Pub Date : 2016-04-11DOI: 10.1080/17498430.2015.1122301
Günhan Caglayan
These classroom notes offer methods of solving the quadrature of lunes, that is, the area of croissant-shaped plane figures bounded by two intersecting non-congruent circular arcs, using Hippocrates of Chios’ area conservation and similarity arguments. I also offer a method of using history in the classroom with students via dynamic geometry snapshots presented in a manner that complements the analytic and the visual approaches.
{"title":"Exploring the lunes of Hippocrates in a dynamic geometry environment","authors":"Günhan Caglayan","doi":"10.1080/17498430.2015.1122301","DOIUrl":"https://doi.org/10.1080/17498430.2015.1122301","url":null,"abstract":"These classroom notes offer methods of solving the quadrature of lunes, that is, the area of croissant-shaped plane figures bounded by two intersecting non-congruent circular arcs, using Hippocrates of Chios’ area conservation and similarity arguments. I also offer a method of using history in the classroom with students via dynamic geometry snapshots presented in a manner that complements the analytic and the visual approaches.","PeriodicalId":211442,"journal":{"name":"BSHM Bulletin: Journal of the British Society for the History of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128166062","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 : 2016-01-02DOI: 10.1080/17498430.2015.1045700
M. Friedman
In this paper we examine two mathematicians who published manuscripts on geometry and origami in the same year, 1893: Hermann Wiener and Sundara Row. The main question that concerns us is to situate these works in the correct historical and mathematical background. We suggest that Wiener and Row offer us, via folding a piece of paper, a new conception of geometry that is very different to the common conception at the time.
{"title":"Two beginnings of geometry and folding: Hermann Wiener and Sundara Row","authors":"M. Friedman","doi":"10.1080/17498430.2015.1045700","DOIUrl":"https://doi.org/10.1080/17498430.2015.1045700","url":null,"abstract":"In this paper we examine two mathematicians who published manuscripts on geometry and origami in the same year, 1893: Hermann Wiener and Sundara Row. The main question that concerns us is to situate these works in the correct historical and mathematical background. We suggest that Wiener and Row offer us, via folding a piece of paper, a new conception of geometry that is very different to the common conception at the time.","PeriodicalId":211442,"journal":{"name":"BSHM Bulletin: Journal of the British Society for the History of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132064383","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 : 2016-01-02DOI: 10.1080/17498430.2015.1080442
Stephanie Crampin
(2nd edition, 2004). Derbyshire, John, Unknown quantity: A real and imaginary history of algebra, Washington: Joseph Henry Press, 2006. Gray, Jeremy J, and Parshall, Karen Hunger (eds), Episodes in the history of modern algebra (180
{"title":"A note on some recent mathematical histories/A curious history of mathematics, by Joel Levy/The universe in zero words, by Dana Mackenzie/17 equations that changed the world, by Ian Stewart/Maths in 100 key breakthroughs, by Richard Elwes","authors":"Stephanie Crampin","doi":"10.1080/17498430.2015.1080442","DOIUrl":"https://doi.org/10.1080/17498430.2015.1080442","url":null,"abstract":"(2nd edition, 2004). Derbyshire, John, Unknown quantity: A real and imaginary history of algebra, Washington: Joseph Henry Press, 2006. Gray, Jeremy J, and Parshall, Karen Hunger (eds), Episodes in the history of modern algebra (180","PeriodicalId":211442,"journal":{"name":"BSHM Bulletin: Journal of the British Society for the History of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115090556","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 : 2016-01-02DOI: 10.1080/17498430.2015.1035582
L. Verburgt
Reverend H F C Logan is put forward as the formerly unidentified figure to which Robert Leslie Ellis referred in a journal entry of 1840 in which he wrote that it was due to his influence that William Whewell came to uphold particular Kantian views on time and space. The historical evidence of Ellis’s early familiarity with, and later commitment to Kant is noteworthy for at least two reasons. Firstly, it puts into doubt the accepted view of the second generation of reformers of British algebra as non-philosophical, practice-oriented mathematicians. Secondly, in so far as Logan was the correspondent of William Rowan Hamilton, it re-emphasizes that the role of Kantianism in the transition from ‘symbolical’ to ‘abstract’ algebra in nineteenth-century British algebra requires closer scrutiny.
