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Historia Mathematica 41 (2014) 400–437
www.elsevier.com/locate/yhmat
How algebra spoiled recreational problems:
A case study in the cross-cultural
dissemination of mathematics
Albrecht Heeffer
Center for History of Science, Ghent University, Faculty of Arts and Letters, LW01,
Blandijnberg 2, B-9000, Ghent, Belgium
Available online 27 August 2014
Abstract
This paper deals with a sub-class of recreational problems which are solved by a simple memorized rule resulting from an elementary
arithmetical or algebraic solution, called proto-algebraic rules. Their recreational aspect is derived from a surprise or trick
solution which is not immediately obvious to the subjects involved. Around 1560 many such problems wane from arithmetic and
algebra textbooks to reappear in the eighteenth century. Several hypotheses are investigated why popular Renaissance recreational
problems lost their appeal. We arrive at the conclusion that the emergence of algebra as a general problem solving method changed
the scope of what is considered recreational in mathematics.
© 2014 Elsevier Inc. All rights reserved.
Sommario
Questo saggio tratta di una sottoclasse di problemi ricreativi risolti tramite memorizzazione di una semplice regola risultante
da una soluzione algebraica o aritmetica, chiamata regola proto-algebraica. L’aspetto ricreativo di questi problemi deriva da una
soluzione a sorpresa o da un trucco non immediatamente ovvi ai soggetti coinvolti. Intorno al 1560 svariati problemi di questo
tipo sparirono dai manuali di algebra e aritmetica, per riapparire nel diciottesimo secolo. Diverse ipotesi sono vagliate sul perché
problemi ricreativi popolari nel Rinascimento persero attrattività, per giungere alla conclusione che l’emergere dell’algebra come
metodo generale di risoluzione di problemi cambiò la portata di ciò che era considerato ricreativo in matematica.
© 2014 Elsevier Inc. All rights reserved.
MSC: 00A08; 97A20
Keywords: Recreational; Problems; Algebra; Cross-cultural
E-mail address: albrecht.heeffer@ugent.be.
http://dx.doi.org/10.1016/j.hm.2014.06.004
0315-0860/© 2014 Elsevier Inc. All rights reserved.
A. Heeffer / Historia Mathematica 41 (2014) 400–437 401
1. Introduction
Anyone engaging in a study of recreational mathematics soon discovers that many of the problems
that are still popular today go back a long time in history. In a recent book on the mathematics behind
card tricks by two professional mathematicians (Diaconis and Graham, 2012, 106–114) the authors traced
the history of a popular three-object divination problem to Prevost (Prevost, 1584) and Bachet (Bachet,
1612). However, after twenty years of research they were surprised to learn that the problem is a frequently
recurring one in abbaco manuscripts of the quattrocento and appears in even earlier sources.
Equally surprising is the fact that the same or similar problems appear in very different cultures and
geographical regions. One such example is the sliding ladder problem about a ladder of a given length which
stands against a wall and which is moved from its original position on the ground. It features in several
Old-Babylonian tablets BM 85196 (Høyrup, 2007, 275–6), BM 34568 (Friberg, 1981, 307–8), in Egyptian
papyri, Cairo JdE. 89127-30 (Parker, 1972, 13–43), in Chinese classics, Ji˘u zh¯ang suàn shù 九章算術, Nine
chapters of the mathematical art (Chemla and Shuchun, 2004), in Sanskrit mathematical texts, Bh¯askara
I. 629. Commentary on the ¯Aryabhat¯ıya, II.16 (Keller, 2006, 79–83) and also in the works by Bh¯askara II,
in Arabic texts, Hasib Tabar¯ı, Mift¯ah. al-mu‘¯amal¯at (Bagheri, 1999), in many abbaco manuscripts, e.g.
Gherardi, Libro di ragioni (Florence, BNCF, Magl. Cl. XI, 86; Arrighi, 1987b) as well as in Fibonacci’s
De Practica Geometrie (Hughes, 2008, 77) and Pacioli’s Summa (Pacioli, 1494, part 2, ff. 54v–55v), and in
seventeenth century works on recreational mathematics (van Etten, 1624, prob. 89; Ozanam, 1725, 320–1,
prob. 42). The interest in this specific problem thus not only spans a period of more than three millennia but
also five very different cultures and mathematical practices! Though conceptualization, contextual meaning
and solution methods may differ amongst these cultures, at least we can induce from this example that
problems we now consider as recreational mathematics are omnipresent in mathematical practice and that
these travel easily between different cultures.
One of the reasons for the multi-cultured aspect of recreational mathematics is that mathematical knowledge
is often embedded in folk stories, riddles, tricks, tangible practices which form the basis for some types
of recreational problems. Solution methods, rules and mnemonic devices are in concert with the problems.
Many recreational problems have been disseminated as folk stories through merchant connections and trade
routes. Embedding mathematics in cultural practices which can be adapted to suit the cultural context allows
problems to cross cultural boundaries. That is the reason why the same problems turn up in such
diverse cultures.
This paper presents a case study on a sub-class of recreational problems which are based on some elementary
arithmetical or algebraic solution. Their recreational aspect stems from a surprise or trick solution
which is not immediately obvious to the subjects involved. The legacy problem about a dying father with an
unknown number of children (discussed below) is a challenge to solve unless you know the simple rule of
thumb which gives you the answer to this and similar problems. The first book to coin the term recreational
mathematics in the title (van Etten, 1624) often uses the qualification “wonder to those that are ignorant in
the cause”.1 As in divination problems, it is important to conceal the mathematical principles behind the
trick or problem to make it surprising and appealing. The title of this paper somewhat polemically states
that this recreational aspect of problems disappears when a general solution method – as is algebra – is
applied to solve such problems. I will indeed demonstrate that this is the case for some specific problems,
while other types of recreational problems are less exposed to algebraic solutions.
I will first discuss the practical context of arithmetical problems. Renaissance recreational problems are
often situated in a practical context to give them some flair or alleged utility. In the next part it is shown
1 I have previously challenged the attribution of this book to the Jesuit Jean Leurechon, and proposed the printer/engraver Jean
Appier Hanzelet as the compiler of the problems (Heeffer, 2006b). For further discussion on the authorship see my forthcoming
critical edition (Heeffer, forthcoming).
402 A. Heeffer / Historia Mathematica 41 (2014) 400–437
how rules or solution recipes are embedded in such problems. I have termed these ‘proto-algebraic’ as the
rule may have been derived by algebra but solving the problem by such rules is not algebraic. Eventually
these problems also turn up in algebra textbooks becoming exercises and cease to be recreational.2 In the
following part I will give two related examples of a problem which was very popular before 1560 and then
virtually disappears from mathematics books for two centuries. Several hypotheses will be discussed to
explain such remarkable discontinuity. In a conclusion, I will provide arguments for the claim in the title
that the scope of recreational mathematics was altered by the advent of algebra.
2. On Renaissance problems and their functions
Records of posing and solving Renaissance problems in the history of mathematics provide us the empirical
grounds for the study of the development of problem-solving methods and techniques as well as the
socio-cultural context in which they were formulated. These problems have functioned as vehicles for the
transmission of new ideas, procedures and concepts in arithmetic, geometry and algebra from Babylonian
times to the age of printing. The concrete setting in which the problems were situated facilitated memory,
education and tradition and provided a justification for the activities involved. Alcuin’s Propositiones ad
Acuendos Juvenes (Propositions for Sharpening Youths) contains 53 problems of which many are repeated
over and over again in medieval and Renaissance works.3 As the title suggests, the problems were intended
to be used for their educational value and to be read aloud for students to copy and solve.4 The long history
and tradition of recreational problems lets us connect sixteenth-century problems with much earlier sources.
Some type of Renaissance problems occur already in the cuneiform texts of the Babylonians dating back
to 1900–1700 BC. Others originated more recently and evolved within decades. Some problems remained
unchanged during several centuries while others developed into more complex and challenging forms over
time.
Most of these problems are formulated in a practical context which gives them a panache and purpose and
make them acceptable within that context. For example, Thomas Digges published a book (Digges, 1579),
reprinted in 1590, named An arithmeticall warlike treatise named Stratioticos compendiously teaching the
science of nombers as well in fractions as integers, and so much of the rules and aequations algebraicall,
and art of nombers cossicall, as are requisite for the profession of a soldier. As the title makes clear, the
book is intended for army officers and every problem is thus set into the context of an army or warfare.
Claiming a practical purpose was an important incentive for selling and producing textbooks on arithmetic
from the sixteenth century onwards.
Many text books on arithmetic of the fifteenth to the seventeenth century carried the term ‘practical’ in
the title; e.g. (Ciruelo, 1505; Peurbach, 1536; Ghaligai, 1521; Feliciano, 1526; Finé, 1532; Glarianus, 1539;
Frisius, 1540; Des Barres, 1545; Morsianus, 1553; Pérez de Moya, 1562; Petri, 1567; Clavius, 1583;
Cortes, 1604; Metius, 1611; Neufville, 1620; Malapert, 1620; Schott, 1662; Coets, 1698). Despite the
advertisement as ‘practical’ by the authors, most of these books were of little practical value for the craftsman
and the merchant class of that time. Notable exceptions are: (Feliciano, 1526) with practical geometry
for surveying, and some problems of (Ghaligai, 1521) that have practical utility. Vernacular books as by
Feliciano and Ghaligai were better suited for practical purposes than the Latin works listed above. They
dealt with subjects such as the Welsche praktijk, methods for facilitating exchange calculations and commercial
transactions (Swetz, 1989, 10–18; Kool, 1999, 157–167). Ad Meskens gives an excellent overview
2 For a survey of the changing function of problems in algebra textbooks see Heeffer (2012a).
3 The attribution to Alcuin (c. 800) is not fully certain. Translations are available only recently: a translation into German (Folkerts
and Gericke, 1993); Hadley provided an English translation, which was published with notes by Singmaster (Hadley and
Singmaster, 1992; Folkerts, 1978) gives a summary of problems from Alcuin in several medieval manuscripts.
4 For the educational context in which these problems were used, see (Bayless, 2002).
A. Heeffer / Historia Mathematica 41 (2014) 400–437 403
of the real mathematical needs for the practice of merchants and craftsmen during the Renaissance: money
exchange, calculation of interests and annuities, navigation, surveying, gauging, fortification and ballistics
(Meskens, 2013). The mathematicians from the Low Countries that contributed most to these practical
problems in the sixteenth century were Valentijn Mennher and Michiel Coignet (Mennher, 1556; Coignet,
1597).5 These authors deal with practical geometrical problems, such as determining the volume of vessel,
which get no treatment in the frequently printed and reissued text books of the Latin tradition (Frisius, 1540;
Finé, 1532). The determination of the practical utility of many of examples and problems in arithmetic
books is problematic. We have to make a distinction between problems and examples that are formulated
with objects and situations common to daily life and those that were intended for real practical use. Embedding
problems in a concrete cultural context had more an educational purpose than a practical one. In
fact, several sixteenth-century authors expressed their doubt about the practical use of some of their problems.
William Kempe, who translated Peter Ramus’ arithmetic book (Ramus, 1569) into English, writes
in the chapter about cistern problems: “Of this sort also are devised divers other questions, which never or
seldome happen in practice, and therefore have scarce anie use or fruite at all” (Kempe, 1592, 52).
3. A preliminary case study of a recreational problem and its solution methods
3.1. The role of recipes
In the beginning of the sixteenth century, we find many determinate problems with one or more equations
and unknowns that can easily be solved algebraically. However, an algebraic solution is less common as
would be expected. Let us look at one specific problem in some detail. The problem has become known as
‘the lazy worker problem’ and deals with a worker who has to perform a job in a number of days. He is
paid a sum for every day he works but has to return a payment for every day he does not work. After the
given period he ends up with a certain sum, in most cases zero. One is asked to find how many days he
worked and did not work. Our first acquaintance with this problem was from the Libro de Abacho, one of
the earliest printed book on arithmetic, shown in Figure 1 (Borghi, 1484, f. 111v, literal translation mine):
Someone wants to do a job and finds a master whom he
promises to conclude the work in 40 days agreeing that he is
paid 20 soldi for every day he works and that he has to pay
back 28 soldi for every day he does not work. When doing his
calculation it was found that the master does not have to pay
anything. The question is how many days have been worked
and how many did he not work.
