
PL. feel free to use the glossary for words you are
either not familiar with or need reassurance regarding the meaning
Vedic
mathematicians in Ancient India (Part II)
Kosla Vepa Ph.D
"practically all topics taught in school
mathematics today are directly derived from the work of mathematicians
originating outside Western Europe before the twelfth century A.D."
Joseph, George Ghevarughese,
"Foundations of Eurocentrism in
Mathematics," Race and Class, XXVII, 3(1987), p.1328.
1
Introduction
We are often told by western historians
and scientists that Indians of the preChristian era were poor historians
and even poorer at record keeping and hence that we know very little of the
identities of the mathematicians and their contributions to the subject of
mathematics. Our contention is precisely the opposite. Not only were the
Indics superb record keepers , but they reported on the discoveries of
their predecessors as well as contemporaries without the slightest sign of
condescension or attempts to purloin the credit for themselves, a trait
that appears to be singularly rare among those studying the field of Indology which seeks to obfuscate anything emanating
out of India. In many cases the Vedic mathematicians were also the
preeminent astronomers of their day. The problem lay not so much in the
much bandied poor record keeping of the Indics, but in the fact that these
records were either ignored or in certain egregious instances were in fact
altered.
In Part I
we furnished sufficient quotes to establish the proposition that throughout
the ancient as well as the medieval eras Mediterranean, Arab and
European savants went to great length to acknowledge the contribution
of the Indics in various fields such as number theory, geometry ,astronomy,
and medicine. It was only in the colonial era beginning with the
discovery of Sir William Jones of the antiquity of Sanskrit, which
had far reaching implications on the roots of their own civilization, that
racial prejudice towards the Indics took on a dominant role and began to
affect the quality as well as the accuracy of the scholarship in
Europe. The roots of this prejudice and its progression over the 2
centuries of British rule in India are chronicled by Thomas Trautmann Everything,
including the truth became subordinate to the paramount goal of
maintaining a dominant role in India and Asia. We do not know for certain
that there was deliberate falsification of records to support their
versions of Indic history, but what we do know is that in certain key
instances, the text has been altered to suit the preconceived notions of
the Europeans on Indic history and that the general approach was to reject
key pieces of data and dub them as being unreliable when it did not fit in
with their overall paradigm of ancient Indic chronology. Before we get into
specific acts of misdating, let us list chronologically the cast of
characters who studied India under the general rubric of Indology
2
List
of Indologists who worked in the area of Indology
Indologists

Sir William Jones
(17461794) the founder of Indology, largely
responsible for postulating a Proto Indo European language for which no
speakers have been found and for misdating the chronology of ancient
India

Hermann George Jacobi
(18501837)was the first to suggest that the Vedic Hymns were collected
around 4500 BCE based on Astronomical observations made by the Vedics

Thomas Babington Macaulay
(18001859) decreed English to be the medium of instruction, drafted the
Indian Penal Code. He did not study the texts himself because he was
ignorant of Sanskrit, but he hired MaxMueller
to do the misrepresentation

Friedrich Maximilian Mueller (18231900)
translated the books of the east. His private views of these books were
vastly at variance with his public pronouncements

Roberto Di Nobili(15771656),Jesuit
Priest, posed as a Brahmana ,posited a
counterfeit Veda, called the Romaka Veda

Rudolf Roth(18211893) studied rare manuscripts
in Sanskrit

Abbe Dubois, Jean Antoine (176518) went to India
to convert the heathen returned discouraged that it was very difficult
too accomplish

William Carey(17611834),Missionary

Sir Charles
Wilkins (17491836)
Translated the Bhagavad Gita in 1785

Colonel Colin Mackenzie (17531821)
Collector of Indian Manuscripts

Henry Thomas
Colebrook (17651837)
Studied Sanskrit from
the Pundits and wrote on the Vedas

Horace Hayman Wilson
(17861860)
First Boden Professor
of Sanskrit at Oxford
U
wrote on the Puranas

August Wilhelm
Schlegel (17671845)
Lecturer in Sanskrit
,Bonn University

Franz Bopp
(17911867)
Did detailed research leading to postulation of
Proto Indo European (PIE)


Arthur Schopenhauer

James Mill
(17731836).Completed The History of British India
in 1817


Sir Monier MonierWilliams
(18191899),Boden Professor of Sanskrit, Oxford

John Playfair

Sir Alexander
Cunningham (18141893), member of Asiatic Society of Bengal

Colonel Boden who
endowed the Boden Chair of Sanskrit Studies in
1811 with the purpose of debunking the Vedas

Frederick Eden Pargiter (18521897) published ‘Purana texts of the Dynasties of the Kali age”

Robert Caldwell (18151891) Collected Sanskrit
manuscripts, a British missionary

Sir Mark Aurel Stein (18621943),Archaeological Survey of India

Vincent Smith(18481920), author of Oxford
History of India

Arthur Barriedale Keith (18791944) published ‘The
religion of and philosophy of the Vedas’ in 2 volumes in 1925,
Cannot be regarded as an authentic or reliable translation

