# Why we have leap days

By Phil Plait | February 29, 2012 4:00 am

Warning: First, this is a somewhat modified repost from — oddly enough — four years ago. Second, this post has math in it. A lot. Some of it might even be correct. If you are mathophobic, then you might want to skip to the end, where I reveal what Rosebud means.

And for those of you who are incredibly anal, yes, I know I kinda lost track of significant digits about 2/3 of the way through this. I was using a calculator, and just used whatever numbers it gave me to the last decimal place, leaving off for the most part trailing 0s. Sue me. I’m free on February 29th, 4800.

When I was a kid, I had a friend whose birthday was on February 29th. I used to rib him that he was only 3 years old, and he would visibly restrain himself from punching me. Evidently he heard that joke a lot.

Of course, he was really 12. But since February 29th is a leap day, it only comes once every four years.

And why is it only a quadrennial event?

Duh. Astronomy!

The Days of Our Lives

We have two basic units of time: the day and the year. Of all the everyday measurements we use, these are the only two based on concrete physical events: the time it takes for the Earth to spin once on its axis, and the time it takes to go around the Sun. Every other unit of time we use (second, hour, week, month) is rather arbitrary. They’re convenient, but not based on independent, non-arbitrary events.

It takes roughly 365 days for the Earth to orbit the Sun once. If it were exactly 365 days, we’d be all set! Our calendars would be the same every year, and there’d be no worries.

But that’s not the way things are. There are not an exactly even number of days in a year; there are about 365.25 days in a year. That means every year, our calendar is off by about a quarter of a day, an extra 6 or so hours just sitting there, left over. After four years, then, the yearly calendar is off by roughly one day:

4 years at 365 (calendar) days/year = 1460 days, but

4 years at 365.25 (physical) days/year = 1461 days.

So after four years the calendar is behind by a day. That means to balance it out again we add that day back in once every four years. February is the shortest month (due to some Caesarian shenanigans), so we stick the day there, call it February 29th, the Leap Day, and everyone is happy.

Integral to the plot

Except…

The year is not exactly 365.25 days long. Our official day is 86,400 seconds long. I won’t go into details on the length of the year itself (you can read a wee bit about it here), but the year we now use is called a Tropical Year and it is 365.242190419 days long. With malice aforethought — my calculator won’t hold that many digits — let’s round it to 365.2421904.

So it’s a bit short of 365.25. That hardly matters, right?

Actually, it does, over time. Even that little bit adds up. After four years, we don’t have 1461 physical days, we have

4 years at 365.2421904 (real) days/year = 1460.968762 days.

That means that when we add a whole day in every four years, we’re adding too much! We should really only add 0.968762 days. But that’s a bit of a pain, so we add in a whole day.

So even though we add a Leap Day in to balance the calendar, it’s still a bit off. It’s a lot better, for sure, but it’s still just a hair out of whack. This time, it’s ahead (since we added a whole day which is too much) by

1 – .968762 days = 0.031238 days, or about 45 minutes.

That’s not a big deal, but you can see that eventually we’ll run into trouble again. The calendar gains 45 minutes every 4 four years. After we’ve had 32 leap years (128 years of calendar time) we’ll be off by a day again!

So we need to adjust our calendar again. But 128 years is hard to remember, so it was decided to round that down to 100 years. After a century, we’ll have added that extra 45 minutes in 25 times (every four years for 100 years = 25 leap years). To be precise, after 100 years the calendar will be off by

25 x 0.031238 days = 0.780950 days.

That’s close enough to a whole day.

Confused yet? Here’s another way to think about. After 100 years, we’ll have had 25 leap years, and 75 non leap years. That’s a total of

(25 leap years x 366 days/leap year) + (75 years x 365 days/year) = 36,525 calendar days.

But in reality we’ve had 100 years of 365.2421904 days, or 36524.2421904 days. So now we’re off by

36,525 – 36524.21904 = .78096

which, within roundoff error, is the number I got above. Woohoo.

So after 100 years, the calendar has gained almost a whole day on the physical number of days in a year. That means we have to stop the calendar and let the spin of the Earth catch up. To do this, every 100 years we don’t add in a leap day! To make it simpler, we only do this in years divisible by 100. So 1700, 1800, and 1900 were not leap years, we didn’t add an extra day, and the calendar edged that much closer to matching reality.

And so we’re good, right? Well…

But notice, he says chuckling evilly, that I didn’t mention the year 2000. Why not?

Because even this latest step isn’t quite enough. Remember, after 100 years, the calendar still isn’t off by a whole number. It’s ahead by 0.78095 days. So when we subtract a day by not having leap year every century, we’re overcompensating; we’re subtracting too much. We’re behind now, by

1 – 0.780950 days = 0.21905 days.

Arg! So every 100 years, the calendar lags behind by 0.21905 days. If you’re ahead of me here (and really, I can barely keep up with myself at this point), you might say "Hey! That number, if multiplied by 5, is very close to a whole day! So we should put the leap day back in every 500 years, and then the calendar will be very close to being right on the money!"

What can I say? My readers are very smart, and you’re exactly correct. So, of course, that’s not how we do things.

Instead, we add the leap day back in every 400 years! Why? Because if there is a stupid way to do something, that’s how it will be done.

After 400 years, we’ve messed up the calendar by 0.21905 days four times (once every 100 years for 400 years), and so after four centuries the calendar is behind by

4 x 0.21905 days = 0.8762 days

and that’s close enough to a whole day. So every 400 years February 29th magically appears on the calendar, and once again the calendar is marginally closer to being accurate.

Sanity check

As a check, let me do the math a second way, in the same method I did for the leap century gambit above. In 400 years we’ve had 303 non-leap years, and 97 leap years. The total number of days is therefore

(97 leap years x 366 days/leap year) + (303 years x 365 days/year) = 146,097 calendar days.

400 x 365.2421904 days = 146096.8762

We can see the remainder is 0.8762 days, which checks with the previous calculation, and so I’m confident I’ve done this right. (phew)

Of course, the calendar’s still not completely accurate at this point, because now we’re ahead again. We’ve added a day, when we should have added only 0.8762 days, so we’re ahead now by

1 – 0.8762 days = 0.1238 days.

