The role of geometrical construction in Theodosius's \textit{Spherics}
type: Item
instance of: scholarly article
title: The role of geometrical construction in Theodosius's \textit{Spherics}
zbMATH Open document ID: 1204.01007
author: Nathan Sidoli
author: Ken Saito
published in: Archive for History of Exact Sciences
publication date: 2009-11-23T00:00:00Z
review text: As is well known, the propositions of Euclid's Elements fall in two classes: theorems and problems, where the latter show how to construct a requested mathematical object, e.g. (I.1), an equilateral triangle with a given side. The proofs of these construction as well as those of many theorems also contain constructions, namely if entities are to be made use of that are not given in or directly implied by the enunciation; often, these two types of constructions have been conflated in discussions. Theodosius's Spherics, a treatise dealing with the geometry of the sphere, starting with elementary propositions and ending with matters of genuine astronomical interest, is similarly structured in theorems and problems, and constructions have the same dual role. However, the particular character of the object a sphere whose interior is not directly accessible for manipulation (if we think with Aristotle of it as the result of an abstraction process, where materiality has been removed from a bronze sphere) allows the authors to make observations that distinguish the constructions that are solutions to problem from those performed in proofs (proofs for theorems as well as proofs for the correctness of problem solutions). The latter may well ask for operations that are purely theoretical (cutting the sphere by a plane though its centre as a way to construct a great circle, etc.), since the function of entities constructed in this way is merely to allow the argument to draw on their well-known properties; the former (with one exception, I.2, ``to find the centre of a given sphere'', used in a particular way in the following and anyhow not possible in a material sphere) always restrict themselves to operations that can be performed with ruler and compass on the surface of the sphere and in a plane support for auxiliary constructions. Once such a problem has been solved, the entity it has shown how to construct (e.g., in I.19, the diameter of a given sphere) can be used freely in…
reviewed by: Jens Høyrup
Mathematics Subject Classification ID: 01A20
zbMATH DE Number: 5637173
zbMATH Keywords: Theodosius
zbMATH Keywords: spherics
zbMATH Keywords: construction
zbMATH Keywords: ancient Greek geometrical
zbMATH Keywords: problems
zbMATH Keywords: ancient Greek mathematics
MaRDI profile type: Publication
full work available at URL: s00407-009-0045-2
OpenAlex ID: W2071143810
cites work: Q4377820
cites work: Apollonius of Perga's \textit{Conica}: text, context, subtext
cites work: Did Euclid's circles have two kinds of radius?
cites work: The concept of existence and the role of constructions in Euclid's \textit{Elements}
cites work: The geometrical works of Abū Sa`īd al-Ḋarīr al-Jurjānī
cites work: Q3715068
cites work: Q5763669
cites work: Q3705389
cites work: Q3261383
cites work: The astronomical content of Menelaos' \textit{Spherics}
cites work: Q4256581
cites work: A History of Ancient Mathematical Astronomy
cites work: On the use of the term diastēma in ancient Greek constructions.
cites work: Heron's Dioptra 35 and Analemma Methods: An Astronomical Determination of the Distance between Two Cities
cites work: Q5447815
cites work: Q4100032
DOI: 10.1007/S00407-009-0045-2
Recommended article: Acts of Geometrical Construction in the Spherics of Theodosios
Recommended article: The Spherics of Theodosios
Recommended article: Graphical choices and geometrical thought in the transmission of Theodosius' Spherics from antiquity to the Renaissance
Recommended article: The definitions and theorems of the \textit{Spherics} of Theodosios
Recommended article: SUR UN THÉORÈME DE GÉOMÉTRIE SPHÉRIQUE: THÉODOSE, MÉNÉLAÜS, IBN ʿIRĀQ ET IBN HŪD
Recommended article: Q2961031
Recommended article: Q3647055
Recommended article: Q2988073
Recommended article: Q4562092
Recommended article: On the homocentric spheres of Eudoxus
description: scientific article; zbMATH DE number 5637173