XI. On the Foundation of Algebra, No. III. By Augustus De Morgan, 

 V.P.R.A.S., F.C.P.S., of Trinity College; Professor of Mathematics in Ufiiversity 

 College, London. 



[Read, Nov. 27, 1843.] 



In two former papers (Vol. vii. pp. 173, and 287) I have described the view which I take 

 of the three fundamental symbols of Algebra, A + B, AB, and A". This third communication is 

 a generalization of the view taken in the second. 



In establishing an independent definition of A", a step which was seen to be indispensable 

 if all its cases are to be explained from the commencement, a preliminary extension of the idea 

 of a logarithm is necessary. I proposed to call the extended logarithm by the name of the 

 logometer. In recapitulation it may be desirable to state, that the complete symbols of the new 

 algebra are all capitals, that the positive and negative quantities of the older algebra are small 

 letters, and that the equation R = (r, p) means that R represents a line of the length r inclined 

 to the unit-line at an angle p. 



Any definition of A^ may be allowed which satisfies the conditions 



A'>A'^ = A"*", A^C = (AC)", iAy = A'''^, 



or rather the proper definition is the most general of those which satisfy these conditions ; unless 

 it should happen that the results of the most general definition can themselves be conveniently 

 expressed in terms of the results of a less general definition. This does happen, as I am about 

 to shew in regard to A"; a wider definition of it even than the one I gave in my last paper is 

 practicable, and must be considered : though it will turn out not to be necessary, because it is 

 capable of expression in terms of its more simple case. 



The function which, whatever be its name (I call it the logometer), plays the part of the 

 logarithm in a complete system of algebra, is fully defined by the equation 



\A + XB = \{AB), or \ {a, a) +\ih, (3) = \{ab, a + fi), 

 and this function being settled, A'^ can be no other than X"' (BX^). In my last paper, I pro- 

 posed as the definition of X (/•, p), tlie line which has the projections log r and p on the unit-line 

 and its perpendicular: so that X (r, p) = log r + jo y/— 1. But it would equally satisfy the 

 fundamental condition if we were to propose as the definition of \{r,p) the following, 



X (c, p) = log r (m + n ^ - 1) + p (^ + V <y - 1), 

 where m, n, n, v are any constants. The geometrical de- 

 finition is as follows. To find the logometer of a line R 

 or (r, p) ; let there be two fixed lines OF and OG 

 (m + « ^/ - 1 and fi. + v \/—^)i which we may call the 

 base of leiigtk and the base of direction : accordingly OF 

 lias y/(ot^ + ri') and OG has \/iiJ? + li') units. Let 

 OP : OF :: log r (the units in the logarithm of the _ 

 length of R) : U and let OQ : OG as p (tlie units in the 

 angle of R) : 1 ; then will OR, the diagonal of the paral- 

 lelogram on OP and OQ, be the logometer of R. 



Vol. VIII. rAin II. T 



