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Mathematical Treasure: Cavalieri's Indivisibles

Frank J. Swetz (The Pennsylvania State University)

Bonaventura Cavalieri (1598-1647) was an Italian mathematician and a Catholic priest of the Order of Jesuati. He is perhaps best remembered for his contributions to the foundations of the calculus through the use of “indivisibles.” His theories on this concept were summarized in his Exercitationes geometricae sex (1647).

Title page from Exercitationes geometricae sex by Bonaventura Cavalieri, 1647

The diagram on page 89 illustrates the concept that would eventually be known as “Cavalieri’s Principle”: if two solids of the same height are bisected by parallel planes, such that each pair of cross-sections resulting are of equal area, then the two solids possess the same volume.

Page 89 of Exercitationes geometricae sex by Bonvaventura Cavalieri, 1647

Cavalieri’s first manuscript containing his ideas on indivisibles was completed in 1627 and published in 1635 (please see “Mathematical Treasure: Cavalieri’s Geometria Indivisibilibus” for more on this earlier publication). Exercitationes geometricae sex was intended to improve the exposition and respond to criticisms of the earlier work.

The images above are provided courtesy of Columbia University Libraries. A copy of this work is also available in its entirety in the Internet Archive.

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Cavalieri's Indivisibles," Convergence (August 2018)