In 1824 a young Norwegian named Niels Henrik Abel proved conclusively that algebraic equations of the fifth order are not solvable in radicals. In this book Peter Pesic shows what an important event this was in the history of thought. He also presents it
Vedi indiceIndex
Abbati, Pietro, 82
Abel, Niels Henrik, 1–3, 85–102,
189n–190n
Abelian Addition Theorem,
151, 200n
abelian equations, 98–100
abelian groups, 112–113
Abelian integrals, 151, 200n
Abel-Ruffini Theorem, 1–3, 89–94,
155–170, 200n
and Cauchy, 87, 93–94, 96
early life, 85–89, 189n
formulas of, 88
and Galois, 105, 108–109,
130–131, 145
and Gauss, 88–89, 95, 151
and Hamilton, 133
illness and death of, 101–102, 190n
notebooks of, 97, 152–153, 200n
and Ruffini, 87–89, 97, 190n
travels in Europe, 95–97
Acad´emie des Sciences, 96, 104–106
Accounting. See Bookkeeping
Algebra
Arabic, 23–28, 45, 54
coefficient, 1, 44–45, 91–93
noncommutative, 131–143
roots, 1, 92, 98
symbolic notation, 40–45, 132
unknown, 43
variable, 1, 44
Algebraic functions, 90–91
al-Khw¯arizm¯ı, Muhammed
ibn-Musa, 25–26, 30, 183n–184n
Alogon, 9
Alternating groups. See Groups
Analytic mathematics, 42, 59
Anrta, 9
Anticommutation, 134–136, 195n
Apollonius of Perga, 42, 51, 59
Aporia, 14, 182n
Archimedes, 32, 60
Area problem, 61–66
Aristotle, 7, 181n
Arithmos, 9, 181n
Athens, 15
Ausdehnungslehre (Grassmann),
135–136
Ayoub, Raymond, 189n, 192n–193n
Babylonian mathematics, 5, 7,
24–25, 30, 183n
Bacon, Francis, 182n–183n
Basham, A. L., 181n
Beaumarchais, Pierre Augustin
de, 200n
206 Index
Bell, Eric Temple, 191n
Bernoulli, Daniel, 65
Bolyai, J´anos, 133, 196n
Bombelli, Raphael, 54–55, 187n
Bookkeeping, double-entry, 27–29,
184n–185n
Boole, George, 132–133, 195n
Boolean algebra, 132–133
Bourbaki, Nicolas (pseudonym),
188n, 195n
Boyer, Carl, 182n–195n
Bring, E. S., 67, 188n
Brioschi, Fernando, 146, 198n
Brizio, Anna Maria, 184n
Brown, R. G., 184n
Bryce, R. A., 189n
Bühler, W. K., 189n
Bulletin (Baron de F´erussac), 96
Bürgi, Jost, 48–49
Burkert, Walter, 1981n
Burn, R. P., 193n
Burnside, William Snow, 190n,
192n–193n
Cajori, Florian, 186n
Calculators, 78, 149–150
Calculus, 60, 62–63
Cantor, Georg, 150, 199n
Cardano, Girolamo, 30–40, 54, 57,
67, 69, 185n
Cartier, Pierre, 197n
Cauchy, Augustin, 83, 87, 96,
104–105
Cauchy’s theorem, 93, 163, 166,
175–180, 201n
and commutativity, 132, 195n
Causality and noncommutativity,
142, 196n
Cayley, Arthur, 112, 126,
136–138, 195n
Cayley numbers, 137
Cayley tables, 112, 119, 122
Cervantes, Miguel de, 23
Charles X, 105
Chatfield, Michael, 185n
Chinese mathematics, 183n
Christiania. See Oslo
Cipher, 27–28, 33, 42
Circle, 62
Code. See Cipher
Coleridge, Samuel Taylor, 135
Commensurable, 7–8, 16
Commercial arithmetic, 27–31
Commutativity, 99, 112
“Completing the cube,” 36–37, 120
“Completing the square,”
25–26, 113
Computers, 147, 197n–198n
Conic sections, 42
Connes, Alain, 197n
Continuum, 11, 18
Cooke, Roger, 200n
Cosa (coss), 27, 44
