Encyclopedia Of Mathematics (Science Encyclopedia) [8 MB].pdf ( PDFDrive )

542 Appendix I

1629 greatest scientific works of all time. Newton outlines
his laws of motion and the INVERSE SQUARE LAW for
French mathematician PIERRE DE FERMAT (1601–65) gravitation.
uses algebra to solve geometric problems but does not
publish his results. French mathematician and philoso- 1693
pher RENÉ DESCARTES (1596–1650) later developed
similar techniques and is today credited as the founder Halley compiles the first set of LIFE TABLES and makes
of this approach. use of STATISTICS to analyze birth and death rates.

1635 1696

Italian mathematician BONAVENTURA CAVALIERI French scholar MARQUIS DE GUILLAUME FRANÇOIS
(1598–1647) introduces a method of indivisibles for ANTOINE L’HÔPITAL (1661–1704) publishes the first
comparing volumes (a precursor to the methods of textbook on CALCULUS.
INTEGRAL CALCULUS) and CAVALIERI’S PRINCIPLE.
1703
ca. 1637
NEWTON is elected president of the ROYAL SOCIETY of
FERMAT introduces modern NUMBER THEORY. He London. Eight years later, after an official investiga-
writes a problem in the margin of a text that confounds tion, the society concludes that Newton, not LEIBNIZ, is
mathematicians for centuries. FERMAT’S LAST THEOREM the true inventor of CALCULUS. It is later revealed that
was finally solved by ANDREW WILES in 1994. Newton, as president, wrote the final proclamation.
The verdict is not considered valid today.
1639
1718
French mathematician GIRARD DESARGUES (1591–1661)
publishes a treatise on his newly discovered PROJEC- French mathematician ABRAHAM DE MOIVRE
TIVE GEOMETRY. The work is essentially ignored for (1667–1754) publishes Doctrine of Chances, the most
200 years. advanced text on the theory of PROBABILITY of its time.
De Moivre later develops the result today known as
1654 STIRLING’S FORMULA.

Mathematician BLAISE PASCAL (1623–62) begins a cor- 1736
respondence with FERMAT about questions of games of
chance. Through five consecutive letters, they together Swiss mathematician LEONHARD EULER (1707–83)
create the theory of PROBABILITY. solves the SEVEN BRIDGES OF KÖNIGSBERG PROBLEM,
thereby establishing the fields of TOPOLOGY and GRAPH
1662 THEORY. Throughout his life Euler also discovers,
among many accomplishments, the number e, his
The ROYAL SOCIETY of London is established. British famous formula relating the trigonometric functions to
mathematician LORD WILLIAM BROUNCKER (1620–84) this number, specific values of the ZETA FUNCTION, and,
is elected as its first president. in geometry, the EULER LINE. Euler also introduces the
notion of a FUNCTION and popularizes the use of the
1666 symbol π for the ratio of the circumference of a circle
to its diameter.
SIR ISAAC NEWTON (1642–1727) develops DIFFEREN-
TIAL and INTEGRAL CALCULUS but does not publish his 1742
results until 1711.
CHRISTIAN GOLDBACH (1690–1764) writes to EULER
1673 posing the problem that has since become known as
GOLDBACH’S CONJECTURE.
German mathematician GOTTFRIED WILHELM LEIBNIZ
(1646–1716) develops DIFFERENTIAL and INTEGRAL 1748
CALCULUS independently of NEWTON. Leibniz begins
publishing his results in 1684, and Newton accuses him MARIA GAËTANA AGNESI (1718–99) publishes her two-
of plagiarism. A bitter dispute between the two men volume survey of elementary and advanced mathematics.
ensues, lasting four decades.
1750
1687
Swiss mathematician GABRIEL CRAMER (1704–52) pub-
Under the urging of astronomer Edmund Halley, New- lishes CRAMER’S RULE.
ton publishes Principia, today considered one of the

Appendix I 543

1767 1822

German mathematician JOHANN HEINRICH LAMBERT French mathematician JEAN BAPTISTE JOSEPH FOURIER
(1728–77) proves that π is irrational. (1768–1830) publishes a treatise on the theory of heat
and develops the notion of a FOURIER SERIES.
1795
1829
France adopts the metric system.
Russian mathematician NIKOLAI IVANOVICH
John Playfair (1748–1819) publishes an equivalent LOBACHEVSKY (1792–1856) and Hungarian mathe-
form of the famous PARALLEL POSTULATE, today known matician JÁNOS BOLYAI (1802–60) independently dis-
as PLAYFAIR’S AXIOM. cover NON-EUCLIDEAN GEOMETRY.

1797 1843

CARL FRIEDRICH GAUSS (1777–1855) proves the FUN- SIR WILLIAM ROWAN HAMILTON (1805–65) discovers
DAMENTAL THEOREM OF ALGEBRA. Throughout his life, QUATERNIONS. Two years later ARTHUR CAYLEY
Gauss, among many accomplishments, derives the (1821–95) discovers octonians.
LEAST SQUARES METHOD, classifies the CONSTRUCTIBLE
regular polygons, and makes fundamental contribu- 1844
tions to NUMBER THEORY, GEOMETRY, STATISTICS,
mathematical physics, and astronomy. JOSEPH LIOUVILLE (1809–82) discovers the first exam-
ple of a TRANSCENDENTAL NUMBER.
1799
1854
Norwegian surveyor Casper Wessel (1745–1818)
publishes the equivalent of an ARGAND DIAGRAM as a British scholar GEORGE BOOLE (1815–64) establishes
means for representing COMPLEX NUMBERS. French the field of symbolic logic with his development of
bookkeeper and mathematician JEAN ROBERT BOOLEAN ALGEBRA.
ARGAND (1768–1822) develops the same method in
1806. German mathematician GEORG FRIEDRICH BERNHARD
RIEMANN (1826–66) offers a universal approach to
1812 geometry. He discovers SPHERICAL GEOMETRY. He later
makes significant advances in the theory of numbers
British mathematician and inventor CHARLES BABBAGE and the study of PRIME numbers.
(1791–1871) constructs the first mechanical calculator.
In 1823 Babbage obtains funds to build the “difference 1858
engine,” the first digital computer, but the project is
never completed. AUGUST FERDINAND MÖBIUS (1790–1868) and Johann
Benedict Listing independently discover the MÖBIUS
ca. 1820 BAND.

Norwegian algebraist NIELS HENRIK ABEL (1802–29) German mathematician JULIUS WILHELM RICHARD
proves that there can be no general formula akin to the DEDEKIND (1831–1916) suggests the notion of a
famous quadratic formula that solves all fifth-degree DEDEKIND CUT as a means to properly define the REAL
polynomial equations. Soon afterward, French mathe- NUMBERS.
matician ÉVARISTE GALOIS (1811–32) begins work to
classify which fifth- and higher-degree equations can be CAYLEY introduces the notion of a MATRIX to the study
so solved and consequently founds the field of GROUP of algebra.
THEORY.
1872
1821
FELIX CHRISTIAN KLEIN (1849–1925) unifies the fields
French mathematician AUGUSTIN-LOUIS CAUCHY of geometry with his “Erlanger program.” He also dis-
(1789–1857) develops the notion of a LIMIT as an covers the KLEIN BOTTLE.
attempt to place CALCULUS on sound mathematical
footing. This idea is later refined by German scholar 1873
KARL THEODOR WILHELM WEIERSTRASS (1815–97).
WILLIAM SHANKS (1812–82) computes, by hand, the
first 607 decimals of π. He is correct up to the 527th
place.

544 Appendix I

1874 lication of their three-volume Principia Mathematica,
an ambitious attempt to derive all mathematics by
GEORG CANTOR (1845–1918) develops SET THEORY logical principles from a small set of beginning
and his theory of CARDINALITY. AXIOMs.

1882 1913

FERDINAND VON LINDEMANN (1852–1939) proves that Indian mathematician SRINIVASA AIYANGAR RAMANU-
π is transcendental and hence that the challenge of JAN (1887–1920) begins a five-year collaboration with
SQUARING THE CIRCLE is impossible. British mathematician GODFREY HAROLD HARDY
(1877–1947).
1883
1921
Françoise-Edouard-Anatole Lucas (1842–91) invents
the TOWER OF HANOI puzzle. German mathematician AMALIE NOETHER (1882–1935)
publishes her theory of RINGs, directing research in alge-
1896 bra away from the study of calculation toward the
study of abstract structures.
JACQUES HADAMARD (1865–1963) and CHARLES DE
LA VALLÉE-POUSSIN (1866–1962) independently prove 1925
the PRIME NUMBER THEOREM first conjectured by
GAUSS in 1792. SIR RONALD AYLMER FISHER (1890–1962) publishes
Statistical Methods for Research Workers, an influen-
1899 tial work that provides the basis for modern experi-
mental design.
German mathematician DAVID HILBERT (1862–1943)
provides a complete axiomatic treatment of EUCLIDEAN 1928
GEOMETRY.
James Alexander (1888–1971) develops the Alexander
1900 polynomial, the first invariant in KNOT THEORY. Subse-
quently, John Conway defined the Conway polynomial
At the International Congress of Mathematicians in in 1968, and Vaughn Jones the Jones polynomial in
Paris, HILBERT presents his famous list of 23 problems 1985.
to challenge scholars of the 20th century.
1931
1901
KURT GÖDEL (1906–78) stuns the mathematical com-
HENRI LÉON LEBESGUE (1875–1941) introduces a revo- munity by establishing a pair of “incompleteness the-
lutionary new approach to INTEGRAL CALCULUS. orems.” Gödel proves that within any logically
rigorous system there will necessarily be statements
1903 that can neither be proved nor disproved. The goal
pursued by RUSSELL and Whitehead is proved
Swedish mathematician Nils Fabian Helge von Koch unattainable.
(1870–1924) introduces the KOCH CURVE, the first
example of an object later to be classified as a FRACTAL. 1938

1904 CLAUDE ELWOOD SHANNON (1916–2001) establishes
that BOOLEAN ALGEBRA can be successfully applied to
JULES-HENRI POINCARÉ (1854–1912) conjectures that computer design. He founds the field of INFORMATION
any three-dimensional object sharing the same topologi- THEORY in 1949.
cal characteristics as a SPHERE must indeed be a sphere.
1944
1905
Hungarian-American mathematician JOHN VON NEU-
ALBERT EINSTEIN (1879–1955) writes five ground- MANN (1903–57) and American economist Oskar Mor-
breaking papers in the field of mathematical physics. genstern (1902–77) found GAME THEORY.
The final two papers develop his famous special theory
of relativity. Einstein publishes his general theory of rel- 1963
ativity in 1916.
Edward Lorenz develops CHAOS theory.
1910

BERTRAND ARTHUR WILLIAM RUSSELL (1872–1970)
and Alfred North Whitehead (1861–1947) begin pub-

Appendix I 545

1976 1999

Using 1,200 hours of computer time, Kenneth Appel Hales proves the “honeycomb conjecture,” which states
and Wolfgang Haken prove the FOUR-COLOR that any partition of the plane into regions of equal area
THEOREM. has perimeter equal to the design of a honeycomb.

1978 Yasumasa Kanada of the University of Tokyo computes
π to 206,158,430,000 decimal places.
Ronald Rivest, Adi Shamir, and Leonard Adleman
develop the RSA public-key encryption system. 2000

1979 Michael Hutchings, Frank Morgan, Manuel Ritoré,
and Antonio Ros prove the outstanding “double bub-
Benoit Mandelbrot discovers the MANDELBROT SET. He ble” conjecture in the theory of SOAP BUBBLES. They
later founds the field of FRACTAL geometry. establish that the design of minimal surface area that
encloses two fixed volumes is indeed the “double bub-
Under the instigation of Pope John Paul II, the Roman ble” configuration one observes in nature.
Catholic Church opens its files on the Galilean trials.
The Church reverses its 17th-century condemnation of 2003
the scholar in 1992.
Tomás Oliveira e Silva verifies that GOLDBACH’S CON-
1994 JECTURE holds true for all even numbers between four
and 6 × 1016.
ANDREW WILES, with the assistance of Richard Taylor,
proves FERMAT’S LAST THEOREM. Michael Shafer discovers that the 6,320,430-digit num-
ber 220,996,011 – 1 is PRIME. It is the largest prime and
1998 the 40th MERSENNE PRIME known to this date.

Thomas Hales uses computer methods to establish 2004
JOHANNES KEPLER’s 1611 conjecture that the cubic
close packing and the hexagonal close packing of Martin Dunwoody announces to the mathematical
spheres are the densest packings of spheres. Mathe- community that he may have proved the Poincaré con-
maticians are unable to verify the proof without the jecture. Mathematicians await the details of his proof.
aid of a computer.

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Singh, Simon. Fermat’s Enigma. New York: Random House, Available online. URL: http://cepa.newschool.edu/het/home.
1997. htm. Accessed on April 20, 2004. This site provides a com-
prehensive review of the development of economic thought
Smart, James R. Modern Geometries. 5th ed. Pacific Grove, and practices.
Calif.: Brooks/Cole, 1998.
Drexel University. “Math Forum.” Available online. URL:
Sondheimer, Ernst, and Alan Rogerson. Numbers and Infin- http://mathforum.org. Accessed on April 20, 2004. This
ity: A Historical Account of Mathematical Concepts. student-friendly site provides an interactive question-and-
Cambridge, U.K.: Cambridge University Press, 1981. answer service for mathematics students and teachers, as
well as a host of mathematical resources, essays, and dis-
Stein, Sherman K., and Anthony Barcellos. Calculus and Ana- cussions listed by topic.
lytic Geometry. 5th ed. New York: McGraw-Hill, 1992.
Frazier, Kendrick. “A Mind at Play: An Interview with Mar-
Stewart, Ian. Concepts of Modern Mathematics. New York: tin Gardner.” Committee for the Scientific Investigation of
Dover, 1995. Claims of the Paranormal. Available online. URL:
www.csicop.org/si/9803/gardner.html. Accessed on March
Stewart, James. Single Variable Calculus. 3rd rd. Pacific 1998. An in-depth interview with recreational mathemati-
Grove, Calif.: Brooks/Cole, 1995. cian Martin Gardner.

Strafflin, P. Game Theory and Strategy. Washington, D.C.: Grissinger, Arthur. “Polya’s Problem Solving Strategy.” Avail-
Mathematical Association of America, 1993. able online. URL: www.lhup.edu/agrissin/polya.htm.
Accessed on April 23, 2004. This site provides a brief
Tanton, James. “A Dozen Questions about a Donut.” Math summary of the problem-solving strategies proposed by
Horizons, (Nov. 1998): 26–31. George Polya.

———. “A Dozen Questions about the Isoperimetric Prob- Joyce, David. “Mathematics in China.” Available online. URL:
lem.” Math Horizons, (Feb. 2003): 23–26. http://aleph0.clarku.edu/~djoyce/mathhist/china.html.
Accessed on September 17, 1995. David Joyce provides a
———. “A Dozen Questions about the Powers of Two.” detailed overview of the development of mathematics in
Math Horizons, (Sept. 2001): 5–10. China.

———. “A Dozen Questions about Squares and Cubes.”
Math Horizons, (Sept. 2000): 29–34.

———. “A Dozen Reasons Why 1 = 2.” Math Horizons,
(Feb. 1999): 21–25.

———. Solve This: Mathematical Activities for Students and
Clubs. Washington, D.C.: Mathematical Association of
America, 2001.

———. “Young Students Approach Integer Triangles.” Focus
22, no. 5 (2002): 4–6.

