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James Gregory and the Pappus-Guldin Theorem

Author(s): 
Andrew Leahy (Knox College)

 

The Geometriae Pars Universalis (GPU) by the Scottish mathematician James Gregory is a 17th century mathematics text which uses geometrical techniques to solve a variety of calculus problems, such as finding tangents, areas, and volumes of revolution. Buried without fanfare right in the middle of the GPU (the 35th of 70 propositions in the work) is a seemingly innocuous geometry result about solids of revolution:

 

Each solid of revolution is equal to a right cylindrical figure whose base is the figure out of the rotation of which the solid is produced and whose altitude is the circumference of a circle in which the center of gravity of the figure is revolved.

 

For our generation of mathematicians, solids of revolution serve primarily as a testing ground for the techniques of integral calculus. So on first glance we might miss the significance of a result which simply gives an equality between the volumes of two three-dimensional objects. However, since the volume of a right cylindrical figure is the area of its base times its height, another way of interpreting Gregory's result is

Volume of Revolution   =
Volume of the cylindrical figure with the same base
= base times height
= distance travelled by its center of gravity.

This rephrasing of Gregory's Proposition 35 may be familiar to those who have seen second semester calculus. It states that the volume of each solid of revolution is equal to the area of its base multiplied by the circumference of the circle in which the center of gravity of that figure is revolved. This is the Theorem of Pappus (or the Pappus-Guldin Theorem). Gregory's geometrical approach toward proving this result and just why this result ended up in Gregory's text in the first place are the subjects of this article.

Dummy View - NOT TO BE DELETED