The Indivisibles of the Continuum

2020 ◽  
pp. 104-122
Author(s):  
Douglas M. Jesseph

This chapter considers some significant developments in seventeenth-century mathematics which are part of the pre-history of the infinitesimal calculus. In particular, I examine the “method of indivisibles” proposed by Bonaventura Cavalieri and various developments of this method by Evangelista Torricelli, Gilles Personne de Roberval, and John Wallis. From the beginning, the method of indivisibles faced objections that aimed to show that it was either conceptually ill-founded (in supposing that the continuum could be composed of dimensionless points) or that its application would lead to error. I show that Cavalieri’s original formulation of the method attempted to sidestep the question of whether a continuous magnitude could be composed of indivisibles, while Torricelli proposed to avoid paradox by taking indivisibles to have both non-zero (yet infinitesimal) magnitude and internal structure. In contrast, Roberval and Wallis showed significantly less interest in addressing foundational issues and were content to maintain that the method could (at least in principle) be reduced to Archimedean exhaustion techniques.

John Wallis (1616-1703), one of the original Fellows of the Royal Society, was a scholar of amazing versatility. Though born into an age of intellectual giants he rapidly acquired a commanding place even among that brilliant group which has made the seventeenth century illustrious in the history of science. More than once he blazed the trail which led to some epoch-making discovery. When Newton modestly declared ‘If I have seen further it is by standing on ye sholders of Giants’, he no doubt had the name of John WalHs well before his mind. Walks was born on 23 November 1616, at Ashford in East Kent, a country town of which his father was rector. On the death of his father, Wallis was sent to school at Ashford. Later he was moved to Tenter den, where he came under the care of Mr James Movat, and even in his earliest years he distinguished himself by that singular aptitude for learning which was to remain with him till the closing years of his life. At the age of fourteen he went to Felsted, and here he acquired a marked proficiency not only in Latin and Greek, but also in Hebrew. From Felsted he entered Emmanuel College, Cambridge, and although his interest in mathematics dates from this period, he gave no evidence of unusual talent for the subject; this, he complains was because there was no one in the University to direct his studies. Divinity was his dominant interest. In 1640 he was ordained, and four years later he was appointed, together with Adoniram Byfield, Secretary to the Assembly of Divines at Westminster. Possibly on account of his ecclesiastical duties, which absorbed much of his time and energy, his early promise as a mathematician still remained unfulfilled.


The Library ◽  
2021 ◽  
Vol 22 (2) ◽  
pp. 177-196
Author(s):  
Samuel V Lemley

Abstract The textual history of the first work on infinitesimal calculus and differentiation in print—Gottfried Wilhelm Leibniz’s ‘Nova methodus pro maximis et minimis …’ in the October 1684 instalment of the Leipzig scientific journal Acta Eruditorum—remains unstudied. Consequent to this inattention, extant copies of Leibniz’s article have been assigned to a single edition and a single press, despite evidence of substantive variation among them. This article examines the typographic and bibliographical evidence across multiple copies of the October 1684 instalment of the Acta to demonstrate that these extant copies in fact represent three separate editions printed on multiple presses over many years. This evidence, in turn, casts new light on both the complex printing history of the Acta Eruditorum in its first decade of publication (1682–93) and the distribution of learned periodicals in the seventeenth century.


1993 ◽  
Vol 26 (3) ◽  
pp. 271-279 ◽  
Author(s):  
Siegmund Probst

Until recently, historians of mathematics usually agreed in refusing to consider the numerous geometrical publications of Thomas Hobbes as a contribution to the development of mathematics in the seventeenth century. From time to time, one could find statements that although Hobbes did not find new theorems he undoubtedly had profound insights into the logical foundations of mathematics, but these occasional remarks did not encourage historians to go deeper into Hobbes's mathematical thought. In the end, the general conclusion was that Hobbes's preoccupation with squaring the circle, doubling the cube (starting when the philosopher was more than forty years of age), and challenging Euclid's definitions were better ignored, at least in the history of science. In particular, his controversy with the Savilian professors Seth Ward and John Wallis was seen as a ‘deplorable affair’, liable only to damage the reputation of the protagonists.


