Archive for History of Exact Sciences最新文献

This paper translates and interprets a neglected and significant original text in Zhu Zaiyu’s 朱載堉 (1536–1611) calendar system. The invention and application of Guo Shoujing’s 郭守敬 (1231–1316) hushi geyuan 弧矢割圓 was a great change in the concept of constructing calendar system. In this text we argue that, by giving the relationship between xian-hu 弦弧 ratio and shi 矢, Zhu Zaiyu provided a new method of hu shi 弧矢新術 with higher accuracy than Guo Shoujing’s. Furthermore, we retrieved the construction of the new method of hu shi and discovered the crucial fact that Zhu Zaiyu created a new interpolation method. According to Zhu Zaiyu, a linear interpolation was used to modify the coefficient in each single step, then by repeating the steps, a high-order polynomial function could be constructed. Zhu Zaiyu’s interpolation was mechanical, and the polynomial function constructed by this method was categorized as the type of Newton interpolation, notably in modern terms. The critical purpose of ancient Chinese calendar compilers in interpolation was to construct higher order polynomial function; crucially, Zhu Zaiyu gave a general solution to this problem.

In this article, I propose a new approach to analyze the interrelations between mathematics and technology. It has the potential to contribute methodologically to both the fields of history of mathematics as well as the study of computational technologies in the current context. Based on the conception of mathematics as a contingent human practice, I claim that the practical engagement with technology not only subjects new fields, materials, and problems to mathematical scrutiny but might even shape mathematics from within. To illustrate my approach and corroborate my thesis, I present a historical case study on the mathematical works of the Swiss clock- and instrument-maker Jost Bürgi (1552–1632). Besides being a practicing artisan, he left three mathematical treatises. The advancements in fine metal working at his time, exemplified in clockwork mechanisms and measuring instruments, not only motivated and directed Bürgi’s mathematical inquiries. Instead, I argue that the interaction with these technical apparatuses in practice has shaped the internal structure and workings of his mathematics, that is, its entities, justifications, presentations, proofs, and procedures. The close analysis of some aspects of his oeuvre, especially his notion(s) of the sine, his way of explaining the occurrence of multiple solutions in algebra, and his visual depiction of the bridging of ten in his logarithmic computational tool, reveals a potential integration of the experience and practical knowledge of a clockmaker into mathematics. I therefore make the point that his mathematical writings portray a clockmaker’s mathematics.

This paper studies the development of intensity schemes within the framework of the old quantum theory. It investigates how these schemes emerged in a complex process involving empirical observation, data analysis and conceptual reconfiguration and became essential tools for predicting the intensities of multiplets in the absence of a well-formed quantum theory of radiation. By applying the concept of paper tools, the study shows how intensity schemes became theoretical representations allowing both the classification and interpretation of observations and the formulation of theoretical predictions. It thereby highlights the importance of representational tools and empirical regularities within the development of the old quantum theory.

This paper provides a comprehensive summary of Johannes Kepler's research during his first tenure at Benatky, from February to June 1600. For the first time, Kepler had unrestricted access to Tycho Brahe's precise Mars observations, enabling him to test and refine his theories of planetary motion. Kepler aimed to resolve inconsistencies in Tycho’s Mars model, particularly its failure to predict parallactic observations accurately. Over the four months, he developed innovative methods, such as combining observations to triangulate distances and employing Tycho’s model as a generator of reliable heliocentric longitudes. Despite numerous mathematical errors and theoretical missteps, Kepler laid the groundwork for the revolutionary ideas he would later present in Astronomia Nova. This paper highlights Kepler’s creative and exploratory approach, his use of Tycho’s data, and the significant progress he made in understanding Mars’ orbit, even as many of his early hypotheses were ultimately discarded.

Although many people have extensively studied the earlier parts of Galois’s testamentary letter, in particular those concerning the Galois theory of algebraic equations and related group theory, it seems that the theory of ambiguity near the end of his letter is less well known and studied, and therefore, remaining somewhat mysterious. One purpose of this paper is to provide an overview of diverse interpretations of Galois’s theory of ambiguity by people such as Lie, Klein, Picard, and Grothendieck. We will discuss how well they fit Galois’s description for this theory and whether they satisfy one important criterion set by him. After a careful analysis of Galois’s statements regarding the theory of ambiguity and the rationale behind them, by taking all Galois’s works into consideration, we will offer our interpretation of it through the theory of monodromy for linear differential equations. Our findings challenge the common perception that Galois could not foresee applications of group theory beyond algebraic equations. Subsequently, we will discuss how these various interpretations have influenced later development of mathematics, particularly their impact on Lie’s idée fixe to develop a theory of transformation groups for differential equations. This analysis also raises doubts about a certain aspect of the commonly accepted narrative regarding the origin of the theory of Lie groups, and provides one important example of theories partially motivated by Galois’s theory of ambiguity. Additionally, we will identify results from works of his near contemporaries such as Riemann, Fuchs, Jordan and later generations such as Siegel, which seem to fit well our rendering of Galois’s description and criterion. This demonstrates the potentially intended broad scope of Galois’s theory of ambiguity. Furthermore, their alignment with our interpretation of Galois’s theory of ambiguity adds feasibility and credibility to the latter. We hope that the analysis in this paper will enhance our understanding of the meaning and impacts of Galois’s theory of ambiguity, reaffirming the profound and broad vision that Galois held for mathematics. Moreover, this paper contributes to an effort to reevaluate some of Galois’s seminal contributions and their impacts on the development of mathematics, transcending the traditional boundaries of algebra and number theory.

