directed to Y, and the two telescopes are then clamped, thus making one observation of the required angle: its value however cannot immediately be obtained as the lower side of the circle is not graduated. The instrument is then turned onwards in its present plane until Qq be directed to X, when Pp will fall into the position Rr; and the instrument is then fixed in position. The telescope Pp is then unclamped, removed from its position Rr, directed to Y, and clamped. The reading on the circle will now be double the value of the angle. The whole circle must then be turned until Pp points to X, and the previous process repeated any even number of times, the circle being always turned to the right through the arc pq, and the two telescopes alternately to the left through the arc rq, or double the former. The final reading is divided by the number of observations to obtain the mean.

The angle having been observed in an oblique plane, the altitudes or elevations of the two objects are then observed or obtained, and the oblique angle is reduced to the horizon by tables prepared for this purpose, or by calculation. (See pp. 52 and 59 for formulæ and example.)

The advantages of this principle of repetition are obtained in the Reflecting Repeating Circle of Troughton, an exceedingly convenient instrument for measuring astronomical altitudes and distances in arc; it must however be noticed that there is always a constant error which cannot be removed by any number of repetitions.

The Surveying- or Box-Sextant is used by surveyors, as a light substitute for a theodolite, for obtaining a few horizontal angles in a small survey, or for one or two casual altitudes, such as heights of buildings, and stationmarks; but it is not a favourite instrument with them,

owing to the disadvantages inseparable from reflecting instruments and the necessary reduction for obtaining horizontal angles from the oblique angles observed; although its accuracy of reading in comparison with magnetic instruments is greatly in its favour.

The principle of this instrument being, like that of the nautical sextant, so well known, it is only necessary to mention its adjustments and tests. They are:

1. That the index-glass shall be truly at right angles to the plane of the instrument.

2. That the horizon-glass shall be truly at right angles to the plane of the instrument.

3. That the line of collimation of the telescope, or line of sight, shall be truly parallel to the plane of the instrument.

The index-glass being a permanent fixture, any error in its position should be rectified by an optician; but its adjustment may be tested by setting the index to 60°, and looking at the reflection of the limb of the instrument in the index-glass, which should appear continuous with the real limb.

The position of the horizon-glass may be corrected by means of the adjusting screws. It can be tested by clamping the index at zero, and observing on a distant well-defined object, whose real and reflected image should coincide or appear perfectly continuous; if it should not be so, a slight motion given to the tangent screw of the index will effect this; the index error resulting should then be noted, and applied to all subsequent observations; this is negative in sign when its amount has to be added as a correction to an angle, and is called positive when its amount has to be subtracted.

The telescope-bearing or attachment being a perma

nent fixture, the correctness of its adjustment may be tested by observation on two objects subtending a right angle and causing the plane of the sextant to revolve. slowly around the axis of observation on the reflected object; this should not disturb the coincidence of the two images.

In surveying sextants there is frequently a supplementary arc and small mirror below the index glass for observing angles above 120°; the size of the brass box constituting the frame of the instrument is generally three or four inches in diameter; the latter is to be preferred, it reads to thirty minutes or to twenty seconds with the aid of the vernier, and with well-defined objects, and by using the small telescope, which can generally be dispensed with, can be employed to a range of threequarters of a mile. The manipulation of these, as of all reflecting instruments, requires some dexterity and practice; the reduction of oblique angles may often be dispensed with by judicious use of a plumb-line.

Troughton's reflecting circle is the best instrument of this type in use in England; its principle and its glass-adjustments are similar to those of the sextant; but as it gives readings to seconds, and has three branches to its index, and admits of reading backwards and forwards, and repetition, the errors of the simple sextant become reduced in it to one-sixth; it will also measure an angle up to 150°.

Its collimation adjustment, or the setting of the line of collimation parallel to the plane of the instrument, is performed with two screws, which also fasten the collar of attachment of the telescope to the upright stem on which it is mounted. The collimation is tested by observing on the sun and the moon, when they are about

130° or more apart, and bringing their limbs in contact at the outside of the wire next to the circle, and afterwards examining under similar conditions at the outside of the other wire. If it answers this test the adjustment is correct.

III. Magnetic Instruments.-The principle applied in using compass bearings for directions in route and small traverse surveys, and in mining operations, and in using differences of bearings as angles for the triangulation and detail of small surveys of any sort, requires no explanation. The forms of compass used are the pivoted needle, and the suspended needle; the former being the most common and convenient generally, while the latter is more accurate and suited to scientific purposes. The graduated arc, on which the needle reads, is, for surveying purposes, divided into degrees and parts from o° to 360°, in such a way that a reading of 90° may be the bearing of an object lying to the magnetic east of the observer. Sights for obtaining the accurate direction of any distant points, and telescopes, may be applied in very various forms. A magnifying prism is frequently used for sighting and reading the bearing simultaneously, in which case the graduations and numbering are reversed in order; and in pivoted compasses a stop or lever may be used for checking the vibration of the needle or throwing it off its pivot to prevent wear. The forms of case or compass box used are also very various. Formerly compasses were rigidly attached to, or formed part of, most theodolites and surveying levels; but as they were seldom used with them, and added to their weight, they are now made to be readily detached or used independently when required. The stand used for

a surveying or miner's compass or circumferenter is generally a light tripod; in the latter case, low for convenience in galleries: in soft ground a spiked rod about five feet long surmounted by a square flat top or resting table is often preferred; and again the compass is sometimes fixed permanently on the top of a surveyor's crosshead, which is used on a spiked rod; while in other instances stands are entirely dispensed with, the box or compass being held in the hand as level as possible.

The four-inch box-compass gives readings to a half or to a third of a degree, larger compasses to a quarter; the many possible causes of disturbance render a further exactitude in reading unnecessary. The neighbourhood of iron, whether in the form of lodes or disseminated iron in surrounding rocks, an iron bridge or structure, or keys and instruments carried about the person, may vitiate observed bearings to a very important extent. Magnetic changes and storms also affect bearings; while the diurnal change itself has been known to amount to as much as a quarter of a degree. Under these circumstances a three-inch compass may be sufficiently accurate.

When a higher degree of accuracy is required, the manifest objections to the supported pivot are obviated by adopting Gauss's principle of suspension, which was applied to surveying or portable purposes by Porro. In Porro's compass the needle is suspended in a tube by a silk thread, which also is in a tube at right angles to the former; a reflector partaking of the movement of the needle is fixed at right angles to it; a half mirror superimposing a scale, seen through the transparent portion of it, shows the reflection of a permanent wire or fine thread, whose oscillation and final position determines