12 



JOURNAL OF THE 



there is no triangle when one acute angle is either o or 

 90°; therefore we can only say that sin x approaches o as 

 its "limit" as x indefinitely diminishes, and tan x in- 

 creases indefinitely as x approaches 90°. 



With this meaning to be given such expressions as 

 sin = 0, tan 90 = cc , they can be safely used (and will 

 be used in what follows), though there is really no sine cor- 

 responding to 0° and no tangent for 90°. 



The next subject treated will be the general one of find- 

 ing the limit of the ratio of two related infinitesimals, 

 which is the principal problem of the differential calculus. 



As a special example, consider 

 the circular arc ABC, fig. i, of ra- 

 dius unity, whose length in circular 

 measure is 2x. Divide it into two 

 equal parts, x = AB = AC and 

 draw tangents AD and CD, inter- 

 secting on radius OB produced. 

 Call the c/wrd AB = c/iord BC = c. 

 Then since the radius is taken as 

 unity, EA = sin x and AD =r tan x. 

 Now by geometry we have, 



AC < 2c < 2x < AD + DC; 

 whence, dividing by 2, we have, 



sin X < r <• X < tan x. 

 Also since, 



;=v 



7./ 



sin X 



sin X 



= cos X 



[im 



= I 



I, as 



tan X tan .i* 



as X indefinitely diminishes, since then, lim. cos x 

 cos X indefinitely approaches unity without ever attain- 

 ing it. 



Now since c and x are always intermediate in value be- 

 tween sin X and tan .r, it follows that the ratio of either to 

 the other, or to sm x or tan .r, approaches indefinitely unity 

 as a limit. 



