But to go back to the mariner on the equator whose latitude and sun’s declination so nearly agree. He is in a predicament should he persist in the plan to determine his whereabouts by position lines of the sun. In such an unusual case, it would be well to resort to some other method or wait until evening and determine the ship’s position by establishing the position line of some star or stars. It will be but a few days before the ship’s progress will cause the sun to leave his right course across the sky and take the hour circles at an angle. Take a case when the sun at noon has a zenith distance of 10°, the change of azimuth during the forenoon is still small, but suppose the bearing was noted 1 hour, or even less, before noon and again in similar amount after noon, a change will be found of perhaps 90°, the difference of moving from the southeast quadrant (if declination is south of latitude) to the southwest quadrant. In this way, a remarkably good cut may be had within a comparatively short time.
The foregoing will convince the reader that he must be governed by the change of bearing and not by time elapsed, in predicting the value of the cut of his position lines.
In the use of position lines, it is necessary to bear in mind, that, when the body’s altitude begins to approach the zenith, or, what is the same thing, when the ship is getting close to the body’s sub-celestial position, the circle is getting proportionately smaller. Under such conditions the arcs of the circle of equal altitudes can no longer be shown as a straight line. The double altitude as it is ordinarily practiced is here impracticable. And even outside this impracticable area, discretion must be shown. The dead reckoning position must be proportionately accurate, and the assumed latitudes must be brought correspondingly close together, in order to have a shorter line of position, because the curvature of the circle is getting sharper as the sub-celestial point is approached. To put it in another way, a smaller arc must be used in order to avoid the error due to excessive curvature.
Very good results can be obtained by noting the time of observation by chronometer (G. M. T.) and correcting it for equation of time in order to get Greenwich apparent time. This, if converted into arc, is the longitude of the sub-solar position. By using the Greenwich mean time to correct the declination taken from the Nautical Almanac for that day, the latitude of the sub-solar position may be obtained. Plot this position on the chart and use it as the center of a circle; then with the zenith distance (90° - altitude) as a radius, draw an arc in the probable position of the vessel. Somewhere along this arc is the ship’s position. The bearing of the sun (rather hard to get so nearly overhead) corrected for compass error, reversed, and laid off from the sub-solar position will give a fair idea of the position of the vessel. Now by waiting a sufficient time for the sun to change its azimuth enough to make a good cut and using its new sub-solar position as a center with the zenith distance of a second observation as a radius, an arc may be drawn which will intersect the first arc at the position of the vessel. The run between the sights will, of course, require the first arc to be carried forward as the first position line in the ordinary double altitude problem.
Johnson’s Method
It is not always found convenient to plot the position lines of a set of observations on a chart; perhaps for lack of a chart of proper scale or possibly for want of the chart itself. Again many navigators do not take kindly to the graphic method, but prefer to solve their latitudes or longitudes by computation. In any event Johnson’s Method comes as a relief to such persons, saving them from the arduous duty of establishing a set of position lines by the chord method of assuming two latitudes to get two longitudes.
Johnson’s Method can be practiced in both the double altitude problem of the sun, where the first sight, or position line, is brought forward to the second sight by correcting it for the intervening run, or where stars are used simultaneously.
Chief among its merits is the saving of figures. It is only necessary to compute two (instead of four) chronometer sights in order to find the ship’s position, thus obtaining a mathematically accurate result by a short cut. But also a great advantage in the Johnson Method is that the resulting longitude is obtained by calculation and it is not necessary to plot the lines upon the chart to secure the position.
In using Johnson’s method it is not absolutely necessary to observe two stars simultaneously as the quick work of a good man is sufficiently close for the practical purposes of navigation.
It becomes evident to anyone reading the foregoing pages that every ordinary time sight places the vessel on a circle of equal altitude, the longitude resulting from the computation, depending on the latitude, by dead reckoning, used. Now rather than work two sights employing two assumed latitudes on either side of the supposed position, make the calculation only once, using the latitude by account.