Tabak J. Mathematics and the Laws of Nature: Developing the Language of Science
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240 MATHEMATICS AND THE LAWS OF NATURE
Moore, Robert Lee 214c
Morita, Kiiti 216c
Morlet, Jean 216c
Moscow State University 158
motion
in Descartes and Fermat’s
studies 74–79
of falling bodies 69, 72
and forces. See mechanics
Galileo’s studies 69,
71–74, 82
of heavens 40
and momentum 80
natural circular 47
Oresme’s graphical
analysis 40–42
planetary. See planetary
motion
prograde 13, 14
projectile 72–73, 74
retrograde 13–14, 28–29
of Saturn 13–14
in Wallis’s studies 80–82
motion, laws of
and combustion 109
and discontinuous
phenomena 155
and discovery of Neptune
95–97
and heat 121–122
limits of 97–101
Newton’s discovery xviii,
88–93
motive power, heat as 121,
125, 140
multi-model ensembles 190
musical scales 66–67
mutations 177, 181
mysticism xvii, 50
N
Nägeli, Karl Wilhelm von
165
Napoléon I Bonaparte
(emperor of the French)
120, 121, 208c, 209c
natural circular motion 47
natural laws 1–7
natural overtone system 67
natural selection, theory of
214c
nature xvii–xviii, 4, 97,
142–146
Navara, Domenico Maria de
44
nebulae 9
negative statements, proving
138
Neptune 52, 93–97, 210c
neutrons, mass of 110, 111
Newcomen, Thomas 116
Newcomen steam engine
116, 206c
new mathematics 207c
Newton, Sir Isaac 85, 207c
belief in four “elements”
53
calculus studies 85–87,
156, 157
Galileo v. 83–84
law of gravity 91–92
laws of motion. See
motion, laws of
Philosophiae naturalis
principia mathematica
88, 206c
Wallis’s inspiration 80
Nicolas Bourbaki group
214c, 217c
Noether, Emmy 60, 147–151,
148, 213c
nondeterministic models 185
non-Euclidean geometry
209c
nonlinear differential
equations 158
nonlinear systems of
equations, initial conditions
for 198
Northern Hemisphere,
apparent movement of stars
9
North Star 9, 10
notation, Leibniz’s 87–88
Nouvelles méthodes pour la
détermination des orbites des
comètes (Legendre) 208c
Nova methodus pro maximis et
minimis (Leibniz) 206c
nuclear physics, conservation
of mass in xviii, 111–113
nuclear reactor 136
numerical analysis 162
O
observations, astronomical.
See astronomical
observations
occultation of stars 44
Ocean Thermal Energy
Conversion (OTEC)
technology 127
octahedron, as Platonic solid
53
octave 66, 67
Oleinik, Olga Arsenievna
158–160, 162–163, 215c
On Floating Bodies, I and II
(Archimedes) 32, 34
“On the Dynamical Theory
of Heat” (Thomson) 210c
On the Equilibrium of Planes
(Archimedes) 35–37, 202c
“On the Hypotheses That
Form the Foundations of
Geometry” (Riemann)
210c
“On the Mode of
Communication of
Cholera” (Snow) 210c
On the Origin of Species by
Means of Natural Selection
(Darwin) 211c
orbital period of planets
51–52, 55
orbits
Apollonius’s model of
planetary motion
28–29
as circles 28–29, 45, 71
in Copernicus’s model of
universe 45
distance from Sun and
orbital period 51–52,
55
law of gravity and path of
92
of Mercury 98
ordinary differential
equations 154
Oresme, Nicholas 40–42, 69,
204c
OTEC (Ocean Thermal
Energy Conversion)
technology 127
oxygen, combustion and
104
P
Padua, University of 44
Papin, Denis 115
Pappus of Alexandria 202c
parabolic path of projectiles
73
Paradoxien des Unendlichen
(Bolzano) 210c
parallelogram of forces
64–65
Index 241
parameterization 191, 192
partial differential equations
154, 158
particulate nature of heredity
