Chapter 1. Classical Greek Mathematics

Chapter 1. Classical Greek Mathematics Greek science and mathematics is distinguished from that of earlier cultures by its desire to know, in contrast to a need to make purely utilitarian advances or improvements. Greek geometry displays abstract and deductive elements which were largely lost during the Dark Ages, following the collapse of the Roman Empire, and only gradually recovered in the 16th and 17th centuries. It must be understood that many of the great discoveries in geometry were made about two and half thousand years ago. Given the difficulty of preserving fragile manuscripts, written on parchment or papyrus, over centuries when warfare could wipe out civilizations, it is not too surprising to find that we do not have many reliable records about the origin of Greek geometry or of its practitioners. We may count ourselves lucky that a few commentaries on Greek geometry, written in the fourth or fifth centuries of the present era, have survived to provide us with what details we have. Greeks from Ionia had settled in Asia Minor and there they had contact with two ancient civilizations, those of Babylon and Egypt. Although knowledge of science was elementary among the Babylonians and Egyptians, nonetheless they supplied the initial impetus that directed the Greeks towards the pursuit of systematic science. All authorities trace the beginnings of Greek geometry to Egypt. The rudimentary study of geometry in Egypt arose out of practical needs. Revenue was raised by taxation of landed prop- erty, and its assessment depended on the accurate fixing of the boundaries of fields in the possession of the landowners. The landmarks were constantly removed by the periodic flooding of the Nile and it became necessary to determine the taxable area by a technique of land surveying. (The Greek historian Herodotus believed that basic knowledge of geometry originated from the recurrent need to measure land after inundation by the Nile. Aristotle, on the other hand, believed that mathematics was the invention of Egyptian priests with the time and leisure to speculate on abstract things.) Egyptian geometry was limited mainly to mensuration of areas and volumes, in addition to what was required to construct pyramids. The written records show a collection of practical rules for measuring the areas of squares, triangles, trapeziums and circles, these rules often being imprecise, and for estimating the volume of various measures of corn in different shapes. The Egyp- tians also required familiarity with the notion of similarity of triangles, to enable them to construct pyramids of different slopes. We have some knowledge of ancient Egyptian mathematics, thanks to the survival of a few papyri. The most complete is the Rhind Mathematical Papyrus, now in the British Museum. It was copied by the scribe Ahmes 1 around 1650 BCE, and was itself copied from documents written before 2000 BCE. It contains 85 mathematical problems and solutions. Another important papyrus relating to mathematics is in Moscow. We now present a little history of those considered to be the founders of Greek mathematics or logical method. • Thales (c. 624-546 BCE) Thales is considered to be the founder of Greek geometry. He was born in Miletus, a town now in modern Turkey (Asia Minor). He was also an astronomer and philosopher. He was held in high regard by the ancient Greeks, and named as one of the seven ‘wise men’ of Greece. (The seven wise men or sages of ancient Greece were held to be: Thales, Solon, Periander, Cleobulus, Chilon, Bias and Pittacus. Various legends grew up about them, and they are mentioned in the writing of Plato. They seem to have been astute politicians and businessmen, rather than great thinkers.) He is said to have made a prediction of a solar eclipse which, according to the famous historian Herodotus, occurred during a battle of the Medes and the Lydians. Modern astronomers have dated this eclipse to 28 May, 585 BCE, which serves to give us some idea of the dates of Thales. While it is doubted if someone could have predicted an eclipse so accurately at the given date, the story of its happening assured his fame. (The Babylonians already knew that the eclipses followed a cycle of 223 lunations, and Thales may have learnt this at some stage in his journeys.) Various stories about Thales have come down to us from historians, especially Dio- genes Laertius. One story relates that he travelled to Egypt, where he became acquainted with Egyptian geometry. While, as we noted above, the Egyptian approach to geometry was essentially practical, Thales’s work was the start of an abstract investigation of geometry. The following discoveries of elementary geometry are attributed to Thales. • A circle is bisected by any of its diameters. • The angles at the base of an isosceles triangle are equal. • When two straight lines cut each other, the vertically opposite angles are equal. • The angle in a semicircle is a right angle. • Two triangles are equal in all respects if they have two angles and one side respec- tively equal. 