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The works of Euler will be cited according to Leonardi Euleri Opera Omnia sub auspiciis Soci-etatis Scientarium Naturalium Helveticae, Zurich und Basel 1911–1986 or by Enestrom index.
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[Euler E343] Euler L (1768) Lettres a une princesse d’Allemagne sur divers sujets de physique &de philosophie, written 1760, presented 1768. In: Opera Omnia III, 11 and 12
[Euler 344] Euler L (1768) Lettres a une princesse d’Allemagne sur divers sujets de physique &de philosophie, written 1760, presented 1768. In: Opera Omnia III, 11 and 12
[Euler E366] Euler L (1763) Institutionum calculi integralis, vol. 2, published 1763 and 1769. In:Opera Omnia I, 11
[Euler E385] Euler L (1770) Institutionum calculi integralis, vol. 3, published 1770. In: OperaOmnia I, 12
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Omnia III, 1[Euler E145] Euler L (1748) Recherches sur les plus grands et plus petits qui se trouvent dans les
actions des forces, written 1748, presented 1748[Euler E146] Euler L (1748) Reflexions sur quelques loix generales de la nature qui s’observent
dans les effets des forces quelconques, written 1748, presented 1748[Euler E176] Euler L (1750) Expose concernant l’examen de la lettre de M. de Leibnitz alleguee
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Index
Absolutemotion, Newton, 235time and space, 3time and space, Euler, 129, 184time and space, Newton, 7
Absolute and relativemotion, Newton, 114, 171time and space, Euler, 16, 49, 129, 165, 184,
207, 252, 264–265time, space and motion, Newton, 66
Absorption and emissionEinstein 1905 light quantum, 1906 specific
heat, 287Heisenberg 1925, 295light, energetic spectrum of atoms and
molecular aggregates, 285Planck 1900, hypothesis of
quantum emission, Hypothese derQuantenemission, 289
Schrodinger 1926, 287Algebra
Algebra Euler E387, 93, 110, 205,207, 226
Algorithmcalculus of differences, Euler E212, 223classical and quantum mechanics,
Wilczek, 10differential calculus, Euler E212,
222, 223fluents and fluxions, Newton, 224Nova methodus, Leibniz, 229–230
Ambiguitydefinition of relative velocity, 241
Analysis infinitorum, Euler E101, E102, 107Analytical expression
function, 151Analytically demonstrated
Euler’s program for mechanics, E015/016,111, 125
Ancient prototypeArchimedes, lever, 21–22geometry, Euclid, Heron, 25–28Klein, Zeuthen, 23Lagrange, limits, 32Leibniz, Specimen dynamicum, 38Newton, Euclid versus Descartes, 26rigour, Euler E212, § 85, 31Taylor on Cavalieri, 77Voltaire, 12–14Zeuthen, Klein, 23–24
Anleitung zur Naturlehre, Euler E842, 1, 8,108, 124, 127, 172, 220, 235, 236, 241,252, 253
Annihilation and creationLeibniz, creation and annihilation of
monads, 201Leibniz, generation of lines, surfaces and
solids, 65Newton, generation of lines, surfaces and
solids, 65Apposition of parts
Cavalieri, 120Leibniz, arithmetical series, 29Newton, excluded the procedure of
apposition of infinitesimal parts, 47Taylor, Method of Increments, 32
Archimedesarchimedean quantities, 22, 24, 135, 170,
191, 193, 200interpretation by Euler, E015/016, lever in
vacuum, 172interpretation by Leibniz, Specimen
dynamicum, 38Aristotle
divisibility in one way, two ways and threeways, 27
point, line, surface, body, 27
325
326 Index
Arithmetical foundation, 91, 93,140, 202
Arithmetical and geometrical ratioscalculus, Euler E212, 5, 12, 31, 65, 69, 102,
104, 112, 138, 148, 153, 159, 179, 199,203, 205, 207, 211, 216, 223, 225, 230,231, 238, 290, 314
rules for arithmetical ratios between finiteand infinitesimal quantities, Euler E212,121–122, 218
rules for geometric ratios between infinitelysmall quantities, the force of thecalculus, 97, 229
Arithmetical operationsaddition, subtraction, multiplication and
division, Algebra E387/388, 73, 85, 104,120, 204, 208, 228
addition, subtraction, multiplication anddivision, Rechenkunst E017, 73, 105,106, 112, 319
Arithmetical progressiondiscrete and continuous variables, Euler
E015/016, E212, 292–293discretization of motion, Euler E015/016,
E212, 133energetic states in quantum mechanics,
118, 275general scheme of increments and
differences, Euler E212, 215mechanics, Euler E015/016, 11, 13, 17, 38,
107, 118, 119, 123, 142, 150, 152, 160,171, 175, 189
with positive and negative numbers, 94, 232without infinitesimal real number zero,
quantum mechanics, 97Arithmetization
arithmetization of the calculus, Leibniz,Johann Bernoulli, Euler E212, 198–202
arithmetization of mathematics, Klein,31–32, 102, 175, 200, 312, 314
arithmetization of mechanics, EulerE015/016, 117–118
arithmetization of mechanics, Lagrange, 118historical development, Euler, Lagrange,
Cauchy, Weierstraß [Klein], 10, 38, 197Arthur
differentials as fictions, 204Leibniz’s differential calculus and theory of
motion between 1672 and 1677, 56, 125Assignable
assignable gaps, 51, 136, 201Assignable and inassignable
