Page:Reflections upon ancient and modern learning (IA b3032449x).pdf/206

 whereby we discover many curious Theorems about the Properties of Numbers, not to be attained by Diophantus's Method; this being the peculiar Advantage of Specious Algebra, first introduced by Vieta, and wonderfully promoted by several worthy Mathematicians since. Beside this intolerable Imperfection of the Ancient Algebra, used by Diophantus, which required as many different Operations as the Problem had different Examples, that is, infinite; all which are included in one general Solution by the Modern Algebra; there is this great Defect in it, that in Undetermined Questions, which are capable of innumerable Solutions, Diophantus's Algebra can seldom find any more than one; whereas, by the Modern Algebra, we can find innumerable, sometimes all in one Analysis; though in many Problems we are obliged to re-iterate the Operation for every new Answer. This is sufficient to let you see, that (even in the Literal Sense) our Algebra does infinitely exceed that of the Ancients. Nor does the Excellency of our Algebra appear less in the great Improvements of Geometry. The reducing all Problems to Analytical Terms, has given Rise to those many excellent