648 



SCIENTIFIC THOUGHT. 



France, somewhat later also in England and Germany. 

 In the latter country, the highly original writings of 

 Abel, and the independent labours of Jacobi, opened out 

 an entirely new branch of higher mathematics, beginning 

 with the discovery of the property of double periodicity of 

 certain functions. 1 This extensive and fruitful province of 

 analysis for a time retarded the revision and extension of 

 the groundwork of mathematical reasoning which Cauchy 

 had begun, and upon which Gauss evidently desired to 

 make the extension of higher mathematics proceed. 2 



1 Before the discovery of the 

 functions with a double period, 

 functions with one period were 

 known : the circular and expon- 

 ential functions the former pos- 

 sessing a real, the latter an imagin- 

 ary, period. The elliptic functions 

 turned out to "share simultaneously 

 the properties of the circular func- 

 tions and exponential functions, and 

 whilst the former were periodical 

 only for real, the latter only for 

 imaginary, values of the argument, 

 the elliptic functions possessed both 

 kinds of periodicity." This great 

 step became clear when it occurred 

 to Abel and Jacobi independently 

 to form functions by inversion of 

 Legendre's elliptic integral of the 

 first kind. The two fundamental 

 principles involved in this new 

 departure were thus the process of 

 inversion and the use of the imagin- 

 ary, as a necessary complement to 

 the real, scale of numbers. The 

 share which belongs independently 

 to Abel and Jacobi has been clearly 

 determined since the publication of 

 the correspondence of Jacobi with 

 Legendre during the years 1827-32 

 (reprinted in Jacobi's ' Gesammelte ' 

 Werke,' ed. Borchardt, vol. i., 

 Berlin, 1881), and of the complete 

 documents referring to Abel, which 

 are now accessible in the memorial 



volume published in 1902. A very 

 lucid account is contained in a 

 pamphlet by Prof. Kbnigsberger, 

 entitled ' Zur Geschichte der Theorie 

 der Elliptischen Transcendenten 

 in den Jahren 1826-29 ' (Leipzig, 

 1879). 



2 Of the four great mathema- 

 ticians who for sixty years did the 

 principal work in connection with 

 elliptic functions viz., Legendre 

 (1752-1833), Gauss (1777-1855), 

 Abel (1802-29), and Jacobi (1804- 

 51), each occupied an independent 

 position with regard to the subject, 

 suggested originally by Euler, and 

 important for the practical applica- 

 tions which it promised. Legendre 

 during forty years, from 1786 on- 

 ward, worked almost alone : he 

 brought the theory of elliptic in- 

 tegrals, which had occurred origin- 

 ally in connection with the compu- 

 tation of an arc of the ellipse, into 

 a system, and to a point beyond 

 which the then existing methods 

 seemed to promise no further ad- 

 vance. This advance was, however, 

 secured by the labours of Jacobi 

 through the introduction of the 

 novel principles referred to in the 

 last note. Two years before Jacobi's 

 publication commenced, Abel had 

 already approached the subject from 

 an entirely different and much more 