greater than any assignable number, EulerE212, 203
Leibniz, 1–32, 33–64, 65–100, 117, 120less than any assignable number, Euler
E212, 91, 219, 220, 222, 225, 231, 232,233, 273
syncategorematic interpretation, Arthur, 58,121, 136, 141, 193, 201, 231
Assignment, 144, 150, 164, 220Auflosungskunst, the art of resolution
1D, 2D and 3D motions, along a straightline, within a plane and in space,respectively, Euler, 131, 154, 155
Background independence, 35Barrow
continual motion, 25and Newton, 26
Berkeleycriticism of the calculus, 67, 96, 100on Newton, 67, 102
Bernoulli, Danielproposals for equation of motion, 38
Bernoulli, JohannBernoulli’s rule in Euler’s representation,
117, 205, 206, 211, 216, 226, 227, 229Brachystochrone problem, 105
Bewegungskrafte, 272, 274, 282Body
change of the state, 18, 20, 41, 52, 53, 76,125, 128, 134, 150, 178, 179, 186, 193,254, 259, 261, 262, 263, 274, 296
elastic, 44, 111, 237, 264extension, 62, 277and ghosts, 162impact, 56, 59, 62, 73, 129, 147, 149, 152,
177, 178, 278impenetrability, 44, 137, 150, 161–176, 188,
189, 253, 254inertia, 44, 55, 61, 64, 122, 124, 151, 160,
173, 189, 191infinitely small, 118, 121, 141, 302, 314influence by God and ghosts, 19interaction, at distance, 260interaction, impact, 24, 41, 55, 56, 59, 60,
73, 129, 136, 147, 176, 177, 178, 235,236, 278
length, breadth, height, 27, 74, 164, 218mass, heavy, 24, 42, 77, 269mass, inert, 44, 55, 61, 76, 150, 189, 299mass point, 133, 238, 274, 278, 282, 298mobility, 161–176, 254preservation of state, 16, 20, 41, 42, 63, 145,
179, 182, 186, 191resistance, 20, 169, 276rigid, 238
Index 327
state, 18, 261, 274steadfastness, 124, 135, 161–176, 254and vacuum, 99, 168, 170
Bohrcomplementarity, 155and Heisenberg, 66
Boltzmann, 285, 293Born
and Heisenberg, 66–67Bos
Leibniz’s foundation of the calculus, 30,81–91, 92, 182
Both . . . andconfigurations, 21, 25, 30, 48, 250, 264,
275–281, 294, 296, 298, 300–304, 314paths, principle of least action, 188, 296paths, Schrodinger, 189, 280, 287, 297, 298rest and motion, 2, 3, 17, 18, 19, 21, 42,
45, 61, 64, 105, 107, 123–129, 146,154, 159, 162, 168, 171, 172, 178, 181,184, 187, 188, 190, 238, 244, 254, 261,271, 274
Brachystochrone problem, 105Breadth
line is without breadth, 27, 78Brevis demonstratio [Leibniz], 34, 38, 73Bundle of paths [Schrodinger], 189, 280, 290,
299, 302
Calculuscriticism, Berkeley, 100criticism, d’Alembert, 141criticism, Lagrange, 224, 228criticism, Nieuwentijt, 36, 37, 38, 67, 86,
92, 100, 153, 201, 205of differences and sums, Leibniz 1680, 4,
66, 85, 88, 199, 206, 233differentialis, Euler 1727, 205, 206,
207–216, 218, 220, 222differentials, 32, 96, 97, 136, 138, 139, 142,
148, 150, 152, 157, 159, 193, 197, 199,200, 201, 203, 205, 206, 207, 208, 213,214, 226, 230, 293, 314
evanescent quantities, 30, 91, 92, 95, 142,203, 213, 227, 275
finite increments, 52, 55, 98, 138, 139, 205,206, 208, 216, 218, 222, 223, 274, 292
first and ultimate ratios, 52, 105, 140fluents and fluxions, 4, 5, 26, 38, 51, 67, 68,
70–77, 80, 81, 85, 113, 115, 132, 198,199, 211, 224, 290
historical development [Klein], 10, 38, 197limits, d’Alembert, 35
“omnis vis calculi versatur”, Euler E212,§ 85, 218
subject of the calculus, Euler E212, § 85, 68,72, 225
sums and differences, 37, 98of zeros, Lagrange, 224, 226
Cantoron actual infinity, 97, 205on differentials, 79, 96, 97, 207on the interpretation of Leibniz, 65, 79, 97,
98, 205Cauchy, 28, 35, 65, 91, 102, 104, 272Causality
classical interpretation, 42, 112, 155, 192,251, 270, 292, 293, 296, 303, 305, 307
quantum mechanical interpretation, 112,288, 295, 298, 300, 309, 314
Cause and effectHelmholtz’s comment on Mayer, 32interpretation of Helmholtz, 238interpretation of Leibniz, 37, 186interpretation of Mayer, 32, 195
Cavalieri, 27, 37, 67, 69, 73, 77, 78, 85, 100,120, 197, 201, 204
Change of motionincrement of velocity, 184increment of velocity is independent
of velocity, 72, 145, 150–153, 252,265, 274
in terms of change of momentum, Newton’s2nd Law, 54
in terms of change of velocity, Euler,Varignon, 55
Change of position, 36, 77, 158, 245, 249Change of state
by an external cause, 146, 171, 178, 187,191, 274, 296
by forces, 28, 41, 55, 57, 64, 116, 161, 162,174, 224, 254, 262
independent of state, Euler, 240, 243,296, 305
independent of state, Helmholtz, 296of rest or uniform motion, 166–167
Change of velocityby an external cause, 296by forces, 137, 141, 146, 262, 263independent of velocity, Euler, 72, 131, 145,
190, 262, 282independent of velocity, Helmholtz, 145,
150–153Cheated by the senses
Chatelet on relative motion, 12, 16–17, 18,22, 25, 64, 118, 182, 185, 248, 251, 252,257, 258
328 Index
Kastner on relative motion, 145, 184, 185,186, 241–243, 251, 252, 253, 261
Cipherdifferent signs to avoid confusion, 224–225,
230, 231Circle
and polygon, 53–54, 84, 144Coeunt, 114, 135, 174, 193Commercium Epistolicum, 69Compraesentia
