World Library  
Flag as Inappropriate
Email this Article


Article Id: WHEBN0001777555
Reproduction Date:

Title: Arithmetica  
Author: World Heritage Encyclopedia
Language: English
Subject: History of algebra, Fermat's Last Theorem, Diophantus, Algebraic number theory, Number theory
Collection: 3Rd-Century Books, Ancient Greek Mathematical Works, History of Algebra
Publisher: World Heritage Encyclopedia


Cover of the 1621 edition, translated into Latin from Greek by Claude Gaspard Bachet de Méziriac.

Arithmetica (Greek: Ἀριθμητικά) is an Ancient Greek text on mathematics written by the mathematician Diophantus in the 3rd century AD.[1] It is a collection of 130 algebraic problems giving numerical solutions of determinate equations (those with a unique solution) and indeterminate equations.

Equations in the book are called Diophantine equations. The method for solving these equations is known as Diophantine analysis. Most of the Arithmetica problems lead to quadratic equations. It was these equations which inspired Pierre de Fermat to propose Fermat's Last Theorem, scrawled in the margins of Fermat's copy of Arithmetica, which states that the equation x^n+y^n=z^n, where x, y, z and n are non-zero integers, has no solution with n greater than 2.

In Book 3, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares or cubes. In book 4, he finds rational powers between given numbers. He also noticed that numbers of the form 4n + 3 cannot be the sum of two squares. Diophantus also appears to know that every number can be written as the sum of four squares. If he did know this result (in the sense of having proved it as opposed to merely conjectured it), his doing so would be truly remarkable: even Fermat, who stated the result, failed to provide a proof of it and it was not settled until Joseph Louis Lagrange proved it using results due to Leonhard Euler.

Arithmetica was originally written in thirteen books, but the Greek manuscripts that survived to the present contain no more than six books.[2] In 1968, Fuat Sezgin found four previously unknown books of Arithmetica at the shrine of Imam Rezā in the holy Islamic city of Mashhad in northeastern Iran.[3] The four books are thought to have been translated from Greek to Arabic by Qusta ibn Luqa (820–912).[2] Norbert Schappacher has written:

[The four missing books] resurfaced around 1971 in the Astan Quds Library in Meshed (Iran) in a copy from 1198 AD. It was not catalogued under the name of Diophantus (but under that of Qust¸a ibn Luqa) because the librarian was apparently not able to read the main line of the cover page where Diophantus’s name appears in geometric Kufi calligraphy.[4]

Arithmetica became known to mathematicians in the Islamic world in the tenth century[5] when Abu'l-Wefa translated it into Arabic.[6]

See also


  1. ^ "Diophantus of Alexandria (Greek mathematician)". Encyclopaedia Britannica. Retrieved 11 April 2013. 
  2. ^ a b Magill, Frank N., ed. (1998). Dictionary of World Biography 1.  
  3. ^  
  4. ^ Schappacher, Norbert (April 2005). "Diophantus of Alexandria : a Text and its History" (PDF). p. 18. Retrieved 2015-10-09. 
  5. ^  
  6. ^  

External links

Diophantus Alexandrinus, Pierre de Fermat, Claude Gaspard Bachet de Meziriac, Diophanti Alexandrini Arithmeticorum libri 6, et De numeris multangulis liber unus. Cum comm. C(laude) G(aspar) Bacheti et observationibus P(ierre) de Fermat. Acc. doctrinae analyticae inventum novum, coll. ex variis eiu. Tolosae 1670, doi:10.3931/e-rara-9423.

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.