Why we have leap days

By Phil Plait | February 28, 2008 12:00 pm

Warning: 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!

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. Convenient, but they are 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.

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.

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.

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.

But we’ve really had

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.

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.

CATEGORIZED UNDER: Astronomy, Cool stuff, Time Sink

Comments (164)

  1. Dark Jaguar

    I was under the impression that the second was the basic unit of time based on something pretty concrete. On Wikipedia:

    “Under the International System of Units, the second is currently 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 caesium-133 atom.[1][2] This definition refers to a caesium atom at rest at a temperature of 0 K (absolute zero). The ground state is defined at zero magnetic field. The second thus defined is equivalent to the ephemeris second, which was based on astronomical measurements. (See Historical origin below.) The international standard symbol for a second is s (see ISO 31-1)”

    Days and years aren’t nearly as precise, as you point out above.

  2. chris

    awesome maths..im blind now, but can anyone explain why its called a ‘leap’ year, what leaps over what why?

  3. You missed my point. The second is an arbitrary length of time, and is now defined based on vibrations of an atomic nucleus. The unit itself has no physical corresponding action. What’s to prevent a second being 10% shorter, or 10% longer? It’s just a definition.

    A day, however, is based on a real event: the rotation of the Earth, just as the year is based on the orbit of the Earth.

  4. PrimeMover

    Speaking of Leap day birthdays & tying in the “birthday paradox”. I was born in 1976 (a leap year). So a large group pf my school mates were born that year also. The interesting thing is that out of a small high school class I graduated with (76 total students) two of them were born on leap day. & I myself only missed it by 5 days (being born last Sunday February 24th). Wonder what the odds are on 2 out of 76 kids being born on leap day. Who is up for some more math?

  5. ioresult

    Happy birthday to TLA

  6. Great post BA! My head hurts, and I actually learned something :-) (I had always thought the top most adjustment on the calendar was 1,000 years).

    My personal take on why Pope Gregory didn’t do the final tweak at 3,200 years is that they were all convinced that the world would end in 2000 (the end of Christ’s millenium on earth). I have a brother who was so convinced to the point that he didn’t participate in the big Y2K party our family had. He and his wife were at home awaiting “the rapture.”

    Besides, we all know that they world isn’t going to end until 2012…

    – Jack

  7. PrimeMover says: “I myself only missed it by 5 days (being born last Sunday February 24th).”

    You’re quite articulate for someone only four days old.

    – Jack

  8. alfaniner

    So then… when did the current Millennium begin, anyway? :)

  9. themadlolscientist

    =LOL= HMS Pinafore, anyone?

    Happy Birthday, Little Astronomer!

  10. Todd

    Phil,

    Very interesting post! I now understand just how strange (and daunting) calendar formation can be. I also understand how to make a person’s brain implode, now.

    But, the real question now is, why the heck do we have one 28-day month (29 every four years, unless divisible by 100, unless divisible by 400), four 30-day months, and seven 31-day months? Why not twelve 28-day months and one 29-day month (or 30-day every 4 years, etc.)?

  11. My nephew was born March 1, 2004, missing a leap-day birthday by only a few hours.

    When the software package for which I manage development introduced an IsLeap() function, we got numerous “bug reports” complaining that it reported 2000 as a leap year. (They remembered the “century years are not leap years” but forgot the “except every 4th century is a leap year” part of the rule.)

    I’ve heard people claim that every 4000 years is not a leap year, bringing the calendar closer to the “right” length. Of course, there is no such rule at this point. And, given that the calendar was developed over 400 years ago, it’s pretty accurate if it’s off by less than a day every 3,200 years.

    You can also find a copy of the text of the British “calendar act” of 1751, which switch the British Empire to the current Gregorian calendar in 1752 here, as well as elsewhere.

  12. Jeremy

    A little bit of nit-picking here:
    —————————————–
    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.

    But we’ve really had

    400 x 365.2421904 days = 146096.8762

    We can see that we’re off by 0.8762 days
    ——————————————-
    You are actually off by 1-0.8762=0.1238 days.

  13. KaiYves

    You never said that “Rosebud” means!

  14. OtherRob

    Hmmm…2012 is a leap year…

  15. Doc

    To add to the fun, the Gregorian calendar was not adopted everywhere at the same time. Only four countries switched over right away, with the rest taking their sweet time. The last country to adopt the calendar was Turkey in 1926.

    I read one article that pointed out how at one point in the 16th century (I think) could take a three or four week trip around northern Europe and pass through three different calendar years.

  16. And I thought I had calendars on the brain!

    I’m a little dizzy now….

  17. BlondeReb3

    The Julian/Gegorian calendar always made history a little more interesting when you got two different death dates for Queen Elizabeth I.

  18. drew terry

    Seconds are arbitrary?

    Is that fact or informed opinion?

    Just curious, thanks.

  19. sirjonsnow

    “Wonder what the odds are on 2 out of 76 kids being born on leap day. Who is up for some more math?”

    The odds that (at least) 2 out of 76 kids were born on the same day (any day) in a leap year is 81%
    The odds of that day being one specific day (in this case Feb 29th) are .2%

  20. Michael

    Considering when they did the work, I think Caesar and Gregory did pretty well.

    And I think we should cite The Pirates of Penzance, not Pinafore….

  21. Mango

    I once calculated the year in which Easter and Christmas will fall on the same day.

    I don’t have the answer because it was a few years ago, and there is no way I’m going to do that again. I may have been wrong, anyway. To my knowledge no one else has worked it out independently.

  22. geomaniac

    Weren’t months originally tied (roughly) to lunar cycles (28 days)?

  23. Adhominem

    The leap days/years I get and actually have no trouble remembering the rule. If you woke me up in the middle of the night and asked me whether any given year was a lep year, I’ld be able to answer. I don’t know why, it’s actually not that relevant to me, but still…

    Anyway, my point is: Leap years I get. What I continue to have a bit of trouble with are leap seconds. Why do we need those again, and how are they caclulated?

  24. bkallee

    My brother’s father in law turns 16 tomorrow. Just think, 8 more years and he’ll be a legal adult and in another 20 years he’ll finally be old enough to drink.

  25. drew terry

    The 400-year cycle of 146,097 days has exactly 20,871 weeks. Therefore, the days of the week in Gregorian year 1603 were exactly the same as for 2003.

    This causes more months to begin on a Friday, and have consequently include more Sundays (688/4800) with more days of church services /offerings, than any other day of the week.

    The least common days are Saturday and Monday, each with 684 out of every 4800 months.

  26. Mark Eret

    Re: PrimeMover

    The probability of those two people having the same birthday is about 18.6%. The formula used to calculate this is p(n)=1-(365^n/366^n). For your particular class, there is a 366 day period in which people are shuffled into your particular grade (with perhaps an exception or two), though that period doesn’t necessarily align with the calendar year, but it doesn’t matter. Assuming people’s birthdays are equally distributed through the year (they’re not), this formula calculates the probability that the n other people in the class DON’T have the same birthday and then subtracts that from one. n, in your particular case, is 75 for this formula; it’s one less than 76 because it’s the probability that the n /other/ people have the same birthday.

    I should note that this is not the birthday paradox. Keep in mind that the “birthday paradox” doesn’t talk about the probability of a collision of a certain date, but rather the probability of a collision of any date. The possibility of any collision in your class of 76 is damned near 100%. It’s well in the 99.9%+ range.

  27. Ian

    “The odds that (at least) 2 out of 76 kids were born on the same day (any day) in a leap year is 81%
    The odds of that day being one specific day (in this case Feb 29th) are .2%”

    That’s not what I get.

