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The Indic Mathematical Tradition 6000 BCE To ?
Moderator note: It would be good if you could define what you mean by "Vedic mathematics". Do you mean mathematics in the Vedic texts? or is it the mathematics of the late achArya bhArati tIrtha? My recommendations for the thread are the following:

1) Ideally broaden the topic to include all Indian mathematics and astronomy, Vedic and post Vedic.

2)If you just want to discuss bhArati tIrtha's mathematics then specifically say so in the title.

3)If you want to just collect links then you may use the link archiving facility instead of a new thread.

If admins permit, I would like to collect information on vedic mathematics in this thread. Thank you


k.ram , I have broadened the title of this thread to include all the sciences of Indian antiquity. For now this will prevent thread proliferation, if we get a lot of contributions on Vedic Mathematics we will change the topic again to Mathematics, If you have a problem with this , email me Kaushal
I have read Shri Bharathi krishna Thirtha's work, but I am not sure if it is a complete work and was also wondering about all the other contributions, hence I just called it vedic mathematics. Please do (renaming or any other advise) as you see fit. Thank you.
k.ram/HH: Don't know if this link is applicable in this thread. You guys decide.

I use Vedic Mathematics methods to do calulation, it is much faster and very simple to use.

Lot of books are available in India.

Vedic Mathematics for schools by J.T.Glover and Dr. L.M.Singhvi -

ISSUE No. 23

Vedic Mathematics is becoming increasingly popular as more and more people are introduced to the beautifully unified and easy Vedic methods. The purpose of this Newsletter is to provide information about developments in education and research and books, articles, courses, talks etc., and also to bring together those working with Vedic Mathematics. If you are working with Vedic Mathematics - teaching it or doing research - please contact us and let us include you and some description of your work in the Newsletter. Perhaps you would like to submit an article for inclusion in a later issue or tell us about a course or talk you will be giving or have given.

If you are learning Vedic Maths, let us know how you are getting on and what you think of this system.


This issue’s article is taken from a longer article by Andrew Nicholas. The full article can be viewed at www.vmacademy.com



In the vedic system, the work is done mentally. This stems from the tradition being an oral one.

In practice, today, the initial problem or question is usually written down and the answer or solution also. The work being done mentally, a one-line answer results. This is the vedic ideal.


But what is this word ‘vedic’? It refers to an ancient period in India’s history. Tradition has it that the system of the vedas covered all branches of knowledge. Originally an oral tradition, it began to be written down around 1600 or 1700BC, according to western scholars. Over the next thousand years four vedas, as they were called, were recorded - rig-veda, yajur-veda, sama-veda and atharva-veda.

An appendix to this last contained a section headed ‘Ganita Sutras’, i.e. mathematical formulae, or principles. In the nineteenth century scholars began to look at it, but could make no sense of what they found there: statements such as, ‘In the reign of King Kamsa, famine, pestilence, and insanitary conditions prevailed.’

Then a brilliant south Indian scholar, Shri Bharati Krishna Tirthaji (1884-1960), began a detailed investigation. He concluded that the above statement about King Kamsa was a cryptic form of the decimal fraction for 1/17, using letters to represent single-digit numbers, much as we might use the letter A to represent 1, and B to represent 2, etc.

Having obtained one clue, further investigation led him to conclude that the whole of mathematics is based on 16 sutras, and he finally wrote 16 volumes on the topic.

Then events intervened. He was virtually forced into becoming a Shankaracharya. Hindu India has four of these top religious leaders - a bit like having four Popes.

The upshot was that he left his beloved vedic mathematics alone for many years. Returning to the subject in the 1950‘s, it emerged that the 16 volumes had been lost. On realising this, he decided to re-write them all, and began by writing a book intended to introduce the whole series. Ill health stopped him from getting any further, and he died in 1960. This introductory book is now all that we have by him. It was first published in 1965.


Western version

When measuring weight, the bigger the number, the greater the weight. Similarly for temperature, length, electric current etc. We are used to the idea that larger numbers are weightier.

Vedic version

In the vedic system, numbers are viewed differently. An analogy is telephone numbers, which we don’t associate with quantity. They are patterns of digits acting as addresses.

Similarly, when working to a base of ten (as we normally do), the vedic system deals with the single-digit numbers 1, 2, 3, 4, up to 9, together with the zero, arranged in different patterns. For example, we don’t divide by 52, we divide by 5, and take account of the 2 afterwards. This shift of focus eliminates the heaviness, or weight, associated with the common view of numbers. The vedic mathematician considers a number such as 52 as 5 and 2 in succession.



To answer the first question first, yes and no. It is used there to some extent. Here is a brief account of the developments.

