Archimedes’ ‘Quadrature of the Parabola’, Prop4

by Corry Shores
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Archimedes

Quadrature of the Parabola

Proposition 4 [quoting]

 

If Qq be the base of any segment of a parabola, and P the vertex of the segment, and if the diameter through any other point R meet Qq in and QP {produced if necessary) in F, then

QV : VO = OF : FR.

Draw the ordinate RW to PV, meeting QP in K.

Then

PV : PW = QV² : RW²;

whence, by parallels,

PQ : PK = PQ² : PF²

In other words, PQ, PF, PK are in continued proportion;
therefore

Hence by parallels

QV : VO = OF : FR.

[It is easily seen that this equation is equivalent to a change of axes of coordinates from the tangent and diameter to new axes consisting of the chord Qq (as axis of x, say) and the diameter through Q (as axis of y).
For, if

where p is the parameter of the ordinates to PV. Thus, if QO = X, and RO = y, the above result gives

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

https://archive.org/details/worksofarchimede00arch