Todd, Deborah. The Facts On File Algebra Handbook. New
York: Facts On File, 2003.

550 Appendix II

Kadon Enterprises, Inc.. “Game Inventor: Martin Gardner.” vides the most comprehensive collection of historical and
Available online. URL:http:// www.gamepuzzles.com/ biographical essays currently available.
martin.htm. Accessed on April 20, 2004. This site pre-
sents a short article about the life and work of Martin Sandifer, C. Edward. “Ed Sandifier’s Home Page.” Available
Gardner. online. URL: http://vax.wcsu.edu/~sandifer/homepage.html.
Accessed January 16, 2004. Ed Sandifer is an expert on the
Kimberling, Clark. “Biographical Studies.” Available online. life and work of Leonhard Euler and provides a wealth of
URL: http://faculty.evansville.edu/ck6/bstud. Accessed on invaluable information about this famous scholar on this
April 20, 2004. Clark Kimberling provides a number of Web site.
in-depth biographical essays on famous mathematicians.
Verrill, H. “Origami Trisection of an Angle.” Available
Loy, Jim. “Trisection of an Angle.” Available online. URL: online. URL: http://hverrill.net/origami/. Accessed on June
http://www.jimloy.com/geometry/trisect.htm. Accessed on 24, 2002. This site provides an accessible overview of the
April 23, 2004. This site presents, with detailed explana- use of mathematics in origami.
tion, the varied techniques that have been proposed
throughout the centuries for trisecting arbitrary angles. Wales, Jimmy, and Larry Sanger, fndrs. “Wikipedia: The Free
Encyclopedia: Mathematics.” Available online. URL:
Math Academy Online. “Platonic Realms Interactive Mathe- http://en.wikipedia.org/wiki/Mathematics. Accessed on
matics Encyclopedia.” Available online. URL: http://www. June 15, 2004. This site serves as a user-friendly resource
mathacademy.com. Accessed on April 23, 2004. This site on mathematical topics.
offers mathematics quotes, brief historical notes, and some
discussion on mathematical concepts. Weisstein, Eric. “Eric Weisstein’s World of Scientific Biogra-
phy.” Available online. URL: http://scienceworld.wol-
Miller, Jeff. “Earliest Known Uses of Some of the Words of fram.com/biography. Accessed on April 20, 2004. This site
Mathematics.” Available online. URL: http://members. provides biographical information on an extensive list of
aol.com/jeff570/mathword.html. Accessed on April 23, famous mathematicians.
2004. This extensive site attempts to pinpoint the first use of
current mathematical terms. ———. “Mathworld.” Available online. URL: http://math-
world.wolfram.com. Accessed on April 22, 2004. Perhaps
———. “Earliest Known Uses of Various Mathematical Sym- the most comprehensive mathematical resource currently
bols.” Available online. URL: http://members.aol.com/ available on the Web, this online encyclopedia provides
jeff570/mathsym.html. Accessed on April 23, 2004. This detailed information on most every aspect of current
site attempts to identify the first use of a given mathemati- mathematical research.
cal symbol and provide the date and name of the docu-
ment in which it first appeared. Wilkins, David R. “The History of Mathematics.” Avail-
able online. URL: http://www.maths.tcd.ie/pub/HistMath.
O’Connor, John, and Edmund Robertson eds. “MacTutor Accessed on April 23, 2004. This site provides a direc-
History of Mathematics Archive.” Available online. URL: tory of Web sites from around the world that relate to
h t t p : / / w w w - g r o u p s . d c s . s t - a n d r e w s . a c . u k / ~ h i s t o r y. the history of mathematics, as well as presents biogra-
Accessed on April 20, 2004. This award-winning site pro- phies of some 17th- and 18th-century mathematicians.

APPENDIX III

ASSOCIATIONS 1400 Washington Avenue, Albany, N.Y. 12222. Tele-
phone: (518) 442 3300. Web site: www.isama.org.
The following organizations provide information about International Statistical Institute, P.O. Box 950, 2270 AZ
mathematics, mathematics research, and mathematics Voorburg, The Netherlands. Telephone: 31 70 3375737.
education of interest to students and teachers. Web site: www.cbs.nl/isi.
London Mathematical Society, De Morgan House, 57–58
American Mathematical Society, 201 Charles Street, Provi- Russell Square, London WC1B 4HS. Telephone: 020 7637
dence, R.I. 02904-2294. Telephone: (800) 321 4267. Web 3686. Web site: www.lms.ac.uk.
site: www.ams.org. Math Circle, P.O. Box 313, Jamaica Plane, Mass. 02130.
Telephone: (617) 519 6397. Web site: www.themathcir-
Association for Women in Mathematics, 4114 Computer and cle.org.
Space Sciences Building, College Park, Md. 20742-2461. Mathematical Association of America, 1529 Eighteenth Street
Telephone: (301) 405 7892. Web site: www.awm-math.org. N.W., Washington, D.C. 20036-1385. Telephone: (800)
741 9415. Web site: www.maa.org.
Canadian Mathematical Society, 577 King Edward, Suite Mathematical Sciences Research Institute, 17 Gauss Way,
109, Ottawa, Ont., Canada K1N 6N5. Telephone: (613) Berkeley, Calif. 94720-5070. Telephone: (510) 642 0143.
562 5702. Web site: www.cms.math.ca. Website: www.msri.org.
Mathematics Foundation of America, 129 Hancock Street,
Clay Mathematics Institute, One Bow Street, Cambridge, Cambridge, Mass. 02139. Telephone: (510) 525 7931.
Mass. 02138. Telephone: (617) 995 2600. Web site: Web site: www.mfoa.org.
www.claymath.org. Math Forum, 3210 Cherry Street, Philadelphia, Penn. 19104.
Telephone: (800) 756 7823. Web site: www.mathforum.org.
European Mathematical Society, Department of Mathemat- National Council of Teachers of Mathematics, 1906 Associa-
ics, P.O. Box 4 (Yliopistonkatu 5), 00014 University of tion Drive, Reston, Va. 20191-1502. Telephone: (703) 620
Helsinki, Finland. Telephone: 3589 1912 2883. Web site: 9840. Web site: www.nctm.org.
www.emis.de. Society for Industrial and Applied Mathematics, 3600 Univer-
sity City Science Center, Philadelphia, Penn. 19104-2688.
Institute for Mathematics and Its Applications, 400 Lind Telephone: (215) 382 9800. Web site: www.siam.org.
Hall, 207 Church Street, S.E., Minneapolis, Minn. St. Mark’s Institute of Mathematics, 25 Marlborough Road,
55455-0436. Telephone: (612) 624 6066. Web site: Southborough, Mass. 01772. Telephone: (508) 786 6126.
www.ima.umn.edu. Web site: www.stmarksschool.org.

Institute of Mathematical Statistics, P.O. Box 22718, Beach-
wood, Ohio 44122. Telephone: (216) 295 2340. Web site:
www.imstat.org.

International Mathematical Union, Institute of Advanced
Study, Einstein Drive, Princeton, N.J. 08540. Telephone:
(609) 683 7605. Web site: www.mathunion.org.

International Society of the Arts, Mathematics, and Architec-
ture, Department of Mathematics, University at Albany,

551

INDEX

Boldface page numbers indicate main entries. Page numbers in italics indicate illustrations or diagrams.

A acute triangle 506 L’Algebra (Bombelli) 47–48 Analyse des infiniment petits
addend 6 Algebra (Simpson) 463 (L’Hôpital) 40, 310
AAA (angle-angle-angle) rule addition 5, 6, 205 algebraic curve 114
1, 463 addition formulae algebraic function 11 analysis 13, 423
algebraic notation 51 Analysis per quantitatum
AAS (angle-angle-side) rule 1 (trigonometry) 6, 41–42 algebraic number 11, 60, 97
abacus 2, 2–3, 56, 74, 260 addition law 344, 367 series (Newton) 353
Abel, Niels Henrik 3, 10, 84, addition-multiplication magic and analytic number theory analysis situs. See topology
360 Analysis situs (Poincaré) 398
215, 241, 405, 431–432 square 327, 328 “The Analyst” (Berkeley) 58
and solutions to polynomial addition rule, in probability algebraic structure 11 analytic engine, Babbage’s 32
equations 471 algebra of logic, Boole’s 48 analytic geometry 13, 347
theory 415–416, 416 Algebra with Arithmetic and analytic number theory 13,
Abelian groups 3, 5, 83, 241 additive function 6
Abel Prize 3 additive identity element (zero) Mensuration (Colebrook) 139–140, 360, 411
abscissa 62 42 analytics 27
absolute convergence 3–4, 6, 257 algorithm 11, 294 anchor ring. See torus
additive inverse, and rings Al-jam’ w’al-tafriq ib hisab al- ancillary theorem 498–499
248, 337 hind (al-Khwa¯rizmı¯) 293 angle 13–15, 14, 276, 441
test for 3–4, 104 448 Almagest (Ptolemy) 421
Adleman, Leonard 111 alphamagic square 328, 328 45º (octant) 363
absolute error 167 affine geometry 6–7, 339 alternate interior/exterior of elevation/depression 14
absolute frequency 206 affine transformation 7 angles 504 of inclination 261
absolute maximum/minimum Agnesi, Maria Gaëtana 7–8, alternating harmonic series angle bisector 46, 260
248, 409 angle brackets 50, 519
330, 331 58 alternating knot 295 angle trisection. See trisecting
absolute value (modulus) 4–5, Ahmes (Ahmose) 155, 447 alternating series 11–12 an angle
Ahmes papyrus. See Rhind alternating-series test 12, Annales de Mathématiques
131, 133, 142, 310, 328–329 104–105 (journal) 25
of complex number 25, 86 papyrus altitude 12–13, 105 Annals of Eugenics (journal)
Alembert, Jean Le Rond d’ 8, 387
abstract algebra 5, 10, 314, of triangle Annals of Mathematics 531
354–355 8–9, 209 concurrency of 12, annulus 15, 89, 94, 177
and Fourier series 202 12–13, 165 anomalous cancellation 56
abstract group 68 and Euler line antiderivative 94–95, 211,
abundant number 389 Alembert’s theorem, d’. See 174–175, 507 360
Académie Française 8 fundamental theorem of antidifferentation 15, 133,
acceleration 523 algebra amicable numbers 13, 18, 272
Accurate Rendering (Ahmes) 423 antinomy. See Russell’s
Ꭽ0 (aleph null) 60–61 paradox
447 Alexander, James Waddell Analemma (Ptolemy) 422
Achilles and the tortoise analog vs. digital 135–136
295–296
(paradox) 532–533 Alexander polynomial
acnode. See isolated point
Acosta, José de 260 295–296
actuarial science 5, 312 algebra 9–11
acute angle 5–6, 14
and geometry 124

552

Index 553

antipodal points (antipodes) Argand, Jean Robert 25, 86 attractor 150 Bernoulli, Johann (II) 40
15, 471, 473 Argand diagram 25, 86, 359 augend 6 Bernoulli, Johann (III) 40
argument 25–26, 86, 199, automaton 30, 30–31, 37, Bernoulli, Nicolaus (I) 40
antisymmetry 365 Bernoulli, Nicolaus (II) 40
apex (apices) 15, 91, 423 524–525 350–351 Bernoulli numbers 40, 435,
Apollonius of Perga 15–16, Aristotle 26–27, 238, 270, automorphic functions 398
average. See mean (average) 489
76, 78, 93, 105, 115, 309, 532 axiom (postulate) 31, 170, Bernoulli’s inequality 267–268
136–137, 189, 239, 252, and formal logic 25–26, Berry’s paradox 374–375
253, 476 199 314 Bertrand, Louis François 40,
axiom of choice 31, 530, 534
and duplicating the cube Aristotle’s wheel paradox 375 71
149 arithmetic 27 B Bertrand’s conjecture 167
Arithmetica (Diophantus) Babbage, Charles 32, 32–33, Bertrand’s paradox 40,
and trisecting an angle 513
Apollonius’s circle 16 137–138, 189, 190, 239, 322 40–41, 75, 374, 417
Apollonius’s theorem 16 253, 525 Babbage’s difference engine 33 “Beweis, dass jede Menge
apothem (short radius) 16, Arithmetica infinitorum Babylonian mathematics 9,
(Wallis) 528 wohlgeordnet werden kann”
403 Arithmetica universalis 33–35, 249, 425, 426, 428 (Zermelo) 533–534
Appel, Kenneth 200–201, 420 (Newton) 353 Bacon, Francis 452 Bha¯skara II (Bhaskaracharya)
applied mathematics 17 Arithmetices principia, nova Bakhshali manuscript 264 10, 17, 41–42, 264, 392
appropriately nice functions methodo exposita (Peano) balance point. See center of
385 and negative numbers 348
315 arithmetic–geometric-mean gravity bias 42
approximation 17 inequality 219, 333 La balancitta (Galileo) 213 biconditional (“if, and only if”
Arabic mathematics 17–18 arithmetic mean 333 Banach, Stefan 24, 35
Arbogast, Louis 249 The Arithmetic of Logarithms Banach-Tarski paradox statement) 42, 91, 514
arc 18, 76 (Briggs) 53–54 bifurcation point 151
Archimedean solids 398 arithmetic sequence 28, 24–25, 35, 374, 526 Bijaganita (Bha¯skara) 41–42
Archimedean spiral 239, 400, 139–140 band ornament. See frieze bijection 209
arithmetic series 28 bimodal distribution 480, 480
474 arrangement. See permutation pattern binary line search. See
Archimedes 18, 18–21, 19, array 28 barber paradox 374–375,
arrow paradox, Zeno’s 533 bisection method
20, 91, 93, 105, 115, 173, Ars conjectandi (Jacob 453–454 binary numbers 30, 42–43,
180, 213, 238–239, 270, Bernoulli) 39–40, 414 Barbier, Joseph 95
272, 476 Ars magna (Cardano) 10, 47, bar chart 479, 479 88, 150
59–60, 112, 190–191, 494 “The Bargaining Problem” binary operation 43, 257, 279
and Archimedean solids Artin, Emil 52
398 A¯ ryabhata (Indian (Nash) 347 cancellation as 56–59
mathematician) 17, 28–29, Barrow, Isaac 35–36, 36, 54 closure as 79
on Fields medal 194 264, 392 Der barycentrische Calcül commutative property as
and his spiral 474 A¯ ryabhatiya (A¯ ryabhata)
and semiregular polyhedra 28–29 (Möbius) 339 83
ASA (angle-side-angle) rule 1 base of exponent 180 Binet’s formula 193
404 “As I Was Going to St. Ives” base of number system 36–38, binomial 43
and sum of first n numbers 156 binomial coefficient 45,
associative property 29, 83 37
489 base of polygon/polyhedron 64–65, 80, 418
and trisecting an angle 513 of addition 5 binomial distribution 43–44,
Archimedes’ water screw 20, in Boolean algebra 49 38, 506
21 and groups 241 base-10 logarithms 53 44, 143, 180
Archytas 21–22, 172 of matrices 223 Bayes, Rev. Thomas 38–39 binomial expansion 44–45
arc length 22, 470 and order of operation Bayes’s theorem 39, 302 binomial theorem 18, 44–45,
area 22–25, 23, 24 bearing 39
of annulus 15 366 Behennde unnd hüpsche 80–81, 219, 309, 364
of basic shapes 23, 23–24, and rings 448 Biometrics (journal) 387
of vector addition 520, Rechnung auf fallen biquadratic equation. See
156, 248, 293 Kauffmannschaften
of circle 76, 222, 392 522 (Widman) 6, 487 quartic equation
of curved figures 172 astroid 115 bel 118 birectangular triangle 474
of ellipsoid 161–162 Astronomiae physicae et Bell, Alexander Graham 118 bisection method
as ill-defined concept bell-shaped distribution 356,
geometricae elementa (D. 480, 480 (dichotomous line search,
22–23, 24, 526 Gregory) 240 Berkeley, George 58 binary line search) 45–46,
of polygon 403 Astronomia nova (Kepler) Bernays, Paul 250 277, 451
of quadrilateral, 292 Bernoulli family 39–40, 50, bisector 46
asymmetrical 490 284, 529 bit 461
Bretschneider’s formula asymptote 29–30, 234, 440 Bernoulli, Daniel 40, 107, 202 Blatzer, Richard 47
for 52–53, 430 asymptotic series 39 Bernoulli, Jacob 39–40, 58, body mass index (BMI). See
of quadrilateral inscribed in atto- (10-18) 465 414 Quételet index
circle, Brahmagupta’s Bernoulli, Jacob (II) 40 Bohlen numbers 195
formula for 51–52 Bernoulli, Jacques 65–66, 489 Bolyai, János 46–47, 171,
of rectangle 443 Bernoulli, Johann 40, 58, 107, 219, 255, 318, 355, 455
of square 474 114, 272, 310, 474, 494 Bolzano, Bernard Placidus 47,
of trapezoid 504 276
of triangle 486 Bolzano’s theorem. See
area hyperbolic functions 281 intermediate-value theorem

