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Archimedes
Quadrature of the Parabola
Proposition 5 [quoting]
If Qq be the base of any segment of a parabola, P the vertex of the segment, and PV its diameter, and if the diameter of the parabola through any other point R meet Qq in O and the tangent at Q in E, then
QO : Oq = ER : RO.
Let the diameter through R meet QP in F.
Then, by Prop. 4,
QV : VO = OF : FR.
Since QV= Vq, it follows that
QV : qO = OF : OR …………………(1).
Also, if VP meet the tangent in T,
PT = PV, and therefore EF = OF.
Accordingly, doubling the antecedents in (1), we have
Qq : qO = OE : OR,whence
QO : Oq = ER : RO.
Archimedes. “Quadrature of the Parabola.” In The Works of Archimedes. Ed. T.L. Heath. Cambridge UP, 1897. Obtained at
https://archive.org/details/worksofarchimede00arch
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