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1. 1. Euclid-The Creation of Mathematics
2. 2. Springer Science+Business Media, LLC
3. 3. Benna Artmann Euclid- The Creation of MatheTIlatics With 116 Illustrations Drawings by the Author , Springer
4. 4. Benno Artmann Pastor Sander Bogen 66 D-37083 Goettingen Germany Mathematics Subject Classifications (1991): OOA05, 01A20, HA05, 51M05 Library of Congress Cataloging-in-Publication Data Artmann, Benno. Euclid-the creation of mathematics / Benno Artmann. p. cm. Includes bibliographical references and index. ISBN 978-1-4612-7134-5 ISBN 978-1-4612-1412-0 (eBook) DOI 10.1007/978-1-4612-1412-0 1. Euclid, Elements. 2. Mathematics, Greek. 1. Title. QA31.A78 1999 510-dc21 98-31042 Printed on acid-free paper. © 1999 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 1999 Softcover reprint ofthe hardcover lst edition 1999 AU rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form ofinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use ofgeneral descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especiaUy identified, is not to be taken as a sign that such names, as understoodby the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by A. Orrantia; manufacturing supervised by Nancy Wu. Photocomposed copy prepared by Bartlett Press, Inc., Marietta, GA. 9 8 7 6 5 4 3 2 (Corrected second prlnting, 2001) ISBN 978-1-4612-7134-5 SPIN 10835465
5. 5. Preface This book is for all lovers of mathematics. It is an attempt to under- stand the nature of mathematics from the point of view of its most important early source. Even if the material covered by Euclid may be considered ele- mentary for the most part, the way in which he presents it has set the standard for more than two thousand years. Knowing Euclid's Elements may be of the same importance for a mathematician today as knowing Greek architecture is for an architect. Clearly, no con- temporary architect will construct a Doric temple, let alone organize a construction site in the way the ancients did. But for the training of an architect's aesthetic judgment, a knowledge of the Greek her- itage is indispensable. I agree with Peter Hilton when he says that genuine mathematics constitutes one ofthe finest expressions ofthe human spirit, and I may add that here as in so many other instances, we have learned that language of expression from the Greeks. While presenting geometry and arithmetic Euclid teaches us es- sential features of mathematics in a much more general sense. He displays the axiomatic foundation of a mathematical theory and its conscious development towards the solution of a specific problem. We see how abstraction works and enforces the strictly deductive presentation of a theory. We learn what creative definitions are and v
6. 6. VI ----=P:..:re:.::::fa=ce how a conceptual grasp leads to toe classification ofthe relevant ob- jects. Euclid creates the famous algorithm that bears his name for the solution ofspecific problems in arithmetic, and he shows us how to master the infinite in its various manifestations. One of the greatest powers of scientific thinking is the ability to uncover truths that are visible only "to the eyes of the mind;' as Plato says, and to develop ways and means to handle them. This is what Euclid does in the case of irrational, or incommensurable, magnitudes. And finally, in the Elements we find so many pieces of beautiful mathematics that are easily accessible and can be studied in detail by anybody with a minimal training in mathematics. Seeing such general phenomena of mathematical thinking that are as valid today as they were at the time of the ancient Greeks, we cannotbut agree with the philosopher Immanuel Kant, who wrote in 1783 in the popular introduction to his philosophy under the heading "Is metaphysics possible after a11?": There is no book at all in metaphysics such as we have in math- ematics. If you want to know what mathematics is, just look at Euclid's Elements. (Prolegomena §4 in free translation) To be sure, some fundamental concepts needed for today's math- ematics are absent from Euclid's Elements, most notably algebraic formulae and the concepts of a function and the real numbers. This does not, however, affect us when we study, for instance, how the exclusion of certain means of construction forces a more rigorous treatment of a piece of mathematics or how generalizations can be very profound on one occasion and shallow on another one. In order to substantiate these general claims, we have to look at Euclid's Elements in detail. Our comments are based on the transla- tion from the Greek original by Th. L. Heath (1908/1926). We will quote many ofEuclid's definitions and theorems, but cite proofs only for a few of the most important results like Pythagoras's theorem or the irrationality ofJ"Z. Many ofEuclid's other proofs are reproduced in their main outlines and in a mildly modernized language to be more easily accessible to the reader. For any questions about specific details the interested reader is advised to have Heath's or any other
7. 7. Preface VII ---------------------- translation into a modern language at hand. All recent translations are reliable, as my friends competent in classical philology tell me. If there are any problems in understanding a passage, they are almost always of a mathematical, and not a linguistic, nature and of minor importance. There are none of the ambiguities in a mathematical text as there are, for instance, in a novel or a philosophical treatise. However, in the Elements we still have a very distant historical text that needs interpretation. Here the present book takes a clear po- sition: The Elements are read, interpreted, and commented upon from the point of view of modern mathematics. I would like to share my joy and enthusiasm in studying Euclid with as many interested readers as possible. The Elements definitely deserve to be more widely known and read, and not just in the narrow circle of specialists in the history of mathematics. Never- theless, I hope that the experts will find one or another aspect of my interpretation new and interesting. Over the years, and during various stages of the preparation of this book, I have been helped and encouraged very generously by many people. Thanks to you all: First and foremost, Maresia Artmann, then U. Artmann, L. Berg- gren, D. Fowler, C. Garner, J. Hainzl, R. Hersh, P. Hilton, K. H. Hof- mann, W. Jonsson, V. Karasmanis, H. Knell, K. Lengnink, A. Mehl, Heike Muller, Ian Mueller, E. Neuenschwander, M. L. Niemann, S. Prediger, H. Puhlmann, L. Schafer, M. Thisbak and his helping hand, S. Unguru, H.-J. Waschkies. Benno Artmann Goettingen, Germany David Hilbert about Euclid: The sequence of our theorems will differ greatly from what one usually finds in textbooks on elementary geometry. It will, how- ever, frequently be the same as in Euclid's Elements. Thus we will be led by our most modern investigations to appreciate the acumen of this ancient mathematician and to admire him in the highest degree.
8. 8. ·..Vll1 Preface (Die Reihenfolge unserer Theoreme wird von der in den Lehrbii- chern der Elementargeometrie iiblichen stark abweichen, sie wird dagegen vielfach iibereinstimmen mit der Reihenfolge bei Euklid. So fiihren uns diese ganz modernen Untersuchungen dazu, den Scharfsinn dieses alten Mathematikers recht zu wiirdigen and aufs hochste zu bewundern.) D. Hilbert: Elemente der Euklidischen Geometrie. Lecture Notes of a course in the winter term 1898/99, p. 44/45. In this course he prepared his famous Grundlagen der Geometrie (1899), but gave many more details than in the published version.
9. 9. Contents Preface v Notes to the Reader xv 1 General Historical Remarks 1 2 The Contents of the Elements 3 2.1 Book I: Foundations of Plane Geometry 3 2.2 Book II: The Geometry of Rectangles 4 2.3 Book III: The Geometry of the Circle . 5 2.4 Book IV: Regular Polygons in Circles 5 2.5 Book V: The General Theory of Magnitudes in Proportion . . . . . . . . . . . . . . . . . . 5 2.6 Book VI: The Plane Geometry of Similar Figures. 6 2.7 Book VII: Basic Arithmetic . . . . . . . . . . . . 7 2.8 Book VIII: Numbers in Continued Proportion. . . 7 2.9 Book IX: Numbers in Continued Proportion; the Theory of Even and Odd Numbers, Perfect Numbers 8 2.10 Book X: Incommensurable Line Segments 8 2.11 Book XI: Foundations of Solid Geometry . . . . . .. 9 IX
10. 10. X Contents 2.12 Book XII: Areas and Volumes; Eudoxus's Method of Exhaustion 9 2.13 Book XIII: The Platonic Solids . . . . . . . . . . 10 3 The Origin of Mathematics 1: The Thstimony of Eudemus 11 Time Thble. . . . . . . . . . . . . . . . . . . . . . . . . . .. 16 4 Euclid Book I: Basic Geometry 17 4.1 The Overall Composition of Book I 17 4.2 Definitions and Axioms. . . . . . . 18 4.3 Book I, Part A: Foundations. . . . . 21 4.4 Book I, Part B: The Theory of Parallels 31 4.5 Book I, Part C: Parallelograms and Their Areas. 36 4.6 Book I, Part D: The Theorem of Pythagoras. . . 42 5 The Origin of Mathematics 2: Parallels and Axioms 47 6 The Origin of Mathematics 3: Pythagoras of Samos 51 7 Euclid Book II: The Geometry of Rectangles 61 8 The Origin of Mathematics 4: Squaring the Circle 73 9 Euclid Book III: About the Circle 79 9.1 The Overall Composition of Book III 79 9.2 The Definitions of Book III . . . . . . 80 9.3 Book III, Part A: Chords in Circles, Circles Intersecting or Thuching Each Other ... 81 9.4 Book III, Part B: Thngents . . . . . . . . . . 82 9.5 Book III, Part Cj : Angles in Segments of Circles 84 9.6 Book III, Part C2: Chords, Arcs, and Angles . . . 86 9.7 Book III, Part C3: More About Angles in Circles 86 9.8 Book III, Part D: Intersecting Chords, Secants, and Thngents . . . . . . . . . . . . . . . . . . . . 88
11. 11. Contents •_______________________ Xl 10 The Origin of Mathematics 5: Problems and Theories 11 Euclid Book IV: Regular Polygons 11.1 The Contents of Book IV . 11.2 The Regular Pentagon . 11.3 Speculations About the Pentagon 12 The Origin of Mathematics 6: The Birth of Rigor 13 The Origin of Mathematics 7: Polygons After Euclid 13.1 What We Missed in Book IV . 13.2 What Euclid Knew ... 13.3 What Archimedes Did . 13.4 What Gauss Proved .. 13.5 How Gauss Did It ... 13.6 The Moral of the Story 13.7 What Plotinus Has to Say About All This 14 Euclid Book V: The General Theory of Proportions 14.1 Proportions Outside of Mathematics. 14.2 General Remarks About Book V . 14.3 Proportions in a Modern Version. 14.4 The Definitions of Book V . 14.5 The Propositions of Book V . 14.6 Proposition V.8 and its Consequences 15 Euclid Book VI:Similarity Geometry 15.1 The Overall Composition of Book VI 15.2 The Basis of Similarity Geometry .. 15.3 The Basic Theorems of Similarity Geometry 15.4 Book VI, Part C: Proportions and Areas (Products) 15.5 Book VI, Part D: Similar Rectilinear Figures. 15.6 Book VI, Part E: The Application of Areas ..... 16 The Origin of Mathematics 8: Be Wise, Generalize 16.1 Extending a Result ... 16.2 Weakened Hypotheses 16.3 Abstraction. . . . . . . 93 97 97 99 102 109 113 113 114 115 116 117 119 120 121 121 123 124 127 130 133 135 135 136 138 141 144 146 151 152 152 153
