130 



What has been done in this effay has been principally done with a 

 view of comparing different folutions of Kepler's Problem. That compa- 

 rifon has led me to point out what I confider as the beft practical folution 

 of the problem, particularly applicable to the planets. This folution is 

 formed by a combination of the folutions of Kepler, Newton, and Caffini. 

 The very fmall fliare I claim in it is from having recommended that com- 

 bination of folutions. The folutions of the two latter have been feparately 

 recommended by writers on aftronomy. Ciiffini has not always been referred 

 to as the author of his method, and Newton rarely. The merit of Caf- 

 fmi's method is derived from its fimplicity, and ready application to the 

 planetary orbits. Newton's folution was the firfl that was applicable to or- 

 bits of every degree of excentricity. 



In addition to the folutions that have been mentioned, it is neceflary 

 confidently with my plan, to notice two others. The one given by Her- 

 man, in 1725,* and the other by Mr. Ivory.f 



The folution of M. Herman had, through inadvertency, efcaped my notice, 

 although referred to by feveral authors. And is given in the fame memoir, 

 together with a conflruftion of the problem. It is in fubftance the fame 

 as the folution of Dr. Matthew Stewart, that has been examined, and there- 

 fore what has been faid of that may fuffice for Herman's folution. The folu- 

 tions only differ by Stewart taking D X perpend, from S on the tang, at 

 D = twice the area DSG, and by Herman's taking SD X perpend, from 

 ^'£- 5- ' G on SD for the fame area. Herman ufes Caffinis approximation without 

 reference, although he had mentioned that folution in the beginning of his 

 memoir. Dalembert, alio, in the Encyclopedie, juftly commending Her- 

 man's Solution, does not notice that the molt valuable part of it was due 

 to Caffini. 



I had not an opportunity of feeing the folution of Mr. Ivory till this ef- 

 fay was nearly printed. It is not, however, neceffary for me to enter into 

 a minute examination of it ; as the ingenious author- has very fully ex- 

 plained 



Comment. Acad. Petrop- Tom. i. p. 142. 

 Edinburgh Tranfaflions, Vol. 5. part ii. 



1 



