A Strange Wilderness
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INDEX
Note: Page numbers in italics include illustrations, photographs, and captions.
Achilles paradox, 19
Ahmes Papyrus, xix
Alberti, Leon Battista, 69
alchemy, 87, 104, 105, 118
Alexandria library, 28–29
Al Gabr Wa’ l Muqabala (al-Khwarizmi), 46, 47
algebra and geometry, 50, 109, 113, 253
Algebra (Khayyam), 50
Algebra (al-Khwarizmi), 73
algorithms, 34, 46
Alhazen, 51, 69
Almagest (Ptolemy), 32, 36
Analysis of the Infinitely Small to Understand Curved Lines (l’Hôpital), 140
Analysis Situs (Poincaré), 239
analytical mechanics, 146
Analytical Mechanics (Lagrange), 166, 168, 170
Anaxagoras of Clazomenae, 19–21, 20
Apollonius of Perga, 31–32, 36
Apprenticeship of a Mathematician (André Weil), 238
Aralar mountain range, ii–iv
Archimedes, 23, 25–26, 27, 33
Aristotle, 5, 46, 51, 74, 117
Arithmetica (Diophantus), 35–36, 48, 112
Arithmetical Investigation (Gauss), 152, 184
armillary sphere, 54–55
Arnauld, Antoine, 119, 122
Aryabhata, 42–43
Aryabhatiya (Aryabhata), 42
Astronomical Canon (Hypatia), 36
astronomy, mathematical, 173
attraction, law of, 131
automorphic functions, 240
Babylonia, xvii, 8
Baez, John C., 195
Ba illet, Adrien, 104, 107, 108
Banti, Cristiano, 94
Barbari, Jacopo de’, 72
Barber of Seville paradox, 212
Bartels, Johann Martin, 149
al-Battani (Albategnius), 48
Beeckman, Isaac, 103, 108
Belaval, Yvon, 116
bell curve, 151
Bell, Eric T., 151
Bernoulli, Daniel, 140, 141, 143
Bernoulli Inequality, 139
Bernoulli, Jacob, 138–39
Bernoulli, John, 139–40, 142–43
Bernoulli, Nicolas (father), 138
Bernoulli, Nicolas (son), 140, 141, 143
binomial theorem, 130, 149
al-Biruni, Abu Rayhan, 48, 49
Boineburg, Johann Christian von, 118, 120, 122
Bolyai, Wolfgang, 152
Bolzano, Bernhard (Bolzano-Weierstrass theorem), 206, 208
Book of Calculation (Fibonacci), 67
Book of Optics (Alhazen), 51, 69
Boole, George (Boolean algebra), 192
Bourbaki, Nicolas (Bourbaki group), 236–39, 242, 244–45, 251, 252
Boyer, Carl, 12, 65
Boyle, Robert, 122, 138
Brahe, Tycho, 88, 89, 90, 91
Brahmagupta, 44–46
Brahmasphuta Siddhanta (Brahmagupta), 44
Broglie, Louis de, 163
Brunelleschi, Filippo, 69
Brunswick, Duke of (Carl Wilhelm Ferdinand), 150, 156
Budget of Paradoxes (Morgan), 146
Buffon, Georges-Louis, 163
Cantor, Georg
background of, 203
death of, 216
education of, 204
in Harz Mountains, 200–201, 213
Leopold Kronecker and, 205, 211, 212, 213, 214
mental health of, 207, 212, 214–16
set theory and, 216
studying continuum hypothesis, 214, 215, 216, 217
studying infinity, 205, 206, 208–11, 212–13, 214
studying numbers theory, 204
teaching at University of Halle, 205, 206, 208, 215, 216
Cardano, Girolamo, 76–80, 79, 81
Carnot, Lazare, 165, 178
Cartan, Henri, 238, 239
Cartesian system, 109, 113, 124
Cartier, Pierre, 244, 246, 251, 254
Catherine II (Catherine the Great) of Russia, 145
Catherine I of Russia, 140, 143
