Pub Date : 2022-09-02DOI: 10.1080/26375451.2022.2115745
J. Oaks
Whether explaining calculations with decimal or sexagesimal notation, arithmetic books composed in Arabic beginning in the ninth century CE consistently describe the zero (ṣifr) as a sign indicating an empty place where there is no number. And yet we find that some arithmeticians explicitly performed operations on this zero. To understand how the zero was conceived and manipulated in medieval Arabic texts we first address the way that numbers themselves were conceived and how ‘nothing’ entered into arithmetical problem-solving. From there we examine arithmetic books for their treatment of zero. We find that there is no inconsistency in operating on what is literally nothing, and thus there was no motive for arithmeticians to regard zero as a number.
{"title":"Zero and nothing in medieval Arabic arithmetic","authors":"J. Oaks","doi":"10.1080/26375451.2022.2115745","DOIUrl":"https://doi.org/10.1080/26375451.2022.2115745","url":null,"abstract":"Whether explaining calculations with decimal or sexagesimal notation, arithmetic books composed in Arabic beginning in the ninth century CE consistently describe the zero (ṣifr) as a sign indicating an empty place where there is no number. And yet we find that some arithmeticians explicitly performed operations on this zero. To understand how the zero was conceived and manipulated in medieval Arabic texts we first address the way that numbers themselves were conceived and how ‘nothing’ entered into arithmetical problem-solving. From there we examine arithmetic books for their treatment of zero. We find that there is no inconsistency in operating on what is literally nothing, and thus there was no motive for arithmeticians to regard zero as a number.","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"37 1","pages":"179 - 211"},"PeriodicalIF":0.4,"publicationDate":"2022-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41832762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-02DOI: 10.1080/26375451.2022.2135064
Bshm Meeting Coordinator, Isobel Falconer
s from past meetings History of mathematics and flight Saturday 2 July 2022 Concorde Centre, Manchester Airport A day of talks about the history of mathematics and flight. Flight was broadly conceived to cover the flight of man-made objects and animals; flight formation, navigation and control. The day included a tour of Concorde. Kate Hindle (St Andrews): D’Arcy Thompson and flight D’Arcy Thompson (1860–1948) is most remembered for his influential book On Growth and Form (1917), which looked to maths to explain why biological creatures take the shapes that they take. In January 1917, a few months before this book was released, Thompson had a letter to the editor published in Nature titled ‘Stability in Flight’. A month later Herbert Maxwell (1845–1937) – a baronet, politician, and fellow of several learned societies – published a letter in Nature as a criticism of Thompson’s work. Thompson reacted to this criticism with a defensive response letter, showing that he was affected by it. This exchange also highlights how Thompson conceptualized advancements in maths as a guiding light for biology, showing how his views on flight coincide with his other biomathematical work. Jane Wess (Independent): Benjamin Robins: Elegant Mathematics Versus Experimental Inconvenience? While academically a constituent of fluid mechanics, practically ballistics was an important area of knowledge for nation states in the eighteenth century. William Mountaine, a mathematics teacher, wrote in 1781 ‘it is not possible in the nature of things for any one kingdom to continue long in a state of peace, the art of gunnery has from time to time engaged the attention of the most eminent mathematicians’. However, the essential nature of the knowledge of the flight of cannon balls did not result in an efficacious mathematical description for a remarkable length of time. Whereas both Huygens and Newton had acknowledged the role of air resistance, textbooks continued to discuss parabolas following Galileo, Torricelli, Halley, and Cotes until the end of the eighteenth century. The obvious question is ‘why?’ There may be several factors at play, including the status of Robins, who challenged the status quo, but it will be argued that beautiful and simple mathematics can be beguiling. As for the case of epicycloidal teeth in gearing, it seems many of those who advocated a mathematical approach were not completely au fait with the most advanced thinking on the topic, in this case by Huygens, Newton, and of course later and most effectively, by Euler. Deborah Kent (St Andrews): A champion’s counterexample? PG Tait and the flight of a golf ball Nineteenth-century mathematician and physicist Peter Guthrie Tait (1831–1901) is well known for the Treatise on Natural Philosophy, which he co-wrote with William Thomson (later Lord Kelvin), and collaborations with James Clerk Maxwell. Less familiar are his aerodynamical studies from the 1890s, which resulted in over a dozen papers on the path of
