An important step in 17th-century research on quadratures involved the use of algebraic procedures. Pietro Mengoli (1625--1686), probably the most original student of Bonaventura Cavalieri (1598--1647), was one of several scholars who developed such procedures. Algebra and geometry are closely related in his works, particularly in Geometriae Speciosae Elementa [Bologna, 1659]. Mengoli considered curves determined by equations that are now represented by y=Kxm(t-x)n. This paper analyzes the interrelation between algebra and geometry in this work, showing the complementary nature of the two disciplines and how their combination allowed Mengoli to calculate quadratures in a new way.
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