Definition:Euclid's Definitions

These definitions appear at the beginning of the various Books of Euclid's.

Book I

 * 1) A point is that which has no part.
 * 2) A line is breadthless length.
 * 3) The extremities of a line are points.
 * 4) A straight line is a line which lies evenly with the points on itself.
 * 5) A surface is that which has length and breadth only.
 * 6) The extremities of a surface are lines.
 * 7) A plane surface is a surface which lies evenly with the straight lines on itself.
 * 8) 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.
 * 9) And when the lines containing the angle are straight, the angle is called rectilineal.
 * 10) 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.
 * 11) An obtuse angle is an angle greater than a right angle.
 * 12) An acute angle is an angle less than a right angle.
 * 13) A boundary is that which is an extremity of anything.
 * 14) A figure is that which is contained by any boundary or boundaries.
 * 15) 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;
 * 16) And the point is called the center of the circle.
 * 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.
 * 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.
 * 19) 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.
 * 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.
 * 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 sides acute.
 * 22) 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.
 * 23) 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.

Book II

 * 1) Any rectangular parallelogram is said to be contained by the two straight lines containing the right angle.
 * 2) And in any parallelogrammic area let any one whatever of the parallelograms about its diameter with the two complements be called a gnomon.

Book III

 * 1) Equal circles are those the diameters of which are equal, or the radii of which are equal.
 * 2) A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.
 * 3) Circles are said to touch one another which, meeting one another, do not cut one another.
 * 4) In a circle straight lines are said to be equally distant from the center when the perpendiculars drawn to them from the center are equal.
 * 5) And that straight line is said to be at a greater distance on which the greater perpendicular falls.
 * 6) A segment of a circle is the figure contained by a straight line and a circumference of a circle.
 * 7) An angle of a segment is that contained by a straight line and a circumference of a circle.
 * 8) An angle in a segment is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line which is the base of the segment, is contained by the straight lines so joined.
 * 9) And, when the straight lines containing the angle cut off a circumference, the angle is said to stand upon that circumference.
 * 10) A sector of a circle is the figure which, when an angle is constructed at the center of the circle, is contained by the straight lines containing the angle and the circumference cut off by them.
 * 11) Similar segments of circles are those which admit equal angles, or in which the angles are equal to one another.