directed to points below the level. Plate 14, fig. 1, is a drawing of a very simple and useful instrument of this kind. ABC is a quadrant mounted upon an axis and pedestal by means of the axis, it may be immediately placed in any vertical position, and the pedestal being moveable in the axis of the circle EF, serves to place it in the direction of any azimuth, or towards any point of the compass. The limb AB, is divided into degrees and halves numbered from A; and upon the radius BC, are fixed two sights, of which B is perforated with a small hole, and is provided with a dark glass to defend the eye from the sun's light; and the other sight Z has a larger hole furnished with cross wires, and also a smaller, which is of use to take the sun's altitude by the projection of the bright image of that luminary upon the opposite sight. From the centre C, hangs a plumb-line, CP. The horizontal circle FE, is divided into four quadrants of 90 degrees; and an arm E, connected with the pedestal, moves along the limb, and consequently shews the position of the plane of the quadrant, as will hereafter be more minutely explained. Lastly, the screws G, H, I, render it very easy to set the whole instrument steadily and accurately in its proper position, notwithstanding any irregularity in the table or stand it may be placed upon.

The rationale of this instrument is very clear and obvious. It is used to measure the angular distance of any body, or appearance, either from the zenith or point immediately above our heads, or from the horizon or level. The plumb-line CP, if continued

upwards from C, would be directed to the zenith Z; and the line CL, supposed to be drawn from the centre of the quadrant to an object L, will form an angle LCZ, which is the zenith distance, and is equal to the angle BCP, formed between the opposite parts of the same lines. We see, therefore, that the degrees on the arc, comprehended on the limb of the quadrant, between the plumb-line and the extremities next the eye, measure the angle of zenith distance.

Again, the line CK, forming a right angle with the perpendicular CZ, is level or horizontal; the angle LCK must therefore be the altitude or elevation of L above the horizon; and this last angle must be equal to the angle measured between the plumb-line and. the end A farthest from the eye; because both these are equal to the quantity which would be left, after taking the zenith distance from a right angle, or the whole quadrant.

The determination of the altitude or zenith dis, tance of an object is not sufficient to ascertain its place, because the object may be placed in any direction with respect to azimuth, or the points of the com. pass, without increase or diminution of its altitude, Hence it is, that an horizontal graduated circle is a necessary addition to a quadrant, which is not intended to be always used in the same plane. The bearing or position of an object relative to the cardinal points; together with the altitude, is sufficient to ascertain the place of any object or phenomenon.

After this short account of the general principles of the quadrant, I shall proceed to shew some of the leading problems resolved by it.

PROBLEM I. To adjust the quadrant for observa

tion.

The quadrant is adjusted for observation when its plane continues perpendicular to the horizon in all positions of the line of sight. To effect this, bring the index to 90° on the horizontal circle, and turn one or both of the screws which are fixed opposite 60°, till the plumb-line lightly touches the plane of the quadrant; then turn the index to 0°, and make the same adjustment by means of the screw at 0°, and the quadrant is ready for observation.

Or otherwise; set the index at 0°, and observe the degree marked by the plumb-line on the limb; then turn the index to the other o°, which is diametrically opposite, and observe the degree marked by the plumb-line: if it be the same as before, there will be no occasion to alter the screws at 60°; but if otherwise, one or both of those screws must be turned till the plumb-line intersects the middle deg. or part, between the two. After this operation, the degree marked by the plumb-line must be observed, as before, by setting the index at both the 90', and the adjustment of the plumb-line to the middle distance must be made by the screw at 0°, taking care not to touch the other screws.

The latter method of adjustment, being more accurate in practice, may be used after the former. The larger or more expensive instruments have apparatus for setting the axis of motion at right

angles to the planes of the horizontal circle and quadrant, the line of sight or collimation parallel to the radius passing through 90°, &c. &c. In small instruments, these adjustments are made by the workmen.

INTRODUCTORY PROBLEMS.

PROBLEM II. To find the distance of an object on the earth by observations made from two stations on the same level.

OBSERVATIONS.

Choose two stations, between which the ground is level, and place a visible mark on each. The distance between them ought not to be less than the seventh or eighth part of the estimated distance of the objects, and neither station ought to be considerably nearer the object than the other. Measure the distance between the stations, by means of measuring poles, a chain, or a piece of stretched cord. From one station direct the quadrant to the object, by looking through the hole in one sight, and moving the upright axis about, till the object is seen through the hole in the other, exactly at the intersection of the cross wires. Observe the degrees and parts shewn by the index on the horizontal circle, then direct the quadrant in the same manner to the mark of the other station, and observe the degrees and parts shewn by the index. The number of degrees and parts intercepted between this and the former position of the index is the an

gle at the first station. The same operations repeated at the second station will give the angle at that station.

Thus, let F, plate 15, fig. 1, be the object, A, B, the two stations 880 feet distant from each other; the angle observed at A found to be 83° 45', that observed at B 85° 15'.

Solution. Take the sum of the two observed angles from 180°, and the remainder will be the angle under which the two station marks would be seen from the object. Let F be the object, A and B the two stations, the angle at A found by observation to be 83° 45', that at B 85° 15', the sum of these two angles is 169°, which taken from 180° gives 11° for the value of angle F.

Then as the sine of angle F,

Solution of the problem by protraction. From a scale of equal parts, lay down a right line to represent the measured base. By means of the protractor, or by the line of chords, draw a line from each extremity of the base, forming angles equal to those actually observed; continue these lines till they intersect.