Page:Reflections upon ancient and modern learning (IA b3032449x).pdf/205

 more curious, and more useful Theorems have been since added to Geometry, is known to every one that is conversant in the afore-mentioned Authors; especially Mr. Newton, Leibnitz and Huygens: To instance particulars, were to transcribe their whole Books and Treatises.'

'Let us, in the next Place, compare the Ancient and Modern Algebra. That the Ancients had some kind of Algebra, like unto ours, is the Opinion of several learned Writers of late: And it is evident from the Seven remaining Books of Diophantus, that it was brought to a considerable Length in his Time. But how infinitely short this was of that Algebra which we now have, since Vieta's Time, will appear to any that considers the different Process of both. For, tho' Diophantus has given us the Solution of a great many hard and knotty Arithmetical Problems, yet the last Step of his Resolution serves only for one particular Example of each Problem: So that for every new Example of the same Question, there must be a new Process made of the whole Analysis. Whereas by our Modern Algebra, the Analysis of any one Case gives a general Canon for all the infinite Cases of each Problem;