# Calculus and Its Origins

This book is an historical approach to calculus. In ten chapters, a rough outline of calculus is given, approaching the topics discussed such as they were originally addressed by their "inventors", but using familiar modern concepts, terminology, and notation. To give an example: chapter 1 discusses infinite sums and how they also show up in the ideas of Archimedes on computing the area of a parabolic segments, filling up the area with smaller and smaller triangles. In a concluding section called "Furthermore" it is again Archimedes who uses successive polygons to approximate the area of a circle and hence the value of π. It is a further challenge/exercise to find a formula that gives the sum of the squares and cubes of the first n integers.

The other chapters follow a similar pattern, each time ending with the "Furthermore" section giving additional information of historical mathematicians, and some (usually rather elementary) exercises. In this style we meet Ibn al-Haytham an Jyesthadeva again on infinite sums. Among others, the reader meets Fermat and Descartes and later Cavalieri and Roberval and their study of curves and the computation of areas and volumes, which bring the reader to the 16th and 17th century. Although the exposition still relies largely in graphics, there is gradually more algebra sneaking in. Still looking at areas under curves like the hyperbola leads to the concept of logarithms and the exponential function as developed by de Sarasa, Brouncker and Wallis. Enter Newton and Leibniz. They lay the foundations of differential calculus, the interpretation of differentials and introduce a notation that looks familiar to a reader of the 21st century. The final chapter is then about continuity as discussed by Bolzano, Weierstrass, Dirichlet, and others. They introduce more rigor and are gradually leaving intuition behind. The few names of mathematicians that I mentioned above are only some examples. Many more that have contributed are mentioned in due course.

The topics are treated in a rather intuitive way, using extensively the many figures of the text. These are manually produced with ruler and pencil, which in time of computers is somewhat surprising, but it blends well with the general approach of the book. There are of course formulas, but no formal proofs are included. So it is not surprising that the book stops with the chapter where the mathematics take off to more abstraction and less intuition.

The book is a collection of topics that more or less follows the evolution of calculus throughout the centuries. It is neither a full history of mathematics or even calculus, nor is it a collection of biographies of prominent mathematicians. It is a mixture of these, with emphasis on the mathematics itself. If used to teach calculus, then it is certainly an unusual approach. Many topics are not discussed and there is no classical set of drilling exercises for derivatives and integrals as is usual in a calculus course. It may be a good source of inspiration to formulate some assignments for homework if used as a textbook besides a more traditional course.

**Submitted by Adhemar Bultheel |

**8 / Feb / 2013