This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1913 edition. Excerpt: ...189). The circles all pass through two other points and therefore form a system of coaxal circles. Since these circles are cut by every transversal in pairs of conjugate points of an involution, the conics in the original figure are all cut in a similar manner by every transversal. (b) Proof by Carnot ...
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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1913 edition. Excerpt: ...189). The circles all pass through two other points and therefore form a system of coaxal circles. Since these circles are cut by every transversal in pairs of conjugate points of an involution, the conics in the original figure are all cut in a similar manner by every transversal. (b) Proof by Carnot's Theorem. This involves an imaginary projection the use of which has not been justified. Therefore Hence B, C; AI, At; Ai, A%' are pairs of conjugate points of an involution. Since B, C and Ai, Ai may be regarded as fixed, any other conic through Bi, t Ci Cj determines on BC pairs of points, which are conjugates in the same involution. Therefore (BCAiA)=(CBA2A2'). Hence B, C; Ai, A; A, A% are pairs of conjugate points of an involution. Since B, C and Al, A% may be regarded as fixed, any other conic touching BBi, BBt, CC, CCt determines by tangents from A pairs of lines, which are conjugates in a given involution. Carnot's, Pascal's and Desargues' Theorems for a pair of lines and their correlatives for a pair of points. In Art. 94 it was pointed out that a pair of straight lines could be regarded as a conic and a pair of points as the correlative of a conic. On reference to the earlicr chapters it will be seen that the theorems proved in this chapter are true, as might be expected, for a pair of lines, and their correlatives for a pair of points. Thus Carnot's theorem for a pair of lines Carnot's theorem for a pair of points follows at once from Menelaus' theorem. follows at once from Ceva's theorem. (Art. 13 (d).) Pascal's theorem for a pair of lines becomes the theorem proved in Art. 36. Desargues' theorem for pairs of lines becomes the Involution Property of a quadrangle proved in Art. 56. (Art. 13 (e).) Brianchon's theorem for a pair of...
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