554 Index

Bombelli, Rafael 47–48, 50, calculator, mechanical 382, Catalan, Eugène Charles 64 Chen, Jing-Run 230
113 383 Catalan conjecture 64 Chêng Ta-wei 325
Catalan numbers 64–65 chessboard puzzle 224
Boole, George 10, 48, 48–49, calculus 7, 8–9, 56, 100, 320 catenary 65–66, 255 chicken (game) 218, 413, 413
199, 309, 460 history of 57–58, 67, 126, Cauchy, Augustin-Louis 47, Ch’in Chiu-shao 72, 73
325, 368 Chinese mathematics 72–74
Boolean algebra 49, 460, and optimization 365 66, 66–67, 103, 287, 529 Chinese proof 425
461–462 and Snell’s law of and definition of limit 269, Chinese remainder theorem
refraction 468–469 312, 459
border square 328, 328 and mean-value theorem 73
Borel, Félix Edouard Émile calculus of variations, 335 chi-squared test 74–75, 98,
Weierstrass’s 284, 529 and number theory 442
216, 307 386, 415
Borromean rings 49, 49 calendar, Chinese 537 Cauchy-Riemann equations choice set 31
Borsuk, Karol 15 cancellation 56–59 67 chord 75, 76, 457, 510, 511
Borsuk-Ulam theorem 15, 471 cancellation law, for fractions Chords in a Circle (Menelaus)
Bosse, Abraham 124 Cauchy-Schwarz inequality
Boss puzzle. See slide 15 puzzle 204–205 268 336
bound 49–50 Canon doctrinae triangulorum chord theorems 78
bounded above/below, Cauchy sequence 442 chord values, Hipparchus’s
(Rhaeticus) 511 Cavalieri, Bonaventura
definition of 50 Canon mathematicus seu ad 239
Bourbaki, Nicolas 50, 168 Francesco 57, 67, 74, 293, Chou pei suan ching
braces 50, 460 triangula (Viète) 525 527, 528
brachistochrone problem 39, Cantor, Georg 11, 59, 203, Cavalieri’s principle 57, (anonymous) 73, 425
67–68, 115, 412, 527, 537 Chudnovsky, Gregory and
114, 310 208, 270, 297, 533 Cayley, Arthur 68, 200
brackets 50–51, 366 on cardinality 60–62, Cayley-Hamilton theorem David 393
Brahe, Tycho 292 129–130, 271, 460–461 68–69 Chuquet, Nicola 526
Brahmagupta 9, 51, 114, 264, and continuum hypothesis Cayley numbers 359, 432 Chu Shih-Chieh (Zhu Shijie)
100 ceiling/floor brackets 51
348, 534 on infinity 47 ceiling function (least-integer 73–74, 75, 383
Brahmagupta’s formula and number theory 63, function) 198 circle 75–76, 92, 136,
359, 367–368, 441, 442 center-limit theorem 69–70
51–52, 53, 249, 264, 430 center of gravity 18–19, 69, 141–142, 166, 188
Brahmasphutasiddhanta Cantor set 203 372, 470 area of 24, 24
Cardano, Girolamo (Jerome center of mass vs. center of diameter of 130
(Brahmagupta) 51, 264 gravity 69 eccentricity of 155
braid 52, 52 Cardan) 10, 47, 59–60, 112, centile. See percentile
braid group 52 190–191, 431, 494 central angle 77, 77 Circles of Proportion and the
breadth 309 central-angle/peripheral-angle Horizontal Instrument
Bretschneider’s formula 52–53 and imaginary numbers theorem 77, 77 (Oughtred) 369
Briggs, Henry 53–54, 320, 87 central-limit theorem 143,
401, 482–483 circle squaring. See squaring
346 and negative numbers the circle
Briggsian logarithms 53 348–349 and normal distribution
Brinkley, John 245 356 circle theorems 75, 76–78,
British Association for the and probability theory 114, 425, 430
413, 414, 417 central projection 418
Advancement of Science 68 central tendency, measures of circular inversion 225
Bromhead, Sir Edward Cardano’s formula 112–113, circumcenter 78, 338, 507
431, 470, 494 480 circumcircle 76, 78, 165,
239–240 centroid 335, 372, 470, 507
Brouillon project d’une cardinality 60–62, 100, 106, Ceva, Giovanni 70 303–304, 321, 354
129–130, 209, 270, 359 Ceva’s theorem 70 circumference
atteinte aux evenemens des and number theory 123, Ceyuan Haijing (Li Ye) 318
recontres du cone avec un 359, 368, 442 chain 365–366 of circle 75–76
plan (Desargues) 123–124 chain rule 70–71, 88, 133, of earth 29
Brouncker, Lord William 54, cardioid 62, 115 circumscribe/inscribe 78
529 Carr, G. C. 435 138, 258, 274, 281, 282, “C1 Isometric Imbeddings”
Brouwer, Luitzen Egbertus Jan “carrying digits” 2, 6 378, 433–434 (Nash) 347
198 Cartesian coordinates change of variable. See cissoid 136–137
Brouwer fixed-point theorem integration: by substitution clarity 66–67, 529–530
198 (orthogonal coordinates) chaos 71, 151, 203 Clavis mathematicae
Brownian motion 158 62–63, 105, 114, 124, 226, chaos game 204 (Oughtred) 344, 370
brute force 54–55, 461 275, 504 characteristica universalis 309 clock math 92, 340–341
Buffon, Georges 55 characteristic polynomial closed curve 114
Buffon-Laplace problem. See of cardioid 62 68–69, 156 closed half-plane 243–244
Buffon needle problem vs. cylindrical coordinates characteristic vector. See closed half-space 243–244
Buffon needle problem 55, 55, eigenvector closed interval 184, 276, 278
302, 393 115 Chebyshev (Tchebyshev), closure property 79
Bürgi, Jobst 320 and direction cosines 138 Pafnuty Lvovich 71, 72, of addition 5
Buridan, Jean 368 vs. polar coordinates 400 167, 415 and groups 241
butterfly effect 151 quadrants in 428 Chebyshev polynomials 71 of matrices 222
of torus 500 Chebyshev’s theorem 71, 72, cluster sampling 407
C Cartesian product (cross 304 coefficient 79, 404, 525
cake cutting. See fair division product, external direct Cogitata physico-mathematica
Calandrini, Giovanni product, product set, set (Mersenne) 338
direct product) 63 Cohen, Paul 101
Ludovico 107 Cartesian space. See Euclidean Colebrook, H. J. 42
space Collatz’s conjecture 79
Cassini, Giovanni Domenico
370
Castillon, Johann 62
casting out nines 63–64

Index 555

The Collected Mathematical in trigonometry 121–122 connected graph 233 cosecant function 510
Papers of Arthur Cayley 68 and zeta function 535–536 consistent 94 cosine function 139, 509, 510
components of a vector 519 constant 94 cosine rule. See law of cosines
Collection (Pappus) 13, 253 composite 87, 186, 409 constant of integration 94–95, “the cossic art” 10, 517
collinear 79–80, 233 composite number 87, 463 cotangent function 510
Colson, John 7–8 composition (of functions) 272, 273 countable 11, 106, 348, 441
combination (selection, 69–70, 87–88, 274, 280 constant width 75, 95, 130 countable number. See natural
composition (of matrices) 330 constructible 95–97, 97, 149,
unordered arrangement) compound interest 153–154, number; whole number
80–82 275–276 218, 238, 403, 475, 476, counterexample 107, 120,
combinatorial coefficient 45, compound statement 514–515 513
384 Comptes Rendu (journal) 66 constructivism 297 229
combinatorics 82, 243 computer 88. See also contact number 472 counterharmonic mean 333
commensurable 82, 172, 238 calculator, mechanical contingency table 98, 98 counting number. See natural
common denominator 82–83 and Boolean algebra 49 continued fraction 98–100,
common difference 28 and critical path 109 487 number
common divisor 83 history of 32–33, 167, Cours d’analyse (Cauchy) 66
common factor 83 calculation of π using 54 Cours d’analyse (Vallée-
common fraction 205 321, 351 and definition of real
common multiple 83, 305 concave 88 Poussin) 518
common notions 170 concave polygon 402 number 441 A Course of Pure Mathematics
common ratio 223 concave up/concave down 89, golden ratio as 231
commutative group 83 and Pythagorean triples (Hardy) 246
commutative property 83 89, 429 cousin primes 515
concentric 89 427 covariance 106, 107, 306
of addition 5 “A Concise Outline of the continuity, defined by Cauchy covering. See tessellation
in Boolean algebra 49 Cramer, Gabriel 107–108
of dot products 148 Foundations of Geometry” 66 Cramer’s rule 107, 108–109,
and groups 241 (Lobachevsky) 318 “Continuity and Irrational
of multiplication 10, 344 conclusion (of syllogism) 27 223, 282
and quaternions 432 concurrent 89, 261, 335, 338, Numbers” (Dedekind) 119 Crelle, August Leopold 3
and rings 448–449 399, 507 continuous function 88, 100, Crelle’s Journal 3
and subtraction 487 conditional (“if . . . then” critical path 109, 109
of vector addition 520, statement) 89–90, 514 184, 276, 386 critical point 331
conditional convergence 4, continuously compounded cross product (vector product)
522 104
commutative ring 449 conditional equation 163 interest 276 63, 110, 520
comparison test 4, 103 conditionally convergent series continuous random variable cryptography 111, 410, 525
compass and straightedge 95 4 cube (hexahedron) 111–112,
complement (set operation) conditional probability 39, 143
90–91, 263 continuum hypothesis 59, 62, 115, 396, 397
460 condition—necessary and cube numbers 195
complementary angles 14 sufficient 91 100–101, 250, 533 cube root 112, 451
complete digraph. See cone 91 contour integral (curvilinear cubic equation 34, 84,
conformal mapping (equi-
tournament angular transformation, integral, line integral) 101, 112–114
complete directed graph 501 isogonal transformation) 91, 239 discriminant of 141
complete graph 233 337, 485 contour line 101 history of 59–60,
complete induction. See congruence 92 contradiction 94, 101, 170, 190–191, 493–494
congruence transformation. 171, 228, 229, 473
induction See isometry contrapositive 101–102, 265 cubic lattice 472
completeness law 367 congruent angles 14 contrapositive reasoning 26 cubit 309
completeness property, of real congruent figures 92, 463 convergence 66, 324 cumulative distribution
congruent triangles 1–2 convergent sequence 102,
numbers 47 conical frustum 207 312, 458 function 143
completing the square 84, 84, conical helix 248 convergent series 3, 12, cuneiform tablets 33, 34
The Conics (Apollonius of 102–105, 182, 240, 260, curl 144
428 Perga) 15–16, 93, 253 409, 459, 530 curve 114, 314
complex conjugate of complex conic sections 18, 76, 92–93, curves, Cramer’s classification
93, 123–124, 149, 238, 419 d’Alembert’s ratio test 8,
number 526 conjecture 31, 499 223 of 108
complex Euclidean space 170 conjugate of complex number curvilinear integral. See
complex fraction 206 86–87 converse 105, 506
complex numbers (C) 85–87, conjugate of sum/difference convex 88 contour integral
131 convex polygon 402 cusp 492, 493
219, 358–359, 408, 522 conjugate of surd 490 Conway, John Horton 296 cycle (graph theory) 233
completing the square and conjunction (“and” statement) cooperate vs. defect, in cyclic group 295
84 93–94, 514 cyclic polygon 114
and Euler’s formula 176 conjunction circuit 94 prisoner’s dilemma 412 cyclic quadrilateral 78, 264,
history of 25, 47–48, 245, connected 94 coordinate geometry 63, 368
398 coordinates 105, 399 430
and intersecting circles 76 coordination number 472 cycloid 114–115, 310
multiplication of 86 Copernican theory 214 cylinder 115, 411
order properties and 367 coprime. See relatively prime cylindrical coordinates 115,
polar coordinates of 86 corner 403
and Pythagorean triples corollary 498–499 504
426 correlation, and chi-squared cylindrical helix 248
reciprocal of 442
test 74–75 D
correlation coefficient 79, data 116, 478
Data (Euclid) 169
106, 107, 306–307, 386,
456–457
corresponding angles 504

