### 13 OCEAN NAVIGATION & PASSAGE PLANNING

#### 55 RHUMB LINE AND GREAT CIRCLE TRACKS

##### 210. THE DIFFERENCE BETWEEN TERRESTRIAL POSITIONS

The equator divides the Earth into the Northern and Southern hemispheres. All places in the Northern hemisphere are said to have a North latitude and all places in the Southern hemisphere, South latitudes. The equator may therefore be defined as the parallel of zero latitudes, i.e., every point on the equator has, a latitude of 000°00.

Small circles on the surface of the Earth which axe parallel to the equator are known as parallels of latitude and every point on a particular parallel of latitude has the same latitude. The latitude of a place on the Earth’s surface is the angle at the Earth’s centre, measured in the plane of a meridian, from the equator to the place. Meridians are semi—great circles which terminate at the Earth’s poles, Thus the latitude of a place may be defined as the arc of a meridian intercepted between the equator and the parallel of latitude on which the place lies.

The most common method Navigators employ to describe a position on the Earth’s surface is to state which parallel of latitude and which meridian pass through the point to be described. The parallel of latitude is denoted by giving the latitude of the place, and the meridian is denoted by giving an angle known as longitude, Whereas the datum parallel from which latitude is, measured is the equator, the datum meridian from which longitude is measured is the meridian on which Greenwich observatory (London) lies. This meridian is known as the Greenwich Meridian, or sometimes as the Prime Meridian.

In fig. 55-1, the angle DOC is the latitude of the point C (and of every other point on the parallel of C’s latitude). If the meridian on which point G lies is the Greenwich meridian, then the West longitude of the point, ‘C’ is given, by the angle EOD. But:

Angle EOD  = angle GFC

= angle EPD

= arc ED

All places which are situated between the Greenwich meridian and the meridian which lies opposite to the Greenwich meridian, are, said to lie in, East or West longitude depending on whether they are East or West of the Greenwich meridian respectively. The longitude of a place is therefore defined as the arc of the equator between the Greenwich meridians and the meridian of the place.

The difference in latitude (d.lat.) between two places is the arc of a meridian intercepted 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.

The difference of longitude (d.long.) between. two places is the shorter arc of the equator contained between the meridians of two places. If the two places are in longitudes of the same name (both East or both West), the d.long. is found by subtracting the lesser from the greater. When the two places are in differently named longitudes (i.e., if the Greenwich meridian lies within the arc of a d.long. between the two places), the d.long. is found. by adding the longitude of the two places. If, however, the l80° meridian lies within the arc of d.long., the d.long. is found by subtracting the sum of the longitudes from 360°.

The following examples illustrate the d.lat. and d.long. between pairs of positions:

#### 211. THE TRUE SHAPE OF THE EARTH

Thus far, the Shape of the Earth has been considered to be perfectly spherical, an assumption which, for normal navigational problems, leads to no appreciable error.

The actual Shape of the Earth, however, is that of an oblate spheroid of revolution. An oblate spheroid is a solid the Shape of which may be swept out by rotating an ellipse about its minor axis.

The ellipticity of the Earth, i.e. the ratio between the difference in lengths of the equatorial and polar radii, and the equatorial radius itself, is approximately 1/300. This very small fraction indicates that the Earth is almost a perfect sphere. The vertical at any place is the direction which is perpendicular to the horizontal plane which touches the Earth’s surface at the place. The angle at the centre of the Earth, measured in the plane of the meridian between the equator and any place is known as the geocentric latitude of the place, whereas the angle between the vertical at a place and the plane of the equator is known as the geo-graphical latitude of the place.

In fig. 55-2 Geocentric latitude of X = ZOX Geographical latitude of X = angle ZYX

Geographical latitude is often referred to as true latitude, and in all navigational problems the term latitude means geographical latitude. The geographical and geocentric latitude of any point on the equator is 00°00‘, and. the geographical and. Geocentric latitude of the pole is 90°00‘. The maximum difference between the geocentric latitude of a place and its geographical latitude occurs when the latitude of the place is 45°N. or S. The geographical latitude of a place is always numerically greater than the geocentric latitude of the place, and for this reason, the geocentric latitude is  sometimes referred to as reduced latitude.

