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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.