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By considering SPc as a plane rightangled triangle, in which P c, the correction to be subtracted, is the cosine of P to radius PS, the latitude is found by plane
trigonometry within a few seconds of the above results.
Method 4th.-By an altitude of the sun, or of a star, out of the meridian, the correct time of observation being known.
By reference to the figure, it will be seen that this method simply involves the solution of the spherical triangle ZPS already alluded to, formed by the zenith, the pole, and the object at the time of observation; of which ZS, the zenith dis- S
tance, PS, the polar distance, and the angle at P are known, and Z P, the co-latitude, is the quantity sought.
The formula given by Baily, for finding the third side, when the other two sides and an angle opposite to one of them are given, is
tan a' = cos given angle × tan adjacent side
and a = (a + a"), which formula is used in the following examples :
On May 4, 1838, the observed altitude of the sun's upper limb at 5h. 47m. 15s. by chronometer was 14° 44′ 58′′. The index error of sextant being 28", and the watch 3m. 34s4 too fast. Barometer 29.9; thermometer 61; required the latitude.
When the sun is the object observed, the hour angle P (as in the last example) is the apparent time from apparent noon at the place of observation, converted into space; but with a star, it is its distance from the meridian, either to the east or west, according as it has or has not come to its culmination; and this angle is simply the sum or difference of the star's right ascension, and the time of the observation converted into sidereal time; to be multiplied by 15 for its conversion into space.
Method 5th.-By two observed altitudes of the sun, or a star, and the interval of time between the observations.
This problem is of importance, as its solution, though long, does not involve a knowledge of the correct time at the place of observation; and the short interval of time can always be measured with sufficient accuracy by any tolerable watch. Various methods have 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. First-In the triangle PS S', the two sides PS and PS' are given, with the included angle at P to find 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 PS Z, 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 Z PS, we have this hour angle ZPS, the polar distance PS, and the coaltitude Z S, to find Z P 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.
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 geodesical 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.
TO 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" Z P S, whose elements have
*A portable transit placed in the plane of the prime vertical, instead of that of the meridian, of course affords the same facility for thus determining the latitude. The stars selected should have their declinations less than the latitude of the place, but by as small a quantity as possible.
Table 7. Baily's Astronomical Tables and Formulæ.