556 Index

days-of-the-week formula denominator 204 difference formula 511–512, disjunction circuit 141
116–118 dense subset 441 512 disjunctive reasoning 26
denumerable (enumerable, dispersion, measures of
decibel 118 difference machine. See
decimal representation 36–38, numerable) 123 difference engine 480–482
denumerable sets 59, 60–62, displacement 141
72–73, 260, 264, 370, 485 difference of two cubes 131 Disquisitiones arithmeticae
declination 261, 261 100, 106 difference of two squares 9,
decomposition 118–119 dependent events 262–263 (Gauss) 218–219
De configurationibus dependent variable 519 131, 257 distance
derangement 390–391, 486 differential 131–132, 132,
qualitatum et motuum De ratiociniis in ludo aleae of point from line 142
(Oresme) 368 310 of point from plane 142
decrement 262 (Huygens) 414 Differential and Integral and velocity 522–523
Decker, Ezechiel de 54 derivative 132, 132, 198, 262, distance formula 22, 75,
Dedekind, Julius Wilhelm Calculus (Babbage) 32 141–142, 310, 520
Richard 58, 119, 120, 271–272, 418 differential calculus 56, distribution 72, 142–143,
172–173, 270, 367, 442 and calculation of velocity 143, 197, 218
Dedekind cut 119–120, and acceleration 523 132–134, 226, 308–309 chi-squared 74–75
172–173, 262, 277, 367 first and second, and vs. integral calculus distributive property
graphing 234 211–212 143–144
and bounded real numbers in Boolean algebra 49
50 Desargues, Girard 123–124, differential equations of dot products 148
270, 391, 418, 419 134–135, 234, 252, 299, and Elizabethan
and definition of real 309, 385, 399
number 442 Desargues’s theorem 124, and partial fractions 380 multiplication 160
270, 391, 419 Pythagoreans and 9
and extreme-value theorem differential geometry 226 and quaternions 432
184 Descartes, René 10, 57, differentiation 133, 309 and rings 448
124–125, 125, 180, 189, digit 135–136 div (divergence operator) 144
De determinantibus 431–432, 519 digital vs. analog 135–136 divergent 144
functionalibus (Jacobi) 287 and amicable numbers 13 digit extraction 183 divergent sequence 102
and analytic geometry 13 dihedral 136 divergent series 240, 459
deductive logic 420 and coordinate geometry dihedral angle 136 divine proportion. See golden
Aristotle and 26 62–63, 226, 235 dilation 225, 278, 317 ratio
and Euler-Descartes dimension 136 divisibility rules 92, 144–146
deductive reasoning 120–121 formula 177 dimension of a matrix. See division 146, 205
defect vs. cooperate, in and imaginary numbers divisor of zero 147
87 order of a matrix Doctrine of Change (De
prisoner’s dilemma 412 Diocles 136–137 Moivre) 122
deferent, of cycloid 115 Descartes’s rule of signs 124, Diophantine equations 54, 99, dodecahedron 396, 397
deficient number 389 125–126 donut. See torus
definite integral 148–149, 275 137, 239, 250, 289–290, 360 dot product (inner product,
deformation 121 descriptive statistics. See Diophantus 9, 137–138, 180, scalar product) 110, 136,
Degen, Ferdinand 3, 5 statistics: descriptive 138, 147–148, 520
degenerate hyperbola 373 189–191, 239, 252–253 double angle formula 525
degenerate quadrics 411 determinant 126–128, 223, directed number. See integer double cusp 492, 493
degree (measure of angle) 14 281–282, 287, 309 directional derivative double integral 148, 239, 400,
degree of a polynomial 121, 473
determinant function 108 138–139, 233 double point 148
404 De triangulis omnimodis direction cosines 138 double root 140, 451
degree of a vertex (valence) direction numbers 138 double torus 501
(Regiomontanus) 444, 511 direction ratios 138 doubling the cube 149
121, 233 deviation 481 directly congruent solids 92 drawer principle. See
degrees of freedom 121 diagonal 128–129, 376, 402, direct proof 139, 265, 420 pigeonhole principle
Dehn, Max 165 direct reasoning 26 duality in projective geometry
Delian altar problem. See 443 directrix 155 124
diagonal argument 59, 60, Dudeney, Henry Ernest 165
duplicating the cube of cone 91 dummy variable 148–149,
deltoid 430, 430 100, 106, 123, 129, of cylinder 115 488
De methodis serierum et 129–130, 359, 441–442 of parabola 373 duplicating the cube 18, 21,
diagonal Latin square 302 Dirichlet, Peter Gustav Lejeune 97, 136–137, 149, 166, 172,
fluxionum (Newton) 352 Dialogo (Galileo) 214 139–140, 172, 208, 337, 394 239, 525
De Moivre, Abraham 85, 107, diameter 75, 76, 130, 471 Dirichlet’s principle. See Dürer, Albrecht 149–150,
diamond 376, 474 pigeonhole principle 231, 232, 325, 391
121–122, 471, 486, 494 dice, tetrahedral 462 disc method 470 dyadic 150
and normal distribution dichotomous line search. See discontinuous functions 100 dyadic fraction 43
356, 414, 486 bisection method Discorsi e dimostrazione dynamical system 71, 88,
dichotomy paradox, Zeno’s matematiche intorno a due 150–151, 203, 296
De Moivre’s formula (De 533 nuove scienze (Galileo) 213 dynamics 299
Moivre’s identity) 85, 122 Dido’s problem 130, 284 discrete 140
Dieudonné, Jean 288 discrete mathematics 167
De Moivre’s quintic 471 “Die von der discrete transformation 140
Démonstration d’une méthode molekularkinetischen discriminant 84, 140–141,
Theorie der Wärme gefurdete 429
pour resoudre les égalitez de Bewegung von in ruhenden discriminant of cubic 113
tous les degrez (Rolle) 449 Flüssigkeiten suspendierten disjoint events. See mutually
“Démonstration d’un Teilchen” (Einstein) 158 exclusive events
théorème sur les fractions difference 130–131, 487 disjunction (“or” statement)
continues périodiques” difference engine, Babbage’s 141, 514
(Galois) 215 32–33, 33
De Morgan, Augustus
122–123, 200, 265, 270, 349
De Morgan’s laws 123, 460,
525

Index 557

E ellipse 92–93, 93, 137, 155, Theory of Electricity and and topology 500
160, 160–161, 239 Magnetism” (Green) 239 and unit circle in
e (eccentricity of conic) 155 reflection property of 365 An Essay on the Principle of
e (Euler’s number) 11, Population (Malthus) 407 trigonometry 511
ellipsoid 161–162, 411 “An Essay Towards Solving a and zeta function 105,
152–154, 154, 173, 283, elliptic cone 411 Problem in the Doctrine of
300, 301, 314 elliptic cylinder 411 Chances” (Bayes) 38 447, 535–536
earth 154–155, 166 elliptic geometry. See spherical Euclid 1, 78, 168–169, 172, Euler circuit 233
eccentric 89 183, 224, 226, 238–239, Euler-Descartes formula. See
eccentricity 155, 161, 254, geometry 252, 253, 265, 270, 314,
373 elliptic paraboloid 374, 411 324, 410 Euler’s theorem
echelon form 220 empty set (null set, Ø) 162, Euler diagrams 26, 362
edge and compass and Eulerian circuit 235–236
257, 267, 460 straightedge 95 Eulerian path 235
on graph 233 Encyclopédie ou dictionnaire Euler line 80, 174, 174–175,
of polyhedron 403 definition of point 399
Egyptian fractions (unit raisonné des sciences, des and formal logic 199, 250 354, 507
fractions) 155, 156, 192, arts, et des métiers and foundations of Euler’s brick 174
206 (d’Alembert) 8 Euler’s constant 174, 175,
Egyptian mathematics endpoints, of curve 114 mathematics 200
155–156, 423, 428, 497 enumerable. See denumerable and fundamental theorem 175, 247, 283, 461
Egyptian multiplication 155, Enumeratio linearum tertii Euler’s formula 85–86, 122,
156 ordinis (Newton) 353 of arithmetic 210
eigenvalue 156–157, 282 envy-free fair division 188 and perfect numbers 389 152, 163, 174, 175–176,
eigenvector (e-vector, latent epicycle 115 and Platonic solids 182, 185, 209, 236, 314
vector, characteristic vector, epicycloid 115
proper vector) 156–157, Epimenides 311 396–397 and logarithms 320
282 equality 162 and pure mathematics 422 Euler’s polygon division
“Eine neue Bestimmung der equal sign (=) 442 and Pythagorean triples
Moleküldimensionen” equating coefficient 162, 380, problem, Catalan numbers
(Einstein) 158 405 426 and 65
Einstein, Albert 157, equating real and imaginary Euclidean algorithm 11, 82, Euler’s polynomial 174
157–159, 227, 255, 398, parts 162–163 Euler squares 303
447, 496, 530 equation 163 159, 169–170, 210, 237 Euler’s theorem (Euler-
elasticity, mathematics of equation of line 163–164 Euclidean geometry 170, 409 Descartes formula) 80, 174,
227–228 equiangular 164–165, 166, 176–177, 177, 233,
“Élémens d’arithmétique 506 axioms in 31 278–279, 358, 404, 472
universelle” (Kramp) 187 equiangular polygon 402 Hilbert on 250 Euler totient function 174
elementary number theory equiangular transformation. vs. non-Euclidean geometry e-vector. See eigenvector
359–360 See conformal mapping even functions 177, 490
elementary operations 364 equidecomposable 165, 165 294 even numbers 177–178
elementary row operation equidistant 76, 78, 165–166, and SAS rule 2 event 178, 413, 417
220–222 260 Euclidean space (Cartesian exa- (1018) 465
Elemente der Mathematik equilateral 164, 166, 285, 506 space, n-space) 170 excircle 261, 354
(Blatzer) 47 equilateral polygon 402 Euclides ab omni naevo excluded middle, law of 304
The Elements (Euclid) 1, 78, “Equilibrium Points in N- vindicatus (Saccheri) 455 Exercitatio geometrica (D.
159, 168–169, 171, 172, person Games” (Nash) 347 Euclid of Megara Gregory) 240
183, 226, 238–239, 252, equipotent (equipollent, (philosopher) 168 Exercitationes geometricae sex
253, 270 equinumerable) sets 60 Euclid’s postulates 6–7, 159, (Cavalieri) 67
compass and straightedge equitangential curve. See 170–171, 226, 255, 455, 474 exhaustive subsets 371
tractrix See also parallel postulate existential quantifier (“there
in 95 equivalence 371 Euclid’s proof of the infinitude exists”) 430–431
and foundations of equivalent knots 295–296 of primes 159, 171–172, expanding brackets 9, 131,
equivalent sets 60 238, 265, 410 162, 179, 179, 404, 431
mathematics 200 Eratosthenes 154, 166–167, Eudoxus 26, 57, 172–173
perfect numbers in 389 462, 510 Euler, Leonhard 10, 107, 120, and binomial theorem 45
Platonic solids in 396–397 Eratosthenes’ sieve 462–463 140, 152, 161–163, 173, and distributive property
Élements de géométrie Erdös, Paul 167, 393 173–174, 175, 180, 182,
(Legendre) 308 Erdös number 167 185, 208, 284, 300, 431, 143
Éléments de mathématiques error 167–168, 197, 451, 457 523–525, 529 and Elizabethan
(Bourbaki) 50 error detection 461–462 and continued fractions 99
Elements of Arithmetic (De error sum of squares 306 and continuously multiplication 160
Morgan) 122 Essai sur une manière de expected value (mean, expecta-
Elements of Geometry représenter les quantités compounded interest 276
(Menelaus) 336 imaginaires dans les and Fermat’s last theorem tion, µ) 179–180, 334
Elements of Geometry constructions géometriques exponent 180–181, 263, 366,
(Playfair) 377 (Argand) 25 190
elimination method. See Essay on Conic Sections and foundations of 475
Gaussian elimination (Pascal) 382 exponential function
Elizabethan multiplication “Essay on the Application of mathematics 200
159–160, 160, 404 Mathematical Analysis to the and geometry 6, 64, 80 152–153, 181, 181–182
ell 309 and Goldbach’s conjecture exponential inequalities 268
exponential notation. See
229–230
and graph theory 235–236 scientific notation
and number theory 13, 18, exponential series 182
exponential time 405
187, 202, 247, 283, 349, expression 182
389, 426 exterior angle 182, 182–183,
and partition function 381
and pi (π) 393 504, 504
exterior-angle theorem 182,

183, 183, 377

558 Index

external direct product. See Fibonacci numbers 192–193, fourth dimension 136 fundamental theorem of
Cartesian product 193, 197, 384, 443, 458 fractal 136, 203–204, 204, isometries 212, 284
and golden ratio 231
extraction 183–184 and polyominoes 406 456, 463 fuzzy logic 212, 305
extrapolation 184, 278, 405 fractal shapes fuzzy-set theory 212
extreme and mean ratio. See field 5, 85, 442
Fields, John Charles 194 and Banach-Tarski paradox G
golden ratio Fields medals 193–194, 531 24 Galilei, Galileo 65, 93, 114,
extreme-value theorem 49, fifteen puzzle. See slide 15
and chaos 71 125, 213–215, 214, 270,
119, 184, 330, 442 puzzle fraction 146, 204–206, 341, 485, 523
extremum 184 figurate numbers 66, 99, 194, Gallai, Tibor 80
358, 439, 444 galley method 159
F 194–195, 287, 381, 389, 423 in binary notation 43 Galois, Évariste 10, 215, 242,
face 185, 403 Fincke, Thomas 511 reciprocal of 442 288, 432, 471
face angle 185 finger multiplication 195–196 Galton, Sir Francis 215–216,
face-centered cubic lattice 472 finite 196, 270 fractional dimensions 136 415
face of graph 185 finite differences 33, 75, fractional exponent 181, 368 Galton graph. See scatter
faceted polyhedron 404 fractional part brackets 51 diagram
factor 185–186 196–197 fractional part function 98, gambler’s ruin 216, 437
factorial 80, 174, 186–187, finite induction. See induction game theory 216–218,
finite projective geometry 419 198 346–348, 350–351, 415
444, 486 finite sets 106 Fraenkel, Adolf 534 gamma function 174, 187
factorization 118–119, 187 Fiore, Antonio Maria 493–494 Français, Jacques, and Gardner, Martin 218
factor theorem 187 first-derivative test 331–332 Garfield, James 425
factor tree 210 Fisher, Sir Ronald Aylmer complex numbers 25 Garnier, René 288
Fagnano, Giovanni 387 Franklin, Benjamin 325 Gauss, Carl Friedrich 119,
fair division (cake cutting) 197–198, 387, 415 freedom equations. See 140, 218–219, 226, 268
five-fold rotational symmetry
187–188 parametric equations and fundamental theorem
fallacy of the converse 26 490 Frege, Gottlob 199 of algebra 209
fallacy of the inverse 26 five stone problem 43 frequency 206
false position, method of 9, fixed point 198 frequency distribution 479 and geometry 47, 97,
floor/ceiling brackets 51 frequency polygon 479, 479 355–356
156 floor function (greatest-integer frequency table 479, 479
Faltings, Gerd 190 frequency theory of and group theory 242
Fano plane 419, 419 function) 198 and knot theory 295
Farey, John 188 fluid mechanics 8, 144, 248, probability, Venn’s 524 and number theory 10, 25,
Farey sequence (series) friendly numbers. See amicable
299 28, 87, 195, 287, 359,
188–189 flux 144 numbers 489
al-Farisi 18 fluxion 57–58, 198, 309, 352 frieze pattern 206, 206–207 and prime numbers 243,
F-distribution 484 focal chord 198 frustum 207 410–411, 518
feedback 150 focal radius 198 F-test 484 and probability 356, 414
femto- (10-15) 465 focus (foci) 155, 199 full linear group 222–223 and topology 500
Ferguson, D. F. 461 full turn 14, 362, 390 Gaussian elimination
Fermat, Pierre de 138, of an ellipse 161 function 139, 173, 207–209, (pivoting) 119, 219–222,
of hyperbola 253, 253–254 282, 288, 438, 491
139–140, 189, 272, 410, 531 of parabola 373 208, 445 Gelfond, Aleksandr 11
and amicable numbers 13 of paraboloid 374 functional analysis 35 general form of an equation
and coordinate geometry foot (unit of length) 309 function of a function. See 222
63, 226, 234–235 force field 101 general linear group 222–223,
and figurate numbers 66, Ford, Lester R. 188 composition 314
287 formal logic (symbolic logic) function of a function rule. See general maximum/minimum
and history of calculus 57 25, 89–90, 94, 101, 199, 331
little theorem of 342 238, 304–305 chain rule general theory of relativity
and probability theory Aristotle and 26 fundamental principle of 158
413, 414, 417 expressions in 182 General trattato di numeri et
founded by Boole 48 counting. See multiplication misure (Tartaglia) 493
Fermat’s last theorem 137, Gödel and 228–229 principle generating circle 500
138, 139–140, 189–190, set theory and 460 fundamental property of least generator 91, 115
227, 531 formula 199–200 time 189 geodesic 219, 223
Formulario mathematico fundamental theorem of geodesic dome 223
Fermat’s little theorem 342 (Peano) 385 algebra 85, 112, 186–187, Geography (Ptolemy) 422
Ferrari, Ludovico 10, 60, foundations of mathematics 209–210, 359, 405, 451 Geometriae pars universalis (J.
47, 200 Gregory) 240
190–191, 431–432, 470, 494 four-color theorem 55, and polynomial equations Geometriae rotundi (Fincke)
Ferro, Scipione del (Ferreo, dal 200–201, 420, 472 471 511
and Möbius band 340 Geometrica organica
Ferro) 10, 59–60, 112, 191, and torus 501 proofs of 10, 25, 218 (Maclaurin) 324
493–494 four elements (earth, air, fire, and zeta function 536 geometric distribution 44
Feuerbach, Karl Wilhelm 354 water) 397 fundamental theorem of geometric mean 333
Fibonacci 10, 155, 156, Fourier, Jean-Baptiste Joseph arithmetic 83, 118–119,
191–192, 409, 511 139, 201, 202, 208 186–187, 210–211, 283,
Fourier series 139–140, 177, 409, 439, 476
and Hindu-Arabic 201–203, 202, 307, 522, 530 proved by Gauss 218
numerals 251 and zeta function 535
fundamental theorem of
and minus sign 487 calculus 133, 153, 211–212,
and zero 534 240, 272, 494
formulated by Leibniz 308
in history of calculus

