The Genius and the Myth

He ranks with Archimedes, Euclid, Isaac Newton and Leonard Euler as one of the greatest mathematicians of the world. He began a new epoch in Indian astronomy and mathematics, that continued for more than a millenium. His book Aryabhateeyam is a masterpiece of brevity and eloquence.

But what did Aryabhata actually do? Aryabhata did NOT invent zero; or gravity; or the heliocentric system. As I wrote in my first essay, even Indian mathematics and Sanskrit scholars are stunningly ignorant of Aryabhata’s actual accomplishments. Since we are equally ignorant of almost all of ancient India’s glories, this is not specifically galling; just generally abysmal. Only Bhaskara was perhaps as popular and admired, but unlike Newton’s apple or Watt’s tea kettle, or the anecdotes of Birbal or Tenali Raman, we don’t even have popular legends about him. But we are so creative, we blame the British for this situation, decades after they left.

Ever computed a square root? Aryabhata.
Cube root? Aryabhata.
Summed up a series of numbers? Aryabhata.
Series of squares? Aryabhata.
Divided by a fraction by multiplying by its inverse? Aryabhata.
Computed the areas of triangles, circles, trapeziums? Aryabhata.
Calculated sines? Aryabhata. 

And that’s just the simple mathematics we learn in school.

Wait! Did he invent ALL of these? Ah, that’s the question. Aryabhata himself claims not a single invention. He explicitly states that “by the grace of Brahma, the precious jewel of knowledge (jnana-uttama-ratnam) has been extracted from the sea of true and false knowledge (sat-asat-jnaana-samudraat), by the boat of my intellect (sva-mati-navaa).” As Euclid compiled five centuries of geometrical discoveries of the Greeks, Aryabhata compiled several centuries of mathematical and astronomical discoveries of Indians.

Sulba sutra and Jain mathematicians knew how to compute, square roots, but Aryabhata was the first to describe the algorithm. We don’t know if cube roots were calculated earlier, his algorithm is the oldest extant. His sine calculations are considered much superior to those listed by Varahamihira. His kuttakara algorithm to find solutions is considered ingenious even today.

It is not feasible to explain his mathematical and astronomical discoveries in a magazine article for the general reader. There are excellent translations, technical papers, books that do that. This essay’s purpose is to provoke you to read them, and marvel at Aryabhata’s sva-mati-navaa. And to place Aryabhata and his work in historical context.

Manuja Grantham

The eighteen siddhantas were attributed to rishis. But every jyotisha siddhanta after Aryabhata and Varahamihira, is attributed only to men, not rishis. These arose from commenting, understanding, questioning, correcting, improving existing siddhantas and inventing or discovering new concepts. There was no fear or taboo against criticizing a mere manuja like Aryabhata or Bhaskara, rather than a rishi. This era of Mathematics and Astronomy is called “Classical” by historians. I prefer VarahaMihira’s phrase Manuja Grantha.

मुनिविरचितमिदमिति यच्चिरन्तनं साधु न मनुजग्रथितम् ।
तुल्येऽर्थेऽक्षरभेदादमन्त्रके का विशेषोक्तिः ॥१–३॥ – बृहत्संहिता

muni-viracitam-idam-iti yat-cirantanam saadhu na manuja-grathitam
tulye-arthe-akshara-bhedaad-amantrake ko viSheshokti – BrihatSamhita 1-3

 

Translation This (idam) is muni-uttered (muni-viracitam) so sacred (cirantanam) and good (saadhu). Not (na) so manuja-grathitam (man-composed) it is said (iti). If it is not a mantra (amantraka), and meaning (artha) is equal (tulye) but words different(akshara-bhedaa), what’s wrong (vishesha) with it?

Philosophically, this verse by Varahamihira, is as insightful and expressive as Kalidasa’s verse puraanamityeva na saadhu sarvam(Not everything is excellent, simply because it is ancient). 

Aryabhateeyam

The phrase Kusumapure abhyaarcitam gnaanam (knowledge respected in Kusumapura), in Aryabhateeyam hints that he lived in Kusumapura (Pataliputra or Patna). No biography or portrait of any Indian astronomer exists. The pictures of Aryabhata pervading the internet, as well as his statue, are merely artists’ imaginations. Almost all we know about him comes from his books and those of his critics and commentators, like Brahmagupta and Bhaskara I, who mentions Pandurangasvami, Latadeva and Nishanku, as pupils of Arybhata.

He composed:

(1) Aryabhateeyam in 499AD when he was 23 years old. Multiple copies survive in full form.

(2) Aryabhata Siddhanta, which is lost, and known only by quotations from commentators. In this book, Arybhata advocated midnight as the starting hour of each day, instead of sunrise, perhaps based on Surya or Romaka Siddhanta. Aryabhateeyam uses sunrise as day-beginning.

