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generally), then however small the excess of the greater over 
the lesser, it can by being continually added to itself be made 
to exceed the greater; in other words, he confirmed the solution 
by the method of exhaustion. One solution by means of 
mechanics is, as we have seen, given in The Method; the 
present treatise contains a solution by means of mechanics 
confirmed by the method of exhaustion (Props. 1-17), and 
then gives an entirely independent solution by means of pure 
geometry, also confirmed by exhaustion (Props. 18-24), 
I. The mechanical solution depends upon two properties of 
the parabola proved in Props. 4, 5. If Qq be the base, and P 
the vertex, of a parabolic segment, P is the point of contact 
of the tangent parallel to Qq, the diameter PV through P 
bisects Qq in V, and, if VP produced meets the tangent at Q 
in T, then TP — PV. These properties, along with the funda 
mental property that QV 2 varies as PV, Archimedes uses to 
prove that, if EO be any parallel to TV meeting QT, QP 
(produced, if necessary), the curve, and Qq in E, F, R, 0 
respectively, then 
QV: VO = OF: FR, 
and QO : Oq = ER: RO. (Props. 4, 5.) 
Now suppose a parabolic segment QR x q so placed in relation 
to a horizontal straight line QA through Q that the diameter 
bisecting Qq is at right angles to QA, i.e. vertical, and let the 
tangent at Q meet the diameter qO through q in E. Produce 
QO to A, making OA equal to OQ. 
Divide Qq into any number of equal parts at 0 1? 0 2 ... 0 n , 
and through these points draw parallels to OE, i. e. vertical 
lines meeting OQ in H x , H 2 , ..., EQ in E x , E 2 , .and the