On the curious historical coincidence of algebra and double-entry bookkeeping

On the curious historical coincidence of algebra
and double-entry bookkeeping
Albrecht Hee?er
November 20, 2009
[To appear in: Karen Fran¸
cois, Benedikt L¨
owe, Thomas M¨
uller, and Bart van
Kerkhove (eds.) Foundations of the Formal Sciences. Bringing together Philos-
ophy and Sociology of Science, College Publications, London.]1
1
Introduction
The emergence of symbolic algebra is probably the most important method-
ological innovation in mathematics since the Euclidean axiomatic method in
geometry. Symbolic algebra accomplished much more than the introduction of
symbols in mathematics. It allowed for the abstraction and generalization of
the concepts of number, quantity and magnitude. It led to the acceptance of
negative numbers and imaginary numbers. It gave rise to new mathematical
objects and concepts such as a symbolic equation and an aggregate of linear
equations, and revealed the relation between coe?cients and roots. It allowed
for an algebraic approach to ancient geometrical construction problems and gave
birth to analytical geometry. Why did this important methodological revolu-
tion happen? Why did it happen in Europe and not in Asia while Indian and
Chinese algebra were more advanced before the fourteenth century? Why did
it happen in the European Renaissance?
We can only touch the surface of possible answers to these fundamental
questions within the scope of this paper. However, we would like to argue that
the answers will involve multiple disciplines and will go beyond the bound-
aries of the history of mathematics. Most historians have taken for granted
that symbolic algebra was an inevitable step within the logical development of
mathematics. But can we speak of a logic of historical necessity? The history
of mathematics at least teaches us that there have been developments within
mathematics that were not in logical sequence. A full notion of the function
concept was developed only after the calculus, while textbooks on calculus ?rst
1Post-doctoral Fellow of Flanders Research Foundation (FWO Vlaanderen). Ghent Uni-
versity, Center for Logic and Philosophy of Science, Blandijnberg 2, B-9000, Gent. This paper
was written while the author was visiting researcher at the Center for Research in Mathematics
Education at Khon Kaen University, Thailand.
1

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introduce functions and then move to di?erentiation and integration. Some con-
cepts emerged within a historical context where no sensible interpretation could
be given to their meaning. A notorious example is imaginary numbers. Still, in
historical accounts these anomalies and anachronisms are considered exceptions,
and exceptio probat regulam. What if there is no such logic of historical develop-
ment? Then all historical questions must be addressed within their cultural and
social-economical context. Answers cannot be found by appealing to the next
logical step in the development, but only in the relation between its practices
and their meaning within the society. In other words, philosophy, history and
sociology of science all contribute to possible answers to the questions we have
raised.
In this text we will ?rst give a short overview of the internalistic approach
to the emergence of symbolic algebra which, as we will show, is present in most
historical accounts on the history of algebra. We will then present some studies
which take a contextual approach to developments in mathematics during the
period we are addressing. We then present our own position that symbolic al-
gebra was made possible by the central idea of value as an objective quantity
in mercantilism. As an illustration of how important developments in mathe-
matics can be matched with macro-economical changes in society we draw the
parallel between symbolic algebra and double-entry bookkeeping. These two
developments of the fourteenth and ?fteenth century were both instrumental
in the objectivation of value and they supported the reciprocal relations of ex-
change on which mercantilism depended. To demonstrate our proposition we
will present a case study to show how symbolic algebra and double-entry book-
keeping function in our understanding of a special class of bartering problems.
It would be wrong to understand a socio-economical account of the history of
mathematics as the right one or the only one. At the contrary, we believe that a
pluralism of explanations leads to a better understanding. However, concerning
the history of European algebra too much emphasis has been put on internal
mechanisms and we present our account as complementary to these approaches.
