The problems of exceptionality: The case of Archimedes and the Greeks

Abstract

The points at which Greek mathematics in general and Archimedes’ contributions in particular are exceptional are here assessed by way of a comparison with the extensive evidence from ancient China. While underlining the need for caution concerning the extent to which concrete conclusions are possible, the outcome is broadly to confirm Netz's argument that a key factor in Archimedes’ success and influence was the way in which in a social and intellectual environment that favoured debate, he was able to contest an assumption found in both the Platonic and Aristotelian traditions. Where they had imagined a sharp division (albeit differently defined) between what they assigned to ‘physics’ and to ‘mathematics’ respectively, Archimedes showed how those two inquiries could be treated as complementary to one another, thereby opening up the possibility of new styles of physical demonstration.

In this inspirational new study, Reviel Netz takes on a topic still redolent with the toxicity of frankly racialist historiography. Although he nowhere mentions the dread expression the ‘Greek miracle’, he knows full well how easily admiration for one or other aspect of Greek civilization has, in the past, turned into perverse celebrations of a presumed intellectual superiority. In 1865 the philologist Ernest Renan stood on the Acropolis at Athens and offered a prayer of thanks for the miracle of Greek rationality. 1 The Greeks alone, on this view, specifically the Athenians, lived by Reason, whereas others, their neighbours in the Mediterranean and beyond, were all more or less deficient in this crucial human cognitive capacity.

Ever since, one may say, the project of coming to terms with the Greek legacy and assessing its exceptionality has continued to haunt historians. Even while most sought to distance themselves from aspects of Renan's extravagant praise, they often remained, wittingly or unwittingly, in his shadow. 2 A flurry of studies written by members the so-called Paris school and others aimed to give us ‘Greeks without miracles’ – to cite the title of a collection of essays by Louis Gernet published posthumously with a preface by Jean-Pierre Vernant. 3 Marcel Detienne especially produced scathing critiques of the thesis that ‘the Greeks are not like others’, while insisting that their assumed ‘incomparability’ did not put them beyond the reach of comparatism. 4

However, the chief focus of interest of Gernet, Vernant, Vidal-Naquet, Loraux and Detienne himself was rather on Greek political and social institutions, and in particular on the impact of the rise of democracy. 5 Reason, they argued, is the child of the city, the polis. Philosophy and science were both implicated, for sure. But none of the principal participants in that debate was a specialist in Greek mathematics. There are a couple of mentions of Archimedes in Gernet, but otherwise he hardly rates a comment. No more did he in the work that was largely instrumental in debunking Greek miracle myths for the English-speaking world, namely Dodds (1951).

Enter Reviel Netz, historian of Greek mathematics, editor of and foremost living authority on Archimedes in particular. His focus is not on Cleisthenes or Pericles or even Plato and Aristotle, but rather on a first phase of the development of Greek mathematics exemplified in the work of Archytas and more especially on the second where the key figure for breadth as well as depth is surely Archimedes himself.

Two words of caution would be salutary at the outset. First, the extent to which any individual's original intellectual achievements can be explained is problematic. They certainly should not be assumed to be fully determined by the circumstances in which they worked. The most we can aim for is to identify certain factors that may have been influential, facilitating that originality or at least removing certain obstacles that it might otherwise have encountered. Conversely, the influence that any given individual exercised on their successors cannot be represented as necessitating those successors’ contributions.

Secondly and more particularly, as with many other areas of Greek thought we are often hampered by the lacunae and bias in all our sources, both Greek and non-Greek. We have barely sufficient evidence for Hippocrates of Chios, Archytas and Eudoxus to gain even a modicum of understanding of their work. As for Archimedes and other denizens of Greek mathematics, Euclid, Apollonius, later Ptolemy, Netz is right to emphasize how fortunate we are to have such substantial portions of their oeuvre, though here too, for sure, gaps remain, making Apollonius in particular hard to evaluate.

So the main two pairs of interrelated problems can be described as follows. First, we have to try to determine what (if anything) is exceptional about Greek mathematics: and when we have identified the explananda we have to attempt some explanatory account, however speculative. Second, what in particular does Archimedes’ exceptionality consist in and how indeed can we begin to account for that? Now we should be wary here, as Netz is, of generalizations about ‘Greek’ mathematics for they tend to imply a greater homogeneity than is warranted (a point also emphasized by Cuomo 2001, for instance).

