degrees and circles

[LORDNELSON]

Why are there 360 degrees in a whole circle? My Grandson wants to know and I cannot think of a reasonable explanation. After all some surveyors use a 400 degree circle so how come most people use 360?

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[stephenh]

It is some 4,000 years since the Babylonians decided on 360 degrees to the circle ,24 hrs to the day and 60 seconds to the minute / hour.
I don't know why they did it but it has outlasted all attempts at change -
100 radians to the circle I am familiar with but 400 dgrees ? Are you sure ? - thats a new one and I am a surveyor !!!

regards
Stephen

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[ccscott49]

The russians used to use 400 degrees, not sure if they still do. But I`m also a downhole surveyor and have never heard of 400 degrees in surveying. Mils were used in the army for artillery gun laying, 500 and odd I seem to remember.

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[LORDNELSON]

Thank you for the Babylonian answer; it does not of course answer the question of why! With regard to the 400 degree circle, I am not sure, but think I came across it in Canada when doing some star shots for locating microwave towers (before the days of DGPS etc!). Needless to say I was using a 360 degree instrument.

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[ubuysa]

I've always believed it's because 360 is wholly divisible by more integers than any(?) other number. We use 12 inches in a foot for the same reason.

Tony C.
<hr width=100% size=1>There are 10 kinds of people, those who understand binary and those who don't.

 

[supermalc]

A quick search has come up with this answer....don't know how much is opinion or fact of course. Interesting question.


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The ancient greeks decided (probably based on experiments: try using a cord
to estimate the circumference of some cylinders) that the perimeter of a
circle (and other symmetric objects) must be a constant multiple of its
diameter. A little later attempts were made to estimate the value of the
constant using the geometry of inscribed polygons. Much later, when
symbolism became more popular in mathematics, "pi" (the first letter in the
greek word for perimeter) started to gain acceptance for representing this
famous constant. Full details on how "they" get it are too long to type
here. Perhaps the earliest estimate for PI was 3. I believe that Archime-
des gets credit for using geometry to improve the estimate to 22/7. The
greeks at that time loved pure fractions and used nothing like our many
decimal representations. Your calculator shows that 22/7 is about 3.142856
or 3.14 rounded to 2 decimal places. Records show that an even better
estimate of 355/113 = 3.1415929 was known to ancient chinese mathemati-
cians. Using computers, estimates for Pi are now correctly known to
millions of decimal places. Look up Wallis' formula for more unusual ways
to write fractions, etc. to Pi.

The origins of 360 degree units around a circle are probably even older. In
Babylonia the number base of 60 was useful for representing fractions
because 60 is divisible by 1 through 6. For example, 1/12 is able to be
written as 0.5 since 5/60 equals 1/12. Could it be that 360 was used to get
even more nice fractional parts of the circle in whole degrees. I will
leave the answer to that for the more learned in the subject of history of
mathematics. Morris Kline's books would be a good place to start.


<hr width=100% size=1>Malcolm.

 

[MainlySteam]

No one has hit the nub of the origin so here is a try without writing an encyclopedia. I had a look on the internet to see if I could find anything helpful to point you at, but not much at all that I could see. I am no expert but the story goes something like the following and springs entirely out of the measurement of time. So we have to start with the story of time a bit.

In Mesopotamian times (2-3000BC) the calendar was based on the moon and the sun with the month starting on the first evening that the moon's crescent came visible and this gave a month of approx 30 days, which they made 30 days (I think we know now that the phases of the moon repeat about every 29-1/2 days). They worked out this happened 12 times in a year (one cycle of the all seasons) so the year was divided into 12 months of 30 days which gave 360 days for the year. They found that every so often they had to chuck an extra month into the year to keep things all tidy (we know that it actually taking 365 and a bit days for the seasons to pass, not 360). At first they just plonked an extra month whenever they felt like it in the year when things were noticably adrift but from a couple of millennia BC they did this in most parts just at the 6th or 12th month. Sometimes they got so far behind that they had to throw in an extra month for a couple of years in a row, but by about 100-500BC they had worked the maths out enough to be able to put the extra months in in an ordered fashion - worked out to needing 7 extra months over each 19 years.

In parallel to all this the Egyptians, again around 2-3000BC developed a 360 day 12 month calender too, but they went on to include divisions in the day based on some complex matter to do with the stars, which is beyond me, which resulted in the division of the dark part of the night into 12 hours and later a 365 day year too. Somehow, I don't know how, they appear to have had shadow clocks way back which divided the day into 12 hours too about when they had the night similalrly divided up - anyway have 12 hour day and 12 hour night now. So we have this 12 number appearing again.

