Project Gutenberg's Five of Maxwell's Papers, by James Clerk Maxwell

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Title: Five of Maxwell's Papers

Author: James Clerk Maxwell

Posting Date: February 25, 2014 [EBook #4908]

Release Date: January, 2004

[This file was first posted on March 24, 2002]

Language: English

Character set encoding: ASCII

*** START OF THIS PROJECT GUTENBERG EBOOK FIVE OF MAXWELL'S PAPERS ***




Produced by Gordon Keener






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.

The main feature, therefore, of Experimental Physics at Cambridge is
the Devonshire Physical Laboratory, and I think it desirable that on
the present occasion, before we enter on the details of any special
study, we should consider by what means we, the University of
Cambridge, may, as a living body, appropriate and vitalise this new
organ, the outward shell of which we expect soon to rise before us.
The course of study at this University has always included Natural
Philosophy, as well as Pure Mathematics.  To diffuse a sound knowledge
of Physics, and to imbue the minds of our students with correct
dynamical principles, have been long regarded as among our highest
functions, and very few of us can now place ourselves in the mental
condition in which even such philosophers as the great Descartes were
involved in the days before Newton had announced the true laws of the
motion of bodies.  Indeed the cultivation and diffusion of sound
dynamical ideas has already effected a great change in the language
and thoughts even of those who make no pretensions to science, and we
are daily receiving fresh proofs that the popularisation of scientific
doctrines is producing as great an alteration in the mental state of
society as the material applications of science are effecting in its
outward life.  Such indeed is the respect paid to science, that the
most absurd opinions may become current, provided they are expressed
in language, the sound of which recals some well-known scientific
phrase.  If society is thus prepared to receive all kinds of
scientific doctrines, it is our part to provide for the diffusion and
cultivation, not only of true scientific principles, but of a spirit
of sound criticism, founded on an examination of the evidences on
which statements apparently scientific depend.

When we shall be able to employ in scientific education, not only the
trained attention of the student, and his familiarity with symbols,
but the keenness of his eye, the quickness of his ear, the delicacy of
his touch, and the adroitness of his fingers, we shall not only extend
our influence over a class of men who are not fond of cold
abstractions, but, by opening at once all the gateways of knowledge,
we shall ensure the association of the doctrines of science with those
elementary sensations which form the obscure background of all our
conscious thoughts, and which lend a vividness and relief to ideas,
which, when presented as mere abstract terms, are apt to fade entirely
from the memory.

In a course of Experimental Physics we may consider either the Physics
or the Experiments as the leading feature.  We may either employ the
experiments to illustrate the phenomena of a particular branch of
Physics, or we may make some physical research in order to exemplify a
particular experimental method.  In the order of time, we should
begin, in the Lecture Room, with a course of lectures on some branch
of Physics aided by experiments of illustration, and conclude, in the
Laboratory, with a course of experiments of research.

Let me say a few words on these two classes of
experiments,--Experiments of Illustration and Experiments of Research.
The aim of an experiment of illustration is to throw light upon some
scientific idea so that the student may be enabled to grasp it.  The
circumstances of the experiment are so arranged that the phenomenon
which we wish to observe or to exhibit is brought into prominence,
instead of being obscured and entangled among other phenomena, as it
is when it occurs in the ordinary course of nature.  To exhibit
illustrative experiments, to encourage others to make them, and to
cultivate in every way the ideas on which they throw light, forms an
important part of our duty.  The simpler the materials of an
illustrative experiment, and the more familiar they are to the
student, the more thoroughly is he likely to acquire the idea which it
is meant to illustrate.  The educational value of such experiments is
often inversely proportional to the complexity of the apparatus.  The
student who uses home-made apparatus, which is always going wrong,
often learns more than one who has the use of carefully adjusted
instruments, to which he is apt to trust, and which he dares not take
to pieces.

