Urban Land Use Planning

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By George Sarton

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Gauss, All these sary 18 26 if men proved 1799). that the fifth postulate is not neces- one accepts another postulate rendering the same Yes, Swiss (Isis 40, 139). ser- EUCLID AND HIS TIME The of any of those alternatives (those quoted above and many others) would, however, increase the difficulty of geometrical teaching; the use of some of them would seem very artificial and would discourage young stuvice. dents. acceptance that a simple exposition is preferable to one difficult; the setting up of avoidable hurdles It is clear which is more would prove the teacher's cleverness and his lack of common Thanks to his genius, Euclid saw the necessity of this postulate and selected intuitively the simplest form of it.

16 All the stars and planets were supposed, a single sphere. That sphere, its central projection on move on for geometrical purposes, to was all right; if a star was not on the it was considered; the angular distances remained the same. 49 ANCIENT SCIENCE AND MODERN CIVILIZATION not give general proofs of them. The projection of all circles are circles (with the apparent exception of circles passing through the pole which are projected as straight lines). The stereographic projection is the only one which is both conformal and perspective, 17 Ptolemy could not have known that unicity, but he had made a good study of projections and was lucky.

The trigonometry is explained in chapters numbered 11 and 13 in Heiberg's edition. Every distance on the sphere is an angular one; the measurement of angles is replaced by the consideration arcs. 8 The chords subtending the corresponding divided into 360 and the diameter into of the circle is 120 parts. Ptolemy used sexagesimal numbers in order to avoid the embarrassment of fractions (that is the way he put Almagest I, 10). Thus, each of the 60 parts of the radius was divided into sixty small parts, and these again were divided into sixty smaller ones.

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