2
Internalist accounts of the history of algebra
Algebra was introduced in medieval Europe through the Latin translations of
Arabic texts between 1145 and 1250 and Fibonacci’s Liber Abbaci (1202) [7]
[54]. Algebraic problem solving was further practiced within the so-called ab-
baco tradition in cities of fourteenth- and ?fteenth century Italy and the south
of France.2 From the sixteenth century, under the in?uence of the humanist
program to provide new foundations to this ars magna, abbaco algebra evolved
to a new logistics of species with Fran¸
cois Vi`
ete [67] as the key ?gure. With
Descartes’s Geometry, this new kind of algebra progressed into our current sym-
bolic algebra. This is a brief characterization of the current view of scholars on
2We follow the convention to name the abbaco or abbacus tradition after Fibonacci’s Liber
abbaci, spelled with double b to distinguish it from the material abbacus. It refers to the
method of counting and calculating with hindu-arabic numerals.
2

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the history of European symbolic algebra.
Most of the studies on the history of algebra provide an internalistic account.
They accept implicitly or more explicitly that the development towards symbolic
algebra was inevitable and depended on some internal mechanisms and intrinsic
processes. Moritz Cantor whose Vorlesungen [13] had an important in?uence
on twentieth-century historians of mathematics, attributes, for the early period,
high importance to the Latin works of Fibonacci and Jordanus. He believed that
the vernacular tradition of practical arithmetic and algebra did not produce any
men capable of understanding the works of these two giants. Cantor assumes
this to be true for most of the 14th and 15th century.3 When dealing with the
sixteenth century, extraordinary importance is attributed to the Arithmetica of
Diophantus [26] [53] [49]. The idea that algebra originated with Diophantus was
fabricated by humanist mathematicians after Regiomontanus’s Padua lecture of
1464 [50]. As a consequence of their reform of mathematics, humanist writers
distanced themselves from “barbaric” in?uences and created the myth that all
mathematics, including algebra descended from the ancient Greeks [31]. Later
writers such as Ramus [47], [48], Peletier [44], Vi`
ete [67] and Clavius [14] par-
ticipated in a systematic program to set up sixteenth-century mathematics on
Greek foundations. The late discovered Arithmetica of Diophantus was taken
as an opportunity by Vi`
ete to restore algebra “which was spoiled and de?led
by the barbarians” to a ?ctitious pure form. To that purpose he devised a new
vocabulary of Greek terms to cover up the Arab roots of algebra “lest it should
retain its ?lth and continue to stink in the old way” [35, p. 318]. The reality
was that, with some exceptions, ancient Greek mathematics was more foreign
to European mathematics than Indian and Arabic in?uences which were well
digested within the vernacular tradition [27].
Contemporary scholars, such as Rafaella Franci and Laura Toti Rigatelli [19]
[20] or Van der Waerden [63] narrate the story of the history of algebra from
their internal dynamics. Early European algebra, as inherited from the Arabs,
recognized six types of equations. Instead of dealing with the general form of
a quadratic equation ax2 + bx + c = 0, the ?rst Latin translations distinguish
three cases depending on the sign of the coe?cients and three cases with one or
two terms missing. Each case had its own solution method. Double solutions
were not recognized except for the case of two positive roots. Early abbacus
masters extended the list of six to include higher degree cases most of which
could be reduced to the original six. During the fourteenth century Maestro
Dardi of Pisa expands this to no less than 198 cases [66] [21]! Already from the
fourteenth century abbacus masters started experimenting with irreducible cases
of higher degree. False rules were given for special cases of the cubic equation
[33]. However, Maestro Dardi gives some examples of cubic equations with
3Cantor [13] II: “Aber die Zeitgenossen der beiden grossen M¨anner waren nicht reif,
deren Schriften vollst¨
andig zu verstehen, geschweige denn sie fortzubilden, und besonders
f¨
ur die eigentlichen Gelehrtenkreise gilt dieses harte Urtheil auch noch im XIV. Jahrhunderte,
w¨
ahrend damals italienische Kau?eute der Algebra so viel Verst¨
andniss entgegenbrachten, dass
wenigstens versucht wurde, Aufgaben zu l¨
osen, welchen die fr¨
uheren Schriftsteller ohnm¨
achtig
gegen¨
uberstanden.
