The navigator has from earliest days depended upon measurements of the altitude of heavenly bodies. At first the latitude alone was calculated, then, with the improvement in time-keepers, the longitude also could be obtained. The latitude was for so long a time all that could be determined that it got to have a peculiar value to navigators, and even after it became possible to obtain the longitude, it was supposed to be more dependable than the longitude though the latitude were itself obtained by methods depending upon time, as the reduction to the meridian and single altitude at a given time. In recent editions of Bow- ditch it is advised that the computed point on the line of position be disregarded in favor of one depending on the dead reckoning latitude, though the computed point is nearer the dead reckoning position than the one recommended and is in a popular sense the “most probable” position. So since the navigator has so long depended upon the measurement of the altitude, he has ignored the possibility of obtaining his position by any other measurement. He has computed his position at sea from a solution of the astronomical triangle using as known quantities the declination obtained from the Nautical Almanac, the altitude, obtained by a measurement with the sextant, and assumed values of either or both the latitude and longitude. The perfection of the circle 1 of reflection and sextant, and the accuracy of observations with them when the horizon is visible, has made this method most satisfactory. The fact that measurements of either of the remain- quantities, if possible, would be equally valuable to him and equally enable him to solve the astronomical triangle and obtain his position is seldom considered, though occasions often arise at sea where measurement of the altitude with accuracy is impossible and that of the azimuth is practicable and convenient.
There is no way of measuring the position angle of a celestial body, but the corresponding angle of a radio compass station on shore is now furnished by radio compass stations along the coast, and vessels are constantly navigating the coasts of the United States, Great Britain and France, using positions obtained by computation or plotting from these angles. An article on the determination of positions by radio compass azimuths and position angles was published on the back of the Pilot Chart for the North Atlantic for January, 1920, reprints of which may be obtained from the Hydrographic Office, Navy Department.
Little attention has been paid to determining positions by measurement of the azimuth of heavenly bodies, though with the improved forms of the azimuth circle the azimuth may be determined with accuracy. The navigator is often able to see the sun or stars overhead when the horizon is entirely obscured by fog, and many devices have been suggested for determining the horizon or a substitute for the horizon so that sextant altitudes can be measured, and thus the method of solution of the astronomical triangle based on the knowledge of the altitude be used. In such cases it is generally possible to obtain fairly accurate measurements of the azimuth arid from a solution of the triangle with the azimuth as a known part obtain points on the line of position more accurately than they could be obtained by computation based on sextant observations of the altitude with uncertain position of the horizon. While the azimuth method is not nearly so accurate as the altitude method, the error of the azimuth method is least when the altitude is great, and as bodies high in altitude are the only ones plainly visible in fog, it is at this time when the usual method fails entirely or is at its worst that the azimuth method is at its best. In the examples given dotted lines are plotted showing the errors in the lines of position due to an error of one degree in the determination of the azimuth.
Since the line of position determined by an observation of the azimuth makes a large angle with the line of position obtained from an altitude of the same body, the two lines may be combined with advantage. As these lines of position are absolutely independent of each other, the position of their intersection is a legitimate fix, subject only to the errors of the two observations. This method of obtaining a fix may be of great value where for any reason the position of the ship is uncertain and an immediate, even if only approximate, fix is desired.
A convenient method for the computation of points on the line of position by the azimuth method, whether the observed body be a wireless station on shore or a celestial body, is as follows:
Let Fig. 1 be a projection of the celestial sphere on the plane of the horizon.
Z, the zenith of the observer.
P, the elevated pole.
M, the observed body or the zenith of the wireless station on shore.
Mm, the perpendicular from M on the meridian of the observer. Then, calling Mm, a, the length of the perpendicular,
Qm,b, the latitude of the foot of the perpendicular,
Zm, C, the zenith distance of the foot of the perpendicular, from Napier’s rules.
Cos t=cot b tan d, whence tan b — tan d sec t; (1)
cos b = tan a cot t, whence tan a=cos b tan t; (2)
sin C = cot Z tan a; (3)
and substituting for tan a from (2) in (3)
sin C=cot Z cos b tan t, (4)
and latitude = b — C when Z< 90°.
= b + C when Z>90°.
To obtain the line of position assume two values of t, the hour angle, and compute the corresponding values of the latitude. This line of position can be used as lines of position obtained in other ways are used.
Where measurements of both altitude and azimuth are made and it is desired to obtain a fix, the line of position depending on the altitude may be obtained by the method of Saint Hilaire or the time sight, or the computation may be combined with the foregoing, this being the ?' ? " method of Bowditch, 1917, page 134, and this line of position, however obtained, plotted on the chart intersecting the line of position obtained from the azimuth will give the required fix. Or a fix may be computed directly aa follows:
sin t cos d = sin Z cos h, whence sin t = sin Z cos h sec d; (1)
cos t = cot b tan d, whence tan b = tan d sec t; (2)
cos Z = tan C tan h, whence cot C=tan h sec Z; (3)
and latitude = b — C when Z<90°
— b + C when Z>90°.
The labor of the computations required in the use of these formulae as well as in other calculations of navigation involving the solution of a spherical triangle may be avoided by the use of a table of right spherical triangles such as that published by the Navy Department in Hydrographic Office Publication No. 200, Table VI. A larger table that would not necessitate interpolation or corrections is much to be desired. Though such a table would be bulky and contain many pages “It is easier to turn pages than it is to interpolate.”