The length of the equator in angular units is 360°, equivalent to 21,600 minutes of arc, these minutes being known as geographical miles. A nautical or sea mile is a if distance which is equal to the length of an arc of a meridian between two places whose geographical latitudes differ by one minute. In fig. 55-2, if the angle between BE and ‘ the plane of the equator exceeds that between AE and the equator by 1‘, then the arc AB is equal to one nautical mile. It will be noticed that a nautical mile is the length of an arc of a meridian between two points whose verticals are inclined to one another to the extent of 1’. Thus if angle AEB is 1′, the arc AB is one nautical mile. The oblateness of the Earth. causes the length of the nautical mile to increase as the latitude increases, so that:

Length of a nautical or sea mile at the equator     = 1843 metres

Length of a nautical or sea mile at the  poles         = 1862 metres

The average length of a nautical mile is therefore 1852.5 metres. In practice, the length is taken as 1852 metres (equivalent to 6016.4 ft.). This distance is sometimes referred to as a standard sea mile.

#### 212. RHUMB-LINE TRACKS

Out of sight of land, vessels are steered from place to place by means of a compass, which indicates a fixed horizontal direction irrespective of the movement of the vessel. Compass points are marked on the outer edge of the compass card and radial lines, ex-tending from the centre of the card to the several points, are known as rhumbs. When a vessel’s head is steadied. on a certain compass point, the fore—and—aft line of the vessel lies in the plane of a rhumb, and it is easy to visualise the path and track of the vessel as an extension of the rhumb. For this reason, a line of constant course is known as a rhumb line.

A rhumb-line is usually defined as an imaginary line on the Earth’s surface which crosses every meridian at the same constant angle. The most convenient path to sail along is the rhumb line which connects the points of departure and destination, w) because the course of the vessel remains constant during the passage. In figs. 55-3a and ﬁgs. 55-3b, FABCT is the rhumb line joining F to T. The angles PFA, PAB, PEC and PET are all equal, and anyone of them may be taken as the course.

Special cases of rhumb lines are the equator, parallels of latitude, and meridians. The equator and all other parallels of latitude are rhumb lines because the course of a vessel sailing along any parallel is constantly 270° or 210°. The meridians are rhumb lines because the course along a meridian is constantly 000° or 180°.

The art of sailing obliquely across the meridians is known as loxodromic’s (from the Greek loxos meaning oblique, and dromos meaning running). For this reason, all rhumb lines other than parallels of latitude and the meridians are sometimes called loxodromic curves. When. a vessel sails along a loxodromic curve, her track is a spiral around the pole (see fig. 55-4) because, although the curve crosses each meridian at the same angle, the meridians close together as the latitude increases. Theoretically, a loxodromic curve never reaches the pole, but continually gets nearer to it.

The ‘Sailings’ comprise the various methods of calculating the course and distance between two points on the Earth’s surface. When the distance a vessel has to sail is not too Great, it is usual, when practicable, to sail along the rhumb-line track, and the course and distance along such a track is usually determined by calculation using either middle Latitude Sailing or Mercator Sailing (described in §s 13-15). For great distances, it may be beneficial to sail along a great circle track because the distance to be sailed would be less than if the rhumb-line were followed. The method of determining the course(s) and distance along a great circle track is called Great Circle Sailing.

The advantage of the rhumb-line sailings is that the course is constant. To navigate along a rhumb-line track, the mariner simply joins the points of departure and destination with a straight line on a Mercator chart and measures the course angle (the inclination of this line to a meridian line). The helmsman is ordered to steer this course, which, if maintained, will bring the vessel to the desired destination.

The distance along a rhumb-line between the points of departure and destination may, if the scale of the chart is sufficiently large, be measured directly with dividers, or if the scale of the chart is too small to permit accurate measurement, can be calculated. The disadvantage of the rhumb-line track is that it does not follow the shortest route between the points of departure and destination.

#### 213. GREAT CIRCLE TRACKS

A straight line is the shortest distance between two points, but it is impossible to draw a straight line on the surface of a sphere. However, the curve which. most nearly approaching a straight line will be the arc of a curve having the greatest radius, and by definition, a great circle fulfils this condition as it has the radius of the sphere itself. Therefore the shortest route on the Earth’s surface between two terrestrial positions is along the shorter arc of the great circle on which the two points lie.