Arthur Anthony McDonnell(18541930), brought
7000 Sanskrit manuscripts from Kashi to Oxford
University


Maurice Bloomfield (18551928), interpreted the
Vedas

Morris Winternitz (18631937), wrote History of Indian
Literature

Sir Robert Erie Mortimer Wheeler(18901976)

Sir John
Hubert Marshall,(18761958) director general Archaeological Survey
of India

Alexander Basham

Edwin Bryant
(PhD Columbia,1997)

Alain Danielou
(19071994)

Heinrich Zimmer
(18901943) author of Philosophies of India "Indian philosophy was at the heart of
Zimmer's interest in oriental studies, and this volume therefore
represents his major contribution to our understanding of Asia.
It is both the most complete and most intelligent account of this
extraordinarily rich and complex philosophical tradition yet written."

Joseph Campbell (19041987) follows in the
tradition of Heinrich Zimmer, albeit he uses the word myth much too
liberally

Among these there are quite a few who neither
harbored preconceived notions nor would they indulge in the dishonest act
of altering documents. Typical of these were names like Playfair,
Jacobi, Schopenhauer, Alain Danielou,
Heinrich Zimmer and Joseph Campbell. But even with the best of
intentions it is difficult to translate accurately from a language and
culture which is alien to ones own. When that language is over 4000 years
old, the difficulties are multiplied in manifold ways. When Europeans
studied Sanskrit and the Vedas the paradigm they followed was that of
studying insects in a jar. The science of studying insects is known as
entomology. The insects for obvious reasons have little say in the matter.
Such was the case also when they studied the civilization of the Indics.
The opinion of the Indics hardly mattered and they were rarely consulted
and in many instances such as that of Max Mueller and Franz Bopp (max Mueller’s professor) they had never set
foot in India
or conversed with a pundit. In just as many instances such as that of max
Mueller they could not converse or chant a single sloka in Sanskrit much
less understand one when it was chanted in front of them. But that did not
stop them from claiming to be Sanskritists of the
first rank. Neither did their dilettante status in Sanskrit stop Bopp and Sir William from deciding that there must have
been an ancestral language (which they called Proto Indo European (PIE for
short) spoken anywhere but in India. Now that I ponder on the reluctance
of Max Mueller to visit India, the suspicion is overwhelming that the
real reason he never wished to set foot in India was that he would thereby
be spared the embarrassment of facing a real pundit in Sanskrit and have to
then admit how shallow his knowledge of Sanskrit was.
3
Motivation
for present dating of Ancients
The original preoccupation of the European (to
some extent true to this day) was to find the roots of his/her own
language, which till the advent of Sanskrit was assumed to have been
derived from Hebrew. The notion of a Hebraic origin was hardly very
popular in Europe steeped as it was in
anti Semitism. Therefore, when Sanskrit was first discovered, the notion
that there was a race of noble Aryans who were their putative ancestors was
then greeted with a great deal of enthusiasm once they had disposed of the
prior suspicion that they were descended from the teeming millions of
India.
The corollary to this proposition was that the
denizens of the Indian subcontinent could not possibly have an antiquity
greater than that of Greece
or that of Pericles and the Shakespearean vision
of the golden age of the Hellenic civilization. It was the conflict with
his strict belief in the Creation theory postulated in the bible that led
Sir William to lop of 1200 years from the Puranic
history of India and to further postulate that the contemporary of Megasthenes , the Greek historian who visited India
around 300 BCE was Chandragupta Maurya and not the Chandragupta of the
Imperial Gupta dynasty.
We have already alluded to the postulate adhered
to by almost all Indologists in the western world, that the Saraswati Sindhu civilization
had little to do with the Vedic civilization in Part I of this series
and we have described in great detail the efforts by the
Colonial Power to undermine the Civilizational
unity of India in the South
Asia File. We wish to emphasize once again that the net result of all
these efforts was to change the complexion of the debate. What was once a
search for the roots of their own languages has now been transformed into,
in their words, an obsession on the part of the Hindu right wing to prove
that the Aryans were indigenous to India.
This is undoubtedly a very astute strategy on their part since it takes the
limelight of their own obsession to find a Urheimat for their group of
languages and the fact that from the inception the postulates of dating
Indic history have been political rather than academic in nature. The
entire dating of the revisionist Indic history by the British and the
Europeans has been a political enterprise right from the start. So, now
when the argument is made that political considerations are driving the
Hindu right wing in their opposition to theories such as the AIT,
regardless of the truth of such an allegation, it ignores the glaring fact
that it has always been so.
It is sad that a section of the Indic
populace has internalized this revisionist view of Indic History propounded
primarily by Europeans and it is important to remember that AIT is crucial
to validate their racial view of civilizations and people and their sense
of self esteem. They had to reconcile what was obviously a vast dependence
on the contributions of the Semitic speaking people primarily along the Mediterranean
Sea. In AIT they saw their opportunity to portray themselves as
the progenitors of a vast Eurasian civilization without aligning themselves
too closely to the brown skinned people of India.
The conclusion is inescapable that while validating the AIT is not crucial
to the pride of the Indic other than that it robbed him of his own
authentic history, the debunking of AIT would have a devastating effect on
the European weltanschauung of the roots of his own civilization. Indics in
general for understandable reasons tend to be Indocentric
and look to their own psyche to comprehend the nature of this paradigm. We,
the Indics, would be much better served if we sought to understand the
motivations and psyche of the European, at least in this instance or in
other words to understand why there is such a constant emphasis on Eurocentrism in the European psyche. See for example
the models of Eurocentrism
currently prevalent.
What do we expect to achieve by removing such
misconceptions created essentially by a Eurocentric world. Every people
need to have pride in their own traditions in order for there to be
harmony, mutual respect and dignity among the peoples and civilizations of
this world . Robbing them of their essential contributions is tantamount to
breeding the seeds of racial superiority. What we are not seeking is
retrospective privileges or an apology for such past misrepresentations,
merely an acknowledgement that legitimate discoveries be attributed accurately
to their rightful civilizations and that no single civilization or group of
civilizations has a monopoly on the capability to advance the sum of human
knowledge.
We will now discuss the anomalies in the
assumptions made by European writers of Indic history and the evidence
supporting our contention and see how devastating they were to a proper
understanding and appreciation of Indic history and to a proper
understanding of the Indic contribution to the sciences of antiquity.
4 Misdating of Aryabhatta the Elder
Aryabhatta is without doubt the
Astronomer/Mathematician nonpareil of the Post
Vedic/Post Epic era in the historical narrative, especially so since his
magnum opus The Aryabhattium, which packs a lot of information in the terse
aphoristic style characteristic, of that era, has survived intact from the
mists of a distant past when he first developed his thesis in 4 Chapters covering
the subject of Mathematics and Astronomy. His work and the prior work in
the Vedic area form an important sheet anchor for the entire chronology
that follows, important by virtue of the fact that it attests to the state
of the language prior to his contribution, and refers to the beginning of Kaliyuga as he reveals his own age. But first we list
the main mathematicians during the period in question
Some of the Vedic personalities that we will meet
here are
Yajnavalkya who wrote the Shatapatha Brahmana ( as well as the Brihadaranyaka
Upanishad)

