DEVELOPMENT OF MATHEMATICAL THOUGHT. 7 17 



There exists, moreover, an analogy between the 

 manner in which these novel and extended ideas have 

 been historically introduced and the mode of reasoning 

 which led Sir W. E. Hamilton to the invention of a new 

 and extended algebra — the algebra of quaternions. This 

 analogy becomes evident if we study the small volume of 

 Hermann Hankel, which appeared about the same time 

 as Riemann's and Beltrami's fundamental geometrical 

 dissertations. 



The extension of Hamilton was only possible by drop- 

 ping one of the fundamental principles of general arith- conceptions, 

 nietic, the commutative principle of multiplication, which 

 is symbolically expressed by saying that a x & is equal to 

 h y.a. By assuming that « x & is equal to — & x a, Hamil- 

 ton founded a new general arithmetic on an apparently 

 paradoxical principle. Similarly Lobatchevski and Bolyai 

 constructed new geometries by dropping the axiom of 

 parallel lines. Hankel made clear the significance of the 

 new algebra, Riemann and Beltrami that of the new geom- 

 etry. The practical performance anticij)ated and led up 

 to the theoretical or philosophical exposition of the under- 

 lying principles. But there was a third instance in 

 which a new science had been created by abandoning the 

 conventional way of looking at things. This was the 

 formation of a consistent body of geometrical teaching 

 by disregarding the metrical properties and studying 

 only the positional or projective properties, following 

 Monge and Poncelet. The two great minds who worked 

 out this geometry independently of the conception of 

 number or measurement, giving a purely geometrical 

 definition of distance and number, were Cayley in Eng- 



