Constructing the Unity of the Cosmic Body in the Timaeus and Archytas's Solution to the Doubling the Cube Problem

Year 2026, Volume: 25 Issue: 1 , 35 - 58 , 29.03.2026

Umut Ayhan

https://doi.org/10.20981/kaygi.1706648

https://izlik.org/JA27GM54TH

Abstract

This paper examines the mathematical model of the Cosmos presented in Plato’s Timaeus, focusing on how geometrical proportions construct the unity of the Cosmos's body. It argues that Plato’s proportions are fundamentally based on magnitudes rather than numbers, a significant point of contention in scholarly interpretations. Engaging with influential scholars such as Heath, Cornford, and Taylor, who contend that Plato's ratios involve numbers due to his Pythagorean influences, this paper challenges these views. It emphasizes Plato's deployment of geometrical concepts, particularly βάθος (depth) in a specific way, which strongly indicates a concern with continuous magnitudes rather than discrete numbers. Crucially, in constructing the body of the Cosmos, Plato appears to allude to Archytas’s solution to the Delian problem. Archytas’s famed solution geometrically determines two mean proportionals between two lines through the intersection of three-dimensional solids. This paper contends that Plato, in his account of the proportional unity of the Cosmos's solid body, draws inspiration from this landmark achievement in stereometry.

Keywords

Archytas , Timaeus , Doubling The Cube , Cosmos , Geometry

Supporting Institution

TÜBİTAK 2219

Evrenin Bedeninin Birliğinin İnşası ve ‘Küpü İki Katına Çıkarma Sorunu’na Arkhutas’ın Çözümü

https://izlik.org/JA27GM54TH

Abstract

Bu makale, Platon’un Timaeus diyalogunda sunulan Kozmos’un matematiksel modelini, geometrik oranların Kozmos’un bedeninin birliğini nasıl inşa ettiğine odaklanarak incelemektedir. Makale, Platon’un oranlarının sayılardan ziyade temelde büyüklüklere dayandığını savunmaktadır ki bu, akademik yorumlarda önemli bir tartışma noktasıdır. Platon’un oranlarının Pythagorasçı etkiler nedeniyle sayıları içerdiğini öne süren Heath, Cornford ve Taylor gibi etkili akademisyenlerle tartışmaya girerek, bu makale söz konusu görüşlere meydan okumaktadır. Platon’un geometrik kavramları, özellikle de ayrık sayılardan ziyade sürekli büyüklüklerle ilgili bir kaygıya güçlü bir şekilde işaret eden βάθος (derinlik) kavramını kullanımını vurgulamaktadır. Daha da önemlisi, Kozmos’un bedenini inşa ederken Platon’un, Archytas’ın Delos problemine getirdiği çözüme atıfta bulunduğu görülmektedir. Archytas’ın meşhur çözümü, üç boyutlu katı cisimlerin kesişimi yoluyla iki doğru arasında iki orta orantılıyı geometrik olarak belirler. Bu makale, Platon’un Kozmos’un katı bedeninin orantısal birliğine dair anlatımında, stereometrideki (katı cisimler geometrisi) bu dönüm noktası niteliğindeki başarıdan ilham aldığını öne sürmektedir.

