To make an intelligent introduction of the solution which is to be disclosed, it will be of advantage to set forth briefly some principles that are well recognized among navigators. Defining the sub-celestial place of any body in the firmament as the point of intersection with the surface of the earth of a straight line drawn from the celestial body to the center of the earth, it will be evident that the geographical position of the sub-celestial place is in a latitude equal to the declination of the celestial body and in a longitude equal to the hour-angle of the celestial body from the prime meridian, and that the celestial body is in the zenith of the sub-celestial place.
In Fig. 1, take A to be the sub-celestial place of a celestial body whose declination is equal to the latitude of A and whose hour-angle at Greenwich is equal to the longitude of A west of the meridian of Greenwich, and B to be any point on the surface of the earth whose great-circle distance from A is z, then to an observer at B the zenith-distance of the celestial body in the zenith of A will also be z, and as B may lie in any direction from A, it follows that the zenith-distance of the celestial body will be the same to an observer situated at any point on a small circle of the earth whose pole is the sub-celestial place and whose circumference is everywhere distant from the sub-celestial place by a great-circle arc equal to the zenith-distance. Therefore, if a navigator, having measured the altitude of a celestial body, should obtain its zenith-distance, he is certain that at the instant of observation his geographical position is somewhere upon the circumference of such a small circle. In practical navigation, the ship's position as given by dead-reckoning always limits the position of the circumference, upon which an observer can possibly be, to such a small arc that, if laid down upon any chart projection, it will not differ sensibly from a straight line. Such a straight line is called a Sumner Line, and, for the same reason that the tangent to the circumference of a circle at any point is at right angles to the radius drawn to that point, it possesses the property of being at right angles to the true bearing of the celestial body upon whose zenith-distance it depends.
Reverting to Fig. I, let E represent the estimated geographical position of the observer at the time when the altitude is measured which gives the zenith-distance z, equal to A B. If the observer should compute or by any other means deduce the great circle distance EA and the bearing of A from E (which represent respectively the zenith-distance and azimuth of the celestial body as they would appear to an observer at E), the portion of the small circle BB', B"B" ' which represents the actual Sumner Line of Position, could at once be laid down on the navigating chart, for it would be at right angles to the bearing of the observed celestial body and removed from the estimated geo graphical position by a perpendicular distance equal to the difference in minutes of arc between the observed and deduced zenith distance, and toward the direction of the observed celestial body or away from it, according as the zenith-distance obtained by observational measurement is less or greater than the zenith distance deduced by dependence on the estimated geographical position.
This process of finding the Sumner Line has come into current use among many navigators under the name of the New Navigation, and the present practice is to compute the zenith distance and the azimuth of the observed celestial body in the following manner:
A latitude and longitude having been assumed within the extreme limits of both as given by the dead-reckoning, from the G. M. T., as known from the chronometer, the hour-angle of the observed celestial body, to an observer situated in this assumed geographical position, is now calculated from the formula:
L. M. T. = G. M. T. ± the assumed longitude expressed in time, the upper sign being used when the longitude is east and the lower sign when it is west of Greenwich; and when the sun is the observed body:
Sun's hour-angle = L. M. T. ± the equation of time; and for any other celestial body:
Star's hour-angle = L. M. T. + R. A. Mean Sun — R. A. Star.
With this hour-angle and the two including sides of the astronomical triangle, viz., the assumed co-latitude of the observer and the known polar distance or co-declination of the observed celestial body, the opposite side of the triangle, which is the zenith-distance or co-altitude of the celestial body, is computed from the formula:
sin x/2 = sin t/2 cos L cos d sec (L—d) (1)
cos z = cos (L—d) cos x
in which x = an auxiliary angle,
t = the hour-angle,
L = the latitude,
d = the declination,
and z = the zenith-distance;
and the true bearing, or angle opposite the side of the astronomical triangle representing the polar distance, is computed from the formula:
sin Az. = sin t cos d cosec z. (2)
There are superior qualities attached to this method of finding the Sumner Line because it may be used in any circumstances so long as the altitude of the observed celestial body does not greatly exceed 8o°, and the results obtained have been demonstrated to possess an enhanced probability of accuracy. The present purpose, however, is not to discuss these advantages which have been expounded by writers upon the subject of The New Navigation, but to disclose a short, simple process of finding the zenith-distance and azimuth without the aid of the extended computations indicated by the formula just referred to, and so to provide for the practice of a method of finding the Line of Position and the Compass Error by inspection.
