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TitleThe History of Mathematics An Introduction
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Table of Contents
Title Page
	ISBN-13: 9780073383156
Chapter 1 Early Number Systems and Symbols
	1.1 Primitive Counting
		A Sense of Number
		Notches as Tally Marks
		The Peruvian Quipus: Knots as Numbers
	1.2 Number Recording of the Egyptians and Greeks
		The History of Herodotus
		Hieroglyphic Representation of Numbers
		Egyptian Hieratic Numeration
		The Greek Alphabetic Numeral System
	1.3 Number Recording of the Babylonians
		Babylonian Cuneiform Script
		Deciphering Cuneiform: Grotefend and Rawlinson
		The Babylonian Positional Number System
		Writing in Ancient China
Chapter 2 Mathematics in Early Civilizations
	2.1 The Rhind Papyrus
		Egyptian Mathematical Papyri
		A Key to Deciphering: The Rosetta Stone
	2.2 Egyptian Arithmetic
		Early Egyptian Multiplication
		The Unit Fraction Table
		Representing Rational Numbers
	2.3 Four Problems from the Rhind Papyrus
		The Method of False Position
		A Curious Problem
		Egyptian Mathematics as Applied Arithmetic
	2.4 Egyptian Geometry
		Approximating the Area of a Circle
		The Volume of a Truncated Pyramid
		Speculations About the Great Pyramid
	2.5 Babylonian Mathematics
		A Tablet of Reciprocals
		The Babylonian Treatment of Quadratic Equations
		Two Characteristic Babylonian Problems
	2.6 Plimpton 322
		A Tablet Concerning Number Triples
		Babylonian Use of the Pythagorean Theorem
		The Cairo Mathematical Papyrus
Chapter 3 The Beginnings of Greek Mathematics
	3.1 The Geometrical Discoveries of Thales
		Greece and the Aegean Area
		The Dawn of Demonstrative Geometry: Thales of Miletos
		Measurements Using Geometry
	3.2 Pythagorean Mathematics
		Pythagoras and His Followers
		Nicomachus’s Introductio Arithmeticae
		The Theory of Figurative Numbers
		Zeno’s Paradox
	3.3 The Pythagorean Problem
		Geometric Proofs of the Pythagorean Theorem
		Early Solutions of the Pythagorean Equation
		The Crisis of Incommensurable Quantities
		Theon’s Side and Diagonal Numbers
		Eudoxus of Cnidos
	3.4 Three Construction Problems of Antiquity
		Hippocrates and the Quadrature of the Circle
		The Duplication of the Cube
		The Trisection of an Angle
	3.5 The Quadratrix of Hippias
		Rise of the Sophists
		Hippias of Elis
		The Grove of Academia: Plato’s Academy
Chapter 4 The Alexandrian School: Euclid
	4.1 Euclid and the Elements
		A Center of Learning: The Museum
		Euclid’s Life and Writings
	4.2 Euclidean Geometry
		Euclid’s Foundation for Geometry
		Common Notions
		Book I of the Elements
		Euclid’s Proof of the Pythagorean Theorem
		Book II on Geometric Algebra
		Construction of the Regular Pentagon
	4.3 Euclid’s Number Theory
		Euclidean Divisibility Properties
		The Algorithm of Euclid
		The Fundamental Theorem of Arithmetic
		An Infinity of Primes
	4.4 Eratosthenes, the Wise Man of Alexandria
		The Sieve of Eratosthenes
		Measurement of the Earth
		The Almagest of Claudius Ptolemy
		Ptolemy’s Geographical Dictionary
	4.5 Archimedes
		The Ancient World’s Genius
		Estimating the Value of π
		The Sand-Reckoner
		Quadrature of a Parabolic Segment
		Apollonius of Perga: The Conics
Chapter 5 The Twilight of Greek Mathematics: Diophantus
	5.1 The Decline of Alexandrian Mathematics
