1843.] Treatment of Geometry as a branch of Analysis. 125 



when d is constant, or for secants through the same point, the rectangle of 

 the segments is a constant quantity, (III. 35, 36). If the point be 

 without the circle, d is greater than r and d 2 — r 2 is tan 2 , therefore 

 ss' = tan 2 (36, 37.) 



24. If two radii be drawn including a given angle at the centre, 

 they determine a certain arc of the circle in length, as well as the 

 sector corresponding to that arc. Denote the former by /, the latter 

 by S ; then (0 standing for the ratio of the angle 9 to the right angle) 



;=*(0)andj=**(0) 



Take p arcs equal to /, we have p angles equal to and p sectors 

 equal to s ; 



•'• HP 6 ) = ~ - P- *(») ^d 1> W) = *£ = P- tf (») 

 The solutions of these equations are (p(0)=^m and ^(0) = n0 I 

 m and n being certain constants, 



.*. I — mr and S = nr 

 Hence if r remains the same, /and S are proportional to 6 9 (VI. 33). 



25. We cannot determine m and rc without the aid of limits, because 

 they involve the comparison of curvilinear length with rectilinear 

 length. If we bisect the arc continually and join the points of bisec- 

 tion, we shall have a series of polygons of chords whose perimeters 

 approximate to the arc without limit, while the areas between them 

 and the radii approximate at the same rate to the sector. Denoting 

 the ratio of the chord of to radius by c ; that of the chord of i 

 by c [J] ; that of chord of ± 6 by c [i] 2 , and so on we have for the 

 perimeters of the successive polygons, 



cr;2c[J]. r ; 2. 2 c [J]. 2 r 2 n c[±y r 



And their areas successively 



~-or/-. 2e[i].r L. 2» c [£].<• r 



r 

 At the limit, therefore, the sector = ~ x arc, and consequently the 



r 

 area of the circle = 5- X circumference. 



Also we calculate the chord of half an arc from that of the whole by 



