147 ■ 



refted by him nearly in thfe fame manner, as Kepler correfts the af- 

 fumed excentric anomaly. Cafllni's correftions of his approximation are 

 very ingenious, but not fuificient in excentric orbits for the nice pur- 

 pofes of modern aftronomy. 



This firft approximation of Caffiini has been adopted by many authors, 

 praftice having flaewn its value; for I know of no one that has at- 

 tempted to fliew its exa(ft and general value. It is Ihewn here, and 

 I think for the firfl time, how clofe an approximation it is, the error 

 depending on the third power of the excentricity. In the orbit of Mars 

 the error is not greater than 10", and in the orbit of Mercury not 

 greater than 5'. The approximation confifts in adding half Seth Ward's 

 anomaly to half the mean anomaly, the fum will be very nearly the 

 excentric anomaly. The angle Cafllni computes is readily fliewn to be 

 equal to Seth Ward's anomaly. 



It is from this approximation that the method recommended, is 

 partly derived. That method is as follows. 



Caflini's firfl approximation, which is equivalent to the fum of half Seth 

 Ward's anomaly, and half the mean anomaly, is taken for the excentric ano- 

 maly. With this excentric anomaly the mean anomaly is computed by 

 Kepler's method. The difference between this computed mean and the true 

 mean anomaly is multiplied by a number taken out of a fmall table. This 

 produft properly applied to the difference, gives the correftion of the ap- 

 proximated excentric anomaly. 



The error of the excentric anomaly fo obtained, is of the fame order 

 as the feventh power of the excentricity, and lefs than a fecond itt all. 

 the planets. 



The formula from which the table is computed, is derived from Sir 

 Ifaac Newton's firfl method. 



If with the correfted approximate excentric anomaly, the operation 



be repeated, the error of the next approximation will be of the fame 



order as the 15th power of the excentricity, and by repeating the pro- 



ceffes, the errors will be of the fame order as the 31ft, 63d, 127th 



powers of the excentricity. 



Machin 



