838 



SCIENCE 



[N. S. Vol. XLIV. No. 1146 



the concrete? Professor A. Voss, of the 

 University of Munich, said in a lecture in 

 1908: 



Our entire present civilization, as far as it de- 

 pends upon the intellectual penetration and utili- 

 zation of nature, has its real foundation in the 

 mathematical sciences. 



You will observe that he does not say in 

 mathematics, but in the sciences which have 

 made use of mathematics in their develop- 

 ment. Let us investigate this a little. 

 Can you realize what would happen, just 

 what stage of civilization we should be in, 

 if all that is developed by the use of mathe- 

 matics could be removed from the world 

 by some magic gesture? Every branch of 

 physics makes use of mathematics ; chem- 

 istry is not free from it; engineering is 

 based upon its developments; sociology, 

 economics and variation in biology make 

 use of statistics and probability. Our sky- 

 scrapers must disappear ; our great bridges 

 and tunnels must be removed; our trans- 

 portation systems, our banking systems, our 

 whole civilization, indeed, must step back- 

 ward many centuries. 



Mathematics and its symbolism appear 

 in rather unexpected places. S. G. Barton, 3 

 of the Flower Observatory, University of 

 Pennsylvania, says that in the Encyclo- 

 paedia Britannica, written not for the spe- 

 cialist so much as for the general reader, 

 there are one hundred four articles which 

 make use of the notation of the infinitesi- 

 mal calculus, of which only about one 

 fourth are pure mathematics. You may be 

 surprised to know that you need the in- 

 finitesimal calculus to read the articles on 

 clock, heat, lubrication, map, power trans- 

 mission, ship building, sky, steam engine 

 and strength of materials. 



Take these sentences from Simon New- 

 comb's article in the Encyclopedia Britan- 

 nica on celestial mechanics : 



a Science, Vol. 40 (1914), p. 697. 



The purpose of the present article is to convey 

 a general idea of the methods by which the re- 

 sults of celestial mechanics are reached, without 

 entering into those tec hni cal details which can be 

 followed only by a trained mathematician. It 

 must be admitted that any intelligent comprehen- 

 sion of the subject requires at least a grasp of the 

 fundamental conceptions of analytical geometry 

 and the infinitesimal calculus, such as only one 

 with some training in these subjects can be ex- 

 pected to have. . . . The non-mathematical reader 

 may possibly be able to gain some general idea, 

 though vague, of the significance of the subject. 



Sir John Herschel in his introduction to 

 his book, "Outlines of Astronomy," says: 



Admission to its (astronomy's) sanctuary and 

 to the privileges and feelings of a votary is only 

 to be gained by one means — sound and sufficient 

 knowledge of mathematics, the great instrument of 

 all exact inquiry, without which no man can ever 

 make such advances in this or any other of the 

 higher departments of science as can entitle him 

 to form an independent opinion on any subject of 

 discussion within their range. 



Professor and Mrs. Mittag-Leffler have 

 given their fortune to the founding of an 

 institute for the promotion of research in 

 pure mathematics in Sweden and the other 

 Scandinavian countries. They say: 4 



Our testament owes its origin to the lively con- 

 viction that a people that does not accord to 

 mathematics a high place in its estimation, will 

 never be in a position to fulfd the most lofty tasks 

 of civilization, and to enjoy in consequence that 

 international consideration which is itself, in the 

 end, an effective means of preserving our place in 

 the world and of safeguarding our right to live 

 our own life. 



I am not claiming any superiority for 

 mathematics over the other sciences. I am 

 trying to emphasize how indispensable 

 mathematics has been in the development 

 of other sciences. Wherein lies its worth? 



Mathematics is an exact science, that is, 

 with the conditions — the postulates — defi- 

 nitely given, the conclusion admits of no 

 doubt, of no variation. The worker in the 



4 Bulletin of the American Mathematical So- 

 ciety, Vol. XXIII., No. 1 (1916), p. 31. 



