The History of Mathematics: An Introduction

by Burton, David M.

• ISBN13: 9780073051895
• ISBN10: 0073051896
• Edition: 6th
• Format: Hardcover
• Copyright: 2005-11-08
• Publisher: McGraw-Hill Science/Engineering/Math

Summary

The History of Mathematics: An Introduction, Sixth Edition, is written for the one- or two-semester math history course taken by juniors or seniors, and covers the history behind the topics typically covered in an undergraduate math curriculum or in elementary schools or high schools. Elegantly written in David Burton's imitable prose, this classic text provides rich historical context to the mathematics that undergrad math and math education majors encounter every day. Burton illuminates the people, stories, and social context behind mathematics' greatest historical advances while maintaining appropriate focus on the mathematical concepts themselves. Its wealth of information, mathematical and historical accuracy, and renowned presentation make The History of Mathematics: An Introduction, Sixth Edition a valuable resource that teachers and students will want as part of a permanent library.

Table of Contents

Preface	xi
Early Number Systems and Symbols	1	(32)
Primitive Counting	1	(8)
A Sense of Number	1	(1)
Notches as Tally Marks	2	(4)
The Peruvian Quipus: Knots as Numbers	6	(3)
Number Recording of the Egyptians and Greeks	9	(11)
The History of Herodotus	9	(2)
Hieroglyphic Representation of Numbers	11	(4)
Egyptian Hieratic Numeration	15	(1)
The Greek Alphabetic Numeral System	16	(4)
Number Recording of the Babylonians	20	(13)
Babylonian Cuneiform Script	20	(1)
Deciphering Cuneiform: Grotefend and Rawlinson	21	(2)
The Babylonian Positional Number System	23	(4)
Writing in Ancient China	27	(6)
Mathematics in Early Civilizations	33	(52)
The Rhind Papyrus	33	(4)
Egyptian Mathematical Papyri	33	(2)
A Key to Deciphering: The Rosetta Stone	35	(2)
Egyptian Arithmetic	37	(9)
Early Egyptian Multiplication	37	(3)
The Unit Fraction Table	40	(3)
Representing Rational Numbers	43	(3)
Four Problems from the Rhind Papyrus	46	(7)
The Method of False Position	46	(3)
A Curious Problem	49	(1)
Egyptian Mathematics as Applied Arithmetic	50	(3)
Egyptian Geometry	53	(10)
Approximating the Area of a Circle	53	(3)
The Volume of a Truncated Pyramid	56	(2)
Speculations About the Great Pyramid	58	(5)
Babylonian Mathematics	63	(10)
A Tablet of Reciprocals	63	(2)
The Babylonian Treatment of Quadratic Equations	65	(5)
Two Characteristic Babylonian Problems	70	(3)
Plimpton 322	73	(12)
A Tablet Concerning Number Triples	73	(4)
Babylonian Use of the Pythagorean Theorem	77	(1)
The Cairo Mathematical Papyrus	78	(7)
The Beginnings of Greek Mathematics	85	(58)
The Geometrical Discoveries of Thales	85	(7)
Greece and the Aegean Area	85	(3)
The Dawn of Demonstrative Geometry: Thales of Miletos	88	(1)
Measurements Using Geometry	89	(3)
Pythagorean Mathematics	92	(15)
Pythagoras and His Followers	92	(4)
Nicomachus's Introductio Arithmeticae	96	(3)
The Theory of Figurative Numbers	99	(4)
Zeno's Paradox	103	(4)
The Pythagorean Problem	107	(15)
Geometric Proofs of the Pythagorean Theorem	107	(2)
Early Solutions of the Pythagorean Equation	109	(2)
The Crisis of Incommensurable Quantities	111	(2)
Theon's Side and Diagonal Numbers	113	(4)
Eudoxus of Cnidos	117	(5)
Three Construction Problems of Antiquity	122	(10)
Hippocrates and the Quadrature of the Circle	122	(4)
The Duplication of the Cube	126	(1)
The Trisection of an Angle	127	(5)
The Quadratrix of Hippias	132	(11)
Rise of the Sophists	132	(1)
Hippias of Elis	133	(3)
The Grove of Academia: Plato's Academy	136	(7)
The Alexandrian School: Euclid	143	(72)
Euclid and the Elements	143	(3)
A Center of Learning: The Museum	143	(2)
Euclid's Life and Writings	145	(1)
Euclidean Geometry	146	(26)
Euclid's Foundation for Geometry	146	(2)
Postulates	148	(1)
Common Notions	148	(2)
Book I of the Elements	150	(8)
Euclid's Proof of the Pythagorean Theorem	158	(3)
Book II on Geometric Algebra	161	(6)
Construction of the Regular Pentagon	167	(5)
Euclid's Number Theory	172	(13)
