Fig. 82.
To lay down [Fig. 82] in this manner, having fixed the first station A, the length of the first primary reference line A B may be laid off upon the meridian, because in this case the bearing being due north, the reference line will be coincident with the meridian. This done, the protractor is to be placed over the plotted point B and the second bearing laid down. Having plotted the length of this line in the point C, the third reference line C A will be determined both in length and direction by the plotted points C A, which should be joined and the line measured to ascertain whether its length corresponds with the measured distance. If these do not correspond, the angle must be replotted and the lengths laid off anew to discover the source of the error. Assuming, however, that the lines close properly, the offsets and other secondary points may be next plotted in the manner previously described.
Angles may be more accurately laid down by means of a table of natural sines and cosines and a linear scale than by means of a protractor. This is especially true when the angles are subtended by long lines, as, for example, lines of 3, 4, and even 6 feet. In such cases, a protractor is of little use. This mode of laying down angles is also convenient in some cases where angles have been taken, but some of the sides not measured. In using the table, it must be remembered that the radius of the sines and cosines is taken as unity; therefore, to find the sine and cosine for any other radius, the sine and cosine in the tables must be multiplied by that radius. To lay down the angle A B C in [Fig. 82], the reduced cosine of the angle should be plotted from B in the point a, according to some scale. The scale length of the reduced sine should then be scribed from a, and the scale length of the radius scribed from B. A line drawn from B through the point of intersection of the scribes will lay down the angle corresponding to the sine and cosine in the table. Suppose the radius chosen to be 5 chains, the angle being 32° 30′. The cosine of 32° 30′ is ·8434, which multiplied by 5, the assumed radius, = 4·2170. Lay off this distance from B on the base A B. The sine of 32° 30′ is ·5373, which multiplied by 5, = 2·6865. From the point a, which is distant 4·2170 chains from B, with a radius equal to 2·6865 chains, describe an arc; and from the point B, with a radius equal to 5 chains, describe another arc. From B draw a line through the intersection of these arcs, and lay off upon it the measured length of 1946 links as recorded in the field-book.
If only the length of the base A B and the magnitudes of the angles A B C and B A C were given, the lengths of the sides B C and A C would have to be calculated by trigonometrical formulæ. This method of calculating the lengths of the sides of triangles and plotting them with the beam compasses, like chained triangles, is the most accurate for laying down the great or primary triangles of a survey.
When it is required to plot according to this principle a solitary angle, as, for example, that between a station line and the meridian, a circle should be drawn with as large a radius as practicable round the station at which the angle is to be laid down. The distance between the points at which the two lines enclosing the angle cut that circle is then found by multiplying the radius by the chord of the angle, that is, by twice the sine of half the angle.
It sometimes happens, particularly in extensive surveys, that all the angular points of some triangles cannot be plotted upon the same sheet of paper. In such cases, the plot of the outlying points and the sides of the triangles may be laid down in the following manner. Plot the intersected triangles independently and trace them on tracing paper. Then, having drawn a fine line upon both sheets to represent the sheet edge, lay the points on the trace corresponding to those already plotted on the first sheet down upon, and make them to coincide with, the latter. Secure the trace in this position and trace the sheet edge line upon it. The intersected lines may now be plotted on the fair sheet with a pricker at points outside the sheet edge line. Next apply the trace to the second sheet and make the sheet edge lines coincide. Having secured the trace in this position, the points and the intersected lines on this second sheet may be plotted upon the fair paper by means of the pricker.
To Plot Traverse Reference Lines.
—In plotting a traverse survey in which the angles have been measured from a fixed line of direction, the magnetic meridian, the direction of the lines may be all laid down at the first angular point. An example will best show the method employed in this case. It is required to lay down the traverse shown in [Fig. 83].
Fig. 83.