274 On reducing the. Lunar Diskmce. 



The construction that has now been described might as!>ist 

 in giving a solution in round numbers of various problems in 

 spherical trigonometry. Thus, having given the latitude of 

 the place, the declination and altitude of the sun to find die 

 time ; in place of the distance, the altitudes of the moon and 

 star, substitute the colatitude, the declination and altitude of 

 the sun respectively; and by attending to the signs of the quan- 

 tities, we shall have cos. declm. x cos. liorary angle. This 

 again might be freed from cos. declin. by a small separate 

 scale, or by such a method as that proposed for the different 

 horizontal parallaxes. 



I have also succeeded in materially improving the construc- 

 tion formerly proposed for showing the effect of refraction ; 

 and in addition to other advantages, it does not now require 

 a ruler laid across in using it. 



Let A B be a line of sines be- 

 ginning at A: draw D A C per- 

 pendicular to it, and join BC ,- 

 draw also a set of straight lines 

 parallel to A B, and reaching 

 from AC to B C : next lay a 

 ruler from D through each de- 

 gree of A B, and draw as many 

 straight lines extending from 

 A B to B C ; these will evidently divide each parallel into 

 a portion of a line of sines, but always to a greater radius, 

 as they are more distant from D. From C draw a set of 

 diverging lines to meet A B, and of course dividing all the 

 other parallels in the same ratio as they do A B. Hence if 

 the greater altitude be sought out on A B, and if from that 

 point E an oblique line be followed till it meet tlie less 

 altitude on some other parallel FG, it is plain diat every 

 oblique line drawn from C must also cut A B and F G in 

 points which denote two altitudes whose sines are in the 

 same ratio as those of the first two. If therefore G H be 

 the effect of refraction in the first case, it will be so for the 

 same distance whenever the sines of the altitudes are in the 

 same ratio; that is, whenever the less altitude falls on FG. 

 Reckoning then the greater altitude always on A B, it follows 

 that the less of every two altitudes whose sines have the same 

 ratio, must fall on the same parallel ; or, that each parallel 

 belongs to a different ratio of the sines of the altitudes. So that 

 a curve may be constructed in the manner formerly directed, 

 of such a nature that when the several parallels are produced 

 to meet it, the part of each produced, or the segment inter- 

 cepted between B C and the curve may always represent the 



eflect 



