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Arabic algebra derives its epistemic value not from proofs but from correctly performing calculations using coequal polynomials. This idea of ‘mathematics as calculation’ had an important influence on the epistemological status of European mathematics until the seventeenth century. We analyze the basic concepts of early Arabic algebra such as the unknown and the equation and their subsequent changes within the Italian abacus tradition. We demonstrate that the use of these concepts has been problematic in several aspects. Early Arabic algebra reveals anomalies which can be attributed to the diversity of influences in which the al-jabr practice flourished. We argue that the concept of a symbolic equation as it emerges in algebra textbooks around 1550 is fundamentally different from the ‘equation’ as known in Arabic algebra.
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A Conceptual Analysis of Early Arabic Algebra
Albrecht Heeffer
Center for Logic and Philosophy of Science
Ghent University, Belgium
Professional address:
Blandijnberg 2,
B-9000 Ghent, Belgium
+32 9 2643979
Second revision: 28/07/06
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1. Introduction...................................................................................................................... 3
2. Starting point .................................................................................................................... 4
2.1. Latin translations of al-Khwārizmī’s Algebra .................................................................4
2.2. Latin translations of other Arabic works ........................................................................5
3. The evolution of the concept of an unknown..................................................................5
3.1. The unknown in early Arabic algebra .............................................................................5
3.1.1. Arabic terminology......................................................................................................................... 5
3.1.2. The ambiguity of māl..................................................................................................................... 6
3.1.3. Conclusion...................................................................................................................................... 8
3.2. Multiple solutions to quadratic problems ........................................................................9
3.2.1. Two positive roots in Arabic algebra............................................................................................. 9
3.2.2. Speculation on the origin of double solutions ............................................................................. 10
3.2.3. Double solutions in the abacus tradition...................................................................................... 13
3.2.4. Double solutions disappearing from abacus algebra ...................................................................13
3.3. The unknown in the abacus tradition ............................................................................14
4. Operations on polynomials ............................................................................................15
4.1. The abacus and cossic tradition......................................................................................15
5. The symbolic equation as a novel concept ....................................................................16
5.1. The concept of an equation in Arabic algebra ..............................................................16
5.2. Operations on “equations” in early Arabic algebra .....................................................17
5.2.1. al-jabr............................................................................................................................................ 18
5.2.2. al-muqābala .................................................................................................................................. 25
5.2.3. al-radd and al-ikmāl...................................................................................................................... 26
5.3. Operations on equations in the abacus tradition ..........................................................26
6. Conclusion......................................................................................................................27
6.1. The expansion of arithmetical operators to polynomials .............................................27
6.2. The expansion of the number concept............................................................................27
6.3. Equating polynomial expressions ...................................................................................28
6.4. Operations on coequal polynomials................................................................................28
6.5. Expansion of arithmetical operators to equations. .......................................................29
6.6. Operations between equations ........................................................................................29
7. Epistemological consequences.......................................................................................29
8. Acknowledgments...........................................................................................................30
9. References....................................................................................................................... 30
9.1. Manuscripts cited.............................................................................................................30
9.2. Primary sources ...............................................................................................................31
9.3. Secundary literature ........................................................................................................31
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1. Introduction
The most common epistemology account of mathematics is based on the idea of apriorism.
Mathematical knowledge is considered to be independent of experience. The fundamental
argument for an apriorist assessment of mathematics is founded on the concept of a formal
proof. Truth in mathematics can be demonstrated by deductive reasoning within an axiomatic
system. All theorems derivable from the axioms have to be accepted solely on basis of the
formal structure. The great mathematician Hardy cogently formulates it as follows (Hardy
It seems to me that no philosophy can possibly be sympathetic to a mathematician which does not
admit, in one manner or another, the immutable and unconditional validity of mathematical truth.
Mathematical theorems are true or false; their truth or falsity is absolute and independent of our
knowledge of them. In some sense, mathematical truth is part of objective reality.
When some years later, Gödel proved that there are true statements in any consistent formal
system that cannot be proved within that system, truth became peremptory decoupled of
provability. Despite the fact that Gödel’s proof undermined the foundament of apriorism it
had little impact on the mainstream epistemological view on mathematics. Only during the
past decades the apriorist account was challenged by mathematical empiricism, through
influential works from Lakatos (1976), Kitcher (1984) and Mancosu (1996). These authors
share a strong believe in the relevance of the history of mathematics for an epistemology of
The apriorist view on mathematics has not always been predominant in western thinking. It
only became so by the growing influence of the Euclidean axiomatic method from the
seventeenth century onwards. With respect to algebra, John Wallis was the first to introduce
the axioms in an early work, called Mathesis Universalis, included in his Operum
mathematicorum (1657, 85). With specific reference to Euclid’s Elements, he gives nine
Axiomata, also called communes notationes, alluding to the function of symbolic rewriting.
From then on, the epistemological status of algebra was transformed into one deriving its truth
from proof based on the axiomatic method. Before the seventeenth century, truth and validity
of an algebraic derivation depended on correctly performing the calculations using an
unknown quantity. While Witgenstein was heavily criticized for his statement that “Die
Mathematik besteht ganz aus Rechnung” (Mathematics consists entirely of calculations),
(1978, 924; 468), his image of mathematics as procedures performed on the abacus, fits in
very well with pre-seventeenth-century conceptions of mathematical knowledge. Algebraical
problem solving consisted of formulating the problem in terms of the unknown and reducing
the form to one of the known cases. Early Arabic algebra had rules for each of six known
cases. While geometrical demonstrations exist for three quadratic types op problems, the
validity of the rules was accepted on basis of their performance in problem solving.
The idea that European mathematics has always been rooted in Euclidean geometry is a myth
cultivated by humanist writings on the history of mathematics. In fact, the very idea that
Greek mathematics is our (Western) mathematics is based on the same myth, as argued by
Jens Høyrup (Høyrup 1996, 103):
According to conventional wisdom, European mathematics originated among the Greeks between
the epochs of Thales and Euclid, was borrowed and well preserved by the Arabs in the early
Middle Ages, and brought back to its authentic homeland by Europeans in the twelfth and
thirteenth century. Since then, it has pursued its career triumphantly.
Høyrup shows that “Medieval scholastic university did produce an unprecedented, and hence
specifically European kind of mathematics” (ibid.). But also outside the universities, in the
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abacus schools of Florence, Siena and other Italian cities, a new kind of mathematics
flourished supporting the practical needs of merchants, craftsman, surveyors and even the
military man.
Symbolic algebra, the Western mathematics par excellence, emerged from algebraic practice
within this abacus tradition, situated broadly between Fibonacci’s Liber Abbaci (1202) and
Pacioli’s Summa (1494). Practice of algebraic problem solving within this tradition grew out
of Arabic sources. The epistemic foundations of a mathematics-as-calculation was formed in
the Arab world. An explicitation of these foundations is the prime motivation of our analysis
of the basic concepts of early Arabic algebra.
2. Starting point
While the original meaning of the Arabic concepts of algebra will be an important guideline
for this study, we relinquish the search for the “exact meaning”. Several scholars have
published studies on the origin of the term algebra, the meaning of al-jabr and al-muchābala
and the Arabic terms for an unknown. Some have done so with the aim of establishing the
correct meaning with the aid of Arabic etymology and linguistics (e.g. Gandz 1926, Saliba
1972, Oaks and Alkhateeb 2005). Strictly taken, the precise meaning of these Arabic terms
and concepts is irrelevant for our study. Even if there would be one exact meaning to be
established, it was not available for practitioners of early algebra in Europe. With a few
exceptions, such as Fibonacci,1 the flourishing of algebraic practice within the abacus
tradition depended on a handful of Latin translations and vernacular interpretations or
rephrasing of these translations. Unquestionably, certain shifts in meaning took place within
the process of interpretation and diffusion during the twelfth and thirteenth centuries. Rather
than the Arabic terms and concepts, the concepts conveyed by the first Latin translations will
be our starting point.
2.1. Latin translations of al-Khwārizmī’s Algebra
Three Latin translations of al-Khwārizmī’s Algebra are extant in sixteen manuscripts (Hughes
1982). These translations have been identified as from Robert of Chester (c. 1145), Gerard of
Cremona (c. 1150) and Guglielmo de Lunis (c. 1215), although there is still discussion
whether the latter translation was Latin or Italian. What became available to the West was
only the first part of al-Khwārizmī’s treatise. The second part on surveying and the third on
the calculation of legacies were not included in these Latin translations. The full text of the
Algebra became first available with the edition of Frederic Rosen (1831) including an English
translation. Rosen used a single Arabic manuscript, the Oxford, Bodleian CMXVIII Hunt.
214, dated 1342. The value of his translation has been questioned by Ruska (1917), Gandz
(1932, 61-3) and Høyrup (1998, note 5). Some years later Guillaume Libri (1838, Note XII,
253-299) published a transcription of Gerard’s translation from the Paris, BNF, Lat. 7377A,
an edition that has been qualified as ‘faulty’ and corrected on eighty accounts by Hughes
(1986, 211, 231). Later during the century, Boncompagni (1850) also edited a Latin
translation from Gerard, but it was later found that this manuscript was not Gerard’s but
Guglielmo de Lunis’ (Hughes 1986). Robert of Chester’s translation was first published with
an English translation by Karpinski (1915). However, Karpinski used a manuscript copy by
Scheubel, which should be seen more as a revision of the original.
It is only during the past decades that critical editions of the three Latin translations have
become available. The translation by Gerard of Cremona was edited by Hughes (1986), based
on seven manuscript copies. Hughes (1989) also published a critical edition of the second
translation from Robert of Chester based on the three extant manuscripts. A third translation
has been edited by Wolfgang Kaunzner (1986). Although this text (Oxford, Bodleian, Lyell
52) was originally attributed to Gerard, it is now considered to be a translation from
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Guglielmo de Lunis (Hughes 1982, 1989). An Italian translation of 1313 from the Latin is
recently published by Franci (2003). It has been argued by several scholars that Gerard of
Cremona’s translation is the best extant witness of the first Arabic algebra (Høyrup 1998).
2.2. Latin translations of other Arabic works
Apart from al-Khwārizmī’s Algebra there have been Latin translations of other works which
contributed to the diffusion of Arabic algebra. The Liber algorismi de pratica arismetrice by
John of Seville (Johannes Hispalensis) precedes the first Latin translations and briefly
mentions algebra (Boncompagni 1857, 112-3). Also of importance is Abū Bakr’s Liber
mensurationum, translated by Gerard of Cremona in the twelfth century (Busard, 1968).
Although this work deals primarily with surveying problems it uses the methods as well as the
terminology of the early Arabic jabr tradition. Jens Høyrup, who named the method “naive
geometry” or “the tradition of lay surveyors”, has pointed out the relation between this work
and Babylonian algebra (Høyrup, 1986, 1990, 1998, 2002). Following Busard, he has
convincingly demonstrated that the operations used to solve these problems are concretely
geometrical. Therefore this work can help us with the interpretation of operations in early
Arabic algebra.
The Algebra of Abū Kāmil was written some decades after that of al-Khwārizmī and bears the
same title Kitab fī al-jābr wa’l-muqābalah. Several versions of the manuscript are extant. An
Arabic version MS Kara Mustafa Kütübhane 379 in Istanbul; a fourteenth-century copy of a
Latin translation at the BNF at Paris, Lat. 7377A, discussed with partial translations by
Karpinski (1914) and published in a critical edition by Sesiano (1993) who attributes the Latin
translation to Guglielmo de Lunis (1993, 322-3). A fifteenth-century Hebrew version with a
commentary by Mordecai Finzi, is translated in German by Weinberg (1935) and n English by
Levey (1966). Levey also provides an English translation of some parts of the Arabic text.
Other texts include Ibn Badr’s Ikhtisār al-jabr wa’l-muqābala which was translated into
Spanish (Sánchez Pérez, 1916) and al-Karajī’s Fakhrī fī al-jābr wa’l-muqābalah with a partial
French translation (Woepcke, 1853).
3. The evolution of the concept of an unknown
3.1. The unknown in early Arabic algebra
The unknown is used to solve arithmetical or geometrical problems. The solution commences
with posing an unknown quantity of the problem as the abstract unknown. By analytical
reasoning using the unknown, one arrives at a value for it. In algebraic problem solving before
Arabic algebra, the abstract unknown is not always the symbolic entity as we now understand.
