THE ELEMENTS, BOOK 1.

...Other quadrilateral figures besides these are to be called trapezia. original: trapezia. In Euclid's system, any four-sided figure that does not fit the specific categories of square, rectangle, rhombus, or rhomboid is grouped under this term.

23. Parallel lines are straight lines which, being positioned in the same plane and produced indefinitely in both directions, do not meet in either direction. original: in infinitum. This definition relies on the concept of lines never intersecting, even if extended forever.

original: Postulata or Aitēmata. These are the "demands" or "petitions" Euclid makes of the reader. He asks us to accept that these five basic actions or truths are possible before he begins his proofs.

1. Let it be granted: to draw a straight line from any point to any point.

2. And to produce a finite straight line continuously in a straight line.

3. And to describe a circle with any center and radius.

4. And that all right angles are equal to one another.

5. And that, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles. This is the famous Parallel Postulate. It describes the condition under which two lines will eventually cross, forming the basis for Euclidean geometry.

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The following notes represent the scholarly apparatus provided by the editor, J.L. Heiberg. They compare different historical manuscripts of Euclid's text, such as those by Proclus, Philoponus, and Boethius.

The accepted order is maintained by V, Proclus, Simplicius, Capella, Boethius, and Campanus. 10. "to produce" original: ekballein V. 11. "to be drawn" original: graphēsthai: all manuscripts and Philoponus; "to draw" original: grapsai adopted from Proclus by August. 13. "to one another" original: allēlais: omitted in V. 15. "a certain straight line" original: eutheia tis P. 17. "less" original: elattonas Proclus page 191, line 18 (but not on page 364). "the two": PBVbp, "two" omitted by F, Proclus twice, Martianus Capella, and Boethius, perhaps correctly. 18. "the straight lines being produced meet at" Proclus page 364. "meet one another" original: sympiptein allēlais PV (corrected to "one another" in manuscript P). 19. "less": Pp, Proclus page 364; "less" variant spelling is common. Then add "angles" FBVb, Philoponus; omitted by Proclus twice and by manuscript Pp. In the Basel edition and in Gregory, Postulates 4 and 5 are read among the common notions (10 and 11) ("all right angles are equal... the lines being produced... will meet"). After Postulate 5, in manuscripts PF and the second hand of V and in Campanus, there follows: "and two straight lines do not enclose a space."