On the native place of aryabhatt I



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FAMOUS ASTRONOMER AND MATHEMATICIAN

ARYABHATT I OF KUSUMPURA

BY
ANAND M. SHARAN
PROFESSOR



FACULTY OF ENGINEERING
MEMORIAL UNIVERSITY OF NEWFOUNDLAND
E-MAIL:
asharan@engr.mun.ca
NOVEMBER 23, 2007

ABSTRACT

In this paper, the place of Aryabhatt I is determined. It involves going over the history of India, and experiments performed by Aryabhatt I. These experiments were performed similar to those done by Eratosthenes in Egypt around 200 BC . Comparing the results and statements of Aryabhatt with those of other scientists in Ancient India it is found that Aryabhatt lived in Kusumpura ( modern Patna ) . This contradicts the findings of K. Chandra Hari5 who shows that Aryabhatt wrote his famous Aryabhatia in Kerala.



1. INTRODUCTION

The golden age of India is considered between 400 AD to 1200 AD1 . In this era, famous mathematicians such as Aryabhatt I ( 476 – 550 AD ) , Brahmgupta ( 628 AD ), Bhaskaracharya I ( 600 AD ) , Bhaskaracharya II ( 1114 AD ) , etc made significant contributions to mathematics.

Chandra Hari writes that Sarma4 describes Āryabhatt I as having flourished at Kusumapura (modern Patna) and explains that the system of Āryabhatia was prevalent in North India owing to the criticisms from later authorities like Brahmagupta ( 628 AD ), Varāhamihira ( 550 AD ), almost a contemporary of Aryabhatt I , and Shripati Mishra (1039 AD). So, there was hardly any one from Kerala or other states of South India.

Kaul10 has looked at the problem of determining the native place of Aryabhatt, and he also believes that it is highly unlikely that Aryabhatt I belonged to Kerala. His explanations from the website are shown in Appendix A.

Chandra Hari also believes that Aryabhatt I was not familiar with exact location of Ujjain whose latitude was calculated by Aryabhatt I as 22°30′ N in place of 23°11′N ( according to modern calculations or 24° N as determined by Brahmgupta ( 628 AD ). Chandra Hari calls this latitude of Ujjain determined by Brahmgupta ( 628 AD ) as prevalent at the time of Aryabhatt I. One can clearly see the fallacy in Chandra Hari’s arguments because Aryabhatt I lived in times when Brahmgupta was not even born. How could have Aryabhatt I known this 24°N latitude of Ujjain ?

Regarding importance given to Aryabhatt I in India, in fact, after the independence when India launched its first satellite, it named it after Aryabhatt I.

There was another Aryabhatt referred to as Aryabhatt II ( 950 AD ).

The place of Aryabhatt I has been accepted as Kusumpura [Georges2 ; Sarma3,4 ]. Similarly, one can cite innumerable sources where the native place of Aryabhatt I has been referred to as Kusumpura ( Modern Patna ).

However, in recent work by Chandra Hari5 , it is claimed that Aryabhatt I was a native of Kerala, and he gives various evidences based on folklore in Kerala; works of Bhaskara-I and his commentators and also of the medieval commentators of Āryabhat I. He writes, “ that commentaries of and works based on Aryabhatīya ( a work of Aryabhatt ), have come largely from South India, from Kerala in particular, certainly constitute a strong argument in favour of Kerala. “

Regarding commentaries about Aryabhatt I’s work not coming from North India – we see that they did come from North India until certain time but not after that . What was the reason ?

Joseph6 writes that the astronomical center had shifted from Pataliputra or Kusumpura to Ujjain after Chandragupta II’s ( 375 AD -415 AD ) annexation of the Saka Kingdom ( Ujjain was in the Saka kingdom ) , and after the Muslim invasion of India ( this can be taken as the invasion of Mohammad Ghazni ( 997 AD – 1026 AD ) - the scientific center started shifting to Kerala from Ujjain . Chandra Hari does not mention the name of any person knowledgeable in astronomy from Kerala who lived during or before Aryabhatt I’s time. He also does not provide any evidence of the existence of any school of astronomy in Kerala either in the time of Aryabhatt I or before his time whereas, Nalanda University near Kusumpura came into existence at the time of Kumargupt ( 415 AD to 455 AD ) before even the birth of Aryabhatt I ( 476 AD -550 AD ).

