8 1. Euclid has already been located, and the “distance” is actually a segment that has already been drawn with the given center as one of its endpoints. Nowhere in these two propositions does he describe what we routinely do with a physical compass: open the compass to the length of a given line segment and then pick it up and draw a circle with that radius some- where else. Traditionally, this restriction is expressed by saying that Euclid’s hypothetical compass is a “collapsing compass”—as soon as you pick it up off the page, it collapses, so you cannot put it down and reproduce the same radius somewhere else. The purpose of Proposition I.3 is precisely to simulate a noncollapsing compass. After Proposition I.3 is proved, if you have a point O that you want to be the center of a circle and a segment AB somewhere else whose length you want to use for the radius, you can draw a segment from O to some other point E (Postulate 1), extend it if necessary so that it’s longer than AB (Postulate 2), use Proposition I.3 to locate C on that extended segment so that OC D AB, and then draw the circle with center O and radius OC (Postulate 3). Obviously, the Greeks must have known how to make compasses that held their sep- aration when picked up, so it is interesting to speculate about why Euclid’s postulate de- scribed only a collapsing compass. It is easy to imagine that an early draft of the Elements might have contained a stronger version of Postulate 3 that allowed a noncollapsing com- pass, and then Euclid discovered that by using the constructions embodied in the first three propositions he could get away with a weaker postulate. If so, he was probably very proud of himself (and rightly so). After Euclid Euclid’s Elements became the universal geometry textbook, studied by most educated Westerners for two thousand years. Even so, beginning already in ancient times, schol- ars worked hard to improve upon Euclid’s treatment of geometry. The focus of attention for most of those two thousand years was Euclid’s fifth pos- tulate, which usually strikes people as being the most problematic of the five. Whereas Postulates 1 through 4 express possibilities and properties that are truly self-evident to anyone who has thought about our everyday experience with geometric relationships, Pos- tulate 5 is of a different order altogether. Most noticeably, its statement is dramatically longer than those of the other four postulates. More importantly, it expresses an assump- tion about geometric configurations that cannot fairly be said to be self-evident in the same way as the other four postulates. Although it is certainly plausible to expect that two lines that start out pointing toward each other will eventually meet, it stretches credulity to ar- gue that this conclusion is self-evident. If the sum of the two interior angle measures in Fig. 1.2 were, say, 179:999999999999999999999999998ı, then the point where the two lines intersect would be farther away than the most distant known galaxies in the universe! Can anyone really say that the existence of such an intersection point is self-evident? The fifth postulate has the appearance of something that ought to be proved instead of being accepted as a postulate. There is reason to believe that Euclid himself was less than fully comfortable with his fifth postulate: he did not invoke it in any proofs until Proposition I.29, even though some of the earlier proofs could have been simplified by using it. For centuries, mathematicians who studied Euclid considered the fifth postulate to be the weakest link in Euclid’s tightly argued chain of reasoning. Many mathematicians tried

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