Book I, definitions, postulates and common notions

Elements is a carefully structured book. At the beginning of (almost) all thirteens books, a number of definitions are given. Most of the definitions are quite obvious to a modern reader, so in the following only the ones believed to be most useful are mentioned.

Book I is unique among the books by the fact that the definitions are followed by a few postulates (axioms that are taken to be true without proof) and some common notions (axioms about dealing with magnitudes). Euclid's fifth postulate, also called the parallel postulate or parallel axiom, is a distinctive property of Euclidean geometry and it cannot be proven from the other postulates. Many have tried, and it has therefore served as a "driving force behind the axiomatization of mathematics" [Artmann, p. 47].

Selected definitions

Definition 10. When a straight line set up on a straight line makes the adjacent angles equal to each other, each of the equal angles is right, and the straight line standing on the other is called perpendicular to that on which it stands.

Definition 11. An obtuse angle is an angle greater than a right angle.

Definition 12. An acute angle is an angle less than a right angle.

Definition 17. 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 straight line also bisects the circle.

Definition 18. A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle.

Definition 19. Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

Definition 20. Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

Definition 21. Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

Definition 22. Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong (a rectangle in modern terms) that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid (a parallelogram in modern terms) that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia (in modern British terms, a trapezium is an quadrilateral with a pair of parallel sides, read more here).

Definition 23. Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Postulates

Postulate 1. It is possible to draw a straight line from any point to any point.

Postulate 2. It is possible to produce a finite straight line continuously in a straight line.

Postulate 3. It is possible to draw a circle with any center and radius.

Postulate 4. All right angles are equal to one another.

Postulate 5. If a straight line falling on two straight lines make 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.

Selected common notions

Common notion 2. If equals be added to equals, the wholes are equal.

Common notion 3. If equals be subtracted from equals, the remainders are equal.

Common notion 5. The whole is greater than the part.