It is very necessary that those who are trying to learn from books the
facts of physical science should be enabled by the help of a few
illustrative experiments to recognise these facts when they meet with
them out of doors.  Science appears to us with a very different aspect
after we have found out that it is not in lecture rooms only, and by
means of the electric light projected on a screen, that we may witness
physical phenomena, but that we may find illustrations of the highest
doctrines of science in games and gymnastics, in travelling by land
and by water, in storms of the air and of the sea, and wherever there
is matter in motion.

This habit of recognising principles amid the endless variety of their
action can never degrade our sense of the sublimity of nature, or mar
our enjoyment of its beauty.  On the contrary, it tends to rescue our
scientific ideas from that vague condition in which we too often leave
them, buried among the other products of a lazy credulity, and to
raise them into their proper position among the doctrines in which our
faith is so assured, that we are ready at all times to act on them.

Experiments of illustration may be of very different kinds.  Some may
be adaptations of the commonest operations of ordinary life, others
may be carefully arranged exhibitions of some phenomenon which occurs
only under peculiar conditions.  They all, however, agree in this,
that their aim is to present some phenomenon to the senses of the
student in such a way that he may associate with it the appropriate
scientific idea.  When he has grasped this idea, the experiment which
illustrates it has served its purpose.

In an experiment of research, on the other hand, this is not the
principal aim.  It is true that an experiment, in which the principal
aim is to see what happens under certain conditions, may be regarded
as an experiment of research by those who are not yet familiar with
the result, but in experimental researches, strictly so called, the
ultimate object is to measure something which we have already seen--to
obtain a numerical estimate of some magnitude.

Experiments of this class--those in which measurement of some kind is
involved, are the proper work of a Physical Laboratory.  In every
experiment we have first to make our senses familiar with the
phenomenon, but we must not stop here, we must find out which of its
features are capable of measurement, and what measurements are
required in order to make a complete specification of the phenomenon.
We must then make these measurements, and deduce from them the result
which we require to find.

This characteristic of modern experiments--that they consist
principally of measurements,--is so prominent, that the opinion seems
to have got abroad, that in a few years all the great physical
constants will have been approximately estimated, and that the only
occupation which will then be left to men of science will be to carry
on these measurements to another place of decimals.

If this is really the state of things to which we are approaching, our
Laboratory may perhaps become celebrated as a place of conscientious
labour and consummate skill, but it will be out of place in the
University, and ought rather to be classed with the other great
workshops of our country, where equal ability is directed to more
useful ends.

But we have no right to think thus of the unsearchable riches of
creation, or of the untried fertility of those fresh minds into which
these riches will continue to be poured.  It may possibly be true
that, in some of those fields of discovery which lie open to such
rough observations as can be made without artificial methods, the
great explorers of former times have appropriated most of what is
valuable, and that the gleanings which remain are sought after, rather
for their abstruseness, than for their intrinsic worth.  But the
history of science shews that even during that phase of her progress
in which she devotes herself to improving the accuracy of the
numerical measurement of quantities with which she has long been
familiar, she is preparing the materials for the subjugation of new
regions, which would have remained unknown if she had been contented
with the rough methods of her early pioneers.  I might bring forward
instances gathered from every branch of science, shewing how the
labour of careful measurement has been rewarded by the discovery of
new fields of research, and by the development of new scientific
ideas.  But the history of the science of terrestrial magnetism
affords us a sufficient example of what may be done by Experiments in
Concert, such as we hope some day to perform in our Laboratory.

That celebrated traveller, Humboldt, was profoundly impressed with the
scientific value of a combined effort to be made by the observers of
all nations, to obtain accurate measurements of the magnetism of the
earth; and we owe it mainly to his enthusiasm for science, his great
reputation and his wide-spread influence, that not only private men of
science, but the governments of most of the civilised nations, our own
among the number, were induced to take part in the enterprise.  But
the actual working out of the scheme, and the arrangements by which
the labours of the observers were so directed as to obtain the best
results, we owe to the great mathematician Gauss, working along with
Weber, the future founder of the science of electro-magnetic
measurement, in the magnetic observatory of Gottingen, and aided by
the skill of the instrument-maker Leyser.  These men, however, did not
work alone.  Numbers of scientific men joined the Magnetic Union,
learned the use of the new instruments and the new methods of reducing
the observations; and in every city of Europe you might see them, at
certain stated times, sitting, each in his cold wooden shed, with his
eye fixed at the telescope, his ear attentive to the clock, and his
pencil recording in his note-book the instantaneous position of the
suspended magnet.

