2. Another differential method has lately been muoh used (at least by continental astronomers), in which the transit instrument alone is employed. [TaaNsrr.1 The axis of the instrument is placed north and south, and carefully levelled, in which ease its line of sight will describe the prime vertical. In the figure, let r be the place of the pole, z the zenith, E z w the prime vertical, which is also the live described by the middle wire of the telescope when it revolves. Let a star, of which the polar distance is well known, be observed at s and s', and the times noted. Then r s, the polar distance of the star, is known, and the angle s r s' is equal to the time between the observa tions, and consequently l s p s', or s P z, is known ; hence we have s r and s r z in the right-angled spherical triangles r z, and tau r z ss tan s x cos L s r z, from which r z, or the co-latitude, is obtained. This is perhaps the most accurate mode of determining the latitude with moderate instrumental means. The transit should be reversed on alternate nights, so as to get rid of the effect of imperfect collimation or unequal pivots, and the level applied repeatedly before and after the observations. The method depends maiuly upon the delicacy of the level and the perfect truth of form in the pivots, and when all precau tions are taken the results are surprisingly good. The supports of the instrument must also be perfectly steady during the levelling and the observation. Differences of latitude may be determined by the transit instrument independently, by observing the same stars at the two stations. In this case any error in the assumed polar distance of the stars will not affect the accuracy of the result.
3. If an observer can carry with him a circle, either an altitude and azimuth, or a repeating circle, he may determine the latitude by cir cumpolar stars independently : but it is better to observe the zenith distances of well known stars several minutes before and after they pass the meridian.* [Clam: ; REPEATINO CIRCLE.) The reduction to the meridian is.easily computed [REPEATINO CincLe], and the places of the stars inserted in the ' Nautical Almanac' are sufficiently accu rate. It is adviiable to observe stars at different zenith distances from 70' on each side, to near the zenith, because if there be any fault iu the instrument which depends on the zenith distance, the stars will be affected similarly on both sides of the zenith, that is, the zenith distances to the north and south will both be too large or too small. Now as the co-latitude is equal to the zenith distance ± the polar distance when the star is north of the zenith ( + when observed above the pole and — when observed below) and is equal to the polar distance —zenith distance when the star is south of the zenith, it is clear that an error in the zenith distance will have precisely opposite effects on the co-latitude deduced from a north and from a south star. Besides, the coincidence or discrepancy of the observations will afford a tolerable notion of the instrument and the observer, and of the value of the final result. The repeating circle was at one time over estimated, and perhaps at present is not quite rated at its true worth. A careful and intelligenC observer will come Very near the truth with it, or with the altitude and azimuth circle. Both are rather troublesome to use, and both require either a very solid support, or a second observer to read the level while the first observer bisects the star and notes the time. Neither can they be considered as portable iu ordinary circumstances, when largo enough for convenient use. The altitude circle should scarcely be less than 12 inches in diameter in either construction. On the whole we are inclined to prefer the repeating circle as a travelling instrument, and the altitude and azimuth for a permanent situation ; but it meet be coulter:4 that few observers have the patience or skill to gee the greatest ptseiblo accuracy out of either. The observe urn should Le °imbued to Mans as neither of thole instruments will ketv Its adjustment* well under the SUM 1%. repeating circle urns Mali by the French astronomers to deter mine the latitudes In their great survey. Since that time the instru
ment has been much better made, and the catalogues of stu-s which bare lamed from Konigsberg. Greenwich, and Cambridge, have supplied more accurate and convenient means of using it. If the levels are very good and sensible, we think that the observations of one fine night, everything being favourable, should bring out the latitude within 2' or 3'.
4. The last class of instruments to be noticed is that of reflecting Instruments', including the reflecting circle of Troughton, the repeating reflecting circle of Bonin, and the sextant of Iledley. These will be described under the article Srsavaar, as the title best suited to their essential quality of reflection. At present we must suppose a general knowledge of their nature. And first we will suppose the observer to have a stand and a mercurial or other horizon. In this case, standard stare should be observed several minutes before and after the meridian passage to the north and south, between the altitudes of 15° and Mil as much as possible In pairs, that is, for each star to the north, a star to the south should be observed about the same altitude, or two, uoe higher and the other lower, so that the mean altitude should nearly correspond. Whatever errors may exist in the division, glasses, &c., will be the same in each star of the pair, and as the error will affect the latitude differently, the mean latitude %i11 be free from the error very nearly. In this way several pairs may be observed, taking the stars of the' Nautical Almanac,' and the mean of the whole will come out very near the truth. With a good sextant or circle, and a mercurial horizon, wo believe that a careful observer would get the latitude within 5', in one fine night. It is, however, supposed that everything is favourable, and especially that the instrument is supported on a stand. Thin is absolutely necessary for the accurate observation of stars, which dance very perplexingly when the instrument is held in the hand, and a high power applied. If a stand cannot be afforded, the run is far the best object to observe with a reflecting instrument. It is always supposed that the observations are made for several minutes before and after the meridian pasesage, and the time noted fur computing the reduction to the meridian. The meridian altitude of the sun, such as it would be if observed on the meridian and freed from instrumental and other errors, is then computed, and as the longitude of the place is known, at least approximately, the declination of the sun at its passage over that meridian may be computed from the ' Nautical Almanac.' The meridian altitude the south declination of the sun, or — the north declination, is, in the northern hemisphere, the co-latitude of the place. With Trougliton's circle, the limbs are alternately obeerved, to get rid of the sun's diameter, and the number of observations, forwards and backwards, should be equal, so as to get rid of the index error. With the repenting reflecting circle, the obser vations should also be of the upper and lower limbs alternately, and should be carried quite round the circle, no as to get rid of eccentricity.. In the sextant the index error should be carefully determined before and after each day's observations, and the alternate limbs observed exactly as with the circles. There is, however, no way of getting rid of excentricity in the sextant by observing one object, and any fault in determining the index error will vitiate the latitude to half its amount. While the circle. will probably give a latitude to nearly 5' with a very careful series of observations of the sun, the (sextant with equal care might be nut 10' or 15'. it is evident therefore, that where accuracy is an object, the observer ought, if possible, either to use a circle or to mount the sextant upon a stand, and °been's, stare as we have above described. It in an additional reason to carry a stand, when practicable, that in low latitudes the sun cannot be observed at all for the latitude, nor any object which is elevated 65' or 70'. in this case stars must be used; and without a stand, the observation, urn; high magnifying is difficult and unsatisfactory.