158 Astronomical and Nautical Cidleclhns. 



radius, or by some other known line, in the proportion of the 

 two intervals of time. Neither of these suppositions is per- 

 fectly correct, and the first is the least accurate ; but either of 

 them enables us to derive the chord, and consequently the whole 

 orbit, from a single distance ; and this distance must be ob- 

 tained by conjecture, or by the " rule of false." These trials 

 must be continued until they become near enough to the truth 

 to enable us to find still more accurate values by interpolation. 

 The same process may be employed in a geometrical construc- 

 tion ; but the complication of a number of discarded lines will 

 be found somewhat inconvenient. 



§ 12. 

 The principal of these indirect methods of construction, or 

 computation, will require to be briefly noticed. Boscovich sup- 

 poses the portion of the orbit to be a right line, described with 

 the velocity appropriate to the middle point, at least if Pingre 

 has quoted him correctly. Lambert employs the supposition, 

 that the chord is divided in the proportion of the times, and 

 compares the magnitude of this chord with the whole time, by 

 means of his own very elegant theorem, Newton, on the other 

 hand, divides the chord much more accurately in the propor- 

 tion of the times than it is divided by the revolving radius ; and 

 compares the length of the chord with the time, by means of 

 a theorem which considerably resembles that of Lambert, al- 

 though it is only an approximation. Hence it arises, that New- 

 ton's construction is the most accurate, Boscovich's the most 

 convenient, and Lambert's holds the middle place in both re- 

 spects. In all of them a distance from the earth is assumed 

 for the middle observation ; the position and length of the 

 chord is then determined by the respective modes of approxi- 

 mation, and the time is compared with the length of the chord, 

 according to the laws of motion in a parabola. Eiiler also has 

 employed the supposition, that the chord is divided in propor- 

 tion to the times ; but he has omitted to compai'e the space de- 

 scribed by the revolving radius, with the whole time ; and, 

 instead of this, he determines an orbit in a conic section from 