Cauchy, Augustin-Louis, 184, 185
Cayley, Arthur, 192–93
Celestial Mechanics (Laplace), 172–73, 174
Cellarius, Andreas, 82–83
Chabauty, Claude, 238
chaos theory, 240
Charraud, Nathalie, 215
Chevalier, Auguste, 191
Chevalley, Claude, 238, 239
Chevreuse, Duke of, 122
Chinese-number theory, 58
Christina of Sweden, 110
Clerselier, Claude, 115–16, 124
Clifford, William Kingdon (Clifford algebra), 193
complex-function theory, 240
complex numbers, 81, 153, 195, 196
Concerning the Hindu Art of Reckoning
(al-Khwarizmi), 46
Condorcet, Marquis de, 145, 147, 163, 173
cones, 31–32
Conics (Apollonius), 36
conic sections, 31–32, 69, 89
conservation laws, 168–69, 227, 232, 233
continuous numbers, 206
continuum hypothesis, 208–11, 214, 215, 216, 217
Copernican heliocentric system, 82–83, 89, 92
counting rods, 57
Craig, John, 88
cube roots, 42
cubes, 23, 24
cubic equations, 68, 73, 74, 75, 76, 78, 80, 220
cycloids, 93
D’Alembert, Jean Le Rond, 171–72
da Vinci, Leonardo, 64–65, 73, 74
De Arte Combinatoria (Leibniz), 117
Dedekind, Richard, 201, 212
De Divina Proportione (Pacioli), 73
Delacroix, Eugène, 180–82
del Ferro, Scipione, 74, 75, 79
Delsarte, Jean, 238, 239
De Morgan, Augustus, 192
derivatives, 123, 124, 140
Descartes, René, 98–99
accused of libel, 110
background of, 100–101
Cartesian system and, 109, 113
Christina of Sweden and, 110
conic sections and, 31–32
death of, 115–16
Discourse on the Method by, 109
finding derivatives, 124
Gottfried Wilhelm Leibniz and, 116, 119, 124
Isaac Beeckman and, 103, 109
Johann Faulhaber and, 104
Marin Mersenne and, 101, 107
as solider, 102, 103, 105, 108
uniting geometry with algebra, 109, 113
using Jupiter symbol, 104–5
The World by, 107, 108
Description of the Marvelous Rule of Logarithms (Napier), 88
descriptive geometry, 176, 177
Diderot, Denis, 146, 171
Dieudonné, Jean, 238, 239
differential calculus, 125, 131
differential equations, 240
differential geometry, 154
dimension irrelevance, 213, 214
Diophantus (Diophantine equations), 35–36, 48, 112
Dirichlet, Peter Gustav Lejeune, 157
Discourse on the Method (Descartes), 109
dodecahedrons, 23, 24
doubling-the-cube problem, 21–22, 69, 182, 191
Dreyfus, Alfred, 243
Dunsink Observatory (Ireland), 195
Dürer, Albrecht, 69, 70, 71
Egypt, xix, 4–7, 9, 10
Ehresmann, Charles, 238, 251
Einhorn, Lutherus, 92
Einstein, Albert, 167, 229–32, 231
Eléments de la philosophie de Newton
(Voltaire), 114–15
Elements (Euclid), 27, 30, 36, 46
Elements of Mathematics (Bourbaki), 244, 245
Elisabeth, Princess of Bohemia, 109–10
ellipses, 31
; elliptic geometry, 152, 153
epicycles, 32
epicycloids, 71
Eratosthenes of Cyrene, 32–34, 33
Erlangen Program, 225
error, law of, 151
Euclid
Elements by, 27, 30, 36, 46
geometry of, 30–31
postulates of, 30–31, 51–52, 152
prime numbers and, 14, 149
Eudoxus of Knidus, 24–25, 26
Euler, Leonhard, 80, 136–37, 142–44, 145, 146, 147, 167
expectations, mathematical, 141
Faulhaber, Johann, 104