169, 174
Pascal, Blaise 205c
Pascaline 205c
patterns in nature xvii–xviii
Peano, Giuseppe 211c
pea plants 167–172
Pearson, Karl 214c
Pennsylvania, University of
215c
Perelman, Grigory 217c
perfect fifth (music) 66
perfect fourth (music) 66
period, orbital 51–52, 55
Persian astronomy 39
perturbations 94, 185
Perugino (Pietro Vannucci)
56
Petrovsky, Ivan G. 158
phase changes 117, 117–118
Philosophiae naturalis principia
mathematica (Newton) 88,
206c
Philosophical Institute of
Olmütz 165
physical sciences, predictions
in 182–183
physics
classical. See classical
physics
nuclear 111–113
Pisa, University of 70
planetary cycle,
Mesopotamian 14
Planetary Hypotheses
(Ptolemy) 29
planetary motion
Apollonius’s model
28–29
Kepler’s laws 50, 50–52,
54–55, 61, 95
Laplace’s model 94–95
and limit of Newton’s
laws of motion 98
Newton’s analysis 92, 93
perturbations 94
of Sun and constellations
10–11
planets. See also specific planets
in Copernicus’s model of
universe 43, 45
gravitational interactions
of 94
in Mesopotamian
astronomy 10–12, 15,
18
orbital period and
distance from Sun
51–52, 55
predicting movement of
12, 18
speed of 45, 54–55
Plato 53, 202c
Platonic solids 49–51, 53,
53–54
Pluto 52
Poincaré, Henri 211c
Poincaré conjecture 217c
Polaris (star) 9, 10
Polish school of topology
213c
pollution, combustion and
109
polygons, regular 48–49
Poncelet, Jean-Victor 209c
population genetics 172–181
population size, gene
frequency and 175–176,
181
position of colliding bodies
91
possible processes, second
law of thermodynamics and
135
post-processing of
mathematical models 192
postulates 2, 36–37
power, motive 121, 125, 140
power-generation,
combustion and 109
prediction(s)
and entropy inequalities
160, 162–163
of event frequencies
184–185
from Kepler’s laws of
planetary motion 52
of Mesopotamian
astronomers 8–9,
12–15, 17
with Mesopotamian
astronomy 12–15
in meteorological
forecasting 195–199
in population genetics
181–183
and probability 164
and randomness 181–186
predictivity of natural laws 6
pressure 66, 115
pressure cooker 115
Princeton Institute for
Advanced Study 149
Principia mathematica
(Newton). See Philosophiae
naturalis principia
mathematica
Principia mathematica (Russell
& Whitehead) 212c
probabilistic forecasting
system 196, 197
probability 164–165, 171–
172, 183–185
probability theory 164–165,
213c
Procedure Texts 18
products, weight of reactants
and 105–106
prograde motion of Saturn
13, 14
projectile motion 72–73, 74
proof(s) 36, 59, 161
protoalgebra 20
protons, mass of 110
Ptolemy, Claudius (Ptolemy
of Alexandria)
Almagest 27, 29, 30, 45,
46, 202c
Copernicus’s model of
universe v. 45–47
influence on science and
mathematics 37
Kepler’s model of plane-
tary motion v. 51, 52
model of the universe
27–32
Planetary Hypotheses 29
pyramids. See Great
Pyramids
Pythagoras of Samos 66, 201c
Pythagoreans 66, 67, 201c
R
randomness 164–186
and differential equations
184–185
and limits of
mathematical models
181–186
Mendel’s heredity studies
165–172
in population genetics
172–181
and probability theory
164–165
242 MATHEMATICS AND THE LAWS OF NATURE
Raphael Sanzio 56
rates of change 79, 153–154,
156, 157
ratios 20–26, 66
reactants, weight of products
and 105–106
reality, in science 6
realizations (experimental
runs) 184
real number line 211c
reasoning 1–6, 74
recessive traits 169, 170
reference frames, inertial
100
Réflexions sur la puissance