in which he describes the motion of the sun and the moon and
advances a 95 year cycle

to synchronize the motions of the sun and the moon

Lagadha who authored the Jyotisha Vedanga

Baudhayana the author of the Sulvasutra
named after him

Apastambha
“

Katyayana
“

Panini the Grammarian for the Indo Europeans

Pingala Binary System of number representation:

Aryabhatta the astronomer laureate of ancient India

Varahamihira who synthesized the knowledge
The author of the Jaina treatises the Suryaprajnapati, Chandraprajnapati and the seventh section of Jambudvipaprajnapati

We will tackle first the misdating of Aryabhatta
for the reasons stated above .
Details of the life and contributions of this veritable genius are
presented in the linked site. What concerns us most here is the episode
relating to the misdating.
"Aryabhatta is
the first famous mathematician and astronomer of Ancient India.
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 A.D 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 B.C. This implies that Aryabhatta was born in 337 Kali
Era or 2765 B.C. 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 B.C.) and Varahamihira
(123 B.C.)."
The
implications are profound, if indeed this is the case. The zero is by
then in widespread use and if he uses Classical Sanskrit then he
antedates Panini
“How the
beginning of Kaliyuga is Linked with the Dates
of Indian astronomers? The ancient Indian astronomers
perhaps purposely linked the determination of their dates of birth,
composition of their works; calculation of number of years elapsed, etc.,
based on two eras Kali and Saka. Therefore,
without the significance of these two eras, the dates cannot be
determined specifically.
Shastabdhanam shastardha vyatitastrashyam yugapadha
Trayadhika vimsatirabdhastdheha mama janmanoatita
"When sixty times
sixty years and three quarter yugas (of the
current yuga) had elapsed, twenty three years had then passed since by
birth" (K. S. Shukla).
"Now when sixty times
sixty years and three quarter Yugas also have passed, twenty increased by
three years have elapsed since my birth" (P. C. Sengupta).
"I was born at the end of
Kali 3600; I write this work when I am 23 years old i.e., at the end of
Kali 3623" “(T. S. Kuppanna Sastry^{11}).
Here, though only Yuga is
mentioned, Kaliyuga is implied and its starting
of 3102 BCE is taken for reckoning purpose. Thus, the date of Aryabhatta
is determined as follows:
The year of birth = 3600
– 3102 = 488 / 499 – 23 = 476 CE. This has been accepted
by most of the scholars and generally considered as accepted date. Had
the commencement year 3102 BCE is a myth or not astronomical one, the
year of Aryabhatta cannot be historical date or could be determined like
this using 3102 BCE.
Bhaskara I in his commentary
to Aryabhatteeyam mentions as follows (Ch.I.verse.9):
Kalpadherabdhnirodhadhayam abdharashiritiritaha:
khagnyadhriramarkarasavasurandhrenadhavaha: te cangkkairapi 1986123730

