24 ) 



II. 



KEPLER'S PROBLEM. 



KEPLER was led, after the discovery of the law which bears 

 his name, to the celebrated problem . which also bears it. 

 Having proved that the squares of the periodic times are as 

 the cubes of the distances, he wished to discover a method of 

 finding the true place of a planet at a given time one of the 

 most important and general problems in astronomy. By a 

 short and easy process of reasoning, he reduced this question 

 to the solution of a transcendental problem ; to draw from a 

 given eccentric point, in the transverse of an ellipse (or the 

 diameter of a circle) a straight line, which shall cut the area 

 of the curve in a given ratio ; or, in the language of astro- 

 nomers, " from the given mean anomaly, to find the anomaly 

 of the eccentric." 



This most important problem is evidently transcendental ; 

 for, in the first place, the curve in question is not quadrable 

 in algebraic terms ; and, in the next place, admitting that it 

 were, the solution cannot be obtained in finite terms. As the 

 general question, for all trajectories, is of vast importance ; 

 and as the paper of Mr. Ivory, in the ' Edinburgh Trans- 

 actions,' contains a most successful application of the utmost 

 resources of algebraic skill to the most important case of it, 

 I shall premise a few remarks upon the problem, when 

 enunciated in different cases. 



Let D 2 be the given area of any curve, whch is the tra- 

 jectory of a planet or other body, or which is to be cut in the 



