98 
TRIGONOMETRY. 
Dem. (By 18.) 
I* -f c 3 — a 5 
C0 ' A =- 2bc ‘ 
From which value, in known terms, for cos a, we may find 
a itself from the tables. Hence, the triangle may be deter¬ 
mined 23, by as we know one of its angles and the 
including sides. Q. E. d. 
06s. We may determine B and c without the help of 23, 
as we determined a, namely by the formula 20. 
cos b — 
a? 4- 
2 ac ’ 
+ c 2 
C05C ~ - 2 a 6 • 
27. Representing the sides and angles as above, then 
any two sides, as a, 6, and the angle, as A, opposite either 
of them, a, being given, the triangle itself may be deter¬ 
mined in two cases, viz., 1°, if the given side not opposite 
the given angle be less than the other given side; 2°, if the 
given side not opposite the given angle be less than the third 
side. 
Dem. (By 18.) 
sin b b . b 
— — —, sin B — —. sin a. 
sin A a a 
In logarithms, 
log. sin b — log. sin a + log. ^ — log. a. 
Consequently sin b is determined, the quantities b, a, and 
sin A beingjknown. But as, (by 6), the sine is the same for 
two supplementary angles, we cannot determine to which 
of these the sine found belongs, unless, in some cases, from 
other considerations. 
1°. If 6 (fig. 18.) beless than a, then b will also be less than 
A. Therefore, if a be acute, b will likewise be acute ; and if 
a be not acute, B will be acute. Hence, in this case b must be 
that supplementary angle, which is less than 90°. 
Thus, if b : a :: 1: 2, and sin az^/3; then, sin b — 
which is the sine either of an angle equal GO 0 , or 120°; 
but as b is less than a the angle b must be acute, and there¬ 
fore must be the lesser of these supplementary angles, i. e. 
60°. 
2°. If 6 be less than c, then by Geom. 106, b will be 
also less than c. Therefore, if c be acute, b will be likewise 
acute; and if c be not acute, b will be acute by 197 Geom. 
Hence, as before, though the value we find for sin b belongs 
indifferently to two supplemental angles, as we know from 
other considerations that b is acute, we know that it must be 
the lesser of these supplementary angles, and therefore can 
determine it. 
Again: having computed b, we obtain c, because c zz 
1 80° — (a + b) ; and having thus found c, we may obtain 
the third side c, of the triangle; for 
c _ sin c m _ sin c 
a sin a’ ’ ’ ' sin a* 
In logarithms, 
log. c — log. a -f- log. sin c — log, sin a. 
Q. E. D. 
We now proceed to the consideration of spherical trigo¬ 
nometry. 
Def. I. A right line is perpendicular to a plane when per¬ 
pendicular to all the straight lines in that plane which pass 
through the point where the line meets the plane. 
Def. II. When two planes cut each other, the angle be¬ 
tween them, or their mutual inclination, is measured by the 
angle between two right lines drawn in those planes from 
the same point of their line of intersection, and perpendicular 
to it. 
Def. III. Two planes are perpendicular to each other 
when the angle between them is measured by a right angle. 
Def. IV. Any figure made where a solid is cut by a sur¬ 
face, is called a Section of that solid. 
Def. V. A sphere is a solid figure bounded by one sur¬ 
face, such, that all right lines drawn from it to one and the 
same point within the figure are equal to one another. 
Def. VI. In a sphere, the point from which the surface is 
every where equally distant is called the centre of the 
sphere. 
Def. VII. A right line drawn from the centre of a sphere, 
and terminated in the surface, is called a Radius of the sphere. 
Def. VIII. A right line drawn through the centre of a 
sphere, and terminated both ways in the surface, is called a 
Diameter of the sphere. 
Def. IX. A great circle of a sphere is that made by a 
plane passing through the centre of the sphere. 
Def. X. A lesser circle of a sphere is that made by a 
plane which does not pass through the centre of the sphere. 
Def. XL A spherical triangle is a spherical surface 
bounded by three arcs of great circles. 
a. The three arcs which bound a spherical triangle are 
called its sides; and are usually considered to be each less 
than a semi-circumference. 
/3. The three angles contained between the planes of the 
three great circles whose arcs form a spherical triangle, are 
called the angles of the triangle. 
y. A spherical triangle is said to be right-angled when 
any of its angles is a right angle. But, unlike a plane right- 
angled triangle, it may have two or three of its angles each 
a right angle. 
1. One portion of a right line cannot be in a plane, and 
another portion out of it. 
2. Three right lines which meet one another are in the 
same plane. 
3. If two planes cut one another, their line of intersection 
will be a right one. 
4. If at the point where two right lines intersect a right 
line stand perpendicular to both, it shall also be perpen¬ 
dicular to the plane in which they lie. 
5. Every section of a sphere, made by a plane, is a circle. 
G. All great circles of the same sphere are equal to each 
other. 
7. Two great circles on the same sphere divide each other 
into two equal parts. 
8. All small circles equally distant from the centre of the 
sphere are equal to each other. 
9. In any right-angled spherical triangle, the product of 
the sine of the side opposite the right angle and the sine of 
either remaining angle is equal to the sine of the side opposite 
the latter angle. 
10. In any right-angled spherical triangle, the tangent of 
either side adjacent to the right angle is equal to the product 
of the tangent of the side opposite the right angle and the 
cosine of the angle between these sides. 
11. In any right-angled spherical triangle, the cosine of 
the side opposite the right angle is equal to the product of the 
cosines of the remaining sides. 
(1.) Let acb (fig. 19.) be a right line, one part of whioh, 
ac, is in the plane defg. Then, no other part of the line 
can be out of the plane defg. 
Demonstration.— Suppose the part cb to lie out of the 
plane defg. Produce the right line ac, in the plane defg, 
to any point i, and let a plane passthrough the points a and 
i. Then the line ai is in this plane. Also let this plane be 
turned about ai until it pass through the point b ; and as the 
points c, b, are in this new plane, the right line cb is also 
in this plane. Now draw ch in this plane, making, with 
ci, a greater angle than bci : consequently, by Geom. 91, the 
angles ach and hci are together = to two right angles. But 
as the line acb is likewise supposed a right one, the angles 
ach and hcb together would be also equal to two right an¬ 
gles. Therefore the angle hci would be equal to the angle 
iicb, the greater to the less,—which is impossible. Hence 
the above supposition is false; cb does not lie out of the 
plane defg. q.e.d. 
(2.) Let the three right lines mn, op, qr, (fig. 20.) meet 
one another in the points a, b, c. Then, these three right 
lines are in one and the same plane. 
Conceive any plane passing through the line bc to be so 
turned 
