SEMINARI D’HISTÒRIA DE LA CIENCIA
UNIVERSITAT POMPEU FABRA
Algebraic operations in an early
sixteenth-century Catalan manuscript:
An approach to the possible sources
Javier Docampo Rey
Preprint 4
Febrer 2008
Algebraic operations in an early sixteenth-century Catalan
manuscript: An approach to the possible sources
Javier Docampo Rey
Grup d’estudis d’Història de la Ciencia, Universitat Pompeu Fabra,
C/ Ramon Trias Fargas, 25-27, 08005 Barcelona, Spain
[email protected]
January 2008
ABSTRACT
This paper adds new evidence to the article “Reading Luca Pacioli’s Summa in Catalonia: An early
16th-century Catalan manuscript on algebra and arithmetic” (Historia Mathematica 33). We will now
focus on the possible sources of the diagrams of coefficients that are used to perform algebraic operations
in this Catalan manuscript. In this research, we will see some of the charts that were used to operate
algebraic expressions in different medieval traditions.
Introduction
Ms.71 of the Monastery of Sant Cugat’s section of the Arxiu de la Corona d’Aragó
(Barcelona) is a Catalan manuscript that was composed around 1520.1 Evidence
strongly suggest that it was written by the Majorcan Joan Ventallol, who is the author of
the Pràtica mercantívol, a commercial arithmetic in Catalan that was published in Lyon
in 1521.
The manuscript has 160 sheets of paper with notes on a wide range of arithmetic and
algebra material that seem to be prepared to be used in teaching. More than half of them
concern the adaptation of some parts of Luca Pacioli’s Summa de Arithmetica,
Geometria, Proportioni et Propotionalità (Venice, 1494), and the rest are devoted to
what we could call “standard” commercial arithmetic.
The diagrams in Ms 71
Of the several interesting features that can be found in the manuscript, surely the most
important one is the presence of a certain kind of diagrams to perform algebraic
operations.2 They contain representations of what we would call polynomials as series
of coefficients that are arranged vertically. Coefficients can be negative or fractional,
1
See [Docampo, 2006] for more information on this manuscript and its authorship. For a detailed study of
all its contents, see [Docampo, 2004, 307-563].
2
There are other representations that are used to perform algebraic operations in the manuscript, like
those for cross multiplication (see below). However, these can be found in Pacioli’s Summa and thus we
will not deal with them here.
-1-
and when one of them is lacking, a zero is written down. Since we have described these
diagrams in a previous article,3 in this section we will just briefly comment some further
examples in order to provide a wider perspective of their use.
These charts illustrate some explanations that appear in the main text concerning the
construction of an equation to solve a given problem and its reduction to one of the 6
basic forms, or at least to a form in which it can be solved by a known rule (see our
previous article). However, the diagrams are not just illustrations but are themselves a
tool to perform those algebraic operations. They are not introduced nor explained in
general terms in any part of the manuscript, and just appear in a few problems.
In a very simple example (f. 105r), we see how one of these diagrams is used to
illustrate that an unknown quantity is divided in three equal parts:
1
1
3
1
3
1
3
0
0
0
0
In modern notation, we would have x on the left and
1
3
x three times inside the
rectangle on the right.
There is one occasion (f. 62r) in which this kind of notation is used in a note on the
right margin of the main text and not in a diagram:
La suma dels dos és
1
0
per lo primer he 2
2
3
.co. me 20 per lo segon, que ajustat fa 3 23 .co. més 20.4
Which, in modern notation would be x + (2 23 x + 20 ) = 3 23 x + 20 .
However, this notation of polynomials as the series of their coefficients is mainly used
to add, subtract and multiply them. In f. 100v we see this diagram to illustrate the steps
that are followed to solve algebraically a partnership problem involving 4 persons: The
solving process is illustrated as follows:5
1 2
0 0
4
2 0
0 0
0 0
4
Sy 0
0
0
400
1
0
4
0
400
0
0 25 0
100
0
0
0
1
0
400 0
0 0
4
0
The first four columns stand for what we could write as x, 2x, 2x2 and 4x3. Then the
necessary operations to solve the problem are performed and the equation that will lead
3
See [Docampo, 2006, 55-57].
4
“The addition of both is
1
0
for the first and 2 23 .co. me 20 for the second, which, after being added, gives
3 23 .co. més 20.”
5
This example is explained in detail in [Docampo 2004, 520-523].
-2-
to the unknown quantity appears in the final columns: 100x3 = 400x and then x3 = 4x.
This is the only case in the manuscript where these diagrams are used and the final
equation is not among the six canonical ones.
In the fifth column we see the word “sy”, which means “if”, marking the initial step of
the application of the rule of three. Literal language is used inside the diagrams in other
occasions. In f. 102v, the following chart is used to solve a commercial problem by
means of a first-degree equation:6
parteix per
paga
1 1
0 23
ve
1/25
1
0
23/25
25
1/25
2/31
2
manco 50
31
és egual a
més 23/25
parteix per
paga
50/31 manco
31
23
2/31
manco 50/31
50
1
25
2
31
resta 19/775
775
50
25
2507
100
1250
50
19638/775 parteix per 19/775
ab
ajusta
ve
25
31
103 196 e tant val la sacha
We see the words paga (he pays), ve (it comes), manco (minus), parteix per (divide
by), etc. In modern terms, the first column stands for x, the second for x + 23, etc.
Finally, in f. 114r we see a diagram containing some slashed-out coefficients:9
ma 1 ma 1
100 220
ma 1 1
220 0
1
220
00
90
ma 1
90
100 9000
1
130
0
9000
220
90
130
This diagram is not related to any problem of the manuscript and appears together
with some arithmetical operations that are not related to it.
6
Fractions in the manuscript are always represented in the form
a
b
. We have used the form a/b inside the
squares in order to facilitate the reproduction of the chart. This example is explained in detail in
[Docampo 2004, 523-526].
7
We have corrected a mistake here: we read 255 in the manuscript instead of 250, and thus 1255 instead
of 1250.
8
We read 1968 in the manuscript, because of the mistake that has been mentioned in the previous note.
The fact that the final result is correctly given suggests that it was copied from the main text instead of
being obtained from the diagram.
9
In this diagram, mª stands for multiplica (multiply in the imperative tense). The lines that are used to
strike out numbers are oblique and not horizontal in the manuscript.
-3-
As we see, the third column is a repetition of the second, perhaps by mistake. It seems
that the elements are slashed out after they are multiplied: in modern terms, after
multiplying x + 220 by x to obtain x2 + 220x, both factors are slashed out. The same
happens when 90 is multiplied by x + 100 to obtain 90x + 9000.
