Draws on new evidence to argue that the Jesuits had plotted Galileo's downfall for reasons other than his beliefs about astronomy.

Pulsing with drama and excitement, Infinitesimal celebrates the spirit of discovery, innovation, and intellectual achievement-and it will forever change the way you look at a simple line. On August 10, 1632, five men in flowing black robes convened in a somber Roman palazzo to pass judgment on a deceptively simple proposition: that a continuous line is composed of distinct and infinitely tiny parts. With the stroke of a pen the Jesuit fathers banned the doctrine of infinitesimals, announcing that it could (...) never be taught or even mentioned. The concept was deemed dangerous and subversive, a threat to the belief that the world was an orderly place, governed by a strict and unchanging set of rules. If infinitesimals were ever accepted, the Jesuits feared, the entire world would be plunged into chaos. In Infinitesimal, the award-winning historian Amir Alexander exposes the deep-seated reasons behind the rulings of the Jesuits and shows how the doctrine persisted, becoming the foundation of calculus and much of modern mathematics and technology. Indeed, not everyone agreed with the Jesuits. Philosophers, scientists, and mathematicians across Europe embraced infinitesimals as the key to scientific progress, freedom of thought, and a more tolerant society. As Alexander reveals, it wasn't long before the two camps set off on a war that pitted Europe's forces of hierarchy and order against those of pluralism and change. The story takes us from the bloody battlefields of Europe's religious wars and the English Civil War and into the lives of the greatest mathematicians and philosophers of the day, including Galileo and Isaac Newton, Cardinal Bellarmine and Thomas Hobbes, and Christopher Clavius and John Wallis. In Italy, the defeat of the infinitely small signaled an end to that land's reign as the cultural heart of Europe, and in England, the triumph of infinitesimals helped launch the island nation on a course that would make it the world's first modern state. From the imperial cities of Germany to the green hills of Surrey, from the papal palace in Rome to the halls of the Royal Society of London, Alexander demonstrates how a disagreement over a mathematical concept became a contest over the heavens and the earth. The legitimacy of popes and kings, as well as our beliefs in human liberty and progressive science, were at stake-the soul of the modern world hinged on the infinitesimal. (shrink)

In Il Saggiatore (1623), Galileo makes a strict distinction between primary and secondary qualities. Although this distinction continues to be debated in philosophical literature up to this very day, Galileo's views on the matter, as well as their impact on his contemporaries and other philosophers, have yet to be sufficiently documented. The present paper helps to clear up Galileo's ideas on the subject by avoiding some of the misunderstandings that have arisen due to faulty translations of his work. In particular, (...) it shows how Galileo's distinction directly implicates a novel understanding of physical bodies, which played an important part in his later condemnation by the Catholic Church. At the same time, the paper also argues that Galileo's distinction can already be found in the text and illustrations of earlier, popular Copernican writings. (shrink)

"The development of the calculus during the 17th century was successful in mathematical practice, but raised questions about the nature of infinitesimals: were they real or rather fictitious? This collection of essays, by scholars from Canada, the US, Germany, United Kingdom and Switzerland, gives a comprehensive study of the controversies over the nature and status of the infinitesimal. Aside from Leibniz, the scholars considered are Hobbes, Wallis, Newton, Bernoulli, Hermann, and Nieuwentijt. The collection also contains newly discovered marginalia of Leibniz (...) to the writings of Hobbes."--BOOK JACKET. (shrink)

The main purpose of this article is to present new material concerning Leibniz's use of indivisibles and infinitesimals in his early mathematical texts. Most of these texts are contained in hitherto unpublished manuscripts and are soon to be printed in volume VII, 4 of the Academy Edition. They present examples which illustrate how Leibniz operated with concepts such as indivisibles and infinitesimals in that period of his development.

Newton and Leibniz had profound disagreements concerning metaphysics and the relationship of mathematics to natural philosophy, as well as deeply opposed attitudes towards analysis. Nevertheless, or so I shall argue, despite these deeply held and distracting differences in their background assumptions and metaphysical views, there was a considerable consilience in their positions on the status of infinitesimals. In this paper I compare the foundation Newton provides in his Method Of First and Ultimate Ratios (sketched at some time between 1671 and (...) 1684, and published in the Principia of 1687) with that provided independently by Leibniz in his unpublished manuscript De quadratura arithmetica (1675-6) as well as in later writings. I argue that both appeal to a version of the Archimedean Axiom to underwrite their use of infinitesimal techniques, which must be interpreted as a shorthand for rigorously finitist methods. (shrink)

Cet article essaie de reconstruire les raisons complexes des changements successifs dans la relation entre le métaphysique de Leibniz dans ses écrits de jeunesse et son intérêt personnel dans l'apologétique et la réunion des Eglises. Je soutiens que, tandis que l'effort primitif de Leibniz pour lier l'intelligibilité de la Philosophie Mécanique au théisme le porta à mettre l'accent sur Dieu comme Moteur Premier, son désir croissant de défendre l'intelligibilité de la Transubstantiation inspira le développement d'analyses métaphysiques plus poussées sur les (...) sources de la substantiate et sur les modes de l'union divine avec le monde. Cependant, l'éveil de préoccupations mathématiques, physiques et théologiques plus amples lors du séjour de Leibniz à Paris crea une tension croissante entre la direaion prise par sa métaphysique et son désir de défendre la Transubstantiation. J'explore les raisons de cette tension et les relie au développement du principe d'équipollence et à la place de plus en plus centrale occupée dans la pensée leibnizienne par l'intelligibilité et l'autonomie de la nature. (shrink)

Traces the development of the integral and the differential calculus and related theories since ancient times.

Weder historisch noch philosophisch kann man zu nur einer einzigen Unterscheidung zwischen primären und sekundären Qualitäten gelangen. Es gibt vielmehr eine große Zahl von Unterscheidungen, die miteinander vermengt und wie eine einzige behandelt werden, und zwar einfach deshalb, weil man bei der Diskussion des Problems wahllos einige Beispiele herausgreift. In diesem Aufsatz werden zweiundzwanzig solcher Unterscheidungen angeführt, wobei Bezug genommen wird auf Philosophen, die diese Unterscheidungen vertreten haben. Im Anschluß daran wird die Vermutung geäußert, daß der Grund für diesen verworrenen (...) Zustand zum Teil im Fehlen einer plausiblen Theorie der Qualitäten zu suchen sei. Da eine solche Theorie fehlt, ist es nicht verwunderlich, daß die Einteilung von Qualitäten und vermeintlichen Qualitäten in primäre und sekundäre Schwierigkeiten bereitet. (shrink)

(I) MATHEMATICAL LECTURES. LECTURE I. Of the Name and general Division of the Mathematical Sciences. BEING about to treat upon the Mathematical Sciences, ...