As it is typically practiced today, the history of ancient Greek mathematics is a history of results and the resources or techniques used to get them, and, when its practitioners do attempt to write about the historical circumstances of the ideas they study, too often they fallaciously confuse their logical reconstructions with past realia. The reason for this, I suspect, is not just that many of the source materials available lack any information about their authors and settings—and so by their nature would seem to direct our attention to results and deductive structure alone—but that many historians of mathematics have not fully separated their subject from mathematics proper. Fortunately, there are recent signs of a major change in how the history of Greek mathematics is to be written. Reviel Netz, for instance, has brought to light valuable information about the cognitive practices constituting what it meant to do mathematics in antiquity by paying close attention to the language in which ancient mathematical argumentation is expressed and the role of diagrams. Serafina Cuomo would have us move even farther from previous work in the field by interpreting ancient mathematical output as a product of human activity with intellectual and social agendas and contexts. The work she analyzes in her excellent book is the Collectio by Pappus of Alexandria (4th century AD a miscellany of eight books on topics ranging from arithmetic and geometry to astronomy and mechanics.
The argument opens with an attempt to define the range of ancient professions that had an interest in mathematics at some level, in the hope of showing that there was a living background or setting for Pappus’ work. As Cuomo would have it, this requires determining who saw their work and expertise as having anything to do with mathematics, and whether they used this connection to construct their professional self-image and demarcate their expertise from that of others. This is a distinct project pursued only in the first chapter of Cuomo’s book and, so far as I am aware, it is new one for historians of mathematics. Indeed, I suspect that fully carried out this project would constitute an important book by itself.
Cuomo has much of value to say about the numerous professions in late antiquity that saw themselves as relying on mathematics and presented this reliance as part of their value, or were seen this way and valued accordingly. But, as it stands, her account needs a more focused elaboration in order to connect Pappus with this living background. After all, if one is going to talk about this kind of setting for Pappus’ work, one ultimately has to concentrate on the fourth century AD when he was writing. And one should give preference to sources concerning Alexandria if we suppose, as Cuomo does (see pp. 5-6), that this is where Pappus was active. This means, in particular, that the diverse materials that Cuomo introduces from the first to third centuries and from the fifth to sixth centuries AD, though extremely interesting and thought-provoking, are not relevant prima facie and, hence, that the validity of their use in understanding Pappus and his circumstances should be carefully demonstrated rather than taken for granted. To give us real information about the relevant contexts in which Pappus was writing, the project announced in this chapter would require us to amplify the remarks drawn from astrologers such Firmicus Maternus (pp. 10-16), the stipulations in Diocletian’s Edict of AD 301 of wages for various professions (pp. 30-31), the laws promulgated in the fourth century AD about the fiscal responsibilities and immunities of these professions (pp. 40-41), and the claims about the importance of mathematics and mathematical education made by philosophers and theologians of that time (pp. 48-50). More too would have to be said about contemporaneous mathematicians (p.52) and the formation of schools and intellectual traditions (pp. 54-55).
In the next chapter, Cuomo turns from this, shall we say, ‘social history project’, to the Collectio itself and what it tells us about how Pappus positions himself in relation to his contemporaries and predecessors. She begins with book 5 of the Collectio, a book which concerns the so-called Platonic solids and issues of isoperimetry, and which is probably addressed to non-mathematicians with a vital interest in these issues, that is, to contemporary philosophers of the Pythagorean and Platonic schools. She argues persuasively that Pappus attempts in this book to present himself as the key representative or authority in a discipline, mathematics, which he regards as superior to philosophy because it proves what philosophers take for granted, paradigmatically, that the sphere is the largest of all regular polyhedra with the same surface area. This agenda, she explains, underlies many features of Pappus’ exposition in book 5, notably his creative manipulation of the works of his predecessors with its resultant formation of a tradition of authority, his claim that the reader need not consult these past works directly but may learn from him, the unusual order of his demonstrations—the lemmata required are often proved after the fundamental result is presented—and his own seemingly professional concern with cases and subcases.
Chapter 3 focuses on the presentation of the science of mechanics in book 8 of the Collectio. Pappus defines mechanics as the science of the rest and motion (both natural and contrary to nature) of bodies in the universe, and distinguishes the theoretical part which embraces the mathematical and natural sciences from the practical part which involves craft. As Cuomo shows, Pappus’ account of mechanics serves to incorporate previous views of what mechanics is in such a way that their differences are at once recognized and mitigated, thus allowing him to present himself as a leading proponent of a science with a tradition that goes back to Archimedes and includes Ptolemy. And this at a time when, in spite of Pappus’ assertion that philosophers hold mechanics in the highest esteem, there was established in some quarters the view that mechanics is inferior to geometry and should be segregated from it. This chapter elaborates the significance of Pappus’ definition of the various mechanical sciences and his systematizing of the fundamental mechanical problems. It concludes with insightful remarks about the virtue of utility that mechanics was thought to have. Particularly helpful on this score is Cuomo’s observation that to the ancients the claim that a mechanical demonstration was useful did not necessarily mean that it was feasible.
