Tabak J. Mathematics and the Laws of Nature: Developing the Language of Science

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230 MATHEMATICS AND THE LAWS OF NATURE

Newton, Isaac. Mathematical Principles of Natural Philosophy. Trans-

lated by Andrew Motte, revised by Florian Cajori. Great Books of

the Western World. Vol. 34. Chicago: Encyclopaedia Britannica,

1952. Some of this book is written for experts, but some is quite

accessible. See especially the section “Axioms or Laws of Motion”

for a classic example of how to apply mathematical methods to the

study of physical science.

Poincaré, Henri. Science and Method. New York: Dover Publica-

tions, 2003. Poincaré was one of the great mathematicians of his

generation. He was also a prominent scientist and popular science

writer. This reprint of the 1914 edition is a nontechnical overview

of Poincaré’s opinions about what science is and how it should

be conducted. The language is old-fashioned but the thinking is

deeply insightful. It is highly recommended.

Ptolemy. The Almagest. Translated by R. Catesby Taliaferro. Great

Books of the Western World. Vol. 16. Chicago: Encyclopaedia Bri-

tannica, 1952. Filled with diagrams, tables, and explanations, this

is one of the most important astronomical texts in history and an

important example of ancient science.

Stevin, Simon. Principal Works. Edited by Ernes Crane et al.;

translated by C. Dikshoorn. 5 vols. Amsterdam: C. V. Swets and

Zeitlinger, 1955–66. Every history of mathematics devotes space

to Simon Stevin, but unlike those of Galileo and Kepler, Stevin’s

writings are difficult to find. This very readable translation is

available in some larger libraries.

Weyl, Hermann. “Symmetry.” In The World of Mathematics. Vol.

1, edited by James R. Newman. New York: Dover Publications,

1956. An extended meditation on a geometric idea that has

become a central organizing principle in contemporary physics by

a pioneer in the subject.

INTERNET RESOURCES

Mathematical ideas are often subtle and expressed in an unfamiliar

vocabulary. Without long periods of quiet reflection, mathemati-

cal concepts are often difficult to appreciate. This is exactly the

type of environment one does not usually find on the Web. To

Further Resources 231

develop a real appreciation for mathematical thought, books are

better. That said, the following sites are some good resources.

Beretta, Marco, Andrea Scotti, Daniele Nuzzo, Pietro Corsi,

Raphaël Bange. The Project, Panopticon Lavoisier. Available

online. URL: http://moro.imss.fi.it/lavoisier/entrance/projbox.

html. Downloaded December 14, 2009. A beautifully crafted

Web site devoted entirely to Lavoisier and his accomplishments.

Eric Weisstein’s World of Mathematics. Available online. URL:

http://mathworld.wolfram.com/. Downloaded December 14,

2009. This site has brief overviews of a great many topics in

mathematics. The level of presentation varies substantially from

topic to topic.

European Centre for Medium-Range Forecasts. About ECMWF.

Available online. URL: http://www.ecmwf.int/about/. Accessed

June 11, 2010. This site provides an overview of the work carried

out at this important meteorological research center that is the

workplace of Dr. Renate Hagedorn, the scientist interviewed in

the Afterword of this volume.

Gangolli, Ramesh. Asian Contributions to Mathematics. Available

online. URL: http://www.pps.k12.or.us/depts-c/mc-me/be-as-ma.

pdf. Downloaded December 14, 2009. As its name implies, this

well-written online book focuses on the history of mathematics in

Asia and its influence on the world history of mathematics. It also

includes information on the work of Asian Americans, a welcome

contribution to the field.

The Math Forum @ Drexel. The Math Forum Student Center.

Available online. URL: http://mathforum.org/students/. Down-

loaded December 14, 2009. Probably the best Web site for infor-

mation about the mathematics that students encounter in their

school-related studies. You will find interesting and challenging

problems and solutions for students in grades K–12 as well as a

fair amount of college-level information.

National Institute of Standards and Technology. Available online.

