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This eBook includes 5 papers or speeches by James Clerk Maxwell.
The contents are:
Foramen Centrale
Theory of Compound Colours
Poinsot's Theory
Address to the Mathematical
Introductory Lecture
On the Unequal Sensibility of the Foramen Centrale to Light of
different Colours.
James Clerk Maxwell
[From the _Report of the British Association_, 1856.]
When observing the spectrum formed by looking at a long ve rtical slit
through a simple prism, I noticed an elongated dark spot running up
and down in the blue, and following the motion of the eye as it moved
_up and down_ the spectrum, but refusing to pass out of the blue into
the other colours. It was plain that the spot belonged both to the
eye and to the blue part of the spectrum. The result to which I have
come is, that the appearance is due to the yellow spot on the retina,
commonly called the _Foramen Centrale_ of Soemmering. The most
convenient method of observing the spot is by presenting to the eye in
not too rapid succession, blue and yellow glasses, or, still better,
allowing blue and yellow papers to revolve slowly before the eye. In
this way the spot is seen in the blue. It fades rapidly, but is
renewed every time the yellow comes in to relieve the effect of the
blue. By using a Nicol's prism along with this apparatus, the brushes
of Haidinger are well seen in connexion with the spot, and the fact of
the brushes being the spot analysed by polarized light becomes
evident. If we look steadily at an object behind a series of bright
bars which move in front of it, we shall see a curious bending of the
bars as they come up to the place of the yellow spot. The part which
comes over the spot seems to start in advance of the rest of the bar,
and this would seem to indicate a greater rapidity of sensation at the
yellow spot than in the surrounding retina. But I find the experiment
difficult, and I hope for better results from more accurate observers.
On the Theory of Compound Colours with reference to Mixtures of
Blue and Yellow Light.
James Clerk Maxwell
[From the _Report of the British Association_, 1856.]
When we mix together blue and yellow paint, we obtain green paint.
This fact is well known to all who have handled colours; and it is
universally admitted that blue and yellow make green. Red, yellow,
and blue, being the primary colours among painters, green is regarded
as a secondary colour, arising from the mixture of blue and yellow.
Newton, however, found that the green of the spectrum was not the same
thing as the mixture of two colours of the spectrum, for such a
mixture could be separated by the prism, while the green of the
spectrum resisted further decomposition. But still it was believed
that yellow and blue would make a green, though not that of the
spectrum. As far as I am aware, the first experiment on the subject
is that of M. Plateau, who, before 1819, made a disc with alternate
sectors of prussian blue and gamboge, and observed that, when
spinning, the resultant tint was not green, but a neutral gray,
inclining sometimes to yellow or blue, but never to green.
Prof. J. D. Forbes of Edinburgh made similar experiments in 1849, with
the same result. Prof. Helmholtz of Konigsberg, to whom we owe the
most complete investigation on visible colour, has given the true
explanation of this phenomenon. The result of mixing two coloured
powders is not by any means the same as mixing the beams of light
which flow from each separately. In the latter case we receive all
the light which comes either from the one powder or the other. In the
former, much of the light coming from one powder falls on particles of
the other, and we receive only that portion which has escaped
absorption by one or other. Thus the light coming from a mixture of
blue and yellow powder, consists partly of light coming directly from
blue particles or yellow particles, and partly of light acted on by
both blue and yellow particles. This latter light is green, since the
blue stops the red, yellow, and orange, and the yellow stops the blue
and violet. I have made experiments on the mixture of blue and yellow
light--by rapid rotation, by combined reflexion and transmission, by
viewing them out of focus, in stripes, at a great distance, by
throwing the colours of the spectrum on a screen, and by receiving
them into the eye directly; and I have arranged a portable apparatus
by which any one may see the result of this or any other mixture of
the colours of the spectrum. In all these cases blue and yellow do
not make green. I have also made experiments on the mixture of
coloured powders. Those which I used principally were "mineral blue"
(from copper) and "chrome-yellow." Other blue and yellow pigments gave
curious results, but it was more difficult to make the mixtures, and
the greens were less uniform in tint. The mixtures of these colours
were made by weight, and were painted on discs of paper, which were
afterwards treated in the manner described in my paper "On Colour as
perceived by the Eye," in the _Transactions of the Royal Society of
Edinburgh_, Vol. XXI. Part 2. The visible effect of the colour is
estimated in terms of the standard-coloured papers:--vermilion (V),
ultramarine (U), and emerald-green (E). The accuracy of the results,
and their significance, can be best understood by referring to the
paper before mentioned. I shall denote mineral blue by B, and
chrome-yellow by Y; and B3 Y5 means a mixture of three parts blue and
five parts yellow.
