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tor, and since (Geom., Prop. 11, Cor. B. 5,) similar arcs are to each other as the radii of the circles to which they belong, we have

CB : CI :: dist. BD : diff. long. IQ. But ce is the cosine of the latitude is to the radius ci, that is,

CB : CI :: cos lat : R The first two terms of these proportions being the same, the last are proportional, and we have cos. lat. : Rad. :: distance : diff. long.

(1) Corollary : hence if the distance between any two meridians, measured on a parallel in latitude be d, and the distance of

L the same meridians, measured on a parallel, in latitude L' be D', we shall have (Geom., Prop. 11, Cor. B. 5,) COS L:D::COS L': D'

(2) By referring to proportion (1) it will be seen that if one of the legs of a right-angled triangle represent the distance run on any parallel, and the adjacent acute angle be equal to the degrees of lat. of that parallel, then the hypothenuse will represent the differ- dist. ence of longitude, since this hypothenuse will be determined by that proportion. It follows, therefore, that any problem in parallel sailing, may be solved by the traverse table, computed to degrees, as a simple case of plane sailing; for by considering the latitude as the course, and the distance as the difference of latitude, the corresponding distance in the table will express the difference of longitude.

diff Tong



A ship from latitude 53° 56' N., longitude 10° 18' E., has sailed due west, 236 miles : required her present longitude.

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Long. left

100 18' E.
Diff. long.= degrees = 6 40 W.


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2. If a ship sail E. 126 miles, from the North Cape, in lat. 71° 10' N., and then due N., till she reaches lat. 73° 26' N.; how far must she sail W. to reach the meridian of the North Cape ?

Here the ship sails on two parallels of latitude, first on the parallel of 71° 10', and then on the parallel of 73° 26', and makes the same difference of longitude on each parallel. Hence, by the corollary,

As cos. lat. 71° 10' arith. comp. 0.491044
: distance 126

:: cos. lat. 73 26



: distance 111.3


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3. A ship in latitude 32° N. sails due east, till her difference of longitude is 384 miles; required the distance run.

325.6 miles. 4. If two ships in latitude 44° 30' N., distant from each other 216 miles, should both sail directly south till their distance is 256 miles, what latitude would they arrive at ?

320 17'


^ 5. Two ships in the parallel of 47° 54' N., have 9° 35' difference of longitude, and they both sail directly south, a distance of 836 miles : required their distance from each other at the parallel left, and at that reached. 6

385.5 miles, and 47.9 miles.

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Middle Latitude Sailing.

100. Having seen how the longitude which a ship makes when sailing on a parallel of latitude may be determined, we come now to examine the more general problem, viz. to find the longitude a ship makes when sailing upon any oblique rhumb:

There are two methods of solving this problem, the one by what is called middle latitude sailing, and the other by Mercator's sailing. The first of these methods is confined in its application, and is moreover somewhat inaccurate even where applicable; the second is perfectly general, and rigorously true; but still there are cases in which it is advisable to employ the method of middle latitude sailing, in preference to that of Mercator's sailing; it is, therefore, proper that middle latitude sailing should be explained, especially since, by means of a correction to be hereafter noticed, the usual inaccutacy of this method may be rectified.

Middle latitude sailing proceeds on the supposition that the departure or sum

of all the meridional distances b'b, c'c,d'd, &c. from A to B, is equal to the distance M'M of the meridians

of A and B, measured

on the middle pa- E



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rallel of latitude be

tween A and B.

This supposition becomes very inaccurate when the course is small, and the distance run great; for it is plain that the middle latitude distance will receive a much greater accession than the departure, if the track A B cuts the successive meridians at a very small angle.

The principal approaches nearer to accuracy as the angle A of the course increases, because then as but little advance is made in latitude, the several component departures lie more in the immediate vicinity of the middle latitude parallel. But

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still, as in very high latitudes, a small advance in latitude makes a considerable difference in meridional distance, this principle is not to be recommended in such latitudes if much accuracy is required.

By means, however, of a small table of corrections, constructed by Mr. Workman, the imperfections of the middle latitude method may be removed, and the results of it rendered in all cases accurate, This table we have given at the end of the present volume.

The rules for middle latitude sailing may be thus deduced.

It has been seen at (Art. 97) that the difference of latitude, departure, and distance, sailed on any

Á oblique rhumb, will be all accurately represented by the sides ab', B'B, AB, of a plane triangle. Now, by the present hypothesis, the departure B's is equal to the middle latitude distance between the meridians of the places sailed from, and arrived at, so that the difference of longitude of the two places of the ship is the same as if it had sailed the distance b'b, on the middle latitude parallel ; the determination of this difference of longitude is, therefore, reduced to a case of parallel sailing, for BB', now representing the distance on the parallel, and an angle A' BB' being made equal to the latitude of that parallel, we shall have the difference of longitude, represented by the hypothenuse A's. We thus have the following theorems, viz., in the triangle A'B'B,

COS A'BB' : BB':: radius ; BA' that is,

1. Cos. mid. lat. : departure :: radius : diff. of long. In the triangle a'B.A,

sin. A' : AB : : sin. A : A'B; that is,

II. Cos. mid. lat. : distance :: sin. course : diff. long. In the triangle ABB', we have the proportion, (Art. 41)

R : tan A :: AB' : BB'


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

AB' : A'B :: cos. A'BB' : tan. A ;

that is,



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 :* Let

d= proper diff. of lat.
D meridional diff. of lat.
m = middle latitude.
M= m + correction.

L=diff. of longitude.
Then, (Art. 100, Form III.)

COS M XL tan course =

d But, (Art. 101, Rule 1.)

rad. XL tan course


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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.

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