་ any arc is equal to the square of radius. (Sin. a + cos.3 a R2.) This is evident from the right-angled triangle CFB, (Fig. 32.) (47.1.) = The square of the tangent + the square of radius is equal to the square of the secant. Tan.3 a + R2 = sec.2 a. (47.1.) Tan. a: R:: R: cotan. a, or tan. a. cot. a = R2. This is evident from the similarity of the triangles ACT and DKC, (Fig. 32,) which give (4.6) AT : AC:: CD: DK. The sine of 30° and the cosine of 60° is each equal to half radius. 113. Geometrical properties most employed in Plane Trigonometry. The angles at the base of an isosceles triangle are equal; and conversely, if two angles of a triangle are equal, the sides which subtend them are equal. (5 and 6.1.) The external angle of a triangle is equal to the two opposite internal ones. (32.1.) The three interior angles of a triangle are equal to two right angles or 180°. (32.1.) Hence, if the sum of two angles be subtracted from 180°, the remainder will be the third angle. If one angle be subtracted from 180°, the remainder is the sum of the other angles. If one oblique angle of a right-angled triangle be subtracted from 90°, the remainder is the other angle. Ꭰ Fig. 11. The sum of the squares of the legs of a right-angled triangle is equal to the square of the hypothenuse. (47.1.) The angle at the centre of a circle is double the angle at the circumference upon the same arc; or, in other words, the angle at the circumference of a circle is measured by half the arc intercepted by its sides. (20.3.) Thus, the angle ADB is half ACB; and is, therefore, measured by one-half of the arc AB. B The sides about the equal angles of equiangular triangles are proportionals. (4.6.) SECTION II. DRAFTING OR PLATTING.* 114. DRAFTING is making a correct drawing of the parts of an object. Platting is drawing the lines of a tract of land so as correctly to represent its boundaries, divisions, and the various circumstances needful to be recorded. It is, in fact, making a map of the tract. It is of great importance to a surveyor to be able to make a correct and neat plat of his surveys. The facility of doing so can only be acquired by practice; the student shou.d, therefore, be required to make a neat and accurate draft of every problem in Trigonometry he is required to solve, and of every survey he is required to calculate. It is not sufficient that he should draw a figure, as he does in his demonstrations in Geometry, that will serve to demonstrate his principles or afford him a diagram to refer to, but he should be obliged to make all parts in the exact proportion given by the data, so that he can, if needful, determine the length of any line, or the magnitude of any angle, by measurement. 115. Straight lines. Straight lines are generally drawn with a straight-edged ruler. If a very long straight line is needed, a fine silk thread may be stretched between the points that are to be joined, and points pricked in the paper at convenient distances; these may then be joined by a ruler. In drawing straight lines, care should be taken to avoid determining a long line by producing a short one, as any variation from the true direction will become more manifest the farther the line is produced. When it is necessary to produce a line, the ruler is fixed with most ease and certainty by putting the points of the compasses into the line to be produced, and bringing the ruler against them. 116. Parallels. Parallels may be drawn as described in * Various hints in this section have been derived from Gillespie's "Land Surveying." Arts. 97, 98. Practically, however, it is better to draw them by some instrument specially adapted to the purpose. Fig. 33. B The square and ruler are very convenient instruments for this purpose. The square consists of two arms, which should be made at right angles to each other, to facilitate the erection of perpendiculars. Let AB (Fig. 33) be A the line to which a parallel is to be drawn through C. Adjust one edge of the square to the line AB, and bring a ruler firmly against the other leg; move the square along the ruler until the edge coincides with C: this edge will then be parallel to the given line. If a square be substituted for a simple right angle, it may be held more firmly against the ruler. Instead of a square, a right-angled triangle is frequently used. The legs should be made accurately at right angles, that it may it is required to draw a Fig. 34. C B parallel. Bring one edge of the triangle accurately to the line, and then place a ruler against one of the other sides. Slide the triangle along the ruler until the point C is in the side which before coincided with the line: this side is then parallel to the given line. The parallel rulers which accompany most cases of instruments are theoretically accurate. They are, however, generally made with so little care that they cannot be depended on where correctness is required; and, even if made true, they are liable to become inaccurate in consequence of wear of the joints. Fig. 35. 117. Perpendiculars. Perpendiculars may be drawn as directed, (Art. 88, et seq.) A more ready means is to place one leg of the square (Fig. 33) upon the line: the other will then be perpendicular to that line. The triangle is another very convenient instrument for this purpose. Let AB (Fig. 35) be the line to which a perpendicular is to be drawn. Place the hypothenuse of the triangle coincident with AB, and bring the ruler against one of the other sides. Remove the triangle and place it with the third C B side against the ruler, as at D: then the hypothenuse will be perpendicular to AB. This method requires the angle of the triangle to be pre cisely a right angle. To test whether it is so, bring one leg against a ruler, as at A, (Fig. 36,) and scribe the other leg. Reverse the triangle, and bring the right angle to the same point A, and Fig. 36. B A again scribe the leg. If the angle is a right angle, the two scribes will exactly coincide. If they do not coincide, the triangle requires rectification. 118. Circles and Arcs. These are generally drawn with the compasses, which should have one leg movable, so that a pen or a pencil may be inserted instead of a point. When circles of long radii are required, the beam compasses should be used. These consist of a bar of wood or metal, dressed to a uniform size, and having two slides furnished with points. These slides can be adjusted to any part of the beam, and clamped, by means of screws adapted to the purpose. The point connected with one of the slides is movable, so that a pencil or drawing pen may be substituted. When the beam compasses are not at hand, a strip of drawing paper or pasteboard may be substituted: a pin through one point will serve as a centre; the pencil point can be passed through a hole at the required distance. 119. Angles. Angles may be laid off by a protractor. This is usually a semicircle of metal, the arc of which is divided into degrees. To use it, place it with the centre at the point at which the angle is to be made, and the straight edge coincident with the given line; then with a fine point prick off the number of degrees required, and join the point thus determined to the centre. The figures on the protractor should begin at each end of the arc, as represented in Fig. 37. 120. By the Scale of Chords. The scale of chords, which is engraved on the ivory scales contained in a box of instruments, may also be used for making angles. For this purpose take from the scale the chord of 60° for a radius. With the point A, at which the angle is to be made, as a centre, and that radius, describe an arc. Take off from the scale the chord of the required number of degrees and lay it on the arc from the given line, join the extremity of the arc thus laid off to the centre, and the thing is done. Thus, if at the point A (Fig. 38) it were required to make an angle BAC of 47°. A Fig. 38. B |