The history of Rolle’s Theorem dates back to the 17th century, when Michel Rolle developed the method of cascades and the roots localization method for the solution of algebraic equations. In the 19th century, these methods provided the basis for Cauchy’s development of the concept of continuity and for Bolzano’s Theorem that initiated a reform of calculus. In modern formulations, the Rolle Theorem and the Bolzano–Cauchy Theorem express the two most important properties of continuous functions. This article explores the history of these developments in fuller detail, including important contributions by Bernard Bolzano, Augustin-Louis Cauchy, Karl Weierstrass, Georg Cantor, and Nikolai Luzin.
A Second Letter from Mr. Colin McLaurin [...] Concerning the Roots of Equations, with the Demonstration of Other Rules in Algebra (1729), Philosophical Transactions of Royal Society of London, vol. 36, pp. 59–96.
Barrow-Green, J. (2009) From Cascades to Calculus: Rolle’s Theorem, in: Robson, E. and Stedall, J. (eds.) The Oxford Handbook of the History of Mathematics. Oxford: Oxford University Press, pp. 737–754.
Bellavitis, G. (1846) Sul più facile modo di trovare le radici reali delle equazioni algebraiche e sopra un nuovo metodo per la determinazione delle radici immaginarie memoria. Venezia: Presso la segreteria dell’Instituto nel palazzo ducale.
Bolzano, B. (1817) Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwei Werthen, die ein entgegengesetzes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege. Prag: Gottlieb Haase.
Bolzano, B. (1955) Chisto analiticheskoe dokazatel’stvo teoremy, chto mezhdu liubymi dvumia znacheniiami, daiushchimi rezul’taty protivopolozhnogo znaka, lezhit po men’shei mere odin veshchestvennyi koren’ uravneniia [Purely Analytic Proof of the Theorem That Between Any Two Values Which Give Results of Opposite Sign, There Lies at Least One Real Root of the Equation], in: Kolman, E. Bernard Boltsano. Moskva: Izdatel’stvo AN SSSR, pp. 170–204.
Campbell, G. (1727/28) A Method for Determining the Number of Impossible Roots in Adfected Aequations, Philosophical Transactions of Royal Society of London, vol. 35, pp. 515–531.
Cantor, G. 1985. Trudy po teorii mnozhestv [The Works on Set Theory]. Moskva: Nauka.
Cauchy, A. (1821) Cours d’analyse de l’École royale polytechnique. Première partie: Analyse algébrique. Paris: L’Imprimerie royale.
Cauchy, A. (1897) Cours d’analyse de l’École royale polytechnique. Première partie: Analyse algébrique, in: Cauchy, A. Oeuvres complètes. Paris: Gauthiers-Villars et fils, 1897, sér. 2, vol. 3.
Clairaut, A. C. (1746) Élémens d’algèbre. Paris: Les frères Guerin.
Dini, U. (1878) Fondamenti per la theorica delle funzioni di variabili reali. Pisa: Tip. T. Nistri.
Drobisch, M. W. (1834) Grundzüge der Lehre von den höheren numerischen Gleichungen. Leipzig: Leopold Voss.
Dugac, P. (1973) Eléments d’analyse de Karl Weierstrass, Archive for History of Exact Sciences, 1973, vol. 10, no. 1– 2, p. 41–176.
Euler, L. (1755) Institutiones calculi differentialis. Petropolis: Academia Imperialis Scientiarum Petropolitana.
Euler, L. (1949) Differentsial’noe ischislenie [Differential Calculus]. Moskva and Leningrad: Gostekhteorizdat.
Fikhtengolts, G. (1968) Osnovy matematicheskogo analiza. 6 izd. [The Foundations of Analysis, 6th ed.]. Moskva: Nauka.
Fontenelle, B. (1719) Eloge de M. Rolle, in: Histoire de l’Académie royale des sciences. Paris: L’Imprimerie royale, p. 94–100.
Heine, H. E. (2012) Lektsii po teorii funktsii [The Lectures on the Theory of Functions] in: Vager, B. G. (ed.) Matematicheskoe modelirovaniie, chislennye metody i kompleksy program [Mathematical Modelling, Calculus of Approximations and Program Systems]. Sankt-Peterburg: SPbGASU, no. 18, pp. 26–46.
Kästner, A. G. (1758–1769) Die mathematischen Anfangsgründe (4 Th. 7 Bd.). Göttingen: Witwe Vandenhoeck.
Kästner, A. G. (1794) Angfangsgründe der Analysis endlicher Grössen. Göttingen: Vandenhoek und Ruprecht.
l’Hôpital, G. F., de (1696) Analyse des infiniment petits pour l’intelligence des lignes courbes. Paris: L’Imprimerie royale.
l’Hôpital, G. F., de (1935) Analiz beskonechno malykh [Infinitesimal Analysis]. Moskva and Leningrad: Gostekhteorizdat.
Lacroix, S. F. (1811) Anfangsgründe der Algebra. Mainz: Kupferberg.
Lacroix, S. F. (1830) Élémens d’algèbre, à l’usage de l’Ecole centrale des Quatre-Nations. 15 ed. Bruxelles: H. Remy.
