Euclidean Geometry

Chapter: Euclid's “Elements”

Introduction

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.Heath²⁷⁷⁵ 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.

References

[626] Callahan, Daniel; Casey, John: “Euclid’s ‘Elements’ Redux”, http://starrhorse.com/euclid/, 2014
[6419] Fitzpatrick, Richard: “Euclid’s Elements of Geometry”, http://farside.ph.utexas.edu/Books/Euclid/Euclid.html, 2007
[2775] 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

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.