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Mathematics of Computation

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Machines for solving algebraic equations


Author: J. S. Frame
Journal: Math. Comp. 1 (1945), 337-353
MSC: Primary 65.0X
DOI: https://doi.org/10.1090/S0025-5718-1945-0011196-2
MathSciNet review: 0011196
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    J. A. de Segner, “Methodus simplex et universalis, omnes omnium aequationum radices detegendi,” Akad. Nauk, S.S.S.R., Leningrad, Novi Commentarii Acad. Sc. Imp. Petrop., v. 7, 1758, p. 211-226. J. Rowning, “Directions for making a machine for finding the roots of equations universally, with the manner of using it,” R. So. London, Trans., v. 60, 1770, p. 240-256. R. Mehmke, “Numerisches Rechnen,” Encykl. d. Math. Wiss., v. 1, part 2, 1902, p. 1067-1073. French ed. by M. d’Ocagne, Encycl. d. Sci. Math., tome 1, v. 4, fasc. 3, 1909, p. 339-340, 432-446. R. E. Moritz, “Some physical solutions of the general equation of the $n$th degree,” Annals Math., s. 2, v. 6, 1905, p. 112-126. I. Ghersi, Matematica dilettevole e curiosa, (Manuali Hoepli), Milano 1913, p. 244-249, “Metodi fisici per la soluzione dei sistemi di equazioni algebriche." P. Riebesell, “Über Gleichungswagen,” Z. Math. Phys., v. 63, 1914, p. 256-274. J. L. Lagrange, Leçons élémentaires sur les Mathématiques, 1795. English translation: Lectures on Elementary Mathematics, Chicago, Open Court, 1898, p. 124-126. E. Lill, “Résolution graphique des équations numériques d’un degré quelconque à une inconnue,” Acad. d. Sci., Paris, Comptes Rendus, v. 65, 1867, p. 854-857; Nouv. Annales Math., s. 2, v. 6, 1867, p. 359-362. L. Cremona, Calcolo Grafico, Turin, 1874. English translation: Graphical Statics, Oxford, 1890, p. 70-76. H. Cunynghame, “On a mechanical method of solving quadratic and cubic equations, whether the roots be real or impossible,” Phil. Mag., s. 5, v. 21, 1886, p. 260-263. R. Mehmke, “Über einen Apparat zur Auflösung numerischer Gleichungen mit vier oder fünf Gliedern,” Z. Math. Phys., v. 43, 1898, p. 338-340. A. B. Kempe, “On the solution of equations by mechanical means,” Messenger Math., s. 2, v. 2, 1873, p. 51-52. T. H. Blakesley, “Logarithmic lazytongs and lattice-works,” Phil. Mag., s. 6, v. 14, 1907, p. 377-381. T. H. Blakesley, “A kinematic method of finding the roots of a rational integral equation . . . ,” Phil. Mag., s. 6, v. 23, 1912, p. 892-900. R. F. Muirhead, “A mechanism for solving equations of the $n$th degree,” Edinburgh Math. So., Proc., v. 30, 1912, p. 69-74. M. Näbauer, “Vorrichtung zur Auflösung eines linearen Gleichungssystems,” Z. Math. Phys., v. 58, 1910, p. 241-246. J. B. Wilbur, “The mechanical solution of simultaneous equations,” Franklin Inst., J., v. 222, 1936, p. 715-724. o J. B. Bérard, Opuscules Mathématiques, Paris, 1810. L. L. C. Lalanne, “Description d’une nouvelle machine à calcul pour résoudre les équations numériques des sept premiers degrés,” Acad. d. Sci., Paris, Comptes Rendus, v. 11, 1841, p. 859-860; and report on this machine by A. Cauchy et al., p. 959-961. É. C. R. Collignon, Traité de Mécanique, v. 2, Statique, Paris, 1873, p. 347-349, 401; second ed. 1886; fourth ed. 1903. o K. Exner, Über eine Maschine zur Auflösung höherer Gleichungen, Progr. Staatsgymnasium, IX. Bezirk, Vienna, 1881. C. V. Boys, “On a machine for solving equations,” Phil. Mag., s. 5, v. 21, 1886, p. 241-245+plate III. G. B. Grant, “Machine for solving numerical equations,” Amer. Machinist, v. 19, 1896, p. 824-826. R. Skutsch, “Über Gleichungswagen,” Z. Math. Phys., v. 47, 1902, p. 85-104. W. Peddie, “A mechanism for the solution of an equation of the $n$th degree,” Int. Cong. Math., Proc., Cambridge, 1912, p. 399-402. A. Demanet, “Résolution hydrostatique de l’équation du troisième degré,” Mathesis, v. 18, 1898, p. 81-83. G. Meslin, “Sur une machine à résoudre les équations,” J. de Phys., s. 3, v. 9, 1900, p. 339-343. Also Acad. d. Sci., Paris, Comptes Rendus, v. 130, 1900, p. 888-891.
  • Arnold Emch, Two Hydraulic Methods to Extract the $n$th Root of any Number, Amer. Math. Monthly 8 (1901), no. 1, 10–12. MR 1515189, DOI https://doi.org/10.2307/2970154
