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Greeks,Practical andTheoretical 69

Planet (or sun)

Epicycle

Earth

Deferent

Fig. 8 The epicycle model. The planet moves clockwise round a small circle (epicycle) whose centre moves anticlockwise round a large circle (deferent) centred on the Earth. This explains why some planets (e.g. Mars) sometimes move backwards in relation to the stars. explanation is not so much to say what ‘really’ happens as to give an accurate description from which you can make predictions. Again, the physics, as we would call it, is not considered. In the first model (‘epicyclic’), the sun moves at constant speed round a small circle or ‘epicycle’; and the centre of the epicycle itself moves at constant speed round the ecliptic, with the Earth as centre (see Fig. 8). The second (‘eccentric’) supposes that the sun is travelling at uniform speed around a circle, but that the Earth is not the centre of the circle. It follows that the sun appears to be travelling more slowly when it is further from the Earth than when it is nearer (see Exercise 8). In Appendix B, I give the detailed working out by , from his observations, of where the centre of the orbit is in relation to the Earth. This is a detailed piece of Greek numerical mathematics, based on the geometric tradition. Is it practical? Very much so, in that it makes possible (the beginnings of ) the calculation of where the sun will be. But it is supported by a formidable theoretical apparatus of results about chords in a circle, , and so on. Considered as a whole—and this is the justification for devoting so much time to his work—Ptolemy’s gives more of an impression of the range and variety of than any other text which we have. Exercise 6. Show that Crd(60 ) 60, and Crd(36 ) 60(( √5 1)/2). ◦ = ◦ = − Exercise 7. How would youfind Crd(θ/2) given Crd(θ)? Exercise 8. Explain the variation in the sun’s apparent speed, on the eccentric hypothesis. Research problem. Find the two reasons why the length of the day varies (a) as we would understand it, (b) as Ptolemy would have put it. (This is called the ‘equation of time’.)

5 On the uncultured Romans

With the Greeks was regarded with the utmost respect and consequently none were held in greater honour than , but we Romans have delimited the size of this art to the practical purposes of measuring and calculating. (Cicero, Tusculan Disputations, tr. Serafina Cuomo, in Cuomo 2001, p. 192)

The above quotation heads Serafina Cuomo’s chapter on the Romans; and her recent book makes thefirst serious attempt to investigate and indeed question a view of their mathematics accepted 70 AHistory ofMathematics since Cicero’s time (first century bce). Briefly, this is that the Romans in contrast to the Greeks made no contribution to mathematics, and that any works of theirs which contain or use it, such as Vitruvius’ architecture, are trivial in comparison with the Greek achievements. The charge has some basis in fact, as Cuomo acknowledges—‘I would be hard put to adduce a equivalent of , or Apollonius’, she admits (p. 194). This seems like an understatement; where are the Latin equivalents of Heron or Nicomachus? True, these second- or third-rate mathematicians worked under Roman rule, and may have been, like St Paul, Roman citizens. However, they wrote in Greek, and in a tradition which was, and continued to be, overwhelmingly Greek. The Romans, with better access to the Greek classics than the ninth-century Arabs or the Renaissance Europeans, never seem to have felt the same need to build on their work and develop it. And while Cuomo also interestingly makes a historicist point about the class contained in Cicero’s statement (ideas of ‘the Roman’ and ‘the Greek’ were marks of different kinds of prestige, while many users of , and land-surveyors in particular, were seen as jumped-up technical upstarts), one is still left with an underlying feeling that it is an ideological statement based on good factual evidence. The most interesting part of her argument is a broader one, and does bear serious consideration: that the practice of mathematical methods of some sophistication pervaded the Roman world from top to bottom. Some of her examples, notably the charioteer whose tomb boasts that he drove chariots for 24 years, ran 4,257 starts and won 1,462 victories, 110 in opening races. In single-entry races he won 1,064 victories, winning 92 major purses, 32 of them (including 3 with six-horse teams) at 30,000 sesterces... (CIL 6.10048 (Rome, 146 ce), tr. in Lewis and Reinhold 1990, pp. 146–7) testify to the power of ‘numbers’ to impress rather than to the ability to do anything with them. However, her study of the practice of the despised land-surveyors (see also Dilke 1971), and of Vitruvius show how an appreciation, and application of classical geometry underlay their prac- tice. Perhaps rather than decrying the ‘low level’ of geometry present in Vitruvius’s architecture, we should think about the fact that it was a Roman, rather than a Greek, who bothered to write such a treatise; the architects of Greek temples were not, it would seem, given to exposition. We have different cultures (cohabiting in the same empire) with different ideas of what a book is for. Similarly, the famous tunnel of Eupalinus in Samos, dated at 550–530 bce, is often cited as an amazing example of very early practical Greek geometry; how did the builders of the tunnel, who started from the two sides of a mountain, contrive to meet so accurately in the middle? The answer is again that we do not know, and no Greek sources seem to have taken the trouble to explain how such a recurrent problem could be solved. The Roman surveyors, however, organized as a profession in which a discipline was transmitted by means of ‘textbooks’, both explained how they did it8 and wrote instructions whose foundation is in their training in some derivative of . This debate is only now beginning; the same applies to the doubts which Cuomo has cast on the idea that Greek mathematics was ‘in decline’ from (say) the time of Ptolemy,if not before. It is not so much a question of rehabilitating the Romans (awarding points to individuals or to civilizations for their excellence in mathematics should not be part of the business of history, though it often is). Rather, as we saw in Chapter 1 with the pre-OB periods, it is a question of looking at practices

