Archive for History of Exact Sciences最新文献

{"title":"Fiction, possibility and impossibility: three kinds of mathematical fictions in Leibniz’s work","authors":"Oscar M. Esquisabel,&nbsp;Federico Raffo Quintana","doi":"10.1007/s00407-021-00277-0","DOIUrl":"10.1007/s00407-021-00277-0","url":null,"abstract":"<div><p>This paper is concerned with the status of mathematical fictions in Leibniz’s work and especially with infinitary quantities as fictions. Thus, it is maintained that mathematical fictions constitute a kind of symbolic notion that implies various degrees of impossibility. With this framework, different kinds of notions of possibility and impossibility are proposed, reviewing the usual interpretation of both modal concepts, which appeals to the consistency property. Thus, three concepts of the possibility/impossibility pair are distinguished; they give rise, in turn, to three concepts of mathematical fictions. Moreover, such a distinction is the base for the claim that infinitesimal quantities, as mathematical fictions, do not imply an absolute impossibility, resulting from self-contradiction, but a relative impossibility, founded on irrepresentability and on the fact that it does not conform to architectural principles. In conclusion, this “soft” impossibility of infinitesimals yields them, in Leibniz view, a presumptive or “conjectural” status.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"75 6","pages":"613 - 647"},"PeriodicalIF":0.5,"publicationDate":"2021-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00407-021-00277-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43936314","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":"The six books of Diophantus’ Arithmetic increased and reduced to specious: the lost manuscript of Jacques Ozanam (1640–1718)","authors":"Francisco Gómez-García,&nbsp;Pedro J. Herrero-Piñeyro,&nbsp;Antonio Linero-Bas,&nbsp;Ma. Rosa Massa-Esteve,&nbsp;Antonio Mellado-Romero","doi":"10.1007/s00407-021-00274-3","DOIUrl":"10.1007/s00407-021-00274-3","url":null,"abstract":"<div><p>The introduction of a new analytical method, due fundamentally to François Viète and René Descartes and the later dissemination of their works, resulted in a profound change in the way of thinking and doing mathematics. This change, known as process of algebrization, occurred during the seventeenth and early eighteenth centuries and led to a great transformation in mathematics. Among many other consequences, this process gave rise to the treatment of the results in the classic treatises with the new analytical method, which allowed new visions of such treatises and the obtaining of new results. Among those treatises is the <i>Arithmetic</i> of Diophantus of Alexandria (approx. 200–284) which was written, using the new algebraic language, by the French mathematician Jacques Ozanam (1640–1718), who in addition to profusely increasing the original problems of Diophantus, solved them in a general way, thus obtaining many geometric consequences. The work is handwritten, it has never been published, it has been lost for almost 300 years, and the known references show its importance. We will show that Ozanam’s manuscript was quoted as an important work on several occasions by others mathematicians of the time, among whom G. W. Leibniz stands out. Once the manuscript has been located, our aim in this article is to show and analyze this work of Ozanam, its content, its notation and its structure and how, through the new algebraic method, he not only solved and expanded the questions proposed by Diophantus, but also introduced a connection between the algebraic solutions and what he called geometric determinations by obtaining loci from the solutions.\u0000</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"75 5","pages":"557 - 611"},"PeriodicalIF":0.5,"publicationDate":"2021-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00407-021-00274-3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43160884","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":"A study of Babylonian records of planetary stations","authors":"J. M. Steele,&nbsp;E. L. Meszaros","doi":"10.1007/s00407-021-00272-5","DOIUrl":"10.1007/s00407-021-00272-5","url":null,"abstract":"<div><p>Late Babylonian astronomical texts contain records of the stationary points of the outer planets using three different notational formats: Type S where the position is given relative to a Normal Star and whether it is an eastern or western station is noted, Type I which is similar to Type S except that the Normal Star is replaced by a reference to a zodiacal sign, and Type Z the position is given by reference to a zodiacal sign, but no indication of whether the station is an eastern or western station is included. In these records, the date of the station is sometimes preceded by the terms <i>in</i> and/or EN. We have created a database of station records in order to determine whether there was any pattern in the use of these notation types over time or an association with any bias in the station date or the type of text the station was recorded in. Predictive texts, which include Almanacs and Normal Star Almanacs, almost always use Type Z notation, while the Diaries, compilations, and Goal-Year Texts use all three types. Type Z records almost never include <i>in</i> or EN, while other types seem to include these interchangeably. When compared with modern computed station dates, the records show bias toward earlier dates, suggesting that the Babylonians were observing dates when the planets appeared to stop moving rather than the true station. Overlapping reports, where a station on the same date was recorded in two or more texts, suggest that predicted station dates were used to guide observations, and that the planet’s position on the predicted stationary date was the true point of the observation rather than the specific date of the stationary point.