Archimedes on the sphere and cylinder
3 February 2026
Max Dehn (1878–1952) said that Archimedes’ (c. 287–212 BCE) discovery that the surface area of a sphere was four times its great circle was the one of the most beautiful results of Greek mathematics.
Archimedes himself had a high opinion of this result and two others in his two books On the Sphere and the Cylinder: that the volume and surface area of a sphere and a cylinder exactly circumscribing it are in the ratio $2 : 3$. One can add a cone fitting inside the cylinder to have ratios $1 : 2 : 3$ (see diagram).
It has been suggested that Archimedes’ conjectures for these ratios may have been guided by a conscious or unconscious search for beautiful integer ratios between geometric configurations. There is no direct evidence for this motivation, but Archimedes’ work seems to exhibit a preference for small integer ratios.
According to Plutarch, Archimedes desired that his tomb should be marked by a cylinder enclosing a sphere and an inscription of the ratio of the one to the other; Cicero related how he had sought out Archimedes’ tomb and found a column just so inscribed.
References
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Archimedes. The Works of Archimedes, vol. I: The Two Books on the Sphere and the Cylinder. Translated into English, together with Eutocius’ commentaries, with commentary, and critical edition of the diagrams. Cambridge University Press, 2004. ISBN: 978-0-521-66160-7
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A.J. Cain. Form & Number: A History of Mathematical Beauty. Lisbon, 2024. pp.106–7.
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Cicero. Tusculan Disputations. Loeb Classical Library, no.141. Cambridge, MA: Harvard University Press, 1927. DOI: 10.4159/DLCL.marcus_tullius_cicero-tusculan_disputations.1927 § V.xxv.
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M. Dehn. ‘The Mentality of the Mathematician: A Characterization’. In: The Mathematical Intelligencer. 5, no.2 (June 1983), pp.21–2. DOI: 10.1007/bf03023621
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Plutarch. ‘Marcellus’. In: Lives, vol.V. Loeb Classical Library, no.87. Cambridge, MA: Harvard University Press, 1917. ISBN: 978-0-674-99097-5. DOI: 10.4159/DLCL.plutarch-lives_marcellus.1917 Stephanus p.307; § XVII.7.
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I. Schneider. Archimedes: Ingenieur, Naturwissenschaftler, Mathematiker. 2nd edition. Mathematik im Kontext. Springer Spektrum. ISBN: 978-3-662-47129-6. pp.89-90
Image credits
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Cone/sphere/cylinder from Form & Number, figure 3.9.
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Paolo Barbotti, Cicero discovering the tomb of Archimedes. [Public domain]
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