The Calculus Story

David Acheson's popular math book 1089 and all that (Oxford U. Press, 2002) was rather successful, and it got with this one a worthy successor. Small size, short chapters, amply illustrated, large font, airy layout, all properties that turn it into a storybook indeed. By storybook, I mean the kind of books you want to read to your children or grandchildren, that keeps their attention and makes them impatient and eager for the adventure to come in the next episode. In this sense, the title is very well chosen. It is indeed a story book, the story of calculus and how it came about, once upon a time...

Since it is still about mathematics, you shouldn't try this on a toddler, but you might catch the attention of any novice to calculus from the age of about twelve or thirteen on. What exactly should be familiar before you can start telling the calculus story? You need some algebra to be able to manipulate equations. Also the concept of a proof, and curves as representations of functions are assumed, but that's about all. The mathematical content of the story is then the same as what you find in a classic textbook: limits, differentiation, integration, and infinite series. In the early chapters of the book Acheson sketches the preliminaries and sheds some light on what one wants to achieve with the next steps about to be taken. And later in the second half of the book, while he got the attention of the reader, Acheson moves on to differential equations, optimisation, complex numbers and chaos.

For reasons of efficiency, a classic textbook will introduce a new concept by following a certain paradigm that consists of giving some motivating examples, followed by a precise and polished definition, or it is done the other way around: first the definition, and then some examples. However, this is not how the concept caught by the definition was developed historically. It is a much more natural approach to follow the historical path. In retrospect, the side tracks that turned out to be dead ends, can be pruned away, but still, letting the concept grow organically usually is a better choice. Decoupling mathematics from its history makes it abstract and dull. Including the history makes it less of a top-down rigid dogmatic doctrine that is forced upon the pupil, but something developed by real people, hence more "human".

These historical elements make Acheson's book a mathematical (his)story(book). For example it is interesting to learn how people struggled with $\infty\times0=?$. Summing infinitely many infinitely small elements was used, and sometimes misused, by the founding fathers such as Kepler, Cavalieri, Wallis, Torricelli and others to compute and area or a volume, hence indirectly summing infinite series. Many of these computations were inspired by physics. The speed and acceleration of falling objects subject to gravity had been investigated but it was Newton who formulated the more general fundamental laws of motion. When applied to the force of gravity, these eventually explained the orbits in our planetary system. Kepler had described the "how" of the orbits, and Newton provided the "why". This has influenced Newton strongly in the way he developed his calculus. He made use of fluxions as it was based on the dynamics of coordinates like an object exposed to some action will move along a path describing its position as a function of time. Independently also Leibniz developed calculus. He used these infinitely small increments. The ratio $\delta y/\delta x$ of small changes gave rise to the notation $dy/dx$ for the derivative that mathematicians still use today, while Newton used $\dot{x}$ for the derivative of $x$, which is more commonly used in physics. With this climax in the calculus story, the birth of calculus, Acheson is about half way in his book.

In the second half, the sine and cosine functions are used to connect an angle to periodic motion and for example to show the Leibniz formula $1−1/3+1/5−1/\cdots=\pi/4$ and other ways to compute π. The Leibniz formula had been hinted to in previous chapters, building up some tension. So, it feels like yet another success of calculus that it can explain why this formula holds. But periodic motion also means differential equations describing the pendulum or a vibrating string. The towering historical mathematician is now Euler. With calculus at our disposal, now topics such as calculus of variations, optimization, logarithms (including e and $i=\sqrt{-1}$), Taylor series expansions, Fourier series and other topics now come in rapid succession. It also requires to reconsider the definition of a limit to the more rigorous $(\epsilon,\delta)$ definition as developed by Weierstraß. Thus it is illustrated how this precise definition of the limit is the endpoint of a whole evolution and perhaps should not be the first definition to which a novice should be exposed. Acheson ends his story with a glimpse on Maxwell's equation of electromagnetism, Schrödinger's equation, and chaos theory.

Thus Acheson introduces the reader with elementary steps to the concepts that matter, creating insight, and answering the why's and how's by calling the historical mathematicians to the stage and by citing from their papers. Thus it is not a bedtime story after all, but it should awaken the interest of the youngsters for the fascinating mathematics that in the end is describing the physics of the world we live in. Even for those students who had to assimilate calculus from a dull textbook, this story underlying all these definitions, theorems and computational rules, may soften their aversive attitude towards the subject. To pull youngsters away from the dark side of mathphobia, this booklet acts as a medicine to be applied to your wishes: preventively, remedially, or supplementary. This calculus story can be applied at all times to create a mathematical success story. 

Adhemar Bultheel
Book details

Acheson uses an historical approach to introduce calculus to beginners. Starting from the most elementary concepts, the reader is exposed to a world of limits, differentiation, integration, and series and their entanglement with physics. Describing our planetary system was an incentive for the development of calculus. A flavour is given of differential equations describing a pendulum or a vibrating string, and there is a preview of Maxwell's and Schrödinger's equations and chaos theory. While previously physics pushed the boundaries of mathematics, it is now mathematics that pushes the limitations of physics.

Author:  Publisher: 
9780198804543 (hbk)
£ 11.99 (hbk)