12. 12. Xl•1· Contents --------------------~- 16.4 The Case of Pythagoras's Theorem 16.5 The Generalization of Ptolemy . 16.6 Marcel Proust on Abstraction .. 153 155 159 17 Euclid Book VII: Basic Arithmetic 161 17.1 The Historical Background. . . 161 17.2 The Overall Composition of Book VII 163 17.3 Definitions. . . . . . . . . . . . . . . 163 17.4 Book VII, Part A: The Euclidean Algorithm 165 17.5 Book VII, Parts Band C: Proportion for Numbers. 167 17.6 Book VII, Part D: Proportions and Products. . .. 171 17.7 Book VII, Part E: The Greatest Common Divisor and Prime Divisors . . . . . . . . . . . . . . . . . . 176 17.8 Book VII, Part F: The Least Common Multiple 182 18 The Origin of Mathematics 9: Nicomachus and Diophantus 183 18.1 Nicomachus: Introduction to Arithmetic 183 18.2 Diophantus: The Arithmetica . . . . . . 188 19 Euclid Book VIII: Numbers in Continued Proportion, the Geometry of Numbers 193 19.1 The Overall Composition of Books VIII and IX 193 19.2 Book VIII, Part A: Numbers in Continued Proportion . . . . . . . . . . . . . . 194 19.3 Book VIII, Part B: The Geometry of Numbers. 195 20 The Origin of Mathematics 10: Thols and Theorems 203 21 Euclid Book IX: Miscellaneous Thpics from Arithmetic 207 21.1 Book IX, Parts B, C, D, and E: More About Numbers in Continued Proportion . . . . . . . . . . . . 207 21.2 Book IX, Part F: The Number of Primes . . . . 209 21.3 Book IX, Part G: Odd and Even Numbers, and Perfect Numbers . . . . . . . . . . . . . . . . . 210
13. 13. _c_on_t_en_ts XlII 22 The Origin of Mathematics 11: Math Is Beautiful 23 Euclid Book X: Incommensurable Magnitudes 23.1 Commensurability and Its Relation to Other Notions . . . . . . . . . . . . . . . . . 24 The Origin of Mathematics 12: Incommensurability and Irrationality 24.1 Arithmetical Proofs: The Square . . 24.2 Arithmetical Proofs: The Pentagon. 24.3 Geometrical Proofs: The Square . . 24.4 Geometrical Proofs: The Pentagon . 24.5 The Mathematician Theodorus of Cyrene . 24.6 Theaetetus of Athens 24.7 Summary . 25 Euclid Book XI: Solid Geometry 25.1 The Overall Composition of Book XI . 25.2 The Definitions of Book XI . . . 25.3 Foundations of Solid Geometry . 25.4 The Affinities of Books I and XI 25.5 The Duplication of the Cube . . 26 The Origin of Mathematics 13: The Role of Definitions 27 Euclid Book XII: Volume by Limits 27.1 The Overall Composition of Book XII 27.2 The Circle . 27.3 The Pyramid . 27.4 Cylinders, Cones, and Spheres 28 The Origin of Mathematics 14: The 'fuming of the Infinite 29 Euclid Book XIII: Regular Polyhedra 29.1 The Overall Composition of Book XIII . 213 223 227 229 231 235 236 238 240 249 251 255 255 256 257 258 263 267 271 271 271 274 277 279 283 283
14. 14. XIV Contents --------------------- 29.2 Division in Extreme and Mean Ratio and the Pentagon . . . . 29.3 About the Decagon . 29.4 The Regular Solids. . . . . . . . . . . . 29.5 The Classification of the Regular Solids 284 286 292 300 30 The Origin of Mathematics 15: Symmetry Through the Ages 303 30.1 Nature .. 304 30.2 Art .... 305 30.3 Philosophy 310 30.4 Mathematics 312 31 The Origin of Mathematics 16: The Origin of the Elements 317 Notes 321 Bibliogxaphy 335 Index 347
15. 15. Notes to the Reader A large part of Euclid's Elements treats a body of mathematics that today would be called elementary geometry and arithmetic. Even today, textbooks for a high-school course in geometry, for example the popular H. R. Jacobs "Geometry" [1987], follow, for large parts, Euclid's Elements rather closely. One can understand, for instance, what is said about similarity in Euclid's Book VI without reading much of the preceding books. While it is advisable to follow Eu- clid step by step, it is not absolutely necessary. A reader with a solid background in high-school geometry can enter the discussion at any particular place to follow a particular interest. Chapter I, con- taining a short description of the contents the thirteen books of the Elements, is included for the convenience of those readers who are interested in special subjects. Hartshorne [2000] gives a radically modern presentation of Books I-VI. In our main text, the description and comments on each of Eu- clid's books will be followed by more general remarks about typical mathematical procedures, subjects of particular historical interest, connections to philosophy, and similar items. The 13 books (= chapters) ofthe Elements are denoted by Roman numerals, the propositions in each Book by Arabic numerals. Hence Prop. II.l4 is Proposition 14 of Book II. Definitions are indicated by Def. 11.2. xv
16. 16. XVI" Notes to the Reader --------------------- Propositions and definitions are quoted from the translation of Euclid by T. L. Heath [1926] or, in a sometimes slightly modernized version, Mueller [1981]. As mentioned in the Preface, I have tried to explain the mathematical content of the Elements to readers with a general interest in mathematics and its history. For more specialized questions one should consult Heath's, Vitrac's or any other transla- tion. Aside from presenting the complete text, Heath discusses a great wealth of historical and other related details. He points out oc- casionallittle gaps in Euclid's arguments, and lists alternative proofs and similar items. He does not, however, speak about the general mathematical outlines of the text; he refrains from any judgment of the relative importance of the various propositions or theories and does not call to the attention of his readers general features of mathematics. All ofthese are ofspecial interest for the present book. Notes for the text are collected at the end of the book. There are very few numbered footnotes in the traditional sense. In the notes relevant books and papers are listed together with a few sup- plementary remarks. Whenever necessary, precise bibliographical information is given in the main text. As the first part of the notes a few books of general interest are listed. Much of the vocabulary used in mathematics stems from Greek origin, words like "theorem" and "orthogonal!' Aside from such fa- miliar terms, there may be a few unfamiliar words. "Scholion" and "anthyphairesis" are the two of them that have to be explained most frequently. Ascholion is a remark to the main text ofthe Elements that was written (mostly in antiquity) in the margin of old manuscripts. Scholia are somewhat like modern footnotes, but they are very valuable because they transmit information from antiquity. Anthy- phairesis (or, equivalently, antaneiresis) is nothingbut the Euclidean algorithm for general magnitudes, mostly for line segments. Finally, the use of the name Euclid is ambiguous. On the one hand, it means the author ofthe Elements who lived about 300 B.C.E. On the other hand, "Euclid" stands for a collection of mostly un- known Greek mathematicians who lived between 500 and 300 B.C.E. and contributed most ofthe material contained in the Elements (and had themselves written "Elements" of some kind before Euclid). Used in that sense, a phrase like "Euclid proved ..." means just "a Greek mathematician proved...."
17. 17. CHAPTER General Historical Remarks The description "Ancient Greece" refers to the period roughly from 800 B.G.E. to 150 B.G.E., from Homer to the time when Rome estab- lished political hegemony over the Greek world. The first Olympic games took place in 776 B.G.E.; democracy was gradually introduced in the political life of the city-states from 600 B.G.E. onwards. The Greeks defended their freedom against the Persians in the "Persian wars" (500-480), after which the great classical period of Greece under the cultural leadership of Athens lasted until the Macedo- nian Kings Philip and Alexander the Great established monarchic rules around 330 and spread Greek culture over the whole ancient world in Hellenistic times, 300-50 B.G.E. Science and philosophy re- mained the domain of Greeks until the end of the Roman empire. Boetius, the "last Roman:' was the first writer to translate mathemat- ical texts from Greek into Latin, about 500 G.E. The Romans ran their Imperium without any mathematics. Greek mathematics and philosophy begins with Thales of Mile- tus about 580 B.G.E., of whom very little is known. Certainly the Greeks learned some mathematical rules and procedures, and espe- cially astronomy, from the old cultures in Egypt and Mesopotamia, but nothing has been found in these sources in the sense of mathe- matics as we know it today and that we encounter in Euclid's work. 1 B. Artmann, Euclid—The Creation of Mathematics © Springer Science+Business Media New York 1999
18. 18. 2 1. General Historical Remarks FIGURE 1.1 Mathematics was invented in Greek's classical period, beginning with Thales and Pythagoras about 530 B.C.E. and finding its final form with Euclid about 300 B.C.E. It stands on equal footing with the great Greek works in literature, sculpture, painting, architecture, historical and political writings, medicine, and philosophy.
19. 19. CHAPTER The Contents ofthe Elements Traditionally, the Elements have been divided into three main parts: 1. Plane geometry, Books I-VI; 2. Arithmetic, Books VII-X; 3. Solid geometry, Books XI-XIII. It will soon be obvious that Books V and X do not really fit into this division, but it is convenient to adhere to it in the following description of the individual books. The size of the books varies between ca. 2.5% ofthe whole for the smallest, Book II, and 25% for Book X. Each of the others is roughly 5-8% of the total. 2.1 Book I: Foundations of Plane Geometry Book I starts with a set of definitions. Basic concepts such as point, line, angle are described in general terms and used to define various sorts oftriangles, quadrangles, etc. The very last definition describes parallel lines in the plane as lines with no common point. After the 3 B. Artmann, Euclid—The Creation of Mathematics © Springer Science+Business Media New York 1999
20. 20. 4 2. The Contents of the Elements -------------------- definitions we find the so-called postulates, which are the axioms of geometry; the fifth and last of these is the famous parallel postu- late. The "common notions" are axioms concerning magnitudes in general, e.g., "things equal to the same thing are equal to each other:' The theorems of Book I can be grouped into four sets, which will be discussed in greater detail below. A. (I.l-26) Fundamental theorems and basic constructions in plane geometry such as the congruence theorems for trian- gles or the bisection of an angle; in part A no use is made of parallel lines. B. (1.27-32) The theory of parallel lines, including the theorem that the sum of the interior angles ofa triangle is equal to two right angles (1.32). C. (1.33-45) The theory of parallelograms; transformation and comparison of areas of parallelograms and triangles. D. (1.46-48) The theorem of Pythagoras. 2.2 Book II: The Geometry of Rectangles Compared to Book I, the second book is very homogeneous. Most of the theorems of Book II can be interpreted as what we might call, in algebraic terms, variations on the theme of the binomial identity: (a +bi = a2 +2ab +b2 • These results are always expressed in the geometric language of subdivisions of rectangles and of the areas of the various parts of the subdivisions. Theorems II.l2 and 13 generalize the theorem of Pythagoras (1.47) to what we would call the law of cosines, and Proposition II.l4 gives the solution of the important problem of constructing a square equal (in area) to a given rectilinear figure.