We face in addition a recurrent problem of disentangling Greek originality from its debts to the mathematics of its neighbours in Egypt and Mesopotamia especially. To what extent, we must constantly ask ourselves, are we being hoodwinked by the fact that many of our sources are Greek into thinking that what they report is indeed (just) the product of their culture. History, we must remind ourselves, is written by and for the victors. If the appropriation by the Greeks of near Eastern mathematical ideas and practices was often comprehensive, allowance for that must certainly be made when assessing ‘achievements’ ‘breakthroughs’ and ‘influences’. To be sure if one mistake to be avoided is to underestimate Greek debts to others, the other extreme to guard against is to be overpersuaded by the converse theme that some ancient Greeks themselves indeed liked to harp on, namely that they took over many ideas and fields of inquiry from those others (the topic recently overplayed in Bernal's volumes of Black Athena 1987–2006 ). Evidently, both the thesis that the Greeks got nothing from other cultures and the opposite extreme that they derived everything of importance from others, are flawed.

The thesis of Greek indebtedness is particularly relevant to our perception of the very early stages of their mathematics, where Netz wisely does not allow himself to be drawn into that question where the ‘Thaleses and Pythagorases’ are concerned. The first investigations for which we have more reliable information relate to later, fifth-century, figures, Hippocrates of Chios, Philolaus, Archytas, then Theaetetus and Eudoxus. If we review the admittedly fragmentary evidence for their work we find ample confirmation of two of Netz's principal theses, the fascination for the strange, and the search for compelling results.

The very first piece of extended Greek mathematical reasoning that we have is the account in Simplicius (Commentary on Aristotle's Physics 54.12-69.34) of Hippocrates’ study of the quadrature of lunes. That interest stems from and develops the simpler problem of ‘squaring the circle’, one of three topics that Knorr (1986) showed account for a large part of what we know of early Greek mathematics (the other two being duplicating the cube and trisecting an angle). We know that many different attempts were made to solve this riddle-like problem, some of which (Antiphon's) later get to be criticized as breaching certain underlying assumptions about the continuum – and which thereby served, in turn, to make those assumptions more explicit. But Hippocrates identifies other curvilinear shapes that can be ‘squared’ and produces clear geometrical constructions and proofs to show how that can be done.

Analogously the initial stimulus to Archytas’ search for two mean proportionals was the problem of duplicating the cube, and in this case, we have not just a solution to a nice puzzle, but a dazzling virtuoso construction involving the intersection of three surfaces of revolution, a right cone, a cylinder and a tore. 6 The elements of surprise are clear. The naive might suppose, like Meno's slave in Plato's dialogue, that doubling an area or a volume might be just a question of constructing such on a doubled side. But not only is that assumption shown to be wrong, but the way to find a correct solution is exhibited in a spectacular fashion.

Now the first question we must press is whether or how far any such cultivation of surprise and displays of virtuosity are to be found in other non-Greek traditions of mathematics, before we go on to evaluate Netz's suggestion that the key background factors for most Greeks were the competitive, open, egalitarian environment in which much intellectual exchange there took place. While I believe specialists in any other tradition of ancient mathematical practice would find it impossible to cite any study that equals Archytas’ duplication procedure in sophistication, we cannot entirely discount the evidence we have that those other traditions too often puzzled over (in)commensurabilities. If squaring the circle did not have quite the same cachet in ancient India and China, the question of the relationship between the area of a circle and its diameter in both cases provoked a good deal of discussion. In China especially we have both a concern to prove that half the diameter times half the circumference yields the area of the circle and also a sequence of more and more accurate approximations to the value of the circumference/diameter ratio. In the latter context, there is a definite sense that later investigators were challenged by the failures of their predecessors to do better. The Chinese commentary tradition very much shares this general characteristic with the Greco-Roman.

