years, and
errs in defect, as it supposes a year of 365 days 5 hours 47 min. 35 sec.

The third, 8/33, gives eight intercalations in thirty-three years or seven
successive intercalations at the end of four years respectively, and the
eighth at the end of five years. This supposes the year to contain 365 days
5 hours 49 min. 5.45 sec.

The fourth fraction, 31/128 = (24 + 7) / (99 + 29) = (3 x 8 + 7) / (3 x 33
+ 29) combines three periods of thirty-three years with one of twenty-nine,
and would consequently be very convenient in application. It supposes the
year to consist of 365 days 5 hours 48 min. 45 sec., and is practically
exact.

The fraction 8/33 offers a convenient and very accurate method of
intercalation. It implies a year differing in excess from the true year
only by 19.45 sec., while the Gregorian year is too long by 26 sec. It
produces a much nearer coincidence between the civil and solar years than
the Gregorian method; and, by reason of its shortness of period, confines
the evagations of the mean equinox from the true within much narrower
limits. It has been stated by Scaliger, Weidler, Montucla, and others, that
the modern Persians actually follow this method, and intercalate eight days
in thirty-three [v.04 p.0991] years. The statement has, however, been
contested on good authority; and it seems proved (see Delambre, _Astronomie
Moderne_, tom. i. p.81) that the Persian intercalation combines the two
periods 7/29 and 8/33. If they follow the combination (7 + 3 x 8) / (29 + 3
x 33) = 31/128 their determination of the length of the tropical year has
been extremely exact. The discovery of the period of thirty-three years is
ascribed to Omar Khayyam, one of the eight astronomers appointed by
Jel[=a]l ud-Din Malik Shah, sultan of Khorasan, to reform or construct a
calendar, about the year 1079 of our era.

If the commencement of the year, instead of being retained at the same
place in the seasons by a uniform method of intercalation, were made to
depend on astronomical phenomena, the intercalations would succeed each
other in an irregular manner, sometimes after four years and sometimes
after five; and it would occasionally, though rarely indeed, happen, that
it would be impossible to determine the day on which the year ought to
begin. In the calendar, for example, which was attempted to be introduced
in France in 1793, the beginning of the year was fixed at midnight
preceding the day in which the true aut