Earth’s Circumference in Indian Astronomy
K. Chandra Hariƒ
Abstract
Earth’s circumference as presented in Indian astronomical texts had been a contentious issue
ever since Brahmagupta commented over Āryabhat̟ a’s value for earth’s circumference. Even
in modern times there have been attempts to configure the astronomical unit in terms of
modern metrological units. A closer look at the rationale suggests that the earth’s
circumference given in Indian astronomy is a derivation from the planetary longitude
computation model. It has been shown that the orbit model of Sūryasiddhānta is essentially
the same as that of the Ārdharātra siddhānta of Āryabhat̟ a and the model seen in Āryabhat̟ iya
is a revision based on a smaller lunar orbit of 216000 yojanas. Computation results were
dependent on the proportions of the size of sun, moon disks and orbits to those of earth and
not on their absolute magnitude in terms of metrological values. All orbital and planetary
dimensions given can be understood in terms of the horizontal parallax of moon and mean
motion of moon as fixed in the model. Use of values that differed by 33% by Āryabhat̟ a
alone in his midnight and sunrise systems convey that the dimensions given in units of
yojanas had to be used with respect to other computational parameters of the associated orbit
model. Practical application of the earth’s circumference value of the models had to be in
terms of the observation of the lunar eclipse at different places and the difference of time
could be converted to yojanas of the treatise for appropriate deśāntara correction.
Prime meridian as stated in Indian astronomical texts by a string of places beginning with the
hypothetical Laňkā also has been examined. Lack of precision in detailing the prime
meridian in later times is illustrated quoting Pr̟ thudaka and also Parameśvara.
I. Introduction
Earth’s circumference in the Indian astronomical tradition had been a subject of controversy
and intricacy not only with ancient astronomers but also from the analytical side of modern
scholarship1. It has been repeatedly attempted to justify the different ancient values with
yojanas scaled to meet the modern value of earth’s circumference without caring to
understand the rationale on which traditions of different values originated.2 Sūryasiddhānta,
regarded as containing the most ancient aspects of Indian astronomy has presented the
ƒ
Chief Geophysicist (Wells), Centre of Excellence in Well Logging Technology, ONGC, Baroda-9
1
dimensions of the earth in terms of diameter equal to 1600 yojanas and circumference as
1600*√10 = 1600*3.1623 = 5060 yojanas.3
ªÉÉäVÉxÉÉÊxÉ ¶ÉiÉÉxªÉ¹]õÉè ¦ÉÚEòhÉÉæ ÊuùMÉÖhÉÉÊxÉ iÉÖ*
iÉi´ÉMÉÇiÉÉä nù¶ÉMÉÖhÉÉi{ÉnÆù ¦ÉÚ{ÉÊ®úÊvɦÉÇ´ÉäiÉÂ**
(I.59)
“Twice eight hundred yojanas are the diameter of the earth; the square root of ten times the
square of that is the earth’s circumference”.
With the correct value of π as 3.1416, the equatorial circumference will be 5027 yojanas.
Burgess’s comments on these values are noteworthy:4
“The astronomical yojana must be regarded as an independent standard of measurement by
which to estimate the value of the other dimensions of the solar system stated in this treatise. To
make the earth’s diameter correct as determined by Sūryasiddhānta, the yojana should equal 4.94
English miles; to make th circumference correct, it should equal 4.91 miles”.
Burgess has also remarked on the use of √10 as approximation to π = 3.1416 in contrast to the
use of the proportion 3438:10800 which gives π as 3.14136 i.e. a better approximation of π in the
determination of the R-sine (Jyā) table.
Further, in IV.1 of Sūryasiddhānta, the mention of the diameters of the sun and moon can be
found:
ºÉÉvÉÉÇÊxÉ ¹É]ÂõºÉ½þ»ÉÉÊhÉ ªÉÉäVÉxÉÉÊxÉ Ê´É´Éº´ÉiÉ&*
ʴɹEò¨¦ÉÉä ¨Éhb÷±ÉºªÉäxnùÉ&ä ºÉɶÉÒÊiɺiÉÖ SÉiÉÖ¶¶ÉÊiÉ**
(IV.1)
“The diameter of the sun’s disk is six thousand and five hundred (6500) yojanas; of the moon’s
four hundred and eighty”
In the discussion following the verse Burgess has presented enlightening details of the orbital
dimensions of Indian astronomy.
