AC (fig. 3) that the sine of c is that of an obtuse angle, and consequently falls upon the diameter BCD, produced without the triangle, then ambiguity in the office is the natural concomitant of such an omission. And this, too, does not represent the full amount of negligence in the field, for in driving either of the two lines from C, the staff are proceeding to a definite point of the compass, one too far asunder from that of the other line to be mistaken for it.

The professional objection of surveyors to treating the ambiguous case as a proposition, either in trigonometry or the mensuration of superficies, is perhaps the most convincing proof that it belongs exclusively to Constructive Geometry. Thus, if admitted into the former, a triangular field would contain either the one or the other of two measures, either the area within the triangle ABC (Example II. Part VIII.), or that within ABC (Example 3); consequently two plans would be required, a greater and a less, for which payment would assuredly be received for neither. Hence the practical con clusion.

In field operations it is always preferable to determine either the three angles and one side or two sides of a triangle, or else two sides and the included angle. But this cannot always be complied with, and the following general rules will assist beginners in making provision for the exception, as by Article 34, Section II. Part VIII.

1. When the sum of the adjacent or opposite angles is equal to a right angle, the other angle is also a right angle.

2. When the sum of the two angles adjacent to any one of the given sides is less than a right angle, the other angle opposite that side is obtuse.

3. When the sum of the two angles adjacent to any one of the two given sides is greater than a right angle, the other angle opposite that side is acute.

4. When the angle included by the two given sides is obtuse, then the other two angles of the triangle are acute.

5. When the complement of the given angle is greater than the angle included by the two given sides, the other remaining angle is obtuse.

6. When the complement of the given angle is less than the angle included by the two given sides, the remaining angle is acute.

7. The nearer the angle opposite the given angle is to a right angle, the greater is the liability to error, in taking approximate estimates of angles.

SECTION VII.

TO SURVEY WITH THE CIRCUMFERENTOR.

The circumferentor is sometimes used in surveying. It consists of a flat bar of brass BB, about fifteen inches long, with sights CC at its opposite ends, and two narrow

slits b, c for taking observations; in the middle of the bar is a circular brass box A, containing a magnetic needle, which, as usual, is covered with glass. The ends of the needle play over a brass circle g, which is divided into 360° in such a manner that the two quadrant s are at right angles to the line drawn through the sights. The instrument is usually supported on a tripod E, and can be turned in any direction by means of its socket-joint. When the magnetic needle is properly balanced, and

m

B

moves freely in its horizontal position, it will retain its position of magnetic north, while the sights are turned from one station to another; consequently the number of degrees which the angle contains can be read of. The great length of the magnetic needle increases the accuracy of the circumferentor, for, if it were short, and consequently the graduations on the brass circle proportionately small, the angle could not be read of with sufficient accuracy.

The instrument is chiefly used in surveying mines, coal-pits, woods, &c.

The circumferentor is sometimes provided with a spirit-level F, with adjusting screws a, a, &c., a tangent screw m, a vernier, &c., in which case it may be made to answer most of the purposes of a theodolite.

The needle should not be suffered to play longer than necessary, but be lifted off its centre, otherwise the delicate point on which it turns would soon be destroyed. The instrument usually has the east and west marked contrary to their true positions, in order that by the reading of the needle the actual direction of the line is shown.

To find the bearing of an object by the circumferentor.

Place the circumferentor over the station, turn N. sight to the object, and looking through the S. sight adjust with the tangentscrew till the hair in the N. sight exactly cuts the object-this part of the operation being the same as in the theodolite. When the needle has perfectly settled, read off the degrees to which its N. end points, from the N. or S. line of the compass-box, accordingly as the N. or S. end of the needle is in the N. or S. part of the compass-box; the angle thus read off is called the bearing of the object.

If the needle stand between two degrees, as between 40° and 41 turn the instrument gently till it stand exactly at 40°, and clamp it; detach the sights, and bring them with the vernier, with which they are connected, carefully back to the object: the number of minutes to be added to the angle will be shown by the first coincidence of a division on the vernier with one on the horizontal plate; if the coincidence take place at the 27 division, the angle will be 40° 27'. The method of taking an angle with this instrument without using the needle, is the same as with the theodolite.

On plotting the bearings taken by the circumferentor, &c.

