Example 7.- April 30th, at oh. 37m. 245. a.m. by ship's watch; lat. 10° 7' N., long. by D.R. 176° 10' W. ; when the mean of a set of times by chronometer was 12h. 17m. 47s., and the mean of a set of observed altitudes of the moon's lower limb was 33° 1' 50" west of the meridian ; the chronometer had been found 2m. 12s. fast on mean noon at Greenwich on March 26th, and losing 7:45. daily ; height of eye 24 feet. Required the longitude of the ship. Example 8.—January 31st, at about 4h. 20m. p.m. at ship; lat. by D.R. 37° 8' S., long. by D.R. 40° E.; the mean of a set of times by chronometer was ih. 39m. 38s., and the mean of a set of observed altitudes of the sun's lower limb was 30° 48' 40" ; height of eye 21 feet; the chronometer had been found 5m. 545. slow on mean noon at Greenwich od December 17th, and losing 3:25. daily. Find the longitude. The Natural Haversine Method introduced into this Example is recommended in preference to any other. D. H. M. S M. S. Lat. 37 loco. 30 II Dec. 14 days. Time by chron. 31 I 39 38 Dec. 17 18 27.6 S Slow I 15.9 45.07 Approx. Gr. time I 45 32 Corr. dec. 17 17 117 3:2 Acc. rate + 2 24 90 00 00 144.224 S. Corr. Gr. time 31 I 47 56 P.D. 72 42 48 Acc. rate 2.24 + Hly. Var. Eq. time 13 43:54 + •64 759.870 + 13 44.18 Obs. alt. 30 48 40 I'15'9 8 Hly. Var, 90 1.8 Co-lat. 52 52 s :6372 33 Parx. + Formula Nat. hav. 0 = nat. hav. 3— nat. hav. (-1) 30 42 43 Semi-diam. + 16 16 L. hav. P=L. cosec. p + L. cosec. l + L. hav. 0—20 Tr. alt. 30 58 59 where P hour angle: 2, the zenith dist.; P, polar dist. and l', the co-lat. Nat. hav. 0.24261 L. cosec. 10.02008 L. cosec. 10.09842 L. bav, 9.32820 90 00 I I 48 00 15 26 44 Hour angle L. hay. 9:44670 47 56 41 14 Longitude 40 18 30 E. Examples for Practice Example 1.–February 26th, at about 7h. a.m. at ship; lat. by D.R. 56° 48' S.; long. by D.R. 135° 30' E.; the mean of a set of times by chronometer was Ioh. 6m. 255., and the mean of a set of altitudes of the sun's lower limb was 15° 5' 10'; height of eye 26 feet; the chronometer had been found 4m. 50s. fast on mean noon at Greenwich on January ist, 1890, and losing 4:8s. daily. Find the longitude. Ans. 136° 7' 30" E. Example 2.-August 23rd, at about 8h. 30m. a.m. at ship ; lat. by D.R. 37° 40' N., long. by D.R. 144o W.; the mean of a set of times by chronometer was 5h. 53m. 16s., and the mean of a set of altitudes of the sun's lower limb was 37° 15' 40"; index error of sextant 2' 15" to subtract; height of eye 20 feet; the chronometer had been found 1777. 305. slow on mean noon at Greenwich on June 30th, and was 18m. 455. slow on mean noon at Greenwich on July 30th. Required the longitude. Ans. 144° 52' 30" W. Example 3.— January 30th, at about 3h. 2om. p.m. at ship; lat. by D.R. 30° 36' N. ; long. 170° E.; the mean of a set of times by chronometer was 4h. om. 155., and the mean of a set of altitudes of the sun's lower limb was 24° 23'; height of eye 25 feet; the chronometer had been found 6m. 3s. fast on mean noon at Greenwich on November 2nd, 1889, and losing 3.5s. daily. Required the longitude. Ans. 169° 31' 15" E. Example 4.-April 15th, at about 3h. 12m. p.m. at ship; lat. by D.R. 44° 58' S., long. by D.R. 73° E.; the mean of a set of times by chronometer was roh. 33m. Is., and the mean of a set of altitudes of the sun's lower limb was 19° 10' 15" ; height of eye 23 feet; the chronometer had been found 3m. 45. slow on mean noon at Greenwich on November 16th, and on January 23rd, 1890, it was 2m. 28. fast on mean noon at Greenwich. Required the longitude. Ans. 73° 26' 30" E. Example 5.—November 29th, at about 3h. 25m. a.m. at ship ; lat. 10° 31' S., long. by D.R. 30° W.; the observed altitude of Aldebaran (a Tauri) west of the meridian was 30° 45' 40"; height of eye 20 feet; time by chronometer was 5h. 29m. 575., which allowing for error and rate) was 3m. 53. slow on mean time at Greenwich. Required the longitude. Ans. 30° 35' 30" W. Example 6.—March 3rd, at about 7h. 20m. p.m. at ship; lat. 8° 58' N., long. by D.R. 60° 30' E.; the observed altitude of Regulus (a Leonis) was 30° 36' 45' E. of the meridian ; height of eye 24 feet; time by chronometer was 3h. 21m. gs., which was iom. 335. fast (allowing for error and rate) on mean time at Greenwich. Required the longitude. Ans. 61° 4' 15" E. Example 7.—May 5th, at about 6h. 40m. a.m. at. ship, in long. by account 140° 40' W., when a chronometer indicated 4h. 3m. 545., which had been found 3m. 4s. fast on mean noon at Greenwich on January 3rd, and on February 28th it was 2m. 45. slow on mean noon at Greenwich; the observed altitude of the sun's lower limb was 20° 14' 40"; height of eye 23 feet. Required the longitude at the time of observation, the latitude at noon on May 4th by observation being 39° 50' N., and the ship has since sailed S. 83° W. (true) 82 miles. Ans. 140° 41' 15" W. SUMNER'S METHOD OF FINDING A SHIP'S POSITION AT SEA Before proceeding with the calculations required in the solution of this problem, it may be as well that the navigator should understand the principle of the problem, and its value as a general method of finding a ship's position at sea. Latitude alone, or Longitude alone, does not indicate the position of a place on the globe. Latitude merely shows that the place is somewhere on a small circle (a parallel) at a definite distance from the equator ; longitude merely shows that the place