Book I · Foundations
Definitions
ὅροι · 23 in all, naming the things the propositions talk about
- 1A point is that which has no part.
- 2A line is breadthless length.
- 3The ends of a line are points.
- 4A straight line is a line which lies evenly with the points on itself.
- 5A surface is that which has length and breadth only.
- 6The edges of a surface are lines.
- 7A plane surface is a surface which lies evenly with the straight lines on itself.
- 8A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.
- 9And when the lines containing the angle are straight, the angle is called rectilineal.
- 10When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
- 11An obtuse angle is an angle greater than a right angle.
- 12An acute angle is an angle less than a right angle.
- 13A boundary is that which is an extremity of anything.
- 14A figure is that which is contained by any boundary or boundaries.
- 15A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.
- 16And the point is called the centre of the circle.
- 17A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.
- 18A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle.
- 19Rectilineal 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.
- 20Of 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.
- 21Further, 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.
- 22Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid 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.
- 23Parallel 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
αἰτήματα · 5 in all; the first three are the straightedge and the compass
- 1To draw a straight line from any point to any point.
- 2To produce a finite straight line continuously in a straight line.
- 3To describe a circle with any centre and distance.
- 4That all right angles are equal to one another.
- 5That, if a straight line falling on two straight lines 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.
Common Notions
κοιναὶ ἔννοιαι · 5 in all, the general truths about equality
- 1Things which are equal to the same thing are also equal to one another.
- 2If equals be added to equals, the wholes are equal.
- 3If equals be subtracted from equals, the remainders are equal.
- 4Things which coincide with one another are equal to one another.
- 5The whole is greater than the part.