Crelle, August, 95–97, 100, 102
Crelle’s Journal, 95, 101
Cross product. See Multiplication
Cube, 5–6
symmetries of, 121
Cyclic groups and symmetries, 113,
117, 122, 195n
d’Alembert, Jean Le Rond, 68
Dance, 111–130
Dauben, Joseph Warren, 199n
Dedekind, Richard, 183n, 199n
Dehn, Edgar, 192n
del Ferro, Scipione, 32–34
del Ferro–Cardano–Tartaglia
method, 32–35, 48–49, 54–55,
77, 174
De Moivre, Abraham, 149, 197n
DeMorgan, Augustus, 132, 195n
Descartes, Ren´e, 50–59, 68, 187n
and conic sections, 64
Index 207
Descartes’s rule of signs, 53
La Geometrie, 50–58
“relativity” of roots, 57, 140
Dickson, Leonard E., 192n
Dimension (algebraic), 50–51
Dirac, Paul, 141
Disquisitiones Arithmeticae (Gauss),
79, 189n
Dodecahedron, 5–6
symmetries of, 124–125, 126–130
Don Quixote (Cervantes), 23
D¨orrie, Heinrich, 188n–190n
Dunham, William, 185n, 199n
Duplication of cube, 196n
e, 28, 150, 184n, 199n
´ Ecole Polytechnique, 105
´ Ecole Pr´eparatoire ( ´ Ecole Normale
Superieure), 105
Edwards, Harold M., 191n
Einstein, Albert, 140, 143
Eisenstein, E. L., 184n
Elements. See Euclid
Equations, algebraic
approximate solutions, 66,
147, 198n
cubic, 3, 28, 30–37, 90, 113–120,
148–149, 185n
general formulation, 1–3
quadratic, 2, 23, 25–26, 64, 90–91,
111–113, 185n
quartic, 2, 35, 38–39, 76–78,
120–122
quintic, 2–3, 77–78, 91–99,
122–129, 198n
roots, 1
Erlangen Program (Felix Klein),
138–140, 196n
Euclid, 5, 17–23, 42, 59, 145–146,
150, 183n
Euclidian geometry, 139
Eudoxus, 17–18, 183m
Euler, Leonhard, 62, 68, 90,
149, 196n
Ewald, William B., 187n, 195n, 199n
Fauvel, John, 184n–185n
Fearnly-Sander, Desmond, 195n
Fermat’s Last Theorem, 87–88,
189n–190n
Ferrari, Ludovico (Luigi), 34–35,
37–39, 57, 69, 76, 122
Fibonacci. See Leonardo of Pisa
Field, J. V., 186n, 199n
Fields (mathematics), 139
Fields (physics), quantum theory
of, 142–143
Fine, Benjamin, 188n
Fine structure constant, 197n
Fontana, Niccol` o. See Tartaglia
Fractions, 7
France, 45, 96–97, 102–108, 190n
Fundamental Theorem of Algebra,
56, 68–73, 79, 146, 188n
Galilei, Galileo, 49–50, 187n
Galois, ´ Evariste, 102–109,
190n–191n
and Abel, 105–106, 108–109,
130–131, 145
and Cauchy, 104–105
death of, 106–108
education of, 102–106, 190n
and his father, 104–105
Galois theory, 125–130, 191n–193n
legend of, 108, 191n
posthumous writings of, 108
and Soci´et´e des Amis du Peuple,
106–107
and St´ephanie Poterin-
Dumotel, 106
G˚arding, Lars, 190n
Gauge fields, nonabelian,
142–143, 196n
208 Index
Gauss, Carl Friedrich, 70–74, 97,
187n–189n
and Abel, 89, 95, 100, 151
and commutativity, 131–132, 195n
and unsolvability of quintic, 79, 88
Gazal´e, Midhat, 183n
Gel’fond, A. O., 197n
Gentzen, Gerhard, 197n
Geometrie, La (Descartes),
50–54, 187n
Geometry, 50, 60, 66
Germain, Sophie, 104
Gibbs, Josiah Willard, 136
Gibbs, W. Wayt, 195n
Gies, J. and F., 184n
Girard, Albert, 51, 56, 68, 187n
Girard’s identities, 61, 92
Gleason, Andrew, 187n
God, 49, 55
