# Why does the amount of days in an year on average of the Gregorian calendar only have 4 decimal places (365.2425)?

Alfonsine tables available at the time of the Gregorian reform provided enough information (however inaccurate) for the calendar to have been designed such that it expressed more precision regarding the amount of days in a year on average (e.g., 365.2425463 instead of 365.2425).

Why isn't the Gregorian calendar as precise as it could have been, with regards to its designation of leap years? Did Pope Gregory XIII just get lazy and think that three rules (every 400 inclusive-or (every 4, but not every 100) years are leap years) were already far too complicated of an algorithm for the average person to handle? Was there some sort of a Christian taboo on writing numbers with more than 4 decimal places? Or perhaps did he think, "We're not really sure if 365.2425463 days in a year is accurate anyway, so we may as well truncate it to 365.2425. Just to be safe."?

• What problem in that era would have been solved by greater precision? What would be the point if investing the extra effort? Back of the envelope, adding one more digit would allow you to be 4 seconds more precise. How many relevant tasks were measured at 4 second precision?
– MCW
Mar 3, 2021 at 12:50
• I may turn this into a proper answer. Three things to remember: 1) At the time, accuracies within a few days were "good enough". 2) Church doctrine was that the world was going to end at some point, probably in a few thousand years at most. 3) Having a correction that was easy for the average person to calculate was important. Mar 3, 2021 at 17:30
• Remember that at the time all computing was done by hand. Now more precision just means typing a couple more of digits, but done by hand it makes calculations harder. Also, a final digit '5' means that when multiplying either the final decimal is either '0' or '5'. Mar 3, 2021 at 17:47
• @nicoty most of those uses were invented within living memory; the pace of change in timekeeping and its applications has recently been very fast. And yet we still had to put in a lot of work to update systems to avoid them breaking in Y2K because of a lack of future-proofing on a timescale of a couple of decades. Calendar drift is a smaller problem than rollover, and wouldn't have been an issue on a timescale of several lifetimes (people might notice the solstices shifting by several days, at most) Mar 4, 2021 at 10:54
• @ChrisH: "Calendar drift … wouldn't have been an issue on a timescale of several lifetimes" Pre-Gregorian calendar drift was already that subtle; the Julian calendar was only drifting by three days out of every four hundred years after all. You'd never notice the drift during a given handful of lifetimes. Without documentation telling us when the solstice was supposed to be, no one would have noticed for several millennia (until we eventually noticed that the seasons themselves were wildly out of place). Mar 4, 2021 at 14:32

Why? Because there was no point.

First, according to more modern astronomical measurements, the current length of the year is closer to about 365.2422 days, so they would've been relatively less accurate had they used a more precise value of 365.2425463 days per year.

Which leads to a very important point about math: you need to be very mindful about how much precision you actually have in your measurements.

Also, let's look at what that very modern document from NASA says:

Before contemplating further corrections to the Gregorian Calendar we must consider how exact the value of 365.2422 is. The length of the average tropical year is now more precisely 365.24219 days but it varies somewhat from year to year and does not track the seasons precisely. Also, because of tiny orbital effects the average tropical year varies by about .00005 days per 1,000 years. Thus correcting any error of this magnitude is probably a waste of time.

That is not a new attitude. Most of the rest of this answer is based on this document. In it, we read about Copernicus:

Copernicus did not believe it was possible to have a perfect calendar, as the solar year was too variable.

So even back then, there was a belief that the calendar would drift in a variable way making too detailed corrections pointless.

Now look at the some measurements that were taken in that era from page 19:

• 1252 Alfonsine 365.24254630
• 1543 Copernicus 365.24269676
• 1551 Prutenic 365.24719907
• 1574-75 Ignazio Danti 365.24166667

We can see pretty clearly that there wasn't a lot of agreement beyond a couple decimal points. As such, it is easy to see why someone might not bother to pay attention to that 0.0000630. They would see it as not a true reflection of reality, just an artifact of the imprecise math and in modern terms, well within the error bars.

It appears that the person ultimately responsible for the calendar was one Aloysius Lilius. From page 20, we see he came up with:

365 +1/4 – 1/100 + 1/400 + 4/100,000

which corresponds to the errors:

• minus 1 day every 4 years;
• plus 1 day every 100 years;
• minus 1 day every 400 years;
• minus 4 days every 100,000 years (that means minus 1 day every 25,000 years).

This was then the basis of the calendar with the last part dropped. We can easily understand why, though. It would not require any change from the Gregorian calendar in another 23,418 years!

At the time, the general view of the age of the Earth was in the thousands of years. In fact, the Alfonsine tables referenced in the question put it at 6984 BC. In addition, the general Catholic belief in the second coming of Christ gave a general expectation that there was an end date, and that it was at most hundreds or thousands of years away. If your worldview has the Earth lasting on the order of 10,000 years, why worry about 25,000?

