At least reformists should be bold enough to abandon Mullerian dates like '600BC' for Apastamba. He was one of the-
-Fathers of Geometry.'SulbaSutra' or 'rules of chords' is the oldest name for geometry
-1st to work with irrational nos
-1st to state Pythagorean theorem with proof
The sutras, though not devoted to pure mathematics, are the world's 1st treatises on it.They contain the seminal discoveries not just of Geometry, but also of Algebra & maybe even the place value. Their deserved recognition in the history of Math is denied
The Sulba Sutras were so inconvenient for the Mullerian gang trying to undermine Vedic culture and make it secondary to Mesopotamia and Greece, they forced so many arbitrary dates on so many CRITICAL texts. And their successors still get away with such gross incompetence.
It is very remarkable that such an archaic work would try to deal with irrational numbers. It was only much later that Aryabhatta would recognize 'irrationality'. Many different approaches are found in the different sutras to obtain pi in a wide range of values 2.99 to 3.2022
It is interesting that humans first stumbled upon irrational numbers when trying to construct a Vedic fire altar. The root of 2 was one that challenged the Vedic Purohits/mathematicians. In the sutras the following solution is found,giving a value 1.412156.. correct to 5 decimals
Though the equation may be used to argue that Apastamba & Manava thought root 2 was an exact number,and didn't know irrationality, they were interested in practical use, not the stone hard modern approach. Many later Indian texts clearly state these cannot be exactly determined
The Pythagorean(or Baudhayana to be truthful) triples obtained in the Sulba sutras are-
5, 12, 13
8, 15, 17
12, 16, 20
12, 35, 37
Contribution to Algebra- The Quadratic equations make their first appearance in literature in the sutras of Baudhayana, Manava and Apastamba- the first men with whose names the history of Mathematics SHOULD start.
The forms ax2 = c and ax2 + bx = c are seen.
Another beautiful achievement of these late Vedic minds is having formulated probably the world's OLDEST formula for calculating square roots. Here, for calculating the irrational root of 10, a is taken as 9 and r is 1. You get 3.162278, correct to 6 decimal places.
There is nothing wrong if the west wishes to call Galileo the 'Father of modern Physics', but refusing to acknowledge that his work is not entirely original and had heavily drawn on the manuscripts of Aryabhata, Bhaskara etc is unacceptably Eurocentric.
Already stated a 100 times, but it's implications cannot be allowed to go unrecognized:- Europe was digesting the manuscripts of the Indian Trio- Aryabhatta, Brahmagupta & Bhaskara for close to 500 years before Galileo threw things down the tower of Pisa & apples fell on Newton.
The Aryabhatiya contains explicit affirmations of the relativity of motion. Aryabhatta was not only fully aware of the concept, he also worked on some of the fledgeling "equations" of modern physics such as relative velocity(gati). Shouldn't his ideas be compared with Galileo's?
Just like Tycho Brahe stole the erroneous planetary model of Neelakanta, Galileo chose to use the exact same analogy used by Aryabhatta to describe relativity- a thousand years later.
Verses 9 and 10 of Aryabhatiya's astronomy section describe it.
"Similar to a person in a boat moving forward who sees the stationary objects
on the bank of the river as moving backwards, the stationary stars at Lanka (equator) are viewed as moving westwards"
Aryabhatta wonderfully describes two things here.
He brings to account the earth's rotation in the same verse. The boat has an independent uniform motion, while the movement of the stars is rotational. People on a boat and all people on earth witness the same principle!
Aryabhatta discusses the opposite experiences of observers in the north and south poles of the earth, variance between latitude and time and even hints the force(Gurutva) which prevents people from falling off the earth- something explicitly done before him in the Surya Siddhanta
Verse 31 is the cherry on the cake. One must remember that this is the 5th century AD(possibly older), when Europe(and most of the world) was contributing nothing just like India after Mohammadan invasions. Imagine the light the Aryabhatiya must have been to Arabs & Europeans
