The regional economies that existed in Europe, or even within a single realm such as France, still hadn’t become well integrated in these pre-modern times, and in general, there were no standardized monies backed by strong governments. Thus, like most Medieval and Renaissance arithmetic texts, the Pamiers manuscript includes many problems dealing with conversions between local currencies.
The practice of minting coins of silver, gold, and other precious metals had arisen many centuries earlier. However, each region tended to have its own coins. Weights and measures also varied from place to place, and even in a given location different units were used for different commodities, as we saw earlier in Section 5 on Barter Transactions. The confusing proliferation of local currencies, weights, and measures in France wasn’t swept away and replaced with a rational, unified, nationwide system until the Revolution of 1789.
Figure 8. “The Moneylender and His Wife” (1514) by Flemish painter Quentin Matsys. Musée du Louvre, Paris. (Image: The Yorck Project / Wikimedia Commons, licensed under the GNU Free Documentation License.)
Many problems of currency conversion, such as the following one from the Pamiers manuscript, are of the type that can be solved by the use of ratio and proportion.
Problem 2. If 3 deniers of Perpignan are worth 5 deniers of Montpellier, and 2 of Montpellier are worth 3 of those of Avignon, I ask how many of those of Perpignan are worth 12 of those of Avignon. (Sesiano 1984, p. 49)
In the above problem there is a twist, because more than just one ratio is given. But this was actually routine for money changers: typically, they knew the exchange rates for neighboring currencies, but to exchange more-distant currencies they had to build a chain out of such pairs of neighbors. Indeed, Perpignan, Montpellier, and Avignon are 3 cities strung along the Mediterranean coast in the order mentioned in the above problem, across a distance of roughly 150 miles, providing a nice geographical interpretation for the concept of “chain.” See if you can solve the problem by one of the methods discussed in Section 4.
Solutions to Problem 2 using composite rule of three, using chain rule, and using algebra
It’s also interesting to note that the term denier mentioned in the above problem goes back to the time of King Charlemagne, who instituted a system of coins, weights, and measures around 800. The term is based on the Latin denarius, an ancient Roman silver coin that had ten times the value of an as, a coin of copper or bronze; the Latin root for “ten” is seen in denarius, as also in English words like decimal and dime. After the Norman conquest of England (1066), Charlemagne’s system of monetary units made its way across the Channel, where the silver denier eventually became known as the penny (plural pennies or pence); this is why the British penny was formerly always abbreviated “d” for denier. In turn, 12 deniers were equal to one sous, a coin mentioned earlier in Section 5 on Barter Transactions.
Monetary questions—including currency exchange, the money supply, and the alloying of precious metals—were complex enough that European mathematicians would sometimes advise rulers on this subject and write treatises about it. The entire Chapter 11 of Fibonacci’s Liber Abaci (1202) is concerned with “the alloying of monies.” Another prime example of this tradition is Nicole d'Oresme (1323-1382), a mathematician and religious philosopher of Norman origin who became both the chaplain and monetary advisor to French King Charles V. By 1355, Oresme had written an important treatise about money, De Origine, Natura, Jure et Mutationibus Monetarum. Paper money did not come into use in England until the 1690s, and in France not until the 1700s.