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PROCEEDINGS 



CAMBRIDGE PHILOSOPHICAL SOCIETY. 



October 28, 1844. 



On the Foundation of Algebra, No. IV. — On Triple Algebra. 

 By Augustus De Morgan, Esq., of Trinity College. 



The extensions which have successively been made in algebraical 

 interpretation have, been consequences of efforts to interpret symbols 

 which presented themselves as necessary parts of the algebraical lan- 

 guage which is suggested by arithmetic. The now well-known sig- 

 nification of a + b^/ — 1 did not yield any new imaginary or unex- 

 plained quantities : and accordingly no effort (within the author's 

 knowledge) was made to produce an algebra which should require 

 three dimensions of space for its interpretation, until Sir William 

 Rowan Hamilton wrote a paper (the first part of which appeared in 

 the Philosophical Magazine * before the present one was begun) on a 

 System of Quaternions. This system, as the name imports, involves 

 four distinct species of units, one of which may by analogy be called 

 real, the three others being imaginaries, as distinct from one another 

 as the imaginary of ordinary algebra is from the real. These ima- 

 ginaries are not deductions, but inventions ; their laws of action on 

 each other are assigned : this idea Mr. De Morgan desires to acknow- 

 ledge as entirely borrowed from Sir William Hamilton. 



Sir William Hamilton has rejected the idea of producing a triple 

 algebra, apparently on account of the impossibility of forming one in 

 which such a symbol as a% + by + c£ represents a line of the length 

 V(a" + b°--\-c <i ). Mr. De Morgan does not admit the necessity of 

 having a symmetrical function of a, b, c, and, throwing away this 

 stipulation, points out a variety of triple systems, partially or wholly 

 interpreted. 



Sir William Hamilton's quaternion algebra is not entirely the 

 same in its symbolical rules as the ordinary algebra : differing in that 

 the equation AB = B A is discarded and AB= — BA supplies its place. 

 Those of Mr. De Morgan's system, which are imperfect, all give 

 AB = BA, but none of them give A(BC) = (AB)C, except in particu- 

 lar cases. 



* Vol xxv. pp. 10, 241. 

 No. II. — Proceedings of the Cambridge Phil. Soc. 



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