and choice of a unit, Leibniz, 48–49and measurement, Leibniz, 48and perception, Leibniz, 48
Condon, 12, 269, 285, 286, 290Configurations
and conservation of energy, 283and extension of a system, 278–281and potential energy, 280of a system, 30, 279, 280, 281, 300–304,
306, 310of a system and paths of bodies, 178
Conservationof center of gravity, 148, 224of direction, 2, 174, 187of energy, 80, 115, 237, 278, 283, 287, 288,
297, 299, 305, 309of living forces, 36, 44, 53, 73, 112, 148,
161, 180, 194, 224, 269, 277, 309of mass, 160of momentum, 114, 153, 160, 258
Constants and variables, 68, 83, 91, 224, 225Contingent
and necessary, Leibniz, 177Continuity
and discrete things, Leibniz, 197of motion, 94, 112, 136, 142, 200, 217
Continuity, principle ofcontinuity of motion, exclusion of
interruption and jumps, Euler, 67continuity of motion, exclusion of leaps,
Leibniz, 217, 289extension to series of discrete numbers by
Euler, 217general form, Leibniz, 81Leibniz, 31, 37, 67, 74, 94, 125, 136, 195,
196, 230, 232, 289Robinson on Leibniz’s principle, the transfer
principle, 230Continuous
bodies as continuous things, Euler, 164creation of lines by the motion of points,
Newton, 74creation of mathematical quantities,
Newton, 7
Continuumcomposition of the continuum, Leibniz, 58divisibility, 164divisibility in infinity, 164indeterminate parts, Leibniz, 120
Contradictionprinciple of contradiction, 177
Countableenergetic values, 112ungereimt, die Mittelorte zahlen zu wollen,
Euler, 112Couturat, 9, 11, 13, 36, 62, 112, 160–161, 180,
237, 269
Dangicourt, 43, 65, 91letter to Dangicourt, Leibniz, 43, 91
Dead forcesand living forces, 15, 21–22, 64, 82, 161measure of dead forces, 60and statics, 22
Decrease and increaseof quantities, unlimited, Euler, 209
Decrements and increments, 77, 224Descartes
external causes, 2laws of nature, 2, 198motion in straight direction, 2principles of motion, 247“quantum in se est”, 3, 7, 20, 129, 170relative motion, 3, 7, 151, 244uniform motion, 239
Differential quotient, 52, 151, 225, 273, 274,275, 276
Differentialsdifferentio-differentials, 84, 86, 96, 136,
138, 159, 199, 219, 272, 290evanescent quantities, 30, 95, 142, 203,
227, 275of first, second and higher order, 214hierarchy, 96magnitude, 273
Divisibilityof bodies, 162, 164of bodies in inifinity, Euler, 162, 164
Divisibleand really divided, Leibniz, 47
Division, 27, 73, 83, 84, 85, 94, 98, 104,120, 123, 142, 166, 200, 204, 273,274, 299
Dynamics and static, 123, 127
Effects and cause, 44, 45, 181, 187, 193Eigenvalues, 287, 295
and quantization, 207
Index 329
Einsteinhypothesis of light quanta, 286quantization as selection problem, 309theory of relativity, 45, 240, 243, 298theory of specific heat, 287, 293
Either . . . orconfigurations, 21, 25, 30, 48, 250, 264,
275–281, 296, 297, 298, 300–310, 314paths, Schrodinger, 287rest and motion, 2, 3, 17, 18, 19, 21, 42, 45,
61, 64, 105, 107, 123–129, 154, 156,159, 162, 165, 168, 171, 172, 178, 181,184, 187, 188, 190, 238, 247, 248, 249,254, 261, 271, 274
Eleganceof the theory of fluents and fluxions,
Newton, 70–72Elements of bodies
Gedancken von den Elementen der Corper,220, 241
Ellipseand parabola, Leibniz, 225
Energykinetic, 35, 115, 149, 276, 277, 278, 279,
280, 281, 297, 301, 303, 304potential, 280, 283, 295, 297, 301, 303, 308total, 276–280, 287, 294, 295, 296, 297,
301, 303Equilibrium
of the lever, 170, 191Errors
and forces and time, Newton, 53, 76, 154Eternal, eternellement, 127, 191Euclid
Axioms, 75Proclus, 25–26
Euleranswer to Daniel Bernoulli, 153–159augmentation to infinity without limit, 216bodies and ghosts, 162bodies, properties, 1, 19, 58, 124, 126,
132, 147, 150, 171, 173, 186, 217,257, 261
calculus of differences, 16, 31, 32, 38, 56,138, 139, 157, 159, 196, 199, 205, 206,215, 217, 222, 223, 233, 290–292, 314
confidence in arithmetics and algorithms,56, 140
confidence in reason in comparison tosenses, 61, 141
criticism of the theory of monads, 107deep insight of Leibniz, 9, 15definition of mass, 25, 41, 148, 161–162
differential calculus, 32, 92, 93, 98, 111,112, 138, 139, 140, 153, 159, 167, 192,195, 205–208, 214, 215, 216, 218, 222,223, 290, 312
equation of motion, 17, 38, 80, 106, 152,183, 184, 190, 198, 214, 236, 238, 240,252, 257–259, 261, 265, 274, 275, 277,279, 281, 296, 297, 305
foundation of the calculus, 4, 11, 25, 30,31, 47, 65–100, 102, 104, 105, 117, 136,140, 141, 142, 156, 182, 195–233, 238,269, 272, 273, 291
foundation of mechanics, 8, 9, 19, 34, 40,56, 140, 161, 168, 175, 180, 181, 194,224, 225, 235, 236, 254, 269
freedom of ghosts, 133ghosts and bodies, 3, 189harmony between Maupertuis’ principles of
rest and motion, 21impenetrability of bodies, 172, 173infinite quantities of different magnitude,
84, 94, 228limit, without limit, 93, 216, 263“mistake of those”, infinite numbers of
different magnitude, 93“Mittelorte”, no counting of intermediate
