FROM ELEMENTARY ,\L(;EBRA TO THE CALCULUS. 415 



Since a" is a continuous function of a (so long as a is -f-), it 

 follows from the above that log a (<? + "') continually increases 

 with a from log 0= - oo through log i=0 to log oo = + oo. 

 Hence there is some value of a (always called^) for which log g = i, 

 and D.e''=e' 



In my view of these functions e is always subordinate to 

 log; in fact, c" is merely a (more convenient) form of \og~^x. 



In the same waycos;c=i (t'^ + t~'^) is to me more satisfactory 

 than I (tJ'-'-l-e"'''). 



V. The Hyperbolic F unctions . — These do not force themselves 

 on our notice until we come to integrate such things as v ( 1+ x"-). 

 It is unsatisfactory to introduce them arbitrarily as inere names 

 for 1 (e' + g-^. 



They originate better thus : — 



If we have to deal with pairs of quantities it,, v the sum of 

 whose squares is i, Pythagoras' Theorem at once suggests the 

 co-ordinates of a point on a unit circle: and the ordinarv Trigono- 

 metry follows inevitably. 



But if we wish to deal with pairs o." quantities the difference 



of whose squares is i, ^r-v^=i leads to " + ^'=-«:^. where «, x 



1 u-v = a ^ are arbitrary. 



Therefore w = i (a* + a~') and v^\ (a' - a'"). 



And if now we differentiate these new functions (of x), 



Du = h (a'^ - a '). Jog a = ■^^ log a; Dv = u. log a. 

 If, therefore, a is taken to be e, Du =v and Dv = u. 

 Thus the only reason for defin-'ng cosh x and sinh x as J {e" ^i^e"') is 

 analogous to that for using circular measure of angles — that 

 D sin X and D cos x may not involve an inconvenient constant 

 factor. 



The essential property of cosh x, sinh x is cosh -x - sinh -x ^ i. 



D (tanh x, etc., sinh"'^;, etc) follow easily. 



VI. Having now obtained the rate of increase of all the known 

 functions and Algebraic combinations of them (including the 

 new functions — logarithmic, exponential, hyperbolic), with the 

 one additional remark that the process called differentiation can 

 be repeated and denoted by D, D-, D\ ... or f'{x), f" {x), . . 

 we pass to Integration. To a great extent integration is the only 

 logical sequel of Newton's and Leibnitz' idea of differentiation ; 

 so that it is natural and desirable to develop and teach integra- 

 tion as soon as differentiation is grasped. 



The ' Summation Theorem ' and the derivation from D{iiv) 



= uDv + vDu of the process known as ' Integration by Parts ' 



as given in the text-books, sound and straightforward. I 



J ' ot propose to comment on them here, but to use them to 



. i^lish Maclaurin's (including Taylor's) Theorem with an exact 



j^^-nder, and to suggest this as the fundamental basis of 



infinit^ series rather than the subtle (and m many ways hazy) 



snecial Processes known as the Binomial, Exponential, logarithmic 