1. Correct ratio of the moon to earth diameters
Modern value of the ratio of earth to moon diameters is 0.27 and the Sūryasiddhānta gives
nearly the same value as 480/1600 = 0.30. Correct estimate of this ratio supposes an equally
correct determination of moon’s horizontal parallax, distance from earth and moon’s
apparent diameter.
2. Moon’s horizontal parallax or the angle subtended by earth’s radius of 800 yojanas at the
centre of moon being fixed as 53’20”, the radius of the moon’s orbit can be derived as
(3438/53.333)*800 =64.5*800 =51570 yojanas. Moon’s orbit therefore will be 2πR =
324000 yojanas. 480 yojanas will therefore be (21600/324000)*480= 32 minutes of arc.
2
Factor 3438/53.33 = 64.5 is the ratio of the radius of moon’s orbit to the earth’s radius and its
modern value is ≈ 60.
51570
M
M
)θ
θ
800
Earth
Moon
Parallax according to Sūryasiddhānta is θ = ASIN(800/51570) =53.33’.
3. Implicit in the above is the equivalence of 15 Yojanas =1minute of arc.
4. Mean daily motion of moon and other planets is 324000/27.32167416 i.e. orbital
yojanas/sidereal period = 11858.717. Orbital circumference can therefore be computed as
11858.717*sidereal period. Solar orbit is obtained as 11858.717*365.25875648 = 4331500
yojanas. Sun’s distance from the earth will therefore be ≈ 689400 yojanas ≈ 862*800 (earth’s
radius) whereas the modern correct value is 24,000. It becomes therefore clear that the
parallax in the case of the sun (3’59.4”) is wrongly observed and the distance etc are different
from actual.
II. Understanding the Differing Picture in Āryabhat̟ īya
Āryabhat̟ īya in a sharp contrast, redefines all the above dimensions as is being enumerated
below:
1. Āryabhat̟ īya gives the diameter of earth as 1050 yojanas, diameter of sun and moon
respectively as 4410 yojanas and 315 yojanas.
2. Horizontal lunar parallax being 52.5’, the radius of moon’s orbit will be (3438/52.5)*525 =
65.485*525 = 34380 yojanas. Moon’s orbit therefore will be 2πR = 216000 yojanas. 315
yojanas diameter will therefore be (21600/216000)*315= 31.5 minutes of arc. Factor
3438/53.33 = 65.485 is the ratio of the radius of moon’s orbit to the earth’s radius and its
modern value is ≈ 60.
3
3. Implicit in the above is the relation 1 minute of arc = 10 yojanas in contrast to the 15 yojanas
used in Sūryasiddhānta.
4.
Mean daily motion of moon and other planets is 216000/27.3216685 i.e. orbital
yojanas/sidereal period = 7905.813. Orbital circumference can therefore be computed as
7905.813*sidereal period. Solar orbit is obtained as 7905.813*365.2586806 = 2887667
yojanas. Sun’s distance from the earth will therefore be ≈ 459585 yojanas ≈ 875 times the
earth radius whereas the modern correct value is 24,000.
34380
M
M
525
)θ
θ
Earth
Moon
5. It becomes therefore clear that the parallax in the case of the sun is wrong in Āryabhat̟ īya
also as the model is derived wholly from moon’s horizontal parallax. Any presumption that
the different dimensions given in Indian astronomical texts have the touch of reality as with
modern values is quite untenable as may be understood from the following table.
Reasonable accuracy can be noted only in the case of the distance to moon. If we take the
distance to sun as given by Āryabhat̟ a as correct, then a yojana has to be 325.5 KM and we get
such ludicrous values with any equivalence drawn between ancient and modern values.
