Reading Euclid 5 Whereas the five postulates express facts about geometric configurations, the common notions express facts about magnitudes. For Euclid, magnitudes are objects that can be compared, added, and subtracted, provided they are of the “same kind.” In Book I, the kinds of magnitudes that Euclid considers are (lengths of) line segments, (measures of) angles, and (areas of) triangles and quadrilaterals. For example, a line segment (which Euclid calls a “finite straight line”) can be equal to, greater than, or less than another line segment two line segments can be added together to form a longer line segment and a shorter line segment can be subtracted from a longer one. It is interesting to observe that although Euclid compares, adds, and subtracts geomet- ric magnitudes of the same kind, he never uses numbers to measure geometric magnitudes. This might strike one as curious, because human societies had been using numbers to mea- sure things since prehistoric times. But there is a simple explanation for its omission from Euclid’s axiomatic treatment of geometry: to the ancient Greeks, numbers meant whole numbers, or at best ratios of whole numbers (what we now call rational numbers). How- ever, the followers of Pythagoras had discovered long before Euclid that the relationship between the diagonal of a square and its side length cannot be expressed as a ratio of whole numbers. In modern terms, we would say that the ratio of the length of the diagonal of a square to its side length is equal to p 2 but there is no rational number whose square is 2. Here is a proof that this is so. Theorem 1.1 (Irrationality of p 2). There is no rational number whose square is 2. Proof. Assume the contrary: that is, assume that there are integers p and q with q ¤ 0 such that 2 D .p=q/2. After canceling out common factors, we can assume that p=q is in lowest terms, meaning that p and q have no common prime factors. Multiplying the equation through by q2, we obtain 2q2 D p2: (1.1) Because p2 is equal to the even number 2q2, it follows that p itself is even thus there is some integer k such that p D 2k. Inserting this into (1.1) yields 2q2 D .2k/2 D 4k2: We can divide this equation through by 2 and obtain q2 D 2k2, which shows that q is also even. But this means that p and q have 2 as a common prime factor, contradicting our assumption that p=q is in lowest terms. Thus our original assumption must have been false. This is one of the oldest examples of what we now call a proof by contradiction or indirect proof, in which we assume that a result is false and show that this assumption leads to a contradiction. A version of this argument appears in the Elements as Proposition VIII.8 (although it is a bit hard to recognize as such because of the archaic terminology Euclid used). For a more thorough discussion of proofs by contradiction, see Appendix F. For details of the properties of numbers that were used in the proof, see Appendix H. This fact had the consequence that, for the Greeks, there was no “number” that could represent the length of the diagonal of a square whose sides have length 1. Thus it was not possible to assign a numeric length to every line segment. Euclid’s way around this difficulty was simply to avoid using numbers to measure magnitudes. Instead, he only compares, adds, and subtracts magnitudes of the same kind.
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