Consider Liu Hui, already the subject of a penetrating comparison with Archimedes in Netz 2018. He was responsible in the third century CE for the first extant commentary on the Jiu Zhang Suan Shu (Nine chapters on mathematical procedures) which itself dates from the turn of the millennium. In that treatise, it is assumed that the circumference/diameter ratio (lü) is 3–1. But in his commentary to 1.32 (Qian 1963, 103–106) Liu Hui shows that to be incorrect for that is the ratio of the inscribed hexagon, smaller than the circle, and he proceeds to give a far more accurate approximation. Nor can there be any serious doubt about the element of competitiveness between mathematicians of different generations. The commentaries on the Nine Chapters envisage getting closer and closer approximations by calculating the areas of regular polygons with up to 3072 sides, while by the time of Zhao Youqin, in the thirteenth century, we find 16,384 sided ones are the target (Volkov 1997). Again while Liu Hui himself was defeated by the problem of the formula for the volume of a sphere, two centuries after him Zu Geng positively crows over his success in cracking the problem (Wagner 1978a and 1978b).

Moreover in studies of the volume of the pyramid, in his commentary on the Nine Chapters Chapter 5, we find Liu Hui having recourse to the manipulation of figures that he himself admits to be very different from any one would be likely to encounter in real life. He sets out to determine the volume of a figure called a yangma (a pyramid with rectangular base and one lateral edge perpendicular to the base). But for this, he needs to use a figure named bienao (a pyramid with right triangular base and a lateral edge perpendicular to that base). But he comments: ‘the object [called] bienao has no practical use … Nevertheless without the bienao there is no way to investigate the number [that is volume] of a yangma; and without the yangma there is no way to know the kinds of zhui and ting [types of shapes connected with cones and truncated pyramids]. These are primary in application’ (ch. 5; Qian 1963, 168.3–4; Wagner 1979, 182; Lloyd 1996, 161, contrast the slightly different translation in Chemla and Guo 2004, 433).

So we find that a certain penchant for the strange, the exotic, the puzzling, the surprising, appears in other traditions besides those of the Greeks and second that there are notable elements of competitiveness between rival would-be experts – and indeed not just in mathematics. At this point then it might look as if the most that we could claim, on those scores, for ancient Greece is a difference in degree rather than one in kind. We shall need to come back to qualify that conclusion in due course, but our next step must be to widen the data under review, especially to include more of the important second, eventually Archimedean, stage in the development of Greek mathematics that plays such a key role in Netz's argument.

Four interrelated developments stand out and deserve detailed discussion. First, there is the systematization of mathematics, begun perhaps by Hippocrates, but instantiated especially in Euclid's Elements. Second, there are the applications of mathematics to such fields as astronomy, harmonics, optics and mechanics. Third, there are developments both in an explicit theory of demonstration and in its practice. Finally, there are developments connected with and stemming from the study of conic sections. Netz himself pays special attention to the last, while not ignoring the others, but if we are to get to grips with Greek exceptionality and its causes we should look carefully at all four, where once again the Chinese experience provides materials for quite nuanced comparisons and contrasts.

First as regards systematization, there is a clear sense, in the Nine Chapters and in Liu Hui, that what they call suan shu (mathematical procedures, dealing both with numbers and with shapes, tu, or blocks, qi) forms a unified subject-matter. We can even detect certain changes in the interests that fall under such a rubric, as evidenced in the earlier compilation of texts known as the suan shu shu. These are sets of bamboo slips recently discovered from a tomb sealed in 186 BCE (Cullen 2004) though their interpretation is complicated both by the disorganized state of the slips and by the fact that they are the work of several authors (Morgan and Chemla 2018). Although right triangles are not completely ignored, the texts appear to antedate the period when the properties of such triangles were the subject of the type of sustained analysis we see in chapter 9 of the Nine Chapters.

