PHYSICAL AND MATHEMATICAL SCIENCES. 215

Now, by the method of Des Cartes , the subject wasgeneralised : — “ Perceiving the importance and powerof the principle which he used in the solution ” (of theproblem of the locus), “ he immediately conceived thenotion of founding upon it the whole geometry of curvelines. By this felicitous application of equations oftwo unknown quantities, the science of geometry wasutterly revolutionised. Every curve described by agiven law, being expressed by an equation between twovariables deducible from that law, was thus broughtunder the dominion of algebra. This equation includ-ing the essence of the curve, its various properties

flowed from it.The immediate consequence

of this memorable discovery was, that geometry at onceoversprang the narrow limits which had circumscribedit for ages, and took a range, the extent of which isliterally infinite. Instead of a few simple and parti-cular curves, which had hitherto constituted the onlyobjects of the science, the geometer discussed the pro-perties of whole classes of curves, distinguished andarranged according to the degrees of the equations whichrepresent them. The variety of curves thus became asinfinite as that of equations.”*

The construction of a curve corresponding to anygiven equation is thus easy to be conceived. Its generalform would, of course, thus be traced out. But it doesnot immediately appear how its various properties arededucible from its equation. The investigations of DesCartes , however, extended in some measure to this partof the subject also. The properties of curves may besaid to depend mainly upon the positions assumed bythose straight lines which are the tangents to them atany points. The determination of these positions, orthe solution of the problem, “ to draw a tangent to acurve,” forms the basis of all such speculations. In par-ticular individual cases, this had been done by theancient geometers.

But the ancient geometers had felt certain difficulties

• Lardner'a Alg, Geom. Pre£ xxii.