6 1. Euclid (In later books, he also compares ratios of such magnitudes.) As mentioned above, it is clear from Euclid’s use of the word “equal” that he always interprets it to mean “the same size” any claim that two geometric figures are equal is ultimately justified by showing that one can be moved so that it coincides with the other or that the two objects can be decomposed into pieces that are equal for the same reason. His use of the phrases “greater than” and “less than” is always based on Common Notion 5: if one geometric object (such as a line segment or an angle) is part of another or is equal (in size) to part of another, then the first is less than the second. Having laid out his definitions and assumptions, Euclid is now ready to start proving things. Propositions Euclid refers to every mathematical statement that he proves as a proposition. This is somewhat different from the usual practice in modern mathematical writing, where a result to be proved might be called a theorem (an important result, usually one that requires a relatively lengthy or difficult proof) a proposition (an interesting result that requires proof but is usually not important enough to be called a theorem) a corollary (an interesting result that follows from a previous theorem with little or no extra effort) or a lemma (a preliminary result that is not particularly interesting in its own right but is needed to prove another theorem or proposition). Even though Euclid’s results are all called propositions, the first thing one notices when looking through them is that, like the postulates, they are of two distinct types. Some propositions (such as I.1, I.2, and I.3) describe constructions of certain geometric config- urations. (Traditionally, scholars of Euclid call these propositions problems. For clarity, we will call them construction problems.) These are usually stated in the infinitive (“to construct an equilateral triangle on a given finite straight line”), but like the first three postulates, they should be read as asserting the possibility of making the indicated con- structions: “[It is possible] to construct an equilateral triangle on a given finite straight line.” Other propositions (traditionally called theorems) assert that certain relationships al- ways hold in geometric configurations of a given type. Some examples are Propositions I.4 (the side-angle-side congruence theorem) and I.5 (the base angles of an isosceles triangle are equal). They do not assert the constructibility of anything. Instead, they apply only when a configuration of the given type has already been constructed, and they allow us to conclude that certain relationships always hold in that situation. For both the construction problems and the theorems, Euclid’s propositions and proofs follow a predictable pattern. Most propositions have six discernible parts. Here is how the parts were described by the Greek mathematician Proclus [Pro70]: (1) Enunciation: Stating in general form the construction problem to be solved or the theorem to be proved. Example from Proposition I.1: “On a given finite straight line to construct an equilateral triangle.” (2) Setting out: Choosing a specific (but arbitrary) instance of the general situation and giving names to its constituent points and lines. Example: “Let AB be the given finite straight line.”
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