ON THE TRIGONOMETRY OP THE PARABOLA. 87 



This suggests the analogous theorem, that if ^ be the angle or amplitude 

 which gives the difference between the parabolic arc and its subtangent, or 

 the tangential difference equal to x times the modulus, or the distance of the 

 focus from the vertex, we shall have 



and 



tan i,'=-x-\ 4- 4- , &0., 



123 ^ 12345 ^ 1234567 



sec^=l + — + — +-^^,&c (34) 



12 ^ 1234 ^ 123456 



But (Lacroix, ' Traite du Calcul Differentiel et du Calcul Integral,' vol. iii. 

 p. 442) the first of these two series is equivalent to 



and the latter to 



iience 



When X is small, tan ^=jr. Let the angle I be divided into an indefinitely 



large number n of parts, so that ?= — -1---1- — ^ to n terms. Then 

 ° ' ^ n n 71 



X XX 



sec - = 1 , tan - = - ; 

 w ' n n 



and as 



sec (« -^ a -i- a -■- to « terms) +tan(a-'-a-'-a-^tow terms) = (sec a + tan a)" 



1+ - j , but sec^+tan^=e*. 

 Hence when n is indefinitely large, 



Tu like manner, 



These theorems, given in Price's * Treatise on the Infinitesimal Calculus,' 

 vol. i. p. 32, are the limiting cases of the very general theorem established 

 in (6). 



XXVIII, To represent the decimal or any other system of logarithms by 

 parabola. 



The parabola which is to give the Napierian system of logarithms being 

 drawn, whose vertical focal distance m is assumed as the arithmetical unit, 

 let another confocal parabola be described having its axis coincident with the 

 former, and such that its vertical focal distance shall be m'. The numbers 

 being set off, as before, on the scalar, which is a tangent to the^ Napierian 

 parabola at its vertex, the differences between the similar parabolic arcs and 