positions, 112necessarily true, 141, 177, 290, 312observers, onlookers, Zuschauer, 17, 107,
183, 240, 255–256, 257, 258, 259origin of forces, E181, 107, 135, 150, 175principle of contradiction, 177principle of sufficient reason, 61, 171, 177,
178, 186, 238, 269quantity, 2, 5, 18, 27, 30, 41, 44, 48, 51, 52,
54, 56, 57, 60, 62, 67, 68, 71, 73, 74,75–97, 121, 144, 159, 160, 165, 174,176, 198, 200, 202, 203, 204, 205, 206,210, 211, 213, 214, 217, 218, 219, 225,226, 227, 228, 229, 230, 240, 263, 265,270, 273, 275, 276, 278, 295, 299, 301,302, 312
relative motion, 3, 7, 12, 16, 17, 19, 22,25, 36, 43, 46, 49, 51, 58, 62, 64, 107,114–118, 124, 145, 151, 169, 171, 174,175, 181–185, 193, 238, 240, 241–263
reliability of reason, 20, 35, 66, 124, 170,177, 190, 195, 196, 198, 202, 309
rigour of the ancients, 21, 77, 196, 314“Standhaftigkeit”, steadfastness, inertia,
126, 162, 167, 173, 253theory of gravitation, 1, 260theory of propagation of light, 19, 109,
163–164, 264
330 Index
theory of propagation of sound, 110,162–163
“Wirksamkeit”, efficiency, 20, 156, 253,254, 269–283
“Zuschauer”, observers, 17, 107, 183, 240,255–256, 256, 257, 258
Eulogyto Euler, Condorcet, 17, 183
Evanescent quantitiesBernoulli, 203Euler, E212, § 87, 30, 275Leibniz, 95, 142, 203, 213, 227, 275relation: “the finite evanesces with respect to
infinite”, 213, 227relation: “the infinitely small evanesces with
respect to finite”, 72, 142Extension
of a body, 59and configurations, 314and energy of the system of bodies,
Helmholtz, 296, 300as a property of bodies, Descartes, 128,
162, 172of a system of bodies, 16, 56, 59, 77, 155,
165, 169, 257, 271, 277–281, 289
Fictitious quantitiesdifferentials, Leibniz, 35, 79, 89, 91, 96,
97, 142Figures
no figures, Euler E212, 199no figures, Lagrange, Mecanique analytique,
118, 167, 175, 236, 313only one geometric line, Newton, 302
Finiteinfinitesimal, finite and infinite quantities,
hierarchy of, 77, 83, 199, 224Fleiss
“mit Fleiss ausgeschlossen”, influence ofghosts is excluded, Euler, 1, 3, 43, 124
Fluents and fluxions, 4, 5, 26, 38, 51, 70–77,80, 81, 85, 113, 115, 132, 199, 211,224, 290
Fluents, fluxions and moments, Newton, 5, 34,66, 71, 72, 202
Fluxcontinuous, Newton, 67, 89, 207, 292lines, surfaces and solid, generated by, 27,
50, 57, 71, 73, 74, 78, 85, 89, 113, 164time, generated by, 25, 49
FluxionsMethod of Fluxions, 25, 26, 34, 37, 47, 71,
72, 75, 76, 77–81, 82, 89, 94, 95, 142,197, 199, 244
Forceactive, 14–15, 20, 25, 61, 64, 82, 99, 125,
144, 173, 179, 185, 285active and passive forces, 14, 61, 64, 125,
173, 179analytical representation, 141, 277dead, 13, 21, 22, 60, 64, 66, 99, 125, 146,
149, 169, 181, 185, 202derivative, 24, 55, 58, 60, 61, 64, 67, 82,
114, 127, 129, 161, 177at distance, 193due to impact, 62, 129, 152, 178due to impenetrability, 177, 254of inertia, 14, 36, 44, 61, 62, 64, 82, 104,
125, 127, 128, 167–172, 173, 181, 184,185, 186, 187, 189, 191
inherent, 61–64, 82, 128, 179, 185, 190living, 2, 5, 7, 12, 13, 15, 21–24, 35, 36, 44,
45, 55, 57–59, 62–67, 79, 82, 99, 110,152, 153, 161, 176, 180, 181, 185, 194,238, 264, 269, 275, 276, 282, 294, 312
minimal forces to avoid impenetrability,principle of least action, 13, 20, 101,126, 150, 165, 188, 189, 254, 297
moving, 7, 14, 24, 36, 41, 45, 59–60, 61, 62,63, 64, 79, 82, 84, 114, 132, 146, 152,154, 155, 157, 165, 171, 175, 185, 187,189, 274
passive, 14, 61, 64, 82, 86, 99, 125, 178, 179primitive, 57, 61, 99, 113, 125, 127, 128,
177, 185primitive and derivative, 61, 64, 99, 125resisting, 61, 179
Forces and motionconstant force model, 53, 137, 282
Forces, origin ofgeneration of forces by bodies, 20, 60,
188, 270Freedom
of ghosts and impenetrability of bodies, 44,118, 133, 137, 171, 172, 189, 254
Functionanalytical expression, 72, 77, 151, 214, 223,
227, 247, 300constants and variables, 72, 79, 83, 95, 208,
225, 231independent and dependent variables, Euler,
5, 30, 51, 74, 79, 86, 115, 138, 142, 144,145, 151, 154, 158, 197, 198, 199, 202,207, 208, 213, 218, 274, 290, 300
GenitumNewton, 94
Index 331
Helmholtzconfigurations of a system, 308differential quotient, 56, 151, 238, 281, 286,
290, 294energy conservation, 80, 275, 277, 287, 288,
297, 299, 305extension of a system, 288–289Leibniz’s living forces, 44, 93potential energy, 295, 297, 301, 303,
306, 308Homogeneity
mathematically defined, 76mechanically defined, 113symbolismus memorabilis, Leibniz,
48, 214Hypotheses, 103, 190
Imaginary numbers, 229, 230Impenetrability
of bodies and freedom of ghosts,Euler, 189
independent of size and shape, 253magnitude of impenetrability, equal for all
bodies, 160, 203Increase and decrease
of quantities, unlimited, Euler, 86Increments and decrements
determinate or of constant magnitude, 273discrete increments and decrements and
continuous variables, 291of a function, 94, 138, 161, 214, 218, 223,
224, 232, 274geometric ratio of the increment of function
