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



In my last paper, the bases of length and direction were on the unit-line and its perpendicular, 



and the lengths of these bases were units. 



If the logometer be p + q^-l, its primitive, {r, p), is found from 



vp - nq mq -np 



log r = , p = • 



° mv-nn mv - n/x 



We are now to express X^': it is convenient to express the radical letter X by its length and 



direction (x, ^), and the exponent by its projections, v + w,^/-l. If X^, which by definition 



is \-'(l'A^), be called Z or («, ^), we have 



log x: = {v - bw) log x - cw^, ^= (v + bw) ^ + aw log ,v, 



m^ + n^ mn + nv /jr + v' 



where a = , b = , c = . 



mv - n/x mv-nn mv - n/x 



If we prefer to express the bases of length and direction by their lengths and directions, as 



m + n^~l = (g, y), fx + V \/ - i = (k, k), 



we have 



g 1 cos ((c - 7) k I 



= -. ; , C = 



k sin ((c - 7) ' sin (/f - 7) ' g sin (< - 7) ' 



which are connected by ac - 6" = 1. 



Some mode of expressing X^ should be contrived, such as X^„^„, which may show its 

 dependence on the arbitrary constants in the bases; this will allow us to reserve X^ for its 

 common signification, as an abridged form of Xj'mi. But, before proceeding further, I may 

 notice that the logometer of my last paper is not as general as it might be, even on the sup- 

 position that X^ is to have no extended meaning. For if k - 7 be a right angle, and if k = g, 

 then a = 1, 6 = 0, c = 1, log x = v log a? — w^, ^ = v^ + w log a?, which two last equations simply 

 express that X^ has the ordinary meaning. That is to say, every result in the last paper remains 

 if, instead of the bases of length and direction being units, they be any equal lines, and if instead 

 of being on the unit-line and its perpendicular, they be on any lines which are at right angles 

 to one another, provided only that the base of direction be a right angle in advance of that of 

 length. 



Returning to the most general definition, we have 



-A,„„fi^ or (,.x, ^)„i„^^ — t 



_ [t— (6-ov'-l)iii]logar + [i.+ (4 + cV-l)K]f V-l _ _j,f- ((.-<! V-l)i(' [f+(»+f\'-l)«']f ^-\ 



Of the three fundamental equations J"^^ = J^^^, A"C'' = (JC)" and (^'')"^=J"^ it is 

 instantly seen that the two first are satisfied by this new signification of the exponent ; and that 

 they are satisfied independently of the relation between a, b, and e, or ac-b'-=\. The third 

 is a little more intricate: the formation of (X" + '"'^-')"'+"'''^-' requires us to write (« - bw) log.r 

 - cwf for log X and (v + bw) ^ + aiv log x for f, v' for v and to' for tv in the first or second of 

 the preceding expressions for X^ . This being done, it is found that in consequence of ac - A^ = 1, 

 the result is precisely the same as if try' - ivtv' had been written for v and vw' + v' w for w, without 

 any substitutes being employed for x and ^. But these last changes turn 

 n + w,y-\ into (v + tv ^ -I) (v + w' ^y-\). 



The theory of quantities once called real admits of no extension ; for if f and «• vanish, 

 x'^^ ^ = g'loe'', or ,v". But the following deductions. 



sV-i _ -be + as^-t ,«v-i _ a a ~ 