Keywords

Arkhutas , Timaios , Küpü İki Katına Çıkarma , Kozmos , Geometri

Supporting Institution

TÜBİTAK 2219

References

• Antonietta, S. M. (2019). The Two Supreme Principles of Plato’s Cosmos, Symmetry, 11, 98, (2019), doi: https://doi.org/10.3390/sym11010098
• Brisson L. (2018). Plato’s Natural Philosophy and Metaphysics, A Companion to Ancient Philosophy (ed. Sean D. Kirkland & Eric Sanday). Illinois: Northwestern University Press.
• Crombie, I. M. (2013). An Examination of Plato’s Doctrines II: Plato on Knowledge and Reality, New York: Routledge.
• Cornford, F. M. (1997). Plato’s Cosmology: The Timaeus of Plato, Indianapolis: Hackett Publishing Company.
• Euclid. (1956). Euclid's Elements, (translated by Sir Thomas Little Heath, New York: Dover.
• Grattan-Guinness, I. (1996). Numbers, Magnitudes, Ratios, and Proportions in Euclid’s Elements, Historia Mathematica 23, (1996), doi: https://doi.org/10.1006/hmat.1996.0038.
• Heath, T., Sir. (1921). A History of Greek Mathematics Volume 1: From Thales to Euclid, Oxford: Clarendon Press.
• Heath, T., Sir. (1908). The Thirteen Books of The Elements (of Euclid), vol 2, Cambridge: Cambridge University Press.
• Huffman, C. A. (2005). Archytas of Tarentum: Pythagorean, Philosopher and Mathematician King, UK: Cambridge University Press.
• Kahn, C. (2013). Plato and the Post-Socratic Dialogue, The Return to The Philosophy of Nature, UK: Cambridge University Press.
• Knorr, W. R. (1975). The Evolution of the Euclidean Elements, Holland:D. Reidel Publishing.
• Knorr, W. R. (1993). The Ancient Tradition of Geometric Problems, New York: Dover Publications.
• Kouremenos, T. (2011). The Tradition of the Delian problem and its origins in the Platonic corpus, Trends in Classics, vol. 3, no. 2, doi: https://doi.org/10.1515/tcs.2011.015
• Netz, R. (2022). A New History of Greek Mathematics, UK: Cambridge University Press.
• Netz, R. (2010). The Works of Archimedes volume 1: The Books on the Sphere and the Cylinder, New York: Cambridge University Press.
• Novak, J. A. (1982). Plato and Irrationals, Apeiron, Vol 16 iss 2, doi: https://doi.org/10.1515/APEIRON.1982.16.2.71.
• Novak, J. A. (1983). ‘’Plato and Irrationals Part 2’’, Aperion, Vol 17 iss 1, doi: https://doi.org/10.1515/APEIRON.1983.17.1.14.
• Plato. (1925). Plato in Twelve Volumes, Vol. 9, London: Harvard University Press, London, 1925.
• Plato. (1968). Plato in Twelve Volumes, Vols. 10 & 11 London: Harvard University Press.
• Plato. (1969). Plato in Twelve Volumes, Vol. 5&6, London: Harvard University Press.
• Plutarch. (1878a). Plutarch’s Morals Vol. 2, Boston:Little, Brown and Company.
• Plutarch. (1878b). Plutarch’s Morals Vol. 4, Boston: Little, Brown and Company, Boston.
• Popper, K. (1952). The Nature of Philosophical Problems and Their Root in Science, The British Journal for the Philosophy of Science, Vol. 3 No. 10.
• Pritchard, P. (1992). The Meaning of Δυναμις at ‘’Timaeus’’ 31c, Phronesis, Vol 35, No. 2.
• Prior W. J. (2013). Unity and Development in Plato’s Metaphysics, New York: Routledge.
• Taylor, A. E. (1926). Forms and Numbers: A Study in Platonic Metaphysics (I.) Mind, Vol 35, no 140, 1926
• Taylor, A. E. (1928). A Commentary on Plato’s Timaeus, Oxford: Clarendon.
• Theon of Smyrna. (1978). Mathematics Useful For Understanding Plato, (translated by Robert and Deborah Lawlor), San Diago: Wizards Bookshelf.
• Wagner, R. & Netz, N. Between music and geometry: a proposal for the early intended application of Euclid Elements Book X, British Journal for the History of Mathematics, 38(2), (2023),doi: https://doi.org/10.1080/26375451.2023.2197351.
• Walter, B. (1972). Lore and Science in Ancient Pythagoreanism, (translated by Edmin L. Minar Jr), Cambridge: Harvard University Press.