Gaps in Euclid’s Arguments 17 Euclid’s Proposition I.12. To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line. A B C D F E‹ G‹ Fig. 1.10. Euclid’s proof of Proposition I.12. Analysis. In this proof, Euclid starts with a line AB and a point C not on that line. He then says, “Let a point D be taken at random on the other side of the straight line AB, and with center C and distance CD let the circle EF G be described.” He is stipulating that the circle should be drawn with D on its circumference, which is exactly what Postulate 3 allows one to do. However, he is also implicitly assuming that such a circle will intersect AB in two points, which he calls E and G. Obviously it is the fact that C and D are on opposite sides of AB that is supposed to guarantee the existence of the intersection points but which of his postulates or previous propositions justifies this conclusion? For that matter, what is “on the other side” supposed to mean? Euclid’s definitions and postulates do not mention “sides” of lines at all, but he regularly refers to them in his proofs. It is clear from the diagrams what he means, but it is not justified by the postulates. Euclid’s Proposition I.16. In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles. A B C D F E Fig. 1.11. Euclid’s proof of Proposition I.16. Analysis. Nowadays, this result is called the exterior angle inequality. Its proof is one of the most subtle and clever in the Elements. It is worth reading it over once or twice to absorb the full impact. It is not easy to see where the gaps are, but there are at least two. After constructing the point E that bisects AC (using Proposition I.10), Euclid then extends BE past E and

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