89 



juftly commended by Lalande,* as affording a fliort and accurate 

 method of computation, in orbits not very excentric, _ It is confiderably 

 more accurate than either the approximations of Boulliald or Mer- 

 cator, and nearly as ready in practice as either of their rules. Simpfon 

 compares his rule with the methods of Boulliald and Ward : but by a 

 reference to the pafTage, it will appear that Simpfon miftook the nature 

 of Boulliald's correftion, and therefore he erroneoufly makes Ward's 

 and Boulliald's methods nearly equal in point of accuracy. 



In his mifcellaneons effays, he computes the correftion to be 

 applied to the mean anomaly, to obtain the angle at the higher focus. 

 His motive for attempting this method of folution feems to have been 

 to fhew how the accuracy of Newton's fecond method of finding the 

 anomaly might be farther extended, and alfo to correct fome miftaken 

 notions that had been entertained refpefting Newton's folution. 



Dr. Matthew Stewart has given alfo a folution of Kepler's problem,! the 

 geometrical elegaiKe of the demonftration of which is to be much ad- 

 mired, as well as the great accuracy of the method, in refpeft to the 

 planetary orbits. His method confifts in obtaining an approximation for 

 the excentric anomaly and then correfting that firfl approximation. 



The firfl approximation is precifely the fame as the above mention- 

 ed one of Caffini. But Stewart's correction of his approxima- 

 tion is more accurate than Caffini's correftion of his approximat'on. 

 For, as is hereafter fliewn, the error of the excentric anomaly deduced 

 by Stewarts's method depends, only on the feventh and higher powers 

 of the excentricity. This method does not appear to furnifh readily a 

 continued approximation the contrary of which is flated in the Life of Stewart, 

 publiflied in the firfl vol. of the Edinburgh Tranfaftions. It may alfo be ob- 

 ferved, that the firfl approximation, correfted by Newton's method, de- 

 pends only in the feventh and higher powers of the excentricity and 

 the procefs is fomewhat more Cmple than by Stewart's method. 



Vol. IX. ( M ) The 



* Lalande's Edit. Halley's Tables, vol. 2. p. 23. 

 t Edinburgh Tranfaftions, 8to, vol, 3, pag. 10;, 120. Stewart's Trafts, 



