Archive for History of Exact Sciences

, Volume 63, Issue 3, pp 325–356 | Cite as

Euler’s beta integral in Pietro Mengoli’s works

  • Ma Rosa Massa Esteve
  • Amadeu Delshams


Beta integrals for several non-integer values of the exponents were calculated by Leonhard Euler in 1730, when he was trying to find the general term for the factorial function by means of an algebraic expression. Nevertheless, 70 years before, Pietro Mengoli (1626–1686) had computed such integrals for natural and half-integer exponents in his Geometriae Speciosae Elementa (1659) and Circolo(1672) and displayed the results in triangular tables. In particular, his new arithmetic–algebraic method allowed him to compute the quadrature of the circle. The aim of this article is to show how Mengoli calculated the values of these integrals as well as how he analysed the relation between these values and the exponents inside the integrals. This analysis provides new insights into Mengoli’s view of his algorithmic computation of quadratures.


Fermat Beta Function Algebraic Expression Combinatorial Number Vacant Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Andersen, Kirsti, 1984/1985. Cavalieri’s method of indivisibles. Archive for History of Exact Sciences 31:291–367.Google Scholar
  2. Auger, Léon (1962) Un savant méconnu: Gilles Personne de Roberval (1602–1675). Libraire A. Blanchard, ParisMATHGoogle Scholar
  3. Baroncini, Gabrielle, and Cavazza, Marta. (1986) La corrispondenza di Pietro Mengoli. Florence, OlschkiGoogle Scholar
  4. Bosmans, Henry. (1924) Sur l’oeuvre mathématique de Blaise Pascal. Mathesis 38: 1–59Google Scholar
  5. Cajori, Florian. 1928–1929. A history of mathematical notations, Vol. 2. Chicago: Open Court (Reprinted by Dover, 1993.)Google Scholar
  6. Calinger, Ronald. (1996) Leonhard Euler: the first St. Petersburg years (1727–1741). Historia Mathematica 23: 121–166MATHCrossRefMathSciNetGoogle Scholar
  7. Cifoletti, Giovanna. 1990. La méthode de Fermat: son statut et sa difussion. In: Cahiers d’histoire et de philosophie des sciences, nouvelle série, Vol. 33. Paris: Société française d’histoire des sciences et des techniques.Google Scholar
  8. Delshams, Amadeu, and Massa, Ma Rosa. (2008) Consideracions al voltant de la funció Beta a l’obra de Leonhard Euler (1707–1783). Quaderns d’Història de l’Enginyeria IX: 59–82Google Scholar
  9. Descartes, René. 1954. The geometry of René Descartes Eds. D.E. Smith and M.L. Latham. New York: Dover.Google Scholar
  10. Edwards, Anthony William F. (2002) (1a ed. 1987) Pascal’s Arithmetical triangle. The story of a mathematical idea. Oxford University Press., New YorkGoogle Scholar
  11. Euler, Leonhard. 1730–1731. De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt. In Euler Opera Omnia 14(I):1–24.Google Scholar
  12. Euler, Leonhard. (1739) De productis ex infinitis factoribus. In: Euler Opera Omnia 14(I): 260–290Google Scholar
  13. Euler, Leonhard. (1753) De expressione integralium per factores. In: Euler Opera Omnia 17(I): 233–267Google Scholar
  14. Fermat, Pierre de. 1891–1922. Oeuvres. Eds. P. Tannery and H. Charles, 4 vols. 7 spp. Paris: Gauthier Villars.Google Scholar
  15. Ferraro, Giovanni. (1998) Aspects of Euler’s theory of series. Historia Mathematica 25: 290–317MATHCrossRefMathSciNetGoogle Scholar
  16. Ferraro, Giovanni. (2000) Functions, functional relations and the laws of continuity in Euler. Historia Mathematica 27: 107–132MATHCrossRefMathSciNetGoogle Scholar
  17. Ferraro, Giovanni. (2004) Differentials and differential coefficients in the Eulerian foundations of the calculus. Historia Mathematica 31: 34–61MATHCrossRefMathSciNetGoogle Scholar
  18. Ferraro, Giovanni. (2008) The integral as anti-differencial. An aspect of Euler’s attempt to transform the calculus into an algebraic calculus. Quaderns d’Història de l’Enginyeria IX: 25–58Google Scholar
  19. Fraser, Craig. (1989) The calculus as algebraic analysis: Some observations on mathematical analysis in the 18th century. Archive for History of Exact Sciences 39: 317–335MATHMathSciNetGoogle Scholar
  20. Giusti, Enrico. (1980) Bonaventura Cavalieri and the theory of indivisibles. Edizioni Cremonese, BolognaMATHGoogle Scholar
  21. Giusti, Enrico. 1991. Le prime ricerche di Pietro Mengoli. La somma delle serie. In Geometry and Complex Variables: Proceedings of an International Meeting on the Occasion on the IX Centennial of the University of Bologna, ed. S. Coen, 195–213. New York: Dekker.Google Scholar
  22. Gray, Jeremy. 1985. Leonhard Euler: 1707–1783. Janus: Revue internationale de l’histoire des sciences, de la médecine, de la pharmacie et de la technique LXXII, 1–3:171–192.Google Scholar
