14 1. Euclid The catch is that one must scrupulously ensure that the proofs of the theorems do not use anything other than what has been assumed in the postulates. If the axioms represent arbitrary assumptions instead of self-evident facts about the real world, then nothing except the axioms is relevant to proofs within the system. Reasoning based on intuition about the behavior of straight lines or properties that are evident from diagrams or common experience in the real world will no longer be justifiable within the axiomatic system. Looking back at Euclid with these newfound insights, mathematicians realized that Euclid had used many properties of lines and circles that were not strictly justified by his postulates. Let us examine a few of those properties, as a way of motivating the more careful axiomatic system that we will develop later in the book. We will discuss some of the most problematic of Euclid’s proofs in the order in which they occur in Book I. As always, we refer to the edition [Euc02]. While reading these analyses of Euclid’s arguments, you should bear in mind that we are judging the incompleteness of these proofs based on criteria that would have been utterly irrelevant in Euclid’s time. For the ancient Greeks, geometric proofs were meant to be convincing arguments about the geometry of the physical world, so basing geometric conclusions on facts that were obvious from diagrams would never have struck them as an invalid form of reasoning. Thus these observations should not be seen as criticisms of Euclid rather, they are meant to help point the way toward the development of a new axiomatic system that lives up to our modern (post-Euclidean) conception of rigor. Euclid’s Proposition I.1. On a given finite straight line to construct an equilateral trian- gle. A B C ? Fig. 1.6. Euclid’s proof of Proposition I.1. Analysis. In Euclid’s proof of this, his very first proposition, he draws two circles, one centered at each endpoint of the given line segment AB (see Fig. 1.6). (In geometric dia- grams in this book, we will typically draw selected points as small black dots to emphasize their locations this is merely a convenience and is not meant to suggest that points take up any area in the plane.) He then proceeds to mention “the point C , in which the circles cut one another.” It seems obvious from the diagram that there is a point where the circles intersect, but which of Euclid’s postulates justifies the fact that such a point always exists? Notice that Postulate 5 asserts the existence of point where two lines intersect under certain circumstances but nowhere does Euclid give us any justification for asserting the existence of a point where two circles intersect. Euclid’s Proposition I.3. Given two unequal straight lines, to cut off from the greater a straight line equal to the less.
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