The Greek Age of Mathematics c Ken W. Smith, 2012 Last modified on February 15, 2012 Contents 2 The Greek Age 2 2.1 Thales, the Pythagoreans and the first Greek proofs . .2 2.1.1 The Pythagoreans . .5 2.1.2 Pythagorean numerology . .5 2.1.3 Commensurable numbers . .5 2.1.4 The impact of noncommensurability . .7 2.1.5 Two types of Pythagorean triples . .8 2.1.6 Polygonal numbers . .8 2.2 The Euclidean tools . 10 2.2.1 The Euclidean calculator . 10 2.2.2 The quadratic formula . 11 2.2.3 The Fundamental Theorem of Algebra and Euclid's lemma . 11 2.2.4 Quadrature . 12 2.3 Euclid's Elements ......................................... 13 2.3.1 The fabric of mathematics . 13 2.3.2 The preamble to the Elements ............................. 14 2.3.3 Propositions from Book I . 15 2.3.4 Euclid's Proposition I.47 . 18 2.4 Elements,PartII......................................... 21 2.4.1 Books II, III, IV . 21 2.4.2 Triangles and their centers . 21 2.4.3 Books V and VI . 22 2.4.4 Books VII, VIII, IX . 22 2.4.5 Book X . 23 2.4.6 Books XI, XII, XIII . 23 2.5 The Greeks after Euclid . 24 2.5.1 Eratosthenes and the Greek view of the universe . 24 2.5.2 Chords and trigonometry . 25 2.5.3 Archimedes . 26 2.5.4 Apollonius and conic sections . 26 2.5.5 Volumes and Areas . 27 2.6 Conclusion of the Greek Age . 34 2.6.1 Diophantus and Diophantine equations . 34 2.6.2 Primitive Pythagorean Triples . 34 2.6.3 The Euclidean algorithm . 35 2.6.4 The three problems of antiquity . 37 2.6.5 Pappus, Theon and the commentators . 37 2.6.6 Truth & Falsehood . 38 2.6.7 References for Greek mathematics . 38 1 2 The Greek Age Following Wikipedia's article on ancient Greece we will set the dates of ancient Greece from 750 BC to 1 592 AD, a period of 13 2 centuries. The most relevant period for our studies { the time of greatest growth { occurs in the classical period of Greek history. During the classical period of Greece (500-323 BC) we see the rise of an intelligentsia of philoso- phers and mathematicians, people who, with leisure time, asked \big questions" about life, science and the universe. During this time, we see the rise of two city-states (\polis") Athens and Sparta. It is in Athens that the philosopher Plato built his Academy and other philosophers such as Socrates, Sopho- cles and Aristotle lectured and built their schools and groups of disciples. Much of western (European and American) civilization owes it worldview and political system to those developed in Athens during the classical period of Greece. The classical period ended with the unification of Greece by Alexander the Great, the creation of the city of Alexandria and then Alexander's death in 323 BC. The Hellenistic period (323-146 BC) extended from the death of Alexander and the division of his empire to the conquest of Greece by the Romans. It was a time of expansion of Greek culture and philosophy. After 146 BC Greece was controlled by Rome but the Greek ideals and Greek language permeated the Roman empire. The Greco-Roman empire persisted through emperors Julius and Augustus Caesar and into the rise of Christianity, until in 330 AD, Constantine changed the empire forever by officially making it \Christian." The new \Christian" culture replaced the Greek culture and the Greek influence declined. (During this time, the female mathematician, Hypatia, was murdered by a Christian mob.) The Greek age ended with the closing of the last neoplatonic academy by the emperor Justinian in 529 AD. Axiomatic systems The Greeks during their time took mathematics to a higher level, far exceeding the works of the Baby- lonians and Egyptians. In particular, the Greeks included mathematics within a broad rational approach to philosophy and scientific investigation. Mathematical \facts" were created by careful reasoning and logical argument. One was allowed to ask, \Why is that true?" and to expect a careful explanation as an answer. A natural result of the Greek rational approach to mathematics was an axiomatic system based on a collection of foundational axioms on which all other arguments rely. The axiomatic system was best displayed by Euclid when (in 300 BC) he wrote a comprehensive treatise on all the areas of mathematics, all the \elements" of mathematics. (We will explore Euclid's Elements in a later section.) Mathematics continues to be based on a careful axiomatic system. The concept of axiomatic system underlies our exploration of Greek history. 