## Thursday, October 22, 2009

### Are the number of days per month rational?

During the last week I have been studying the computation of the Julian Day Number that astronomers use to keep track of time. To do this one needs to know how to convert from month and day to day of year. The number of days in a month is quite irregular and one wonders how this came about. Most of the features of our present calendar are due to the reforms of Julius Caesar in 46 BC. Caesar turned to an Egyptian astronomer, Sosigenes of Alexandria, for assistance in correcting the errors in the Roman calendar at that time. The lengths of the months and leap day date from that time. At first the length of the months seem quite arbitrary but if one considers the period from March to the following February the pattern if more regular. Placing the leap day at the end of this period is consistent with December being the "10th month."

The problem of designing a calendar with twelve month and 365 days is how to distribute the odd 5 days. The pattern seems to be consistent with using multiples of 30 7/12 rather than the more obvious 30 5/12 as seen in the calculation below. An irregularity is that the sequence is shifted by one month.

In order to do this one needs to be able to perform integer division and this can be done quite easily using multiplication tables and Sosigenes would have been quite capable of doing this. This gives us the sums of the days of the months. The formula for computing the sums turns out to be rather simple and allows us to derive a formula for January through December counting January as the first month. This simplifies converting month and day to day of year with the inclusion of a leap day in leap years.