Euclid's Elements

mathematical treatise by Euclid

Euclid's Elements (sometimes: The Elements, Greek: Στοιχεῖα Stoicheia) is a large set of math books about geometry and numbers, written by the ancient Greek mathematician known as Euclid (c.325 BC–265 BC) in Alexandria (Egypt) circa 300 BC.[1] The set has 13 volumes, or sections, and has been printed often as 13 physical books (numbered I-XIII), rather than one large book. It has been translated into Latin, with the title "Euclidis Elementorum". It is the most famous mathematical text from ancient times.[1]

The title page of Sir Henry Billingsley's first English version of Euclid's Elements, in 1570.

Euclid collected together all that was known of geometry in his time. His Elements is the main source of ancient geometry. Textbooks based on Euclid have been used up to the present day. In the book, he starts out from a small set of axioms (that is, a group of things that everyone thinks are true). Euclid then shows the properties of geometric objects and of whole numbers, based on those axioms.

The Elements also includes works on perspective, conic sections, spherical geometry, and possibly quadric surfaces. Apart from geometry, the work also includes number theory. Euclid came up with the idea of greatest common divisors. They were in his Elements. The greatest common divisor of two numbers is the greatest number that can divide evenly into both of the two numbers.

The geometrical system described in the Elements was long known simply as "geometry" and was considered to be the only geometry possible. Today, that system is referred to as Euclidean geometry, to distinguish it from other so-called non-Euclidean geometries which mathematicians discovered in the 19th century.

Added volumes XIV and XV


Occasionally in ancient times, writings were attributed to celebrated authors but were not written by them. It is in this way that the apocryphal books XIV and XV of the Elements were sometimes included in the collection.[2] The spurious Book XIV was probably written by Hypsicles on the basis of a treatise by Apollonius of Perga. The book continues Euclid's comparison of regular solids inscribed in spheres. The chief result is that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes.

The spurious Book XV was probably written, at least in part, by Isidore of Miletus. This book covers topics such as counting the number of edges and solid angles in the regular solids, and finding the measure of dihedral angles of faces that meet at an edge.[2]


The Italian Jesuit Matteo Ricci (left) and the Chinese mathematician Xu Guangqi (right) published the Chinese edition of Euclid's Elements (幾何原本) in 1607.
  • 1460s, Regiomontanus (incomplete)
  • 1533, editio princeps by Simon Grynäus
  • 1557, by Jean Magnien and Pierre de Montdoré, reviewed by Stephanus Gracilis (only propositions, no full proofs, includes original Greek and the Latin translation)
  • 1572, Commandinus
  • 1574, Christoph Clavius


  • 1505, Bartolomeo Zamberti (Latin)
  • 1543, Venturino Ruffinelli (Italian)
  • 1555, Johann Scheubel (German)
  • 1557, Jean Magnien and Pierre de Montdoré, reviewed by Stephanus Gracilis (Greek to Latin)
  • 1562, Jacob Kündig (German)
  • 1564, Pierre Forcadel de Béziers (French)
  • 1570, Henry Billingsley (English)
  • 1576, Rodrigo de Zamorano (Spanish)
  • 1594, Typografia Medicea (edition of the Arabic translation of Nasir al-Din al-Tusi)
  • 1607, Matteo Ricci, Xu Guangqi (Chinese)
  • 1660, Isaac Barrow (English)
  • 1720s Jagannatha Samrat (Sanskrit, based on the Arabic translation of Nasir al-Din al-Tusi)[3]
  • 1738, Ivan Satarov (Russian from French)
  • 1780, Baruch Ben-Yaakov Mshkelab (Hebrew)[4]
  • 1807, Józef Czech (Polish based on Greek, Latin and English editions)

Currently in print


Euclid's Elements – All thirteen books in one volume, based on Heath's translation, Green Lion Press, ISBN 978-1-888009-18-7.

The Elements: Books I-XIII-Complete and Unabridged (2006), translated by Sir Thomas Heath, Barnes & Noble, ISBN 978-0-7607-6312-4.

  1. 1.0 1.1 "Euclid's Elements of Geometry",, February 2, 2011, web: UT-Euclid. Archived 2014-04-19 at the Wayback Machine
  2. 2.0 2.1 Boyer (1991). Euclid of Alexandria. pp. 118–119.
  3. K. V. Sarma (1997), Helaine Selin (ed.), Encyclopaedia of the history of science, technology, and medicine in non-western cultures, Springer, pp. 460–461, ISBN 978-0-7923-4066-9
  4. JNUL Digitized Book Repository,, 2011, web: [1]. Archived 2009-06-22 at