Cambridge Philosophical Society. 



convey with it the expression of my own opinion that never was a 

 Medal more worthily earned. Permit me, Sir, at the same time to 

 remark, that the character of the Council is most deeply pledged in 

 this award, and that I trust that, at no distant period, it will be re- 

 deemed by such communication of the details of the observations as 

 ■will enable the Council to refer other inquirers to publications that 

 are within the reach of all for a sufficient justification of this judge- 

 ment. 



The following Fellows were elected Officers and Council for the 

 ensuing year, viz. — 



PrMjVew^.— Captain W. H. Smyth, R.N., K.S.F., D.C.L., F.R.S. 

 — Vice-Presidents. George Biddell Airy, Esq., M.A., F.R.S. , Astro- 

 nomer Royal; Samuel H. Christie, Esq., M.A., F.R.S.; Bryan 

 Donkin, Esq., F.R.S.; Thomas Galloway, Esq., M.A., F.R.S.— 

 Treasurer. George Bishop, Esq. — Secretaries^ Rev. Robert Main, 

 M.A. ; William Rutherford, Esq. — Foreign Secretary. Rev. Richard 

 Sheepshanks, M.A., F.R.S. — Council. George DoUond, Esq., 

 F.R.S.; Solomon M. Drach, Esq.; Lieut.-Col. George Everest, 

 F.R.S.; Rev. George Fisher, M.A., F.R.S.; Manuel J. Johnson, 

 Esq., M.A. ; John Lee, Esq., LL.D., F.R.S. ; Edward Riddle, Esq. ; 

 Richard W. Rothman, Esq. ; Lieutenant William S. Stratford, 

 R.N., F.R.S.; The Right Hon. Lord Wrottesley, M.A., F.R.S. 



CAMBRIDGE PHILOSOPHICAL SOCIETY. 

 [Continued from p. 60.] 



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 effiarts 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^ — l 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^-\- bi) +cX represents a line of the length 



V(a'^+b'^+c'^). Mr. De Morgan does not admit the necessity of 



* Vol. XXV. pp. 10, 241. 



