296 PROFESSOIl DE MORGAN, ON THE 



( { 1 + ,.}", ndp) = (1 +m)" • (1 + dp J- 1)" (A). 



Now let n be infinite, and let (1 + n)" = r, ndp = p, from which we find 



= f, (l + dpj-l)" = e<>^, 

 M ftp 



all following from the assigned laws of operation. Hence re p ^~ x is the 

 representative of a line r inclined at an angle p ; while log r and p, each 

 in its own way, and to a different radix, may be considered as a register 

 of the number of transitions by which we pass from (1,0) to (r, p). 

 The term logarithm itself, as is well known, is a consequence of a similar 

 notion of comparison of numbers by the registration of the 'numbers 

 of the ratios' by which we pass from unity to those numbers. The ratiun- 

 cula? u and dp must be in the proportion of log r and p, and the line 

 formed by adding WU and UV, or n + dpj—l, is one which gives these 

 two ratiuncula? for its two projections. This line repeated n times gives 

 log r + 9J—1, the logometer of (r, 9) as I have called it. Should any 

 objection be taken to that term, perhaps the words compound logarithm 

 might be preferable. Observe, that this derivation of the logometer is 

 independent of the second side of {A), and might be introduced pre- 

 viously to the expression of (r, 9) in the form re'v'- 1 . 



I have throughout avoided considering the ambiguous values of sym- 

 bols, a thing for which there is no necessity, as has been frequently 

 shewn, and as I noted in my last Paper. The more I think on this 

 subject, the better satisfied do I feel, that the new algebra should have 

 no symbols of double or multiple value whatsoever; that is, that the 

 meaning of each elementary symbol should not be considered as com- 

 plete, unless it expresses the amount of revolution from the unit line 

 by which it is to be made to attain its direction, as well as that direction 

 itself. Undoubtedly, after a time, the student should be shewn how to 

 drop this part of the definition ; but this he will better be able to do 

 than to take it up after a previous training, which has never intro- 

 duced it. This is an important point to those who believe as I do, that 

 it will not be long before the new algebra is introduced into elementary 

 instruction ; and it is the more important, because there are some new 

 species of ambiguities altogether peculiar to the most general view, and 

 which must remain such until further inquiry points out the mode of 



