Gaps in Euclid’s Arguments 15 A B D E? F Fig. 1.7. Euclid’s proof of Proposition I.3. Analysis. In his third proof, Euclid implicitly uses another unjustified property of circles, although this one is a little more subtle. Starting with a line segment AD that he has just constructed, which shares an endpoint with a longer line segment AB, he draws a circle DEF with center A and passing through D (justified by Postulate 3). Although he does not say so explicitly, it is evident from his drawing that he means for E to be a point that is simultaneously on the circle DEF and also on the line segment AB. But once again, there is nothing in his list of postulates (or in the two previously proved propositions) that justifies the claim that a circle must intersect a line. (The same unjustified step also occurs twice in the proof of Proposition I.2, but it is a little easier to see in Proposition I.3.) Euclid’s Proposition I.4. If two triangles have the two sides equal to two sides respec- tively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend. A B C D E F Fig. 1.8. Euclid’s proof of Proposition I.4. Analysis. This is Euclid’s proof of the well-known side-angle-side congruence theorem (SAS). He begins with two triangles, ABC and DEF , such that AB D DE, AC D DF , and angle BAC is equal to angle EDF . (For the time being, we are adopting Euclid’s convention that “equal” means “the same size.”) He then says that triangle ABC should be “applied to triangle DEF .” (Some revised translations use “superposed upon” or “su- perimposed upon” in place of “applied to.”) The idea is that we should imagine triangle ABC being moved over on top of triangle DEF in such a way that A lands on D and the segment AB points in the same direction as DE, so that the moved-over copy of ABC occupies the same position in the plane as DEF . (Although Euclid does not explicitly mention it, he also evidently intends for C to be placed on the same side of the line DE as F , to ensure that the moved-over copy of ABC will coincide with DEF instead of being a mirror image of it.) This technique has become known as the method of superposition.

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