Reading Euclid 7 (3) Specification: Announcing what has to be constructed or proved in this specific in- stance. Example: “Thus it is required to construct an equilateral triangle on the straight line AB.” (4) Construction: Adding points, lines, and circles as needed. For construction prob- lems, this is where the main construction algorithm is described. For theorems, this part, if present, describes any auxiliary objects that need to be added to the figure to complete the proof if none are needed, it might be omitted. (5) Proof: Arguing logically that the given construction does indeed solve the given problem or that the given relationships do indeed hold. (6) Conclusion: Restating what has been proved. A word about the conclusions of Euclid’s proofs is in order. Euclid and the classical mathematicians who followed him believed that a proof was not complete unless it ended with a precise statement of what had been shown to be true. For construction problems, this statement always ended with a phrase meaning “which was to be done” (translated into Latin as quod erat faciendum, or q.e.f.). For theorems, it ended with “which was to be demonstrated” (quod erat demonstrandum, or q.e.d.), which explains the origin of our traditional proof-ending abbreviation. In Heath’s translation of Proposition I.1, the conclusion reads “Therefore the triangle ABC is equilateral and it has been constructed on the given finite straight line AB. Being what it was required to do.” Because this last step is so formulaic, after the first few propositions Heath abbreviates it: “Therefore etc. q.e.f.,” or “Therefore etc. q.e.d.” We leave it to you to read Euclid’s propositions in detail, but it is worth focusing briefly on the first three because they tell us something important about Euclid’s conception of straightedge and compass constructions. Here are the statements of Euclid’s first three propositions: Euclid’s Proposition I.1. On a given finite straight line to construct an equilateral trian- gle. Euclid’s Proposition I.2. To place a straight line equal to a given straight line with one end at a given point. Euclid’s Proposition I.3. Given two unequal straight lines, to cut off from the greater a straight line equal to the less. One might well wonder why Euclid chose to start where he did. The construction of an equilateral triangle is undoubtedly useful, but is it really more useful than other funda- mental constructions such as bisecting an angle, bisecting a line segment, or constructing a perpendicular? The second proposition is even more perplexing: all it allows us to do is to construct a copy of a given line segment with one end at a certain predetermined point, but we have no control over which direction the line segment points. Why should this be of any use whatsoever? The mystery is solved by the third proposition. If you look closely at the way Postulate 3 is used in the first two propositions, it becomes clear that Euclid has a very specific inter- pretation in mind when he writes about “describing a circle with any center and distance.” In Proposition I.1, he describes the circle with center A and distance AB and the circle with center B and distance BA and in Proposition I.2, he describes circles with center B and distance BC and with center D and distance DG. In every case, the center is a point that
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