The best books on The History of Mathematics

Why should we be interested in the history of mathematics?

Mathematics, like painting, music, literature, has a long history. Indeed, it’s longer than most, since the first writing is believed to be numerical. It’s also multicultural, with its historical origins in Africa, the Middle East and Asia. The history of mathematics also involves particular individuals who are part of our world culture. Perhaps more school children would be interested in maths if it were taught more from a historical point of view. For example, how often do they learn that quadratic equations have been solved for 4000 years, having their origins in clay tablets discovered in what is now Iraq, an area that features in our daily newspapers?

Who are some of the most significant mathematicians in that shared culture?

First and foremost are probably Archimedes, who contributed to so many different areas of mathematics, both theoretical and practical, and Sir Isaac Newton, who was the co-inventor, with Leibniz, of the calculus, and whose ‘Principia Mathematica’ explained how the planets move under the universal law of gravitation. Many people would then add Carl Friedrich Gauss, while others might mention Euclid of Alexandria, whose ‘Elements’, sometimes credited as the most published book of all time after the Bible, was studied for some 2000 years. My own list would be Archimedes, Newton and Euler.

In your new book you call Euler’s equation ‘the most beautiful theorem in mathematics’. You’re clearly quite a fan!

Leonhard Euler was an 18th-century Swiss mathematician who spent most of his career in the scientific academies at St Petersburg and Berlin. The most prolific mathematician of all time, Euler published over 800 books and papers in over 70 volumes. Ranging across almost all branches of mathematics and physics at the time, these amounted to about one-third of all the physical science publications of the 18th century.

Euler’s equation elegantly and profoundly combines five of the most important numbers in mathematics: 1 (the basis of our counting system), 0 (the ‘nothingness’ number), π (the basis for our study of circles), e (the number associated with exponential growth), and i (the ‘imaginary’ square root of minus 1): it states that if we raise e to the power (i times π), and add 1, we get 0. It was Euler who showed how the introduction of imaginary numbers connects these seemingly unrelated numbers.

The equation is also of importance in physics and engineering, where it features in such subjects as quantum mechanics and image processing. As a physicist was moved to remark: ‘What could be more mystical than an imaginary number interacting with real numbers to produce nothing?’

In my book, I devote a chapter to each number, describing how they developed historically, before showing in the final chapter how they were combined into ‘Euler’s pioneering equation’.

Your first choice is Ronald Calinger’s biography of Euler. What is particularly valuable about this book?

There have been several biographies of Euler. Some have concentrated on his life without attempting to come to grips with his mathematical contributions. Others, written mainly for mathematics students, have been more mathematical in nature and don’t give a thorough or accurate idea of his life. Ron Calinger’s substantial book of over 650 pages aims to introduce the reader to both aspects, and is full of useful and interesting material.

For some time I’ve been working on a much shorter book about Euler, designed to present both the biography and the mathematics to an interested general public. Calinger’s book is a most useful reference work for this purpose, and I certainly recommend it as the book on Euler for anyone interested in finding out more about this fascinating mathematician.

Dirk Struik’s A Concise History of Mathematics is your second choice. What singles this book out for you in what is, presumably, a large field?

I once attended a lecture by Dirk Struik, delivered with great clarity and without notes – a remarkable performance, given that the speaker was then 104 years old! Although there are many fine books on the history of mathematics, such as the excellent texts of Victor Katz (‘History of Mathematics: An Introduction’) and David Burton (‘The History of Mathematics’), I’ve always had a soft spot for Struik’s down-to-earth and straightforward paperback, which first appeared in 1948 and which I regard as a classic. Indeed, when I once taught a history of mathematics course in the USA, I was both surprised and delighted when some students declared that they preferred it to the fuller and more up-to-date treatments of the subject to be found in more recent works.

What distinguishes your third choice, The History of Mathematics: A Reader by J. Fauvel and J. J. Gray?

