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
18881807 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
(twothirds 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 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
x
(1 + ^{1}/_{3}
+ ^{1}/_{12}) = (^{1}/_{12})^{2}.
This has the solution
x = 1/(3 4
34) which is approximately
0.002450980392. We now have a square the length of whose sides
is
1 +
1/3 + 1/(3 4)  1/(3
4
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
x
(1 + ^{1}/_{3}
+ ^{1}/_{12})  x^{2} = (^{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://wwwhistory.mcs.standrews.ac.uk/HistTopics/Indian_sulbasutras.html]