    Odds that (at least) 2 out of 76 kids were born on the same day (any day) in a leap year: 1 – 366! / (366-76)! / (366 ^ 76) = 0.99977184479623038
    Odds that (at least) 2 out of 76 kids born in a leap year were born on Feb 29th: 1 – ((365/366) ** 76 + 76 * (1/366) * (365/366) ** 75) = 0.018611569699379826

  28. I just remember that presidential elections take place in leap years. Yes, I know 1900 had a presidential election yet was NOT a leap year, but I wasn’t born yet, so it doesn’t count.

    Dark Jaguar: I would guess that seconds have been around a lot longer than the decay rate of caesium-133 atom has been known.

    I also find it interesting that the number 12 factors into so many things. For example, it’s the number of months in a year; 12=one dozen; a gross is 12 dozen; 12 inches in a foot; 4 times 12=28, which is the number of days in a lunar month; 12 is a factor of 60, as in 60 seconds=one minute and 60 minutes=one hour; 12 hours on the face of a clock; 12 is also a factor of 5280, which is the number of feet in a mile; and most important of all, there are 12 ounces in a standard can of beer.

    In fact, I think I’ll go have cold one now.

  29. Aerimus

    BA: “Because if there is a stupid way to do something, that’s how it will be done.”

    Ain’t it the truth. I cracked up when I read that. You almost made Mountain Dew come out my noise.

  30. Ian

    “The probability of those two people having the same birthday is about 18.6%. The formula used to calculate this is p(n)=1-(365^n/366^n).”

    Not quite. That’s the probability of at least *one* person out of n having been born on a specific day.

  31. PrimeMover

    So now I am getting conflicting numbers between two responses. sirjonsnow & Mark Eret are reaching different conclusions to the same problem. A good effort but now I am more confused.

  32. Let’s have a metric year; redefine the length of seconds, minutes and hours with 100 seconds per minute, 100 minutes per hour, 10 hours per day, 10 days in a week, 10 weeks in a month and 10 months in a year. The new months would be Oneuary, Twouary, Threeuary etc. and seasons would have little to do with the month, but it would be a change of pace.

  33. Mark Eret

    Re: Todd

    First off, don’t let anyone tell you it has to do with Julius Caesar and Augustus Caesar taking days from February. It’s complete bunk and easily disproven by the fact that February had 28 days long before the Caesars.

    The most likely explanation is that the Roman Calendar originally had ten months. Ten months? What?!? The earliest Roman calendars only had the months March through December and they just didn’t keep track of time all that much during the intervening period. The calendar was mostly used as a tool to aid harvesting and there wasn’t a whole lot of farm-related anything going on between December and March. Along came Numa Pompilius, an 8th Century Roman King, who decided that the calendar needed to be improved. He decided the Romans needed to keep track of those dreary winter months after all and added January and February. He made the months 29 and 31 days long (even numbers were unlucky), but found that he needed one even-lengthed month. He stuck February with that task and made it slightly shorter. Why February? Who knows. If you invent a time machine, go back and ask Numa himself.

  34. Mark Eret

    Hey, Mr. BA,

    How do leap seconds factor into all of this?

  35. Todd –

    I like the 13-month calendar. And if you treated the one extra day/year and the second extra day/leap year as special holidays – i.e., not assigned to a day of the week – then you have some real benefits:

    every month would have exactly four weeks (great for accountants)
    every date would fall on the same day (e.g., the 7th would always be Saturday)
    a 13-month calendar would strike a blow against astrologers (though they seem to have dealt with the demotion of Pluto fairly well)

    Are there any reasons besides tradition and transistion (which are valid reasons, of course) against a 13-month calendar?

  36. Todd

    @Mark Eret

    That still doesn’t quite get at why the some months between March and December have 30 days, while others have 31.

  37. Jeremy, good catch. That is a nitpick, but a good one, so I corrected it (but didn’t bother keeping the old text; this is confusing enough!).

  38. Todd

    @Mango

    Easter and Christmas on the same day? But I thought they calculated Christmas as always on Dec. 25, and Easter as always after March 20? I mean, I know it’s a little weird with Easter floating around all the time, but by that much?

  39. Aerimus

    @idahogie:

    I don’t that there’s a really good reason to not accept a 13 month calendar other than people just wouldn’t accept it. I mean, we’re still using English measures [in the US].

  40. Mark Eret

    Re: Todd

    Beats me. I know pre-Julian reform, the calendar looked like this:
    29, 28, 31, 29, 31, 29, 31, 29, 29, 31, 29, 29

    I guess Julius decided to rearrange some of the months further. Who knows why he did it.

  41. Blinky

    Leap seconds are added (or subtracted) to compensate for changes in the earths axial rotation due to its wobble.

  42. Ian

    idahogie:

    You mentioned transition as a reason, but I want to stress that it’s a big one. There’s a lot of software that assumes a 12-month calendar, and updating all that would be a huge amount of work — much more than was needed to fix the Y2K bugs or to update timezone data after the recent change in daylight saving time dates.

    I also foresee a psychological problem with making leap day a holiday. People like holidays, and every 3 out of 4 century years there might be a lot of upset people who don’t understand why they don’t get their usual quadrennial holiday.

  43. Kendall in MD

    HEY Uncle Flip!! You know what,
    TMI!!!! sheesh… my BRAIN hurts now!! Thanks alot!!

  44. Mango

    @Todd: Easter is the Sunday following the first full moon after the vernal equinox.

    The equinox gradually moves backward in the calendar year because of the skew BA mentions.

    In the tens of thousands of years it will take for the spring equinox to move back into the previous year, it seems likely there may be some calendar adjustments (and maybe the rise and fall of several advanced civilizations; possibly a human extinction, hard to say). But on our current calendar you can figure out when that will happen.

  45. Mark Eret

    Re: PrimeMover

    I did make a slight mistake. I forgot to factor in the probability that the person had the birthday in the first place.

    It’s P(A)*P(B), where A is the probability of a birthday collision and B is the probability of someone having the leap day birthday in the first place.

    P(A) = .01855
    P(B) = 76*(1/366) = .207

    The actual probability is .0385 or 3.9% or so.

  46. Todd

    @Mango

    I came across a web site that mentioned that the calculation for Easter Sunday comes from the book of prayer, and isn’t based on the Vernal Equinox. *Shrug*

  47. TLA:
    Happy Birthday to you!
    Happy Birthday to you!
    Happy Birthday dear TLA,
    Happy Birthday to you!
    And many mooooooore!

    Here’s your present:
    http://stason.org/TULARC/science-engineering/space/102-Acronym-List-For-Space-And-Astronomy.html
    Rich :-D

  48. Mango

    I’ve always felt that we should have one day at some point in the year that are not assigned weekdays, and leap days should be unassigned too.

    Imagine December 31 isn’t any particular day of the week. December 30 is a Saturday, then January 1 is a Sunday. Then weeks would align with the calendar year, so any given day would be on the same day of the week every year.

    Election day in the US would always be on the same day. So would all holidays that by law must end up on a Friday or a Monday.

    Wouldn’t that be so much simpler?

  49. Todd:
    > Easter as always after March 20?

    No, Easter is the first Sunday after the first full moon after the vernal equinox.

    If it weren’t for leap years (or the century-correction to the leap years, or the 4-century-correction to the century-correction), Easter would slip a little each year, just as would the equinoxes and solstices.

    And, even with the leap-year corrections, things will still slip ever so slightly, as BA pointed out with the off-by-a-day-every-3200-years part of the math. (Unless he was wrong, as Jeremy pointed out.)

