Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T02:58:10.227Z Has data issue: false hasContentIssue false

A mathematician among the molasses barrels: Maclaurin's unpublished memoir on volumes Introduction: Maclaurin's memoir and its place in eighteenth-century Scotland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
Affiliation:
Pitzer College Claremont, California 91711, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Copyright
Copyright © Edinburgh Mathematical Society 1996

References

REFERENCES

1. Anonymous, “The Second Paper of Memoirs of Mr MacLaurin beginning 1725”, Glasgow University Library MS Gen 1378/2 [in an 18th-century hand; catalogue says “Contains material not used by Patrick Murdoch in his account of Maclaurin's life”].Google Scholar
2. Atton, Henry and Holland, Henry Hurst, The King's Customs: An Account of Maritime Revenue and Contraband Traffic in England, Scotland and Ireland from the Earliest Times to the Year 1800 [1908] (London, Frank Cass & Co., 1967).Google Scholar
3. Clarkson, A., The Pre-Industrial Economy in England, 1500–1750 (London, B. T. Batsford, 1971).Google Scholar
4. Christie, John R. R., The Origins and Development of the Scottish Scientific Community, 1680–1760, History of Science, 1974, pp. 122141.CrossRefGoogle Scholar
5. Crouch, Henry A Complete View of the British Customs (London, J. Osborn and W. Bell, 1724).Google Scholar
6. Daiches, David Glasgow (London, Andre Deutsch, 1977).Google Scholar
7. Gregory, David A Treatise of Practical Geometry. In Three Parts [c. 1685]. Tr. from the Latin, with Additions (Edinburgh, W. and T. Ruddimans, 1745).Google Scholar
8. Hamilton, Henry An Economic History of Scotland in the Eighteenth Century (Oxford, Clarendon Press, 1963).Google Scholar
9. Hoon, Elizabeth Evelynola, The Organization of the English Customs System, 1696–1766 (Newton Abbott, David and Charles, 1968).Google Scholar
10. Hutcheson, Francis An Inquiry concerning Beauty, Order, Harmony, Design (ed. Kivy, Peter, The Hague, Nijhoff, 1933).Google Scholar
11. Leadbetter, Charles The Royal Ganger, or Gauging Made Perfectly Easy, As It Is Actually Practiced by the Officers of His Majesty's Revenue of Excise, London 1739; 4th ed. (London, Wicksteed, 1755).Google Scholar
12. Lenman, Bruce An Economic History of Modern Scotland, 1660–1976 (London, B. T. Batsford, 1977).Google Scholar
13. Maclaurin, Colin, Dissertatio De Ortu Geometriae, Habita in Auditorio publico Collegii novi Abredonensis, Febr. 10. 1722, Aberdeen University Library, MS 34 [only pages 1–6, 9, and 10, which ends in mid-sentence, survive from this manuscript]. Quoted from at length in [21], p. 95 ff.Google Scholar
14. Maclaurin, Colin, Memorial offered to the Honourable Commissioners of Excise concerning the Mensuration of Tuns or Backs that have some irregularity in the Figure and Situation of the Bottom, and in the Height and Position of the Staves: and of ascertaining a just place for taking the dry Inches in such a Vessel: To which is added a Method of correcting the common Tables, and some new Theorems concerning the Mensuration of regular Solids. Manuscript, National Library of Scotland Adv MS 23.1.13. Published immediately below.Google Scholar
15. Maclaurin, Colin, Memorial offer'd to the Honble Commrs of Excise…, manuscript, Edinburgh University Library DC. 1.17; content and wording virtually identical to [14] although pagination, spelling, and punctuation differ slightly.Google Scholar
16. Maclaurin, Colin A Treatise of Fluxions in Two Books (Edinburgh, Ruddimans, 1742).Google Scholar
17. Mccusker, John J. and Menard, Russell R., The Economy of British America, 1607–1789 (Chapel Hill and London, University of North Carolina Press, 1985).Google Scholar
18. Mills, Stella The Collected Letters of Colin MacLaurin (Nantwich, Shiva Publishing, 1982).Google Scholar
19. Murdoch, Patrick, An Account of the Life and Writings of the Author, in Colin Maclaurin, Account of Sir Isaac Newton's Philosophical Discoveries, London, 1748, pp. ixx; reprinted (New York, Johnson Reprint Corp., 1968).Google Scholar
20. Oakley, A., The Second City (London and Glasgow, Blackie and Son, 1946).Google Scholar
21. Sageng, Erik Lars, Colin MacLaurin and the Foundations of the Method of Fluxions, unpublished Ph.D. Dissertation, Princeton, 1989.Google Scholar
22. Scott, Richard, The Politics and Administration of Scotland, 1725–1748, Unpublished Ph.D. Thesis, University of Edinburgh, 1981.Google Scholar
23. Scottish Record Office RH 2/4/446, 382.Google Scholar
24. Shammas, Carole The Pre-Industrial Consumer in England and America (Oxford, Clarendon Press, 1990).Google Scholar
25. Sheridan, Richard B., Sugar and Slavery: An Economic History of the British West Indies, 1623–1775 (Barbados, Caribbean Universities Press, 1974).Google Scholar