"Since the beginning
of the current Kalpa, the number of years
elapsed is this: zero, three, seven, three, twelve, six, eight, nine, one
(proceeding from right to left) years. The same (years) in figures are
1986123730"
"The time
elapsed, in terms of years, since the commencement of the current kalpa is zero, three, seven, three, twelve, six,
eight, nine, one (years written in figures) are 1986123730".
Aryabhata gives the number of years
elapsed since the beginning of the current yuga
= 6 Manus + 27 ¾ yugas
= 6 x 72 yugas + ¾ yugas
= 6 x 72 + 27 ¾ ) x
43,20,000 years
= (1866240000 +
119880000) years
= 198612000 years.
From this, we can calculate
the number of years elapsed since Bhaskara wrote his commentary
= 1986123730
–198612000 = 3730 years
= 3730 –
3102
= 628 / 629 CE.
Bhaskaracharya and others too imply Kaliyua/ era in their works
as revealed through commentators, as their dates are determined with the
calculations reckoning the date of starting of Kaliyuga
/ era as 3102 BCE.
Mahabharata and
Kaliyuga: The starting of Kaliyuga
has been associated with the following events^{12}:
The end of Mahabharat war.
The death of Sri
Krishna.
The deluge, which
made Dwaraka, submerged.
Coronation of Yudhistira.
The renouncement of Yuddhistira.
In Indian
astronomical works including Tantras
and Karanas, the word
yuga has been taken as Kaliyuga
for calculating, illustrative and explanatory purposes. In a Tantra, the epoch is the beginning of Kaliyuga or 3102 BCE. In a Karana, any convenient epoch is selected by
the astronomer.
"The Saka year (when the civil days are required) added to
3179 gives the solar years elapsed since the beginning of the Kaliyuga" (Sisyadhivrddhida Tantra^{13}
– hereinafter mentioned as ST  of Lalla.I.12).
Here, that the Saka year began 3179 after the beginning of the Kaliyuga is specifically mentioned. Moreover, in the
calculation of days elapsed, solar years elapsed, Suddhi
ertc.,
Kaliyuga is repeatedly mentioned and used for
illustrations.
"…the
solar months elapsed since the beginning of the Kaliyuga
multiplied by 22,26,389 and divided by 21,60,0000 give the corresponding
lunar months" (ST.I.15).
The commentators
Bhaskara I, Somesvara,
Suryadeva Yajvan and others have
pointed out the relation between Mahabharata and Kaliyuga.
Mahabharata and
Kali Era:
The usage of Kali era by the astronomers with the Mahabharata,
that too, with Mahabharat
war in particular, has been consistent. Many astronomers mention Kali era
and Saka era together.
"Since the
birth of Brahma up to the beginning of the Saka
era, 8 ½ years (of Brahma), ½ month (of Brahma), 6 Manus of the (current)
day (of Brahma), 27 ¾ yugas, and 3179 years of
the (current) Kali era had gone by" (Vatesvara Siddhanta^{14}.I.10,
K. S. Shukla).
Here, the number of
years 3179 specifically mentioned is to obtain any year in terms of Saka, but it has been derived from the Kali era i.e, 3102 / 3101 + 78 =
3180 / 3179. As Vatesvara
(c.880960 CE) uses the notation, it is evident that even during the 9^{th}
century it had been very popular among the astronomers and established
one. He also records his year of birth in that fashion as explained below.
Kali Era and Saka Era: After Aryabhata,
astronomers use the computation of years in Saka
and as well as Kali Eras. The number of years to be reckoned in Saka with respect to Kali is given as 3179 and this
is obtained by adding 78 to 3101 / 3102, thus, 3101 + 78 = 3179.
Thus, the Tantra directs: Navadhrirupagniyuttam mahibhujam shakendratnam gatavarshadaraham
(I.4) meaning, "Add 3179 to the Saka years
elapsed, the Kali years elapsed are obtained". Thus, it is evident
that such method of reckoning of years in Saka
Era related to Kali Era and vice versa had been in vogue before 6^{th}
century.
Vateswara says: "When 802 years had elapsed since
the commencement of the Saka era, my birth took
place; and when 24 years had passed since my birth, this Siddhanta was
written by me by the grace of the heavenly bodies" (I.21).
Thus, the year of
birth = 802 + 78 = 880 CE and that of his work = 880 + 24 = 904 CE.
The astronomers use
certain Sakas as
illustrative examples in their works. For example Mallikarjuna Suri and Candesvara, an astronomer of Mitthila use 1100 and 1107 Sakas for illustrating
rules.
Mahabharata and
3102 BCE:
The above discussion about Kaliyuga and Kali
era amply proves its astronomical importance in time reckoning. It also
points to the well established date of such reckoning starting with 3102
BCE and its connection with Mahabharat.
The Saka Era has also been associated with it
as starting 3179 after the starting of Kali Era. The date 3102 / 3101 BCE
is very often used by the epigraphists, numismatists, archaeologists,
historians, astronomers and others, but, assert that Kali Era / Yuga is a
myth! Therefore, the application of 3102 BCE to determine and calculate
other dates should be explained properly, because, the modern scholars
use the same date, but even condemn and criticize it unwittingly at many
places.
.
Bhaskara mentions the names of Latadeva,
Nisanku and Panduranga
Svami as disciples of Aryabhatta.

