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The algebraic contents
of
Bento Fernandes’ Tratado da arte de arismetica (1555)1, ∗
Maria do Céu Silva¶
April 27, 2006
¶ Centro de Matemática da Universidade do Porto (CMUP)
Rua do Campo Alegre, 687 4169-007 Porto
Portugal
Abstract
The principal aim of this paper is to shed some light on the algebraic content of the Tratado da arte
de arismetica, written by Bento Fernandes and published in Porto, in 1555. As it is the first treatise
of a Portuguese author that has come down to us where algebra is studied, it deserves special
attention since it constitutes a testimony of the state of development of algebra in Portugal, in the
middle of 16th century. At a time where Pacioli’ Summa, the first printed text that includes some
algebra, was already so diffuse, it is surprising that it has not been the source of the algebraic
material of Bento Fernandes. The comparative study we do between the Tratado da arte de
arismetica and other abacus’ books from the 14th and the 15th centuries shows that Bento Fernandes
did not know the work of Luca Pacioli and that portuguese algebra has, probably, it’s origin in
some abacus manuscripts antedating the Summa. 1 I would like to thank Antoni Malet and António Machiavelo for their helpful advice and discussions. It is also a pleasure to thank Rafaella Franci and Jens Høyrup for thoughtful and detailed suggestions and for the bibliographical documents sent, which have greatly improved the paper. Javier Docampo Rey provided me with a copy of his Ph.D. dissertation on La Formación matemática del mercader catalán 1380-1521. Análisis de fuentes manuscritas. Fabrizia Tazón, librerian of the Dipartimento di Matematica, Universita’Degli Studi di Parma, supplied me the Anónimo, Libro dí contí e mercatanzíe, a cura e com introduzione di Silvano Gregori e Lucia Grugnetti (1998), and Trattato d’abaco AA.VV. (XV Secolo), a cura e com introduzione de Silvano Gregori e Lucia Grugnetti (2001). ∗ Work partially funded by Fundação para a Ciência e Tecnologia (FCT), through the Centro de Matemática da Universidade do Porto (CMUP) and the programme POCI/MAT/61237/2004.
1
Resumo
O principal objectivo deste artigo é dar visibilidade ao conteúdo algébrico do Tratado da Arte de
Arismética, escrito por Bento Fernandes e publicado no Porto, em 1555. Sendo o primeiro tratdo de
autor português, que chegou até nós, em que a álgebra é estudada, ele merece uma atenção especial,
pois constitui um testemunho do estádio de desenvolvimento da álgebra em Portugal, em meados
do século XVI. Numa época em que a Summa de Pacioli, a primeira obra impressa contendo
álgebra, estava já bastante difundida, é surpreendente que não tenha sido ela a fonte do material
algébrico de Bento Fernandes. O estudo comparativo entre o Tratado da Arte de Arismética e
alguns tratados de ábaco dos séculos XIV e XV mostra que Bento Fernandes não conhecia a
Summa e que a álgebra portuguesa tem, provavelmente, a sua origem em textos manuscritos de
ábaco anteriores à época de Luca Pacioli.
2
Introduction
The Tratado da arte de arismetica, of Bento Fernandes1, is one of the first Portuguese arithmetic
works. It was printed2 in Porto, in 1555, by Francisco Correa3. It is a very rare4 treatise written in
Portuguese, in the style of the mercantile arithmetic5 belonging to the vernacular tradition. Its
intended readers are, as the author explicitly mentions, all persons that want to know arithmetic,
mainly the merchants and the contractors, that will find in this text many questions and rules
related to their trade; but the work also includes “the rules of «cousa» which are more important to
the curious persons and the experts on the art”6. Generally, we can say that this is a work specially
directed to problem solving, which includes the explanation and uses of the indo-arabic numeration
system, as well as the usual operations (addition, subtraction, multiplication, division, and root
extraction) with integers and fractions. Many of the proposed problems are of a practical nature and
arise, in general, in business activities: they concern cost of goods, calculation of interest and
discount, exchange rates, alloys, for example. There are also recreational problems, and purely
abstract problems about numbers. In order to solve all this variety of questions several rules are
described, including the rules of three and five, different rules of companies, the rule of a quarter
and a twentieth [a regra de quarto e vintena], the rule of the reckoning of Flanders [conta de
frandres] and the rule of barters [baratas]. The rules of false position and double false position are
often used. This work also includes a part that deals with several algebraic concepts.
Taking into account the description made by several experts7 on the basic characteristics of abacus
treatises, in their different variants, we can say that the Tratado da arte de arismetica, from Bento
Fernandes (1555) can be considered a work of this kind. It is the first one that includes some
algebra written by a Portuguese author.
Before the Tratado da arte de arismetica, it had been published in Portugal the Tratado da Pratica
d’Arismétyca (1519), by the Portuguese Gaspar Nicolás, reedited several times between 1530 and
1541. Another work published in Portugal before 1555 was Ruy Mendes’ Pratica d’Arismetica
3
(1540). However, these works do not include algebra8. Moreover, in spite of our search in the
archives of Biblioteca Nacional de Lisboa, Arquivo Nacional da Torre do Tombo, and Biblioteca
do Palácio da Ajuda; Biblioteca Pública Municipal do Porto and Arquivo Histórico do Porto;
Biblioteca Pública de Braga and Arquivo Distrital de Braga; Biblioteca Pública de Évora and
Biblioteca Geral da Universidade de Coimbra, we have no notice of any manuscript treatise, in the
Portuguese language, up to 1555, that includes algebra.
It is quite possible that these treatises existed in Portuguese for reasons analogous to the ones that J.
Docampo Rey9 gives to justify the existence of algebraic manuscripts in the Iberian Peninsula
before the appearance of the Marco Aurel’s Libro primero de Arithmética Algebrática10. He finds it
odd that there are no known texts prior to 1500 with algebraic contents either in Catalan or
Castilian. Even with those mercantile arithmetic books known in the Iberian Peninsula up to 16th
century that do not deal with algebra, the situation is very similar. We must, nevertheless, indicate
the following works: the Summa de l’art d’Aritmètica, of Francesc Santcliment, printed in
Barcelona in 1482, studied by A. Malet11, and El Arte del Alguarismo, an anonymous manuscript
written shortly after 1400, studied by B. Caunedo12. Because of his recent investigations, J. D. Rey
found three arithmetical manuscripts written in Catalan, only one of them being considered
complete13. These texts are from the 14th century and from the 15th century and they belong to the
Biblioteca de Catalunya (Barcelona), the Biblioteca degl’Intronati (Siena) and the Arxiu del Regne
de Mallorca (Palma de Maiorca).
Recently, J. D. Rey14 presented a study of the algebraic contents included in an anonymous Catalan
manuscript of the early 16th century.
Van Egmond15 refers that about three hundred manuscripts dealing with arithmetic and algebra
survived in Italy written prior to 1500. Raffaella Franci reinforces this information when she says
that there were known16 (in 1992) “about three hundred manuscripts from the 14th and 15th
centuries with abacus’ treatises or practical geometry, in usually italian had been found, and even
being very different in its form and contents, they usually have the following structure: “I-
4
Introduction, II- Presentation of the indo-arabic numerical system, III- «indigitação» [this means to
count with the fingers]; IV – arithmetical operations with integer numbers, V – calculations with
fractions, VI ─ rule of three and of false position, VII ─ mercantile problems (exchange of monies,
weights and mensurations, company, cheap, exchanges, etc.), III ─ practical geometry, IX ─
recreational mathematics, X ─ algebra. Sometimes, these subjects are not all present in particular,
cannot exist parts II, III, IV, IX e XI, and other times the problems of recreational mathematics are
mixed with commercial problems. It is just the appearance of these last problems that characterizes
a text as an abacus’ treatise.”
This quantity of manuscripts texts largely surpasses the known twenty-five or so17 French
arithmetic mercantile texts written between 1300 and 1500; and obviously that of the Iberian
arithmetic works from the same period that have been found so far18.
Mercantile problems in the «Tratado da arte de arismetica »
In Tratado da arte de arismetica, as in almost all Italian abacuses’ treatises, the algebra chapter is
only introduced after the study of the various arithmetic rules. These arithmetic rules — as the rule
of three and of five, the rules of company, the rule of the quarter and twentieth, the rule of the conta
de frandres and the rule of barters — are used to solve almost all the problems proposed.
For Raffaella Franci19 “Algebra was not a necessary tool to solve these problems”; she notes that
algebra entered in abacus curricula most probably because “it was the subject of a long chapter of
Liber abaci (1202) of Leonardo de Pisa, which was the most important source for abacus teaching
in Italy”. Høyrup does not agree with this argument. He points out20 that this belief “is probably
the principle of the great book” that is “the belief that everything in a book, if not an innovation,
must be derived from a famous book that is known to us”.
5
It is therefore surprising that Bento Fernandes uses the rule of the thing [regra da cousa] for
solving some problems, even before the chapter where he presents this subject. We are quite sure
that this was not accidental because he states21: “by the rule of the thing about which I will later on
show you many and very ingenious rules”. His purpose was perhaps to motivate the reader to study
a new subject whose operationality was guaranted. Bento Fernandes uses the rule of the thing to
solve two business problems from the chapter of barters, and two abstract problems by the rule of
opposition [oposiçã].
The first problem relates to commerce. It is the following22: “Two merchants are bartering. One has
cloth from London and the other has cotton. The price in cash of the cloth from London is 19
cruzados of gold, and in the barter it is at a certain amount that we do not know and 41 in cash. The
value of the quintal [a measure of weight] of cotton is 10 cruzados in cash, and in the barter it is 13
cruzados, and it will be equal in the barter. I ask at what price we must put the piece of London in
the barter.”23 In the resolution, Bento Fernandes24 supposes that the price of the cloth from London
is one thing [cousa], and using basic operations with things, he reaches an ‘equation’, which he
solves. It is clear that he knew well the basic rules of algebra which he applies to solve equations.
In modern symbolic notation, his procedure is the following: after obtain the equation
xxxx217
413247 +=⎟
⎠⎞
⎜⎝⎛ +− , he says: “undo the debit and lend 3 things and a
41 to each part and
that part where it is less undo it debit and that part where it is 7 things and 21 will be 10 things and
43 ”. Afterwards, using the rule of the thing [regra da cousa] he divides the number, 247, by the
things,4310 and gets the solution, 22 cruzados and
4342 of cruzado.
The resolution of the second problem of barter by the rule of the thing25 uses the equation
96
624620
=−−
xx , which is also of the first degree, but it requires one more operation to be transformed
6
in one of the type . Bento Fernandes solves the problem by the same procedure as we solve
it nowadays, although using a rhetorical language.
bax =
Abstract problems in «Tratado da arte de arismetica»
It is also in the context of the applications of the arithmetical rules to the resolution of problems, in
the chapter “Here I will show to you many questions and reasons to get numbers by the rule of the
«oposição» given before”26, that Bento Fernandes uses algebra to solve two problems, but now on
abstract numbers.
