thing which may tend to shorten his other operations in the survey, or will assist him in drawing his plans. When he has settled, by the cross, the place of an offset or short perpendi cular, it will be easiest to measure the length of it as he goes along, to save the time and trouble of returning to the place a second time.

It is proper to remark, that the plan ought to be drawn upon paper, with horizontal distances only; otherwise it will be impossible to join several fields together without distortion. For when several lines are to be joined together, a small error in the lengths of some of them will alter the position of others; a circumstance which has a greater tendency to distort the plan, than even the lengths of the lines themselves. It is, however, impossible for a surveyor to ascertain the exact level of every elevation and depression of his lines; but it would be of great advantage to him to take a level at that part which he judges to have a mean inclination. This may be done with the offset-staff thus:-Having laid the chain along that part, place one end of the offset-staff at the uppermost of 10 links on it, and let the assistant take the other end, and a line and plummet hung exactly over the other end of the 10 links on the chain, and let the surveyor apply a pocket or other level to the staff; and when it is level, the line of the plummet will point out on the staff the horizontal length of the 10 links of the chain. Consequently, by using a diagonal scale of 10 to a link, it will point out how much the line is diminished to get the horizontal length of it.

OF THE INSTRUMENTS USED FOR TAKING ANGLES.

be

Angles in the field are taken either in a vertical or in a horizontal plane. The former are measured by a Quadrant, and the latter by a Theodolite or Circle.

A QUADRANT is the fourth part of a circle of any conve nient radius. It is made of brass or wood, and the arc is divided into 90 degrees, and each degree is subdivided into smaller parts. The degrees are numbered from one extremity, called the beginning of the arc, to the other extremity or end of it.

The most simple quadrant, ABC, has a line with a plummet suspended from its centre, as AD, which, when hanging freely, is always perpendicular to the horizon; and sights, or a telescope, is affixed to the radius AB, which passes through the 90th degree, or end of the arc, to direct the eye in a straight line towards the object.

B

C

Sometimes an index AD, with telescopic sights, is made to revolve round the centre A; in which case a spirit-level is fixed to the radius AC, which passes through the beginning of the arc. The telescope is placed along AD. But sometimes the degrees are numbered from B, and a telescope is fixed at D, perpendicular to the index AD.

F

G

The THEODOLITE is the most complete instrument for surveying. It consists of a brass circular plate, the circumference of which is divided into 360 degrees, or twice 180 degrees, and each degree is subdivided into smaller parts. An index with a compass on it is fixed to the centre, and revolves round it; and on it is erected a semicircle, perpendicular to the plane of the instrument, furnished with a telescope perpendicular to the index of it, which moves round its centre. The use of the circle is for taking horizontal angles, and that of the semicircle is for taking vertical ones. The instrument is furnished with two spirit-levels for placing the plate, and the telescope, when at the top of the semicircle, in a horizontal direction; in subservience to which, the tripod upon which the instrument stands has four screws, &c. A more particular description of this instrument, in its most improved state, would scarcely be intelligible to a learner, without seeing and using it; and it is therefore omitted here.

The CIRCUMFERENTER is a circle, on the centre of which a large compass; and the circumference is divided, not only into points and quarters, but also into degrees and parts of a degree. An index or two is moveable about the centre. Its use is the same with that of the theodolite; only, when using it, greater reliance is placed upon the compass. It is chiefly used for surveying mines.

Large LEVELS, with telescopic sights, are often requisite for finding the elevation of one place above another in feet, &c. And the surveyor ought also to be possessed of several pocketlevels, to be applied when occasion requires them.

Each of the indices of these instruments has a NONIUS, for enabling the artist to read off minutes. The nonius is a scale on which the number of divisions is greater by one than the number in the same space upon the arc. If the nonius occupy the space of 29 divisions on the arc, it is divided into 30 equal parts, by which means each division will exceed one on the nonius by of a division on the arc; so that, by moving forward the index of a division of the arc, the

a compass is fastened to one of the sides of the table. There is a loose index to be used with it, having a telescope placed parallel to its fiducial side; and there are several plane scales upon the index, for laying down the measured distances. A sheet of paper, moistened equally with a sponge, is spread upon the table, and the frame pressed down upon it to keep it fixed. The paper will become smooth when it is dry, and it will then be fit for drawing the plan upon.

An angle may be measured with the plane-table, by placing that side of the frame uppermost which has degrees on it, and proceeding as with the theodolite. Or the angle may be draw on the table, by directing the index to marks in the sides of the angle in the field; and, in like manner, a given angle may be formed in the field by the table. Also, a perpendi cular may be drawn in the field with it, by placing the centre of the instrument at the given point, and turning it, till the index, while cutting the same divisions on opposite sides of the frame, is in the direction of the given line: then, if the inder be made to cut similar divisions on the other sides of the table, it will give the direction of the perpendicular.

OF HEIGHTS AND DISTANCES.

PROB. VIII. To find the height of an object A. when the point B on the level ground, directly below it, is accessible.

On the level ground measure any distance BC in a straight line, and at C take the angle of elevation ACB with a quadrant.

