Torricelli’s Computation of the Area Under the Cycloid

Abstract

Like many mathematicians of his era, Evangelista Torricelli (1608-1647) studied the cy- cloid. This paper examines Torricelli’s proof that the area under the cycloid (the quadrature) is three times the area of the generating circle. Torricelli proved this result in three distinct ways. It is worth noting that he carried on an extensive mathematical correspondence with Cavalieri and it is clear that Cavalieri’s techniques influenced his proofs. What follows is a discussion of Torricelli’s proofs with an emphasis on the use of symmetry and geometric reasoning, as viewed through the lens of graphical representations of the proofs.