90 A Method for ascertaining Leaiitude and Time 
Thus the formulas(11), (12), (13),14, (15)and (16), which 
contain the practical solution of the problem in M. Gauss’s 
method, and to w hich he has attained by a very dexterous 
analysis, aie the same as are taken at sight in the three tri- 
ariales that have been resolved at all times by astronomers. 
So that it is here demonstrated, ipso facto, that analysis 
can by its own strength lead to the same result as geome- 
trical consideration. ~ But it is not the less clear, that these 
jast have the advantage of facility and conciseness ; and we 
may add that it is always pleasing for the calculator to 
understand what he goes through; he sees that x and x are 
the first segments of “the two bases: 90 —é+ 2, and 90 —? 
— 2, the second seoments of the same bases ; W, V and w 
angles comprised “between known sides. These notions 
would suffice for him to find all the formalas he wants, with- 
out having recourse to any book ; while it is, as I may say; 
impossible to grave in our memory the six analytical for- 
mulas; so that we are obliged to follow them blindly, be- 
cause we cannot divine what are the subsidiary ares F, W, 
“V, wand G, nor (F — 4) and (G — 8) in this problem. 
From the demonstrated identity of the two methods, it 
is easy to conceive that in point of conciseness, the dif- 
ference cannot be great. In the angle W alone, to ascer- 
tain which the processes are not quite the same, the tri- 
gonometrical method is somewhat shorter, as will be seen 
by the calculated example. 
M. Gauss lastly inquires eo differentiation of funda- 
mental formulas, what may be the influence of the errors 
ih and dh’ of the two altitudes observed on the latitude aiid 
the horary angle A. I have expressed this same effect by 
different formulas, with which I shall begin. 
In the first place : : It is clear that the errors dh. and dh’ 
can bave no influence on the first three formulas which 
gave x, V aud D, for those quantities only depend on the 
two stars. 
The triangle to the zenith gives the equation cos W cos 
Asin D = sin A’—sin h cos D. Whence dW sin W cos h 
sin D—dh sinh cos W sin D= dh’ cosh’ —dh cos li 
cos D. We shall suppose D constant — dW = dh 
sin h. cos W sin D — cosh cos D dh’ cos h’ 
( sin W cos /sin- ae ) ‘cosh sin D sin "a 
dh cos hi k dh cos h cos D ; hr 
cothsinDsin W sin W sin D cosh © dh tang h cos W 
: cos hcosD — sin / isin D dh’ cos h’ 
as = dh {te coshsinDsinW ee Ww) rey y 
he 
cos hv sin D sinW 
+ dW 
