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.