but if not, unscrew the two screws which carry the frame of the cross wires, and move the frame till the intersection appears to lie on a new object, half way between the object first observed, and that to which the wires are applied in the last position. Return the semicircle of altitude to its original position: if the intersection of the wires be then found to be on the object to which they were last directed, the line of sight is truly adjusted; but if not, the frame must be again altered as before; and the same general operation must be repeated, till the cross wires in both positions apply to the same object.

Besides this adjustment of the centre of intersection, it is necessary that one of the wires should be in the plane of the declination semicircle, and the other at right angles to that plane. As the wires are fixed at right angles to each other, the adjustment of one of them will be sufficient. For this purpose, observe any small object on one of the wires; if it be the vertical wire, move the index of the semicircle of declination; or, if the other, move the last mentioned semicircle on the axis of the equatorial circle. In either case, the object will coincide with the wire during its motion, if the position be right; if not, alter that position, taking care not to Jisplace the centre from its adjustment,

To adjust the piece which carries the hole for forming the solar spot, direct the sights to the sun, so that the centre of the luminous circle, formed by the aperture which carries the cross wires, may fall

precisely on the upper sight-hole. Then move the frame, with the small perforation, till the solar spot falls exactly on the lower sight-hole.

Thirdly, to find the correction to be applied to observations by the semicircle of altitude. Set the nonius on the declination semicircle to O, and the nonius on the horary circle to XII; direct the sights to any fixed and distant object, by moving the horizontal circle and semicircle of altitude, and nothing else; note the degree and minute of altitude or depression; reverse the declination semicircle, by di. recting the nonius on the horary circle to the opposite XII; direct the sight again to the same object, by means of the horizontal circle and semicircle of altitude, as before. If its altitude or depression be the same as was observed in the other position, no correction will be required; but if otherwise, half the difference of the two angles is the correction to be added to all observations or rectifications made with that quadrant, or half of the semicircle, which shew the least angle; or to be subtracted from all observations or rectifications made with the other quadrant, or half.

When the level and cross wires are once truly set, they will preserve their adjustment a long time, if not deranged by violence; and the correction to be applied to the semicircle of altitude is a constant quantity.

PROBLEM XIX. To measure angles either of azimuth altitude, or depression.

Set the middle mark of the nonius on the declination at O, and fix it by means of the milled screw behind. Set the horary circle at XII on the equator, and the instrument (previously adjusted) is ready for observation. Then if the sights be directed successively to any two objects, the degrees and minutes contained between the two positions of the nonius, on the limb of the horizontal circle, will shew the horizontal angle in the same manner as has been described at prob. ii. of the quadrant. And likewise, if the sights be directed to any object, by moving the horizontal circle and semicircle of altitude, the degree and minute marked by the nonius on the last-mentioned semicircle will be the angle of altitude, if on the quadrant or part next the eye; or of depression, if on the remoter quadrant.

Remark. It is proper in this place to describe the nature and use of the admirable contrivance commonly called a nonius. It depends on the simple circumstance, that if any line be divided into equal parts, the length of each part will be greater, the fewer the divisions; and contrariwise, it will be less in proportion as those divisions are more numeThus it may be observed, that the distance between the two extreme strokes on the nonius, in the equatorial before us, is exactly equal to eleven degrees on the limb, but that it is divided into

rous.

twelve equal parts. Each of these last parts will therefore be shorter than the degree in the proportion of 11 to 12; that is to say, it will be onetwelfth part, or five minutes, shorter. Consequently, if the middle stroke be set precisely opposite to any degree, the relative position of the nonius and the limb must be altered five minutes of a degree, before either of the two adjacent strokes next the middle, on the nonius, can be brought to coincide with the nearest stroke of a degree; and so likewise the second strokes on the nonius will require a change of ten minutes, the third of fifteen, and so forth to thirty, when the middle line of the nonius will be seen to be equidistant between two of the strokes on the limb; after which, the lines on the opposite side of the nonius will coincide in succession with the strokes on the limb.

It is clear from this, that whenever the middle stroke of the nonius does not stand precisely opposite to any degree, the odd minutes, or distance between it and the degree immediately preceding, may be known by the number of the strokes on the nonius which coincides with any of the strokes on the limb. It must be observed, however, that as the degrees in several quadrants are reckoned in opposite diretions, so likewise the nonius has two sets of numbers for the use of which it need only be remembered, that they always begin from the middle, and go to 30 minutes, and thence from the opposite 30 minutes in the same direction to the middle; and

that they must always be reckoned in the opposite direction to the degrees on the limb.*

PROBLEM XX. To find the distance of an object on the earth, by observations made at two stations.

This may be done by measuring a base line and the horizontal angles, and proceeding as directed at Problem ii. But as the equatorial measures angles of depression as well as elevation, the stations may not only be on the same level, but may be vertically the one above the other. For example, if the altitude of any object be taken from a lower window of any building, and its depression from a window immediately above, and the distance of the two stations of the instrument be accurately measured. Then,

As the sine of the sum of the angles of altitude and depression, (or of the difference, if both be altitude or both depression)

Is to the sine of the angle at the upper station;

So is the distance between the stations

To the distance of the object from the lower

station.

* In the instrument they must be read in the opposite direction; but when the nonius plate has its divisions fewer than the number of parts on the limb to which it is equal, they coincide successively in the same direction as that of the motion of the index.