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the difference of the sums of the readings in the two cases, divided by the whole number of readings, will be the index error; while the sum of all the readings, divided by their number, will be the sun's diameter.

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The readings on the arc of excess being less than those on the arc of the instrument, the index error, 2' 49", is to be subtracted from all the readings of the instrument.

NOTE. In taking off the readings on the arc of excess, the vernier must be read backwards; that is, the division read off on the limb, being the division next to the left of the zero of the vernier, the divisions of the vernier to be added must be reckoned from the other end of the vernier to the division coinciding with a division upon the limb; or the reading of the vernier forwards, according to the usual method, may be subtracted from 10', the limb being divided to 10', and the remainder added to the reading of the division upon the limb next to the left of the zero of the vernier, as before.

The manner of observing with the sextant has been already explained, when treating of the pocket sextant (p. 120).

TROUGHTON'S REFLECTING CIRCLE.

In this instrument, which is the same in principle as the sextant, the limb is a complete circle, L. LL. It has three verniers, v v v, one of which is furnished with the clamp and tangent screw, s s, for regulating the contacts; and the verniers are read by a magnifier, M, which may be applied successively to all the verniers. In the middle of the frame, and attached to it by a broad base or flanch, is a hollow center, upwards of two inches long, in which an axis revolves. The triple vernier bar, I II, is attached at one end of the axis, and the index glass at the other, so that both turn together, but on opposite sides of the instrument. A secondary frame, B B, carries the telescope, T, the horizon glass, and the dark glasses. HH are two handles, one of them bent, and passing round to the center of the instrument on the other side; and there is a third handle,

which can be screwed on perpendicular to the plane of the instrument, either into the handle at c, or upon the other side of the instrument, at its center. The adjustments and manner of observing with the instrument are explained by the inventor, Mr. Troughton, as follows:

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Directions for observing with Troughton's Reflecting Circle.-" Prepare the instrument for observation by screwing the telescope into its place, adjusting the drawer to focus, and the wires parallel to the plane, exactly as you do with a sextant: also set the index forwards to the rough distance of the sun and moon, or moon and star; and, holding the circle by the short handle, direct the telescope to the fainter object, and make the contact in the usual way. Now read off the degree, minute, and second, by that branch of the index to which the tangent screw is attached; also, the minute and second shown by the other two branches; these give the distance taken on three different sextants; but as yet it is only to be considered as half an observation: what remains to be done, is to complete the whole circle, by measuring that angle on the other three sextants. Therefore set the index back wards nearly to the same distance, and reverse the plane of the instrument, by holding it by the opposite handle, and make the contact as above, and read off as before what is shown on the three several branches of the index. The mean of all six is the true apparent distance, corresponding to the mean of the two times at which the observations were made.

"When the objects are seen very distinctly, so that no doubt whatever remains about the contact in both sights being perfect, the above may safely be relied on as a complete set; but if, from the haziness of the air, too much motion, or any other causes, the observations have been rendered doubtful, it will be advisable to make more: and if, at such times, so many readings should be deemed troublesome, six observations, and six readings, may be conducted in the manner following:-Take three successive sights forwards, exactly as is done with a sextant; only take care to read them off on different

branches of the index. Also make three observations backwards, using the sane caution; a mean of these will be the distance required. When the number of sights taken forwards and backwards are unequal, a mean between the means of those taken backwards and those taken forwards, will be the true angle.

"It need hardly be mentioned, that the shades, or dark glasses, apply like those of a sextant, for making the objects nearly of the same brightness; but it must be insisted on, that the telescope should, on every occasion, be raised or lowered, by its proper screw, for making them perfectly so."

The foregoing instructions for taking distances apply equally for taking altitudes by the sea or artificial horizon, they being no more than distances taken in a vertical plane. Meridian altitudes cannot, however, be taken both backwards and forwards the same day, because there is not time; all, therefore, that can be done is, to observe the altitude one way, and use the index error; but, even here, you have a mean of that altitude, and this error taken on three different sextants. Both at sea and land, where the observer is stationary, the meridian altitude should be observed forwards one day, and backwards the next, and so on alternately from day to day; the mean of latitudes, deduced severally from such observations, will be the true latitude; but in these there should be no application of index error, for that being constant, the result would in some measure be vitiated thereby.

"When both the reflected and direct images require to be darkened, as is the case when the sun's diameter is measured, and when his altitude is taken with an artificial horizon, the attached dark glasses ought not to be used: instead of them, those which apply to the eye-end of the telescope will answer much better; the former having their errors magnified by the power of the telescope, will, in proportion to this power, and those errors, be less distinct than the latter.

"In taking distances, when the position does not vary from the vertical above thirty or forty degrees, the handles which are attached to the circle are generally most conveniently used; but in those which incline more to the horizontal, that handle which screws into a cock on one side, and into the crooked handle on the other, will be found more applicable.

"When the crooked handle happens to be in the way of reading one of the branches of the index, must be removed, for the time, by taking out the finger screw, which fastens it to the body of the circle.

"If it should happen that two of the readings agree with each other very well, and the third differs from them, the discordant one must not on any account be omitted, but a fair mean must always be taken.

