Born in Milan, Cavalieri joined the rather small Jesuit religious order as a boy of 15. At the age of 23, he took up an intensive study of mathematics, having been told that it was an effective remedy against depression. He then visited with Galileo at Padua and for a short while was a lecturer in mathematics at Pisa. Cavalieri always considered himself a disciple of Galileo, writing the great man 112 letters between 1619 and 1641. In 1629, probably through Galileo’s influence, Cavalieri was made a professor at the University of Bologna, where he remained until his death.

Cavalieri’s method of "indivisibles" was a crude and naive foreshadowing of the integral calculus. The method rests on the assumption that a line is composed of an infinite number of successive points, a plane figure of an infinite number of parallel line segments, and a solid body of an infinite number of parallel plane areas. The determination of, say, a plane area was obtained by dividing it up into equally spaced parallel lines, arbitrarily close together, and summing up "all the lines" (that is, the space between them) as the number of lines becomes infinitely large. We know that these ideas were developed in 1629, for Cavalieri wrote Galileo about them; but they did not appear in print until the publication of his Geometria indivisibilibus (1635). The work was roundly criticized and later expanded into the influential Exercitationes geometricae of 1647. Although Cavalieri’s techniques yielded correct results -- such as the area of an ellipse and the volume of a sphere -- they lacked a mathematical foundation.