While Euclid’s list of axioms in the “Elements” is not exhaustive, it represents the most important principles. His proofs often invoke axiomatic notions, which were not originally presented in his list of axioms. Later editors have interpolated Euclid’s implicit axiomatic assumptions in the list of formal axioms.
For example, in the first construction of Book 1, Euclid uses a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points. Later, in the fourth construction, he uses superposition (moving the triangles on top of each other) to prove that if two sides and their angles are equal then they are congruent. During these considerations, he uses some properties of superposition, but these properties are not constructed explicitly in the treatise. If superposition is to be considered a valid method of geometric proof, all of the geometry would be full of such proofs. For example, propositions 1.1 – 1.3 can be proved trivially by using superposition.
Mathematician and historian W. W. Rouse Ball puts these criticisms in perspective, remarking that “the fact that for two thousand years The Elements was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose.”
The following table lists results from this book which are also known in modern mathematics, but which were proven by Euclid purely geometrically about 2500 years ago:
|Euclid's Elements||Corresponding Contemporary Results|
|Prop 1.04||Congruent triangles "Side-Angle-Side"|
|Prop 1.08||Congruent triangles "Side-Side-Side"|
|Prop 1.26||Congruent triangles "Angle-Side-Angle" and "Angle-Angle-Side"|
|Prop 1.20||triangle inequality.|
|Prop 1.32||Sum of angles in a triangle in plane geometry|
|Prop 1.47 and Prop 1.48||Pythagorean theorem and its converse|
Moreover, this book contains the following compass and ruler constructions: