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Return To Mathematical Circles, By Howard W. Eves, Copyright 1988 by PWS-Kent Publishing Company
354º Cistern problems. Many type-problems of elementary algebra have enjoyed long histories. In Item 13º of In Mathematical Circles, we described a type-problem, still familiar today, which has been traced back in time at least to the Rhind papyrus of about 1650 B.C. Another type-problem with a long history is the so-called cistern problem, originally concerned with filling cisterns by means of pipes having given rates of flow.
The cistern problem seems to have first appeared, in definite form, in Heron’s Metrica of about A.D. 100. It is next found in the works of Diophantus of about A.D 25 and among the Greek epigrams attributed to Metrodorus of about A.D. 500. Soon after, it became common property in both the East and West. It was found in the list of problems attributed to Alcuin (ca. 800), in the Indian classic Lilãvati of Bhãskara (ca. 1150), and in subsequent Arabian arithmetics. When books began to be printed, the cistern problem was among the stock problems of such early writers as Tonstall (1522), Frisius (1540), and Recorde (ca. 1540).
Originally, the cistern problems reflected an observation of daily life; anyone living along the Mediterranean coast saw cisterns that were filled by pipes of various diameters. But there is an interesting law of textbook writers—that it is quite all right to steal from one another with almost no scruples provided the theft is thinly veiled. Accordingly, the cistern problem went through a number of metamorphoses.
Thus, starting in the fifteenth century, we find variations involving a lion, a dog, and a wolf, or other animals, eating the carcass of a sheep. In the sixteenth century we find further variations involving men building a wall or a house—problems of the form: “If A can do a piece of work in 4 days, and B in 3 days, how long will it take if both men work together?”
In a work of Frisius (1540), the problem becomes a ridiculous drinking problem: “A man can drink a cask of wine in 20 days, but if his wife drinks with him it will take only 14 days—how long would it take for the wife alone?” Under the growth of commerce we also find the case of a ship with three sails, by the aid of the largest of which a voyage can be made in 2 weeks, with the next in size in 3 weeks, and with the smallest in 4 weeks—find the time if all three sails are used. Here the problem, that in its original cistern form had a practical aspect, has become unrealistic, as it ignores the matter of one sail blanketing another and the fact that the speed of the ship is not proportional to the area of sail. Probably the height of absurdity was reached when one writer proposed: “If one priest can pray a soul out of purgatory in 5 hours, while it takes a second priest 8 hours, how long will it take if the two priests prayed together?”
Since the solution of a cistern problem involves a special procedure, it is quite certain that problems of this genre will continue to be found among the story problems of our elementary algebra textbooks.
Return To Mathematical Circles, By Howard W. Eves, Copyright 1988 by PWS-Kent Publishing Company
354º Cistern problems. Many type-problems of elementary algebra have enjoyed long histories. In Item 13º of In Mathematical Circles, we described a type-problem, still familiar today, which has been traced back in time at least to the Rhind papyrus of about 1650 B.C. Another type-problem with a long history is the so-called cistern problem, originally concerned with filling cisterns by means of pipes having given rates of flow.
The cistern problem seems to have first appeared, in definite form, in Heron’s Metrica of about A.D. 100. It is next found in the works of Diophantus of about A.D 25 and among the Greek epigrams attributed to Metrodorus of about A.D. 500. Soon after, it became common property in both the East and West. It was found in the list of problems attributed to Alcuin (ca. 800), in the Indian classic Lilãvati of Bhãskara (ca. 1150), and in subsequent Arabian arithmetics. When books began to be printed, the cistern problem was among the stock problems of such early writers as Tonstall (1522), Frisius (1540), and Recorde (ca. 1540).
Originally, the cistern problems reflected an observation of daily life; anyone living along the Mediterranean coast saw cisterns that were filled by pipes of various diameters. But there is an interesting law of textbook writers—that it is quite all right to steal from one another with almost no scruples provided the theft is thinly veiled. Accordingly, the cistern problem went through a number of metamorphoses.
Thus, starting in the fifteenth century, we find variations involving a lion, a dog, and a wolf, or other animals, eating the carcass of a sheep. In the sixteenth century we find further variations involving men building a wall or a house—problems of the form: “If A can do a piece of work in 4 days, and B in 3 days, how long will it take if both men work together?”
In a work of Frisius (1540), the problem becomes a ridiculous drinking problem: “A man can drink a cask of wine in 20 days, but if his wife drinks with him it will take only 14 days—how long would it take for the wife alone?” Under the growth of commerce we also find the case of a ship with three sails, by the aid of the largest of which a voyage can be made in 2 weeks, with the next in size in 3 weeks, and with the smallest in 4 weeks—find the time if all three sails are used. Here the problem, that in its original cistern form had a practical aspect, has become unrealistic, as it ignores the matter of one sail blanketing another and the fact that the speed of the ship is not proportional to the area of sail. Probably the height of absurdity was reached when one writer proposed: “If one priest can pray a soul out of purgatory in 5 hours, while it takes a second priest 8 hours, how long will it take if the two priests prayed together?”
Since the solution of a cistern problem involves a special procedure, it is quite certain that problems of this genre will continue to be found among the story problems of our elementary algebra textbooks.
