VOL. XXXVIII.] PHILOSOPHICAL TRANSACTIONS. 675 



distance between the two observations 8 hours, as before, and the arch aaa, 

 Ex. 2, described by the zenith distance of the first observation from the centre 

 c, and the angle cab, 40°, is the angle between the ship's way and the azimuth 

 of the sun continued, given by the azimuth compass; and that during the 8 

 hours the ship has made one degree, or 6o minutes, from a to b, or from the 

 sun; then, as radius is to the cosine of cab, 40°, so is ab, 6o' to ac 46'; add 

 46' to the zenith distance ca; and with k, the point of the beam-compass set 

 at that distance, describe the arch cbe; then with the zenith distance of the 

 last observation, whose centre is d, draw the arch ff; the point where it cuts 

 the arch cbe, is the place where the ship was last; and its distance, taken on 

 the meridian from the equator, shows its latitude; the minutes reckoned on 

 the equator from the meridian to d, the time of the last observation, show the 

 hour, or its distance from I'Z o'clock. 



Ca?e 2. If the ship had sailed from « to p, or towards the sun : the cosine of 

 the angle Pay, or of the angle between the ship's way and the sun, must be 

 subtracted from the zenith distance of the first observation. 



N. B. Only the two arches cbe, ff, are to be drawn on the globe, the rest 

 being added liere, to show the reason of the construction. 



Case 3. To find the latitude of the first place, from the equator, with a pair 

 of compasses, take the distance sailed 6o'; and with one foot in the intersec- 

 tion of the arches be, ff, the place found before, put the other in the arch 

 aaa, the zenith distance of the first observation, and in this instance, on the 

 left hand of the azimuth of the sun, this is the place sought; and its distance 

 taken on the meridian from the equator, shows the latitude; and the minutes, 

 reckoned on the equator, from the meridian to c, the time of the first obser- 

 vation, show the hour. The interval, in time or degree, between the two 

 places, shown by the index g, is the difference of longitude. Those observa- 

 tions are best, whose arches cross each other almost at right angles. 



Prop. 11. — The zenith, distances of two stars observed at the same time, 

 their declination, and right ascension being known; to find the latitude of the 

 place of observation. — Fix the centre of the beani-compass to the declina- 

 tion of either of the stars, and with the zenith distance of that star describe 

 an arch; move the meridian as many hours farther as is the diflTerence of right 

 ascension of the other star; and fix the centre of the beam-compass to its 

 declination, and with its zenith distance cross the first arch; the intersection 

 shows the latitude of the place of observation, and also the distance of the 

 right ascension of the zenith from that of either of the stars; by which means 

 the hour may be known. 



When a celestial globe is used, then place the centre of the beam-compass 



