350 



SCIENCE 



[N. S. Vol. XXXVIII. No. 976 



PKECISION OP DEFINITIONS 



First, in regard to two matters which il- 

 lustrate how we are forced by physical 

 problems into abstract inquiries. It is a 

 constantly recurring need of science to re- 

 consider the exact implication of the terms 

 employed; and as numbers and functions 

 are inevitable in all measurement, the pre- 

 cise meaning of number, of continuity, of 

 infinity, of limit, and so on, are funda- 

 mental questions ; those who will receive 

 the evidence can easily convince them- 

 selves that these notions have many pit- 

 falls. Such an imperishable monument as 

 Euclid's theory of ratio is a familiar sign 

 that this has always been felt. The last 

 century has witnessed a vigorous inquiry 

 into these matters, and many of the results 

 brought forward appear to be new; nor is 

 the interest of the matter by any means ex- 

 hausted. I may cite, as intelligible to all, 

 such a fact as the construction of a func- 

 tion which is continuous at all points of a 

 range, yet possesses no definite differential 

 coefficient at any point. Are we sure that 

 human nature is the only continuous vari- 

 able in the concrete world, assuming it be 

 continuous, which can possess such a vacil- 

 lating character? Or I may refer to the 

 more elementary fact that all the rational 

 fractions, infinite in number, which lie in 

 any given range, can be enclosed in inter- 

 vals whose aggregate length is arbitrarily 

 small. Thus we could take out of our life 

 all the moments at which we can say that 

 our age is a certain number of years, and 

 days, ajid fractions of a day, and still have 

 appreciably as long to live ; this would be 

 true, however often, to whatever exactness, 

 we named our age, provided we were quick 

 enough in naming it. Though the recur- 

 rence of these inquiries is part of a wider 

 consideration of functions of complex vari- 

 ables, it has been associated also with the 

 theory of those series which Fourier used so 



boldly, and so wickedly, for the conduction 

 of heat. Like all discoverers, he took much 

 for granted. Precisely how much is the 

 problem. This problem has led to the pre- 

 cision of what is meant by a function of 

 real variables, to the question of the uni- 

 form convergence of an infinite series, as 

 you may see in early papers of Stokes, to 

 new formulation of the conditions of inte- 

 gration and of the properties of multiple 

 integrals, and so on. And it remains still 

 incompletely solved. 



CyLCULUS OF VARIATIONS 



Another case in which the suggestions of 

 physics have caused grave disquiet to the 

 mathematicians is the problem of the varia- 

 tion of a definite integral. No one is likely 

 to underrate the grandeur of the aim of 

 those who would deduce the whole physical 

 history of the world from the single prin- 

 ciple of least action. Every one must be in- 

 terested in the theorem that a potential 

 function, with a given value at the boun- 

 dary of a volume, is such as to render a cer- 

 tain integral, representing, say, the energy, 

 a minimum. But in that proportion one 

 desires to be sure that the logical processes 

 employed are free from objection. And, 

 alas! to deal only with one of the earliest 

 problems of the subject, though the finally 

 sufficient conditions for a minimum of a 

 simple integral seemed settled long ago, 

 and could be applied, for example, to New- 

 ton 's celebrated problem of the solid of 

 least resistance, it has since been shown to 

 be a general fact that such a problem can 

 not have any definite solution at all. And, 

 although the principle of Thomson and 

 Dirichlet, which relates to the potential 

 problem referred to, was expounded by 

 Gauss, and accepted by Riemann, and re- 

 mains to-day in our standard treatise on 

 National Philosophy, there can be no doubt 

 that, in the form in which it was originally 