As we know, the rule of oposiçam27, also called rule of false position, was an arithmetic procedure
often used to solve problems, before the development of the algebraic methods28. It usually applies
to problems of the first degree, in one or more unknowns. To Bento Fernandes this is a very
important tool. He carefully explains29 what the rule of one false position is, justifies its name and
applies it often to different types of problems.
Bento Fernandes uses the method of false position to solve problems of linear equations on one
unknowns30, of the second degree as and 31bax =2 bxax =2 and of the first degree on two or
more than two unknown32. When he cannot solve a problem by the rule of false position, as occurs
in problems of the second degree in two unknown33, he uses the rule of two false positions. As he
says, the rule of two false positions is a new and important tool to solve problems34 that allows us
to solve problems, which we cannot solve by any other rule. In this chapter Bento Fernandes also
applies the rule of the thing to one equation of the second degree in one unknown, of the type
, and to one equation of the second degree in two unknownsbxax =2 35.
Let us look to the first problem mentioned above: “What is the number that subtracted of its 31 and
41 and the rest multiplied against by itself, makes as much as the number?” 36
7
The algebraic procedure used is correctly structurated and the computation is well done, in spite of
the difficulty of the work with fractional numbers. The equation that solves the problem has also
the solution zero, but the author does not consider it, and only points out the positive solution,
because it was the “real” solution. This follows the tradition in Italian abacus’s treatise of the
Middle Ages Renaissance37.
In his study of abstract problems, under the title “Another of numbers by the rule of the thing”
[Outra de numeros pela regra da cousa], Bento Fernandes also proposes the following problem38,
“Make me two parts of ten so that when each of them is multiplied against itself the difference
from one part to the other, that is to say, from the less to the larger, makes 50”39. To solve it, he
uses essentially the procedure we use nowaday and obtains a correct solution. However, he does it
in a rhetorical form. He assumes that “the first part is one thing”, and therefore the other must be
“10. minus one thing”. Squaring each one of these parts and subtracting the first part from the
second he obtains a first degree equation40 whose solution is 212 . Therefore, the value of the thing
being 212 , the parts of ten which solve the problem are
212 and
217 . Bento Fernandes verifies that
this is a correct solution, arguing that the sum of the numbers is 10, and multiplying each one
against itselves and subtracting them, one indeed obtains the result is 50.
The explanation of the solution of such a problem without the resources provided by the symbolic
notation41 is a hard work. Nevertheless, Bento Fernandes does it well, detailing the procedure and
explaining it very carefully. He does this as if he was manipulating algebraic expressions, operating
correctly with monomials and binomials.
Bento Fernandes points out that it is also possible to solve this problem by the rule of double false
position (and even solves it), but he puts the emphasis in the algebraic procedure. Perhaps his idea
8
was to motivate the most expert readers to a new tool to solve problems: the algebra. A common
attitude in the abacus’s schools was to teach the resolution of problems by the rule of double false
position, the rule of algebra being reserved to the “students which were more dotated and had
desire of penetrate deeply in the mathematic studies”42.
Calculation with radicals
The algebraic sections in Bento Fernandes’ treatise is precedeed by the study of the roots and
calculation with radicals, developed in folios fol. 80 [v] to 82 [v]. Bento Fernandes defines43 six
different types of roots, which he presentes by the following order: square root [raiz quadra], cubic
root [raiz cubica], root of root [raiz de raiz], root of promica root, relata root [raiz relata] and surd
root [raiz sorda]. Raiz sorda means the root that cannot be found exactly; the terms quadra and
cubica have the same meaning that nowadays, and the raiz de raiz is the fourth root. Besides, a is
the root of the promica root of n if naa =+2 , and a is the relata root of n if naa =×2 . To
illustrate these two last definitions, Bento Fernandes provides the following examples: 9 is the root
of the promica root of 84, and 4 is the relata root of 32. To Pacioli the square and the cube roots
have the same meaning that for Bento Fernandes, but he uses the term relata to stand for fifth root.
He also considers a type of root analogous to the root of the promica root mentioned above, which
he names pronica root44.
Bento Fernandes teaches rules to calculate square roots (and cubic roots too) when the radicands
are perfect squares numbers (respectively, perfect cubes) and he explains how to get an
approximate value of the root, in the other cases. He presents some rules on radicals: namely how
to sum and subtract, to multiply and divide them. They are, in modern symbolic notation, the
following:
1) ( ) abbaba 4++=+ 2) ( ) abbaba 4−+=−
9
3) baba ×=× 4) 22 bababa ×=×=×
5) aaa =× 6) ba
ba=
7) b
ab
ab
a 22== 8)
22 b
a
b
aba
== .
The calculations made in this part of the treatise only deal with quadratic radicals, but some folios
after, when Bento Fernandes deals with equations, he also calculates with cubic radicals and
radicals of radicals. He never uses any abbreviation or symbol with the meaning of radical though
this already was a common practice between the abacists, even of the previous centuries.
The Algebra in the «Tratado da arte de arismetica»
Bento Fernandes’ account of algebra is given in folios 83 [r] to 94 [v]. He begins by asserting the
following: “Here I will show you the rules of «zibra moquavel», commonly called rules of «cousa»
and «censo», and of the « soe~ç » and «cubi» and root of the «cousa» and root of the number and all
kinds of rules and reasons for who has fine talent and memory.”45 These words are reinforced when
he says46: “Although the rules and questions are to people with good memory and knowledge, and
not to all the people, I wanted to put it here to profit those ones who want to do it”. These two
phrases reavel well the difficulty he attributes to this matter. According to Franci47, only about a
third of the abacus treatises contain a chapter devoted to algebra.
From the introductory words of the algebraic section, we can conclude that Bento Fernandes
considers three parts on the study of algebra:
1. Calculation with powers;
2. General rules in the study of equations;
10
3. Study of several particular types of equations, providing the rule of resolution and some
applications for each case that is presented.
Some rules on manipulation of the different quantities envolved in the calculations are not
included. Bento Fernandes never says, for example, that “plus times minus makes minus” or that
“minus times minus makes plus”, or that “plus added to minus means subtract and take the sign of
the larger of the both”. Some authors of abacus’ treatises, as Pacioli48, explain these rules in detail.
Bento Fernandes does not do this, but we are sure that he was closely acquainted with these rules,
because he applies them correctly49.
Calculation with powers
To Bento Fernandes the calculation with powers is an indispensable knowledge to whoever wants
to understand the study of equations, and he advises the reader to memorize some of rules. He
says50: “and first I will show to you one table [tavoada] that you must keep in your memory, as I
said about the little arithmetical table refered to before”.
We note that this analogy between the arithmetical table and the little arithmetical table that
concerns to the numbers is very interesting because the author points out the basis on both
arithmetic and algebra, for understanding these matters.
Let us see how Bento Fernandes presentes the tavoada. He says51:
“You must know that multiplying one thing by one thing that makes a square.
And multiplying one square by one square makes a square of a square.
And one thing times [via] one square makes a cube.
And one cube times [via] one cube makes a cube of cube.
And one thing times [via] one cube makes a square of square.
And one square times [via] one cube makes a square of cube.”
11
The table above (Table 1) gives us the rules to find some powers of the unknown, which we write
below in modern symbolic notation, adding the name that Bento Fernandes gives to them :
1) [censo] 2xxx =×
2) [censo de censo] 422 xxx =×
3) [cubo] 32 xxx =×
4) [cubo de cubo] 633 xxx =×
5) [censo de censo] 43 xxx =×
6) [censo de cubo] 532 xxx =×
These operations are presented without any previous reference to the meaning of the elements
involved. He uses the terms “cousa”, that is thing, and “çeço”, that is square, to mean the unknown
and the square of the unknown, respectively. The cube of the unknown is called “cubo” (which
plural is “cuby” or “cubi”), and the next powers are the square of square [“censo de censo”], the
square of cube [“censo de cubo”] and the cube of cube [“cubo de cubo”]. The highest power
mentioned is just the cube of cube. We think that the words cousa and “çeço” used by Bento
Fernandes, are possibly the translation to Portuguese of the words cosa (or chosa) and censo, used
in Italian algebraic tradition. We note that Franci and Rigatelli52 say that “Leonardo begins his
treatment [of algebra] by specifying that ‘in the composition of “algebra” and “almuchabala”’
three kinds of quantities must be considered: radix, quadratus, numerus simplex. The word radix
(the author also call it “res”, means thing) is the unknown; quadratus means the square of the
thing, that is, the square of the unknown; numerus simplex is a numerical quantity; nevertheless,
Gandz53 observes that the use of these terms shows the influence of the Arabic mathematics.
Bento Fernandes does not give a geometrical interpretation for the construction of the powers of
the unknown; which is quite common in other abacus’ treatise. For example, in the Dardi de Pisa’
12
Aliabraa-Argibra, the first abacus’ treatise only devoted to the algebra54, thing [cosa], square
[censo], cube [cubo] and square of square [censo di censo], are defined as follows: “The thing is a
length and it is the root of the square, and its name is thing [cosa] because this word can be
attributed to all the things in the world. Usually, the square is an area [anpiezza superficiale], and it
is the square of the thing … and the square of the square is the proportional middle between the
thing and the cube. The cube is a solid quantity [grossezza chorporale] that includes the length of
the thing and the area of the square….And with more order we can reach the rules on which we
will formulate the questions that we write here…”55.
Cajori56 pointed out two distinct principles to construct the higher powers in combining the
symbols of the lower powers: “One was the additive principle of the Greeks in combining powers;
the other was the multiplicative principle of the Hindus.” He gives as examples Diophantus and
Bhaskara saying that “Diophantus expressed the fifth power of the unknown by writing the
symbols for and , one following the other; the indices 2 and 3 were added” and on the other
hand, “Bhaskara writes his symbols for and in the same way, but lets the two designate, not
, but ; the indices 2 and 3 are multiplied.” Among the authors that used the additive
principle, Cajori mentions
2x 3x
2x 3x
5x 6x
57 Leonardo de Pisa, who calls cubus cubus, or else, census census
census; and between those who employed the multiplicative principle, Cajori mentiones
6x
58 Pacioli to
whom is the censo de cubo. To Bento Fernandes the fifth power is the censo de cubo or the
cubo de censo and the sixth power is the cubo de cubo, thus we can say that he uses the additive
principle. In our opinion, this is an important difference between his algebra and the algebra of
Luca Pacioli.
6x
The multiplicative principle has the disadvantage that it does not allow the construction, by a
recurrent procedure, of the powers whose exponents are prime numbers; and it implies the use of
special names for these powers. Pacioli uses primo relato for , secundo relato for , etc. 5x 7x
13
Although in his History of Mathematical Notations, Cajori gives several examples of authors who
adopt each one of these principles, he does not mention any algebraic Italian treatise from the 14th
or 15th centuries where the additive principle was used. However, Franci and Rigatelli59 mention
that the skilled algebraist Antonio de’Mazzinghi (Florence 1353, 1383) “presents only the first six
powers of the unknown, which are also called ‘terms of algebra’ (termini d’algebra). They are
named respectively cosa, censo, cubo, censo di censo, cubo relato or duplici cubo, cubo di cubo”.
This leads us to think that Antonio de’Mazzinghi was using the additive principle.