Bacon's conception of "Experiments in concert" was thus realised, the
scattered forces of science were converted into a regular army, and
emulation and jealousy became out of place, for the results obtained
by any one observer were of no value till they were combined with
those of the others.

The increase in the accuracy and completeness of magnetic observations
which was obtained by the new method, opened up fields of research
which were hardly suspected to exist by those whose observations of
the magnetic needle had been conducted in a more primitive manner.  We
must reserve for its proper place in our course any detailed
description of the disturbances to which the magnetism of our planet
is found to be subject.  Some of these disturbances are periodic,
following the regular courses of the sun and moon.  Others are sudden,
and are called magnetic storms, but, like the storms of the
atmosphere, they have their known seasons of frequency.  The last and
the most mysterious of these magnetic changes is that secular
variation by which the whole character of the earth, as a great
magnet, is being slowly modified, while the magnetic poles creep on,
from century to century, along their winding track in the polar
regions.

We have thus learned that the interior of the earth is subject to the
influences of the heavenly bodies, but that besides this there is a
constantly progressive change going on, the cause of which is entirely
unknown.  In each of the magnetic observatories throughout the world
an arrangement is at work, by means of which a suspended magnet
directs a ray of light on a preparred sheet of paper moved by
clockwork.  On that paper the never-resting heart of the earth is now
tracing, in telegraphic symbols which will one day be interpreted, a
record of its pulsations and its flutterings, as well as of that slow
but mighty working which warns us that we must not suppose that the
inner history of our planet is ended.

But this great experimental research on Terrestrial Magnetism produced
lasting effects on the progress of science in general.  I need only
mention one or two instances.  The new methods of measuring forces
were successfully applied by Weber to the numerical determination of
all the phenomena of electricity, and very soon afterwards the
electric telegraph, by conferring a commercial value on exact
numerical measurements, contributed largely to the advancement, as
well as to the diffusion of scientific knowledge.

But it is not in these more modern branches of science alone that this
influence is felt.  It is to Gauss, to the Magnetic Union, and to
magnetic observers in general, that we owe our deliverance from that
absurd method of estimating forces by a variable standard which
prevailed so long even among men of science.  It was Gauss who first
based the practical measurement of magnetic force (and therefore of
every other force) on those long established principles, which, though
they are embodied in every dynamical equation, have been so generally
set aside, that these very equations, though correctly given in our
Cambridge textbooks, are usually explained there by assuming, in
addition to the variable standard of force, a variable, and therefore
illegal, standard of mass.

Such, then, were some of the scientific results which followed in this
case from bringing together mathematical power, experimental sagacity,
and manipulative skill, to direct and assist the labours of a body of
zealous observers.  If therefore we desire, for our own advantage and
for the honour of our University, that the Devonshire Laboratory
should be successful, we must endeavour to maintain it in living union
with the other organs and faculties of our learned body.  We shall
therefore first consider the relation in which we stand to those
mathematical studies which have so long flourished among us, which
deal with our own subjects, and which differ from our experimental
studies only in the mode in which they are presented to the mind.

There is no more powerful method for introducing knowledge into the
mind than that of presenting it in as many different ways as we can.
When the ideas, after entering through different gateways, effect a
junction in the citadel of the mind, the position they occupy becomes
impregnable.  Opticians tell us that the mental combination of the
views of an object which we obtain from stations no further apart than
our two eyes is sufficient to produce in our minds an impression of
the solidity of the object seen; and we find that this impression is
produced even when we are aware that we are really looking at two flat
pictures placed in a stereoscope.  It is therefore natural to expect
that the knowledge of physical science obtained by the combined use of
mathematical analysis and experimental research will be of a more
solid, available, and enduring kind than that possessed by the mere
mathematician or the mere experimenter.

