https://ima.org.uk/11565/enlightening-symbols-a-short-history-of-mathematical-notation-and-its-hidden-powers/

From the outset let me say that this is a good book worthy of the attention of anyone interested in the history of mathematical symbols. Joseph Mazur is an experienced author who writes confidently so that the book reads more like a novel than a textbook. It is full of interesting facts about how the symbols used in mathematics have arisen and it would certainly interest anyone who studies the history of mathematics. One way of avoiding the obvious danger that the book becomes just a list of facts is to inject stories and the personality and opinion of the author into the book, and this Mazur does. Some may find parts of his prose just a little strident in places and take issue with the style and the content, but this reviewer thinks he has the balance just right. Joseph Mazur achieves this mostly through his engaging style but is also helped by breaking the book into short chapters, 24 of them, and avoiding being completely chronological. The first collection of chapters concerns the origins of the numerals themselves, some origins are of course shrouded in obscurity and controversy and where this is the case the alternatives are honestly treated by the author. One is left wondering how on earth civilization did not catch on to placement arithmetic and indeed the existence of zero before the Liber Abaci in the 12th century when they seemed to be surrounded by it for over a century; one can feel the exasperation of the author, a testament to the excellence of the writing. The unfortunate story of the eminent French mathematician Michel Chasles who was duped into believing that thousands of obviously fraudulent documents purporting to originate from such luminaries as Cleopatra, Alexander the Great and Isaac Newton (all writing in French!) is recounted. Michel Chasles’ difficulty determining what was true or false, hindered by his reluctance to credit India with anything original because in his 19th century eyes that part of the world was second or third rate, is evident. The second collection is about algebra and is consequently more about the culture of development and use. This contains a wealth of information and strays into anthropology and religion including more of the beliefs of the author. Researching some of the origins of algebra remains an active area and new forensic techniques are helping to disentangle who did what when. There is a considerable amount of detail here, and once again the author is not shy of pointing out what he sees as the mistakes in others and criticising, gently, the wrong turns they made. The final part on the development of calculus is really superb, avoiding the usual fatuous guesses of who might have done what first it concentrates on a description of the personalities and details of the notation. The third and final section on the power of symbols is more philosophical and is mainly about how we think when we see and do mathematics. A lot is about the psychology of perception and may not be to everyone’s taste. It is still a good read though. What we have here is a scholarly book written by an author in command of his subject. The book is well structured and the short chapters make it a useful reference book for students and scholars alike. It is written primarily for the layman, but some knowledge of algebra and complex numbers is useful; the lack of which would not prevent understanding or enjoyment but might discourage potential readers from picking this as a holiday read. However doubtless there will be readers already steeped in the history of mathematics that will find new material here amongst the wealth of detail that the author presents. Phil Dyke FIMA Book review first published in Mathematics Today June 2018 Published 28th March 2019