General rules in the study of equations
After presenting Table 1, and without any other explanation, Bento Fernandes introduces another
table with the following collection of rules60 (that we numbered from [1] to [15]):
“And you must note that dividing number by thing it makes number” [1]
“And dividing number by square makes root” [2]
“And dividing thing by square makes number” [3]
“And dividing number by cube makes cubic root” [4]
“And dividing thing by cube makes root” [5]
“And dividing square by cube makes number” [6]
“And dividing number by square of square makes root of root” [7]
“And dividing thing by square of square makes cubic root” [8]
“And dividing square by square of square makes root” [9]
“And dividing cube by square of square makes number” [10]
“And dividing number by cube of cube makes cubic root of cubic root” [11]
“And dividing thing by cube [of cube] makes root of cubic root61 [de cubo]” [12]
“And dividing square by cube of cube makes root of root” [13]
“And dividing cube by cube of cube makes cubic root” [14]
“And dividing square of square by cube of cube makes root” [15].
14
In fact, this is a very ‘strange’ collection of rules. A first glance these sentences (that we numbered
from [1] to [15]) seem to be absurd; but its meaning becomes clear when farther on the author
studies the different cases of equations.
The following example illustrates what we have just said: “When the squares are equal to the
number we must divide the number by the square and make the root of the result, and this is the
value of the thing”62. This must mean that, if we have an equation as and we divide [both
members of it] by a, we will have
bax =2
abx = .
In symbolic modern notation, the rules above are the following:
[1] abxbax =⇒= [2]
abxbax =⇒= 2 [3]
abxxbax =⇒= 2
[4] 33 abxbax =⇒= [5]
abxbxax =⇒= 3 [6]
abxbxax =⇒= 23
[7] 44 abxbax =⇒= [8] 34
abxbxax =⇒= [9]
abxbxax =⇒= 24
[10] abxbxax =⇒= 34 [11] 66
abxbax =⇒= [12] 56
abxbxax =⇒=
[13] 426 abxbxax =⇒= [14] 336
abxbxax =⇒= [15]
abxbxax =⇒= 46 .
We note that some of these rules are latter studied in detail, but our author never anticipates this
fact.
Particular study of several types of equations
15
Bento Fernandes presents twenty-seven types of equations in the same order in which we are going
to list them here (but without numbering them). For each one of these equations, he gives the rule
of resolution and an example (except in the cases 1) and 4), where he presents two examples for
each one).
The types of equations and the corresponding formula for their resolution are, in modern symbolic
notation, the following:
1) ; bax =abx = 2) ; 2 bax =
abx =
3) ; 2 bxax =abx = 4) ; 2 cbxax =+
ab
ac
abx
22
2−+⎟
⎠⎞
⎜⎝⎛=
5) ; 2 cbxax +=a
bac
abx
22
2++⎟
⎠⎞
⎜⎝⎛=
6) ; 2 caxbx +=a
bac
abx
22
2+−⎟
⎠⎞
⎜⎝⎛=
7) ; 3 cax = 3 acx = 8) 3 cax = ; 3
2
a
cx =
9) ; 3 bxax =abx = 10) ; 23 bxax =
abx =
11) ; 23 cxbxax +=a
bac
abx
22
2++⎟
⎠⎞
⎜⎝⎛=
12) ; cbxax += 3a
bac
abx
22
2++⎟
⎠⎞
⎜⎝⎛=
13) ; cbxax += 23a
bac
abx
22
2++⎟
⎠⎞
⎜⎝⎛=
14) ;dcxbxax ++= 23a
ba
dca
bx22
2
++
+⎟⎠⎞
⎜⎝⎛=
16
15) ; cxaxbx =+ 32a
bac
abx
22
2−+⎟
⎠⎞
⎜⎝⎛=
16) ; 23 bxcxax =+a
bac
abx
22
2+−⎟
⎠⎞
⎜⎝⎛=
17) ; 4 dax =adx = 18) cxax = ; 4 3
acx =
19) ; 24 cxax =acx = 20) ; 34 cxax =
acx =
21) ; eaxcx += 42ae
ac
acx −⎟
⎠⎞
⎜⎝⎛−=
2
22
22) cax =2 ; 2
a
cx = 23) ccax +=2 ; ac
acx +=
24) ; daxbxcx =++ 32
abac
ad
ab
ac
x −+
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
= 3 3
3
25) ; eaxbxcxdx =+++ 432
abad
ae
abad
x −+⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=
2
26) eaxbxcx =++ 432 ;
abac
a
e
abac
x −+⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=2
2
27) 342 bxeaxdxcx +=++ ;
aba
dab
ab
aba
d
x 244
2
2
−+⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+=
We observe that some of these formulas are not correct, and others are true only in particular cases.
We also note that Bento Fernandes essentially repeats the type 1) after presenting the first three
17
types. However, he proposes the same rule for its resolution. In this case, the example has very
different features, as we can see in the Appendix.
Twenty-one problems are of a theoretical type, seven concern business, and one is devoted to
geometry, related with a triangle. The theoretical problems require the discovery of one, two or
three numbers satisfying some conditions. Problems numbers 1), 2), 6) and 27) are about finding a
single number. Problems number 3), 9), 10), 12) 13) 18) 19) 20) 21) 22) and 23) deal with two
numbers. Problems 7), 8), 11), 14), 15) and 16) concern three numbers. The resolutions presented
by Bento Fernandes use algebraic calculations but in some cases there is evidence of the
reminiscences of the rule of false position, with what the author intents to simplify calculations.
This is attested, for example, when he chooses 2x or 3x, and not x, to the value of the unknowns.
He does this in the problems 7), 10), 14), 15), 16), 18), 22) and 23). Other authors of abacus’s
treatises, as Gerardi, also do this63. We note that these problems deal with numbers under certain
proportions, although Bento Fernandes, as well as Gerardi, do not study in their treatises the notion
of ratio64.
We summarize below some comments on the rules presented by Bento Fernandes grouping them
according to some particular features. For each group we selected the most important examples in
order to give an overview of the choices made by Bento Fernandes to illustrate those rules. We
include all the others in an Appendix.
(1) The first six cases — usual equations of first and second degree.
We notice that in all the cases, except the first, the coefficient of the highest power of the thing is
not 1 and in the first case the rule refers to the normalized case. We also point out that all the
formulas presented are correct but, in the sixth case, where the equation is studied,
only one solution is given. Other authors, even from former times, as Jacopo de Firenze and Paolo
Gerardi, gave two solutions
2 caxbx +=
65. Bento Fernandes’ rule for this 6th case is: “When the cousas are
18
equal to the censo and the number, you must divide by the censo and then halve the cousa and
multiply it by itself and you take away the number and the root of what remains plus the half of the
cousa this will be the equal the cousa”66. Gerardi also adds: “Or half of the cosa minus the root of
what remains after the number has been taken away from the multiplication of half the cosa, and
this much will equal the cosa”67.
(2) Cases 7) to 10) — equations of the third degree with only two terms.
Bento Fernandes gives correct formulas to all the equations proposed. We just note that in order to
solve 3 cax = (case 8)) he writes “the root of the cubic root” instead of writing “the cubic root of
the root”. This implies the knowledge that the square root of a cubic root is equal to the cubic root
of the square root, which however he never states explicitly.
Let us see the illustrating problem for this case: “Find me three numbers which are in relation
together the first to the second as 2 is to 3 and the second with the third as 3 to 4, and multiplying
the first number by the second and the result by the third, makes the root of 12.” To solve this
problem, Bento Fernandes begins by proposing one number to be 2 cousas, the other 3 cousas and
the last 4 cousas, that is, in symbolic notation, the numbers are and . The equation of
the problem,
x,x 3 2 x4
1224 3 =x , is successively transformed into 24123 =x ,
576123 =x and
4813 =x . Therefore, 48
4813 ==x , and the numbers are 66
311
4812 =× , or 6
163
15 , and
63185 , that is, both the procedure and the solution are correct.
(3) Cases 11) to 16) — equations of third degree with at least three terms.
The formula presented to solve each one of these cases is correct only if zero is a root of the
corresponding equation, that is, when it is possible to reduce it to a quadratic equation. Bento
19
Fernandes gives a wrong rule for the cases , and
to which he extends those rules used for the quadratic equations.
cbxax += 3 23 cbxax += dcxbxax ++= 23
It is curious to observe how Bento Fernandes applies the erroneous formula to the 12th problem:
“Find me two numbers which are such parts, the one of the other, as 2 of 3 and multiplying the first
by itself and then by the number, makes as much as adding the two numbers and adding them to
16”. As we see, he wants to find two numbers in the ratio 2:3 such that the cube of the first equals
the sum of the two numbers plus 16. He begins the procedure by assigning the values of the two
numbers as 2 things and 3 things, respectively and he obtains the equation . After
normalizing it, he applies the proposed rule and obtains
( ) 1652 3 += xx
165
256252 +=x ; in consequence, the
numbers are 85
64258 + and
1615
25622518 + .
The result is obviously wrong but the author does not make any comments on it, and he finishes
with the words “as you can prove”.
In the other two cases were the formula is also wrong, Bento Fernandes does the same, that is, he
does not check the solution. On the contrary, he usually checks the solutions he obtains, even when
he is working with approximate values. This leads us to think that he surely knew that the solution
was wrong!
(4) Cases 17) to 21) — equations of the fourth degree.
We point out that the first four cases are binomial equations and the last can be reduced to a
quadratic equation. All the formulas proposed to solve these equations are correct, although Bento
Fernandes only mentioned the positive solution. Here, as well as in other cases, he does not provide
the condition under which the equation has solutions.
20
From all the provided problems, the 17th is the most interesting, being entirely different from the
other examples of Bento Fernandes algebra, because he deals with a geometric subject. It is the
following: “It is a shield that has 3 faces, that it has the same (measures) on one face as in another
and one does not know how many «braças»68 it has; however one knows that the shield is «quadro
100 braças»69. I ask how much its face has.” To solve it, Bento Fernandes proposes the following
procedure: “do as follows: put that the shield be one thing for each face. Because by the rule you
always do this way, equal or not equal, you will add that which the shield is for each face. Now add
one thing for each face, it makes 3 things. In addition, these 3 things you must multiply by21 ; it
makes one and 21 thing. In addition, you will say as follows, from one thing to one thing and
21
thing there is 21 thing. Now multiply
21 thing by one and
21 thing and is makes
43 square [censo].
Moreover, say furthermore, from one thing to one and 21 thing, there is
21 thing, and multiply
21
thing by 43 square, it makes
83 cube. Now repeat, and again, for the third face and you will say, as
follows, from one thing to one and 21 thing there is
21 thing, and multiply this
21 thing by
83 cube, it
makes 163 square of square. Thus the root of
163 square of square is equal to 100 «braças». In
addition, multiply 100 «braças» by itself, it makes 10000 «braças» [squared]. These you must
divide by the square of square, which is 163 and it comes 53333 and
31 . So you will say that the
root of the root of 53333 and 31 will be the value of the thing. Moreover, since you putted that
shield was one thing for each face, you will say that the shield is for each face the root of the root
of 53333 and31 , as you can verify. The closest, because one cannot take the square root, you will
find that the root of the root of the previously mentioned 53333 and 31 will be 15 and
51 , so that the
shield is of 15 braças and 51 of braça for each face, and this taking the surd root.”
