Archimedes’ Works in Conoids as a Basis for the Development of Mathematics

IF 0.3 Q4 MATHEMATICS

Mathematics Enthusiast Pub Date : 2021-01-01 DOI:10.54870/1551-3440.1519

Kenton Ke

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引用次数: 0

Abstract

This paper explores Archimedes’ works in conoids, which are three dimensional versions of conic sections, and will discuss ideas that came up in Archimedes’ book On Conoids and Spheroids. In particular, paraboloids, or three dimensional parabolas, will be the primary focus, and a proof of one of the propositions is provided for a clearer understanding of how Archimedes proved many of his propositions. His main method is called method of exhaustion, with results justified by double contradiction. This paper will compare the ideas and problems brought up in On Conoids and Spheroids and how they relate to modern day calculus. This paper will also look into some basic details on the method of exhaustion and how it allowed the ancient Greek mathematicians to prove propositions without any knowledge of calculus. In addition, this paper will discuss some mathematical contributions made by Arabic mathematicians such as Ibn alHaytham and how his work connects to mathematics in the seventeenth Century regarding sums of powers of whole numbers and the Basel Problem. Complicated forms of conoids such as hyperbolic paraboloids and other shapes that came after Archimedes will not be covered.

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阿基米德的圆锥体著作是数学发展的基础

本文探讨了阿基米德关于圆锥体的著作，它是圆锥曲线的三维版本，并将讨论阿基米德在《论圆锥体和椭球体》一书中提出的观点。特别是，抛物面，或三维抛物线，将是主要的焦点，并提供一个命题的证明，以便更清楚地了解阿基米德如何证明他的许多命题。他的主要方法被称为穷尽法，其结果被双重矛盾所证明。本文将比较《论圆锥体与椭球体》中提出的思想和问题，以及它们与现代微积分的关系。本文还将探讨穷竭法的一些基本细节，以及它如何使古希腊数学家在没有任何微积分知识的情况下证明命题。此外，本文将讨论阿拉伯数学家如Ibn alHaytham所做的一些数学贡献，以及他的工作如何与17世纪关于整数幂和和巴塞尔问题的数学联系起来。复杂形式的圆锥体，如双曲抛物面和阿基米德之后出现的其他形状将不会被涵盖。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematics Enthusiast MATHEMATICS-

CiteScore

1.40

自引率

0.00%

发文量

期刊介绍： The Mathematics Enthusiast (TME) is an eclectic internationally circulated peer reviewed journal which focuses on mathematics content, mathematics education research, innovation, interdisciplinary issues and pedagogy. The journal exists as an independent entity. The electronic version is hosted by the Department of Mathematical Sciences- University of Montana. The journal is NOT affiliated to nor subsidized by any professional organizations but supports PMENA [Psychology of Mathematics Education- North America] through special issues on various research topics. TME strives to promote equity internationally by adopting an open access policy, as well as allowing authors to retain full copyright of their scholarship contingent on the journals’ publication ethics guidelines. Authors do not need to be affiliated with the University of Montana in order to publish in this journal. Journal articles cover a wide spectrum of topics such as mathematics content (including advanced mathematics), educational studies related to mathematics, and reports of innovative pedagogical practices with the hope of stimulating dialogue between pre-service and practicing teachers, university educators and mathematicians. The journal is interested in research based articles as well as historical, philosophical, political, cross-cultural and systems perspectives on mathematics content, its teaching and learning. The journal also includes a monograph series on special topics of interest to the community of readers.