Figure 1. Illustration of the lazy worker problem from Borghi (1484).
Let us look at the solution in some detail. Borghi gives a solution which applies “always when you have
similar problems”. In a loose translation the proposed method is as in Figure 2:
5 Specific for interest calculation we can add Jean Tenchant, Martinus Wentsel and Simon Stevin. See the dissertation of Waller
Zeper for a comprehensive overview (Waller Zeper, 1937).
404 A. Heeffer / Historia Mathematica 41 (2014) 400–437
Put first how many soldi he earns for the days that he has worked
and then how many soldi for the days he has not worked and
then proceed as I will demonstrate. Thus for the days that he
earns money we put 20 soldi and for the days that he loses
money 28 soldi. Thus we put against the days that he works
28 soldi in such a way against the days that he does not work,
20 soldi. Together 28 soldi and 20 soldi makes 48 and this is the
work he has done in 48 days of which 28 days he has worked
and 20 days he has not worked, and has received nothing. But
because the problem tells us that the work is completed in 40
days, say thus that in 48 days he worked 28 and as he should
have worked 40 days, subtract from the days that he should have
worked 23 1/3. Thus to know how many days he did not work:
subtract 23 1/3 from 40 which leaves 16 2/3 and this is the
number of days he did not work. From (Borghi, 1484, f. 111v).
Figure 2. Borghi’s solution to the lazy worker problem.
The problem can be solved by two linear equations in two unknowns, x for the days working and y for
the days absent:
x + y = c
ax − by = d
with solutions x =
bc + d
a + b
and y =
ac − d
a + b
(0.1)
The problem values will be denoted as (a,b,c,d)
However, an algebraic solution with two equations in two unknowns, as represented here, turns up only in
the eighteenth century!6 Vogel, in a section on Der Arbeiter im Weinberg (Vogel, 1954, 222–3), Tropfke in
Der faule Arbeiter (Tropfke, 1980, 603) and Singmaster (Singmaster, 2004, section 7.AK) give a history of
the problem. Sesiano calls the problem l’ouvrier paresseux (Sesiano, 1984, 147). The oldest documents in
which the problem can be located is in Arabic texts such as the Fakhr¯ı by Al-Karkh¯ı from the early eleventh
century. Al-Karkh¯ı gives three instances of the problem set in the same context of a day-worker (Sections 2,
12, 13 and 14), Woepcke (Woepcke, 1853, 83) lists three instances of the problems with values
10
30
,
6
30
,30,0 ,
10
30
,
6
30
,30,4 and
10
30
,
6
30
,30,−2 .
Unfortunately, Woepcke does not include a transcription of the solution text, only the equations as below:
1
3
x =
1
5
(30 − x)
1
3
x = 4 +
1
5
(30 − x)
1
3
x + 2 =
1
5
(30 − x)
The most likely route of transmission to Europe seems to go through Fibonacci who gives two instances
of the problem in his Liber Abbaci (Boncompagni, 1857, 160–1 and 323; Sigler, 2002, 250–1 and 453–4)
and through the abbaco tradition with La practica di geometria, by Gherardo de Dino (Florence, Biblioteca
6 Algebraic solutions using one unknown are discussed below.
A. Heeffer / Historia Mathematica 41 (2014) 400–437 405
Riccardiana, codice 2186; Arrighi, 1966, f. 323). The problems are, as with Al-Karkh¯ı, formulated as a
payment per month, instead of per day. The first problem De laboratore laborante in quodam opere (7/30,
4/30, 30, 1), is solved by what Fibonacci calls “the method of companies”. This rather unique procedure
is illustrated in the Liber Abbaci by a marginal drawing as in Figure 3.7 In order to explain the method,
Figure 3 replaces the quantities of the problems (using the wages per month) by the symbolic expressions
from (0.1). In this way we can demonstrate that adding c to a, b and d is part of the solution procedure.
d + b
a + c
a − d
c − b
d + c
Figure 3. A marginal illustration of Fibonacci’s ‘method of companies’.
The ‘method of companies’ is a term used for proportional division. It is derived from the principle that
business companions should receive a part of the profit proportional with their contribution to the capital.
In this case, proportional division refers to c, the numbers of days, to be divided in proportion to the parts
b + d and a − d. Thus dividing c = 30 (not accidently, the number of days in a month) in the proportion of
b + d = 5 to a − d = 6, he thus arrives at the solution
13
7
11
and 16
4
11
.
Except for a numerical test he gives no further argumentation why the method is correct.8 Adding b + d
and a − d gives a + b, thus using proportional division we get:
x = c
b + d
a + b
and y = c
a − d
a + b
.
Dividing the monthly wages by c, we should arrive at
a
c
x −
b
c
y = d or a
b + d
a + b
− b
a − d
a + b
= d
and indeed (a+b)d
a+b = d.
The second, De laboratore, question notabilis (7/30, 4/30, 30, 30), is solved by the general method
of double false position, which he borrows from the Arabs by the name elchataym. We find solutions to
this problem with double false position over the next centuries: in Munich, Bayerischen Staatsbibliothek,
Cod. Lat. Mon. 14908, f. 48r (Curtze, 1895c, 41), in the Summa (Pacioli, 1494, f. 99r), the very rare
Compendion de l’Abaco (Pellos, 1492, f. 71v–f. 72r), the second Rechenbuch of Adam Ries (Ries, 1522;
1574, ff. 60v–61r), in the General Trattato (Tartaglia, 1551–1560, Bk. I, f. 275r), and Robert Recorde
(Recorde, 1552, 312–6). It is important to note that in both cases, Fibonacci does not give an algebraic
solution. It might seem plausible that Fibonacci has the problem from Al-Karkh¯ı, because we know that
several of his problems from the last chapter are taken from this source.9
7 In the English translation (Sigler, 2002, 624), a footnote is added: “Leonardo uses 37 down to 26 instead of 7 down to −4 in
order to avoid negative numbers”. However, this interpretation makes no sense.
8 A numerical test involves checking the correctness of the solution by placing the calculated values back in the problem enunci-
ation.
9 This has been argued by (Woepcke, 1853).
406 A. Heeffer / Historia Mathematica 41 (2014) 400–437
There could be an alternative path of transmission. A Greek manuscript, Paris, BNF, Cod. suppl. Gr. 387,
written c. 1308, has the same problem with values (10, 4, 30, 132) on f. 127v (Vogel, 1968, 69). The μέθοδος
in the manuscript gives the following recipe:
Add 10 and 4, which makes 14. Also takes four times 30 days, makes 120. This added to 132 gives 252.
From this you take the 14th part which gives 18. That is the number of days he worked, and 12 days he did
not work.
The recipe thus corresponds with the procedure
x =
bc + d
a + b
.
This rule is applied to problems without any derivation or justification. Although this manuscript is dated
after the Liber abbaci, the Byzantine tradition preceding Pachymeres and Planudes, could be a plausible
source for several of Fibonacci’s problems. His use of bezants as currency unit, in this and many other
problems, gives strong support to a Byzantine influence. Kurt Vogel, who has studied and published several
Byzantine arithmetic manuscripts, has always stressed the role of Byzantium as an intermediary in the
transmission of Ancient and Arabic mathematics to the West.10 Fibonacci, as often in the Liber Abbaci, is
not contended with the application of meaningless recipes. The application of “the method of companies”
is most likely an original solution method of his own.
3.2. Identifying the solution type
We have shown three different methods for solving the lazy worker problem in some early sources: the
solution from the Liber Abbaci using proportional division, the resolution of an expression in one unknown
by the rule of double false position and the Byzantine recipe. These solution types can now be used for
further identification of influences. Before doing so, we have to add a simplified recipe, specifically for the
case in which d = 0. This type of problems is found in some early Italian and German manuscripts. An
anonymous manuscript of c. 1290, New York, Columbia University, X511, solves an instance of the problem
with values (5, 9, 30, 0) as well as (4, 7, 30, 1) (Vogel, 1977, 86–7).11 In our symbolic representation
this leads to ax = by,x + y = c. The solution then amounts to the proportional division of c into a and b.
This can be accomplished as follows:
x = c
b
a + b
,
y = c
a
a + b
.
Representing the procedure in this way, we can see that the recipe can be understood as an application of
the rule of three: if someone earns a + b in c days, in how many days does he earn b? The abbaco text
indeed formulates the procedure referring to the rule of three.12
10 In fact, the last part of this sentence is the title of one of his articles (Vogel, 1978). See also (Vogel, 1962, 1964, 1968).
11 Vogel dates the manuscript at 1350 but Høyrup has argued that 1288–1290 is a better estimate (Høyrup, 2007, 31, note 70).
This makes the treatise the second oldest extant abbaco text.
12 Columbia, X511 (Vogel, 1977, 86–7): “Queste la sua diritta reghola, chomo si dei fare, che tur dei dire 5 e 9 fanno 14 e chosi
di, ch’ello die sia 14 ore che le 9 ore lavora e le 5 si posa. Si voli fare in 30 die si die: se 14 fusse 30, che seria 9? Di: vua 30 fanno
270 a partire 14, che nne viene 19 die e 4/14 di die e tanto lavorào e si noi [volemo] savere, quanto [no a] lavorato, si di: se 14
A. Heeffer / Historia Mathematica 41 (2014) 400–437 407
It is difficult to establish if the recipes were derived from the rule of three or that the rule of three was
given as a justification for the validity of an earlier formulated recipe. Given that the more general form
of the recipe appears in previous Byzantine manuscripts, the second seems more plausible. An additional
argument is that the author makes some mistakes in applying the recipe. Namely, in the second example
with d = 1, he again applies the recipe and obviously reaches the wrong solution.13
Tartaglia approaches the case with d = 0 with the recipe but solves the two cases with d = 0 by double
false position (problem 39 and 42) (Tartaglia, 1551–1560, Bk. I, f. 275r). Also manuscripts from German
monasteries use this very recipe. A collection of 350 problems is found in the so-called Algorismus Ratisbonensis
written c. 1450 and preserved in several Munich manuscripts. Number 183 is our problem with
values (10, 12, 40, 0) in Munich, Bayerischen Staatsbibliothek, Cod. Lat Mon. 14908, f. 100v.14 The recipe
also refers to the rule of three:
Do as such: add 10 and 12, arriving at 22, the divisor. Say 22 gives 40, what shall give 10? Makes 18 days
and 2 hours. When a day makes 11 hours, this is how much he took free.
Exactly the same problem and the solution is repeated in the Bamberger Rechenbuch of 1483 (see Figure 4).
In some cases the rationale of the proportional division is missing and only the recipe survives. This the
case in the Memoriale of Francesco Bartoli (c. 1420, Avignon, Archives départementales du Vaucluse, cote
1F 54) who applies it to the case (9, 11, 30, 0) (Sesiano, 1984, 137). This notebook of an Italian business
man travelling between Italy and the south of France at the beginning of the fifteenth century gives us an
unique insight into the paths of transmission of problems and solution methods throughout Europe.
Figure 4. The lazy worker problem from the Bamburger Rechenbuch, 1483, f. 46v.
fusso 30 che seria 5? Di: 5 via 30 fanno 150, parti in 14 che nne vie[ne] 10 die 10/14 e die e tanto [no] lavora e per questo modo
si fa tutte le altre die più o di meno”.
13 Vogel speculates that the author jumbles up several problems and that this solution corresponds with 7x − 4y = 15 (Vogel,
1977, 87). Instead, we believe that he did not have a recipe for the case that d = 0. Apparently he divides the surplus florin into
two and adds and subtracts the halves from the florins per hour, which are our coefficients (“vedi che tenne per florino l’ora che in
unna ora”, f. 32r).
14 (Vogel, 1954, 86): “Machβ also: addir 10 et 12, erit 22, divisor. Dic 22 dant 40, quod dat 10, facit 18 dies 2 horas, hoc est,
quando dies est 11 horarum und alz vil hat er gefeirt”.
408 A. Heeffer / Historia Mathematica 41 (2014) 400–437
In addition to arithmetic tools such as the rules of exchange, multiplication tables, the itinerary from
Florence to Avignon and price lists, it contains a collection of thirty problems of the recreational sort.
