326 AN ACCOUNT OF THE 
each of them we obtain three angles, which are those of a plane triangle, 
having it’s sides respectively equal to those oi the original spherical trian- 
gie. The application of this rule is so simple as to require no explana- 
tien, all that is necessary being a table of the spherical excess as it-is 
called, which being very small in most cases, and proportional to the area 
of the triangle, may be determined sufficiently near a priori. This is given 
in table 4. | 
5. ALTHOUGH this theorem be a very convenient one, yel it is not by 
any means indispensible. It is easy to apply the common spherical ana- 
logies to small triangles, and this without any extension of the tables; by 
considering t < sines and tangents as referred to a radius whose length is. 
equal to that of the sphere, and expressed in the same measure as the sides of 
the triangle. The sines and tangents of small arcs, differ so little from the, 
arcs themselves, that it appears to be the most direct as well as the easiest 
way to find them by means of those differences. Thus the logarithmic 
sine, (BennycasTLz’s Trigonometry )=Log. arc — = sgt +a5r’ &e.) 
and Log. tang. = Log. are + = (+ ist 5r* &c.) The first 
terms of these series are sufficient for our purpose, and taking these 
it is evident that the difference of the tangent from the arc is double 
-the difference of the sine; that in the former case it is additive, in 
the latter subtractive. All that is necessary then, is to calculate ad sm Be 
being the radius of the earth, and = being the reciprocal of the logarith- 
mic modulus = 2°302581. Table 5, gives this correction for the proba- 
ble distances of the survey. For sines, half of it is to be subtracted from 