motrice du feu (Carnot) 121,
209c
refrigerators 137, 138, 142
Regnault, Henri-Victor
138–139
regular polygons 48–49
Reign of Terror 105, 208c
relativity xviii, 98–101, 149,
210c
representative samples 184,
185
reproducibility of
experiments 62
reproductive rates,
differential 177
reservoirs, thermal 123–127
resistors 128, 129
resolution of meteorological
forecasting models
194–195
retrograde motion 13–14,
28–29
revolution of Earth 9–10, 45
Rhind papyrus 201c
Riemann, Georg Friedrich
Bernhard 210c
right triangles 22–23, 41, 42
rotation 27, 31, 46, 47
The Rubáiyát (Omar
Khayyám) 203c
Rumford, Count. See
Thompson, Benjamin
Russell, Bertrand 212c
S
sail-powered wagon 64
Saturn 10, 12–14, 45, 48, 52
Savery, Thomas 115–116,
206c
Saxena, Nitin 217c
scales (music) 66–67
science. See also specific fields
geometry in 147
mathematics as language
for 40
mathematics v. 1–7
mysticism v. xvii
seasons, in Greek astronomy
29
secant line 77, 78–79
second law of
thermodynamics 135–140,
142, 159–160
segregation of traits 171
sensible heat 118
Shannon, Claude 215c
Shewhart, Walter 215c
shock waves 151–153
sickle-cell anemia 179–180
Sierpi´nski, Wacław 213c
Sierpi´nski gasket 213c
simplicity of natural laws 145
sky, color of 131
slope 78–79
smallpox 207c
small-scale processes,
parameterization of 192
Smoluchowski, Marian 212c
Snow, John 210c
Sobolev, Sergey L’vovich 152
computers in studies 161
and discontinuous
phenomena 151–153
election to USSR
Academy of Sciences
214c–215c
functional analysis
155–156
generalized derivatives
156–157
higher dimension studies
156
and mathematics for
discontinuous processes
154–155
and Oleinik’s studies 160
Sobolev spaces 156
Socrates 201c
solar system 47–50, 49,
94–95. See also specific
planets
sonic booms 151–152
Sorbonne 102
Soviet Union (USSR) 213c
Soyuz rocket 89
spaces, in functional analysis
155–156
special theory of relativity
xviii, 98–101
specific heat 118
speed. See also velocity
Earth’s 16
of light 98–101
of planets 45, 54–55
Sun’s 16–17, 29, 45
and weight of falling
bodies 69
spheres
celestial 27, 46
inscribed and
circumscribed by a
cube 50
in Kepler’s model of
universe 48–49
in Ptolemy’s model of
universe 29–30
universe as rotation of 27
Stalin, Joseph 155
stars 9–10, 12, 44, 46
statics, Stevin’s studies of
64–69
station of Saturn 13, 14
statistical downscaling 193
statistical reasoning 74
Staudt, Karl Georg Christian
von 210c
steady-state flow 58
steam engines 115, 115–116,
119, 121, 123–124, 206c. See
also heat engines
Stevin, Simon “Stevinus” 32,
60
De la Spartostatique 62
La Disme 63, 204c
early life and inventions
63–64
falling-body experiment
69
hydrostatics studies
65–69
infinitesimal analysis
66–69
musical contributions
66–67
parallelogram properties
of forces 64–65
scientific approach 61–63
vector properties of
forces 65, 90, 91
Sumerians 7
Sun
in Copernicus’s model of
universe 45, 46
Index 243
distance from Earth 15,
22–24, 202c
distance from Uranus 95
ecliptic path 11, 11,
16–17, 29
gravitational field 93–
94
in Kepler’s laws of
planetary motion
51–52, 55
in Mesopotamian
astronomy 10–18
predicting movement of
12, 16–18
relative size of 24
Saturn’s orbit of 13
speed of 16–17, 29
Sun Microsystems 216c
surface of discontinuity
152–153
survival rates, differential
177
Syene, Egypt 24
symmetry 36, 150–151
T
tangents 76–79
telegraph cables, Thomson’s
work with 139
