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infinity, that lines radiating from this "cæcus focus" are parallel and have no other point at infinity.

The Stereometria led Cavalieri, an Italian Jesuit, to the consideration of infinitely small quantities. Bonaventura Cavalieri (1598–1647), 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 method of exhaustion of the Greeks and the methods of Newton and Leibniz. He considers lines as composed of an infinite number of points, surfaces as composed of an infinite number of lines, and solids of an infinite number of planes. 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 finds the sum of the squares of all lines making up a triangle equal to one-third the sum of the squares of all lines of a parallelogram of equal base and altitude; for if in a triangle, the first line at the apex be 1, then the second is 2, the third is 3, and so on; and the sum of their squares is

$\scriptstyle {1^{2}+2^{2}+3^{2}+\cdots +n^{2}=n(n+1)(2n+1)\div 6}$.

In the parallelogram, each of the lines is n and their number is n; hence the total sum of their squares is $\scriptstyle {n^{2}}$. The ratio between the two sums is therefore

$\scriptstyle {n(n+1)(2n+1)\div 6n^{3}={\frac {1}{3}}}$,

since n is infinite. From this he concludes that the pyramid or cone is respectively $\scriptstyle {\frac {1}{3}}$ of a prism or cylinder of equal base and altitude, since the polygons or circles composing the former decrease from the base to the apex in the same way as the squares of the lines parallel to the base in a triangle decrease from base to apex. By the Method of Indivisibles, Cavalieri