		The Waning of the Golden Age
		The Spread of Christianity
		Constantinople, A Refuge for Greek Learning
	5.2 The Arithmetica
		Diophantus’s Number Theory
		Problems from the Arithmetica
	5.3 Diophantine Equations in Greece, India, and China
		The Cattle Problem of Archimedes
		Early Mathematics in India
		The Chinese Hundred Fowls Problem
	5.4 The Later Commentators
		The Mathematical Collection of Pappus
		Hypatia, the First Woman Mathematician
		Roman Mathematics: Boethius and Cassiodorus
	5.5 Mathematics in the Near and Far East
		The Algebra of al-Khowârizmî
		Abû Kâmil and Thâbit ibn Qurra
		Omar Khayyam
		The Astronomers al-Tûsî and al-Kashî
		The Ancient Chinese Nine Chapters
		Later Chinese Mathematical Works
Chapter 6 The First Awakening: Fibonacci
	6.1 The Decline and Revival of Learning
		The Carolingian Pre-Renaissance
		Transmission of Arabic Learning to the West
		The Pioneer Translators: Gerard and Adelard
	6.2 The Liber Abaci and Liber Quadratorum
		The Hindu-Arabic Numerals
		Fibonacci’s Liber Quadratorum
		The Works of Jordanus de Nemore
	6.3 The Fibonacci Sequence
		The Liber Abaci’s Rabbit Problem
		Some Properties of Fibonacci Numbers
	6.4 Fibonacci and the Pythagorean Problem
		Pythagorean Number Triples
		Fibonacci’s Tournament Problem
Chapter 7 The Renaissance of Mathematics: Cardan and Tartaglia
	7.1 Europe in the Fourteenth and Fifteenth Centuries
		The Italian Renaissance
		Artificial Writing: The Invention of Printing
		Founding of the Great Universities
		A Thirst for Classical Learning
	7.2 The Battle of the Scholars
		Restoring the Algebraic Tradition: Robert Recorde
		The Italian Algebraists: Pacioli, del Ferro, and Tartaglia
		Cardan, A Scoundrel Mathematician
	7.3 Cardan’s Ars Magna
		Cardan’s Solution of the Cubic Equation
		Bombelli and Imaginary Roots of the Cubic
	7.4 Ferrari’s Solution of the Quartic Equation
		The Resolvant Cubic
		The Story of the Quintic Equation: Ruffini, Abel, and Galois
Chapter 8 The Mechanical World: Descartes and Newton
	8.1 The Dawn of Modern Mathematics
		The Seventeenth Century Spread of Knowledge
		Galileo’s Telescopic Observations
		The Beginning of Modern Notation: François Viéta
		The Decimal Fractions of Simon Stevin
		Napier’s Invention of Logarithms
		The Astronomical Discoveries of Brahe and Kepler
	8.2 Descartes: The Discours de la Méthode
		The Writings of Descartes
		Inventing Cartesian Geometry
		The Algebraic Aspect of La Géométrie
		Descartes’s Principia Philosophiae
		Perspective Geometry: Desargues and Poncelet
	8.3 Newton: The Principia Mathematica
		The Textbooks of Oughtred and Harriot
		Wallis’s Arithmetica Infinitorum
		The Lucasian Professorship: Barrow and Newton
		Newton’s Golden Years
		The Laws of Motion
		Later Years: Appointment to the Mint
	8.4 Gottfried Leibniz: The Calculus Controversy
		The Early Work of Leibniz
		Leibniz’s Creation of the Calculus
		Newton’s Fluxional Calculus
		The Dispute over Priority
		Maria Agnesi and Emilie du Châtelet
Chapter 9 The Development of Probability Theory: Pascal, Bernoulli, and Laplace
	9.1 The Origins of Probability Theory
		Graunt’s Bills of Mortality
		Games of Chance: Dice and Cards
		The Precocity of the Young Pascal
		Pascal and the Cycloid
		De Méré’s Problem of Points
	9.2 Pascal’s Arithmetic Triangle
		The Traité du Triangle Arithm etique
