Page:The Rhind Mathematical Papyrus, Volume I.pdf/61

Rh called the heliacal rising of Sirius, its rising with the sun, and this day marked the beginning of a kind of year that would end when the day came for the same event to occur again. This Sirius year, like the solar year, was a quarter of a day longer than the Egyptian calendar year, and its beginning, like the rise of the Nile, occurred one day later every four years.

The siderial year is longer than the solar year. The vernal equinox moves along the ecliptic at the rate of about 50″ a year, and when the sun reaches the vernal equinox it has still 50″ to go to reach the point where it passed the vernal equinox the year before. If a star is on the ecliptic the time from the moment when the sun passes it to the moment when this event occurs again will be exactly a siderial year. We might suppose that the same would be true for any ﬁxed star. If we think of the great circle of the ecliptic with the earth as a point at its center, we should say that the sun passes any ﬁxed star when they have the same longitude, longitude being measured along the ecliptic. But the apparent daily revolution of the sun and stars around the earth is a motion parallel to the equator, and when the observer thinks of a star as overtaking and passing the sun, the passing that he thinks of is when they have the same right ascension, for right ascension is measured along the equator.

The heliacal rising of Sirius as observed by the Egyptians was not, indeed, the moment when Sirius passed the sun, even in this sense, for the star would have to rise before the upper edge of the sun in order to be visible. The refraction of the atmosphere and other conditions must be taken account of in determining this moment, and especially the fact that Sirius in the south reaches the horizon farther along on its circle of rotation than the sun. These conditions, however, for the most part, are the same from year to year and would not aﬁect the length of time between two such events. But the rate of increase in the right ascension of the star varies with the position of the equator, and the star year is not usually the same as the siderial year. The difference is, however, a periodic one, with a period of over 25,000 years. During one part of this period the star year is longer than the siderial year, and during the other part it is shorter. It simply happens that during practically all of Egyptian history the Sirius year was almost exactly $365 1/4$ days. The