Euclid, Geometry. Book I, Proposition 1

In this lesson, we study the demonstration of the first proposition in Euclid’s Geometry (Proposition 1.1). The analysis of the proposition is provided below. To complete the objectives of this lesson, complete the following tasks:

1. Review the memorization of the elements of Book I.
2. Study the proposition for mastery.
3. Identify the parts of the proposition.
4. Complete the lesson assessment.

Lesson

1. Identify the Proposition studied in this lesson.
   In this proposition, we are studying Proposition 1 of Book I, i.e., Proposition 1.1.
2. Summarize what has been studied in recent/previous propositions to prepare for this proposition.
   This is the first proposition.
3. State the proposition. Note that this should be memorized.
   On a given finite straight line to construct an equilateral triangle. (Memorize)
4. Is this proposition a problem or a theorem? Prove your answer.
   This proposition is a problem because something is to be done.
5. Give the exposition of this proposition–if there is one. This should clarify what is given.
   Let AB be the given finite straight line.
6. Give the determination of this proposition–if there is one. This should clarify what is sought.
   It is required to construct an equilateral triangle on the straight line AB.
7. Give the construction of this proposition–if there is one. There is often no construction for theorems.
   1. With centre A and distance AB let the circle BCD be described (Post. 3); 
   2. Again, with centre B and distance BA let the circle ACB be described (Post. 3).[efn_note]Footnote content.[/efn_note]
   3. From the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined. (Post. 1)
8. Give the demonstration of this proposition, separating each syllogism if possible.
   1. Since the point A is the centre of the circle CDB, AC is equal to AB. (Def. 15).
   2. Since the point B is the centre of the circle CAB, BC is equal to BA. (Def. 15).
   3. Things which are equal to the same thing are also equal to one another; (C.N. 1).  
   4. Therefore the three straight lines CA, AB, BC are equal to one another. 
9. Give the conclusion of this proposition.
   The triangle ABC is equilateral and it has been constructed on the given finite straight line AB. Which was to be done. (Q.E.F.)

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