21
It is curious to see that to solve an equation like bax =4 , Bento Fernandes takes 2b instead of
b, and only after this the roots disappear by the equality of the radicands.
The calculations show that Bento Fernandes uses the theorem of Heron70, which he applies to the
particular case of an equilateral triangle. Nevertheless, again, in this example, as in other cases, he
is concerned with the generalization of the process to any other triangle, equilateral or not, as we
can see by the following sentence: “Because for general rule you will always make in this way,
[either] equal or not equal”71.
(5) Cases 22) and 23) — equations of the second degree, again.
The equations studied in these cases are of the type cax =2 , and ccax +=2 . As these are two
second-degree equations we would expect that they were presented as particular cases of
or, at least, studied immediately after them (as occurs, for example, in the case 2 bax =
3 cax = , which follows immediately ). This does not happen. The explanation resides
perhaps in the fact that its resolution deals with the calculation of a “root of root”.
cax =3
Bento Fernandes gives a correct rule only to the equation of the type 22. The rule for the other case
is wrong, because he takes the “sum of the roots” instead of the “root of the sum”. The examples
presented in both cases are analogous, as well as the procedure used is its resolution.
For the problem 23, “ Give me two numbers that are such part one of the other like 3 is of 4, and
multiplying one against the other will make 10 and root of 10”, Bento Fernandes supposes that the
numbers are and , and solving (by the given rule) the equation x3 x4 101012 2 +=x he obtains
the wrong solution 4125
65 +=x . The remainder computations based on this value (that is, the
22
computation of and ) are well done, what means that Bento Fernandes knew that x3 x4
( ) 4 424 cabacba +=+× , even in the cases where b and c are not integers.
(6) Cases 24) and 25) — Complete equations of the third and fourth degree.
These are particularly interesting because the rules presented are wrong, but they do get he right
results when applied to the particular problems choosed.
As we can easily see, the first member of the equation can be completed to a
cube
daxbxcx =++ 32
3
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+
abac
x , if ac
a
b=
2
2
3, and then one obtains
abac
ad
ab
ac
x −+
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
= 3 3
3
. With the equation
the situation is analogous: if eaxbxcxdx =+++ 432bd
ab
×= 4 and bd
ac
×= 6 , the first
member can be completed to
4
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+
abad
x and the solution will then be
abad
ae
abad
x −+⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=
2
.
Bento Fernandes does not mentiones these constraints, although he applies his rules to two similar
problems satisfying the given conditions above made explicit.
Let us see in detail the illustrating problem 24: “One merchant gave another 100 livras de
grossos72, using the same sort of exchanges related to Anvers73 for 3 years, and that each year he
must tell him profit and investment and at the end of those 3 years the profit and the investment
was 150 livras de grossos. I ask which monthly rate did this merchant give to the other, since in 3
years with 100 livras de grossos he makes 50 livras de grossos.”
23
Setting the unknown x equal to the number of dinheiros that one livra earns in one month (that is
x201 of the livra74), Bento Fernandes successively calculates the profit for the first year —
x201100× livras, the profit and investment after one year — x5100 + , the profit for the second
year — ( xx 5100201
+ ) , the profit and investment after two years — ( )xxx 51002015100 +++ , the
profit for the third year — ⎟⎠⎞
⎜⎝⎛ ++ 2
4110100
201 xxx , and the profit and investment in the end of the
third year — 32801
4315100 xxx +++ . Therefore, the equation of the problem is
150801
4315100 32 =+++ xxx . According to the given rule, he divides by
801 and obtains
whose solution is 4000601200 32 =++ xxx 20120003 − . It results that the merchant makes the
profit 20120003 − dinheiros by one livra at each month, wich is the correct solution.
It is curious to observe that, after reaching this solution, Bento Fernandes takes 191722 + as
approximate value of 3 12000 and uses it to obtain, for the profit, the aproximate value 19172 .
After, using a little arrangement on the successive calculations, he verifies that the solution is
correct.
In the case 25) Bento Fernandes proposes an analogous illustrating example (see Appendix) but the
investment is made for four years. Assuming that x is the number of dinheiros that one livra earns
in one month, the equation that expresses the problem is the following
. 9600080240032000 432 =+++ xxxx
(7) Case 26)
24
This is, perhaps, the most interesting case among those studied by Bento Fernandes. In the
confused rhetorical explanation we can ‘guess’ that the author intends to solve the equation
432 eaxbxcx =++ , which is a fourth-degree equation, but we see with surprise that, instead
of working with the fourth root, the rule proposed relates to the square root. On the other hand, the
illustrating problem deals with an equation like eaxbxcx =++ 432 , and although the rule is
wrong it yields the right answer in the particular problem proposed.
Let us see how Bento Fernandes expounds the rule: “When the censo and the cuby and the censo de
censo are equal to the root of the number you must divide by the censo de censo but reducing the
root to a censo de censo; and after you must divide the censo by the cubo and multiply it on itself;
and the result you put on the root of the number and of what you see equal. You must take the root
of the difference [recolhimento] minus the result of the division of the censo by the cubo and that
will be the value of the cousa.”
The illustrating example is once more about commerce, and it is analogous to the problems given
to cases 24) and 25). It says: “One merchant gave to another 50 livras de grossos to invest, for 2
years, making profits of profits; and in the previously mentioned 2 years, the merchant made of the
investment and profit 50 livras de grossos and the root of 484 livras. I ask which monthly rate this
merchant gave to the other, since in 2 years with the investment of 50 livras de grossos he has as
profit the root of 484 livras de grossos.”
The procedure is also similar to that he used in the previouses examples and takes him to the
equation 4845081550 2 +=++ xx . As we see, this is a second-degree equation, in which the
second member is an integer number. Therefore, it would be natural, and even easier, that Bento
Fernandes used the stated rule for the quadratic equation, which he knew very well. Instead of this,
25
he works with the equation 484644
525 4
32 =++xxx , put it on the form
30976801600 432 =++ xxx and applies the proposed rule to this case. He obtains the
solution 2030976400 −+=x .
We point out that Bento Fernandes of course knew that 17630976 = , and in consequence, the
solution to the problem is 30976400 +=x , that is just 24=x . Therefore, we may ask why he
preferred a more complicated procedure. We can argue that, perhaps he, or the author by whom he
copied, believes that it was possible to “create” a rule for a new type of equation based on the
solution already obtained using the quadratic equation rule. This is, at least, very curious!
(8) Case 27)
This case is related to a complete fourth degree equation of the type . However,
Bento Fernandes does not interpret it as such.
342 bxeaxdxcx +=++
Let us see how he formulates the rule for solving this equation. He writes: “When the root of the
censo and the thing and the censo of censo is equal to the root of the number and cubos, you always
must divide the cubo by 4, but that be roots equal to root; and you must divide the thing by 2; and
divide the result by what, in the beginning, was the cubo and keep the result in your memory. After
that you must multiply the result of the cubo by 4 and you put it on the result of the division of the
thing by 2 and on the initial value of the cubo; and the root of the result and that which comes from
the division of the cubo by 4 minus the root of the result of the thing divided by 2 and after on that
was in the beginning the cubo, this will be the cousa.”
26
If one tries to understand the procedure described by Bento Fernandes, indeed a hard task, we
conclude that he uses the formula
aba
dab
ab
aba
d
x 24
2
42 −+
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+= to solve 342 bxeaxdxcx +=++ . This
formula is obviously wrong, in particular, because the formula does not depend on e. Why did
Bento Fernandes does this? Høyrup has given, to a similar case, an interesting and convincing
explanation. He writes75: “non-valid rules for non-reducible cubics and quartics were produced and
transmitted; they proliferated and remained alive throughout the fifteenth century. The reason is
double: the abbaco masters used them to impress their public and the municipal authorities that
might employ them; and solutions contained intricate expressions involving roots, whence fallacies
were difficult to expose”.
In the present case, the illustrating problem proposed is the following: “Give me two parts of 10
that multiplied, the greater against the lesser, and after divided this product by the difference of the
two parts, makes the root of 18.” Using modern notation, and setting the parts as x (the lesser) and
(the greater), one solution of the equation x−10 ( )( ) 1810
10=
−−−
xxxx gives the answer to the
problem. Indeed, the equation ( xxx 2101810 2 −=− ) has the solutions 43185 −+=x and
43185 ++=x , but only the first satisfies the problem, because only positive values to the
numbers are allowed. Therefore, if Bento Fernandes appllied the formula to the second-degree
equation, he would obtain the correct solution to the proposed problem, that is 43185 −+ to
the lesser number, and 43185 +− to the greater. Instead of this, he worked with the squares of
each one of the members76, and he obtained the equation 342 2072082 xxxx =++ [the text has,
in this part, two mistakes because Bento Fernandes forgets the number 1800, and he uses 320x
instead of ]. However, if he applied his wrong formula to the right equation, which is
, or, the equivalent equation,
320 x
2432 72720180020100 xxxxx +−=+−
27
18002072082 342 +=++ xxxx , he would get, by coincidence, the value 1852518 −++ to
one of the parts (which is not the correct value because this is not the lesser part!). Bento Fernandes
does not notice the mistake, and his confused rhetorical description leads us to think that he did a
bad copy of another work.
As in others problems, our author proves (or tries to prove!) that the obtained solution is correct,
but here he does it by an incorrect procedure. Taking 536 and
414 as approximate values to 43
and 18 , respectively, he obtains 2013218435 =+− and
207718543 =−+ , which sums
exactly 10. However, he forgot to verify that the two numbers satisfy the condition
( )( ) 1810
10=
−−−
xxxx , and this does not occur. Therefore, the conclusion is obviously wrong.
The algebraic sources of the «Tratado da arte de arismetica»: looking for analogies and
differences
Let us start by comparing Bento Fernandes’ algebra with the major algebra book printed when
Bento Fernandes was working in his treatise, the Pacioli’s Summa.
Bento Fernandes goal to introduce algebra in his Tratado da arte de arismetica is very different
from that of Pacioli in his Summa, not only on what concerns the organization but also the contents.
Moreover, if we observe the setting up and the solution of equations, we will see further important
differences between both works.
Both Bento Fernandes and Pacioli present the usual basic six equations as well as the rules to solve
them. However, we point out that they use a different sequence in their texts, and provide different
examples for all sorts of equations. For each one of the six cases, the questions proposed in Summa
28
deal with numbers while in the Tratado da arte de arismetica some examples concern commercial
problems. On the other hand, Pacioli gives geometrical proofs for the three complete equations
while Bento Fernandes does not do that.
These are some of the minor differences between the algebraic part in the two treatises. Perhaps
more relevant is the fact that Bento Fernandes gives false rules for non-reducible cubics and
quartics and Pacioli, in his Summa77, adverts that it is not possible to give general rules, in these
cases.
All different features in both texts of algebra reveal that the source of Bento Fernandes’ algebra
was not the Summa de Arithmetica Geometria Proportioni et Proportionalità, de Luca Pacioli.
