Article ID: CBB000932080

Euler's Beta Integral in Pietro Mengoli's Works (2009)

unapi

Abstract 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.

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Authors & Contributors
Massa i Esteve, Maria Rosa
M. Rosa Massa-Esteve
Jens Lemanski
Mellado-Romero, Antonio
Linero-Bas, Antonio
Herrero-Piñeyro, Pedro J.
Concepts
Mathematics
Algebra
Calculus
Integral equations
Mathematical analysis
Visual representation; visual communication
Time Periods
17th century
18th century
19th century
Renaissance
16th century
Places
England
Poland
Italy
Germany
France
Europe
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