EXAMPLES FOR PRACTICE.

1. 1887, January 2nd, A.M. at ship, latitude 36° 59′ S., observed altitude sun's L.L. 49° 10′, index correction — 2′ 40′′, height of eye 14 feet, time by a chronometer id 19h 8m 50a (being 78m 50o A.M. at Greenwich), which was slow 18m 2" for mean noon at Greenwich, November 30th, 1886, and on December 7th, 1886, was 19m 106 slow for mean time at Greenwich: required the longitude.

2. 1887, February 19th, A.M. at ship, latitude 38° 18′ S., observed altitude sun's L.L. 21° 30′ 40′′, index correction 6′ 45′′, height of eye 14 feet, time by a chronometer 18d 19h 53m 37.6 (being 7h 53m 376 A.M. at Greenwich), which was 4m 166 fast for mean noon at Greenwich, January 23rd, and on January 30th was 5m 98 fast for mean time at Greenwich.

3. 1887, March 28th, P.M. at ship, latitude 20° 19' S., observed altitude sun's L.L. 30° 14', index correction 2′ 10′′, height of eye 30 feet, time by chronometer 28d oh 10m (being oh 10m P.M. at Greenwich), which was 54m 488 fast for mean noon at Greenwich, October 20th, 1886, and on December 2nd, 1886, was 51m 56s fast for mean noon at Greenwich.

4. 1887, April 15th, A.M. at ship, latitude 53° 5' N., observed altitude sun's L.L. 16° 8' 40", index correction - 40", height of eye 15 feet, time by a chronometer 15d 6h 18m 49° (being P.M. at Greenwich), which was om 44 slow for mean noon at Greenwich, February 11th, and on March 11th was 2m 38 fast for mean noon at Greenwich.

5. 1887, May 19th, P.M. at ship, latitude 2° 58′ S., observed altitude sun's L.L. 30° 30′, index correction + 52′′, height of eye 19 feet, time by chronometer 19d oh 23m 58', which was 28 fast for mean noon at Greenwich, January 3rd, and on January 31st was 42 slow on mean time at Greenwich.

6. 1887, June 15th, A.M. at ship, latitude 12° 11′N., observed altitude sun's L.L. 39° 39′ 40′′, index correction + 20", height of eye 17 feet, time by a chronometer 14a 17h 59m 30o (being 5h 59m 30° A.M. at Greenwich), which was slow 5m 563 for mean time at Greenwich, April 20th, and on May 12th was 2m 29.5 slow for mean noon at Greenwich.

7. 1887, July 5th, A.M. at ship, latitude 23° 48′ N., observed altitude sun's L.L. 48° 36′ 50′′, index correction 50", height of eye 17 feet, time by chronometer 5d oh 42m 38′ (being oh 42m 38° P.M. at Greenwich), which was fast 4m 47-8 for mean noon at Greenwich, May 6th, and on June 1st was fast 6m 50 for mean noon at Greenwich.

8. 1887, August 13th, A.M. at ship, latitude 30° 46′ S., observed altitude sun's L.L. 27° 15′, index correction — 1′ 15′′, height of eye 21 feet, time by a chronometer 13d 2h om (being ah om P.M. at Greenwich), which was slow 26m 76 for mean noon at Greenwich, April 1oth, and on May 1st was slow 25m 13° for mean noon at Greenwich.

9. 1887, September 1st, P.M. at ship, latitude 35° 49′ N., observed altitude sun's L.L. 44° 32′ 10′′, index correction + 1′ 46′′, height of eye 20 feet, time by chronometer August 31d 19h 24m 57o (being 7h 24m 57a A.M. at Greenwich), which was fast 11m 574 for mean noon at Greenwich, July 3rd, and on July 31st was fast 12m 17 for mean noon at Greenwich.

10. 1887, October 25th, P.M. at ship, latitude 51° 30′ S., observed altitude sun's L.L. 40° 22′, index correction 1′ 50′′, eye 20 feet, time by chronometer 25d 8h 22m 1o (or 8h 23m 1o P.M.), which on July 20th was slow 21m 19 for mean noon at Greenwich, and gaining 47 daily.

11. 1887, November 27th, A.M. at ship, latitude 39° 20′ S., observed altitude sun's L.L. 34° 37′ 55′′, index correction +1'15", eye 18 feet, time by a chronometer 27d 7h 41m 30o (being P.M. at Greenwich), which on November 9th was 29m 40′ fast on mean noon at Greenwich, and losing 67 daily.

12. 1887, December 25th, A.M. at ship, latitude 9° 59' S., observed altitude sun's L.L. 10° 38′ 45′′, index correction 3' 12", eye 18 feet, time by a chronometer 24d 18h 6m (being A.M. at Greenwich) which was slow 4m 2014 for mean noon at Greenwich, July 1st, and on July 29th showed mean noon at Greenwich.