In fig. 55-5(a), A and B are two points in the Northern Hemisphere, and in fig. 55-5(b) point A lies in the Northern hemisphere while point B lies in the Southern hemisphere. In both cases, the great circle arc AB is the shortest route between A and B, and NA and NB are arcs of the meridians through A and B and are, therefore, also arcs of great circles. The triangle NAB is a spherical triangle the solution of which will provide data on the required great circle track AB since this forms one side of the triangle NAB. This triangle has three known quantities:

NA                    = co-lat. of A.

NB                    = co-lat. of B

ANGLE ANB  = d.long. between A and B.

Unlike a rhumb-line track, the angle which a great circle track makes with the meridians is constantly changing (unless the great circle track is also a rhumb-line track). Thus, if it is desired to sail along the shortest track from one place to another, the great circle track must be followed, in which case the course must be constantly changed. The correct course to be steering when following a great circle track is the true bearing of the point of destination. In fig. 55-6, the true bearing of T from F is the angle between the meridian through F and the great circle joining F and T, measured clockwise from the meridian, i.e., the angle PFT. This angle represents the initial course to be steered by a vessel sailing on a great circle track from F to T. At any intermediate point between F and T, say G in fig. 55-6, the true by bearing of T is the angle PGT, and this is not equal to the angle PFT’. To a navigator moving along the great circle T from F to T, the true bearing of T changes continuously.

The difference between the distances along the rhumb—line track and the great circle track between any two points depends on:-

Distance between the points;

D.long. between the points;

and Latitude of the points.

A considerable difference results when the d.long. between two places is great and the latitude of the places are high. If the d.long. is small, the rhumb—line track is almost due North or South, in which case it approximates to the great circle track. If the latitudes of the two places are low, the rhumb—line track is almost due East or West along the equator, where it will again approximate to the great circle track. Some idea of the difference between rhumb—line and great circle distances may be seen from the following examples:

Because of the distortion of a Mercator chart, the great—circle track between two places appears as a curved line which is convex to the equator, whereas the rhumb-line appears as a straight line. The two tracks drawn together on a Mercator chart give the false impression that the rhumb-line distance is shorter than the great circle distance, but if drawn on a terrestrial globe, it would readily be seen that the great circle track is shorter than the rhumb-line track.

Fig. 55-7(a) shows both the rhumb—line and great circle tracks between places A and. B drawn on the terrestrial sphere, while fig. 55-7(b) shows y the same two tracks drawn on a Mercator chart. In both figures, the rhumb-line makes a constant angle with the meridian, whereas the direction of the great circle constantly changes.

On the globe, as depicted in fig. 55-7(a), the rhumb-line appears as a curved line, while the great circle arc AB (when viewed in its plane) appears as a straight line (N.B.; in the diagram, the rhumb-line AB and the great circle arc AB intersect at the equator, but this is not to be taken as the general case). On a Mercator chart as depicted in fig. 55-7(b), the rhumb-line track AB appears as a straight line, and the great circle track appears as a curved line.

Since the angle a great circle makes with the meridian is continually changing, to follow a great circle track a vessel would have to alter course continuously. In practice this is impossible, because a vessel must hold a steady course until a definite alteration is made. Consequently the vessel is steered along several short rhumb—line tracks which commence and terminate at positions on the great circle track. This is sometimes called “Approximate Great Circle Sailing”, but is more generally known simply as ‘Great Circle Sailing‘. In fig. 55-8, the straight pecked lines FA, AB, BC and CT represent rhumb—line tracks which approximate to the great circle track from F to T.

The navigator would alter course at A, B, and C, and would choose the lengths FA, AB, etc. to suit his convenience. For example, one navigator may choose to alter course after every 12 or 24 hours run, while another navigator may choose to alter course when the track crosses meridians either 5° or 10° apart.

214. PLOTTING GREAT CIRCLE TRACKS

In order to assist the navigator in finding the great circle track between two places, charts are constructed so that any straight line drawn on them represents a great circle. Such charts are known as gnomonic charts because they are constructed on the gnomonic projection, in which the Earth’s surface is projected outwards from the Earth’s centre onto a plane which is tangential to the sphere at any convenient point. The tangential point chosen is usually at the centre of the area to be represented by the chart.