Given Colour. Standard Colours. Coefficient
V. U. E. of brightness.
B8 , 100 = 2 36 7 ............ 45
B7 Y1, 100 = 1 18 17 ............ 37
B6 Y2, 100 = 4 11 34 ............ 49
B5 Y3, 100 = 9 5 40 ............ 54
B4 Y4, 100 = 15 1 40 ............ 56
B3 Y5, 100 = 22 - 2 44 ............ 64
B2 Y6, 100 = 35 -10 51 ............ 76
B1 Y7, 100 = 64 -19 64 ............ 109
Y8, 100 = 180 -27 124 ............ 277
The columns V, U, E give the proportions of the standard colours which
are equivalent to 100 of the given colour; and the sum of V, U, E
gives a coefficient, which gives a general idea of the brightness. It
will be seen that the first admixture of yellow _diminishes_ the
brightness of the blue. The negative values of U indicate that a
mixture of V, U, and E cannot be made equivalent to the given colour.
The experiments from which these results were taken had the negative
values transferred to the other side of the equation. They were all
made by means of the colour-top, and were verified by repetition at
different times. It may be necessary to remark, in conclusion, with
reference to the mode of registering visible colours in terms of three
arbitrary standard colours, that it proceeds upon that theory of three
primary elements in the sensation of colour, which treats the
investigation of the laws of visible colour as a branch of human
physiology, incapable of being deduced from the laws of light itself,
as set forth in physical optics. It takes advantage of the methods of
optics to study vision itself; and its appeal is not to physical
principles, but to our consciousness of our own sensations.
On an Instrument to illustrate Poinsot's Theory of Rotation.
James Clerk Maxwell
[From the _Report of the British Association_, 1856.]
In studying the rotation of a solid body according to Poinsot's
method, we have to consider the successive positions of the
instantaneous axis of rotation with reference both to directions fixed
in space and axes assumed in the moving body. The paths traced out by
the pole of this axis on the _invariable plane_ and on the _central
ellipsoid_ form interesting subjects of mathematical investigation.
But when we attempt to follow with our eye the motion of a rotating
body, we find it difficult to determine through what point of the
_body_ the instantaneous axis passes at any time,--and to determine its
path must be still more difficult. I have endeavoured to render
visible the path of the instantaneous axis, and to vary the
circumstances of motion, by means of a top of the same kind as that
used by Mr Elliot, to illustrate precession*. The body of the
instrument is a hollow cone of wood, rising from a ring, 7 inches in
diameter and 1 inch thick. An iron axis, 8 inches long, screws into
the vertex of the cone. The lower extremity has a point of hard
steel, which rests in an agate cup, and forms the support of the
instrument. An iron nut, three ounces in weight, is made to screw on
the axis, and to be fixed at any point; and in the wooden ring are
screwed four bolts, of three ounces, working horizontally, and four
bolts, of one ounce, working vertically. On the upper part of the
axis is placed a disc of card, on which are drawn four concentric
rings. Each ring is divided into four quadrants, which are coloured
red, yellow, green, and blue. The spaces between the rings are white.
When the top is in motion, it is easy to see in which quadrant the
instantaneous axis is at any moment and the distance between it and
the axis of the instrument; and we observe,--1st. That the
instantaneous axis travels in a closed curve, and returns to its
original position in the body. 2ndly. That by working the vertical
bolts, we can make the axis of the instrument the centre of this
closed curve. It will then be one of the principal axes of inertia.
3rdly. That, by working the nut on the axis, we can make the order of
colours either red, yellow, green, blue, or the reverse. When the
order of colours is in the same direction as the rotation, it
indicates that the axis of the instrument is that of greatest moment
of inertia. 4thly. That if we screw the two pairs of opposite
horizontal bolts to different distances from the axis, the path of the
instantaneous pole will no longer be equidistant from the axis, but
will describe an ellipse, whose longer axis is in the direction of the
mean axis of the instrument. 5thly. That if we now make one of the
two horizontal axes less and the other greater than the vertical axis,
the instantaneous pole will separate from the axis of the instrument,
and the axis will incline more and more till the spinning can no
longer go on, on account of the obliquity. It is easy to see that, by
attending to the laws of motion, we may produce any of the above
effects at pleasure, and illustrate many different propositions by
means of the same instrument.
* _Transactions of the Royal Scottish Society of Arts_, 1855.
Address to the Mathematical and Physical Sections of the British
Association.
James Clerk Maxwell
[From the _British Association Report_, Vol. XL.]
[Liverpool, _September_ 15, 1870.]
At several of the recent Meetings of the British Association the
varied and important business of the Mathematical and Physical Section
has been introduced by an Address, the subject of which has been left
to the selection of the President for the time being. The perplexing
duty of choosing a subject has not, however, fallen to me.