21. 21. 2.5. Book V: The General Theory of Magnitudes in Proportion 5 2.3 Book III: The Geometry of the Circle Book III has neither the obvious subdivisions of Book I nor the ho- mogeneous structure of Book II. After some definitions it presents the basic geometrical facts about circles, tangents, and circles in con- tact. The second half of Book III is in part concerned with what one could call the theory of quadrangles and circles, including Propo- sition III.2I, which asserts the equality of all angles in the same segment of a circle. 2.4 Book IV: Regular Polygons in Circles Book IV is by far the most homogeneous and tightly constructed book of the Elements. The following four problems are treated systematically: (i) to inscribe a rectilinear figure in or (ii) circumscribe it about a given circle; (iii) to inscribe a circle in or (iv) circumscribe it about a given rectilinear figure. These problems are solved for triangles in general, squares, regular pentagons, hexagons and IS-gons. - 2.5 Book V: The General Theory of Magnitudes in Proportion Book V is the most abstract book in the Elements and is independent ofthe precedingbooks. Whereas the other books are concerned with either geometrical objects or numbers, this one treats "magnitudes," which include, according to Aristotle, numbers, lines, solids, and
22. 22. 6 2. The Contents of the Elements -------------------- times. In VI.33 angles are treated as magnitudes, and plane areas figure as magnitudes in VI.1, XII.1, and XII.2 (areas of circles), as well as in many other places. This generality makes the theory of proportions applicable throughout mathematics, justifying Eratos- thenes' statement that it is lithe unifying bond of the mathematical sciences!' Various sources indicate that Eudoxus (about 400-350) created the theory in Book Y. Some of the theorems in Book V, such as Y.16. a: b = c : d => a : C = b : d, were almost certainly used (with different definitions and proofs) long before the time of Eudoxus. 2.6 Book VI: The Plane Geometry of Similar Figures In its composition and general outlook Book VI is close to Book I. In fact, Books I, III, and VI represent the core of plane geometry, and their overall organization gives the impression of a standard treatment of geometry that has been reworked several times. The whole edifice ofBook VI is based on Theorem VI.1 and its immediate consequence, VI.2, the fundamental theorem on the proportionality of line segments. One of the main theorems connects lines and areas of similar triangles (and polygons) (VI 19,20): If triangles are similar with a similarity factor k for lines, then the factor for the corresponding areas is k2 . The concluding section of Book VI deals with what Euclid calls lithe application of areas," which in modern terms amounts to the geometric solution of quadratic problems. It can be translated into a modern treatment of quadratic equations. For this reason it has been called "geometric algebra" by some authors.
23. 23. 2.8. Book VIII: Numbers in Continued Proportion 7 -------------"------------ 2.7 Book VII: Basic Arithmetic With Book VII, Euclid starts afresh. Nothing from the preceding books is used, particularly none ofthe general theory ofproportions of Book V. The definitions at the beginning of Book VII are intended to serve for all of VII-IX. Proportionality of numbers is defined in Def. 20 with no connection to Book V. Euclidean arithmetic is founded on the Euclidean algorithm for determining whether two numbers are prime to one another (VII.l- 4). In fact, the Euclidean algorithm gives the greatest common divisor (gcd) of any two numbers a, b. The next part of Book VII uses Def. 20 to establish the fundamental properties of proportions for numbers. 'Ib a great extent the more general theorems estab- lished in Book V are proved again for numbers. The mathematical core of Book VII is the theory of the gcd (VII.20-32), which has its counterpart in the theory of the least common multiple Ocm; VII.33-39). The important subject of prime numbers in Euclid's arithmetic will be discussed in detail below. 2.8 Book VIII: Numbers in Continued Proportion Whereas Book VII, like Book I, has a clear internal structure, Book VIII and the first part of IX are entangled like Book III. Numbers in continued proportion are the main focus of Book VIII.l-lO. The second part (VIII.1l-27) is concerned more with special types of numbers "in geometrical shape," such as squares and cubes. One important question in this context is how to characterize numbers a, b for which there exists a mean proportional x, i.e., an x such that a: x = x: b.
24. 24. 8 2. The Contents of the Elements 2.9 Book IX: Numbers in Continued Proportion; the Theory of Even and Odd Numbers, Perfect Numbers There is no break in subject matter between Books VIII and IX be- cause IX.l picks up where Book VIII ends. Euclid's other books have well-defined subjects, but in this case the division between VIII and IX looks artificial. Even more curious is that after IX.20 Euclid turns to a completely new subject, the theory of even and odd numbers (lithe even and the odd:' as Plato says), which has no connection with what precedes, but rests only on Definitions 6-10 of Book VII. This theory culminates in the construction ofeven perfect numbers (IX.36). 2.10 Book X: Incommensurable Line Segments Book X is the most voluminous book of the Elements, occupying about one quarter of the whole work. In it, the Euclidean algorithm of Book VII is applied to general magnitudes in order to obtain the criterion for commensurability: X.S. Commensurable magnitudes have to one another the ratio which a number has to a number. X.6. Iftwo magnitudes have to one another the ratio which a num- ber has to a number, the magnitudes will be commensurable. In X.9 Euclid states as an immediate consequence that the side of a square of area n is incommensurable with the side of a square of area 1 when n is not a square number. The bulk of the material of Book X, up to Proposition llS, consists in a careful study of vari- ous types of incommensurable lines and is beyond the scope of our intentions.
25. 25. 2.12. Book XII: Areas and Volumes; Eudoxus's Method of Exhaustion 9 Historically, the discovery of incommensurable lines, or, as we would say today, irrational numbers, seems to have been of paramount importance. We will give an up-to-date description of what is known about this problem. 2.11 Book XI: Foundations of Solid Geometry Book XI begins with a long list ofdefinitions for Books XI-XIII. There are no postulates of the kind we find in Book I, so that there is no axiomatic foundation for Euclid's deductions at the beginning of Book XI. The general composition of Book XI is closely parallel to that of Book I. It has the following sections A. (XI.l-19) Fundamentals of solid geometry (lines, planes, parallelism, and orthogonality). B. (XI.20-23) Solid angles, their properties and construction. C. (XI.24-37) Parallelepipedal solids. 2.12 Book XII: Areas and Volumes; Eudoxus's Method of Exhaustion Some infinitesimal methods are needed to determine the area of a circle in relation to a square, or the volume of a pyramid. The method of exhaustion, which Euclid employs, is said to have been first applied rigorously by Eudoxus, to whom most of the contents of Book XII are attributed. The method of proof is quite different from-and much more intricate than-anything in the preceding geometrical books, with the exception of Book V.
26. 26. 10 2. The Contents of the Elements 2.13 Book XIII: The Platonic Solids The first part of Book XIII, that is, XIII.l-12, consists of various planimetric propositions. Some of them are evidently lemmas for the subsequent theory of the regular polyhedra, and some others are concerned more generally with the division of a straight line in extreme and mean ratio (much later called the golden section). Because division in extreme and mean ratio is indispensable for the constructions of the icosahedron and the dodecahedron, and these are the subject of the last part of Book XII, we may regard the whole first part of Book XIII as preparatory for the second. Each of the regular solids is treated in a separate two-part theorem: (i) To construct the solid and to comprehend it in a given sphere. (ii) To compare the diameter of the sphere with the side of the polyhedron, in the sense of the classification in Book X. Euclid's treatment of the regular polyhedra is especially impor- tant for the history of mathematics because it contains the first example of a major classification theorem. Moreover, the regular polyhedra have always been of interest for mathematicians. They also playa major role in Plato's philosophy. These topics will be discussed in an appendix to Book XIII.
27. 27. CHAPTER The Origin of Mathematics 1 The testimony of Eudemus Our most important source about the history of Greek mathemat- ics before Euclid originates from Eudemus of Rhodes, a student of Aristotle, who lived from about 350 to 300 B.C.E. He wrote a book on the history of mathematics, which has, however, been lost except for a few passages quoted by other authors. The following one was preserved in Proclus's Commentary on Euclid. Proclus (410-485 C.E.) is writing about the origin and development of geometry, and the Eudemus passage starts with the second paragraph: Limiting our investigation to the origin of the arts and sciences in the present age, we say, as have most writers of history, that geometry was first discovered among the Egyptians and origi- nated in the remeasuring of their lands. This was necessary for them because the Nile overflows and obliterates the boundary lines between their properties. It is not surprising that the dis- 11 B. Artmann, Euclid—The Creation of Mathematics © Springer Science+Business Media New York 1999
28. 28. 12 3. Origin 1: Thstimony of Eudemus covery of this and the other sciences had its origin in necessity, since everything in the world of generation proceeds from im- perfection to perfection. Thus they would naturally pass from sense-perception to calculation and from calculation to reason. Just as among the Phoenicians the necessities of trade and ex- change gave the impetus to the accurate study of number, so also among the Egyptians the invention ofgeometry came about from the cause mentioned. Thales, who had travelled to Egypt, was the first to introduce this science into Greece. He made many discoveries himself and taught the principles for many others to his successors, attack- ing some problems in a general way and others more empirically. Next after him Mamercus, brother of the poet Stesichorus, is re- membered as having applied himself to the study of geometry; and Hippias of Elis records that he acquired a reputation in it. Following upon these men, Pythagoras transformed mathemati- cal philosophy into a scheme of liberal education, surveying its principles from the highest downwards and investigating its the- orems in an immaterial and intellectual manner. He it was who discovered the doctrine of proportionals and the structure of the cosmic figures. After him Anaxagoras ofClazomenae applied him- selfto many questions in geometry, and so did Oenopides ofChios, who was a little younger than Anaxagoras. Both these men are mentioned by Plato in the Erastae as having got a reputation in mathematics. Following them Hippocrates ofChios, who invented the method ofsquaring lunules, and Theodorus ofCyrene became eminent in geometry. For Hippocrates wrote a book on elements, the first of whom we have any record who did so. Plato, who appeared after them, greatly advanced mathemat- ics in general and geometry in particular because of his zeal for these studies. It is well known that his writings are thickly sprin- kled with mathematical terms and that he everywhere tries to arouse admiration for mathematics among students of philoso- phy. At this time also lived Leodamas of Thasos, Archytas of Threntum, and Theaetetus ofAthens, by whom the theorems were increased in number and brought into a more scientific arrange- ment. Younger than Leodamas were Neoclides and his pupil Leon, who added many discoveries to those of their predecessors, so that Leon was able to compile a book of elements more care-
