THE STOMACHION: MOTIVATING THE DISCUSSION

My moment of revelation – indeed, the starting point for writing this book – was while trying to make sense of Archimedes' Stomachion. This treatise, surviving on a single parchment leaf containing the introduction, a preliminary proof, and one stump of a proof – all mutilated and difficult to read – has gained little scholarship since its first publication by Heiberg in 1915. I would have never paid it much attention myself – it did not appear to be a “serious” work – but it is after all a page out of the Archimedes Palimpsest, and just looking at the parchment one could not resist the temptation to work on it. The page looked to be in such a bad shape, surely Heiberg did not manage to read it satisfactorily!

My reading did not add many words to those read by Heiberg. But I was probably the first person in many years to have read, slowly and attentively, the introduction to the Stomachion. I quote a tentative translation:

As the so-called Stomachion has a variegated theoria of the transposition of the figures from which it is set up, I deemed it necessary: first, to set out in my investigation of the magnitude of the whole figure each of the <figures> to which it is divided, by which <number> it is measured; and further also, which are <the> angles, taken by combinations and added together; <all of the above> said for the sake of finding out the fitting-together of the arising figures, whether the resulting sides in the figures are on a line or whether they are slightly short of that <but so as to be> unnoticed by sight. […]

The overall structure of the argument is simple. First, we have covered, in the first three chapters of this book, a substantial body of evidence pointing to a certain style of Hellenistic mathematics. Its three major constituents are: (1) mosaic composition, (2) narrative surprise, (3) generic experimentation (a more specialized phenomenon is that of the carnival of calculation). Second, we have briefly noted, in chapter 4, how such stylistic features may also be typical of the major literary works of the same period (with the carnival of calculation paralleled by the carnival of erudition). The minimal claim of the book, then – the one backed up by evidence – is of a certain homology of style between the exact sciences and poetry in the Hellenistic world. In the conclusion to the preceding chapter I have already pointed beyond, to much more tentative claims. It is tempting to postulate a historical force underlying the homology. More than this: if indeed we suggest that a certain historical process led to a Hellenistic interest in generic experimentation, then it becomes very tempting to suggest that the rise of the exact sciences as a major cultural phenomenon should be seen as part and parcel of this practice of experiment in genre, where a hitherto minor genre suddenly gains in prominence.

All of this, however, is highly tentative, largely because our evidence can support such historical interpretations only with difficulty. In this section I acknowledge and address the limitations imposed by our evidence.

The claim of the following chapter is twofold, looking at how poetic practices are (i) complementary to those of the exact sciences, and (ii) parallel to them. Section 4.1 makes the case for (i) complementarity. It shows how Greek poets turned to scientific concepts and contents, weaving science into their poetry just as the scientists were weaving poetry into their science. Sections 4.2–4.3 make the case for (ii) parallelism. Section 4.2 takes a central example of the practices of mythography in Hellenistic poetry as a starting-point for an analysis of the familiar role of “erudition” in this poetry – now considered in light of the scientific practices discussed in this book. Section 4.3 broadens the discussion to look at the poetic parallels to the scientific practices seen throughout this book, as a whole: the narrative surprise and the mosaic text. Of course, the complementarity and parallelism are tightly connected. Section 4.4 offers a brief summary, with some tentative conclusions.

The Hellenistic world was, for generations, the least intensively studied of all ancient periods, its culture alien and uninviting for classicists inspired by Greek glory or by Roman grandeur. With changes in contemporary taste, as well as with the overall explosion of academic writing, considerable and sophisticated studies of Hellenistic civilization have appeared over the last couple of decades. Even this scholarship, however – as is not surprising – concentrates on Hellenistic literature to the exclusion of science.

Euclid's Elements stand out, among Hellenistic mathematical works, in their pedagogic intent. Yet their very end – book xiii – already suggests the ludic, and at the very end is a theorem, attached as a kind of appendix, that would have been worthy of Archimedes. The theorem is often considered to have been discovered early (though its form may be due to Euclid himself, or even to some later reader of him). However this may be, it may serve as an example of an important compositional phenomenon: the mosaic proof. Here then is the proof that there are exactly five regular solids (adapted from Heath's translation):

This, my third study on Greek mathematics, serves to complete a project. My first study, The Shaping of Deduction in Greek Mathematics (1999) analyzed Greek mathematical writing in its most general form, applicable from the fifth century bc down to the sixth century ad and, in truth, going beyond into Arabic and Latin mathematics, as far as the scientific revolution itself. This form – in a nutshell, the combination of the lettered diagram with a formulaic language – is the constant of Greek mathematics, especially (though not only) in geometry. Against this constant, the historical variations could then be played. The historical variety is formed primarily of the contrast of the Hellenistic period (when Greek mathematics reached its most remarkable achievements) and Late Antiquity (when Greek mathematics came to be re-shaped into the form in which it influenced all of later science). My second study, The Transformation of Mathematics in the Early Mediterranean (2004), was largely concerned with the nature of this re-shaping of Greek mathematics in Late Antiquity and the Middle Ages.

