79 
by Means of two Altitudes of the Sun. 
latitudes, we may extend the limits above specified a few degrees 
farther from the meridian, without offering any material violence 
to the theory, as it has hitherto been explained; and that, on 
the other hand, when the declination and latitude are nearly 
equal and of the same denomination, it will be expedient to 
confine our observations within a much shorter distance from 
noon. But it will afterwards be demonstrated that, whatever be 
the magnitude of the hour-angles, or however nearly the lati- 
tude and declination may approach towards each other, we 
can always secure, with very little additional trouble, an exact 
conclusion. 
We may remark that the latitude, determined in this man- 
ner, will be nearly equivalent, in point of accuracy, to the mean 
result of two meridian altitudes. For we know that the incre- 
ment of latitude : increment of altitude : : radius : cosine of azi- 
muth; and, since the cosine of a small angle differs so little 
from the radius, this may be considered, within the limits 
which I have prescribed, as a ratio of equality. If, therefore, 
one altitude of the sun were taken, and we could ascertain the 
error in time arising from an error in the assumed latitude 
without the aid of a second observation, the latitude would be 
discovered with nearly the same precision as if it had been de- 
duced from the meridian altitude. But, by means of a second 
observation made on a different side of noon, we obtain a 
second error in time of the same kind ; and this being added 
to the former, and their sum divided in a just proportion be- 
twixt the two observations, the same effect will be produced, 
with respect to the accuracy of the result, as if two latitudes 
had been deduced from meridian altitudes, and a mean betwixt 
them had been taken. 
