May 27, 1909] 



NATURE 



38J 



connected logically by no means implies that this connec- 

 tion is obvious, nor does it preclude their being discovered, 

 even in a correct form, by the exercise of what is popularly 

 called ■' intuition." 



By the side of this ever-deepening investigation into the 

 principles of mathematics went on an inquiry, carried on 

 by entirely different men, into the nature and purposes of 

 our conceptions in physics. Through the worl-c of these 

 men, the true relation of mathematics to physical science, 

 which had been a subject on which there had been until 

 then much confusion of thought, appeared clearly. We 

 will glance at the history of mathematics and of the 

 application of mathematics to physics. 



From the earliest times until the seventeenth century 

 mathematicians were chiefly occupied with particular ques- 

 tions — the properties of particular numbers and the geo- 

 metrical properties of particular figures, together with 

 simple mechanical questions concerning centres of gravity, 

 the lever, and so on. The only exception to this was 

 afforded by algebra, in which symbols (like our present 

 v and y) took the place of numbers, so that, what is a 

 great advance in economy of thought and other labour,' 

 a part of calculation could be done with symbols instead 

 of numbers, so that the one result stated a proposition 

 valid for a whole class (often an infinity) of different 

 numbers. Such a result is that which we now write : — 



(a+bY = a' + 3a-b+3ab' + h\ 



which remains valid when we substitute anv particular 

 numbers for a and b, and labour in calculation is often 

 saved by the formula, even in this very simple case. 



The great revolution in mathematical thought brought 

 about by Descartes in 1637- consists in the application of 

 this general algebra to geometry by the verv natural 

 thought of substituting the numbers expressing the lengths 

 of straight lines for those lines. Thus a point in a plane 

 (for instance) is determined in position by two " co- 

 ordinates " or numbers denoted by x and y, x denoting 

 the distance from a fixed point along a fi.xed straight line 

 (the abscissa) to a certain point, and y denoting the 

 distance from this last point along a perpendicular (an 

 " ordinate ") to the abscissa to the point in question deter- 

 mined by X and y. As the point in question varies in 

 position, X and y both vary ; to every x belongs, in 

 general, one or rnore y's, and we arrive at the most 

 beautiful idea of a single algebraical equation between .1; 



' In T/ie Xnu Quarterly !or October. 1908 (vol. i., p. 4g8), Mr. N. R. 

 Campbell has objected to the idea of Mach that " economy of thought " 

 is the end for which scientific theories are formed, for reasnns based, 

 It seems to me, on a misunderstanding of what Mach 

 Perhaps the phrase " e'-onomy of thought " is not wel 

 may lead to such misunderstandings ; for the principle d 

 to a rule of scientific method which can be readily admiltet 

 the goal of science, as guided by this principle, will ni 

 attained when its students have ceased to think." Thi; 

 thus described. As science advances, besides actually 1 

 obstacle, it, consciously or unconsciously, leaves marks of euidance" for 

 those who come after ; so that those obstacles which required great genius to 

 overcome in the first inst.ince aferwards became quite easily so. This 

 is necessary in order that our energies may not be spent by the time 

 that we reach a new obstacle not hitherto surmounted; and "economy 

 of thought " means that we are to be spared waste of the energy of 

 thought whilst treading the path already trodden by our predecessors, 

 so that we may keep it for the really important new problems— not 

 that we may cease to think about problems, new or old. 



And thus we have legacies left by great men, such as Lagrange's 

 analytical mechanics and Fourier's theory of the conduction of heat, 

 which are merely inventories of extensive classes of facts, arranged 

 with wonderful compactness. In this description of an infinity, perhaps, 

 of facts by a fe-M formula, there is undoubtedly an Esthetic mitive 

 and value ; but, apart from this, there is this important economical 

 aspect, that a multitude of particular facts and " laws," which we had 

 "ficially (in a note-book or library), 

 ■ symbolical formulae, which only 



sy\\re losical development to get at the particular cases. From this 

 that "economy of thought" 

 The solution of the paradox 



ally 





id certainly 



hitherto to 



is, in the theory, comprised 



logical development to get at the 

 point of view we get the apparent parado: 

 leads to the replacing of memory by reason. 

 is that logical development can be made 

 memory, and that thus thought is spared 

 it on the unsolved problems which are al 



that 



nechanical 



•adv 



The tendency to economy of thought, which is shown in the gi 

 of physics— for example, in the inclusion of the particular Biot's 

 of the distribution of temperatures in Fourier's theory— may als 

 seen in the symbolism of pure mathematics. 