URL: http://www.nist.gov/index.html. Accessed June 11, 2010.

This site is an excellent source of information about one of the

232 MATHEMATICS AND THE LAWS OF NATURE

premier U.S. research laboratories. Research at NIST generally

incorporates sophisticated models and is conducted at the highest

level.

Netz, Reviel, and William Noel. Archimedes: The Palimpsest Project.

Available online. URL: http://www.archimedespalimpsest.org.

Downloaded August 15, 2010. An excellent history of Archime-

des’ famous work, The Method, this site also contains images of the

original text and a great deal of additional information about this

important work.

O’Connor, John L., and Edmund F. Robertson. The MacTutor His-

tory of Mathematics Archive. Available online. URL: http://www–

gap.dcs.st-and.ac.uk/~history/index.html. Downloaded December

14, 2009. This is a valuable resource for anyone interested in

learning more about the history of mathematics. It contains an

extraordinary collection of biographies of mathematicians in dif-

ferent cultures and times. In addition it provides information about

the historical development of certain key mathematical ideas.

PERIODICALS, THROUGH THE MAIL

AND ONLINE

Scientific American

415 Madison Avenue

New York, NY 10017

A serious and widely read monthly magazine, Scientific American

regularly carries high-quality articles on mathematics and mathe-

matically intensive branches of science. This is the one “popular”

source of high-quality mathematical information that you will

find at a newsstand.

ScienceNews

1719 N. Street NW

Washington, D.C. 20036

This weekly magazine carries short and medium-length articles

on mathematics and science. The well-written articles are a good

way to stay current.

233

i n d e x

Abel, Niels Henrik 209c

Absolute Science of Space

(Bolyai) 209c

abstraction of entropy

162–163

abstract sets, theory of

measure of 212c

Academy of Sciences 102

acceleration 40–42, 72, 73,

“Account of Carnot’s Theory

of the Motive Power of

Heat” (Thomson) 139, 140

accuracy 145, 195–199

Adams, John Couch 96

advertising, representative

samples in 184

aether 53–54

Africa, sickle-cell anemia in

180

Agrawal, Manindra 217c

Ahmes 201c

Ahmes papyrus 201c

air, chemical composition of

104, 116

Aldebaran (star) 44

Alembert, Jean le Rond d’ 77,

207c

Alexandria, Egypt 24, 202c

algebra 20, 149

algebraic equations 58–59,

75–76

algorithms, for infinite

processes 68

alleles 175–176, 179–180

Almagest (Ptolemy) 27, 29,

30, 45, 46, 202c

Altdorf, University of 87

American Institute of Public

Opinion 214c

American Mathematical

Society 214c

American Revolutionary War

119

analysis 85, 193

Analysis situs (Poincairé) 211c

The Analyst (Berkeley) 207c

analytical engine 209c

analytic geometry, funda-

mental principle of 75–76

angles of Platonic solids 53

Apollonius of Perga 28–29,

72, 73, 202c

Arabic astronomy 39

arbitrariness, in nature 4–5,

187

Archimedes of Syracuse 24,

32–39, 61, 109, 202c

arcs of stars 9, 10

area 57–60

Argand, Jean Robert 208c

Argand diagram 208c

Aristarchus of Samos 22–24,

26, 47, 202c

Aristotle 26, 35, 69

arithmetic, in Mesopotamian

astronomy 20

Arithmetica (Diophantus)

202c

Arithmetica infinitorum

(Wallis) 80, 84, 205c

Ars conjectandi (Bernoulli)

206c

Ars magna (Cardano) 204c

Artis analyticae praxis

(Harriot) 205c

Aryabhata 47, 203c

The Assayer (Galileo) 83

astrology 40

Astronomia nova (Kepler)

