Math 2.0

History and biography Object

Euclid's Elements

Builds on Pythagorean theoremEuclidean algorithmEuclid's theoremPlatonic solid

Description

The Elements is a mathematical treatise in thirteen books, written in Greek by Euclid of Alexandria around 300 BC. Books I to VI treat plane geometry, and Books VII to IX treat the arithmetic of whole numbers. Book X treats incommensurable magnitudes, and Books XI to XIII treat solid geometry. Book I starts from definitions, five postulates and a list of common notions, and derives every proposition from these and from earlier propositions. The fifth postulate of Book I is the parallel postulate. Book I ends with the Pythagorean theorem and its converse. Book VII contains the Euclidean algorithm, Book IX contains Euclid's theorem that there are infinitely many primes, and Book XIII constructs the five Platonic solids.

Remarks

The book is the model of the axiomatic method: a short list of first principles and a chain of proofs. Its axioms are not complete by modern standards; for example, the proof of its first proposition assumes that two circles meet without an axiom that says so. Modern axiom systems such as Hilbert's axioms fill these gaps.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Angle at a point, concept, no evidence levelAngle at a pointconceptDivisibility, definition, no evidence levelDivisibilitydefinitionEuclidean algorithm, concept, no evidence levelEuclidean algorithmconceptEuclidean distance, concept, no evidence levelEuclidean distanceconceptPlatonic solid, concept, no evidence levelPlatonic solidconceptTriangle, concept, no evidence levelTriangleconceptPrime number, definition, no evidence levelPrime numberdefinitionPythagorean theorem, statement, no evidence levelPythagorean theoremstatementExistence of a prime divisor, statement, no evidence levelExistence of a prime …statementEuclid's theorem, statement, no evidence levelEuclid's theoremstatementEuclid's Elements, object, no evidence levelEuclid's Elementsobject
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Angle at a point concept
  2. Divisibility definition
  3. Euclidean algorithm concept
  4. Euclidean distance concept
  5. Platonic solid concept
  6. Triangle concept
  7. Prime number definition
  8. Pythagorean theorem statement
  9. Existence of a prime divisor statement
  10. Euclid's theorem statement
  11. Euclid's Elements object

Further reading

  1. Web https://en.wikipedia.org/wiki/Euclid%27s_Elements
About this page
Tag83WHDAV68J
URIm2://83WHDAV68J
Kindobject
WritingW0 stub
MSC01A20, 51M04, 51-03
UsesPythagorean theorem, Euclidean algorithm, Euclid's theorem, Platonic solid
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