THE ELEMENTS, BOOK 1.

...is contained by the diameter and the arc cut off by it. The center of the semicircle is the same as that of the circle. This concludes Definition 18. A semicircle is the area trapped between a circle's diameter and its curved edge.

19. Rectilinear figures are those contained by straight lines. Trilateral three-sided figures are those contained by three straight lines, quadrilateral four-sided by four, and multilateral figures are those contained by more than four straight lines.

20. Among trilateral figures, an equilateral triangle is that which has its three sides equal. An isosceles triangle original: aequicrurius, meaning equal-legged is that which has only two sides equal. A scalene triangle is that which has its three sides unequal.

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. An acute-angled triangle is that which has its three angles acute. In Euclid's system, a triangle is named for its most prominent angle: a right angle is exactly 90 degrees, an obtuse angle is wider, and an acute angle is narrower.

22. Among quadrilateral figures, a square is that which is both equilateral and right-angled. An oblong original: parte altera longius, literally "longer on the other part" is that which is right-angled but not equilateral. A rhombus is that which is equilateral but not right-angled. A rhomboid is that which has its opposite sides and angles equal to one another, but which is neither equilateral nor right-angled. Re... The text breaks off here as the definition continues on the next page.

The following are scholarly notes regarding variations found in ancient manuscripts of Euclid.

Boethius omits both this and the definition from Proclus; the evidence regarding Heron is not clear. 3 rectilinear figures [found in manuscripts] P, b, and p, and Proclus; usually written as figures rectilinear. "is" [appears in manuscripts] P and F. Definition 19 is commonly divided into four; manuscript V omits the numbers here. 3 straight lines in Proclus and Boethius. 6 four in manuscript B. straight lines [written as] sides in Proclus and Boethius. 8 "is" [appears in manuscripts] P and F. 9 the two [written as] two in manuscript b and Proclus. only in Proclus. 10 sides omitted in Proclus. Definition 20 is commonly divided into three. 11 but [found in manuscript] P and Proclus; omitted in manuscript b; usually written as and. 12 "is" [appears in manuscripts] P and F. having one [found in] V margin by the first hand, Proclus, and Psellus. 13 having one [found in] Proclus and Psellus; one angle [found in] V margin by the first hand. that which has [line 14] but [found in] the margin of manuscript B by the same hand. acute-angled in manuscript phi. 16 which is both equilateral and in Proclus. is, which [is] equilateral and omitted in manuscript phi. oblong [written] twice in manuscript phi. 17 which [written as] the in Proclus. 20 which omitted in manuscripts F, b, and p. neither [written as] but neither in manuscripts F, b, and p. "is" omitted in Proclus.