Euclid Book 1, Proposition 1
(with definitions, and demarcation visible)

Enunciation –  “The enunciation states what is given and what is being sought from it.”

[Given] On a given finite straight line [Sought] to construct an equilateral triangle.

                          

————————————

a given finite straight line

Slide Two: Exposition

Enunciation –  “The enunciation states what is given and what is being sought from it.”

[Given] On a given finite straight line [Sought] to construct an equilateral triangle.

Exposition – “The exposition takes separately what is given and prepares it in advance for use in the investigation.”

Let AB be the given finite straight line.

 

A—————————————— B

Slide Three: Specification

Enunciation –  “The enunciation states what is given and what is being sought from it.”

[Given] On a given finite straight line [Sought] to construct an equilateral triangle.

Exposition – “The exposition takes separately what is given and prepares it in advance for use in the investigation.”

Let AB be the given finite straight line.

Specification – “The specification takes separately the thing that is sought and makes clear precisely what it is.”

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

A—————————— B

Slide Four: Construction

Enunciation –  “The enunciation states what is given and what is being sought from it.”

[Given] On a given finite straight line [Sought] to construct an equilateral triangle.

Exposition – “The exposition takes separately what is given and prepares it in advance for use in the investigation.”

Let AB be the given finite straight line.

Specification – “The specification takes separately the thing that is sought and makes clear precisely what it is.”

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

Construction –  “The construction adds what is lacking in the given for finding what is sought.”

With centre A and distance AB let the circle BCD be described; 

again, with centre B and distance BA let the circle ACE be described; 

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

Slide Five: Proof

Enunciation –  “The enunciation states what is given and what is being sought from it.”

[Given] On a given finite straight line [Sought] to construct an equilateral triangle.

Exposition – “The exposition takes separately what is given and prepares it in advance for use in the investigation.”

Let AB be the given finite straight line.

Specification – “The specification takes separately the thing that is sought and makes clear precisely what it is.”

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

Construction –  “The construction adds what is lacking in the given for finding what is sought.”

With centre A and distance AB let the circle BCD be described; again, with centre B and distance BA let the circle ACE be described; and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined.

Proof –  “The proof draws the proposed inference by reasoning scientifically from the propositions that have been admitted.”

Now, since the point A is the centre of the circle CDB, AC is equal to AB.

Again, since the point B is the centre of the circle CAE, BC is equal to BA.

. 

But CA was also proved equal to AB; therefore each of the straight lines CA, CB is equal to AB.

And things which are equal to the same thing are also equal to one another; therefore CA is also equal to CB.

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

Slide Six: Conclusion

Enunciation –  “The enunciation states what is given and what is being sought from it.”

[Given] On a given finite straight line [Sought] to construct an equilateral triangle.

Exposition – “The exposition takes separately what is given and prepares it in advance for use in the investigation.”

Let AB be the given finite straight line.

Specification – “The specification takes separately the thing that is sought and makes clear precisely what it is.”

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

Construction –  “The construction adds what is lacking in the given for finding what is sought.”

With centre A and distance AB let the circle BCD be described; again, with centre B and distance BA let the circle ACE be described; and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined.

Proof –  “The proof draws the proposed inference by reasoning scientifically from the propositions that have been admitted.”

Now, since the point A is the centre of the circle CDB, AC is equal to AB. Again, since the point B is the centre of the circle CAE, BC is equal to BA. But CA was also proved equal to AB; therefore each of the straight lines CA, CB is equal to AB. And things which are equal to the same thing are also equal to one another; therefore CA is also equal to CB. Therefore the three straight lines CA, AB, BC are equal to one another.

Conclusion – “The conclusion reverts to the enunciation, confirming what has been proved.”

Therefore the triangle ABC is equilateral; 

and it has been constructed on the given finite straight line AB.  

(Being) what it was required to do.

Copyright 2012 by David Hirsch and Dan Van Haften; all rights reserved.