324. -Mr. Woodhouse on the Independence of the 
can be adequately and unambiguously stated in the general 
terms of algebra, then deduction with such terms may be 
strictly made, and expeditiously ; since it is to be made accord- 
ing to a known and established process. I have shewn at some 
length, that reasoning may be conducted with terms which 
separately cannot be arithmetically computed: for the mere 
process of deduction, it is not necessary to have distinct and 
complete notions of the things signified by the general terms. 
The principal object of the present paper is to shew, that the 
analytical calculus needs no aid from geometry, and ought to 
reject it, relying entirely on its own proper -resources. By this 
means, it would gain perspicuity, precision, and conciseness; 
advantages not to be lightly estimated, by any one who has a 
regard to certainty and demonstration, or considers the bulk to 
which scientific treatises have of late years swelled. 
In order to prove and illustrate the opinion I wished to 
establish, I directed my search to those cases which have been 
always thought to require the aid of the geometrical method. 
By a purely analytical process, I have traced the origin and 
derivation of certain fluxionary expressions, usually referred to 
logarithms and circular arcs. I have given demonstrations of 
the series for the sine of an arc in terms of the arc ; of the ana- 
lytical formula for the root of a cubic equation in the irreducible 
case ; of the resolution of x n =f= a n into quadratic factors ; of the 
series for the chord, &c. of a multiple arc in terms of the 
simple arc, &c. which demonstrations, with as much confi- 
learning half a series of truths by one method, and half by another. These considera- 
tions, however, depreciate the value of the geometrical method only in one point of 
view i for, after all, the finest exemplar of clear and accurate reasoning is contained la 
the works of E u c n u . 
