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why does a circle have 360 degrees?


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Showing 76-87 of 87 posts in this discussion
In reply to an earlier post on Apr 24, 2010, 6:49:19 PM PDT
Customer says:
But we don't care about 'easy even divisions' any longer. We have calculators and computers. When's the last time you cared about 'round decimals'?

In reply to an earlier post on Apr 24, 2010, 6:51:54 PM PDT
Customer says:
There's evidence that some culture used base 6?

Why? Can you give an example? I need to learn base 6.

In reply to an earlier post on Apr 24, 2010, 8:07:51 PM PDT
Omnireader says:
Golly, jb I was wondering, what do think the title of this discussion means?

Posted on Apr 24, 2010, 10:48:32 PM PDT
Okay... a lot of this 360 degree stuff came out of the Greeks when they were meticulously calculating chord lengths in a circle. The reason for using a base 60 is that it's more convenient to calculate with. They didn't even use unit circles back then and they did calculate things down to the seconds. The early tables go by 5 degrees around the entire circle. (I want to say it was Aristarchus who worked on this stuff.) In any event, that's where it comes from and that's why we use it.

The convenience of a 60 base system surely came from the Sumerian's, but for the application of making 360 degrees in a circle, that came from the Greeks. They were the first to sit down and really map it all out. It was after all the theory was done that Astronomers took it and applied stuff like Trigonometry.

If you want to see some REAL impressive calculations without calculators look into the history of Logarithms and look up Napier... the guy was nuts.

In reply to an earlier post on Apr 24, 2010, 11:40:27 PM PDT
Last edited by the author on Apr 24, 2010, 11:49:34 PM PDT
iceman says:
J. black says:

But we don't care about 'easy even divisions' any longer. We have calculators and computers. When's the last time you cared about 'round decimals'?

even with a calculator, you can see the DIFFERENCE between 0.250 and 0.33333333333333...

and if you truncate the latter after 3 significant figures, every one KNOWS you have only an APPROXIMATION.

you can never get the PRECISION of a 12-base system with just a 10-base system.

whoever first came up with the idea of using 360 degrees to mark off circles, obviously knew the ADVANTAGE of a 12-base (or 6-base) system.

marking a circle off into 100 'degrees' would lack precision. and even 400 'degrees' despite the finer divisions, would be nowhere as elegant as a 360 degree division of circles, and fractional arcs of them.

In reply to an earlier post on Apr 25, 2010, 3:42:50 PM PDT
Last edited by the author on Apr 25, 2010, 3:43:48 PM PDT
'probabilist says:
http://en.wikipedia.org/wiki/Base_6

http://en.wikipedia.org/wiki/Babylonian_numerals

In reply to an earlier post on Apr 25, 2010, 3:48:14 PM PDT
Last edited by the author on Apr 25, 2010, 3:50:16 PM PDT
'probabilist says:
J. black asked:

> When's the last time you cared about 'round decimals'?

This week. When you display a double precision floating point number on a 32-bit computer, it contains far more digits than you care about.

All the best,

P

___________________________

> options( digits = 16 )

> pi
[1] 3.141592653589793

# significant digits:
> options( digits = 7 )

> pi
[1] 3.141593

> args( round )
function(x, digits = 0.)

# digits past the decimal:
> round( pi, digits = 5 )
[1] 3.14159

In reply to an earlier post on Apr 25, 2010, 3:54:53 PM PDT
'probabilist says:
Hi, Adam -

> trigonometry

My understanding is that trigonometry (using the trigonometric ratios we know as sines, cosines, tangents and so on) was first developed in the Indian subcontinent, and made its way to Europe via mathematicians writing in Arabic in the medieval Islamic world. The ancient Greeks knew chords of circles, but did little (if anything) with the trigonometric ratios.

All the best,

P

Posted on Apr 25, 2010, 3:56:20 PM PDT
'probabilist says:
A History of Mathematics
by Carl B. Boyer

In reply to an earlier post on Apr 25, 2010, 4:36:52 PM PDT
Omnireader says:
Thanks!
Most writing began as the keeping of numerical texts. It seems that accounting for numbers came before the accounting of letters to form written text in most cultures.

Interesting how widespread the 'Pythagorean' theory was in the middle East about 2000 BCE; much earlier than Pythagorus lived.
During ancient Grecian times, one of the Pythagorean cult mysteries had people entering, while turning in circles. to indicate the planets rotated. I wonder how much has been lost, and is still being re-discovered.

Posted on Apr 25, 2010, 6:01:08 PM PDT
I read a pretty good book about this.... Kind of a simple book to understand and relatively short, called "Sacred Geometry." On of the things the author mentioned is the use of fractions in the past and the relative ease that using a 6, 12, or 60 based system lent to the use of fractions. Also, one of the other considerations was the amount of factors a number had, and since 60 is divisible by (besides 1) 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, it made sense to use it since to use 60 for ease of mental math. Also, this lends itself to the use of 360 degrees, which is also a very "mentally usable" number. I personally believe the system of counting in the use of degrees is secondary to the ease of use and the availability of measuring tools or methods. There are many measurements that have been used in the past that we no longer use today. Thinking logically, before the advent of calculators, most people would use units that would result in whole numbers or fractions. For instance, we may measure Stonehenge and find that it is 68.329 feet in diameter (this is a fictional number... Just giving an example.), or, 22.2143 meters in diameter (also fictional... I think). However, ancient mathematicians were not likely to have used this method. They probably divided the distance with a unit of measurement that resulted in a whole number and logically divisible fractions, say, 30 cubits, for example. Halfway is 30/2, or 15. One quarter is 30/4, or 7 and 1/2. And the rest follows from there.

