A History of Mathematical Notations

Florian Cajori’s A History of Mathematical Notations may well be one of those books so well known that it doesn’t need a review. Originally published in two volumes in 1928–29, the book is an attempt to collect all the information available on various symbols and notations. The two volumes have been reprinted in one big book (without any sort of repagination) by Dover. Both volumes are now in the public domain and scans can almost certainly be found online, though scans are often hard to use. The archive.org site has the first volume (only) in pdf format.

Cajori explains that he has tried

to give not only the first appearance of a symbol and its origin (whenever possible), but also to indicate the competition encountered and the spread of the symbol among writers in different countries.

The first volume covers numeration, symbols in elementary algebra and arithmetic, and symbols used in elementary (Euclidean) geometry. The second opens with more advanced symbols from algebra and arithmetic, symbols from analysis, and finally with notations from more advanced parts of geometry. It is not always obvious, however, which symbols will appear where. For example, \(e\) and \(\pi\) are in the second volume, as is the dollar sign. Trigonometry is also “more advanced.”

In a book like this the detailed table of contents and the index are essential. Unfortunately, there are two of each, one for each volume. In this edition the contents at the beginning are those of the first volume, and after the 451 pages of that volume one finds the contents of the second volume. I have resorted to leaving a card in the spot where the first volume ends, so that I can quickly locate the index to volume 1 (just before) and the contents of volume 2 (just after). Users should note that the contents and indices do not refer to pages, but rather to the book's numbered paragraphs. It's a bit confusing at first, but has the advantage that the paragraphs are numbered sequentially throughout the two volumes and the pages are not. Most paragraphs are shorter than a page, so references to them are more precise than page references.

Most of the book is arranged topically. For example, the section of volume 1 discussing arithmetical and algebraic symbols deals in turn with “addition and subtraction”, then “multiplication,” and so on. But other parts seem to have required different arrangements. The chapter on numeration has two distinct parts, one on the number systems of various ancient cultures, the other on the history of the “Hindu-Arabic” numerals. The first volume has a long section that treats the symbols used by individual writers from Diophantus to Leibniz, a method that is repeated in the second volume’s discussion of the symbols of mathematical logic.

The book concludes with a long section on “the teachings of history,” where Cajori tried to figure out whether he has learned something from gathering this enormous amount of information. That he had some such goal in mind is signaled already in the introduction:

If the object of this history of notations were simply to present an array of facts, more or less interesting to some students of mathematics — if, in other words, this undertaking had no ulterior motive — then indeed the wisdom of preparing and publishing so large a book might be questioned. But the author believes that this history constitutes a mirror of past and present conditions in mathematics which can be made to bear on the notational problems now confronting mathematics. The successes and failures of the past will contribute to a more speedy solution of the notational problems of the present time.

I had never heard anyone say that mathematics in the early twentieth century had gone through some sort of notational crisis. What was Cajori worried about? The section on the “teachings of history” is supposedly his answer. It opens with a collection of quotations from various authors on the need for a more detailed study of the history of mathematical notations and for a clearer awareness of the benefits of good and consistent notations. Then comes a section called “Empirical Generalizations on the Growth of Mathematical Notations,” an attempt to categorize and understand notations and their role. After a look at other disciplines, we finally get Cajori’s conclusion. The title of the section is “Agreements to be reached by international committees the only hope for uniformity in notation.” So in a way the whole book is an argument, a plea for some group to sit down and agree upon a uniform system of notation.

It never did happen, though many notations have been made uniform in other ways: some were adopted by influential writers (notably Bourbaki), others became agreed-upon first within certain research communities and then disseminated via a particularly influential textbook, and many were forced upon mathematicians simply by the need to understand each other.

Is Cajori’s book still useful, almost a century later? In many ways, it is. One of the great virtues of the book is that it reproduces, photographically or in typeset form, many of the notations it discusses. These samples remain very useful. On the other hand, one cannot take Cajori’s word any more when he asserts something about ancient numeration or about the first known use of a symbol. Cajori provides a starting point, from which one must go on to look at the literature and at original sources. Unfortunately, no more recent book covers the whole range, though some sections of the book can be replaced by more modern surveys. In the case of numeration, for example, there is Stephen Chrisomalis’s Numerical Notation: A Comparative History (Cambridge, 2010). As always when it comes to historical books, we must trust but verify.

Fernando Q. Gouvêa wonders where Cajori found the time.