Abbati, Pietro, 82 Abel, Niels Henrik, 1–3, 85–102, 189N–190N

Index

Abbati, Pietro, 82 symbolic notation, 40–45, 132 Abel, Niels Henrik, 1–3, 85–102, unknown, 43 189n–190n variable, 1, 44 Abelian Addition Theorem, Algebraic functions, 90–91 151, 200n al-Khw¯arizm¯ı, Muhammed abelian equations, 98–100 ibn-Musa, 25–26, 30, 183n–184n abelian groups, 112–113 Alogon, 9 Abelian integrals, 151, 200n Alternating groups. See Groups Abel-Ruffini Theorem, 1–3, 89–94, Analytic mathematics, 42, 59 155–170, 200n Anrta, 9 and Cauchy, 87, 93–94, 96 Anticommutation, 134–136, 195n early life, 85–89, 189n Apollonius of Perga, 42, 51, 59 formulas of, 88 Aporia, 14, 182n and Galois, 105, 108–109, Archimedes, 32, 60 130–131, 145 Area problem, 61–66 and Gauss, 88–89, 95, 151 Aristotle, 7, 181n and Hamilton, 133 Arithmos, 9, 181n illness and death of, 101–102, 190n Athens, 15 notebooks of, 97, 152–153, 200n Ausdehnungslehre (Grassmann), and Ruffini, 87–89, 97, 190n 135–136 travels in Europe, 95–97 Ayoub, Raymond, 189n, 192n–193n Académie des Sciences, 96, 104–106 Accounting. See Bookkeeping Babylonian mathematics, 5, 7, Algebra 24–25, 30, 183n Arabic, 23–28, 45, 54 Bacon, Francis, 182n–183n coefficient, 1, 44–45, 91–93 Basham, A. L., 181n noncommutative, 131–143 Beaumarchais, Pierre Augustin roots, 1, 92, 98 de, 200n 206 Index

Bell, Eric Temple, 191n Cervantes, Miguel de, 23 Bernoulli, Daniel, 65 Charles X, 105 Bolyai, János, 133, 196n Chatfield, Michael, 185n Bombelli, Raphael, 54–55, 187n Chinese mathematics, 183n Bookkeeping, double-entry, 27–29, Christiania. See Oslo 184n–185n Cipher, 27–28, 33, 42 Boole, George, 132–133, 195n Circle, 62 Boolean algebra, 132–133 Code. See Cipher Bourbaki, Nicolas (pseudonym), Coleridge, Samuel Taylor, 135 188n, 195n Commensurable, 7–8, 16 Boyer, Carl, 182n–195n Commercial arithmetic, 27–31 Bring, E. S., 67, 188n Commutativity, 99, 112 Brioschi, Fernando, 146, 198n “Completing the cube,” 36–37, 120 Brizio, Anna Maria, 184n “Completing the square,” Brown, R. G., 184n 25–26, 113 Bryce, R. A., 189n Computers, 147, 197n–198n Bühler, W. K., 189n Conic sections, 42 Bulletin (Baron de Férussac), 96 Connes, Alain, 197n Bürgi, Jost, 48–49 Continuum, 11, 18 Burkert, Walter, 1981n Cooke, Roger, 200n Burn, R. P., 193n Cosa (coss), 27, 44 Burnside, William Snow, 190n, Crelle, August, 95–97, 100, 102 192n–193n Crelle’s Journal, 95, 101 Cross product. See Multiplication Cajori, Florian, 186n Cube, 5–6 Calculators, 78, 149–150 symmetries of, 121 Calculus, 60, 62–63 Cyclic groups and symmetries, 113, Cantor, Georg, 150, 199n 117, 122, 195n Cardano, Girolamo, 30–40, 54, 57, 67, 69, 185n d’Alembert, Jean Le Rond, 68 Cartier, Pierre, 197n Dance, 111–130 Cauchy, Augustin, 83, 87, 96, Dauben, Joseph Warren, 199n 104–105 Dedekind, Richard, 183n, 199n Cauchy’s theorem, 93, 163, 166, Dehn, Edgar, 192n 175–180, 201n del Ferro, Scipione, 32–34 and commutativity, 132, 195n del Ferro–Cardano–Tartaglia Causality and noncommutativity, method, 32–35, 48–49, 54–55, 142, 196n 77, 174 Cayley, Arthur, 112, 126, De Moivre, Abraham, 149, 197n 136–138, 195n DeMorgan, Augustus, 132, 195n Cayley numbers, 137 Descartes, René, 50–59, 68, 187n Cayley tables, 112, 119, 122 and conic sections, 64 Index 207

