"There aren't 360 Degrees in a Circle"

I shall now go through my opponent's arguments and clarify my own. 1. The Arbitrary Nature of Definitions.My opponent continues to provide examples on the arbitrariness of the definition of a Degree. For instance, he starts talking about alternate realities, where a degree, a centimeter or a mile is defined differently. However, I have addressed this argument in detail in my 2nd round. I can only reiterate what I said: the arbitrary nature of Mathematical definitions has nothing to do with the Resolution of this debate. We are talking about a specific definition of a Degree, and we are using this definition to discuss a specific geometric concept - a Circle. My opponent does not address the issue with those arguments. He completely ignores my rebuttal and goes on asserting the same thing. This is a logical fallacy, known as Argumentum ad nauseam [1] and is also a red herring. [2]2. Misunderstanding of the Nature of an Angular Measurement.My opponent claims that in order to measure an angle, you "need to define the number of degrees in a straight line". This is simply not true. A degree is an angular measurement, which looks as follows (it is represented by θ): A measurement of a straight line is carried out completely differently (i.e. the distance between points A and B): I feel that it is obvious, from a mathematical standpoint, that an angular measurement, and a distance measurement, are completely different terms. Somehow, my opponent is trying to conflate them here. He needs to provide justification for his claim that a Degree can be represented as a distance measurement on a straight line. As far as I know, this would be impossible by definition.3. Internal Angle is not a Central Angle.Again, my opponent completely drops my rebuttal in the 2nd Round, and continues to use Internal angles as arguments for his Resolution. As I have clearly stated in the previous round, we need to be talking about Central angles, as that is exactly what the wording of the resolution suggests.My opponent is talking about the Internal angles of polygons, which look like this (i.e. angles A, B and C): However, I have asked my opponent to provide an argument, using the Central angle (i.e. θ): It seems that my opponent is trying to infer something about the Central angle of a Circle, using Polygons and their Internal angles as examples. While this is not necessarily a fallacious method, my opponent has not provided any mathematical justification for his inferences, especially since he has misunderstood the concept of a Central angle altogether.My opponent also shows a misunderstanding of Polygons and Limits. He states that "a circle only has one side and thus one angle, so it can't have an infinitely large angle". His argument can be represented by a syllogism:P1. A Circle has one EdgeP2. A Circle has one Internal Angle.C1. Therefore, a Circle cannot have an infinitely large Angle.All of the premises are demonstrably false.Let's assume that a Circle is a Polygon. Then, it is a Polygon with infinite edges (or "sides"), as I have demonstrated with my Limit formula in the previous round. This follows from the fact that in order to construct a Circle via the Polygon iteration process, the amount of Edges has to be increased to infinity (while keeping the Perimeter constant). [3] Secondly, an Edge cannot be curved, as it is by definition a line segment, which is always straight. [4] Therefore, P1 is completely and utterly false.The second premise of my opponent can be negated by observing that each Internal angle of a regular polygon corresponds to a Vertex. A Vertex is located at the intersection point of two Edges. [5] Thus, if we assume that a Circle is a regular polygon with infinite Edges, it follows that it has infinite vertices, and, therefore, infinite internal Angles.Although my opponent's Conclusion never follows from his incorrect premises, it is mathematically true. Observe that for any regular n-sided polygon, its internal angle is calculated as follows [3]: Thus, if we take a limit of this, as n-approaches infinity (which is essentially a Circle), then we see that the internal angle of a Circle has a measure of exactly 180 degrees, not infinity. Moreover, if we add up all these angles, we get infinity, because a circle has an infinite amount of angles.However, any of this has nothing to do with the central angle, and thus my opponent's whole argument is negated.4. Final Rebuttals.My opponent asserts that "angles will be enclosed by an arch unless they are right angles", concluding that this somehow shows that "we are taught to accept this idea [of 360 degrees in a Circle]".I fail to see any logical or mathematical coherence within this argument. I might just negate the whole premise by showing you a right angle that is enclosed by an arc: Secondly, my opponent disputes my calculations in Round 1 by stating that my conversions "do not prove anything". I suppose he is talking about the conversion from Radians to Degrees, but I fail to see his justification for this assertion. All of my conversions adhered to the definitions of both Radians and Degrees. Therefore, they are consistent within Mathematics, and successfully show that there are 360 Degrees in a Circle. I challenge my opponent to point out a specific flaw in those calculations. He has to show that they are mathematically or logically inconsistent, but he hasn't done so yet.Thirdly, my opponent says that I have demonstrated that "the internal angle of a circle is 180 degrees, not 179.9... degrees" but that I have failed to "show how 360 is derived".To start with, I didn't show that "the internal angle of a circle is 180 degrees, not 179.9... degrees". I actually showed that the internal angle of a circle is both 180 and 179.999... degrees, which are mathematically equivalent expressions, thus, not contradictory. [6] The reason why I didn't derive any "360" from this, is because I have explicitly stated that the argument from Internal degrees had nothing to do with the Central angle of a circle. Therefore, my whole refutation was simply due to my mathematical interest in clearing up any confusion. It was never supposed to be an argument, because my opponent's argument was a fallacious red herring in the first place. I demonstrated the derivation of 360 degrees in my 1st Round arguments A and B, which my opponent has still not managed to refute mathematically or logically.Fourthly, my opponent concedes that Pi is not needed to calculate the angle of a circle. Thus, I stand by my assertion that Babylonians could have used the same argument, which I used in my 1st Round (Argument A), since they were the ones who defined a Degree.Finally, my opponent challenges me to prove that "there are 360 degrees in a circle without making any assumptions". I have done so in Round 1, with both argument A and B. I did not make any unnecessary assumptions, but I had to make assumptions of basic mathematical axioms, such as Peano axioms. I sincerely hope that my opponent will not dispute these, as this is, by nature, a mathematical debate, thus, fundamental axioms should be assumed.I have demonstrated how, using the definition of a Degree, one may conclude that there are 360 degrees in a Circle (Round 1, Argument A). Also, I have demonstrated how, using the definition of a Radian, and the relation of Radians and Degrees, that there are 360 degrees in a Circle (Round 1, Argument B).Conclusion so farMy opponent has failed to satisfactorily challenge any of my arguments. Therefore, I conclude that my opponent has still not met his burden of proof, while my arguments still stand strong and unrefuted.Sources:[1] http://rationalwiki.org...[2] http://rationalwiki.org...[3] https://en.wikipedia.org...[4] https://en.wikipedia.org...[5] https://en.wikipedia.org...[6] https://en.wikipedia.org......