2.1 Thales, the Pythagoreans and the first Greek proofs Beginning around 750 BC, the Greeks asked philosophical questions. \Why?" \Is it always this way? Is there a pattern? Is there an underlying principle?" The philosopher Thales (b. before 600 BC in Miletus?) is the first recorded philosopher. His emphasis on reasoning and concepts made mathematics part of his philosophical thinking. He was followed by Pythagoras (possibly a student of Thales) whose teachings on geometry and number permanently changed our understanding of mathematics. Pythagoras probably spent some time in Egypt, learning and then extending the Egyptian understanding of mathematics. He started a school in southern Italy, a school that was both mathematical and cultic, with rigid religious beliefs and se- cret rituals. His followers, who worshiped mathematics (\All is number!"), significantly extended the 2 understanding of mathematics, creating a foundation of theorem and proof. Thales proved a number of geometrical theorems. Five theorems attributed to Thales (see Eves p. 73, Burton, p. 87) are: 1. A circle is bisected by its diameter. 2. The base angles of an isoceles triangle are equal. 3. Vertical (opposite) angles formed by intersecting lines are equal. 4. The ASA congruence rule for triangles: If two triangles agree on two angles and an included side then the triangles are congruent. 5. Any angle inscribed in a semicircle is a right angle. Let us prove the third and fifth of Thales' theorems. 3. Theorem. Vertical (opposite) angles formed by intersecting lines are equal. Proof. In modern terms, we label the four angles formed when two lines cross. Call the angles A; B; C and D. (See the figure below.) (This figure is from the Wikipedia webpage on vertical angles. It is in the public domain.) We seek to prove that angles A and B are equal. However, the sum of angles A and C is a \straight line" as is the sum of angles B and C. Thus A is a \straight line" less the value of C and B is also a \straight line" less the value of C. Therefore A and B are equal. In modern notation \A + \C = π = \B + \C: Therefore \A + \C = \B + \C: and subtracting \C from both sides gives \A = \B: 2 What results did we assume along the path of this proof? We assumed the \geometric" property that all straight lines give the same angle and the \logic" property that \equals subtracted from equals gives equals." 3 5. Theorem. Any angle inscribed in a semicircle is a right angle. (This result was apparently known to the Babylonians. But did they have a proof?) Proof. Consider a triangle inscribed in a semicircle. Label the vertices of the triangle A; B and C with A and C on the diameter of the circle. Let O represent the center of the circle. By result # 2 above (on isosceles triangles), the angles \ABO and \BAO are congruent. (Let's identify the magnitude of these angles by α: By a similar argument, the angles \BCO and \CBO are congruent. (Let's identify the magnitude of these angles by β:) (This figure is from this Wikipedia webpage on \Thales'" theorem. It is released into the public domain by inductiveload.) o Now we look at the angles \AOB and \BOC. These add to a straight line (that is, π or 180 :) That is, \AOB + \BOC = π: (1) But the sum of the angles of a triangle is \a straight line" also, that is 2α + \AOB = π and 2β + \BOC = π Therefore, adding these last two equations we have 2α + 2β + \AOB + \BOC = 2π (2) Substitution equation 1 into equation 2 we have 2α + 2β + π = 2π and so 2α + 2β = π and upon dividing by two we have α + β = π=2: But α + β is the magnitude of the angle C and so C is a right angle. 2 4 2.1.1 The Pythagoreans Pythagoras followed Thales and may have been a student. Apparently Pythagoras traveled from Samos to Egypt and then to Crotona in southern Italy where he started a commune or cult which worshipped number. Many early mathematical statements are attributed to the Pythagoreans including the main theorem named after Pythagoras. Along with worshipping "number", the Pythagoreans held a variety of mystical (religious) beliefs including the transmigration of the soul and possible release from the perpetual cycle of reincarnation through various purification rites. They were vegetarians (apparently for a variety of reasons) and rec- ognized the relationship between numbers and music. The Pythagoreans were the first to prove the theorem now named after them. They may have had several different proofs. A dissection proof of the Pythagorean theorem appears on page 81 of Eve's book and page 105 of Burton. (See also this website.) There are many proofs of the Pythagorean theorem at the cut-the-knot webpages. The Pythagorean Theorem is equivalent (logically) to the parallel postulate. (More on the parallel postulate later.) Some of the main subjects of the Pythagoreans show up in the medieval collection of subjects in early universities.