Euclid and Geometry

The Bakhshali Manuscript circa 300 AD, containing algorithms for computing square roots etc.

Did you know? In Bengal we worship that form of the Mother Goddess, where she was born as the daughter of a mathematician. Mā Durgā as Kātyāyanī was the child of Kātyāyana, a pre-Euclid era geometer and linguist. He is one of the most notable contributors to the collection of Sulbasutras, or earliest extant geometry texts in the world.

    To clarify: Euclid (circa 300 BC) was an Alexandrian mathematician commonly misnomed ‘Father of Geometry’. His step-by-step approach forms the basis of how Geometry is introduced in schools to students as the first course in mathematics and forms a prominent evaluation aspect of the Math Olympiad exams. In India, we have Geometry texts called ‘Śulbasūtras’ written well before Euclid, at least as early as a millennium Before Christ. While Kātyāyana was blessed with an incarnation of the Mother Goddess as a daughter, a temple dedicated to Swami Baudhāyana (another important author of Śulbasūtras) exists at his birthplace somewhere in the state of Bihar (India).

    In fact, Śulbasūtras form quite a superset of Euclidean knowledge. After the fall of Alexandria after Arab invasions, geometers and number theorists are said to have migrated with their cults in Byzantium and India, and the second wave of Indian mathematics flourished in the stable political scenario post Greek and Saka invasions. Brahmagupta composed his advanced geometry texts around that time, which definitely had some Euclidean touch.

    Essentially, as Cāṇakya compiled all existing knowledge of Economics and SrīKrsṇa of Theology, Euclid was a compiler— as were perhaps many other working mathematicians (compare with Dummit & Foote's treatise on Abstract Algebra or Melvyn Nathanson's texts on Additive Number Theory). Then, is Euclid of no importance? Well, claiming that ‘everything is made in India’ is equally problematic as ‘the Mughals gave India national structure’.

    It's not impossible, while extremely unlikely, for one single person to build up a science so strong and concrete like Geometry, all by himself and out of nothing. So, while we didn't ``import'' Geometry from foreigners, it doesn't also make Euclid's mathematical contributions any less; and I believe quite firmly that Brahmagupta was enriched by Euclid's texts. Ample evidence exists about Euclidean geometry in practice elsewhere long before Euclid. Pyramids of Egypt are one such contrapositive example of engineering wonder. Come to Pythagoras, who was himself appalled by the theorem named after him implying the irrationality of √2— to the extent that he gave up mathematics altogether. A clay tablet list of Pythagorean triplets from Mesopotamia (Iraq) suggests that they must’ve had the concrete formulation of them; else, one can’t just ‘come across’ Pythagorean triplets of the order of 10⁵. The solar doorframes of the Stonehenge have definitely been constructed with very precise calculation of angles. And, we have Śulbasūtras.

    While Geometry itself was amply in practice in India and elsewhere long before Euclid wrote his textbooks, Euclid pioneered something else which an English mathematician championed in the 20th century. It won’t take an actual practicing mathematician much cranial effort to make out what Euclid is important for. He is not the ‘Father of Geometry’; he is the ‘Father of Mathematical Rigour’.* And Indians might feel just a little elated on hearing that the man who had placed Euclidean rigour on firm grounds of permanent validity, Prof. G. H. Hardy, was awestruck in the face of the divine ‘intuitions’ of an Indian mathematician; while Ramanujan claimed that his simple unexplained scribblings of the most lofty mathematical results had been prompted by the Indian Goddess Namagiri from his hometown’s Temple, the atheist Hardy hadn’t a more credible explanation.

You can further read:

1. Kalika Purana

2. Śulbasūtras

3. Euclid's Elements

4. Simon Singh, Fermat's Last Theorem

5. Robert Kanigel, The Man Who Knew Infinity

6. Brahma-sphuta-siddhanta

7. Dr. Manjul Bhargava's lectures

[*The distinction between Euclidean Geometry and Euclidean Rigour first struck me during a lecture by Prof. P. Mukhopadhyay at RKMRC.]