Olbers on Comets. 159 



each trial, although still very far from the truth, without limit- 

 ing it to a parabolic form, but proceeds to compute a fourth 

 observation from the elements thus found, in order to judge of 

 the accuracy of the first assumption. A labour so enormous, 

 that no astronomer appears to have followed his steps in it ; 

 nor, indeed, has he himself adhered to this method, although the 

 approximation which he employed in his researches on the 

 comet of 1769 is little more convenient, and requires no further 

 notice at present, any more than Newton's first method, pub- 

 lished in his early work De Mundi Systemate, which it may be 

 safely asserted that Newton had never attempted to reduce to 

 practice in a single instance. 



§ 13. 



All these methods have some advantage over that of Lacaille, 

 since the suppositions which they adopt supersede the necessity 

 of some of the trials : for when a distance is once found which 

 gives the whole time correctly, the middle observation,with which 

 these calculations begin, must necessarily agree pretty accurately, 

 without the repetitions which Lacaille's method requires for 

 adjusting it. On the other hand, this method may be rendered 

 ultimately more accurate, since (1) the supposition of the divi- 

 sion of the chord is never mathematically correct: and (^2) the 

 observations employed can only be at small intervals from each 

 other, for, otherwise, the error of the suppositions must be very 

 considerable ; hence the errors of observation will materially 



affect the results. 



§ 14. 



In order to supersede the necessity of these repeated conjec- 

 tural trials, all the acuteness of talent, and all the artifices of 

 calculation, possessed by the greatest mathematicians, have 

 been repeatedly employed, and solutions of the problem have 

 been successively made public by Lambert, Boscovich, Hennert, 

 Dusejour, Lagrange, and Laplace. 



§ 15. 

 Lambert thought it possible to reduce the whole to an equa- 

 tion of the sixth degree ; l)ut the equation is properly, as La- 