So in summary:

1. Their measurements weren't good enough to get that kind of precision required for a more accurate calendar
2. They had reason to believe that the year was variable enough to make more precision impossible
3. If the "best guess" was right, it would be trivial to fix on the year 25,000
4. They had good reason (in their view) that the year 25,000 would not happen
• "The Rapture" had nothing to do with it -- you had a great answer going before you wandered off in that direction! Mar 3, 2021 at 22:11
• Yes, I was going to suggest replacing "The Rapture" with "The Second Coming of Christ" which is a very different thing, but believed by basically all Christian denominations (as opposed to the Rapture), and still works for the point you are making. Mar 3, 2021 at 22:32
• I don't think Catholicism has any formal opinion about when the end of the world is likely to happen. Your final point needs evidence. Mar 3, 2021 at 23:09
• The revisions have improved your answer! One more comment: You say, "If your worldview has the Earth lasting on the order of 10,000 years, why worry about 25,000?" If you are living in a Medieval world with comparatively short lifespans et al, why worry about even a thousand years? Yet they did. (For that matter, why should you or I care about creating a calendar accurate 25,000 years into the future?) Mar 4, 2021 at 1:17
• I doubt it's a matter of thinking the world would never reach the year 25,000. Rather that's so far away, further away than recorded history, we have plenty of time to tweak the system. Speaking as a programmer who worked through y2k, that is a perfectly reasonable amount of "good enough". I cannot tell you how much code didn't include the 400 year rule; the lesser known y2k bug. Mar 4, 2021 at 5:03

The Gregorian Calendar was introduced (to the Catholic World) in 1582, the result of preparation over the preceding five or so years.

However the popularization of decimal fractions would wait another three years until the publication of La Thiende [The Tenth] in 1585 by the Flemish mathematician Simon Stevin. Though not the inventor of a decimal representation of fractions, the publication in 1585 of both La Thiende and La Disme [The Decimal]) popularized them and explained their use.

An important consideration in the preparation and adoption of the Gregorian Calendar was that it be easily understood by those not of great mathematical sophistication. Using a description reliant on an unfamiliar mathematical notation would not have aided the cause.

Note that our entire concept of "number" has changed dramatically over recorded history. A striking observation from Euclid's Elements is how the portion dealing with what we now term Number Theory is discussed entirely in terms of "length of a line segment or arc". That was "number" to Euclid and contemporaries.

The very acceptance (in Europe) of negative numbers dates only to the 13th century (and may comingle with the simultaneous development of double-entry bookkeeping) - so is barely 300 years old at the time of the Gregorian calendar reforms. One must take great care in interpreting historical concept of "number" to not overlay our modern understanding and interpretation.

• I don't think a relevant consideration at the time was that the calendar be easily understood. I would think even nowadays that only a minority can describe any aspect of the Gregorian calendar without reference to a calendar, and at the time of its introduction, you needed a licence from the Pope himself to print such a calendar! Mar 3, 2021 at 21:10
• I think the point here is that the people who made the calendar would not be thinking in terms of decimal numbers. Also, I would expect the people making the calendar would want priests to be able to understand it, and priests likely had little mathematical training Mar 4, 2021 at 0:48
• The distinction between decimals and fractions is a good consideration, but this answer is lacking without some discussion about what fractional framing would have meant for the accuracy question. For example, they might not have expressed it as 365.2425463, but if they could calculate it as 365 and 303183/1250000 (or whatever), they could certainly have seen that a calendar with an average of 365 and 97/400 doesn't match, and the core of the question remains.
– R.M.
Mar 4, 2021 at 20:20
• @Steve: The exclusive calendar license issued on 3 April 1582 to Antoni Lilio was revoked barely 5 months later, on 20 September 1582, due to his inability to keep up with demand. Dec 21, 2022 at 2:09
• @PieterGeerkens, do you mean it was changed from an exclusive licence to just one of multiple licences issued, or do you mean the licensing system was abolished completely? Dec 21, 2022 at 10:45

The exercise of reforming a calendar was intended to make a calendar more accurate, but was not intended to eliminate the need for any theoretical future adjustment.

The Gregorian reform was inspired by the Julian reform, and the Julian reform of the Roman calendar was merely the last of many "consular" modifications of the calendar. Under the Roman calendar system, it was known and accepted that imperfections in the length of a year would accumulate over time and need to be corrected for; the Julian reform was meant to make those corrections necessary much less frequently, but did not pretend that it would make them unnecessary forever.

The Gregorian reform was a further improvement - but if it too was imperfect, and would require someone far in the future to add an intercalary day to a year to bring the calendar back into alignment with the true year...well, that was some future Pope's problem.

The other answers have touched some very good points, but there is a mathematical point I would like to make as well: The way the leap year rules are designed is fundamentally incompatible with the decimal expansion. Specifically, rules of the kind "leap/no leap year every X years" don't really care about the number of digits, they are more of a form of continued fractions.