But what will be the effect on the University, if men Pursuing that
course of reading which has produced so many distinguished Wranglers,
turn aside to work experiments?  Will not their attendance at the
Laboratory count not merely as time withdrawn from their more
legitimate studies, but as the introduction of a disturbing element,
tainting their mathematical conceptions with material imagery, and
sapping their faith in the formulae of the textbook?  Besides this, we
have already heard complaints of the undue extension of our studies,
and of the strain put upon our questionists by the weight of learning
which they try to carry with them into the Senate-House.  If we now
ask them to get up their subjects not only by books and writing, but
at the same time by observation and manipulation, will they not break
down altogether?  The Physical Laboratory, we are told, may perhaps be
useful to those who are going out in Natural Science, and who do
not take in Mathematics, but to attempt to combine both kinds of study
during the time of residence at the University is more than one mind
can bear.

No doubt there is some reason for this feeling.  Many of us have
already overcome the initial difficulties of mathematical training.
When we now go on with our study, we feel that it requires exertion
and involves fatigue, but we are confident that if we only work hard
our progress will be certain.

Some of us, on the other hand, may have had some experience of the
routine of experimental work.  As soon as we can read scales, observe
times, focus telescopes, and so on, this kind of work ceases to
require any great mental effort.  We may perhaps tire our eyes and
weary our backs, but we do not greatly fatigue our minds.

It is not till we attempt to bring the theoretical part of our
training into contact with the practical that we begin to experience
the full effect of what Faraday has called "mental inertia"--not only
the difficulty of recognising, among the concrete objects before us,
the abstract relation which we have learned from books, but the
distracting pain of wrenching the mind away from the symbols to the
objects, and from the objects back to the symbols.  This however is
the price we have to pay for new ideas.

But when we have overcome these difficulties, and successfully bridged
over the gulph between the abstract and the concrete, it is not a mere
piece of knowledge that we have obtained: we have acquired the
rudiment of a permanent mental endowment.  When, by a repetition of
efforts of this kind, we have more fully developed the scientific
faculty, the exercise of this faculty in detecting scientific
principles in nature, and in directing practice by theory, is no
longer irksome, but becomes an unfailing source of enjoyment, to which
we return so often, that at last even our careless thoughts begin to
run in a scientific channel.

I quite admit that our mental energy is limited in quantity, and I
know that many zealous students try to do more than is good for them.
But the question about the introduction of experimental study is not
entirely one of quantity.  It is to a great extent a question of
distribution of energy.  Some distributions of energy, we know, are
more useful than others, because they are more available for those
purposes which we desire to accomplish.

Now in the case of study, a great part of our fatigue often arises,
not from those mental efforts by which we obtain the mastery of the
subject, but from those which are spent in recalling our wandering
thoughts; and these efforts of attention would be much less fatiguing
if the disturbing force of mental distraction could be removed.

This is the reason why a man whose soul is in his work always makes
more progress than one whose aim is something not immediately
connected with his occupation.  In the latter case the very motive of
which he makes use to stimulate his flagging powers becomes the means
of distracting his mind from the work before him.

There may be some mathematicians who pursue their studies entirely for
their own sake.  Most men, however, think that the chief use of
mathematics is found in the interpretation of nature.  Now a man who
studies a piece of mathematics in order to understand some natural
phenomenon which he has seen, or to calculate the best arrangement of
some experiment which he means to make, is likely to meet with far
less distraction of mind than if his sole aim had been to sharpen his
mind for the successful practice of the Law, or to obtain a high place
in the Mathematical Tripos.

I have known men, who when they were at school, never could see the
good of mathematics, but who, when in after life they made this
discovery, not only became eminent as scientific engineers, but made
considerable progress in the study of abstract mathematics.  If our
experimental course should help any of you to see the good of
mathematics, it will relieve us of much anxiety, for it will not only
ensure the success of your future studies, but it will make it much
less likely that they will prove injurious to your health.