3

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a correct solution derived from numerical examples. These histories of algebra
then focus on the sixteenth-century breakthrough of Scipione del Ferro in solving
the depressed cubic and the feud between Tartaglia and Cardano for publishing
a general solution for the cubic equation. It then moves to symbolism introduced
by Vi`
ete and the general approach to problems. With a mention of Girard, these
developments culminate into the quest for the fundamental theorem of algebra
moving well into the eighteenth century. All this is presented as a continuous
?ow of necessary logical development. Each step is the necessary next move
in the logical puzzle of the history of symbolic algebra. Van der Waerden goes
to great lengths of demonstrating this continuity back to the earliest Greek
mathematics. In a earlier publication, van der Waerden [62, p. 116], uses the
so-called Bloom of Thymaridas to connect algebra with the Pythagoreans. He
goes as far as to claim that “we see from this that the Pythagoreans, like the
Babylonians, occupied themselves with the solution of systems of equations with
more than one unknown”. We have elsewhere demonstrated that these claims
cannot be sustained and that these should be understood as a by “humanist
education deeply inculcated prejudice that all higher intellectual culture, in
particular all science, is risen from Greek soil” [29].
Jacob Klein, a student of Heidegger and interpreter of Plato, wrote a long
treatise in 1936 on the number concept starting with Plato and the development
of algebra from Diophantus to Vi`
ete [35]. It became very in?uential for the his-
tory of mathematics after its translation into English in 1968. Klein goes even
further than Van der Waerden or Franci and Rigatelli in their internalistic ac-
count of history. For Klein it is not the evolution of solution methods for solving
equations which follows some logical path but the ontological transformation of
the underlying concepts within an ideal Platonic realm. He restricts all other
possible understandings of the emergence of symbolic algebra by formulating
his research question as follows: “What transformation did a concept like that
of arithmos have to undergo in order that a ‘symbolic’ calculating technique
might grow out of the Diophantine tradition?” [35, 147]. According to Klein it
is ultimately Vi`
ete who “by means of the introduction of a general mathematical
symbolism actually realizes the fundamental transformation of conceptual foun-
dations” ([35] 149). Klein places the historical move towards the use of symbols
with Vi`
ete and thus ignores important contributions by the abbaco masters, by
Michael Stifel [57] [58], Girolamo Cardano [11] [12] and the French algebraists
Jacques Peletier [44], Johannes Buteo [9] and Guillaume Gosselin [23]. The
new environment of symbolic representation provides the opportunity to “the
ancient concept of arithmos” to “transfer into a new conceptual dimension” [35,
p. 185]. As soon as this happens, symbolic algebra is born: “A soon as ‘general
number’ is conceived and represented in the medium of species as an ‘object’ in
itself, that is, symbolically, the modern concept of ‘number’ is born”[35, p. 175].
It is hard to understand why a philosophy like this, rooted in German idealism,
where concepts realize themselves with the purpose to advance mathematics, is
so appealing to modern historians looking for an explanation for the emergence
of symbolic algebra.
The three di?erent approaches to the history of algebra are exempli?ed by
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three historians of mathematics. Cantor believes in a continuous development
from ancient Greek notions of number and proof to modern algebra, only ob-
scured during the medieval period in which the old masters were not fully un-
derstood. Van der Waerden and Franci see a historic realization of the logical
development from quadratic equations to cubic and higher degree ones towards
a theory of the structure of equations. Klein discerns a realization of symbolic
algebra in a necessary ontological transformation of the underlying number con-
cept. All three share the idea that there is some internal necessity and logic in
the historical move towards symbolic algebra. But all pass by at the fundamen-
tal historical changes that took place in the context in which medieval algebra
matured: the mercantile centers of northern Italy and the French Proven¸
ce re-
gion. We will now look at some contextual explanations and further demonstrate
that the emergence of symbolic algebra cannot be understood without account-
ing for the socio-economical context of that time. We will illustrate this with
a speci?c class of bartering problems which were discussed in arithmetic and
algebra books during several centuries. We do not want to present such socio-
economical interpretation as ‘the right one’, but at least as a complementary one
to the one-sided internatistic interpretation so dominant in the historiography
of algebra.