Hindu numerals and calculation where associated to the takht when they entered Arab
mathematics. The takht was a board that was covered with sand or fine dust over which
all calculations were performed using the fingers or with a stylus. In Damascus at least,
attempts started to be made (in the mid-tenth century) to adapt Indian arithmetic to
paper and ink, and thus avoiding some of the disadvantages of the dust board.10 There
were some later developments, and counters that were arranged in lines were used, so
working on dust was not necessary. These new tool is commonly known as the “lines
abacus”, and it was used by authors like the Western Muslim Ibn al-Bannā (1256–
1321), and it does not appear in Eastern Arabic texts.11 On the other hand, another
calculation board was used in the Maghreb at least from the twelfth century. It was a
wooden board with a plate of clay over which operations were performed with ink using
a stylus, rubbing out being made with wet clay.12
Leonardo Pisano often uses representations of operations inside rectangles in his
famous Liber abaci.13 In an example to explain the cross multiplication method,
numbers are written inside a rectangle that is said to represent a tabula dealbata, which
reminds us clearly of the Arab calculation boards.14 Here we see how the steps of the
product of 12 by itself are represented, illustrating the explanation of the main text: 15
10
One of these disadvantages appeared when the calculations had been rubbed out and the scribe needed
to repeat his work.
11
See [Saidan, 1978, 351-352].
12
See [Abdeljaouad, 2002, 20-21, 60], [Lamrabet, 1994, 203].
13
This work is edited in [Boncompagni, 1857] and translated into English in [Sigler, 2002].
14
Leonardo Pisano writes “in tabula dealbata in qua littere leviter deleantur” [Boncompagni, 1857, 7].
Thus he refers to a “whitened board” where numbers could easily be rubbed out. This expression reminds
us of the dust board: the board would be whitened by the sand or dust that was spread over it. However,
the clay of the board that was used in the Maghreb to perform operations at that time was also white (see
[Lamrabet, 1994, 203]), and thus we can not discard that Leonardo was referring to this clay board. On
the other hand, it has been stated [Saidan, 1978, 351, note 4] that the origin of the title Liber abaci has to
do with the fact that even when the takht was being left for paper and ink, books on calculation with the
Hindu arithmetic continued to be written under the name Hisāb al-Takht. Calculation over dust or clay
during the medieval period is also examined in [Ifrah, 1997, 1289-1290, 1298-1305].
15
[Boncompagni, 1857, 7].
-4-
descriptio
prima
4
12
12
Secunda 44
12
12
Vltima 144
12
12
When some of the methods that were performed in calculation boards were transferred
to paper and ink, numbers were slashed out because they could not be erased easily.
Then, it is reasonable to wonder if the slashed-out coefficients that we have seen in our
last example may indicate some kind of relation of the charts in Ms. 71 with dust of clay
boards. It is interesting that the author of the manuscript often refers to the diagrams
with expressions like la pràctica és tal (“the practice is such”) or com veus afigurat (“as
you see it arranged in a figure”) or la pràtica es com veus dalt afigurat (“the practice is
as you see above in the figure”) or per posar la qüestió en pràtica (“to put the question
in practice”). The terms figures or pràticas afigurades are also used in commercial
arithmetics like Francesc Santcliment’s Summa de l’art d’Aritmètica (Barcelona,
1482)16 to refer to any representation of the steps of an arithmetic problem that is
explained in a basically rhetorical text. It must be noted that the expression Wa
Suratuhu Hakadha (“its image is represented like this”) was used in Arab manuscripts
to refer to those representations whose purpose was to illustrate a rhetorical text but
which are not essential for its understanding.17 Sometimes, these illustrations showed
what the calculator would actually see on the dust board, “much like a diagram in a
geometrical argument”.18 Analogous expressions are very common in Latin versions of
al-Khwārizmī’s arithmetic to refer to representations of numbers or operations into
rectangles that illustrate what is mentioned in the main text, and it seems clear that they
also refer to calculation boards.19 The same situation can be also found occasionally in
Italian abacus literature.20
16
See [Malet, 1998, 220, 222-223, 230 (f. 69v, 70v-71r, 74v)] for some examples.
See [Abdeljaouad, 2002, 8, 60-61].
18
See [Berggren, 1986, 32].
19
See, for instance, [Allard, 1992, 8]: “Et hec est eorum figura” (“And this is its figure”) or, in p. 119, “Et
hec est facte divisiones figura” (And the figure of the performed division is the following) . The
rectangles are often used to indicate the steps that must be followed to perform operations. See, for
instance, ibid., 28–36, 155–174. In another page we find the expression “Cumque uolueris multiplicare
numerum in altero, pone unum ex eis secundum quantitatem mansionum eius in tabula uel in qualibet re
alia quam uolueris” (If you want to multiply a number by another, put one of them, according to its ranks,
on a table or on any other material that you want, p. 9). On other pages there are explicit references to the
erasure of the numbers that appear on the tables (the tables being represented by the rectangles) (ibid., 11,
17
-5-
It is known that the costume of representing operations in rectangles was kept in some
treatises, even when operations were already performed using paper and ink instead of
calculation boards (for instance, in the Liber abaci). Thus it is reasonable to think that
diagrams like those in Ms. 71 can be influenced by representations of either the dust
board, the clay board that was used in the Maghreb, or even another kind of tool.
However, we can not discard the possibility that algebraic operations appearing in the
diagrams of the Catalan manuscript were actually performed in one of these kinds of
calculation boards: we must remember that Western Muslim authors like al-Qalasādī
(1412-1486) and Ibn Ghāzī (1437-1513) do mention calculation boards in examples of
operations with polynomial expressions.21
Algebraic operations in the works of al-Karajī and as-Samaw’al 22
The first elements of a theory of polynomials are found in the Eastern Arab author alKarajī (d. 1029), who was active in Baghdad at the end of the tenth century and the
beginning of the eleventh.23 In his al-Kitāb al-fakhrī fi l-jabr wa l-muqābala24 and in his
al-Kitāb al-badī c fil-hisab25 we can find the generalization of the monomials with the
unknown as the starting point and not from the combination of squares and cubes, as
had been done before.26 This is explained in this way by J.L.Berggren: «His [alKarajī’s] point of view is that unknown quantities, whether absolute numbers or
geometrical magnitudes, can be a “root”, “side” or “thing” (both corresponding to our
“x”), or they can be māl (x2), cube (x3), māl māl (x4), māl cube (x5), etc., where each
member is the product of “thing” by the previous member.»27 It is interesting that alKarajī calls an “order” to each of these species, the same term he uses for the places of
the different powers of ten in decimal arithmetic (in fact, he explicitly compares these
species with units, tens, hundreds, etc.). The notion of power is also extended to the
93, 100, etc.). Thus it seems clear that, again those illustrative rectangles are related with tables on which
numbers were written and could be easily erased.
20
In a 15th-c. manuscript that is preserved in Siena (Cod. L. VI. 46 of the Biblioteca degl’Intronati di
Siena) numerical operations are represented inside rectangles that are referred to with expressions of the
kind “come ti mostra la presente figura qui di sotto” (“as it is shown by the figure below”). See [Arrighi,
2004, 116].
21
These authors used Maghrebian symbolism to represent algebraic expressions. See [Abdeljaouad, 2002,
14-15, 21].
22
I am very grateful to Marco Panza for turning again my attention to al-Karajī’s work and for putting me
in touch with Pascal Crozet.
23
For an introduction to al-Karajī’s life and works, see [Rashed, 1970-1980].
24
See [Woepcke, 1853] for detailed information on this work.
25
Modern edition in [Anbouba, 1964], with a description in French.
