"There aren't 360 Degrees in a Circle"

I will now defend my position and rebut my opponent's claims. Definition of a DegreeMy opponent contests the first definition of a Degree that I have provided, i. e. :Degree - (in Geometry) the 360th part of a complete angle or turn, often represented by the sign °, as in 45°, which is read as 45 degrees. He states that it is "very specific" and not as "accurate" as the Oxford Dictionary Definition. I disagree with him. Firstly, we cannot use any definition of a degree, as the debate we are currently having focuses on Mathematics, specifically, Geometry. Therefore, there is a need of a specific definition, which is applicable to the field of Geometry, and which is widely accepted by mathematicians. It seems that most, if not all serious dictionaries and sources provide a geometrical definition of a degree, as the 360th part of a complete circle. My opponent claims that The Oxford Dictionary defines a degree as "a unit for measuring angles". This is not true, as this is not the whole definition. The whole definition of a degree, provided by the Oxford Dictionary in the context of Geometry, goes like this:Degree - unit of measurement of angles, one ninetieth of a right angle or the angle subtended by one three-hundred-and-sixtieth of the circumference of a circle. [1]The library version of the Oxford Dictionary provides a definition as follows:Degree - Geom. (Astron. , Geogr. , etc. ) A unit of measurement of angles or circular arcs, being an angle equal to the 90th part of a right angle, or an arc equal to the 360th part of the circumference of a circle (which subtends this angle at the centre). [2]Therefore, it seems that my opponent is simply ignoring an important part of the definition.1. Defense of Argument A. My opponent claims that "The definitions you have used assume it is impossible to calculate the number of degrees in a circle. " I fail to see any kind of justification for this claim. The definition that I have provided clearly defines a degree as being the 1/360th part of the circle. Thus, it is trivially easy to calculate the amount of degrees in a circle. By simple mathematics,If 1/360 of a Circle corresponds to 1 degree,Then 1 (full) Circle corresponds to 1 * 360 = 360 degrees. My opponent also talks about the Babylonians, claiming that they could not have calculated the degrees in a circle without knowing the value of Pi. But my opponent fails to realize that Pi is not needed to calculate the angle of a circle, especially if one defines a degree as the Babylonians did, that is, the 1/360th part of a circle. The Babylonians would have perfectly been capable of calculating the amount of degrees in a circle, as I did above. We know how many degrees there are in a circle, because we have defined a degree in this way. There is no need for any polygons or complex measurements when the answer simply comes straight out of the definition of the word.2. Defense of Argument B. My opponent completely drops the argument. He says that my second definition of a degree, which relates degrees and radians, is insufficient proof for the conclusion that there are 360 degrees in a circle. He does not dispute any of my premises or calculations; he completely ignores my whole proof and effectively ignores the whole argument I have provided. He then provides a completely irrelevant comment about the arbitrariness of a degree, claiming that "I could just as easily create my own unit and call it a 'klark' and say 1 degree equals 1.6. .. 'klarks'"I do not dispute that. I completely agree that the definition of a degree, or, indeed, the definitions of all units in Mathematics are completely arbitrarily defined. However, they are defined nonetheless, and those definitions have been used extensively in mathematics, and they are accepted. If the mathematical community accepts your definition of a 'klark' I will see no problem in measuring angles in klarks. We are not discussing whether the definition of a degree is arbitrary. We are discussing whether a degree, as it is defined, leads to the conclusion that there are 360 degrees in a circle. Both of my arguments, which support this conclusion, are still standing at this point.3. My opponent's Argument. My opponent provides an iterative process, showing how increasing the number of sides of a regular polygon increases its internal angle. Please remember that this debate is about the central angle of a circle. Therefore, whatever my opponent has written here is actually irrelevant to the debate. Nevertheless, I will go through his argument to clear any misconceptions. He shows that a hexadecagon has an internal angle of 157.5 degrees, that a chiliagon has an angle of 0.36 degrees and so forth. He makes the assumption that a circle is a polygon with infinite sides. He then states that by his proof, the internal angle of a circle is equal to 179.9. .. degrees. Indeed, if one assumes a circle to be a polygon, the same rule applies. It can actually be expressed by the following formula: My opponent seemingly arrived at the answer 179.9. .. , which he claims to be contradictory. But he failed to realize, that this is a repeating decimal, that is, 179.999. .. , which can be written as 179. (9), which is exactly equal to 180. Thus, if a circle is defined as a polygon, it can be concluded that its internal angle is 180 degrees. The sum of its internal angles is equal to infinity. This is to be expected, as a circle has no corners, or infinite sides. However, my opponent is incorrect in assuming that such a circle would have no curvature. That would only be the case if the radius of such a circle would be infinitely large, but that would be a special case known as an Apeirogon. [3]Anyway, as the proof goes, please remember that we are debating the central angle of a circle, not the internal one, as that's exactly the angle we mean when we say "360 degrees in a circle". I ask my opponent to refrain from conflating the terms. The Debate so farAt the moment, I am still waiting for my opponent to refute my both arguments. I. e. he should show that the definition of a degree does not lead to the conclusion of a 360-degree circle. Also, he should point out flaws in my second argument, which he hasn't done yet. Also, I would advice my opponent to produce a constructive case, which would be based on the central angle of a circle. Otherwise, my opponent might fail to meet his burden of proof. Sources:[1] . http://www.oxforddictionaries.com...[2] . http://www.oed.com... (See 9th definition; Library access or subscription will be required)[3] . https://en.wikipedia.org...