G¨odel, Kurt, 197n
“Golden ratio,” 28
Goldstine, Herman H., 198n
Gonz´alez de Posada,
Francisco, 198n
Gorman, Peter, 181n
Grafton, Anthony, 185n
Grassmann, Hermann,
135–136, 195n
Gray, Jeremy, 184n–185n, 197n
Great Art (Cardano), 30–40, 185n
Greek mathematics, 5–21
Greene, Brian, 196n
Gregory, Duncan, 132, 195n
Grossmann, Israel, 193n
Groups, 109, 111–130, 138–140,
193n–195n
A3, 118–120
A4, 121–122
A5, 123–129, 139
abelian, 112–113, 129
continuous, 140
cosets, 176
cyclical, 113, 117, 122,
175–180, 195n
definition of, 125–126
identity, 112, 119, 125
invariant subgroups, 119, 129
Lagrange’s Theorem, 128, 175–176
Lorentz, 196n
monster group, 130, 195n
nonabelian, 118, 129, 142–143
normal subgroups, 119, 129,
193n–195n
order, 176
and permutations, 175–180
philosophical aspects, 193n
quotient, 130, 176–177, 193n–194n
S2, 112–113
S3, 113–120, 139
S4, 120–122, 139
S5, 122–124
simple groups, 130
solvable chains of, 130, 194n–195n
V, 122
visualization of, 193n
Gu´erard, Albert, 190n
Hadlock, Charles Robert, 192n
Hamilton, William Rowan,
133–136, 196n
Hankins, Thomas L., 196n
Harmony of the World (Kepler), 48,
121, 124, 186n
Hartshorne, Robin, 181n,
191n, 194n
Heath, Thomas, 183n
Heisenberg uncertainty
principle, 141
Hellman, Morton J., 185n
Henry IV, 45
Heptagon, 48, 186n–187n
Hermite, Charles, 146, 150, 198n
Herrstein, I. N., 192n
Hexagon, 48
Index 209
Hilbert, David, 197n
Hippias of Mesopontum, 10
Hirano, Yo¨ıchi, 193n
Hoe, J., 183n
H¨older, Otto, 130–131, 177, 194n
Holmboe, Berndt Michael, 87, 97,
190n, 200n
Holy Spirit, 55
Huffman, C. A., 181n
Huntley, H. E., 181n
Hypergeometric functions,
198n–199n
Icosahedron, 5–6
symmetries of, 123–129
Incommensurability, 7–14
Indian mathematics, 9
Indistinguishability of quanta, 142
Infinity, 22, 146, 148, 151, 153
Institut de France, 100, 106
Invariance, 113, 139
Invariant subgroups. See Groups
Irrational magnitudes, 7–14, 19–21,
145–146, 183n
Irreducible case (cubic
equations), 54
Irreversibility, 141
Isograph, 147
Jacobi, Carl Gustav Jacob, 100, 146
Jacobson, Nathan, 192n
Jerrard, George B., 67, 133,
188n, 195n
Johnston, K. S., 184n
Jordan, Camille, 130–131, 133, 146,
177, 194n
“July monarchy,” 105–106
Kabbalists, 48
Kaku, Michio, 196n
Kant, Immanuel, 200n
Karl XIII, 85
Kemp, Christine, 96, 101–102
Kepler, Johannes, 48–49, 121, 124,
186n–187n
Khayy¯am, Omar, 30, 184n
Kiernan, B. Melvin, 193n
King, R. Bruce, 192n
Klein, Felix, 138–140, 143, 189n,
191n, 196n, 199n
Klein, Jacob, 182n, 187n
Kline, Morris, 182n, 192n
Knorr, Wilbur Richard, 182n
Kronecker, Leopold, 146, 190n, 198n
Lafayette, General, 105
La Geometrie (Descartes), 50–58
Lagrange, Joseph-Louis, 73–83,
87, 188n
Lagrange resolvent, 74–79
Lagrange’s Theorem, 128,
175–176, 194n
Lalanne, Leon, 147
La Nave, Federica, 187n
Laplace, Pierre Simon, 51, 80
Legendre, Adrien-Marie, 96,
100–101
Leibniz, Gottfried Wilhelm, 55,
65–67, 183n, 187n–188n
Le Lionnais, Fran¸cois, 199n
Le Mariage de Figaro