Ferdinand, Carl Wilhelm. See Brunswick, Duke of (Carl Wilhelm Ferdinand) Fermat, Pierre de, 31–32, 111, 112, 191
Ferrari, Lodovico, 80
Fibonacci (Fibonacci spiral, sequence), 65–69, 66
Fields Medal, 229, 253
Fior, Antonio Maria, 75–76
Flower, The (Fibonacci), 68
fluid dynamics, 146
fluxions method, 130
Fontana, Niccolò. See Tartaglia (Niccolò
Fontana) Fontenelle, Bernard le Bovier de, 125
forms theory, 193
Fourier, Joseph (Fourier analysis), 174–76
fractals, 242
fractions, 15, 18–19, 53, 67, 86
Fraenkel, Abraham (Zermelo-Fraenkel set theory), 216
Francesca, Piero della, 71
Franco, Francisco, 248, 249
Fréchet, Maurice, 242
Frederick II of Germany, 67–68, 103
Frederick the Great of Prussia, 140, 144–45, 169
French Revolution, 140, 162, 169, 173, 174, 177, 182, 188
Freundlich, Erwin, 230
Friedrich, Hans. See Hanover, Duke of (Hans Friedrich) Fuchs, Lazarus, 240
Fundamental Theorem of Algebra, 153, 154
Galileo (Galileo Galilei), 92–93, 94, 95, 99, 206
Galois, Évariste, 22, 182–84, 183, 185–92, 189, 229, 233
Gauss, Carl, 147–54, 148, 153, 155–57
general covariance, 232
gnomon, 45
golden ratio, 66, 73
GPS (Global Positioning System), 8, 113
gravity, 92, 130, 131, 133, 134, 172, 173, 230
Great Art (Cardano), 78, 79, 80, 81
Great Internet Mersenne Prime Search, 108
Grossmann, Marcel, 230
Grothendieck, Alexander
background of, 246–47
in concentration camps, 250, 254
elusiveness of, 234–35, 250, 256, 257
father of, 246–50
Harvesting and Sowing by, 246
mother of, 248–50
political activism of, 254–55
schemes and, 253
topological groups and, 250–51
topos and, 252
group theory, 182, 191, 224, 225, 226, 244
Gsell, Georg, 143–44
Hadamard, Jacques (Hadamard inequality), 242–43
Halley, Edmond, 131, 166
Hamilton, William Rowan, 194–97, 196
Hanover, Duke of (Hans Friedrich), 123, 126
Hardy, G. H., 221–24, 222
Harmonia Cosmographica (Cellarius), 82–83
harmonic series, 139
Harvesting and Sowing (Grothendieck), 246
Heath, Thomas, 8, 12
heat study, 175, 176
Henri IV of France, 84, 85, 86
Hilbert, David, 93, 216, 226, 228, 230–32
Hippasus of Metapontum, 18–19
Hooke, Robert, 132, 138
Horologium Oscillarium (Huygens), 123
House of Wisdom, 46, 47, 48
Hugo, Victor, 190
Humboldt, Alexander von, 156
Husson, Raoul, 236
Huygens, Christiaan, 123, 126, 132
Hypatia, 35, 36–37
hyperbolas, 31
hyperbolic geometry, 152, 153
hypergeometric series, 222
Iamblichus, 9
Ibn Qurra, Thabit, 41
Ibn Sina (Avicenna), 48
Ibn Yunus, 49, 88
icosahedrons, 23, 24
imaginary numbers, 50, 81, 137
inclined planes, 79
Indica (al-Biruni), 48
Infinite Hotel, 93–94, 206
infinite series, 132, 139, 149, 150, 212
infinite sets, 93–94
infinitesimal calculus, 124
Infinity and the Unconscious: An Essay on Georg Cantor (Charraud), 215
infinity study, 205, 206, 208–11, 212–13
Inquisition, 94, 95, 101, 107, 108, 206
integral calculus, 25, 27, 91, 92, 131, 139
intercept theorem, 6
invariants, 228, 229, 232, 233
irrational numbers, 19, 68, 202, 212
