Definition:Euclid's Definitions - Book I

These definitions appear at the start of Book I of by Euclid.


 * 1) A 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.
 * 2) And when the lines containing the angle are straight, the angle is called rectilineal.
 * 3) When 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.
 * 4) An obtuse angle is an angle greater than a right angle.
 * 5) An acute angle is an angle less than a right angle.
 * 6) A boundary is that which is an extremity of anything.
 * 7) A figure is that which is contained by any boundary or boundaries.
 * 8) A 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;
 * 9) And the point is called the center of the circle.
 * 10) 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.
 * 11) 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.
 * 12) Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral figures those contained by four, and multi-lateral those contained by more than four straight lines.
 * 13) 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.
 * 14) 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 sides acute.
 * 15) Of 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 equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.
 * 16) Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in either direction, do not meet one another in either direction.
 * 1) 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.
 * 2) 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.
 * 3) Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral figures those contained by four, and multi-lateral those contained by more than four straight lines.
 * 4) 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.
 * 5) 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 sides acute.
 * 6) Of 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 equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.
 * 7) Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in either direction, do not meet one another in either direction.