14 ms·

Full Circle: The complicated history of why there are 360 degrees in a circle

135 points by Hooke 7y ago

blauditore 7y ago
I always thought this was somehow motivated by great divisibility, since prime factors of 360 are 2, 2, 2, 3, 3, 5. This means you can work with integers most of the time when dividing a circle, since all but four numbers in 1-15 are divisors.
- roelschroeven 7y ago
  That large number of divisors was the reason for the Babylonians to use base-60, I've always been thaught. I always assumed the the 360 degree circle was a consequence of that, but this article contradicts that. Actually I don't think the article does a great job proving its thesis ("In school we learn there are 360 degrees in a circle, but where did the 360 come from? When it is pointed out that the Babylonians counted to base-60, rather than base-10 as we do, people often ask if there is a connection. The short answer is no."). The article's conclusion is "So, although angles come from the Greeks, the 360 degrees comes from Babylonian astronomy". And it explains, more-or-less, how the Babylonians came to use 360 degrees for astronomy: because they had a habit of dividing things into 12 part, subdivided in 30. But that doesn't preclude they choose 12 and 30 (instead of, say, 13 and 28) because both are divisors of their base number 60. Wikipedia points in that direction (but admits no-one really knows): "Some ancient calendars, such as the Persian calendar, used 360 days for a year.[citation needed] The use of a calendar with 360 days may be related to the use of sexagesimal numbers." "Another theory is that the Babylonians subdivided the circle using the angle of an equilateral triangle as the basic unit and further subdivided the latter into 60 parts following their sexagesimal numeric system.[7][8] The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle. A chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard sexagesimal divisions, was a degree." Another Wikipedia-quote agrees with your point of view: "Another motivation for choosing the number 360 may have been that it is readily divisible: 360 has 24 divisors,[note 1] making it one of only 7 numbers such that no number less than twice as much has more divisors" "
  - herodotus 7y ago
    I think 30 comes from the Synodic month - the time from first crescent to first crescent of the moon, which varies between 29.18 and 29.93 days. There of 12 of these months within a year (and a few days left over).
  - twodave 7y ago
    I think it more likely that both the 360-day year and the base-60 numbering system came from a common ancestor (the Babylonian's study of astronomy). Perhaps they just took the lunar year and "rounded" it to correspond more consistently with their view of the stars, which put them on the path to both things.
    - simiones 7y ago
      The (approximately) 360-day year is a property of nature (the ratio between the Earth's revolution speed and the Earth's rotation speed is approximately 360 if expressed in base 10). It is a discovered number, not invented/chosen. It is possible that the base-60 numbering comes from this basic astronomical observation, or they may be mostly unrelated.
  - im3w1l 7y ago
    An arc-minute is a 60th of a degree. And an arc-second is a 60th of an arc-minute. This strongly points to a base-60 origin imo.
    - gumby 7y ago
      The minute ("small part") and second ("even smaller part") come from the era of clockmaking as they were pointless when your timekeeping resolution was crude.* We are quite lucky that hours and then minutes were divided into 60 sections rather than 10, 12, or 100. This may have come from the system you describe (arc minutes, seconds, thirds, etc) which is much older. * or not just crude but completely different; e.g. Babylonian and Roman hours varied in length each day as the time from sunrise to sunset was divided into twelve equal sections and sunset to sunrise implicitly the same. IIRC the constant-duration hour goes back only to marine navigation a few hundred years ago
- pkulak 7y ago
  Yeah, I think I'm going to refuse to believe that it's just some coincidence that 360 is a highly-composite number AND a specially chosen number. Especially with how societies have made use of all the others below it throughout history (12, 24, 60, etc). Extraordinary claims require extraordinary evidence, and all.
Koshkin 7y ago
Theories abound: Babylonian math has roots in the numeric system started by the Sumerians, a culture that began about 4000 BCE in Mesopotamia, or southern Iraq, according to USA Today. “The most commonly accepted theory holds that two earlier peoples merged and formed the Sumerians,” USA Today reported. “Supposedly, one group based their number system on 5 and the other on 12. When the two groups traded together, they evolved a system based on 60 so both could understand it.” https://www.thoughtco.com/why-we-still-use-babylonian-mathematics-116679 https://www.thoughtco.com/why-we-still-use-babylonian-mathem...
deleted 7y ago
[deleted]
donquichotte 7y ago
Ah, measures of angle. One of my favourite is the Warsaw Pact Millirad, of which there are 6000 in a full circle, a full 6.25% less than the 6400 NATO mils.
amelius 7y ago
But why are there 2*pi radians in a circle, instead of defining it as pi? EDIT: I meant redefining pi instead of the radian.
