' Nonagesimal degree of the ecliptic. 303 
ulas for Tang. F and Tang. G, furnish in logarithms the rule above 
given for calculating F and G to be substituted in the value of L= 
90° + F + G, 
When R is in the ascending signs, as in Fig. 2, and Z is situated 
without the polar circles, D, Sand }'T must be acute. In this case 
the formula for Tang. G and Tang. F become negative, consequent- 
ly G, F will be obtuse. In Fig. 3, R is in the descending signs, and 
Z without the polar circles, D and S are acute, 3 T obtuse, and its 
tangent becomes negative; hence, by the formula tang. G and tang. 
F are positive, and F,G acute. Consequently G and F are of a dif- 
ferent affection from } T agreeable to the rule. | 
If the polar distance P Z (Fig. 2, 3) decrease and become equal to 
Pp, D, B and F will be=0. By decreasing farther the value of P Z, 
the point Z will fall within the north polar circle, P Z will be less than 
P p, and D, B will become negative, and F change its sign. Hence 
to make use of the formula L=90+F +G, it will be necessary, in 
this case, to write 360°—F instead of F. On the contrary, if the 
polar distance P Z (Fig. 2, 3.) be supposed to increase, D and S will 
remain acute until P Z = 180° — P g, then S = 90° and its cosine = 0, 
consequently A and tang. G will become infinite and G=90°. Be. 
yond that point, within the south polar circle, S will exceed 90°, its 
cosine will be negative, A, B, will be negative, and tang. G will 
ehange sign, consequently the supplement of its former value must be 
taken. These agree with the rule and include all the cases. 
Again (by § 10 page 653 of the Navigator) we have in the tri- 
angle P p Z, Cos. 3 (Pp Z—P Zp): Cos.3(PpZ+PZp):: tang 
(Pp + PZ) : tang. 3 p Z, which in symbols is Cos. (180° — F) : Cos. 
(180° — G) : : tan. S : tang. 2 H, or by reduction cos. F : cos. G:: 
tang. S : tang. 3 H, whence the rule for finding the half altitude of the 
Nonagesimal is easily deduced. It may be observed that this rule 
