KEPLEB'S PEOBLEM. 29 



resolution before adverted to, will not apply to those more 

 complicated problems. But the reader will find a variety 

 of examples of this species of proposition in the ' Philoso- 

 phical Transactions of the Eoyal Society of London for 1798,' 

 which were investigated chiefly in the manner above de- 

 scribed.* It is evident that the application of such problems 

 to physics does not proceed so far ; for we have never yet 

 discovered an example of a central force acting in a curvi- 

 linear direction. 



The solutions now described, of Kepler's problem, and of 

 severalj problems of a more general sort, are of a theoretical 

 nature. They exhibit the mode of expressing by curve lines, 

 or imaginary relations of known quantities, the relation 

 required of the quantities given ; they rather vary the diffi- 

 culty, or simplify the relation, than remove the impediments 

 to practical measurement. If it be required to exhibit the 

 anomaly of the eccentric, we may indeed adopt the solution 

 given by Sir Isaac Newton (Principia, lib. i. prop. 31, and 

 Schol.), or that hinted at by Kepler himself. The Newtonian 

 solution proceeds upon the description of a cycloid, and an 

 easy construction, by which the point required is found in 

 the intersection of a straight line with the given trajectory. 

 In the tract referred to, a solution is given more directly, by 

 the intersection of a species of a cycloid of easy description, 

 with the given curves, without any subsequent construction. 

 But these solutions, though more pleasing and beautiful in 

 theory, are useless, when it is required to exhibit a value 

 of the abscissa corresponding to the anomaly of the eccentric, 

 or its supplement, in such a manner that a comparison may 

 be made of this line with some known measure of length. It 

 becomes necessary, in this case, to find a numerical value of 

 the quantity in question. Now, this can only be done by a 

 series ; and the two great objects in finding such a series are, 

 first, to give one which may be regulated by a simple law ; 

 and, secondly, to give one which may converge rapidly : so 



* The Paper is given in this volume : it is the First Tract. 



