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a first Approximation to the Orbit of a Comet. 
If we reflect on the suppositions made by Newton for sim- 
plifying the method of finding a comet's trajectory, it must be 
admitted that they seem judicious, and that they are founded 
in fact. In the usual circumstances in which terrestrial obser- 
vations are made, there is no doubt that the real path of a 
comet, for a short time at least, differs little from a straight 
line, and that the motion is nearly uniform. Yet the problem 
grounded on these suppositions, although it has been often 
applied, has never, in any instance, led to a satisfactory deter- 
mination of a comet's orbit. The reason of this was first 
noticed by Boscovich, who shewed that in the actual state of 
the data, owing to the earth and the comet being both in mo- 
tion, the problem is as indeterminate when four observations 
are employed, as we have already remarked that it is in the 
case of three observations only. 
In order to explain what has just been said, it must be ob- 
served that, for a short time, the earth as well as a comet, may 
be supposed to move in a straight line with a uniform velocity. 
If we draw a tangent to that point of the earth's orbit which 
the planet would occupy at the middle instant between, the 
two extreme observations, the supposition of a uniform motion 
in the tangent will not much displace the earth from its real 
positions, and will produce but little alteration in the apparent 
places of a comet. Suppose then that the earth moves with a 
uniform motion in the straight line AD, (fig. 2. PI. IV.) so that 
the parts AB, BC, CD, are proportional to the intervals of time 
between four observations of a comet : also let the four observed 
places of the comet in the ecliptic be represented by the points 
P, E, F and Q lying in one straight line of which the parts PE, 
EE, FQ, are proportional to the same intervals of time, and 