Since a great circle is formed by the intersection of a plane through the Earth’s centre with the Earth’s surface, and as one plane will always cut another in a straight line, all great circles will appear on a gnomonic chart as straight lines, But the meridians will not be parallel unless the tangent point is on the equator, nor will rhumb lines be straight. Angles are also distorted except at the tangent point. It is therefore impossible to take courses, bearings or distances from. A gnomonic chart. Positions, however, may be lifted with ease, so that a gnomonic chart has great value to the navigator as an auxiliary to a Mercator chart.

When it is required to lay off a great circle track on a Mercator chart, the track is first drawn on a gnomonic chart. Suppose it is required to find the great circle track from A in Lat. 40°N., Longitude 55°W. to B in Lat. 65°N. Longitude 10°W., then a straight line would be drawn on the gnomonic chart (fig.55-9a) between points A and B. Positions of several points along this track, say where the track crosses every whole 10° of longitude at a, b, c and d, are then lifted and transferred to the corresponding Mercator chart (fig. 55-9b). A fair curve would then be drawn through these plots. Positions on the Mercator chart, and this curve is the great circle track required. It can be seen in fig.55-9(b) how much this differs from the rhumb-line track AB.

It will be noticed in figs. 55-9(a) – (b) that the nearest approach to the pole of the great circle track AB is at position d. At this point, the track cuts the meridian at an angle of 90°. To the eastwards of point d, the course at any point on the great circle track is south-easterly, while to the westwards of point d, the course is north-easterly. The point on a great circle track where the course changes from northerly to southerly (or vice versa) are known as the vertex of the great circle. At this point, the great circle cuts the meridian through the vertex at 90° and the course at the vertex is due East or due west. Every great circle has two vertices, one in the northern hemisphere and one in the southern hemisphere.

Since the curve of the great circle track is greatest near the vertex, the rhumb—line tracks which are plotted to approximate to the great circle track should be shorter, and the course altered more frequently, in this area than at other parts of the track. In fig. 55-10, the rhumb lines AB, BC, CD, etc, approximate to the great circle track from A to G, but where the curve of the great circle track is greatest (between C and G) the rhumb lines are shorter and the course altered more frequently.

Gnomonic charts are available for the major oceans of the world and for the polar regions, but for small craft with space and cost problems, undoubtedly the most useful is the Meade Great-Circle Diagram (Admiralty Chart 5029) on which any great circle may be plotted. This is a gnomonic chart in which the tangent point is on the equator and in which the graticule is symmetrical about the meridian through this tangent point, which is independent of the longitude so that tile longitude scale can be adjusted to fit the navigator‘s convenience.

Fig. 55-14 illustrates the principle of the Meade Great Circle Diagram while fig. 55-11 shows a reduced version of the diagram itself. In fig. 55-11, the tangent point is in Lat. 0°, Longitude 0°.  The diagram shows a series of vertical lines representing meridians. These are all parallel to each other but are not equally spaced. The parallels of latitude appear as hyperbolic curves. To transfer a great circle track such as FT in fig. 55-11 from the Meade Great Circle Diagram to a Mercator chart, first calculate the mid-longitude between F and T to the nearest 10°, and pencil this figure in under the central meridian on the Diagram. Designate the meridians on either side of the central meridian with values corresponding to every + 1O°and – 10° to the value of the central meridian. For example, if the mid-long between the departure and destination positions on the Great Circle is 110% E., the central meridian would be designated 110°E., the printed meridians to the right of this would be designated 120° E., 130° E., 140° E., etc., and the printed meridians to the left of the central meridian would be designated 100° E., 90° E., 80°., etc., In fig. 55-11, the mid-long between F and T happens to be the Greenwich meridian, and for the sake of clarity only the meridians at 20° intervals either side of this have been shown.

Having joined F and T with a straight line which represents the great circle track required, pick off the latitude and longitude of convenient points A, B. C, etc on the line FT, and mark these points on the corresponding Mercator chart. Strictly, the points should be joined by a smooth curve on the Mercator chart (as shown on fig. 55-9b), but in practice- each point is usually joined with a straight line representing the rhumb line course to be steered to each successive point, course being altered at each to fetch the next. When the departure and destination points of a great circle lie on opposite sides of the equator, the Meade Great-Circle Diagram can still be used because a gnomonic chart of both hemispheres must be symmetrical about the equator when the tangent point is on the equator.