Fig. 2 represents a stereographic projection of the celestial sphere on the plane of the meridian.
If the latitude of the observer be laid off along the bounding meridian at L, and the declination of the observed celestial body be laid off at M along a meridian making an angle with the bounding meridian equal to the hour-angle of the observed celestial body, an astronomical triangle will be formed in which the known parts are the two sides PL and PM, representing respectively the co-latitude and co-declination, and their included angle LPM, which is the hour-angle of the observed celestial body. Two of the unknown parts of this triangle are the azimuth, PLM, and the co-altitude, LM, of the observed celestial body. If the triangle, PLM, were revolved about the central point of the projection, with the side PL, kept in coincidence with the bounding meridian, until the point L is brought to the Position of the point P, the latter would then occupy the position P', and the point M would fall at M', so that the unknown side of the triangle, representing the co-altitude, would lie along some meridian, and could be measured from the graduation on the projection, and the unknown angle, representing the azimuth, would become an included angle between two meridians, which could likewise be measured from the graduations of the projection. And thus the altitude and azimuth of any observed celestial body could be simultaneously determined from the diagram with any degree of precision that the scale of the projection might permit. To obviate the necessity for actual revolution of the triangle, as described above, a series of equally spaced concentric circumferences and a series of equally-spaced radial lines have been drawn over the projection in lines of dashes. For the purposes of identification, the overlaid system of concentric circumferences is numbered serially from the center of the projection outwards to the bounding meridian, and the radials are also marked by numbers indicating their angular distance in minutes of arc counted in a clockwise direction from some fixed origin, like the line OS; so that, having plotted the declination and hour-angle of the observed celestial body at M, it is only necessary to note the number of the circumference and the number of the radial which pass through this position, and then, adding the co-latitude expressed in minutes to the number of the radial, find the intersection, M', of the noted circumference with the radial whose number is the sum just found, and read off from the graduated arcs of the projection the altitude and azimuth of this point of intersection.
In order that the required results might be found to the nearest minute of arc, the stereographic projection was constructed with a diameter of twelve feet; but this, in one continuous sheet, was obviously of unmanageable size for ordinary use, and had to be changed to convenient form by being cut up into the overlapping sections which form the pages of this book. In this form the projection can be used with the same facility as if preserved in one continuous sheet, for it has been pointed out that, in effecting the required solution, only those parts of the projection are involved which lie in the immediate vicinity of the points whose co-ordinates are to be plotted or read off.
To complete the illustration of the method it will be sufficient to treat one example, inasmuch as the process of solution is identical in every case.
EXAMPLE.
At sea, April 2, 1902, about 6h. 35m. p. m., in latitude 39' 16' N. and longitude 60° 00' W. by estimation observed a Aurigae, bearing N. 60° W. per compass, to be in altitude 66° 29' 25" with the observer's eye at a height of 31 feet above the sea-level, the index-correction of the sextant being I' 30". Chronometer time of observation 6h. 28 m. 34s.: chronometer slow on G. M. T. 4h. 2m. 18.2s. Required the Sumner Line of geographical position and the Total Error of the Compass.
NOTE.