Descartes’s rule of signs, 53 Euler, Leonhard, 62, 68, 90, La Geometrie, 50–58 149, 196n “relativity” of roots, 57, 140 Ewald, William B., 187n, 195n, 199n Dickson, Leonard E., 192n Dimension (algebraic), 50–51 Fauvel, John, 184n–185n Dirac, Paul, 141 Fearnly-Sander, Desmond, 195n Disquisitiones Arithmeticae (Gauss), Fermat’s Last Theorem, 87–88, 79, 189n 189n–190n Dodecahedron, 5–6 Ferrari, Ludovico (Luigi), 34–35, symmetries of, 124–125, 126–130 37–39, 57, 69, 76, 122 Don Quixote (Cervantes), 23 Fibonacci. See Leonardo of Pisa Dörrie, Heinrich, 188n–190n Field, J. V., 186n, 199n Dunham, William, 185n, 199n Fields (mathematics), 139 Duplication of cube, 196n Fields (physics), quantum theory of, 142–143 e, 28, 150, 184n, 199n Fine, Benjamin, 188n Ecole´ Polytechnique, 105 Fine structure constant, 197n Ecole´ Préparatoire (Ecole´ Normale Fontana, Niccolò. See Tartaglia Superieure), 105 Fractions, 7 Edwards, Harold M., 191n France, 45, 96–97, 102–108, 190n Einstein, Albert, 140, 143 Fundamental Theorem of Algebra, Eisenstein, E. L., 184n 56, 68–73, 79, 146, 188n Elements. See Euclid Equations, algebraic Galilei, Galileo, 49–50, 187n approximate solutions, 66, Galois, Evariste,´ 102–109, 147, 198n 190n–191n cubic, 3, 28, 30–37, 90, 113–120, and Abel, 105–106, 108–109, 148–149, 185n 130–131, 145 general formulation, 1–3 and Cauchy, 104–105 quadratic, 2, 23, 25–26, 64, 90–91, death of, 106–108 111–113, 185n education of, 102–106, 190n quartic, 2, 35, 38–39, 76–78, and his father, 104–105 120–122 Galois theory, 125–130, 191n–193n quintic, 2–3, 77–78, 91–99, legend of, 108, 191n 122–129, 198n posthumous writings of, 108 roots, 1 and Société des Amis du Peuple, Erlangen Program (Felix Klein), 106–107 138–140, 196n and Stéphanie Poterin- Euclid, 5, 17–23, 42, 59, 145–146, Dumotel, 106 150, 183n G˚arding, Lars, 190n Euclidian geometry, 139 Gauge fields, nonabelian, Eudoxus, 17–18, 183m 142–143, 196n 208 Index

