METCALFE'S PROTRACTOR.

101

"As when away from home, it seldom happens that the surveyor can obtain a good drawing board, or even a table, with a good straight-edge, I fix a flat ruler, A, Fig. 72 to the table, B B, by means of a pair of clamps, CD, and against this ruler I work the pattern square, E, one side of which has the stock flush with the blade; or if a straight edged board be at hand, then the square may be turned over, and used against that edge instead of the ruler A. Here, then, is the most perfect kind of parallel ruler that art can produce, capable of carrying the protractor over the whole of a sheet of plotting-paper of any size, and may be used upon a table of any form. It is convenient to suppose the north on the left hand, and the upper edge of the blade to represent the meridian of the station.

"This protractor is held in the hand while the vernier is set, which is an immense comfort to the sight; and it will be seen that as both sides of the arm are parallel with the zero and centre, the angle may be drawn on the paper on either side, as the light or other circumstances may render desirable.

"From this description, and a mere glance at the plate, it is clear that angles taken with the theodolite can be transferred to the plot as accurately as the protractor can be set, namely, to a single minute, and that, too, in a rapid and pleasant manner.*

By means of the notch at the end of the arm, this instrument may be used in the manner of a circular protractor, should a square not be at hand."

The above method of Mr. Howlett's is generally a very valuable one, and leaves nothing to be desired in ordinary cases.

There is another kind of semicircular protractor, a modification of " Howlett's," and known as Metcalfe's, illustrated at Fig. 73. The principal addition to this protractor is a contrivance at A, by means of which the radius is turned into a scale, so that distances may be set off at the same time that an angle is protracted; there is a scale on each side, and the vernier is double so as to apply to each. For extensive general surveys, plotted to scales of three or four inches to a mile, this is a decided addition: when the instrument is used for this purpose, first adjust the vernier to the scale according to distance, then adjust the vernier according to the given angle, and, laying the instrument along the T square, adjust the zero of the scale to the station point on the plan. Diagonal scales, corresponding to those on the arm of the instrument, are engraved upon it.

It need scarcely be observed that in using either of the above protractors for plotting a traverse, the first bearing or angle being

* With a seven or eight-inch protractor we can read much closer than this.

protracted, the line is drawn in, and by a scale made the length of the chain line; the blade of the T square is then pushed to the station thus found, the next bearing set off, &c.; and this operation is repeated until the whole of the lines are laid down, when the corresponding offsets are afterwards plotted also, and this part of the survey is complete.

Although for most ordinary cases these semicircular protractors, used in conjunction with the T square, comply with most requirements, it will be observed that there is no check as to the instrument being exactly over the centre of the station, such as is afforded by the circular protractor when both arms are used. Also, when a long line has to be laid down, we have only two points to do so by, instead of three, which is a decided advantage in the circular instrument; where, for engineering purposes, plans of townlands are often required to scales of 30 and 40 feet to an inch, lines of great length on paper have often to be laid down, in doing which, unless great care is observed, a mistake, amounting to six inches or a foot on the ground, will soon creep in, and occasion considerable trouble. Much the same observations hold good with regard to the bases for a working survey for a canal or railway.

Protracting by Chords.-For laying down the sides of triangles, the protraction of angles by chords is often very convenient where angles have been taken, but some of the sides not measured. An example will best illustrate the manner of doing this-From the base A B, Fig. 74, measuring 210 chains, to fix on paper the objects C, D and E, the angles at A being C A B = 41; DA B=75°-10'; EA B=110° 12'; and CBA.= 43°; DBA = 48°; EBA = 20°.

Tabulate your angles in the following manner in pairs :

NORTHINGS AND SOUTHINGS.

103

By means of any protractor whatever, or even by guess, set out lines approximating to the angles required; set the beamcompasses to 210 chains, and from A and B, as centres, describe portions of arcs opposite the lines set off as the approximate angles. Next set your compasses to 147.08, and from B, as a centre, set off this first chord, and mark the intersection as No. 1; next set your compasses to 153.93, and from A make the intersection No. 2; draw in the lines A 1 and B 2, and their intersection will give the point C, and the angles CAB and CBA.

Next set the compasses to 256-16, and from B find intersection No. 3; and with 170-81 in the compasses describe from A the intersection No. 4; draw in A 3 and B 4, produced so as to get the intersection D, and the angles D A B and D BA. With regard to the triangle A E B, observe that the obtuse angle 110° 12' is equal to 90° + 20° 12, and that we have only to set out the chord of 20° 12′ beyond 90°, to obtain the desired intersection; twice the sine of 45° x radius = 296-98; from B, with this distance in the compasses, set off the intersection 90°, and from it, with the length 73.67, mark off the intersection No. 5; also from A, with the distance 72.91, set off intersection No. 6, to find the point E, and the angles E A B and E B A.

