## Lesson ι': Definitions

The masculine noun o/(ros (rough breathing) means boundary or limit (when neuter, it means mountain, and has soft breathing). Definitions limit the meaning of a word, so that it can be used with precision in the reasoning to follow. It is exceedingly difficult to define the fundamental terms used in geometry, and there are difficulties with Euclid's definitions. However, any alternatives are usually worse, and never any better. Some seem not to have been properly understood by commentators not acquainted with doing things in the real world. The definition of a straight line as lying equally with the points on it is perfectly clear to anyone who has ever tried to file a straight edge, or to true up a bent spoke. Other definitions concisely state an essential property, such as a point having no parts. The notions of point, line and plane are evident to our perception, so that a definition need only bring out those properties that will be important in our reasoning. A word defined need not correspond to an actual object, nor need it give only a minimum of properties. The definition of a square as a figure with four equal straight sides and four equal internal angles is not faulty. When Euclid shows how to construct a square, he also shows that not only are the four sides equal, but the four angles as well. One could give a name to a triangle with two right angles, but it would never correspond to anything real, and could not be constructed.

Book I begins with the fundamental definitions for geometry. Subsequent books give the additional definitions necessary for them. The Greek text will be shown in red, with suggestions of the meanings of the words beneath. Work out the cases of the nouns, and the tenses, moods, voices, person and number of the verbs. Most of the words have been already presented in previous lessons, to which recourse should be made for further information. Additional explanations will be given here. Let us begin with the first nine definitions.

The second accent on Shmei~on is for the benefit of the enclitic e)sti, which has a ν-movable before the following vowel. Think of tou~ shmei~ou to understand the relative pronoun; the accent distinguishes it from the negative ou). The relative pronoun is just the definite article without the τ. ou)qe/n is from ou)-d-e/v, "not one", the δ for euphony. A form of "is" is omitted for brevity. Euclid's original definitions were quite concise, but later writers have added additional explanation, whether or not it was necessary.

In a)pla/tes we see the α-privative, and the adjective modifies neuter mh~kos. Again, "is" has been omitted but the de/ gives continuity. The next definition expands on the definition of a point by saying the ends of a line are points. Next a straight line is defined, by the method mentioned above, as one which lies equally with the points on it. This separates such a line from a general curved line, and is a good expression of straightness, which is difficult to define. The word h)/tis combines the relative pronoun η and the indefinite pronoun ti/s, and means the line considered is chosen from lines in general: it is "some" or "a certain" line. Note the elision of ι and the change of π to φ before rough breathing in e)f = e)pi. The verb kei~mai, to lie, is in the middle mood. The line acts on nothing, nor is acted upon, but the state of lying is something it does for itself, and this is excellently expressed in Greek by the middle mood. Some forms of this verb are shown in the box. We have contraction again, and the circumflex is evidence for it. Note that the circumflex disappears in the future, where there is no contraction, and we see the full endings -otai and ontai. You now know how to recognize the present middle of any verb.

In the definition of a surface, the use of the neuter relative pronoun may be confusing. Actually, it does not refer to the feminine noun, but rather means "that which". Only after we know that the thing has only length and breadth does it get a feminine name. The sixth definition is precisely analogous to the third, and "is" is omitted. E)pi/pedos is an adjective using the same forms for masculine and feminine, as so many technical terms do. Here, the relative pronoun does refer to a feminine antecedent, and has the tis that makes it indefinite: a certain surface. Flatness is expressed like straightness, but with reference to lines in the surface.

Now we turn to angles. The definition of angle was considerably lengthened by later commentators, to the present form. Euclid merely stated that it was the inclination of two lines to each other where they met. a)ptome/nwn is a present active participle, genitive plural, agreeing with the lines, as is keime/nwn. A kli/sij, -ewj is an inclination, and a/)ptw means to join or fasten. The conjunction o/(tan requires the subjunctive. The present subjunctive of ei/)mi is shown in the box. You should be getting used to the ν-movable by now, and we will not mention it further. The verb perie/cw combines the preposition pe/ri with the familiar verb e/cw: around + have = to surround or encompass. Be on the lookout for this kind of word-building in Greek; it is much used.