Les commentaires d'al-Māhānī et d'un anonyme du Livre X des Éléments d'Euclide

1999 ◽  
Vol 9 (1) ◽  
pp. 89-156 ◽  
Author(s):  
Marouane Ben Miled
Keyword(s):  
Negative Numbers ◽  
Irrational Numbers ◽  
Quadratic Equations ◽  
First Time

This paper presents the first edition, translation and analyse of al-Māhānī’s commentary of the Book X of Euclid’s Elements (9th century, the most ancient to have reached us) and of an anonymous’ one (prior to 968, among the first algebraic commentaries). For the first time, irrational numbers are defined and classified. The algebraisation of Elements’ X-91 to 102, on the basis of al-Khwārizmī’s Algebra, shows irrational numbers as solution to algebraic quadratic equations. The algebraic calculus makes here the first steps. On this occasion, negative numbers and their calculation rules appears. Simplifications imposed by the algebraic writings are sometimes in opposition with the conclusions of propositions conceived in a purely geometrical framework, revealing a contradiction between geometrical and algebraic goals. It will be resolved by the independant way algebra will take with mathematicians belonging to the tradition of al-Karajī and al-Samaw’al from the 11th-12th centuries on.

The Great Art

2016 ◽  
Author(s):  
Joseph Mazur
Keyword(s):  
Eleventh Century ◽  
Negative Numbers ◽  
Square Roots ◽  
Quadratic Equations ◽  
Indian Mathematician ◽  
Northern India ◽  
First Millennium

This chapter discusses the origins of the art of algebra, beginning with the possibility that it may have come from the Greeks or from the Hindus. However, the Brahmins of northern India had some idea of algebra long before the Arabians learned it, contributed to it and brought that art to Spain in the late eleventh century. The Brahmasphutasiddhanta, written by the Indian mathematician Brahmagupta in 628, not only advanced the mathematical role of zero but also introduced rules for manipulating negative and positive numbers, methods for computing square roots, and systematic methods of solving linear and limited types of quadratic equations. The chapter also considers the contriburions of Abu Jafar Muhammad ibn Musa al-Khwārizmī and suggests that negative numbers originated in China, where they had been used since the beginning of the first millennium.

Six Rhetoric Quadratic: the Six Types of Quadratic Equations Presented by Abū Ja’far Muḥammad ibn Mūsā al-Khwārizmī in al-Kitāb al-Mukhtaṣor fī Ḥisāb al-Jabr wa al-Muqōbala

2019 ◽  
Author(s):  
Adib Rifqi Setiawan
Keyword(s):  
Middle East ◽  
Positive Solutions ◽  
Current View ◽  
Negative Numbers ◽  
Quadratic Equations ◽  
Mathematical Treatise

This work explains the six types of quadratic equations presented by Abū Ja’far Muḥammad ibn Mūsā al-Khwārizmī (Arabic: أبو جعفر محمد بن موسی الخوارزمی) in al-Kitāb al-Mukhtaṣor fī Ḥisāb al-Jabr wa al-Muqōbalah (Arabic: الكتاب المختصر في حساب الجبر والمقابلة). In this mathematical treatise written approximately 820 CE, equations are verbally described in terms of “squares” (Arabic: مربع الجذر; what would today be “x^2”), “roots” (Arabic: الجذور; what would today be “x”) and “numbers” (Arabic: الأعداد; “constants”: ordinary spelled out numbers, like ‘twenty-six’). The six types equations are:[1] <المربعات تساوي الجذور> or squares equal roots or with current notations ax^2=bx;[2] <المربعات تساوي الأعداد> or squares equal number or ax^2=c;[3] <الجذور تساوي الأعداد> or roots equal number or bx=c;[4] <المربعات والجذور تساوي الأعداد> or squares and roots equal number or ax^2+bx=c;[5] <المربعات والأعداد تساوي الجذور> or squares and number equal roots or ax^2+c=bx; and[6] <الجذور والأعداد تساوي المربعات> or roots and number equal squares or bx+c=ax^2.Al-Kitāb al-Mukhtaṣor fī Ḥisāb al-Jabr wa al-Muqōbala thoroughly rhetorical, with the syncopation that is the numbers were written out in words rather than symbols. However, in author’s day, most of this notation had not yet been invented, so he had to use ordinary text to present problems and their solutions. The types of problems which the book discusses reveals middle east mathematicians didn’t deal with negative numbers at all. Hence an equation like bx+c=0 doesn’t appear in the classification, because it has no positive solutions if all the coefficients are positive. Similarly equation types 4, 5 and 6, which look equivalent to our current view, were distinguished because the coefficients must all be positive.

