Lecture 10. The Elements of Euclid Four periods in the history of Greece The ancient Greek period can be divided into four periods: • The Greek Dark Ages (1100-750 B.C.), Greeks used geometric designs on pottery. • The Archaic period (750-480 B.C.), artists made larger free-standing sculptures in stiff, hieratic poses with the dreamlike `archaic smile.'1 • The Classical period (500-323 B.C.) is characterized by a style which was considered by later observers to be exemplary. • The Hellenistic period (or the Alexandrian period)(323-146 B.C.) is when Greek culture and power expanded into the near and middle east. This period begins with the death of Alexander and ends with the Roman conquest. Figure 10.1 Euclid Euclid Very little is known about Euclid, even less than that of Pythagoras. All that we know is that he flourished around 300 B.C. and taught in Alexandria, the Greek city 1The Archaic period is often taken to end with the overthrow of the last tyrant of Athens in 510 B.C. 63 in Egypt founded by Alexander the Great in 322 B.C. His own education was probably received in Plato's Academy. All the information we have on Euclid's personal life comes from a one-paragraph passage in Proclus' Commentary.2 Euclid and Apollonius (see next Lecture) both lived in Alexandrian period. Even though their time was after the classical period, people still identify Euclid's work with the classical period because his work is an organization of the separate discoveries by Greek mathemati- cians during the classical period. Also, Apollonius' work was done during the Alexandrian period but the content and spirit of his major work, Conic Sections, is that of the classical period. Two stories are told about Euclid. The first states that King Potemy I asked Euclid if there was a shorter way to learn geometry than through the Elements, Euclid told the King: \there is no royal road to geometry." The second story concerns a student who asked the perennial question: \What shall I gain from learning mathematics?" Euclid called his slave and said: \Give him a coin if he must profit from what he learns." Figure 10.2 Euclid's `Elements, a fragment of a papyrus (left), about 100 AD, found among the remarkable rubbish piles of Oxyrhynchus in 1896-97 by the renowned expedition of B.P. Grenfell and A. S. Hunt. It is now located at the University of Pennsylvania. Another copy(right), Constantinople, 888 AD (Greek). MS at the Bodleian Library, Oxford University. The influence of Euclid's Elements Euclid's most famous work is the Elements, which is a mathematical and geometric treatise consisting of 13 books. It comprises a collection 2Proclus, (410 - 412), was a Greek Neo Platonist philosopher, one of the last major Classical philosophers. 64 of definitions, postulates (axioms), propositions (theorems and constructions), and mathe- matical proofs of the propositions. With the exception of Autolycus' On the Moving Sphere, the Elements is one of the oldest existing Greek mathematical treatises and it is the oldest existing axiomatic deductive treatment of mathematics. The Elements proved to exert a heavy influence on the development of Western mathe- matics for the next 2000 years. Figure 10.3 Euclid's Elements, 1482 printed by Erhard Ratdolt in Venice (left). The first English version in 1570 (right). Euclid's Elements is the most successful and influential textbook ever written. Being first set in type in Venice in 1482, it is one of the very earliest mathematical works to be printed after the invention of the printing press and is second only to the Bible in the number of editions published. It was used as the basic text on geometry throughout the Western world for over 2,000 years. Euclid undoubtedly owes much of his material to the Platonists with whom he studied. Certainly the elementary geometry of triangles and circles was known prior to Euclid's time. A number of the most sophisticated parts of the Elements can also be credited to earlier mathematicians. Proclus says that Euclid put into his Elements many of Eudoxus' 65 theorems on irrationals and the \method of exhaustion", and the theory of regular polyhedra of Theartetus. The structure of Euclid's Elements In the Elements, Euclid began with a limited number of assumptions (definitions, 5 postulates and 5 axioms)3 and sought to prove all other results (propositions) through these assumptions. For examples of definitions, it was defined that a point is that of which there is no part, and that a line is a length without breadth. The most famous part of The Elements contains the following five postulates (i.e., ax- ioms): 1. A straight line segment can be drawn from any point to another point. 2. Any straight line segment can be extended indefinitely in a straight line. 3. A circle can be drawn with any center and radius. 4. That all right angles are equal to each other. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.4 The 5th axioms are also called 5 common notions: 1. Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part 3In modern language, a theorem is a mathematical statement that is proved using rigorous mathematical reasoning, and a proposition is a proved and often interesting result, but generally less important than a theorem. 4This postulate is equivalent to what is known as the parallel postulate. 66 The most famous proposition (i.e., theorem) is the Pythagorean theorem (Proposition 47): In right-angled triangles, the square on the side subtending the right-angle is equal to the (sum of the) squares on the sides containing the right-angle. Figure 10.4 Euclid's Elements, 1776 version (left); and 1847 version (right) Even though the form of the individual propositions did not originate with Euclid5, the form of presentation of the entire work is his own: • the statement of all the axioms at the outset; • the explicit statement of all definitions; • the orderly chain of theorems; • the theorems are arranged to go from the simple to complicated. The Elements was so successful that it displaced all previous texts on geometry. The thirteen books Euclid's \Elements" consists of 13 books. Book 1 contains the basic propositions of geometry: the Pythagorean theorem, equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are \equal" (have the same area). Book 2 is commonly called the \book of geometrical algebra," in which the material can be easily interpreted in terms of algebra. 5See the end of Lecture 9. 67 Book 3 deals with circles and their properties: inscribed angles, tangents, and the power of a point. Book 4 is about inscribing and circumscribing triangles and regular polygons. Books 5 through 10 introduce ratios and proportions. Book 6 applies proportions to geometry: Thales' theorem, and similar figures. Book 7 deals strictly with elementary number theory: divisibility, prime numbers, great- est common divisor, and least common multiple. Book 8 deals with proportions in number theory and geometric sequences. Book 9 applies the results of the preceding two books: the infinitude of prime numbers, the sum of a geometric series, and perfect numbers. Book 10 attempts to classify incommensurable (in modern language, irrational) magni- tudes by using the method of exhaustion, a precursor to integration. Book 11 generalizes the results of plane to space: perpendicularity, parallelism, volumes of parallelepipeds. Book 12 calculates areas and volumes by using the method of exhaustion: cones, pyra- mids, cylinders, and the sphere. Book 13 generalizes Book 4 to space: golden section, the five regular Platonic solids inscribed in a sphere. Euclid's Elements was used as the main mathematics book in schools in Europe, West Asia, and America for two thousand years until the 20th century. Even today there are still some mathematicians who begin studying geometry from Euclid's books. Euclidean geometry has been regarded as the unique way to observe our universe. The philosopher Immanuel Kant (1724-1804) once said: \information from our sense is organized exclusively along Euclidean templates before it is recorded in our consciousness." This view had not be changed until the non-Euclidean geometry was born (see Lectture 33). Figure 10.5 Euclid in the School of Athens (left); statue of Euclid at the Oxford University. 68.