142 Mr. DE morgan, ON THE FOUNDATION OF ALGEBRA. 



I rather expected to find something of the sort in the present inquiry : remembering that the first 

 great diflficultv arose from the inverse process to addition, the next from an inverse process to 

 multiplication, I should not have been surprised to have found a third in the most general direct 

 and inverse consideration of J^. But though we are not to look for any new inexplicables 

 from A + B, AB, or A", it should be remembered that there is a scale of ascent in the funda- 

 mental mode of deriving them from one another which does stop anywhere. Addition being 

 obtained, and the general notion of operation, the solution of (p{cc + l) = cpx + c gives (pj; = c.r, 

 and introduces multiplication. Next (p{a; + 1) = c(p<v gives <p,v = c\ and introduces involution. 

 But (p(,i' + 1) = c*', the solution of which gives the next step, gives for (h.i' a function which has 

 not been considered ; though its particular cases 



(pl=a, ^2 = c", (pS^c'", (j)i = c'-''"°\ &c. 



are known. If (px could be completely inverted, new inexplicables might, and perhaps would 

 arise, either from this or some succeeding case. 



A. DE MORGAN. 



University College, Lokdon, 

 October 7, 184.3. 



