of reducing the Lunar Distances. 179 
gtaphical means, I have often thought it would be much more 
convenient if a general method could be found in which all the 
lines were ready drawn without requiring scales, compasses, or 
even different plates; for although in Margett’s tables and some 
others, no new lines are wanted, still each plate only serving for 
a small part of the various cases that may occur, many plates are 
necessary to comprehend the whole; and in the different editions 
of La Caille’s method, some moveable parts are required, as well 
as several accurate measurements with compasses, &c., operations 
not only troublesome, but in unskilful hands they are apt to pro- 
duce errors in abundance. 
What I was in quest of, therefore, was the construction of a 
general plate by which all the cases might be solved without the 
aid of any thing else, except a common ruler to lay across the 
plate ; and my researches on the subject have upon the whole 
been fully as successful as I had at first expected; but it proba- 
bly admits of still further improvement. 
The outline of all the other methods that I have seen for solv- 
ing the problem by projection, especially so far as relates to pa- 
rallax, is the same; viz. the orthographical projection of the 
spherical triangle formed by the distance and the complements of 
the altitudes, upon the plane of the circle of which the distance 
forms a part. ‘The one I am about to describe, however, is en- 
tirely different. In this there is given a separate contrivance for 
the effect of refraction and another for that of parallax: but both 
ean still be conveniently put in the same plate without enlarging 
its size. I once intended to have combined the two in one cor- 
rection*, which was not impossible; but afterwards thought it 
better to abandon that idea, hecause it led to some inaccuracy, 
arising from the necessity of varying the effect of refraction in 
the same ratio as the horizontal parallax; and I wished to give 
a method founded on principles admitting of some degree of ex- 
actness, should I afterwards have occasion to construct it on a 
large scale. . 
I shall now proceed to explain the principle on which the cor- 
rection for refraction is founded :—Assuming the refraction as 
the cotangent of the altitude, it may easily’ be shown that, on a 
given distance, the effect of refraction is the same for any two 
altitudes whose sines have the same ratio. If neither altitude is 
under 10° this assumption cannot materially err; but if we aug- 
ment each apparent altitude by nearly three times the corre- 
* Of the methods in which all the lines are to be drawn for each case, 
Dr. Kelly's perhaps comes as near the truth as any similar one could re- 
quiring so little labour. In it the refraction is combined with parallax. An 
improved method of the same kind has lately been given in Professor 
Brande’s Journal. 
Z2 sponding 