Table-1: Impossibility of Equivalence between Ancient and Modern
Astronomical Dimensions
Graha
Moon
Ancient Orbit
in Yojanas
216000.000
Ancient Orbit
Modern value
Distance/
Yojana KM
Radius = Distance Distance -KM Earth’s radius for each case
34377.387
384400
65.48
11.18
Sun
2887666.800
459585.371
149,600,000
875.40
325.51
Mars
5431291.460
864414.862
227900000
1646.50
263.65
Jupiter 34250133.368
5451065.280
778,300,000
10382.98
142.78
Saturn 85114493.163
13546360.638
1,427,000,000
25802.59
105.34
4
6. Āryabhat̟ a’s ingenuity had been to keep the observations on moon as basis and to define the
lunar orbit in terms of astronomical numbers such as 216000 and 34380 etc so that the
observed mean motion of moon forms the basis of his planetary model. Also he fixed the
diameter of moon in terms of the observed disk of 31.5 minutes as 315 yojanas. As the radius
of the solar orbit is 13.369 times the lunar orbit, the diameter of sun got fixed as 4410
yojanas to conform to be observed disk of 33 minutes of arc. As the distance to moon had
been 10*R, horizontal parallax = Earth’s radius/Distance to Moon = Earth’s radius/10 in
minutes of arc. Defining the moon’s orbit with the unit 1 minute of arc = 10 yojana gave the
circumference of the sky lighted by sun as 12474720576000 and with 1577917500 days in a
Mahāyuga, the mean longitude of sun for example is obtained as –
(12474720576000*Number of days)/(1577917500*2887666.8) = Mean longitude.
or 7905.813/2887666.8 = Moon’s mean motion in minutes*10/ length of solar orbit =
0.985603 degree per day.
7. Sūryasiddhānta and Āryārdharātra siddhānta had used the lunar orbit as 15 yojanas per
minute of arc and so the dimensions differed. Principle was the same i.e. all planets had the
same speed and sizes of the orbits were proportional to the periods of the planets. Āryabhat̟ a
had put forth these principles in Kālakriyā pāda verses 12-14.
¹É¹]ÂõªÉÉ ºÉÚªÉÉǤnùÉxÉÉÆ |É{ÉÚ®úªÉÎxiÉ OɽþÉ ¦É{ÉÊ®úhÉɽÆþ*
Ênù´ªÉäxÉ xɦÉ&{ÉÊ®úÊvÉÆ ºÉ¨ÉÆ §É¨ÉxiÉ& º´ÉEòIªÉɺÉÖ **
(12)
The Planets moving with equal linear velocity in their own orbits complete (a distance equal to)
the circumference of the sphere of the asterisms in a period of 60 solar years and the
circumference of the sky in a Yuga.
¨Éhb÷±É¨É±{ɨÉvɺiÉÉiÉ EòɱÉäxÉɱ{ÉäxÉ {ÉÚ®úªÉÊiÉ SÉxpù&*
={ÉÊ®ú¹]õÉiÉ ºÉ´Éæ¹ÉÉÆ ¨É½þSSÉ ¨É½þiÉÉ ¶ÉxÉè¶SÉÉ®úÒ**
(13)
Moon completes its lowest and smallest orbit in the shortest time while Saturn takes the longest
time being in the highest and longest orbit.
+±{Éä ʽþ ¨Éhb÷±Éƒ±{ÉÉ ¨É½þÊiÉ ¨É½þÉxiɶSÉ ®úɶɪÉÉä& YÉäªÉÉ&*
+ƶÉÉ& Eò±ÉɺiÉlÉè´ÉÆ Ê´É¦ÉÉMÉiÉÖ±ªÉÉ& º´ÉEòIªÉɺÉÖ**
(14)
The linear dimensions of the rāśis are proportonal to the orbit lengths, smaller in small orbits and
larger in large orbits; so are the lengths of the degree, minute etc. But angular divisions are the
same for all planets.
5
Table-2 below is illustrative of the rationale underlying the Indian astronomical concept of
Yojana – role that the metrological unit played in astronomy in defining a planetary model for
computation of planetary longitudes.