Tirthaji died in 1960

‘Vedic Mathematics’ was published in 1965

Before going to India in 1981 I wrote to all Indian universities to find out what more was known about the subject. About 30% of them replied. No one could tell me anything more about it. Evidently the subject was being neglected. However, one or two letters pointed me to Tirthaji’s last residence and ashram in Nagpur. Visiting there, I was invited to return the following year to teach a fortnight’s course.

These days, the subject can be taught in schools, alongside the conventional system. Where this is done, I am told, the pupils have no problem with learning the two approaches side-by-side - the western and the vedic.

There is also a passionate debate raging about the status of Tirthaji’s system. Some argue that it is historically accurate, despite the lack of normal historical evidence. Others argue that, lacking evidence for its historical validity, it should be dismissed - despite the fact that, mathematically, it works.

My view (which I am not alone in holding) is that it is a reconstruction. At present we are unable to say for sure that it is historically accurate - nor to prove that it is not. This is because we are dealing with an oral tradition, and it is no surprise that written evidence may not be available.


Tirthaji points out that it normally takes about 16 years to go from first steps in mathematics to a Degree in the subject. (e.g. from age 5 to age 21). But he states that with the vedic system the course in its entirety could be done in about two years! Of course, at present we don’t have all the material that’s needed available.

Needless to say, however, this would benefit everybody - not least those who are not interested in mathematics and would prefer to spend less time on it!

I think, myself, that once vedic mathematics begins to win general acceptance it will lead people to question other academic disciplines. Are rapid methods available in other subjects? If so, are they being used, and if not can they be developed?




New course in london

Following the successful recent course at Imperial College another introductory course is to take place at the Regency Hotel, Queen’s Gate, London, SW7 5AG, on five Mondays from 29th April 2002. Time: 7.00 to 8.30 pm. Course fee: 30 pounds (20 pounds, students and concessions). Enquiries: tel. 020 8688 2642. Topics covered: Squares and Cubes, pi and the Vedic numeral code, Easy Calculus, Fibonacci within Nature, Mathematics and Mind.


“Business India” has published an interesting article by Chetan Dalal entitled “Practical application of Vedic mathematics – Vedic mathematics has certain visual solutions which could be applied in problem solving”. This is on the application of Anurupye Shunyam Anayat (zero value of one of the variables in Simultaneous equations where the other variables are in perfect proportion to constants) illustrated in an Insurance claim. The article ends:

“This illustration . . . emphasizes on the simplicity of the tenets of the sutras of vedic mathematics. Perhaps research and intensive study of vedic scriptures might reveal even more advanced applications. What is illustrated above is a very elementary application of the sutra. The depth and richness of the vedic knowledge is beyond description. Greater research and more teamwork in sharing of ideas and interpretation may provide revolutionary results.”

It would be good to see more such applications of the Vedic Sutras.


A lot of interest was taken in the article in the last newsletter. Mr. Carlos Javier Maya from Mexico has given an idea for doing multiplications of 2 digit by 2 digit on the hands when the multiplier is 19. Mrs Sharma is developing the methods further and is currently conducting a series of courses on Vedic Mathematics.

Dr Abhijit Das in Mumbai, India, has also been researching this area, but without using fingers. We hope to have an article by him for the next newsletter.


As stated in the last newsletter this Vedic Maths course, that covers the National Curriculum for England and Wales, can now be obtained. In India you can purchase whatever you need from Motilal Banarsidass shops and presumably from other bookshops.

The ISBN’s are as follows:

Full set: 81-208-1871-7

Book 1: 81-208-1862-8

Book 2: 81-208-1863-6

Book 3: 81-208-1864-4

Teachers Guide: 81-208-1865-2

Answer Book 1: 81-208-1866-0

To purchase the course in the UK contact:

Motilal Books, PO Box 324, Borehamwood, WD6 1NB

Tel: 0208 905 1244


Price 39.75 pounds

For the USA contact:

THE SACRED SCIENCE INSTITUTE who have the books on order.

Address: PO Box 3617, Idyllwild, CA 92549-3617



Tel: +1 (909) 659-8181

Fax: +1 (909) 659-8383



If you want to know about Vedic Mathematics Workshops or research in India send an email to Mr R. P. Jain at mlbd@vsnl.com




First of all I am thankful to those who are behind this effort of rejuvenating vedic science or mathematics.

I have learned very few mathema-tactics when I was giving some scholarship exams in 4 th standard. These were taught to me by my Nanny at that time. I could score 99/100 in that exam. and much of it due to use of vedic maths. But afterwards I never pursued it. I have done engineering and after so much of years have passed now I have decided to study vedic maths from scratch. I have done tutorials from your site and they are simply best to add my interest. So please subscribe me as student and pls. guide me what next I should do.


Your comments about this Newsletter are invited.

If you would like to send us details about your work or submit an article for inclusion please let us know on news@vedicmaths.com

Previous issues of this Newsletter can be copied from the Web Site: www.vedicmaths.org

Issue 1: An Introduction

Issue 2: "So What's so Special about Vedic Mathematics?"