This study, finally, is concerned with the nature of Greek mathematics in the Hellenistic period itself. Throughout, my main concern is with the form of writing: taken in a more general, abstract sense, in the first study, and in a more culturally sensitive sense, in the following two.

The three studies were not planned together, but the differences between them have to do not so much with changed opinions as with changed subject matter.

In May 2003, from the Baikanur launchpad in the Central Asian deserts of Kazakhstan, British scientists fired a Russian Soyuz-Fregat rocket to launch a probe called the Mars Express, intended to determine whether recognizable chemical signs of life could be found in the thin atmosphere and dusty rocks of the red planet. In 1971, the Soviets had been the first to land a probe on Mars, and they were followed by the American Viking missions in 1976. In January 2004, the U.S. National Aeronautics and Space Administration (NASA) landed the mobile rovers Spirit and Opportunity on Mars. These represented huge and dangerous efforts. Of thirty previous missions to Mars, twenty had gone seriously wrong. In 2003, a British probe intended to explore the Martian surface, called – significantly – Beagle-2, failed to arrive on the surface. The European mission cost 300 million euros and the American mission ten times as much. Behind all these efforts lies the necessity of securing wide political and public support. Thus, the space missions are performed in “full view of the public.” As Alan Wells, director of space research at the University of Leicester, put it, “We are breaking new ground in the public presentation of space science.” His duty, in his words, is to be a professor of public relations as well as planetary science.

Among the modern life sciences, physiology trails only the evolutionary sciences in the attention it has received from historians. Lamarck, Darwin, and Mendel may be better known than the heroes of modern physiology, but names such as François Magendie (1783–1855), Johannes Müller (1801–1858), Claude Bernard (1813–1878), Hermann von Helmholtz (1821–1894), Ivan Pavlov (1849–1936), and Charles Sherrington (1857–1952) require little introduction for those who read more than occasionally in the history of science. Physiology may have attracted such attention because it has been widely viewed as the first of the modern biological disciplines to emerge from traditional approaches to the phenomena of life embodied in medicine and natural history. Furthermore, physiology allowed historians of science of the first generation after World War II to develop a series of narratives that reflected their broader concerns about the nature and significance of modern science and about how to write its history. If the historiography of the physical sciences in the 1950s and 1960s found its normative models in the “Scientific Revolution” of the sixteenth and seventeenth centuries, so, too, in those decades did historians of the life sciences locate their normative models in nineteenth-century physiology.

Buoyed by the combination of optimism of understanding the natural world from Isaac Newton’s version of the mechanical philosophy and the excitement of discovering natural artifacts of the natural world from naturalists such as Carl Linnaeus, Abraham Werner, and Georges Buffon, natural philosophers turned increasingly to studying nature in nature by the end of the eighteenth century and the beginning of the nineteenth century. Certainly the maturation of the cabinet tradition in the form of emerging national museums (Muséum d’Histoire Naturelle, British Museum) and national botanical gardens (Royal Botanical Gardens at Kew) at this same time underscores the importance of learning from the natural world. Furthermore, continued overseas expansion and exploration, especially in North America, the Indian subcontinent of Asia, and Australia, heightened European interests in this direction.

Many of these same eighteenth-century motivations continued into the nineteenth century and, moreover, may be described after the model of scientific transmission and development offered by George Basalla, which he developed by examining the early history of American science vis-à-vis science in England. It is certainly appropriate to borrow from and to expand on Basalla, for much of the eighteenth-century interest in the natural world was exhibited by Europeans who observed nature outside of Europe, primarily within their colonial holdings. They collected specimens on voyages of discovery and recruited local colonialists to collect specimens that could later be sent back to European museums and universities following the return of the imperial explorers to their mother country (see MacLeod, Chapter 3, this volume).