Euclidean Divisibility Properties	172	(3)
The Algorithm of Euclid	175	(4)
The Fundamental Theorem of Arithmetic	179	(3)
An Infinity of Primes	182	(3)
Eratosthenes, the Wise Man of Alexandria	185	(11)
The Sieve of Eratosthenes	185	(3)
Measurement of the Earth	188	(2)
The Almagest of Claudius Ptolemy	190	(2)
Ptolemy's Geographical Dictionary	192	(4)
Archimedes	196	(19)
The Ancient World's Genius	196	(4)
Estimating the Value of π	200	(5)
The Sand-Reckoner	205	(2)
Quadrature of a Parabolic Segment	207	(2)
Apollonius of Perga: The Conics	209	(6)
The Twilight of Greek Mathematics: Diophantus	215	(56)
The Decline of Alexandrian Mathematics	215	(4)
The Waning of the Golden Age	215	(2)
The Spread of Christianity	217	(2)
Constantinople, A Refuge for Greek Learning	219	(1)
The Arithmetica	219	(6)
Diophantus's Number Theory	219	(3)
Problems from the Arithmetica	222	(3)
Diophantine Equations in Greece, India, and China	225	(9)
The Cattle Problem of Archimedes	225	(2)
Early Mathematics in India	227	(3)
The Chinese Hundred Fowls Problem	230	(4)
The Later Commentators	234	(6)
The Mathematical Collection of Pappus	234	(1)
Hypatia, the First Woman Mathematician	235	(2)
Roman Mathematics: Boethius and Cassiodorus	237	(3)
Mathematics in the Near and Far East	240	(9)
The Algebra of al-Khowarizmi	240	(4)
Abu Kamil and Thabit ibn Qurra	244	(5)
Omar Khayyam	249	(2)
The Astronomers al-Tusi and al-Karashi	251	(3)
The Ancient Chinese Nine Chapters	254	(8)
Later Chinese Mathematical Works	262	(9)
The First Awakening: Fibonacci	271	(32)
The Decline and Revival of Learning	271	(8)
The Carolingian Pre-Renaissance	271	(3)
Transmission of Arabic Learning to the West	274	(2)
The Pioneer Translators: Gerard and Adelard	276	(3)
The Liber Abaci and Liber Quadratorum	279	(10)
The Hindu-Arabic Numerals	279	(3)
Fibonacci's Liber Quadratorum	282	(3)
The Works of Jordanus de Nemore	285	(4)
The Fibonacci Sequence	289	(6)
The Liber Abaci's Rabbit Problem	289	(2)
Some Properties of Fibonacci Numbers	291	(4)
Fibonacci and the Pythagorean Problem	295	(8)
Pythagorean Number Triples	295	(4)
Fibonacci's Tournament Problem	299	(4)
The Renaissance of Mathematics: Cardan and Tartaglia	303	(36)
Europe in the Fourteenth and Fifteenth Centuries	303	(12)
The Italian Renaissance	303	(2)
Artificial Writing: The Invention of Printing	305	(4)
Founding of the Great Universities	309	(3)
A Thirst for Classical Learning	312	(3)
The Battle of the Scholars	315	(7)
Restoring the Algebraic Tradition: Robert Recorde	315	(2)
The Italian Algebraists: Pacioli, del Ferro, and Tartaglia	317	(4)
Cardan, A Scoundrel Mathematician	321	(1)
Cardan's Ars Magna	322	(8)
Cardan's Solution of the Cubic Equation	322	(5)
Bombelli and Imaginary Roots of the Cubic	327	(3)
Ferrari's Solution of the Quartic Equation	330	(9)
The Resolvant Cubic	330	(3)
The Story of the Quintic Equation: Ruffini, Abel, and Galois	333	(6)
The Mechanical World: Descartes and Newton	339	(100)
The Dawn of Modern Mathematics	339	(25)
The Seventeenth Century Spread of Knowledge	339	(2)
Galileo's Telescopic Observations	341	(6)
The Beginning of Modern Notation: Francois Vieta	347	(3)
The Decimal Fractions of Simon Stevin	350	(2)
Napier's Invention of Logarithms	352	(5)
The Astronomical Discoveries of Brahe and Kepler	357	(7)
Descartes: The Discours de la Methode	364	(18)
The Writings of Descartes	364	(4)
Inventing Cartesian Geometry	368	(5)
The Algebraic Aspect of La Geometrie	373	(3)
Descartes's Principia Philosophiae	376	(3)
Perspective Geometry: Desargues and Poncelet	379	(3)
Newton: The Principia Mathematica	382	(28)
The Textbooks of Oughtred and Harriot	382	(2)
Wallis's Arithmetica Infinitorum	384	(3)
The Lucasian Professorship: Barrow and Newton	387	(6)
Newton's Golden Years	393	(6)
The Laws of Motion	399	(6)
Later Years: Appointment to the Mint	405	(5)
Gottfried Leibniz: The Calculus Controversy	410	(29)
The Early Work of Leibniz	410	(4)
Leibniz's Creation of the Calculus	414	(3)
Newton's Fluxional Calculus	417	(8)
The Dispute over Priority	425	(5)
Maria Agnesi and Emilie du Chatelet	430	(9)