Kaye7 writes that after Ptolemy (87 – 150 AD ), the Indian astronomy was heavily influenced by Greek astronomy. All the 7 tables in Chandra Hari’s paper are shown in Appendix B. Greek astronomy’s influence can be seen in Chandra Hari’s Table 1 ( based on Suryasiddhānta ) where the circumference of the earth has been taken as the same as that determined in Egypt ( Table 7 of Chandra Hari’s paper ). The date of Suryasiddhānta can be taken as 408 AD8 .

Now, we come to the two main points that Chandra Hari writes - “we meet in the history of Indian astronomy two specific conflicts in which Āryabhata is involved: (i) Conflict of the latitude of Ujjayinī and (ii) Conflict of the earth’s circumference “ .

Another important point that Chandra Hari writes ( he lays importance to the place of observation and its relationship to the circumference of the earth mentioned in all his 7 tables ) : “at the place of the observer the circumference shall be an integer multiple of 360, so that the distance in yojanas for 1/6th of the nazhika (1/360) or per degree longitude could be represented by an integer.

In order to determine the place of birth or the native place of Aryabhatt -There are some important points to note before we go through the analysis of Chandra Hari’s paper:

1. We will see extensive discussions about the circumference of the spherical shaped earth but very little details about how the Indian scientists found the circumference ?

2. Secondly, does the place of observation mean that the scientist was born and raised there?

3. Thirdly, does the place of observation mean automatically that a school of astronomy existed there?



2. DETERMINATION OF LATITUDE OF UJJAIN AND THE CIRCUMFERENCE OF EARTH

Let us see what had already happened in Egypt at the time of Eratosthenes who had also measured earth’s circumference from his observations between Alexandria (31.5° N ) and Syenne ( at Tropic of Cancer at 24°N ) on the summer solstice day in 200 BC. This experiment can be understood by looking at Fig. 1. The angle a at the center of the earth would be equal to



( 1 )

where, R is the radius of the earth. Then, the circumference, C , would be equal to



( 2 )

The distance between the two places S was known to be 5000 studies . From this information, one can calculate C using

( 3 )

The value of C comes out to be 250, 000 stadias. The latitude of the place Ѳ

can be determined by an experiment performed at noon on the equinox day by using a vertical gnomon shown in Fig. 4. Here, we have

( 4 )
Coming to Aryabhatt’s value of the circumference, he was quite familiar with Madhya Pradesh where there are many astronomical sites on the Tropic of Cancer ( Ujjain, Udayagiri near Sanchi, and Eran near Sagar ) which were used during the Gupta Period9. The author and Dass, M. have identified the site at Eran as astronomical and belonging to the time of Aryabhatt. Similarly, there is a paper available by Sharan, and Balasubramaniam about Udayagiri11. For other works on the Udayagiri site one has to search for papers written by either R. Balasubramanium or M. Dass as co-authors.

We have to bear in mind that Ujjain had been an important place in the Indian History since India’s Bronze Age ( 1800 BC to 1400 BC ) after the Indus Valley Civilization days. Emperor Ashoka was the Viceroy at Ujjain , and had married the daughter ( mother of Samghamitra and Mahendra ) of a business man at Vidisha near Udayagiri. Udayagiri finds mention in Kalidasa’s works, and also in the Betaal Kathaa. There are some pictures of Ujjain, Udayagiri, and Eran shown in this work starting with Fig. 6. During the Samudra Manthan, the Amrit fell at four places – Haridwar, Prayaga ( Allahabad ), Ujjain, and Nasik. So, these places have become the sites of the Kumbha Mela held at the interval of 12 years – the period of planet Jupiter.



