For Star Identification, Great-Circle Sailing, True Bearings of Heavenly Bodies, Hour Angle and Altitude of Bodies on the Prime Vertical, and, in general, for the Approximate Solution of Spherical Problems.
Introduction.
The appended diagram is a stereographic projection of the sphere containing great circles and parallels, at intervals of one degree, for the measurement of spherical coordinates. The novelty of the graphical method consists in the adaptation of this single projection, or system of circles, to the measurement of more than one system of coordinates in the solution of the same problem.
In the figure on the next page let us suppose that the full lines, consisting of a system of great circles and parallels, with M’ and M” as the poles of the latter, have served to project the points M and JV, the coordinates of which are given.
Let us suppose, further, that M is one of the poles of a similar set of parallels in combination with a similar set of great circles, and that to this system of dotted lines —which serve for the measurement of another system of spherical coordinates—the point JV is to be referred for the solution of a problem.
To make the explanation more practical, let us assume, 1st, that the full lines consist of vertical circles, or circles of azimuth, and parallels of altitude; the primitive, or bounding circle, being the meridian of the observer, M’ his zenith, and that, having the azimuth and altitude of a heavenly body given, we have projected its place, N, upon the figure; 2d, that M is the elevated celestial pole, the dotted lines hour circles and parallels of declination, and that we require the hour angle and the declination of the body at N.
If the graduations of the primitive and of the dotted diameter or equinoctial, CD, were properly numbered we would simply have to note the parallel of declination and the hour circle passing through the point N, follow one to the primitive and the other to the equinoctial, and take readings, in order to find the declination and the hour angle of the body, that is to say, the coordinates of the point N according to the system of coordinates measured by the dotted lines.
So long as the relative positions of the two points, M and M’, remained the same, such a figure would serve for the solution of similar problems involving any other position of an interior point N; but, since the relative positions of M and M’ are constantly changing in practice, no two sets of lines similar to those of the figure and printed upon a single sheet can be of universal application in the manner described. The object of the writer’s method is to overcome this difficulty by adapting the system of full lines to serve the purpose of both systems for all positions of M and N.
The position of N with respect to the dotted lines is defined by its position relative to the points M and O. Since the two systems of line are similar, if we transfer N to N’ so that N’ shall have the same position with respect to M’ and O that N has with respect to M and O, then N’ will have the same relation to the full lines that N has to the dotted lines, and we may therefore let N’ represent N and the system of full lines, in connection therewith, represent the system of dotted lines.
Briefly, then, the method is to first assume that the full lines represent a system of spherical coordinates to correspond with given data, and then to project M and N; next to transfer N to N”, and then to assume that the same lines represent another system of spherical coordinates to which it is necessary to refer N’ for a solution. Article I explains the graphical method.
While stereographic projection has long been used for the graphical solution of problems in nautical astronomy, by making a special projection of each case, the writer is not aware that any method has ever before been devised by which a single projection is made generally applicable for solutions.
Methods which are similar, to the extent that they adapt prepared stereographic projections to general use to avoid drawing, have been devised by Chauvenet and by Saxby, but both of these methods employ more than one projection or diagram for the purpose.
The writer has never seen Saxby’s Spherograph, but it is described in several successive numbers of the Nautical Magazine (English) for 1856. It consists of two concentric projections, each capable of being revolved about a common pivot at the center, one being transparent. It appears that different sets are employed for different purposes. Judging from the description, it is a less perfect device than Chauvenet’s.
Chauvenet’s Great-Circle Protractor was formerly issued by the U.S. Hydrographic Office, but it has now become rare. It consists of two concentric projections, one fixed and the other revolving about a pivot at the center. The revolving one is transparent. These projections, which are precisely the same, are on the plane of the meridian like that employed by the writer. The transparent card is objectionable, because it becomes more or less opaque and very brittle in time. In some relative positions of the two projections the maze of lines is such as doubtless to prove an obstacle to its use by those who do not understand the principles on which the Protractor is based, otherwise it is difficult to conjecture why a device so simple in conception has fallen so completely into disuse. Strange to say, Chauvenet gave an imperfect rule for finding the vertex of a great circle upon his Protractor; he overlooked the fact that the vertex is at 90° difference of longitude from the point where the great circle crosses the equator.
The writer has adapted his own method to orthographic as well as stereographic projection, not with a view of publication, but simply as an interesting study. In fact, he had no intention to publish it in any shape until urged to such action by some of his brother officers.
The method now published is exceedingly simple, and is correct in principle; but accuracy of result depends upon the precision attained in projecting points and in reading from the graduated scales. A person of only ordinary skill will doubtless be able to project and read to one-quarter of a degree in most cases, which may perhaps be regarded as excessive accuracy for great circle courses, and sufficient for true bearings. The diagram in the size given does not serve to find the longitude from a “time sight,” because a result to the nearest minute is sought; but for the partial determination of Sumner lines it suffices, as will be shown. For star identification it is especially well adapted, as the stars used by navigators are separated by considerable angular distances. It is cheaper, more compact and more easily lighted than a celestial globe, and affords more accurate measurements than ordinary globes of the same diameter.
