How did the integer degrees angles counting being first adopted in geometry and mathematics? [duplicate]

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• How were irrational numbers that are not constructible accepted by mathematicians? 4 answers
• Origin of 360 degrees? 1 answer

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

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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

Answer 1

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.

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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 $\pi=3$.

Answer 2

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.