30 KEPLER'S PROBLEM. 



that its denominators rapidly increasing, the quantities may 

 soon become so small, as not to deserve attention in our com- 

 putations. 



The approximation given by Mr. Ivory in his paper in the 

 ' Edinburgh Transactions,'* deserves the first place among 

 those of which we are in possession, whether we consider its 

 simplicity, universality, or accuracy. The series is of easy 

 management, applies to the most eccentric orbits, as well as 

 to those approaching nearer to the circle, and to all degrees 

 of eccentricity in the given point, the centre of forces. It 

 has the benefit, too, of a most rapid convergence. 



He first gives a very simple and elegant geometrical method 

 of approximation, by an application of the rectangular case 

 of the general problem de inclinationibus of the ancient geo- 

 meters. But as this is by no means satisfactory to the 

 practical calculator (for reasons before assigned), he proceeds 

 next to the algebraic solution. 



He begins with investigating the series for the eccentric 

 anomaly when the mean anomaly is a right angle. It con- 

 verges quickly, and the terms err alternately, by defect and 

 excess, the difference growing continually less and less. 



He then proceeds to the investigation of a similar series, 

 found in the same manner, for the other cases of the mean 

 anomaly. I should in vain attempt to give the reader a 

 more minute idea of this solution, without a detail as full 

 as the paper now before us, and shall only note an erratum 

 that has crept into the twelfth article. After putting tan. A 



sin -ft . A 



= e x cos. m x sec. 45, he infers that sin. = tan. 



2 



X 45; it should be sin. ~ = tan. x sin. 45. 

 2i 2i 



He next gives two examples of the application of his 

 method to geometric problems, concerning the circle. The 

 one, is to bisect a given semicircular area by a chord from a 



* Vol. v. p. 111. 1802. 