2020 ◽  
Vol 54 (4) ◽  
pp. 403-431
Author(s):  
Bulat R. Rakhimzianov

Abstract This article explores relations between Muscovy and the so-called Later Golden Horde successor states that existed during the fifteenth and sixteenth centuries on the territory of Desht-i Qipchaq (the Qipchaq Steppe, a part of the East European steppe bounded roughly by the Oskol and Tobol rivers, the steppe-forest line, and the Caspian and Aral Seas). As a part of, and later a successor to, the Juchid ulus (also known as the Golden Horde), Muscovy adopted a number of its political and social institutions. The most crucial events in the almost six-century-long history of relations between Muscovy and the Tatars (13–18th centuries) were the Mongol invasion of the Northern, Eastern and parts of the Southern Rus’ principalities between 1237 and 1241, and the Muscovite annexation of the Kazan and Astrakhan khanates between 1552 and 1556. According to the model proposed here, the Tatars began as the dominant partner in these mutual relations; however, from the beginning of the seventeenth century this role was gradually inverted. Indicators of a change in the relationship between the Muscovite grand principality and the Golden Horde can be found in the diplomatic contacts between Muscovy and the Tatar khanates. The main goal of the article is to reveal the changing position of Muscovy within the system of the Later Golden Horde successor states. An additional goal is to revisit the role of the Tatar khanates in the political history of Central Eurasia in the fifteenth and sixteenth centuries.


Author(s):  
Karel Schrijver

This chapter describes how the first found exoplanets presented puzzles: they orbited where they should not have formed or where they could not have survived the death of their stars. The Solar System had its own puzzles to add: Mars is smaller than expected, while Venus, Earth, and Mars had more water—at least at one time—than could be understood. This chapter shows how astronomers worked through the combination of these puzzles: now we appreciate that planets can change their orbits, scatter water-bearing asteroids about, steal material from growing planets, or team up with other planets to stabilize their future. The special history of Jupiter and Saturn as a pair bringing both destruction and water to Earth emerged from the study of seventeenth-century resonant clocks, from the water contents of asteroids, and from experiments with supercomputers imposing the laws of physics on virtual worlds.


Author(s):  
Jonathan Hehn

This chapter outlines the history of Presbyterian worship practice from the sixteenth century to the present, with a focus on North American Presbyterians. Tracing both their hymnody and their liturgy ultimately to John Calvin, Presbyterian communions have a distinct heritage of worship inherited from the Church of Scotland via seventeenth-century Puritans. Long marked by metrical psalmody and guided by the Westminster Directory, Presbyterian worship underwent substantial changes in the nineteenth century. Evangelical and liturgical movements led Presbyterians away from a Puritan visual aesthetic, into the use of nonscriptural hymnody, and toward a recovery of liturgical books. Mainline North American and Scottish Presbyterians solidified these trends in the twentieth and twenty-first centuries; however, conservative North American denominations and some other denominations globally continue to rely heavily on the use of a worship directory and metrical psalmody.


2020 ◽  
Vol 14 (2) ◽  
pp. 241-259
Author(s):  
Dirk Werle ◽  
Uwe Maximilian Korn

