From twice the log. of the base subtract the log. of the perpendicular, and add the corresponding natural number to the perpendicular; then, to the log. of this sum add the log. of the perpendicular, and half the sum of these two logs. will be the log. of the hypothenuse. As thus:

Solution of Oblique-angled Plane Triangles by Logarithms.

PROBLEM I.

Given the Angles and One Side of an Oblique-angled Plane Triangle, to find the other Sides.

RULE.

As the Log. sine of any given angle, is to its opposite given side; so is the log. sine of any other given angle to its opposite side.

Note. When a log. sine, or log. co-sine, is the first term in the proportion, the arithmetical complement thereof may be taken directly from the Table of secants by using a log. co-secant in the former case, and a log. secant in the latter.

PROBLEM II.

Given two Sides and an Angle opposite to one of them, to find the other Angles and the third Side.

RULE.

As any given side of a triangle is to the log. sine of its opposite given angle, so is any other given side to the log. sine of the angle opposite thereto.

The angles being thus found, the third side is to be computed by the preceding Problem.

Example.

Let the side A B, of the triangle A B C, be 436.7, the side A C 684. 5, and the angle B 100:7:35"; required the angles A and C, and the side BC?

=

Note. The angle A 100:7:35" + the angle C 38:54:22" 139:1:57"; and 180 - 139:1:57" the angle A = 40:58:37

=

Remark. An angle found by this rule is ambiguous when the given side opposite to the given angle is less than the other given side; that is, the angle opposite to the greater side may be either acute or obtuse: for trigonometry only gives the sine of an angle, which sine may either represent the measure of the angle itself, or of its supplement to 180 degrees. But when the given side opposite to the given angle is greater than the other given side, then the angle opposite to that (other given) side is always acute, as in the above example.

PROBLEM III.

Given two Sides and the included Angle, to find the other Angles and the third Side.

RULE.

Find the sum and difference of the two given sides; subtract the given angle from 180; take half the remainder, and it will be half the sum of the unknown angles; then say,

As the sum of the sides is to their difference; so is the log. tangent of half the sum of the unknown angles, to the log. tangent of half their difference.

Now, half the difference of the angles, thus found, added to half their sum, gives the greater angle, or that which is opposite to the greater side; and being subtracted, leaves the angle opposite to the less side.

The angles being thus determined, the third side is to be computed by Problem I., page 177.

C

160.2

180°

the angle B 110:1:20% 69:58:402 34:59:20% =

half the sum of the angles A and C.

So is sum of angles 34:59:20" Log. tang. =

7.431212

50.1 Log. =

9.845048

Given the three Sides of a Plane Triangle, to find the Angles.

RULE.

Add the three sides together, and take half their sum; the difference between which and the side opposite to the required angle call the remainder; then,

To the arithmetical complements of the logs. of the other two sides, add the logs. of the half sum and of the remainder : half the sum of these four logs. will be the log. co-sine of an arch; which, being doubled, will give the required angle.

Now, one angle being thus found, either of the other two angles may computed by Problem II., page 178.

Example.

Let the side A B, of the triangle ABC,

be 260. 1, the side AC 190. 5, and the side
BC 140. 4; required the angles A, B, and C? A<

100.5

140.4

be

Now, angle C 102:34:44" + angle B 45:37:45 148:12:29"; and 180 148:12:29 = 31:47:31" the angle A.

THE RESOLUTION OF THE DIFFERENT PROBLEMS, OR CASES, IN SPHERICAL TRIGONOMETRY, BY LOGARITHMS.

Spherical Trigonometry is that branch of the mathematics which shows how to find the measures of the unknown sides and angles of spherical triangles from some that are already known. It is divided into three parts; viz., right-angled, quadrantal, and oblique-angled.

A right-angled spherical triangle has one right angle; the sides including the right angle are called legs, and that opposite thereto the hypothenuse.

A quadrantal spherical triangle has one side equal to 90°, or the fourth part of a circle.

An oblique-angled spherical triangle has neither a side nor an angle equal to 90:

A spherical triangle is formed by the intersection of three great circles on the surface of the sphere.