As an essential part of the analytical reasoning, it is an entity related to the context of the
problem and the model used for problem solving. For Babylonian algebra, it is shown by
Høyrup (2002) that the model was a geometrical one. The unknown thus refers to geometrical
elements such as the sides of a rectangle or a surface. In Indian algebra we find the unknown
(or unknowns) used for monetary values or possessions as in the rule of gulikāntara
(Colebrooke 1817, 344). The terms used in Arabic algebra reflect both the geometrical
interpretation of the unknown as well as the one of a possession. We will argue that the
difficulties and confusions in the understanding of the concept of the Arabic unknown are
induced by diverse influences from Babylonian and Indian traditions.
3.1.1. Arabic terminology
The central terms in Arabic algebra are māl, shay’ and jidhr. In addition, the monetary unit
dirham is also used in problems and in their algebraic solutions. It is generally accepted that
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the term māl refers to possession, or wealth or even a specific sum of money. The shay’ is
translated as ‘thing’ ever since the first commentators wrote about it (Cossali 1797-9). From
the beginning, shay’ was considered the unknown (Colebrooke 1817, xiii).
The difficulties of interpretation arise when we translate māl by ‘square’ and shay’ by ‘root’.
Rosen (1831) and Karpinski (1915) both use ‘square’ for māl on most occasions. Karpinski
even uses the symbolic x2. However, when the problem can be stated without the use of a
square term, they both change the interpretation of the māl. For example in problem III.11,
Rosen uses ‘number’ and Karpinski employs x instead of x2 as used for the other problems.2
This already contributes to the confusion as the Latin translation uses the same word in both
cases. Moreover the choice of the word ‘square’ is misleading. Neither the geometrical
meaning of ‘square’, nor the algebraical one, e.g. x2, are adequate to convey the meaning of
māl.3 For the geometrical problems, al-Khwārizmī elaborates on the use of māl for the
algebraic representation of the area of a geometrical square. If the meaning of māl would be a
square, why going through the argumentation of posing māl for the area?4 The algebraic
interpretation of a square is equally problematic. If māl would be the same as the square of the
unknown then jidhr or root would be the unknown. However, this is in contradiction with the
original texts in which māl, if not the original unknown by itself, is at least transformed into
the unknown. Høyrup (1998, 8) justly uses the argument that māl is used in linear problems in
al-Karajī’s Kāfī (Hochheim 1878, iii, 14). This corresponds with the use of a possession in
Hindu algebra, in formulating algebraic rules for linear problems, such as the gulikāntara.
māl shay’ jidhr dirham ‘adad mufrat
ﺪﺪﻋ ﺪﺮﻔﻤ
Hispalensis (none) res radix (none) numerus
Robert substancia res radix dragma numerus
Gerard census res radix dragma numerus simplex
Guglielmo census res radix dragma numerus
Abū Kāmil (latin) census res radix dragma numerus simplex
Table 1: The terms used in early Latin translations of Arabic texts
3.1.2. The ambiguity of māl
The interpretation of māl as the unknown, pure and simple, is not as straightforward as often
presented. While māl (in Robert’s translation substancia and in Gerard’s census) is used to
describe the problem, the algebraic derivation depends on operations on other terms than the
original ‘possession’. Also Hughes points out the problem in his commentary of Robert of
Chester’s edition:5
Terminology also must have jolted Robert’s readers. In problems four and six of Chapter I and in
five, ten, and thirteen of Chapter II, substancia in the statement of the initial equation becomes res
or radix in its solution. Excursions such as these must have challenged the reader”
Let us look more closely at problem III.13, as it is instructive to point out what constitutes a
transformation in the original concept of māl:
Karpinski 1930, 118 Hughes 1989, 61
I multiply a square by two-thirds of
itself and have five as a product.
Explanation. I multiply x by two-thirds
x, giving 2/3 x2, which equals five.
Substanciam in eius duabus terciis sic
multiplico, ut fiant 5. Exposicio est, ut rem in
duabus terciis rei multiplicem, et erunt 2/3
unius substancie 5 coequancia. Comple ergo
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Complete 2/3 x2 by adding to it one-
half of itself, and one x2 is obtained.
Likewise add to five one-half of itself,
and you have 7 1/2, which equals x2.
The root of this, then, is the number
which when multiplied by two-thirds of
itself gives five.
2/3 substancie cum similitudine earum medii,
et erit substancia. Et similiter comple 5 cum
sua medietate, et erit habebis substanciam vii
et medium coequantem. Eius ergo radix est res
que quando in suis duabus terciis multiplicata
feurit, ad quinarium excrescet numerum.
Substancia here is used in the problem text as well as the solution. But clearly it must have a
different meaning in these two contexts. In the beginning of the derivation substancia is
replaced by res. In the English translation, Karpinski switches from ‘square’ to x. By
multiplying the two res terms, x and 2/3x, two thirds of a new substancia is created. This
second substancia is an algebraic concept where the first one, in the problem text, is a
possession and may refer to a sum of money. While Gerard of Cremona uses census instead of
substancia, his translation has the same ambiguity with regard to census.6 The root of real money
This anomaly of Arabic algebra is discussed now for almost two centuries. Libri (1838),
Chasles (1841, 509), and others have noticed the problem. Some have chosen to ignore it
while others pointed out the inconsistency, but did not provide any satisfactory answer. Very
recently, two analyses have reopened the discussion. In the yet to be published Høyrup (2006)
and Oaks and Alkhateeb (2005) the double meaning of the māl is prominently present in their
interpretation of early Arabic algebra.7 Høyrup (2006) adequately describes the anomaly as
“the square root of real money”. As māl or census originally is understood as a possession,
and the unknown is designated by shay’ or res, which is the root of the census, problems
looking for the value of a possession thus deal with the root of real money when they use the
shay’ in their solution. According to Høyrup the difference between the two was already a
formality for al-Khwārizmī. Abū Kāmil towards a resolution of the ambiguity
We find the anomaly also in the algebra of Abū Kāmil, almost a century later. But Abū Kāmil
is the first to point out that the transformation of a value or possession into an algebraic
quantity is an arbitrary choice. His double solution to problem 52 is very instructive in this
respect. The problem commences as follows (translation from the Arabic text, f. 48v; Levey
1966, 164, note 167):
If one says to you that there is an amount [māl] to which is added the root of its ½. Then the sum is
multiplied by itself to give 4 times the first amount. Put the amount you have equal to a thing and
to it is added the root of its ½ which is a thing plus the root of ½ a thing, (then multiply it by itself)
[sic]. It gives a thing plus the root of ½ a thing. Then one multiplies it by itself to give a square plus
½ a thing plus the root of 2 cubes [ka‘abin, a dual of ka‘ab] equal to 4 things.
The Latin translation makes the anomaly apparent (Sesiano 1993, 398, 2678-2683):
Et si dicemus tibi: Censui adde radicem medietatis eius; deinde duc additum in se, et provenie[n]t
quadruplum census. Exemplum. Fac censum tuum rem, et adde ei radicem medietatis eius, et
[prov-] erunt res et radix ½ rei. Que duc in se, et provenie[n]t census et ½ rei et radix 2 cuborum,
equales 4 rebus.
In symbolic representation the solution depends on:
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As is common, the translator uses census for the possession or amount of money in the
problem formulation. The solution starts by stating literally ‘make from the census your res
(“Fac censum tuum rem”) which could easily be misinterpreted as “make x from x2”. In the
rest of the solution, res is used as the unknown.
Abū Kāmil adds a second solution: “You might as well use census for the possession”, he
reassures the reader (Sesiano 1993, 399, 2701-2705),
Et, si volveris, fac censum tuum censum, et adde ei radicem medietatis ipsius, et erunt census et
radix medietatis census, equales radici 4 cens[ibus]uum, [et] quia di[x]cis: “Quando ducimus
e[umJa in se, [erit] proveniet quadruplum census”. Est ergo census et radix ½ census, equales radici
4pli censu[um]s. Et hoc est 2 res.
Here, the symbolic translation would be:
22 2
The census is now used for the possession. But there is still a difference between the census of
the problem formulation and the census of the problem solution. “Fac censum tuum censum”
should here be understood as “put the amount you have equal to the square of a thing”. What
Abū Kāmil seems to imply by providing alternative solutions to a single problem, is that there
are several ways to ‘translate’ a problem into algebraic form. The possession in the problem
text is not necessarily the unknown. You can use the unknown for the possession, but you
might as well use the square of the unknown. In the abacus tradition from the thirteenth to the
sixteenth century, this freedom of choice was highly convenient for devising clever solutions
to problems of growing complexity. The ambiguity in the concept of māl, by many
understood as a nuisance of Arabic algebra, could have facilitated the conceptual advance to
the more abstract concept of an algebraic quantity.
3.1.3. Conclusion
There is definitely an anomaly with the original concept of an unknown in early Arabic
algebra. One the one hand, māl is used as the square term in quadratic problems of the type
māl and roots equal number’ such as the prototypical case four from al-Khwārizmī
210 39xx+=
Early Arabic algebra provides procedures for problems which can be reduced to one of the six
standard types. On the other hand, māl is also used for describing the quantity of a problem,
mostly a sum of money or a possession. Possibly, at some time before al-Khwārizmī’s
treatise, these two meanings were contained in a single word and concept. As problems
dealing with possessions were approached by algebraic method from the al-jabr tradition, a
transformation of the concept māl became a necessity. We notice in al-Khwārizmī’s Algebra
and all the more in that of Abū Kāmil, a shift towards māl as an algebraic concept different
from a possession or a geometrical square. The confusion and discontent expressed by several
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twentieth-century scholars with terminology in early Arabic algebra stems from a failure to
see the conceptual change of the māl.
We do not know much about the origin of the al-jabr tradition, preoccupied with quadratic
problems and their ‘naive’ geometric demonstrations. Jens Høyrup (1994, 100-2) speculates
on a merger of two traditions. The first is the class of calculators employing the hisāb for
arithmetical problem solving. The second stems from the tradition of surveyors and practical
geometers, going back to Old Babylonian algebra. We would like to add the possible
influence from Hindu algebra. While the al-jabr tradition is definitely different from the
Indian one in methods and conceptualization, the type of problems dealing with possessions
are likely to have been imported from the Far East. The ambiguities within the concept of māl
reflects the variety of influences.
3.2. Multiple solutions to quadratic problems
A second particularity of Arabic algebra is the acceptance of double solutions for one type of
quadratic problems. The recognition that every quadratic equation has two roots is generally
considered as an important conceptual advance in symbolic algebra. We find this insight in
the mostly unpublished works of Thomas Harriot of the early seventeenth century. More
influential in this respect, is Girard’s Invention Nouvelle en Algebre, published in 1629.
However, it is less known that early Arabic algebra fully accepted two positive solutions to
certain types of quadratic problems. It is significant that this achievement of Arabic algebra
has largely been neglected during the abacus tradition, while it might have functioned as a
stepping stone to an earlier structural approach to equations. We believe there is an
explanation for this, which is related to the concept of an unknown of the abacus masters. Let
us first look at the first occurrence of double solutions in early Arabic algebra.
3.2.1. Two positive roots in Arabic algebra
Two positive solutions to quadratic problems are presented in al-Khwārizmī’s fifth case of the
quadratic problems of “possession and number equal to roots”. This problem, in symbolic
form, corresponds with the normalized equation
221 10
x+= .
al-Khwārizmī talks about addition and subtraction leading to two solutions in the following
rule for solving the problem:
From Robert’s translation
(Hughes 1989, 34):
The rule from the Arabic manuscript
(Rosen 1931, 42):
Primum ergo radices per medium
dividas et fient 5. Eas ergo in se
multiplica et erunt 25. Ex hiis ergo
21 diminuas quem cum substancia
iam pretaxauimus, et remanebunt 4.
Horum ergo radicem accipias id est
2, que ex medietate radicum id est 5
diminuas et remanebunt tria, vnam
radicem huius substancie constituen-
cia, quam scilicet substanciam
novenus complet numerus. Et si
volueris ipsa duo que a medietate
radicum iam diminuisti, ipsi
When you meet with an instance which refers you to
this case, try its solution by addition, and if that do not
serve, then subtraction certainly will. For in this case
both addition and subtraction may be employed, which
will not answer in any other of the three cases in which
the number of the roots must be halved. And know,
that, when in a question belonging to this case you
have halved the number of the roots and multiplied the
moiety by itself, if the product be less than the number
of dirhems connected with the square, then the instance
is impossible; but if the product be equal to the dirhems
by themselves, then the root of the square is equal to
the moiety of the roots alone, without either addition or
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medietati id est 5 ad 20 dicias, et
fient 7.
