12 Mr. P. Newton on the Triscction of an Arc. 



circle in L. Draw the sine Lx; the chord LA ; and, 

 through s, the extremity of the semi- radius F^, and in a di- 

 rection parallel to the diameter AB draw the chord YZ. But 

 the sine L.r, and the semi-radius Fi', being perpendiculars to 

 the parallels AB, YZ, are themselves parallels, and from the 

 nature of parallel lines the sine L.r is longer than Ys. And 

 because the third part of a quadrant =30 degrees, and the 

 sine of 30 degrees = half the radius, the sine L.r, being longer 

 than the half radius F5, exceeds its proper magnitude. The 

 sine L.r is here considered as = the sine of ^d part of the 

 quadrantal arc AH or HB augmented by an arc commencing 

 at L, and described from the centre c, with the radius Lc, part 

 of the chord LcH. With cL, therefore, as radius, and centre c, 

 describe the arc L/rMN6, forming a lune with the given circle, 

 and meeting or intersecting the given circle in the points I^, 

 aiid h, at equal distances from N, the extension of the diame- 

 ter AB. With the chord AL as a distance, and L as a 

 centre, describe an arc at M. With the same distance AL, 

 and centre N, describe an arc at k. The arc NM = the arc 

 A-L. From which I infer that if we apply the chord AL, from 

 N to^, on the lunar arc LAMN6, the point k fixes the situation 

 of the sine kix\ drawn of course perpendicularly from the 

 point/-, or parallel to the sine L.r; and the point i, in which 

 the sine kiv intersects the given circle, is one extremity of the 

 true sine ; the other extremity v is consequently on the dia- 

 meter; and zi; = F5 is = the sine of ^d })art of the quadrant 

 AH, or HB. 



For the sake of illustration, let us suppose the radius cL, 

 the sine L.r, and the chord LA, to be composed ofj or repre- 

 sented by, three inflexible rods. Let the extremity c of the 

 radius Lc be a fixed point, and let the chord LA, the sine 

 La', and the radius Lr, all meeting in the point L, be so joined 

 or attached to one another in this point, that the end A of 

 the chord or rod LA shall, in quitting the arc LA, fall upon 

 the lunar arc L/lMN^ at M, and that while the extremity L 

 of the lunar radius cL, bearing with it the sine L.r, shall move 

 along the arc L/!MN/j, from L to k-. the extremity M of the 

 chord or rod LM shall be driven along the same arc L/tMN6, 

 from M to N, and shall lie in the direction /tN, during which 

 time and motion the sine L.r, preserving its perpendicular 

 direction, is carried into the situation of the sirie kiv; for the 

 arc MN is = the arc k\u. It hence appears that the chord 

 LA, applied to the arc LZ-MN6, leaves a remainder of this 

 arc MN, or /I^, = the intervening arc which .separates the 

 sine L.i- from the sine kiv\ the part iv of the latter of which 



is 



