Euclid, Elements
Book I, Propositions 1 through 3

The following is Adapted from Euclid, The Thirteen Books of Euclid's Elements, Translated by Thomas L. Heath. This is how Euclid's masterwork begins.

Preliminaries

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.

5. A surface is that which has length and breadth only.

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

9. And when the lines containing the angle are straight, the angle is called rectilineal.

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.

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 are equal to one another;

16. And 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, 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. Rectilineal 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.

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.

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. To draw a [unique] 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 center 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.

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.

Proposition 1

On a given finite straight line [it is possible] to construct an equilateral triangle.

Let AB be the given finite straight line.

Thus, it is required to construct an equilateral triangle on the straight line AB.

With center A and distance AB, let the circle be described [Postulate (Post.) 3].

Again, with center B and distance BA, let [a second] circle be described [Post. 3].

And from the point C, in which the circles cut one another, to the points A and B let the straight lines CA and CB be joined [Post. 1 ].

Now, since the point A is the center of the circle CDB, AC is equal to AB [Definition (Def.) 15].

Again, since the point B is the center of the circle CAE, BC is equal to BA [Def. 15].

But CA was also proved equal to AB.

Therefore, each of the straight lines CA and CB is equal to AB.

And things which are equal to the same thing are also equal to one another [Common Notion (CN) 1].

Therefore, CA is also equal to CB.

Therefore, the three straight lines CA, AB and BC are equal to one another.

Therefore, the triangle ABC is equilateral; and it has been constructed on the given finite straight line AB — what it was required to do.

Proposition 2

To place at a given point (as an extremity) a straight line equal to a given straight line.

Let A be the given point; and BC the given straight line.

Thus, it is required to place at the point A (as an extremity) a straight line equal to the given straight line BC.

From the point A to the point B let the straight line AB be joined [Post. 1].

And on it, let the equilateral triangle DAB be constructed [Proposition (Prop.) 1].

Let the straight lines AE and BF be produced in a straight line with DA and DB [Post. 2].

And with center B and distance BC, let the circle be described [Post. 3].

[Let G be the point of intersection of the circle and line DF.]

And again, with center D and distance DG, let [a second] circle be described [Post. 3].

[Let H be the point of intersection of the second circle and line DE.]

Then, since the point B is the center of the [first] circle, BC is equal to BG.

Again, since the point D is the center of the [second] circle DH is equal to DG.

And in these, DA is equal to DB.

Therefore, the remainder AH is equal to the remainder BG [CN 3].

But BC was also proved equal to BG.

Therefore, each of the straight lines AH and BC is equal to BG.

And things which are equal to the same thing are also equal to one another [CN 1].

Therefore, AH is also equal to BC.

Therefore, at the given point A the straight line AH is placed equal to the given straight line BC —

what it was required to do.

Proposition 3

Given two unequal straight lines, to cut off from the greater a straight line equal to the less.

Let AB and CD be the two given unequal straight lines, and let AB be the greater of them.

Thus it is required to cut off from AB, the greater, a straight line equal to CD, the less.

At the point A, let AE be placed equal to the straight line CD [Prop. 2].

And with center A and distance AE let the circle be described [Post. 3].

[Let the point F be the intersection of the circle and line AB.]

Now, since the point A is the center of the circle, AF is equal to AE [Def. 15].

But CD is also equal to AE.

Therefore, each of the straight lines AF and CD is equal to AE; so that AF is also equal to CD [CN 1].

Therefore, given the two straight lines AB and CD, from AB, the greater, AE has been cut off equal to CD, the less — what it was required to do.