204c

astronomical calculations,

Mesopotamian 15–19

astronomical diaries,

Mesopotamian 8, 14, 15

astronomical observations

of Brahe 51, 54

of Copernicus 44

of Galileo 70–71

Mesopotamian 8–11

astronomy

Copernicus’s model of

universe 45–48

Copernicus’s studies

42–47

Galileo’s studies 70–71

Kepler’s model of solar

system 48–50

Kepler’s studies 48–52,

54–55

Laplace’s contributions

94–95, 97

Mesopotamian. See

Mesopotamian

astronomy

Ptolemy’s model of

universe 27–32

ratios and measuring

distances 20–26

and rotation of Earth 31

atmosphere 108, 121,

191–192

atomic nuclei, conservation

of mass and 110–113

autumnal equinox 29

averages, in weather

forecasting 193

axioms

in 19th-century

mathematics 4

Galileo’s 72

in Greek mathematics 2,

and laws of nature 4–5,

in mathematical systems

3–4

Newton’s 88

Page numbers in italic indicate illustrations;

page numbers followed by c indicate chronology entries.

234 MATHEMATICS AND THE LAWS OF NATURE

Babbage, Charles 209c

Baghdad 203c

Banach, Stefan 213c

Banach spaces 213c

base 2 number system 87

base 60 number system 19

Basel, University of 107

Bayes, Thomas 207c

Bell Laboratories 215c, 216c

Berkeley, George 207c

Berlin, University of 131

Bernoulli, Jacob 206c

Bhaskara II 195c

Black, Joseph 116–120, 122,

125, 131–132, 207c

blended inheritance 166–

167

boats, buoyancy force and 33

Bologna, University of 44

Bolyai, János 209c

Bolzano, Bernhard 210c

Boole, George 210c

Borel, Émile 212c

Botticelli, Sandro 56

Bourbaki group 215c, 217c

Boyle, Robert 115

Brahe, Tycho 51, 54, 92

Brahmagupta 203c

Brahma Sphuta Siddhanta

(Brahmagupta) 203c

Breslau, Germany 206c

“A Brief Account of

Microscopical

Observations” (Brown)

209c

Brouwer, Luitzen E. J. 213c

Brown, Robert 209c

Brownian movement 209c,

212c

Bryn Mawr College 149

Buffon, Georges-Louis

Leclerc, comte de. See

Leclerc, Georges-Louis

buoyancy, Archimedes

principle of 32–35, 39, 61,

109

butterfly effect 198

C (computer language) 216c

calculus

d’Alembert’s studies 77

derivatives 156–157

fundamental theorem of

and infinitesimal analysis

Newton and Leibniz’s

studies 85–88, 156, 157

and Newton’s laws of

motion 91

rates of change in 153–

154, 156, 157

calendar, Mesopotamian 12

caloric theory of heat 118–

120, 122–125, 128, 130, 135,

136

Cambridge University 84, 138

Canada, sickle-cell anemia in

180

cannons, boring 119–120

canon law 44

Canon mathematicus (Viète)

204c

Cantor, Georg 211c

carbon dioxide 116

Cardano, Giralamo 204c

Carnot, Lazare 120

Carnot, Sadi 120–127

and caloric theory of heat

122–124

Carnot engine 124–127

early life and family

120–121

and first law of

thermodynamics 125,

130

motive power of heat 140

Réflexions sur la puissance

motrice du feu 121, 209c

view of flow of heat 135

Carnot engine 124–127, 135,

209c

Cartan, Henri 217c

catastrophe theory 216c

category theory 215c

Catholic Church 40, 42, 44,

cause and effect 163, 164, 186

celestial sphere 27, 46

chance 183. See also

randomness

Charles V (king of France)

