the quotient of the available heat _Q'_ transformed into work
divided by the sum of the transformed and transferred heats (the total
sum used), the so-called _economical coefficient_ of the process, is,

_(T₁-T₂)/T₁_.


IV. THE CONCEPTIONS OF HEAT.

When a cold body is put in contact with a warm body it is observed that
the first body is warmed and that the second body is cooled. We may say
that the first body is warmed _at the expense of_ the second body. This
suggests the notion of a thing, or heat-substance, which passes from the
one body to the other. If two masses of water _m_, _m'_, of unequal
temperatures, be put together, it will be found, upon the rapid
equalisation of the temperatures, that the respective changes of
temperatures _u_ and _u'_ are inversely proportional to the masses and
of opposite signs, so that the algebraical sum of the products is,

_mu + m'u' = 0_.

Black called the products _mu_, _m'u'_, which are decisive for our
knowledge of the process, _quantities of heat_. We may form a very clear
_picture_ of these products by conceiving them with Black as measures of
the quantities of some substance. But the essential thing is not this
picture but the _constancy_ of the sum of these products in simple
processes of conduction. If a quantity of heat disappears at one point,
an equally large quantity will make its appearance at some other point.
The retention of this idea leads to the discovery of specific heat.
Black, finally, perceives that also something else may appear for a
vanished quantity of heat, namely: the fusion or vaporisation of a
definite quantity of matter. He adheres here still to this favorite
view, though with some freedom, and considers the vanished quantity of
heat as still present, but as _latent_.

The generally accepted notion of a caloric, or heat-stuff, was strongly
shaken by the work of Mayer and Joule. If the quantity of heat can be
increased and diminished, people said, heat cannot be a substance, but
must be a _motion_. The subordinate part of this statement has become
much more popular than all the rest of the doctrine of energy. But we
may convince ourselves that the motional conception of heat is now as
unessential as was formerly its conception as a substance. Both ideas
were favored or impeded solely by accidental historical circumstances.
It does not follow that heat is not a substance from the fact that a
mechanical equivalent exists for quantity of heat. We will make this
clear by the following question which bright students have sometimes put
to me. Is there a mechanical equivalent of electricity as there is a
mechanical equivalent of heat? Yes, and no. There is no mechanical
equivalent of _quantity_ of electricity as there is an equivalent of
_quantity_ of heat, because the same quantity of electricity has a very
different capacity for work, according to the circumstances in which it
is placed; but there _is_ a mechanical equivalent of electrical energy.

Let us ask another question. Is there a mechanical equivalent of water?
No, there is no mechanical equivalent of quantity of water, but there is
a mechanical equivalent of weight of water multiplied by its distance
of descent.

When a Leyden jar is discharged and work thereby performed, we do not
picture to ourselves that the quantity of electricity disappears as work
is done, but we simply assume that the electricities come into different
positions, equal quantities of positive and negative electricity being
united with one another.

What, now, is the reason of this difference of view in our treatment of
heat and of electricity? The reason is purely historical, wholly
conventional, and, what is still more important, is wholly indifferent.
I may be allowed to establish this assertion.

In 1785 Coulomb constructed his torsion balance, by which he was enabled
to measure the repulsion of electrified bodies. Suppose we have two
small balls, _A_, _B_, which over their whole extent are similarly
electrified. These two balls will exert on one another, at a certain
distance _r_ of their centres, a certain repulsion _p_. We bring into
contact with _B_ now a ball _C_, suffer both to be equally electrified,
and then measure the repulsion of _B_ from _A_ and of _C_ from _A_ at
the same distance _r_. The sum of these repulsions is again _p_.
Accordingly something has remained constant. If we ascribe this effect
to a substance, then we infer naturally its constancy. But the essential
point of the exposition is the divisibility of the electric force _p_
and not the simile of substance.

In 1838 Riess constructed his electrical air-thermometer (the
thermoelectrometer). This gives a measure of the quantity of heat
produced by the discharge of jars. This quantity of heat is not
proportional to the quantity of electricity contained in the jar by
Coulomb's measure, but if _Q_ be this quantity and _C_ be the capacity,
is proportional to _Q_²/2_C_, or, more simply still, to the energy of
the charged jar. If, now, we discharge the jar completely through the
thermometer, we obtain a certain quantity of heat, _W_. But if we make
the discharge through the thermometer into a second jar, we obtain a
quantity less than _W_. But we may obtain the remainder by completely
discharging both jars through the air-thermometer, when it will again be
proportional to the energy of the two jars. On the first, incomplete
discharge, accordingly, a part of the electricity's capacity for work
was lost.

When the charge of a jar produces heat its energy is changed and its
value by Riess's thermometer is decreased. But by Coulomb's measure the
quantity remains unaltered.

Now let us imagine that Riess's thermometer had been invented before
Coulomb's torsion balance, which is not a difficult feat, since both
inventions are independent of each other; what would be more natural
than that the "quantity" of electricity contained in a jar should be
measured by the heat produced in the thermometer? But then, this
so-called quantity of electricity would decrease on the production of
heat or on the performance of work, whereas it now remains unchanged;
in that case, therefore, electricity would not be a _substance_ but a
_motion_, whereas now it is still a substance. The reason, therefore,
why we have other notions of electricity than we have of heat, is purely
historical, accidental, and conventional.

This is also the case with other physical things. Water does not
disappear when work is done. Why? Because we measure quantity of water
with scales, just as we do electricity. But suppose the capacity of
water for work were called quantity, and had to be measured, therefore,
by a mill instead of by scales; then this quantity also would disappear
as it performed the work. It may, now, be easily conceived that many
substances are not so easily got at as water. In that case we should be
unable to carry out the one kind of measurement with the scales whilst
many other modes of measurement would still be left us.

In the case of heat, now, the historically established measure of
"quantity" is accidentally the work-value of the heat. Accordingly, its
quantity disappears when work is done. But that heat is not a substance
follows from this as little as does the opposite conclusion that it is a
substance. In Black's case the quantity of heat remains constant because
the heat passes into no _other_ form of energy.

If any one to-day should still wish to think of heat as a substance, we
might allow that person this liberty with little ado. He would only have
to assume that that which we call quantity of heat was the energy of a
substance whose quantity remained unaltered, but whose energy changed.
In point of fact we might much better say, in analogy with the other
terms of physics, energy of heat, instead of quantity of heat.

When we wonder, therefore, at the discovery that heat is motion, we
wonder at something that was never discovered. It is perfectly
indifferent and possesses not the slightest scientific value, whether we
think of heat as a substance or not. The fact is, heat behaves in some
connexions like a substance, in others not. Heat is latent in steam as
oxygen is latent in water.


V. THE CONFORMITY IN THE DEPORTMENT OF THE ENERGIES.

The foregoing reflexions will gain in lucidity from a consideration of
the conformity which obtains in the behavior of all energies, a point to
which I called attention long ago.[56]

A weight _P_ at a height _H₁_ represents an energy _W₁ = PH₁_. If we
suffer the weight to sink to a lower height _H₂_, during which work is
done, and the work done is employed in the production of living force,
heat, or an electric charge, in short, is transformed, then the energy
_W₂ = PH₂_ is still _left_. The equation subsists

_W₁/H₁ = W₂/H₂_, (2)

or, denoting the _transformed_ energy by _W' = W₁-W₂_ and the
_transferred_ energy, that transported to the lower level, by _W = W₂_,

_W'/(W' + W) = (H₁-H₂)/H₁_, (3)

an equation in all respects analogous to equation (1) at page 165. The
property in question, therefore, is by no means peculiar to heat.
Equation (2) gives the relation between the energy taken from the higher
level and that deposited on the lower level (the energy left behind); it
says that these _energies_ are proportional to the _heights of the
levels_. An equation analogous to equation (2) may be set up for _every_
form of energy; hence the equation which corresponds to equation (3),
and so to equation (1), may be regarded as valid for every form. For
electricity, for example, _H₁_, _H₂_ signify the potentials.

When we observe for the first time the agreement here indicated in the
transformative law of the energies, it appears surprising and
unexpected, for we do not perceive at once its reason. But to him who
pursues the comparative historical method that reason will not long
remain a secret.

Since Galileo, mechanical work, though long under a different name, has
been a _fundamental concept_ of mechanics, as also a very important
notion in the applied sciences. The transformation of work into living
force, and of living force into work, suggests directly the notion of
energy--the idea having been first fruitfully employed by Huygens,
although Thomas Young first called it by the _name_ of "energy." Let us
add to this the constancy of weight (really the constancy of mass) and
we shall see that with respect to mechanical energy it is involved in
the very definition of the term that the capacity for work or the
potential energy of a weight is proportional to the height of the level
at which it is, in the geometrical sense, and that it decreases on the
lowering of the weight, on transformation, proportionally to the height
of the level. The zero level here is wholly arbitrary. With this,
equation (2) is given, from which all the other forms follow.

When we reflect on the tremendous start which mechanics had over the
other branches of physics, it is not to be wondered at that the attempt
was always made to apply the notions of that science wherever this was
possible. Thus the notion of mass, for example, was imitated by Coulomb
in the notion of quantity of electricity. In the further development of
the theory of electricity, the notion of work was likewise immediately
introduced in the theory of potential, and heights of electrical level
were measured by the work of unit of quantity raised to that level. But
with this the preceding equation with all its consequences is given for
electrical energy. The case with the other energies was similar.

_Thermal_ energy, however, appears as a special case. Only by the
peculiar experiments mentioned could it be discovered that heat is an
energy. But the measure of this energy by Black's quantity of heat is
the outcome of fortuitous circumstances. In the first place, the
accidental slight variability of the capacity for heat _c_ with the
temperature, and the accidental slight deviation of the usual
thermometrical scales from the scale derived from _the tensions of
gases_, brings it about that the notion "quantity of heat" can be set up
and that the quantity of heat _ct_ corresponding to a difference of
temperature _t_ is nearly proportional to the energy of the heat. It is
a quite accidental historical circumstance that Amontons hit upon the
idea of measuring temperature by the tension of a gas. It is certain in
this that he did not think of the work of the heat.[57] But the numbers
standing for temperature, thus, are made proportional to the tensions of
gases, that is, to the work done by gases, with otherwise equal changes
of volume. It thus happens that _temperature heights_ and _level heights
of work_ are proportional to one another.

If properties of the thermal condition varying greatly from the tensions
of gases had been chosen, this relation would have assumed very
complicated forms, and the agreement between heat and the other energies
above considered would not subsist. It is very instructive to reflect
upon this point. A _natural law_, therefore, is not implied in the
conformity of the behavior of the energies, but this conformity is
rather conditioned by the uniformity of our modes of conception and is
also partly a matter of good fortune.


VI. THE DIFFERENCES OF THE ENERGIES AND THE LIMITS OF THE PRINCIPLE OF
ENERGY.

Of every quantity of heat _Q_ which does work in a reversible process
(one unaccompanied by loss) between the absolute temperatures _T₁_,
_T₂_, only the portion

_(T₁-T₂)/T₁_

is transformed into work, while the remainder is transferred to the
lower temperature-level _T₂_. This transferred portion can, upon the
reversal of the process, with the same expenditure of work, again be
brought back to the level _T₁_. But if the process is not reversible,
then more heat than in the foregoing case flows to the lower level, and
the surplus can no longer be brought back to the higher level _T₂_
without some _special_ expenditure. W. Thomson (1852), accordingly, drew
attention to the fact, that in all non-reversible, that is, in all real
thermal processes, quantities of heat are lost for mechanical work, and
that accordingly a dissipation or waste of mechanical energy is taking
place. In all cases, heat is only partially transformed into work, but
frequently work is wholly transformed into heat. Hence, a tendency
exists towards a diminution of the _mechanical_ energy and towards an
increase of the _thermal_ energy of the world.

For a simple, closed cyclical process, accompanied by no loss, in which
the quantity of heat _Q₁_ is taken from the level _T₁_, and the quantity
_Q₂_ is deposited upon the level _T₂_, the following relation, agreeably
to equation (2), exists,

_-(Q₁/T₁) + (Q₂/T₂) = 0_.

Similarly, for any number of compound reversible cycles Clausius finds
the algebraical sum

_[sum]Q/T = 0_,

and supposing the temperature to change continuously,

_[integral]dQ/T = 0_ (4)

Here the elements of the quantities of heat deducted from a given level
are reckoned negative, and the elements imparted to it, positive. If the
process is not reversible, then expression (4), which Clausius calls
_entropy_, increases. In actual practice this is always the case, and
Clausius finds himself led to the statement:

1. That the energy of the world remains constant.

2. That the entropy of the world tends toward a maximum.

Once we have noted the above-indicated conformity in the behavior of
different energies, the _peculiarity_ of thermal energy here mentioned
must strike us. Whence is this peculiarity derived, for, generally every
energy passes only partly into another form, which is also true of
thermal energy? The explanation will be found in the following.

Every transformation of a special kind of energy _A_ is accompanied with
a fall of potential of that particular kind of energy, including heat.
But whilst for the other kinds of energy a transformation and therefore
a loss of energy on the part of the kind sinking in potential is
connected with the fall of the potential, with heat the case is
different. Heat can suffer a fall of potential without sustaining a loss
of energy, at least according to the customary mode of estimation. If a
weight sinks, it must create perforce kinetic energy, or heat, or some
other form of energy. Also, an electrical charge cannot suffer a fall of
potential without loss of energy, i. e., without transformation. But
heat can pass with a fall of temperature to a body of greater capacity
and the same thermal energy still be preserved, so long as we regard
_every quantity_ of heat as energy. This it is that gives to heat,
besides its property of energy, in many cases the character of a
material _substance_, or quantity.

If we look at the matter in an unprejudiced light, we must ask if there
is any scientific sense or purpose in still considering as energy a
quantity of heat that can no longer be transformed into mechanical work,
(for example, the heat of a closed equably warmed material system). The
principle of energy certainly plays in this case a wholly superfluous
rôle, which is assigned to it only from habit.[58] To maintain the
principle of energy in the face of a knowledge of the dissipation or
waste of mechanical energy, in the face of the increase of entropy is
equivalent almost to the liberty which Black took when he regarded the
heat of liquefaction as still present but latent.[59] It is to be
remarked further, that the expressions "energy of the world" and
"entropy of the world" are slightly permeated with scholasticism. Energy
and entropy are _metrical_ notions. What meaning can there be in
applying these notions to a case in which they are not applicable, in
which their values are not determinable?

If we could really determine the entropy of the world it would represent
a true, absolute measure of time. In this way is best seen the utter
tautology of a statement that the entropy of the world increases with
the time. Time, and the fact that certain changes take place only in a
definite sense, are one and the same thing.



VII. THE SOURCES OF THE PRINCIPLE OF ENERGY.

We are now prepared to answer the question, What are the sources of the
principle of energy? All knowledge of nature is derived in the last
instance from experience. In this sense they are right who look upon the
principle of energy as a result of experience.

Experience teaches that the sense-elements [alpha beta gamma delta ...]
into which the world may be decomposed, are subject to change. It tells
us further, that certain of these elements are _connected_ with other
elements, so that they appear and disappear together; or, that the
appearance of the elements of one class is connected with the
disappearance of the elements of the other class. We will avoid here the
notions of cause and effect because of their obscurity and
equivocalness. The result of experience may be expressed as follows:
_The sensuous elements of the world ([alpha beta gamma delta ...]) show
themselves to be interdependent._ This interdependence is best
represented by some such conception as is in geometry that of the mutual
dependence of the sides and angles of a triangle, only much more varied
and complex.

As an example, we may take a mass of gas enclosed in a cylinder and
possessed of a definite volume ([alpha]), which we change by a pressure
([beta]) on the piston, at the same time feeling the cylinder with our
hand and receiving a sensation of heat ([gamma]). Increase of pressure
diminishes the volume and increases the sensation of heat.

The various facts of experience are not in all respects alike. Their
common sensuous elements are placed in relief by a process of
abstraction and thus impressed upon the memory. In this way the
expression is obtained of the features of _agreement_ of extensive
groups of facts. The simplest sentence which we can utter is, by the
very nature of language, an abstraction of this kind. But account must
also be taken of the _differences_ of related facts. Facts may be so
nearly related as to contain the same kind of a [alpha beta gamma ...],
but the relation be such that the [alpha beta gamma ...] of the one
differ from the [alpha beta gamma ...] of the other only by the number
of equal parts into which they can be divided. Such being the case, if
rules can be given for deducing _from one another_ the numbers which are
the measures of these [alpha beta gamma ...], then we possess in such
rules the _most general_ expression of a group of facts, as also that
expression which corresponds to all its differences. This is the goal of
quantitative investigation.

If this goal be reached what we have found is that between the [alpha
beta gamma ...] of a group of facts, or better, between the numbers
which are their measures, a number of equations exists. The simple fact
of change brings it about that the number of these equations must be
smaller than the number of the [alpha beta gamma ...]. If the former be
smaller by one than the latter, then one portion of the [alpha beta
gamma ...] is _uniquely_ determined by the other portion.

The quest of relations of this last kind is the most important function
of special experimental research, because we are enabled by it to
complete in thought facts that are only partly given. It is self-evident
that only experience can ascertain that between the [alpha beta gamma
...] relations exist and of what kind they are. Further, only experience
can tell that the relations that exist between the [alpha beta gamma
...] are such that changes of them can be reversed. If this were not the
fact all occasion for the enunciation of the principle of energy, as is
easily seen, would be wanting. In experience, therefore, is buried the
ultimate well-spring of all knowledge of nature, and consequently, in
this sense, also the ultimate source of the principle of energy.

But this does not exclude the fact that the principle of energy has also
a logical root, as will now be shown. Let us assume on the basis of
experience that one group of sensuous elements [alpha beta gamma ...]
determines _uniquely_ another group [lambda mu nu ...]. Experience
further teaches that changes of [alpha beta gamma ...] can be
_reversed_. It is then a logical consequence of this observation, that
every time that [alpha beta gamma ...] assume the same values this is
also the case with [lambda mu nu ...]. Or, that purely _periodical_
changes of [alpha beta gamma ...] can produce no _permanent_ changes of
[lambda mu nu ...]. If the group [lambda mu nu ...] is a mechanical
group, then a perpetual motion is excluded.

It will be said that this is a vicious circle, which we will grant. But
psychologically, the situation is essentially different, whether I think
simply of the unique determination and reversibility of events, or
whether I exclude a perpetual motion. The attention takes in the two
cases different directions and diffuses light over different sides of
the question, which logically of course are necessarily connected.

Surely that firm, logical setting of the thoughts noticeable in the
great inquirers, Stevinus, Galileo, and the rest, which, consciously or
instinctively, was supported by a fine feeling for the slightest
contradictions, has no other purpose than to limit the bounds of thought
and so exempt it from the possibility of error. In this, therefore, the
logical root of the principle of excluded perpetual motion is given,
namely, in that universal conviction which existed even before the
development of mechanics and co-operated in that development.

It is perfectly natural that the principle of excluded perpetual motion
should have been first developed in the simple domain of pure mechanics.
Towards the transference of that principle into the domain of general
physics the idea contributed much that all physical phenomena are
mechanical phenomena. But the foregoing discussion shows how little
essential this notion is. The issue really involved is the recognition
of a general interconnexion of nature. This once established, we see
with Carnot that it is indifferent whether the mechanical laws are
broken directly or circuitously.

The principle of the excluded perpetual motion is very closely related
to the modern principle of energy, but it is not identical with it, for
the latter is to be deduced from the former only by means of a definite
_formal conception_. As may be seen from the preceding exposition, the
perpetual motion can be excluded without our employing or possessing the
notion of _work_. The modern principle of energy results primarily from
a _substantial_ conception of work and of every change of physical
condition which by being reversed produces work. The strong need of such
a conception, which is by no means necessary, but in a formal sense is
very convenient and lucid, is exhibited in the case of J. R. Mayer and
Joule. It was before remarked that this conception was suggested to both
inquirers by the observation that both the production of heat and the
production of mechanical work were connected with an expenditure of
substance. Mayer says: "Ex nihilo nil fit," and in another place, "The
creation or destruction of a force (work) lies without the province of
human activity." In Joule we find this passage: "It is manifestly
_absurd_ to suppose that the powers with which God has endowed matter
can be destroyed."

Some writers have observed in such statements the attempt at a
_metaphysical_ establishment of the doctrine of energy. But we see in
them simply the formal need of a simple, clear, and living grasp of the
facts, which receives its development in practical and technical life,
and which we carry over, as best we can, into the province of science.
As a fact, Mayer writes to Griesinger: "If, finally, you ask me how I
became involved in the whole affair, my answer is simply this: Engaged
during a sea voyage almost exclusively with the study of physiology, I
discovered the new theory for the sufficient reason that I _vividly felt
the need of it_."

The substantial conception of work (energy) is by no means a necessary
one. And it is far from true that the problem is solved with the
recognition of the need of such a conception. Rather let us see how
Mayer gradually endeavored to satisfy that need. He first regards
quantity of motion, or momentum, _mv_, as the equivalent of work, and
did not light, until later, on the notion of living force (_mv²/2_). In
the province of electricity he was unable to assign the expression which
is the equivalent of work. This was done later by Helmholtz. The formal
need, therefore, is _first_ present, and our conception of nature is
subsequently gradually _adapted_ to it.

The laying bare of the experimental, logical, and formal root of the
present principle of energy will perhaps contribute much to the removal
of the mysticism which still clings to this principle. With respect to
our formal need of a very simple, palpable, substantial conception of
the processes in our environment, it remains an open question how far
nature corresponds to that need, or how far we can satisfy it. In one
phase of the preceding discussions it would seem as if the substantial
notion of the principle of energy, like Black's material conception of
heat, has its natural limits in facts, beyond which it can only be
artificially adhered to.

  FOOTNOTES:

  [Footnote 39: Published in Vol. 5, No. I, of _The Monist_, October,
  1894, being in part a re-elaboration of the treatise _Ueber die
  Erhaltung der Arbeit_, Prague, 1872.]

  [Footnote 40: _On Matter, Living Force, and Heat_, Joule:
  _Scientific Papers_, London, 1884, I, p. 265.]

  [Footnote 41: "Atqui hoc si sit, globorum series sive corona eundem
  situm cum priore habebit, eademque de causa octo globi sinistri
  ponderosiores erunt sex dextris, ideoque rursus octo illi
  descendent, sex illi ascendent, istique globi ex sese _continuum et
  aeternum motum efficient, quod est falsum_."]

  [Footnote 42: "A igitur, (si ullo modo per naturam fieri possit)
  locum sibi tributum non servato, ac delabatur in _D_; quibus positis
  aqua quae ipsi _A_ succedit eandem ob causam deffluet in _D_, eademque
  ab alia istinc expelletur, atque adeo aqua haec (cum ubique eadem
  ratio sit) _motum instituet perpetuum, quod absurdum fuerit_."]

  [Footnote 43: "Accipio, gradus velocitatis ejusdem mobilis super
  diversas planorum inclinationes acquisitos tunc esse aequales, cum
  eorundum planorum elevationes aequales sint."]

  [Footnote 44: "Voi molto probabilmente discorrete, ma oltre al veri
  simile voglio con una esperienza crescer tanto la probabilità, che
  poco gli manchi all'agguagliarsi ad una ben necessaria
  dimostrazione. Figuratevi questo foglio essere una parete eretta
  all'orizzonte, e da un chiodo fitto in essa pendere una palla di
  piombo d'un'oncia, o due, sospesa dal sottil filo _AB_ lungo due, o
  tre braccia perpendicolare all'orizzonte, e nella parete segnate una
  linea orizontale _DC_ segante a squadra il perpendicolo _AB_, il
  quale sia lontano dalla parete due dita in circa, trasferendo poi il
  filo _AB_ colla palla in _AC_, lasciata essa palla in libertà, la
  quale primieramente vedrete scendere descrivendo l'arco _CBD_, e di
  tanto trapassare il termine _B_, che scorrendo per l'arco _BD_
  sormonterà fino quasi alla segnata parallela _CD_, restando di per
  vernirvi per piccolissimo intervallo, toltogli il precisamente
  arrivarvi dall'impedimento dell'aria, e del filo. Dal che possiamo
  veracemente concludere, che l'impeto acquistato nel punto _B_ dalla
  palla nello scendere per l'arco _CB_, fu tanto, che bastò a
  risospingersi per un simile arco _BD_ alla medesima altezza; fatta,
  e più volte reiterata cotale esperienza, voglio, che fiechiamo nella
  parete rasente al perpendicolo _AB_ un chiodo come in _E_, ovvero in
  _F_, che sporga in fuori cinque, o sei dita, e questo acciocchè il
  filo _AC_ tornando come prima a riportar la palla _C_ per l'arco
  _CB_, giunta che ella sia in _B_, inoppando il filo nel chiodo _E_,
  sia costretta a camminare per la circonferenza _BG_ descritta in
  torno al centro _E_, dal che vedremo quello, che potrà far quel
  medesimo impeto, che dianzi concepizo nel medesimo termine _B_,
  sospinse l'istesso mobile per l'arco _ED_ all'altezza
  dell'orizzonale _CD_. Ora, Signori, voi vedrete con gusto condursi
  la palla all'orizzontale nel punto _G_, e l'istesso accadere,
  l'intoppo si metesse più basso, come in _F_, dove la palla
  descriverebbe l'arco _BJ_, terminando sempre la sua salita
  precisamente nella linea _CD_, e quando l'intoppe del chiodo fusse
  tanto basso, che l'avanzo del filo sotto di lui non arivasse
  all'altezza di _CD_ (il che accaderebbe, quando fusse più vicino al
  punto _B_, che al segamento dell' _AB_ coll'orizzontale _CD_),
  allora il filo cavalcherebbe il chiodo, e segli avolgerebbe intorno.
  Questa esperienza non lascia luogo di dubitare della verità del
  supposto: imperocchè essendo li due archi _CB_, _DB_ equali e
  similmento posti, l'acquisto di momento fatto per la scesa nell'arco
  _CB_, è il medesimo, che il fatto per la scesa dell'arco _DB_; ma il
  momento acquistato in _B_ per l'arco _CB_ è potente a risospingere
  in su il medesimo mobile per l'arco _BD_; adunque anco il momento
  acquistato nella scesa _DB_ è eguale a quello, che sospigne
  l'istesso mobile pel medesimo arco da _B_ in _D_, sicche
  universal-mente ogni memento acquistato per la scesa d'un arco è
  eguale a quello, che può far risalire l'istesso mobile pel medesimo
  arco: ma i momenti tutti che fanno resalire per tutti gli archi
  _BD_, _BG_, _BJ_ sono eguali, poichè son fatti dal istesso medesimo
  momento acquistato per la scesa _CB_, come mostra l'esperienza:
  adunque tutti i momenti, che si acquistano per le scese negli archi
  _DB_, _GB_, _JB_ sono eguali."]

  [Footnote 45: "Constat jam, quod mobile ex quiete in _A_ descendens
  per _AB_, gradus acquirit velocitatis juxta temporis ipsius
  incrementum: gradum vero in _B_ esse maximum acquisitorum, et suapte
  natura immutabiliter impressum, sublatis scilicet causis
  accelerationis novae, aut retardationis: accelerationis inquam, si
  adhuc super extenso plano ulterius progrederetur; retardationis
  vero, dum super planum acclive _BC_ fit reflexio: in horizontali
  autem _GH_ aequabilis motus juxta gradum velocitatis ex _A_ in _B_
  acquisitae in infinitum extenderetur."]

  [Footnote 46: "Si gravitas non esset, neque aër motui corporum
  officeret, unumquodque eorum, acceptum semel motum continuaturum
  velocitate aequabili, secundum lineam rectam."]

  [Footnote 47: "Si pondera quotlibet, vi gravitatis suae, moveri
  incipiant; non posse centrum gravitatis ex ipsis compositae altius,
  quam ubi incipiente motu reperiebatur, ascendere.

  "Ipsa vero hypothesis nostra quominus scrupulum moveat, nihil aliud
  sibi velle ostendemus, quam, quod nemo unquam negavit, gravia nempe
  sursum non ferri.--Et sane, si hac eadem uti scirent novorum operum
  machinatores, qui motum perpetuum irrito conatu moliuntur, facile
  suos ipsi errores deprehenderent, intelligerentque rem eam mechanica
  ratione haud quaquam possibilem esse."]

  [Footnote 48: "Si pendulum e pluribus ponderibus compositum, atque e
  quiete dimissum, partem quamcunque oscillationis integrae
  confecerit, atque inde porro intelligantur pondera ejus singula,
  relicto communi vinculo, celeritates acquisitas sursum convertere,
  ac quousque possunt ascendere; hoc facto centrum gravitatis ex
  omnibus compositae, ad eandem altitudinem reversum erit, quam ante
  inceptam oscillationem obtinebat."]

  [Footnote 49: "Notato autem hic illud staticum axioma etiam locum
  habere:

    "Ut spatium agentis ad spatium patientis
    Sic potentia patientis ad potentiam agentis."]

  [Footnote 50: "Cependant, comme dans cet ouvrage on ne fut d'abord
  attentif qu'à considérer ce beau développement de la mécanique qui
  semblait sortir tout entière d'une seule et même formule, on crut
  naturellement que la science etait faite, et qu'il ne restait plus
  qu'à chercher la démonstration du principe des vitesses virtuelles.
  Mais cette recherche ramena toutes les difficultés qu'on avait
  franchies par le principe même. Cette loi si générale, où se mêlent
  des idées vagues et étrangères de mouvements infinement petits et de
  perturbation d'équilibre, ne fit en quelque sorte que s'obsurcir à
  l'examen; et le livre de Lagrange n'offrant plus alors rien de clair
  que la marche des calculs, on vit bien que les nuages n'avaient paru
  levé sur le cours de la mécanique que parcequ'ils étaient, pour
  ainsi dire, rassemblés à l'origine même do cette science.

  "Une démonstration générale du principe des vitesses virtuelles
  devait au fond revenir a établir le mécanique entière sur une autre
  base: car la demonstration d'une loi qui embrasse toute une science
  ne peut être autre chose qua la reduction de cette science à une
  autre loi aussi générale, mais évidente, ou du moins plus simple que
  la première, et qui partant la rende inutile."]

  [Footnote 51: _Traité de la lumière_, Leyden, 1690, p. 2.]

  [Footnote 52: "L'on ne sçaurait douter que la lumière ne consiste
  dans le _mouvement_ de certaine matière. Car soit qu'on regarde sa
  production, on trouve qu'içy sur la terre c'est principalement le
  feu et la flamme qui l'engendrent, lesquels contient sans doute des
  corps qui sont dans un mouvement rapide, puis qu'ils dissolvent et
  fondent plusieurs autres corps des plus solides: soit qu'on regarde
  ses effets, on voit que quand la lumière est ramasseé, comme par des
  miroires concaves, elle a la vertu de brûler comme le feu.
  c-est-à-dire qu'elle desunit les parties des corps; ce qui marque
  assurément du _mouvement_, au moins dans la _vraye Philosophie_,
  dans laquelle on conçoit la cause de tous les effets naturels par
  des raisons de _mechanique_. Ce qu'il faut faire à mon avis, ou bien
  renoncer à tout espérance de jamais rien comprendre dans la
  Physique."]

  [Footnote 53: _Sur la puissance motrice du feu_. (Paris, 1824.)]

  [Footnote 54: "On objectra peut-être ici que le mouvement perpétuel,
  démontré impossible par les _seules actions mécaniques_, ne l'est
  peut-être pas lorsqu'on emploie l'influence soit de la _chaleur_,
  soit de l'électricité; mais pent-on concevoir les phénomènes de la
  chaleur et de l'électricité comme dus à autre chose qu'à des
  _mouvements quelconques des corps_ et comme tels ne doivent-ils pas
  être soumis aux lois générales de la mécanique?"]

  [Footnote 55: By this is meant the temperature of a Celsius scale,
  the zero of which is 273° below the melting-point of ice.]

  [Footnote 56: I first drew attention to this fact in my treatise
  _Ueber die Erhaltung der Arbeit_, Prague, 1872. Before this, Zeuner
  had pointed out the analogy between mechanical and thermal energy. I
  have given a more extensive development of this idea in a
  communication to the _Sitzungsberichte der Wiener_ _Akademie_,
  December, 1892, entitled _Geschichte und Kritik des Carnot'schen
  Wärmegesetzes_. Compare also the works of Popper (1884), Helm
  (1887), Wronsky (1888), and Ostwald (1892).]

  [Footnote 57: Sir William Thomson first consciously and
  intentionally introduced (1848, 1851) a _mechanical_ measure of
  temperature similar to the electric measure of potential.]

  [Footnote 58: Compare my _Analysis of the Sensations_, Jena, 1886:
  English translation, Chicago, 1897.]

  [Footnote 59: A better terminology appears highly desirable in the
  place of the usual misleading one. Sir William Thomson (1852)
  appears to have felt this need, and it has been clearly expressed by
  F. Wald (1889). We should call the work which corresponds to a
  vanished quantity of heat its mechanical substitution-value; while
  that work which can be _actually_ performed in the passage of a
  thermal condition _A_ to a condition _B_, alone deserves the name of
  the _energy-value_ of this change of condition. In this way the
  _arbitrary_ substantial conception of the processes would be
  preserved and misapprehensions forestalled.]



THE ECONOMICAL NATURE OF PHYSICAL INQUIRY.[60]


When the human mind, with its limited powers, attempts to mirror in
itself the rich life of the world, of which it is itself only a small
part, and which it can never hope to exhaust, it has every reason for
proceeding economically. Hence that tendency, expressed in the
philosophy of all times, to compass by a few organic thoughts the
fundamental features of reality. "Life understands not death, nor death
life." So spake an old Chinese philosopher. Yet in his unceasing desire
to diminish the boundaries of the incomprehensible, man has always been
engaged in attempts to understand death by life and life by death.

Among the ancient civilised peoples, nature was filled with demons and
spirits having the feelings and desires of men. In all essential
features, this animistic view of nature, as Tylor[61] has aptly termed
it, is shared in common by the fetish-worshipper of modern Africa and
the most advanced nations of antiquity. As a theory of the world it has
never completely disappeared. The monotheism of the Christians never
fully overcame it, no more than did that of the Jews. In the belief in
witchcraft and in the superstitions of the sixteenth and seventeenth
centuries, the centuries of the rise of natural science, it assumed
frightful pathological dimensions. Whilst Stevinus, Kepler, and Galileo
were slowly rearing the fabric of modern physical science, a cruel and
relentless war was waged with firebrand and rack against the devils that
glowered from every corner. To-day even, apart from all survivals of
that period, apart from the traces of fetishism which still inhere in
our physical concepts,[62] those very ideas still covertly lurk in the
practices of modern spiritualism.


By the side of this animistic conception of the world, we meet from time
to time, in different forms, from Democritus to the present day, another
view, which likewise claims exclusive competency to comprehend the
universe. This view may be characterised as the _physico-mechanical_
view of the world. To-day, that view holds, indisputably, the first
place in the thoughts of men, and determines the ideals and the
character of our times. The coming of the mind of man into the full
consciousness of its powers, in the eighteenth century, was a period of
genuine disillusionment. It produced the splendid precedent of a life
really worthy of man, competent to overcome the old barbarism in the
practical fields of life; it created the _Critique of Pure Reason_,
which banished into the realm of shadows the sham-ideas of the old
metaphysics; it pressed into the hands of the mechanical philosophy the
reins which it now holds.

The oft-quoted words of the great Laplace,[63] which I will now give,
have the ring of a jubilant toast to the scientific achievements of the
eighteenth century: "A mind to which were given for a single instant all
the forces of nature and the mutual positions of all its masses, if it
were otherwise powerful enough to subject these problems to analysis,
could grasp, with a single formula, the motions of the largest masses as
well as of the smallest atoms; nothing would be uncertain for it; the
future and the past would lie revealed before its eyes." In writing
these words, Laplace, as we know, had also in mind the atoms of the
brain. That idea has been expressed more forcibly still by some of his
followers, and it is not too much to say that Laplace's ideal is
substantially that of the great majority of modern scientists.

Gladly do we accord to the creator of the _Mécanique céleste_ the sense
of lofty pleasure awakened in him by the great success of the
Enlightenment, to which we too owe our intellectual freedom. But to-day,
with minds undisturbed and before _new_ tasks, it becomes physical
science to secure itself against self-deception by a careful study of
its character, so that it can pursue with greater sureness its true
objects. If I step, therefore, beyond the narrow precincts of my
specialty in this discussion, to trespass on friendly neighboring
domains, I may plead in my excuse that the subject-matter of knowledge
is common to all domains of research, and that fixed, sharp lines of
demarcation cannot be drawn.

The belief in occult magic powers of nature has gradually died away, but
in its place a new belief has arisen, the belief in the magical power of
science. Science throws her treasures, not like a capricious fairy into
the laps of a favored few, but into the laps of all humanity, with a
lavish extravagance that no legend ever dreamt of! Not without apparent
justice, therefore, do her distant admirers impute to her the power of
opening up unfathomable abysses of nature, to which the senses cannot
penetrate. Yet she who came to bring light into the world, can well
dispense with the darkness of mystery, and with pompous show, which she
needs neither for the justification of her aims nor for the adornment of
her plain achievements.

The homely beginnings of science will best reveal to us its simple,
unchangeable character. Man acquires his first knowledge of nature
half-consciously and automatically, from an instinctive habit of
mimicking and forecasting facts in thought, of supplementing sluggish
experience with the swift wings of thought, at first only for his
material welfare. When he hears a noise in the underbrush he constructs
there, just as the animal does, the enemy which he fears; when he sees a
certain rind he forms mentally the image of the fruit which he is in
search of; just as we mentally associate a certain kind of matter with a
certain line in the spectrum or an electric spark with the friction of a
piece of glass. A knowledge of causality in this form certainly reaches
far below the level of Schopenhauer's pet dog, to whom it was ascribed.
It probably exists in the whole animal world, and confirms that great
thinker's statement regarding the will which created the intellect for
its purposes. These primitive psychical functions are rooted in the
economy of our organism not less firmly than are motion and digestion.
Who would deny that we feel in them, too, the elemental power of a long
practised logical and physiological activity, bequeathed to us as an
heirloom from our forefathers?

Such primitive acts of knowledge constitute to-day the solidest
foundation of scientific thought. Our instinctive knowledge, as we shall
briefly call it, by virtue of the conviction that we have consciously
and intentionally contributed nothing to its formation, confronts us
with an authority and logical power which consciously acquired knowledge
even from familiar sources and of easily tested fallibility can never
possess. All so-called axioms are such instinctive knowledge. Not
consciously gained knowledge alone, but powerful intellectual instinct,
joined with vast conceptive powers, constitute the great inquirer. The
greatest advances of science have always consisted in some successful
formulation, in clear, abstract, and communicable terms, of what was
instinctively known long before, and of thus making it the permanent
property of humanity. By Newton's principle of the equality of pressure
and counterpressure, whose truth all before him had felt, but which no
predecessor had abstractly formulated, mechanics was placed by a single
stroke on a higher level. Our statement might also be historically
justified by examples from the scientific labors of Stevinus, S. Carnot,
Faraday, J. R. Mayer, and others.

All this, however, is merely the soil from which science starts. The
first real beginnings of science appear in society, particularly in the
manual arts, where the necessity for the communication of experience
arises. Here, where some new discovery is to be described and related,
the compulsion is first felt of clearly defining in consciousness the
important and essential features of that discovery, as many writers can
testify. The aim of instruction is simply the saving of experience; the
labor of one man is made to take the place of that of another.

The most wonderful economy of communication is found in language. Words
are comparable to type, which spare the repetition of written signs and
thus serve a multitude of purposes; or to the few sounds of which our
numberless different words are composed. Language, with its helpmate,
conceptual thought, by fixing the essential and rejecting the
unessential, constructs its rigid pictures of the fluid world on the
plan of a mosaic, at a sacrifice of exactness and fidelity but with a
saving of tools and labor. Like a piano-player with previously prepared
sounds, a speaker excites in his listener thoughts previously prepared,
but fitting many cases, which respond to the speaker's summons with
alacrity and little effort.

The principles which a prominent political economist, E. Hermann,[64]
has formulated for the economy of the industrial arts, are also
applicable to the ideas of common life and of science. The economy of
language is augmented, of course, in the terminology of science. With
respect to the economy of written intercourse there is scarcely a doubt
that science itself will realise that grand old dream of the
philosophers of a Universal Real Character. That time is not far
distant. Our numeral characters, the symbols of mathematical analysis,
chemical symbols, and musical notes, which might easily be supplemented
by a system of color-signs, together with some phonetic alphabets now in
use, are all beginnings in this direction. The logical extension of what
we have, joined with a use of the ideas which the Chinese ideography
furnishes us, will render the special invention and promulgation of a
Universal Character wholly superfluous.

The communication of scientific knowledge always involves description,
that is, a mimetic reproduction of facts in thought, the object of which
is to replace and save the trouble of new experience. Again, to save the
labor of instruction and of acquisition, concise, abridged description
is sought. This is really all that natural laws are. Knowing the value
of the acceleration of gravity, and Galileo's laws of descent, we
possess simple and compendious directions for reproducing in thought all
possible motions of falling bodies. A formula of this kind is a complete
substitute for a full table of motions of descent, because by means of
the formula the data of such a table can be easily constructed at a
moment's notice without the least burdening of the memory.

No human mind could comprehend all the individual cases of refraction.
But knowing the index of refraction for the two media presented, and the
familiar law of the sines, we can easily reproduce or fill out in
thought every conceivable case of refraction. The advantage here
consists in the disburdening of the memory; an end immensely furthered
by the written preservation of the natural constants. More than this
comprehensive and condensed report about facts is not contained in a
natural law of this sort. In reality, the law always contains less than
the fact itself, because it does not reproduce the fact as a whole but
only in that aspect of it which is important for us, the rest being
either intentionally or from necessity omitted. Natural laws may be
likened to intellectual type of a higher order, partly movable, partly
stereotyped, which last on new editions of experience may become
downright impediments.

When we look over a province of facts for the first time, it appears to
us diversified, irregular, confused, full of contradictions. We first
succeed in grasping only single facts, unrelated with the others. The
province, as we are wont to say, is not _clear_. By and by we discover
the simple, permanent elements of the mosaic, out of which we can
mentally construct the whole province. When we have reached a point
where we can discover everywhere the same facts, we no longer feel lost
in this province; we comprehend it without effort; it is _explained_ for
us.

Let me illustrate this by an example. As soon as we have grasped the
fact of the rectilinear propagation of light, the regular course of our
thoughts stumbles at the phenomena of refraction and diffraction. As
soon as we have cleared matters up by our index of refraction we
discover that a special index is necessary for each color. Soon after we
have accustomed ourselves to the fact that light added to light
increases its intensity, we suddenly come across a case of total
darkness produced by this cause. Ultimately, however, we see everywhere
in the overwhelming multifariousness of optical phenomena the fact of
the spatial and temporal periodicity of light, with its velocity of
propagation dependent on the medium and the period. This tendency of
obtaining a survey of a given province with the least expenditure of
thought, and of representing all its facts by some one single mental
process, may be justly termed an economical one.