Professor Sylvester, the President of Section A at the Exeter Meeting,
gave us a noble vindication of pure mathematics by laying bare, as it
were, the very working of the mathematical mind, and setting before
us, not the array of symbols and brackets which form the armoury of
the mathematician, or the dry results which are only the monuments of
his conquests, but the mathematician himself, with all his human
faculties directed by his professional sagacity to the pursuit,
apprehension, and exhibition of that ideal harmony which he feels to
be the root of all knowledge, the fountain of all pleasure, and the
condition of all action. The mathematician has, above all things, an
eye for symmetry; and Professor Sylvester has not only recognized the
symmetry formed by the combination of his own subject with those of
the former Presidents, but has pointed out the duties of his successor
in the following characteristic note:--
"Mr Spottiswoode favoured the Section, in his opening Address, with a
combined history of the progress of Mathematics and Physics; Dr.
Tyndall's address was virtually on the limits of Physical Philosophy;
the one here in print," says Prof. Sylvester, "is an attempted faint
adumbration of the nature of Mathematical Science in the abstract.
What is wanting (like a fourth sphere resting on three others in
contact) to build up the Ideal Pyramid is a discourse on the Relation
of the two branches (Mathematics and Physics) to, their action and
reaction upon, one another, a magnificent theme, with which it is to
be hoped that some future President of Section A will crown the
edifice and make the Tetralogy (symbolizable by _A+A'_, _A_, _A'_,
_AA'_) complete."
The theme thus distinctly laid down for his successor by our late
President is indeed a magnificent one, far too magnificent for any
efforts of mine to realize. I have endeavoured to follow Mr
Spottiswoode, as with far-reaching vision he distinguishes the systems
of science into which phenomena, our knowledge of which is still in
the nebulous stage, are growing. I have been carried by the
penetrating insight and forcible expression of Dr Tyndall into that
sanctuary of minuteness and of power where molecules obey the laws of
their existence, clash together in fierce collision, or grapple in yet
more fierce embrace, building up in secret the forms of visible
things. I have been guided by Prof. Sylvester towards those serene
heights
"Where never creeps a cloud, or moves a wind,
Nor ever falls the least white star of snow,
Nor ever lowest roll of thunder moans,
Nor sound of human sorrow mounts to mar
Their sacred everlasting calm."
But who will lead me into that still more hidden and dimmer region
where Thought weds Fact, where the mental operation of the
mathematician and the physical action of the molecules are seen in
their true relation? Does not the way to it pass through the very den
of the metaphysician, strewed with the remains of former explorers,
and abhorred by every man of science? It would indeed be a foolhardy
adventure for me to take up the valuable time of the Section by
leading you into those speculations which require, as we know,
thousands of years even to shape themselves intelligibly.
But we are met as cultivators of mathematics and physics. In our
daily work we are led up to questions the same in kind with those of
metaphysics; and we approach them, not trusting to the native
penetrating power of our own minds, but trained by a long-continued
adjustment of our modes of thought to the facts of external nature.
As mathematicians, we perform certain mental operations on the symbols
of number or of quantity, and, by proceeding step by step from more
simple to more complex operations, we are enabled to express the same
thing in many different forms. The equivalence of these different
forms, though a necessary consequence of self-evident axioms, is not
always, to our minds, self-evident; but the mathematician, who by long
practice has acquired a familiarity with many of these forms, and has
become expert in the processes which lead from one to another, can
often transform a perplexing expression into another which explains
its meaning in more intelligible language.
As students of Physics we observe phenomena under varied
circumstances, and endeavour to deduce the laws of their relations.
Every natural phenomenon is, to our minds, the result of an infinitely
complex system of conditions. What we set ourselves to do is to
unravel these conditions, and by viewing the phenomenon in a way which
is in itself partial and imperfect, to piece out its features one by
one, beginning with that which strikes us first, and thus gradually
learning how to look at the whole phenomenon so as to obtain a
continually greater degree of clearness and distinctness. In this
process, the feature which presents itself most forcibly to the
untrained inquirer may not be that which is considered most
fundamental by the experienced man of science; for the success of any
physical investigation depends on the judicious selection of what is
to be observed as of primary importance, combined with a voluntary
abstraction of the mind from those features which, however attractive
they appear, we are not yet sufficiently advanced in science to
investigate with profit.
Intellectual processes of this kind have been going on since the first
formation of language, and are going on still. No doubt the feature
which strikes us first and most forcibly in any phenomenon, is the
pleasure or the pain which accompanies it, and the agreeable or
disagreeable results which follow after it. A theory of nature from
this point of view is embodied in many of our words and phrases, and
is by no means extinct even in our deliberate opinions.
It was a great step in science when men became convinced that, in
order to understand the nature of things, they must begin by asking,
not whether a thing is good or bad, noxious or beneficial, but of what
kind is it? and how much is there of it? Quality and Quantity were
then first recognized as the primary features to be observed in
scientific inquiry.