We can assume that Bartoli was just one of the many links in the trade routes by which the tradition of
arithmetical problem solving was passed from Italy to France and the Low Countries.
In conclusion we can state that a single type of recreational problems as the lazy worker problem can be
solved by different means: solution recipes for specific cases or general rules such as the rule of three and
the rule of false position. We can identify Borghi’s solution as an application of the recipe of proportional
division given the case that d = 0. Although this simplified recipe, which appears first around 1350, could
have been derived from an algebraic solution, the method applied by Borghi is not algebraic. Vogel makes
a similar observation in relation to the Byzantine text.15
3.3. Algebraic solutions
As the main thesis of this paper is that some types of recreational problems lost their appeal once they
became solved by algebra, we will now look at some early algebraic solutions to the lazy worker problem.
Given that algebra was introduced in Western Europe since the twelfth century, it is rather surprising that
an algebraic approach to such simple problem appears only in the late fifteenth century. The earliest Italian
source is Piero della Francesca’s Trattato d’abaco, written in 1480, Florence, Biblioteca Medicea Laurenziana,
Ashburn. 359∗ (Arrighi, 1970). Piero has several linear problems which he solves by La regula de
la positione, but he also adds some algebraic solutions. The lazy worker problem is presented as follows,
Florence, Biblioteca Medicea Laurenziana, Ashburn. 359∗, f. 38v (Arrighi, 1970, 99):
Uno dà a fare uno suo lavoro e fa pacto cello maestro de darli 20 soldi el dì che lui lavora, et il dì che non
lavora vole che il maestro dia 16 soldi a lui, e sono d’acordo, et che il maestro de’ fenire i’ lavoro in 30 dì.
Et capo de 30 dì, il maestro à finito il lavoro; fano ragione, nisuno à dare l’uno a l’altro. Domando quanti dì
lavoro, et quanti non lavorè.
Piero uses the unknown for the days worked and thus arrives at
20x − 16(30 − x) = 0 (without actually writing the zero).
This leads to 36x = 480 or,
13
1
3
for the days worked and 16
2
3
for the days not worked.
He adds a numerical test as a proof of the validity of the solution.
But Piero was not the first. The earliest algebraic solution we could find is by Frederic Amann who wrote
an algebraic text in 1461, preserved in Munich, Bayerischen Staatsbibliothek, Cod. Lat. Mon. 14908.16
After solving some problems by the regula falsi, he adds algebraic solutions to the same problems. Our
problem with values (5, 3, 28, 0) is formulates as follows (Curtze, 1895a, 68):
Quidam conventus est ad laborandum per 28 dies. Die laboris 5 habet, die vacacionis restituet 3; in fini
nullum lucrum habuit. Quot dies laboravit?
15 (Vogel, 1968, 149): “Und doch benützt der Verfasser in zahlreichen Fällen rezeptmässichen Formeln, die aber nur in vorhergehenden
algebraischen Operationen gefunden sein konnten, wobei er sich keinerlei Gedanken über deren Herleitung macht”. The
question of how these recipes could have been derived from algebraic solution is discussed in (Heeffer, 2006a).
16 (Tropfke, 1980, 603) only mentions the solution with regula falsi, (Singmaster, 2004) has neither.
A. Heeffer / Historia Mathematica 41 (2014) 400–437 409
Using the coss x for the days worked, the earning is 5x. The money returned is the number of days not
worked (28 − x) times 3. Both are equal and Amann constructs the equation ‘84 minus 3x equande 5x’,
which gives the solution 10 1/2.
The problem with (7, 5, 40, 0) appears with an algebraic solution in the margins of the Dresden codex
C80. The text, written in 1481 is attributed to the monk, Aquinas Dacus.17 The marginal notes appear to be
of the hand of Johannes Widmann. His solution is the same as that of Amann on f. 355r (Wappler, 1899,
543):
Pone eum laborasse ad 1x, quare 1x per 7 multiplica, et erunt 7x, et 40 dies ocij −1x per 5 multiplica, et
erunt 200−5x, que ex ypothesi equivalent. Restaura ubique 5x addendo, et stabunt 12x ex una parte 200 Ø,
quare iuxta primi capituli preceptum operare, et patet valor x, scilicet laboravit 16 diebus 2/3, vacauit ad 23
dies 1/3 mervit 116 2/3, et tantum opportuit ipsum restituere.
Adam Ries writes in his Coss (Dresden, Sächsische Landesbibliothek, Codex C80) that he found the
example in the gloss of ‘ein alten Buch’, identified by Berlet as from Andreas Alexander (Berlet, 1860,
20). This is in fact, Munich, Bayerischen Staatsbibliothek, Cod. Lat. Mon. 14908, discussed above.18
Ries gives an algebraic solution for the problem with values (3,5,28,0) (problem 37) and another one
with (11,8,36,−12) in problem 110 (Berlet, 1860, 29). Christoff Rudolff solves the problem with values
(7,5,30,6) using a single unknown as 7x − 5(30 − x) = 6 and arrives at 13 and 17 (Rudolff, 1525, f. Oiir;
Stifel, 1553, ff. 278r–278v). He then discusses the cases with d = −30 and d = 0. The youngest source in
which we find the problem is treated algebraically is from the sixteenth century by Caspar Peucer19:
Oeconomus conducit operarium in dies triegenta, ea conditione, ut promittat soluturum se ei in dies singulos
7 detracturum se rursus minetur 5 mumulos in singulos dies, quibus opere intermisso cessarit. Circumactis
30 diebus neuter elteri quidquam debet. Quaeritur ergo quot diebus vacarit operi imposito, quot cessarit?
This is Rudolff’s third case with values (7,5,30,0). Peucer’s solution is noteworthy as he combines algebra
with the rule of three. He uses x for the number of days worked, thus the number of days not working is
30 − x. As one day work earns 7, x days must earn 7x. Also, if one day not working pays 5, 30 − x days
not working results in 150 − 5x. As both these are equal he constructs the equation 7x = 150 − 5x and
arrives at the solution 12 1/2 and 17 1/2.
3.4. Proto-algebraic rules
We have looked at one problem from Borghi’s book which corresponds with the arithmetic of determining
the value of x in
x =
bc + d
a + b
.
17 Little is known about this intriguing figure. Aquinas Dacus was a monk of the Prediger order (Vogel and Stein, 1981, 10).
Regiomontanus mentions him in his letter of 4 July 1471 to Christian Roder: “Multa equidem de tua excellentia cum ex aliis
plerisque omnibus Erfordia venientibus, tum ex fratre Aquino volupe intellexi” (Curtze, 1902, 325). He was a teacher of Andreas
Alexander and Andreas Stöberl (Stiborius) and put his knowledge of algebra available in exchange for money.
18 Not only problem 37 of Ries, but also his previous one nicely correspond with the gloss from the Clm 14908.
19 From (Peucer, 1556, ff. Tvr–Tvv), not mentioned by (Tropfke, 1980) or (Singmaster, 2004). Peucer’s Logistice has received
some attention only very recently (Meiner and Deschauer, 2005) with a fairly complete description and a German translation of
some fragments.
410 A. Heeffer / Historia Mathematica 41 (2014) 400–437
The procedure in itself is not algebraic but depends on the application of a prescribed recipe. Given the
right recipe for the right type of problem, the method guarantees a solution. We have also seen that in
some cases the application of a recipe to a different problem type results in errors. Such recipes have
been called ‘proto-algebraic rules’, because these rules can be and could have been derived algebraically
though its application as such does not constitute algebra (Heeffer, 2007). The notebook of Bartoli is a rare
historical record that shows how problems and their corresponding proto-algebraic rules disseminate over
Europe during the Renaissance. The many instances we have found for this problem and its solution by a
proto-algebraic rule show the popularity of the problem type and the widespread use of solution recipes.
The many authors we have listed seemed to have found pleasure in presenting and solving this problem in
their treatises.
In the later part of this problem’s life cycle we find more and more algebraic solutions in textbooks. After
Peucer, the problem disappears from textbooks for almost two centuries. We have even evidence of a case
in which Bombelli, some years later, decided to retract this and similar problems from a book published
in 1572 (discussed below). In order to understand why such problem disappears, let us see if this was a
systematic trend at the second half of the sixteenth century.
4. Some instances of discontinuity
In order to demonstrate that several types of recreational problems virtually disappear from arithmetic
and algebra books by the end of the sixteenth century, we will now give two typical examples. It is difficult
to prove that there are no instances to be found of these problems between the 1560s and the eighteenth
century. Possibly, some versions of the problems here discussed do appear somewhere in books we have not
consulted. However, we have traced the most representative ones and our findings can be confirmed from
the existing classifications (Tropfke, 1980; Singmaster, 2004). In fact, Tropfke’s lists of these problems
ends around the 1560s, an indication that little was found after this date. To show that some other types of
problems did continue to be discussed after 1560, we will conclude with one counterexample.
4.1. The monkey and coconut problem
One type of a problem that was popular before the end of the sixteenth century is called Schachtelaufgaben
(Tropfke, 1980, 592) but better known under its twentieth-century name “monkey and coconut
problem” (see Table 1). The name is derived from a short story by Ben Ames Williams, published in a
newspaper in 1926.20 It is about five sailors that are shipwrecked on an island. They gather coconuts all
day. During the night each of the sailors takes one fifth of the remaining coconuts and gives one to the
monkey. In the morning they each get one fifth, again leaving one coconut for the monkey. The question
is how many coconuts they had gathered. This type of problem developed from a simpler version, popular
during the Renaissance. For example, a Flemish manuscript by van Varenbrakens of 1532 gives the
following formulation21:
There was a hermit who was in serious need for money. This hermit went to the church to stand for Saint
Paul and prayed him to double the money in his purse in exchange for six groat. Saint Paul agreed to this.
Then he went to Saint Peter and also asked him to double what was left in his purse. Saint Peter agreed, and
the hermit gave him six groat. Then he went to Saint Francis and also asked him to double what was left
in his purse in exchange for six groat, which he agreed to. In the end he was left with nothing because six
groat was all that he had. The question is: how much had the hermit in his purse to begin with.
20 The story about the publication and the many reactions it provoked is described by Martin Gardner (Gardner, 1958). See also
(Singmaster, 1997, 2004).
21 (van Varenbrakens, 1532, Ghent, Universiteitsbibliotheek, Hs. 214, f. 158). Transcription by Kool (Kool, 1988). English translation
mine.
A. Heeffer / Historia Mathematica 41 (2014) 400–437 411
Table 1
An overview of the Monkey and Coconut problem, showing the discontinuity of its treatment in arithmetic books.
