Many of the story problems in the Pamiers text reflect the emergence of the company (companhia in Languedocian), a partnership of men banded together in an ongoing money-making enterprise.
Typically these entrepreneurs were already wealthy men, such as successful merchants. They would each advance some of their own money to help establish the company, in hopes that it would later gain additional money or profit (called in Languedocian gasanh, literally “gain” or “winnings”).
One of the key types was the “company of limited duration,” where merchants invested varying amounts of money and time in the enterprise. As exemplified by the problem below, the profit was divided, or parceled out, to each investor in proportion both to the money that he’d advanced and to the time that he’d waited before getting his return.
Figure 10. Three business partners divide their proceeds in this drawing from an illuminated manuscript version of Filippo Calandri’s Trattato di Arithmetica, an abbaco treatise with many problems similar to those of the Pamiers manuscript. It was originally printed in Florence in 1491. (Source: Biblioteca Riccardiana (Florence, Italy), Ricc. 2669, page 64 recto. Image used by permission; further reproduction is prohibited.)
Problem 5. Three merchants formed a company together. The first advanced 200 motos and stayed for 15 months, the second advanced 94 motos and stayed for 17 months, the third advanced 38 motos and stayed for 10 months; and at the end they realized a profit of 400 [motos]. I ask how it should be divided [among them]. (Sesiano 1984, p. 47)
Just like today, to have one’s money tied up in an investment for a period of time was seen as an imposition, and the burden was proportional both to the amount of money and the amount of time. Time and money alike were quantities that should redound to the person who ventured to invest them. Thus, in the above problem, the shares of profit need to be reckoned based on the product of each merchant’s time and money. The Treviso Arithmetic, a book mentioned in Section 5, Barter Transactions, contains two problems very similar to this one (see Swetz 1987, pp. 138-151, 236-238).
Solution to Problem 5 by proportion and gelosia multiplication
The formation of such companies was clearly an embryonic shoot of capitalism, a stage of society that emerged in late Medieval times in Italy and then also coalesced in neighboring countries. The characteristic feature of capitalism is that goods and services are produced not for their intrinsic value but for their usefulness in making money. In a capitalist enterprise, the profits made are more important than the products made; that is why in story problems like the one above, the nature of the company is not mentioned.
The most powerful early capitalists were merchants, like those mentioned in the preceding three story problems. But the dominance of merchants was eventually replaced by that of factory owners and then bankers. Our words factory and manufacture arose because French and Italians referred to a craftsman as a factor, as seen in the next problem. Factor was originally a Latin term for “doer,” “maker,” “producer.” In other words, factors made a product. This terminology is also seen in mathematics more directly: for example, the factors 2 and 3 make the product 6.
In manufacturing, a common type of association that arose at this time was the commande or commenda contract, in which merchants invested capital only, whereas factors invested their labor and possibly some capital as well. The money value of the craftsman’s labor, agreed upon ahead of time, would play a key role when it came time to divide the profits or losses— even if things didn’t go as planned, as was the case in this next problem.
Problem 6. A merchant gave 600 liras to a factor who had 200 liras of his own, by such arrangement that he work with that 800 liras for 5 years, and at the end of that time [said the merchant] we will divide in half the principal and the profit. It so happened that the factor spent none of the 200 liras, but had made use of the 600 liras of the merchant; and at the end of 5 years had realized 2400 liras, counting principal and profit. I ask how the division should be carried out— considering that the factor spent none of what he should have spent— in order that the merchant not be deceived. (Sesiano 1984, p. 48)
One can imagine various ways to resolve this problem, but in the real world it was solved by a strictly proportional division of the proceeds. For five years, the merchant had no access to any of the money he’d invested, whereas the factor was free to use his 200 liras in any way he liked. Thus, “in order that the merchant not be deceived,” it should be reckoned that in this case, the factor’s only investment was the value of his labor. (Hint: first determine the agreed-upon value of the factor’s five years of labor, in liras.)
Solution to Problem 6 by proportion
The above problem suggests that from the earliest days of capitalism, whenever labor and capital came together to produce things, there were issues of judgment, law, ethics, and potential conflict that were involved. First, the value of labor, in liras or other monetary units, had to be negotiated ahead of time. Second, all sorts of unexpected things could happen before the goods were produced and sold.