chemical reactions, mass-

energy conversion and

112–113

chemical revolution 104

chemistry 4, 53

Chinese mathematics 39,

201c, 202c

chromatic scale 67

chromosomes 171

circles 28–29, 45, 52, 54, 71

circular motion, natural 47

circumference of Earth

24–26

circumscribed sphere of a

cube 50

classical physics. See also

thermodynamics

Archimedes’s studies 32,

axioms 4

chance in 183

and Einstein’s theory of

relativity 100–101, 149

of falling bodies 69

interactions of forces in

64–65

Newton’s laws of motion

Renaissance scientists 32

Clausius, Rudolph 130–138,

141, 155, 210c

Collection (Pappus) 202c

Collège Mazarin 102

colliding bodies 80–81, 90,

color of sky 131

combustion 104, 109

completeness of laws of

nature 144

compounds (Lavoisier’s

definition) 104

computers 161–162, 182

conclusions 5, 6

Conics (Apollonius) 202c

conservation laws 57–60,

154–155, 159. See also

mathematical study of

conservation laws

conservation of caloric

118–120

conservation of energy

149–151

conservation of mass, law of

xviii, 102–113, 208c

conservation of momentum,

law of 83–101

and calculus 85–88

and conservation of mass

107

and discovery of Neptune

93–97

limits of Newton’s laws

of motion 97–101

Index 235

Newton’s laws of motion

88–93

and Newton v. Galileo

83–84

consistency, logical 3

consistency of laws of nature

143

constellations, motion of Sun

and 10–11

context, for laws of nature 6

continuity 57–60, 152–153

control forecast 195

coordinate systems of

Descartes and Fermat 75

Copernican revolution 48

Copernicus, Nicolaus 42–47

De hypothesibus motuum

coelestium a se constitutes

commentariolus 44–45

De revolutionibus orbium

coelestium 42, 43, 45,

47, 204c

early life 42, 44

Kepler’s model v. 51, 52,

54, 55

model of universe 45–48,

coupled reactions,

mathematical models for

109

A Course in Pure Mathematics

(Hardy) 174

Cramer, Gabriel 207c

Cramer’s rule 207c

cryptography 80

cube 50, 53

cuneiform tablets,

Mesopotamian 7–8, 8, 18,

curves 75–76, 79, 156

cyclic motion of heavens 40

Darwin, Charles 211c

Daubechies, Ingrid 216c

deceleration 40–42

The Decimal. See La Disme

(Stevin)

Dedekind, Richard 211c

deductive model of nature

97, 142–146

deductive reasoning 1–5

De hypothesibus motuum

coelestium a se constitutes

commentariolus

(Copernicus) 44–45

De la Spartostatique (Stevin)

deleterious genes 179–181

DEMETER (Development

of a European Multimodel

Ensemble System for

Seasonal to Interannual

Prediction) 189, 190

Deming, William Edwards

214c

density, buoyancy force and

deoxyribonucleic acid. See

DNA

De ratiociniis in ludo aleae

(Huygens) 206c

De revolutionibus orbium

coelestium (Copernicus) 42,

43, 45, 47, 204c

derivatives

in calculus 85

of curves 79, 156

functions from 92–93

generalized. See general-

ized derivatives

and infinite processes

of mass 106

of momentum 91, 92

in Sobolev’s studies

156–157

of tangents 76–78

Desargues, Gérard 205c

Descartes, René 74–79, 82,

205c

determinants, theory of 206c

deterministic models 185,

195–197

Development of a European

Multimodel Ensemble

System for Seasonal to

Interannual Prediction

(DEMETER) 189, 190

Dialogue Concerning the Two

Chief World Systems

(Galileo) 71

Dialogues Concerning Two New

Sciences (Galileo) 71–74,

205c

Differential Analyzer 214c

differential calculus 85, 86

differential equations

for classical conservation

laws 144–145

differential inequalities v.

