2. Given A = 34° 15′ 03′′, B = 42° 15′ 13′′, and c = 76° 35' 36", to find C, a, and b.

Ans. C 121° 36′ 12′′, a = 40° 0′ 10′′, b = 50° 10′ 30′′.

3. Given B = 82° 24', C = 120° 38', and a 75° 19', to find A, b, and c.

Ans. A 73° 31′ 13′′, b = 90° 50′ 50′′, c = 119° 46′ 22′′.

CASE V.

Given the three sides, to find the remaining parts.

89. The angles may be found by means of formula (3), Art. 81; or, one angle being found by that formula, the two others may be found by means of Napier's Analogies.

Examples.

1. Given a = 74° 23', b 35° 46' 14", and 39', to find A, B, and C.

c = 100°

Applying logarithms to formula (3), Art. 81, we have

log cos A 10+ [log sin is + log sin (†s — a)

+(a. c.) log sin b + (a. c.) log sin c

20];

is

Using the same formula as before, and substituting B for A, b for a, and a for b, and recollecting that s − b = 69° 37′ 53′′, we have

Using the same formula, substituting C for A, c for a, and a for c, recollecting that sc 4° 45′ 07′′, we have

.. C = 67° 52′ 25′′, and C = 135° 44′ 50′′.

=

2. Given a 56° 40', b = 83° 13', and c = 114° 80', to find A, B, and C.

Ans. A 48° 31' 18", B = 62° 55' 44", C = 125° 18'56".

3. Given a = 115° 15', b = 125° 30', and c = 110° 15',

to find A, B, and C.

Ans. A = 145° 15′ 04′′, B = 149° 07′ 52, C = 143° 45' 10".

CASE VI.

The three angles being given, to find the sides.

90. The solution in this case is entirely analogous to the preceding one.

Applying logarithms to formula (2), Art. 82, we have

In the same manner as before, we change the letters, to suit each case.

Examples.

4

=

1. Given A = 48° 30′, B = 125° 20′, ar C 62° 54′, to find a, b, and c.

Ans. a 56° 39′ 30′′, b = 114° 29′ 58′′ c = 83° 12′ 06′′.

2. Given A = 109° 55′ 42′′, B = 116° 38' 33", and C = 120° 43′ 37′′, to find a, b, and c.

Ans. a = 98° 21′ 40′′, b = 109° 50′ 22′′, c = 115° 13′ 28′′.

3. Given A 160° 20', B = 135° 15', and C = 148° 25', to find a, b, and c.

Ans. a 155° 56′ 10′′, b = 58° 32′ 12", c = 140° 36' 48".

MENSURATION.

91. MENSURATION is that branch of Mathematics which treats of the measurement of Geometrical Magnitudes.

92. The measurement of a quantity is the operation of finding how many times it contains another quantity of the same kind, taken as a standard. This standard is called the unit of measure.

93. The unit of measure for surfaces is a square, one of whose sides is the linear unit. The unit of measure for volumes is a cube, one of whose edges is the linear unit.

If the linear unit is one foot, the superficial unit is one square foot, and the unit of volume is one cubic foot. If the linear unit is one yard, the superficial unit is one square yard, and the unit of volume is one cubic yard.

94. In Mensuration, the expression product of two lines, is used to denote the product obtained by multiplying the number of linear units in one line by the number of linear units in the other. The expression product of three lines, is used to denote the continued product of the number of linear units in each of the three lines.

Thus, when we say that the area of a parallelogram is equal to the product of its base and altitude, we mean that the number of superficial units in the parallelogram is equal to the number of linear units in the base, multiplied by the number of linear units in the altitude. In

like manner, the number of units of volume, in a rectangular parallelopipedon, is equal to the number of superficial units in its base multiplied by the number of linear units in its altitude, and so on.

95,

MENSURATION OF PLANE FIGURES.

To find the area of a parallelogram.

From the principle demonstrated in Book IV.,

Prop. V., we have the following

RULE.-Multiply the base by the altitude; the product will be the area required.

Examples.

1. Find the area of a parallelogram, whose base is 12.25, and whose altitude is 8.5. Ans. 104.125.

2. What is the area of a square, whose side is 204.3 feet? Ans. 41738.49 sq. ft.

3. How many square yards are whose base is 66.3 feet, and altitude

there in a rectangle 33.3 feet?

Ans. 245.31 sq. yds.

4. What is the area of a rectangular board, whose length is 12 feet, and breadth 9 inches? Ans. Ans. 9 sq. ft.

5. What is the number of square yards in a parallelogram, whose base is 37 feet, and altitude 5 feet 3 inches? Ans. 21.

To find the area of a plane triangle.

96. First Case. When the base and altitude are given.