Chapter: Euclid's “Elements”

Euclid's “Elements” is a mathematical and geometric treatise comprising about 500 pages and consisting of 13 books written by the ancient Greek mathematician Euclid in Alexandria ca. 300 BC. It is a collection of definitions, postulates (axioms), common notions (unproved lemmata), propositions and lemmata (i.e. theorems and constructions), corollaries (for which in some editions the Greek word "porisms" is used) and mathematical proofs of the propositions. The 13 books cover geometry, now known as Euclidean, and the ancient Greek version of elementary number theory. The work also includes an algebraic system that has become known as algebraic geometry, which is powerful enough to solve many algebraic problems, including the problem of finding the square root of a number.

The “Elements” are still considered a masterpiece in the application of logic in mathematics. In historical context, the work has proven enormously influential in many areas of science. Scientists like Mikołaj Copernicus, Johannes Kepler, Galileo Galilei, and Isaac Newton were all influenced by the “Elements”, and applied their knowledge of this work to their work. Mathematicians and philosophers, such as Bertrand Russell (1872 - 1970), Alfred Whitehead (1861 - 1947), and Baruch Spinoza, have attempted to create their own foundational “Elements” for their respective disciplines by adopting the axiomatized deductive structures that Euclid’s work introduced.

Over the years, the “Elements” have been copied, recopied, modified, commented upon and interpreted unceasingly. Only the painstaking comparison of all available sources allowed Heiberg in 1888 to essentially reconstruct the original version. The most important source (M.S. 190 ; this manuscript dates from the 10th century) was discovered in the treasury of the Vatican when Napoleon’s troops invaded Rome in 1809. Heiberg’s text has been translated into all scientific languages.

An important English translation is a translation by Sir Thomas L.Heath in 1908 (second enlarged edition 1926). However, for a modern reader, it is hard to read because it uses an archaic state of the English language. A more modern English translation[^6419] was done by Prof. Richard Fitzpatrick (University of Texas at Austin) in 2007, and other considerable efforts to use a more modern mathematical language for the “Elements” have been made by other authors like Daniel Callahan[^626], or the authors of proofwiki, who, for instance, contributed titles to the propositions and definitions, missing in the original version. Those titles have also been used in the following edition.

A unique feature of this online edition of the “Elements” BookofProofs is that it combines the modern English translation1 of the original text[^6419] with supplemental modern versions of the theorems and definitions. We hope that this unique feature of this online edition will help the reader to better understand and appreciate the timeless achievements of Euclid's original work.

  1. Section: Book 01: Fundamentals of Plane Geometry Involving Straight Lines
  2. Section: Book 02: Fundamentals of Geometric Algebra
  3. Section: Book 03: Fundamentals of Plane Geometry Involving Circles
  4. Section: Book 04: Circles: Inscription and Circumscription
  5. Section: Book 05: Proportion
  6. Section: Book 06: Similar Figures
  7. Section: Book 07: Elementary Number Theory
  8. Section: Book 08: Continued Proportion
  9. Section: Book 09: Applications of Number Theory
  10. Section: Book 10: Incommensurable Magnitudes
  11. Section: Book 11: Elementary Stereometry
  12. Section: Book 12: Proportional Stereometry
  13. Section: Book 13: Platonic Solids

Chapters: 1 2
Parts: 3 4 5 6
Persons: 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Theorems: 24

Thank you to the contributors under CC BY-SA 4.0!



Adapted from CC BY-SA 3.0 Sources:

  1. Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014


  1. Health, T.L.: "The Thirteen Books of Euclid's Elements - With Introduction and Commentary by T. L. Health", Cambridge at the University Press, 1968, Vol 1, 2, 3

Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from (Public Domain)

  1. Heiberg, I.L, Menge H: "Euclidis Opera Omnia" 1883


  1. Please note that Prof. Fitzpatrick's original translation is not (!) licensed under "CC BY-SA 4.0", however, extracts from this translation have been compiled in this online edition which is (!) made available under the "CC BY-SA 4.0" license with the kind permission of Prof. Richard Fitzpatrick.