analytical and geometrical Methods of Investigation . 117 
by equations, and the order of classing them, considered as 
generated by description. Moreover, he animadverts on the 
custom of confounding the two sciences of algebra and geo- 
metry ; * and, if any authority is attached to his assertion, that 
the two sciences ought not to be confounded together, the 
separation of geometry from algebra will thereby be equally 
urged as the separation of algebra from geometry. And it can- 
not be said with greater truth, that the simplicity of geometry 
is vitiated with algebraic equations,, than that the simplicity of 
analysis is vitiated with geometrical forms and expressions. Ih 
fact, each science ought to be kept distinct ; and be made to 
derive its riches from its proper sources. 
XXV. It will not demand much meditation to be assured of 
this truth, that, in any mathematical investigation, the geome- 
trical method, properly so called, is not essentially or absolutely 
necessary. The properties of extension and figure, to which this 
method has been especially appropriated, may be analytically 
tieated; and here it is proper to state a distinction neces- 
sary to be made, between what may be called analytical geo- 
metry, and- the application of analysis to geometry. The first 
does not suppose or require the existence of such a method as 
the geometrical ; but, from a few fundamental principles, analy- 
tically investigates the properties of extension ; whereas, in the 
latter, analysis is applied to propositions already established by 
the geometrical method: so that, strictly, to shew the justness 
and propriety of the application, a separate investigation is, 
* « Multiplications, divisions, et ejusmodi computa, in geometriam recens intro- 
* e aucta sunt : idque iriconsulto, et contra prinrum institutum scientbe hujus, &c„ 
“ ' Proinde ha dus scientias confundi non debent, &c, Et recentes, utramque 
“ confundendo > araiserunt simplicitatem in qua geometric elegantia omnis consistit,T 
