analytical and geometrical Methods of Investigation. 121 
lines and mixtilinear triangles therein exhibited cannot be called 
natural signs, since they are only imperfect and inadequate 
representations of other imaginary lines and triangles, of which 
the mind must form what notion it can. Not, however, to infer 
want of perspicuity from inefficiency in the cause assigned, if 
we employ the geometrical method, or view its employment in 
investigation, concerning motion, curves, &c. it will not appear 
a perspicuous method ; and, if instances of its obscurity were re- 
quired of me, I could find them, even in the immortal work of 
the Principia. Whether we consider the fact, or speculate about 
the cause, I think the geometrical method can only be allowed 
to have superior evidence in investigations of a simple nature. 
That the analytical calculus is more commodious for the de- 
duction of truth than the geometrical, will not perhaps be con- 
tested; and, an examination into its nature, would shew why it 
is so well adapted for easy combination and extensive gene- 
ralization. No language like the language of analysis, one of 
the greatest of modern mathematicians has observed, is capable 
of such elegance as flows from the developement of a long 
series of expressions connected one with the other, and all de- 
pendent on the same fundamental idea. 
If we view what has been respectively done by each method, 
111 the explanation of natural phenomena, the superiority of the 
one above the other will appear immense : yet the cultivators 
of geometry were men of consummate abilities, and possessed 
this great advantage, that the method or instrument of thought 
and reasoning which they employed had, during preceding 
times, received the greatest improvement. The analytical cal- 
culus, which has verified the principle of gravitation, was a 
hundred years ago in its infancy. 
Mpcccir. R 