    Leap seconds

    My understanding about leap seconds is that, due to some wobble in the Earth itself, some years are longer than others, and when the “extra” time in the longer years add up to a second, they slip in a leap-second. (And let’s not get into that discussion again as to “what is a day/year?”)

  50. Doc

    @Psyberdave

    The French tried that back in the late 18th century. It didn’t go over very well.

    Wikipedia has a decent article on that whole mess.

    http://en.wikipedia.org/wiki/Decimal_time

  51. Mango

    Todd, was it a religious website?

    Are some evangelicals trying to disguise the fact that the date of Easter is based on really old Pagan tradition?

    Here’s the whole story:
    http://aa.usno.navy.mil/faq/docs/easter.php

  52. Ian

    Mark Eret:

    “It’s P(A)*P(B), where A is the probability of a birthday collision and B is the probability of someone having the leap day birthday in the first place.”

    P(B) is the probability that one or more persons out of 76 have the leap day birthday. I’m not convinced that that’s independent of P(A), so I’ll stand by my own result of about 1.9%.

  53. Greg in Austin

    Lugosi said, “4 times 12=28,”

    But I think you meant “4 times 12 = 48.”

    That’s the only nit I have to pick. ;)

  54. alfaniner

    Trivia:

    There are only 14 unique calendar configurations.

  55. Mango

    @Alfaniner:

    Trivia: But the smallest repeating cycle of calendar configurations is 2800 years long.




    I think.

  56. Ian

    Mango:

    No, a single 400-year cycle has 146097 days, which is divisible by 7, so in fact the cycle of calendar configurations is also 400 years.

  57. Just as an aside – did anyone catch the promo vid for the new Microsoft Worldwide Telescope? See it at Gizmodo

    http://gizmodo.com/361728/microsoft-worldwide-telescope-in-awesome-video-action-verdict-my-god-its-full-of-stars

    And also I see NASA is about to ram the moon, causing untold damage, knocking it out of alignment, destroying the earth. Like the old TV series Space 1999! :)

    http://gizmodo.com/361995/nasa-to-intentionally-smash-two-spacecraft-into-the-moon

    Wicked!

  58. Ian

    I wrote:

    “P(B) is the probability that one or more persons out of 76 have the leap day birthday. I’m not convinced that that’s independent of P(A), so I’ll stand by my own result of about 1.9%.”

    Actually, I think it is independent. But I can see no other reason why we should have gotten different results. Any idea?

  59. Mango

    Awesome. Thanks Ian.

    Not sure how I got a different number the first time…

  60. Dr. Trintignant

    “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. ”

    Why is this a stupid way of doing things?

    They could instead have said “Every 4 years is a leap year, unless the year is divisible by 128″. This would have given very good precision and not needed any additional exceptions. But the calculation is much harder in a decimal system (easy in binary, though!). I can tell you immediately and with almost no math skills whether a given year will be a leap year or not, but I’d have to break out a pen and paper to do the alternate calculation.

    I think it’s a brilliant way of doing things, even today, but particularly in an era without pocket calculators or widespread math education.

  61. Ian

    Okay, simulation using several hundred thousand repeated trials estimates the probability to be about 1.87% +- 0.02%. But I still can’t see what’s wrong with Mark Eret’s calculation.

  62. There’s a really awesome clock in Besancon, France which was put together a couple of hundred years ago. It’s a real engineering marvel, thoroughly impressive, and from memory it sits in a little room at the back of an old cathedral.

    It has about 70 dials, showing (amongst other things) local time in various parts of the world, tides at major French ports, and times of sunrise and sunset. The part that’s relavent to this discussion is the face that tracks leap year cycles.

    When I was about 9 (mid 90’s) I remember being taken there to visit the clock, walking around it to look at all the little cogs, the many hands ticking away. The guide who was telling us all about it showed us how little figures at the top did a dance every hour, driven by the clock’s mechanism, but they weren’t quite sure what would happen when they got to a leap century (2000) and were waiting with baited breath to find out. I always imagine lots of curious people packed into that room at midnight on December the 31st 1999 holding their breath to see what the clock would do to mark the significant event.

    http://www.tp178.com/mh/besancon/besancon.html

  63. Mango

    Now that I’ve proven my incompetence at math, I’ll weigh in on the birthday problem.

    The formula I’d use is:

    (76 choose 2) * (1/366 * 1/366 * (365/366)^74))

    The first part is 76! / (74! * 2!), representing the number of pairs of students born during a leap year who could possibly have been born on Feb 29. The second part is the probability each given pair is born on Feb 29, and no one else in the class was.

    So I get 0.01737, or about 1.7%. I suspect Ian’s calculation is also right, but the different approaches are creating floating point rounding error.

  64. rosebud

    If you really want to explain what I mean, you’ll have to clear it with my wife. She’s usually the one stuck with that job.

  65. Edward

    Could a clock be devised to account for the 1/4 day?

    As for the different lengths of months, I was taught that Julius
    and Augustus Caesar did this this for an ego trip. Taking a day from
    some other month and assigning it to a month with their name.
    July and August. Octavius may have done the same for the month
    of October.

  66. Mango

    Ian, it may be that Mark Eret is not discounting the probability of more than 2 people being born on Feb 29, whereas you calculation does.

    I’m pretty sure my number is off due to calc.exe rounding error.

  67. Ian

    Mango:

    Actually, mine is the probability of at least two people, not exactly two. I think that’s actually what accounts for most of the difference between your calculation and mine, not rounding error. I redid your calculation using arbitrary-precision ints as much as possible to reduce rounding error, and I got about the same result (0.01738).

    Also, if I modify my formula to calculate P(>=2) – P(>=3) to get P(=2), I get your result.

  68. Ian

    Okay, I think I see the subtle fallacy in Mark Eret’s calculation. The P(A) * P(B) formulation is intended to calculate the odds of the following algorithm successfully finding a pair of students whose birthdays are both on Feb 29:

    1) Check each student until you find one whose birthday is Feb 29 [P(B)].
    2) Check each remaining student until you find another one whose birthday is Feb 29 [P(A)].

    However, the algorithm it actually tests is more like this:

    1) Check each student until you find one whose birthday is Feb 29.
    2) Check each student until you find another one whose birthday is Feb 29 including the students that have already been checked, with the assumption that their birthdays might have changed.

    So the calculation of P(A) is a bit over-optimistic, and I was initially right in my hunch that P(A) and P(B) are not independent.

    Phew, I’m glad to put that behind me. Maybe now I can get back to work.

  69. The Western calendar is a minefield of historical facts and myths (or a treasure trove, depends on your mindset). I did a lot of research on it a few years ago, trying to work out how many days it has been since the founding of Rome in 753 BC. I dug up a huge amount of useless information in the process, and this was mostly before I had access to the Internet! (It turns out that there have been roughly a million days, but that could be out by a few thousand either way.) I was inspired to do this after reading “Day Miillion” by Frederick Pohl.

    For example: Did you know that although February has 29 days on a leap year, the actual day added is NOT the 29th? (Wiki it.)

    The original Roman calendar had only nine months, since it was based on the time between conception and birth. The 90 day Winter period was originally not marked, and the year started with the Spring Equinox in March.

    When the Gregorian calendar was introduced in 1753, people rioted in the streets demanding their “11 days” back (Octber 4th was followed by Oct 15th that year.) This has been made to show how stupid some people were at the time, but actually it was a genuine complaint as landlords were charging a full month’s rent, but employers were not paying a full month’s pay. Simple really.

