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. [Illustration: 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.