A Comparative Study on Three Pioneer Methods for the Square Roots

During the classical period, the South Asian region had an illustrious history of mathematics, and it was regarded as fertile ground for the birth of pioneer mathematicians that produced a wide range of mathematical ideas and creations that made significant contributions. Among them, three creative personalities Bhaskaracarya, Gopal Pande and Bharati Krishna Tirthaji and their specific methods to find square roots are focused on this study. The analytical study of their methods is expressing in comparison with similarities, variety and simplicity. Each of the three mathematical treatise has its own formula for calculating the square roots. The Lilavati seems to have some effect upon the Vedic and Pande’s systems. In spite of having influenced by Lilavati, Gopal Pande disagreed on the problems regarding square roots and cube roots. To prove his point, Gopal Pande used the unitary method against the method described in Bhaskaracarya's famous book Lilavati. In the case of practicality and simplicity, the Vedic method is more practical, interesting and simpler to understand for the mathematics learners in comparison to the other two methods.

Disseminação do estudo de Análise Matemática e a Repercussão de Instituzioni Analitiche de Maria Gaetana Agnesi

ResumoNeste artigo apresentamos alguns desdobramentos relativos à divulgação e repercussão da obra Instituzioni Analitiche ad uso dela giuveniu italiana, por ocasião de sua publicação em Milão, em 1748, e nos cinquenta anos posteriores, sobretudo em função do direcionamento dado por Maria Gaetana Agnesi (1718-1799) ao seu tratado matemático. Para compreender o lugar ocupado pela estudiosa e sua obra na história da matemática, é essencial situá-la em malhas contextuais mais amplas, de modo a abarcar parte do processo de circulação dos discursos e da divulgação da álgebra e do cálculo, naquele contexto. Mediante este enfoque, a abordagem metodológica adotada neste trabalho se baseou em uma análise documental, a partir da articulação das esferas epistemológica, historiográfica e contextual, na concepção de Alfonso-Goldfarb e Ferraz. Considerando que uma interligação entre tais esferas constituí uma empreitada não trivial, nosso corpus é composto pela obra matemática Instituzioni Analitiche, as correspondencias de Agnesi com seus contemporâneos, além de alguns trabalhos de estudiosos que se debruçaram sobre a vida e obra da estudiosa, como forma de trazer à luz indícios de que houve interesse e comprometimento de Agnesi em divulgar seu tratado para além do solo milanês, e à vista disso, este teve ampla repercussão, a despeito de ter sido esquecido, em sua maioria, sob muitos aspectos, pelos historiadores da matemática.Palavras-chave: História da matemática, Educação matemática, Maria Gaetana Agnesi, Análise matemática, História das ciências.AbstractIn this article, we present some developments related to the dissemination and repercussion of the work Instituzioni Analitiche ad usage della giuveniu Italiana, on its publication in Milan, in 1748, and the following fifty years, mainly due to the direction given by Maria Gaetana Agnesi (1718- 1799) to her mathematical treatise. To understand the place occupied by the scholar and her work in the history of mathematics, it is essential to place it in broader contextual networks, to cover part of the process of circulation of discourses and the dissemination of algebra and calculus in that context. From this perspective, the methodological approach adopted in this work was based on a documentary analysis, from the articulation of the epistemological, historiographical, and contextual spheres, in Alfonso-Goldfarb and Ferraz’s conception. Considering that an interconnection between such spheres constitutes a non-trivial endeavour, our corpus is composed of the mathematical work Instituzioni Analitiche, Agnesi’s correspondence with contemporaries, and some studies based on her life and work, bringing to light evidence that she was interested in and committed to having her treaty publicised beyond Milanese lands, which gave it extensive repercussion. However, despite her importance, Agnesi has been forgotten, in many aspects, by the historians of mathematics.Keywords: History of mathematics, Mathematics education, Maria Gaetana Agnesi, Mathematical analysis, History of sciences.ResumenEn este artículo presentamos algunos desarrollos relacionados con la difusión y repercusión de la obra Instituzioni Analitiche ad use della giuveniu Italiana, en el marco de su publicación en Milán, en 1748, y los siguientes cincuenta años, principalmente debido a la dirección que Maria Gaetana Agnesi (1718-1799) dió a su tratado matemático. Para comprender el lugar que ocupa la académica y su obra en la historia de las matemáticas, es fundamental ubicarla en redes contextuales más amplias, para abarcar parte del proceso de circulación de los discursos y la difusión del álgebra y el cálculo en ese contexto. Desde esta perspectiva, el enfoque metodológico adoptado en este trabajo se basó en un análisis documental, a partir de la articulación de los ámbitos epistemológico, historiográfico y contextual, en la concepción de Alfonso-Goldfarb y Ferraz. Considerando que la interconexión entre tales esferas constituye un esfuerzo no trivial, nuestro corpus está compuesto por el trabajo matemático Instituzioni Analitiche, la correspondencia de Agnesi con sus contemporáneos, y algunos estudios basados en su vida y obra, sacando a la luz evidencias de su interés y comprometimiento a que su tratado se publicitara más allá de las tierras milanesas, lo que le dio una amplia repercusión. Sin embargo, a pesar de su importancia, Agnesi ha sido olvidada, en muchos aspectos, por los historiadores de las matemáticas.Palabras clave: Historia de las matemáticas, Educación matemática, Maria Gaetana Agnesi, Análisis matemático, Historia de las Ciencias.

