ABSTRACT
So how do you measure distance? With a ruler. But how do you
calibrate the ruler?
In Euclidean geometry, these questions are answered by the Pythagorean
Theorem. In infinitesimal form,1 and in standard rectangular coordinates,
the Pythagorean Theorem tells us that
ds2 = dx2 + dy2 (1.1)
as shown in Figure 1.1. This version of the Pythagorean Theorem not only
tells us about right triangles, but also how to measure length along any
curve: Just integrate (the square root of) this expression to determine the
arclength.