I confine this essay to Aryabhateeyam. It consists of two parts. The first, Dasha Geetika (Ten Songs), lists astronomical constants:

·        Orbital periods and Diameters of Sun, Moon, Planets

·        Number of years in a yuga, yugas in a kalpa, kalpas in a manu

·        Deviation of planets from the ecliptic

·        Epicycles, in different quadrants

·        Table of Sine differences.

 

His first verse is a salutation to Brahma - he was a scientist, but not an atheist. Almost every jyotisha who followed him begins his work with a salutation to his favorite God. Jain mathematician Mahavira begins with an invocation to his namesake, the tirthankara Vardhamana Mahavira. It may also indicate that he was updating the Paitamaha (Brahma) siddhanta, some of whose data, had become obsolete.

The second part, called AryaAshataShatam (i.e The 108 Arya verses) consits of three chapters – Ganita (Mathematics), Kaala Kriyaa (Calculating Time), and Gola (Sphere – i.e. Celestial, Sphere meaning the visible universe).

The siddhantas of later jyotishas were each nearly a thousand verses long. What Aryabhata summaries in one or two verses is explained by them with whole chapters. So cryptic and compact was Aryabhateeyam, it was impossible to understand without bhashyaas (commentaries); such was its impact, that bhaashyaas were written on it centuries after others improved upon his methods. Telugu Marathi and Malayalam commentaries followed those in Sanskrit, Arabic etc; and English translations in the colonial period, which range in appreciation from astonishment to incredulity to calumny.

1.    Ganita - Mathematics

The mathematics set forth by Aryabhata is mostly practical, not theoretical: its primary purpose is astronomy. I mention only simpler concepts in this essay.

It also varies from extremely simple to extremely complex statements, hypotheses, and algorithms.

We must understand that mathematics was not taught to school children, then as it is today; it was perhaps the most advanced of technical subjects and confined to specialists.  Arithmetic symbols familiar to us like + - x ÷ = were only introduced in fifteenth century Europe. Mathematics was not expressed in equations, but in slokas.

Aryabhata gives two line slokas like this:

त्रिभुजस्य फल शरीरं समदलकोटी भुजार्ध संवर्गः

Tribhujasya phala shareeram samadalakoti bhujaardha samvargaH.

 

Bhuja means Arm. Tribhuja means three-armed or Triangle.

Translation “Multiplication (SamvargaH) of perpendicular(Samadalakoti) and half (ardha) the base(Bhuja) results (phala) in Triangle’s (Tribhuja-sya) area(Shareeram).”

A similar verse(sloka) defines the area of a circle as its half-perimeter (or half-circumference) multiplied by its half-diameter (radius) 

This is a simple algorithm, just a formula really, to calculate one value, based on known parameters. A more complex version is his algorithm for summation of a series, which includes several calculations, including for the mean of the series, and encoding an alternate algorithm! This way of stating multiple mathematical formulae is called muktaka by Bhaskara I.

Kaalakriyaa – Time

Aryabhata divided time and circles  with the same geometric units as earlier siddhantas. His major departure, was to define the four yugapadas namely krta, treta, dvaapara and kali, as of equal time; and as the time it took all the nine planets to align, or complete an integral number of revolutions around the earth. He included a biographical note, that 3600 years passed between the beginning of Kali yuga (end of Mahabharata war) and the twenty-third year of his birth. This implies that the constants in DashaGitike were based on his personal observations in that year.

This differed from the smriti definition of the first three yugapadas as four, three and two times as long as the kaliyuga, and offended the orthodox of everyone. Even his followers didn’t accept this division, but they followed his computations and algorithms, as they were significantly better than those of earlier siddhantas.

Gola – Celestial Sphere

Arybahata states that Solar and Lunar eclipses are shadows of the Moon on Earth and Earth on the Moon, respectively. He also stated that the  Sun is the only source of light, and not just planets, but even the stars only reflect sunlight.

Kadamba flower

Aryabhata used the metaphor of a kadamba-pushpa-grantha,  to explain how people and creatures in all parts of the world believe they are standing on top of the world. He introduced another metaphor, for Earth’s rotation: consider a boat-rider on the Ganga, who feels trees on the shore pass him by; whereas, in reality it is the boat that is moving. Similarly Aryabhata suggested, the earth actually rotates, and like trees on a river bank, the stars seem to revolve around it. But it was only a metaphor, not a proof.

He also explains such concepts as Ascencions of the Zodiac, Sine of Ecliptic etc. which are too technical for this essay.

The impact of Aryabhata was phenomenal. Even fervent critics could not ignore him or his works. But he launched an era of manuja grantham, and he was followed by a long line of brilliant scholars, whom we will discuss next.