  23. Gregory, James. 1939. Tercentenary memorial volume, ed. H. W. Turnbull, London: Bell.Google Scholar
  24. Hérigone, Pierre. 1642–1644. Cursus mathematicus, Nova, Brevi et clara methodo demonstratus, Per NOTAS reales & universales, citra usum cuiuscunque idiomatis, intellectu faciles. (Second editions), 6 vols. Paris: Simeon Piget.Google Scholar
  25. Hobson, Ernest William. (1913) Squaring the circle. A history of the problem. Cambridge University, CambridgeGoogle Scholar
  26. Jami, Catherine. (1988) Une histoire chinoise du nombre pi. Archive for History of Exact Sciences 38 1: 39–50CrossRefGoogle Scholar
  27. Leibniz, Goffried, W. 1672–1676. Sämtliche Schriften und Briefe, series VII: Mathematische Schriften, Vol. III: Differenzen-Folgen-Reihen, Berlin, 2003, 735–748.Google Scholar
  28. Mahoney, Michael Sean. 1973. The mathematical career of Pierre de Fermat. Princeton: Princeton University (reprinted 1994).Google Scholar
  29. Malet, Antoni. (1996) From indivisibles to infinitesimals, Studies on seventeenth century mathematizations of infinitely small quantities. Universitat Autònoma de Barcelona, BarcelonaMATHGoogle Scholar
  30. Massa-Esteve, Ma Rosa (1994) El mètode dels indivisibles de Bonaventura. Cavalieri Butlletí de la Societat Catalana de Matemàtiques 9: 68–100Google Scholar
  31. Massa-Esteve, Ma Rosa. (1997) Mengoli on “Quasi Proportions”. Historia Mathematica 24(2): 257–280CrossRefMathSciNetGoogle Scholar
  32. Massa-Esteve, Ma Rosa. 1998. Estudis matemàtics de Pietro Mengoli (1625–1686): Taules triangulars i quasi proporcions com a desenvolupament de l’àlgebra de Viète.Tesi Doctoral. Barcelona: Universitat Autònoma de Barcelona.
  33. Massa-Esteve, Ma Rosa. (2003) La théorie euclidienne des proportions dans les “Geometriae Speciosae Elementa” (1659) de Pietro Mengoli. Revue d’Histoire des Sciences 56(2): 457–474CrossRefMathSciNetGoogle Scholar
  34. Massa-Esteve, Ma Rosa. (2006a) Algebra and geometry in Pietro Mengoli (1625–1686). Historia Mathematica 33: 82–112MATHCrossRefMathSciNetGoogle Scholar
  35. Massa-Esteve, Ma Rosa. 2006b. L’Algebrització de les Matemàtiques. Pietro Mengoli (1625–1686). Barcelona: Institut d’Estudis Catalans.Google Scholar
  36. Massa-Esteve, Ma Rosa. 2006c. La algebrización de las Matemáticas. L’algèbre de Pierre Hérigone (1580–1643). In Actas del IX Congreso de la Sociedad Española de Historia de las Ciencias y de las Técnicas, Tomo 1, Cádiz, 183–197.Google Scholar
  37. Massa-Esteve, Ma Rosa. 2007. Leonhard Euler (1707–1783): L’home, el creador i el Mestre. Mètode, 35–38. Valencia: Universitat de València.Google Scholar
  38. Massa-Esteve, Ma Rosa. (2008) Symbolic language in early modern mathematics: the algebra of Pierre Hérigone (1580–1643). Historia Mathematica 35: 285–301CrossRefMathSciNetGoogle Scholar
  39. Mengoli, Pietro. 1650. Novae Quadraturae Arithmeticae seu de Additione Fractionum, Bologna.Google Scholar
  40. Mengoli, Pietro. 1659. Geometriae Speciosae Elementa. Bologna.Google Scholar
  41. Mengoli, Pietro. 1672. Circolo. Bologna.Google Scholar
  42. Natucci, A. 1970–1991. Mengoli. In Dictionary of scientific biography, ed. C.C. Gillispie, Vol. 9, 303–304. New York: Scribner’s.Google Scholar
  43. Pascal, Blaise. (1954) Oeuvres complètes. Gallimard, ParisGoogle Scholar
  44. Probst, Siegmund. 2009. The Reception of Pietro Mengoli’s Work on Series by Leibniz (1672–1676). In Proceedings of Joint International Meeting UMI-DMV, Perugia, 18–22 June 2007.Google Scholar
  45. Seidenberg, Abraham. (1981) The ritual origin of the Circle and Square. Archives for History of Exact Sciences 25: 269–327MATHCrossRefGoogle Scholar
  46. Stedall, Jacqueline (2001) The discovery of wonders: reading between the lines of John Wallis’s arithmetica infinitorum. Archive for History of Exact Sciences 56: 1–28MATHCrossRefMathSciNetGoogle Scholar
  47. Volkov, Alexei (1997) Zhao Youqin and his calculation of pi. Historia Mathematica 24: 301–331MATHCrossRefMathSciNetGoogle Scholar
  48. Walker, Evelyn. (1986) A study of the Traité des Indivisibles of... Roberval. Columbia University, New YorkGoogle Scholar
  49. Wallis, John. 1972. Opera mathematica, 3 vols. Arithmetica infinitorum, Vol. 1. New York: Olms.Google Scholar
  50. Youschkevitch, Adolf P. 1970–1991. Euler. In Dictionary of scientific biography, ed. C.C. Gillispie Vol. 9, 467–484. New York: Scribner’s.Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada IUniversitat Politècnica de CatalunyaCataloniaSpain
  2. 2.Centre de Recerca per a la Història de la TècnicaUniversitat Politècnica de CatalunyaCataloniaSpain

Personalised recommendations