Classical Greek Geometry - 1
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.
We cannot give a systematic account of how Greek geometry came into existence and how it was perfected, so we must confine ourselves to describing a few generally agreed highlights. Thales (c. 624-546 BCE) 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. 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.
Various stories about Thales have come down to us from historians. One story relates that he travelled to Egypt, where he became acquainted with Egyptian geometry. While 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 respectively equal.
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. 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. With regard to the state of Egyptian geometry, 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. There is controversy among modern historians of mathematics about the extent of Thales’s discoveries. It is first noted that 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.
The next major figure in the history of Greek geometry is Pythagoras. He is thought to have been 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, and the Pythagoreans published none of their work. Thus it is not possible to ascribe accurately any given work to Pythagoras himself.
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 arithmetic 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.
• Proclus says that “the word ‘mathematics’ originated with the Pythagoreans.” (The word ‘mathematics’ means ‘that which is learned’, with connotations of knowledge and skill.)
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 equilateral triangles, squares, or regular hexagons.
• Eudemus states that the Pythagoreans discovered the five regular solids.
• Heron of Alexandria and Proclus ascribe to Pythagoras a method of constructing right–angled triangles whose sides have integer length.
• Eudemus ascribes the discovery of irrational quantities to Pythagoras.
We should now address some of the issues that arise from these claims, as they are not accepted by all commentators and, indeed, some are implausible. The theorem that the sum of the angles in a triangle is two right angles is not provable without recourse to Euclid’s fifth or parallel postulate. This is a highly subtle point and any proof by the Pythagoreans that the sum is constant must have had some implied appeal to the parallel postulate. The Greek historian Plutarch tells us that the Egyptians knew of the right-angled triangle whose sides have lengths equal to 3, 4, and 5 units, and that in this case they observed that the square of the hypotenuse equals the sum of the squares of the other two sides. Other versions of this arithmetical construction seem to have been known earlier in Babylon. More generally, positive integers a, b and c are said to form a Pythagorean triple (a, b, c) if a2 + b2 = c2. It seems to have become known at some time that such Pythagorean triples may be used to form the sides of a right angled triangle, where the hypotenuse has length c units, and so on. Proclus has described a method of finding such Pythagorean triangles using an odd integer m, which he attributes to Pythagoras. We take an odd integer m and set
Note that both b and c are integers, because m is odd. It is straightforward to verify that (a, b, c) is a Pythagorean triple, and this is the method used by Pythagoras to generate such triples. There seems to be agreement that what we know as the theorem of Pythagoras concerning right-angled triangles is not due to Pythagoras or the Pythagoreans. A proof of the general theorem is found as Proposition 47 in Book 1 of Euclid’s Elements, but is more complicated than the proof that would be given nowadays, using the theory of similar triangles.
We will have more to say about the five regular or Platonic solids later in this chapter. Suffice it to say here that the five regular solids are the tetrahedron, cube, octahedron, dodecahedron and icosahedron. The regular tetrahedron, cube, octahedron are of ancient origin and may be seen in Egyptian architecture. Thus it cannot be said that the Pythagoreans discovered these solids. Specialists now agree that it is unlikely that the Pythagoreans discovered the other two regular solids, the dodecahedron and icosahedron,
whose constructions are less obvious. Instead, it seems that Theaetetus, an Athenian who died in 369 BCE, discovered the other two regular solids and wrote a study of all five solids. It is possible that he proved that only five different types of regular solid exist (this is a theorem in Euclid’s Elements). Theaetetus was associated with Plato and his Academy in Athens, and his death was commemorated by Plato’s dialogue entitled Theaetetus. This dialogue also contains information on irrational numbers, which had recently been discovered and had caused a furore in mathematical and philosophical circles of the time. Theaetetus was associated with some of this work on irrationals. The regular solids are also called Platonic solids, because of the importance they held in the teaching of Plato. He used the solids to explain various scientific phenomena. Indeed, the four elements (earth, air, fire and water) were associated with the five regular solids in a cosmic scheme that fascinated thinkers well into Renaissance times.