How many rugby balls can you fit in a minibus?

1991 ◽  
Vol 75 (471) ◽  
pp. 32-39
Author(s):  
Nigel Walkey ◽  
Gerald Goodall
Keyword(s):  
National Curriculum ◽  
Growing Up ◽  
Mathematics Lesson ◽  
Quadratic Equations ◽  
University Mathematics ◽  
First Time ◽  
Sixth Form

GCSE mathematics and Sixth Form Mathematics are now out of step. Sixth Form Mathematics and University Mathematics may also soon be out of step. These are two far reaching statements—but what has this got to do with rugby balls and a minibus?By now many of us currently or recently involved in GCSE mathematics, whether as students or teachers, have become used to mathematical investigations, applied problems and project write ups all lasting about two weeks. How times have changed. It is now quite common for 15 and 16 year olds in their GCSE mathematics lesson to work in groups on open ended problems. Gone are the days (familiar to some of us!) of solving endless lists of quadratic equations and stopping without hesitation the first time b2 is less than 4ac. Generations of pupils will now, if the National Curriculum does not completely move “the goal posts”, be growing up on a new range of discoveries made possible by the introduction of fresh ideas and new starting points. “222 and all that” and “Watch out” for example are two standard pieces of coursework encountered by many pupils in their GCSE training.

Irrational Numbers, Square Roots, and Quadratic Equations

2015 ◽  
Vol 20 (8) ◽  
pp. 468-474
Author(s):  
Gorjana Popovic
Keyword(s):  
Irrational Numbers ◽  
Square Roots ◽  
Quadratic Equations

How do we introduce irrational numbers without simply telling students that they are irrationals?

scholarly journals BPS black hole entropy and attractors in very special geometry. Cubic forms, gradient maps and their inversion

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Bert van Geemen ◽  
Alessio Marrani ◽  
Francesco Russo
Keyword(s):  
Black Hole ◽  
Black Hole Entropy ◽  
Extremal Black Hole ◽  
Inhomogeneous System ◽  
Homogeneous Polynomials ◽  
Quadratic Equations ◽  
Cubic Form ◽  
Black Hole Background ◽  
Cubic Forms ◽  
First Time

Abstract We consider Bekenstein-Hawking entropy and attractors in extremal BPS black holes of $$ \mathcal{N} $$ N = 2, D = 4 ungauged supergravity obtained as reduction of minimal, matter-coupled D = 5 supergravity. They are generally expressed in terms of solutions to an inhomogeneous system of coupled quadratic equations, named BPS system, depending on the cubic prepotential as well as on the electric-magnetic fluxes in the extremal black hole background. Focussing on homogeneous non-symmetric scalar manifolds (whose classification is known in terms of L(q, P, Ṗ) models), under certain assumptions on the Clifford matrices pertaining to the related cubic prepotential, we formulate and prove an invertibility condition for the gradient map of the corresponding cubic form (to have a birational inverse map which is given by homogeneous polynomials of degree four), and therefore for the solutions to the BPS system to be explicitly determined, in turn providing novel, explicit expressions for the BPS black hole entropy and the related attractors as solution of the BPS attractor equations. After a general treatment, we present a number of explicit examples with Ṗ = 0, such as L(q, P), 1 ⩽ q ⩽ 3 and P ⩾ 1, or L(q, 1), 4 ⩽ q ⩽ 9, and one model with Ṗ = 1, namely L(4, 1, 1). We also briefly comment on Kleinian signatures and split algebras. In particular, we provide, for the first time, the explicit form of the BPS black hole entropy and of the related BPS attractors for the infinite class of L(1, P) P ⩾ 2 non-symmetric models of $$ \mathcal{N} $$ N = 2, D = 4 supergravity.

Annulate lamellae in the Sertoli cells of guinea pig testis after unilateral torsion of the spermatic cord

1984 ◽  
Vol 42 ◽  
pp. 196-197
Author(s):  
J. Chakraborty ◽  
A. P. Sinha Hikim ◽  
J. S. Jhunjhunwala
Keyword(s):  
Sertoli Cells ◽  
Guinea Pigs ◽  
Spermatic Cord ◽  
Present Report ◽  
Cell Types ◽  
Annulate Lamellae ◽  
Spermatogenic Cells ◽  
Electron Microscopic ◽  
Structural Details ◽  
First Time

Although the presence of annulate lamellae was noted in many cell types, including the rat spermatogenic cells, this structure was never reported in the Sertoli cells of any rodent species. The present report is based on a part of our project on the effect of torsion of the spermatic cord to the contralateral testis. This paper describes for the first time, the fine structural details of the annulate lamellae in the Sertoli cells of damaged testis from guinea pigs.One side of the spermatic cord of each of six Hartly strain adult guinea pigs was surgically twisted (540°) under pentobarbital anesthesia (1). Four months after induction of torsion, animals were sacrificed, testes were excised and processed for the light and electron microscopic investigations. In the damaged testis, the majority of seminiferous tubule contained a layer of Sertoli cells with occasional spermatogonia (Fig. 1). Nuclei of these Sertoli cells were highly pleomorphic and contained small chromatinic clumps adjacent to the inner aspect of the nuclear envelope (Fig. 2).