But why should we labour to prove the advantage of practical science
to the University?  Let us rather speak of the help which the
University may give to science, when men well trained in mathematics
and enjoying the advantages of a well-appointed Laboratory, shall
unite their efforts to carry out some experimental research which no
solitary worker could attempt.

At first it is probable that our principal experimental work must be
the illustration of particular branches of science, but as we go on we
must add to this the study of scientific methods, the same method
being sometimes illustrated by its application to researches belonging
to different branches of science.

We might even imagine a course of experimental study the arrangement
of which should be founded on a classification of methods, and not on
that of the objects of investigation.  A combination of the two plans
seems to me better than either, and while we take every opportunity of
studying methods, we shall take care not to dissociate the method from
the scientific research to which it is applied, and to which it owes
its value.

We shall therefore arrange our lectures according to the
classification of the principal natural phenomena, such as heat,
electricity, magnetism and so on.

In the laboratory, on the other hand, the place of the different
instruments will be determined by a classification according to
methods, such as weighing and measuring, observations of time, optical
and electrical methods of observation, and so on.

The determination of the experiments to be performed at a particular
time must often depend upon the means we have at command, and in the
case of the more elaborate experiments, this may imply a long time of
preparation, during which the instruments, the methods, and the
observers themselves, are being gradually fitted for their work.  When
we have thus brought together the requisites, both material and
intellectual, for a particular experiment, it may sometimes be
desirable that before the instruments are dismounted and the observers
dispersed, we should make some other experiment, requiring the same
method, but dealing perhaps with an entirely different class of
physical phenomena.

Our principal work, however, in the Laboratory must be to acquaint
ourselves with all kinds of scientific methods, to compare them, and
to estimate their value.  It will, I think, be a result worthy of our
University, and more likely to be accomplished here than in any
private laboratory, if, by the free and full discussion of the
relative value of different scientific procedures, we succeed in
forming a school of scientific criticism, and in assisting the
development of the doctrine of method.

But admitting that a practical acquaintance with the methods of
Physical Science is an essential part of a mathematical and scientific
education, we may be asked whether we are not attributing too much
importance to science altogether as part of a liberal education.

Fortunately, there is no question here whether the University should
continue to be a place of liberal education, or should devote itself
to preparing young men for particular professions.  Hence though some
of us may, I hope, see reason to make the pursuit of science the main
business of our lives, it must be one of our most constant aims to
maintain a living connexion between our work and the other liberal
studies of Cambridge, whether literary, philological, historical or
philosophical.

There is a narrow professional spirit which may grow up among men of
science, just as it does among men who practise any other special
business.  But surely a University is the very place where we should
be able to overcome this tendency of men to become, as it were,
granulated into small worlds, which are all the more worldly for their
very smallness.  We lose the advantage of having men of varied
pursuits collected into one body, if we do not endeavour to imbibe
some of the spirit even of those whose special branch of learning is
different from our own.

It is not so long ago since any man who devoted himself to geometry,
or to any science requiring continued application, was looked upon as
necessarily a misanthrope, who must have abandoned all human
interests, and betaken himself to abstractions so far removed from the
world of life and action that he has become insensible alike to the
attractions of pleasure and to the claims of duty.

In the present day, men of science are not looked upon with the same
awe or with the same suspicion.  They are supposed to be in league
with the material spirit of the age, and to form a kind of advanced
Radical party among men of learning.

We are not here to defend literary and historical studies.  We admit
that the proper study of mankind is man.  But is the student of
science to be withdrawn from the study of man, or cut off from every
noble feeling, so long as he lives in intellectual fellowship with men
who have devoted their lives to the discovery of truth, and the
results of whose enquiries have impressed themselves on the ordinary
speech and way of thinking of men who never heard their names?  Or is
the student of history and of man to omit from his consideration the
history of the origin and diffusion of those ideas which have produced
so great a difference between one age of the world and another?

