CALCULUS Understanding Its Concepts and Methods

Bonaventura Cavalieri (1598--1647) --- Historical Sketch

In 1635, a mathematical work of notable importance---Geometria Indivisibilibus Continuorum---was published. Its author was Francesco Bonaventura Cavalieri, and even not knowing Latin, one can discern from the title of this work that it relates to fundamental ideas of calculus.

Cavalieri was born into aristocracy, but not wealth, in Milan, Italy. In 1615 (or possibly two years earlier) he became a member of the Jesuates, a Catholic religious order not to be confused with the Jesuits. He taught theology before turning his attention to mathematics. He attended the University of Pisa and became a professor of mathematics at the University of Bologna in 1629. Three years later, he published Directorium universale uranometricum, which contained tables of logarithms and trigonometric functions, all to eight decimal places---a truly monumental undertaking at the time. Cavalieri taught at Bologna until his untimely death.

Though he is today best known for what is called Cavalieri's principle, his most important work was the foreshadowing of integral calculus. In his geometric efforts, he developed what would today be written as an equation you will encounter when learning about natural logarithms, where this integral plays a prominent role. Cavalieri based his work on a concept he called indivisibles of geometry, in which the indivisibles of a line or line segment are points, the indivisibles of a plane or a plane figure are lines or line segments, and indivisibles of a solid figure are plane figures. By means of his concept, he developed fundamental ideas of length, area, and volume, all topics of what came to be and is today known as integral calculus.

As for Cavalieri's principle, sometimes called Cavalieri's theorem in solid geometry books, it is often stated something like this:

• For two solids of equal height, if parallel planes intersect the solids in equal cross-sectional areas, then the volumes of the solids are equal.

But this can be made more general:

• If two solids have equal heights, and if parallel planes intersect the solid in areas of a given ratio, then the volumes of the solids are also in this ratio.

The modern generalization of Cavalieri's principle is called volume by known cross-sectional area, and is generally written as where is the cross-sectional area of the solid at height from the plane on which the solid sits.

Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.