Interestingly, a much better approximation of Pi is found a fraction that was used much more often in ancient times than the decimal approximation we used today. 22/7, on the new Windows 7 scientific calculator, gives Pi to about 24 decimal places. My TI-84 plus silver edition only gives about 9, I think. 22/7 is still an approximation, but, considering the significance of the number "7" in ancient times and likewise today, this fraction is significant. And, in all honesty, it would be much easier to count out a distance, divide it into seven equal parts, and then measure out twenty-two of these, and you can pretty much get the ratio of the diameter of a circle to its circumference almost perfectly.

Anyway, I'm not a mathematics expert. I have an MBA, and two undergrad degrees, one in IS and another in Business. However, I am crazy about history AND mathematics, and enjoy just reading and studying these things for myself. Anyway, I hope this helps a little. Check out the book. You would probably enjoy it.

Posted on Apr 25, 2010, 10:55:30 PM PDT
Omnireader says:
Stonehenge is 320 feet in diameter.

There is an article which makes it clear that the ancient tribes in the area built Stonehenge, and not space aliens.
From: scienceblogs.com/goodmath/2006/06/nutty_numerology_and_stoneheng.php
[ Research has revealed that before the Sarsen Circle of upright stones was erected, a 285 foot diameter circle of 56 chalk holes, 3 feet in diameter, was created. (These are called the Aubrey Holes, in honor of John Aubrey).
A CBS TV program in the 1960's ran a computer analysis of the Aubrey circle. They declared that Stonehenge's location--latitude 51 degrees 11 minutes, was a very special location for eclipses of the moon. This location produces moon eclipses in the repeating sequence of 19 years, 19 years, and 18 years.

Adding 19+19+18=56. Thus if the white 3 foot diameter chalk holes were covered by a black stone, that was moved around the circle in synch with the passage of moon cycles, the black stone would arrive at the heel stone position, on the exact day when a moon eclipse would occur. (Eclipse computer.) (S.I.D.)]

And continuing from the same source:
[As they point out, lunar eclipses occur in a regular pattern at this location. This fact is an implication of the relationships of the positions and motions of celestial bodies. But you don't need to know the positions and velocities of the bodies: you need to know the observable relationships between their motions. And that is something that is easily observable.

To give a simple example of this kind of thing: There's a right triangle whose sides have lengths 1, 2, and the square root of three.
To draw a 1,2, square root 3 right triangle, you could start with a horizontal line 1 inch long, and then draw a vertical line whose height is sqrt (3) inches, and then draw the hypotenuse. To do this, you need to be able to compute the square root of three, which is not the easiest thing to do. You clearly need to be able to do something beyond simple arithmetic to be able to compute and measure the square root of three without using a geometric relationship. On the other hand, you could draw a horizontal line of length 1; then draw a long vertical line from its endpoint; and then take a ruler, and rotate it until the distance from the endpoint of the horizontal line to an intersection with the vertical line was 2 inches. The second way doesn't require you to be able to compute roots.

Following the stone computer, came the erection of the 30 upright stones that formed the Sarsen Circle, 100 feet in diameter.

(My question was why 30? I divided 360 degrees by 30 and discovered the number 12.
The number 12 is one of the most important numbers in the Anunnaki civilization...their Pantheon consisted of the Twelve Great Anunnaki gods, they declared 12 months in one year-2 twelve hour parts of each day, they created the 12 signs of the Zodiac.]

{Jeremy Black and Anthony Green offer a slightly different perspective on the Igigi and the Anunnaki, writing that "lgigu or Igigi is a term introduced in the Old Babylonian Period as a name for the (ten) 'great gods'. While it sometimes kept that sense in later periods, from Middle Babylonian times on it is generally used to refer to the pods of heaven collectively, just as the term Anunnakku (Anuna) was later used to refer to the gods of the underworld. In the Epic of Creation, it is said that there are 300 lgigu of heaven."}See wiki anunnakku

[These Sarsen uprights are harder than granite and weigh 25 tons each. They were quarried at Marlborough Downs using tools not locally available at that time, and then transported these huge stones over 20 miles to this site.] No space aliens needed.
From: scienceblogs.com/goodmath/2006/06/nutty_numerology_and_stoneheng.php

The basic geometry of Stonehenge is on YouTube; youtube.com/watch?v=6WbUZi863iU

Other interesting geometric models are found in a PDF file :www.tifr.res.in/~vahia/Astronomy_in_stones.pdf

And a site that copies Gerald Hawkins 1963 schematic of Stonehenge, showing the basic alignments. See circulartimes.org/Stonehenge%20Revisited%20Colette%20Dowell%20CT.htm
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