The Elements

[Left Page]

Geometric diagrams illustrating definitions: a circle with its center and a radius; three parallel vertical lines; two lines intersecting at an acute angle; and a single horizontal line segment at the bottom.

[illegible marginal notes, likely citations or references to other parts of the text]

... to describe a circle. ...?
... parallel straight lines are those which ...?

1. Things equal to the same thing are also equal to one another.
2. And if equals be added to equals, the wholes are equal.
3. And if equals be subtracted from equals, the remainders are equal.
4. And if equals be added to unequals, the wholes are unequal.
5. And things which are double of the same thing are equal to one another.
6. And things which are halves of the same thing are equal to one another.
7. And things which coincide with one another are equal to one another.
8. And the whole is greater than the part.
9. And two straight lines do not contain a space.

1. Let it be requested to draw a straight line from any point to any point.
2. And to produce a finite straight line continuously in a straight line.
3. And to describe a circle with any center and radius.

[Right Page]

3
1

Geometric diagram for Euclid's Elements, Book I, Proposition 1. It shows two intersecting circles of equal radius (AB), each passing through the center of the other. An equilateral triangle ΑΒΓ is inscribed using the centers Α and Β and one of the intersection points Γ.

A circular library stamp or institutional seal is visible in the center of the page.

4. And all right angles are equal to one another.
5. And if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on which are the angles less than the two right angles.

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Proposition 1.

On a given finite straight line to construct an equilateral triangle.

Let AB be the given finite straight line. It is required to construct an equilateral triangle on the straight line AB. Let the circle BΓΔ be described with center A and distance AB, and again let the circle ΑΓΕ be described with center B and distance BA. And from the point Γ, at which the circles cut one another, let the straight lines ΓΑ and ΓΒ be joined to the points A and B.

And since the point A is the center of the circle BΓΔ, ΑΓ is equal to ΑΒ. Again, since the point B is the center of the circle ΑΓΕ, ΒΓ is equal to ΒΑ. But ΓΑ was also shown to be equal to ΑΒ. Therefore each of ΓΑ and ΓΒ is equal to ΑΒ. And things equal to the same thing are also equal to one another. Therefore ΓΑ is also equal to ΓΒ. Therefore the three lines ΓΑ, ΑΒ, and ΒΓ are equal to one another.

Therefore the triangle ΑΒΓ is equilateral; and it has been constructed on the given finite straight line ΑΒ.