From ab014e37079f8a84087c63f072ef5bf8599b3bbf Mon Sep 17 00:00:00 2001 From: nathan Date: Sat, 26 Mar 2016 15:52:54 -0400 Subject: [PATCH] finished toolbox, attempted to learn lambda functions and list comprehensions --- frequency.py | 76 +- pg4908.txt | 1955 ++++++++++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 2015 insertions(+), 16 deletions(-) create mode 100644 pg4908.txt diff --git a/frequency.py b/frequency.py index acaa449..6e476b5 100644 --- a/frequency.py +++ b/frequency.py @@ -4,21 +4,65 @@ import string def get_word_list(file_name): - """ Reads the specified project Gutenberg book. Header comments, - punctuation, and whitespace are stripped away. The function - returns a list of the words used in the book as a list. - All words are converted to lower case. - """ - pass + """ Reads the specified project Gutenberg book. Header comments, + punctuation, and whitespace are stripped away. The function + returns a list of the words used in the book as a list. + All words are converted to lower case. + """ + #open and read file + f = open(file_name, 'r') + text = f.read() + + #create word list which we will return + word_list = [] + + #make lowercase + text = string.lower(text) + + #replace punctuation with spaces + for char in string.punctuation: + text = string.replace(text, char, ' ') + + for line in text.split('\n'): + for word in line.split(' '): + word_list.append(word) + + #don't include lines with only empty or \r + word_list = [x for x in word_list if x != ''] + word_list = [x for x in word_list if x != '\n'] + word_list = [x for x in word_list if x != '\r'] + + return word_list + def get_top_n_words(word_list, n): - """ Takes a list of words as input and returns a list of the n most frequently - occurring words ordered from most to least frequently occurring. - - word_list: a list of words (assumed to all be in lower case with no - punctuation - n: the number of words to return - returns: a list of n most frequently occurring words ordered from most - frequently to least frequentlyoccurring - """ - pass \ No newline at end of file + """ Takes a list of words as input and returns a list of the n most frequently + occurring words ordered from most to least frequently occurring. + + word_list: a list of words (assumed to all be in lower case with no punctuation + n: the number of words to return + returns: a list of n most frequently occurring words ordered from most + frequently to least frequentlyoccurring + """ + histogram = [] + + for uniq in set(word_list): + counter = 0 + for word in word_list: + if uniq == word: + counter += 1 + histogram.append((uniq, counter)) + histogram.sort(key=lambda tup: tup[1], reverse=True) + return histogram[0:n] + + + +file = 'pg4908.txt' + +word_list = get_word_list(file) +top_words = get_top_n_words(word_list, 20) + +for item in top_words: + print 'word = {}, number of times = {}'.format(item[0], item[1]) +#it appears that something is going wrong with the text parsing. I get some sort of space +#or new line in the histogram. diff --git a/pg4908.txt b/pg4908.txt new file mode 100644 index 0000000..ed629fe --- /dev/null +++ b/pg4908.txt @@ -0,0 +1,1955 @@ +Project Gutenberg's Five of Maxwell's Papers, by James Clerk Maxwell + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org/license + + +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 + + +*** 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. + + + + + + + + +End of Project Gutenberg's Five of Maxwell's Papers, by James Clerk Maxwell + +*** END OF THIS PROJECT GUTENBERG EBOOK FIVE OF MAXWELL'S PAPERS *** + +***** This file should be named 4908.txt or 4908.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/4/9/0/4908/ + +Produced by Gordon Keener + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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