Mr. Lax's Method of finding the Latitude of a Place , &c. 75 
pared with x itself, is inconsiderable. Hence, if the abscisse of 
the curve ABCD (fig. 1.) be always pro- Fic> I# 
portional to z, and its ordinate to T, the area 
GB intercepted betwixt any two of these or- 
dinates may represent the increment of the 
latitude corresponding to the increment of the 
time EG. Let abed (fig. 2.) be another 
curve, whose abscisse ae is always equal to 
AE in the preceding figure, but whose ordi- 
nate eb is proportional to t, the tangent of 
the hour-angle ; then will the area g b vary / 
as GB, at small distances from the meridian, 
and, of course, may represent the increment 
of the latitude. Now, to prove this, we have 
only to shew that T and t, when both are 
small, bear to each other a given ratio. a ey & e £S 
Let S and £ be the sine and cosine of the azimuth : s and cr 
Fig. 2. 
the sine and cosine of the angle at the pole ; then will ~ = Z , 
- I - 4 r T , and j- = zy — 1 ; Z = J-, and z = -L. But, since the 
complements of the declination and altitude remain constant, 
whilst the latitude is made to vary, S will be to s as S to s ; 
and, therefore, we shall have : 7- : : - x 1 ^ - T - : -f— x 1 ■ 
: : 1 + T* : 1 -f : : the square of the secant of the azi- 
muth : the square of the secant of the hour-angle, which may 
be considered as a ratio of equality, when the angles are very 
small. The fluxions, therefore, of the tangents are as the tan- 
gents themselves ; and, consequently, they must always preserve 
the same ratio towards each other. Let us now suppose that 
L 2 