Gauss, Carl Friedrich, 70–74, 97, cyclical, 113, 117, 122, 187n–189n 175–180, 195n and Abel, 89, 95, 100, 151 definition of, 125–126 and commutativity, 131–132, 195n identity, 112, 119, 125 and unsolvability of quintic, 79, 88 invariant subgroups, 119, 129 Gazalé, Midhat, 183n Lagrange’s Theorem, 128, 175–176 Gel’fond, A. O., 197n Lorentz, 196n Gentzen, Gerhard, 197n monster group, 130, 195n Geometrie, La (Descartes), nonabelian, 118, 129, 142–143 50–54, 187n normal subgroups, 119, 129, Geometry, 50, 60, 66 193n–195n Germain, Sophie, 104 order, 176 Gibbs, Josiah Willard, 136 and permutations, 175–180 Gibbs, W. Wayt, 195n philosophical aspects, 193n Gies, J. and F., 184n quotient, 130, 176–177, 193n–194n Girard, Albert, 51, 56, 68, 187n S2, 112–113 Girard’s identities, 61, 92 S3, 113–120, 139 Gleason, Andrew, 187n S4, 120–122, 139 God, 49, 55 S5, 122–124 Gödel, Kurt, 197n simple groups, 130 “Golden ratio,” 28 solvable chains of, 130, 194n–195n Goldstine, Herman H., 198n V, 122 González de Posada, visualization of, 193n Francisco, 198n Guérard, Albert, 190n Gorman, Peter, 181n Grafton, Anthony, 185n Hadlock, Charles Robert, 192n Grassmann, Hermann, Hamilton, William Rowan, 135–136, 195n 133–136, 196n Gray, Jeremy, 184n–185n, 197n Hankins, Thomas L., 196n Great Art (Cardano), 30–40, 185n Harmony of the World (Kepler), 48, Greek mathematics, 5–21 121, 124, 186n Greene, Brian, 196n Hartshorne, Robin, 181n, Gregory, Duncan, 132, 195n 191n, 194n Grossmann, Israel, 193n Heath, Thomas, 183n Groups, 109, 111–130, 138–140, Heisenberg uncertainty 193n–195n principle, 141 A3, 118–120 Hellman, Morton J., 185n A4, 121–122 Henry IV, 45 A5, 123–129, 139 Heptagon, 48, 186n–187n abelian, 112–113, 129 Hermite, Charles, 146, 150, 198n continuous, 140 Herrstein, I. N., 192n cosets, 176 Hexagon, 48 Index 209

Hilbert, David, 197n Kemp, Christine, 96, 101–102 Hippias of Mesopontum, 10 Kepler, Johannes, 48–49, 121, 124, Hirano, Yo¨ıchi, 193n 186n–187n Hoe, J., 183n Khayy¯am, Omar, 30, 184n H¨older, Otto, 130–131, 177, 194n Kiernan, B. Melvin, 193n Holmboe, Berndt Michael, 87, 97, King, R. Bruce, 192n 190n, 200n Klein, Felix, 138–140, 143, 189n, Holy Spirit, 55 191n, 196n, 199n Huffman, C. A., 181n Klein, Jacob, 182n, 187n Huntley, H. E., 181n Kline, Morris, 182n, 192n Hypergeometric functions, Knorr, Wilbur Richard, 182n 198n–199n Kronecker, Leopold, 146, 190n, 198n

Icosahedron, 5–6 Lafayette, General, 105 symmetries of, 123–129 La Geometrie (Descartes), 50–58 Incommensurability, 7–14 Lagrange, Joseph-Louis, 73–83, Indian mathematics, 9 87, 188n Indistinguishability of quanta, 142 Lagrange resolvent, 74–79 Inﬁnity, 22, 146, 148, 151, 153 Lagrange’s Theorem, 128, Institut de France, 100, 106 175–176, 194n Invariance, 113, 139 Lalanne, Leon, 147 Invariant subgroups. See Groups La Nave, Federica, 187n Irrational magnitudes, 7–14, 19–21, Laplace, Pierre Simon, 51, 80 145–146, 183n Legendre, Adrien-Marie, 96, Irreducible case (cubic 100–101 equations), 54 Leibniz, Gottfried Wilhelm, 55, Irreversibility, 141 65–67, 183n, 187n–188n Isograph, 147 Le Lionnais, Fran¸cois, 199n Le Mariage de Figaro Jacobi, Carl Gustav Jacob, 100, 146 (Beaumarchais), 200n Jacobson, Nathan, 192n Lemniscate, 65, 152–153, 200n Jerrard, George B., 67, 133, Leonardo da Vinci, 6, 28, 184n 188n, 195n Leonardo of Pisa (Fibonacci), 27–28, Johnston, K. S., 184n 30, 184n Jordan, Camille, 130–131, 133, 146, Lieber, Lillian R., 192n 177, 194n Lindemann, Ferdinand, 150, 199n “July monarchy,” 105–106 Liouville, Joseph, 133 Littlewood, D. E., 192n Kabbalists, 48 Locus problem, 57, 59 Kaku, Michio, 196n Logos, 9 Kant, Immanuel, 200n Louis XVI, 104 Karl XIII, 85 Louis XVIII, 104–105 210 Index

