Sharaf al-Dīn al-Ṫūsī on the number of positive roots of cubic equations

Hogendijk, Jan P.

doi:10.1016/0315-0860(89)90099-2

Cited by 22 publications

(9 citation statements)

References 3 publications

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“…The conic section based solution of cubic equations by Al-Khayyām (Woepcke 1851) and the foreshadowing of the Ruffini-Horner scheme by Al-Ṭūsī (Rashed 1986;Hogendijk 1989), are irrelevant both in terms of their form and their date (11 th and 12 th -century; see also Rashed 1994, Ch. 3).…”

Section: Transmission Outside India?mentioning

confidence: 99%

Citrabhānu's Twenty-One Algebraic Problems in Malayalam and Sanskrit

Wagner

2015

Historia Mathematica

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This paper studies the Sanskrit and Malayalam versions of Citrabhānu's Twenty-One problems: a discussion of quadratic and cubic problems from 16 th -century Kerala. It reviews the differences in the approaches of the two versions, highlighting the distinction between the Sanskrit indeterminate integer-remainder arithmetical techniques and the Malayali fixed point iterations. The paper concludes with some speculations on the possible transmission of algebraic knowledge between Kerala and the west.

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Section: Transmission Outside India?mentioning

confidence: 99%

Citrabhānu's Twenty-One Algebraic Problems in Malayalam and Sanskrit

Wagner

2015

Historia Mathematica

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“…In discussing the possibilities of solution of a cubic equation of the form f (x) ϭ x 3 ϩ ax 2 ϩ bx ϭ c, Sharaf al-Dīn exhibits a quadratic polynomial equivalent to fЈ(x) and shows that if a is the root of fЈ(x) ϭ 0 that yields the larger value of f (a) then f (x) ϭ c has no root for c Ͼ f (a) and one root if c ϭ f (a). 6 Roshdi Rashed feels that this is evidence that Sharaf al-Dīn had recognized the derivative, but Jan Hogendijk [64] 7 argues that this is not the case and that one can obtain Sharaf al-Dīn's conditions by arguments that do not go beyond the mathematics of Elements, II. The continuing efforts to understand Sharaf al-Dīn's motivation in more modern terms, found in Nicholas Farè s [50] and Christian Houzel [73], show that this controversy has not yet been settled.…”

Section: Al-khwā Rizmī and His Timesmentioning

confidence: 99%

“…7 This paper also contains a useful and concise summary of the contents of Sharaf's treatise. 8 It is regrettable that Farè s makes no reference to Hogendijk [64].…”

Section: Al-khwā Rizmī and His Timesmentioning

confidence: 99%

Mathematics and Her Sisters in Medieval Islam: A Selective Review of Work Done from 1985 to 1995

Berggren

1997

Historia Mathematica

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This paper surveys work done over the past decade, largely in Western Europe and North America, in the history of the mathematical sciences as practiced in medieval Islam from central Asia to Spain. Among the major topics covered, in addition to the usual branches of mathematics, are mathematical geography, astronomy, and optics. We have also given accounts of some current debates on the interpretation of important texts and, in addition, we have surveyed some of the literature dealing with the interrelation of mathematics and society in medieval Islam. © 1997 Academic Press Cet exposé pré sente un rapport de recherche fait au cours de la derniè re dé cennie en histoire des sciences mathé matiques durant le Moyen Â ge. Il é value l'influence islamique de cette pé riode dans le territoire qui s'é tend de l'Asie centrale jusqu'en Espagne. Parmi les sujets traité s, on y trouve la gé ographie, l'astronomie, et l'optique en plus des branches habituelles des mathé matiques. Nous donnons un compte rendu des dé bats actuels en ce qui concerne l'interpré tation de publications islamiques de cette é poque. De plus à partir des textes litté raires, on a examiné la relation entre les mathé matiques et la socié té islamique du Moyen Â ge.

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“…Although he never put a name to it in Arabic, which like Latin later on in Europe was the language of scholarship in the Muslim world, he also used the derivative to find the maxima of polynomials, which until recently historians of mathematics had attributed to the 16th century French mathematician François Vi'ete [15,19]. Sharaf al-din Tusi who died only six years before the cataclysmic Mongol invasion of Iran in 1219 is now considered to be the forerunner of algebraic geometry.…”

Section: Introductionmentioning

confidence: 99%

A Continuous Derivative for Real-Valued Functions

Edalat

2007

Lecture Notes in Computer Science

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We develop a notion of derivative of a real-valued function on a Banach space, called the L-derivative, which is constructed by introducing a generalization of Lipschitz constant of a map. As with the Clarke gradient, the values of the L-derivative of a function are non-empty weak* compact and convex subsets of the dual of the Banach space. The L-derivative, however, is shown to be upper semi continuous, a result which is not known to hold for the Clarke gradient. We also formulate the notion of primitive maps dual to the L-derivative, an extension of Fundamental Theorem of Calculus for the L-derivative and a domain for computation of real-valued functions on a Banach space with a corresponding notion of effectivity. For real-valued functions on finite dimensional Euclidean spaces, the L-derivative can be obtained within an effectively given continuous domain. We also show that in finite dimensions the L-derivative and the Clarke gradient coincide thus providing a computable representation for the latter in this case.This paper is dedicated to the historical memory of Sharaf al-din Tusi (d. 1213), the Iranian mathematician who was the first to use the derivative systematically to solve for roots of cubic polynomials and find their maxima.

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Sharaf al-Dīn al-Ṫūsī on the number of positive roots of cubic equations

Cited by 22 publications

References 3 publications

Citrabhānu's Twenty-One Algebraic Problems in Malayalam and Sanskrit

Citrabhānu's Twenty-One Algebraic Problems in Malayalam and Sanskrit

Mathematics and Her Sisters in Medieval Islam: A Selective Review of Work Done from 1985 to 1995

A Continuous Derivative for Real-Valued Functions

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