Keywords
Chebyshev
Russian mathematics
Segner machine
steam engine
geography
artillery
ballistics
gunnery
cartography
elliptic integrals
continued fractions
distribution of primes
How to Cite
Abstract
On the occasion of Chebyshev’s twohundredth anniversary, I review part of his work, showing that in several respects he was the heir of Euler. In doing this, I consider the works of Euler and Chebyshev on three topics in applied science: industrial machines, ballistics and geography, and then on three topics in pure mathematics: integration, continued fractions and number theory, showing that in each filed the two mathematicians were interested in the same kind of questions.
References
Papadopoulos A. Euler and Chebyshev: From the sphere to the plane and backwards. Proceedings in Cybernetics. 2016;2:56-70.
Chebyshev P. L. Œuvres (French) and Sochinenia (Russian), 2 volumes, edited by A. Markov and N. Sonin. Saint Petersburg; Imprimerie de l’Acad´emie Imp´eriale des Sciences, 1899-1907.
Chebyshev P. L. Complete Collected Works (Russian), 5 volumes, edited by I. I. Artobolevsky, S. N. Bernstein, N. G. Bruevich, I. M. Vinogradov, B. N. Delone, V. L. Goncharov, A. N. Kolmogorov, A. Krylov, L. S. Leibenson, V. I. Smirnov and S. L. Sobolev. Moscow-Leningrad; Izdatel’stvo Akad. Nauk SSR, 1946-1951.
Vucinich A. Science in Russian culture: A history to 1860. London; Peter Owen, 1965.
Euler L. Mechanica, Vol. 1, St. Petersburg: Imperial Academy of Sciences, Volume 1, p. 1-480. Opera Omnia, Series 2, Volume 1, p. 1-407.
Euler L. Mechanica, Vol. 2, St. Petersburg: Imperial Academy of Sciences, Volume 2, p. 1-500. Opera Omnia, Series 2, Volume 2, p. 1-459.
Euler L. Discussion plus particuli`ere de diverses mani`eres d’´elever de l’eau par le moyen des pompes avec le plus grand avantage. M´emoires de l’Acad´emie des Sciences de Berlin. 1754;8:149-184. Opera Omnia, Series 2, Volume 15, p. 251-280.
Euler L. Recherches plus exactes sur l’effet des moulins a` vent. M´emoires de l’Acad´emie des Sciences de Berlin. 1758;12:165-234. Opera Omnia: Series 2, Volume 16, p. 65-125.
Euler L. Sur l’action des scies. M´emoires de l’Acad´emie des Sciences de Berlin. 1758;12:267-291. Opera Omnia, Series 2, Volume 17, p. 66-88.
Euler L. Scientia navalis, Vol. 1. St. Petersburg: Imperial Academy of Sciences. 1738;1:1-444. Opera Omnia, Series 2, Volume 18, p. 1-426.
Euler L. Scientia navalis, Vol. 2. St. Petersburg: Imperial Academy of Sciences. 1749;2:1-534. Opera Omnia, Series 2, Volume 19, p. 1-459.
Euler L. Recherches sur l’effet d’un machine hydraulique propos´ee par M. Segner, professeur `a G¨ottingue. M´emoires de l’acad´emie des sciences de Berlin. 1752;6:311-354. Opera Omnia, Series 2, Volume 15, p. 1-39.
Euler L. Application de la machine hydraulique de M. Segner `a toutes sortes d’ouvrages et de ses avantages sur les autres machines hydrauliques dont on se sert ordinairement. M´emoires de l’acad´emie des sciences de Berlin. 1753;7:271-304. Opera Omnia, Series 2, Volume 15, p. 105-133.
Euler L. Th´eorie plus complette des machines qui sont mises en mouvement par la r´eaction de l’eau. M´emoires de l’acad´emie des sciences de Berlin. 1756;10:227-295. Opera Omnia Citation, Series 2, Volume 15, p. 157-218.
Euler L. De motu et reactione aquae per tubos mobiles transfluentis. Novi Commentarii academiae scientiarum Petropolitanae. 1761;6:312-337. Opera Omnia, Series 2, Volume 15, p. 80-104.
