Cubes, Conic Sections, and Crockett Johnson - Constructions in Euclid’s Elements

Stephanie Cawthorne (Trevecca Nazarene University) and Judy Green (Marymount University)

Many of Euclid’s propositions in his Elements were actually not what we now think of as theorems, but were “Greek problems” as described by Knorr.  In fact, such a problem appears in the very first proposition, Proposition I.1, of the Elements:  “On a given finite straight line to construct an equilateral triangle” [Euclid 1956, I:241].  Euclid's justification of Proposition I.1, as translated by Heath, reads as follows [Euclid 1956, I:241-42]:

    Let AB be the given finite straight line.

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


With centre A and distance AB let the circle BCD be described; [Postulate 3]
  again, with centre B and distance BA let the circle ACE be described; [Postulate 3]
  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. [Postulate 1]
  Now, since the point A is the centre of the circle CDB, AC is equal to AB. [Definition 15]
  Again, since the point B is the centre of the circle CAE, BC is equal to BA. [Definition 15]
  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; [Common Notion 1]
  therefore CA is also equal to CB.

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

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.

It is clear that Euclid thought of propositions based on constructions somewhat differently than other propositions since they were ended with “being what it was required to do” or in the Latin translation QEF, Quod erat faciendum (that which was to be done), rather than the more well-known QED, Quod erat demonstrandum (that which was to be proven).  It is also clear that the meaning of Euclid's equality of “straight lines” is what we now refer to as the equality of the lengths of line segments.


The diagram above appears in Heath's text [Euclid 1956, I:241], and was recreated using Geometer's Sketchpad.