RHUMB

, Rumb, or Rum, in Navigation, a vertical circle of any given place; or the intersection of a part of such a circle with the horizon. Rhumbs therefore coincide with points of the world, or of the horizon. And hence mariners distinguish the Rhumbs by the same names as the points and winds. But we may observe, that the Rhumbs are denominated from the points of the compass in a different manner from the winds: thus, at sea, the north-east wind is that which blows from the north-east point of the horizon towards the ship in which we are; but we are said to sail upon the north-east Rhumb, when we go towards the north-east.

They usually reckon 32 Rhumbs, which are represented by the 32 lines in the rose or card of the compass.

Aubin defines a Rhumb to be a line on the terrestrial globe, or sea-compass, or sea-chart, representing one of the 32 winds which serve to conduct a vessel. So that the Rhumb a vessel pursues is conceived as its route, or course.

Rhumbs are divided and subdivided like points of the compass. Thus, the whole Rhumb answers to the cardinal point. The half Rhumb to a collateral point, or makes an angle of 45 degrees with the former. And the quarter Rhumb makes an angle of 22° 30′ with it. Also the half-quarter Rhumb makes an angle of 11° 15′ with the same.

For a table of the Rhumbs, or points, and their distances from the meridian, see Wind.

Rhumb-Line, Loxodromia, in Navigation, is a line prolonged from any point of the compass in a nautical chart, except the four cardinal points: or it is the line which a ship, keeping in the same collateral point, or rhumb, describes throughout its whole course.

The chief property of the Rhumb-line, or loxodromia, and that from which some authors define it, is, that it cuts all the meridians in the same angle.

This angle is called the angle of the Rhumb, or the loxodromic angle. And the angle which the Rhumbline makes with any parallel to the equator, is called the complement of the Rhumb.

An idea of the origin and properties of the Rhumbline, the great foundation of Navigation, may be conceived thus: a vessel beginning its course, the wind by which it is driven makes a certain angle with the meridian of the place; and as we shall suppose that the vessel runs exactly in the direction of the wind, it makes the same angle with the meridian which the wind makes. Supposing then the wind to continue the same, as each point or instant of the progress may be esteemed the beginning, the vessel always makes the same angle with the meridian of the place where it is each moment, or in each point of its course which the wind makes.

Now a wind, for example, that is north-east, and which consequently makes an angle of 45 degrees with the meridian, is equally north-east wherever it blows, and makes the same angle of 45 degrees with all the meridians it meets. And therefore a vessel, driven by the same wind, always makes the same angle with all the meridians it meets with on the surface of the earth.

If the vessel sail north or south, it describes the great circle of a meridian. If it runs east or west, it cuts all the meridians at right angles, and describes either the circle of the equator, or else a circle parallel to it.

But if the vessel sails between the two, it does not then describe a circle; since a circle, drawn obliquely to a meridian, would cut all the meridians at unequal angles, which the vessel cannot do. It describes theresore another curve, the essential property of which is, that it cuts all the meridians in the same angle, and it is called the loxodromy, or loxodromic curve, or Rhumbline.

This curve, on the globe, is a kind of spiral, tending continually nearer and nearer to the pole, and making an infinite number of circumvolutions about it, but without ever arriving exactly at it. But the spiral Rhumbs on the globe become proportional spirals in the stereographic projection on the plane of the equator.

The length of a part of this Rhumb-line, or spiral, then, is the distance run by the ship while she keeps in the same course. But as such a spiral line would prove very perplexing in the calculation, it was necessary to have the ship's way in a right line; which right line however must have the essential properties of the curve line, viz, to cut all the meridians at right angles. The method of effecting which, see under the article Chart.

The are of the Rhumb-line is not the shortest distance between any two places through which it passes; for the shortest distance, on the surface of the globe, is an arc of the great circle passing through those places; so that it would be a shorter course to sail on the arc of this great circle: but then the ship cannot be kept in the great circle, because the angle it makes with the meridians is continually varying, more or less.

Let P be the pole, RW the equator, ABCDEP a spiral Rhumb, divided into an indefinite number of equal parts at the points B,C,D, &c; through which are drawn the meridians, PS, PT, PV, &c, and the parallels FB, KC, LD, &c, also draw the parallel AN. Then, as a ship sails along the Rhumbline towards the pole, or in the direction ABCD &c, from A to E, the distance sailed AE | is made up of all the small equal parts of the Rhumb AB + BC + CD + DE; and the sum of all the small differences of latitude AF + BG + CH + DI make up the whole difference of latitude AM or EN; and the sum of all the small parallels FB + GC + HD + IE is what is called the departure in plane sailing; and ME is the meridional distance, or distance between the first and last meridians, measured on the last parallel; also RW is the difference of longitude, measured on the equator. So that these last three are all different, viz, the departure, the meridional distance, and the difference of longitude.

If the ship sail towards the equator, from E to A; the departure, difference of latitude, and difference of longitude, will be all three the same as before; but the meridional distance will now be AN, instead of ME; the one of these AN being greater than the departure FB + GC + HD + IE, and the other ME is less than the same; and indeed that departure is nearly a mean proportional between the two meridional distances ME, AN. Other properties are as below.

1. All the small elementary triangles ABF, BCG, CDH, &c, are mutually similar and equal in all their parts. For all the angles at A, B, C, D, &c are equal, being the angles which the Rhumb makes with the meridians, or the angles of the course; also all the angles F, G, H, I, are equal, being right angles; therefore the third angles are equal, and the triangles all similar. Also the hypotenuses AB, BC, CD, &c, are all equal by the hypothesis; and consequently the triangles are both similar and equal.

2. As radius : distance run AE :: sine of course [angle]A : departure FB + GC &c, :: cosin. of course [angle]A : dif. of lat. AM. For in any one ABF of the equal elementary triangles, which may be considered as small right-angled plane triangles, it is, as rad. or sin. [angle]F : sin. course A :: AB : FB :: (by composition) the sum of all the distances AB + BC + CD &c : the sum of all the departures FB + GC + HD &c.

And, in like manner, as radius : cos. course A :: AB : AF :: AB + BC &c : AF + BG &c.

Hence, of these four things, the course, the difference of latitude, the departure, and the distance run, having any two given, the other two are found by the proportions above in this article.

By means of the departure, the length of the Rhumb, or distance run, may be connected with the longitude and latitude, by the following two theorems.

3. As radius : half the sum of the cosines of both the latitudes, of A and E :: dif. of long. RW : departure.

Because RS : FB :: radius : sine of PA or cos. RA, and VW : IE :: radius : sine of PE or cos. EW.

4. As radius : cos. middle latitude :: dif. of longitude : departure.—Because cosine of middle latitude is nearly equal to half the sum of the cosines of the two extreme latitudes.