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It has been already observed that when the plan consists of more than one field, the computation of the whole should be compared with the computation of the enclosures taken separately, and then added together. The computing book should be kept in such manner that the calculations may be referred to at a future time.

Where an extensive area has been triangulated, the whole should be first computed from the triangles, making such additions and deductions as may be rendered necessary by certain portions of the area under survey being cut off by the sides of the triangles, or by other portions outside of the survey proper being included within the triangles.

To calculate the area of a triangle by means of the lengths of the sides : take half the sum of the three sides added together; from such half sum subtract the three sides separately; the square root of the three remainders and the half multiplied consecutively together, is equal to the area of the triangle; let A, B, and C represent the three sides of a triangle, and Sthe sum of the three sides ; and therefore I S, half the sum; the area of the triangle will be equal to

N{Sx (I S-A) (1 S-B) (H S-C). The fences of fields are generally more or less crooked or uneven, and one of the first operations, before commencing the computations, is to reduce them to straight lines, which is done by a system of "give and take;" to effect this, draw a fine line on a very clear bit of tracing-paper, some six inches long, and lay this on the fence so that you can see when the line takes as much in on one side as it leaves out on the other ; then prick through at both ends, and draw in the line in pencil, which will equalize the fence; or lay down the edge of rule, guessing at the equalization, and draw in a pencil line ; if, when the rule is removed, it does not appear quite correct, rub it out, and draw in another, until there appears to be as much left out as taken in, so as to balance; a little practice will soon make this right.

CHAPTER IV.

SURVEYING INSTRUMENTS.

The Cross Staff and Optical Square.The Prismatic Compass.

-The Sertant and Box-Sextant.-The Transit Theodolite.

-The Ordinary Theodolite.-Colonel Everest's Theodolite. The Cross Staff and Optical Square are used for setting out lines perpendicular to others, and which seldom exceed five or six chains in length. The cross-staff, though rather old-fashioned and cumbersome, is still very commonly used for this purpose, and it is made two different ways. One shape consists of four sights, fixed at right angles upon a brass cross, which, when in use, is fixed to the top of a staff. To set out a perpendicular to a given line with this instrument, and at a given point, thrust the staff into the ground at this point, so that through one pair of sights you can see both ends of your main line by moving from one side to the opposite one, which will ensure your being exactly on the line, and your instrument in the proper position; now move round to the cross-sights, and in line with them stick up a pole, you will thus have set off your perpendicular.

The second shape consists of a hollow brass cylinder, about three or four inches in diameter, and about as many in depth ; through this are pierced sights at right angles to each other, and it is adapted to a staff. This is to be used exactly in the same manner as the other.

Thc optical square is very superior to the above for convenience, dispatch, and portability. It is a shallow, circular, brass box, containing two mirrors fixed at an angle of 45° to each other, so as to reflect a right angle or of 90. Fig. 46 explains the construction and use of this little instrument. A and B are two mirrors, fixed at an angle of 45° to each other; E is supposed to be the place of the eye, and the line FE is perpendicular to the line D E. A ray of light from the object F falling upon the mirror A at the above angle will be reflected at the same angle on to B, from whence it will be reflected at the same angle again to the observer's eye at E. The mirror B has

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only the upper half of its surface silvered, so that the eye looking along the line ED through an aperture in the box, sees D by direct vision through the lower and unsilvered portion of the mirror B; and any object seen in the upper or silvered portion of it, and coinciding so as it were to make one with the object D, is on a line perpendicular to E D at the position E of the ob

The reader will see how rapidly this instrument is used by standing at any point on a line, and holding it in the hand, looking towards an object at the end of the line, and sighting an assistant with a rod until it is seen to coincide with the first object; a perpendicular is then set off from the point where the observer is standing.

In the same manner, if a perpendicular is to be made to fall from a given point on to a given line, it is only necessary to walk along the line until you come to a point from which you see one end of your line and an object at the given point coinciding together in the mirror D, when you will be standing at the spot on which the perpendicular will fall.

It is on a similar principle that the box-sextant, one of our most valuable surveying instruments, is constructed, to which we shall come presently

THE PRISMATIC COMPASS.

This pocket instrument, shown at Fig. 47, is contained in a shallow brass box about two inches in diameter; it is used for measuring horizontal angles, which it will do with a considerable degree of accuracy considering the nature and size of the instrument, and as this may be done by merely holding the instrument in the hand, the work is carried on with much rapidity; for an approximate survey, or for filling in where the principal lines and stations have been previously fixed, and for taking the angles on lines not exceeding fifteen or twenty chains in length, it is very useful, more particularly where there are not many angles to take before we chain into a point previously fixed; for this purpose of filling in it is daily used to a very considerable extent.

