# Advanced Arithmetic from Twelfth-Century al-Andalus, Surviving Only (and Anonymously) in Latin Translation? A Narrative That Was Never Told

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## Abstract

As Ahmed Djebbar has pointed out, 11th-century and earlier al-Andalus produced a “solid research tradition in arithmetic”. So far, no continuation of this tradition has been known, but analysis of three sections of two Latin works suggest that they borrow material that can hardly come from elsewhere:
1. The Liber mahameleth, likely to be a more or less free translation made by Gundisalvi or somebody close to him of an Arabic original presenting “mu a¯mala¯t mathematics vom höheren Standpunkt aus” contains systematic variations, for instance of proportions p:g :: P:G (claimed to deal with prices and goods), where the givens may be sums, differences, products, sums or differences of square roots, etc., solved sometimes by means of algebra,
sometimes with appeals to Elements II.5–6, often after reduction by means of proportion techniques.
2. A passage in Chapter 12 of the Liber abbaci first presents the simple version of the recreational problem about the “unknown heritage” (likely to be of late Ancient or Byzantine origin): a father leaves to his first son 1 monetary unit and 1/n of what remains, to the second 2 units and 1/n of what remains, etc.; in the end, all get the same, and nothing remains. Next it goes on with complicated cases where the arithmetical series is not proportional to 1 – 2 – 3 ..., and the fraction is not an aliquot part. Fibonacci gives an algebraic solution to one variant and also general formulae for all variants – but these do not come from his algebra, and he thus cannot have derived them himself. A complete survey of occurrences once again points to al-Andalus.
3. Chapter 15 Section 1 of Fibonacci’s Liber abbaci mainly deals with the ancient theory of means though not telling so. If M is one such mean between A and B, it is shown systematically how each of these three numbers can be found if the other two are given – once more by means of algebra, Elements II.5–6, and proportion techniques. The lettering shows that Fibonacci uses an Arabic or Greek source, but no known Arabic or Greek work contains anything similar. However, the structural affinity suggests inspiration from the same environment as produced the Liber mahameleth. So, this seems to be a non-narrative, a story that was not revealed by the participants, and was not discovered by historians so far.
Originalsprog Engelsk Premodern Translation : Comparative Approaches to Cross-Cultural Transformations Sonja Brentjes, Alexander Fidora Belgien Brepols Publishers 2021 33-61 978-2-503-59097 978-2-503-59098-1 Udgivet - 2021
Navn Contact and Transmission 2