Tuesday, February 1, 2011

Notes on February

February, by my calculation, celebrates its two thousand seven hundred and twenty-fourth birthday today, the month having been introduced into the Roman calendar during the reign of, and by, Numa Pompilius (753-673 BC), the second king of Rome and the successor of Romulus. In 713 BC approximately, Numa introduced calendar reforms and added two months to a year that, until then, in the West, had only had ten months. The story is told by Livy (59 BC-17 AD) in these words—the only authority we have for this event. The quoted text is taken from here taken from a work titled Books From the Founding of the City, usually rendered Livy’s History of Rome.

And first of all he divided the year into twelve months, according to the revolutions of the moon. But since the moon does not give months of quite thirty days each, and eleven days are wanting to the full complement of a year as marked by the sun’s revolution, he inserted intercalary months in such a way that in the twentieth year the days should fall in with the same position of the sun from which they had started, and the period of twenty years be rounded out. [7] He also appointed days when public business might not be carried on, and others when it might, since it would sometimes be desirable that nothing should be brought before the people. [Livy, Ab urbe condita libri, Book 1, Chapter 19]
The two new months, January and February, were added, one at the beginning and one at the end of the year, thus ending up in reverse order, all in efforts more or less to bring the months into conformity with the lunar calendar. The solar year’s length, 365 days, was already known in Numa’s time. If we divide that number by 29 days for the lunar cycle (in actuality 29.5306), we get 12.36 months, hence adjustments must be made. Herewith some other arithmetic:

365 / 29.5306 days = 12.36 months — more than a third of a month too many.

365 - 29 = 336 days

336 days / 11 month = 30.5 days. From this we get the alternating 30 and 31 days for months, but with seven months having 31 and four having 30 days, for a total of 337 days. That leaves 28 days for the 12th months, February, which under Numa was also the last month. Now since the year is actually 365.25 days long, every four years an extra day is added—again in February, the odd lady out, as it were. We owe the observation of that quarter day slip to the Julian calendar, instituted by Julius Caesar (100-44 BC).

Now another problem arises. The actual solar year, the time of the earth’s actual orbit around the sun, is precisely measured to be 365.2425 days in length, thus not quite a quarter of a day a year longer than 365 days. The upshot is that adding a day to February every four years adds too much time to the calendar, about 11 minutes every fourth (leap) year. For this reason, and to compensate, we omit the leap year every 100 years—unless the year is evenly divisible by 400. The exception is that if a year is divisible both by 400 and 100 evenly, the 400 carries the day and the 100 year omission of the leap year is therefore not observed. Five examples:

YearDivided by 400By 100By 4Leap Year

In the above I highlight the computation that determines the leap year itself. The rules: If evenly divisible by 400, always a leap year. If evenly divisible by 100, never a leap year except when also divisible by 400. If evenly divisible by 4, always a leap year unless also divisible by 100.

February is the oddest month—because it varies. And it was in that year that changes were made because it used to be the last month too. That changed in 452 BC when the then ruling powers in Rome, the decimvirs (“the Ten Men”—no feminism then) switched the positions of January and February—and the last became the second. The Ten Men, by the way, were a kind of constitutional committee with time-limited rule. When their time came to leave, they didn’t want to (surprise!) and had to be forced out of office…

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