# Significant Figures

Ian Stewart has found yet another way to bring mathematics to a broad public. After over 40 books in his well known entertaining style, he is now writing a selective history of mathematics, not using the numbers or the mathematics as the main players, but this time it are the mathematicians of all ages that are the significant figures in this case.

He has selected 25 mathematicians starting with Archimedes and ending with William Thurston. The thread connecting both consist of an impressive list of names all of them identifying famous mathematicians. Since most popular books on the history of mathematics are European-centered, perhaps the first names following Archimedes are a bit less familiar. The jump is usually made from the Greek whose work came to us in the form of Arab translations to Fibonacci. The latter also popularized the positional number system brought to us by work of al-Khwarizmi. This Fibonacci is just a stepping stone to bring us to the Renaissance with Italian, French, and German mathematicians whose more familiar names are echoing in the formulas and theorems that are still in use today.

However, what the Arabs brought was not only the Greek tradition. They also inherited from Chinese and Indian mathematical culture. Hence Stewart rightfully introduces two exponents of these cultures too. Liu Hui (3rd century) was one of the most important Chinese mathematicians of his time. The Chinese also had geometry, including the Pythagorean Theorem and they knew a rather accurate approximation of *π*. Al-Khwarizmi lived around 780-850 and his *al-jabr* is the origin of our name for algebra. Even though that stands nowadays mainly for symbolic manipulation of mathematical quantities, the al-jabr was verbal and arguments were mostly geometric. India is represented by Madhava of Sangamagrama (1350-1425). He is the founder of the Kerala school, in the South-West of India. They had trigonometry and infinite series. Kerala was a common stopping place for long distance navigation. There is no evidence though that their mathematical ideas were directly brought to Europe by sea travelling traders. In any case, they had results that were only discovered by European mathematicians much later.

The early mathematics of the East and their influence on European mathematics were exposed only recently by the books of George Gheverghese Joseph, in particular *The Crest of the Peacock* (2010). Of course the names discussed in Stewart's book are just representing a whole culture and he does not restrict to just these particular men, but also comments on their background, some of their contemporaries, and their heritage. The same holds for the other "figures" in the chain connecting Archimedes and Thurston. The account given for each of them is forced to be fragmentary. With an average of about 10 pages for each, not much room is given for an extensive biography and a discussion of their mathematical contribution. So we get some executive summary for Cardano, Fermat, Newton, Euler, Fourier, Gauss, Lobachevsky, Galois, Ada Lovelace, Boole, Riemann, Cantor, Sofia Kovalevskaia, Poincaré, Hilbert, Emmy Noether, Ramanujan, Gödel, Turing, Mandelbrot, and ending with Thurston.

This list of names is disputable of course. Every selection is subject to controversy. And what is told about each of them is again just a selection, because there exist much more extensive biographies for each of them. Stewart lists them at the end of the book as references for further reading. The mathematician is situated, sometimes introduced with a short sketch (Gauss deciding to choose for mathematics instead of languages after detecting how to divide a circle in 17 equal pieces with ruler and compass, the newspaper announcing the death of Galois after a duel, Hardy receiving his first letter from Ramanujan,...), followed by a short biographic summary, and some discussion of his or her work. In some cases, for the more prolific specimen, discussing only some particular element of it.

There is a lot of folklore floating around about historical facts. Stewart is very good in busting several of these myths. In this respect he discusses the motive and the opponent in Galois' duel about which there is some controversy. He unravels the dispute about the priority of discovering and the disclosure of the formula for solving the cubic equation between Cardano and Niclolo Fontana (known as Tartaglia, the stammerer). There is also the story about the taxicab number. Hardy claimed that 1729, the number of his taxi, was boring and thus a bad omen. But Ramanujan immediately recognized it as the first number that can be written as the sum of two cubes in two different ways. Stewart claims that this was probably a set-up by Hardy, trying to cheer up his sick friend. For a mathematician, especially for a number theorist like Ramanujan, it would not be difficult to immediately recognize 1728 as the cube of 12 and that this number is also 1000 (10 cubed) plus 729 (9 cubed). And Stewart places question marks after some other myths.

Of course all of these figures stand out in one way or another. After reading what they have achieved, one can only sit back in awe. There are throughout the ages mathematicians of all sorts. Some were poor, some were rich, some were religious, other were politically engaged, some were child prodigies, and some blossomed at later age. Some were not even professional mathematicians like Fermat, who was a lawyer, just fond of mathematics. Some were very applied, others worshipped the pure stuff. With this sample (although limited) of great mathematical minds it is tempting to ask whether there is some common ground, some way of stimulating the development of extraordinary mathematical skills. Stewart concludes there is none that we could influence. Some just have it, others don't.

This is a wonderful read authored by one of the best in this genre. Mathematical knowledge is not explicitly needed, but the reading will be best appreciated if there is a minimal background (certainly for the mathematicians active in the 19th and 20th century) but with some love for mathematics and a bit of interest in its history you will savour the text from the first till the last page.

**Submitted by Adhemar Bultheel |

**20 / Jul / 2017