In laying down the bearings, in cases where the angles are taken from the magnetic meridian of the circumferentor, that meridian is first drawn on the plan, and from a point in it the bearing of the first line from it is laid off with the protractor; the length of the line is next laid off, and through its extremity another meridian is drawn parallel to the former; the second bearing, and the length of the second line, are then laid off in the same manner; and so on till the work be completed, and the variation of the compass being known, the true meridian may then be drawn on the plan, that the work may have its proper position on the finished plan.

The above method is the same as that of laying down a wood, or road survey, by the theodolite, except in laying off the angles.

When the angles are taken with the circumferentor without reference to the magnetic needle, the method of laying down the work is the same as in theodolite surveying.

NOTE. In coal-mines, when a pit or shaft is required to be sunk from the surface to a given point in the works below, it is usual to take the several bearings and distances with this instrument, from the bottom of an existing shaft along the passages in the mine, to the required point; and then to repeat the same operation on the surface of the ground, without plotting the work on paper; by which means complete accuracy is frequently obtained.

RAILWAY SURVEYING.

A complete survey for an intended railway contains six problems, three of which are field operations, and the remainder office work. They are as follows:

Prob. I. To range the station poles that form the chain and other lines.

Prob. II. To measure with the chain the chain lines; with the theodolite the angles which they form, also the angles of depression, to obtain data for the reduction of the hypotenusal planes of the earth's surface to a common plane; and with levelling instruments to take the levels of stations.

The field notes under this problem are in three field-books; the first for the chain, the second for the theodolite, and the third for the levelling instruments.

Prob. III. Given the field notes, to plot the plan and sections. Prob. IV. To apportion the land required for the intended railway.

Prob. V. To find the field notes required for setting out the railway estate and the line, including the diversion of roads, &c.

Prob. VI. To set out the railway estate and line, including roads, &c., whose diversion is necessary.

In a preliminary survey, and in a parliamentary survey, and also in surveying a railway which passes through an estate being surveyed, only a portion of the above field and office operations are required.

In the following directions for a preliminary survey, the lines are ranged for laying down the railway from a series of connected baselines. It is seldom, however, that lines thus ranged can be measured by the chain with sufficient accuracy, and when such is the case, even ground adjacent must be selected for base-lines, so as to obtain the true lengths of the former by trigonometry.

Another method is to range the right lines of the projected railway, and either to join them by the conveying tangents from the two tangential station-poles, or else by the chord-line that lies between them (the two tangential stations), and then to take the bearings of these stations from the adjacent base-stations, so as to find their reduced distances or base-lengths required for plotting; and when an adjacent base-line is not perfectly level, its hypotenusal length must be reduced to base-measure, otherwise the field notes, both for plotting and setting out the railway, will be erroneous. (See ADVERTISEMENT, p. viii., under RAILWAY SURVEYING.)

Illustrated general directions for a preliminary survey of a portion of a projected railway.

Landowners (resident and absentee), the executors of minors, mortgagees, and capitalists, often order plans purposely to ascertain how far projected railways will interfere with their respective interests involved, and whether they would be justified in farther supporting such projects.

Plans of this kind are of the simplest character, consistent with the purpose they are intended to serve. The several resident agents in charge of estates proposed to be intersected watch such railway movements closely, and never fail to apply to their employers for instructions how to act. They are often by profession landsurveyors themselves, and, when such, they send to their absent employers a sketch of the projected line on tracing-paper, taken from the plans of their estates; or when otherwise, they employ their surveyors to do so for them, who may give a general sketch of the whole intended line upon an ordnance map.

Again, several parties may join and order a survey and plan of the whole district through which the projected railway is intended to pass, or only of that portion of it in which they are more immediately interested.

The annexed figure, already referred to, may be presumed to represent a plan of this latter kind. It is, in point of fact, a copy of a plan of a portion of a district surveyed for the purpose of the projected railway RST, the base-line AC being ranged so as to avoid Lynch Wood. This line terminates at +C, partly on account of its diverging too far from the line of the railway, and partly to avoid the River Ouse beyond + C. The next base-line commences at + B in AC, and runs close to the railway at T. Beyond +D (on the complete survey of the whole line) it continues the same direction till its deviation requires another base-line.

To range the base-lines.

The initiatory step in such a survey is to range the base-lines. This is done by going over the ground and carefully examining it from points where the projected railway may be presumed to be on a level with the surface. Such points are generally determined by the amount of excavation, on the one hand, that is required to fill up the hollows on the other. An experienced eye will approximate very closely to such data. Having come to a definite conclusion on these points, and the curvature to which they may give rise, the flagstaffs are then set so as to avoid obstructions, and be visible from each