So, just like in Egypt, one can select Ujjain at the Tropic of Cancer and one other location either X1 ( to the south ) or X2 ( to the North ), and carry out experiments just like Eratosthenes. The relationship between the arc distance along the earth’s surface, and the angle at the center of the earth are shown in Figs 3A, and 3B respectively.

Thus, Aryabhatt could determine the circumference of the earth just like Eratosthenes. As pointed out earlier – using Fig. 4, he knew the latitude of Ujjain, and it could have been quite accurate because he was familiar with issues involving accuracies. He had determined the value of to four decimal places. He had himself come up with trigonometric functions or tables.

Now, let us look at the first part of the statement – “From the centre of the land and water, at a distance of one-quarter of the earth’s circumference lies Laňkā”.

Looking at the Figs 3A or 3B, he is referring to the arc distance from the South Pole to the point on the Equator ( Lanka ) which is equal to C / 4. The value of C was determined by him to be equal to 3299 Yojanas using Eq. ( 3 ).

Next, let us look at the next part of the statement – “and from Laňkā at a distance of one-fourth thereof, exactly northwards, lies Ujjayinī

Here, without knowing the latitude of Ujjain, he could not have come up with this statement but, in both parts of the statements, he is using simple fractions. Therefore, the question of Aryabhatt not being aware of the location of Ujjain does not arise at all.


Aryabhatt must have performed an experiment just like Eratoshenes . This is quite clear from his description – “ From the centre of the land and water, at a distance of one-quarter of the earth’s circumference lies Laňkā; and from Laňkā at a distance of one-fourth thereof, exactly northwards, lies Ujjayinī “ .

If we refer to Figs. 2, it shows the prime meridian through Ujjain in India at his time. He could have measured the latitude in Fig. 4 quite accurately but, while making this statement in a simplest possible form – he expresses it as one sixteenth of the circumference. The circumference is a very large number. When we are dealing with numbers so large as the circumference of the earth, the percentage error or relative error due to a slightly simple statement would be small.

This author believes that - One has to look at the context in which a statement is made

One should not think that the best possible angle that he ( Aryabhatt ) measured was one sixteenth of 360 degrees which is 22.5 degrees.

Similarly, when he said or expressed any where that in the use of gnomon in Fig 4, he come up with 5:12 ratio in a simple manner , we should not infer that his measurements turned out to be integers or whole numbers. These are just rounded figures expressed as integers, and it is highly unlikely that in his measurements also – he had 5:12 ratio.

Now, let us look at Fig. 5 which shows two concentric spheres of radii R1, and R2 respectively. For a given angle , the radii of circles for these two spheres would be



( 5 )

and

( 6 )

So, for those two circles of radii r1, and r2 respectively, we can write the relation between their respective arc distances , and per degree as



( 7 )

( 8 )

Now, let us look at values of C that Chandra Hari uses. What he must mean is that the scientists at first computed C1, and the corresponding value of was a non integer. So, they rounded it () to an integer value of at their respective places of observations and found C2 which is another sphere at their respective latitudes.

He believes that Aryabhatt I was born in Kerala at Ponnāni (10°51′N) because Aryabhatt adjusted the value of C to get an integer value of 9. He has presented a table ( Table 7 ) showing the integer values obtained by various researchers at the place of their observation.
Chandra Hari does not even state in his paper as to how these scientists in India could get the value of C by observation at one point only ? Obviously, there had to be another point.
Chandra Hari does not question the values of C obtained by those researchers which show enormous variations from the true value obtained by Aryabhatt I. Obviously, to prove his point, he does not need to question that even though he has criticized Aryabhatt for his negligible inaccuracy ( in comparison to the magnitude of inaccuracy in the C values of other scientists ) regarding the latitude of Ujjain. Chandra Hari concludes from this inaccuracy of Aryabhatt that he did not know the location of Ujjain.
Brahmagupta’s results at latitude 26°61′N in Table 3 do not follow this ( integer rule ) nor Vateśvara’s value in Table 6 corresponding to Ujjain . Bhaskar II shows two different values of C ( Tables 4, and 5 ) which are quite different from each other.