{"title":"BSHM meeting news","authors":"Bshm Meeting Coordinator, Isobel Falconer","doi":"10.1080/26375451.2022.2135064","DOIUrl":"https://doi.org/10.1080/26375451.2022.2135064","url":null,"abstract":"s from past meetings History of mathematics and flight Saturday 2 July 2022 Concorde Centre, Manchester Airport A day of talks about the history of mathematics and flight. Flight was broadly conceived to cover the flight of man-made objects and animals; flight formation, navigation and control. The day included a tour of Concorde. Kate Hindle (St Andrews): D’Arcy Thompson and flight D’Arcy Thompson (1860–1948) is most remembered for his influential book On Growth and Form (1917), which looked to maths to explain why biological creatures take the shapes that they take. In January 1917, a few months before this book was released, Thompson had a letter to the editor published in Nature titled ‘Stability in Flight’. A month later Herbert Maxwell (1845–1937) – a baronet, politician, and fellow of several learned societies – published a letter in Nature as a criticism of Thompson’s work. Thompson reacted to this criticism with a defensive response letter, showing that he was affected by it. This exchange also highlights how Thompson conceptualized advancements in maths as a guiding light for biology, showing how his views on flight coincide with his other biomathematical work. Jane Wess (Independent): Benjamin Robins: Elegant Mathematics Versus Experimental Inconvenience? While academically a constituent of fluid mechanics, practically ballistics was an important area of knowledge for nation states in the eighteenth century. William Mountaine, a mathematics teacher, wrote in 1781 ‘it is not possible in the nature of things for any one kingdom to continue long in a state of peace, the art of gunnery has from time to time engaged the attention of the most eminent mathematicians’. However, the essential nature of the knowledge of the flight of cannon balls did not result in an efficacious mathematical description for a remarkable length of time. Whereas both Huygens and Newton had acknowledged the role of air resistance, textbooks continued to discuss parabolas following Galileo, Torricelli, Halley, and Cotes until the end of the eighteenth century. The obvious question is ‘why?’ There may be several factors at play, including the status of Robins, who challenged the status quo, but it will be argued that beautiful and simple mathematics can be beguiling. As for the case of epicycloidal teeth in gearing, it seems many of those who advocated a mathematical approach were not completely au fait with the most advanced thinking on the topic, in this case by Huygens, Newton, and of course later and most effectively, by Euler. Deborah Kent (St Andrews): A champion’s counterexample? PG Tait and the flight of a golf ball Nineteenth-century mathematician and physicist Peter Guthrie Tait (1831–1901) is well known for the Treatise on Natural Philosophy, which he co-wrote with William Thomson (later Lord Kelvin), and collaborations with James Clerk Maxwell. Less familiar are his aerodynamical studies from the 1890s, which resulted in over a dozen papers on the path of","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"37 1","pages":"258 - 265"},"PeriodicalIF":0.4,"publicationDate":"2022-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45547182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-02DOI: 10.1080/26375451.2022.2106061
R. Siegmund‐Schultze
This paper provides some methodological, didactical, and historiographical reflections on Egyptian pyramid volume formulas, responding to suggestions by Paul Shutler from 2009. These suggestions partly reiterate a historically documented proof by the Chinese Liu Hui (third century CE), although Lui Hui’s contribution was apparently unknown to Shutler. The latter came forward, in addition, with intuitive arguments which might have been used by the Egyptians to convince themselves of the correctness of their formula for the volume of the full pyramid. In a broad sense, the reflections in this paper may contribute to the use of history in the mathematical classroom. As a cautionary note: The paper is an abridged version of a longer manuscript that contains detailed explanations and discussions of historical secondary sources. Since the paper is somewhat outside the usual canon of mathematics historiography, I have deposited the longer manuscript on .
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Pub Date : 2022-07-10DOI: 10.1080/26375451.2022.2082158
Ana Patrícia Martins, Teresa Sousa
The inclusion–exclusion principle is a simple, intuitive, and extremely versatile result. It is one of the most useful methods for counting and it can be used in different areas of mathematics. In the eighteenth century, the first uses of this result that appear in the literature are related to the study of problems of games of chance. However, the first formulations of this principle appear, independently by several authors, only in the nineteenth century. In this article, we study the formulations obtained by Adrien-Marie Legendre, Daniel Augusto da Silva, James Joseph Sylvester, and Henri Poincaré. We highlight the contribution of the Portuguese mathematician Daniel Augusto da Silva, since his formulation can be applied to different problems of number theory, whenever collections of numbers satisfying certain properties are involved, and this is the reason why his formulation stands out compared with all the others.