26
[Djebbar, 2005, 54].
27
[Berggren, 1986, 112].
-6-
“parts” of these “orders” ( 1x ,
1
x2
1
x3
,
, …).28 Al-Karajī gives general rules to add, subtract
and multiply polynomials, and also to divide a polynomial by a monomial.29 He also
gives a general rule to calculate the square root of a polynomial with rational positive
coefficients.30
We must note that his algebra is basically rhetorical, so these
developments were explained in words.
Al-Karajī initiated a tradition in which algebra underwent a process of
“arithmetization”. This term is used by Roshi Rashed to refer to the fact that the
operations of elementary arithmetic and algorithms, like the extraction of the square
root, were extended to algebraic expressions, and specially to polynomials.31
As-Samaw’al (d. 1175),32 who was born in Baghdad, further developed al-Karajī’s
work. He writes about “operating on the unknowns using all the arithmetical tools, such
as the arihtmeticien operates on the known terms”.33 In his al-Kitāb al-bāhir fī l-jabr wa
l-muqābala [The Resplendent book in algebra]34 we find methods that could be used to
explain the basic principles that underlie diagrams of coefficients in Ms. 71.
The handling of the rules of signs in all generality allowed as-Samaw’al to
give general rules to divide polynomials and to extract the square root of any
polynomial with rational coefficients. In fact, he operates expressions that could be
n
∑a
expressed in modern terms as
k =− m
rational number for every k.
35
k
x k , m and n being two positive integers, and a k a
Thus in this section, the term polynomial should be
understood as any expression of this kind.
Many of these procedures would have been very difficult to perform without the use
of the table symbolism in which each polynomial is represented by the sequence of its
coefficients, written in Hindu numerals. Apart from the symbols of the numeration
system, these tables are the only form of symbolism that we find in the works that were
composed in the Arab East of that time.36 As-Samaw’al called polynomials
“expressions with known images”. The “known images” would be the coefficients of
28
See [Woepcke, 1853, 48].
[Rashed, 1984, 45–46].
30
See [Anbouba, 1964, 40-41].
31
See [Rashed, 1984, 10].
32
For an introduction to the life and works of as-Samaw’al, see [Anbouba, 1970-1980].
33
See [Rashed, 1984, 27].
34
Modern edition in [Ahmad, Rashed, 1972].
35
See [Rashed, 1984, 46–47].
36
[Djebbar, 2005, 55].
29
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the polynomial that were placed on the takht, and were copied on the manuscript
afterwards.37
As-Samaw’al is able to enunciate a rule that is equivalent, in modern terms, to state
that x m . x n = x m+ n with m, n ∈ℤ. This is possible because he previously introduces the
zero power: in modern terms, x0 =1 when x ≠ 0.38 In order to explain the rules of
multiplication of powers of the unknown, he includes a table whose upper part can be
represented in this way:39
9
8
7
6
5
4
3
2
1
0
1
x9
x8
x7
x6
x5
x4
x3
x2
x
1
1
x
2
1
x2
3
1
x3
4
1
x4
5
1
x5
6
1
x6
7
1
x7
8
1
x8
9
1
x9
In his explanation, he states that “the distance of the order of the product of the two
factors from the order of one of the two factors is equal to the distance of the order of
the other factor from the unit. If the factors are in different directions then we count (the
distance) from the order of the first factor towards the unit; but, if they are in the same
direction, we count away from the unit.”40 While the “order” of an power is indicated by
numbers in the head line, the “distance” is given by the number of places that one has to
move along the diagram. The previous fragment is a way of expressing the law of
exponents that we mentioned before. For example, if we multiply the second power by
the third power, to obtain the result we would have to count three places from the
second order as we move away from the unit. Then we would get to the fifth power.
Thus, this can be expressed as x 2 . x 3 = x 2+3 = x 5 . Also, if we multiply the third power
by “the part of the fourth power” ( x 3 . x14 ), we must count four places from the third
place on the left towards the unit, and we would get to the place of “the part of the
unknown” ( 1x ).
37
[Abdeljaouad, 2002, 14].
[Rashed, 1984, 125].
39
In the original text, numbers heading the columns are not in Hindu numerals but Arabic alphabetic
numerals, the powers of the unknown are expressed in words, and the table includes two more lines with
the corresponding powers of 2 and 3 as numerical examples. See [Ahmad, Rashed, 1972, 16-19 of the
French introduction, 21 of the Arab text]. This table is analysed in [Rashed, 1984, 125] and [Berggren,
1986, 113-115]. We have kept the order as it appears in the Arabic text, in order to better compare it with
other diagrams.
40
We have taken this translation from [Berggren, 1986, 114]. To see the way in which as-Samaw’al
justifies the rule, see [Ibid., 115].
38
-8-
After giving the mentioned rule, as-Samaw’al gives some examples of its application.
Then he is able to explain how to multiply two composite expressions (each of two or
more orders), by multiplying each term of one of them by all the terms of the other, and
then adding up the results. Combinations of powers are added – or subtracted – by
adding – or subtracting – their corresponding terms.41
These rules show how polynomials could have been operated in the diagrams of Ms.
71.42 If the same rule that was given by as-Samaw’al was the one actually used by the
author, then the lowest row in the diagrams of Ms. 71 (which is occupied by the simple
numbers) would represent what we could call a “zero rank”. Then, to obtain the position
of the product of two coefficients, we should add their “ranks”. If that lowest row was
considered to be of rank or degree one, then the position of the product of two
coefficients would be obtained by adding their ranks and subtracting 1.
He have not found charts of coefficients that are devoted to add, subtract or multiply
two given polynomials in the al-Bāhir. They are used in the far more complex
operations of division and square root extraction of polynomials,43 which are not
performed in the charts of Ms. 71. Even when charts of coefficients in this manuscript
and those in the al-Bāhir are not comparable, the idea of moving along the chart when
two powers of the unknown are multiplied also underlies those diagrams in the Catalan
treatise.
From what we have seen, it is reasonable to think that the diagrams of coefficients in
Ms. 71 can be a late product derived from the tradition that was initiated by al-Karajī,
with as-Samaw’al’s table symbolism playing a key role. However, the author of the
Catalan manuscript was active more than three centuries after as-Samaw’al composed
his al-Bāhir, and the authors that could have inspired him (or a previous scholar from
whose work he took that technique) to develop these diagrams are diverse. Now we will
wonder about some of these possible sources.
A West Muslim source?
Even when there is still no clear evidence of the circulation of the works of al-Karajī
and as-Samawa’l towards al-Andalus and the Maghreb, it seems that the use of the
41
See [Berggren, 1986, 113, 115].
See, for instance, the example that is explained in [Docampo, 2006, 56]. We must note, however, that
negative powers of the unknown never appear in the diagrams of coefficients of the Catalan manuscript.
43
For the method to divide polynomials that was used by as-Samaw’al, see [Ahmad, Rashed, 1972, 22-27
of the French introduction] or [Berggren, 1986, 115–118]. For square root extraction, see [Ahmad,
Rashed, 1972, 28-34 of the French introduction].