3
Contextual approaches
From the 1920’s, history of science began to account for contextual aspects of
the society in which science develops and is practiced. As a reaction to the ro-
mantic narratives of Great Men making Great Discoveries in science communist
historians of science pointed out the role of social and economic conditions in
the emergence and development of science. Gary Werskey [69] describes how
Soviet historians irritated Charles Singer, the chairman of the Second Interna-
tional Congress in the History of Science and Technology in London in 1931,
by repeatedly asking questions about socio-economical in?uences on the evolu-
tion of science. But this conference was a historical one making an impact on
the thought of many young scholars with socialist sympathies such as Joseph
Needham and Lancelot Hogben. Beginning with Boris Hessen’s The Social and
Economic Roots of Newton’s Principia [30], several papers and books were pub-
lished, placing the achievements of individual scientists within the context of
social superstructures. Speci?c histories of mathematics based on an analysis
of socio-economical conditions appeared much later. Dirk Struik was a con-
vinced Marxist who wrote a widely read A Concise History of Mathematics
[59]. Although the book cannot be considered a Marxist analysis, his vision of
mathematics as a product of culture and evolving within a dialectic process was
having an impact on other historians.
Only a limited number of authors focused on the mathematical sciences dur-
ing the period that symbolic algebra developed in Europe, being 1300 – 1600.
Michael Wol? [70] in a comprehensive study of the concept of impetus argues
that the “new physics” of the fourteenth century developed from contemporary
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social thought. The scienti?c revolution basically was a revolution in socio-
economical ideas. Drawing upon the theories of Marx and Borkenau, Richard
Hadden [25] develops the idea that practitioners of commercial arithmetic, as
a consequence of their social relations, delivered the new concept of “general
algebraic magnitude” to the new mechanics. Joel Kaye [34] argues that the
transformation of the model of the natural world of the Oxford and Paris schol-
ars such as Thomas Bradwardine, John Buridan, and Nicole Oresme during the
fourteenth century follows the rapid monetization of the European society. This
transformation happened beyond the university and outside the culture of the
book.
We would like to argue that symbolic algebra functioned together with
double-entry bookkeeping as the main instruments for the determination of
objective value, the basic idea of the mercantile society. The foundations for
symbolic algebra were laid within the abbaco tradition. While scholars on this
tradition, such as Jens Høyrup [32] maintain that the problem solving treatises
written by abbaco masters served no practical purpose whatsoever, we argue
that their activities and writings delivered an essential contribution to Renais-
sance mercantilism in the creation of objective, computable value. According
to Foucault [18, p. 188] the essential aspect for the process of exchange in the
Renaissance is the representation of value. “In order that one thing can rep-
resent another in exchange, they must both exist as bearers of value; and yet
value exists only within the representation (actual or possible), that is, within
the exchange or the exchangeability”. The act of exchanging, i.e. the basic op-
eration of merchant activity, both determines and represents the value of goods.
To be able to exchange goods, merchants have to create a symbolic representa-
tion of the value of their goods. All merchants involved must agree about this
common model to complete a successful transaction. As such, commercial trade
can be considered a model-based activity. Given the current global ?nancial
market and the universal commensurability of money we pass over the common
symbolic representation as an essential aspect of trade. However, during the
early Renaissance, the value of money depended on the coinage, viz. the pre-
cious metals contained in the coins which di?ered between cities, and varied in
time. As the actions and reciprocal relations of merchants, such as exchange,
allegation of metals and bookkeeping became the basis for the symbolic and
abstract function of money, so did the operations and the act of equating poly-
nomials lead to the abstract concept of the symbolic equation. Both processes
are model-based and use the symbolism as the model. Therefore, we have to
understand the emergence of symbolic algebra within the same social context
as the emergence of double-entry bookkeeping.
