FOUNDATION OF ALGEBRA. 185 



in perfect fulfilment of all the fundamental conditions which prescribed 

 definitions impose. The assumption gives 



(aE e )^~ l = J27V=i , 6 iog«.v=r 5 



where E e ^~ l must be a symbol of length, and e lo s<>V=i of a unit inclined 

 at the angle log a. Consequently e 6 ^ 1 must signify a unit inclined at an 

 angle 9. 



It might be asked whether there is anything in the preceding process 

 which restricts us to the use of the base e rather than any other, I answer, 

 nothing whatever : but at the same time there is nothing which binds 

 us to the use of any particular method of measuring angles. It may be 

 deduced from the preceding that the base e must be used co-ordinately 

 with that mode of measurement which I call theoretical*. This connexion 

 depends entirely upon the purely numerical process by which the equation 

 6 2irv/=5 = 1 is proved to be satisfied when e and ■*■ have their usual meanings. 

 If for any reason we prefer the base a, the measure of two right angles 

 must be tx {log e to the base a\. 



I think it cannot be disputed that interpretation should be avoided 

 where explanation can be given. If where the latter cannot be obtained 

 suggestion upon such analogies as present themselves were to take its 

 place, the former would be also replaced by verification. In the present 

 instance, the attainment of 



6 <V=i = cos 9 + </~-\ sin 9 from E 6 = cos 9 + \/^l sin 9 



is the verification. 



1 now come to the theory of logarithms. It is a circumstance which 

 I hold to be not a little remarkable, that the ancient form of algebra was 

 only saved from being convicted of incapacity to produce its own legitimate 

 results, but very little time before such an escape would have been 

 rendered impossible by its receiving the necessary accession from the more 

 extended form. Mr Graves has admitted that his view of the new 

 logarithms should rather have been that of an extension imperatively 



* In those works on Trigonometry which use the arc and angle indiscriminately, this mode 

 of measurement is said to be in parts of the radius. A term is much wanted which shall not 

 imply this confusion between arcs and angles ; and I propose that the angle which subtends an 

 arc equal to the radius shall be called the theoretical unit. 

 Vol. VII. Part II. A A 



