A Euclidean Approach to the FTC - Other Resources

Gregory wasn’t alone in using Euclidean methods to prove theorems in calculus. In 1670, Cambridge mathematician Isaac Barrow published the Lectiones Geometricae , which also proved many results from Calculus—including the FTC—using only geometry. This work is presented in a highly-annotated (and somewhat controversial) book entitled The Geometrical Lectures of Isaac Barrow published in 1916 by J.M. Child (Open Court Publishing Company).

For a general introduction to the work of Gregory and Barrow (along with the work of many other contributors to the foundations of calculus), see The Origins of the Infinitesimal Calculus by Margaret E. Baron, available (cheaply!) from Dover Publications. Victor Katz also has a good discussion of Barrow's geometrical proof of the FTC on pages 500–501 of his book A History of Mathematics: An Introduction (2nd edition) (Addison-Wesley).