Calculus and Its Origins begins with ancient questions that had been explored long before calculus was born, and details the remarkable story of how subsequent scholars wove these inquiries into a unified theory. Other books have traveled these paths, but they presuppose knowledge of calculus. This book requires only a basic knowledge of geometry and algebra (similar triangles, polynomials, factoring), and a willingness to treat the infinite as metaphor.
1. The Ancients
2. East of Greece
6. The Fundamental Theorem of Calculus
9. Zero over zero
About the Author
2.2 Porphyry, Pappus, and Bhaskara count. The Syrian scholar Porphyry (born c. 234), in his commentary on the works of Aristotle (Greece, born 384 BCE), did not resist a detour into mathematics when the opportunity arose. Porphyry wished to explain the differences between five ‘qualities’ that catergorize all real things (such as horses, the number 10, Aristotle himself, and so on). This ambitious undertaking prompted Porphyry to contrast each of the five qualities with each of the others.
(a) Rather than simply begin, Porphyry paused to argue that there are ten such comparisons. Choose any of the five qualities; we may compare it to each of the other four. Now when we consider pairing a second quality to the others, we do not want to reconsider the first quality, which has already been compared to the second; rather, we pair the second to the remaining three. In this manner, Porphyry claimed, we see that the total number of pairings is 4 + 3 +2 +1 = 10. (This is the fourth triangular number.)
Use induction to argue that the number of ways to pair n objects is the triangular number Tn-1.
(b) The Greek mathematician Pappus (born c. 290) set his mind to many problems of geometry, one of which led him to consider the same problem as Porphyry. Pappus counted the crossing points of lines that are drawn so that no two are parallel and no three intersect at the same point. Explain the connection between this problem and that of Porphyry.
David Perkins earned his doctorate in mathematics at the University of Montana (advised by P. Mark Kayll). He taught at Houghton College for ten years before moving to Luzerne County Community College to live and work with his wife Michelle LaBarre. In 2004, he became an ‘orange dot’ as a member of ProjectNExT.
The goal of this book is to “teach calculus as the culmination of an intellectual pursuit and place the discovery of calculus at the end.” The author succeeds admirably in describing this pursuit and its culmination in such a short book. It is less clear whether it is possible to teach calculus from this book. (I expect that the author’s own calculus course is wonderful for those who are interested, but I’m less sure that anyone else will be able to teach a calculus course directly from the book).
Nevertheless, this review is about the book, not about the course, and the book is a very good and original addition to the calculus literature. The prose is very agreeable. It looks very nice too: all figures are wonderful hand-drawn pictures. One wonders whether they could be brought to life in the classroom with some computer animation. Continued...