Author Date Book Problem Page
Gherardi 1328 Libro di Ragioni (unnumbered) 47–48, 100
Anonymous 1330 Libro d’Abaco (Lucca, Biblioteca
Statale, Ms. 1754)
(unnumbered) 26r–27r, 59v
Paolo dell’abbaco 1339 Trattato di Tutta l’Arte dell’Abacho (unnumbered) 25r–25v
Biagio 1340 Trattato di praticha d’arismetricha
Gilio 1384 Arithmetica e geometrica 40, 41 145r–146r
Anonymous 13xx D’astucie algorismi 6 8r–11r
BL, Royal 12 C XII
Anonymous 13xx Quaestiones arithmeticae 3 17v–21r
BL, Cotton Cleop. B IX
Anonymous 1417 Turin, N.III.53 (unnumbered) 11v
Munich Clm 14684 14xx Incipiunt subtilitates enigmatum 5, 6, 36 30r, 32v
Bartoli 1420 Memoriale 9 75v
Paolo Dagomeri 1440 Trattato d’arithmetica 47, 71 44, 65–7
Fridericus Gerhart 1450 Algorismus Ratisbonensis 185, 187
Pierro della Francesca 1480 Trattato d’abaco 73, 74, 97, 98, 104 23r, 37v, 41v
Chuquet 1484 Triparty 30–33
Johannes Widmann 1489 Behende und hubsche Rechenung 3 problems 97v–98v
Calandri 1491 Arimethrica (unnumbered) 66v
Luca Pacioli 1494 Summa di Arithmetica et Geometria 22 150v
Luca Pacioli 1500 De viribus quantitates 67 120v
Johannes Köbel 1514 Rechenbuechlein auf den linien mit
Rechenpfeningen
(unnumbered) 89r
Ghaligai 1521 Practica d’Arithmetica 29 66v
Tunstall 1522 De Arte Supputandi 43, 44 173–4
Adam Riese 1524 Die Coss 53, 58, 65, 66, 92, 94,
109, 110
Christoff Rudolff 1525 Coss 5
van Varenbrakens 1532 Die Edel Conste Arithmetic
vanden Hoecke 1537 Een sonderlinghe boeck
Nicolo Tartaglia 1556 General Trattato (Bk, par) 12, 34;
16, 47; 17, 9; 17, 20
199v, 113–6, 246r,
253v–254r
Jean Trenchant 1558 L’arithmétique, Book 3 6 (1618) 321
Johannes Buteo 1559 Logistica 6, 13, 19, 20, 21 334–350
Humfrey Baker 1562 Well spring of sciences 7, 8 193r–193v
Jacques Ozanam 1725 Recreations Mathematiques 28 211–2
Thomas Simpson 1745 A treatise of algebra XV, XXVII 86–89
Leonhard Euler 1770 Vollständige Anleitung zur Algebra added in (Euler, 1822) 204
412 A. Heeffer / Historia Mathematica 41 (2014) 400–437
This formulation as an oblation by a poor man or hermit originated in the fifteenth century and was popular
until the early sixteenth century but the problem is still older. Other versions include a merchant who gains
and spends money at different places and a girl who collects apples in a garden but has to give the guards a
share of the apples. Singmaster (2004) gives a comprehensive history of this type of problems, which also
includes indeterminate versions of the forms
xi+1 =
n − 1
n
xi and xi+1 =
n − 1
n
(xi − 1).
Its history goes back to Babylonian, Chinese and Indian sources, but several formulations of the problem
appear in medieval Europe until halfway the sixteenth century. A Byzantine source lists two versions
(Vogel, 1968, 109 & 135, problem 89 and 119). The Libro d’Abaco of c. 1330 has three versions, Lucca,
Biblioteca Statale, Ms. 1754 (Arrighi, 1973, 64–5 & 135). The Trattato d’Aritmetica of the early fourteenth
century gives the apple garden version twice (see Figure 5).22
Figure 5. Three porters in an apple garden (from the Trattato d’Arithmetica, c. 1340).
Folkerts found 17 instances of this type of problems in fourteenth and fifteenth-century manuscripts
(Folkerts, 1968). The Algorismus Ratisbonensis (c. 1450), has 2 examples of the problem (Vogel, 1964,
prob. 185 and 187). All important Renaissance treatises deal with the problem: Bartoli’s Memoriale
(Sesiano, 1984, 138 & 148), Munich, Bayerischen Staatsbibliothek, Cod. Lat. Mon. 14684 (Curtze, 1895a,
prob. 5), (Chuquet, Paris, BNF, Français 1346, prob. 30–33; Calandri, 1491, f. 66v & 74r; Pacioli, 1494,
prob. 22; Köbel, 1514; Ghaligai, 1521; Tunstall, 1522, question 43 and 44; Riese, Dresden, Sächsische
Landesbibliothek, Codex C80, aufgabe 53; Tartaglia, 1551–1560, Book 17, art 9 and 20; Trenchant, 1558,
prob. 6; Buteo, 1559, prob. 21). The problem, as given here, can easily be solved by reversing the order
and calculating backwards. In Indian mathematics the method was called vipar¯ıtakarma or Rule of inversion.
¯Aryabhata I prescribes the vipar¯ıtakarma for the four arithmetical operations, but this was extended
to include squaring and roots by Brahmagupta and Bh¯askara. The Bakhsh¯al¯ı manuscript has examples
formulated as business trips, c. 700 (Hayashi, 1995, S¯utra C1).
22 BNCF, Magl. XI, 86 (Arrighi, 1964, 44 & 65–7), problem 47 and 71. Arrighi attributes the manuscript to Paolo dell’Abaco but
this is disputed (Van Egmond, 1977; 1980, 114–5).
A. Heeffer / Historia Mathematica 41 (2014) 400–437 413
Fibonacci gives over twenty variations of the problem under the heading De Viagiorum propositionibus
(business trips). He solves the problem by calculating the fractional parts (Sigler, 2002, 372–383) and
gives an alternative solution using the rule of double false position (Sigler, 2002, 460). In the Algorismus
Ratisbonensis the problem is also solved by the double regula falsi (Vogel, 1954, Prob. 185).
The inversion method is also found in later European texts. Widman gives a problem of business trips
by three times halving and adding four, leaving 3 1/2 (f. 97v), and another one adding two, leaving one
(Widman, 1489, f. 100r). He coins the name Regula transversa for the rule of calculating backwards.23
Pacioli discusses the possible solutions in his unpublished manuscript De viribus Quantitates (see Figure
6).24 According Pacioli, the problem can be solved, (1) by a standard rule, (2) in the ordinary way, (3)
by the rule of double false position and (4) by algebra. As he has “no intention to fatigue” his audience,
he proposes the “general rule which can be applied without any mental exertion”. His general rule is calculating
backwards. The ordinary way is presumably the calculation of the fractions. Some years before,
he discussed two versions of the problem in the Summa (Pacioli, 1494, f. 105v, problem 21 and 22). For
the first he calculates the fractions and the second he solves by calculating backwards. Four more complex
versions are given in the distinction (i.e. chapter) dealing with business trips (Pacioli, 1494, ff. 187r–188r).
These are solved by algebra. For example, problem 10 concerns an unknown number of business trips in
which each trip contributes 20% to the starting capital. At the end, it is found that he the capital has become
the square of the starting capital.
Figure 6. Pacioli discusses the possible solutions to the monkey and coconut problem.
Pacioli was not the first to apply algebra to the problem with business trips. Such problems and their
algebraic solution appear regularly in the Trattato di praticha d’arismetricha of Biagio The Old, c. 1340,
Siena, Biblioteca Comunale, L.IV.21, and the Arithmetica e geometrica of Maestro Gilio, 1384, Siena,
Biblioteca Comunale, L.IX.28.25
Algebraically, our problem from van Varenbrakens results in a simple determinate equation with one
unknown:
2 2(2x − 6) − 6 − 6 = 0 with solution x = 5 1/4.
23 An alternative name for the Regula transversa is the (first) Regula pulchra. For further discussion on these rules, see below.
24 Bologna, Biblioteca universitaria, no. 250, f. 120v. A comprehensive study of Pacioli’s De Viribus can be found in (Amedeo
Agostini, 1924) and a transcript is published by (Peirani, 1997). Dario Uri has pictures of the complete manuscript on-line and
is in the process of doing the transcription. See http://www.uriland.it/matematica/. A recent publication includes a facsimile and
commentaries (Honsell and Bagni, 2009).
25 See Heeffer (2006a) for further references. The problems are BTP058, BTP065, BTP066, BTP092, BTP094, BTP109, BTP110,
GQA030-33, GQA040 and GQA041.
414 A. Heeffer / Historia Mathematica 41 (2014) 400–437
However, at the beginning of the sixteenth century a straightforward translation into such an equation
is still uncommon. A manuscript of the early fifteenth century gives a rare algebraic solution to a simpler
version of a man making two business trips, c. 1417, Turin, Bibliotheca Nazionale, N. III; 53, f. 37r (Rivoli,
1983, 53–4). At the first, he doubles his money and spend 8 denari, at the second he makes 3 denari from 2,
and spend 10. He is left with nothing. The anonymous author uses a cosa for the starting capital and arrives
at 2x – 8 after the first trip. After the second trip he is left with 3x − 12 − 10 or 3x − 22 = 0 (“Avemo che
3 cose meno d. 22 sono equale a niente, perchè tu voi che non li resta nula”), with the solution x = 7 1/3.
Ries has nine instances of the problem in his Coss. One is alike van Varenbrakens’, but doubles and
spends 12. Ries proceeds as follows26:
Suppose they had x and thus at first they had 2x from which they gave away 12 d. thus they kept 2x − 12 Ø.
Now, in the second this amount to 4x − 24 Ø. Gives away 12 d. leaves 4x − 36 Ø. In the third they arrive at
8x − 72 Ø, give away 12 and keep therefore 8x − 84 Ø. This is the amount they ended up with, thus 0. This
gives two parts that are equal, one 8x and the other 84 Ø. Divide, and so is 10 1/2 the amount they started
with.
Although the solution applies algebra, it is different from what would become common from the eighteenth
century. Instead of translating the problem in symbolic form, the unknown is used to build an argumentation
which ultimately leads to an equation. The same approach is taken in (Rudolff, 1525; Stifel, 1553, f. 183r,
problem 5).
Formulated as a symbolic expression, it becomes understandable that an algebraic formulation of the
problem spoils the recreational aspect.27 In contrast with some other problems, this type did indeed disappear
by the end of the sixteenth century. Baker was the last one to be “troubled” by this problem (Baker,
1562).28 The development of algebra changed the scope of recreational mathematics. Problems that can
be easily solved algebraically loose their appeal in the seventeenth century. Only from the early eighteenth
century onwards, the problem reappears as an easy exercise in works on algebra (Euler, 1770, 204; Wingate
and Kersey, 1650, 397; Kersey, 1673, 67).
4.2. Similar problem but all getting the same
In the early thirteenth century a problem very similar to the monkey and coconut problem appears in
which the shares are of equal value. Curiously, the condition that the shares are all equal – which is essential
for solving the problem – is often concealed or implicit. The problem cannot be solved by calculating
backwards.29 Here is an example from a Byzantine arithmetic book of c. 130830:
26 Dresden C80. My translation from the transcription (Berlet, 1860, 22). A possible source for Ries is the problem from the
Vienna codex 7255, f. 14r.
27 I presented this problem as an informal test to twelve test subjects. All of them solved the problem algebraically and not a single
person used the rule of inversion which is actually the easiest method in this case. This is an illustration of how symbolic algebra
strongly influences and narrows our scope of possible problem solving methods.
28 (Baker. 1562, ff. 193r–193v): “A merchant did ride into three severall fayres: at the first hee doubled his money and spent 10
crowns, at the second he did also double his money and spent 10 crowns. And likewise at the third fayre, he did double his money
and spent 10 crowns, and in the ende, hee founde that he had remaining but 2 crowns”.
29 For an extensive discussion of this problem see (Høyrup, 2008). Høyrup notes that the problem can be solved directly, without
algebra, from the equality of the last two shares, but this is on the condition that one believes that there is a solution. However, at
the time that the problem was formulated this was not understood in this way.
30 Cod. Par. Suppl. Gr. 387, f. 135r, My translation from (Vogel, 1968, 103–104).
A. Heeffer / Historia Mathematica 41 (2014) 400–437 415
At breakfast apples are served. The first person gets 1 apple and the 7th part of the rest of the apples, the
second gets two apples and the 7th part of the rest, the third, three apples and the 7th part of the rest, the
fourth, four apples and the 7th part of the rest, and the others also according to this order. One will answer
how many there were at breakfast and how many apples have been eaten.
The problem could have evolved from an older version, possibly of Arabic origin, where the number of
parts is given. In Liber augmenti et diminutionis we find a chapter on division problems Capitulum de
pomis, with a division of apples leaving one.31 Høyrup however dismisses an Arabic origin and believes
that the evidence of advanced arithmetical and algebraic experimentation in the Liber mahamaleth and the
Liber abbaci makes it more likely to have been created in the Maghreb or al-Andalus of the 12th century
(Høyrup, 2008).
The same problem is often formulated as a legacy problem in which a dying man distributes gold pieces
to his children, all receiving the same amount. He dies before finishing his sentence. With i children, each
child gets ai plus 1/nth of the rest (with a being an arbitrary factor). The question is how many gold pieces
he possessed and how many children he had. The problem is known in German writing by the name “Der
unbekannte Erbschaft” (Figure 7) (Vogel, 1968, 173; Tropfke, 1980, 586). Fibonacci is our earliest source
to this version of the problem. Although we find the problem treated in chapter 12 of the Liber Abbaci on
elchataym, it is not solved by double false position but by the following rule: if the rest is divided by seven,
subtract one from seven and square the result, which is 36. This is the original amount (Boncompagni, 1857,
279; Sigler, 2002, 399). The solution thus depends on the following rule of thumb, without any explanation
provided:
x = (n − 1)2
.
