## Shears

A shear is a transformation of a rectangle into a parallelogram which preserves one base and the corresponding height.

One fundamental fact about shears is that

Shears preserve area.

Since a shear takes a rectangle into a parallelogram, this asserts also that:

The area of a parallelogram is equal to the product of its base and height.

This is Proposition I.35 of Euclid. It can be demonstrated in several ways, some of which are suggested by the following pictures. The first few are what might be called static arguments.

The figure above proves the assertion by a kind of subtraction of geometric figures.

Euclid's proof above also uses subtraction.

The basic idea here is to partition the rectangle and its transform so as to match up congruent pieces. This is complicated only because the number of pieces grows as the shear becomes more extended.

The dynamic argument is perhaps more intuitive. We can think of the rectangle as being made up of an infinite number of thin slices, none of which changes shape in the course of the shear. A shear thus acts like sliding a deck of cards along horizontally. A rigourous version of this argument naturally involves limits.