Louis-Philippe I, 105–106 lemma 28, 61–66, 148 Lycée Louis-le-Grand, 104 Newton’s identities, 60–61 Newton’s method, 66 Macve, Richard, 184n Newtonian dynamics, 136, 141 Magnitudes, 7–8, 23 Niven, Ivan, 199n Magnus, Wilhelm, 193n Nonabelian gauge fields, Malfatti, Gianfrancesco, 77, 82 142–143, 196n Maor, Eli, 184n, 197n, 199n Nonabelian groups. See Groups Marinoni, Augusto, 184n Noncommutative geometry, MathematicaTM, 198n 143, 197n Matrix, 136–138 Noncommutativity, 99–100, Maxfield, John E. and Margaret W., 131–143, 195n 191n–192n Normal subgroups. See Groups Maxwell, James Clerk, 135–136 Norway, 85 Maxwellian dynamics, 141 Numbers Mayer, Uwe F., 188n algebraic, 146, 150, 197n Mazur, Barry, 187n complex and imaginary, 54–56, 70, Meno, 13–14, 182n 148–149, 187n Mercantile Arithmetic (Widman), 29 counting, 9 Merzbach, Uta C., 182n–195n in Greek mathematics, 9 Minkowski, Hermann, 140 irrational magnitudes, 7–8, 18–19, Mitchell, David, 193n 23, 146 Modular functions, 198n line, 51 Monster. See Groups negative, 51–54, 187n Montucla, Jean Etienne,´ 79, 189n octonions (Cayley numbers), 137 Morduhai-Boltovsky, D., 197n place value, 24 Multiplication quaternions, 134–135, 196n commutativity of, 131–132 rational, 7–8, 146 Grassmann algebra, 135–136 sexagesimal, 24–25 matrix, 136–138 transcendental, 62, 66, 150, quaternion, 134 197n, 199n scalar product, 135 ultraradical, 146, 150, 197n vector product, 135 Music, 7, 19–20, 48, 183n Octahedron, 5–6 Mutafian, Claude, 192n symmetries of, 121 Octonions. See numbers Nahin, Paul J., 187n Ore, Øystein, 185n, 189n Napoleon, 104, 108 Oslo, 87, 101 Needham, Joseph, 183n Oval, 61–66 Newton, Isaac, 59–66, 149–150, 187n–188n Pacioli, Luca, 6, 28–30, 184n–185n and Descartes, 59, 66 Panton, Arthur William, 190n, 192n Index 211

Pappus, 10–11, 42, 57, 182n and Galois theory, 140, 196n–197n Parabola, 65 general, 143, 196n Parshall, Karen Hunger, 184n–185n of roots, 57, 140 Pascal, Blaise, 147 special, 140 Peacock, George, 132, 195n of space-time, 140 Pentagon, 49 Republic (Plato), 15, 182n Permutations, 75–77, 82, 108–109, Resolvent, see Lagrange resolvent 111–130, 175–180 Richard, Louis-Paul-Emile,´ 104–105 Pesic, Peter, 142, 182n–183n, 186n, Roman law, 43 188n, 196n–197n Roots of unity, 74, 97 Peterson, Mark, 185n Rosen, Michael, 190n, 200n Pi (π), 62, 150, 199n Rosenberger, Gerhard, 188n Pierce, Benjamin, 138, 195n Rothman, Tony, 191n Pierce, C. S., 138, 195n Royal Frederick’s University, Piero della Francesca, 28, 30, 185n Christiania (Oslo), 87 Pierpont, J., 190n Rta, 9, 181n Planck, Max, 141, 196n Ruffini, Paolo, 80–83 Plato, 11–17, 44, 140–141, 182n Platonic solids, 5–6, 122, 138, Sacrifice, 10, 46 143, 193n Saigey, Jaques Frédéric, 96 Poisson, Siméon–Denis, 101 Scalars, 135 Postnikov, M. M., 192n Second Law of Poterin-Dumotel, Stéphanie, 106 Thermodynamics, 141 Pourciau, Bruce, 188n Seventeen-sided polygon, 70, Principia (Newton), 59–66, 187n 74, 189n Pycior, Helena M., 186n–187n Shanker, S. G., 197n Pythagoras, 5–11, 46, 181n Shurman, Jerry, 192n Pythagorean theorem, 11 Shylock, 27, 184n Pythagoreans, 5–11, 15 Singh, Simon, 190n Skau, Christian, 190n Quantum theory, 141–143, 196 Smale, Steve, 197n Quaternions. See numbers Société des Amis du Peuple, 106–107 Radicals, 2, 35 Socrates, 13–17, 182n Ralph, Leslie, 181n Solomon, Ron, 195n Rashed, Roshdi, 184n Solution in radicals, 2 Raspail, Fran¸cois-Vincent, 97, 106, Space 108, 191n four-dimensional 135, 197n Rational magnitudes, 9 n-dimensional, 135–136, 138 Ratios, 7 three-dimensional, 139–141, 143 Reductio ad absurdum, 7–8, 64, 90 Spearman, Blair K., 198n Relativity Species, logic of, 44, 132 212 Index

