Although the usual methods of determining local civil time from the time diagram are entirely satisfactory, the better-known methods for finding the corresponding date are not simple and are capable of improvement. It is the object of this note to show how the date is readily determined from the time diagram by employing the lower branch of the hour circle of the (mean) sun.
In the time diagram (Fig. 1) the circle represents the equinoctial; G represents
the Greenwich meridian, with g its lower branch; M and M' represent arbitrary meridians, with m and m’ their lower branches; and S represents the hour circle of the mean sun, with s its lower branch. At the instant indicated by the time diagram (and at any instant), there are two and only two (half-) meridians on the earth
at which the date is changing, namely the international date line (indicated by g) and the meridian on which L.C.T. is midnight (indicated by s). Consequently the two arcs of the equinoctial terminated by the points g and s represent arcs throughout each of which the date is unchanged. The date is the same at all points of the arc g m S m' G M s, and is the same at all points of the arc s M' g.
In order to compare the dates at a given instant at two places, say M and M', we need merely ask whether or not M and M' lie on the same one of the two arcs of the equinoctial bounded by g and s. Thus, in Fig. 1, the points M and M' lie on different arcs, and their dates at this instant are different; on the other hand, the two points M and G lie on the same arc of the equinoctial bounded by g and s, and therefore at this instant have the same date.
The additional question, as to which is the later date, is easily answered by consideration of the points of the equinoctial near s. In fact, the point s can be interpreted as midnight, moving around the equinoctial from E to W with the usual speed of 360° per 24 hours. At points just eastward of s, it is now slightly past midnight; at points just westward of s, it is now nearly midnight; consequently the date at the former points is one day later than at the latter points. A special case of Fig. 1 occurs at noon G.C.T.; at this instant the two points g and s coincide; there exists but a single arc of the equinoctial bounded by g and s; the date is the same at all points of the earth’s surface.
The general method set forth in the present note can be summarized as above in terms of the arcs into which g and s divide the equinoctial, and can alternatively be expressed in terms of the concept of separation. If, as in Fig. 2, the four points A, B, C, D lie on a circle successively in that order, the pair A and C is said to separate the pair B and D. But the pair A and B does not separate the pair C and D. To revert now to Fig. 1, the dates at two arbitrary meridans M and M' are the same if M and M' are not separated by g and s, and are different if M and M' are separated by g and s.