Definitions

Definitions

1. A point has no part.

2. A line is a length with no width.

3. The ends 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 width only.

6. The edges 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 2 lines in a plane which meet one another and do not lie in a straight line.

9. When the lines containing the angle are straight, the angle is called rectilinear.

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 shape 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 on it from one point among those lying within the figure equal one another.

16. 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. Such a straight line also bisects the circle.

18. A semicircle is the shape contained by the diameter and the circumference cut off by it. The semicircle’s center is the same as that of the circle.

19. Rectilinear shapes are those which are contained by straight lines

• Trilateral shapes being those contained by 3
• Quadrilateral shapes are those contained by 4
• Multilateral shapes are those contained by more than 4 straight lines

20. Of trilateral shapes:

• an equilateral triangle has its 3 sides equal
• an isosceles triangle has 2 of its sides alone equal
• a scalene triangle has its 3 sides unequal

21. Of trilateral shapes:

• a right-angled triangle has a right angle
• an obtuse-angled triangle has an obtuse angle
• an acute-angled triangle has its 3 angles acute.

22. Of quadrilateral shapes:

• a square is both equilateral and right-angled
• an oblong is right-angled but not equilateral
• a rhombus is equilateral but not right-angled
• a rhomboid has its opposite sides and angles equal to one another but is neither equilateral nor right-angled.
• a trapezia is quadrilateral other than these.

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

Let the following be postulated:

1. A straight line connects any point to any point.

2. A finite straight line can be extended continuously.

3. A center and a radius draws a circle.

4. All right angles equal one another.

5. If a straight line that falls on 2 straight lines makes the interior angles on the same side less than 2 90 degree angles, then the 2 straight lines going indefinitely will meet on that side which have angles less than 2 90 degrees.

Common Notions

1. Things which equal the same thing also equal one another.

2. If equals are added to equals, then the wholes are equal.

3. If equals are subtracted from equals, then the remainders are equal.

4. Things which coincide with one another equal one another.

5. The whole is greater than the part.