Popular scientific lectures
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Title: Popular scientific lectures
Author: Ernst Mach
Translator: Thomas J. McCormack
Release date: April 22, 2012 [eBook #39508]
Language: English
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*** START OF THE PROJECT GUTENBERG EBOOK POPULAR SCIENTIFIC LECTURES ***
Produced by Anna Hall, Albert László and the Online
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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
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