23c 



A NEW and UNIVERSAL SOLUTION 



The infpeftion of this table fliews, that the error of the firft 

 approximation, obtained by the fuppolition of ^ = e, in all the 

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

 neglected on mofl occafions. 



It is to be recolledled, that the rule we have inveftigated for 

 computing the eccentric anomaly, would give a rigorous refult, 

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

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

 calculation is neceflary to corredl the firfl: affumed value of e^ 

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

 The method of proceeding that is dire<5ted above, viz. to afTume 

 at firfl: f = £, 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 circumfl;ances 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 firfl; affumed value of e 

 be greater than its true value : and, if the firfl 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 defed. Therefore, in applying the general method, 

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

 cular. 



