194 A Method for ascertaining Latitude and Time 
‘which Mr, Gauss has given, and may be easily deduced 
from my formule, night be still more easily demonstrated 
by the figure which has supplied the fundamental equation 
of the problem: they show evidently that if A’= A, dh 
will necessarily be = dh’; for the second members being 
equal, the first must be so likewise. If A’= 180’ + A, 
di’ will be equal to —dh. In fact, in the first case, h’ = 
h + d, andsin the second case 4 + A’ + D = 180. Thus, 
if the first altitude A had been observed, it would only be 
required to set down the time when /! would be equal to 
h+D, or where 4) = 1680 —h —D; but it would be 
necessary to be assured that the second star would pass by 
the same vertical as the first: but if two complete obser- 
vations of altitudes and times are made, it is possible that 
dh' and dh will be such that the altitudes & and A’ will no 
longer agree with the time elapsed, with the two declina- 
tions and with the distance D. Thus, the suppositions it 
is necessary to admit no longer agree with what passes in 
the heavens; and in order that the differential formule 
could be established, either the declinations or the angle 4 
should have’been made to vary. 
If A’ differs not much from A, the two equations will 
not differ much from each other. And as it is from their 
differences that the values of dg and da are computed; in 
order properly todetermine those two quantities, an extreme 
precision is requisite in ascertaining the values of A’ and 
A, a precision which can be expected neither from the ob- 
servation nor the calculation. These differential formule 
are therefore only approximations, on which dependence 
should not always be placed, which, bowever, does not im- 
ply that dg and da will in that case be very considerable, 
If the cosines of the azimuths are small, it nay be ex- 
pected that the latitude will not be well determined ; if the 
sines of the azimuths are small, the hour will not be well 
known; but if one of the observations is made towards 
the first vertical, and the othér towards the meridian, all 
the precision will be obtained that can be expected from 
this method, which may be of great value, when no other 
is feasible, bat which can never be compared with those 
that separately determine the latitude and the time. 
Mr. Van Beek Calkoen, director of the observatory _ 
roval at Utrecht, has just given a different solution of this 
problem, with a shght change, which was not even indis- 
pensable. It consists in reducing to the same moment the 
observations of the two altitudes. For this purpose Mr. 
Calkoen commences by observing one of the two stars 
immediately 