The greatest perfection of mental economy is attained in that science
which has reached the highest formal development, and which is widely
employed in physical inquiry, namely, in mathematics. Strange as it
may sound, the power of mathematics rests upon its evasion of
all unnecessary thought and on its wonderful saving of mental
operations. Even those arrangement-signs which we call numbers are a
system of marvellous simplicity and economy. When we employ the
multiplication-table in multiplying numbers of several places, and so
use the results of old operations of counting instead of performing the
whole of each operation anew; when we consult our table of logarithms,
replacing and saving thus new calculations by old ones already
performed; when we employ determinants instead of always beginning
afresh the solution of a system of equations; when we resolve new
integral expressions into familiar old integrals; we see in this simply
a feeble reflexion of the intellectual activity of a Lagrange or a
Cauchy, who, with the keen discernment of a great military commander,
substituted for new operations whole hosts of old ones. No one will
dispute me when I say that the most elementary as well as the highest
mathematics are economically-ordered experiences of counting, put in
forms ready for use.

In algebra we perform, as far as possible, all numerical operations
which are identical in form once for all, so that only a remnant of work
is left for the individual case. The use of the signs of algebra and
analysis, which are merely symbols of operations to be performed, is due
to the observation that we can materially disburden the mind in this way
and spare its powers for more important and more difficult duties, by
imposing all mechanical operations upon the hand. One result of this
method, which attests its economical character, is the construction of
calculating machines. The mathematician Babbage, the inventor of the
difference-engine, was probably the first who clearly perceived this
fact, and he touched upon it, although only cursorily, in his work, _The
Economy of Manufactures and Machinery_.

The student of mathematics often finds it hard to throw off the
uncomfortable feeling that his science, in the person of his pencil,
surpasses him in intelligence,--an impression which the great Euler
confessed he often could not get rid of. This feeling finds a sort of
justification when we reflect that the majority of the ideas we deal
with were conceived by others, often centuries ago. In great measure it
is really the intelligence of other people that confronts us in science.
The moment we look at matters in this light, the uncanniness and magical
character of our impressions cease, especially when we remember that we
can think over again at will any one of those alien thoughts.

Physics is experience, arranged in economical order. By this order not
only is a broad and comprehensive view of what we have rendered
possible, but also the defects and the needful alterations are made
manifest, exactly as in a well-kept household. Physics shares with
mathematics the advantages of succinct description and of brief,
compendious definition, which precludes confusion, even in ideas where,
with no apparent burdening of the brain, hosts of others are contained.
Of these ideas the rich contents can be produced at any moment and
displayed in their full perceptual light. Think of the swarm of
well-ordered notions pent up in the idea of the potential. Is it
wonderful that ideas containing so much finished labor should be easy to
work with?

Our first knowledge, thus, is a product of the economy of
self-preservation. By communication, the experience of _many_ persons,
individually acquired at first, is collected in _one_. The communication
of knowledge and the necessity which every one feels of managing his
stock of experience with the least expenditure of thought, compel us to
put our knowledge in economical forms. But here we have a clue which
strips science of all its mystery, and shows us what its power really
is. With respect to specific results it yields us nothing that we could
not reach in a sufficiently long time without methods. There is no
problem in all mathematics that cannot be solved by direct counting. But
with the present implements of mathematics many operations of counting
can be performed in a few minutes which without mathematical methods
would take a lifetime. Just as a single human being, restricted wholly
to the fruits of his own labor, could never amass a fortune, but on the
contrary the accumulation of the labor of many men in the hands of one
is the foundation of wealth and power, so, also, no knowledge worthy of
the name can be gathered up in a single human mind limited to the span
of a human life and gifted only with finite powers, except by the most
exquisite economy of thought and by the careful amassment of the
economically ordered experience of thousands of co-workers. What strikes
us here as the fruits of sorcery are simply the rewards of excellent
housekeeping, as are the like results in civil life. But the business of
science has this advantage over every other enterprise, that from _its_
amassment of wealth no one suffers the least loss. This, too, is its
blessing, its freeing and saving power.

The recognition of the economical character of science will now help us,
perhaps, to understand better certain physical notions.

Those elements of an event which we call "cause and effect" are certain
salient features of it, which are important for its mental reproduction.
Their importance wanes and the attention is transferred to fresh
characters the moment the event or experience in question becomes
familiar. If the connexion of such features strikes us as a necessary
one, it is simply because the interpolation of certain intermediate
links with which we are very familiar, and which possess, therefore,
higher authority for us, is often attended with success in our
explanations. That _ready_ experience fixed in the mosaic of the mind
with which we meet new events, Kant calls an innate concept of the
understanding (_Verstandesbegriff_).

The grandest principles of physics, resolved into their elements, differ
in no wise from the descriptive principles of the natural historian. The
question, "Why?" which is always appropriate where the explanation of a
contradiction is concerned, like all proper habitudes of thought, can
overreach itself and be asked where nothing remains to be understood.
Suppose we were to attribute to nature the property of producing like
effects in like circumstances; just these like circumstances we should
not know how to find. Nature exists once only. Our schematic mental
imitation alone produces like events. Only in the mind, therefore, does
the mutual dependence of certain features exist.

All our efforts to mirror the world in thought would be futile if we
found nothing permanent in the varied changes of things. It is this that
impels us to form the notion of substance, the source of which is not
different from that of the modern ideas relative to the conservation of
energy. The history of physics furnishes numerous examples of this
impulse in almost all fields, and pretty examples of it may be traced
back to the nursery. "Where does the light go to when it is put out?"
asks the child. The sudden shrivelling up of a hydrogen balloon is
inexplicable to a child; it looks everywhere for the large body which
was just there but is now gone.

Where does heat come from? Where does heat go to? Such childish
questions in the mouths of mature men shape the character of a century.

In mentally separating a body from the changeable environment in which
it moves, what we really do is to extricate a group of sensations on
which our thoughts are fastened and which is of relatively greater
stability than the others, from the stream of all our sensations.
Absolutely unalterable this group is not. Now this, now that member of
it appears and disappears, or is altered. In its full identity it never
recurs. Yet the sum of its constant elements as compared with the sum of
its changeable ones, especially if we consider the continuous character
of the transition, is always so great that for the purpose in hand the
former usually appear sufficient to determine the body's identity. But
because we can separate from the group every single member without the
body's ceasing to be for us the same, we are easily led to believe that
after abstracting all the members something additional would remain. It
thus comes to pass that we form the notion of a substance distinct from
its attributes, of a thing-in-itself, whilst our sensations are regarded
merely as symbols or indications of the properties of this
thing-in-itself. But it would be much better to say that bodies or
things are compendious mental symbols for groups of sensations--symbols
that do not exist outside of thought. Thus, the merchant regards the
labels of his boxes merely as indexes of their contents, and not the
contrary. He invests their contents, not their labels, with real value.
The same economy which induces us to analyse a group and to establish
special signs for its component parts, parts which also go to make up
other groups, may likewise induce us to mark out by some single symbol a
whole group.

On the old Egyptian monuments we see objects represented which do not
reproduce a single visual impression, but are composed of various
impressions. The heads and the legs of the figures appear in profile,
the head-dress and the breast are seen from the front, and so on. We
have here, so to speak, a mean view of the objects, in forming which the
sculptor has retained what he deemed essential, and neglected what he
thought indifferent. We have living exemplifications of the processes
put into stone on the walls of these old temples, in the drawings of our
children, and we also observe a faithful analogue of them in the
formation of ideas in our own minds. Only in virtue of some such
facility of view as that indicated, are we allowed to speak of a body.
When we speak of a cube with trimmed corners--a figure which is not a
cube--we do so from a natural instinct of economy, which prefers to add
to an old familiar conception a correction instead of forming an
entirely new one. This is the process of all judgment.

The crude notion of "body" can no more stand the test of analysis than
can the art of the Egyptians or that of our little children. The
physicist who sees a body flexed, stretched, melted, and vaporised, cuts
up this body into smaller permanent parts; the chemist splits it up into
elements. Yet even an element is not unalterable. Take sodium. When
warmed, the white, silvery mass becomes a liquid, which, when the heat
is increased and the air shut out, is transformed into a violet vapor,
and on the heat being still more increased glows with a yellow light. If
the name sodium is still retained, it is because of the continuous
character of the transitions and from a necessary instinct of economy.
By condensing the vapor, the white metal may be made to reappear.
Indeed, even after the metal is thrown into water and has passed into
sodium hydroxide, the vanished properties may by skilful treatment still
be made to appear; just as a moving body which has passed behind a
column and is lost to view for a moment may make its appearance after a
time. It is unquestionably very convenient always to have ready the name
and thought for a group of properties wherever that group by any
possibility can appear. But more than a compendious economical symbol
for these phenomena, that name and thought is not. It would be a mere
empty word for one in whom it did not awaken a large group of
well-ordered sense-impressions. And the same is true of the molecules
and atoms into which the chemical element is still further analysed.

True, it is customary to regard the conservation of weight, or, more
precisely, the conservation of mass, as a direct proof of the constancy
of matter. But this proof is dissolved, when we go to the bottom of it,
into such a multitude of instrumental and intellectual operations, that
in a sense it will be found to constitute simply an equation which our
ideas in imitating facts have to satisfy. That obscure, mysterious lump
which we involuntarily add in thought, we seek for in vain outside the
mind.

It is always, thus, the crude notion of substance that is slipping
unnoticed into science, proving itself constantly insufficient, and ever
under the necessity of being reduced to smaller and smaller
world-particles. Here, as elsewhere, the lower stage is not rendered
indispensable by the higher which is built upon it, no more than the
simplest mode of locomotion, walking, is rendered superfluous by the
most elaborate means of transportation. Body, as a compound of light and
touch sensations, knit together by sensations of space, must be as
familiar to the physicist who seeks it, as to the animal who hunts its
prey. But the student of the theory of knowledge, like the geologist and
the astronomer, must be permitted to reason back from the forms which
are created before his eyes to others which he finds ready made for
him.

All physical ideas and principles are succinct directions, frequently
involving subordinate directions, for the employment of economically
classified experiences, ready for use. Their conciseness, as also the
fact that their contents are rarely exhibited in full, often invests
them with the semblance of independent existence. Poetical myths
regarding such ideas,--for example, that of Time, the producer and
devourer of all things,--do not concern us here. We need only remind the
reader that even Newton speaks of an _absolute_ time independent of all
phenomena, and of an absolute space--views which even Kant did not shake
off, and which are often seriously entertained to-day. For the natural
inquirer, determinations of time are merely abbreviated statements of
the dependence of one event upon another, and nothing more. When we say
the acceleration of a freely falling body is 9·810 metres per second, we
mean the velocity of the body with respect to the centre of the earth is
9·810 metres greater when the earth has performed an additional 86400th
part of its rotation--a fact which itself can be determined only by the
earth's relation to other heavenly bodies. Again, in velocity is
contained simply a relation of the position of a body to the position of
the earth.[65] Instead of referring events to the earth we may refer
them to a clock, or even to our internal sensation of time. Now, because
all are connected, and each may be made the measure of the rest, the
illusion easily arises that time has significance independently of
all.[66]

The aim of research is the discovery of the equations which subsist
between the elements of phenomena. The equation of an ellipse expresses
the universal _conceivable_ relation between its co-ordinates, of which
only the real values have _geometrical_ significance. Similarly, the
equations between the elements of _phenomena_ express a universal,
mathematically conceivable relation. Here, however, for many values only
certain directions of change are _physically_ admissible. As in the
ellipse only certain _values_ satisfying the equation are realised, so
in the physical world only certain _changes_ of value occur. Bodies are
always accelerated towards the earth. Differences of temperature, left
to themselves, always grow less; and so on. Similarly, with respect to
space, mathematical and physiological researches have shown that the
space of experience is simply an _actual_ case of many conceivable
cases, about whose peculiar properties experience alone can instruct us.
The elucidation which this idea diffuses cannot be questioned, despite
the absurd uses to which it has been put.

Let us endeavor now to summarise the results of our survey. In the
economical schematism of science lie both its strength and its weakness.
Facts are always represented at a sacrifice of completeness and never
with greater precision than fits the needs of the moment. The
incongruence between thought and experience, therefore, will continue to
subsist as long as the two pursue their course side by side; but it will
be continually diminished.

In reality, the point involved is always the completion of some partial
experience; the derivation of one portion of a phenomenon from some
other. In this act our ideas must be based directly upon sensations. We
call this measuring.[67] The condition of science, both in its origin
and in its application, is a _great relative stability_ of our
environment. What it teaches us is interdependence. Absolute forecasts,
consequently, have no significance in science. With great changes in
celestial space we should lose our co-ordinate systems of space and
time.

When a geometer wishes to understand the form of a curve, he first
resolves it into small rectilinear elements. In doing this, however, he
is fully aware that these elements are only provisional and arbitrary
devices for comprehending in parts what he cannot comprehend as a whole.
When the law of the curve is found he no longer thinks of the elements.
Similarly, it would not become physical science to see in its
self-created, changeable, economical tools, molecules and atoms,
realities behind phenomena, forgetful of the lately acquired sapience of
her older sister, philosophy, in substituting a mechanical mythology for
the old animistic or metaphysical scheme, and thus creating no end of
suppositious problems. The atom must remain a tool for representing
phenomena, like the functions of mathematics. Gradually, however, as the
intellect, by contact with its subject-matter, grows in discipline,
physical science will give up its mosaic play with stones and will seek
out the boundaries and forms of the bed in which the living stream of
phenomena flows. The goal which it has set itself is the _simplest_ and
_most economical_ abstract expression of facts.

       *       *       *       *       *

The question now remains, whether the same method of research which till
now we have tacitly restricted to physics, is also applicable in the
psychical domain. This question will appear superfluous to the physical
inquirer. Our physical and psychical views spring in exactly the same
manner from instinctive knowledge. We read the thoughts of men in their
acts and facial expressions without knowing how. Just as we predict the
behavior of a magnetic needle placed near a current by imagining
Ampère's swimmer in the current, similarly we predict in thought the
acts and behavior of men by assuming sensations, feelings, and wills
similar to our own connected with their bodies. What we here
instinctively perform would appear to us as one of the subtlest
achievements of science, far outstripping in significance and ingenuity
Ampère's rule of the swimmer, were it not that every child unconsciously
accomplished it. The question simply is, therefore, to grasp
scientifically, that is, by conceptional thought, what we are already
familiar with from other sources. And here much is to be accomplished. A
long sequence of facts is to be disclosed between the physics of
expression and movement and feeling and thought.

We hear the question, "But how is it possible to explain feeling by the
motions of the atoms of the brain?" Certainly this will never be done,
no more than light or heat will ever be deduced from the law of
refraction. We need not deplore, therefore, the lack of ingenious
solutions of this question. The problem is not a problem. A child
looking over the walls of a city or of a fort into the moat below sees
with astonishment living people in it, and not knowing of the portal
which connects the wall with the moat, cannot understand how they could
have got down from the high ramparts. So it is with the notions of
physics. We cannot climb up into the province of psychology by the
ladder of our abstractions, but we can climb down into it.

Let us look at the matter without bias. The world consists of colors,
sounds, temperatures, pressures, spaces, times, and so forth, which now
we shall not call sensations, nor phenomena, because in either term an
arbitrary, one-sided theory is embodied, but simply _elements_. The
fixing of the flux of these elements, whether mediately or immediately,
is the real object of physical research. As long as, neglecting our own
body, we employ ourselves with the interdependence of those groups of
elements which, including men and animals, make up _foreign_ bodies, we
are physicists. For example, we investigate the change of the red color
of a body as produced by a change of illumination. But the moment we
consider the special influence on the red of the elements constituting
our body, outlined by the well-known perspective with head invisible, we
are at work in the domain of physiological psychology. We close our
eyes, and the red together with the whole visible world disappears.
There exists, thus, in the perspective field of every sense a portion
which exercises on all the rest a different and more powerful influence
than the rest upon one another. With this, however, all is said. In the
light of this remark, we call _all_ elements, in so far as we regard
them as dependent on this special part (our body), _sensations_. That
the world is our sensation, in this sense, cannot be questioned. But to
make a system of conduct out of this provisional conception, and to
abide its slaves, is as unnecessary for us as would be a similar course
for a mathematician who, in varying a series of variables of a function
which were previously assumed to be constant, or in interchanging the
independent variables, finds his method to be the source of some very
surprising ideas for him.[68]

If we look at the matter in this unbiassed light it will appear
indubitable that the method of physiological psychology is none other
than that of physics; what is more, that this science is a part of
physics. Its subject-matter is not different from that of physics. It
will unquestionably determine the relations the sensations bear to the
physics of our body. We have already learned from a member of this
academy (Hering) that in all probability a sixfold manifoldness of the
chemical processes of the visual substance corresponds to the sixfold
manifoldness of color-sensation, and a threefold manifoldness of the
physiological processes to the threefold manifoldness of
space-sensations. The paths of reflex actions and of the will are
followed up and disclosed; it is ascertained what region of the brain
subserves the function of speech, what region the function of
locomotion, etc. That which still clings to our body, namely, our
thoughts, will, when those investigations are finished, present no
difficulties new in principle. When experience has once clearly
exhibited these facts and science has marshalled them in economic and
perspicuous order, there is no doubt that we shall _understand_ them.
For other "understanding" than a mental mastery of facts never existed.
Science does not create facts from facts, but simply _orders_ known
facts.

Let us look, now, a little more closely into the modes of research of
physiological psychology. We have a very clear idea of how a body moves
in the space encompassing it. With our optical field of sight we are
very familiar. But we are unable to state, as a rule, how we have come
by an idea, from what corner of our intellectual field of sight it has
entered, or by what region the impulse to a motion is sent forth.
Moreover, we shall never get acquainted with this mental field of view
from self-observation alone. Self-observation, in conjunction with
physiological research, which seeks out physical connexions, can put
this field of vision in a clear light before us, and will thus first
really reveal to us our inner man.

Primarily, natural science, or physics, in its widest sense, makes us
acquainted with only the firmest connexions of groups of elements.
Provisorily, we may not bestow too much attention on the single
constituents of those groups, if we are desirous of retaining a
comprehensible whole. Instead of equations between the primitive
variables, physics gives us, as much the easiest course, equations
between _functions_ of those variables. Physiological psychology teaches
us how to separate the visible, the tangible, and the audible from
bodies--a labor which is subsequently richly requited, as the division
of the subjects of physics well shows. Physiology further analyses the
visible into light and space sensations; the first into colors, the last
also into their component parts; it resolves noises into sounds, these
into tones, and so on. Unquestionably this analysis can be carried much
further than it has been. It will be possible in the end to exhibit the
common elements at the basis of very abstract but definite logical acts
of like form,--elements which the acute jurist and mathematician, as it
were, _feels_ out, with absolute certainty, where the uninitiated hears
only empty words. Physiology, in a word, will reveal to us the true real
elements of the world. Physiological psychology bears to physics in its
widest sense a relation similar to that which chemistry bears to physics
in its narrowest sense. But far greater than the mutual support of
physics and chemistry will be that which natural science and psychology
will render each other. And the results that shall spring from this
union will, in all likelihood, far outstrip those of the modern
mechanical physics.

What those ideas are with which we shall comprehend the world when the
closed circuit of physical and psychological facts shall lie complete
before us, (that circuit of which we now see only two disjoined parts,)
cannot be foreseen at the outset of the work. The men will be found who
will see what is right and will have the courage, instead of wandering
in the intricate paths of logical and historical accident, to enter on
the straight ways to the heights from which the mighty stream of facts
can be surveyed. Whether the notion which we now call matter will
continue to have a scientific significance beyond the crude purposes of
common life, we do not know. But we certainly shall wonder how colors
and tones which were such innermost parts of us could suddenly get lost
in our physical world of atoms; how we could be suddenly surprised that
something which outside us simply clicked and beat, in our heads should
make light and music; and how we could ask whether matter can feel, that
is to say, whether a mental symbol for a group of sensations can feel?

We cannot mark out in hard and fast lines the science of the future, but
we can foresee that the rigid walls which now divide man from the world
will gradually disappear; that human beings will not only confront each
other, but also the entire organic and so-called lifeless world, with
less selfishness and with livelier sympathy. Just such a presentiment as
this perhaps possessed the great Chinese philosopher Licius some two
thousand years ago when, pointing to a heap of mouldering human bones,
he said to his scholars in the rigid, lapidary style of his tongue:
"These and I alone have the knowledge that we neither live nor are
dead."

  FOOTNOTES:

  [Footnote 60: An address delivered before the anniversary meeting of
  the Imperial Academy of Sciences, at Vienna, May 25, 1882.]

  [Footnote 61: _Primitive Culture._]

  [Footnote 62: Tylor, _loc. cit._]

  [Footnote 63: _Essai philosophique sur les probabilités_. 6th Ed.
  Paris, 1840, p. 4. The necessary consideration of the initial
  velocities is lacking in this formulation.]

  [Footnote 64: _Principien der Wirthschaftslehre_, Vienna, 1873.]

  [Footnote 65: It is clear from this that all so-called elementary
  (differential) laws involve a relation to the whole.]

  [Footnote 66: If it be objected, that in the case of perturbations
  of the velocity of rotation of the earth, we could be sensible of
  such perturbations, and being obliged to have some measure of time,
  we should resort to the period of vibration of the waves of sodium
  light,--all that this would show is that for practical reasons we
  should select that event which best served us as the _simplest_
  common measure of the others.]

  [Footnote 67: Measurement, in fact, is the definition of one
  phenomenon by another (standard) phenomenon.]

  [Footnote 68: I have represented the point of view here taken for
  more than thirty years and developed it in various writings
  (_Erhaltung der Arbeit_, 1872, parts of which are published in the
  article on _The Conservation of Energy_ in this collection; _The
  Forms of Liquids_, 1872, also published in this collection; and the
  _Bewegungsempfindungen_, 1875). The idea, though known to
  philosophers, is unfamiliar to the majority of physicists. It is a
  matter of deep regret to me, therefore, that the title and author of
  a small tract which accorded with my views in numerous details and
  which I remember having caught a glance of in a very busy period
  (1879-1880), have so completely disappeared from my memory that all
  efforts to obtain a clue to them have hitherto been fruitless.]



ON TRANSFORMATION AND ADAPTATION IN SCIENTIFIC THOUGHT.[69]


It was towards the close of the sixteenth century that Galileo with a
superb indifference to the dialectic arts and sophistic subtleties of
the Schoolmen of his time, turned the attention of his brilliant mind to
nature. By nature his ideas were transformed and released from the
fetters of inherited prejudice. At once the mighty revolution was felt,
that was therewith effected in the realm of human thought--felt indeed
in circles far remote and wholly unrelated to the sphere of science,
felt in strata of society that hitherto had only indirectly recognised
the influence of scientific thought.

And how great and how far-reaching that revolution was! From the
beginning of the seventeenth century till its close we see arising, at
least in embryo, almost all that plays a part in the natural and
technical science of to-day, almost all that in the two centuries
following so wonderfully transformed the facial appearance of the earth,
and all that is moving onward in process of such mighty evolution
to-day. And all this, the direct result of Galilean ideas, the direct
outcome of that freshly awakened sense for the investigation of natural
phenomena which taught the Tuscan philosopher to form the concept and
the law of falling bodies from the _observation_ of a falling stone!
Galileo began his investigations without an implement worthy of the
name; he measured time in the most primitive way, by the efflux of
water. Yet soon afterwards the telescope, the microscope, the barometer,
the thermometer, the air-pump, the steam engine, the pendulum, and the
electrical machine were invented in rapid succession. The fundamental
theorems of dynamical science, of optics, of heat, and of electricity
were all disclosed in the century that followed Galileo.

Of scarcely less importance, it seems, was that movement which was
prepared for by the illustrious biologists of the hundred years just
past, and formally begun by the late Mr. Darwin. Galileo quickened the
sense for the simpler phenomena of _inorganic_ nature. And with the same
simplicity and frankness that marked the efforts of Galileo, and without
the aid of technical or scientific instruments, without physical or
chemical experiment, but solely by the power of thought and observation,
Darwin grasps a new property of _organic_ nature--which we may briefly
call its _plasticity_.[70] With the same directness of purpose, Darwin,
too, pursues his way. With the same candor and love of truth, he points
out the strength and the weakness of his demonstrations. With masterly
equanimity he holds aloof from the discussion of irrelevant subjects and
wins alike the admiration of his adherents and of his adversaries.

Scarcely thirty years have elapsed[71] since Darwin first propounded the
principles of his theory of evolution. Yet, already we see his ideas
firmly rooted in every branch of human thought, however remote.
Everywhere, in history, in philosophy, even in the physical sciences, we
hear the watchwords: heredity, adaptation, selection. We speak of the
struggle for existence among the heavenly bodies and of the struggle for
existence in the world of molecules.[72]

The impetus given by Galileo to scientific thought was marked in every
direction; thus, his pupil, Borelli, founded the school of exact
medicine, from whence proceeded even distinguished mathematicians. And
now Darwinian ideas, in the same way, are animating all provinces of
research. It is true, nature is not made up of two distinct parts, the
inorganic and the organic; nor must these two divisions be treated
perforce by totally distinct methods. Many _sides_, however, nature has.
Nature is like a thread in an intricate tangle, which must be followed
and traced, now from this point, now from that. But we must never
imagine,--and this physicists have learned from Faraday and J. R.
Mayer,--that progress along paths once entered upon is the _only_ means
of reaching the truth.

It will devolve upon the specialists of the future to determine the
relative tenability and fruitfulness of the Darwinian ideas in the
different provinces. Here I wish simply to consider the growth of
natural _knowledge_ in the light of the theory of evolution. For
knowledge, too, is a product of organic nature. And although ideas, as
such, do not comport themselves in all respects like independent organic
individuals, and although violent comparisons should be avoided, still,
if Darwin reasoned rightly, the general imprint of evolution and
transformation must be noticeable in ideas also.

I shall waive here the consideration of the fruitful topic of the
transmission of ideas or rather of the transmission of the aptitude for
certain ideas.[73] Nor would it come within my province to discuss
psychical evolution in any form, as Spencer[74] and many other modern
psychologists have done, with varying success. Neither shall I enter
upon a discussion of the struggle for existence and of natural selection
among scientific theories.[75] We shall consider here only such
processes of transformation as every student can easily observe in his
own mind.

       *       *       *       *       *

The child of the forest picks out and pursues with marvellous acuteness
the trails of animals. He outwits and overreaches his foes with
surpassing cunning. He is perfectly at home in the sphere of his
peculiar experience. But confront him with an unwonted phenomenon; place
him face to face with a technical product of modern civilisation, and he
will lapse into impotency and helplessness. Here are facts which he
does not comprehend. If he endeavors to grasp their meaning, he
misinterprets them. He fancies the moon, when eclipsed, to be tormented
by an evil spirit. To his mind a puffing locomotive is a living monster.
The letter accompanying a commission with which he is entrusted, having
once revealed his thievishness, is in his imagination a conscious being,
which he must hide beneath a stone, before venturing to commit a fresh
trespass. Arithmetic to him is like the art of the geomancers in the
Arabian Nights,--an art which is able to accomplish every imaginable
impossibility. And, like Voltaire's _ingénu_, when placed in our social
world, he plays, as we think, the maddest pranks.

With the man who has made the achievements of modern science and
civilisation his own, the case is quite different. He sees the moon pass
temporarily into the shadow of the earth. He feels in his thoughts the
water growing hot in the boiler of the locomotive; he feels also the
increase of the tension which pushes the piston forward. Where he is not
able to trace the direct relation of things he has recourse to his
yard-stick and table of logarithms, which aid and facilitate his thought
without predominating over it. Such opinions as he cannot concur in, are
at least known to him, and he knows how to meet them in argument.

Now, wherein does the difference between these two men consist? The
train of thought habitually employed by the first one does not
correspond to the facts that he sees. He is surprised and nonplussed at
every step. But the thoughts of the second man follow and anticipate
events, his thoughts have become adapted or accommodated to the larger
field of observation and activity in which he is located; he conceives
things as they are. The Indian's sphere of experience, however, is quite
different; his bodily organs of sense are in constant activity; he is
ever intensely alert and on the watch for his foes; or, his entire
attention and energy are engaged in procuring sustenance. Now, how can
such a creature project his mind into futurity, foresee or prophesy?
This is not possible until our fellow-beings have, in a measure,
relieved us of our concern for existence. It is then that we acquire
freedom for observation, and not infrequently too that narrowness of
thought which society helps and teaches us to disregard.

If we move for a time within a fixed circle of phenomena which recur
with unvarying uniformity, our thoughts gradually adapt themselves to
our environment; our ideas reflect unconsciously our surroundings. The
stone we hold in our hand, when dropped, not only falls to the ground in
reality; it also falls in our thoughts. Iron-filings dart towards a
magnet in imagination as well as in fact, and, when thrown into a fire,
they grew hot in conception as well.

The impulse to complete mentally a phenomenon that has been only
partially observed, has not its origin in the phenomenon itself; of this
fact, we are fully sensible. And we well know that it does not lie
within the sphere of our volition. It seems to confront us rather as a
power and a law imposed from without and controlling both thought and
facts.

The fact that we are able by the help of this law to prophesy and
forecast, merely proves a sameness or uniformity of environment
sufficient to effect a mental adaptation of this kind. A necessity of
fulfilment, however, is not contained in this compulsory principle which
controls our thoughts; nor is it in any way determined by the
possibility of prediction. We are always obliged, in fact, to await the
completion of what has been predicted. Errors and departures are
constantly discernible, and are slight only in provinces of great rigid
constancy, as in astronomy.

In cases where our thoughts follow the connexion of events with ease,
and in instances where we positively forefeel the course of a
phenomenon, it is natural to fancy that the latter is determined by and
must conform to our thoughts. But the belief in that mysterious agency
called _causality_, which holds thought and event in unison, is
violently shaken when a person first enters a province of inquiry in
which he has previously had no experience. Take for instance the strange
interaction of electric currents and magnets, or the reciprocal action
of currents, which seem to defy all the resources of mechanical science.
Let him be confronted with such phenomena and he will immediately feel
himself forsaken by his power of prediction; he will bring nothing with
him into this strange field of events but the hope of soon being able
to adapt his ideas to the new conditions there presented.

A person constructs from a bone the remaining anatomy of an animal; or
from the visible part of a half-concealed wing of a butterfly he infers
and reconstructs the part concealed. He does so with a feeling of
highest confidence in the accuracy of his results; and in these
processes we find nothing preternatural or transcendent. But when
physicists adapt their thoughts to conform to the dynamical course of
events in time, we invariably surround their investigations with a
metaphysical halo; yet these latter adaptations bear quite the same
character as the former, and our only reason for investing them with a
metaphysical garb, perhaps, is their high practical value.[76]

Let us consider for a moment what takes place when the field of
observation to which our ideas have been adapted and now conform,
becomes enlarged. We had, let us say, always seen heavy bodies sink when
their support was taken away; we had also seen, perhaps, that the
sinking of heavier bodies forced lighter bodies upwards. But now we see
a lever in action, and we are suddenly struck with the fact that a
lighter body is lifting another of much greater weight. Our customary
train of thought demands its rights; the new and unwonted event likewise
demands its rights. From this conflict between thought and fact the
_problem_ arises; out of this partial contrariety springs the question,
"Why?" With the new adaptation to the enlarged field of observation, the
problem disappears, or, in other words, is solved. In the instance
cited, we must adopt the habit of always considering the mechanical work
performed.

The child just awakening into consciousness of the world, knows no
problem. The bright flower, the ringing bell, are all new to it; yet it
is surprised at nothing. The out and out Philistine, whose only thoughts
lie in the beaten path of his every-day pursuits, likewise has no
problems. Everything goes its wonted course, and if perchance a thing go
wrong at times, it is at most a mere object of curiosity and not worth
serious consideration. In fact, the question "Why?" loses all warrant in
relations where we are familiar with every aspect of events. But the
capable and talented young man has his head full of problems; he has
acquired, to a greater or less degree, certain habitudes of thought, and
at the same time he is constantly observing what is new and unwonted,
and in his case there is no end to the questions, "Why?"

Thus, the factor which most promotes scientific thought is the gradual
widening of the field of experience. We scarcely notice events we are
accustomed to; the latter do not really develop their intellectual
significance until placed in contrast with something to which we are
unaccustomed. Things that at home are passed by unnoticed, delight us
when abroad, though they may appear in only slightly different forms.
The sun shines with heightened radiance, the flowers bloom in brighter
colors, our fellow-men accost us with lighter and happier looks. And,
returning home, we find even the old familiar scenes more inspiring and
suggestive than before.

Every motive that prompts and stimulates us to modify and transform our
thoughts, proceeds from what is new, uncommon, and not understood.
Novelty excites wonder in persons whose fixed habits of thought are
shaken and disarranged by what they see. But the element of wonder never
lies in the phenomenon or event observed; its place is in the person
observing. People of more vigorous mental type aim at once at an
_adaptation of thought_ that will conform to what they have observed.
Thus does science eventually become the natural foe of the wonderful.
The sources of the marvellous are unveiled, and surprise gives way to
calm interpretation.

Let us consider such a mental transformative process in detail. The
circumstance that heavy bodies fall to the earth appears perfectly
natural and regular. But when a person observes that wood floats upon
water, and that flames and smoke rise in the air, then the contrary of
the first phenomenon is presented. An olden theory endeavors to explain
these facts by imputing to substances the power of volition, as that
attribute which is most familiar to man. It asserted that every
substance seeks its proper place, heavy bodies tending downwards and
light ones upwards. It soon turned out, however, that even smoke had
weight, that it, too, sought its place below, and that it was forced
upwards only because of the downward tendency of the air, as wood is
forced to the surface of water because the water exerts the greater
downward pressure.

Again, we see a body thrown into the air. It ascends. How is it that it
does not seek its proper place? Why does the velocity of its "violent"
motion decrease as it rises, while that of its "natural" fall increases
as it descends. If we mark closely the relation between these two facts,
the problem will solve itself. We shall see, as Galileo did, that the
decrease of velocity in rising and the increase of velocity in falling
are one and the same phenomenon, viz., an increase of velocity towards
the earth. Accordingly, it is not a place that is assigned to the body,
but an increase of velocity towards the earth.

By this idea the movements of heavy bodies are rendered perfectly
familiar. Newton, now, firmly grasping this new way of thinking, sees
the moon and the planets moving in their paths upon principles similar
to those which determine the motion of a projectile thrown into the air.
Yet the movements of the planets were marked by peculiarities which
compelled him once more to modify slightly his customary mode of
thought. The heavenly bodies, or rather the parts composing them, do not
move with constant accelerations towards each other, but "attract each
other," directly as the mass and inversely as the square of the
distance.

This latter notion, which includes the one applying to terrestrial
bodies as a special case, is, as we see, quite different from the
conception from which we started. How limited in scope was the original
idea and to what a multitude of phenomena is not the present one
applicable! Yet there is a trace, after all, of the "search for place"
in the expression "attraction." And it would be folly, indeed, for us to
avoid, with punctilious dread, this conception of "attraction" as
bearing marks of its pedigree. It is the historical base of the
Newtonian conception and it still continues to direct our thoughts in
the paths so long familiar to us. Thus, the happiest ideas do not fall
from heaven, but spring from notions already existing.

Similarly, a ray of light was first regarded as a continuous and
homogeneous straight line. It then became the path of projection for
minute missiles; then an aggregate of the paths of countless different
kinds of missiles. It became periodic; it acquired various sides; and
ultimately it even lost its motion in a straight line.

The electric current was conceived originally as the flow of a
hypothetical fluid. To this conception was soon added the notion of a
chemical current, the notion of an electric, magnetic, and anisotropic
optical field, intimately connected with the path of the current. And
the richer a conception becomes in following and keeping pace with
facts, the better adapted it is to anticipate them.

Adaptive processes of this kind have no assignable beginning, inasmuch
as every problem that incites to new adaptation, presupposes a fixed
habitude of thought. Moreover, they have no visible end; in so far as
experience never ceases. Science, accordingly, stands midway in the
evolutionary process; and science may advantageously direct and promote
this process, but it can never take its place. That science is
inconceivable the principles of which would enable a person with no
experience to construct the world of experience, without a knowledge of
it. One might just as well expect to become a great musician, solely by
the aid of theory, and without musical experience; or to become a
painter by following the directions of a text-book.

In glancing over the history of an idea with which we have become
perfectly familiar, we are no longer able to appreciate the full
significance of its growth. The deep and vital changes that have been
effected in the course of its evolution, are recognisable only from the
astounding narrowness of view with which great contemporary scientists
have occasionally opposed each other. Huygens's wave-theory of light was
incomprehensible to Newton, and Newton's idea of universal gravity was
unintelligible to Huygens. But a century afterwards both notions were
reconcilable, even in ordinary minds.

On the other hand, the original creations of pioneer intellects,
unconsciously formed, do not assume a foreign garb; their form is their
own. In them, childlike simplicity is joined to the maturity of manhood,
and they are not to be compared with processes of thought in the average
mind. The latter are carried on as are the acts of persons in the state
of mesmerism, where actions involuntarily follow the images which the
words of other persons suggest to their minds.

The ideas that have become most familiar through long experience, are
the very ones that intrude themselves into the conception of every new
fact observed. In every instance, thus, they become involved in a
struggle for self-preservation, and it is just they that are seized by
the inevitable process of transformation.

Upon this process rests substantially the method of explaining by
hypothesis new and uncomprehended phenomena. Thus, instead of forming
entirely new notions to explain the movements of the heavenly bodies and
the phenomena of the tides, we imagine the material particles composing
the bodies of the universe to possess weight or gravity with respect to
one another. Similarly, we imagine electrified bodies to be freighted
with fluids that attract and repel, or we conceive the space between
them to be in a state of elastic tension. In so doing, we substitute
for new ideas distinct and more familiar notions of old
experience--notions which to a great extent run unimpeded in their
courses, although they too must suffer partial transformation.

The animal cannot construct new members to perform every new function
that circumstances and fate demand of it. On the contrary it is obliged
to make use of those it already possesses. When a vertebrate animal
chances into an environment where it must learn to fly or swim, an
additional pair of extremities is not grown for the purpose. On the
contrary, the animal must adapt and transform a pair that it already
has.

The construction of hypotheses, therefore, is not the product of
artificial scientific methods. This process is unconsciously carried on
in the very infancy of science. Even later, hypotheses do not become
detrimental and dangerous to progress except when more reliance is
placed on them than on the facts themselves; when the contents of the
former are more highly valued than the latter, and when, rigidly
adhering to hypothetical notions, we overestimate the ideas we possess
as compared with those we have to acquire.

The extension of our sphere of experience always involves a
transformation of our ideas. It matters not whether the face of nature
becomes actually altered, presenting new and strange phenomena, or
whether these phenomena are brought to light by an intentional or
accidental turn of observation. In fact, all the varied methods of
scientific inquiry and of purposive mental adaptation enumerated by John
Stuart Mill, those of observation as well as those of experiment, are
ultimately recognisable as forms of one fundamental method, the method
of change, or variation. It is through change of circumstances that the
natural philosopher learns. This process, however, is by no means
confined to the investigator of nature. The historian, the philosopher,
the jurist, the mathematician, the artist, the æsthetician,[77] all
illuminate and unfold their ideas by producing from the rich treasures
of memory similar, but different, cases; thus, they observe and
experiment in their thoughts. Even if all sense-experience should
suddenly cease, the events of the days past would meet in different
attitudes in the mind and the process of adaptation would still
continue--a process which, in contradistinction to the adaptation of
thoughts to facts in practical spheres, would be strictly theoretical,
being an adaptation of thoughts to thoughts.

The method of change or variation brings before us like cases of
phenomena, having partly the same and partly different elements. It is
only by comparing different cases of refracted light at changing angles
of incidence that the common factor, the constancy of the refractive
index, is disclosed. And only by comparing the refractions of light of
different colors, does the difference, the inequality of the indices of
refraction, arrest the attention. Comparison based upon change leads the
mind simultaneously to the highest abstractions and to the finest
distinctions.

Undoubtedly, the animal also is able to distinguish between the similar
and dissimilar of two cases. Its consciousness is aroused by a noise or
a rustling, and its motor centre is put in readiness. The sight of the
creature causing the disturbance, will, according to its size, provoke
flight or prompt pursuit; and in the latter case, the more exact
distinctions will determine the mode of attack. But man alone attains to
the faculty of voluntary and conscious comparison. Man alone can, by his
power of abstraction, rise, in one moment, to the comprehension of
principles like the conservation of mass or the conservation of energy,
and in the next observe and mark the arrangement of the iron lines in
the spectrum. In thus dealing with the objects of his conceptual life,
his ideas unfold and expand, like his nervous system, into a widely
ramified and organically articulated tree, on which he may follow every
limb to its farthermost branches, and, when occasion demands, return to
the trunk from which he started.

The English philosopher Whewell has remarked that two things are
requisite to the formation of science: facts and ideas. Ideas alone lead
to empty speculation; mere facts can yield no organic knowledge. We see
that all depends upon the capacity of adapting existing notions to fresh
facts.

Over-readiness to yield to every new fact prevents fixed habits of
thought from arising. Excessively rigid habits of thought impede freedom
of observation. In the struggle, in the compromise between judgment and
prejudgment (prejudice), if we may use the term, our understanding of
things broadens.

Habitual judgment, applied to a new case without antecedent tests, we
call prejudgment or prejudice. Who does not know its terrible power! But
we think less often of the importance and utility of prejudice.
Physically, no one could exist, if he had to guide and regulate the
circulation, respiration, and digestion of his body by conscious and
purposive acts. So, too, no one could exist intellectually if he had to
form judgments on every passing experience, instead of allowing himself
to be controlled by the judgments he has already formed. Prejudice is a
sort of reflex motion in the province of intelligence.

On prejudices, that is, on habitual judgments not tested in every case
to which they are applied, reposes a goodly portion of the thought and
work of the natural scientist. On prejudices reposes most of the conduct
of society. With the sudden disappearance of prejudice society would
hopelessly dissolve. That prince displayed a deep insight into the power
of intellectual habit, who quelled the loud menaces and demands of his
body-guard for arrears of pay and compelled them to turn about and
march, by simply pronouncing the regular word of command; he well knew
that they would be unable to resist that.

Not until the discrepancy between habitual judgments and facts becomes
great is the investigator implicated in appreciable illusion. Then
tragic complications and catastrophes occur in the practical life of
individuals and nations--crises where man, placing custom above life,
instead of pressing it into the service of life, becomes the victim of
his error. The very power which in intellectual life advances, fosters,
and sustains us, may in other circumstances delude and destroy us.

       *       *       *       *       *

Ideas are not all of life. They are only momentary efflorescences of
light, designed to illuminate the paths of the will. But as delicate
reagents on our organic evolution our ideas are of paramount importance.
No theory can gainsay the vital transformation which we feel taking
place within us through their agency. Nor is it necessary that we should
have a proof of this process. We are immediately assured of it.