So the question is which version of the sloka in Aryabhattium is the correct one
Is it this one?
Shastyabdanam Shadbhiryada
vyateetastra yascha yuga padah.”
“Shastyabdanam Shadbhi” means 60 x 6
= 360. Which places his birth at 2765 BCE (360 23 – 3102)
Or this one
Shastabdhanam shastardha vyatitastrashyam yugapadha
Trayadhika vimsatirabdhastdheha mama janmanoatita
Shastabdhanam shastardha means 60 x 60 = 3600
Which places his birth at (3600
– 23 – 3102) = 475 CE
The resulting shift in the date of Aryabhatta of 3240
years, makes him roughly 3 times higher in chronology , and has profound
consequences for the Indic contributions relative to those of Babylonian
mathematicians
5 The Vedanga
period (Vyakarana, Jyotisha,
Chandas, Kalpa Sutra)
The Vedanga (IAST
veda?ga,
"member of the Veda") are six auxiliary disciplines for the
understanding and tradition of the Vedas.
 Shiksha (sik?a): phonetics
and phonology
(sandhi)
 Chandas (chandas): meter Pingala
 Vyakarana (vyakara?a): grammar
Panini
 Nirukta (nirukta): etymology
Yaska
 Jyotisha (jyoti?a): astrology
Lagadha
 Kalpa (kalpa): ritual
Apastambha,
Baudhayana, Katyayana , Manava
The Vedangas are first mentioned in the Mundaka Upanishad
as topics to be observed by students of the Vedas. Later, they developed
into independent disciplines, each with its own corpus of Sutras.
During this period we will consider the timeline
of the following contributors
Panini  Vyakarana
(Grammar, Place value system) was one of the many all time great savants
that India
has produced in such abundance. The scope and scale of his vast
contributions to language, grammar, computing science, and place value
system is mind numbing at a time when scripts were in an embryonic stage of
development. It is no hyperbole to say that there was no study of grammar
as a codified set of rules until the west discovered Pannini’s
Ashtadhyayi
Pingala
Yaska
Apastambha
Baudhayana
Katyayana
Ashvalayana
6
TimeLine
according to MaxMueller

