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The Sutra era of ancient Hindu Mathematics

 

The first glimpses of Mathematical activity in the Indian peninsula appear in the Sulva Sutras. The commonly accepted dates for  such a dating are not without controversy. These conventional dates place the sutras, about 800 years before the common era.  Here is one alternative chronology compiled by Lakshmikantham and Leela;

Proposed skeleton of  Indian Chronology

3374 BCE

Event  Individual Date
Birth

Birth

Birth

Birth

Coronation of

Coronation of

Reign of

Coronation of

Pancha Siddhantas, authored by

Reign of

Birth of

Era of

Writings

 

 

Veda Vyaasa

Apasthambha

Baudhayana

Aryabhatta

Gautama Buddha

Chandragupta Maurya

Ashoka Maurya

Kanishka

Chandragupta of Imperial Gupta dynasty

Varahamihira

Vikramaaditya

Brahmagupta

Salivahana

Bhaskara II

Siddhanta Siromani

 

3374BCE

 

3200 BCE

2765 BCE

1888-1807 BCE

1554 BCE

1472 BCE

1294 - 1234 BCE

327 BCE

123 BCE

102 BCE to 78 CE

30 BCE

78 CE 

486 CE

486 CE


The politically correct date of the birth of Aryabhatta is 476 CE. Why the difference of 23 centuries. Apparently this resulted from a mistranslation of a Sanskrit word,where in fact one entire word has been replaced with another.

To quote Lakshmikantham and Leela,  Page 27,Chapter 4, Chronological Aspects

Was Sir William indulging in subterfuge or was he woefully incompetent  that he did not recognize the difference between shasti and shabdi

In the modern period of Bharat (India), Aryabhatta is the first famous mathematician and astronomer. In his book Aryabhatteeyam, Aryabhatta clearly provides his birth data. In the 10th stanza, he says that when 60 x 6 = 360 years elapsed in this Kali Yuga, he was 23 years old. The stanza of the sloka starts with “Shastyabdanam Shadbhiryada vyateetastra yascha yuga padah.” “Shastyabdanam Shadbhi” means 60 x 6 = 360. While printing the manuscript, the word “Shadbhi” was altered to “Shasti”, which implies 60 x 60 = 3600 years after Kali Era.  As a result of this intentional arbitrary change, Aryabhatta’s birth time was fixed as 476 CE Since in every genuine manuscript, we find the word “Shadbhi” and not the altered “Shasti”, it is clear that Aryabhatta was 23 years old in 360 Kali Era or 2742 BCE. This implies that Aryabhatta was born in 337 Kali Era or 2765 BCE. and therefore could not have lived around 500 A.D., as manufactured by the Indologists to fit their invented framework. Bhaskara I is the earliest known commentator of Aryabhatta’s works. His exact time is not known except that he was in between  Aryabhatta (2765 BCE.) and Varahamihira (123 BCE.).

Is this another case of Napoleon''s Dictum 'attribute not to malice that which can be explained away by sheer incompetence. You be the judge.

The authors feel that such an error could not have crept in ( remember it was a oral tradition) until the wording was changed after the British arrived in India and may have been done at the behest of Sir William

 The Indian Sulvasutras (also pronounced Sulbasutras)

(based on the Wiki)


 

c

Cut the remaining parts (two-thirds of a strip) into eight equal strips and arrange them around the square we are constructing as in the diagram. We have now used all the parts of the second square but the new figure we have constructed is not quite a square having a small square corner missing. It is worth seeing what the side of this "not quite a square" is. It is

1 + 1/3 + 1/(3 cross 4)

which, of course, is the first three terms of the approximation. Now Datta argues in [1] that to improve the "not quite a square" the Sulbasutra authors could have calculated how broad a strip one needs to cut off the left hand side and bottom to fill in the missing part which has area (1/12)2. If x is the width one cuts off then

2 cross x cross (1 + 1/3 + 1/12) = (1/12)2.

This has the solution x = 1/(3 cross 4 cross 34) which is approximately 0.002450980392. We now have a square the length of whose sides is

1 + 1/3 + 1/(3 cross 4) - 1/(3 cross 4 cross 34)

which is exactly the approximation given by the Apastamba Sulbasutra.

Of course we have still made an approximation since the two strips of breadth x which we cut off overlapped by a square of side x in the bottom left hand corner. If we had taken this into account we would have obtained the equation

2 cross x cross (1 + 1/3 + 1/12) - x2 = (1/12)2

for x which leads to x = 17/12 - v2 which is approximately equal to 0.002453105. Of course we cannot take this route since we have arrived back at a value for x which involves v2 which is the quantity we are trying to approximate!

In [4] Gupta gives a simpler way of obtaining the approximation for v2 than that given by Datta in [1]. He uses linear interpolation to obtain the first two terms, he then corrects the two terms so obtaining the third term, then correcting the three terms obtaining the fourth term. Although the method given by Gupta is simpler (and an interesting contribution) there is certainly something appealing in Datta's argument and somehow a feeling that this is in the spirit of the Sulbasutras.

Of course the method used by these mathematicians is very important to understanding the depth of mathematics being produced in India in the middle of the first millennium BC. If we follow the suggestion of some historians that the writers of the Sulbasutras were merely copying an approximation already known to the Babylonians then we might come to the conclusion that Indian mathematics of this period was far less advanced than if we follow Datta's suggestion.

 

Article by: J J O'Connor and E F Robertson


November 2000

MacTutor History of Mathematics
[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_sulbasutras.html]



 

 

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