		Mathematical Induction
		Francesco Maurolico’s Use of Induction
	9.3 The Bernoullis and Laplace
		Christiaan Huygens’s Pamphlet on Probability
		The Bernoulli Brothers: John and James
		De Moivre’s Doctrine of Chances
		The Mathematics of Celestial Phenomena: Laplace
		Mary Fairfax Somerville
		Laplace’s Research in Probability Theory
		Daniel Bernoulli, Poisson, and Chebyshev
Chapter 10 The Revival of Number Theory: Fermat, Euler, and Gauss
	10.1 Marin Mersenne and the Search for Perfect Numbers
		Scientific Societies
		Marin Mersenne’s Mathematical Gathering
		Numbers, Perfect and Not So Perfect
	10.2 From Fermat to Euler
		Fermat’s Arithmetica
		The Famous Last Theorem of Fermat
		The Eighteenth-Century Enlightenment
		Maclaurin’s Treatise on Fluxions
		Euler’s Life and Contributions
	10.3 The Prince of Mathematicians: Carl Friedrich Gauss
		The Period of the French Revolution: Lagrange, Monge, and Carnot
		Gauss’s Disquisitiones Arithmeticae
		The Legacy of Gauss: Congruence Theory
		Dirichlet and Jacobi
Chapter 11 Nineteenth-Century Contributions: Lobachevsky to Hilbert
	11.1 Attempts to Prove the Parallel Postulate
		The Efforts of Proclus, Playfair, and Wallis
		Saccheri Quadrilaterals
		The Accomplishments of Legendre
		Legendre’s Eléments de géométrie
	11.2 The Founders of Non-Euclidean Geometry
		Gauss’s Attempt at a New Geometry
		The Struggle of John Bolyai
		Creation of Non-Euclidean Geometry: Lobachevsky
		Models of the New Geometry: Riemann, Beltrami, and Klein
		Grace Chisholm Young
	11.3 The Age of Rigor
		D’Alembert and Cauchy on Limits
		Fourier’s Series
		The Father of Modern Analysis, Weierstrass
		Sonya Kovalevsky
		The Axiomatic Movement: Pasch and Hilbert
	11.4 Arithmetic Generalized
		Babbage and the Analytical Engine
		Peacock’s Treatise on Algebra
		The Representation of Complex Numbers
		Hamilton’s Discovery of Quaternions
		Matrix Algebra: Cayley and Sylvester
		Boole’s Algebra of Logic
Chapter 12 Transition to the Twentieth Century: Cantor and Kronecker
	12.1 The Emergence of American Mathematics
		Ascendency of the German Universities
		American Mathematics Takes Root: 1800–1900
		The Twentieth-Century Consolidation
	12.2 Counting the Infinite
		The Last Universalist: Poincaré
		Cantor’s Theory of Infinite Sets
		Kronecker’s View of Set Theory
		Countable and Uncountable Sets
		Transcendental Numbers
		The Continuum Hypothesis
	12.3 The Paradoxes of Set Theory
		The Early Paradoxes
		Zermelo and the Axiom of Choice
		The Logistic School: Frege, Peano, and Russell
		Hilbert’s Formalistic Approach
		Brouwer’s Institutionism
Chapter 13 Extensions and Generalizations: Hardy, Hausdorff, and Noether
	13.1 Hardy and Ramanujan
		The Tripos Examination
		The Rejuvenation of English Mathematics
		A Unique Collaboration: Hardy and Littlewood
		India’s Prodigy, Ramanujan
	13.2 The Beginnings of Point-Set Topology
		Frechet’s Metric Spaces
		The Neighborhood Spaces of Hausdorff
		Banach and Normed Linear Spaces
	13.3 Some Twentieth-Century Developments
		Emmy Noether’s Theory of Rings
		Von Neumann and the Computer
		Women in Modern Mathematics
		A Few Recent Advances
General Bibliography
Additional Reading
The Greek Alphabet
Solutions to Selected Problems
Document Text Contents
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The History of