The question is: Could Aryabhatt I not have gone to Ponnani which was at the maximum possible distance on the land ( refer to Fig. 2 ) to get better accuracy in his results ? After all, even in Egypt, the distance used was fairly long for the size of the country ? This being so, can one conclude that because Aryabhatt I went to Kerala to perform one experiment – he was born and raised in Kerala ? Such conclusions are not being drawn about Bhaskara II ( about the place of birth ) ?


Hence, even after looking at Table 7, one can not draw a conclusion that Aryabhatt I was born, and raised in Kerala and studied astronomy there. This goes against all evidences including Aryabhatt’s own statement – “Āryabhatta sets forth here the knowledge honoured at Kusumapura” . These are his own words. Was Aryabhat I not intelligent enough to write this statement ?
It is mind boggling to me that any one would dismiss the words of a brilliant person like Aryabhatt and start changing it around.

Regarding no discussions on longitudes by Aryabhatt I, he did not need to do it since he was carrying out experiment on the prime meridian only.


3. CONCLUSIONS

In the present work one can find the following:

1. From the review of the Ancient History of India , it was shown that, at first, the work of Aryabhatt I was commented on exclusively by the scientists coming from North India. After the Muslim invasion, the center of the scientific work had shifted to Kerala. Only then the scientists from South India have written commentaries

2. From the statements of Aryabhatt about the latitude of Ujjain, and the circumference of earth, it was found here that he had made these statements in an approximate , and simple manner.

3. The reason behind these statements was found to be experiments performed by Aryabhatt which were similar to those carried out in Egypt in 200 BC.

4. It was found that it is possible that Aryabhatt might have gone to Kerala along the longitude of Ujjain to perform the experiments due to the integer value appearing in the circumference expressed per degree at the Kerala site.

5. Based on the statements of Aryabhatt about his famous work - Aryabhatia, and other evidences presented in this work, it was conclusively shown that he belonged to Kusumpura and not Kerala.

4. REFERENCES

1. http://www.geocities.com/dipalsarvesh/mathematics.html#a6

2. Georges, I., 2005, The Universal History of Numbers – II, Penguin Books India, New Delhi, 2005, p. 182.

3. Sarma, K. V., Doctoral thesis, Panjab University, 1977, vol. I, pp. 6–8.

4. Ibid, p. xviii.

5. Chandra Hari, K. , “ http://www.ias.ac.in/currsci/oct252007/1177.pdf “

6. Joseph, G, G., “The Crest of the Peacock: Non - Europeans Roots of Mathematics “,Princeton,U. S. A , 2000, Chs. 8, and 9.

7. Kaye , G. R. , 1981 , " Hindu Astronomy " , Cosmo Publications , New Delhi, India , p 39

8 Abhyankar , K. D. , and Sidharth, Editors , 1993 , " Treasures of Ancient Indian Astronomy " , Ajanta Publications Delhi , India , pp 47 - 77

9 Sharan, A. M., and Dass, M. , http://www.engr.mun.ca/~asharan/ERAN/ARYABHATT_ERAN_V1.htm

10 Kaul, A. K. , Message #23202 of 23208 (http://groups.yahoo.com/group/hinducivilization/ ) NOVEMBER 14, 2007.

11 Sharan, A. M., and Balasubramanium, R., 2004, "Date of Sanakanika Inscriptions and Its Astronomical Significance for Archaelogical Structures at Udayagiri.", Current Science, Vol. 87, No 11 , pp. 1562 - 1566 .



APPENDIX A

For the convenience of the readers, the following text is reproduced below in the words of Kaul A. K.



http://groups.yahoo.com/group/hinducivilization/
Message #23202 of 23208
NOVEMBER 14, 2007

--- In hinducivilization@yahoogroups.com, "Avtar Krishen Kaul"


Dr. Anand M. Sharan ji,
Namaskar!
I have been pondering on the "native place" of Aryabhata for quite some time. However, whichever way we look at it, it appears quite doubtful that Aryabhata's native place was Kerala, though there is a remote possibility that he may have visited Kerala at a later date.

