to the best determinations of modern astronomy (Le Verrier's _Solar
Tables_, Paris, 1858, p. 102), the mean geocentric motion of the sun in
longitude, from the mean equinox during a Julian year of 365.25 days, the
same being brought up to the present date, is 360 deg. + 27".685. Thus the mean
length of the solar year is found to be 360 deg./(360 deg. + 27".685) x 365.25 =
365.2422 days, or 365 days 5 hours 48 min. 46 sec. Now the Gregorian rule
gives 97 intercalations in 400 years; 400 years therefore contain 365 x 400
+ 97, that is, 146,097 days; and consequently one year contains 365.2425
days, or 365 days 5 hours 49 min. 12 sec. This exceeds the true solar year
by 26 seconds, which amount to a day in 3323 years. It is perhaps
unnecessary to make any formal provision against an error which can only
happen after so long a period of time; but as 3323 differs little from
4000, it has been proposed to correct the Gregorian rule by making the year
4000 and all its multiples common years. With this correction the rule of
intercalation is as follows:--

Every year the number of which is divisible by 4 is a leap year, excepting
the last year of each century, which is a leap year only when the number of
the century is divisible by 4; but 4000, and its multiples, 8000, 12,000,
16,000, &c. are common years. Thus the uniformity of the intercalation, by
continuing to depend on the number four, is preserved, and by adopting the
last correction the commencement of the year would not vary more than a day
from its present place in two hundred centuries.

In order to discover whether the coincidence of the civil and solar year
could not be restored in shorter periods by a different method of
intercalation, we may proceed as follows:--The fraction 0.2422, which
expresses the excess of the solar year above a whole number of days, being
converted into a continued fraction, becomes

1
-----
4 + 1
-----
7 + 1
-----
1 + 1
-----
3 + 1
-----
4 + 1
-----
1 +, &c.

which gives the series of approximating fractions,

1/4, 7/29, 8/33, 31/128, 132/545, 163/673, &c.

The first of these, 1/4, gives the Julian intercalation of one day in four
years, and is considerably too great. It supposes the year to contain 365
days 6 hours.

The second, 7/29, gives seven intercalary days in twenty-nine