Page:Reflections upon ancient and modern learning (IA b3032449x).pdf/201

 tings upon the most difficult Parts of Geometry, for such are the Quadratures of Curve Lines, will be sufficient Vouchers for his Skill in these Things. I shall set down what he says, in his own Words.

'If we take a short View of the Geometry of the Ancients, it appears, that they considered no Lines, except Streight Lines, the Circle, and the Conick Sections: As for the Spiral, the Quadratrix, the Conchoid, the Cissoid, and a few others, they made little or no Account of them. It is true, they have given us many excellent and useful Theorems concerning the Properties of these others; but far short of what has been discovered since. Thus the Quadrature of the Circle, which did so much exercise and perplex the Thoughts of the Ancients; How imperfect is that of Archimedes, in comparison of that exhibited by Van Ceulen? And every Body knows how this is exceeded by the later Performances of Mr. Newton, and Monsieur Leibnitz. Archimedes, with a great deal of Labour, has given us the exact Quadrature of the Parabola; but the Rectification of the Parabolick Line, depending on the Quadrature of the Hyperbola, is the Invention of this last Age. The rare Properties of the Conick Se-