##### Document Text Contents

Page 2

burton-title-page.eps

Revised Con rming Pages

bur83155 fm i-xii.tex i 01/13/2010 16:12

The History of

Mathematics

AN INTRODUCTION

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

2

r

:

But if T is the time of one revolution, then

v D 2³r

T

:

On substituting this value of v, one gets

F D 4³

2mr

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

F D

�

4³2m

c

�

1

r2

:

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:

bur83155 ch08 337-438.tex 396 12/30/2009 11:28

Page 410

Revised Con rming Pages

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

P

R

Earth

Moon’s orbit

Moon

Q

S

P'

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)

2

PP 0

D (173; 930)

2³

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.

bur83155 ch08 337-438.tex 397 12/30/2009 11:28

Page 818

Con rming Pages

M ATH EM ATICAL GENERAL

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

(Oxford)

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

(Cambridge)

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

bur83155 end 1-4.tex 3 12/07/2009 18:12

Page 819

Con rming Pages

M ATH EM ATICAL GENERAL

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

railway

1826–1866 Bernhard Riemann 1871 German empire

1831–1916 Richard Dedekind 1876 Bell’s telephone

1845–1918 Georg Cantor 1878 American Journal of

Mathematics

1848–1925 Gottlob Frege 1879 Edison’s electric lamp

1849–1925 Felix Klein 1894 American Mathematical

Society

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

bur83155 end 1-4.tex 4 12/07/2009 18:12

burton-title-page.eps

Revised Con rming Pages

bur83155 fm i-xii.tex i 01/13/2010 16:12

The History of

Mathematics

AN INTRODUCTION

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

2

r

:

But if T is the time of one revolution, then

v D 2³r

T

:

On substituting this value of v, one gets

F D 4³

2mr

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

F D

�

4³2m

c

�

1

r2

:

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:

bur83155 ch08 337-438.tex 396 12/30/2009 11:28

Page 410

Revised Con rming Pages

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

P

R

Earth

Moon’s orbit

Moon

Q

S

P'

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)

2

PP 0

D (173; 930)

2³

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.

bur83155 ch08 337-438.tex 397 12/30/2009 11:28

Page 818

Con rming Pages

M ATH EM ATICAL GENERAL

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

(Oxford)

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

(Cambridge)

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

bur83155 end 1-4.tex 3 12/07/2009 18:12

Page 819

Con rming Pages

M ATH EM ATICAL GENERAL

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

railway

1826–1866 Bernhard Riemann 1871 German empire

1831–1916 Richard Dedekind 1876 Bell’s telephone

1845–1918 Georg Cantor 1878 American Journal of

Mathematics

1848–1925 Gottlob Frege 1879 Edison’s electric lamp

1849–1925 Felix Klein 1894 American Mathematical

Society

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

bur83155 end 1-4.tex 4 12/07/2009 18:12