Speed of light, 140, 142, 196n Third Law of Planetary Motion Square, 7–14 (Kepler), 49 Square roots, sound of, 20 Tignol, Jean–Pierre, 192n Squaring the circle, 150, 196n Time, irreversibility of, 141 Stahl, Saul, 191n, 195n Topology, 72 “Standard theory” (physics), Torres Quevedo, Leonardo, 142, 196n 147, 198n Stein, Howard, 183n Torture, 16–17, 182n–183n Steiner, George, 191n Toti Rigatelli, Laura, 190n–192n Stewart, Ian, 192n, 197n Transcendental. See Number Stillwell, John, 193n “Treviso arithmetic,” 29 Stubhaug, Arild, 189n–191n, 200n Trial, 18 Subgroups. See Groups Triangle, symmetries of, 113–120 Suleiman II, 153 Trigonometry, 47, 149–150, 186n Summary of Arithmetic (Pacioli), Trisection of angle, 187n, 196n 28–29 Tschebotaröw, N., 192n Swetz, Frank, 185n Sylvester, James Joseph, Universal Arithmetic (Newton), 60 136–138, 195n Uspensky, J. V., 185n, 188n Symmetric groups. See Groups Symmetry, 113 Vandermonde, Alexandre- in algebraic expressions, 60 Théophile, 75, 188n of fundamental particles, van der Waerden, B. L., 183n–184n, 142–143 188n, 193n of polyhedra (see Triangle; van Roomen, Adriaan, 45–47, 186n Tetrahedron; Cube; Vectors, 135 Dodecahedron; Icosahedron; Vernier, Hippolyte Jean, 104 Octahedron) Verriest, G., 192n of three-dimensional space, 124 Vetter, Guido, 186n Synthetic mathematics, 42–43 Viète, Fran¸cois, 41–47, 56–58, 73, 132, 151, 186n Tartaglia, 32–34, 185n Voltaire, Fran¸cois Marie Taton, René, 191n Arouet de, 85 Taylor, R. Emmett, 184n von Fritz, Kurt, 182n Tetractys, 10 von Tschirnhaus, Count Ehrenfried Tetrahedron, 5–6 Walter, 66–68, 188n symmetries of, 120–121 Tschirnhaus transformation, 68 Theaetetus, 15–17, 20, 182n Theodorus, 17, 182n Walker, D. P., 186n Theology, Christian, 55 Wallis, John, 186n Thermodynamics, 141 Weil, Simone, 183n Theta functions, 146, 198n West, M. L., 183n Index 213

Weyl, Hermann, 140, 193n, 196n Widman, Johann, 29 Wiles, Andrew, 189n Williams, Kenneth S., 198n Wilson, Curtis, 193n Winternitz, Emmanuel, 184n Wussing, Hans, 188n–189n, 193n

Xenophon, 182n

Yaglom, I. M., 192n, 195n–196n Yamey, B. S., 184n

Zammattio, Carlo, 184n Zero, 9, 43