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ease there will be no triangle, or the conditions are impossible.

To determine the value of Cd in terms of the given parts, we have the proportion sin. 90°, or R : AC :: sin. A : Cd, AC şin. A

orlog. Cd=log. AC-+log. sin. A-10.

R So that, when the logarithm of the opposide side CB is equal to log. AC+log. sin. A-10, there is one solution; when less, the problem is impossible ; when greater, but less than the logarithm of AC, two solutions; and when greater than the logarithm of AC, but one.

Ex. 3.—Given two angles of a plane triangle 22° 37' and 134 46', and the contained side 351 : required the remaining parts.

Answer.-Angle 22° 37'; sides 351 and 648.

Example 4.–Given two sides of a plane triangle 50 and 40 respectively, and the angle opposite the latter equal to 32° ; to determine the triangle.

Answer. If the angle opposite to the side 50 be acute, it is = 41° 28', the third angle = 106°32', and the remaining side = 72.36. If the angle opposite to side 50 be obtuse, it is =138° 32', the other angle 9° 28', and the side = 12.415.

CASE II. 60. When two sides and the included angle are given.

The solution is made by Art. 44. Take the given angle from 180°, the remainder is the sum of the other two angles, which being divided by 2, gives half their sum. Then find half their difference: their half difference being added to half their sum, gives the greater angle, and being subtracted from it, gives the less.*

And as the greater angle is opposite the greater side, and the less angle opposite the less side, the three angles and two sides of the triangle become known: the third side may then be known by Case I.

62. Example 1.-Let there be given in any plane triangle ABC, AC=450, BC=540, and the included angle C=80°, to find the other angles and the remaining side.

ist Let atb=s, and a-b=d: then by adding 2a=std, or a=istid; and

BC+AC=990, BC - AC=90, 180°-C=100
As BC +AC, 990

To BC-AC, 90

1.954243 So is tang. }(A+B), 50° 10.076186

To tang. } (A-B), 6° 11' 9.034793
Hence 50° +6° 11' 56° 11'=A; and 50-6° 11'=43° 49
Then to find the third side AB.

As sin. B, 43° 49' 9.840328
To AC, 450

So is sin. C 80° 9.993351

To AB, 640.08 2.806236 Example 2. Given two sides of a plane triangle, 1686 and 960, and their included angle 128° 4': to find the other parts.

Answer.—Angles 33° 34' 39", 18° 21' 21", side 2400.03.

CASE III. 62. When the three sides of a plane triangle are given, to find the angles. Let fall a perpendicular from the angle opposite the greater side, dividing the given triangle into two right-angled triangles ; then find the difference of the segments by Art. 45. Half the difference being added to half the base, gives the greater segment; and, being subtracted from half the base, gives the less segment. Then since the greater segment belongs to the right-angled triangle having the greatest hypothenuse, we have the sides and right angle of two right-angled triangles, to find the acute angles.

Example 1.-The sides of a plane triangle (Pl. I. Fig. 4), are AC=40, AB=34, and BC=25. Required the angles. As AC : AB+BC :: AB-BC : AD-CD,

59 x9 That is 40 :




= 26.6375=AD:


As AB, 34

1.531479 | As CB, 25 To sin. D, 90° 10.000000 To sin D, 90° So is AD, 26.6375 1.425494 So is DC, 13.3625

1.397940 10.000000 1.125887

; , }

To sin. ABD, 51

To sin. CBD, 32° 9.894014

9.727947 34' 40"

18' 35" Hence 90°-51° 34' 40"=38° 25' 20"=A; 90°-32° 18' 35" =57° 41' 25"=C; and 51: 34' 40'+32° 18' 35"=83° 53' 15" =ABC.

Example 2.-.When the sides are 4, 5, and 6, what are the angles ?

Answer.-41° 24' 35", 55° 46' 16", and 82° 49 9".

SOLUTION OF RIGHT-ANGLED TRIANGLES. 63. The unknown parts of a right-angled triangle may be found by either of the three last cases: or, if two of the sides be 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 Art. 46.

Example 1.-In the right-angled triangle ABC, there are given the hypothenuse AC=25, and the base AB=24. Required the other parts. As AC, 25 1.397940 | As radius

10.000000 To sin. B, 90° 10.000000 To AB, 24

1.380211 So is AB, 24 1.380211 So is tang. A, 16°


15' 37" To sin. C, 73°44' 23" 9.982271 90° -73° 44' 23"=16° 15' 37" To BC, 7.00004 40.845100


Example 2.-In a right-angled triangle, there are given one leg equal to 384, and its opposite angle 53° 8, to find the other leg and the hypothenuse.

Answer.--Leg 280, hypothenuse 480,

Example 3.- In a right-angled triangle, are given the base 195, its adjacent angle 47° 55' to find the rest.

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64. SURVEYING, in its most extensive signification, comprises all the operations necessary for determining the area, or content of any portion of the earth's surface, the lengths and direction of the bounding lines, and their accurate delineation on paper.

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

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

67. The radius of the earth, however, being large, the curvature of its surface is small, and when the measurement is limited to inconsiderable portions, the error is not sensible in supposing the surface a plane. This method of measurement and computation, is called Plane Surveying, which will be alone treated of in these Elements.

68. 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 named horizontal planes.

69. A plane perpendicular to a horizontal plane, is called a vertical plane.

70. The lines of a horizontal plane, as well as all lines which are parallel to it, are named horizontal lines.

71. Lines which are perpendicular to a horizontal plane, are called vertical lines, and lines which are inclined to it, oblique lines.

72. The horizontal distance between two points, is the horizontal line intercepted between the two vertical lines passing through those points.

73. A horizontal angle is one whose sides are horizontal ; its plane also is horizontal. A horizontal angle may also be defined, the angle included between two vertical planes passing through the angular point, and the two objects which subtend the angle.

74. A vertical angle is one whose plane is vertical.

75. An angle of elevation, is a vertical angle having one of its sides horizontal, and the inclined side above the horizontal side.

76. An angle of depression, is a vertical angle having one of its sides horizontal, and the inclined side under the horizontal side.

77. An oblique angle, is one whose plane is oblique to the horizontal plane.

78. All lines, which can be the object of measurement, must belong to one of the three classes above named: that is, they are either horizontal, vertical, or oblique. The angles also are distributed into three classes ; horizontal angles, vertical angles, and oblique angles: the class of vertical angles being subdivided into angles of elevation, and angles of depression.

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