Funny thing is, no one worries about that. There is no official rule for leap days with cycles bigger than 400 years. I think this is extremely ironic, because the amount we are off every 400 years is almost exactly 1/8th of a day! So after 3200 years, we’ve had 8 of those 400 year cycles, so we’re ahead by

8 x 0.1238 days = 0.9904 days.

If we then left leap day off the calendars again every 3200 years, we’d only be behind by 0.0096 days! That’s phenomenally accurate. I can’t believe we stopped at 400 years.

OK, so how does all work again?

But despite that, we’re done! We can now, finally, see how the Leap Year Rule works:

What to do to figure out if it’s a leap year or not:

We add a leap day every 4 years, except for every 100 years, except for every 400 years. In other words…

If the year is divisible by 4, then it’s a leap year, UNLESS

it’s also divisible by 100, then it’s not a leap year, UNLESS FURTHER

the year is divisible by 400, then it is a leap year.

So 1996 was a leap year (The Little Astronomer was almost born on leap day that year, in fact). 1997, 1998, and 1999 were not. 2000 was a leap year, because even though it is divisible by 100 it’s also divisible by 400.

1700, 1800, and 1900 were not leap years, but 2000 was. 2100 won’t be, nor 2200, nor 2300. But 2400 will be.

This whole 400-year thingy was started in the year 1582 by Pope Gregory XIII. That’s close enough to the year 1600 (which was a leap year!), so in my book, the year 4800 should not be a leap year.

But who listens to me? If you’ve gotten this far without blowing out your cerebrum, then I guess you listen to me. All this is fun, in my opinion, and if you have gotten this far you know as much about leap years as I do.

Which is probably too much. All you really need to know is that this year is a leap year, and we’ll have plenty more for some time. You can go through my math and check me if you’d like…

Or you can just believe me. Call it a leap of faith.

Related Posts:

- Another orbit? Why, you don’t look a rotation older than 4.56 billion years!
- Wait just a (leap) second
- Take a flying leap second
- Followup: leap seconds

CATEGORIZED UNDER: Astronomy, Cool stuff, Time Sink, Top Post
MORE ABOUT: Leap Day, Leap Year

1. hhEb09'1

“Instead, we add the leap day back in every 400 years! Why? Because if there is a stupid way to do something, that’s how it will be done.”

That’s not why!

It’s because 146097 (the number of days in 400 years, including 97 leap days) is divisible by 7. We can re-use the calendars every 400 years.

2. Jeffrey Shallit

You might want to read my paper, “Pierce expansions and rules for the determination of leap years”.

http://www.fq.math.ca/Scanned/32-5/shallit.pdf

Regards – Jeffrey Shallit

3. tudza

February had 28 days well before the Julian calendar.

4. Andreas H

While this would have no big influence on leap days I would love if we would add the “world calender” or a similar fixed calendar.

But once more religion is holding us back… pretty much any argument against such a calendar system is religiously motivated

Before I forget there is one thing that would change with a fixed calendar for leap days, they would always be in the middle of a 3 day weekend (between Sat and Sun, in most proposals in the middle of the year) how can anyone be against that?

5. I have always said that the universe has no reason to follow our desires, even when it comes to the orbit of our planet. Love the explanation!

6. You can go through my math and check me if you’d like…

Your maths is much more likely to be right than mine is!

Yes, I trust you BA – good article – thanks.

By the way for those who haven’t heard yet, with this February going, going alll-most gone, the IAU are determined to make sure it stays gone and gets leapt over every year from now on! Click on my name for details or see :

‘Astronomers declare February no longer a month’

Posted on the BA blog on the 7th of March , 2009 at 8:44 am.
(An oldie but a goodie.)

@ 1. hhEb09’1

We can re-use the calendars every 400 years.

Oh yeah, gee that’s so handy , now then where did I put my old calander from 1612!

PS. Am I the only one who is currently missing the usual “older posts” button on the end of the BA blog main page and thus can’t easily go back to check older threads?

7. Jeffrey Shallit

Actually, there are only 14 different calendars over all years, not 400. A yearly calendar is determined by (a) the day of the week the year starts on and (b) whether the year is a leap year or not. This gives 7 * 2 = 14 possibilities.

8. chief

How about the changes in the length of the day due to earthquakes and tidal effects. We need to calculate this in as well. Better get out the calculator again.

9. dcsohl

Actually, before Caesar redid the calendar, the Roman year had 355 days in it, and they would add an entire leap month (called Mercedonius) into the calendar and they did this at a very odd place – in the middle of February. Specifically…

They didn’t count days like we do. Instead of today being February 29th they would say today is the 2nd day before the Kalends of March. (The Kalends of March is March 1st and is simultaneously the Kalends and the day before the Kalends… very confusing.) So Mercedonius would be inserted after the 7th day before the Kalends and before the 6th day before the Kalends of March.

When Caesar rejiggered the calendar, that was the natural place to put his leap day. So the calendar thus went “8th day before the Kalends, 7th day before the Kalends, extra 6th day before the Kalends, normal 6th day before the Kalends, 5th day…”

In latin, the term for the normal 6th day before the Kalends of March was “ante diem sextile Kalendas Martias”, and the extra 6th day got called “ante diem bis sextile Kalendas Martias”, and this is where we get the word “bissextile” from. Don’t believe me? Look it up in a dictionary!

So sorry, Phil, today’s not leap day. The actual leap day was last Friday the 24th. Sorry you missed it!

10. Chris

I always find it funny when creationists claim that proof that God exists is that the moon is exactly the same size as the sun as viewed from Earth (which it isn’t), however they never mention how the Earth takes 365 days plus a little bit more which plays havoc with our timekeeping. If there was a nice God, it’s be a nice integer number. So this must mean either there is no God, or it is a very sadistic God (probably the same one who put those fossils in the ground to confuse us) who keeps laughing at these puny humans trying to get their calendars straight.

11. terryp

this isn’t *real* math, it’s only arithmetic.

12. Daniel J. Andrews

If you are mathophobic, then you might want to skip to the end, where I reveal what Rosebud means.

You didn’t reveal the Rosebud secret. It could be anything from a type of Rosetta stone to something ridiculous, like a sled.