Table-2: Orbits and Planetary Periods
Moon
Mercury
Orbit
in Yojanas
216000.000
695473.416
Relative size
of orbits
0.075
0.241
Sidereal
Period
27.32167
87.96988
Length in Yojanas
Rāśī
Degree minute
18000
600
10
57956
1932
32
Venus
1776421.436
0.615
224.69814
148035
4935
82
Sun
2887666.800
1.000
365.25868
240639
8021
134
Mars
Jupiter
Saturn
5431291.460
34250133.368
85114493.163
1.881
11.861
29.475
Graha
686.99974
452608 15087
4332.27217 2854178 95139
10766.06466 7092874 236429
251
1586
3940
8. It is therefore evident that Yojana had specific roles in astronomy and the earth’s
circumference as given in Indian astronomy is a derived value based on lunar parallax that
fitted the computational model.
III.
Ārdharātrikā paks̟ a of Āryabhat̟ a
As has become known through commentators, Āryabhat̟ a had written two treatises on astronomy
viz., Audayikā siddhānta or Āryabhat̟ īya and the Ārdharātra siddhānta. Sunrise and Midnight
systems of Yuga model have been giving entirely different dimensions of the orbital parameters
as may be noted from the Mahābhāskarīya account of the midnight system.5
1. Diameter of the earth is 1600 yojanas, of sun and moon respectively are 6480 and 480
yojanas.
2. Mean distance of the sun and moon are respectively 689358 and 51566 yojanas.
It is therefore evident that the yojana unit we find in Āryabhat̟ īya is 1.5 times the yojana of
Ārdharātrikā system. As observed by Sengupta6, the Sūryasiddhānta elements underwent
revision with the adoption of the elements of Ārdharātrikā siddhānta of Āryabhat̟ a and hence the
common values between the two ancient works.
6
3. Use of two differing values as 1050 and 1600 for earth’s diameter by an astute astronomer
like Āryabhat̟ a alone is sufficient to draw the inference that the absolute magnitude of yojana
as a metrological unit had no relevance in astronomy.
4. Use of slightly different values by astronomers and the rationale of having integer yojanas at
their latitudes has been already brought out in earlier works of the present author.7 Not only
Āryabhat̟ a but also Manjula and Bhāskara-II too had been using differing values of earth’s
circumference for computational convenience.
IV.
Use of Prime Meridian in Indian Astronomy
Planetary longitudes in Indian astronomy were computed for mean sunrise at Laňkā and the
customization of the treatises to specific locations demanded appropriate corrections to changes
in latitude and longitude. Correction for change in longitude had the traditional name ‘Deśāntara’
defined as equal to the distance from Ujjayinī in yojanas or the longitudinal difference with
Ujjayinī divided by 6 ghat̟ ikas. By rule of three the daily motion of planets is then applied to
account for the Deśāntara ghat̟ ikās. Knowledge of the prime meridian for which a treatise was
given shape to was therefore essential for practical observations and verification of computed
positions at different places. Bhāskara-I (629 AD) had defined the prime meridian in terms of a
string of places as –8
±ÉRÂóEòÉiÉ& JÉ®úxÉMÉ®Æú ʺÉiÉÉä¯ûMÉä½Æþ {ÉÉhÉÉ]õÉä ʨÉʺÉiÉ{ÉÖ®úÒ iÉlÉÉ iÉ{ÉhÉÔ*
=kÉÖMÆ ÉκºÉiÉ´É®úxÉɨÉvÉäªÉ ¶Éè±ÉÉä ±ÉI¨ÉÒ´Éi{ÉÖ®ú¨ÉÊ{É ´ÉÉiºªÉMÉÖ±¨ÉºÉÆYÉÆ**
Ê´ÉJªÉÉiÉÉ ´ÉxÉúxÉMÉ®úÒ iÉlÉÉ Á´ÉxiÉÒ ºlÉÉxÉä¶ÉÉä ¨ÉÖÊnùiÉVÉxɺiÉlÉÉ SÉ ¨Éä¯û&*
String begins with the hypothetical Laňkā (0N0, 75E45) and towards north, Khara-nagara,
Sitorugeha, Ujjayinī, Tāneśvar etc. Many places of the list are beyond identification.