159

initial conditions for

160, 162

linear and nonlinear 158

for mass 106

for momentum 86–87

for motion of fluids

107–108

partial and ordinary 154,

158

and random processes

184–185

solving, with calculus

154

differential inequalities 159

differential reproductive

rates 177

differential survival rates

177

differentiation 86, 92

“digester” 115

dimension (Brouwer’s

definition) 213c

dimension theory 213c

Dimension Theory (Hurewicz

& Wallman) 215c

Diophantus of Alexandria

202c

discontinuous phenomena

151–160, 162–163

“Discontinuous Solutions to

Nonlinear Differential

Equations” (Oleinik) 158–

159, 215c

Discours de la méthode

(Descartes) 205c

discovery, mathematical 3

La Disme (Stevin) 63, 204c

Disquisitiones arithmeticae

(Gauss) 208c

dissipation of heat 140

distance(s)

in Archimedes’s law of

levers 36

of Earth to Sun and

Moon 22–24

and force in work 129

and law of gravity 92

Oresme’s representation

204c

of planets as Platonic

solids 49–50

from Sun and orbital

periods 51–52, 55

Sun to Uranus 95

DNA (deoxyribonucleic acid)

171, 172, 182

HOM MathLaws-dummy.indd 235 4/1/11 4:49 PM

236 MATHEMATICS AND THE LAWS OF NATURE

The Doctrine of Chances

(Moivre) 207c

dodecahedron, as Platonic

solid 53

Domesday Book 203c

dominant traits 169, 170

Dürer, Albrecht 204c

Earth

circumference of 24–26

Copernicus’s model of

universe 45, 46

Eudoxus’s celestial sphere

Kepler’s studies of solar

system 48

observations of Saturn

from 13–14

orbital path 15–16

orbital period and

distance from Sun 52

orbital speed 16

relative distances to

Moon and Sun 15,

22–24, 202c

relative size 24

revolution 9–10, 45

rotation 31, 46, 47

ecliptic path of Sun 11, 11,

16–17, 29

ECMWF. See European

Centre for Medium-Range

Weather Forecasts

École Polytechnique 121

Edinburgh, University of 116

efficiency 115–116, 123–127

Egyptian mathematics 201c

Eighty Years’ War 63

Eilenberg, Samuel 215c

Einstein, Albert xviii,

98–101, 111, 112, 149, 212c

Electromagnetic Theory

(Heaviside) 211c

Electronic Numerical

Integrator and Calculator

(ENIAC) 215c

elements 53–54, 104

Elements (Euclid) 1–2, 36, 72,

84, 202c

Éléments de géométrie

(Legendre) 208c

Éléments de mathématique

(Nicolas Bourbaki group)

214c

ellipse 15–16, 51, 52, 54–55

Encyclopédie (d’Alembert and

Diderot) 207c

energy. See also conservation

of energy

conservation of 132–134

and entropy 141

mass-energy conversion

111–112

and second law of

thermodynamics 142

and thermodynamics 114

engineering 1–7

ENIAC (Electronic

Numerical Integrator and

Calculator) 215c

Ensemble Prediction System

(EPS) 191, 195, 196

ensembles 184, 185, 190

entropy 141, 159–160,

162–163

entropy condition 162–163

entropy inequalities 160,

162–163

epidemiological studies of

disease 210c

EPS. See Ensemble

Prediction System

equilibrium 36–37, 64–65,

175

equitempered tuning 67

Eratosthenes of Cyrene

24–26, 202c

Erlangen, University of 147,

148

Erlanger Programme 211c

errors, probability theory and

164

Les espaces abstraits et leur

théorie considérée comme

introduction à l’analyse

générale (Frechet) 213c

Essai sur les coniques (Pascal)