Issue 3: Sri Bharati Krsna Tirthaji: More than a Mathematical Genius

Issue 4: The Vedic Numerical Code

Issue 5: "Mathematics of the Millennium"- Seminar in Singapore

Issue 6: The Sutras of Vedic Mathematics

Issue 7: The Vedic Square

Issue 8: The Nine Point Circle

Issue 9: The Vedic Triangle

Issue 10: Proof of Goldbach's Conjecture

Issue 11: Is Knowledge Essentially Simple?

Issue 12: Left to Right or Right to Left?

Issue 13: The Vinculum and other Devices

Issue 14: 1,2,3,4: Pythagoras and the Cosmology of Number

Issue 15: A Descriptive Preparatory Note on the Astounding Wonders of Ancient

Indian Vedic Mathematics

Issue 16: Vedic Matrix

Issue 17: Vedic Sources of Vedic Mathematics

Issue 18: 9 by 9 Division Table

Issue 19: “Maths Mantra”

Issue 20: Numeracy

Issue 21: Only a Matter of 16 Sutras

Issue 22: Multiplication on the Fingertips

To subscribe or unsubscribe to this Newsletter simply send an email to that

effect to news@vedicmaths.com

Please pass a copy of this Newsletter on (unedited) to anyone you think may

be interested.

Editor: Kenneth Williams

Visit the Vedic Mathematics web site at



15th April 2002



Vedic Math: Crunch more in less time!

SHRADDHA KAMDAR | Wednesday, October 15, 2003 12:6:50 IST

Thousand-year-old 'sutras' can put the modern math methods to shame

Scientific techniques developed thousands of years ago, mentioned in

the Vedas, are now being utilised by America's National Aeronautic &

Space Administration (NASA) and form a part of the curriculum in

European schools.

Sadly, however, they have lost importance in the country of origin.

Vedic Math, as it is called, is extracted from 16 'sutras' and 13

sub-'sutras' of the Vedas. It is a set of techniques, which can be

applied to a wide spectrum of mathematic topics, to reduce the

calculation time to one-tenth

of the actual time of any traditional method.

Professor Atul Gupta, an IIT engineer who chanced upon a book on

Vedic Math about a decade ago, was intrigued by it and learnt from

it. The process was long, but nonetheless interesting and fruitful.

Later, he thought of sharing his knowledge. Prof Gupta now has

school students, IIT aspirants, housewives and retired persons who

are simply math enthusiasts learning from him. "It is so

fascinating, it has turned math-haters into math-lovers," claimed

the professor.

Useful for Arithmetic, Algebra, Calculus, Trigonometry and

Astromomy, the techniques are easy to learn and remember. The

professor had this reporter so awe-struck with the methods, that it

was difficult to wind up the interview and move out of his class.

And he had not even touched the tip of the ice-berg. "Now you can

imagine what a treasure this is. It should be passed on to our

future generations," said Prof. Gupta. In that regard, he has

already conducted several workshops with school children.

"These techniques are very helpful for IIT aspirants, as the

entrance exam papers are full of such questions. If they save even

about 10 minutes over all, imagine how many more questions they'd be

able to attempt!" said Prof. Gupta for whom clearly Vedic Math is

not just something he teaches, but is also a passion.

(For more details, contact 2551-3728, 2557-7553).

How long does it take to divide 257910 by 9?

Using a Vedic Math technique, the answer can be arrived at in a

couple of seconds! How?

It's simple.

Add all the digits of the number 257910 and reduce it to a single

digit: 2+5+7+9+1+0=24. Reduce 24 further -- 2+4=6, which is the


Another way is by removing the digit 9 while adding. Or even the

digits that add up to 9. For example, in 257910 don't use 9, 2 and

7. By adding the remaining digits, you still get the correct answer,

i.e. 5+1+0=6.

This technique, called 'Navashesh', is applicable to any number, but

only while dividing it by 9. It has a wide range of applications, to

check humongous multiplications and additions.

Another technique is finding the square of a number ending in 5. For

example, for squaring 85, all you have to do is take the square of

5, i.e. 25, at the end, and multiply 8 by the next arithmetic digit,

9 (8x9=72) and the answer is 7225
k.ram: Sometime back I had math book on [url="http://www.amazon.com/exec/obidos/tg/detail/-/0313232008/ref=pd_sbs_b_1/103-4184427-1687010?v=glance&s=books&n=507846"]Trachtenberg Speed System[/url] which basically thought same things. Don't have that book with me anymore to check if this guy lifted these techniques from Vedic Mathematics.
Viren, I used to have that book too long time ago. Yes some techniques do seem similar. I would not speculate further than that. <img src='http://www.india-forum.com/forums/public/style_emoticons/<#EMO_DIR#>/wink.gif' class='bbc_emoticon' alt='Wink' />
[url="http://www.amazon.com/exec/obidos/tg/detail/-/812081777X/103-9194566-2699851?v=glance"]Lilavati of Bhaskaracharya[/url]
I too have read a bit of Trachtenberg Speed System for basic mathematics and if I recall correctly the author wrote it while in jail. In the appendix he gives mathematical proofs (distributive law, etc. IIRC) of the techniques he used.

