ground, or was specially uncertain in its direction when observed. The inaccuracy must then be distributed among all the lines in proportion to their length. Each point in the figure, B, C, D, E, must be moved in a direction parallel to EA, by a certain distance which is obtained thus. Multiply the distance EA by the distance AB, and divide by the sum of all the courses. The quotient will be the distance BB'. To get CC', multiply EA by AB+ BC, and divide the product by the same sum of all the courses. To get DD', multiply EA by AB+BC+ CD, and divide as before. So for any course, multiply by the sum of the lengths of that course and of all those preceding it, and divide as before. Join the points thus obtained, and the closed polygon AB'C'D'A will thus be formed, and will be the most probable plat of the given survey."

The method of Latitudes and Departures, to be explained hereafter, is, however, the best for effecting this object.

(269) Field Platting. It is sometimes desirable to plat the courses of a survey in the field, as soon as they are taken, as was mentioned in Art. (247), under the head of "Keeping the fieldnotes." One method of doing this is to have the paper of the Field-book ruled with parallel lines, at unequal distances apart, and to use a rectangular pro

tractor (which may be made. of Bristol-board, or other stout drawing paper,) with lines ruled across it at equal distances. of some fraction of an inch. A bearing having been taken and noted, the protractor is laid on

Fig. 184.

90°

the paper and its centre placed at the station where the bearing is to be laid off. It is then turned till one of its cross-lines coincides with some one of the lines on the paper, which represent East and West lines. The long side of the protractor will then be on a meridian and the proper angle (40° in the figure) can be at once marked off. The length of the course can also be set off by the equal spaces between the cross-lines, letting each space represent any convenient number of links.

* This was demonstrated by Dr. BOWDITCH, in No. 4, of "The Analyst."

(270) A common rectangular protractor without any cross-lines,

or a semi-circular one, can also

Fig. 185.

180

be used for the same purpose. The parallel lines on the paper (which, in this method, may be equi-distant, as in common ruled writing paper) will now represent meridians. Place the centre of the protractor on the meridian nearest to the station at which the angle is to be laid off, and turn it till the given number of degrees is cut by the meridian. Slide the protractor up or down the meridian (which must continue to pass through the centre and the proper degree) till its edge passes through the station, and then draw by this edge a line, which will have the bearing required.

Fig. 186.

C

B

(271) Paper ruled into squares, (as are sometimes the righthand pages of surveyors' field-books), may be used for platting bearings in the field. The lines running up the page may be called North and South lines, and those running across the page will then be East and West lines. Any course of the survey will be the hypothenuse of a right-angled triangle, and the ratio of its other two sides will determine the angle. Thus, if the ratio of the two sides of the right-angled triangle, of which the line AB in the figure is the hypothenuse, is 1, that line makes an angle of 45° with the meridian. If the ratio of the long to the short side of the right-angled triangle of which the line AC is the hypothenuse, is 4 to 1,

the line AC makes an angle

A

E

D

of 14° with the meridian. The line AD, the hypothenuse of an

equal triangle, which has its long side lying East and West, makes likewise an angle of 14° with that side, and therefore makes an angle of 76° with the meridian.*

To facilitate the use of this method, the following table has been prepared.

TABLE FOR PLATTING BY SQUARES.

To use this table, find in it the ratio corresponding to the angie which you wish to plat. Then count, on the ruled paper, any number of squares to the right or to the left of the point which represents the station, according as your bearing was East or West; and count upward or downward according as your bearing was North or South, the number of squares given by multiplying the first number by the ratio of the Table. Thus; if the given bearing from A in the figure, was N. 200 E. and two squares were counted to the right, then 2 × 2.75 = 51 squares, should be counted upward, to E, and AE would be the required course.

(272) With a paper protractor. Engraved paper protractors may be obtained from the instrument-makers, and are very conve

This and all the following ratios may be obtained directly from Trigonome trical Tables; for the ratio of the long side to the short side, the latter being taken as unity, is the natural cotangent of the angle.

Fig. 187.

N

30°

nient. A circle of large size, divided into degrees and quarters, is engraved on copper, and impressions from it are taken on drawing paper. The divisions are not numbered. Draw a straight line to represent a meridian, through the centre of the circle, in any convenient direction. Number the degrees from 0 to 90°, each way from the ends of this meridian, as on the compass-plate. The protractor is now ready for use. Choose a convenient point for the first station. Suppose the first bearing to be N. 30° E. The line passmg through the centre of the circle and through the opposite points N. 30° E. and S. 30 W. has the bearing required. But it does not pass

16

W

09

30°

S

09

E

through the station 1. Transfer it thither by drawing through station 1 a line parallel to it, which will be the course required, its proper length being set off on it from 1 to 2. Now suppose the bearing from 2 to be S. 60° E. Draw through 2 a line parallel to the line passing through the centre of the circle and through the opposite points S. 60° E., and N. 60° W., and it will be the line desired. On it set off the proper length from 2 to 3, and so proceed.

When the plat is completed, the engraved sheet is laid on a clean one, and the stations "pricked through," and the points thus obtained on the clean sheet are connected by straight lines. The pencilled plat is then rubbed off from the engraved sheet, which can be used for a great number of plats.

If the central circle be cut out, the plat, if not too large, can be made directly on the paper where it is to remain.

The surveyor can make such a paper protractor for himself, with great ease, by means of the Table of Chords at the end of this volume, the use of which is explained in Art. (275). The engraved ones may have shrunk after being printed.

Such a circle is sometimes drawn on the map itself. This will be particularly convenient if the bearings of any lines on the map,

not taken on the ground, are likely to be required. If the map be very long, more than one may be needed.

(273) Drawing-Board Protractor. Such a divided circle, as has just been described, or a circular protractor, may be placed on a drawing board near its centre, and so that its 0° and 90° lines are parallel to the sides of the drawing board. Lines are then to be drawn, through the centre and opposite divisions, by a ruler long enough to reach the edges of the drawing board, on which they are to be cut in, and numbered. The drawing board thus becomes, in fact, a double rectangular protractor. A strip of white paper may have previously been pasted on the edges, or a narrow strip of white wood inlaid. When this is to be used for platting, a sheet of paper is put on the board as usual, and lines are drawn by a ruler laid across the 0° points and the 90° points, and the centre of the circle is at once found, and should be marked . The bearings are then platted as in the last method.

C

B

Fig. 188.

E

(274) With a scale of chords. On the plane scale contained in cases of mathematical drawing instruments will be found a series of divisions numbered from 0 to 90, and marked CH, or C. This is a scale of chords, and gives the lengths of the chords of any arc for a radius equal in length to the chord of 60° on the scale. To lay off an angle with this scale, as for example, to draw a line making at A an angle of 40° with AB, take, in the dividers, the distances from 0 to 60 on the scale of chords; with this for radius and A for centre, describe an indefinite arc CD. Take the distance from 0 to 40 on the same scale, and set it off on the arc as a chord, from C to some point D. Join AD, and prolong it. BAE is the angle required.

A

40

09

The Sector, represented on page 36, supplies a modification of this method, sometimes more convenient. On each of its legs is a scale marked C, or CH. Open it at pleasure; extend the compass from 60 to 60, one on each leg, and with this radius describe Then extend the compasses from 40 to 40, and the dis

an arc.