Reading Euclid 3 Postulates It is in the postulates that the great genius of Euclid’s achievement becomes evident. Although mathematicians before Euclid had provided proofs of some isolated geometric facts (for example, the Pythagorean theorem was probably proved at least two hundred years before Euclid’s time), it was apparently Euclid who first conceived the idea of ar- ranging all the proofs in a strict logical sequence. Euclid realized that not every geometric fact can be proved, because every proof must rely on some prior geometric knowledge thus any attempt to prove everything is doomed to circularity. He knew, therefore, that it was necessary to begin by accepting some facts without proof. He chose to begin by postulating five simple geometric statements: Euclid’s Postulate 1: To draw a straight line from any point to any point. Euclid’s Postulate 2: To produce a finite straight line continuously in a straight line. Euclid’s Postulate 3: To describe a circle with any center and distance. Euclid’s Postulate 4: That all right angles are equal to one another. Euclid’s Postulate 5: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. The first three postulates are constructions and should be read as if they began with the words “It is possible.” For example, Postulate 1 asserts that “[It is possible] to draw a straight line from any point to any point.” (For Euclid, the term straight line could refer to a portion of a line with finite length—what we would call a line segment.) The first three postulates are generally understood as describing in abstract, idealized terms what we do concretely with the two classical geometric construction tools: a straightedge (a kind of idealized ruler that is unmarked but indefinitely extendible) and a compass (two arms connected by a hinge, with a sharp spike on the end of one arm and a drawing im- plement on the end of the other). With a straightedge, we can align the edge with two given points and draw a straight line segment connecting them (Postulate 1) and given a previously drawn straight line segment, we can align the straightedge with it and extend (or “produce”) it in either direction to form a longer line segment (Postulate 2). With a compass, we can place the spike at any predetermined point in the plane, place the drawing tip at any other predetermined point, and draw a complete circle whose center is the first point and whose circumference passes through the second point. The statement of Postu- late 3 does not precisely specify what Euclid meant by “any center and distance” but the way he uses this postulate, for example in Propositions I.1 and I.2, makes it clear that it is applicable only when the center and one point on the circumference are already given. (In this book, we follow the traditional convention for referring to Euclid’s propositions by number: “Proposition I.2” means Proposition 2 in Book I of the Elements.) The last two postulates are different: instead of asserting that certain geometric con- figurations can be constructed, they describe relationships that must hold whenever a given geometric configuration exists. Postulate 4 is simple: it says that whenever two right angles have been constructed, those two angles are equal to each other. To interpret this, we must address Euclid’s use of the word equal. In modern mathematical usage, “A equals B” just means the A and B are two different names for the same mathematical object (which could

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