# New PDF release: Geometry, a comprehensive course

By Dan Pedoe

*Mathematics Gazette*

*n*dimensions; the projective iteration of conics and quadrics; Moebius tetrahedra; the tetrahedral advanced; the twisted cubic curve; the cubic floor; orientated circles; and advent to algebraic geometry.

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**Sample text**

7, 8; for the point is determined by drawing from F and G, on the opposite side to that where X is, straight lines FY, C Y making with FD angles equal to the angles DFX, DGX respectively. Hence the two circles will have at least three points common: which· is impossible. " Therefore GD cannot be greater than GH; accordingly GD must be either equal to, or less than, GH, and Euclid's proof is valid. The particular hypothesis in which FG is supposed to be in the same straight line with A but G is on the side of Faway from A is easily disposed of, and would in any case have been left to the reader by Euclid.

13 of Heiberg's text Prop. 12, and so on through the Book. What was said in the note on the last proposition applies, mutatis muta1zdis, to this. Camerer proceeds in the same manner as before; and we may use the same alternative argument in this case also. Euclid's proof is valid provided only that, if FG, joining the assumed centres, meets the circle with centre Fin C and the other circle in D, C is not within the circle ADE and D is not within the circle ABC. ) Now, if C is within the circle ADE III.

He then gives substantially the proof and figure of III. 1 Z. It seems clear that neither Heron nor an-NairIzi had III. 12 in this place. Campanus and the Arabic edition of Na~Iraddin at-Tlls1 have nothing more of III. 1 Z than the following addition to III. 1 I. ) It is most probable that Theon or some other editor added Heron's proof in his edition and made Prop. 12 out of it (an-NairlzI, ed. Curtze, pp. 1ZI-Z). An-Nairizi and Campanus, conformably with what has been said, number Prop. 13 of Heiberg's text Prop.