Euclid's Elements

Euclid's Elements (Greek Στοιχεία) is a mathematical and geometric treatise, consisting of 13 books, written by the Greek mathematician Euclid around 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems) and proofs thereof. Euclid's books are in the fields of Euclidean geometry, as well as the ancient Greek version of number theory. The Elements is one of the oldest extant axiomatic deductive treatments of geometry, and has proved instrumental in the development of logic and modern science.

Later axiomizations

Surprisingly, mathematicians in the nineteenth century discovered that Euclid's proofs require additional assumptions, ones not stated among either his postulates or his common notions. For example:

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• One of his theorems is to the effect that any line segment is part of a triangle, which he constructs in the usual way, by drawing circles around both endpoints and taking their intersection and them as three corners. However, his axioms do not guarantee that the circles actually do intersect.
• It is proven by superposition that two triangles are congruent if two of their sides and the angle between them are respectively equal. The method relies on three assumptions: that the straight line making a given angle with another on one side is unique, that the straight line connecting two points is unique, and that the properties of a figure are independent of where it is situated.

David Hilbert gave a revised list containing no fewer than 23 separate axioms.

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