In chapter 4 Cuomo considers how Pappus situates himself in relation to other mathematicians and their work. Her opening text is book 3 of the Collectio. This book, which is addressed to a mathematician named Pandrosion, concerns one of her pupils who sent Pappus outlines of solutions to three problems for criticism. Apparently, this (unnamed) pupil had sent these outlines—constructions accompanied perhaps by descriptions but no proofs—to others as well: Pappus indicates that some were quite impressed with the outlines and that others, particularly, some associates of the philosopher Hierius, had undertaken to examine them. Eventually, it seems, Pappus decided to stop waiting for the author to publish the proofs; and on the ground that his opinion had been asked in the first place, he made known his views of their worth. The fundamental judgment, of course, is that they are unprofessional. But, as Cuomo shows, Pappus tells his reader much about himself and proper mathematical argumentation in the course of conveying this negative judgment.
After Pappus has disposed of Pandrosion’s pupil—and Pandrosion too, one fears—he turns to a history of proper solutions to one of the problems the pupil raised, namely, the problem of finding two mean proportionals to two given lines. (Solving this problem is necessary if one is to solve the Delian Problem of duplicating a cube.) By comparing Pappus’ history of the solutions to the Delian Problem with that offered by Eutocius in the sixth century AD, Cuomo demonstrates effectively that Pappus has adapted his sources by reporting only certain features of selected solutions and omitting, for example, their criticisms of one another’s efforts and any indication that the Delian Problem might not be a problem in pure mathematics. Pappus’ history culminates with his own solution to the problem. And so in book 3 he presents himself as a custodian of proper geometry and a distinguished successor to Eratosthenes, Nicomedes, and Hero.
In the second part of chapter 4, Cuomo turns to book 4 of the Collectio and its discussion of linear curves. These curves were not well defined as a group in antiquity. Indeed, as Cuomo shows by considering how Hero and Proclus classify curves such as the spiral and quadratrix, they were difficult to incorporate in a tidy, systematic analysis of mathematical entities, and theorists tended to view them as somewhat odd. Pappus limits his attention to three linear curves: the spiral, the quadratrix, and the cochloid. In each case, after describing how the curve is produced and identifying its characteristic property, he demonstrates its use in solving geometrical problems. This treatment, Cuomo argues, goes beyond previous accounts by making what initially seemed an odd group of entities more homogeneous and intelligible. After all, the curves are each described as outcomes of the motions of points and lines, though it seems that they had not always been presented this way; their defining properties are succinctly identified; and their utility is shown by their application to classic, familiar problems of Greek geometry. Thus, as Cuomo surmises, one of Pappus’ aims in book 4 is to rehabilitate linear curves as respectable objects of mathematical study.
In the concluding chapter, Cuomo analyzes what have proven to be persistent features in the books of the Collectio and might well be described as characteristics of Pappus’ mathematical style. Such, for example, are his concern with the conditions of solutions and with whether a construction is possible and, if so, in how many ways, as well as his preference for mechanical procedures, and his interest in cases (distinct variations in the placement of points, lines and so forth, in constructions). Cuomo focuses on three features—Pappus’ frequent generalizations of well known or important theorems, his distinction of cases and subcases that have sometimes been neglected, and his use of arithmetical particularizations to facilitate the understanding of a geometrical argument—and shrewdly sees these as ways in which Pappus both introduces and engages the past. But, as she shows by examining carefully what Pappus says about his mathematical predecessors and contemporaries and the ways in which he dispenses criticism and praise, he is no mere slave to the past. What Pappus actually does in the Collectio is to create a mathematical tradition by positing a community that is committed to the extension of knowledge and bound by respect for those who contribute to this knowledge, both past and present. In short, he imagines himself as a (leading) member of a community devoted to a special knowledge that is both useful and ennobling, while presenting this community as an embodiment of social and political virtues valued in his day. According to Cuomo, this is the vision that governs much of what transpires in the Collectio.
As I said at the outset, Cuomo’s book takes the study of the history of ancient Greek mathematics in a very new and promising direction. This is a stimulating book that will be an excellent model for research on ancient works in and about mathematics. Indeed, a similar study in the cases of Proclus and Eutocius could well be revolutionary by challenging the naïve reading of these authors that underlies the standard account of ancient Greek mathematics.