Citrabhānu’s cubes

The Hindus unlike their yavana cousins preferred algebra to geometry. Yet on occasions they could indulge in geometric games for demostrating proofs of algebraic relations. We see a bit of this in the Āryabhaṭa school and the great Bhāskara-II, but one of the masters of the self-evident geometric demonstration was Citrabhānu the nambūtiri brāhmaṇa poet, mathematician and astronomer from the cera country (1500s). His student (also that of one of India’s greatest pre-modern scientists Nīlakaṇṭha somayājin) was the ritual assistant Śaṃkara Vāriyār. In his Kriyakramakārī he records some of these self-evident proofs of his teacher. Now one such question Citrabhānu poses is to show without any algebra that:

\dfrac{a^3-b^3-(a-b)^3}{3(a-b)}=ab

Of course, this is simple 7th class algebra in our old school system. My father probably taught it to my while I was in 4th class. Yet it is some fun to show without using any algebra at all.

chitrabhanu

Citrabhānu has several such ‘wordless’ proofs that delve into figurate numbers. Of course this was not a late medieval innovation suddenly dawning on Citrabhānu. It has a long Hindu tradition going back to the Āryabhaṭa school and from there to figurate numbers of the Vedic ritual altar. Thus, it is another example of the continuity of Hindu science with the Vedic tradition of which Citrabhānu, like his coethnics, was a major practitioner. Finally, we may note that Citrabhānu seems to have had a particularly fertile abstract conception of higher dimensional space in the context of figurate numbers and what we show above is only the simplest of algebra.

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