The 12 signs of the zodiac from Mesopotamia were known in rising Greek civilisation from around 5-600 BC and they used the 12 months of 30 days, with an extra 7 months chucked in over every 19 years too, and that all obviously came from Mesopotamia. It was at that same time that the development of geometry occurred in Greece associated with a model of the universe which had the earth in the middle and everything else sun, moon and stars, rotating around it on the surface of multiple spheres with earth at the centre. The sun was on sphere which rotated with the others but which also itself rotated once every year. It is then possible to see a link between the yearly rotation of the suns own sphere and dividing it into 360 daily bits like the degrees we use.

But very little evidence exits for what was happening in this department between then and around 2nd century BC. It is believed that the Greek Hipparchus (around 2nd century BC) was the first to formally use in a geometry (and trigonometry) a division of the circle into 360 degrees. Hardly any of his work survives but that of Ptolemy (2nd century AD, Egypt ) does and is known to be based on and preserving of Hipparchus's work. So at that point we formally have the circle divided into 360 degrees.

Hipparchus and Ptolemy (the AD one not the BC dynasty of the same name) went on to do great things in working out models of the motions of the heavens, distances between places on the known earth using the first kind of longitudes and latitudes together with the time of astonomic events, and the diameter of the earth. All their stuff is on the internet so will leave that up to you to hunt out, but in the end considering the base they had to work from they were obviously real clever guys.

John

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[Talbot]

Could be that the first mathematician to actually work all this out was deformed and had an extra digit on each hand, thus naturally worked to a base 6 or 12 rather than 5 or 10. /forums/images/icons/smile.gif

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[gjeffery]

2 pi radians to the circle.

400 grades to the circle. Some theodolites have a switch that switches to this system, but before the advent of electronic calculators, I never saw reduction tables to enable this sustem to be used.

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[Jacket]

Napoleon introduced 400 degrees for a circle- if you look at French charts from that period the compas roses are all 400 degrees. I think France stuck with 400 degrees until fairly recently- certainly I've seen some plans for a french bridge from the 30's that used this form of measurement (and wasted 3 days of work before I realised what was going on).

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[charles_reed]

The first mathematicians and earliest astronomers were the Babylonians.

Their math was based upon a duodecimal system, (base 12 instead of 10). Time and arc still use their nomenclature.

In fact the Romans used to only have 10 months in their year (hence December, the 10th month) only adding an extra 2 months in immediate pre-Republican times.

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[robind]

It couldn`t be that the circumference of the circle can be devided up by the radius six times, Could it?

Regards

Rob

<hr width=100% size=1><A target="_blank" HREF=http://www.rejuvanu.com>RejuVanu</A>

 

[Guest]

Time .... 60 x 60 x 60 etc.

Remember the world graduated in degrees of longitude .... .... dear old Father Time .....

<hr width=100% size=1>Nigel ...
So WHAT does the EU really stand for ????/forums/images/icons/cool.gif

 

[Guest]

circ / radius x6

2 pi ..... is approximately 6.3 ...... not 6

Sorry about that ....

<hr width=100% size=1>Nigel ...
So WHAT does the EU really stand for ????/forums/images/icons/cool.gif<P ID="edit"><FONT SIZE=-1>Edited by nigel_luther on 10/01/2004 18:56 (server time).</FONT></P>

 

[Alisdairjohnston]

Unless it's changed in the last year, there are 6400 mils to a circle. or there were last time I looked at my (the Queen's) prismatic compass!

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[peterb]

The 400 unit circle was introduced by the French after the French Revolution, at the same time as they introduced the metric system. Their unit (I believe it was called a grad or grade) could be subdivided into a hundred smaller units (mils?), in the same way that we subdivide a degree into minutes. That means that there were 10000 of the smaller unit to the right-angle. The kilometre was defined as 1/10000 of the distance from equator to pole, so that it was a direct equivalent to the nautical mile.

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[MainlySteam]

The Babylonians, who were one of the inhabitants of the Mesopotamia I mentioned (the Assyrians also were important in astronomy) actually, as SuperMalc says, used 60 as the base for astronomical work - they may have also used 12 as a base for other arithmetic, I don't know, but was certainly 60 for astronomy.

Their astronomy/time work from around 500 BC is beyond my understanding /forums/images/icons/frown.gif and is based on period relations, linear zig zag and step functions relating astronomical events. But with it they had worked out the (visible) planetary periods, etc. They must have spent many long nights and centuries, and maintaining the knowledge across generations nutting it out in my view. The 19 year cycle I mentioned still today applies to the calculation of Easter and Passover.

John

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[supermalc]

Just as an overview, it appears this all came about because of the calender of the world. We have day and night, the moon appears approxomately once a month, and twelve times a year, or the space between hot and cold, the 4 seasons we know, well in the UK.

So cleverer people, and better mathmaticians than me converted to the units we know today.

Does that make sense?

<hr width=100% size=1>Malcolm.

 

[LORDNELSON]

Mnay thanks everyone! My grandson, Petar, is overwhelmed. He certainly started a good discussion!

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[supermalc]

Have to agree about the discussion....informative and very interesting.


Pity more people couldn't take peasure from such things.

<hr width=100% size=1>Malcolm.