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"75 4","pages":"415 - 438"},"PeriodicalIF":0.5,"publicationDate":"2021-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00407-021-00272-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44676940","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":"Einstein on involutions in projective geometry","authors":"Tilman Sauer,&nbsp;Tobias Schütz","doi":"10.1007/s00407-020-00270-z","DOIUrl":"10.1007/s00407-020-00270-z","url":null,"abstract":"<div><p>We discuss Einstein’s knowledge of projective geometry. We show that two pages of Einstein’s Scratch Notebook from around 1912 with geometrical sketches can directly be associated with similar sketches in manuscript pages dating from his Princeton years. By this correspondence, we show that the sketches are all related to a common theme, the discussion of involution in a projective geometry setting with particular emphasis on the infinite point. We offer a conjecture as to the probable purpose of these geometric considerations.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"75 5","pages":"523 - 555"},"PeriodicalIF":0.5,"publicationDate":"2021-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00407-020-00270-z","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44262958","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":"A study of Babylonian planetary theory III. The planet Mercury","authors":"Teije de Jong","doi":"10.1007/s00407-020-00269-6","DOIUrl":"10.1007/s00407-020-00269-6","url":null,"abstract":"&lt;div&gt;&lt;p&gt;In this series of papers I attempt to provide an answer to the question how the Babylonian scholars arrived at their mathematical theory of planetary motion. Papers I and II were devoted to system A theory of the outer planets and of the planet Venus. In this third and last paper I will study system A theory of the planet Mercury. Our knowledge of the Babylonian theory of Mercury is at present based on twelve &lt;i&gt;Ephemerides&lt;/i&gt; and seven &lt;i&gt;Procedure Texts&lt;/i&gt;. Three computational systems of Mercury are known, all of system A. System A&lt;sub&gt;1&lt;/sub&gt; is represented by nine &lt;i&gt;Ephemerides&lt;/i&gt; covering the years 190 BC to 100 BC and system A&lt;sub&gt;2&lt;/sub&gt; by two &lt;i&gt;Ephemerides&lt;/i&gt; covering the years 310 to 290 BC. System A&lt;sub&gt;3&lt;/sub&gt; is known from a &lt;i&gt;Procedure Text&lt;/i&gt; and from Text M, an &lt;i&gt;Ephemeris&lt;/i&gt; of the last evening visibility of Mercury for the years 424 to 403 BC. From an analysis of the Babylonian observations of Mercury preserved in the &lt;i&gt;Astronomical Diaries&lt;/i&gt; and &lt;i&gt;Planetary Texts&lt;/i&gt; we find: (1) that dates on which Mercury reaches its stationary points are not recorded, (2) that Normal Star observations on or near dates of first and last appearance of Mercury are rare (about once every twenty observations), and (3) that about one out of every seven pairs of first and last appearances is recorded as “omitted” when Mercury remains invisible due to a combination of the low inclination of its orbit to the horizon and the attenuation by atmospheric extinction. To be able to study the way in which the Babylonian scholars constructed their system A models of Mercury from the available observational material I have created a database of synthetic observations by computing the dates and zodiacal longitudes of all first and last appearances and of all stationary points of Mercury in Babylon between 450 and 50 BC. Of the data required for the construction of an ephemeris synodic time intervals Δt can be directly derived from observed dates but zodiacal longitudes and synodic arcs Δλ must be determined in some other way. Because for Mercury positions with respect to Normal Stars can only rarely be determined at its first or last appearance I propose that the Babylonian scholars used the relation Δλ = Δt −3;39,40, which follows from the period relations, to compute synodic arcs of Mercury from the observed synodic time intervals. An additional difficulty in the construction of System A step functions is that most amplitudes are larger than the associated zone lengths so that in the computation of the longitudes of the synodic phases of Mercury quite often two zone boundaries are crossed. This complication makes it difficult to understand how the Babylonian scholars managed to construct System A models for Mercury that fitted the observations so well because it requires an excessive amount of computational effort to find the best possible step function in a complicated trial and error fitting process with four or five free parameters. ","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"75 5","pages":"491 - 522"},"PeriodicalIF":0.5,"publicationDate":"2021-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00407-020-00269-6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50444750","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":"The gravitational influence of Jupiter on the Ptolemaic value for the eccentricity of Saturn","authors":"Christián C. Carman","doi":"10.1007/s00407-020-00271-y","DOIUrl":"10.1007/s00407-020-00271-y","url":null,"abstract":"<div><p>The gravitational influence of Jupiter on Saturn produces, among other things, non-negligible changes in the eccentricity of Saturn that affect the magnitude of error of Ptolemaic astronomy. The value that Ptolemy obtained for the eccentricity of Saturn is a good approximation of the real eccentricity—including the perturbation of Jupiter—that Saturn had during the time of Ptolemy's planetary observations or a bit earlier. Therefore, it seems more probable that the observations used for obtaining the eccentricity of Saturn were done near Ptolemy’s time, and rather unlikely earlier than the first century AD. Even if this is not quite a demonstration that Ptolemy used observations of his own, my argument increases its probability and practically discards the idea that Ptolemy borrowed values or observations from astronomers further back than the first century AD, such as Hipparchus or the Babylonians.