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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

albrecht.heeffer@ugent.be

Second revision: 28/07/06

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

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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

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

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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

1929):

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

mathematics.

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

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

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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

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

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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

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

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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

Arab

لﺎﻤ

ﻲﺸﺀﱞ

ﺮﺬﺠ

ﺮﺪﻢه

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

3.1.2.1. 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.

3.1.2.2. 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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2

14

2

x

xx

⎛⎞

+=

⎜⎟

⎜⎟

⎝⎠

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:

2

22 2

14

2

x

xx

⎛⎞

+=

⎜⎟

⎜⎟

⎝⎠

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

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.

subtraction.

The procedure thus corresponds with the following formula:

2

1,2 22

bb

x

c

⎛⎞

=± −

⎜⎟

⎝⎠ .

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

234

x

x+=

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

2

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):

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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:

1012

64

010

ya v ya ru

ya v ya ru

literally transcribed:

22

10120 0

64

xxxx

+

+= ++

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:

()

2

32 256x−=

132

016

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:

22

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.

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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

equation

221 10

x

x+=

which is given in its direct form, corresponds remarkably well with a standard type of

problem from Babylonian algebra:

10

21

ab

ab

+=

=

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

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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

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

2910

x

x+=

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

2nn

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

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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.

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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

()()

8095xx++

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

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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

x

x+=

The binomial 221x

+

is coequal with the monomial 10

x

, 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

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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

5.2.1.1. 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

x

xx=−

. In this interpretation the al-jabr is understood

as the addition of 2

4

x

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

x

x=+

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

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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

x

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,

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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

x

xx=− Deinde restaurabis quadraginta per

quattuor census. Post hoc addes census

censui

247-

248

VI.3 410

x

x=− Restaura itaque decem per rem, et adde

ipsam quattuor.

248

VI.5 2

100 2 20 55xx+−=

Restaura ergo centum et duos census per

res que fuerunt diminute, et adde eas

quinquaginta octo.

249

VII.1 2

10 21xx−= Restaura igitur decem excepta re per

censum, et adde censum viginti uno.

250

VII.4 22

21

21 2 100 2 2

36

x

xx x x+− −= + −

Restaura ergo illud, et adde duos census

et sextam centum et duobus censibus

exceptis viginti rebus

251

VII.5 21

100 20 2

x

xx+− = Restaura igitur centum et adde viginti

res medietati rei.

252

VII.6 2

100 20 81

x

xx+− = Restaura ergo centum, et adde viginti

radices octoginta uni.

252

2

11

52 10 10

22

x

xx−=−

Restaura ergo quinquaginta duo et semis

per decem radices et semis, et adde eas

decem radicibus excepto censu.

253

VII.8

2

11

52 20

22

x

x=−

Deinde restaura eas per censum et

adde censum quinquaginta duobus et

semis.

253

VIII.1 2

100 20 81xx+− = Restaura ergo centum et adde viginti

radices octoginta uni et erunt centum et

census,

257

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

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 21 -

demonstration depends on the completion of the polynomial 2

(5)25x

+

− 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

2

111

120

12 3 4

xxx++ + = and 2

124

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.

5.2.1.2. 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

x2

5x

5x

māl

Figure 2: completing the square (from al-

Khwārizmī)

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 22 -

22

11

42 4 100 2 20

24

x

xxx−=+−

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

x

+, the second to 1

62 2

x

.

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 10x – x2 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

equation.

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):

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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

to

22

100 20 8 40xxx+− − = .

So, now the question is, in al- Karkhī’s terminology: what is restored here, the composed

number 2

100

x

+ or the simple number 2

x

? The text of Abū Kāmil leaves no doubt: “Restaura

ergo 100 et censum cum 20 [radicibus] rebus et radice 8 censuum”. Thus the polynomial

2

100

x

+ is restored by 2

20 8

x

x+.

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.

5.2.1.3. 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:

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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

2

1

3

x

to 22

1

3

x

x−

is basically the same act as restoration

2

2

3

x

back to the form 2

x

. 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:

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 25 -

• An operation aiming at the restoration of a defected quantity to its original

completeness.

• 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

x

xx

+

−+

as

2

100 2 20

x

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

coequancia.

Thus al-Khwārizmī applies al-jabr to 2

100 2 20 58xx

+

−=

in order to restore 2

100

x

+,

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

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

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

below.

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.

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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

Rosen

(from Arab)

reduce

separate

reduce reduce complete

Robert of Chester restaurare

complere

opponere converte complere

Karpinski

(from Robert)

restore

complete

by opposition reduce complete

Gerard restaurare opponere reducere reintegrare

Guglielmo restaurare eicere reducere restaurare

reliquitur

integer

Abū Kāmil restaurare

reintegrare

opponere reducere complere

(geometrical)

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

equation.

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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

other.

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

polynomials.

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.

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

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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

Practicae

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,

241).

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.

A CONCEPTUAL ANALYSIS OF EARLY 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

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Milan, Ambrosiano P 81 sup (transcription and Italian translation by Picutti, 1983)

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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

.

34

xx

x

=

⎛⎞⎛⎞

⎜⎟⎜⎟

⎝⎠⎝⎠ , while the form

22

2

.

34

xx

x

=

⎛⎞⎛⎞

⎜⎟⎜⎟

⎝⎠⎝⎠

would be more consistent with his interpretation of the māl.

3 Also argued by Høyrup (1998, note 11).

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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 http://mathforum.org/kb/forum.jspa?forumID=149&start=0

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

72

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

A CONCEPTUAL ANALYSIS OF EARLY ARABIC ALGEBRA

- 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).