Vol I. Howard Eves. In Mathematical Circles Quadrants I, II, III, IV. Published and Distributed by The Mathematical Association of America

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1 Vol I Howard Eves In Mathematical Circles Quadrants I, II, III, IV Published and Distributed by The Mathematical Association of America

2 PREFACE xiii QUADRANT ONE THE ANIMAL WORLD, REAL AND IMAGINARY 1 A Scotch crow, 3 2 The solitary wasp, 4 3 The Harvard katydid, 4 4 The ingenious honey bees, 4 5 A classification of mathematicians, 5 6 Logarithms and multiplication, 5 7 Good induction versus bad induction, 8 The mathematical horse, 6 PRIMITIVE MAN 9 Two plus two, 7 10 Addition of vectors, 7 11 The great size of three, 8 12 GogandGug, 8 PRE-HELLENIC MATHEMATICS 13 Problem 79 of the Rhind papyrus, The pyramid of Gizeh, The greatest Egyptian pyramid, Squaring the circle, Plimpton 322, The angular degree, / 7 19 Magic squares, The triangle problem, 19 xv

3 A FEW LATER CHINESE STORIES 21 The Drunken Dragon loses his hair, Huai-Wen calculates the dates on a tree, I-Hsing finds his teacher, 21 THALES 24 How to become rich, The recalcitrant mule, Why Thales never married, Thales as a stargazer, Credit where credit is due, Moral advice, An incongruity, The Thales puzzle, Thales, the engineer, Thales, the astronomer, Thales, the statesman, 26 PYTHAGORAS 35 The lure of geometry, The first recorded facts in mathematical physics, A hecatomb of oxen, A play on words, A philosopher, Friendship, The marriage of Pythagoras, Pythagorean teaching, Pythagoras's golden thigh, The end of Pythagoras, Pythagoras's proof of his theorem, 30 THE PYTHAGOREAN BROTHERHOOD 46 Motto of the Pythagorean Brotherhood, Himself said it, Brotherhood loyalty, Damon and Phintias, The three questions, 33 xvi

4 PYTHAGOREANISM 51 The transmigration of souls, Number rules the universe, Amicable numbers, Deficient, perfect, and abundant numbers, Pythagorean philosophy and geometry at stake, Pythagoras justified, The case for Pythagoreanism, 37 PLATO 58 Plato's motto and the transfer of training, Michel Chasles and the forged autograph letters, Some particularly elusive Platonic numerology, The five Platonic solids, Kepler's explanation of the Timaeus associations, The Platonic solids in nature, The most extraordinary application of the Platonic solids to a scientific problem, Some problems concerning the Platonic solids, The Delian problem, 45 EUCLID 67 The royal road in geometry, Euclid and the student, Euclid's Elements compared with Newton's Principia, The most famous single utterance in the history of science, The most fruitful single utterance in the history of science, 48 ARCHIMEDES 72 Archimedes' boast, Archimedes'defense of Syracuse, The fraudulent goldsmith, The Archimedean screw, The stomach of Archimedes, The death of Archimedes, The questionable mosaic, The tomb of Archimedes, 52 xvn

5 ERATOSTHENES AND APOLLONIUS 80 Eratosthenes' measurement of the earth, The death of Eratosthenes, The nicknames of Eratosthenes and Apollonius, The names "ellipse," "parabola," and "hyperbola," 56 DlOPHANTUS 84 Diophantus's personal life, The syncopation of algebra, A diophantine riddle, The Greek meaning of "arithmetic," 62 THE END OF THE GREEK PERIOD 88 Some famous inequalities, Pappus's extension of the Pythagorean Theorem, The first woman mathematician, 66 QUADRANT TWO HINDU MATHEMATICS 91 KingAsoka, Inversion, The rule of three, Hindu syncopation of algebra, Bhaskara's daughter, Behold! Hindu embroidery, Buddha's examination, False position in the Bakshali manuscript, Contrast between Greek and Hindu mathematics, Srinivasa Ramanujan, 78 ARABIAN MATHEMATICS 102 Arabian names in astronomy, The origin of our word "algebra," The origin of our word "algorithm," The origin of our word "zero," The origin of our word "sine," 80 xvin

6 Alhazen's madness, The three students, Omar's roses, CONTENTS THE RETURN OF MATHEMATICS TO WESTERN EUROPE 110 Gerbert, Pope Sylvester II, The century of translators, The Norman kingdom of Sicily, The Italian commercial centers, From rabbits to sunflowers, A mathematical tournament, The blockhead, Finger numbers, The eulogist of mathematics, Submathematical analysis, 93 THE FOURTEENTH, FIFTEENTH, AND SIXTEENTH CENTURIES 120 The mechanical eagle, Introduction of + and-, The cossic art, Leonardo da Vinci's proof of the Pythagorean Theorem, The stone upon which one may sharpen his wits, The origin of our equal sign, The death of Robert Recorde, Adam Riese, Nicolaus Copernicus, Michael Stifel, Theartofbeasting, 100 THE EPISODE OF CUBIC AND QUARTIC EQUATIONS 131 The story of the algebraic solution of cubic equations, Girolamo Cardano, Tartaglia, The story of the algebraic solution of quartic equations, 104 FRANQOIS VIETE 135 The origin of a friendship, Christian versus unchristian, Work unfit for a Christian, 105 xix

7 SIMON STEVIN, JOHN NAPIER, AND HENRY BRIGGS 138 A multiple reputation, Napier's misjudgment of himself, The science fiction writer of his day, Exposing a thief, Impounding pigeons, The meeting, Some terminology, Laplace's statement, A historical curiosity, Napierian logarithms versus natural logarithms, 109 THOMAS HARRIOT AND WILLIAM OUGHTRED 148 Harriot in America, On the origin of > and <, The teacher of giants, The invention of the slide rule, Oughtred's longevity, 114 GALILEO GALILEI AND JOHANNES KEPLER 153 The oscillating lamp, Falling bodies, The telescope, and further trouble, The inquisition and the unhappy end of a great scholar, Authority versus reasoning in science, Galileo's reconciliation of science and Scripture, Some Galileo-Kepler correspondence, Tycho Brahe's golden nose, Kepler's pertinacity, The rarity of problem solvers, Pure versus applied mathematics, A life of misfortune, Numerology and theology, 122 GERARD DESARGUES AND BLAISE PASCAL 166 Desargues' forgotten book, The precocity of Pascal, The greatest" might-have-been" in the history of mathematics, Pascal's " mystic hexagram " theorem, 125 xx

8 170 Lovis de Montalte and Amos Dettonville, Two very practical contributions, A specious use of probability, 127 RENE DESCARTES AND PIERRE DE FERMAT 173 A challenge problem and a friendship, The birth of an idea, Descartes' advice, Two significant contributions to mathematical notation, The death of Descartes, The Fermat numbers, Fermat's method of infinite descent, The most tantalizing marginal note in the history of mathematics, 134 xxi

Syllabus for MTH U201: History of Mathematics

Syllabus for MTH U201: History of Mathematics Instructor: Professor Mark Bridger Office: 527 Nightingale; ext. 2450 Hours: M,W,Th, 1:00-1:30, and by appointment e-mail: bridger@neu.edu Text: The History