been devised to shorten the calculation of "double altitudes" by tables formed for the purpose, one of which may be found at page 231 of "Riddle's Navigation"; but the direct method by spherical trigonometry is most readily understood and easily followed.

Let S and S represent the places of the object at the times of the two several observations, (and they may be on different sides of the meridian, or as in the figure, both on the same side); ZS and ZS then, are their respective zenith distances, and PS and PS' their polar distances, SPS' being the hour angle observed.

P

First-In the triangle PSS', the two sides PS and PS' are given, with the included angle at P tofind SS and the angle PSS'. Again in the triangle ZSS', we have the three sides to find the angle ZSS which, taken from PSS' just found, leaves the remaining angle PSZ. Lastly—in the triangle PSZ we have PS, ZS, and the angle PSZ, to find PZ, the co-latitude sought. In a similar manner the latitude may be found by simultaneous altitudes of different stars, the difference of their right ascensions giving the angle SPS, without the use of a watch. Tables have been calculated by Dr. Brinkley, from which the distance SS' can be obtained by inspection, (allowing for the change in the right ascension of the stars after any long interval,) and the calculation is thus considerably abridged. By an azimuth and altitude instrument, the latitude may also be found by the two altitudes, and the difference or sum of the observed azimuths of the sun or star.

Equal altitudes of the same star on different sides of the meridian, with the interval of sidereal time, between the observations, also furnish the means of ascertaining the latitude, and this method is most useful in a perfectly unknown country. The hour angle, east or west, will evidently be measured by half the elapsed interval of time; and in the triangle ZPS, we have this hour angle ZPS, the polar distance PS,and the co-altitude ZS,to find ZP the co-latitude;

moreover, the hour angle being known, and also the right ascension of the star, the point of the equinoctial which is on the meridian, and consequently the local sidereal time is determined, from which the mean time can be deduced.

The latitude may likewise be ascertained independently of the graduation of the instrument, by placing the axis of the telescope of an altitude and azimuth circle due north and south, so that the vertical circle shall stand east and west. The observations of the two moments T and T' (in sidereal time), in which the star passes the wire of the telescope, will give the latitude from the following formula.

Cos. L cot. declination X cos. (T-T')

If a chronometer set to mean time is used, the interval (T—T') must be multiplied by 1.0027379, or the value corresponding to the interval, found in the table for converting mean into sidereal time, must be added.*

The accuracy of this method depends upon the correctness of the tabulated declination of the star, but a slight error in this will not affect the difference of latitude between two places, thus found. By observing on following days with the axis reversed, and taking the mean of the observations, any error in the instrument is corrected; this method is particularly recommended by Mr. Baily for adoption in geodetical operations, as the difference of latitude of two stations is obtained almost independently of the declination of the star, and the only material precaution to be taken is in levelling the axis of the telescope, which should be one of very good quality.

PROBLEM IV.

ΤΟ FIND THE TIME.

Method 1st. From single or absolute altitudes of the sun, or a star, whose declination is known, as also the latitude of the place. This problem is solved by finding the value of the horary angle P, in the same "astronomical triangle" ZPS, whose elements have already been described. In this case, the three sides, viz., the co-latitude, the zenith, and polar distances, are given to find the hour

* Table 7.-Baily's Astronomical Tables and Formulæ.

angle P, which, when the sun is the object observed, will (as was explained in page 156) be the apparent time from apparent noon at the place of observation; and it is converted into mean time by applying to it the equation of time with its proper sign. In the case of a star, it will denote its distance in time from the meridian, which being added to its right ascension, if the observation be made to the westward of the meridian, or subtracted from the right ascension (increased by 24 hours if necessary) if to the eastward, will give the sidereal time, to be converted into mean solar time, if required.

A simple formula for finding the angle of a spherical triangle sin. (S-c) (sin. S—b) whose three sides are given is sin. x = 2 sin. c. sin. b.

where S denotes the sum of the three sides a, b, and c, of which a is assumed as the one opposite the required angle. In the present case a represents the co-altitude or zenith distance; b the co-declination, or polar distance; and c the co-latitude.

EXAMPLE.

Observed altitude Ō on May 4th, 1838, was 14° 44′ 58" at 5h 47m 15 by chronometer; latitude 51° 23′ 40"; longitude 2m 21.5o east; index error of sextant 28".

Method 2nd.-From equal altitudes of a star or the sun, and the interval of time between the observation.

If a star is the object observed, it is evident that half the interval of time elapsed between its returning to any observed altitude, after its culmination, will give the moment of its passing the meridian without any correction, from whence the error of the clock or chronometer is at once found. But with regard to the sun, there is a correction to be applied to this half interval, on account of his constant change of declination. From mid-winter to mid-summer the sun gradually approaches the North Pole, and therefore a longer period will intervene after than before noon,—between the sun's descent to the same altitude in the evening, as at the morning observation: and the reverse takes place from mid-summer to midwinter. The amount of this correction depends partly upon the change of declination, proportional to the interval of time on the day of observation, and partly upon the latitude of the place; and the difference of the sun's horary angles at the morning and afternoon observations is easily calculated by the following formula of Mr. Baily's:—

* The most favourable time for observing single or absolute altitudes of the sun or a star, to obtain the local time, is when they are on or near the prime vertical, since their motion in altitude is then most rapid, and a slight error in the assumed latitude is not of so much consequence. The corrections for the refraction however, are then considerable. The same observation will of course give the azimuth Z, and also the variation of the needle, if the magnetic bearing of the star, or of either limb of the sun, is taken by another observer at the same moment as the altitude. This will be further explained in a subsequent chapter.

Y