22 MATHEMATICS. 
two angles or arcs, which, when added together, make 180°, have equal sines. 
CosyiAe 
sin. @ 
whence there may be two values for B—one above, the other under, 90°. 
In fact, pl. 3, fig. 116, shows that the two triangles, bac and ba''c, havea side, 
bc, common, and the angles opposite to A equal (since the angles bac and 
ba''c are equal), while all the remaining parts of the one triangle are supple- 
ments (180 — the part) of the corresponding parts in the other. 
In the solution of acute angled spherical triangles, two cases occur in 
which the results of trigonometrical calculations are ambiguous: 1, when 
two sides and the angle opposite the smaller of these are given ; 2, when two 
angles and the side opposite the smaller one are given. Fig. 117 illustrates 
the latter case. If, in the triangle abc, we have given the angles abe and 
acb, and the side, ac, opposite the smaller angle, then a second and entirely 
different triangle, acb’’, may be constructed, of very different parts, provided 
that ab’ is so taken that its prolongation ad— ab, and consequently abc = 
adc — ab''c. 
In astronomy, it is frequently desirable to ascertain what effect a very 
slight alteration of one part (a side or angle) of a triangle produces on 
another part, all the rest remaining unchanged. These effects may be often 
determined by geometrical considerations, as, for instance, when the change 
sought is that which alteration of an angle of a spherical triangle produces 
on the opposite side. In fig. 118, convert the triangle acb into acb" by a 
slight alteration of the angle acb, and indicate the change of the angle 
c by dc; that of the opposite side, c’ by oc’. If we let fall from b on ab”, 
the perpendicular, bz, we may take ax = ab, and b’x= 0c’, and we will 
have dc’ = sin. abc, sin. a’, Oc. 
The application of trigonometry, both plane and spherical, to geodesy, is 
of great importance. The piece of land to be surveyed is divided into 
triangles whose corners are indicated by signals; of the sides of these trian- 
gles only one need be measured, as a basis from which, with the help of the 
observed angles, to calculate the remaining sides. In this respect, some 
special formule are still necessary, of which we here give but one example: 
—given the angular interval of two signals of moderate height above the 
horizon, to deduce the horizontal angle of the two points of the horizontal 
plane on which the signals are erected. In fig. 119, let a, b, be the 
signals observed from o; and let the angle aob be measured. If we suppose 
a sphere constructed with o as the centre, and from z, the vertical point or 
zenith of 0, the great circles zac, zbd, described, cod being the horizontal 
plane, cod or czd will be the horizontal angle sought. If we make the 
angle aob = m, cod or ced—=m-+2x; ac=h, bd=N’, then the correction 
of the measured angle m is z =14 ({h +h’'} tang. }:m—[h-h’} cot. 3 m). 
For the solution of triangles which, supposing the earth to be a perfect 
sphere, may be taken for spherical, three methods are principally used: 
they may be either considered as spherical triangles, in which case the 
central angle corresponding to each side is deduced from the known radius 
of the earth; or from the angles of the spherical triangle, the angles of their 
22 
Thus, if in the triangle ABC, A and a are given, we have sin. B= 