13. 1887, January 1st, P.M. at ship, latitude 38° 28′ S., observed altitude sun's L.L. 39° o', index correction 2′ 25′′, eye 12 feet, time by chronometer id 11h 58m 29° (being P.M. at Greenwich), which was slow 1h 49m 19 for mean noon at Greenwich, September 12th, 1886, and on October 13th was 1h 52m 53° slow for mean noon at Greenwich.

14. 1887, February 11th, A.M. at ship, latitude 53° 12' N., observed altitude sun's L.L. 12° 10′, index correction — 49", eye 12 feet, time by chronometer rod 22h 22m 223 (being A.M. at Greenwich), which was fast 34m 417 for mean noon at Greenwich, October 31st, and on December 1st, 1886, was fast 38m 59* for mean noon at Greenwich.

15. 1887, October 26th, A.M. at ship, latitude 28° 10' N., observed altitude sun's U.L. 25° 32′ 20′′, index correction o", eye 17 feet, time by chronometer oh 24m 6" (being P.M. at Greenwich), which was fast 1m 31a on mean time at Greenwich, August 1st, and on Sept. 4th was fast om 6a for mean noon at Greenwich.

16. 1887, February 6th, P.M. at ship, latitude 6° 58′ N., observed altitude sun's U.L. 21° 43′ 40′′, index correction o", eye 18 feet, time by a chronometer 11h 40m 26a (being a.M. at Greenwich), which was slow 16m 48 on mean noon at Greenwich, January 2nd, and on January 20th was slow 17m 42o on mean noon at Greenwich.

17. 1887, May 1st, P.M. at ship, latitude 21° 8' N., observed altitude sun's L.L. 28° 5′ 30′′, index correction +2′ 50′′, height of eye 16 feet, time by a chronometer April 3cd 18h 50m 29°4 (being 6h 50m 29°4 A.M. at Greenwich), which was 10m 12' slow for mean noon at Greenwich, December 31st, 1886, and on February 17th, 1887, was 7m 336 slow for mean noon at Greenwich.

18. 1887, April 21st, P.M. at ship, latitude at noon o° 20′ N., observed altitude sun's U.L. 32° 21' 10", index correction I' 10", eye 12 feet, time by a chronometer 3h 44m 1 (being A.M. at Greenwich), which was slow 9 71 for mean noon at Greenwich, November 14th, 1886, and on January 11th, 1887, was slow 7 342 for mean noon at Greenwich, course since noon S.W. by W. (true), distance 36 miles: required the longitude at the time of observation, and also at noon.

19. 1887, August 21st, A.M. at ship, latitude at noon o° 20′ S., observed altitude sun's L.L. 33° 49′, index correction + 2' 10", eye 15 feet, time by chronometer 8h 14m 0 (being P.M. at Greenwich), which was slow 4TM 40o for mean noon at Greenwich, March 13th, and on April 30th was slow 5m 40 for mean noon at Greenwich, course till noon S. W. by W., distance 36 miles: required longitude at time of sights, and also at noon.

20. 1887, March 20th, A.M. at ship, latitude o°, observed altitude sun's L.L. 28° 50' 10", index correction +1', eye 23 feet, time by chronometer 20d 1h 35m (being P.M. at Greenwich), which was 1m 599 fast for mean noon at Greenwich, February 1st, and on February 28th was fast 2m 8 for mean noon at Greenwich.

21. 1887, September 23rd, A.M. at ship, latitude oo, observed altitude sun's L.L. 45° 21′ 10", index error 1′ 7′′, height of eye 17 feet, time by chronometer September 22d 20h 56m 13o, which was slow 156 on Greenwich mean time, April 30th, and on June 1st was fast 10o on Greenwich mean time.

SUMNER'S METHOD.

ON FINDING SHIP'S LINE OF POSITION OR PARALLEL OF EQUAL ALTITUDE.

Problem I.

TO FIND THE SHIP'S CURVE OF POSITION OR PARALLEL OF EQUAL ALTITUDE.

177. In the preceding chapters of this work we have treated of methods of finding the position of a point on the earth's surface by the two co-ordinates latitude and longitude; and, therefore, in all these methods the required position is determined by the intersection of two circles, one a parallel of latitude, and the other a meridian. It is evident that if we could find any two other lines passing through the ship, their intersection would also determine the position of the ship, and thus answer the same purpose as parallels of latitude and meridians. In the following method the position of the ship is determined by circles oblique to the parallels of latitude and the meridian. The principle which underlies the method has often been applied, but its value as a practical nautical method was first clearly shown by Capt. T. H. Sumner.

178. Let an altitude of the sun, or any other object, be observed at any time, the time being noted by a chronometer regulated to Greenwich time.

Fig. 55.

