The publication is a vital source of information for historians, mathematicians, and researchers interested in infinitesimal calculus. The book reviews the infinitesimal methods in England and Low Countries and rectification of arcs. Topics include the link between differential and integral processes, concept of tangent, first investigations of the cycloid, and arithmetization of integration methods. The manuscript then examines infinitesimals and indivisibles in the early 17th century and further advances in France and Italy. Discussions focus on the growth of kinematics in the West, latitude of forms, influence of Aristotle, axiomatization of Greek mathematics, theory of proportion and means, method of exhaustion, discovery method of Archimedes, and curves, normals, tangents, and curvature. The publication first ponders on Greek mathematics, transition to Western Europe, and some center of gravity determinations in the later 16th century. The Origins of Infinitesimal Calculus focuses on the evolution, development, and applications of infinitesimal calculus. 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. 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. 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. 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. 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. Philosophers, scientists, and mathematicians across Europe embraced infinitesimals as the key to scientific progress, freedom of thought, and a more tolerant society. Indeed, not everyone agreed with the Jesuits. 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. If infinitesimals were ever accepted, the Jesuits feared, the entire world would be plunged into chaos. 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. With the stroke of a pen the Jesuit fathers banned the doctrine of infinitesimals, announcing that it could never be taught or even mentioned. 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. 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.

Publisher: Scientific American / Farrar, Straus and Giroux

The arithmetic of infinitesimals