ii8 Mr. Woodhouse on the Independence of the 
necessary. We find, however, in general, a vague analogy sub- 
stituted, as a connecting principle between the two methods. 
XXVI. The application of algebra to geometry, gives to 
Descartes the fairest title to fame for mathematical invention ; 
yet the cause and nature of the benefit conferred on science by 
that application, seems to be indistinctly apprehended.* For, the 
Analytical Calculus, when applied to geometry, was not en- 
riched with the truths of the latter science, because some con- 
necting principle had been discovered, or some process invented 
by which the property of the two methods became common, 
and might, from one to the other, without formality be trans- 
ferred ; but because the investigation of certain properties could 
not proceed, without first improving the means by which they 
were to be investigated. These means Descartes improved : 
he found, when certain conditions in problems concerning ex- 
tension were translated into the language of algebra, that the 
process of deduction with the general terms was slow and in- 
commodious, because, such was the low state of the algebraic 
Calculus, the relation between the general terms had not been 
established. The aim and merit of Descartes's speculations is 
to have established this relation. If illustration were needed to 
make my meaning clear, I should say that Descartes, New- 
ton, and D’Alembert, benefited science precisely after the 
same manner. The first applied the analytical Calculus to 
extension ; the second to motion ; the third to the equilibrium, 
resistance, &c. of fluids. As the object of investigation became 
* Thus far was the Analytical Calculus benefited by the existence of the geome- 
trical method : certain properties of figure and extension, discovered by the latter* 
became to the former, objects of investigation. 
