August i8, 1904] 



NA TURE 



575 



physicists, we shall find, I think, that the prevalent con- 

 ception of the world was that it was constructed on some 

 sort of absolute geometrical plan, and that the changes in 

 it proceeded according to precise laws ; that, although the 

 principles of mechanics might be imperfectly stated in our 

 text-books, at all events such principles existed, and were 

 ascertainable, and, when properly formulated, would possess 

 the definiteness and precision which were held to 

 characterise, say, the postulates of Euclid. Some writers 

 have maintained, indeed, that the principles in question 

 were finally laid down by Newton, and have occasionally 

 used language which suggests that any fuller understand- 

 ing of them was a mere matter of interpretation of the text. 

 But, as Hertz has remarked, most of the great writers on 

 Dynamics betray, involuntarily, a certain malaise when ex- 

 plaining the principles, and hurry over this part of their 

 task as quickly as is consistent with dignity. They are 

 not really at their ease until, having established their equa- 

 tions somehow, they can proceed to build securely on these. 

 This has led some people to the view that the laws of Nature 

 are merely a system of differential equations ; it may be 

 remarked in passing that this is very much the position 

 in which we actually stand in some of the more recent 

 theories of Electricity. As regards Dynamics, when once 

 the critical movement had set in, it was easy to show that 

 one presentation after another was logically defective and 

 confused ; and no satisfactory standpoint was reached until 

 it was recognised that in the classical Dynamics we do not 

 deal immediately with real bodies at all, but with certain 

 conventional and highly idealised representations of them, 

 which we combine according to arbitrary rules, in the hope 

 that if these rules be judiciously framed the varying com- 

 binations will image to us what is of most interest in some 

 of the simpler and more important phenomena. The 

 changed point of view is often associated with the publi- 

 cation of Kirchhoff's lectures on Mechanics in 1876, where 

 it is laid down in the opening sentence that the problem 

 of Mechanics is to describe the motions which occur in 

 Nature completely and in the simplest manner. This state- 

 ment must not be taken too literally ; at all events, a fuller, 

 and I think a clearer, account of the province and the 

 method of -Abstract Dynamics is given in a review of the 

 second edition of Thomson and Tait, which was one of the 

 last things penned by Maxwell, in 1879 (Nature, vol. xx. 

 p. 213; Scientific Papers, vol. ii. p. 776). A "complete" 

 description of even the simplest natural phenomenon is an 

 obvious impossibility ; and, were it possible, it would be 

 uninteresting as well as useless, for it would take an in- 

 calculable time to peruse. Some process of selection and 

 idealisation is inevitable if we are to gain any intelligent 

 comprehension of events. Thus, in Astronomy we replace 

 a planet by a so-called material particle — i.e., a mathe- 

 matical point associated with a suitable numerical coefficient. 

 .\11 the properties of the body are here ignored except those 

 of position and mass, in which alone we are at the moment 

 interested. The whole course of phvsical science and the 

 language in which its results are expressed have been 

 largely determined by the fact that the ideal images of 

 Geometry were already at hand at its service. The ideal 

 representations have the advantage that, unlike the real 

 objects, definite and accurate statements can be made about 

 them. Thus two lines in a geometrical figure can be pro- 

 nounced to be equal or unequal, and the statement is in 

 either case absolute. It is no doubt hard to divest oneself 

 entirely of the notion conveyed in the Greek phrase oel d 

 $fhs yfaiifTpii, that definite geometrical magnitudes and 

 delations are at the back of phenomena. It is recognised 

 indeed that all our measurements are necessarily to some 

 degree uncertain, but this is usually attributed to our own 

 limitations and those of our instruments rather than to the 

 ultimate vagueness of the entity which it is sought to 

 measure. Everyone will grant, however, that the distance 

 between two clouds, for instance, is not a definable magni- 

 tude ; and the distance of the earth from the sun, and even 

 the length of a wave of light, are in precisely the same 

 case. The notion in question is a convenient fiction, and is 

 a striking testimony to the ascendency which Greek Mathe- 

 matics have gained over our minds, but I do not think that 

 more can be said for it. It is, at any rate, not verified by 

 the experience of those who actually undertake physical 

 measurements. The more refined the means employed, the 



NO. I 8 16, VOL. 70] 



more vague and elusive does the supposed magnitude 

 become ; the judgment flickers and wavers, until at last in 

 a sort of despair some result is put down, not in the belief 

 that it is exact, but with the feeling that it is the best we 

 can make of the matter. .\ practical measurement is in 

 fact a classification ; we assign a magnitude to a certain 

 category, which may be narrowly limited, but which has 

 in any case a certain breadth. 