I bought it for a few rupees from a footpath hawker in Connaught Place in Delhi, but the amazon page linked to above shows a different cover and a price of $68.95. Amazing!
[quote name='Jaspreet' date='Oct 16 2003, 02:59 PM'] I bought it for a few rupees from a footpath hawker in Connaught Place in Delhi, but the amazon page linked to above shows a different cover and a price of $68.95. Amazing! [/quote]

Had same observation Jaspreet. I must have paid less than Rs 25 in Mumbai for that.
I bought Lilavati in a flea market in Florida (about 10 yrs ago) for 25 cents...
Not about vedic math, but a great mathematical genius. Graduate

Quote:The man who 'discovered' Everest

   By Soutik Biswas

BBC News Online 

One day in 1852 in British-ruled India, a young man burst into an office in the northern Dehra Dun hill town and announced to his boss: "Sir, I have discovered the highest mountain in the world!"

Radhanath Sickdhar was an intrepid mathematician from Calcutta

After four long and arduous years of unscrambling mathematical data, Radhanath Sickdhar had managed to find out the height of Peak XV, an icy peak in the Himalayas.

The mountain - later christened Mount Everest after Sir George Everest, the surveyor general of India - stood at 29,002 feet (8,840 metres).

Sickdhar's feat, unknown to many Indians, is now part of the Great Arc Exhibition in London's vibrant Brick Lane.

The Indian Government-sponsored exhibition celebrates 200 years of the mapping of the Indian subcontinent.

The exercise, which was called "one of the most stupendous works in the whole history of science" was begun by William Lambton, a British army officer, in Madras in 1802.

The survey involved several thousand Indians and was named the Great Trigonometrical Survey (GTS) in 1819.

Mount Everest, the world's highest mountain, is 29,035 ft high today

It covered more than 1,600 miles and countless people died during the work. Tigers and malaria were the main causes of death.

Sickdhar, who was 39 when he made his discovery, was one of the survey's largely unsung heroes.

The man from Calcutta was called a "computer" since he worked on computation of data collected by survey parties.

He was promoted to the position of "chief computer" because of his good work.

'Rare genius'

"Mathematical skills were essential for Sickdhar's work and he was acknowledged by George Everest as a mathematician of rare genius," British historian John Keay, author of two books on the subject, told BBC News Online.

"His greatest contribution to the computation was in working out and applying the allowance to be made for a phenomenon called refraction - the bending of straight lines by the density of the Earth's atmosphere," said Mr Keay.

"Like George Everest himself, [Sickdhar] may have never seen [Mount Everest]."

It was first identified as a possible contender for the world's highest peak in 1847 when surveyors glimpsed it from near Darjeeling.

Sir George Everest found Sickdhar a rare mathematical genius

Several observations were recorded over the next three years by different survey parties.

But the announcement that it was the highest - thanks to Sickdhar's efforts - was delayed until 1856 as calculations had to be checked repeatedly.

Sickdhar, the son of a Bengali Brahmin, was born in October 1813 in Jorasanko, Calcutta's old city.

He studied mathematics at the city's renowned Hindoo College and had a basic knowledge of English.

A workaholic, Sickdhar never married, instead dedicating his life to knotty mathematical calculations.

George Everest was always full of praise for the number-crunching genius.

He wrote that Sickdhar was a "hardy, energetic young man, ready to undergo any fatigue, and acquire a practical knowledge of all parts of his profession.

"There are a few of my instruments that he cannot manage; and none of my computations of which he is not thoroughly master. He can not only apply formulate but investigate them."

Mount Everest has risen higher since Sickdhar's findings.

In 1955, the mountain "grew" by 26 ft to 29,028 ft (or by 8 m to 8,848 m).

Mount Everest grew another 7 ft (2 m) in 1999 after researchers analysed fresh data from the mountain.

Today, the world's highest mountain stands 29,035 ft (8,850 m) high.

Not about vedic mathematics exclusively but has lot of tidbits about the extent of knowledge and the sciences in ancient India and in antiquity

[url="http://www.cdacindia.com/html/pdf/ramdasi.pdf"]Visualizing Indian heritage Digital Library Metaphor[/url]
[url="http://india.coolatlanta.com/GreatPages/sudheer/astro.html"]Ancient India's Contribution to ASTRONOMY[/url]
The latest India Abroad has a 3 page article by Dr Jayant Narlikar on this subject. A very good read. Anyone here with OCR scanner?
The Vedic Concept of Time

The measurement of time in the West is restricted to a second at the lowest

level and a century at the highest level.However, ancient Vedic scholars had

defined time from a very minute part of a second to a large multiple of century.

