Impact of Aryabhata on science and humanity

In the context of my latest novel The Aryabhata Clan, it's doubtless that Aryabhata, the legendary maverick Indian mathematician, would play a significant role. What it's about Aryabhata that's there in the novel may be figured out only after reading the novel and that's what I would request everyone to do. But it might be relevant to talk about Aryabhata in general.

I talked about Aryabhata in the book launch at the New Delhi World Book Fair 2018, on 10th January. Below is the video of the conversation about Aryabhat and its transcript.

Later, there's a small article about Aryabhata highlighting his works and influence.

Transcript

Anubha: Tell us about Aryabhata, the mathematician, and the role that he plays in the book. After all the book is called The Aryabhata Clan. 

Sudipto: What role he plays? if I’ve to tell that, then I’ve to divulge a lot about the climax of the book. I won’t say what role Aryabhata plays in the book. But I can certainly say about Aryabhata [in general]. Aryabhata may be the second global Indian after Gautam Buddha who had such a big impact on the world. Buddha, we all know [about his ideas about peace and nonviolence. He has made a huge mark on the world, socially and politically. We all know about that. We talk about Yoga, and so many things about India. But the impact of Aryabhata is often lost in the narrative. Just to give you a small example: Aryabhata is contemporary to Kalidasa. He lived around 1500 years ago. Thousand years later, when Amir Khusrau was writing the first Hindu poem – he is often credited to have written the first Hindi poem – he also mentioned the impact of Hindustan, and he [implicitly] mentioned about Aryabhata and the impact of the decimal place value system. The entire mathematics is around the place value system. We’ve read since kindergarten – 10, 100, 1000, everything increases of decreases in orders of 10s. The entire concept is so natural to us. Aryabhata didn’t invent the decimal place value system. But he was the first person who popularized it and theorized it in a proper scientific manner. From Aryabhata it went to Arab when a lot of Arabic mathematicians were translating Indian works to Arabic, and from there it went to Europe – it was translated to Latin. Someone commented that if there were no decimal place value system, there wouldn’t have been any Newton too. All the scientific advancements happened 15th century onward – Newton, Laplace, whatever you’ve read in mathematics, trigonometry, calculus, everything was born only after Aryabhata’s place value system had been transported to Europe via Arab. So, from that point of view, I think, the impact of Aryabhata on modern science is immense. Without Aryabhata there wouldn’t be computer, there wouldn’t be Newton, no gravitational laws, we wouldn’t have anything to read in our physics books. Kids would have been happier. They would have lot less to learn. But the world would have been a much different place to live in. 

Not much later than the advent of the Islam came the cultural and scientific renaissance, in the eighth century, in the entire Islamic world. It was spearheaded by the Arabs under the Abbasid Caliphs, heralded by none other than al Mansur himself, the second Caliph of the Abbasid clan. Al Mansur was perhaps the first Caliph to have heard of the stupendous developments made by the Indians in the fields of mathematics and astronomy. He ordered a recent Indian treatise on mathematics – it was perhaps Brahmagupta’s most celebrated work Brahmasphuta Siddhanta – to be translated to Arabic. And that opened the flood gates of Indian wisdom to the Arabs and brought the illustrious al-Arjabad, aka Aryabhata, to them.

Aryabhata’s equivalent of the modern sine - jyardha, half-chord, or just jya in short – was so much a talk of the time that the Arabs picked it up very soon. Jya was transliterated in Arabic to a meaningless jiab, which was later changed to jaib, which is a real word in Arabic. It’s akin to the Persian and Hindi-Urdu jeb, meaning pocket, or a fold in the garment. Years later, in the 12^th century, when Gherardo of Cremona was translating these Arabic texts into Latin, he used the word sinus, which means curve and fold in Latin – thus came about sine to refer to Aryabhata’s half-chord. It may not be a mere coincidence that in Arabic jaib also means bosom, whose “curve” is not much different from that of a sine – Gherardo was doubtless an admirer or curves.

Following is the 12^th verse from Ganitapada, Mathematics, the second chapter of his seminal work, Aryabhatiya, written some 1500 years ago. He gives out the secret sauce for creating the first ever “sine” table in the world.

Prathamat chapa-jya-ardhat yair unam khanditam dvitiya-ardham |

Tat prathama-jya-ardha-amshais tais tair unani sheshani || 2.12, Aryabhatiya

In the above verse, jya-ardha or the half chord is the equivalent of the modern-day sine. But what’s interesting is that the term khandita, meaning segmented, could be very easily deduced to be implicitly referring to the “delta” of calculus, of the dy/dx fame. So jya-ardha is sine = x, and khandita jya-ardha, referred to her as khandita ardha for the sake of brevity, is delta sine or dx. So, Aryabhata was talking about derivatives, though not directly, more than a millennium before Newton and Leibniz. Even more surprisingly, it can be construed from the above verse that not only derivative, Aryabhata was also aware of second derivative and that he knew the second derivative of sine is sine.

Talking about the Lord of the Chords and his usage of the concepts which much later came to be known as Calculus, there’s another equally interesting observation, as interesting as his referring to derivatives as “segments”.

Two verses before the one we’ve discussed earlier – the tenth one of the second chapter of Aryabhatiya – the maverick mathematician said the following:

Chatur-adhikam shatam-ashta-gunam dva-shashtis-tatha sahasranam |

Ayuta-dvaya-vishkambhasya-asanno vritta-parinahah ||

It literally translates to:

Four more (of) hundred, times eight, likewise (more) of sixty-two thousand,

Nears the circle-circumference of diameter for 20000.