Eventually, as Liu Hui puts it in his important Preface to the Nine Chapters (Qian 1963, 91; Chemla and Guo 2004, 127) the investigations he labels as shu 數 or suan 筭 form, as noted and as he emphasizes, a coherent whole. He connects their origin to the trigrams and comments on their role in the study of the calendar and the pitch-pipes and the search for the Dao. They belong to the six arts and were used to assess talent and they have an educational value. As for their unity, ‘although there are different branches, they all have the same trunk/root’. Yet neither there nor elsewhere in ancient Chinese mathematical texts is there any sense that that unity depends on the whole constituting a single axiomatic-deductive system. The distinct sections of the Nine Chapters illustrate a single principle, the same pattern or patterns (gangji), but there is no suggestion that the whole forms a single structure deducible from a limited number of indemonstrable primary premisses (definitions, postulates and ‘common opinions’ in Euclid's case). We shall need to come back to the fortunes of any such ambition later. To see ‘mathematics’ as a unified discipline is not unique to the Greeks in the ancient world, but that should not lead us to ignore the point that the ‘unity’ they were held to possess takes different forms.

The second item on our agenda is particularly interesting since it too yields quite diverse findings. On the one hand, the Greek construction of complex geometrical models reducing the motions of the sun, moon and planets to combinations of circular movements has no parallel in other ancient civilizations, including that of China down to the time of Shen Gua (eleventh century CE) who at least shows some concern to model the shapes of the retrograde arcs of the planets (Sivin 1995: chs. II and III). 7 On the other, the Chinese certainly had developed interests in the mathematical analysis of musical harmonies, where the second century BCE compendium, the Huainanzi, ch. 3 tianwenxun (Major 1993) produces a complex numerical analysis of the relations between the 12 pitch pipes forming what we would call the 12-tone scale. Netz rightly criticizes some of the excessively optimistic judgements scholars such as Needham have given about the mathematical analysis of mechanical and optical phenomena in China. At the same time, the case of harmonics shows that Chinese interests in such analyses could be more than merely sketchy and superficial.

We have accordingly to be wary about generalizations that would purport to sum up a uniform Chinese attitude to the applicability of mathematics to various perceptible phenomena. To make sense of both where Chinese theorists did and where they did not pursue such connections it is essential to consider the underlying values in play. It is, one may conjecture, largely because harmonics provides such a fine example of the interplay of yin and yang and the need for balance between these two that Chinese investigators engaged in such analyses, where we should recall that according to Liu Hui's Preface (Qian 1963, 91) mathematics itself enabled him to study the role of those two cosmic principles. But evidently, we should be careful to resist attempts to generalize globally over where any given cultural tradition stood on the question of how, when and where ‘mathematics’ is to be ‘applied’ to ‘physical’ problems. This is, however, a topic to which we shall need to return.

Our next topic, the theory and practice of proof, has in recent years been the topic of sustained scholarly debate (Chemla 2012 especially). We have particularly detailed analysis of what is needed for the most rigorous kind of proof in Aristotle who points out, among other things, that not everything can be proved. Any claim to that effect involves an infinite regress. The strictest mode of proof proceeds from primary premisses that themselves are self-evident, definitions, postulates and ‘common opinions’. As already noted that triad differs slightly from that used by Euclid in his Elements while that text provided the model for the conditions that a systematic axiomatic-deductive demonstration of a body of knowledge should meet.

However, it is important not to forget that Aristotle allowed less rigorous modes of proof as well, especially those he labelled ‘rhetorical demonstrations’ (rhetorikai apodeixeis), where the requirement for the premisses to be necessary was relaxed. In the Nicomachean Ethics 1094b25ff. he explicitly contrasts the rigour one expects from a mathematician with the plausibility of an orator's arguments. Their proofs were adequate provided the premisses are generally accepted or at least accepted by your opponent in debate. They should be endoxa as he called them, defined as the reputable opinions commonly held by most people or the wise, where the latter characterization suggests an elitist, while the former rather a more democratic, criterion.

