THE ELEMENTS, BOOK 1.

...nor right-angled. Let other quadrilateral figures besides these be called trapezia. original: trapezia. In Euclid's classification, a trapezium is any four-sided shape that is not a square, rectangle, rhombus, or rhomboid. This is broader than the modern definition of a trapezoid.

23. Parallel lines are straight lines which, being in the same plane and being produced indefinitely original: eis apeiron, meaning "into the infinite." In geometry, "produced" means extending a line further than its current endpoints. in both directions, do not meet one another in either direction.

Postulates

original: Aitēmata. In ancient Greek mathematics, a postulate is a "petition" or "demand." The author is asking the student to grant that these basic constructions and truths are possible.

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.
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3. And to describe a circle with any center and distance. In this context, "distance" refers to the radius of the circle.
4. And that all right angles are equal to one another.
5. And that, if a straight line falling on two straight lines 15 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 are the angles less than the two right angles. This is the famous Parallel Postulate. It states that if the sum of the inner angles is less than 180 degrees, the lines are tilted toward each other and will eventually cross.

23. Hero of Alexandria, definition 71. Philoponus in his commentary on the Posterior Analytics, Book 2, folio 18 verso. Psellus, page 35. Martianus Capella, Book 6, section 712. Boethius, page 376, line 23. Postulates 1 to 5: Martianus Capella, Book 6, section 722. Boethius, page 377, line 4. Aspasius, as quoted by Simplicius in his commentary on Aristotle's On the Heavens, folio 149, mentions "the five postulates." 1. Philoponus in Posterior Analytics, Book 2, folios 9 verso, 10, and 29. 2. Simplicius in his commentary on the Physics, folio 119. 3. Philoponus in Posterior Analytics, Book 2, folio 10 and 29. 4. Philoponus, ibid., folio 10. 5. Philoponus, ibid., folio 10 and 29. Proclus, page 364, line 14. These references track where these specific definitions and postulates appear in the works of other ancient and medieval scholars.

1. Manuscript B reads "squares." 2. Manuscript b uses a variant spelling for "trapezia." Editors commonly divide Definition 21 into three parts and Definition 22 into five parts. 3. Manuscript B adds "but" or "and" after "parallel." Proclus and Psellus read "Parallel lines are..." 4. Manuscript V uses a different form of the word for "into." 5. Manuscript P uses the infinitive form "to meet." Manuscript F omits the phrase "one another." 6. Manuscript V reads "five postulates." Manuscripts B and F, and a later correction in Manuscript b, read "there are five postulates." Manuscript F omits the numbering. 9. Manuscripts P, B, F, b, and p place the phrase "in a straight line" before the word "continuously." This section, the "critical apparatus," notes small differences found in the various historical handwritten copies of Euclid's text, such as Manuscript B or Manuscript V.