telescope, Galileo’s 70
temperature
and efficiency of Carnot
engine 126–127
and heat 116–118, 140
Kelvin scale 127, 139
and surface of
discontinuity 152–153
in thermodynamics
114–115
tetrahedron, as Platonic solid
53
Thales of Miletus 20–22, 24,
201c
Théorie analytique des
probabilités (Laplace) 209c
theory, experiments and 39,
62–63
thermal reservoirs 123–127
thermodynamics, laws of
114–146
and caloric theory of heat
114–120
and Carnot engine’s
efficiency 126–127
in Carnot’s studies
120–127
and deductive model of
nature 142–146
and entropy 141
first 130–134
in Joule’s studies
128–130
second 135–140, 142
thermometer, Galileo’s
114–115
Thom, René 216c
Thompson, Benjamin (Count
Rumford) 119–120, 208c
Thomson, James 138
Thomson, William (first
baron Kelvin)
computer 211c
explanation for color of
sky 131
“On the Dynamical
Theory of Heat” 210c
second law of
thermodynamics 135,
138–140, 142
time 41, 42, 101, 150–151
Topologie (Kuratowski) 214c
training data, in
meteorological forecasting
193–194
Traité de la section perspective
(Desargues) 205c
Traité des propriétés des figures
(Poncelet) 209c
traits, genes and 169–171
translational symmetry of
energy conservation
150–151
triangles 22–23, 41, 42
Trinity College 84
Truesdell, Clifford 160
Tübingen, University of 48
turbines 122–123
turbulent flow 183
Turing, Alan 214c
U
“Über eine Eigenschaft des
Inbegriffes aller reelen
algebraischen Zahlen”
(Cantor) 211c
uncertainty, in meteorology
197–200
Union of Soviet Socialist
Republics (USSR) 213c
unison 66
United States Census Bureau
211c, 214c, 215c
Universal Automatic Com-
puter (UNIVAC I) 215c
universe 27–32, 45–48. See
also solar system
University College, London
214c
Uranus 52, 95–96, 208c,
210c
Urysohn, Paul Samuilovich
213c
USSR. See Union of Soviet
Socialist Republics (USSR)
USSR Academy of Sciences
214c
V
Vannucci, Pietro (Perugino)
56
variability 166–167, 170
vectors, forces as 65, 90, 91
velocity. See also speed
acceleration as derivative
of 76
in Leonardo’s
conservation law 57–58
molecular 210c
and momentum 80
in Newton’s laws of
motion 89–90
in Oresme’s graphical
analysis of motion
41–42
and special theory of
relativity 98–101
velocity-area law 57–60
Venus 10, 14, 45, 48, 52
vernal equinox 29
Verrier, Urbain-Jean-Joseph
Le 96–98, 210c
Verrocchio, Andrea 56, 57
“Versuche über
Pflanzenhybriden”
(Mendel) 211c
Vienna, University of 165
Viète, François 84, 204c
Vija-Ganita (Bhaskara II)
203c
Vincenzo Galilei 67
volcanoes 122
volume 34, 58, 60, 143
volumetric flow 58, 60
W
Wallis, John 205c
Arithmetica infinitorum
80, 84, 205c
244 MATHEMATICS AND THE LAWS OF NATURE
conservation of
momentum 80–82, 90
integrals in studies 85
in Newton’s studies 84
Wallman, Henry 215c
water 58–59, 65–69, 145
Waterloo, battle of 209c
water turbines 122–123
Watt, James 116, 119, 207c
Watt steam engine 116, 119
Watzenrode, Lukasz 42, 44
“wavelets” 216c
weather 121–122, 189–200
Weierstrass, Karl 211c
weight 33, 34, 36, 69, 105–106
Weinberg, Wilhelm 173
Wessel, Caspar 208c
Whitehead, Alfred North
212c
Wiles, Andrew 216c
work
in Carnot engine 124,
126
conversion of heat to
132–133
and dissipation of heat
140
force and distance in 129
and heat 114, 121, 128
and translational
symmetry of energy
conservation 151
World Meteorological
Organization 190
World War I 213c
World War II 155, 214c
X
x-axis 41
Y
y-axis 41
Z
Zeno of Elea 201c
Zermelo, Ernst 212c