Euclid's Book on Divisions of Figures by Raymond Clare Archibald

By Raymond Clare Archibald

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Then the quadrilaterals Ibcd, tbed are equal and the construction has been made as required ; for ct :ta = ez\ zi= quadl. Ibcd: Adal. And if bd intersect ta [a similar construction may be given to divide the given quadrilateral, by a line through dy into a quadrilateral and triangle in the required ratio]. Leonardo then gives the construction for dividing a quadrilateral in a given ratio by a line drawn through a point which divides a side of the quadrilateral in the given ratio. 37] PROPOSITIONS PROPOSITION 47 14-16 16.

Part of Leonardo 27s6, p . 1 2 7 , 11. 2 - 3 . " At any rate Leonardo gives constructions for the cases when the given point is on any side. These I shall take up successively. The figure is made from more than one of Leonardo's, and there is a slight change in the lettering. Let ad be the shorter of the parallel sides ad, bg, which are bisected in t and k respectively. Join tk. Then if bt, gt be joined, [it is clear, from triangles on equal bases and between the same parallels, that tk bisects the trapezium].

Suppose however that the given point were at b. Join bd. Through n draw ne parallel to bd. Join be. Then be divides the trapezium in the required ratio. Similarly for the p o i n t y and for any point between b and n, or between g and /. Some of the parts which I have filled in above are covered by the general final statement: " nee non et diuidemus ipsum quadrilaterum ab omni puncto dato super aliquod laterum ipsius " (Page 134, 11. I O - I I . ) PROPOSITION 10. 31. " [Leonardo 18, p. 124, 11.

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