Papers by Sammarchi Eleonora
Arabic Sciences and Philosophy, 2025
In Arabic treatises on algebra, Book II of Euclid’s Elements quickly became a
traditional work of... more In Arabic treatises on algebra, Book II of Euclid’s Elements quickly became a
traditional work of reference, especially for justifying quadratic equations. However, in
many of these treatises we find a representation of Euclid’s notions that deviates from
the “original Euclid.” In this article, I focus on the way in which propositions of Book II
were understood and reported by al-Karaǧī (11th c.) in two of his algebraic writings.
Inspired by the variety of arithmetical practices of his time, al-Karaǧī transposed these
Euclidean propositions from geometrical objects to numbers and applied them to an
algebraic context. This allowed him to combine various argumentative strategies deriving
from different fields. Building upon al-Karaǧī’s work, al-Zanǧānī (13th c.) no longer
needed to mention Euclid and instead conceived of a justification of quadratic equations
(the “cause” of the equation) which is completely internal to algebra. These case
studies provide evidence for the use of the Elements as a toolbox for the development of
algebra. More importantly, they shed further light upon a typical feature of medieval
mathematics, namely the existence of a plurality intrinsic in the name “Euclid.”
Historia Mathematica, 2024
In the texts of the arithmetical-algebraic tradition initiated by al-Karajī (11th c.), the terms ... more In the texts of the arithmetical-algebraic tradition initiated by al-Karajī (11th c.), the terms of a composite algebraic expression are considered to be positive or negative only in the sense of relational entities. Indeed, within the rules for operating with these algebraic expressions, additives and subtractives represent operands, and tabular methods allow one to compute with subtractives more freely. The strategies adopted for impossible problems further confirm the relational nature of these entities. In my article, I examine these aspects by considering the work of al-Zanjānī (13th c.) and of his predecessors al-Karajī and al-Samaw'al.
Atti del Convegno della Società Italiana di Storia della Scienza, 2023
Within medieval and early modern mathematics, arithmetic and algebra were closely related to each... more Within medieval and early modern mathematics, arithmetic and algebra were closely related to each other in terms of rules, methods, and problems.As a result, they were characterized by several types of interactions-or contaminations-. In this paper, I focus on one of these contaminations, related to the study of numbers. I take into account several case studies from Mediterranean medieval mathematics, belonging to two different traditions of arithmetician-algebraists. The first is the Arabic tradition of the ḥussāb (calculators), represented here by the mathematician Abū Bakr al-Karajī (d. beg. 11th c.) and his successors al-Samaw'al (d. 12 th c.) and al-Zanjānī (d. mid 13 th c.). The second is the Italian tradition of the maestri d'abaco, within which I focus on two authors-Paolo Gherardi and an anonymous author-of the 14th century. Within my investigation of the notion of number, I concentrate on integers and fractions and-only in relation to the Arabic context-on expressible and surd numbers (i.e. rational and irrational numbers). First, I clarify the polysemic nature of the term "arithmetic" and mention some aspects related to the historical context that characterized my authors. Second, I analyze the primary sources and present some further observations that build upon earlier studies dedicated to these same sources. My main goal is to show that the development of algebra contributes to the expansion of the domain of numbers, and hence to the expansion of arithmetic.