Now consider the following statement: The emergence of double-entry book-
keeping by the end of the ?fteenth century was a consequence of the transforma-
tion from the traveling to the sedentary merchant, primarily in the wool trade
situated in Italy and Flanders [16] [64]. Given the vast body of evidence from
Renaissance economic history and the evident causal relationship, not many will
contest the relevance of merchant activities on the emergence of bookkeeping.
What about the mitigated statement: “The emergence of symbolic algebra in
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the sixteenth century is to be situated and understood within the socio-economic
context of mercantilism”?. Philosophers of mathematics who believe in an in-
ternal dynamics of mathematics will not accept decisive social in?uences as an
explanation for the emergence of something as fundamental as symbolic algebra.
At best, they will accept social factors in the acceleration or impediment of what
they consider to be a necessary step in the development of mathematics. Also it
seems di?cult to pinpoint direct causal factors within economic history for ex-
plaining new developments in mathematics. However, the relationship between
bookkeeping and symbolic algebra is quite remarkable. Many authors who have
published about bookkeeping also wrote on algebra. The most notorious exam-
ple is Pacioli’s Summa, which deals with algebra as well as bookkeeping, and
the book had an important in?uence in both domains. But there are several
more coincidences during the sixteenth century. Grammateus [22] gives an early
treatment of algebra together with bookkeeping. The Flemish reckoning master
Mennher published books on both subjects including one treating both in the
same volume [42]. So did Petri in Dutch [46]. Simon Stevin wrote an in?uential
book on algebra [55] and was a practicing bookkeeper who wrote a manual on
the subject [56]. In Antwerp, Mellema published a book on algebra [40] as well
as on bookkeeping [41]. While there is no direct relationship between algebra
and bookkeeping, the teaching of the subjects and the books published often
addressed the same social groups. Children of merchants were sent to reckoning
schools (in Flanders and Germany) or abbacus schools (in Italy) where they
learned the skills useful for trade and commerce. There is probably no need for
algebra in performing bookkeeping operations but for complex bartering oper-
ations or the calculation of compound interest, basic knowledge of arithmetic
was mandatory and knowledge of algebra was very useful.
4
Case study: Bartering with cash values
In an interesting article in the Journal of the British Society for the History of
Mathematics, John Mason expresses his surprise at the solution method adopted
for bartering problems which involve cash. He cites a problem from Piero della
Francesca in a translation by Judith Field [17, p. 17]:
Two men want to barter. One has cloth, the other wool. The piece
of cloth is worth 15 ducats. He puts it up for barter at 20 and 1/3
in ready money. A cento of wool is worth 7 ducats. What price for
barter so that neither is cheated?
Mason originally expected the solution to be based on the proportion of the
barter value to the original value with the barter value being “either 20 + 20/3
ducats or to 20 + 15/3 ducats, depending on which value the 1/3 is intended to
act upon” However Piero’s solution appears to be di?erent [39, p. 161]:
This computation intrigued me because I was astonished at the se-
quence of calculations: ?rst reduce by the ready money paid (as a
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fraction of the barter price), and only then compare barter prices. It
seemed to me that in a modern economy it would be more natural to
carry out one of the calculations I considered, since the ready money
to be paid is a cash value, and the bartering in?ation refers to the
noncash-traded amounts.
Thus Piero subtracts one third of 20 from 20, which leaves 13 1/3 and the
same value from 15 which becomes 8 1/3. The proportion of these two values
is hence the fair barter pro?t to be applied by both parties. Though Mason
lists several other examples which follow the same solution method, he does
not provide an explanation why this particular method is adopted in abbaco
treatises and in later printed books. Given that this way of calculating was in
use for over two centuries, not only in Italy but in several European countries,
this particular bartering practice needs an explanation. We will demonstrate
that his astonishment is based on a wrong interpretation and even more so, a
wrong translation of the original problem. We will provide an explanation by
placing these early bartering problems within the speci?c context of Medieval
Italian merchant practices.
