X48 



Machin has alio given a foluti>ii of Kepler's problem,* remarkable 

 for its ingenuity. His motive for attempting the folution was, as he 

 tells us, to give one which might be general. None of the methods, 

 according to him, being applicable to excentric orbits, and all of them 

 requiring fome rule or hypothefis to begin the computation. He himfelf was 

 aware that his method was too intricate for common praftice in regard to the 

 planetary orbits, and had he confidered Newton's firfl method, he would not 

 have afferted that no rule was fubfifling, but what was abfolutely ufelefs in 

 the elliptical orbits of comets. With refpe<3: to a rule for beginning 

 the computation it may be obferved that if the mean anomaly be af- 

 fumed for the firfl excentric anomaly, the error of the firfl correfted 

 anomaly will be of the fame order as the third power of the excen- 

 triciy ; the fecond of the feventh power, &c. and therefore in the mofl 

 excentric orbits, fuch an aifumption would be fufficient for beginning 

 the computation. Machin's method however, is peculiarly applicable to 

 very excentric orbits. It confifls of two parts. By the firfl is obtained 

 an approximation for beginning the computation. The rule, although 

 intricate in practice, is as accurate for very excentric orbits, as for orbits 

 of fmall excentricity, and in this the merit of the method confifls. By 

 the fecond part, the approximated excentric anomaly is correfted. His 

 method of correftion is, as may be readily {hewn, deduced from a 

 combination of Kepler's and Newton's firfl method ; but the author 

 has not acknowledged th's circumflance. 



Thomas Simpfon, who ranks fo high among the Britifh mathematici- 

 ans, exerted himfelf on this problem with his ufual ingenuity.! His two 

 firfl methods may be confidered as illuflrations of Newton's firfl me- 

 thod. In his third method he obtains the true anomaly, by reverfing 

 the ferics for expreffing the mean in terms of the true ; and from this 

 folution he deduces a fliort approximation for the true anomaly, but 

 limited, as he obferves, in point of exaftnefs. This practical rule is 



jufUy 



» Phil. Tranf. Vol. 40. Abiidg. vol. 8. 

 ■j SimpfoD's Effays, p. 41. MifcdI. Trafls, p. 4G. 



