* Apology for the Postscript on the Refractions,” Bc. 17 
not therefore have been better to refer to some work of undoubted 
reputation with the public, than to have attempted to explain 
‘them by calculations, of which it cannot be said that any one 
result is accurate? But in this manner his readers would have 
been better able to judge of his consistency and fairness of 
arguing, when they found him affirming gravely that there is no 
want of conyergency of the series, at the very time the default 
of convergency obliges him to employ subsidiary expedients. 
By taking the whole values of the variable quantities at two 
intervals, he seems to have considerably diminished the error 
arising from the want of convergency of the series, But, how 
many intervals must be taken in order to exhaust it completely ? 
We thus fall upon the same discussions agitated from the origin 
of the science. At any rate it appears necessary that he push 
his calculations up to the mark of truth, at least in some one 
instance, before the methods he recommends can be fairly com- 
pared with those usually followed. But, however this be, it must 
not be forgotten that the method of calcnlating by intervals, has 
nothing to do with the construction of the table in the Nautical 
Almanack. 
The formula used in the construction of the table contains 
four terms; and the horizontal refraction in the table, is imme- 
diately found by solving the proper equation. But when we take 
a case of real theory; that is, one proceeding upon a given hy- 
pothesis of density, by which means the coefficients of the series 
are taken out of the clutches of the computer, and are derived 
solely from the nature of the case; then six terms of the series, 
not to say four, are totally inadequate for finding the refraction 
with the requisite exactness. What is the reason of this? Is 
it not that, in the.one case, the coefficients are so adjusted as to 
bring out the desired result; while, in the other case, the ex- 
_pectation of the computer is balked, because the modelling of 
the series is placed out of his power? 
The coefficient of the first of the four terms is unavoidably de- 
. termined by the nature of the case, or by the differential equa- 
tion: the other three are empirical. Nor will much be abated 
from this, if it be allowed that some assistance has been derived 
from a small exertion of the reasoning faculty in fixing the form 
of the coefficients, while their quantity is obtained entirely by a 
tentative method aiming at given results. Nothing in the apology 
is contrary to what is here advanced. It is admitted that the 
. formula is partly empirical, and we are referred to Euler’s Lunar 
Theory, as a parallel case. This instance is not very much to 
the point: for although the immensity of the calculations, and 
the impracticability of performing them, made it necessary to 
seek from observation what could not be found by theory, vet 
Vol. 59. No, 285, Jan. 1822. C this 
