Skip to main content Accessibility help
×
Home
Hostname: page-component-55597f9d44-rn2sj Total loading time: 0.544 Render date: 2022-08-18T20:51:36.898Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Ex aequali Ratios in the Greek and Arabic Euclidean Traditions

Published online by Cambridge University Press:  24 October 2008

Gregg De Young
Affiliation:
The American University in Cairo, 113 Sharia Kasr el Aini, P.O. Box 2511, Cairo, Egypt

Abstract

Euclid discusses the ex aequali relationship twice in the Elements. The first is in Book V (based on definitions 17 and 18, propositions 22 and 23), during his discussion of arithmetical relations between mathematical magnitudes in general. The second is in Books VII–IX (developed using proposition VII,14), where he focuses on arithmetical relations in the case of numbers only. Although the distinction between mathematical magnitudes in general and numbers in particular often seems somewhat forced to contemporary philosophers, it was apparently very real to Euclid. Because Euclid seemed so conscious of the differences between the subject matter of Book V (magnitudes) and Books VII–IX (numbers), he was not much troubled by the differences between his treatment of ex aequali ratios in these two contexts. Later generations of mathematicians, however, found these differences less acceptable and tried to minimize them in various ways. This paper summarizes Euclid's use of the ex aequali relation in developing his mathematics. The paper then outlines the fate of the post-Theonine Greek attempts to “improve” the Euclidean discussion when the Elements entered the Arabic/Islamic intellectual tradition. The study concludes with the attempts by Ibn al-Hayṯam and Ibn al-Sarī to improve the parallelism between the discussions of ex aequali ratios in Book V and Book VII.

Euclide discute de la relation ex aequali deux fois dans les Éléments: la première fois dans le Livre V (fondé sur les définitions 17 et 18, les propositions 22 et 23), au cours de sa discussion des relations arithmétiques entre les grandeurs mathématiques en général; la seconde fois dans les Livres VII–IX (développement qui utilise la proposition VII, 14), où il se concentre sur les relations arithmétiques dans le cas des nombres seulement. Bien que la distinction entre grandeurs mathématiques en général et nombres en particulier paraisse souvent quelque peu forcée aux yeux de philosophes contemporains, elle fut apparemment très réelle pour Euclide. Parce qu'Euclide semblait être aussi conscient des differences entre le sujet du Livre V (grandeurs) et les Livres VII–IX (nombres), il ne fut pas très gêné par les différences entre son traitement des rapports ex aequali en ces deux contextes. Cependant des générations ultérieures de mathématiciens trouvèrent ces différences moins acceptables et essayèrent de les minimiser de diverses manières. Cet article présente en résumé l'utilisation par Euclide du rapport ex aequali dans le développement de ses mathématiques. Ensuite l'article met en évidence le destin des tentatives grecques post-Théonines visant à “améliorer” la discussion euclidienne quand les Éléments firent leur entrée dans la tradition intellectuelle arabo-musulmane. L'étude conclut avec les tentatives d'Ibn al-Haytham et d'Ibn al-Sarī visant à améliorer le parallélisme entre les discussions des rapports ex aequali dans le Livre V et le Livre VII.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)
2
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Ex aequali Ratios in the Greek and Arabic Euclidean Traditions
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Ex aequali Ratios in the Greek and Arabic Euclidean Traditions
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Ex aequali Ratios in the Greek and Arabic Euclidean Traditions
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *