History of Exact Sciences

Poncelet’s porism: a long story of renewed discoveries, II Abstract The first part of the article appeared in the previous issue of this journal. It deals with the history of the research on Poncelet’s porism, and related subjects, which were developed from the middle of the eighteenth century until the end of the nineteenth century. In this second part, we take the research developed in the twentieth century. We also offer a comparison of the main works on the subject, and we draw some conclusions.

Gearing up for Lagrangian dynamics Abstract James Clerk Maxwell’s 1865 paper, “A Dynamical Theory of the Electromagnetic Field,” is usually remembered as replacing the mechanical model that underpins his 1862 publication with abstract mathematics. Up to this point historians have considered Maxwell’s usage of Lagrangian dynamics as the sole important feature that guides Maxwell’s analysis of electromagnetic phenomena in his 1865 publication. This paper offers an account of the often ignored mechanical analogy that Maxwell used to guide him and his readers in the construction of his new electromagnetic equations. The mechanical system consists of a weighted flywheel geared into two independently driven crank wheels in what amounts to a mechanical differential. I will demonstrate how Maxwell made use of the analogy between his flywheel system and electromagnetic induction to ground his study of electromagnetism in clear mechanical conceptions and to structure the derivation of the equations that together are now recognized as Maxwell’s equations for electrodynamics. By reconceiving specific components of his model in electromagnetic terms, while at the same time retaining many of the relations between concepts in the mechanical case, Maxwell gradually assembled increasingly generalized equations for electromotive force. Maxwell thus realized a much sought after balance between physical analogy and abstract mathematics in this, the last of his three seminal papers on electromagnetism.

Trigonometric tables: explicating their construction principles in China Abstract The trigonometric table and its construction principles were introduced to China as part of calendar reform, spear-headed by Xu Guangqi (1562–1633) in the late 1620s to early 1630s. Chinese scholars attempted and succeeded in uncovering how the construction principles were established in the seventeenth century and then in the eighteenth century expanded to include more algorithms to compute the values of trigonometric lines. Successful as they were in discoursing the construction principles, most Chinese scholars did not actually construct trigonometric tables anew. In the early nineteenth century, a revolutionary approach was developed, which resembles computing a finite sum of power series to trigonometric functions of an arbitrary arc less than a one-half circle. Though hailed by many modern historians as Chinese achievements in developing “infinite series” of trigonometric functions, this approach was viewed by the actors at the time as a quick means to construct trigonometric tables. Interestingly, even with these “quick” methods, no trigonometric table was constructed. Besides the fact that constructing a trigonometric table afresh is a time-consuming business, the classification of the trigonometric table and their construction principles into different genres of knowledge by scholars offers an additional explanation of drastically uneven treatment of trigonometric tables and their construction principles.

Robert Boyle’s mechanical account of hydrostatics and pneumatics: fluidity, the spring of the air and their relationship to the concept of pressure Abstract This article in an attempt to identify the precise way in which Robert Boyle provided a mechanical account of the features that distinguish liquids and air from solids and from each other. In his pneumatics, Boyle articulated his notion of the ‘spring’ of the air for that purpose. Pressure appeared there only in a common, rather than in a technical, sense. It was when he turned to hydrostatics that Boyle found the need to introduce a technical sense of pressure to capture the fluidity of water which, unlike air, lacked a significant spring. Pressure, understood as representing the state of a liquid within the body of it rather than at its surface, enabled Boyle to trace the transmission of hydrostatic forces through liquids and thereby give a mechanical account of that transmission according to his understanding of the term. This was a major step towards the technical sense of pressure that was to be adopted in Newton’s hydrostatics and in fluid mechanics thereafter.

Poncelet’s porism: a long story of renewed discoveries, I Abstract In 1813, J.-V. Poncelet discovered that if there exists a polygon of n-sides, which is inscribed in a given conic and circumscribed about another conic, then infinitely many such polygons exist. This theorem became known as Poncelet’s porism, and the related polygons were called Poncelet’s polygons. In this article, we trace the history of the research about the existence of such polygons, from the “prehistorical” work of W. Chapple, of the middle of the eighteenth century, to the modern approach of P. Griffiths in the late 1970s, and beyond. For reasons of space, the article has been divided into two parts, the second of which will appear in the next issue of this journal.

