A Euclidean Approach to the FTC - Some Definitions and Background

Gregory's proof is not without its limitations. To begin with, it assumes that the curve OFL passes through the origin and is increasing, so it is only valid for a specific case of the FTC. More importantly, at several crucial junctures Gregory relies heavily on geometrical intuition to an extent that would make most mathematicians today uneasy. The most obvious geometrical assumption is that the curve OEK as defined point-wise above actually has a tangent line at each of its points. Indeed, Gregory not only assumes this, but his figure (see below) indicates that he knew the indefinite integral of an increasing function must be concave up.

The second geometrical assumption is about the nature of a tangent line. In Gregory's proof, the crucial idea is to suppose that \(C\) is the point on the segment \(OA\) such that \(IK/IC = IL\) but that \(KC\) is not tangent to the curve \(OEK\). The rest of the proof consists of using this claim to derive a contradiction. But what does it mean for \(KC\) not to be tangent to \(OEK\)? For someone trained in classical Euclidean geometry, the definition of a tangent line was inherited from Greek mathematicians such as Euclid (in the Elements III.16) and Apollonius (in the Conics I.33, et al.). For these geometers, a line is tangent to a curve when (i) the line touches the curve at one point and (ii) all other points on the line lie on one side of the curve.

This definition is slightly different from the definition we use today and has the drawback of only working for curves where the concavity doesn't change sign. For instance, the tangent line to \(f(x) = x^3\) at \(x = 0\) would not satisfy this definition. But it has the benefit that a line will not be tangent to a curve precisely when the part of the line on one side of its intersection with the curve is above the curve, while the other part is below.