\u0000</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"75 4","pages":"439 - 454"},"PeriodicalIF":0.5,"publicationDate":"2021-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00407-020-00271-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50444810","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":"The development of the concept of uniform convergence in Karl Weierstrass’s lectures and publications between 1861 and 1886","authors":"Klaus Viertel","doi":"10.1007/s00407-020-00266-9","DOIUrl":"10.1007/s00407-020-00266-9","url":null,"abstract":"<div><p>The history of uniform convergence is typically focused on the contributions of Cauchy, Seidel, Stokes, and Björling. While the mathematical contributions of these individuals to the concept of uniform convergence have been much discussed, Weierstrass is considered to be the actual inventor of today’s concept. This view is often based on his well-known article from 1841. However, Weierstrass’s works on a rigorous foundation of analytic and elliptic functions date primarily from his lecture courses at the University of Berlin up to the mid-1880s. For the history of uniform convergence, these lectures open up an independent branch of development that is disconnected from the approaches of the previously mentioned authors; to my knowledge, Weierstraß never explicitly referred to Cauchy’s continuity theorem (1821 or 1853) or to Seidel’s or Stokes’s contributions (1847). In the present article, Weierstrass’s contributions to the development of uniform convergence will be discussed, mainly based on lecture notes made by Weierstrass’s students between 1861 and the mid-1880s. The emphasis is on the notation and the mathematical rigor of the introductions to the concept, leading to the proposal to re-date the famous 1841 article and thus Weierstrass’s first introduction of uniform convergence.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"75 4","pages":"455 - 490"},"PeriodicalIF":0.5,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00407-020-00266-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50506818","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":"BM 76829: A small astronomical fragment with important implications for the Late Babylonian Astronomy and the Astronomical Book of Enoch","authors":"Jeanette C. Fincke,&nbsp;Wayne Horowitz,&nbsp;Eshbal Ratzon","doi":"10.1007/s00407-020-00268-7","DOIUrl":"10.1007/s00407-020-00268-7","url":null,"abstract":"<div><p>BM 76829, a fragment from the mid-section of a small tablet from Sippar in Late Babylonian script, preserves what remains of two new unparalleled pieces from the cuneiform astronomical repertoire relating to the zodiac. The text on the obverse assigns numerical values to sectors assigned to zodiacal signs, while the text on the reverse seems to relate zodiacal signs with specific days or intervals of days. The system used on the obverse also presents a new way of representing the concept of numerical ‘zero’ in cuneiform, and for the first time in cuneiform, a system for dividing the horizon into six arcs in the east and six arcs in the west akin to that used in the Astronomical Book of Enoch. Both the obverse and the reverse may describe the periodical courses of the sun and moon, in a similar way to what is found in astronomical texts from Qumran, thus adding to our knowledge of the scientific relationship between the two cultures.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"75 3","pages":"349 - 368"},"PeriodicalIF":0.5,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00407-020-00268-7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50501813","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":"Back to the roots of vector and tensor calculus: Heaviside versus Gibbs","authors":"Alessio Rocci","doi":"10.1007/s00407-020-00264-x","DOIUrl":"10.1007/s00407-020-00264-x","url":null,"abstract":"<div><p>In June 1888, Oliver Heaviside received by mail an officially unpublished pamphlet, which was written and printed by the American author Willard J. Gibbs around 1881–1884. This original document is preserved in the Dibner Library of the History of Science and Technology at the Smithsonian Institute in Washington DC. Heaviside studied Gibbs’s work very carefully and wrote some annotations in the margins of the booklet. He was a strong defender of Gibbs’s work on vector analysis against quaternionists, even if he criticised Gibbs’s notation system. The aim of our paper is to analyse Heaviside’s annotations and to investigate the role played by the American physicist in the development of Heaviside’s work.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"75 4","pages":"369 - 413"},"PeriodicalIF":0.5,"publicationDate":"2020-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00407-020-00264-x","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47448391","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":"Correction to: What Heinrich Hertz discovered about electric waves in 1887–1888","authors":"Jed Buchwald,&nbsp;Chen-Pang Yeang,&nbsp;Noah Stemeroff,&nbsp;Jenifer Barton,&nbsp;Quinn Harrington","doi":"10.1007/s00407-020-00267-8","DOIUrl":"10.1007/s00407-020-00267-8","url":null,"abstract":"","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"75 2","pages":"173 - 173"},"PeriodicalIF":0.5,"publicationDate":"2020-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00407-020-00267-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50446410","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}