Maecu 13, 1903.] 



SCIENCE. 



405 



for the interests of science in general that 

 there should be a strong bodj' of men thor- 

 oughly possessed of the scientific method 

 in both its inductive and its deductive 

 forms. "We are confronted with the ques- 

 tions : What is science ? What is the scien- 

 tific method? What are the relations be- 

 tween the mathematical and the natural 

 scientific processes of thought? As to 

 these questions, I refer to articles and ad- 

 dresses of Poincare,* Boltzmannf and 

 Burihardt,! and to Mach's 'Science of 

 Mechanics' and Pearson's 'Grammar of 

 Science.' 



Without elaboration of metaphysical or 

 psychological details, it is sufficient to refer 

 to the thought that the individual, as eon- 

 fronted with the world of phenomena in 

 his effort to obtain control over this 

 world, is gradually forced to appreciate a 

 Iniowledge of the usual coexistences and 

 sequences of phenomena, and that science 

 arises as the body of formulas serving to 

 epitomize or summarize conveniently these 

 usual coexistences and sequences. These 

 formulas are of the nature of more or less 

 exact descriptions of phenomena ; they are 

 not of the nature of explanations. Of all 

 the relations entering into the formulas of 

 science, the fundamental mathematical no- 

 tions of number and measure and form 

 were among the earliest, and pure mathe- 

 matics in its ordinary acceptation may be 

 understood to be the systematic develop- 

 ment of the properties of these notions, in 

 accordance with conditions prescribed by 



* In addition to those already cited: 'On the 

 Foundations of Geometry,' The Monist, vol. 9, 

 October, 1898, pp. 1-43. ' Sur les principes de la 

 m^canique,' BihUothique du Congrcs Interna- 

 tional de Philosophic, vol. 3, pp. 4.57-494:. 



t ' Ueber die jSIethoden der theoretischen Physik,' 

 Dyck's Eatalog mathematischer und mathemat- 

 isch-physikalischer Modellc, Apparate und Instni- 

 mciitr, pp. 89-98, jMunich, 1892. 



t ' Mathematisches und naturwissenschaftliches 

 Denken,' Jahresbericht der Deutscheii Math.-Ter., 

 vol. 11 (1902), pp. 49-57. 



physical phenomena. Arithmetic and 

 geometry, closely united in mensuration 

 and trigonometry, early reached a high 

 degree of advancement. But after the de- 

 velopment of the generalizing literal nota- 

 tions of algebra, and largely in response 

 to the insistent demands of mechanics, 

 astronomy and physics, the seventeenth 

 century, binding together arithmetic and 

 geometry infinitely more closely, created 

 aualj'tic geometry and the infinitesimal 

 calculus, those mighty methods of research 

 whose application to all branches of the 

 theoretical and practical physical sciences 

 so fundamentally characterizes the civiliza- 

 tion of to-day. 



The eighteenth century was devoted to 

 the development of the powers of these 

 new instruments in all directions. While 

 this development continued during the 

 nineteenth century, the dominant note of 

 the nineteenth century was that of crit- 

 ical I'corganization of the foundations of 

 pure mathematics, so that, for instance, 

 the majestic edifice of analysis was seen to 

 rest upon the arithmetic of positive in- 

 tegers alone. This reorganization and the 

 consequent course of development of pure 

 mathematics were independent of the ques- 

 tion of the application of mathematics to 

 the sister sciences. There has thus arisen 

 a chasm between pure mathematics and 

 applied mathematics. There have not been 

 lacking, however, infiuences making toward 

 the bridging of this chasm; one thinks es- 

 pecially of the whole influence of Klein in 

 Germany and of the Ecole Polytechnique 

 in France. As a basis of union of the pure 

 mathematicians and the applied mathema- 

 ticians, Klein has throughout emphasized 

 the impoi-tance of a clear understanding 

 of the relations between those two parts 

 of mathematics which are conveniently 

 called 'mathematics of precision' and 

 'mathematics of approximation,' and I re- 

 fer especially to his latest work of this 



