Tangent

Abstract

In an important theorem in Book III of his Elements, Euclid proves that the straight line drawn at right angles to any diameter BA of a circle at one of its extremities A falls outside the circle (except where it touches it), and that no other straight line can be interposed into the space HAE between the circumference and the straight line, and further that the “horn-like angle” HAE that the circumference makes with EA is less than any rectilineal angle.¹ The line EA is the tangent to the circle at A (Fig. 50), and, as Euclid’s results indicate, it lies closer to the circumference than any other straight line through A. The fact that Euclid felt compelled to prove this result, which most of us would regard as “obvious”, attests to the high level of rigor that permeated Greek mathematics in the days of Plato’s Academy.