Caesar!

Picture this: it is January 48BC. Caesar crossed the Rubicon nearly an year ago. Then he proceeded to march on Rome then go to Spain, destroy the forces loyal to the Senate there and come back. Now he is in Italy while Pompey and most of the Senate are in Greece assembling an army to meet him. There is a real possibility that Pompey will use the resources of the richer Roman East to outnumber and defeat Caesar. Furthermore, the Senate’s navy, led by Bibulus, controls the Mediterranean, and are blockading the Strait of Otranto.

However, Caesar knows something Bibulus doesn’t. The roman calendar calendar is a real mess. It is a lunar calendar with common years of 355 days and leap years of 377 or 378. The Pontifex Maximus, that is Caesar, was responsible for manually adding in leap years to keep the calendar in sync. But Caesar had been busy conquering Gaul for the last ten years so the calendar had drifted so far that it said January, but it was actually early Autumn. It was too dangerous to sail in winter, so Bibulus assumed Caesar wouldn’t dare a crossing. Caesar, of course, did dare the crossing. Then he proceeded to destroy Pompey’s army at Pharsalus. Pompey fled to Egypt, where king Ptolemy XIII had him beheaded. I think the lesson here is not to mess with the guy who owns the calendar.

After winning the civil war, Caesar proceeded to fix the calendar, so nobody could pull this trick ever again. The calendar he instituted, known as the Julian calendar, had the familiar 365 day regular years and 366 day leap years every 4 years, so an year length of 365.25. It was used into the XXth century in places like Russia and the Ottoman Empire. But in the XVIth century, Pope Gregory XIII noticed the calendar was 10 days out of sync which was messing up the calculation of the date of Easter. The Pope didn’t use this knowledge to win any battles, but he did fix the calendar to the precision of a day for the next 7700 years, which seems good enough. The Gregorian calendar has an year length of 365.2425 days. In the XIXth century John Herschel proposed modifying the Gregorian calendar by removing a leap year every 4000 years which would make it yet closer to the true tropical year. I will propose a better solution.

The true length of a tropical year is 365.242189 days. And the continued fraction of that is

    \[365.242189=365+\frac 1{4+\frac 1{7+\frac 1{1+\frac 1{3+\frac 1\ddots}}}}=[365;4,7,1,3,40,2,3,5].\]

This means that we can crop the continued fraction and get a good approximation of the length of the year. Cropping just before a large coefficient yields an especially good approximation compared to the size of the denominator, so I propose a calendar with year length [365;4,7,1,3]=365+\frac{31}{128}=365+\frac{1}{4}-\frac 1{128}, which corresponds to common years of 365, intercalated with leap years of 366 days every 4 years, except for once every 128 years which is a regular year instead. This is simpler than the rules for the Gregorian calendar and yet it achieves a smaller error

    \begin{equation*} \begin{split} 365.242189-365-\frac{1}{4}+\frac{1}{128} & = \phantom{-}0.0000015 \\ 365.242189-365-\frac{1}{4}+\frac{1}{100}-\frac 1{400} & = -0.000311 \end{split} \end{equation*}

So we’d be on the correct date for the next six hundred thousand years, which is close to the maximum length of time we can predict that sort of thing over anyway.