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straight lines, showing the application of algebra to suchquestions. He then proceeds to the more immediatesubject of his researches, in which the powers of themodern analysis are so conspicuously displayed. AVewill endeavour briefly to explain it.

AVe have, in a former section, mentioned the problemcalled “ locus ad quatuor rectos,” proposed and par-tially discussed by Apollonius and Pappus. To this,Des Cartes , in the first instance, turned his attention,and soon effected a complete algebraic solution of it. Thenature of the problem may be thus stated in a generalway: — Four lines are given in position; and it is requiredto determine a point such, that if from it perpendicularsbe drawn to the four lines, a certain complex combin-ation of the rectangles or products of the perpendicularsshall be of a constant magnitude. The given lines may, infact, be more than four, and the other lines may be in-clined at any given angles. It is, of course, impossibleto give a more exact idea of the case without enteringinto algebraic details; it must suffice to say that DesCartes readily showed that the problem is indeterminate;that an infinite number of points fulfil the condition;and these, being laid down and connected together, forma locus, which he found to be always one of the curvesof the conic sections when the prescribed combinationsof the quantities are products of only two factors, or, inthe language of algebra, rise only to two dimensions.AV'hen these are of higher dimensions, correspondingcurves of different kinds result: when of only one di-mension, the locus is a straight line. The problem, asoriginally proposed, is involved in much greater com-plexity than is at all necessary for the purpose of thistheory. And one of the main improvements consistedin taking, as the fundamental lines to which the wholeconstruction is referred, only two lines, inclined at anygiven angle. These are called the axes; and uponthese, or upon lines respectively parallel to them, aremeasured the corresponding values of the variables, inthe indeterminate equation, when solved for one inp 3