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
me...
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.
Citation
Massa, M.; Delshams, A. Euler's beta integral in Pietro Mengoli's works. "Archive for history of exact sciences", Març 2009, vol. 63, núm. 3, p. 325-356.