Let AB be the given finite straight line.
It is required to construct an equilateral triangle on the straight line AB.
Describe the circle BCD with center A and radius AB. Again describe the circle ACE with center B and radius BA. Join the straight lines CA and CB from the point C at which the circles cut one another to the points A and B.
Now, since the point A is the center of the circle CDB, therefore AC equals AB. Again, since the point B is the center of the circle CAE, therefore BC equals BA.
But AC was proved equal to AB, therefore each of the straight lines AC and BC equals AB.
And things which equal the same thing also equal one another, therefore AC also equals BC.
Therefore the three straight lines AC, AB, and BC equal one another.
Therefore the triangle ABC is equilateral, and it has been constructed on the given finite straight line AB.
Euclid could have chosen proposition I.4 to come first, since it doesn’t logically depend on the previous three, but there are some good reasons for putting I.1 first. For one thing, the Elements ends with constructions of the five regular solids in Book XIII, so it is a nice aesthetic touch to begin with the construction of a regular triangle. More important, though, is I.1 is needed in I.2, and that in I.3. Propositions I.2 and I.3 give constructions for moving lines, and I.4, although not logically dependent on I.2 or I.3, does use the concept of superposition which involves, in some sense, moving points and lines.
Sometimes the justification is quoted in full as C.N.1 is here, but usually it is left to the reader to determine the justification.
The rest of the proofs end with Q.E.D. instead, an abbreviation for quod erat demonstrandum which means “which was to be demonstrated.” It’s convenient to have a standard way to indicate the end of a proof. These Latin abbreviations are a bit of an anachronism. It would be less of an anachronism to use abbreviations for the original Greek phrase, or abbreviations for a modern English phrase since the rest of this version of the Elements is in English. But by now, Q.E.F. and Q.E.D. are traditional. In recent decades a small square has become common as a symbol to indicate the end of a proof.
Why does the point C exist? Near the beginning of the proof, the point C is mentioned where the circles are supposed to intersect, but there is no justification for its existence. The only one of Euclid’s postulate that says a point exists the parallel postulate, and that postulate is not relevant here. Indeed, some postulate is needed for that conclusion, such as “If the sum of the radii of two circles is greater than the line joining their centers, then the two circles intersect.” Such a postulate is also needed in Proposition I.22. There are models of geometry in which the circles do not intersect. Thus, other postulates not mentioned by Euclid are required. In Book III, Euclid takes some care in analyzing the possible ways that circles can meet, but even with more care, there are missing postulates.
Why is ABC a plane figure? After concluding the three straight lines AC, AB, and BC are equal, what is the justification that they contain a plane figure ABC? Recall that a triangle is a plane figure bounded by contained by three lines. These lines have not been shown to lie in a plane and that the entire figure lies in a plane. It is proposition XI.1 that claims that all parts of a line lie in a plane, and XI.2 that claims that the entire triangle lie in a plane. Logically, they should precede I.1. The reason they don’t, of course, is that those propositions belong to solid geometry, and plane geometry is developed first in the Elements, also, no doubt, plane geometry developed first historically.
Why does ABC contain an equilateral triangle? Proclus relates that early on there were critiques of the proof and describes that of Zeno of Sidon, an Epicurean philosopher of the early first century B.C.E. (not to be confused with Zeno of Elea famous of the paradoxes who lived long before Euclid), and whose criticisms, Proclus says, were refuted in a book by Posidonius. The critique is sound, however, and the refutation faulty.
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Zeno of Sidon criticized the proof because it was not shown that the sides do not meet before they reach the vertices. Suppose AC and BC meet at E before they reach C, that is, the straight lines AEC and BEC have a common segment EC. Then they would contain a triangle ABE which is not equilateral, but isosceles.
Zeno recognized that in order to destroy his counterexample it was necessary to assume that straight lines cannot have a common segment. Proclus relates a supposed proof of that statement, the same one found in proposition XI.1, but it is faulty. Proclus and Posidonius quoted properties of lines and circles that were never proven and never explicitly assumed as postulates. |
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The possibilities that haven’t been excluded are much more numerous than Zeno’s example. The sides could meet numerous times and the region they contain could look like a necklace of bubbles. What needs to be shown (or assumed as a postulate) is that two infinitely extended straight lines can meet in at most one point.