The transformation of ideas thus appears as a part of the general
evolution of life, as a part of its adaptation to a constantly widening
sphere of action. A granite boulder on a mountain-side tends towards the
earth below. It must abide in its resting-place for thousands of years
before its support gives way. The shrub that grows at its base is
farther advanced; it accommodates itself to summer and winter. The fox
which, overcoming the force of gravity, creeps to the summit where he
has scented his prey, is freer in his movements than either. The arm of
man reaches further still; and scarcely anything of note happens in
Africa or Asia that does not leave an imprint upon his life. What an
immense portion of the life of other men is reflected in ourselves;
their joys, their affections, their happiness and misery! And this too,
when we survey only our immediate surroundings, and confine our
attention to modern literature. How much more do we experience when we
travel through ancient Egypt with Herodotus, when we stroll through the
streets of Pompeii, when we carry ourselves back to the gloomy period of
the crusades or to the golden age of Italian art, now making the
acquaintance of a physician of Molière, and now that of a Diderot or of
a D'Alembert. What a great part of the life of others, of their
character and their purpose, do we not absorb through poetry and music!
And although they only gently touch the chords of our emotions, like the
memory of youth softly breathing upon the spirit of an aged man, we have
nevertheless lived them over again in part. How great and comprehensive
does self become in this conception; and how insignificant the person!
Egoistical systems both of optimism and pessimism perish with their
narrow standard of the import of intellectual life. We feel that the
real pearls of life lie in the ever changing contents of consciousness,
and that the person is merely an indifferent symbolical thread on which
they are strung.[78]

We are prepared, thus, to regard ourselves and every one of our ideas as
a product and a subject of universal evolution; and in this way we shall
advance sturdily and unimpeded along the paths which the future will
throw open to us.[79]

  FOOTNOTES:

  [Footnote 69: Inaugural Address, delivered on assuming the Rectorate
  of the University of Prague, October 18, 1883.

  The idea presented in this essay is neither new nor remote. I have
  touched upon it myself on several occasions (first in 1867), but
  have never made it the subject of a formal disquisition. Doubtless,
  others, too, have treated it; it lies, so to speak, in the air.
  However, as many of my illustrations were well received, although
  known only in an imperfect form from the lecture itself and the
  newspapers, I have, contrary to my original intention, decided to
  publish it. It is not my intention to trespass here upon the domain
  of biology. My statements are to be taken merely as the expression
  of the fact that no one can escape the influence of a great and
  far-reaching idea.]

  [Footnote 70: At first sight an apparent contradiction arises from
  the admission of both heredity and adaptation; and it is undoubtedly
  true that a strong disposition to heredity precludes great
  capability of adaptation. But imagine the organism to be a plastic
  mass which retains the form transmitted to it by former influences
  until new influences modify it; the _one_ property of _plasticity_
  will then represent capability of adaptation as well as power of
  heredity. Analogous to this is the case of a bar of magnetised steel
  of high coercive force: the steel retains its magnetic properties
  until a new force displaces them. Take also a body in motion: the
  body retains the velocity acquired in (_inherited_ from) the
  interval of time just preceding, except it be changed in the next
  moment by an accelerating force. In the case of the body in motion
  the _change_ of velocity (_Abänderung_) was looked upon as a matter
  of course, while the discovery of the principle of _inertia_ (or
  persistence) created surprise; in Darwin's case, on the contrary,
  _heredity_ (or persistence) was taken for granted, while the
  principle of _variation_ (_Abänderung_) appeared novel.

  Fully adequate views are, of course, to be reached only by a study
  of the original facts emphasised by Darwin, and not by these
  analogies. The example referring to motion, if I am not mistaken, I
  first heard, in conversation, from my friend J. Popper, Esq., of
  Vienna.

  Many inquirers look upon the stability of the species as something
  settled, and oppose to it the Darwinian theory. But the stability of
  the species is itself a "theory." The essential modifications which
  Darwin's views also are undergoing will be seen from the works of
  Wallace [and Weismann], but more especially from a book of W. H.
  Rolph, _Biologische Probleme_, Leipsic, 1882. Unfortunately, this
  last talented investigator is no longer numbered among the living.]

  [Footnote 71: Written in 1883.]

  [Footnote 72: See Pfaundler, _Pogg. Ann., Jubelband_, p. 182.]

  [Footnote 73: See the beautiful discussions of this point in
  Hering's _Memory as a General Function of Organised Matter_ (1870),
  Chicago, The Open Court Publishing Co., 1887. Compare also Dubois,
  _Ueber die Uebung_, Berlin, 1881.]

  [Footnote 74: Spencer, _The Principles of Psychology_. London,
  1872.]

  [Footnote 75: See the article _The Velocity of Light_, page 63.]

  [Footnote 76: I am well aware that the endeavor to confine oneself
  in natural research to _facts_ is often censured as an exaggerated
  fear of metaphysical spooks. But I would observe, that, judged by
  the mischief which they have wrought, the metaphysical, of all
  spooks, are the least fabulous. It is not to be denied that many
  forms of thought were not originally acquired by the individual, but
  were antecedently formed, or rather prepared for, in the development
  of the species, in some such way as Spencer, Haeckel, Hering, and
  others have supposed, and as I myself have hinted on various
  occasions.]

  [Footnote 77: Compare, for example, _Schiller, Zerstreute
  Betrachtungen über verschiedene ästhetische Gegenstände_.]

  [Footnote 78: We must not be deceived in imagining that the
  happiness of other people is not a very considerable and essential
  portion of our own. It is common capital, which cannot be created by
  the individual, and which does not perish with him. The formal and
  material limitation of the _ego_ is necessary and sufficient only
  for the crudest practical objects, and cannot subsist in a broad
  conception. Humanity in its entirety may be likened to a
  polyp-plant. The material and organic bonds of individual union
  have, indeed, been severed; they would only have impeded freedom of
  movement and evolution. But the ultimate aim, the psychical
  connexion of the whole, has been attained in a much higher degree
  through the richer development thus made possible.]

  [Footnote 79: C. E. von Baer, the subsequent opponent of Darwin and
  Haeckel, has discussed in two beautiful addresses (_Das allgemeinste
  Gesetz der Natur in aller Entwickelung_, and _Welche Auffassung der
  lebenden Natur ist die richtige, und wie ist diese Auffassung auf
  die Entomologie anzuwenden_?) the narrowness of the view which
  regards an animal in its existing state as finished and complete,
  instead of conceiving it as a phase in the series of evolutionary
  forms and regarding the species itself as a phase of the development
  of the animal world in general.]



ON THE PRINCIPLE OF COMPARISON IN PHYSICS.[80]


Twenty years ago when Kirchhoff defined the object of mechanics as the
"description, in complete and very simple terms, of the motions
occurring in nature," he produced by the statement a peculiar
impression. Fourteen years subsequently, Boltzmann, in the life-like
picture which he drew of the great inquirer, could still speak of the
universal astonishment at this novel method of treating mechanics, and
we meet with epistemological treatises to-day, which plainly show how
difficult is the acceptance of this point of view. A modest and small
band of inquirers there were, however, to whom Kirchhoff's few words
were tidings of a welcome and powerful ally in the epistemological
field.

Now, how does it happen that we yield our assent so reluctantly to the
philosophical opinion of an inquirer for whose scientific achievements
we have only words of praise? One reason probably is that few inquirers
can find time and leisure, amid the exacting employments demanded for
the acquisition of new knowledge, to inquire closely into that
tremendous psychical process by which science is formed. Further, it is
inevitable that much should be put into Kirchhoff's rigid words that
they were not originally intended to convey, and that much should be
found wanting in them that had always been regarded as an essential
element of scientific knowledge. What can mere description accomplish?
What has become of explanation, of our insight into the causal connexion
of things?

       *       *       *       *       *

Permit me, for a moment, to contemplate not the results of science, but
the mode of its _growth_, in a frank and unbiassed manner. We know of
only _one_ source of _immediate revelation_ of scientific facts--_our
senses_. Restricted to this source alone, thrown wholly upon his own
resources, obliged to start always anew, what could the isolated
individual accomplish? Of a stock of knowledge so acquired the science
of a distant negro hamlet in darkest Africa could hardly give us a
sufficiently humiliating conception. For there that veritable miracle of
thought-transference has already begun its work, compared with which the
miracles of the spiritualists are rank monstrosities--_communication by
language_. Reflect, too, that by means of the magical characters which
our libraries contain we can raise the spirits of the "the sovereign
dead of old" from Faraday to Galileo and Archimedes, through ages of
time--spirits who do not dismiss us with ambiguous and derisive
oracles, but tell us the best they know; then shall we feel what a
stupendous and indispensable factor in the formation of science
_communication_ is. Not the dim, half-conscious _surmises_ of the acute
observer of nature or critic of humanity belong to science, but only
that which they possess clearly enough to _communicate_ to others.

But how, now, do we go about this communication of a newly acquired
experience, of a newly observed fact? As the different calls and
battle-cries of gregarious animals are unconsciously formed signs for a
common observation or action, irrespective of the causes which produce
such action--a fact that already involves the germ of the concept; so
also the words of human language, which is only more highly specialised,
are names or signs for universally known facts, which all can observe or
have observed. If the mental representation, accordingly, follows the
new fact at once and _passively_, then that new fact must, of itself,
immediately be constituted and represented in thought by facts already
universally known and commonly observed. Memory is always ready to put
forward for _comparison_ known facts which resemble the new event, or
agree with it in certain features, and so renders possible that
elementary internal judgment which the mature and definitively
formulated judgment soon follows.

Comparison, as the fundamental condition of communication, is the most
powerful inner vital element of science. The zoölogist sees in the
bones of the wing-membranes of bats, fingers; he compares the bones of
the cranium with the vertebræ, the embryos of different organisms with
one another, and the different stages of development of the same
organism with one another. The geographer sees in Lake Garda a fjord, in
the Sea of Aral a lake in process of drying up. The philologist compares
different languages with one another, and the formations of the same
language as well. If it is not customary to speak of comparative physics
in the same sense that we speak of comparative anatomy, the reason is
that in a science of such great experimental activity the attention is
turned away too much from the _contemplative_ element. But like all
other sciences, physics lives and grows by comparison.

       *       *       *       *       *

The manner in which the result of the comparison finds expression in the
communication, varies of course very much. When we say that the colors
of the spectrum are red, yellow, green, blue, and violet, the
designations employed may possibly have been derived from the technology
of tattooing, or they may subsequently have acquired the significance of
standing for the colors of the rose, the lemon, the leaf, the
corn-flower, and the violet. From the frequent repetition of such
comparisons, however, made under the most manifold circumstances, the
inconstant features, as compared with the permanent congruent features,
get so obliterated that the latter acquire a fixed significance
independent of every object and connexion, or take on as we say an
_abstract_ or _conceptual_ import. No one thinks at the word "red" of
any other agreement with the rose than that of color, or at the word
"straight" of any other property of a stretched cord than the sameness
of direction. Just so, too, numbers, originally the names of the fingers
of the hands and feet, from being used as arrangement-signs for all
kinds of objects, were lifted to the plane of abstract concepts. A
verbal report (communication) of a fact that uses only these purely
abstract implements, we call a _direct description_.

The direct description of a fact of any great extent is an irksome task,
even where the requisite notions are already completely developed. What
a simplification it involves if we can say, the fact _A_ now considered
comports itself, not in _one_, but in _many_ or in _all_ its features,
like an old and well-known fact _B_. The moon comports itself as a heavy
body does with respect to the earth; light like a wave-motion or an
electric vibration; a magnet, as if it were laden with gravitating
fluids, and so on. We call such a description, in which we appeal, as it
were, to a description already and elsewhere formulated, or perhaps
still to be precisely formulated, an _indirect description_. We are at
liberty to supplement this description, gradually, by direct
description, to correct it, or to replace it altogether. We see, thus,
without difficulty, that what is called a _theory_ or a _theoretical
idea_, falls under the category of what is here termed indirect
description.

       *       *       *       *       *

What, now, is a theoretical idea? Whence do we get it? What does it
accomplish for us? Why does it occupy a higher place in our judgment
than the mere holding fast to a fact or an observation? Here, too,
memory and comparison alone are in play. But instead of _a single_
feature of resemblance culled from memory, in this case _a great system_
of resemblances confronts us, a well-known physiognomy, by means of
which the new fact is immediately transformed into an old acquaintance.
Besides, it is in the power of the idea to offer us more than we
actually see in the new fact, at the first moment; it can extend the
fact, and enrich it with features which we are first induced to _seek_
from such suggestions, and which are often actually found. It is this
_rapidity_ in extending knowledge that gives to theory a preference over
simple observation. But that preference is wholly _quantitative_.
Qualitatively, and in real essential points, theory differs from
observation neither in the mode of its origin nor in its last results.

The adoption of a theory, however, always involves a danger. For a
theory puts in the place of a fact _A_ in thought, always a _different_,
but simpler and more familiar fact _B_, which in _some_ relations can
mentally represent _A_, but for the very reason that it is different, in
other relations cannot represent it. If now, as may readily happen,
sufficient care is not exercised, the most fruitful theory may, in
special circumstances, become a downright obstacle to inquiry. Thus, the
emission-theory of light, in accustoming the physicist to think of the
projectile path of the "light-particles" as an undifferentiated
straight-line, demonstrably impeded the discovery of the periodicity of
light. By putting in the place of light the more familiar phenomena of
sound, Huygens renders light in many of its features a familiar event,
but with respect to polarisation, which lacks the longitudinal waves
with which alone he was acquainted, it had for him a doubly strange
aspect. He is unable thus to grasp in abstract thought the fact of
polarisation, which is before his eyes, whilst Newton, merely by
adapting to the observation his thoughts, and putting this question,
"_Annon radiorum luminis diversa sunt latera?_" abstractly grasped
polarisation, that is, directly described it, a century before Malus. On
the other hand, if the agreement of the fact with the idea theoretically
representing it, extends further than its inventor originally
anticipated, then we may be led by it to unexpected discoveries, of
which conical refraction, circular polarisation by total reflexion,
Hertz's waves offer ready examples, in contrast to the illustrations
given above.

Our insight into the conditions indicated will be improved, perhaps, by
contemplating the development of some theory or other more in detail.
Let us consider a magnetised bar of steel by the side of a second
unmagnetised bar, in all other respects the same. The second bar gives
no indication of the presence of iron-filings; the first attracts them.
Also, when the iron-filings are absent, we must think of the magnetised
bar as in a different condition from that of the unmagnetised. For, that
the mere presence of the iron-filings does not induce the phenomenon of
attraction is proved by the second unmagnetised bar. The ingenuous man,
who finds in his will, as his most familiar source of power, the best
facilities for comparison, conceives a species of _spirit_ in the
magnet. The behavior of a warm body or of an _electrified_ body suggests
similar ideas. This is the point of view of the oldest theory,
_fetishism_, which the inquirers of the early Middle Ages had not yet
overcome, and which in its last vestiges, in the conception of forces,
still flourishes in modern physics. We see, thus, the _dramatic_ element
need no more be absent in a scientific description, than in a thrilling
novel.

If, on subsequent examination, it be observed that a cold body, in
contact with a hot body, warms itself, so to speak, _at the expense_ of
the hot body; further, that when the substances are the same, the cold
body, which, let us say, has twice the mass of the other, gains only
half the number of degrees of temperature that the other loses, a wholly
new impression arises. The demoniac character of the event vanishes, for
the supposed spirit acts not by caprice, but according to fixed laws. In
its place, however, _instinctively_ the notion of a _substance_ is
substituted, part of which flows over from the one body to the other,
but the total amount of which, representable by the sum of the products
of the masses into the respective changes of temperature, remains
constant. Black was the first to be _powerfully_ struck with this
resemblance of thermal processes to the motion of a substance, and under
its guidance discovered the specific heat, the heat of fusion, and the
heat of vaporisation of bodies. Gaining strength and fixity, however,
from these successes, this notion of substance subsequently stood in the
way of scientific advancement. It blinded the eyes of the successors of
Black, and prevented them from seeing the manifest fact, which every
savage knows, that heat is _produced_ by friction. Fruitful as that
notion was for Black, helpful as it still is to the learner to-day in
Black's special field, permanent and universal validity as a _theory_ it
could never maintain. But what is essential, conceptually, in it, viz.,
the constancy of the product-sum above mentioned, retains its value and
may be regarded as a _direct description_ of Black's facts.

It stands to reason that those theories which push themselves forward
unsought, instinctively, and wholly of their own accord, should have the
greatest power, should carry our thoughts most with them, and exhibit
the staunchest powers of self-preservation. On the other hand, it may
also be observed that when critically scrutinised such theories are
extremely apt to lose their cogency. We are constantly busied with
"substance," its modes of action have stamped themselves indelibly upon
our thoughts, our vividest and clearest reminiscences are associated
with it. It should cause us no surprise, therefore, that Robert Mayer
and Joule, who gave the final blow to Black's substantial conception of
heat, should have re-introduced the same notion of substance in a more
abstract and modified form, only applying to a much more extensive
field.

Here, too, the psychological circumstances which impart to the new
conception its power, lie clearly before us. By the unusual redness of
the venous blood in tropical climates Mayer's attention is directed to
the lessened expenditure of internal heat and to the proportionately
lessened _consumption of material_ by the human body in those climates.
But as every effort of the human organism, including its mechanical
work, is connected with the consumption of material, and as work by
friction can engender heat, therefore heat and work appear in kind
equivalent, and between them a proportional relation must subsist. Not
_every_ quantity, but the appropriately calculated _sum_ of the two, as
connected with a proportionate consumption of material, appears
_substantial_.

By exactly similar considerations, relative to the economy of the
galvanic element, Joule arrived at his view; he found experimentally
that the sum of the heat evolved in the circuit, of the heat consumed in
the combustion of the gas developed, of the electro-magnetic work of
the current, properly calculated,--in short, the sum of all the effects
of the battery,--is connected with a proportionate consumption of zinc.
Accordingly, this sum itself has a substantial character.

Mayer was so absorbed with the view attained, that the indestructibility
of _force_, in our phraseology _work_, appeared to him _a priori_
evident. "The creation or annihilation of a force," he says, "lies
without the province of human thought and power." Joule expressed
himself to a similar effect: "It is manifestly absurd to suppose that
the powers with which God has endowed matter can be destroyed." Strange
to say, on the basis of such utterances, not Joule, but Mayer, was
stamped as a metaphysician. We may be sure, however, that both men were
merely giving expression, and that half-unconsciously, to a powerful
_formal_ need of the new simple view, and that both would have been
extremely surprised if it had been proposed to them that their principle
should be submitted to a philosophical congress or ecclesiastical synod
for a decision upon its validity. But with all agreements, the attitude
of these two men, in other respects, was totally different. Whilst Mayer
represented this _formal_ need with all the stupendous instinctive force
of genius, we might say almost with the ardor of fanaticism, yet was
withal not wanting in the conceptive ability to compute, prior to all
other inquirers, the mechanical equivalent of heat from old physical
constants long known and at the disposal of all, and so to set up for
the new doctrine a programme embracing all physics and physiology;
Joule, on the other hand, applied himself to the exact verification of
the doctrine by beautifully conceived and masterfully executed
experiments, extending over all departments of physics. Soon Helmholtz
too attacked the problem, in a totally independent and characteristic
manner. After the professional virtuosity with which this physicist
grasped and disposed of all the points unsettled by Mayer's programme
and more besides, what especially strikes us is the consummate critical
lucidity of this young man of twenty-six years. In his exposition is
wanting that vehemence and impetuosity which marked Mayer's. The
principle of the conservation of energy is no self-evident or _a priori_
proposition for him. What follows, on the assumption that that
proposition obtains? In this hypothetical form, he subjugates his
matter.

I must confess, I have always marvelled at the æsthetic and ethical
taste of many of our contemporaries who have managed to fabricate out of
this relation of things, odious national and personal questions, instead
of praising the good fortune that made _several_ such men work together
and of rejoicing at the instructive diversity and idiosyncrasies of
great minds fraught with such rich consequences for us.

We know that still another theoretical conception played a part in the
development of the principle of energy, which Mayer held aloof from,
namely, the conception that heat, as also the other physical processes,
are due to motion. But once the principle of energy has been reached,
these auxiliary and transitional theories discharge no essential
function, and we may regard the principle, like that which Black gave,
as a contribution to the _direct description_ of a widely extended
domain of facts.

It would appear from such considerations not only advisable, but even
necessary, with all due recognition of the helpfulness of theoretic
ideas in research, yet gradually, as the new facts grow familiar, to
substitute for indirect description _direct_ description, which contains
nothing that is unessential and restricts itself absolutely to the
abstract apprehension of facts. We might almost say, that the
descriptive sciences, so called with a tincture of condescension, have,
in respect of scientific character, outstripped the physical expositions
lately in vogue. Of course, a virtue has been made of necessity here.

We must admit, that it is not in our power to describe directly every
fact, on the moment. Indeed, we should succumb in utter despair if the
whole wealth of facts which we come step by step to know, were presented
to us all at once. Happily, only detached and unusual features first
strike us, and such we bring nearer to ourselves by _comparison_ with
every-day events. Here the notions of the common speech are first
developed. The comparisons then grow more manifold and numerous, the
fields of facts compared more extensive, the concepts that make direct
description possible, proportionately more general and more abstract.

First we become familiar with the motion of freely falling bodies. The
concepts of force, mass, and work are then carried over, with
appropriate modifications, to the phenomena of electricity and
magnetism. A stream of water is said to have suggested to Fourier the
first distinct picture of currents of heat. A special case of vibrations
of strings investigated by Taylor, cleared up for him a special case of
the conduction of heat. Much in the same way that Daniel Bernoulli and
Euler constructed the most diverse forms of vibrations of strings from
Taylor's cases, so Fourier constructs out of simple cases of conduction
the most multifarious motions of heat; and that method has extended
itself over the whole of physics. Ohm forms his conception of the
electric current in imitation of Fourier's. The latter, also, adopts
Fick's theory of diffusion. In an analogous manner a conception of the
magnetic current is developed. All sorts of stationary currents are thus
made to exhibit common features, and even the condition of complete
equilibrium in an extended medium shares these features with the
dynamical condition of equilibrium of a stationary current. Things as
remote as the magnetic lines of force of an electric current and the
stream-lines of a frictionless liquid vortex enter in this way into a
peculiar relationship of similarity. The concept of potential,
originally enunciated for a restricted province, acquires a
wide-reaching applicability. Things as dissimilar as pressure,
temperature, and electromotive force, now show points of agreement in
relation to ideas derived by definite methods from that concept: viz.,
fall of pressure, fall of temperature, fall of potential, as also with
the further notions of liquid, thermal, and electric strength of
current. That relationship between systems of ideas in which the
dissimilarity of every two homologous concepts as well as the agreement
in logical relations of every two homologous pairs of concepts, is
clearly brought to light, is called an _analogy_. It is an effective
means of mastering heterogeneous fields of facts in unitary
comprehension. The path is plainly shown in which _a universal physical
phenomenology_ embracing all domains, will be developed.

In the process described we attain for the first time to what is
indispensable in the direct description of broad fields of fact--the
wide-reaching _abstract concept_. And now I must put a question smacking
of the school-master, but unavoidable: What is a concept? Is it a hazy
representation, admitting withal of mental visualisation? No. Mental
visualisation accompanies it only in the simplest cases, and then merely
as an adjunct. Think, for example, of the "coefficient of
self-induction," and seek for its visualised mental image. Or is,
perhaps, the concept a mere word? The adoption of this forlorn idea,
which has been actually proposed of late by a reputed mathematician
would only throw us back a thousand years into the deepest
scholasticism. We must, therefore, reject it.

The solution is not far to seek. We must not think that sensation, or
representation, is a purely passive process. The lowest organisms
respond to it with a simple reflex motion, by engulfing the prey which
approaches them. In higher organisms the centripetal stimulus encounters
in the nervous system obstacles and aids which modify the centrifugal
process. In still higher organisms, where prey is pursued and examined,
the process in question may go through extensive paths of circular
motions before it comes to relative rest. Our own life, too, is enacted
in such processes; all that we call science may be regarded as parts, or
middle terms, of such activities.

It will not surprise us now if I say: the definition of a concept, and,
when it is very familiar, even its name, is an _impulse_ to some
accurately determined, often complicated, critical, comparative, or
constructive _activity_, the usually sense-perceptive result of which is
a term or member of the concept's scope. It matters not whether the
concept draws the attention only to one certain sense (as sight) or to a
phase of a sense (as color, form), or is the starting-point of a
complicated action; nor whether the activity in question (chemical,
anatomical, and mathematical operations) is muscular or technical, or
performed wholly in the imagination, or only intimated. The concept is
to the physicist what a musical note is to a piano-player. A trained
physicist or mathematician reads a memoir like a musician reads a score.
But just as the piano-player must first learn to move his fingers singly
and collectively, before he can follow his notes without effort, so the
physicist or mathematician must go through a long apprenticeship before
he gains control, so to speak, of the manifold delicate innervations of
his muscles and imagination. Think of how frequently the beginner in
physics or mathematics performs more, or less, than is required, or of
how frequently he conceives things differently from what they are! But
if, after having had sufficient discipline, he lights upon the phrase
"coefficient of self-induction," he knows immediately what that term
requires of him. Long and thoroughly practised _actions_, which have
their origin in the necessity of comparing and representing facts by
other facts, are thus the very kernel of concepts. In fact, positive and
philosophical philology both claim to have established that all roots
represent concepts and stood originally for muscular activities alone.
The slow assent of physicists to Kirchhoff's dictum now becomes
intelligible. They best could feel the vast amount of individual labor,
theory, and skill required before the ideal of direct description could
be realised.

       *       *       *       *       *

Suppose, now, the ideal of a given province of facts is reached. Does
description accomplish all that the inquirer can ask? In my opinion, it
does. Description is a building up of facts in thought, and this
building up is, in the experimental sciences, often the condition of
actual execution. For the physicist, to take a special case, the
metrical units are the building-stones, the concepts the directions for
building, and the facts the result of the building. Our mental imagery
is almost a complete substitute for the fact, and by means of it we can
ascertain all the fact's properties. We do not know that worst which we
ourselves have made.

People require of science that it should _prophesy_, and Hertz uses that
expression in his posthumous _Mechanics_. But, natural as it is, the
expression is too narrow. The geologist and the palæontologist, at times
the astronomer, and always the historian and the philologist, prophesy,
so to speak, _backwards_. The descriptive sciences, like geometry and
mathematics, prophesy neither forward or backwards, but seek from given
conditions the conditioned. Let us say rather: _Science completes in
thought facts that are only partly given_. This is rendered possible by
description, for description presupposes the interdependence of the
descriptive elements: otherwise nothing would be described.

It is said, description leaves the sense of causality unsatisfied. In
fact, many imagine they understand motions better when they picture to
themselves the pulling forces; and yet the _accelerations_, the facts,
accomplish more, without superfluous additions. I hope that the science
of the future will discard the idea of cause and effect, as being
formally obscure; and in my feeling that these ideas contain a strong
tincture of fetishism, I am certainly not alone. The more proper course
is, _to regard the abstract determinative elements of a fact as
interdependent_, in a purely logical way, as the mathematician or
geometer does. True, by comparison with the will, forces are brought
nearer to our feeling; but it may be that ultimately the will itself
will be made clearer by comparison with the accelerations of masses.

If we are asked, candidly, when is a fact _clear_ to us, we must say
"when we can reproduce it by very _simple_ and very familiar
intellectual operations, such as the construction of accelerations, or
the geometrical summation of accelerations, and so forth." The
requirement of _simplicity_ is of course to the expert a different
matter from what it is to the novice. For the first, description by a
system of differential equations is sufficient; for the second, a
gradual construction out of elementary laws is required. The first
discerns at once the connexion of the two expositions. Of course, it is
not disputed that the _artistic_ value of materially equivalent
descriptions may not be different.

Most difficult is it to persuade strangers that the grand universal laws
of physics, such as apply indiscriminately to material, electrical,
magnetic, and other systems, are not essentially different from
descriptions. As compared with many sciences, physics occupies in this
respect a position of vantage that is easily explained. Take, for
example, anatomy. As the anatomist in his quest for agreements and
differences in animals ascends to ever higher and higher
_classifications_, the individual facts that represent the ultimate
terms of the system, are still so different that they must be _singly_
noted. Think, for example, of the common marks of the Vertebrates, of
the class-characters of Mammals and Birds on the one hand and of Fishes
on the other, of the double circulation of the blood on the one hand and
of the single on the other. In the end, always _isolated_ facts remain,
which show only a _slight_ likeness to one another.

A science still more closely allied to physics, chemistry, is often in
the same strait. The abrupt change of the qualitative properties, in all
likelihood conditioned by the slight stability of the intermediate
states, the remote resemblance of the co-ordinated facts of chemistry
render the treatment of its data difficult. Pairs of bodies of different
qualitative properties unite in different mass-ratios; but no connexion
between the first and the last is to be noted, at first.

Physics, on the other hand, reveals to us wide domains of _qualitatively
homogeneous_ facts, differing from one another only in the number of
equal parts into which their characteristic marks are divisible, that
is, differing only _quantitatively_. Even where we have to deal with
qualities (colors and sounds), quantitative characters of those
qualities are at our disposal. Here the classification is so simple a
task that it rarely impresses us as such, whilst in infinitely fine
gradations, in a _continuum of facts_, our number-system is ready
beforehand to follow as far as we wish. The co-ordinated facts are here
extremely similar and very closely affined, as are also their
descriptions which consist in the determination of the numerical
measures of one given set of characters from those of a different set by
means of familiar mathematical operations--methods of derivation. Thus,
the common characteristics of all descriptions can be found here; and
with them a succinct, comprehensive description, or a rule for the
construction of all single descriptions, is assigned,--and this we call
_law_. Well-known examples are the formulæ for freely falling bodies,
for projectiles, for central motion, and so forth. If physics apparently
accomplishes more by its methods than other sciences, we must remember
that in a sense it has presented to it much simpler problems.

The remaining sciences, whose facts also present a physical side, need
not be envious of physics for this superiority; for all its acquisitions
ultimately redound to their benefit as well. But also in other ways this
mutual help shall and must change. Chemistry has advanced very far in
making the methods of physics her own. Apart from older attempts, the
periodical series of Lothar Meyer and Mendelejeff are a brilliant and
adequate means of producing an easily surveyed system of facts, which by
gradually becoming complete, will take the place almost of a continuum
of facts. Further, by the study of solutions, of dissociation, in fact
generally of phenomena which present a continuum of cases, the methods
of thermodynamics have found entrance into chemistry. Similarly we may
hope that, at some future day, a mathematician, letting the
fact-continuum of embryology play before his mind, which the
palæontologists of the future will supposedly have enriched with more
intermediate and derivative forms between Saurian and Bird than the
isolated Pterodactyl, Archæopteryx, Ichthyornis, and so forth, which we
now have--that such a mathematician shall transform, by the variation of
a few parameters, as in a dissolving view, one form into another, just
as we transform one conic section into another.

Reverting now to Kirchhoff's words, we can come to some agreement
regarding their import. Nothing can be built without building-stones,
mortar, scaffolding, and a builder's skill. Yet assuredly the wish is
well founded, that will show to posterity the complete structure in its
finished form, bereft of unsightly scaffolding. It is the pure logical
and æsthetic sense of the mathematician that speaks out of Kirchhoff's
words. Modern expositions of physics aspire after his ideal; that, too,
is intelligible. But it would be a poor didactic shift, for one whose
business it was to train architects, to say: "Here is a splendid
edifice; if thou wouldst really build, go thou and do likewise".

The barriers between the special sciences, which make division of work
and concentration possible, but which appear to us after all as cold and
conventional restrictions, will gradually disappear. Bridge upon bridge
is thrown over the gaps. Contents and methods, even of the remotest
branches, are compared. When the Congress of Natural Scientists shall
meet a hundred years hence, we may expect that they will represent a
unity in a higher sense than is possible to-day, not in sentiment and
aim alone, but in method also. In the meantime, this great change will
be helped by our keeping constantly before our minds the fact of the
intrinsic relationship of all research, which Kirchhoff characterised
with such classical simplicity.

  FOOTNOTES:

  [Footnote 80: An address delivered before the General Session of the
  German Association of Naturalists and Physicians, at Vienna, Sept.
  24, 1894.]



THE PART PLAYED BY ACCIDENT IN INVENTION AND DISCOVERY.[81]


It is characteristic of the naïve and sanguine beginnings of thought in
youthful men and nations, that all problems are held to be soluble and
fundamentally intelligible on the first appearance of success. The sage
of Miletus, on seeing plants take their rise from moisture, believed he
had comprehended the whole of nature, and he of Samos, on discovering
that definite numbers corresponded to the lengths of harmonic strings,
imagined he could exhaust the nature of the world by means of numbers.
Philosophy and science in such periods are blended. Wider experience,
however, speedily discloses the error of such a course, gives rise to
criticism, and leads to the division and ramification of the sciences.

At the same time, the necessity of a broad and general view of the world
remains; and to meet this need philosophy parts company with special
inquiry. It is true, the two are often found united in gigantic
personalities. But as a rule their ways diverge more and more widely
from each other. And if the estrangement of philosophy from science can
reach a point where data unworthy of the nursery are not deemed too
scanty as foundations of the world, on the other hand the thorough-paced
specialist may go to the extreme of rejecting point-blank the
possibility of a broader view, or at least of deeming it superfluous,
forgetful of Voltaire's apophthegm, nowhere more applicable than here,
_Le superflu--chose très nécessaire_.

It is true, the history of philosophy, owing to the insufficiency of its
constructive data, is and must be largely a history of error. But it
would be the height of ingratitude on our part to forget that the seeds
of thoughts which still fructify the soil of special research, such as
the theory of irrationals, the conceptions of conservation, the doctrine
of evolution, the idea of specific energies, and so forth, may be traced
back in distant ages to philosophical sources. Furthermore, to have
deferred or abandoned the attempt at a broad philosophical view of the
world from a full knowledge of the insufficiency of our materials, is
quite a different thing from never having undertaken it at all. The
revenge of its neglect, moreover, is constantly visited upon the
specialist by his committal of the very errors which philosophy long ago
exposed. As a fact, in physics and physiology, particularly during the
first half of this century, are to be met intellectual productions
which for naïve simplicity are not a jot inferior to those of the Ionian
school, or to the Platonic ideas, or to that much reviled ontological
proof.

Latterly, there has been evidence of a gradual change in the situation.
Recent philosophy has set itself more modest and more attainable ends;
it is no longer inimical to special inquiry; in fact, it is zealously
taking part in that inquiry. On the other hand, the special sciences,
mathematics and physics, no less than philology, have become eminently
philosophical. The material presented is no longer accepted
uncritically. The glance of the inquirer is bent upon neighboring
fields, whence that material has been derived. The different special
departments are striving for closer union, and gradually the conviction
is gaining ground that philosophy can consist only of mutual,
complemental criticism, interpenetration, and union of the special
sciences into a consolidated whole. As the blood in nourishing the body
separates into countless capillaries, only to be collected again and to
meet in the heart, so in the science of the future all the rills of
knowledge will gather more and more into a common and undivided stream.

It is this view--not an unfamiliar one to the present generation--that I
purpose to advocate. Entertain no hope, or rather fear, that I shall
construct systems for you. I shall remain a natural inquirer. Nor expect
that it is my intention to skirt all the fields of natural inquiry. I
can attempt to be your guide only in that branch which is familiar to
me, and even there I can assist in the furtherment of only a small
portion of the allotted task. If I shall succeed in rendering plain to
you the relations of physics, psychology, and the theory of knowledge,
so that you may draw from each profit and light, redounding to the
advantage of each, I shall regard my work as not having been in vain.
Therefore, to illustrate by an example how, consonantly with my powers
and views, I conceive such inquiries should be conducted, I shall treat
to-day, in the form of a brief sketch, of the following special and
limited subject--of _the part which accidental circumstances play in the
development of inventions and discoveries_.

       *       *       *       *       *

When we Germans say of a man that he was not the inventor of
gunpowder,[82] we impliedly cast a grave suspicion on his abilities. But
the expression is not a felicitous one, as there is probably no
invention in which deliberate thought had a smaller, and pure luck a
larger, share than in this. It is well to ask, Are we justified in
placing a low estimate on the achievement of an inventor because
accident has assisted him in his work? Huygens, whose discoveries and
inventions are justly sufficient to entitle him to an opinion in such
matters, lays great emphasis on this factor. He asserts that a man
capable of inventing the telescope without the concurrence of accident
must have been gifted with superhuman genius.[83]

A man living in the midst of civilisation finds himself surrounded by a
host of marvellous inventions, considering none other than the means of
satisfying the needs of daily life. Picture such a man transported to
the epoch preceding the invention of these ingenious appliances, and
imagine him undertaking in a serious manner to comprehend their origin.
At first the intellectual power of the men capable of producing such
marvels will strike him as incredible, or, if we adopt the ancient view,
as divine. But his astonishment is considerably allayed by the
disenchanting yet elucidative revelations of the history of primitive
culture, which to a large extent prove that these inventions took their
rise very slowly and by imperceptible degrees.

A small hole in the ground with fire kindled in it constituted the
primitive stove. The flesh of the quarry, wrapped with water in its
skin, was boiled by contact with heated stones. Cooking by stones was
also done in wooden vessels. Hollow gourds were protected from the fire
by coats of clay. Thus, from the burned clay accidentally originated the
enveloping pot, which rendered the gourd superfluous, although for a
long time thereafter the clay was still spread over the gourd, or
pressed into woven wicker-work before the potter's art assumed its final
independence. Even then the wicker-work ornament was retained, as a sort
of attest of its origin.

We see, thus, it is by accidental circumstances, or by such as lie
without our purpose, foresight, and power, that man is gradually led to
the acquaintance of improved means of satisfying his wants. Let the
reader picture to himself the genius of a man who could have foreseen
without the help of accident that clay handled in the ordinary manner
would produce a useful cooking utensil! The majority of the inventions
made in the early stages of civilisation, including language, writing,
money, and the rest, could not have been the product of deliberate
methodical reflexion for the simple reason that no idea of their value
and significance could have been had except from practical use. The
invention of the bridge may have been suggested by the trunk of a tree
which had fallen athwart a mountain-torrent; that of the tool by the use
of a stone accidentally taken into the hand to crack nuts. The use of
fire probably started in and was disseminated from regions where
volcanic eruptions, hot springs, and burning jets of natural gas
afforded opportunity for quietly observing and turning to practical
account the properties of fire. Only after that had been done could the
significance of the fire-drill be appreciated, an instrument which was
probably discovered from boring a hole through a piece of wood. The
suggestion of a distinguished inquirer that the invention of the
fire-drill originated on the occasion of a religious ceremony is both
fantastic and incredible. And as to the use of fire, we should no more
attempt to derive that from the invention of the fire-drill than we
should from the invention of sulphur matches. Unquestionably the
opposite course was the real one.[84]

Similar phenomena, though still largely veiled in obscurity, mark the
initial transition of nations from a hunting to a nomadic life and to
agriculture.[85] We shall not multiply examples, but content ourselves
with the remark that the same phenomena recur in historical times, in
the ages of great technical inventions, and, further, that regarding
them the most whimsical notions have been circulated--notions which
ascribe to accident an unduly exaggerated part, and one which in a
psychological respect is absolutely impossible. The observation of steam
escaping from a tea-kettle and of the clattering of the lid is supposed
to have led to the invention of the steam-engine. Just think of the gap
between this spectacle and the conception of the performance of great
mechanical work by steam, for a man totally ignorant of the
steam-engine! Let us suppose, however, that an engineer, versed in the
practical construction of pumps, should accidentally dip into water an
inverted bottle that had been filled with steam for drying and still
retained its steam. He would see the water rush violently into the
bottle, and the idea would very naturally suggest itself of founding on
this experience a convenient and useful atmospheric steam-pump, which by
imperceptible degrees, both psychologically possible and immediate,
would then undergo a natural and gradual transformation into Watt's
steam-engine.

But granting that the most important inventions are brought to man's
notice accidentally and in ways that are beyond his foresight, yet it
does not follow that accident alone is sufficient to produce an
invention. The part which man plays is by no means a passive one. Even
the first potter in the primeval forest must have felt some stirrings of
genius within him. In all such cases, the inventor is obliged _to take
note_ of the new fact, he must discover and grasp its advantageous
feature, and must have the power to turn that feature to account in the
realisation of his purpose. He must _isolate_ the new feature, impress
it upon his memory, unite and interweave it with the rest of his
thought; in short, he must possess the capacity _to profit by
experience_.

The capacity to profit by experience might well be set up as a test of
intelligence. This power varies considerably in men of the same race,
and increases enormously as we advance from the lower animals to man.
The former are limited in this regard almost entirely to the reflex
actions which they have inherited with their organism, they are almost
totally incapable of individual experience, and considering their simple
wants are scarcely in need of it. The ivory-snail (_Eburna spirata_)
never learns to avoid the carnivorous Actinia, no matter how often it
may wince under the latter's shower of needles, apparently having no
memory for pain whatever.[86] A spider can be lured forth repeatedly
from its hole by touching its web with a tuning-fork. The moth plunges
again and again into the flame which has burnt it. The humming-bird
hawk-moth[87] dashes repeatedly against the painted roses of the
wall-paper, like the unhappy and desperate thinker who never wearies of
attacking the same insoluble chimerical problem. As aimlessly almost as
Maxwell's gaseous molecules and in the same unreasoning manner common
flies in their search for light and air stream against the glass pane of
a half-opened window and remain there from sheer inability to find their
way around the narrow frame. But a pike separated from the minnows of
his aquarium by a glass partition, learns after the lapse of a few
months, though only after having butted himself half to death, that he
cannot attack these fishes with impunity. What is more, he leaves them
in peace even after the removal of the partition, though he will bolt a
strange fish at once. Considerable memory must be attributed to birds of
passage, a memory which, probably owing to the absence of disturbing
thoughts, acts with the precision of that of some idiots. Finally, the
susceptibility to training evinced by the higher vertebrates is
indisputable proof of the ability of these animals to profit by
experience.

A powerfully developed _mechanical_ memory, which recalls vividly and
faithfully old situations, is sufficient for avoiding definite
particular dangers, or for taking advantage of definite particular
opportunities. But more is required for the development of _inventions_.
More extensive chains of images are necessary here, the excitation by
mutual contact of widely different trains of ideas, a more powerful,
more manifold, and richer connexion of the contents of memory, a more
powerful and impressionable psychical life, heightened by use. A man
stands on the bank of a mountain-torrent, which is a serious obstacle to
him. He remembers that he has crossed just such a torrent before on the
trunk of a fallen tree. Hard by trees are growing. He has often moved
the trunks of fallen trees. He has also felled trees before, and then
moved them. To fell trees he has used sharp stones. He goes in search of
such a stone, and as the old situations that crowd into his memory and
are held there in living reality by the definite powerful interest which
he has in crossing just this torrent,--as these impressions are made to
pass before his mind in the _inverse order_ in which they were here
evoked, he invents the bridge.

There can be no doubt but the higher vertebrates adapt their actions in
some moderate degree to circumstances. The fact that they give no
appreciable evidence of advance by the accumulation of inventions, is
satisfactorily explained by a difference of degree or intensity of
intelligence as compared with man; the assumption of a difference of
kind is not necessary. A person who saves a little every day, be it ever
so little, has an incalculable advantage over him who daily squanders
that amount, or is unable to keep what he has accumulated. A slight
quantitative difference in such things explains enormous differences of
advancement.

The rules which hold good in prehistoric times also hold good in
historical times, and the remarks made on invention may be applied
almost without modification to discovery; for the two are distinguished
solely by the use to which the new knowledge is put. In both cases the
investigator is concerned with some _newly observed_ relation of new or
old properties, abstract or concrete. It is observed, for example, that
a substance which gives a chemical reaction _A_ is also the cause of a
chemical reaction _B_. If this observation fulfils no purpose but that
of furthering the scientist's insight, or of removing a source of
intellectual discomfort, we have a discovery; but an invention, if in
using the substance giving the reaction _A_ to produce the desired
reaction _B_, we have a practical end in view, and seek to remove a
source of material discomfort. The phrase, _disclosure of the connexion
of reactions_, is broad enough to cover discoveries and inventions in
all departments. It embraces the Pythagorean proposition, which is a
combination of a geometrical and an arithmetical reaction, Newton's
discovery of the connexion of Kepler's motions with the law of the
inverse squares, as perfectly as it does the detection of some minute
but appropriate alteration in the construction of a tool, or of some
appropriate change in the methods of a dyeing establishment.