57–58

Index 559

geometric progression. See Greek alphabet 237, 237 harmonic series 4, 104–105, Hydrodynamica (Daniel
geometric sequence Greek mathematics 9, 174, 247–248, 368, 409 Bernoulli) 40

geometric sequence 223–224, 237–239 Haros, C. 188 hydrodynamics 140
489 Green, George 239–240 hatcheck problem 391 hydrostatics 18–19, 20
Gregory, David 240 al-Haytham 18 Hypatia 239, 252–253
geometric series 103, 163, Gregory, James 57, 240, 241, Heawood, Percy 200–201 hyperbola 30, 92–93, 93, 137,
409, 441, 488 height
476–477, 494, 496 155, 189, 239, 253,
geometric transformation Gregory series 11–12, 241, of cone 91 253–254, 513
224–225, 258, 316, 503 of cylinder 115 hyperbolic cosine 65–66,
325, 393, 461 of pentagram 388 254–255
La géométrie (Descartes) 63, Grelling’s paradox 374–375 of prism 412 hyperbolic cylinder 411
124–125, 226 The Grounde of Artes of tetrahedron 497 hyperbolic functions 254,
of triangle 506 254–255, 369
“Geometrische (Recorde) 443 Heisenberg, Werner 68 hyperbolic geometry 219,
Untersuchungen” group 59, 241–242, 252, 279, helix 114, 248 227, 255, 318, 355–356, 455
(Lobachevsky) 318 hemisphere 471 hyperbolic paraboloid 374,
398, 490 Hermite, Charles 11, 154, 411
geometry 124, 125, 225–228, group theory 5, 10, 215, 242, 301, 314 hyperbolic sine 254
238–239 Herodotus 498 hyperbolic spiral 474
288, 299, 350, 432 Heron 248–249 hyperboloid 255–256, 256
Geometry (Simpson) 463 Grundbegriffe der Heron’s formula 52–53, 78, hyperboloid of one/two sheets
Germain, Marie-Sophie 190, 248, 249, 336, 486, 506, 508 411
Wahrscheinlichkeitsrechnung Heron’s method 34–35, 183, hypercomplex numbers 432
226–228 (Kolmogorov) 296 248, 249, 285, 353, 476 hypercube 112, 256
giga- (109) 465 Grundlagen der Geometrie hertz (Hz) 206 hypersphere 136, 471
Girard, Albert 209, 398 (Hilbert) 171, 250 Hertz, Heinrich 158 hypocycloid 115
glide reflection 206, 225, 228 Grundlagen der Mathematik Heumann, Casper 312 hypotenuse 77, 256, 509
global maximum/minimum (Hilbert and Bernays) 250 hexadecimal notation 135 Hypothesis physica nova
gry 310 hexagonal lattice 472 (Leibniz) 309
330 Gunter, Edmund 369, 466 hexahedron. See cube hypothesis testing 143
gnomon 228 hierarchy 454
gnomonic projection 485 H Hieronymus 497–498 I
Gödel, Kurt 31, 101, 199, haberdasher’s puzzle 165 higher arithmetic. See number i (square root of –1) 85, 173,
Hadamard, Jacques 243, 411, theory
200, 228–229, 453, 458 higher derivative 249 358–359, 529
Gödel’s incompleteness 518 highest common factor. See i (unit vector) 520
Haidao suanjing (Liu Hui) 73 greatest common divisor Ibrahim ibn Sinan 18
theorems 101, 199, 229, hairy ball theorem 472, 521 Hilbert, David 137, 170, 171, icosahedron 232, 396, 397
250, 458 Haken, Wolfgang 200–201, 199, 229, 249–250, 355, 533 “Idealtheorie in
Goldbach, Christian 229, Hilbert’s axioms 171
230, 410 420 Hilbert’s infinite hotel paradox Ringbereichen” (Noether)
Goldbach’s conjecture half-chord 511 250–251, 269 355
229–230, 250, 359, 410 half-cone 91–93, 473 Hilbert space 250 identity 163, 257, 304
golden ratio 193, 230–231, half-line. See ray Hindu-Arabic numerals identity element 257, 258,
388–389 half-open interval 278 191–192, 251, 263, 370, 451 279, 295, 366
golden rectangle 231–232, 232 half-plane 94, 235, 243–244, Hipparchus 510
golden triangle 232 Hippasus 172, 283 additive 534
Goodwin, E. J. 394 267, 473 Hippocrates 251–252, multiplicative 344
googol/googolplex 232, 389 half-space 243–244 322–323, 476, 513 identity matrix (unit matrix)
Gosset, William Sealy 232, Hall, Monty 343 Hippocrates lune 251 128, 257, 257–258, 281
387, 415, 483 Hall, Philip 244 Hisab al-jabr w’al muqa¯bala identity property 222, 241
grad 233 Halley, Edmund 5, 312, 353, (al-Khwa¯rizmı¯) 10, 17, 251, image (range) 207, 258, 481
grade. See slope 293, 517 imaginary numbers 87. See
gradian 14 445 histogram 143, 479, 479 also complex numbers (C)
gradient 139, 233, 466–468, Halley’s comet 445 Historia natural y moral de las implicit differentiation
467 Hall’s matching (marriage) Indias (Acosta) 260 258–259
Graeco-Latin square 302–303 HOMFLYPT polynomial 296 implicit function 259
Grandi, Guido 451 theorem 244 homogeneous 252 improper factor 186
graph (network) 94, 233, 233, halting problem 88, 244 homomorphism 252 improper fraction 205
501 Hamilton, Sir William Rowan homotopy 398–399 improper integral (unrestricted
graphical solution 163, 234, honeycomb 292, 372 integral, infinite integral)
491 10, 87, 236, 244–245, 359, Horner, William George 72 104, 144, 259–260
graph of function 114, 207, 432, 520 How to Solve It (Pólya) 401 In artem analyticam isagoge
234–235, 275, 368 Hamiltonian circuit 236, 505 hundkurve 503 (Viète) 525
graph theory 82, 174, Hamiltonian path 236, 502 Hutchings, Michael 469 Incan mathematics 260
235–236, 245, 505 ham-sandwich theorem 245 Huygens, Christiaan incenter 260
Graunt, John 312, 415 hand (unit of length) 309 114–115, 180, 414, 503 inch 309
gravitation, theory of handshake lemma 121, 236, incircle 260–261, 354
399–400 245 inclination 261, 261
great circle 223, 314 handshakes across a table
greatest common divisor problem 65
(greatest common factor) Hardy, Godfrey Harold 246,
82, 169–170, 236–237, 290, 381, 435–436
305, 446 harmonic function 246
greatest-integer function. See harmonic mean 21, 333
floor function harmonic sequence
(progression) 246–247

560 Index

inclined plane 261 vs. differential calculus An Investigation of the Laws Julia, Gaston Maurice
inclusion-exclusion principle 211–212 of Thought (Boole) 48 203–204

48–49, 261, 391, 460 and volume calculation irrational numbers 59, 238, Julia set 203–204
incomposite numbers 409 470, 527 283, 358, 423 Jungius, Joachim 65
increasing/decreasing 89, and definition of real
integral domain 449 number 441–442 K
261–262 “Intégrale, longeur, aire” Kronecker’s doubts about k (unit vector in three-
increment 262 297
indefinite integral 95, 272, (Lebesgue) 307 Theodorus’s work on 498 dimensional space) 520
Intégrales de Lebesgue (Vallée- al-Kashi, Jamshid Mas’ud 18,
275 irreducible polynomial 186
independent. See pairwise Poussin) 518 Isagoge ad locos planos et 291, 392
integral test, for convergent Kasner, Edward 232
disjoint solidos (Fermat) 189, 226 Kempe, Alfred Bray 200
independent axiom 171, 262 series 104 isogonal transformation. See Kendall, M. G. 438
independent events 91, 178, integrand 148–149, 273, 274, Kendall’s coefficient 438
conformal mapping Kendall’s method 438
262–263, 417 360 isolated point (acnode) 283 Kepler, Johannes 93, 291–293,
independent variable 519 integration 15, 133, 272 isometry (congruence
indeterminate. See unknown 292, 396, 397, 472, 528
indeterminate equation 263 by parts 95, 273–274, transformation) 140, Kepler’s laws, Newton and
index (indices) 263. See also 443, 486 283–284
isomorphism 284 353
exponent by substitution (change of isoperimetric inequality 268 The Key to Arithmetic (al-
index of summation 488 variable, substitution rule isoperimetric problems 130,
Indian mathematics 263–265 for integration) 274–275, 284–285, 469, 472, Kashi) 291
indirect proof (proof by 287 474–475, 484–485, 529 Khandakhadyaka
isosceles trapezoid (trapezium)
contradiction, reductio ad intercept 275 285, 504 (Brahmagupta) 51
absurdum) 139, 171, 265, intercept form 164, 275 isosceles triangle 238, 285, Khayyám, Omar. See Omar
336, 420 interest 275–276 285, 506
induction 122, 126, 176–177, interior angle 276, 504, 504 “Ist die Trägheit eines Körpes Khayyám
261, 265–267, 270, 420 intermediate-value theorem von seinem Energieinhalt al-Khwa¯rizm¯ı, Muhammad
inductive definition. See abhängig?” (Einstein) 158
recursive definition (Bolzano’s theorem) 100, Istituzioni analitiche (Agnesi) ibn Mu¯ sa¯ 9, 10–11, 17, 251,
inductive reasoning 120–121 184, 276–277, 367, 442 7 293–294, 517, 534
Indus inch 263 iterated integral 148 kilo- (103) 465
inequality 267–268 bisection method and 46 iteration 150–151, 285–286 kinematics 27, 213
inference 268, 415 Bolzano’s proof of 47 Iverson, Kenneth 198 kite 430, 430
inferential statistics. See and Dedekind cut 119 Klarner, David 406
statistics: inferential and ham-sandwich theorem J Klarner’s constant 406
infinite 196 j (engineers’ i) 85 Klein, Felix Christian 294,
infinite integral. See improper 245 j (unit vector) 520 314, 355, 473–474
integral in history of calculus 58 Jacobi, Carl Gustav Jacob Klein bottle 200–201,
infinite order 406 and logarithmic function 294–295, 339–340
infinite product 144, 287–288 Klein-four group (viergruppe)
268–269, 312, 459 320 Jacobian determinants 287 295
infinite series 102, 324, 351, interpolation 184, 405. See Jacquard, Joseph-Marie 33 knot theory 295, 295–296
368, 439 jerk 523 Koch, Nils Fabian Helge von
infinitesimal 47, 57–58, 67, also Lagrange’s formula Jiuzhang suanshu 203
269, 271, 533 intersection 144, 460 Koch curve 203
infinity (∞) 269–270, 529 interval 278 (anonymous) 73, 226 Kolmogorov, Andrey
inflection (inflexion) point 89, Introduction à l’analyse des Jones, Vaughn 295–296 Nikolaevich 296
89, 270, 492 Jones, William 393 Kramp, Christian 187
information theory 271 lignes courbes algébriques Jones polynomial 296 Kremer, Gerhard (Mercator)
injective function 209 (Cramer) 107–108 Jordan, Marie Ennemond 337
inner product. See dot product “Introduction to Mathematical Kronecker, Leopold 296–297
inradius 260 Philosophy” (Russell) 453 Camille 288, 307 Kruskal’s count 297–298
inscribe/circumscribe 78 An Introduction to Jordan, Wilhelm 288 Kulik, Yakov 463
inscribed-angle theorems 77, Mathematics (Whitehead) Jordan canonical form 288
77–78 530 Jordan curve theorem 76, 288 L
inscribed circle 387 “Introduction to the Doctrine Josephus problem 288–289 Laczovich, Miklov 165
instantaneous value 271 of Fluxions” (Bayes) 38–39 Jourdain’s paradox 289 laddered exponents 65
instantaneous velocity 523 invariant 278–279 Journal de Mathématiques Lafforgue, Laurent 194
integer (directed number, inverse element 279–280 Lagrange, Joseph-Louis 99,
signed number) 271, 358 inverse function (inverse Pures et Appliqués
integer triangle 507–508 mapping, reverse function) (Liouville’s journal) 317 201, 249, 278, 299–300,
integral 66, 180, 380, 447, 280–281 Journal for Pure and Applied 379, 399, 414
523 inverse hyperbolic functions Mathematics (Crelle’s
integral calculus 22, 24, 56, 281 journal) 3 and group theory 242
133, 271–273, 272, 308–309 inverse mapping. See inverse J-shaped distribution 480, and history of calculus 58
function 480 and mean-value theorem
and Archimedes’ method of inverse matrix 222, 280, jug-filling problem 137, 170,
exhaustion 18 281–282 289–290 334
inverse property 5, 49, 223, and square numbers 195
241 and Taylor series 494
inverse square law 282
inverse trigonometric functions
280–281, 282–283
inversion in a circle 225