Although only the general knowledge of nautical astronomy possessed by every navigator is requisite to solve the problems given herein, a better acquaintance with that subject will enable those using the method to greatly extend its application upon the diagram.
The writer does not, in a general sense, recommend his method as a substitute for computation; he simply submits it as a legitimate means of solving certain problems which may be more acceptable to those who prefer graphical methods when they can be employed to advantage, and also as offering in its applications a wide range for study. He believes, however, that a beginner in navigation, especially one with but little acquaintance with mathematics, can gain an intelligent working knowledge of the problems given incomparably quicker by his method than by computation.
A diagram like that given herewith is being prepared for publication by the U, S. Hydrographic Office, under the direction of Commander J. R. Bartlett, U.S.N., Hydrographer, who has consented to this advanced publication. That diagram will be printed for issue on both sides of a heavy Bristol board, and the board will then be coated with a thin, transparent mixture known as “ivorine.” Pencil marks may be easily erased from the “ivorine” surface. The following pages are copied from the manuscript prepared for the Hydrographic Office. Although much of the matter is elementary to members of the Naval Institute, it is deemed best to treat the subject as if for general publication.
General Graphical Method.
Article I. The following elementary, graphical process forms the basis of solutions; its special application to each case is explained in its proper place. See sheet containing the diagram.
Figure 1. Having projected upon the diagram two points, as M and N, given in position, one upon the primitive, or bounding circle, and the other within, conceive a sector MOC whose radii OM and OC shall include these points. Conceive this imaginary sector to be revolved about O until M coincides with some other given point upon the primitive, as M’; then find KP the revolved position of N. The radii need never be drawn.
There are various ways of finding N’ but the following are suggested. The first is always available, and involves marking points only upon the diagram; the second requires a piece of tracing paper, but makes no marks whatever upon the diagram.
Since one case embraces all, let it be required to revolve the sector MOC about O until M coincides with M’ and find N’. Since M will traverse the arc MM’ the point C will traverse an equal arc CC.
1st Method. Align a straight-edge on O and N to find the point C. Make the arc CC equal to the arc MM’ either by the divisions upon the scale of the primitive, or by transferring the chord MM’ to CC with a slip of paper. Align a slip of paper on O and C and mark upon it the points O, N and C Then align the slip on O and C so that its marks O and C shall coincide with O and C of the diagram, respectively. The point N of the slip will be N’, the revolved position of N.
2d Method. Lay a piece of tracing paper (or, as a makeshift, writing paper greased and rubbed thoroughly dry) upon the diagram and trace the points M, O and N. Revolve the tracing about O until M coincides with M’; the traced point N will fall at N’.
In the explanatory Figures 1 to 8, lines of the diagram are represented by full lines, imaginary and traced lines by dotted lines. Although letters of reference arc used the rules are general.
The diagram should be kept dry because it is intended to be an exact circle. Should it become eccentric so that one radial line, as ONC (Fig. 1) will be of a different length from another, as OC, to
find upon the latter the revolved position N’ of the intermediary point N, proceed as follows, referring to the figure below : from some point, as O, on a slip of paper draw lines duplicating the lines ONC and OC of the diagram, making any convenient angle with each other. Join C and C. and through N draw a parallel to CC cutting OC in N’. Then use the line ON’ C instead of the line ONC.
To Find the Name of an Observed Star.
Article II. Frequently a star favorable for observation cannot be referred to the surrounding constellations because of cloudiness, and therefore cannot be identified. In such a case measure the altitude as usual, noting the Greenwich time. At the same time, or immediately afterwards, take a compass bearing of the star. Correct the observed bearing for variation and deviation, and reckon it from north in north latitude and south in south latitude; this will give the star’s azimuth or true bearing near enough for our purpose. Correct the altitude. Find the latitude and longitude by account, and the local mean time of observation by applying the longitude in time to the Greenwich time of observation. A navigator will always know his latitude and longitude near enough.
Graphical Solution.
The diagram is first assumed to be a projection of the celestial sphere composed of azimuth circles, or vertical circles, and parallels of altitude. M’ is the zenith, and the line AB the celestial horizon. The primitive is the celestial meridian of the observer.
Figure 3. With the azimuth inspect the scale of AB, and reckoning from either extremity, find the star’s azimuth circle.
With the true altitude inspect the scale of the primitive, and reckoning from either extremity of AB, upwards, find the star’s parallel of altitude.
The projected place, N, of the star is at the point where its azimuth circle intersects its parallel of altitude.
Reckoning from the same extremity of AB as in finding the azimuth circle, find upon the primitive, above AB, the point, M, corresponding to the latitude. M is the place of the elevated celestial pole.
