ELEMENTS, BOOK 1.

...of the same circumference. And the center of the semicircle is the same as that of the circle. This concludes Definition 18. A semicircle is defined as the space enclosed by the diameter and the part of the circle's edge it cuts off.

19. Rectilinear figures are those contained by straight lines. Trilateral figures are those contained by three straight lines, quadrilateral by four, and multilateral 5 figures are those contained by more than four straight lines. Trilateral and quadrilateral are the formal terms for triangles and four-sided shapes like squares or rectangles.

20. Among trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle is that which has only two of its sides equal, and a scalene triangle is 10 that which has its three sides unequal. Isosceles literally means "equal legs" in Greek. Scalene means "uneven" or "slanting."

21. Further, among trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle is that which has an obtuse angle, and an acute-angled triangle is that which has its three angles acute. A right angle is 90 degrees. An obtuse angle is wider than a right angle. An acute angle is narrower.

15 22. Among quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong is that which is right-angled but not equilateral; a rhombus is that which is equilateral but not right-angled; and a rhomboid is that which has its opposite sides and 20 angles equal to one another, but which is neither equilateral nor... original: "ἑτερόμηκες" (heteromēkes). This refers to a rectangle that is not a square. Euclid distinguishes between figures where all sides are equal (square and rhombus) and those where only opposite sides are equal (oblong and rhomboid).

19. Philoponus, Commentary on the Posterior Analytics, Book 2, folio 39 recto; compare On the Soul. Boethius pages 375, 14 to 21. 20. Hero, Definitions 43, 44, 45. Psellus page 36. Boethius page 376, 2. 21. Hero, Definitions 46, 48, 47. Philoponus, Commentary on the Posterior Analytics, Book 2, folio 39 recto. Psellus page 37. Boethius page 376, 6. 22. Psellus page 37. Martianus Capella Book 6, 712. Boethius page 376, 14. Rhombus: Galen Volume 18, Part 1, page 466.

1. "of it" [feminine]: "of it" [masculine] in manuscript B. "of the circumference": "circumference of the circle" in manuscripts P, B, F, V, but "of the circle" is omitted in b, p, Proclus, Hero, Capella, and Boethius. "And the center... is" [lines 1 to 2]: August added this from Proclus page 160, having rejected Definition 3, 6, which all manuscripts provide in this place as follows: "A segment of a circle is the figure contained by a straight line and a circumference of a circle, either greater or less than a semicircle" ("of the circle is" omitted in φ; for the first "or" in B, F, V there is "either"; "lesser" in P). Campanus has the same; on the other hand Capella... This section is an apparatus criticus, a scholarly record of how different ancient and medieval copies of Euclid's text vary. It shows how editors like Heiberg decided which words were original and which were added later by teachers or scribes.