Leap Years and SuchAuthor(s): Martin CooperSource: The Arithmetic Teacher, Vol. 27, No. 5 (January 1980), pp. 26-27Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41191669 .

Accessed: 12/06/2014 14:06

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Arithmetic Teacher.

http://www.jstor.org

This content downloaded from 185.44.78.31 on Thu, 12 Jun 2014 14:07:00 PMAll use subject to JSTOR Terms and Conditions

Leap Years and Such

By Martin Cooper - jr>

26 Arithmetic Teacher

How many days are in a year? For many people, the answer would be 365. But is it? With a calculator in hand, look at a 1979 calendar and total the numbers of days in each month. Do the same thing with a 1980 calendar. What and where is the difference? Why is there a difference? The year 1980 is a leap year. The years 1976 and 1972 were also leap years. The year 1984 will be a leap year, but what many

Martin Cooper is a professor of education at the University of New South Wales in Kensington in Australia. Prior to his present position, he was a member of the Measurement and Experiementa- tion Department of the Faculty of Education at the University of Ottawa. He has taught both mathe- matics and physics in Canada and in Australia.

people do not realize is that every fourth year is not a leap year. The his tory of the calendar is an interesting one and it offers some practical exer- cises for calculator computations.

The solar year, the time taken for the earth to travel once around the sun is 365.2422 days, or 365 days 5 hours 48 minutes 46 seconds. This period is, of course, short of 365 Va days by 1 1 minutes 14 seconds, or 0.0078 year. Taking a leap year every fourth year would therefore gradually put the cal- endar ahead ornature; the calendar would gain a day on nature every 128.19 calendar years.

In 45 B.c., the Julian system was in- stituted. This called for calendar years of 365 days in general, with an extra day added to each fourth year. The av- erage length of a year under this sys- tem was 365.25 days. By the sixteenth century, when astronomers had be- come able to measure the passage of time with reasonable accuracy, they re-

alized that the calendar had indeed run ahead ornature. Accordingly, Pope Gregory XIII ordered that the year 1582 be shortened by ten days. This was done by the omission of the days from 5 October through 14 October, the day following 4 October being called 15 October.

Let us pause in order to do some cal- culations. In the period from 45 B.C. to 1581 A.D., there were about 1626 Julian years - we say "about" because of un- certainty as to whether or not there was a year "zero". These years would ac- count for 593 896.5 days, assuming that each fourth year was a leap year. In the same period, the earth made 1626.0347 revolutions around the sun. Since 1626 solar years are equivalent to 593 883.8 days, the calendar was 12.7

This content downloaded from 185.44.78.31 on Thu, 12 Jun 2014 14:07:00 PMAll use subject to JSTOR Terms and Conditions

days ahead ornature. It thus seems that Pope Gregory's revision did not correct the calendar back to 45 B.c., but to some later date.

The Gregorian revision was adopted by Catholic countries, but the Greek and Protestant churches refused to rec- ognize the Pope's authority. In Britain and its colonies, the calendar contin- ued to gain on nature until a British Act of Parliament caused eleven days to be dropped from the year 1752, which would otherwise have had 366 days. This was 130 years after the Gre- gorian revision and resulted in bring- ing the calendars of the Catholic and Protestant countries in line with each other.

In order to prevent the calendar from running away from nature in an

unchecked fashion, other modifica- tions were made to the Julian system. These included the rule that century years would not be leap years unless they were divisible by 400. In effect, this is equivalent to dropping three leap years every 400 years, so that there are, in fact, ninety-seven leap years every four centuries.

Let us now do some calculations for the modified Julian system, which is referred to as the "Gregorian system." In 400 calendar years, there are 303 years of 365 days and 97 years of 366 days making 146 097 days altogether. Dividing this number of days by 365.2422, we obtain 400.000 32. Thus, 400 calendar years are equivalent to 400.000 32 solar years.

A period of 400 solar years accounts for 146 096.88 days, so that in four cen- turies the calendar is 0. 12 days ahead ornature. Thus, the calendar will be three days ahead in 10 000 years, or one day in about 3000 years.

We shall conclude with the state- ment of an equation in which the num- ber of days in a solar year are analyzed into components: 365.2422

= 365.25 -0.01+ 0.0025 - 0.0003

The first term on the right-hand side of the equation corresponds to the Jul- ian system of having one leap year in every fouryears. The second term re- fers to the dropping of a leap year each century and the third corresponds to the retention of a leap year each fourth century. The last term represents the error in the Gregorian system, about three days in ten thousand years.

Every fourth year is a leap year; right? Wrong! П

January 1980 27

This content downloaded from 185.44.78.31 on Thu, 12 Jun 2014 14:07:00 PMAll use subject to JSTOR Terms and Conditions