Chandas Rg Veda 1200 to
1000 BCE

Mantras later Vedas 1000 to 800 BCE

Brahmanas
800 to 600 BCE

Sutras
600 to 200 BCE


7 Timeline
according to Keith
Taittiriya Samhita
500 BCE

Baudhayana
400 BCE

Ashvalayana
350 BCE

Sankhayana
350 BCE

Yaska
300
BCE

Apastambha
300 BCE

Pratisakhya
300 BCE

Panini
250 BCE

Katyayana
800to 600 BCE


8 Conventional
Timeline of Mathematicians in the ancient world (Wiki)

 ca. 70,000 BC
South Africa ochre rocks adorned with scratched geometric patterns [1]
 ca. 35,000 BC to 20,000 BC Africa & France,
earliest known prehistoric attempts to quantify time (references: [2], [3], [4])
 ca. 20,000 BC
Nile Valley, Ishango
Bone: earliest known prime number sequences and also Egyptian multiplication
 ca. 3400 BC Mesopotamia the Sumerians invent the first numeral system, a system of weights and measures, and are the first to construct
cities
 ca. 3100 BC Egypt,
earliest
known decimal system allows indefinite counting by way of
introducing new symbols, [5]
 ca. 2800 BC Indus Valley Civilization
on the Indian subcontinent, earliest use of decimal fractions in a uniform system of ancient weights and measures, the smallest unit of
measurement used is exactly 0.001704 meters and the smallest unit of
mass used is exactly 0.028 kg
 2800 BC
The Lo Shu Square,
the unique normal magic square of order three, was discovered in China
 ca. 2700 BC Indus Valley Civilization,
the earliest use of negative numbers (see Negative Number: History)
 2700 BC
Egypt,
precision surveying
 2600 BC
Indus Valley Civilization
objects,
streets, pavements, houses, and multistoried buildings are
constructed at perfect rightangles, with each brick having exactly
the same dimensions
 2400 BC
Mesopotamia,
the
Babylonians invent the earliest calculator, the Abacus
 2400 BC
Egypt,
precise Astronomical Calendar, used even in the Middle Ages for its mathematical regularity
 ca. 2000 BC Mesopotamia,
the
Babylonians use a base60 decimal system, and
compute the first known approximate value of p at 3.125
 1800 BC
Moscow Mathematical Papyrus,
generalized formula for finding volume of frustums, [6]
 1800 BC
Berlin Papyrus shows that the ancient
Egyptians knew how to solve 2nd order algebraic equations: [7].
 ca. 1800 BC Vedic India Yajnavalkya writes the Shatapatha Brahmana,
in which he describes the motions of the sun and the moon, and
advances a 95year cycle to synchronize the motions of the sun and the
moon
 ca. 1800 BC the Yajur Veda,
one of
the four Hindu Vedas, contains the earliest concept of infinity, and states that "if you remove a part
from infinity or add a part to infinity, still what remains is
infinity"
 1650 BC
Rhind
Mathematical Papyrus, copy of a lost scroll from
around 1850 BC, the scribe Ahmes
presents one of the first known approximate values of p at 3.16, the first attempt at squaring the circle, earliest known use of a sort of
cotangent, and knowledge of solving first order
linear equations
 1350 BC
Indian astronomer Lagadha writes the "Vedanga Jyotisha",
a Vedic text on astronomy that describes rules for tracking the
motions of the sun and the moon, and uses geometry and trigonometry for astronomy
9 Conventional
Timeline of Mathematicians during the 1st millennium BCE (Wiki)
 ca 1000 BC  Vulgar fractions used by the Egyptians.
 800 BC  Baudhayana, author of the Baudhayana Sulba Sutra, a Vedic Sanskrit geometric text, contains the first
use of the Pythagorean theorem, quadratic equations, and calculates the square root of 2 correct to five decimal places
 600 BC  Apastamba, author of the Apastamba Sulba Sutra, another Vedic Sanskrit geometric text, makes an attempt at squaring the circle and also calculates the square root of 2 correct to five decimal places
 ca. 600 BC  the other Vedic "Sulba Sutras"
("rule of chords" in Sanskrit) contain the first use of irrational numbers, the use of Pythagorean triples, evidence of a number of
geometrical proofs, and approximation of p at 3.16
 530 BC  Pythagoras studies propositional geometry and vibrating lyre strings; his group also
discover the irrationality of the square root of two,
 ca. 500 BC  Indian grammarian Panini,
considered the father of computing machines, writes the Astadhyayi, which contains
the use of metarules, transformations and recursions, originally for the purpose of
systematizing the grammar of Sanskrit
 ca. 400 BC  Jaina mathematicians in India write the "Surya
Prajinapti", a mathematical text which
classifies all numbers into three sets: enumerable, innumerable and infinite. It also recognizes five different types of
infinity: infinite in one and two directions,
infinite in area, infinite everywhere, and infinite perpetually.
 300s BC  Indian texts use the Sanskrit word "Sunya"
to refer to the concept of 'void' (zero)
 370 BC  Eudoxus states the method
of exhaustion for area determination,
 350 BC  Aristotle discusses logical reasoning in Organon,
 300 BC  Jaina mathematicians in India write the "Bhagabati
Sutra", which contains the earliest information on combinations
 300 BC  Euclid in his Elements studies geometry as an axiomatic system, proves the infinitude of prime numbers and presents the Euclidean algorithm; he states the law of reflection
in Catoptrics, and he proves the fundamental theorem of arithmetic
 ca. 300 BC  Brahmi numerals, the first
base10 numeral system, is conceived in India
 ca. 300 BC  Indian mathematician Pingala writes the "Chhandahshastra",
which contains the first use of zero (indicated by a dot) and also presents the
first description of a binary numeral system, along with the first use of Fibonacci numbers and Pascal's triangle
 260 BC  Archimedes computes p to two decimal places using inscribed and
circumscribed polygons and computes the area under a parabolic segment,
 ca. 250 BC  late Olmecs had already begun
to use a true zero (a shell glyph) several centuries before Ptolemy in the New World. See 0 (number).
 240 BC  Eratosthenes uses his sieve algorithm to quickly isolate prime numbers,
 225 BC  Apollonius of Perga writes
On Conic Sections and names the ellipse, parabola, and hyperbola,
 150 BC  Jain mathematicians in India write the "Sthananga
Sutra", which contains work on the theory of numbers,
arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic
equations, and permutations and combinations
 140 BC  Hipparchus develops the
bases of trigonometry,
 50 BC  Indian numerals, the first positional notation base10 numeral system, begins developing in India
10 Proposed
Timeline of Mathematicians in the ancient world up to 1000 BCE
Era