Incidentally, if people are a mathophobic, they need to get over it. Put some work into getting familiar with it. Saying “I can’t do math” is like saying “I can’t do reading”. Being innumerate is as bad as being illiterate. Saying “I can’t do x” whether x is math or drawing or whatever is just a cop-out not to not even try. People will never get good at something if they don’t try so if they’re bad at math (or drawing) it is because they’ve made a self-fulfilling prophecy by saying “I can’t do…”

Stepping off soapbox now and going quietly away. I’m a math dunce, by the way, far behind classmates I’ve had from grade school to university stats. I had to retake numerous math classes (summer school) till I finally grasped the concepts, and in my spare time I read books on how to do math quickly in your head. Now people think I’m good at math, but it was just me getting so ticked off at my brain for not getting it like others that I became obsessive about learning it.

13. amphiox

But 128 years is hard to remember, so it was decided to round that down to 100 years.

Hmm. It may not be long now, thanks to the digital age, that binary numbers like 128 won’t be harder to remember than 100, and perhaps one day we’ll change our calender and simply things!

I think this is extremely ironic, because the amount we are off every 400 years is almost exactly 1/8th of a day! So after 3200 years, we’ve had 8 of those 400 year cycles, so we’re ahead by

Oh, I see where you’re going with this. You want us moderns to establish a new Long Count calendar! So the people of 3412 will get the pleasure of dealing with another doomsday cult…. Sneaky.

14. Jon F

One would hope that by the year 4800 we’ll be using a different time system that makes more sense than the one we have now.

15. Cindy

Phil,

You forgot to mention that today’s Julian Date is: 2455562

16. Great article! It’s fun to see the whole thing worked out with all the numbers. But one statement is partly incorrect: “Every other unit of time we use (second, hour, week, month) is rather arbitrary. They’re convenient, but not based on independent, non-arbitrary events.” The month (“moonth”) is based on the Moon’s orbit around Earth, which takes about 29.5 days from full Moon to full Moon (but only about 27.3 days with respect to the stars). That’s why our months tend to be about 30 days each. And if you look at the lunar calendars of yore, you can go through the same type of math that’s in this article to explain why every now and then the ancients had to add an extra (“intercalary”) month to keep their lunar calendars from going out of whack. Fun stuff!

17. Renee Marie Jones

Pay attention. This is all vitally important if you happen to be a pirate’s apprentice.

18. Eric

I love irony.

“Or you can just believe me. Call it a leap of faith.”

ROTFL!

19. I listen to you. And I totally agree; “If there is a stupid way to do something, that’s how it will be done.”

Describes politics too.

20. Anthony

Bonus trivia: the proper term for “leap day” is “intercalary day,” likewise with intercalary month and year, and adding leap days/months/years is called intercalation.

More bonus math: how to convert from Julian to Gregorian, so you can apply the above algorithm to any date!

21. Kirk Aplin

I’ve known all this for years, but I still think “Leap Year” is a misnomer.

22. George Martin

@2 Tudza said:

February had 28 days well before the Julian calendar.

That was true until the Roman calendar reform instigated by Julius Caesar. From the link Phil gave early in his article, February in the initial Julian calendar had 29 days for a short time. Since the Roman month of Quintilis Julius (July) was named for Julius Caesar, the Roman Senate decided to renamed the month Sextillus to Augustus, after Octavian/Augustus. Also since July had 31 days, the Senate August should also have 31 days instead of its current 30 days. They took the extra day from February, returning it to its original 28 days prior to the Julian calendar reform.

George

23. George Martin

@2 Tudza said:

February had 28 days well before the Julian calendar.

That was true until the Roman calendar reform instigated by Julius Caesar. From the link Phil gave early in his article, February in the initial Julian calendar had 29 days for a short time. Since the Roman month of Quintilis was named Julius (July) for Julius Caesar, the Roman Senate decided to renamed the month Sextillus to Augustus, after Octavian/Augustus. Also since July had 31 days, the Senate decided August should also have 31 days instead of its then current 30 days. They took the extra day from February, returning it to its original 28 days prior to the Julian calendar reform.

George

24. SkyGazer

So first you set him up…
“And why is it only a quadrennial event?
Duh. Astronomy!”

And than you bullied him!
“I used to rib him that he was only 3 years old”

Kudo´s for cunning.

25. Rafael Wambier dos Santos

Small correction here (well, HUGE, considering this is a blog about astronomy : A “week” is not an arbitrary unit, it is the time it takes for the Moon to change from one given phase into another.

26. Kevin Hines

Who originally figured all this leap day stuff out? Has his/her name been lost to history?

27. @Daniel J. Andrews,

I’m the opposite of a mathophobe. My idea of fun back in middle school and high school was posing a math problem to myself (e.g. What is 2 to the hundredth power?) and then figuring it out. With paper and pen, mind you, not a calculator or computer. Yes, I was (and still am) a math geek.

I’m happy to report that my oldest son is following in my footsteps. He might only be in third grade now, but he likes working math equations into everything he does.

28. “We have two basic units of time: the day and the year. Of all the everyday measurements we use, these are the only two based on concrete physical events: the time it takes for the Earth to spin once on its axis, and the time it takes to go around the Sun. Every other unit of time we use (second, hour, week, month) is rather arbitrary.” — Actually, the length of time it takes the moon to orbit is also non-arbitrary.

The length of time it takes the moon to orbit is a “month” (moon), but we have to fudge the length of months in order to get 12 of them into a year. 1/12th of a year is 30.44 days, which is why months have to constantly flip between being 30 and 31 days.

According to Wikipedia, the moon’s synodic period is actually 29.530589 days, which means there’s 12.37 “moons” in a year.

29. OneofNone

So after four years the calendar is behind by a day.

IMO not true. The calendar is ahead.
The calendar is already in its next year, but the earth did not yet finish the fourth way around the sun. Like my watch: if it is ahead it shows leisure time, but I still have to do some work on the job.

By adding one day to the calendar, we slow it down. So the earth can catch up with the calendar.

my calculator won’t hold that many digits

Why not use that one shipped with Windows?

As I understand “Rosebud” means ‘older posts’. Is this correct?

30. Every other unit of time we use (second, hour, week, month) is rather arbitrary. They’re convenient, but not based on independent, non-arbitrary events.

Not quite true… the month is loosely based on one cycle of the moon. (In fact, they share the same etymological roots) The exact delineations are arbitrary, of course, but the approximate duration is not.

