As a youth, Torricelli took courses in mathematics and philosophy with the Jesuits in Faenza, Italy. They noticed his outstanding promise and sent him for further education to a school in Rome run by a former student of Galileo’s. Torricelli himself may be viewed as Galileo’s last pupil, for he came to live with the blind and ill Galileo in 1641. They had only a little time to work together, for the aged scholar died within three months.

Appointed to the chair of mathematics in Florence, the position left vacant by Galileo, Torricelli’s own career was cut short when he died suddenly, probably of typhoid fever, five years later at the age of 39. He is often remembered today for his demonstration of the weight of air. The demonstration consisted of taking a long tube filled with mercury and sealed at one end, and inverting it into a basin of mercury; the changing pressure of air on the free surface of mercury in the basin made the level in the tube stand higher on some occasions than on others.

Torricelli was a mathematician of considerable accomplishment. Using Cavalieri’s method of indivisibles, he solved the famous problem of finding the area under one arch of the cycloid; later, he determined the length of the infinitely many revolutions of the logarithmic spiral (in polar coordinates, In 1641, he established a result so astonishing that mathematicians of the day thought it to be impossible: there is a geometric solid which is infinitely long, but nonetheless has a finite volume. The body, which he called "the acute hyperbolic solid," is generated when the region bounded by a branch of the hyperbola y = 1/x, the line x = 1 and the x-axis is revolved around the x-axis. Its finite volume is given in modern notation by the integral

When he communicated his discovery to the French geometers in 1644, Torricelli’s status changed from being a virtual unknown to one of the most acclaimed mathematicians in Europe. The proof itself constituted the high point in the Opera geometrica (1644), the only work of Torricelli to be published in his lifetime.