Introduction

Long before the calculus arrived a medieval philosopher, Nicole Oresme, developed what he called the latitude of forms, a graphical representation that sheds light on the fundamental connection between area and what we now call the integral. In a calculus course, the latitude of forms can be used to introduce the idea of the integral as area, while simultaneously introducing the idea that the distance traveled is the integral of velocity. Of course the two ideas can be addressed separately, if you prefer. In that case, the latitude of forms might be used to connect the two. In any event, you will be reviewing some simple geometry that students have often forgotten.

At the risk of being untrue to the original, I have modernized my presentation. The Commentary section will attempt to partially correct this distortion.

Historical Background

Scholastic philosophers, following Aristotle, were greatly interested in explaining the workings of the natural world. In this sense they appear to our eyes as scientists. They also were interested in precise definitions, careful distinctions between cases, and rigorous logical deduction. To us they appear to be mathematicians and analytic philosophers. Yet when we read their works, we can see they were also trying to explain why things work, and seeing how well their explanations fit their theology. Thus to us they appear to be trying to tackle everything at once.

This article focuses on Nicole Oresme (c. 1323–1382), who was born in Normandy and studied at the University of Paris. Immediately upon receiving his doctorate he became grand master of the University of Navarre.