Seventh Edition

David M. Burton
University of New Hampshire

Page 409

Revised Con rming Pages

396 C h a p t e r 8 T h e M e c h a n i c a l W o r l d : D e s c a r t e s a n d N e w t o n

with velocity v in a circular orbit of radius r , the centrifugal force is

F D mv


But if T is the time of one revolution, then

v D 2³r

On substituting this value of v, one gets

F D 4³

T 2

as an expression for the constant force required to hold the planet in its circular orbit.
By Kepler’s third law, T 2=r3 D c, where c is a constant; whence






Thus, if the earth’s gravity provided the force maintaining the moon in its orbit, this
force would be inversely proportional to the square of the separating distance.

During the plague years, Newton carried out a calculation to see whether a force of
attraction that varied with the inverse square of the distance between two bodies would
account for the motion of the moon around the earth. Unfortunately, Fate played a trick on
Newton in this enterprise, and his test was at rst a disappointment. Through Galileo’s
experiments with falling bodies, afterward repeated more accurately by others, it was
generally known that the rate of fall at the surface of the earth was 16 Ð 602 feet in one
minute. The accepted value for the distance of the moon from the earth was 60r , where r
was the earth’s radius. Hence, if the inverse-square law held, the gravitational attraction
the earth exerted on the moon would be 1=602 of the attraction the earth exerted on an
object on its own surface. The moon would therefore descend a distance of 16 feet in
one minute toward the center of the earth.

The next stage in Newton’s calculation, determining the distance over which the
moon actually fell in one minute toward the earth’s center, required an accurate value of
the earth’s radius and the mean time of the moon’s revolution around the earth. The value
of the latter was very nearly 27 days, 7 hours, and 43 minutes, or 39,343 minutes. “Being
away from books,” Newton took for his calculations the standard local estimate, used by
seamen and old geographers, that there were 60 miles to a degree of latitude along the
earth’s equator. This led him to infer that the circumference of the earth was 60 Ð 360
miles, so that its radius would be 60 Ð 360=2³ , or 3438 miles. The moon’s distance from
the center of the earth was known to be 60 times the earth’s radius, so the moon’s orbit,
taken to be circular, would be 602 Ð 360 miles long. If this were assumed to be the usual
statute mile, whose length had been de ned in 1593 to be 5280 feet, then the orbit would
be 602 Ð 360 Ð 5280 feet long. (Some historians argue that Newton was more likely to
have set a mile equal to 5000 feet.) Hence the moon’s velocity in feet per minute at any
point such as P would be

602 Ð 360 Ð 5280 feet
39; 343 minutes

D 173; 930 feet=minute:

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N e w t o n : T h e P r i n c i p i a M a t h e m a t i c a 397

Constrained to follow its curved path, the moon would have traveled from P to Q, an arc
of length 173,930 feet, in one minute. Moreover, in that time it had “fallen” the distance
PS D RQ toward the earth. Because triangles PSQ and QSP 0 are similar, it follows that

PS Ð SP 0 D (SQ)2:

Fall of moon in one minute



Moon’s orbit




If PP 0 is used as an approximation to SP 0, and arc PQ as an approximation to SQ, this
last relation becomes

PS D (arc PQ)

PP 0
D (173; 930)

60 Ð 60 Ð 360 Ð 5280 feet;

or PS D RQ D 13:89 feet (or 14.67 feet if one takes a mile to be 5000 feet). Thus there
was a serious discrepancy between the two values for the fall of the moon—the 16 feet
per minute as determined from the inverse-square law of gravity, and on the other hand,
the 13.89 feet per minute as deduced from the moon’s mean period and the size of the
orbit. Although Newton said that he found his calculations to “answer pretty nearly,”
they did not match well enough to be convincing. Somewhat discouraged that the results
did not answer expectation, he abandoned all work on the gravitational problem, without
bothering to publish any account of it.

During the dozen years from 1667 to 1679, when Newton pushed the idea of gravi-
tation to the back of his mind, others began to duplicate much of his rst work. In 1673
the great Dutch scientist Christiaan Huygens published his mathematical analysis of the
motion of the pendulum, Horologium Oscillatorium sive de Motu Pendulorum, a work in
which he derived the law of centrifugal force for uniform circular motions. As a result,
the inverse-square law for gravitational attraction was formulated independently by that
versatile but jealous physicist Robert Hooke, by the astronomer Edmund Halley, by the
architect and astronomer Christopher Wren, and by Huygens himself. Hooke, Halley and
Wren—all brilliant young members of the newly founded Royal Society—were greatly
interested in the problem of gravitation for noncircular orbits but were unable to han-
dle the mathematics involved. Hooke, acting formally in his new capacity as secretary
of the society, wrote (1679) a conciliatory letter to Newton, begging him to renew his
correspondence with its members on scienti c matters. Stimulated by Hooke’s opinions
on the dynamics of planetary motions, Newton’s attention was drawn to his calculations
on the moon that had lain neglected for twelve years.

