The Calendar Blog

The Ides of March

2012-03-13T08:00:00.000+01:00

2055 years ago, on 15 March 44 BC, just a little over a year after the introduction of the calendar that bears his name, the Roman leader Julius Caesar was murdered.

In Shakespeare's famous play Julius Caesar a soothsayer warns the Roman leader about the assassination with the words, "Beware the Ides of March." Why is that?

The Romans referred to 15 March as "Idus Martiae" or "Idus Martii," and that term was anglicized as "Ides of March." So the soothsayer is really saying, "Beware the 15th of March," but that sounds much more mundane than Shakespeare's words.

The term is frequently misunderstood, and you occasionally hear people ask, "What are the eyes of March?" What indeed?

Best wishes,
Claus

P.S. The Calendar FAQ has a section about how the Romans numbered the days of the month.

What is a tropical year?

2012-02-27T08:00:00.000+01:00

In my last post I explained that at tropical year is the time between, for example, two spring equinoxes. This means that a tropical year is 365.2422 days. Our calendar is based on the tropical year.

But that definition of the tropical year is not quite accurate. The earth's orbit around the sun is elliptic rather than circular, and this means that the speed of the earth on its travel around the sun is not constant. Therefore the time between two spring equinoxes is not the same as the time between two autumn equinoxes, nor is it the same as the time between two summer or winter solstices.

I did a little calculation on the times for equinoxes and solstices in the years 2000-2020. On the average the time between...
    two March equinoxes is 365.2422 days,
    two September equinoxes is 365.2418 days,
    two June solstices is 365.2415 days,
    two December solstices is 365.2425 days.

Of these, the time between March equinoxes fits the tropical year quite nicely, but that's merely a coincidence. When astronomers calculate the length of the tropical year, they have to average the time not just between equinoxes and solstices but also between the recurrence all other positions in between.

Also, the time between, say, two March equinoxes varies from year to year by as much as 20 minutes because the gravity of the other planets affect the speed of the earth.

This, of course, makes an exact definition of a tropical year quite unwieldy. But if you say that a tropical year is the time between two March equinoxes, you're not far from the truth.

Best wishes,
Claus

P.S. You can find dates and times for equinoxes and solstices at the US Naval Oceanographic Portal.

What is a year?

2012-02-09T08:10:00.000+01:00

Seasons (Photo by ganzoman)

What is a year? Many people would answer simply, "A year is the time it takes for the earth to travel around the sun." Few people realize how vague that answer is.

The difficult question is: How do we know that the earth has completed an orbit around the sun? We cannot simply leave a marker in space and then see when we come across that marker again.

There are (at least) three important – but different – definitions of what a year is:

Imagine that you are sitting on a distant star and observing the earth through a very powerful telescope. You make a note of the earth's position relative to the sun, and then you wait for the earth to be in the same position again. This takes a sidereal year, which is 365.2564 days.

From your distant location you notice that the earth's orbit is not completely circular, but slightly elliptic. The earth is closest to the sun in early January and furthest from the sun in early July. You also notice that as time passes, the earth's orbit itself moves. The point in space where the earth is closest to the sun moves a little each year. The period between each time the earth is closest to the sun is an anomalistic year, which is 365.2596 days or about five minutes longer than the sidereal year.

But the axis of the earth also moves. The poles remain in place, of course, but the whole earth "wobbles" like a spinning top. This means that at spring equinox this year the earth will be at a slightly different position on its orbit that at spring equinox last year. The time between two equinoxes is a tropical year, which is 365.2422 days or about 20 minutes shorter than the sidereal year.

Since the orientation of the earth's axis with respect to the sun controls our seasons, we base our calendar on the tropical year.

Oh, by the way: What I've said above about the tropical year and the equinoxes is only an approximation. I'll say more about that next time.

Best wishes,
Claus

A Curious Number

2012-01-28T08:00:00.000+01:00

A square root (Photo: Ben Tesch)

Last time I wrote about 2012 being a double leap year with both an extra day and an extra second. This leads me to the question: How many seconds are there in a leap year?

The answer is, of course, easy to calculate: Multiply 366 with the number of seconds in a day. The result is 31,622,400 seconds (plus an extra second in the case of 2012).

A mathematician who sees this number might well exclaim, "What an extraordinary sequence of digits!" The thing is that the digits 3 1 6 2 2 are the first digits of the square root of 10.

It turns out that with an error of only 0.001%, the number of seconds in a leap year is the square root of 1,000,000,000,000,000.

So now you know!

Best wishes,
Claus

A Double Leap Year

2012-01-14T08:00:00.000+01:00

Digital clock showing a leap second

2012 will be a double leap year. Its length will not be 366 days, but approximately 366.00001 days.

