Comments by "EebstertheGreat" (@EebstertheGreat) on "The Discovery That Transformed Pi" video.

  1. Archimedes' approach to this problem was interesting, because at the time, many philosophers considered it to be impossible to measure the length of a curve. The reason for this is rather subtle, because they didn't have a problem with the idea of measuring the area of a curved region like a disk. This is because a bounded region must be greater in area than any inscribed shape and less in area than any circumsribed shape. Like, if I can draw a square entirely inside some curvy region and another square entirely outside it, then the curvy region must have an area between those two squares. But for arc lengths, that reasoning does not apply. Imagine drawing a very spiky star inside a circle. That star has a way larger perimeter than the circumference of the circle. Archimedes reasoned--correctly--that the same containment ought to apply for convex regions. Today, there is no surviving ancient proof that the circumference and area of a circle depend only on their radius and radius squared, respectively. In other words, there is no surviving proof from the time that π is a constant independent of the circle. This fact cannot be proved from Euclid's axioms, because arc length cannot even be defined using those axioms except for piecewise-straight lines. Euclid's Elements presents Eudoxus of Cnidus's proof that A/r² is constant for all circles based on the axiom that "the whole is greater than the part," applied to areas, but he never once mentions a formula for the circumference of a circle. David Richeson speculates that Archimedes provided a proof that C = 2A/r (i.e. C = 2πr) in his work Measurement of a Circle, but that work only survives in fragmentary form, and such a theorem is absent. However, the theorem can easily be proved from the two additional postulates Archimedes introduces in his work On the Sphere and the Cylinder which include measurements of the surface areas and volumes of spheres and cylinders. These two postulates are: (1) A straight line has the least length of any curve connecting two given points, and (2) If a curve concave in one direction connects two points and is entirely between another curve concave in the same direction connecting those points and a straight line connecting those points, then its length is strictly between the lengths of the other curve and the straight line. Thus, we can sandwich a circle between regular polygons and determine its circumference the same way we determined its area. These postulates become theorems in modern geometry which use some sort of completeness axiom to prove the existence of limiting points necessary to do calculus.
    17