145 



the true and excentric anomalies deduced from the indireft methods 

 •examined. 



Kepler's ftatement of his problem, and his own folution arc then no- 

 ticed. 



Boulliald's firfl: hypothefis, and Seth Ward's funplification of that hy- 

 pothefis are next in order confidered. Seth Ward's fimplification, which 

 is better known by the name of Seth Ward's hypothefis, or the fimple 

 elliptic hypothefis, is particularly examined, and the anomaly as deduced 

 thereby, is compared generally with the true anomaly. The feries ex- 

 prefljng his anomaly is remarkable by its fimplicity, and may be of 

 ufe in other enquiries. 



The examination of Boulliald's correflion of Ward's hypothefis fol- 

 lows next. It is Ihewn that this correftion, contrary to what has been 

 generally fuppofed, at lead in more modern times, was a real con- 

 reftion, and aiforded a very fhort and exaft method of deriving the 

 true from the mean anomaly. The error is of the fame order 

 as the third power of the excentricity, and the error of Ward's hy- 

 pothefis is of the fame order as the facond power of the excentricity. 

 Obfervations made at that tim.e, could not have detected any error in 

 the places of Venus, the Sun, Jupiter and Saturn ; and even in Mars 

 the error, when at its maximum, could fcarcely have been noticed, ex- 

 cept when Mars was then in oppofition. 



In the year 1664, Mercator propofed a new hypothefis,* which he, 

 having compared it with forty-two obferved places of Mars, afl'erted to 

 be more ready in practice, and more accurate than any before given. 

 His hypothefis is here fhewn to be fomewhat lefs accurate than Boul- 

 liald's correction of Ward's, although as ready in praftice. 



Accurately fpeaking, the above mentioned folutions of Ward, Boulliald 

 and Mercator ought not to be called folutions of Kepler's Problem. 

 Kepler, refting upon obfervation, afferted that the orbit of a planet 

 was an ellipfe, the fun being in one of the foci, and alfo that the law 

 of the elliptical motion was fuch that equal areas were defcribed in equal 



times, 



* Aftr. Edit. 167 1, p. 163, 164. 