As science has been developed, the domain of quantity has everywhere
encroached on that of quality, till the process of scientific inquiry
seems to have become simply the measurement and registration of
quantities, combined with a mathematical discussion of the numbers
thus obtained. It is this scientific method of directing our
attention to those features of phenomena which may be regarded as
quantities which brings physical research under the influence of
mathematical reasoning. In the work of the Section we shall have
abundant examples of the successful application of this method to the
most recent conquests of science; but I wish at present to direct your
attention to some of the reciprocal effects of the progress of science
on those elementary conceptions which are sometimes thought to be
beyond the reach of change.
If the skill of the mathematician has enabled the experimentalist to
see that the quantities which he has measured are connected by
necessary relations, the discoveries of physics have revealed to the
mathematician new forms of quantities which he could never have
imagined for himself.
Of the methods by which the mathematician may make his labours most
useful to the student of nature, that which I think is at present most
important is the systematic classification of quantities.
The quantities which we study in mathematics and physics may be
classified in two different ways.
The student who wishes to master any particular science must make
himself familiar with the various kinds of quantities which belong to
that science. When he understands all the relations between these
quantities, he regards them as forming a connected system, and he
classes the whole system of quantities together as belonging to that
particular science. This classification is the most natural from a
physical point of view, and it is generally the first in order of
time.
But when the student has become acquainted with several different
sciences, he finds that the mathematical processes and trains of
reasoning in one science resemble those in another so much that his
knowledge of the one science may be made a most useful help in the
study of the other.
When he examines into the reason of this, he finds that in the two
sciences he has been dealing with systems of quantities, in which the
mathematical forms of the relations of the quantities are the same in
both systems, though the physical nature of the quantities may be
utterly different.
He is thus led to recognize a classification of quantities on a new
principle, according to which the physical nature of the quantity is
subordinated to its mathematical form. This is the point of view
which is characteristic of the mathematician; but it stands second to
the physical aspect in order of time, because the human mind, in order
to conceive of different kinds of quantities, must have them presented
to it by nature.
I do not here refer to the fact that all quantities, as such, are
subject to the rules of arithmetic and algebra, and are therefore
capable of being submitted to those dry calculations which represent,
to so many minds, their only idea of mathematics.
The human mind is seldom satisfied, and is certainly never exercising
its highest functions, when it is doing the work of a calculating
machine. What the man of science, whether he is a mathematician or a
physical inquirer, aims at is, to acquire and develope clear ideas of
the things he deals with. For this purpose he is willing to enter on
long calculations, and to be for a season a calculating machine, if he
can only at last make his ideas clearer.
But if he finds that clear ideas are not to be obtained by means of
processes the steps of which he is sure to forget before he has
reached the conclusion, it is much better that he should turn to
another method, and try to understand the subject by means of
well-chosen illustrations derived from subjects with which he is more
familiar.
We all know how much more popular the illustrative method of
exposition is found, than that in which bare processes of reasoning
and calculation form the principal subject of discourse.
Now a truly scientific illustration is a method to enable the mind to
grasp some conception or law in one branch of science, by placing
before it a conception or a law in a different branch of science, and
directing the mind to lay hold of that mathematical form which is
common to the corresponding ideas in the two sciences, leaving out of
account for the present the difference between the physical nature of
the real phenomena.
The correctness of such an illustration depends on whether the two
systems of ideas which are compared together are really analogous in
form, or whether, in other words, the corresponding physical
quantities really belong to the same mathematical class. When this
condition is fulfilled, the illustration is not only convenient for
teaching science in a pleasant and easy manner, but the recognition of
the formal analogy between the two systems of ideas leads to a
knowledge of both, more profound than could be obtained by studying
each system separately.
There are men who, when any relation or law, however complex, is put
before them in a symbolical form, can grasp its full meaning as a
relation among abstract quantities. Such men sometimes treat with
indifference the further statement that quantities actually exist in
nature which fulfil this relation. The mental image of the concrete
reality seems rather to disturb than to assist their contemplations.
But the great majority of mankind are utterly unable, without long
training, to retain in their minds the unembodied symbols of the pure
mathematician, so that, if science is ever to become popular, and yet
remain scientific, it must be by a profound study and a copious
application of those principles of the mathematical classification of
quantities which, as we have seen, lie at the root of every truly
scientific illustration.
There are, as I have said, some minds which can go on contemplating
with satisfaction pure quantities presented to the eye by symbols, and
to the mind in a form which none but mathematicians can conceive.
There are others who feel more enjoyment in following geometrical
forms, which they draw on paper, or build up in the empty space before
them.
