2 1. Euclid Reading Euclid Before going any further, you should take some time now to glance at Book I of the Ele- ments, which contains most of Euclid’s elementary results about plane geometry. As we discuss each of the various parts of the text—definitions, postulates, common notions, and propositions—you should go back and read through that part carefully. Be sure to ob- serve how the propositions build logically one upon another, each proof relying only on definitions, postulates, common notions, and previously proved propositions. Here are some remarks about the various components of Book I. Definitions If you study Euclid’s definitions carefully, you will see that they can be divided into two rather different categories. Many of the definitions (including the first nine) are de- scriptive definitions, meaning that they are meant to convey to the reader an intuitive sense of what Euclid is talking about. For example, Euclid defines a point as “that which has no part,” a line as “breadthless length,” and a straight line as “a line which lies evenly with the points on itself.” (Here and throughout this book, our quotations from Euclid are taken from the well-known 1908 English translation of the Elements by T. L. Heath, based on the edition [Euc02] edited by Dana Densmore.) These descriptions serve to guide the reader’s thinking about these concepts but are not sufficiently precise to be used to justify steps in logical arguments because they typically define new terms in terms of other terms that have not been previously defined. For example, Euclid never explains what is meant by “breadthless length” or by “lies evenly with the points on itself” the reader is expected to interpret these definitions in light of experience and common knowledge. Indeed, in all the books of the Elements, Euclid never refers to the first nine definitions, or to any other descriptive definitions, to justify steps in his proofs. Contrasted with the descriptive definitions are the logical definitions. These are def- initions that describe a precise mathematical condition that must be satisfied in order for an object to be considered an example of the defined term. The first logical definition in the Elements is Definition 10: “When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.” This describes angles in a particular type of geometric configuration (Fig. 1.1) and tells us that we are entitled to call an angle a right angle if and only if it occurs in a configuration of that type. (See Appendix E for a discussion about the use of “if and only if in definitions.) Some other terms for which Euclid provides logical definitions are circle, isosceles triangle, and parallel. Fig. 1.1. Euclid’s definition of right angles.
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