## Lesson ιζ': Postulates

The postulates are the basis on which the whole structure of the Elements is built, and they must be carefully constructed. Euclid gave five postulates, of which the first three are of remarkable simplicity. The final two actually contain the essential specification of Euclidean space. Many commentators have moved them erroneously to common notions, or have tried to prove them from the other postulates. These are misguided efforts that miss the point completely. Euclid's geometry is not the only geometry. Even Euclid knew that geometry on the surface of a sphere was different, but he had a concrete idea of three-dimensional space that his geometry was to represent. In fact, it does so very well. The departures from Euclidean geometry in the neighbourhood of the Earth are extremely small, less than those arising from drawing figures on a portion of the Earth's surface close to us. Euclid's results, are, eminently, true in a very practical sense.

The fourth postulate is equivalent to a statement that angles and distances are unchanged by an arbitrary rotation or translation in space, and the fifth postulate that space is "flat" in a sense well-known in Riemannian geometry. These postulates are stated by Euclid in a form that is applicable to the course of reasoning in the Elements, and allow the proof of the necessary results. It is idle to try to prove these postulates; non-Euclidean geometries are well-known.

The ai)th/mata are the things demanded, the postulates. The third-person imperative has no expressed subject; it means "let it be conceded that". The verb a/)gw is, unusually, reduplicated in the aorist, a characteristic which is more often the sign of the perfect. Here, the verb means "to draw", like the Latin ducere, extended from its basic meaning of "to lead", as to lead the pen. The infinitive is aorist active, as can be detected by the circumflex on the ultima. The second postulate also contains an aorist active infinitive, e)k + ba/llw, to "throw out". In the third postulate, the infinitive is aorist passive, and ku/klon is its object. In the fourth, you may ask why the angles are in the accusative. Here, they are the object of the infinitive ei/)nai. That the verb is intransitive does not matter; objects of infinitives are always in the accusative, even when they are actually subjects.

The fifth postulate is quite different from the others. The reason why Euclid stated it in this form can be seen when we come to the proposition in which it is used, to prove that if two lines are parallel, then the sum of the interior angles is two right angles. The converse is easily proved by contradiction, and, in fact, Euclid does this first. But now, if we have two parallel lines by hypothesis, we cannot suppose that they meet and prove a contradiction, because parallel lines never meet, by definition. Hence, the need for this postulate. There are other ways out of this difficulty, but none as clear and straightforward as Euclid's. The e)a/n calls for the subjunctive, which is the poi/h. There is a present active participle, e)pi/ptousa. The ε is not an augment, but part of the verb stem. E)la/sswn is spelled with -σσ- instead of -ττ- here. Analyze this sentence until you can put the words together in good English.