Jacobi, Carl Gustav, 186
Jantar Mantar observatory, 43, 44
Johannes of Palermo, 68
al-Karkhi, 48
al-Kashi, Jamshid, 52–53
Keill, John, 126
Kepler, Johannes, 24, 89–92, 90, 91, 130
Key to Arithmetic (al-Kashi), 53
Khayyam, Omar, 46, 50, 51
al-Khwarizmi, Mohammad ibn, 46–47, 65, 66, 67, 73
Kitab al-Tafhim (al-Biruni), 49
Klein, Felix (Klein bottle), 224–27, 225, 232, 240
Koenig, Jules C., 216
Kovalevskaya, Sofia, 202
Kronecker, Leopold, 202, 205, 206, 211, 212, 214
Kublai Khan, 59, 60
Kummer, Ernst Eduard, 206
Lacroix, Sylvestre, 187–88
Lagrange, Joseph (Lagrangian function), 166–70, 168, 179, 232, 233
Laplace, Pierre-Simon (Laplacian operator), 171–74, 172, 179
La Rochelle siege, 108, 109, 120
Last Supper (da Vinci), 73
Last Thoughts (Poincaré), 240–41
Lavoisier, Antoine, 169, 170
Laws of Thought (Boole), 192
least-squares method, 150, 164
Legendre, Adrien-Marie, 164, 179
Leibniz, Gottfried Wilhelm, 123
background of, 116
calculus and, 122, 124–26
Cartesian system and, 124
Christiaan Huygens and, 123, 126
Christopher Wren and, 122
conic sections and, 31–32
De Arte Combinatoria by, 117
death of, 127
De Principio Individui thesis of, 117
derivatives and, 123
Duke of Hanover and, 123, 126
Henry Oldenburg and, 123
inventions of, 120
Isaac Newton and, 114–15, 124–26, 133, 134, 135
Jacob Bernoulli and, 139
Johann Christian von Boineburg and, 118, 120, 122
John Bernoulli and, 139, 140
John Pell and, 122
monads and, 117
Nature’s Testimony Against the Atheists by, 119
New Methods for Finding Maxima and Minima by, 125
Philosopher’s Creed by, 123
philosophy of, 118, 119
in politics, 118
religious-unification plan of, 120–22, 127
René Descartes and, 116, 119, 124
Robert Boyle and, 122
studying areas, 92, 123
Theoria Motus Abstracti by, 119
Theoria Motus Concreti by, 119
Leonard of Pisa. See Fibonacci (Fibonacci spiral, sequence) Lévi-Strauss, Claude, 243
Lévy, Paul, 242
l’Hôpital, Marquis de, 139, 140, 163
Liberty Leading the People (Delacroix), 180–82
Lie, Sophus (Lie groups), 168, 194, 197, 229, 233
linear proportion, 71
linear transformations, 193
Liouville, Joseph, 191
Li Zhi, 59–61
logarithms, 87, 88
Lorentz, Hendrik (Lorentz group), 229
Louis Philippe of Orleans, 182, 186, 187
Louis XIII
of France, 108, 120
Louis XIV of France, 120–22, 121, 161, 162
Louis XVI of France, 169, 170
magic squares, 56–57, 70
al-Mamun, 45–46
Mandelbrojt, Szolem, 242
Mathematical Analysis of Logic (Boole), 192
Mathematical Canon (Viète), 84
Mathematical Principles of Natural Philosophy (Newton), 132–33, 134
Mathematician’s Apology (Hardy), 223
matrices theory, 192, 193
Melencolia I (Dürer), 70
Mercator, Nicholas, 132
Mersenne, Marin, 101, 107–8
Method of Mechanical Theorems
(Archimedes), 33
metric system, 164–65
Minkowski, Hermann, 228, 229
Mittag-Leffler, Gosta, 214
Möbius strip, 225
Mochus, 10
Moerbeke, Willem van, 79
Mögling, Daniel, 104
monads, 117
Monge, Gaspard, 176–79, 178