This method will appear tedious, but where the angles are subtended by lines of three and four feet, and sometimes even of twice and three times this length, ordinary protractors are of no use, and this method by chords is extremely useful.

Plotting a Traverse by Northings and Southings, Eastings and Westings, or by Difference of Latitude and Departure.

Reference to Figs. 68 and 69, or a glance at Fig. 75, will show that in going round the perimeter of either of the above figures, it will be necessary to travel as much north as south, and as much east as west; that is, that starting northwards from any point on the perimeter or outline of the figure, and then returning to it, the distance travelled south must be equal to that travelled north before the starting point can be again reached, and the same as regards east and west; thus in Fig. 66, going north from A to b= going south from 6 to A; and going east from b to B= going west from B to b; so that in going from A to B, and returning to A, the distance travelled north = that travelled south, and the distance east distance west; or in other words, the northings the southings, and the eastings=

=

the westings; similarly as to all the other sides of the polygon, so that all the northings all the southings, and all the eastings = all the westings.

As regards the angles formed by the bearings with the meridians we may consider the northings and southings as the cosines of the angles, and the eastings and westings as their sines; thus Ab is the cosine, and Bb is the sine of the angle b A B, and Bc is the cosine, and Cc the sine of the angle N' B C, or its supplement C B c;t and d C is the cosine, and Dd the sine of the angle D Cd; and De is the cosine, and Fe the sine of the angle N" D F, or its supplement F De.

Taking, therefore, every bearing formed with meridian as the angle, and each side of the polygon or " traverse," as radius, we can by means of the tables of sines and cosines ascertain the northings or southings, and the eastings or westings.

When the bearing is not above 90°, enter the reading as N.E. or N.W., in the column allotted to bearings, in the following form; if greater than 90°, subtract it from 180°, and enter the remainder in the column of bearings; if greater than 180°, and less than 270°, subtract from 270°, and enter the remainder; if greater than 270°, and less than 360°, then subtract from 360°, to find the proper entry. From the tables of sines and cosines ⚫ enter in the column of signs the cosine and sine for each angle.

The northings and southings are thus the products of the chainage by the cosines of the angles, and the eastings and westings are the products by the sines. By means of these a most accurate method is obtained for plotting a traverse. On the assumed meridian, as N A W, plot the lengths due to the northings and southings; through the points thus found, draw in lines perpendicular to the meridian, as through b, c, d', e' and g, on which set off the eastings and westings, as b B; c C, both eastings; c C-d D, westing; e F, easting; e' F-g G westing; ob

*These meridians we may, in practice, assume to be parallel as far as all proceedings in engineering field-work are concerned.

Should the reader be totally unacquainted with trigonometry, we must refer him to the beginning of the chapter on "Setting out Curves."

APPLICATION OF NORTHINGS AND SOUTHINGS.

105

serving that whilst the eastings are positive, the westings are negative, or quantities to be subtracted from the eastings; for accuracy this method leaves nothing to be desired, and is undeniably the best for an extensive traverse, for instance, a mile or two across; but unless one is in the constant practice, it takes up a little time, and requires the work to be first plotted roughly to a small scale; a mere sketch in fact, for mistakes are not possible, as in adding up the columns of northings and southings, the totals must be equal, as also those of eastings and westings.

Some years since, before protractors were brought to the perfection they have since attained, this method of plotting a traverse was very usual amongst the best surveyors; but it is less commonly resorted to at present. Accurary, however, is seldom to be attained by applying one universal method to everything under any circumstances; the reader is by this time supposed to have acquired considerable insight into the practice of land-surveying, for, as far as the writer is aware, no detail of practice has been neglected in the foregoing pages, even at the risk of appearing tedious; the student should, therefore, now be able to lay down two or three traverses on a large sheet of plan paper, and judge for himself how far one system is preferable to another under a variety of conditions.

Setting aside the plotting of a traverse in the above manner, considerable advantage may be derived from the calculation of northings and southings, eastings and westings, in exploring a wild country, in which a general, and, perhaps, very indifferent map is the only guide at hand. After being obliged, by the nature of the country, to abandon any direction first taken, and to traverse round before it is practicable to return to it, it is easy, after a time, to ascertain from a certain point how far we have deviated from our first original route, and therefore how far we have to steer east or west, north or south, in order to return to it.

Our meaning will be best illustrated by means of Fig. 74, where A B is the direction of a line pursued from some distance. On reaching the point A the nature of the country is found to be such that it is no longer possible to do so, and a deviation becomes necessary; let this commence at A, in the direction of the lines A c, cd, de, ef, the bearings of which are all carefully taken, and the lines chained. On reaching the point f, we find the country clear in a direction towards the original route, and we desire to return to it; find the northings and eastings made by the above lines to the point f, by using the theodolite in the manner already described, and sketch the work in the field-book as you go on, and enter the bearing opposite each station.