    The longest year on record was in 44BC when Julius Caesar introduced his new calendar. To get it in line with reality, he had to add two intercessionary months plus extra days which made that year 445 days long!

    And so on… The calendar really can be fascinating (if you’re a nerd like me!)

  70. Wonder what the odds are on 2 out of 76 kids being born on leap day. Who is up for some more math?

    Well, the odds of 1 kid out of 76 being born on that day are about 5%, so i’d estimate the odds of 2 kids out of 76 being born on that day somewhere in the 2 to 3 percent range.

  71. Of course, it’s been a long time since I did statistics…

  72. GAZZA

    Presumably leap seconds are added to make up the remaining time? Does anyone know how often we get a leap second, and whether they’re scattered evenly throughout the year or if they fall on a particular day?

    Oh, and all proper Unix users know about September 1752, right?

  73. Rosebud burned up in the fire. :)

  74. Inertially Guided

    Ugh. Math hurt Mongo’s Head. Mongo Not Like Math. Mongo Just Pawn In Great Game of Life.

    Thanks for a stimulating post, BA! And Happy Bird-Egg to TLA!

  75. 1752 of course, not 1753. It’s late here in England & I’m tired.

    Thanks for putting me right. And of course I meant intercalary, not intercessionary. Where was my brain at?

  76. Ed Davies

    Drew Terryon:

    > Seconds are arbitrary?

    > Is that fact or informed opinion?

    24 hours in a day, 60 minutes in an hour and 60 seconds in a minute is pretty arbitrary. Even if you take those numbers as being, somehow, convenient and so not arbitrary the choice of the day used to define the second is a bit arbitrary, too.

    This discussion is also relevant to the whole leap second thing.

    The rotation of the Earth is slowly slowing down as tidal effects transfer angular momentum to the Moon. The Moon is moving away from us at a rate of a couple of cm per year and the length of the day is increasing at a rate of, IIRC, about 1.8 milliseconds per century. Various other effects (like the leaves falling off the trees in the autumn, earthquakes, icecaps melting and so on but mostly changes in the circulation in the Earth’s core, as I understand it) mean that there are other shorter term variations in the length of the day as well.

    Originally, an hour was a 24th of a day, a minute a 60th of an hour and a second a 60th of a minute. When it came to defining the second precisely in, round about, the 1920s they had to decide on which day length to use. A weighted average of the last hundred or so year’s observations were used meaning that the second was defined from the day length in about 1870 or some such. This definition was refined and carried forward to form the modern atomic definition of the second.

    In this sense, the definition of the second is arbitrary – if technology had been developed at a different pace a different year and so a different day length would have been used in the definition of the second.

  77. JB of Brisbane

    @Michael – don’t forget Dionysius, who got to within four (4) years of the year Jesus was supposed to have been born from six hundred years down the road.

    @Psyberdave and all – does anybody remember hearing of the World Calendar? That was the one where the 365th day of the normal year was called “Worldsday” instead of one of the seven usual days of the week. This would keep each day of the year occuring on the same day of the week from year to year. Of course, there would also have to be a day called “Leap Year Day” every four years, or as defined above. I remember reading about this in an old “How And Why Wonder Book” on time.

  78. Roy

    What math? There was only multiplication, addition, subtraction, and division. You didn’t even use trigonometry!

  79. Awesome rundown Phil. I don’t think that math would be daunting to anyone. The trick is to make sure folks don’t get intimidated by all the numbers being thrown about. Especially all the long partial numbers. Those always seem to freak people out.

  80. Mark Eret

    Re: Ian

    I had a feeling there was something kind of tricky in there and I wasn’t quite convinced that I was right. Thanks for finding my error!

  81. There were varying responses to the probability that 2 of your 76 classmates had the same birthday on Feb 29, so I figured I’d calculate it myself and see where we were.

    Some assumptions: There were no twins, and no crazy event that involved lots of women getting pregnant 9 months before Feb 29. Also, all 76 members of your class were born in 1976, which is highly unlikely, but the calculation is generic enough you can rerun the numbers.

    The binomial theorem tells us that (A + B)^n = summation from i=0 to n of A^(n-i) * B^i * (n choose i). Specifically, when 0 <= A,B <= 1, and A+B = 1, this gives us a probability distribution. Take A = 365/366, the probability a given child was not born on Feb 29, and B = 1/366, the probability a given child was born on Feb 29, with n = 76. We are interested in the term where i = 2, call it p2, which is the probability that exactly 2 children were born on Feb 29.

    for i = 0, p0 = (365/366)^76 * (1/366)^0 * 1 = 81.22%
    for i = 1, p1 = (365/366)^75 * (1/366)^1 * 76 = 16.91%
    for i = 2, p2 = (365/366)^74 * (1/366)^2 * 76 * 75 / 2 = 1.74%

    p0 is the probability none were born on Feb 29, p1 the probability that 1, and p2 the probability that 2, which is 1.74%. Ignoring some significant figures, there is around a .13% chance for all numbers greater than 2, that is the odds that more than 2 children were born on Feb 29 is .13%.

    This number agrees with Mango, so it is correct.

    To go even deeper into the math, what is the expected number of children such that there is greater than 50% chance that there are 2 born on Feb 29?

    (365/366)^n * (1/366)^2 * 76 * 75 / 2 = .5
    (365/366)^n = .5 * 2 * 366^2 / (76 * 75) = 23.501
    n * log(365/366) = log(23.501)
    n = log(23.501) / log(365/366) = 428.98, so around 429 children.

  82. MandyDax

    @PrimeMover
    “The interesting thing is that out of a small high school class I graduated with (76 total students) two of them were born on leap day. Wonder what the odds are on 2 out of 76 kids being born on leap day.”

    Probability makes my brain hurt, but for your particular class of 76, you collapsed the wave function and we find there were exactly 2 students with a Leap Day birthday. 100%.

    See? Quantum physics isn’t hard, as long as you’re being observant. ;)

  83. Efogoto

    I have seen the objection raised that having the days hit the same day of the week every year would mean that if you were born on Tuesday, you’d never get a weekend birthday. Now *there’s* a reason to forgo calendar adjustment.

  84. Hairy Doctor Professor

    Miss M: and were waiting with baited breath to find out.

    A worm on your tongue?

    (Sorry. It’s all I got tonight. Did two 75-minute lectures and an unexpected lab today. Most of the numbers look good from a quick scan, but I’m too tired to go into much depth.)

  85. @ Hairy:
    Bated breath is the way it’s actually spelled.
    http://www.phrases.org.uk/meanings/56550.html
    Rich

  86. Kimpatsu

    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.
    Very Pirates of Penzance; but did you know another great man born on the bisextile day is actor Joss Acland, who is either 80 or 20 today, depending upon your viewpoint.

  87. George E. Martin

    Mango said:

    @Todd: Easter is the Sunday following the first full moon after the vernal equinox.

    Actually the definition for the date of Easter is that it falls on the first Sunday which follows the *ecclesiastical* full moon that falls on or after March 21st.

    March 21st might be called the ecclesiastical vernal equinox which obviously is different than the astronomical vernal equinox, this year being March 20th unless you are in the U.S. Mountain Time Zone of farther west when the astronomical event happens on the 19th. Because we had a leap year in 2000, March 21st will drift farther away from the astronomical vernal equinox, the year 2100 bringing things back.

    Of course the Gregorian calendar reform was put in to bring back into alignment the astronomical and ecclesiastical vernal equinoxes.

    The ecclesiastical full moon comes from tables in which the 14th day is considered the full moon. Every once in a while, but not too often, that definition gives a different day than when the astronomical full moon occurs.

