analytical and geometrical Methods of Investigation . 8§ 
rads, j , sin. x ; but nothing is gained by this ; since, in order to 
find the arc of a circle, x' ( 1 — x 2 )~ i is expanded, and the inte- 
grals of the several parts taken and added together. To shew 
(if it is necessary to add any thing more on so clear a point] 
that fx° | l = arc circle, is merely a mode of expres- 
sion borrowed from geometry ; suppose the investigation of the 
properties of motion to have been prior to the investigation of 
the properties of extension, for, that the science of geometry was 
first invented is properly an accidental circumstance, then, such 
an expression as fx' 1 1 — x* j“-f might have occurred, and its 
value must have been exhibited as it really is now, that is, by 
expanding it, and integrating the several terms. 
IV. It is an objection certainly against these modes of ex- 
pression, that they are foreign, and tend to produce confused 
and erroneous notions ; for the student may be led by them to 
believe, that the determination of the values of certain analytical 
expressions, essentially require the existence of certain curves, 
and the investigation of their properties. But there is, a more 
valid objection against them, which is, that they divert the mind 
from the true derivation of such expressions as x- ( 1 — f, 
&c. and consequently tend to produce ambiguity and indirect 
methods ; for although, in order to obtain approximately the 
numerical value of / -,/*• (i— x 4 )~f, &c. it is convenient to 
expand the expressions, and to take the integrals of the result- 
ing terms, yet, if the symbol / denotes a reverse operation, 
/■p J x ‘ i 1 are not properly and by strict inference equal 
to (x-i) — i + ^ [x &c. and x+ — -4- 
*2 ^5 ■ ^ 
JT +> But, in order to explain clearly what I mean, it is 
MDCCCII. ]SJ 