The original problem by Piero, in Gino Arrighi’s transcription from the
manuscript, is formulated as follows (f. 8r; [4, p. 49]):
Sono doi che voglano baractare, l’uno `
a panno e l’altro `
a lana. La
pe¸
c¸
ca del panno vale 15 ducati et mectela a baracto 20 et s`? ne vole
1/3 de contanti; et il cento de la lana vale 7 ducati a contanti. Che
la d`
ei mectere a baracto a ci`
o che nisuno non sia ingannato?
A literal translation of the medieval Italian would be as follows:4
There are two [men] that want to barter. One has cloth, the other
has wool. The piece of cloth is worth 15 ducats. And he puts this
to barter [at] 20 and of this he wants 1/3 in cash. And a hundred of
wool is worth 7 ducats in cash. What shall they put for barter so
that not one of them is being cheated?
Formulated this way, there is little room for doubt. The one with the cloth
wants 20 ducats per piece of cloth, of which one third in cash. Obviously then,
a third of the value refers to the barter value of 20. The amount of cash per
piece is thus 20/3. To know the barter value of the cloth without the cash one
has to subtract the cash from it, being 13 1/3. That Mason wants to add one
third of the value rather than subtracting it stems from the wrong translation
of “et s`? ne vole 1/3 de contanti”.
Is this interpretation the correct one for all bartering problems of this type
in abbaco treatises? Let us look for further clues. Mason provides pointers to
several abbaco treatises in which bartering problem appear with a cash value.
4My translation. For a discussion on the translation of abbaco texts see a recent critical
edition of a ?fteenth-century treatise on algebra [28, p. 132]
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The earliest he discusses are problems 33, 86, and 87 of Paolo Dagomari’s Trat-
tato d’aritmetica, written in 1339. He describes problem 86 as a problem which
“involves grain to be bartered at 15s but valued at 12, with one-third in ready
money, in exchange for orzo (?) at 10s.”. The word orzo should pose no prob-
lems as it is the modern Italian word for barley. In our translation:5
There are two that want to barter together. The one has grain and
the other has barley. And the one with the grain which is valued
at 12 s. puts it in barter at 15 s. per bushel. And he wants from
the one with the barley one third of the value in cash. And from
what remains he will get barley. And a bushel of barley values 10
s. Asked is what they arrive at in this barter so that none is left
cheated.
Here also, the meaning of the problem is di?erent from the one paraphrased
by Mason. It is not the person with the grain who puts in the cash, but the
other one. Furthermore, the enunciation clearly speci?es that the second person
should deliver one third of the value in cash and the rest in barley and this
conforms with our interpretation.
4.1
First occurrences of bartering with cash
Was Paolo the ?rst to deal with cash values in bartering problems? We checked
all available transcriptions of abbaco treatises before Paolo’s Trattato. The earli-
est one is probably the Columbia algorismus (Columbia, X 511, A l3) published
by Vogel [68]. Vogel himself dated the manuscript in the second half of the 14th
century. However, a recent study of the coin list contained in the manuscript
is dated between 1278 and 1284, which makes it the earliest extant treatise
within the abbaco tradition [61, pp. 88-92]. Høyrup suspects it “likely to be
a copy of a still earlier treatise” [32, p. 31]. It contains two barter problems
(19 and 20) but none involves money. The anonymous Livero del l’abbecho is
dated c.1289-1290 and has also two bartering problems without money [6, p.