Index 561

Lagrange’s formula 120, 278, Legendre polynomials 308 linear algebra 314, 491 M
300, 405 Legendre symbol 308 linear-congruence method
Lehmer, D. H. 436 Maclaurin, Colin 107–108,
Lagrange’s theorem 299 Leibniz, Gottfried Wilhelm 436–437 324, 496
Lagrangian (unit) 300 linear equation 34, 156, 163,
Lagrangian description 300 122, 208, 308, 308–309, Maclaurin series. See Taylor
Lagrangian multipliers 300 523–525 314–315 series
Lambert, Johann Heinrich linear interpolation 277–278
and calculus notation 249, linearly dependent/ Madhava of Sangamagramma
283, 300–301, 392, 477 488 264, 324–325
Landa, Diego de 332 independent 315
Langlands, Robert 194 and differential 132 Linear Perspective (Taylor) magic constant 325
Laplace, Pierre-Simon, and discovery of calculus magic division square 327,
494
marquis de 69, 245, 57–58, 211, 227, 240, linear programming 316 327
301–302, 356, 399, 414 272, 352, 494 linear transformation 7, magic multiplication square
Laplace operator 302 and double integrals 148
Laplace transform 302 and formal logic 199, 228 316–317 327, 327
latent vector. See eigenvector Leibniz’s series. See Gregory line integral. See contour magic rectangle 328, 328
lateral surface 115 series magic square 149, 149–150,
Latin square 174, 302–303 Leibniz’s theorem 418 integral
latitude 105, 154, 337 lemma 498–499 Liouville, Joseph 11, 59, 215, 325, 325–328, 326, 327
lattice 366 length 309–310 magnitude 328–329
lattice method 159 lens 76 317–318 al-Mahani 18
lattice point 164, 166 Leonardo of Pisa. See Fibonacci Liouville’s constant (Liouville’s major arc 18
lattice polygon 164–165, 166 Les méthodes nouvelles de la major premise 27
law of averages 303, 304 méchanique céleste number) 11, 317 Malthus, Thomas 407
law of conservation of energy (Poincaré) 399 Listing, Johann 295, 339, 340 Mandelbrot, Benoit 204
309 Let’s Make a Deal! (game Littlewood, John 246 Mandelbrot set 204
law of continuity 309 show) 343 Liu Hui 73 Manière universelle de Mr.
law of cosines 5–6, 16, 78, letterbox principle. See Li Ye (Li Chi, Li Zhi) 73, 318
303, 303–304, 506 pigeonhole principle Lobachevskian geometry. See Desargues (Bosse) 124
law of excluded middle 304 “A Letter on Asymptotic mapping. See function
law of falling bodies 213 Series” (Bayes) 39 hyperbolic geometry Markov, Andrei 415
law of identity 304 Letters to a German Princess Lobachevsky, Nikolai Markov chains 296
law of large numbers 71–72, (Euler) 174 The Mathematical Analysis of
303, 304, 399, 414–415, 436 levers and pulleys 19–20 Ivanovich 46–47, 171, 219,
law of noncontradiction 304 L’Hôpital, Guillaume François, 227, 255, 318, 355, 377, Logic (Boole) 48
law of sines 1–2, 78, 303, marquis de 40, 58, 310 455 “Mathematical Contribution
303–304, 463, 507 L’Hôpital’s rule 40, 310–311 local maximum/minimum
law of the lever 18, 69, 304 Lho shu square 325, 326 330–331, 331, 468 to the Theory of Evolution”
laws of gravity 353 Lhuilier, Simon 474 locus (loci) 319 (Pearson) 387
laws of planetary motion liar’s paradox 289, 311, 374, logarithm 247, 252, 284, Mathematical Dissertations
291–293 458 319–320, 342, 411, 461, 466 (Simpson) 464
laws of thought 26, 304–305 Liber abaci (Fibonacci) 156, mathematical induction. See
lawyer paradox 374–375 191–192 history of 53–54, 55, induction
leading coefficient 305 Liber de ludo aleae (Cardano) 291–293, 345–346 “The Mathematical Theory of
least common denominator 414 Communication” (Shannon
83 Liber quadratorum (Fibonacci) of matrix 157 and Weaver) 461
least common multiple 83, 192 logarithmic differentiation A Mathematician’s Apology
305 life tables (mortality tables) (Hardy) 246
least-integer function. See 311–312, 415 258–259 Mathematische Grundlagen
ceiling function ligancy 472 logarithmic function der Quantenmechanik (von
least squares method 107, likelihood 197 Neumann) 351
218, 302, 305–307, 308, 445 Lilavati (Bha¯skara) 41–42 152–153, 153, 320–321, 337 matrix (matrices) 109, 119,
Lebesgue, Henri-Léon 58, 307 limit 312–313 logarithmic graph 235 126–128, 221–222, 314,
LeBlanc, Louis 227 d’Alembert and 8–9 logarithmic scale 118, 321, 329
Leçons sur les séries and analytic number theory matrix addition 329–330
trigonométriques (Lebesgue) 13 369, 466 matrix algebra 68, 458
307 Bolzano’s work on 47 logarithmic spiral 39, 474 matrix inverse. See inverse
Lectiones geometricae and chain rule 70–71 Logarithmorum chilias prima matrix
(Barrow) 36 ε - δ definition of 529 matrix multiplication 241,
Lectures on Quaternions and definition of tangent (Briggs) 54 330
(Hamilton) 520 492 logically equivalent 515 matrix operations 329–330
left derivative 307–308 derivative as 132 logic gates 88 Matyasevic, Yuri 137
left-handed system 369, 448 devised by Cauchy 66 Logic of Chance (Venn) 524 maximin strategy 217
left identity 257 in history of calculus 58 long division 37, 37–38, 38, maximum/minimum 89, 133,
left/right, limit from the 313 Lindemann, Carl Louis 310, 330–332, 331
Legendre, Adrien-Marie 187, Ferdinand von 11, 297, 146, 349, 404, 433 Mayan mathematics 332–333
308 313–314, 393, 477 longitude 105, 154–155, 337, McManus, Chris 118
line 314, 315 mean (average) 44, 333–334,
445 480, 537
long radius 321 mean. See expected value
Loomis, Elisha Scott 425 mean value 334
loop (graph theory) 233 mean-value theorem 22,
Lorentz, Hendrik 398 94–95, 119, 184, 211, 262,
Lovelace, Augusta Ada 33, 311, 334–335, 442, 450

321–322, 322
lowest terms. See reduced form
Loyd, Sam 465–466
lozenge 376, 474
Lucas, Edouard 502
lune 251–252, 322, 322–323
Lyapunov, Aleksandr

Mikhailovich 69, 356, 415

562 Index

Measurement of a Circle minimum. See Mysterium cosmographicum Newton number 472
(Archimedes) 18 maximum/minimum (Kepler) 292 Newton quotient 132, 360
Newton’s method 46, 249,
measures of central tendency minor arc 18 N
480 minor premise 27 Nakayama, Kazuhiko 393 286, 353–354, 451
minuend 487 nano- (10-9) 465 Newton’s third law of motion
measures of dispersion minus sign 131, 398, 486, Napier, John 74, 293,
480–482 9
487 319–320, 345, 345–346, 466 A New Treatise of Fluxions
measure theory 58, 307 minute (measure of angle) 14 Napier’s bones (rods) 74, 346,
Mécanique analytique Mirifici logarithmorum (Simpson) 464
346, 347 Neyman, Jerzy 354
(Lagrange) 299 canonis descriptio (Napier) Napier’s formulae 346 n-fold rotational symmetry
Mécanique céleste (Laplace) 346 Napier’s inequality 268
Miscellanea analytica (De Napoléon’s theorem 507 490
67, 245 Moivre) 122 nappe 91–93, 473 n-gon 97, 402
Mechanica (Euler) 173, 393 mixed strategy (game theory) Nash, John 346–348, 415 Nicomachos of Gerasa 389
mechanics 8 217–218 Nash equilibrium 413 nine-point circle 354
median 174–175, 480 mixed surd 490 Natural and Political node 148, 233, 493
median of triangle 335, Miyoshi, Kazunori 393 Noether, Amalie (Emmy) 10,
Möbius, August Ferdinand Observations Made upon the
335–336, 338, 507 338–339, 340 Bills of Mortality (Graunt) 354–355
Meditationes algebraicae Möbius band (strip) 294–295, 312 n-omino 406
339–340, 340, 369 natural number 106, 129, noncontradiction, law of 304
(Waring) 229 Möbius function 339 348, 358, 385 “Non-Cooperative Games”
mega- (106) 465 Möbius inversion formula
Melancholia (Dürer) 149, 325 339 See also whole number (Nash) 347
“Mémoire sur les équations mode 480 naught. See zero nondegenerate quadrics 411
modular arithmetic 59, Nave, Hannibal 191 nondeterministic polynomial
algébriques” (Abel) 5 71–72, 92, 147, 174, 218, n-dimensional kissing number
“Mémoire sur quelques 340–342, 436 time 357
modulus. See absolute value 472 nonempty set 162
propriétés remarquables des monic 305 n-dimensional space 68, 472 non-Euclidean geometry 159,
quantités transcendantes monohedral tessellation nearest-neighbor algorithm
circulaires et logarithmiques” 496–497 171, 227, 255, 314,
(Lambert) 300 monomial 342 505 355–356, 377, 473–474
“A Memoir on the Geometric monomino 406 negation (‘not’ statement)
Representation of Imaginary Monte Carlo method discovery of 46, 318, 455
Numbers” (Français) 25 342–343 348, 514 vs. Euclidean geometry
Memoir on the Theory of Monty Hall problem 343 negative coordinates 63
Matrices (Cayley) 68 Mordell conjecture 190 negative infinity 121 294
Menabrea, Luigi 322 Morgan, Frank 469 negatively oriented 369, 448 nonmeasurable sets 24
Menelaus 17, 336 Morgenstern, Oskar 216, negatively skewed 465, 480, nonnegative 408
Menelaus’s theorem 70, 336, 350–351 nonpositive 408
336 Morley, Frank 343 480 nonrepeating decimals 441
Mercator (cartographer). See Morley’s theorem 343 negative numbers 48, 51, nontrivial solution 514
Kremer, Gerhard Morrison, Nathan 296 nonzero-sum game theory 347
Mercator, Nicolaus mortality tables. See life tables 348–349, 358, 408 normal distribution 69–70,
(mathematician) 337 multi-choosing 81–82 negative slope 467
Mercator’s expansion 4, 337 multiplicand 344 neo-Pythagorean mean 333 72, 122, 143, 356, 356–357,
Mercator’s projection 91, 337 multiplication 5, 86, 86, 146, nested multiplication 349, 414, 482, 483, 486
Méré, Chevalier de 414 205, 343–344, 370 normal to a curve 357
meridian 154–155 multiplication law 367 349–350, 350, 405 normal to a plane 357
Mersenne, Marin 189, multiplication principle net (accounting) 350 normal to a surface 357
337–338, 389 (fundamental principle of net (geometry) 350, 350 “Note on the Application of
Mersenne prime 338, 389, counting) 344 net (of hypercubes) 256 Machinery to the
410 net weight 350 Computation of
Metaphysics (Aristotle) 498 and permutations 391 network. See graph Astronomical and
meter 465 multiplication rule Neumann, John von 350–351, Mathematical Tables”
method. See algorithm (Babbage) 32–33
method of exhaustion 18, 20, for exponents 180–181 393 “Note sur une équation aux
57, 67, 172 in probability theory 416, neutral element. See identity differences finie” (Catalan)
method of indivisibles 67 64
Methodus incrementorum 416–417 element Notions sur la machine
directa et inversa (Taylor) multiplicative identity 257 New Experiments concerning analytique de Charles
494 multiplicative inverse 449 Babbage (Menabrea) 322
Metrica (Heron) 248, 249 multiplier 344 Vacuums (Pascal) 382 Nova methodus pro maximis
micro- (10-6) 465 multivariate analysis 197 Newton, Sir Isaac 93, 122, et minimis (Leibniz) 309
middle-square method 436 mutually exclusive. See Nova stereometria doliorum
midpoint 76, 338 125, 132, 169, 180, 282, (Kepler) 292–293
midrange 480 pairwise disjoint 292, 324, 351–353, 352, Novum Organum (Bacon) 452
Mihailescu, Preda 64 mutually exclusive events 452, 523, 528 NP complete 88, 357–358
mile 309–310 n-polyomino 406
milli- (10-3) 465 (disjoint events) 344 and binomial theorem 45 n-space. See Euclidean space
minimax strategy 217 mutually orthogonal 302 and discovery of calculus nth dihedral group 242
minimax theorem 218 nth-order general linear group
57–58, 211, 227, 240, (GLn) 222–223
272, 308–309, 352, 494 nth-order root 451
and fluxions 198
and polar coordinates 63,
400
and three-dimensional
kissing number 472

Index 563

nth-order symmetric group “On the Problem of the Most osculation (tacnode) 148 of helixes 95
490 Efficient Tests of Statistical osculinflection 492, 493 of Möbius band 340
Hypotheses” (Neyman) 354 Oughtred, William 344, of torus 500
nth root 112 of unit circle 517
nth root of unity 358 “On the Propagation of Heat 369–370, 370, 466 parentheses 50–51, 519
nth-term test 103 in Solid Bodies” (Fourier) outlier 370 parity 378–379
n-tuple 358 201 oval 370 Parmenides 532
null angle 14 ovals of Cassini 370 Parmenides (Plato) 532
null set. See empty set On the Sphere and Cylinder Ozanam, Jacques 449 partial derivative 138, 233,
number 358–359 (Archimedes) 18, 173 239, 259, 379
number-base machine 30 P partial differential equations,
number line 359, 441, 442 “On the Theory of Groups packing spheres 472 d’Alembert and 8–9
number-naming puzzle 54–55 Depending on the Symbolic paddle wheel 174 partial fractions 162,
number systems 358–359 Equation θn = 1” (Cayley) Paganini, Nicolò 13 379–380
number theory 82, 139–140, 68 pairwise disjoint (independent, partially ordered set 365–366
partial sum 102, 380, 459
159, 167, 191, 337–338, “onto” function 208 mutually exclusive) 371 and limit 312
359–360, 429–430 open half-plane 243–244 Pappus 13, 57, 239, 253, partition 358, 380–381, 436
numerable. See denumerable open half-space 243–244 partition function 381
numerator 204 open interval 278 371–372, 470 Pascal, Blaise 74, 189,
numerical differentation 360 operation 364 Pappus’s problem 372 381–382, 382, 383
numerical integration 360–361 operational precedence. See Pappus’s theorems 372–373, on cycloid 310
and history of calculus 57
O order of operation 470, 471 and Pappus’s theorem
obelus (%) 146 operations research (OR) 33, and torus 500
oblate spheroid 154, 362 372–373
oblique 362 364 parabola 92–93, 93, 137, and probability theory
oblique angle 362 opposite 364 155, 189, 239, 373, 373,
oblique cone 362 opposite angles 364 429 413, 414, 417
oblique cylinder 115 oppositely congruent solids 92 as trajectory 213, 503 Pascal’s distribution 44
oblique pyramid 423 Optica promota (J. Gregory) Pascal’s triangle 45, 65, 195,
obtuse angle 14, 362 parabolic cylinder 411
obtuse triangle 362, 506 240 paraboloid 101, 235, 196, 382–385, 384, 497
obverse 362 Optics (Euclid) 169 and combination 80–81
octahedron 396, 397 Optics (Ptolemy) 422 373–374
octant 363 optics, Newton and 353 paradox 374–375, 412, 532 pathological functions 529
octonions 359, 432 Optiks (Newton) 353 Pathwaie to Knowledge
odd functions 177, 490 optimization 133, 161, 285, and pure mathematics
odd numbers 177–178 422–423 (Recorde) 443
odds (probability) 363, 439 316, 364, 364–365 payoff matrix 217
Oeuvres completes d’Augustin and history of calculus 57 “Paradoxien des Unendlichen” Peano, Giuseppe 265, 385
and inscribed triangles 387 (Bolzano) 47 Peano’s curve 385, 385–386
Cauchy 67 and Snell’s law of Peano’s postulates 31, 265,
Oeuvres de Camille Jordan refraction 468 parallel 375–376, 376, 377
Steiner and 484–485 parallelepiped (parallelopiped) 271, 385, 386
288 von Neumann and 350 Pearson, E. S. 354
officer problem 302–303 376 Pearson, Karl 106, 107,
Omar Khayyám (Umar al- OR. See operations research and scalar triple product of
orbit (sequence of iterates) three vectors 512 215–216, 386, 386–387,
Kha¯yyam¯ı) 18, 227, 415
363–364 150 parallelogram 376, 430, 474 peasant multiplication. See
“On a Curious Property of order. See permutation area of 23, 23 Russian multiplication
Vulgar Fractions” (Farey) order, of polyomino 406 cross (vector) product as pedal circle, of triangle 387
188 ordered partition 380–381 110 pedal triangle 387
“On a New Method in ordered set 365–366, 367, Pell, John 54
Elementary Number Theory” parallelogram law Pell’s equation 54
(Erdös) 167 530 (parallelogram rule) 376, pencil-turning trick 505–506,
On Burning Mirrors (Diocles) order of a matrix (dimension 376–377 506
136 pendulums 114–115, 213
On Conoids and Spheroids of a matrix) 258, 366 parallelotope 376 Penny Cyclopedia 122
(Archimedes) 18 order of group 366 parallel postulate (Euclid’s fifth pentagon 285
On Divisions (Euclid) 169 order of magnitude 329 pentagram (pentacle,
“one-to-one” function 209 order of operation (opera- postulate) 159, 227, 239, pentalpha, pentangle) 230,
On Floating Bodies 255, 375, 377, 377–378, 387–388, 388, 424
(Archimedes) 18–19 tional precedence) 366–367 378, 473 percentage 388
On Reasoning in a Dice Game order properties 267, 367 percentage error 388–389
(Huygens) 180 ordinal numbers 367–368 and altitudes of triangle percentile (centile, quartile)
On Spirals (Archimedes) 474 ordinate 62 12–13 389
On the Economy of Oresme, Nicole 63, 208, 234, perfect number 18, 174, 338,
Machinery and Manufactures and angles of triangle 505 389, 423–424
(Babbage) 33 368 converse of exterior-angle perigon (round angle) 14,
orientation 369 362, 390
origami 513 theorem 183 perimeter 390, 403, 443
origin 105, 359 and Euclidean vs. non- period doubling 151
orthant 363
orthic triangle 387 Euclidean geometry
orthocenter 12, 354, 387, 507 170–171, 318, 355–356
orthogonal 369, 391 Omar Kayyám and 18,
orthogonal. See Cartesian 364
parallel projection 418
coordinates parametric equations (freedom
Osborne’s rule 369 equations) 164, 378, 386
and arc length of curve 22
of cardioid 62
of circle 75, 95