EMS Newsletter, 2019
Al-Karajı − 's school of arithmetician-algebraists One of the major accomplishments of ninth-cent... more Al-Karajı − 's school of arithmetician-algebraists One of the major accomplishments of ninth-century algebraists such as al-Khwa − rizmı − and Abu − Ka − mil was the elaboration of a theory for second-degree equations that could be applied in order to solve both geometrical and arithmetical problems. Once this theory was established, algebraists redirected their interest to new topics. By the end of the tenth century, the mathematician al-Karajı − chose to investigate the interaction between arithmetic and algebra, and began to create a coherent and exhaustive system of rules for calculating with algebraic entities. His work gave rise to a new tradition of arithmetician-algebraists, whose aim was to improve algebra with the help of arithmetic and viceversa. This tradition focused on the notion of operation, and its aim was to make the algebraist able to manipulate unknown quantities as the arithmetician manipulates known ones. Al-Karajı − 's research was then improved upon by the twelfth-century scholar al-Samaw'al (d. 1135). In the middle of the thirteenth century, the Persian mathematician al-Zanja − nı − followed this same tradition, and his Qist • a − s al-mu'a − dala fı − 'ilm al-jabr wa'l-muqa − bala (Balance of the equation in the science of algebra and muqa − bala) accurately recalls and elaborates upon al-Karajı − 's work. Algebraic powers In the presentation of the rules for algebraic operations, unknown quantities are considered as either simple or composed entities. The basic terms are the algebraic powers. These are defined by al-Zanja − nı − as follows: A thing (shay') multiplied by itself is called a root, and the result [of the multiplication] is a square […] The product of the root by the square is a cube, and by the cube a square-square, and by the square-square a square-cube, and by the square-cube a cube-cube, and by the cube-cube a square-square-cube, and so on. If the root is two, the square is four, the cube is eight, the square-square is sixteen, the square-cube is thirty-two, the cube-cube is sixty-four and the square-squarecube a hundred and twenty-eight. 2
Médiévales (77), 2019
À partir de la fin du Xe siècle, une nouvelle école d’algébristes se constitue
autour de la figur... more À partir de la fin du Xe siècle, une nouvelle école d’algébristes se constitue
autour de la figure du mathématicien al-Karajī. Elle vise à explorer les rapports entre arithmétique et algèbre. Le livre Qisṭās al-mu‘ādala fī ‘ilm al-jabr wa al-muqābala (Balance de l’équation sur la science de la restauration et de la comparaison) du mathématicien al-Zanjānī (première moitié du XIIIe siècle) est un traité sur le calcul algébrique qui s’inscrit clairement dans cette tradition de recherche arithmético-algébrique. Il se compose d’une partie théorique et de deux longues collections de problèmes. Afin d’étudier les deux collections il est indispensable de considérer, au préalable, l’histoire de la transmission de ces problèmes, en examinant les continuités et les discontinuités entre le recueil d’al-Zanjānī et ceux des auteurs qui ont constitué pour lui une référence directe ou indirecte. Outre cet héritage,
l’étude révèle également que c’est dans les problèmes que la théorie du
calcul algébrique s’explique et se concrétise. Ainsi, les problèmes montrent
l’algèbre d’al-Zanjānī en action. Ils peuvent également offrir à l’algébriste la
possibilité de repérer des nouvelles branches de sa propre discipline, comme
le cas de l’analyse indéterminée le témoigne. Dans cet article, nous nous
concentrons sur la structure qui caractérise Qisṭās al-mu‘ādala et détaillons la résolution de quelques exemples de problèmes déterminés et indéterminés contenus dans l’ouvrage.
Book Reviews by Sammarchi Eleonora
Revue d’Histoire des Sciences , 2019
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Papers by Sammarchi Eleonora
traditional work of reference, especially for justifying quadratic equations. However, in
many of these treatises we find a representation of Euclid’s notions that deviates from
the “original Euclid.” In this article, I focus on the way in which propositions of Book II
were understood and reported by al-Karaǧī (11th c.) in two of his algebraic writings.