The disclosure of new provinces of facts before unknown can only be
brought about by accidental circumstances, under which are _remarked_
facts that commonly go unnoticed. The achievement of the discoverer here
consists in his _sharpened attention_, which detects the uncommon
features of an occurrence and their determining conditions from their
most evanescent marks,[88] and discovers means of submitting them to
exact and full observation. Under this head belong the first disclosures
of electrical and magnetic phenomena, Grimaldi's observation of
interference, Arago's discovery of the increased check suffered by a
magnetic needle vibrating in a copper envelope as compared with that
observed in a bandbox, Foucault's observation of the stability of the
plane of vibration of a rod accidentally struck while rotating in a
turning-lathe, Mayer's observation of the increased redness of venous
blood in the tropics, Kirchhoff's observation of the augmentation of the
_D_-line in the solar spectrum by the interposition of a sodium lamp,
Schönbein's discovery of ozone from the phosphoric smell emitted on the
disruption of air by electric sparks, and a host of others. All these
facts, of which unquestionably many were _seen_ numbers of times before
they were _noticed_, are examples of the inauguration of momentous
discoveries by accidental circumstances, and place the importance of
strained attention in a brilliant light.

But not only is a significant part played in the beginning of an inquiry
by co-operative circumstances beyond the foresight of the investigator;
their influence is also active in its prosecution. Dufay, thus, whilst
following up the behavior of _one_ electrical state which he had
assumed, discovers the existence of _two_. Fresnel learns by accident
that the interference-bands received on ground glass are seen to better
advantage in the open air. The diffraction-phenomenon of two slits
proved to be considerably different from what Fraunhofer had
anticipated, and in following up this circumstance he was led to the
important discovery of grating-spectra. Faraday's induction-phenomenon
departed widely from the initial conception which occasioned his
experiments, and it is precisely this deviation that constitutes his
real discovery.

Every man has pondered on some subject. Every one of us can multiply the
examples cited, by less illustrious ones from his own experience. I
shall cite but one. On rounding a railway curve once, I accidentally
remarked a striking apparent inclination of the houses and trees. I
inferred that the direction of the total resultant _physical_
acceleration of the body reacts _physiologically_ as the vertical.
Afterwards, in attempting to inquire more carefully into this
phenomenon, and this only, in a large whirling machine, the collateral
phenomena conducted me to the sensation of angular acceleration,
vertigo, Flouren's experiments on the section of the semi-circular
canals etc., from which gradually resulted views relating to sensations
of direction which are also held by Breuer and Brown, which were at
first contested on all hands, but are now regarded on many sides as
correct, and which have been recently enriched by the interesting
inquiries of Breuer concerning the _macula acustica_, and Kreidel's
experiments with magnetically orientable crustacea.[89] Not disregard of
accident but a direct and purposeful employment of it advances research.

The more powerful the psychical connexion of the memory pictures
is,--and it varies with the individual and the mood,--the more apt is
the same accidental observation to be productive of results. Galileo
knows that the air has weight; he also knows of the "resistance to a
vacuum," expressed both in weight and in the height of a column of
water. But the two ideas dwelt asunder in his mind. It remained for
Torricelli to vary the specific gravity of the liquid measuring the
pressure, and not till then was the air included in the list of
pressure-exerting fluids. The reversal of the lines of the spectrum was
seen repeatedly before Kirchhoff, and had been mechanically explained.
But it was left for his penetrating vision to discern the evidence of
the connexion of this phenomenon with questions of heat, and to him
alone through persistent labor was revealed the sweeping significance of
the fact for the mobile equilibrium of heat. Supposing, then, that such
a rich organic connexion of the elements of memory exists, and is the
prime distinguishing mark of the inquirer, next in importance certainly
is that _intense interest_ in a definite object, in a definite idea,
which fashions advantageous combinations of thought from elements before
disconnected, and obtrudes that idea into every observation made, and
into every thought formed, making it enter into relationship with all
things. Thus Bradley, deeply engrossed with the subject of aberration,
is led to its solution by an exceedingly unobtrusive experience in
crossing the Thames. It is permissible, therefore, to ask whether
accident leads the discoverer, or the discoverer accident, to a
successful outcome in scientific quests.

No man should dream of solving a great problem unless he is so
thoroughly saturated with his subject that everything else sinks into
comparative insignificance. During a hurried meeting with Mayer in
Heidelberg once, Jolly remarked, with a rather dubious implication, that
if Mayer's theory were correct water could be warmed by shaking. Mayer
went away without a word of reply. Several weeks later, and now
unrecognised by Jolly, he rushed into the latter's presence exclaiming:
"Es ischt aso!" (It is so, it is so!) It was only after considerable
explanation that Jolly found out what Mayer wanted to say. The incident
needs no comment.[90]

A person deadened to sensory impressions and given up solely to the
pursuit of his own thoughts, may also light on an idea that will divert
his mental activity into totally new channels. In such cases it is a
psychical accident, an intellectual experience, as distinguished from a
physical accident, to which the person owes his discovery--a discovery
which is here made "deductively" by means of mental copies of the world,
instead of experimentally. _Purely_ experimental inquiry, moreover, does
not exist, for, as Gauss says, virtually we always experiment with our
thoughts. And it is precisely that constant, corrective interchange or
intimate union of experiment and deduction, as it was cultivated by
Galileo in his _Dialogues_ and by Newton in his _Optics_, that is the
foundation of the benign fruitfulness of modern scientific inquiry as
contrasted with that of antiquity, where observation and reflexion
ofttimes pursued their respective courses like two strangers.

We have to wait for the appearance of a favorable physical accident. The
movement of our thoughts obeys the law of association. In the case of
meagre experience the result of this law is simply the mechanical
reproduction of definite sensory experiences. On the other hand, if the
psychical life is subjected to the incessant influences of a powerful
and rich experience, then every representative element in the mind is
connected with so many others that the actual and natural course of the
thoughts is easily influenced and determined by insignificant
circumstances, which accidentally are decisive. Hereupon, the process
termed imagination produces its protean and infinitely diversified
forms. Now what can we do to guide this process, seeing that the
combinatory law of the images is without our reach? Rather let us ask,
what influence can a powerful and constantly recurring idea exert on the
movement of our thoughts? According to what has preceded, the answer is
involved in the question itself. The _idea_ dominates the thought of the
inquirer, not the latter the former.

Let us see, now, if we can acquire a profounder insight into the process
of discovery. The condition of the discoverer is, as James has aptly
remarked, not unlike the situation of a person who is trying to remember
something that he has forgotten. Both are sensible of a gap, and have
only a remote presentiment of what is missing. Suppose I meet in a
company a well-known and affable gentleman whose name I have forgotten,
and who to my horror asks to be introduced to some one. I set to work
according to Lichtenberg's rule, and run down the alphabet in search of
the initial letter of his name. A vague sympathy holds me at the letter
_G_. Tentatively I add the second letter and am arrested at _e_, and
long before I have tried the third letter _r_, the name "Gerson" sounds
sonorously upon my ear, and my anguish is gone. While taking a walk I
meet a gentleman from whom I receive a communication. On returning home,
and in attending to weightier affairs, the matter slips my mind.
Moodily, but in vain, I ransack my memory. Finally I observe that I am
going over my walk again in thought. On the street corner in question
the self-same gentleman stands before me and repeats his communication.
In this process are successively recalled to consciousness all the
percepts which were connected with the percept that was lost, and with
them, finally, that, too, is brought to light. In the first case--where
the experience had already been made and is permanently impressed on our
thought--a _systematic_ procedure is both possible and easy, for we know
that a name must be composed of a limited number of sounds. But at the
same time it should be observed that the labor involved in such a
combinatorial task would be enormous if the name were long and the
responsiveness of the mind weaker.

It is often said, and not wholly without justification, that the
scientist has solved a _riddle_. Every problem in geometry may be
clothed in the garb of a _riddle_. Thus: "What thing is that _M_ which
has the properties _A_, _B_, _C_?" "What circle is that which touches
the straight lines _A_, _B_, but touches _B_ in the point _C_?" The
first two conditions marshal before the imagination the group of circles
whose centres lie in the line of symmetry of _A_, _B_. The third
condition reminds us of all the circles having centres in the straight
line that stands at right angles to _B_ in _C_. The _common_ term, or
common terms, of the two groups of images solves the riddle--satisfies
the problem. Puzzles dealing with things or words induce similar
processes, but the memory in such cases is exerted in many directions
and more varied and less clearly ordered provinces of ideas are
surveyed. The difference between the situation of a geometer who has a
construction to make, and that of an engineer, or a scientist,
confronted with a problem, is simply this, that the first moves in a
field with which he is thoroughly acquainted, whereas the two latter are
obliged to familiarise themselves with this field subsequently, and in a
measure far transcending what is commonly required. In this process the
mechanical engineer has at least always a definite goal before him and
definite means to accomplish his aim, whilst in the case of the
scientist that aim is in many instances presented only in vague and
general outlines. Often the very formulation of the riddle devolves on
him. Frequently it is not until the aim has been reached that the
broader outlook requisite for systematic procedure is obtained. By far
the larger portion of his success, therefore, is contingent on luck and
instinct. It is immaterial, so far as its character is concerned,
whether the process in question is brought rapidly to a conclusion in
the brain of one man, or whether it is spun out for centuries in the
minds of a long succession of thinkers. The same relation that a word
solving a riddle bears to that riddle is borne by the modern conception
of light to the facts discovered by Grimaldi, Römer, Huygens, Newton,
Young, Malus, and Fresnel, and only by the help of this slowly developed
conception is our mental vision enabled to embrace the broad domain of
facts in question.

A welcome complement to the discoveries which the history of
civilisation and comparative psychology have furnished, is to be found
in the confessions of great scientists and artists. Scientists _and_
artists, we might say, for Liebig boldly declared there was no essential
difference between the two. Are we to regard Leonardo da Vinci as a
scientist or as an artist? If the artist builds up his work from a few
motives, the scientist discovers the motives which permeate reality. If
scientists like Lagrange or Fourier are in a certain measure artists in
the presentation of their results, on the other hand, artists like
Shakespeare or Ruysdael are scientists in the insight which must have
preceded their creations.

Newton, when questioned about his methods of work, could give no other
answer but that he was wont to ponder again and again on a subject; and
similar utterances are accredited to D'Alembert and Helmholtz.
Scientists and artists both recommend persistent labor. After the
repeated survey of a field has afforded opportunity for the
interposition of advantageous accidents, has rendered all the traits
that suit with the mood or the dominant thought more vivid, and has
gradually relegated to the background all things that are inappropriate,
making their future appearance impossible; then from the teeming,
swelling host of fancies which a free and high-flown imagination calls
forth, suddenly that particular form arises to the light which
harmonises perfectly with the ruling idea, mood, or design. Then it is
that that which has resulted slowly as the result of a gradual
selection, appears as if it were the outcome of a deliberate act of
creation. Thus are to be explained the statements of Newton, Mozart,
Richard Wagner, and others, when they say that thoughts, melodies, and
harmonies had poured in upon them, and that they had simply retained the
right ones. Undoubtedly, the man of genius, too, consciously or
instinctively, pursues systematic methods wherever it is possible; but
in his delicate presentiment he will omit many a task or abandon it
after a hasty trial on which a less endowed man would squander his
energies in vain. Thus, the genius accomplishes[91] in a brief space of
time undertakings for which the life of an ordinary man would far from
suffice. We shall hardly go astray if we regard genius as only a slight
deviation from the average mental endowment--as possessing simply a
greater sensitiveness of cerebral reaction and a greater swiftness of
reaction. The men who, obeying their inner impulses, make sacrifices for
an idea instead of advancing their material welfare, may appear to the
full-blooded Philistine as fools; yet we shall scarcely adopt Lombroso's
view, that genius is to be regarded as a disease, although it is
unfortunately true that the sensitive brains and fragile constitutions
succumb most readily to sickness.

The remark of C. G. J. Jacobi that mathematics is slow of growth and
only reaches the truth by long and devious paths, that the way to its
discovery must be prepared for long beforehand, and that then the truth
will make its long-deferred appearance as if impelled by some divine
necessity[92]--all this holds true of every science. We are astounded
often to note that it required the combined labors of many eminent
thinkers for a full century to reach a truth which it takes us only a
few hours to master and which once acquired seems extremely easy to
reach under the right sort of circumstances. To our humiliation we learn
that even the greatest men are born more for life than for science. The
extent to which even they are indebted to accident--to that singular
conflux of the physical and the psychical life in which the continuous
but yet imperfect and never-ending adaptation of the latter to the
former finds its distinct expression--that has been the subject of our
remarks to-day. Jacobi's poetical thought of a divine necessity acting
in science will lose none of its loftiness for us if we discover in this
necessity the same power that destroys the unfit and fosters the fit.
For loftier, nobler, and more romantic than poetry is the truth and the
reality.

  FOOTNOTES:

  [Footnote 81: Inaugural lecture delivered on assuming the
  Professorship of the History and Theory of Inductive Science in the
  University of Vienna, October 21, 1895.]

  [Footnote 82: The phrase is, _Er hat das Pulver nicht erfunden_.]

  [Footnote 83: "Quod si quis tanta industria exstitisset, ut ex
  naturae principiis at geometria hanc rem eruere potuisset, eum ego
  supra mortalium sortem ingenio valuisse dicendum crederem. Sed hoc
  tantum abest, ut fortuito reperti artificii rationem non adhuc satis
  explicari potuerint viri doctissimi."--Hugenii Dioptrica (de
  telescopiis).]

  [Footnote 84: I must not be understood as saying that the fire-drill
  has played no part in the worship of fire or of the sun.]

  [Footnote 85: Compare on this point the extremely interesting
  remarks of Dr. Paul Carus in his _Philosophy of the Tool_, Chicago,
  1893.]

  [Footnote 86: Möbius, _Naturwissenschaftlicher Verein für
  Schleswig-Holstein_, Kiel, 1893, p. 113 et seq.]

  [Footnote 87: I am indebted for this observation to Professor
  Hatscheck.]

  [Footnote 88: Cf. Hoppe, _Entdecken und Finden_, 1870.]

  [Footnote 89: See the lecture "Sensations of Orientation," p. 282 et
  seq.]

  [Footnote 90: This story was related to me by Jolly, and
  subsequently repeated in a letter from him.]

  [Footnote 91: I do not know whether Swift's academy of schemers in
  Lagado, in which great discoveries and inventions were made by a
  sort of verbal game of dice, was intended as a satire on Francis
  Bacon's method of making discoveries by means of huge synoptic
  tables constructed by scribes. It certainly would not have been
  ill-placed.]

  [Footnote 92: "Crescunt disciplinae lente tardeque; per varios
  errores sero pervenitur ad veritatem. Omnia praeparata esse debent
  diuturno et assiduo labore ad introitum veritatis novae. Jam illa
  certo temporis momento divina quadam necessitate coacta emerget."

  Quoted by Simony, _In ein ringförmiges Band einen Knoten zu machen_,
  Vienna, 1881, p. 41.]



ON SENSATIONS OF ORIENTATION.[93]


Through the co-operation of a succession of inquirers, among whom are
particularly to be mentioned Goltz of Strassburg and Breuer of Vienna,
considerable advances have been made during the last twenty-five years
in our knowledge of the means by which we ascertain our position in
space and the direction of our motion, or orient ourselves, as the
phrase goes. I presume that you are already acquainted with the
physiological part of the processes with which our sensations of
movement, or, more generally speaking, our sensations of orientation,
are connected. Here I shall consider more particularly the physical side
of the matter. In fact, I was originally led to the consideration of
these questions by the observation of extremely simple and perfectly
well-known physical facts, before I had any great acquaintance with
physiology and while pursuing unbiasedly my natural thoughts; and I am
of the conviction that the way which I have pursued, and which is
entirely free from hypotheses, will, if you will follow my exposition,
be that of easiest acquisition for the most of you.

No man of sound common sense could ever have doubted that a pressure or
force is requisite to set a body in motion in a given direction and that
a contrary pressure is required to stop suddenly a body in motion.
Though the law of inertia was first formulated with anything like
exactness by Galileo, the facts at the basis of it were known long
previously to men of the stamp of Leonardo da Vinci, Rabelais, and
others, and were illustrated by them with appropriate experiments.
Leonardo knew that by a swift stroke with a ruler one can knock out from
a vertical column of checkers a single checker without over-throwing the
column. The experiment with a coin resting on a piece of pasteboard
covering a goblet, which falls into the goblet when the pasteboard is
jerked away, like all experiments of the kind, is certainly very old.

With Galileo the experience in question assumes greater clearness and
force. In the famous dialogue on the Copernican system which cost him
his freedom, he explains the tides in an unfelicitous, though in
principle correct manner, by the analogue of a platter of water swung to
and fro. In opposition to the Aristotelians of his time, who believed
the descent of a heavy body could be accelerated by the superposition
of another heavy body, he asserted that a body could never be
accelerated by one lying upon it unless the first in some way impeded
the superposed body in its descent. To seek to press a falling body by
means of another placed upon it, is as senseless as trying to prod a man
with a lance when the man is speeding away from one with the same
velocity as the lance. Even this little excursion into physics can
explain much to us. You know the peculiar sensation which one has in
falling, as when one jumps from a high springboard into the water, and
which is also experienced in some measure at the beginning of the
descent of elevators and swings. The reciprocal gravitational pressure
of the different parts of our body, which is certainly felt in some
manner, vanishes in free descent, or, in the case of the elevator, is
diminished on the beginning of the descent. A similar sensation would be
experienced if we were suddenly transported to the moon where the
acceleration of gravity is much less than upon the earth. I was led to
these considerations in 1866 by a suggestion in physics, and having also
taken into account the alterations of the blood-pressure in the cases in
question, I found I coincided without knowing it with Wollaston and
Purkinje. The first as early as 1810 in his Croonian lecture had touched
on the subject of sea-sickness and explained it by alterations of the
blood-pressure, and later had laid similar considerations at the basis
of his explanation of vertigo (1820-1826).[94]

Newton was the first to enunciate with perfect generality that a body
can change the velocity and direction of its motion only by the action
of a force, or the action of a second body. A corollary of this law
which was first expressly deduced by Euler is that a body can never be
set _rotating_ or made to cease rotating of itself but only by forces
and other bodies. For example, turn an open watch which has run down
freely backwards and forwards in your hand. The balance-wheel will not
fully catch the rapid rotations, it does not even respond fully to the
elastic force of the spring which proves too weak to carry the wheel
entirely with it.

Let us consider now that whether we move ourselves by means of our legs,
or whether we are moved by a vehicle or a boat, at first only a part of
our body is directly moved and the rest of it is afterwards set in
motion by the first part. We see that pressures, pulls, and tensions are
always produced between the parts of the body in this action, which
pressures, pulls, and tensions give rise to sensations by which the
forward or rotary movements in which we are engaged are made
perceptible.[95] But it is quite natural that sensations so familiar
should be little noticed and that attention should be drawn to them only
under special circumstances when they occur unexpectedly or with unusual
strength.

[Illustration: Fig. 45.]

Thus my attention was drawn to this point by the sensation of falling
and subsequently by another singular occurrence. I was rounding a sharp
railway curve once when I suddenly saw all the trees, houses, and
factory chimneys along the track swerve from the vertical and assume a
strikingly inclined position. What had hitherto appeared to me perfectly
natural, namely, the fact that we distinguish the vertical so perfectly
and sharply from every other direction, now struck me as enigmatical.
Why is it that the same direction can now appear vertical to me and now
cannot? By what is the vertical distinguished for us? (Compare Figure
45.)

The rails are raised on the convex or outward side of the track in order
to insure the stability of the carriage as against the action of the
centrifugal force, the whole being so arranged that the combination of
the force of gravity with the centrifugal force of the train shall give
rise to a force perpendicular to the plane of the rails.

Let us assume, now, that under all circumstances we somehow sense the
direction of the total resultant mass-acceleration whencesoever it may
arise as the vertical. Then both the ordinary and the extraordinary
phenomena will be alike rendered intelligible.[96]

I was now desirous of putting the view I had reached to a more
convenient and exact test than was possible on a railway journey where
one has no control over the determining circumstances and cannot alter
them at will. I accordingly had the simple apparatus constructed which
is represented in Figure 46.

In a large frame _BB_, which is fastened to the walls, rotates about a
vertical axis _AA_ a second frame _RR_, and within the latter a third
one _rr_, which can be set at any distance and position from the axis,
made stationary or movable, and is provided with a chair for the
observer.

[Illustration: Fig. 46.

From Mach's _Bewegungsempfindungen_, Leipsic, Engelmann, 1875.]

The observer takes his seat in the chair and to prevent disturbances of
judgment is enclosed in a paper box. If the observer together with the
frame _rr_ be then set in uniform rotation, he will feel and see the
beginning of the rotation both as to direction and amount very
distinctly although every outward visible or tangible point of reference
is wanting. If the motion be uniformly continued the sensation of
rotation will gradually cease entirely and the observer will imagine
himself at rest. But if _rr_ be placed outside the axis of rotation, at
once on the rotation beginning, a strikingly apparent, palpable,
actually visible inclination of the entire paper box is produced, slight
when the rotation is slow, strong when the rotation is rapid, and
continuing as long as the rotation lasts. It is absolutely impossible
for the observer to escape perceiving the inclination, although here
also all outward points of reference are wanting. If the observer, for
example, is seated so as to look towards the axis, he will feel the box
strongly tipped backwards, as it necessarily must be if the direction of
the total resultant force is perceived as the vertical. For other
positions of the observer the situation is similar.[97]

Once, while performing one of these experiments, and after rotating so
long that I was no longer conscious of the movement, I suddenly caused
the apparatus to be stopped, whereupon I immediately felt and saw myself
with the whole box rapidly flung round in rotation in the opposite
direction, although I knew that the whole apparatus was at rest and
every outward point of reference for the perception of motion was
wanting. Every one who disbelieves in sensations of movement should be
made acquainted with these phenomena. Had Newton known them and had he
ever observed how we may actually imagine ourselves turned and displaced
in space without the assistance of stationary bodies as points of
reference, he would certainly have been confirmed more than ever in his
unfortunate speculations regarding absolute space.

The sensation of rotation in the opposite direction after the apparatus
has been stopped, slowly and gradually ceases. But on accidentally
inclining my head once during this occurrence, the axis of apparent
rotation was also observed to incline in exactly the same manner both as
to direction and as to amount. It is accordingly clear that the
acceleration or retardation of rotation is felt. The acceleration
operates as a stimulus. The sensation, however, like almost all
sensations, though it gradually decreases, lasts perceptibly longer than
the stimulus. Hence the long continued apparent rotation after the
stopping of the apparatus. The organ, however, which causes the
persistence of this sensation must have its seat in the _head_, since
otherwise the axis of apparent rotation could not assume the same motion
as the head.

If I were to say, now, that a light had flashed upon me in making these
last observations, the expression would be a feeble one. I ought to say
I experienced a perfect illumination. My juvenile experiences of vertigo
occurred to me. I remembered Flourens's experiments relative to the
section of the semi-circular canals of the labyrinths of doves and
rabbits, where this inquirer had observed phenomena similar to vertigo,
but which he preferred to interpret, from his bias to the acoustic
theory of the labyrinth, as the expression of painful auditive
disturbances. I saw that Goltz had nearly but not quite hit the bull's
eye with his theory of the semi-circular canals. This inquirer, who,
from his happy habit of following his own natural thoughts without
regard for tradition, has cleared up so much in science, spoke, as early
as 1870, on the ground of experiments, as follows: "It is uncertain
whether the semi-circular canals are auditive organs or not. In any
event they form an apparatus which serves for the preservation of
equilibrium. They are, so to speak, the sense-organs of equilibrium of
the head and indirectly of the whole body." I remembered the galvanic
dizziness which had been observed by Ritter and Purkinje on the passage
of a current through the head, when the persons experimented upon
imagined they were falling towards the cathode. The experiment was
immediately repeated, and sometime later (1874) I was enabled to
demonstrate the same objectively with fishes, all of which placed
themselves sidewise and in the same direction in the field of the
current as if at command.[98] Müller's doctrine of specific energies now
appeared to me to bring all these new and old observations into a
simple, connected unity.

[Illustration: Fig. 47.

The labyrinth of a dove (stereoscopically reproduced), from R. Ewald,
_Nervus Octavus_, Wiesbaden, Bergmann, 1892.]

Let us picture to ourselves the labyrinth of the ear with its three
semi-circular canals lying in three mutually perpendicular planes (Comp.
Fig. 47), the mysterious position of which inquirers have endeavored to
explain in every possible and impossible way. Let us conceive the nerves
of the ampullæ, or the dilated extensions of the semi-circular canals,
equipped with a capacity for responding to every imaginable stimulus
with a sensation of rotation just as the nerves of the retina of the eye
when excited by pressures, by electrical or chemical stimuli always
respond with the sensation of light; let us picture to ourselves,
further, that the usual excitation of the ampullæ nerves is produced by
the inertia of the contents of the semi-circular canals, which contents
on suitable rotations in the plane of the semi-circular canal are left
behind in the motion, or at least have a tendency to remain behind and
consequently exert a pressure. It will be seen that on this supposition
all the single facts which without the theory appear as so many
different individual phenomena, become from this single point of view
clear and intelligible.

I had the satisfaction, immediately after the communication in which I
set forth this idea,[99] of seeing a paper by Breuer appear[100] in
which this author had arrived by entirely different methods at results
that agreed in all essential points with my own. A few weeks later
appeared the researches of Crum Brown of Edinburgh, whose methods were
even still nearer mine. Breuer's paper was far richer in physiological
respects than mine, and he had particularly gone into greater detail in
his investigation of the collateral effects of the reflex motions and
orientation of the eyes in the phenomena under consideration.[101] In
addition certain experiments which I had suggested in my paper as a test
of the correctness of the view in question had already been performed by
Breuer. Breuer has also rendered services of the highest order in the
further elaboration of this field. But in a physical regard, my paper
was, of course, more complete.

In order to portray to the eye the behavior of the semi-circular canals,
I have constructed here a little apparatus. (See Fig. 48.) The large
rotatable disc represents the osseous semi-circular canal, which is
continuous with the bones of the head; the small disc, which is free to
rotate on the axis of the first, represents the mobile and partly liquid
contents of the semi-circular canal. On rotating the large disc, the
small disc as you see remains behind. I have to turn some time before
the small disc is carried along with the large one by friction. But if I
now stop the large disc the small disc as you see continues to rotate.

[Illustration: Fig. 48.

Model representing the action of the semi-circular canals.]

Simply assume now that the rotation of the small disc, say in the
direction of the hands of a watch, would give rise to a sensation of
rotation in the opposite direction, and conversely, and you already
understand a good portion of the facts above set forth. The explanation
still holds, even if the small disc does not perform appreciable
rotations but is checked by a contrivance similar to an elastic spring,
the tension of which disengages a sensation. Conceive, now, three such
contrivances with their mutually perpendicular planes of rotation joined
together so as to form a single apparatus; then to this apparatus as a
whole, no rotation can be imparted without its being indicated by the
small mobile discs or by the springs which are attached to them.
Conceive both the right and the left ear equipped with such an
apparatus, and you will find that it answers all the purposes of the
semi-circular canals, which you see represented stereoscopically in Fig.
47 for the ear of a dove.

Of the many experiments which I have made on my own person, and the
results of which could be predicted by the new view according to the
behavior of the model and consequently according to the rules of
mechanics, I shall cite but one. I fasten a horizontal board in the
frame _RR_ of my rotatory apparatus, lie down upon the same with my
right ear upon the board, and cause the apparatus to be uniformly
rotated. As soon as I no longer perceive the rotation, I turn around
upon my left ear and immediately the sensation of rotation again starts
up with marked vividness. The experiment can be repeated as often as one
wishes. A slight turn of the head even is sufficient for reviving the
sensation of rotation which in the perfectly quiescent state at once
disappears altogether.

We will imitate the experiment on the model. I turn the large disc until
finally the small disc is carried along with it. If, now, while the
rotation continues uniform, I burn off a little thread which you see
here, the small disc will be flipped round by a spring into its own
plane 180°, so as now to present its opposite side to you, when the
rotation at once begins in the opposite direction.

We have consequently a very simple means for determining whether one is
actually the subject or not of uniform and imperceptible rotations. If
the earth rotated much more rapidly than it really does, or if our
semi-circular canals were much more sensitive, a Nansen sleeping at the
North Pole would be waked by a sensation of rotation every time he
turned over. Foucault's pendulum experiment as a demonstration of the
earth's rotation would be superfluous under such circumstances. The only
reason we cannot prove the rotation of the earth with the help of our
model, lies in the small angular velocity of the earth and in the
consequent liability to great experimental errors.[102]

Aristotle has said that "The sweetest of all things is knowledge." And
he is right. But if you were to suppose that the _publication_ of a new
view were productive of unbounded sweetness, you would be mightily
mistaken. No one disturbs his fellow-men with a new view unpunished. Nor
should the fact be made a subject of reproach to these fellow-men. To
presume to revolutionise the current way of thinking with regard to any
question, is no pleasant task, and above all not an easy one. They who
have advanced new views know best what serious difficulties stand in
their way. With honest and praiseworthy zeal, men set to work in search
of everything that does not suit with them. They seek to discover
whether they cannot explain the facts better or as well, or
approximately as well, by the traditional views. And that, too, is
justified. But at times some extremely artless animadversions are heard
that almost nonplus us. "If a sixth sense existed it could not fail to
have been discovered thousands of years ago." Indeed; there was a time,
then, when only seven planets could have existed! But I do not believe
that any one will lay any weight on the philological question whether
the set of phenomena which we have been considering should be called a
sense. The phenomena will not disappear when the name disappears. It was
further said to me that animals exist which have no labyrinth, but which
can yet orientate themselves, and that consequently the labyrinth has
nothing to do with orientation. We do not walk forsooth with our legs,
because snakes propel themselves without them!

But if the promulgator of a new idea cannot hope for any great pleasure
from its publication, yet the critical process which his views undergo
is extremely helpful to the subject-matter of them. All the defects
which necessarily adhere to the new view are gradually discovered and
eliminated. Over-rating and exaggeration give way to more sober
estimates. And so it came about that it was found unpermissible to
attribute all functions of orientation exclusively to the labyrinth. In
these critical labors Delage, Aubert, Breuer, Ewald, and others have
rendered distinguished services. It can also not fail to happen that
fresh facts become known in this process which could have been predicted
by the new view, which actually were predicted in part, and which
consequently furnish a support for the new view. Breuer and Ewald
succeeded in electrically and mechanically exciting the labyrinth, and
even single parts of the labyrinth, and thus in producing the movements
that belong to such stimuli. It was shown that when the semi-circular
canals were absent vertigo could not be produced, when the entire
labyrinth was removed the orientation of the head was no longer
possible, that without the labyrinth galvanic vertigo could not be
induced. I myself constructed as early as 1875 an apparatus for
observing animals in rotation, which was subsequently reinvented in
various forms and has since received the name of "cyclostat."[103] In
experiments with the most varied kinds of animals it was shown that, for
example, the larvæ of frogs are not subject to vertigo until their
semi-circular canals which at the start are wanting are developed (K.
Schäfer). A large percentage of the deaf and dumb are afflicted with
grave affections of the labyrinth. The American psychologist, William
James, has made whirling experiments with many deaf and dumb subjects,
and in a large number of them found that susceptibility to giddiness is
wanting. He also found that many deaf and dumb people on being ducked
under water, whereby they lose their weight and consequently have no
longer the full assistance of their muscular sense, utterly lose their
sense of position in space, do not know which is up and which is down,
and are thrown into the greatest consternation,--results which do not
occur in normal men. Such facts are convincing proof that we do not
orientate ourselves entirely by means of the labyrinth, important as it
is for us. Dr. Kreidl has made experiments similar to those of James and
found that not only is vertigo absent in deaf and dumb people when
whirled about, but that also the reflex movements of the eyes which are
normally induced by the labyrinth are wanting. Finally, Dr. Pollak has
found that galvanic vertigo does not exist in a large percentage of the
deaf and dumb. Neither the jerking movements nor the uniform movements
of the eyes were observed which normal human beings exhibit in the
Ritter and Purkinje experiment.

After the physicist has arrived at the idea that the semi-circular
canals are the organ of sensation of rotation or of angular
acceleration, he is next constrained to ask for the organs that mediate
the sensation of acceleration noticed in forward movements. In
searching for an organ for this function, he of course is not apt to
select one that stands in no anatomical and spatial relation with the
semi-circular canals. And in addition there are physiological
considerations to be weighed. The preconceived opinion once having been
abandoned that the _entire_ labyrinth is auditory in its function, there
remains after the cochlea is reserved for sensations of tone and the
semi-circular canals for the sensation of angular acceleration, the
vestibule for the discharge of additional functions. The vestibule,
particularly the part of it known as the sacculus, appeared to me, by
reason of the so-called otoliths which it contains, eminently adapted
for being the organ of sensation of forward acceleration or of the
position of the head. In this conjecture I again closely coincided with
Breuer.

That a sensation of position, of direction and amount of
mass-acceleration exists, our experience in elevators as well as of
movement in curved paths is sufficient proof. I have also attempted to
produce and destroy suddenly great velocities of forward movement by
means of various contrivances of which I shall mention only one here.
If, while enclosed in the paper box of my large whirling apparatus at
some distance from the axis, my body is in uniform rotation which I no
longer feel, and I then loosen the connexions of the frame _rr_ with _R_
thus making the former moveable and I then suddenly stop the larger
frame, my forward motion is abruptly impeded while the frame _rr_
continues to rotate. I imagine now that I am speeding on in a straight
line in a direction opposite to that of the checked motion.
Unfortunately, for many reasons it cannot be proved convincingly that
the organ in question has its seat in the head. According to the opinion
of Delage, the labyrinth has nothing to do with this particular
sensation of movement. Breuer, on the other hand, is of the opinion that
the organ of forward movement in man is stunted and the persistence of
the sensation in question is too brief to permit our instituting
experiments as obvious as in the case of rotation. In fact, Crum Brown
once observed while in an irritated condition peculiar vertical
phenomena in his own person, which were all satisfactorily explained by
an abnormally long persistence of the sensation of rotation, and I
myself in an analogous case on the stopping of a railway train felt the
apparent backward motion in striking intensity and for an unusual length
of time.

There is no doubt whatever that we feel changes of vertical
acceleration, and it will appear from the following extremely probable
that the otoliths of the vestibule are the sense-organ for the
_direction_ of the mass-acceleration. It will then be incompatible with
a really logical view to regard the latter as incapable of sensing
horizontal accelerations.

In the lower animals the analogue of the labyrinth is shrunk to a little
vesicle filled with a liquid and containing tiny crystals, auditive
stones, or otoliths, of greater specific gravity, suspended on minute
hairs. These crystals appear physically well adapted for indicating both
the direction of gravity and the direction of incipient movements. That
they discharge the former function, Delage was the first to convince
himself by experiments with lower animals which on the removal of the
otoliths utterly lost their bearings and could no longer regain their
normal position. Loeb also found that fishes without labyrinths swim now
on their bellies and now on their backs. But the most remarkable, most
beautiful, and most convincing experiment is that which Dr. Kreidl
instituted with crustaceans. According to Hensen, certain Crustacea on
sloughing spontaneously introduce fine grains of sand as auditive stones
into their otolith vesicle. At the ingenious suggestion of S. Exner, Dr.
Kreidl constrained some of these animals to put up with iron filings
(_ferrum limatum_). If the pole of an electro-magnet be brought near the
animal, it will at once turn its back away from the pole accompanying
the movement with appropriate reflex motions of the eye the moment the
current is closed, exactly as if gravity had been brought to bear upon
the animal in the same direction as the magnetic force.[104] This, in
fact, is what should be expected from the function ascribed to the
otoliths. If the eyes be covered with asphalt varnish, and the auditive
sacs removed, the crustaceans lose their sense of direction utterly,
tumble head over heels, lie on their side or their back indifferently.
This does not happen when the eyes only are covered. For vertebrates,
Breuer has demonstrated by searching investigations that the otoliths,
or better, statoliths, slide in three planes parallel to the planes of
the semi-circular canals, and are consequently perfectly adapted for
indicating changes both in the amount and the direction of the
mass-acceleration.[105]

I have already remarked that not every function of orientation can be
ascribed exclusively to the labyrinth. The deaf and dumb who have to be
immersed in water, and the crustaceans who must have their eyes closed
if they are to be perfectly disorientated, are proof of this fact. I saw
a blind cat at Hering's laboratory which to one who was not a very
attentive observer behaved exactly like a seeing cat. It played nimbly
with objects rolling on the floor, stuck its head inquisitively into
open drawers, sprang dexterously upon chairs, ran with perfect accuracy
through open doors, and never bumped against closed ones. The visual
sense had here been rapidly replaced by the tactual and auditive senses.
And it appears from Ewald's investigations that even after the
labyrinths have been removed, animals gradually learn to move about
again quite in the normal fashion, presumably because the eliminated
function of the labyrinth is now performed by some part of the brain. A
certain peculiar weakness of the muscles alone is perceptible which
Ewald ascribes to the absence of the stimulus which is otherwise
constantly emitted by the labyrinth (the labyrinth-tonus). But if the
part of the brain which discharges the deputed function be removed, the
animals are again completely disorientated and absolutely helpless.

It may be said that the views enunciated by Breuer, Crum Brown and
myself in 1873 and 1874, and which are substantially a fuller and richer
development of Goltz's idea, have upon the whole been substantiated. At
least they have exercised a helpful and stimulative influence. New
problems have of course arisen in the course of the investigation which
still await solution, and much work remains to be done. At the same time
we see how fruitful the renewed co-operation of the various special
departments of science may become after a period of isolation and
invigorating labor apart.

I may be permitted, therefore, to consider the relation between hearing
and orientation from another and more general point of view. What we
call the auditive organ is in the lower animals simply a sac containing
auditive stones. As we ascend the scale, 1, 2, 3 semi-circular canals
gradually develop from them, whilst the structure of the otolith organ
itself becomes more complicated. Finally, in the higher vertebrates, and
particularly in the mammals, a part of the latter organ (the lagena)
becomes the cochlea, which Helmholtz explained as the organ for
sensations of tone. In the belief that the entire labyrinth was an
auditive organ, Helmholtz, contrary to the results of his own masterly
analysis, originally sought to interpret another part of the labyrinth
as the organ of noises. I showed a long time ago (1873) that every tonal
stimulus by shortening the duration of the excitation to a few
vibrations, gradually loses its character of pitch and takes on that of
a sharp, dry report or noise.[106] All the intervening stages between
tones and noises can be exhibited. Such being the case, it will hardly
be assumed that one organ is suddenly and at some given point replaced
in function by another. On the basis of different experiments and
reasonings S. Exner also regards the assumption of a special organ for
the sensing of noises as unnecessary.

If we will but reflect how small a portion of the labyrinth of higher
animals is apparently in the service of the sense of hearing, and how
large, on the other hand, the portion is which very likely serves the
purposes of orientation, how much the first anatomical beginnings of the
auditive sac of lower animals resemble that part of the fully developed
labyrinth which does not hear, the view is irresistibly suggested which
Breuer and I (1874, 1875) expressed, that the auditive organ took its
development from an organ for sensing movements by adaptation to weak
periodic motional stimuli, and that many apparatuses in the lower
animals which are held to be organs of hearing are not auditive organs
at all.[107]

This view appears to be perceptibly gaining ground. Dr. Kreidl by
skilfully-planned experiments has arrived at the conclusion that even
fishes do not hear, whereas E. H. Weber, in his day, regarded the
ossicles which unite the air-bladder of fishes with the labyrinth as
organs expressly designed for conducting sound from the former to the
latter.[108] Störensen has investigated the excitation of sounds by the
air-bladder of fishes, as also the conduction of shocks through Weber's
ossicles. He regards the air-bladder as particularly adapted for
receiving the noises made by other fishes and conducting them to the
labyrinth. He has heard the loud grunting tones of the fishes in South
American rivers, and is of the opinion that they allure and find each
other in this manner. According to these views certain fishes are
neither deaf nor dumb.[109] The question here involved might be solved
perhaps by sharply distinguishing between the sensation of hearing
proper, and the perception of shocks. The first-mentioned sensation may,
even in the case of many vertebrates, be extremely restricted, or
perhaps even absolutely wanting. But besides the auditive function,
Weber's ossicles may perfectly well discharge some other function.
Although, as Moreau has shown, the air-bladder itself is not an organ of
equilibrium in the simple physical sense of Borelli, yet doubtless some
function of this character is still reserved for it. The union with the
labyrinth favors this conception, and so a host of new problems rises
here before us.

I should like to close with a reminiscence from the year 1863.
Helmholtz's _Sensations of Tone_ had just been published and the
function of the cochlea now appeared clear to the whole world. In a
private conversation which I had with a physician, the latter declared
it to be an almost hopeless undertaking to seek to fathom the function
of the other parts of the labyrinth, whereas I in youthful boldness
maintained that the question could hardly fail to be solved, and that
very soon, although of course I had then no glimmering of how it was to
be done. Ten years later the question was substantially solved.

To-day, after having tried my powers frequently and in vain on many
questions, I no longer believe that we can make short work of the
problems of science. Nevertheless, I should not consider an
"ignorabimus" as an expression of modesty, but rather as the opposite.
That expression is a suitable one only with regard to problems which are
wrongly formulated and which are therefore not problems at all. Every
real problem can and will be solved in due course of time without
supernatural divination, entirely by accurate observation and close,
searching thought.

  FOOTNOTES:

  [Footnote 93: A lecture delivered on February 24, 1897, before the
  _Verein zur Verbreitung naturwissenschaftlicher Kenntnisse in
  Wien_.]

  [Footnote 94: Wollaston, _Philosophical Transactions, Royal
  Society_, 1810. In the same place Wollaston also describes and
  explains the creaking of the muscles. My attention was recently
  called to this work by Dr. W. Pascheles.--Cf. also Purkinje, _Prager
  medicin_. _Jahrbücher_, Bd. 6, Wien, 1820.]

  [Footnote 95: Similarly many external forces do not act at once on
  all parts of the earth, and the internal forces which produce
  deformations act at first immediately only upon limited parts. If
  the earth were a feeling being, the tides and other terrestrial
  events would provoke in it similar sensations to those of our
  movements. Perhaps the slight alterations of the altitude of the
  pole which are at present being studied are connected with the
  continual slight deformations of the central ellipsoid occasioned by
  seismical happenings.]

  [Footnote 96: For the popular explanation by unconscious inference
  the matter is extremely simple. We regard the railway carriage as
  vertical and unconsciously infer the inclination of the trees. Of
  course the opposite conclusion that we regard the trees as vertical
  and infer the inclination of the carriage, unfortunately, is equally
  clear on this theory.]

  [Footnote 97: It will be observed that my way of thinking and
  experimenting here is related to that which led Knight to the
  discovery and investigation of the geotropism of plants.
  _Philosophical Transactions_, January 9, 1806. The relations between
  vegetable and animal geotropism have been more recently investigated
  by J. Loeb.]