If I wanted to go way out on a limb, I could almost argue that the week is only partially arbitrary. One lunar cycle is 29.53 days. If you approximate that with 29, that’s a prime number. If you approximate it with 28, the prime factorization is 2*2*7, so if you want to subdivide it evenly, you can only have 2-day, 4-day, 7-day, or 14-day “weeks”. Of course, that account doesn’t explain why we wouldn’t just have five 6-day weeks or six 5-day “weeks” per month… Meh, so I guess I’m just blowing smoke in this paragraph.

You’ll get absolutely no quarrel from me on seconds and hours. And my logic regarding weeks is rather tortured — I don’t really buy it myself. But you can’t really say that months are not even based on something non-arbitrary.

31. Nothing in this article makes any sense at all. I’ve been using a metric calendar for at least a deciliter.

32. Matt T

Every other unit of time we use (second, hour, week, month) is rather arbitrary. They’re convenient, but not based on independent, non-arbitrary events

Well someone obviously never came to my planetarium show ::sob:: I’d say that the month is definitely *based* on an independent event (the orbit of the moon). It’s definitely heavily rounded, but it’s not a completely arbitrary unit. And the week… well, that’s harder to argue that it’s not arbitrary, but I still think it’s supremely cool that it comes from the “planets”.

And speaking of rounding errors and astronomical motions, I’m slightly ashamed to admit how long it took me to realize the connection between 360 degrees in a circle and 365.mumble days in a year.

The fact that all our everyday (haha) timekeeping units originate with astronomical motions gives me a huge astro-brainer.

*EDIT TO ADD*: James Sweet beat me to it. The use of 7, as I understand it, derives from the 7 visible “planets” (non-fixed lights): Sun, Moon, Mercury, Venus, Mars, Jupiter, Saturn.

Also, I am surprised that nobody — myself included — has yet taken you (Dr P) to task because the week is actually based on the independent, non-arbitrary event of GODZ SPESHUL CREASHUN in 6000BC!!1!zeropointninenine!

33. Chris A.

@James Sweet (#22):
The reason we have a seven day week is because of the seven “planets” recognized by the ancients (Mercury, Venus, Mars, Jupiter, Saturn, the Sun, and the Moon). The days of the week, after all, are named after them.

The order (Sun day, Moon day, Mars day, Mercury day, Jupiter day, Venus day, and Saturn day) is due to an arcane belief that each hour of the day was ruled by a planet, in decreasing order of their synodic periods. Since seven doesn’t divide evenly into twenty-four, each day began with an hour ruled by a different planet (starting with Saturn day, originally), in the order we now use.

34. Tara Li

Personally, I’m a fan of the 13 months x 28 days calendar, with non-numbered year-end holidays at the end of the year (“New Year Day” and “Peace Day” for the leap years).

Then again, I’m a weirdo who likes the UNIFON alphabet, so … meh. Your kilometerage may vary.

35. Special One

A few historical corrections to the linked article on Julius Caesar and Augustus;

Unfortunately, Caesar himself was only able to enjoy one July during his life—the very first July, in 45 B.C. The following year he was murdered on the Ides of March.

Not quite. Although the Julian Calendar was instituted during his lifetime, Quintilus wasn’t renamed to July until after Caesar’s death.

After Julius’s grandnephew Augustus defeated Marc Antony and Cleopatra, and became emperor of Rome, the Roman Senate decided that he too should have a month named after him.

Well, yes. But the month was renamed in 8 BCE, 22 years after the defeat of Antony and Cleopatra. It commemorated that event, which occurred during the month of Sextilis (Antony died on August 1, Cleopatra on August 12, 30 BCE), but also commemorated the Battle of Pharsalus (Aug 9, 48 BCE), Octavian’s first consulship (Aug 19, 43 BCE) and the birth of Claudius (Aug 1, 10 BCE). It also happened to be the month Augustus died (Aug 19, 14 CE) which explains, perhaps, why it survived when Neronis (April) and Claudionis (May) did not.

Not only did the Senate name a month after Augustus, but it decided that since Julius’s month, July, had 31 days, Augustus’s month should equal it …

Sextilis had 31 days in the original Julian Calendar of 45 BCE.

36. Charlie

So how many years will we have to wait for tidal drag to get us to exactly 366 days in a solar year?

37. Eric Smith

We have two basic units of time: the day and the year. Of all the everyday measurements we use, these are the only two based on concrete physical events: the time it takes for the Earth to spin once on its axis, and the time it takes to go around the Sun.

Actually the second was defined in 1967 as “the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom”. That’s pretty much based on a concrete physical events . And it’s also why the day is no longer 86400 seconds long, which is why we need leap seconds (like the one coming up this June) from time to time.

39. JC

One funny bit of trivia, is that in 1582, October 4 was followed by October 15th to correct for the accumulated error of the Julian calendar. You’re not giving the 1582′s astronomers a lot of credit, it’s amazing how accurately they measured the length of a year. The reason why we don’t have a 3200 year rule is because of an error in the measurement.

41. fredR

“If the year is divisible by 4, then it’s a leap year, UNLESS
it’s also divisible by 100, then it’s not a leap year, UNLESS FURTHER
the year is divisible by 400, then it is a leap year.”

I find it clearer to reverse the order you check. I don’t like the “Unless”es. I think like a programmer:

If the year is divisible by 400
it is a leap year.
else if it is divisible by 100
it is not a leap year
else if it is divisible by 4
it is a leap year
else
it is not a leap year

The advantage here is that once you find a ‘true’ condition, there is no reason to check anything any further.

42. Stephen

“I always find it funny when creationists claim that proof that God exists is that the moon is exactly the same size as the sun as viewed from Earth (which it isn’t), however they never mention how the Earth takes 365 days plus a little bit more which plays havoc with our timekeeping.”

I always find it funny when atheists turn every perfectly nice article about mathematics or science into a Christianity-bashing exercise. They’re like that guy who inserts the fact that he doesn’t own a TV (an “idiot box”, “boob tube”, etc.) into every conversation.

43. Florian

Well fortunately we still have about 2770 years left to refine the rules for improved accuracy.

44. Joe Moore

fredR, there’s a much easier algorithm for programmers:
if year is between 1900 and 2100, if it’s divisible by 4, it’s a leap year, else not.
if it’s not in that range, I don’t need to worry about it because I’ll be long-retired before then.