29. 29. 133. Origin 1: Thstimony of Eudemus fully designed to take account of the number of propositions that had been proved and of their utility. He also discovered diorismi, whose purpose is to determine when a problem under investiga- tion is capable of solution and when it is not. Eudoxus of Cnidus, a little later than Leon and a member of Plato's group, was the first to increase the number of the so-called general theorems; to the three proportionals already known he added three more and multiplied the number of propositions concerning the "sec- tion" which had their origin in Plato, employing the method of analysis for their solution. Amyclas of Heracleia, one of Plato's followers, Menaechmus, a student of Eudoxus who also was asso- ciated with Plato, and his brother Dinostratus made the whole of geometry still more perfect. Theudius of Magnesia had a reputa- tion for excellence in mathematics as in the rest ofphilosophy, for he produced an admirable arrangement ofthe elements and made many partial theorems more general. There was also Athenaeus of Cyzicus, who lived about this time and became eminent in otherbranches ofmathematics and most ofall in geometry. These men lived together in the Academy, making their inquiries in common. Hermotimus of Colophon pursued further the inves- tigations already begun by Eudoxus and Theaetetus, discovered many propositions in the Elements, and wrote some things about locus-theorems. Philippus of Mende, a pupil whom Plato had en- couraged to study mathematics, also carried on his investigations according to Plato's instructions and set himself to study all the problems that he thought would contribute to Plato's philosophy. All those who have written histories bring to this point their account of the development of this science. Not long after these men came Euclid, who brought together the Elements, systematiz- ing many ofthe theorems ofEudoxus, perfecting many ofthose of Theaetetus, and putting in irrefutably demonstrable form proposi- tions that had been rather loosely established by his predecessors. He lived in the time of Ptolemy the First ... (Proclus-Morrow pp. 51-56) Let us underline some of the specific characteristics of this quo- tation. At the beginning, the text acknowledges the foreign origin of mathematics: The Greeks learned geometry ("land measurement") from the Egyptians and arithmetic from the Phoenicians. The al-
30. 30. 14 3_._O_r--'igJ.~·_n_l_:_Th..:...8;...:.t1_·m_o;:,..n-"y:-o..:...f_E_u;,...d;,...e;,...m_u--.:.8 leged Egyptian origin ofgeometry strikes us as implausible: We don't know any Egyptian text that would substantiate this claim, but it does occur several times in Greek sources. Thales, one of the proverbial wise men of Greece, is characterized as the intermediator between the "barbarians" and the Greeks. Pythagoras gets the credit for the de- cisive transformation of mathematics into an abstract science and, moreover, into a subject of liberal education, but there are doubts about the origin of the passage about the doctrine of proportion- als and the cosmic figures, Le., the regular solids. This may be an interpolation by Proclus. The very strong relations between mathematics and philosophy are stressed in the remarks about Plato. He is described as a sort of research director, who instructs the mathematicians what to do. Both Theaetetus and Leon are praised for bringing theorems "into a more scientific arangement:' Theudius of Magnesia became famous because he "made many partial theorems more genera!:' At the beginning of the last paragraph there is the hint that the passage is by Eudemus, though he is not mentioned by name, and the remarks about Euclid seem to be by Proclus himself. Some spe- cific details of this report are disputed, especially in its tendency to overestimate the role of Pythagoras and Plato. On the other hand, we would know virtually nothing about many ofthe mathematicians mentioned had Proclus not preserved this passage for us. For the moment let us note that Eudemus mentions some writers of "Elements" earlier than Euclid, the first of whom is Hippocrates of Chios about 440-430 B.C.E. (He is not to be confused with the medical man Hippocrates of KOs.) Leon compiled the textbook for Plato's Academy (370 B.C.E.?), and Theudius "produced an admirable arrangement of the elements:' In due course we will discuss the contribution of various mathematicians to Euclid's Elements. Neither Eudemus nor any other Ancient writer tells us anything about one invention of singular importance by the earliest Greek geometers: the lettered diagram. Except for the simplest cases a mean- ingful mathematical discourse about a diagram without denoting points, lines, and so on, by letters is not possible. It is to the credit of Reviel Netz [1999] to have directed our attention to this most impor- tant step in the early development ofmathematics. One can still see how the letters were introduced into some diagrams in the process
31. 31. What Does the Word "MATHEMATICS" Mean? 15 of writing down the proof. For instance, in Proposition 1.44, A and B are the end points of a line; C is (the area of) a triangle; D denotes an angle; and E, F are again points. Another equally elusive historical problem is the invention ofthe formalistically structured mathematical language of Euclid and his predecessors. M. Asper [2001] suggests to seek prototypes for this in the early standardized legal texts with their precise definitions and discussions of cases of which a great many have survived engraved into the walls of temples or other public buildings. What Does the Word 44MATHEMATICS" Mean? This whole book is about mathematics, but here we are looking only at the etymological side of the question. The Greek word math- ema (IJ-CX()TJI.UX) originally means "that which is learned, learning, science" and was first used in this sense by Plato and, probably, the Pythagoreans. The associated verb is the Greek manthanein, to learn. The word is derived from the Indo-European root mendh- "to have one's mind aroused, apply oneself to." Here are some related words from other languages: English: mind German: munter (awake, lively, merry, vigorous) Middle High German: Minne (love) Gothic: munda (to aim) Old Slavic: modru (wise, sage) Sanskrit: man (to think) Latin: mens (mind) Greek: mantis (a seer), and possibly even the Greek muse and Prometheus. We mathema- ticians find ourselves among a host of very sympathetic counter- parts.
32. 32. 16 3. Origin 1: Thstimony of Eudemus TIME TABLE (all dates are B.C.E.) General history, related to mathematics Mathematics 900-600 Geometric period of Greek art 600 Money is invented Thales of Miletus (~ 580) (first coins) Pythagoras (~ 570-490) 500 Persian Wars (~ 500-480) ~ 460 Temple of Zeus in Pythagoreans in Olympia, Proportions 2 : 1 southern Italy 450 ~ 450-430 Pericles, "high classics" ~ 440 Parthenon temple Hippocrates of Chios writes in Athens, proportions first Elements ~ 430 9 : 4 = length: breadth = breadth : height Socrates ~ 470-399 Theodorus of Cyrene ~ 460-390 400 Plato 428-348 Theaetetus ~ 415-370 Leon writes new Elements Aristotle 384-322 in Plato's Academy Eudoxus ~ 410-355 350 Alexander the Great Various other writers of 356-323 mathematical treatises, e.g., Menaechmus: Conics 300 Euclid: Elements Alexandria is the cultural center of the Hellenistic 250 Apollonius of Perga, world 300-50 Archimedes of Syracuse
33. 33. Euclid Book I CHAPTER Basic Geometry 4.1 The Overall Composition of Book I Definitions 1-23 Postulates 1-5 Common Notions 1-5 1-26 27-32 33-45 46-48 Basic concepts are described or defined Postulates and Common Notions are the axioms of plane geometry A: Foundations of plane geometry without using parallels B: The theory of parallel lines, angles in a triangle C: The theory ofparallelograms and their areas D: The theorem of Pythagoras 17 B. Artmann, Euclid—The Creation of Mathematics © Springer Science+Business Media New York 1999
34. 34. 18 4. Euclid Book I 4.2 Definitions and Axioms Euclid created the model ofa mathematical text: Start with explicitly formulated definitions and axioms, then proceed with theorems and proofs. Unlike modern authors, who do not pretend to know what a set is, Euclid wants to say what he is talking about, or to give some sort of description of the objects of geometry. He does this in the first group of definitions, 1-9. Definitions Def. 1. A point is that which has no part. Def. 2. A line is breadthless length. Def. 3. The extremities ofa line are points. Def. 8. A plane angle is the inclination to one another oftwo lines in a plane which meet one another and do not lie in a straight line. Def. 9. And when the lines containing the angle are straight, the angle is called rectilinear. It has often been observed that Euclid makes no use ofthese def- initions in his subsequent proofs. They are explications that should clarify the significance of a term to the reader but play no formal rule in deductions. In Def. 8, the lines forming an angle may be curved. In Book III Euclid occasionally uses angles between circles and straight lines, but in our discussion of Proposition 1.5 we will find indications for a greater popularity of angles between curved lines in pre-Euclidean times. Most ofthe following definitions are abbreviations in the modern manner, for instance: trilateral figures are those contained by three straight lines ... Oftrilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle which has two ofits sides alone equal, and a scalene triangle that which has its three sides unequal. Def. 20. Definitions Def. 19.
35. 35. 4.2. Definitions and Axioms 19 In a modern formal sense, an equilateral triangle is isosceles as well, but not so for Euclid. Similarly, in Def. 22, a rectangle (called "oblong" there) is not a square. This agrees with a more colloquial modern use: Ifa rectangle is spoken of, in most cases this means "not a square;' because otherwise one could be more specific. Obviously, from a logical point of view, it is much more convenient to include the squares with the rectangles. After the definitions, Euclid proceeds to state his famous postu- lates. Modern axioms of geometry resemble these postulates rather closely. Postulates 1. Let it be postulated to draw a straight line from any point to any point, and 2. to produce a limited straight line in a straight line, 3. to describe a circle with any center and distance, 4. that all right angles are equal to each other. S. [The parallel postulate will be discussed belowJ Thday Postulates 1 and 2 would be expressed in a way like "given any two distinct points, there is a unique line passing through them:' Euclid's emphasis is more on construction than on "existence," more a difference in style than in substance. The geometric postulates are followed by what Euclid calls "com- mon notions." These are axioms about the behavior of general magnitudes, not only geometric objects. 1. Things equal to the same thing are also equal to one another. 2. Ifequals are added to equals the wholes are equal. 3. Ifequals are subtracted from equals the remainders are equal. 4. Things which coincide with one another are equal to one another. S. The whole is greater than the part. Many authors have noted the incompleteness of Euclid's axioms in comparison to modern foundations of geometry. The most ob- vious point is the absence of any thought of the ordering of points on a line or the concept of "betweenness:' Euclid uses all assertions about ordering on an intuitive basis. These objections concern rel- atively minor points and do not in any way diminish Euclid's basic
36. 36. 20 4. Euclid Book I achievement: In mathematics, one has to start from explicitly stated first principles and deduce all following assertions from these prin- ciples. (About ordering, see Notes. About modern axiomatics, see Hartshorne [2000].) Historically, the idea of stating axioms seems to be rather new in Euclid's time when compared to the definitions. In a very thorough and penetrating investigation, Mueller [199Ib] examines the starting points ofmathematical theories as preserved in the writings ofPlato, Aristotle, and Euclid. Mueller summarizes on p. 63: However, if we look at the Elements, although we find at the be- ginning ofbook I definitions, postulates, and common notions ... at the beginning of the remaining books we find only definitions. I believe there are two related inferences we can draw from this: (1) Euclid did not believe that proportion theory, number theory, or solid geometry required its own postulates; (2) at the end of the fourth century there were no accepted presentations of these theories which included postulates, and probably no such presen- tations at all, presumably because no mathematician recognized the need for them. A further inference I draw is that the idea of such presentations ofany mathematical theory was relatively new in Euclid's time, i.e., did not precede Plato's maturity. I believe the evidence suggests that Euclid himself is responsible for the pos- tulates, but for the moment I will only say that, even if they are thought to predate, say, Plato's Republic, they should still be seen as the exception rather than the rule by Euclid's time. The rule in the Elements and, I am suggesting, earlier in the history ofGreek mathematics is a theory, the only explicit starting points ofwhich are definitions. These definitions are, for the most part, either explications, which perhaps clarifY the significance of a term to the reader but play no formal role in subsequent arguments, or abbreviations in the modern manner. In spite of this rather diverse historical picture, Euclid's axioms have been of utmost importance for the development of mathemat- ics because, as it was said at the beginning, with them he created the model of a mathematical theory.
