A Synthesis of Mediterranean Methods

The Pamiers text, like other examples of the Medieval algorisme tradition, represented a synthesis of Mediterranean culture, including the commercial practices and mathematical techniques of the region. It is written in Languedocian, an old Romance language of southern France that has now almost disappeared (see Appendix A). Sesiano (1984) provides extensive excerpts from the original Languedocian text, and his own commentary in modern French. (Here on these web pages, all 11 problem statements and other quotations drawn from Sesiano’s redaction have been translated by me from Languedocian into English with the help of a glossary of key terms that Sesiano provides.)

The existing manuscript was handwritten on just over 100 leaves of paper and parchment. It includes no date, title, or author, but it is known to be a copy made in about 1446 of a text originally composed in Pamiers in the 1420s or 1430s, a decade or two before Gutenberg invented his printing press. Copies of books like this had to be made one at a time in the old-fashioned way; special clerks called copyists were employed for this purpose. Spiesser notes that Catholic religious orders, which would have included copyist monks, had a strong presence in Pamiers at the time, and the town also had a university faculty; she speculates that the manuscript’s author might have been a cleric well-versed in the theory and practical techniques of arithmetic (2002, p. 289).

The book opens with benedictions to “our Lord Jesus Christ, merciful and compassionate,” to the Virgin Mary, and to Antonin, patron saint of Pamiers. Then it praises a Muslim scholar from whose name the term algorisme (algorithm) is derived, Muhammad al-Khwārizmī, whom the author described as “a man strongly learned and enlightened, and expert in four sciences [the quadrivium], knowing arithmetic, geometry, music, and astronomy” (Sesiano 1984, pp. 31-32). As inherited from Greek tradition, these were considered the four branches of mathematics when al-Khwārizmī and other scientists flourished in Baghdad around 825 AD.

Figure 4. First page of the Pamiers manuscript. Line 2 of the text begins with an abbreviation for Jesus Christ; line 9 with an underlined name of the town, Pamias (Pamiers); and line 12 with algorisme, which is referred to both as an art and as a real person (i.e., al-Khwārizmī). The markings at the top of the page indicate that the manuscript was once item no. 5194 in the collection of the Bibliothèque de Colbert, and the mark at the bottom identifies it as belonging later to the Bybliothèca Regia (Royal Library). (Image is from Sesiano 1984, p. 30.)

Consistent with its intended use for training in commercial arithmetic, the book consists largely of worked-out examples and story problems, sometimes with little or no justification as to why the methods used are valid. As is true of Medieval works of mathematics in general, no mathematical symbols are used other than the digits for writing numerals.

The manuscript is divided into three sections. The first explains Arabic numerals and the decimal place-value system, then covers the basic arithmetical operations, followed by arithmetic and geometric progressions, and square and cube roots. The second section covers the same topics for fractions, and the third section covers applications of these arithmetical procedures, including roughly 100 story problems.

The pen-and-paper algorithms of arithmetic covered by the Pamiers text were made possible by decimal place-value numeration, which arose in India and then was adapted and advanced by scholars in the Middle East. Many of the algorithms have remained essentially unchanged to this day, while others are somewhat different from those with which we are most familiar. To multiply multi-digit numbers, for example, the Pamiers text recommends the lattice technique, which in Italian is known as multiplication “by gelosia,” a word for the latticed grille on a window. You can probably figure out how the method is carried out by inspecting this example from the manuscript, illustrating the multiplication 437 × 345 = 150765.

Figure 5. Gelosia (lattice) multiplication (437 x 345) from the Pamiers manuscript. (Image on left is from Sesiano 1984, p. 34.)

The mechanics of this method are shown in a video by PedagoNet, and are explained in Math Forum 1996 as well as Swetz 1987, pp. 80-84, 205-209. The method is also illustrated as part of the solutions to Problem 5 (see Appendix B). Smith (1968, pp. 114-117) notes that lattice multiplication was brought to Italy from the Middle East. The method is still being taught today (see Boag 2007, Nugent 2007).