The Art Of Euclid — страница 2

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gives the reader no doubt that the proof will work. Euclid also uses a proposition proven by a common notion to prove a later proposition. For example, propositions four and ten are correlated in this manner. Proposition four, which deals with congruent sides and their included angle, is used to prove proposition ten, which is used to bisect a given finite straight line. Euclid also proves propositions in succession, proving one using the propositions that directly precedes it. An example of this is propositions eighteen, nineteen, and twenty, which deal with greater angles subtending greater sides. He does this because he is confident that by using a proposition proven by a common notion, which has to be true, the later proposition that is based upon the earlier also has to be

true. Not only is Euclid confident when he uses this reasoning, but so is the reader who is persuaded by reference to an earlier common notion. Euclid?s writing has many stylistic aspects that help prove his theories of triangles and parallel areas. In using the various stylistic devices in his Elements, especially the use of common notions and postulates, Euclid systematically explains each step of his propositions with a reference each time to either a common notion or a postulate, or some other form. Since almost all of the propositions contain either a postulate or a common notion, Euclid persuades the reader that he is right because of the acceptance of postulates and common notions as true.

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