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The Mean Value Theorem is the midwife of calculus -- not very important or glamorous by itself, but often helping to delivery other theorems that are of major significance.
Calculus with Analytic Geometry, 5th edition, Englewood Cliffs, NJ: Prentice Hall, 1987.
'He Advanced Him 200 Lambs of Gold': The Pamiers Manuscript
A Source of the Two-Trains Problem?
As with a modern textbook, the Pamiers text contains a number of story problems that, while challenging, are of dubious practical value. They are contrived examples dressed up as real-world applications. The real purpose of these “recreational” puzzles is to further sharpen the mathematical skills of the students.
Among the examples given in the Pamiers text are several that remind us of classic problems still found in modern textbooks. Consider the following rate-and-time problem.
Figure 12. A wine cask with three taps, one bigger than the other two, appears 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 108 verso. Image used by permission; further reproduction is prohibited.)
We are given to assume that the draining speed of each tap is constant over time.
In its modern form, such a problem might be phrased: “Michael can mow the entire lawn in 3 hours, Ellen in 4 hours, and Art in 6 hours….” Although the story is different, the mathematics is exactly the same! This suggests that some types of problems are old standards that have persisted for centuries, their form gradually evolving over time (see Sanford 1972). In fact, more than 200 years earlier, in Chapter 12 of Fibonacci’s Liber Abaci, we find almost this same example, but instead of “three spigots,” the vat has “four holes at the bottom” (Sigler 2002, p. 281).
The following problem, although nearly six centuries old, is highly reminiscent of a modern classic: “Two trains leave their stations in Detroit and Chicago at the same time…,” but instead of waving to each other from passing trains, the two men involved see each other on horseback.
We are given to assume that the speeds of both men are constant over time. And if you think about it, Problem 10 is mathematically of the same type as Problem 9: instead of three taps working together to drain an entire wine cask, two men are working together to cover an entire distance of 63 leagues.
The Treviso Arithmetic contains an almost identical problem (see Swetz 1987, pp. 158-160). The men are couriers sent between Rome and Venice, a distance of 250 Roman miles, but the travel times are still 7 and 9 days.
Schwartz, Randy K., "'He Advanced Him 200 Lambs of Gold': The Pamiers Manuscript," Loci (July 2012), DOI: 10.4169/loci003888