The procedure thus corresponds with the following formula:
1,2 22
=± −
⎝⎠ .
al-Khwārizmī states that the problem becomes unsolvable when the discriminant becomes
negative. When the square of b/2 equals the number (of dinars) there is only one solution
which is half the number of roots. The gloss in Gerard’s translation of problem VII.1 gives a
geometric demonstration with the two solutions. This problem from al-Khwārizmī is also
treated by Abū Kāmil (Karpinski 1914, 42-3; Sesiano 1993, 330-6). A lesser known Arabic
manuscript, which most likely predates al-Khwārizmī, also has the geometric demonstration
with double solutions (Sayili 1985, 163-5).
Chasles (1841, 504) mentions a Latin translation of Gerard (Paris, BNF, anciens fonds 7266)
from a treatise on the measurement of surfaces, by an Arab called Sayd. A problem of the
same type, corresponding with the symbolic equation
is solved by addition and subtraction (“Hoc namque est secundum augmentum et
diminutionem”), referring to the values x = 2 + 1 and x = 2 – 1, resulting in the double
solution x = 3 and x = 1.
In conclusion: double positive solutions to one type of quadratic problems were fully accepted
in the earliest extant sources of Arabic algebra.
3.2.2. Speculation on the origin of double solutions
Dealing with quadratic problems, Diophantus never arrives at double solutions. If the problem
has two positive solutions, he always finds the larger one (Nesselmann 1842, 319-21; Tropfke
1933-4, 45). So, where do the double solutions of Arabic type V problems originate from? If
not from Greek descent, the most likely origin would be Hindu algebra. However, Rodet
(1878) was the first to critically investigate the possible influence of Hindu sources on Arabic
algebra. One of his four arguments against such lineage is the difference in approach to
double solutions of the quadratic equation. As the Hindus accepted negative values for roots
and numbers they had one single format for complete quadratic equations, namely
ax bx c±=±
whereas the Arabs had three types. The Hindu procedure for solving complete quadratic
problems accounts for double solutions as stated by Bhāskara (and his predecessors): 8
If the root of the absolute side of the equation be less than the number, having the negative sign,
comprised in the root of the side involving the unknown, then putting it negative or positive, a two-
fold value is to be found of the unknown quantity: this [holds] in some cases.
The “root of the absolute side of the equation” refers to the ± c. The Hindu procedure to find
the roots of a quadratic equation can be illustrated by the following example (Bhāskara stanza
139; Colebrooke 1817, 215-6):
- 11 -
The eighth part of a troop of monkeys, squared, was skipping in a grove and delighted with their
sport. Twelve remaining were seen on the hill, amused with chattering to each other. How many
were they in all?
Using the unknown ya 1 for the number of monkeys, Bhāskara solves the problem as follows:
ya v ya ru
ya v ya ru
literally transcribed:
10120 0
+= ++
164 0
0 0 768
ya v ya ru
ya v ya ru
bringing to the same denominator:
264 768xx−=
making the left side a perfect square:
32 256x−=
ya ru
ya ru
extracting the root results in:
32 16x
The two solutions thus become x = 48 and x = 16.
In the next problem the acceptance of two solutions is more challenging (Bhāskara stanza
140; Colebrooke 1817, 216):
The fifth part of the troop [of monkeys] less three, squared, had gone to a cave; and one monkey
was in sight, having climbed on a branch. Say how many they were.
This leads to the equation:
155 0
0 0 250
ya v ya ru
ya v ya ru
literally transcribed:
1 55 0 0 0 250xx xx−+=+
with solutions x = 50 and x = 5. Bhāskara has some reservations about the second solution
because one fifth of five minus three becomes negative.
The very different approach towards quadratic problems and the acceptance of negative roots
in Hindu algebra makes it an improbable source for the double solutions of type V problems
in Arabic algebra.
If not from Greek or Indian origin, there is only one candidate left. Solomon Gandz, in an
extensive, and for that time, exhaustive comparison of solutions to quadratic problems from
Babylonian, Greek and Arabic origin concluded (Gandz 1937, 543):
Greek and Arabic algebra are built upon the rock of the old Babylonian science and wisdom. It is
the legacy of the old Babylonian schools which remain the very foundation and cornerstone of both
the Greek and Arabic systems of algebra. The origin and early development of the science cannot
be understood without the knowledge of this old Babylonian legacy.
- 12 -
Although the relation should now be qualified and differentiated more cautiously, recent
studies, such as the groundbreaking and novel interpretation by Høyrup (2002) endorse some
line of influence. If we look again at the type V problem from al-Khwārizmī, the resulting
221 10
which is given in its direct form, corresponds remarkably well with a standard type of
problem from Babylonian algebra:
The important difference between the two is that Babylonian algebra uses a geometrical
model for solving problems. The two parts a and b are represented as the sides of a rectangle
ab and they function as two unknowns in the meaning we have defined elsewhere.9 Arabic
algebra uses geometry only as a demonstration of the validity of the rules and its analytic part
is limited to reducing a problem to one of the standard forms using a single unknown. al-
Khwārizmī systematically uses the unknown for the smaller part. Thus in problem VII he
proceeds as follows (de Lunis; Kaunzner 1989, 78):
Ex quarum unius multiplicatione per alteram 21 proveniant. Sit una illarum res, altera 10 minus re,
ex quarum multiplicatione proveniunt 10 res minus censu, que data sunt equalia 21. Per
restaurationem igitur diminuti fiunt 10 res censui ac 21 equales ecce quintus modus, resolve per
eum et invenies partes 3 et 7.
al-Khwārizmī multiplies x with 10 – x, with
value 21. After “restoration” this leads to the
standard form of the equation above. While the
rule for type V prescribes trying addition first
and then subtraction (in the Robert translation),
the solutions arrived at here are 3 first and then
7. We believe that the recognition of two
solutions to this type of quadratic problem is a
direct relic of the Babylonian solution method.
Although the geometric proof for this problem,
present in the Arabic texts and the three Latin
translations, does not correspond with any
known Babylonian tablets, some of al-
Khwārizmī’s geometric demonstrations ought
to be placed within the surveyor’s tradition
which descends in all probability from Old
Babylonian algebra.
Høyrup (2002, 412-4) points out that al-
Khwārizmī’s provides two rather different
geometrical demonstrations to the case
“possessions and roots equal number”. Only
one corresponds with the procedure described
- 13 -
in the text. The other, shown in Figure 1,
corresponds remarkably well with the
Babylonian table BM 13901, nr. 23.
Figure 1: A geometrical demonstration by
al-Khwārizmī (from Rosen 1841, 10).
According to Høyrup, al-Khwārizmī’s proof must have been derived from this tradition. This
way of demonstrating may then have been more familiar than the al-jabr itself.
3.2.3. Double solutions in the abacus tradition
We continue to find double solutions in the early abacus tradition. The first vernacular algebra
by Jacopa da Firenze (1307) mentions double solutions to the fifth type, both in the rules and
in the corresponding examples. Maestro Dardi (1344, van Egmond 1983), in an extensive
manuscript some decades later, continues to account for double solutions (Franci 2001, 83-4).
Significantly, he leaves out the second positive solution for the geometrical demonstration of
221 10
x+= which is copied from the Arabic texts. Later treatises gradually drop the second
solution for this type of problem. For example, the anonymous Florence Fond. Prin. II.V.152,
later in the fourteenth century, has an intermediate approach. The author writes that:
In some cases you have to add half the number of cosa, in others you have to subtract from half the
number of cosa and there are cases in which you have to do both.
However, when applying the rule to an example with two positive solutions, he proceeds to
perform only the addition.10 For the equation
he gives the solution x = 9 and does not mention the second root x = 1. Also Maestro Biagio
mentions addition and subtraction in his sixth rule but only applies the addition operation, as
in problem 3 where two positive solutions are possible (Pieraccini 1983, 3).
Later abacus masters abandon the second solution altogether. For example the Riccar. 2263
gives only one solution to the problem 10, 22ab ab
== (Simi 1994, 33). Pacioli only uses
addition for the fifth case of the quadratic problems (Pacioli 1494, 145). Maestro Gori, in the
early sixteenth century, generalizes his rules to a form where the powers of the unknown are
relative to each other. The Arabic rule V corresponds with his rule 4 in which “one finds three
terms in continuous proportion of which the major and the minor together equal the middle
one” (Siena L.IV.22, f. 75r; Toti Rigatteli 1984, 16). This corresponds with the equation type
ax c bx+=
Here Gori is in complete silence about a second possible solution, in the explanation of the
rule, as well as in the examples given.
3.2.4. Double solutions disappearing from abacus algebra
Why do we see these double solutions for quadratic problems fading away during algebraic
practice in the abacus tradition? It could be interpreted as an achievement of Arabic algebra
which becomes obscure in vernacular writings. In our understanding, the abandonment of
double solutions has to be explained through the rhetorical structure employed by abacus
writers. The strict, repetitive and almost formalized structure of the problem solution text is a
striking feature of many of the algebraic manuscripts in Italian libraries. The solution always
starts with a hypothetical reformulation of the problem text by use of an unknown. For
example, Gori, as an illustration of the rule cited above, selects a division problem of ten into
- 14 -
two parts with certain conditions given. The solution commences in the typical way “suppose
that the smaller part equals one cosa” (“pongho la minor parte sia 1 co.”, ibid. p. 17). One
particular value of the problem is thus represented by the unknown. The unknown here is no
indeterminate as in later algebra; it is an abstract representation for one specific quantity of
the problem. Given that this recurring rhetoric structure, which is so important for the abacus
tradition, commences by posing one specific value, it makes no sense to end up with two
values for the unknown. For the type of division problems which have descended from
Babylonian algebra the quadratic expression leads to the two parts of the division. However,
if one starts an argumentation stating that the cosa represents the smaller part, one does not
expect to end up with the value of the larger part. The concept of an unknown in the abacus
tradition is closely connected with this rhetorical structure in which the choice of the
unknown excludes double solutions by definition.
3.3. The unknown in the abacus tradition
With al-Khwārizmī’s treatise and more so with Abū Kāmil’s Algebra, the unknown became a
more abstract concept, independent of a geometrical interpretation. While the unknown in one
type of quadratic problems allowed for double solutions, this was gradually reduced to a
single value through the rhetorical structure of abacus treatises. Let us now summarize the
development of the concept unknown within the abacus tradition.
The ambiguity of the māl was carried over, to some degree, from the Arabic texts to the
abacus tradition by Fibonacci. Høyrup (2000, 22-3) has pointed out the inconsistent use of
Latin words for shay’ and māl by Fibonacci.11 For most of the algebra part, Fibonacci uses the
res and census terminology of Gerard of Cremona. However, in the middle of chapter 15 he
switches from census to avere for māl (Sigler 2002, 578-601). For Høyrup this is an
indication that vernacular treatises may have been circulating around 1228, the time of the
second edition of the Liber abbaci. The Milan Ambrosiano P 81 sup, (fols. 1r-22r) is a later
revision of Gerard’s translation. Here the author uses cosa for res (Hughes 1986, 229). While
this manuscript is probably of later origin, the use of the vernacular cosa rather than census or
res is characteristic for the abacus tradition. With the first vernacular algebra extant, by
Jacopa da Firenze in 1308, the use of cosa removed most of the original ambiguities. Where
the conversion from the māl as a possession to the māl as an algebraic entity will have defied
the student of Arabic algebra, the vernacular tradition eliminated these difficulties. When
Jacopa provides the solution to a problem on loan interest calculation he commences as
follows (Høyrup 2000, 30):
Fa così: pone che fusse prestata a una cosa el mese de denaro, sì che vene a valere l’anno la libra 12
cose de denaro, che 12 cose de denaro sonno el vigensimo de una libra, sì che la libra vale l’anno
1/20 [de cosa] de una libra.
By posing that the loan was lent at one cosa in denaro a month, the calculation can be done in
libra leading to a quadratic equation with a standard solution. The rhetorical structure of the
solution text starts from a conversion of a quantity of the problem, in this case the number
denari lent, to an unambiguous unknown cosa. Reformulating the problem in terms of the
unknown, the problem can be solved by reducing the formulation to a known structure. In this
case to censi and cosa equal to numbers.
This was basically the function and meaning of the cosa for the next two centuries within the
abacus tradition. In all, the notion of the unknown in the abacus tradition was fairly constant
and unproblematic.