On 5 January I received an e-mail from the IERS (see below) informing me that they had decided that a leap second should be inserted in the final minute of 30 June 2012 UTC, making that minute 61 seconds long.

What are leap seconds? Well, the Earth's rotation is not completely uniform. Sometimes it speeds up a little, sometimes it slows down a little. We're only talking about small fractions of a second here, but these small changes in speed mean that corrections are needed to ensure that midnight remains the middle of the night.

In order to keep midnight where it should be, we occasionally need to insert or remove a second. This is the so-called leap second.

Leap days follow a well-defined pattern with one leap day inserted every fourth year (with a few exceptions). But leap seconds follow no pattern. They are inserted when required as decided by the kind people at the Paris Observatory.

The Paris Observatory is home to the IERS. This abbreviation used to stand for the hilariously named "International Earth Rotation Service". (You almost feel they should have a motto such as, "We make the earth move for you.") But in 2003 the name was changed to the much less amusing "International Earth Rotation and Reference Systems Service", although they retained the abbreviation IERS.

The IERS keeps very accurate track of the rotation of the earth, and twice a year they decide if a leap second should be inserted or not. Since the invention of the leap second in 1972 a total of 25 leap seconds have been inserted. So far we have only had positive leap seconds (that is, an added second), but in theory it is also possible to have negative leap seconds (that is, a removed second).

For clock makers, leap seconds are not easy to handle because they occur in an irregular pattern. And does it really matter if midnight drifts a minute or so every century? Many people find the leap second unnecessary, and there is currently a proposal on the table to abolish them. If the proposal is approved, there will be no leap seconds after 2017.

Best wishes,
Claus

Twelve Years

2011-12-31T08:00:00.000+01:00

The first 12 years of the 2000s have gone. What calendrical significance does a 12-year-period have?

Some of you may remember that 2000 started on a Saturday, and these days we see that 2012 starts on a Sunday. Every 12 years, the weekdays shift one position.

It is well-known that if a certain date falls on a Monday in one year, it will fall on a Tuesday in the following year, except if there is an intervening leap day, in which case the date will fall on a Wednesday in the following year. This is a fairly simple rule, but it is quite complicated to apply across a span of several years.

This is where the 12-year-rule comes in handy: If a certain date falls on a Monday in one year, it will fall on a Tuesday 12 years later.

For longer time spans there is a 28-year-rule: If a certain date falls on a Monday in one year, it will also fall on a Monday 28 years later.

There are no exceptions to these rules in the years 1901-2099, which means that they apply to most current human lifespans.

Happy New Year,
Claus

On the 59th day of Christmas my true love gave to me...

2011-12-17T08:00:00.000+01:00

If advertisements and shop windows are anything to go by, Christmas appears to start in early November. But, of course, the first day of Christmas is actually 25 December.

But how long does Christmas last? In English, the expression "The Twelve Days of Christmas" is used, referring to the days from 25 December to 5 January. On 6 January we have Epiphany, the celebration of the magis visiting the newborn Jesus.

So does Christmas last for twelve days? Well, that depends. In Sweden, Epiphany is called "trettondedag jul" (literally, "Thirteenth day of Christmas"), so the Swedes include that extra day. Other countries have longer Christmasses. Some churches celebrate 40 days of Christmas starting on 25 December and ending on 2 February, which is known as Candlemas or the Purification of the Blessed Virgin Mary.

A popular Danish Christmas song includes the words, "Christmas lasts 'till Easter. No, that's not true: before Easter there is Lent." Well, this winter Lent starts on 22 February, so that should give us 59 days of Christmas.

May the blessing that started in Bethlehem be with you all,
Claus

The Shortcomings of a Standard

2011-12-03T08:00:00.000+01:00

9/11 or 11/9?

What happened on 9/11?

That date makes most Americans think about September 11, 2001 and the horrible terrorist attacks on New York and Washington.

To a European, the date may bring memories of 9 November 1989 when the Berlin Wall came down.

So does 9/11 mean September 11 or 9 November? Well, that obviously depends on your cultural background. When nations disagree, standards are needed. Therefore the International Organization for Standardization (ISO) in their infinite wisdom have standardized a date format that consists of the year followed by the month followed by the day. So the two important dates mentioned above are 2001-09-11 and 1989-11-09, respectively. This is known as the ISO 8601 standard.

So far so good. And yet, something is missing. If the day is unknown, the standard allows us to abbreviate the date. For example, September 2001 is simply 2001-09. You can also omit the month and just refer to the year (for example 2001), and you may even just write the first two digits of the year (20 to indicate the current century).

But suppose somebody asks you, "When is Christmas Day?" What is the answer? Neither 12-25 nor 25-12 are legal representations of a date within the framework of ISO 8601. You must specify a year. But since Christmas comes once a year, that's clearly not possible.

