654 



SCIENTIFIC THOUGHT. 



20. 

 Quater- 

 nions. 



this country the labours of De Morgan and of Sir William 

 Eowan Hamilton gave the matter a further and very- 

 important extension.^ It was also in this country that 

 the second problem, tlie 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 reason- 

 ing 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- 

 mitted ever since Sir William Eowan Hamilton, after 

 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 



^ Far more important than 

 the suggestions or artifices men- 

 tioned in the foregoing note, and 

 which since the time of Argand 

 and Gauss have been variously 

 modified, is the conception that 

 our common numbers do not form 

 a complete system without the 

 addition of the imaginary unit, 

 but that with the introduction 

 of a second unit " uumbei's 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 

 .start with them"' (Cajdey in art. 

 "Equation," ' Ency. Brit.'; 'Coll. 



Works,' vol. xi. p. 50.3). There 

 seems little doubt that this con- 

 ception was first clearly established 

 in the mind of Gauss, and that 

 none of the contemporary writers 

 can be shown to have had a 

 similarly clear insight. Since this 

 has become generallj' recognised — 

 and we owe this recognition 

 probably to the independent 

 labours of Gra'^smann and Rie- 

 mann — the discussion of the whole 

 subject 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. 

 E. Study in vol. i., 'Encyk. Math. 

 AViss.,' pp. 147-184. 



