this correction however, to the circle readings, it will be additive, or subtractive, according as, by the construction of the circle, increasing readings represent increasing or decreasing zenith dis tances. Half the difference of two corrected readings in opposite positions of the instrument is the star's apparent zenith distance on the meridian; or the mean of all the observations in one position may be compared with the mean of all those in the other, and half their sum is the zenith point. To this zenith distance add the correction for refraction, taking into consideration the readings of the thermometer and barometer, and apply the star's declination for the day (from the Nautical Almanac) for the latitude. The above instructions* apply only to stars observed near the meridian. The latitude can however, be obtained by similar observations of stars situated very far from the meridian, though this method would very seldom be resorted to. When the sun is the object observed, a further correction must be made on account of the change in declination during the time occupied by the observation, which is expressed by S being the change of declination in one minute of time, minus when decreasing. .E the sum of the horary angles observed to the east, expressed in minutes of time, and considered as integers. W their sum to the west, and In the number of these observations. When a star is the object observed, and the time is noted by a chronometer, regulated to mean time, the value of A must be multiplied by 1.0054762. Also, if the clock does not keep its rate either of sidereal or mean time accurately, a further correction is imperative; and A must be multiplied by 1 + 0002315 r, where denotes the daily rate of the clock in seconds, minus when gaining, and plus when losing. * See "Corps Papers," vol. iii. page 328, where will also be found examples worked out in detail, of latitudes thus obtained on the survey of the North American Boundary. EXAMPLE. On March 8, 1837, the following observations were taken, with a sextant, the chronometer being fast 9m 16s; index error of sextant, 1' 20"; barometer, 29.54; thermometer, 50°. Method 3rd.-By the altitude of the pole star, at any time of the day*. If the altitude of the pole star can be taken when on the meridian, its polar distance, either added to, or subtracted from, the altitude, gives at once the latitude; and when observed out of the meridian, as at the point S or S' in the figure, the latitude can be easily obtained, as follows: Let ZPO represent the meridian, Z the zenith, P the pole, and a Sa' the circle described by the polar star S, at its polar distance P S. The star's horary angle ZP S, or Z P S', is evidently the difference between its right ascension and the sidereal time of observation; and in the spherical triangle ZPS (or ZPS') we have ZS, PS, and the angle ZPS, to find ZP, the co-latitude. The result may be obtained with almost equal accuracy by considering PSc as a plain right-angled triangle, of which Pc is the cosine of the angle cPS to radius PS; the distance Pc thus found is to be added to, or subtracted from, the altitude HS, according as the star is above or below the pole, which is thus ascertained :—If the angle ZPS' be less than 6, or more than 18 hours, the star is above the pole, as at S'; if between 6 and 18 hours, it is below the pole, as at S. By the tables given in the Nautical Almanac, the solution is even more easy, and has the advantage of not requiring any other reference. The rule is as follows: : 1st. From the corrected altitude subtract l'. * This of course is only applicable to northern latitudes. In the southern hemisphere there is no star sufficiently near to the south pole to be made available in thus determining the latitude. 2nd. Reduce the mean time of observation at the place to the corresponding sidereal time. 3rd. With this sidereal time take out the first correction from Table I., with its proper sign, to be applied to the altitude for an approximate latitude. 4th. With this approximate latitude and sidereal time take out from Table II. the second correction; and with the day of the month and the same sidereal time take from Table III. the third correction. These are to be always added to the approximate latitude for the latitude of the place. EXAMPLE. On Oct. 26, 1838, the double altitude of Polaris, observed with a repeating circle, at 11h 55m 30s mean time, was 105° 44′ 63′′, the barometer standing at 29-8; thermometer, 50°. Required the latitude of the place of observation. By the method given in the Nautical Almanac,— |