16 1. Euclid This is an intuitively appealing argument, because we have all had the experience of moving cutouts of geometric figures around to make them coincide. However, there is nothing in Euclid’s postulates that justifies the claim that a geometric figure can be moved, much less that its geometric properties such as side lengths and angle measures will remain unchanged after the move. Of course, Propositions I.2 and I.3 describe ways of constructing “copies” of line segments at other positions in the plane, but they say nothing about copying angles or triangles. (In fact, he does prove later, in Proposition I.23, that it is possible to construct a copy of an angle at a different location but that proof depends on Proposition I.4!) This is one of the most serious gaps in Euclid’s proofs. In fact, many scholars have inferred that Euclid himself was uncomfortable with the method of superposition, because he used it in only three proofs in the entire thirteen books of the Elements (Propositions I.4, I.8, and III.23), despite the fact that he could have simplified many other proofs by using it. There is another important gap in Euclid’s reasoning in this proof: having argued that triangle ABC can be moved so that A coincides with D, B coincides with E, and C coincides with F , he then concludes that the line segments BC and EF will also coincide and hence be equal (in size). Now, Postulate 1 says that it is possible to construct a straight line (segment) from any point to any other point, but it does not say that it is possible to construct only one such line segment. Thus the postulates provide no justification for concluding that the segments BC and EF will necessarily coincide, even though they have the same endpoints. Euclid evidently meant the reader to understand that there is a unique line segment from one point to another point. In a modern axiomatic system, this would have to be stated explicitly. Euclid’s Proposition I.10. To bisect a given finite straight line. A B C D‹ Fig. 1.9. Euclid’s proof of Proposition I.10. Analysis. In the proof of this proposition, Euclid uses another subtle property of inter- sections that is not justified by the postulates. Given a line segment AB, he constructs an equilateral triangle ABC with AB as one of its sides (which is justified by Proposition I.1) and then constructs the bisector of angle ACB (justified by Proposition I.9, which he has just proved). So far, so good. But his diagram shows the angle bisector intersecting the segment AB at a point D, and he proceeds to prove that AB is bisected (or cut in half) at this very point D. Once again, there is nothing in the postulates that justifies Euclid’s assertion that there must be such an intersection point.

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