#### 60 REVISION OF BASIC NAVIGATIONAL CONCEPTS

##### 231. THE EARTH AND ITS POSITION ON ITS SURFACE

A Sphere is a surface every point on which is equidistant from one and the same point, called the centre. The distance of the surface from the centre is called the radius of the sphere. The Earth, the planet on which we live, is an oblate spheroid, which means it is a sphere slightly flattened in the Polar regions. This flattening is so small that, for the purposed of practical Navigation, the Earth is considered to be a perfect sphere. Errors arising from this assumption are negligible.

A Great Circle. If we slice a section off a sphere, the flat surface of the section, where it intersects with the sphere’s surface, always forms a circle. If this flat surface, or plane of the circle, passes through the centre of the sphere, the circle is called a Great Circle. (fig. 60-2(a)).

A Small Circle is a circle on a sphere’s surface whose plane does not pass through the centre of the sphere (fig. 60-2(b)).

Position. In order to describe the position of a point on the Earth’s surface, it is only necessary to quote its shortest angular distance from two lines, perpendicular to each other, on the surface of the Earth. The two reference lines are the Equator and the meridian of Greenwich (fig. 60-3).

The Equator is a particular great circle on the Earth’s surface whose plane is perpendicular to the Earth’s axis of rotation (i.e., the line joins the Poles). It divides the Earth into the Northern and Southern Hemispheres, and is the reference line from which latitude is measured.

The Greenwich Meridian is a semi-great circle, which passes through Greenwich Observatory (London, England), and is the reference line from which longitude is measured. It is sometimes called the Prime Meridian.

Parallels of Latitude are all small circles parallel to the Equator, named North or South according to the hemisphere in which they are situated (fig. 60-4(a)).

Meridians of Longitude are all semi-great circles, terminating at the Poles, and named East or West according to whether they lie East or West of the Greenwich Meridian (fig.60-4(b). The continuation of the Greenwich Meridian beyond the Poles on the opposite side of the Earth is the meridian of 180º longitude.

Latitude of a certain place is the arc of a meridian between the Equator and the Parallel of Latitude on which the place lies. Note that the latitude of the Pole is 90º. Latitude is named north (N.) or south (S.) depending on whether the place described is north or south of the Equator.

Longitude of a certain place is the arc of the Equator between the meridian of Greenwich and the meridian on which the place lies. All places which are situated between the Greenwich meridian and the meridian which lies opposite to the Greenwich meridian (180º) are said to lie East or West longitude depending on whether they are east or west of the Greenwich meridian respectively.

The concepts derived above will perhaps become cleared by careful study of fig. 60-5.

In the figure, which represents the Earth, N and S are the North and South Poles respectively. The Great Circle WFDGE is the Equator, and the semi-great circle NBCDS is the Meridian of Greenwich. O is the Centre of the Earth, in the very heart of the sphere. X is a place on a particular parallel of N. latitude (XBJ) and on a particular meridian west of the Greenwich meridian (NXHFS).

The latitude of place X (by the definition given above) is the arc of this meridian between the Equator and the parallel of latitude on which X lies, so on our figure this is the arc FX. It will be seen that this arc is the same as the angle FOX in the vertical plane at the centre of the Earth. If this angle were, say, 55º, then we would say that place X is in Latitude 55º N., this being its angular distance north of the Equator.

The longitude of place X (by the definition given above) is the arc of the Equator between the Greenwich meridian and the meridian on which X lies. The longitude of X is therefore the arc DF, which is the same as the angle DOF in a horizontal plane at the centre of the Earth. If this angle were, say, 70º, then we would say that place X is in longitude 70º W., because this is its angular distance west of the Greenwich meridian.

Similarly, the latitude of place Y is the arc GY, which is equal to the angle GOY at the centre of the Earth, and as Y lies north of the Equator it is named “north latitude”. The longitude of Y is the arc of the Equator DG, which is equal to the angle DOG at the centre of the Earth. Place Y, however, is east of the Greenwich meridian whereas place X was to the west of it, so the longitude of Y would be quoted as “so-many degrees East”.

It is frequently necessary in Navigation to know the difference of latitude and the difference of longitude between two places.

The Difference of Latitude (usually abbreviated to “d.lat.”) between two places is the arc of a meridian cut off between the parallels of latitude of the two places. If the two places are in the same hemisphere, the d.lat. is found by subtracting the “lower” from the “higher” latitude. If the two places are in different hemispheres, the d.lat. is found by adding the two latitudes. In fig. 60-5, the d.lat. between places X and Y is the arc of any meridian cut off by the parallels of latitude through X and Y. The d.lat. is therefore any of the arcs XH, BC or JY. Its value would be the latitude of Y subtracted from the latitude of X because both are in the northern hemisphere.

The Difference of Longitude (usually abbreviated to d.long.) between two places is the shorter arc of the Equator between the meridians of the two places. If the two places are in longitudes of the same name (i.e., north are east longitude or both are west longitude), the d.long. is found by subtracting the lesser from the greater. When the two places are in differently named longitudes (i.e., one is east and the other is west), the d.long. is found by adding the longitude of the two places. If, however, the result of this addition is greater than 180, it should be subtracted from 360 to give the correct d.long. (a d.long. cannot be greater than 180, the meridian on the opposite side of the Earth to the Greenwich meridian). In fig. 60-5, the d.long. between X and Y is the shorter arc of the Equator between the meridians of X and Y, that is the arc FG. Its value would be the sum of the longitudes of X and Y because X is in W. Longitude and Y is in E. Longitude The longitude of X is arc DF and the longitude of Y is arc DG, so the d.long. is arc DF+DG=FG.

