The Method of Mechanical Theorems - Curvilinear Shapes With Rational Volumes

Curvilinear Shapes With Rational Volumes

One of the remarkable things about the Method is that Archimedes finds two shapes defined by sections of cylinders, whose volume does not involve π, despite the shapes having curvilinear boundaries. This is a central point of the investigation—certain curvilinear shapes could be rectified by ruler and compass, so that there are nontrivial rational relations between the volumes defined by the intersections of geometrical solids.

Archimedes emphasizes this in the beginning of the treatise, and invites the reader to try to reproduce the results by some other method. Unlike the other examples, the volume of these shapes is not rigorously computed in any of his other works. From fragments in the palimpsest, it appears that Archimedes did inscribe and circumscribe shapes to prove rigorous bounds for the volume, although the details have not been preserved.

The two shapes he considers are the intersection of two cylinders at right angles, which is the region of (x, y, z) obeying:

(2Cyl)

and the circular prism, which is the region obeying:

(CirP)

Both problems have a slicing which produces an easy integral for the mechanical method. For the circular prism, cut up the x-axis into slices. The region in the y-z plane at any x is the interior of a right triangle of side length whose area is, so that the total volume is:

(CirP)

which can be easily rectified using the mechanical method. Adding to each triangular section a section of a triangular pyramid with area balances a prism whose cross section is constant.

For the intersection of two cylinders, the slicing is lost in the manuscript, but it can be reconstructed in an obvious way in parallel to the rest of the document: if the x-z plane is the slice direction, the equations for the cylinder give that while, which defines a region which is a square in the x-z plane of side length, so that the total volume is:

(2Cyl)

And this is the same integral as for the previous example.