This Aristotelian theory of strict demonstration and the mathematical practices that appear to follow its model are, as we said, without parallel in other ancient cultures, Babylonian, Egyptian, Indian or Chinese. So they certainly earn a place in the explananda that we have to tackle when confronting apparent Greek exceptionality though any talk of miracles is out of the question if we bear in mind how such a development responded to a perceived need to see off the opposition in the cut and thrust of debate. But a lack of a notion of axiomatic-deductive demonstration certainly did not and does not mean that other cultures lacked any notion of proof. In China, in particular, to take our best-documented example, there is close attention to proving in the sense of checking results, both in the context of confirming particular results and in establishing that the procedures to get them are correct – points on which Chemla has repeatedly and rightly insisted. That is so not just with the solutions to particular problems but with the algorithms applied to groups of them. In a very simple example, Liu Hui's discussion of the procedure for the addition of fractions (Qian 1963, 95–96; Chemla and Guo 2004, 159) he first shows how a fraction may be expressed in more, or less, simple terms and then he names the two procedures that enable them to be manipulated. ‘Every time denominators multiply a numerator which does not correspond to them, we call this “homogenize” (qi). Multiplying with one another the set of denominators, we call this “equalize” (tong)’. But once cross-multiplication has been carried out, the fractions can be added together and he explicitly remarks that ‘the procedures cannot have lost the original quantities’.

We should note first the careful explication of key terms which are in effect defined even though Liu Hui does not offer any commentary on what counts as a definition let alone claim for such the status of a primary self-evident premiss. Secondly these procedures of ‘homogenizing’ and ‘equalizing’ are not confined to the particular problem in focus here, that of the manipulation of fractions. The same terms and procedures appear in other stretches of the text and Liu Hui's commentary on it. As already noted he is on the look-out for the ‘guiding principles’, the gangji, that are applicable generally. If we turn to our second main source for classical Chinese mathematics, the cosmographical treatise Zhoubi suanjing, it too states as its principal aim to find the methods that are ‘concisely worded but of broad application’ (Qian 1963, 24; Cullen 1996, 177).

Obviously, the conclusion we have to draw is not that the Greeks were the only mathematicians who sought and found proofs for their results. The claim has to be far more restricted, though remarkable enough, namely that only in Greece, among ancient civilizations, do we find an explicit theory of strict deductive demonstration proceeding from primary indemonstrable premisses via valid arguments to incontrovertible conclusions, a theory formulated by Aristotle but exemplified in Euclid, and, yes, Archimedes among others especially. However, as I have repeatedly stressed elsewhere (Lloyd 2006a, 2006b, 2006c, 2014), Greek attempts to follow such a model in such fields as medicine or physiology (Galen) let alone theology (Proclus) showed how misguided such an ambition could be, for the requirement that the study should proceed from primary premisses (axioms) that are self-evident was far harder to meet than was generally imagined.

But if that identifies one phenomenon that we have to endeavour to comment on if not explain, there is then the particular point that Netz's paper highlights, the star role (if one may put it like that) played by conic sections, a recurrent preoccupation in Archimedes’ oeuvre and one that, in Netz's view, had the most profound implications for the future development not just of mathematics, but of science as a whole, in the West.

Archimedes himself did not invent conic sections, to be sure. We hear of a treatise by Euclid, no longer extant, on the topic. Earlier still Archytas’ virtuoso investigation of two mean proportionals incorporates a cone among the solids manipulated to give his result. Archimedes himself still uses the traditional way of referring to the different conic sections, classifying them in terms of the cones used to derive the curves in question. Thus the ellipse is a section of a right circular cone where the vertical angle at the apex is acute, the parabola one where that angle is a right angle, and the hyperbola one where the angle at the apex is obtuse. The terminology from which our terms are derived was certainly known to, and according to some commentators was invented by, Archimedes’ slightly younger contemporary Apollonius of Perga.

Moving forward, the investigation of conic sections, in Archimedes and his successors, enabled or at least accompanied a sheaf of fundamental advances, including the method of indivisibles set out in the Method. Of course no one can claim, and Netz certainly does not, that there was no further work to be done to get to the theory of the calculus as in Newton or Leibniz, or even to Cavalieri's theory of indivisibles. Questions of generalizability and explicitness had to be sure to be resolved before that became part of the toolbox (as Netz would say) available to any mathematician. At the same time while some might insist on what was still lacking in Archimedes’ presentation and use, we should not deny that the Quadrature of the Parabola and eventually, when it was rediscovered, the Method opened up a range of possible subjects for mathematical analysis that had simply not been on the horizon of any mathematician, Greek or non-Greek, previously.