    Of mild interest is the fact this year Easter, 23 March, misses by one day being the earliest possible date for Easter.

    See:

    http://aa.usno.navy.mil/faq/docs/easter.php

    George

  88. antaresrichard

    On second thought, I happened on this over at Wikipedia…

    “Vote to stop leap seconds

    At the 47th meeting of Civil Global Positioning System Service Interface Committee (CGSIC) at Fort Worth, Texas it was announced that a mailed vote would go out on stopping leap seconds. The plan for the vote is [7]:

    * April 2008: ITU Working Party 7A will submit to ITU Study Group 7 project recommendation on stopping leap second[s]
    * During 2008 Study Group 7 will conduct a vote through mail among member states
    * 2011: if 70% [of] member states agree[, the] World Radio Conference will approve [the] recommendation
    * 2013: application of leap second[s] will stop and UTC will become a continuous time scale”

  89. Keith Harwood

    I was under the impression that there was a rule that years divisible by 4000 were not leap years, called the Lillian correction (though I have no idea who Lillius was), but that it wasn’t worthwhile making it official. Presumably the thought was, “let the people of 4000CE sort it out for themselves, I can’t be bothered”.

    Edward: October isn’t named after Octavian. Octavian was Augustus so he already had a month named after him.

    Years ago Punch magazine had a competition: tell the stories of the gods that the months September to December are named after.

  90. eyesoars

    So why do we care exactly how long a year is? The only advantages are to astronomers. The major inconvenience to people is that the seasons will wander around the calendar year somewhat.

    Which is apparently what happened: the original calendar was started with the Vernal equinox occurring on March 1, and then the extra day was added at the end of the year (Feb. 29) when it was needed. Clearly things had drifted a bit by the time we started using the Gregorian calendar. Somewhere along the way January became the first year, messing up the names of some of the later months (October , November, December (not the 8th, 9th, or 10th months of our calendar, respectively).

    There are in fact “better” calendars (better being defined as tracking the sidereal year better), e.g., the Persian calendar, which has 8 leap years every 33 years:

    http://en.wikipedia.org/wiki/Persian_calendar#Seasonal_error

    A diagram at that URL (“The Jalaali Calendar”) shows the seasonal errors in that calendar; for comparison, the year to year error in the Gregorian calendar can be found at:
    http://en.wikipedia.org/wiki/Gregorian_calendar#Calendar_seasonal_error

  91. defectiverobot

    Phil,

    With due respect to PZ, I thought it was funny.

    However, the banana guy’s explanation of shooting down the satellite was funnier.

  92. defectiverobot

    And yes, I am a brown-noser (though I prefer the term obsequious!).

  93. Awrang

    Phil, I have to disagree with you.

    A day is also an arbitrary unit of time, and 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″ is also a real, physical event, and by the definition has “physical corresponding action.”

    You are right that there is nothing to prevent someone from changing the value of a second by 10%, but there is nothing to prevent someone from defining a day to be equivalent to the rotation of the earth on its axis 5 times.

  94. zeb

    Don’t forget that gravitational and tidal forces change the length of the year and day over the millenia. By time 3200 years have passed, we may find that we’ll have to adjust the rule.

    Also, within those 3200 years we’ll likely have colonized a lot of the moons and planets in our solar system (and probably a number of worlds in other systems), and they’ll all have unique rules as well! Arg!

  95. Awrang, our day is based on a pre-existing event. So is the year. The second, the minute, the hour, the week: those were made up. They’re arbitrary. We now have a definition of a second, but it was made after we had already established how long a second was, and that is an arbitrary length of time.

  96. Awrang

    Phil,

    I agree that a day is a pre-existing event, but to use it as a unit of time is completely arbitrary–no less arbitrary than the pre-existing cycles of the cesium atom. In a nutshell, a day as a unit of time is entirely a human invention just like the second. Just my two-cents:)

  97. Ian

    I’m inclined to agree with Awrang. Consider for example that the Roman bissextile day was legally considered to be the second half of a single 48-hour “day”.

  98. Tim G

    It’s February’s Fifth Friday!

    This should be a holiday of some sort. One that occurs once every 28 years or so.

  99. Coucherias

    @ Todd et al:

    There was an attempt at a quasi-decimal calendar in France after their revolution. There were twelve months of 30 days each, with five or six days at the end of the year to round it off. The link below explains things pretty simply and clearly, except for translating the names of the months. The months were named after the principal agricultural activity or meteorological characteristic of that part of the year. Day 1 corresponded to September 22:

    Vendémiaire — from vendange, grape harvest
    Brumaire — from brume, light fog
    Frimaire — from frimas, freezing foggy weather
    Nivôse — from the Latin nivosus (French neige), snowy weather
    Pluviôse — from pluvieux, rainy weather
    Ventôse — from venteux, windy
    Germinal — from the Latin germen (French germination, same in English)
    Floréal — from the Latin floreus (French fleuri), or flowered
    Prairial — from prairie, same as in English
    Messidor — from Latin messis (French moisson), cereal harvest
    Thermidor — from Greek thermon, summer heat
    Fructidor — from Latin fructus (French fruit, same in English)

    Voilà.

    http://webexhibits.org/calendars/calendar-french.html#SECTION00710000000000000000

  100. themadlolscientist

    @ Michael: Of course you’re right. My bad. Having posted that comment midafternoon, I can’t even blame it on the lateness of the hour. Damn. :-(

  101. Nick

    @Psyberdave – No. Kill decimal time. It’s a bad idea from start to finish.

    The division of the day into 24 hours, and further divisions into 60 minutes per hour and 60 seconds per minute may seem arbitrary but they work extremely well in what they’re intended to do, which is divide up the period of one day. 24 is divisible by 2, 3, 4, 6, 8 and 12. 60 is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30. By contrast, you could only divide metric units by products of 1, 2 and 5.

    By the way, decimal time is very different to the metric system or decimal currency. Those are measurements, i.e. more to do with adding, so it’s simpler to say “35cm + 65cm = 1m” than “2ft 4in + 1ft 9in = 1yd 1in”. Seconds, minutes and hours, however, are more about dividing up the day so you can say “1/4 of a day is 6 hours”.

    By the way, regarding the calendar: I reckon we should have 12 months of 30 days but begin the year with a monthless 5-day (or 6-day on leap years) period of New Years celebration.

  102. Coucherias

    The use of base six for measurement of time is not arbitrary. From James I. Nienhuis at http://www.author-me.com/nonfiction/base6.htm : “The ancient Sumerians (Babylon) used a Base Six number system because they knew that the length of one side of a six-sided polygon (that is a hexagon) is the same length as is the radius of the circle that circumscribes this hexagon. They also knew that the constellations of the zodiac seem to rotate along the horizon like a merry-go-round at a rate of 72 years per degree (per one degree of 360 degrees) of rotation which is known as the precession of the stars. This precession results from the slow wobble of the earth’s axis (like a gyroscope) which would wobble once through one full 360 degree precession in 25,920 years (as 72 years x 360 degrees equals 25,920 years/360 degrees of precession).”

    This base-six system of counting was applied to time and has lasted to this day.

  103. Nigel Depledge

    BA, just a stylistic comment:

    The BA wrote:
    “There are not an exactly even number . . .”

    “There are not an” jars with me. It sounds awkard. How about “The number of days per year is not exactly even.” instead?

