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By Hedberg L.I.

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Again we are led to the suggestion that the original development of the construction of the pentagon was done in the manner shown; the mathematician involved simply had all his efforts concentrated on the frame defined by the isosceles triangle ofIV, 10, and he was led to a construction in which the bisection of the doubled angles gave equal angles, with the actual magnitudes of the arcs or central angles simply playing no role. B. the result is not used in the Elements. Of course this does not, a priori, exclude the result having some source or use exterior to the Elements.

ABE -... tlABH. ABE = ~ REA = ~BAH. From this we have EB:BA = AB:BH or, since by above BA = EH, BE:EH = EH:HB which is the definition of DEMR. Since BE is bigger than EH, V,14 tells us that HE is larger than HB - that is, HE, which was shown to be equal to the side of the pentagon, is the larger segment. Uses: 1,32; 111,28; IV,14; VI,33 ... Used in: XIII,11; XIII,17[a],corollary. Discussed in: Sections 2,C; 22,A. THEOREM XIII,9. Take any circle and let CD = a6 and Be = a 10. Then the line BD formed by adding segments Be and CD together is divided in EMR at C and CD = aa is the larger segment.

I wish to emphasize that in this section I remain entirely within the confines of the Euclidean text. There is no drawing upon any other material such as an ancient text which indicates the state of the art at a certain time or an historical comment on some mathematician. In later sections I will present all this material and inferences obtained from them by other authors as well as myself. It should be noted that in what follows a statement such as ~ 'Euclid showed ... " is to be interpreted as referring either to Euclid or to the text from which he took his material.

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