Referring again to fig. 55-11, suppose F lies north of the equator and T lies south of the equator, then the following geometrical construction will suffice to transfer the required great circle to a Mercator chart:-

• Mark the position of F on the chart in the usual way.
• Mark the position of T as if it were in the northern hemisphere.
• Join F to K, the point on the equator which has T’s longitude.
• Join T to H, the point on the equator which has F’s longitude.
• Drop a perpendicular RQ on to the equator from R, the point where FK cuts TH
• Draw FQ and QT. Then FQ is the great-circle track in the northern hemisphere, and QT is the reflection of its continuation south of the equator. Points on QT may therefore be treated as if they were in the southern hemisphere.

#### 215. COMPOSITE GREAT CIRCLE TRACKS

It will have been noticed from fig. 55-9(b) that every point on a great circle track is in a higher latitude than the point on the rhumb—line track which lies on the same meridian.

The great circle always carries a vessel into higher latitudes than does the rhumb—line track between the same two places. Many great circle tracks climb into extremely high latitudes, where bad weather and/or ice is likely to be encountered, and are not, therefore suitable to navigate along. Other great circle tracks may be similarly unsuitable because they cross land.

Should a navigator, after considering the great circle track from one place to another, find that it would take his vessel into too high latitude and therefore decide not to follow it, he may choose what is to be his vessel’s maximum safe latitude and plot a modified great circle track known as a composite track. This will not be the shortest possible route to his destination, but it will be the shortest possible track and is formed by two great circle arcs joined by an arc of the limiting or safe parallel of latitude.

Note carefully that a composite great circle track consists of great circle arcs joined by the arc of the limiting parallel. Composite sailing is not a case of plotting a great circle direct from departure to destination; sailing along it until the maximum latitude is reached, then proceeding along that parallel of latitude until picking up and following the arc of the same great circle at the other end. A composite track involves one great circle from the initial position to the limiting parallel, thence along the parallel, and finally along a second great circle to the destination. The distance to sail along the limiting parallel is from the vertex of the first great circle to the vertex of the second great circle, i.e., the vertices of the two great circle tracks lie on the limiting parallel.

In fig. 55-12, FLVMT is the great circle joining F and T. Latitudes higher than the parallel of LM is assumed to be dangerous. The vessel cannot, therefore, follow the great circle arc LVM. Nor would she go from F to L, along to M and then down to T. The shortest safe route she can take is FABT, where FA and BT are great circle arcs tangential to the safe parallel at A and B, and FABT is therefore the composite track in this example. It is the shortest route because, if L and M are taken as any points on the parallel outside the part AB, then (FL + LA) is greater than FA and (BM + MT) is greater than BT. Moreover, since A is the point nearest the pole on the great circle of which FA is an arc, any other great circle from F to a point between A and B would ~ the parallel between L and A and so carry the vessel into danger.

Fig. 55-13 illustrates the composite great circle track from Port Lyttelton (New Zealand) to Valparaiso (Chile) with the limiting parallel of 50° S. The arcs LV1 and V2V are two separate great circles whose vertices lie at V, and V2   respectively on the limiting parallel of 50° S.

The use of the Meade Great Circle Diagram (Admiralty Chart 5029) for plotting both great circle and composite great circle tracks is illustrated in fig. 55-14. To plot the great circle track from Juan de Fuca Strait (Vancouver Island) to Honolulu, first find the mid—longitude between the two places to the nearest 10° thus:—

Pencil in, under the printed central (0°) meridian on the Diagram, the mid-long. 140°, and pencil in under the adjacent meridians the corresponding figures 130° and 120° to the right and 150° and 160° to the left. Using the printed latitudes and the pencilled longitudes, plot the initial and destination positions off Vancouver and Honolulu respectively and join them with a straight line. This represents the required great circle track from which the ringed positions would be lifted and transferred to the corresponding navigational (Mercator) chart as previously described.

In the case of a passage from Yokohama to Vancouver, when the great circle track is plotted on the Diagram (peeked in fig. 55-l4) it can be seen that the track would run up to almost 54° N., north of the Aleutian Islands. Inspection of Admiralty Sailing Directions and the routeing chart shows that the northerly limit should be about 50° N. A composite great circle track is therefore required and would be plotted as follows:

As before, calculate to the nearest whole 10°, the mid—longitude (in this case l7O° W.), designate this to the central meridian of the Diagram (top of Diagram in fig. 55-14), and pencil in the values of each 10° of longitude either side of this.

