analytical and geometrical Methods of Investigation. 87 
demonstration prolix, by rendering it less direct, and has made it 
deficient in precision and exactness, by diverting the mind 
from the true source and derivation of analytical expression. 
The expressions and formulas alluded to are geometrical, 
that is, taken from the language of geometry, and established 
by its rules : they are to be found mixed with analytical* ex- 
pressions and reasonings, in all works on abstract science ; and, 
as they are certainly foreign and circumlocutory, if it can be 
shewn that they are not essentially necessary, there will exist 
an argument for their exclusion, especially if it appears that in 
analytical investigation they are productive of the evils above 
mentioned. 
That, in algebraical calculation, geometrical expressions and 
formulas are not essentially necessary, perhaps this short and 
easy consideration may convince us ; since algebra is an uni- 
versal language, it ought surely to be competent to express the 
conditions belonging to any subject of inquiry ; and, if adequate 
expressions be obtained, then there is no doubt that with such, 
reasoning or deduction may be carried on. 
All expressions and formulas, such as, sin. x, cos. x, hyp. 
log. x, sin. n x = 2 cos. .r. sin. (n — 1 ) x — sin. ( n — 2) x. 
* The terms analysis, analytical, algebra, algebraical, have been so often distin- 
guished, and so often confounded, that I shall not take the trouble again to distinguish 
them. I use the words analytical, -algebraical, indifferently, in contradistinction to 
geometrical. The first relates to an arbitrary system of characters ; the latter to a system 
of signs, that are supposed to bear a resemblance to the things signified, and in which 
system, lines and diagrams are used as the representatives of quantity : and I am prin- 
cipally induced to use the words indifferently, because, if analytical were properly 
defined, another word with a sufficient extent of meaning could not be found ; for, 
by an improper limitation, the word algebraical has not an extensive signification, 
being frequently used in contradistinction to transcendental, exponential, &c. 
