Looking back is, of course, what historians do. The title of this book is a play on that, because in addition to looking back on history it also gives us a “look back” at the author’s own history as one of our best historians of mathematics. It contains an updated version of her first book, The Calculus as Algebra, plus several articles on the history of calculus, mostly focused on the crucial transition to “rigor” in the late eighteenth and early nineteenth centuries. Grabiner has always felt that it was part of her mission as historian to communicate with mathematicians, both to teach them about the history of their field and to offer pedagogical reflections inspired by that history. This desire seems to be the major criterion for choosing the articles included in the book.
The first one hundred pages of A Historian Looks Back contain The Calculus as Algebra, which is based on Grabiner’s 1966 Ph.D. thesis. It focuses on the work of J. L. Lagrange, who tried to “reduce the calculus to algebra” and so free it of “any consideration of infinitely small quantities or evanescents, of limits or of fluxions,” all of which Lagrange considered vague and ill-founded. The central revelation of Grabiner’s analysis is that, far from being a failure, Lagrange’s “algebraic” calculus turned out to offer crucial ideas for Cauchy’s work. In particular, Lagrange’s work contains the “theorem” that Cauchy and others would later recast as a definition: the derivative of a function \(f\) at a point \(a\) is the unique real number such that \( f(a+h)\approx f(a)+f'(a)h\) with error that goes to zero faster than \(h\). In fact, Grabiner finds many of the techniques of epsilon-delta calculus in Lagrange, ready there for Cauchy (and later Weierstrass) to use them.
It is hard to find this information in the book, but the version of The Calculus as Algebra reproduced here seems to be the one from the edition published by Garland in 1990. There are some strange glitches, however. The “Preface to the Garland Edition” says that “a new bibliography, updating that in the 1966 version, follows this preface.” But it doesn’t. In fact, it seems to appear after the original bibliography, starting on page 119. Close inspection, however, shows that this cannot be the bibliography in the Garland edition either, because it cites books from after that date. So the bibliography has in fact been updated further, perhaps to 2010. The footnotes also seem to have been updated to refer to this bibliography. Bibliographers and historians of the history of mathematics will find all this confusing.
The book itself, however, is well worth reading. Grabiner is attentive to mathematical details and techniques, which makes her account of this period particularly interesting. She also pays close attention to context, filling in the background in illuminating ways. I found it particularly interesting to note how often Lagrange gets in trouble for failing to note the difference between \(<\) and \(\leq\), or how much he is hampered by the lack of a good notation for absolute values. Some points I would like to examine further, such as Grabiner’s contention that the meaning of “convergence,” as applied to series in the 18th century, is not the same as the modern meaning.
The Calculus as Algebra is followed by a selection of articles from various periods in Grabiner’s career. Here too there is a weird production glitch: “The Mathematician, the Historian, and the History of Mathematics,” (reproduced from Historia Mathematica) contains references to a non-existent bibliography that left me frustrated. It’s a very interesting article, nicely bookended by another article on the relationship between historians and mathematicians, “Why Should Historical Truth Matter to Mathematicians? Dispelling Myths while Promoting Maths.” Since “The Mathematician etc” is from 1975 and “Why Should etc” is from 2007, they provide an interesting frame for the other articles.
Most of the articles clearly fit into the same overall project as The Calculus as Algebra, focusing on the historical “transition to rigor” in the calculus. The main characters are Lagrange, Cauchy, and Maclaurin. Grabiner is always very interested in dispelling historical myths. Her main targets here are “Lagrange was a formalist” and “Maclaurin’s Treatise of Fluxions was a dead end.” The first is effectively disposed of in The Calculus as Algebra, but gets mentioned several other times in the articles; the second is the focus of several more recent articles on Maclaurin. Alas, Grabiner’s article on Maclaurin among the molasses barrels is only mentioned, not included. It is worth seeking out.
The last article seems somewhat out of place here. “Why Did Lagrange ‘Prove’ the Parallel Postulate?” focuses on the history of geometry and on the philosophical question of the relationship between geometry and physical space rather than the history of the calculus. Uncharacteristically, Grabiner leaves some issues unclarified in this one. For example, she never explains why Lagrange’s argument is wrong, speaking instead of the consensus, at the time, that physical space must be infinite, homogeneous, and isotropic. Standard hyperbolic space is infinite, homogeneous, and isotropic, so that cannot be the issue here. On the other hand, the article is very good on the cultural background and on the philosophical positions that drove the conversation.
This book is not just for historians: there is much to learn here that can affect how we teach our students. It has already done that for me, in fact. It will now find its place next to Grabiner's The Origins of Cauchy’s Rigorous Calculus, another excellent book.
Several of these articles received awards and are both well-known and widely reprinted. There are probably many people, however, who still do not know “Who Gave You the Epsilon?” and are unfamiliar with “The Changing Concept of Change,” who have never asked themselves “Was Newton’s Calculus a Dead End?” They can now find these articles here, and others similar to them, all worth reading and re-reading.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College.