It is true that the history of science is very different from the
science of history.  We are not studying or attempting to study the
working of those blind forces which, we are told, are operating on
crowds of obscure people, shaking principalities and powers, and
compelling reasonable men to bring events to pass in an order laid
down by philosophers.

The men whose names are found in the history of science are not mere
hypothetical constituents of a crowd, to be reasoned upon only in
masses.  We recognise them as men like ourselves, and their actions
and thoughts, being more free from the influence of passion, and
recorded more accurately than those of other men, are all the better
materials for the study of the calmer parts of human nature.

But the history of science is not restricted to the enumeration of
successful investigations.  It has to tell of unsuccessful inquiries,
and to explain why some of the ablest men have failed to find the key
of knowledge, and how the reputation of others has only given a firmer
footing to the errors into which they fell.

The history of the development, whether normal or abnormal, of ideas
is of all subjects that in which we, as thinking men, take the deepest
interest.  But when the action of the mind passes out of the
intellectual stage, in which truth and error are the alternatives,
into the more violently emotional states of anger and passion, malice
and envy, fury and madness; the student of science, though he is
obliged to recognise the powerful influence which these wild forces
have exercised on mankind, is perhaps in some measure disqualified
from pursuing the study of this part of human nature.

But then how few of us are capable of deriving profit from such
studies.  We cannot enter into full sympathy with these lower phases
of our nature without losing some of that antipathy to them which is
our surest safeguard against a reversion to a meaner type, and we
gladly return to the company of those illustrious men who by aspiring
to noble ends, whether intellectual or practical, have risen above the
region of storms into a clearer atmosphere, where there is no
misrepresentation of opinion, nor ambiguity of expression, but where
one mind comes into closest contact with another at the point where
both approach nearest to the truth.


I propose to lecture during this term on Heat, and, as our facilities
for experimental work are not yet fully developed, I shall endeavour
to place before you the relative position and scientific connexion of
the different branches of the science, rather than to discuss the
details of experimental methods.

We shall begin with Thermometry, or the registration of temperatures,
and Calorimetry, or the measurement of quantities of heat.  We shall
then go on to Thermodynamics, which investigates the relations between
the thermal properties of bodies and their other dynamical properties,
in so far as these relations may be traced without any assumption as
to the particular constitution of these bodies.

The principles of Thermodynamics throw great light on all the
phenomena of nature, and it is probable that many valuable
applications of these principles have yet to be made; but we shall
have to point out the limits of this science, and to shew that many
problems in nature, especially those in which the Dissipation of
Energy comes into play, are not capable of solution by the principles
of Thermodynamics alone, but that in order to understand them, we are
obliged to form some more definite theory of the constitution of
bodies.

Two theories of the constitution of bodies have struggled for victory
with various fortunes since the earliest ages of speculation: one is
the theory of a universal plenum, the other is that of atoms and void.

The theory of the plenum is associated with the doctrine of
mathematical continuity, and its mathematical methods are those of the
Differential Calculus, which is the appropriate expression of the
relations of continuous quantity.

The theory of atoms and void leads us to attach more importance to the
doctrines of integral numbers and definite proportions; but, in
applying dynamical principles to the motion of immense numbers of
atoms, the limitation of our faculties forces us to abandon the
attempt to express the exact history of each atom, and to be content
with estimating the average condition of a group of atoms large enough
to be visible.  This method of dealing with groups of atoms, which I
may call the statistical method, and which in the present state of our
knowledge is the only available method of studying the properties of
real bodies, involves an abandonment of strict dynamical principles,
and an adoption of the mathematical methods belonging to the theory of
probability.  It is probable that important results will be obtained
by the application of this method, which is as yet little known and is
not familiar to our minds.  If the actual history of Science had been
different, and if the scientific doctrines most familiar to us had
been those which must be expressed in this way, it is possible that we
might have considered the existence of a certain kind of contingency a
self-evident truth, and treated the doctrine of philosophical
necessity as a mere sophism.

