Archimedes’ ‘Quadrature of the Parabola’, Prop5

by Corry Shores
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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