Euclid’s Kinds and (Their) Attributes

Relying upon a very close reading of all of the definitions given in Euclid’s Elements, I argue that this mathematical treatise contains a philosophical treatment of mathematical objects. Specifically, I show that Euclid draws elaborate metaphysical distinctions between (i) substances and non-substantial attributes of substances, (ii) different kinds of substance, and (iii) different kinds of non-substance. While the general metaphysical theory adopted in the Elements resembles that of Aristotle in many respects, Euclid does not employ Aristotle’s terminology, or indeed, any philosophical terminology at all. Instead, Euclid systematically uses different types of definition to distinguish between metaphysically different kinds of mathematical object.

MARIA GAETANA AGNESI, ENFANT PRODIGE, MATEMATICA, PIA NOBILDONNA

Maria Gaetana Agnesi (1718-1799) was a prominent figure in eighteenth-century Milan. A child prodigy, and an attraction in the scientific and philosophical disputes organized in the paternal home, she was the first woman to publish a mathematical treatise, the Instituzioni analitiche ad uso della gioventù italiana (1748), a clear and systematic presentation of both Cartesian geometry and infinitesimal analysis. Among the curves studied in that work is the versiera, (witch) the cubic curve that is still associated with her name. Appointed by Pope Benedict XIV in 1750 on the chair of mathematics at the University of Bologna, she did not accept that assignment. After her father’s death in 1752, she left mathematics to devote herself entirely to pious works, and to taking care of poor and infirm women in the Pio Albergo Trivulzio, where she spent the last fifteen years of life.

Practices of reasoning: persuasion and refutation in a seventeenth-century Chinese mathematical treatise of “linear algebra”

ArgumentThis article documents the reasoning in a mathematical work by Mei Wending, one of the most prolific mathematicians in seventeenth-century China. Based on an analysis of the mathematical content, we present Mei’s systematic treatment of this particular genre of problems, fangcheng, and his efforts to refute the traditional practices in works that appeared earlier. His arguments were supported by the epistemological values he utilized to establish his system and refute the flaws in the traditional approaches. Moreover, in the context of the competition between the Chinese and Western approaches to mathematics, Mei was motivated to demonstrate that the genre of fangcheng problems was purely a “Chinese” achievement, not discussed by the Jesuits. Mei’s motivations were mostly expressed primarily in the prefaces to his works, in his correspondence with other scholars, in synopses of his poems, and in biographical records of some of his contemporaries.

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

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.

Mathematical Treatise to Model Dihedral Energy in the Multiscale Modeling of Two-Dimensional Nanomaterials

An approximate mathematical treatise is proposed to improve the accuracy of multiscale models for nonlinear mechanics of two-dimensional (2D) nanomaterials by taking into account the contribution of dihedral energy term in the nonlinear constitutive model for the generalized deformation (three nonzero components of each strain and curvature tensors) of the corresponding continuum. Twelve dihedral angles per unit cell of graphene sheet are expressed as functions of strain and curvature tensor components. The proposed model is employed to study the bending modulus of graphene sheets under finite curvature. The atomic interactions are modeled using first- and second-generation reactive empirical bond order (REBO) potentials with the modifications in the former to include dihedral energy term for accurate prediction of bending stiffness coefficients. The constitutive law is obtained by coupling the atomistic and continuum deformations through Cauchy–Born rule. The present model will facilitate the investigations on the nonlinear mechanics of graphene sheets and carbon nanotubes (CNTs) with greater accuracy as compared to those reported in the literature without considering dihedral energy term in multiscale modeling.

From Religion to Dialectics and Mathematics

Hermann Grassmann is known to be the founder of modern vector and tensor calculus. Having as a theologian no formal education in mathematics at a university he got his basic ideas for this mathematical innovation at least to some extent from listening to Schleiermacher’s lectures on Dialectics and, together with his brother Robert, reading its publication in 1839. The paper shows how the idea of unity and various levels of reality first formulated in Schleiermacher’s talks about religion in 1799 were transformed by him into a philosophical system in his dialectics and then were picked up by Grassmann and operationalized in his philosophical-mathematical treatise on the extension theory (German: Ausdehnungslehre) in 1844.