James Gregory and the Pappus-Guldin Theorem - Historical Background: Pappus

The Pappus-Guldin Theorem is simultaneously one of the last great results in Greek mathematics and one of the first novel results in the 16th and 17th century renaissance in European mathematics. It was first stated by Pappus of Alexandria, who flourished around 300-350 CE, in Book Seven of his Collection. In the midst of a discussion of the Conics of Apollonius, Pappus launches into an emotional lament on the state of mathematical learning in his time:

When I see everyone occupied with the rudiments of mathematics and of the material for inquiries that nature sets before us, I am ashamed; I for one have proved things that are much more valuable and offer much application. In order not to end my discourse declaiming this with empty hands, I will give this for the benefit of the readers . . . [1, p. 122]

He then presents the following as evidence that the glory of Greek mathematics has not faded completely during his time:

The ratio of solids of complete revolution is compounded of (that) of the revolved figures and (that) of the straight lines similarly drawn to the axes from the centers of gravity in them . . .

Several additional results follow, but all are variations on the first one.

Like Gregory's statement of his result, it's necessary to peel away some geometrical ideas (this time in Euclidean proportion theory) to find the familiar result. If AB is a figure with center of gravity a which is revolved around the axis CD, let radius(a) denote the distance between a and CD. Similarly, if EF is another figure with center of gravity e which is rotated around the axis GH, let radius(e) denote the corresponding distance.

Let area(AB) and area(EF) denote the areas of the two planar figures AB and EF, and let rev(AB) and rev(EF) denote the volumes of the solids obtained by revolving AB and EF around the axes CD and GH, respectively. If, as is customary, we interpret ratios as fractions and compounding of ratios as multiplying fractions, then Pappus' result becomes:

Multiply the top and the bottom of the equation by 2π to get

Here circum(a) and circum(e) denote circumferences of a circle of radius(a) and radius(e), respectively. This is not quite our formula, but to a mathematician brought up in Euclidean proportion theory this proportional expression of the result is perfectly natural and analogous to the formulas for areas and volumes occurring in Euclid's Elements (e.g., Euclid VI.23, or Euclid XII,18).