Page:Mind (New Series) Volume 8.djvu/355

THE PHILOSOPHY OF SPINOZA AND LEIBNIZ. 341 of the problem of the relation between a straight line and a curve, a problem insoluble by Euclid because he postu- lated them independently ; but the solution had still to be worked out, the unity of which the straight line and curve are immediate differences had still to be determined. The solution was obtained in connexion with the problem of drawing a tangent to a curve. If the method of limits is followed, the tangent is the limit of a secant cutting the curve in two points, when these two points are brought infinitely near to one another, i.e., when they are separated from one another by less than any assignable distance. But even in the limit case we have still two points and a line, an infinitely little line, it is true, but yet a line. The infinitely little distance is regarded as real but as negligible. Now just about the time of Leibniz another step forward was taken. 1 In connexion with the fact that finite numbers may be resolved into infinite series, it was contended that the finite line rests upon the infinitely little, that the in- finitely little is really its generating principle. Every line has length and direction. An infinitely little line has infinitely little length ; but no reduction in its length can make any alteration in its direction. Accordingly the infinitely little line means really the direction, which is the essence or generating principle of the line. Given the direction, the line may be drawn to any length, great or small. The essence of every line is thus its direction, that is its quality or characteristic and not its quantity as the distance between two points. The points presuppose the line. Thus, if we regard a curve as generated by the motion of a point, the tangent to the curve at any point will simply be the direction of motion at that point. The direction of the moving point changes continuously and, in the case of a regular curve, uniformly, in accordance with a law which is characteristic of the particular curve. Accordingly, in general, the straight line and the curve are essentially varieties of direction in space, the straight line being a continuous uniform direction, while the curve is a continuously varying direction of more or less complexity. And the direction of a curve at any point must be regarded as a ratio between two infinitely small quantities, because change of direction in a plane is relative to two axes and continuous change of direction means infinitely small variation from point to point. It was the solution of problems resulting from such conceptions as these that led to the discovery of the Infinitesimal Calculus. lr The advance was made by Eoberval (1602-1675).