01(03): Do leap years always occur every four years?

While learning about watches, I chanced upon the fancy feature of ‘perpetual calendars’. However, before talking about perpetual calendars in watches, I should also discuss annual calendars in watches and watches without either feature. In mechanical/automatic watches without either the annual or perpetual calendar feature, there is a date wheel that adjusts every 24 hours and runs from 1 to 31. The watch owner is expected to manually correct the date on the first of the month following any month that has fewer than 31 days (5 times a year: February, April, June, September, and November).

Annual calendars keep track of months with 30 or 31 days. This means that the owner needs to adjust the date on 1 March each year since annual calendars are unable to handle months with fewer than 30 days (*cough* February).

Perpetual calendars not only take into account that all months can have a different number of days (it is usually either 30 or 31 days in a month; however, there are usually 28 days in February), they are also able to take into account the fact that February has 29 days in leap years. So, this means the date on a watch with the ‘perpetual calendar’ feature will always been correct, right? After all, the word ‘perpetual’ suggests eternal adherence. To fully grasp this issue, we need to examine the history of our calendar.

The calendar that is ubiquitous in today’s world is known as the Gregorian calendar. It is named as such because the calendar is named after Pope Gregory XIII, who introduced it in October 1582. Before talking about the Gregorian calendar, let us discuss the calendar it replaced – the Julian calendar.

The Julian calendar got its name from Julius Caesar, who introduced it in 46 BC. It was a reform of the Roman calendar, and it took effect on 1 January 45 BC. The Julian calendar has two types of years: ‘normal’ years with 365 days and ‘leap’ years of 366 days. There exists a simple cycle of three ‘normal’ years followed by a leap year and this pattern continues forever without exception. The Julian year is therefore, on average, 365.25 solar days long.

At this point, some readers might be confused. ‘Isn’t this the way our calendar works?’ you might ask. What’s wrong with this? Why did we need to switch to the Gregorian calendar? Doesn’t the leap year every four years ensure that the average year is close enough to the actual tropical year? Strictly speaking, the tropical year is about 20 minutes shorter than the solar/sidereal year due to the seasonal cycle not remaining exactly synchronised with the position of the Earth in its orbit around the Sun resulting from the precession of the equinoxes. But, hey, let’s take the tropical year and solar year to be basically synonymous.

Surprise, surprise. One tropical year is 365.24217 solar days long, implying that the mean tropical year is approximately 365 days, 5 hours, 48 minutes, 45 seconds. To clarify, 1 solar day in this context is NOT 86,400 seconds (ie 24 hours). A day that lasts exactly 24 hours is known as an ephemeris day (which is not the kind of ‘day’ we are talking about).

As we established three paragraphs earlier, the Julian calendar is 365.25 solar days. 365.25 is not the same as 365.24217. Although this difference is miniscule over a matter of a few years, you can imagine how this error will accumulate over many centuries. Although Greek astronomers had known, at least since Hipparchus, a whole century before the Julian reform, that the tropical year was slightly shorter than 365.25 solar days, the Julian calendar did not compensate for this difference. As a result, the Julian calendar year gains about three days every four centuries compared to observed equinox times and the seasons.

So, how do you fix a discrepancy of three days over four centuries?

Something absolutely brilliant.

The difference between the Gregorian and Julian calendars is that in the Gregorian calendar, year numbers evenly divisible by 100 are not leap years, except that those evenly divisible by 400 remain leap years. What does this mean? That means 2000 was a leap year, whereas 2100, 2200, and 2300 will not be leap years!! The mean Gregorian year is thus 365.2425 days long, which is much closer to the tropical year (difference of 0.0008 solar days) than the Julian calendar (difference of 0.0083 solar days).

At the time of the Gregorian reform, there was a 10-day difference between the Julian and Gregorian calendar. To deal with the difference that this drift had already reached, the date was advanced so that 4 October 1582 was followed by 15 October 1582. There was no discontinuity in the cycle of weekdays (i.e. 4 October 1582 was a Thursday and 15 October 1582 was a Friday).

Of course, the Gregorian calendar isn’t perfect either. The Gregorian calendar diverges from astronomical observations by one day in 3,030 years. Perhaps we could consider every year that is a multiple of 3032 to not be a leap year? It shall be the Nicholasian calendar. HAHA,

What does this mean then for the ‘perpetual calendars’ in watches that I mentioned earlier? They are perpetual only in the Julian Calendar! Watches with ‘perpetual calendars’ will erroneously consider 2100, 2200, and 2300 to be leap years. Thus, such watches aren’t, strictly speaking, perpetual.

I hope that you, dear reader, have found this article to be insightful.

Yours faithfully,
Nic
27 January 2019

References:
https://en.wikipedia.org/wiki/Julian_calendar
https://en.wikipedia.org/wiki/Gregorian_calendar
https://slideplayer.com/slide/4160613/
https://www.history.com/news/5-things-you-might-not-know-about-julius-caesar

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