Others, again, are not content unless they can project their whole
physical energies into the scene which they conjure up. They learn at
what a rate the planets rush through space, and they experience a
delightful feeling of exhilaration. They calculate the forces with
which the heavenly bodies pull at one another, and they feel their own
muscles straining with the effort.
To such men momentum, energy, mass are not mere abstract expressions
of the results of scientific inquiry. They are words of power, which
stir their souls like the memories of childhood.
For the sake of persons of these different types, scientific truth
should be presented in different forms, and should be regarded as
equally scientific whether it appears in the robust form and the vivid
colouring of a physical illustration, or in the tenuity and paleness
of a symbolical expression.
Time would fail me if I were to attempt to illustrate by examples the
scientific value of the classification of quantities. I shall only
mention the name of that important class of magnitudes having
direction in space which Hamilton has called vectors, and which form
the subject-matter of the Calculus of Quaternions, a branch of
mathematics which, when it shall have been thoroughly understood by
men of the illustrative type, and clothed by them with physical
imagery, will become, perhaps under some new name, a most powerful
method of communicating truly scientific knowledge to persons
apparently devoid of the calculating spirit.
The mutual action and reaction between the different departments of
human thought is so interesting to the student of scientific progress,
that, at the risk of still further encroaching on the valuable time of
the Section, I shall say a few words on a branch of physics which not
very long ago would have been considered rather a branch of
metaphysics. I mean the atomic theory, or, as it is now called, the
molecular theory of the constitution of bodies.
Not many years ago if we had been asked in what regions of physical
science the advance of discovery was least apparent, we should have
pointed to the hopelessly distant fixed stars on the one hand, and to
the inscrutable delicacy of the texture of material bodies on the
other.
Indeed, if we are to regard Comte as in any degree representing the
scientific opinion of his time, the research into what takes place
beyond our own solar system seemed then to be exceedingly unpromising,
if not altogether illusory.
The opinion that the bodies which we see and handle, which we can set
in motion or leave at rest, which we can break in pieces and destroy,
are composed of smaller bodies which we cannot see or handle, which
are always in motion, and which can neither be stopped nor broken in
pieces, nor in any way destroyed or deprived of the least of their
properties, was known by the name of the Atomic theory. It was
associated with the names of Democritus, Epicurus, and Lucretius, and
was commonly supposed to admit the existence only of atoms and void,
to the exclusion of any other basis of things from the universe.
In many physical reasonings and mathematical calculations we are
accustomed to argue as if such substances as air, water, or metal,
which appear to our senses uniform and continuous, were strictly and
mathematically uniform and continuous.
We know that we can divide a pint of water into many millions of
portions, each of which is as fully endowed with all the properties of
water as the whole pint was; and it seems only natural to conclude
that we might go on subdividing the water for ever, just as we can
never come to a limit in subdividing the space in which it is
contained. We have heard how Faraday divided a grain of gold into an
inconceivable number of separate particles, and we may see Dr Tyndall
produce from a mere suspicion of nitrite of butyle an immense cloud,
the minute visible portion of which is still cloud, and therefore must
contain many molecules of nitrite of butyle.
But evidence from different and independent sources is now crowding in
upon us which compels us to admit that if we could push the process of
subdivision still further we should come to a limit, because each
portion would then contain only one molecule, an individual body, one
and indivisible, unalterable by any power in nature.
Even in our ordinary experiments on very finely divided matter we find
that the substance is beginning to lose the properties which it
exhibits when in a large mass, and that effects depending on the
individual action of molecules are beginning to become prominent.
The study of these phenomena is at present the path which leads to the
development of molecular science.
That superficial tension of liquids which is called capillary
attraction is one of these phenomena. Another important class of
phenomena are those which are due to that motion of agitation by which
the molecules of a liquid or gas are continually working their way
from one place to another, and continually changing their course, like
people hustled in a crowd.
On this depends the rate of diffusion of gases and liquids through
each other, to the study of which, as one of the keys of molecular
science, that unwearied inquirer into nature's secrets, the late Prof.
Graham, devoted such arduous labour.
The rate of electrolytic conduction is, according to Wiedemann's
theory, influenced by the same cause; and the conduction of heat in
fluids depends probably on the same kind of action. In the case of
gases, a molecular theory has been developed by Clausius and others,
capable of mathematical treatment, and subjected to experimental
investigation; and by this theory nearly every known mechanical
property of gases has been explained on dynamical principles; so that
the properties of individual gaseous molecules are in a fair way to
become objects of scientific research.
Now Mr Stoney has pointed out[1] that the numerical results of
experiments on gases render it probable that the mean distance of
their particles at the ordinary temperature and pressure is a quantity
of the same order of magnitude as a millionth of a millimetre, and Sir
William Thomson has since[2] shewn, by several independent lines of
argument, drawn from phenomena so different in themselves as the
electrification of metals by contact, the tension of soap-bubbles, and
the friction of air, that in ordinary solids and liquids the average
distance between contiguous molecules is less than the
hundred-millionth, and greater than the two-thousand-millionth of a
centimetre.