37. 37. 4.3. Book I, Part A: Foundations 4.3 Book I, Part A: Foundations 21 The essential contents of Part A of Book I are first the basic congruence theorems and elementary constructions such as bi- secting angles and segments, and second some propositions about "greater" relations ofangles and sides of triangles, based on 1.16 and culminating with the triangle inequality 1.20. The very first propositions show how to construct an equilateral triangle and how to copy segments without moving them. The deli- cate constructions in 1.2, 3 are based directly on the Postulates I, 2, and 3. Proposition 1.4 is the first substantial theorem, the congruence theorem "side-angIe-side;' for short, SAS. Euclid states it like this: Prop.1.4. If two triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend. For the proof see Fig. 4.1: Let ABC, DEF be two triangles having the two sides AB, AC equal to the two sides DE, DF respectively, namely AB to DE and AC to DF, and the angle BAC equal to the angle EDF. I say that the base BC is also equal to the base EF, the triangle ABC will be equal to the triangle DEF, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal side subtend, that is, the angle ABC to the angle DEF, and the angle ACB to the angle DFE. Before looking into the proof, we observe some peculiarities of Euclid's style. He always states his theorems in two ways: at first in general words, and then a second time in a more specific way indi- cating points, lines, angles, and so on by various letters.l Very often I This is very much like today's usage: Theorem: A continuous real function maps closed intervals onto closed intervals. Let [a, bjbe a closed interval andf : [a, bj- R be continuous ...
38. 38. 22 8 'v:A FIGURE 4.1 4. Euclid Book I the theorem is accompanied by a suitable diagram. One particular phrase needs explanation: *The triangle ABC will be equal to the tri- angle DEF.* This is clarified by later use of the same expression: It simply means "the triangles have equal areas." Euclid uses the word "area"(or its Greek equivalent) only occasionally.2 The Proof of Prop. 1.4 The method of proof of 1.4 stands in strong contrast to the meticu- lous constructions in 1.1-3. Euclid just takes the triangle ABC and superimposes it on triangle DEF in a such way that A is placed on D, Bon E, and C on F. From this he easily derives his assertions. On the one hand, this method ofsuperposition clearly has no ba- sis in Euclid's axioms, but on the other hand, practically nothing can be done in elementary geometry without the congruence theorems. (For the congruence theorem side-side-side, SSS, in 1.8 he uses the same method.) In fact, what we see here is another axiom. Modern axiomatic studies by Hilbert and others have shown that there is no way to resolve this dilemma: Either SAS has to be used as an axiom or one has to use superposition in a modern version by postulating the existence ofcertain rigid motions ofthe plane. (For more details, see Hartshorne [2000], Section 17, esp. p. 154.) 2The Greeks knew perfectly well how to measure their properties, and they knew that Pharaoh's taxation office measured the fields of the Egyptian peasants to their disadvantage. In mathematics they avoid the concept of"area," using instead phrases like the one above, e.g., "this rectangle is equal to that rectangle" and similarly.
39. 39. 4.3. Book I, Part A: Foundations 23 ---"------------------- .J) FIGURE 4.2 In the next pair 1.5/6 ofpropositions, Euclid proves a fundamen- tal lemma about isosceles triangles that is used frequently in Books I- VI. In the statement and proof of 1.5 we ignore Euclid's assertion about outer angles. 1.6 is the converse of 1.5. Prop. 1.5 In isosceles triangles the angles at the base are equal to one another. Let ABC be an isosceles triangle having the side AB equal to the side AC; and let the straight lines BD, CE be produced further in a straight line with AB, AC. I say that the angle ABC is equal to the angle ACB and the angle CBD to the angle BCE [Fig. 4.2). Prop. 1.6. If in a triangle two angles are equal to one another, the sides which subtend the equal angles will also be equal to one another. For the proofof1.5, Euclid first constructs two auxiliary triangles BFC and CGB [Fig. 4.3]: Let a point F be taken at random on BD; from AE the greater let AG be cut off equal to AF the less; and let the straight lines FC, GB be joined. In the next two steps he first shows the congruence of the tri- angles 6.AFC and 6.AGB by using SAS, and then again by SAS the congruence 6.BFC ~ 6.CGB:
40. 40. 24 4. Euclid Book I------------_....:..:......===-=~:...:: o FIGURE 4.3 (1) We have LFAC = LGAB, and AF = AG by construction, and AC =AB, hence l::.AFC;: l::.AGB, and especially BG = CF and LBFC =LCGB. (2) From the construction we get BF = CG; moreover, we know from the outset BC = CB, and from (l) we have LFBC = LCGB; hence l::.CGB ;: l::.BFC by SAS. Now Euclid concludes: Therefore the angle FBC is equal to the angle GCB, and the angle BCF to the angle CBG. Accordingly, since the whole angle ABG was proved equal to the angle ACF, and in these the angle CBG is equal to the angle BCF, the remaining angle ABC is equal to the remaining angle ACB; and they are at the base of the triangle ABC. QE.D. All the steps ofthis proofare justified by Euclid's axiomatic base. (For details, see the analysis in Hartshorne [2000], Section 10.) We will direct our attention to another question, which has fre- quently baffled students of Euclid: How can anybody understand the introduction of his auxiliary points, lines, and triangles at the beginning of his proof? In this particular instance, 1.5, we are in the lucky position ofhav- ing a historical predecessor of Euclid's proofthat explains the initial construction. It comes from Aristotle's Prior Analytics, we quote it from Heath's commentary on 1.5. It makes use of mixed angles be- tween circular arcs and straight lines in the following way: (a) the
41. 41. 4.3. Book I, Part A: Foundations 25 ---'------------------ angles of semicircles (called AC and BD), that is, between a diam- eter and the circumference, are equal, and (b) the two angles in a segment, that is, between a chord and the circumference, are equal. Aristotle uses the proof in his discussion ofsome logical points. (See Fig. 4.4.) For let A, B be drawn [Le. joined] to the center. If then, we assumed (1) that the angle AC is equal to the angle BD without asserting generally that the angles of semicircles are equal, and again (2) that the angle Cis equal to the angle D without making the further assumption that the two angles ofall segments are equal, and ifwe then inferred, lastly, that, since the whole an- gles are equal, and equal angles are subtracted from them, the angles which remain, namely E, F are equal. We should com- mit a petitio principii, unless we assumed [generally] that, when equals are subtracted from equals, the remainders are equal. (Prior Analytics 41 b 13-22) First observe a basic similarity in Aristotle's and Euclid's proofs: We have two equal big angles, from which two smaller equal angles are subtracted, resulting in the desired equality of the base angles. In Euclid's time mixed angles were no longer acceptable; he does not use them save on a few minor occasions in Book III. A transition from Aristotle's proof to one without mixed angles can be explained in a plausible way. (We use Euclid's notation; see Fig. 4.5) FIGURE 4.4
42. 42. 26 4. Euclid Book I ------------_--.:.-~_-:..=~ FIGURE 4.5 First replace the "angle of the semicircle" by the right angle be- tween the radius AB and the tangent BG, similarly for AC and CF. The congruence theorem ASA (With Ct, side AC = side AB, and the right angles) would be needed in order to show 6.ABG ~ 6.ACF. Hence the "big" right angles L.ABG and L.ACF are equal, and the result would follow as in Euclid's proof. Note that the symmetri- cally situated small triangles 6.BFC and 6.CBG replace the intuitively symmetric segment. However, Euclid cannot use tangents and ASA at this stage of Book I, so he disposes with the right angles between radii and tan- gents and cleverly provides himselfwith the equal sides AF and AG in a direct way so that he can use SAS instead ofASA. (E and Dare merely auxiliary points for the prolongation of the sides.) Here, I think, we have found a natural explanation for Euclid's construction, albeit a hypothetical one. Observe that Fig. 4.5 (b) ap- pears in Book III.17 where Euclid constructs the tangent to a circle and, in a less obvious way, in 1.2 as well. Aristotle lived 384-322 and was a member of Plato's Academy 367-348 when Plato died. It seems very likely that he got his math- ematical education in the Academy, and so it is possible that in looking at his proof we see a small fragment of Leon's "Elements:' the textbook of Plato's Academy. Propositions 7-15. In Propositions 7 and 8 Euclid proves the congruence theorem side-side-side (SSS), using the method of su- perposition for the second time. Propositions 9-15 are devoted
43. 43. 4.3. Book I, Part A: Foundations 27 A FIGURE 4.6 to the common auxiliary constructions and initial propositions of plane geometry: bisecting angles and segments, constructing perpendiculars, supplementary and vertical angles. Prop. 1.16. Ifone ofthe sides ofany triangle is produced, the exterior angle is greater than each ofthe interior and opposite angles. Claim. angle a < angle <5 (Fig. 4.6). Constrnction. Bisect AC at E, draw BE and extend it to F such that BE = EF, join C and F, let a' = angle ECF (Fig. 4.7). PlOof (i) Triangle ABE is congruent to triangle CFE by the congruence theorem SAS. Hence a = a'. (ii) But a' is a part of O. Hence a = a' < 0 by common notion 5, QE.D. C D FIGURE 4.7
44. 44. 28 B A F .B FIGURE 4.8 4. Euclid Book I F c If Euclid had the theory ofparallels at his disposal right here, the claim of1.16 would be a trivial consequence of1.32 about the sum of the angles in a triangle. He explicitly states in 1.32 that the exterior angle is the sum of the two interior and opposite angles. Hence we see a conscious composition at work. Before discussing this, we will try to understand the genesis of the proof of 1.16 with the help of parallels. What can be said about this proof? It is ingenious, and one can see how its author hit upon his idea: Just add the line AF to the figure (Fig. 4.8). All ofa sudden, we see a parallelogram ABGF "behind" the proof of 1.16. At this stage, we may use parallels and have a = a' because of alternate angles; AG is a transversal of the two parallel lines AF and BG. Furthermore, E will be the intersection of the diagonals of this parallelogram. However, and this is the essential idea, in order to prove 1.16 it is possible to avoid parallels and use the congruence theorem I, 4 instead. Further evidence ofthe mathematical competence ofthe author of1.16 is his ability to connect 1.16 with its consequences, the impor- tant theorems 1.20, the triangle inequality, and 1.27, the existence of parallels. (After all, deductive structures are what mathematics is all about.) On the other hand, there is a weak spot in the proof. The asser- tion "a' is a part of 8/1 has no base in Euclid's axioms. It is just read off from the diagram. This has often been observed: Compare, for instance, Heath's commentary. (Heath confuses the "Riemann hy-
45. 45. 4.3. Book I, Part A: Foundations 29 ---'-------------------- FIGURE 4.9 pothesis" with Riemannian non-Euclidean geometry, but otherwise he is mathematically correct.) Proposition I.16 is not true in the so- called elliptical (spherical) geometries, which satisfy all of Euclid's axioms except the parallel postulate. A counterexample to I.16 is eas- ily drawn on the sphere. In Fig. 4.9, supplied by E. Hartmann, the point F will be in the southern hemisphere; hence (x' > ~. It should, however, be clear that any Greek mathematician would reply to this objection that he was dealing with plane, not spherical, geometry. Certainly a man like Menelaus of Alexandria (about 100 C.E.), who wrote about spherical geometry, knew the phenomenon. It seems that nobody noted the error before the end of the nine- teenth century, when non-Euclidean geometries and order-relations in geometry came to the attention of mathematicians. The likely reason for Euclid's neglect of questions about the ordering of points on a line (or betweenness) may be that he regarded it as a part of logic-or just took it for granted. In fact, I.16 remains valid in the second class of non-Euclidean geometries, the so-called hyperbolic geometries, which can be defined over ordered fields. The reader interested in more details about order relations in geometry should consult Hartshorne, Chapters 1.3 and 3.15. Propositions 1.17-20. Proposition I.17 is a direct consequence of I.16. It is again a weak variant of 1.32 about the sum of the angles in a triangle:
46. 46. 30 4. Euclid Book I -----------_-.-...:~=:..:..-::...:...:..::....: Prop. 1.17. In any triangle two angles taken together in any manner are less than two right angles. Proposition 1.18 says that in any triangle the greater side sub- tends the greater angle, and 1.19 is its converse. These propositions lead to Prop. 1.20. In any triangle two sides taken together in any manner are greater than the remaining one. This is the famous triangle inequality. Proclus comments on this: The Epicureans are want to ridicule this theorem, say it is evident even to an ass and needs no proof ... they make [this] out from the observation that, ifhay is placed at one extremity of the sides, an ass in quest of provender will make his way along the one side and not by way of the two others. (Produs-Morrow p. 251) (The Epicureans of today might as well add that one could see the proof on every campus where people completely ignorant of mathematics traverse the lawn in the manner of the ass.) Proclus replies rightly that a mere perception of the truth of a theorem is different from a scientific proof of it, which moreover gives reason why it is true. (Which means, to embed it into a deductive structure.) In the case of Euclid's geometry, the triangle inequality can indeed be derived from the other (equally plausible) axioms. On the other hand, the Epicureans win in the modern theory of metric spaces, where the triangle inequality is the fundamental axiom ofthe whole edifice. Propositions 1.21-26. Three ofthe remaining propositions ofpart A are about "greater" relations for sides and angles oftriangles (21,24, 25). Prop. 22 gives the construction of a triangle from its sides, pro- vided that the triangle inequality is valid. Using this, Euclid shows in Prop. 23 how to copy an angle. The combined congruence theorems ASA and AAS are tagged on in 1.26 as a sort of loose end.