42
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unknown to express all the powers instead of combining squares and cubes was
common among some algebraists from the area at least from the 14th century.44
There is an early example of an West Muslim author who showed a special interest in
algebraic operations: Abū l-Qāsim al-Qurashī (12th c.) was a teacher from Al-Andalus
that lived in Bugia (in Central Maghreb) and taught the science of heritages there. He
composed a commentary (that is now lost) on Abū Kāmil’s algebra. We know from
references by other authors that he introduced some changes in it, and among them is
that he starts introducing the operations on monomials and polynomials before starting
to deal with the resolution of equations. Al-Qurashī’s treatise was studied in the
Maghreb at least until the fourteenth century.45
Leonardo Pisano (ca. 1170 – d. after 1240)46 studied mathematics in Bugia, and
further increased his knowledge during several business trips to Egypt, Syria, Greece,
Sicily and Provence. It has been suggested that his stay in Egypt and Syria could have
made him aware of algebra contributions inscribed in al-Karajī’s tradition, from which
no clear traces have been found in West Muslim mathematical texts.47 However, it is
clear that diagrams in Ms. 71 are not related to any known work by Fibonacci.
During a good part of the medieval period, a certain kind of algebraic symbolism was
used in many West Muslim texts.48 Furthermore, it has been suggested that contacts
with the Maghreb could have inspired the first traces of symbolization in European
algebra.49 Here we see a couple of examples of Maghrebian notations, with their
equivalences in modern notation:50
3x 2 + 5 x = 7
7‫ ل‬5 3
3x 2 − 5 x = 7
7‫ ل‬5 3
From the point of view of the operations with polynomials, it is important to give a
brief account on Al-Qatrawānī (15th c. or earlier). He was of Egyptian origin but lived
in Tunis for some time and seems to have taught there. He composed a treatise titled
44
See [Djebbar, 2005, 95].
See [Djebbar, 2005, 81].
46
A introduction to the life and works of Leonardo Pisano (also known as Fibonacci) can be seen in
[Vogel, 1970-1980].
47
See [Djebbar, 2005, 111-112]. For an approach to the possible connections between some parts of
Fibonacci’s works and al-Karajī, see [Allard, 1997, 228].
48
See [Djebbar, 2005, 91–92]. For a detailed account on Maghrebian symbolism, see [Abdeljaouad,
2002].
49
See [Abdeljaouad, 2002, 61] and [Høyrup 2002, pp. 7, 10].
50
We have taken these examples from [Djebbar, 2005, 92].
45
- 10 -
Rashfat ar-rudāb can thughūr a cmāl al-hisāb, which is a handbook on arithmetic and
algebra. In f. 122, before calculating the square of 2x2 + 8x + 4, he represents this
polynomial it in this way: 2 .˙. 8 .˙. 4. The three dots just have the function of
separating the coefficients, since they are also used to separate the numerals in the
chapter on the root extraction of
integer numbers. This resemblance between the
representation of integers and polynomials can also be found in as-Samaw’al. However,
al-Qatrawānī presents the resulting polynomial 16 + 64x + 80x2 + 32x3 + 4x4 using
Maghrebian symbolism, while as-Samaw’al gives results using tables.51
Al-Qatrawānī also deals with the extraction of the square root and the cubic root of a
polynomial.52 Before extracting its square root, he writes the polynomial 81x6 + 72x5 +
106x4 + 184x3 +89x2 + 80x + 64 using the literal Maghrebian symbolism:53
81 72 106 184 89 80 64
In fact, he uses Maghrebian symbols to give the initial polynomials, the results and the
presentation of the multiplication that is used to extract the root.54 When a power is
lacking in the initial polynomial, he indicates that a zero must be written down.55
As far as we know, and even when he does it in a very restricted way, as we have
seen, al-Qatrawānī is the only author of the medieval Muslim West that writes down
some polynomials as the series of their coefficients occasionally. However, he generally
uses the Maghrebian symbolism and not the table symbolism used by as-Samawa’l.
From the evidence that we have seen, it is difficult to justify any connection of diagrams
in Ms. 71 with al-Qatrawānī’s work. On the other hand, his procedures require a
considerable skill in the operations with polynomials. Then, it seems reasonable to
suspect that at least some algebraists of the Muslim West had at least a partial
knowledge of the developments that had been made by as al-Karajī and as-Samaw’al,
like the extension of all the arithmetic operations to polynomials. Furthermore, Ahmed
Djebbar has stated that the existence of the works of al-Qatrawānī, together with those
of other West Muslim authors, should at least make us wonder about the possibility that
51
See [Abdeljaouad, 2002, 14].
He is the only known medieval Arab author to have dealt with the calculation of the cubic root of a
polynomial. For more information about al-Qatrawānī and how he extracts square and cubic roots of
polynomials, see [Djebbar, 2005, 97–100] and [Lamrabet, 1994, 113–114, 226–234].
53
[Djebbar, 2005, 98–99].
54
[Djebbar, 2005, 100].
55
See [Lamrabet, 1994, 226–234].
52
- 11 -
the first elements of a theory of polynomials were developed in Al-Andalus and then in
the Maghreb with the only knowledge of Abū Kāmil’s tradition as the starting point.56
So far, we have not found texts from the Muslim West with similar diagrams to those
in Ms. 71. But the fact that operations with polynomials were handled skilfully by some
authors in that area must be had into account for two reasons. The first one is that there
still are a lot of Arabic mathematical texts to be studied. The second one is that
geographical proximity, as well as important commercial and cultural relations of alAndalus and the Maghreb with the Christian kingdoms of the Iberian Peninsula57 made
the West Mediterranean area a very favourable scenery for scientific exchange.
A Latin source?
In 1851, Baldassarre Boncompagni published the transcription of a Latin version of
al-Khwārizmī’s algebra, and presented it as being made by Gerardo of Cremona in the
twelfth century.58 In fact, this version seems to descend from Gerardo of Cremona’s
translation,59 but its authorship is still not clear.60 The copy that was used by
Boncompagni was written in the fourteenth century61 and contains some verses to
explain the three algorithms that are used to solve the compound canonical equations.62
Their style is that of the didactic poem (Urjuza) that was composed by the Western
Muslim Ibn-Yāsamīn (d.1204).63
We can find three diagrams representing polynomial expressions up to the second
degree in this Latin manuscript. They appear in a chapter named “How to represent the
56
See [Djebbar, 2005, 100]. Driss Lamrabet has also mentioned the possibility of an independent
development [Lamrabet, 1994, 226].
57
Just to mention an example, we should regard the strong presence of merchants from the Catalanoaragonese kingdom, and specially from Majorca, in the most important commercial centers of Central
Maghreb during the fourteenth century. See [López Pérez, 2002-2003, 366].
58
See [Boncompagni, 1851].
59
See [Allard, 1997, 221].
