FOUNDATION OF ALGEBRA. 177 



Algebra of the same author*. But in this branch of logical algebra 

 the lines must be all in one plane, or at least affected by only one modifi- 

 cation of direction : the branch which shall apply to a line drawn in any 

 direction from a point, or modified by two distinct directions, is yet to be 

 found. 



It is obvious that our power of making the preceding application 

 of algebra is co-ordinate with that of assigning a symbol Q, such that 



a + bQ — «, + b^Q gives a — a x and b = b u 



An extension to geometry of three dimensions is not practicable until 

 we can assign two symbols, Q and w, such that 



a + bQ + cm — a, + b x Q + c x w gives a — « 1} b = b x and c = c, : 



and no definite symbol of ordinary algebra will fulfil this condition. 

 Again, in passing from a; to — x by two operations, we make use in 

 ordinary algebra of one particular solution of 



<p 2 x = — x, namely cpx = \/ - 1 . x. 



An extension to three dimensions would require a solution of the equation 

 <p*x = — x, containing an arbitrary constant, and leading to a function of 

 triple value, totally unknown at present. 



A general solution of cp^x = ax can be expressed when any particular 

 solution -&X is known. For if f-arf-^x be the general solution, we have 



cpx =f , sr !! f- ] x =faf~ l x — ax, or fax = afx: 



so that it is only necessary that f and a should be convertible. Since then 

 ( - l)^x is a particular solution of (jy'x = — x, a general solution is 



f\ - 1 \f~ x x} where f(-x) = -fx. But with our very limited knowledge 

 of the laws of inversion, no solution which we can now express in finite 

 terms will afford any help. Our means of expression must be augmented 

 before we can hope to overcome this difficulty : or, as in most other cases 



* Professor Peacock is the first, I believe, who distinctly set forth the difference between 

 what I have called the technical and the logical branches of algebra. The second term, I am 

 aware, is a very bad one, and I should be glad to see a better one proposed ; but I prefer 

 technical to symbolical, because the latter word does not distinguish the use of symbols from 

 the explanation of symbols. 



Vol. VII. Part II. Z 



