Ultimately, however, the accuracy of the watch, and its indispensableness in the perfected form of the chronometer, having been fully demonstrated, the method of lunars became practically obsolete and the mariner was able to determine his longitude with the aid of sextant, chronometer, and _Nautical Almanac_, by means of direct observation of the altitude of the sun by day and of sundry fixed stars by night, a much simpler calculation sufficing than was required by the older method. As the sun is the chief time-measurer, whose rate of passage in a seeming circumnavigation of the heavens is recorded by the dial of clock, watch, or chronometer, it would seem as if the simplest possible method of determining longitude would be found through observation of the sun's meridian passage. The user of the sextant on shipboard always makes, if weather permits, a meridian observation of the sun, and such observation gives him an accurate gauge of the altitude of the sun at its highest point and hence of his own latitude. By adjusting the arm of the sextant with which this observation is made, the observer is able to determine the exact point reached by the sun in its upward course with all requisite accuracy. But, unfortunately for his purpose, the sun does not poise for an instant at the apex of its upward flight and then begin its descent. On the contrary, its orbit being circular, the course of the sun just at its highest point is approximately horizontal for an appreciable length of time, and while the observer therefore has adequate opportunity to measure with accuracy the highest point reached, he cannot possibly make sure, within the limits of a considerable fraction of a minute, as to the precise moment when the center of the sun is on the meridian. He can, indeed, determine this point with sufficient accuracy for rough calculations, but modern navigation demands something more than rough calculations, inasmuch as a variation in time of one minute represents one-quarter of a degree of longitude, or fifteen nautical miles at the equator, and such uncertainty as this would imply can by no means be permitted in the safe navigation of a ship that may be passing through the water at the rate of a nautical mile in less than three minutes. It follows that meridian observation of the sun, owing to the necessary inaccuracy of such observation, is not the ideal method. In point of fact the sun may be observed for this purpose to much better advantage when it is at a considerable distance from the meridian, since then its altitude above the horizon at a given moment is the only point necessary to be determined. The calculation by which the altitude of the sun may be translated into longitude is indeed more complicated in this case; but while spherical trigonometry is involved in the calculation, the tables supplied by the _Nautical Almanac_ enable the navigator to make the estimate without the use of any knowledge beyond that of the simplest mathematics.