Fibonacci adds two more problems after the unknown heritage problem which Sigler places under the
heading “On the Separation of a Number into Parts” (Sigler, 2002, 399). These are discussed by Høyrup
and provide a solution to the same problem using the regula recta, a form of linear algebra using ‘the thing’
(Høyrup, 2008). Interestingly, the problems that are presented as abstract partitioning problems are solved
by algebra, while the ‘recreational’ version of the heritage problem is solved by the recipe. This seems to
indicate that while all three problems are mathematically equivalent, Fibonacci already makes a distinction
between a recreational version of the problem to be solved by a recipe and an abstract version to be solved
by algebra.
The same rule of thumb for the identical problem is repeated in Planudes’ Ψηφηφoρια κατ’ Ivδoυς η
Λεγoμεvη Μεγαλη of c. 1300 (Allard, 1981, 192–194; 233–234). Also in Paolo Dagomeri (Arrighi, 1964,
140) with 1000 and 1/10th and another one with 1/6th, Libro d’Abaco of c. 1330 (Arrighi, 1973, 199). In
the Trattato d’Abacho of Maestro Benedetto, c. 1470, Florence, BNCF, Magl. Cl. XI, 76, f. 53r (Arrighi,
1987a) the problem appears with 1000 and 1/10th.
In the Algorismus Ratisbonesis (c. 1450) three instances of the problem are solved with the same rule of
thumb (Vogel, 1954, problem 114, 115 and 352). The rule continues to be prescribed well into the sixteenth
century and it takes some time before algebra is applied to the problem. Ghaligai (Ghaligai, 1521, Chap. 9,
problem 25, 26) and Tartaglia (Tartaglia, 1551–1560, book 9, quest. 2, 98r–98v) do not solve the problem
31 Paris, BNF, Lat. 7377A (Libri, 1838, 336–337): “Quod si quis dixerit quidam vir intravit viridarium, et collegit in eo poma;
viridarium vero habebat tres portas, quarum quamque hostiarius custodiebat. Vir ergo ille partitus est poma cum primo, et insuper
donavit ei duo, et partitus est cum secundo et donavit ei duo, et partitus est cum tertio et donavit ei duo, et egressus est habens
unum: quantus ergo fuit numerus pomorum que collegit?” The resulting equation is: x − x/2 − 2 − (x − x/2 − 2)/2 − (x − x/2 −
2 − (x − x/2 − 2)/2)/2 = 1. The original author of this treatise according to (Hughes, 1994, 2001) is Abraham ben Meir ibn Ezra.
This attribution is disputed by Jens Høyrup. The problem is solved by double false position.
416 A. Heeffer / Historia Mathematica 41 (2014) 400–437
algebraically. Buteo, who solves 67 recreational problems by the use of algebra in book V, lists it in the
fourth book on arithmetical problems instead of the algebra section (Buteo, 1559, quaestio 78, 286–288).
The earliest algebraic solution I could find (apart from
the partioning problem of Fibonacci) is from Rudolff
for the problem with n = 10 (Rudolff, 1525; 1553, 252r,
RBH1110). Using the unknown for the legacy, he expresses
the share of the first person as x−1
10 + 1 and
calculates the share of the second as 9x−29
100 + 2. As all
shares are equal, the two can be equated and he arrives
at a solution of 81. Cardano gives the solution by rule of
thumb but adds “potest etiam fieri per algebra” (it can
also be done by algebra). He adds an algebraic solution
to the problem with first giving 1/7th of the remainder
and then 100 (Cardano, 1539, Chap. 66, problem 65).
Figure 7. The unknown legacy problem from (Widmann, 1489, f. 97v).
He equates the share of the first son with that of the second and arrives at the equation:
1
7
x + 100 =
1
7
6
7
x − 100 + 200
which is easily solved to x = 4200.
We find the algebraic solution reproduced by Valentin Mennher in his revised Practicque pour brievement
apprendre à ciffrer (Mennher, 1556, 21b).32 After Mennher the problem virtually vanishes from
arithmetic books. One notable exception is Bachet’s Problèmes plaisants, can be considered as the first
book fully devoted to recreational mathematics (Bachet, 1612, 149–154). The problem gets a full algebraic
treatment in the later editions of Bachet’s works (Lasbosne, 1874, 1879). With the further exception
(Ozanam, 1725, 67–68) the problem reappears only with Euler (Euler, 1770, 227). Euler treats many problems,
which we now consider as recreational, as exercises on algebra. Euler considers the legacy problem
to be of particular interest.33
32 (Chuquet, Paris, BNF, Français 1346) gives a general solution based on a progression. He does not formulate the problem as an
equation which is solved algebraically. See (Flegg et al., 1985, 225).
33 (Euler, 1770): “Diese Frage ist von einer gantz besondern Art und verdienet deswegen bemercket zu werden” (Weber, 1911,
227).
A. Heeffer / Historia Mathematica 41 (2014) 400–437 417
It appears frequently in nineteenth-century textbooks, not always with the most elegant solution.
A French classic is Bourdon who gives a general solution with legacy x and the nth child getting na
plus one nth of the rest (Bourdon, 1840, 62–3). The first child gets
a +
x − a
n
or
an + x − a
n
the second part then becomes
x − 2a −
(an + x − a)
n
divided by n, or
nx − 3an − x + a
n2
thus, the second child gets
2a +
nx − 3an − x + a
n2
or
2an2 + nx − 3an − x + a
n2
equating the share of the first and that of the second gives
an + x − a
n
=
2an2 + nx − 3an − x + a
n2
which leads to x = an2
− 2an + a
substituting this in the expression for the part of the first child gives
an + an2 − 2an + a − a
n
which simplifies to a(n − 1).
As all the shares are the same, dividing the total by the share of the first child gives the number of heirs:
an2 − 2an + a
an − a
which results in n − 1.
Hackley, possibly misunderstanding the original problem, goes as far as expanding the expression for the
third child, subtracts the three parts from x and then equates to zero. He thus arrives at the value of the
legacy as (Hackley, 1846, 164–6):
(6n2 − 4n + 1)a
n2 − 2n + 1
.
In contrast, Euler’s solution is refreshingly simple and elegant. He uses the example in which the ith child
gets 100.i + 1/10th of the rest. He notices that the differences between subsequent parts are the same and
can be expressed as:
100 −
x + 100
10
.
But as all children get an equal part, these differences must be zero, therefore
1000 − x − 100 = 0 or x = 900.
The problem of the unknown legacy is a suitable example to illustrate our case: It has been very popular
for several centuries. It was commonly solved by some trick or rule particular to the problem. A general
418 A. Heeffer / Historia Mathematica 41 (2014) 400–437
solution became available in books on algebra around 1560. It looses its appeal after an algebraic solution
emerged, to return later as an exercise in elementary algebra.
4.3. A counter example
We have discussed three types of problems for which we observe a discontinuity in arithmetic and
algebra books during the second half of the sixteenth century. All the three types have a history which
goes back at least to the fourteenth century, but often even earlier sources can be given. These problems
have been very popular and we have shown how they appear widespread in medieval and Renaissance
texts. Then, around 1560 they virtually disappear from the books only to turn up again about two centuries
later. Given all this evidence one might suspect that this is a general phenomenon, that all such recreational
problems lost their appeal or function from the late sixteenth century. We will now demonstrate that this is
not the case and that the observed discontinuity only occurs for some particular types of problems. Other
problems continue to get treated both arithmetically and algebraically in the many textbooks of the late
sixteenth and seventeenth centuries.
Cistern problems are a type of division and sharing problems which is as old as mathematics itself. The
name of the problem refers to a fountain or cistern which filled by several pipes with a different flow rate.
Given the rates of each pipe and the volume, the question is how long it takes to fill the cistern. The older
versions deal with a task to be performed by several labourers with different working rates.
Table 2
Meeting and overtaking problems between 1560 and 1700.
Author Date Book Problem Page
Humfrey Baker 1562 Well spring of sciences Prob. 3 and 4 36v–37v
Vander Gucht 1569 Cijfer bouck 2 problems 99v
Dionigi Gori 1571 Libro di arimetricha 3 problems 81v–82r
Christian Wurstisen 1579 Elementa Aritmeticae 2 problems
Nicolaus Petri 1583 Practicque om te leeren rekenen 2 problems 133v
Giambattista Benedetti 1585 Diversum Speculationum Mathematicarum Theorema CVI–CXIIII 68–77
Christoph Clavius 1583 Epitome arithmeticae practicae Quest. 20 (1607) 256
Thomas Hood 1596 The elements of arithmeticke 2 problems 110, 119
Christoph Clavius 1608 Algebra Cap. 31, 2, 3 306–307
Denis Henrion 1634 Cursus Mathematicus, II Algebra Quest. 19 172
John Kersey 1650 Arithmetic made easy Quest. 14 361
Jonas Moore 1660 Moores arithmetick, vol. 2 Algebra Quest. 1 55
William Leybourn 1667 Arithmetical recreations 16, 17, 18 41–44
Edmund Wingate 1668 Wingate’s Arithmetick 14–18, 23 450–453
Juan Caramuel 1670 Mathesis Biceps 81, 82 139–140
Edward Cocker 1678 Arithmetic 32 181
William Leybourn 1694 Pleasure with profit 8 38
Edward Wells 1698 Elementa arithmeticae numerosae 109, 116 207–208
Peter Lauremberger 1698 Institutiones Arithmeticae 12.XII 196
Isaac Newton 1707 Arithmetica Universalis Prob. 5 81–85
Thomas Simpson 1745 A treatise of algebra XIX, L, LXIII, LI 89, 113–116
Leonhard Euler 1770 Vollständige Anleitung zur Algebra 16, 25 205–6
Several Babylonian tablets before 1500 BC describe such problems: YBC 7673, 7164 and 4669, BM
85196 (Neugebauer and Sachs, 1945). We find them also in another form in the earliest Chinese writings
of the second century BC, e.g. the Suàn shù sh¯u 筭數書, Writings on Reckoning (Cullen, 2004, 54–6) and
A. Heeffer / Historia Mathematica 41 (2014) 400–437 419
they are discussed by the main authors of Hindu algebra. Such problems have always existed and have been
solved by different means. Their popularity does not seem to be affected during the seventeenth century.34
Even more pronounced is the continuous attention for meeting and overtaking problems. Several versions
of the problem exist. Two travellers travel towards each other with different speed. Given the distance, the
question is when do they meet? Variations to the problem use the same direction with different starting
times and add travelling rates in arithmetic or geometric progression. Our first sources of such problems
are Chinese, the Suàn shù sh¯u and the Ji˘u zh¯ang suàn shù 九章算術, Nine chapters of the mathematical
art (Vogel, 1968; Chemla and Shuchun, 2004). They appear very frequently in Indian mathematics and
are omnipresent in the West after Fibonacci. As a prime example of a recreational problem which retained
its popularity, while many other disappeared, we have collected a list of references from books published
between 1560 and 1770 (see Table 2).
5. Possible explanations of discontinuities
Several authors have traced the history and evolution of recreational problems, either as a whole (Smith,
1925, II, 532–594; Tropfke, 1980; Singmaster, 2004) or for some particular types of problems (Curtze,
1895a, 1895b; Vogel, 1954, 1968; Folkerts, 1978). Only Vera Sanford, a student of David Eugene Smith,
paid specific attention to the phenomenon of discontinuity.35 She formulates four possible explanation for
the disappearance of certain problems (Sanford, 1927, 79–94):
Problems have been eliminated from elementary textbooks when the business and economic conditions
upon which they are based have become obsolete; when their subject matter has lost interest; when the
mathematical ideas they illustrate have been forgotten; and when the invention of better mathematical tools
makes them trivial or causes their transfer to more advanced work.
Let us look at this fourfold vindication more closely.
