The variation from the true tropical year brought about by the processional movement of Sirius or any other star, however, can be watched by noting its heliacal rising in relation to any physical phenomenon which marks the true length of the tropical year. Such a phenomenon we have in the solstice and in the rising of the Nile, which, during the whole course of historical time, has been found to rise and fall with constancy in each year, the initial rise of the waters, some little way above Memphis, taking place very nearly at the Summer Solstice.
Again, M. Biot has made a series of calculations from which we learn that the heliacal rising of Sirius AT THE SOLSTICE occurred on July 20 (Julian) in the year 3285 B.C., and that in the year 275 B.C., the solstice occurred on June 27 (Julian), while the heliacal rising of Sirius took place, as before, on July 20 (Julian), so that in Ptolemaic times, at Memphis, there was a difference of time of about 24 days between the heliacal rising of Sirius and the solstice, and therefore the beginning of the Nile flood in that part of the river. This, among other things, is shown on the next page.
CONDITIONS OF THE HELIACAL RISING OF SIRIUS FROM 4000 B.C. TO 600 A.D.
The diagram shows (1) by white horizontal lines the Gregorian and Julian dates for the rising at Thebes and at Memphis; (2) by the full diagonal line the Julian date of the solstice or beginning of the inundation in each century, at a point of the river near Memphis. The fainter lines show the Julian dates for other places where the time of the beginning of the flood differs by three days from the Memphis dates. The interval between each line represents a difference of three days in the arrival of the flood; (3) the interval in days between the heliacal rising and the inundation at different periods and at different points of the river. This can be determined for each century by noticing the interval between the proper diagonal line and that indicating the heliacal rising; (4) by dots at the top of the diagram the commencement of the Sothic period as determined by Oppolzer, Biot, and the author.
We learn from the work of Biot and Oppolzer, then, that the precessional movement of the star caused successive heliacal risings of Sirius at the solstice to be separated by almost exactly 365¼ days—that is, by a greater period than the length of the true year. So that, in relation to this star, two successive heliacal risings at the 1st of Thoth vague are represented by a period of (365¼ × 4 =) 1461 years, while in the case of the solstices we want 1506.
Now in books on Egyptology the period of 1461 years is termed the Sothic period, and truly so, as it very nearly correctly measures the period elapsing between two heliacal risings at the solstice (or the beginning of the Nile flood) on the 1st of Thoth in the vague year.
But it is merely the result of chance that 365¼ × 4 represents it. It was not then known that the precessional movement of Sirius almost exactly made up the difference between the true length of the year and the assumed length of 365¼ days. It has been stated that this period had not any ancient existence, but was calculated back in later times. This seems to me very improbable. I look upon it rather as a true result of observation, the more so as the period was shortened in later times, as Oppolzer has shown.
It will be seen that our investigations land us in several astronomical questions of the greatest interest, and that the study is one in which modern computations, with the great accuracy which the work of Leverrier and others gives to them, can come to the rescue, and eke out the scantiness of the ancient records.
To consider the subject further, we must pass from the mere question of the year to that of chronology generally.