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46 THE NATURE OF MATHEMATICS Descartes and many of his followers adopted different forms of definition which really involve the idea of a limit, an idea which appears boldly in the infinitesimal calculus. A tangent is the limit of a secant as the points of intersection approach infinitely near to one another; it is a produced side of the polygon with infinitesimal sides that the curve is supposed to be ; it is the direction of motion at an instant of a point moving in the curve considered. The equation got from that of the curve by substituting for y from the equation of the intersecting straight line has, if this straight line is a tangent, two equal roots. In the above case, this equation was quadratic. In the case of a circle, we can easily deduce the well-known property of a tangent of being perpendicular to the radius ; and see that this property has no analogue in the case of other curves. We must remember that, just as -plane, curves determine and are determined by equations with two independent variables x and y, so surfaces spheres for instance in three-dimensional space determine and are determined by equations with three independent variables, x, y, and z. Here x, t/, and z are the co-ordinates of a point in space ; that is to say, the numerical measures of the distances of this point from three fixed planes at right angles to each other. Thus, the equation of a sphere of radius d and centre at (a, 6, c) is (a; a) 2 + (y 6) 2 + (z-c) 2 = d 2 . We may look at analytical geometry from another point of view which we shall find afterwards to be important, and which even now will suggest to us some interesting thoughts. The essence of Descartes' method also appears when we represent loci by the method. Consider a circle ; it is the locus of a point (P) which moves in a plane so as to preserve a constant distance from a fixed point (0). Here we may think of P as varying in position, and make up a very striking picture of what we call a variable in mathematics. We must, however, remember that, by what we call a " variable " for the sake of picturesqueness, we do not necessarily mean some- thing which varies. Think of the point of a pen as it moves over a sheet of writing paper ; it occupies different positions with respect to the paper at different times, and we under- standably say that the pen's point moves. But now think of a point in space. A geometrical point which is not the bit
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