A Mathematician among the Molasses Barrels: Maclaurin's Unpublished Memoir on Volumes

Suppose we are given a solid of revolution generated by a conic section. Slice out a frustum of the solid [14, diagrams pp. 77, 80]. Then, construct a cylinder, with the same height as the frustum, whose diameter coincides with the diameter of the frustum at the midpoint of its height. What is the difference between the volume of the frustum and the volume of this cylinder? Does this difference depend on where in the solid the frustum is taken? The beautiful theorems which answer these questions first appear in a 1735 manuscript by Colin Maclaurin (1698-1746). This manuscript [14], the only original mathematical work by Maclaurin not previously printed, is published here for the first time, with the permission of the Trustees of the National Library of Scotland. (An almost identical copy [15] exists in the Edinburgh University Library.) In this work, Maclaurin proved that the difference between the cylinder constructed as above and the frustum of the given solid depends only on the height of the frustum, not the position of the frustum in the solid. When the solid is a cone, Maclaurin showed that its frustum exceeds the corresponding cylinder by one fourth the volume of a similar cone with the same height. For a sphere, the cylinder exceeds the frustum by one half the volume of the sphere whose diameter is equal to the height of the frustum; this holds, he observed, for all spheres. He derived analogous results for the ellipsoid and hyperboloid of revolution. Finally, for the paraboloid of revolution, he proved that the cylinder is precisely equal to the frustum. These results resemble some propositions from classical geometry, especially Archimedes' theorem that the difference between the volume of a cylinder with height and diameter equal and the volume of a sphere with the same diameter is a cone with that height and diameter. The volumes of the cone, sphere, and cylinder stand to one another in the ratio 1:2:3. Maclaurin's friend, the philosopher of aesthetics Francis Hutcheson, gave Archimedes' result as a key example of mathematical beauty [10, p. 49]. In fact Maclaurin later proved his theorems geometrically in his Treatise of Fluxions [16, pp. 24-27], where he extended them to frustums terminated by planes oblique to the axis of the solid. But these theorems originated in a far different setting: the present manuscript, a memoir addressed to the Commissioners of Excise for …