"Surya Siddhanta" is an ancient Vedic text dealing with the astronomical

configurations of the zodiac.

Time in West is measured in the order of seconds, minutes, hours, days, weeks,

fortnights (biweekly), months, years, decades and centuries. Though scientists

divide seconds into milli, micro or nano seconds, the nomenclature still

assumes a second as the basic unit.

As per the Surya Siddhanta, the smallest measurement of time is Truti, which is

equivalent to one 3,240,000th part of a second. The time taken to pin a padma

patra (lotus leaf) with a needle is called Truti.

60 such Trutis are equal to one Renuvu, which make it one 54,000th part of a

second. 60 Renuvus comprise one Lavamu, thereby, one Lavamu is a 900th part of a

second. 60 Lavamus make one Leekshakamu. Hence one Leekshakamu is one 15th of a

second. 60 Leekshakamus equal to one Pranamu, accordingly one Pranamu is 4

seconds long. It is also stated that one Pranamu is the time taken to enunciate

ten long syllables.

As per Surya siddhanta (from 10th verse), 6 Pranamus are equal to one

Vighadiya. One Vighadiya is 24 seconds long. 60 Vighadiyas make Ghadiya, which

therefore is equal to 24 minutes. In other words, one minute consists of 2.5

Vighadiyas whereas one hour has 2.5 Ghadiyas.

60 Ghadiyas are said to equal one Stellar day, technically called one

Nakshatra Ahoratram. 60 Ghadiyas of the Hindu systema re equal to 24 hours of

the Western system. 30 stellar days make one Nkshatra Masam (month).

A day is counted by Hindus from one sunrise to the next, while Westerners

treat the time between two consecutive midnights as one day. A month is defined

by Hindus in four different ways. One is stellar month, as described in the

first paragraph. Second kind is called Saavana month;The time between two

consecutive sunrises is considered as one Saavana day and 30 such days

constitute Saavana month. The third type of the month is Lunar month.The time

between two consecutive new moon days is called lunar month, and it consists of

30 lunar tithis. The fourth type of month is a solar month. The zodiac is

treated as a circle having 360 degrees. The time taken by the sun to travel 1

degree of this circular zodiac is called one solar day. 30 such solar days

combine to make one solar month. The zodiac is divided into 12 (Rasi) signs,

namely Mesha, Vrishaba... and so on. Each sign occupies 30 degrees. The sun's

movement thru' one Rasi (sign)is completed on one solar month which is also

referred to as one Sankramana. Thus Sun's passage thru' Mesha is called Mesha


A solar year consists of 12 solar months. One solar year is said to be equal

to one day of the Devatas or one Divine day. Thus 360 solar years are equal to

one Divine year. 12,000 Divine years form one Maha Yuga. Thus one Maha Yuga

contains 4,320,000 solar years. A Maha Yuga contains 4 yugas, namely Krita Yuga

(1,728,000 solar years), Treta Yuga (1,296,000 solar years), Dwapara Yuga

(864,000 solar years) and Kali Yuga (432,000 solar years) in that order. Each

yuga is again divided into 4 equal Padas (Quadrants).

71 Maha Yugas are collectively termed as a Manvantra, which thereby consists

of 306,720,000 solar years. After each Manvantra, it is said that there is a

Sandhi Kala of the duration of one Krita Yuga (1,728,000 solar years). It is

also stated that the entire earth is submerged under water during such a Sandhi


A Manvantra, along with its Sandhi Kala, is jointly considered as a unit

(308,448,000 solar years). 14 such units combine to form a Kalpa (4,318,272,000

solar years). At the beginning of each Kalpa is included an Adi Sandhi period,

again of the duration of one Krita Yuga (1,728,000 solar years). Thus one Kalpa

is equal to 4,320,000,000 (432 crores or 4.32 Billion) solar years, that is 1000

Maha Yugas.

One Kalpa is said to be half a day for Lord Brahma. Hence one day of Lord

Brahma is equal to 864 crore ( 8.64 Billion) solar years. The name of the

present Kalpa is Sweta Varaha. In the present Kalpa, six Manvantaras plus

Sandhis have been completed and the seventh Manvantara by name Vyvaswata

Manvantara is running. In this Manvantara, 27 Maha Yugas are over and we are in

the twentieth (28). In this 28th Maha Yuga, three Yugas are past and the fourth,

that is Kali Yuga's first quadrant started about 5005 years ago. This is the

reason why before commencing any puja, ritual or vrata (worship), we start

invoking the Gods, reminding ourselves of the time elapsed to date, by chanting

the Sankalpa thus.