Some authors78 make, in their treatises, references to other books they consulted, and sometimes
they even present details of those texts. Bento Fernandes does not quote any author or text, and this
obviously makes it difficult to determine the sources he uses in the algebraic chapters of his
treatise. However, the total absence of symbolic notation, even in abbreviation of some names, led
us to look for clues in some old abacus manuscripts.
The article of Raffaella Franci “Trends in Fourteenth-Century Italian Algebra”79, the results
presented by Jens Høyrup in “Jacopo da Firenze and the beginning of italian vernacular algebra”80,
and the text of Warren van Egmond “The Earliest Vernacular Treatment of Algebra: The Libro di
Ragioni of Paolo Gerardi (1328)” provided us with very important informations to look for the
possible sources of Bento Fernandes’ algebra.
Besides those works, we consulted different abacus’s treatises81 of the 14th and 15th centuries, in
order to find similarities and differences between Bento Fernandes’ Tratado da arte de arismetica
and those works. In all these texts and in the mentioned articles, we turned our attention to the
different cases of equations presented, to the formulation of each one, to the rules themselves and
29
even to the examples that their authors provide to illustrate them. Among all these texts, we think
that three of them are the most important to our purposes. They are the following: Jacopo of
Florentia’s treatise, Vatican Ms. Vat. Lat. 4826 (identical82 to Jacopo’ treatise, written in
Montpellier in 1307); Paolo Gerardi’ Libro di ragioni, written in Montpellier in 1328; an
anonymous Parma’s manuscript, Líbro dí contí e mercatanzíe83 (perhaps84 from the 15th century).
We compared all these treatises with the Tratado da arte de arismetica published in Porto, in 1555.
In order to facilitate the comparison between Bento Fernandes’ algebra and the algebraic chapters
in the mentioned works, we constructed the scheme below, which summarizes relevant
information. The first column shows the different kinds of equations presented in the Tratado da
arte de arismetica. The second has the formula used by Bento Fernandes to describe the
corresponding rule. In some cases, there is the mark (w) which correspond to a wrong formula and
the mark (p) in which cases the formula is not correct but it works on the proposed example.
In the third column, the number following the letter B refers to the order of the equation in the
collection presented by Bento Fernandes; the pair “En” is indicating the number of examples given
by the author. The same applies to the following three columns that refer to anonymous Parma’s
manuscript (P), Gerardi (G) and Jacopo (J), already mentioned. Besides, we include the indication
En-m, which signifies that the author gives “n” examples, “m” of which are exactly those of Bento
Fernandes, an asterisk (*) being added if the numerical values do not correspond to those of Bento
Fernandes.
We would like to note that we took the information included in the column J from Jens Høyrup’s
paper85 mentioned above.
bax =
abx =
B1
E1
P1
E1-1*
G1
E1-1*
J1
E2-1*
2 bax = abx =
B2 P2 G2 J2
30
E1 E1-1* E1-1* E1
2 bxax = abx =
B3
E1
P3
E1-1*
G3
E1-1*
J3
E1-1*
2 cbxax =+
ab
ac
abx
22
2−+⎟
⎠⎞
⎜⎝⎛=
B4
E2
P4
E2-2
G4
E1-1
J4
E2
2 cbxax +=
ab
ac
abx
22
2++⎟
⎠⎞
⎜⎝⎛=
B5
E1
P6
E2-1
G6
E1-1
J6
E1
2 caxbx +=
ab
ac
abx
22
2+−⎟
⎠⎞
⎜⎝⎛=
B6
E1
P5
E2-1
G5
E1-1
J5
E3
3 cax = 3 acx =
B7
E1
P7
E1
G7
E1-1
J7
----
3 cax = 3
2
a
cx = B8
E1
P8
E1-1
G8
E1-1
---
3 bxax = abx =
B9
E1
P9
E1-1
G9
E1-1
J8
----
23 bxax = abx =
B10
E1
P10
E1-1
G10
E1-1
J9
---
23 cxbxax +=
ab
ac
abx
22
2++⎟
⎠⎞
⎜⎝⎛=
B11
E1
P11
E1-1
G11
E1-1
J12
---
cbxax += 3
ab
ac
abx
22
2++⎟
⎠⎞
⎜⎝⎛= (w)
B12
E1
P12
E1-1
G12
E1-1
---
23 cbxax +=
ab
ac
abx
22
2++⎟
⎠⎞
⎜⎝⎛= (w)
B13
E1
P13
E1-1
G13
E1-1
---
23 dcxbxax ++=
ab
adc
abx
22
2+
++⎟
⎠⎞
⎜⎝⎛= (w)
B14
E1
P14
E1-1
G14
E1-1
---
31
32 cxaxbx =+
ab
ac
abx
22
2−+⎟
⎠⎞
⎜⎝⎛=
B15
E1
P15
E1-1
G15
E1-1
J10
---
23 bxcxax =+
ab
ac
abx
22
2+−⎟
⎠⎞
⎜⎝⎛=
B16
E1
P16
E1-1
--- ----
4 dax =
adx =
B17
E1
P17
E1-1
--- J13
---
cxax =4 3acx =
B18
E1
P18
E1-1
--- J14
---
24 cxax = acx =
B19
E1
P19
E1-1
--- J15
----
34 cxax = acx =
B20
E1
P22
E1-1
--- J16
---
eaxcx += 42
ae
ac
acx −⎟
⎠⎞
⎜⎝⎛−=
2
22
B21
E1
P20
E1-1
--- ---
cax =2 2
a
cx = B22
E1
P21
E1-1
--- ---
ccax +=2
ac
acx += (w)
B23
E1
P23
E1-1
--- ---
32 daxbxcx =++
abac
ad
ab
ac
x −+
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
= 3 3
3
(p)
B24
E1
P24
E1-1
--- ---
432 eaxbxcxdx =+++
abad
ae
abad
x −+⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=
2
(p)
B25
E1
P25
E1-1
--- ---
32
432 eaxbxcx =++
abac
a
e
abac
x −+⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=2
2
(p)
B26
E1
--- --- ----
342 bxeaxdxcx +=++
aba
dab
ab
aba
d
x 244
2
2
−+⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+=
B27
E1
--- --- ---
We observe that in the last file of this table, Bento Fernandes applies the presented rule to the
equation and not to 342 bxexdxcx +=++ 342 bxeaxdxcx +=++ .
To simplify our analysis of the results included in the above scheme, we divide the discussion of
the formulas into the following five topics:
a) On the six first cases of equations.
b) The study of cases 7) to 15), in which are included the incorrect rules.
c) On the cases 16) to 23), and also on the two particular cases 24) and 25).
d) The odd case 26).
e) On the case 27).
a) On the six first cases of equations
All the mentioned authors in the table above study the six first types of equations and indicate
correct formulas to solve them. We already said that both Gerardi and Jacopo consider the
possibility of existence of two solutions. This is also “the only case where Gerardi recognizes the
possibility of two solutions to a quadratic, because it is the only case where both roots might be
positive”86. However, in the illustrating problem, Gerardi does not consider both possibilities, and
he only gives one solution. We return to this subject to point out that the anonymous author of the
33
Parma’s manuscript also recognizes the possibility of the existence of two solutions but, like
Gerardi, he does not consider it in the illustrating example.
Bento Fernandes presents these equations in a different order from that of the other three authors —
for example, he studies before — and his illustrating problems are also
different. But, as these matters were very well known at the time, we think that these differences
are not relevant. The first and second degree equations mentioned are already included in the Liber
abaci
2 cbxax += caxbx += 2
87 and, in addition, as Høyrup88 observes they were familiar since the
balaha wa'l-muqb al-jabr aKit of al- rizmiaKhw .
b) The study of cases 7) to 15) in which are included the incorrect rules
Both the Libro di ragioni and the Líbro dí contí e mercatanzíe contain the study of these cases,
which are also included in the Tratado da arte de arismetica, and their order is also the same.
Besides, the three authors give exactly the same rules to solve them. The rhetoric language used by
Gerardi, Bento Fernandes and the author of the Parma’s manuscript is very similar and the
illustrating problems are almost the same. We only note one difference related to the example
provided in case 7. The anonymous author of the Parma treatise uses a business problem, which
involves exchanges of money89, while Bento Fernandes and Gerardi take up a problem on abstract
numbers.
On the other hand, we note two particularities that can be important to establish the genealogy of
Bento Fernandes’ treatise. The first is that Gerardi uses the abbreviated notation R, for radix, and
neither Bento Fernandes nor the author of the Parma’s manuscript uses it. The other, concerns the
solution of the problems. Bento Fernandes is the only one who checks repeatedly the solution and
provides approximate rational values when it is an irrational number. This last particularity reveals
that to the author it was important to have an idea of the magnitude of the result number (cf. case
34
7). We think that, perhaps, this is related with the main purpose of his work — the commerce.
However, we should stress that he knew that the approximate values could not be used to check the
exactness of the answer (cf. case 5).
The presence of the same wrong rules in both Bento Fernandes’ Tratado da arte de arismetica and
Paolo Geradi’s Libro di ragioni, and of the same examples to all the cases 4) to 15), does not
necessary mean that Bento Fernandes must have known the work of the Italian abbacist. As Franci
and Rigatelli90 observe, the wrong rules used by Paolo Gerardi in the cases ,
and were very common and are included in several manuscript
texts from the 14
cbxax += 3
cbxax += 23 dcxbxax ++= 23
th and the 15th centuries91. On this subject, we can only say that Bento Fernandes
and Paolo Gerardi had, almost surely, a common source.
c) On the cases 16) to 23), and also on the two particular cases 24) and 25)
Gerardi did not study any more types of equations behond the fifteenth mentioned above. In what
concerns the cases now cited, Jacopo only studies the binomial equations of the fourth degree
(although in a different order from Bento Fernandes). This collection of equations included in the
Tratado da arte de arismetica appears in the Líbro dí contí e mercatanzíe, but in a slightly different
order. However, the problems are the same in both treatises as well as the procedure used to solve
them.
In particular, Bento Fernandes and the author of Parma’s text, introduce algebraic matters in a
strikingly similar way. Before beginning the study of algebra, Bento Fernandes points out the
difficulty of the subject. He says, “Here I will show to you rules of «zibra moquavel» which we
usually call rules of «cousa» and «censo» and of the « soe~ç » and «cubi» and root of the «cousa»
and root of the number and all kind of rules and reasons for whoever has fine reasoning and
memory.” The anonymous author of the Libro dí contí e mercatanzíe uses almost the same words
35
to introduce his algebraic chapter. He says92, “We will write in this notebook the rule and examples
of «Gibra mocabile», which we usually call rule of the «cosa and «censi» and of the «cubi»
[number and of the censi of censi] and root of «cosa» and root of number and all kind of rule and
reasons.” Besides, in this context, they both use similar phrases like “In nomine domini amen” and
“In nome de dio amen”, respectively included in Tratado da arte de arismetica93 and in Libro dí
contí e mercatanzíe94. However, once again, we think that these are not relevant differences
because we know that other authors used it; for example in the beginning of the geometry, Jacopo
de Firenze says, as Bento Fernandes, “In nominee Domine amen” 95.