and the increment of variable, 225indeterminate or variable, 99, 223, 224,
241, 242and indivisibles, 78, 100, 211relation (geometric ratio) between
increments/decrements of a function anda variable, 79, 100, 149, 165, 214, 221,222, 225, 231, 232, 240, 259
of variables, 214, 274Increment of velocity
is independent of velocity, Euler, 72, 131,145, 150–153, 190, 262
is independent of velocity,Helmholtz, 282
Indeterminacyin the choice of independent variables,
75, 193Indivisibles, 21, 27, 28, 47, 51, 58, 66, 67, 69,
73, 75, 77, 100, 120, 136, 197, 200, 201,204, 211
Infinite degrees, 205
Infinitely little quantityone infinitely little quantity, O, Newton, 5,
67, 75, 197Infinite quantities
of different magnitude, Euler E387, xivInfinitesimal
quantities of different magnitude, 83, 93, 96,228, 231
Infinitum continuum vel discretumLeibniz, 88
Infinityas limit, Lagrange, 54, 223, 236
Instantaneous actionof forces, 137
Instant of time, 91Integration constants
mathematical interpretation, Euler,Wirksamkeit, 100
mechanical interpretation, Helmholtz,“living forces”, “work” and “totalenergy”, 277
Jesseph, 39, 90, 99, 200
Kastneron Euler’s mechanics, 122–123on relative motion, 180
Kepler, 43, 61, 75, 78, 102, 130, 143, 159, 162,168, 171, 172, 197, 244, 270
Keynes“magic was forgotten”, 37on Newton, 37, 40
Kinematicsand dynamics, 272–275
Kleinarithmetization of mathematics, 10, 24,
31–32, 99, 102, 196–200, 207arithmetization of mechanics, 28–31, 141,
175, 176, 195“Bahnbrecher”, break through by Newton
and Leibniz, 10, 31calculus of differences (Taylor, Euler) and
differential calculus, 215differentials, “less than any finite quantity,
but different from zero”, 230on Euler, Lagrange, Cauchy and
Weierstraß, 272on Euler’s concept of function, “like a
ferment”, 32formal foundation of mathematics by
Leibniz, 31–32logical consistency, 28naive intuition versus logical
consistency, 272
332 Index
polygon and circle, abstract logical point ofview, 39
principle of continuity, 31proposals to made use of Hamilton’s
theory, 63
Labyrinthlabyrinth of the contiuum, 201
Lacroixdifferential calculus and calculus of
differences, 206Lagrange
algebraic foundation of calculus, 119calculus of zeros, 224, 226foundation of mechanics by variational
principles, 12, 148mechanique analytique, 103, 167, 175,
196, 199mechanics as for dimensional geometry, 144
Lambert, 105, 176Language, 9, 14, 15, 16, 39, 42, 81, 90, 93,
99, 103, 119, 128, 162, 177, 182, 190,198, 208
Least partsof bodies, 141, 164, 166
Leibnizanswer to Nieuwentijt, 38best of all possible worlds, criticism by
Voltaire and Mach, 4, 180body is an extended resisting thing, 59dead and living forces, 13, 21, 64, 66, 82,
99, 125, 146, 181, 185differentials as fictitious quantities, 96,
136, 205evanescent quantities, 142excitation of internal forces by external
action of forces, 63fictitious quantities, 35, 79, 89, 91, 96, 136generation of lines by motion of points, 201on geometry and metaphysical
principles, 57impressions of dead forces, 82“Leges metaphysicas extensionis legibus
addendo”, 6, 20, 180living forces, measure of, 2, 5, 7, 12,
93, 149mechanics and phenomena described by
geometry and metaphysical principles, 3,37–38, 42, 43, 56, 58
monads, 43, 58, 107, 133, 141, 165, 174,239, 249
motion and time do not exist aswholes, 56
perspectives, 63, 174, 242, 249
possible and real worlds, 4, 188principles, 201rules for differentiation, 90, 210Specimen dynamicum, 38, 230Theoria motus abstracti, 38, 55, 136vacuum is an extended thing without
resistance, 59, 154, 165, 168Lever
Archimedes, 21, 22, 23, 24, 25, 36, 127Euler, 24, 147, 168, 170, 190, 191
Lichtenbergon Euler’s theory of mass points, 13
Lightvelocity, Einstein, 51, 239, 243velocity, Euler, 261, 266
Light quantumEinstein, 286, 299Planck, 286
Limitabsence of limit, Euler, 178approach to a limit, Lagrange, 35, 54,
122, 141d’Alembert, 122, 140Euler, 141, 226increase without limit, 278Lagrange, 228, 239polygon and circle as a model, 53–54,
84, 144veritables limites, Lagrange, 54, 226
LineEuclid, 26, 28, 73, 201Heron, generation of a line, 25Leibniz, generation of a line, 25“Mittelorte”, no counting of intermediate
positions, Euler, 112Newton, generation of a line, 73, 120one, two, three and four points defining
different geometrical objects, 134and plane, 27and point, 27and surface, 27
Living forcesand dead forces, 13, 21, 22, 62, 66, 82, 99,
125, 146, 181, 185and dynamics, 22, 276measure of living forces, Leibniz, 5, 12, 13,
93, 149
Machacceleration and mass, 114criticism of absolute space and time, 40mass, definition, “gegenuberstehende
Korper”, 181Newton’s physical principles, 260
Index 333
Massacceleration and mass, 114, 160critcism of metaphysics, 140definition of mass, Newton, 161mass, definition, “gegenuberstehende
Korper”, Mach, 181“the only exception is Euler”, Jammer, 160operational definition of mass, Euler,
159–160Mass point