The Development of Probability Theory: Pascal, Bernoulli, and Laplace	439	(58)
The Origins of Probability Theory	439	(17)
Graunt's Bills of Mortality	439	(4)
Games of Chance: Dice and Cards	443	(3)
The Precocity of the Young Pascal	446	(6)
Pascal and the Cycloid	452	(2)
De Mere's Problem of Points	454	(2)
Pascal's Arithmetic Triangle	456	(12)
The Traite du Triangle Arithmetique	456	(5)
Mathematical Induction	461	(2)
Francesco Maurolico's Use of Induction	463	(5)
The Bernoullis and Laplace	468	(29)
Christiaan Huygens's Pamphlet on Probability	468	(2)
The Bernoulli Brothers: John and James	470	(7)
De Moivre's Doctrine of Chances	477	(1)
The Mathematics of Celestial Phenomena: Laplace	477	(4)
Mary Fairfax Somerville	481	(1)
Laplace's Research in Probability Theory	482	(7)
Daniel Bernoulli, Poisson, and Chebyshev	489	(8)
The Revival of Number Theory: Fermat, Euler, and Gauss	497	(64)
Marin Mersenne and the Search for Perfect Numbers	497	(14)
Scientific Societies	497	(2)
Marin Mersenne's Mathematical Gathering	499	(3)
Numbers, Perfect and Not So Perfect	502	(9)
From Fermat to Euler	511	(28)
Fermat's Arithmetica	511	(5)
The Famous Last Theorem of Fermat	516	(4)
The Eighteenth-Century Enlightenment	520	(4)
Maclaurin's Treatise on Fluxions	524	(3)
Euler's Life and Contributions	527	(12)
The Prince of Mathematicians: Carl Friedrich Gauss	539	(22)
The Period of the French Revolution: Lagrange and Monge	539	(4)
Gauss's Disquisitiones Arithmeticae	543	(6)
The Legacy of Gauss: Congruence Theory	549	(6)
Dirichlet and Jacobi	555	(6)
Nineteenth-Century Contributions: Lobachevsky to Hilbert	561	(92)
Attempts to Prove the Parallel Postulate	561	(21)
The Efforts of Proclus, Playfair, and Wallis	561	(3)
Saccheri Quadrilaterals	564	(5)
The Accomplishments of Legendre	569	(3)
Legendre's Elements de geometrie	572	(10)
The Founders of Non-Euclidean Geometry	582	(20)
Gauss's Attempt at a New Geometry	582	(5)
The Struggle of John Bolyai	587	(4)
Creation of Non-Euclidean Geometry: Lobachevsky	591	(5)
Models of the New Geometry: Riemann, Beltrami, and Klein	596	(5)
Grace Chisholm Young	601	(1)
The Age of Rigor	602	(22)
D'Alembert and Cauchy on Limits	602	(6)
Fourier's Series	608	(4)
The Father of Modern Analysis, Weierstrass	612	(3)
Sonya Kovalevsky	615	(3)
The Axiomatic Movement: Pasch and Hilbert	618	(6)
Arithmetic Generalized	624	(29)
Babbage and the Analytical Engine	624	(3)
Peacock's Treatise on Algebra	627	(1)
The Representation of Complex Numbers	628	(3)
Hamilton's Discovery of Quaternions	631	(6)
Matrix Algebra: Cayley and Sylvester	637	(6)
Boole's Algebra of Logic	643	(10)
Transition to the Twentieth Century: Cantor and Kronecker	653	(60)
The Emergence of American Mathematics	653	(13)
Ascendency of the German Universities	653	(2)
American Mathematics Takes Root: 1800--1900	655	(7)
The Twentieth-Century Consolidation	662	(4)
Counting the Infinite	666	(24)
The Last Universalist: Poincare	666	(2)
Cantor's Theory of Infinite Sets	668	(5)
Kronecker's View of Set Theory	673	(3)
Countable and Uncountable Sets	676	(6)
Transcendental Numbers	682	(5)
The Continuum Hypothesis	687	(3)
The Paradoxes of Set Theory	690	(23)
The Early Paradoxes	690	(4)
Zermelo and the Axiom of Choice	694	(3)
The Logistic School: Frege, Peano, and Russell	697	(4)
Hilbert's Formalistic Approach	701	(3)
Brouwer's Institutionism	704	(9)
Extensions and Generalizations: Hardy, Hausdorff, and Noether	713	(30)
Hardy and Ramanujan	713	(8)
The Tripos Examination	713	(1)
The Rejuvenation of English Mathematics	714	(3)
A Unique Collaboration: Hardy and Littlewood	717	(1)
India's Prodigy, Ramanujan	718	(3)
The Beginnings of Point-Set Topology	721	(6)
Frechet's Metric Spaces	721	(2)
The Neighborhood Spaces of Hausdorff	723	(2)
Banach and Normed Linear Spaces	725	(2)
Some Twentieth-Century Developments	727	(16)
Emmy Noether's Theory of Rings	727	(6)
Von Neumann and the Computer	733	(2)
Women in Modern Mathematics	735	(2)
A Few Recent Advances	737	(6)
General Bibliography	743	(3)
Additional Reading	746	(1)
The Greek Alphabet	747	(1)
Solutions to Selected Problems	748	(15)
Index	763