41-4 Addilionat Notes. 



from the ends of the arch, and then k will be sufficiently exact 

 to bring the spot and horizon in a right line, on any part of the 

 horizon-rane, by moving the vanes nearer together or flirther 

 apart, 4he middle of the horizon-vane being parallel to the hori- 

 zon, then the zenith distance will be the sum of the distances of 

 the vanes from the end of the quadrant arch. ' For, putting r = 

 the radius of the quadrant, a — the distance of the spot from the 

 middle of the horizon-vane, 5= the sine, and c = the cosine of 

 half the suns altitude, unity being radius, the sine of the errour 



2 sua 

 will be nearly equal to c 4-4; x "rr" j ?-nd, tlierefore, when greatest 



(which is when the zenith distance is 0000, or 47 degrees 45 

 minutes), of the distance of ^ of the radius of the quadrant from 

 tlie middle of the horizon-Vaue, it is but 1 '30^ I would advise 

 to bring the upper or lower edge of the spot, and not the middle 

 and horizon, on a right line, and then subtract or add sixteen 

 minutes for the sun's semidiameter from or to the zenith distance 

 given by the vane. 



N. B. There should be an allowance for the obser\-er s height 

 above the surface of the sea, by subtracting four, five, or six mi- 

 nutes. A table of this kind would not be amiss on the back of 

 the quadrant. 



There may be some graduations put on the staff, near the centre,, 

 to be cut by a plumb-line hung on a pin put into a small hole for 

 land observations. One of these quadrant-., being eighteen inches 

 and two feet radius, if well graduated, will be sufficient to take 

 tlie sun's zenith distance within two or three minutes. 



Succeeding so well witli the sun, encouraged mc to take what 

 appeared a more difficult task, the finding some way to take the 

 altitude of the stars at sea (\\hen the horizon may be seen) belter 

 than by the fore-staff, which I concluded must be by bringing the 

 two objects, horizon and star, together. I first considered one re- 

 flexion ; but the faults of Davis's quadrant were here enlargc^d, 

 which is chiefly tlie flying of the objects from each other, by the 

 least motion of the instrument. I then examined what two re- 

 flexions would do, which perfectly answered my desire, being 

 equally useful in taking the distance of stars from each other, 

 and also from the moon, and I believe practicable at sea ; for I 

 found that when one star was made to coincide" by two reflex- 

 ions with another, the distance of these stars would be double 

 the inclination of the reflecting planes, as may be easily demon- 

 strated. 



