d CD, &c. is called the course from A to E. Let G be the situation of Greenwich, then G H is the latitude of Greenwich, G P its colatitude, A K the latitude of the point A, or of any place on the parallel AI; FK, or its equal E O, is the latitude of F, or of E, or of any place on the parallel FE; FA or E I is the difference of latitude of the points A and E, or of the parallels À I and E F, or of any places on those parallels.

As P G H is the first meridian, the longitude of G, or of any place on the meridian P G H, is nothing; the arc T H, or the angle TPH, which T H measures, is the longitude of the meridian P T, or of any place on that meridian; the arc H K, or the angle HPK, is the longitude of A, or of F, or of any place on the meridian PK; the arc H O, or the angle H P O, is the longitude of the points O, I, S, and E, or of any place on the meridian PO; the arc K O, which is the difference of H O and H K, is the difference of the longitudes of the meridians P K and P O, or of any two places on those meridians ; and T O, the sum of T H and H O, is the difference of longitude of the meridians P T and P O.

If a ship sail from A to E, E F will be her meridian distance; but if she sail from E to A, A I will be her meridian distance.

If A B, B C, C D, &c. be conceived to be equal, and indefinitely small, and their number indefinitely great; then the triangles A B b, B c C, &c. may be considered as indefinitely small right angled plane triangles. And as the angles B A b, C B c, &c. are equal, and the right angles A b B, B c C, &c. are equal, the remaining angles A Bb, BC c, &c. are equal; and as the sides A B, BC, &c. are also equal, these elementary triangles A B b, BC c, C D d, &c. will be all identical triangles; therefore A E will be the same multiple of A B, that the sum of A b, B c, C d, &c. is of A b; and that the sum of B b, C c, D d, &c. is of B b. But the sum of A b, B c, C d, &c. is equal to the whole difference of latitude A F, or E I, and the sum of B b, Cc, Dd, &c. is what is called the departure. Hence as A B b is a right angled plane triangle, a straight line equal to A E, and a straight line equal to A F, will form two sides of a right angled plane triangle, of which the third side will be equal to the sum B b, C c, D d, &c., and the angle included by the lines equal to A E and A F will be equal to the angle B A b, made by the rhumb line with the meridian on the globe.

Therefore when any two of these four elements are given, viz. the difference of latitude, course, nautical distance, and departure, the other two may be found by the resolution of a right angled plane triangle.

It may be observed here, that when the rhumb line on which a ship sails coincides with a meridian, the difference of latitude will be the nautical distance, and the ship will change her latitude only, not her

longitude; and when the rhumb line cuts the meridians at right angles, the nautical distance will be equal to the meridian distance, and the ship will change her longitude only, and not her latitude; but in every other direction of the rhumb line both the latitude and longitude will be changed.

We have seen however that with respect to course, nautical distance, difference of latitude, and departure, the results would be correct even when applied to the globe, if the computations were made on the supposition that the earth is a plane, the meridian parallel straight lines, and the rhumb lines also straight lines. This is called PLANE SAILING, and its usefulness is hence very obvious, But we have now to consider how the longitude of a ship may be computed when she sails a given distance from a known place, and on a given rhumb line; or generally to shew how a ship's change of longitude is connected with her change of place on the globe,

E

I

H

C

We shall begin with the most simple case of the general problem, viz. that in which a ship sails on a parallel of latitude, or when she changes her longitude only. Let PA and PB be two meridians, A B the arc of the equator which they intercept, or the difference of their longitudes; let the planes of the meridians intersect each other in P C, (C being the centre of the globe,) and let ACB be the plane of the equator, and D H E, F I G the planes of any two parallels of latitude, of which the parts corresponding to the difference of longitude A B, are D E and F G respectively. Then as D H and HE are respectively parallel to A C and B C (for each of these lines is perpendicular to P C) the angle DHE is equal to the angle ACB, and consequently the arc D E is the same portion of the whole parallel of which it is a part, that the arc A B is of the equator.

B

Hence D E will be to A B, as D H, the radius of the parallel, is to A C, the radius of the equator, or the radius of the globe. But D H is the sine of D P, or cosine of D A, the latitude of the parallel; therefore as the cosine of any given latitude is to any portion of a parallel in that latitude, so is radius to the corresponding arc of the equator, or the difference of longitude.

But in any right angled plane triangle, the base is to the hypothenuse as the cosine of the acute angle at the base is to radius; therefore if the angle at the base of a right angled plane triangle be made equal to any given latitude, and the length of the base equal to that of any given portion of a parallel in that latitude, the length of the hypothenuse will be equal to the arc of the equator, or the difference of longitude, corresponding to the given meridian distance, or the given arc of the parallel.

Again it may be shewn, in the same way, that D H is to F I as DE is to F G; or as the cosine of any latitude is to any portion of the parallel in that latitude, so is the cosine of any other latitude to the corresponding portion of its parallel. Hence if two right angled plane triangles have a common hypothenuse, equal to the difference of longitude of any two meridians, and the angles at the base of the triangles be respectively equal to any given latitudes, their bases will represent the arcs of two parallels which correspond, in their respective latitudes, to the proposed difference of longitude.

The above are the elementary principles of what is called PARALLEL SAILING; we have next to consider how the difference of longitude may be determined, when a ship sails upon an oblique rhumb.

