ii6 



the fame order as i. Therefore the error of Stewart's corredled approxhnation 

 depends only in the feventh and higher powers of the excentricity. 



Hence this method gives the excentric anomaly with great precifion 

 for all the planets, and even for orbits more excentric ; but the prafticc 

 of the method is Icfs fimple than Caffini's firft approximation correfted 

 by Newton's firft method, or by the method hereafter given. It alfo 

 does not readily afford a farther approximation. 



* On Simpfon^s praBkal Rule. 



Simpfon having deduced a feries for the equation, finds the fum 

 of that feries, as far as the third power of the excentricity inclufive, 



2«, m 



nearly equivalent to an arch, the fine of which is diminiflied 



i+lecs, m 



aes, m \' 

 by J of the arch the fine of which is -^ . The praftical rule thence 



I + tecs, m 

 arifing is full as eafy as the rule by the firft approximation of Cafilni ; but 

 then no convenient method oiFers itfelf for extending the approximation by 

 Simpfon's method. Simpfon's anomaly computed by this rule, eafily ap- 

 pears to be, regarding only the third power of the excentricity. 



5 .> 7 ; 1073 



m — 2«, 7«H e s, 2m e s, m-\ e s, yn 



4 32 96 



comparing this with the feries for the true anomaly, the correftion appears 



I 3 I 3 

 to be e s, m e s, ^m, which is a maximum when m==$5° 16' 



52 32 



nearly. And therefore in the orbit of Mercury the error from this term 

 when a max. is about 57", in the orbits of the other planets the errors are 

 as the author rightly obferves very fmall. 



On 



* Effajrs, p. 47. 