2 He is also credited with a method for finding the distance to a ship at sea, and a method to determine the height of a pyramid by means of the length of its shadow. It is not certain whether this implies that he understood the theory of similar (equiangular) triangles. Thales may be considered to have originated the geometry of lines, which forms a basic part of elementary geometry. We also see in the theorems of Thales the first attempt to find order and constancy in the midst of geometric data. It seems that he passed on no written work to later generations, so we must rely on traditional stories, not all likely to be true, for our information about him. The commentator Proclus (whom we will discuss in more detail later), writing almost one thousand years after the time in which Thales flourished, says that Thales first brought knowledge of geometry into Greece after his time spent in Egypt. There is controversy among modern historians of mathematics about the extent of Thales’s discoveries, for as we observed, Egyptian geometry was rudimentary, had no theoretical basis, and consisted mainly of a few techniques of mensuration. It is also considered unlikely that Thales could have obtained theoretical proofs of the theorems attributed to him, but he may guessed the truth of the results on the basis of measurements in particular cases. • Pythagoras (c. 582–c. 500 BCE) It is believed that Pythagoras was born around 582 BCE, in Samos, one of the Greek islands. He had a reputation of being a highly learned man, a reputation that endured for many centuries. He is said to have visited Egypt and possibly Babylon, where he may have learnt astronomical and mathematical information, as well as religious lore. He emigrated around 529 BCE to Croton in the south of Italy, where a Greek colony had earlier been founded. He became the leader there of a quasi-religious brotherhood, who aimed to improve the moral basis of society. After opposition developed to the influence of his followers, he moved to Metapontum, also in the south of Italy, where he is thought to have died around 500. While geometry was introduced to Greece by Thales, Pythagoras is held to be the first to establish geometry as a true science. It is difficult to distinguish the work of the followers of Pythagoras (the Pythagoreans, as they are called) from that of Pythagoras himself. Part of the problem is that later Pythagoreans tended to refer everything to the master, remarking that he himself has said it. Furthermore, the Pythagorean school only transmitted their knowledge orally and left no written record of their work. By the time 3 of Aristotle (4th cent BCE), precise knowledge of any aspects of Pythagorean ethical or physical theories was lacking. The Pythagorean school was effectively divided into two groups. There was an inner (esoteric) circle, whose followers had learnt the Pythagorean theory of knowledge in its entirety. They were called mathematikoi (mathematicians). The outer (exoteric) circle, who knew only the Pythagorean rules of conduct, were called akousmatikoi (hearers). The word mathematics derives from the Greek mathema (µαθηµα), which means that which is learnt. In Plato’s writing, this word is used for any subject of study or instruction, although with a tendency to restrict it to the studies now called mathematics. A number of statements regarding the Pythagoreans have been transmitted to us, among which are the following. • Aristotle says “the Pythagoreans first applied themselves to mathematics, a science which they improved; and penetrated with it, they fancied that the principles of mathematics were the principles of all things.” • Eudemus, a pupil of Aristotle, and a writer of a now lost history of mathematics, states that “Pythagoras changed geometry into the form of a liberal science, regarding its principles in a purely abstract manner, and investigated its theorems from the immaterial and intellectual point of view.” • Aristoxenus, who was a musical theorist, claimed that Pythagoras esteemed arith- metic above everything else. (“All is number” is a motto attributed to Pythagoras.) • Pythagoras is said to have discovered the numerical relations of the musical scale. Concerning the geometric work of the Pythagoreans we have the following testimony. • Eudemus states that the theorem that the sum of the angles in a triangle is two right angles is due to the Pythagoreans and their proof is similar to that given in Book 1 of Euclid’s Elements. • According to Proclus, they showed that space may be uniformly tesselated by equi- lateral triangles, squares, or regular hexagons. • Eudemus states that the Pythagoreans discovered the five regular solids.

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