Lagrange, J.-L. (1879) Traité de la resolution des équations numériques de tous les degrés, avec des notes sur plusieurs points de la théorie des équations algébriques, in: Lagrange, J.-L. Oeuvres complètes. Paris: Gauthier-Villars, vol. 8, pp. 11–367.
Luzin, N. N. (1946) Differentsialnoe ischislenie [Differential Calculus]. Moskva: Sovetskaia nauka.
Mordukhai-Boltovskoi, D. D. (1952) Iz proshlogo analiticheskoi geometrii [From the Past of Analytical Geometry], in: Trudy Instituta istorii estestvoznaniia, vol. 4, pp. 217–235.
Newton, I. (1707) Arithmetica universalis; sive De compositione et resolutione arithmetica liber. Cantabrigia: Typus academicis.
Newton, I. (1711) De analysi per aequationes numero terminorum infinitas, in: Newton, I. Analysis per quantitatum series, fluxiones, ac differentias: cum enumeratione linearum tertii ordinis. Londinium: Ex officina Pearsoniana.
Newton, I. (1736) The Method of Fluxions and Infinite Series; with Its Application to the Geometry of Curve-Lines. London: Henry Woodfall.
Newton, I. (1948). Vseobshchaia arifmetika, ili kniga ob arifmeticheskom sinteze i analize [Universal Arithmetic; Or, a Treatise of Arithmetical Composition and Resolution]. Moskva: Nauka.
Raphson, J. (1702) Analysis аequationum universalis. 2nd ed. Londinium.
Reyneau, Ch.-R. (1708) Analyse démontrée, ou la Méthode de résoudre les problèmes de mathématiques. 2 vols. Paris: J. Quillau.
Rolle, M. (1690) Traité d’algèbre, ou principes généraux pour résoudre les questions de mathématique. Paris: Estienne Michallet.
Rolle, M. (1691) Démonstration d’une méthode pour résoudre les égalités de tous les degrés. Paris: Jean Cusson.
Rösling, Ch. L. (1805) Grundlehren von den Formen, Differenzen, Differentialien und Integralien der Functionen. Erlangen: J. J. Palm.
Rykhlik, K. (1958) Teoria veshchestvennykh chisel v rukopisnom nasledii Boltsano [Theory of Real Numbers in Bolzano’s Manuscripts], Istoriko-matematicheskie issledovaniia, vol. 11, pp. 515–532.
Shain, J. (1937) The Method of Cascades, American Mathematical Monthly, vol. 44, pp. 24–29.
Shatunovskii, O. (1923) Vvedenie v analiz [Introduction to Analysis]. Odessa: Mathesis.
Simpson, T. (1740) Essays on Several Curious and Useful Subjects, in Speculative and Mix’d Mathematicks. London: Henry Woodfall.
Sinkevich G. I. (2012). Genrikh Eduard Geine. Teoriia funktsii [Heinrich Eduard Heine. Function Theory], in: Vager, B. G. (ed.) Matematicheskoe modelirovaniie, chislennye metody i kompleksy programm [Mathematical Modelling, Calculus of Approximations and Program Systems]. SanktPeterburg: SPbGASU, no. 18, pp. 6–26.
Sinkevich, G. I. (2012). K istorii epsilontiki [On the History of Epsilontics], Matematika v vysshem obrazovanii, no. 10, pp. 149–166.
Sinkevich, G. I. (2013) Poniatie nepreryvnosti u Dedekinda i Kantora [A Notion of Continuity in Dedekind’s and Cantor’s Works], in: Afanasiev V. V. (ed.) Trudy XI Mezhdunarodnykh Kolmogorovskikh chtenii. Iaroslavl’: IaSPU, pp. 336–347.
Sinkiewich, G. I. (2013) Historia dwóch twierdzeń analizy matematycznei: M. Rolle, B. Bolzano, A. Cauchy, in: Więsław, W. (ed.) Dzieje matematyki polskiej II. Wrocław: Sowa, pp. 165–181.
Stolz, O. B. (1881) Bolzano’s Bedeutung in der Geschichte der Infinitesimalrechnung, Mathematische Annalen, vol. 18, pp. 255–279.
Wallis, J. (1685) A Treatise of Algebra, Both Historical and Practical: Shewing the Original, Progress, and Advancement Thereof, from Time to Time, and by What Steps It Hath Attained to the Heighth at Which Now It Is; With Some Additional Treatises. London; Oxford: John Playford.
Weierstrass, K. (1988) Ausgewählte Kapitel aus der Funktionenlehre. Vorlesung, gehalten in Berlin 1886. Mit der akademischen Antrittsrede, Berlin 1857, und drei weiteren Originalarbeiten von K. Weierstrass aus den Jahren 1870 bis 1880/86, in: Siegmund-Schultze, R. (ed.) Teubner-Archiv zur Mathematik. Leipzig: B. G. Teubner, vol. 9.
Yanovskaya, S. (1947) Michel’ Roll’ kak kritik analiza beskonechno malykh [Michel Rolle as a Critic of Infinitesimal Analysis], Trudy instituta istorii estestvoznaniia, vol. 1, pp. 327–346.
Yushkevich, A. (ed.) (1977) Khrestomatiia po istorii matematiki. Matematicheskii analiz [Reading Book on the History of Mathematics. Analysis]. Moskva: Prosveshcheniie.
