230 A NEW and UNIVERSAL SOLUTION 







Limit of 





Value of £ 



the error 

 of p 



Mercury, 



- 2 °5S I 3 



I; 46". O 



Venus, 



.006885 



O .0 



Earth, 



.016814 



0.0 



Mars, 



.093088 



9.8 



Jupiter, 



.048077 



I -3 



Saturn, 



.056223 



2 .1 



Georgium Sidus, 



.046683 



1 .2 



The inflection of this table fhews, that the error of the fir ft 

 approximation, obtained by the fuppofition of e — s, in all the 

 planetary orbits, is a very fmall quantity, and fuch as may be 

 neglected on moft occafions. 



It is to be recollected, that the rule we have invefligated for 

 computing the eccentric anomaly, would give a rigorous refult, 

 provided the exact: value of e were known. But, as that value 

 cannot be deduced directly from the data, the repetition of the 

 calculation is neceffary to correct the firft affumed value of e, 

 and to make it approach nearer and nearer to the true value. 

 The method of proceeding that is directed above, viz. to afiume 

 at firft e =: s, and from thence to deduce a feries of approxima- 

 tions to the arch fought, is perhaps the only one that will apply 

 univerfally, and in all circumftances of the problem. But it is 

 to be obferved, that the reafoning in Art. 4, 5, and 6. will re- 

 main the fame, provided only that the firft affumed value of e 

 be greater than its true value : and, if the firft affumed value of 

 e be lefs than its true value, the reafoning will not thereby be 

 effentially altered ; all the change that will take place, is, that 

 the error of the approximations will now. be alternately in ex- 

 cefs and in defect. Therefore, in applying the general method, 

 If we can take hold of any circumftances, peculiar to the parti- 

 cular. 