Suppose X is in Lat. 55º 16’N., Longitude 70º 49’W. and Y is in Lat. 33º 52’N., Longitude 29º 18’E., then the d.lat. and the d.long. between the two places would be:

Lat.                                                                      Long.

X          55º 16’N.          ) Same names           X              70º 49’W.    ) Different names

Y .        33º 52’N.          ) SUBTRACT               Y               29º 18’E.     ) ADD

d.lat.   21º 24’                                                  d.long.  100º 07’

#### 232. RHUMB LINE AND GREAT CIRCLE TRACKS

The most convenient course for a ship to steer is a steady course, one along which the bearing of her head remains constant for a given period of time. Her track will then cut all meridians at the same angle. Ships are steered from place to place by means of a compass, and the radial lines extending from the centre of the compass card to the various “points” of the compass are known as “rhumbs”; when a ship’s head is steadied on a certain compass point, the fore and aft line of the ship lies in the plane of the rhumb, and for this reason, a line of constant course is called a rhumb-line.

Because the angle each rhumb-line makes with each meridian is constant, and the meridians converge towards the Pole (fig. 60-6).

Line X to Y. The angles PXA, PAB, PBC and PCY are all equal and any one of them may be taken as the rhumb-line course. When this constant angle is 90º, the rhumb-line is either a parallel of latitude or the Equator itself and when the angle is 0º, the rhumb-line coincides with the meridian.

The “rhumb-line course” is always referred to as “the Course”, any meridian and the rhumb-line joining the points of departure and destination. When the distance to be sailed by a vessel is not too great, it is usual, when practical, to sail along a rhumb-line track. To do this, the navigator simply joins the points of departure and destination with a straight line on a Mercator Chart and measures the course angle (which is the inclination of the parallel ruler laid along the rhumb-line with a meridian). The helmsman is ordered to steer this course, which, if maintained, will bring the vessel to her desired destination.

The disadvantage of a rhumb-line track is that it is not the shortest route between the points of departure and destination. A straight line is the shortest distance between two points, but when those two points lie on the surface of a sphere, the arc of the Great Circle joining them is the curve that most nearly approaches a straight line, because it has the greatest radius and therefore the least curvature (fig 60-7). The shortest route between two positions on the Earth’s surface is thus along the shorter arc of the Great Circle on which the two points lie, and the Great Circle track is the one top follow when great distances are involved, such as on trans-ocean passages.

In order to measure courses along a great-circle track it is necessary to plot the track on a Mercator Chart, but because of the distortion of the Mercator projection, the great-circle track between two places appears as a curved line because the angle at which the track makes with the meridians is constantly changing, whereas the rhumb-line track appears as a straight line. The two tracks together on a Mercator chart give the false impression that the rhumb-line distance is shorter than the great-circle distance, but on a terrestrial globe it would readily be seen that the great-circle track is shorter than the rhumb-line track.

Fig. 60-9 illustrates the two tracks (a) on the sphere, and (b) on a Mercator chart. Notice that the rhumb-line makes a constant angle with the meridian, while the direction of the great circle is constantly changing. On the globe, as depicted in (a), the rhumb line appears as a curved line and the great circle, when viewed in its plane, appears as a straight line. On the Mercator chart, as depicted in (b), the rhumb line appears as a straight line and the great-circle track as a curved line.

Since the angle a great circle makes with the meridian is continually changing, a vessel following a great-circle track would have to alter its course continuously. In practice, this is impossible, since a vessel must hold a steady course until a definite alternation is made. Consequently, she sails a series of rhumb lines between successive points on the great circle, and makes what is called “approximate great-circle sailing”.

Fig. 60-10 shows such an approximate great-circle sailing as it would appear on a Mercator chart. The navigator would alter course at A, B and C and would choose the lengths XA, AB, BC and CY to suit his convenience. XA, for example, might be a twelve-hour run. Or A, B and C might be meridians which cross the great circle at 10º intervals.

In order to assist the navigator to find a great-circle track between two places, charts are constructed so that any straight line drawn on them represents a great circle. These are known as gnomonic charts.

Although great circles appear on gnomonic charts as straight lines, rhumb lines appear as curved lines, the meridians are not parallel, angles are distorted, and it is impossible to take courses and distances from such charts.

Fig. 60-11 shows a great-circle track drawn on a gnomonic chart. A gnomonic chart is nevertheless of great value an auxiliary to a Mercator chart. A navigator wishing to sail a great-circle track first joins the points of departure and destination with a straight line on a gnomonic chart. Then he selects convenient points on this line (A, B, C, etc., in fig. 60-10) and transfers them to a Mercator chart.

A fair curve through these plotted positions on a Mercator chart is the great-circle track required, but in practice, the points would be joined by a series of rhumb lines as shown in fig. 60-11.

Although the practical operation of laying down both rhumb-line and great-circle tracks on a Mercator chart is as has been described above, owing to the small scale of charts used for long-distance navigation it is not usually possible to measure courses and distances along these tracks with any degree of accuracy, so this must be done by calculation. The various methods of calculating the courses and distances between two points on the Earth’s surface are known in navigation as “The Sailings”. These are described in the next chapters.

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