5.1. Conditions that have become obsolete
The conditions that have become obsolete deal with problems such as “the assize of bread”, in German
Brotordnung. This problem asks for the calculation of the size or weight of a bread according to the cost of
wheat, for a given price, or the calculation of the price of a bread for a given weight. In order to facilitate
such calculations, books where published with tables working our the different combinations of weight,
cost of wheat and price of bread (Riese, Dresden, Sächsische Landesbibliothek, Codex C80; Gebhardt,
2004). Sanford shows how such problems materialize in arithmetic books to disappear by the end of the
sixteenth century when prices stabilized and the problem became a less pressing one. Other types of problems
discussed by Sanford, which met the same fate as the Brotordnung, include a specific contract type
for shepherds and a type of pasturage problem (Sanford, 1927, 81–86). She also mentions alligation, but
this is disputable as the rules for solving alligation problems apply to other contexts as well.
We can add tollet reckoning as a forgotten practice. The word ‘tollet’ is most likely derived from the
Italian tavola, in Venetian dialect toleta (Cantor, 1880–1908, II, 222). An example is shown in Figure 8.
34 (Singmaster, 2004, 7.H) lists seven references between 1570 and 1700, but several more arithmetic books of the seventeenth
century have versions of the problem.
35 This book is based on extensive research in Plimpton’s vast collection of rare books and manuscripts, now part of Columbia
University. Smith himself added more than thirteen thousand volumes on the history of mathematics and astronomy, including
some unique manuscripts such as Omar Khayyam’s treatise on algebra and trigonometry. Sanford’s book is often quoted as it
contains many problem texts from the original sources, which are difficult to find.
420 A. Heeffer / Historia Mathematica 41 (2014) 400–437
Tollet reckoning assisted in the complex calculations of exchange between the currencies of different cities
and countries which used coins of different gold and silver contents. The letters M, C, X, lb, mr, x, lot and
quint were used for denominations of marks, the unit of weight.
In any case, the type of recreational problems we have discussed above, is not dependent on such socioeconomical
conditions and therefore this explanation provided by Sanford does not help us any further.
Figure 8. A layout of a table for tollet reckoning, from (Apianus, 1527, f. Aaiiijv).
5.2. Subject matter that has lost its interest
For the problems whose subject matter has lost its interest, Sanford refers to Biblical themes such as the
computation of the number of angels in heaven. This medieval theme appeared regularly within writings
of the magic tradition, such as Albertus Magnus who conjectured their number at 6.6662 and cabbalist
writings where angels were determined to number exactly 301.655.722. These calculations became part of
arithmetical exercises in textbooks such as the General Trattato (Tartaglia, 1551–1560, I. f. 261v). We could
add Stifel’s preoccupation with Biblical word reckoning. Using the Roman numerals in Latin expressions,
numbers can be created. For example using I (= J), V (= U), X, L, C, D and M in “Jesus Nazarenus Rex
Judeorum” from his Endchrist, allows to reconstruct the publication year as MDXVVVVII (Stifel, 1532;
Reich, 2004, 334). As with the obsolete conditions, the recreational problems we have discussed do not
touch on subject matters that have lost their interest as these biblical themes.
5.3. Mathematical concepts now forgotten
The third explanation provided by Sanford refers somewhat awkwardly to “problems based on mathematical
concepts now forgotten”. As an example, she gives the kind of testament problems known from
Arabic algebra.36 Arabic law prescribes certain shares of a legacy for sons, daughters and the mother, when
36 Such problems appear in the third part of al-Kw¯arizm¯ı’s Algebra, in English translation by (Rosen, 1831). For a devastating
critique of Rosen’s interpretation, see (Gandz, 1937) who interprets the problems within the context of Arab inheritance law.
A. Heeffer / Historia Mathematica 41 (2014) 400–437 421
the father dies. This gives rise to rather complex calculations of proportional division. Such problems became
mixed with the rather hypothetical problem of posthumous twins known from Roman law (Cantor,
1880–1908, I, 522–25). Such problems appear commonly between the thirteenth and sixteenth centuries.
Sanford cites Frisius, Ortega, Borghi, Tonstall and Buteo. Let us look at one typical instance from the
pseudo Paolo dell’abacco (c. 1440) (see Figure 9)37:
One makes a testament, and if the [pregnant]
wife gives birth to a male child,
she will have one third of what he
leaves, and his son the other two parts;
and if the wife gives birth to a female
child, the wife will receive two thirds
and the girl one third. It now happens
that the woman bears twins, one male
and one female. The question is how
much comes to the woman, the male
and female child when the legacy values
70 lire.
Figure 9. Illustration of the posthumous twins problem (from Magliab. XI, 86).
One wonders what “mathematical concepts now forgotten” Sanford has in mind. The problem is usually
solved by establishing the relative portions of the legacy. As the son will get twice the part of his mother and
the mother twice the part of her daughter the estate is to be divided in the proportion son : wife : daughter
as 4 : 2 : 1. With a legacy of 70 lire, this nicely divides as 40, 20 and 10 lire.38 Later treatments of the
problem such as (Buteo, 1559, 341–2) are algebraical. Therefore the problem does not depend on specific
mathematical concepts. We even question that this type of problem lost its appeal. It appears continuously in
arithmetic books after the sixteenth century (Vander Schuere, 1600, f. 98; Schott, 1674, 559–60; Leybourn,
1694, 39–40; Ozanam, 1725, 179) and further on during the eighteenth century.
5.4. The invention of new mathematical tools
The final reason for the disappearance of certain problems is ascribed by Sanford to the invention of
new mathematical tools. Throughout the book she stresses the role of the new symbolism in the change
of the way problems are approached. The best formulation is from her discussion of recreational problems
(Sanford, 1927, 54):
These problems lose their mystery when the algebraic relations they represent are written out. It is significant
that the problems shifted from the main body of the work into the group of recreations when algebraic
symbolism was sufficiently improved to show this clearly.
37 From Florence, BNCF, Magliab. XI, 86 (Arrighi, 1964, 85): “Uno fa testamento e llascia che sse ila donna fae fanciullo maschio,
che ella abbi 1/3 di ciò ch’egl’à e il maschio l’altre due partj; e anchora lascia che xxe la donna xua fae fanciulla femjna, che la
donna abbj 2/3 di ciò ch’egl’à e ila fanciulla abbj l’altro terzo. Ora adiviene che lla donna sua partoriscie e fae uno fanciullo
maschio e una femjna; adomando che ttoccherà alla donna e che ttoccherà al fanciullo e che ttoccherà alla fanciulla, valendo il suo
70 lire”, translation mine.
38 (Arrighi, 1964, 85): “Fa’ choxj. Lo intendimento del padre si fue che ila donna avexxi 2 chotantj che ila fanciulla e che il
masc[h]io avexxi 2 chotantj che lla donna; adunque puoj dire choxì: avendo la fanciulla i arebbe la donna 2 e avendo la donna 2
arebbe il maschio 4”.
422 A. Heeffer / Historia Mathematica 41 (2014) 400–437
This is the explanation we will pursue further. But before going into the role of symbolic algebra, let us
explore some alternative explanations which are not discussed by Sanford.
5.5. The discovery of Diophantus
The disappearance of some standard arithmetical problems after 1560 coincides with the discovery and
dissemination of the Arithmetica of Diophantus. Antonio-Mario Pazzi, who thought mathematics at the
University of Bologna, contacted Rafael Bombelli in the late 1560s about the Greek manuscript of the
Arithmetica at the Vatican library. Together they set to translating the six extant books.39 Unfortunately
their translation is lost. However, Bombelli included 143 problems from the Arithmetica in his Algebra
published in 1572. The history of Bombelli’s Algebra has been traced by Bortolotti after discovering the
manuscript in a Bologna library in 1923.40 The original edition consisted of five books, most likely completed
before 1560.41 Books I and II are included in the printed edition. Book IV and V on geometry have
been omitted. The missing books were published in 1929 (Bortolotti, 1929) and later a complete edition
was published (Bombelli, 1966). Book III has only later been analysed by Jayawardene (1973). It contains
156 practical and recreational problems solved by algebra, including “men finding a purse” (problems 40
and 41) and “monkey and coconuts” problems, formulated as business trips (problems 101, 112, 113, 133,
135, 147). Jayawardene argues that although the printed book show little signs of the influence of practical
arithmetic on the printed Algebra, the manuscript certainly does. So after discovering Diophantus,
Bombelli decided to replace the practical and recreational problems by 143 more abstract problems from
the Arithmetica as a better illustration of algebraic practice. This highly unusual decision to leave practical
problems “to the dignity of arithmetic” (Bombelli, 1572, 317) is significant. It was common practice in
algebraic works before 1560 to include many practical and recreational problems, partitioned by type of
equation, as exercises in mastering the “new art”. Bombelli clearly distanced himself from this tradition in
an attempt to restore the analytic approach of the Greeks.
The disaffection with practical arithmetic was further pronounced by Stevin. In his L’Arithmétique
(Stevin, 1585), he chose to deal with the theoretical issues first and treated the arithmetic (La practique
d’Arithmétique) with its mercantile rules and practical problems last. The part on arithmetic was even preceded
by his own free translation of Diophantus’ Arithmetica from Xylander’s latin version (Xylander,
1575). Stevin had principle reasons do so. He believed that is was necessary to fully master the theoretical
foundation of arithmetic before applying them. Explaining the rule of algebra using practical examples
would be illicit from didactic point of view.42 Although he does not apply algebra to practical problems
their influence is clearly present in his choice of 27 problems illustrating the rule of algebra. Let us look
at Stevin’s 24th problem: “Find four numbers such that the sum of the first, second and third be 10, the
second, third and fourth equals 14, the third, fourth and first 13 and the fourth, first and second 11, leading
to the equations43:
39 Bombelli mentions seven books, but this is considered a mistake. Ver Eecke identifies Bombelli source based on the missing
proposition 28, as Codex Vaticana Gr. 200, which contains the known six books (Ver Eecke, 1926, lxiv).
40 Bologna, Bibliotheca Comunale dell’Archiginnasio, Codex B. 1569. Another copy of this manuscript is in the University library
of Bologna MS 595 (0)J2.
41 According to Bortolotti before 1556, but Jayawardene writes “between 1557 and 1560” (Jayawardene, 1973, p. 521).
42 As common at that time, Stevin considered algebra as a rule rather than a general problem solving method. He literally used the
term “la reigle d’Algebre”.
43 (Stevin, 1585), question XXIV, 424–425; (Struik, vol. IIB, 689; 707–708). For a earlier history of this type of problem see
Heeffer (2010).
A. Heeffer / Historia Mathematica 41 (2014) 400–437 423
x + y + z = 10
y + z + u = 14
x + z + u = 13
x + y + u = 11
with the solution: x = 2, y = 3, z = 5, u = 6.
This type of problem was at that time commonly set in a concrete situation and appears for example in
the correspondence between Regiomontanus and Giovanni Bianchini,44 and in Peletier’s Algebre.45 A more
complex problem with 5 men is found in 15th century Italian manuscripts.46 However, Stevin was not the
first to strip the practical context of problems and present them in a purer form. Estienne de la Roche,
whose Larismetique of 1520 depended heavily on the manuscript of Chuquet (Marre, 1881), treats classic
problems such as “If I had one from you...” or Geben und Nehmen in Tropfke’s terminology, as purely
arithmetical problems.47
Practical or recreational problems are completely absent in the works of Viète. In fact, Viète is more
interested in the structure of an equation and the relations between equations than in the solution to algebraic
problems. In Viète we observe a complete shift in interest in objects of analysis. Despite the postscript
“Nullum non problema solvere” (to leave no problem unsolved), Viète in In artem analyticam isagoge, does
not solve a single one of the many problems that he must have found in the many books on arithmetic of
his time (Viète, 1591). In the Zetetics (Viete, 1593) he uses 24 problems from Diophantus as an illustration
of how to translate a given problem in algebraic terms, while the application of this new art to practical
problems would demonstrate its power and usefulness more convincingly.
The explanation that some recreational problems disappeared with the discovery of Diophantus apparently
has some value. At least Bombelli and Stevin have consciously chosen to treat abstract problems
from the Arithmetica at the expense of the more common practical and recreational problems of their era.