Conceive a sector, MOC, whose radii shall include M and N. Revolve the sector about O until M coincides with M’. Find N’, the revolved position of N (Article I).
The diagram is now assumed to be composed of hour circles and parallels of declination. M’ is the elevated pole, AB the equinoctial, or the celestial equator, and N’ the place of the star.
Read the declination upon the primitive at the parallel of declination passing through N’, reckoning from the nearer extremity of AB. When the parallel is above AB, the declination is of the same name as the latitude; when it is below AB, the declination is of the contrary name to the latitude.
Take the reading upon AB at the hour circle passing through N’ reckoning from the extremity of AB opposite to that from which the azimuth was reckoned. Convert the degrees and minutes into time. This time will be the hour angle of the star east or west of the meridian according as the star is observed east or west of the meridian.
Having the hour angle, find the star’s right ascension by the rule given below, then scan the star tables of the Nautical Almanac and find the name of the star having (approximately) the given right ascension and declination.
To find the Star’s Right Ascension. To the right ascension of the mean sun (Nautical Almanac, page II), corrected for the Greenwich mean time of observation, add the local mean time. The result will be the sidereal time. If the hour angle of the star is east of the meridian, add the hour angle to the sidereal time, or if west of the meridian subtract it from the sidereal time, and we have the R. A. of the star.
Example.
The method of solution is shown in Figure 3, which represents the diagram.
At sea May 10, 1884, P.M.; latitude 27° 15’ N., longitude 89° 30’ W. True altitude 43° 15’, corrected compass bearing or azimuth N. 110° W., Greenwich M. time of observation i3h. 12m. Right ascension of the mean sun for Greenwich time, 3h. i6m. 56s. The star is Procyon, whose tabulated right ascension is 7h. 33m., declination 5° 30’ N.
In Figure 3, AE is made equal to the azimuth of the star, and the arc BL to its altitude; then M’E is the azimuth circle of the star, LL is its parallel of altitude, and JV is its projected place, AM is made equal to the latitude, then M is the place of the elevated pole. MOC is the sector.
M’OC is the revolved position of the sector, N’ the revolved position of the star, and M’ the revolved position of the elevated pole. M’F is then the hour circle and BD the parallel of declination of the star. BF is the measure of the hour angle BM’F, and BD of the declination.
To Find the Hour Angle and Altitude of a Heavenly Body on the Prime Vertical.
Article III. If its declination is greater than the latitude, a body does not cross the prime vertical ; if less, it crosses above the horizon if of the same name as the latitude, below if of the opposite name. Any case may be solved by the diagram; but the passage above the horizon being the one of practical use to navigators, the directions which follow are intended for that case.
The data required for effecting a solution are the approximate latitude and longitude of the place of observation and the approximate declination of the body.
Hour Angle. The angle obtained from the diagram is the hour angle of the body east or west of the meridian, according as the time of east or west passage is sought. In the case of the sun, therefore, when converted into time, it is the local apparent time of west passage; and its difference from 12 hours is the local apparent time of east passage. In the case of any other body, to find the time of passage apply its hour angle in time to the time of the body’s meridian passage.
Graphical Solution.
The diagram is first assumed to be a projection of the celestial sphere composed of hour circles and parallels of declination. M’ is the elevated pole, and AB the equinoctial, or the celestial equator. The primitive is the celestial meridian of the observer.
Figure 4. With the declination inspect the scale of the primitive, and, reckoning from either extremity of AB upwards, find the parallel of declination of the body. Similarly, find the point M corresponding to the latitude. M will be the zenith of the observer.
A right line OM will be the upper branch of the prime vertical, and its intersection, N, with the parallel of declination will be the point of transit.
Read the hour angle upon AB at the hour circle passing through N, reckoning from that extremity of AB which is adjacent to M.
Since the time of passage is now known approximately, if the latitude and longitude are accurately known, greater accuracy of solution may be attempted if desired. Having the local apparent time and the longitude in time, find the Greenwich Mean, or apparent time, to which reduce the declination of the body. With the corrected declination, repeat the operation upon the diagram. This repetition is not likely to be needed excepting when the latitude and declination are nearly equal, especially when both are small; i. e. when the prime vertical OM intersects the parallel of declination at an acute angle.
To find the altitude, revolve the imaginary line OM about O until M coincides with M’, and find N’, the revolved position of N; that is, make ON’ equal to ON with a slip of paper.
Now assume that the projection is composed of vertical circles and parallels of altitude. AB is the celestial or rational horizon, M’ the zenith, M’M” the prime vertical, and N’ the point of transit.
Read the true altitude upon the primitive at the parallel of altitude passing through N’, reckoning from the nearer extremity of AB.
Example.
The method of solution is shown in Figure 4, which represents the diagram.
Required the sun’s altitude on the prime vertical and the times of east and west passage in latitude 37° N., when the declination is I7°15’ N. By computation the true altitude is 29°31’; time of west passage 4h. 22m. 40s., east passage 7h. 37m. 20s.