"Divide the distance between the 2 bodies moving in the opposite
directions by the sum of their speeds, and the distance between the two bodies moving in the same direction by the difference; this will give the time
elapsed since the two met or before they will meet"
A. Parakh and Roopa H Narayan drew attention to the seminal ideas of physics contained in the Aryabhatiya and their probable influence on later day Europe.
How did the Aryabhatiya and its path breaking algebra, trigonometry and astronomy enter the Arab and western world?
According to the Tabqatul-Umam written by Said al-Andalusi(1029–1070AD) a scientific delegation led by an Indian astronomer Kanaka visited Baghdad itself!
He carried only a small collection- but it contained the most catalytic of all combinations. The most advanced manuscripts on earth then. He gave this priceless trilogy to the Caliph of Islam.
-Surya siddhānta(Author unknown)
-Brāhmasphuṭasiddhānta(Of Brahmagupta)
But can we call him a traitor? Sharing knowledge freely was in the nature of the Vedic culture he came from. Never in his wildest dreams would he have imagined that the Arabs and the Latins would scavenge on it, plagiarize it & later put a racial identity & a price tag on science
Arab historian Al-Qifti(~1200AD) reports that the Caliph(or more likely his Persian astronomers) was so amazed that ordered them to be translated to Arabic. Al-Fazari did that job. And he became the first 'Arab astronomer' of renown. His famous work is Zīj al-Sindhind(siddhānta).
The sindhind spread all over the muslim world, incl Spain. In the library of Toledo was the great repository of centuries of manuscripts, translations and commentaries. By 1100AD, all of it fell into the hands of the church. Mass Latin translations and appropriation began.
The Indian astronomical tables had been compiled by the astronomers at Toledo and was known as the 'Toledo tables' which accurately predicted motions of celestial bodies. By 1270, Alfonso of Castile turned them into 'Alfonsine tables'. Vedic origins vanished there-and remains so.
Sanskrit jyā & koti-jyā became sine & cosine.
Trigonometry was attributed to Greeky Ptolemy
Algebra was bestowed on Diophantus
Geometry was blessed on the holy Euclid of Alexandria
Based on what?Primary manuscripts?At least a copy?Or EVEN 1 primary reference to any of them?
One must admit, the Arab/Persian scholars were far more intellectually honest than the Nicky Copernican & Christo Clavian gangs.
Aryabhatta is mentioned by name as 'Al-Arjhabhar' in multiple Arab texts by multiple authors
-Al Ahwazi & Abu Mashar~800AD
-Al Hashmi~850
-Al Biruni
Kanaka led the scientific delegation to Baghdad in the year 771 AD, a hundred years after Brahmagupta died. That was the year the baton of reason was given to foreigners.
In a way it was good. Indian mahaviharas & institutions were going to be destroyed in the next few centuries.
But that does not mean the story was over in India. Part 2 was beginning.The Arabs and Latins were not capable of advancing science to the next level- The analytical level of calculus. Because they simply did not have the cultural/philosophical sophistication to conceive of it.
They used the numerals, place value and the basic math for trade, navigation & astronomy. The clumsy Roman numerals were wiped out of use, though tally sticks had to be forcibly burned even in 18th CE. A commendable achievement was the standardisation of heliocentrism by Keplar.
While the Church scholars of the 12th-13th CE were busy translating and manufacturing manuscripts to attribute everything to Greeks, in India, the Trilogy had already reached the South, where Bhaskara and the Kerala mathematicians would painstakingly expand them & invent calculus
One wonders-If archaeological evidence for such math & technology in India is found only in the Harappan era and if literary attestation is found only in Vedic SulbaSutras-What is difficult in accepting they both represent the same culture?
Baudhayana & Apasthamba were Harappans!
Lets also remember on his birthday..
Who was Leonhard Euler?
He- like Newton, Gregory etc- 'independently rediscovered' a 1000 year old Indian mathematical method. This time from Aryabhatta himself. The "Euler's method" for ordinary differential equations
In Ganita 12 , Aryabhatta mentions the use of the numerical method of sine differences, though in a cursory verse. It was elaborated further by Brahmagupta.