(Beaumarchais), 200n
Lemniscate, 65, 152–153, 200n
Leonardo da Vinci, 6, 28, 184n
Leonardo of Pisa (Fibonacci), 27–28,
30, 184n
Lieber, Lillian R., 192n
Lindemann, Ferdinand, 150, 199n
Liouville, Joseph, 133
Littlewood, D. E., 192n
Locus problem, 57, 59
Logos, 9
Louis XVI, 104
Louis XVIII, 104–105
210 Index
Louis-Philippe I, 105–106
Lyc´ee Louis-le-Grand, 104
Macve, Richard, 184n
Magnitudes, 7–8, 23
Magnus, Wilhelm, 193n
Malfatti, Gianfrancesco, 77, 82
Maor, Eli, 184n, 197n, 199n
Marinoni, Augusto, 184n
MathematicaTM, 198n
Matrix, 136–138
Maxfield, John E. and Margaret W.,
191n–192n
Maxwell, James Clerk, 135–136
Maxwellian dynamics, 141
Mayer, Uwe F., 188n
Mazur, Barry, 187n
Meno, 13–14, 182n
Mercantile Arithmetic (Widman), 29
Merzbach, Uta C., 182n–195n
Minkowski, Hermann, 140
Mitchell, David, 193n
Modular functions, 198n
Monster. See Groups
Montucla, Jean ´ Etienne, 79, 189n
Morduhai-Boltovsky, D., 197n
Multiplication
commutativity of, 131–132
Grassmann algebra, 135–136
matrix, 136–138
quaternion, 134
scalar product, 135
vector product, 135
Music, 7, 19–20, 48, 183n
Mutafian, Claude, 192n
Nahin, Paul J., 187n
Napoleon, 104, 108
Needham, Joseph, 183n
Newton, Isaac, 59–66, 149–150,
187n–188n
and Descartes, 59, 66
lemma 28, 61–66, 148
Newton’s identities, 60–61
Newton’s method, 66
Newtonian dynamics, 136, 141
Niven, Ivan, 199n
Nonabelian gauge fields,
142–143, 196n
Nonabelian groups. See Groups
Noncommutative geometry,
143, 197n
Noncommutativity, 99–100,
131–143, 195n
Normal subgroups. See Groups
Norway, 85
Numbers
algebraic, 146, 150, 197n
complex and imaginary, 54–56, 70,
148–149, 187n
counting, 9
in Greek mathematics, 9
irrational magnitudes, 7–8, 18–19,
23, 146
line, 51
negative, 51–54, 187n
octonions (Cayley numbers), 137
place value, 24
quaternions, 134–135, 196n
rational, 7–8, 146
sexagesimal, 24–25
transcendental, 62, 66, 150,
197n, 199n
ultraradical, 146, 150, 197n
Octahedron, 5–6
symmetries of, 121
Octonions. See numbers
Ore, Øystein, 185n, 189n
Oslo, 87, 101
Oval, 61–66
Pacioli, Luca, 6, 28–30, 184n–185n
Panton, Arthur William, 190n, 192n
Index 211
Pappus, 10–11, 42, 57, 182n
Parabola, 65
Parshall, Karen Hunger, 184n–185n
Pascal, Blaise, 147
Peacock, George, 132, 195n
Pentagon, 49
Permutations, 75–77, 82, 108–109,
111–130, 175–180
Pesic, Peter, 142, 182n–183n, 186n,
188n, 196n–197n
Peterson, Mark, 185n
Pi (π), 62, 150, 199n
Pierce, Benjamin, 138, 195n
Pierce, C. S., 138, 195n
Piero della Francesca, 28, 30, 185n
Pierpont, J., 190n
Planck, Max, 141, 196n
Plato, 11–17, 44, 140–141, 182n
Platonic solids, 5–6, 122, 138,
143, 193n
Poisson, Sim´eon–Denis, 101
Postnikov, M. M., 192n
Poterin-Dumotel, St´ephanie, 106
Pourciau, Bruce, 188n
Principia (Newton), 59–66, 187n
Pycior, Helena M., 186n–187n
Pythagoras, 5–11, 46, 181n
Pythagorean theorem, 11
Pythagoreans, 5–11, 15
Quantum theory, 141–143, 196
Quaternions. See numbers
Radicals, 2, 35
Ralph, Leslie, 181n
Rashed, Roshdi, 184n