is somewhere on a great circle (a meridian) that makes a definite angle with another great circle which passes through a fixed conventional place of reference. To know the exact position of a place it is necessary to determine the point of the intersection of these two circles—that is, of the meridian with the parallel ; but this cannot always be done at sea, at any given or required instant, by any of the ordinary rules of nautical astronomy. The position may, however, be found by a combination of rules, or partly by computation and partly by projection; or where a good point cannot be ascertained as that on which the ship is, a line may be found on or near to which she is known to be, and this at the time may be priceless : the position of the ship is thus determined by a method of utilising parts of circles which, in their completeness, would be oblique to the parallels and meridians. When the declination of a celestial object coincides in amount and name with the latitude of a place on the terrestrial sphere, it must at some time, during the earth's rotation on its axis, appear in the zenith of that place; it will do so when the object's hour-angle for the place is oh., that is, when it is on the meridian. When this occurs, the Greenwich time by chrono meter being known, let it be. taken as granted that the object is above the horizon of another place; that its altitude is observed, and its zenithdistance consequently known. In Fig. I an object is vertical to the point S on the globe ; with S as a pole, and the observed zenith-distance S A as a polar s'. distance, describe a small circle: this bl is a circle of position, on some point of š which the observation has been made, al for from every point within or without this small circle a less or greater zenithdistance than S A would be observed at the instant of the object being at S. Fig.1 “ If, then,” as Chauvenet says, navigator can project this small circle upon an artificial globe, or chart, the knowledge that he is upon this circle will be just as valuable to him in enabling him to avoid dangers as the knowledge of either his latitude alone or his longitride B “ the T alone ; since one of the latter elements only determines a point to be in a certain circle, without fixing upon any particular point of that circle." The altitude of another celestial object S', taken at the same time as the former, gives a second circle of position (see Fig. 1, B). The observer being in the circumference of each of these circles, must be at one of their points of intersection, at P or P': there will be no difficulty in ascertaining which point is to be taken, as it will be sufficiently indicated by the dead reckoning The circles to which reference has been made are such as they would appear when represented on the spherical surface of a globe, and they illustrate the principle of the problem. On a Mercator's chart, where the distance between the parallels is considerably augmented in the higher latitudes—in order to preserve the proportion that exists at different parts of the earth's surface between the meridians and the parallels—circles of position would be represented as elliptical figures (Fig. 2); perhaps we had better say as curves of position, which, to delineate properly, would require to be computed for every 5 cr 10 degrees : happily, in the projection of the problem, we only require a very small part of these curves, for Fig. 2. which we assume two latitudes a few miles on each side of the latitude by dead reckoning. We may take the tangent (T) to the curve, or the chord (C); but our computation and projection will be the more perfect the more closely the chord and the tangent coincide-in fact, we shall then have the best line of position. The data for the problem are (1) the correct Greenwich Date by chronometer ; (2) simultaneous altitudes of two stars, or of a star and planetwhich are by far the best objects to give the ship's position; when the sun alone is the object there must be an earlier and later altitude, with the course and distance carefully noted in the interval of the observations; (3) two assumed latitudes, the basis of which must be the latitude by D.R.; and, finally, (4) the elements from the Nautical Almanac, respecting which there is no excuse for taking them out inaccurately. SIMULTANEOUS ALTITUDES Simultaneous altitudes of two celestial objects are unquestionably the best for determining the position of a ship—which is thus got at once without any change of place, or interval of time for which to allow. With a good knowledge of the stars and planets two objects can be selected at pleasure, and in such relation to each other that the angle between their verticals shall be the best possible—something between 60° and 120°—and so develop a good point of intersection. If there be any doubt, a third star will give, with the two others, a space or triangle of certainty, within which the ship must be. Taken in the twilight—and how often may this be done when no sun has been visible all day—the altitudes, by a practised hand, ought to be obtained within a limit of 2' to 3', less rather than more, When assuming the two latitudes it is generally sufficient to select them about 30' or less on each side of the latitude by D.R. ; but this will inuch depend upon what length of time has elapsed since the ship's position had |