The zenith distance is ordinarily too large to become a radius for such use on the chart. The circles of equal altitude are in practice so large that 10 to 40 mile arcs in the vicinity of the vessel are treated by her navigator as straight lines, known as Sumner or position lines. These lines are, theoretically, chords or tangents according to the method employed in establishing the line, but in practice the divergence from the circle is negligible, excepting always when the body is too close to the zenith. The establishment of the position line has been done in several ways for many years until the advent of this new and more expeditious method.
The altitude of a body at any selected time for an assumed position can be readily calculated. If this altitude does not agree (and it seldom does) with the altitude measured simultaneously with the sextant, corrected for the usual errors, the assumed position is not coincident with the actual position of the vessel. The navigator now proceeds to lay off from the assumed position, the line of azimuth of the body taken from the azimuth tables, Weir’s Azimuth Diagram, or determined by observation. On this line the distance between the observed and computed altitude, expressed in minutes of arc, is measured, towards the body if the observed altitude is greater, and away from it if less, than the computed altitude. The point thus indicated is a position on a circle of equal altitudes, the arc in the immediate vicinity of the computed point being, approximately, the position line. This line is at right angles to the azimuth for the reason that a tangent is at right angles to the radius of a circle at a given point.
It is now known that the ship is somewhere on this line of position, and it is necessary to cut it with another such line to determine definitely her position. If the sun is the body being observed, it becomes necessary, in order to provide a good angle of intersection, to wait until the azimuth changes at least 30°, when the observation is repeated, a second line established, and the first line brought forward in exact accordance with the ship’s run. The interval required naturally depends upon the latitude of the ship and the declination of the sun. The intersection of the lines will be the position of the ship at the time of the second sight.
The use of stars has a decided advantage in that there are always some of these bodies available for observation lying in various azimuths; it is practicable, with a well-defined horizon, to observe simultaneously two or more of these bodies whose bearings show that they would produce desirable position lines. From the resulting intersections the position of the ship is secured. This obviates waiting for the second line, a feature that is always inconvenient and sometimes, perhaps, dangerous.
The calculation of the altitude is accomplished by the solution of the spherical triangle in which we have given the co-latitude (90° - assumed latitude), the polar distance and the hour angle of the meridian of the assumed position. Thus with two sides and the included angle, the third side or the zenith distance (90° - altitude) is easily determined by either of several formulas.
With the use of this method all the formulas that formerly, and still, often puzzle the navigator to remember can be reduced to this one sight. One of the most important features it possesses is that it can be utilized regardless of the altitude of the body (except when very high), its azimuth, or its hour angle, all of which are elements that have to be used under certain favorable circumstances in order to get accurate results from the older forms. The navigator is now given a greater freedom in choosing bodies to observe than is found in any other method.
The mariner to-day has been almost entirely relieved from the labor of computing position at sea, should he care to avail himself of a set of altitude tables, several excellent ones have made their appearance on the market, among them Hydrographic Office Publication No. 200. From them the altitude can be selected corresponding to the conditions of any particular observation. With a set of these tables a navigator is no longer required to be a mathematician or to remember the forms of a half dozen sights. Thus in this wonderful age the mariner’s utopian dream of obtaining position at sea by inspection, is, in a way, realized.
In order to illustrate the practical working of a problem by this method, the following example is taken up point by point:
Early on the morning of May 21, 1899, while in the assumed position of latitude 55° 00´ N., longitude 112° 08´ E. observed the true altitude of the star Arcturus to be 37° 14´ 50´´, bearing west of the meridian. The chronometer carrying Greenwich mean time read 20 d. 6 h. 20 m. 03 s. The observer desired his position.