88 , Mr. Woodhouse on the Independence of the 
= circular arc, fx- = elliptical arc,&c. 
are geometrical, or involve geometrical language : they suppose 
the existence of a particular system of signs, and method of de- 
duction ; and relate to certain theorems, established conformably 
to such system and method. 
I. Sin. x, cos. x, tang. x } &c. These expressions are borrowed 
from geometry ; but, analytically, denote certain functions of x. 
Typographically considered, these expressions are more commo- 
dious than ( 2 \/ —i)"" 1 1 — g x ^ 1 1 j-, (s) -1 j -f- 
e—xV~ j, &c. but this is the sole advantage ; for, all analytical 
operations with these latter signs are much easier, and more 
expeditious, than with the former ; since they are carried on 
after a manner analogous to that by which operations with 
similar expressions are. But, if the geometrical expressions be 
retained, then, in order to calculate with them, recourse must 
be had to the geometrical method, proceeding 4 by the similarity 
of triangles, the doctrines of proportions, and of prime and ulti- 
mate ratios ; so that, in the same investigation, two methods of 
deduction, between which there is no similarity, must be em- 
ployed. 
II. The value of f x' ( l -f- x)— 1 , is said to be a portion of the 
area of .an hyperbola intercepted between two ordinates to its 
assymptotes ; but this is a foreign and circumlocutory mode of 
expression; since, to find the value of the area, x\ (l -]~ x) ~ r - 
must be expanded, and the integrals of the several terms taken; 
and this same operation must have taken place, in order to ap- 
proximate to the value of f x‘ ( l -j- at)— 1 , if no such curve as the 
hyperbola had ever been invented. 
III. i — ufj ~ i is said to equal the arc of the circle 