- 15 -
4. Operations on polynomials
Most current textbooks on the history of algebra consider operations on polynomial
expressions as natural to a degree that they do not question the circumstances in which these
operations emerged. This is rather peculiar as most algebra textbooks, from the late abacus
tradition onwards, explain these operations at length in their introduction. Focusing on the
operations which have led to the formation of new concepts, we consider operations on
polynomials crucial in the understanding of the equation as a mathematical concept. A
possible reason for this neglect of conceptual innovations is the structural equivalence of
algebraic operations with arithmetical or geometrical ones.
al-Khwārizmī (c. 850) introduces operations on polynomials in the Arabic version of his
Algebra after the geometrical proofs and before the solution to problems. Strangely, he treats
multiplication first, to be followed by a section on addition and subtraction, and he ends with
division. Algebraic and irrational binomials are discussed interchangeably. Geometrical
demonstrations are provided for the irrational cases. This order is followed in the three Latin
translations. Abū Kāmil in his Algebra (c. 910) extends the formal treatment of operations on
polynomials from al-Khwārizmī with some geometrical demonstrations and some extra
examples, and moves division of surds to the first part. al-Karkhī (c. 1000) improves on the
systematization, but still follows the order of multiplication, division, root extraction, addition
and subtraction. He treats surds after algebraic polynomials.
Hindu and Arabic treatments of operations on polynomials differ too widely to suspect any
influence from either side. The order of operations and the way negative terms are treated are
systematically dissimilar in both traditions. Nonetheless, there is the historical coincidence in
the introduction of operations on polynomials in two dispersed traditions.
4.1. The abacus and cossic tradition
Although Fibonacci’s algebraic solutions to problems use operations on polynomials
throughout chapter 15, he does not formally discuss the subject as known in Arabic algebra.
Typically, such preliminaries are skipped in early abacus writings and the authors tend to
move directly to their core business: problem solving. A formal treatment of operations on
polynomials is found gradually from the fourteenth century onwards.
Maestro Dardi in his Aliabraa argibra commences his treatise with an extensive section
dealing with operations on surds (1344, Siena I.VII.17, fols. 3v-14r; Franci 2001). A short
paragraph deals with the multiplication of algebraic binomials in between the geometrical
demonstrations and the problems (ibid. f. 19v). This is the location where we found the subject
in al-Khwārizmī’s Algebra. As far as we know, the anonymous Florence Fond. prin. II.V.152
dated 1390, is the first abacus text which has a comprehensive treatment on the multiplication
of polynomials (ff. 145r-152r; Franci and Pancanti 1988, 3-44). It provides numerous
examples with binomials and trinomials, including roots and higher powers of the unknown.
Some curious examples are
a form including a zero term we are familiar with from Hindu algebra, and the complex form
5432 5432
666666666666xxxxx xxxxx+++++ +++++
Still, the examples are limited to multiplying polynomials.
During the fourteenth century, such introduction becomes more common and with the
anonymous Modena 578 (1485, van Egmond 1986) we find a more systematic treatment of
- 16 -
the addition, subtraction and multiplication of unknowns and polynomials. Finally, Pacioli
(1494) raises the subject to the level of an algebra textbook.
5. The symbolic equation as a novel concept
5.1. The concept of an equation in Arabic algebra
Because the following paragraphs will deal with operations on equations, we have to make
clear what the meaning is of an equation in early Arabic algebra. In fact, there are no
equations in Arabic algebra as we currently know them. However, some structures in Arabic
algebra can be compared with our prevailing notion of equations. Many textbooks dealing
with the history of algebraic equations go back to Babylonian algebra. So, if there are no
equations in Arabic algebra, what are they talking about? Let us therefore try and interpret the
concept within Arabic algebraic treatises more positively.
Some basic observations on early Arabic algebra should not be ignored:
The Latin translations do not talk about equations but about rules for solving certain
types of quadratic problems. This terminology is used throughout: “the first rule”,
“demonstration of the rules”, “examples illustrating the rules”, “applying the fourth
rule”, etc. Apparently, these rules can be transformed directly into symbolic equations,
but this is true for many other rules which cannot even be considered algebraic, such
as medieval arithmetical solution recipes.12
There is no separate algebraic entity in al-Khwārizmī’s treatise which corresponds
with an equation. The closest we get to an entity are “modes of equating” or “the act
of equating”, referring to actions, not to a mathematical entity. The best way to
characterize a mathematical entity is by the operations which are allowed on it. In
early Arabic algebra there are no operations on equations. On the other hand, there are
operations on polynomials. al-Khwārizmī has separate chapters on these operations.
Early Arabic algebra is preoccupied with quadratic problems. Although linear
problems are later approached algebraically by al-Karkhī, no rules are formulated for
solving linear problems, as common in Hindu algebra. Therefore, if we consider the
rules for solving quadratic problems equations, then there is no analogous case for
linear problems.
The correct characterization of the Arabic concept of an equation is the act of keeping related
polynomials equal. Guglielmo de Lunis and Robert of Chester have a special term for this:
coaequare. In the geometrical demonstration of the fifth case, de Lunis proves the validity of
the solution for the “equation”
221 10
The binomial 221x
is coequal with the monomial 10
, as both are represented by the
surface of a rectangle (Kaunzner 1989, 60):
Ponam censum tetragonum abgd, cuius radicem ab multiplicabo in 10 dragmas, quae sunt latus be,
unde proveniat superficies ae; ex quo igitur 10 radices censui, una cum dragmis 21, coequantur.
Once two polynomials are connected because it is found that their arithmetical value is equal,
or, in this case, because they have the same geometrical interpretation, the continuation of the
derivation requires them to be kept equal. Every operation that is performed on one of them
should be followed by a corresponding operation to keep the coequal polynomial arithmetical
- 17 -
equivalent. Instead of operating on equations, Arabic algebra and the abacus tradition operate
on the coequal polynomials, always keeping in mind their relation and arithmetical
equivalence. At some point in the history of algebra, coequal polynomials will transform into
an equation. Only by drawing the distinction, we will be able to discern and understand this
important conceptual transformation. We will now investigate how and when this
transformation took place.
5.2. Operations on ‘equations’ in early Arabic algebra
Much has been written about the origin of the names al-jabr and al-muchābala, and the
etymological discussion is as old as the introduction of algebra into Western Europe itself. We
are not interested in the etymology as such (as does for example Gandz 1926) but in the
concepts designated by the terms. The older writings wrongly refer to the author or inventor
of algebra by the name Geber. Several humanist writers, such as Ramus, chose to neglect or
reject the Arabic roots of Renaissance algebra altogether (Høyrup 1998). Regiomontanus’s
Padua lecture of 1464 was probably the most damaging for a true history of algebra. John
Wallis, who was well-informed on Arabic writings through Vossius, attributes the name
algebra to al-jabr w’al-muchābala in his Treatise on Algebra and points at the mistaken
origin of Geber’s name as common before the seventeenth century (Wallis 1685, 5). He
interprets the two words as operations and clearly not as Arabic names:13
The Arabic verb Gjābara, or, as we should write that found in English letters, jābara (from whence
comes the noun al-gjābr), signifies, to restore … The Arabic verb Kābala (from whence comes the
noun al-mulābala) signifies, to oppose, compare, or set one thing against another.
Montucla (1799, I, 382) repeats Wallis’ comments on Geber by Wallis but seems to interpret
al-muchābala as the act of equating itself:
Suivant Golius, le mot arabe, gebera ou giabera, s’explique par religavit, consolidavit; et mocabalat
signifie comporatio, oppositio. Le dernier de ces mots se rapporte assez bien à ce qu’on fait en
algèbre, dont une des principales opérations consiste à former une opposition ou comparaison à
laquelle nous avons donné le nom d’équation.
We want to understand the concept of the ‘equation’ within the context of the dissemination
of early Arabic algebra in Western Europe. We will approach this conceptual reconstruction
from the operations that were performed on the structures we now call equations. Changes in
the operations on these structures will allow us to understand the changes in the concept of an
equation. In a fairly recent publication, Saliba (1972) analyzed the possible meanings of al-
jabr and other operations in the Arabic text of the Kitāb al-muhtasar fi hisāb al-jabr wa’l-
muqābalah (c. 860) by al-Khwārizmī, but also the lesser-known works Kitāb al-Badī‘ fī al-
Hisāb (Anbouba, 1964), Kitāb al-kafi si’l’hisāb by al-Karajī (c. 1025), and its commentaries,
the Kitāb al-bāhir ficilm al-hisāb by Ibn cAbbās (12th century) and the Kitāb fi’l-jabr wa’l-
muqābalah from Ibn cAmr al-Tannūkhī al-Macarrī. Concerning the use of operations, Saliba
concludes (1972, 190-1):
We deduce from them the most common definitions of the algebraic operations commonly denoted
in those texts by the words jabr, muqābala, radd and īkmal.
The understanding of the precise meaning of these operations is an ongoing debate since the
last century and earlier. There are basically two possible explanations. Either the Arabic
authors of algebra treatises terms used the term inconsistently, or there are fundamental
difficulties in understanding their meaning. Saliba is clearly convinced of the former, and
- 18 -
seizes every opportunity to point at differences in interpretation and double uses of some
terms. Others believe that there are no inconsistent uses at all and attempt to give an
interpretation of their own. A recent discussion, on the Historia Mathematica mailing list, has
raised the issue of interpretation once again.14 Jeffrey Oaks writes that:
the words used to describe the steps of algebraic simplification, ikmāl (completion), radd
(returning), jabr (restoration) and muqābala (confrontation), are not technical terms for specific
operations, but are non-technical words used to name the immediate goals of particular steps. It
then follows, contrary to what was previously thought, that al-Khwārizmī and other medieval
algebraists were not confusing and inconsistent in their uses of these words.
We do not want to be unsporting by claiming that a middle position is here more appropriate.
We tend to defend the latter position. While there may be some inconsistent uses of the terms
between authors and possibly even within a single treatise, the proper meaning of the
operations can be well established within the context in which they occur. We will show that
some confusions can be explained by translating or scribal errors and that a symbolic
interpretation of the operations as Saliba’s is highly problematic. We found out that while our
interpretation of the al-jabr operation is new with respect to most twentieth-century
discussions, it is not divergent from nineteenth-century studies, as Chasles’ (1841) and
Rodet’s (1878).
5.2.1. al-jabr Early occurrences
The jabr operation is commonly interpreted as “adding equal terms to both sides of an
equation in order to eliminate negative terms”.15 It appears first in al-Khwārizmī’s book in the
first problem for the ‘equation’ 22
40 4
. In this interpretation the al-jabr is understood
as the addition of 2
to both parts of the equation in order to eliminate the negative term in
the right-hand part. As a typical symbolical interpretation we give the description from Saliba
(1972, 192):
If f(x) – h(x) = g(x), then f(x) = g(x) + h(x); which is effected by adding h(x) to both sides of the
equation and where f(x), h(x), g(x) are monomials. E.g. if 210 19xx
= then 219 10
Saliba (1972) points out that the Arabic root jabara has a double meaning. On the one hand
‘to reduce a fracture’, on the other ‘to force, to compel’. He believes the second interpretation
is justified as it corresponds with his mathematical understanding. We will argue the contrary.
Surprisingly, the symbolic interpretation such as van der Waerden’s and Saliba’s has, until
very recently, never been challenged. The rule corresponds with one of the later axioms of
algebra: you may add the same term to both sides of an equation.16 As such, the rule seems to
be in perfect correspondence with our current understanding of algebra. However, we will
show this is not the case.
Let us follow the available translations of the original text. The first of al-Khwārizmī’s
illustrative problems is formulated as the division of 10 into two parts such that one part
multiplied by itself becomes four times as much as the two parts multiplied together. Using
the unknown for one of the parts, the other is 10 minus the unknown. al-Khwārizmī proceeds
as follows (Rosen 1831, 35-6):
Then multiply it by four, because the instance states “four times as much”. The result will be four
times the product of one of the parts multiplied by the other. This is forty things minus four
- 19 -
squares. After this you multiply thing by thing, that is to say one of the portions by itself. This is a
square, which is equal to forty things minus four squares. Reduce it now by the four squares; and
add them to the one square. Then the equation is: forty things are equal to five squares; and one
square will be equal to eight roots, that is, sixty-four; the root of this is eight, and this is one of the
two portions, namely, that which is to multiplied by itself.
The jabr operation is thus described by “reduce it now by the four squares, and add them to
the one square”. Remark that this description is somewhat odd. The operation here seems to
consist of two steps, first reducing the four squares from it and secondly, adding them to the
one square. For the second problem, Rosen (1831, 37) also uses the term reduce in the context
“Reduce it to one square, through division by nine twenty-fifths”, which is clearly a different
type of operation of division by a given factor. On most other occasions Rosen translates the
jabr operation as “separate the <negative part> from the <positive part>”.17 Karpinski’s
translation gives a different interpretation. He used Scheubel’s copy of the Latin translation
by Robert of Chester and translates the passage as “Therefore restore or complete the number,
i.e. add four squares to one square, and you obtain five squares equal to 40x” (Karpinski 1915,
105). Karpinski does not use ‘restore’ in the second sense. In his view, restoring describes a
one-step operation. The addition of the four squares to the one square explains the act of
restoration. Can we find this interpretation confirmed by the first Latin translations?