{"title":"Robert Leslie Ellis, William Whewell and Kant: the role of Rev H F C Logan","authors":"L. Verburgt","doi":"10.1080/17498430.2015.1035582","DOIUrl":"https://doi.org/10.1080/17498430.2015.1035582","url":null,"abstract":"Reverend H F C Logan is put forward as the formerly unidentified figure to which Robert Leslie Ellis referred in a journal entry of 1840 in which he wrote that it was due to his influence that William Whewell came to uphold particular Kantian views on time and space. The historical evidence of Ellis’s early familiarity with, and later commitment to Kant is noteworthy for at least two reasons. Firstly, it puts into doubt the accepted view of the second generation of reformers of British algebra as non-philosophical, practice-oriented mathematicians. Secondly, in so far as Logan was the correspondent of William Rowan Hamilton, it re-emphasizes that the role of Kantianism in the transition from ‘symbolical’ to ‘abstract’ algebra in nineteenth-century British algebra requires closer scrutiny.","PeriodicalId":211442,"journal":{"name":"BSHM Bulletin: Journal of the British Society for the History of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124291725","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 : 2016-01-02DOI: 10.1080/17498430.2015.1048640
Christopher D. Hollings
Throughout E T Bell’s writings on mathematics, both those aimed at other mathematicians and those for a popular audience, we find him endeavouring to promote abstract algebra generally, and the postulational method in particular. Bell evidently felt that the adoption of the latter approach to algebra (a process that he termed the ‘arithmetization of algebra’) would lend the subject something akin to the level of rigour that analysis had achieved in the nineteenth century. However, despite promoting this point of view, it is not so much in evidence in Bell’s own mathematical work. I offer an explanation for this apparent contradiction in terms of Bell’s infamous penchant for mathematical ‘myth-making’.
{"title":"A tale of mathematical myth-making: E T Bell and the ‘arithmetization of algebra’†","authors":"Christopher D. Hollings","doi":"10.1080/17498430.2015.1048640","DOIUrl":"https://doi.org/10.1080/17498430.2015.1048640","url":null,"abstract":"Throughout E T Bell’s writings on mathematics, both those aimed at other mathematicians and those for a popular audience, we find him endeavouring to promote abstract algebra generally, and the postulational method in particular. Bell evidently felt that the adoption of the latter approach to algebra (a process that he termed the ‘arithmetization of algebra’) would lend the subject something akin to the level of rigour that analysis had achieved in the nineteenth century. However, despite promoting this point of view, it is not so much in evidence in Bell’s own mathematical work. I offer an explanation for this apparent contradiction in terms of Bell’s infamous penchant for mathematical ‘myth-making’.","PeriodicalId":211442,"journal":{"name":"BSHM Bulletin: Journal of the British Society for the History of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126546252","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 : 2016-01-02DOI: 10.1080/17498430.2015.1055156
S. Negrepontis, G. Tassopoulos
An ‘infinite decreasing sequence of Gnomons’ is characteristic, according to Proclus, of incommensurability, hence David Fowler's idea to reconstruct Theodorus’ proofs of incommensurabilities, reported in the Theaetetus147d, employing Gnomons, is attractive and solidly based. The ‘preservation of the shape of the Gnomons’ is a form of the Pythagorean principle of the Limited according to Aristotle. In the present paper we propose a reconstruction that employs Gnomons but is free of the drawbacks present in Fowler's reconstruction.
{"title":"Theodorus’ proofs of incommensurabilities with Gnomons","authors":"S. Negrepontis, G. Tassopoulos","doi":"10.1080/17498430.2015.1055156","DOIUrl":"https://doi.org/10.1080/17498430.2015.1055156","url":null,"abstract":"An ‘infinite decreasing sequence of Gnomons’ is characteristic, according to Proclus, of incommensurability, hence David Fowler's idea to reconstruct Theodorus’ proofs of incommensurabilities, reported in the Theaetetus147d, employing Gnomons, is attractive and solidly based. The ‘preservation of the shape of the Gnomons’ is a form of the Pythagorean principle of the Limited according to Aristotle. In the present paper we propose a reconstruction that employs Gnomons but is free of the drawbacks present in Fowler's reconstruction.","PeriodicalId":211442,"journal":{"name":"BSHM Bulletin: Journal of the British Society for the History of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134351734","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 : 2016-01-02DOI: 10.1080/17498430.2015.1035588
Richard Decesare
You will not see Robert Patterson’s name mentioned in many mathematics books. While his mathematical works survive, his name is more likely to appear in American history books dealing with the Colonial period, given his associations with the most influential men of that time. In this article, we will examine his mathematical work, as well as his contributions to a newly-formed nation. Most of what we know about Robert Patterson’s ancestors and life is due to his grandson, William Ewing DuBois, who wrote a family history in 1847. For other information, I have drawn upon diaries and a great many letters. All spelling and syntax are copied exactly as they appear.