BD is made equal to the declination and BM to the latitude, then DD is the sun’s parallel of declination, M the zenith, OM the upper branch of the prime vertical, N the point of transit, M’E the hour circle through N, and BE the measure of the hour angle BM’E.
M’O is the revolved position of MO and N’ the revolved position of N. N’L is the parallel of altitude through N’ and AL is the measure of the true altitude ON’.
To Find the True Bearing or Azimuth of a Heavenly Body.
Article IV. True bearings of heavenly bodies for determining compass error, or the direction of Sumner Lines of Position, may be found from the diagram when either the latitude, declination and hour angle, or the latitude, declination, and altitude are given. The application to Sumner Lines is explained in Article V.
The total compass error is found by comparing the compass bearing of any body with its true bearing at the same time. The deviation of the compass for the direction of the ship’s head at the time of observation is found by applying to the total compass error the magnetic variation for the place of observation. The magnetic variation is commonly given by the sailing chart.
Note the Greenwich time of observation. If the latitude and longitude are not known by observation, find them by account. Reduce the declination for the Greenwich time, and correct the altitude when the latter is used.
Hour Angle. For use with the diagram, reckon the hour angle both east and west through 12 hours. The hour angle of the sun west of the meridian is the local apparent time of observation, found by applying the longitude in time, and the equation of time to the Greenwich mean time of observation; the supplement (to 12 hours) of the local apparent time is the hour angle of the sun when east of the meridian.
The hour angle of the moon, or a planet, or a fixed star for the time and place of observation is found by the following formula:
H. A. of body=Sidereal Time of Observation—R. A. of body, or H. A. of body=R. A. of mean sun +L. M. T.— R. A. of body.
When the angle found by the formula exceeds 12 hours, its difference from 24 hours must be used.
Convert the hours and minutes of the hour angle into degrees and minutes.
Graphical Solution.
Case I.— When the Latitude, Declination and Hour Angle are known.
The diagram is first assumed to be a projection of the celestial sphere, composed of hour circles and parallels of declination. M’ is the elevated pole, and AB is the equinoctial, or the celestial equator. The primitive is the celestial meridian of the observer.
Figure 5. With the hour angle inspect the scale of AB, and, reckoning from either extremity, find the hour circle of the body.
With the declination inspect the scale of the primitive, and reckoning from either extremity of AB, find the parallel of declination of the body, which must be sought above AB when the declination is of the same name as the latitude, but below when of the contrary name.
Mark the place, N, of the body where its hour circle intersects its parallel of declination.
Reckoning upwards from that extremity of AB which is opposite the one from which the hour angle was reckoned, find upon the primitive a point M corresponding to the latitude.
Conceive a sector, M’OC, formed by radii to include M’ and N. Revolve the sector about O until M’ coincides with M, and find N’ the revolved position of N. (Art. I.)
Now assume that the projection is composed of azimuth circles and parallels of altitude. M is the elevated pole, M’ the zenith, AB the celestial horizon, and N’ the place of the body.
Upon AB, reckoning from that extremity adjacent to the elevated pole, read the true bearing or azimuth at the azimuth circle passing through N’. Reckon the bearing from north in north latitude and from south in south latitude, and towards the east or west as the body is east or west of the meridian or as the hour angle is reckoned east or west of the meridian.
Expeditious method of finding a series of true bearings of the same body.—When a ship is maneuvered to get a series of compass bearings of a heavenly body, for deviation on the several points of the compass, the time thus occupied will generally be so short that the change in declination during the interval may be disregarded, and the declination as well as the latitude assumed to be constant throughout the series. The true bearings of the body corresponding to all the observations of the series may then be found at one operation upon the diagram.
The graphical solution is effected as follows: Mark the place M as before, then lay a piece of tracing paper upon the diagram, and project upon the former, N1, N2, N3, etc., the intersections of the several hour circles with the parallel of declination of the body. Trace the points O and M’.
Revolve the tracing about O until the traced point M’ coincides with M. Read the several true bearings upon AB at the azimuth circles passing through N1’, N2’, N3’, etc., the revolved positions of N1, N2, N3, etc., respectively.
Example under Case I.
The method of solution is shown in Figure 5, which represents the diagram.
In latitude 34° 30’ S., find the true bearing of a body when its declination is 10° north, and its hour angle 3h. 10m. (47° 30’) east of the meridian. The true bearing by computation is south 125° 31’ E. or N. 54° 29’ E.
In Figure 5, AD is made equal to the declination (of the contrary name to the latitude), and AE to the hour angle, then DD is the parallel of declination of the body, M”E the hour circle and AM”E the hour angle. N is the place of the body and M’OC the imaginary sector.
The arc BM being made equal to the latitude, M is the position of the elevated pole after the sector is revolved. MOC is the revolved position of the sector, N’ the revolved position of N, M’E the azimuth circle passing through N’, and BE the measure of the azimuth or true bearing BM’E.
Graphical Solution.
