Pub Date : 2023-12-14DOI: 10.1080/26375451.2023.2258331
Jorge Nuno Silva, P. Freitas
{"title":"The\u0000 Loterias Lisbonenses\u0000 of Francisco Giraldes Barba","authors":"Jorge Nuno Silva, P. Freitas","doi":"10.1080/26375451.2023.2258331","DOIUrl":"https://doi.org/10.1080/26375451.2023.2258331","url":null,"abstract":"","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"36 6","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138971289","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 : 2023-09-21DOI: 10.1080/26375451.2023.2250980
David Acheson
An elegant geometrical argument concerning Dido’s problem, traditionally credited to Jakob Steiner in about 1840, appears to have been invented by Thomas Simpson, almost 100 years earlier. The argument itself uses only the idea of triangle area and the converse of Thales’ theorem about the angle in a semicircle. It appears in Simpson’s book Elements of Geometry, published in 1760, as part of a highly original section on ‘The Maxima and Minima of Geometrical Quantities’, but traces of the argument can even be found in an earlier edition dated 1747.
{"title":"Thomas Simpson and Dido’s problem","authors":"David Acheson","doi":"10.1080/26375451.2023.2250980","DOIUrl":"https://doi.org/10.1080/26375451.2023.2250980","url":null,"abstract":"An elegant geometrical argument concerning Dido’s problem, traditionally credited to Jakob Steiner in about 1840, appears to have been invented by Thomas Simpson, almost 100 years earlier. The argument itself uses only the idea of triangle area and the converse of Thales’ theorem about the angle in a semicircle. It appears in Simpson’s book Elements of Geometry, published in 1760, as part of a highly original section on ‘The Maxima and Minima of Geometrical Quantities’, but traces of the argument can even be found in an earlier edition dated 1747.","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136130701","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 : 2023-09-02DOI: 10.1080/26375451.2023.2248446
Peter Cameron
{"title":"Graph theory in America: the first hundred years","authors":"Peter Cameron","doi":"10.1080/26375451.2023.2248446","DOIUrl":"https://doi.org/10.1080/26375451.2023.2248446","url":null,"abstract":"","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49195763","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 : 2023-06-09DOI: 10.1080/26375451.2023.2215652
Daniel F. Mansfield
{"title":"Mesopotamian square root approximation by a sequence of rectangles","authors":"Daniel F. Mansfield","doi":"10.1080/26375451.2023.2215652","DOIUrl":"https://doi.org/10.1080/26375451.2023.2215652","url":null,"abstract":"","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42128980","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 : 2023-05-04DOI: 10.1080/26375451.2023.2222414
Zakkai Goriely
In the eighteenth century, a rift opened between the mathematical communities in Britain and the European continent. Following the Newton-Leibniz controversy on the priority of the calculus’s discovery, most British mathematicians pledged loyalty to Newton, and by extension, to his fluxional calculus (Guicciardini 2009, 1). However, by the early nineteenth century, several British mathematicians noticed the progress made on the European continent following the work of Leibniz, to which Britain had been mostly blind. These mathematicians dedicated themselves to the reform of British mathematics by circulating differential calculus. The main difference between Newton and Leibniz’s calculus is that Newton’s ‘fluxional calculus’ rested on geometrical and physical intuition (see Stedall 2008; Guicciardini 2009; Kline 1990). His method of finding the ‘fluxion’ of a function was to consider the velocity of a particle moving along a curve. In modern terms, if a particle moves in the plane with position (x(t), y(t)), its velocity is expressed in terms of x’(t) and y’(t). Newton (1669, 178) examined the limit of the ratio of these derivatives and wrote ẋo for x’(t), where ẋ is the fluxion, or instantaneous velocity, of x, and o is an infinitely small time interval (Guicciardini 2009). By contrast, Leibniz’s ‘differential calculus’ had an algebraic basis. Leibniz directly examined the infinitely small increments in x and y (differentials) and determined their relationship without the need for physical intuition. Instead, he used infinite sums with a notation close to what we use today (Leibniz 1682; translated in Stedall 2008); in place of Newton’s ẋo, o and ẋ, Leibniz used dx, dt, and dx dt respectively (Guicciardini 2009, 3).
