The purpose of this question is trying to know originally how did counting in integer degrees angles from (one degree to $360$ degrees) being adopted basically in geometry, despite the impossibility or the great difficulty of constructing exactly (without any approximation methods) those many integer degrees angles that aren't divisible by $3$,

So, I thought that there may be a very good unknown historical reasons (mainly by the ancient Babilions or Egyptians) to adopt such systems instead of using the exactly constructible angles defined essentially by the full angle of twice $\pi$

Thanking your opinions

Editing: October 4, 2018

No, the point of my question is quite different from the answers provided that well-explain the historical details about the issue, for the basic reason that many angels are impossible construction, where this strictly includes a subset of majority (or exactly two thirds) of integer degree angles of the form ($3n +/- 1$), where $n$ is a natural number

The other more important point about the true reason of the impossibility of constructing exactly certain angels, which was well proven by Wantzel in 1837, regarding the angle $\pi/9 = 20$ degrees, where despite the fact of the proof, people (including the proofers) still think that such angle exists but impossible to construct

And finally the most important point that might seem shocking and not at all acceptable by the mainstream academic mathematicians (for the first while), which states that such angles as $\pi/9 = 20$ degrees, doesn’t exist at all, same like all those integer degrees angles of the above-described form $3n +/- 1$,

So, to say why the angle of $20$ degrees is impossible construction is simply because it doesn’t exist at all Otherwise, how can an angle or a real number exist but can’t be exactly constructed? Wonder! To explain this simple fact, it is impossible existence for a triangle with exactly three known sides to have exactly an angle of $20$ degrees, or $19$ degrees or $22$ degrees, or generally any integer degrees angle which isn’t divisible by $3$, of course one shouldn’t use any approximation methods and concludes or claims the exact construction by means of limits or convergence or cuts or else, since all those methods are basically and strictly with endless number of steps (theoretically unfinished operation) So, the hope lies upon the mathematicians to give themselves a little chance not to miss this very important and new point of view

However, the whole topic is deeply related to real number concept that had been extended wrongly beyond the real positive constructible numbers (which are the only true real existing numbers), where this had been well-proved and were also published publically in other forums and also some here on SE, but were deleted from SE sections by moderators since they so, unfortunately, contradict strictly many common beliefs among mathematicians

EDIT: NOV. 25th, 2018 Adding reference: https://www.quora.com/Do-most-of-the-named-angles-in-mathematics-truly-exist

I thought of not adding my content here about my earlier claims regarding the absolute non-existence of majority of well-known angles in current and old mathematics as well since truly most likely it would be immediately deleted as were done to many of my other deleted claims here and generally at SE

But the fact which is more important than all of us, for sure

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    $\begingroup$ @MauroALLEGRANZA that should be posted as an answer :-) $\endgroup$ Oct 3 '18 at 13:23
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    $\begingroup$ I'm not sure those ancient civilizations had discovered $\pi$ but would be interested to know if they did. $\endgroup$ Oct 3 '18 at 13:26
  • $\begingroup$ @CarlWitthoft, I'm afraid to say that $\pi$ (considering it as a real number) haven't been discovered yet, nor if it would be ever discovered, but $\pi$ as an angle was truly discovered by the ancient $\endgroup$ Oct 4 '18 at 9:22
  • $\begingroup$ "the angle of 20 degrees doesn’t exist at all" !!!! Obviously from ancient Babylonian astronomers to today, we can measure an angle of $20°$ with a reasonable accuracy, in any case with an accuracy sufficient to : compute time, predict eclipses, launch rockets landing on the Moon, manage Space Station orbiting around the Earth and many many others. As you can see from the answer below, for some old "applications" (ancient Babylonian and Egypt surveyors) an approximation of $\pi$ with $3$ were enough. $\endgroup$ Oct 4 '18 at 12:59
  • $\begingroup$ @MauroALLEGRANZA, of course, we can construct what might we believe as 20-degrees angle with reasonable accuracy as we can but not absolutely, but we can't construct it exactly like many other integer degrees angles which are divisible by $3$, and the real reason behind this fallacy is its non-existence, and then it becomes so simple to understand the impossibility of constructing such angles like $\pi/9$, which was never known before or being understood correctly as simple as that, and this wasn't fully comprehended even for the popovers of impossibility of constructing such angles $\endgroup$ Oct 6 '18 at 7:30

See : Bartel van Der Waerden, Science Awakening, Oxford UP (original ed, 1950), page 37 :

The sexagesimal system was taken over by the Semitic Babylonians from their predecessors, the Sumerians. This remarkable cultural group, which had also invented the cuneiform script, dominated southern Mesopotamia during the third millenium B.C.

Regarding $\pi$, see the History of $\pi$ for some references to ancient (Egyptian and Babylonian) approximation of $\pi$.

See also : Bartel van der Waerden, Geometry and Algebra in Ancient Civilizations, Springer (1983), page 170 for An Ancient Egyptian Rule for Squaring the Circle.

The method used in Problem 50 of the Rhind mathematical papyrus (around 1550 BCE) amounts to assuming :

$\pi = (\dfrac {16}{9})^2$.

See also page 173 :

The Babylonians too used to compute area and circumference of the circle by means of $C=3d$. We can express the same facts by saying that the Babylonians and the Egyptian author of Problem 32 [of the Cairo papyrus (first century BCE)] assumed $\pi=3$.


Counting angles in degrees does not really belong to "geometry and mathematics". It was invented in astronomy by the Babylonians. Later, the degrees were used in geography and geodesy. And not only whole degrees were used, but when needed, fractions of a degree. Either simple fractions of a degree, or minutes. In mathematics, there is a more natural unit of angles: the radian.

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    $\begingroup$ This is interesting but doesn't address the question: why 360? (to which the answer is that the Babylonians already had a base-60 numbering system) $\endgroup$ Oct 3 '18 at 13:25
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    $\begingroup$ And there is a reason for the base-60 numbering system as well. That without having an arithmetic of fractions 60 was a much more comfortable set of units then 100, since it allowed for many more integer fractions (1/7 being the first difficult one). $\endgroup$ Oct 3 '18 at 18:04

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