We seek in the Traverse Table till the diff. of lat. 136-6, and dep. 180-5, are found opposite each other, in their respective columns; the nearest to these are 1805 and 1360, which give the course (at the bottom of page, dep. being the most) S. 53° W., and distance 226". This is an illustration of the remark, No. 141, page 89, that when the departure exceeds the difference of latitude, the course is more than 45°.

Lat. left

D. Lat. 136'6

38° 25' N. = 2 17 S.

The lat. in is found according to
Rule XLVI, page 93.

Lat. in (or arrived at)

36 8 N.

Ex. 3. A ship from lat. 37° 24′ S., sails the following true courses:-S.W. by S., 20 miles; West, 16 miles; N.W. by W., 28 miles; S.S.E., 32 miles; E.N.E., 14 miles; S.W., 36 miles required the lat. in, also the course and distance made good.

DIFF. LAT.

DEPARTURE.

COURSES.

DIST.

N.

S.

E.

W.

S. 3 W.

20

16.6

II'I

We seek in the several pages of the Traverse Table II, for the diff. lat. 507; and dep. 507; the nearest found to these are diff. lat. 50°9, dep. 50'9, give course S. 45° W., distance 72 miles.

The diff. lat. and dep. being of equal amount, the course is 45°, or 4 points, which illustrates the remark, No. 141, page 89.

Ex. 4.

Lat. left.
Diff. lat. 507 =
Lat. arrived at

37°24' S.
51 S.

38 15 S.

The lat. sailed from being South, and the ship having sailed South, the ship has evidently increased her South lat., whence the sum of lat. from and diff. lat. is taken to obtain

lat. in.-(See Rule XLVI, 1°, page 93).

A ship from lat. 20° 56′ N. sails (all true courses) N.W. by N., 20 miles; S.W., 40 miles; N.E. by E., 60 miles; S.E., 55 miles; W. by S., 41 miles; E.N.E., 66 miles: required the latitude in, also the course and distance made good.

Course due East, and dist. 702, the same as the departure. (See No. 141, page 89). The Traverse Table being filled up, the sums of the Northings and Southings are both 75'2, and being of contrary directions, show that the ship has returned to the same parallel of latitude which she sailed from. The sun of the Eastings is 149'8, and that of the Westings 796; their difference, 70-2, shows that the ship has gained so much to the Eastward, that being the greater. Consequently the Course is due East, and the Distane 70'2, the same as the departure.

1

Ex. 5. A ship sails from a place in lat. 1° 15′ N., the following true courses: -S.W. by W., 45 miles; E.S.E., 50 miles; S.W., 30 miles; S.E. by E., 60 miles; S.W. & S., 63 miles: required the latitude in, also the course and distance made good.

DIFF. LAT.

DEPARTURE.

The Traverse Table being completed, the sum of the Southings is 149.2 miles, and to that amount the ship has altered her latitude. The miles of departure in the East column are 96'1, and those in the West column are also 961; but as the East and West departures destroy one another, there is no resulting departure; and therefore, it is not necessary to refer to the Traverse Table. The ship is under the same meridian as she sailed from; consequently, the course is due South, and the distance sailed is equal to the diff. of lat., viz., 149 2. This is according to No. 141, page 89.

Ex. 6. A ship from latitude 46° 10' N., sails as follows: S. 48° E., 25 miles; S. 51° E., 18.9 miles; N. 87° E., 12.4 miles; S. 70° E., 14'5 miles; S. 68° E., 21'6 miles; N. 25° W., 16.4 miles; N. 8° E., 7.8 miles; N. 19° E., 13.7 miles; N. 76° E., 39.6 miles; required the lat. in, also the course and distance made good.

Explanation.-The course 48° (found at the bottom of one of the pages in Table I), and dist. 25 (in dist. column), opposite this last stands 16-7 diff. lat., and 18.6 dep., and as the ship is sailing on a S. and E. course, the diff. lat. is written in the diff, lat. column, and the dep. in the East column.

To take out the next course and distance we proceed thus:-51° and dist. 18:9, taken as 189, give diff. lat. 1189, and dep. 146'9, now removing the decimal point in each, one place to the left we have diff. lat. 118-9 = 11.89, and dep. 146′9 14'69; we do not require to use both the decimal places, but if, as in the case with both diff. lat. and dep., the second decimal figure amounts to 5, we add 1 to the first, and the diff. lat. thus becomes 119, and the dep. 147.

The third course is N. 87° E., and distance 12'4; then 87° and dist. 124 (omitting the decimal point), give diff. lat. 06'5, and dep. 123-8; now dropping the tenths in each, viz., the 5 and the 8, and increasing the preceding figures by 1 in each case, as the tenths exceed 5, we have, after removing the decimal point one place to the left, diff. lat. o'7, and dep. 124.

Proceed in this way with the remaining courses, except the last, in which case the distance being more than 300, we proceed as follows:

Course 76°, and dist. 300 give diff. lat. 72'6 and dep. 291'1

Now, cutting off the last figure in each, the 8 and the 2, and removing the decimal point one place to the left, we have diff. lat. 9'6 (not 9'5, as the figure 8 which is dropped exceeds 5, one is added to ihe tenths), and dep. 38.4.

I.

A ship from the Texel in lat. 52° 58′ N., sails W. by N., 44 miles; S. by E., 45 miles; W. by S., 35 miles; S.S.E., 44 miles; W.S.W. W., 42 miles; find diff. lat. and dep., the course and dist, made good, also the lat. arrived at.

