Our thesis explores aspects of the geometrical work and thought of Isaac Newton in order to better understand and re-evaluate his approach to geometry, and specifically his synthetic methods and the organic description of plane curves. In pursuing this research we study Newton's geometrical work in the context of the changing view of geometry between the late 16th and early 18th centuries, a period defined by the responses of the early modern geometers to a new Latin edition of Pappus' Collectio. By identifying some of the major challenges facing geometers of this period as they attempted to define and practice geometry we are able to contrast Newton's own approach to geometry. The themes emerging from the geometrical thought of early modern geometers provide the mathematical context from which to understand, interpret and re-evaluate the approach taken by Newton. In particular we focus on Newton's profound rejection of the new algebraic Cartesian methods and geometrical philosophies, and the opportunity to focus more clearly on some of his most astonishing geometrical contributions. Our research highlights Newton's geometrical work and examines specific examples of his synthetic methods. In particular we draw attention to the significance of Newton's organic construction and the limitations of Whiteside's observations on this subject. We propose that Newton's organic rulers were genuinely original. We disagree with Whiteside that they were inspired by van Schooten, except in the loosest sense. Further, we argue that Newton's study of singular points by their resolution was new, and that it has been misunderstood by Whiteside in his interpretation of the transformation effected by the rulers. We instead emphasise that it was the standard quadratic transformation. Overall we wish to make better known the importance of geometry in Newton's scientific thought, as well as highlighting the mathematical and historical importance of his organic description of curves as an example of his synthetic approach to geometry. This adds to contemporary discourse surrounding Newton's geometry, and specifically provides a foundation for further research into the implications of Newton's geometrical methods for his successors.

A presente pesquisa, de cunho quantitativo, tem como propósito responder a seguinte questão: De que modo e em que alcance o trabalho pedagógico articulado com a história, geometria e meio computacional tem refletido sobre posturas e caminhos que levassem os alunos a se envolver com o conhecimento matemático? Desse modo, fizemos uma investigação e análise sobre os efeitos de uma articulação entre o ensino da história da matemática e o uso de ferramentas computacionais como solução para as dificuldades apresentadas no Ensino de Geometria, principalmente no Ensino Médio. Utilizamos a obra de Lakatos e a primeira proposição (do livro 1) de Euclides para realizar a verificação de sua demonstração através de um software de Geometria dinâmica. Os resultados serão utilizados para a construção de um novo software que envolva o ensino e aprendizagem de história da matemática e geometria. Outros objetivos podem ser assim colocados: Refletir sobre as condições e viabilidade da integração de recursos computacionais para o ensino da Matemática no âmbito Ensino Médio em especial a partir do produtos/softwares propostos para a educação matemática; Compreender o potencial de softwares de geometria dinâmica para a educação matemática escolar; Analisar as necessidades matemáticas de uma instrumentação eficaz, a partir da história da Matemática, para compreender a Matemática como um 9 processo em construção, em especial no âmbito das relações geométricas. Para isso foram retiradas vivências do cotidiano das aulas de Matemática para a reflexão sobre a geometria, e os resultados foram que a história da Matemática junto as novas tecnologias podem mudar as concepções de conhecimento da Matemática, pois através do professor ela pode chegar à sala de aula e transformar a prática pedagógica. / The current study focused on quantity, aims to answer the following question: How and to what extent the educational work linked to the story, using computational geometry and has reflected on postures and paths that could lead students to engage with the mathematical knowledge? Thus, we developed a research and analysis on the effects of an articulation between the teaching of mathematics and the use of computational tools as a solution to the problems presented in the Teaching of Geometry, especially in high school. We used the work of Lakatos and the first proposition (Book 1) Euclid to perform the verification of his statement through a dynamic geometry software. Results will be used for the construction of a new software that involves teaching and learning of history of mathematics and geometry. Other goals may be well placed: Reflecting on the conditions and feasibility of integrating computing resources - to the teaching of mathematics in high school - in particular from the product / software proposed for mathematics education; understand the potential of software for geometry dynamic for mathematics education; analyze the needs of a mathematical effective instrumentation, from the history of mathematics, to understand mathematics as an ongoing process, particularly in the context of geometric relationships. For that were withdrawn daily experiences of mathematics lessons to reflect on the geometry and the results 11 were that the history of mathematics with the new technologies may change the concepts of knowledge of mathematics, because through it the teacher can get to the room transform the classroom and practice teaching.

