. A treatise on surveying and navigation: uniting the theoretical, the practical, and the educational features of these subjects. (2).b & Hence, . sin.^4 : : b \ O F D Or, . a : 5=sin A : sin. B) Scholium 1. When either angle is 90°, its sine is radius. Scholium 2. When CB is less than A C, and the angle B, acute,the triangle is represented by A CB. When the angle B becomesB, it is obtuse, and the triangle is A CB; but the proportion isequally true with either triangle; for the angle CBD= CBA,and the sine of CBD is the same as the sine of ABC. In prac-tice we can determin
. A treatise on surveying and navigation: uniting the theoretical, the practical, and the educational features of these subjects. (2).b & Hence, . sin.^4 : : b \ O F D Or, . a : 5=sin A : sin. B) Scholium 1. When either angle is 90°, its sine is radius. Scholium 2. When CB is less than A C, and the angle B, acute,the triangle is represented by A CB. When the angle B becomesB, it is obtuse, and the triangle is A CB; but the proportion isequally true with either triangle; for the angle CBD= CBA,and the sine of CBD is the same as the sine of ABC. In prac-tice we can determine which of these triangles is proposed by theside AB, being greater or less than AC; or, by the angle at thevertex C, being large as ACB, or small as ACB. In the solitary case in which A C, CB, and the angle A, are given,and CB less than A C, we can determine both of the As A CBand ACB; and then we surely have the right one. PROPOSITION 5. If from any angle of a triangle, a perpendicular be let fall on theopposite side, or base, the tangents of the segments of the angle are to one another as the segments of the 54 SURVEYING. Let ABC be the triangle. Let fall theperpendicular CD, on the side AB. Take any radius, as Cn, and describethe arc which measures the angle n, draw qnp parallel to AB. Thenit is obvious that np is the tangent of theangle D CB, and nq is the tangent of the angle A CD. Now, by reason of the parallels AB and qp, we have,qn : np=AD : DB That is, tan.^t CD : CB=AD : DB Q. E. D. PROPOSITION 6. If a perpendicular be let fall from any angle of a triangle to its op-posite side or base, this base is to the sum of the other two sides, as thedifference of the sides is to the difference of the segments of the base.(See figure to proposition 5.) Let AB be the base, and from C, as a center, with the shorterside as radius, describe the circle, cutting AB in 0,AC in F, andproduce AC to E. It is obvious that AE is the sum of the sides A C and CB, andAF i
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Keywords: ., boo, bookcentury1800, booksubjectnavigation, booksubjectsurveying
