Algebraic Equations East and West until the Middle Ages (original) (raw)
I sketch here a history of quadratic equation aiming at illustrating the potential fruitfulness of a non-linear history of algebraic equations. I show how a combination of three approaches is required to conduct such an inquiry in that particular case. Comparative history: in Babylonian, Chinese, and Greek sources, what we might recognize as quadratic equations manifests itself in very different ways depending on the corpus of texts considered. Conceptual history: this remark highlights that, in Antiquity, there were different concepts of equation available. We find respectively equation seen as problems (Mesopotamian sources), as operation depending on root extraction (Chinese sources), and as assertion of an equality (al-Khwarizmi). Historical documents show that, instead of one eliminating the other concepts, the history of algebraic equations evidences moments of synthesis between them. In particular, the work On equations by Sharaf al-Din al-Tusi at the end of the 12th century,...
Related papers
The Chinese algebraic method, the tian yuan shu, was developed during Song period (960-1279), of which Li Ye's works contain the earliest testimony. Two 18th century editors commentated on his works: the editor of the Siku quanshu and Li Rui, the latter responding to the former. Korean scholar Nam Byeong-gil added another response in 1855. Differences can be found in the way these commentators considered mathematical objects and procedures. The conflicting nature of these commentaries shows that the same object, the quadratic equation, can beget different interpretations, either a procedure or an assertion of equality. Textual elements in this paper help modern readers reconstruct different authors' understandings and reconsider the evolution of the definition of the object we now call 'equation'.
Diophantus and premodern algebra: New light on an old image
As a theme of historical research Diophantus’ work raises two main issues that have been intensely debated among researchers of the period: (i) The proper understanding of Diophantus’ practice; (ii) the recognition of the mathematical tradition to which this practice belongs. The traditional answer to this range of questions – since medieval Islam, through the Renaissance and the Early Modern period, up to the leading historians of mathematics of the 20th century – was that Diophantus’ book is a book on algebra. This traditional approach has been criticized recently by some historians of mathematics who point out the anachronistic methodology that historians in the past often were using in analyzing ancient texts. But, criticizing the methodology by which one defends a historical claim does not mean necessarily that the claim itself is wrong. The paper discusses some crucial issues involved in Diophantus’ problem-solving, thus, giving support to the traditional image about the algebraic character of Diophantus’ work, but put in a totally new framework of ideas.
Diophantus, al-Karaji, and quadratic equations
pp. 271-294 in: Michalis Sialaros, ed., Revolutions and Continuity in Greek Mathematics, 2018
In the beginning of his Arithmetica, Diophantus promises to show how to solve three-term quadratic equations. The rules are not in the extant part of his work, but al-Karajī (early eleventh century) provides clues as to how Diophantus may have derived and presented them. In his algebra book al-Fakhrī al-Karajī gives both the standard Arabic rule for solving three-term equations and the variation practiced by Diophantus. After proofs of the rules he gives a derivation that he calls "the method of Diophantus". By examining proofs and problems in several of al-Karajī's works we propose that Diophantus derived his rules for solving three term equations by this "method of Diophantus", and that al-Karajī borrowed both the derivations and the rules from the Arithmetica. That we are able to speak of Diophantus and al-Karajī both solving quadratic equations is due to the common conceptual foundations and procedures shared by Greek and Arabic algebra. Continuity from one to the other derives not from any chain of texts, but rather from oral tradition that apparently flourished across the old world, and which has left traces in our surviving texts. It is within this conceptual setting that a variation in rules for solving equations enables us to sort out al-Karajī's rules and postulate Diophantus's lost exposition.
2008
Algebra as we encounter it in or Descartes (1637) looks wholly different from what we know from al-Khwārizmī and Fibonacci. Indeed, early Modern algebra did not build on these: its foundation was the algebra of the Italian Abbacus school. The paper follows the development of this tradition from 1307 onward, in particular the appearance of abbreviations, the naming of powers and roots, formal calculations, schemes, and the solution of higher-degree equations.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.