About the beginning of this century, the properties of bodies were
investigated by several distinguished French mathematicians on the
hypothesis that they are systems of molecules in equilibrium.  The
somewhat unsatisfactory nature of the results of these investigations
produced, especially in this country, a reaction in favour of the
opposite method of treating bodies as if they were, so far at least as
our experiments are concerned, truly continuous.  This method, in the
hands of Green, Stokes, and others, has led to results, the value of
which does not at all depend on what theory we adopt as to the
ultimate constitution of bodies.

One very important result of the investigation of the properties of
bodies on the hypothesis that they are truly continuous is that it
furnishes us with a test by which we can ascertain, by experiments on
a real body, to what degree of tenuity it must be reduced before it
begins to give evidence that its properties are no longer the same as
those of the body in mass.  Investigations of this kind, combined with
a study of various phenomena of diffusion and of dissipation of
energy, have recently added greatly to the evidence in favour of the
hypothesis that bodies are systems of molecules in motion.

I hope to be able to lay before you in the course of the term some of
the evidence for the existence of molecules, considered as individual
bodies having definite properties.  The molecule, as it is presented to
the scientific imagination, is a very different body from any of those
with which experience has hitherto made us acquainted.

In the first place its mass, and the other constants which define its
properties, are absolutely invariable; the individual molecule can
neither grow nor decay, but remains unchanged amid all the changes of
the bodies of which it may form a constituent.

In the second place it is not the only molecule of its kind, for there
are innumerable other molecules, whose constants are not
approximately, but absolutely identical with those of the first
molecule, and this whether they are found on the earth, in the sun, or
in the fixed stars.

By what process of evolution the philosophers of the future will
attempt to account for this identity in the properties of such a
multitude of bodies, each of them unchangeable in magnitude, and some
of them separated from others by distances which Astronomy attempts in
vain to measure, I cannot conjecture.  My mind is limited in its power
of speculation, and I am forced to believe that these molecules must
have been made as they are from the beginning of their existence.

I also conclude that since none of the processes of nature, during
their varied action on different individual molecules, have produced,
in the course of ages, the slightest difference between the properties
of one molecule and those of another, the history of whose
combinations has been different, we cannot ascribe either their
existence or the identity of their properties to the operation of any
of those causes which we call natural.

Is it true then that our scientific speculations have really
penetrated beneath the visible appearance of things, which seem to be
subject to generation and corruption, and reached the entrance of that
world of order and perfection, which continues this day as it was
created, perfect in number and measure and weight?

We may be mistaken.  No one has as yet seen or handled an individual
molecule, and our molecular hypothesis may, in its turn, be supplanted
by some new theory of the constitution of matter; but the idea of the
existence of unnumbered individual things, all alike and all
unchangeable, is one which cannot enter the human mind and remain
without fruit.

But what if these molecules, indestructible as they are, turn out to
be not substances themselves, but mere affections of some other
substance?

According to Sir W. Thomson's theory of Vortex Atoms, the substance of
which the molecule consists is a uniformly dense _plenum_, the
properties of which are those of a perfect fluid, the molecule itself
being nothing but a certain motion impressed on a portion of this
fluid, and this motion is shewn, by a theorem due to Helmholtz, to be
as indestructible as we believe a portion of matter to be.

If a theory of this kind is true, or even if it is conceivable, our
idea of matter may have been introduced into our minds through our
experience of those systems of vortices which we call bodies, but
which are not substances, but motions of a substance; and yet the idea
which we have thus acquired of matter, as a substance possessing
inertia, may be truly applicable to that fluid of which the vortices
are the motion, but of whose existence, apart from the vortical motion
of some of its parts, our experience gives us no evidence whatever.

It has been asserted that metaphysical speculation is a thing of the
past, and that physical science has extirpated it.  The discussion of
the categories of existence, however, does not appear to be in danger
of coming to an end in our time, and the exercise of speculation
continues as fascinating to every fresh mind as it was in the days of
Thales.








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