Archive for History of Exact Sciences最新文献

{"title":"Hero and the tradition of the circle segment","authors":"Henry Mendell","doi":"10.1007/s00407-023-00308-y","DOIUrl":"10.1007/s00407-023-00308-y","url":null,"abstract":"<div><p>In his <i>Metrica</i>, Hero provides four procedures for finding the area of a circular segment (with <i>b</i> the base of the segment and <i>h</i> its height): an Ancient method for when the segment is smaller than a semicircle, <span>((b + h)/2 , cdot , h)</span>; a Revision, <span>((b + h)/2 , cdot , h + (b/2)^{2} /14)</span>; a quasi-Archimedean method (said to be inspired by the quadrature of the parabola) for cases where <i>b</i> is more than triple <i>h</i>, <span>({raise0.5exhbox{$scriptstyle 4$} kern-0.1em/kern-0.15em lower0.25exhbox{$scriptstyle 3$}}(h , cdot , b/2))</span>; and a method of Subtraction using the Revised method, for when it is larger than a semicircle. He gives superficial arguments that the Ancient method presumes <span>(pi = 3)</span> and the Revision, <span>(pi = {raise0.5exhbox{$scriptstyle {22}$} kern-0.1em/kern-0.15em lower0.25exhbox{$scriptstyle 7$}})</span>. We are left with many questions. How ancient is the Ancient? Why did anyone think it worked? Why would anyone revise it in just this way? In addition, why did Hero think the Revised method did not work when <span>(b &gt; 3;h)</span>? I show that a fifth century BCE Uruk tablet employs the Ancient method, but possibly with very strange consequences, and that a Ptolemaic Egyptian papyrus that checks this method by comparing the area of a circle calculated from the sum of a regular inscribed polygon and the areas of the segments on its sides as determined by the Ancient method with the area of the circle as calculated from its diameter correctly sees that the calculations do not quite gel in the case of a triangle but do in the case of a square. Both traditions probably could also calculate the area of a segment on an inscribed regular polygon by subtracting the area of the polygon from the area of the circle and dividing by the number of sides of the polygon. I then derive two theorems about pairs of segments, that the reviser of the Ancient method should have known, that explain each method, why they work when they do and do not when they do not, and which lead to a curious generalization of the Revised method. Hero’s comment is right, but not for the reasons he gives. I conclude with an exploration of Hero’s restrictions of the Revised method and Hero’s two alternative methods.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"77 5","pages":"451 - 499"},"PeriodicalIF":0.5,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00407-023-00308-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41439424","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Auerbach, Lotka, and Zipf: pioneers of power-law city-size distributions","authors":"Diego Rybski,&nbsp;Antonio Ciccone","doi":"10.1007/s00407-023-00314-0","DOIUrl":"10.1007/s00407-023-00314-0","url":null,"abstract":"<div><p>Power-law city-size distributions are a statistical regularity researched in many countries and urban systems. In this history of science treatise we reconsider Felix Auerbach’s paper published in 1913. We reviewed his analysis and found (i) that a constant absolute concentration, as introduced by him, is equivalent to a power-law distribution with exponent <span>(approx 1)</span>, (ii) that Auerbach describes this equivalence, and (iii) that Auerbach also pioneered the empirical analysis of city-size distributions across countries, regions, and time periods. We further investigate his legacy as reflected in citations and find that important follow-up work, e.g. by Lotka (Elements of physical biology. Williams &amp; Wilkins Company, Baltimore, 1925) and Zipf (Human behavior and the principle of least effort: an introduction to human ecology, Martino Publishing, Manfield Centre, CT (2012), 1949), does give proper reference to his discovery—but others do not. For example, only approximately 20% of city-related works citing Zipf (1949) also cite Auerbach (Petermanns Geogr Mitteilungen 59(74):74–76, 1913). To our best knowledge, Lotka (1925) was the first to describe the power-law rank-size rule as it is analyzed today. Saibante (Metron Rivista Internazionale di Statistica 7(2):53–99, 1928), building on Auerbach and Lotka, investigated the power-law rank-size rule across countries, regions, and time periods. Zipf’s achievement was to embed these findings in his monumental 1949 book. We suggest that the use of “Auerbach–Lotka–Zipf law” (or “ALZ-law”) is more appropriate than “Zipf’s law for cities”, which also avoids confusion with Zipf’s law for word frequency. We end the treatise with biographical notes on Auerbach.