Paul Guldin's critique of Bonaventura Cavalieri's indivisibles is contained in the fourth book of his De Centro Gravitatis (also called Centrobaryca), published in 1641. Cavalieri's proofs, Guldin argued, were not constructive proofs, of the kind that classical mathematicians would approve of. This was undoubtedly true: in the conventional Euclidean approach, geometric figures are constructed step-by-step, from the simple to the complex, with the aid of only a straight edge and a compass, for the construction of lines and circles, respectively. Every step in a proof must involve such a construction, followed by a deduction of the logical implications for the resulting figure.Cavalieri, however, proceeded the other way around: he began with ready-made geometric figures such as parabolas, spirals, and so on, and then divided them up into an infinite number of parts.Such a procedure might be called "deconstruction" rather than "construction," and its purpose was not to erect a coherent geometric figure but to decipher the inner structure of an existing one. m at h e m at i c s
the secret spiritual history of calculus
Integral calculus originated in a 17th-century debate that was as religious as it was scientific