47. 47. 4.4. Book I, Part B: The Theory of Parallels 4.4 Book I, Part B: The Theory of Parallels Euclid defines in Def 1.23: Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. 31 For short, parallels in a plane are nonintersecting straight lines. This has remained unchanged in the modern theory of incidence geometry (cf. Hartshorne, [2000], Section 6 and many other places). In so-called affine planes, parallel lines are those that have no point in common. The modern parallel axiom in an affine plane is this: Given a line g and a point P not on g, there exists one and only one line h passing through P that does not meet g. This axiom really has two parts: (1) The parallel h to g through P exists. (2) It is unique ("only one line"). (Part (2) is sometimes called "Playfair's axiom:') Euclid's geometry is richer than the theory of affine planes: He has the congruence axioms and-implicitly-the conditions of or- dering and betweenness. Via ordering he got 1.16 and from this he derives the "existence" part for parallels in 1.27. For uniqueness, he has to introduce a special axiom, the famous parallel postulate. He uses it in 1.29 in order to prove a property of parallels that immedi- ately provides uniquenes (without saying so). We quote Postulate 5 below, but first 1.27 together with its proof. Prop. 1.27 Ifa straight line falling on two straight lines makes the alternate angles equal to oneanother, the straight lines will beparallel to one another. For let the straight line EF falling on the two straight lines AB, CD make the alternate anglesAEF, EFD equal to one another [Fig. 4.10]; I say that AB is parallel to CD.
48. 48. 32 c FIGURE 4.10 J) 4. Euclid Book I For, if not, AB, CD when produced will meet either in the direction ofB, D or towards A, C. Let them be produced and meet, in the direction ofB, D at G. Then, in the triangle GEF, the exterior angle AEF is equal to the interior and opposite angle EFG: which is impossible. [1.16] Therefore AB, CD when produced will not meet in the direction of B,D. Similarly it can be proved that neither will they meet towards A, C. But straight lines which do not meet in either direction are parallel; [Def. 23] therefore AB is parallel to CD. Before going on we introduce some convenient notation: and gllh for parallel lines g and h 2R for two right angles (or 180°). Proposition 28 is a useful variant of 27. It says, with the notation taken from Fig. 4.11, Ci= y~g II h, f3 + Y =2R ~ g II h. For the proof of Prop. 29, we need the parallel postulate. Postulate 5. Let it be postulated: That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines,
49. 49. 4.4. Book I, Part B: The Theory of Parallels FIGURE 4.11 FIGURE 4.12 33 ifproduced indefinitely, meet on that side on which are the angles less than the two right angles. [Fig. 4.12] Prop. 1.29. A straight line falling on parallel straight lines makes the altemate angles equal to one another, the exterior angle equal to the interior and oppo- site angle, and the interior angles on the same side equal to two right angles. We will abbreviate the proof by using the notation from Fig. 4.13(b) and dealing with the main case of alternating angles a, ponly. The claim is: If g II h, then a=p. This is the converse of 1.27.
50. 50. 34 4. Euclid Book I ------------------ A o--~--__oB C~--_cr---o.D F (4.) FIGURE 4.13 (b) The proof is by contradiction, or rather by proving the logically equivalent statement If not a = f3, then not gil h. If not a = f3, then one of them is greater, say a > f3: f3 < a ~ f3 + y < a + y. But a+ y = 2R, whence f3+y< 2R=a+y. Now, Postulate 5 says that g and h have to meet, that is, f3 + y < 2R, and Post. 5 ~ notg II h. (Ifyou want to see a contradiction like the one in Euclid's proof, continue: but g II h by hypothesis....) Propositions 27 and 29 together give us the fundamental property of parallels (PP), again with the notation of figure 13(b): (PP) (for alternating angles). g II h has the intuitive meaning "g and h have no common point," a property that has to be checked from here to infinity. On the other side, a = f3 can be verified locally and is a most useful practical device that connects the intuitive notion with the other concepts of
51. 51. _4_.4_._B_o_o_k_I!-.P_ar~t...:.B...:.:_Th_e~Th----:..eo...:.ry..iL....:0_f_P_ar_all~el_s 35 congruence geometry. This is a very common feature ofmathemat- ics: Define a concept by an intuitive meaning (if possible), and then prove that this is equivalent to a technically useful other statement. Another striking example of this is the definition of a tangent to a circle in Book III. In modern mathematics, unfortunately, the tech- nical devices are often put in the foreground at the expense of the intuitive meaning. Proposition 30 shows the transitivity of parallelism, and Propos- tion 31 exploits (PP) for the construction of parallels by alternate angles. Prop. 1.32. In any triangle, ifone ofthe sides is produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles. Again, we will take the convenient notation from Fig. 4.14 for Euclid's proof. Prolong the line BC to the point D. Let EC be parallel to AB. Then, by Prop. 29, we have a = a' and f3 = f3'. Hence the exterior angle a' +f3' is equal to a+f3, and because of y +a' +f3' = 2R, we have a+f3+y=2R. Euclid emphasizes the exterior angle a' +f3' because he will use it on several later occasions. The sum of the angles of a triangle is the most important and fundamental invariant in elementary geometry. No matter what the E l> FIGURE 4.14
52. 52. 36 4. Euclid Book I ----------_------::.:.......==.==:..=..::..:=-= shape ofthe triangle is, its angles will invariably add up to two right angles (or 180 degrees, or nJ. This is used so often that one is prone to forget its significance. One first immediate consequence is the formula for the sum of the (interior) angles of a convex polygon. If it has n vertices, it can be dissected into n - 2 triangles and has (n - 2)2R (or, expressed in another way, (n - 2)n') as the sum of its angles. Proc1us proves this in his comment on 1.32. (Proc1us-Morrow p. 301.) Proc1us proceeds (p. 302) to state that I/the property ofhaving its interior angles equal to two right angles is an essential property ofthe triangle as such." He refers to Aristotle for the meaning ofl/es- sential property:' In the words of today this means that the triangle is characterized (among convex polygons) by the sum of its angles: A convex polygon is a triangle if and only if the sum of its interior angles equals two right angles. Theorem 1.32 has played its role in philosophy later on as well. For Immanuel Kant it is the quintessen- tial example ofwhat he calls I/a synthetic a priorijudgment;' that is, a statement of absolute certainty (not depending on experience) that adds to our knowledge (Critique ofPure Reason B 744-746). One ofthe far-reaching consequences ofl.32 was found by Jacob Steiner (1796-1863). He used the formula (n - 2)7l' for the sum of the interior angles ofa polygon for a simple proofofEuler's formula for convex polyhedra: If such a polyhedron has v vertices, e edges, and f faces, then I/invariably" v - e+ f =2. Thus the simple invariant of triangles goes as far as proving one of the most important invariants of modern algebraic topology, the Euler characteristic, in its first significant case ofconvex polyhedra. 4.5 Book I, Part C: Parallelograms and Their Areas In part C we find a systematic study of the interrelations between the concepts of I/parallelism" and I/of equal content."
53. 53. 4.5. Book I, Part C: Parallelograms and Their Areas 37 Euclid defines various types of "quadrilaterial figures" in Def. 22 at the beginning of Book I, but not the parallelograms that figure so prominently in this section C. Instead, he introduces them together with their basic symmetry properties in Propositions 33 and 34. Prop. 1.33. The straight lines joining equal and parallel straight lines (at the extrem- ities which are) in the same directions (respectively) are themselves equal and parallel. Prop. 1.34. In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas. In Prop. 34 Euclid speaks about halving the "area" of a parallelo- gram, but he does not use this word in the subsequent propositions, which are-in our understanding-equally statements about areas. In daily life, the Greeks measured their properties, and in fact the very word "geometry" means "measuring the fields!' Measuring a field means attaching a number to it; it measures so and so many square feet. In mathematical language this amounts to a function that associates numbers to certain (polygonic) plane surfaces. But the concept of a function is alien to the Elements. Euclid does not use it, and moreover, he does not use any formulas that in effect would define functions. For a modern description of what Euclid does, we quote Hartshorne [2000], Section 3 (but see also Section 22) about Euclid's notion of "equal figures": So what did Euclid have in mind? Since he does not define it, we will consider this new equality as an undefined notion, just as the notions of congruence for line segments and angles were undefined. We will call this new notion equal content, to avoid confusion with other notions ofequality or congruence. We do not want to use the word area, because this notion is quite different from our common understanding ofarea as a function associating a number to each figure. From the way Euclid treats this notion, it is clear that he re- gards it as an equivalence relation, satisfYing the common notions. In particular (a) Congruent figures have equal content.