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"77 6","pages":"601 - 613"},"PeriodicalIF":0.5,"publicationDate":"2023-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00407-023-00314-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47063522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An early system A-type scheme for Saturn from Babylon","authors":"John Steele,&nbsp;Teije de Jong","doi":"10.1007/s00407-023-00311-3","DOIUrl":"10.1007/s00407-023-00311-3","url":null,"abstract":"<div><p>In this paper we publish three fragments of a cuneiform tablet that, when complete, contained the dates and zodiacal positions of Saturn’s synodic phenomena for roughly 60 years. The text is unique in containing comparisons of computed data with observations. Through an analysis of the preserved data we propose that the dates and positions were computed by an otherwise unknown two-zone System A-type scheme and show that the computed data in the tablet can be dated to the fourth century BC. This early date and the comparisons with observations suggest that the text was produced during the period of active development of the planetary systems.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"77 5","pages":"501 - 535"},"PeriodicalIF":0.5,"publicationDate":"2023-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00407-023-00311-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44377583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Eudoxus’ simultaneous risings and settings","authors":"Francesca Schironi","doi":"10.1007/s00407-023-00309-x","DOIUrl":"10.1007/s00407-023-00309-x","url":null,"abstract":"<div><p>The article provides a reconstruction of Eudoxus' approach to simultaneous risings and settings in his two works dedicated to the issue: the <i>Phaenomena</i> and the <i>Enoptron</i>. This reconstruction is based on the analysis of Eudoxus’ fragments transmitted by Hipparchus. These fragments are difficult and problematic, but a close analysis and a comparison with the corresponding passages in Aratus suggests a possible solution.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"77 4","pages":"423 - 441"},"PeriodicalIF":0.5,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46893922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Geometry and analysis in Anastácio da Cunha’s calculus","authors":"João Caramalho Domingues","doi":"10.1007/s00407-023-00313-1","DOIUrl":"10.1007/s00407-023-00313-1","url":null,"abstract":"<div><p>It is well known that over the eighteenth century the calculus moved away from its geometric origins; Euler, and later Lagrange, aspired to transform it into a “purely analytical” discipline. In the 1780 s, the Portuguese mathematician José Anastácio da Cunha developed an original version of the calculus whose interpretation in view of that process presents challenges. Cunha was a strong admirer of Newton (who famously favoured geometry over algebra) and criticized Euler’s faith in analysis. However, the fundamental propositions of his calculus follow the analytical trend. This appears to have been possible due to a nominalistic conception of variable that allowed him to deal with expressions as names, rather than abstract quantities. Still, Cunha tried to keep the definition of fluxion directly applicable to geometrical magnitudes. According to a friend of Cunha’s, his calculus had an algebraic (analytical) branch and a geometrical branch, and it was because of this that his definition of fluxion appeared too complex to some contemporaries.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"77 6","pages":"579 - 600"},"PeriodicalIF":0.5,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00407-023-00313-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48266395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Measurements of altitude and geographic latitude in Latin astronomy, 1100–1300","authors":"C. Philipp E. Nothaft","doi":"10.1007/s00407-023-00312-2","DOIUrl":"10.1007/s00407-023-00312-2","url":null,"abstract":"<div><p>This article surveys measurements of celestial (chiefly solar) altitudes documented from twelfth- and thirteenth-century Latin Europe. It consists of four main parts providing (i) an overview of the instruments available for altitude measurements and described in contemporary sources, viz. astrolabes, quadrants, shadow sticks, and the torquetum; (ii) a survey of the role played by altitude measurements in the determination of geographic latitude, which takes into account more than 70 preserved estimates; (iii) case studies of four sets of measured solar altitudes in twelfth-century Latin sources; (iv) an in-depth discussion of the evidence relating to altitude measurements performed in Paris in the period 1281–1290. The findings from the last part indicate that by the end of the thirteenth century Parisian astronomer had developed rigorous standards of observational practice in which altitudes were typically measured to a precision of minutes of arc and with a level of accuracy higher than ± 0;5°, and sometimes exceeding ± 0;1°.