[1] _Phil. Mag._, Aug. 1868.
[2] _Nature_, March 31, 1870.
These, of course, are exceedingly rough estimates, for they are
derived from measurements some of which are still confessedly very
rough; but if at the present time, we can form even a rough plan for
arriving at results of this kind, we may hope that, as our means of
experimental inquiry become more accurate and more varied, our
conception of a molecule will become more definite, so that we may be
able at no distant period to estimate its weight with a greater degree
of precision.
A theory, which Sir W. Thomson has founded on Helmholtz's splendid
hydrodynamical theorems, seeks for the properties of molecules in the
ring vortices of a uniform, frictionless, incompressible fluid. Such
whirling rings may be seen when an experienced smoker sends out a
dexterous puff of smoke into the still air, but a more evanescent
phenomenon it is difficult to conceive. This evanescence is owing to
the viscosity of the air; but Helmholtz has shewn that in a perfect
fluid such a whirling ring, if once generated, would go on whirling
for ever, would always consist of the very same portion of the fluid
which was first set whirling, and could never be cut in two by any
natural cause. The generation of a ring-vortex is of course equally
beyond the power of natural causes, but once generated, it has the
properties of individuality, permanence in quantity, and
indestructibility. It is also the recipient of impulse and of energy,
which is all we can affirm of matter; and these ring-vortices are
capable of such varied connexions and knotted self-involutions, that
the properties of differently knotted vortices must be as different as
those of different kinds of molecules can be.
If a theory of this kind should be found, after conquering the
enormous mathematical difficulties of the subject, to represent in any
degree the actual properties of molecules, it will stand in a very
different scientific position from those theories of molecular action
which are formed by investing the molecule with an arbitrary system of
central forces invented expressly to account for the observed
phenomena.
In the vortex theory we have nothing arbitrary, no central forces or
occult properties of any other kind. We have nothing but matter and
motion, and when the vortex is once started its properties are all
determined from the original impetus, and no further assumptions are
possible.
Even in the present undeveloped state of the theory, the contemplation
of the individuality and indestructibility of a ring-vortex in a
perfect fluid cannot fail to disturb the commonly received opinion
that a molecule, in order to be permanent, must be a very hard body.
In fact one of the first conditions which a molecule must fulfil is,
apparently, inconsistent with its being a single hard body. We know
from those spectroscopic researches which have thrown so much light on
different branches of science, that a molecule can be set into a state
of internal vibration, in which it gives off to the surrounding medium
light of definite refrangibility--light, that is, of definite
wave-length and definite period of vibration. The fact that all the
molecules (say, of hydrogen) which we can procure for our experiments,
when agitated by heat or by the passage of an electric spark, vibrate
precisely in the same periodic time, or, to speak more accurately,
that their vibrations are composed of a system of simple vibrations
having always the same periods, is a very remarkable fact.
I must leave it to others to describe the progress of that splendid
series of spectroscopic discoveries by which the chemistry of the
heavenly bodies has been brought within the range of human inquiry. I
wish rather to direct your attention to the fact that, not only has
every molecule of terrestrial hydrogen the same system of periods of
free vibration, but that the spectroscopic examination of the light of
the sun and stars shews that, in regions the distance of which we can
only feebly imagine, there are molecules vibrating in as exact unison
with the molecules of terrestrial hydrogen as two tuning-forks tuned
to concert pitch, or two watches regulated to solar time.
Now this absolute equality in the magnitude of quantities, occurring
in all parts of the universe, is worth our consideration.
The dimensions of individual natural bodies are either quite
indeterminate, as in the case of planets, stones, trees, &c., or they
vary within moderate limits, as in the case of seeds, eggs, &c.; but
even in these cases small quantitative differences are met with which
do not interfere with the essential properties of the body.
Even crystals, which are so definite in geometrical form, are variable
with respect to their absolute dimensions.
Among the works of man we sometimes find a certain degree of
uniformity.
There is a uniformity among the different bullets which are cast in
the same mould, and the different copies of a book printed from the
same type.
If we examine the coins, or the weights and measures, of a civilized
country, we find a uniformity, which is produced by careful adjustment
to standards made and provided by the state. The degree of uniformity
of these national standards is a measure of that spirit of justice in
the nation which has enacted laws to regulate them and appointed
officers to test them.
This subject is one in which we, as a scientific body, take a warm
interest; and you are all aware of the vast amount of scientific work
which has been expended, and profitably expended, in providing weights
and measures for commercial and scientific purposes.
