Distributed for EPFL Press
Mathematical Recreations in the Middle Ages
A whimsical examination of math in the Middle Ages.
This book brings together many unusual and entertaining examples of Medieval mathematical exercises. Medieval treatises on arithmetic and algebra frequently include along with the usual commercial and daily life applications, mathematical games, and recreational problems. The latter, while still on the subject of trade or daily life, involve conditions that are quite unrealistic if not totally absurd. As this book demonstrates, the authors of these problem sets hoped to show that mathematics could both be fun and useful.
This book brings together many unusual and entertaining examples of Medieval mathematical exercises. Medieval treatises on arithmetic and algebra frequently include along with the usual commercial and daily life applications, mathematical games, and recreational problems. The latter, while still on the subject of trade or daily life, involve conditions that are quite unrealistic if not totally absurd. As this book demonstrates, the authors of these problem sets hoped to show that mathematics could both be fun and useful.
Table of Contents
Chapter I. Sharing liquid by decanting
1. First appearance
2. Decanting operations
3. Other examples
4. Solving conditions
5. Generalization
Chapter II. Sharing liquid and vessels
1. First appearance and solution
2. Further examples
3. Solving conditions according to Bachet de Méziriac
4. Generalization
Chapter III. Minimal set of weights
1. Introduction
2. Arabic references
3. Finding the solution
4. Mediaeval examples
5. Generalization
Chapter IV. Successive distributions
1. Doorkeepers
2. Giving to the church
3. The last will
Chapter V. Mutual borrowing
1. Moschos’ problem
2. Findingapurse
3. Buyingahorse
4. The dishonest partners
5. Examples
Chapter VI. Filling and emptying cisterns
1. Mediaeval examples
2. Related problems
Chapter VII. Messengers
1. Introduction
2. Progress of a single person
3. Pursuit
4. Meeting
5. Dog and hare
6. Moving back and forth
Chapter VIII. Large numbers
1. First occurrences
2. Selected mediaeval examples
3. Doubling on the chessboard
4. Earth’s size, ancient and mediaeval estimates
5. Some illustrations of the result of doubling
6. Estimating the earth’s weight
7. Perfect numbers
Chapter IX. Arrangements
1. Musical chairs
2. The broken eggs
3. Unwanted passengers
4. The nuns’ cells
Chapter X. Crossing a river
1. Goat, wolf and cabbages
2. Two couples and a boat with two places
3. Three couples and a boat with two places
4. Four couples and a boat with two places
5. Five or six couples and a boat with two places
6. Four couples and a boat with three places
7. Five couples and a boat with three places
Chapter XI. Miscellanea
1. Sharing with a traveller
2. Multiplication of rabbits
3. The dishonest servant
4. The future heirs
5. Sharing camels
6. Striking the hour
7. Two workers, same work
8. Same earnings from same items in different quantities
Chapter XII. Family relationships
1. Two men marry each other’s sister
2. Two men marry each other’s mother
3. Two men marry each other’s daughter
4. Of two men, one marries the other’s mother and the second the other’s daughter
5. Father and son marry mother and daughter
6. Father and son marry daughter and mother
7. Three generations
Chapter XIII. The knight’s tour
1. Euler’s study
2. Mediaeval tours
Chapter XIV. Magic squares
1. Early history
2. Arrival of magic squares in late mediaeval Europe
3. First attempts at reconstruction
Chapter XV. Infinite sets
1. Natural numbers and rational fractions
2. Finite and infinite in Greece and the Middle Ages
3. Examples of comparing infinites in the Middle Ages
4. Ancient and mediaeval root approximations
Chapter XVI. Geometrical recreations
1. The tower and the river
2. Measuring an inaccessible height
3. The ladder leaning against a wall
4. The two towers
5. Two towers and a fountain
6. Further problem of two towers
7. The falling tree
8. The stone thrown into a cistern
Chapter XVII. Hidden numbers
1. The walnuts
2. The novenary
3. The unknown date
4. The unknown sum
5. The hidden ring
6. Two people and two things
7. Three people and three things
8. The game with coins
Appendices
A. Brief outline of the history of mathematics
1. Arithmetic and its applications
2. Progressions and series.
3. Algebra
4. Symbolism
1. First appearance
2. Decanting operations
3. Other examples
4. Solving conditions
5. Generalization
Chapter II. Sharing liquid and vessels
1. First appearance and solution
2. Further examples
3. Solving conditions according to Bachet de Méziriac
4. Generalization
Chapter III. Minimal set of weights
1. Introduction
2. Arabic references
3. Finding the solution
4. Mediaeval examples
5. Generalization
Chapter IV. Successive distributions
1. Doorkeepers
2. Giving to the church
3. The last will
Chapter V. Mutual borrowing
1. Moschos’ problem
2. Findingapurse
3. Buyingahorse
4. The dishonest partners
5. Examples
Chapter VI. Filling and emptying cisterns
1. Mediaeval examples
2. Related problems
Chapter VII. Messengers
1. Introduction
2. Progress of a single person
3. Pursuit
4. Meeting
5. Dog and hare
6. Moving back and forth
Chapter VIII. Large numbers
1. First occurrences
2. Selected mediaeval examples
3. Doubling on the chessboard
4. Earth’s size, ancient and mediaeval estimates
5. Some illustrations of the result of doubling
6. Estimating the earth’s weight
7. Perfect numbers
Chapter IX. Arrangements
1. Musical chairs
2. The broken eggs
3. Unwanted passengers
4. The nuns’ cells
Chapter X. Crossing a river
1. Goat, wolf and cabbages
2. Two couples and a boat with two places
3. Three couples and a boat with two places
4. Four couples and a boat with two places
5. Five or six couples and a boat with two places
6. Four couples and a boat with three places
7. Five couples and a boat with three places
Chapter XI. Miscellanea
1. Sharing with a traveller
2. Multiplication of rabbits
3. The dishonest servant
4. The future heirs
5. Sharing camels
6. Striking the hour
7. Two workers, same work
8. Same earnings from same items in different quantities
Chapter XII. Family relationships
1. Two men marry each other’s sister
2. Two men marry each other’s mother
3. Two men marry each other’s daughter
4. Of two men, one marries the other’s mother and the second the other’s daughter
5. Father and son marry mother and daughter
6. Father and son marry daughter and mother
7. Three generations
Chapter XIII. The knight’s tour
1. Euler’s study
2. Mediaeval tours
Chapter XIV. Magic squares
1. Early history
2. Arrival of magic squares in late mediaeval Europe
3. First attempts at reconstruction
Chapter XV. Infinite sets
1. Natural numbers and rational fractions
2. Finite and infinite in Greece and the Middle Ages
3. Examples of comparing infinites in the Middle Ages
4. Ancient and mediaeval root approximations
Chapter XVI. Geometrical recreations
1. The tower and the river
2. Measuring an inaccessible height
3. The ladder leaning against a wall
4. The two towers
5. Two towers and a fountain
6. Further problem of two towers
7. The falling tree
8. The stone thrown into a cistern
Chapter XVII. Hidden numbers
1. The walnuts
2. The novenary
3. The unknown date
4. The unknown sum
5. The hidden ring
6. Two people and two things
7. Three people and three things
8. The game with coins
Appendices
A. Brief outline of the history of mathematics
1. Arithmetic and its applications
2. Progressions and series.
3. Algebra
4. Symbolism