And while that’s going to result in another Y2K-like issue, we have enough date troubles (Mayan long count rollover, UNIX epoch rollover, etc) between now and then that noone will notice

45. Thomas Brandon

“We have two basic units of time: the day and the year. Of all the everyday measurements we use, these are the only two based on concrete physical events”
Sorry but I think you’re a little out of date there. The time it takes the Earth to rotate around the sun is not measurable to the degree needed by numerous current technologies. In 1967 the second was officially defined as “the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom”. This is now the only, suitably accurate, “concrete physical event” that underlies time. We still track ‘mean solar time’ but this is a sideshow compared to the real time we keep. Those ‘tiny’ fractions your calculator skipped when calculating leap years are vital to the functioning of GPS and numerous other technologies we depend on. In fact the computer you used to write this email almost certainly uses leap-seconds in order to keep a rough sync between mean solar years and the ‘real’ time.

46. dELLROY

as always, there is some atheist (see #10) popping up to make fun of creation.
And, while you’re at it…

……WHOM Do YE SERVE??

47. OneofNone

@38 Charlie:

This can not happen.
The drag makes the month longer, because the moon gets farther out.
But it also makes the days longer, because the Earth rotates slower.

As result there will be fewer days in a year, not more.
In the past we (well, not actually *we*) had years with 400 or more days. I remember the very young earth rotated in just about 6 hours. But I’m not sure this is precise.

48. RM

fredR The advantage here is that once you find a ‘true’ condition, there is no reason to check anything any further.

The disadvantage is that you front-load the rare cases, and cause the common cases to be the most work. Most (75%) of the cases are regular non-leap years, but by doing the tests your way, you have to do three divisibility checks on each of those. Better to deal with the most common cases first off, and let the rarely encountered cases be the ones you spend time on.

If you were truly thinking like a programmer, you’d realize that jump-on-true is entirely the same as jump-on-false. So you’d really want:

If the year isn’t divisible by 4, it’s not a leap year
else if it isn’t divisible by 100 it is a leap year,
else if it isn’t divisible by 400 it isn’t a leap year
else it is a leap year.

…. which is more or less what the Bad astonomer said, after correcting the programmer’s double negatives to conventional English.

49. JA

In modern times we use a solar year in which the year is based on the time it takes the Earth to go around the Sun. But we should remember that many cultures in the past used a lunar year based on the moon’s cycles (months).

50. Paul Hannah

Daniel J Andrews: There may be other reasons aside from laziness that people don’t ‘get’ mathematics and arithmetic. I am numerically dyslexic, so, while I do quite well in language based tasks, numbers get mixed up ‘in my head’. I can’t remember numbers, frequently misdial phone numbers and make too many mistakes in my workshop.
Like blind people who develop extraordinary hearing, this disability has given me an unusual talent – I can scan a page, previously unread, sometimes from quite a distance and pick a spelling error. Stranger still, this does not exempt me from making such errors myself.
Even though I have this disability, I read and understood the post by not absorbing the numbers per se, but by noting their relative size to each other e.g. ‘this is a bit more than that’ etcetera.

51. KevinC

To be a nerd…

The Earth does not take 24 hours for one complete full revolution – but rather slightly shorter at approximately 23 hours 56 minutes (or is it 4 minutes more?). The additional 4 minutes are required because the Earth has moved 1/365 of the way around the Sun in the meantime, and it therefore has to rotate approximately 1 degree more (less?) to have the same spot face directly to the sun.

24 hours * 1/361 = approx 4 minutes

52. Ken Mitchell

The best explanation of leap years was published about 250 years ago, by a very bright guy named Ben Franklin. It’s in one of Poor Richard’s Almanacs.

53. Phaldor

@OneofNone & @ 38 Charlie

This is going to chap your hide, but if you consider it closely, even though the earth slows down in rotation, the time it takes to travel around the sun doesn’t. So what will end up happening is a redefinition of the time measured during a day (adding an hour?, stretching a second?). What you take away has to be added back somewhere.

Although I enjoyed reading the article, I found I was somewhat dissapointed that there was no mention of why we occationally add “leap seconds” to the clock. By the way, this year we will add one, I believe June 30th. If I remember correctly, these extra seconds are necessary due to the natural decline or incline of the earths rotation following an earthquake. The Japan earthquake did mess with the rotation, so this year’s leap second might be the correction necessary to compensate for that.

54. Majikjon

@Kevin Hines @27:

Very likely the Astronomer Royal.

55. K Scharf

The length of a week at 7 days comes from the bible. It took God six days to create the universe and on the seventh day he rested. The number 7 has always had ‘magical’ properties. There are 7 openings in the human body (unless you’ve been injured). Radio waves circle the globe seven times in one second therefore the earth / ionosphere form a resonant cavity at seven hertz. Many others.

56. @Renee Marie Jones Says:

“Pay attention. This is all vitally important if you happen to be a pirate’s apprentice.

Not too many Geeks apprecicate G&S, or even know who G&S were! ;^)

I used to tell my C Programming students about one C compiler years ago with an Object Module Librarian, named “Marion”! In 8 years only one student ever got that joke as well!

Hapy Birthday to my cousin!

57. UNLESS FURTHER
the year is divisible by 400, then it is a leap year.

Yup. You’d be surprised (okay, probably not) at how many people forget that part. We received plenty of bug reports on our software package, because it (correctly) reported that 2000 would be a leap year. People liked to point out that “years divisible by 100 aren’t leap years”, thinking that we had forgotten that part.

58. Of course, while the Roman Catholic Church adopted the Gregorian calendar in 1582, Great Britain (and the colonies that would become The United States) did not do so until 1752.

And, for the sake of geekiness, here’s the actual wording:

[...]

Be it further enacted by the Authority aforesaid,

That the several Years of our Lord, 1800, 1900, 2100, 2200, 2300, or any other hundredth Years of our Lord, which shall happen in Time to come, except only every fourth hundredth Year of our Lord, whereof the Year of our Lord 2000 shall be the first, shall not be esteemed or taken to be Bissextile or Leap Years, but shall be taken to be common Years, consisting of 365 Days, and no more;

[Years Bissextile of 366 Days.]

and that the Years of our Lord 2000, 2400, 2800, and every other fourth hundred Year of our Lord, from the said Year of our Lord 2000 inclusive, and also all other Years of our Lord, which by the present Supputation are esteemed to be Bissextile or Leap Years, shall for the future, and in all Times to come, be esteemed and taken to be Bissextile or Leap Years, consisting of 366 Days, in the same Sort and Manner as is now used with respect to every fourth Year of our Lord.