difference between mass point andgeometrical point, Euler, 144
Maupertuisanalysis and criticism of the mechanical
principles of Descartes, Newton,Huygens, Leibniz and Euler, 11, 15
Euler’s representation and comments, 12,15, 21, 128
harmony between principles for rest andmotion, Euler, 184
minimal forces and least action, Euler, 13principle of least action, 106, 162, 163, 184,
236, 254, 296principle for rest, 184principle for rest and motion, 21
Measurementarithematics, representation of results by
arithmetical progression, Galileo, 29arithmetical differences between the terms
of the series, Galileo, 157division of distances and time intervals into
equal parts, Euler, 208division of distances and time intervals into
equal parts, Galileo, 28unit, choice of the unit, Leibniz, 48unit and object, Leibniz, 48unit and object, simultaneously present and
perceived, Leibniz, 48–49Mechanics
analytical, 116, 167, 215Eulerian and non-Eulerian, 262, 299
Melion the model of motion, 136
Methoddirect and indirect, 152, 188
Method of increments, 32, 208,215, 216
Modellingby thought experiments, 188
Modelsof the body, Euler, 132of the body, infinitely small body,
Euler, 118of the lever, Archimedes, 21–22
physics of models, Schrodinger, 285–310reaction of Lichtenberg to Euler’s model of
mass points, 13of the world consisting of a finite number of
bodies, Euler, 190of the world consisting of moving bodies
and moving observers, Euler, 261of the world to demonstrate mechanical
laws, Euler, 115Momentary increments
and differentials, Nova methodus, Leibniz,5, 85, 197, 210
Momentsfluents, fluxions and moments, Newton, 202
Monadscomment and reinterpretation by Euler, 133criticism by Euler, 107, 141distinguishing parts of the plenum, 164–165internal state of successions, 133least indivisible elements, 141, 239perspectives, 249
Motionabsolute, 17, 46, 47, 115–116, 118,
129, 142, 151, 169, 171, 182, 183,235, 238–241, 242, 252, 253, 256,259, 260
along a straight line, in a plane, in space, 80analytical representation of motion, 125,
133, 141–145, 155, 258–259apparent, “scheinbar”, 241change of position, 36, 77, 245change of state of rest or uniform motion,
21, 41, 53, 128, 150, 178, 190,247, 263
complete knowledge of motion, E842,§§ 21–25, 144
generates motion, Leibniz, 5, 57, 168,247, 251
non-uniform motion, 18, 118, 121, 134,136, 142, 143, 145, 152, 166, 167,256, 257
relative, 3, 7, 12, 16, 17–19, 22–25, 36, 43,46, 47, 49–51, 58, 62, 64, 107, 114–116,124, 145, 151, 159, 169, 171, 174–176,181–182, 184–185, 193, 238–259,261, 263
and rest, 16, 64, 114, 116, 125in terms of change of momentum, Newton’s
2nd Law, 54in terms of change of velocity, Euler,
Varignon, 154and time being no wholes, 132, 143, 240true, “wahre”, 253, 256
334 Index
uniform motion, 16, 18, 29, 31, 36, 40,41, 45, 47, 53, 54, 59, 64, 87, 88, 96,113, 114, 118, 121, 127, 131, 134,142, 143, 146, 150, 152, 157, 165,166–167, 174, 178, 190, 208, 238, 239,257, 263
Necessaryand contingent, Leibniz, 56, 139necessarily true, “non solum verum, sed
etiam necessario verum”, Euler, 141truth, geometric truth, Euler, 190–191
Neither . . . norpaths, 280rest and motion, 172, 187, 190
Newtonaccount of Commercium, 69activation of inertia by impressed
forces, 179axioms, 3, 29, 35, 40, 41, 45, 48, 97,
131, 140, 149, 160, 167, 178, 188,220, 296
comment on Leibniz’s calculus, 35Commercium, 34, 52, 66, 69, 79, 98,
120, 197on Descartes and Euclid, 6, 25–28, 39, 73,
74, 102on Descartes, Westfall, 1–32, 39, 41, 42,
59–60, 99, 112, 115, 130, 178, 179, 186errors, times and forces, 52–53fluents and fluxions, 4, 5, 26, 38, 70–72,
80, 81, 85, 113, 115, 132, 198,199, 211
force of inertia, 14, 36, 42, 62, 64, 82, 160,168, 171, 173, 185, 186
generation of lines by motion of points, 201genitum, 94impressed moving force, 44, 45, 59, 60, 63,
76, 79, 82, 113, 129, 140, 147, 167, 169,181, 185, 213, 269
on infinite numbers of different magnitude,93–94
integrated program of science and religion,Snobelen, 40
on “men of recent times”, 3, 39, 74“more elegant”, only ”one infinitely small
quantity, O, 5phenomena and forces, 43, 177, 244polygon and circle, modelling of the action
of central forces, 130ratios, first and last, 93rules for differentiation, 5, 214very beginning of motion, 51–53, 66,
81, 146
Nieuwentijtcriticism of the calculus, 37rejection of higher order differentials, 86Verelst, interpretation of Nieuwentij’s
theory, 96Nonstandard analysis
extension principle, 231rules for infinitesimal, finite and infinite
quantities, 122standard part principle, 231, 314transfer principle, 231
Nothingzeros, Euler, 223
Observersand bodies, 245and different perspectives of the world,
Leibniz, 245distinction between bodies and observers,
Euler, 252–253distinction between substances, Descartes,
4, 17–20making use of light for communication,
Euler, 263–266moving relatively to each other, Euler, 247relatively to each other resting and moving