For this purpose let us recur to the figure on page 148. In that figure, the elementary and equal parts of the departure B b, C c, D d, &c. are severally less than the corresponding parts of the parallel A I, but greater than those of the parts which correspond to them on the parallel FE. If however R S be the arc of the middle parallel between F E and A I, the elementary parts of the departure on one side of R S will exceed the corresponding parts of that line, by nearly as much as the elementary parts of the departure on the other side of RS are less than the corresponding parts of that line. Hence RS, the meridian distance in the middle latitude, will be nearly equal to the departure, or the sum of the elementary meridian distances that arise in sailing on an oblique rhumb. Therefore, considering the departure as a meridian distance on the middle parallel, between the latitude left and the latitude arrived at, the difference of longitude may be computed in the same manner as in parallel sailing,

For if the angle at the base of a right angled plane triangle be taken equal to the middle latitude, and the base equal to the departure, the hypothenuse will be nearly equal to the difference of longitude; not exactly equal, for though, in places near the equator, or indeed for such short distances as an ordinary day's run in any situation, except in very high latitudes, the meridian distance in the middle parallel may, without any important sacrifice of accuracy, be taken for the departure; they are in no case exactly equal,

This method of connecting the change in longitude with a ship's change of place, is called MIDDLE Latitude Sailing,

To exhibit more plainly the practical application of what has been here said, let DE (see the last figure) a given meridian distance, in a given latitude Ą D, be represented in the annexed figure, by the line B C; then if the angle A B C in the annexed figure be measured by the latitude Ą D, in the former one, A B in this figure will be equal to A B, the difference of longitude, in the other figure.

Again if A E (see figure, page 148) be represented by

B

D B in this figure, and A F in that figure by D C in this, then the angle BCD will be a right angle, and the side B C, the departure, will be correctly equal to the sum of B b, C c, &c. and nearly equal to R S, the meridian distance in the middle latitude. If therefore in the right angled triangle B C A, the angle C B A be made equal to the middle latitude, or to the latitude of the parallel R S, then A B, the hypothenuse, will be nearly equal to K O (see figure, page 148) the difference of longitude which the ship has made in sailing from A to E. If therefore, in these two connected right angled triangles, D C be the difference of latitude, B D the nautical distance, and C B A the middle latitude, then C D B will be the course, C B the departure, or the meridian distance in the middle latitude nearly, and A B will be the difference of longitude nearly.

The solution of the leading problems in navigation being thus in practice reduced to the computation of the different parts of two right angled plane triangles, as D B C, BCA, having the meridian distance in the middle latitude for a common side, and forming together one triangle, as A B D, the different parts of those triangles may be determined from each other by trigonometry, inspection, or otherwise; but there are one or two useful relations among the parts of the triangles, to which it may be well to draw the student's particular attention. 1. In the oblique angled triangle A B D

sin A: BD :: sin D: A B

But the angle A is the complement of the angle A B C, the middle latitude, therefore this proportion is

cos. mid lat : dist :: sin. course: diff long 2. In the right angled triangles D C B and B C A

and

Hence

and

Therefore

or

DC: CB:: rad: tan BDC
CB: BA:: cos C BA : rad
DC. tan DBC=CB. rad
BA. cos CBA = C B. rad
D C. tan DBC =
BA. COS CBA
DC: BA:: cos C BA: tan D B C

that is, diff lat: diff long :: cos mid lat: tan course.

These two proportions may be varied according to the data which may be given.

There is however another, and a very ingenious method, by which the connection between a ship's change in longitude and her change of place may be determined, called, from its inventor, MERCator's SAILING. In this method the globe is conceived to be so projected on a plane, that the meridians are parallel lines, and the elementary parts of the meridians and parallels bear, in all latitudes, the same proportion to each other that they do upon the globe.

Now, as the meridians are all great circles, and the parallels are all small circles of the globe, it is evident that, except at the equator,

no given part of a meridian can be equal to a like part of its parallel; and as the parallels diminish towards the poles, their like parts will also diminish; and, consequently, any given portion of a meridian will differ more from a like portion of a parallel, according as the parallel is at a greater distance from the equator.

But as in Mercator's projection of the globe the meridians are parallel to each other, the distance of any two meridians will, in all latitudes, be the same, and equal also to the difference of longitude of those meridians. Hence as the parallels are less than the equator, but in the projection they are all made equal to it, they are all in the projection increased beyond their relative magnitudes on the globe; those which are least in themselves, or whose latitudes are greatest, being increased most. If therefore the elementary parts of the meridians bear the same proportion to the like parts of their parallels in the projection, that they do to each other on the globe, the elementary parts of the meridian must, in the projection, be increased in the same proportion as those of the parallels are; and as the parallels distant from the equator are most increased, the elementary parts of the projected meridian must increase more and more as they are at a greater distance from the equator.

To investigate the proportion in which this increase must take place, we have already seen that, cosine latitude : radius, or radius: sect lat any portion of a parallel a like portion of the equator. But the equator and meridian are equal circles, therefore radius : sect lat any portion of the parallel a like portion of the

meridian.

:

Now in Mercator's projection the parallels are all equal to the equator, or to the meridian on the globe; therefore, in this projection, radius sect lat: an elementary part of a meridian on the globe the length of that elementary part in the projection. Or if we take radius equal to unity, and consider l' as an elementary portion of the meridian, we have rad (1): sect lat : 1': the length of 1' in the projection. Hence the length of l' in the projection in any latitude is equal to the natural secant of that latitude, the radius being unity.

Therefore in lat l', 1' of lat = sec l' in the projection

And consequently sec l' + sec 2' + sec 3' is the distance of the third minute of the meridian from the equator in the projection; and this sum, which is found in Table 3., is called the meridional parts of 3'.

In this manner may the meridional parts corresponding to every portion of the meridian be computed; but the smaller the part is