824 
TRIGONOMETRY. 
T R t 
greatest angle, and the smallest side opposite the 
smallest angle. 
3. Any two sides taken together are greater 
than the third. 
4. If the three angles are all acute, or all right, 
or all obtuse; the three sides will be, accord- 
ingly, all less than 90°, or equal to 90°, or 
greater than 90° ; and vice versa. 
5. If from the three angles A, B, C, (fig. 253,) 
of a triangle ABC, as poles, there are described, 
upon the surface of the sphere, three arches of 
a great circle DE, DF, FE, forming by their in- 
tersections a new spherical triangle DF.F ; each 
side of the new triangle will be the supplement 
of the angle at its pole ; and each angle of the 
same triangle will be the supplement of the side 
opposite to it in the triangle ABC. 
6. In any triangle ABC, (fig. 2 55,) or AiC, 
right-angled in A, 1st, The angles at the hypo- 
thenuse are always of the same kind as their 
opposite sides ; 2dly, The hypothenuse is less or 
greater than a quadrant, according as the sides 
including the right angle are of the same or dif- 
ferent kinds ; that is to say, according as these 
same sides are either both acute or both obtuse, 
or as one is acute and the other obtuse. And 
vice versa, 1st, The sides including the right 
angle are always of the same kind as their op- 
posite angles : 2dly, The sides including the right 
angle will be of the same or different kinds, ac- 
cording as the hypothenuse is less or more than 
90° ; but one at least of them will be of 90°, if 
the hypothenuse is so. 
The Solution of the Cases of right-angled Spherical Triangles (fig. 252.). 
Case. 
Given. 
Sought. 
Solution. 
1 
The hyp. AC and one 
angle A 
The opposite leg 
BC 
As radius * sine hyp. AC ** sine A * sine 
BC (by the former part of theor. 1.) 
2 
The hyp. AC and one 
angle A 
The adjacent leg 
AB 
As radius * co-sine of A [ [ tang. AC ‘ tang. 
AB (by the latter part of theor. 1.) 
3 
The hyp. AC and one 
angle A 
The other angle 
C 
As radius ‘ co-sine of AC [ * tang. A * co- 
tang. C (by theorem 4.) 
4 
The hyp. AC and one 
leg AB 
The other leg 
BC 
As eo-sine AB radius * * co-sine AC [ co- 
sine BC (by theorem 2.) 
5 
The hyp. AC and one 
leg AB 
The opposite angle 
C 
As sine AC ) radius ] * sine AB * sine C (by 
the former part of theorem I.) 
6 
The hyp. AC and one 
leg AB 
The adjacent angle 
A 
As tang. AC * tang. AB ** radius * co-sine 
A (by theorem 1.) 
7 
One leg AB and the 
adjacent angle A 
The other leg 
BC 
As radius ' sine AB * * tang. A * tang. BC 
(by theorem 4.) 
8 
One leg AB and the 
adjacent angle A 
The opposite angle 
C 
As radius * sine A " m " co-sine of AB * co- 
sine of C (by theorem 3.) 
9 
One leg AB and the 
adjacent angle A 
The hyp. 
AC 
As co-sine of A * radius ) * tang. AB ; tang 
AC (by theorem 1.) 
10 
One leg BC and the 
opposite angle A 
The other leg 
AB 
As tang. A * tang. BC • ) radius * sine AB 
(by theorem 4.) 
11 
One leg BC and the 
opposite angle A 
The adjacent angle 
C 
As co-sine BC * radius * * co-sine of A * 
sine C (by theorem 3.) 
12 
One leg BC and the 
opposite angle A 
The hyp. 
AC 
As sine A ’ sine BC * * radius [ sine AC (by 
theorem 1.) 
13 
Both legs 
AB and BC 
The hyp. 
AC 
As radius ] co-sine AB * " co-sine BC [ co- 
sine AC (by theorem 2.) 
14 
Both legs 
AB and BC 
An angle, suppose 
A 
As sine AB * radius * * tang. BC * tang. A 
(by theorem 4.) 
15 
Both angles 
A and C 
A leg, suppose 
AB 
As sine A ) co-sine C * * radius * co-sine 
AB (by theorem 3.) 
