In the calendar, the duration of a year is taken as 365 days 5 hours 48 minutes 46 seconds. Express this in days, expand the measurement into a continued fraction, 1] then form the approximate fractions of the continued fraction. Explain the measures regarding leap years in the Julian and Gregorian calendars based on the approximate fractions.

[^0]:    ${ }^{1}$ For the necessary concepts regarding continued fractions, see the footnote of problem 1283, K. M. L. 29 (1964) p. 67.
The part of the year exceeding 365 days is $(5 \cdot 60 + 48) \cdot 60 + 46 = 20926$ seconds. On the other hand, 1 day $= 86400 \text{ s}$, so the fractional part, expressed in days, and then as a continued fraction,

$$
t = \frac{20926}{86400} = \frac{1}{4 + \frac{1348}{10463}} = \frac{1}{4 + \frac{1}{7 + \frac{1027}{1348}}} = \ldots
$$

The digits (partial denominators) of the continued fraction are sequentially $4, 7, 1, 3, 5, 64$, and the approximate fractions are:

$$
1 / 4, \quad 7 / 29, \quad 8 / 33, \quad 31 / 128, \quad 163 / 673
$$

The Julian calendar, established by Julius Caesar, took the fraction to be $1 / 4$ of a day, adding the accumulated fractional parts every four years as one leap day, and called these years leap years.

Given the duration of the year mentioned above, we now see that this method added 11 minutes and 14 seconds more to each year than necessary, causing the calendar to lag behind by 44 minutes and 56 seconds (almost 3/4 of an hour) every four years. This lag accumulates to one day every 32 years, so the next logical adjustment would have been to skip one leap year every 32 years. This is also indicated by the 4th approximate fraction, which suggests that out of 128 years, it would be more appropriate to have 31 leap years instead of 32. However, since the duration of the year was not known with the above precision in Caesar's time, and thus this instruction could not be given, it would also have been difficult to keep track of the omission due to the non-round number 128.

Similarly, it would still be confusing today to have 7 leap days every 29 years or 8 leap days every 33 years, as suggested by the 2nd and 3rd approximate fractions. The advice based on the 3rd approximate fraction can be seen as a combination of the first two: in the first 29 years of a 33-year period, insert 7 leap days, and in the remaining 4 years, insert 1 leap day. Alternatively, since 33 years is nearly a third of 100 years, this can also be stated as: every 100 years, have $3 \cdot 8 = 24$ leap days, one less than the Julian calendar suggests.

This is almost identical to the instruction of the Gregorian calendar reform (1582), which stipulates that in every 400 years, 3 out of 4 years ending in 00 are considered common years, and a year ending in 00 is a leap year only if the part before 00 is divisible by 4. Thus, in every 400 years, we have 97 leap days. This easily memorable and long-lasting solution can also be explained by the combination of the 4th and 1st approximate fractions: in a round 400 years, 3 periods of 128 years fit, and the remaining 16 years can be filled with 4 four-year periods, thus yielding $3 \cdot 31 + 4 \cdot 1 = 97$ leap years. The fortunate accuracy of this combination is enhanced by the fact that $1 / 4$ is an upper and 31/128 is a lower approximate fraction. Similarly, the 5th approximate fraction approximates from above and, combined with the 4th, suggests 194 leap days in 801 years.

The 400-fold of the fractional day of the year is 2000 hours, 19200 minutes, or 1152000 seconds, which is just under 3 hours less than the 97 leap days inserted. In other words: on average over 400 years, we take the year to be

$$
365 + \frac{97}{400} = 365 + \frac{20952}{86400}
$$

days, or 26 seconds longer than necessary. Both methods show that the difference only grows to one day after 3000 years.

In this context, the Gregorian calendar reform can be considered very fortunate, especially considering that the duration of the year was not known with the above precision in 1582, as Huygens only invented the pendulum clock in the 17th century, and the mean solar day as a unit of time only emerged in the 19th century.

It is also worth noting that the reform had another, much more important measure at the time, as the adjustment discussed above was only first applied in 1700. It was observed that the weather and equinoxes were about 10 days ahead of the old records in the calendar. The reform corrected this by writing October 15, 1582, immediately after October 4, 1582, thus setting the calendar so that the vernal equinox would again fall on March 21 or 22.