The earth has been measured as a basis for a permanent standard of
length, and every property of metals has been investigated to guard
against any alteration of the material standards when made. To weigh
or measure any thing with modern accuracy, requires a course of
experiment and calculation in which almost every branch of physics and
mathematics is brought into requisition.
Yet, after all, the dimensions of our earth and its time of rotation,
though, relatively to our present means of comparison, very permanent,
are not so by any physical necessity. The earth might contract by
cooling, or it might be enlarged by a layer of meteorites falling on
it, or its rate of revolution might slowly slacken, and yet it would
continue to be as much a planet as before.
But a molecule, say of hydrogen, if either its mass or its time of
vibration were to be altered in the least, would no longer be a
molecule of hydrogen.
If, then, we wish to obtain standards of length, time, and mass which
shall be absolutely permanent, we must seek them not in the
dimensions, or the motion, or the mass of our planet, but in the
wave-length, the period of vibration, and the absolute mass of these
imperishable and unalterable and perfectly similar molecules.
When we find that here, and in the starry heavens, there are
innumerable multitudes of little bodies of exactly the same mass, so
many, and no more, to the grain, and vibrating in exactly the same
time, so many times, and no more, in a second, and when we reflect
that no power in nature can now alter in the least either the mass or
the period of any one of them, we seem to have advanced along the path
of natural knowledge to one of those points at which we must accept
the guidance of that faith by which we understand that "that which is
seen was not made of things which do appear."
One of the most remarkable results of the progress of molecular
science is the light it has thrown on the nature of irreversible
processes--processes, that is, which always tend towards and never
away from a certain limiting state. Thus, if two gases be put into
the same vessel, they become mixed, and the mixture tends continually
to become more uniform. If two unequally heated portions of the same
gas are put into the vessel, something of the kind takes place, and
the whole tends to become of the same temperature. If two unequally
heated solid bodies be placed in contact, a continual approximation of
both to an intermediate temperature takes place.
In the case of the two gases, a separation may be effected by chemical
means; but in the other two cases the former state of things cannot be
restored by any natural process.
In the case of the conduction or diffusion of heat the process is not
only irreversible, but it involves the irreversible diminution of that
part of the whole stock of thermal energy which is capable of being
converted into mechanical work.
This is Thomson's theory of the irreversible dissipation of energy,
and it is equivalent to the doctrine of Clausius concerning the growth
of what he calls Entropy.
The irreversible character of this process is strikingly embodied in
Fourier's theory of the conduction of heat, where the formulae
themselves indicate, for all positive values of the time, a possible
solution which continually tends to the form of a uniform diffusion of
heat.
But if we attempt to ascend the stream of time by giving to its symbol
continually diminishing values, we are led up to a state of things in
which the formula has what is called a critical value; and if we
inquire into the state of things the instant before, we find that the
formula becomes absurd.
We thus arrive at the conception of a state of things which cannot be
conceived as the physical result of a previous state of things, and we
find that this critical condition actually existed at an epoch not in
the utmost depths of a past eternity, but separated from the present
time by a finite interval.
This idea of a beginning is one which the physical researches of
recent times have brought home to us, more than any observer of the
course of scientific thought in former times would have had reason to
expect.
But the mind of man is not, like Fourier's heated body, continually
settling down into an ultimate state of quiet uniformity, the
character of which we can already predict; it is rather like a tree,
shooting out branches which adapt themselves to the new aspects of the
sky towards which they climb, and roots which contort themselves among
the strange strata of the earth into which they delve. To us who
breathe only the spirit of our own age, and know only the
characteristics of contemporary thought, it is as impossible to
predict the general tone of the science of the future as it is to
anticipate the particular discoveries which it will make.
Physical research is continually revealing to us new features of
natural processes, and we are thus compelled to search for new forms
of thought appropriate to these features. Hence the importance of a
careful study of those relations between mathematics and Physics which
determine the conditions under which the ideas derived from one
department of physics may be safely used in forming ideas to be
employed in a new department.
The figure of speech or of thought by which we transfer the language
and ideas of a familiar science to one with which we are less
acquainted may be called Scientific Metaphor.
Thus the words Velocity, Momentum, Force, &c. have acquired certain
precise meanings in Elementary Dynamics. They are also employed in
the Dynamics of a Connected System in a sense which, though perfectly
analogous to the elementary sense, is wider and more general.
These generalized forms of elementary ideas may be called metaphorical
terms in the sense in which every abstract term is metaphorical. The
characteristic of a truly scientific system of metaphors is that each
term in its metaphorical use retains all the formal relations to the
other terms of the system which it had in its original use. The
method is then truly scientific--that is, not only a legitimate
product of science, but capable of generating science in its turn.
