632 



SCIENTIFIC THOUGHT. 



Twofold 

 interest 

 in mathe- 

 matics. 



inasmuch as number and form are considered to be 

 the highest categories of human thought, or likewise as 

 the ultimate elements of all reality. These two interests 

 existed already in antiquity,^ as the word " geometry " 



much with the question. See the 

 following works : ' Die AUgeraeine 

 Functionentheorie,' part i., Tiib- 

 ingen, 1882; ' Ueber die Gruud- 

 lagen der Erkenntniss in den ex- 

 acten "Wissenschaften,' Tubingen, 

 1890; and his paper "Ueber die 

 Paradoxien des Infiuitiircalciils " 

 ('Mathematische Annalen,' vol. ix. 

 p. 149). In addition to the two 

 main interests which attach to 

 mathematical research, and which 

 I distinguish as the practical and 

 the philosophical, a third point of 

 view has sprung up iu modern 

 times which can be called the 

 purely logical. It proposes to 

 treat any special development of 

 mathematical research with the 

 aid of a definite, logically con- 

 nected complex of ideas, and not 

 to be satisfied to solve definite 

 problems with the help of any 

 methods which may casually pre- 

 sent themselves, however ingenious 

 they may be. In this way the 

 gi-eat geometrician, Jacob Steiner, 

 e.g., refused the assistance of ana- 

 lysis in the solution of geometrical 

 problems, conceiving geometry as 

 a complete organism which should 

 Solve its problems by its own 

 means. This view has been much 

 .strengthened by the development 

 in modern times of the theory of 

 Groups ; a group of operations 

 being ^defined as a sequence of such 

 operations as alwaj^s lead back 

 again to operations of the same 

 kind. Mathematical rigorists in 

 this sense would look upon the 

 use of mixed methods or opera- 

 tions not belonging to the same 

 group with that kind of disfavour 

 with which we should regard an 



essaj'ist who could not express his 

 ideas in pure English, but was 

 obliged to import foreign words 

 and expressions. It is interesting 

 to see that the country which has 

 offended most by the importation 

 of foreign words — namely, Ger- 

 many — is that in which this purism 

 in mathematical taste has found 

 the most definite expression. (See, 

 inter alia, Prof. Friedrich Engel's 

 Inaugural Lecture, " Der Gesch- 

 mack ill der neueren Mathematik," 

 Leipzig, 1890, as also Prof. F. 

 Klein's suggestive tract, ' Ver- 

 gleichende Betrachtungen iiber 

 neuere Geometrische Forschungen,' 

 Erlangen, 1872.) 



■* The literature of this subject 

 is considerable. I confine myself 

 to two works. The late eminent 

 mathematician, Hermann Hankel, 

 of whom more in the sequel of 

 this chapter, besides showing much 

 originalitj' in the higher branches 

 of the science, took great interest 

 in its philosophical foundations 

 and historical beginnings. In 1870 

 he published a small but highly 

 interesting volume, ' Zur Ge- 

 schichte der Mathematik in Alter- 

 thum und Mittelalter' (Leipzig, 

 Teubner). We have, besides, the 

 great work of Prof. Moritz Cantor, 

 ' Vorlesungen iiber Geschichte der 

 Mathematik,' in three large volumes 

 (Leipzig, Teubner). It brings the 

 history down to 1758. Referring 

 to the two interests which led to 

 mathematical investigations, Hankel 

 says (p. 88) : " From the moment 

 that Greek philosophers begin to 

 attract our attention through their 

 mathematical researches, the as- 

 pect which mathematics present 