Adya Brahmanah, Dwiteeya Parardhe, Sri Sweta Varaha Kalpe, Vyvaswata Manvantare,

Kali Yuge, Prathama Pade, Jambu Dweepe, Bharata varshe,..

and so on.

The author (Sri Rao) concludes:

I wish to impress upon the readers that in the Vedic system, the measurement of

time ranges from the smallest Truit (3,240,000th part of a second) to a Day of

Lord Brahma (8.64 Billion solar years). This detailed visualization of time is

unique to the Hindu system, unequalled by any other system in the world. I

salute to the intelligence of the ancient scholars of our country.
The philosophy of pa~nchshikha asurAyana a~ngirasa and sulabhA prAdhani

The essence of hindu atomism is seen in sulabhA's discourse , to the king of mithilA, dharmadhvaja jAnaka. sulabhA was a student of he student of pa~nchshika asurAyaNa.

Mbh. shAntiparvA 309.113-125 (Pune edition); 321 (Bengal edition)

avyaktaM prakR^iti.n tvAsAM kalAnAM kash chidichchhati .

vyakta.n chAsAM tathaivAnyaH sthUladarshI prapashyati .. 113..

Some believe that all existence emerges solely from the unmanifest prakR^iti, while others believe that it emerges from the always manifest prakR^iti.

avyakta.n yadi vA vyakta.n dvayImatha chatuShTayIm .

prakR^iti.n sarvabhUtAnAM pashyantyadhyAtmachintakAH .. 114..

Whether it emerges from the manifest, or the unmanifest, or a combination of two principles or of four principles, the knowers term the basis of all existence to be prakR^iti

seyaM prakR^itiravyaktA kalAbhirvyaktatA.n gatA .

aha.n cha tvaM cha rAjendra ye chApyanye sharIriNaH .. 115..

Myself and yourself, oh indra of the rajanyas, and other entities that occupy space are as result of prakR^iti.

bindunyAsAdayo.avasthAH shukrashonita sambhavAH .

yAsAmeva nipAtena kalalaM nAma jAyate .. 116..

A point formed by the union the spermatic and female principles is beginning of our existence. It then transforms itself into the primary embryonic globule called kalala

kalalAdarbudotpattiH peshI chApyarbudodbhavA .

peshyAstva~NgAbhinirvR^ittirnakharomANi chA~NgataH .. 117..

The kalala divides to a state with numerous bubbles called (arbuda), and the the arbuda gives rise to an elongated sheath like form (peshi). From the peshi, emerges the a~NgA, where the rudiments of all the organs are seen, from this form differentiates the external elements like hair and nails.

sampUrNe navame mAse jantorjAtasya maithila .

jAyate nAma rUpatva.n strI pumAnveti li~NgataH .. 118..

Upon the expiration of the 9th month emerges a neonate, Oh Maithila. It is female or male depending on the li~nga it had developed.

jAtamAtra.n tu tadrUpaM dR^iShTvA tAmranakhA~Nguli .

kaumAra rUpamApanna.n rUpato na palabhyate .. 119..

When a being issues forth from the womb it has nails and fingers of the hue of blood. The next step is infancy when the form seen at birth changes.

kaumArAdyauvana.n chApi sthAviryaM chApi yauvanAt .

anena kramayogena pUrvaM pUrvaM na labhyate .. 120..

from infancy, youth is reached and from that old age is reached. As the creature advances from stage to stage irreversibly, there is constant change in the body.

kalAnAM pR^ithagarthAnAM pratibhedaH kShaNe kShaNe .

vartate sarvabhUteShu saukShmyAttu na vibhAvyate .. 121..

This is because the body is made of minute particles which serve diverse functions and undergo changes, that are hard to perceive from second to second.

na chaiShAmapyayo rAja.NllakShyate prabhavo na cha .

avasthAyAmavasthAyA.n dIpasyevArchiSho gatiH .. 122..

The birth and death of these particles is continous, with one leading to the other, just as the changes in flame of a burning lamp, oh King.

tasyApyevaM prabhAvasya sadashvasyeva dhAvataH .

ajasra.n sarvalokasya kaH kuto vA na vA kutaH .. 123..

Thus is the existence of the body of all creatures, with their elements constantly in rapid locomotion like a good horse in run. Ever state comes and goes.

kasyeda.n kasya vA nedaM kuto vedaM na vA kutaH .

sambandhaH ko.asti bhUtAnA.n svairapyavayavairiha .. 124..

Who then has come whence? from what? what has it not come from? what is the relationship between the elements and the body of a creature?

yathAdityAnmaNeshchaiva vIrudbhyashchaiva pAvakaH .

bhavetyeva.n samudayAtkalAnAmapi jantavaH .. 125..