16 
Both angles 
A and C 
The hyp. 
AC 
As tang. A * co-tang. C * * radius * co-sine 
AC (by theorem 4.) 
Note, The 10th, 11th, and 12th cases are ambiguous; since it cannot be determined by the 
«kta, whether ABC, and AC, are greater or less than 90° each. 
In any spherical triangle, the area, or surface 
inclosed by its three sides upon the surface of 
the globe, will be found by this proportion : 
As 8 right angles, or 720°, 
Are to the whole surface of the sphere ; 
Or, as 2 right angles, or 180°, 
To one great circle of the sphere ; 
So is the excess of the 3 angles above 2 right 
angles, 
* To the area of the spherical triangle. 
Hence, if a denotes .7854, 
d — diam. of the globe, and 
s — sum of the 3 angles of the triangle; 
, j — 180 „ , , . , 
then add x = area of the spherical 
180 1 
triangle. 
Hence also, if r denotes the radius of the 
sphere, and c its circumference ; then the area 
of the triangle will thus be variously expressed ; 
. j — 180 s — 189 
viz., Area ea ad 2 x = cd x 
180 720 
^ - 180 
‘ r X ~ 3GO " 5 ° r barel 7 
r X s — 180 9 , 
in square degrees, when the radius r is* estimated 
in degrees ; for then the circumference c is = 
360°. 
Farther, because the radius r, of any circle, 
when estimated in degrees, is = — . 
° 3.14159 &c. 
= 57.2957795, the last rule r x j — 180, for 
expressing the area A of the spherical triangle, 
in square degrees, will be barely 
A — 57.2957795 s — 10313.24 — 
— ~ 10313J very nearly. 
Hence may be found the sums of the three 
angles in any spherical triangle, having its area 
A known ; for the last equation give the sum 
= 4 + 130 =-/ 
57.29 &C. 
+ 180= 
9683 
-f 180. 
So that, for a triangle On the surface of the 
earth, whose three sides are known ; if it is but 
small, as of a few miles extent, its area may be 
found from the known length of its sides, con- 
sidering it as a plane triangle, which gives the 
value of the quantity A ; and then the last rule 
above will give the value of s, the sum of the 
three angles ; which will serve to prove whether 
those angles are nearly exact, that have been 
taken with a very nice instrument, as in large 
and extensive measurements on the surface of 
the earth. 
Spherical Polygon's a-figure of more than 
three sides, formed on the surface of a globe by 
the intersecting arcs of great circles. 
The area of any spherical polygon will be 
found by the following proportion, viz. 
As 8 right angles, or 720*, 
To the whole surface of the sphere; 
Or, as 2 right angles, or 180°, 
To a great circle of the sphere ; 
So is the excess of all the angles above the 
product of 180, and 2 less than the number 
of angles, 
To the area of the spherical polygon. 
That is, putting n = the number of angles, 
i = sum of all the angles, 
d = diam. of the sphere, 
a = .78539 &c. ; 
T1 . , 2 r — (n — 2) 180 
Then A ~ ad~ x — = the area 
loO 
of the spherical polygon. 
Hence other rules might be found, similar t© 
those for the area of the spherical triangle. 
Hence also, the sum s of all 'the angles of any 
spherical polygon, is always less than 180a, but 
greater than 180 (a — 2) ; that is, less than n 
times 2 right angles, but greater than n — 2 
times 2 right angles. 
This will be deemed sufficient on the subject 
as an introduction to trigonometry, and we can 
with great satisfaction refer our readers for far- 
ther information to Bonnycastle’s “ Treatise on 
Plane and Spherical Trigonometry, with their 
most useful Practical Applications,” which is un- 
questionably the best book on the subject in the 
English language. 
TRIGUERA, a genus of the pentandria 
monogynia class and order of plants. The 
corolla is bell-shaped ; nect. short ; berry 
four-celled, two seeds in each cell. There 
are two species, of no note. 
TRIH I LATJE, from tres, “ three,” and 
hilum, “ an external mark on the seed the 
name of the 23d class in Linnaeus’s Fragments 
of a Natural Method ; consisting of plants 
