Teacher Supplement: Muses on the Gregorian calendar

  1. Easter generally falls on the first Sunday, after the first full moon after the equinox.
  1. The equinox occurs twice a year (March 20 and September 23) when the Sun crosses the celestial equator. At that time, the day has 12 hours of daylight.
  1. The Julian calendar assumed a leap year every 4 years because it was thought that there were 365 ¼ days in the tropical year. However the correct value was 365.24219 days. The difference of 0.00781 days meant that the Julian calendar over-reached the tropical year by 11 minutes and 15 seconds. This over-reach was first noticed by the venerable Bede in AD 725, who noted that the full moon was ahead of its tabulated date.
  1. 11 minutes and 15 seconds over-reach turns into 45 minutes in any four year period. It turns into a full day over-reach in 128 years, and 75 hours (3 days and 3 hours) in every 400 years. To fix any future over-reach, a new calendar was envisagedthat omitted 3 days every 400 years. In 1582, Pope Gregory heralded in the new calendar, but it took another 170 years for the UK and its colonies to switch over. They enacted the change in 1751. The new calendar simply replaced 3 particular leap years with 3 common years every 400 years. The most obvious time for the 3 switches were the end of centuries, so now end-of-century years which are not divisible by 400, are not leap years (eg 1800,1900,2100,2200, etc). The astronomer who came up with the idea was Aluise Baldassar Lilio (1510 – 1576), the true inventor of the Gregorian calendar.
  1. For the UK to fix the existing damage, 11 days had to be cut out of the calendar to bring the paschal full moon back to its tabulated date. So Thursday 14 September 1752 became the day after Wednesday 2 September 1752. Furthermore the Julian calendar, by convention, had the last day of the year as the 24th March. New Years Daywas alwaysLady’s Day on the 25thMarch each year and this is why the month prefixes for September, October, November and December look so strange in today’s calendar. The new Gregorian calendar changed this to the 31st December, so that when the changes were introduced in 1751, the year ended early on 31 December 1751 with the next day being 1 January 1752.

Three interesting people

  1. Christian Zeller (1822-1899) in 1883 discovered an amazing formula that allowed the calculation of the number of elapsed days in a given common year. He noted that every month had at least 28 days, and if we number the months from 1 to 12, we can create a table of excess days as follows:

Month / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12
Excess / 3 / 0 / 3 / 2 / 3 / 2 / 3 / 3 / 2 / 3 / 2 / 3
Total / 29 / 29 / 3 / 5 / 8 / 10 / 13 / 16 / 18 / 21 / 23 / 26

Zeller’s formula works from the 1st March and this is why the third row shows the accumulation from March (shaded to emphasise this) To see how it works, take any particular month number and add 9 (let’s take 7 so the total is 16), then discard the complete multiples of 12 to collect the left overs only (so 16 reduces to 4). Multiply these left overs by 13 and add 2 (so 4 times 13 plus 2 is 54), before finally dividing the result by 5 (this gives 10.4). The whole number part of the answer will be the cumulative total of excess days from the 1st of March to the day prior to month number you originally chose. Hence the whole part of our example is 10, and you can see from the table that 10 excess days have accrued prior to beginning the 7th month. Take another example, say November. 11 plus 9 is 20, so 8 remain when the twelve is removed. Multiply by 13 to give 104 and add 2 to make 106. Now divide 106 by 5 to give 21.2. Look back at the 3rd row in the table under month 10 to find 21 days.

Now it’s a short step from this formula to find the total elapsed days from the 1 January to any date in a common year. For example, suppose we wanted to find the total number of elapsed days to the 6th November. The elapsed days from 1st March must be 8 lots of 28 (224 days) plus Zeller’s 21 days plus the 59 days of January and February – that’s 304 days!

  1. Lewis Carroll, writer of Alice in Wonderland was born 27 January 1832 and died 14 January 1898. His real name was Charles Dodgson and he was a clever mathematician. So he invented a clever way to determine the day of the week for any given day in the Gregorian calendar. This is described in the student handout.
  1. John Conway, professor of mathematics at Princeton University, discovered, in his investigations, another amazing fact about the Gregorian calendar. In any year (leap or common) the following dates will always occur on the same day of the year:

Fourth day of the fourth month (4 April)

Sixth day of the sixth month (6 June)

Eighth day of the eighth month(8 August)

Tenth day of the tenth month(10 October)

Twelfth day of the twelfth month (12 December)

Fifth day of the ninth month and the ninth day of the fifth month

Seventh Day of the eleventh month and the eleventh day of the seventh month

And the last day in February

[The special day (the doomsday as Conway calls it) for 2014 is FRIDAY]

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