Propositiones ad acuendos iuvenes

The Propositiones ad acuendos iuvenes ( Latin for tasks to sharpen the mind of the youth ) are an early medieval collection of mathematical puzzles . It is attributed to Alcuin ; the oldest surviving manuscript is from the late ninth century. It is the first such collection in Latin.

Text transmission

A collection of mathematical problems is first mentioned in a letter that Alcuin wrote to Charlemagne in 799 or 800 . There it says: “ Misi excellentiae vestrae… aliquas figuras arithmeticae subtilitatis, laetitiae causa ” ( Alcuin : Ep. 172, German: “I have sent your highness ... some figures of arithmetic subtlety for your entertainment.”) However, the tasks are missing.

The complete edition of Alkuin's works contains a version that contains 53 exercises. Another version can be found in Beda Venerabilis , with three additional tasks, two after task 11, one after task 33.

A modern edition was not made until 1978 by Menso Folkerts . He found twelve manuscripts, the oldest dating from the late ninth century and already containing the three additional tasks of the Bede text, but is incomplete.

A translation into English was made by John Hadley in 1992 under the title Problems to Sharpen the Young with notes by David Singmaster in The Mathematical Gazette . A year later a German translation was published with comments by Folkerts and Helmuth Gericke .

The following manuscripts are known:

Surname	Age	origin	current location
R ₁	End of the 9th century	St. Denis Monastery near Paris	Vatican Apostolic Library
O	Late 10th century	West Germany / East France	Vatican Apostolic Library
A.	Late 10th century	Reichenau Monastery	Badische Landesbibliothek , Karlsruhe
W.	around 1010	Sankt Mang Monastery , Füssen	Austrian National Library , Vienna
M ₂	around 1020	St. Emmeram Monastery / Chartres	Bavarian State Library , Munich
V	1025	St. Martial Abbey , Limoges	Leiden University Library
B.	first half of the 11th century	West Germany / East France	British Museum , London
M.	first half of the 11th century	Eastern France	Montpellier University Library
R.	11th century	St. Mesmin near Orléans	Vatican Apostolic Library
M ₁	12th Century	St. Emmeram	Bavarian State Library, Munich
C.	13th Century	St Albans Abbey	British Museum, London
S.	15th century	Buckfast Abbey	British Museum, London

content

The tasks wrap a mathematical question in a short framework that mostly arises from everyday life. In some tasks, as in fables, animals are the characters.

Depending on the issue, the solutions can be found directly after the tasks or collected in a separate section. In most cases these only give the result, there is no solution. It is only checked that the given result is correct. Some solutions are incomplete, some even wrong.

A walker

The second task dresses a linear equation in the following story: On his way a walker saw a group of people coming towards him and said: “I wish you were more, namely once more as many as you are, plus a quarter of this sum and that half of that extra. With me we would be a hundred. ”How many people did the walker see?

The task leads to the equation with the solution . This number indicates Alkuin without a solution and confirms the result with a sample. ${\ displaystyle x + x + {\ tfrac {2x} {4}} + {\ tfrac {2x} {8}} + 1 = 100}$ ${\ displaystyle x = 36}$

Further tasks of this type can be found in numbers 3, 4, 36, 40, 44, 45 and 48.

Goat, wolf and cabbage

Task 18 is the well-known problem of wolf, goat and cabbage: a man has to cross a river with a wolf, a goat and a cabbage. The only boat can only carry one other passenger besides him. How can he cross the river without the wolf eating the goat or the goat eating the cabbage?

Solution: The man first leaves the wolf and cabbage and rows the goat to the other bank. There he turns around and brings the wolf over. On the way back he takes the goat with him, which he leaves on the original bank to bring the cabbage over. Finally he brings the goat back to the other bank.

Other tasks also deal with such river crossings : In the 17th problem, three men want to cross a river with their sisters in a boat for two, without one of the women having to fear that another man will violate her in the absence of her brother. The 19th task is a family made up of a father, mother and two children. Only the children are so light that they can sit together in the boat without going under. The same task follows again, with the only difference that it is now a family of hedgehogs.

A hundred pigs

A man wants to buy 100 pigs with his 100 denarii . A boar costs 10 denarii, a sow 5 denarii, and a pair of piglets costs one denarius.

Solution: The man buys a boar, nine sows and 90 piglets.

