Archive for History of Exact Sciences最新文献

{"title":"A comparative study between the approaches of Archimedes and Euclid to the sphere","authors":"Guilherme Luiz Grüdtner","doi":"10.1007/s00407-026-00380-0","DOIUrl":"10.1007/s00407-026-00380-0","url":null,"abstract":"<div><p>The aim of this article is to study the result of the sphere demonstrated by Euclid in <i>Elements</i> XII, 16–18 and the result of the sphere demonstrated by Archimedes in <i>Sphere and Cylinder</i>, I, 33–34. The focus of this work is to offer a structural reconstruction of both proofs and, based on this, to conduct a comparative reading of the mathematical techniques and logical structure used. Finally, based on this structural reconstruction and historical and philological arguments, a correlative reading among these demonstrations will be proposed, in which there is a suggestion that Archimedes adapted the techniques of Euclid.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"80 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2026-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148614768","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":"Two old Babylonian multiplication tables","authors":"Mohannad Khalaf Jamen Al Shamari,&nbsp;Cécile Michel","doi":"10.1007/s00407-026-00376-w","DOIUrl":"10.1007/s00407-026-00376-w","url":null,"abstract":"<div><p>Old Babylonian cuneiform tablets containing multiplication tables were an essential tool in the education of scribes. Along with other numerical lists, these tables helped students to learn arithmetic and the sexagesimal (base-60) system. The two tablets published here, which are now in the Iraq Museum, contain multiplication tables for 3:20 and 8, respectively. They belong to Type III of the pedagogical material. Despite their provenance being unknown, stylistic and textual clues suggest that the tablets probably originated in southern Mesopotamia, possibly in Larsa. Overall, the tablets provide valuable insights into ancient Babylonian mathematical training and writing conventions.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"80 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2026-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148614769","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":"Hankel’s principle as an anti-Kantian program","authors":"Iulian D. Toader","doi":"10.1007/s00407-026-00382-y","DOIUrl":"10.1007/s00407-026-00382-y","url":null,"abstract":"<div><p>Hankel used his <i>principle of the permanence of formal laws</i> (PFL) as a guide for the extension of number systems and as a necessary condition for the legitimacy of their formal theories. He acknowledged that these applications have important limitations, evidenced by the extension to hypercomplex numbers and by what he saw as the unavoidable inconsistency of a formal theory of irrational numbers. Yet, intriguingly enough, he remained fully committed to the PFL. I argue that this was due to his understanding it as an expression of a conservative strategy, inherited from Peacock and Hamilton, which permits the revision of the basic laws of arithmetic if there are reasons for revision that are found, upon deliberation, to outweigh the reasons for their preservation. Then I discuss criticisms by Schubert and Pringsheim, who reformulated the PFL to align it with their own anti-revisionary conservative strategy, at the cost of relinquishing parts of modern mathematics. I conclude by emphasizing the deep philosophical difference between these kinds of conservatism in mathematics.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"80 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2026-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13400678/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148594396","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":"Correction: Analysing the number concept in Weierstraß’ lecture notes: conceptions disregarded and remained unknown","authors":"Gert Schubring,&nbsp;Débora Ferreira,&nbsp;Michel Salazar","doi":"10.1007/s00407-026-00381-z","DOIUrl":"10.1007/s00407-026-00381-z","url":null,"abstract":"","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"80 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2026-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00407-026-00381-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148466092","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":"Universal mathematics and the new algebra: Maurolico, van Roomen, Descartes","authors":"Jeffrey Oaks","doi":"10.1007/s00407-026-00371-1","DOIUrl":"10.1007/s00407-026-00371-1","url":null,"abstract":"<div><p>In the first part of this study, we compare the different approaches Francesco Maurolico (1557) and Adriaan van Roomen took to develop a universal mathematics. Maurolico sought to ground practical numerical calculation of lines, surfaces, volumes, weights, times, etc., in Euclidean theory while van Roomen sought to establish a general theory of ratio and proportion that precedes arithmetic and geometry. One key difference is that Maurolico’s numbers measure the various species of quantity via a posited unit while van Roomen’s numbers measure their ratios. In the second part, we compare two unfinished drafts by van Roomen (ca. 1598–99) and Descartes (the <i>Regulae</i>, ca. 1628–31) which propose a universal algebra that can model all quantities, including numbers and geometric magnitudes. In fact, there can be no such algebra, since the arithmetical unit as a multiplicative identity is incompatible with the heterogeneous dimensions of Euclidean geometry. Thus, van Roomen abandoned his draft. Descartes, whose interest lay with numerical problem-solving, initially avoided the problem by positing a unit for each species of quantity. But once he became acquainted with theoretical, unitless geometry through the Pappus problem he, too, abandoned his draft and worked with two algebras in <i>La géométrie</i> (1637).