Ibn al-Kammād’s Muqtabis zij and the astronomical tradition of Indian origin in the Iberian Peninsula Abstract In this paper, we analyze the astronomical tables in al-Zīj al-Muqtabis by Ibn al-Kammād (early twelfth century, Córdoba), based on the Latin and Hebrew versions of the lost Arabic original, each of which is extant in a unique manuscript. We present excerpts of many tables and pay careful attention to their structure and underlying parameters. The main focus, however, is on the impact al-Muqtabis had on the astronomy that developed in the Iberian Peninsula and the Maghrib and, more generally, on the transmission and diffusion of Indian astronomy in the West after the arrival of al-Khwārizmī’s astronomical tables in al-Andalus (Muslim Spain) in the tenth century. This tradition of Indian origin competed with the Greek tradition represented by al-Battānī’s astronomical tables and was much more alive in the Iberian Peninsula and the Maghrib than previously thought. From Spain, the Indian tradition entered mainstream European astronomy, where we find echoes of it in all versions of the Alfonsine Tables, both in manuscript and in the printed editions (beginning in 1483), as well as in Copernicus’s De Revolutionibus, published in 1543.

Masters, questions and challenges in the abacus schools Abstract The mathematical scenario in Italy during the Late Middle Ages and the Renaissance is mainly dominated by the treatises on the abacus, which developed together with the abacus schools. In that context, between approximately the last thirty years of the fourteenth century and the first twenty years of the sixteenth century, the manuscript and printed tradition tell us of queries and challenges, barely known or totally unknown, in which the protagonists were abacus masters. We report in this work on the most significant examples and draw out interesting cues for thoughts and remarks of a scientific, historical and biographical nature. Five treatises, written in the fifteenth and sixteenth century, have been the main source of inspiration for this article: the Trattato di praticha d’arismetricha and the Tractato di praticha di geometria included in the codices Palat. 573 and Palat. 577 (c. 1460) kept in the Biblioteca Nazionale of Florence and written by an anonymous Florentine disciple of the abacist Domenico di Agostino Vaiaio; another Trattato di praticha d’arismetrica written by Benedetto di Antonio da Firenze in 1463 and included in the codex L.IV.21 kept in the Biblioteca Comunale of Siena; the Tractatus mathematicus ad discipulos perusinos written by Luca Pacioli between 1477 and 1480, manuscript Vat. Lat. 3129 of the Biblioteca Apostolica Vaticana; and Francesco Galigai’s Summa de arithmetica, published in Florence in 1521.

Variations on a theme: Clifford’s parallelism in elliptic space Abstract In 1873, W. K. Clifford introduced a notion of parallelism in the three-dimensional elliptic space that, quite surprisingly, exhibits almost all properties of Euclidean parallelism in ordinary space. The purpose of this paper is to describe the genesis of this notion in Clifford’s works and to provide a historical analysis of its reception in the investigations of F. Klein, L. Bianchi, G. Fubini, and E. Bortolotti. Special emphasis is placed upon the important role that Clifford’s parallelism played in the development of the theory of connections.

L’arithmétique des fractions dans l’œuvre de Fibonacci: fondements & usages

Solar and lunar observations at Istanbul in the 1570s Abstract From the early ninth century until about eight centuries later, the Middle East witnessed a series of both simple and systematic astronomical observations for the purpose of testing contemporary astronomical tables and deriving the fundamental solar, lunar, and planetary parameters. Of them, the extensive observations of lunar eclipses available before 1000 AD for testing the ephemeredes computed from the astronomical tables are in a relatively sharp contrast to the twelve lunar observations that are pertained to the four extant accounts of the measurements of the basic parameters of Ptolemaic lunar model. The last of them are Taqī al-Dīn Muḥammad b. Ma‘rūf’s (1526–1585) trio of lunar eclipses observed from Istanbul, Cairo, and Thessalonica in 1576–1577 and documented in chapter 2 of book 5 of his famous work, Sidrat muntaha al-afkar fī malakūt al-falak al-dawwār (The Lotus Tree in the Seventh Heaven of Reflection). In this article, we provide a detailed analysis of the accuracy of his solar (1577–1579) and lunar observations.