Renowned Indian astronomer and mathematician Nīlakaṇṭha Somayājī is well known for his innate ability to provide ingenious proofs. In his elaborate commentary on Āryabhaṭīya called Āryabhaṭīyabhāṣya, we find elegant upapattis or rationales for three algebraic identities involved in calculating cubes and cube-roots. In this paper, we detail these upapattis which may be called dissection proofs in the modern parlance. Incidentally, Nīlakaṇṭha’s simple, yet concise and convincing demonstrations are pertinent in the context of mathematics pedagogy as well.

中文翻译：

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Nīlakaṇṭha 的 Āryabhaṭīyabhāṣya 中代数恒等式的优雅剖析证明

著名的印度天文学家和数学家 Nīlakaṇṭha Somayājī 以其与生俱来的提供巧妙证明的能力而闻名。在他对Āryabhaṭīya称为Āryabhaṭīyabhāṣya 的详尽评论中，我们找到了优雅的upapattis或基本原理，用于计算立方体和立方根的三个代数恒等式。在本文中，我们详细介绍了这些upapattis，在现代术语中可以称为解剖证明。顺便说一下，Nīlakaṇṭha 简单但简洁且令人信服的演示也适用于数学教学法。