There are in these two works some other important similitudes. For example, we observe that
Bento Fernandes’ equations cax =2 and ccax +=2 (which are the 21st and the 23rd in the
Líbro dí contí e mercatanzíe) are introduced as cases on its own, though they are particular cases of
. On the other hand, the two authors commit the same error in the rule that they give to
solve
2 bax =
ccax +=2 : they take the square root of a sum as the sum of the square roots of its terms.
Perhaps more relevant will be the fact that both authors present the same illustrating example to the
equation (case 17) although this problem is absolutely outside the context of the others.
Also, we have not seen it in any other treatise.
4 dax =
The rules for cases 24) and 25) studied in the Tratado da arte de arismetica are those presented in
the Líbro dí contí e mercatanzíe. The illustrating examples and the procedure used by both Bento
Fernandes and the anonymous author are still the same. These are very special rules because they
apply to complete equations of the third and fourth degree, and work correctly in the given
problems. [We note that the formula that Bento Fernandes gives to solve the equation
— that is also complete and of the third degree — is wrong]. dcxbxax ++= 23
36
The Aliabraa Argibra also includes these rules and his author, Dardi96, illustrates them with the
same problems and uses the same procedures used by Bento Fernandes. This does not surprise us,
because the knowledge of these rules was widely spread in the 14th and 15th and the rules
themselves are included in many abacus texts97, often followed by the very same problems. We
note that, although Dardi and Bento Fernandes procedures are very similar, Dardi also presents an
alternative way of solving the problems based in the rule of three.
In what concerns the equations and
Franci remarks
4000120060 23 =++ xxx
9600032000240080 234 =+++ xxxx 98 that they can be written as
and respectively, and she observes( ) 120020 3 =+x ( ) 25600020 4 =+x 99 that this may explain
how the rules presented appears. As we can see, this happens because we are working with
problems of the form P20
1I =⎟⎠⎞
⎜⎝⎛ +
nx , where I is the initial investment, P is the total final investment
and profit, and n is the number of years.
Høyrup comments on these two cases of equations studied in the Parma’s manuscript. He says100,
“The rules are still false, [as happens on the cases 12, 13 and 14] but they are not copied from rules
for second-degree cases — and they work for examples that are given.” After that he points out
that, “The former example [that is problem 24] coincides with Jacopo’s example (4a), with the
difference that the 100 libre are lent for three, not for two years – but the capital still grows to 150
libre, which speaks in favour of inspiration from Jacopo’s or same related text”. In fact, Jacopo’s
example 4 (a) is the following: “Someone lent to another 100 libre at the term of two years, to
make (up at) the end of year”101; and as we can see, it is of the form P20
1I =⎟⎠⎞
⎜⎝⎛ +
nx , with . 2n =
37
From the foregoing observations we conclude that the algebraic section of Bento Fernandes
Tratado da arte de arismetica is closer to Líbro dí contí e mercatanzíe than to Gerardi’ text and we
arevconvinced that Bento Fernandes did not saw Dardi’s treatise.
d) The odd case 26)
Neither Jacopo nor Gerardi present this case in their treatises. The anonymous author of the Parma’
manuscript also does not study it. Moreover, it is not included in the extensive list of 198 equations
of Dardi’s Aliabraa argibra102, which Høyrup103 shows to be “the first full-scale vernacular algebra
that does not depend on Jacopo”.
This case is only present in a manuscript text of the Biblioteca Comunale of Palermo that Raffaella
Franci104 quotes, but until the moment, we had not yet the opportunity to consult it105.
However, we should like to observe that the illustrating problem presented by Bento Fernandes is
very similar to the problem 4(a) of Jacopo’ treatise, studied by Jens Høyrup106; the only differences
are in the investment (Bento Fernandes starts with 50 livras and Jacopo starts with 100 libre) and in
the return ( 48450+ in the first case and 150 in the other). Besides, in this problem, Bento
Fernandes uses the square root of an amount of real money (that indeed is an integer number,
although presented as an irrational number!) and Jacopo does the same with the root of 36 fiorini
(which is 6 fiorini!)
e) On the case 27)
None of the other three authors quoted in our scheme includes in their treatises this case of
complete fourth degree equation, studied by Bento Fernandes. However, Dardi of Pisa presented it
in his Aliabraa Argibra107, and therefore as we said elsewhere, the rule to solve the equation
38
342 bxeaxdxcx +=++ was known since early times. The corresponding solution formula is
ab
ad
ab
aea
c
x×
−++⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=244
4
2
, as one can deduce from the rhetorical text that states it. The
example that Dardi proposes is the same that we already saw in the Tratado da arte de arismetica,
but Bento Fernandes’ procedure is wrong, unlike Dardi who proceeds correctly. Supposing that x is
the larger part, the problem leads to the following equation ( ) 18102
10=
−−
xxx . By squaring both
sides, this quadratic equation converts in a quartic, and the square root disappears. This case, as
well as , that Dardi also solves by the same rule, arises from problems that
requires the division of 10 into two parts, x and
324 bxcxeaxdx ++=+
x−10 , such that ( ) Nx
xx=
−−102
10 , where N is the
given number. Franci108 observes that the resulting equation is different according to or
, and we can see that in the two problems proposed by Dardi, which are the ones for
and for
25<N
25>N
18=N 28=N , he obtains exactly equations of those types, that is,
and . 324 20180072082 xxxx +=++ 324 201228001120 xxxx ++=+
Many algebra treatises of the 14th and 15th centuries include these rules109, often followed by the
same numerical examples; however, Bento Fernandes did not know them.
Concluding remarks
Bento Fernandes’ Tratado da arte de arismetica, a Portuguese algebraic vernacular text, is written
in a rhetorical style in which everything is set out by using complete words. This applies in
particular to its algebraic contents ─ equations, formulas for solutions, procedures ─ all described
just in words. The only symbols used are those for natural numbers and fractions, and even
abbreviations for words are not present.
39
A detailed analysis of the algebraic prerequisites to Bento Fernandes’ study of equations shows a
not very well structured work, the concepts being usually introduced in the development of the
procedure of resolution of the questions proposed. We note, in particular, that Bento Fernandes
never mentions the method he uses to work with polynomials, and yet he applies correctly the
rules. On the other hand, the only matter that he specifies is the product of powers of the thing for
which he gives a table that must be memorized. We also point out the level and contents of the
problems presented, which are very similar to those of the Italian abacus’ tradition.
The study of equations presented in the Tratado da arte de arismetica is very peculiar: Bento
Fernandes states a rule for a particular equation type and applies it to an illustrating problem, and
sometimes to more than one. Each problem and his worked solution are presented in the following
standard pattern: First, he enunciates the problem. After, he assigns the values of the unknown, that
is the cousa, and of its powers, the censo and the cubo. Third, he formulates the equation according
the conditions of the problem. Fourth, he solves the equation using the stated rule. Finally, he gives
the solution to the proposed problem, and many times he verifies that the solution is correct. Unlike
Leonardo of Pisa and al-Khwarizmi – that solve by algebra only numerical problems on numbers –
Bento Fernandes applies algebra also to commercial problems. This particularity, and the total
absence of geometrical justifications to the rules for the second-degree equations, suggests that the
Tratado da arte de arismetica does not descend directly from the algebras written by these authors.
As we noted, the fifteen cases of equations studied by Paolo Gerardi in his Libro di Ragioni are all
included in the Tratado da arte de arismetica. The rhetoric language used by both authors is very
similar, and the illustrating problems are almost the same (only excluding the examples for the
most usual cases). They even present the same wrong rules to the cubic non-reducible equations.
This could lead us to the conclusion that Bento Fernandes necessarily knew the work of the Italian
abbacist. However, Bento Fernandes includes in his text many more rules than Gerardi does, and
also he does not uses Gerardi’s notation for the root of a number. Therefore, we only can conclude
40
that Bento Fernandes and Paolo Gerardi had surely a common source, while it is not probable that
Bento Fernandes had seen the Libro di ragioni.
The use of the mentioned incorrect rules places the sources of the Portuguese author in early
algebra texts, and as we have already said it is clear that the algebra contained in Bento Fernandes’
treatise has not its origins in Luca Pacioli’ Summa. On the other hand, Bento Fernandes does not
study some cases of equations included in Jacopo’s work or in Dardi’s treatise, and he gives a
wrong rule to solve a quadratic equation, which do not happen with those authors; and this led us to
suppose that he did not know their works.
In the twenty-five cases of equations presented in the anonymous manuscript Libro de conti e
mercatanzíe we pointed out the presence of various common features with the Tratado da arte de
arismetica. Of all the algebraic contents to which we have compared the work of Bento Fernandes,
this is the undoubtedly most similar one. However, his author uses some abbreviations110 that are
not included in Bento Fernandes’treatise. This convinces us that Bento Fernandes was not
acquainted with this manuscript text from the fifteenth century111 , but that these works belong to a
common tradition.
Finally, we would like to observe that although we have not obtained any concluding results about
the sources that may have been the basis to the algebra of Bento Fernandes, the data we have
presented point to the close relationship between the Tratado da arte de arismetica with some
Italian manuscripts of the 14th and 15 th centuries.
Appendix illustrating proposed problems for different cases of equations
This appendix contains the statement of all problems provided by Bento Fernandes to illustrate the
rules to the proposed equations, and not yet included above. As we have already said, the Tratado
da arte de arismetica is a very rare book and, on the other hand, it is writen in the Portuguese
41
language, and these facts can make the access to the problems quite difficult. Therefore, we think
that it may be useful put them here, in one English translation.
In all problems presented, we keep the numeration according the number of the corresponding case
of equation. When more than an example is given, we add the letters a, b, etc, to the mentioned
number.
1. a. Make two parts of 12 for me, so that, when the larger is divided by the smaller 7 results on the
partition.
b. A merchant has 3 pieces of cloth to sold, which value is 100 coroas. He says, I do not know
what the value of the first cloth is, but I know that the second costs twice the first and more 7
coroas, and the third costs the same as the first and second added, and one coroa less. I ask what the
value of each piece is.
2. Find me a number which taking away 31 and
51 and multiplying the remainder by itself makes
28.
3. Find me two numbers that are such parts, the one of the other, as 3 of 5, and multiplying the one
against the other makes as much as when they are joined.
4. a. A merchant gave to another 20 libras at the term of 2 years to make (up at) the end of year.
When it came to the end of the two years, the other gives back to him 30 libras. I ask at which rate
was lent the libra a month, knowing that on 2 years with 20 libras he gains 10 libras?
b. A merchant lends another a sum of money, but he did not know how much nor at which
monthly rate he gives him the libra; and after one year the merchant made of investment and profit
42
80 libras de grossos. After, he gives to the other the 80 libras de grossos at the term of one year at
the same rate, and one more dinheiro by each libra lent than in the first year. At the end of the two
years, he observed that the gain of the second year when multiplied by 8 was equal to the sum that
he gives him in the first year. I ask how much money did the merchant gave him in the first year,
and at which rate by month does he give the libra?
5. I divide 100 by a quantity and I remember what results, and then I divide by 5 more than the first
time and add together what came first with what came next, and it makes 20. I ask what I divided
by the first time and on second time.
