The Beauty of the Gregorian Calendar

The argument for an improved value of the length of the month, or the
number of months per year N = Y/M, is analogous to the above argument
for the length of the year. The principle of secularity implies that corrections
to the Metonic cycle must be of the form
235
19
+ ε/30
100 S2
(19)
where S2 is the number of secular years in which ε resets of the epact are
made. The divisor 30 enters because a unit reset of the epact effects an
advancement or a retardation of the calendar moon by 1/30 lunations. For
a more detailed discussion see [7]. To obtain an optimal rational value for
ε/S2, we equate (19) to the astronomical value of N, and find
or
ε
S2
ε/30
S2
�
= 30 1236.8266 −
= 100 N −
23 500
19
(20)
23 500
�
= −0.465... = −[0, 2, 6, 1, ...]. (21)
19
The rational continued fraction approximations are −1/2, −6/13, −7/15.
This reasoning could not have been familiar to the makers of the Gregorian
calendar because (19) was not explicitly known to them. Their estimate
ε/S2 ≈ −(75 − 32)/100 = −43/100 is nevertheless a remarkably good guess,
only 3 lunar resets off the best value in 10 000 years. The number NG given
in (13) is obtained with this estimate:
NG = 235
19
43 1
− · . (22)
10 000 30
Considering the values YG and NG in (12) and (13), we see immediately
that the Gregorian calendar has a period of PG = 5 700 000 years. This is
because the solar period of 400 years divides the lunar period PG, and the
number of days in a solar period happens to be a multiple of 7: 146 097 days
are 20 871 weeks.
An analogous argument for the Julian values YJ and NJ, see (10) and
(11), shows that
76 YJ = 940 MJ = 27 759 days. (23)
This is the so called Callippic cycle known in astronomy from antiquity, and
named after Callippos of Kyzikos, about 330 BC. As 27 759 is not a multiple
of 7, the period of the lunisolar Julian calendar is seven times the Callippic
cycle,
532 YJ = 6580 MJ = 194 313 days. (24)
10