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HISTORICAL VIEW OP THE

this, of course cut the cone, and gave rise to the curve.If the cone had a right angle at its summit, the curvewas a parabola: if the vertical angle was less than aright angle, an ellipse: if greater, an hyperbola. Adifferent species of cone was thus supposed necessary togive each different curve. A century elapsed before itwas seen that they might all be obtained from one andthe same cone of any species, by merely altering the in.clination of the cutting plane. Of the high importanceof these curves in the researches of modern physicalscience, this is not the place to speak. We must hereconfine ourselves to noticing that the ancients happilyattached due value to them, and continued with unre-mitting diligence to investigate their various properties.In these researches, none was more eminent than Me-nechme, the friend and disciple of Plato .

But the Platonic school was scarcely less distinguishedfor originating other important branches of mathematicalspeculation. Among these, the most remarkable, perhaps,was the geometrical analysis. This invention is ex-pressly ascribed to Plato himself, by Proclus . Anygeometrical question, whether problem or theorem,being submitted to analysis, is assumed as solved, or astrue. From this assumption, a chain of consequencesis drawn, which, by the ingenuity of the geometer, iscontinued, until he arrives at some proposition knownto be true or false, possible or impossible. The finalconsequence points out whether the question be true orpossible ; and by retracing the steps, a synthetic proofor solution may be found.

Another class of speculations commenced and pursuedin this school, was the geometric loci. These, in theirsimplest form, arose out of problems where it was re-quired to find a point determined, for instance, by theintersection of two lines under certain given conditions,and where it was found that there were an infinitenumber of points fulfilling the requisition, but eachrestricted to a certain position: so that if they were allassigned, they would all lie in a certain line or locus,