"... a pupil of Galileo and professor at Bologna, is celebrated for his Geometria indivisibilibus continuorum nova quadam ratione promota 1635. This work expounds his method of Indivisibles, which occupies an intermediate place between the of the Greeks and the methods of Newton and Leibniz. Indivisibles were discussed by Aristotle and the scholastic philosophers. They commanded the attention of Galileo. Cavalieri does not define the term. He borrows the concept from the scholastic philosophy of Bradwardine and Thomas Aquinas, in which a point is the indivisible of a line, a line the indivisible of a surface, etc. Each indivisible is capable of generating the next higher continuum by motion; a moving point generates a line, etc. The relative magnitude of two solids or surfaces could then be found simply by the summation of series of planes or lines. For example... he concludes that the pyramid or cone is respectively 1/3 of a prism or cylinder of equal base and altitude... By the Method of Indivisibles, Cavalieri solved the majority of the problems proposed by Kepler. Though expeditious and yielding correct results, Cavalieri's method lacks a scientific foundation. If a line has absolutely no width, then the addition of no number, however great, of lines can ever yield an area; if a plane has no thickness whatever, then even an infinite number of planes cannot form a solid. Though unphilosophical, Cavalieri's method was used for fifty years as a sort of integral calculus. It yielded solutions to some difficult problems. [Paul] Guldin made a severe attack on Cavalieri... [who] published in 1647... a treatise entitled Exercitationes geometriece sex in which he replied to the objections of his opponent and attempted to give a clearer explanation of his method. ...A revised edition of the Geometria appeared in 1653."