  [Footnote 98: This experiment is doubtless related to the
  galvanotropic experiment with the larvæ of frogs described ten years
  later by L. Hermann. Compare on this point my remarks in the
  _Anzeiger der Wiener Akademie_, 1886, No. 21. Recent experiments in
  galvanotropism are due to J. Loeb.]

  [Footnote 99: _Wiener Akad._, 6 November, 1873.]

  [Footnote 100: _Wiener Gesellschaft der Aerzte_, 14 November, 1874.]

  [Footnote 101: I have made a contribution to this last question in
  my _Analysis of the Sensations_, (1886), English translation, 1897.]

  [Footnote 102: In my _Grundlinien der Lehre von den
  Bewegungsempfindungen_, 1875, the matter occupying lines 4 to 13 of
  page 20 from below, which rests on an error, is, as I have also
  elsewhere remarked, to be stricken out. For another experiment
  related to that of Foucault, compare my _Mechanics_, p. 303.]

  [Footnote 103: _Anzeiger der Wiener Akad._, 30 December, 1875.]

  [Footnote 104: The experiment was specially interesting for me as I
  had already attempted in 1874, although with very little confidence
  and without success, to excite electromagnetically my own labyrinth
  through which I had caused a current to pass.]

  [Footnote 105: Perhaps the discussion concerning the peculiarity of
  cats always falling on their feet, which occupied the Parisian
  Academy, and, incidentally, Parisian society a few years ago, will
  be remembered here. I believe that the questions which arose are
  disposed of by the considerations advanced in my
  _Bewegungsempfindungen_ (1875). I also partly gave, as early as
  1866, the apparatus conceived by the Parisian scientists to
  illustrate the phenomena in question. One difficulty was left
  untouched in the Parisian debate. The otolith apparatus of the cat
  can render it no service in _free_ descent. The cat, however, while
  at rest, doubtless knows its position in space and is instinctively
  conscious of the amount of movement which will put it on its feet.]

  [Footnote 106: See the Appendix to the English edition of my
  _Analysis of the Sensations_, Chicago, 1897.]

  [Footnote 107: Compare my _Analysis of Sensations_, p. 123 ff.]

  [Footnote 108: E. H. Weber, _De aure et auditu hominis et
  animalium_, Lipsiae, 1820.]

  [Footnote 109: Störensen, _Journ. Anat. Phys._, London, Vol. 29
  (1895).]



ON SOME PHENOMENA ATTENDING THE FLIGHT OF PROJECTILES.[110]


     "I have led my ragamuffins where they were peppered."--_Falstaff._

     "He goes but to see a noise that he heard."--_Midsummer Night's
     Dream._

To shoot, in the shortest time possible, as many holes as possible in
one another's bodies, and not always for exactly pardonable objects and
ideals, seems to have risen to the dignity of a duty with modern men,
who, by a singular inconsistency, and in subservience to a diametrically
contrary ideal, are bound by the equally holy obligation of making these
holes as small as possible, and, when made, of stopping them up and of
healing them as speedily as possible. Since, then, shooting and all that
appertains thereto, is a very important, if not the most important,
affair of modern life, you will doubtless not be averse to giving your
attention for an hour to some experiments which have been undertaken,
not for advancing the ends of war, but for promoting the ends of
science, and which throw some light on the phenomena attending the
flight of projectiles.

Modern science strives to construct its picture of the world not from
speculations but so far as possible from facts. It verifies its
constructs by recourse to observation. Every newly observed fact
completes its world-picture, and every divergence of a construct from
observation points to some imperfection, to some lacuna in it. What is
seen is put to the test of, and supplemented by, what is thought, which
is again naught but the result of things previously seen. It is always
peculiarly fascinating, therefore, to subject to direct verification by
observation, that is, to render palpable to the senses, something which
we have only theoretically excogitated or theoretically surmised.

In 1881, on hearing in Paris the lecture of the Belgian artillerist
Melsens, who hazarded the conjecture that projectiles travelling at a
high rate of speed carry masses of compressed air before them which are
instrumental in producing in bodies struck by the projectiles certain
well-known facts of the nature of explosions, the desire arose in me of
experimentally testing his conjecture and of rendering the phenomenon,
if it really existed, perceptible. The desire was the stronger as I
could say that all the means for realising it existed, and that I had in
part already used and tested them for other purposes.

And first let us get clear regarding the difficulties which have to be
surmounted. Our task is that of observing a bullet or other projectile
which is rushing through space at a velocity of many hundred yards a
second, together with the disturbances which the bullet causes in the
surrounding atmosphere. Even the opaque solid body itself, the
projectile, is only exceptionally visible under such circumstances--only
when it is of considerable size and when we see its line of flight in
strong perspective abridgement so that the velocity is apparently
diminished. We see a large projectile quite clearly when we stand behind
the cannon and look steadily along its line of flight or in the less
pleasant case when the projectile is speeding towards us. There is,
however, a very simple and effective method of observing swiftly moving
bodies with as little trouble as if they were held at rest at some point
in their path. The method is that of illumination by a brilliant
electric spark of extremely short duration in a dark room. But since,
for the full intellectual comprehension of a picture presented to the
eye, a certain, not inconsiderable interval of time is necessary, the
method of instantaneous photography will naturally also be employed. The
pictures, which are of extremely minute duration, are thus permanently
recorded and can be examined and analysed at one's convenience and
leisure.

With the difficulty just mentioned is associated still another and
greater difficulty which is due to the air. The atmosphere in its usual
condition is generally not visible even when at rest. But the task
presented to us is to render visible masses of air which in addition
are moving with a high velocity.

To be visible, a body must either emit light itself, must shine, or must
affect in some way the light which falls upon it, must take up that
light entirely or partly, absorb it, or must have a deflective effect
upon it, that is, reflect or refract it. We cannot see the air as we can
a flame, for it shines only exceptionally, as in a Geissler's tube. The
atmosphere is extremely transparent and colorless; it cannot be seen,
therefore, as a dark or colored body can, or as chlorine gas can, or
vapor of bromine or iodine. Air, finally, has so small an index of
refraction and so small a deflective influence upon light, that the
refractive effect is commonly imperceptible altogether.

A glass rod is visible in air or in water, but it is almost invisible in
a mixture of benzol and bisulphuret of carbon, which has the same mean
index of refraction as the glass. Powdered glass in the same mixture has
a vivid coloring, because owing to the decomposition of the colors the
indices are the same for only one color which traverses the mixture
unimpeded, whilst the other colors undergo repeated reflexions.[111]

Water is invisible in water, alcohol in alcohol. But if alcohol be mixed
with water the flocculent streaks of the alcohol in the water will be
seen at once and _vice versa_. And in like manner the air, too, under
favorable circumstances, may be seen. Over a roof heated by the burning
sun, a tremulous wavering of objects is noticeable, as there is also
over red-hot stoves, radiators, and registers. In all these cases tiny
flocculent masses of hot and cold air, of slightly differing
refrangibility, are mingled together.

In like manner the more highly refracting parts of non-homogeneous
masses of glass, the so-called striæ or imperfections of the glass, are
readily detectible among the less refracting parts which constitute the
bulk of the same. Such glasses are unserviceable for optical purposes,
and special attention has been devoted to the investigation of the
methods for eliminating or avoiding these defects. The result has been
the development of an extremely delicate method for detecting optical
faults--the so-called method of Foucault and Toepler--which is suitable
also for our present purpose.

[Illustration: Fig. 49.]

Even Huygens when trying to detect the presence of striæ in polished
glasses viewed them under oblique illumination, usually at a
considerable distance, so as to give full scope to the aberrations, and
had recourse for greater exactitude to a telescope. But the method was
carried to its highest pitch of perfection in 1867 by Toepler who
employed the following procedure: A small luminous source _a_ (Fig. 49)
illuminates a lens _L_ which throws an image _b_ of the luminous source.
If the eye be so placed that the image falls on the pupil, the entire
lens, if perfect, will appear equally illuminated, for the reason that
all points of it send out rays to the eye. Coarse imperfections of form
or of homogeneity are rendered visible only in case the aberrations are
so large that the light from many spots passes by the pupil of the eye.
But if the image _b_ be partly intercepted by the edge of a small slide,
then those spots in the lens as thus partly darkened will appear
brighter whose light by its greater aberrations still reaches the eye in
spite of the intercepting slide, while those spots will appear darker
which in consequence of aberration in the other direction throw their
light entirely upon the slide. This artifice of the intercepting slide
which had previously been employed by Foucault for the investigation of
the optical imperfections of mirrors enhances enormously the delicacy of
the method, which is still further augmented by Toepler's employment of
a telescope behind the slide. Toepler's method, accordingly, enjoys all
the advantages of the Huygens and the Foucault procedure combined. It is
so delicate that the minutest irregularities in the air surrounding the
lens can be rendered distinctly visible, as I shall show by an example.
I place a candle before the lens _L_ (Fig. 50) and so arrange a second
lens _M_ that the flame of the candle is imaged upon the screen _S_. As
soon as the intercepting slide is pushed into the focus, _b_, of the
light issuing from _a_, you see the images of the changes of density and
the images of the movements induced in the air by the flame quite
distinctly upon the screen. The distinctness of the phenomenon as a
whole depends upon the position of the intercepting slide _b_. The
removal of _b_ increases the illumination but decreases the
distinctness. If the luminous source _a_ be removed, we see the image of
the candle flame only upon the screen _S_. If we remove the flame and
allow _a_ to continue shining, the screen _S_ will appear uniformly
illuminated.

[Illustration: Fig. 50.]

After Toepler had sought long and in vain to render the irregularities
produced in air by sound-waves visible by this principle, he was at last
conducted to his goal by the favorable circumstances attending the
production of electric sparks. The waves generated in the air by
electric sparks and accompanying the explosive snapping of the same, are
of sufficiently short period and sufficiently powerful to be rendered
visible by these methods. Thus we see how by a careful regard for the
merest and most shadowy indications of a phenomenon and by slight
progressive and appropriate alterations of the circumstances and the
methods, ultimately the most astounding results can be attained.
Consider, for example, two such phenomena as the rubbing of amber and
the electric lighting of modern streets. A person ignorant of the myriad
minute links that join these two things together, will be absolutely
nonplussed at their connexion, and will comprehend it no more than the
ordinary observer who is unacquainted with embryology, anatomy, and
paleontology will understand the connexion between a saurian and a bird.
The high value and significance of the co-operation of inquirers through
centuries, where each has but to take up the thread of work of his
predecessors and spin it onwards, is rendered forcibly evident by such
examples. And such knowledge destroys, too, in the clearest manner
imaginable that impression of the marvellous which the spectator may
receive from science, and at the same time is a most salutary
admonishment to the worker in science against superciliousness. I have
also to add the sobering remark that all our art would be in vain did
not nature herself afford at least some slight guiding threads leading
from a hidden phenomenon into the domain of the observable. And so it
need not surprise us that once under particularly favorable
circumstances an extremely powerful sound-wave which had been caused by
the explosion of several hundred pounds of dynamite threw a directly
visible shadow in the sunlight, as Boys has recently told us. If the
sound-waves were absolutely without influence upon the light, this could
not have occurred, and all our artifices would then, too, be in vain.
And so, similarly, the phenomenon accompanying projectiles which I am
about to show you was once in a very imperfect manner incidentally seen
by a French artillerist, Journée, while that observer was simply
following the line of flight of a projectile with a telescope, just as
also the undulations produced by candle flames are in a weak degree
directly visible and in the bright sunlight are imaged in shadowy waves
upon a uniform white background.

_Instantaneous illumination_ by the electric spark, the method of
rendering visible small optical differences or striæ, which may hence be
called the _striate_, or _differential_, method,[112] invented by
Foucault and Toepler, and finally the _recording_ of the image by a
_photographic_ plate,--these therefore are the chief means which are to
lead us to our goal.

I instituted my first experiments in the summer of 1884 with a
target-pistol, shooting the bullet through a striate field as described
above, and taking care that the projectile whilst in the field should
disengage an illuminating electric spark from a Leyden jar or Franklin's
pane, which spark produced a photographic impression of the projectile
upon a plate, especially arranged for the purpose. I obtained the image
of the projectile at once and without difficulty. I also readily
obtained, with the still rather defective dry plate which I was using,
exceedingly delicate images of the sound-waves (spark-waves). But no
atmospheric condensation produced by the projectile was visible. I now
determined the velocity of my projectile and found it to be only 240
metres per second, or considerably less than the velocity of sound
(which is 340 metres per second). I saw immediately that under such
circumstances no noticeable compression of the air could be produced,
for any atmospheric compression must of necessity travel forward at the
same speed with sound (340 metres per second) and consequently would be
always ahead of and speeding away from the projectile.

I was so thoroughly convinced, however, of the existence of the supposed
phenomenon at a velocity exceeding 340 metres per second, that I
requested Professor Salcher, of Fiume, an Austrian port on the Gulf of
Quarnero, to undertake the experiment with projectiles travelling at a
high rate of speed. In the summer of 1886 Salcher in conjunction with
Professor Riegler conducted in a spacious and suitable apartment placed
at their disposal by the Directors of the Royal Imperial Naval Academy,
experiments of the kind indicated and conforming in method exactly to
those which I had instituted, with the precise results expected. The
phenomenon, in fact, accorded perfectly with the _a priori_ sketch of it
which I had drafted previously to the experiment. As the experimenting
was continued, new and unforeseen features made their appearance.

It would be unfair, of course, to expect from the very first experiments
faultless and highly distinct photographs. It was sufficient that
success was secured and that I had convinced myself that further labor
and expenditure would not be vain. And on this score I am greatly
indebted to the two gentlemen above mentioned.

The Austrian Naval Department subsequently placed a cannon at Salcher's
disposal in Pola, an Adriatic seaport, and I myself, together with my
son, then a student of medicine, having received and accepted a
courteous invitation from Krupp, repaired to Meppen, a town in Hanover,
where we conducted with only the necessary apparatus several experiments
on the open artillery range. All these experiments furnished tolerably
good and complete pictures. Some little progress, too, was made. The
outcome of our experience on both artillery ranges, however, was the
settled conviction that really good results could be obtained only by
the most careful conduct of the experiments in a laboratory especially
adapted to the purpose. The expensiveness of the experiments on a large
scale was not the determining consideration here, for the size of the
projectile is indifferent. Given the same velocity and the results are
quite similar, whether the projectiles are large or small. On the other
hand, in a laboratory the experimenter has perfect control over the
initial velocity, which, provided the proper equipment is at hand, can
be altered at will simply by altering the charge and the weight of the
projectile. The requisite experiments were accordingly conducted by me
in my laboratory at Prague, partly in conjunction with my son and partly
afterwards by him alone. The latter are the most perfect and I shall
accordingly speak in detail here of these only.

[Illustration: Fig. 51.]

Picture to yourself an apparatus for detecting optical striæ set up in a
dark room. In order not to make the description too complicated, I shall
give the essential features only of the apparatus, leaving out of
account altogether the minuter details which are rather of consequence
for the technical performance of the experiment than for its
understanding. We suppose the projectile speeding on its path,
accordingly, through the field of our differential optical apparatus.
On reaching the centre of the field (Fig. 51) the projectile disengages
an illuminating electric spark _a_, and the image of the projectile, so
produced, is photographically impressed upon the plate of the camera
behind the intercepting slide _b_. In the last and best experiments the
lens _L_ was replaced by a spherical silvered-glass mirror made by K.
Fritsch (formerly Prokesch) of Vienna, whereby the apparatus was
naturally more complicated than it appears in our diagram. The
projectile having been carefully aimed passes in crossing the
differential field between two vertical isolated wires which are
connected with the two coatings of a Leyden jar, and completely filling
the space between the wires discharges the jar. In the axis of the
differential apparatus the circuit has a second gap _a_ which furnishes
the illuminating spark, the image of which falls on the intercepting
slide _b_. The wires in the differential field having occasioned
manifold disturbances were subsequently done away with. In the new
arrangement the projectile passes through a ring (see dotted line, Fig.
51), to the air in which it imparts a sharp impulse which travels
forward in the tube _r_ as a sound-wave having the approximate velocity
of 340 metres per second, topples over through the aperture of an
electric screen the flame of a candle situated at the other opening of
the tube, and so discharges the jar. The length of the tube _r_ is so
adjusted that the discharge occurs the moment the projectile enters the
centre of the now fully clear and free field of vision. We will also
leave out of account the fact that to secure fully the success of the
experiment, a large jar is first discharged by the flame, and that by
the agency of this first discharge the discharge of a second small jar
having a spark of very short period which furnishes the spark really
illuminating the projectile is effected. Sparks from large jars have an
appreciable duration, and owing to the great velocity of the projectiles
furnish blurred photographs only. By carefully husbanding the light of
the differential apparatus, and owing to the fact that much more light
reaches the photographic plate in this way than would otherwise reach
it, we can obtain beautiful, strong, and sharp photographs with
incredibly small sparks. The contours of the pictures appear as very
delicate and very sharp, closely adjacent double lines. From their
distance from one another, and from the velocity of the projectile, the
duration of the illumination, or of the spark, is found to be 1/800000
of a second. It is evident, therefore, that experiments with mechanical
snap slides can furnish no results worthy of the name.

[Illustration: Fig. 52.]

Let us consider now first the picture of a projectile in the rough, as
represented in Figure 52, and then let us examine it in its photographic
form as seen in Figure 53. The latter picture is of a shot from an
Austrian Mannlicher rifle. If I were not to tell you what the picture
represented you would very likely imagine it to be a bird's eye view of
a boat _b_ moving swiftly through the water. In front you see the
bow-wave and behind the body a phenomenon _k_ which closely resembles
the eddies formed in the wake of a ship. And as a matter of fact the
dark hyperboloid arc which streams from the tip of the projectile really
is a compressed wave of air exactly analogous to the bow-wave produced
by a ship moving through the water, with the exception that the wave of
air is not a surface-wave. The air-wave is produced in atmospheric space
and encompasses the projectile in the form of a shell on all sides. The
wave is visible for the same reason that the heated shell of air
surrounding the candle flame of our former experiments is visible. And
the cylinder of friction-heated air which the projectile throws off in
the form of vortex rings really does answer to the water in the wake of
a vessel.

[Illustration: Fig. 53. Photograph of a blunted projectile.]

Now just as a slowly moving boat produces no bow-wave, but the bow-wave
is seen only when the boat moves with a speed which is greater than the
velocity of propagation of surface-waves in water, so, in like manner,
no wave of compression is visible in front of a projectile so long as
the speed of the projectile is less than the velocity of sound. But if
the speed of the projectile reaches and exceeds the velocity of sound,
then the head-wave, as we shall call it, augments noticeably in power,
and is more and more extended, that is, the angle made by the contours
of the wave with the direction of flight is more and more diminished,
just as when the speed of a boat is increased a similar phenomenon is
noticed in connexion with the bow-wave. In fact, we can from an
instantaneous photograph so taken approximately estimate the speed with
which the projectile is travelling.

The explanation of the bow-wave of a ship and that of the head-wave of a
body travelling in atmospheric space both repose upon the same
principle, long ago employed by Huygens. Conceive a number of pebbles to
be cast into a pond of water at regular intervals in such wise that all
the spots struck are situate in the same straight line, and that every
spot subsequently struck lies a short space farther to the right. The
spots first struck will furnish then the wave-circles which are widest,
and all of them together will, at the points where they are thickest,
form a sort of cornucopia closely resembling the bow-wave. (Fig. 54.)
The resemblance is greater the smaller the pebbles are, and the more
quickly they succeed each other. If a rod be dipped into the water and
quickly carried along its surface, the falling of the pebbles will then
take place, so to speak, uninterruptedly, and we shall have a real
bow-wave. If we put the compressed air-wave in the place of the
surface-waves of the water, we shall have the head-wave of the
projectile.

[Illustration: Fig. 54.]

You may be disposed to say now, it is all very pretty and interesting to
observe a projectile in its flight, but of what practical use is it?

It is true, I reply, one cannot _wage war_ with photographed
projectiles. And I have likewise often had to say to medical students
attending my lectures on physics, when they inquired for the practical
value of some physical observation, "You cannot, gentlemen, cure
diseases with it." I had also once to give my opinion regarding how much
physics should be taught at a school for millers, supposing the
instruction there to be confined _exactly_ to what was necessary for a
miller. I was obliged to reply: "A miller always _needs_ exactly as much
physics as he _knows_." Knowledge which one does not possess one cannot
use.

Let us forego entirely the consideration that as a general thing every
scientific advance, every new problem elucidated, every extension or
enrichment of our knowledge of facts, affords a better foundation for
practical pursuits. Let us rather put the special question, Is it not
possible to derive some really practical knowledge from our theoretical
acquaintance with the phenomena which take place in the space
surrounding a projectile?

No physicist who has ever studied waves of sound or photographed them
will have the least doubt regarding the sound-wave character of the
atmospheric condensation encompassing the head of a flying projectile.
We have therefore, without ado, called this condensation the head-wave.

Knowing this, it follows that the view of Melsens according to which the
projectile carries along with it masses of air which it forces into the
bodies struck, is untenable. A forward-moving sound-wave is not a
forward-moving mass of matter but a forward-moving form of motion, just
as a water-wave or the waves of a field of wheat are only forward-moving
forms of motion and not movements of masses of water or masses of wheat.

By interference-experiments, on which I cannot touch here but which will
be found roughly represented in Figure 55, it was found that the
bell-shaped head-wave in question is an extremely thin shell and that
the condensations of the same are quite moderate, scarcely exceeding
two-tenths of an atmosphere. There can be no question, therefore, of
explosive effects in the body struck by the projectile through so slight
a degree of atmospheric compression. The phenomena attending wounds from
rifle balls, for example, are not to be explained as Melsens and Busch
explain them, but are due, as Kocher and Reger maintain, to the effects
of the impact of the projectile itself.

[Illustration: Fig. 55.]

A simple experiment will show how insignificant is the part played by
the friction of the air, or the supposed conveyance of the air along
with the moving projectile. If the photograph of the projectile be
taken while passing through a flame, i. e., a visible gas, the flame
will be seen to be, not torn and deformed, but smoothly and cleanly
perforated, like any solid body. Within and around the flame the
contours of the head-wave will be seen. The flickering, the extinction
of the flame, etc., take place only after the projectile has travelled
on a considerable distance in its path, and is then affected by the
powder gases which hurry after the bullet or by the air preceding the
powder-gases.

The physicist who examines the head-wave and recognises its sound-wave
character also sees that the wave in question is of the same kind with
the short sharp waves produced by electric sparks, that it is a
_noise_-wave. Hence, whenever any portion of the head-wave strikes the
ear it will be heard as a report. Appearances point to the conclusion
that the projectile carries this report along with it. In addition to
this report, which advances with the velocity of the projectile and so
usually travels at a speed greater than the velocity of sound, there is
also to be heard the report of the exploding powder which travels
forward with the ordinary velocity of sound. Hence two explosions will
be heard, each distinct in time. The circumstance that this fact was
long misconstrued by practical observers but when actually noticed
frequently received grotesque explanations and that ultimately my view
was accepted as the correct one, appears to me in itself a sufficient
justification that researches such as we are here speaking of are not
utterly superfluous even in practical directions. That the flashes and
sounds of discharging artillery are used for estimating the distances of
batteries is well known, and it stands to reason that any unclear
theoretical conception of the facts here involved will seriously affect
the correctness of practical calculations.

It may appear astonishing to a person hearing it for the first time,
that a single shot has a double report due to two different velocities
of propagation. But the reflexion that projectiles whose velocity is
less than the velocity of sound produce no head-waves (because every
impulse imparted to the air travels forward, that is, ahead, with
exactly the velocity of sound), throws full light when logically
developed upon the peculiar circumstance above mentioned. If the
projectile moves faster than sound, the air ahead of it cannot recede
from it quickly enough. The air is condensed and warmed, and thereupon,
as all know, the velocity of sound is augmented until the head-wave
travels forward as rapidly as the projectile itself, so that there is no
need whatever of any additional augmentation of the velocity of
propagation. If such a wave were left entirely to itself, it would
increase in length and soon pass into an ordinary sound-wave, travelling
with less velocity. But the projectile is always behind it and so
maintains it at its proper density and velocity. Even if the projectile
penetrates a piece of cardboard or a board of wood, which catches and
obstructs the head-wave, there will, as Figure 56 shows, immediately
appear at the emerging apex a newly formed, not to say newly born,
head-wave. We may observe on the cardboard the reflexion and diffraction
of the head-wave, and by means of a flame its refraction, so that no
doubt as to its nature can remain.

[Illustration: Fig. 56.]

Permit me, now, to illustrate the most essential of the points that I
have just adduced, by means of a few rough drawings taken from older and
less perfect photographs.

In the sketch of Figure 57 you see the projectile, which has just left
the barrel of the rifle, touch a wire and disengage the illuminating
spark. At the apex of the projectile you already see the beginnings of
a powerful head-wave, and in front of the wave a transparent fungiform
cluster. This latter is the air which has been forced out of the barrel
by the projectile. Circular sound-waves, noise-waves, which are soon
overtaken by the projectile, also issue from the barrel. But behind the
projectile opaque puffs of powder-gas rush forth. It is scarcely
necessary to add that many other questions in ballistics may be studied
by this method, as, for example, the movement of the gun-carriage.

[Illustration: Fig. 57.]

A distinguished French artillerist, M. Gossot, has applied the views of
the head-wave here given in quite a different manner. The practice in
measuring the velocity of projectiles is to cause the projectile to pass
through wire screens placed at different points in its path, and by the
tearing of these screens to give rise to electro-magnetic time-signals
on falling slabs or rotating drums. Gossot caused these signals to be
made directly by the impact of the head-wave, did away thus with the
wire screens, and carried the method so far as to be able to measure the
velocities of projectiles travelling in high altitudes, where the use of
wire screens was quite out of the question.

The laws of the resistance of fluids and of air to bodies travelling in
them form an extremely complicated problem, which can be reasoned out
very simply and prettily as a matter of pure philosophy but practice
offers not a few difficulties. The same body having the velocity 2, 3, 4
... displaces in the same interval 2, 3, 4 ... times the same mass of
air, or the same mass of fluid, and imparts to it _in addition_ 2, 3, 4
... times the same velocity. But for this, plainly, 4, 9, 16 ... times
the original force is required. Hence, the resistance, it is said,
increases with the square of the velocity. This is all very pretty and
simple and obvious. But practice and theory are at daggers' points here.
Practice tells us that when we increase the velocity, the law of the
resistance changes. For every portion of the velocity the law is
different.

The studies of the talented English naval architect, Froude, have thrown
light upon this question. Froude has shown that the resistance is
conditioned by a combination of the most multifarious phenomena. A ship
in motion is subjected to the friction of the water. It causes eddies
and it generates in addition waves which radiate outward from it. Every
one of these phenomena are dependent upon the velocity in some different
manner, and it is consequently not astonishing that the law of the
resistance should be a complicated one.

The preceding observations suggest quite analogous reflexions for
projectiles. Here also we have friction, the formation of eddies, and
the generation of waves. Here, also, therefore, we should not be
surprised at finding the law of the resistance of the air a complicated
one, nor puzzled at learning that in actuality the law of resistance
changes as soon as the speed of the projectile exceeds the velocity of
sound, for this is the precise point at which one important element of
the resistance, namely, the formation of waves, first comes into play.

No one doubts that a pointed bullet pierces the air with less resistance
than a blunt bullet. The photographs themselves show that the head-wave
is weaker for a pointed projectile. It is not impossible, similarly,
that forms of bullets will be invented which generate fewer eddies,
etc., and that we shall study these phenomena also by photography. I am
of opinion from the few experiments which I have made in this direction
that not much more can be done by changing the form of the projectile
when the velocity is very great, but I have not gone into the question
thoroughly. Researches of the kind we are considering can certainly not
be detrimental to practical artillery, and it is no less certain that
experiments by artillerists on a large scale will be of undoubted
benefit to physics.

No one who has had the opportunity of studying modern guns and
projectiles in their marvellous perfection, their power and precision,
can help confessing that a high technical and scientific achievement has
found its incarnation in these objects. We may surrender ourselves so
completely to this impression as to forget for a moment the terrible
purposes they serve.

Permit me, therefore, before we separate, to say a few words on this
glaring contrast. The greatest man of war and of silence which the
present age has produced once asserted that perpetual peace is a dream,
and not a beautiful dream at that. We may accord to this profound
student of mankind a judgment in these matters and can also appreciate
the soldier's horror of stagnation from all too lengthy peace. But it
requires a strong belief in the insuperableness of mediæval barbarism to
hope for and to expect no great improvement in international relations.
Think of our forefathers and of the times when club law ruled supreme,
when within the same country and the same state brutal assaults and
equally brutal self-defence were universal and self-evident. This state
of affairs grew so oppressive that finally a thousand and one
circumstances compelled people to put an end to it, and the cannon had
most to say in accomplishing the work. Yet the rule of club law was not
abolished so quickly after all. It had simply passed to other clubs. We
must not abandon ourselves to dreams of the Rousseau type. Questions of
law will in a sense forever remain questions of might. Even in the
United States where every one is as a matter of principle entitled to
the same privileges, the ballot according to Stallo's pertinent remark
is but a milder substitute for the club. Nor need I tell you that many
of our own fellow-citizens are still enamored of the old original
methods. Very, very gradually, however, as civilisation progresses, the
intercourse of men takes on gentler forms, and no one who really knows
the good old times will ever honestly wish them back again, however
beautifully they may be painted and rhymed about.

In the intercourse of the nations, however, the old club law still
reigns supreme. But since its rule is taxing the intellectual, the
moral, and the material resources of the nations to the utmost and
constitutes scarcely less a burden in peace than in war, scarcely less a
yoke for the victor than for the vanquished, it must necessarily grow
more and more unendurable. Reason, fortunately, is no longer the
exclusive possession of those who modestly call themselves the upper ten
thousand. Here, as everywhere, the evil itself will awaken the
intellectual and ethical forces which are destined to mitigate it. Let
the hate of races and of nationalities run riot as it may, the
intercourse of nations will still increase and grow more intimate. By
the side of the problems which separate nations, the great and common
ideals which claim the exclusive powers of the men of the future appear
one after another in greater distinctness and in greater might.

  FOOTNOTES:

  [Footnote 110: A lecture delivered on Nov. 10, 1897.]

  [Footnote 111: Christiansen, _Wiedemann's Annalen_, XXIII. S. 298,
  XXIV., p. 439 (1884-1885).]

  [Footnote 112: The German phrase is _Schlierenmethode_, by which
  term the method is known even by American physicists. It is also
  called in English the "shadow-method." But a term is necessary which
  will cover all the derivatives, and so we have employed
  alternatively the words _striate_ and _differential_. The etymology
  of _schlieren_, it would seem, is uncertain. Its present use is
  derived from its technological signification in glass-manufacturing,
  where by _die Schlieren_ are meant the wavy streaks and
  imperfections in glass. Hence its application to the method for
  detecting small optical _differences_ and faults generally.
  Professor Crew of Evanston suggests to the translator that
  _schlieren_ may be related to our _slur_ (L. G., _slüren_, to trail,
  to draggle), a conjecture which is doubtless correct and agrees both
  with the meaning of _schlieren_ as given in the large German
  dictionaries and with the intransitive use of our own verb _slur_,
  the faults in question being conceived as "trailings," "streakings,"
  etc.--_Trans._]



ON INSTRUCTION IN THE CLASSICS AND THE SCIENCES.[113]


Perhaps the most fantastic proposition that Maupertuis,[114] the
renowned president of the Berlin Academy, ever put forward for the
approval of his contemporaries was that of founding a city in which, to
instruct and discipline young students, only Latin should be spoken.
Maupertuis's Latin city remained an idle wish. But for centuries Latin
and Greek _institutions_ exist in which our children spend a goodly
portion of their days, and whose atmosphere constantly surrounds them,
even when without their walls.

For centuries instruction in the ancient languages has been zealously
cultivated. For centuries its necessity has been alternately championed
and contested. More strongly than ever are authoritative voices now
raised against the preponderance of instruction in the classics and in
favor of an education more suited to the needs of the time, especially
for a more generous treatment of mathematics and the natural sciences.

In accepting your invitation to speak here on the relative educational
value of the classical and the mathematico-physical sciences in colleges
and high schools, I find my justification in the duty and the necessity
laid upon every teacher of forming from his own experiences an opinion
upon this important question, as partly also in the special circumstance
that in my youth I was personally under the influence of school-life for
only a short time, just previous to my entering the university, and had,
therefore, ample opportunity to observe the effects of widely different
methods upon my own person.

Passing now, to a review of the arguments which the advocates of
instruction in the classics advance, and of what the adherents of
instruction in the physical sciences in their turn adduce, we find
ourselves in rather a perplexing position with respect to the arguments
of the first named. For these have been different at different times,
and they are even now of a very multifarious character, as must be where
men advance, in favor of an institution that exists and which they are
determined to retain at any cost, everything they can possibly think of.
We shall find here much that has evidently been brought forward only to
impress the minds of the ignorant; much, too, that was advanced in good
faith and which is not wholly without foundation. We shall get a fair
idea of the reasoning employed by considering, first, the arguments that
have grown out of the historical circumstances connected with the
original introduction of the classics, and, lastly, those which were
subsequently adduced as accidental afterthoughts.

       *       *       *       *       *

Instruction in Latin, as Paulsen[115] has minutely shown, was introduced
by the Roman Church along with Christianity. With the Latin language
were also transmitted the scant and meagre remnants of ancient science.
Whoever wished to acquire this ancient education, then the only one
worthy of the name, for him the Latin language was the only and
indispensable means; such a person had to learn Latin to rank among
educated people.

The wide-spread influence of the Roman Church wrought many and various
results. Among those for which all are glad, we may safely count the
establishment of a sort of _uniformity_ among the nations and of a
regular international intercourse by means of the Latin language, which
did much to unite the nations in the common work of civilisation,
carried on from the fifteenth to the eighteenth century. The Latin
language was thus long the language of scholars, and instruction in
Latin the road to a liberal education--a shibboleth still employed,
though long inappropriate.

For scholars as a class, it is to be regretted, perhaps, that Latin has
ceased to be the medium of international communication. But the
attributing of the loss of this function by the Latin language to its
incapacity to accommodate itself to the numerous new ideas and
conceptions which have arisen in the course of the development of
science is, in my opinion, wholly erroneous. It would be difficult to
find a modern scientist who had enriched science with as many new ideas
as Newton has, yet Newton knew how to express those ideas very correctly
and precisely in the Latin language. If this view were correct, it would
also hold true of every living language. Originally every language has
to adapt itself to new ideas.

It is far more likely that Latin was displaced as the literary vehicle
of science by the influence of the nobility. By their desire to enjoy
the fruits of literature and science, through a less irksome medium
than Latin, the nobility performed for the people at large an
undeniable service. For the days were now past when acquaintance with
the language and literature of science was restricted to a caste, and in
this step, perhaps, was made the most important advance of modern times.
To-day, when international intercourse is firmly established in spite of
the many languages employed, no one would think of reintroducing
Latin.[116]

The facility with which the ancient languages lend themselves to the
expression of new ideas is evidenced by the fact that the great majority
of our scientific ideas, as survivals of this period of Latin
intercourse, bear Latin and Greek designations, while in great measure
scientific ideas are even now invested with names from these sources.
But to deduce from the existence and use of such terms the necessity of
still learning Latin and Greek on the part of all who employ them is
carrying the conclusion too far. All terms, appropriate and
inappropriate,--and there are a large number of inappropriate and
monstrous combinations in science,--rest on convention. The essential
thing is, that people should associate with the sign the precise idea
that is designated by it. It matters little whether a person can
correctly derive the words _telegraph_, _tangent_, _ellipse_, _evolute_,
etc., if the correct idea is present in his mind when he uses them. On
the other hand, no matter how well he may know their etymology, his
knowledge will be of little use to him if the correct idea is absent.
Ask the average and fairly educated classical scholar to translate a few
lines for you from Newton's _Principia_, or from Huygens's _Horologium_,
and you will discover at once what an extremely subordinate rôle the
mere knowledge of language plays in such things. Without its associated
thought a word remains a mere sound. The fashion of employing Greek and
Latin designations--for it can be termed nothing else--has a natural
root in history; it is impossible for the practice to disappear
suddenly, but it has fallen of late considerably into disuse. The terms
_gas_, _ohm_, _Ampère_, _volt_, etc., are in international use, but they
are not Latin nor Greek. Only the person who rates the unessential and
accidental husk higher than its contents, can speak of the necessity of
learning Latin or Greek for such reasons, to say nothing of spending
eight or ten years on the task. Will not a dictionary supply in a few
seconds all the information we wish on such subjects?[117]

It is indisputable that our modern civilisation took up the threads of
the ancient civilisation, that at many points it begins where the latter
left off, and that centuries ago the remains of the ancient culture were
the only culture existing in Europe. Then, of course, a classical
education really was the liberal education, the higher education, the
ideal education, for it was the _sole_ education. But when the same
claim is now raised in behalf of a classical education, it must be
uncompromisingly contested as bereft of all foundation. For our
civilisation has gradually attained its independence; it has lifted
itself far above the ancient civilisation, and has entered generally new
directions of progress. Its note, its characteristic feature, is the
enlightenment that has come from the great mathematical and physical
researches of the last centuries, and which has permeated not only the
practical arts and industries but is also gradually finding its way into
all fields of thought, including philosophy and history, sociology and
linguistics. Those traces of ancient views that are still discoverable
in philosophy, law, art, and science, operate more as hindrances than
helps, and will not long stand before the development of independent and
more natural views.

It ill becomes classical scholars, therefore, to regard themselves, at
this day, as the educated class _par excellence_, to condemn as
uneducated all persons who do not understand Latin and Greek, to
complain that with such people profitable conversations are not to be
carried on, etc. The most delectable stories have got into circulation,
illustrative of the defective education of scientists and engineers. A
renowned inquirer, for example, is said to have once announced his
intention of holding a free course of university lectures, with the word
"frustra"; an engineer who spent his leisure hours in collecting insects
is said to have declared that he was studying "etymology." It is true,
incidents of this character make us shudder or smile, according to our
mood or temperament. But we must admit, the next moment, that in giving
way to such feelings we have merely succumbed to a childish prejudice. A
lack of tact but certainly no lack of education is displayed in the use
of such half-understood expressions. Every candid person will confess
that there are many branches of knowledge about which he had better be
silent. We shall not be so uncharitable as to turn the tables and
discuss the impression that classical scholars might make on a scientist
or engineer, in speaking of science. Possibly many ludicrous stories
might be told of them, and of far more serious import, which should
fully compensate for the blunders of the other party.

The mutual severity of judgment which we have here come upon, may also
forcibly bring home to us how really scarce a true liberal culture is.
We may detect in this mutual attitude, too, something of that narrow,
mediæval arrogance of caste, where a man began, according to the special
point of view of the speaker, with the scholar, the soldier, or the
nobleman. Little sense or appreciation is to be found in it for the
_common_ task of humanity, little feeling for the need of mutual
assistance in the great work of civilisation, little breadth of mind,
little truly liberal culture.

A knowledge of Latin, and partly, also, a knowledge of Greek, is still a
necessity for the members of a few professions by nature more or less
directly concerned with the civilisations of antiquity, as for lawyers,
theologians, philologists, historians, and generally for a small number
of persons, among whom from time to time I count myself, who are
compelled to seek for information in the Latin literature of the
centuries just past.[118] But that all young persons in search of a
higher education should pursue for this reason Latin and Greek to such
excess; that persons intending to become physicians and scientists
should come to the universities defectively educated, or even
miseducated; and that they should be compelled to come only from schools
that do _not_ supply them with the proper preparatory knowledge is going
a little bit too far.

       *       *       *       *       *

After the conditions which had given to the study of Latin and Greek
their high import had ceased to exist, the traditional curriculum,
naturally, was retained. Then, the different effects of this method of
education, good and bad, which no one had thought of at its
introduction, were realised and noted. As natural, too, was it that
those who had strong interests in the preservation of these studies,
from knowing no others or from living by them, or for still other
reasons, should emphasise the _good_ results of such instruction. They
pointed to the good effects as if they had been consciously aimed at by
the method and could be attained only through its agency.

One real benefit that students might derive from a rightly conducted
course in the classics would be the opening up of the rich literary
treasures of antiquity, and intimacy with the conceptions and views of
the world held by two advanced nations. A person who has read and
understood the Greek and Roman authors has felt and experienced more
than one who is restricted to the impressions of the present. He sees
how men placed in different circumstances judge quite differently of the
same things from what we do to-day. His own judgments will be rendered
thus more independent. Again, the Greek and Latin authors are
indisputably a rich fountain of recreation, of enlightenment, and of
intellectual pleasure after the day's toil, and the individual, not less
than civilised humanity generally, will remain grateful to them for all
time. Who does not recall with pleasure the wanderings of Ulysses, who
does not listen joyfully to the simple narratives of Herodotus, who
would ever repent of having made the acquaintance of Plato's Dialogues,
or of having tasted Lucian's divine humor? Who would give up the glances
he has obtained into the private life of antiquity from Cicero's
letters, from Plautus or Terence? To whom are not the portraits of
Suetonius undying reminiscences? Who, in fact, would throw away _any_
knowledge he had once gained?

Yet people who draw from these sources only, who know only this culture,
have surely no right to dogmatise about the value of some other culture.
As objects of research for individuals, this literature is extremely
valuable, but it is a different question whether it is equally valuable
as the almost exclusive means of education of our youth.

Do not other nations and other literatures exist from which we ought to
learn? Is not nature herself our first school-mistress? Are our highest
models always to be the Greeks, with their narrow provinciality of mind,
that divided the world into "Greeks and barbarians," with their
superstitions, with their eternal questioning of oracles? Aristotle with
his incapacity to learn from facts, with his word-science; Plato with
his heavy, interminable dialogues, with his barren, at times childish,
dialectics--are they unsurpassable?[119] The Romans with their apathy,
their pompous externality, set off by fulsome and bombastic phrases,
with their narrow-minded, philistine philosophy, with their frenzied
sensuality, with their cruel and bestial indulgence in animal and man
baiting, with their outrageous maltreatment and plundering of their
subjects--are they patterns worthy of imitation? Or shall, perhaps, our
science edify itself with the works of Pliny who cites midwives as
authorities and himself stands on their point of view?

Besides, if an acquaintance with the ancient world really were attained,
we might come to some settlement with the advocates of classical
education. But it is words and forms, and forms and words only, that are
supplied to our youth; and even collateral subjects are forced into the
strait-jacket of the same rigid method and made a science of words,
sheer feats of mechanical memory. Really, we feel ourselves set back a
thousand years into the dull cloister-cells of the Middle Ages.

This must be changed. It is possible to get acquainted with the views of
the Greeks and Romans by a shorter road than the intellect deadening
process of eight or ten years of declining, conjugating, analysing, and
extemporisation. There are to-day plenty of educated persons who have
acquired through good translations vivider, clearer, and more just views
of classical antiquity than the graduates of our gymnasiums and
colleges.[120]

For us moderns, the Greeks and the Romans are simply two objects of
archæological and historical research like all others. If we put them
before our youth in fresh and living pictures, and not merely in words
and syllables, the effect will be assured. We derive a totally different
enjoyment from the Greeks when we approach them after a study of the
results of modern research in the history of civilisation. We read many
a chapter of Herodotus differently when we attack his works equipped
with a knowledge of natural science, and with information about the
stone age and the lake-dwellers. What our classical institutions
_pretend_ to give can and actually will be given to our youth with much
more fruitful results by competent _historical_ instruction, which must
supply, not names and numbers alone, nor the mere history of dynasties
and wars, but be in every sense of the word a true history of
civilisation.

