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A sense of duty is useful in work but offensive in personal relations. Certain characteristics of the subject are clear. To begin with, we do not, in this subject, deal with particular things or particular properties: we deal formally with what can be said about "any" thing or "any" property. We are prepared to say that one and one are two, but not that Socrates and Plato are two, because, in our capacity of logicians or pure mathematicians, we have never heard of Socrates or Plato. A world in which there were no such individuals would still be a world in which one and one are two. It is not open to us, as pure mathematicians or logicians, to mention anything at all, because, if we do so we introduce something irrelevant and not formal.
In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster, 1956.
'He Advanced Him 200 Lambs of Gold': The Pamiers Manuscript
The town of Pamiers in the far south of France lies on the Ariège River, north of where it tumbles down from the Pyrenees on its route to join the Garonne (see map in Figure 1b). In modern times the picturesque site has sometimes been an overnight stop in the Tour de France bicycle race. But few visitors would guess that Pamiers, having less than 20,000 people and situated in a rural department, was the town where an important text in the history of mathematics was written nearly six centuries ago.
Figure 1a. The beautiful town of Pamiers, France, in 2000. Some of the buildings here were already standing by the 15th century. (Photograph by M. Bastien, public domain via Wikimedia Commons.)
One of the striking features of the Pamiers manuscript is the fact that it includes the world’s earliest known instance in which a negative number was accepted as the answer to a problem for purely mathematical reasons. The fact that this occurred in the context of a commercial arithmetic, rather than a more scholastic or theoretical work, is a surprise.
The Pamiers text is similar to many modern textbooks of practical mathematics in one respect: it is filled with story problems. From these, we can get a feel not only for the mathematics of late Medieval and early Renaissance Europe, but also, and especially, for how this mathematics was shaped by commerce and other aspects of culture.
In what follows, I summarize the abbaco tradition of which the Pamiers text was a part, cover some of the arithmetical techniques used in the text, and then present 11 representative story problems from it as a way to explore the mathematical applications that these students were being taught. The 11th problem is the path-breaking one involving negative numbers. I challenge you, the reader, to solve as many of the story problems as you can. You might want to use algebra, matrix methods, or else the purely arithmetical methods covered in the Pamiers text, such as ratio and proportion, inversion (working backward), or false position. Sample solutions are provided in Appendix B.
Figure 1b. Map of France with arrow indicating the town of Pamiers. (Image adapted from David Monniaux / Wikimedia Commons.)
Table Of Contents
Schwartz, Randy K., "'He Advanced Him 200 Lambs of Gold': The Pamiers Manuscript," Loci (July 2012), DOI: 10.4169/loci003888