7 February 2026
Leonardo Pisano (c. 1170–after 1240), dubbed ‘Fibonacci’, thought that Archimedes’ proof that $\pi$ was between $3\frac{10}{71}$ and $3\frac{1}{7}$ was beautiful [pulcra].1
Archimedes’ proof proceeds by calculating approximate ratios of the perimeters of $96$-gons circumscribed about and inscribed in a circle to the diameter of that circle, implicitly starting with dodecagons and repeatedly bisecting edges to obtain $24$-, $48$-, and then $96$-gons (see diagram).2
Fibonacci’s judgement seems to be the earliest extant description of a proof as beautiful in the European tradition. [Al-Nasawī ( fl. 1029–44) had earlier described a proof as beautiful.3]
But Fibonacci had not really understood Archimedes’ proof. Archimedes knew that $256/153 < \sqrt{3} < 151/78$ and used the appropriate bound as an approximation when calculating the perimeters of the circumscribed and inscribed polygons, so he knew that his results would be upper and lower bounds for the actual perimeters. But in his own account of Archimedes’ work, Fibonacci used the same approximation for both and rounded up or down to the nearest number at each stage.
Later, Luca Pacioli (c. 1445–1517 called Archimedes’ result a ‘beautiful and subtle [bella e sotile]’ discovery. He followed Fibonacci’s exposition, but mangled the line of Archimedes’ thought even more than Fibonacci had, ignoring the whole idea of finding bounds.4
But there is a twist in the story, for Fibonacci had not really understood Archimedes’ proof. Archimedes knew that $256/153 < \sqrt{3} < 151/78$ and used the appropriate bound as an approximation when calculating the perimeters of the circumscribed and inscribed polygons, so he knew that he would obtain upper and lower bounds for the actual perimeters. But in his own account of Archimedes’ work, Fibonacci used the same approximation for both and rounded up or down to the nearest number at each stage.
Later, Luca Pacioli (c. 1445–1517) called Archimedes’ result a ‘beautiful and subtle [bella e sotile]’ discovery. He followed Fibonacci’s exposition, but mangled the line of Archimedes’ thought even more than Fibonacci had, ignoring the whole idea of finding bounds.5
Fibonacci. De Practica Geometrie. Ed. by B. Hughes. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, 2008. ISBN: 978-0-387-72930-5. ch. 3, §§ 194–200, pp. 154–8. ↩
E. J. Dijksterhuis. Archimedes. Princeton University Press, 1987. ISBN: 978-0-691-08421-3. pp. 224–9. ↩
A. J. Cain. Form & Number: A History of Mathematical Beauty. Lisbon, 2024. pp. 179–80. ↩
M. Clagett. Archimedes in the Middle Ages, vol. 3, pt 3: The Fate of the Medieval Archimedes: Part III: The Medieval Archimedes in the Renaissance, 1450–1565. Memoirs of the American Philosophical Society, no. 125, pt B. Philadelphia: American Philosophical Society, 1978. ISBN: 978-0-87169-125-5. p. 432. ↩
L. Pacioli. Summa de arithmetica, geometria, proportioni et proportionalita. Toscolano: Paganino Paganini, 1523. DOI: 10.3931/e-rara-9150. pt II (new pagination), fol. 31r. ↩