in the Calculus of Functions^ noticed by Mr. Babbage. 4-49 



with ei+A/-i2m^^ one of its values equal to ^, and is Me 7«os^ 

 general exponential form that possesses this property; a pro- 

 perty which seems to me, even according to ordinary accep- 

 tation,^ confer on every quantity included in the formula 



1 4- V^- i 2nnr . , 



■. , /— r' a i''ght to the appellation r-Iog of e. Here 



let me remark that I cannot conceive how any difference in re- 

 sults can be obtained from operating correctly on two sfrictly 

 equivalent algebraic forms such as cos fl and cos (2ew + 5). 

 It IS true that one may suggest what we have to recollect with 

 respect to the other, and it is true that in the treatment of 

 such forms there are many specious fallacies to be guarded 

 against. Thus, it would not be correct to reason as though 

 cos {c(2 It: ± fl)} were the same function of cos (2 iit + 5) 

 that cos(cfl2 ^^"^^^^ ^' if 5 denote a particular individual 

 value of cos-' cos fl. On account of the preceding considera- 

 tions, among others, instead of obtaining my formula for x, 

 «ie general logarithm of 3/, by the method' stated by Professor 

 De Morgan as substantially the same as mine, viz. by setting 



out at once with the unelementary definition t>(' + ^-J^ "i=^> 

 = ely+V-xin^^ J gi^^yij prefer building, as I have done, 

 on received principles of analogy, which, 1 think, would na- 

 turally entide 4/j/ to the name of an e-log of j/, if any value 

 of a"^^ were equal to y, especially if we found that ^y pos- 

 sessed the property ^y + '\>y = ■^yy. I do not meet in books 

 any explicit exclusion of 4 from the name of logarithm of — 10 

 to the base 100 on the ground that —10 is not what is called 



the arithmetical value of 1 00-, and, in legal phrase, I sub- 

 mit that the onus of making out their case lies on those who 

 advocate such an exclusion. It would refuse the name of 

 logarithm to any function whatever o\' y, where 3/ and the base 

 a were not real and positive, or would require some definition 

 of what is meant by the arithmetical value of a-^' for all values 

 of a and x, as well imaginary as real. Now, for some values 

 of a, it may be matter of arbitrary decision to determine 

 which one of a certain pair of values of a^ is to be considered 

 as correspo7iding to that which is the arithmetical value of a', 



when a and x are real and positive. Is V— ^ o^' / — l^^ 



arithmetical value of ( — 1)5? For some values of « and j-, 

 «"■ has more than one real positive value, anti for some, 

 again, a' has a single real positive value corresponding to 

 Third Scries. Vol. 9. No. 56. Dec. I83G. 3 U 



