by Means of two Altitudes of the Sun. 85 
for the area gc become ~ ; and, consequently, (since the tan- 
gents of the azimuth and hour-angle vanish in the ratio of 
their sines, or of the sines of the opposite sides in the triangle 
alluded to before,) we shall have the area GC = when the 
sun is upon the meridian. But this area is always the same 
when the latitude is given, whatever be the sun’s declination, 
and therefore may always be represented by = — x -7- 
== ; and the area gc will be generally expressed by — x 
cos, of mend, aitit. w j len t i ie hour-ande does not exceed the limits 
cos. of declin. 7 0 
which have been recommended. Hence, if we add together 
the constant log. 3,1015, the log. radius, and the log. cosine of 
the merid. altitude, and subtract from their sum the log. cosines 
of the latit. and declin. we shall obtain the log. value of gc. 
It will be necessary, perhaps, to meet an objection which 
some may be inclined to urge against the method of deducing 
the hour-angle in terms of the cosine, when this angle is very 
small. But it should be recollected, that with the angle itself 
we have no immediate concern, the accuracy of our conclusion 
depending entirely upon the accuracy with which the area cor- 
responding to any particular increment of time can be deter- 
mined. Now this area, whatever be the sun’s distance from 
the meridian, will be nearly proportional to the increment of the 
latit. and, consequently, its magnitude is totally unconnected 
with that of the hour-angle. A given error in the quantity 
which expresses this area will equally affect our conclusion, 
whether the angle be 2, or whether it be 20 degrees. But let 
us inquire what effect will actually be produced, by admitting 
