BOOK I

DEFINITIONS

1. A point is that which has no parts.
2. A line is that which has length without breadth.
3. The limits of a line are points.
4. A straight line is that which lies equally to the points on it.
5. A surface is that which has only length and breadth.
6. The limits of a surface are lines.
7. A plane surface is that which lies equally to the straight lines on it. . . .

Let it be granted:

1. That a straight line may be drawn from any one point to any other point.
2. And that a finite straight line may be extended to any length in that straight line.
3. And that a circle may be drawn from any centre, at any distance from that centre.
4. And that all right angles are equal to each other.
5. And that if a straight line crosses two straight lines in such a way that the interior angles on one side of it add up to less than two right angles, these straight lines will meet if extended to infinity on the side in which the angles add up to less than two right angles.

1. Things which are equal to the same thing are equal to one another.
2. And, if equals are added to equals, the wholes are equal.
3. And, if equals are taken from equals, the remainders are equal.
4. And, if equals are added to unequals, the wholes are unequal.
5. And doubles of one and the same thing are equal to each other.
6. And halves of one and the same thing are equal to each other.
7. And magnitudes which can be made to coincide with one another are equal.
8. And the whole is greater than the part.
9. And two straight lines cannot enclose a space.

1. To draw an equilateral triangle on a given finite straight line.

Suppose there is given a finite straight line AB. The task is to construct an equilateral triangle on the finite straight line AB. Draw a circle BCD, with its centre at A, and a radius of AB. Then draw another circle ACE, with its centre at B. and a radius of BA. Then draw straight lines, CA and CB, from C, which is the point of intersection between the two circles.

Now, since the point A is the centre of the circle CDB, AC will be equal to AB. Again, since the point B is the centre of the circle CAE, BC will be equal to BA. But it has been proved that CA is also equal to AB. So both CA and CB are equal to the straight line AB. But things which are equal to one and the same thing are also equal to each other [Axiom 1]. Therefore CA is also equal to CB. Therefore CA, AB, and BC are equal. Therefore the triangle ABC is equilateral, and it has been constructed on the given finite straight line AB. This was the task.

[And so on with another 47 propositions in Book I alone.]

Go to Index to Euclid