A presente pesquisa, de cunho quantitativo, tem como propósito responder a seguinte questão: De que modo e em que alcance o trabalho pedagógico articulado com a história, geometria e meio computacional tem refletido sobre posturas e caminhos que levassem os alunos a se envolver com o conhecimento matemático? Desse modo, fizemos uma investigação e análise sobre os efeitos de uma articulação entre o ensino da história da matemática e o uso de ferramentas computacionais como solução para as dificuldades apresentadas no Ensino de Geometria, principalmente no Ensino Médio. Utilizamos a obra de Lakatos e a primeira proposição (do livro 1) de Euclides para realizar a verificação de sua demonstração através de um software de Geometria dinâmica. Os resultados serão utilizados para a construção de um novo software que envolva o ensino e aprendizagem de história da matemática e geometria. Outros objetivos podem ser assim colocados: Refletir sobre as condições e viabilidade da integração de recursos computacionais para o ensino da Matemática no âmbito Ensino Médio em especial a partir do produtos/softwares propostos para a educação matemática; Compreender o potencial de softwares de geometria dinâmica para a educação matemática escolar; Analisar as necessidades matemáticas de uma instrumentação eficaz, a partir da história da Matemática, para compreender a Matemática como um 9 processo em construção, em especial no âmbito das relações geométricas. Para isso foram retiradas vivências do cotidiano das aulas de Matemática para a reflexão sobre a geometria, e os resultados foram que a história da Matemática junto as novas tecnologias podem mudar as concepções de conhecimento da Matemática, pois através do professor ela pode chegar à sala de aula e transformar a prática pedagógica. / The current study focused on quantity, aims to answer the following question: How and to what extent the educational work linked to the story, using computational geometry and has reflected on postures and paths that could lead students to engage with the mathematical knowledge? Thus, we developed a research and analysis on the effects of an articulation between the teaching of mathematics and the use of computational tools as a solution to the problems presented in the Teaching of Geometry, especially in high school. We used the work of Lakatos and the first proposition (Book 1) Euclid to perform the verification of his statement through a dynamic geometry software. Results will be used for the construction of a new software that involves teaching and learning of history of mathematics and geometry. Other goals may be well placed: Reflecting on the conditions and feasibility of integrating computing resources - to the teaching of mathematics in high school - in particular from the product / software proposed for mathematics education; understand the potential of software for geometry dynamic for mathematics education; analyze the needs of a mathematical effective instrumentation, from the history of mathematics, to understand mathematics as an ongoing process, particularly in the context of geometric relationships. For that were withdrawn daily experiences of mathematics lessons to reflect on the geometry and the results 11 were that the history of mathematics with the new technologies may change the concepts of knowledge of mathematics, because through it the teacher can get to the room transform the classroom and practice teaching.

This thesis examines Dar&uuml / lf&uuml / nun Konferanslari which consists of a series of lectures that were delivered by Salih Zeki in 1914 &ndash / 1915 in Ottoman State. These lectures were on geometry, its history and especially on the discovery of non-Euclidean geometries. And the purpose of this thesis is to propose the sufficiency and the legitimacy of these lectures as an account on the history of geometry. As a preliminary to analyzing Salih Zeki&rsquo / s lectures, different views on geometry&rsquo / s history and progress will be analyzed and compared. The results of this comparison will be the guide by means of which Dar&uuml / lf&uuml / nun Konferanslari will be examined. This thesis also serves as a source that makes Salih Zeki&rsquo / s ideas accessible, by presenting an English summary of his lectures which were originally published in Ottoman Turkish.

No description available.

La parution en 1750 de l'Introduction à l'analyse des lignes courbes algébriques, unique ouvrage publié de Gabriel Cramer, professeur de mathématiques à l'Académie de Genève, est un jalon important dans l'histoire de la géométrie des courbes et de l'algèbre. Il s'est passé près de dix années entre le moment où le Genevois a écrit les premières lignes de son traité des courbes, à l'automne 1740, et sa publication effective : il s'agit de l'œuvre d'une vie.Ce livre a une histoire, à la fois intellectuelle et matérielle, qui s'inscrit dans les contextes scientifiques, professionnels, académiques et sociaux dans lesquels évoluent son auteur puis ses lecteurs. De la genèse d'un texte manuscrit en devenir dont nous suivrons les évolutionsau cours du processus d'écriture et au gré des événements de la vie de son auteur, aux différentes lectures mathématiciennes et historiennes du texte publié qui en sont faites dans les quelque deux siècles qui ont suivi sa publication, je souhaite ici écrire, pour autant que cette expression ait un sens - ce que je m'attacherai à démontrer - une « biographie » de l'Introduction de Gabriel Cramer / The publication in 1750 of the Introduction à l'analyse des lignes courbes algébriques, the only published work by Gabriel Cramer, professor of mathematics at the Geneva Academy, is an important milestone in the history of geometry of curves and algebra. Almost ten years passed between the time when the Genevan wrote the first lines of his treatise on curves in the autumn of 1740 and its actual publication, making it a lifetime achievement.This book has a history, both intellectual and material, which takes place in the scientific, professional, academic and social contexts in which evolve its author and its readers. From the genesis of a handwritten text as a work in progress of which we will follow the evolutions during the process of writing and according to the events of its author's life, to the various mathematicians and historians' readings of the published text which are made in the two centuries following its publication, I would like to write here, insofar as this expression makes sense - which I shall endeavour to demonstrate - a « biography » of Gabriel Cramer's Introduction

