Elements of GeometryEuclid
About Elements of Geometry
Euclid compiled the around 300 BC in Alexandria. The work is not a collection of original discoveries but a systematic organization of Greek mathematical knowledge into a deductive structure so rigorous that it served as the model of demonstrative reasoning for over two thousand years.
The begins with definitions, postulates, and common notions, then derives propositions by logical necessity. Book I establishes the basic theorems of plane geometry, culminating in the Pythagorean theorem. Books II through IV treat geometric algebra, circles, and inscribed figures. Books V and VI present Eudoxus's theory of proportion, which handles incommensurable magnitudes without the concept of irrational numbers. Books VII through IX develop number theory, including the proof that primes are infinite. Book X classifies irrational magnitudes. Books XI through XIII construct the geometry of solids, ending with the proof that there are exactly five regular polyhedra.
The fifth postulate (the parallel postulate) attracted special attention because it seemed less self-evident than the others. Attempts to derive it from the remaining postulates continued for centuries and failed. Their failure led, in the nineteenth century, to the discovery of non-Euclidean geometries, which revealed that Euclid's system describes one possible geometry among others, not the necessary structure of space itself.