124 Treatment of Geometry as a branch of Analysis. [No. 134. 



= x, the reverse is 2rr — x ; hence their halves or the angles at the 

 circumference on opposite sides of the same chord are -J x and it — 1 x 

 their sum is therefore it (III. 22.) 



If the angle at the circumference stand on the chord c, the radius 

 being r, the angle at the centre is 20 and (by *y in art. 12) it is seen 

 c = 2 r sin 0. I assume the formulae of trigonometry here, as they 

 are all deducible independently by help of y. Hence if c and r be 

 constant, is constant; or if r and be constant,*; is constant, (III. 26, 

 27, 28, 29), Also c is a maximum with sin 0, i. e. when = 1 7r 

 (III. 15). 



22. Now as to lines intersecting a circle. Let P be a point whose 

 distance from the centre is d, and p a secant through it inclined to d at 

 an angle 0. Then p, d and r (the radius) form a triangle, the two for- 

 mer including ; hence 



r 2 = p« + S— 2 pd. cos 

 or p 2 — 2 p d cosO = r 2 — d 2 

 The quadratic form shews that there are two roots only. Hence 

 the line cuts the circle in two points at most. The solution of the qua- 

 dratic is 



p = d cos 6 ± Jr^^d^Jsinef 

 If the point be within the circle, r 7 d; and the roots are both always 

 possible since sin 0.^ 1. If = L 7r, the two values of/o become 

 equal; which with its converse is (III. 3). The increase of } dimi- 

 nishing d cos 6 and increasing d sin 0, will diminish p ; the maximum 

 of p being when 0=0 and the minimum when = tt (III. 7). 

 If be measured negatively and the secant called R, we shall have, 



R - d cos (— 0) ± V r* — d 2 (sin — 0)2 



=- d cos ± Jr 2 —d 2 (smO) 2 

 which shews an equal secant on the opposite side of the diameter, 

 (HI. 7). 



The same is true if the point be beyond the circle, but as d is then 

 7 r, the line p will only cut the circle while d sin is less than r, 

 (III. 8). When d sin = r, p = d cos ; since there is only one 

 value the line p is a tangent and for that value T* 2 + (tan) 2 = c? 2 or 

 the tangent is perpendicular to the radius through the point of contact, 

 (111.17,18,19). 



23. By the theory of equations, if s and s' be the segments of p 

 between the point P and the circumference, ss' = d? — rz . Hence 



