and on Algebra as the Science of Pure Time. 413 
In general, if we can discover any law of dependence of one couple ® (4, @,), upon 
another (4, a,), such that fer all values of the numbers a, a, 4, b, the condition 
© (4, 4) ® (%, b= (a, + A, b, + b) (128.) 
is satisfied, then, whether this function-couple © (@%, 4) be or be not coincident with 
the particular function-couple F (4, 4), we may call it (by the same analogy of defi- 
nition) an eaponential functien-couple, calling the particular couple (1, 0) its base, or 
dase-couple ; and may call the couple (@,, ¢,), when considered as depending inversely 
on ® (4, @,), a logarithmic function, or function-covple, which we may thus denote, 
(a, 7) =b—'@,, 4), if ©, 4,)=o (a, a). (129.) 
12. We have shown that the particular exponential function-couple (4,, 4.) = 
F(a, a) is always possible and determinate, whatever determinate couple (a, a2) 
may be ; let us now consider whether, inversely, the particular logarithmic function- 
couple (a, a.) =F7' (4,, 6.) is always possible and determinate, for every determined 
couple (4, &.). By the exponential properties of the function r, we have 
(4, &J=FCa, a)=E(am, 0) F(O, w)=EF( a) F(O, a) 
(130.) 
: sin a»), 
=(e"'cos ay, e* 
if we define the functions cosa and sina, or more fully the cosine and sine of any 
number a, to be the primary and secondary numbers of the couple F (0, a), or the 
numbers which satisfy the couple-equation, 
F (0, a) =(cos a, sina). {131.) 
Vrom this definition, or from these two others which it includes, namely from the fol- 
lowing expressions of the functions cosine and sine as limits of the sums of series, 
which are already familiar to mathematicians, 
Ph aa ai 2. a &c 
= 1x2 1x2x3x4 i 
(132.) 
: a a & 
$1 = _ Se 
ae ¢ iKaKS 1 LKDKONANS ¢ 
it is possible to deduce all the other known properties of these two functions ; and 
especially that they are periodical functions, in such a manner that while the variable 
