+ a 
- none is used 
by means of Two known Stars. 83 
this analogy requires five logarithms, which being united 
to the teh or cleven preceding will make fifteen or sixteen 
at will, 
Let u= V + W, uw will then be the angle to the first star 
between the vertical and the circle of declination. In the same 
manner we should have w= V ++ W’ for the second star. 
Knowing the angle x, you will endeavour to find the 
“horary angle 4 and the latitude g by means of the follows 
ing formulas, which are all similar to the formulas (1), (2) 
and (3). 
tang uw sin 
cos (3-4) 
cos A tang (?+4+2), (5), (6) and (7), by changing @ into &’, A 
into A’, and A into a’, and reciprocally, you will obtain 
the horary angle of the second star, by means of the se- 
cond altitude. Ya 
Instead of the last formula, we might. bave sin ¢= 
sin h sin (+2). 
cos % 
fang x = cos ucot h; tanga = 
. fang 6 = 
---.(7*). This third triangle therefore re- 
‘quires ten or eleven more logarithms. 
The entire solution thus requires five-and-twenty or sevens 
and-twenty logarithms, but the method requiring only fives 
d ore than once, 
These formulas are easy to establish, and require no effort 
and-twenty. spay alsegve be preferred. Of these logarithms, 
; Of nEOy tees might obtain the angle W by the fur- 
, ; sin ’ — sin h-cos D sin }! 
et) ..+.(4%), but it would require seven logarithms in- 
stead of five: in the case h = h’ or of two equal heights, 
the formulas are on the contrary reduced to cos-W = 
sin h (1—cos D) _ tang 42sin° 5D _ ; Bi 
cos hsinD ~~ 2 sin }D cos 4D ~ tang A tang 4D, which 1; 
only requires three logarithms “instead of five. 3 
* Cos W may belong to a negative as well as to a positive — 
arc. It is this uncertainty which has already been re- 
» marked on the subject of sin? 4W.. It is common to all 
methods, 
It is not likely that a more easy or shorter solution can 
ve found than the one contained in the foregoing formulas. 
Let us now examine M. Gauss’s, giving to the analyti- 
cal calculations a more direct and elementar ‘form, - 
M. Gauss first draws from spherical trigonometry the 
two fundamental equations; 
Sin h = sin 3 sin g + cos 3 cos 9 cos Nistnip ssn (de 
Sin /’ = sin &’ sin g + cos & cos 9 cos (A—4) ...., -(2). 
ane He 
