KEPLER'S PROBLEM. 31 



given point in the circumference. The results of the series 

 which he gives for the eccentric anomaly are as follows : 



Eccent. anom. = 47 4' (first value, and less than the truth), 

 ,, = 47 40' 14" (second value, and greater than 



the truth). 

 ,, = 47 39' 12" (third value, and less than the 



truth). 



From this example, may be perceived the excellence of the 

 method ; for, whereas the first two terms differ by nearly 36', 

 the second and third differ only by 1' 2" ; or, in other words, 

 while, by the two first trials, we come to a space of above 

 half a degree, in some part of which the point required is to 

 be found ; by the second and third trials, we obtain a space 

 of about the sixtieth part of a degree, in some" part of which 

 lies the result. By the third term of the series, then, we 

 obtain a solution not more than 31" distant from the truth, 

 and this in circumstances the least favourable. 



The other example is a solution of the problem " to draw 

 from a point in the circumference two chords which shall 

 trisect the circular area." Here the 



Eccent. anom. = 30 33' (first value less). 



= 304 4' 11" (second greater). 



Euler's solution (Analysis, Inf. XI. 22) differs litte more 

 than 30" from this solution, given by Mr. Ivory's second term. 



This specimen will sufficiently show the superior excel- 

 lency of Mr. Ivory's method. Former analysts have only 

 resolved the case within the eccentricity is small : his solu 

 tion extends to comets as well as planets. For the planets, 

 his rules apply with peculiar accuracy and ease ; and his series 

 converges with extreme rapidity ; so much so, that we may 

 consider the approximation of one term sufficient for practice. 

 He has given a table of the values of the errors (or differences) 

 for the different planets computed in this way. He adds an 

 exemplification for the famous comet of 1682, supposed to be 

 the same which reappeared in 1759. His first approximation 