Following are the main reasons for my such an assessment:


1. "Commenting on this stanza, Bhasakara-I writes:'Kusumapura is Patliputra. Aryabhata sets forth the knowledge honoured there. This is what one hears said : Indeed this Svayambhuva-sidhanta was honoured by the learned people of Kusumapura (Patliputra), although the Paulisha, Romaka, Vasishtha and Saura Sidhanta were also known there." --- "Aryabhatiya" (page 33) edited by K. S. Shukla and K V Sarma, published by INSA

The following points emerge from this statement:


  1. Bhaskara-I (629 AD) was the first and the earliest commentator of Aryabhatiya, hardly a century away from Aryabhata, who must have passed away at least around 530 AD presuming that he lived only for about fifty-three years. Bhaskara-I, as such, could have been more aware of the nativitiy of Aryabhata. Thus if Bhaskara-I says that Aryabhata was a native of Patliputra, we have no reason to doubt him. We cannot "accuse" him of being parochial either since Bhaskara-I was from somewhere in Kathiawar whereas Patliputa was/is in Bihar!


ii) The word Kusumapura is much more akin to Sanskrit names from Northern and Central India than to any area in Kerala. For example, we had Padmapura in Kashmir which is known as Pampore these days!

iii) Pitamaha Brahma is worshipped more in Northern India (though we have just one temple for Him at Pushkar!)than in Kerala. As such, if Aryabhata had been a Keralite, he would have preferred to pay obecience to Kartikeya or even Shiva than any other deity.

iv)"Ashmaka" could mean a place where stones were being excavated, since "ashma" means "stone" in Sanskrit. It could be that Aryabhata was born in some village or suburb where stone excavation/crushing was going on and later he had shifted to Patiliputa where he had
learnt astronomy. But Ashmaka as well certainly could not mean a place in Kerala since if he had been a Keralite, he would have tried to learn astronomy there itself instead of coming to Patliputra.

2. Varahamihira (about 505 AD) was almost a contemporary of Aryabhata. As per S B Dikshit, Varahamihira has referred to Aryabhata in his Panchasidhantika. If Aryabhata had been a Keralite, his works would not have come to the attention of VM that fast in those days.


3. Arya-sidhanta of Aryabhata has/had given (lifted) virtually the mean elements of all the planets as given by VM in the Surya Sidhanta of his Pancha-sidhantika! This system of Aryabhata is/was known as "ardharatrika" system. Panchasidhantika does not appear to have been in vogue in Kerala in about fifth century AD. As such, it is impossible that Aryabhata could have copied the Mean elements of planets from the Surya Sidhanta of Panchasidhantika if he were in Kerala at that time, though he could have done it much easily at


Patliputra.

4. Aryabhatiya also gives all the Mean elements of planets of the Surya Sidhanta of anchasidhantika with the only difference that they have been manipulated to make them yield zero degrees mean longitudes at 6 am, Ujjain Meantime (and not the sunrise time, as presumed wrongly!) of Feburary 18, 3102 BCE, instead of mean midnight of Feb 17/18, 3102 BCE. Since, as already stated, Panchasidhantika does not appear to have been available in Kerala then, it means that even Aryabhaitya could not have been compiled if Aryabhata was in Kerala then!

5. If Aryabhata had been a native of Kerala, the chances are that he would have compiled his works in Malayalam language than in Sanskrit since I am not aware that we have any Sanskrit wroks from South India of that period i.e. fifth century AD. At least, a Malayalam version or a Malayalam commentary of Aryabhati could have been prepared by Aryabhata himself.

6. Even the first available commetnary on Aryabhatiya is not in Malayalam but Sanskrit.

7. In his paper about the determination of Eclipse by Aryabhata in 5th century in Kerala, K. Chandra Hari has given heresay as proofs that Aryabhata had observed that eclipse there. And as hearsays are hearsays, we can be sure that they are just wishful thinking!