Case II.— When the Latitude, Decimation and Altitude are known.
First assume the diagram to be composed of hour circles and parallels of declination, as in Case I.
Figure 6. Mark the point M upon the diagram as before. (In Fig. 6 the assumed data are not the same as in Fig. 5.)
Lay a piece of tracing paper upon the diagram and trace the parallel of declination of the body—or the essential part of it—and the points O and M’.
Revolve the tracing about O until the traced point M’ coincides with M.
Now assume that the diagram is composed of azimuth circles and parallels of altitude, as in Case I. M is the elevated pole, and the traced parallel, in its revolved position, is still the parallel of declination of the body.
With the altitude inspect the scale of the primitive, and, reckoning from AB upwards, find the parallel of altitude of the body.
N’, the point of intersection of this parallel of altitude and the traced parallel of declination, is the place of the body.
Find the true bearing upon AB at the azimuth circle passing through N’; reckon it from the side adjacent to M, and otherwise as directed in Case I.
Example under Case II.
The method of solution is shown in Figure 6, which represents the diagram.
In latitude 49° 30’ N., find the true bearing of a heavenly body when its altitude is 40° 15’ east of the meridian, and its declination 20° N. The true bearing, by computation, is N. 107° 32’ E. or S. 72° 28’ E.
In Figure 6, AD is made equal to the declination (of the same name as the latitude), then DD is the parallel of declination of the body, which is to be traced. The arc BM being made equal to the latitude, M is the position of the elevated pole alter the tracing is revolved about O.
D’N’D’ is the revolved position of the traced parallel of declination. AL is made equal to the altitude, then LN’L is the parallel of altitude of the body when M is the pole. N’ is the place of the body. M’F is the azimuth circle passing through N’ and BF is the measure of the true bearing or azimuth BM’F.
Partial Determination of Sumner Lines.
Article V. The Sumner Line of Position is perpendicular to the true bearing, or line of bearing, of the observed body. From this property, if we compute two points in the line of position and project the line upon the sailing chart, we may find from it the true bearing; or if we compute one point in the line and find, by any means, the true bearing or line of bearing of the body for the time and place of observation, we may determine the line of position upon the chart by projecting the point and drawing through it a right line perpendicular to the line of bearing. In the latter case the diagram will give the true bearing with the accuracy required for navigation.
An error of a whole degree in the true bearing—an error greater than is at all likely to occur in using the diagram —produces an error of only one mile in the line of position upon the chart at a distance of sixty miles from the established point, while close to the latter it is practically nothing.
The true bearing may be found as described in Case I or Case II of Article IV, according to the data used.
The line of position may then be projected upon the sailing chart by means of a parallel rule and the compass rose of the chart, or by the following method which will generally give greater accuracy:
By means of a protractor draw somewhere near the established point, a right line making with a parallel of latitude an angle equal to the azimuth or true bearing, and in a direction to be perpendicular to the line of bearing of the body. With the parallel rule draw a parallel to this line through the established point. The second line will be the Sumner Line of Position.
Great-Circle Sailing.
General Remarks.
Article VI. The many conditions necessary to be considered in the selection of an extended ocean route demand for each case a special judgment, but, since in nearly all cases the chief object in view is to shorten the voyage or passage, whatever scheme tends to this end should be weighed according to its merits. It cannot be said that this is commonly done in respect to great-circle sailing.
There are three routes which offer great advantage over the simple rhumb route, and one or another of which may generally be adopted: 1st, the great-circle or direct route; 2d, the meteorological route, often circuitous but passing through regions of favorable winds and currents; 3d, a compound route, made up of a meteorological route embracing stretches upon great circles.
A glance at a globe makes it apparent that the shortest distance between any two places upon the surface of the sphere is upon the great circle which joins them, and that it is only while maintaining her great-circle course that a vessel heads for her port as if it were in sight. Excepting when sailing along a meridian or the equator the course upon a great circle changes continuously with the advance of the vessel, but so slowly that in practice it need be changed, generally, only for each 100 or 200 miles of distance made good. Since the great-circle course for any position of the vessel is quickly found, a necessity for a change of the course is made known. From what has been said, it is seen that a vessel in going over a great-circle route actually sails upon a series of rhumbs closely approximating to a great circle.
It is, unfortunately, too common a practice amongst navigators to accept the straight line of the Mercator chart as a direct route. Apart from magnetic variation, this practice offers a theoretically constant course; but since magnetic variation must be taken into account, and since it is impracticable to long maintain the original straight line, a constant course is seldom realized excepting for short distances.
The rhumb line of the sphere is a spiral which has the property of making a constant angle with meridians, but upon the Mercator chart it projects as a straight line and thus presents a fallacious appearance of minimum distance. The chart only serves to direct a vessel’s course and to mark her progress; the vessel actually sails upon the sphere, and, when her course is shaped by the rhumb, she approaches her port or place upon a spiral.