</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"80 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2026-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148323711","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":"Saturn observations and measurements at the Maragha observatory: error, success, and comparative study in historical context","authors":"Seyyed Mohammad Mozaffari","doi":"10.1007/s00407-026-00363-1","DOIUrl":"10.1007/s00407-026-00363-1","url":null,"abstract":"<div><p>This paper investigates Muḥyī al‑Dīn al‑Maghribī’s measurements of Saturn at the Maragha observatory (northwestern Iran, 13^th century), preserved in Chapters 2–5 of Book VIII of his <i>Talkhīṣ al‑Majisṭī</i> (<i>The Compendium of the</i> Almagest). Employing Ptolemy’s iterative three‑point method as described in <i>Almagest</i> X and XI, he measured Saturn’s orbital elements (eccentricity and the longitude of the apogee) from three observations conducted near the planet’s opposition to the mean Sun on 25 October 1263, 9 December 1266, and 27 February 1273 (with longitudinal errors not exceeding 1/6°). He obtained an eccentricity of 3;15 (radius of orbit = 60), a value already encountered in Ptolemy’s <i>Canobic Inscription</i>, and a longitude of the apogee of 258;38° in 1273. A calculational mistake prevented him from measuring the more accurate value of 3;23 for the eccentricity; nevertheless, his apogee longitude stands as the most precise determination in the Islamic astronomical tradition. The accuracy of his values for the Ptolemaic parameters of Saturn and the quantitative outcomes of his theory are examined comprehensively and comparatively with all other known theories established in the late Islamic period, especially Saturn’s theory laid out in the <i>Īlkhānī zīj</i>. As a technical case study, this investigation furnishes new insights into the nature of observational practice and theoretical innovation in the medieval Islamic period.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"80 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2026-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148323058","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":"Combining electrodynamics with hydrodynamics: the origins and development of magnetohydrodynamics","authors":"Alexander S. Blum,&nbsp;Arnab Rai Choudhuri","doi":"10.1007/s00407-026-00373-z","DOIUrl":"10.1007/s00407-026-00373-z","url":null,"abstract":"<div><p>The 1908 discovery of magnetic fields in sunspots raised the question of how such magnetic fields arise in astronomical bodies. The realization that the material in the Sun is in the plasma state made Larmor (1919) speculate that motions in the plasma may generate or sustain such magnetic fields. The first steps towards magnetohydrodynamics (MHD), which combines electrodynamics and hydrodynamics, were taken by Cowling (1933) and Ferraro (1937). These first results raised serious doubts whether MHD can explain the origin of the magnetic fields of the earth and of sunspots. The MHD equations first appeared in their full form in the works of Alfvén (1942a, 1943c), who used these equations to postulate what he called ‘electromagnetic-hydrodynamic waves’ (now known as Alfvén waves), which were discovered in the laboratory only about a decade later. Alfvén’s work was initially met with skepticism (and his theory of sunspots was rejected), and MHD was taken up very slowly at first. With the astrophysical interest generated by the discovery of magnetic fields in other astronomical systems from around 1950, the methods of MHD started spreading rapidly, leading to first solutions of the so-called ‘dynamo problem’—to explain how magnetic fields arise in astrophysical systems. The concept of ‘flux freezing’, first put forth by Alfvén (1943c), led to a new viewpoint that magnetic fields can be considered more fundamental than electric currents, which made MHD a more coherent and accessible theory. Curiously, Alfvén himself opposed this viewpoint and, although he was given the 1970 Nobel Prize “for fundamental work and discoveries in magnetohydrodynamics”, he became a harsh critic of MHD in later life.