7 



Mr. P. Newton on the Trisection of an Arc. 13 



is the true sine. For, produce now the sine iv, to meet the 

 given arc in a on the opposite side of the diameter AB. Then 

 because the chord of" an arc = double sine of half that arc, 

 the chord ia ■=■ double the sine iv, or the sine av = sine iv, 

 and the arc A/ = the arc Ka. Draw gH, joining a, the up- 

 per extremity of the sine a v, with H, the termination of the 

 quadrant AH; and parallel to a H draw t^wi, connecting the 

 lower extremity i', of the sine av., with ?h, the termination of 

 the quadrant cnm described with half radius. Join «F = ra- 

 dius of the given circle. Bisect the sine av in p, and parallel 

 to the diameter AB draw -pno, passing through the point n 

 of intersection of the radius «F with the line vm. 



Because aF = radius of given circle, and ?jF = half radius; 

 therefore an ■=■ half radius or = nY. And because the sine 

 au is bisected in p, and pn is perpendicular to av, the sides 

 ap, -pn, of the triangle apn are respectively equal to the sides 

 vp, p n, of the triangle vp 7i, and they include equal angles 

 ap7i, vpn. Consequently the side vn =. the side an, or = 

 nF. The line jmo being parallel to the diameter AB, the 

 arc rec = the arc oD, and the co-arc 71711 = the co-arc 7no. 

 Consequently the Z nYc = the Z oFD. And because the 

 triangle vYn is isosceles, and the side V7i ■= the side 7«F, the 

 Z.7ivY = the Z wFu = the Z oFD, or by sim. As. = the Z 

 m n o. But the Z ?« n o at the circumference = only half the 

 Z 7nYo at the centre. Therefore the arc 7no (= the arc 7im) 

 is double the arc oD, or double the arc nc. See Leslie's Geo- 

 metrical Analysis, book i. prop. 31st, prefixed to his Geometry 

 of Curve Lines. Or, since the line _/;o, which bisects the sine 

 av in p), bisects its equal ??«F in tt', the Z mnvo or Z ?«wo = 

 Z nVc; because the sine 7711x1 of the former Z is = the sine 

 wF of the latter Z. But because the Z 7n7io is an angle at 

 the circumference, the arc nio, or its equal the arc 71171, is 

 double the arc nc, the Z «Fc being an Z at the centre. 

 Again, because the circumferences of circles have the same 

 ratio to each other as their radii, and the radius AF, or «F, 

 is double the radius 71Y or (F, the arcs Ka and «H are re- 

 spectively double of the arcs C7i and 7im, and the arc «H is 

 double the arc A« : or this latter arc Aa is = jd of the arc 

 AH, or = idof HB. 



In a siuiihvr manner we may find the third of any fractional 

 part of the quadrant, as well as a third of the whole, because 

 for every variable position of the chord HL we describe a 

 new circle with the varying radius cL. That is, when HB 

 ceases to be the arc of a right angle, or, which is the same 

 tiling, when HC moves into the jjosition ol Kr, or when, in- 



steail 



