Euclid index

An introduction to Euclid - Definitions

Index --- introduction --- definitions --- axioms and postulates --- propositions --- other

Before describing a mathematical system, you must define the words that you are going to use.

Euclid's definitionComment
1. A point is that which has no part.It has zero dimensions.
2. A line is breadthless length. It has one dimension.
3. The ends of a line are points.
4. A straight line is a line which lies evenly with the points on itself.Not quite sure what this means, but see Postulate 1.
5. A surface is that which has length and breadth only. It has two dimensions.
6. The edges of a surface are lines.
7. A plane surface is a surface which lies evenly with the straight lines on itself.Not quite sure what this means, , but see Postulate 5.
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. So the lines need not be straight.
9. And when the lines containing the angle are straight, the angle is called rectilinear.This website will refer to rectilinear angles as angles, since the lines are all straight.
10. When a straight line standing 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 equal one another.
16. And the point is called the centre 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. Rectilinear 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. Another name for trilateral figures is triangles.
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 angles 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 and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.'Right-angled' means all angles are right angles. Some of the definitions are different to modern meanings. We now use 'rectangle' rather than 'oblong'. A 'parallelogram' (not defined by Euclid, although used by him) has opposite sides and angles equal to one another, like the 'rhomboid', except we consider squares, rectangles and rhombuses to be parallelograms as well. The modern meaning for the trapezium (or trapezoid in US) has one pair of sides parallel. 'Quadrilaterals other than these' are just called quadrilaterals!
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.

I use certain other words in this website - click here to see their definitions.