60
Ms Vat. Lat. 4606, f. 72r-77r and Oxford, Bodl. Lyell 52, fol. 42r-49v are the two only known copies
of this version. The former (which was the one used by Boncompagni) was made around the middle or
the second half of the fourteenth century, while the second one dates from the beginning of that century
(see [Hissette, 2003]). Barnabas Hughes (see [Hughes, 1986]) attributed this version to Guglielmo de
Lunis (ca. 1250). Further still, there is an algebraic manuscript in Italian (ms Vat. Urb. lat. 291, f. 34r-42r)
of the first half of the fifteenth century that is mostly based on Vat. Lat. 4606. Even when it has been
suggested that Guglielmo de Lunis could have written both the Latin and vernacular versions, this point is
not clear (see [Hissette, 1997 and 2003]). The Oxford Latin manuscript and the Italian version are edited,
respectively, in [Kaunzer, 1986] and [Franci, 2003].
61
[Hissette, 2003, 137].
62
These three cases can be written in modern notation, as x2 + px = q, x2 + q = px and x2 = px + q. See
[Boncompagni, 1851, 415-418].
63
See [Abdeljaouad, 2002, 9-10, 15].
- 12 -
census, the radices and the dragme”.64 The author announces that he will give some
precepts so it will be “lighter” for the learner to make the necessary calculations to
arrive to one of the six canonical equations. They are precepts “to write and to multiply”
what we would call today second-degree polynomial expressions.65 Then he explains
how to use the notations that are exemplified by the three following diagrams (they
appear in the margin of the manuscript, each one next to its explanation):
2
c
3
r
4
d
2
3
4
3
4
5
c
r
d
2
2
5
5
c
c
r
r
3
4
2
4
r
d
c d
˙
˙
˙
˙
In modern terms, the first diagram corresponds to 2x2 + 3x + 4 and the second one to
2
3
x 2 + 34 x + 54 . The third diagram represents four expressions: 2x2 – 3x, 2x2 – 4, 5x – 2x2
and 5x – 4. So the straight line under a letter stands for “plus” and a dot stands for
“minus”. Each of the three charts is referred to respectively with the expression sic
figurentur, hoc modo figurentur and sic notantur. There are no other diagrams like these
in the rest of the manuscript.66
While in the last diagram that appears in the Latin text coefficients are arranged
vertically, as happens in Ms. 71, those representations in the charts of the Catalan
manuscript differ essentially from those in Ms Vat. Lat. 4606: in the former, the
denomination of each coefficients is given by the position that it occupies in a sequence
and not by the presence of an additional symbol (like c, r or d in the Latin manuscript).
The fact that subtractive terms have to be on the bottom of the diagrams in the Latin text
(see the third rectangle) shows that a positional system of the coefficients was not
applied to operate the polynomials. However, the analogy that is made between
arithmetical and algebraic operations (just in very simple cases) in the Latin manuscript
64
“Qualiter figurentur census radices et dragme”. This chapter and the diagrams can be seen in
[Boncompagni, 1851, 420-422].
65
“Porro omnis compotus qui in restauratione diminuti vel particione superhabundans exercetur, ad
aliquod horum sex capitulorum convertibilis est. Quod ut levius fiat discenti: quedam scribendi et
multiplicandi precepta damus, quibus integer res ad invicem nec non res, quibus diminuitur vel
superhabundat numerus, aut que diminuuntur vel superhabundant numero, multiplicetur hoc presupposito,
quod ex ductu rei in rem provenit tantum census, et ex ductu rei in numerum, non nisi rerum multitudo.”
[Ibid., 420].
66
In the Oxford manuscript (see above, note 60), the simple numbers (dragme) in the third diagram do
not have any d underneath so the dots appear under the 4 in the second and fourth columns. Furthermore,
the diagrams appear inside the main text and not in the margin. See [Kaunzer, 1986, 64].
- 13 -
is interesting: after the third example, the text goes on explaining how to multiply firstdegree polynomials.67 Here the terms res and numerus are used instead of radix and
dragma.68 The author explains that one has to follow the same method that is used to
multiply combinations (by addition or subtraction) of a digitus (one-digit number) and
an articulus (ten),69 so he is indicating explicitly that an arithmetical operation in a
particular case is also valid for certain algebraic expressions. When these combinations
of digitus and articulus are additions, the procedure is the same as cross multiplication
of two-digit numbers.70 The author describes a procedure that can be explained in this
way: let d1 and d2 be digitus and a1, a2 articulus. The following multiplications must be
made and then results must be added:71
a1
d1
a2
d2
Two examples are given: (10 + 2).(10 + 3) and (10 − 2).(10 − 3) . In the first case, these
calculations are made: 2.3 = 6; 10.10 = 100; 2.10 = 20; 3.10 = 30; 6+100+20+30 = 156.
Obviously, in the second case the rules of signs must be used. 72
After these examples, the author gets to the multiplication of algebraic expressions
proper, under the title “How multiplication is made among census, radices and
dragme”.73 Making reference to the previous explanations, he now includes the
examples (10 − 2 x).(10 − 3 x) and (10 + 2 x).(10 − 3 x) . The method is completely
analogous to that used in the numerical examples.74
67
[Boncompagni, 1851, 422].
For the different terms that used in early Arabic and Latin algebra texts and their significance, see [Puig
and Rojano, 2004].
69
“Deinceps multiplicatione rerum quibus auctus vel diminutus proponitur numerus, aut que augentur vel
diminuuntur a numero id ipsum quod in multiplicatione digitorum et articulorum observamus, ut res si
tamquam articulus superhabundans vero vel diminutum tamquam digitus.” [Boncompagni, 1851, 422].
70
Cross multiplication was known as multiplication per crocetta in the vernacular Italian texts. See
[Swetz, 1987, 203-205].
71
“Si enim proponatur digitus et articulus in digitum et articulum multiplicandi, quadripartimur totam
multiplicationem per quartum secundum elementorum, scilicet ut digitus in digitum, articulus in
articulum. Subalterne digitus in articulum, articulus in digitum contraditorie multiplicetur, ut diminutus si
diminutum multiplicet, vel superhabundans superhabundans, inde productum multiplicatione reliquorum
coacervatum totius proposite multiplicationis summa ostendit.” [Boncompagni, 1851, 422].
72
In the second case: “Proponantur quoque 10 minus 2 in 10 minus 3 inter puncta ducenda.
Multiplicamus 2 diminuta in 3 diminuta, et erunt 6 addenda, 10 in 10 et erunt 100, 2 diminuta in 10, et
erunt 20 minuenda, 10 in 3 diminuta et erunt 30 minuenda; ergo reliquorum summa 56 ex multiplicatione
provenit.” [Ibid.].
73
“Qualiter census radices et dragme inter se multiplicentur”[Ibid., 423].
74
In the first example, we read “Verbi gratia, ponantur 10 dragme minus duabus rebus multiplicanda in
10 dragmas minus tribus rebus. Multiplicamus duas res diminutas in 3 res diminutas; et erunt 6 census
addendi. Item 10 dragms in 10 fiunt 100, 2 res diminutas in 10 dragmas, fiunt 20 res minuende, 10 in 3
res diminutas, fiunt 30 res minuende. Summa igitur huius multiplicationis est 6 census et 100 dragme
68
- 14 -
Having in mind what we have seen about representations of numbers and operations
inside rectangles in medieval treatises, it would not be surprising that diagrams in Ms.
4606 showed a kind of notation that was used on a calculation board.