observers, Euler, 261resting observers, 246, 248, 249, 256, 263separation of bodies and observers,
245, 249taking into account the finite velocity of
light, retardation of the observation ofevents, Euler, 264
union of bodies and observers, Euler’srelativistic theory, 241–243
union of bodies and observers, Leibniz,248–249
Occupation of placesby bodies, exclusion principle, 191
Orderof arithmetical progressions, 138, 208, 292of numbers, 50, 66of successions and coexistences, time and
space, 50Origin of forces
avoiding penetration, 53, 254, 297change of the state, 41, 52, 53, 126,
145–150impenetrability, 172minimal forces, 152, 262
Parabolacomposition of motions, Galileo, 28–31and ellipse, Leibniz, 225
Index 335
Pembertonon Newton’s studies of Descartes and
Euclid, 6, 26, 38Perpetual motion
Leibniz on perpetual motion, 45, 55Planck
action parameter, 289action parameter and classical physics, 289entropy and probability, 289, 293, 295hypothesis of quantum emission, 289scientific autobiography, 290universal parameter, 293
Planeforces, geometrically related to planes,
Euler, 156planes and straight lines, Euler, 155planes and straight lines in spaces of
different dimension, Ehrenfest, 155stability of motion and trajectories in 3D
space, Ehrenfest, 155Points
and line, surface and solid, 26, 27mathematical and physical points,
Euler, 165Polygon
and circle, Lagrange, 54, 129and circle, Leibniz, 75and circle as a model for straight and
curved, 84and circle, Newton, 129, 136circumscribed and inscribed, 54, 129–130
Positionabsolute, 242of a body, 113, 131, 252limited space, 113, 115relative, 49, 56, 124, 241, 249, 276
Potentiaconservation, neither less nor more potency
in cause and effect, Leibniz, 44, 55,153, 181
Potential energy, 254, 278, 279, 280, 295, 297,301, 306, 308
Principlesof contradiction, 16, 182, 177, 312external and internal, Euler E289, 20,
132–135external, 20, 44, 46, 97, 118, 126, 127,
145–153, 173, 247, 252general and fundamental principle of
mechanics, Euler E177, 178, 192internal, 44, 46, 114, 117, 118, 123–129,
132–133, 147, 149, 150–152, 170, 173,247, 252, 254
of least action, 13, 21, 53, 103, 106, 107,125, 148, 162, 163, 168, 184, 188, 224,236, 296
of sufficient reason, 61, 171, 177, 178, 186,238, 269
Proclusthe origin of the concept of flux, 25, 26–27
Quantification and ordertime and space, 50–51
Quantitas motus, 60, 62Quantities of the same sort
analytical treatment, Euler, 253, 276, 277space and time, Euler, 33, 45
Quantitydeterminate, 205, 224, 225finite, 11, 52, 54, 67, 71, 74, 79, 85, 86, 91,
95, 96, 98, 136, 140, 144, 148, 193, 196,202, 205, 206, 211, 215, 216–220, 225,227, 231, 265, 302, 312, 314
indeterminate, 91, 159, 223, 295infinite, 54, 73, 84, 96, 140, 196, 202, 204,
216–220, 223, 227, 228, 302infinite of different magnitude, 83, 93, 204,
217, 228, 232infinitesimal, 216–220infinitesimal of different magnitude, 83, 93,
94, 96, 204, 217, 232measurement, 218, 302
Quantizationaction parameter, Planck, 125, 240, 286,
287, 289, 295Bohr and Sommerfeld, 112differential equation, 262, 275, 286, 305as eigenvalue problem, Schrodinger, 287,
295, 309of energy, 308of energy or frequencies, 290matrix mechanics, 305as selection problem, Einstein, 295–296,
301, 306, 309Quantum numbers, 301
Ratiosarithmetic, 96, 225, 229first, 92geometric, 68, 72, 73, 79, 97, 121, 138, 139,
143, 218, 219, 225, 228, 229, 275, 302ultimate, last, 51, 52, 93, 92, 105, 136, 140,
197, 292Reaction principle, 181, 237Reason
sufficient, principle of sufficient reason, 61,171, 177, 178, 186, 238, 269
336 Index
RechenkunstRechenkunst and algebra, Euler, 104
Reichenbachon Leibniz and Newton, 16, 45, 46, 49, 240
Restabsolute, 47, 48, 64, 169, 170, 238, 244, 250change of rest and change of motion,
122, 127and dead and living forces, 13, 21, 64, 66,
82, 99, 125, 146, 181as evanescent motion, Leibniz, 72, 114, 125,
142, 213and motion, 2–3, 17–19, 21, 42, 45, 61, 64,
105, 107, 116, 117, 118, 119, 121–127,145, 146, 154, 159, 162, 165, 167, 168,171, 172, 178, 181, 184, 187, 188, 190,238, 247–249, 254, 261, 271, 274
relative, 22, 47, 64, 123, 169, 181, 184, 238,249, 250, 252, 257, 265
Rigourof the Ancients, Euler, 226, 269, 270of the Ancients, Lagrange, 196of the Ancients, Leibniz, 22, 57, 59,
125, 269of the Ancients, Newton, 3, 4, 5, 39, 74,
77, 269of the Ancients, Taylor, 216arithmetic, 196geometric, 196, 226
Robinsonon Leibniz’s principle of continuity, 37non-standard analysis, 65
Russelon the “popular” and the “esoteric” Leibniz,
13, 36
Schrodingeramplitude equation, 293, 294, 305, 306amplitude function, 281, 294configuration space, but not pq-space, 294,
305, 306“full force of logical opposition”, 189,
297, 298on intuition and methods of transcendental
algebra, 103quantization of energy or frequencies, 294quantization as selection problem, 304–310on recommendations concerning Hamilton’s
theory by Klein, 63, 298relation to Heisenberg’s matrix
