712 



SCIENTIFIC THOUGHT. 



53. 

 Non- 

 Euclidean 

 geometry. 



an introduction to the advanced theories of Gauss and 

 Kiemann ; and for this purpose he went back to the 

 unnoticed labours of Grassmann in Germany, to the 

 writings of Peacock and De Morgan in England, and 

 incidentally introduced into Germany the elaborate 

 algebra of quaternions, invented and practised by 

 Hamilton twenty years before that time. The papers 

 of Eiemann and Helmholtz similarly showed the neces- 

 sity of a thorough investigation of the principles and 

 foundations of ordinary or Euclidean geometry, and 

 showed how consistent systems of geometry could be 

 elaborated on other than Euclidean axioms. Only 

 from that moment, in fact, did it become generally 

 recognised that already, a generation before, two in- 

 dependent treatises on elementary geometry had been 

 published in which the axiom of parallel lines was 

 dispensed with and consistent geometrical systems 

 developed. These were contained — as already stated 

 — in the ' Kasan Messenger,' under date 1829 and 



ing ouly a short paper by Dr F. 

 Gregorj' on Symbolical Algebra 

 in the Edinburgh ' Transactions.' 

 Whilst Hankel was delivering 

 lectures on these fundamentals, 

 Weierstrass in Berlin was likewise 

 in the habit of introducing his 

 lectures on the Theory of Analytic 

 Functions by a discussion of the 

 theory of Complex Numbers. This 

 introduction was published, with 

 Weierstrass's permission, in the year 

 1872 by Dr E. Kossak (in a pro- 

 gramme of the Friedrichs-Werder 

 Gymnasium), after lectures de- 

 livered by Weierstrass in 1865-66. 

 To what extent Hankel may have 

 been influenced by Weierstrass's 

 lectures, which he seems to have 

 attended after leaving Gottingen, 



is uncertain, for in spite of his very 

 extensive references he does not 

 mention Weierstrass. In Kossak's 

 ' Elemente der Arithmetik ' the 

 term " permanence of formal rules " 

 is not used, but the treatment of 

 the extended arithmetic is carried 

 on along the same lines — i.e., not 

 by an attempt to represent the 

 complex quantities, but on the 

 ground of maintaining the rules 

 which govern the arithmetic of 

 ordinary numbers. Great im- 

 portance is also attached to the 

 principle of inversion as having 

 shown itself of value in the theory 

 of elliptic functions, and being not 

 less valuable in arithmetic. As 

 stated above (p. 640, note), this prin- 

 ciple is also insisted on by Peacock. 



