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In the triangle BAD, to find the angle BAD. As AB. . 25. .

ar. comp.

8.602060 : 13.3625

1.125887 sin D. 900

10.000000 : sin BAD. 320 18' 35"

9.727947 Hence 90° — DAC=90°-51° 34' 40"=38° 25' 20"=C

and 900 — BAD=90° - 32° 18' 35"=57° 41' 25"=B and BAD+DAC=51° 34' 40'+32° 18' 35"=83° 53' 15"=A.

2. In a triangle, in which the sides are 4, 5 and 6, what are the angles. ?

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


67. 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 other two sides. Or the parts may be found by Theorem V.


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


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To find the angle B. As BC. 250

ar. comp.

7.602060 : AC • 240

2.380211 :: sin A

10.000000 : sin B 73° 44' 23"

9.982271 But C=90°- B=90° -73° 44' 23"=16° 15' 37":


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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'.



Definitions and Introductory Remarks.

68. Surveying, in its most extensive signification, comprises all the operations necessary for finding,

Ist. The area or content of any portion of the surface of the earth;

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

3d. The accurate delineation of the whole on paper.

69. The earth being spherical, its surface is curved, and every line traced on its surface is also curved.

If large portions of the surface are to be measured, such as states and territories, the curvature must be taken into account; and very material errors will arise if it be neglected. When the curvature is considered, the method of measurement and computation is called Geodesic Surveying.

The radius of the earth, however, being large, the curvature of its surface is small, and when the measurement is limited to small portions of the surface, the error becomes insensible, if we consider the surface a plane. This method of measurement and computation, is called Plane Surveying, and is the only kind that will be treated of in these Elements.

70. If at any point of the surface of the earth, a plane be drawn perpendicular to the radius passing through this point, such plane is tangent to the surface, and is called a horizontal plane. All planes parallel to such a plane, are also called horizontal planes.

71. A plane which is perpendicular to a horizontal plane 72. All lines of a horizontal plane, and all lines which are parallel to it, are called horizontal lines.

73. Lines which are perpendicular to a horizontal plane, are called vertical lines ; and all lines which are inclined to it, are called oblique lines. Thus, AB and DC are hori

D zontal lines; BC and AD are vertical lines ; and AC and BD are oblique lines.

74. The horizontal distance between two points, is the horizontal 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 the points B and D.

75. A horizontal angle is one whose sides are horizontal; its plane is also horizontal.

A horizontal angle may also be defined to be, the angle included between two vertical plunes passing through the angular point, and the two objects which subtend the angle.

76. A vertical angle is one, the plane of whose sides is vertical.

77. 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.

78. An angle of depression, is a vertical angle having one of its sides horizontal, and the inclined side under the horizontal side. Thus, DCA is the angle of depression from C to A.

79. An oblique angle is one, the plane of whose sides is oblique to the horizontal plane.

80. 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. All the angles may also be divided into three classes, viz. : 2d. Vertical angles; which may be again divided into angles of elevation and angles of depression : and

3d. Oblique angles,


Of the measurement and calculation of Lines and Angles,

81. It has been shown (Art. 62), 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 caleulations, two things are necessary.

1st. To measure certain lines on the ground; and also, as many angles as may be necessary to render at least three parts of every triangle known: and

21. To calculate, by trigonometry, the other sides and angles that may be required. Our attention, then, is di. rected,

1st. To the measurement of lines ;
23. To the measurement of angles; and

3d. To the calculations for the unknown and required parts.

82. Any tape, rod, or chain, on which equal parts are marked, may be used as a measure ; and one of the equal parts into which the measure is divided, is called the unit of the measure,

The unit of a measure may be a foot, a yard, rod, or any other ascertained distance.

83. 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. Every tenth link from either end, is marked by a small attached brass plate, which is notched, to designate its number from the end. The division of the chain into 100 equal parts, is a very convenient one, since the divisions or links, are decimals of the whole chain, and in the calculations may be treated as such,

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