By Roger Herz-Fischler

ISBN-10: 0486400077

ISBN-13: 9780486400075

A complete research of the old improvement of department in severe and suggest ratio ("the golden number"), this article lines the concept's improvement from its first visual appeal in Euclid's *Elements* during the 18th century. The coherent yet rigorous presentation bargains transparent reasons of DEMR's historic transmission and contours quite a few illustrations.

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This is isosceles which means that (1,5) 1:-FAB = 1:ABF. Thus (1,32) exterior angle BFG = 2·~FAB. Hence 2·1:-BFM= ~BFG = 2-~FAB and 1-BFM = ~FAB. Ifwe now look at ~ABF and 6BFN, we see that they are similar because two angles are equal and 1:ABF is common. By proportionality and VI,17 we have (1) o F,GURE 1-30. XIII,9 Proof: Draw the diagram as shown. Since BC is the side of the decagon arc (ACB) = 5 -arc (BC) so that S(BF) = R(AB,BN). Nowwetumourattentionto~KNAand~KBA and show that these are similar.

In case the given figure is not a triangle, one may use techniques similar to those used to prove 1,45 from 1,42,44 and VI,25, as it stands, from 1,44,45. THEOREM VI,27. Let AB be any line with C its midpoint and let CBED be an arbitrary parallelogram. Let KBHF (III) be a similar parallelogram and construct the figure shown. Then the area of parallelogram ACDL-which is the same as that of the parallelogram CBED-is greater than that of parallelogram AKFG. ) THEOREM E 1-19. VI,27 Proof: I = II (1,43), therefore I + III = II + III.

For a discussion, see Taisbak [1982, 63] and Mueller [1981, 262]. THEOREM FIGURE 1-33. XIII,13'(i) Following is a proof of (i) which emulates that proof of (ii) found in the Elements. On AB construct square THE EUCLIDEAN TEXT ABGE. Then, since rectang~es which have the same altitudes have their areas in the same ratio as the bases (this is VI,I), we have AB: Be = R(AB,AE): R(BC,AE) = R(AB,AB): R(BC,AB) = S(AB): R(BC,AB). However, from VI,S AB: BD = BD: Be and VI,17 tells us R(BC,AB) = S(BD). THEOREM XIII,14.

### A Mathematical History of the Golden Number (Dover Books on Mathematics) by Roger Herz-Fischler

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