Inspired by the variety of arithmetical practices of his time, al-Karaǧī transposed these
Euclidean propositions from geometrical objects to numbers and applied them to an
algebraic context. This allowed him to combine various argumentative strategies deriving
from different fields. Building upon al-Karaǧī’s work, al-Zanǧānī (13th c.) no longer
needed to mention Euclid and instead conceived of a justification of quadratic equations
(the “cause” of the equation) which is completely internal to algebra. These case
studies provide evidence for the use of the Elements as a toolbox for the development of
algebra. More importantly, they shed further light upon a typical feature of medieval
mathematics, namely the existence of a plurality intrinsic in the name “Euclid.”
autour de la figure du mathématicien al-Karajī. Elle vise à explorer les rapports entre arithmétique et algèbre. Le livre Qisṭās al-mu‘ādala fī ‘ilm al-jabr wa al-muqābala (Balance de l’équation sur la science de la restauration et de la comparaison) du mathématicien al-Zanjānī (première moitié du XIIIe siècle) est un traité sur le calcul algébrique qui s’inscrit clairement dans cette tradition de recherche arithmético-algébrique. Il se compose d’une partie théorique et de deux longues collections de problèmes. Afin d’étudier les deux collections il est indispensable de considérer, au préalable, l’histoire de la transmission de ces problèmes, en examinant les continuités et les discontinuités entre le recueil d’al-Zanjānī et ceux des auteurs qui ont constitué pour lui une référence directe ou indirecte. Outre cet héritage,
l’étude révèle également que c’est dans les problèmes que la théorie du
calcul algébrique s’explique et se concrétise. Ainsi, les problèmes montrent
l’algèbre d’al-Zanjānī en action. Ils peuvent également offrir à l’algébriste la
possibilité de repérer des nouvelles branches de sa propre discipline, comme
le cas de l’analyse indéterminée le témoigne. Dans cet article, nous nous
concentrons sur la structure qui caractérise Qisṭās al-mu‘ādala et détaillons la résolution de quelques exemples de problèmes déterminés et indéterminés contenus dans l’ouvrage.
Book Reviews by Sammarchi Eleonora
traditional work of reference, especially for justifying quadratic equations. However, in
many of these treatises we find a representation of Euclid’s notions that deviates from
the “original Euclid.” In this article, I focus on the way in which propositions of Book II
were understood and reported by al-Karaǧī (11th c.) in two of his algebraic writings.
Inspired by the variety of arithmetical practices of his time, al-Karaǧī transposed these
Euclidean propositions from geometrical objects to numbers and applied them to an
algebraic context. This allowed him to combine various argumentative strategies deriving
from different fields. Building upon al-Karaǧī’s work, al-Zanǧānī (13th c.) no longer
needed to mention Euclid and instead conceived of a justification of quadratic equations
(the “cause” of the equation) which is completely internal to algebra. These case
studies provide evidence for the use of the Elements as a toolbox for the development of
algebra. More importantly, they shed further light upon a typical feature of medieval
mathematics, namely the existence of a plurality intrinsic in the name “Euclid.”
autour de la figure du mathématicien al-Karajī. Elle vise à explorer les rapports entre arithmétique et algèbre. Le livre Qisṭās al-mu‘ādala fī ‘ilm al-jabr wa al-muqābala (Balance de l’équation sur la science de la restauration et de la comparaison) du mathématicien al-Zanjānī (première moitié du XIIIe siècle) est un traité sur le calcul algébrique qui s’inscrit clairement dans cette tradition de recherche arithmético-algébrique. Il se compose d’une partie théorique et de deux longues collections de problèmes. Afin d’étudier les deux collections il est indispensable de considérer, au préalable, l’histoire de la transmission de ces problèmes, en examinant les continuités et les discontinuités entre le recueil d’al-Zanjānī et ceux des auteurs qui ont constitué pour lui une référence directe ou indirecte. Outre cet héritage,
l’étude révèle également que c’est dans les problèmes que la théorie du
calcul algébrique s’explique et se concrétise. Ainsi, les problèmes montrent
l’algèbre d’al-Zanjānī en action. Ils peuvent également offrir à l’algébriste la
possibilité de repérer des nouvelles branches de sa propre discipline, comme
le cas de l’analyse indéterminée le témoigne. Dans cet article, nous nous
concentrons sur la structure qui caractérise Qisṭās al-mu‘ādala et détaillons la résolution de quelques exemples de problèmes déterminés et indéterminés contenus dans l’ouvrage.