The history of mathematics can be studied and taught in different ways. In the past many people took the traditional ‘who-did-what-and-when?’ approach, but more recently there’s been an increased emphasis on putting mathematics into the context of the time. This was the approach taken by the Open University’s pioneering course MA290: ‘Topics in the History of Mathematics’, which ran from 1987 to 2007, and which was heavily based on translated original source material from this source book, edited by two distinguished members of the course team. Over the years there have been several good source books in the history of mathematics, but I particularly like this one, especially for the earlier material, and another one called ‘Mathematics Emerging’ by Jacqueline Stedall, covering the period from 1540 to 1900.

How does it help students to read original source material, presented in volumes like these?

Many courses on the history of mathematics describe what mathematical results have been discovered, but the student has little chance to explore these discoveries ‘from the inside’. A good source book provides a wide range of original sources (usually in translation and edited as necessary) which enable us to see the problems solved in the context of their time.

Let me give you a couple of examples. First, the origin of quadratic equations. It’s all very well to say that a Mesopotamian (Babylonian) clay tablet 4000 years ago ‘solved these equations’ (a statement that’s regularly heard), but until students work through the statements of the problems and the details of the calculations on the tablet, they don’t really understand exactly what problems were being solved, and why, and what form the solution took.

Second, when undergraduate students first study mathematical analysis (sometimes described as ‘calculus done properly’), they often find it difficult to see the need for introducing the particular technicalities involved (such as ‘epsilons and deltas’). Looking at the original works of such mathematicians as D’Alembert, Cauchy, and Bolzano, helps us to understand how and why this particular approach arose.

Other reasons for studying original sources are that they’re fun to try to sort out, and that they provide contextual interest for students learning the material in their mathematics courses. Jackie Stedall’s source book, ‘Mathematics Emerging’, in particular, reproduces the original works as they first appeared, followed by a translation into English (where needed) and a commentary.

Your fourth choice is ‘The Mathematical Pamphlets of Charles Lutwidge Dodgson and Related Pieces’. Where have I heard that name?

Charles Dodgson is better known as ‘Lewis Carroll’ the author of the ‘Alice’ books, and also as one of the best photographers of the Victorian era than as a mathematician. But his day job for 25 years was as a lecturer in mathematics at Christ Church. Oxford. as I explain in my book Lewis Carroll in Numberland, which written for a general audience and is one of the books I’ve most enjoyed writing, Dodgson’s mathematical work covers several different areas. He was a great expert on, and enthusiast for, Euclid’s geometry, spending much of his time in teaching it to undergraduates and championing it in his popular book Euclid and his Modern Rivals.

Dodgson was also keen on algebra, writing an innovative book on the theory of determinants, and one of his main ideas has become a topic of current research. He always enjoyed introducing his ideas on symbolic logic to adults and children alike. And he was a pioneer in the theory of voting, on which he made substantial contributions which we would do well to adopt today. A supporter of proportional representation, he illustrated the deficiencies in our first-past-the-post and other voting systems, and his studies would have usefully informed the ‘alternative vote’ discussions in Britain a few years ago. His mathematical work in all of these areas is the basis of a multi-authored volume which I’m currently in the process of editing.

I chose this book of Mathematical Pamphlets because it includes a wide range of his mathematical writings and gives one a good idea of his approach to the subject. It is also a fun book to dip into!

Your final choice is ‘Mathematical Models’ by H. M. Cundy and A. P. Rollett. Why does this book stand out for you?

I first came across this book more than fifty years ago when I was still at school. It introduced me to a world of ‘fun’ mathematics, such as the different types of tessellation (tiling), the nets for making polyhedra, the sketching of interesting curves, and much else besides. It proved particularly valuable when I had to organise speech-day mathematical exhibitions for the visiting school parents. Even now, I sometimes go back and re-read it just for fun – and if mathematics isn’t fun, then it’s not worth doing!