[...]

59. Steve Metzler

@MTU #6:

PS. Am I the only one who is currently missing the usual “older posts” button on the end of the BA blog main page and thus can’t easily go back to check older threads?

No, not just you. But I know how to get you there. The URL format for accessing older entries looks like this:

Bit of a pain, but it works. Obviously, substitute other numbers in for the ’2′ to go further back. My best guess on this is that Phil didn’t want any more inane responses to the Heartland Institute/Peter Gleick thread

@OneofNone #31:

As I understand “Rosebud” means ‘older posts’. Is this correct?

See:

en.wikipedia.org/wiki/Citizen_Kane#Rosebud

60. Illectro

I remember Duncan Steel presenting a pretty credible theory that involved a plan hatched by Henry the 8ths advisors to change England over the the Persian calendar (which uses a 33 year Cycle) to highlight imperfections in the Gregorian calendar and thus win propaganda points against the catholic church. The proponents pointed out that if you measured time from the right location then you could keep the date of the equinox on the same day, and by extension, remove a lot of unecessary complications in determining the date of easter.

http://www.hermetic.ch/cal_stud/gods_longitude.htm

61. Jeffersonian

But wasn’t it Augustus who instituted the 4-year Leap Day that we have today?
The Julian reform created a 3-year leap day that once again required adjustment.
Funny we credit gregory with our calendar (the 100-year adjustment was the addition) when we’re realistically using the Roman calendar. (Granted Caesar got the 365-day idea from Egypt).

62. If you like Calendars, this one takes a slightly different approach to leap days:

Alpo Balognia: Calendar A

http://ancestorsnow.com/press/news.php?item.34.1

63. decora

the ancient Hindus had a lot of this figured out since their time scale was based on a period of roughly 4 million years. but whatever.

64. Michelle M

The Persian solar calendar is more accurate than ours.

65. Joe Ridge

I want this article corrected. The definition of a day is just plain factually wrong. If a day were the length of time for the earth to complete a rotation about it’s access then daytime would swap with with night time every 1/2 year as the sun moves around it’s orbit.

The day is the average amount of time it takes for the earth to return to the same orientation relative to the sun, which is not one revolution, because the earth moves relative to sun as it spins.

66. Meskine

I’ll have to second Andreea H’s (#4) suggestion of adding a third day to a weekend instead of simply tacking the leap day on the end of February. I’m on salary and always joke the I’m working for free on February 29 (conveniently forgetting that my check is the same whether a month has 30 or 31 days).

67. jyyh

This has been worrying me ever since I learned about this. How are the ecologists, tv-weathermen, psychologists and sleep deprivation scientists ever going to find out anything when they have a poor frame of reference that make the calculations of the diurnal cycles hard? So make the leap day 23 hours 15 minutes and 1 second. Still there would be a 33minute displacement of the sun time and the clock time in the third year of the cycle, but this would be easier to correct for. Since everything is supposed to be a bit twisted on leap day, why not even the lenght of the day? Additionally all those sleep deprivation scientists could then compare their results in 4 year cycles! I’d rather move all my clocks the ~45 minutes back in every 4 years than the double movements of daylight savings.

68. jyyh

After 6521,73913 years one would then have to add an additional minute on the cycle (if I got that correctly). This is not even close to the 12 Bak’tun cycle (which is actually the 20 Bak’tun cycle), but rather near 16,54 Bak’tuns bringing the Mayan long count and the shortened leap day calendar within a week each other again.

But I guess it makes too much sense.

69. tudza

@Jeffersonian #66

No, the Julian calendar was indeed meant to have an extra day every four years according to the expert astronomer Julius Caesar called in. They simply implemented it incorrectly by doing what apparently was a common Roman thing, counting inclusively. Augustus had to re-adjust the calendar and get them to follow the plan correctly.

70. @63. Steve Metzler :

@MTU #6: “PS. Am I the only one who is currently missing the usual “older posts” button on the end of the BA blog main page and thus can’t easily go back to check older threads?”
No, not just you. But I know how to get you there. The URL format for accessing older entries looks like this: blogs(dot)discovermagazine (dot)com / badastronomy / page / 2 /

Cheers for that! Much appreciated.

Bit of a pain, but it works. Obviously, substitute other numbers in for the ’2′ to go further back. My best guess on this is that Phil didn’t want any more inane responses to the Heartland Institute/Peter Gleick thread.

That one was still going!? Thought it would’ve finished several days ago!

If that was the case couldn’t the BA “lock” or close down that specific thread as happens on some forums / fora? Couldn’t he have said so himself? Oh well, guess not.

71. Strange leap year rules. Why? Because if there is a stupid way to do something, that’s how it will be done.:(

72. #27 Kevin:
It depends whether you mean “who figured out the problem”, or “who figured out how to solve it”.
Long before the Romans, the Egyptians used a 365 day calendar; they used 12 equal months of 30 days each, with the extra 5 days tacked onto the end. They also realised that the actual length of a year was roughly 365.25 days, but strangely, they never bothered to correct for the quarter day. They just accepted that the calendar moved out of step with the seasons by a day per four years – coming back into step after a very long cycle of 1461 years ( 4 x 365.25 = 1461 ).
As for who thought of the solution; when Julius Caesar introduced the system of leap years, it was on the advice of an astronomer named Sosigenes.
When Pope Gregory XIII further reformed the calendar in 1582, he was advised by astronomer Christopher Clavius, who realised that the residual error of 0.0078 days per year could be ( almost ) resolved by omitting three leap years per 400 years.

73. #59 K Scharf:
I presume you’re joking – but just in case you’re not…
The concept of the seven day week predates the Bible by at least a couple of millennia. As a couple of people have already commented, it originated from a supposed association of the days with the seven so-called “planets” of ancient astrology ( Sun, Moon, Mercury, Venus, Mars, Jupiter and Saturn ). The days are still named after those “planets” to this day – though in English, we use the equivalent Nordic names.
So the “on the seventh day, he rested” myth originated from the already existing concept of the seven day week – not the other way around.

