TRIGONOMERY, SPHERICAL. 
Having divided the riglit-angled triangle 
into two right-angled triangles, the hypo- 
thenuses and bases of which are given, to 
find the angles by Gunter. 1. The extent 
from 105 to 135 will reach from 35 to 45 
on the line of sines. 2. The extent from 
85 to 75, on the line of numbers, will reach 
from radius to 61° 56', the angle B D A on 
the line of sines. 3. The extent from 50 
to 30, on the line of numbers, will reach 
from radius to angle ADC 36° 53', on the 
line of sines. 
Trigonometry, spherical, relates to tri- 
angles, or figures which are reducible to 
triangles, whose sides are segments of cir- 
cles. Thus, if we describe a triangle on 
any spherical body, say a globe, it is evi- 
dent that all the sides must be composed 
of curved lines ; and it is the same in the 
case of a series of circles, or of orbits, in- 
tersecting each other. When two equal 
circles intersect, they will giye a parabolic 
spindle; more or less acute, according, as 
the centres of the two circles may be more 
or less distant. When three circles mu- 
tually intersect, there will be formed a great 
variety of spherical triangles, of which the 
areas, and the properties, could not be 
ascertained by plane-trigonometry ; but 
come under consideration as parts of sphe- 
rical surfaces. The following definitions 
should be clearly understood ; they are sim- 
ple in tile extreme, but highly important : 
1st. The poles of a sphere are two points 
in the superficies of the sphere, that are the 
extreme of the axis. 2d. 'fhe pole of a 
circle in a sphere is a point in the super- 
ficies of the sphere, from which all right 
lines that are drawn to the circumference 
of the circle are equal to one another. 3d. A 
great circle ip a sphere, is that whose plane 
passes through the centre of the sphere; 
and whose centre is the same as that of the 
sphere. 4th. A spherical triangle is a figure 
comprehended under the arcs of three great 
circles in a sphere. 5th. A spherical angle 
is that which, in the superficies of the 
sphere, is contained under two arcs of great 
circles ; and this angle is equal to the in- 
clinations of the planes of the said circles. 
It is particularly to be held in mind, that 
although we can, upon any actual sphere, 
describe triangles at pleasure, which may 
nearly embrace the whole circumference, 
yet that such cannot be laid down, so as to 
be represented on paper ; for every side of 
a spherical triangle is less than a semi-circle. 
With respect to spherical triangles, the 
leaincv may generally entertain a correct 
opinion of their value, if he considers that 
every arc or segment of a circle may have 
a chord drawn from one to the other extre- 
mity; and that the triangle which can be 
contained within such arc or segment, taking 
the chord for a hypothenuse, will determine 
how much of that circle has been cut oflT, and is 
included between the extremes of the seg- 
ment. It is utterly impossible to produce 
any two measurable segments taken from 
two different circles, which, having chords 
of equal length, will contain the same angle. 
A semicircle, having the diameter for its 
chord, will give a right angle ; for if to any 
point within that semicircle two lines be 
drawn, from the ends of the chord respec- 
tively, their union at such assumed point 
will form a right angle. In proportion as 
the chord is less than a diameter, so must 
the segment be a less part of the whole 
circle, and the angle contained therein will 
be more acute. Spherical triangles may 
be acute, right-angled, or obtuse, the same 
as on plane-trigonometry. In all right- 
angled spherical triangles, the sign of the 
hypothenuse : radius sine of a leg : sine 
of its opposite angle. And the sine of the 
leg ; radius :: tangent of the other leg : 
tangent of its opposite angle. In any right- 
angled spherical triangle, AB C (fig. 25), it 
will be as radius is to the co sine of one 
leg, so is the co sine of the other leg to 
the co-sine of the hypothenuse. Hence, if 
two right-angled spherical triangles ABC, 
C B D (fig. 26), have the same perpendi- 
cular B C, the co-sines of their hypothenuses 
will be to each other directly as the co- 
sines of their bases. In any spherical tri- 
angle it will be, as radius is to the sine of 
cither angle, so is the co-sine of the adja- 
cent leg to the co-sine of the opposite angle. 
Hence, in right-angled spherical triangles, 
having the same perpendicular, the co-sines 
of the angles at the base will be to each 
other, directly, as the sines of the vertical 
angles. In any right-angled spherical tri- 
angle it will be, as radius is to the co-sine 
of the hypothenuse, so is the tangent of 
either angle to the co-tangent of the other 
angle. As the sum of the sines of two un- 
equal arches is to their dilference, so is the 
tangent of half the sum of those arches to 
the tangent of half their difference : and as 
the sum of their co-sines is to their difference, 
so is the co-tangent of half the sum of the 
arches to the tangent of half the difference 
of the same arches. In any spherical tri- 
angle ABC (fig. 27), it will be, as the co- 
tangent of half the sum of the angles, at the 
