H Mr. P. Newton on the Trheciion of an Arc. 



stead of a right angle, the given arc becomes only = RB, 

 then cL varies to cq, which now becomes a substituting radius 

 for f L. 



To find one third part of the arc AE, or of RB. — Draw 

 the line R/F?- through the centre F of both the concentric 

 circles. Through the point c, which bisects the radius AF, 

 draw the chord Rcy, meeting the given circle in q. With 

 the radius qc, and centre <", describe the lunar arc 5'Pfl', inter- 

 secting or meeting the given circle in the points q and d points 

 equidtstant from P, the extension of the diameter AB. To pre- 

 vent a confusion of sines, we will pass to the other side of the 

 diameter AB. Through the point of intersection d draw the 

 augmented sine dh. Apply the chord of the arc Ac? to the 

 arc PrZ (as for the quadrant) from P to r, and through this 

 latter point c draw the sine eg, intersecting the given circle in 

 the point u. The sine ug is the true sine, or is the sine of 

 ^d of the arc AE, or of g^d of the arc RB, and consequently 

 the arc Ku = ^d of the arc AE, or ^d RB. Make the arc 

 Ae = the arc An, then will the arc Ae = ^d of the arc AI], 

 or = ^d of RB. If the upper extremity u of the sine iig 

 were connected with R, the termination of the arc RB, by 

 means of a straight line ; and if a straight line were drawn from 

 g, the low^er extremity of the sine ug, to t, the termination of 

 the arc D/ ; and if, moreover, the sine itg were bisected in a 

 manner similar to that employed for the right angle, a similar 

 proof woidd follow ; viz. that the arc Au = ^d of the arc AE, 

 or = ^d of the arc RB. The straight line drawn from g would 

 fall on the arc cfi between c and ?/, and would be = vn, or 

 = 7i¥, and on being produced would pass through the point 

 of intersection made by the other two lines ; drawn from 21, 

 and from the bisection of the sine ug. 



Since the arc Aa = -^d of the arc AH, or = ^d of HB,and 

 the arc Am = ^d of tlie arc AE, or = ^d of RB; the dif- 

 ference between these thirds, viz. the ai'c na, or the arc ei, = 

 ^d of the co-arc EH, or of HR. Make, next, the arc iQ = 

 the arc A?, then will the arc eQ = ^d of the arc EB, or = ^d 

 of the arc AR. 



Scholium. — When the proposed arc is less than half a qua- 

 drant, as the arc EH or HR; the complement AE, or RB, 

 may be trisected, and the diflerence between ^d of this com- 

 plement, and ^d of a quadrant as the arc ua, will be = ^d of 

 the proposed arc EH or HR. 



P. N. 



V. On 



