654 SCIENTIFIC THOUGHT. 



this country the labours of De Morgan and of Sir William 

 Kowan Hamilton gave the matter a further and very 

 important extension. 1 It was also in this country that 

 the second problem, the critical examination of the 

 principles which underlie the process of legitimate 

 generalisation of algebra, received distinct attention. To 

 George Peacock, and to the school of algebraists which 

 followed him, is due the merit of having brought out 

 clearly the three fundamental laws of symbolical reasoning 

 now generally admitted in text -books on the subject 

 the associative, distributive, and commutative principles. 

 That these principles were to a great extent conventional, 

 or empirically adopted from ordinary arithmetic, and in 

 consequence not necessarily indispensable for a consistent 

 system of symbolical reasoning, has been generally ad- 

 20. mitted ever since Sir William Eowan Hamilton, after 



Quater- 

 nions, ten years of labour, succeeded in establishing a new 



calculus :the method of quaternions, in which the com- 

 mutative principle of multiplication is dropped. This 



1 Far more important than the Works,' vol. xi. p. 503). There 

 suggestions or artifices mentioned seems little doubt that this con- 

 in the foregoing note, and which i ception was first clearly established 

 since the time of Argand and in the mind of Gauss, and that 



Gauss have been variously modified, 

 is the conception that our com- 

 mon numbers do not form a 

 complete system without the ad- 

 dition of the imaginary unit, but 

 that with the introduction of a 

 second unit ' ' numbers form a 

 universe complete in itself, such 

 that, starting in it, we are never 

 led out of it. There may very well 

 be, and perhaps are, numbers in a 

 more general sense of the term ; 

 but in order to have to do with 

 such numbers (if any) we must 



none of the contemporary writers 

 can be shown to have had a 

 similarly clear insight. Since this 

 has become generally recognised 

 and we owe -this recognition prob- 

 ably to the independent labours 

 of Grassmann and Riemann 

 the discussion of the whole sub- 

 ject has been raised to a much 

 higher level, as may be seen by 

 comparing the Report of Peacock, 

 quoted above, with the discussion 

 of Hankel (loc. cit,), and still more 

 with the exhaustive article by Prof. 



start with them" (Cayley in art. E. Study in vol. i., ' Encyk. Math. 

 " Equation," ' Ency. Brit. '; 'Coll. Wiss.,' pp. 147-184. 