AbstractResearch on the history of fiction of the early modern period has up to now taken primarily the novel into consideration and paralleled the rise of the novel as the leading genre of narrative literature with the development of the modern consciousness of fictionality. In the present essay, we argue that contemporary reflections on fictionality in epic poetry, specifically, the carmen heroicum, must be taken into account to better understand the history of fiction from the seventeenth century onwards. The carmen heroicum, in the seventeenth century, is the leading narrative genre of contemporary poetics and as such often commented on in contexts involving questions of fictionality and the relationship between literature and truth, both in poetic treatises and in the poems themselves. To reconstruct a historical understanding of fictionality, the genre of the epic poem must therefore be taken into account.The carmen heroicum was the central narrative genre in antiquity, in the sixteenth century in Italy and France, and still in the seventeenth century in Germany and England. Martin Opitz, in his ground-breaking poetic treatise, the Buch von der Deutschen Poeterey (1624), counts the carmen heroicum among the most important poetic genres; but for poetry written in German, he cites just one example of the genre, a text he wrote himself. The genre of the novel is not mentioned at all among the poetic genres in Opitz’ treatise. Many other German poetic treatises of the seventeenth century mention the importance of the carmen heroicum, but they, too, provide only few examples of the genre, even though there were many Latin and German-language epic poems in the long seventeenth century. For Opitz, a carmen heroicum has to be distinguished from a work of history insofar as its author is allowed to add fictional embellishments to the ›true core‹ of the poem. Nevertheless, the epic poet is, according to Opitz, still bound to the truthfulness of his narrative.Shortly before the publication of Opitz’ book, Diederich von dem Werder translated Torquato Tasso’s epic poem Gerusalemme liberata (1580); his translation uses alexandrine verse, which had recently become widely successful in Germany, especially for epic poems. Von dem Werder exactly reproduces Tasso’s rhyming scheme and stanza form. He also supplies the text with several peritexts. In a preface, he assures the reader that, despite the description of unusual martial events and supernatural beings, his text can be considered poetry. In a historiographical introduction, he then describes the course of the First Crusade; however, he does not elaborate about the plot of the verse epic. In a preceding epyllion – also written in alexandrine verse – von dem Werder then poetically demonstrates how the poetry of a Christian poet differs from ancient models. All these efforts can be seen as parts of the attempt to legitimate the translation of fictional narrative in German poetry and poetics. Opitz and von dem Werder independently describe problems of contemporary literature in the 1620s using the example of the carmen heroicum. Both authors translate novels into German, too; but there are no poetological considerations in the prefaces of the novels that can be compared to those in the carmina heroica.Poetics following the model established by Opitz develop genre systems in which the carmen heroicum is given an important place, too; for example, in Balthasar Kindermann’s Der Deutsche Poet (1664), Sigmund von Birken’s Teutsche Rede- bind- und Dicht-Kunst (1679), and Daniel Georg Morhof’s Unterricht von der Teutschen Sprache und Poesie (1682). Of particular interest for the history of fictionality is Albrecht Christian Rotth’s Vollständige Deutsche Poesie (1688). When elaborating on the carmen heroicum, Rotth gives the word ›fiction‹ a positive terminological value and he treats questions of fictionality extensively. Rotth combines two contradictory statements, namely that a carmen heroicum is a poem and therefore invented and that a carmen heroicum contains important truths and is therefore true. He further develops the idea of the ›truthful core‹ around which poetic inventions are laid. With an extended exegesis of Homer’s Odyssey, he then illustrates what it means precisely to separate the ›core‹ and the poetic embellishments in a poem. All these efforts can be seen as parts of the attempt to legitimize a poem that tells the truth in a fictional mode.The paper argues that a history of fictionality must be a history that carefully reconstructs the various and specifically changing constellations of problems concerning how the phenomenon of fictionality may be interpreted in certain historical contexts. Relevant problems to which reflections on fictionality in seventeenth-century poetics of the epic poem and in paratexts to epic poems react are, on the one hand, the question of how the genre traditionally occupying the highest rank in genre taxonomy, the epic, can be adequately transformed in the German language, and, on the other hand, the question of how a poetic text can contain truths even if it is invented.


Religions ◽  
2021 ◽  
Vol 12 (8) ◽  
pp. 570
Author(s):  
James W. Watts

Leviticus 25:39–46 describes a two-tier model of slavery that distinguishes Israelites from foreign slaves. It requires that Israelites be indentured only temporarily while foreigners can be enslaved as chattel (permanent property). This model resembles the distinction between White indentured slaves and Black chattel slaves in the American colonies. However, the biblical influence on these early modern practices has been obscured by the rarity of citations of Lev. 25:39–46 in sixteenth- and seventeenth-century sources about slavery. This article reviews the history of slavery from ancient Middle Eastern antiquity through the seventeenth century to show the unique degree to which early modern institutions resembled the biblical model. It then exposes widespread knowledge of Leviticus 25 in early modern political and economic debates. Demonstrating this awareness shows with high probability that colonial cultures presupposed the two-tier model of slavery in Leviticus 25:39–46 to naturalize and justify their different treatment of White indentured slaves and Black chattel slaves.


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