516 POPULAR SCIENCE MONTHLY. 



which it is sought to measure. Every one will grant, however, that the 

 distance between two clouds, for instance, is not a definable magnitude ; 

 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 mathematics 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 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. A practical measurement is in fact a classifica- 

 tion; we assign a magnitude to a certain category, which may be nar- 

 rowly limited, but which has in any case a certain breadth. 



By a frank process of idealization a logical system of abstract dy- 

 namics 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 can not 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 con- 

 ceptions are to be traced to an empirical source was received with some- 

 thing 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 approximately 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 tbe 

 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 frag- 

 ments 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 geo- 



