The Delian Problem, its Two Pre-Euclidean Solutions and Plutarch’s Assessment

Year 2025, Issue: 63, 81 - 99, 07.01.2026

Umut Ayhan

https://doi.org/10.26650/arcp.1618657

https://izlik.org/JA98DC37TC

Abstract

The Delian Problem—doubling a cube's volume—was a significant challenge in preEuclidean mathematics, necessitating the geometric construction of the cube root of 2. This paper analyzes two pivotal solutions: Archytas's method using intersecting 3D solids and Menaechmus's use of conic sections. Plutarch later claimed Plato condemned these solutions as "mechanical," arguing they undermined pure mathematical inquiry. This study reevaluates Plutarch's assertion by examining Plato's dialogues, including the Republic, Sophist, and Timaeus. It argues for a more nuanced interpretation of Plato's stance, suggesting that while he valued mathematical purity, he likely recognized the utility of diagrams and geometric constructions as tools for accessing Ideal Forms. The solutions by Archytas and Menaechmus, though potentially imperfect by Platonic standards, could thus have been seen as valuable intellectual advancements. The paper posits that Plutarch’s account may not fully capture Plato’s intricate and possibly evolving perspective on these innovative mathematical methods. Instead of outright rejection, Plato might have viewed these solutions as important, albeit incomplete, steps in the ongoing pursuit of mathematical understanding, reflecting both his philosophical ideals and his appreciation for ingenuity in geometry.

Keywords

Delios , Plato , Archytas , Menaechmus , Plutarch

Delos Sorunu’nun Öklid Öncesi Döneme Ait İki Çözümü ve Plutarkhos’un Değerlendirmesi

https://izlik.org/JA98DC37TC

Abstract

Delos sorunu, yani bir küpün hacminin iki katına çıkarılması, Öklid öncesi dönemin önemli bir matematik problemidir ve 2 sayısının küp kökünün geometrik inşasını gerektirir. Bu çalışma, Arkhutas’ın üç boyutlu cisimlerin kesişimini kullanan ve Menaihmos’un konik kesitlere başvuran çözümlerini inceler. Bu çözümlerin soyut yapısına karşın, Plutarkhos, Platon’un bu yöntemleri “mekanik” bularak eleştirdiğini iddia etmiştir. Bu çalışma, Plutarkhos’un bu iddiasının güvenilirliğini, Platon’un Devlet, Sofist ve Timaios gibi temel diyalogları çerçevesinde yeniden değerlendirmektedir. Platon’un matematiksel bilginin ideal saflığını ön planda tutarken, geometrik inşaların ve diyagramların Platonik İdealara ulaşmada araçsal bir rol oynayabileceğini kabul etmiş olabileceği savunulmaktadır. Bu bağlamda, Arkhutas ve Menaihmos’un çözümleri, Platon için mutlak mükemmellikte olmasalar da entelektüel ilerleme yolunda değerli adımlar olarak görülebilir. Sonuç olarak, Plutarkhos’un aktarımlarının, Platon’un bu yenilikçi geometrik yöntemlere dair karmaşık ve muhtemelen zamanla değişen bakış açısını tam olarak yansıtmadığı; Platon'un bu çözümleri toptan reddetmek yerine, onları matematiksel anlayışın gelişiminde önemli, ancak ideal hedefe ulaşmada belki de ara basamaklar olarak değerlendirmiş olabileceği ileri sürülmektedir.

