194 ■ Mr. Ivory on a new Method of deducing 
which continue visible for a short time only, and in a small 
part of their orbit. 
In order to evade the difficulties attending a direct consi- 
deration of the problem, and to obtain an approximate solution 
at least, Sir Isaac Newton proposed to take a small portion 
of the orbit fora straight line described with a uniform motion. 
On this supposition the projections of the comet on the plane 
of the ecliptic will lie in one straight line as well as the real 
places in the heavens; and the several parts of botli lines will 
have the same proportions as the intervals of time between 
the observations : so that in order to find the projection of a 
comet's trajectory on the plane of the ecliptic, we have only to 
draw a straight line which shall cut the several straight lines 
whose positions are determined by the observed longitudes 
in such a manner that the intercepted segments shall have 
given proportions. With three observations only this problem 
is indeterminate, or admits of innumerable solutions: for, let 
AP-, BE, CQ, (fig. 1. PI. IV.) drawn in the plane of the ecliptic, 
represent the directions in which the comet is seen from the 
earth at the three observations ; then, having assumed any 
point at pleasure, as E, in one of those lines, ^ we can draw, 
through that point, a straight line PO to terminate in the other 
two lines AP and CQ, so that PE and EQ shall have the same 
proportion as the intervals of time between the observations. 
When four observations are employed, the problem, gene- 
rally speaking, is determinate, and is easily solved.-f 
* Prin. Math. Lib. 3, Lem. 7. 
f Arith. Univ. Prob. 52. Prin. Math. Lib. 1, Lem. 27 Cor. Tho. Simpson’s 
Elem. Geom. 3d edit, p. 353. Dav. Greg. Astr. Phys. et Geo. Elem. Lib. 5, Prop. 1 1, 
Edin. Tran. Vol. HI. 
