# Problems with the Julian Calendar

The Julian calendar was a major improvement over the older Roman calendar. The rule for leap years was simple enough that anyone could keep track of it, and the calendar stayed pretty much in time with the solar year. But just how accurate was the Julian calendar? With the quadrennial leap year, the mean calendrical year is 365.25 days. As we have seen, the actual length of the mean tropical year is closer to 365.2422 days. So the civil year on average is too long by about 11 minutes. Over a long period of time, this discrepancy means the seasons will slowly fall behind the calendar date. If we calculate a value Y, which equals the number of calendar years it takes for one day of "seasonal slip" to occur, we have a way to compare the relative accuracy of various calendars.

For those interested, the rough formula is Y = 1/(MCY - 365.2422), where MCY is the mean civil year length in days. To even more precise, we could take into account the gradually decreasing length of the tropical year. That requires we consider the calendar over a specific span of years and replace the approximate value for the tropical year with: 365.24231533 - (y1 + y2) * 3.06713e-8, where y1 is the start year and y2 the stop year in astronomical years (= Common Era, but for BCE dates, use negative number and add one, e.g., 1 BC = 0, 2 BC = -1, etc.). This kind of precision will be irrelevant unless we have precise astronomical observations in the relevant years against which to compare it. Otherwise, rounding the number to the nearest 10th gives a good value for comparison.

A similar value, M, could be generated to show the number of months it takes for a lunar calendar to slip 1 day, assuming the calendar works by rule rather than direct observation.

In the Egyptian calendar, Y = 4.1; clearly, no one who used this calendar was seriously concerned to keep it aligned with the seasonal year. Using the approximate formula, Y for the Julian calendar is 128.2; the precise formula, considered over the span -44 to 1582, gives 129.3.

These numbers should suggest why the Julian calendar remained unreformed for so long. Within any one person's lifespan, only an astronomer would notice the difference. Even if a culture keeps records over a long period of time, the change is not all that great: less than 8 days in a millennium. While an extremely long period of time (about 27,700 years) might change "the darling buds of May" into the darling buds of December (in the northern hemisphere, of course), it hardly seems like a pressing problem.

The fact is, however, that from the beginning of the 13th century, there was a constant call among leading intellectuals (most notably Roger Bacon in 1267) for a modification to correct this drift. The press of other urgent matters and general inertia delayed matters until the late 16th century, but eventually a change was made. To understand why, we first need to look at the calculation of Easter.

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