Page:Reflections upon ancient and modern learning (IA b3032449x).pdf/207

 Methods, whereby we have advanced Geometry infinitely beyond the Limits assigned to it by the Ancients. To this we owe, (1.) The Expressing all Curves by Equations, whereby we have a View of their Order, proceeding gradually on in infinitum. (2.) The Method of Constructing all Problems of any Assignable Dimension; whereas the Ancients never exceeded the Third. Nay, from the Account which Pappus gives us of the afore-mentioned Question, it is evident, that the Ancients could go no further than Cubick Equations: For he says expresly, they knew not what to make of the continual Multiplication of any Number of Lines more than Three; they had no Notion of it. (3.) The Method of Measuring the Area's of many Infinities of Curvilinear Spaces; whereas Archimedes laboured with great Difficulty, and wrote a particular Treatise of the Quadrature of only one (l), which is the simplest and easiest in Nature. (4.) The Method of Determining the Tangents of all Geometrick Curve Lines; whereas the Ancients went no further than in determining the Tangents of the Circle and Conick Sections. (5.) The Method of Determining the Lengths of an infinite Num-