Why is a Circle 360 Degrees?

Date: 2 Jan 1995 15:16:33 -0500
From: Roy P. Sachs
Subject: Origin of degrees

Over the holiday, the question came up in a family discussion, and could 
easily come up in one of my geometry classes.  What is the origin/basis 
of the degree measurement? Why is a circle divided into 360 degrees 
rather than some other number?  Any light that you could shed would be 
appreciated.

Roy Sachs
rsachs@umd5.umd.edu

Date: 2 Jan 1995 16:34:20 -0500
From: Dr. Ken
Subject: Re: Origin of degrees

I'm glad you asked this question, because I've been wondering it myself.  I
figured it had something to do with the Babylonians, who used a base 60
number system.  But it sure took a lot of digging in several books to find
out anything concrete about it.

I finally found what I was looking for in a book called "A History of Pi" by
Petr Beckmann, a mathematician from Czechoslovakia.  Here's the passage:

  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.

So that's who gave us the 360 degrees in the circle.  See, assignment of
degree-measure to angles is somewhat arbitrary.  Some choices are more
natural than others, though, and when you're working in base 60, 6x60 is a
pretty natural choice. 

As a sidenote, the actual ratio that the Babylonians talk about is
6r/(2r*Pi) = 3/Pi, which is about 0.95493.  They say it's 24/25 = .96. 
And you might ask why we chose Pi as the letter to represent the 
number 3.141592..., rather than some other Greek letter like 
Alpha or Omega.  Well, it's Pi as in Perimeter - the letter Pi 
in Greek is like our letter P.

I hope this helps answer your question.  Write back if you have 
more!

-Doctor Ken,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 8/28/96 at 16:58:39
From: Herbert S. Friedman
Subject: Re: Origin of degrees

There is more to this than the six sixes for the 360 from
the Babylonians.  It has to do with Claudius Ptolemy (100-170 AD), 
who divided the circle into 360 parts for his sine table. 
He actually used the length of the chord for each central 
angle in steps of 1/2 degree in a circle of radius 60 rather 
than sines.  See p. 212, problem 4 of Burton's "The History of 
Mathematics" (1985, Allyn and Bacon).  I still have the feeling 
that the Babylonians were partially responsible.