{"title":"Robert Patterson: American ‘revolutionary’ mathematician","authors":"Richard Decesare","doi":"10.1080/17498430.2015.1035588","DOIUrl":"https://doi.org/10.1080/17498430.2015.1035588","url":null,"abstract":"You will not see Robert Patterson’s name mentioned in many mathematics books. While his mathematical works survive, his name is more likely to appear in American history books dealing with the Colonial period, given his associations with the most influential men of that time. In this article, we will examine his mathematical work, as well as his contributions to a newly-formed nation. Most of what we know about Robert Patterson’s ancestors and life is due to his grandson, William Ewing DuBois, who wrote a family history in 1847. For other information, I have drawn upon diaries and a great many letters. All spelling and syntax are copied exactly as they appear.","PeriodicalId":211442,"journal":{"name":"BSHM Bulletin: Journal of the British Society for the History of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124966481","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 : 2016-01-02DOI: 10.1080/17498430.2015.1080441
A. Rice
{"title":"Taming the unknown: a history of algebra from antiquity to the early twentieth century, by Victor J Katz and Karen Hunger Parshall","authors":"A. Rice","doi":"10.1080/17498430.2015.1080441","DOIUrl":"https://doi.org/10.1080/17498430.2015.1080441","url":null,"abstract":"","PeriodicalId":211442,"journal":{"name":"BSHM Bulletin: Journal of the British Society for the History of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133378794","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 : 2016-01-02DOI: 10.1080/17498430.2015.1046037
A. Rice, Ezra Brown
This two-part paper investigates the discovery of an intriguing and fundamental connection between the famous but apparently unrelated mathematical work of two late third-century mathematicians, a link that went unnoticed for well over 1500 years. In this, the first installment of the paper, we examine the initial chain of mathematical events that would ultimately lead to the discovery of this remarkable link between two seemingly distinct areas of mathematics, encompassing contributions by a variety of mathematicians, from the most distinguished to the relatively unknown.
{"title":"Commutativity and collinearity: a historical case study of the interconnection of mathematical ideas. Part I","authors":"A. Rice, Ezra Brown","doi":"10.1080/17498430.2015.1046037","DOIUrl":"https://doi.org/10.1080/17498430.2015.1046037","url":null,"abstract":"This two-part paper investigates the discovery of an intriguing and fundamental connection between the famous but apparently unrelated mathematical work of two late third-century mathematicians, a link that went unnoticed for well over 1500 years. In this, the first installment of the paper, we examine the initial chain of mathematical events that would ultimately lead to the discovery of this remarkable link between two seemingly distinct areas of mathematics, encompassing contributions by a variety of mathematicians, from the most distinguished to the relatively unknown.","PeriodicalId":211442,"journal":{"name":"BSHM Bulletin: Journal of the British Society for the History of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128196136","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 : 2015-09-02DOI: 10.1080/17498430.2015.1055089
T. Crilly
