This angle is acute like its opposite side (see p. 196). IV. TO FIND THE TWO UNKNOWN ANGLES OF THE TRIANGLE ZSP. * This sign is employed to express the difference between two quantities, which ever may be the greater. Upon the same principles may the latitude be determined from the altitudes of two fixed stars, taken at the same time; in this case s, s', in the preceding figure, will represent the two stars: PS, Ps', their known polar distances, and the angle sPs', the difference of their right ascensions; the same quantities are therefore given as in the case of the sun, but, as in the case of two stars, PS, Ps', may differ very considerably, ss' cannot be considered as the base of an isosceles triangle, but must be computed from the other two sides and their included angle. For other modes of determining the latitude, see the next Appendix. ON FINDING THE LONGITUDE. The determination of the longitude of a place always requires the solution of these two problems, viz.: 1st, to determine the time at the place at any instant; and, 2d, to determine the time at the first meridian, or that from which the longitude is estimated, at the same instant; for the difference of the times converted into degrees, at the rate of 15° to an hour, will obviously give the longitude. When the latitude of the place is known (and it may be found by the methods already explained), the time may be computed from the altitude of any celestial object whose declination is known; for the coaltitude, codeclination, and colatitude, will be three sides of a spherical triangle given to find the hour angle, comprised between the codeclination and the colatitude. (See Art. 84.) The following example will illustrate the mode of proceeding. At Columbia College, January 13th, 1850, the double altitude of the sun's lower limb was observed by reflexion from Mercury to be 42° 29'. Thermometer 40°, and Barometer 30 in. Time by the watch, 10' 6m 10′ A.M. Longitude from Greenwich in time, 4′ 56′′ 4′. Required correct time of Observation and error of the watch. Equation of time at ap. noon, January 13th, 1850, 9' 00" 27. Sun's declination at ap. noon, January 13th, 1850, 21° 29′ 19' 5 S. Index error of sextant. Additive (see 2d note, p. 290). Double altitude corrected for index error, Altitude, xx., Refraction (Th. 40), (B. 30), Table XXX., 's true altitude. corrected for refraction, parallax, o's zenith distance (90° - Altitude) Approx. time at station, Longitude from Greenwich in time, Time at Greenwich, Equation of time, subtractive, Time after apparent noon at Greenwich, 's declination at time of observation†, This must be multiplied by the time after apparent noon at Greenwich, found below, reduced to hours and decimals of an hour, and the product added to the equation of time at noon above, to obtain the equation of time at the instant of observation. + This is computed from the data in the third and fourth lines from the top of the page, in the same manner as the equation of time. Ps or z='s N.P.D=(90°+D)=111° 28' 4" 18 ar. co. log. sin 0.03122 P Hour angle in time, or apparent time before noon, 24 3m 52 •1 2. Ship Admiral at sea, March 5th, 1850, at 10 o'clock, A. M. 9 56 7.9 10 10 6 10 59 02: March 6th, o's semidiameter, 16' 7"-9; eq. of time, 3. March 12th, at 4 P.M. Astronomical Account. Alt. of the sun's lower limb, Time at Greenwich by mean of chronometers, Lat. of ship at time of obs. To be subtracted from mean time. + The dec. is of course diminishing till the equinox March 21st. 27° 44 12 41 12 49° 54' 00" 11m 30*98* 11 16 .56 5° 40' 48' 3 S. 18° 42' 40" 7 51m 30° 41° 41' 30" To find the time at Greenwich requires the aid of additional data, besides those furnished by observations made at the place. The Greenwich time may, indeed, be obtained at once, independently of any observations at the place, by means of a chronometer, carefully regulated to Greenwich time, provided it be subject to no irregularities after having been once properly adjusted. A ship furnished with such a timepiece always carries the Greenwich time with her, and the longitude then becomes reduced to the problem of finding the time at the place. EXAMPLE. Time computed by an altitude of the sun, as at p. 288, Chronometer showed Greenwich time at the instant of observation to be Difference of longitude of the place of observation from Gr. in time, To convert time to space multiply by. Longitude of the ship west of Greenwich, The same method applies to examples 2 and 3, p. 289. 46° 17' 30' Still, however, as the most perfect contrivance of human art is subject to accident, and the more delicate the machine the more liable is it to disarrangement, from causes which we may not be able to control, it becomes highly desirable, in so important a matter as finding the place of a ship at sea, to be possessed of methods altogether beyond the influence of terrestrial vicissitudes, and such methods the celestial motions alone can supply. The angular motion of the moon in her orbit is more rapid than that of any other celestial body, and sufficiently great to render the portion of her path passed over in so short a time as two or three seconds, a measurable quantity even with a small portable instrument (the sextant).f In computing the D's semi-diameter and parallax, second differences need not be used, the consequent error being less than 0''•1. + THE SEXTANT is constructed upon the same principles as the quadrant. It consists of a graduated |