Every great circle intersects the equator at the extremities of a diameter of the sphere, and is divided at the equator into two equal parts or semicircles. The vertex of each semicircle is that point upon it which is highest in latitude. The two vertices are diametrically opposite points of the sphere; they have the same latitude, but of opposite names, and are at 90° difference of longitude from the points where the great circle and the equator intersect. At a vertex the course is east or west. If in approaching a vertex there is northing in the great-circle course there will be southing after passing it, and vice versa. In this graphical method no use is made of a vertex excepting when it occurs on the route to be sailed over.
The saving in distance which a great circle route offers as compared with the corresponding rhumb route varies greatly, of course, amounting sometimes to hundreds of miles. Between Yokohama, Japan, and Cape Flattery, Washington Territory, the example given herewith, it is 268 miles. A comparison may be made for any two places by subtracting the distance found upon the diagram from the rhumb distance computed by the rule of Mercator’s sailing.
A knowledge of the great-circle course is of importance in working to windward, for in blindly following the rhumb the vessel may even be sailed away from her place of destination. The great-circle course frequently varies three or four points from the rhumb course. On the route from Yokohama (Cape King) to Cape Flattery, the great circle course at Cape King is NE., while the rhumb course is E. by N., a difference of three points. In this case, for a wind directly ahead on the rhumb an uninformed commander would lay his vessel on either tack indifferently; if on the port tack, and the vessel work in twelve points, she would head SE. by S., nine points away from her great-circle course, the only course on which she would head directly for her place of destination. On the starboard tack she would head N. by E., only three points away from her great-circle course. It is quite certain that this vessel would be laid on the unfavorable tack for a wind whose direction was between the great circle and the rhumb courses. Less extreme cases may be assumed and the possibilities investigated by any seaman. It does not suffice to say that the great-circle course is to the northward of the rhumb in north latitude and to the southward in south latitude; in order to know when to go about and make the very best of his way to windward a commander should know the amount of variation of the two courses.
When a great-circle route has been decided upon, the whole route should be projected upon the sailing chart, either by a continuous line, or by frequent points, that it may be subjected to examination for general direction, obstructions, meteorological conditions, etc. When the vessels falls off the original great circle it is not attempted to regain it, for the shortest distance then is upon the great circle which joins her actual position and her place of destination. Her course is always the great-circle course at her actual position, and this may be found from the diagram, or the general direction of the original projected great circle may be followed by shaping rhumb courses by the Mercator Chart; the former method is preferable because more exact.
Graphical Solutions.
Let M be the place of departure, and N the place of destination.
To find the great-circle course. First assume the diagram to be a projection of the terrestrial sphere, composed of parallels of latitude and meridians of longitude. M’ is the North Pole, M” the South Pole and AB the Equator. The primitive is always the meridian of the place of departure.
Figure 1. Project M upon the primitive in its proper latitude—north or south as the case may be—on the right side if it is the eastern place, on the left side if it is the western place.
Project N in its proper latitude, and upon a meridian whose difference of longitude from M is that of the two places.
Conceive a sector, MOC, formed by radii, to include M and N (Article I). Note, by a glance simply, if N would fall above or below AB, if the sector were revolved so as to make M coincide with that extremity of AB which is adjacent to M. If above, reckon the course from north; if below, from south.
Revolve the sector about O until M coincides with M or M’—the nearer extremity of M’M”—and find N’ the revolved position of M.
Now assume M’ or M”—whichever is the revolved position of M—to be the place of departure and JV’ the place of destination. The former meridians then become great circles through the place of departure, and the parallels are parallels of great-circle distance from the same place. The scale of AB gives the angle which each great circle makes with the primitive, the meridian of the place of departure, and hence the course.
The great circle passing through M’, N’ and M” is the required great circle: read the course at its intersection with AB, reckoning from the nearer extremity of AB. Having the course, reckon it from north or south as previously found, and towards the east or the west as the place of destination is to the eastward or westward.
To find the great-circle distance. Find the great-circle distance upon the primitive at the parallel of distance passing through N’ reckoning it from M’ or M”—the place of departure—by taking the complement of the reading. Multiply the degrees by sixty and add the minutes; the result will be the distance in nautical miles.
To find the vertex and other points upon the great circle. The quickest method is by means of tracing paper. Trace the required great circle through N’, and revolve the tracing about O until M’ or M”—whichever is the revolved position of M—coincides with M. The traced great circle will then pass through M and N.
If the vertex is of any use it falls upon the diagram, and it is found upon a meridian at 90° difference of longitude from the point where the traced and revolved great circle intersects AB, the equator.
Take points upon the revolved great circle at 5° or 10° intervals of longitude from M towards TV—or, if desired, on both sides of the vertex when it falls between M and N—and find the latitude and longitude of each, measuring latitudes upon the primitive and differences of longitude from M upon the scale of AB. Transfer the points to the sailing chart and adjust or “fair” a curve to them.