Plotting the declination and hour-angle roughly with reference to the parallels and meridians (counted from the left-hand bounding meridian) of the Index to Plates, we find that the position of the observed body falls on Plate 51 approximately at the intersection of circumference 17.4 with radial 8400, and adding to 8400 the co-latitude expressed in minutes, which is 3044, we find the approximate place of the revolved position to be at the intersection of circumference 17.4 with radial 8400 3044 = 11444, which intersection falls within the limits of Plate 136. Turning now to Plate 51 and plotting the hour angle and the declination to the nearest minute we find the position of the observed body to fall at the intersection of circumference 495.6 with radial 8410.1. Adding the co-latitude expressed in minutes to the number of this radial we obtain 8410.1 + 3044 = 11454.1 as the number of the radial at whose intersection with circumference 495.6, on Plate 136, the solution is to be found by reading off the altitude with reference to the parallels and the azimuth with reference to the meridians counting from the right-hand bounding meridian.
COURSES AND DISTANCES IN GREAT-CIRCLE SAILING.
In Fig. 2, if L be taken to represent the latitude of the point of departure laid off along the bounding meridian and M the latitude of the point of destination laid off along a meridian making an angle with the bounding meridian equal to the difference of longitude between the point of departure and the point of destination, then the length of the side LM of the spherical triangle PLM will be the great-circle distance between the point of departure and the point of destination and the angle PLM will be the initial course for a vessel proceeding along the great-circle track between L and M. It will therefore be apparent that the course and distance in great-circle sailing may be found with very great facility by the method of solution which has just been described.
The altitude represented as having been measured in the foregoing problem is subject, like all other observed altitudes that navigators are accustomed to take for the purpose of fixing a position, to the uncertainties introduced into instrumental measurement by the extraordinary deflection of the rays of light coming from the horizon, which frequently occurs in every part of the world to such an extent as to produce a displacement of the horizon of large magnitude extending to five, ten, and even fifteen minutes of arc. And, perhaps among the greatest advantages that may be claimed for the foregoing method of solution is the facility with which it lends itself to a deduction of Wirtz's locus of geographical position by constant difference of altitude, of two observed celestial bodies with their difference of altitude which is found by comparing the measured differences of altitude as deduced by the foregoing method of finding altitude by inspection. Such a line of position is free from the usual uncertainties in the dip of the horizon caused by abnormal refraction, as well as from the additional uncertainties of the height of the eye, the index correction of the sextant, the want of parallelism in the mirrors, and to the errors of defective centering of the sextant.
9 point of this horizon-free line of position, as well as its azimuth may be speedily deduced by the method already described, as will be seen from the following example:
At sea, September 26, 1899, in estimated geographical position 30 40' south, and 320 34 west of Greenwich:
Chronometer slow on Greenwich Mean Time 2 Min. 39 Sec.
The hour-angle of a Arietis is +2h. 59m. 17s. or 44°49' 15" W., and we are enabled to find by inspection that a body in this hour-angle with a declination of + 22° 59' .5 must be in altitude 38° 48' .5 and azimuth N. 57° W., to an observer in the stated geographical position.
The hour-angle of a Canis Major is 2h. urn. 24s. or 32° 51' E., and we are enabled to find by inspection that a body in this hour-angle with a declination of +5° 29' must be in altitude 55° 56' .2 and azimuth N. 74° E., to an observer in the stated geographical position.
The difference between these two deduced altitudes is 17° 7' .7 and this subtracted from the measured difference of altitude set down in the statement of the problem gives the means, when taken in connection with the two deduced azimuths, of reducing the estimated place of observation to a point on the required line of position.
Having performed this computation, we find the value of d.i to be which applied to —320 34', the longitude of the assumed geographical position, becomes 320 36'.7, which is the longitude of a point of the horizon-free line of position. The latitude of this point is the same as the latitude of the assumed or estimated geographical position. The co-ordinates of the required point of the line of position are, therefore, ç = -3° 40', y =-32° 23.7’. The azimuth of this line of position will be the mean of the azimuths of the two observed stars, which in this case is 1880.5 counting from north through east, south and west.
Obviously, if the two differences of altitude be measured, i. e., if three different stars have been observed in quick succession, two lines of position will be obtained, and their intersection is the true geographical position of the observer.
