Mr. Ivory on the astronomical refractions. 457 
he has no intimation, and for which he can make no allowance. 
In comparison of this great source of error we may reckon 
as of small account, the inaccuracies that are owing to the 
neglect of the moisture diffused in the atmosphere, or to our 
want of an exact knowledge of the law of density in regard 
to temperature. There can hardly be any other remedy 
than that of which astronomers so often avail themselves, 
whenever an ignorance of the real causes obliges them to 
assimilate the phenomena to the effect of chance ; namely, to 
multiply observations in different circumstances, with the 
view of making the inequalities of an opposite description 
compensate one another. 
From the foregoing discussion we may draw this conclu- 
sion : that an atmosphere constituted like that of the earth, 
must have an altitude of at least 25 miles, in order that the 
refractions from the zenith to the horizon be such as they 
are actually observed to be. But an atmosphere agreeing 
with nature in the quantity of the refractions may be found, 
that shall have any proposed altitude greater than the mini- 
mum quantity. 
We may infer from the duration of the twilight, that the 
atmosphere of the earth must have an altitude equal to 50 
miles, or even more ; which corresponds to the supposition 
of m equal to, or greater than 10. But all these cases are so 
little different, as to the refractions, from the extreme case 
when m is infinitely great, that we may suppose them to 
coincide with it. The most probable supposition with respect 
to the mean law of density, seems therefore to be contained 
in these equations, viz. 
— u 
1 — U = C 
s = C 1 — f) u i- 2/(1 —r" M ) ; 
3 N 
MDCCCXXIII. 
