Historia Mathematica 34 (2007) 344–358
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Giochi matematici alla corte di Carlomagno. Problemi per rendere acuta la mente dei giovani
Translation and Commentary by Raffaella Franci. Pisa (Edizioni ETS). 2005. ISBN 88-467-1351-6. Illustrations,
141 pp. + Index. €12.00
In company with the Rhind Papyrus, the Greek Anthology, and the Haidao suanjing (Sea Island Mathematical
Manual), the Propositiones ad acuendos juvenes (Problems to Sharpen the Minds of Teenagers) of Alcuin of York
(735–804) has been a popular source for challenging problems of recreational mathematics. Using the critical Latin
edition (1978) of Menso Folkerts, which accompanies each problem, Franci has supplied readers of Italian with an
engaging introduction to, translation of, and commentary upon the 53 riddles, 5 of which have two solutions. An
instructive Appendix elaborates upon two better-known problems. A helpful bibliography completes the work.
The five-part Introduction opens with a sufficiently detailed vita of Alcuin of York, tracing his early career as
director of the cathedral school in York, his commission by Charlemagne to lead the palace school, his dissatisfaction
with court life, his accomplishments in both places, his brief return to York, the books he wrote, and the final challenge
as Abbot of the Abbey of Saint Martin in Tours where he died. In the second part Franci offers a synopsis of medieval
mathematics education, singling out the seven liberal arts as the basic curriculum. The quadrivium, however, was
hardly more than a foray into theory of numbers, enough music to sing well, an acquaintance with astronomy to
understand a horoscope, and adequate geometry for construction and land measurement. The only other mathematical
chore was to compute the date of Easter; hence, there was a course called computus. The third introductory section
provides an overview of other collections of problems, one from ancient Egypt (Rhind Papyrus), three from China
(The Nine Chapters of the Mathematical Art, The Mathematical Manual of Master Sun, and The Mathematical Manual
of Zhang Qiuijiang), and one Hellenic (The Greek Anthology). Franci’s purpose is to point out the two avenues
along which mathematics developed, one theoretical, the other practical. The collections of problems, those cited
here together with the Propositiones, belong to the practical or popular form of mathematics, and as the next sections
Reviews / Historia Mathematica 34 (2007) 344–358 345
illustrates, provide a different venue for learning mathematics. To substantiate this position, she notes the use of the
method of double false position and the Chinese Remainder Theorem.
A generous overview of the Problems composes the fourth part. Here Franci classifies the problems: 6 solved by
simple arithmetic, 10 about quantities solved by first-degree equations, a single problem about giving and receiving
something of value, 7 problems about birds or food that require equations in several unknowns, a problem about two
people or animals traveling at different speeds, a single inheritance problem, a problem about buying and selling for a
profit, 2 problems on dividing a quantity of liquid among several people, 9 problems that require thought without any
computations, and 12 problems about pieces of material or sections of land that are to be divided into so many smaller
equal sections. This part concludes with remarks about similar problems elsewhere, noting that 23 of the problems are
completely new, not found in any of the cited works. In the final part of the Introduction Franci discusses the source
for her translation and lists pertinent medieval units of measurement.
The Appendix focuses on two easily recognized problems. The first, number 18 in the text, may be called “The Safe
Crossing Problem.” A man must transport a wolf, a goat, and some cabbage across a river, but the boat can hold only
two of the three. How does he convey the three across two at a time without damage to goat or cabbage? Franci walks
the reader through the history of the problem in its several later appearances, from a German abbot in the early 13th
century, to Luca Pacioli in the 15th, Niccolò Tartaglia in the 16th, Eduard Lucas and De Fonteney in the late 19th, and
Kordenski in the 20th century. Over the centuries the character have changed from jealous husbands accompanying
their wives to teenage girls chaperoned by their fathers, but there is always a river to be crossed and a boat that cannot
hold all the persons or objects at once. The second problem (number 52) has been rephrased in modern military terms
during or following World War II. A jeep has to cross the Sahara Desert. Its fuel tank cannot hold enough gas for
the whole trip. Where along the road should it deposit supplies of gas? Franci analyzes several published solutions,
among which the most recent may be by Oberschelp [1998]. In Alcuin’s version, a camel must transport 90 bags of
grain a distance of 30 leagues. The job will take three trips of 30 bags for a distance of the first 20 leagues. For every
league outward, the camel will eat one bag of grain. How many bags are left after the third trip? How many remain
after the final 10 leagues?
Gracing her work are colored illustrations, one of Alcuin of York from a medieval manuscript and seven picturing
puzzles from a later Italian manuscript. Raffaella Franci has made a praiseworthy addition to the translations into
English of the Propositiones by David Singmaster [1998] and into German by Menso Folkerts and Helmut Gericke
[1993].
References
Folkerts, M., Gericke, H., 1993. Die Alcuin zugeschriebenen Propositiones ad acuendos juvenes (Aufgaben zur Schärfung
der Geistes der Jugend). In: Butzer, P.L., Lohmann, D. (Eds.), Science in Western and Eastern Civilization in
Carolingian Times. Birkhäuser, Basel, pp. 283–362.
Oberschelp, W., 1998. Alcuin’s camel and the jeep problem. In: Butzer, P.L., Jongen, H.Th., Oberschelp, W. (Eds.),
Charlemagne and His Heritage: 2000 Years of Civilization and Science in Europe. Vol. II. Mathematical Arts.
Brepols, Turnbout, pp. 411–421.
Singmaster, D., 1998. The history of some of Alcuin’s problems. In: Butzer, P.L., Jongen, H.Th., Oberschelp, W.
(Eds.), Charlemagne and His Heritage: 2000 Years of Civilization and Science in Europe. Vol. II. Mathematical
Arts. Brepols, Turnbout, pp. 11–29.
Barnabas B. Hughes
Department of Secondary Education,
California State University, Northridge,
Northridge, CA 91330-8265, USA
E-mail address: barnabashughes@hotmail.com
Available online 27 October 2006
10.1016/j.hm.2006.09.001