intercalation had been followed without correction, and the
cycle been perfectly exact; but as neither of these suppositions is true,
two equations or corrections must be applied, one depending on the error of
the Julian year, which is called the solar equation; the other on the error
of the lunar cycle, which is called the lunar equation. The solar equation
occurs three times in 400 years, namely, in every secular year which is not
a leap year; for in this case the omission of the intercalary day causes
the new moons to arrive one day later in all the following months, so that
the moon's age at the end of the month is one day less than it would have
been if the intercalation had been made, and the epacts must accordingly be
all diminished by unity. Thus the epacts 11, 22, 3, 14, &c., become 10, 21,
2, 13, &c. On the other hand, when the time by which the new moons
anticipate the lunar cycle amounts to a whole day, which, as we have seen,
it does in 308 years, the new moons will arrive one day earlier, and the
epacts must consequently be increased by unity. Thus the epacts 11, 22, 3,
14, &c., in consequence of the lunar equation, become 12, 23, 4, 15, &c. In
order to preserve the uniformity of the calendar, the epacts are changed
only at the commencement of a century; the correction of the error of the
lunar cycle is therefore made at the end of 300 years. In the Gregorian
calendar this error is assumed to amount to one day in 3121/2 years or eight
days in 2500 years, an assumption which requires the line of epacts to be
changed seven times successively at the end of each period of 300 years,
and once at the end of 400 years; and, from the manner in which the epacts
were disposed at the Reformation, it was found most correct to suppose one
of the periods of 2500 years to terminate with the year 1800.

The years in which the solar equation occurs, counting from the
Reformation, are 1700, 1800, 1900, 2100, 2200, 2300, 2500, &c. Those in
which the lunar equation occurs are 1800, 2100, 2400, 2700, 3000, 3300,
3600, 3900, after which, 4300, 4600 and so on. When the solar equation
occurs, the epacts are diminished by unity; when the lunar equation occurs,
the epacts are augmented by unity; and when both equations occur together,
as in 1800, 2100, 2700, &c., they compensate each other, and the epacts are
not changed.

In consequence of the solar and lunar equations, it is evident that the
epact or moon's age at the beginning of