Electron spectroscopic imaging and diffraction

1991 ◽  
Vol 49 ◽  
pp. 706-707
Author(s):  
M. Rühle ◽  
J. Mayer ◽  
J.C.H. Spence ◽  
J. Bihr ◽  
W. Probst ◽  
...  
Keyword(s):  
Materials Science ◽  
Electron Spectroscopic Imaging ◽  
Magnetic Filter ◽  
Magnetic Imaging ◽  
Illumination System ◽  
Probe Size ◽  
Diffraction Patterns ◽  
Fritz Haber ◽  
Koehler Illumination ◽  
First Time

A new Zeiss TEM with an imaging Omega filter is a fully digitized, side-entry, 120 kV TEM/STEM instrument for materials science. The machine possesses an Omega magnetic imaging energy filter (see Fig. 1) placed between the third and fourth projector lens. Lanio designed the filter and a prototype was built at the Fritz-Haber-Institut in Berlin, Germany. The imaging magnetic filter allows energy-filtered images or diffraction patterns to be recorded without scanning using efficient area detection. The energy dispersion at the exit slit (Fig. 1) results in ∼ 1.5 μm/eV which allows imaging with energy windows of ≤ 10 eV. The smallest probe size of the microscope is 1.6 nm and the Koehler illumination system is used for the first time in a TEM. Serial recording of EELS spectra with a resolution < 1 eV is possible. The digital control allows X,Y,Z coordinates and tilt settings to be stored and later recalled.

A Novel TEM Technique for Characterizing As-Grown CVD Diamond Films

1991 ◽  
Vol 49 ◽  
pp. 956-957 ◽  
Author(s):  
Z.L. Wang ◽  
J. Bentley ◽  
R.E. Clausing ◽  
L. Heatherly ◽  
L.L. Horton
Keyword(s):  
Diamond Films ◽  
Chemical Vapor ◽  
Growth Direction ◽  
Dark Field ◽  
Growth Surface ◽  
Specimen Preparation ◽  
Transmission Electron ◽  
Diffraction Patterns ◽  
First Time ◽  
Microstructural Studies

Microstructural studies by transmission electron microscopy (TEM) of diamond films grown by chemical vapor deposition (CVD) usually involve tedious specimen preparation. This process has been avoided with a technique that is described in this paper. For the first time, thick as-grown diamond films have been examined directly in a conventional TEM without thinning. With this technique, the important microstructures near the growth surface have been characterized. An as-grown diamond film was fractured on a plane containing the growth direction. It took about 5 min to prepare a sample. For TEM examination, the film was tilted about 30-45° (see Fig. 1). Microstructures of the diamond grains on the top edge of the growth face can be characterized directly by transmitted electron bright-field (BF) and dark-field (DF) images and diffraction patterns.

A microscopic marker experiment study on high temperature oxidation of Ni-Al alloy

1986 ◽  
Vol 44 ◽  
pp. 514-515
Author(s):  
Shou-kong Fan
Keyword(s):  
Transport Mechanism ◽  
High Temperature Oxidation ◽  
Transverse Section ◽  
Al Alloy ◽  
Metal Interface ◽  
Electron Microscopic ◽  
Temperature Oxidation ◽  
Analytical Electron ◽  
Microscopic Studies ◽  
First Time

Transmission and analytical electron microscopic studies of scale microstructures and microscopic marker experiments have been carried out in order to determine the transport mechanism in the oxidation of Ni-Al alloy. According to the classical theory, the oxidation of nickel takes place by transport of Ni cations across the scale forming new oxide at the scale/gas interface. Any markers deposited on the Ni surface are expected to remain at the scale/metal interface after oxidation. This investigation using TEM transverse section techniques and deposited microscopic markers shows a different result,which indicates that a considerable amount of oxygen was transported inward. This is the first time that such fine-scale markers have been coupled with high resolution characterization instruments such as TEM/STEM to provide detailed information about evolution of oxide scale microstructure.

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