- leetcrew 7y ago
  because then the formula for area would be (pi/2)r^2?
  - SamBam 7y ago
    Indeed, that's an advantage. If you use tau=2pi, you'd then you'd see how the area = 1/2 τr^2 is actually conceptually similar to KE = 1/2 mv^2 (and similar for PE, distance fallen in gravity, etc. That 1/2 is actually important.)
    - deleted 7y ago
      [deleted]
- Ono-Sendai 7y ago
  You mean pi should have been defined as the ratio of the circumference of a circle to the radius, instead of to the diameter? Probably, yeah :) Better yet, I think the unit of angle we should use is the 'full rotation' ('rot'?) So instead of 360 degrees, or 2pi radians, we say 1 rot. The many factors of 2pi in common equations imply we got this wrong.
  - martincmartin 7y ago
    I think OP is suggesting that we change the definition of radian, not Pi.
    - Ono-Sendai 7y ago
      it's a bit ambiguous. Of course the radian has that definition for a particular reason, 1 radian being the angle such that the arc length equals the radius.
LarryDarrell 7y ago
So time is a flat circle then?
- greggyb 7y ago
  More of a Jeremy Bearimy sort of shape.
playworker 7y ago
My Dad was once gifted a Silva compass which he gave to me to take on a Boy Scout expedition, everyone was baffled by the fact it had 400 "degrees" marked on it. Maybe we need to go metric with our circles too: https://en.wikipedia.org/wiki/Gradian https://en.wikipedia.org/wiki/Gradian
- fanf2 7y ago
  The nautical mile is the length of an arc minute on the equator, the circumference of the earth divided by 360 * 60 The kilometre is the length of a centigrad of arc of the meridian through Paris, the circumference of the earth divided by 4 * 100 * 100
- sllabres 7y ago
  Surveyors seems to use Gradian/gon often. There is still another circle measurement, the mil which is 6400 for one turn. But I just learned by reading the wikipedia page, that there is not only the 6400 NATO mils I knew, but also 6000 Warsaw Pact mils or 6300 Swedish "streck"
  - BurningFrog 7y ago
    Swedish "streck" = (short) line.
adelHBN 7y ago
This was awesome. Thanks for sharing it. Amazing how much has come from the Fertile Crescent.
rendall 7y ago
I also note that History Today, as an English company, is no longer GDPR compliant. "Take this tracking or get out. Honey badger don't care"
kens 7y ago
One circle unit that strikes me as strange is the "mils" used for military targeting: there are 6400 mils in a circle. The benefit of this is that 1 meter wide at a distance of 1 kilometer corresponds to 1 mil. So if your tank misses the target by 5 meters and the target is 1 km away, you adjust the angle by 5 mils, nice and easy to compute. A bit of math shows that this unit is a milliradian (since sin 1/1000 ≈ 1/1000). The strange part is that there should be 6283.18... mils in a circle but that's too inconvenient, so they round it to 6400. In other words, they are using milliradians with the value of 3.2 for π.
- dresstotheleft 7y ago
  And that is just the NATO convention. There are several others in use across the globe - https://en.wikipedia.org/wiki/Milliradian#Definitions_for_maps_and_artillery https://en.wikipedia.org/wiki/Milliradian#Definitions_for_ma...
- BurningFrog 7y ago
  Sounds like a reasonable battlefield optimization.
everyone 7y ago
I'm confused by their description of the ecliptic.. " For them, Venus was a single object and they observed its changing position, along with the other planets and the moon. These positions all lie on the same great circle, called the ecliptic, defined by the apparent motion of the sun as seen from the earth during the course of a year. " Isnt the ecliptic basically a line (or very very skinny ellipse) from Earths perspective as we are on that great disc ourselves? Also "For them, Venus was a single object" so for us its not??
- T-hawk 7y ago
  Venus: some ancient cultures recognized its "evening star" and "morning star" appearances as the same object; some thought they were two different phenomena. The ecliptic, physically, is the plane of Earth's orbit around the Sun. It is so named because that is where eclipses happen, when the Moon crosses that plane. (Or other planets; we call that event a transit, but that's the same as a very annular eclipse.) The projection of that plane as viewed on the sky from Earth is a great circle (or slightly offset, because the observer is on Earth's surface not at its center.) If the Earth were transparent, you could see the entire circle. In reality, you see a segment of it from horizon to horizon, as a line. All the planets and the moon seem to stay near the ecliptic in the sky, because their orbits are within a few degrees of coplanar with Earth's.
  - everyone 7y ago
    I'm realising I look at this differently to the article writer + u. I would say, the ecliptic is a great disc. But when your on a disc, from your perspective that disc looks like a straight line, not a circle...
adelHBN 7y ago
It's amazing that Babylonians did this without any modern equipments. Recently I learned that 2500 years ago Persians (the Persian New Year was last week) predicted the Spring Equinox with a certainty of 5 minutes. This is crazy. Very impressive. Sad how the Middle East lost its technological and scientific edge over the centuries.