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Early Modern Period (Seventeenth and Eighteenth Centuries)

1588–1648 Marin Mersenne 1607 Jamestown founded
1591–1661 Gérard Desargues 1608 Telescope invented
1596–1650 René Descartes 1611 King James Bible
1601–1665 Pierre de Fermat 1618–1648 Thirty Years’ War
1608–1647 Evangelista Torricelli 1619 Savilian Professorship

1616–1703 John Wallis 1620 Landing of Pilgrims
1623–1662 Blaise Pascal 1632–1723 Christopher Wren
1629–1695 Christiaan Huygens 1636 Harvard College founded
1630–1677 Isaac Barrow 1642–1649 English Civil War
1635–1703 Robert Hooke 1658 Death of Cromwell
1642–1727 Isaac Newton 1662 Royal Society of London
1646–1716 Gottfried Leibniz 1663 Lucasian Professorship

1654–1705 James Bernoulli 1666 Académie des Sciences
1656–1742 Edmond Halley 1682 Acta Eruditorum
1661–1704 Marquis de l’Hospital 1683 Turks defeated at Vienna
1667–1733 Girolamo Saccheri 1687 Newton’s Principia
1667–1748 John Bernoulli 1694–1778 Voltaire
1667–1754 Abraham DeMoivre 1712–1786 Frederick the Great
1685–1731 Brook Taylor 1737–1794 Edward Gibbon
1690–1764 Christian Goldbach 1751 Diderot’s Encyclopédie
1707–1783 Leonhard Euler 1769 James Watt’s steam engine
1717–1783 Jean le Rond d’Alembert 1769–1821 Napoleon Bonaparte
1718–1799 Maria Agnesi 1770–1827 Ludwig van Beethoven
1728–1777 Johann Lambert 1776–1783 American Revolution
1736–1813 Joseph Louis Lagrange 1789 Washington President
1749–1827 Pierre Simon Laplace 1789 French Revolution
1752–1833 Adrien-Marie Legendre 1798 Eli Whitney’s cotton gin
1768–1830 Joseph Fourier 1798 Ecole Normale founded
1777–1855 Carl Friedrich Gauss 1798 Bonaparte in Egypt
1781–1848 Bernhard Bolzano 1799 Rosetta Stone

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Modern Period (Nineteenth and Twentieth Centuries)

1789–1857 Augustine Louis Cauchy 1801 Ceres discovered
1793–1856 Nicolai Lobachevsky 1809–1882 Charles Darwin
1802–1829 Niels Henrik Abel 1812–1814 War of 1812
1802–1860 John Bolyai 1812–1870 Charles Dickens
1804–1851 Carl Gustav Jacobi 1818–1883 Karl Marx
1805–1865 William Rowan Hamilton 1820–1910 Florence Nightingale
1805–1859 P. G. Lejune Dirichlet 1823 Monroe Doctrine
1809–1882 Joseph Liouville 1825 Erie Canal opens
1810–1893 Ernst Eduard Kummer 1826 Crelle’s Journal
1811–1832 Evariste Galois 1836 First telegraph
1814–1897 James Joseph Sylvester 1846 Discovery of Neptune
1815–1897 Karl Weierstrass 1858 Transatlantic cable
1821–1895 Arthur Cayley 1861–1865 American Civil War
1823–1891 Leopold Kronecker 1869 American transcontinental

1826–1866 Bernhard Riemann 1871 German empire
1831–1916 Richard Dedekind 1876 Bell’s telephone
1845–1918 Georg Cantor 1878 American Journal of

1848–1925 Gottlob Frege 1879 Edison’s electric lamp
1849–1925 Felix Klein 1894 American Mathematical

1850–1891 Sonya Kovalesky 1895 Discovery of X-rays
1852–1939 Ferdinand Lindemann 1903 First powered air ight
1858–1932 Giuseppe Peano 1914 Completion of Panama Canal
1862–1943 David Hilbert 1914–1918 First World War
1868–1942 Felix Hausdorff 1917 Bolshevik Revolution
1872–1970 Bertrand Russell 1927 Lindberg’s ight to Paris
1877–1947 Godfrey Harold Hardy 1929 The Great Depression
1882–1966 L. E. J. Brouwer 1933 Hitler becomes Chancellor
1882–1935 Amalie Emmy Noether 1939–1945 Second World War
1887–1920 Srinivasa Ramanujan 1963 Kennedy Assassinated
1903–1957 John von Neumann 1969 Landing on the moon
1906–1978 Kurt Gödel 1989 Berlin Wall dismantled

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