205c

“Essay Towards Solving a

Problem in the Doctrine of

Chances” (Bayes) 207c

Euclidean geometry 2, 4,

72–73, 212c

Euclid of Alexandria

Elements 1–2, 36, 202c

and Galileo’s studies of

projectile motion 72

influence on science and

mathematics 37

logic of 4

in Newton’s studies 84

Eudoxus of Cnidas 27, 201c

Euler, Leonhard 107–108,

110, 207c

European Centre for

Medium-Range Weather

Forecasts (ECMWF) 189–

191, 194, 197, 199

European philosophy/

philosophers 30, 32, 39–40

European science in Middle

Ages 39–60

Copernicus’s studies

42–47

force in 61

Kepler’s studies 48–52,

54–55

Leonardo da Vinci’s

equation of continuity

55–60

Oresme’s studies 40–42

and Platonic solids 53–

proof of equation of

continuity 59

evaporation, heat and

121–122

everything, theory of 98

Experimental Mathematics

(journal) 216c

experimental science 61–82

in Descartes and Fermat’s

studies 74–79

in Galileo’s studies

70–74

and rigorous mathematics

61–63

and Stevin’s musical

contributions 66–67

in Stevin’s studies 63–69

in Wallis’s studies 80–82

experiments

realization of 184

reproducibility of 62

testing conclusions with

and theory 39, 62–63

verification of natural

laws 5, 146

explosions 153

faces of Platonic solids 53

falling bodies 69, 72, 76, 79

Fermat, Pierre de 74–79

analysis of 82

coordinate system 75

Index 237

derivatives in studies

76–78, 85

last theorem 205c, 216c

in Newton’s studies 84

Ferrara, University of 44

Fibonacci, Leonardo 204c

fifth (musical interval) 66

first law of thermodynamics

125, 130–134, 155

Fisher, Ronald Aylmer 214c

flow 58–60, 145–146

fluid dynamics, Euler’s

studies 107–108, 110

fluids

buoyancy force for 34, 35

classical conservation

laws in study of 144

Stevin’s studies 65–69

turbulent 183

working 124

foci of an ellipse 52, 54

focus groups 184

Fontana, Niccolò (Tartaglia)

204c

force(s)

buoyancy 33–34

in European science of

Middle Ages 61

on fluids 107–108, 110

Galileo’s studies 73

in law of gravity 92

and motions. See

mechanics

and Newton’s laws of

motion 90, 91

Stevin’s studies 64–69,

90, 91

on Uranus 96

as vectors 65, 90, 91

of water exerted on a wall

65–69

work 129

FORTRAN (FORmula

TRANslator) 215c

fossils 178

Fourier, Jean-Baptiste-Joseph

138

fourth (musical interval) 66

The Fractal Geometry of

Nature (Mandelbrot) 216c

Franco-Prussian War 131

Frechet, Maurice 213c

freefall, bodies in. See falling

bodies

French Revolution 105, 120,

208c

friction, heat and 120, 130

function(s)

continuous nowhere

differentiable 211c

from derivatives 92–93

in functional analysis

155–156

graphical representations

of 161–162

Sobolev’s definition 215c

functional analysis 155–156

fundamental property of

natural laws 143

Galilei, Galileo 32, 60, 70,

204c

The Assayer 83

astronomical

observations 70–71

Copernicus’s influence

on 47

Dialogue Concerning Two

Chief World Systems 71

Dialogues Concerning Two

New Sciences 71–74,

205c

falling-body experiment

father of 67

and Kepler 70

motion studies 69, 71–74,

Newton v. 83–84

and prediction of

patterns in nature 97

scientific approach

61–63

thermometer 114–115

Galle, Johann G. 97

Gallup, George Horace

214c

Galois, Évariste 209c

Ganita Sara Sangraha

(Mahavira) 203c

gases 105, 210c

Gauss, Carl Friedrich 208c

gene frequencies 172–178,

181

generalizability of natural

laws 143

generalized derivatives 156–

160, 162–163

General Theory of Measure and

Probability Theory

(Kolmogorov) 213c

general theory of relativity

149

genes 171

deleterious 179–181

frequencies of. See gene

frequencies

mutations 177, 181

and traits 169–171

Genetical Theory of Natural

Selection (Fisher) 214c

genetic diseases 179–180

genetics 182. See also

heredity

geometric model of universe

27–32

Geometrie der Lage (Staudt)

