410 Mr. Ivory on the astronomical refractions. 
medium, and it is computed by the most elementary principles 
of optics. This hypothesis, although extremely simple, leads 
to a rule for the refractions which, to a certain extent, is as 
accurate as any other. Perhaps it is owing to its great sim- 
plicity, that the method of Cassini seems not to have met 
from astronomers with the attention it deserves. Another 
hypothesis attributes a variable density to the atmosphere, 
but assumes that the rate of decrease is exactly proportional 
to the height ascended. This supposition is in some degree 
less inaccurate than that of Cassini. Most of the formulae 
for the refractions that have obtained any extensive use in 
astronomy may be deduced from it. Kramp took a more 
extended view of the problem, and one less exceptionable, as 
approaching nearer to nature. He conducted his calculations 
by the real laws that regulate the density of the air, namely, 
the pressure and temperature. Laplace coincides with Kramp 
in the general view he takes of this theory ; but, in treating 
it, he has given new proofs of that sagacity and mathematical 
skill, which has enabled him to accomplish so much in phy- 
sical science. The table, computed by the theory of Laplace, 
first published in 1806, perhaps at this day gives the law of 
the mean refractions with greater accuracy than any other, 
whether founded on theory or observation. 
In the hypothesis of Cassini, the atmosphere extends no 
higher than five miles above the earth's surface. On the 
supposition of a density decreasing at an equable rate, its 
height is limited to ten miles ; but in the view of the problem 
taken by Kramp and Laplace, the atmosphere extends in- 
definitely into space. In the two first hypotheses, the hori- 
zontal refraction is considerably less than the observed 
