# Frank Morgan's Math Chat - Dear Marilyn, About Leap Years

October 14, 1998

From: Frank Morgan Frank.Morgan@williams.edu
Subject: Correction on skipping leap years
September 29, 1998

Dear Marilyn:

Your explanation [in Parade Magazine] of September 13, 1998 that we may need to skip a leap year EVERY FEW THOUSAND YEARS because the tropical year (about 365.2422 days) is a bit shorter than the average calendar year (365.2425 days) neglects CHANGES in the length of the day and year, which are more important on this time scale. . . .

Many thanks for all you are doing to give the country a better understanding and appreciation of mathematics!

Sincerely yours,

Frank Morgan

IS THE YEAR 2000 A LEAP YEAR? Yes indeed, as you would expect, since 2000 is divisible by four. Nevertheless, this is an excellent question, since the century years 1900 and 2100, although divisible by four, are NOT leap years. Why?

The reason we have leap years in the first place is that the astronomical "tropical year," which determines the seasons, is not exactly 365 days long, but closer to 365.25. So to stay in tune with the seasons, we have to add a leap day every four years. Unfortunately the year is not exactly 365.25 days long either, but closer to 365.2422 days long. Our current calendar approximates the tropical year better by canceling century leap years unless they are divisible by 400. That is why 1900 is not a leap year but 2000 is.

On the average, our current calendar has 365.2425 days per year, with an error of about .0003 days per year, or about one day every 3000 years.In her column in Parade Magazine on September 13, Marilyn Vos Savant concludes that we may need to skip a leap year every few thousand years. Her logic is right, but the specific conclusion is wrong,because on the scale of thousands of years there are other more important factors.

First of all, the length of the astronomical year itself is changing. As a spinning top running down, as the earth's revolution slows down,the wobbling or precession of its axis speeds up. Since our seasons are caused by whether the axis tilts towards or away from the sun, this increasing precession causes the tropical year to speed up, currently at about one day every 167,000 years. This error accumulates, and would cause an error in the calendar of one day after about 600 years.

There is another big effect, which all texts I have seen seem to overlook: the DAY is getting longer, and longer days mean fewer days per year. Current controversial measurements suggest we are now losing about one day per 150,000 years, causing an accumulated error in the calendar of one day after about 550 years.

Recently a controversy has raged on the internet over a peculiar definition usually used by astronomers. They mark the tropical year by the beginning of spring, while they probably should average over all the seasons. It makes a difference because the earth's orbit is not perfectly round, but a bit elliptical. As far as I can tell, the effects of this peculiar definition currently may be canceling out the other effects and keeping our calendar in almost perfect agreement with the tropical year, at least for the coming millennia.

NEW CHALLENGE (Steve Jabloner). In the lobby of a hotel there are three on/off switches. One is attached to a light bulb on the third floor; the other two are attached to nothing at all. You may work the three switches as often as you please. How many times do you need to visit the third floor to determine which switch is attached to the light bulb?