92 Sir I S A A C N E W T O NTs BookI.

kings wrote treatises upon it. Thales brought the principles ofit into Greece , where it was so diligently cultivated that theelementary part was soon compleated, and was so highlyesteemed as to have the appellation of the mathemata in amanner appropriated to it. An oracle appointing the cubicalaltar of Apollo to be doubled was, we presume, os greater ad-vantage to geometry than to the Athenians then afflicted withthe plague ; as it gave occasion to Plato to consider the famousproblem of the duplication of the cube, and produced the solidgeometry. It afterwards received great improvements from theincomparable Archimedes , who squared the area of the parabola ,made some progress in the mensuration of the circle, and en-riched this science with many discoveries worthy of lo excellenta genius.

It appears that it advanced but by degrees, and sometimes byvery stow steps : one, we are told, discovered that the threeangles of an equilateral triangle were equal to two right ones ;another went farther, and shewed the same thing of those thathave two sides equal and are called isosceles triangles ; and itwas a third who found that the theorem was general, and ex-tended it to triangles of all sorts *. In like manner, when thescience was farther advanced, and they came to treat of theconic sections, the plane of the section was always supposedperpendicular to the side of the cone ; the parabola was the »only section that was considered in the right-angled cone, theellipse in the acute-angled cone, and the hyperbola in the ob-tuse-angled. From these three sorts of cones, the figuresof the sections had their names, for a considerable time ; till atlength Apollonius shewed how they might be all cut out of any

* Prtcli Comment, in Euclidem'. ' \

one