Concerning irrational quantities, we encounter some problems about the Greek concepts of magnitude and number. Magnitudes are what we would call continuous quantities, such as lengths of lines or areas of plane figures. Number is a discrete quantity, such as an integer. Aristotle made a distinction between these two quantities. A magnitude is that which is divisible into divisibles that are infinitely divisible, while the basis of number is the indivisible unit. The Pythagoreans did not make such a distinction, as they considered number to be the basis of everything and believed that everything can be counted. To count a length, one needed a unit of measure. Once this unit was chosen, it was indivisible. They then assumed that it is possible to choose a unit so that the diagonal and side of a square can both be counted. This was eventually shown to be untrue–the precise time is uncertain. As was said in Greek mathematics, the lengths of the diagonal and the side of a square are incommensurable–they do not possess a common unit of measure. Nowadays we would say that, in its initial stages, the Greek theory of numbers essentially held that all numbers are rational. The consensus now is that the discovery of incommensurable magnitudes, or equivalently, of irrational quantities, is not due to Pythagoras or the group associated with him, but to later Pythagoreans, around 420 BCE.
The modern approach to the question is quite straightforward. Suppose that we have a unit square. Then by the Pythagoras theorem, if c is the length of the diagonal, c2 = 2. Now we claim that c cannot be a rational number, that is, it cannot be expressed as a quotient of two integers. For suppose thatwhere r and s are integers. We can
assume that r and s have no common factors. Then, on squaring, we obtain
2s2 = r2.
Since 2s2 is an even integer, r2 is even and thus r is even. Thus we can write r = 2t, where t is an integer. Substituting, we obtain s2 = 2t2, and hence s is also even. This contradicts the assumption that r and s have no common factor. Rather similar arguments can be used to prove that several other square roots of integers are irrational. This was already known in the circles around Plato. Indeed, in his dialogue Theaetetus, Plato says that his teacher, Theodorus of Cyrene, also the teacher of Theaetetus, had proved that the square root of any non-square integer between 3 and 17 is irrational. From the modern point of view, this is easy to prove, as the the square root of any non-square integer is irrational. Presumably the method used by Theodorus involved specific arguments that missed the full mathematical generality.
The personality and thought of Plato play a large role in the history of Greek mathematics, so we should say something about his life and influence. Plato (427-347 BCE) is known primarily as a philosopher but he was an important promoter of mathematics, especially geometry. He founded the famous Academy in Athens, around 380 BCE, which became a centre where specialists met and discussed intellectual topics. Innovative mathematicians, including Theodorus of Cyrene, Eudoxus of Cnidus, Theaetetus and Menaechmus, are closely associated with the Academy. Plato himself made no significant contribution to creative mathematics, but he inspired others to ground-breaking work and guided their activity. It is said that over the doors of his school the motto “Let no one ignorant of geometry enter” was written. The authenticity of this claim is doubtful, as the earliest reference to it occurs in the sixth century CE, but nonetheless it encapsulates the spirit of his Academy.
We know much detail about Plato’s life and career, and virtually all his writings have survived. The source for much of our information about Plato, and indeed about many other philosophers, is “Lives of the Philosophers” by Diogenes Laertius (3rd century CE?). Diogenes has been described as a mere compiler and anecdote-monger, and his testimony cannot always be trusted, but he seems to be reliable on many aspects relating to Plato (he provides, for example, Plato’s will).
Plato became associated with Socrates, close to the trial and execution of the latter for impiety in 399. Plato was impressed by Socrates’s use of the art of argument and his
search for truth, but we should note that Socrates was himself no enthusiast for mathematics. Plato felt that it was his duty to defend Socratic ideas and methods, and conceived the notion of training the young men of Athens in the discipline of mathematics and then, when mentally ready, in Socratic interrogation. This was to counteract what he saw as the problem of young people bewildering themselves in philosophical enquiry at too early an age.
Around the year 390, Plato visited Sicily, where he came under the influence of Archytas of Tarentum, a follower of the Pythagoreans. Archytas studied, among other mathematical topics, the theory of those means that are associated with Greek mathematics: the arithmetic, geometric and harmonic means. Plato returned to Athens in 388, and in the next twenty years, his Academy came into existence. The purpose of the Academy was to train young people in the sciences (mathematics, music and astronomy) before they undertook careers as legislators and administrators. The two main interests of the Academy were mathematics and dialectic (the Socratic examination of the assumptions made in reasoning). While Plato regarded the study of mathematics as preparatory to the study of dialectic, he nonetheless believed that the study of arithmetic and plane geometry, as well as the geometry of solids, must form the basis of an education leading to knowledge, as opposed to opinion. Plato’s teaching at the Academy was assisted by Theaetetus, whom we have mentioned above. Eudoxus of Cnidus, a pupil of Archytas and an important contributor to the emerging Greek theory of magnitude and number, also taught from time to time at the Academy. Plato’s role in the teaching at the Academy was probably that of an organizer and systematizer, and he may have left the specialist teaching to others. The Academy may be seen as a place where selected sciences were taught and their foundations examined as a mental discipline, the goal being practical wisdom and legislative skill. Clearly, this has relevance to the nature of university learning nowadays, especially as it relates to the conflict between a liberal education, as espoused by Plato, and vocational education with some special aim or skill in mind.
