ELEMENTS, BOOK 1.

...makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing upon the other is called a perpendicular to that on which it stands. This concludes Definition 10 from the previous page. It establishes that a "right" angle is defined by two lines meeting such that the angles on both sides are perfectly balanced.

XI. An obtuse angle is one which is greater than a right angle. In Greek, "ambleia gōnia," literally meaning a blunt or dull angle.

XII. An acute angle is one which is less than a right angle. In Greek, "oxeia gōnia," literally meaning a sharp or pointed angle.

XIII. A boundary is that which is the extremity of anything. original: "Terminus." This refers to the outer limit of a point, line, or surface.

XIV. A figure is that which is contained by any boundary or boundaries.

XV. A circle is a plane figure contained by a single line [which is called the circumference], such that all straight lines falling upon it from a single point among those lying within the figure are equal to one another. Euclid defines the circle not by its roundness, but by the mathematical property that every point on its edge is the same distance from its center.

XVI. And that point is called the center of the circle.

XVII. A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a line also divides the circle into two equal parts. This definition includes the first mention of symmetry, noting that a line through the center bisects the area.

XVIII. A semicircle is the figure which...

The following notes refer to variations found in historical manuscripts and commentaries on the text above.

1 Regarding the phrase "each is right": some manuscripts (B, F, and V) and commentators like Simplicius and Philoponus omit the word "is." However, the standard reading is found in other manuscripts and the works of Proclus and Hero.
2 The word "equal" is omitted by several ancient authors including Ammonius and Martianus Capella. In line 2, some versions use "line" instead of "straight line." Some scholars, such as Hero and Ammonius, swap the order of Definitions 11 and 12.
6 In manuscript V, Definition 13 is labeled as 14, and the numbering continues shifted from there. The philosopher Boethius swaps the order of Definitions 13 and 14.
7 A minor grammatical variation in the word "is" appears in some early editions.
10 The phrase "which is called the circumference" is absent in the commentaries of Proclus, Taurus, and Sextus Empiricus, though it is present in most standard manuscripts and the works of Hero and Campanus.
12 The phrase "to the circumference of the circle" is missing in many early academic sources, including Proclus and Hero, but is preserved in the main manuscript tradition.
13, 19, 20 These notes refer to minor grammatical variations and spelling differences in the Greek and Latin verbs and conjunctions across different manuscript families.