How do we know Euler was studying Indian manuscripts?
He published an elaborate article on the Indian calender in 1700.
Correction-The article was published in the 1730s. Its amazing that despite obvious evidences of circumstance, precedence, motive, access & even admission, these mass appropriations are still called "Independent rediscoveries"
From 'Euler: Genius in the Enlightenment' by Ronald S
Read this thread to understand how Vedic trigonometry was imported to Europe by Jesuit missionaries. Europeans did not have the mathematical wherewithal for it. Their numerals simply couldn't handle such fractions!
How Europe learned celestial navigation-
This #PiApproximationDay let's remember the forgotten Vedic story of Pi

Yajnavalkya~2000BCE- 3.125
Baudhayana~1800BCE- 3.088
Manava~1500BCE- 3.1604
Aryabhata-499AD- 3.1416
Madhava-1370AD- 3.1415926535897
Neelakantha- 1480AD- 'Folks! Pi is irrational! 🤣'
He gets even bolder/dumber and adds-

"...whilst there is a complete absence of any recorded mention of the Kerala School..."

In other words, he implies that plagiarism can be established only if the plagiarists self damningly gives attribution to their true sources.

Just wow.
How do we know that the Portuguese/Jesuit missionaries were amassing manuscripts from Kerala?

1.The marauder Vasco da Gama, ignorant of Vedic celestial navigation, used the services of Indian navigator Kanha to sail from Kenya to Goa

Yet Vaso 'discovered' the sea route to India
In his diary Vasco records that he took the instruments used by Kanha to graduate it to inches

The 'rapalagai' of clear south Indian make was a wooden board with a graduated string to measure the pole star altitude & hence the latitude

No wonder he returned to terrorize kerala
That was the very first contact of Europe with advanced celestial navigation, a turning point in history.

It can truly be called the beginning of full blown colonialism.

It was also the very first event in the century long story of plagiarism of Calculus from Kerala.
The Portuguese were defeated and expelled by the Zamorin, the Maharaja of Calicut. But they were welcomed by the raja of Kochi, his rival.

The kochiraja can rightfully be called the ManSingh/Jaichand of South India. His actions eventually plunged Kerala into colonial disaster.
The Kochiraja patronized Portuguese merchants & allowed Jesuit missionaries to build their institutions- including the famous Cochin college which was used SPECIFICALLY to translate malayalam & sanskrit texts to Latin

Sadly, Madhava & his disciples lived in the Kingdom of Kochi.
Even worse! The Kerala school Mathematician of the time- Sankara Varier(1500-1560) served at the Kochiraja's court.

He authored Yuktidipika- the clearest manuscript of Calculus at the time. He also possessed ALL THE WORKS of his predecessors- Madhava, Jyesthadeva & Nilakanta.
You'd think it can't get any worse

Christopher Clavius, under orders of Pope Gregory, sent one of his best trained Jesuit mathematicians to Kochi. His was Matteo Ricci

Clavius & Ricci had just learned the(rudimentary) decimal system & was struggling to learn Indian trigonometry
Dear reader, let that sink in.

The most intelligent man on earth at that time- Sankara Variar-was serving a king who was literally enthralled to the Portuguese

At a time when the Jesuits were manuscript hunting to learn Indian trigonometry just to reform the Julian calendar.
2. Coming back, let us give @thewire_in the 2nd DOCUMENTED evidence of European contact with KSOM.

"I am trying to learn the methods of reckoning time from an intelligent Brahmin."
-Matteo Ricci wrote to Petri Maffei in Dec 1581

That Brahmin was probably a disciple of Sankara
The original words in Portuguese-

“Com tudo não me parece que sera impossivel saberse, mas has de ser por via
d`algum mouro honorado ou brahmane muito intelligente que saiba as cronicas dos tiempos, dos quais eu procurarei saber tudo”

Reproduced in Documenta Indica, XII (p 474)
Now we can TRULY appreciate how UTTERLY wrong the article's 'sequential development' statement is

Clavius whose lifelong work was decimal system & Indian trigonometric tables died in 1612. Newton was born in 1642

From decimals to calculus within a century!