  104. Nike

    First of all, the Gregorian calendar is NOT based upon the Tropical Year, which is an error that is commonly repeated, even by professional astronomers. However, Gregory’s Inter Gravissimas (http://www.bluewaterarts.com/calendar/NewInterGravissimas.htm) makes it clear that it was based upon the vernal equinox. The average period between successive vernal equinoxes is about 365.24237 days, resulting in an error of one day in 8000 years, although the values of the VE year and the mean solar day are not stable over that length of time, and the error will probably decrease in the future.

    See http://www.hermetic.ch/cal_stud/cassidy/err_trop.htm

    Also, the leap day was not originally intended to be added at the end of the month of February. The extra day was inserted after a religious festival which was held on the 23rd day of February. The Romans (and Popes) did not count days of the month as we do, from the first day to the last, but counted BACKWARDS to the following kalends, nones or ides. (Remember the ides of March?) The kalends of March is March 1st. February 23 is day seven before the kalends of March (VII Kalendas Martii), so the leap day was the day six before the kalends of March (VI Kal. Mart.) or the 24th day of February by our counting. (They counted inclusively, so the last day of the month was counted as day two before kalends.) The leap day was called bissextum, which means “twice six”, because the inserted day was given the same number, so basically 48 hours was counted as a single day. In leap years (called bissextile years) the last four days of February were shifted by one day (in our counting) so that saints’ days, birthdays, debt payments, etc., would occur one day later when counting from the beginning of the month. For instance, if your birthday was the 26th day of February in a common year, it would be the 27th on a leap year. If you were born on the leap day (bissextum), that was VI Kal. Mart., so you would celebrate it on VI Kal. Mart., the 24th, every February, leap year or not.

    Apparently, the Church recently changed their saints’ days to reflect modern counting, and the EU also recently changed the law to match, so I guess that we can legitimately say now that the leap day is the 29th, but we already missed the real bissextum.

  105. John

    What I find interesting is that despite all the changes made to the calendar over the centuries, that there hasen’t been a missed weekday. That is, the series Sun-Mon-Tue-Wed-Thu-Fri-Sat was uninterrupted.

  106. JoakimR

    Ed Davies:
    > Originally, an hour was a 24th of a day

    No. Originally as hour was a 12th of the time between sunrise and sunset. It was when we started making mechanical clocks we switched to a more practical fixed-length hour.

  107. hheb09'1

    “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.”

    Just a little cavil: without the leap day, the calendar would be counting years too fast, so we’d say that it was ahead, wouldn’t we? Like a watch that was too fast, reading five minutes ahead.

    Without a leap day, the day might read March 2. Adding in the leap day, it now reads March 1. So, it had been ahead, I think.

  108. defectiverobot

    Awrang,

    I get what your saying, but I think you’re overthinking. By “non-arbirtrary,” Phil means that the definition of a day arose organically, derived from a natural occurence. A day was a day back when we were all primordial soup, and we didn’t decide to define it by some other arbitrary means back when man put their collective, lice-ridden, unbarbered coifs together to think about such things. It just seemed a sensible duration and there was no reason to mess with it.

    Same thing for the definition of a day. Or for base-10 math.

    However, that a day is 24 hours is completely arbitrary. We took the natural duration of a day and carved it up into 24 ticks on a clock face (well, 12, but I think the first clocks actually designated 24 hours). Same is true for hours and seconds. What genius decided on 60? Why not 50, so that it’s easier for us near-innumerates to divide? Why not 10 so we can keep track of time with our fingers?

    Anyway, the point is that days and years were essentially pre-determined by naturally occurring phenomena, while months, hours, and seconds were devised in the minds imbeciles.

  109. CGM3

    Listening to the morning radio show hosts nattering about Leap Year on the way to work this morning, a thought popped into my head: how do creationists explain why the year is not an exact number of days? Eclipses have been hailed as “proof” of divine creation — the argument being that the likelihood of the sun and the moon having the same apparent size is too small to be coincidental — but I don’t know if they’ve ever addressed this.

    Being a generous soul, I’ll offer a few possible explanations:

    “God’s timer was off that day.”

    “God was feeling mischievous.”

    “It was God’s time of the millenium, and She was too cranky to do the fine tuning.”

  110. What I find interesting is that despite all the changes made to the calendar over the centuries, that there hasn’t been a missed weekday. That is, the series Sun-Mon-Tue-Wed-Thu-Fri-Sat was uninterrupted.

    Having the sabbath on the 7th day is more important to the judeo-christian-islamic tradition then any other issue about counting time.
    That’s why we won’t see a ‘metric’ calender that doesn’t have a perpetual 7 day week count any time soon.

  111. Nike

    Hours predate clocks by quite a bit. People used to tell time with sundials or similar devices. These were divided in some places into twelve parts, which meant that in the summer, hours were longer than in the winter.

    I think that minutes and seconds were introduced in the Middle Ages, after clocks started being used. If you have a 12-hour clock and add a hand that rotates once per hour, each hour is automatically divided into 12 parts by the hour markers; put four marks between each hour marker, and you get 60 parts per hour. Minutes and seconds dividing degrees of arc angle by 60 and 3600 have been used since Babylonian days, so the same units were used to divide hours. Minute comes from the Latin for “(first) small part” and second from “second small part”. Thirds (1/60 sec), fourths, etc., have also been used.

    As stated earlier, during the French Revolution they replaced 7-day weeks with 10-day “décades”, with days named Primidi, Duodi, etc., and divided the day into 10 decimal hours, each hour into 100 decimal minutes, each minute into 100 decimal seconds. Decimal clocks flopped, but décades were used until Napolean made a treaty with the Church in 1802. However, astronomers use decimal time today, in the decimal fraction of Julian Dates. MJD 54526.56789 would be 5 h 67 m 89 s (GMT) on a Revolutionary clock. (54526 = Decadi 10 Ventôse, an CCXVI on the Republican Calendar.)

  112. On a post a few days ago (the Windmill post) I said something to the effect that sometimes we need to take the hit (inconvenience and economic) of ending a mistake.

    I think this is one of them. I understand it’s a HUGE BIG DEAL but it’s just going to keep throwing us off for eternity until it is taken care of.

    Why are we so important that we can’t deal with the consequences of fixing things like this?

    My belief is that we should adjust the “second” and every other unit so that we don’t have these bizarre time units.

    In fact, why does it make make sense at all to deal in the base units of our time system???

    60 seconds in a minute. 60 minutes in an hour. 24 hours in a day. 7 days in a week. 52 weeks in a years. Why do we continue a nonsensical system like that?

    Anyway, I’m done with it. I’m getting a metric clock.

  113. anon1234

    Further leap-year information, from Digital Equipment, can be found at:

    http://www.interesting-people.org/archives/interesting-people/200002/msg00089.html

  114. Mango writes:

    [[Are some evangelicals trying to disguise the fact that the date of Easter is based on really old Pagan tradition?]]

    Actually, I think it was based on really old Jewish tradition.

  115. Dunc

    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.

    Well, if we’re still using the same calendar in 3200 years, we’ll call you. You might like to spend the intervening time working out how to explain leap seconds… ;)

  116. TSK

    Phil, you are wrong, the second wasn’t arbitrary. Humans will choose smallest timescales which they are able to discern. Movements quicker than 1/5 second will begin to blur and with 1/20 second no difference between following images cannot be made out. The jewish time has as smallest unit the “regaim” (tick) equalling 1/22 second which is pretty close to the limit. Dividing a day in 1/1000 seconds is senseless.
    Only the dividing between day and second is arbitrary. What you didn’t mention is that the second has *two* definitions with the advent of atomic timescales:

    a) The citizen/astronomic version: The second is 1/86400 of the mean timespan between two middays.

    b) The atomic version with caesium transitions.