Region

What was
discovered or developed and by whom






Ca 4000 BCE

Vedic India

Rg Veda Mandalas
composed over a 500 year period


ca. 4000 BCE

Vedic India

Yajnavalkya writes the Shatapatha Brahmana,
in which he describes the motions of the sun and the moon, and advances
a 95year cycle to synchronize the motions of the sun and the moon


ca. 4000 BCE

the Yajur Veda,

one of the four Hindu Vedas, contains the earliest concept of infinity, and states that "if you remove a part
from infinity or add a part to infinity, still what remains is
infinity"


Ca 3500 BCE – 4000 BCE

Post Vedic India
Vedanga Vyakarana

Panini develops Sanskrit
Grammar. considered the father of computing machines, writes the Astadhyayi, which contains
the use of metarules, transformations and recursions, originally for the purpose of
systematizing the grammar of Sanskrit Enunciated PaniniBackus form and is
the father of Classical sanskrit as we know
it today distinguished from Vedic sanskrit.
There are increasing inferences that Panini developed the place value
system and is the inventor of the decimal system as we know it today


ca. 3400 BCE

Mesopotamia

the Sumerians use a system of weights and measures, and construct cities






3102 BCE

India

Beginning of Kali Yuga,
sheet anchor for dating Indic civilization


ca. 3100 BCE

Egypt,

earliest known decimal system allows indefinite counting by way of
introducing new symbols, [5]






Ca. 3000 BCE

Post Vedic India
Vedanga Jyotisha

Earliest astronomical text
in India.
astronomer Lagadha writes the "Vedanga Jyotisha",
a Vedic text on astronomy that describes rules for tracking the
motions of the sun and the moon, and uses geometry and trigonometry for astronomy


Ca. 2700 BCE

Post Vedic Vedanga KalpaSutra

Apastambha Sulva sutra


Ca. 2700 BCE

Post Vedic Vedanga Kalpa Sutra

Baudhayana Sulva Sutra


Ca.2700BCE

Post Vedic Vedanga Kalpa Sutra

Ashvalayana Sulva sutra


ca. 2800 BCE

Saraswati Sindhu
Civilization

on the Indian subcontinent, earliest use of decimal fractions in a uniform system of ancient weights and measures, the smallest unit of
measurement used is exactly 0.001704 meters and the smallest unit of
mass used is exactly 0.028 kg


2800 BCE

The Lo Shu Square,

the unique normal magic square of order three, was discovered in China


ca. 2700 BCE

Saraswati Sindhu
Civilization,

the earliest use of negative numbers (see Negative Number: History)


2700 BCE

Egypt,

precision surveying


2600 BCE

Saraswati Sindhu
Civilization

objects, streets,
pavements, houses, and multistoried buildings are constructed at
perfect rightangles, with each brick having exactly the same
dimensions


2576 BCE

Post Vedic Classical
Sanskrit

Astronomer Laureate of India
Aryabhatta the Elder, postulates Heliocentric model of the solar system


2400 BCE

Mesopotamia,

the Babylonians invent the earliest calculator, the Abacus


2400 BCE

Egypt,

precise Astronomical Calendar, used even in the Middle Ages for its mathematical regularity


ca. 2000 BCE

Mesopotamia,

the Babylonians use a base60 decimal system, and compute
the first known approximate value of p at 3.125


1800 BCE

Moscow Mathematical Papyrus,

generalized formula
for finding volume of frustums, [6]


1800 BCE

Berlin Papyrus

Shows that the ancient Egyptians
knew how to solve 2nd order algebraic equations: [7].










1650 BCE

Rhind
Mathematical Papyrus,

copy of a lost scroll from
around 1850 BCE, the scribe Ames presents one of the first known approximate
values of p at 3.16, the first attempt at squaring the circle, earliest known use of a sort of cotangent, and knowledge of solving first order
linear equations




















