NOTE.-To construct the above table, suppose the base AB, in the preceding triangle to be 99.5, and the hypotenuse AC=100; then by Trig. as 100: 1 :: 99.5 995, the nat. co-sine of the angle BAC-5° 44'. In the same manner, the rest of the angles are obtained, by different operations, accounting the base 99 in finding the second angle, 98 in finding the third, &c.

Quadrants differ in construction, but are similar in principle. The instrument is for taking vertical angles. In surveying, it has been nearly superseded by the semicircle of the theodolite and spiritlevel, and in astronomy by the mural circle.

1. The annexed diagram ABC represents a quadrant of the simplest mechanism. It is suspended from C either by the hand or top of a staff, the plumb-line CG being the index. The sights through which the object at D is viewed are two small holes drilled in the brass plates a and b on the radius AC.

2. Another example is similar to the last, a telescope being substituted for the brass sights on AC.

3. A third kind has a spirit-level on the radius BC, which is placed horizontally on CE; the quadrant or are BA is towards D; and a telescope is affixed longitudinally on the revolving index CG, through which the object is viewed.

4. A fourth construction has a spirit-level on AC, which is placed horizontally on CE; BC is towards the object; and the telescope is now fixed tangentially upon the revolving index at the cireumference of the are, the line of collimation thus forming a tangent.

To take the Altitude of a Hill with the Quadrant.

Upon the top of the hill fix an object, exactly as high as your eye will be from the ground, in taking the observation. At the bottom of the hill fix the quadrant staff perpendicularly to the horizon, which may be easily done by means of the plummet. Then with one eye at A, the other being closed, look through the sights, turning the quadrant until you perceive the object at D; so will the arc BG, cut off by the plumb-line CG, be the measure of the angle DCE, or the altitude of the hill, in degrees, above the horizon.

To take the Altitude of a Steeple, &c. with the Quadrant.

Screw the quadrant fast to its staff, so that the plummet may hang exactly at 45° when the staff is perpendicular to the horizon. Then move the staff backward or forward (always keeping it perpendicular), until you can see the top of the object through both the sights. Measure the distance between the bottom of the staff and that of the object, which being added to the height of your eye, will give the altitude required.

METHOD II.

As the foregoing method of reducing hypotenusal to horizontal lines can only be applied, with accuracy, when hills are of a regular slope, surveyors, in general, elevate the chain, as they ascend or descend a hill, in order to preserve the horizontal line.

Suppose the lines AB and BC to represent the acclivity and de'clivity of an irregular hill, it is required to measure them, and to preserve the horizontal line AC.

From A stretch the chain toward B, and suppose it to reach to a; the same extent, upon the base, will evidently reach from A to g; and a perpendicular erected from g will intersect the line AB in d; hence the distance Ad, upon the hypotenuse, will make one chain upon the base. At A, stick your offset-staff into the ground, perpendicularly to the horizon, and let your assistant hold the chain, suppose at the twenty-fifth link, close to the surface of the hill, as at b; at the same time you must elevate the end of the chain to c, forming the horizontal line cb; then move forward to b, at which place fix your staff again, as before. Let your assistant hold the fiftieth link at p, while you elevate the twenty-fifth to n, forming the horizontal line np. Again, fixing your staff at p, elevate the fiftieth link to m, while your assistant holds the seventy-fifth at e. Lastly, put down the staff at e, and elevate the seventy-fifth link to r, while the hundredth is held by your assistant at d. There he must put down an arrow, and thus you must proceed until you arrive at B, where you will have obtained the horizontal line AD.

In descending from B to C, let your assistant hold one end of the chain at B, while you elevate, suppose, the fiftieth link to n, forming the horizontal line Bn; then fix the staff at a, perpendicularly to the horizon, and touching the chain at n. Next, let your assistant hold the fiftieth link at a, while you elevate the hundredth to m, and put down the staff at r, as before. In this manner, having arrived at c, you will have obtained the horizontal line DC, which being added to AD, will give the base or horizontal line AC, as required.

NOTE 1. If you wish to obtain the hypotenusal, as well as the horizontal line, divide your field-book into four columns, in one of which you must enter the number of links between a and d, &c., which being added to the horizontal will give the hypotenusal line.

2. When the ascent or descent of a hill is great, you will not be able to elevate more than 10 or 15 links of the chain at one time; for, in such cases, if you attempt to elevate 20 or 30 links, you will find that the perpendiculars ac, bn, &c. will exceed your own height before you can form the horizontal lines cb, np, &c. (See the last Figure.)

METHOD III.

Hypotenusal lines may likewise be reduced to horizontal ones during the survey by quadrants constructed for the purpose. Of these two examples are subjoined, viz. King's Quadrant and Nesbit's Quadrant.

KING'S QUADRANT.

Description.

The quadrant is fitted to a wooden square, which slides upon an offset-staff, and may be fixed at any height by means of a screw, which draws in the diagonal of the staff, thus embracing the four sides, and keeping the limb of the square perpendicular to the staff. The staff should be pointed with iron to prevent wear. When the staff is fixed in the ground on the station-line, the square answers the purpose of a cross staff, and may, if desired, have sights fitted to it. The quadrant is three inches radius, of brass, is furnished with a spirit-level, and is fastened to a limb of the square by means of a

screw.

6

When the several lines on the limb of the quadrant have their first division coincident with their respective index-divisions, the axis of the level is parallel to the staff.

The first line next the edge of the quadrant is numbered from right to left, and is divided into 100 parts, showing the number of links in the horizontal line which are completed in 100 links on the hypotenusal line, and in proportion for any smaller number.

The second, or middlemost line, shows the number of links the chain is to be drawn forward, to render the hypotenusal measure the same as the horizontal.

The third, or uppermost line, gives the perpendicular height, when the horizontal line is equal to 100.'

K

Use.

'Lay the staff along the chain-line on the ground, so that the plane of the quadrant may be upright; then move the quadrant till the bubble stands in the middle, and on the several lines you will have-1. The horizontal length gone forward in that chain; 2. The links to be drawn forward to complete the horizontal chain; 3. The perpendicular height or descent made in going forward one horizontal chain.

The first two lines are of the utmost importance in surveying land, which cannot possibly be planned with any degree of accuracy without having the horizontal line; and this is not to be obtained by any instrument in use without much loss of time to the surveyor; whilst with this he has only to lay his staff along the ground, and set the quadrant till the bubble is in the middle of the space, which is very soon performed. And he saves by it more time in plotting his survey than he can lose in the field; for as he completes the horizontal chain as he goes forward, the offsets are always in their right places, and the field-book being kept by horizontal measure, his lines are sure to close.

'If the superficial content, by the hypotenusal measure, be required for any particular purpose, he has that likewise, by entering in the margin of the field-book the links drawn forward in each chain, having thus the hypotenusal and horizontal length of every line.

The third line, which is the perpendicular height, may be used with success in finding the height of timber. Thus, measure with a tape of 100 feet the surface of the ground from the root of the tree, and find by the second line how much the tape is to be drawn forward to complete the distance of 100 horizontal feet; and the line of perpendiculars shows how many feet the foot of the tree is above or below the place where the 100 feet distance is completed. Then, inverting the quadrant by means of sights fixed on the staff, place the staff in such a position as to point to that part of the tree whose height you want; and sliding the quadrant till the bubble stands level, you will have on the line of perpendiculars on the quadrant the height of that part of the tree above the level of the place where you are; to which add or substract the perpendicular height of the place from the foot of the tree, and you obtain the height required.'