Plot on the Diagram the positions of departure and destination.

From the Yokohama, point draw a straight line at a tangent to the curve of the 50° parallel (just touching it at about 165° W.)

From the Juan de Fuca Strait point draw another tangent to the 50° parallel (just touching it at about 145° W.)

The required composite great circle track would then be:-

a great circle track from Lat.34°40′ N. Long.140° E. (Yokohama) to Lat. 50° N. Long. 165° W

a parallel sailing track from Lat.50° N. Long.165° W. to Lat. 50° N. Long. 145° W.

a great circle track from Lat. 50° N. Long. 145° W. to Lat. 48° N. Long. 125° W. (Juan de Fuca Str.)

Intermediate points for plotting on the appropriate Mercator chart(s) and for altering course are shown ringed on the Diagram. This composite great circle track is 4,110 miles in length, which is only 30 miles more than the (impossible) great circle of 4,060 mile – compared with the rhumb-line track distance of 4,345 miles – an, appreciable saving of 241 miles with winds mostly abaft the beam throughout.

#### 216. THE DETERMINATION OF GREAT CIRCLE DISTANCE

Neither rhumb-line nor great circle distances can be measured on a small-scale Mercator chart (and all ocean charts are small-scale) with any degree of accuracy, and students may be wondering how the distances quoted in the preceding paragraph have been arrived at. In fact, all the factors required in both great circle and composite great circle sailing, i.e. distance, initial and final courses, positions at selected points along the track, positions of vertices, etc. can be calculated by trigonometrical methods, but the graphical methods of plotting the track described in this Lesson are quite adequate for all practical purposes except for measuring the distance.

Great circle distance, however, can be determined by a tabular method using Sight Reduction Tables for Marine Navigation NP 401, similar to the method used for sight reduction, with sufficient accuracy for practical purposes. The reason for this will become evident from a study of figs. 55-15(a) and 55-15(b).

Figures 55-15 (a) (the Earth) and 43-15 (b) (the Celestial Sphere) show that:

EARTH                                                                                           CELESTIAL SPHERE

Latitude of Departure Point (F)  corresponds to                   D.R. Latitude

Latitude of Destination Point (T)                                              Declination

d.long. between F and T                                                            Local Hour Angle (L.H.A.)

Great Circle Distance                                                                 Zenith Distance (or 90° – Alt.)

Initial Great Circle                                                                       Azimuth

Thus, by entering the Sight Reduction Tables with latitude of departure point as latitude, latitude of destination as declination, and difference of longitude as L.H.A., the altitude (Hc) and azimuth (Z) angle may be extracted and converted to distance and initial course. The great circle distance is obtained by subtracting the tabulated altitude (Hc) from 90° and converting this to minutes (= nautical miles). The initial great circle course is the tabulated azimuth (Z) converted to true bearing in the same way as for sight reduction.

Since in finding the great circle distance, the latitude of destination becomes the declination entry, and all declinations appear on every page of N.P. 401, the great circle distance can always be found from the volume which covers the latitude belt containing the latitude of departure. Furthermore, since the distance must obviously be the same in either direction, where necessary the points of departure and destination can be reversed when using the tables if this is more convenient. For instance, to find the great circle distance between Yokohama in Lat.34°40′ N., Longitude 140° E. to the Juan de Fuca Strait in Lat.48° N., Longitude 125° W. in that order would require Vol. III of N.P. 401 and complicated interpolation with the departure latitude, but by finding the same distance from Juan de Fuca Strait to Yokohama, we can use our Volume 1V of N.P. 401 and only our usual interpolation for the minutes in the destination latitude (as for declination increment in sight reduction).

