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Innovative Teaching Exchange: Volumes and History

Volumes and History:

A Calculus Project Involving Reading an Original Source

Elyn Rykken
Mathematical Sciences Department
Muhlenberg College
Allentown, PA 18104
and Jody Sorensen
Department of Mathematics and Statistics
Grand Valley State University
Allendale, MI 49401

While conducting a survey of old calculus books in the rare book room at American University, we came across James Hodgson's The Doctrine of Fluxions. While the book was originally published in 1726, we looked at a posthumous 1756 edition. We were looking for various calculus books' presentations of volumes of solids of revolution. While Hodgson's notation and language are awkward to our modern eyes, his geometric explanations are clear. Furthermore, he employs a wide variety of techniques that are still taught today. These include the methods using "disks", "washers", and "shells". We decided to use several pages of this text as a reading assignment for our students. What follows is a summary of what we hope to achieve by the project, along with links to the text, including both images of the actual book and our transcribed copy of the pages. We've also included the assignment we give to our students, a summary of the difficulties they have encountered, and their reactions.

When teaching our students how to find volumes of solids of revolution, we emphasize visualizing the geometry that is involved. We teach them to consider a representative slice of the region which is to be rotated, and then to imagine what shape this slice will generate when rotated. This naturally leads to the use of disks, washers or shells, depending on the particular slice that is considered. When we discovered Hodgson's treatment of this topic in his text, we were pleased to see that he uses the same approach. We decided that reading this text would be a good exercise for our students. It would reinforce the idea of a representative slice, and it would serve as an introduction to the history of mathematics as a discipline, and to the history of calculus in particular.

James Hodgson lived in England from 1672 to 1755. He was master of the Royal Mathematical School in Christ-Hospital in London, and was a member of the Royal Society. In addition to mathematics, he wrote books on navigation, chronology, annuities and astronomy. Here is a copy of the title page of Hodgson's text and a page showing his calculation of the volume of a sphere. We have also included our transcription of four pages from his book in both dvi format and pdf format.

Hodgson's calculus text, The Doctrine of Fluxions founded on Sir Isaac Newton's Method, is clearly on the British side of the Newton-Leibniz controversy. He uses fluxions and fluents instead of differentials and integrals, and defends his choice early on in the text. This allows us to discuss several of the important figures in the development of calculus and the controversy that arose. It also allows us to discuss our current notation for integration in contrast to Newton's. In addition, Hodgson's text is one of the earlier calculus texts. At the time it was published, there was no uniform symbol for pi and no uniform coordinate system. Our students are usually surprised by this and come to appreciate the clarity and conciseness of our modern symbols and notation as well as our coordinate system.

We have used this project in several second semester calculus courses. Here is the assignment we give our students. They are allowed to work together but each turns in an individual write up. We give them about two weeks to complete the assignment, which counts for about five percent of their final grade.

Our students' initial reaction is always that the assignment is impossible and that they will never be able to read Hodgson's text. A week after assigning the project, as a class, we work through the example of the calculation of the volume of the sphere. We discuss the notation used, the use of pi, the coordinates and formulas and other possible sticking points. After this, the students are better prepared to tackle the remaining examples on their own, and are usually able to complete the assignment to our satisfaction. On rare occasions, we have even had students look up other original source material to further their understanding of the differences between Newton's and Leibniz's methods.

Our students often have the impression that mathematics never changes. We reinforce this notion when we explain that the mathematics of the Greeks is still used and accepted as true today. What can be lost is any sense of how mathematics evolves. Since its invention in the 1670's, the presentation of calculus has changed greatly. Function notation, a uniform coordinate system, a symbol for pi, and new notation for derivatives and integrals have transformed how we express the ideas of calculus. This project lets students discover these changes, and yet appreciate the fact that the concepts themselves have not changed in 300 years.

This project was inspired by our experiences at the Institute on the History of Mathematics and Its Use in Teaching (IHMT), an NSF sponsored program whose goal is to increase the presence of history in mathematics classrooms. We would like to thank its organizers, Fred Rickey and Victor Katz, for their hard work and support.