The view still widely prevails that although all "higher, ideal
culture," all extension of our view of the world, is acquired by
philological and in a lesser degree by historical studies, still the
mathematics and natural sciences should not be neglected on account of
their usefulness. This is an opinion to which I must refuse my assent.
It were strange if man could learn more, could draw more intellectual
nourishment, from the shards of a few old broken jugs, from inscribed
stones, or yellow parchments, than from all the rest of nature. True,
man is man's first concern, but he is not his sole concern.

In ceasing to regard man as the centre of the world; in discovering that
the earth is a top whirled about the sun, which speeds off with it into
infinite space; in finding that in the fixed stars the same elements
exist as on earth; in meeting everywhere the same processes of which the
life of man is merely a vanishingly small part--in such things, too, is
a widening of our view of the world, and edification, and poetry. There
are here perhaps grander and more significant facts than the bellowing
of the wounded Ares, or the charming island of Calypso, or the
ocean-stream engirdling the earth. He only should speak of the relative
value of these two domains of thought, of their poetry, who knows both.

The "utility" of physical science is, in a measure, only a _collateral_
product of that flight of the intellect which produced science. No one,
however, should underrate the utility of science who has shared in the
realisation by modern industrial art of the Oriental world of fables,
much less one upon whom those treasures have been poured, as it were,
from the fourth dimension, without his aid or understanding.

Nor may we believe that science is useful only to the practical man. Its
influence permeates all our affairs, our whole life; everywhere its
ideas are decisive. How differently does the jurist, the legislator, or
the political economist think, who knows, for example, that a square
mile of the most fertile soil can support with the solar heat annually
consumed only a definite number of human beings, which no art or science
can increase. Many economical theories, which open new air-paths of
progress, air-paths in the literal sense of the word, would be made
impossible by such knowledge.

       *       *       *       *       *

The eulogists of classical education love to emphasise the cultivation
of taste which comes from employment with the ancient models. I candidly
confess that there is something absolutely revolting in this to me. To
form the taste, then, our youths must sacrifice ten years of their life!
Luxury takes precedence over necessity. Have the future generations, in
the face of the difficult problems, the great social questions, which
they must meet, and that with strengthened mind and heart, no more
important duties to fulfil than these?

But let us assume that this end were desirable. Can taste be formed by
rules and precepts? Do not ideals of beauty change? Is it not a
stupendous absurdity to force one's self artificially to admire things
which, with all their historical interest, with all their beauty in
individual points, are for the most part foreign to the rest of our
thoughts and feelings, provided we have such of _our own_. A nation that
is truly such, has its own taste and will not go to others for it. And
every individual perfect man has his own taste.[121]

And what, after all, does this cultivation of taste consist in? In the
acquisition of the personal literary style of a few select authors! What
should we think of a people that would force its youth a thousand years
from now, by years of practice, to master the tortuous or bombastic
style of some successful lawyer or politician of to-day? Should we not
justly accuse them of a woful lack of taste?

The evil effects of this imagined cultivation of the taste find
expression often enough. The young _savant_ who regards the composition
of a scientific essay as a rhetorical exercise instead of a simple and
unadorned presentation of the facts and the truth, still sits
unconsciously on the school-bench, and still unwittingly represents the
point of view of the Romans, by whom the elaboration of speeches was
regarded as a serious scientific (!) employment.

       *       *       *       *       *

Far be it from me to underrate the value of the development of the
instinct of speech and of the increased comprehension of our own
language which comes from philological studies. By the study of a
foreign language, especially of one which differs widely from ours, the
signs and forms of words are first clearly distinguished from the
thoughts which they express. Words of the closest possible
correspondence in different languages never coincide absolutely with the
ideas they stand for, but place in relief slightly different aspects of
the same thing, and by the study of language the attention is directed
to these shades of difference. But it would be far from admissible to
contend that the study of Latin and Greek is the most fruitful and
natural, let alone the _only_, means of attaining this end. Any one who
will give himself the pleasure of a few hours' companionship with a
Chinese grammar; who will seek to make clear to himself the mode of
speech and thought of a people who never advanced as far as the analysis
of articulate sounds, but stopped at the analysis of syllables, to whom
our alphabetical characters, therefore, are an inexplicable puzzle, and
who express all their rich and profound thoughts by means of a few
syllables with variable emphasis and position,--such a person, perhaps,
will acquire new, and extremely elucidative ideas upon the relation of
language and thought. But should our children, therefore, study Chinese?
Certainly not. No more, then, should they be burdened with Latin, at
least in the measure they are.

It is a beautiful achievement to reproduce a Latin thought in a modern
language with the maximum fidelity of meaning and expression--for the
_translator_. Moreover, we shall be very grateful to the translator for
his performance. But to demand this feat of every educated man, without
consideration of the sacrifice of time and labor which it entails, is
unreasonable. And for this very reason, as classical teachers admit,
that ideal is never perfectly attained, except in rare cases with
scholars possessed of special talents and great industry. Without
slurring, therefore, the high importance of the study of the ancient
languages as a profession, we may yet feel sure that the instinct for
speech which is part of every liberal education can, and must, be
acquired in a different way. Should we, indeed, be forever lost if the
Greeks had not lived before us?

The fact is, we must carry our demands further than the representatives
of classical philology. We must ask of every educated man a fair
scientific conception of the nature and value of language, of the
formation of language, of the alteration of the meaning of roots, of the
degeneration of fixed forms of speech to grammatical forms, in brief, of
all the main results of modern comparative philology. We should judge
that this were attainable by a careful study of our mother tongue and of
the languages next allied to it, and subsequently of the more ancient
tongues from which the former are derived. If any one object that this
is too difficult and entails too much labor, I should advise such a
person to place side by side an English, a Dutch, a Danish, a Swedish,
and a German Bible, and to compare a few lines of them; he will be
amazed at the multitude of suggestions that offer themselves.[122] In
fact, I believe that a really progressive, fruitful, rational, and
instructive study of languages can be conducted only on this plan. Many
of my audience will remember, perhaps, the bright and encouraging
effect, like that of a ray of sunlight on a gloomy day, which the meagre
and furtive remarks on comparative philology in Curtius's Greek grammar
wrought in that barren and lifeless desert of verbal quibbles.

       *       *       *       *       *

The principal result obtained by the present method of studying the
ancient languages is that which comes from the student's employment with
their complicated grammars. It consists in the sharpening of the
attention and in the exercise of the judgment by the practice of
subsuming special cases under general rules, and of distinguishing
between different cases. Obviously, the same result can be reached by
many other methods; for example, by difficult games of cards. Every
science, the mathematics and the physical sciences included, accomplish
as much, if not more, in this disciplining of the judgment. In addition,
the matter treated by those sciences has a much higher intrinsic
interest for young people, and so engages spontaneously their attention;
while on the other hand they are elucidative and useful in other
directions in which grammar can accomplish nothing.

Who cares, so far as the matter of it is concerned, whether we say
_hominum_ or _hominorum_ in the genitive plural, interesting as the fact
may be for the philologist? And who would dispute that the intellectual
need of causal insight is awakened not by grammar but by the natural
sciences?

It is not our intention, therefore, to gainsay in the least the good
influence which the study of Latin and Greek grammar _also_ exercises on
the sharpening of the judgment. In so far as the study of words as such
must greatly promote lucidity and accuracy of expression, in so far as
Latin and Greek are not yet wholly indispensable to many branches of
knowledge, we willingly concede to them a place in our schools, but
would demand that the disproportionate amount of time allotted to them,
wrongly withdrawn from other useful studies, should be considerably
curtailed. That in the end Latin and Greek will not be employed as the
universal means of education, we are fully convinced. They will be
relegated to the closet of the scholar or professional philologist, and
gradually make way for the modern languages and the modern science of
language.

Long ago Locke reduced to their proper limits the exaggerated notions
which obtained of the close connexion of thought and speech, of logic
and grammar, and recent investigators have established on still surer
foundations his views. How little a complicated grammar is necessary for
expressing delicate shades of thought is demonstrated by the Italians
and French, who, although they have almost totally discarded the
grammatical redundancies of the Romans, are yet not surpassed by the
latter in accuracy of thought, and whose poetical, but especially whose
scientific literature, as no one will dispute, can bear favorable
comparison with the Roman.

Reviewing again the arguments advanced in favor of the study of the
ancient languages, we are obliged to say that in the main and as
applied to the present, they are wholly devoid of force. In so far as
the aims which this study theoretically pursues are still worthy of
attainment, they appear to us as altogether too narrow, and are
surpassed in this only by the means employed. As almost the sole,
indisputable result of this study we must count the increase of the
student's skill and precision in expression. One inclined to be
uncharitable might say that our gymnasiums and classical academies turn
out men who can speak and write, but, unfortunately, have little to
write or speak about. Of that broad, liberal view, of that famed
universal culture, which the classical curriculum is supposed to yield,
serious words need not be lost. This culture might, perhaps, more
properly be termed the contracted or lopsided culture.

       *       *       *       *       *

While considering the study of languages we threw a few side glances at
mathematics and the natural sciences. Let us now inquire whether these,
as branches of study, cannot accomplish much that is to be attained in
no other way. I shall meet with no contradiction when I say that without
at least an elementary mathematical and scientific education a man
remains a total stranger in the world in which he lives, a stranger in
the civilisation of the time that bears him. Whatever he meets in
nature, or in the industrial world, either does not appeal to him at
all, from his having neither eye nor ear for it, or it speaks to him in
a totally unintelligible language.

A real understanding of the world and its civilisation, however, is not
the only result of the study of mathematics and the physical sciences.
Much more essential for the preparatory school is the _formal_
cultivation which comes from these studies, the strengthening of the
reason and the judgment, the exercise of the imagination. Mathematics,
physics, chemistry, and the so-called descriptive sciences are so much
alike in this respect, that, apart from a few points, we need not
separate them in our discussion.

Logical sequence and continuity of ideas, so necessary for fruitful
thought, are _par excellence_ the results of mathematics; the ability to
follow facts with thoughts, that is, to observe or collect experiences,
is chiefly developed by the natural sciences. Whether we notice that the
sides and the angles of a triangle are connected in a definite way, that
an equilateral triangle possesses certain definite properties of
symmetry, or whether we notice the deflexion of a magnetic needle by an
electric current, the dissolution of zinc in diluted sulphuric acid,
whether we remark that the wings of a butterfly are slightly colored on
the under, and the fore-wings of the moth on the upper, surface:
indiscriminately here we proceed from _observations_, from individual
acts of immediate intuitive knowledge. The field of observation is more
restricted and lies closer at hand in mathematics; it is more varied and
broader but more difficult to compass in the natural sciences. The
essential thing, however, is for the student to learn to make
observations in all these fields. The philosophical question whether our
acts of knowledge in mathematics are of a special kind is here of no
importance for us. It is true, of course, that the observation can be
practised by languages also. But no one, surely, will deny, that the
concrete, living pictures presented in the fields just mentioned possess
different and more powerful attractions for the mind of the youth than
the abstract and hazy figures which language offers, and on which the
attention is certainly not so spontaneously bestowed, nor with such good
results.[123]

Observation having revealed the different properties of a given
geometrical or physical object, it is discovered that in many cases
these properties _depend_ in some way upon one another. This
interdependence of properties (say that of equal sides and equal angles
at the base of a triangle, the relation of pressure to motion,) is
nowhere so distinctly marked, nowhere is the necessity and permanency of
the interdependence so plainly noticeable, as in the fields mentioned.
Hence the continuity and logical consequence of the ideas which we
acquire in those fields. The relative simplicity and perspicuity of
geometrical and physical relations supply here the conditions of natural
and easy progress. Relations of equal simplicity are not met with in
the fields which the study of language opens up. Many of you, doubtless,
have often wondered at the little respect for the notions of cause and
effect and their connexion that is sometimes found among professed
representatives of the classical studies. The explanation is probably to
be sought in the fact that the analogous relation of motive and action
familiar to them from their studies, presents nothing like the clear
simplicity and determinateness that the relation of cause and effect
does.

That perfect mental grasp of all possible cases, that economical order
and organic union of the thoughts which comes from it, which has grown
for every one who has ever tasted it a permanent need which he seeks to
satisfy in every new province, can be developed only by employment with
the relative simplicity of mathematical and scientific investigations.

When a set of facts comes into apparent conflict with another set of
facts, and a problem is presented, its solution consists ordinarily in a
more refined distinction or in a more extended view of the facts, as may
be aptly illustrated by Newton's solution of the problem of dispersion.
When a new mathematical or scientific fact is _demonstrated_, or
_explained_, such demonstration also rests simply upon showing the
connexion of the new fact with the facts already known; for example,
that the radius of a circle can be laid off as chord exactly six times
in the circle is explained or proved by dividing the regular hexagon
inscribed in the circle into equilateral triangles. That the quantity of
heat developed in a second in a wire conveying an electric current is
quadrupled on the doubling of the strength of the current, we explain
from the doubling of the fall of the potential due to the doubling of
the current's intensity, as also from the doubling of the quantity
flowing through, in a word, from the quadrupling of the work done. In
point of principle, explanation and direct proof do not differ much.

He who solves scientifically a geometrical, physical, or technical
problem, easily remarks that his procedure is a _methodical_ mental
quest, rendered possible by the economical order of the province--a
simplified purposeful quest as contrasted with unmethodical,
unscientific guess-work. The geometer, for example, who has to construct
a circle touching two given straight lines, casts his eye over the
relations of symmetry of the desired construction, and seeks the centre
of his circle solely in the line of symmetry of the two straight lines.
The person who wants a triangle of which two angles and the sum of the
sides are given, grasps in his mind the determinateness of the form of
this triangle and restricts his search for it to a certain group of
triangles of the _same form_. Under very different circumstances,
therefore, the simplicity, the intellectual perviousness, of the
subject-matter of mathematics and natural science is felt, and promotes
both the discipline and the self-confidence of the reason.

Unquestionably, much more will be attained by instruction in the
mathematics and the natural sciences than now is, when more natural
methods are adopted. One point of importance here is that young students
should not be spoiled by premature abstraction, but should be made
acquainted with their material from living pictures of it before they
are made to work with it by purely ratiocinative methods. A good stock
of geometrical experience could be obtained, for example, from
geometrical drawing and from the practical construction of models. In
the place of the unfruitful method of Euclid, which is only fit for
special, restricted uses, a broader and more conscious method must be
adopted, as Hankel has pointed out.[124] Then, if, on reviewing
geometry, and after it presents no substantial difficulties, the more
general points of view, the principles of scientific method are placed
in relief and brought to consciousness, as Von Nagel,[125] J. K.
Becker,[126] Mann,[127] and others have well done, fruitful results will
be surely attained. In the same way, the subject-matter of the natural
sciences should be made familiar by pictures and experiment before a
profounder and reasoned grasp of these subjects is attempted. Here the
emphasis of the more general points of view is to be postponed.

Before my present audience it would be superfluous for me to contend
further that mathematics and natural science are justified constituents
of a sound education,--a claim that even philologists, after some
resistance, have conceded. Here I may count upon assent when I say that
mathematics and the natural sciences pursued alone as means of
instruction yield a richer education in matter and form, a more general
education, an education better adapted to the needs and spirit of the
time,--than the philological branches pursued alone would yield.

But how shall this idea be realised in the curricula of our intermediate
educational institutions? It is unquestionable in my mind that the
German _Realschulen_ and _Realgymnasien_, where the exclusive classical
course is for the most part replaced by mathematics, science, and modern
languages, give the _average_ man a more timely education than the
gymnasium proper, although they are not yet regarded as fit preparatory
schools for future theologians and professional philologists. The German
gymnasiums are too one-sided. With these the first changes are to be
made; of these alone we shall speak here. Possibly a _single_
preparatory school, suitably planned, might serve all purposes.

Shall we, then, in our gymnasiums fill out the hours of study which
stand at our disposal, or are still to be wrested from the classicists,
with as great and as varied a quantity of mathematical and scientific
matter as possible? Expect no such proposition from me. No one will
suggest such a course who has himself been actively engaged in
scientific thought. Thoughts can be awakened and fructified as a field
is fructified by sunshine and rain. But thoughts cannot be juggled out
and worried out by heaping up materials and the hours of instruction,
nor by any sort of precepts: they must grow naturally of their own free
accord. Furthermore, thoughts cannot be accumulated beyond a certain
limit in a single head, any more than the produce of a field can be
increased beyond certain limits.

I believe that the amount of matter necessary for a useful education,
such as should be offered to _all_ the pupils of a preparatory school,
is very small. If I had the requisite influence, I should, in all
composure, and fully convinced that I was doing what was best, first
greatly curtail in the lower classes the amount of matter in both the
classical and the scientific courses; I should cut down considerably the
number of the school hours and the work done outside the school. I am
not with many teachers of opinion that ten hours work a day for a child
is not too much. I am convinced that the mature men who offer this
advice so lightly are themselves unable to give their attention
successfully for as long a time to any subject that is new to them, (for
example, to elementary mathematics or physics,) and I would ask every
one who thinks the contrary to make the experiment upon himself.
Learning and teaching are not routine office-work that can be kept up
mechanically for long periods. But even such work tires in the end. If
our young men are not to enter the universities with blunted and
impoverished minds, if they are not to leave in the preparatory schools
their vital energy, which they should there gather, great changes must
be made. Waiving the injurious effects of overwork upon the body, the
consequences of it for the mind seem to me positively dreadful.

I know of nothing more terrible than the poor creatures who have learned
too much. Instead of that sound powerful judgment which would probably
have grown up if they had learned nothing, their thoughts creep timidly
and hypnotically after words, principles, and formulæ, constantly by the
same paths. What they have acquired is a spider's web of thoughts too
weak to furnish sure supports, but complicated enough to produce
confusion.

But how shall better methods of mathematical and scientific education be
combined with the decrease of the subject-matter of instruction? I
think, by abandoning systematic instruction altogether, at least in so
far as that is required of _all_ young pupils. I see no necessity
whatever that the graduates of our high schools and preparatory schools
should be little philologists, and at the same time little
mathematicians, physicists, and botanists; in fact, I do not see the
possibility of such a result. I see in the endeavor to attain this
result, in which every instructor seeks for his own branch a place apart
from the others, the main mistake of our whole system. I should be
satisfied if every young student could come into living contact with
and pursue to their ultimate logical consequences merely a _few_
mathematical or scientific discoveries. Such instruction would be mainly
and naturally associated with selections from the great scientific
classics. A few powerful and lucid ideas could thus be made to take root
in the mind and receive thorough elaboration. This accomplished, our
youth would make a different showing from what they do to-day.[128]

What need is there, for example, of burdening the head of a young
student with all the details of botany? The student who has botanised
under the guidance of a teacher finds on all hands, not indifferent
things, but known or unknown things, by which he is stimulated, and his
gain made permanent. I express here, not my own, but the opinion of a
friend, a practical teacher. Again, it is not at all necessary that all
the matter that is offered in the schools should be learned. The best
that we have learned, that which has remained with us for life, outlived
the test of examination. How can the mind thrive when matter is heaped
on matter, and new materials piled constantly on old, undigested
materials? The question here is not so much that of the accumulation of
positive knowledge as of intellectual discipline. It seems also
unnecessary that _all_ branches should be treated at school, and that
exactly the same studies should be pursued in all schools. A single
philological, a single historical, a single mathematical, a single
scientific branch, pursued as common subjects of instruction for all
pupils, are sufficient to accomplish all that is necessary for the
intellectual development. On the other hand, a wholesome mutual stimulus
would be produced by this greater variety in the positive culture of
men. Uniforms are excellent for soldiers, but they will not fit heads.
Charles V. learned this, and it should never be forgotten. On the
contrary, teachers and pupils both need considerable latitude, if they
are to yield good results.

With John Karl Becker I am of the opinion that the utility and amount
for individuals of every study should be precisely determined. All that
exceeds this amount should be unconditionally banished from the lower
classes. With respect to mathematics, Becker,[129] in my judgment, has
admirably solved this question.

With respect to the upper classes the demand assumes a different form.
Here also the amount of matter obligatory on all pupils ought not to
exceed a certain limit. But in the great mass of knowledge that a young
man must acquire to-day for his profession it is no longer just that ten
years of his youth should be wasted with mere preludes. The upper
classes should supply a truly useful preparation for the professions,
and should not be modelled upon the wants merely of future lawyers,
ministers, and philologists. Again, it would be both foolish and
impossible to attempt to prepare the same person properly for all the
different professions. In such case the function of the schools would
be, as Lichtenberg feared, simply to select the persons best fitted for
being drilled, whilst precisely the finest special talents, which do not
submit to indiscriminate discipline, would be excluded from the contest.
Hence, a certain amount of liberty in the choice of studies must be
introduced in the upper classes, by means of which it will be free for
every one who is clear about the choice of his profession to devote his
chief attention either to the study of the philologico-historical or to
that of the mathematico-scientific branches. Then the matter now treated
could be retained, and in some branches, perhaps, judiciously
extended,[130] without burdening the scholar with many branches or
increasing the number of the hours of study. With more homogeneous work
the student's capacity for work increases, one part of his labor
supporting the other instead of obstructing it. If, however, a young man
should subsequently choose a different profession, then it is _his_
business to make up what he has lost. No harm certainly will come to
society from this change, nor could it be regarded as a misfortune if
philologists and lawyers with mathematical educations or physical
scientists with classical educations should now and then appear.

       *       *       *       *       *

The view is now wide-spread that a Latin and Greek education no longer
meets the general wants of the times, that a more opportune, a more
"liberal" education exists. The phrase, "a liberal education," has been
greatly misused. A truly liberal education is unquestionably very rare.
The _schools_ can hardly offer such; at best they can only bring home to
the student the necessity of it. It is, then, his business to acquire,
as best he can, a more or less liberal education. It would be very
difficult, too, at any one time to give a definition of a "liberal"
education which would satisfy every one, still more difficult to give
one which would hold good for a hundred years. The educational ideal, in
fact, varies much. To one, a knowledge of classical antiquity appears
not too dearly bought "with early death." We have no objection to this
person, or to those who think like him, pursuing their ideal after their
own fashion. But we may certainly protest strongly against the
realisation of such ideals on our own children. Another,--Plato, for
example,--puts men ignorant of geometry on a level with animals.[131]
If such narrow views had the magical powers of the sorceress Circe, many
a man who perhaps justly thought himself well educated would become
conscious of a not very flattering transformation of himself. Let us
seek, therefore, in our educational system to meet the wants of the
present, and not establish prejudices for the future.

But how does it come, we must ask, that institutions so antiquated as
the German gymnasiums could subsist so long in opposition to public
opinion? The answer is simple. The schools were first organised by the
Church; since the Reformation they have been in the hands of the State.
On so large a scale, the plan presents many advantages. Means can be
placed at the disposal of education such as no private source, at least
in Europe, could furnish. Work can be conducted upon the same plan in
many schools, and so experiments made of extensive scope which would be
otherwise impossible. A single man with influence and ideas can under
such circumstances do great things for the promotion of education.

But the matter has also its reverse aspect. The party in power works for
its own interests, uses the schools for its special purposes.
Educational competition is excluded, for all successful attempts at
improvement are impossible unless undertaken or permitted by the State.
By the uniformity of the people's education, a prejudice once in vogue
is permanently established. The highest intelligences, the strongest
wills cannot overthrow it suddenly. In fact, as everything is adapted to
the view in question, a sudden change would be physically impossible.
The two classes which virtually hold the reins of power in the State,
the jurists and theologians, know only the one-sided, predominantly
classical culture which they have acquired in the State schools, and
would have this culture alone valued. Others accept this opinion from
credulity; others, underestimating their true worth for society, bow
before the power of the prevalent opinion; others, again, affect the
opinion of the ruling classes even against their better judgment, so as
to abide on the same plane of respect with the latter. I will make no
charges, but I must confess that the deportment of medical men with
respect to the question of the qualification of graduates of your
_Realschulen_ has frequently made that impression upon me. Let us
remember, finally, that an influential statesman, even within the
boundaries which the law and public opinion set him, can do serious harm
to the cause of education by considering his own one-sided views
infallible, and in enforcing them recklessly and inconsiderately--which
not only _can_ happen, but has, repeatedly, happened.[132] The monopoly
of education by the State[133] thus assumes in our eyes a somewhat
different aspect. And to revert to the question above asked, there is
not the slightest doubt that the German gymnasiums in their present
form would have ceased to exist long ago if the State had not supported
them.

All this must be changed. But the change will not be made of itself, nor
without our energetic interference, and it will be made slowly. But the
path is marked out for us, the will of the people must acquire and exert
upon our school legislation a greater and more powerful influence.
Furthermore, the questions at issue must be publicly and candidly
discussed that the views of the people may be clarified. All who feel
the insufficiency of the existing _régime_ must combine into a powerful
organisation that their views may acquire impressiveness and the
opinions of the individual not die away unheard.

I recently read, gentlemen, in an excellent book of travels, that the
Chinese speak with unwillingness of politics. Conversations of this sort
are usually cut short with the remark that they may bother about such
things whose business it is and who are paid for it. Now it seems to me
that it is not only the business of the State, but a very serious
concern of all of us, how our children shall be educated in the public
schools at _our_ cost.

  FOOTNOTES:

  [Footnote 113: An address delivered before the Congress of Delegates
  of the German Realschulmännerverein, at Dortmund, April 16, 1886.
  The full title of the address reads: "On the Relative Educational
  Value of the Classics and the Mathematico-Physical Sciences in
  Colleges and High Schools."

  Although substantially contained in an address which I was to have
  made at the meeting of Natural Scientists at Salzburg in 1881
  (deferred on account of the Paris Exposition), and in the
  Introduction to a course of lectures on "Physical Instruction in
  Preparatory Schools," which I delivered in 1883, the invitation of
  the German Realschulmännerverein afforded me the first opportunity
  of putting my views upon this subject before a large circle of
  readers. Owing to the place and circumstances of delivery, my
  remarks apply of course, primarily, only to German schools, but,
  with slight modifications, made in this translation, are not without
  force for the institutions of other countries. In giving here
  expression to a strong personal conviction formed long ago, it is a
  matter of deep satisfaction to me to find that they agree in many
  points with the views recently advanced in independent form by
  Paulsen (_Geschichte des gelehrten Unterrichts_, Leipsic, 1885) and
  Frary (_La question du latin_, Paris, Cerf, 1885). It is not my
  desire nor effort here to say much that is new, but merely to
  contribute my mite towards bringing about the inevitable revolution
  now preparing in the world of elementary instruction. In the opinion
  of experienced educationists the first result of that revolution
  will be to make Greek and mathematics alternately optional subjects
  in the higher classes of the German Gymnasium and in the
  corresponding institutions of other countries, as has been done in
  the splendid system of instruction in Denmark. The gap between the
  German classical Gymnasium and the German Realgymnasium, or between
  classical and scientific schools generally, can thus be bridged
  over, and the remaining inevitable transformations will then be
  accomplished in relative peace and quiet. (Prague, May, 1886.)]

  [Footnote 114: Maupertuis, _Oeuvres_, Dresden, 1752, p. 339.]

  [Footnote 115: F. Paulsen, _Geschichte des gelehrten Unterrichts_,
  Leipsic, 1885.]

  [Footnote 116: There is a peculiar irony of fate in the fact that
  while Leibnitz was casting about for a new vehicle of universal
  linguistic intercourse, the Latin language which still subserved
  this purpose the best of all, was dropping more and more out of use,
  and that Leibnitz himself contributed not the least to this result.]

  [Footnote 117: As a rule, the human brain is too much, and wrongly,
  burdened with things which might be more conveniently and accurately
  preserved in books where they could be found at a moment's notice.
  In a recent letter to me from Düsseldorf, Judge Hartwich writes:

  "A host of words exist which are out and out Latin or Greek, yet are
  employed with perfect correctness by people of good education who
  never had the good luck to be taught the ancient languages. For
  example, words like 'dynasty.' ... The child learns such words as
  parts of the common stock of speech, or even as parts of his
  mother-tongue, just as he does the words 'father,' 'mother,'
  'bread,' 'milk.' Does the ordinary mortal know the etymology of
  these Saxon words? Did it not require the almost incredible industry
  of the Grimms and other Teutonic philologists to throw the merest
  glimmerings of light upon the origin and growth of our own
  mother-tongue? Besides, do not thousands of people of so-called
  classical education use every moment hosts of words of foreign
  origin whose derivation they do not know? Very few of them think it
  worth while to look up such words in the dictionaries, although they
  love to maintain that people should study the ancient languages for
  the sake of etymology alone."]

  [Footnote 118: Standing remote from the legal profession I should
  not have ventured to declare that the study of Greek was not
  necessary for the jurists; yet this view was taken in the debate
  that followed this lecture by professional jurists of high standing.
  According to this opinion, the preparatory education obtained in the
  German Realgymnasium would also be sufficient for the future jurists
  and insufficient only for theologians and philologists. [In England
  and America not only is Greek not necessary, but the law-Latin is so
  peculiar that even persons of _good_ classical education cannot
  understand it.--_Tr._]]

  [Footnote 119: In emphasising here the weak sides of the writings of
  Plato and Aristotle, forced on my attention while reading them in
  German translations, I, of course, have no intention of underrating
  the great merits and the high historical importance of these two
  men. Their importance must not be measured by the fact that our
  speculative philosophy still moves to a great extent in their paths
  of thought. The more probable conclusion is that this branch has
  made very little progress in the last two thousand years. Natural
  science also was implicated for centuries in the meshes of the
  Aristotelian thought, and owes its rise mainly to having thrown off
  those fetters.]

  [Footnote 120: I would not for a moment contend that we derive
  exactly the same profit from reading a Greek author in a translation
  as from reading him in the original; but the difference, the excess
  of gain in the second case, appears to me, and probably will to most
  men who are not professional philologists, to be too dearly bought
  with the expenditure of eight years of valuable time.]

  [Footnote 121: "The temptation," Judge Hartwich writes, "to regard
  the 'taste' of the ancients as so lofty and unsurpassable appears to
  me to have its chief origin in the fact that the ancients were
  unexcelled in the representation of the nude. First, by their
  unremitting care of the human body they produced splendid models;
  and secondly, in their gymnasiums and in their athletic games they
  had these models constantly before their eyes. No wonder, then, that
  their statues still excite our admiration! For the form, the ideal
  of the human body has not changed in the course of the centuries.
  But with intellectual matters it is totally different; they change
  from century to century, nay, from decennium to decennium. It is
  very natural now, that people should unconsciously apply what is
  thus so easily seen, namely, the works of sculpture, as a universal
  criterion of the highly developed taste of the ancients--a fallacy
  against which people cannot, in my judgment, be too strongly
  warned."]

  [Footnote 122: English: "In the beginning God created the heaven and
  the earth. And the earth was without form and void; and darkness was
  upon the face of the deep. And the spirit of God moved upon the face
  of the waters."--Dutch: "In het begin schiep God den hemel en de
  aarde. De aarde nu was woest en ledig, en duisternis was op den
  afgrond; en de Geest Gods zwefde op de wateren."--Danish: "I
  Begyndelsen skabte Gud Himmelen og Jorden. Og Jorden var ode og tom,
  og der var morkt ovenover Afgrunden, og Guds Aand svoevede ovenover
  Vandene."--Swedish: "I begynnelsen skapade Gud Himmel och Jord. Och
  Jorden war öde och tom, och mörker war pä djupet, och Gods Ande
  swäfde öfwer wattnet."--German: "Am Anfang schuf Gott Himmel und
  Erde. Und die Erde war wüst und leer, und es war finster auf der
  Tiefe; und der Geist Gottes schwebte auf dem Wasser."]

  [Footnote 123: Compare Herzen's excellent remarks, _De
  l'enseignement secondaire dans la Suisse romande_, Lausanne, 1886.]

  [Footnote 124: _Geschichte der Mathematik_, Leipsic, 1874.]

  [Footnote 125: _Geometrische Analyse_, Ulm, 1886.]

  [Footnote 126: In his text-books of elementary mathematics]

  [Footnote 127: _Abhandlungen aus dem Gebiete der Mathematik_,
  Würzburg, 1883.]

  [Footnote 128: My idea here is an appropriate selection of readings
  from Galileo, Huygens, Newton, etc. The choice is so easily made
  that there can be no question of difficulties. The contents would be
  discussed with the students, and the original experiments performed
  with them. Those scholars alone should receive this instruction in
  the upper classes who did not look forward to systematical
  instruction in the physical sciences. I do not make this proposition
  of reform here for the first time. I have no doubt, moreover, that
  such radical changes will only be slowly introduced.]

  [Footnote 129: _Die Mathematik als Lehrgegenstand des Gymnasiums_,
  Berlin, 1883.]

  [Footnote 130: Wrong as it is to burden future physicians and
  scientists with Greek for the sake of the theologians and
  philologists, it would be just as wrong to compel theologians and
  philologists, on account of the physicians, to study such subjects
  as analytical geometry. Moreover, I cannot believe that ignorance of
  analytical geometry would be a serious hindrance to a physician that
  was otherwise well versed in quantitative thought. No special
  advantage generally is observable in the graduates of the Austrian
  gymnasiums, all of whom have studied analytical geometry. [Refers to
  an assertion of Dubois-Reymond.]]

  [Footnote 131: Compare M. Cantor, _Geschichte der Mathematik_,
  Leipsic, 1880, Vol. I. p. 193.]

  [Footnote 132: Compare Paulsen, _l. c._, pp. 607, 688.]

  [Footnote 133: It is to be hoped that the Americans will jealously
  guard their schools and universities against the influence of the
  State.]



APPENDIX.

I.

A CONTRIBUTION TO THE HISTORY OF ACOUSTICS.[134]


While searching for papers by Amontons, several volumes of the Memoirs
of the Paris Academy for the first years of the eighteenth century, fell
into my hands. It is difficult to portray the delight which one
experiences in running over the leaves of these volumes. One sees as an
actual spectator almost the rise of the most important discoveries and
witnesses the progress of many fields of knowledge from almost total
ignorance to relatively perfect clearness.

I propose to discuss here the fundamental researches of Sauveur in
Acoustics. It is astonishing how extraordinarily near Sauveur was to the
view which Helmholtz was the first to adopt in its full extent a hundred
and fifty years later.

The _Histoire de l'Académie_ for 1700, p. 131, tells us that Sauveur had
succeeded in making music an object of scientific research, and that he
had invested the new science with the name of "acoustics." On five
successive pages a number of discoveries are recorded which are more
fully discussed in the volume for the year following.

Sauveur regards the _simplicity_ of the ratios obtaining between the
rates of vibration of consonances as something universally known.[135]
He is in hope, by further research, of determining the chief rules of
musical composition and of fathoming the "metaphysics of the agreeable,"
the main law of which he asserts to be the union of "simplicity with
multiplicity." Precisely as Euler[136] did a number of years later, he
regards a consonance as more perfect according as the ratio of its
vibrational rates is expressed in smaller whole numbers, because the
smaller these whole numbers are the oftener the vibrations of the two
tones coincide, and hence the more readily they are apprehended. As the
limit of consonance, he takes the ratio 5:6, although he does not
conceal the fact that practice, sharpened attention, habit, taste, and
even prejudice play collateral rôles in the matter, and that
consequently the question is not a purely scientific one.

Sauveur's ideas took their development from his having instituted at
all points more exact quantitative investigations than his predecessors.
He is first desirous of determining as the foundation of musical tuning
a fixed note of one hundred vibrations which can be reproduced at any
time; the fixing of the notes of musical instruments by the common
tuning pipes then in use with rates of vibration unknown, appearing to
him inadequate. According to Mersenne (_Harmonie Universelle_, 1636), a
given cord seventeen feet long and weighted with eight pounds executes
eight visible vibrations in a second. By diminishing its length then in
a given proportion we obtain a proportionately augmented rate of
vibration. But this procedure appears too uncertain to Sauveur, and he
employs for his purpose the beats (_battemens_), which were known to the
organ-makers of his day, and which he correctly explains as due to the
alternate coincidence and non-coincidence of the same vibrational phases
of differently pitched notes.[137] At every coincidence there is a
swelling of the sound, and hence the number of beats per second will be
equal to the difference of the rates of vibration. If we tune two of
three organ-pipes to the remaining one in the ratio of the minor and
major third, the mutual ratio of the rates of vibration of the first two
will be as 24: 25, that is to say, for every 24 vibrations to the lower
note there will be 25 to the higher, and one beat. If the two pipes give
together four beats in a second, then the higher has the fixed tone of
100 vibrations. The open pipe in question will consequently be five feet
in length. We also determine by this procedure the absolute rates of
vibration of all the other notes.

It follows at once that a pipe eight times as long or 40 feet in length
will yield a vibrational rate of 12½, which Sauveur ascribes to the
lowest audible tone, and further also that a pipe 64 times as small will
execute 6,400 vibrations, which Sauveur took for the highest audible
limit. The author's delight at his successful enumeration of the
"imperceptible vibrations" is unmistakably asserted here, and it is
justified when we reflect that to-day even Sauveur's principle, slightly
modified, constitutes the simplest and most delicate means we have for
exactly determining rates of vibration. Far more important still,
however, is a second observation which Sauveur made while studying
beats, and to which we shall revert later.

Strings whose lengths can be altered by movable bridges are much easier
to handle than pipes in such investigations, and it was natural that
Sauveur should soon resort to their use.

One of his bridges accidentally not having been brought into full and
hard contact with the string, and consequently only imperfectly impeding
the vibrations, Sauveur discovered the harmonic overtones of the string,
at first by the unaided ear, and concluded from this fact that the
string was divided into aliquot parts. The string when plucked, and
when the bridge stood at the third division for example, yielded the
twelfth of its fundamental note. At the suggestion of some
academician[138] probably, variously colored paper riders were placed at
the nodes (_noeuds_) and ventral segments (_ventres_), and the division
of the string due to the excitation of the overtones (_sons
harmoniques_) belonging to its fundamental note (_son fondamental_) thus
rendered visible. For the clumsy bridge the more convenient feather or
brush was soon substituted. . While engaged in these investigations
Sauveur also observed the sympathetic vibration of a string induced by
the excitation of a second one in unison with it. He also discovered
that the overtone of a string can respond to another string tuned to its
note. He even went further and discovered that on exciting one string
the overtone which it has in common with another, differently pitched
string can be produced on that other; for example, on strings having for
their vibrational ratio 3:4, the fourth of the lower and the third of
the higher may be made to respond. It follows indisputably from this
that the excited string yields overtones simultaneously with its
fundamental tone. Previously to this Sauveur's attention had been drawn
by other observers to the fact that the overtones of musical instruments
can be picked out by attentive listening, particularly in the
night.[139] He himself mentions the simultaneous sounding of the
overtones and the fundamental tone.[140] That he did not give the proper
consideration to this circumstance was, as will afterwards be seen,
fatal to his theory.

While studying beats Sauveur makes the remark that they are
_displeasing_ to the ear. He held the beats were distinctly audible only
when less than six occurred in a second. Larger numbers were not
distinctly perceptible and gave rise accordingly to no disturbance. He
then attempts to reduce the difference between consonance and dissonance
to a question of beats. Let us hear his own words.[141]

     "Beats are unpleasing to the ear because of the unevenness of the
     sound, and it may be held with much plausibility that the reason
     why octaves are so pleasing is that we never hear their beats.[142]

     "In following out this idea, we find that the chords whose beats we
     cannot hear are precisely those which the musicians call
     consonances and that those whose beats are heard are the
     dissonances, and that when a chord is a dissonance in one octave
     and a consonance in another, it beats in the one and does not beat
     in the other. Consequently it is called an imperfect consonance. It
     is very easy by the principles of M. Sauveur, here established, to
     ascertain what chords beat and in what octaves, above or below the
     fixed note. If this hypothesis be correct, it will disclose the
     true source of the rules of composition, hitherto unknown to
     science, and given over almost entirely to judgment by the ear.
     These sorts of natural judgment, marvellous though they may
     sometimes appear, are not so but have very real causes, the
     knowledge of which belongs to science, provided it can gain
     possession thereof."[143]

Sauveur thus correctly discerns in beats the cause of the disturbance
of consonance, to which all disharmony is "probably" to be referred. It
will be seen, however, that according to his view all distant intervals
must necessarily be consonances and all near intervals dissonances. He
also overlooks the absolute difference in point of principle between his
old view, mentioned at the outset, and his new view, rather attempting
to obliterate it.

R. Smith[144] takes note of the theory of Sauveur and calls attention to
the first of the above-mentioned defects. Being himself essentially
involved in the old view of Sauveur, which is usually attributed to
Euler, he yet approaches in his criticism a brief step nearer to the
modern theory, as appears from the following passage.[145]

     "The truth is, this gentleman confounds the distinction between
     perfect and imperfect consonances, by comparing imperfect
     consonances which beat because the succession of their short
     cycles[146] is periodically confused and interrupted, with perfect
     ones which cannot beat, because the succession of their short
     cycles is never confused nor interrupted.

     "The _fluttering roughness_ above mentioned is perceivable in all
     other perfect consonances, in a smaller degree in proportion as
     their cycles are shorter and simpler, and their pitch is higher;
     and is of a _different kind_ from the _smoother beats_ and
     undulations of _tempered consonances_; because we can alter the
     rate of the latter by altering the temperament, but not of the
     former, the consonance being perfect at a given pitch: And because
     a judicious ear can often hear, at the same time, both the
     flutterings and the beats of a tempered consonance; sufficiently
     distinct from each other.

     "For nothing gives greater offence to the hearer, though ignorant
     of the cause of it, than those rapid, piercing beats of high and
     loud sounds, which make imperfect consonances with one another. And
     yet a few slow beats, like the slow undulations of a close shake
     now and then introduced, are far from being disagreeable."

Smith is accordingly clear that other "roughnesses" exist besides the
beats which Sauveur considered, and if the investigations had been
continued on the basis of Sauveur's idea, these additional roughnesses
would have turned out to be the beats of the overtones, and the theory
thus have attained the point of view of Helmholtz.

Reviewing the differences between Sauveur's and Helmholtz's theories, we
find the following:

1. The theory according to which consonance depends on the frequent and
regular coincidence of vibrations and their ease of enumeration, appears
from the new point of view inadmissible. The simplicity of the ratios
obtaining between the rates of vibration is indeed a _mathematical_
characteristic of consonance as well as a _physical_ condition thereof,
for the reason that the coincidence of the overtones as also their
further physical and physiological consequences is connected with this
fact. But no _physiological_ or _psychological_ explanation of
consonance is given by this fact, for the simple reason that in the
acoustic nerve-process nothing corresponding to the periodicity of the
sonant stimulus is discoverable.

