Books: Five of Maxwell's Papers
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James Clerk Maxwell >> Five of Maxwell's Papers
This eBook was produced by Gordon Keener.
This eBook includes 5 papers or speeches by James Clerk Maxwell.
Each is separated by three asterisks ('***').
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 vertical 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.