    1887 the second was defined according to definition a). 1956 a committee nailed it down to 100/315569259747 parts of the tropical year 1900. The current version was adopted 1967.

    And this is the cause of the leap *second*. The earth rotation (the day) is over long timescales slowing down over time due to tidal friction, but in shorter timescales the earth rotation speeds up or slows down to unknown reasons. If we adopt atomic time, the midday would slowly go out of sync: 12:01, 12:10, 14:00 and so on.
    So the people decided to implement a hybrid: If the difference of the atomic time and citizen time goes over 0,9 seconds, a second is inserted or deleted from a day of the year.

  117. Aerimus

    “Only the dividing between day and second is arbitrary. What you didn’t mention is that the second has *two* definitions with the advent of atomic timescales:

    a) The citizen/astronomic version: The second is 1/86400 of the mean timespan between two middays.

    b) The atomic version with caesium transitions.”

    Actually, BA mentioned in a earlier post about seconds being arbitrary, even when you add the second definition, since the second definition came into existence as a more standard way to reach the first definition. In other words, the only reason why a second is defined as “the atomic version with caesium transitions” is because “the atomic version with caesium transitions” is roughly one second, as a second is defined in definition (a).

  118. rhr

    The 400 year cycle is better than it looks initially compared to a 500 year cycle. The 100 year system becomes off by a day after 456 years, so 500 isn’t that much better to begin with. But after adding a leap day every 500 years (121/500), it will be off by a day after ~5000 years. But then you need to *add* a leap day again (where do you put it?) for 1211/5000. And that will be off by a day after ~100ka. On the other hand, with the 400 year cycle the next correction is to skip a day again at 3200, and 775/3200 happens to be really close. You’re good for ~350ka then. If you add a day at 320k you’re good for almost 5Ma (by which time tidal perturbation will have changed the length of the day and made this system irrelevant).

  119. Then, there are some who would end the Leap Year
    (This is obviously tongue-in-cheek, especially note the location of Venus in the graphic near the beginning)

    Gee, I hope this works!

    J/P=?

  120. ^%#$^*\

    Okay, let’s try this:
    LeapYear

  121. Irishscribe

    I have a headache

  122. George E. Martin

    Nike said:

    “However, astronomers use decimal time today, in the decimal fraction of Julian Dates. MJD 54526.56789 would be 5 h 67 m 89 s (GMT) on a Revolutionary clock. (54526 = Decadi 10 Ventôse, an CCXVI on the Republican Calendar.)”

    I’m not quite sure what you mean. The “.56789″ in the MJD is the fraction of the day. Thus 54526.56789 is 2008 March 1 13:37:46 UTC. (It’s been years since GMT was used.)

    The MJD, Modified Julian Day , is the Julian day number – 2400000.5. The Julain day number is the continuous count of days since noon on Monday, 1st January of year 4713 B.C.E. The Julian day was defined to begin at noon on the Greenwich meridian so that a given night would always have the same day number, at least for astronomers in Western Europe.

    The Julian in the name has nothing to do with Julius Caesar or the Julian calendar.

    See for example:

    http://www.fourmilab.ch/documents/calendar/

  123. Dark Jaguar

    I get ya now. You meant arbitrary as in how it was originally defined, not how it is presently defined. But, that said, the second now is much more rigid than all the rest. That is, how that atom behaves won’t change, even after the earth is burnt away by the sun. Really though even in the short term days and years are only as good as the planet you happen to be sitting on. Seconds will be good for wherever you find yourself. Pfft, humans always think in such small terms… building their cities RIGHT where glaciers will come screaming through in a mere 200,000 years.

    But that said, months were originally based on a real event too weren’t they?

  124. Nike

    George E. Martin said: “I’m not quite sure what you mean. The “.56789? in the MJD is the fraction of the day. Thus 54526.56789 is 2008 March 1 13:37:46 UTC. (It’s been years since GMT was used.)”

    What I mean is that the fraction of the day in the MJD corresponds with the time displayed by the decimal clocks built in France in the 1790s, when those clocks are set to mean solar time at Greenwich. 0.56789 is displayed on a decimal clock as 5 h 67 m 89 s. That is to say, the hour hand is between 5 and 6, the minute hand between 67 and 68, and the second hand points at 89. See http://www.antiquorum.com/html/vox/vox2004/revolutionary.htm for photos of decimal clocks.

    I used MJD instead of JD because the latter is a half-day off from GMT, but MJD corresponds exactly. I referred to GMT because I was talking about French Revolutionary clocks, which were built before UTC existed. I meant literally mean solar time at Greenwich, to distinguish it from the time in Paris, where the clocks were primarily used, and which is on the mean 9m21s ahead of GMT. Although UTC has replaced GMT for most purposes, GMT has not simply ceased to exist; GMT is still the mean time at Greenwich, and a perfectly valid term, especially in historical context. In fact, GMT is still officially in use in the UK, which is where Greenwich is. BBC’s World Service still announces GMT every day. I do not know if they are actually referring to UTC or UT1, but it hardly matters here.

    You are wrong about Julian the name. The name comes from the fact that it is based upon the Julian Period, which was named by Joseph Scaliger. There is a myth that he named it after his own father. However, Scaliger, himself, stated, “We have called it Julian merely because it is accommodated to the Julian year.” By Julian year he meant a year in the Julian calendar, which was named for Julius Caesar. For more information, see the Wikipedia article on Julian Day. (Disclaimer: I contributed to this article, but original sources are referenced there.)

  125. Nike

    George E. Martin said: “I’m not quite sure what you mean. The “.56789? in the MJD is the fraction of the day. Thus 54526.56789 is 2008 March 1 13:37:46 UTC. (It’s been years since GMT was used.)”

    What I mean is that the fraction of the day in the MJD corresponds with the time displayed by the decimal clocks built in France in the 1790s, when those clocks are set to mean solar time at Greenwich. 0.56789 is displayed on a decimal clock as 5 h 67 m 89 s. That is to say, the hour hand is between 5 and 6, the minute hand between 67 and 68, and the second hand points at 89. See http://www.antiquorum.com/html/vox/vox2004/revolutionary.htm for photos of decimal clocks.

    I used MJD instead of JD because the latter is a half-day off from GMT, but MJD corresponds exactly. I referred to GMT because I was talking about French Revolutionary clocks, which were built before UTC existed. I meant literally mean solar time at Greenwich, to distinguish it from the time in Paris, where the clocks were primarily used, and which is on the mean 9m21s ahead of GMT. Although UTC has replaced GMT for most purposes, GMT has not simply ceased to exist; GMT is still the mean time at Greenwich, and a perfectly valid term, especially in historical context. In fact, GMT is still officially in use in the UK, which is where Greenwich is. BBC’s World Service still announces GMT every day. I do not know if they are actually referring to UTC or UT1, but it hardly matters here.

    You are wrong about Julian the name. The name comes from the fact that it is based upon the Julian Period, which was named by Joseph Scaliger. There is a myth that he named it after his own father. However, Scaliger, himself, stated, “We have called it Julian merely because it is accommodated to the Julian year.” By Julian year he meant a year in the Julian calendar, which was named for Julius Caesar. For more information, see the Wikipedia article on Julian Day. (Disclaimer: I contributed to this article, but original sources are referenced there.)

  126. Nike

    George E. Martin said: “I’m not quite sure what you mean. The “.56789″ in the MJD is the fraction of the day. Thus 54526.56789 is 2008 March 1 13:37:46 UTC. (It’s been years since GMT was used.)”

