The Elements

Book I, the opening

Whatever is true of the figure must follow from the postulates, and from nothing else.

Euclid of Alexandria wrote the Elements around 300 BCE. It is not a textbook of results — it is a method: name the things, grant five permissions, and construct the rest. Two thousand years of geometry were spoken in this voice.

1. 1The straightedgeGiven two points, we may join them with a unique straight line.
2. 2Produce the lineA segment may be extended as far as we need, in either direction.
3. 3The compassGiven a center and a radius, we may draw the circle they determine.
4. 4Right anglesEvery right angle is the same size. There is one rightness.
5. 5The parallel postulateIf a transversal cuts two lines so that the interior angles on one side add up to less than 180°, those two lines will meet on that side.

• I.1The equilateral triangleOn a given finite straight line, to construct an equilateral triangle.
• I.5Pons asinorumIn isosceles triangles the angles at the base are equal to one another.
• I.15Vertical anglesIf two straight lines cut one another, they make the vertical angles equal.
• I.32The angle sumIn any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles are equal to two right angles.
• I.47The Pythagorean theoremIn right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

1. I

   Foundations of the plane

2. II

   Geometric algebra

3. III

   The circle

4. IV

   Regular figures

5. V

   Proportion

6. VI

   Similarity

7. VII

   The numbers

8. VIII

   Continued proportion

9. IX

   Number and infinity

10. X

    Incommensurables

11. XI

    Solids

12. XII

    Method of exhaustion

13. XIII

    The cosmic figures