So it is particularly important to examine precisely what this break-through may have involved or consisted in. The extant remains of classical Chinese mathematics do not include any treatise specifically devoted, as Apollonius's was, to conic sections. Yet we have already noted that in apologizing for invoking figures such as the yangma and the bienao Liu Hui explicitly defends his study of them because it is useful for that of the zhui and the ting, that is a cone and a truncated pyramid with a circular base. These shapes appear in Chapter 5 of the Nine Chapters where however the main focus of attention is finding the areas of rectilinear rather than curvilinear figures.

But if it is certainly the case that, compared with the ancient Greeks, the Chinese showed comparatively little interest in conic sections, it has plausibly been argued (e.g. Wagner 1978a, 1979; cf. Netz 2018) that the Chinese arrived at something like the equivalent of Cavalieri's principle by a different route, the investigation of the areas of complex solids. That principle states that solids of equal height in which all corresponding cross sections are of equal area are of equal volume. That in turn depended on the idea that a solid could be treated as the sum of an indefinite number of (indivisible) slices. But it is precisely that idea that underlies the inquiry into the volume of the yangma in the Nine Chapters and then in the subsequent commentaries not just of Liu Hui but also of Zu Geng. Indeed the Chinese call what is known in the West as Cavalieri's principle the principle of Zu Geng.

It is clear, therefore, that certain methods and procedures were available to early Chinese mathematicians although their field of application was much more limited than was eventually the case with Archimedes and his later Greek successors, let alone to those who extended the inquiries so dramatically from the sixteenth century onwards. In particular, as already noted, there was nothing in China that resembled the successive attacks on the problem of the geometrical modelling of the movements of the heavenly bodies – where Kepler's eventual recourse to ellipses marked a significant turning point and the real end (as Netz argues) to the use of Greek Ptolemaic mechanisms (which still dominate in Copernicus).

So it is fair to say that theses concerning massive contrasts between Greek and Chinese turn out, time and again, to need qualification in the face of what we find in the Nine Chapters themselves, in Liu Hui (especially) and in later commentators such as Zu Geng. Nevertheless contrasts there are, both before Archimedes and more especially once he came on the scene. Among the points of commonality between our two sets of data, we may start most obviously with plenty of practical mathematics that would be of interest to anyone from farmers to tax collectors (the point that has often obsessed historians of Chinese mathematics and given rise to a quite biased picture of their accomplishments). Second, we must add that there are abundant signs of theoretical concerns, as we have seen with closer and closer approximations to the circumference/diameter ratio, the recognition of incommensurabilities, and the explorations of strange and exotic shapes to resolve the puzzles they set. Yet on the other hand, on the side of contrasts, we do find exceptional Greek preoccupations, with defining and practising strict axiomatic-deductive demonstration, with a far greater range of geometric conundra, and with the applications of geometry in constructing astronomical models especially.

If the ‘that’ can be thus far at least pinned down, what of the ‘why’ or ‘wherefore’, where Netz is not afraid to try out a sheaf of provocative ideas while acknowledging the speculative nature of many of them. In the period up to Archimedes, one of his arguments is the symbiosis of mathematics and philosophy. To be sure neither of those disciplines was rigidly defined before Plato and in fact practising mathematicians often combined interests in philosophical areas, Eudoxus in ethics, notably, and Archytas himself in politics. He was a successful statesman and general himself, a model, according to one line of argument that I would endorse, for Plato's philosopher-kings, even though Archytas lived in democratic Tarentum and won office by election – which Plato certainly considered to be anything but the right way to go about choosing a leader. The thesis of a covert rivalry between would-be intellectuals of various types in classical Greece has a good deal going for it. Yet that, to be sure, does not discriminate between the situation there and what obtained in early China, both before and after the unification under Qin Shi Huang Di. Both before the Qin and afterward there were advisers competing for the ear of rulers or ministers, travelling persuaders (you shi) as they were known (Lloyd 1996: ch. 2). There are, for instance, sustained attacks on other intellectuals in Xunzi who devotes a whole chapter (6) to attacking, among others, Mencius, the leading proponent of the teachings of Confucius in the previous generation.