24, 28]. The Tractatus Algorismi by Jacopo da Firenze is extant in an earliest
version of 1307. It is the subject of a recent comprehensive study of the abbaco
tradition by Jens Høyrup [32]. However this extensive treatise does not contain
any bartering problems. The next available transcription is the Liber habaci,
dated by van Egmond (1980) to 1310 [65], and is the ?rst to involve cash in a
bartering transaction. The enunciation of the single bartering problem is more
elaborate and functions as a prototype for later reformulations by Paolo and
Piero:6
5From Arrighi’s transcription [1, p. 75]: “E’ xono due che barattano insieme, l’uno `ae
grano e Il’altro `
ae orzo; e quello che `
a grano gli mette in baratto lo staio del grano 15 s., che
vale 12 s., e vuole il terzo da quello dell’orzo di ci`
o che monta il suo grano di chontantj; e
dell’avanzo se ne togle orzo. Ello staio dell’orzo vale 10 s., adornando quanto glele chonter`
a
in questo baratto acci`
o che no‘ rrimangha inghannato”.
6From Liber habaci, Biblioteca Nazionale Centrale Firenze, Magl. Cl. XI, 88, transcription
by Arrighi (1987, p. 147): “Sono due merchatanti che volglono barattare insieme, l’uno si `
a
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There are two merchants who want to barter together. The one
has wool and the other has cloth. The one with the wool tells the
one with the cloth: “how much do you want for the channa of your
cloth”. And he says: “I want 8 lb. (and he knows well that it
values not more than 6 lb.) and I want one quarter in cash and I
want three quarters in wool. And the wool is valued at 20 lb. per
hundred. Asked is what suits him to sell the wool per hundred so
that he is not being cheated.
We ?nd here all the elements of the later bartering problems. The problem
clearly speci?es that one party will deliver one quarter of the value in cash and
three quarters in merchandize. The reference to a fair deal becomes a standard
formulation in abbaco bartering problems. The solution recipe is the standard
formula adopted in later treatises as discussed by Mason:7
You shall do as such, one quarter is asked in cash, say as such: one
quarter of eight is 2. The rest until eight is 6. From 2 until 6 is 4,
therefore say as such: for every 4 lb. I get 6 lb., how much do I get
for 20 lb.? Multiply 20 lb. against 6 lb. this makes 120 lb. Divide
by 4 and 30 lb. results from it. This is how much it suits him to get
per hundred for this wool.
We have now found an adequate interpretation for the subtraction of the
cash value from the barter price, but why is this cash value also subtracted
from the original value? This example from the Liber habaci already gives us
an insight. Obviously, if one takes into account the barter value minus the cash
value (here 6 lb.) something also has to be done with the original value of the
merchandize (also 6 lb.). In this example these values are the same and there
would be no pro?t ratio. However, adapting Mason’s original reasoning to the
new interpretation, one could still compare the total barter value (here 8 lb.)
with the original value (6 lb.) and use this as a pro?t ratio. Why is it not done
this way?
4.2
Early Italian merchant practices
To answer that question we must look at Italian merchant practices at the begin-
ning of the fourteenth century. One important breakthrough took place around
that time: the introduction of double-entry bookkeeping. Records of stewards
of authorities of Genoa in 1278 show no trace of this kind of bookkeeping while
lana e l’altro si `
a pannj; dice quellj ch’`
a lla lana a quellj del pannj: che vuo’ tu della channa
del panno? E que’ dice: io ne volglo lb. viij (e sa bene che non vale pi`
u di lb. vj) e volglo il
quarto i’ d. chontanti e tre quarti volglo in lana. El centinaio della lana vale lb. xx, adomando
che lgli chonviene vendere il centinaio di questa lana acci`
o che non sia inghannato”. A channa
is a unit of length of about 2 m.
7Ibid: “De’ chos`? fare. E’ domanda il quarto in danari, diray chos`?: il quarto d’otto si `e ij.
Insino inn otto si `
a vj, da ij insino vj si iiij or diray chos`?: ongnj iiij lb. mi mette lb. vj, che
mi metter`
a lb. xx? Multipricha lb. xx via lb. vj far`
a lb. Cxx, dividi per iiij ne viene xxx lb.:
chotanto gli chonviene mettere il centinaio di questa lana”.
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