'He Advanced Him 200 Lambs of Gold': The Pamiers Manuscript - Acceptance of Negative Solutions

Figure 13. Pieces of fine cloth are being sold in this scene from one of the annual international Fairs of Champagne in medieval northern France. The engraving is from a 19^th-century book. (Image: Wikimedia Commons.)

The Pamiers manuscript also includes recreational problems in which a group of men consider pooling their money to buy an expensive item, such as a horse or a piece of fine cloth. Each man tells the others what fraction of their money they would need to give him in order for him to afford the purchase.

Problem 11. There are 5 men who [each] want to purchase a piece of cloth in such manner [i.e., at such a cost] that the first asks of all the others 1/2 of all the gold and silver they carry, the second asks 1/3, the third asks 1/4, the fourth [asks] 1/5 and the fifth asks 1/6. I ask what the piece cost and what each of them carried. (Sesiano 1984, p. 52)

In this rather contrived problem, which involves a series of linear relationships, no actual monetary amount is specified. As in all such problems, it turns out that what matters is only the ratios among the unknowns. Thus, there are infinitely many solutions, but all of them are multiples of one another. The author used a rote procedure to find the solutions, but supplied no justification for the procedure.

Solutions to Problem 11 by rote procedure (followed by justification for the rote procedure) and by matrix algebra

This genre of problem, despite its complexity, was already very old by the 1400s. Fibonacci, for example, had solved quite a few of these in Chapters 12 and 13 of his Liber Abbaci; they always involve a group of men in a commercial situation, such as chancing upon a purse with money, or pooling their money to buy a horse, etc. Going back further, this type of problem is also found in treatises from the Middle East in the 10^th century, and even older Chinese and Byzantine works (Sesiano 1985, p. 112).

What makes the discussion in the Pamiers text exceptional, even within this centuries-old tradition, is the fact that in the case of the above problem, one quantity turns out to be negative in each solution, and the author accepts this as completely legitimate without further comment. He derives his result without making a fuss about the negative quantity, which he records matter-of-factly as 10 et ¾ mens de non res (“10¾ less than nothing”). Unlike Fibonacci, he makes no attempt to interpret what it would mean for a man to pool his “negative” gold and silver with that of his companions in order to afford a piece of cloth. Clearly, in this instance at least, the anonymous author was more interested in the mathematics itself than in its practical application. In the whole of world history, this is the earliest known instance in which a negative final answer—not just a negative intermediate result—was accepted for purely mathematical reasons, i.e., because it satisfied the constraints of the problem (Sesiano 1984, pp. 28, 51-53; Sesiano 1985, pp. 133-134, 148).

The receptivity of the text’s author to the concept of negative numbers also seems to underlie his highly unusual, but quite perceptive, characterization of subtraction as the opposite of addition (sostrayre es lo contrari de aiustar, Sesiano 1984, p. 32), and similarly of division as the opposite of multiplication. This realization would not have been feasible without a full acceptance of negatives (Spiesser 2002, p. 290-291).

The new attitude toward negative quantities was one that had to swim against the prevailing tide of mathematical practice—but ultimately it won out, marking a turning point in the history of mathematics. Some of the first southern-French algorisme texts written after the appearance of that of Pamiers would copy verbatim many of the latter’s problems of this genre, while omitting this particular problem, apparently a sign of their authors’ reluctance to accept negative solutions (Spiesser 2002, p. 297). However, by the end of the century, such solutions had been accepted in at least five other treatises, most of which came to be far better known than the Pamiers work:

• Lo Compendion de lo Abaco (probably c. 1460), written by Francès Pellos in the Niçois dialect of the Occitan language. Later, in the year 1492, this work would be published in Turin, the first book—of any kind—printed in Niçois. (Sesiano 1985, p. 134)
• The Compendy de la Praticque des Nombres (probably c. 1471), an abbaco manuscript written in French by a Dominican friar in southern France, Barthélemy de Romans. His negative numbers arose in the same type of problems discussed by others (horse purchases, etc.), and he used the same term “less than nothing” (meins de riens) as had been used in the Pamiers manuscript. (Spiesser 2004)
• The Tractatus Mathematicus ad Discipulos Perusinos (1478), a commercial arithmetic that Luca Pacioli wrote in Italian for his students at the University of Perugia. (Sesiano 1985, pp. 142-146)
• The Triparty en la Science des Nombres (1484), written in French by Nicolas Chuquet in Lyon, which included negative solutions for a wider range of story problems and also for some abstract problems. (Sesiano 1985, pp. 134-142)
• Pacioli’s Summa de Arithmetica, Geometria, Proportioni et Proportionalità (1494), a textbook in Latin published in Venice. (Sesiano 1985, pp. 146-147)

Of course, the eventual acceptance of negative numbers was inevitable, and their use would prove to be of immense importance in mathematics. It furthered techniques that are needed to solve many of the more advanced practical problems, and it led to a great expansion in the scope of mathematical theory. We can see this impact in such topics as the algebraic solution of polynomial equations, in the now-ubiquitous use of complex numbers, and in notions of identity, inverse, and algebraic structure.

The Pamiers text helps us see that European mathematics advanced to new heights because of constant interaction between its theory and its practical applications, and also because of the rich interaction among many different peoples, cultures, and practices, not all of them native to Europe.