lg6 ■ On the ORIGIN and 



any two of the lines AB, CD, &c. proportionally, will alfo cut 

 the others proportionally, and will be cut by them into feg- 

 ments having the fame ratio to one another, with the fegments 

 of the lines AO and BL. 



From this it follows, that the aftronomical problem, above 

 mentioned, becomes a Porifm, and is indeterminate, in the cafe 

 when the obfervations of the comet are not very diftant from 

 one another. For on this fuppofition, the arches defcribed by 

 the earth, and by the comet during the time in which the ob- 

 fervations are made, will not differ much from two ftraight 

 lines ; and thefe lines will be divided fimilarly to one another, be- 

 caufe each of them will be divided into parts, proportional to the 

 intervals of time between the obfervations. The places of the 

 earth, at the times of the obfervations, may therefore be nearly 

 reprefented by the points A, C, E and O, in the ftraight line 

 AO, and thofe of the comet by the points B, D, F and L, in 

 the ftraight line BL, thefe lines AO and BL being divided both 

 into parts having the fame ratios. The polition of BL there- 

 fore is not given, fince, by the Porifm, it may be any line 

 whatever, which cuts the two lines, AB and OL, in a certain ratio. 



It is alfo to be remarked, that in order to render this, or any 

 other geometrical problem, of no ufe in queftions where the 

 data are furniftied by obfervation, and are confequently liable 

 to fome inaccuracy, it is not neceffary, that the problem mould 

 be reduced exactly to the porifmatic cafe ; for even on a near 

 approach to that cafe, a very fmall error in the data will pro- 

 duce fo great an error in the conclunon, that no dependence 

 can be had upon its accuracy. 



This will be made evident in the prefent inftance, by confi- 

 dering how the conftruclion of the Porifm is fubfervient to the 

 folution of the other cafes of the problem. Suppofe that four lines, 

 AB, CD, EF, RS, (fig. 1 1 .) are given in pofition, and that it is re- 

 quired to draw a ftraight line that fliall be divided by thefe 

 lines into parts having the ratios of the given lines a/3, /3y, y%. 



Let 