2. A ship from Heligoland, lat. 54° 12' N., sails W.S.W., 12 miles; N.W., 24 miles; S. by W., 20 miles; N.W. by W., 32 miles; S. by E., 36 miles; W. by N. N., 42 miles; S.S.E.E., 16 miles; W. N., 45 miles: required diff. lat. and dep., course and dist. made good, also the lat. arrived at.

3. A ship sails from lat. 3° 50' N., sails S.S.W., 112 miles; S. by E., 86 miles; S.S.E., 112 miles; S. by W., 86 miles: find diff. lat. and dep., the course and dist., made good, also the lat. arrived at.

4. Yesterday we were in lat. 19° S., and since then have sailed S.E. S., 13 miles; S. by E., 19 miles; S.E. by E., 22 miles; E. by S. S., 32 miles; N.N.E., 20 miles; N. by W. W., 27 miles; N.E. by E. E., 24 miles; S.W. S., 10 miles.

5. A ship from lat. 1° N., sails East, 8 miles; E. & N., 20 miles; S.E. by E., 33 miles; S. & W., 31 miles; N.E. § N., 43 miles; South, 28 miles; S. † E., 21 miles; S. by W. & W., 12 miles required diff. lat. and dep., course and dist. made good, and also the lat. in.

6. A ship from lat. 1° 10′ S., sails E. by N. § N., 56 miles; N. 4 E., 80 miles; S. by E. E., 96 miles; N. E., 68 miles; E.S.E., 40 miles; N.N.W. W., 86 miles; E. by S., 65 miles find diff. lat. and dep., course and dist. made good, also the lat. in.

7. A ship from lat. 47° 12' N., sails S. 31° W., 16 miles; N. 72° E., 13'1; S. 52° W., 15' ; S. 44° E., 15''1; N. 44° W., 19''7; N. 77° E., 11''4; S. 40° W., 16'; S. 14° E., 6'; required the course and dist. made, the lat. arrived at, and the dep. made.

8. Since leaving lat. 34° 11' N., we have sailed the following courses :-N. 36° W., 27'; N. 24° E., 30'; S. 75° W., 47′; S. 80° W., 29'; N. 72° W., 42'; N. 78° W., 34'; S. 12° E., 28'; required the course and dist. made, the lat. arrived at, and the dep. made.

9.

Since leaving lat. 36° 35′ S., the ship has sailed N. 84° W., 18'; N. 89° W., 30'4; N. 67° W., 29''9; N. 39° W., 33′ ́9; N. 8° W., 25′′9; N. 73° W., 34'9; N. 86° W., 44′′7 ; S. 65° E., 56'; required the lat. arrived at, and the course and dist. made good.

10. A ship sails from lat. 1° 46' N., on the following compass courses, viz., S.W. W., 62 miles; S. by W., 16 miles; W. S., 40 miles; S. W. & W., 29 miles; S. by E., 30 miles; and S. E., 14 miles: required the lat. arrived at, and the course and dist. made good, the variation of the compass being 21° W.

PARALLEL SAILING.

248. When two places lie on the same parallel of latitude, or due east or west of each other, the distance between them estimated along a parallel, or E. and W. (which is all departure) is converted into difference of longitude; or, on the other hand, the difference of longitude is converted into distance by Parallel Sailing.

Since the meridians are all parallel at the equator and meet at the poles, the distance between any two meridians, measured east and west, is less as the latter is greater—that is, the absolute number of miles, or of feet, in a degree of longitude, is less as the latitude in which they are measured is greater. Hence, also, a given number of miles between two meridians corresponds to a greater difference of longitude, as the latitude in which they are measured is greater. Eor example, two places in lat. 10° and distant 60 miles east and west from each other, have 60''9 diff. long. In lat. 60° N. or S., two places similarly situated have 2° o' diff. long., while at 73° the diff. long. is 3° 25'. Questions of this kind are solved by Parallel Sailing.

249. Given the departure made good on a given parallel of latitude, to find the diff. of long. corresponding thereto.

RULE LXIX.

1°. Take out of the Tables the log. secant of latitude (rejecting 10 from index), and the log. of departure made good.

2o. Add these logs. together, and find the nat. number corresponding thereto. The result is the difference of longitude required.

250. In parallel sailing the latitude being constant, the difference of longitude bears a constant ratio to the distance, and all problems may be completely solved by the solution of a right-angled plane triangle, and therefore by inspection of the Traverse Table by

RULE LXX.

With the latitude of the parallel as a course, and the distance sailed on it as difference of latitude, the corresponding distance, in the Traverse Table, is the difference of longitude.

In each of the following examples the difference of longitude is required :

251. The method of parallel sailing will apply correctly enough for all practical purposes to cases where the course is nearly east and west (true). In latitudes not higher than 5o, when the distance does not exceed 300 miles, the departure may be be used at once for the difference of longitude, the resulting error scarcely exceeding one mile.

252. Given the difference of longitude of two places on the same parallel, to find their distance as measured along the parallel.

RULE LXXI.

To the log. of the diff. of long. add the cosine of lat.; the sum (neglecting 10) is log. of the distance required.

EXAMPLE.

Ex. 1. Required the distance between St. Abb's Head, in latitude 55° 55′ N., longitude 2° 10′ W., and Uraniberg in the same latitude, but in longitude 12° 52′ E.