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QUADRATURE OF THE PARABOLA.

"Archimedes to Dositheus greeting.

"When I heard that Conon, who was my friend in his lifetime, was dead, but that you were acquainted with Conon and withal versed in geometry, while I grieved for the loss not only of a friend but of an admirable mathematician, I set myself the task of communicating to you, as I had intended to send to Conon, a certain geometrical theorem which had not been investigated before but has now been investigated by me, and which I first discovered by means of mechanics and then exhibited by means of geometry. Now some of the earlier geometers tried to prove it possible to find a rectilineal area equal to a given circle and a given segment of a circle; and after that they endeavoured to square the area bounded by the section of the whole cone* and a straight line, assuming lemmas not easily conceded, so that it was recognised by most people that the problem was not solved. But I am not aware that any one of my predecessors has attempted to square the segment bounded by a straight line and a section of a rightangled cone [a parabola], of which problem I have now discovered the solution. For it is here shown that every segment bounded by a straight line and a section of a right-angled cone [a parabola] is four-thirds of the triangle which has the same base and equal height with the segment, and for the demonstration of this property the following lemma is assumed: that the excess by which the greater of (two) unequal areas exceeds the less can, by being added to itself, be made to exceed any given finite area. The earlier geometers have also used this lemma; for it is by the use of this same lemma that they have shown that circles are to one another in the duplicate ratio of their diameters, and that spheres are to one another in the triplicate ratio of their diameters, and further that every pyramid is one third part of the prism which has the same base with the pyramid and equal height; also, that every cone is one third part of the cylinder having the same base as the cone and equal height they proved by assuming a certain lemma similar to that aforesaid. And, in the result, each of the aforesaid theorems has been accepted* no less than those proved without the lemma. As therefore my work now published has satisfied the same test as the propositions referred to, I have written out the proof and send it to you, first as investigated by means of mechanics, and afterwards too as demonstrated by geometry. Prefixed are, also, the elementary propositions in conies which are of service in the proof {arot-^ela Kcoviko, xpeiav e%ovTa e's rdv dvoBei^iv). Farewell."

* There appears to be some corruption here: the expression in the text is rat oXou Too Kuvov To/iai, and it is not easy to give a natural and intelligible meaning to it. The section of 'the whole cone' might perhaps mean a section cutting right through it, i.e. an ellipse, and the ' straight line' might be an axis or a diameter. But Heiberg objects to the suggestion to read Toi i^uywlov Kizvov rofiat, in view of the addition of xal euitlat, on the ground that the former expression always signifies the whole of an ellipse, never a segment of it iQutirstionet Archimedeae, p. 149).

Proposition 1.

If from a point on a parabola a straight line be drawn which is either itself the axis or parallel to the axis, as PV, and if QQ' be a chord parallel to the tangent to the parabola at P and meeting PV in V, then

QV = VQ'.

Conversely, if QV= VQ', the chord QQ' will be parallel to the tangent at P.

* The Greek of this passage is: avfifiaivci Si ruv Trpofipriuifuv Beupri/iiTuv luaaTov H7)5h rjtraov Tuv &vtv roirov Tou Xt)/x//otos ivoSeSuyiiivuv TcmirrcvKivai. Here it would seem that vtrurrtvKivai must be wrong and that the passive should have been used.

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Proposition 2.

If in a parabola QQ' be a chord parallel to the tangent at P, and if a straight line be drawn through P which is either itself the axis or parallel to the axis, and which meets QQ' in V and the tangent at Q to the parabola in T, then

PV= PT.

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Proposition 3.

If from a point on a parabola a straight line be drawn which is either itself the axis or parallel to the axis, as PV, and if from two other points Q, Q' on the parabola straight lines be drawn parallel to the tangent at P and meeting PV in V, V respectively, then

PV: PV' = QV* : Q'V\ "And these propositions are proved in the elements of conies.*"

Proposition 4.

// 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 O and QP (produced if necessary) in F, then

QV:VO = OF: FR.

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

* i.e. in the treatises on conies by Enclid and Aristaeus.

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

therefore

PQ : PF = PF: PK

= PQ±PF:PF±PK

= QF: KF. 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).

2

For, if QV = a, PV=—, where p is the parameter of the ordinates to PV.

Thus, if QO = x, and RO y, the above result gives
a OF

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Proposition 5.

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.

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