22 1. Euclid In addition, the following terms that Euclid uses without explanation are also used by modern writers, so you don’t necessarily need to change them but make sure that you know what they mean and that the meanings will be clear to your readers: the base of an isosceles triangle, to bisect a line segment or an angle, vertical angles, adjacent angles, exterior angles, interior angles. Euclid sometimes writes “I say that [something is true],” which is a phrase you will seldom find in modern mathematical writing. When you see this phrase in Euclid, think about how it fits into the logic of his proof. Is he saying this is a statement that follows from what he has already proved? Or a statement that he thinks is obvious and does not need proof? Or a statement that he claims to be true but has not proved yet? Or something else? How might a modern mathematician express this? Finally, after you have rewritten each proof, write a short discussion of the main features of the proof, and try to answer these questions: Why did Euclid construct the proof as he did? Were there any steps that seemed superfluous to you? Were there any steps or justifications that he left out? Why did this proposition appear at this particular place in the Elements? What would have been the consequences of trying to prove it earlier or later? 1D. Identify the fallacy that invalidates the proof of the “fake theorem” that says every triangle has two equal sides, and justify your analysis by carefully drawing an example of a nonisosceles triangle in which that step is actually false. [Hint: The problem has to do with drawing conclusions from the diagrams about locations of points. It’s not enough just to find a step that is not adequately justified by the axioms you must find a step that is actually false.] 1E. Find a modern secondary-school geometry textbook that includes some treatment of axioms and proofs, and do the following: (a) Read the first few chapters, including at least the chapter that introduces triangle congruence criteria (SAS, ASA, AAS). (b) Do the homework exercises in the chapter that introduces triangle congruence criteria. (c) Explain whether the axioms used in the book fill in some or all of the gaps in Euclid’s reasoning discussed in this chapter.
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