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

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

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

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