44« . AfETHOD OF CLEARING OBSERVED LUNAR DISTANCES 



through thofe points before the cutting of the fphere. Farther, 

 let the chord AB be parallel to the chord DC, and fince the 

 four points A, B, C, D, lie in the fection of a fphere, that is, 

 in the circumference of the fame circle, the chord AD will be 

 / equal to the chord BC, and the diagonal BD to the diagonal 

 AC. For the fame reafon (becaufe ABCD is a quadrilateral 

 figure infcribed in a circle) the rectangle under the diagonals 

 BD and AC is equal to the fum of the rectangles under the 

 oppofue fides AD, BC, and AB, DC ; or, becaufe BD and 

 AC are equal, as alfo AD and BC, the fquare of BD is equal 

 to the fquare of BC, together with the rectangle under AB 

 and DC, , 



Now let the point B reprefent the moon's apparent place, 

 and D the apparent place of the fun or a ftar, and let the chords 

 AB and DC be parallel to the horizon ; then is BD the chord 

 of the arc of the apparent diftance, and AD or BC are the 

 chords of the arcs of azimuth circles palling through the fun 

 or a ftar and the moon, intercepted between the parallels of 

 apparent altitude ; of in other words, AD or BC is the chord 

 of the difference of altitude. The length of the chord AB is 

 to that of DC in the ratio of the cofine of the altitude of the 

 object at B to the cofine of the altitude of the object at D, 

 and the rectangle under the chords AB and DC will be to the 

 rectangle under any two affumed chords parallel to AB and 

 DC, and terminated by the fame azimuth circles which pafs 

 through the points B and D, as the rectangle under the cofines 

 of the altitudes of the points B and D is to the rectangle under 

 the cofines of the altitudes of the two affumed chords. And 

 fince BD and BC are had by obfervation, the rectangle under 

 AB and DC is alfo known, being equal to BD 2, — i BC*, or 

 to BD -f BC x BD — BC, 



Make b the moon's true place, fituated fome where in the 

 fame azimuth circle that palled through B, and d the fun, or 

 flar's true place, fituated in the fame azimuth circle that paffed 

 through D ; and analagous to A B C D draw the quadrilateral 

 Figure abed, which will have all the properties of the former, 

 and ad or be will be the chord of the arc of the difference of 

 the true altitudes, and b d the chord of the arc of the true 

 diitance. And it has been already (hewn, that the rectangle 

 under the cofines of the apparent altitudes is to the rectangle 

 under the cofines of the true altitudes, as AB x DC is to 