There are certain electrical phenomena, again, which are connected
together by relations of the same form as those which connect
dynamical phenomena. To apply to these the phrases of dynamics with
proper distinctions and provisional reservations is an example of a
metaphor of a bolder kind; but it is a legitimate metaphor if it
conveys a true idea of the electrical relations to those who have been
already trained in dynamics.
Suppose, then, that we have successfully introduced certain ideas
belonging to an elementary science by applying them metaphorically to
some new class of phenomena. It becomes an important philosophical
question to determine in what degree the applicability of the old
ideas to the new subject may be taken as evidence that the new
phenomena are physically similar to the old.
The best instances for the determination of this question are those in
which two different explanations have been given of the same thing.
The most celebrated case of this kind is that of the corpuscular and
the undulatory theories of light. Up to a certain point the phenomena
of light are equally well explained by both; beyond this point, one of
them fails.
To understand the true relation of these theories in that part of the
field where they seem equally applicable we must look at them in the
light which Hamilton has thrown upon them by his discovery that to
every brachistochrone problem there corresponds a problem of free
motion, involving different velocities and times, but resulting in the
same geometrical path. Professor Tait has written a very interesting
paper on this subject.
According to a theory of electricity which is making great progress in
Germany, two electrical particles act on one another directly at a
distance, but with a force which, according to Weber, depends on their
relative velocity, and according to a theory hinted at by Gauss, and
developed by Riemann, Lorenz, and Neumann, acts not instantaneously,
but after a time depending on the distance. The power with which this
theory, in the hands of these eminent men, explains every kind of
electrical phenomena must be studied in order to be appreciated.
Another theory of electricity, which I prefer, denies action at a
distance and attributes electric action to tensions and pressures in
an all-pervading medium, these stresses being the same in kind with
those familiar to engineers, and the medium being identical with that
in which light is supposed to be propagated.
Both these theories are found to explain not only the phenomena by the
aid of which they were originally constructed, but other phenomena,
which were not thought of or perhaps not known at the time; and both
have independently arrived at the same numerical result, which gives
the absolute velocity of light in terms of electrical quantities.
That theories apparently so fundamentally opposed should have so large
a field of truth common to both is a fact the philosophical importance
of which we cannot fully appreciate till we have reached a scientific
altitude from which the true relation between hypotheses so different
can be seen.
I shall only make one more remark on the relation between Mathematics
and Physics. In themselves, one is an operation of the mind, the
other is a dance of molecules. The molecules have laws of their own,
some of which we select as most intelligible to us and most amenable
to our calculation. We form a theory from these partial data, and we
ascribe any deviation of the actual phenomena from this theory to
disturbing causes. At the same time we confess that what we call
disturbing causes are simply those parts of the true circumstances
which we do not know or have neglected, and we endeavour in future to
take account of them. We thus acknowledge that the so-called
disturbance is a mere figment of the mind, not a fact of nature, and
that in natural action there is no disturbance.
But this is not the only way in which the harmony of the material with
the mental operation may be disturbed. The mind of the mathematician
is subject to many disturbing causes, such as fatigue, loss of memory,
and hasty conclusions; and it is found that, from these and other
causes, mathematicians make mistakes.
I am not prepared to deny that, to some mind of a higher order than
ours, each of these errors might be traced to the regular operation of
the laws of actual thinking; in fact we ourselves often do detect, not
only errors of calculation, but the causes of these errors. This,
however, by no means alters our conviction that they are errors, and
that one process of thought is right and another process wrong. I
One of the most profound mathematicians and thinkers of our time, the
late George Boole, when reflecting on the precise and almost
mathematical character of the laws of right thinking as compared with
the exceedingly perplexing though perhaps equally determinate laws of
actual and fallible thinking, was led to another of those points of
view from which Science seems to look out into a region beyond her own
domain.
"We must admit," he says, "that there exist laws" (of thought) "which
even the rigour of their mathematical forms does not preserve from
violation. We must ascribe to them an authority, the essence of which
does not consist in power, a supremacy which the analogy of the
inviolable order of the natural world in no way assists us to
comprehend."
Introductory Lecture on Experimental Physics.
James Clerk Maxwell
The University of Cambridge, in accordance with that law of its
evolution, by which, while maintaining the strictest continuity
between the successive phases of its history, it adapts itself with
more or less promptness to the requirements of the times, has lately
instituted a course of Experimental Physics. This course of study,
while it requires us to maintain in action all those powers of
attention and analysis which have been so long cultivated in the
University, calls on us to exercise our senses in observation, and our
hands in manipulation. The familiar apparatus of pen, ink, and paper
will no longer be sufficient for us, and we shall require more room
than that afforded by a seat at a desk, and a wider area than that of
the black board. We owe it to the munificence of our Chancellor,
that, whatever be the character in other respects of the experiments
which we hope hereafter to conduct, the material facilities for their
full development will be upon a scale which has not hitherto been
surpassed.