Yet if we look a little closer an important difference does open up. Both ancient civilizations practised hard-hitting debate, not just on policy matters, on ethics and the law, but also on technical subjects such as the determination of the calendar. But the difference, certainly one of degree and in places one of kind, lies in context, as Netz 2018 already adumbrated. Most Greek debates were in the public domain, some held before a lay audience. Sometimes it was that audience who decided who had the better of the argument, in the political and legal domain by voting on that: sometimes it was the debaters themselves. But in ancient China the prime target to be persuaded was more often the king or his ministers: the debate was more likely to be a court matter and one adjudicated by the state authorities, certainly never by taking a vote on the topic (cf. Cullen 2007). True the commentators on the Nine Chapters treat one another as peers across the generations. But even mathematicians had one eye on the political importance of their work as we can tell from Liu Hui's autobiographical Preface.

Not all Greek mathematicians worked in democracies, for sure, and increasingly, as time went on, as we see in the case of Archimedes himself, they included kings among their addressees. Yet Netz is surely right to insist that the model in the background continued to be that of a public debate between peers. Certainly one difference that made was with regard to a certain pressure to show off: the more outlandish the philosophical thesis, the stranger the mathematical object, the more those who propounded them imagined that their success would redound to their credit. So too did Chinese thinkers, though in court they could not afford to forget that they should not waste the ruler's time. As we saw, when Liu Hui discusses his strange mathematical objects, it was not to glory in his cleverness, but to apologize for them and claim that, despite appearances, they were useful.

It is part of Netz's overall story that the situation facing Greek mathematicians in the Hellenistic period was in certain respects a new one. The important interactions and rivalries were not between mathematicians and philosophers, but among those who all recognized themselves as mathematicians. Although where actual political regimes were concerned democracies came to lose out to autocracies, there continued to be a sense of rivalry among equals as Archimedes sent out his challenges to his correspondents. Of course, if the ruler was convinced that the presence of spectacular intellects at their court added to their glory, both sides to the relationship benefited – at least so long as the ruler's beneficent mood persisted. The unreliability of their mood is a recurrent concern in both China and Hellenistic Greece. So if the argument that democratic models of discussion still operated in the background in the latter case has some plausibility, it must be acknowledged to be subject to notable qualifications. It is the combination of a traditional background of debate and a continuing ideal of intellectual autonomy, however hard that was to achieve in the actual political situation that individuals faced, that has to do the chief work of explaining the distinctive characteristics of Hellenistic mathematics.

As with most other extraordinarily original thinkers, we must generally admit defeat if asked to give some account of just how it was that they achieved what they did. That is not just a phenomenon associated with some of the Great Names who have figured in triumphalist Greek stories, for there are plenty of geniuses including mathematical ones in other cultures. Islamic ones such as Khayyam and Al-Khwarizmi are brought into the Greek narrative in Netz's account. But there was no Greek connection where the Song mathematicians of the eleventh and twelfth centuries are concerned. Netz does not go into details, but the case for Shen Gua is overwhelming (Fu Daiwie 1993-4; Sivin 1995: ch. III, Sivin 2015).

Yet Great Genius history is just an admission of defeat. What remains of the previous exercise of examination of the what and the why is that just a little light is shed on the factors in play. We are not in a position to say precisely how Archimedes came to do the work he did. But we can at least relate parts of it to his intellectual background (the work of his predecessors) and part again to his social and institutional situation. At no point does it help to say that he was Greek. It is just as well to remember that there were plenty of Greeks (and not just the Roman soldier who killed him at the end of the siege of Syracuse) who could not make head nor tail of his work. But it was not just in ancient Greece that mathematicians considered their studies to be neglected, for once again we find that in his Preface Liu Hui registers just that point. Netz's work reconstructing just how many or rather how few were the mathematicians in any one generation in the Greco-Roman world is then crucially important, for it was always the case throughout antiquity that the transmission of mathematical knowledge hung by a thread. The question ‘if Archimedes had not existed, what would have been different’, should be extended. We must ask what difference it would have made if the texts of Archimedes had not survived or if the balance as between more theoretical and more applied works had been very different. One then goes on to think immediately of those whose achievements have been completely lost, where counterfactual piles upon counterfactual.