包容-排斥原理是一个简单、直观、非常通用的结果。它是最有用的计数方法之一，可用于数学的不同领域。在18世纪，这一结果的首次应用出现在文献中，与研究机会游戏的问题有关。然而，这一原理的第一个表述，仅在19世纪由几位独立的作者出现。在这篇文章中，我们研究了Adrien-Marie Legendre, Daniel Augusto da Silva, James Joseph Sylvester和Henri poincar得到的公式。我们强调葡萄牙数学家丹尼尔·奥古斯托·达席尔瓦的贡献，因为他的公式可以应用于数论的不同问题，只要涉及满足某些性质的数字集合，这就是为什么他的公式与所有其他公式相比脱颖而出的原因。
{"title":"Formulations of the inclusion–exclusion principle from Legendre to Poincaré, with emphasis on Daniel Augusto da Silva","authors":"Ana Patrícia Martins, Teresa Sousa","doi":"10.1080/26375451.2022.2082158","DOIUrl":"https://doi.org/10.1080/26375451.2022.2082158","url":null,"abstract":"The inclusion–exclusion principle is a simple, intuitive, and extremely versatile result. It is one of the most useful methods for counting and it can be used in different areas of mathematics. In the eighteenth century, the first uses of this result that appear in the literature are related to the study of problems of games of chance. However, the first formulations of this principle appear, independently by several authors, only in the nineteenth century. In this article, we study the formulations obtained by Adrien-Marie Legendre, Daniel Augusto da Silva, James Joseph Sylvester, and Henri Poincaré. We highlight the contribution of the Portuguese mathematician Daniel Augusto da Silva, since his formulation can be applied to different problems of number theory, whenever collections of numbers satisfying certain properties are involved, and this is the reason why his formulation stands out compared with all the others.","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"37 1","pages":"212 - 229"},"PeriodicalIF":0.4,"publicationDate":"2022-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46934351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-04DOI: 10.1080/26375451.2022.2086386
Bshm Meeting Coordinator, Isobel Falconer, Daniel Antonio Di Liscia Munich, Stefano Gulizia Milan, Thomas Henderson Durham, H. Gropp, Eleonora Sammarchi, Suzanne Aigrain University of Oxford, M. Jauzac, Agathe Keller
Maths city. A snapshot of Abstract: I will explore the various spaces and practices for Athenian numeracy in the 5th and 4th century BCE, and discuss rates of numeracy, and also sketch a pro fi le of who may Abstract: Thirteenth-century Chinese mathematical works attest to two interesting innovations. Qin Jiushao ’ s Mathematical Work in Nine Chapters ( Shushu Jiuzhang 數 書 九 章 , 1247) describes an algorithm solving congruence equations in ways related to the so-called ‘ Chinese remainder theorem ’ . Moreover, Li Ye ’ s 李 冶 Measuring the Circle on the Sea-Mirror ( Ceyuan haijing , 1248) shows how to use polynomial algebra to establish algebraic equations solving mathematical problems. Both authors make use of the same technical expression: ‘ one establishes one heavenly source/origin as
{"title":"BSHM Meeting News","authors":"Bshm Meeting Coordinator, Isobel Falconer, Daniel Antonio Di Liscia Munich, Stefano Gulizia Milan, Thomas Henderson Durham, H. Gropp, Eleonora Sammarchi, Suzanne Aigrain University of Oxford, M. Jauzac, Agathe Keller","doi":"10.1080/26375451.2022.2086386","DOIUrl":"https://doi.org/10.1080/26375451.2022.2086386","url":null,"abstract":"Maths city. A snapshot of Abstract: I will explore the various spaces and practices for Athenian numeracy in the 5th and 4th century BCE, and discuss rates of numeracy, and also sketch a pro fi le of who may Abstract: Thirteenth-century Chinese mathematical works attest to two interesting innovations. Qin Jiushao ’ s Mathematical Work in Nine Chapters ( Shushu Jiuzhang 數 書 九 章 , 1247) describes an algorithm solving congruence equations in ways related to the so-called ‘ Chinese remainder theorem ’ . Moreover, Li Ye ’ s 李 冶 Measuring the Circle on the Sea-Mirror ( Ceyuan haijing , 1248) shows how to use polynomial algebra to establish algebraic equations solving mathematical problems. Both authors make use of the same technical expression: ‘ one establishes one heavenly source/origin as","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"37 1","pages":"164 - 169"},"PeriodicalIF":0.4,"publicationDate":"2022-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48178248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-04DOI: 10.1080/26375451.2022.2092370
Stefano Gulizia
This paper suggests that layered ontology is important within Kepler’s method, and that it developed at least partially in response to a disciplinary and religious crisis. As such, and despite Platonic allegiances, it places him in a longue durée of geometrical constructivism in seventeenth-century Europe. After introducing the pivotal role of Mysterium cosmographicum (1596) and how Kepler’s career may be seen in the context of courtly bricolage, the exposition realigns De nive sexangula (1611) with the mathematical communities of its time and argues in Reviel Netz’s tradition that its cognitive practices enact a ludic style of demonstration. Kepler’s essay on crystallography is an epistemological improvement on previous types of natural jokes.
