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(diagram 2 = general diagram)
The purpose of this theorem, and as with others in the group is to establish a method of compression. Obviously as H gets nearer to E the difference between Z and L also gets smaller.
(diagram 1) 10. Again, let ABF be a balance and its middle B, and let BDHK be a trapezoid having right angles at points B, H, but side KD converging to G, and let trapezoid BDHK have this ratio to L that which AB has to BH, and let trapezoid BDHK be suspended from the balance at points B, H, and let area Z also be suspended at A and let it incline equally to trapezoid BDKH holding as it is now positioned. I say that area Z is less than area L.
(diagram 2) For let AG be cut at E in such a way that EH have this ratio to BE, that which the double of DB and KH have a ratio to the double of KH and BD, and let EN, after being drawn through E parallel to BD, be bisected at Q. Q is, in fact, the center of the weight of trapezoid BDHK. For this has been proved in the Mechanics.(*) And so if trapezoid BDHK is suspended at E and if it is released from points B, H, it maintains the same situation for the same (reasons) as those before and inclines equally to area Z. (diagram 3) And so since trapezoid BDHK when suspended at E inclines equally to area Z when suspended at A, as AB will be to BE trapezoid BDHK will be to area Z. Therefore, trapezoid BDHK will have a larger ratio to Z than to L, since AB will also have a larger ratio to BE than to BH. Thus Z will be smaller than L.
(*) This book does not survive, but in his Mechanica (II 35), Heron of Alexandria ascribes certain fundamental theorems to Archimedes, However, he only provides a general strategy for finding the center of weight of a quadrilateral (Mechanica II 36). The theorem in question is proved in Archimedes, Equilibria of Planes I 15. or another reference to the Mechanica see Quadrature of the Parabola 6.