Olbevs on Cornels. 157 



obtain at last even a linear equation, since the problem is more 

 than sufficiently determined by three observations. 



^ 10. 

 In the actual state of the problem, it has been necessary to 

 have recourse to approximations and hypotheses. The method, 

 particularly termed by Pingre that of false suppositions, which 

 seems to have been first circumstantially indicated by Lacaille,may 

 be mentioned, as the most inartificial, in the first place. A distance 

 from the earth or the sun is arbitrarily assumed for the first ob- 

 servation ; a distance is then found by trials for the third, so re- 

 lated to the former, that the time required for the description of 

 the intervening space may agree with the observed time ; we have 

 then to find the place of the comet in the orbit thus determined at 

 the time of the intermediate observation, and to compare it with 

 the true place; and the whole operation must be repeated with new 

 values for the first distance, until the whole of the results are found 

 to correspond with the observations. This method has also been 

 particularly explained by Pingre and Lalande, and it was gene- 

 rally employed in France until Laplace's solution was made 

 public ; but the German mathematicians have always thought 

 it very tedious, and circuitous, and fatiguing. It must, how- 

 ever, be confessed that it is not extremely inconvenient, provided 

 that the quantities assumed are tolerably near the truth ; and 

 it may be remarked, that the operation might be considerably 

 shortened, if the theorem of Lambert were employed in it 

 which does not appear to have been hitherto done. 



§ 11. 

 All other mathematicians, who have attempted the indirect 

 solution of the problem, have employed some approximate hy- 

 pothesis for reducing the problem to the investigation of one 

 unknown quantity : for instance, a true or a curtate distance : 

 and for this purpose two suppositions have been principally 

 adopted, either (1) the portion of the orbit between the three 

 observations has been considered as a straight line, described 

 with a uniform velocity ; or (2) it has simply been assumed that 

 the chord of this portion of the orbit is divided by the revolving 