This Latin source presents two features that are interesting for our research. One of
them is that, though in a very restrictive way, certain algebraic expressions are operated
explicitly using arithmetical operations as a model. The other is that the notations that
appear in the diagrams are said to be introduced in order to facilitate the reduction of an
equation into one of the six canonical cases. These are ideas that also underlie the charts
in Ms. 71. However, the positional value of the coefficients that is crucial in the
diagrams of the Catalan manuscript can not be found in those charts appearing in the
Latin text.
An Italian source?
The possible Latin influence on Ms. 71 could turn into an Italian influence if the
author knew the Ms. Vat. Urb. lat. 291 (see note 60), which contains the same diagrams
as Ms. Vat. Lat. 4606.75 However, there is at least another Italian work that must be
considered. The Regole di Geometria e della cosa (ca. 1460) is an anonymous
Florentine manuscript that seems to consist of a set of personal notes and deals with
algebra and geometry (as can be drawn from its title).76 The part that is devoted to
algebra contains a section on multiplication of algebraic expressions that starts with a
series of examples of products of monomials. Then, some of the products of
polynomials and also of monomials by polynomials are included. In the three cases
where two polynomials are multiplied, we find some diagrams that are derived from
arithmetic methods of multiplication: they are the cross multiplication (in the first and
minus 50 rebus.” [Ibid.]. In modern terms, the result is 6x2 – 50x + 100. The author also indicates that the
method can also be applied to multiply combinations of whole numbers and fractions, and includes the
examples (2 + 13 ).(2 + 14 ) and (2 − 13 ).( 2 − 14 ) .
75
See [Franci, 2003, 37 – 39]. The Italian translator uses s, c and d instead of c, r and d in the diagrams
because of the vernacular usual terminology (c for cosa and s for censo, in this second case in order to
differentiate the letter – “per lo cienso scrivete s, cioè per li sensi, dicendo che senso sia cienso” [Ibid.,
38]). On the other hand, he does not include the rule to multiply first-degree polynomials in general, but
just gives the numerical examples. A similar symbolism (at least in the case of positive terms) is used
inside the main text in an Italian manuscript of the first half of the fifteenth century (Ms. Vittorio
Emanuele 379, Biblioteca Nazionale Centrale di Roma). See [Arrighi, 1965].
76
Codice Palatino 575 (Biblioteca Nazionale di Firenze). We have used the edition in [Simi, 1992].
- 15 -
third examples) and the per scachiero (chessboard) method77 (in the second example)
Two of the three examples are multiplications of a polynomial by itself.78
The first example shows the multiplication that can be expressed in modern terms as
(4 + 4 x) . (4 + 4 x) = 32 x + 16 + 16 x 2 . The left part of the illustrative diagram and the
explanation that precedes it79 are similar to others that can be found in other Italian
treatises when cross multiplication is performed, including Luca Pacioli’s Summa, and
also appear in Ms. 71.80 However, the right part of the diagram shows that coefficients
of the result are placed in what seems to be a table.
nu
4
cose
4
nu
4
cose
4
co
32
nu
16
censi
16
The chart in the second example is closer to the diagrams in Ms. 71. It is devoted to
the multiplication of 3 12 + 3 14 x + 3 15 x 2 by itself. In modern terms:
61 2
(3 12 + 3 14 x + 3 15 x 2 ) . (3 12 + 3 14 x + 3 15 x 2 ) = 12 14 + 22 34 x + 21 80
x + 10 256 x 4 + 10 52 x 3
Following the explanation in the manuscript, [after placing the coefficients in the table
that can be seen on the left], one must multiply each of the three coefficients in the top
row by each of the coefficients on the left part of the table, the results being placed in
the corresponding columns inside the table. However, only five of the nine products that
must be performed are explicitly mentioned in the main text, and the columns that
would correspond to cubi (x3) and censi di censi (x4) do not appear in the table.81 After
that, the result is presented in a table with each term in a square.
77
See [Swetz, 1987, 205-208] for more information about this method.
We have taken the diagrams and their accompanying texts from [Simi, 1992, 29-31]. We have changed
the way in which fractions and mixed numbers are represented in the main text of this edition: we use ab
78
instead of a/b and c
a
b
instead of c.a/b.
79
“Multiprica 4 numeri e 4 cose via 4 numeri e 4 cose; farai così: multiprica lo primo 4 numeri via lo
secondo 4 cose, fae 16 cose e poi multiprica 4 numeri via 4 cose, fa 16 cose e poi multiprica 4 cose via 4
cose, fa 16 censi e poi 4 numeri via 4 numeri via 4 numeri, fa 16 numeri.” [Simi, 1992, 29].
80
See [Simi, 1994, 49; Pacioli, 1494, ff. 92v-93r, 94r; Ms. 71, f. 20v]. Representations of this kind were
also used in products of irrational binomials. See, for instance, [Simi, 1994, 19–21] and Ms. 71, f. 61r.
81
Multiprica numeri 3 12 , 3 cose 14 , 3 censi e 15 via 3 numeri 12 , 3 cose e 14 , 3 censi e 15 . Farai così:
incomincia dalli 3 numeri
1
2
ch’à capo e multipricali via 3 censi e
socto el suo nome. Poi rimultiprica lo decto 3 numeri
loro nome, poi multiprica 3 cose
1
4
via 3 cose
1
4
1
2
1
5
via 3 numeri
, fanno censi 11 15 e poni censi 11 15
1
2
, fa numeri 12 14 , mectili socto lo
, fanno censi 10 169 di censi, ponili. Poi multiprica 3 cose
- 16 -
/
co
n
1
2
3 14
n
cose
3
via
censi
1
5
3
/
c
3 15
12 14
co
1
4
3
11 83
11 15
11 83
10 169
n~u
12
co
1
4
22
3
4
/
//
//
c
cc
cu
21
61
80
10
6
25
10 25
/
n
1
2
3
Finally, in the third example we see the multiplication of two different first-degree
polynomials: (12 12 + 12 x) . (1 12 x + 12 ) = 6 14 + 19 x + 34 x 2 . As happened in the first example,
the cross multiplication method is used.82 But now the diagram consists of three
rectangles. First of all, coefficients of the initial polynomials are written down in the
upper rectangle. Then the partial results are placed in columns in the central part of the
chart, and the corresponding terms are added to obtain the final result, which appears in
the last rectangle. This example is closer to the diagrams in Ms. 71 than the other two,
even when coefficients are written down horizontally and, far more important,
multiplication is still not performed on a positional basis (a principle that underlies
diagrams in the Ms. 71 and in other works), but follows the cross multiplication pattern
that is commonly used in other Italian algebra treatises.
1
4
via 3 numeri 12 , fanno cose 11 83 , poni cose 11 83 . Poi multiprica 3 censi e 15 via 3 censi e 15 , fanno 10
censi di censi e
3
4
[read
1
4
6
25
, mectili socto lo suo nome e è multipricato. Ora ne fa’ una somma. Mecti 12 numeri e
] soto lo suo nome, poi ragiugni insieme le cose, cioè 11 83 e 11 83 , fanno 22 34 , mectili socto
61
lo suo nome. Poi somma li censi, cioè 11 15 e 10 169 , fanno 21 80
, mectili socto lo suo nome, poi mecti 10
cubi e
2
5
socto lo suo nome, poi mecti 10 censi di censi e
6
25
e 10 cubi e 25 e sta bene e per la prova del
7 si porria provare. [Simi, 1992, 29-30]. In the manuscript, terms on the left side of the first table are
written down upwards.
