# How did the ancient cultures determine that the year was actually a fraction of an extra day beyond 365 days?

Google says the year is exactly 365.2422 days, and so they make a leap year every 4th year, but that ends up being 365.25 days per year on average. So every 100th year they don't have a leap year, but on the 400th year they do have a leap year. And it can get more refined than that, such as skipping a leap year every 4,000 years, making it accurate up to 20k years. Could ancient people pre-civilization have known this level of detail?

I also saw this, but didn't answer the question: To several decimal places, how many days are in one year?

With my little knowledge of astronomy or history, I asked ChatGPT how they did it, and it said by carefully observing:

1. Sunrises
2. Sunsets
3. Equinoxes: The days when the day and night are approximately equal, which happens when the sun rises exactly in the east and sets exactly in the west.
4. Solstices: The days when the sun reaches its northernmost or southernmost extremes (longest and shortest days).

How do you observe those, and come to the conclusion of 365.25 days (or more accurate, 365.2422 days)? How do those 4 events (sunrises/sunsets/equinoxes/solstices) change or how does the sun position slightly differ from year to year to tell there is an extra 1/4 of a day? How accurate can you get with no tools, or just primitive tools (origin of civilization or before)? What exactly do you see with the sun (and these 4 types of events) that you can keep track of to make this detailed calculation?

• As to timing, Dante (c.1310) already talks of the year not being exactly 365¼: “the hundredth of a day which men neglect”. Paradiso, xxvii, 142. Commented Apr 28 at 19:25
• That tropical year length of 365.2422 days is an average approximation. The actual time from one March equinox to the next varies in a fairly complicated way. i.sstatic.net/4CHtg.png Commented Apr 28 at 20:36
• "How did the ancient cultures determine that the year was actually a fraction of an extra day beyond 365 days?" --> 1: demarcate with markers the winter solstice (for northern hemisphere people, notice the most southerly sunrise or sunset. Somewhat easy - not a year-round activity. After doing this for 10, 100, ... years (10, 100, someone is going to ask how many days between solstices? After 10 years of counting (assuming our astronomers are fed well enough for such an activity), ones going to get values of 365 and sometimes 366. More annual counts, the mean value will get more precise. Commented Apr 29 at 13:03
• I'm going to doubt all of the answers below relying on shadows. The Sun is 1/2 degree in apparent size, and does not cast a fine shadow. The answer is more likely realet to the answer linked below about how the equinox was found in ancient times. astronomy.stackexchange.com/questions/49816/… Commented Apr 29 at 16:02
• Just a reminder that relying on an AI for an answer is even sketchier than wikipedia - It might be correct, but it might also have just pulled from incorrect internet sources. Or forgotten how to count. Commented Apr 29 at 18:57

This is not so hard to do. All you really need is a bit of care, a few sticks and a convenient place to observe the sunrise or sunset.

So note that there is a cycle of the year and you want to know more. You know that if you stand next to the big rock by the village, then in winter the sun rises in the direction of the old oak tree on the hill and in summer it rises in the direction of the other village that we fight against (in modern language these directions would be called "south-east" and "north-east". You decide to mark the exact direction that the sun rises each day so on the first day you stand by the big rock and your assistant places a stick in the ground in the direction of the sunrise. The next day you do the same and you are please to note that the sun rise is in a slightly different position.

Repeating this over the year you discover that after 365 days the position of the sun rise is almost back to the start... but not quite... after 730 days it's out by a bit more... but after 1461 days the sun rises in exactly the the same place as on the first day!

Conclusion, it takes 1461/4 = 365¼ days for the sun to go through a full cycle.

The sunrise direction cycle is just one annual cycle that can be easily observed, but not the only one. Of particular importance in Egypt (for example) was the first day on which the star Sirius was visible, when it rose just before the sun. This happens annually on the 15th of July (it varies from place to place) and was considered to be a harbinger of the Nile floods.

• This sounds reasonable but my quick approximation gives be around 0.5° change per day, so if you want to measure the difference of a 1/4 day that effect is only 1/10 of a degree, well beyond what one can do with pen and paper today or what I believe feasible with sticks on the ground. Commented Apr 29 at 10:44
• @quarague you forget that it's way easier to do with sticks in the ground. Because even minor angle changes become relevant over larger distances Commented Apr 29 at 12:05
• Observing the annual cycle of which fixed stars are visible in the sky just after sunset or just before sunrise strikes me as much easier than tracking the bearing on which sunrise happens with sticks, so the last paragraph is probably the crucial bit. Commented Apr 29 at 13:08
• Legend has it that people in the past used to pile up big rocks in a circle to measure such things. Commented Apr 29 at 18:53
• All I can see if a cloudy day on day 1461 and having to wait 4 more years. Commented Apr 30 at 2:32

It's easier than you think to notice the fraction... simply because the rounding error compounds over the years! After a good amount of years, you notice that solstices are drifting from their predicted dates, hence a need for correction.

That's how the civilisations: Roman civilisation aka "Julian calendar", then Christian civilisation aka "Gregorian calendar" gradually refined the leap years.

