Last week, in Fundamental Understanding of Mathematics XLVII, I mistitled the diary as number 47 rather than number 68, which would be LXVIII. In addition to making mistakes with roman numerals, we took a look at extending our discovery about triangles to cover parallelograms.
This week, we will make one more extension to that idea, which was codified into a mathematical principle by a gentleman named Bonaventura Cavalieri, a contemporary of Galileo Galilei, the notorious 17th century heretic.
It is thus known as Cavalieri's Principle, because he wrote about it in Latin.
Cavalieri's Principle has a two dimensional and a three dimensional version. The two dimensional version says
Two figures with horizontal cross sections of the same length have the same area.
The three dimensional version says
Two objects with equal corresponding cross-sectional areas have the same volume
So, we have a new idea: a cross section. What is it?
If we take some figure, and draw a horizontal line across it, the length of the line inside the figure (shown in yellow) is the cross section at that height.
By calling this shape a "figure" we mean that it is a flat two dimensional shape, and it has an area, but it does not have a volume. The length of the yellow line is the "horizontal cross section" at either Height H1 or Height H2. We can, of course, pick any height, draw another horizontal line, and find a new cross section.
Cavalieri says that if we tweak the shape in such a way that all the cross sections remain the same length
then the new shape will have the same area as the original shape.
We can show that this is true for regular figures by cutting the figure into parallelograms and triangles
Since the individual pieces keep the same area when deformed this way (same base and height means same area) when you add all the little areas up they will add up to the same larger area for the whole figure.
Cavalieri's insight was that he could take some arbitrary shaped figure
And slice it into parallelograms and triangles
and if he cut into small enough pieces
the odd shaped pieces at the edge would be indistinguishable from parallelograms or triangles, so a cross section preserving tweak would produce a figure with the same area.
Cavalieri was basically inventing calculus from a geometric point of view.
Have fun in the comments.