So where I must ask in conclusion have we got to with Netz's question? His exploration of the development of Greek mathematics through all the changes of fortune, intellectual and institutional, that it suffered, serves to highlight how very variegated what passes as the story of that development has been. That underlines how there never was any excuse to generalize over that whole history, to treat it as monolithic, let alone as the product of a single overriding causal factor. Different influences are detectable at different junctures. There are exceptional features that cannot be paralleled in other cultures, but also ones where Greek exceptionality is at most a matter of degree not of kind. Where the latter is the case appeal to some particular quality of Greek rationality is a mistaken explanation of a miscued explanandum. Even when we find exceptionality the factors we can identify in play are complex.

As for Archimedes, he was in no sense inevitable. What he took over from his predecessors gets to be transformed in his hands. Could someone else, a non-Greek for instance, have achieved that all-important step of opening up the study of conic sections? The only sensible argument to mount on that score proceeds by way of an investigation into the types of problems that attracted major attention in any given culture at any time – an investigation of the values of the group in question. Even then two reservations need to be added. The first is to make allowance for the possibility of inquiry conducted just for its own sake, for example in the spirit of playfulness that Netz showed is such a powerful force in Hellenistic culture. It was not because he was Greek that Archimedes was able to do what he did, but because he was a Greek at a certain time and place where such playfulness was not just tolerated but won you a reputation. And the second is to note that his influence in the West took a long time in coming, a matter as Netz shows of many ramifications in many fields, where conic sections in particular finally achieve something like their apotheosis with Kepler.

So at the risk now of some repetitiveness let me endeavour to draw up a balance sheet of our findings. First of all, there can be no question of treating ancient Greece, or China, or India, or Egypt, or Babylonia, as if they were each uniform cultural entities. To do so would be to elide all the great variety to be found in each in different contexts and at different periods. It follows that it is doubly unacceptable to treat Greece and China in particular as polar opposites. The important commonalities we find between their histories include the existence of a plurality of independent polities, at least before the rise of the Roman empire and the unification of China by Qin Shi Huang Di. Autocracies on smaller or larger scales were common in both the Greco-Roman and the ancient Chinese worlds, but some were, others were not, keen to foster intellectual pluralism, cherished, when it was cherished, more for reasons of prestige than an unalloyed commitment to research.

Where cultural as opposed to purely political matters are concerned, we find a penchant for the exotic, the paradoxical, the riddling, at different times in both Greece and China (Netz 2004). Again both societies exhibit intense rivalry between would-be intellectual leaders who engage in competitive display and often hard-hitting debate whether in the written or the oral mode, whether with their contemporaries or with past figures of renown. Those debates generally concerned issues of political or moral importance, but sometimes also technical or practical ones, including even mathematics.

Against that background, just two main examples of occasional Greek exceptionality stand out. In Greek democratic regimes, where they existed however impermanently, decisions were taken by majority vote. That provoked a negative reaction among some Greek intellectual leaders who protested that in matters to do with philosophy or science or mathematics the issues were not to be decided by counting heads. How then were they to be resolved? The answer (again exceptional) offered by some Greeks was by securing the truth by incontrovertible methods, in particular by valid arguments from self-evident premisses, an ambition that has no parallel in other ancient societies. Yet as I have also insisted, such a method could be, and was, a quite misleading model for the investigation of many subjects.

Archimedes was the inheritor and beneficiary of such an ideal. The development of the method where conic sections were concerned had the most far-reaching consequences, for its application there was to prove crucial not just within mathematical investigations but in cosmological ones. Where Plato among others had driven a wedge between what they called mathematics and the study of nature (‘physics’), it can be argued that it was largely thanks to Archimedes’ conic sections that those two disciplines could eventually be successfully reunited, providing an altogether more robust basis for claims to understand reality, and in particular the movements of the heavenly bodies, as a whole. 8 Archimedes himself was not to proclaim his achievement in quite those terms: indeed the full force of his contribution was (as Netz said) not realized until long after his death. Yet the explanation of that achievement should not be sought in some miraculous blinding revelation: rather it was a felicitous combination of already existing potentialities in the recognizably exceptional features of the cultural background he inherited.