Philosophy and Geometry

Ezequias Cassela , 2021

This article aims to study the ellipse from the perspective of pure or synthetic geometry to the representation of points on a plane through the use of real numbers, as well as the representation and classification of this conic curve through the use of equations. The perspective developed in this article is based on the view of René Descartes, in considering that "the algebraic steps in a demonstration should really correspond to a geometric representation." The relevance of this article is to bring a reflection that eliminates the study of Analytical Geometry through ready-made and finished formulas, without satisfactory justification and without a logical chain that gives a greater meaning to the studied concepts. In general, the study developed in this article emphasizes the demonstration of results based on propositions adapted a priori, whose ability to be developed is aimed at establishing an "if...then" type of reasoning, making conjectures involving various knowledge already acquired and confirming such truths from a logical system, using definitions and propositions. Therefore, the demonstrations made in the scope of Synthetic Geometry will help to establish a connection with the equations obtained from the perspective of Analytical Geometry, serving as a consultation for students and professors of Analytical Geometry, thus avoiding sudden transitions between contents of degrees of distinct difficulties.

Rangel Cerceau Netto, 2020

Resumo: O artigo constitui um estudo sobre o Ensino de Geometria na Academia Real de Matemática de Barcelona entre 1720 a 1803. A contribuição busca refletir sobre a pesquisa da técnica e da arte no universo dos tratados de Geometria e a sua circulação no que tange aos conhecimentos aplicados na construção de edificações. Neste sentido a pesquisa reflete como o conhecimento matemático era fundamental para a arquitetura de fortificações e, sobretudo para uma cultura do conhecimento que refletia na visão espacial arquitetônica do mundo hispânico. Palavra-chaves: Arquitetura de Fortificações; Geometria; Academia Real de Matemática; Barcelona; Abstract: The article constitutes a study on the Teaching of Geometry at the Royal Academy of Mathematics in Barcelona between 1720 and 1803. The contribution seeks to reflect on the research of technique and art in the universe of treatises on Geometry and its circulation with respect to knowledge applied in the construction of buildings. In this sense, the research reflects how mathematical knowledge was fundamental for the architecture of fortifications and, above all, for a culture of knowledge that reflected in the architectural spatial vision of the Hispanic world. Keywords: Fortification Architecture; Geometry; Royal Academy of Mathematics; Barcelon Recebido em: 13/07/2020 – Aceito em 25/07/2020

The Mathematical Gazette, 2004

Let History into the Mathematics Classroom, 2017

This chapter is concerned with an important question in geometry: the division (or dissection) of 2D figures. This question has been represented and developed in numerous mathematical traditions since Mesopotamian times. In particular, the great geometers of ancient Greece such as Euclid or Hero of Alexandria each tackled it in their own way. Many other developments were achieved in Islamic countries both in the East and West. Based on extracts from medieval Latin literature of the twelfth and thirteenth centuries with Plato of Tivoli, Fibonacci and Jordanus de Nemore, this chapter deals with just one basic geometric problem: 'dividing a triangle into two equal parts', from which the geometric constraints evolve. A broad historic introduction allows these problems to be placed in context; it is a great opportunity to introduce an historic perspective into mathematics teaching. When an author is quoted for the first time, their name is followed, in brackets, by the date of death when known or the proven period when they were active. When too little viable information is known, only the century will be given.

2015

"Elementary Geometry" received its name from Euclid's "Elements", it deals with "elements", i.e. points, lines, circles, planes and "relations", i.e. incidences, proportions, lengths and angles. While in former times Elementary Geometry constituted a fixed part in mathematics syllabuses, providing a trainings field for logic reasoning and coordinate free proving methods, the topic now has little scientific esteem and seems to vanish completely in Maths education. The paper tries to show that Elementary Geometry still has its merits and even opens up for new fields in Mathematics. Connecting Elementary Geometry with Projective Geometry and Circle and Chain Geometry extends it to "Advanced Elementary Geometry". These extensions sometimes give better insight into the nature of a basic theorem and it justifies the preoccupation with that subject. The paper aims at providing teachers with perhaps fascinating geometric problems. Most of the presented material stems from other publications of the author and is nothing but a "closer look" to very basic elementary geometric theorems. Thereby iteration and generalization as the standard methods of research in mathematics are applied and often lead to astonishing facts and incidences. Modern dynamic visualization tools together with automatized theorem proving software quickly allow to formulate statements and theorems and therefore are a great stimulus to do research at an early stage of mathematics education.

2015

Usually the evaluations at the end of the geometry education are, aimed to measure geometric knowledge and ability. But in this study we tried to investigate how the geometric thinking should be, which is more important than knowledge and ability. Teaching at this direction we aim; improving attention, imagine, creative and criticizing thought, with additional geometric drawings agility, different perspectives and relating between forms. The main objective is focused on geometric thoughts improvement in geometric knowledge and ability. At this direction some geometry questions have been solved, without changing their data’s, but putting them to different geometric thought base. With these solutions some activities were presented about degree, length, domain and circle to find answers of the following questions “How should the geometric thought be? How should it be improved?”

Springer Undergraduate Mathematics Series, 2011

Proceedings of the 13th International Congress on Mathematical Education, 2017

2002

2019

This volume consists of a collection of essays on geometry from a historical viewpoint, addressed to the general mathematical community interested in the history of ideas and their evolution. In planning this book as Editors we went with the conviction that writing on the basic geometrical concepts from a historical perspective is an essential element of our science literature, and that geometrico-historical surveys constitute an important ingredient, alongside purely mathematical and historical articles, for the development of the subject. We have also held, for many years, that we need such articles written by mathematicians, dealing with topics and ideas that they are directly engaged with and consider as fundamental, and it was our endeavour to put together an ensemble of articles of this variety on a broad spectrum of topics that constitute the general area of Geometry. The authors of the various chapters are A. A'Campo-Neuen, V. N. Berestovskii, M. Chaperon, A. Chenciner, ...