2. In the recognition of beats as a disturbance of consonance, both
theories agree. Sauveur's theory, however, does not take into account
the fact that clangs, or musical sounds generally, are composite and
that the disturbance in the consonances of distant intervals principally
arise from the beats of the overtones. Furthermore, Sauveur was wrong in
asserting that the number of beats must be less than six in a second in
order to produce disturbances. Even Smith knows that very slow beats are
not a cause of disturbance, and Helmholtz found a much higher number
(33) for the maximum of disturbance. Finally, Sauveur did not consider
that although the number of beats increases with the recession from
unison, yet their _strength_ is diminished. On the basis of the
principle of specific energies and of the laws of sympathetic vibration
the new theory finds that two atmospheric motions of like amplitude but
different periods, _a_ sin(_rt_) and _a_ sin[(_r_ + [rho])(_t_ +
[tau])], cannot be communicated with the same amplitude to the same
nervous end-organ. On the contrary, an end-organ that reacts best to the
period _r_ responds more weakly to the period _r_ + [rho], the two
amplitudes bearing to each other the proportion _a_: [phi]_a_. Here
[phi] decreases when [rho] increases, and when [rho] = 0 it becomes
equal to 1, so that only the portion of the stimulus [phi]_a_ is subject
to beats, and the portion (1-[phi])_a_ continues smoothly onward without
disturbance.

If there is any moral to be drawn from the history of this theory, it is
that considering how near Sauveur's errors were to the truth, it
behooves us to exercise some caution also with regard to the new theory.
And in reality there seems to be reason for doing so.

The fact that a musician will never confound a more perfectly consonant
chord on a poorly tuned piano with a less perfectly consonant chord on a
well tuned piano, although the roughness in the two cases may be the
same, is sufficient indication that the degree of roughness is not the
only characteristic of a harmony. As the musician knows, even the
harmonic beauties of a Beethoven sonata are not easily effaced on a
poorly tuned piano; they scarcely suffer more than a Raphael
drawing executed in rough unfinished strokes. The _positive
physiologico-psychological_ characteristic which distinguishes one
harmony from another is not given by the beats. Nor is this
characteristic to be found in the fact that, for example, in sounding a
major third the fifth partial tone of the lower note coincides with the
fourth of the higher note. This characteristic comes into consideration
only for the investigating and abstracting reason. If we should regard
it also as characteristic of the sensation, we should lapse into a
fundamental error which would be quite analogous to that cited in (1).

The _positive physiological_ characteristics of the intervals would
doubtless be speedily revealed if it were possible to conduct aperiodic,
for example galvanic, stimuli to the single sound-sensing organs, in
which case the beats would be totally eliminated. Unfortunately such an
experiment can hardly be regarded as practicable. The employment of
acoustic stimuli of short duration and consequently also free from
beats, involves the additional difficulty of a pitch not precisely
determinable.

  FOOTNOTES:

  [Footnote 134: This article, which appeared in the Proceedings of
  the German Mathematical Society of Prague for the year 1892, is
  printed as a supplement to the article on "The Causes of Harmony,"
  at page 32.]

  [Footnote 135: The present exposition is taken from the volumes for
  1700 (published in 1703) and for 1701 (published in 1704), and
  partly also from the _Histoire de l'Académie_ and partly from the
  _Mémoires_. Sauveur's later works enter less into consideration
  here.]

  [Footnote 136: Euler, _Tentamen novae theoriae musicae_, Petropoli,
  1739.]

  [Footnote 137: In attempting to perform his experiment of beats
  before the Academy, Sauveur was not quite successful. _Histoire de
  l'Académie_, Année 1700, p. 136.]

  [Footnote 138: _Histoire de l'Académie_, Année 1701, p. 134.]

  [Footnote 139: _Ibid._, p. 298.]

  [Footnote 140: _Histoire de l'Académie_, Année 1702, p. 91.]

  [Footnote 141: From the _Histoire de l'Académie_, Année 1700, p.
  139.]

  [Footnote 142: Because all octaves in use in music offer too great
  differences of rates of vibration.]

  [Footnote 143: "Les battemens ne plaisent pas à l'Oreille, à cause
  de l'inégalité du son, et l'on peut croire avec beaucoup d'apparence
  que ce qui rend les Octaves si agréables, c'est qu'on n'y entend
  jamais de battemens.

  "En suivant cette idée, on trouve que les accords dont on ne peut
  entendre les battemens, sont justement ceux que les Musiciens
  traitent de Consonances, et que ceux dont les battemens se font
  sentir, sont les Dissonances, et que quand un accord est Dissonance
  dans une certaine octave et Consonance dans une autre, c'est qu'il
  bat dans l'une, et qu'il ne bat pas dans l'autre. Aussi est il
  traité de Consonance imparfaite. Il est fort aisé par les principes
  de Mr. Sauveur qu'on a établis ici, de voir quels accords battent,
  et dans quelles Octaves au-dessus on au-dessous du son fixe. Si
  cette hypothèse est vraye, elle découvrira la véritable source des
  Règles de la composition, inconnue jusqu'à présent à la Philosophie,
  qui s'en remettait presque entièrement au jugement de l'Oreille. Ces
  sortes de jugemens naturels, quelque bisarres qu'ils paroissent
  quelquefois, ne le sont point, ils ont des causes très réelles, dont
  la connaissance appartient à la Philosophie, pourvue qu'elle s'en
  puisse mettre en possession."]

  [Footnote 144: _Harmonics or the Philosophy of Musical Sounds_,
  Cambridge, 1749. I saw this book only hastily in 1864 and drew
  attention to it in a work published in 1866. I did not come into its
  actual possession until three years ago and then only did I learn
  its exact contents.]

  [Footnote 145: _Harmonics_, pp. 118 and 243.]

  [Footnote 146: "Short cycle" is the period in which the same phases
  of the two co-operant tones are repeated.]



II.

REMARKS ON THE THEORY OF SPATIAL VISION.[147]


According to Herbart, spatial vision rests on reproduction-series. In
such an event, of course, and if the supposition is correct, the
magnitudes of the residua with which the percepts or representations are
coalesced (the helps to coalescence) are of cardinal influence.
Furthermore, since the coalescences must first be fully perfected before
they make their appearance, and since upon their appearance the
inhibitory ratios are brought into play, ultimately, then, if we leave
out of account the accidental order of time in which the percepts are
given, everything in spatial vision depends on the oppositions and
affinities, or, in brief, on the qualities of the percepts, which enter
into series.

Let us see how the theory stands with respect to the special facts
involved.

1. If intersecting series only, running anteriorly and posteriorly, are
requisite for the production of spatial sensation, why are not analogues
of them found in all the senses?

2. Why do we measure differently colored objects and variegated objects
with one and the same spatial measure? How do we recognise differently
colored objects as the same in size? Where do we get our measure of
space from and what is it?

3. Why is it that differently colored figures of the same form reproduce
one another and are recognised as the same?

Here are difficulties enough. Herbart is unable to solve them by his
theory. The unprejudiced student sees at once that his "inhibition by
reason of form" and "preference by reason of form" are absolutely
impossible. Think of Herbart's example of the red and black letters.

The "help to coalescence" is a passport, so to speak, made out to the
name and person of the percept. A percept which is coalesced with
another cannot reproduce all others qualitatively different from it for
the simple reason that the latter are in like manner coalesced with one
another. Two qualitatively different series certainly do not reproduce
themselves because they present the same order of degree of coalescence.

If it is certain that only things simultaneous and things which are
alike are reproduced, a basic principle of Herbart's psychology which
even the most absolute empiricists will not deny, nothing remains but to
modify the theory of spatial perception or to invent in its place a new
principle in the manner indicated, a step which hardly any one would
seriously undertake. The new principle could not fail to throw all
psychology into the most dreadful confusion.

As to the modification which is needed there can be hardly any doubt as
to how in the face of the facts and conformably to Herbart's own
principles it is to be carried out. If two differently colored figures
of equal size reproduce each other and are recognised as equal, the
result can be due to nothing but to the existence in both series of
presentations of a presentation or percept which is qualitatively _the
same_. The colors are different. Consequently, like or equal percepts
must be connected with the colors which are yet independent of the
colors. We have not to look long for them, for they are the like effects
of the muscular feelings of the eye when confronted by the two figures.
We might say we reach the vision of space by the registering of
light-sensations in a schedule of graduated muscle-sensations.[148]

A few considerations will show the likelihood of the rôle of the
muscle-sensations. The muscular apparatus of _one_ eye is unsymmetrical.
The two eyes together form a system which is vertical in symmetry. This
already explains much.

1. The _position_ of a figure influences its view. According to the
position in which objects are viewed different muscle-sensations come
into play and the impression is altered. To recognise inverted letters
as such long experience is required. The best proof of this are the
letters d, b, p, q, which are represented by the same figure in
different positions and yet are always distinguished as different.[149]

2. It will not escape the attentive observer that for the same reasons
and even with the same figure and in the same position the fixation
point is also decisive. The figure seems to change _during_ the act of
vision. For example, an eight-pointed star constructed by successively
joining in a regular octagon the first corner with the fourth, the
fourth with the seventh, etc., skipping in every case two corners,
assumes alternately, according to where we suffer the centre of vision
to rest, a predominantly architectonic or a freer and more open
character. Vertical and horizontal lines are always differently
apprehended from what oblique lines are.

[Illustration: Fig. 58.]

3. The reason why we prefer vertical symmetry and regard it as something
special in its kind, whereas we do not recognise horizontal symmetry at
all immediately, is due to the vertical symmetry of the muscular
apparatus of the eye. The left-hand side _a_ of the accompanying
vertically-symmetrical figure induces in the left eye the same muscular
feelings as the right-hand side _b_ does in the right eye. The pleasing
effect of symmetry has its cause primarily in the repetition of muscular
feelings. That a repetition actually occurs here, sometimes sufficiently
marked in character as to lead to the confounding of objects, is proved
apart from the theory by the fact which is familiar to every one _quem
dii oderunt_ that children frequently reverse figures from the right to
the left, but never from above downwards; for example, write [epsilon]
instead of 3 until they finally come to notice the slight difference.
Figure 50 shows how pleasing the repetition of muscular feelings may be.
As will be readily understood, vertical and horizontal lines exhibit
relations similar to symmetrical figures which are immediately disturbed
when oblique positions are chosen for the lines. Compare what Helmholtz
says regarding the repetition and coincidence of partial tones.

[Illustration: Fig. 59.]

I may be permitted to add a general remark. It is a quite universal
phenomenon in psychology that certain qualitatively quite different
series of percepts mutually awaken and reproduce one another and in a
certain aspect produce the appearance of sameness or similarity. We say
of such series that they are of like or of similar form, naming their
abstracted likeness _form_.

     1. Of spatial figures we have already spoken.

     2. We call two melodies like melodies when they present the same
     succession of pitch-ratios; the absolute pitch (or key) may be as
     different as can be. We can so select the melodies that not even
     two partial tones of the notes in each are common. Yet we recognise
     the melodies as alike. And, what is more, we notice the form of the
     melody more readily and recognise it again more easily than the key
     (the absolute pitch) in which it was played.

     3. We recognise in two different melodies the same rhythm no matter
     how different the melodies may be otherwise. We know and recognise
     the rhythm more easily even than the absolute duration (the tempo).

These examples will suffice. In all these and in all similar cases the
recognition and likeness cannot depend upon the qualities of the
percepts, for these are different. On the other hand recognition,
conformably to the principles of psychology, is possible only with
percepts which are the same in quality. Consequently there is no other
escape than to imagine the qualitatively unlike percepts of the two
series as necessarily connected with other percepts which are
qualitatively alike.

Since in differently colored figures of like form, like muscular
feelings are necessarily induced if the figures are recognised as alike,
so there must necessarily lie at the basis of all forms also, and we
might even say at the basis of all abstractions, percepts of a peculiar
quality. And this holds true for space and form as well as for time,
rhythm, pitch, the form of melodies, intensity, etc. But whence is
psychology to derive all these qualities? Have no fear, they will all be
found, as were the sensations of muscles for the theory of space. The
organism is at present still rich enough to meet all the requirements of
psychology in this direction, and it is even time to give serious ear to
the question of "corporeal resonance" which psychology so loves to dwell
on.

Different psychical qualities appear to bear a very intimate mutual
relation to one another. Special research on the subject, as well also
as the demonstration that this remark may be generally employed in
physics, will follow later.[150]

  FOOTNOTES:

  [Footnote 147: This article, designed to illustrate historically
  that on Symmetry, at page 89, first appeared in Fichte's
  _Zeitschrift für Philosophie_, for 1865.]

  [Footnote 148: Comp. Cornelius, _Ueber das Sehen_; Wundt, _Theorie
  der Sinneswahrnehmung_.]

  [Footnote 149: Comp. Mach, _Ueber das Sehen von Lagen and Winkeln_.
  _Sitzungsb. der Wiener Akademie_, 1861.]

  [Footnote 150: Comp. Mach, _Zur Theorie des Gehörorgans_.
  _Sitsungsber, der Wiener Akad._, 1863.--_Ueber einige Erscheinungen
  der physiolog. Akustik._ _Ibid._, 1864.]



INDEX.


  Absolute, temperature, 162;
    time, 204;
    forecasts, have no signification in science, 206.

  Abstract, meaning of the term, 240.

  Abstraction, 180, 200, 208, 231.

  Acceleration, organ for forward, 299 et seq.

  Accelerations, 204, 216, footnote, 225-226, 253.

  Accident, logical and historical, in science, 160, 168, 170, 213;
    in inventions and discoveries, 262 et seq.

  Accord, the pure triple, 46.

  Accumulators, electrical, 125 et seq.;
    132, footnote.

  Acoustic color, 36.

  Acoustics, Sauveur on, 375 et seq.

  Action and reaction, importance of the principle of, 191.

  Adaptation, in organic and inorganic matter, 216, 229;
    in scientific thought, 214-235.

  Æsthetics, computation as a principle of, 34;
    researches in, 89, footnote;
    repetition, a principle of, 91.

  Africa, 186, 234, 237.

  Agreeable effects, due to repetition of sensations, 92, 97 et seq.

  Agriculture, transition to, 265.

  Air-gun, 135.

  Alcohol and water, mixture of oil and, in Plateau's experiments, 4.

  Algebra, economy of, 196.

  Alien thoughts in science, 196.

  All, the, 88.

  Amontons, 174, 346.

  Ampère, the word, 314.

  Ampère's swimmer, 207.

  Analogies, mechanical, 157, 160;
    generally, 236-258.

  Analogy, defined, 250.

  Analysis, 188.

  Analytical geometry, not necessary to physicians, 370, footnote.

  Anatomic structures, transparent stereoscopic views of, 74.

  Anatomy, character of research in, 255.

  Andrieu, Jules, 49, footnote.

  Animals, the psychical activity of, 190, 231;
    the language of, 238;
    their capacity for experience, 266 et seq.

  Animism, 186, 187, 243, 254.

  Anisotropic optical fields, 227.

  Apparatus for producing movements of rotation, 287 et seq.

  Arabesque, an inverted, 95.

  Arabian Nights, 219.

  Arago, 270.

  Aral, the Sea of, 239.

  Archæopteryx, 257.

  Archimedes, 4, 237.

  Arcimboldo, Giuseppe, 36.

  Area, principle of least superficial, 10 et seq.

  Ares, the bellowing of the wounded, 272.

  Aristotelians, 283.

  Aristotle, 348, 296.

  Art, development of, 28 et seq.

  Artillery, practical, 334-335.

  Artistic value of scientific descriptions, 254.

  Arts, practical, 108.

  Ascent, heights of, 143-151.

  Asia, 234.

  Assyrians, the art of, 79.

  Astronomer, measures celestial by terrestrial distances, 136.

  Astronomy, antecedent to psychology, 90;
    rigidity of its truths, 221.

  Atomic theories, 104.

  Atoms, 207.

  Attention, the rôle of, in sensuous perception, 35 et seq.

  Attraction, generally, 226;
    of liquid particles, 13-14;
    in electricity, 109 et seq.

  Aubert, 298.

  Audition. See _Ear_.

  Austrian gymnasiums, 370, footnote.

  Axioms, instinctive knowledge, 190.


  Babbage, on the economy of machinery, 196.

  Bach, 20.

  Backwards, prophesying, 253.

  Bacon, Lord, 48, 280.

  Baer, C. E. von, 235.

  Balance, electrical, 127, footnote;
    torsion, 109, 168.

  Balloon, a hydrogen, 199.

  Barbarism and civilisation, 335 et seq.

  Bass-clef, 101.

  Bass, fundamental, 44.

  Beats, 40-45, 377 et seq.

  Beautiful, our notions of, variable, 99.

  Beauty, objects of, in nature, 91.

  Becker, J. K., 364, 369.

  Beethoven, 39, 44.

  Beginnings of science, 189, 191.

  Belvedere Gallery at Vienna, 36.

  Bernoulli, Daniel, on the conservation of living force, 149;
    on the vibrations of strings, 249.

  Bernoulli, James, on the centre of oscillation, 149.

  Bernoulli, John, on the conservation of living force, 149;
    on the principle of virtual velocities, 151.

  Bible, parallel passages from, for language study, 356.

  Binocular vision, 66 et seq.

  Black, his theory of caloric, 138, 162;
    on quantity of heat, 166, 174;
    on latent heat, 167, 178;
    researches in heat generally, 244.

  Blind cat, 303.

  Bodies, heavy, seek their places, 224 et seq.;
    rotating, 285.

  Body, a mental symbol for groups of sensations, 200-203;
    the human, our knowledge of, 90.

  Boltzmann, 236.

  Booth, Mr., 77.

  Borelli, 217.

  Boulder, a granite, 233.

  Bow-wave of ships and moving projectiles, 323 et seq.

  Boys, 317.

  Bradley, 273.

  Brahman, the, 63.

  Brain, localisation of functions in, 210.

  Breuer, 272, 282 et seq., 293, 298, 300, 301, 303, 306.

  Brewster, his stereoscope, 73.

  Bridge, invention of the, 264, 268.

  British Association, 108.

  Brooklyn Bridge, 75, footnote.

  Brown, Crum, 293, 301.

  Building, our concepts directions for, 253;
    facts the result of, 253;
    science compared to, 257.

  Building-stones, metrical units are, 253.

  Busch, 328.

  Business of a merchant, science compared to the, 16.

  Butterfly, a, 22.


  Calculating machines, their economical character, 196.

  Caloric, theory of, stood in the way of scientific advancement, 138, 167.

  Calypso, the island of, 351.

  Canterbury, Archbishop of, 39.

  Cantor, M., 361, footnote.

  Capacity, electrical, 116 et seq., 123;
    thermal, 123;
    specific inductive, 117.

  Capulets and Montagues, 87.

  Cards, difficult games of, 357.

  Carnot, S., excludes perpetual motion in heat, 156, 162;
    his mechanical view of physics, 156;
    on thermodynamics, 160 et seq.;
    his principle, 162;
    also, 191.

  Carus, Dr. Paul, 265, footnote.

  Casselli's telegraph, 26.

  Cassini, 51.

  Cauchy, character of the intellectual activity of a, 195.

  Causal insight, awakened by science, 357.

  Causality, 157-159, 190, 198 et seq., 221 et seq., 237, 253, 254.

  Cause and effect, 198 et seq. See also _Causality_.

  Centimetre-gramme-second system, 111.

  Centre of gravity, must lie as low as possible for equilibrium to
  subsist, 15;
    Torricelli's principle of, 150 et seq.

  Centre of oscillation, 149.

  Change, method of, in science, 230.

  Changeable character of bodies, 202.

  Changes, physical, how they occur, 205.

  Character, a Universal Real, 192.

  Character, like the forms of liquids, 3;
    persons of, 24.

  Charles the Fifth, 369.

  Chemical, elements, 202;
    symbols, 192;
    current, 118.

  Chemistry, character of research in, 255;
    the method of thermodynamics in, 257.

  Child, a, modes of thought of, 223;
    looking into a moat, 208.

  Child of the forest, his interpretation of new events, 218-219.

  Childish questions, 199-200.

  Children, the drawings of, 201-202.

  Chinese language, economy of, 192;
    study of, 354.

  Chinese philosopher, an old, 186.

  Chinese, speak with unwillingness of politics, 374;
    the art of, 79-80.

  Chosen, many are called but few are, 65.

  Christ, saying of, 65.

  Christianity, Latin introduced with, 311.

  Christians and Jews, monotheism of the, 187.

  Church and State, 88.

  Cicero, 318.

  Circe, 372.

  Circle, the figure of least area with given periphery, 12.

  Circular polarisation, 242.

  Civilisation and barbarism, 335 et seq.

  Civilisation, some phenomena of, explained by binocular vision, 74.

  Civilised man, his modes of conception and interpretation, 219.

  Clapeyron, 162.

  Class-characters of animals, 255.

  Classical, culture, the good and bad effects of, 347;
    scholars, not the only educated people, 345.

  Classics, on instruction in, 338-374;
    the scientific, 368.

  Classification in science, 255.

  Clausius, on thermodynamics, 165;
    on reversible cycles, 176.

  Claviatur, Mach's, 42-43.

  Club-law, 335.

  Cochlea, the, a species of piano-forte, 19.

  Cockchafer, 86.

  Coefficient of self-induction, 250, 252.

  Colophonium, solution of, 7.

  Color, acoustic, 36.

  Color-sensation, 210.

  Color-signs, their economy, 192.

  Colors, origin of the names of, 239.

  Column, body moving behind a, 202.

  Communication, its functions, import and fruits, 197, 238 et seq.;
    by language, 237;
    high importance of, 191 et seq.

  Comparative physics, 239.

  Comparison in science, 231, 238 et seq.

  Computation, a principle of æsthetics, 34.

  Concepts, abstract, defined, 250-252;
    metrical, in electricity, 107 et seq.

  Conceptual, meaning of the term, 240.

  Conceptual thought, 192.

  Concha, 18.

  Condensers, electrical, 125 et seq. 132, footnote.

  Conductors and non-conductors. See _Electrical_, etc.

  Conformity in the deportment of the energies, 171-175.

  Confusion of objects, cause of, 95.

  Conic sections, 257.

  Conical refraction, 29, 242.

  Conservation of energy, 137 et seq. See _Energy_.

  Conservation of weight or mass, 203.

  Consonance, connexion of the simple natural numbers with, 33;
    Euclid's definition of, 33;
    explanation of, 42;
    scientific definition of, 44;
    and dissonance reduced to beats, 376, 370, 383.

  Consonant intervals, 43.

  Constancy of matter, 203.

  Constant, the dielectric, 117.

  Constants, the natural, 193.

  Continuum of facts, 256 et seq.

  Cornelius, 388, footnote.

  Corti, the Marchese, his discovery of minute rods in the labyrinth of
  the ear, 19.

  Coulomb, his electrical researches, 108, 109, 113;
    his notion of quantity of electricity, 173;
    his torsion-balance, 168.

  Crew, Prof. Henry, 317, footnote.

  Criticism, Socrates the father of scientific, 1, 16.

  _Critique of Pure Reason_, Kant's, 188.

  Crucible, derivation of the word, 49, footnote.

  Crustacea, auditory filaments of, 29, 272, 302.

  Cube of oil, 5.

  Culture, ancient and modern, 344.

  Currents, chemical, 118;
    electrical, 118;
    galvanic, 132;
    measurement of electrical, 135-136;
    of heat, 244, 249-250;
    strength of, 250.

  Curtius, 356.

  Curved lines, their asymmetry, 98.

  Curves, how their laws are investigated, 206.

  Cycles, reversible, Clausius on, 176.

  Cyclical processes, closed, 175.

  Cyclops, 67.

  Cyclostat, 298.

  Cylinder, of oil, 6;
    mass of gas enclosed in a, 179.


  D'Alembert, on the causes of harmony, 34;
    his principle, 142, 149, 154;
    also 234, 279.

  Danish schools, 338, footnote.

  Darwin, his study of organic nature, 215 et seq.;
    his methods of research, 216.

  Deaf and dumb, not subject to giddiness, 299.

  Deaf person, with a piano, analyses sounds, 27.

  Death and life, 186.

  Definition, compendious, 197.

  Deiters, 19.

  Delage, 298, 301, 302.

  Democritus, his mechanical conception of the world, 155, 187.

  Demonstration, character of, 362.

  Deportment of the energies, conformity in the, 171-175.

  Derivation, laws only methods of, 256.

  Descent, Galileo's laws of, 193;
    generally, 143 et seq., 204, 215.

  Description, 108, 191, 236, 237;
    a condition of scientific knowledge, 193;
    direct and indirect, 240;
    in physics, 197, 199.

  Descriptive sciences, their resemblance to the abstract, 248.

  Determinants, 195.

  Diderot, 234.

  Dielectric constant, the, 117.

  Difference-engine, the, 196.

  Differential coefficients, their relation to symmetry, 98.

  Differential laws, 204.

  Differential method, for detecting optical imperfections, 317.

  Diffraction, 91, 194.

  Diffusion, Fick's theory of, 249.

  Discharge of Leyden jars, 114 et seq.

  Discoveries, the gist of, 270, 375.

  Discovery and invention, distinction between, 269.

  Dissonance, explanation of, 42;
    definition of, 33, 44. See _Consonance_.

  Distances, estimation of, by the eye, 68 et seq.

  Dogs, like tuning-forks, 23;
    their mentality, 190.

  Domenech, Abbé, 92.

  Dramatic element in science, 243.

  Drop of water, on a greased plate, 8;
    on the end of a stick, 8;
    in free descent, 8.

  Dubois, 218.

  Dubois-Reymond, 370, footnote.

  Dufay, 271.

  Dynamics, foundations of, 153 et seq.


  Ear, researches in the theory of, 17 et seq.;
    diagram of, 18;
    its analysis of sounds, 20 et seq.;
    a puzzle-lock, 28;
    reflected in a mirror, 93;
    no symmetry in its sensation, 103.

  Earth, its oblateness not due to its original fluid condition, 2;
    rotation of, 204;
    internal disturbances of, 285.

  Economical, nature of physical inquiry, 186;
    procedure of the human mind, 186;
    order of physics, 197;
    schematism of science, 206;
    tools of science, 207;
    coefficient of dynamos, 133.

  Economy, of the actions of nature, 15;
    the purpose of science, 16;
    of language, 191 et seq.;
    of the industrial arts, 192;
    of mathematics, 195-196;
    of machinery, 196;
    of self-preservation, our first knowledge derived from, 197;
    generally, 186 et seq., 269.

  Education, higher, 86;
    liberal, 341 et seq., 371.

  Efflux, liquid, 150.

  Ego, its nature, 234-235.

  Egypt, 234.

  Egyptians, art of, 78 et seq., 201.

  Eighteenth century, the scientific achievements of, 187, 188.

  Eleatics, on motion, 158.

  Electrical, attraction and repulsion, 109 et seq., 168;
    capacity, 116 et seq.;
    force, 110, 119, 168;
    spark, 117, 127, 132, 133, 190;
    energy, measurement of, 128 et seq., 169;
    currents, conceptions of, 118, 132, 135-136, 226-227, 249, 250;
    fluids, 112 et seq., 228;
    pendulums, 110;
    levels, 173;
    potential, 121 et seq.;
    quantity, 111, 118, 119.

  Electricity, as a substance and as a motion, 170;
    difference between the conceptions of heat and, 168 et seq.,
    rôle of work in, 120 et seq.;
    galvanic, 134.
    See _Electrical_.

  Electrometer, W. Thomson's absolute, 127, footnote.

  Electrometers, 122, 127.

  Electrostatic unit, 111.

  Electrostatics, concepts of, 107 et seq.

  Elements, interdependence of the sensuous, 179;
    of bodies, 202;
    of phenomena, equations between, 205;
    of sensations, 200;
    used instead of sensations, 208-209.

  Ellipse, equation of, 205;
    the word, 342.

  Embryology, possible future state of, 257.

  Energies, conformity in the deportment of, 171-175;
    differences of, 175.

  Energy, a metrical notion, 178;
    conservation of, 137 et seq.;
    defined, 139;
    metaphysical establishment of the doctrine of, 183;
    kinetic, 177;
    potential, 128 et seq.;
    substantial conception of, 164, 185, 244 et seq.;
    conservation of, in electrical phenomena, 131 et seq.;
    limits of principle of, 175;
    principle of, in physics, 160-166;
    sources of principle of, 179, 181;
    thermal, 177;
    Thomas Young on, 173.

  Energy-value of heat, 178, footnote.

  Enlightenment, the, 188.

  Entropy, a metrical notion, 178.

  Environment, stability of our, 206.

  Equations for obtaining facts, 180;
    between the elements of phenomena, 205.

  Equilibrium, conditions of, in simple machines, 151;
    figures of liquid, 4 et seq.;
    general condition of, 15;
    in the State, 15.

  Etymology, the word, misused for entomology, 316.

  Euclid, on consonance and dissonance, 33;
    his geometry, 364.

  Euler, on the causes of harmony, 34;
    impression of the mathematical processes on, 196;
    on the vibrations of strings, 249, 285, 376.

  Euler and Hermann's principle, 149.

  Euthyphron, questioned by Socrates, 1.

  Evolute, the word, 342.

  Evolution, theory of, as applied to ideas, 216 et seq.

  Ewald, 298, 304.

  Excluded perpetual motion, logical root of the principle of, 182.

  Exner, S., 302, 305.

  Experience, communication of, 191;
    our ready, 199;
    the principle of energy derived from, 179;
    the wellspring of all knowledge of nature, 181;
    incongruence between thought and, 206.

  Experimental research, function of, 181.

  Explanation, nature of, 194, 237, 362.

  Eye, cannot analyse colors, 20;
    researches in the theory of the, 18 et seq.;
    loss of, as affecting vision, 98.

  Eyes, purpose of, 66 et seq.;
    their structure symmetrical not identical, 96.


  Face, human, inverted, 95.

  Facts and ideas, necessary to science, 231.

  Facts, description of, 108;
    agreement of, 180;
    relations of, 180;
    how represented, 206;
    reflected in imagination, 220 et seq.;
    the result of constructions, 253;
    a continuum of, 256 et seq.;
    equations for obtaining, 180.

  Falling bodies, 204, 215;
    Galileo on the law of, 143 et seq., 284.

  Falling, cats, 303, footnote.

  Falstaff, 309.

  Familiar intermediate links of thought, 198.

  Faraday, 191, 217, 237;
    his conception of electricity, 114, 271.

  Fechner, theory of Corti's fibres, 19 et seq.

  Feeling, cannot be explained by motions of atoms, 208 et seq.

  Fetishism, 186, 243, 254;
    in our physical concepts, 187.

  Fibres of Corti, 17 et seq.

  Fick, his theory of diffusion, 249.

  Figures, symmetry of, 92 et seq.

  Figures of liquid equilibrium, 4 et seq.

  Fire, use of, 264.

  Fishes, 306.

  Fixed note, determining of a, 377.

  Fizeau, his determination of the velocity of light, 55 et seq.

  Flats, reversed into sharps, 101.

  Flouren's experiments, 272, 290.

  Flower-girl, the baskets of a, 95.

  Fluids, electrical, 112 et seq.

  Force, electric, 110, 119, 168;
    unit of 111;
    living, 137, 149, 184;
    generally 253.
    See the related headings.

  Forces, will compared to, 254.

  Foreseeing events, 220 et seq.

  Formal conceptions, rôle of, 183.

  Formal need of a clear view of facts, 183, 246;
    how far it corresponds to nature, 184.

  Formative forces of liquids, 4.

  Forms of liquids, 3 et seq.

  Forward movement, sensation of, 300.

  Forwards, prophesying, 253.

  Foucault, 57, 70, 296.

  Foucault and Toepler, method of, for detecting optical faults, 313
  et seq., 320.

  Foundation of scientific thought, primitive acts of knowledge, the, 190.

  Fourier, on processes of heat, 249, 278.

  Fox, a, 234.

  Franklin's pane, 116.

  Frary, 338, footnote.

  Fraunhofer, 271.

  Freezing-point, lowered by pressure, 162.

  Fresnel, 271.

  Fritsch, 321.

  Frogs, larvæ of, not subject to vertigo, 298.

  Froude, 333.

  Frustra, misuse of the word, 345.

  Future, science of the, 213.


  Galileo, on the motion of pendulums, 21;
    his attempted measurement of the velocity of light, 50 et seq.;
    his exclusion of a perpetual motion, 143;
    on velocities acquired in free descent, 143-147;
    on the law of inertia, 146-147;
    on virtual velocities, 150;
    on work, 172;
    his laws of descent, 193;
    on falling bodies, 225;
    great results of his study of nature, 214 et seq.;
    his rude scientific implements, 215;
    selections from his works for use in instruction, 368;
    also 105, 182, 187, 237, 272, 274, 283.

  Galle, observes the planet Neptune, 29.

  Galvanic, electricity, 134;
    current, 132;
    dizziness, 291;
    vertigo, 298.

  Galvanoscope, 135.

  Galvanotropism, 291.

  Garda, Lake, 239.

  Gas, the word, 264;
    mass of, enclosed in a cylinder, 179.

  Gases, tensions of, for scales of temperature, 174.

  Gauss, on the foundations of dynamics, 154;
    his principle, 154;
    also, 108, 274.

  Genius, 279, 280.

  Geography, comparison in, 239.

  Geometers, in our eyes, 72.

  Geotropism, 289.

  German schools and gymnasiums, 372, 373, 338, footnote.

  Ghosts, photographic, 73.

  Glass, invisible in a mixture of the same refrangibility, 312;
    powdered, visible in a mixture of the same refrangibility, 312.

  Glove, in a mirror, 93.

  Goethe, quotations from, 9, 31, 49, 88;
    on the cause of harmony, 35.

  Goltz, 282, 291.

  Gossot, 332.

  Gothic cathedral, 94.

  Gravitation, discovery of, 225 et seq.

  Gravity, how to get rid of the effects of, in liquids, 4;
    also 228.

  Gray, Elisha, his telautograph, 26.

  Greased plate, drop of water on a, 8.

  Great minds, idiosyncrasies of, 247.

  Greek language, scientific terms derivedfrom, 342-343;
    common words derived from, 343, footnote;
    still necessary for some professions, 346;
    its literary wealth, 347-348;
    narrowness and one-sidedness of its literature, 348-349;
    its excessive study useless, 349-350;
    its study sharpens the judgment, 357-358;
    a knowledge of it not necessary to a liberal education, 371.

  Greeks, their provinciality and narrow-mindedness, 349;
    now only objects of historical research, 350.

  Griesinger, 184.

  Grimaldi, 270.

  Grimm, 344, footnote.

  Grunting fishes, 306.


  Habitudes of thought, 199, 224, 227, 232.

  Haeckel, 222, 235.

  Hamilton, deduction of the conical refraction of light, 29.

  Hankel, 364.

  Harmonics, 38, 40.

  Harmony, on the causes of, 32 et seq.;
    laws of the theory of, explained, 30;
    the investigation of the ancients concerning, 32;
    generally, 103.
    See _Consonance_.

  Harris, electrical balance of, 127, footnote.

  Hartwich, Judge, 343, 353, footnote.

  Hat, a high silk, 24.

  Hats, ladies', development of, 64.

  Head-wave of a projectile, 323 et seq.

  Hearing and orientation, relation between, 304 et seq.

  Heat, a material substance, 177;
    difference between the conceptions of electricity and, 168 et seq.;
    substantial conception of, 243 et seq.;
    Carnot on, 156, 160 et seq.;
    Fourier on the conduction of, 249;
    not necessarily a motion, 167, 170, 171;
    mechanical equivalent of, 164, 167;
    of liquefaction, 178;
    quantity of, 166;
    latent, 167, 178, 244;
    specific, 166, 244;
    the conceptions of, 160-171;
    machine, 160;
    a measure of electrical energy, 133 et seq.;
    mechanical theory of, 133;
    where does it come from? 200.

  Heavy bodies, sinking of, 222.

  Heights of ascent, 143-151.

  Helm, 172.

  Helmholtz, applies the principle of energy to electricity, 184;
    his telestereoscope, 84;
    his theory of Corti's fibres, 19 et seq.;
    on harmony, 35, 99;
    on the conservation of energy, 165, 247;
    his method of thought, 247;
    also 138, 305, 307, 375, 383.

  Hensen, V., on the auditory function of the filaments of Crustacea, 29,
  302.

  Herbart, 386 et seq.

  Herbartians, on motion, 158.

  Herculaneum, art in, 80.

  Heredity, in organic and inorganic matter, 216, footnote.

  Hering, on development, 222;
    on vision, 210.

  Hermann, E., on the economy of the industrial arts, 192.

  Hermann, L., 291.

  Herodotus, 26, 234, 347, 350.

  Hertz, his waves, 242;
    his use of the phrase "prophesy," 253.

  Herzen, 361, footnote.

  Hindu mathematicians, their beautiful problems, 30.

  Holtz's electric machine, 132.

  Horse, 63.

  Household, physics compared to a well-kept, 197.

  Housekeeping in science and civil life, 198.

  Hudson, the, 94.

  Human beings, puzzle-locks, 27.

  Human body, our knowledge of, 90.

  Human mind, must proceed economically, 186.

  Humanity, likened to a polyp-plant, 235.

  Huygens, his mechanical view of physics, 155;
    on the nature of light and heat, 155-156;
    his principle of the heights of ascent, 149;
    on the law of inertia and the motion of a compound pendulum, 147-149;
    on the impossible perpetual motion, 147-148;
    on work, 173;
    selections from his works for use in instruction, 368;
    his view of light, 227-228, 262.

  Huygens, optical method for detecting imperfections in optical glasses
  313.

  Hydrogen balloon, 199.

  Hydrostatics, Stevinus's principle of, 141.

  Hypotheses, their rôle in explanation, 228 et seq.


  Ichthyornis, 257.

  Ichthyosaurus, 63.

  Idea? what is a theoretical, 241.

  Idealism, 209.

  Ideas, a product of organic nature, 217 et seq.;
    and facts, necessary to science, 231;
    not all of life, 233;
    their growth and importance, 233;
    a product of universal evolution, 235;
    the history of, 227 et seq.;
    in great minds, 228;
    the rich contents of, 197;
    their unsettled character in common life, their clarification in
    science, 1-2.

  Ideography, the Chinese, 192.

  Imagery, mental, 253.

  Imagination, facts reflected in, 220 et seq.

  Inclined plane, law of, 140-141.

  Incomprehensible, the, 186.

  Indian, his modes of conception and interpretation, 218 et seq.

  Individual, a thread on which pearls are strung, 234-235.

  Industrial arts, economy of the, E. Hermann on, 192.

  Inertia, law of, 143 et seq., 146 et seq., 216, footnote, 283 et seq.

  Innate concepts of the understanding, Kant on, 199.

  Innervation, visual, 99.

  Inquirer, his division of labor, 105;
    compared to a shoemaker, 105-106;
    what constitutes the great, 191;
    the true, seeks the truth everywhere, 63 et seq.;
    the, compared to a wooer, 45.

  Instinctive knowledge, 189, 190.

  Instruction, aim of, the saving of experience, 191;
    in the classics, mathematics, and sciences, 338-374;
    limitation of matter of, 365 et seq.

  Insulators, 130.

  Integrals, 195.

  Intellectual development, conditions of, 286 et seq.

  Intentions, acts of nature compared to, 14-15.

  Interconnexion of nature, 182.

  Interdependence, of properties, 361;
    of the sensuous elements of the world, 179.

  Interference experiments with the head-wave of moving projectiles,
  327-328.

  International intercourse, established by Latin, 341.

  International measures, 108.

  Invention, discovery and, distinction between, 269.

  Inventions, requisites for the development of, 266, 268 et seq.

  Iron-filings, 220, 243.

  Italian art, 234.


  Jacobi, C. G. J., on mathematics, 280.

  James, W., 275, 299.

  Java, 163.

  Jews and Christians, monotheism of the, 187.

  Jolly, Professor von, 112, 274.

  Joule, J. P., on the conservation of energy, 163-165, 167, 183;
    his conception of energy, 245;
    his metaphysics, 183, 246;
    his method of thought, 247;
    also 137, 138.

  Journée, 317.

  Judge, criminal, the natural philosopher compared to a, 48.

  Judgment, essentially economy of thought, 201-202;
    sharpened by languages and sciences, 357-358;
    also 232-233, 238.

  Juliet, Romeo and, 87.

  Jupiter, its satellites employed in the determination of the velocity of
  light, 51 et seq.

  Jurisprudence, Latin and Greek unnecessary for the study of, 346,
  footnote.


  Kant, his hypothesis of the origin of the planetary system, 5;
    his _Critique of Pure Reason_, 188;
    on innate concepts of the understanding, 199;
    on time, 204;
    also footnote, 93.

  Kepler, 187, 270.

  Kinetic energy, 177.

  Kirchhoff, his epistemological ideas, 257-258;
    his definition of mechanics, 236, 258, 271, 273.

  Knight, 289.

  Knowledge, a product of organic nature, 217 et seq., 235;
    instinctive, 190;
    made possible by economy of thought, 198;
    our first, derived from the economy of self-preservation, 197;
    the theory of, 203;
    our primitive acts of the foundation of science, 190.

  Kocher, 328.

  Koenig, measurement of the velocity of sound, 57 et seq.

  Kölliker, 19.

  Kopisch, 61.

  Kreidl, 299, 302, 306;
    his experiments, 272.

  Krupp, 319.


  Labels, the value of, 201.

  Labor, the accumulation of, the foundation of wealth and power, 198;
    inquirer's division of, 105, 258.

  Labyrinth, of the ear, 18, 291, 305.

  Lactantius, on the study of moral and physical science, 89.

  Ladder of our abstraction, the, 208.

  Ladies, their eyes, 71;
    like tuning-forks, 23-24.

  Lagrange, on Huygens's principle, 149;
    on the principle of virtual velocities, 150-155;
    character of the intellectual activity of a, 195, 278.

  Lake-dwellers, 46, 271.

  Lamp-shade, 70.

  Lane's unit jar, 115.

  Language, knowledge of the nature of, demanded by a liberal education,
  356;
    relationship between, and thought, 358;
    communication by 237;
    economy of, 191 et seq.;
    human its character, 238;
    of animals, 238;
    instruction in, 338 et seq.;
    its methods, 192.

  Laplace, on the atoms of the brain, 188;
    on the scientific achievements of the eighteenth century, 188;
    his hypothesis of the origin of the planetary system, 5.

  Latent heat, 167, 178, 244.

  Latin city of Maupertuis, 339.

  Latin, instruction in, 311 et seq.;
    introduced with the Christian Church, 340;
    the language of scholars, the medium of international intercourse,
    its power, utility, and final abandonment, 341-347;
    the wealth of its literature, 348;
    the excessive study of, 346, 349, 354, 355;
    its power to sharpen the judgment, 357-358.

  Lavish extravagance of science, 189.

  Law, a, defined, 256;
    a natural, not contained in the conformity of the energies, 175.

  Law-maker, motives of not always discernible, 9.

  Layard, 79.

  Learning, its nature, 366 et seq.

  Least superficial area, principle of, accounted for by the mutual
  attractions of liquid particles, 13-14;
    illustrated by a pulley arrangement, 12-13;
    also 9 et seq.

  Leibnitz, on harmony, 33;
    on international intercourse, 342, footnote.

  Lessing, quotation from, 47.

  Letters of the alphabet, their symmetry, 94, 97.

  Level heights of work, 172-174.

  Lever, a, in action, 222.

  Leverrier, prediction of the planet Neptune, 29.

  Leyden jar, 114.

  Liberal education, a, 341 et seq., 359, 371.

  Libraries, thoughts stored up in, 237.

  Lichtenberg, on instruction, 276, 370.

  Licius, a Chinese philosopher, 213.

  Liebig, 163, 278.