Euler L. D´etermination de l’effet d’un machine hydraulique invent´ee par M. Segner, professeur `a G¨ottingue, written in 1752, Euler’s Opera Postuma. 1862;2:146-173. Opera Omnia, Series 2, Volume 15, p. 40-79.
Euler L. Maximes pour arranger le plus avantageusement les machines destin´ees `a ´elever de l’eau par le moyen des pompes. M´emoires de l’Acad´emie des Sciences de Berlin. 1754;8:185-232. Opera Omnia, Series 2, Volume 15, p. 281-318.
Sadovsky V. N., Kelle V. V. Tektology. Book 1. Center for System studies, University of Hull, 1996.
Vassilief A. P. Tch´ebychef et son oeuvre scientifique. I Turin; Bollettino di bibliografia e storia delle scienze matematiche, 1898.
Peaucellier A. Note sur une question de g´eom´etrie de compas. Nouv. Ann. de Math. 2e s´erie. 1873;12:71-72.
Lipkin L. Dispositif articul´e pour la transformation rigoureuse du mouvement circulaire en mouvement rectiligne. Revue des Mines et de la M´etallurgie de Li`ege. 1871;30, 4e livraison:149-150.
Sossinsky A. Configuration spaces of planar linkages. Handbook of Teichmu¨ller theory. Vol. VI (ed. A. Papadopoulos). Zu¨rich; European Mathematical Society Publishing House, 2016. P. 335-373.
Robins B. Remarks on Mr. Euler’s Treatise of motion, Dr. Smith’s complete system of optics, and Dr. Jurins essay upon distinct and indistinct vision. London, 1739. Reprinted in: Mathematical tracts of the late Benjamin Robins. Esq. London, 1761. Vol. II.
Robins B. New principles of gunnery: containing the determination of the force of gunpowder, and an investigation of the difference in the resisting power of the air to swift and slow motions. London, 1742.
Euler L. New principles of gunnery. London, 1742. German edition: Neue Grundsa¨tze der Artillerie. 1745. Opera Omnia, Series 2, Volume 14, p. 1-409.
Brown H. The true principles of gunnery investigated and explained, comprehending translations of professor Euler’s observations upon the new Principles of Gunnery, published by the late Mr. Benjamin Robins, and that celebrated author’s Discourse upon the Track described by a body in a resisting medium, inserted in the Memoirs of the Royal Academy of Berlin, for the year 1753, to which are added many necessary explanations and remarks, together with tables calculated for practice, the use of which is illustrated by proper examples; with the method of solving that capital problem, which requires the elevation of the greatest range with any given initial velocity. London; J. Nourse, 1777.
Euler L. Nouveaux principes d’artillerie de M. Benjamin Robins, comment´es par M. Leonard Euler, translated into French by J. L. Lombard, Frantin, Dijon et Jombert, Paris, 1783.
Euler L. Meditatio in experimenta explosione tormentorum nuper instituta. Opera Postuma 2, 1862, p. 800-804. Opera Omnia, Series 2, Volume 14, p. 468-477.
Euler L. Recherches sur la v´eritable courbe que d´ecrivent les corps jet´es dans l’air ou dans un autre fluide quelconque. M´emoires de l’acad´emie des sciences de Berlin. 1755;9:321-352. Opera Omnia, Series 2, Volume 14, p. 413-447.
Euler L. Tentamen explicationis phaenomenorum aeris. Commentarii academiae scientiarum Petropolitanae, Volume 2, p. 347-368. Opera Omnia, Series 2, Volume 31, p. 1-18.
Prokhorov V. Pafnuty Chebyshev: To mark the 175th anniversary of his birth. Teor. Veroyatnost. i Primenen. 1996;41(3):613-627. English translation: Theory Probab. Appl. 1996;41(3):519-530.
Atlas Russicus mappa una generali et undeviginti specialibus vastissimum Imperium Russicum adjacentibus regionibus secundum leges geographicas, et recentissimas observationes delineatum exhibens. Academia imperialis Scientiarum Petropolitanae, 1745.
