564 ■ I'HILOSOPHICAL TRANSACTIONS. [aNNO 1732. 



pleiTient of the distance sb. But an is the chord of the arch ahn of the great 

 circle ban, equal to the translation of the point a, or double the inclination of 

 the specula ; and an is the chord of the arch ahn, of a great circle measuring 

 the angle ann, by which the point a appears removed by the tivo reflections, to 

 an eye placed in the centre r. Therefore the translation, or apparent change 

 of place, of the point a, is measured by an arch of a great circle, whose chord 

 is to the chord of the arch ahn (equal to double the inclination of the specula) 

 as the sine complement of its distance from the great circle ban, is to the 

 radius. 



From any point c of the circumference obc, draw the chords cm and cm, to 

 the same side of the point c, and equal to the chords an and an respectively ; 

 draw the radius rm ; and from r and m, draw kq and mp, both perpendicular 

 to CM, and cutting it in q and p. Ra is the sine complement, and cm double 

 the sine of half the angle mrc, or arn, or of the angle of inclination of the 

 specula. The little arch Mm will represent the difference of tlie apparent trans- 

 lations of the objects in a and a ; and if it be very small, may be considered as 

 a straight line ; and the small mixed triangle Mmp as a rectilinear one, which 

 will be similar to rmp, because rm is perpendicular to Mm, and rq to cm, and 

 the angles at q and p right angles. The line cp may be taken as equal to cm, 

 and MP as the difference of the lines cm and cm. Therefore the small arch 

 Mm, is to the line mp, nearly as rm to rq: but cm (i. e. an) was to cm (i. e. an), 

 as BR to br, and the difference mp, of cm and cm, to the difference bd, of br 

 and br, as cm to br. Therefore Mm, the difl^erence of the apparent translations, 

 is to BD, the versed sine of the distance nb, or to an arch equal to it, in the 

 compound ratio of rm the radius, to rq the sine complement of the angle of 

 inclination of the specula, and cm double the sine of the same, to br the 

 radius, i. e. as cm to rq. 



The observation may be corrected by one easy operation in trigonometry, as 

 will appear from the first part of this corollary, viz. by taking the half of the angle 

 observed, and then finding another angle, whose sine is to the sine of that half, 

 as the sine complement of the distance sb. is to the radius : this angle doubled, 

 will be the true distance of the objects. But as this operation, though easy, 

 will require the use of figures, Mr. H. rather chooses the method of approxi- 

 mation, because by that the observer, retaining in his memory the proportions 

 of the sines of a few particular arches to the radius, may easily estimate the 

 correction without figures, when the angle is not great, and by a line of artifi- 

 cial numbers and sines, may always determine it with greater exactness than 

 will ever be necessary. 



When the angle observed is very near 1 80 degrees, the correction may be 



