60 AHistory ofMathematics including Archimedes, Heron of Alexandria, and Ptolemy.All were tremendously influential in the later development of mathematics, and all raise interesting questions about the varieties of Greek mathematics which were (one might guess) competing for influence over the long historical period which concerns us. At the same time, because of the length of the period, and the variety of the work produced, it would be impractical in a book of this kind to try to cover everything. In particular for the important work of Apollonius, Diophantus, and Pappus, you will have to look elsewhere. The remarks on sources made in the previous chapter apply on the whole. The major historian who has recently concentrated attention on the late period is Cuomo, to whose works (2000) and (2001) we shall return in due course. Exercise 1. Check that Menaechmus’s construction does give a line of length a√3 2. How would you generalize it to solve the problem of increasing the volume of the cube by a factor m? Exercise 2. Hippocrates of Chios (fifth century bce) showed that the general problem (multiplying a cube by m) can be solved if, between two given linesA,B, with the ratioB:A m, one can construct two ‘mean = proportionals’C,D; that is so that the ratiosA:C,C:D,D:B are equal. Why is this true? 2 Archimedes Archimedes is one of the most heroizedfigures in the history of science; but unlike Galileo and Newton, whose lives are available in minute detail, we know rather little about him. There is a growing literature on him; not so much ‘biographical’ as an attempt to understand him from his works. True, his life is better documented than that of any other Greek mathematician (with the possible exception of Hypatia), but that is not saying much. The chief sources tend to concentrate on a few memorable events—the ‘Eureka story’, his role in the siege of Syracuse, and his death at the hands of a Roman soldier. His works have always been seen as uniquely brilliant and difficult, and perhaps his portrait has been constructed tofit them; though unusually, there are letters introducing several of the writings which are ‘personal’ as not much else is in Greek mathematics. A late portrait of Archimedes as the absent-minded pure researcher is given in Plutarch’s Life of Marcellus, and for whatever reason it has become influential. In line with a Platonic propagandist viewpoint, Plutarch (while crediting Archimedes with major military inventions), claims that such practical considerations were unimportant to him. Yet Archimedes possessed so high a spirit, so profound a soul, and such treasures of scientific knowledge, that though these inventions had now obtained him the renown of more than human sagacity, he yet would not deign to leave behind him any commentary or writing on such subjects; but, repudiating as sordid and ignoble the whole trade of engineering, and every sort of art that lends itself to mere use and profit, he placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life; studies, the superiority of which to all others is unquestioned, and in which the only doubt can be whether the beauty and grandeur of the subjects examined, or the precision and cogency of the methods and means of proof, most deserve our admiration. It is not possible tofind in all geometry more difficult and intricate questions, or more simple and lucid explanations. Some ascribe this to his natural genius; while others think that incredible effort and toil produced these, to all appearances, easy and unlaboured results. No amount of investigation of yours would succeed in attaining the proof, and yet, once seen, you immediately believe you would have discovered it; by so smooth and so rapid a path he leads you to the conclusion required. (Plutarch, in Fauvel and Gray 4.B.1) Greeks,Practical andTheoretical 61 One suspects that Plutarch had not read the works which he describes as ‘smooth and rapid’, since later generations have found them impressive but difficult. The geometrical core, which includes the Measurement of a Parabola and On the Sphere and the Cylinder carries on, with great ingenuity, from the harder parts of Euclid; we shall not deal with them here, but there are good extracts in Fauvel and Gray (see also Archimedes 2002). There is, however, more to Archimedes than these works suggest, and some of his other surviving works contradict Plutarch’s image of the ‘pure’ mathematician. The Statics and On Floating Bodies are the most serious works of theoretical physics, outside the framework of Aristotle’s thought, in the Greek tradition; and as such, they had a great influence in the Renaissance, particularly on Galileo—see Chapter 6. Further evidence of a mechanical tendency in Archimedes is provided by the strange document called the ‘Method’. Extravagant claims have been made for this manuscript,2 for example, that it contains a version of the calculus, and that the course of history would have been changed if it had not been ‘lost’. There is no need for such exaggeration; The Method is, so far as we know, a very unusual work which had no imitators, and for good reason. In his introductory letter to Eratosthenes, Archimedes describes what he is doing, and why: Seeing moreover in you, as I say, an earnest student, a man of considerable eminence in philosophy, and an admirer [of mathematical inquiry], I thoughtfit to write out for you and explain in detail in the same book the peculiarity of a certain method, by which it will be possible for you to get a start to enable you to investigate some of the problems in mathematics by means of mechanics. This procedure is, I am persuaded, no less useful even for the proof of the theorems themselves; for certain thingsfirst became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the same method did not furnish an actual demonstration. (From Archimedes tr. Heath, in Fauvel and Gray 4.A9 (a)) The ‘Method’ referred to consists of measuring the areas of bodies (e.g. a segment of a parabola) by ‘balancing’ them against simpler bodies (e.g. a triangle), using a division into infinitely thin slices. (See Fauvel and Gray 4.A9(a) for an example.) Two things are striking here:first, the use of weighing as a guide to understanding, presumably inspired by the work in the Statics—this is the ‘applied side’ of Archimedes; and second, the insistence, in the letter quoted above, that this is not a proof, but that a proof has to be constructed once you have found the answer. (And, in some sense, that it clarifies why the answer is what it is.) I should stress that the fact that The Method is an applied work does not make it an easy read; if it had been, perhaps it would have been preserved and quoted more. To describe it as ‘lost’ is only partly accurate; someone in the ninth century, and various others before that, must have known it and thought it of enough interest to be worth copying. However, it had no influence on the later traditions, either through Byzantium or the Islamic world, so far as we know; and this although some Islamic mathematicians had a great respect for Archimedes and worked hard to reconstruct alleged works of his which they did not have. In contrast, one work of Archimedes had tremendous influence, and still does. This was his Measurement of a Circle. It is very short—it is thought that it is only part of a longer work of which the rest has been lost; but what remained was found immensely useful by much more simple- minded mathematicians. The three theorems which it contains are worth quoting in full, as a typically Greek way of approaching what we would call the problem of calculatingπ: Proposition 1. The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius and the other to the circumference of the circle. 2. Discovered by Heiberg in Istanbul in 1906, then lost again, but recently rediscovered, sold at Christie’s for $2 m., and subjected to modern scientific reading methods. 62 AHistory ofMathematics Proposition 2. The area of any circle is to the square on its diameter as 11 is to 14. Proposition 3. The circumference of any circle exceeds three times the diameter by a quantity that is less than one-seventh of the diameter but greater than ten parts in seventy-one. It is clear that Proposition 2 is both wrongly placed (it depends on Proposition 3) and probably not as Archimedes stated it (it claims as exact what is recognized in Proposition 3 to be an approxi- mation). This of course added to the confusion of medieval readers, who tended to go for the more usable Proposition 2; but at different times, all three parts were found useful.3 Thefirst states (in 1 our terms) that the areaA is 2 rC, wherer is the radius andC is the circumference. The Greeks sometimes worried, as we would not, whether this implied the necessary existence of a straight 11 2 22 2 line whose length was equal to the curved lineC. The second states thatA 14 (2r) ( 7 r ). 1 = = The third also gives the approximation 3 7 for the ratio ofC to 2r which is still used after over 2000 years, and was gladly taken as the ‘right’ answer by calculators who had no use for Archimedes’ more precise formulation: 10 C 1 3 < <3 71 2r 7 1 When we use the approximation 3 7 forπ, we are therefore indebted to Archimedes, although we probably know nothing of his methods.