Euler L. Preface of the Atlas Geographicus omnes orbis terrarum regiones in XLI tabulis exhibens. Acad´emie Royale des Sciences et Belles-Lettres de Prusse, Berlin, 1753. Opera Omnia, Series 3, Volume 2, p. 305-317.
Euler L. Methodus viri celeberrimi Leonhardi Euleri determinandi gradus meridiani pariter ac paralleli telluris, secundum mensuram a celeb. de Maupertuis cum sociis institutam. Commentarii Academiae Scientiarum Petropolitanae. 1750;12:224-231. Opera Omnia, Series 2, Volume 30, p. 73-88.
Euler L. De attractione corporum sphaeroidico-ellipticorum. Commentarii academiae scientiarum Petropolitanae. 1747;10:102-115. Opera Omnia, Series 2, Volume 6, p. 175-188.
Euler L. El´emens de la trigonom´etrie sph´ero¨ıdique tir´es de la m´ethode des plus grands et plus´ petits. M´emoires de l’acad´emie des sciences de Berlin. 1755;9:258-293. Opera Omnia, Series 1, Volume 27, p. 309-339.
Euler L. De la parallaxe de la lune tant par rapport `a sa hauteur qu’`a son azimuth, dans l’hypoth`ese de la terre sph´ero¨ıdique. M´emoires de l’acad´emie des sciences de Berlin. 1751;5:326-338. Opera Omnia, Series 2, Volume 30, p. 140-150.
Euler L. Von der Gestalt der Erden, Anmerckungen u¨ber die Zeitungen. St. Petersburg, 3 April 1738 – 25 December 1738. Opera Omnia, Ser. III, Vol. 2, p. 325-346.
Euler L. De repraesentatione superficiei sphaericae super plano. Acta Academiae Scientiarum Imperialis Petropolitanae. 1777, p. 107-132. Opera Omnia, Series 1, Volume 28, p. 248-275.
Euler L. De proiectione geographica superficiei sphaericae. Acta Academiae Scientiarum Imperialis Petropolitanae. 1777, p. 133-142. Opera Omnia, Series 1, Volume 28, p. 276-287.
Euler L. De proiectione geographica De Lisliana in mappa generali imperii russici usitata. Acta Academiae Scientiarum Imperialis Petropolitanae. 1777, p. 143-153. Opera Omnia, Series 1, Volume 28, p. 288-297.
De Lagrange J.-L. Sur la construction des cartes g´eographiques, Nouveaux m´emoires de l’Acad´emie Royale des Sciences et Belles-lettres de Berlin, ann´ee 1779, Premier m´emoire: Œuvres compl`etes, tome 4, p. 637-664. Second m´emoire: Œuvres compl`etes, tome 4, p. 664-692.
Caddeo R., Papadopoulos A. (ed.) Geography in the eighteenth century: Euler, Lambert and Lagrange. Springer, to appear in 2021.
Leonhard Euler, Opera Omnia, Series quarta A, commercium epistolicum, Volumen quartum, Pars prima, (M. Mattmu¨ller, F. Lemmermeyer, eds.). Birkh¨auser, Basel, 2015.
Papadopoulos A. Quasiconformal mappings, from Ptolemy’s geography to the work of Teichmu¨ller. Uniformization, Riemann-Hilbert Correspondence, Calabi-Yau Manifolds, and Picard-Fuchs Equations. Ed. L. Ji and S.-T. Yau. International Press and Higher Education Press, ALM 42, 2018, p. 237-315.
Papadopoulos A. Maps with least distortion between surfaces: from geography to brain warping. Notices of the American Mathematical Society. November 2019:1628-1639.
Papadopoulos A. Mathematics and map drawing in the eighteenth century. Gan.ita Bh¯ara¯t¯ı (Indian Mathematics). 2019;41(1-2):1-37.
Papadopoulos A. Map drawing and foliations of the sphere. Russian Journal of Cybernetics. 2020;1(1):42-49.