Enter the Sight Reduction Tables (Vol.4) with an L.H.A. of 95° (corresponding to the d.long) and under latitude 48° (lat. of departure point) and Declination 34° (whole degree below latitude of destination) extract the altitude (Hc), declination difference (d) and azimuth (Z) from the Latitude ~ name as Declination page (because both the departure and destination latitudes are North). This gives:-

Enter the Sight Reduction Tables (Vol.4) with an L.H.A. of 95° (corresponding to the d.long) and under latitude 48° (lat. of departure point) and Declination 34° (whole degree below latitude of destination) extract the altitude (Hc), declination difference (d) and azimuth (Z) from the Latitude ~ name as Declination page (because both the departure and destination latitudes are North). This gives:-

N.B. The interpolated Z value (62.0) when converted to a true bearing by Zn = 360° Z = 298° gives the initial course from Vancouver to Yokohama 298° T., or the final course from Yokohama to Vancouver 118° T. (its reciprocal).

In cases where the departure and destination positions are in different hemispheres the procedure is exactly the same except that the page Latitude CONTRARY name to Declination is entered. Where the great circle distance is greater than 90° (5400 miles) in the same hemisphere enter the tables on the CONTRARY page and use 180° — d.long as the L.H.A., adding 90° to the extracted altitude. Where the great circle distance is greater than 90° (5400 miles) and the latitudes of departure and destination are in different hemispheres, enter the tables on the SAME name page, use 180° – d.long. as the L.H.A. and add 90° to the extracted altitude. The rules for following these procedures will be found on page xxvi of the Introduction to Sight Reduction Tables for Marine Navigation.

N.P. 401 but note that in certain copies of Vol.4, in the fourth line of Case IV on this page, ‘opposite name‘ should be corrected to ‘same name‘. Even the Hydrographic Department can make mistakes!

The data from NP.401 are applicable to the rapid solution of the distance in composite great circle sailing. The complete solution consists in finding the combined length of two great circle arcs and the length of the intervening limiting parallel. To find the first great circle distance, enter the tables with L.H.A. = 90°, and with the latitude of the limiting parallel, find that declination for which the altitude is equal to the latitude of departure, when 90° – Dec. is equal to the distance from the point of departure to the point of tangency of the great circle with the parallel of limiting latitude, and the azimuth angle is the difference in longitude between the point of departure and the point of tangency. At the same page opening the corresponding quantities for the second great circle from the limiting parallel to the destination can be found.

The intervening distance along the limiting parallel is found simply by converting the d.long. between the point of tangency (in minutes) into departure (nautical miles) using the Traverse Table (in all volumes of Nautical Tables) entered with the whole number of degrees of the limiting latitude, or the formula Departure (distance in nautical miles along the parallel) = d.long. between points of tangency X cosine of limiting latitude. We can now calculate the distance along the composite great circle track between Yokohama and Vancouver plotted in fig. 55-14.

2. Find the distance along the composite great circle track between Yokohama (Lat.34° 40’N.. Long. 140°E) and Juan de Fuca Strait.   Vancouver (Lat.48° N., Long. 125° W.) with the limiting parallel of Lat.50° N.

Enter the N.P. 401 tables with L.H.A. = 90°, same name (page 182) and the latitude of the limiting parallel (50°). Travel down the 50° latitude column until an altitude (Hc) is found equal to the latitude of departure (34° 40′). By mental interpolation the corresponding declination is found to be 47 9°, and the azimuth (Z) is 54.5°

Distance from Departure point to point or” tangency with 50° parallel = 90° – 47.9° = 42.1“.

Distance = 42.1° x 60′ = 2526 miles (first great circle)

D.long between Departure point and point of tangency (Z) = 54.5° to be added to longitude of Yokohama (140°E.) thus giving point of tangency as Lat.50°N., Longitude 165°30W. (the small—scale plot in fig. 55-14 gave 165° W. but the calculation is more accurate).

On the same entry page and under the latitude of the limiting parallel (50°) find the altitude (He) equal to the destination latitude (48°) and extract the corresponding Dec. (76°) and Azimuth (Z) 21.2°, thus:

Distance from Destination to point of tangency with 50° parallel = 90° – 76° = 14°

Distance = 14° x 60’= 840 miles (second great circle)

D.long between Destination and point of tangency = Z = 21.2° = 21°12‘ to add to the longitude of Vancouver = 125°W. + 21° 12‘ = Longitude146° 12’W. (we got 145°W. from the small-scale plot in fig. 55-14) but this calculation is more accurate).

