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By Arthur Wouk
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Additional resources for A course of applied functional analysis
PROPOSITION IS. OJ straight lines in a circle the diameter is greatest, and oj the rest the 'nearer to the centre is always greater tlzan the more remote. Let ABCD be a circle, let AD be its diameter and E the centre; and let BC be nearer to the diameter AD, and FG more remote; I say that AD is greatest and BC greater than FG. For from the centre E let EH, EK be drawn perpendicular to BC, FG. Then, since BC is nearer to the centre and E G more remote, EK is greater than EH. [III. Def. sJ Let EL be made equal to EH, through L let L/lf b~ drawn at right angles to EK and earned through to N, and let ME, EN, FE, EG be joined.
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. 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.
2] but it fell within the circle ABCD and outside A CK [m. Def. 3]: which is absurd. 35 Therefore a circle does not touch a circle externally at more points than one. And it was proved that neither does it so touch it internally. Therefore etc. Q. E. D. f, 27, 30, 33· ABDC. Euclid writes ABeD (here and in the next proposition), notwithstanding the order in which the points are placed in the figure. 25, 37. does it so touch it. vros) leaves to be lmderstood. The difficulties which have been felt in regard to the proofs of this proposition need not trouble us now, because they have already been disposed of in the discussion of the more crucial propositions III.
A course of applied functional analysis by Arthur Wouk