Although we find in Hispalensis (Boncompagni 1857, 112-3) a corrupted version of the title
of al-Khwārizmī’s book, “Exceptiones de libro qui dicitur gleba mutabilia”, al-jabr is not
further discussed.18 The jabr operation is most commonly translated into Latin by the verb
restaurare and appears only once in Robert of Chester’s translation for this problem (Hughes
1989, 53): “Restaura ergo numerum et super substanciam 4 substancias adicias” which
literally means “Therefore restore the number and to the square term add 4 square terms”.
The other occurrence is in the title Liber Algebre et Almuchabolae de Questionibus
Arithmetic(i)s et Geometricis. In nomine dei pii et misericordis incipit Liber Restauracionis et
Opposicionis Numeri quem edidit Mahumed filius Moysi Algaurizmi. Robert also uses the
verb complere twice as an alternative translation for al-jabr (Hughes 1989, 56:1, 57:21).
The second Latin translation by Gerard of Cremona (c. 1150) uses restaurare eleven times.
For the first problem Gerard formulates the jabr operation as “deinde restaurabis quadraginta
per quatuor census. Post hoc addes census censui, et erit quod quadraginta res erunt equales
quinque censibus”.19 Thus, the two Latin translations agree. Translated in symbolic terms,
when given 22
40 4
xx−=, the 40x is restored by the 4x2 and only then, post hoc, the 4x2 is
added to the x2. If we look at the actual text used by Karpinski (published by Hughes 1989,
53) “Restaura ergo numerum et super substancia, 40 rebus absque 4 substancias adicias,
fientque 40 res 5 substancias coequentes”, the same interpretation can be justified. The al-jabr
or restoration operation consists of completing the original term 40x. It is considered to be
incomplete by the missing four censi. The addition of the four censi to the census is a second
step in the process, basically different from the al-jabr operation. The other occurrences of the
operations within the problem sections are listed in the Table 2.
With this exhaustive list of all occurrences of the jabr operation in al-Khwārizmī's’ Algebra
we can now draw an interpretation for the meaning of the operation:20
The restoration is an operation which reinstates a polynomial to its original form. We
use polynomial as a generalization of the several cases. In VI.4 it is a simple number
which is being restored. Also cases VII.5 and VII.6 refer to the single number 100,
- 20 -
instead of 100 + x2. However in problem VI.5 is the binomial 100 + 2x2 which is
restored. This is consistent with the other Latin translations.
The restoration consists of adding (back) the part which has been diminished (“que
fuerunt diminute”) to the polynomial. The restoring part can itself be a polynomial, as
in problem VII.4 with 2x2 – 1/6 as the restoring part.
The restoration operation is always followed by the addition of the restoring part to the
other (coequal) polynomial.
Prob Meta-description Actual text pp.
VI.1 22
40 4
xx=− Deinde restaurabis quadraginta per
quattuor census. Post hoc addes census
VI.3 410
x=− Restaura itaque decem per rem, et adde
ipsam quattuor.
VI.5 2
100 2 20 55xx+−=
Restaura ergo centum et duos census per
res que fuerunt diminute, et adde eas
quinquaginta octo.
VII.1 2
10 21xx−= Restaura igitur decem excepta re per
censum, et adde censum viginti uno.
VII.4 22
21 2 100 2 2
xx x x+− −= + −
Restaura ergo illud, et adde duos census
et sextam centum et duobus censibus
exceptis viginti rebus
VII.5 21
100 20 2
xx+− = Restaura igitur centum et adde viginti
res medietati rei.
VII.6 2
100 20 81
xx+− = Restaura ergo centum, et adde viginti
radices octoginta uni.
52 10 10
Restaura ergo quinquaginta duo et semis
per decem radices et semis, et adde eas
decem radicibus excepto censu.
52 20
Deinde restaura eas per censum et
adde censum quinquaginta duobus et
VIII.1 2
100 20 81xx+− = Restaura ergo centum et adde viginti
radices octoginta uni et erunt centum et
Table 2: All the occurrences of the restoration operation in al-Khwārizmī's’ Algebra in the
Latin translation by Robert of Chester
In such interpretation of Arabic algebra, the basic operation of al-jabr, from which the name
of algebra is derived, does not consist of adding a negative term to the two parts of an
equation. Instead, it refers to the completion of a polynomial which is considered incomplete
by the presence of what we now would call, a negative term. An understanding of al-jabr in
early Arabic algebra is inextricably bound with a geometric interpretation. We conjecture the
al-jabr operation to be a generalization of the basic geometrical acts like cutting and pasting
as we know them from Babylonian algebra. The original use of the restoration may refer to
the restoration of a geometrical square. As we have discussed above, the māl as the Arabic
concept of the unknown is a mixture of the meaning of possession, known from Hindi sources
and from the geometrical square. While the original form of the jabr operation may have been
purely geometrical, the operation can easily be generalized to simple numbers or polynomials.
The demonstration of the solution to the quadratic problems in chapter 7 of al-Khwārizmī’s
Algebra gives us the most likely context of interpretation. Given that 210 39xx+=
, the
- 21 -
demonstration depends on the completion of the polynomial 2
with value 39 (see
Figure 2). The jabr operation restores the māl, the square term, in the polynomial. Hence, the
value of the completed square 2
(5)x+ can be determined through a separate operation of
adding 25 to 39. Also the third translation, by Guglielmo de Lunis (c. 1215), uses restauracio.
In eight problems the operation is applied in the same meaning as the two other translations.
We will therefore not discuss these further.21
However, in two similar problems, 4 and 6, restaurare is also used for a different kind of
operation. This happens in situations where an expression involves a fraction of the māl as in
12 3 4
xxx++ + = and 2
12 xx
In these two cases restaurare consists of multiplying the polynomials by 12. This operation is
called al-ikmāl in Arabic and will be discussed below. al-jabr in later Arabic sources
Let us verify if this new interpretation of Arabic algebra can be sustained in later texts.
Abū Kāmil uses the term restaurare (as the Latin translation for jabare) forty times in his
Algebra. On other occasions he uses reintegrare or ikmāl as a synonym for restaurare. All
occurrences have the same meaning as with al-Khwārizmī and can be reconciled with our new
interpretation. For the third problem Abū Kāmil constructs the ‘equation’ 4x = 10 – x and
proceeds “Restaura ergo 10 per rem cum re, et appone adde rem 4 rebus; et erunt 5 res,
equales 10 dragmis” (Sesiano 1993, 361:1117). 22 Also here the restoration consists of
completing the 10 and the following step is adding x to the 4x. As with one case of al-
Khwārizmī, the jabr operation with Abū Kāmil frequently refers to the restoration of a
polynomial. For example the coequal polynomials
Figure 2: completing the square (from al-
- 22 -
42 4 100 2 20
are restored as follows (Sesiano 1993, 365:1285-91):
Restaura ergo 100 dragmas et 2 census cum 20 radicibus, et adde illas ad 42 res et ½ rei diminutis 4
censibus et ¼; et erunt 62 res et ½ rei diminutis 4 censibus et ¼ census, equales 100 dragmis et 2
censibus. Restaura item 62 res et ½ rei cum 4 censibus et 1/4 , et adde illos 100 dragmis et 2 et
censibus; et erunt 100 dragme et 6 census et ¼ census, que equantur 62 rebus et ½ rei.
The first restoration refers to the 2
100 2
+, the second to 1
62 2
Interestingly, the critical edition adds some omissions in the Latin translation which are
present in an Arabic copy of the original. In this case the original had “Restaura ergo 100
dragmas et 2 census diminitus 20 rebus cum 20 radicibus”. This reaffirms our interpretation of
restoration as “restore <the defected polynomial> with <the part that was diminished>”.
Jeffrey Oaks and Haitham Alkhateeb defend the position on the Historia Mathematica forum,
that the al-jabr operation for 2
10 21xx
= should be interpreted as follows:
Think of 10xx2 as a diminished 10x. Its identity as 10x is retained even though x2 has been taken
away from it. Its restoration to its former self is accomplished by adding x2 to the other side of the
This was answered by Luis Puig, who apparently raised the issue in a publication
previously.23 In Puig’s reconstruction of the al-jabr operation for the same problem, it is the
10x which is restored: “Restaura luego las diez cosas del tesoro [substraído] y añádelo a
veintiuno. Resulta entonces diez cosas, que igualan veintiún dirhams y un tesoro” (Puig
1998). On the discussion forum, Puig refers to the distinction made by al-Karkhī between
nombres simples and nombres composés. This distinction is indeed quite relevant for an
interpretation of the al-jabr operation. In the Al-Fakhrī, partially translated by Woepcke, al-
Karkhī gives an introduction to algebra treating the multiplication of polynomials. A marginal
comment on the distinction of the two types of ‘numbers’ is as follows (Woepcke 1853, 50):
Il y a des personnes qui sont d’avis que ce nombre (10 – a) est composé, puisqu’il est formé par
deux expressions d’un ordre différent. Mais il n’est pas ainsi, parce que en disant : dix moins chose,
vous indiquez un seul nombre de l’ordre des unités ; si, au lieu de cela, il y avait eu : dix plus
chose, cela aurait été composé. Cependant, placez les expressions de ce genre dans quelle catégorie
vous voudrez, cela ne change rien aux principes du calcul.
The special status of ‘incomplete’ or ‘defected’ simple numbers can further explain the nature
of the al-jabr operation. As the bone surgeon, algebrista in old Spanish, splints a broken leg,
so does the al-jabr operation restore an incomplete number.24 While a negative term is
considered a defect, the addition of a positive term is considered a constructive step for a
composed number. It also explains that we should not consider the – x2 in 10 – x2 as a
negative term, but as the defect of the incomplete number 10. While al-Karkhī’s distinction
between simple and composed numbers is essential in contextualizing the al-jabr operation, it
cannot be stated that al-jabr refers to the completion of simple numbers only. In a problem of
Abū Kāmil’s Algebra, we find an interesting case in which the ‘defected polynomial’ consists
of four terms (Sesiano 1993, 390-1):
- 23 -
Et si dicemus tibi: Divisi 10 in duas partes, et multiplicavi unam [in aliam] duarum partium in se et
aliam in radicem 8; deinde proieci quod [agregatum] productum fuit ex multiplicatione unius
duarum partium in radicem 8 ex eo quod provenit ex multiplicatione (alterius) in se, et remanserunt
40 dragme. Exemplum. Faciamus unam duarum partium rem, reliquam vero 10 diminuta re. Et
ducamus 10 diminuta re in se, et erunt 100 dragme et census diminutis 20 rebus. Deinde multiplica
rem in radicem de 8, et proveniet radix 8 censuum. Quam prohice ex 100 dragmis et censu
diminutis 20 rebus, et remanebunt 100 dragme et census 20 [radicibus] rebus diminutis et diminuta
radice 8 censuum, que equantur 40 dragmis. Restaura ergo 100 et censum cum 20 [radicibus] rebus
et radice 8 censuum, et adde (eas) ad 40 dragmas. Et habebis 100 dragmas et censum, que equantur
40 dragmis et 20 rebus et [rei] radici 8 censuum.
This solution of a division problem can be described symbolically as follows. Consider the
two parts to be x and 10 – x. Multiplying the second by itself and the first by the root of 8, the
difference equals 40. Thus:
(10 )(1 0 ) 8 40xxx−−=
Expanding the square of the second part and bringing the x within the square root, this leads
100 20 8 40xxx+− = .
So, now the question is, in al- Karkhī’s terminology: what is restored here, the composed
number 2
+ or the simple number 2
? The text of Abū Kāmil leaves no doubt: “Restaura
ergo 100 et censum cum 20 [radicibus] rebus et radice 8 censuum”. Thus the polynomial
+ is restored by 2
20 8
After that, the two terms are added to 40.