6. A merchant went on two voyages to Frãdes [Flandres] and in the first voyage, he earned 12
cruzados, and in the second voyage he earned the same rate of soldo to libra as he made in the first
voyage; and when his voyages were completed, he found himself with 100 cruzados, earned and
capital together. I want to know with how much money he set out from his house in the first
voyage, and in addition in the second.
7. Find me three numbers which are in relation together the first to the second as 2 is to 3 and the
second with the third as 3 to 4, and multiplying the first number by the second, and the result by the
third, makes 96.
9. Find me two numbers which are such parts, the one of the other, as 2 of 3 and multiplying the
first number by itself and then by the number makes as much as the second number.
10. Find me two numbers which are such parts, the one of the other, as 3 of 4 and multiplying the
first number by itself and then by the number makes as much as the second number multiplied by
itself.
43
11. Find me three numbers which are in relation together as 3 is to 4 and 4 to 5, and multiplying the
first by itself and then by the number makes as much as the second number multiplied by itself and
added onto the third number.
13. Find me two numbers which are such parts, the one of the other, as 2 of 3 and multiplying the
first by itself and then by the number, makes as much as the second multiplied by itself and adding
them onto 12.
14. Find me three numbers which are in relation together as 2 is to 3 and 3 to 4, and multiplying the
first by itself and then by the number makes as much as the second number multiplied by itself and
added onto the third number and then added to 12.
15. Find me three numbers which are in relation together as 2 is to 3 and 3 to 4, and multiplying the
first by itself and then by the number, and on the other hand, the second number multiplied by
itself, and added both products makes the same as the third number.
16. Find me three numbers which are in relation together as 3 is to 4 and 4 to 5, and multiplying the
first by itself, and then by the number, and that multiplication added onto the second number,
makes as much as the third number multiplied by itself.
18. Find me two numbers which are such parts, the one of the other, as 2 of 3 and multiplying the
first by itself, and then by the number, makes as much as the second number.
19. Find me two numbers, which are such parts, the one of the other, as 3 of 4 and multiplying the
first by itself, and then by the number, makes as much as the second multiplied by it self.
44
20. Find me two numbers that are such parts, the one of the other, as 4 of 5 and multiplying the first
by it self and then by the number, makes as much as the second multiplied by itself, and after
multiplied by the number.
21. Find me two numbers that are such parts, the one of the other, as 1 of 3 and multiplying the first
by itself and after one more by itself and adding them onto 20, makes as much as the second
multiplied by itself.
22. Give me two numbers that are such part one of the other like 2 is of 3, and multiplying one
against the other will make the root of 12.
25. One merchant gave to another 100 livras de grossos to invest, for 4 years, making profit and
investment [which one understands by making profits of profits]; and in the aforesaid 4 years the
merchant made of the investment and profit 160 livras de grossos. I ask which monthly rate did this
merchant give to the other, since in 4 years with 100 livras de grossos he makes 60 livras de
grossos.
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Mathematica, 33 (2006), pp. 4-42.
Leonardo Pisano. Liber abbaci. Codice Magl., C. I., 2626, Badia Fiorentinna, nº 73. Biblioteca
Nazionale di Firenze. Trad in Sigler, L. (trad.) [2002].
Malet, A. (1990). “Changing Notions of Proportionality in Pre-modern Mathematics”, in Asclepio,
42, pp. 183-212.
Malet, A. (1998). Francesc Santcliment, Summa de l’art d’Aritmètica, 1482. Introducció,
transcriptió i notes. Textos d’Història de la Ciência. Eumo Editorial.
Malet, A. (2003). “Algebra as language: Wallis and Condillac on the nature of algebra”, in Cronos.
Cuadernos Valencianos de Historia de la Medicina y de la Ciencia. Vol. 5 -6 (2002 – 2003), pp. 5-
24.
Marco Aurel (1552). Libro Primero de Arithmetica Algebratica, enel qual se contiene el arte
Mercantivol, com otras muchas Reglas del arte menor, y la Regla del Álgebra, vulgarmente
llamada Arte mayor, o Regla de la cosa: sin la qual no se podra a entender el decimo de Euclides,
47
ni otros muchos primores, assi en Aritmética como en Geometría: compuesto, ordenado, y hecho
Imprimir por Marco Aurel, natural Aleman: Intitulado, Despertador de ingenios. Va dirigido al
muy magnifico señor mossen Bernardo Cimon, Ciudadano dela dela muy insigne y coronada
Ciudad de Valencia. Valencia, En casa de Joan de Mey, Flandro.
Luca Pacioli (1494). Somma di aritmetica, geometria, proporzioni e proporzionalità Su[m]ma de
Arithmetica Geometria Proportioni [et] Proportionalità. Reproducción digital de la edición de
Venezia, Paganino de Paganini, 1494. Biblioteca Virtual Miguel de Cervantes, 2003.
Rankin, F. K. C. (1992). The Arithmetic and Álgebra of Luca Pacioli (c. 1445-1517). Tesis.
London University.
Rey, J. D. (2004). La Formación matemática del mercador catalán 1380-1521. Análisis de fuentes
manuscritas. Tese doctoral sob a direcção de A. Malet e J. Cajaraville Pegito. Universidade de
Santiago de Compostela. Departamento de Didáctica das Ciencias Experimentais.
Rey, J. D. ([2005]). “Reading Luca Pacioli’s Summa in Catalonia: An early 16th-century Catalan
manuscript on algebra and arithmetic”, in Historia Mathematica, Article in Press, pp. 1-20.
Sigler, L. (2002, trad.). Liber abaci. A Translation into Modern English of Leonardo Pisano’s Book
of Calculation. New York, Springer.
Simi, A. (1994). “Anonimo (sec. XIV) Trattato dell’Alcibra Amuchabile, dal Codice Ricc. 2263
della Biblioteca Riccardiana di Firenze”. Quaderni del Centro Studi della matemática Medioevale
(22). A cura e com Introduzione de Analiza Simi. Universitá di Siena.
Van Egmond, W. (1978). “The Earliest Vernacular Treatment of Algebra: The Libro di Ragioni of
Paolo Gerardi (1328)”, in Physis, Anno XX, pp. 155-189.
Van Egmond, W. (1980). Practical Mathematics in the Italian Renaissance. A catalog of Italian
abacus manuscripts and printed to 1600, in Sep. Annali dell’Istituto e Museo di Storia della
Scienza, Firenze.
Van Egmond, W. (1983). “The Algebra of Master Dardi of Pisa”, in Historia Mathematica, 10,
399-421.
48
Van Egmond, W. (1988). “How algebra came to France”, in Mathematics from manuscript to print.
Cynthia Hay (ed.). Oxford: Clarendon, pp. 127-144.
49
1 Not much is known about Bento Fernandes’s life. According to Diogo Barbosa Machado (Bibliotheca Lusitana: história, crítica e cronológica, 4 tomos, Tomo I, p. 501), he was born in Porto where he was a merchant and “he was one of the most famous Arithmeticians of his time”. We had the opportunity to see in the Arquivo Distrital do Porto (3ª secção, nos 4 and 5, fl 298) a document from 22 of July of 1552 who proves that, in that year, Bento Fernandes resided in the street “Ponte de S. Domingos”, in Porto, where he had his dwelling-house and a workshop. 2 See references to this edition in Innocêncio da Silva (Diccionario Bibliographico Portuguez, 1858, Lisboa, Imprensa Nacional, p. 344), and in António Ribeiro dos Santos (“Memórias sobre alguns matemáticos portugueses e estrangeiros domiciliados em Portugal ou nas conquistas” in Memorias de Literatura Portugueza, Lisboa, Academia Real das Sciencias, 1812, t. VIII, Parte I, pp. 108). The historian Marques de Almeida (Almeida, A. A. Marques (1994, p. 92)) refers a first edition, from 1541, entitled Arte de Arismetica. He says that he never saw any copy of it, although he accepts its existence. 3 In the colophon of the treatise we can see the following information: “Foy impresso ho presente tractado da arte de arismetica. Em a muy nobre τ sempre leal cidade do Porto de portugal per Francisco correa impressor. E acabou se aos 20 dias do mes de Fevereiro no ano de 1555”. 4 Ricardo Jorge (“O Primeiro Arithmetico do Porto”, in O Tripeiro, pp. 67-68) relates the rarity of this text in the following words, “a Guttemberg almost rudimental, whose destination was a dark corner from Mercadores’ street” and he indicates the three known volumes, which belong to the libraries of Porto, Évora and to the private Library of the King. 5 The contents of the Tratado da arte de arismetica are the following: (1) Prologo; (2) Tavoadas; (3) Numerar; (4) Regra de assomar inteiros; (5) Regra de demenuir; (6) Regra de multipricar; (7) Regra de repartir inteiros; (8) Regra de três por muitos modos; (9) Regra de cinquo de toda a sorte; (10) Regra de companhias chãs; (11) Companhias com tempo; (12) Companhias com tempo e à razão de tanto por cento; (13) Declaração das provas reaes; (14) Prova real nas regras de três; (15) Regra de companhias de meyo e terço e quarto e quinto; (16) Regra de companhias diferentes; (17) Regra dassomar quebrados de todo o género; (18) Regra deminuir quebrados de toda a sorte; (19) Regra multipricar quebrados de toda a sorte; (20) Regra de repartir quebrados de todo modo; (21) Regra de três de quebrados de toda a sorte; (22) Regra de três com tempo de quebrados; (23) Regra de companhias de quebrados de toda a sorte; (24) Regra de companhias com tempo de quebrado; (25) Regra de companhias com tempo e rezam de tanto por cento de quebrado; (26) Regra de companhias diferentes; (27) Regra da menos deminuição; (28) Regra de quarto e vintena; (29) Regra da conta de frandres; (29) Baratas; (30) Rezões da regra de progressão; (31) Regra de pagamentos em deferentes moedas; (32) Rezão de desconto reduzido a um dia; (33) Rezões de mercadores e perguntas necessárias ao trato da mercancia; (34) Rezões da regra de duas falsas oposições de todo o género; (35) Rezões da regra de hua falsa oposição; (36) Preguntas e rezões de tirar números pela regra da oposição; (37) Regra de tirar raízes quadras e cúbicas e sordas e promicas e relatas e raiz de raiz de todo género e modo de raiz; (38) As quatro regras de raízes: assomar, demenuir, multipricar e repartir; (39) Regra da zibra moquavel a qual vulgarmente se chama regra da cousa e de censo de censo e de cubi e raiz da cousa he raiz de numero e de toda valia de regra y rezão pêra quem tever delicado engenho e memoria; (40) Regras de três deferentes com tempo; (41) Preguntas; (42) Rezões e preguntas da liga da prata e do ouro. 6 Bento Fernandes (1555, 1st not numbered sheet). 7 For a description of the main features of these treatises, see Malet, A. (1998, p. 18), Franci, R (1994, 64), Caunedo, F. and Córdoba, R. (2000, p. 56-57) and Rey, J. D. (2004, 51). 8 Almeida, A. A. M. (1992, p. 92) describes the contents of the arithmetical treatises published in Portugal from 1519 to 1679. About the Tratado da arte de arismetica (1555), Marques de Almeida says: “But Bento Fernandes closely follows Gaspar Nicolás, from whose book he took a great deal of information to write his work. There is, however one aspect in which he supersedes Gaspar Nicolás: it is in the rule of the thing (that is, the regla da cosa). Although he uses a rudimentary computation method, he proceeds to solve first and second degree equations. Here, he outdoes Nicolás, who had forget, in his Prática d’Arismétyca the algebraic lessons of the italian friar. This fact documents the dissemination of the Summa in Portugal in the first half of the 16th century, because it was, without any doubt, the source of algebraic material used by Bento Fernandes, after all, the same that Gaspar Nicolás disdained thirty years before”. As we will see in this paper, this paragraph contains several inexactitudes.