Made available in DSpace on 2016-06-02T20:29:26Z (GMT). No. of bitstreams: 1 6170.pdf: 1990132 bytes, checksum: 74a0095a1e3b0461556172eb75b7403e (MD5) Previous issue date: 2014-07-28 / Financiadora de Estudos e Projetos / A course of Professional Qualification in Boiler Tracing aims to develop the capacities to calculate, chart, plan parts, prepare technical design, modeling sets and subsets. However the aim of this work is not training in Tracing the Boilers, but working some topics of boiler tracing in order to bring the student to see and better understand some basic concepts of geometry inherent to high school instruction, since in this way he will have the opportunity of a practical activity. We begin by presenting a brief history of geometry, delineating how it has being developed and crafted for thousands of years, and how the evolution of mathematics developed by great characters influenced the development and growth of societies as well as humanity as a whole, and further, how geometry is still crafted in society in general. In this work we examine how and to what extent a more technical classroom professionalizing activity can assist the needs of the curriculum of high school and even more, wondering whether the high school curriculum meets the needs of the contemporary challenges of the 21st. century society, which is increasingly characterized by intensive use of knowledge, either to work, socialize or exercise citizenship. The research methodology used was the Didactical Engineering, that guided us in the creation and use of a planned teaching sequence, in a group of students, for the teaching and learning of geometric concepts, drawn from the observation of the difficulty faced by students to understand these concepts. / Um curso de Qualificação Profissional Traçador de Caldeiraria tem por objetivo o desenvolvimento das capacidades de calcular, traçar, planificar peças, elaborar desenho técnico de modelagem de conjuntos e subconjuntos, porem o objetivo deste trabalho não é a formação em Traçagem de Caldeiraria, mas trabalhar alguns tópicos de traçagem de caldeiraria com o intuito de levar o aluno do ensino médio a enxergar e entender melhor alguns conceitos básicos de geometria do ensino médio, visto que desta forma ele terá a oportunidade de vivenciar uma atividade prática, dando assim maior significado ao seu aprendizado. Como introdução, é delineado um resumo da história da geometria, a forma com que ela vem sendo desenvolvida e trabalhada por milhares de anos, e como a evolução da matemática desenvolvida por grandes personagens influenciaram o desenvolvimento e crescimento de sociedades assim como da humanidade como um todo, e mais ainda, como a geometria ainda hoje é trabalhada na sociedade de forma geral. Neste trabalho procuramos analisar de que forma e até que ponto uma aula mais técnica e voltada para uma atividade profissionalizante pode auxiliar nas necessidades do currículo do Ensino Médio e mais ainda, se o currículo do Ensino Médio atende as necessidades dos desafios contemporâneos da sociedade do século XXI que cada vez mais é caracterizada pelo uso intensivo do conhecimento, seja para trabalhar, conviver ou exercer a cidadania. A metodologia de pesquisa utilizada foi a Engenharia Didática que nos orientou na confecção e aplicação planejada de uma sequência didática em um grupo de alunos no ensino e aprendizagem de conceitos geométricos, elaborado a partir da constatação da dificuldade que os alunos enfrentam para compreender esses conceitos.

The central question of this thesis is: what is mathematics about? The answer arrived at by the thesis is an unsettling and unsatisfying one. By examining two of the most promising contemporary accounts of the nature of mathematics, I conclude that neither is as yet capable of giving us a conclusive answer to our question. The conclusion is arrived at by a combination of historical and conceptual analysis. It begins with the historical fact that, since the middle of the nineteenth century, mathematics has undergone a radical transformation. This transformation occurred in most branches of mathematics, but was perhaps most apparent in geometry. Earlier images of geometry understood it as the science of space. In the wake of the emergence of multiple distinct geometries and the realization that non-Euclidean geometries might lay claim to the description of physical space, the old picture of Euclidean geometry as the sole correct description of physical space was no longer tenable. The first chapter of the dissertation provides an historical account of some of the forces which led to the destabilization of the traditional picture of geometry. The second chapter examines the debate between Gottlob Frege and David Hilbert regarding the nature of geometry and axiomatics, ending with an argument suggesting that Hilbert’s views are ultimately unsatisfying. The third chapter continues to probe the work of Frege and, again, finds his explanations of the nature of mathematics troublingly unsatisfying. The end result of the first three chapters is that the Frege-Hilbert debate leaves us with an impasse: the traditional understanding of mathematics cannot hold, but neither can the two most promising modern accounts. The fourth and final chapter of the thesis investigates mathematical structuralism—a more recent development in the philosophy of mathematics—in order to see whether it can move us beyond the impasse of the Frege-Hilbert debate. Ultimately, it is argued that the contemporary debate between ‘assertoric’ structuralists and ‘algebraic’ structuralists recapitulates a form of the Frege-Hilbert impasse. The ultimate claim of the thesis, then, is that neither of the two most promising contemporary accounts can offer us a satisfying philosophical answer to the question ‘what is mathematics about?’.