The standard also allows esoteric date/time indications such as 2012-W23-4T09:00, which means Thursday in week 23 of 2012 at 9 o'clock. But there is no way to specify, say, that a company holds a weekly meeting Thursdays at 9 o'clock.

Best wishes,
Claus

Calendar reform? No, thank you

2011-11-19T08:00:00.000+01:00

Simplicity or variety?

Over the years, many attempts have been made to change the structure of our calendar.

One of the most promising ones was the World Calendar, which in the 1930s was seriously considered by the League of Nations as a possible replacement for our current calendar.

The main goal of most calendar reform proposals is to make all years identical. Since a week has 7 days and a year has 365 or 366 days, which cannot be divided by 7, this is a difficult goal to achieve. The World Calendar solved this problem by proposing that a year should include one or two days that were neither Mondays, nor Tuesdays, nor Wednesdays, etc.

After World War II, the United Nations took over the calendar work from the League of Nations, but in the 1950s the project died. The main problem was that the seven-day-week has religious significance for both Jews, Christians and Muslims. If the World Calendar were introduced, there would occasionally be eight days between two Sundays, and thus the seven day cycle would be broken – something that has probably never happened in the past 3500 years. For this reason, some calendar reform proposals are based on a leap week rather than a leap day. One such proposal is the Symmetry454 calendar.

But I do not agree with the whole purpose of a calendar reform! Why should all years be the same? Why is simplicity better than variety? I would not be pleased if my birthday always fell on a Monday! The quaint peculiarities of our calendar contribute in a small way to making our lives a little more interesting. And more importantly, the fact that we today are using a calendar with a structure that was introduced by Julius Caesar in 45 BC helps us feel connected to the history of the human race.

Best wishes,
Claus

P.S. If you want to know more about calendar reform proposals, you will find some interesting information on the Home Page for Calendar Reform.

Once in a century

2011-11-10T08:00:00.000+01:00

Tomorrow is 11/11/11. Six identical digits. Since neither 00/00/00 nor 22/22/22 nor any other similar combination are real dates, 11/11/11 is truly a once in a lifetime event for most of us.

Enjoy it.
Claus

Frequency of Easter Dates

2011-10-27T19:28:00.000+02:00

Which date for Easter Sunday is the most frequent?

When I asked myself that question I assumed that the there would be several days with almost equal frequency, but it turns out that 19 April is noticeably more frequent than any other Easter date.

In the Gregorian calendar the dates for Easter have a cycle of 5,700,000 years, so I wrote a small program to find the frequency of each Easter date in a period of 5,700,000 years. (What? Are you saying that calculating Easter more than five million years into the future is silly? Well, never mind that!)

The chart above shows the frequency of each Easter date. The horizontal axis shows the dates for Easter Sunday, the vertical axis shows the number of occurrences of each date in a 5,700,000 year period.

It is not surprising that the earliest and latest possible Easter dates are fairly rare, with 22 March being the rarest, occurring in less than 0.5% of the years. But it struck me as surprising that the chart has a obvious peak at 19 April.

The explanation for this phenomenon is, of course, found in the way Easter is calculated. Take a look at the table showing the relationship between Epacts and the Paschal full moon. Each Epact gives rise to a total of seven possible Easter dates, depending on when the Sunday falls. For example, an Epact of 13 can cause Easter Sunday to fall on any of the dates 1-7 April. Conversely, most Easter dates can be triggered by seven different Epacts (for example, any of the Epacts 13-19 can cause Easter Sunday to fall on 1 April). But because of the peculiar rule for Epact 25, an Easter date of 19 April can be caused by eight different Epacts (namely, 1 and 24-30), hence its higher frequency.

Best wishes,
Claus

The Julian Calendar Today

2011-10-15T10:12:00.000+02:00

Foula Island (Photo: Lis Burk)

Although the Gregorian calendar is now official in almost every country in the world, there are a few areas where the old Julian calendar is still in use. Below, I will cover four such areas.

Firstly, and most importantly, the majority of Greek Orthodox Churches still use the Julian calendar. This means that countries with a predominantly Orthodox population frequently celebrate Easter later than other countries do. Christmas is a bit more tricky. 25 December (Julian) currently corresponds to 7 January (Gregorian), and the Orthodox churches in Russia do indeed celebrate Christmas Day on 7 January. But in many parts of the Greek Orthodox world, Christmas has never been a very important feast – Easter is far more important. This means that Christmas these days is more of a commercial event than a religious one. So if you visit the Greek part of Cyprus, for example, you will find that Easter is celebrated according to the Julian calendar, but Christmas is celebrated according to the Gregorian calendar because the Christmas celebration is mainly dictated by western commercialism. (I’m not sure when Greece celebrates Christmas. Can anybody help?)