  • J. Massau, “MĂ©moire sur l’intĂ©gration graphique et ses applications,” Assoc. d. IngĂ©nieurs sortis d. Écoles SpĂ©ciales de Gand, Annales, v. 2, 1878, p. 13-15, 203-281 (figs. 1-65); v. 7, 1884, p. 53-132 (figs. 66-104); v. 10, 1886, p. 1-535+12 plates (figs. 105-310). Reprinted. “Appendice,” v. 12, 1889, p. 185-457+6 plates (figs. 311-350); reprinted, Paris, Gauthier Villars, 1890, 264 p. This is a complete list of references to Massau’s work; the particular pages discussing the apparatus in question were not identified. The reference given by Encycl. d. Sci. Math..$^{3}$, p. 436, to “v. 10 (1886/7), p. 58 et planche hors texte fig. 26” is certainly incorrect. K. Fuchs, “Näherungsweise Elimination durch Mittelwerte,” Archiv. Math. Phys., s. 3, v. 17, 1910, p. 103-105; Jahrb. Fort. Math., v. 41, p. 197. o K. Fuchs, “Eine Gleichungsmaschine aus kommunizierenden Gefassen,” Oesterr. Z.f. Vermessungsw., v. 10, 1912, p. 325-329; J. Fort. Math., v. 43, p. 151. K. Fuchs, “Hydrostatische Gleichungsmaschinen,” Z. Math. Phys., v. 63, 1914, p. 203-214.
  • T. E. W. Schumann, The principles of a mechanical method for calculating regression equations and multiple correlation coefficients and for the solution of simultaneous linear equations, Philos. Mag. 29 (1940), 258–273. MR 0001531
  • F. Lucas, “GĂ©nĂ©ralisation du thĂ©orème de Rolle,” Acad. d. Sci., Paris, Comptes Rendus, v. 106, 1888, p. 121-122. F. Lucas, “DĂ©termination Ă©lectrique des racines rĂ©elles et imaginaires de la dĂ©rivĂ©e d’un polynĂ´me quelconque,” loc. cit., p. 195-197. F. Lucas, “RĂ©solution Ă©lectrique des Ă©quations algĂ©briques,” loc. cit., p. 268-270. F. Lucas, “DĂ©termination Ă©lectrique des lignes isodynamiques d’un polynĂ´me quelconque,” loc. cit., p. 587-589. F. Lucas, “RĂ©solution immĂ©diate des Ă©quations au moyen de l’électricitĂ©,” loc. cit., p. 645-648. F. Lucas, “RĂ©solution des Ă©quations par l’électricitĂ©,” loc, cit., p. 1072-1074. F. Lucas, “RĂ©solution Ă©lectromagnĂ©tique des Ă©quations,” Acad. d. Sci., Paris, Comptes Rendus, v. 111, 1890, p. 965-967. o J. C. Jamin, Cours de Physique de l’École Polytechnique, Paris, v. 4, fourth ed. by E. Bouty, 1888-91, p. 170, figs. 2-3. Also P. É. A. GuĂ©bhard, J. d. Physique, s. 2, v. 1, 1882, p. 205. L. Kann, “Zur mechanischen Auflösung von Gleichungen. Eine elektrische GleichungsMaschine,” Z. Math. Phys., v. 48, 1902, p. 266-272. A. Russell & A. Wright, “The Arthur Wright electrical device for evaluating formulae and solving equations,” Phil. Mag., s. 6, v. 18, 1909, p. 291-308. A. Russell & J. N. Alty, “An electromagnetic method of studying the theory of and solving algebraical equations of any degree,” Phil. Mag., s. 6, v. 18, 1909, p. 802-811. R. R. M. Mallock, “An electrical calculating machine,” R. So. London, Proc., v. 140A, 1933, p. 457-483. H. C. Hart & I. Travis, “Mechanical solution of algebraic equations,” Franklin Inst., J., v. 225, 1938, p. 63-72.
  • Harold D. Green, Square root extractor, Rev. Sci. Instruments 11 (1940), 262–264. MR 2487
  • R. L. Dietzold, “The isograph—a mechanical root-finder,” Bell Laboratories Record, v. 16, 1937, p. 130-134. R. O. Mercner, “The mechanism of the isograph,” loc. cit., p. 135-140. “Mechanical aids to mathematics: Isograph for the solution of complex polynomials,” Electronics, v. 11, Feb., 1938, p. 54.
  • S. Leroy Brown, A mechanical harmonic synthesizer-analyzer, J. Franklin Inst. 228 (1939), 675–694. MR 776, DOI https://doi.org/10.1016/S0016-0032%2839%2990460-6
  • S. Leroy Brown and Lisle L. Wheeler, A mechanical method for graphical solution of polynomials, J. Franklin Inst. 231 (1941), 223–243. MR 3587, DOI https://doi.org/10.1016/S0016-0032%2841%2990048-0
  • D. Baxandall, “Calculating machines,” EncyclopĹ“dia Britannica, fourteenth ed., v. 4, 1929, p. 551-553. Note bibliography at end. P. Werkmeister, “Die Auflösung eines Systems linearer Gleichungen mit Hilfe der Rechenmaschine ’Hamann-Automat’,” Z.f. Instrumententechnik, v. 51, 1931, p. 490.
  • S. Lilley, Mathematical machines, Nature 149 (1942), 462–465. MR 6254, DOI https://doi.org/10.1038/149462a0
  • V. Bush, “The differential analyzer. A new machine for solving differential equations,” Franklin Inst., J., v. 212, 1931, p. 447-488. References to unreviewed items are as follows: G. RosĂ©n, “Eine elektromechanische ’Gleichungswage,”’ Elektrotechn. Z., v. 50, 1929, p. 1726-1727. G. Revessi, “Verso soluzioni meccaniche ed elettriche dei sistemi di equazioni lineari,” L’Elettrotecnica, v. 12, 1925, p. 550-553.

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Article copyright: © Copyright 1945 American Mathematical Society