276 



On redutiiiif the Lunar Distance. 



ready method could be found for dividing tUe distance into 

 the two proper segments, it would aftbrd the means of obtain- 

 ing the effect of refraction with considerable facility. This 

 division has sometimes been accomplished by calculation, as 

 in Witchel's method: and can likewise be done by projection; 

 but at present, I wish to show how a very simple diagram may 

 be constructed to serve the purpose, as well as to give the ef- 

 fect of parallax. 



Let A D be a square of which the two sides A B, C D are 

 lines of sines whose zeros are at A and C ; the corresponding 

 degrees being joined by straight lines which are of course pa- 

 rallel to each other ; and these again are to be divided into 

 sixty equal parts by a set of straight lines parallel to A B, 

 which in their turn are divided by the former into lines of sines 

 the same as A B. The lines parallel to A C ai'e also cut by ii 

 curve in such a manner that the segment of each reckoned from 

 D C may represent the refraction corresponding to that alti- 

 tude, and the same lines are further cut by a circle described 

 from A to D about the centre C, so that the segments will al- 

 ways represent the parallax in altitude for a horizontal paral- 

 lax of 60': each of the 60 equal parts being counted one mi- 

 nute of parallax ; but to enlarge the scale as much as possible, 

 it is proposed to reckon each of the equal parts to be only 10" 

 of refraction, and the whole line only ten minutes. From C 

 draw a set of divei'ging lines to spread over the whole square; 

 these will obviously cut the vertical lines proportionally, so 

 that if the segments of two vertical lines cut by one oblique 

 line be the sines of the altitudes, the cosines of the segments 

 of the distance will also be segments of the same verticals 

 when both are cut by another oblique line, since these cosines 

 are proportional to the sines of the altitudes. 



To find the effect of refraction by this method : Look for the 

 greater altitude in A B, and from that point follow down an 

 oblicjue line till it meet the less al- 

 titude in some other vertical ; then 

 follow up at sam.i time the two ver- 

 tical lines in which the altitudes 

 were found till some oblique line 

 cut them both in such a manner diat 

 the sum or difference of the remain- 

 ing zenith distances may equal the 

 distance ; and the sum or difference 

 of the corresponding refractions will 

 be the effect required, which is al- 

 ways additive. There is no ambi- 

 guity as to whether the sum or difference of the zenith di- 

 stances 



