Another method exists for determining the latitude of a place by taking the elevations of a circumpolar star at its upper and lower culmination. When such observations are made, half the sum of the elevations cleared of index error and refraction will be the latitude required. This is a method however which cannot always be conveniently practised in this country, on account of the small altitude of the circumpolar star at its lower meridional passage, when it will be often involved in the mist which surrounds the horizon.

For observations of this nature the Reflecting Circle is an instrument which is frequently employed in India as possessing very superior qualities over the Sextant, but which having been inadvertently omitted to be noticed in its proper place in Part 2, the following description of its use and adjustments is taken from "Simms on Mathematical Instruments."

The above figure represents this instrument, which in principle and use is the same as the sextant. It has three vernier readings, A B C, moving round the same centre as the index-glass, E, which is upon the opposite face of the

instrument. One of the verniers, B, carries the clamp and tangent-screw. D represents the miscroscope for reading the verniers; it is similar to the one used in reading the sextant, and is adapted to each index-bar, by slipping it on a pin placed for that purpose, as shown in the figure. The horizonglass is shown at F. The barrel, G, contains the screws for giving the upand-down motion to the telescope; it is put in action by turning the milled head under the barrel. H is the telescope, adapted to the instrument in a manner similar to that of the sextant. I and J are two handles fixed parallel to the plane of the circle, and a third handle, K, is screwed on at right angles to that plane, and can be transferred to the opposite face of the instrument by screwing it into the handle, I; the use of this extra handle is for convenience in reading and in holding the instrument, when observing angles that are nearly horizontal; it can be shifted, according as the face of the instrument is held upwards or downwards. The requisite dark glasses are attached to the framework of the circle, to be used in the same manner and for the same purposes as those of the sextant. With respect to the adjustments and application of this instrument, we cannot do better than use the words of the inventor, Mr. TROUGHTON, contained in a paper which he calls "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 backwards 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 cause, 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 same 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 these 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 indexerror; 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, it must be removed, for the time, by taking out the finger-screw, which fastens it to the body of the circle.

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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 still of more consequence, the error of the centre 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 onesixth 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 instrument, 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 horizonglass 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 the same, just at the outside of the other wire: its being so is the proof of adjustment.

CHAPTER IX.

ON LONGITUDE.

THE Longitude of a place may be defined to be an arc of the equator intercepted between the first meridian, and that passing through the given place. The selection of the first meridian from which longitudes are measured, is entirely arbitrary. The English use the meridian of the Royal Observatory at Greenwich as the first meridian, and reckon all longitudes to the east and west thereof. On the other hand, le premier meridien of the French Geographers passes through the Paris Observatory, which is 2° 20′ 15′′ east of the former.

The longitude of a place is as often given in space as in time, the reduction of one measure to the other, being made at the rate of 15° an hour. Thus the longitude of the Dome of the Government House in Calcutta may be indifferently stated at 88° 23′ 27′′ or 5h. 53m. 33.8s. east of Greenwich, and in this proportion of space into time, the Tables in Chapter II. have been constructed.

The reason of reckoning the longitude in time, is this; suppose of two places A and B, the time at A is given, it is required to determine the corresponding time at B. This is a problem of great astronomical importance, and when the difference of longitude between A and B is given in time, it may be easily solved in this way: According as B is to the east or west of A, add the difference of longitude to, or subtract it from, the given time at A, the sum or difference so obtained, will be the required time at B. Thus when it is 7h. 22m.