18 : Reply to the 
this must be considered as an imperfection and a blemish, if we 
may be allowed to use such words in speaking of a matter that 
so highly concerned the benefit of mankind. A few years ago, 
the Academy of Sciences proposed, for their prize-question, the 
Construction of Lunar Tables by Theory alone, the fortunate 
competitors being M. Damoiseau and MM. Plana and Carlini. 
But, in the case of the refractions, we are desired to hold a re- 
trograde course, aud are required to re-compute by an, empirical 
formula the very same numbers already calculated by theory. 
The author of the Apology misquotes my words, and slurs over 
the question of the identity of his table with that of the French. 
It is not enough to say that they agree in all ordinary cases ; for 
there is no difference between them in the mean refractions. 
This isa fact of which any one may satisfy himself by reducing 
both tables to bar. 30, or both to bar, 29-93, the mean tempera- 
ture being the same in both cases. The slight differences that 
occur will generally be found less than the discrepancies arising 
in solving over again the equations of the new method. 
As there is no particular hypothesis of density adopted, the 
theory of the formula, if there be any, can be nothing but the 
general consideration that the density of the air, being a function 
of the refraction, may be developed in a series of the powers of 
that quantity. It therefore became necessary to prove not only 
that the series converged in every possible hypothesis of density, 
but that it converged so fast as to permit the rejecting of all the 
terms after the four first. Now this is not only not done, but 
it is not true. 
But, it may be asked, how then does it happen that the for- 
mula represents the French mean refractions so exactly? Now 
even this question may, I think, be answered in a satisfactory 
manner. By adopting the hypothesis of-a density decreasing 
uniformly, we obtain an exact solution of the problem of refrac- 
tions in the form of an equation containing the two first powers 
of the quantity sought. The rules of Bradley, Mayer, &c. are 
all equivalent to the solution of a quadratic equation®*. In their 
original form these rules can be applied only to compute the re- 
fractions at altitudes greater than 12°, or 14°; nearer the hori- 
zon they diverge from the truth. But if we relax from the 
strictly theoretical quantities, and determine the coefficients so 
as to represent the refractions at the horizon and at 45° from 
the zenith, we obtain empirical formule that apply with consi- 
derable exactness even at low altitudes. Now if to the two terms 
of such a formula, two more be added, so as to have three terms 
with indeterminate coefficients, a great latitude of calculation , 
" Kramp, Ref. Ast., p. 164. 
, will 
