Fibonacci’s treatment of so-called recreational problems in his Liber Abbaci has been interpreted as an early episode both in the history of systems of linear equations, and in the history of negative numbers. However, these problems are also interesting in their own right. We discuss some of the conventions which seem to have governed these problems. By considering certain pairs of problems, where one problem is unsolvable and its partner is solvable, we show that Fibonacci went to a significant effort (...) to conform to these conventions. We also examine the methods which he could have used to construct his problems. (shrink)

In a famous passage from his al-Bāhir, al-Samaw’al proves the identity which we would now write as (ab)n=anbn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(ab)^n=a^n b^n$$\end{document} for the cases n=3,4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=3,4$$\end{document}. He also calculates the equivalent of the expansion of the binomial (a+b)n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a+b)^n$$\end{document} for the same values of n and describes the construction of what we now call the Pascal Triangle, showing (...) the table up to its 12th row. We give a literal translation of the whole passage, along with paraphrases in more modern or symbolic form. We discuss the influence of the Euclidean tradition on al-Samaw’al’s presentation, and the role that diagrams might have played in helping al-Samaw’al’s readers follow his arguments, including his supposed use of an early form of mathematical induction. (shrink)