Vera Sanford’s A Short History of Mathematics: A Mathematical Treasure – The Textbook

Sanford’s A Short History of Mathematics is a textbook covering the history of elementary mathematics. The book is part of a series on mathematical topics that was published by Houghton Mifflin under the editorship of Dartmouth mathematician John Wesley Young [Simons 1931, p. 124]. The edition we examined was printed in 1930 by Houghton Mifflin. There were printings by at least two different publishers: Houghton Mifflin for the United States and George G. Harrap & Co. Ltd. for the rest of the world, but the contents of the two different printings appear to be the same, and both were printed in the United States.

The book begins with a short introduction by David Eugene Smith. In this introduction, Smith described the mathematics curriculum as a “flowing stream,” and he described the goal of the book as helping those teaching it to identify from the past the things that are relevant in the present and will be relevant in the future.

The introduction is followed by a short preface by Sanford. She mentioned that the book was primarily directed toward teachers and included illustrations, maps, and chronologies that would be useful to those teaching mathematics. The rest of the preface was used to acknowledge those who aided in the preparation of the text. Interestingly, in neither the preface nor the introduction is there an indication of how the book would be expected to be used, nor whether certain chapters were dependent on others.

The main text of the book begins with a lengthy chapter presenting a chronology of “Men who Made Mathematics.” Sanford focused this almost exclusively on Western mathematics and mathematicians. The exceptions are short discussions of Indian mathematicians and Islamic mathematicians.^[2] These primarily emphasized those cultures’ contributions to numeration, with brief mention of their work with quadratics and simultaneous linear equations. Sanford also noted stylistic differences between the writings of the Indian mathematicians, whose works she described as containing poetic quality and fanciful problems, and the Western mathematicians who came after them [Sanford 1930, pp. 18, 144, 211].

Next, Sanford turned her attention to specific mathematical topics of interest, which are described in various levels of detail. She devoted the most space to discussing arithmetic (including the development of numeration systems), commercial mathematics, verbal problems, algebra, and geometry. Given that her presumed audience was pre-service and in-service teachers, her concentration on the development of school mathematics makes sense. She also included some reflections on the role of mathematics in the school curriculum, but it is surprising that she did not spend more time discussing this, given her audience. Sanford also included extensive material on calculating devices that would have been of practical interest to her teacher audience.

The volume’s other topics comprised calculus, number theory, and weights and measures. Sanford concluded the book with a list of references to more extensive general histories of mathematics and, finally, a chronological outline of mathematical history.

A Short History of Mathematics contained numerous illustrations, maps, portraits of mathematicians, and tables, many of which came from the collection of David Eugene Smith (see, for example, Figures 5, 7, and 8). The images below come from the copy owned by co-author Toke Knudsen; they are available for use in your mathematics classroom with credit to photographer Kento Igarashi.

Figure 3. Finger notation from Luca Pacioli’s 1494 Summa de arithmetica, geometria, proportioni et proportionalita [Sanford 1930, pp. 76–77]. Photograph by Kento Igarashi. Used by permission.

Figure 4. Comparison of Hindu-Arabic numeral symbols, reprinted from Francis George Hill’s 1915 The Development of Arabic Numerals in Europe [Sanford 1930, pp. 94–95]. Photograph by Kento Igarashi. Used by permission.

Figure 5. Late use of Roman numerals in a 1602 commercial account book from England.
According to one of the reviews (see below), this is the first appearance of the figure
in scholarly literature [Sanford 1930, pp. 96–97]. Photograph by Kento Igarashi. Used by permission.

Figure 6. The first appearance in print, in Peter Apian’s 1527 treatise on arithmetic, of an illustration of
Pascal’s triangle [Sanford 1930, pp. 178–179]. Photograph by Kento Igarashi. Used by permission.

Figure 7. Photographs from Smith’s collection, included in the discussion of astrolabes [Sanford 1930, pp. 242–243]. Photograph by Kento Igarashi. Used by permission.

Figure 8. Mathematical tools: compasses and Napier’s rods [Sanford 1930, pp. 338–339].
Photograph by Kento Igarashi. Used by permission.

We found four published reviews of the textbook [Cowley 1931; Gilham 1931; Simons 1930; Simons 1931]. All three reviewers—Lao Genevra Simons (1870–1949), another student of Smith and notable historian of mathematics in her own right, wrote two separate reviews—were generally positive, assessing the book as filling its own niche among the existing textbooks at the time. They all mentioned that A Short History of Mathematics would be a good book for teacher training as well as for use by practicing teachers who were looking to further their insight into the history of mathematics. All four reviews also discussed the abundance of illustrations used in the book. Simons [1930] disclosed the fact that several of the illustrations used in the book were published here for the first time (see, for example, Figure 5). Some of the unique features of the book that were pointed out by both Cowley and Simons were the inclusion of content covering commercial mathematics, verbal problems, and mathematics in the school curriculum.

A Short History of Mathematics was reprinted in 1958 and 1960, and copies of the printings—which were distributed in the United States and internationally—can still be found in many libraries today. This suggests it was widely read. At our institution, Sanford set the tone for mathematics education through her work in the department, her work with the elementary mathematics curriculum in New York State, and the publication of her books. SUNY Oneonta’s significant collection of American mathematics textbooks is named after her (the Vera Sanford Collection of Mathematical Works). The collection has expanded to include a broader range of historical mathematics books, as subsequent generations of professors have helped to augment it, in part with the assistance of V. Frederick Rickey and Florence Fasanelli, who for the past fifty years have shown mathematicians how to employ history of mathematics in the teaching of mathematics. We encourage readers to pursue historical approaches in their own courses; this journal, MAA Convergence, is of course a great resource, as are organizations such as the MAA’s History of Mathematics Special Interest Group (HOMSIGMAA) and the International Study Group on the relations between History and Pedagogy of Mathematics (often identified as HPM). Finally, we hope instructors and students will look into the histories of their own departments to investigate the legacies that are offered there—you are likely to find a surprising number of enterprising women and other mathematicians from underrepresented groups among your predecessors!