Intermediaries between Abū Kāmil's and Fibonacci's algebras – lost but leaving indubitable traces

It is an oft-repeated claim that Leonardo Fibonacci's algebra borrows from Abū Kāmil. A thorough analysis of the Liber abbaci as a whole and, in particular of course, of its chapter 15 part 3, the algebra, confirms this; but it also shows that at least many, plausibly all of the borrowings are indirect, and almost certainly that at least one group of borrowings go via a Latin intermediary which is not itself a direct translation of Abū Kāmil's algebra.

In order to see that one has, firstly, to know whether Fibonacci was faithful or creative when borrowing. Elsewhere in the Liber abbaci he demonstrates to be deliberately faithful. Taking that into account in the analysis of his algebra confirms that a number of problems come from Abū Kāmil but not directly. Since, moreover, a cluster of borrowings use the translation avere (a loanword from a Romance vernacular meaning “possession”) for an initial non-algebraic unknown number (sometimes represented afterwards by a res, “thing” in the ensuing algebraic solution, sometimes by a census), the ultimate intermediary can be seen to have been Latin: there is no reason an Arabic treatise should borrow a Romance-vernacular term, and if against all odds this should have happened, then its orthography would hardly have survived the transcription into Arabic and Fibonacci's ensuing retranscrpition undistorted.

The terminological innovation demonstrates metamathematical acumen on part of the intermediary: neither al-Khwārizmī nor Abū Kamil make this distinction between a non-algebraic and an algebraic māl.