135 points by Hooke 7y ago

I always thought this was somehow motivated by great divisibility, since prime factors of 360 are 2, 2, 2, 3, 3, 5. This means you can work with integers most of the time when dividing a circle, since all but four numbers in 1-15 are divisors.

roelschroeven 7y ago
That large number of divisors was the reason for the Babylonians to use base-60, I've always been thaught. I always assumed the the 360 degree circle was a consequence of that, but this article contradicts that. Actually I don't think the article does a great job proving its thesis ("In school we learn there are 360 degrees in a circle, but where did the 360 come from? When it is pointed out that the Babylonians counted to base-60, rather than base-10 as we do, people often ask if there is a connection. The short answer is no."). The article's conclusion is "So, although angles come from the Greeks, the 360 degrees comes from Babylonian astronomy". And it explains, more-or-less, how the Babylonians came to use 360 degrees for astronomy: because they had a habit of dividing things into 12 part, subdivided in 30. But that doesn't preclude they choose 12 and 30 (instead of, say, 13 and 28) because both are divisors of their base number 60. Wikipedia points in that direction (but admits no-one really knows): "Some ancient calendars, such as the Persian calendar, used 360 days for a year.[citation needed] The use of a calendar with 360 days may be related to the use of sexagesimal numbers." "Another theory is that the Babylonians subdivided the circle using the angle of an equilateral triangle as the basic unit and further subdivided the latter into 60 parts following their sexagesimal numeric system.[7][8] The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle. A chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard sexagesimal divisions, was a degree." Another Wikipedia-quote agrees with your point of view: "Another motivation for choosing the number 360 may have been that it is readily divisible: 360 has 24 divisors,[note 1] making it one of only 7 numbers such that no number less than twice as much has more divisors" "
- herodotus 7y ago
  I think 30 comes from the Synodic month - the time from first crescent to first crescent of the moon, which varies between 29.18 and 29.93 days. There of 12 of these months within a year (and a few days left over).
- twodave 7y ago
  I think it more likely that both the 360-day year and the base-60 numbering system came from a common ancestor (the Babylonian's study of astronomy). Perhaps they just took the lunar year and "rounded" it to correspond more consistently with their view of the stars, which put them on the path to both things.
  - simiones 7y ago
    The (approximately) 360-day year is a property of nature (the ratio between the Earth's revolution speed and the Earth's rotation speed is approximately 360 if expressed in base 10). It is a discovered number, not invented/chosen. It is possible that the base-60 numbering comes from this basic astronomical observation, or they may be mostly unrelated.
- im3w1l 7y ago
  An arc-minute is a 60th of a degree. And an arc-second is a 60th of an arc-minute. This strongly points to a base-60 origin imo.
  - gumby 7y ago
    The minute ("small part") and second ("even smaller part") come from the era of clockmaking as they were pointless when your timekeeping resolution was crude.* We are quite lucky that hours and then minutes were divided into 60 sections rather than 10, 12, or 100. This may have come from the system you describe (arc minutes, seconds, thirds, etc) which is much older. * or not just crude but completely different; e.g. Babylonian and Roman hours varied in length each day as the time from sunrise to sunset was divided into twelve equal sections and sunset to sunrise implicitly the same. IIRC the constant-duration hour goes back only to marine navigation a few hundred years ago
pkulak 7y ago
Yeah, I think I'm going to refuse to believe that it's just some coincidence that 360 is a highly-composite number AND a specially chosen number. Especially with how societies have made use of all the others below it throughout history (12, 24, 60, etc). Extraordinary claims require extraordinary evidence, and all.

Koshkin 7y ago

Theories abound: Babylonian math has roots in the numeric system started by the Sumerians, a culture that began about 4000 BCE in Mesopotamia, or southern Iraq, according to USA Today. “The most commonly accepted theory holds that two earlier peoples merged and formed the Sumerians,” USA Today reported. “Supposedly, one group based their number system on 5 and the other on 12. When the two groups traded together, they evolved a system based on 60 so both could understand it.” https://www.thoughtco.com/why-we-still-use-babylonian-mathematics-116679 https://www.thoughtco.com/why-we-still-use-babylonian-mathem...

deleted 7y ago

[deleted]

donquichotte 7y ago

Ah, measures of angle. One of my favourite is the Warsaw Pact Millirad, of which there are 6000 in a full circle, a full 6.25% less than the 6400 NATO mils.

amelius 7y ago

But why are there 2*pi radians in a circle, instead of defining it as pi? EDIT: I meant redefining pi instead of the radian.