It's perfectly explained here: https://www.britannica.com/story/ten-days-that-vanished-the-switch-to-the-gregorian-calendar

When the drift became more and more noticeable, the calendar was revised and the year's fraction was measured more precisely. The last adjustment in 1582 caused 10 days to be skipped to re-align the solstices.

• I think you meant 1582, not 1852... also interesting fact is that not everybody actually switched their calendars at the same time. Commented May 1 at 14:42
• @Michael thanks, fixed it! Commented May 1 at 23:13

The ancient Egyptians appeared to know that the year was 365.25 days. This was established by observing on what day the heliacal rising of Sirius occurred (i.e. on what day that it first becomes visible above the eastern horizon just before sunrise.

The Egyptians had a 365 day calendar, which "slipped" because of the 0.25 day discrepancy. That meant the heliacal rising of Sirius gradually occurred later and later in the year, coming back to its starting data at the end of the 1461 year Sothic cycle.

Schaefer (2000) considers a variety of uncertainties in judging the date of the heliacal rising of Sirius and concludes that from year-to-year the likely uncertainty would be something like $$\pm 2$$ days, due mainly to variations in atmospheric extinction. Therefore after about 10 years you would establish that the year was longer than 365 days, but it would take measurements over about 100 years to establish the extra fraction with 10% precision.

• IIRC, the "Egyptians appeared to know that the year was 365.25 days. " is more like "Egyptians appeared to know that the year was nearly 365.25 days. " as they had enough historical data, using " heliacal rising of Sirius" to know it was shy of 365.25 days. IOWs, I thought they had enough data to get to the 3rd digit of average days/years via this method. Commented Apr 29 at 12:51
• @chux Maybe. ;) According to en.wikipedia.org/wiki/Egyptian_calendar#Civil_calendar the Egyptians were using the 365 day civil calendar by the 25th century BC, if not earlier. So they'd observed at least one full Sothic cycle by the time of the (unsuccessful) Decree of Canopus by Ptolemy III in 238 BC, which proposed adding a leap day every 4 years. Commented Apr 29 at 13:32

For the most part, they didn't. Calendars have historically been a practical device. They were used to measure times of events, such as the planting and harvesting of crops. They only needed to be as accurate as that topic required. Adding days here and there was acceptable.

Fast forward a bunch to 1079, Khyayyam determined the length of a year to be 365.24219858156 days. Take that accuracy with a grain of salt,1 as the length of the year changes in the 6th digit within a lifetime, but he basically nailed the time you got from Google.

So what happened in between? A useful line in the sand shows up somewhere in the 2000-1000BC era, when the Persians used a solar calendar. Previous calendars were typically lunar (based on the moon's cycles) or lunisolar (based on the moon's cycles with additional days to bring it into alignment with the solar year). It's not until one uses a solar calendar that it's really important to know how many days are in a year.

This is not exactly an answer to your question. You ask how could a pre-civilization group compute the length of a year. Several answers suggest how it could be done. But practically speaking per-civilization groups are not believed to have done such calculations. It just wasn't as useful as it seems to our modern calendar-driven culture.

1. Actually, a whole salt lick block may be called for, not just a single grain. See the comments below which link to a History of Science and Mathematics answer which calls this whole claim into question. The paper linked therin claims there's no historical evidence for the cycles, and argues they were conceived in the mid 20th century. Thanks Stack Exchange, for teaching me a bit more about calendars!

• See here for more discussion about Khayyam's calculation and estimates of its precision Commented Apr 29 at 10:04
• "to be 365.24219858156 days" is more like "1,029,983 days made 2,820 years". Using a rounded 14-digit number of the quotient is certainly excessive. Why stop at 14? How about 365.24219858156028368794326241135...? IMO, assuming a +/-0.5 day error in the day sum, a days/year of 365.242 is a more reasonable quotient. Commented Apr 29 at 12:44
• @ChrisH Thank you for that link! I learned quite a lot from it! That just goes to show you what happens when your source is Wikipedia. I should have been more judicious in selecting my sources! Commented Apr 29 at 14:52
• @chux-ReinstateMonica And thank you too (Stack Exchange only lets me tag one person at a time. The above thanks goes to you as well!) Commented Apr 29 at 14:52
• Isn't this also why we have January and February, two months that didn't exist before (and December wasn't the 12th month but the 10th 'deca' month)? In the winter there was no calendar. Commented Apr 29 at 22:14

There is a good article in the Britannica. It seems the Sumerians used a 360-day year, and stuck an extra month in when the seasons got too far out of whack.

The present year is a leap year(sic). The coming month shall therefore be recorded a Elul II--Hammurabi, c.a. 1750BCE (cited in National Geographic Volume XCIC, January 1951.

The 365¼-day year (1 leap year ever 4 years, including end of century) dates from Caesar's calendar reforms of 46 BCE: the calendar was a bit of a mess, so the Caesar, as pontifex maximus, sorted it out.

