Archive for History of Exact Sciences最新文献

{"title":"An alternative interpretation of BM 76829: astrological schemes for length of life and parts of the body","authors":"John Steele","doi":"10.1007/s00407-021-00279-y","DOIUrl":"10.1007/s00407-021-00279-y","url":null,"abstract":"<div><p>In this paper I present an alternative reading and interpretation of the cuneiform tablet BM 76829. I suggest that the obverse of the tablet contains a simple astrological scheme linking the sign of the zodiac in which a child is born to the maximum length of life, and that the reverse contains a copy of a scheme relating parts of the body to the signs of the zodiac.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"76 1","pages":"1 - 14"},"PeriodicalIF":0.5,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00407-021-00279-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50499848","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":"Mathématiques en perspective: Desargues, la Hire, le Poîvre","authors":"Jean-Yves Briend","doi":"10.1007/s00407-021-00275-2","DOIUrl":"10.1007/s00407-021-00275-2","url":null,"abstract":"<div><h2>Résumé</h2><div><p>Il est tentant de considérer l’œuvre mathématique de Girard Desargues, plus particulièrement son <i>Brouillon Project</i> sur les coniques, comme un travail de mathématiques appliquées à l’art de la perspective. Nous voudrions montrer dans cet article qu’il est sans doute plus pertinent de considérer que Desargues fait des mathématiques en <i>praticien de la perspective</i> ou, plus précisément, que son œuvre peut être lue comme un travail de perspective appliquée à la géométrie. Nous allons analyser quelques passages de l’œuvre du Lyonnais en adoptant ce point de vue perspectiviste afin de montrer comment ce parti pris permet d’éclairer les aspects novateurs d’un contenu mathématique parfois difficile à saisir dans le style touffu de l’auteur. Nous montrerons ensuite comment cette manière de faire de Desargues peut se retrouver chez Philippe de la Hire et Jacques-François le Poîvre, ce qui les a menés à l’idée nouvelle de considérer une <i>transformation</i> du plan dans lui-même comme objet explicite de la géométrie.</p></div></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"75 6","pages":"699 - 736"},"PeriodicalIF":0.5,"publicationDate":"2021-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00407-021-00275-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48708649","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":"David Hilbert and the foundations of the theory of plane area","authors":"Eduardo N. Giovannini","doi":"10.1007/s00407-021-00278-z","DOIUrl":"10.1007/s00407-021-00278-z","url":null,"abstract":"<div><p>This paper provides a detailed study of David Hilbert’s axiomatization of the theory of plane area, in the classical monograph <i>Foundation of Geometry</i> (1899). On the one hand, we offer a precise contextualization of this theory by considering it against its nineteenth-century geometrical background. Specifically, we examine some crucial steps in the emergence of the modern theory of geometrical equivalence. On the other hand, we analyze from a more conceptual perspective the significance of Hilbert’s theory of area for the foundational program pursued in <i>Foundations</i>. We argue that this theory played a fundamental role in the general attempt to provide a new independent basis for Euclidean geometry. Furthermore, we contend that our examination proves relevant for understanding the requirement of “purity of the method” in the tradition of modern synthetic geometry.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"75 6","pages":"649 - 698"},"PeriodicalIF":0.5,"publicationDate":"2021-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00407-021-00278-z","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49196755","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":"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}