Kepler’s laws provided sufficient geometry and kinematics to strengthen astronomers’ preference for heliocentrism. While Kepler outlined some dynamic arguments, they were not rigorous enough to turn his laws into kinematic tools. As a result, some astronomers found ways to reconcile Kepler’s findings with geo-heliocentrism. One of these was the Jesuit astronomer Giovanni Battista Riccioli, who proposed a method known as the “epic-epicycle” (Riccioli, Almagestum novum, 1651). This paper will explore how Riccioli received and interpreted Kepler’s first and second laws within his own astronomical framework. This analysis will include a discussion of how Riccioli understood the concept of “physics” in his work, beginning with a study of the Sun’s motion (Riccioli, Astronomia reformata, 1665).

The stadion is the unit of length by which distances are reported in ancient Greek geographical sources. The itinerary indications in stadia can be found in several texts, but no specific unit values are given in the ancient geographers’ surviving works. However, the notion of a vaguely quantified, non-metrological itinerary unit is contradicted by the presence, since Hellenistic times, of road marker stones bearing distance indications along major ancient roads. The key assumption in this study is that, whatever the unit involved, main roads were actually measured to the best of capabilities, and distance measurements in ancient works did refer to some specific metrological system. Some well-known Greek languagecxesst sources are analyzed with the support of archeologic information obtained from a small number of pre-Roman road markers, and from modern reports of investigations about ancient roads and sites. Based on the evidence, it is shown that two different stadion values were most often used as itinerary units in the Greek and Hellenistic world, namely 177 m and 210 m, that can be traced respectively to the so-called Attic foot and Philetaeric (Ionic/Samian) foot. Conversion among units did also occur, and this may offer explanations for supposed textual inconsistencies that have so far proved hard to understand.

In ancient China, astronomers attempted to correct mean geocentric longitude of the inner planets using planetary “Expansion–Contraction Difference” (yingsuo cha 盈縮差) to obtain the true geocentric longitude. They used the “Limit Degree” (xiandu 限度) as the independent variable for the “Expansion–Contraction Difference”. Although this idea was relatively ideal and operationally simple, the algorithm of the “Expansion–Contraction Difference” designed by ancient Chinese astronomers had significant flaws in terms of its actual computational effectiveness for calculating the position of the inner planets. The reason for this flaw is that, based on its intended purpose, the “Expansion–Contraction Difference” should be a three-variable function, including the planetary equation of center, the solar equation of center and the phase angle of the planet, and each variable has different independent variables. However, ancient Chinese astronomers attempted to simplify this complex three-variable function into a single-variable function, and such simplification was unsuccessful. Further research indicates that the starting point of the expansion phase in the “Table of Expansion–Contraction Difference” (Yingsuo Li 盈縮曆) for the inner planets in ancient Chinese astronomical systems did not accurately depict the position of the planetary perihelion. Adjusting the starting point based on the longitude can improve the accuracy to some extent. Although the special coefficients of “double it for Venus, triple it for Mercury” can enhance the accuracy of calculations on the position of the inner planets, the result is not as ideal as expected. This study highlights that within the framework of ancient Chinese planetary theory, the algorithm of the “Expansion–Contraction Difference” for the inner planets possessed inherent and irreparable flaws, resulting in significant errors in the calculation of the apparent position of the inner planets.

During the Renaissance, several scholars worked to revive the contents and methods developed by the ancient Greek mathematicians. They began their research by studying the Latin editions of the Greek classics. The problem of Apollonius is a significant case study that sheds light on the recovery and re-appropriation of the solution methods employed by Greek mathematics. In this article, I will explore both the manuscript sources and the printed editions used by the Urbino School (Federico Commandino and Guidobaldo del Monte) to solve the problem of Apollonius.