Hence, 90° DAC 90°

51° 34' 40"

=

38° 25′ 20′′ = C,

B,

and, 90° - BAD 90° 32° 18′ 35′′ = 57° 41′ 25′′ =

and, BAD + DAC = 51° 34′ 40′′ + 32°

= 83° 53′ 15′′ = A.

18′ 35′′ =

2. In a triangle, of which the sides are 4, 5, and 6, what

are the angles?

Ans. 41° 24′ 35′′; 55° 46′ 16′′; and 82° 49′ 09′′.

SOLUTION OF RIGHT-ANGLED TRIANGLES.

73. The unknown parts of a right-angled triangle may be found by either of the four last cases; or, if two of the sides are given, by means of the property that the square of the hypothenuse is equal to the sum of the squares of the two other sides. Or, the parts may be found by Theorems IV. and V.

EXAMPLES.

1. In a right-angled triangle BAC, there are given the hypothenuse BC= 250, and the base AC 240: required the other parts.

To find the angle B.

By Theorem I., we have,

α : b sin A : sin B.

Applying logarithms, we have,

But,

C = 90° B = 90° 73° 44′ 23" = 16° 15' 37".

A

2. In a right-angled triangle BAC, there are given,

AC = 384, and B = 53° 08';

required the remaining parts.

Ans. AB 287.96; BC= 479.979; C = 36° 52'.

BOOK II.

PLANE

SURVEYING.

SECTION I.

MEASUREMENT OF LINES AND ANGLES.

1. SURVEYING, in its most extensive signification, comprises all the operations necessary for finding:

1st. The area or contents of any portion of the surface of the earth;

2d. The lengths and directions of the bounding lines; and,

3d. For making, on paper, an accurate delineation, both of the surface and boundaries; which delineation is called a Map. or Plot.

2. PLANE SURVEYING is that branch in which the curvature of the earth is neglected; as it may be when the survey is limited to small portions of the surface.

3. GEODESIC SURVEYING is when the curvature of the earth is taken into account, as it must be in all extensive surveys.

4. A HORIZONTAL PLANE, is a plane parallel to the waterlevel. If the plane passes through a point on the surface of the earth, it is tangent to the surface, and also perpendicular to the radius passing through the point of contact.

i

5. A VERTICAL PLANE, is a plane perpendicular to a hori

zontal plane.

6. A HORIZONTAL LINE, is a line parallel to the waterlevel, or parallel to a horizontal plane.

7. A VERTICAL LINE, is a line perpendicular to a hori zontal plane.

8. OBLIQUE LINES, are those which are inclined to a hori zontal plane.

Thus, AB and DC are horizontal lines; BC and AD are vertical lines; and AC and BD are oblique lines.

D

A

B

9. THE HORIZONTAL DISTANCE between two points, is the borizontal line intercepted between the two vertical lines passing through those points. Thus, DC or AB, is the horizontal distance between the two points A and C, or between the points B and D.

10. A HORIZONTAL ANGLE, is one, whose sides are horizontal: its plane is also horizontal. A horizontal angle is always equal to the angle included between two vertical planes passing through the angular point and the two objects which subtend the angle.

11. A VERTICAL ANGLE, is one, the plane of whose sides is vertical.

12. AN ANGLE OF ELEVATION, is a vertical angle having one of its sides horizontal, and the inclined side above the horizontal side. Thus, in the last figure, BAC is the angle of elevation from A to C.

13. AN ANGLE OF DEPRESSION, is a vertical angle having

horizontal side. Thus, DCA is the angle of depression from C to A.

14. AN OBLIQUE ANGLE, is one, the plane of whose sides is oblique to a horizontal plane.

15. All lines, which can be the object of measurement, must belong to one of the classes above named, viz.:

1st. Horizontal lines; 2d. Vertical lines; 3d. Oblique lines. 16. All angles may also be divided into three classes, viz.:

1st. Horizontal angles; whose sides are horizontal.

2d. Vertical angles; which include angles of elevation and angles of depression; and,

3d.

Oblique angles, or those included by oblique lines.

MEASUREMENT OF LINES AND ANGLES.

17. It has been shown (Bk. I., Art. 45), that at least one side and two of the other parts of a plane triangle must be given, or known, before the remaining parts can be found, by calculation.

When, therefore, distances are to be found, by trigonometrical calculations, two preliminary steps are necessary:

1st. To measure certain lines on the ground; and, 2d. To measure the necessary angles.

MEASUREMENT OF DISTANCES.

18. Any tape, rod, or chain, divided into equal parts, may be used as a measure; and this is called the unit of measure. The unit of measure may be a foot, a yard, a rod, or any other ascertained distance.

The measure in general use, is a chain of four rods or sixty-six feet in length; it is called Gunter's chain, from the name of the inventor. This chain is composed of 100 links.