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Bonaventura Cavalieri and the theory of indivisibles. (English) Zbl 0933.01003

Bologna: Edizioni Cremonese, 104 p. (1980).
The reprint of Cavalieri’s work “Exercitationes geometricae sex” (see Zbl 0933.01004) is especially valuable because the editor added a booklet consisting of five sections: 1. A short account of Cavalieri’s life, 2. The theory of indivisibles, 3. The controversy with Guldin and the genesis of the Exercitationes, 4. The Exercitationes geometricae sex: Content and comments, 5. A list of Cavalieri’s published works.
Cavalieri detached the joint tyranny of the indirect proof and of the special case with his mathematical ideas. His purposes were the calculation of areas and volumes, not the composition of the continuum. Giusti clarifies the nature of Cavalieri’s new objects (all the lines, all the planes) and the meaning and procedures of the operations on them. Neither the concept of set nor that of integral is adequate for a complete description of the new objects.
It is a question of the ancient and ambiguous category of magnitudes as it may be found in its final form in its fifth book of Euclid’s Elements. Giusti outlines the results that Cavalieri was able to deduce. While Cavalieri looked at the indivisibles from the point of view of a mathematician, Galileo looked at them from that of a philosopher. Cavalieri is busy distinguishing between the infinity of the number of indivisibles and the finiteness of “all the lines”. Though Galileo denied the possibility of measuring the continuum by means of its indivisibles, he did not deprive himself of the advantages that stem from the reasoning by indivisibles in the actual proofs. Cavalieri’s contemporaries were extremely reluctant to enter in questions of principle concerning the metaphysics of the theory, but at the same time very open-minded to use the method in geometrical research.

MSC:

01A40 History of mathematics in the 15th and 16th centuries, Renaissance

Biographic References:

Cavalieri, Bonaventura

Citations:

Zbl 0933.01004
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