General Structure of Euclid’s Proofs

The following is taken from an exercise of the first section of Hartshorne’s Geometry: Euclid and Beyond.

Proclus describes six parts of a theorem: the enumeration, which states what is given and what is sought, the exposition, which says again what is given, often in a more specific form; the specification, which makes clear what is sought; the construction, which adds what is needed; the proof, which infers deductively what is sought from what has been previously demonstrated; and the conclusion, which confirms what has been proved.¹

We will try to identify these parts of proofs within Euclid’s Elements.²

Identifier	Term	Description
(1)	Enumeration	Euclid’s propositions always state the enumeration in the first sentence. Constructions usually take the form of “given … to construct …“, whereas theorems usually take the form of “if … then …“.
(2)	Exposition	The second paragraph/sentence usually deals with exposition and defines the relevant, given shapes via “Let.”
(3)	Specification	The third paragraph/sentence usually deals with the specification and states the goal starting with “Thus, it is required that.”
(4)	Construction	After (3), usually in the fourth paragraph, Euclid constructs a figure, one case of which is usually illustrated besides the proposition, using his postulates and previously proven propositions. This part is usually concise and lacks any explanation.
(5)	Proof	After (4), usually in the fifth paragraph, Euclid goes back to the constructed figure and demonstrates the conclusion using previously proven propositions and definitions/common notions. It utilizes the construction to argue the point.
(6)	Conclusion	After the proof, Euclid restates the conclusion/goal/proposition usually beginning with “Therefore.” After the statement, he sometimes includes “(Being) what was required to do,” “Q. E. D.,” or “Q. E. F.3

And we will look at Proposition 1 as an example.

Proposition 1

Part	Statement
(1)	On a given finite straight line to construct an equilateral triangle.
(2)	Let \(AB\) be the given finite straight line.
(3)	Thus, it is required to construct an equilateral triangle on the straight line \(AB\).
(4)	With centre \(A\) and disatnce \(AB\) let the circle \(BCD\) be described; [Post. 3] again, with centre \(B\) and distance \(BA\) let teh circle \(ACE\) be described; [Post. 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. [Post. 1]
(5)	Now, since the point \(A\) is the centre of the circle \(CDB\), \(AC\) is euqal to \(AB\). [Def. 15] Again, since the point \(B\) is the centre of the circle \(CAE\), \(BC\) is equal to \(BA\). [Def. 15] But \(CA\) was also proved equal to \(AB\). And things which are equal to the same thing are also equal to one another; therefore \(CA\) is also equal to \(CB\). [Common Notion 1] Therefore, the three straight lines \(CA, AB, BC\) are equal to one another.
(6)	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.4

1. Robin Hartshorne, “Euclid’s Geometry,” in Geometry: Euclid and Beyond, ed. K.A. Ribet, F. W. Gehring, and S. Axler (New York City, NY: Springer, 2000), 7-18. ↩

2. Euclid, Thomas Little Heath, and Dana Densmore, “BOOK I,” in Euclid’s Elements: All Thirteen Books Complete in One Volume: the Thomas L. Heath Translation (Santa Fe, NM: Green Lion Press, 2002). ↩

3. QED stands for “quod erat demonstrum,” which is Latin for “that which was to be demonstrated,” and is a phrase that is usually placed to mark the end of mathematical proofs. Whereas QEF stands for “quod erat faciendum,” which is Latin for “that which has to be done” (somewhat akin to “(being) what was required to do”), and is a translation of the greek works used by Euclid to mark the end of a construction. ↩

4. Following figure created using Geogebra and rendered using Tikz (LaTeX). ↩