Convert 1158′ d.long. into departure at Lat. 50° N. either by Traverse Table or by the formula Dep. = d.long X cos.lat.  = 1158’ x cos. 50°

The Total Distance along the composite track can now be calculated as follows:

#### 217. THE USE OF GREAT CIRCLE TRACKS IN SAILING CRAFT

On ocean passages of several thousand miles a great circle track can be appreciably shorter than the rhumb line track in certain cases, and large commercial vessels tend to follow great circle tracks whenever possible. In the case of yachts, whether sail or power, other factors have to be taken into consideration. A sailing yacht usually Shapes a course which will take her through areas where winds and currents are most likely to be favourable, avoiding areas where bad weather or head winds can be expected. The moderate sized motor yacht is also more interested in hospitable seas and winds, and both types will often reach their destination more quickly and more comfortably, by selecting courses based on the expectations of wind, weather and currents, as indicated by routeing charts and sailing directions.

In sailing craft it is possible to follow the Great Circle track throughout, except in the region of one of the world’s great permanent winds. A sailing vessel, for example bound from Durban (South Africa) to Fremantle (Western Australia) should follow the great circle track because in addition to covering the shortest possible distance, she would probably get plenty of westerly wind in the Roaring Forties. On the other hand, bound from Fremantle to Durban, the great circle track would still, of course, be the shortest distance, but it would be impractical to follow it on account of the strong headwinds. By coming across the Indian Ocean on a track a long way to the northward of the great circle she would cover more distance but would make a better passage by taking advantage of the SE Trade wind.

The real use of great circle sailing in sailing vessels is to show which is the more favourable tack in turning to windward. If a sailing craft is endeavouring to follow a great circle track and the wind is dead ahead it is immaterial which tack she sails on so far as reducing the distance to her destination is concerned, but if the wind is blowing obliquely to the great circle track then it is desirable, so far as possible, to keep her on that tack which will enable her to head as near to the great circle course as the adverse wind will permit. This practice is called ‘Windward Great Circle Sailing‘.

Even when a sailing craft is not following a complete great circle track, the principle of windward great circle sailing can be extremely advantageous, if not fundamental to good seamanship. In the absence of any special reason to the contrary, a vessel should be put on that tack which brings her nearer to the great circle track (even though she is following a rhumb line track), because any shift of wind will then either‘ bring her up towards the G.C. track or will make the unfavourable tack less unfavourable. For example, suppose a yacht can sail a good full at 5 points from the wind, is in Lat. 40° N., Longitude 60° W. and bound for the English Channel. The rhumb line course to Bishop’s Rock lighthouse is 075° T. and suppose the wind is ENE. The skipper might argue that on the port tack he would keep in the favourable current of the Gulf Stream, or on the starboard tack he would have more chance of finding a westerly wind farther north, but that in either case he would be sailing 5 points off the rhumb line course, so that there is very little difference. But, if he consults the gnomonic chart or calculates the azimuth or true bearing of his destination, he will find that the initial great circle course from his position is 057° T., therefore on the starboard tack he will be less than 3½ points from this course and making good nearly 8 miles for each 10 miles sailed, but on the port tack he will be more than 6 points from this course and will make good only 2½ miles for each 10 miles sailed.

The ‘ABC Table‘ in all volumes of Nautical Tables (normally used as an alternative method of determining Azimuths for compass error and deviation) provides a quick and easy method of finding the true bearing of one’s destination or initial great circle course if you consider:-

• Present latitude as ‘Latitude’ in Table A
• Destination latitude as ‘Declination’ in Table B
• d.long. between present and final position as H.A. in both tables A and B.
• The azimuth found in Table C is the initial course or true bearing of destination.

Find the true bearing of a destination in position Lat. 52° 00‘ N. Longitude 55° 00‘ W. from a present position in Lat. 51° 10′ N., Longitude 10° 00‘ W.

True Bearing or Initial Course = N 70.4° W. or 289.6° T.

An easy method of following a great circle track, adopted by some navigators, is to work out and steer each day a new initial course to the same destination point using the ABC Table as described above. In practising this method, it is well to remember that each new course, if made good, will take the vessel further away (on the poleward side) from the original great circle, and the total distance could be considerably increased thereby. Some compensation for this possibility should be made when setting each new course – at least by steering “nothing to the nor‘ard” in north latitudes. The figure obtained by adding the distance to go, each day, to the total distance made good (from point of departure on the great circle) will indicate any distance being lost.

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