{"title":"Kepler’s snow: the epistemic playfulness of geometry in seventeenth-century Europe","authors":"Stefano Gulizia","doi":"10.1080/26375451.2022.2092370","DOIUrl":"https://doi.org/10.1080/26375451.2022.2092370","url":null,"abstract":"This paper suggests that layered ontology is important within Kepler’s method, and that it developed at least partially in response to a disciplinary and religious crisis. As such, and despite Platonic allegiances, it places him in a longue durée of geometrical constructivism in seventeenth-century Europe. After introducing the pivotal role of Mysterium cosmographicum (1596) and how Kepler’s career may be seen in the context of courtly bricolage, the exposition realigns De nive sexangula (1611) with the mathematical communities of its time and argues in Reviel Netz’s tradition that its cognitive practices enact a ludic style of demonstration. Kepler’s essay on crystallography is an epistemological improvement on previous types of natural jokes.","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"37 1","pages":"117 - 137"},"PeriodicalIF":0.4,"publicationDate":"2022-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45384672","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-26DOI: 10.1080/26375451.2022.2052631
M. Friedman
In 1932, the mathematician Hans Hahn delivered a lecture titled ‘The crisis of intuition’, held within a lecture series called ‘Crisis and Reconstruction in the Exact Sciences’, organized by Karl Menger. In order to account for the various crises, Hahn and his colleagues employed various metaphors. That being said, the dominant metaphor was architectural. Why was this particular metaphor used? And were there other metaphors that were equally important? In this paper, I aim not only to answer these questions, taking into account the image of mathematics and of the mathematician which was conveyed by those metaphors, but also to examine how the various crises were considered via these metaphorical reactions.
{"title":"Metaphorical reactions in 1932: from the mathematical ‘crisis of intuition’ to ‘reconstruction in the exact sciences’","authors":"M. Friedman","doi":"10.1080/26375451.2022.2052631","DOIUrl":"https://doi.org/10.1080/26375451.2022.2052631","url":null,"abstract":"In 1932, the mathematician Hans Hahn delivered a lecture titled ‘The crisis of intuition’, held within a lecture series called ‘Crisis and Reconstruction in the Exact Sciences’, organized by Karl Menger. In order to account for the various crises, Hahn and his colleagues employed various metaphors. That being said, the dominant metaphor was architectural. Why was this particular metaphor used? And were there other metaphors that were equally important? In this paper, I aim not only to answer these questions, taking into account the image of mathematics and of the mathematician which was conveyed by those metaphors, but also to examine how the various crises were considered via these metaphorical reactions.","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"37 1","pages":"138 - 161"},"PeriodicalIF":0.4,"publicationDate":"2022-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46014483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-12DOI: 10.1080/26375451.2022.2056968
Deborah Kent
{"title":"Einstein, Eddington, e o/and the Eclipse: Impressões de Viagem/Travel Impressions","authors":"Deborah Kent","doi":"10.1080/26375451.2022.2056968","DOIUrl":"https://doi.org/10.1080/26375451.2022.2056968","url":null,"abstract":"","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"37 1","pages":"162 - 163"},"PeriodicalIF":0.4,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46880295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-16DOI: 10.1080/26375451.2022.2040897
P. Kidwell
The objects shown in the exhibits or stored in the cabinets of museums and mathematics departments—or used in mathematical research and teaching—rarely convey a sense of crisis. However, crises create new roles, mix cultures, bring about new needs, make unexpected use of time (and sometimes free time from usual duties), and generate fear. All of these changes have shaped these now-placid objects. Examination of a few instruments, considering them as part of the lives of the mathematicians and others associated with them, suggests such connections.
{"title":"Mathematical instruments from times of crisis","authors":"P. Kidwell","doi":"10.1080/26375451.2022.2040897","DOIUrl":"https://doi.org/10.1080/26375451.2022.2040897","url":null,"abstract":"The objects shown in the exhibits or stored in the cabinets of museums and mathematics departments—or used in mathematical research and teaching—rarely convey a sense of crisis. However, crises create new roles, mix cultures, bring about new needs, make unexpected use of time (and sometimes free time from usual duties), and generate fear. All of these changes have shaped these now-placid objects. Examination of a few instruments, considering them as part of the lives of the mathematicians and others associated with them, suggests such connections.","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"37 1","pages":"103 - 116"},"PeriodicalIF":0.4,"publicationDate":"2022-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45600259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-02DOI: 10.1080/26375451.2021.2009720
N. Biggs
{"title":"William Morgan, Eighteenth-century actuary, mathematician and radical","authors":"N. Biggs","doi":"10.1080/26375451.2021.2009720","DOIUrl":"https://doi.org/10.1080/26375451.2021.2009720","url":null,"abstract":"","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"37 1","pages":"81 - 82"},"PeriodicalIF":0.4,"publicationDate":"2022-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48848182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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