Source / 来源: Project Gutenberg

The Elements: Book I - Foundations

EuclidTranslated by Thomas L. Heath (1908)

Definitions

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.

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.

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.

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.

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.

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Postulates

Let the following be postulated:

1. To draw a straight line from any point to any point.

2. To produce a finite straight line continuously in a straight line.

3. To describe a circle with any centre and distance.

4. That all right angles are equal to one another.

5. That, if a straight line falling on two straight lines make 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.

Note: This is the famous Parallel Postulate. In modern notation: If , then the lines intersect.

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Common Notions

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

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

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

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

5. The whole is greater than the part.