196. 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 pofition of BL there- 
fore is not given, fince, by the Porifm, it may be any lne 
whatever, which cuts the two lines, AB and OL, ina certain ratio. 
Ir 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 furnifhed by obfervation, and are confequently liable 
to fome inaccuracy, it is not neceflary, that the problem fhould 
be reduced exaétly to the porifmatic cafe; for even on a near 
approach to that cafe, a very fimall error in the data will pro- 
duce fo great an error in the conclufion, that no dependence 
can be had upon its accuracy. 
Turs will be made evident in the prefent inftance, by confi- 
dering how the conftruction of the Porifm is fubfervient to the 
folution of the other cafes of the problem. Suppofe that four lines, 
AB, CD, EF, RS, (fig. tr.) are given in pofition, and that it is re- 
quired to draw a ftraight line that fhall be divided by thefe 
lines into parts having the ratios of the given lines «@, By, 7 
Let 