The more exact method is to take the intervals from M towards N, for the points will then fall upon printed meridians of the diagram. The advantage of measuring from the vertex is that points equally distant in longitude on either side have the same latitude.
Note.—The vertex and other points may also be found without tracing paper, as follows, taking the case projected in Figure 1.
The Vertex. Figure 7. Observe which semi-diameter, OA or OB, bisects the required great circle M’N’M”, and conceive it to be revolved in the direction in which the sector MOC (of Fig. i) was revolved, and through an equal angle at O. The revolved semi-diameter being the semi-equator when the place of departure is at M’ (or M”), find its intersection, E, with the required great circle. The vertex, V, of the latter is upon a parallel at 90° from the parallel of E, reckoning in the opposite direction to that in which the semi-diameter was revolved. When the rule fails the vertex is not upon the diagram and is of no use.
Conceive a radius, OR, through V. Revolve the sector M’OR (or M” OR, as M’ or M” is the place of departure) about O until M’ (or M”) coincides with M. Then V’ the revolved position of V, will be the vertex of the required great circle through M.
Other points. Figure 8. Mark the points X, V, Z, etc., in which the required great circle, M”N’M”, is intersected by parallels of distance at intervals of 5° or 10° from the revolved place of departure at M’ or M”.
Conceive radii, OR, OS, OT, etc., through the points X, V, Z, etc., respectively, and when the sector M’OC’ (or M’WC’) of Fig. 1 is revolved back to MOC, revolve these radii severally in the same direction and through an equal angle at O. Find the revolved positions, X’, V, Z’, etc., of the points X, Y, Z, etc. The revolved points will be points upon the required great circle through M; upon the sphere they will be separated by equal distances.
Example.
In this example the results are obtained by computation that they may be compared with those obtained from the diagram. This particular great circle is not practicable because it passes north of the Aleutian Group. It is introduced that it may be compared with a composite route between the same places, given in Article VII. Figure 1 represents the diagram and contains the projection of the unbroken route.
Great circle from M, a position in latitude 34° 50’ N., longitude 140° 00’ E., near Yokohama, Japan, to N a position in latitude 48° 30’ N., longitude 125° 00’ W., near Cape Flattery, Washington Territory.
Position of the Vertex, latitude 54° 08’ N., longitude 160° 12’ W.
First great-circle course, N. 45° 32’ E., true.
Constant rhumb course, N. 79° E., true.
Distance on great circle, 4058 miles.
Distance on rhumb, 4326 miles.
Difference of distances, 268 miles.
Points Upon the Great Circle
Long. From Vertex | Latitudes | Longitudes | Longitudes |
5° | 54° 02’ N | 165° 12’ W | 155° 12’ W |
10° | 53° 43’ N | 170° 12’ W | 150° 12’ W |
15° | 53° 11’ N | 175° 12’ W | 145° 12’ W |
20° | 52° 26’ N | 179° 48’ E | 140° 12’ W |
25° | 51° 25’ N | 174° 48’ E | 135° 12’ W |
30° | 50° 09’ N | 169° 48’ E | 130° 12’ W |
40° | 46° 40’ N | 164° 48’ E |
|
45° | 44° 22’ N | 154° 48’ E |
|
50° | 41° 39’ N | 149° 48’ E |
|
55° | 38° 26’ N | 144° 48’ E |
|
In Figure I, AM is the latitude of M, and BL, or HN, the latitude of N. AH measures the difference of longitude of M and N. MOC is the sector.
M’OC’ is the revolved position of the sector, N’ the revolved position of N, and M’N’M” the required great circle. The arc M’D measures the great circle distance M’N’, and BG measures the course of BM’G.
MVE is the revolved or true position of the required great circle, E the point where it intersects the equator, EI the measure of 90° of longitude from E, IVM’ the meridian of the vertex, V the vertex, AK, or IV, the latitude of the vertex, and AI measures the difference of longitude of M and V.
Composite Great-Circle Sailing.
General Remarks.
Article VII. When the great circle joining the two places ascends into higher latitudes than it is prudent or possible to penetrate, a modified or composite great-circle route may be taken between them. Having determined the highest latitude to which he will sail, the navigator composes his route of an arc of the limiting parallel, one great circle passing through his place of departure and the other through his place of destination. He then sails upon the first component great circle until its vertex is reached, thence along the limiting parallel to the vertex of the second component great circle, and finally upon the second great circle to his place of destination. This composite route is the shortest possible under the restriction of limited latitude. Along the limiting parallel of latitude the course is shaped upon the Mercator chart by the common method of rhumb sailing.
Note.—For an example of a composite great-circle route, projected stereographically, see under Professional Notes, “A Fallacy in Composite Great-Circle Sailing.’’
To find the Vertex, V1 , and other points upon the first component great circle. Let M be the place of departure and N the place of destination.
Figure 2. Project M upon the primitive in its proper latitude as in Article VI.