Papadopoulos A. Pafnuty Chebyshev and geography. To appear in the Mathematical Intelligencer, 2021.
Papadopoulos A. On Chebyshev’s work on geography. Mathematicians from Saint Petersburg and their theorems, ed. Nikita Kalinin, St. Petersburg State University Publishing House, to appear in 2022.
Euler L. Institutionum calculi integralis. St. Petersburg, Imperial Academy of Sciences. 1768;1:1-542. Opera Omnia, Series 1, Volume 11, p. 1-462.
Euler L. Institutionum calculi integralis. St. Petersburg, Imperial Academy of Sciences. 1769;2:1-542. Opera Omnia, Series 1, Volume 12, p. 1-542.
Euler L. Institutionum calculi integralis. St. Petersburg, Imperial Academy of Sciences. 1770;3:1-639. Opera Omnia, Series 1, Volume 13, p. 1-508.
Euler L. Specimen de constructione aequationum differentialium sine indeterminatarum separation. Commentarii academiae scientiarum Petropolitanae. 1738;6:168-174. Opera Omnia, Series 1, Vol. 20, p. 17.
Euler L. Observationes circa integralia formularum ∫︀ xp−1dx(1 − xn)q/n−1 posito post integrationem x = 1. Novi Commentarii academiae scientiarum Petropolitanae. 1761;6:37-57. Opera Omnia, Series 1, Volume 20, p. 58-79.
Euler L. Observationes circa integralia formularum ∫︀ xp−1dx(1 − xn)q/n−1 posito post integrationem x = 1 (Observations concerning the integrals of formula∫︀ xp−1dx(1 − xn)q/n−1 setting x = 1 after integration). M´elanges de philosophie et de la math´ematique de la soci´et´e royale de Turin. 1766;3:156-177. Opera Omnia, Series 1, Volume 17, p. 268-288.
Euler L. Speculationes super formula integrali , ubi simul egregiae observationes circa fractiones continuas occurrunt. Acta Academiae Scientarum Imperialis Petropolitinae. 1786:62-84. Opera Omnia, Series 1, Volume 18, p. 244-264.
De fractionibus continuis dissertation. Commentarii academiae scientiarum Petropolitanae. 1744;9:98-137. Opera Omnia, Series 1, Volume 14, p. 187-216.
Euler L. Introductio in analysin infinitorum, vol. 1, 1748. Lausanne: Marcum-Michaelem Bousquet, Volume 1, p. 1-320. Opera Omnia, Series 1, Volume 8, p. 1-392.
Euler L. Introduction to Analysis of the Infinite. Trans. J. D. Blandon. New York and Berlin; Springer Verlag, 1988.
De fractionibus continuis observations. Commentarii academiae scientiarum Petropolitanae. 1750;11:32-81. Opera Omnia, Series 1, Volume 14, p. 291-349.
Euler L. D´ecouverte d’une loi tout extraordinaire des nombres par rapport `a la somme de leurs diviseurs. Biblioth`eque imp´eriale. 1751;3:10-31. Opera Omnia, Series 1, Vol. 2, p. 241-253.
Atlas Geographicus omnes orbis terrarum regiones in XLIV tabulis exhibens. Acad´emie Royale des Sciences et Belles Lettres de Prusse, Berlin, 1753. Opera Omnia, Series 3, Volume 2, p. 305-317.
Euler L. Variae observationes circa series infinitas. Commentarii academiae scientiarum Petropolitanae. 1744;9:160-188. Opera Omnia, Series 1, Vol. 14, p. 217-244.
Euler L. Index syst´ematique et raisonn´e des oeuvres arithm´etiques d’Euler par M.M. Bouniakovsky et Tch´ebychef. Leonh. Euleri Commentationes arithmeticae collectae, vol. I and II, 1849.
Euler L. De repraesentatione superficiei sphaericae super plano. Acta Academiae Scientarum Imperialis Petropolitinae. 1778:107-132. Opera Omnia, Series 1, Volume 28, p. 248-275.