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 
.
|
Total(1997) |
|
Japanese sites Mathematical Formulas| Kodawari House| Pinpoint StreetView| Excel VBA Techniques| Excel Formula Analysis| |