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"80 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00407-026-00373-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148236442","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":"Analysing the number concept in Weierstraß’ lecture notes: conceptions disregarded and remained unknown","authors":"Gert Schubring,&nbsp;Débora de M. L. Ferreira,&nbsp;Michel Salazar","doi":"10.1007/s00407-026-00369-9","DOIUrl":"10.1007/s00407-026-00369-9","url":null,"abstract":"<div><p>The motivation for undertaking this research came from reading Kossak’s 1872 publication about Weierstraß’ conception of numbers. It became clear that Weierstraß was developing a general notion of complex numbers that aligned closely with the concepts found in the eighteenth-century French arithmetic textbooks, an area already investigated by one of us. This prompted a deeper examination of the arithmetic framework that Weierstraß had introduced as the foundation for his lectures on analytic function theory. Our inquiry led to a comprehensive assessment of the lecture notes produced by students who attended these courses, which constitute the essential sources for understanding the development of Weierstraß’ foundational approach from the early 1860s to 1886. Rigorous notions of irrational numbers were developed in the second half of the nineteenth century by one French mathematician and three German mathematicians: Charles Méray, Georg Cantor, Richard Dedekind, and Carl Weierstraß. Nowadays, Dedekind’s approach to irrational numbers largely dominates the modern understanding of the real number system, while Weierstraß’ conception has been mostly forgotten. Weierstraß did not develop his theory with the aim of establishing a rigorous notion of real numbers, as is commonly reported by those who are aware of his achievements. Instead, his goal was to create a new notion of complex numbers—not merely the “standard” complex numbers introduced by Gauß in 1831, but a more general concept. To achieve this, he introduced his own method for constructing irrational numbers. This broader conception, which is little known today, provided the numerical foundation for Weierstraß’ preferred field of mathematical research: elliptic and Abelian functions. His primary concern, which he reformulated over time, came from Gauß’ 1831 observation about the impracticality of extending the number system to what Weierstraß called <i>general complex numbers</i>, later known as <i>hypercomplex numbers</i>.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"80 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00407-026-00369-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148236558","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":"Sunrise and sunset times in the Chongxiu-Daming-li revisited","authors":"Dongbin Kim,&nbsp;Junhyeok Jeon","doi":"10.1007/s00407-026-00374-y","DOIUrl":"10.1007/s00407-026-00374-y","url":null,"abstract":"<div><p>We investigated the interpolation methods used in <i>Chongxiu-Daming-li</i> (重修大明曆), focusing on the principles behind the tables for sunrise times (Sunrise Parts 日出分) through the divided difference method. Our analysis compared these methods with modern polynomial interpolation techniques. Our findings show that the latitude derived from <i>Chongxiu-Daming-li</i> closely matches historical locations, though discrepancies suggest possible over- or underestimation of sunrise times. While the maximum interpolation error for sunrise times is approximately 15 s, discrepancies between computed and actual sunrise times can reach up to 4 min. Comparisons with earlier calendars, such as <i>Xuanming-li</i> (宣明曆), reveal that fundamental observational techniques saw limited advancement. Despite the use of advanced interpolation methods, practical accuracy in traditional Chinese calendars did not significantly improve until the late 13th century with Guo Shoujing (郭守敬)’s <i>Shoushi-li</i> (授時曆).</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"80 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148172947","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":"Excavated nine-nines rhymes from ancient China: a critical edition of the texts with commentary","authors":"Xulin Zhou,&nbsp;Joseph W. Dauben","doi":"10.1007/s00407-026-00360-4","DOIUrl":"10.1007/s00407-026-00360-4","url":null,"abstract":"<div><p>Since the early twentieth century, numerous complete or partial Chinese nine-nines rhymes dating from the mid-Warring States period to the eleventh century <span>ce</span> have been excavated and published in China. This article surveys all of the as-yet known examples of such nine-nines rhymes, each of which is described, transcribed, collated, annotated, and translated into English, providing a comprehensive overview of what they have in common, as well as how they differ in certain important respects. As a result, the evolution of the nine-nines rhymes can be traced over a period of nearly a millennium-and-a-half down to the Tang (618–907 <span>ce</span>) and Southern Song (1127–1279 <span>ce</span>) dynasties. Thus, it will be possible for the first time to provide a general picture of how the nine-nines rhyme changed from pre-Qin times to survive in the form it usually assumes today.</p></div>","PeriodicalId":50982,"journal":{"name":"Archive for History of Exact Sciences","volume":"80 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148009168","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}