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 (
1
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
2
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?
3
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
 
S
R
(1)
where, R is the radius of the earth. Then, the circumference, C , would be equal to
C2  R
(2)
The distance between the two places S was known to be 5000 studies . From this information,
one can calculate C using
C 360

S

(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
p
arctan 
b
(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
4
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 onefourth 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
5
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
r1R1 cos (  )
(5)
r2R2 cos (  )
(6)
and
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
1

cos (  )
360
C1
(7)
1

cos (  )
360
C2
(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
6
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
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
7
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.
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"
<jyotirved@... wrote:
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:
9
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 Svayambhuvasidhanta 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:
i) 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
10
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.
11
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
12
APPENDIX B
TABLES OF RESULTS AS THEY APPEAR IN5
Table 1. Equatorial circumference of
Sūryasiddhānta
Latitude
Circumference Integer yojanas
(φ)
at 0°N (C)
per degree (Y)
cos φ= Y
(degrees)
yojanas
×360/C
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.
Circumference
at 0°N (C)
yojanas
3299
3299
3299
3299
3299
3299
Equatorial circumference of
Integer
yojanas
per degree
(Y)
5
6
7
8
9
10
Latitude
(φ)
(degrees)
cos φ= Y
×360/C
0.55
0.65
0.76
0.87
0.98
1.09
56.93
49.10
40.19
29.19
10.85
–
13
Table 3. Brahmagupta latitude 26°61′N
Integer
Latitude
Circumference
yojanas
(φ)
at 0°N (C)
per degree
cos φ= Y
(degrees)
yojanas
(Y)
×360/C
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
–
Circumference
at 0°N (C)
yojanas
4967
4967
4967
4967
4967
4967
Table 4. Bhaskara II, φ= 19.57°
Integer
Latitude
yojanas
(φ)
per degree
cos φ= Y
(degrees)
(Y)
×360/C
9
0.65
49.28
10
0.72
43.55
11
0.80
37.13
12
0.87
29.57
13
0.94
19.57
14
1.01
–
Second value of Bhaskara-II at φ=
23°55′
Integer
Latitude
yojanas
(φ)
per degree
cos φ= Y
(degrees)
(Y)
×360/C
9
0.83
34.41
10
0.92
23.55
11
1.01
–
Table 5.
Circumference
at 0°N (C)
yojanas
3927
3927
3927
Table 6.
14
reference
Circumference
at 0°N (C)
yojanas
3311.24
3311.24
3311.24
3311.24
3311.24
Integer
yojanas
per degree
(Y)
7
8
8.5
9
10
Latitude
(φ)
(degrees)
cos φ= Y
×360/C
0.76
0.87
0.92
0.98
1.09
40.44
29.57
22.46
11.91
–
Table 7. Data summary of the places of astronomers
Yojanas
2πrat
cos φ=
per
Latitude Place of
Astronomer
0°N (C)
Y
degree
(φ°)
choice/native
yojanas
×360/C
Y at φ
Eratosthenes
4320
12
0.86
31.00
Alexandria
5040
14
1.00
0.00
Equator
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
3600
8 (3240)
0.9
25.84
Prakāśa
4800
12
0.9
25.84
25°36′N
(4320)
15
16
17
18
19
20
21
22
23
24