leetcrew 7y ago
because then the formula for area would be (pi/2)r^2?
- SamBam 7y ago
  Indeed, that's an advantage. If you use tau=2pi, you'd then you'd see how the area = 1/2 τr^2 is actually conceptually similar to KE = 1/2 mv^2 (and similar for PE, distance fallen in gravity, etc. That 1/2 is actually important.)
  - deleted 7y ago
    [deleted]
Ono-Sendai 7y ago
You mean pi should have been defined as the ratio of the circumference of a circle to the radius, instead of to the diameter? Probably, yeah :) Better yet, I think the unit of angle we should use is the 'full rotation' ('rot'?) So instead of 360 degrees, or 2pi radians, we say 1 rot. The many factors of 2pi in common equations imply we got this wrong.
- martincmartin 7y ago
  I think OP is suggesting that we change the definition of radian, not Pi.
  - Ono-Sendai 7y ago
    it's a bit ambiguous. Of course the radian has that definition for a particular reason, 1 radian being the angle such that the arc length equals the radius.

LarryDarrell 7y ago

So time is a flat circle then?

greggyb 7y ago
More of a Jeremy Bearimy sort of shape.

playworker 7y ago

My Dad was once gifted a Silva compass which he gave to me to take on a Boy Scout expedition, everyone was baffled by the fact it had 400 "degrees" marked on it. Maybe we need to go metric with our circles too: https://en.wikipedia.org/wiki/Gradian https://en.wikipedia.org/wiki/Gradian

fanf2 7y ago
The nautical mile is the length of an arc minute on the equator, the circumference of the earth divided by 360 * 60 The kilometre is the length of a centigrad of arc of the meridian through Paris, the circumference of the earth divided by 4 * 100 * 100
sllabres 7y ago
Surveyors seems to use Gradian/gon often. There is still another circle measurement, the mil which is 6400 for one turn. But I just learned by reading the wikipedia page, that there is not only the 6400 NATO mils I knew, but also 6000 Warsaw Pact mils or 6300 Swedish "streck"
- BurningFrog 7y ago
  Swedish "streck" = (short) line.

adelHBN 7y ago

This was awesome. Thanks for sharing it. Amazing how much has come from the Fertile Crescent.

rendall 7y ago

I also note that History Today, as an English company, is no longer GDPR compliant. "Take this tracking or get out. Honey badger don't care"

kens 7y ago

One circle unit that strikes me as strange is the "mils" used for military targeting: there are 6400 mils in a circle. The benefit of this is that 1 meter wide at a distance of 1 kilometer corresponds to 1 mil. So if your tank misses the target by 5 meters and the target is 1 km away, you adjust the angle by 5 mils, nice and easy to compute. A bit of math shows that this unit is a milliradian (since sin 1/1000 ≈ 1/1000). The strange part is that there should be 6283.18... mils in a circle but that's too inconvenient, so they round it to 6400. In other words, they are using milliradians with the value of 3.2 for π.

dresstotheleft 7y ago
And that is just the NATO convention. There are several others in use across the globe - https://en.wikipedia.org/wiki/Milliradian#Definitions_for_maps_and_artillery https://en.wikipedia.org/wiki/Milliradian#Definitions_for_ma...
BurningFrog 7y ago
Sounds like a reasonable battlefield optimization.

everyone 7y ago

I'm confused by their description of the ecliptic.. " For them, Venus was a single object and they observed its changing position, along with the other planets and the moon. These positions all lie on the same great circle, called the ecliptic, defined by the apparent motion of the sun as seen from the earth during the course of a year. " Isnt the ecliptic basically a line (or very very skinny ellipse) from Earths perspective as we are on that great disc ourselves? Also "For them, Venus was a single object" so for us its not??

T-hawk 7y ago
Venus: some ancient cultures recognized its "evening star" and "morning star" appearances as the same object; some thought they were two different phenomena. The ecliptic, physically, is the plane of Earth's orbit around the Sun. It is so named because that is where eclipses happen, when the Moon crosses that plane. (Or other planets; we call that event a transit, but that's the same as a very annular eclipse.) The projection of that plane as viewed on the sky from Earth is a great circle (or slightly offset, because the observer is on Earth's surface not at its center.) If the Earth were transparent, you could see the entire circle. In reality, you see a segment of it from horizon to horizon, as a line. All the planets and the moon seem to stay near the ecliptic in the sky, because their orbits are within a few degrees of coplanar with Earth's.
- everyone 7y ago
  I'm realising I look at this differently to the article writer + u. I would say, the ecliptic is a great disc. But when your on a disc, from your perspective that disc looks like a straight line, not a circle...

adelHBN 7y ago

It's amazing that Babylonians did this without any modern equipments. Recently I learned that 2500 years ago Persians (the Persian New Year was last week) predicted the Spring Equinox with a certainty of 5 minutes. This is crazy. Very impressive. Sad how the Middle East lost its technological and scientific edge over the centuries.