comparing this with the first proportion above, observing that the extremes of this are the means of that, we have

that is,

AB' A'B COS. A'BB': tan. a;

111. Diff. lat. diff. long. : : cos. mid. lat. : tan. course.

These three proportions comprise the theory of middle latitude sailing, and when to the middle latitude the proper correction, taken from Mr Workman's table, is added, these theorems will be rendered strictly accurate.

This is Table VI; the middle latitude is to be found in the first column to the left; in a horizontal line with which, and under the given difference of latitude, is inserted the proper correction to be added to the middle latitude to obtain the latitude in which the meridian distance is accurately equal to the departure. The formula for constructing this table is obtained as follows :*

1. A ship, in latitude 51° 18' N., longitude 22° 6' W., is

*The investigation of this formula should be postponed till after reading the next article, and may be omitted entirely.

bound to a place in the S. E. quarter, 1024 miles distant, and in lat. 37° N.: what is her direct course and distance, as also the difference of longitude between the two places?

1024 3.010300 cos. mid. lat. 44° 9' ar. comp. 0.144167

For the course.

As distance

: radius

10.

: tan.course 33 5 2.933487:: diff. lat. 858

2.891553

:: cos. course 33° 5' 9.923187: diff. long. 779

In this operation the middle latitude has not been corrected, so that the difference of longitude here determined is not without error. To find the proper correction, look for the given middle latitude, viz. 44° 9' in the table of corrections, the nearest to which we find to be 45o ; against this and under 14° diff. of lat. we find 27', also under 15° we find 31', the difference between the two being 4'; hence corresponding to 14° 18' the correction will be about 28'. Hence the corrected middle latitude is 44° 37', therefore,

cos. corrected mid. lat. 44o 37' ar. comp. 0.147629

therefore, the error in the former result is about 6 miles.

2. A ship sails in the N.W. quarter, 248 miles, till her departure is 135 miles, and her difference of longitude 310 miles required her course, the latitude left, and the latitude come to.

Course N. 32° 59′ W.; lat. left 62° 27′ N.; lat. in 65° 55′ N.

3. A ship, from latitude 37° N., longitude 9° 2′ W., having sailed between the N. and W., 1027 miles, reckons that she has made 564 miles of departure; what was her direct course, and the latitude and longitude reached ?

Course N. 33° 19' W. or N. W. by N. nearly; lat. 51° 18′
N.; long. 22° 8' W.

4. Required the course and distance from the east point of St. Michael's, lat. 37° 48′ N., long. 25o 13′ W., to the Start Point, lat. 50o 13' N., long. 3° 38′ W., the middle latitude being corrected by Workman's Table.

Course N. 51° 11' E.; distance 1189 miles.

Mercator's Sailing.

101. It has been already seen that when a ship sails on any oblique rhumb, the difference of latitude, the departure, and the distance run, are truly represented by the sides of a right-angled plane triangle. The departure B'B

small

ช

B

represents the sum of all the very meridian distances, or elementary departures, b'b, c'c, &c. in the diagram, at Art. 100, the difference of latitude AB' represents the sum of all the corresponding small differences in the figure referred to; and the distance AB, the sum of all the distances to which these several departures and differences belong, and each of these elements is supposed to be taken so excessively small as to form on the sphere a series of triangles, differing insensibly from plane triangles. Let Ab'b in the annexed diagram represent one of these elementary triangles, b'b will be one of the elements of the departure, and Ab', the corresponding difference in latitude; and as b'b is a small portion of a parallel of latitude, it will be to a similar portion of the equator, or of the meridian, as the cosine of its latitude to radius (Art. 99). This similar portion of the equator, or of the meridian, being the difference of longitude between b' and b. Suppose now the distance Ab prolonged to p, till the departure p'p is equal to the difference of longitude of b', and b, then b'b will be to p'p as the cosine of the latitude of b'b to the radius; but b'b: p'p : Ab' Ap'; hence the proper difference of latitude ab' is t the increased difference ap' as the cosine of the latitude of

to the radius. Calling, therefore, the proper difference of latitude d, the increased difference of latitude D, the latitude of b'b, l, and the radius 1, which it is in the table of natural sines, we have

The ship, therefore, having made the small departure b'b, and the difference of latitude ab', must continue her course till the difference of latitude becomes D, in order that her departure may become equal to the difference of longitude corresponding to b'b. Conceiving all the elementary distances to be in this manner increased, the sum of all the corresponding increased departures will necessarily be the whole difference of longitude made by the ship during the course; to represent, therefore, the difference of longitude due to the departure Â'Â, and difference of latitude AB', we must prolong AB' till Ac' is equal to the sum of all the elementary differences increased as above, and the departure c'c, due to this difference of latitude, will represent the difference of longitude actually made in sailing from a to B. The determination of Ac' requires the previous determination of all its elementary parts; if d be taken equal to 1', each of these parts will be expressed by D= 1′ sec. l, or D = sec. l, it being understood that sec. I expresses so many minutes or geographic miles, from which equation the values of n, corresponding to every minute of 1, from the equator to the pole, may be calculated; and by the continued addition of these there will be obtained, in succession, the values of the increased latitude correspondto 1', 2', 3', &c. of proper latitude; these values are called the meridional parts, corresponding to the several proper latitudes, and when registered in a table, form a table of meridional parts, given in all books on Navigation.

The following may serve as a specimen of the manner in which such a table may be constructed, and, indeed, of the manner in which the first table of meridional parts was actually formed by Mr. Wright, the proposer of this ingenious and valuable method.

Mer. pts. of 1'=nat. sec. 1'.

Mer. pts. of 2'-
Mer. pts. of 3'
Mer. pts. of 3'
sec. 4',

nat. sec. 1' + nat. sec. 2'.

=

nat. sec. 1' + nat. sec. 2' + nat. sec. 3'.
nat. sec. 1' + nat. sec. 2′ + nat. sec. 3'+nat.

Hence, by means of a table of natural secants, we have

There are other methods of construction, but this is the most simple and obvious. The meridional parts, thus determined, are all expressed in geographical miles, because in the general expression D1' sec. 7, 1' is a geographical mile.

Having thus formed a table of meridional parts, (Table VII. at the end,) if we enter it with the latitudes sailed from, and come to, and take the difference of the corresponding parts in the table, the remainder will be the meridional difference of latitude, or the line ac' in the preceding diagram, and the difference of longitude c'c will then be obtained by this proportion, viz.

1. As radius is to the tangent of the course, so is the meridional difference of latitude to the difference of longitude; or if the departure be given instead of the course, then the proportion will be

2. As the proper difference of latitude is to the departure, so is the meridional difference of latitude to the tangent of the course. Other proportions immediately suggest themselves from the preceding figure.

102. As an example of Mercator's, or more properly of Wright's, sailing, let us take the following:

1. Required the course and distance from the east point of St. Michael's to the Start Point.