Machines for solving algebraic equations
HTML articles powered by AMS MathViewer
- by J. S. Frame PDF
- Math. Comp. 1 (1945), 337-353 Request permission
References
-
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 10.2307/2970154 J. Massau, “Mémoire sur l’intégration graphique et ses applications,” Assoc. d. Ingénieurs sortis d. Écoles Spéciales 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, DOI 10.1080/14786444008521266 F. Lucas, “Généralisation du théorème de Rolle,” Acad. d. Sci., Paris, Comptes Rendus, v. 106, 1888, p. 121-122. F. Lucas, “Détermination électrique des racines réelles et imaginaires de la dérivée d’un polynôme quelconque,” loc. cit., p. 195-197. F. Lucas, “Résolution électrique des équations algébriques,” loc. cit., p. 268-270. F. Lucas, “Détermination électrique des lignes isodynamiques d’un polynôme quelconque,” loc. cit., p. 587-589. F. Lucas, “Résolution immédiate des équations au moyen de l’électricité,” loc. cit., p. 645-648. F. Lucas, “Résolution des équations par l’électricité,” loc, cit., p. 1072-1074. F. Lucas, “Résolution électromagnétique des équations,” 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. Guébhard, 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, DOI 10.1063/1.1751703 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 10.1016/S0016-0032(39)90460-6
- 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 10.1016/S0016-0032(41)90048-0 D. Baxandall, “Calculating machines,” Encyclopœdia 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 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. Rosén, “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.
Additional Information
- © Copyright 1945 American Mathematical Society
- 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