ways greater than the error of the firfl: approximated excentric anomaly 

 computed by the above method. When the mean anomaly is 180% s 

 is greateft, and then this quantity =1° 40' nearly. The aftual error 

 by computation appears to be 1°. i3\ When the excentric anomaly 

 is lefs than 230, the above limit of the error will be lefs than a fe- 

 cond ; fo that in every elliptic orbit, when the excentric anomaly is 

 ^efs than 23°, the above method will fufEce, without further correc- 

 tion, to find the excentric anomaly to lefs than a fecond.* 



The demonftration of the rule for a farther correftion ad libitum, the 

 author has not given, nor fliewn the convergency of the fucceffive corrcftions. 



To demonftrate the rule. Let m be the mean anomaly computed from 



the approximated excentric anomaly c let m+m and c-\-c be the accu- 



r \ 



rate mean and excentric anomalies. 



\ ■ 

 \ / 1 \ / 



Then m+m=c+c—es, c+c and m=c^-es, c therefore 



}n=c--ceci, c+i<: es, c nearly, ncglefting the powers of c above the fecond. 



\c*e5y c 



Or c= '_ -j -nearly or c — " but i — ecs, c=the planets 



I \ I \ \ 



i—ecsc 1 — ecsc i — ecs, c 



diftance from the fun. Hence Machin's rule. The error of c from neg- 



2 / 



letting ic es, c muft always be very fmall, becaufe as was fliewn above 



i—ecs, c 



c is 



* But it ought to be obferved that a fmall error, in the excentric anomaly, oc- 

 cafions a great error in the true anomaly computed from thence, when] the orbit is tery 

 excentric, and the body near perihelion. 



