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A mathematician among the molasses barrels: Maclaurin's unpublished memoir on volumes Introduction: Maclaurin's memoir and its place in eighteenth-century Scotland

Published online by Cambridge University Press:  20 January 2009

Judith V. Grabiner
Judith V. Grabiner
Affiliation:
Pitzer College Claremont, California 91711, U.S.A.
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Abstract

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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.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 39 , Issue 2 , June 1996 , pp. 193 - 240
Copyright
Copyright © Edinburgh Mathematical Society 1996

References

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