However, I shall write separately about "legends" and "hearsays" while discussing the feasibility of Arybhata having "observed" that eclipse in Kerala.


Thus we can say conlusively that Aryabhata was not a native from Kerala but Patliputra.


With regards,


Avtar Krishen Kaul

APPENDIX B

TABLES OF RESULTS AS THEY APPEAR IN5


Table 1.

Equatorial

circumference of Sūryasiddhānta

Circumference

Integer yojanas




Latitude (φ)

at 0°N (C) yojanas

per degree (Y)

cos φ= Y ×360/C

(degrees)

5040

8

0.57

55.15

5040

9

0.64

49.99

5040

10

0.71

44.42

5040

11

0.79

38.21

5040

12

0.86

31.00

5040

13

0.93

21.79

5040

14

1.00

0.00

5040

15

1.07



Note: 4230 yojanas at the latitude of Alexandria became 5040 yojanas at the equator.

Greeks had been smaller units like stadia 4320 yojanas = 4320 × 50 = 216,000 stadia

and 5040 yojanas = 5040 × 50 = 252,000 stadia.



Table 2.

Equatorial circumference of Āryabhatīyam

Circumference

Integer yojanas




Latitude (φ)

at 0°N (C) yojanas

per degree (Y)

cos φ= Y ×360/C

(degrees)

3299

5

0.55

56.93

3299

6

0.65

49.10

3299

7

0.76

40.19

3299

8

0.87

29.19

3299

9

0.98

10.85

3299

10

1.09





Table 3.

Brahmagupta latitude 26°61′N

Circumference

Integer yojanas




Latitude (φ)

at 0°N (C) yojanas

per degree (Y)

cos φ= Y ×360/C

(degrees)

5000

9

0.65

49.61

5000

10

0.72

43.95

5000

11

0.79

37.63

5000

12

0.86

30.23

5000

13

0.94

20.61

5000

14

1.01





Table 4.

Bhaskara II, φ= 19.57°

Circumference

Integer yojanas




Latitude (φ)

at 0°N (C) yojanas

per degree (Y)

cos φ= Y ×360/C

(degrees)

4967

9

0.65

49.28

4967

10

0.72

43.55

4967

11

0.80

37.13

4967

12

0.87

29.57

4967

13

0.94

19.57

4967

14

1.01





Table 5.

Second value of Bhaskara-II at φ= 23°55′

Circumference

Integer yojanas




Latitude (φ)

at 0°N (C) yojanas

per degree (Y)

cos φ= Y ×360/C

(degrees)

3927

9

0.83

34.41

3927

10

0.92

23.55

3927

11

1.01






Table 6.

Vateśvara: Ujjayinīas reference

Circumference

Integer yojanas




Latitude (φ)

at 0°N (C) yojanas

per degree (Y)

cos φ= Y ×360/C

(degrees)

3311.24

7

0.76

40.44

3311.24

8

0.87

29.57

3311.24

8.5

0.92

22.46

3311.24

9

0.98

11.91

3311.24

10

1.09





Table 7.

Data summary of the places of astronomers

Astronomer

rat 0°N (C) yojanas

Yojanas per degree Y at φ

cos φ= Y ×360/C

Latitude (φ°)

Place of choice/native

Eratosthenes

4320

12

0.86

31.00

Alexandria




5040

14

1.00

0.00

Equator

Āryabhata

3299

9

0.98

10.85

Ponnāni




4948

13.5

0.98

10.82

Ponnāni

Brahmagupta

5000

13

0.94

20.61

Bhilmala

Bhāskara-II

4967

13

0.94

19.57

Bid.




3927

10

0.92

23.55

Ujjayinī

Vateśvara

3311.24

8.5

0.92

22.50

Ujjayinī

Varāhamihira

3200

8

0.90

25.84

Kusumapura

Manjula

3600

8 (3240)

0.9

25.84

Prakāśa




4800

12 (4320)

0.9

25.84

25°36′N



























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