EXHAUSTING THE CIRCLE - ON MATHEMATICS AND HISTORY OF RECTIFICATION

摘要

We describe the history of the formulae 2πr and πr~2 for the circle's perimeter and area, respectively, a full understanding of which turns out to demand profound reflection that most students are not aware of. We thereby explain the difficulty why the ancients fell short of defin-ing circular arc-lengths, like us moderns, as limit of inscribed polygons with infinitely many sides (as observed by Chasles above, and later on by Dieudonne). The appendices contain a comprehensive history of rec-tification~2, a notion far more delicate than area and volume, with key observations why the method of "indivisibles" (in the phraseology of the 17th century), when applied to rectification, was fully rigorized only by the end of the 19th century (by L. Scheeffer and C. Jordan).