74. John

“If we then left leap day off the calendars again every 3200 years, we’d only be behind by 0.0096 days! That’s phenomenally accurate. I can’t believe we stopped at 400 years.”

This is off. The correct figure is not the mean tropical year of 365.2421904 days, as is usually claimed, but the mean interval between vernal equinoxes, 365.242374 d. This is much closer to the average Gregorian year of 365.2425 d, which would add up to about a day in 8000 years, rather than 3200. However, the precise period varies a bit over centuries, so the exact year cannot be accurately predicted.

The Wikipedia article on Leap year contains the correct figure, and discussions on the issue can be found elsewhere on the Internet. But it is very easy for anyone to determine the figure simply by looking at the average difference between the date/time of equinoxes for two different years. It’s also clearly stated in Pope Gregory’s decree that he based his calendar on the equinoctial year.

I pointed this out four years ago, so I expect the error will be repeated in 2016.

75. AC

this ain’t math in here, it’s numbery

76. Fundamentally, February is the short month (and the month that gets leap days, for that matter) because (prior to the adoption of the Gregorian calendar) it was the last month. January and February were added as months 11 and 12 when the military calendar (which ran from the first march in March until the end of fall — the Roman army was inactive in winter) was adapted for civilian use. February, being last, got whatever was left over before the beginning of the next March.

77. Scott

“I always find it funny when atheists turn every perfectly nice article about mathematics or science into a Christianity-bashing exercise.”

I always find it funny that when someone is mocking religious fundamentalists, the religious fundamentalists assume you are only bashing *their* particular religion (not surprising, since their world view is so blinkered).

Also, as far as discrimination goes in modern western society, athiests *still* get the worse end of that stick. E.g. Proportion of athiests in North America? At least 15%. Proportion of athiests among our elected officials? Far less than 15%. Proportion of our countries’ leaders who have been openly athiest? Usually 0%. (If there have been any, they have had to remain in the closet). Tyranny of the majority indeed.

And finally, I reserve the right to point out the ridiculousness of any beliefs wherever I many find them, though they *are* most often found among fundamentalist religious types.

78. OneofNone

80. John said:

It’s also clearly stated in Pope Gregory’s decree that he based his calendar on the equinoctial year.

Another proof of this is the skipping of 10 days when the calendar was first made official in 1582. Thursday October 4 was followed by Friday October 15, to bring Easter back to where it belonged. The date of Easter (its shifting) was the main problem here.

79. OneofNone

@57. Phaldor said:

even though the earth slows down in rotation, the time it takes to travel around the sun doesn’t. So what will end up happening is a redefinition of the time measured during a day (adding an hour?, stretching a second?). What you take away has to be added back somewhere.

That’s true, and is the base of my statement.
It does not matter in what units you measure the length of a day, as long as it is equivalent to “from Sun in Zenit/South/North to Sun in Zenit/South/North”. When the earth slows down the rotation, there are less rotations in a travel around the sun.
So unless you define a day other than above, you have less days in a year. Now we have less than 366 in a year, in the future we will not get back to that. Catastrophes like “Theia hitting the Earth” ignored of course.

By the way, there is a more direct way to get to older posts:

80. Jeff

Neil:

“As a couple of people have already commented, it originated from a supposed association of the days with the seven so-called “planets” of ancient astrology ( Sun, Moon, Mercury, Venus, Mars, Jupiter and Saturn ). The days are still named after those “planets” to this day – though in English, we use the equivalent Nordic names.

and leap day 2012 just after sunset, all were visible to naked eye easily with the exception of Saturn. What a night.

81. Prof. Burnin

Here is my version of the same math. It is definitely very politically incorrect. If you loved George Bush, Don’t read this! And if you find it insulting, please note that it is humor, not truth. However, the math is absolutely accurate.

George Bush and the Global Slowing Conspiracy:

http://db.tt/nOKq3dRE

82. Matthew

Anyone who programs in C knows this already. It’s one of the first algos in the ‘C’ book.

83. It’s complicated, but nothing to worry about… after so many years we can eliminate that one day excess and February will be 28 days only… May I ask: Why is February so short when they could reduce July one day and give it to February… Some people born in July 31 will be complaining for they won’t be able to celebrate their Birthday, excep they move it to the 30 th. In the benefit for February…

84. Jeffersonian

jeff,
I’ve been looking for Mercury all week. Can’t see it!

85. Ben

@38 Charlie, @51 OneofNone:

Years are gradually getting longer, so a Leap Second (a minute that is 61 seconds in length) is added once or twice a year.

When are they added? It’s determined by measurement – not by a rule. Wikipedia’s article on Leap Second has a good explanation. It’s very very different from the Leap-year.

86. FrankH.

Re, leap years and calendar systems, an interesting and surprising treatise on those is found in J.R.R. Tolkien’s “Lord of the Rings” – Appendix D. It maintains the idea of evolving calendar systems within the imaginary history, with the best/final system having a “millenial deficit of 4 hours, 46 minutes, 40 seconds”. That’s in fact better than the Gregorian calendar.

The Gregorian calendar is, as you say, ever so slightly ahead and improving on it would need skipping a leap year every every 3200 years, Tolkien’s “King’s Reckoning” is slightly behind and requires adding a leap year every ~5000 years.

For the inclined, if you don’t own LoTR, a reference of the calendar system is at http://lalaith.vpsurf.de/Tolkien/Time.html – I wonder if any other SF/fantasy author ever did the same …

87. MacBruce

One undisputed aspect of the calender is that it can only be increased or decreased by a whole day. The day is the minimum quanta of the calendar. Therefore, a leap day is a true “quantum leap”. Has someone calculated what color are the photons released by this event?

Thanks Phil for such an entertaining post.

88. Nigel Depledge

Chris A (35) said:

The reason we have a seven day week is because of the seven “planets” recognized by the ancients (Mercury, Venus, Mars, Jupiter, Saturn, the Sun, and the Moon). The days of the week, after all, are named after them.

Let’s see . . .

Sunday for the sun.
Monday for the moon.
Tuesday for …. er … Tyr (or Tiw)
Wednesday for Wodin (or Woden)
Thursday for Thor
Friday for Frige (or Freya)
Saturday for Saturn.

