entered on the field-book. The 'bisection' of the object and the 'reading' should be verified by observation before unclamping the vertical limb or the upper plate of the horizontal limb. After cautiously unclamping these parts or members, the telescope should be directed to a well-defined object having the vertical of its centre in the right-hand line of the angle, by turning the instrument gently on its vertical axis without disturbing the clamp of the lower plate or the other adjustments. The upper plate should be reclamped, and the ‘bisection' made by working the slow-motion screw to this plate; then the 'reading' and verification should be made as above directed. The measure of the angle, or the arc passed over by the zero of the vernier, will be found by taking the difference of readings on a vernier, if the reading for the right-hand object be greater than that for the left-hand object. If the reading for the righthand object be less than the reading for the left-hand object, 360° less by the latter or greater reading must be added to the former to find the measure of the contained angle. A mean of the measured angles thus reduced for each vernier will be more correct than the angle reduced from readings on a single vernier. If it be necessary to obtain the angle with greater accuracy, the errors due to observation and unequal graduations should be removed by measuring the 'arc' or angle several times on different parts of the graduated circle, and taking a mean, first of the measured angle on each arc, reduced from the readings of the verniers as above; and secondly, by taking a mean of the angles so corrected, to obtain the true measure of the angle.

Angles in the Vertical Plane.

For this purpose the adjustments and observation should be made as above directed, and the vernier to the vertical limb read and entered for elevation or depression, as the case may be. If the observed object be above the level of the theodolite, the angle will be an angle of elevation, but if below that level, it will be an angle of depression. If the vernier reads zero when the bubble of the level tube shall be at the middle of its run, and the line of collimation in the horizontal plane, the reading, for an observed object, will give the vertical angle. If the vernier be not so adjusted as to read zero when the line of collimation shall be adjusted in the horizontal plane, the reading in this case will have an index error, which should be noted and properly applied to the readings for vertical angles to find the correct measure of these angles.

THE VERNIER.

The vernier is a contrivance for reading off more accurately than is practicable with a simple pointer. The graduations or divisions on a line, whether circular or straight, must be of uniform equality to admit of reading with a vernier scale minute subdivisions of a graduation. The graduations of the vernier scale must be also of uniform equality, and so spaced on the graduated circle common to both scales, that an odd number of divisions on the primary scale shall be equal to the next even number and extreme division on the vernier scale. It may be seen by a little consideration, and an inspection of the illustration (fig. 84), that a division of the primary scale will be divided by the zero line of the vernier scale into

as many equal parts as there are divisions in this scale,

30

Fig. 84.

Vernier Scale

5 19 15 20

40

50

Primary Scale

by the successive coincidence of the graduations on the latter with those of the former scale.

This may be stated, generally, as follows:-Let n represent the number of divi

sions on the vernier scale, then n−1 will represent the number on the primary scale equal to n. Let p' and p represent a graduation on the primary and vernier scales respectively; then—

np=(n−1) p', and p' —p=P'.

n

19

200 scale.

inches, the extent of a division on the vernier

PRISMATIC COMPASS.

The prismatic compass is an improvement on the wellknown miner's dial. With the improved instrument, the 'bisection' of the object and the graduations on the card, expressing the magnetic angle, may be observed at the same time. The instrument consists of a light brass cylindrical box, open at the top, having a hinged sight vane at one end of a diameter, and a glass prism attached to a sliding piece of brass at the opposite end of the diameter. The illustration (fig. 85) shows the arrangement of the parts of the instrument; s is the sight vane, p the prism, b the box, and c a graduated compass card, securely attached to a magnetic bar,

suitably supported on the fine point of a strong central pin, securely fixed on the bottom of the box; m is a hinged plane reflector, revolving on the same axis as s, so as to reflect in the diametric plane sp. By this arrangement the vertical range of the instrument is considerably increased. There is an eye hole and vane wire slot in the back of the slide piece, to which the

prism p is securely attached.

A fine platina wire is fitted in the sight vane so as to be in a diametric plane perpendicular to the face of the instrument, in which the centre of the eye hole shall also be. In the lower part of the box there is a damper spring for checking the oscillations of the card and supporting it above the pin point, when angles are not being measured with the instrument.

In measuring angles with the prismatic compass, the instrument, held in the hand or supported on a stand, should be so held as to be approximately horizontal, that the diametric sight vane plane shall be approximately in a

vertical plane. In this position the eye should be applied to the eye hole, and the piece p raised or lowered till the graduations on the card and the platina wire shall be in apparent contact. The line of sight being in the diametric plane, objects in any given line bisected by the wire will be in this plane, the magnetic angle of which is expressed by reading the graduation in apparent contact with the wire. The zero of the graduations is adjusted by the maker to be in the magnetic meridian-magnetic vertical plane.

THE BOX-SEXTANT.

This instrument, as usually made, is of cylindrical form. It is about three inches in diameter, and one and a half

Quadrant, from which it differs only in the construction of its parts. The following brief explanation of the principles of construction will assist in explaining and making the adjustments easily understood.

It is shown in catoptrics that a ray from A (fig. 87) will be reflected to a along the lines, Ab, bc, cd making the angle Aa c = 2 × b a' c in the same plane. The planes of the reflectors bc, and the axes on which they revolve, should be perpendicular to the plane of reflection. The angles made by the ray on each side of b b', or c c'— the normals to the mirrors b, c,-are equal. These equal