right lines or axes. These distances varied with every change of position in the point. This geometric idea of co-ordinate representation, together with the algebraic idea of two variables in one equation having an indefinite number of simultaneous values, furnished a method for the study of loci, which is admirable for the generality of its solutions. Thus the entire conic sections of Apollonius is wrapped up and contained in a single equation of the second degree.
The Latin term for "ordinate" used by Descartes comes from the expression lineæ ordinatæ, employed by Roman surveyors for parallel lines. The term abscissa occurs for the first time in a Latin work of 1659, written by Stefano degli Angeli (1623–1697), a professor of mathematics in Rome.3
Descartes' geometry was called "analytical geometry," partly
because, unlike the synthetic geometry of the ancients, it is
actually analytical, in the sense that the word is used in logic; and partly because the practice had then already arisen, of designating by the term analysis the calculus with
general quantities.
The first important example solved by Descartes in his geometry is the "problem of Pappus"; viz. "Given several
straight lines in a plane, to find the locus of a point such that the perpendiculars, or more generally, straight lines at given angles, drawn from the point to the given lines, shall satisfy the condition that the product of certain of them shall be in a given ratio to the product of the rest." Of this celebrated problem, the Greeks solved only the special case when the number of given lines is four, in which case the locus of the point turns out to be a conic section. By Descartes it was solved completely, and it afforded an excellent example of the use which can be made of his analytical method in the study of loci. Another solution was given later by Newton
in the Principia.
The methods of drawing tangents invented by Roberval
and Fermat were noticed earlier. Descartes gave a third method. Of all the problems which he solved by his geometry, none gave him as great pleasure as his mode of constructing tangents. It is profound but operose, and, on that account, inferior to Fermat's. His solution rests on the method of Indeterminate Coefficients, of which he bears the honour of
invention. Indeterminate coefficients were employed by him also in solving biquadratic equations.
The essays of Descartes on dioptrics and geometry were sharply criticised by Fermat, who wrote objections to the former, and sent his own treatise on "maxima and minima"
to show that there were omissions in the geometry. Descartes thereupon made an attack on Fermat's method of tangents.
Descartes was in the wrong in this attack, yet he continued