So, if close reading of the original text provides us with this divergent interpretation of the
basic operation of Arabic algebra, why did scholars, proficient in Islam sciences and algebra
fail to see it? Take for example Solomon Gandz, the leading expert on Arabic and Babylonian
algebra in the early days of Isis and Osiris. Devoting an article on “The origin of the term
‘Algebra’”, Gandz (1926, 440) concludes that the al-jabr wa’l-muqābalah “ought to be
rendered simply as Science of equations”. Arguing against the older interpretation of
restoration, he raises an intriguing question: “Why should we use an artificial surgical term
for a mathematical operation, when there are such good plain words as zāda and tamma for
the operation of addition and completion?” (ibid., 439). This should indeed ring a bell. Maybe
al-jabr is not just “a mathematical operation” as we tend to see it. Maybe the operation is
something very different from addition. The specific choice of the term al-jabr instead of
other “good plain words” deserves an explanation within the context of early Arabic algebra
and is no argument against an interpretation as restoration. Older interpretations
Troubled by the question why the interpretation of al-jabr, as the restoration of a defected
polynomial, is virtually absent in the twentieth century, we looked at some earlier studies. In
Chasles (1841, 605-616) we recognize several important aspects of our interpretation:
- 24 -
Quand, dans un membre d’une équation, une quantité positive est suivie ou affectée d’une quantité
négative, on restaure la quantité positive, c’est-à-dire qu’on la rétablit dans son intégralité. Pour
cela on ajoute aux deux membres de l’équation une quantité égale, au signe près, à la quantité
négative. Dans le langage de notre algèbre actuelle, nous dirions qu’on fait passer la quantité
négative, du membre où elle se trouve, dans l’autre membre. Mais les Arabes ne pouvaient
s’éxprimer ainsi, parce qu’ils ne considéraient pas de quantités négatives isolément. Quoi qu’il en
soit, c’est, à mon sens,, cette opération de restauration, telle que je viens de la définir, que les
Arabes ont appelée jebr, et les traducteurs algebra.
He considers al-jabr as a restoration of a positive quantity to its original integrity. In doing so,
one must “add an equal quantity to the two members of the equation”. Chasles rightly adds
that isolated negative quantities are not recognized in Arabic algebra.
Woepcke (1854, 365) is less concerned with the aspect of restoration and considers al-jabr as
“the action of removing a negative particle and consequently replacing it at the other member
to conserve the equality”.25 Rodet (1878, 38), based on the authority of Freytag (1830) for a
translation of jabara as “post paupertalum ditivait”, uses enrichissant. Thus he interprets the
restoration of 100 – 20x = 40 by al-Khwārizmī as:
Il commence par faire disparaître le terme négatif – 20x, en enrichissant, comme il dit, les 100
unités de déficit que leur a causé la soustraction des 20x. Pour compenser cet enrichissant, il doit
naturallement ajouter 20x dans le second membre de l’equation.
Carra de Vaux (1897) wrote a short note on the meaning of al-jabr in Bibliotheca
Mathematica after inspecting a manuscript of Ibn El-Hāim in the Ambrosiano Library in
Milan (&, 64, sup. f. 28r). In that text the term is also applied to the restoration of a quantity
with a missing fraction: “Thus to make 5/6 equal to one whole, you divide 1 by 5/6 which
leads to 1 + 1/5 and then multiply it with 5/6. Otherwise, you can take the difference of 1 –
5/6 and 5/6 which is 1/5 and this you add to 5/6 to obtain one”. There is one occasion in
al-Khwārizmī’s problems in which the same operation is performed. In problem III.13,
discussed above in §3.1 complere was used in the same way. By using the same term for the
operation, al-Khwārizmī shows that adding
to 22
is basically the same act as restoration
back to the form 2
. 26
Carra de Vaux’s note also includes a reference to the encyclopedia of the Turkish historian
Hādjī Khalīfa (c. 1650). Here a definition of djebr is given strong support for our favored
interpretation: “le djebr c’est ajouter ce qui manque à l’une des deux quantités mises en
équation pour qu’elle devienne égale à l’autre”.27
It is with some surprise that we have to admit the relevance of the nineteenth-century analyses
in the current discussions on the interpretation of Arabic algebra. It seems that with Hankel
and Cantor the interpretation as adding the term to both parts of an equation, was generally
accepted.28 Many twentieth-century authors have neglected to look up the studies of
nineteenth-century scholars and missed their valuable comments.29
In summary, we believe that the al-jabr operation in early Arabic algebra can be characterized
as follows:
- 25 -
An operation aiming at the restoration of a defected quantity to its original
The restored quantity could initially have been a simple number in the sense of al-
Karkhī, but for Abū Kāmil it also applies to polynomials.
The operation is probably derived from or to be interpreted in a geometrical sense.
The operation is not performed on an equation but on the affected part of one of two
coequal polynomials.
The addition of the defected part to the coequal polynomial is not a part of but a
consequence of the operation.
5.2.2. al-muqābala
The second operation, al-muqābala, is generally understood as the addition of homogeneous
terms in a polynomial. So the operation allows to rewrite 22
100 20
100 2 20
x+− (from al-Khwārizmī’s third problem, Hughes 1989, 58). The Latin word for
this is simply summa and derived from its geometrical interpretation of adding areas together.
A second, equally important meaning of al-muqābala is the elimination of a term by
subtracting it from the coequal polynomial. The latin term for this is opponere and is used in
problem III.5 of al-Khwārizmī’s Algebra (Hughes 1989, 56:3):
habebis 100 et duas substancias absque 20 radicibus 58 coequantes. Comple igitur 100 et 2
substancias cum re quam diximus et adde eam super 58, et fient 100 et due subtancie, 58 et 20 res
coequancia. Hoc igitur oppone id est ex numero 29 proicias et remanebunt 21 et substancia 10 res
Thus al-Khwārizmī applies al-jabr to 2
100 2 20 58xx
in order to restore 2
translated on this occasion by complere. Omitted here by the scribe is a step which divides
both polynomials by two to arrive at the coequal 2
50 29 10
x+=+ . Then he applies al-
muqābala to eliminate the number 29 from the second polynomial by subtracting it from the
first, resulting in 2
21 10
x+= . Hughes (1989, 20) understands the division by two as
complere, but we believe this to be mistaken, as complere is also used, in the meaning
described here, in problem two of the second chapter “habebis 40 et 20 res 100 coequantes.
Hec ergo centeno opponas numero et 40 ex 100 auferas et remanebunt 60, 20 res coequancia”
(Hughes 1989, 57/23). Rosen (1838, 40), who used the Arabic manuscript, does include the
missing step as “Reduce this to one square, by taking the moiety of all you have. It is then:
fifty dirhems and a square, which are equal to twenty-nine dirhems and ten things”. The Latin
translation of Abū Kāmil’s Algebra paraphrases muqābala as mukabala or mucabele and
explains it as oppositio (Sesiano 1993, lines 527 and 532), but does not use the term within
the problems. The verb complere only appears in its strict geometrical sense. Saliba (1972,
199) finds only one occasion in which al-Karkhī uses muqābala in the same sense as al-
Khwārizmī. He believes that al-Karkhī also uses muqābala for the two operations discussed
While our interpretation of al-jabr considers the operation of completion as distinct from the
subsequent step of adding the completed part to the coequal polynomial, al-muqābala appears
to operate on the coequal polynomials within the same operation.
- 26 -
5.2.3. al-radd and al-ikmāl
The last two operations called al-radd and al-ikmāl are less controversial. They normally refer
respectively the division or to the multiplication of coequal polynomials by a constant.
However, in some cases ikmāl is used synonymously with jabr by Abū Kāmil and tama (to
complete) for the ikmāl operation.
al-jabr al-muqābala al-radd al-ikmāl Arab
ﺮﺑﺠﻠا ﺔﻠﺑﺎﻘﻣﻟا ﱞﺪﺮﻟا لﺎﻤﻜﻺا
(from Arab)
reduce reduce complete
Robert of Chester restaurare
opponere converte complere
(from Robert)
by opposition reduce complete
Gerard restaurare opponere reducere reintegrare
Guglielmo restaurare eicere reducere restaurare
Abū Kāmil restaurare
opponere reducere complere
Table 3: terms for the basic operations of Arabic algebra in the main Latin translations
The best reference problem is problem III.5, as it combines the first three operations in a
single problem solution. While Robert leaves out the al-radd step, he uses the verb converte
for reducing the square term in problems III.3 and III.12 (“ergo ad unam converte
substanciam”). The completion of the square term appears in problems III.4 and III.6.
5.3. Operations on equations in the abacus tradition
In the course of the fourteenth century, the original context of al-jabr as restoring a defected
or incomplete quantity was almost entirely abandoned. The initial al-jabr operation, acting on
a single quantity was extended by Abū Kāmil to be applied on polynomials. While the Arabic
understanding of the operation continues to be present in some Latin treatises, we witness a
clear shift in meaning of the operation.
With Fibonacci’s Liber Abbaci and the early vernacular algebra texts, the operation acts
simultaneously on two coequal polynomials. The relation between the words used for
restoration and its etymological root becomes disconnected. In the beginning of the fourteenth
century, restoration involves both the addition and the subtraction of a term to coequal
polynomials, sometimes within the same derivation. With maestro Biagio, from the fourteenth
century onwards, the terminology discards all references to the restoring aspect and simply
operates on both parts in order ‘to level out’ the positives as well as the negatives. The
simultaneous operation on coequal polynomials is the beginning of what constitutes an
algebraic equation. We cannot yet consider ragguagliare as an operation on an equation, but
the simultaneous addition, subtraction, division and multiplication of coequal polynomials by
some quantity contributes to the further transformation of this structure into a symbolic
- 27 -
6. Conclusion
The symbolic equation has resulted from a series of developments in algebraic practice
spanning a period of three centuries. The concept of a symbolic equation as it emerges in
algebra textbooks around 1550 is fundamentally different from the ‘equation’ as known
before the sixteenth century. This transformation of the equation concept was completed
through the practice of algebraic problem solving. We can distinguish several phases of
development which were necessary to realize the modern concept of an equation. We will
now summarize these developments as discussed here, and place them within a broader
framework. We will present them in logical order which does not perforce coincide with
consecutive historical events. Several of these developments overlap and have reinforced each
6.1. The expansion of arithmetical operators to polynomials
A process of expansion and generalization has allowed applying the operations of addition,
subtraction, division and multiplication to other entities than natural numbers. This expansion
process can be looked at from the viewpoint of the objects as well as of the operators.
Operations on polynomial terms emerged as an expansion of the operators. These were
introduced in Hindu texts around 600 and in Arabic algebra before 800. Essential differences
in approach suggest an independent development in these two traditions. The presentation of
operations on polynomials together with or following the operations on irrational binomials
provides strong support for a historic process of generalization from irrationals to algebraic
polynomials. We have written evidence that operations on polynomials were introduced in
Europe through the Latin translations of Arabic works on algebra. Possibly there has been
some influence too from Hindu algebra through sub-scientific traditions. The abacus tradition
paid little attention to a formal treatment of operations on polynomials. Only from the end of
the fourteenth century some abacus treatises devote a section to the multiplication of
binomials or trinomials. Early German cossist texts of the fifteenth century were the first to
formally introduce these operations. They reflect the structure of an algorism applied to terms
involving unknowns. By the beginning of the sixteenth century every serious work on algebra
has an introduction explaining at least addition, subtraction and multiplication of algebraic
6.2. The expansion of the number concept
The process of applying arithmetical operations on terms with unknowns invoked an
expansion of the number concept. The cossist tradition forwards the idea, which later becomes
omnipresent in algebra textbooks, that cossic numbers are some kind of number, next to
whole numbers, fractions and surds. Systematic treatments of arithmetic and algebra typically
include binomials in the exposition of the numeration, the types of numbers in arithmetic.
This evolution culminates in the Arithmetica of Cardano (1539). Cardano departs from the
prevailing structure and treats the operators one by one. For each operation he discusses its
application to whole numbers, fractions, irrationals and polynomial expressions. Polynomials,
which he calls de numeratione denominationem, are thus presented as part of the number
concept. The idea of polynomials as numbers is abandoned by the end of the sixteenth
century. Later interpretations of higher-order polynomials with multiple roots and the
unknown as a variable are in direct contradiction with a cossic number having one
determinate arithmetical value.
- 28 -
6.3. Equating polynomial expressions
The very idea of an equation is based on the act of equating polynomial expressions. In fact,
the Latin terms aequatio and aequationis refer to this action. Also the Sanskrit words
samīkarana, samīkarā, or samīkriyā, used in Hindu algebra can be interpreted in this way.
The word sama means ‘equal’ and kri stands for ‘to do’. The meaning of an equation in the
first Latin texts is most correctly conveyed by the terminology used by Guglielmo de Lunis
and Robert of Chester. The term coaequare denotes the act of keeping related polynomials
equal. The whole rhetoric of abacus texts is based on the reformulation of a problem using the
unknown and the manipulation of coequal polynomials to arrive at a reducible expression in
the unknown. One looks in vain for equations in abacus texts. Every reference to an equation
is purely rhetorical, meaning that the only equation discussed is that <coequal polynomial 1>
equals <coequal polynomial 2>. If the manuscript contains illustrations or marginal
comments then these are always polynomials or operations on polynomials. Only by the end
of the fifteenth century do we find equations in the non-rhetorical meaning. They first appear
in German texts such as the Dresden C 80. Apparently Italian algebra was too dependent on a
rigid rhetorical structure to view an equation as a separate entity. Pacioli’s Summa (1494), full
of marginal illustrations, does not give a single equation.30 In Rudolff (1525) and Cardano
(1539) we find the first illustrations of an equation in print. Both in the literal and the
historical sense, we find the construction of an equation by equating polynomials (see Figure
3, from Cardano 1539, 82).
