Florian Cajori: A History Of Mathematical Notations

I just noticed that over at archive.org they have Volume I of Florian Cajori’s A History Of Mathematical Notations. There’s a fair chance this means nothing to you, but, Dr Cajori did a great deal of work in writing the history of mathematics in the early 20th century, and with a scope and prose style that still leaves me a bit awed. (He also wrote a history of physics; I remember reading the book, originally written in the mid-1920s, with his description of one of the mysteries of the day. With the advantage of decades on my side I knew this to be the Zeeman effect, a way that magnetic fields affect spectral lines.)

Archive.org has several of Cajori’s books, including the histories mentioned, but Mathematical Notations I like as it’s an indispensable reference. It describes, with abundant examples, the origins of all sorts of the ways we write out mathematical ideas, from numerals themselves to the choices of symbols like the + and x signs to how we got to using letters to represent quantities to something called alligation which was apparently practiced in 15th-century Venice.

Unfortunately archive.org hasn’t yet got Volume II, which includes topics like where the $ symbol for United States currency came from — Cajori had some strong opinions about this, suggesting he was tired of tracking down false leads — but it’s a book you can feel confident in leafing through to find something interesting most any time. I think his description of the way historical opinions had changed particularly fascinating, and recommend particularly Paragraph 96 (pages 64 through 68 of the book, and not one enormous block of text), describing “Fanciful hypotheses on the origins of the numeral forms”, many of them based on ideas that the symbols for numbers contain the number of vertices or strokes or some other mnemonic to how big a number is represented. Of those hypothesis formers he says, “Nor did these writers feel that they were indulging simply in pleasing pastimes or merely contributing to mathematical recreations. With perhaps only one exception, they were as convinced of the correctness of their explanations as are circle-squarers of the soundness of their quadratures”.

Dover publishing, of course, reprints the entire book on paper if you want Volumes I and II together. I admit that’s the form I have, and enjoy, since it becomes one of those books you could use to beat off an intruder if need be.


Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

3 thoughts on “Florian Cajori: A History Of Mathematical Notations”

  1. Thanks – this is very interesting! I think the significance of notation’s details and it interdependence with culture is often underrated. I have recently learned, for example, that in the pre-letterpress times scribes used to write without blanks… sort of a mapping of spoken languages (without blanks between the words) to paper. I can vaguely recall some stories about the competing notations of Newton and Leibniz… and I wondered if it changes our understanding of calculus whether we write little dots or curly d’s.


    1. Yeah, my understanding was that spacing between words (and, for that matter, including vowels in the written text) are relatively new compared to the history of all writing, and is part of what makes deciphering older texts challenging. (I’m too cowardly to face original manuscripts, myself.)

      I am curious how the different calculus notations change our understanding of what’s represented. There are a couple of formats — I’m thinking particularly of like \partial_x u — that I’ve never warmed to but I do see people feeling comfortable with.

      For my money, though, the most fascinating parts of Cajori’s book are the experimental phases in notation. The attempts to write decimals, particularly, had fascinating quirks, such as (I think) Stettinus’s notation of writing, for the digits past the decimal point, a circled number showing how many digits past the decimal you were. So, for example, he’d write “2.7182” with a circled (1) above the 7, and a (2) above the 1, and a (3) above the 8, and a (4) above the 2. That’s an awfully longwinded notation, and it seems to make sense mostly as an educational one, showing what it is you mean this thing to represent, than as something you’d do for practical purposes. It might still be useful as an educational tool, come to think of it.


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