8. See Cuomo (2001, p. 158) for a surveyor’s account of how he helped the weeping villagers whose tunnel had manifestly gone badly wrong. Greeks,Practical andTheoretical 71 which have been dismissed as trivial or non-mathematical and seeing if they do in fact belong in our history—and if so, where.

6

Hypatia, the daughter of Theon the , was initiated in her father’s studies; her learned commentaries have illuminated the geometry of Apollonius and , and she publicly taught, both at Athens and Alexandria, the philosophy of and Aristotle. In the bloom of beauty, and in the maturity of wisdom, the modest maid refused her lovers and instructed her disciples; the persons most illustrious for their rank and merit were impatient to visit the female philosopher; and Cyril beheld with a jealous eye the gorgeous train of horses and slaves who crowded the door of her academy. (Gibbon, n.d. chapter XLVII) Hypatia was born in the later part of the Roman Empire, an era when women were not free to pursue careers. This was a time when orthodox belief effectively wiped out centuries of scientific discovery. works were torched and scholars were murdered. Hypatia was the last proprietor of the Hellenic Age wonder, the . She is portrayed as a young adult facing the issues of a changing world. The reader will discover uncanny parallels to many current situations within the United States and, indeed, the world. Hypatia, a real, historically documented heroine, is afind for today’s young adults who are searching for strong, non-fiction role models. (From review of a novel, ‘Dear Future People’, at www.erraticimpact. com/˜feminism/html/women_hypatia.htm)

