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.13-28.

1         Introduction

 

We are often told by western historians and scientists that Indians of the pre-Christian 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 pre-eminent 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[1]  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 pre-conceived 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 (1746-1794) 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  (1850-1837)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 (1800-1859) 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 (1823-1900) translated the books of the east. His private views of these books were vastly at variance with his public pronouncements

Roberto Di Nobili(1577-1656),Jesuit Priest, posed as a Brahmana ,posited a counterfeit Veda, called the Romaka Veda

Rudolf Roth(1821-1893) studied rare manuscripts in Sanskrit

Abbe Dubois, Jean Antoine (1765-18) went to India to convert the heathen returned discouraged that it was very difficult too accomplish

William Carey[2](1761-1834),Missionary

Sir  Charles Wilkins (1749-1836)

Translated the Bhagavad Gita  in 1785

Colonel Colin Mackenzie (1753-1821)

Collector of Indian Manuscripts

Henry Thomas Colebrook (1765-1837)

Studied Sanskrit from the Pundits and wrote on the Vedas

Horace Hayman Wilson (1786-1860)

First Boden Professor of Sanskrit at Oxford U

wrote on the Puranas

August Wilhelm Schlegel (1767-1845)

Lecturer in Sanskrit ,Bonn University

Franz Bopp (1791-1867)

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

 

Arthur Schopenhauer

James Mill (1773-1836).Completed The History of British India in  1817

 

Sir Monier Monier-Williams (1819-1899),Boden Professor of Sanskrit, Oxford

John Playfair

Sir Alexander Cunningham (1814-1893), 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 (1852-1897) published ‘Purana texts of the Dynasties of the Kali age”

Robert Caldwell (1815-1891) Collected Sanskrit manuscripts, a British missionary

Sir Mark Aurel Stein (1862-1943),Archaeological Survey of India

Vincent Smith(1848-1920), author of Oxford History of India

Arthur Barriedale Keith (1879-1944) 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(1854-1930), brought 7000 Sanskrit manuscripts from Kashi to Oxford University

 

Maurice Bloomfield (1855-1928), interpreted the Vedas

Morris Winternitz (1863-1937), wrote History of Indian Literature

Sir Robert Erie Mortimer Wheeler(1890-1976)

Sir John Hubert  Marshall,(1876-1958) director general Archaeological Survey of India

Alexander Basham

Edwin Bryant (PhD Columbia,1997)

Alain Danielou (1907-1994)

Heinrich Zimmer (1890-1943) 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 (1904-1987) 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 non-pareil 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:[3]

 

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 Sastry11).

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 events12:

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 -Tantra13 – 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 Siddhanta14.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.880-960 CE) uses the notation, it is evident that even during the 9th 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 6th 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.

  1. Shiksha (sik?a): phonetics and phonology (sandhi)
  2. Chandas (chandas): meter  Pingala
  3. Vyakarana (vyakara?a): grammar  Panini
  4. Nirukta (nirukta): etymology           Yaska
  5. Jyotisha (jyoti?a): astrology           Lagadha
  6. 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[4]

 

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[5]

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)

9         Conventional Timeline of Mathematicians during the 1st millennium BCE (Wiki)

 

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 95-year 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 Panini-Backus 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 multi-storied buildings are constructed at perfect right-angles, 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 base-60 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, 97--119. (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 Chinese-like) mathematics. He suggests that Old-Babylonian 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 Chinese-like 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'ung-Chih, and then further improved by the Tsu Ch'ung-Chih's son Tsu Keng-Chih). 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 Old-Babylonian 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, Cavalieri-type, 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(1/2s)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, 193--220. (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


 

[1] Trautmann,Thomas R(1997) Aryans and British India, University of California Press, Berkeley

[2]

[3] A Mathematician named Pingala (c. 100BC) developed a system of binary enumeration convertible to decimal numerals [See 3]. He described the system in his book called Chandahshaastra. The system he described is quite similar to that of Leibnitz, who was born in the 17th century.[3]

[4] Friedrich MaxMueller (1968) A History of  Ancient Sanskrit Literature,Varanasi,Chowkambika Sanskrit Series Office

[5] Keith,Arthur Barriedale, various works, see Rajaram (1995)

 

 

 

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