1. Suppose BC 236 feet, and ACB 35° 48'. In the triangle ABC, right-angled at B, are given BC 236, and ACB 35° 48'. To find AB.

B

Ans. R tan. C :: CB: BA 170-208 feet. NOTE. The height thus obtained is that above the level of the eye of the observer, and must be increased by the height of the eye, to have its height above the level ground. The same is to be done in all the observations on heights.

2. From the bottom of a steeple I measured upon a level plane a straight line 136 feet, and at its extremity I took the elevation of the top of the steeple 47° 25'. Required the height of the steeple. Ans. 147.98 feet.

3. The elevation of a wall, taken from the edge of the ditch 18 feet wide, was 62° 40'. Required the height of the wall, and the length of a ladder to reach the top of it.

Ans. Height 34-824, ladder 39-2014 feet.

4. At 85 feet from the bottom of a tower, the angle of its elevation was 52° 30'. Required its altitude.

Ans. 110-744 feet.

5. Near the bottom of a hill I took the elevation of its top 54° 40′, and the altitude of the hill was 1156 feet. Required the distance of my station from its top. Ans. 1417-01 feet.

PROB. IX. From the top of a known height AB, to find the distance of an object C, on the plane below.

Take the angle of depression CAD; then, in the triangle ABC, right-angled at B, are given AB, suppose 83 feet, and the angle ACB =DAC, suppose 23° 37'. To find AC or BC.

D

B

Ans. Sin. C: R:: AB: AC 207.181 feet, and tan, C: R ::AB: BC 189.829.

feet, and the Required the

NOTE. If AC be given, AB and BC may be found from it. 2. Let the sloping side of a hill AC be 268 angle of depression at its top DAC, be 33° 45'. base BC, and its perpendicular height AB.

Ans. BC 222-834, AB 148.893 feet. 3. From the top of a mast 80 feet high, the angle of depression of another ship's hull was 20°. Required their distance. Ans. 219.798 feet. 4. From the top of a tower 120 feet high I took the depression of two trees 57° and 25° 30'. Required their distances from the tower and from each other.

Ans. 77.93 feet, and 251.58 feet, and 173.65 feet. 5. Suppose the mean semidiameter of the sun subtends at the earth an angle of 16′ 7′′; what is his distance from the earth? Ans. 213.2379 semidiameters. 6. From the top of a lighthouse 110 feet high I observed two ships in a straight line from it, and took the angles of depression of their hulls 56° 44′ and 18° 26'. Required their distance from the lighthouse.

Ans. 72.165 feet, and 330·032 feet.

PROB. X. To measure an inaccessible height AB.

On the level ground measure any distance CD, in a straight line towards the height, and at C and D take the angles of elevation ACB and ADB; their difference is CAD. Let CD be 248 yards,

D

ACB 23° 30', ADB 37° 24'; then CAD is 13° 54'.

Ans. Sin. CAD: sin. ACD :: CD: DA, and R : sin. D:: DA: AB 250-026 yards; that is, sin. C x sin. D x CD÷ sin. (D - C) = AB.

Or the difference of the natural cotangents of C and D is to the radius as CD to AB.

2. Sailing in a boat, a hill was observed, and the elevation of its top above the level of the sea was 27° 38'. After sailing 540 fathoms, each 5 feet, directly towards the hill, the eleva tion of its top was 35° 28'. Required the height of the hill above the level of the sea. Ans. 1066-26 fathoms 3. The elevation of a hill at the bottom of it was 46°, and at 100 yards distance 31°. Required the height of it. Ans. 143 145 yards.

4. The angle of elevation of a tower was 26° 30′, and, 75 yards nearer to it, the elevation was 51° 26'. Required its height and distance. Ans. Height 6197, dist. 49-294 yards. 5. Measured 149 yards towards a hill, and at the extre mities of the line the elevations of its top were 29° 17′ and 39° 25'. Required its height. Ans. 263-02 yards.

PROB. XI. To measure a height which has no level ground before it.

G

Take two stations C and D, in a vertical plane, and measure CD, and at C take the elevation of D above C, viz. GCD 31° 26′, and the elevations or depressions of the top and bottom of the height, viz. ACF 53° 26', and BCF 18° 32', and at D take the elevation of the top ADE 22° 30, and let CD be 286 feet. Since EDC = : DCG, ADC=ADE+ DCG = 53° 56, and DAC = ACF - ADE. The triangle ADC has two angles; and the side CD given, to find AC. Then in the triangle ACB are given ACB = ACFBCF, and B=90° ±BCF, and AC; to find AB. Ans. Sin. DAC: sin. ADC :: DC: CA, and sin. B : sin. ACB :: CA : AB 271:39.

sum

NOTE 1. If DE be above A, the angle DAC is the of ACF and ADE; otherwise it is their difference. Also, in this case ADC is the difference of DCG and ADE; otherwise it is their sum. Also, when F is below B, the angle ACB is the difference of ACF and BCF; otherwise it is their

sum.

Nore 2. If the stations C and D cannot be conveniently taken in a vertical plane, they may be taken anywhere, and then the angles ADC and ACD must be measured with a sextant, and the triangle ACD will give the side AC.