564 Index

periodic function 390 Poisson, Siméon-Denis 44, 71, prime numbers 186, 389 progression. See harmonic
peripatetics 27 399–400, 414–415 and Bertrand’s conjecture sequence
peripheral angle 77, 77 71, 167
permillage 388 Poisson distribution 44, vs. composite numbers 87 projected vector 418–419
permutation 82, 127, 186, 399–400 and factorization 186–187 projection 337, 391–392,
and fundamental theorem
209, 390–391, 490 polar-coordinate graph 235 of arithmetic 83, 418–419
permutation group 68 polar coordinates 22, 39, 63, 186–187 projective geometry 123–124,
perpendicular 369, 391, 468 infinitude of 171–172
perpendicular bisector 46, 400, 400–401, 504 Mersenne’s work on 338 270, 339, 372, 382, 409,
of cardioid 62 418, 419
174–175, 338, 391, 507 of complex numbers 86, prime-number theorem 71, prolate spheroid 362
perspective 149–150, 86 219, 410, 410–411, 447, 535 proof 420
of logarithmic spiral 474 proofs of 167, 243, 518 proof by contradiction. See
391–392, 419 of rose 451 indirect proof
peta- (1015) 465 principal axes 411 proofs, defective
Phaedo (Plato) 396 polar form 176 Principia (Newton) 293,
phyllotaxis 193 Polignac, Alphonse de 107 that 0 = 1 29
Physics (Aristotle) 532 Pólya, George 401–402 351–353, 352 that 1 = 2 4, 248, 337
physics, Aristotelian 27 polychoron 403 Principia Mathematica that 1 is the largest integer
pi (π) 55, 392–394 polygon 119, 149, 402–403,
(Russell and Whitehead) 284
approximations of 20, 403 453, 530 that all horses are the same
28–29, 291, 537 area of 23, 24 Principia philosophiae
circumcircle of 78 (Descartes) 125 color 266–267
irrational 283, 300, 477 convexity of 88 The Principle of Relativity that the moon is made of
not constructible 97 exterior angles of 182 (Whitehead) 530
transcendental 11, 301, and tessellation 496–497 The Principles of Empirical cheese 90
Logic (Venn) 524 proper factor 186
313–314, 393, 477 polygon division problem 64 Principles of Mathematics proper fraction 205
Piazzi, Giuseppe 219 polyhedral angle 469 (Russell) 453 properly divergent 144
pico- (10-12) 465 polyhedron 88, 128–129, Principles of the Art of proper vector. See eigenvector
pie chart (graph) 479, 479 Weighing (Stevin) 485 proportion 149, 230–232, 370
pigeonhole principle 394–395 176–177, 185, 236, Prior and Posterior Analytics proportional 420–421, 439
Pitiscus, Bartholomaeus 511 403–404 (Aristotle) 27 proposition 421
pivoting. See Gaussian polynomial 404–405 prism 403, 411–412 Protagoras 374
polynomial equations 163, prismatoid 412 p-series test 104
elimination 215 prismoid 412 pseudoprime numbers 410
place-value system 6, 51, 155, polynomial time 405 prisoner’s dilemma 218, 412, pseudosphere 300
polyomino 405–406, 406 412–413 Ptolemy 239, 253, 392, 421,
192, 251, 359, 487 polytope 403 probability 122, 143, 189,
planar curve 114 Poncelet, Jean Victor 419 243, 262–263, 413–418 421–422, 510
planar graph 233 population and sample Ptolemy’s theorem 52, 422,
Planck, Max 158 406–407 and harmonic functions
plane 80, 315, 395 population mean 483 246 422
plane geometry 170 population model 182, public-key cryptography 111
Plane loci (Apollonius) 189 407–408 Kolmogorov’s work on pure mathematics 159, 168,
Plato 166, 238, 395–396, position vector 395, 408 296
positive 408 422–423
397, 498, 532 positively oriented 369, 448 Monte Carlo method pure surd 490
Platonic solids 159, 232, 238, positively skewed 465, 480, 342–343 pyramid 403, 423
480 pyramidal frustum 207
291–292, 395, 396, positive slope 467 Monty Hall problem 343 Pythagoras 9, 57, 172, 230,
396–398, 404, 424 postage-stamp problem 446 and mutually exclusive
Platonicus (Eratosthenes) 166 postulate (axiom) 31, 170, 238, 387, 396, 423,
Playfair, John 171, 227, 355, 409 events 344 423–424, 425
376, 377 potential 239 odds and 363 Pythagoras’s theorem 159,
Playfair’s axiom 46, 171, 227, power series 121, 324, 399, Pascal and 382 165–166, 208, 226, 238,
355, 376, 377 409, 495, 530 and permutations 391 256, 314, 424, 424–426,
plot 398 P problems vs. NP problems Poisson and 399–400 425, 506
Plouffe, Simon 183 357–358 Quételet and 432–433
plus 398 Practica geometriae and random walks 437 in Babylonian mathematics
plus/minus symbol 398 (Fibonacci) 511 and set theory 178 34–35
plus sign 398, 486 precision 66–67, 409, and two-card puzzle 515
Pneumatica (Heron) 248–249 529–530 probability density function in Chinese mathematics 73
Poincaré, Jules Henri 255, “Preliminary and Elementary 143 converse of 506, 506–507
314, 398–399, 500, 530 Essay on Algebra as the probability models 416 and distance formula 141
Poincaré disk 255 Science of Pure Time” probability tree 415–416, 416 and law of cosines 303
Poincaré’s conjecture 399 (Hamilton) 245 procedure. See algorithm and polar coordinates 400
point 399 premise 409 Proclus 171, 239, 497 and Pythagorean triples
point of contact (tangency primary data 116 product of matrix 330
point) 399 prime 409–410, 461 product rule 133, 273, 309, 426
point of inflection. See prime factorization 187 418, 433–434 tessellation and 474, 475,
inflection point product set. See Cartesian
point-slope form, of equation product 497
of line 164 and vectors 520
The Pythagorean Proposition
(Loomis) 425
Pythagorean triples 99, 137,
162–163, 190, 226, 360,
422, 426–427

Index 565

Q rational numbers (Q) 106, regular polyhedron. See Ritoré, Manuel 469
QED 159, 428 129, 358, 440–441, 443 Platonic solid Rivest, Ron 111
QEF/QEI 428 vs. algebraic numbers 11 Roberval, Gilles Personne de
QI. See Quételet index definable as ratio 439 regular tessellation 496
Qin Jiushao. See Ch’in Chiu- and definition of real “Rein Analytischer Beweis” 57
number 441–442 Rolle, Michel 449
shao denumerability of 123 (Bolzano) 47 Rolle’s theorem 184, 335,
quadrangle. See quadrilateral discreteness of 140 Reinhardt, Karl 497
quadrant 363, 428 Relación de las cosas de 449–450
quadratic 428–430 ratio test 103, 182, 439 Roman numerals 155, 192,
quadratic curve 430 ratio theorem 439 Yucatán (Landa) 332
quadratic equations 163, 373, raw data 116 relation (relationship) 445 251, 450–451
ray (half-line) 441 relative complement 131 root (zero) 451, 534
443 “Real Algebraic Manifolds” relative error 445 root test 103–104
quadratic form 430 relative frequency 206 Ros, Antonio 469
quadratic formula 84, 200, (Nash) 347 relatively prime 139, 172, rose 451
real line. See number line rotation 206, 225, 278,
451 real numbers (R) 106, 174, 446
quadrature of the circle. See relative maximum/minimum 283–284, 317, 490
129–130, 358, 441–442 round angle. See perigon
squaring the circle and Dedekind cut 330–331, 331 rounding 451–452
Quadrature of the Parabola 119–120 remainder 433 round-off (rounding) error
and Euclid 172 remainder theorem 187, 446
(Archimedes) 18 nondenumerability of 123 removable discontinuity 440 452
quadrilateral 88, 376, 430, repeating decimals 440–441, round-robin tournament. See
receiver 501–502
430 “Recherches sur diverses 526 tournament
quadruple (4-tuple) 358 retrograde motion 173 Royal Society of London 452
quantifier 199, 430–431 applications de l’analyse Reuleaux triangle 95 RSA encryption method 111
quantum (quanta) 158 infinitésimale à la théorie des reverse function. See inverse rubber-band problem
quantum mechanics 159 nombres” (Dirichlet) 140
quartic equation 59–60, 84, reciprocal 442 function 247–248
reciprocal equation 442 reverse J-shaped distribution rubber-sheet geometry. See
190–191, 431–432 Recorde, Robert 162,
quartile. See percentile 442–443 480, 480 topology
quaternions 241, 244–245, rectangle 185, 376, 443, 474 Rhaeticus, Georg Joachim Rudolff, Christoff 475
Ruffini, Paolo 10, 72, 431
359, 432 area of 23, 23 511 rule, 68-95-99.7 356
Quételet, Lambert Adolphe vs. trapezoid 504 Rhind, Alexander Henry 446 Russell, Bertrand Arthur
rectangular (Cartesian) Rhind papyrus 9, 155–156,
Jacques 415, 432–433 coordinates 62–63, 105 William 199, 229, 452,
Quételet index (QI) 433 recurrence relation (recursive 392, 446–447, 476, 510 452–453, 460–461, 530, 533
quintessence 397 relation, reduction formula) rhomboid 376 Russell’s paradox (antinomy)
quintic equation 3 443, 458 rhombus (rhomb) 376, 474 200, 374, 453–454, 460–461
quipu 260 recursive definition (inductive Richter, Charles 321 Russian multiplication
quotient 85, 146, 204, 433 definition, recursion) 444 Richter scale 321 (peasant multiplication) 43,
quotient rule 133, 433–434 reduced form (lowest terms) Riemann, Georg Friedrich 58, 156, 454, 454
444
R reductio ad absurdum. See 227, 270, 272, 355, S
radian 14, 115, 465, 477 indirect proof 447–448, 473–474, 485, 535 Saccheri, Girolamo 171, 227,
radian measure 473, 509 reduction formula. See Riemann hypothesis 243, 246,
radical sign 475 recurrence relation 250, 447 455
radius 76 re-entrant angle 276 Riemannian geometry. See saddle point 217, 492
radius-chord theorem 78, 78 reflection 198, 206, 283–284, spherical geometry salient 276
radius of convergence 495 490 Riemann integral 58, 447 same side interior/exterior
radix 36–38 in a circle 225 Riemann sphere 485
Rahn, Johann Heinrich 146 and isometry 212 Riemann’s zeta function. See angle 504
Ramanujan, Srinivasa 246, in a line 224–225 zeta function sample 143
in a point 225 right angle 14, 226, 448 sample mean 333
381, 435–436, 477 reflection property right circular cone 473 sample space 178, 413, 417,
randomness 41 of ellipse 365 right cylinder 115
random numbers 436–437 of parabola 373, 373 right derivative 307–308 455
random sampling 407 reflex angle 14, 362 right-handed system 369, 448 The Sand Reckoning
random walk 246, 437 “Réflexions sur la cause right-hand rule 110
range. See image générale des vents” right identity 257 (Archimedes) 19
rank 437–438 (d’Alembert) 8 right/left, limit from the 313 sandwich result. See squeeze
rank correlation 438–439 reflexivity 365 right pyramid 423
ranunculoid 115 Regiomontanus 444–445, 511 right quandrangular prism rule
ratio 146, 420, 439 regression 107, 445 411 SAS (side-angle-side) rule 1–2,
rational function (expression) regression line 306 right spherical triangle 474
regular polygon 164, 166, right triangle 506 463, 506
439–440 402 right-triangle principle. See Scalar 456
rationalizing the denominator Pythagoras’s theorem scalar multiplication 329, 520
ring 5, 144, 252, 271, scalar product. See dot
85, 440, 442, 490 354–355, 448–449
Ringel, Gerhard 200 product
rise over run (slope) 467 scalar triple product 512
scale 36–38, 456
scale factor 463
scalene 506
scatter 481

