A first course in projective geometry . (a). Fig. 95. For if C be the centre and CN, On perpendicular to PQ, pq^AP = CN and aj) = Cn. .. CN = Cn and the chords QQ, qq are equal. .*. PQ and pq, being the differences between the halves ofthese chords and the radius, are equal to one another. Hence the application of the above definition gives a resultin accordance with the obvious fact that the curvature of thecircle is the same at every point on it. TANGENT AND NORMAL PROPERTIES 179 Moreover, in diflferent circles the curvatures are inverselyas the radii. For let the circles touch at A; also le
A first course in projective geometry . (a). Fig. 95. For if C be the centre and CN, On perpendicular to PQ, pq^AP = CN and aj) = Cn. .. CN = Cn and the chords QQ, qq are equal. .*. PQ and pq, being the differences between the halves ofthese chords and the radius, are equal to one another. Hence the application of the above definition gives a resultin accordance with the obvious fact that the curvature of thecircle is the same at every point on it. TANGENT AND NORMAL PROPERTIES 179 Moreover, in diflferent circles the curvatures are inverselyas the radii. For let the circles touch at A; also let AP, PQ as aboverefer to the one circle and AP, P^^ to the other. Let PQ, Pqproduced cut their respective circles in Q, q (Fig. ^bh). Then PQ. PQ = AP2 = Pq . Pq. Hence PQ^P^l P^ PQ Ultimately, when P moves up to A, PQ, Pq become the diameters of the circles. POHence the ultimate value of the ratio —: is in the inverse ratio of the radii. ^ This also agrees with common experience, for of circlestouching at a common point, that with the l
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
