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  • MS. D'Orville 301, fol. 113v
    Bodleian Libraries, University of Oxford     Order image

  • MS. D'Orville 301, fol. 113v
    Bodleian Libraries, University of Oxford     Order image

  • MS. D'Orville 301, fol. 114r
    Bodleian Libraries, University of Oxford     Order image

  • MS. D'Orville 301, fol. 114r
    Bodleian Libraries, University of Oxford     Order image

Euclid, Elements of geometry

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AD 888

Manuscript

This is the oldest surviving manuscript of what would become the standard version of Euclid’s Elements, as re-edited in the fourth century AD by Theon of Alexandria. Theon standardized Euclid’s style and vocabulary, and expanded numerous passages with a view to helping the reader. The manuscript is open to show Euclid’s definition of the golden ratio, or divine proportion: ‘A straight line is said to have been to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.’ For millennia, the unique properties of this ratio have been explored by mathematicians, biologists, artists, architects, musicians and, more recently, economists.

Excerpt of the Elements of Euclid (in Greek), read by Cressida Ryan

Excerpt of the Elements of Euclid (in English), read by Cressida Ryan

To cut a given finite straight line in extreme and mean ratio. Let AB be the given finite straight line; thus it is required to cut AB in extreme and mean ratio. On AB let the square BC be described; and let there be applied to AC the parallelogram CD equal to BC and exceeding by the figure AD similar to BC. [VI. 29] Now BC is a square; therefore AD is also a square. And, since BC is equal to CD, let CE be subtracted from each; therefore the remainder BF is equal to the remainder AD. But it is also equiangular with it; therefore in BF, AD the sides about the equal angles are reciprocally proportional; [VI. 14] therefore, as FE is to ED, so is AE to EB. But FE is equal to AB, and ED to AE. Therefore, as BA is to AE, so is AE to EB. And AB is greater than AE; therefore AE is also greater than EB.

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Comments

What makes this a treasure?

Thank you for noticing this! I've replaced the recording on this page with the correct one but you can still hear the Sappho in Greek at http://treasures.bodleian.ox.ac.uk/Fragment-of-a-Lyric-by-Sappho.

Posted by Bodleian Libraries

On 28/05/2013

The excerpt of the Elements of Euclid (in Greek), read by Cressida Ryan, is in fact fragment No.16 of Sappho! But please let it there! Its beautiful!

Posted by Georg Joachim Schmitt

On 26/05/2013

Greek maths is one of those things we don’t study as undergraduates but repays the effort to engage with it. Using the extended alphabet the Greek system is much more subtle and refined than the Roman one, although resting on diacritical marks for interpretation, it’s still very hard to unravel. Words not pictures give us geometry, demonstrating a sensitivity to the spoken word that we might have lost. The golden mean is born from Greek concepts of balance, going back to the famous saying ‘nothing in excess’, and this has been translated into mathematical terms. Understanding ratios informed the development of music, for working out the lengths of strings and value of harmonics and intervals. This oldest edition of Euclid preserves great understanding and a fascinating system in a beautiful script.

Posted by Cressida Ryan

On 29/09/2011

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