Mr. Ivory on the astronomical refractions. 411 
quantity ; in the last, it is much greater than the truth. Now 
we may suppose an infinite number of atmospheres, gradually 
increasing in height, to be interposed between the two ex- 
treme cases ; and as the horizontal refraction appears to in 
crease with the height, there must be some intermediate case 
which will quadrate with observation in this respect. If we 
reflect that all these atmospheres will agree in giving the 
refractions actually observed by astronomers as far as 70° 
or 8o° from the zenith, it is natural to think that the one 
which likewise coincides with nature at the horizon, will 
deviate but little from the truth in the intermediate io°. At 
any rate we may conjecture, that the height of the atmosphere 
is an element in the problem that ought not to be neglected. 
It may be argued indeed that the infinite atmosphere con- 
sidered by Kramp and Laplace, will hardly be different, 
mathematically speaking, from one of such considerable alti- 
tude as we must suppose in the case of the earth ; and that 
in reality, all very high atmospheres may be reckoned as form- 
ing only one case, or at least as leading to results differing 
from one another only by insensible shades, that may safely 
be neglected in practice. This observation is probably well 
founded ; and, beyond a certain limit, it must undoubtedly 
be true ; but in a problem of such capital importance in as- 
tronomy, the point deserves at least to be examined ; more 
especially as it may lead to some more certain knowledge 
than we have yet acquired, with respect to the extent and 
constitution of our atmosphere. 
We have no direct knowledge of the height of the atmo- 
sphere, except what is derived from the duration of the twi- 
light, and from the great elevation at which meteors are 