Raspail, Fran¸cois-Vincent, 97, 106,
108, 191n
Rational magnitudes, 9
Ratios, 7
Reductio ad absurdum, 7–8, 64, 90
Relativity
and Galois theory, 140, 196n–197n
general, 143, 196n
of roots, 57, 140
special, 140
of space-time, 140
Republic (Plato), 15, 182n
Resolvent, see Lagrange resolvent
Richard, Louis-Paul- ´ Emile, 104–105
Roman law, 43
Roots of unity, 74, 97
Rosen, Michael, 190n, 200n
Rosenberger, Gerhard, 188n
Rothman, Tony, 191n
Royal Frederick’s University,
Christiania (Oslo), 87
Rta, 9, 181n
Ruffini, Paolo, 80–83
Sacrifice, 10, 46
Saigey, Jaques Fr´ed´eric, 96
Scalars, 135
Second Law of
Thermodynamics, 141
Seventeen-sided polygon, 70,
74, 189n
Shanker, S. G., 197n
Shurman, Jerry, 192n
Shylock, 27, 184n
Singh, Simon, 190n
Skau, Christian, 190n
Smale, Steve, 197n
Soci´et´e des Amis du Peuple,
106–107
Socrates, 13–17, 182n
Solomon, Ron, 195n
Solution in radicals, 2
Space
four-dimensional 135, 197n
n-dimensional, 135–136, 138
three-dimensional, 139–141, 143
Spearman, Blair K., 198n
Species, logic of, 44, 132
212 Index
Speed of light, 140, 142, 196n
Square, 7–14
Square roots, sound of, 20
Squaring the circle, 150, 196n
Stahl, Saul, 191n, 195n
“Standard theory” (physics),
142, 196n
Stein, Howard, 183n
Steiner, George, 191n
Stewart, Ian, 192n, 197n
Stillwell, John, 193n
Stubhaug, Arild, 189n–191n, 200n
Subgroups. See Groups
Suleiman II, 153
Summary of Arithmetic (Pacioli),
28–29
Swetz, Frank, 185n
Sylvester, James Joseph,
136–138, 195n
Symmetric groups. See Groups
Symmetry, 113
in algebraic expressions, 60
of fundamental particles,
142–143
of polyhedra (see Triangle;
Tetrahedron; Cube;
Dodecahedron; Icosahedron;
Octahedron)
of three-dimensional space, 124
Synthetic mathematics, 42–43
Tartaglia, 32–34, 185n
Taton, Ren´e, 191n
Taylor, R. Emmett, 184n
Tetractys, 10
Tetrahedron, 5–6
symmetries of, 120–121
Theaetetus, 15–17, 20, 182n
Theodorus, 17, 182n
Theology, Christian, 55
Thermodynamics, 141
Theta functions, 146, 198n
Third Law of Planetary Motion
(Kepler), 49
Tignol, Jean–Pierre, 192n
Time, irreversibility of, 141
Topology, 72
Torres Quevedo, Leonardo,
147, 198n
Torture, 16–17, 182n–183n
Toti Rigatelli, Laura, 190n–192n
Transcendental. See Number
“Treviso arithmetic,” 29
Trial, 18
Triangle, symmetries of, 113–120
Trigonometry, 47, 149–150, 186n
Trisection of angle, 187n, 196n
Tschebotar¨ow, N., 192n
Universal Arithmetic (Newton), 60
Uspensky, J. V., 185n, 188n
Vandermonde, Alexandre-
Th´eophile, 75, 188n
van der Waerden, B. L., 183n–184n,
188n, 193n
van Roomen, Adriaan, 45–47, 186n
Vectors, 135
Vernier, Hippolyte Jean, 104
Verriest, G., 192n
Vetter, Guido, 186n
Viète, Fran¸cois, 41–47, 56–58, 73,
132, 151, 186n
Voltaire, Fran¸cois Marie
Arouet de, 85
von Fritz, Kurt, 182n
von Tschirnhaus, Count Ehrenfried
Walter, 66–68, 188n
Tschirnhaus transformation, 68
Walker, D. P., 186n
Wallis, John, 186n
Weil, Simone, 183n
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