Mathematicians, like engineers and physicists, have very rarely been women—the rarity is far more serious than (for example) for poets and painters. As a result, the study of ‘women mathematicians’ faces a serious difficulty in even getting off the ground, since there has until the twentieth century been no continuing tradition from which to construct a history. The feminist historian (of whom Margaret Alic 1986 was a pioneer) therefore necessarily (a) points out the existence of numerous such women of whom we know little or nothing, and (b) attempts to weave the major lives of which we do know something into some kind of thread. The philosopher and mathematician Hypatia, who was stoned to death by a Christian mob in 415 ce, (this much is undoubtedly historical fact) is the most important early, perhaps the foundingfigure for the tradition, and the questions which surround her life and activity may illustrate the wider problem. For if one were to take any other figure from Greek mathematics as ‘representative’ of something (Nicomachus as a Palestinian, Ptolemy as an African,...), generalizations based on the little that is known of their lives would be hard, although the works at least give some basis for building theories. With Hypatia, the difficulty is the opposite one. Her life is unusually well documented, in the general context of Greek mathematics, as the result of friendly and hostile accounts by later Christian writers. Most particularly, her devoted pupil Synesius, bishop of Cyrene in Libya, wrote a of letters to her and about her which give substantial detail about her life and teaching, if from a particular viewpoint. On the other hand, although her ability as a mathematician is well documented and at least the titles of some works have been preserved, there is no extant text attributed to Hypatia, no ‘Hypatia’s theorem’, no discovery which tradition assigns to her. With Heron, as was noted above, we know the works but nothing of who he was; with Hypatia, it is the other way round. On her life and her philosophy, for which the sources are good (Synesius does not appear to have been so interested in mathematics), Maria Dzielska’s monograph (1995) is a recent excellent source. Dzielska begins with an account of the myths which have built up around her as an iconic figure since the seventeenth century, and which my opening quotes illustrate. She was a victim of Christianity and symbol of the death of the ancient learning at the hands of the new ignorance (Gibbon); a feminist icon and precursor of (for example) Marie Curie (Margaret Alic); a symbol of 72 AHistory ofMathematics the imposition of European rule on Africa (Martin Bernal). Dzielska produces a convincing picture of Hypatia as an influential teacher of mathematics, , and neoplatonic9 philosophy to a circle of initiates, both pagan and Christian.

Around their teacher these students formed a community based on the Platonic system of thought and interpersonal ties. They called the knowledge passed on to them by their ‘divine guide’ mysteries. They held it secret, refusing to share it with people of lower social rank, whom they regarded as incapable of comprehending divine and cosmic matters ... Hypatia’s private classes and public lectures also included mathematics and astronomy, which primed the mind for speculation on higher epistemological levels. (Dzielska 1995, p. 103)

She shows Hypatia becoming involved in a power struggle between factions in Alexandria in the years following 410 which led to a witch-hunt, and eventually to her death There was, indeed, a careful line to be drawn in late Roman times between the praiseworthy pursuit of geometry (ars geometriae) and the damnable art of astrology (confusingly, ars mathematica)—see Cuomo (2000, p. 39); and Hypatia was probably not the only scholar to be caught on the wrong side of the line. Dzielska further establishes, fairly convincingly, that her age at the time was about 60 (demolishing the image of a beautiful maiden cut down in the bloom of youth), and points out that her death was far from marking the end of learning in Alexandria, or the Greek world generally—or even of paganism. Pagan religiosity did not expire with Hypatia, and neither did mathematics and Greek philosophy. (Dzielska 1995, p. 105) She also unearths a number of other references to women in the late Greek philosophical world, which show Hypatia’s example to be not so unusual as had been thought. This is helpful, but of course the historian of mathematics would like to have more, and here as so often we enter the world of more or less ingenious conjecture. As Gibbon states, her father was the mathematician Theon who has not been highly estimated in recent times (‘a competent but unoriginal mathematician’, Calinger 1999, p. 219). However, like many others of the period he studied and commented the difficult works of his predecessors, and edited the text of Euclid in a version which was almost the only one to survive. The titles of several of Hypatia’s works—mainly commentaries, for example, on Ptolemy and Diophantus, are known from later bibliography, and Synesius, her student, was a philosopher not a mathematician; and by the time when Islamic scholars recovered and translated Greek works, none of them bore her name. The scholar who wishes to study her as a mathematician, supposing it possible, has to use a certain amount of imaginative reconstruction. Nonetheless, in line with the revival of neglected women in antiquity, she is given two pages in Calinger’s general history (1999), and that common and convenient view which dismisses Theon’s works as pedestrian and second-rate attempts to pick out the more interesting parts of them and ascribe them to Hypatia.10 Following the initial stage of ‘recovery’, where the aim was to point out Hypatia’s status and relative neglect as a mathematician, Dzielska’s work has been well received as perhaps marking the start of a second period in the study of women mathematicians, still rather in its infancy: an attempt to place them in a historical context, even when (as with Byzantine Alexandria) that