82
Multiprica 12 numeri 12 e 12 cosa via una cosa 12 e 12 numero. Porrai come sta in margine. Dì così: 1
cosa
1
2
via 12 numeri 12 fa 18 cose e
multipricha 12 numero via 12 cosa, fa
3
4
1
4
. Poi dì: 1 cosa
1
2
via
1
2
cosa, fa
3
4
di censo. Poi
di cosa. Somma in tucto 6 numeri 14 , cose 19, censi
[Simi, 1992, 30].
- 17 -
3
4
e tanto fa.
n~u
cos a
1
2
12 14
1
no
1
2
6
18 34
cos a
1
2
n~u
1
4
cos e
1
4
n~u
cos e
6 14
19
censi
3
4
censi
3
4
In all these examples all coefficients are positive, and the same happens in the rest of
the section on multiplication. The notations in these charts remind us of the Maghrebian
symbolism, and probably the diagrams in this Italian manuscript were adapted (or at
least got some inspiration) from a West Muslim source.
As it can be read in the main text of the second example, each coefficient of the result
is placed “under its name” (socto lo suo nome) after it is obtained. We see how a
positional principle is used for addition, but not in the factors when a multiplication
must be performed, nor the coefficients are ordered in the first two examples. Then, the
rule that was given by as-Samaw’al, in which one could move along the chart to know
where the product of two terms should be placed is not applied here.
Some features in Ms. 71 suggest that Luca Pacioli’s Summa was not the only Italian
treatise that was known by the author.83 It would not be impossible that diagrams like
those in the Regole di Geometria e della cosa had inspired some competent algebraist to
develop the charts in the Ms. 71. However, the lack of a positional principle in
multiplications in these Italian diagrams makes this an unlikely possibility.
A Jewish source?84
Between the twelfth and the fourteenth centuries, several texts attest to the diffusion
of Arab mathematical knowledge among Jewish intellectual circles in the Iberian
Peninsula, Italy, Provence and Sicily. The Epistle of the Number (Iggeret ha-mispar) is
a mathematical treatise in Hebrew that was composed by a well known Jewish
astronomer: Isaac ben Salomon ben al-Ahdab (ca 1350-ca 1429).85 He was born in
Castile and lived there during most of the second half of the fourteenth century,
83
[Docampo, 2006, 51, note 53].
I am very grateful to Pascal Crozet for telling me about Ilana Wantenberg’s research and putting me in
touch with her.
85
We have taken most of the information about this work from [Lévy, 2003a]. I am very grateful to Ilana
Wartenberg for telling me about this reference and also for her comments on this treatise. The Iggeret hamispar is been studied in a PhD dissertation that she has recently presented (see [Wartenberg, 2007]).
84
- 18 -
although in the last years of that century he was in Sicily, where he composed the
Epistle. This treatise is based on the Talkhīs a‘māl al-hisāb of Ibn al-Bannā (1256-
1321), but Isaac ben Salomon added commentaries and material from other sources.
Two different books are found in the Epistle: the first one deals with basic arithmetic
operations (including square root extraction), and the second one includes the rule of
three, the rule of double false position and algebra. This second book contains a long
description of the basic operations with “compound” expressions, which are what we
would call monomials, binomials, trinomials, etc.86 However, it is in the case of
multiplication that Isaac ben Salomon uses tables. For instance, in the multiplication of
3x3 + 7x2 + 10x by 9x3 + 6x2 + 5x, he uses the following two tables:
1
10
5
1
2
7
6
2
3
3
9
3
0
1
2
50
3
35
60
50
95
4
15
42
90
147
5
18
63
6
27
81
27
We have used bold type for numbers that represent the exponents of the monomials
(these numbers are written in red in the manuscript). The first table contains both factors
of the product with their coefficients arranged in a increasing order of the corresponding
powers. On the other hand, the second table contains the partial products of the
coefficients. Even when in this example there are not what we could call zero-degree
nor first-degree monomials in the product, two columns are left in prevision. As the
author himself writes: “at the end, we have marked zero (sifra); in effect, this is what is
convenient when a number appears”.87
In modern terms, we could write:
86
Here it is important to clarify the use of this terminology: Tony Lévy explains that considering the way
in which that Isaac ben Salomon deals with expressions that we would today call polynomials, it would
be more correct to talk about “algebraic expressions” instead of “polynomials”. See [Lévy, 2003a, 289290 (note 51), 292 (note 56)]. While this is also the case in several treatises that are mentioned in this
article, we have sometimes used modern terms like “monomial” and “polynomial” in order to facilitate
reference.
87
For all the example and its explanation, we have followed [Lévy, 2003a, 297-298].
- 19 -
10x + 7x2 + 3x3
5x + 6x2 + 9x3
50x2 + 35x3 + 15x4
60x3 + 42x4 + 18x5
90x4 + 63x5 + 27x6
50x2+ 95x3 +147x4 +81x5 +27x6
Two more diagrams of coefficients appear in the Epistle, both of them involving
negative as well as positive integer coefficients.88
Even those sources of the Epistle that are different from the Talkhīs are probably Arab
(among them, there are the commentaries on the Talkhīs), and the influence of the
tradition that was started by al-Karajī can be seen in the algebra chapter.89 It is very
interesting to see how the addition of the exponents in the first table indicates where
each of the resulting coefficients should be placed in the second. Thus, even when the
appearance of this chart and those in the Ms. 71 is different, this property shows a
conceptual connection between both kinds of diagram. Having this into account, it is
reasonable to think that, even when a direct influence from Isaac ben Salomon’s treatise
on the Ms. 71 seems not very likely, the sources that inspired both works were very
close to each other and probably circulated in the same environment. This environment
was probably that of Jewish intellectual circles in the Iberian Peninsula or Provence,
this thesis being supported by the existence of another source that circulated in these
circles and is relevant for the history of decimal fractions.
Immanuel ben Jacob Bonfils (fl. 1340-1377) was a translator from Latin into Hebrew,
as well as a mathematician, astronomer and astrologer who worked in Provence, mainly
in Tarascon but also in Avignon and Orange.90 Among the works that he wrote in
Hebrew, there is a set of arithmetical rules that involve decimal fractions,91 a version of
which, together with some of his astronomical and astrological writings, can be found in
a manuscript that is preserved in Paris.92 This version, which in fact could have been
made by one of his pupils,93 contains a table with the multiplication 4541,321×3135,432
= 14 239 003, 185 672 (if numbers are written in our decimal notation). This table
appears in f. 138r in the following form:94
88
Personal communication of Ilana Wantemberg [9-5-2007].
[Lévy, 2003a, 300].
90
See [Gandz, 1936, 17].
91
See [Gandz, 1936] and [Lévy, 2003b].
92
BNF, Ms heb. 903. We have obtained all the information about this manuscript from [Lévy, 2003b].
93
See [Lévy, 2003b, 292–293].
