A Short Account of the History of Mathematics

How can a 500 page book lay claim to be a “short account” of anything? Well, if we consider the sweep of mathematics through recorded history, it starts to look like a reasonable appellation, especially if we factor in that the author is a Victorian don from Trinity College, Cambridge, i.e., a scholar from the old school, when academic publications did not appear at a pace and in such numbers as to boggle the mind of even the narrow specialist. Consider in this connection, for example, that a Google search for “history of mathematics” yields on the order of 16,800,000 hits: enough to make a Victorian don take up … well, how about beekeeping?

Anyway, the book under review, suggestively titled A Short Account of the History of Mathematics, being the Dover rendering of the 1908 fourth edition of this book, whose first edition appeared two decades earlier, is an example of both scholarship and pedagogy of an age gone by.

The first edition was substantially a transcript of some lectures which I [W. W. Rouse Ball, the author] delivered in the year 1888 with the object of giving a sketch of the history, previous to the nineteenth century, that should be intelligible to any one acquainted with the elements of mathematics. In the second edition, issued in 1893, I rearranged parts of it, and introduced a good deal of additional matter.

Well, this is actually an understatement: the new “scheme of arrangement,” as WWRB puts it in his Preface, engenders a split of the 500 pages of the book into a first half devoted to the era up to and including the Renaissance, with the remaining 250 pages or so devoted to the history of the subject from Huygens and Descartes through the 19th century. In fact, this second half, titled “Third Period. Modern Mathematics,” is really worth the price of admission in its own right (and, given that we’re dealing with Dover, it’s a steal!), in that it includes as its last chapter an appraisal of the mathematics of the 19th century done at its close, by a major scholar in the history of the discipline: it doesn’t get any better than that!

Specifically, Chapter XIX starts with a survey of number theory with Gauss, Dirichlet, Eisenstein, Smith, and Kummer; proceeds to automorphic functions (“Development of the Theory of Functions of Multiple Periodicity”) featuring Abel, Jacobi, and Riemann; then goes to the theory of functions and “Higher Algebra” with Cauchy, Hamilton, Grassmann, Galois, Cayley and Sylvester, Lie, and Hermite. And there’s more: “Synthetic Geometry” (where we meet Steiner and von Staudt), non-Euclidean geometry, and then, in proper 19th century British style, a good deal of applied mathematics. (Where’s Dedekind? Well, he does appear, but doesn’t get headline billing: perhaps this just underscores the fact that in certain major ways he was really a 20th century figure.)

Now, in point of fact, WWRB’s 19th chapter weighs in at less than 60 pages! How does one cram this many players’ contributions into such a compact orbit? Well, it’s a “short account,” after all, and we find, e.g., the following on Galois in toto:

A new development of algebra — the theory groups of substitutions — was suggested by Evariste Galois, who promised to be one of the most original mathematicians of the nineteenth century, born at Paris on October 26, 1811, and killed in a duel on May 30, 1832, at the early age of 20. The theory of groups, and of invariant subgroups, has profoundly modified the treatment of the theory of equations. An immense literature has grown up on the subject. The modern theory of groups originated with the treatment by Galois, Cauchy, and J. A. Serret (1819–1885), professor at Paris; their work is mainly concerned with finite discontinuous substitution groups. This line of investigation has been pursues by M.E. C.Jordan (1838-1922) of Paris and E. Netto of Strasbourg. The problem of operations with discontinuous groups, with applications to the theory of functions, has been further taken up by (among others) F. G.Frobenius of Berlin, F. C. Klein of Göttingen, and W. Burnside formerly of Cambridge and now of Greenwich.

Taking this entry as an exemplar, we can easily infer one of the proper uses for the book under review in today’s university classroom: it could obviously serve as a source book of sketches for more expansive treatments done elsewhere, e.g. in such texts as Boyer-Merzbach or Katz.

However, there’s much more to WWRB’s Short Account than that. The earlier eighteen chapters of the book correspond, after all, to the author’s Cambridge lectures on the subject and, accordingly, are much more than telegraphic sketches: consider, for instance, that the 16th chapter is devoted to Newton, and the 17th to Leibniz and “the mathematicians of the first half of the eighteenth century.” Obviously today’s professor faced with pre-19th century material to present to his history class can do a great deal, building on what Rouse Ball presents here. To be sure, we do not encounter as many worked out examples and specially devised problem sets in these pages as we’d find in more recent books, but we do find a marvelous scholarly discussion of themes of major interest. (And WWRB does present us with various interesting illustrations; consider, e.g. his marvellous discussion of Archimedes’ quadrature of the parabola on pp. 67–70, in the dozen page subsection of Chapter 4 devoted to Archimedes.)

Thus, A Short Account of the History of Mathematics is a treasure trove for a number of reasons. It certainly serves two roles in connection with our classroom work in the subject: wonderful renderings of some of the older themes, and good sketches of the 19th century material. Indeed, as one marches back in time, leafing through the book’s pages, the amount of detail the author includes seems to increase proportionally. Well, it can’t really be otherwise for a historian, I suppose.

In addition it is worth noting that the book is in its own way somewhat encyclopædic: Chapter 1 covers Egypt and Phoenicia; “First Period” covers the Greeks and their considerable influence; “”Second Period” deals with the Middle Ages and the Renaissance; “Third Period” is suggestively titled “Modern Mathematics.” This clearly displays the high quality of Rose Ball’s scholarship and illustrates the persistent utility of the book in today’s Mathematics departments. A Short Account of the History of Mathematics should serve well as a supplementary text in any history of mathematics course — like the one I am teaching in the Fall.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.