    What I mean is that the fraction of the day in the MJD corresponds with the time displayed by the decimal clocks built in France in the 1790s, when those clocks are set to mean solar time at Greenwich. 0.56789 is displayed on a decimal clock as 5 h 67 m 89 s. That is to say, the hour hand is between 5 and 6, the minute hand between 67 and 68, and the second hand points at 89. See http://www.antiquorum.com/html/vox/vox2004/revolutionary.htm for photos of decimal clocks.

    I used MJD instead of JD because the latter is a half-day off from GMT, but MJD corresponds exactly. I referred to GMT because I was talking about French Revolutionary clocks, which were built before UTC existed. I meant literally mean solar time at Greenwich, to distinguish it from the time in Paris, where the clocks were primarily used, and which is on the mean 9m21s ahead of GMT. Although UTC has replaced GMT for most purposes, GMT has not simply ceased to exist; GMT is still the mean time at Greenwich, and a perfectly valid term, especially in historical context. In fact, GMT is still officially in use in the UK, which is where Greenwich is. BBC’s World Service still announces GMT every day. I do not know if they are actually referring to UTC or UT1, but it hardly matters here.

    You are wrong about Julian the name. The name comes from the fact that it is based upon the Julian Period, which was named by Joseph Scaliger. There is a myth that he named it after his own father. However, Scaliger, himself, stated, “We have called it Julian merely because it is accommodated to the Julian year.” By Julian year he meant a year in the Julian calendar, which was named for Julius Caesar. For more information, see the Wikipedia article on Julian Day. (Disclaimer: I contributed to this article, but original sources are referenced there.)

  127. Nigel Depledge

    Drew Terry said:
    “Seconds are arbitrary?

    Is that fact or informed opinion?

    Just curious, thanks.”

    It is both, Drew. The second is an arbitrary division of time. Its definition is not based on any physical phenomenon, even though it is now defined by the caesium time standard.

    To turn it around, think of these questions: What physical phenomenon takes exactly one second? What physical phemonena might possibly take exactly one second?

  128. B Fenerty

    Nigel asked: […] What physical phenomena might possibly take exactly one second?

    How about the time it took Welles to say “rosebud”?

    If so, we then have the OWRSS, the Orson Welles Rosebud Standard Second! I bet Phil you subconsciously had this very thing in mind when you mentioned his famous quote at the top!

    Sliding along, the above are what leaped into my thoughts on this burning question. Got to run now, there are only so many OWRSS in a busy day, so will just close by asking: any second opinions?)

  129. Enok

    “What physical phenomenon takes exactly one second? What physical phemonena might possibly take exactly one second?”

    A heartbeat.

  130. drew terry

    It seems to me what is arbitrary is the blind faith in the “year” equal to exactly how long it takes the Earth to revolve around the Sun; so what does it matter?

    Besides planting and harvesting, and taxes, does it really matter how long is a year?

    If the day equal to 24 hours is arbitrary, and if the 1 hour equal to 60 minutes is arbitrary, then is it also arbitrary that 3600 seconds = 60 minutes while 86400 seconds equal 24 hours?

    Think about what did numbers mean and for what were they used? They measured circles long before there was a clock on the wall or 24 hours to the day or 3600 seconds to 1 hour; these are all terms of navigation.

    3×2=6; 6×4=24; 24×3=72;

    72/2=36; 72/3=24; 3/2=36/24;

    384-24=360; 360/10=36

    If you think the metric system would be easier for time, consider these simple math functions as base-10 logarithmic functions:

    60-24=3.6e1
    36/24=1.5
    36-24=1.2e1
    12/36=3.3333333333e-1
    3/36=8.3333333333e-2
    3/12=2.5e-1
    1/4/12=2.0833333333e-2
    1/6/12=1.3888888889e-2

    Now here they are as fractions:
    60-24=36
    36/24=3/2
    36-24=12
    12/36=1/3
    3/36=1/12
    3/12=1/4
    1/4/12=1/48
    1/6/12=1/72

    Anything but arbitrary.

  131. “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. Convenient, but they are not based on independent, non-arbitrary events.”

    The year and the day are not the only two measurements of time based on physical events. A month is also based on a physical event, or was in earlier calendars which used 28 days for one month. Now a month is only based on a lunar day (which is also a lunar year because the Moon’s year and day are both 28 days long. This is due to Earth’s gravity, which tidally locks one side of the moon to always face the Earth.)

    Something just occurred to me to wonder about: does anyone know what the Star Trek system of time measurement is based on? Never did figure that out.

  132. gazza666

    As far as I know the ST time was DELIBERATELY chosen to be indecipherable. They were aware that the Einsteinian equations would mean that UTC would prove problematic (to say nothing of the fact that it would seem too Earth centric) so they invented “Stardates” which apparently compensated for the effects of relativity and so forth.

    That being said, in the Next Generation era, the second digit of the star date tended to increase by 1 every season, but I don’t think it was ever established that this was canonical.

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  137. CuriosButNotGeorge

    So, I understand -I think- about leap years and other calendar systems.
    My question is related to this extraction of your text: “Every other unit of time we use (second, hour, week, month) is rather arbitrary.”
    So, how was it decided (arbitrary) that a minute is a 60th of and hour, and a second a 60th of a minute but then they stopped there?
    Why did the second not get divided using the sexagesimal system and they used the decimal system instead (10ths, 100ths, 1000ths of a second … and so on…?

  138. Dave

    60 x 60 x 24 x 365 x 4 + 864000 = 126230400 / 4/365 /24 / 60/60 = 1.000684931506849. Is this wrong?

  139. Mavis mabry

    Question? If we have 7 days in a week and 52 weeks in a year, how can we have 365.25 days in a year. 7 times 52 is 364 not 365.25. We are teaching our students in school wrong. How do we rectify this?

  140. donald hilton

    My brother was born on a leap year feb 29th, and he has twins born on a leap year feb 29th exactly 29 minutes apart in a hospital that was completed and opened on feb 29th exactly 29 miles from my brothers house. Now what are the odds of that???

  141. Steve

    Come on people! The pope fixed it up best he could back then, I say lets start planning on making this whole thing better. If we can speed up the Earth in it’s orbit slight faster, we can cut off the extra .24 day in a year and synchronize the number of days so it fits to EXACTLY one year in the Earth’s orbit.

    That will of course make the Earth slightly closer to the sun – hmm, maybe a bad idea. How about instead we slow the Earth’s orbit down, so every year is 366 days, and every year has a Feb 29th! By precisely putting the Earth into it’s correct orbit – we can get rid of this whole nonsense of leap days and years of different lengths. And – the Earth would be slightly further away from the sun, maybe helping this global warming thing a tiny bit.

    If we make the year 372 days long, each month could also be 31 days each, further clearing up this nonsense of months of different lengths. The Earth would be slightly more away from the sun, thus cooling it off a little more.

    So simply moving the Earth in it’s orbit will fix a whole bunch of problems! Of course, all computer programs would need to be rewritten to know about the new calendar, this would make the Y2K issue look like nothing!

    If this all works, then we can start messing with the moon’s orbit, slow it down so each lunar month is also 31 days to match our new calendar. With a little bit of work, we could also move it in it’s orbit a bit, so the full moon is always on the 1st day of each month. Tide tables would be greatly simplified, since they’d be the same each month!

  142. john

    WHY DON’T WE NOTICE THE 6 HOURS WE GAIN EVERY YEAR? Wouldn’t the sunset a lot later? Like instead of setting at 5:30 pm in the winter it would set at 11:30 p.m.. We would notice the progression.

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