# 360 degrees

#### whozum

Why was the number '360' chosen to be the number of degrees in a complete revolution? How far back does the number go?

I'd think it was something to do with the number of days in the year that they believed was correct at the time. Also it is easy to divide it up I guess...

I'm not too good on the history of science...

#### whozum

There are easier numbers to divide. I just dont see any logical sense as to why 360 was chosen.

I still think its something to do with the number of days in the year...

#### Huckleberry

A site I checked out claimed this exerpt was from a book called 'The History of Pi' by Petr Beckman.

In 1936, a tablet was excavated some 200 miles from Babylon. Here one
should make the interjection that the Sumerians were first to make one of
man's greatest inventions, namely, writing; through written communication,
knowledge could be passed from one person to others, and from one
generation to the next and future ones. They impressed their cuneiform
(wedge-shaped) script on soft clay tablets with a stylus, and the tablets
were then hardened in the sun. The mentioned tablet, whose translation
was partially published only in 1950, is devoted to various geometrical
figures, and states that the ratio of the perimeter of a regular hexagon
to the circumference of the circumscribed circle equals a number which in
modern notation is given by 57/60 + 36/(60^2) (the Babylonians used the
sexagesimal system, i.e., their base was 60 rather than 10).

The Babylonians knew, of course, that the perimeter of a hexagon is
exactly equal to six times the radius of the circumscribed circle, in fact
that was evidently the reason why they chose to divide the circle into 360
degrees (and we are still burdened with that figure to this day). The
tablet, therefore, gives ... Pi = 25/8 = 3.125.

cool :D - well at least I learned something new staying up this late revising....

#### whozum

So the 360 is somewhat arbitrary, and could be replaced by any figure, or does it truly represent something about a circle (or an angle)?

#### Huckleberry

It seems to be arbitrary. The Babylonians chose 60 as their numeric base rather than 10. 60 can be divided without remainder by 1,2,3,4,5 and 6. 60x6=360. Seems that avoiding decimals is the main reason it was put into use.

It was during the reign of Nebuchadnezzar (605-562 BC) in the Chaldean dynasty in Babylon that the circle was divided into 360 degrees. This was because the Chaldeans had calculated by observation and inference that a complete year numbered 360 days. The basis of angular measure for the mathematicians of Babylon was the angle at each of the corners of an equilateral triangle. They did not have decimal fractions and thus found it difficult to deal with remainders when doing division. So they agreed to divide the corner of an equilateral triangle into 60 degrees, because 60 could be divided by 2, 3, 4, 5 and 6 without remainder. Each degree was divided into 60 minutes and each minute into 60 seconds. If the angles at the corners of six equilateral triangles are placed together they form the angle formed by a complete circle. It is for this reason that there are six times 60 degrees of arc in the complete circle.

#### whozum

Interesting stuff, thanks alot.

#### Night Owl

Whoa. The babylonians used...base 60? That means they'd need 60 individual symbols for the numbers 0 through 59, right? Last edited:

#### whozum

That sounds okay, but then there are numbers that have a more factors than 60. why was it just sixty? For example, setting it at 120 would give them 1,2,3,4,5,6,8,10.

#### Huckleberry

Finally we should look at the question of why the Babylonians had a number system with a base of 60. The easy answer is that they inherited the base of 60 from the Sumerians but that is no answer at all. It only leads us to ask why the Sumerians used base 60. The first comment would be that we do not have to go back further for we can be fairly certain that the sexagesimal system originated with the Sumerians. The second point to make is that modern mathematicians were not the first to ask such questions. Theon of Alexandria tried to answer this question in the fourth century AD and many historians of mathematics have offered an opinion since then without any coming up with a really convincing answer.
This comes from the site posted above. They go over some possible explanations, but none are conclusive. There may not be an answer.

Looking at their system of numerals it looks like using a base of 120 would give them one more numeral that would not result in decimals and would create much more difficulty in expressing those numbers. The example in the text states that 424000 would be written in Sumerian by using the numbers 1,57,46,40. They would have to calculate 1 x 60^3 + 57 x 60^2 + 46 x 60 + 40 = 424000. With a base of 120 they would need over twice as many numerals and would need to calculate by hand much larger numbers. That is my guess.

#### whozum

But then one could argue that a number smaller than 60, perhaps 30, would make that calculation even easier. I wonder what their real reason was.

#### Huckleberry

Very true. Your guess is as good as mine, probably better. My understanding of math sucks.

#### whozum

Thanks for the help man.