The work in which this was contained is a tract of no more than 106 quarto pages ; and there is probably no book of the same sire which has conferred so much and so just celebrity on its author. It was first publishedin 1637.
In the first of the three books into which the tract just mentioned is divided, the author begins with the consideration of such geometrical problems as may be resolved by circles and straight lines ; and explains the method of contracting algebraic formulas, or of translating a truth from the language of algebra into that of geometry. He then proceeds to the consideration of the problem, known among the ancients by the name of the locus ad quatuor rectos, and treated of by Apollonius and Pappus. The algebraic analysis afforded a method of resolving this problem in its full extent ; and the consideration of it is again resumed in the second book. The thing required is, to find the locus of a point, from which, if perpendiculars be drawn to four lines given in position, a given function of these perpendiculars, in which the variable quantities are only of two dimen sions, shall be always of the same magnitude.' Descartes shows the locus, on this hypo ' thesis, to be always a conic section ; and he distinguishes the cases in which it is a circle, -an ellipsis, a parabola, or a hyperbola. It was an instance of the most extensive investiga tion which had yet been undertaken in geometry, though, to render it a complete solu pon of the problem, much more detail was doubtless necessary. The investigation is ex tended to the cases where the function, which remains the same, is of three, four, or five dimensions, and where the locus is a line of a higher order, though it may, in certain circumstances, become a conic section. The lines given in position may be more than four, or than any given number ; and the lines drawn to them may either be perpendi culars, or lines making given angles with them. The same analysis applies to all the cases ; and this problem, therefore, afforded an excellent example of the use of algebra in the, investigation of geometrical propositions. The author takes notice of the unwilling
ness of the ancients to transfer the language of arithnietic into geometry, so that they were forced to have recourse to very circuitous methods of expressing those relations of quan tity in which powers _beyond the third are introduced. Indeed, to deliver investigation from those modes of expression which involve the composition of ratios, and to substitute in their room the multiplication of the numerical measures, is of itself a very great advan tage, arising from the introduction of algebra into geometry.
In this book also an ingenious method of drawing tangents to curves is proposed by Descartes, as following from his general principles, and- it is an invention with which he appears to have been particularly pleased. He says, " Nec verebor dicere problema hoc non mode eorum, gum scio, utilissimum et generalissimum ease, sed etiam eorum quie in geometria scire unquam desideraverim." This passage is not a little characteristic of Descartes, who was very much disposed to think well of what he had done himself, and even to suppose that it could not easily be rendered more perfect. The truth, however, is, that his method of drawing tangents is extremely operose, and is one of those hasty views which, though ingenious and even profound, require to be vastly simplified, before they can be reduced to practice. Fermat, the rival and sometimes the superior 'of Des cartes, was far more fortunate with regard to this problem, and his method of drawing tangents to curves, is the same in effect that has been followed by all the geometers since his tiine,—while that of Descartes, which could only be valued when the other was un known, has been long since entirely abandoned. The remainder of the second book is occupied with the consideration of the curves, which have been called the ovals of Des cartes, and with some investigations concerning the centres of lenses ; the whole indicat ing the hand of a great master, and deserving the most diligent study of those who would become acquainted with this great enlargement of mathematical science.