4 1. Euclid be a number, an angle, a triangle, a polynomial, or whatever). But Euclid uses the word differently: when he says that two geometric objects are equal, he means essentially that they have the same size. In modern terminology, when Euclid says two angles are equal, we would say they have the same degree measure when he says two lines (i.e., line seg- ments) are equal, we would say they have the same length and when he says two figures such as triangles or parallelograms are equal, we would say they have the same area. Thus Postulate 4 is actually asserting that all right angles are the same size. It is important to understand why Postulate 4 is needed. Euclid’s definition of a right angle applies only to an angle that appears in a certain configuration (one of the two ad- jacent angles formed when a straight line meets another straight line in such a way as to make equal adjacent angles) it does not allow us to conclude that a right angle appearing in one part of the plane bears any necessary relationship with right angles appearing else- where. Thus Postulate 4 can be thought of as an assertion of a certain type of “uniformity” in the plane: right angles have the same size wherever they appear. Postulate 5 says, in more modern terms, that if one straight line crosses two other straight lines in such a way that the interior angles on one side have degree measures adding up to less than 180ı (“less than two right angles”), then those two straight lines must meet on that same side of the first line (Fig. 1.2). Intuitively, it says that if two lines start out ? Fig. 1.2. Euclid’s Postulate 5. “pointing toward each other,” they will eventually meet. Because it is used primarily to prove properties of parallel lines (for example, in Proposition I.29 to prove that parallel lines always make equal corresponding angles with a transversal), Euclid’s fifth postulate is often called the “parallel postulate.” We will have much more to say about it later in this chapter. Common Notions Following his five postulates, Euclid states five “common notions,” which are also meant to be self-evident facts that are to be accepted without proof: Common Notion 1: Things which are equal to the same thing are also equal to one another. Common Notion 2: If equals be added to equals, the wholes are equal. Common Notion 3: If equals be subtracted from equals, the remainders are equal. Common Notion 4: Things which coincide with one another are equal to one an- other. Common Notion 5: The whole is greater than the part.
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