CHAPTER III.

Descartes, relative to the analytical representation of geometrical figures: it is this which forms the special object of this chapter. I will continue to consider at first, for more facility, only geometry of two dimensions, which alone was treated by Descartes; and will afterwards examine separately, under the same point of view, the theory of surfaces and curves of double curvature.

PLANE CURVES.

Expression of Lines by Equations. In accordance with the manner of expressing analytically the position of a point on a plane, it can be easily established that, by whatever property any line may be defined, that definition always admits of being replaced by a corresponding equation between the two variable co-ordinates of the point which describes this line; an equation which will be thenceforth the analytical representation of the proposed line, every phenomenon of which will be translated by a certain algebraic modification of its equation. Thus, if we suppose that a point moves on a plane without its course being in any manner determined, we shall evidently have to regard its co-ordinates, to whatever system they may belong, as two variables entirely independent of one another. But if, on the contrary, this point is compelled to describe a certain line, we shall necessarily be compelled to conceive that its co-ordinates, in all the positions which it can take, retain a certain permanent and precise relation to each other, which is consequently susceptible of being expressed by a suitable equation; which will become the very clear and very rigorous analytical definition of the line under consideration, since it will express an algebraical property belonging exclusively to the co-ordinates of all the points of this line. It is clear, indeed, that when a point is not subjected to any condition, its situation is not determined except in giving at once its two co-ordinates, independently of each other; while, when the point must continue upon a defined line, a single co-ordinate is sufficient for completely fixing its position. The second co-ordinate is then a determinate function of the first; or, in other words, there must exist between them a certain equation, of a nature corresponding to that of the line on which the point is compelled to remain. In a word, each of the co-ordinates of a point requiring it to be situated on a certain line, we conceive reciprocally that the condition, on the part of a point, of having to belong to a line defined in any manner whatever, is equivalent to assigning the value of one of the two co-ordinates; which is found in that case to be entirely dependent on the other. The analytical relation which expresses this dependence may be more or less difficult to discover, but it must evidently be always conceived to exist, even in the cases in which our present means may be insufficient to make it known. It is by this simple consideration that we may demonstrate, in an entirely general manner—independently of the particular verifications on which this fundamental conception is ordinarily established for each special definition of a line—the necessity of the analytical representation of lines by equations.

Expression of Equations by Lines. Taking up again the same reflections in the inverse direction, we could show as easily the geometrical necessity of the representation of every equation of two variables, in a determinate system of co-ordinates, by a certain line; of which such a relation would be, in the absence of any other known property, a very characteristic definition, the scientific destination of which will be to fix the attention directly upon the general course of the solutions of the equation, which will thus be noted in the most striking and the most simple manner. This picturing of equations is one of the most important fundamental advantages of analytical geometry, which has thereby reacted in the highest degree upon the general perfecting of analysis itself; not only by assigning to purely abstract researches a clearly determined object and an inexhaustible career, but, in a still more direct relation, by furnishing a new philosophical medium for analytical meditation which could not be replaced by any other. In fact, the purely algebraic