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Geometry

Antiquity And The Middle Ages

Unfortunately there is little evidence of the transmission of geometrical knowledge from either Egypt or Babylonia to the emerging Greek culture. Significantly, the Greeks seem to have been interested in proof, and the nature of mathematical knowledge, in a way that these other cultures were not. Plato's dialogues display these features in a dramatic way. In the Meno, for example, Plato has Socrates ask a slave boy about the diagonal of a square. What Socrates elicits is a comparison between the square of the diagonal and the square on the side; not a numerical answer, and not an approximation to 2, but an argument accompanied by a proof.

The advent of proof permitted an important discovery: 2 is what we would call an irrational number: there are no integers p and q such that 2 p/q. Historians used to present this discovery as momentous. Allegedly the mathematics of earlier, Pythagorean, times was based on the idea that everything could be counted, any two lengths could be regarded as multiples of a common measure. The incommensurability of the side and diagonal of a square put an end to that belief and caused a crisis in Greek mathematics. In the late twentieth century, however, historians retreated from this position. The only evidence for it is very late, and no contemporaneous evidence suggests a crisis. Rather, as Plato's dialogues suggest, there might have been surprise and excitement. The slave boy, after all, gave an acceptable answer. The existence of incommensurable pairs of lengths greatly complicated the theory of proportion, which is attributed to Eudoxus of Cnidus (c. 400–c. 350 B.C.E.) and presented in books 5 and 6 of Euclid's (fl. c. 300 B.C.E.) Elements, but again there is no suggestion of a crisis.

The paradoxes of Zeno.

Further evidence of the sophistication of Greek thought is found in Zeno of Elea's (c. 495–c. 430 B.C.E.) paradoxes, which survive only in the form of a somewhat hostile account by Aristotle (384–322 B.C.E.). Zeno aimed to show that the analysis of motion led inevitably to contradictions. Achilles may never catch a tortoise, because each time he runs to where the tortoise was it is still ahead, committing Zeno to an infinite chase. Indeed, by a somewhat similar argument he cannot get started. An arrow cannot move in an instant; therefore, it is at rest in every instant of its flight and therefore always at rest. Whatever Zeno's intention in proclaiming them, his paradoxes testify to a deep-seated interest in logical reasoning, and they continued to attract interest.

The notion of proof.

Much of Greek mathematics would be impossible without good notions of proof. The simplest form of proof was proof by showing, in which arrangements of pebbles were used to show such results as the sum of two odd numbers is even. Zeno's paradoxes display another form of reasoning, called reductio ad absurdum, in which a proposition is refuted by showing that it leads logically to a self-contradiction or other evident impossibility. This method was used extensively by Archimedes in his estimation of areas and volumes, and also earlier by Euclid in his Elements, for example when he showed that there are infinitely many prime numbers. For, if there are not, then there are only finitely many prime numbers, p1, p2, …, pn say, in which case the number p1 p2pn 1 is larger than any of these, so it cannot be prime, and yet it is divisible by none of them, so it must be prime.

Proofs in geometry turn approximate estimates based on a finite number of cases into certain knowledge. For example, the assumptions made at the start of Euclid's Elements, including the parallel postulate as described below, permitted Euclid to prove that the angle sum of a triangle is exactly two right angles by exhibiting a suitable pair of parallel lines, to prove Pythagoras's theorem by moving areas around, and, ultimately, to show that there are exactly five regular solids.

Euclid's Elements and the axiomatization of geometry.

The most impressive form of proof, however, in Greek mathematics is the axiomatic method, developed at length in Euclid's Elements. The aim, not perfectly honored but impressively so, was to state definitions of the fundamental terms, gives rules for what may be said about them, and then to derive truths successively from this base of assumptions (the axioms). The result is that later propositions in each book of the Elements depend in an elaborate, tree-like way, on the earlier ones, and confidence in these results depends on the transparency of the proofs and the quality of the original axioms.

Apollonius and Archimedes.

One of the intellectual high points of Greek mathematics is undoubtedly Apollonius of Perga's (c. 262–c. 190 B.C.E.) theory of conic sections. It is forbiddingly austere, but it goes a long way to creating a unified theory of all (nondegenerate) plane sections of a cone: the ellipse, parabola, and hyperbola. The names derive from the way their construction is shown to produce an area that falls short, is equal to, or exceeds another area (compare the terms for figures of speech: elliptical, or of few words; a parable is exact, hyperbole an exaggeration). The comparisons of areas yield a proportion, which is modernized as the equation of the curves, and Apollonius shows in some detail how the equation may be simplified by suitable geometric choices and how properties of the conic sections may be obtained, such as the focal properties of conics and the construction of tangents.

Archimedes (c. 287–212 B.C.E.), a near contemporary of Apollonius, has earned a reputation as the greatest of the Greek geometers not only for the brilliance of his achievements, but also perhaps because they are easier to admire. He found volumes of sections of cones and various solid figures, he was the first to show that the constant that enters the formulas for Figure 1. The parallel postulate the circumference and the area of a circle is in fact the same, and he also made a number of practical and mechanical discoveries. He also left a unique account, known as the Method, of how he came to some of his discoveries by heuristic means, regarding areas as made up of lines that could be moved around. A tenth-century copy of this account was discovered in 1906 in a monastery in Istanbul. It was then lost again, but reappeared in 1998, when it was put on auction and sold for the surprisingly small sum of $2.2 million.

Arabic and Islamic work on geometry.

Islamic scholars did much more than simply transmit Greek ideas to the later West, as some accounts have suggested. They far surpassed all previous cultures in geometric design. They also produced the most penetrating analyses of the single most obvious weakness in all of Euclid's Elements: the parallel postulate. Euclid had assumed that if two lines m and n cross a third, k, and the angles and the lines m and n make with k are less than two right angles on one side of the line (in the figure 2 right angles) then the lines will meet on that side of the line if they are produced sufficiently far (see fig. 1).

Unlike all Euclid's other assumptions, the parallel postulate makes claims about what happens arbitrarily far away and so could be false. However, very few theorems can be proved unless the parallel postulate is known, so mathematicians were in a quandary. Greek and still more Islamic commentators took the view that it would be better to drop the parallel postulate from the list of axioms, and to derive it instead from the other axioms as a theorem.

Remarkably, from Thabit ibn Qurrah (c. 836–901) to Nasir ad-Din at-Tusi (1201–1274), they all failed. To give just one example, Ibn al-Haytham (Alhazen; 965–1039) assumed that if a segment of fixed length and perpendicular to a given line moves with one endpoint on the line then the other end point draws a straight line, parallel to the given line. Certainly, the parallel postulate follows as a theorem if one may make this assumption, and the parallel postulate implies it, but this only invites the question: how can the assumption itself be proved, or is it merely an alternative assumption to the parallel postulate? Some years later, Omar Khayyám (1048?–?1131) objected to the assumption on just these grounds, arguing that it was an illegitimate use of motion in geometry to attempt to define a curve this way, still more to assume that the curve so produced was a straight line.

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