Also, the Greek Orthodox state of Mount Athos in north eastern Greece uses the Julian calendar.

Secondly, the Julian calendar is sometimes used in astronomy. This may seem surprising. Of course modern astronomers use the Gregorian calendar to date events, but many astronomical formulas are based on a Julian year (365.25 days) or a Julian century (36,525 days). Also, a light-year is defined as the distance that light travels in the course of a Julian year. This means that a light-year is exactly 9,460,730,472,580.8 kilometres, because – by definition – a metre is the distance travelled by light in 1/299,792,458 seconds.

Thirdly, the tax year of the United Kingdom starts on 6 April. Before 1752, the English calendar year started on 25 March (Julian). When England moved the start of the calendar year to 1 January and later converted to the Gregorian calendar, the tax year was not moved. In the 1700s, 25 March (Julian) corresponded to 5 April (Gregorian), so the tax year started on 5 April. In the year 1800 the difference between the Julian and Gregorian calendar grew one day larger, so the start of the British tax year was moved to 6 April (Gregorian) to keep it on 25 March (Julian). But when the difference between the two calendars grew again in 1900, no change was made to tax year.

So today the British tax year still starts on 6 April as a legacy of the Julian calendar and an old new year’s day.

Fourthly, the island of Foula in the Scottish Shetland Islands still uses the Julian calendar – sort of. They never converted to the Gregorian calendar, but nevertheless they omitted a leap year in 1900, which means that today they are one day ahead of the Julian calendar, but 12 days behind the Gregorian. If you compare this to the information about the British tax year mentioned above, you find that 6 April (Gregorian) still corresponds to 25 March (Foula calendar).

Best wishes,
Claus

Equivalent Months

2011-10-08T08:00:00.000+02:00

In non-leap years, January is identical to October; in leap years January is identical to July.

All other months either start on different days of the week or have different lengths. For example, in non-leap years, February, March and November all start on the same day of the week, but they have different lengths.

Best wishes,
Claus

Friday the 13th

2011-10-01T08:00:00.000+02:00

A common supersition states that Friday the 13th of any month is an unlucky day. Well, if that's true, I've got some bad news for you.

How frequently does the 13th day of a month fall on a Friday?

In the old Julian calendar, the answer was easy: The 13th day of a month would fall on any day of the week with equal frequency. So the risk of the 13th falling on a Friday was exactly ¹/₇.

But in the Gregorian calendar, which we use today, things are different. The leap year cycle of our calendar is 400 years long. During those 400 years we have 97 leap years, which means that the total number of days in 400 years is 400×365+97=146,097. Curously enough, this number can be divided by 7, which means that the week day pattern repeats itself every 400 years. So in order to find out how frequently the 13th falls on a Friday, all we need to do is check the months in any 400 year period.

The results are shocking (if you are superstitious). In any 400 year period, the 13th falls on

a Monday	685 times
a Tuesday	685 times
a Wednesday	687 times
a Thursday	684 times
a Friday	688 times
a Saturday	684 times
a Sunday	687 times

So the 13th day of a month falls on a Friday more frequently than on any other day!

Best wishes,
Claus

Fifteen Years Ago

2011-09-28T10:34:00.002+02:00

What happened in 1996? The Summer Olympics were held in Georgia, USA. Prince Charles and PrincessDiana divorced. Boris Yeltsin was reelected President of Russia. Bill Clinton was reelectedPresident of the United States. And the first version of the Calendar FAQ was published on Usenet.

With the help of many kind people over the years, the Calendar FAQ has grown a lot. Working with theFAQ has been extremely rewarding for me, and based on the many positive emails I have received, Ibelieve that the FAQ has been useful for others as well.

Times change, and so does the Internet. Today, few people know what “Usenet” is. Also,the amount of information on the World Wide Web has grown enourmously. Especially the advent of suchinformational giants as Wikipedia makes me wonder: What purpose can the Calendar FAQ serve in thefuture?

One of the problems has been that the process of updating the document has been quite slow. Itherefore think that a more dynamic and modern approach is required.

Starting today, the Calendar FAQ will no longer be published on Usenet, nor will it be available asa downloadable PDF file. Instead, the FAQ will be available as a set of web pages. I hope to be ableupdate and expand these pages continally, thus ensuring that the Calendar FAQ becomes a more organicdocument.

Furthermore, I have started this blog, where I will publish small tidbits of information of acalendrical nature, and also provide information about the development of the Calendar FAQ itself.This blog is also a means for you to provide feedback and suggestions for the document.

I am looking forward to this. I hope you will find it useful.

As always, the Calendar FAQ is available here: http://www.tondering.dk/claus/calendar.html.

Best wishes,
Claus Tøndering (claus@tondering.dk)