Assumptions
The Sulvasutras precede the developments
in Babylon and Egypt
and must therefore date at least to 2000 BCE.
The RgVeda must have been fully composed
prior to 2000 BCE because of the drying up of the Saraswati
River prior to 1900 BCE
The Vedangas indicate a knowledge of the
use of zero and the place value system
Seidenberg on the significance of the Sulvasutras
Seidenberg, A. On the volume of a
sphere. Arch. Hist. Exact Sci.
39 (1988), no. 2, 97119. (Reviewer: K.B. Gundlach.) SC: 01A20 (01A15
01A17 01A25 01A32), MR: 89j:01012.
Abraham Seidenberg argues that there is a
common source for Pythagorean and Chinese (or Chineselike) mathematics.
He suggests that OldBabylonian mathematics is a derivative of a more
ancient mathematics having a much clearer geometric component (p. 104),
and is "in some respects ... is derivative of a Chineselike
mathematics" (p. 109). Van der Waerden holds a similar
view on this, and tells us that the mathematics of the Chiu Chang Suan
Shu represents the common source
more faithfully than the Babylonian does. Seidenberg
believes that the common source is most similar to the Sulvasutras.
He discusses how questions of the sphere and the circle were treated by
the Greeks, Chinese, Egyptians, and to a lesser extent Indians. He
discusses the some similarities and differences in the work on the sphere
in Greece (Archimedes, with a very brief account of the application of
his Method), and in
Chinese (first in the Chiu Chang Suan Shu,
improved by Liu Hui
or perhaps Tsu Ch'ungChih, and then
further improved by the Tsu
Ch'ungChih's son Tsu KengChih). He believes that the problem of
the volume of a sphere goes back to the common source, to the first part
of the second millennium B.C. or earlier. An interesting and
related topic is the topic of the equality of the proportionality
constants pi that occur in the formulas for the area and circumference
of a circle. Seidenberg examines the Moscow Papyrus, Chinese sources, and
an OldBabylonian text and finds that this fact seemed to be recognized
in all three groups. He argues that the Egyptian, Babylonian, and Chinese
approaches to the volume of a truncated pyramid may have derived from the
same common source. He believe that the common
source also used infinitesimal, Cavalieritype,
arguments as well. It is interesting as well that Heron, who as Seidenberg
notes is sometimes considered to be continuing the Babylonian tradition,
gives the formula ^{1}/_{2}(s+p)p+^{1}/_{14}(^{1}/_{2}s)^{2} for the area of a
segment of a circle with chord s
and height (sagita,
arrow) p (with an
Archimedean value of ^{22}/_{7} for pi), and "that the 'ancients'
took [the area as] ^{1}/_{2}(s+p)p and even conjectured that they
did so because they took pi
= 3." The paper is also interesting in that he discusses the
development of some of his ideas from his early papers in the 60s until
much later (the paper was received soon before his death). Closely
related topics: The Sphere,
The Circle,
The
Pythagoreans, China, The
Chiu Chang Suan Shu (Nine
Chapters on the Mathematical Art), Sumerians
and Babylonians, The Sulvasutras, Archimedes,
Archimedes'
Method, The
Moscow Mathematical Papyrus, and Heron.
Mathews, Jerold. A Neolithic oral tradition for
the van der Waerden/Seidenberg origin of mathematics. Arch. Hist.
Exact Sci. 34 (1985), no. 3, 193220. (Reviewer: M. Folkerts.) SC: 01A10
(01A25), MR: 88b:01005.
Abraham Seidenberg advanced a theory that
mathematics arose from a common origin, and that some the mathematics was
preserved by an oral tradition, and very likely a religious tradition,
perhaps one like the one seen in the Indian Sulvasutras. Van der Waerden's
book Geometry and Algebra in Ancient
Civilizations takes a similar views,
and in fact van der Waerden credits Seidenberg for making him look
at the history of mathematics a new way. As Mathews
notes, the Chinese Chiu Chang Suan Shu is
very important in van der Waerden's work. Mathews relies heavily on this
work as well to "give a small, coherent, and basic core of geometry
concerning rectangles and their parts, ..., which may serve as what van der Waerden
has called an 'oral tradition current in the Neolithic age.'" He
states the he hoped "to give this hypothesized ancient core some
credence through its relation to the Chiu
Chang and its explanatory power. After giving a thorough
discussion of this geometry, he then carefully analyzes the ninth chapter
of the Chiu Chang in
terms of this core. He is able to find a strong match, though his
conclusions on one of the problems (Problem 20) are not consistent with
those of some other researchers, who find in problem 20 instead
suggestions of something like Horner's method. A very interesting
article. Hopefully future papers will discuss how well the author's
geometry agrees with the ancient geometry of other cultures. As he notes,
"Until I can thoroughly test his conjecture on, say, the Babylonian
corpus, I can argue for the merits of my conjecture only on such grounds
as the simplicity of explanation it allows, or its congruence with
received results or figures." Closely related topics: The
Neolithic Era, Religion,
Geometry,
The
Chiu Chang Suan Shu (Nine
Chapters on the Mathematical Art), and Abraham
Seidenberg.
11 Summary
In Part II we have laid the groundwork for a new chronology. In
subsequent chapters we will present the basis for the new chronology.
Some of the topics we will be discussing are
The Date of the Rg Veda (based on Astronomical observations and
the precession of the Equinoxes)
The extent of Astronomical Knowledge of the Vedics and their
successors inn the immediate Post Vedic era
The Significance of the Sulvasutras
The contributions of Panini
The contributions of Aryabhatta and other Mathematicians
Please click
on
Vedic Mathematicians in Ancient India Part III
for the next
chapter in this series



