(no subject)

[ewx]

After seeing this, I spent some time looking for web pages concerning the history of the degree as 1/360 of a circle. The most widespread answer seems to be that the Babylonians invented it, full stop. But other pages suggest that the notion of a 360-day year goes back to the Sumerians, with Greek mathematicians inventing the concept of a degree for measuring angles and Seleucid mathematicians (i.e. Mesopatamia after the Greek conquest) coming up with the further subdivison into arc-minutes and arc-seconds.

(Specifically, they called arc-minutes "the first part" and arc-seconds "the second part"; and these were translated into Latin as pars minuta prima and pars minuta secunda, hence the modern English names. The word "degree" is borrowed from Latin de gradus which allegedly derives [but see comments below] from the Arabic daraja, the Greeks using an entirely different word μοιρα.)

David Ewing Duncan, in his book The Calendar, also credits the 360-day year to the Sumerians, with the Babylonians inheriting it from them. It also notes that the Jewish calendar required months to be intercalated from time to time; and that the Egyptians had a 365-day year around 4000BCE, and even noticed their quarter-day error, but didn't incorporate it into their calendar until forced to by Augustus Caesar. This tremendous time-span with a broken calendar is long enough for any given day to have drifted through the full year twice.

In passing I also came across a rather wacky web page based on a mention in the Bible of a sundial doing something rather odd: going backwards by 10 steps. The author thinks this corresponded to a change in the Earth's orbit which changed the year from 360 days to its current length! (Never mind that other counters already had a 365-day year millenia previously or that people routinely inserted extra days and even months to make their calendars fit the solar year back then.)

Anyway. None of this gets us very close to the origin of the degree. Everything I can find just states that "the Babylonians" invented the 360-degree circle, without referring to any evidence or even usually clearly distinguishing between the Seleucids or Hammurabi's rather more ancient Babylon (though I'd normally interpret it as meaning the latter, absent any qualification). The 1936 Susa tablet, dating from around 2000BCE, relates the perimeter of a hexagon to that of a circumscribed circle, which might well have involved a numerical concept of angle, but nobody seems to report the workings.

Most of the most promising-looking references seem to go back to David Eugene Smiths's History Of Mathematics. So, I've ordered a copy of that (it looks interesting anyway, I'm not just buying it for the answer to this one question!)

My cold is somewhat better than yesterday, and I seem to have slept a bit better. Hopefuly back to work tomorrow.

Comments

Calendar geekery

[lethargic-man.livejournal.com]

The word "degree" is borrowed from Latin de gradus which allegedly derives from the Arabic daraja, the Greeks using an entirely different word μοιρα.

<geek type="linguistics">Not so. De gradus means "about the step", and comes from Latin de plus the past participle of gradior. I don't have my etymological dictionary here at work, but the Net of a Thousand Lies says (http://www.valtozovilag.hu/ab/nap0328.htm):

The Latin root derives from Proto-Indo-European ghredh- "to walk, step, move," also the source of Sanscrit kra-, kram "to go" and German schreiten "to step" and Schritt "step." The Latin root is also found in English congress (come together), ingredient (what goes in), transgress (to step across the line). We also have "grade," "gradual," and "graduate" from related Latin gradus "step," the noun from gradior.

Besides, Arabic is a Hamito-Semitic language, unrelated to the Indo-European Latin, and when gradior entered the Latin language, Latin was spoken only by the inhabitants of the Latium plain, a thousand miles away from Arabic speakers in the Arabian peninsula (though it is true that the (Semitic-speaking) Phoenicians could have acted as a bridge between the two.)</geek>

It also notes that the Jewish calendar required months to be intercalated from time to time; and that the Egyptians had a 365-day year around 4000BCE, and even noticed their quarter-day error, but didn't incorporate it into their calendar until forced to by Augustus Caesar. This tremendous time-span with a broken calendar is long enough for any given day to have drifted through the full year twice.

The Jewish calendar is a luni-solar one, i.e. its months are lunar months but its years roughly aligned with the solar year. This is in contradistinction to the solar Gregorian calendar, whose year is exactly aligned with the solar year but whose months bear no relation to the phase of the moon; and the Muslim calendar, with a year of twelve lunar months which precesses quite fast around the solar year. The Jewish calendar has been fixed by calculation, rather than by observation, since the time of Hillel the Younger (fourth century), and is more accurate than the Gregorian calendar, which it antedates by over a thousand years.

None of this gets us very close to the origin of the degree. Everything I can find just states that "the Babylonians" invented the 360-degree circle, without referring to any evidence

Ancient Iraq by Georges Roux would probably be able to answer your question, but I can't quote from it to you, as I've lent my copy to livredor. However, I do remember it presented evidence for the Mesopotamians, from the time of the Sumerians (3000 BCE) to the Persian conquest (C6 BCE), as being essentially a single civilisation; and added that they were good at "what?" science, but had no interest in "how?" or "why?".

Re: Calendar geekery

[ewx.livejournal.com]

I think that the source I was using there was claiming not that the word was borrowed, but rather that the use of a pre-existing word meaning 'step' to refer to a unit angle was copied from one to the other. Perhaps 'derives' is the wrong term to use to mean that.

[bellinghman.livejournal.com]

The Jewish Calendar (http://www.jewfaq.org/calendar.htm) is indeed strange for those used to the western calendar.

I suspect a lot of it (strange calendars in general) comes from trying to make regularity from the day/moon/year cycles. Since it's a lot easier to work out when you are in the lunar cycle than in the solar cycle, you end up with the concept of the 7 day week (7 days being the closest you get to the new-half-full-half-new moon intervals), then collecting 4 weeks into a month with an optional day or two to resync, and then 12 or 13 into a year.

A relatively close match to the year is the 7 day week, 4 week month, 13 month year with an extra day, and a leap day every four years. But that causes the month to slip against the moon, and that may be a problem.

Re: Calendar geekery

[vyvyan.livejournal.com]

Ah! I was going to comment about that in the same way. "Derivation" does imply to me (in a historical linguistic context) either genetic-style inheritance or perhaps borrowing; combined with the superficial similarity of "de gradus" and "daraja" I also assumed that was what you meant. "Semantic influence" seems to be the intended meaning, then.

[teleute.livejournal.com]

i know I've read that the Babylonians had a counting system based on base-60. Which is why they chose 360. Quite why 6 * 60 is so exciting my book never made clear, perhaps they hit the same dead spot you've just found.

Re:

[ewx.livejournal.com]

The Susa tablet seems to be reasonable evidence for the base-60 thing, from the descriptions of it I've seen, and even points in the direction of 6*60 since it concerns the relation between a circle and an inscribed hexagon. What I'm after is the evidence for who made the next step - a general concept of angle, measured in terms of 360ths of a full rotation - and when they did so.

Re:

[teleute.livejournal.com]

Have you found in your travels anything about who started the radian thing? and when and why? I'm vaguely interested in that...

[ewx.livejournal.com]

Why seems obvious given (i) the relationship between radius and circumference (and that looks suspiciously like it motivates the name, too) and (ii) the way they make derivatives of trig functions sensible.

This article (http://members.aol.com/jeff570/r.html) dates it to the early 1870s, with several mathematicians working on the concept about the same time, using various similar names, and James Thomson (Lord Kelvin's brother) being the first to use "radian" in print. The article in which he used it would be the next interesting step on this particular road...