The Ptolemaic Cosmology was published c.a. 150 BCE. Assuming that Zeus got the measurements right when the Universe was set up, it wouldn't be difficult to convince oneself that 365¼ days/year was exact.

• From en.wikipedia.org/wiki/Pontifex_maximus#Duties "Under his authority as pontifex maximus, Julius Caesar introduced the calendar reform that created the entirely solar Julian calendar, with a fault of less than a day per century". Commented Apr 29 at 5:49
• @PM2Ring You are probable right. The pontifex maximus was the appropriate authority. Caesar was dictator several times, but he may have been in between dictatorships when he reformed the calendar. I'll edit my answer accordingly. Commented Apr 29 at 7:31

How do you observe those

Sunrises and sunsets are pretty obvious. However, in order to accurately measure the length of day versus night, you need a type of clock that's not a sundial. A water clock is a simple way to achieve this.

Solstices can be determined by measuring the position of the sun (or rather, the tip of a shadow cast by a sundial, obelisk, or other structure) and noting the northernmost and southernmost points throughout the year. With enough knowledge, you can build structures that align with the equinoxes or solstices, such as Chichen Itza.

Or you can observe the position of stars and constellations throughout the year. As @ProfRob's answer has mentioned, the Egyptians measured the Sothic cycle by observing the star Sirius, which was considered to herald the annual flooding of the Nile.

and come to the conclusion of 365.25 days (or more accurate, 365.2422 days)?

Well, the ancient world's calendars weren't that accurate.

A solar year is between 365.2416 and 365.2427 days, depending on which astronomical event you use to define it.

• The average Julian calendar year is 365.25 days, which is in error by about one day every 130 years.
• The average Hebrew calendar year, defined as 235/19 months (Metonic cycle) of 765433/25920 days, which works out to an average year length of 35975351/98496 (≈ 365.246822) days, in error by about one day every 200 years.
• The Mayans didn't bother with leap years at all, defining their solar year as a whole 365 days. Perhaps aligning the calendar with the seasons wasn't important to them because their homeland doesn't really have seasons, being hot and humid year-round.

But as for how you arrive at an approximation, it's just a matter of keeping detailed observations over a long enough period of time.

For example, if one observer notes that the summer solstice of year X+50 occurs 18262 days later than in year X, then that works out to an average year length of 365.24 days. If you assume an error of up to 1 day in the 50-year difference, then you know that a year is between 365.22 and 365.26 days.

Precision can be improved simply by having more people make more observations over a longer period of time.

The Julian calendar had been in use for an impressive 1626 years at the time that Pope Gregory XIII decided to reform it. That was enough time to notice that the date of the spring equinox had drifted 12 or 13 days since Julius Caesar's day, necessitating a 0.007-0.008 day reduction in the average year length.

# Count the days. Many days.

You cannot expect high accuracy if you measure the thickness of a single sheet of paper with a ruler. You can improve it greatly if you stack 1000 sheets, measure the total height and divide it by 1000.

Patterns in star position will repeat after approximately 365 days, but they will repeat almost exactly after 4 revolutions in 1461 days, which gives $$\frac{1461}{4} = 365.25$$. With enough time and dedication, it's possible to improve the accuracy even further. You just need to count the days, possibly over multiple generations.

• The Babylonians made ~7 centuries of daily astronomical (and weather) observations from around 700 BC. en.wikipedia.org/wiki/Babylonian_astronomical_diaries So they had lots of data. That data was the basis of the astronomical tables of Hipparchus, which Ptolemy used in creating the formulae of celestial motions in the Almagest. I'm rather impressed that we still have a significant chunk of that ancient data. Commented Apr 30 at 20:18
• @PM2Ring: Thanks for the info. 7 centuries? That's really impressive. Even just recognizing 2 seasons per year would already lead to a very precise measurement of the length of a year after such a long time. Commented Apr 30 at 20:44

For long-term consistent time measurements there would usually be markings put on the walls of irrigation canals which would cast a sharp shadow against the opposing wall of the stone-walled canal or channel, and because the channels were kept clear and were immovable and also not easily accessible the markings would be left undisturbed and serve as a fixed sundial. Water levels were also tracked with markings on the side of the irrigation channels and the two measurements together also added some additional short term measurement of the length of the daylight hours, but being measured on the side of the canal was primarily because it would be undisturbed and was itself a natural sundial because edge of the canal would cast a shadow on the opposite wall.

When we think of a quarter of a day being spread out through an entire year and imagine trying to measure that it of course sounds impossible with a sundial, but if we imagine that on day one we got a perfect measurement of the start of the year precisely at sunrise, then on day 365 we would see that our sundial marking for sunrise on the first day of the year was off by 6 hours, which is a substantial amount of time compared to the day.

So you would take your 6 hour offset at the end of the year, and then do the math of:

(6 / 365) * 60

Six hours divided by 365 days is 0.016438 35616 hours per day offset, and we multiply by 60 to get 0.9863013699 minutes of offset or just under one minute, and there you can get an idea of how they would get high precision numbers from primitive measuring instruments.