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 Ptolemy, 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, angles, and so on. Considered as a whole—and this is the justification for devoting so much time to his work—Ptolemy’s Almagest gives more of an impression of the range and variety of Greek mathematics 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 geometry was regarded with the utmost respect and consequently none were held in greater honour than mathematicians, 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 Latin equivalent of Euclid, Archimedes 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 angle contained in Cicero’s statement (ideas of ‘the Roman’ and ‘the Greek’ were marks of different kinds of prestige, while many users of numbers, 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 Euclidean geometry. 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 Hypatia, the daughter of Theon the mathematician, was initiated in her father’s studies; her learned commentaries have illuminated the geometry of Apollonius and Diophantus, and she publicly taught, both at Athens and Alexandria, the philosophy of Plato 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. Ancient Greek works were torched and scholars were murdered. Hypatia was the last proprietor of the Hellenic Age wonder, the Library of Alexandria. 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.