  In his famous work, "Measurement of a Circle," Archimedes described a
procedure for measuring both the circumference of a circle and the area it
bounds. Implicit in his work is the idea that his procedure defines these
quantities. Modern approaches for defining $\pi$ eschew his method and instead
use arguments that are easier to justify, but they involve ideas that are not
elementary. This paper makes Archimedes' measurement procedure rigorous from a
modern perspective. In so doing, it brings a rigorous and geometric treatment
of the differential properties of the trigonometric functions into the purview
of an introductory calculus course.
