There are four different lines of research evident in the works of Torricelli, which cover almost all the subjects treated by Galileo, with the exception of astronomy. In the first - Geometry - Torricelli obtained remarkable results, in particular the quadrature of the cycloid curve and the cubature of the hyperboloid of revolution (acute hyperbolic solid). He was the first to use "curved indivisibles", contributing in this way to the "method of indivisibles" introduced into geometry by Bonaventura Cavalieri. The close scientific collaboration between the two outstanding mathematicians, also extremely close friends, is documented by a rich scientific correspondence.
The second direction of research, to maintain this system of classification, consists of the application of geometry to the study of motion.
The barometric experiment, which led to the invention of the mercury barometer, and the development of techniques for producing telescope lenses are two final directions of research in which Torricelli demonstrated an ability comparable to that of a skilled artisan.
Torricelli's work on geometry and its applications displayed his brilliant intellectual capacities most clearly. His interest in abstract reasoning was combined with the refusal, expressed on numerous occasions, to allow himself to be imprisoned within the narrow limits of a particular physical phenomenon. "I propose", he wrote to his friend Michelangelo Ricci in a letter of February 1646, "or suppose that a body or point will move down and up with the said proportion and horizontally with equal motion. If this is the case, I say that all of what Galileo has said and I have restated will follow. If, then, balls of lead, iron, or stone do not observe the supposed proportion, that's their bad luck and we will say that we are not speaking of them".
Coming from a scientist who spent much of his time constructing instruments designed for the study of natural phenomena, these observations might seem rather surprising at first glance. In reality, Torricelli's interests were not solely directed to abstract reasoning. One could almost say that he contained two different individuals: the technician, who perfected the practical methods for making telelscope lenses without worrying about the theoretical aspects of the problem, and the motion theorist who was unwilling to seek experimental proof, perhaps because he did not believe in the irrefutable proof furnished by the direct observation of phenomena.