54. 54. 38 4. Euclid Book I------------_-=.:......~::::..:.-:::....::.....:..::....= (b) Iftwo figures each have equal content with a third, they have equal content. (c) If pairs of figures with equal content are added in the sense of being joined without overlap to make bigger figures, then these added figures have equal content. (d) Ditto for subtraction, noting that equality of content of the difference does not depend on where the equal pieces were removed. (e) Halves of figures of equal content have equal content (used in the proof of 1.37). (f) The whole is greater than the part, which in this case means ifone figure is properly contained in another, then the two figures cannot have equal content (used in the proof of 1.39). In terms of the axiomatic development of the subject, at this point Euclid is introducing a new undefined relation, and taking an the properties just listed as new axioms governing this new relation. In the next propositions, 35-41, Euclid achieves more flexibility in handling the concept of equal content, or equality, as he says. Prop. 1.35. Parallelograms which are on the same base and in the same parallels are equal to one another. Prop. 1.36. Parallelograms which are on equal bases and in the same parallels are equal to one another [Fig. 4.15). ~' < •.., A ' ' A2.,·' .' ,."'.. FIGURE 4.15 !A'l
55. 55. 4.5. Book I, Part C: Parallelograms and Their Areas 39 FIGURE 4.16 Propositions 37-40 say similar things for triangles, and Proposi- tion 41 associates parallelograms and triangles. (The parallelogram situated as in Fig. 4.16 is the double of the triangle.) At this point the theory of equal content branches out in two di- rections. The first branch leads directly to the theorem ofPythagoras (1.46-48), which in any case is a goal in its own right; and the second one leads via 1.42-45 and the theorem of Pythagoras to the impor- tant result 11.14: It is possible to construct a square ofcontent equal to that ofany rectilinear figure. Or shorter: Any rectilinear figure can be squared. We stick to Euclid's sequence and discuss 1.42-45, which will find their sequels in Book II. Prop. 1.42. To construct, in a given rectilineal angle, aparallelogram equal to agiven triangle. The construction is easy enough, compare Fig. 4.17, where 6.ABC and the angle 0 are given and E is the midpoint ofBC. FIGURE 4.17
56. 56. 40 4. Euclid Book I H A ~__~~:---:-----:- -.,qD B FIGURE 4.18 F c The "given angle" will be a right angle in Euclid's subsequent ap- plications. So we might as well specialize it to this case in the next propositions. (The generalization from rectangles to parallelograms is easy enough because shears preserve areas.) We replace "parallel- ogram" by "rectangle" and "given angle" by "right angle" in Euclid's Props. 1.43-45. Book II is about rectangles throughout. Prop. 1.43. In any rectangle the complements of the rectangles about the diagonal are equal to one another. Figure 4.18 shows a diagram that is used over and over again in the Elements. In several contexts Euclid simply calls it "the schema." The point K is on the diagonal of the rectangle DABCD, and the lines EF, GH are parallel to the sides. Euclid denotes DBGKE by BK, and DKFDH by KD. These latter rectangles are the "so-called complements:' (The reader may want to look ahead at Props. VI.16, 24,26.) We have to prove: K is on AC ~ DBK and DKD are of equal content (are equal). By 1.34, the triangle f::lABC is equal to f::lDAC. For the same reason, f::lGCK and f::lFKC as well as f::lAEK and f::lKHA are equal. Subtracting the two smaller triangles from the large one on each side of the diagonal gives the result. Simple as it is, Prop. 43 has very many useful consequences. The next proposition is the first one. By "applying" a figure C to a line (segment) AB Euclid means to construct a rectangle with one side AB of equal content with figure C.
57. 57. 4.5. Book I, Part C: Parallelograms and Their Areas H FIGURE 4.19 41 Prop. 1.44. Th agiven straight line to apply a rectangle equal to agiven triangle. Construction. Let /:::,.C and line AB be the given figures (Fig. 4.19). Construct a rectangle DBF ofequal content with /:::,.C via 1.42. Place AB so that it prolongs side EB and construct DBH. Prolong FE and HB until they meet in K. (Euclid shows that they will meet by means of Post. 5.) Complete the figure as shown in Fig. 4.19. DBL has one side AB and is of equal content with DBF by Prop. 1.43. Prop. 1.45. Th construct a rectangle equal to a given rectilinear figure. For the given figure Euclid takes a quadrangle, dissects it into two triangles, and transforms these by Prop. 1.44 into two rectangles with a common side. By combining the two rectangles with the common side he gets one rectangle as desired. The proofis done meticulously by justifying every single step. In spite of the general assertion Euclid, selects a quadrangle for the proof. But the procedure is quite transparent, and it is obvious how to proceed in the general case. This way of handling proofs, which today might be done by mathematical induction, is quite typical for Euclid. We will see it now and then on other occasions. 4.5.1 Comment on Props. 1.44/45 We will for a moment use modern formulas. The area A ofa rectangle with sides (of length) a, b is given by A = abo In 1.44, let R be the
58. 58. 42 4. Euclid Book I given rectangle and a be the given side. In terms of these formulas, the problem of 1.45 amounts to the solution of the linear equation R=ax, where x is the second side of the desired new rectangle. Seen this way, 1.45 is algebra in geometric disguise, and hence it has been interpreted as "geometric algebra." Historians have said that this in- terpretation is not justified and an anachronism. Mathematicians have replied that the formulas represent an isomorphic image of the geometric situation and hence are the correct modern way of describing Euclid's procedures. The same problem arises again in Book VI, where the geometric equivalent of quadratic equations is treated. Because the geometric version is quite sufficient for the understanding of Euclid's text, we will leave the formulas aside. Oc- casionally we will use them in order to facilitate understanding for the modern reader. 4.6 Book I, Part D: The Theorem of Pythagoras In proposition 1.46 Euclid shows how to construct a square on a given line; 1.47 is the famous theorem of Pythagoras, and 1.48 is its converse. We quote 1.47 and its proof verbatim. Prop. 1.47. In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. Let ABG be a right angled triangle having the angle BAG right; I say that the square on BG is equal to the squares on BA, AG. For let there be described on BG the square BDEG, and on BA, AG the squares GB, HG; through A let AL be drawn parallel to either BD or GE, and let AD, FG be joined [Fig. 4.20]. Then, since each ofthe angles BAG, BAG is right, it follows that with a straight line BA, and at the point A on it, the two straight lines AG, AG not lying on the same side make the adjacent angles equal to two right angles;
59. 59. 4.6. Book I, Part D: The Theorem of Pythagoras 43 f k 1> o-;..~L---O E FIGURE 4.20 therefore CA is in a straight line with AG. For the same reason BA is also in a straight line with AH. And, since the angle DBC is equal to the angle FBA: for each is right: let the angle ABC be added to each; therefore the whole angle DBA is equal to the whole angle FBC. And, since DB is equal to BC, and FB to BA, the two sides AB, BD are equal to the two sides FB, BC respectively, and the angle ABD is equal to the angle FBC; therefore the base AD is equal to the base FC, and the triangle ABD is equal to the triangle FBC. [1.4] Now the parallelogram BL is double of the triangle ABD, for they have the same base BD and are in the same parallel BD, AL. [1.41] And the square GB is double ofthe triangle FBC, for they again have the same base FB and are in the same parallels FB, GC. [I. 41] Therefore the parallelogram BL is also equal to the square GB. Similarly, if AE, BK are joined, the parallelogram CL can also be proved equal to the square HC; therefore the whole square BDEC is equal to the two squares GB, HC. And the square BDEC is described on BC, and the squares GB, HC onBA, AC. Therefore the square on the side BC is equal to the squares on the sides BA, AC.
