Torricelli / Infinitesimal geometry

Torricelli(1608-1647) had contributed to development of mathematics, physics, optics, dynamics, ballistics and meteorology. Among them, he is especially famous in 'Torricelli's experiment' that created the first vacuum of human beings.

A basic concept of Torricelli is a "non-separable method". He improved the method for application to some curves and curved surfaces.

[Area of circle]

Let represent a straight line that equals to length of the circle of radius . We draw a circle having a radius at a center , passing a point on the radius . We also draw a straight line in parallel with .
Length of a line segment equals to that of a circle of radius as triangles and is similar.
A set of non-separable circles corresponds to a set of non-separable triangles. Areas of the circles equal to those of the respective triables.
Therefore, .

[Sharp-pointed body of hyperboloid]

Volume of an infinitely extended solid is calculated below. The volume is that of a body of rotation with a center on axis of a hyperbolic curve . That can be expressed as .

As seen from the figure above, a figure extending infinitely has finite volume. This is the first discovery at the time of Torricelli. He showed that the non-separable amount of the solid was formed of surface of a circular cylinder having sides .
Let surface area of the circular cylinder be , length of the diameter of the bottom of the circular cylinder be , and height be .
Area of the circle of the diameter in the circular cylinder equals to surface area of the circular cylinder . Therefore, area of the sharp-pointed hyperboloid body equals to non-separable area of the circular cylinder . Volume of the sharp-pointed body of hyperboloid equals to that of the circular cylinder .

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