302 Mr. Bowditch on the altitude and longitude of the 
Demonstration. dd 
Let EQ (Plat. 2, Fig. 2) be the equator, ¥3 2 the ecliptic, + the 
point of Aries, p the north pole of the ecliptic, P the north pole of the 
equator, Z the zenith of the place of observation, P / the obliquity of 
the ecliptic, P Z the north polar distance of the point Z.. Thenasin 
President Willard’s paper, in the first volume of the Memoirs of 
the Academy, p Z is the altitude of the nonagesimal = H; and the 
angle * p Zits longitude=L. The angle » P Z is equal to the right 
ascension of the meridian R, found by adding the apparent time to 
the sun’s right ascension. a 
In the spherical triangle p P Z putS=i(P Z + Pp), D=i(PZ— 
P p), the anglep P Z = 90°+ R=T, G= 180° —3 (PpZ+pZP) 
F=180°—i(PpZ—pZP). Hence Pp Z = 360° — F — Gand 
90°—P pZ =~ p Z = longitude of the nonagesimal L becomes L’= 
90° + F + G, rejecting-as usual the 360°. Then, by the noted rules 
of Napier (marked (7) (8) in page 653 of the third edition of the Nav- 
igator) we have Cos. S : Cos. D:: Cot. 1T : Tang. (180°—G) and 
Sine S : Sine D :: Cot. 2T : Tang. (180°—F). Dividing the 
terms of the last analogy by the corresponding ones of the former, and 
en Sine D a 
putting Gos, 5 — Lang. 8, Cos. D = Lang: D, and noting the signs 
of the terms in the usual way, in order to ascertain the affection of F 
and G, by putting Tang. (180° — G) — — Tang. G, Tang. (180°— 
F) =— Tang. F, we have Cos.S : Cos. D :: Cot.iT ; —Tang: 
. . = aoe! Py D:: Tang. G : Tang. F. Or, in putting 
Con So Tang. S = Tang. G = — A Cot. 2 T, Tang: F= 
B Tang. G, : 
The quantities A, B, are evidently the natural numbers corres- 
ponding to the logarithms A, B, of the preceding table. The form- 