B is the brass box containing the compass card C, to the under side of wbich is attached a magnetic needle, turning upon an agate centre fixed to the bottom of the box. The compass card is usually divided to thirty minutes, according to the size of the instrument, and therefore to half a degree ;' a more minute division must be appreciated by the eye. The sight vane V is mounted upon a hinge, so that when the instrument is not in use it may be turned down flat on the glass. Along the opening in the

THE PRISMATIC COMPASS.

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sight vane is stretched a fine thread, which is made to intersect the object on which the observation is taken. P is a prism attached to a plate sliding up and down in a socket S, so that it may be raised or lowered to admit of distinct view of the divisions on the compass card ; this also is supplied with a hinge joint, by which means it can be turned down into the box. In the plate attached to the prism is a narrow aperture, to which the eye of the observer is applied when making an observation ; there is a button attached to a spring, and which on being slightly pressed checks the vibrations of the compass card so as to bring it sooner to rest when an observation is being taken. There is a stop in the side of the box B, by touching which the compass is thrown off its centre when the astrument is not in use; particular care should always be taken to do this after making observations, or the constant vibrations of the needle will soon wear off the delicate finish of the agate point and sensibly impair the value of the instrument. A brass cover fits on to the top of the box, so that it may be carried in the pocket, or packed in a leather case, which by means of a belt may be strapped to the side, which is much the safest plan. Besides the above, Messrs. Elliott make a larger instrument of the same description with a silver compass ring, and which is a superior instrument; this as well as the above is provided, if desired, with a stand for the instrument to be placed upon when observations are being taken ; as this, however, is rather inconvenient to carry about in the fields, the prismatic compass when in use is generally held in the hand.

On looking through the aperture above mentioned, and raising or lowering the prism in its socket, distinct vision of the graduations on the compass card is soon obtained, as they vibrate under the sight vane; seen through the prism they appear as prolongations of the thread as each successively comes into coincidence with it, the latter being seen by direct vision through the part of the aperture projecting beyond the prism.

The following shows the method of using the instrument: turn up the sight vane V and the prism P on their hinges, and hold the instrument as nearly horizontal as possible, so that the needle may play freely; raise the prism in its socket until the divisions on the compass card are distinctly seen through the prism. Turn the instrument round until through the portion of the aperture projecting beyond the prism the object to be observed is seen to coincide with the thread of the sight vane ; now bring the card to rest by touching the spring; the reading at the graduation on the compass card which appears as a prolongation of the thread gives the angle made by the line on which

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observing with the magnetic meridian. The bearing of a second object from the same place being made in the same manner, the difference between the two bearings gives the angle subtended by the two objects observed. When vision is obstructed by the sun, a dark glass G is interposed.

By repeating an observation two or three times, and taking the mean, the bearing will be obtained more accurately, and it will be still more so if the bearing is again similarly taken at the other end of the line, and the mean taken again between the bearing at one end and at the other. This will also check any local influence on the needle.

M is a mirror which is made to slide on or off the sight vane, and may if required have its face turned downwards, or it may be inclined at an angle by means of its joint h; it will remain stationary on any part of the sight vane by the friction of its slides. This is applied for the purpose of taking the bearing of objects much above or below the level of the point of observation; it is put on with its face upward or downwards accordingly as the objects observed are much above or below the level of the observer.

The circle being divided to 360°, the compass card is also so divided, and, as we have said before, each degree is again subdivided. In some instruments the degrees are numbered from the North point round by East, South, West, and to the North again up to 360° consecutively, in others, and as we think in a better way, the instruments are numbered from North round by East to the South Pole as 180o and from the South round by West to North again as 180° more, completing the 360°; the readings in this mauner are much simplified, for if we have a reading so many degrees N.E. we know it will be the same number of degrees S.W.; for instance, in Fig. 48, where the circle is divided into twice 180°, the one reading from North round by East to South, and the other from South round by West to North, a bearing is taken to the left of the North, and reads 13.5° West from the South Pole; then we know at once that it will read 135° East reading from the North Pole. This renders unnecessary any mention of North and South in reading a bearing, as we know at once that all bearings reading East go progressively from North to South, and all bearings reading West go progressively from South to North.

The extent to which this instrument may be used will depend on many circumstances, such as the ability of the observer, the quality and object of the survey, &c. As there are no means of reading correctly to within fifteen minutes, it of course only gives approximations; but then something of the same kind

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