[chatur adhikam shatam = four more of hundred = 4 + 100 = 104

ashta gunam = eight times

chatur adhikam shatam ashta gunam = (4 + 100) x 8

dva-shashtih sahasra = sixty-two thousand = 62000

ayuta dva = 2 ten-thousand = 20000

viskambha = diameter

vritta = circle

parinaha = circumference

asanna = near]

Put simplistically, it says 8 x (4 + 100) + 62000 = 62832 is “near” (asanna) to the circumference of a circle with diameter of 20000. Considering the circumference of circle is  x diameter, the verse indirectly says that the value of  “nears” or approaches 62832 / 20000 = 3.1416. In calculus parlance, this is nothing but limit  à 3.1416. The word “asanna” may also imply that Aryabhata might have had some inkling of the irrationality of , something which was proved formally by Laplace some 1300 years later.

In the second verse of Ganitapada or Mathematics, the second chapter of Aryabhatiya, the Lord of the Chords said the following:

Ekam dasha cha shatam cha sahasram ayuta niyute tatha prayutam |

Koti arbudam cha vrindam sthanat sthanam dasha-gunam syat ||

One (eka), ten (dasha) and hundred (shata) and thousand (sahasra), ten thousand (ayuta), hundred thousand (niyuta), likewise million (prayuta),

Ten million (koti), hundred million (arbuda) and billion (vrinda), from place to place is ten times from that (preceding).

This is a formal definition – doubtless the earliest one to have been recorded anywhere in the world – of the decimal place value system. But the system has been in vogue in India for a very long time. The Rig Veda, the first ever book written by mankind, sometime around 1500 BC, has the expression “shashti shata shat sahasra shashtih shad” – sixty hundred, (or) six thousand, (and) sixty-six –, in the 7.18.14^th verse. This keeps no room for any confusion that during that time too, the Indians used the same decimal place value system which Aryabhata formalized in the 5^th century, 2000 years later.

The 17.2^nd verse of the Shukla Yajur Veda, written few centuries later, has the expression, “eka cha dasha cha dasha cha shatam cha shatam cha sahasram cha sahasram cha ayutam cha ayutam cha niyutam cha niyutam cha prayutam cha arbudam cha nyarbudam cha samudram cha madhyam cha antam cha parardham” – one (eka) and ten (dasha), and ten and hundred (shata), and hundred and thousand (sahasra), and thousand and ten thousand (ayuta), and ten thousand and hundred thousand (niyuta), and hundred thousand and million (prayuta) and ten million (arbuda) and hundred million (nyarbuda) and billion (samudra) and ten billion (madhya) and hundred billion (anta) and trillion (parardha). This one, though not meant to be a definition of the place value system, sounds very similar to Aryabhata’s verse.

So, it’s true that Aryabhata didn’t discover, per se, the place value system. Neither did he use the ten Indian numerals in his treatise. But it can be asserted that his effort and influence in formalizing the both were indeed quite significant. Immediately after him, both the things appear very prominently in the works of all mathematicians and astronomers in India, and very soon, among the Arabs, through translations.

Al Khwarizmi, perhaps the most influential among the mathematicians and astronomers of the Islamic renaissance period – the word “algorithm” derives from algorizmi and algoritmi, the Latin corruptions of his name – wrote a book named Kitāb al-jam wa’l-tafrīq bi-hisāb al-Hind, the book (kitab) of Addition (jam) and Subtraction (tafriq) according to Hindu calculation (hisab), in the eighth century. The book exists only in its Latin translations, which took Al Khwarizmi’s descriptions of the Indian (Hindu) numerals and the decimal place value system formalized by Aryabhata to Europe and the rest of the world.

Al Khwarzimi’s most famous book, Al-kitāb al-mukhtaṣar fī hīsāb al-gabr wa’l-muqābala, the small (mukhtasar) book on calculation (hisab) by Completion (al-jabr) and Balancing (wa’l muqabla), considered to be the earliest treatise on Algebra (from al-jabr), mentions the same value of  as given by al-Arjabad, Aryabhata, the Lord of the Chords.

Such was the aura of the Indian numerals and the place value system of Aryabhata, that almost 800 years later, when Amir Khusrow was writing Nuh Sipihr, Nine Skies, he speaks of them as India’s gift to the world – the other gifts he mentioned are the Panchatantra tales and chess.

India has had cultural and trade relations with many countries around the world since many millennia, since the times of the Indus Valley Civilization. Elements of Indian culture and wisdom have transcended the boundaries of the Indian subcontinent and reached shores afar. But Aryabhata was perhaps the first Indian export to the world in such a mammoth scale, in the global sense, per se. It may be matter of conjecture what would have become of mathematics and science in the world without him, and of course without his formalization of the Indian numerals – commonly known as the Hindu-Arabic Numerals – and the place value system. Without either of these, the modern science wouldn’t be the same. It can be argued that had it not been for him, someone else would have been born and done the same. But then, it’s like saying, without Newton, we might have still had the Laws of Gravitation.

The overall importance and influence of Aryabhata is best summed by the French Mathematician Laplace, when he said:

It is India that gave us the ingenious method of expressing all numbers by the means of ten symbols, each symbol receiving a value of position, as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit, but its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions, and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity.