According to Biot the heliacal rising of Sirius _at the solstice_ took place on July 20 (Julian), in the year 3285 B.C.; and according to Oppolzer it took place on July 18 (Julian), in the year 3000 B.C. But this is too general a statement, and it must be modified here. There was a difference of seven days in the date of the heliacal rising, according to the latitude, from southern Elephantine and Philæ, where the heliacal rising at the solstice was noted first, to northern Bubastis. There was a difference of four days between Memphis and Thebes, so that the connection between the heliacal rising and the solstice depended simply upon the latitude of the place. The further south, the earlier the coincidence occurred. Here we have an _astronomical_ reason for the variation in the date of New Year's Day. There no doubt was a time when the Egyptian astronomer-priests imagined that, by the introduction of the 365-days year, marking its commencement, as I have said, by the rising of one of the host of heaven, they had achieved finality. But, alas, the dream must soon have vanished. Even with this period of 365 days, the true length of the year had not been reached; and soon, whether by observations of the beginning of the inundation, or by observations of the solstice in some of the solar temples when these had been built, it was found that there was a difference of a day every four years between the beginning of the natural and of the newly-established year, arising, of course, from the fact that the true year is 365 days _and a quarter of a day_ (roughly) in length. With perfectly orientated temples they must have soon found that their festival at the Summer Solstice--which festival is known all over the world to-day--did not fall precisely on the day of the New Year, because, if 365 days had exactly measured the year, that flash of bright sunlight would have fallen into the sanctuary just as it did 365 days before. But what they must have found was that, after an interval of four years, it did not fall on the first day of the month, but on the day following it. Recurrent solstices │ │ │ │ │ │ │ │ │ Recurrent 1st. of Thoth │ │ │ │ │ │ │ │ │ │ The true year and the newly-established year of 365 days, then, behaved to each other as shown in the following diagram, when the solstice, representing the beginning of the calendar year, occurred on the 1st Thoth of the newly-established calendar year. We should have, in the subsequent years, the state of things shown in the diagram. The solstice would year by year occur _later_ in relation to the 1st of Thoth. The 1st of Thoth would occur _earlier_, in relation to the solstice; so that in relation to the established year the solstice would sweep forwards among the days: in relation to the true year the 1st of Thoth would sweep backwards.