Method of indivisibles

Bonaventura Cavalieri's method of indivisibles, published in 1635, provided a technique for calculating areas and volumes that would become the foundation for integral calculus. The idea was geometrically intuitive: a plane figure could be imagined as composed of infinitely many parallel lines, and its area could be compared to another figure by comparing their cross-sections.

The principle, now called Cavalieri's principle, states that if two solids have equal cross-sectional areas at every height, they have equal volumes. A cylinder and a prism with equal base areas and heights have the same volume because every horizontal slice has the same area. No ancient mathematics could prove this rigorously, but Cavalieri's approach let mathematicians calculate volumes that had resisted two millennia of geometric attack.

The method built on Archimedes' quadrature techniques but went further. Where Archimedes had exhausted figures with polygons of increasing complexity, Cavalieri treated areas as actually composed of indivisible lines. This was philosophically troubling—how can lines without breadth sum to areas with breadth?—but mathematically productive.

Kepler had used similar infinitesimal reasoning in his astronomical work and his treatise on wine barrel volumes. Galileo explored related ideas. The conceptual framework for treating quantities as composed of infinitely small parts was spreading through 17th-century mathematics, preparing the ground for the calculus that Newton and Leibniz would formalize.

The method of indivisibles demonstrated that infinite processes could yield finite, useful results. Integration—summing infinitely many infinitely small quantities—is precisely what Cavalieri's technique accomplished, even if the logical foundation remained unclear until later centuries.

Cavalieri's principle survives in every calculus textbook. The integral sign that Leibniz would introduce is an elongated S for "summa"—a sum of indivisibles. The method that appeared scandalously unrigorous in 1635 became, after refinement, one of mathematics' most powerful tools.