This article is dedicated to all those students for whom the limit is their most bitter memory of calculus. I grant that the limit really is the heart of calculus and one of the most powerful ideas in mathematics today. I also grant that the story of how the limit evolved from being a logically-suspect proving tool in the hands of analysts like Newton and Leibniz in the 17th century into a highly-polished mathematical definition articulated by Cauchy in the 19th century is one of the more interesting stories in the history of mathematics. But it turns out that many results from calculus--including the pivotal fundamental theorem of calculus--can be proved without any notion of the limit whatsoever. In fact, my aim is to present a proof of the FTC using only mathematical tools that were available to Euclid nearly 2000 years before mathematicians began wrestling seriously with the idea of the limit.
What's especially interesting about this proof is that it is not new. It's part of an often-overlooked chapter in the history of mathematics in which mathematicians endeavored to answer questions our students first see in calculus using the well-worn proving techniques of Euclidean geometry instead of the analytic techniques developed in the 17th century. Years before Newton and Leibniz published the results that eventually grew into the calculus we learn today, the proof I will present appeared in slightly modified form buried in the proof of a proposition found in The Universal Part of Geometry, a geometry book published in 1668 by the Scottish mathematician James Gregory.