94
We have taken the table from [Lévy, 2003b, 292]. Bold type is used for the two factors and the final
result, and underlined powers are the denominators of the decimal fractions. In the original manuscript,
89
- 20 -
106
105
104
103
102
101
unit
101
102
103
104
105
106
107
1
2
3
1
4
5
4
2
3
4
5
3
1
3
2
3
4
5
3
1
3
4
6
8
0
7
2
6
6
9
2
6
0
4
9
2
3
4
5
3
1
3
8
2
7
1
4
5
2
1
0
6
1
7
7
6
5
1
8
2
7
1
4
5
2
1
2
7
6
5
8
1
3
0
0
9
3
2
4
1
If we consider that any positive number n can be written as an expression like
∑ a 10
k
k
, with a k ≥ 0 and k ∈ ℤ, we can call coefficients to those one-digit numbers
that form the decimal expression of n. To perform the operation in the previous table,
each coefficient of the first factor is multiplied by all coefficients of the second one and
the result is written down in a column. After this is made with all the coefficients of the
first factor, the resulting columns are added. Bonfils unifies the scales of the powers of
ten and its inverses by the assignment of a “name”: with the notion of decimal fraction
(the tenth of a unit, the tenth of the tenth, …) as the starting point, he uses for them a
terminology that is typical of sexagesimal fractions in Hebraical mathematical literature:
“prime” fractions, “seconds”, “thirds”, etc. Then he extends this terminology to the
powers of ten, and thus tens, hundreds and thousands become “prime” integers,
“seconds”, “thirds”,(…). To complete this unification, zero is the “name” that should be
given to the unit. Even when this last step is not made by Immanuel, this fact does not
prevent the methods of multiplication and division from working.95 These operations
require, respectively, addition and subtraction of “names” (more precisely, of the
numbers that are implicitly associated to them).
numbers are written down in alphabetic numerals, the two first columns are struck out and the decimal
fractions and positive powers of ten on the left are indicated by their “names” (orders or ranks): units,
prime fractions, second fractions, etc., and prime integers, second integers, etc. respectively. There are a
lot of operational mistakes in the original table. In his article, Tony Lévy indicates mistakes and
omissions of numbers, but here we have just kept the corrected part.
95
See [Lévy, 2003b, 286–290].
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Roshi Rashed has showed how the theory of decimal fractions was already developed
within the tradition of al-Karajī and as-Samaw’al.96 However, we still do not know
which were the sources used by Immanuel Bonfils. Tony Lévy considers that the
question of finding them is a complex one and he has reserved it for further research,
although he gives some clues of which could they be: works on sexagesimal fractions in
Hebraic astronomical literature or in Latin texts like the Liber Ysagogarum Alchorismi
(12th century) and the theory of decimal fractions that was already developed in the
12th century in the Arab East.97
Apart from the fact that numbers are written down upwards, Bonfils’ table clearly
reminds us of multiplication of polynomials in the diagrams of Ms. 71, specially in
some examples where we see how the partial results are placed and added (for instance,
in the second example that we explained in our previous article).98 However, when the
product of two coefficients is greater than ten in Bonfils’ chart, the number of tens must
be “carried” to be added to the next row. Furthermore, the diagrams in Ms. 71 contain
negative numbers (and thus require the use of the rules of signs) as well as fractional
coefficients.
It is very likely that the diagrams in Ms. 71 and those in the manuscript of Immanuel
Bonfils have their origin in an Arab, Hebraical or even Latin common source. However,
the author of the Catalan manuscript or whoever developed the diagrams could also
have used Immanuel’s work itself. The access to it, as in the case of Isaac ben
Salomon’s Epistle, was probably not too difficult for a scholar who had moved in
fifteenth-century Jewish intellectual circles. We must also have into account that a lot of
Jews from Southern France migrated to the land of the Crown of Aragon during the
fourteenth century because of the expulsions of 1290, 1306, 1322 and 1394.99 In fact,
Jewish communities of both the North-East of the Iberian Peninsula and Southern
France played a key role in the production of Hebraical mathematical literature between
the 11th and the 14th centuries.100
96
See [Rashed, 1978] (also in [Rashed, 1984, 93–145]).
See [Lévy, 2003b, 284, 294].
98
See [Docampo, 2006, 57]
99
See [Hinojosa, 2001, 320].
100
On the other hand, Abraham ibn 'Ezra (ca. 1089-1164), whose Book of number (Sefer ha-Mispar) was
carefully studied by Immanuel Bonfils [Lévy, 2003b, 294], was born in Toledo (or perhaps in Tudela,
Navarre). See [Lévy, 1998]. A recent research has shown certain connections between algebra that was
transmitted in Hebrew and algebra that circulated in Castilian in the late middle ages (see [Lévy, 2007]).
This again reminds us of the importance of considering those works in Hebrew that have been preserved
when studying mathematical literature in vernacular Iberian languages (and vice versa).
97
- 22 -
Conclusion
We have seen West Muslim, Latin, Italian and Hebrew treatises that could show us
which could have been the techniques that directly inspired the charts in Ms. 71. After
examining these texts, it seems reasonable to assume that the Jewish intellectual circles
of the Iberian Peninsula or the Provence are the most likely environment where the
direct sources we are looking for circulated. But wherever the direct influences could
come from, if we go further back in time it is difficult not to see al-Karajī’s tradition as
the mainstream that allowed these developments, with as-Samaw’al playing an essential
role.
However, several features are, as far as we know, only found in the diagrams of Ms.
71. One of them is that they are an integral technique that is applied to equations, and
not just polynomials. Another one is that they give an overview of the algebraic
operations that are performed in the solving process of a problem, some of them even
containing short commands in words and arithmetical operations that are performed into
the diagram itself. Finally, it must also be noted that no indication (a letter, an
abbreviation nor a number) indicates the denomination of each monomial, which is only
indicated by the position of its coefficient in the diagram.
The possibility that these diagrams were independently developed by the author of the
Catalan manuscript or any predecessor in the vernacular tradition of commercial
arithmetic can not be completely discarded. After all, these charts do not contain
negative powers of the unknown, nor divisions or root extractions of polynomials, and
involve relatively simple algebraic expressions. However, the use of the table
symbolism and the existence of previous works like those that we have seen in this
paper, makes this, in our opinion, a quite unlikely hypothesis.
The appearance of further studies on medieval algebra, together with the finding of
further sources, will perhaps allow us to find clear traces of the influence of the tradition
initiated by al-Karajī in the Muslim West, the Jewish circles, and Latin and abacus
traditions. As we have seen, the diagrams of coefficients in Ms. 71 could be a late
testimony of that influence in Europe.
- 23 -
Acknowledgements
First of all, I am very grateful to Marco Panza, Pascal Crozet and Ilana Wantemberg,
because their help has been crucial for me to write this paper.
I also would like to thank Angela Martí for her help with Latin texts and Tony Lévy
for giving me information on some points about Isaac ben Salomon’s treatise.
Julio Samsó, Roland Hissette, Uwe Burkheiser and Rafaella Franci kindly helped me
to get some of the references that have been necessary for this article. Many thanks are
also due to them.
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