Nope, I think you must be thinking in French, where we have:
Lundi for Monday (La Lune being the moon)
Mardi (Tuesday) named for Mars
Mercredi (Wednesday) named for Mercury
Jeudi (Thursday) named for Jove or Jupiter
Vendredi (Friday) named for Venus
Samedi (Saturday) named for the Sabbath
Dimanche (Sunday) named for the Lord’s day

Which also doesn’t quite fit. Ironic that the only two days that in French are not named for celestial bodies are two of the only three days that in English are named for celestial bodies.

89. Nigel Depledge

Stephen (46) said:

I always find it funny when atheists turn every perfectly nice article about mathematics or science into a Christianity-bashing exercise. They’re like that guy who inserts the fact that he doesn’t own a TV (an “idiot box”, “boob tube”, etc.) into every conversation.

Wow, this sounds like you’re a bit touchy.

And wrong.

First, laughing at creationists is not “Christianity-bashing”, because (a) not all creationists are Christian, and (b) plenty of Christians are not creationists.

Second, the comment to which you object is germane to the discussion because the mathematics of matching up our units of timekeeping to the celestial movements on which they are nominally based illustrates how imperfect our solar system is, in relation to the common claim made by creationists that the “perfection” of the solar system is in some way proof of something or other.

Third, it’s not “bashing” to point out that a piece of evidence disproves a claim that someone has made.

90. Nigel Depledge

dELLROY (50) said:

as always, there is some atheist (see #10) popping up to make fun of creation.

Tell us one reason why creationists deserve not to be made fun of?

Erm . . . in case you had not noticed, this is a scepticism blog.

You are aware, are you not, that Phil is an ex-president of JREF, right?

And, while you’re at it…

……WHOM Do YE SERVE??

Why does that commenter have to serve anyone?

And while you’re at it, check your CapsLock key before typing, yeah?

91. Nigel Depledge

Phaldor (57) said:

Although I enjoyed reading the article, I found I was somewhat dissapointed that there was no mention of why we occationally add “leap seconds” to the clock. By the way, this year we will add one, I believe June 30th.

Er, dude, three of the BA’s “related posts” links deal with leap seconds.

92. Modest_Proposer

The obvious solution to the leap year problem would be to alter the orbit of the earth so that a year is EXACTLY 360 days long. With this one minor change we would no longer need to bother with all of this calendar malarky.

Additionally, if we were feeling OCD we could also nudge the orbit of the moon to be EXACTLY 30 days too. This would have the benefit of eliminating any uncertainty about the date of certain events like easter and the lunar new year.

93. #94 Nigel:
But those Nordic names are those of the equivalent gods and goddesses to the Roman ones after whom the planets are named. e.g. Thor was the equivalent of Jupiter, so the French have Jeudi, and we have Thursday. Freya was the equivalent of Venus, so the French have Vendredi, and we have Friday.
The astrological origin of the week, and the association of the days with the seven “planets”, is well established. Duncan Steel wrote an excellent article on it some years ago, in one of Sir Patrick’s Astronomy Yearbooks.

94. Nigel Depledge

@ Neil Haggath (101) -

Aw, you had to go and spoil a perfectly good nitpick . . .

95. Nigel Depledge

K Scharf (59) said:

There are 7 openings in the human body (unless you’ve been injured).

Wait, what?

Ears (2); eyes (2); nostrils (2); mouth (1) – that’s 7 in the head.

I count at least 2 other openings in the body.

96. Nigel Depledge

Joe Ridge (71) said:

I want this article corrected. The definition of a day is just plain factually wrong. If a day were the length of time for the earth to complete a rotation about it’s access then daytime would swap with with night time every 1/2 year as the sun moves around it’s orbit.

The day is the average amount of time it takes for the earth to return to the same orientation relative to the sun, which is not one revolution, because the earth moves relative to sun as it spins.

This really is splitting hairs.

Whether you measure the Earth’s rotation against the Sun or the stars is simply the difference between a solar day and a sidereal day. As it happens we use days that are pretty close in length to the Mean Solar Day (hence leap seconds are not all that common).

Don’t forget that the stars, too, are moving, and that the Solar System is moving relative to them. Perhaps our days should be measured relative to the black hole at the centre of the galaxy, hmmm? (I guess this would be a “Galactic” day). That, too, would be ever so slightly different from a sidereal day.

And that’s the worst mis-spelling of “axis” I have ever seen.

97. Nigel Depledge

Ben (91) said:

Years are gradually getting longer, so a Leap Second (a minute that is 61 seconds in length) is added once or twice a year.

This is not the reason for Leap seconds.

If years were genuinely getting longer, then every now and then we would need to add an extra second to the year permanently. But this does not happen.

Leap Seconds are one-off additions because our year is not a whole number of seconds long. The correspondence between our calendar and the solar year gradually drifts off, and the leap seconds are added to bring them back into closer alignment.

98. Nigel Depledge

Modest_Proposer (98) said:

Additionally, if we were feeling OCD we could also nudge the orbit of the moon to be EXACTLY 30 days too. This would have the benefit of eliminating any uncertainty about the date of certain events like easter and the lunar new year.

We only need to wait a while. Tidal interactions between the Earth and Moon mean that the moon is taking some of the Earth’s rotational velocity and turning it into orbital velocity. When you increase the energy of an orbit, you increase the distance from the focus and hence increase its period.

The moon is moving away from the Earth at a couple of cm per year, and Earth’s rotation about its axis is decreasing at the same time. Sooner or later, as the moon’s orbital period increases, there will come a month when the moon’s orbital period is exactly 30 days.

99. The moon is moving away from the Earth at a couple of cm per year, and Earth’s rotation about its axis is decreasing at the same time. Sooner or later, as the moon’s orbital period increases, there will come a month when the moon’s orbital period is exactly 30 days.

Then creationists will have proof, for that could only be true if an Intelligent Designer made it that way. Just like he made the sun and moon the same apparent size in today’s sky.

100. Matt B.

@2. Jeffrey Shallit

Big deal, I’ve worked out a Hebrew-style calendar wherein the month averages to the resonance cycle of Martian and Tellurian days.

Kidding. That’s some interesting math, and it adds to what I’ve discovered in playing around with calendars since junior high. I currently favor a calendar (for Earth) that uses weeks instead of months. Using the normal procedure of adding in leap weeks and then taking some out, then putting some of those back in, it still ends up with a grand cycle of 400 years.

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