The task leads to a linear system of equations with two equations in three variables and the additional condition that the solution must be whole numbers . It is not a matter of course that there is a clear solution in this case. This type of task can already be found in fifth century China, where there are a hundred birds. In the time of Alcuin it was known all over the world thanks to the Indians and Arabs. In the Propositiones there are eight exercises of this type: 5, 32, 33, 33a, 34, 38, 39 and 47. It is possible that exercise 53 was also intended in this form, but it is severely distorted by spelling mistakes.

Area calculations

Exercises 21 to 31 deal with problem 26 with area calculation . It starts with a rectangular field measuring 200 by 100 feet.

The next task already deals with a field of irregular shape: It is 100 rods ( pertica ) long and 50 rods wide at both ends, but 60 rods in the middle. Alkuin determines the mean width of the field as the simple arithmetic mean of 50, 60 and 50, and receives a rounded width of 53 rods, which he then multiplies by the length. Since the exact shape remains unclear, no formula can be given for the area, but if it is a double trapezoid, the mean width would have to be determined as an average of 50 and 60, i.e. 55 rods.

The next task does not describe the shape of the field either, but only names its side lengths 30, 34, 32 and 32 rods. Alkuin again determines the mean width or length as the mean value of the opposite sides, i.e. 31 and 33 rods, which he multiplies again. Its result is therefore greater than the largest possible area that can be achieved by the quadrilateral .

This is followed by a triangular field with edges of 30, 30 and 18 rods. Alkuin again determines medium widths, namely 30 rods for the two legs and 9 rods as half of the base. As the area, he then specifies the product that lies above the correct content.

The field of the next task is a circle of 400 rods in circumference. The correct solution would be . There are two different solutions in the manuscripts: The first solution includes the calculation , the second includes . ${\ displaystyle {\ tfrac {400 ^ {2}} {4 \ pi}}}$ ${\ displaystyle \ pi = 4}$ ${\ displaystyle \ pi = 3}$

The ladder with the hundred rungs

Task 42 tells of a ladder with a hundred rungs. A pigeon sits on the first rung, two on the second, three on the third, and so on. How many pigeons are sitting on the ladder in total?

The solution derives the equation, later known as the Gaussian empirical formula , by combining the single pigeon with the 99 on the penultimate rung, the two with the 98, etc., so that 49 pairs of 100 pigeons result, plus another 50 pigeons and the 100 of the last rung, so a total of 5050.

The pigs

Exercise 43 reads as follows: A man has 300 pigs and orders them to all be killed on the following three days, an odd number each day. How many pigs should be killed on each day?

Mathematically, the problem cannot be solved because the sum of three odd numbers must be odd and therefore cannot add up to 300.

In the English-speaking world, this task circulates as a joke question , based on the fact that odd not only means odd, but also strange, and thus you can kill a pig on the first two days and the remaining 298 on the third, which is not odd, but is a very strange number of pigs to be killed in a day.

literature

Menso Folkerts: The oldest collection of mathematical exercises in Latin: The PROPOSITIONES AD ACUENDOS IUVENES ascribed to Alkuin. Austrian Academy of Sciences, Mathematical and Natural Science Class, Memorandum 116, 1978, pp. 13–80.
John Hadley, David Singmaster: Problems to Sharpen the Young. In: The Mathematical Association: The Mathematical Gazette. Vol. 76, no. 475, The Use of the History of Mathematics in the Teaching of Mathematics, March 1992, pp. 102-126. ( JSTOR 3620384 )
Menso Folkerts, Helmuth Gericke: The "Propositiones ad acuendos iuvenes" attributed to Alcuin. In: Paul Leo Butzer, Dietrich Lohrmann: Science in Western and Eastern Civilization in Carolingiam Times. Birkhäuser, Basel 1993.

Web links

Wikisource: Propositiones ad acuendos iuuenes - Sources and full texts (Latin)

The anti PISA campaign Charlemagne (PDF; 678 kB)

Individual evidence

↑ ^a ^b ^c The anti PISA campaign Charlemagne (PDF; 678 kB)
^ ^A ^b John Hadley, David Singmaster: Problems to Sharpen the Young.

[jensditthardt-1] The anti PISA campaign Charlemagne (PDF; 678 kB)

[hadley-singmaster-2] A ^b John Hadley, David Singmaster: Problems to Sharpen the Young.