1879.] 



281 



[Chase. 



Approximate Quadrature of the Circle. By Pliny Earle Cham, LL.D . 

 (Bead before the American Philosophical Society, June 20th, 1879.) 

 Y„ 



AB = 3 ; AC = 20 



AD = 3 AB ; AX = 3 AC 



BE parallel to CD 



EY=:AC 



XY = 3.141585 AC 



The deviation from perfect accuracy is less than -^^^ of one per cent, 

 which would give an error of less than ^ of an inch per mile. For all 

 practical purposes the construction may be regarded as exact, for the error 

 would be inappreciable in any mechanical work. 



Haverford Colleoe, June 16th, 1879. 



Note. — July 16, 1879. My attention has been called to the following 

 more-complicated construction, and closer approximation, in Perkins's 

 Geometry (D. Appleton & Co., 1853). 



On an indefinite straight line A N, take AB = BD = DE.-1; atE 

 erect a perpendicular EG = 2AB = 2EF; on EN take EH = HK 

 = A G, K L (towards A) = A F, L M (towards N)=DG, MN^DF; 

 bisect E N at P, E P at R, A B at C ; trisect E R at T. Then C T = 

 3.1415922. 



The author calls this method "very simple," and says, that a better one 

 "can hardly be expected, or even desired." But the approximation of 

 Adrian Metius, f||, is still closer, and the following construction of his 

 ratio is simpler. 



b — c 



On A B =1 erect the perpendicular BC = 8; extend C B to D, making 

 B D = 9 ; on A D erect the perpendicular D F = 15 ; take A E — A C, and 



AF 



— = -J- 5| _ 3.1415929, the true ratio be- 



Then 



draw E G parallel to F C. 

 ing3.141592G+. 



The error of this construction is less than tt^ootj °f one percent. Per- 

 kins's error is more than T oooo °f one P er cent. Neither method is so 

 simple, nor so desirable for practical purposes, as the one which I commu- 

 nicated to the Society at its June meeting. 