But there are some problems with this line of reasoning. It overestimates the importance of Diophantus on
the vernacular arithmetic books from the abbaco tradition, the reckoning schools and the rechenmeister in
Germany and the low countries. Diophantus might have had a profound influence on Bombelli and Viète
but the Arithmetica did not touch the practioners of arithmetic who were more interested in recipes and
algorithms for solving problems of daily concern.
Another problem with the explanation of the discovery of Diophantus is that it was actually discovered
much earlier. Copies of the manuscript of the Arithmetica were, as all editions of Greek mathematics,
eagerly collected by the Italian humanists of the Quattrocento. One of the most industrious was Cardinal
Bessarion living in Venice. He succeeded in collecting over 500 Greek manuscripts before his death in
1472.48 The 1468 catalogue includes a copy of the Arithmetica, now known as MS Venice, Biblioteca
Marciana, Graeci 308. Regiomontanus had developed a friendly relationship with Bessarion and had the
opportunity to study the Greek text around 1463. He reported his find of the six books of the Arithmetica
44 (Curtze, 1902, 291): “Tres socii sunt, et quilibet per se habet denarios in marsupio. Duo primi absque tertio habent ducatos 30;
duo etiam secundi absque primo habent ducatos 42; sed duo alii absque secundo habent 54: queritur, quot quilibet per se habeat”.
45 (Peletier, 1554, 103–105): “Quatre hommes ont chacun certeine somme d’Ecuz. Le premier, second e tiers, ont ensamble 149.
Le second, tiers e quart ont 110. Le tiers, quart e premier ont ansamble 125. Le quart premier e second ont ansamble 138. Quele ét
la somme particuliere de tous?”. In modern notation: x + y + z = 149, y + z + u = 110, x + z + u = 125, x + y + u = 138 with
solution x = 64, y = 49, z = 36, u = 25.
46 In particular, manuscripts by Giovanni di Bartolo and Maestro Benedetto, see (Franci and Rigatelli, 1985, 54–55).
47 For example, problem 70 from Chuquet (Marre, 1881, 432; Flegg et al., 210–211) is stripped from its concrete context: “Trouves
troys nombres tels que leves 12 du second et du tiers et les adiouster au premier, le premier sera le double des aultres +6. Et leves
13 du tiers et du premier et les adiouster au second. Le second sera le quadruple des aultres deux +2 et leves 11 du premier et
second. Et les adiouster au tiers. Le tiers sera le triple des aultres deux +2”. In (de la Roche, 1520, f. 43v) the problem is reproduced
literally, but uses a different solution method. For more on the relation of de la Roche and Chuquet see (Heeffer, 2012b).
48 See (Rose, 1975, 44–46 and 90–109), for the role of Bessarion in the transmission of Greek mathematics to Renaissance Italy.
424 A. Heeffer / Historia Mathematica 41 (2014) 400–437
in a letter to Giovanni Bianchini (Curtze, 1902, 256–7). He was then already well-acquainted with the
Arab algebra and owned a copy of the manuscript on algebra by Al-Kw¯arizm¯ı, possibly from his own
pen (New York, Columbia University, Plimpton 188).49 Very receptive to the processes of transmission of
mathematical ideas, he found in Diophantus a source of inspiration for Arab algebra. In his Oratio, a series
of lectures at the University of Padua in 1464, he was the first to propose the view that Arabic algebra
descended from the Arithmetica of Diophantus (Regiomontanus, 1537). Having discovered Diophantus,
Regiomontanus’ interest in practical and recreational problems does not seem to be affected as is shown in
his correspondence with Giovanni Bianchini, Jacob von Speier and Christian Roder.
A final argument against the role of Diophantus on the disappearance of recreational problems is that only
some types vanished while others continued to flourish and be used in arithmetic books. An explanation for
the lack of interest in some types of problems should take the particular structure of those problems into
account.
5.6. The replacement of the oral tradition by printing
The specific structure and function of recreational problems could lead us to another possible explanation
why many types disappeared in the sixteenth century. Rhyme and cadence functioned as mnemonic aids
and facilitated the oral tradition of asking riddles and telling stories. Many of the older problems are put in
verse. Some best known examples are “Going to St-Yves” using the geometric progression 7+72 +73 +74
(Tropfke, 1980, 629). We know also many problems in rhyme from Greek epigrams50 such as Archimedes
cattle problem (Hillion and Lenstra, 1999), the ass and mule problem from Euclid (Singmaster, 1999) and
age problems (Tropfke, 1980, 575–576). During the middle ages complete algorisms were written this way,
taking over 500 verses (Waters, 1929). Even without rhyme, problems were cast into a specific cadence
to make it easier to learn by heart. The 53 problems of Alcuin early 9th century clearly show a character
of declamation, specific for the medieval system of learning by rote.51 Medieval students were required
to calculate the solution to problems mentally and to memorize rules and examples. Mnemonic aids were
not only used in formulating problems, but also in remembering solutions. Memorizing a trick solution,
a mathematical short cut, is also typical for many recreational problems. The solution to the problem of
Christians and Turks, also called Josephus or survival problem, was commonly learned by heart by use of
a vowel mnemonic.52 To avoid to be thrown overboard, the men have to be arranged into a specific order.
This is remembered as a sequence of men of the same type, represented by vowels a, e, i, o, u. Rex angli
cum veste bona dat signa serena represents the sequence 2, 1, 3, 5, 2, 2, 4, 1, 1, 3, 1, 2, 2, 1, saving the
15 Christians from 15 Jews counted by 10, Algorismus Ratisbonensis (Vogel, 1954, 52).
And then, with the emergence of printed books, our collective memory was completely transformed.
Collection of problems and recipes for solutions became available in printed form for everyone to read
and study. And they were popular indeed! The Latin book of Gemma Frisius on practical arithmetic
49 Menso Folkerts has since long announced a transcription which will be published in the series Algorismus by Erwin Rauner
Verlag in Augsburg as Algebraischen Studien des Regiomontanus. Edition eines Textes aus der Handschrift Plimpton 188.
50 The most comprehensive collection of Greek epigrams is in The Greek Anthology (Paton, 1979). A representative selection of
these problems appear in the Récréation Mathématique of 1624.
51 As an example, Proposition 5, propositio de emptore denariorum: “Dixit quidam emptor: Volo de denariis C porcos emere; sic
tamen, ut verres X denariis emantur; scrofa autem V denariis; duo vero porcelli denario uno. Dicat, qui intelligit, quot verres, quot
scorfae, quotve porcelli esse debeant, ut in neutris numerus nec superabundet, nec minuatur? (Folkerts, 1978).
52 For a comprehensive history of the problem and list vowel mnemonics for the Arab, Latin, English, French and German language
see (Singmaster, 2004). One not in his list is: “Rex Darius gentes profanas in mare legat”, for 15 Turks and 15 Christians counted
by 10, from (van Varenbrakens, Ghent, Universiteitsbibliotheek, Hs. 214, f. 148v).
A. Heeffer / Historia Mathematica 41 (2014) 400–437 425
went through 64 editions before 1600.53 Vernacular works such as the second arithmetic book of Adam
Ries (Ries, 1522) saw no less than 99 editions in the sixteenth century (Gebhardt and Rochhaus, 1996).
Mnemonic aids became less relevant for the continued existence of recreational problems and the problems
itself were not the only means to convey arithmetical knowledge. A possible hypothesis is therefore that the
replacement of oral traditions by printed works caused the disappearance of recreational problems. However,
there are some problems with this line of explanation. Again, not all recreational problems disappeared
at the end of the sixteenth century, only certain types of problems. We would expect printing technology
to affect all types of problems equally. Secondly, this explanation somewhat forces the timing. I consider
the period between 1540 and 1560 to be pivotal for the disappearance of problems after 1560. It is true that
the growing popularity of arithmetic books lagged behind that of printed bibles and calendars. However,
its effect should have manifested some decades earlier. Thirdly, and most importantly, the oral tradition
was not at all replaced by printed works. Many recreational problems, such as the river crossing problem
are part of our cultural heritage. They have been told over and over again since Alcuin and probably much
earlier.54 I learned them as a child at the camp fire as I continued the tradition by posing the riddles to my
children.
5.7. The replacement of standard recipes by algebra
We now come to our explanation for the disappearance of certain types of problems during the second
half of the sixteenth century. Sanford has already set the line of explanation by stating that recreational
problems lose their mystery when expressed by algebraic equations (Sanford, 1927). We have discussed
the example of monkey and coconut problem, reduced to a simple equation as 2(2(2x − 6) − 6) − 6 = 0.
Indeed, the availability of a general method for solving linear problems by 1560 made some types of
standard Renaissance problems less appealing for use in arithmetic and algebra textbooks. Problems for
which the solution was based on some standard recipe, or rule of thumb, became trivial with the advent of
symbolic algebra. The new art of algebra provided a general solution method which rendered the knowledge
of standard recipes or specific rules superfluous.
Arithmetic books before the sixteenth century contained a great many recipes for a wide variety of
problems. General methods for solving arithmetical problems were limited to the regula falsi, or rule of
double false position. This rule applied to simple linear problems of the type ax +b = c. Problems involving
more unknown quantities or specific linear problems such as ax + b = cx + d were solved through a recipe
that applied only to one type of problem. Such recipes appear scattered through arithmetic books and ought
to be understood from examples. No formal explanation or justification is given, except for a numerical test
of the solution, at least in Medieval and Renaissance texts. By the end of the fifteenth century, some authors
felt the need for a more systematic exposition of the vast repository of the rules of mercantile arithmetic.
Two noteworthy examples are (Widmann, 1489) and (Pacioli, 1494). They both give a treatment of many
rules which have become known under certain Latin names during the fourteenth and fifteenth centuries.
Both did not hesitate to coin new names using Latin expressions which they considered appropriate for the
procedure described by the recipe. Table 3 gives an overview of the names of rules used by Widmann in his
arithmetic of 1489.55
53 Not counting French and Italian translations. Van Ortroy (1920) compiled a list with 62 Latin editions appearing before 1600.
Two more editions not included in his listing are Georg Hantzsch, Leipzig, 1558 and Johannes Rhamba, Leipzig, 1566.
54 Alcuin (c. 800) is our first written source. He has four versions of the problem (propositio 17–20), with the wolf, goat and
cabbage as the best known formulation today (Folkerts, 1978).
55 (Widmann, 1489), reprinted in Pforzheim, 1508. Our table is compiled from (Treutlein, 1879, 63), Leipzig, Codex 1470
(Kaunzner, 1968), (Tropfke, 1980) and the original text. There are some additional rules formulated by Widmann in some
manuscripts: the Leipzig Codex 1470 contains Regula usuri (f. 448v), Regula leporiss (f. 448v), Regula camby (f. 534v) and
Regula de (tali) tela (f. 502v), the last one also appears in the Dresden C80, (f. 287v) see (Kaunzner, 1968).
426 A. Heeffer / Historia Mathematica 41 (2014) 400–437
Some of the names have been coined previously. For example the regula equalitatis also appears in the
Vorlesung at Erfurt by Gottfried Wolack (1467–8) and the Algorithmus from Peurbach (published 1510)
(Tropfke, 1980, 600). But as others do not appear in previous manuscripts, Widmann must have devised
many of the names by himself. Typical medieval problems using standard solution recipes, such a woman
handing out figs to children, have risen in esteem through the use of Latin names, such as Regula augmenti
et decrementi, here given by Widmann. But the large number of rules made life not easier for the typical
student of vernacular arithmetic of the sixteenth century.
For several of these rules the principle behind it remains a mystery, and in several cases it is hard to
understand what the rules are about.56 What to think of the problem “when 3 times 3 equals 10, how much
will 4 time 4 be?”, which is “solved” by the Regula suppositionis (Widmann, 1489, f. 117r).57 Even Widmann
himself loses track, as he uses one name for several different rules such as the Regula pulchra and
also uses different names for the same rule, such as Regula augmenti.58 We cannot withhold the suspicion
that the fabrication of new rules or devising fancy names for existing recipes was motivated by the commercial
aims of selling books. We do not know much about Widmann’s life but at least some rechenmeister
after him, such as Adam Ries made a fortune from selling arithmetic books. Presenting problem solutions
by a great number of rules with Latin names legitimized the method as well as the author. The use of many
names for different methods of solving arithmetical problems reinforced the efficacy of reckoning school
and the value of textbooks.