WHAT sequence?!!
The 3rd documentary evidence of Jesuit translation activities

Meet Roberto de Nobili. Fluent in Sanskrit & Tamil. So confident of his Sanskrit, he wrote a nasty criticism on Vedanga Jyotisha in 1608 CE

This genius chose the oldest & thus the simplest astronomical text to target
The Vedanga Jyotisha(~1370 BCE)was already declared outdated by Varahamihira 1000 yrs before Nobili

Obviously neither Nobili nor his master Clavius was capable of digesting the latest, most advanced Indian texts-Yuktibhasa & Yuktidipika

That required the services of Newton & Co
Let us look at individual instances of plagiarism, sticking to chronology as far as possible.

To realize how ludicrous the 'rediscovery' claim is in the face of such massive evidence, we shall 1st discuss the Indian mathematician, his discovery & its primacy, then the Euro claim
The Pingala series
Every third number is the sum of the first two. The Indian mathematicians who developed the series are-

Pingala-~400 BC (Chandaḥśāstra)
Virahanka~700AD( work is lost)
Gopāla (c.1135)
Hemachandra (c.1150)

Now known after Leonardo of Piza or Fibonacci.
'Meruprastara'- Triangle of binomial coefficients, an extention of pingala series

Mahavira-850AD- Multiplication formula for binomial coefficients
Utpala~1000AD-Equated additive & Multiplicative formulas

Now known as 'Pascals triangle'
Now this might seem too simple & boring. But I start with the pingala series for a specific reason.

Who was Fibonacci?(c.1175 –1250)
He was a first generation RECIPIENT of Latin translations of Arab works from Toledo library(described above), including Al Khwarizmi's.
Fibonacci wrote the 'Liber abbaci' in which he introduced the Indian numerals, use of zero , place value system and explained the superiority of 'Modus Indorum' to the clumsy Roman numerals. It widely circulated.

He was the first and last European to give credit to Indians.
What is clear is that he was dependant on the flood of translated latin works flowing out of Toledo library.

The greatest of the 12th-century translators of Toledo was Gerard of Cremona (1114-87)

Fibonacci used Gerard's translation of the 'Algebra' of al-Khwarizmi
Now, one by one, formula by formula, name by name, we shall appreciate the endless list of intellectual thefts that went on unabated for the next 600 years after Fibonacci.

To be continued.
1. Algebra-The true history

We have seen how the following forms is studied in the sulba sutras(2nd millenium BCE)
ax²+ bx+ c= 0
ax²= c

The Jain mathematicians continued the work with cubic & quartic equations(from 300BCE)

The next great leap came with the Bakhshali manuscript
The Bakhshali is an extremely important evidence devastating to Eurocentric history. Many tried to push it's date to the 6-8th century- where it is harmless to Europe.

But C14 dating has given us AD 224–383 for its oldest pages

A VERY dangerous century for an Algebraic text
Some of its results:-

Solution to the quadratic equation(At last!)
Sol of linear equations with upto 5 unknowns
Quadratic indeterminate eq (ax/c = y)
Simultaneous eq

G.R Kaye, the terrified translator, tried to date it to the 12th CE- to prove Greek/Arab influence!
What makes the Bakhshali manuscript truly troublesome is that it appears to be a COMMENTARY on an OLDER text which is lost. Algebra was already developing at full pace by 100-200 AD or earlier. It is an irrefutable documented evidence- a kind which Europe does NOT have for Greeks
What did Daniel Bernouli(1700-1782 CE) plagiarize from Narayana Pandit(1340-1400 CE) indirectly through Jesuit translations of Kerala manuscripts?

What did 17th century European Mathematicians plagiarize from Bhaskara the IInd(1114-1185 CE)?
What did 17th century Europe plagiarize from Madhava of Sangamagrama(1340-1425 CE) ?
What did 17th century Europe plagiarize from Nilakanta Somayaji(1444~1544 CE)?
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