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This essay was first published as part of a series in Swarajya
For the entire series click this link --> Indian Astronomy and Mathematics   

References

1.      The Aryabhateeyam by Walter Eugene Clark, University of Chicago, 1930.

2.      Aryabhatiyam, translated by KV Sarma and KC Sukla, Indian National Science Academy, New Delhi, 1976.

3.      Facets of Indian Astronomy, KV Sarma, Madras.

Aryabhata's ghana citi for 2025

Most of us studied not only numbers, but also series of numbers and sums of series in school in mathematics. I am sure most people remember, that the sum of the series from 1 to any number N is given by the formula N*(N+1)/2.

In other words, 1+2+3+4…..+N =  (N * (N+1))  /2

In some school mathematic text books, the name of Carl Friedrich Gauss, the great German genius, is mentioned in association with this series. It seems a German teacher asked his class of nine or ten year olds, what is the sum of the first hundred natural numbers. And Gauss, who was in this class quickly responded, 5050. When asked how we calculated it so quickly, Gauss responded, that he added the smallest and largest numbers 1 and 100, which came to 101; then he added 2 and 99 the second smallest and second largest numbers, which also came to 101; next 3 and 98, then 4 and 97, also each adding to 101. He realized that there were 50 such pairs, each adding to 101, so the sum is 50 times 101 which is 5050. Gauss went on to do amazing things in mathematics, and became one of the greatest mathematical geniuses the world ever saw.

Nice story. Every student and teacher can relate to it. Why don’t we have such stories about Indian mathematicians, except for the famous taxi number 1729 of Ramanujan?

While some formulae in mathematics have names attributed to their inventors or discoverers, there are several mathematical formulae that remain anonymous. All the formula that have names in either physics or mathematics have European or American scientist or mathematician’s names. So, we have Pythagoras theorem, Newton’s formula, Einstein’s formula, Euler identity, and so on. Have you ever wondered why? Why is this Gauss story told without mentioning that Aryabhata gave us this formula for sum of series?

We also learnt that Indians invented zero – we are wrongly told that Aryabhata invented zero. No, Aryabhata did not invent zero. Zero was at least a few hundred years old before Aryabhata was even born. Besides Aryabhata at least Bhaskara is famous as a great mathematician in India. Why do we never learn about some Aryabhata theorem or Bhaskara formula.

Also, even if Aryabhata discovered or invented zero, he must have invented something else also?

Let us discuss one set of things Aryabhata presented, which are given in school textbooks throughout India and the world without mentioning his name. Aryabhata gave not just the formula for the sum of series of numbers, he gave formula for the sum of series of squares and the sum of series of cubes.

In Sanskrit books, the word citi is used for series. Citi  (Sanskrit चिति Tamil சிதி) is literally the word for series of bricks with which a yagna or fire altar for Vedic rituals is made. Aryabhata uses these terms for these formulae

citi: for sum of series of numbers  (1+2+3+4… +N) = N*(N+1)/2

varga citi: for sum of series of squares (1^2 + 2^2 + 3^2+ 4^2 …. + N^2) = N*(N+1)*(2N+1)/6

ghana citi: for sum of series of cubes (1^3 + 2^3 + 3^3+ 4^3 …. + N^3) = (N*(N+1)/2)^2

Varga (वर्ग வர்க) and ghana (घन கன)are the words used in most Indian languages for square and cube. Varga moola and ghana moola are the words used for square root and cube root – incidentally Aryabhata also gave us algorithms to calculate varga moola and ghana moola, but that is a topic for another day.

We learn these formulae in school with Greek notations, invented by European mathematicians in the 18^th and 19^th century like sigma for sum.

Interestingly the sum of the series of the cubes upto 9, that is, 1^3 + 2^3 + 3^3+ 4^3+…9^3 is equal to 2025, which is the Christian year that comes up shortly. I am sure social media will be full of posters and jpegs and gifs and short videos telling you this interesting fact, and perhaps bated breath narrations of Gauss. And zero mention of Aryabhata. So, here is Aryabhata wishing you a happy 2025.

Ironically Aryabhata knew nothing about this Christian calendar adopted in Constantinople and the Roman empire, a few decades before he was born. He used the Kali Yuga notation in his book on astronomy, giving his own year of birth as 23 years before the 3600^th year of the Kali calendar. As 499 AD is Kali year 3600, historians of mathematics believe he was born in 473 AD. In the Kali yuga calendar 2025 is the year 5126 – I am sure enterprising mathematicians will come up with interesting ways to compute this number using Aryabhata’s various formulae.

Related Links

Other essays about Indian Mathematics and Astronomy

My essay in The week magazine on Aryabhata

My essay in Swarajya magazine about Aryabhata

Aryabhata -  CSIR NiScPR Posters