Lay a piece of tracing paper upon the diagram and trace the limiting parallel of latitude (or a small part of it about where it is judged the vertex will fall), and the points and M. Revolve the tracing about O until M coincides with M’ or M”—whichever is the nearer.
The first component great circle is that which is tangent to the traced parallel in its revolved position. Read the first course upon AB at the tangent great circle. Trace this great circle and revolve the tracing about O until M comes back to its original position.
Obtain points for transfer to the sailing chart as described in Art. VI, and find the vertex, V1, upon the limiting parallel at 90° from the point where the traced and revolved great circle intersects AB.
To find any course and distance upon the first component great circle. Having found the vertex V1, treat it as a place of destination, or as N, for finding all courses and distances upon the first component great circle, following the method of Article VI.
To find the vertex V1, and other points upon the second component great circle. Proceed by the rule prescribed for the first component great circle, but as if sailing from the place of destination towards the limiting parallel. That is, treat the place of destination as M; project it upon the primitive on its own side and in its proper latitude; trace the limiting parallel; revolve the tracing about O, and find the tangent great circle as before.
To find the course and distance upon the second component great circle. Treat the place of the ship as M and the place of destination as N. Then proceed as prescribed in Article VI.
To find the distance upon the limiting parallel. Proceed by the rules for parallel sailing as given in books on practical navigation, or inspect the table on page 268 of Bowditch, new edition, or page 64, old edition.
Example.
In the case given in the example under Article VI, assume 51° north as the limiting latitude, and find the composite route between the two places. The following results are obtained by computation.
Figure 2 represents the diagram, and contains the projection of the first component great circle.
Vertex V1, latitude 51° N., longitude 164° 18’ W. First course, N: 50° 04’ E.
Vertex V1, latitude 51° N., longitude 148° 45’ W.
Distance from M to V1, 2562 miles.
Distance from V2 to N, 929
Distance from V1 to V2 on the parallel of 51° N. by parallel sailing, 587
Total distance by composite route, 4078
Total distance by rhumb, 4326
Difference, 248
Rhumb course N. 79° E.
In Figure 2, AM is the latitude of M, AK the limiting latitude, and ATT the limiting parallel. M’” is the revolved position of M’ when M is revolved to M’ and K’K’ is the revolved position of the traced parallel KK. M’ GM” is the first component great circle tangent to K’K’, and BG measures the first course BM’ G.
MV’E is the, traced and revolved great circle when M arrives back at its original position, E the point where it intersects the equator, IE the measure of 90° of longitude from E, M’I the meridian of V1 the vertex, and AI the measure of the difference of longitude of M and V1.
Having found V1, the tracing is again revolved about O until M arrives at M’, then V1’ is the revolved position of V1, and M’D the measure of the distance of V1 from M.
APPENDIX.
Simple Orthographic Construction for Star Identification.
When the stereographic projection, or diagram, is not at hand, the simple special construction shown in the accompanying figure may be used for star identification, when the data prescribed in Article II is given. The construction will be greatly facilitated by the use of draftsman’s triangles.
Describe any circle, h’d’h’a, about o as a centre, and draw a diameter h’li’. Make the arc h’p’ equal to the latitude, h’l’ equal to the altitude, and h’c’a equal to the azimuth. Draw the radii oa, op’ and oe’ , making oe’ perpendicular to op’. Draw l’l’ parallel and l’l perpendicular to h’h’. With centre o and radius ol describe the arc Is, cutting oa in s. Draw s^ perpendicular to h’h’ and cutting I’ I’ in s’.
Through s’ draw c’d’ perpendicular to op’ , cutting op’ in b’ and the original circle in cf and d’. With center b’ and radius b’d’ describe a semicircle d’s”c’. Draw s’s” perpendicular to db’ d’ , cutting the semicircle in s”.
The angle s”b’d’ is equal to the hour angle of the star east or west of the meridan, as the case maybe. The angle d’oe’ is equal to the declination of the star. When d’ falls on the same side of e’ as p’, the declination is of the same name as the latitude; when it falls on the opposite side of e’, the declination is of the opposite name to the latitude. Having the declination and hour angle, find the star’s right ascension as directed in Article II.
Explanation of the Figure. The figure is a projection of one-half of the visible celestial sphere, h’d’h’ on the plane of the observer’s meridian, and hah on the plane of his horizon. On the vertical plane h’d’h’ is the upper meridian, h’h’ the horizon, l’l’ the parallel of altitude of the star,p’ the elevated pole, op’ the semi-axis (or the upper branch of the 6 o’clock hour circle), oe’ the semi-equinoctial, s’ the place of the star, c’d’ the parallel of declination of the star, and d’ the point of transit over the upper meridian. The semicircle c’s”d’ is the parallel of declination, revolved about its diameter c’d’ into the vertical plane, and s” is the revolved position of s’. In the horizontal projection h’ah’ is the horizon, oa the azimuth circle, and is the parallel of altitude of the star, s is the place of the star. This same construction may be used for other problems.