T he BSHM was co-sponsor of this De Morgan meeting commemorating the 150th anniversary of the setting up of the London Mathematical Society (LMS). Augustus De Morgan was its first president, the society inspired by students A C Ranyard and George De Morgan, the ‘mathematical son’ of Augustus. The meeting began with a welcome from the two current presidents, Terry Lyons (for the LMS) and Philip Beeley (for the BSHM). It was fitting that the De Morgan meeting took place in ‘his’ building, located not more than half a mile from the place where the inaugural meeting took place, the meeting at which he presided. We were reminded of the traditions of the society when two participants signed the LMS book and formally became members. Following an overview of De Morgan’s life, we were treated to lectures on aspects of his life in mathematics and logic. He was a great letter writer, and there were talks on his correspondence with George Boole and Ada Lovelace, and lectures on his role in setting up the LMS, his writing of the Budget of Paradoxes and his influence on modern thought. As an introduction, Adrian Rice spoke on De Morgan’s life and work. As he noted De Morgan is perhaps best remembered for the laws that bear his name: the famous De Morgan laws linking union, intersection, and complement of sets, the laws which appear in just about all modern textbooks. That De Morgan was the first discoverer of these is open to historical dispute, and given this tenuous link, what else should De Morgan be remembered for? In recent decades, research has shed light on forgotten aspects of his life and associations giving us a more complete picture of the range and diversity of his mathematical activities, and not least his personality. He always insisted on being De Morgan (and not de Morgan), declined membership of the Royal Society, and, in an age when the Anglican Church was all powerful, described himself as a ‘Christian unattached’. As the meeting progressed De Morgan emerged as a man of singular character. The lecture by Chris Hollings delved into De Morgan’s correspondence with Ada King, Countess of Lovelace, the daughter of Lord Byron. Previous readings of this correspondence, in which De Morgan adopted the role of teacher, have resulted in wildly differing assessments of her mathematical abilities. Here it was argued that she lay somewhere around half-way between the extremes of ‘hopeless’ and ‘mathematical genius.’ Of De Morgan’s publications on general mathematics, his Budget of Paradoxes was best known in his day, and it is still a good read. It was a natural outgrowth of his book reviews published in the weekly London-based literary magazine, The Athenæum. Sloan Despeaux’s talk included sample episodes associated with the amateurs and cranks of Victorian England bound together for all to enjoy. Top of the pile must be James Smith, a Liverpool merchant and Arch-Paradoxer, the king of the circle-squarers. He spent much of his retirement insisting that t
BSHM是纪念伦敦数学学会(LMS)成立150周年的德摩根会议的共同赞助者。奥古斯都·德·摩根(Augustus De Morgan)是该协会的首任主席，该协会的灵感来自学生A·C·兰亚德和奥古斯都的“数学之子”乔治·德·摩根(George De Morgan)。会议以两位现任主席特里·莱昂斯(LMS的主席)和菲利普·比利(BSHM的主席)的欢迎开始。德·摩根的会议在“他的”大楼里举行是再合适不过的了，这座大楼离他主持的就职会议举行的地方不到半英里。当两名参与者在LMS书上签名并正式成为会员时，我们想起了这个协会的传统。在概述了德·摩根的生平之后，我们听取了他在数学和逻辑方面的生平讲座。他是一个伟大的书信作家，有关于他与乔治·布尔和阿达·洛夫莱斯的通信的讲座，也有关于他在建立LMS中的作用的讲座，他写的《悖论预算》以及他对现代思想的影响。作为开场白，阿德里安·赖斯讲述了德·摩根的生活和工作。正如他所指出的，德·摩尔根最为人所铭记的也许是以他的名字命名的定律:著名的德·摩尔根定律，它将集合的并集、交集和补集联系起来，这些定律几乎出现在所有现代教科书中。德·摩尔根是这些理论的第一个发现者，这在历史上是有争议的，鉴于这种微弱的联系，德·摩尔根还应该被人们记住什么呢?近几十年来，研究揭示了他的生活和交往中被遗忘的方面，使我们更全面地了解了他的数学活动的范围和多样性，尤其是他的个性。他一直坚持自己是德·摩根(而不是德·摩根)，拒绝成为皇家学会的会员，在英国国教势力强大的时代，他称自己为“独立的基督徒”。随着会谈的进行，德·摩根成为了一个性格独特的人。克里斯·霍林斯的讲座深入探讨了德·摩根与拜伦勋爵的女儿、洛夫莱斯伯爵夫人艾达·金的通信。在之前的阅读中，德摩根扮演了老师的角色，对她的数学能力产生了截然不同的评价。在这里，人们认为她介于“绝望”和“数学天才”这两个极端之间。在德·摩尔根关于一般数学的著作中，他的《悖论预算》在他那个时代最为著名，至今仍是一本好书。这是他在伦敦文学周刊《the Athenæum》上发表书评的自然结果。斯隆·德斯波的演讲包含了一些与维多利亚时代英国的业余爱好者和怪人有关的例子，供所有人欣赏。排在第一位的一定是詹姆斯·史密斯，他是利物浦商人，也是最大的悖论者，是圆方之王。他退休后的大部分时间都坚持认为p的真正价值是
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