As fire come from the contact of wood pieces or sun's (rays) through bead, some combinations of the fundamental elements gives rise to the constituents of bodies.
I found this contribution by Prasenjit medhi on the net.

I have included a few important details about just a few of the most famous ancient Indian mathematicians from past years. To my mind, the most important and most influential of these figures were Aryabhatta and Panini. Aryabhatta had an excellent understanding of the Keplerian Universe more than a thousand years before Kepler, while Panini made a remarkable analysis of language, namely Sanskrit, which was not matched for 2,500 years, until the modern Bacchus form, in the 20th century.

***Aryabhata the Elder


Born: 476 in Kusumapura (now Patna), India

Died: 550 in India

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Aryabhata wrote Aryabhatiya , finished in 499, which is a summary of Hindu mathematics up to that time, written in verse. It coveres astronomy,spherical trigonometry, arithmetic, algebra and plane

trigonometry.Aryabhata gives formulas for the areas of a triangle and a circle which are correct, but the formulas for the volumes of a sphere and a pyramid are wrong.

Aryabhatiya also contains continued fractions, quadratic equations, sums ofpower series and a table of sines. Aryabhata gave an accurate approximation for pi (equivalent to 3.1416) and was one of

the first known to use algebra. He also introduced the versine ( versin = 1 - cos) into trigonometry.

Aryabhata also wrote the astronomy text Siddhanta which taught that the apparent rotation of the heavens was due to the axial rotation of the Earth. The work is written in 121 stanzas. It gives a

quite remarkable view of the nature of the solar system.

Aryabhata gives the radius of the planetary orbits in terms of the radius of the Earth/Sun orbit as essentially their periods of rotation around the Sun. He believes that the Moon and planets shine

by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains the causes of eclipses of the Sun and the Moon.

His value for the length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since the true value is less than 365 days 6 hours.

References (4 books/articles) References for Aryabhata the Elder


1.Dictionary of Scientific Biography 2.Biography in Encyclopaedia Britannica 3.B Datta, Two Aryabhatas of al-Biruni, Bull. Calcutta Math. Soc. 17 (1926), 59-74. 4.H-J Ilgauds, Aryabhata I, in H

Wussing and W Arnold, Biographien bedeutender Mathematiker (Berlin, 1983).



Born: 1114 in Biddur, India

Died: 1185 in Ujjain, India

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Bhaskara represents the peak of mathematical knowledge in the 12th Century and reached an understanding of the number systems and solving equations which was not to be reached in Europe for several

centuries. Baskara was head of the astronomical observatory at Ujjain, the leading mathematical centre in India at that time.He understood about 0 and negative numbers. He knew that x^2 = 9 had two

solutions. He gives the formula <Picture: sqrt>(a<Picture: + or - ><Picture: sqrt><img src='http://www.india-forum.com/forums/public/style_emoticons/<#EMO_DIR#>/cool.gif' class='bbc_emoticon' alt='B)' /> = <Picture: sqrt>((a+<Picture: sqrt>(a<Picture: ^2>-<img src='http://www.india-forum.com/forums/public/style_emoticons/<#EMO_DIR#>/cool.gif' class='bbc_emoticon' alt='B)' />)/2) <Picture: + or - > Picture: sqrt>((a-<Picture:

sqrt>(a<Picture: ^2>-<img src='http://www.india-forum.com/forums/public/style_emoticons/<#EMO_DIR#>/cool.gif' class='bbc_emoticon' alt='B)' />)/2).Baskara also studied Pell's equation x^2=1+py^2 for p=8, 11, 32, 61 and 67.When p = 61 he found the solutions x =1776319049, y = 22615390. He studied many Diophantine


Bhaskara's mathematical works include Lilavati (The Beautiful) and Bijaganita (Seed Counting) while he also wrote on astronomy, for example Karanakutuhala (Calculation of Astronomical Wonders).

References (3 books/articles) References for Bhaskara


1.Dictionary of Scientific Biography 2.Biography in Encyclopaedia Britannica 3.B Datta, The two Bhaskaras, Indian Historical Quarterly 6 (1930), 727-736.



Born: 598 in (possibly) Ujjain, India

Died: 670 in India

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Brahmagupta was head of the astronomical observatory at Ujjain which was the foremost mathematical centre of ancient India.

He wrote important works on mathematics and astronomy. He wrote Brahmasphuta- siddhanta (The Opening of the Universe), in 21 chapters, at Bhillamala in 628. His second work on mathematics and

astronomy is Khandakhadyaka written in 665.Brahmagupta's understanding of the number systems was far beyond others of the period. He developed some algebraic notation. He gave remarkable formulas

for the area of a cyclic quadrilateral and for the lengths of the diagonals in terms of the sides.