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"77 6","pages":"537 - 577"},"PeriodicalIF":0.5,"publicationDate":"2023-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00407-023-00312-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50456450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Jeffreys–Lindley paradox: an exchange","authors":"Jeremy Gray,&nbsp;Joshua L. Cherry,&nbsp;Eric-Jan Wagenmakers,&nbsp;Alexander Ly","doi":"10.1007/s00407-023-00310-4","DOIUrl":"10.1007/s00407-023-00310-4","url":null,"abstract":"<div><p>This Editorial reports an exchange in form of a comment and reply on the article “History and Nature of the Jeffreys–Lindley Paradox” (Arch Hist Exact Sci 77:25, 2023) by Eric-Jan Wagenmakers and Alexander Ly.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"77 4","pages":"443 - 449"},"PeriodicalIF":0.5,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41682671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Federico Commandino and the Latin edition of Apollonius’s Conics (1566)","authors":"Argante Ciocci","doi":"10.1007/s00407-023-00307-z","DOIUrl":"10.1007/s00407-023-00307-z","url":null,"abstract":"<div><p>Federico Commandino’s Latin editions of the mathematical works written by the ancient Greeks constituted an essential reference for the scientific research undertaken by the moderns. In his Latin editions, Commandino cleverly combined his philological and mathematical skills. Philology and mathematics, moreover, nurtured each other. In this article, I analyze the Greek and Latin manuscripts and the printed edition of Apollonius’ <i>Conics</i> to highlight in a specific case study the role of the editions of the classics in the renaissance of modern mathematics.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"77 4","pages":"393 - 421"},"PeriodicalIF":0.5,"publicationDate":"2023-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00407-023-00307-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41707906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Ptolemy’s treatise on the meteoroscope recovered","authors":"Victor Gysembergh,&nbsp;Alexander Jones,&nbsp;Emanuel Zingg,&nbsp;Pascal Cotte,&nbsp;Salvatore Apicella","doi":"10.1007/s00407-022-00302-w","DOIUrl":"10.1007/s00407-022-00302-w","url":null,"abstract":"<div><p>The eighth-century Latin manuscript Milan, Veneranda Biblioteca Ambrosiana, L 99 Sup. contains fifteen palimpsest leaves previously used for three Greek scientific texts: a text of unknown authorship on mathematical mechanics and catoptrics, known as the <i>Fragmentum Mathematicum Bobiense</i> (three leaves), Ptolemy's <i>Analemma</i> (six leaves), and an astronomical text that has hitherto remained unidentified and almost entirely unread (six leaves). We report here on the current state of our research on this last text, based on multispectral images. The text, incompletely preserved, is a treatise on the construction and uses of a nine-ringed armillary instrument, identifiable as the “meteoroscope” invented by Ptolemy and known to us from passages in Ptolemy's <i>Geography</i> and in writings of Pappus and Proclus. We further argue that the author of our text was Ptolemy himself.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"77 2","pages":"221 - 240"},"PeriodicalIF":0.5,"publicationDate":"2023-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00407-022-00302-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48791537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Felix Klein, Sophus Lie, contact transformations, and connexes","authors":"L. D. Kay","doi":"10.1007/s00407-023-00305-1","DOIUrl":"10.1007/s00407-023-00305-1","url":null,"abstract":"<div><p>Much of the mathematics with which Felix Klein and Sophus Lie are now associated (Klein’s Erlangen Program and Lie’s theory of transformation groups) is rooted in ideas they developed in their early work: the consideration of geometric objects or properties preserved by systems of transformations. As early as 1870, Lie studied particular examples of what he later called <i>contact transformations</i>, which preserve tangency and which came to play a crucial role in his systematic study of transformation groups and differential equations. This note examines Klein’s efforts in the 1870s to interpret contact transformations in terms of <i>connexes</i> and traces that interpretation (which included a false assumption) over the decades that follow. The analysis passes from Klein’s letters to Lie through Lindemann’s edition of Clebsch’s lectures on geometry in 1876, Lie’s criticism of it in his treatise on transformation groups in 1893, and the careful development of that interpretation by Dohmen, a student of Engel, in his 1905 dissertation. The now-obscure notion of connexes and its relation to Lie’s <i>line elements</i> and <i>surface elements</i> are discussed here in some detail.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"77 4","pages":"373 - 391"},"PeriodicalIF":0.5,"publicationDate":"2023-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00407-023-00305-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42745210","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}