9. was not simply a revival of , but had elements of ; as such it played a semi-religious part in the late Roman empire. 10. For these and similar arguments see Knorr (1989), who also suggests that, since Diophantus’s as it survives contains comments, the edition itself in the form we have it may have been prepared by Hypatia. Doubts are expressed by Cameron (1990). Greeks,Practical andTheoretical 73 context is remote and offers few opportunities for identifying role models. An assessment of her life and works, if any can be reliably ascribed to her, while sympathizing with her difficult, ultimately tragic situation, is not dependent either on approving her obvious ability and charisma, or on disapproving of the élitism, and that belief that the state would be better off if run by philosophers which she shared with other Neoplatonists.

Appendix A. From Heron’s Metrics

[Introductory note. The standard translation (e.g. the one which you willfind in Fauvel and Gray) has been changed so that there is as little as possible ‘modernization’ of Heron’s language, and no explicit algebra. This is more difficult than one might imagine, since (a) Heron does think of lengths as numbers, and multiply them—this happens in thefirst part, (b) in the geometric proof the kind of straightforward and perhaps over-simple statements I have made about Euclid (areas of rectangles are just areas, not products of the lengths of sides etc.) are no longer clearly true, and it is possible that something like algebra, of an embryonic form, was in Heron’s mind even if you cannot see it on the page. Prepositions like ‘on’ and ‘by’ indicate areas of rectangles or multiplication, and it is unclear which is being used. Most unexpectedly, ‘the on ABC’ means the productAB timesBC, not the area of a triangle.] There is a general method forfinding, without drawing a perpendicular, the area of any triangle whose three sides are given. For example, let the sides of the triangle be 7, 8, 9 units. Add together the 7 and the 8 and the 9; the result is 24. Take half of this; the result is 12. Take away the 7 units; the remainder is 5. Again take away from the 12 the 8; the remainder is 4. And then the 9; the remainder is 3. Multiply the 12 by the 5; the result is 60. and this by 4; the result is 240. And this by 3; the result is 720. Take the side [= square root] of this, and it will be the area of the triangle. Since 720 does not have a rational square root, we shall reach a different [number] close to the root as follows. Since the square nearest to 720 is 729 and it has a root 27, divide the 27 into the 720; the result is 26 and two thirds. Add the 27; the result is 53 and two thirds. Take half of this; the 1 1 5 result is 26 2 3 [This is the ‘Egyptian way’ of writing 26 6 .] Therefore the square root of 720 will 1 1 1 1 1 be very near to 26 2 3 . For 26 2 3 multiplied by itself gives 720 36 ; so that the difference is a 36th part of a unit. If we wish to make the difference less than the 36th part, instead of 729 we shall 1 take the number now found 720 36 , and by the same method we shallfind a difference much less 1 than 36 . The geometrical proof of this is this (Fig. 9): In a triangle whose sides are given tofind the area. Now it is possible to draw a perpendicular and calculate its magnitude and sofind the area of the triangle, but let it be required to calculate the area without drawing the perpendicular. Let the given triangle be ABC and let each ofAB,BC, CA be given; tofind the area. Let the circle DEZ be inscribed in the triangle with centreH [Euclid IV.4], and letAH,BH,CH,DH,EH,ZH be joined. Then the [rectangle]BC timesEH is twice the triangle BHC [Euclid I.41], and CA times ZH is twice the triangle AHC, andAB timesDH is twice the triangle ABH. So the perimeter of the triangle ABC timesEH, that is, the [radius] of the circle DEZ, is twice the triangle ABC. LetCB be produced, and letBF be made equal toAD; then CBF is half the perimeter of the triangle ABC becauseAD is equal toAZ andDB toBE andZC toCE. SoCF timesEH is equal to the triangle ABC. ButCF timesEH is the side of the [square] onCF times the one onEH.