Archimedes’ [P21] ‘Quadrature of the Parabola’, Proposition 21

by Corry Shores
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Archimedes

Quadrature of the Parabola

Proposition 21 [quoting]

Proposition 21.

P21. If Qq be the base, and P the vertex, of any parabolic segment, and if R be the vertex of the segment cut off by PQ, then

ΔPQq = 8ΔPRQ.

The diameter through R will bisect the chord PQ, and therefore also QV, where PF is the diameter bisecting Qq. Let the diameter through R bisect PQ in Y and QVinM. Join P3I. By Prop. 19,

Also, if RW, the ordinate from R to PV, be produced to meet the curve again in r,

RW = rW,

and the same proof shows that

ΔPQq = 8ΔPRQ.

 

Archimedes. “Quadrature of the Parabola.” In The Works of Archimedes. Ed. T.L. Heath. Cambridge UP, 1897. Obtained at

https://archive.org/details/worksofarchimede00arch