Figure 3: Cardano’s construction of an equation by equating polynomial expressions.
6.4. Operations on coequal polynomials
The concept of an equation is shaped by the operations on coequal polynomials. The early
development of the equation concept is determined by the first Arabic texts on algebra. Arabic
algebra emerged from several competing traditions which are reflected in the meaning of the
unknown and the operations allowed on coequal polynomials. These influences are most
likely the ‘high’ tradition of calculators and the ‘low’ tradition of practical surveyors. A third
influence of solving recreational problems concerning possessions may stem from Indian
practice. The conceptual ambiguity of the māl, the unknown in Arabic algebra, can be
explained through this diversity of influences. Also the al-jabr, the basic operation of Arabic
algebra is challenging for a modern interpretation. Early Arabic texts interpret al-jabr as the
restoration of a defected polynomial. The restoration of such polynomial to its integral
(positive) form requires the subsequent step of adding the restored term to the coequal
polynomial. This operation has transformed into the more general addition of terms to coequal
polynomials. The characterization of the al-jabr as the restoration of one defected polynomial
depends on the distinction made between co-equal polynomials and equations. When viewing
Arabic algebra as operating on equations, such an interpretation would be meaningless.
Other operations such as bringing together homogeneous terms and dividing or multiplying
coequal polynomials by a common factor can be related directly to their Arabic archetypes.
These operations have been applied and discussed only implicitly in abacus problem solving.
An explicit or formal exposition of the possible operations on coequal polynomials is first
seen by the end of the fifteenth century in Germany. The formulation of rules and making
- 29 -
these operations explicit contributed to the idea of operating on a single algebraic entity. It
will take two more centuries to formulate these rules as axioms of algebra.
6.5. Expansion of arithmetical operators to equations
The transformation of operations on coequal polynomials to operations on equations is a
subtle one. Only by making the distinction between the two can we understand and discern
the changes in the concept of an equation.
The first explicit use of a multiplication of an
equation is found in Cardano (1539, f. HH1r)
where he uses two unknowns to solve a linear
problem. Eliminating one unknown, he arrives at
an equation, expressed in the second unknown,
which he multiplies with 35, as shown in Figure
4. Operating on equations here is closely
connected with the use of the second unknown.
Figure 4: First operation on an
equation in Cardano’s Arithmetica
6.6. Operations between equations
The second unknown has been the driving force behind the introduction of operations
between equations. Cardano (1545) not only performs operations on equations but also he was
the first two subtract equations in order to eliminate one of the unknowns (Opera Omnia, III,
Using Cardano’s method of
eliminating a second unknown from
the Ars Magna and Stifel’s
extension of algebraic symbolism
for multiple unknowns, Jacques
Peletier (1554) operates on an
aggregate of linear equations.
Figure 5: Cardano (1545) subtracts the first
equation from the second to result in the third
He adds and subtracts pairs of equations in a systematic way to solve a set of linear equations.
Buteo’s text (1559 corresponds closely with our meta-description in modern symbolism. The
concept of a symbolic equation can thus be regarded as completed. The method was further
refined by Gosselin (1577) from which we know that he had some influence on Viète
(Cifoletti 1993).
7. Epistemological consequences
We have presented a detailed analysis of the basic concepts of algebra since the first extant
texts in the Arab world and their subsequent introduction in Western Europe. The basic
concepts of algebra are the unknown and the equation. We have demonstrated that the use of
these concepts has been problematic in several aspects. Arabic algebra texts reveal anomalies
which can be attributed to the diversity of influences from which the al-jabr practice emerged.
We have characterized a symbolic equation as a later development which builds upon the
basic Arabic operations on coequal polynomials. The concept of an equation can be
considered as a solidification of the possible operations on coequal polynomials. In this way,
the equation sign, as it was introduced by Robert Recorde (1557), represents not only the
arithmetical equivalence of both parts, but at the same time symbolizes the possible
operations on that equation. The equation, the basis of symbolic algebra, emerged from the
basic operations on pre-symbolic structures, as we have studied them within Arabic algebra.
- 30 -
The equation became epistemological acceptible by the confidence in the basic operations it
represented. Knowledge depending on this new concept, such as later algebraic theorems or
problems solved by algebra, derived their credibility from the operations accepted as valid for
the concept. This new mathematics-as-calculation, derived from Arabic algebra, became the
interpretation of mathematical knowledge in the sixteenth and seventeenth centuries. The
introduction of symbolism allowed for a further abstraction from the arithmetical content of
the algebraic terms. Operating on and between equations became such powerful tool that it
standed as a model for a mathesis universalis, a normative discipline of arriving at certain
knowledge. This is the function Descartes describes in Rule IV of his Regulae. Later, Wallis
(1657) uses Mathesis Universalis as the title for his treatise on algebra. As a consequence, the
study of algebra delivered natural philosophers of the seventeenth century a tool for correct
reasoning in general. In the early modern period, algebra functioned as a model for analysis,
much more than Euclidean geometry did.
8. Acknowledgments
This paper is a shortened and slightly revised version of chapter 6 of my PhD dissertation,
From Precepts to Equations: The Conceptual Development of Symbolic Algebra in the
Sixteenth Century, supervised by Diderik Batens, Ghent University Belgium. Funding for this
research was provided by the research project G.0193.04 from the Flemish fund for scientific
research (FWO Vlaanderen). I would like to thank Jan Hogendijk and Saskia Willaert for their
comments on an earlier version of this text and Joris van Winckel for his help in the
understanding and transcription of some central Arabic terms and their lexical intricacies.
9. References
9.1. Manuscripts cited
Florence, Biblioteca Riccardiana, 2263 (transcription by Simi, 1994)
Florence, BNCF, Fond. princ. II.V.152 (fols. 145r-180v, transcription by Franci and Pancanti,
Istanbul, Kara Mustafa Kütübhane 379 (partial transcription by Levey, 1966)
Madrid, Codex Escurialensis, fondo Arabe 936 (transcription and Spanish translation by
Sánchez Pérez, 1916)
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- 31 -
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1 Although it has been argued that Fibonacci used a Latin translation of al-Khwārizmī’s Algebra, particularly
Gerard of Cremona’s translation (Miura 1981, 60; Allard 1996, 566), one has to account for the fact that he had
direct access to Arabic sources. Leonardo was educated in Bugia, at the north of Africa, now Bejaje in Algeria,
and travelled to several Arabic countries. He writes in his prologue of the Liber Abbaci that he “learnt from
them, whoever was learned in it, from nearby Egypt, Syria, Greece, Sicily and Provence, and their various
methods, to which locations of business I travelled considerably afterwards for much study” (Sigler 2002, 15-6).
2 al-Khwārizmī’s Algebra contains several problems which have been numbered in some translations. We will
use the part numbers of the treatise as Roman numerals, followed by the sequence number and refer to the Latin
translation if the problem numbering differs. Problem III.11 in Robert’s translation is as follows: ‘Terciam
substancie in eius quartam sic multiplico, ut tota multiplicacionis summa ipsi coequetur substancie’ (Hughes
1989, 61). The problem is given by Karpinski in modern symbolism as
⎝⎠⎝⎠ , while the form
would be more consistent with his interpretation of the māl.
3 Also argued by Høyrup (1998, note 11).
- 37 -
4 For a representation of māl as a geometrical square see Figure 2 in the discussion on al-jabr below.
5 Hughes (1989, 18-9). Apparently Hughes mixes up the chapter numbering. Read instead “problems four and
six of part II and in five, ten, and thirteen of part III”.
6 This problem is numbered 14 in chapter VIII of the Gerard’s translation (Hughes 1986, 260).
7 A preliminary version of both these articles came to our attention when most of this chapter was already
written. The analysis of Oaks and Alkhateeb (2005) and especially their section on ‘the deliberate shift from the
original māl to the algebraic māl’ agrees with our observation. In fact, they discern three different meanings for
māl. For the third meaning, they refer to the “division rule”. If the result of the division of a by b is c, then the
value of the māl a can be “recovered” by multiplying b and c.
8 We will follow the analysis of Rodet (1878, 84-8). The English translation is from Colebrooke (1817, 208).
9 Heeffer, A.: “The Regula Quantitatis: From the Second Unknown to the Symbolic Equation”, forthcoming.
10 f. 155v; Franci and Pancanti, 1988, 54: “Quando le chose sono iguali a censi ed al numero prima si parta ne’
censi e poi si dimezi le chose e l’una metà si multripica per se medesimo e di quella multripicazione si tralgha il
numero, la radice del rimanente agiunto overo tratto dall’altra metà delle chose, chotanto varà la chosa e tieni a
mente che sono quistioni dove di bisogno agiugnere la metà delle chose e sono di quelle che àno bisogno di
trarre del la metà delle chose e sono di quelle che per l’uno e per l’al tro si solvono. Esenpro al'agiugnere, prima
dirò chosì”.
11 Fibonacci, Liber Abbaci, second edition of 1228, on which Boncompagni’s transcription is based. Høyrup
(2002) suggests that the inconsistencies stem from the later additions and believes there must have existed an
Italian vernacular text from before 1228 in which the term avere was used.
12 Although Hughes (1986, 1989) consistently talks about equations, he implicitly agrees with this position when
he writes that Gerard “uses the word questio to signify our term equation” (Hughes 1986, 214).
13 From the English edition, Wallis 1685, 2. Chasles (1841, 612) critizises Wallis for the algebraic interpretation
of the terms al-jabr and al-muchābala as synthesis and analysis. However, Chasles has been very selective in his
reading of the Treatise on Algebra.
14 The discussion has been archived at
15 From van der Waerden (1980, 4). Compare with “Addition gleicher Terme zu beiden Seiten einer Gleichung,
um subtraktive Glieder zu elimineren”, Alten e.a. (2003, 162) and “to add the absolute value of a negative term
from one side of an equation to itself and to the other side”, Hughes 1986, 218. Hughes (1989, 20) defines the
synonymous Latin term complere as “to transfer a term from one side of the equation to another”.
16 Axioms play a role in the formulation of algebraic theory only from the seventeenth century. See chapter 8 for
a further discussion on this.
17 Rosen 1831, 42, 43, 47, 48, 52, 52, for the problems discussed below. Problems of section VIII (in Gerard’s
translation) do not appear in the Arabic manuscript.
18 There exist two copies of an Arabic manuscript by Abd al Hamīd ibn Wāsic ibn Turk, called Logical
Necessities in Mixed Equations, studied by Sayili (1985). There are good reasons to believe that this work on
algebra predates the one of al-Khwārizmī’s. Interestingly, except for the title, there is no reference to al-jabr.
19 This is the same formulation as the version of Libri (1938, I, 275), from the Paris Latin 7377A.
20 Some clarifications may be necessary. The solution to VII.1 possibly contains a scribal error. Before the
restoration step, (10 – x) is multiplied with x. Consistent with the other cases, the restoration thus refers to 10x,
instead of 10 – x as in the text. Problem VI.5 refers to “the roots that have been diminished”, thus 20x
21 Problems 1, 3, 5, 7, 8, 9, 10 and 11 in the numbering by Kaunzner (1986).
22 The line numbers from the Sesiano transcription are given after the column. Some other examples from
Sesiano (1993): “Restaura ergo eas cum 9 rebus” (1132), “Restaura 10 radices per censum” (1174), “Restaura
igitur 100 dragmas cum 20 rebus” (1243).
23 Puig 1998, 16, discussed in the Historia Mathematica mailing list.
24 For the meaning of algebrista see Smith (1958, II, 389). For a quotation from Don Quixote see Cantor 1907, I,
679, note 3, and Kline 1964, 95).
25 Woepcke 1854, 365 : “Algèbre signifie dans la langue technique l’action, d’ôter la particule de la négation et
ce qui la suit, et de reporter, en conservant l’égalité dans l’autre membre”.
26 Hughes (1989, 18-9) misses the point when he writes in his commentary that al-Khwārizmī “does not use the
multiplicative inverse to obtain 21
x=” and that this “must have jolted Robert’s readers”. However, the
performed operation is prefectly comprehensible given our interpretation of al-jabr.
