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Elegant dissection proofs for algebraic identities in Nīlakaṇṭha’s Āryabhaṭīyabhāṣya

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Abstract

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.

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  • 30 December 2021

    Due to a font problem some mistakes occurred that have been corrected.

Notes

  1. As the adage goes:

    अर्धमात्रालाघवेन पुत्रोत्सवं मन्यन्ते वैयाकरणाः।

  2. In the commentary of the sūtra कृत्यल्युटो बहुलम् (3.3.113) we find the vārtika कृतो बहुलम् पादहारकाद्यर्थम्।

  3. Some of the verses are in Gīti meter which essentially comes under the Āryā class of moric meters.

  4. This is not to undermine the significance of the commentary of Bhāskara I (7th cent.), which is also an extremely important and elaborate commentary. What may be worth noting is the fact that the nature, style and emphasis of the two commentaries widely vary from one another.

  5. The following statement of Nīlakaṇṭha appears in his commentary on verse 26 of the Gaṇita section (Āryabhaṭīya of Āryabhaṭācārya 1930, p. 156):

    ... मयाद्य प्रवयसा ज्ञाता युक्तीः प्रतिपादयितुं भास्करादिभिः अन्यथा व्याख्यातानां कर्माण्यपि प्रतिपादयितुं यथाकथञ्चिदेव व्याख्यानमारब्धम्।

    ... somehow, I have started the commentary today at my ripe age, in order to present the rationales that have been understood by me, and also to describe the procedures explained differently by Bhāskara, etc.

  6. This is as per the sūtra of Pāṇini:

    प्रैषातिसर्गप्राप्तकालेषु कृत्याश्च (3.3.163).

  7. Here the word ghane is to be understood as ghanākhye gaṇitakarmaṇi (in the mathematical procedure for determining cubes).

  8. वायुकोणे इत्यर्थः।

  9. In the only edition of the text that is currently available, the first two words have been clubbed together and printed as षडेतेनैव। Such a reading could thoroughly confuse the readers as they may be tempted to split the word षट्+एतेन+एव, which would lead to completely different meaning that does not make any sense in the present context.

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Acknowledgements

One of the authors Dr. K. Mahesh would like to place on record his sincere gratitude to the project on History of Mathematics in India (HoMI), IIT Gandhinagar, which generously funded this study. The other authors would like to acknowledge MHRD for the generous support extended to them to carry out research activities on Indian science and technology by way of initiating the Science and Heritage Initiative (SandHI) at IIT Bombay. The authors are also thankful for the valuable inputs from the referee of this journal.

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  1. Cell for Indian Science and Technology in Sanskrit, Indian Institute of Technology Bombay, Mumbai, India

    K. Mahesh, D. G. Sooryanarayan & K. Ramasubramanian

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  1. K. Mahesh
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  2. D. G. Sooryanarayan
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  3. K. Ramasubramanian
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Mahesh, K., Sooryanarayan, D.G. & Ramasubramanian, K. Elegant dissection proofs for algebraic identities in Nīlakaṇṭha’s Āryabhaṭīyabhāṣya. Indian J Hist. Sci. 56, 71–84 (2021). https://doi.org/10.1007/s43539-021-00017-x

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