566 Index

scatter diagram (scatter plot, sigma notation 459, 488 Spearman, Charles 438 Steiner, Jakob 284, 484
Galton graph) 106, 456–457 sigma squared 481 Spearman’s coefficient Steiner point 484–485
signed number. See integer Steiner surface 484
Schickard, Wilhelm 382 significant figures 168, 409 438–439 stellated polyhedron 404
Schnirelmann, L. 230 similar figures 456, 463 Spearman’s method 438 stem-and-leaf plot 479, 479
schubfachprinzip. See similar triangles 1–2 special theory of relativity 158 steradian 15, 465, 469
simple fraction 205 specific gravity 18 stereographic projection 485
pigeonhole principle simple interest 275 speed vs. velocity 519 Stevin, Simon 485–486
Schwarz, Hermann 469 simple root 451 Sphaerica (Menelaus) 336 Stifel, Michael 180, 348–349
scientific notation 125, 168, simplification, by cancellation sphere 76, 136, 166, 471–472 Stirling’s formula 122, 486
straight angle 14, 362
457 56 connectedness of 94 straightedge and compass 95
screw propeller 174 Simpson, Thomas 463–464 diameter of 130 stratified sampling 407
secant 457, 493, 510 Simpson’s rule 361, 464 four-color map on strictly increasing/decreasing
secant theorem 1, 457, 457 simultaneous linear equations
second (measure of angle) 14 200–201 261–262
secondary data 116 219–220, 234, 282, 315, great circle as “straight” Student’s t-distribution 484
second-derivative test 332 464, 486, 491 Student’s t-test 415, 484
segment bisector 46 314 subfactorial 486
Segner, J. A. 64 and determinant of matrix volume of, vs. cylinder 20, substitution 486–487, 491
selection. See combination 126–128 substitution rule for integra-
self-reference 244, 457–458 91, 115
semi-magic square 327, 458 solved by Cramer’s rule sphere packing 292, 472 tion. See integration: by
semimajor axis 161 108–109, 128 spherical coordinates substitution
semiminor axis 161 subtraction 146, 205, 487
semiregular polyhedron 404 sine function 291, 509, 472–473, 504 subtrahend 487
semiregular tessellation 496 510–511, 511 spherical excess 474 successive doubling 156
sequence (progression) history of 28–29, 51, spherical geometry 225–226, Sulbasutram 425
444–445 summation (Σ) 173, 487–488
458–459 227, 270, 323, 355–356, summation, index of 148
sequence (totally ordered set, sine rule. See law of sines 472, 473–474 summation problem 65
single cusp 492, 493 spherical helix 248 sums of powers 51, 488,
chain) 365–366 singularity 464 spherical segment 472 488–490
series 459–460, 488 singular point 464 spherical triangle 472, 474 supplementary angles 14
set complement 460 sink 501–502 spheroid 161 surd 490
set direct product. See Sirotta, Milton 232 spiral of Archimedes 372, surjective function 208
SI units 464–465 474, 476 “Sur la décomposition des
Cartesian product 68-95-99.7 rule 356 spirals in nature 193 ensembles de points en
set intersection 460 Siyuan Yujian (Chu Shih- splitting game 279, 279 partiens respectivement
set operations 524 square 430, 474–475 congruent” (Banach, Tarski)
set theory 144, 422–423, Chieh) 383 square brackets 50–51 35
skew curve 114 square matrix 329 Sur l’homme et le
460–461 skew lines 465 square model 416, 416–417 developpement de ses
and axiom of choice 31 skewness 465 square numbers 194, 194–195 facultés, essai d’une physique
as Boolean algebra 49 Skolem, Thoralf 534 square pyramid 423 sociale (Quételet) 432–433
and De Morgan’s laws 123 slide 15 puzzle (Boss puzzle, square root 157, 249, 451, “Sur l’intégration des
difference in 131 475–476, 486 functions discontinues”
and probability 178 fifteen puzzle) 391, squaring the circle (circle (Lebesgue) 307
and Russell’s paradox 465–466, 466 squaring, quadrature of the Su-yuan yu-chien (Chu Shih-
453–454 slide rule 369–370, 466 circle) 97, 238–239, Chieh) 73–74, 75
von Neumann and 350 slope (grade, gradient) 132, 251–252, 313–314, 393, syllogism 27
Zermelo’s attempt to 466–468, 467 476–477 symbolic logic. See formal
axiomatize 534 slope-intercept form 164 squeeze rule (sandwich result) logic
smooth vs. non-smooth 16, 321, 477, 477–478, 526 Symbolic Logic (Venn) 524
set-theory paradoxes 530 surface 357 SSS (side-side-side) rule 1–2, symmetric difference 131
set union 460 Snell, Willebrord van Roijen 463 symmetry 206–207, 490
seven bridges of Königsberg 468–469 Stäckel, Paul 515 Synagoge (Pappus) 239,
Snell’s law of refraction 365, stadium paradox 533 371–372
problem 235, 235, 500 468, 468–469 stair climbing problem 65 Synopsis of Elementary
sexagesimal numbers 33–34 soap bubbles 469, 472 standard deviation 44, 386, Results in Pure Mathematics
shadows 149 Socrates 374 481–482 (Carr) 435
Shamir, Adi 111 solar system, mathematical Synopsis palmariorum
Shanks, William 392, 461 stability of 296 vs. mean and z-score 537 matheseos (Jones) 393
Shannon, Claude Elwood solid angle 15, 469 standard form. See scientific Syntaxis mathematic (Ptolemy)
solid geometry 170 421–422
271, 461–462 solid of revolution 57, 372, notation systematic sampling 407
short radius. See apothem 382, 469–470, 500 Statistical Methods for system of equations 121, 491
Shushu jiuzhang (Ch’in) 72 solution by radicals 3, 215, “Systems of Right Lines in a
Siamese method 327 470–471 Research Workers (Fisher) Plane” (Hamilton) 245
Sicherman, Col. George 462 Soma cube 218 197, 415
Sicherman dice 462, 462 “Some Properties of statistics 414, 437, 478
Sidereus nuncius (Galileo) 213 Bernoulli’s Numbers”
Sierpinski, Vaclav 203 (Ramanujan) 435 descriptive 116, 478–482
Sierpinski’s triangle 203–204, source 501–502 inferential 143, 268, 354,
space-filling curve 385
204 478, 482–484
sieve of Eratosthenes 410, Bayes and 38–39
and central-limit
462–463 theorem 69–70

Index 567

T theorem of Thales 77 Traité de mécanique céleste Le triparty en la science des
Theorems Stated by (Laplace) 301 nombres (Chuquet) 526
Tabulae (Regiomontanus) 444
tacnode (osculation) 148 Ramanujan (Watson) 436 Traité des proprietés des triple (3-tuple) 358
tacpoint 493 Theoria motus corporum figures (Poncelet) 419 triple integral 148
Tait, Peter 295 triple torus 501
tangency point. See point of coelestium (Gauss) 219 Traité des substitutions et des triple vector product 376,
Théorie analytique de la equations algebraique
contact (Jordan) 288 448, 512
tangent (geometric) 492, 493 chaleur (Fourier) 201 trirectangular triangle 474
tangent (trigonometric) 132, Théorie analytique des trajectory 503 trisecting an angle 97, 239,
transcendental curve 114
492, 510 probabilités (Laplace) 301, transcendental numbers 11, 291, 372, 512, 512–513, 525
tangent plane 492 415 Tristram Shandy paradox
tangent theorems 76–77, 77 Theorie der Parallellinien 59, 297, 301, 317–318, 477
Tarski, Alfred 24, 35 (Lambert) 300 transfinite numbers 368 269, 513
Tartaglia, Niccolò 10, 60, “Théorie des opérations transformation 503–504 trivial 183, 514
linéaires” (Banach) 35 transitive law 367 trivial ring 449
112, 190–191, 493, 493–494 The Theory of Games and transitive reasoning 26 trivial solution 513–514
task diagram 109 Economic Behavior (von translation 206, 225, tromino 406
tautochrone property of Neumann and Morgenstern) truncation 452
216, 350–351 283–284 truth table 25, 514, 514–515,
pendulums 114–115, 299 The Theory of Numbers transmitter 501–502
tautology 494, 515 (Hardy and Wright) 246 transpose of matrix 330 515
Taylor, Brook 240, 274, 324, “Theory of Systems of Rays” transposition 390 for biconditionals 42
(Hamilton) 245 transversal 336, 375, 377, for conjunction 94
325, 494, 495 three-body problem 299 for contrapositives
Taylor, Richard 190, 531 three-dimensional coordinates 504, 504 101–102
Taylor series 152, 182, 63, 363 trapezoid (trapezium) 23, 24, for disjunction 141
three-dimensional vectors 520 for hypothetical 90, 90
240–241, 268, 271, 315, three-fold rotational symmetry 430, 504 for negation 348
324–325, 337, 409, 494–496 490 trapezoidal rule for integration
three-utilities problem 233, Tsu Chung Chi. See Zu
and approximation 16 499, 499 360–361 Chongzhi
convergence of 45 tiling. See tessellation traveling-salesman problem
Timaeus (Plato) 396, 397 Turing, Alan 244
Chebyshev’s work on time, as fourth dimension 55, 505 turning point 515
71 136, 256 traverse 504, 504 twin primes 410, 515
time-series graph 479, 479 Treatise of Fluxions twisted curve 114
and Euler’s formula 175 topological space 500 two-card puzzle 91, 515
and general binomial topology 121, 307, 339, (Maclaurin) 324 two-chord theorem 78, 78
499–500 A Treatise on Algebra two-dimensional vectors 520
theorem 45 Torricelli, Evangelista 114 two-pancake theorem 245,
and history of calculus 57 torus 94, 399, 470, 500–501, (Maclaurin) 324
and permutations 391 501 Treatise on Algebra (Wallis) 277
of sine function 536 two-point form 164
Tchebyshev. See Chebyshev Euler’s formula for 236 529 two-secant theorem 457, 457
technique. See algorithm Euler theorem and 177 Treatise on Demonstration of type, in set theory 454
tensor 447, 496 and Jordan curve theorem
The Tenth (Stevin) 485 Problems of Algebra (Omar U
tera- (1012) 465 76, 288 Khayyám) 363 “Über die Addition transfiniter
tessellation (covering, tiling) and three-utilities problem Treatise on Human
403, 430, 475, 496–497, Proportions (Dürer) 231 Cardinalzahlen” (Zermelo)
497, 507 499 The Treatise on the Chord and 533–534
tesseract 256 totally ordered set 365–366 Sine (al-Kashi) 291 “Über die Hypothesen welche
“The Testing of Statistical tournament (round-robin Treatise on the Equilibrium of der Geometrie zu Grunde
Hypotheses in Relation to Liquids (Pascal) 382 liegen” (Riemann) 227
Probabilities A Priori” tournament, complete Treatise on Universal Algebra “Über die Stabilität des
(Neyman) 354 digraph) 501, 501–502 (Whitehead) 530 Gleichgewichts” (Dirichlet)
tetragon. See quadrilateral towers of Hanoi (Brahma) tree (graph theory) 233 140
tetrahedral dice 462 443, 502–503 triangle (trigon) 114, 505–508 Über die Zahl (Lindemann)
tetrahedral numbers 195, 497 tractrix (equitangential curve, 314
tetrahedron 165, 396, 397, tractory) 503 area of 23, 23, 249 “Über einen die Erzeugung
403, 423, 497 Traité analytique des sections diameter of 130 und Verwandlung des Lichtes
tetromino 406 coniques (L’Hôpital) 310 Euler line in 174–175 betreffenden heuristischen
Teutsche algebra (Rahn) 146 Traité d’algèbre (Rolle) 449 exterior angles of 182–183 Gesichtspunkt” (Einstein)
Thabit ibn Qurra 18 Traité de dynamique triangle inequality 267, 425, 158
Thales 238, 423, 497–498, (d’Alembert) 8 506, 508 Ulam, Stanislaw 15
510 Traité de la résolution des triangular numbers 194, unary operation 364, 516
Theaetetus (Plato) 498 équations numériques de 194–195, 349, 405, 497 unbounded interval 278
The Grammar of Science tous les degrés (Lagrange) trichotomy law 31, 365, 367 uncertainty, and error
(Pearson) 387 299 trigon. See triangle 167–168
Theodorus 82, 283, 475–476, Traité de l’équilibre et du Trigonometria (Pitiscus) 511 undefined fraction 205–206
498 mouvement des fluides trigonometric functions 254, unexpected quiz (hanging)
Theon of Alexandria 252–253 (d’Alembert) 8 369 paradox 374–375, 375
Theon of Smyrna 99 trigonometry 16, 28, 239,
theorem (proposition) 444–445, 508–512, 516–517
498–499 Trigonometry (Simpson) 463
Trigonometry and Double
Algebra (De Morgan) 122
trinomial 512

568 Index

uniform distribution 480, 480 vector field 101, 521 vortex theory, Descartes’s 125 yotta- (1024) 465
uniform motion 522 vector multiplication 329, 520 vulgar fraction 205 Youngs, J. W. T. 200
union (set operation) 144, vector operations 520
vector product. See cross W Z
460 Waerden, Bartel Leendert van Zadeh, Lofti 212
unique factorization theorem. product Zahlbericht (Hilbert) 250
vector space 5, 71, 201, 243, der 355 Zeno 57, 238, 270, 532
See fundamental theorem of al-Wafa, Abu 510 Zeno’s paradoxes 269,
arithmetic 315, 520, 521–522 Wallace, William 165
unique solution 516 vector subtraction 520 wallet paradox 375 532–533
unitary ratio 439 vector triple product of three Wallis, John 63, 180, 269, zepto- (10-21) 465
unit circle 427, 516–517 Zermelo, Ernst Friedrich
unit denominator rule 204 vectors 512 370, 528, 528–529
unit fraction. See Egyptian velocity 132, 408, 519, and complex numbers 25 Ferdinand 31, 350, 461,
fractions 530, 533–534
unit matrix. See identity 522–523 Wallis’s product 269, 393, zero (naught) 5, 6, 41, 51,
matrix Venn, John 523–524 486, 528, 529, 536 332, 534–535
unit sphere 469 Venn diagram 123, 524,
universal calculus 199 wallpaper pattern 207 in Boolean algebra 49
universal language 199 524–525 Wantzel, Pierre Laurent 149, as exponent 534
universal mathematics 199 Verhulst, Pierre-François 408 factorial of 534
universal quantifier (“for all”) vertex 91, 233, 403, 505 513 on number line 359
430–431 vertical angles 364 Waring, Edward 229 as number vs. placeholder
unknown (indeterminate, vibrating strings 8, 299 warping of space 447
variable) 517 viergruppe. See Klein-four Watson, George Neville 436 51, 293
unordered arrangement. See weak transitivity 365 and rings 448
combination group Weaver, Warren 461 zero fraction 205
unordered partition 381 Viète, François (Franciscus Weierstrass, Karl Theodor zero matrix 147, 534
unrestricted integral. See zero of function. See root
improper integral Vieta) 10, 113–114, Wilhelm 5, 58, 284, 297, zero-sum game 217, 534
“Untersuchungen über ein 348–349, 511, 525 312, 529–530 zero-th term, of series 223
Problem der Hydrodynamik” Viète’s formula (Vieta’s Weierstrass’s product zeta function 105, 174, 243,
(Dirichlet) 140 formula) 231, 393, 511, inequality 268 246, 250, 393, 447–448,
525–526 well-ordered set 530, 533 489, 535–536
V vinculum 81, 526 Wessel, Casper 25, 86 and proofs of prime-
valence. See degree of a vertex Vinogradoff, Ivan 230 The Whetstone of Witte
Vallée-Poussin, Charles-Jean Vlacq, Adriaan 54, 320 (Recorde) 443 number theorem 518
Voevodsky, Vladimir 194 Whitehead, Alfred North 199, Zeteticorum libri quinque
de la 411, 518 volume 526–527, 527 229, 452–454, 530
Varahamihira 264 Whitworth, W. Allen 486 (Viète) 525
variable 517, 518–519 of cone 91, 173, 293, 527 whole number 443, 530–531. zetta- (1021) 465
variance 106, 306, 481, 484 of cube 111 See also natural number Zhu Shijie. See Chu Shih-
Variorum de rebus of cylinder 115, 156 Widman, Johannes 6, 131,
of frustum 207, 527 398, 487 Chieh
mathematicis responsorum as ill-defined concept 526 Wiles, Andrew John 190, zone 472
(Viète) 525 of parallelepiped 376 531 z-score (z-value) 357, 483,
Veblen, Oswald 288 of Platonic solids 397 Wilson’s theorem 299
vector 14, 142, 147, 164, of prism 412 word (braids) 52 537
310, 519–520 of pyramid 173, 293, 423 Wright, E. M. 246 Zu Chongzhi (Tsu Chung Chi)
of solids of equal height
and collinearity 80 Y 74, 392, 537
decomposition of 119 67–68 Yang Hui 325 “Zur Elektrodynamik
vector addition 376, 520 of sphere 293 year, length of 333, 363
vector equation of sphere vs. cylinder 20, Yi, Tien-Lien 71 bewegter Körper” (Einstein)
of line 520–521 yocto- (10-24) 465 158
of plane 521 91, 115 Yorke, James 71 Zur Theorie der Abelschen
of tetrahedron 497 Functionen (Weierstrass)
von Neumann, John 216, 529
218, 415, 436
Vorlesungen über
Zahlentheorie (Dirichlet)
140