2 We need hardly point out that this change was not suHden- 

 Descartes's "G^ometrie " was not a "proles sine maire creata," but 

 here, as everywhere, the development of r ' 

 principle of continuity. 



and y representing the whole of a curve — the one equation, 

 called the " equation of the curve," expressing the general 

 law by which, given any particular x out of an infinity 

 of them, the corresponding y or y's can be found. Thus 

 y = 3.i: + 2 gives one y for each x, y" = 3.v-|-2, or, more 

 generally, y- = mx + n, where m and n stand for any fixed 

 numbers, gives two y's, one positive and one negative 

 (above and below the abscissa respectively), for each x, ex- 

 cept when x is zero. 



The problem of drawing a tangent — the limiting posi- 

 tion of a secant, when the two meeting points approach 

 indefinitely close to one another — at any point of a curve 

 came into prominence as a result of Descartes 's work, and 

 this, together with the allied conceptions of velocity and 

 acceleration " at an instant " ' which appeared in Galilei's 

 classical investigation, published in iti38, of the law 

 according to which freely falling bodies move, gave rise 

 at length to the powerful and convenient " infinitesimal 

 calculus of Leibniz and the " calculus of fluxions " of 

 Newton. It is now clearly established that those two 

 methods, which are theoretically — but not practically — the 

 same, were discovered independently; Newton discovered 

 his first, and Leibniz published his first, in 1684. The 

 finding of the areas of curves and of the shapes of the 

 curves which moving particles describe under given forces 

 showed themselves, in this calculus, as results of the 

 inverse process to that of the direct process which serves 

 to find tangents and the law of attraction to a given 

 point from the datum of the path described by a particle. 

 The direct process is called "differentiation," the inverse 

 process " integration." 



Newton's fame is chiefly owing to his application of 

 this method to the solution, which, in its broad outlines, 

 he gave, of the problem of the motion of the bodies in 

 the solar system, which includes his discovery of the law 

 according to which all matter gravitates towards (is 

 attracted by) other matter. This was given in his " Prin- 

 cipia " of ibSj; and, for more than a century afterwards, 

 mathematicians were occupied in extending and applying 

 the calculus. 



Of the great mathematicians of this time — the brothers 

 Bernoulli, Euler, Clairaut, d'.Membert, Maclaurin, 

 Lagrange, Laplace, Legendre, Fourier, Poisson, and others 

 — most were Frenchmen ; and the successful application of 

 mathematics to celestial and molecular mechanics, to 

 hydrodynamics, to the theory of the conduction of heat, 

 and to electricity and magnetism, brought about, in a 

 great measure, that enthusiastic trust in science, that 

 faith that the whole mystery of life and of our lives was 

 about to be uncovered by it, and that waning of faith in 

 religion, which are so characteristic of France in the 

 eighteenth century, and which are met with in the highest 

 degree in Laplace. 



\\'hether or not it was due to the indirect influence of 

 Kant, whose " Critique of the Pure Reason " first appeared 

 in 1781, an increasing tendency towards critical examina- 

 tion into the validity and the limits of validity of mathe- 

 matical conceptions and methods appeared in the mathe- 

 matics of the nineteenth century. First of all we must 

 mention Gauss, who, in an unexampled degree, combined 

 the power of discovery and profound critical insight, so 

 that in the seven volumes of his publications, in the col- 

 lected edition of his works, there is hardly a page which 

 is not both important in the history of matherriatics and ' 

 free from error. But perhaps of still greater influence was 

 the work of the French mathematician Cauchy ; it is he 

 who must be regarded as the chief inspirer — perhaps in- 

 direct — of Weierstrass ; it is Weierstrass who was the chief 

 inspirer of Georg Cantor, and it is to the influence of 

 Cantor and Dedekind, most of all, that we owe thht trend 

 of thought which, with modern mathematical logicians, 

 has resulted in the great discovery of the logical nature of 

 mathematics. 



Of course, in this short description there is no implica- 

 tion th.it the nineteenth century has been poor in the more 

 technical achievements or physical applications of mathe- 

 matics ; in England alone the names of Stokes, Thomson 

 t Mathemaficallv. the finding of the tanpent at a point of a curve, 

 and finding the velocity of a particle describing this curve when it rets 

 to tb^t point, are identical problems. Thev are expressed as finding 

 the '' differential coefficient," or the " fluxion " at the point. 



NO. 2065, VOL. 80] 