mechanics, 305on the relations between classical and
quantum mechanics, 151, 286, 297“selbsttatig aussondern”, 307
transfer of classical concepts into quantummechanics, 31
undulatory mechanics, 294, 299, 304wave equation, 287, 300, 306wave function, 267, 280, 287, 294, 296, 297,
300, 301, 303, 312Selection problem
in classical mechanics, 297energetic spectrum of molecular aggregates,
quantization as selection problem,Einstein, 304
in quantum mechanics, 298Senses and ratio
cheated by our senses, Chatelet, 248reliability of ratio, Euler, 127
Shadowspace separated from time, 144, 239, 267time and space separated from each other
are mere shadows, Minkowski, 267the union of space and time,
Minkowski, 144Smolin
background independence, 154Sound
propagation of sound and propagation oflight, 163
Space and timeabsolute, 245, 260absolute, Newton, 45, 239, 240, 258,
261–263Leibniz, 9, 16, 17, 35, 47–52, 48, 51–52,
188, 214and motion, 8, 16, 35, 40, 46, 49, 64, 116,
181, 236, 239, 241as orders, 49, 89quantification and order, 50–51quantities of the same sort, Euler, 48relational definition, Leibniz, 45, 240, 248relational theory, “newly come”, Leibniz,
50, 249, 251relative, Newton, 115spatiolum, Euler, Wolfers, 275
Spatiolumand tempusculum, 75, 224, 293“Zeittheilchen”, 275
Stadium, 49, 159, 242, 244, 246State
changed by an external cause, 2, 127, 135,146, 150, 151, 171, 178, 186, 187, 190,191, 274, 296
changed by forces, 126, 193, 194, 262change of the state, mutually and
simultaneously, 191
Index 337
change of velocity is independent ofvelocity, Euler, 72, 131, 145
change of velocity is independent ofvelocity, Helmholtz, 273, 274, 282
conservation of state and change of position,170, 187
of rest and motion, 2, 3, 18, 19, 21, 42,117, 121, 123–127, 132, 145, 146,190, 274
the smallest body can change the state of thelargest body, 203
Staticsancient prototype of science of
equilibrium, 59and dynamics, 22, 24, 118, 123, 127, 146
Sufficient reasonfor changes in nature, 61principle, 61, 171, 177, 178, 182, 186, 238,
269, 312Syncategorematic, 58, 121, 135, 136, 141, 193,
201, 231
Tayloron Cavalieri and the perfection of ancient
geometers, 77“Grenzubergang von unerhorter Kuhnheit”
[Klein], 32making use of infinitesimal quantities, 197method of increments, 216
Tempusculumand spatiolum, 75, 224, 275, 293“verschwindendes Zeittheilchen”,
Helmholtz, 275“Zeittheilchen”, 275
thought experimentsEuler’s models, 188
Timeabsolute, 3, 7, 36, 45, 46, 74, 76, 115, 129,
181, 184, 235, 238–240, 252, 258, 260,261, 263, 264, 265, 270
and motion, 47, 89, 239, 297relational definition, Leibniz, 45–46and space, 3, 5, 7, 12, 16, 25, 36, 43, 45,
46–49, 50, 59, 87, 89, 115, 119, 128,129, 132, 143, 165, 166, 174, 183, 184,207, 225, 235, 238–240, 252, 257, 260,265, 267, 274
tempusculum, Euler, Wolfers, 275generated by a continual flux, Newton, 25as order of successions, Leibniz, 9, 89Planck’s time defined by universal
parameters, 287–288in quantum mechanics, 302Zeittheilchen, Wolfers, Helmholtz, 275
Time and spaceand geometry, Euler, 132mechanics as a four dimensional geometry,
Lagrange, 144Transcreation, 51, 132, 201Transfer principle
Robinson on Leibniz’s principle ofcontinuity, the transfer principle,230–231
Translationand motion, 113, 121
TropfkeGeschichte der Elementarmathematik, 200
Truncationof a system of differential equations,
mathematical procedure and mechanicalinterpretation, 159, 208, 306
Truthcontingent and necessary, 177
Unassignable, 51, 90–91, 112, 136, 199–200,218
Uncertainty relation, 19Units, 48, 212, 264, 273
Vacuumbody and vacuum, 99vacuum as a part of a mechanical
model, 157vacuum and plenum, 12
Variables and constantsvariables and functions, 224
Varignoninfinitesimals and uniform motion, 225time, space, motion and forces, 34, 38, 56
Velocitychanged by an external cause, 127changed by forces, 133, 139, 141,
259, 261change of velocity is independent of
velocity, Euler, 72change of velocity is independent of
velocity, Helmholtz, 273as a fluent, 42, 76, 77,
79, 80relative velocity, 22, 247,
249, 256, 257, 258, 263,264, 265
Verelstinterpretation of Nieuwentijt’s
theory, 96Vermeulen, 96, 204Veteres
rigour, Cavalieri, 197
338 Index
rigour, Euler, 226rigour, Lagrange, 196rigour, Leibniz, 57rigour, Newton, 77rigour, Taylor, 196rigour, Voltaire, 180static, 270
Voltairecriticism of metaphysics, 9a Frenchman in London, 12on Newton and Leibniz, 8
Wallis, 38, 201Westfall
Newton’s relation to Descartes, 3Weyl
on the relations between mathematics andphysics, 236
Whittakertreatment of differential equations, 307–308
Wilczekalgorithms in classical and quantum
mechanics, 288missing algorithms in classical
mechanics, 35
Zeittheilchen“verschwindende Zeittheilchen”, Helmholtz,
274Wolfers, 275
Zeno, 42, 49, 93, 96, 159, 165, 242, 243, 244,246
Zerocalculus of zeros, 224, 226
Zeuthen, 23
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