Derivatives or differential coefficients. We return to the consideration of the properties which we naturally associate with the notion of a curve. The first and most obvious property is, as we saw in the last chapter, that which gives a curve its appearance of connectedness, and which we embodied in our definition of a continuous function.

The ordinary curves which occur in elementary geometry, such as straight lines, circles and conic sections, have much more ‘regularity’ than is implied by mere continuity. In particular they have a definite direction at every point; there is a tangent at every point of the curve. The tangent to a curve at P is defined, in elementary geometry, as ‘the limiting position of the chord PQ, when Q moves up towards coincidence with P’. Let us consider what is implied in the assumption of the existence of such a limiting position.

In the figure (Fig. 34) P is a fixed point on the curve y = ϕ(x), and Q a variable point; PM, QN are parallel to OY and PR to OX. We denote the coordinates of P by x, y and those of Q by x + h, y + k: h will be positive or negative according as N lies to the right or left of M.

We have assumed that there is a tangent to the curve at P, or that there is a definite ‘limiting position’ of the chord PQ. Suppose that PT, the tangent at P, makes an angle ψ with OX.