210c

geometry

in astronomy 39

axioms in 2

descriptions of nature

with 61

of ellipses 52, 54

in Greek astronomy 20

heavenly 27

in Kepler’s studies of

solar system 48–52

in science 147

symmetry in 150

Ghirlandajo, Domenico 56

Glasgow, University of 116,

138

goal-year text 14

Gödel, Kurt 214c

Göttingen University 148,

149

Grand Coulee Dam 122

Graunt, John 206c

gravitational field of Sun

93–94

gravity, law of 91–92, 97

Great Pyramid at Giza 21,

22, 24

Great Pyramid at Khufu

201c

Greek mathematics and

science 20–38, 203c

Archimedes’s buoyancy

studies 32–35

curves in 75–76

deductive reasoning in

1–4

European view of 30, 32,

39–40

law of the lever 35–38

Platonic solids in 53

238 MATHEMATICS AND THE LAWS OF NATURE

Ptolemy’s model of the

universe 27–32

ratios and measuring

astronomical distances

20–26

and rotation of Earth 31

Grundzüge der Mengenlehre

(Hausdorff) 213c

Guild of Saint Luke 57

Hagedorn, Renate 189,

189–200

Halle University 131

Halley, Edmund 206c

Han dynasty 202c

Hardy, Godfrey Harold 173,

174

Hardy-Weinberg law 173–

181, 212c

Harriot, Thomas 205c

Hausdorff, Felix 213c

heat

Black’s studies 116–119

in Carnot’s studies 121

Clausius’s theory of 210c

dissipation of 140

and evaporation 121–122

and friction 130

latent 118, 132, 133, 207c

mechanical equivalent of

129–130, 209c

as motive power 121,

125, 140

and phase changes 117,

117–118

sensible 118

specific 118

and temperature 116–

118

and work 114, 121, 128

heat death 138, 140

heat engines

“backward”-flowing heat

in 136

of Carnot 124–127

and latent heat 132, 133

OTEC technology 127

in second law of

thermodynamics 142

heavenly geometry 27

Heaviside, Oliver 211c

height, of Great Pyramid at

Giza 21

heliocentric model of

universe 47

heliostatic model of universe

heredity 166–172, 174

heredity, laws of 165, 168

Herschel, William 95, 208c

Hilbert, David 148, 149, 212c

Hipparchus 29

Hisab al-jabr wa’l muqabala

(al-Khwarizmi) 203c

Histoire naturelle (Leclerc)

207c

Hollerith tabulator 211c

Hooke’s law 97

House of Wisdom 203c

Hurewicz, Witold 215c

Huygens, Christiaan 84, 115,

206c

hydrostatics, Stevin’s studies

of 65–69

Hypatia 203c

icosahedron, as Platonic solid

impossible processes, second

law of thermodynamics and

135

inclined plane, law of 63

incommensurability of

motion of heavens 40

incompressibility of water 58

Indian astronomy 39

individuals, in population

genetics 173

inductive reasoning 5–6

Industrial Revolution 116

inequalities 137, 159, 160,

162–163

inertial reference frames 100

infinite processes, derivatives

and 78

infinitesimal analysis 66–69

inheritance, theory of

166–167

inheritance by blood 166–167

initial conditions

for differential equations

160, 162

in meteorological

forecasting 191

for natural phenomena

145–146

uncertainty about

183–184

in weather forecasting

198–200

inscribed sphere of a cube 50

Institute for Marine Sciences

189

integral calculus 85, 86

integration 86, 92, 212c

international financial

system, mathematical

models and 217c

Introductio (Euler) 207c

inverse operations 86, 92

irreversible processes 114

isolated systems 80–81, 90,

91, 106, 141

Al-jabr w’al muqabala (Omar

Khayyám) 203c

Janiszewski, Zygmunt 213c

Japanese school of topology

216c

JAVA (computer language)

216c

Joule, James Prescott 128–

130, 209c

Jupiter 10, 14–15, 19, 45, 48,

Kayal, Neeraj 217c

Kelvin, Lord. See Thomson,

William (1st baron Kelvin)

