Method Of Indivisibles Research Articles

On Indivisibles and Infinitesimals: A Response to David Sherry, “The Jesuits and the Method of Indivisibles”

In “The Jesuits and the Method of Indivisibles” David Sherry criticizes a central thesis of my book Infinitesimal: that in the seventeenth century the Jesuits sought to suppress the method of indivisibles because it undermined their efforts to establish a perfect rational and hierarchical order in the world, modeled on Euclidean Geometry. Sherry accepts that the Jesuits did indeed suppress the method, but offers two objections. First, that the book does not distinguish between indivisibles and infinitesimals, and that whereas the Jesuits might have reason to object to the first, the second posed no problem for them. Second, seeking an alternative explanation for the Jesuits’ hostility to the method, he proposes that its implicit atomism conflicted with the Catholic doctrine of the sacrament of the Eucharist, and was therefore heretical. In response to Sherry’s first criticism I point out that the Jesuits objected to all forms of the method of indivisibles, and that the distinction between indivisibles and infinitesimals consequently cannot explain the fight over the method in the seventeenth century. With regards to the Eucharist, I agree with Sherry that the Jesuits were indeed concerned about the method’s affinity to atomism and materialism, though for a different reason: these doctrines were antithetical to their efforts to impose divine hierarchy and order on the world. In as much as the technical details of the miracle of the Eucharist mattered, they provided no grounds for objecting to a mathematical (rather than physical) doctrine.

Geometric disintegration and star-shaped distributions

Geometric and stochastic representations are derived for the big class of p-generalized elliptically contoured distributions, and (generalizing Cavalieri’s and Torricelli’s method of indivisibles in a non-Euclidean sense) a geometric disintegration method is established for deriving even more general star-shaped distributions. Applications to constructing non-concentric elliptically contoured and generalized von Mises distributions are presented.AMS subject classification Primary 60E05; 60D05; secondary 28A50; 28A75; 51F99

Cavalieri's method of indivisibles

C., sa vie, son oeuvre. La theorie des indivisibles dans Geometria. Application de cette theorie aux sections coniques. Les reactions des mathematiciens de l'epoque (Toricelli, Galilee...).

The Chinese concept of Cavalieri's principle and its applications

Bonaventura Cavalieri (1598–1647) was noted for his method of indivisibles which led to the principle which bears his name. In the third century, while attempting to derive the volume of a sphere, Liu Hui applied a similar principle to determine the ratio of the volumes of a sphere and a solid circumscribing the sphere. This solid is formed by the intersection of two perpendicular cylinders circumscribing the sphere and is called mou he fang gai . Liu Hui left unresolved the problem of finding the volume of the mou he fang gai . In the fifth century Zu Geng, also applying Cavalieri's principle, solved this problem and thus derived the volume of a sphere. The influence of Zu Geng's method on later mathematicians is discussed in the latter part of the article. Bonaventura Cavalieri (1598–1647) est connu pour sa méthode des indivisibles dont découle le principe qui porte son nom. Au troisième siècle, Liu Hui, cherchant à calculer le volume d'une sphère, a employé un principe similaire pour établir le rapport entre le volume de la sphère et celui d'un solide circonscrit à celle-ci. Ce solide, appelé mou he fang gai , est l'intersection de deux cylindres circonscrits placés perpendiculairement l'un par rapport à l'autre. Liu Hui laisse en suspend le la détermination du volume du mou he fang gai . Au cinquième siècle, Zu Geng résolut ce problème en employant le principe de Cavalieri, et put de la sorte déterminer le volumed de la sphère. Nous terminons en analysant l'influence ultérieure de la méthode de Zu Geng's. Bonaventura Cavalieri (1598–1647) ist durch seine Indivisibelnmethode bekannt, die ihn zum sog. Cavalierischen Prinzip führte. Im 3. Jh. verwandte Liu Hui ein ähnliches Prinzip, um das Verhältnis des Kugelvolumens zum Volumen eines gewissen umbeschriebenen Körpers zu bestimmen. Dieser Körper entsteht bei der Durchdringung zweier zueinander senkrechtstehender Zylinder, die der Kugel umbeschrieben sind, und wird mou he fang gai genannt. Das Problem der Bestimmung des Volumens dieses Körpers ließ Liu Hui offen. Unter Anwendung des sog. Cavalierischen Prinzips löste im 5. Jh. Zu Geng dieses Problem und bestimmte somit auch den Kugelinhalt. Es wird ferner im vorliegenden Beitrag auch der Einfluß der von Zu Geng verwendeten Methode auf spätere Mathematiker diskutiert.

Memorabilia Mathematica

Prior to the year 1650, the state of mathematics was at a low ebb. But the middle of the seventeenth century may be said to have ushered in a new era in mathematical learning. On the Continent, Descarte's geometry appeared in 1637, and Cavalieri's method of indivisibles, introduced a year or two later, presaged the blossoming of the calculus. In England, John Wallis of Oxford and Isaac Barrow of Cambridge were to form a connecting link between Descartes and Cavalieri, on the one hand, and Isaac Newton on the other.