And then algebra became popular as the general method for solving all arithmetical problems. The
recognition of the universal applicability of algebra did not happen in a single day. Within the abbaco
tradition in Italy and starting from the second half of the fifteenth century in Germany and France, problems
which used to be solved by standard recipes, entered the reach of the new art of algebra. Some problems
became an easier prey than others.
Age problems and problems solved by the regula augmentati are instances of problems that were subject
to an early approach by algebra.59 For others, such as the legacy problem discussed before, it was not
before the sixteenth century that we find the first algebraic solutions. With the wide spread of printing in
the sixteenth century, the algebraic method for solving classic problems became known all over Europe.
A handful of books laid the foundations for algebraic problem solving, initially, Pacioli’s Summa, although
the major part of his work also deals with typical mercantile recipes (Pacioli, 1494). This was followed
by Ghaligai who included a chapter on algebra with almost fifty problems frequently appearing within the
abbaco tradition (Ghaligai, 1521). In Germany, the first book covering algebra was (Grammateus, 1518)
which demonstrated the general applicability of algebra as an alternative to the regula falsi for 15 examples
of linear problems. But Grammateus was overshadowed in scope and method by Christoff Rudolff who
compiled a repository of four hundred problems from fifteenth-century manuscripts and solved all of them
by algebra (Rudolff, 1525). Imagine the impact this must have had on its target public in the early sixteenth
century.
Having learned and practiced mercantile recipes for years, there comes a book in which all these different
type of problems, on profit and loss, alligation, alloys, exchange, barter, company and company with
time, are solved using a single rule (the resolution of linear equations).The power and applicability of the
56 Witness the comments from Kaunzner’s treatment of Widmann, as on the Regula positionis: “Wenn kein Beispiele dabeistünde
könnten wir mit dieser Erläuterung nichts anfangen” (Kaunzner, 1968, 77).
57 (Høyrup, 2007, 67) coined the term “counterfactual calculations” for these problems in arithmetic.
58 See also (Eneström, 1908, 197–8) for a critique on Cantor’s treatment of Widmann’s rules (Cantor, 1880–1908, II, 234). Widmann’s
use of the rules is also discussed by Kaunzner (Kaunzner, 1968, 76).
59 For the early age problems see (Tropfke, 1980, 575) and the sixteenth century versions are covered by (Singmaster, 2004, 7X).
Their most common form, as in the Greek Anthology, is a1x + x
a2
+ x
a3
··· + x
an
= b. Algebraic solutions to problems that fall
under the regula augmenti or augmentationis will are discussed extensively in Heeffer (2007).
A. Heeffer / Historia Mathematica 41 (2014) 400–437 427
Table 3
An overview of the rules from Widmann’s mercantile arithmetic.
Rule Purpose 1489 1508
Regula residui x − x
a = b 52r
Regula reciprocationis
(x − x
a1
− ··· − x
an
)2 = x 53v, 55v
Regula excessus
Regula divisionis x + y = a, x
y = b 55r
Regula inventionis Variation on Regula detri
76r 57r
Regula resolutionis
Regula fusti Accounting for tainted food 96r 65r
Regula pulchra I
Calculating backwards
97v 66r
Regula transversa 100r
Regula detri conversa Variation on Regula detri 98v 66v
Regula ligar Alloys and mixtures 102r 68v
Regula positionis ax = by = cz,dx + ey + f z = g 103v 70r
Regula pulchra II ax + b(x + e) + c(x + f ) = d 105v 71r
Regula equalitatis ax + bx + cx = d 107r 72r
Regula legis ax + bx + cx = d,x + y + z = e 108v 73r
Regula augmenti ax + b = cx + d 110v 74r
Regula augmenti + Decrementi ax + b = cx − d 112r 75v
Regula plurima ax + b = d − cx 114r 76r
Regula pulchra III ax + b = cx − d 114v 77r
Regula sententiarum 1
a (1
b x) = c 115v 77v
Regula suppositionis Variation on Regula detri 117r 78v
Regula residui II Profit and loss reckoning 117v 79r
Regula excessus II x + y = a
xy = b
118v 79v
Regula collectionis x + x
a1
+ x
a2
··· + x
an
= b 119v 80r
Regula pulchra IV x + a = b(y − a),y + b = c(x − b) 120v 81r
Regula quadrata Pythagorean theorem 122v 82r
Regula cubica Cubic root approximation 124r 82v
Regula bona Double other’s money problem 125v 84r
Regula lucri Calculation of interest 127v 85r
(no name) x + c = ay,y + c = bx 146v 98v
Regula pagamenti Money exchange 76r 107r
Regula alligationis Also known as Regula virginum 159r 107v
algebraic method is so evident in Rudolff’s book that it could not come without a drastic effect on existing
mercantile rules and methods. A few years before, a book by de la Roche (1520) was published in
Lyon, France’s center for commerce and trade in the early sixteenth century. Not only the maître, but also
the banker, and the merchant in the wool trade or maritime trade, benefited from the way many practical
problems were solved by algebra in this Larismethique (de la Roche, 1520).60
These few books were sufficient to initiate an important change in the selection of problems in the books
published during the following years. It is not so that from now on all problems were solved by algebra.
Arithmetic books in which problems were abundantly approached by standard recipes and purely arithmetic
means continued to be published for centuries. But the simpler type of problems, whose answer depends
on knowledge of some rule of thumb, became unattractive knowing that they had a trivial solution in books
such as (Rudolff, 1525).
60 The influence of Chuquet and de la Roche on German cossic algebra may have been more important than previously considered,
as argued in (Heeffer, 2012b).
428 A. Heeffer / Historia Mathematica 41 (2014) 400–437
During the next years the plethora of mercantile rules for arithmetic problems became an embarrassment.
Their use became a subject of ridicule and their recommendation was criticized by authors such as Cardano.
Both in his Arithmetica practicae and in his Ars magna, Cardano hits on Pacioli, who presented many of
these rules in their works. In response to the way these authors churned out mercantile rules, Cardano
himself devised the Regula de modo. This has been misinterpreted as just another rule for solving linear
problems with two unknowns.61 Instead, Cardano presents the method as “the rule, from which all rules of
the previous chapters can be extracted” (“regula per quam extrahuntur omnes haec regulae ex superioribus
capitulis vocatur regula de modo”) (Cardano, 1663, IV, 79). These previous chapters of the Arithmetica
practicae treat mercantile rules as well as the rule of double false position. “One can fill a book full of these
rules in one month” Cardano sneers. He exposes the method by which Pacioli can fabricate new rules. “Just
solve the problems by algebra, then omit the unknown and keep the operations on the term together and
there you have a general rule”.62 In the Ars magna he spends a full chapter on the Regula de modo and
writes63:
This rule of method (so called because it shows the method for fabricating as many mercantile rules as
you wish) is very useful to teachers of arithmetic in teaching that art, certain easier [methods] having been
discovered [by means of it]. With its help, we constructed the greater part of the sixth book. This, then, is
the rule: Solve any given problem by any means you can, either [by using] an unknown or with the help of
the sixth book. Then leave out the unknown and the other rules and use those operations which you can best
use, looking always for brevity, and you will have the rule for the method of [solving] any similar problem.
This is a clear statement on the redundancy of mercantile rules. Algebra, as a general problem solving
method, subsumes a great number of individual rules. It even allows to generate as many mercantile rules as
you wish. Here, Cardano not only reveals the secret of the reckoning masters of the early sixteenth century,
he also uncovers the probable origin of some of these mercantile rules. Most likely they are relics from
previous algebraic solutions. We tend to use the term fossils as a characterization of the rules, as fossilized
imprints of previous algebraic practice. Cardano depicts the process as starting from an algebraic solution,
leaving out the unknown, formulating the operations of the derivation as a rule and you can apply it to
similar problems. Many of the mercantile recipes curiously correspond with rules described by ¯Aryabhata,
Br¯ahamagupta and Bh¯askara.64
6. Conclusion
As we have shown by several examples, the development of symbolic algebra between (Cardano, 1539)
and (Buteo, 1559) is precursory to the disappearance of certain types of problems which became part of
recreational mathematics. Curiously, several works after this period refer to the art of solving problems
without algebra. A German book by Zacharias Lochner makes explicit reference in the title, that it contains
61 (Tropfke, 1980, 402): “Cardano nennt die lösung eines Systems von zwei linearen Gleichungen mit zwei Unbekannten Regula
de Modo”. This is apparently an interpretation from the editors Vogel, Reich and Gericke, because Tropfke in his original edition
(Tropfke, 1930–1940, 45, note 228) points out the special status of this rule, and cites and translates the relevant fragment.
62 (Cardano, 1663, IV, 79–80): “Est etiam regula de Modo a me appellata, quoniam ex ipsa habent regulae infinitae in rebus maxime
mercantilibus, et potes replere librum ex ipsis in uno mense, diversarum operationum, quae omnes regulae diversae videbuntur:
et ita Frater Lucas, Borgias, Fortunatus, fecerunt libros per Neotericis instruendis, et ita tu lector poteris quotidie novas regular et
inusitatis fabricare. Modus est talis solve quaestionem quamvis per algebra deinde detrahe la co. et serva operationes easdem in
terminis suis, et erit regula generalis”.
63 (Cardano, 1663, IV, 272–3). This translation is slighly adapted from (Witmer, 1968, 180).
64 For a systematic study of a remarkable parallel of three typical problems in Renaissance arithmetic books and their Indian
counterparts see (Heeffer, 2007).
A. Heeffer / Historia Mathematica 41 (2014) 400–437 429
problems “with the advantage that they can be solved without the rule of algebra” (Lochner, 1583). Hans
Sybrandt advertises his book with solutions to hundred geometrical questions, that they can all be solved
“some with lines and some by numbers (though all without cossic numbers)” (Sybrandt, 1612, foreword).
John Speidell writes on logarithms recommending that “they require not at all any skill in algebra, or cossike
numbers, but may be used by every one that can onely adde and subtract, in whole numbers, according
to the common or vulgar arithmeticke, without any consideration or respect of + and –” (Speidell, 1624).
Webster (1722) advertises his book by asserting that all demonstrations can be performed by the principles
of arithmetick, without the use of algebra. Thomas Rudd gives a selection of “hundred geometricall questions,
with their solutions and demonstrations, some of them being performed arithmetically, and others
geometrically, yet all without the help of algebra” (Rudd, 1650). Advertising books on arithmetic which
do not use algebra may be prompted by the fear for the difficulties of learning algebra. However, there is
something more going on when William Leybourn (1667) for his Arithmetical recreations, uses the subtitle:
“Enchiridion of arithmetical questions both delightful and profitable, all of them performed without
algebra”.65
Clearly, symbolic algebra, as a general problem solving method for arithmetical problems, spoils their
recreational aspect. Mathematical problems lose their recreational nature if they are easily solved by a
general rule or method. We can also turn this reasoning backwards: recreational mathematics functions
best in solving problems for which no general solution method is readily available.
Acknowledgments
This paper grew out of a presentation with the title “How Algebra Spoiled Renaissance Practical and Recreational
Problems” presented at the conference The Origins of Algebra: From al-Khw¯arizm¯ı to Descartes at the Universitat
Pompeu Fabra, Barcelona, March 27–29, 2003. Several contributions to this conference were published in a special
issue of Historia Mathematica in 2006, vol. 33 (1). At that time I was not fully ready with the subject and later
expanded the text to chapter 3 of my unpublished PhD dissertation From Precepts to Equations: The Conceptual
Development of Symbolic Algebra in the Sixteenth Century, Ghent University, 2006. I would like to thank Jean Paul
Van Bendegem for assuring me in 2002 that recreational mathematics is a subject worthwhile exploring. I also thank
several participants of that conference for their feedback, including Jens Høyrup, Staffan Rodhe and Henk Bos. The
final paper benefitted from extensive comments by Karine Chemla and Jens Høyrup. My research during this period
of the past ten years has been generously funded by the Research Foundation Flanders (FWO Vlaanderen) (Grant
No. 3G002713).
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