A

a

P

Suppose that at this Greenwich time the sun is vertical to an observer at the point S of the globe (Fig. 55), having there an altitude of 90°, and if a small circle A A'A" be supposed to be described about the point S as a pole, with a distance SA, equal to the zenith distance or complement of It is the altitude of the sun. evident that at all places within this circle an observer would, at the given time, observe a smaller zenith distance, and at all places without this circle a greater zenith distance; and, therefore, the observation fully determines the observer to be on the circum

ference of the small circle A A'A".

a

A"

If, then, the seaman can project this small circle upon an artificial globe, or

a chart, the knowledge that he is upon this circle will be just as valuable to him to avoid dangers as the knowledge of either his latitude alone, or his longitude 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 small circle of the globe, described from the projection of the celestial object as a pole, we shall call a circle of position.

179. A small portion of this circle passing through two positions near together may be considered as a straight line, and this line is evidently perpendicular to the direction of the observed object, and it is this line which has been called the line of position.

180. The altitude of another celestial object S' being taken at the same time (or a second altitude of the same body after a suitable change of bearing) gives a second circle of position (Fig. 55, dotted circle). The observer being on the circumference of each of these circles must be at one of their points of intersection P or P. In good observations, selected within proper limits, these points, P and P', lie so far apart that, with a very rough latitude by account, there can be no difficulty in chosing the right one.

181. In finding the longitude by chronometer, an error in the latitude from which the time is computed, affects the resulting longitude in a corresponding degree. If several latitudes, not differing very greatly from the truth, be successively employed in calculating the longitude, each latitude will give a different longitude; and if with each latitude and the longitude deduced from it a point is found on the chart, it will be seen that these points lie very nearly in a straight line,* whose direction is the complement to the bearing of the observed object at the time of observation; in other words, the line of position is perpendicular to the direction of the object at the time of observation.

182. Two assumed latitudes, with the resulting longitudes, will be sufficient to determine the line of position, provided the latitudes do not greatly differ from the truth. Lay off on the chart these two positions, and join them by a straight line, somewhere on this line, or very near it, the observer's place is to be found; and thus, if this line (being produced) passes through a point of land or other object, the bearing of such object is known, though the ship's place on the line of its direction is not known.

183. After the observation the ship changes her place, which is usually the case at sea; but, if from any point in the line of position the ship's run

The direction of this line at any point is the direction of the tangent to the curve at this point. The direction of this tangent, as just stated, will be at right-angles to the bearing of the sun at that point. Hence, by a line of position we may determine the azimuth of the sun. By reference to the principles of construction of the Mercator's chart, it will be seen that only the loxodromic curve plots as a straight line. The circle of position would plot as an irregular figure, its greatest diameter coinciding with the arc of the meridian. The whole figure could not indeed be plotted on ordinary charts unless the zenith distances MA, M A', &c., were very small.

It is customary to plot only the small portion of the curve lying between the assumed latitudes, as that is all that is required. For small differences of latitude, as before observed, this would be practically a straight line.

be laid off in proper direction according to the ship's course, and through the point thus found a line be drawn parallel to the line of position, the ship's actual place is still somewhere on this line.

184. If a second observation of the same or any other celestial body be taken, separated from the first by a suitable difference of bearing, and treated in the same manner, another line of position may be drawn, which will also pass through the place of observation; and, therefore, the intersection of the two lines will give the ship's place, which may thus be obtained by projection on the chart, without the computation of a double altitude. The intersection is most distinctly marked when the lines cross at rightangles, and they must not in any case cut too acutely or the unavoidable errors of observation will be considerably magnified. The difference of bearings should not be less than 45° nor greater than 135°.

The foregoing remarks contain the elementary principles of Sumner's Method of finding the ship's position at sea.

185. The latitude, longitude, and time at ship being uncertain, it is required, from an altitude of the sun and the Greenwich time, as found from the chronometer, to project on Mercator's chart a line of position or a parallel of equal altitude, which shall pass through the position of the ship, and show-

1st-The bearing of the land.

2nd-The sun's true bearing or azimuth.

3rd-The error in the longitude corresponding to any error in the latitude.

186. The method of finding the approximate projection of a circle of position on the Mercator's chart is as follows:

SOLUTION BY CONSTRUCTION ON MERCATOR'S CHART.*

The Observation.—Take an altitude of the sun and note the corresponding time by chronometer.

RULE LXXV.

1o. The Calculation.-To the time by chronometer apply its original error and accumulated rate, as directed in Rule LXXI, page 218; the result is the Greenwich date at the instant of observation.

2°. Take out of the Nautical Almanac, page II, the sun's declination and the equation of time for the noon of Greenwich date, and the corresponding hourly difference for each; also, take out the sun's semidiameter.

3°. Reduce the sun's declination and equation of time to the Greenwich time, either by multiplying each hourly difference by the hours and decimals of an hour in the Greenwich time; or, by multiplying each hourly difference by the hours of

• Charts for this problem, price 6d. each, may be had of T. L. AINSLEY, South Shields, by Post, yd.