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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
could you explain a geometric proof of the first result?
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