Brahmagupta also studied arithmetic progressions, quadratic equations,theorems on right-angled triangles, surfaces and volumes.

The remaining chapters deal with solar and lunar eclipses, planetary conjunctions and positions of the planets. Brahmagupta believed in a static Earth and he gave the length of the year as 365 days

6 hours 5 minutes 19 seconds in the first work, changing the value to 365 days 6 hours 12 minutes 36 seconds in the second book. This second values os not, of course, an improvement on the first

since the true length of the years if less than 365 days 6 hours.

One has to wonder whether Brahmagupta's second value for the length of the year is taken from Aryabhata since the two agree to within 6 seconds, yet are about 24 minutes out.

References (4 books/articles) References for Brahmagupta


1.Dictionary of Scientific Biography 2.Biography in Encyclopaedia Britannica 3.B Datta, Brahmagupta, Bull. Calcutta Math. Soc. 22 (1930),39-51. 4.H T Colebrooke, Algebra, with Arithmetic and

Mensuration from the Sanscrit of Brahmagupta and Bhaskara (1817).



Born: about 520 BC in India

Died: about 460 BC in India

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The dates given for Panini are pure guesses. Experts give dates in the 4th,5th and 6th century BC.

Panini was a Sanskrit grammarian who gave a comprehensive and scientific theory of phonetics, phonology, and morphology. Sanskrit was the classical literary language of the Indian Hindus.

In a treatise called Astadhyayi Panini distinguishes between the language of sacred texts and the usual language of communication. Panini gives formal production rules and definitions to describe

Sanskrit grammar. The construction of sentences, compound nouns etc. is explained as ordered rules operating on underlying structures in a manner similar to modern theory.

Panini should be thought of as the forerunner of the modern formal language theory used to specify computer languages. The Backus Normal Form was discovered independently by John Backus in 1959,

but Panini's notation is equivalent in its power to that of Backus and has many similar properties.

Reference (One book/article) References for Panini


1.P Z Ingerman, 'Panini-Backus form' suggested, Communications of the ACM 10 (3)(1967), 137.



Born: 1019 in (probably) Rohinikhanda, Maharashtra, India

Died: 1066

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Sripati wrote on astronomy and mathematics. His mathematical work is undertaken with applications to astronomy in mind, for example a study of spheres.His works include Dhikotidakarana (1039), a

work on solar and lunareclipses, Dhruvamanasa (1056), a work on calculating planetary longitudes,eclipses and planetary transits, Siddhantasekhara a major work on astronomy in 19 chapters. The

titles of Chapters 13, 14, and 15 are Arithmetic, Algebra and On the Sphere. Sripati obtained more fame in astrology than in other areas.

Reference (One book/article)

***Cadambathur Tiruvenkatacharlu Rajagopal


Born: 1903 in Triplicane, Madras, IndiaDied: 25 April 1978 in Madras, India


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Rajagopal was educated in Madras, India. He Graduated in 1925 from the Madras Presidency College with Honours in mathematics.

He spent a short while in the clerical service, another short while teaching in Annamalai University then, from 1931 to 1951, he taught in the Madras Christian College. Here he gained an

outstanding reputation as a teacher of classical analysis.

In 1951 Rajagopal was persuaded to join the Ramanujan Institute of Mathematics then, four years later, he became head of the Institute. Under his leadership the Institute became the major Indian

mathematics research centre.

Rajagopal studied sequences, series, summability. He published 89 papers in this area generalising and unifying Tauberian theorems.

He also studied functions of a complex variable giving an analogue of a theorem of Landau on partial sums of Fourier series. In several papers he studied the relation between the growth of the mean

values of an entire function and that of its Dirichlet series.

A final topic to interest him was the history of medieval Indian mathematics. He showed that the series for tan^-1 (x) discovered by Gregory and those for sin x and cos x discovered by Newton were

known to the Hindus 150 years earlier. He identified the Hindu mathematician Madhava as the first discoverer of these series.

Rajagopal is described in [1] as follows:-

Rajagopal was a teacher par excellence and a reliable and inspiring research guide. No words can adequately describe his modesty. Rational thinking and interest in psychic studies were two

attributes which he imbibed with pride from his teacher Ananda Rau.

References (4 books/articles)

References for Cadambathur Tiruvenkatacharlu Rajagopal


1.C T Rajagopal, Bull. London Math. Soc. 13 (5) (1981), 451-458. 2.C T Rajagopal: September 8, 1903, to April 25, 1978, J. Anal. 1 (1993), vii. 3.Y Sitaraman, Professor C T Rajagopal (1903-1978),

J. Math. Phys. Sci. 12(5) (1978), i-xvi. 4.M S Rangachari, Prof. C T Rajagopal, Indian J. Math.22 (1) (1980), i-xxix.
Some more interesting data on Ancient Indain mathematics


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