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 Nicolaus Copernicus, Johannes Kepler, Galileo Galilei, and Sir 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, Alfred North Whitehead, 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 the 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 English language.
A more modern English translation6419 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 Callahan626 and authors of the proofwiki.
In BookOfProofs, (as a still on-going project – please feel free to contribute), the “Elements” are presented combining both forms:
- Including the English translation of the original text1, based on the version provided by Prof. Richard Fitzpatrick6419, with his kind permission.2
- Whenever appropriate, a modern formulation is added, based on “CC BY-SA 3.0”-licenced work available from Daniel Callahan626, authors of proofwiki, the public domain version of the commented translation of Sir Health2775, and the own work of authors of BookOfProofs.
1 The Greek text of J.L. Heiberg (1883–1885) from Euclidis Elementa, edidit et Latine interpretatus est I.L. Heiberg, in aedibus B.G. Teubneri.
2 Please note that his original translation work6419 is not (!) “CC BY-SA 3.0”-licenced, however, extracts compiled from his work in BookOfProofs are under this licence, with the kind permission of Prof. Richard Fitzpatrick.
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6291.Book 01: Fundamentals of Plane Geometry Involving Straight-Lines
6302.Book 02: Fundamentals of Geometric Algebra
6323.Book 03: Fundamentals of Plane Geometry Involving Circles
6334.Book 04: Circles: Inscription and Circumscription
6355.Book 05: Proportion
6376.Book 06: Similar Figures
18567.Book 07: Elementary Number Theory
18578.Book 08: Continued Proportion
18589.Book 09: Applications of Number Theory
185910.Book 10: Incommensurable Magnitudes
186011.Book 11: Elementary Stereometry
186112.Book 12: Proportional Stereometry
186213.Book 13: Platonic Solids
This work is a derivative of:
 Callahan, Daniel: “Euclid’s ‘Elements’ Redux”, http://starrhorse.com/euclid/, 2014
 Fitzpatrick, Richard: “Euclid’s Elements of Geometry”, http://farside.ph.utexas.edu/Books/Euclid/Euclid.html, 2007
 Heiberg, I.L, Menge H: “Euclidis Opera Omnia”, https://archive.org/details/euclidisoperaomn01eucluoft, 1883
Bibliography (further reading)
 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