18 1. Euclid uses Proposition I.3 to choose a point F on that line such that EF is the same length as BE. Here is where the first problem arises: although Postulate 3 guarantees that a line segment can be extended to form a longer line segment containing the original one, it does not explicitly say that we can make the extended line segment as long as we wish. As we mentioned above, if we were working on the surface of a sphere, this might not be possible because great circles have a built-in maximum length. The second problem arises toward the end of the proof, when Euclid claims that angle ECD is greater than angle ECF . This is supposed to be justified by Common Notion 5 (the whole is greater than the part). However, in order to claim that angle ECF is “part of” angle ECD, we need to know that F lies in the interior of angle ECD. This seems evident from the diagram, but once again, there is nothing in the axioms or previous propositions that justifies the claim. To see how this could fail, consider once again the surface of a sphere. In Fig. 1.12, we have illustrated an analogous configuration, with A at the north pole and B and C both on the equator. If B and C are far enough apart, it is entirely possible for the point F to end up south of the equator, in which case it is no longer in the interior of angle ECD. (Fig. 1.13 illustrates the same configuration after it has been “unwrapped” onto a plane.) A B C D F E Fig. 1.12. Euclid’s proof fails on a sphere. A B C D F E Fig. 1.13. The same diagram “unwrapped.” Some of these objections to Euclid’s arguments might seem to be of little practical consequence, because, after all, nobody questions the truth of the theorems he proved. However, if one makes a practice of relying on relationships that seem obvious in diagrams, it is possible to go wildly astray. We end this section by presenting a famous fallacious “proof” of a false “theorem,” which vividly illustrates the danger. The argument below is every bit as rigorous as Euclid’s proofs, with each step justi- fied by Euclid’s postulates, common notions, or propositions and yet the theorem being proved is one that everybody knows to be false. This proof was first published in 1892 in a recreational mathematics book by W. W. Rouse Ball [Bal87, p. 48]. Exercise 1D asks you to identify the incorrect step(s) in the proof. Fake Theorem. Every triangle has at least two equal sides. Fake Proof. Let ABC be any triangle, and let AD be the bisector of angle A (Proposition I.9). We consider several cases. Suppose first that when AD is extended (Postulate 2), it meets BC perpendicularly. Let O be the point where these segments meet (Fig. 1.14(a)). Then angles AOB and AOC are both right angles by definition of “perpendicular.” Thus the triangles AOB and

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