Kelvin temperature scale

127, 139

Kepler, Johannes

Astronomia nova 204c

and Copernicus 47

early life 48

and Galileo 70

laws of planetary motion

50, 50–52, 54, 55, 61,

model of solar system 49,

49–50

and Newton 84, 92, 93

Platonic solids in studies

49–51

Khayyám, Omar 203c

Khwarizmi, Mohammed ibn-

Musa al- 203c

Klein, Felix 148, 211c

Kolmogorov, Andrey

Nikolayevich 213c, 215c

Kolmogorov complexity

215c

Kraków, University of 44

Kuratowski, Kazimierz 214c

Index 239

Laplace, Pierre-Simon

94–95, 97, 121, 165, 209c

latent heat 118, 132, 133,

207c

Lavoisier, Antoine-Laurent

102–106, 103, 116, 208c

laws. See specific laws, e.g.:

conservation of mass, law

Laws of Thought (Boole)

210c

Leaning Tower of Pisa 69

least squares, method of

208c

Lebesgue, Henri-Léon

212c

Leclerc, Georges-Louis

(comte de Buffon) 207c

Legendre, Adrien-Marie (or

Le Gendre) 208c

Leibniz, Gottfried Wilhelm

87–88, 156, 157, 206c, 207c

Lenin, Vladimir Ilyich 213c

Leonardo da Vinci 56–61,

143, 204c

Leonardo of Pisa (Fibonacci)

204c

lever, law of the 35–38

Liber abaci (Fibonacci) 204c

Liber de ludo aleae (Cardano)

204c

lightning 153

Lilavati (Bhaskara II) 203c

linear differential equations

158

Lobachevsky, Nikolai

Ivanovich 209c

logic 1–3, 6, 77

logical consistency of axioms

logical independence of

axioms 3

London Bills of Mortality

206c

Lorenz, Ed 198

Mac Lane, Saunders 215c

Mahavira 203c

malaria 179, 180

Al-Ma’mun 203c

Mandelbrot, Benoît 216c

Markov, Andrey Andreyevich

212c

Mars 10, 28, 28, 45, 48, 52

mass 59–60, 80, 105–106,

111–112. See also conserva-

tion of mass, law of

Massachusetts Institute of

Technology 214c

Mästlin, Michael 48

mathematical models

of conservation laws 159

conservation of mass in

112–113

conservation of

momentum in 97–98

for coupled reactions 109

and international

financial system 217c

Mesopotamian 16–19

for meteorology 190–196

for natural phenomena

142–146

and randomness 181–186

of random phenomena

185

mathematical study of

conservation laws 147–163

with computers 161–162

Noether’s studies 147–151

Oleinik’s studies 158–

160, 162–163

Sobolev’s studies

151–157

mathematical systems

axioms in 3–4

for flow of air/water

145–146

initial conditions in

nonlinear 198

for natural phenomena

145

nature v. 4

“A Mathematical Theory of

Communication”

(Shannon) 215c

Maurice, prince of Nassau

63, 64

Maxwell, James Clerk 210c

measure of abstract sets,

theory of 212c

mechanical equivalent of heat

129–130, 209c

mechanics 74–82

Menaechmus 202c

Mendel, Gregor 166, 211c

early life 165

genetic change and

studies of 174

heredity studies 167–172

laws of heredity 165, 168

and theory of inheritance

166–167

“Versuche über Pflanzen-

hybriden” 211c

Menger, Karl 213c

Mercury 99

Copernicus’s model of

universe 45

Kepler’s studies of solar

system 48

Mesopotamian

astronomy 10, 14

orbital period and

distance from Sun 52

orbit of 98

Mesopotamian astronomy

7–19

calculations 15–19

Greek astronomy v. 20

observations 8–15

and predicting the future

12–15

tablets 18, 19

written records 7–8

meteorological forecasting

189–200

The Method (Archimedes) 33,

37–38, 202c

Middle Ages. See European

science in Middle Ages

migration, in population

genetics 176–177

Milky Way 187

model runs 185

models. See mathematical

models

Moivre, Abraham de 207c

molecular genetics 172, 173

molecular velocities in gases

210c

momentum 80–82, 91, 92.