In particular the fact that the decimal expansion has only finitely many digits is both a happy accident in the first place and an unintended consequence of a deliberate choice made afterwards:

1. The error made when setting the length of a year to 365 days is very close to 1/4 of a day, so for the Julian calendar the choice was made to include a leap year every four years, leading to 365.25 days per year on average. If it had been close to 1/3 instead, they would have chosen a leap year every three years, leading to 365.333333333...., an expansion with infinitely many digits. The same for any other factor that has prime divisors other than 2 or 5. In fact you can have a look at the Hebrew calendar, which is based on a complicated 19 year cycle for a real world example of this.

2. When the Julian calendar then was updated to the Gregorian calendar, a deliberate choice was made to keep the rules simple to calculate for a given year. Dropping a leap year every 132 years, would have had almost the same effect as the current rule of dropping one every 100 and adding 1 every 400, with an average day of 365.2424242424...., again with infinitely many digits. However such a rule would have made it incredible tedious to calculate if a given year is a leap year or not, while the current rule only involves calculations that can be made mentally within seconds. But those calculations again are easy because the numbers involved only have prime factors 2 and 5, which again result in a finite number of digits. (Another happy accident here by the way, something like 1 in 300 would have lead to infinitely many digits again.)

• This touches on the concept of (so-called) Egyptian fractions, by which any rational number can be expressed as a sum of unit fractions (ie ones where the denominator is constrained to be 1). Hence 1/4 - 1/100 + 1/400 - 1/4000. With the further constraint that each denominator have only 2 and 5 as prime factors, one ensures an exact decimal representation. Jul 31, 2023 at 17:50

This question was answered in a 1986 paper by the late professor Noel Swerdlow [1].

Briefly, the 1577 proposal for calendar reform was based on a lost book by then-already-dead astronomer Aloysius Lilius (c. 1510 – 1576), in which he proposed the intercalation of 97 days in 400 years. Moreover, Lilius was not the first to propose that intercalation system, for it had already been proposed in 1560 by Petrus Pitatus of Verona. In his work, Pitatus stated that his proposal was consistent with the tropical year of the Alphonsine tables and the mean tropical year of Copernicus' "De revolutionibus" and Reinhold's "Prutenic tables". How could that consistency be achieved? Simply by expressing the fractional part of those values in sexagesimal notation and rounding it to seconds (i.e. 1/3600) of a day:

``````Alphonsine tables: 365;14,33,9,57

De revolutionibus: 365;14,33,12,54

Prutenic tables..: 365;14,33,9,24

Rounded to 1/3600: 365;14,33
``````

From that:

``````14 x 60 + 33    873    97
------------ = ---- = ---
3600       3600   400
``````

Notably, the 1577 proposal to the Pope contained two possible intercalation schemes: the 97/400 and a variable one, in which the decision on whether the first year of each coming century would be leap or regular would be taken on the go as that century approaches. The latter was rejected on the grounds that it was both astronomically uncertain and superfluous to the calendar in any case.

Therefore the adoption of a 97/400 intercalation system was quite reasonable, for the purpose of accuracy, on the basis of the current state of knowledge of Western astronomy, and actually more accurate than the precise value of the mean length of the tropical year in that Western state of knowledge.

In contrast, the 8/33 intercalation system proposed independently by the Syrian ex-patriarch Ignatius N'amat allah who stood for the oriental church in Pope Gregory's calendar commission [2] and by John Dee in England, while closer to the real mean length of the vernal equinoctial year, was not supported by contemporary Western science but by the work of Omar Khayam.

Moreover, replacing the Julian calendar by the Gregorian and not by a radically different one was most logical in view of its foremost purpose, which was not civil but ecclesiastical: the determination of the date of Easter. Since the Church had an elaborate system for that purpose (the Paschal “computus”) which was based on the Julian calendar and lunar tables, it made perfect sense to adopt a calendar that was as close as possible to the Julian. Quoting [1]:

The Gregorian Calendar is not really a new calendar [...] but a number of small “corrections” (aequationes) to the julian calendar and to the Nicaean Ecclesiastical Calendar, constructed on the julian calendar and already in use for over a thousand years at the time of the calendar reform. The object of the reform was (a) to correct errors that had accumulated since the Council of Nicaea and (b) to prevent errors from accumulating again in the future, while altering as little as possible the Nicaean Ecclesiastical Calendar and the method of determining the date of Easter (*)

(*) Note: I object to calling the Ecclesiastical Calendar "Nicaean", as all that Council said was that all churches should celebrate Easter on the same date as the Romans and the Alexandrians, without providing any rules to determine that date.

References

[1] Swerdlow, Noel M. “The length of the year in the original proposal for the Gregorian calendar.” Journal for the History of Astronomy, Vol. 17, no 2 (1986): 109-118. https://legacy.adsabs.harvard.edu/full/1986JHA....17..109S

[2] Bien, Reinhold. “Viète’s Controversy with Clavius Over the Truly Gregorian Calendar.” Archive for History of Exact Sciences Vol 61, no. 1 (2007): 39–66. http://www.jstor.org/stable/41134239.