  Life and death, 186.

  Light, history of as elucidating how theories obstruct research, 242;
    Huygens's and Newton's views of, 227-228;
    its different conceptions, 226;
    rectilinear propagation of, 194;
    rôle of, in vision, 81;
    spatial and temporal periodicity of, explains optical phenomena, 194;
    numerical velocity of, 58;
    where does it go to? 199;
    generally, 48 et seq.

  Like effects in like circumstances, 199.

  Likeness, 388, 391.

  Lilliput, 84.

  Lines, straight, their symmetry, 98;
    curved, their asymmetry, 98;
    of force, 249.

  Links of thought, intermediate, 198.

  Liquefaction, latent heat of, 178.

  Liquid, efflux, law of, 150;
    equilibrium, figures of, 4 et seq.;
    the latter produced in open air, 7-8;
    their beauty and multiplicity of form, 7, 8;
    made permanent by melted colophonium, 7.

  Liquids, forms of, 1-16;
    difference between, and solids, 2;
    their mobility and adaptiveness of form, 3;
    the courtiers _par excellence_ of the natural bodies, 3;
    possess under certain circumstances forms of their own, 3.

  Living force, 137, 184;
    law of the conservation of, 149.

  Lloyd, observation of the conical refraction of light, 29.

  Lobster, of Lake Mohrin, the, 61.

  Localisation, cerebral, 210.

  Locke, on language and thought, 358.

  Locomotive, steam in the boiler of, 219.

  Loeb, J., 289, 291, 302.

  Logarithms, 195, 219;
    in music, 103-104.

  Logical root, of the principle of energy, 181;
    of the principle of excluded perpetual motion, 182.

  Lombroso, 280.

  Lucian, 347.


  _Macula acustica_, 272.

  Magic lantern, 96.

  Magic powers of nature, 189.

  Magical power of science, belief in the, 189.

  Magnet, a, 220;
    will compared to the pressure of a, 14;
    coercive force of a, 216.

  Magnetic needle, near a current, 207.

  Magnetised bar of steel, 242-243.

  Major and minor keys in music, 100 et seq.

  Malus, 242.

  Man, a fragment of nature's life, 49;
    his life embraces others, 234.

  Mann, 364.

  Manuscript in a mirror, 93.

  Maple syrup, statues of, on Moon, 4.

  Marx, 35.

  Material, the relations of work with heat and the consumption of, 245
  et seq.

  Mathematical methods, their character, 197-198.

  Mathematics, economy of, 195;
    on instruction in, 338-374;
    C. G. J. Jacobi on, 280.

  Matter, constancy of, 203;
    its nature, 203;
    the notion of, 213.

  Maupertuis, his Latin city, 338.

  Maximal and minimal problems, their rôle in physics, 14, footnote.

  Mayer, J. R., his conception of energy, 245, 246;
    his methods of thought, 247;
    on the conservation of energy, 163, 164, 165, 167, 183, 184;
    his metaphysical utterances, 183, 246;
    also 138, 184, 191, 217, 271, 274.

  Measurement, definition of, 206.

  Measures, international, 108.

  Mécanique céleste, 90, 188;
    sociale, and morale, the, 90.

  Mechanical, conception of the world, 105, 155 et seq., 188, 207;
    energy, W. Thomson on waste of, 175;
    analogies between ---- and thermal energy, 17 et seq.;
    equivalent of heat, electricity, etc., 164, 167 et seq.;
    mythology, 207;
    phenomena, physical events as, 182;
    philosophy, 188;
    physics, 155-160, 212;
    substitution-value of heat, 178, footnote.

  Mechanics, Kirchhoff's definition of, 236.

  Medicine, students of, 326.

  Melody, 101.

  Melsens, 310, 327.

  Memory, a treasure-house for comparison, 230;
    common elements impressed upon the, 180;
    its importance, 238;
    science disburdens the, 193.

  Mendelejeff, his periodical series, 256.

  Mental, adaptation, 214-235;
    completion of phenomena, 220;
    imagery, 253;
    imitation, our schematic, 199;
    processes, economical, 195;
    reproduction, 198;
    visualisation, 250.

  Mephistopheles, 88.

  Mercantile principle, a miserly, at the basis of science, 15.

  Mersenne, 377.

  Mesmerism, the mental state of ordinary minds, 228.

  Metaphysical establishment of doctrine of energy, 183.

  Metaphysical spooks, 222.

  Metrical, concepts of electricity, 107 et seq.;
    notions, energy and entropy are, 178;
    units, the building-stones of the physicist, 253.

  Metronomes, 41.

  Meyer, Lothar, his periodical series, 256.

  Middle Ages, 243, 349.

  Midsummer Night's Dream, 309.

  Mill, John Stuart, 230.

  Millers, school for, 326.

  Mill-wheel, doing work, 161.

  Mimicking facts in thought, 189, 193.

  Minor and major keys in music, 100 et seq.

  Mirror, symmetrical reversion of objects in, 92 et seq.

  Miserly mercantile principle at the basis of science, 15.

  Moat, child looking into, 208.

  Modern scientists, adherents of the mechanical philosophy, 188.

  Molecular theories, 104.

  Molecules, 203, 207.

  Molière, 234.

  Momentum, 184.

  Monocular vision, 98.

  Monotheism of the Christians and Jews, 187.

  Montagues and Capulets, 87.

  Moon, eclipse of, 219;
    lightness of bodies on, 4;
    the study of the, 90, 284.

  Moreau, 307.

  Mosaic of thought, 192.

  Motion, a perpetual, 181;
    quantity of, 184;
    the Eleatics on, 158;
    Wundt on, 158;
    the Herbartians on, 158.

  Motions, natural and violent, 226;
    their familiar character, 157.

  Mountains of the earth, would crumble if very large, 3;
    weight of bodies on, 112.

  Mozart, 44, 279.

  Müller, Johann, 291.

  Multiplication-table, 195.

  Multiplier, 132.

  Music, band of, its _tempo_ accelerated and retarded, 53;
    the principle of repetition in, 99 et seq.;
    its notation, mathematically illustrated, 103-104.

  Musical notes, reversion of, 101 et seq.;
    their economy, 192.

  Musical scale, a species of one-dimensional space, 105.

  Mystery, in physics, 222;
    science can dispense with, 189.

  Mysticism, numerical, 33;
    in the principle of energy, 184.

  Mythology, the mechanical, of philosophy, 207.


  Nagel, von, 364.

  Nansen, 296.

  Napoleon, picture representing the tomb of, 36.

  Nations, intercourse and ideas of, 336-337.

  Natural constants, 193.

  Natural law, a, not contained in the conformity of the energies, 175.

  Natural laws, abridged descriptions, 193;
    likened to type, 193.

  Natural motions, 225.

  Natural selection in scientific theories, 63, 218.

  Nature, experience the well-spring of all knowledge of, 181;
    fashions of, 64;
    first knowledge of, instinctive, 189;
    general interconnexion of, 182;
    has many sides, 217;
    her forces compared to purposes, 14-15;
    likened to a good man of business, 15;
    the economy of her actions, 15;
    how she appears to other animals, 83 et seq.;
    inquiry of, viewed as a torture, 48-49;
    view of, as something designedly concealed from man, 49;
    like a covetous tailor, 9-10;
    magic powers of, 189;
    our view of, modified by binocular vision, 82;
    the experimental method a questioning of, 48.

  Negro hamlet, the science of a, 237.

  Neptune, prediction and discovery of the planet, 29.

  New views, 296 et seq.

  Newton, describes polarisation, 242;
    expresses his wealth of thought in Latin, 341;
    his discovery of gravitation, 225 et seq.;
    his solution of dispersion, 362;
    his principle of the equality of pressure and counterpressure, 191;
    his view of light, 227-228;
    on absolute time, 204;
    selections from his works for use in instruction, 368;
    also 270, 274, 279, 285, 289.

  Nobility, they displace Latin, 342.

  Notation, musical, mathematically illustrated, 103-104.

  Numbers, economy of, 195;
    their connexion with consonance, 32.

  Numerical mysticism, 33.

  Nursery, the questions of the, 199.


  Observation, 310.

  Observation, in science, 261.

  Ocean-stream, 272.

  Oettingen, Von, 103.

  Ohm, on electric currents, 249.

  Ohm, the word, 343.

  Oil, alcohol, water, and, employed in Plateau's experiments, 4;
    free mass of, assumes the shape of a sphere, 12;
    geometrical figures of, 5 et seq.

  One-eyed people, vision of, 98.

  Ophthalmoscope, 18.

  Optic nerves, 96.

  Optimism and pessimism, 234.

  Order of physics, 197.

  Organ, bellows of an, 135.

  Organic nature, results of Darwin's studies of, 215 et seq.
    See _Adaptation_ and _Heredity_.

  Oriental world of fables, 273.

  Orientation, sensations of, 282 et seq.

  Oscillation, centre of, 147 et seq.

  Ostwald, 172.

  Otoliths, 301 et seq.

  Overtones, 28, 40, 349.

  Ozone, Schöbein's discovery of, 271.


  Painted things, the difference between real and, 68.

  Palestrina, 44.

  Parameter, 257.

  Partial tones, 390.

  Particles, smallest, 104.

  Pascheles, Dr. W., 285.

  Paulsen, 338, 340, 373.

  Pearls of life, strung on the individual as on a thread, 234-235.

  Pencil surpasses the mathematician in intelligence, 196.

  Pendulum, motion of a, 144 et seq.,
    increased motion of, due to slight impulses, 21;
    electrical, 110.

  Percepts, of like form, 390.

  Periodical, changes, 181;
    series, 256.

  Permanent, changes, 181, 199;
    elements of the world, 194.

  Perpetual motion, a, 181;
    defined, 139;
    impossibility of, 139 et seq.;
    the principle of the, excluded, 140 et seq.;
    excluded from general physics, 162.

  Personality, its nature, 234-235.

  Perspective, 76 et seq.;
    contraction of, 74 et seq.;
    distortion of, 77.

  Pessimism and optimism, 234.

  Pharaohs, 85.

  Phenomenology, a universal physical, 250.

  Philistine, modes of thought of, 223.

  Philology, comparison in, 239.

  Philosopher, an ancient, on the moral and physical sciences, 89.

  Philosophy, its character at all times, 186;
    mechanical, 155 et seq., 188, 207, 259 et seq.

  Phonetic alphabets, their economy, 192.

  Photography, by the electric spark, 318 et seq.

  Photography of projectiles, 309-337.

  Photography, stupendous advances of, 74.

  Physical, concepts, fetishism in our, 187;
    ideas and principles, their nature, 204;
    inquiry, the economical nature of, 186;
    research, object of 207, 209.

  Physical phenomena, as mechanical phenomena, 182;
    relations between, 205.

  Physico-mechanical view of the world, 155, 187, 188, 207 et seq.

  Physics, compared to a well-kept household, 197;
    economical experience, 197;
    the principles of, descriptive, 199;
    the methods of, 209;
    its method characterised, 211;
    comparison in, 239;
    the facts of, qualitatively homogeneous, 255;
    how it began, 37;
    helped by psychology, 104;
    study of its own character, 189;
    the goal of, 207, 209.

  Physiological psychology, its methods, 211 et seq.

  Physiology, its scope, 212.

  Piano, its mirrored counterpart, 100 et seq.;
    used to illustrate the facts of sympathetic vibration, 25 et seq.

  Piano-player, a speaker compared to, 192.

  Picture, physical, a, 110.

  Pike, learns by experience, 267.

  Pillars of Corti, 19.

  Places, heavy bodies seek their, 224 et seq.

  Planetary system, origin of, illustrated, 5.

  Plasticity of organic nature, 216.

  Plateau, his law of free liquid equilibrium, 9;
    his method of getting rid of the effects of gravity, 4.

  Plates of oil, thin, 6.

  Plato, 347, 371.

  Plautus, 347.

  Playfair, 138.

  Pleasant effects, cause of, 94 et seq.

  Pliny, 349.

  Poetry and science, 30, 31, 351.

  Poinsot, on the foundations of mechanics, 152 et seq.

  Polarisation, 91;
    abstractly described by Newton, 242.

  Politics, Chinese speak with unwillingness of, 374.

  Pollak, 299.

  Polyp plant, humanity likened to a, 235.

  Pompeii, 234;
    art in, 80.

  Popper J., 172, 216.

  Potential, social, 15;
    electrical, 121 et seq.;
    measurement of, 126;
    fall of, 177;
    swarm of notions in the idea of, 197;
    its wide scope, 250.

  Pottery, invention of, 263.

  Prediction, 221 et seq.

  Prejudice, the function, power, and dangers of, 232-233.

  Preparatory schools, the defects of the German, 346-347;
    what they should teach, 364 et seq.

  Pressure of a stone or of a magnet, will compared to, 14;
    also 157.

  Primitive acts of knowledge the foundation of scientific thought, 190.

  Problem, nature of a, 223.

  Problems which are wrongly formulated, 308.

  Process, Carnot's, 161 et seq.

  Projectiles, the effects of the impact of, 310, 327-328;
    seen with the naked eye, 311, 317;
    measuring the velocity of, 332;
    photography of, 309-337.

  Prony's brake, 132.

  Proof, nature of, 284.

  Prophesying events, 220 et seq.

  Psalms, quotation from the, 89.

  Pseudoscope, Wheatstone's, 96.

  Psychology, preceded by astronomy, 90;
    how reached, 91 et seq.;
    helps physical science, 104;
    its method the same as that of physics, 207 et seq.

  Pully arrangement, illustrating principle of least superficial area,
  12-13.

  Purkinje, 284, 285, 291, 299.

  Purposes, the acts of nature compared to, 14-15;
    nature pursues no, 66.

  Puzzle-lock, a, 26.

  Puzzles, 277.

  Pyramid of oil, 6.

  Pythagoras, his discovery of the laws of harmony, 32, 259.


  Quality of tones, 36.

  Quantitative investigation, the goal of, 180.

  Quantity of electricity, 111, 118, 119, 167-170, 173;
    of heat, 166, 167-171, 174, 177, 244;
    of motion, 184.

  Quests made of the inquirer, not by him, 30.

  Quételet, 15, footnote.


  Rabelais, 283.

  Raindrop, form of, 3.

  Rameau, 34.

  Reaction and action, principle of, 191.

  Reactions, disclosure of the connexion of, 270 et seq.

  Realgymnasien, 365.

  Realschulen, 365, 373.

  Reason, stands above the senses, 105.

  Reflex action, 210.

  Reflexion, produces symmetrical reversion of objects, 93 et seq.

  Refraction, 29, 193, 194, 208, 230, 231.

  Reger, 328.

  Reliefs, photographs of, 68.

  Repetition, its rôle in æsthetics, 89, footnote, 91 et seq., 97, 98
  et seq., 390.

  Reproduction of facts in thought, 189, 193, 198, 253.

  Repulsion, electric, 109 et seq., 168.

  Research, function of experimental 181;
    the aim of, 205.

  Resemblances between facts, 255.

  Resin, solution of, 7.

  Resistance, laws of, for bodies travelling in air and fluids, 333 et seq.

  Resonance, corporeal, 392.

  Response of sonorous bodies, 25.

  Retina, the corresponding spots of 98;
    nerves of compared to fingers of a hand, 96 et seq.

  Reversible processes, 161 et seq., 175, 176, 181, 182.

  Rhine, the, 94.

  Richard the Third, 77.

  Riddles, 277.

  Riders, 379.

  Riegler, 319.

  Riess, experiment with the thermo-electrometer, 133 et seq., 169.

  Rigid connexions, 142.

  Rind of a fruit, 190.

  Rings of oil, illustrating formation of rings of Saturn, 5.

  Ritter, 291, 299.

  Rods of Corti, 19.

  Rolph, W. H., 216.

  Roman Church, Latin introduced with the, 340 et seq.

  Romans, their provinciality and narrow-mindedness, 270.

  Romeo and Juliet, 87.

  Römer, Olaf, 51 et seq.

  Roots, the nature of, in language, 252.

  Rosetti, his experiment on the work required to develop electricity, 131.

  Rotating bodies, 285.

  Rotation, apparatus of, in physics, 59 et seq.;
    sensations of, 288 et seq.

  Rousseau, 336.

  Rubber pyramid, illustrating the principle of least superficial area,
  10-11.

  Ruysdael, 279.


  Sachs, Hans, 106.

  Salcher, Prof. 319.

  Salviati, 144.

  Saturn, rings of, their formation illustrated, 5.

  Saurians, 257.

  Sauveur, on acoustics, 34, 375 et seq.

  Savage, modes of conception and interpretation of a, 218 et seq.

  Schäfer, K., 298.

  _Schlierenmethode_, 317.

  Schönbein's discovery of ozone, 271.

  School-boy, copy-book of, 92.

  Schoolmen, 214.

  Schools, State-control of, 372 et seq.

  Schopenhauer, 190.

  Schultze, Max, 19.

  Science, a miserly mercantile principle at its basis, 15;
    compared to a business, 16;
    viewed as a maximum or minimum problem, 16, footnote;
    its process not greatly different from the intellectual activity of
    ordinary life, 16, footnote;
    economy of its task, 16;
    relation of, to poetry, 30, 31, 351;
    the church of, 67;
    beginnings of, 189, 191;
    belief in the magical power of, 189;
    can dispense with mystery, 189;
    lavish extravagance of, 189;
    economy of the terminology of, 192;
    partly made up of the intelligence of others, 196;
    stripped of mystery, 197;
    its true power, 197;
    the economical schematism of, 206;
    the object of, 206;
    the tools of, 207;
    does not create facts, 211;
    of the future, 213;
    revolution in, dating from Galileo, 214 et seq.;
    the natural foe of the marvellous, 224;
    characterised, 227;
    growth of, 237;
    dramatic element in, 243;
    described, 251;
    its function, 253;
    classification in, 255, 259 et seq.;
    the way of discovery in, 316.
    See also _Physics_.

  Sciences, partition of the, 86;
    the barriers and relations between the 257-258;
    on instruction in the, 338-374.

  Scientific, criticism, Socrates the father of, 1, 16;
    discoveries, their fate, 138;
    knowledge, involves description, 193;
    thought, transformation and adaptation in, 214-235;
    thought, advanced by new experiences, 223 et seq.;
    thought, the difficulty of, 366;
    terms, 342-343;
    founded on primitive acts of knowledge, 190.

  Scientists, stories about their ignorance, 342.

  Screw, the, 62.

  Sea-sickness, 284.

  Secret computation, Leibnitz's, 33.

  Seek their places, bodies, 226.

  Self-induction, coefficient of, 250, 252.

  Self-observation, 211.

  Self-preservation, our first knowledge derived from the economy of, 197;
    struggle for, among ideas, 228.

  Semi-circular canals, 290 et seq.

  Sensation of rounding a railway curve, 286.

  Sensations, analysed, 251;
    when similar, produce agreeable effects, 96;
    their character, 200;
    defined, 209;
    of orientation, 282 et seq.

  Sense-elements, 179.

  Senses, theory of, 104;
    the source of our knowledge of facts, 237.

  Seventh, the troublesome, 46.

  Shadow method, 313 et seq., 317 footnote.

  Shadows, rôle of, in vision, 81.

  Shakespeare, 278.

  Sharps, reversed into flats, 101.

  Shell, spherical, law of attraction for a, 124, footnote.

  Shoemaker, inquirer compared to, 105-106.

  Shooting, 309.

  Shots, double report of, 229 et seq.

  Similarity, 249.

  Simony, 280.

  Simplicity, a varying element in description, 254.

  Sines, law of the, 193.

  Sinking of heavy bodies, 222.

  Sixth sense, 297.

  Smith, R., on acoustics, 34, 381, 383.

  Soap-films, Van der Mensbrugghe's experiment with, 11-12.

  Soapsuds, films and figures of, 7.

  Social potential, 15.

  Socrates, the father of scientific criticism, 1, 16.

  Sodium, 202.

  Sodium-light, vibrations of, as a measure of time, 205.

  Solidity, conception of, by the eye, 71 et seq.;
    spatial, photographs of, 73.

  Solids, and liquids, their difference merely one of degree, 2.

  Sonorous bodies, 24 et seq.

  Soret, J. P., 89.

  Sounds, symmetry of, 99 et seq.;
    generally, 22-47, 212.

  Sound-waves rendered visible, 315 et seq.

  Sources of the principle of energy, 179 et seq.

  Space, 205;
    sensation of, 210.

  Spark, electric, 117, 127, 132, 133, 190.

  Spatial vision, 386.

  Species, stability of, a theory, 216.

  Specific energies, 291.

  Specific heat, 166, 244.

  Specific inductive capacity, 117.

  Spectral analysis of sound, 27.

  Spectrum, mental associations of the, 190.

  Speech, the instinct of, cultivated by languages, 354.

  Spencer, 218, 222.

  Sphere, a soft rotating, 2;
    the figure of least surface, 12;
    electrical capacity of, 123 et seq.

  Spherical shell, law of attraction for 124, footnote.

  Spiders, the eyes of, 67.

  Spirits, as explanation of the world 186, 243.

  Spiritualism, modern, 187.

  Spooks, metaphysical, 222.

  Squinting, 72.

  Stability of our environment, 206.

  Stallo, 336.

  Stars, the fixed, 90.

  State, benefits and evils of its control of the schools, 372 et seq.;
    the Church and, 88.

  Statical electricity, 134.

  Stationary currents, 249.

  Statoliths, 303.

  Steam-engine, 160, 265.

  Steeple-jacks, 75.

  Stereoscope, Wheatstone and Brewster's, 73.

  Stevinus, on the inclined plane, 140;
    on hydrostatics, 141;
    on the equilibrium of systems, 142;
    discovers the principle of virtual velocities, 150;
    characterisation of his thought, 142;
    also 182, 187, 191.

  Stone Age, 46, 321.

  Störensen, 306.

  Stove, primitive, 263.

  Straight line, a, its symmetry, 98.

  Straight, meaning of the word, 240.

  Street, vista into a, 75.

  Striae, in glass, 313.

  Striate method, for detecting optical imperfections, 317.

  Striking distance, 115, 127.

  Strings, vibrations of, 249.

  Struggle for existence among ideas, 217.

  Substance, heat conceived as a, 177, 243 et seq.;
    electricity as a, 170;
    the source of our notion of, 199;
    rôle of the notion of, 203, 244 et seq.;
    energy conceived as a, 164, 185, 244 et seq.

  Substitution-value of heat, 178, footnote.

  Suetonius, 348.

  Sulphur, specific inductive capacity of, 117.

  Sun, human beings could not exist on, 3.

  Swift, 84, 280.

  Swimmer, Ampère's, 207.

  Symmetry, definition of, 92;
    figures of, 92 et seq.;
    plane of, 94;
    vertical and horizontal, 94;
    in music, 99 et seq.

  Sympathetic vibration, 22 et seq., 379.


  Tailor, nature like a covetous, 9-10.

  Tangent, the word, 263.

  Taste, doubtful cultivation of, by the classics, 352-353;
    of the ancients, 353.

  Taylor, on the vibration of strings, 249.

  Teaching, its nature, 366 et seq.

  Telegraph, the word, 263.

  Telescope, 262.

  Telestereoscope, the, 84.

  Temperament, even, in tuning, 47.

  Temperature, absolute, 162;
    differences of, 205;
    differences of, viewed as level surfaces, 161;
    heights of, 174;
    scale of, derived from tensions of gases, 174.

  Terence, 347.

  Terms, scientific, 342-343.

  Thales, 259.

  Theories, their scope, function, and power, 241-242;
    must be replaced by direct description, 248.

  Thermal, energy, 174, 177;
    capacity, 123, footnote.

  Thermodynamics, 160 et seq.

  Thermoelectrometer, Riess's, 133, 169.

  Thing-in-itself, the, 200.

  Things, mental symbols for groups of sensations, 200-201.

  Thomson, James, on the lowering of the freezing-point of water by
  pressure, 162.

  Thomson, W., his absolute electrometer, 127, footnote;
    on thermodynamics, 162;
    on the conservation of energy, 165;
    on the mechanical measures of temperature, 174, footnote;
    on waste of mechanical energy, 175;
    also 108, 173, footnote.

  Thought, habitudes of, 199, 224, 227, 232;
    relationship between language and, 329;
    incongruence between experience and, 206;
    luxuriance of a fully developed, 58;
    transformation in scientific, 214-235.

  Thoughts, their development and the struggle for existence among them,
  63;
    importance of erroneous, 65;
    as reproductions of facts, 107.

  Thread, the individual a, on which pearls are strung, 234-235.

  Tides, 283.

  Timbre, 37, 38, 39.

  Time, 178, 204, 205, footnote.

  Toepler and Foucault, method of, for detecting optical faults, 313
  et seq., 320.

  Tone-figures, 91.

  Tones, 22-47, 99 et seq., 212.

  Torsion, moment of, 132.

  Torsion-balance, Coulomb's, 109, 168.

  Torricelli, on virtual velocities, 150;
    his law of liquid efflux, 150;
    on the atmosphere, 273.

  Tourist, journey of, work of the inquirer compared to, 17, 29, 30.

  Transatlantic cable, 108.

  Transformation and adaptation in scientific thought, 214-235.

  Transformation of ideas, 63.

  Transformative law of the energies, 172.

  Translation, difficulties of, 354.

  Tree, conceptual life compared to a, 231.

  Triangle, mutual dependence of the sides and angles of a, 179.

  Triple accord, 46.

  Truth, wooed by the inquirer, 45;
    difficulty of its acquisition, 46.

  Tumblers, resounding, 23.

  Tuning-forks, explanation of their motion, 22 et seq.

  Tylor, 186.

  Tympanum, 18.

  Type, natural laws likened to, 193;
    words compared to, 191.


  Ulysses, 347.

  Understanding, what it means, 211.

  Uniforms, do not fit heads, 369.

  Unique determination, 181-182.

  Unison, 43.

  Unit, electrostatic, 111.
    See _Force_ and _Work_.

  United States, 336.

  Universal Real Character, a, 192.

  Utility of physical science, 351.


  Variation, the method of, in science, 230;
    in biology, 216.

  Velocity, of light, 48 et seq.;
    of the descent of bodies, 143 et seq.;
    meaning of, 204;
    virtual, 149-155.

  _Verstandesbegriffe_, 199.

  Vertical, perception of the, 272, 286 et seq.;
    symmetry, 389.

  Vertigo, 285, 290.

  Vestibule of the ear, 300.

  Vibration, 22 et seq.

  Vibration-figures, 91.

  Vinci, Leonardo da, 278, 283.

  Violent motions, 225.

  Virtual velocities, 149-155.

  Visibility, general conditions of, 312.

  Vision, symmetry of our apparatus of, 96.
    See _Eye_.

  Visual nerves, 96.

  Visualisation, mental, 250.

  Volt, the word, 343.

  Volta, 127, footnote, 134.

  Voltaire, 260.

  Voltaire's ingènu, 219.

  Vowels, composed of simple musical notes, 26.


  Wagner, Richard, 279.

  Wald, F., 178, footnote.

  Wallace, 216.

  War, and peace, reflexions upon, 309, 335 et seq.

  Waste of mechanical energy, W. Thomson on, 175.

  Watches, experiment with, 41;
    in a mirror, 93.

  Water, jet of, resolved into drops, 60;
    free, solid figures of, 8;
    objects reflected in, 94, 191;
    possible modes of measurement of, 170.

  Watt, 266.

  Wealth, the foundation of, 198.

  Weapons, modern, 335.

  Weber, 108, 306.

  Weight of bodies, varies with their distance from the centre of the
  earth, 112.

  Weismann, 216.

  Wheatstone, his stereoscope, 73;
    his pseudoscope, 96;
    also 59.

  Wheel, history and importance of, 61 et seq.

  Whewell, on the formation of science, 231.

  Whole, the, 204, footnote.

  Why, the question, 199, 223.

  Will, Schopenhauer on the, 190;
    man's most familiar source of power, 243;
    used to explain the world, 186;
    forces compared to, 254;
    compared to pressure, 14.

  Windmill, a rotating, 53.

  Wire frames and nets, for constructing liquid figures of equilibrium,
  4 et seq.

  Witchcraft, 187.

  Wollaston, 284, 285.

  Wonderful, science the natural foe of the, 224.

  Woods, the relative distance of trees in, 68.

  Wooer, inquirer compared to a, 45.

  Words and sounds, 343.

  Words, compared to type, 191.

  Work, of liquid forces of attraction, 14;
    in electricity, 173;
    measure of, 119 et seq., 130, 223;
    relation of, with heat, 162, 245 et seq.;
    amount required to develop electricity, 131 et seq.;
    produces various physical changes, 139;
    substantial conception of, 183-184.
    See _Energy_.

  World, the, what it consists of, 208.

  World-particles, 203.

  Wronsky, 172.

  Wundt, on causality and the axioms of physics, 157-159; 359 footnote.


  Xenophon, 49, footnote.


  Young, Thomas, on energy, 173.


  Zelter, 35.

  Zeuner, 171.

  Zoölogy, comparison in, 239.



THE WORKS OF ERNST MACH.

THE SCIENCE OF MECHANICS.

A CRITICAL AND HISTORICAL EXPOSITION OF ITS PRINCIPLES.

By DR. ERNST MACH.

PROFESSOR OF THE HISTORY AND THEORY OF INDUCTIVE SCIENCE IN THE
UNIVERSITY OF VIENNA.

Translated from the Second German Edition By THOMAS J. McCORMACK.


250 Cuts. 534 Pages. Half Morocco, Gilt Top, Marginal Analyses.

Exhaustive Index. Price $2.50.



TABLE OF CONTENTS.


STATICS.

     The Lever.

     The Inclined Plane.

     The Composition of Forces.

     Virtual Velocities.

     Statics in Their Application to Fluids.

     Statics in Their Application to Gases.


DYNAMICS.

     Galileo's Achievements.

     Achievements of Huygens.

     Achievements of Newton.

     Principle of Reaction.

     Criticism of the Principle of Reaction and of the Concept of Mass.

     Newton's Views of Time, Space, and Motion.

     Critique of the Newtonian Enunciations.

     Retrospect of the Development of Dynamics.


THE EXTENSION OF THE PRINCIPLES OF MECHANICS.

     Scope of the Newtonian Principles.

     Formulæ and Units of Mechanics.

     Conservation of Momentum, Conservation of the Centre of Gravity,
     and Conservation of Areas.

     Laws of Impact.

     D'Alembert's Principle.

     Principle of _Vis Viva_.

     Principle of Least Constraint.

     Principle of Least Action.

     Hamilton's Principle.

     Hydrostatic and Hydrodynamic Questions.


FORMAL DEVELOPMENT OF MECHANICS.

     The Isoperimetrical Problems.

     Theological, Animistic, and Mystical Points of View in Mechanics.

     Analytical Mechanics.

     The Economy of Science.


THE RELATION OF MECHANICS TO OTHER DEPARTMENTS OF KNOWLEDGE.

     Relations of Mechanics to Physics.

     Relations of Mechanics to Physiology.



PRESS NOTICES.


"The appearance of a translation into English of this remarkable book
should serve to revivify in this country [England] the somewhat
stagnating treatment of its subject, and should call up the thoughts
which puzzle us when we think of them, and that is not sufficiently
often.... Professor Mach is a striking instance of the combination of
great mathematical knowledge with experimental skill, as exemplified not
only by the elegant illustrations of mechanical principles which abound
in this treatise, but also from his brilliant experiments on the
photography of bullets.... A careful study of Professor Mach's work, and
a treatment with more experimental illustration, on the lines laid down
in the interesting diagrams of his _Science of Mechanics_, will do much
to revivify theoretical mechanical science, as developed from the
elements by rigorous logical treatment."--Prof. A. G. Greenhill, in
_Nature_, London.

"Those who are curious to learn how the principles of mechanics have
been evolved, from what source they take their origin, and how far they
can be deemed of positive and permanent value, will find Dr. Mach's able
treatise entrancingly interesting.... The book is a remarkable one in
many respects, while the mixture of history with the latest scientific
principles and absolute mathematical deductions makes it exceedingly
attractive."--_Mechanical World_, Manchester and London, England.

"Mach's Mechanics is unique. It is not a text-book, but forms a useful
supplement to the ordinary text-book. The latter is usually a skeleton
outline, full of mathematical symbols and other abstractions. Mach's
book has 'muscle and clothing,' and being written from the historical
standpoint, introduces the leading contributors in succession, tells
what they did and how they did it, and often what manner of men they
were. Thus it is that the pages glow, as it were, with a certain
humanism, quite delightful in a scientific book.... The book is
handsomely printed, and deserves a warm reception from all interested in
the progress of science."--_The Physical Review_, New York and London.

"Mr. T. J. McCormack, by his effective translation, where translation
was no light task, of this masterly treatise upon the earliest and most
fundamental of the sciences, has rendered no slight service to the
English speaking student. The German and English languages are generally
accounted second to none in their value as instruments for the
expression of scientific thought; but the conversion bodily of an
abstruse work from one into the other, so as to preserve all the meaning
and spirit of the original and to set it easily and naturally into its
new form, is a task of the greatest difficulty, and when performed so
well as in the present instance, merits great commendation. Dr. Mach has
created for his own works the severest possible standard of judgment. To
expect no more from the books of such a master than from the elementary
productions of an ordinary teacher in the science would be undue
moderation. Our author has lifted what, to many of us, was at one time a
course of seemingly unprofitable mental gymnastics, encompassed only at
vast expenditure of intellectual effort, into a study possessing a deep
philosophical value and instinct with life and interest. 'No profit
grows where is no pleasure ta'en,' and the emancipated collegian will
turn with pleasure from the narrow methods of the text-book to where the
science is made to illustrate, by a treatment at once broad and deep,
the fundamental connexion between all the physical sciences, taken
together."--_The Mining Journal_, London, England.

"As a history of mechanics, the work is admirable."--_The Nation_, New
York.

"An excellent book, admirably illustrated."--_The Literary World_,
London, England.

"Sets forth the elements of its subject with a lucidity, clearness, and
force unknown in the mathematical text-books ... is admirably fitted to
serve students as an introduction on historical lines to the principles
of mechanical science."--_Canadian Mining and Mechanical Review_,
Ottawa, Can.

"A masterly book.... To any one who feels that he does not know as much
as he ought to about physics, we can commend it most heartily as a
scholarly and able treatise ... both interesting and profitable."--A. M.
Wellington, in _Engineering News_, New York.

"The book as a whole is unique, and is a valuable addition to any
library of science or philosophy.... Reproductions of quaint old
portraits and vignettes give piquancy to the pages. The numerous
marginal titles form a complete epitome of the work; and there is that
invaluable adjunct, a good index. Altogether the publishers are to be
congratulated upon producing a technical work that is thoroughly
attractive in its make-up."--Prof. D. W. Hering, in _Science_.

"There is one other point upon which this volume should be commended,
and that is the perfection of the translation. It is a common fault that
books of the greatest interest and value in the original are oftenest
butchered or made ridiculous by a clumsy translator. The present is a
noteworthy exception."--_Railway Age_.

"The book is admirably printed and bound.... The presswork is
unexcelled by any technical books that have come to our hands for some
time, and the engravings and figures are all clearly and well
executed."--_Railroad Gazette_.



TESTIMONIALS OF PROMINENT EDUCATORS.


"I am delighted with Professor Mach's _Science of Mechanics_."--_M. E.
Cooley_, Professor of Mechanical Engineering, Ann Arbor, Mich.

"You have done a good service to science in publishing Mach's _Science
of Mechanics_ in English. I shall take every opportunity to recommend it
to young students as a source of much interesting information and
inspiration."--_M. I. Pupin_, Professor of Mechanics, Columbia College,
New York.

"Mach's _Science of Mechanics_ is an admirable ... book."--_Prof. E. A.
Fuertes_, Director of the College of Civil Engineering of Cornell
University, Ithaca, N. Y.

"I congratulate you upon producing the work in such good style and in so
good a translation. I bought a copy of it a year ago, very shortly after
you issued it. The book itself is deserving of the highest admiration;
and you are entitled to the thanks of all English-speaking physicists
for the publication of this translation."--_D. W. Hering_, Professor of
Physics, University of the City of New York, New York.

"I have read Mach's _Science of Mechanics_ with great pleasure. The book
is exceedingly interesting."--_W. F. Magie_, Professor of Physics,
Princeton University, Princeton, N. J.

"The _Science of Mechanics_ by Mach, translated by T. J. McCormack, I
regard as a most valuable work, not only for acquainting the student
with the history of the development of Mechanics, but as serving to
present to him most favorably the fundamental ideas of Mechanics and
their rational connexion with the highest mathematical developments. It
is a most profitable book to read along with the study of a text-book of
Mechanics, and I shall take pleasure in recommending its perusal by my
students."--_S. W. Robinson_, Professor of Mechanical Engineering, Ohio
State University, Columbus, Ohio.

"I am delighted with Mach's 'Mechanics.' I will call the attention to it
of students and instructors who have the Mechanics or Physics to study
or teach."--_J. E. Davies_, University of Wisconsin, Madison, Wis.

"There can be but one opinion as to the value of Mach's work in this
translation. No instructor in physics should be without a copy of
it."--_Henry Crew_, Professor of Physics in the Northwestern University,
Evanston, Ill.



POPULAR SCIENTIFIC LECTURES.

A PORTRAYAL OF THE SPIRIT AND METHODS OF SCIENCE.

By DR. ERNST MACH.

PROFESSOR OF THE HISTORY AND THEORY OF INDUCTIVE SCIENCE IN THE
UNIVERSITY OF VIENNA.

Translated by THOMAS J. McCORMACK.

_Third Edition, Revised Throughout and Greatly Enlarged._


Cloth, Gilt Top. Exhaustively Indexed. Pages, 415. Cuts, 59. Price,
$1.50.



TITLES OF THE LECTURES.


     The Forms of Liquids.

     The Fibres of Corti.

     On the Causes of Harmony.

     On the Velocity of Light.

     Why Has Man Two Eyes?

     On Symmetry.

     On the Fundamental Concepts of Static Electricity.

     On the Principle of the Conservation of Energy.

     On the Economical Nature of Physical Inquiry.

     On the Principle of Comparison in Physics.

     On the Part Played by Accident in Invention and Discovery.

     On Sensations of Orientation.

     On the Relative Educational Value of the Classics and the
     Mathematico-Physical Sciences.

     A Contribution to the History of Acoustics.

     Remarks on the Theory of Spatial Vision.

     On Transformation and Adaptation in Scientific Thought.


PRESS NOTICES.

"A most fascinating volume, treating of phenomena in which all are
interested, in a delightful style and with wonderful clearness. For
lightness of touch and yet solid value of information the chapter 'Why
Has Man Two Eyes?' has scarcely a rival in the whole realm of popular
scientific writing."--_The Boston Traveller_.

"Truly remarkable in the insight they give into the relationship of the
various fields cultivated under the name of Physics.... A vein of humor
is met here and there reminding the reader of Heaviside, never offending
one's taste. These features, together with the lightness of touch with
which Mr. McCormack has rendered them, make the volume one that may be
fairly called rare. The spirit of the author is preserved in such
attractive, really delightful, English that one is assured nothing has
been lost by translation."--Prof. Henry Crew, in _The Astrophysical
Journal_.

"A very delightful and useful book.... The author treats some of the
most recondite problems of natural science, in so charmingly untechnical
a way, with such a wealth of bright illustration, as makes his meaning
clear to the person of ordinary intelligence and education.... This is a
work that should find a place in every library, and that people should
be encouraged to read."--_Daily Picayune_, New Orleans.

"In his translation Mr. McCormack has well preserved the frank, simple,
and pleasing style of this famous lecturer on scientific topics.
Professor Mach deals with the live facts, the salient points of science,
and not with its mysticism or dead traditions. He uses the simplest of
illustrations and expresses himself clearly, tersely, and with a
delightful freshness that makes entertaining reading of what in other
hands would be dull and prosy."--_Engineering News_, N. Y.

"The general reader is led by plain and easy steps along a delightful
way through what would be to him without such a help a complicated maze
of difficulties. Marvels are invented and science is revealed as the
natural foe to mysteries."--_The Chautauquan_.

"The beautiful quality of the work is not marred by abstruse discussions
which would require a scientist to fathom, but is so simple and so clear
that it brings us into direct contact with the matter treated."--_The
Boston Post_.

"A masterly exposition of important scientific truths."--_Scotsman_,
Edinburgh.

"These lectures by Dr. Mach are delightfully simple and frank; there is
no dryness or darkness of technicalities, and science and common life do
not seem separated by a gulf.... The style is admirable, and the whole
volume seems gloriously alive and human."--_Providence Journal_, R. I.

"The non-scientific reader who desires to learn something of modern
scientific theories, and the reasons for their existence, cannot do
better than carefully study these lectures. The English is excellent
throughout, and reflects great credit on the translator."--_Manufacturer
and Builder_.

"We like the quiet, considerate intelligence of these
lectures."--_Independent_, New York.

"Professor Mach's lectures are so pleasantly written and illumined with
such charm of illustration that they have all the interest of lively
fiction."--_New York Com. Advertiser_.

"The literary and philosophical suggestiveness of the book is very
rich." _Hartford Seminary Record_.

"All are presented so skilfully that one can imagine that Professor
Mach's hearers departed from his lecture-room with the conviction that
science was a matter for abecedarians. Will please those who find the
fairy tales of science more absorbing than fiction."--_The Pilot_,
Boston.

"Professor Mach ... is a master in physics.... His book is a good one
and will serve a good purpose, both for instruction and
suggestion."--Prof. A. E. Dolbear, in _The Dial_.

"The most beautiful ideas are unfolded in the exposition."--_Catholic
World_, New York.



THE ANALYSIS OF THE SENSATIONS

By DR. ERNST MACH.

PROFESSOR OF THE HISTORY AND THEORY OF INDUCTIVE SCIENCE IN THE
UNIVERSITY OF VIENNA.


Pages, 208. Illustrations, 37. Indexed.

(Price, Cloth, $1.25.)



CONTENTS.


     Introductory: Antimetaphysical.

     The Chief Points of View for the Investigation of the Senses.

     The Space-Sensations of the Eye.

     Space-Sensation, Continued.

     The Relations of the Sight-Sensations to One Another and to the
     Other Psychical Elements.

     The Sensation of Time.

     The Sensation of Sound.

     Influence of the Preceding Investigations on the Mode of Conceiving
     Physics.


"A wonderfully original little book. Like everything he writes a work of
genius."--_Prof. W. James_ of Harvard.

"I consider each work of Professor Mach a distinct acquisition to a
library of science."--_Prof. D. W. Hering_, New York University.

"There is no work known to the writer which, in its general scientific
bearings, is more likely to repay richly thorough study. We are all
interested in nature in one way or another, and our interests can only
be heightened and clarified by Mach's wonderfully original and wholesome
book. It is not saying too much to maintain that every intelligent
person should have a copy of it,--and should study that copy."--_Prof.
J. E. Trevor_, Cornell.

"Students may here make the acquaintance of some of the open questions
of sensation and at the same time take a lesson in the charm of
scientific modesty that can hardly be excelled."--_Prof. E. C. Sanford_,
Clark University.

"It exhibits keen observation and acute thought, with many new and
interesting experiments by way of illustration. Moreover, the style is
light and even lively--a rare merit in a German prose work, and still
rarer in a translation of one."--_The Literary World_, London.


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  Transcriber's note:

  _Underscores_ have been used to indicate _italic_ fonts.





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