"It should be stated, that when the angle is about thirty degrees, neither the distance of the sun and moon, nor an altitude of the sun, with the sea horizon, can be taken backwards; because the dark glasses at that angle prevent the reflected rays of light from falling on the index glass; whence it becomes necessary, when the angle to be taken is quite unknown, to observe forwards first, where the whole range is without interruption; whereas in that backwards you will lose sight of the reflected image about that angle. But in such distances, where the sun is out of the question, and when his

altitude is taken with an artificial horizon (the shade being applied to the end of the telescope), that angle may be measured nearly as well as any other; for the rays incident on the index glass will pass through the transparent half of the horizon glass without much diminution of their brightness. "The advantages of this instrument, when compared with the sextant, are chiefly these: the observations for finding the index error are rendered useless, all knowledge of that being put out of the question, by observing both forwards and backwards. By the same means the errors of the dark glasses are also corrected; for if they increase the angle one way, they must diminish it the other way by the same quantity. This also perfectly corrects the errors of the horizon glass, and those of the index glass very nearly. But what is of still more consequence, the error of the center is perfectly corrected by reading the three branches of the index; while this property, combined with that of observing both ways, probably reduces the errors of dividing to one-sixth part of their simple value. Moreover, angles may be measured as far as one hundred and fifty degrees, consequently the sun's double altitude may be observed when his distance from the zenith is not less than fifteen degrees; at which altitude the head of the observer begins to intercept the rays of light incident on the artificial horizon; and, of course, if a greater angle could be measured, it would be of no use in this respect.

"This instru:nent, in common with the sextant, requires three adjustments. first, the index glass perpendicular to the plane of the circle. This being done by the maker, and not liable to alter, has no direct means applied to the purpose; it is known to be right when, by looking into the index glass, you see that part of the limb which is next you reflected in contact with the opposite side of the limb as one continued arc of a circle: on the contrary, when the arc appears broken where the reflected and direct parts of the limb meet, it is a proof that it wants to be rectified. The second is, to make the horizontal glass perpendicular. This is performed by a capstan screw, at the lower end of the frame of that glass; and is known to be right when by a sweep of the index the reflected image of any object will pass exactly over, or cover the image of that object seen directly. The third adjustment is for making the line of collimation parallel to the plane of the circle. This is performed by two small screws, which also fasten the collar into which the telescope screws to the upright stem on which it is mounted; this is known to be right when the sun and moon, having a distance of one hundred and thirty degrees, or more, their limbs are brought in contact, just at the outside of that wire which is next to the circle, and then examining if it be just the same at the outside of the other wire: its being so is the proof of adjustment."

Another instrument of Troughton's construction upon the principle of the sextant is the dip sector, for measuring the dip of the horizon. Any person who is thoroughly acquainted with the sextant will find no difficulty in using it, after a few words of explanation from the maker.

THE TRANSIT INSTRUMENT.

The reflecting instruments, which we have just described, from their portability and the promptitude and facility with which they may be used in all situations, and upon all occasions, are most useful instruments to the surveyor. The sextant or reflecting circle, with an artificial horizon, and a

good chronometer, forms, in fact, a complete observatory, with which the latitudes and longitudes of places may be determined to a great degree of accuracy; while to the navigator a reflecting instrument is indispensable; all other instruments requiring to be supported upon a stand perfectly at rest*, while the sextant and similar instruments are held in the hand, and perform their duty well on the deck of a rolling. ship. In permanent observations, however, the capital angular instruments are placed permanently in the plane of the meridian, and the measurements sought for by their aid are the exact times at which the observed objects pass the meridian, and their angular altitudes or zenith distances, when upon the meridian. The instrument with which the first of these measurements are obtained is called a transit instrument, transit telescope, or merely a transit. Transits of portable dimensions, besides their use in small or temporary observatories, are also found serviceable to the surveyor, for determining, with the greatest possible accuracy, the true north point, and thence setting out a line in any required direction; and to the scientific traveller, for determining the longitude of any place from astronomical observations, and for adjusting his time-keepers with greater accuracy than can be obtained by his sextant or reflecting circle. The annexed figure represents a portable transit.

TT is a telescope formed of two parts, connected by a spherical center-piece, into which are fitted the larger ends of two cones, the common axis of which is placed at right angles to the axis of the telescope, to serve as the horizontal axis of the instrument. The two small ends of these cones are ground into two perfectly equal cylinders, called pivots. The pivots rest upon angular bearings or Ys. The Ys are supported upon the standards E and w, of which E may be called the eastern, and w the western standard; and one of the Ys is fixed in a horizontal groove, on the western standard, so that, by means of the screw s, one end of the axis may be pushed a little forwards or backwards, and a small motion in azimuth be thus communicated to the telescopet. The standards, E

* In observatories the instruments are supported by stone walls, or pillars, which pass below the flocrs, without touching them, or any part of the building, and are consequently independent of any tremor, communicated to the floor or walls of the buildings. It was considered that the passage of a railway through Greenwich Park would impair the observations at the Royal Observatory, by communicating a tremor to the ground.

The large transits in permanent observations have their Ys placed in two dove-tailed grooves, one horizontal, and the other vertical. By means

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