By a frank process of idealisation a logical system of 

 .Abstract Dynamics can doubtless be built up, on the lines 

 sketched by Maxwell in the passage referred to. Such 

 difficulties as remain are handed over to Geometry. But 

 we cannot stop in this position ; we are constrained to 

 examine the nature and the origin of the conceptions of 

 Geometry itself. By many of us, I imagine, the first 

 suggestion that these conceptions are to be traced to an 

 empirical source was received with something of indignation 

 and scorn ; it was an outrage on the science which we had 

 been led to look upon as divine. Most of us have, however, 

 been forced at length to acquiesce in the view that Geometry, 

 like Mechanics, is an applied science ; that it gives us merely 

 an ingenious and convenient symbolic representation of the 

 relations of actual bodies ; and that, whatever may be the 

 a priori forms of intuition, the science as we have it could 

 never have been developed except for the accident (if I may 

 so term it) that we live in a world in which rigid or approxi- 

 mately rigid bodies are conspicuous objects. On this view 

 the most refined geometrical demonstration can be resolved 

 into a series of imagined experiments performed with such 

 bodies, or rather with their conventional representations. 



It is to be lamented that one of the most interesting 

 chapters in the history of science is a blank ; I mean that 

 which would have unfolded the rise and growth of our 

 system of ideal Geometry. The finished edifice is before 

 us, but the record of the efforts by which the various stones 

 were fitted into their places is hopelessly lost. The few 

 fragments of professed history which we possess were edited 

 long after the achievement. 



It is commonly reckoned that the first rude beginnings, 

 of Geometry date from the Egyptians. I am inclined to 

 think that in one sense the matter is to be placed much 

 further back, and that the dawn of geometric ideas is to 

 be traced among the prehistoric races who carved rough 

 but thoroughly artistic outlines of animals on their weapons. 

 I do not know whether the matter has attracted serious 

 speculation, but I have myself been led to wonder how rnen 

 first arrived at the notion of an outline drawing. The 

 primitive sketches referred to immediately convey to the 

 experienced mind the idea of a reindeer or the like ; but in 

 reality the representation is purely conventional, and is ex- 

 pressed in a language which has to be learned. For nothing 

 could be more unlike the actual reindeer than the few 

 scratches drawn on the surface of a bone : and it is of course 

 familiar to ourselves that it is only after a time, and by an 

 insensible process of education, that very young children 

 come to understand the meaning of an outline. Whoever 

 he was, the man who first projected the world into two 

 dimensions, and proceeded to fence off that part of it which 

 was reindeer from that which was not, was certainly under 

 the influence of a geometrical idea, and had his feet in the 

 path which was to culminate in the refined idealisations of 

 the Greeks, .^s to the manner in which these latter were 

 developed, the only indication of tradition is that some 

 propositions were arrived at first in a more empirical or 

 intuitional, and afterwards in a more intellectual way. So 

 long as points had size, lines had breadth, and surfaces 

 thickness, there could be no question of exact relations 

 between the various elements of a figure, any more than 

 is the case with the realities which they represent. But the 

 Greek mind loved definiteness, and discovered that if we agree 

 to speak of lines as if they had no breadth, and so on, exact 

 statements became possible. If any one scientific invention 

 can claim pre-eminence over all others, I should be inclined 

 myself to erect a monument to the inventor of the mathe- 

 matical point, as the supreme type of that process of abstrac- 

 tion which has been a necessary condition of scientific work 

 from the very beginning. 



It is possible, however, to uphold the importance of the 

 part which Abstract Geometry has played, and must still 

 play, in the evolution of scientific conceptions, without com- 

 mitting ourselves to a defence, on all points, of the 



