Archimedes, Quadrature of the Parabola (Heath 1897)
Source: Internet Archive
worksofarchimede00arch; local OCR excerpt.txt, lines 21260–22152 of the archive’s_djvu.txt, retrieved 2026-09-08. Metadata: The Works of Archimedes, edited in modern notation with introductory chapters by T. L. Heath (Cambridge: Cambridge University Press, 1897), “Quadrature of the Parabola”, pp. 233–252. Extraction: archive.org OCR of the 1897 printing; diagrams and Greek quotations are lost or garbled, the English prose is clean. Rights: public domain.
Source digest
The treatise proves that a parabolic segment is four-thirds of the triangle with the same base and equal height. Proposition 17 gives the mechanical proof by balancing, and Proposition 24 the geometrical proof by exhaustion. The prefatory letter to Dositheus states that the theorem was first discovered by mechanics and then demonstrated by geometry.
For this project, the constant-force lens area of C001 is exactly this quantity. With \(x=v\tau\), \(y=F\tau^2/(2m)\) and apex at the half-time point, the inscribed triangle has area \(Fv\tau^3/(16m)\), and four-thirds of it is \(A_{\rm lens}=Fv\tau^3/(12m)\); the identity was checked symbolically on 2026-09-08. The letter’s discovery-then-proof theme also parallels Newton’s account of analysis and synthesis in NATP00385.
Passage anchors
- Excerpt line 1: heading, then “Archimedes to Dositheus greeting”.
- Archive line 21286: the statement that the segment “is four-thirds of the triangle which has the same base” in the letter.
- Archive lines 21841 and 22082: the same statement at Propositions 17 and 24.
Coverage and limits
Reading level passage for the letter and Propositions
16–17 and 24 on OCR text; metadata for the intermediate
lemmas. Figures require the scan.