{"title":"British mathematical reformers in the nineteenth century: motivations and methods Winner of the BSHM undergraduate essay prize 2022","authors":"Zakkai Goriely","doi":"10.1080/26375451.2023.2222414","DOIUrl":"https://doi.org/10.1080/26375451.2023.2222414","url":null,"abstract":"In the eighteenth century, a rift opened between the mathematical communities in Britain and the European continent. Following the Newton-Leibniz controversy on the priority of the calculus’s discovery, most British mathematicians pledged loyalty to Newton, and by extension, to his fluxional calculus (Guicciardini 2009, 1). However, by the early nineteenth century, several British mathematicians noticed the progress made on the European continent following the work of Leibniz, to which Britain had been mostly blind. These mathematicians dedicated themselves to the reform of British mathematics by circulating differential calculus. The main difference between Newton and Leibniz’s calculus is that Newton’s ‘fluxional calculus’ rested on geometrical and physical intuition (see Stedall 2008; Guicciardini 2009; Kline 1990). His method of finding the ‘fluxion’ of a function was to consider the velocity of a particle moving along a curve. In modern terms, if a particle moves in the plane with position (x(t), y(t)), its velocity is expressed in terms of x’(t) and y’(t). Newton (1669, 178) examined the limit of the ratio of these derivatives and wrote ẋo for x’(t), where ẋ is the fluxion, or instantaneous velocity, of x, and o is an infinitely small time interval (Guicciardini 2009). By contrast, Leibniz’s ‘differential calculus’ had an algebraic basis. Leibniz directly examined the infinitely small increments in x and y (differentials) and determined their relationship without the need for physical intuition. Instead, he used infinite sums with a notation close to what we use today (Leibniz 1682; translated in Stedall 2008); in place of Newton’s ẋo, o and ẋ, Leibniz used dx, dt, and dx dt respectively (Guicciardini 2009, 3).","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"38 1","pages":"159 - 167"},"PeriodicalIF":0.4,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43093420","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 : 2023-05-04DOI: 10.1080/26375451.2023.2215650
Melinda Lanius
Fostering students' relational understanding of the equals sign is a challenge for math educators that begins in the primary levels and persists into tertiary education. In this paper, I develop an entry point, especially for students who only have an operational understanding of the equals sign, to the core idea of equivalence in linear algebra. My approach is informed by the history of mathematics: In the 17th and 18th centuries, mathematics research underwent an algebraicization, with mathematicians replacing their classical geometric questions with novel algebraic investigations. In this paper, I will offer geometric interpretations of two operations developed at the precipice of this monumental shift: algebraic substitution and Gaussian elimination. I will then utilize Lakoff & Johnson's theory of conceptual metaphor to compare and contrast this historically-grounded geometric re-interpretation of modern linear algebra to the direct algebraic interpretation taken in most modern textbooks.
{"title":"Through the looking glass, and what algebra found there: historically informed conceptual metaphors of algebraic substitution and Gaussian elimination","authors":"Melinda Lanius","doi":"10.1080/26375451.2023.2215650","DOIUrl":"https://doi.org/10.1080/26375451.2023.2215650","url":null,"abstract":"Fostering students' relational understanding of the equals sign is a challenge for math educators that begins in the primary levels and persists into tertiary education. In this paper, I develop an entry point, especially for students who only have an operational understanding of the equals sign, to the core idea of equivalence in linear algebra. My approach is informed by the history of mathematics: In the 17th and 18th centuries, mathematics research underwent an algebraicization, with mathematicians replacing their classical geometric questions with novel algebraic investigations. In this paper, I will offer geometric interpretations of two operations developed at the precipice of this monumental shift: algebraic substitution and Gaussian elimination. I will then utilize Lakoff & Johnson's theory of conceptual metaphor to compare and contrast this historically-grounded geometric re-interpretation of modern linear algebra to the direct algebraic interpretation taken in most modern textbooks.","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"38 1","pages":"141 - 157"},"PeriodicalIF":0.4,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41458979","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 : 2023-05-04DOI: 10.1080/26375451.2023.2224132
B. Wardhaugh
In late 1683, the physician and F.R.S. Martin Lister displayed to the Royal Society a new way of recording barometric observations, which amounted in all but name to the construction of line graphs. The innovation was communicated to the Oxford Philosophical Society, where Robert Plot used the method to display a year’s observations, published in the Philosophical Transactions early in 1685. At the Dublin Philosophical Society, William Molyneux displayed a month’s worth of observations kept in the same way during May 1684. Lister’s own engraved forms—what amounted to graph paper—circulated among these groups but are not known to survive; Molyneux had forms of his own engraved. Finally, the London instrument maker John Warner engraved his own version of a similar form for recording weather observations and offered it to Plot. Two exemplars survive, but neither Warner’s offer nor this graphical method itself seem to have been more widely taken up in this period. This paper reviews the evidence for this early interest in and promotion of line graphs and graph paper, a century before the wider uptake of these technologies.
