'He Advanced Him 200 Lambs of Gold': The Pamiers Manuscript - Proportionality Techniques

The author of the Pamiers text wrote that his goal in the third and final section was to describe the application of some general rules for “the capable and savvy man to be able to calculate rapidly so as to manage the buying and selling of goods and the forming of companies” (Sesiano 1984, p. 45). The author focuses especially on a handful of algorithms that were widely used in practical settings: the rule of three and methods of false position. These techniques allowed tradesmen to rely on arithmetic alone, in situations that might otherwise have required the use of algebra.

The rule of three is an ancient rote procedure for solving problems involving ratio and proportion. Given three of the four quantities that appear in a standard proportion, the fourth can be calculated using one multiplication and one division. For instance, to cite an example from the Pamiers text (Sesiano 1984, p. 47), “A linen rope that measures 11 and 3/4 cannas in length costs 5 and 2/3 florins, what will one cost that is 5 cannas and 3 and 1/2 pams?,” where 10 pams is 1 canna or about 2 meters. The problem amounts to solving a proportion,

(11 and 3/4) : (5 and 2/3) :: (5 and 7/20) : ?

Those using the rule of three memorized that once the problem was set up in this way, it could be solved by multiplying the two middle terms and dividing the result by the initial term. Doing so here, we get:

(5 and 7/20) × (5 and 2/3) ÷ (11 and 3/4) = 2 and 409/705, or about 2 and 3/5 florins.

The rule of three is discussed further in the Dibner Institute / WGBH website under Toolkit; Goetzmann 2005, pp. 130-132; and Swetz 1987, pp. 101-134, 224-232, 241-242. It is also illustrated as part of the solutions to Problems 1, 5, and 6 (see Appendix B).

In many problem situations, two or more ratios have to be “composed,” also called “compounded.” A typical situation is that of currency conversion. To take a simple example from U.S. experience: if 5 pennies are worth 1 nickel, and 5 nickels are worth one quarter, and 4 quarters are worth 1 dollar, how many pennies are worth 3 dollars? This amounts to solving a proportion,

1 dollar : 100 pennies :: 3 dollars: ?.

In our modern way of thinking, we might treat the ratios as fractions, and multiply them by one another in chainlike fashion, using the units as a guide: \[{\frac{5\,\,{\rm pennies}}{1\,\,{\rm nickel}}}\times{\frac{5\,\,{\rm nickels}}{1\,\,{\rm quarter}}}\times{\frac{4\,\,{\rm quarters}}{1\,\,{\rm dollar}}}\times 3\,\,{\rm dollars}=300\,\,{\rm pennies.}\]

In Medieval times, however, ratios were not thought of as fractions. Instead, such a proportion was solved by an extension of the rule of three, known as the composite rule of three. The pairs of numbers (called antecedents and consequents) from the given ratios were placed alternately on the upper or lower of two rows, laid out successively from left to right:

Figure 6.

Practitioners of this rule memorized that one then multiplies the antecedents and divides by the consequents, as guided by the zigzag lines in Figure 6:

(5 × 5 × 4 × 3) ÷ (1 × 1 × 1) = 300.

As employed by money changers, this procedure was known as the “chain rule,” and just as with a real chain it could be continued indefinitely for whatever number of ratios needed to be composed. For more details, see Hughes 2008, pp. 57-59, 99-100; Sigler 2002, pp. 179-211; or Smith 1968, pp. 572-573. The chain rule is also illustrated as part of the solutions to Problem 5 in Appendix B (page 13).

Methods of false position were used for solving problems involving linear relationships. A guess or supposition is made as to the value of the unknown, and based on the degree to which the outcome is “false” (i.e., different from the target value), the supposed value is adjusted to the true position, i.e., the correct answer.

Specifically, problems involving a relationship of the form \(ax = b\) could be solved by single false position. (This makes sense graphically, since a line through the origin is determined by a single additional point.) An input number \(x^{\prime}\) was guessed as the value of the unknown quantity, \(x,\) and the resulting output value \(ax^{\prime} = b^{\prime}\) was found. The resulting proportion, \(b^{\prime}:b\,\,::\,\,x^{\prime}:\,?,\) was then solved by the rule of three. The method is illustrated as part of the solutions to Problems 9 and 10 (see Appendix B).

Problems involving a relationship of the form \(ax + b = c\) could be solved by the rule of double false position. (This, too, makes sense graphically, since an arbitrary line is determined by two distinct points; the method amounts to linear interpolation or extrapolation.) The author of the Pamiers text wrote that the technique is

quite marvelous, finding the truth from two falsehoods. […] We will give several examples and several further instructions that in diverse ways assist in doing various quite difficult calculations, which would without the rule be most fatiguing. (Sesiano 1984, p. 54)

The rule of double false position had been borrowed from the Arab world and introduced to Europe by scholars such as Fibonacci. The latter called it the rule of elchataym, his transliteration of the Arabic name al-khata’ayn, “two falsehoods,” and he devoted Chapter 13 of his Liber Abbaci to it (Sigler 2002, pp. 447-487). Details of the technique can be found in the Dibner Institute / WGBH website under Toolkit; in Schwartz 2010; or in Schwartz 2011. The method is also illustrated as part of the solutions to Problems 1, 3, 4, 7, and 8 (see Appendix B).