Keywords

Delos , Platon , Arkhutas , Menaihmos , Plutarkhos

References

• Adam, James. The Republic of Plato II. Cambridge. Cambridge University Press. google scholar
• Burnyeat, M. F. “Platonism and Mathematics, a Prelude to Discussion.’’ Mathematics and Metaphysics in Aristotle, Derleyen: Andreas Graeser, Swirzerland: Paul Haupt Berne, 1987. google scholar
• Euclid. The Thirteen Books of The Elements Vol 2. Çeviren Sir Thomas L. Heath. Cambridge: Cambridge University Press, 1908. google scholar
• Euclid. The Thirteen Books of The Elements Vol 3. Çeviren Sir Thomas L. Heath. Cambridge: Cambridge University Press, 1908. google scholar
• Heath, Thomas. A History of Greek Mathematics Vol 1. UK: Oxford, 1921. google scholar
• Huffman, Carl A. Arkhutas of Tarentum: Pythagorean, Philosopher and Mathematician King. UK: Cambridge, 2005. google scholar
• Robins, Ian. Mathematics and the Conversion of the Mind. Ancient Philosophy 15, 2, (1995). google scholar
• Knorr, W. Richard. The Ancient Tradition of Geometric Problems. New York: Dover Publications, 1993. google scholar
• Knorr, Wilbur. Textual Studies in Ancient and Medieval Geometry. Birkhauser: Boston, 1999. google scholar
• Kouremenos, Theokritos. ‘’The Tradition of the Delian problem and its origins in the Platonic corpus’’ Trends in Classics, vol. 3.2. 2011. google scholar
• Kouremenos, Theokrıtos. "Solıd Geometry, Astronomy And Constructıons In Plato's Republıc" Philologus 148, No. 1 (2004): 34-49. google scholar
• Lunsford, Matt D. “The Delian problem, platonic solids and Finite Fields.’’ Mathematics Magazine. (2019). google scholar
• Little, John B. “A Mathematician Reads Plutarch: Plato's Criticism of Geometers of His Time.” Journal of Humanistic Mathematics, 7, 2 (2017). google scholar
• Masia, Ramon. “A new reading of Arkhutas’ doubling of the cube and its implications.” Archive for History of Exact Sciences 70 (2) (2016). google scholar
• Netz, Reviel. The Works of Archimedes Volume 1: The Two Books on the Sphere and the Cylinder. UK: Cambridge University Press, 2004. google scholar
• Netz, Reviel. A New History of Greek Mathematics. Cambridge University Press, 2022 google scholar
• Reviel Netz. The Shaping of Deduction in Greek Mathematics, A Study in Cognitive History. UK: Cambridge University Press, 1999. google scholar
• Plato. “Republic.’’ Plato in Twelve Volumes, Vols. 6 içinde. Çeviren Paul Shorey. London: Harvard University Press, 1969. google scholar
• Plato, “Euthydemus.’’ Plato in Twelves Volumes, Vol 3 içinde. Çeviren W. R. M. Lamb, Cambridge: Harvard University Press, 1967. google scholar
• Plato. “Philebus.” Plato in Twelves Volumes, Vol 9 içinde, Çeviren. Harold N. Fowler, Cambridge: Harvard University Press, 1921. google scholar
• Plato. “Sophist.’’ Plato in Twelves Volumes. Vol 12 içinde. Çeviren. Harold N. Fowler. Cambridge: Harvard University Press, 1921. google scholar
• Plato. “Timaeus.’’ Plato in Twelves Volumes. Vol 9 içinde. Çeviren. W. R. M. Lamb. Cambridge: Harvard University Press, 1925. google scholar
• Plutarch. Plutarch’s Moralia in Fifteen Volumes Vol. 7. Çeviren Lacy and Einarson. London: Harvard University Press, 1959. google scholar
• Plutarch, Plutarch’s Morals Vol 3. Düzenleyen. William W. Goodwin. Boston: Little, Brown and Company, 1878). google scholar
• Plutarch, Plutarch’s Lives Vol 5. Çeviren Bernadotte Perrin. Cambridge: Harvard University Press, 1961. google scholar
• Riginos, A. S. Platonica: the anecdotes concerning the life and writings of Plato. Leiden: Brill. 1976. google scholar
• Waerden, B. L. Van Der. Science Awakening I. Çeviren Arnold Dresden. Kluwer Academic Pub, 1975. google scholar
• Theon of Smyrna. Mathematics Useful for Understanding Plato. Çeviren Robert and Deborah Lawlor. San Diago: Wizards Bookshelf, 1979. google scholar
• Zhmud, Leonid. “Plato as ‘Architect of Science’.” Phronesis 43, 3 (1998). google scholar