Plato’s enthusiasm for mathematics is described by Eudemus, writing some time after the death of Plato:
• Plato . . .caused the other branches of knowledge to make a very great advance through his earnest zeal about them, and especially geometry: it is very remarkable how he crams his essays throughout with mathematical terms and illustrations, and everywhere tries to arouse an admiration for them in those who embrace the study
Aristotle (384-322 BCE), the famous philosopher and logician, came to Athens in 367 and became a member of Plato’s Academy. He remained there for twenty years, until Plato’s death in 347. As we noted above, in Plato’s time, dialectic was of primary importance at the Academy, with mathematics an important prerequisite. Aristotle held that the mathematical method then being developed was to be a model for any properly organized science. Greek mathematics at the time was distinguished by its axiomatic method, and sequence of reasoning, from which irrefutable theorems are derived. Aristotle required that any science should proceed as mathematics does, and the mathematical method should be applied to all sciences.
Aristotle is important for laying down the working method for each demonstrative science. Writing in his Posterior Analytics, he says:
• By first principles in each genus I mean those the truth of which it is not possible to prove. What is denoted by the first terms and those derived from them is assumed; but, as regards their existence, this must be assumed for the principles but proved for the rest. Thus what a unit is, what the straight line is, or what a triangle is must be assumed, but the rest must be proved. Now of the premises used in demonstrative sciences some are peculiar to each science and others are common to all . . .Now the things peculiar to the science, the existence of which must be assumed, are the things with reference to which the science investigates the essential attributes, e.g. arithmetic with reference to units, and geometry with reference to points and lines. With these things it is assumed that they exist and that they are of such and such a nature. But with regard to their essential properties, what is assumed is only the meaning of each term employed: thus arithmetic assumes the answer to the question what is meant by ‘odd’ or ‘even’, ‘a square’ or ‘a cube’, and geometry to the question what is meant by ‘the irrational’ or ‘deflection’ or the so-called ‘verging’ to a point.
Aristotle notes that every demonstrative science must proceed from indemonstrable principles; otherwise, the steps of demonstration would be endless. This is especially apparent in mathematics. He discusses the nature of what is an axiom, a definition, a postulate and a hypothesis. It is quite difficult to distinguish between a postulate and a hypothesis. All these terms play a leading role in Euclid’s Elements.
Aristotle’s influence on later European thought was immense. For many centuries,
virtually all Greek learning, except that of Aristotle, fell into oblivion. Aristotle was held to be the basis of all knowledge. Universities and grammar schools were founded with the study of Aristotle as their main intellectual activity. We see the extent of his influence even now by noting how many Aristotelian words have survived in modern use, for example: principle, maxim, matter, form, energy, quintessence, category, and so on. It was really only in the Renaissance that the authority of Aristotle was questioned and supplanted.
We have little reliable knowledge about the lives of the early Greek geometers, and our best sources are the Alexandrian mathematician Pappus (exact dates unknown, probably third century CE) and the Byzantine Greek mathematician Proclus (410-485 CE), who both lived many centuries after the golden age of Greek geometry had ended. Proclus, who wrote a commentary on the first book of Euclid’s Elements, is our main authority on Euclid. He states that Euclid lived in the time of Ptolemy I, king of Egypt, who reigned 323-285 BCE, and that Euclid was younger than the associates of Plato (active around 350 BCE) but older than Eratosthenes (276-196 BCE) and Archimedes (287-212 BCE). Euclid is said to have founded the school of mathematics in Alexandria, a city that was becoming a centre of commerce, and of learning, following its foundation around 330 BCE. Proclus has preserved a famous incident relating to Euclid. On being asked by Ptolemy whether he might learn geometry more easily than by studying the Elements, Euclid replied that “there is no royal road to geometry”. The exact dates of Euclid, his place of birth, and details of his life are not known, but we can say that he flourished around 300 BCE.