Cavalieri's Principle

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Cavalieri’s Principle

 

If the areas of the cross sectionsof two solids by any plane parallel to a given plane are invariablyequal, then the two solids have the same volume. This proposi-tion (and the analogous one for plane figures), which was alreadywell-known to ancient Greek mathematicians, is usually calledCavalieri’s principle, although the Italian mathematicianF. B. Cavalieri in his Geometry (1635) does not take it as aprinciple but proves it.

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On the other side of the question were leading intellectuals of the early modern age, including Galileo, whose final book in 1633 expounded the theory of "indivisibles"; Bonaventura Cavalieri, whose name and books on geometry were often cited by later mathematicians; Evangelista Torricelli, who provided a rigorous defense and series of proofs using indivisibles; and John Wallis, who sparred with Hobbes's philosophies and mathematical claims for several decades through a series of books and pamphlets.
In the early seventeenth century, Bonaventura Cavalieri and his circle also looked inside geometric shapes, arguing that you could establish the equivalence of two triangles, for example, by adding all the lines together that made them up.