60. 60. 44 4. Euclid Book I ------------_--=.:~=::....=.=..:..::::...= Let us recapitulate the main points of the proof. The right angle at A guarantees that G, A, C are in a straight line parallel to FB. This is the decisive point. For the equality ofDGB and DBL Euclid has to resort to their respective halves, /:::,.FBA (which is not shown) and /:::,.BDL (also not shown). By 1.41 these are equal to (i.e., are of equal content with) /:::,.FBC and /:::"BDA. These two triangles are congruent by SAS, and we are done. Comment There are many dozens of different proofs of Pythago- ras's theorem. Proclus credits Euclid personally with this one. It is a marvellous piece ofmathematics, and I personally like it better than any other proof. There is no special trick or need of a formula, one sees in such a clear way how the square OGB is transformed into the rectangle DBL, and in spite of its simplicity the argument is in no way trivial. The theorem of Pythagoras is as fundamental for mathematics today as it was in Euclid's time. It is the progenitor of all the dif- ferent kinds of metrics and of quadratic forms, and of theorems like sin2 a + cos2 a = 1. Via its generalization, the law of cosines, and the corresponding scalar product in vector spaces, it pervades mathematics as far as the eye can see. Prop. 1.48 Ifin a triangle the square on one ofthe sides is equal to the squares on the remaining two sides ofthe triangle, the angle contained by the remaining two sides ofthe triangle is right. FIGURE 4.21
61. 61. 4.6. Book I, Part D: The Theorem of Pythagoras 45 Proof Let 6ABC be the given triangle. We take the notation from Fig. 4.21 and abbreviate. Let AD be equal to AB and fP be a right angle, then by 1.47 we have [2 = d2 + b2 = c2 +b2 , which by assumption is equal to a2 . Hence [ is equal to a. (Here is a little gap. This implication has not been proved before.) Now, by the congruence theorem SSS the two triangles 6ABC and 6ADC are congruent, and hence a = fP is a right angle. Propositions 1.47 and 48 combined are the full theorem of Pythagoras. We conclude this chapterby quoting a fine sonnet by the German poet Adelbert von Chamisso, translated by Max Delbriick, together with a nice remark by C. L. Dodgson. According to a legend from antiquity, Pythagoras sacrificed a hundred (a hecatomb, or, in an- other version, only one) oxen to the gods after he had discovered his theorem. (The German original of the poem can be found in the notes.) Adelbert von Chamisso: The'Ih1th (1tanslated by Max Delbriick) The TRUTH: her hallmark is ETERNITY As soon as stupid world has seen her light PYTHAGORAS' theorem today is just as right As when it first was shown to the FRATERNITY. The GODS who sent to him this ray of light to them PYTHAGORAS a token sacrificed: One hundred oxen, roasted, cut, and sliced Expressed his thank to them, to their delight. The oxen, since that day, when they surmise That a new truth may be unveiling Forthwith burst forth in fiendish railing. PYTHAGORAS forever gives them jitters - Quite powerless to stem the thrust of such emitters of LIGHT, they tremble and they close their eyes. But neither thirty years, not thirty centuries, affect the clearness, or the charm, ofGeometrical truths. Such a theorem as "the square
62. 62. 46 4. Euclid Book I of the hypotenuse of a rightangled triangle is equal to the sum of the squares of the sides" is as dazzlingly beautiful now as it was in the day when Pythagoras first discovered it, and celebrated its advent, it is said, by sacrificing a hecatomb of oxen - a method of doing honor to Science that has always seemed to me slightly exaggerated and uncalled-for. One can imagine oneself, even in these degenerate days, marking the epoch ofsome brilliant scien- tific discovery by inviting a convivial friend or two, to join one in a beefsteak and a bottle of wine. But a hecatomb of oxen! It would produce a quite inconvenient supply ofbeef. C. L. Dodgson (Lewis Carroll)
63. 63. CHAPTER The Origin of Mathematics 2 Parallels and Axioms The discussion of the parallel axiom has been the driving force behind the axiomatization of mathematics. We will sketch the development in the history of mathematics. (For more detailed information see the Notes.) Historians generally assume that the introduction of Post. 5 and the theory of parallels was fairly recent at Euclid's time. Before that, there may have been a definition of parallels comprising properties like the definition 1.17 of the diameter of a circle: "A diameter of the circle is a straight line drawn through the center ... and such a straight line also bisects the circle:' A similar definition of parallels may have said that they exist, are unique, and can be constructed using alternate angles, or more simply, right angles. (Possibly even their property of being equidistant was stated.) These aspects of parallels were quite natural for architects and stone masons, who worked with parallel layers of stones all the time and were used to very precise measurements. A faint hint ofthe origin ofthe parallel axiom (Post. 5 for short in the following text) may be a remark by Proclus in his discussion of 47 B. Artmann, Euclid—The Creation of Mathematics © Springer Science+Business Media New York 1999
64. 64. 48 5_._Th_e_O_r.....igtL·n_o_f_M_a_th_e_ma_t_ic_s_2 FIGURE 5.1 Post. 5: "That there are lines that approach each other indefinitely but never meet seems implausible and paradoxical, yet it is nev- ertheless true and has been ascertained for other species of lines" (Produs-Morrow 151). The most prominent example of such lines are a hyperbola and its asymptotes (Fig. 5.1). Admitting angles be- tween curved (and straight) lines, this is a "counterexample" to Post. 5, or at least it could draw the attention of mathematicians to the problem. The first to write about conics was Menaechmus, who may have lived about 380-320B.C.E. Ifthe speculation above is right, this would put the introduction of Post. 5 at a time around 340. Other historians think it may have been introducedby Euclid himself. Aristotle never mentions the postulates of Book I. Compared to the other postulates, the parallel postulate is a rather complicated statement and not as obvious as, say, the equal- ity of all right angles. Therefore, mathematicians of antiquity tried to eliminate it, either by proving it or by replacing it by a sim- pler and more intuitively convincing axiom. However, none ofthem succeeded. The problem was taken up again in the seventeenth century. John Wallis gave a public talk in Oxford in 1663 about the subject and proved the following: If there are similar triangles of different content, then Post. 5 is true. This result replaces Post. 5 by a more plausible one. Several other mathematicians tried the following way: Start with the negation of Post. 5 and try to find consequences that contradict established theorems. The most prominent among these
65. 65. 5. The Origm· of Mathematics 2 49-------><----------------- was the Jesuit Girolamo Saccheri, who in 1733 published a book entitled Euclid liberated from every defect. Although it was largely for- gotten, this book contained a key distinction. Saccheri looked at a quadrangle that was constructed with 3 right angles. Then there are three alternatives for the remaining fourth angle: It might be ob- tuse, right, or acute. The hypothesis of the right angle is equivalent to Post. 5. Then he managed to find a contradiction derived from the hypothesis of the obtuse angle. (It contradicted the existence of ines of infinite length. This is the situation on the sphere.) From the hypothesis of the acute angle he drew many conse- quences but could not find a contradiction. (He believed he had one, but there was a mistake.) In 1766 Johann Heinrich Lambert wrote a paper on "The Theory of Parallel Lines;' which was pub- lished posthumously in 1786. It seems that he thought about a valid geometry that could be derived from Saccheri's hypothesis of the acute angle. He points to Euclid's 1.16/17 as evidence that Euclid held similar opinions. Finally, around 1830 three mathematicians were thoroughly con- vinced of the existence of a valid "non-Euclidean" geometry: Carl Friedrich Gauss, Janos Bolyai, and Nikolai Ivanovitch Lobachevsky. But Gauss did not publish what he had found, and the papers by Bolyai (1832) and Lobachevsky (1835) were hard to read. So the sub- ject lay dormant until the second half of the 1860s, when Gauss's (who had died in 1855) private letters were published and the sit- uation changed dramatically. Some papers by Riemann on abstract geometries in the 1850s, the development of differential geometry, and the prestige of Gauss made non-Euclidean geometry one of the hottest research topics in mathematics. The theory derived from the "hypothesis of the acute angle" was now called "hyperbolic geome- try." The possibility of a (still) hidden contradiction was ruled out by Beltrami (1868), who interpreted it as the geometry on a surface with constant negative curvature. Felix Klein (1871) found the now so-called Cayley-Klein model in projective geometry, and in 1882 Poincare placed it in the context of complex function theory. The question ofthe parallel axiom was finally settled by 1880: There are three types of plane geometry that satisfy Euclid's other axioms: el- liptic geometry (on the sphere, with antipodal points identified, the hypothesis of the obtuse angle) with no parallels at all; Euclidean geometry with Post. 5 (hypothesis of the right angle), where paral-
66. 66. 50 5_._T_b_e_O_r...,!;igti!...·n_o_f_M_a_tb_e_ma_ti_·c_s_2 leIs exist and are unique; and hyperbolic geometry (hypothesis of the acute angle), where parallels exist but are not unique. In elliptic geometry the sum of the angles in a triangle is greater then 180°, in Euclidean geometry it is 180°, and in hyperbolic geometry it is smaller than 180°. But there is more to the story ofaxioms in geometry. Euclid had employed the notions of ordering and betweenness of points on a line intuitively. In effect, he had ruled out the hypothesis of the ob- tuse angle by these means. The subject ofordering was taken up by Moritz Pasch in a book Lectures on Recent Geometry (Vorlesungen uber neuere Geometrie) in 1882. His axioms about betweenness essentially completed the axiomatization of plane geometry. (Pasch had such an acute sense for fine points in logic that in his later days he was the foremost expert on the bylaws of his university in Giessen. His son-in-law C. Thaer produced the standard German translation of Euclid.) For Pasch, geometry was still the science of physical space. This last barrier was broken by David Hilbert (1899) in his Foundations of Geometry (Grundlagen der Geometrie). Hilbert explicitly says that the objects of geometry are called points, lines, planes by convention, but they could be called by other, fancy, names just as well. These objects are defined only "implicitly" by what is said about them in the axioms. (For instance, a point might be a pair of real numbers.) The subject of geometry has changed from a study of space to the study ofthe logical interdependence ofcertain statements about otherwise undefined objects. Hilbert grouped the axioms under five headings: axioms of in- cidence like "Two different points lie on a unique line," axioms of ordering, axioms of congruence, the parallel axiom, and finally, axioms of continuity. The axioms of continuity guarantee that the points ofa line may be identified with the real numbers. This enables him to prove his main theorem: The five groups ofaxioms determine the Euclidean plane (up to isomorphism) uniquely. It may be viewed as the plane ofanalytic geometry over the field ofreal numbers. In an appendix to hisbook Hilbert presents the first axiom system for the real numbers. Starting from this point, the axiomatic method conquered the whole of mathematics in the twentieth century.
67. 67. CHAPTER The Origin of Mathematics 3 Pythagoras ofSamos Pythagoras lived about 570-490 B.C.E. The only roughly determined date in his life is ~ 530, when he left Samos to settle in Crotona, in southern Italy. At Crotona he founded a religious and philosoph- ical society that soon came to exert considerable political influence in the Greek cities of southern Italy. He was forced to leave Cro- tona about 500 and retired to Metapontum, where he died (see Fig. 6.1). The Pythagoreans, as his followers were called, continued to ex- ert political power until sometime in the middle or late fifth century, when a democratic revolution occurred and they were forced to leave the Greek cities of southern Italy. Some of them went to Sicily and others to the Greek mainland, where they found new centers for their activities. The last of the Pythagoreans were known in about 350 B.C.E. as poor vegetarian wandering pilgrims. The city (and island) of Samos together with its close neighbors Miletus and Ephesus on what is now the Thrkish coast were booming economic and intellectual centers in the sixth century B.C.E. Thales and his student Anaximandros taught in Miletus in the first half 51 B. Artmann, Euclid—The Creation of Mathematics © Springer Science+Business Media New York 1999
68. 68. 52 6. The Origin of Mathematics 3 a b c FIGURE 6.1 (a) Coin from Metapontum with a "pentagram" com- posed of grains of barley (about 440 B.C.E.?). (b) Coin from Abdera (430 B.C.E.?) showing an idealized portrait of (n) YeArOPH~, that is, (P)YTHAGORES. (c) Coin from Melos (before 420 B.C.E.) with pentagram of the sixth century. The philosopher Heraclitus of Ephesus was a contemporary of Pythagoras. 1Wo examples of practical geometry will illustrate the atmo- sphere of the city of Pythagoras's youth. Outside of Samos was the ancient sanctuary ofthe goddess Hera. In 570-560 the city commis- sioned the architects Rhoikos and Theodoros with the construction of a new temple of dimensions hitherto unheard of. Its ground plan was 100 x 200 cubits (52.5 x 105 meters); its 104 columns were 18 meters high. The bases of the columns had a diameter of up to 1.80 meters and weighed about 1500 kg each. In spite of these extraordi- nary dimensions, they were turned on lathes! This temple was to be the prototype of the Ionian style in architecture. It was constructed during Pythagoras's youth. Just completed, it was destroyed in a rev- olution in the 530s, which brought the tyrant Polykrates into power. Polykrates immediately ordered the construction ofa new and even bigger temple. (See Figs. 6.5 and 6.6 at the end ofthis Section.) The second example ofhigh technology in the second half ofthe sixth century in Samos is the tunnel of Eupalinus. This tunnel, for a new water main, was about 1 km long and, this is the noteworthy point, was built from both sides ofthe mountain! The diggers met in the middle ofthe mountain with a deviation of about 10 meters. The people who gave the money for the construction did indeed trust geometry and geodesics. (See notes for more details.) Such was the background of Pythagoras's youth. He may have traveled to Egypt and Babylon for studies, but Samos was where the action was in his time.

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