50
9 Rey, J. D. (2004, p. 547). 10 Marco Aurel (1552). 11 Malet, A. (1998). 12 Caunedo, B. and Córdoba , R. (2000). 13 Rey, J. D. (2004, p.141). 14 Rey, J. D. (2006). 15 Van Egmond, W. (1988, p. 129). 16 Free tanslation from Franci, R. (1992, pp. 59-60). 17 Van Egmond, W. (1980, p. 129). 18 Malet, A. (1998, pp. 16-17). 19 Franci, R. (2002, p. 81). 20 Høyrup, J. (2005, p. 26). 21 Bento Fernandes (1555, fo. 74 [v]). 22 Bento Fernandes (1555, fo. 44). 23 Here and everywhere, the translations into English are our own, if nothing else is indicated. We thank Antonio Machiavelo for his help in this task. 24 Bento Fernandes (1555, fo. 44). 25 “Dois mercadores baratam, um lã e outro seda. O quintal da lã vale a dinheiro de contado a 20 coroas e no barato se mete a 24 coroas e a libra de seda vale a dinheiro de contado a 6 coroas e no barato se mete a 9 coroas. Pergunto, que parte houveram em dinheiro de contado e qual das partes o houve.” Bento Fernandes (1555, fo. 44-44 [v]). 26 “Aqui vos mostrarei muitas perguntas e razões [raciocínios] de tirar números pela regra da oposição que atrás ficou declarada”. Bento Fernandes (1555, fo. 70 [v]). 27 Bento Fernandes uses the terms «oposiçam», «oposiçã» and «o posiçã», indistinctly. 28 Malet, A. (1998, p. 31). 29 Bento Fernandes (1555, fo. 67 [v]). 30 See the problems denoted by 2, 3, 4, 5, 6, 7, 13, 14, 15, 16, 22, 24, 25, 26 in Bento Fernandes (1555, fo. 70-75). 31 See the problems 9, 10, 11, 12 and 17 in Bento Fernandes (1555, fo.72–73). 32 See the problems 35, 36, 37, 42, 53 in Bento Fernandes (1555, fol. 76-80). 33 See the problem 43 in Bento Fernandes (1555, fo. 78). 34 “A Regra de duas falssas oposiçoes he hua das mais sustanciaes regras q nesta arte vos tehno mostrado por q per ela podereis alc∗çar τ saber rezoes τ pergutas muy sotys τ de gr∗de sust∗cia as quaes p outra nenhua regra se pode fazer sen∗ p esta”. Bento Fernandes (1555, fo. 59). 35 See the problem 24 and the problem 43 in Bento Fernandes (1555, fo. 74 and fo. 78-79). 36 Bento Fernandes (1555, fo. 74 [v]). 37 Franci, F. and Toti Rigatelli, L. (1983, pp. 299-300). 38 Wich is problem nº 43 in our enumeration. 39 Bento Fernandes (1555, fo. 78 -78 [v]). 40 In the middle of the arguments there ais a small error! Bento Fernandes says “100 things are equal to 50 and 20 things”, instead of saying “100 are equal to 50 and 20 things”, but this error as no implications on the remainder of the arguments.
51
soe
41 In the article “Algebra as language: Wallis and Condillac on the nature of algebra”, Malet (2003, p. 9) makes important reflections about the «language» of algebra. 42 Franci, F. and Toti Rigatelli, L. (1983, p. 310). 43 These subjects are present under the title “Here I will show to you the rule of extract square roots and cubic and surd [sordas] and promicas, and relatas and the root of the root of all the kind and mode of root” Bento Fernandes (1555, fo. 80 [v]). 44 Rankin, F. K. C. (1992, p. 274). 45 “Aqui vos mostrarei regras da zibra moquavel a qual vulgarmente se chama regra da cousa e de ~ç
soe, de
~ç
( )( )( )
e de cubi e raiz da cousa e raiz de numero e de toda valia de regra e razão para quem tiver delicado engenho e memoria”. Bento Fernandes (1555, fo. [83]). 46 Bento Fernandes (1555, fo. [83]). The author already expressed this idea at the beginning of his work, when, on the title of his work, he says: “E as regras da cousa que sam de mais sustancia pera pessoas curiosas τ experimentadas na arte”. 47 Franci, R. (2002, p. 83). 48 Rankin, F. K. C. (1992, p. 269). 49 See for exemple, the problem 43 proposed and solved by Bento Fernandes (1555, fo. 78). 50 “E primeiramente vos mostrarei aqui uma tavoada que é necessária para terdes em vossa memória e a saberdes de cor, assim como vos disse da tavoada pequena que atrás fica”, in Fernandes, Bento (1555, fo. 83). 51 Bento Fernandes (1555, fo. 83) 52 Franci, R. and Toti Rigatelli, L. (1985, p. 19). 53 Gandz, S. (1936, p. 273). 54 Franci, R. and Toti Rigatelli, L. (1985, p. 36). 55 Franci, R. and Toti Rigatelli, L. (1985, p. 36). 56 Cajori, F. (1993, p. 342) 57 Cajori, F. (1993, p. 90) 58 Cajori, F. (1993, p. 108) 59 Franci, R. and Toti Rigatelli, L. (1985, pp. 40-41) 60 Bento Fernandes (1555, fo. 83) 61 The author only writes cubo but according the whole text we must read “cubo of cubo”. Bento Fernandes (1555, fo. [83]). 62 Bento Fernandes (1555, fo. [83]. 63 Franci, R. (2002, p. 89, footnotes 14 and 15). 64 Malet, A. (1990, p. 186) points out that the maestri of abaco “were interested in proportionality only as a relationship linking three or more numbers. So, while dealing with proportionality as a series of numerical rules computationally useful, they did not use, nor study, the notion of ratio in itself”. 65 Franci, R. (2002, p. 86). 66 Bento Fernandes (1555, fo. 86). 67 Van Egmond, W. (1978, p. 177). 68 It is an old unit mesure. 69 This means: it’s area is 100 [squared] braças. 70 This theorem relates the area, A, of a triangle in function to the half-perimeter and each one of its sides:
csbsassA −−−= , where s is the half-perimeter and a, b and c are the sides.
52
71 Bento Fernandes (1555, fo. 90). 72 It is an old unity of money. 73 Anvers is now Antuerpy. 74 As one livra is equal to 240 dinheiros, x dinheiros is equal to x
201
320x
of the livra. (Bento Fernandes, 1555,
fo. 40). 75 Høyrup, J. (2002, p. 9). 76 In this part, the text has two mistakes, because Bento Fernandes forgets the number 1800 and instead of saying he says ‘the root’ of this. 77 Luca Pacioli (1494, fol 150). 78 This happens, for example, with Maestro Benedetto of Firenze, who in his Trattato di praticha d’arismetica (1463, ms. L.IV.21 Municipal Library, Siena) refers to Piero Chalandri, Leonardo Pisano, Maestro Biagio and his Trattato di Praticha, and so on (Franci, F. and Toti Rigatelli, L. 1983). 79 Franci, R. (2002, pp. 81-105). 80 Høyrup, J. (2006, pp. 1-39)). 81 As Mº Gilio, Questioni d’Algebra, dal codice L.IX.28 della Biblioteca Comunale di Siena; Anónimo (séc. XIV). Il Trattato d’Algibra, dal Manoscrito Fond. Prin. II.V.152 della Biblioteca Nazionale di Firenze; Anonimo (sec. XIV) Trattato dell’Alcibra Amuchabile, dal Codice Ricc. 2263 della Biblioteca Riccardiana di Firenze; AA.VV. (XV secolo) Trattato d’Abaco, dal Manoscritto Parmense 78 della Biblioteca Palatina di Parma; Anónimo (sec. XV) Líbri di contí e marcatanzíe, dal Ms. Pal. 312 della Biblioteca Palatina di Parma. 82 This treatise “is identical with Jacopo’s treatise from 1307”, written in Montpellier (Høyrup, J. [2005], p. 3). 83 I would like to thank to Dipartimento di Matematica of the Universita’Degli Studi di Parma for sending me the following books: Anónimo, Libro dí contí e mercatanzíe, a cura e com introduzione di Silvano Gregori e Lucia Grugnetti (1998), and Trattato d’abaco AA.VV. (XV Secolo), a cura e com introduzione de Silvano Gregori e Lucia Grugnetti (2001). 84 Gregori, S. and Grugnetti, L. (1998). 85 Høyrup (2006, p. 20). 86 Van Egmond, W. (1978, p. 184). 87 Leonardo Pisano (2002, pp. 554 -558). 88 Høyrup, J. (2006, p. 8). 89 See Gregori, S. and Grugnetti, L. (1998, pp. 104-105). 90 Franci, R. and Toti Rigatelli, L. (1985, p. 31). 91 Franci, R. and Toti Rigatelli, L. (1985, p. 72, note 38). 92 “Qui apresso scriviremo in questo quaderno le regole e asempri della Gibra mocabile, le quali si chiamano vulgarmente regole della cosa e di ciensi e di cubi [numeri e di censi di censi] e radicie di cosa e radice di numeri e tucte baile di raguaglamenti di ragiony d’ogni modo”, in Gregori, S. e Grugnetti, L (1998, p. 98). 93 Bento Fernandes (1555, fo. 83). 94 Gregori,S. and Grugnetti, L (1998, p. 98). 95 Høyrup, J. (2006, p.14, Note 22). See also Barbieri, F., Franci, R.and Toti Rigatelli, L., 2004, p. 79. 96 Franci, R. (2001, p. 270-271). 97 See Franci, F. and Toti Rigatelli (1988, p. 19 and p. 25). Also in Franci, R. (1985, p. 39). 98 Franci, R. (2002, p. 95).
53
99 Franci, R. (1985, p. 38). 100 Høyrup, J. (2006, p. 23). 101 Høyrup, J. (2006, p. 9). 102 See Van Egmond, W. (1983). 103 Høyrup, J. (2003, p. 21). 104 Franci, R. (2002, p. 97). 105 At the moment, we are still waiting a copy of it, from the Biblioteca of Palermo. 106 Høyrup, J. ([2005], pp. 9-10). 107 Franci, R. (2001, p. 272). 108 Franci, R. (2002, p. 95). 109 Franci, R. and Toti Rigatelli, L (1985, p. 74 note 67); also in Franci, R (2002, p. 96). 110 Gregori, S. and Grugnetti, L. (1988, p. XIV). 111 Gregori, S. and Grugnetti, L. (1998, Introduzione).
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