Mr, Cockle on the Symbols of Algebra, 407 



ideal) quantity is the oiFspring of arbitrary ultra-algebraic 

 definition*. 



I should propose to apply the term hyper-algebraic to typal 

 and ideal quantities, and to confine it to those quantities. 

 Ought we to apply the word hyper-algebraic to impossible 

 quantity ? I think not. If I may be permitted to use the term 

 possible so as to include under it not only real quantities but 

 also the unreal quantities of ordinary algebra, I would suggest 

 that it is only in respect of certain anomalous resultsf (re- 

 sults, however, that do not defy explanation J) that impossible 

 differs from possible quantity, and consequently that impossible 

 quantity must be regarded as algebraic. It is unquestionably 



* I must not be understood as desiring to underrate these symbolic 

 children of definition. So far from it, I think it within the limits of pos- 

 sibility so to define symbols as that they may have their prototypes in na- 

 ture, and serve to expound other of her phaenomena than those to which 

 symbols have been yet applied. For instance, might not arbitrary symbols 

 be made the representatives of chemical phaenomena ? Mr. Boole's Ma- 

 thematical Analysis of Logic is a step out of the beaten track which sym- 

 bolic science has hitherto persevered in j and although to pass from mental 

 to chemical phaenomena may not authorise us to hope that such sciences 

 as chemistry may be rendered symbolic, yet I cannot help thinking, that by 

 a proper notation for affinity, &c. chemical decompositions might be repre- 

 sented : at any rate it may be worth a trial. 



t Vide supra, pp. 41, 42. It is not a little singular, that if we abandon 

 the principle laid down, supra, p. 39, note -j-, we have 



(i+Vi)a-Vi)=i-v'/=i-i=0; 



and also that if we preserve that principle, we derive from the equation (1.), 

 supra, p. 39, the following : 



(-1)4=(+ ^jf={ /;X ^j)x(Vjx VJ)=JxJ, 

 or 



1=/. 

 On the other hand, although from the relation 



il+k)^=2k=z2ij 

 we may deduce 



still, seeing, from the relation {supra, p. 40) 



the anomalous nature of the evolution of impossibles, we must not attempt 

 to express \/J as a linear function of i,J, k. 



I may add, that in previous papers (supra, pp. 45, and 135,) the radius 

 of the (larger) sphere is supposed to be unity. 



I See paragraph 8 (and 10) of the Rev. Prof. Charles Graves's paper on 

 Triple Algebra (supra, p. 119-126). I propose to call Mr. C. Graves's 

 system a trinar, and Mr. De Morgan's a ternary algebra; the latter form 

 of name being given to the quadratic system, in which the square of the 

 imaginary is negative. And hence the respective terras trine and temion 

 suggest themselves as distinctive^ 



