COR. 1 The length of a degree of longitude in any parallel is as the cosine of the latitude,

COR. 2. As the cosine of any parallel is to the cosine of any other parallel, so is the length of any arc of the former to the length of the corresponding are of the latter.

COR. 3. From this theorem is derived the method of constructing the line of longitude, in the Plane Scale.*[See Plane Scale.]

EXAMPLES.

1. A ship in latitude 44° 12 N. sails E. 79 miles. Required her difference of longitude.

Construction.

place C. Therefore, as the circumference of the equator: the circumference of the parallel :: radius : the cosine of the latitude.

But similar parts are as the wholes. Therefore, the difference of longitude between any two meridians is to the distance be tween those meridians, measured on any parallel of latitude, as radius to the cosine of the latitude. Which was to be proved.

* The truth of the process may be easily shewn. For, since AF, equal to radius CE, is to the cosines A50, A4c, A30, &c. as 60 to 50, 40, 30, &c. respectively, an equatorial degree, or 60 miles, must have the same ratios to the degrees of the parallels, of whose latitudes the lines are the cosines. But E50, E40, E30, &c. are the chords of those latitudes. Therefore, the latitude being taken on the chord line, the corresponding number on the line of longitude shews the length of a degree on that parallel, in 60th parts of an equatorial degree, or in nautical miles.

BE in E. Join BE, and the right line BE is the dif ference of longitude.

This construction is a geometrical solution of the proportion in the theorem, taken in an inverted order. That is, in the similar triangles ACD, ABE, cosine latitude AC radius AB :: departure, or meridian distance CD : difference of longitude BE.

Another Construction.

Make an angle CAB of as many degrees and minutes as the latitude, From the angular point A set off AC79 miles the departure or meridional distance. From C erect the perpendicular CB, meeting A AB in B. AB will be the difference of longitude required.

B

This construction exhibits the same proportion as the foregoing. For, in the right-angled triangle ABC, cosine latitude radius: departure AC: difference of longitude AB.

Computation.

Opposite to 44

the latitude on the line of chords, stands 43 on the line of longitude, which is, therefore, the number of miles in a degree of longitude, in that latitude. Whence 43: 60 :: 79: 110 miles, or the difference of longitude.

Solution by the Traverse Table.

With the colatitude as a course, and the meridional distance as departure, find the corresponding distance. This distance is the difference of longitude. That is, with 46° at the bottom of the page, seek 79 in the column of departure, and the distance 110 is the difference of longitude.

The reason of this operation is clear from considering,: that in the right-angled triangle ABC, the angle B, or the colatitude, is opposed to the departure AC, exactly the same as if it were an angle of course, and that the difference of longitude AB is the side opposite to the right angle, in like manner as-the distance in plane sailing.

2. A ship in latitude 46° 5′ N. is bound to Gibraltar, which lies on the same parallel, and in 4° 46′ W. longitude. By an eclipse of the moon, she finds her present longitude to be 13° 8′ W. What is her meridional dis

tance?

Construction.

Construction.

The difference of longitude, or difference between 4 46 and 13° 8', is 8° 22′, or 502 miles.

With the radius AB, or sine 90°, describe the arc BE, on which set off BE

HA

Solution by the Line of Longitude..

Opposite to 46° the latitude on the line of chords, stands 41, the number of miles in a degree of longitudein that latitude. Therefore, 60: 41: miles, the meridional distance or departure.

By the Traverse Table.

502 348

With the colatitude 44° as a course, and, half the

difference of longitude

251 as distance, is found 174

departure, which doubled is

348, the meridional dis

tance or departure.

MIDDLE LATITUDE SAILING.

MIDDLE LATITUDE SAILING is a method of finding a ship's place on the globe, by applying the principles of parallel sailing to a course made good on an oblique rhumb.

When a ship sails on an oblique rhumb, that is, on a course between the meridian and parallel, she alters at the same time both her latitude and meridional distance. But the departure, found by plane sailing, will not be her meridional distance, either at the latitude sailed from or come to. For, at the greater latitude it will be too great, because the meridians converge toward the poles; and for a contrary reason, it will be too small at the less latitude.. Whence it follows, that the departure is the true meridional distance, measured on a parallel, which lies between the two extreme parallels, namely, that sailed from and that come to. In middle latitude sailing, the departure is taken for the meridional distance, measured on the paral

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VOL. II.