534 



SCIENCE. 



[N. S. Vol. XI. No. 275. 



this thought Lie once said, " What do the 

 natural phenomena present to us if it is 

 not a succession of infinitesimal transforma- 

 tions of which the laws of the universe are 

 the invariants? " 



It need scarcely be added that all mathe- 

 matical thought even on the same subject 

 is not running in the same channel. Kleia 

 has divided mathematicians into three main 

 categories,* viz, the logicians, the formalists 

 and theintuitionists. The term logician is 

 " intended to indicate that the main strength 

 of the men belonging to this class lies in 

 their logical and critical powers, in their 

 ability to give strict definitions and to de- 

 rive rigid deductions therefrom. The great 

 and wholesome influence exerted in Ger- 

 many by Weierstrass in this direction is 

 well known. The formalists among the 

 mathematicians excel mainly in the skillful 

 formal treatment of a given question, in 

 divising for it an algorithm. Gordon, or 

 let us say Cayley and Sylvester, must be 

 ranged in this group. To the intuitionists, 

 finally, belong those who lay particular 

 stress on geometrical intuition, not in pure 

 geometry only, but in all branches of mathe- 

 matics. What Benjamin Peirce has called 

 ' geometrizing a mathematical question ' 

 seems to express the same idea. Lord Kel- 

 vin and von Standt may be mentioned as 

 types of this category." 



In his address before the Zurich Inter- 

 national Congress Poincare says,t "Mathe- 

 matics has a triple end. It should furnish 

 an instrument for the study of nature. 

 Furthermore, it has a philosophic end, and, 

 I venture to say an esthetic end. It ought 

 to incite the philosopher to search into the 

 notions of number, space and time ; and 

 above all, adepts find in mathematics de- 

 lights analogous to those that painting and 

 music give. They admire the delicate 

 harmony of numbers and of forms ; thej-^ 



* The Ei-anston CuUoquium, p. 2. 



tiSeKue Generale des Sciences, vol. 8, p. 857. 



are amazed when a new discovery discloses 

 for them an unlocked for perspective; and 

 the joy they experience has it not the 

 esthetic character although the senses take 

 no part in it? Only the privileged few are 

 called to enjoy it fully, it is true, but is it 

 not the same with all the noblest arts? 

 Hence I do not hesitate to say that mathe- 

 matics deserves to be cultivated for its own 

 sake and that the theories not admitting of 

 application to physics deserve to be studied 

 as well as the others. Moreover, a science 

 produced with a view single to its applica- 

 tions is impossible ; truths are fruitful only 

 if they are concatenated ; if we cleave to 

 those only of which we expect immediate 

 results the connecting links will be lacking, 

 and there will be no longer a chain." 



In closing we may remark that no effort 

 has been made to mention all the new fields 

 of mathematical thought. Mathematics, 

 like the other sciences, seems to offer inex- 

 haustible fields of investigation. As it ex- 

 pands its perimeter increases and hence 

 there is a continually increasing demand 

 for investigators. The fields that have 

 been examined present many difficulties 

 which cannot at present be surmounted- 

 Some of the old difficulties are being re- 

 moved by the light of the new decoveries. 

 Still we know only a few things even about 

 the fields which have been investigated. 

 It is the exception that something can be 

 done by known methods, the rule is that it 

 cannot yet be done. 



When we study the literature of some of 

 the older subjects we are sometimes sur- 

 prised by the large number of known facts^ 

 but when we come to study the subjects 

 themselves and ask independent questions 

 we are generally sui-prised to learn that so 

 few properties are known. Hence it seems 

 very desirable that the advanced student, at 

 least, should study subjects rather than the 

 known facts in regard to these subjects. 

 In this way a more accurate idea of the 