The formula that has been employed for reducing the estimated geographical position of the observer to the line of position or locus of constant difference of altitude, by computing the difference of longitude between the two points, is most appropriate in all cases in which the mean azimuth (A2+A1)/2 lies between 315° and 45° and between 135° and 225°.
When the mean azimuth lies outside these limits, or in other words when it lies between 45° and 135° and between 225° and 3150, it is better to reduce the estimated geographical position to a point on the required line of position by computing the difference of latitude between the two points.
To FIND THE NAME OF AN OBSERVED STAR.
Frequently a star that is favorably placed for observation can not be identified because clouds obscure the surrounding parts of the sky. If, when the altitude of such a star is measured, its compass bearing be observed and the approximate true azimuth be obtained by correcting the bearing for the variation and deviation of the compass, the identity of the star may at once be ascertained by reversing the order of proceedings that has been described for finding the altitude and azimuth from the declination and hour-angle. That is, having plotted the corrected altitude of the star on the meridian of the projection which makes an angle with the right-hand bounding meridian equal to the star's azimuth counted from the north pole, note the number of the radial and the number of the circumference that pass through the point so plotted, and, having subtracted the co-latitude of the place of observation expressed in minutes from the number of the noted radial, find the intersection of the noted circumference with the radial whose number is the remainder thus found by subtraction, and read from the graduations of the projection the declination of this point and its hour-angle from the left-hand bounding meridian. The hour-angle of the observed star thus found must be converted into right ascension by the following rule:
From the local mean time of observation and the star's hour-angle, to find the star's right ascension. To the right ascension of the mean sun (or sidereal time, American Nautical Almanac, page II, last column, also British Nautical Almanac, page II, last column), corrected for the Greenwich mean time of observation. The result will be the local sidereal time of observation. If the star is east of the meridian, add its hour-angle to the local sidereal time of observation; if west of the meridian, subtract its hour-angle therefrom. The result will be the right ascension of the star.
Then scan the star tables of the Nautical Almanac and find the name of the star whose tabulated right ascension and declination come nearest to the values of the right ascension and declination that have been deduced. The stars that are of a sufficient magnitude to be observed by navigators are so widely separated that there will be no difficulty in making the selection from the tables, even when we proceed no further than the use of the Index to Plates in effecting the required solution, as has been done in the following example:
At sea, February 26, 1901, local mean time (L. M. T.) 6h.-30m. p. m. Weather overcast and cloudy. Observed the altitude of an unknown star through a break in the clouds to be 29°-30' (true), bearing N. 740 W. (true). What is the name of the star? Ship's position by dead-reckoning Lat. 350 N. Long. 600 W.
SOLUTION.
On the Index to Plates, plot the altitude, 29°-30' on the meridian making an angle equal to the true bearing or azimuth ('°), with the right-hand bounding meridian, and note the number of the circumference and the number of the radial which pass through this point. They are circumference 72 and radial 12360. Subtract the co-latitude (90° — 35° = 550) expressed in minutes, which is 55° X 6o' = 3300, from the number of the radial, and find the intersection of the radial whose number is the remainder (12360 — 3300 =) 9060 with the above-noted circumference 772. Read for the graduations of the projection the declination of this point and its hour-angle reckoned from the left-hand branch of the bounding meridian. They are declination 28°-4o North and hour-angle 72°= 4h.-50m.-. From the hour-angle thus deduced by inspection, we proceed to find the unknown star's right-ascension as follows:
L.M.T 6h 30m 00s.
R.A.M.S 22 22 33
Cor. G.M.T 1 43
Local Sidereal Time. 4 54
Star's hour-angle 4 50
Star's R.A. oh 4m.
The right ascension and declination of the unknown star, as we have approximately found them by inspection, are R. A. = oh.-4m. and Dec. = 28°-4o' N. The star is therefore Andromedw, whose tabulated R. A. = oh.-o3m.-16s. and Dec. = 82°-32' N.