127 
a first Approximation to the Orbit of a Comet. 
consequently BE will cutpq in e^ and CF will cut it in/. Thus 
it is proved that innumerable lines may be drawn that will all 
be cut by the four lines AP, BE, CF, DO into segments hav- 
ing the same proportions as the intervals of time between four 
observations of a comet. The problem is therefore not more 
determinate with four, or even a greater number of, observa- 
tions, than it is with three : and it is totally unfit for finding 
the lengths of the lines AP, BE, CF, DQ, or the comet's dis- 
tances from the earth. The shorter we make the intervals 
of time between the observations, and the more exactly the 
suppositions made by Sir Isaac Newton are fulfilled, the less 
fitted is the problem to answer the purposes of astronomy, 
because it approaches more nearly to the conditions which 
make it indeterminate.'^ 
Besides tlie method of which we have hitherto been speak- 
ing, Sir Isaac Newton likewise gave another very able solu- 
tion of the problem,/ which however is too tedious and 
laborious in practice to fulfil the wishes of astronomers, and 
it is now disused. Dr. Halley, who studied the astronomy 
of comets with much diligence, has no where explained the 
method he followed in determining tlTeir orbits. Since his time 
this problem has not been much discussed in England ; but it 
is the subject of a great number of researches scattered in the 
academical collections of the learned societies on the conti- 
nent. All the resources of the modern mathematics have 
been directed to overcome the peculiar difficulties of this in- 
tricate investigation. Yet it must be confessed that practical 
astronomy has not reaped so much benefit as might justly 
* Vide a Memoir by Professor Platfair in the 3d Vol. of the Edin. Trans. 
f Prin. Math. Lib. 3, Prop. 41. 