27 Translation by Carra de Vaux, from Flügel 1835-58, II, 582.
28 Cantor (1907, I, 676) uses Wiederherstellung as the German translation of al-jabr and defines it as follows:
“Wiederherstellung ist genannt, wenn eine Gleichung der Art geordnet wird, dass auf beiden Seiten des
Gleichheitszeichens nur positive Glieder sich finden”. This is a curious definition as the equation sign appeared
- 38 -
only in Recorde (1557). Hankel even cites the Arithmetica of Diophantus as a source for the al-jabr of the Arabs:
“’Wenn aber auf der einen oder auf beiden Seiten negative Grössen vorkommen, so muss man diese auf beiden
Seiten addiren, bis man auf beiden Seiten positive Grössen erhält’ und das ist al gebr”. The quotation is taken
from the Bachet (1621), Diophanti Alexandrini Arithmeticorum, p. 11.
29 A notable exception is Tropfke (1933, II, 66): “In dem Beispiele 13x – 5 = 7x + 4 ist die linke Seite
unvolständig, da ein fehlendes Glied vorkommt; si muβ also mit 5 ergänzt werden, die dann auch rechts
hinzuzufügen ist”. This interpretation is not respected by the editors of the 1980 edition.
30 Except for the standard rules of algebra, the six Arabic types and two impossible cases (Pacioli 1494, f. 149r).
... 3. manipulation of polynomials: using the unknown, the problem text is formulated in terms of coequal polynomials and manipulated in such a way that these are kept equal. The vernacular terms ristorare and later ragguagliare are used for both the restoration and opposition operations, known from Arabic algebra [13]. 4. reduction to a canonical form: the purpose of manipulating the polynomials is to reduce them to a form in which a standard rule applies. ...
... Therefore, the concept of an equation, as it emerged in Europe by the end of the sixteenth century, is absent from abbaco algebra. We previously coined the term 'co-equal polynomials' as a suitable substitute for the modern term 'equation' [13]. Less obvious may be the terms for abbaco operations which we too easily take for granted as modern-day operations: adding, dividing, subtracting, etc. Høyrup notices the subtle distinction between 'partire per' and 'partire in' in the formulation of the rules of algebra in early abbaco treatises [10]. ...
In this paper we present a chapter on algebra from an abbaco treatise on arithmetic. The abbaco tradition of teaching arithmetic and algebraic problem solving is situated between two major works of the Italian Middle Ages: the Liber Abbaci of Fibonacci (1202) and the Summa di Arithmetica et Geometria of Lucca Pacioli (1494). Peculiar of abbaco texts is their strong similarities and coherence within that period of almost three centuries. We will argue that this feature stems from the way texts were produced and appropriated. Problems and problem solving play a central function in abbaco treatises and the way problems were ’invented’ and adapted determines this process of text production and appropriation. We will illustrate this with one chapter from a family of several manuscript copies of a single treatise, providing a critical edition and an English translation. We will also discuss the relation with other abbaco texts before and after the creation of our text. With the possible exception of Høyrup’s recent book on the abbaco tradition, who calls his transcription semicritical, all publications of abbaco texts have been based on a single manuscript though several copies are usually available. We believe that a critical edition in line with the Latin scholarly tradition provides us with the necessary insights in the way the production of abbaco texts functioned.
... As argued in Heeffer (2008), the correct characterization of the Arabic concept of an equation is the act of keeping related polynomials equal. Two of the three translators of al-Khw¯ arizm¯ ı's algebra, Guglielmo de Lunis and Robert of Chester use the specific term coaequare. ...
... Such a notion is intimately re- lated with the al-jabr operation in early Arabic algebra. As is now generally acknowledged ( Oaks and Alkhateeb, 2007;Heeffer 2008;Hoyrup 2010, note 7), the restoration operation should not be interpreted as adding a term to both sides of an equation, but as the repair of a deficiency in a polynomial. Once this polynomial is restored -and as a second step -the coequal polynomial should have the same term added. ...
Full-text available
The symbolic equation slowly emerged during the course of the sixteenth century as a new mathematical concept as well as a mathematical object on which new operations were made possible. Where historians have of-ten pointed at FrançoisVì ete as the father of symbolic algebra, we would like to emphasize the foundations on whichVì ete could base his logistica speciosa. The period between Cardano's Practica Arithmeticae of 1539 and Gosselin's De arte magna of 1577 has been crucial in providing the necessary build-ing blocks for the transformation of algebra from rules for problem solving to the study of equations. In this paper we argue that the so-called "second unknown" or the Regula quantitates steered the development of an adequate symbolism to deal with multiple unknowns and aggregates of equations. Dur-ing this process the very concept of a symbolic equation emerged separate from previous notions of what we call "co-equal polynomials". L'histoire de la résolution de equation a plusieurs inconnues n'a pas encore donné lieù a un travail d'ensemble satisfaisant, qui donnerait d'ailleurs lieù a d'assez longues recherches. Il est intimement lié auxprog es des notations algébriques. J'ai appelé l'attention sur leprobì eme de la resolution des equations simultanées, chaque fois que je l'ai rencontré, chez les auteurs de la fin du XVIe et du commencement du XVIIe sì ecle. (Bosmans, 1926, 150, footnote 16).
... Dalam buku Hisab al-jabr w'al-muqabala, al-Khwarizmi memulai idenya dengan mendefinisikan istilah-istilah dasar matematika yang sangat penting dalam aljabar seperti: "sesuatu"(variabel), "suatu hal yang tidak diketahui", dan "kuadrat" [12]. Setelah itu, ia akan membawanya kedalam enam bentuk persamaan( 2 = , 2 = , = , 2 + = , 2 + = , 2 = + ). ...
Penerapan sejarah matematika dalam pembelajaran merupakan salah satu alat (history as a tool) atau strategi untuk membangun pembelajaran yang bermakna dan sarat dengan nilai. Sejarah matematika memberikan landasan pemahaman yang mendalam tentang evolusi konsep matematika, memahami kenapa dan bagaimana konsep matematika dikembangkan selama bertahun-tahun dengan kerja keras, belajar sejarah matematika bisa meningkatkan minat dan mengembangkan sikap positif siswa terhadap matematika. Sejarah matematika menjelaskan, sebagai contoh, bagaimana Al-Khawarizmi mengembangkan metode kuadrat sempurna dalam menyelesaikan persamaan kuadrat. Melalui sejarah matematika, kerja keras para matematikawan dalam menemukan dan mengembangkan suatu konsep atau penyelesaian suatu permasalahan bisa menjadi kisah inspiratif. Penelitian ini bertujuan agar pendidik mata pelajaran matematika dapat meningkatkan kualitasnya . Selain itu, penelitian ini juga bermanfaat bagi siswa agar lebih mudah dalam memahami mata pelajaran matematika pada umumnya dan materi aljabar pada khususnya, serta mengurangi kecemasan yang terjadi dalam mempelajari aljabar karena keabstrakan aljabar. Metode yang digunakan dalam penelitian ini adalah studi literatur, yaitu mengkaji penelitian-penelitian sebelumnya yang relevan dan menyimpulkan berdasarkan hasil yang diperoleh. Hasil dari penelitian ini adalah membantu pendidik mengatasi miskonsepsi pada tahap awal belajar aljabar, ketika aljabar dijelaskan dengan menggunakan sistem yang familiar atau natural language maka akan lebih mudah dipahami daripada menggunakan sistem yang tidak dikenal, pendidik dapat memiliki variasi cara mengajar agar sesuai dengan kapasitas peserta didiknya, siswa dapat lebih mudah pula untuk dapat meningkatkan kemampuan berpikir siswa, meningkatkan motivasi, merangsang penalaran dan mengembangkan sikap positif siswa terhadap matematika, serta dapat merangsang penalaran dan kegiatan berpikir siswa.
... The word "al-muqabala" in the title of the book refers to the balancing operation. The word algebra comes from the "al-Jabr" (literally, to impose) in the title of his book, which al Khwarizmi uses to describe the solving operation (Heeffer, 2008). In the second part of the book, al Khwarizmi writes about measurement. ...
Full-text available
The development of computer technology to its current level of sophistication is not separate from the contribution of a Muslim scholar who was focused on the field of mathematics. The scholar was named Abu Ja'far Muhammad bin Musa al-Khawarizmi or better known as Al Khawarizmi. This research used a qualitative research method with a literature review design. In using the literature review research method, the researcher will go through stages consisting of identifying the topic and problem to be studied, searching for sources related to the discussed and determined problem, selecting sources relevant to the topic and problem being studied, collecting data from the selected sources, analyzing the collected data, and writing a report on the results of the literature review. The conclusion of this research is that Al Khawarizmi did play a role in the development of current computer technology. This is evidenced by several discoveries made by Al Khawarizmi which are actually used in the field of computer science, including algorithms and the number zero. The algorithm has a very important role in computer programs.
... We here summerize the basic arguments. For an extensive discussion of this subject see my[11]. ...
Abbaco algebra is a coherent tradition of algebraic problem solving mostly based in the merchant cities of fourteenth and fifteenth-century Italy. This period is roughly situated between two important works dealing with algebra: the Liber Abbaci by Fibonacci (1202) and the Summa di Arithmetica et Geometria by Lucca Pacioli (1492). Such continuous tradition of mathematical practice was hardly known before the first transcriptions of extant manuscripts by Gino Arrighi from the 1960’s and the ground-breaking work by Warren van Egmond (1980). After some decades of manuscript study and the recent assessment of Jens Høyrup (2007) we now have a better understanding of this tradition. In this paper we provide an overview of the basic characteristics of the abbaco tradition and discuss the role it played towards the new symbolic algebra as it emerged in sixteenth-century Europe. We argue that its influence on the sixteenth century has largely been ignored and that the new ars analytica from the French algebraists should be understood as establishing new foundations for the general practice of abbaco problem solving.
... However, during the abbacus tradition it gradually disappears. InHeeffer (2007a) we argue that this evolution is invoked by the specific rhetoric of abbacus problem solving.15 Pacioli 1494, f. 91 r : "Questa solverai per la 14 a chiave. ...
... The rhetorical structure compensated for the lack of symbolism but prohibited, for example, multiple solutions to quadratic problems. Where two positive solutions for certain type of quadratic problems were accepted in Arabic algebra, their recognition gradually fades within the abacus tradition (Heeffer, 2006). The reason is that the analytical structure of the problem solving process poses one specific arithmetical value as the cosa, the unknown. ...
Full-text available
When dealing with abductive reasoning in scientific discovery, historical case studies are focused mostly on the physical sciences, as with the discoveries of Kepler, Galilei and Newton. We will present a case study of abductive reasoning in early algebra. Two new concepts introduced by Cardano in his Ars Magna, imaginary numbers and a negative solution to a linear problem, can be explained as a result of a process of abduction. We will show that the first appearance of these new concepts fits very well Peirce's original description of abductive reasoning. Abduction may be regarded as one important strategy for the formation of new concepts in mathematics.
In this paper, we analyze the symbolic language used in some algebraic works of the second half of 16th century in the Iberian Peninsula. Examples show that in some cases the use of specific symbols was due to the constrictions of typography. In other cases symbolic language was used only as a simplification to contribute to a better understanding of rhetoric reasoning, and finally it was used as a part of symbolic reasoning, thereby moving forward from operations with numbers to operations with new objects of algebra. The works we refer to in this paper were written by the Iberian authors, Marco Aurel (fl. 1552), Juan Pérez de Moya (c. 1513-c.1597), Antic Roca (c. 1530-1580), Pedro Nunez (1502-1578) and Diego Pérez de Mesa (1563- c. 1633). We focus both on the symbolism for the unknowns and on the symbols to indicate the operations, as well as on the way these authors solved the systems of equations. This analysis will contribute to a better understanding of the status of algebra in works from the Iberian Peninsula in the 16th century.
This book presents contributions of mathematicians covering topics from ancient India, placing them in the broader context of the history of mathematics. Although the translations of some Sanskrit mathematical texts are available in the literature, Indian contributions are rarely presented in major Western historical works. Yet some of the well-known and universally-accepted discoveries from India, including the concept of zero and the decimal representation of numbers, have made lasting contributions to the foundation of modern mathematics. Key topics include: The work of two well-known Indian mathematicians: Brahmagupta and Bhaskaracharya; The relationship of Indian mathematics to the mathematics of China and Greece; The transmission of mathematical ideas between the Western and non-Western world; A study of Keralese mathematics and coverage of the techniques used in the Śulbasūtras; The calendrical calculations, complete with computer programs, enabling readers to determine Indian dates. Ancient Indian Leaps into Mathematics examines these ancient mathematical ideas that were spread throughout India, China, the Islamic world, and Western Europe. Through a systematic approach, it gives an historical account of ancient Indian mathematical traditions and their influence on other parts of the world. © Springer Science+Business Media, LLC 2011. All rights reserved.