128 Mr. Ivory on a new Method of deducing ^ 
have been expected from th^ labours of so many eminent 
men. In most of the solutions that have been proposed we 
find certain coefficients which are not only extremely small 
when reduced into numbers, but they likewise suffer great 
alterations in their relative magnitudes when small changes, 
within the limits of the probable errors, are made in the quan- 
tities furnished by observation. In such circumstances the 
results obtained are often wide of the truth, and in no case caq 
entire dependence be had on their accuracy. The fact is that the 
solutions here spoken of, are uncertain and useless in practice 
for the very same reason as the one already mentioned, which 
supposes the orbit to be rectilineal, and proposes to deduce it 
from four observations Nor is this to be wondered at. We 
have already shewn that it is not any inaccuracy in the sup- 
positions made by Sir Isaac Newton, which renders his first 
method insufficient and of no use in practical astronomy; nor 
is this owing to any defectiveness in the geometrical construc- 
tion ; it arises from his having overlooked that connection 
between the motion of the earth and the motion of the comet, 
which makes the data approach so near the indeterminate case 
of the problem that the conclusion becomes quite uncertain. 
We fall upon the indeterminate case when we suppose both 
the earth and the comet to move in straight lines with uniform 
velocities ; and the very same hypothesis will be found to 
make the small coefficients alluded to, accurately evanescent.* 
The inconvenience is inherent in the quantities obtained by 
* The small coefficients here mentioned, are exactly evanescent when all the three 
observed places of a comet are in one great circle of the heavens ; but it is easy to 
prove that if two celestial bodies be supposed to move uniformly in straight lines, one 
of them will be seen from the other to describe constantly the same great circle. 
