By Roger Herz-Fischler
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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Extra info for A Mathematical History of the Golden Number (Dover Books on Mathematics)
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