{"title":"Graphs in the 1680s: Martin Lister, Robert Plot, William Molyneux and John Warner","authors":"B. Wardhaugh","doi":"10.1080/26375451.2023.2224132","DOIUrl":"https://doi.org/10.1080/26375451.2023.2224132","url":null,"abstract":"In late 1683, the physician and F.R.S. Martin Lister displayed to the Royal Society a new way of recording barometric observations, which amounted in all but name to the construction of line graphs. The innovation was communicated to the Oxford Philosophical Society, where Robert Plot used the method to display a year’s observations, published in the Philosophical Transactions early in 1685. At the Dublin Philosophical Society, William Molyneux displayed a month’s worth of observations kept in the same way during May 1684. Lister’s own engraved forms—what amounted to graph paper—circulated among these groups but are not known to survive; Molyneux had forms of his own engraved. Finally, the London instrument maker John Warner engraved his own version of a similar form for recording weather observations and offered it to Plot. Two exemplars survive, but neither Warner’s offer nor this graphical method itself seem to have been more widely taken up in this period. This paper reviews the evidence for this early interest in and promotion of line graphs and graph paper, a century before the wider uptake of these technologies.","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"38 1","pages":"97 - 106"},"PeriodicalIF":0.4,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46048846","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 : 2023-05-04DOI: 10.1080/26375451.2023.2197351
Roy Wagner, R. Netz
This paper attempts a new interpretation of Euclid’s Elements Book X. This study of irrational lines has long been viewed as an anomaly within the Euclidean corpus: it includes a tedious and seemingly pointless classification of lines, known as ‘the cross of mathematicians’. Following Ken Saito’s toolbox conception, we do not try to reconstruct the book’s mathematical process of discovery, but, instead, the kind of applications for which it serves as a toolbox. Our claim is that the book provides tools for solving questions about proportional lines inspired by results in music theory and a context of Pythagorean-Platonic interest in proportions. We show that the entire content of Book X can indeed be accounted for as a set of tools for these questions, augmented by the general editorial norms that govern the Elements. We conclude by explaining why the purpose of Book X as reconstructed here has disappeared from mathematical memory.
{"title":"Between music and geometry: a proposal for the early intended application of Euclid’s Elements Book X","authors":"Roy Wagner, R. Netz","doi":"10.1080/26375451.2023.2197351","DOIUrl":"https://doi.org/10.1080/26375451.2023.2197351","url":null,"abstract":"This paper attempts a new interpretation of Euclid’s Elements Book X. This study of irrational lines has long been viewed as an anomaly within the Euclidean corpus: it includes a tedious and seemingly pointless classification of lines, known as ‘the cross of mathematicians’. Following Ken Saito’s toolbox conception, we do not try to reconstruct the book’s mathematical process of discovery, but, instead, the kind of applications for which it serves as a toolbox. Our claim is that the book provides tools for solving questions about proportional lines inspired by results in music theory and a context of Pythagorean-Platonic interest in proportions. We show that the entire content of Book X can indeed be accounted for as a set of tools for these questions, augmented by the general editorial norms that govern the Elements. We conclude by explaining why the purpose of Book X as reconstructed here has disappeared from mathematical memory.","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"38 1","pages":"69 - 96"},"PeriodicalIF":0.4,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42082233","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 : 2023-05-04DOI: 10.1080/26375451.2023.2224600
Brigitte Stenhouse
{"title":"BSHM Meeting news","authors":"Brigitte Stenhouse","doi":"10.1080/26375451.2023.2224600","DOIUrl":"https://doi.org/10.1080/26375451.2023.2224600","url":null,"abstract":"","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"38 1","pages":"158 - 158"},"PeriodicalIF":0.4,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45531391","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 : 2023-05-04DOI: 10.1080/26375451.2023.2186648
Domingo Martínez-Verdú, M. Massa-Esteve, A. Linero-Bas
During the Spanish eighteenth century, a process of modernization took place in scientific knowledge, partly driven by the circulation and appropriation of new scientific ideas. In this context, the Spanish mathematician Benito Bails (1731–1797) published his course Elementos de Matemática (Elements of Mathematics) consisting of ten volumes (1779–1799), in which, among other subjects, he presented one of the most complete mathematical developments of logarithmic calculation methods of his time, by using the infinity through infinite series. The aim of our article is to demonstrate how algebraic analytical reasoning enabled Bails to obtain new and more efficient infinite algorithms that converge more quickly in the computation of logarithms in any system. We show how Euler's number ‘e’ is calculated, probably for the first time in Spanish teaching, in an eighteenth century mathematical text. Our analysis concludes that Bails’ course constituted an innovation and provides evidence of its creativity, originality and ingenuity.
{"title":"Infinite analytical procedures for the computation of logarithms in works by Benito Bails (1731–1797)","authors":"Domingo Martínez-Verdú, M. Massa-Esteve, A. Linero-Bas","doi":"10.1080/26375451.2023.2186648","DOIUrl":"https://doi.org/10.1080/26375451.2023.2186648","url":null,"abstract":"During the Spanish eighteenth century, a process of modernization took place in scientific knowledge, partly driven by the circulation and appropriation of new scientific ideas. In this context, the Spanish mathematician Benito Bails (1731–1797) published his course Elementos de Matemática (Elements of Mathematics) consisting of ten volumes (1779–1799), in which, among other subjects, he presented one of the most complete mathematical developments of logarithmic calculation methods of his time, by using the infinity through infinite series. The aim of our article is to demonstrate how algebraic analytical reasoning enabled Bails to obtain new and more efficient infinite algorithms that converge more quickly in the computation of logarithms in any system. We show how Euler's number ‘e’ is calculated, probably for the first time in Spanish teaching, in an eighteenth century mathematical text. Our analysis concludes that Bails’ course constituted an innovation and provides evidence of its creativity, originality and ingenuity.","PeriodicalId":36683,"journal":{"name":"British Journal for the History of Mathematics","volume":"1 1","pages":"107 - 140"},"PeriodicalIF":0.4,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60143797","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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