Histories of mathematics have been written to serve various purposes. Some are directed toward use in the classroom. Others serve as reference works. In this latter category resides The History of Mathematics by Florian Cajori (1859-1930). That Cajori’s book has enjoyed numerous reprints and editions over a time span of nearly one hundred years speaks to its continuing value as a resource for mathematicians and historians of mathematics.
The original book appeared in 1893 (436 pages). A second, revised and enlarged edition (524 pages) covering the period from antiquity to the close of World War I was published in 1919, and was reprinted in 1931, 1942, 1948, 1949 and 1960. Subsequent editions are all based on the second and have approximately the same number of pages. A third edition, published in 1980, included new information and notes, as well as additions to the index. The fourth and fifth editions (1985, 1991), edited by “A. G.”, correct some errors in earlier versions and offer a revised chapter on Babylonian mathematics with references to several texts by O. Neugebauer and A. Sachs.
The first hundred or so pages of the fifth edition cover the mathematics of the Babylonians, Egyptians, Greeks, Romans, Mayans, Chinese, Japanese, Hindus and Arabs. Some sections try to provide an overview that spans antiquity to the early twentieth century, while others treat more restrictive periods.
The next one hundred or so pages are devoted to the contributions of European mathematicians from the middle ages through the end of the eighteenth century. The last two hundred or so pages partition nineteenth and early twentieth century mathematics into the following categories: definition of mathematics; synthetic geometry; analytic geometry; algebra; analysis; theory of functions; theory of numbers; and applied mathematics. Ten pages of editor’s notes include clarifications, corrections, and updates to the author’s material.
The 27 page index, which consists largely of names, gives the reader a sense of the breadth of research done by Cajori. There is no bibliography because cited works are listed within the text. They can be referenced in the index by author’s name.
The reception of the different editions of the book is interesting to trace, not only to understand the critiques of the work in the context of different eras, but also to obtain a glimpse of the interactions among mathematicians and historians of mathematics of the times. Miller (1921) addresses some problems in Cajori’s second edition and compares Cajori’s work to a comparable history written by Cajori’s contemporary W. W. R. Ball (1850-1925).
A search of JSTOR produces reviews of every edition of Cajori’s book except the third. The review of the first edition by Halsted (1984) should be followed by a reading of Cajori’s obituary for Halsted (1930) in which the author related some of his experiences with Halsted. A nice discussion of the second edition by Cajori’s contemporaries Mordell (1921) and Smith (1920) explain the additions to the first edition, and reveal topics of concern during that era.
To Ashurst’s nice review of the fifth edition (1994), in which he calls it a “gem of a book”, this reviewer would only comment on the following useful feature of The History of Mathematics: Cajori’s extensive research appeals to the publications of his era (rather than his search of original documentation). Because resources are cited within the discussion of a specific topic, the reader can easily connect various treatments and get a sense of the prevailing understanding of the topic at that time.
Archibald, Raymond Clare. Florian Cajori 1859-1930. Isis Vol 17, No. 2, (1932), pp. 384-407.
Ashurst, F. Gareth. Review [Untitled] Reviewed Work(s): A History of Mathematics (fifth edition) by Florian Cajori. The Mathematical Gazette, vol. 78, No. 483 (Nov., 1994), pp. 361-362.
Ball, W. W. R. A Short Account of the History of Mathematics. New York: The MacMillian Company, 1888.
Cajori, Florian. George Bruce Halsted. The American Mathematical Monthly. Vol. 29, No. 9 (Oct., 1922), pp. 338-340.
Halsted, George Bruce. Review: Cajori’s History of Mathematics. Reviewed Work(s): A History of Mathematics (first edition) by Florian Cajori. The Mathematical Gazette. No. 3. (Dec., 1894), pp. 19-20.
Miller, G. A. Different Types of Mathematical History. Isis. Vol. 4, No. 1 (May, 1921), pp 5-12.
Mordell, L.J. Mathematicians and their Work. The Mathematical Gazette, Vol. 10, No. 154. (Oct., 1921), pp. 321-323.
Pritchard, Chris. Review: [untitled] Reviewed Work(s): History of Mathematics (fourth edition) by Florian Cajori. The Mathematical Gazette, Vol. 72, No. 463. (Mar., 1989) pp. 66-67.
Smith, David Eugene. Review: Cajori’s History of Mathematics. Reviewed Work(s): A History of Mathematics (second edition) by Florian Cajori The American Mathematical Monthly. Volume 27, No. 3 (Mar., 1920), pp. 120-127.
Taylor, James M. Review: [untitled] Reviewed Work(s): A History of Mathematics (first edition) by Florian Cajori. The School Review, Vol. 2, No. 8 (Oct., 1894), pp. 513-514.
Elena A. Marchisotto is Professor of Mathematics at California State University, Northridge. She is the author, with James T. Smith, of The Legacy of Mario Pieri in Geometry and Arithmetic.