Preprint
Negative Größen bei Diophant?
Abstract
In this paper we champion Diophantus of Alexandria and Isabella Basmakova against Norbert Schappacher. In two publications ([46] and [47]) he puts forward inter alia two propositions: Questioning Diophantus' originality he considers affirmatively the possibility, that the Arithmetica are the joint work of a team of authors like Bourbaki. And he calls Basmakova's claim (in [5]), that Diophantus uses negative numbers, a "nonsense", reproaching her for her "thoughtlessness". First, we disprove Schappacher's Bourbaki thesis. Second, we investigate the semantic meaning and historical significance of Diophantus' keywords leipsis and mparxis. Next, we discuss Schappacher's epistemology of the history of mathematics and defend Basmakova's methods. Furthermore, we give 33 places where Diophantus uses negative quantities as intermediate results; they appear as differences a - b of positive rational numbers, the subtrahend b being bigger than the minuend a; they each represent the (negative) basis (pleyra) of a square number (tetragonos), which is afterwards computed by the formula (a - b)^2 = a^2 + b^2 - 2ab. Finally, we report how the topic "Diophantus and the negative numbers" has been dealt with by translators and commentators from Maximus Planudes onwards.
Citation
@article{urn:nbn:de:hebis:34-200604048979,
author={Barner, Klaus},
title={Negative Größen bei Diophant?},
year={2005}
}
0500 Oax 0501 Text $btxt$2rdacontent 0502 Computermedien $bc$2rdacarrier 1100 2005$n2005 1500 1/ger 2050 ##0##urn:nbn:de:hebis:34-200604048979 3000 Barner, Klaus 4000 Negative Größen bei Diophant? / Barner, Klaus 4030 4060 Online-Ressource 4085 ##0##=u http://nbn-resolving.de/urn:nbn:de:hebis:34-200604048979=x R 4204 \$dPreprint 4170 Mathematische Schriften Kassel ;; 05, 04 7136 ##0##urn:nbn:de:hebis:34-200604048979
2006-04-04T10:12:57Z 2006-04-04T10:12:57Z 2005 urn:nbn:de:hebis:34-200604048979 http://hdl.handle.net/123456789/200604048979 4587692 bytes application/pdf ger Universität Kassel, FB 17, Mathematik/Informatik Urheberrechtlich geschützt https://rightsstatements.org/page/InC/1.0/ Historia <Diophant> Mathematik / Geschichte Gericke, Helmuth Diophantus <Alexandrinus> Negative Zahlen Geschichte der Zahlen Bashmakowa, Isabella Schappacher, Norbert 510 Negative Größen bei Diophant? Preprint In this paper we champion Diophantus of Alexandria and Isabella Basmakova against Norbert Schappacher. In two publications ([46] and [47]) he puts forward inter alia two propositions: Questioning Diophantus' originality he considers affirmatively the possibility, that the Arithmetica are the joint work of a team of authors like Bourbaki. And he calls Basmakova's claim (in [5]), that Diophantus uses negative numbers, a "nonsense", reproaching her for her "thoughtlessness". First, we disprove Schappacher's Bourbaki thesis. Second, we investigate the semantic meaning and historical significance of Diophantus' keywords leipsis and mparxis. Next, we discuss Schappacher's epistemology of the history of mathematics and defend Basmakova's methods. Furthermore, we give 33 places where Diophantus uses negative quantities as intermediate results; they appear as differences a - b of positive rational numbers, the subtrahend b being bigger than the minuend a; they each represent the (negative) basis (pleyra) of a square number (tetragonos), which is afterwards computed by the formula (a - b)^2 = a^2 + b^2 - 2ab. Finally, we report how the topic "Diophantus and the negative numbers" has been dealt with by translators and commentators from Maximus Planudes onwards. open access Barner, Klaus Mathematische Schriften Kassel ;; 05, 04 Mathematische Schriften Kassel 05, 04
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