Lalla has been quoted as giving a different string of places in the Mahābhāskarīya commentary
of Shukla. In fact Lalla gives no such list and he mentions only Ujjayinī-Laňkā connecting
meridian:
Gò¨ÉähÉ ±ÉRÂóEòÉäVVÉʪÉxÉÒ Ê½þ¨ÉÉSÉ±É |ɤÉrù®úäJÉÉʴɹɪÉä¹ÉÖ ¨ÉvªÉ¨ÉÉ&*
¦É´ÉÎxiÉ {ÉÚ´ÉÉÇ{É®ú{ÉkÉxÉ乴ɨÉÒ iÉiɶSÉ näù¶ÉÉxiÉ®úEò¨ÉǺÉƺEÞòiÉÉ&**
“The Laňka-Ujjayinī line (longitude) passing through Himalayas in the north to the pole is the
meridian for which the mean longitudes are correct. For either side of this line, the mean
longitudes become correct when deśāntara is applied”
7
List of places credited to Lalla in fact belongs to Mallikarjuna Sūri’s commentary which gives
the string of places, Kānchī, Kumārī, Vatsgulmam, Māhis̟ matī, Kuruks̟ etra etc.
Based on the information provided by Rai9 in his study, the places on the prime meridian as
given in Siddhāntic texts can be identified in terms of their modern latitude and longitude as
follows:
Table-3: Indian Prime Meridian as in Siddhāntic Texts
Lanka
Kanyakumari
Kanchi
0
08 N 00
12N 50
75 E 45
77 E 40
79 E 45
Kharanagar (Nasik)
19 N 58
73 E 48
Vatsyagulma
20 N 05
77 E 10
Mahishmati
22 N11
75 E 37 or 76 E 11
Ujjayini – Āryabhat̟ a
22 N 50
75 E 45
24 N 00 or 23 N 09
XX N YY
24N 40
27 N 00
28 N 54
29N 50
29N 54
75 E 45
75 E 22
77 E 40
75 E 45
76 E 38
76 E 56
76 E 56
Ujjayini
Vananagar
Tumain
Malavanagar
Rohtak
Thanesvar
Kurukshetra
It can be easily understood that the places have been enumerated as on the prime meridian
passing through Ujjain without any precise astronomical observations and verification. It is
worth remembering in this context that Āryabhat̟ a had been observing the skies at 10N51, 75E45
– precisely on the Ujjayinī meridian but by the time of Paramesvāra this fact was forgotten and
he had declared his place almost on the Ujjayinī longitude as 18 yojanasa west of the prime
meridian.
ºÉ¨É®äúJÉɪÉÉ& {ɶSÉÉnù¹]õÉnù¶É ªÉÉäVÉxÉÉxiÉ®äú OÉɨÉä*
º´É®úEÞòiɹÉ]Âõ iÉÖʱÉiÉäƒIÉä ´ÉºÉiÉÉ ¶ÉÉEäòƒIɹÉÎ]ÂõjÉSÉxpùʨÉiÉä**
18 yojanas to the west of prime meridian at the sine latitude of 647 (10N51) is the description
that Parameśvara gives to his place Aśvattha grāma or Ālattūr. If we take the Ujjayinī meridian
passing through Camravattam as reference, Ālattūr is hardly 25 KM east and there cannot be any
place west of the meridian as 10N51, 75E45 marks the point where the Ujjayinī meridian
intercepts the west coast. As a result, Parameśvara’s verse had been a source of confusion as may
8
be noted in the earlier works of Sarma10, present author’s study on eclipse observations of
Parameśvara11 etc. Parameśvara’s statement can be correct only if we assume that the prime
meridian for him had been passing through Kanyākumāri which is at modern longitude of 77E40
i.e. 2 degree east of Ujjayinī meridian (75E45) that passed very close to the place of
Parameśvara.
18 Yojanas for 2 degree longitude suggest that the earth’s circumference at the latitude of
Parameśvara had been 3240 (360x9) yojanas.
Discrepancy in the definition of prime meridian by the string of places had been in the
knowledge of ancient Indian astronomers as may be understood from the Pr̟ thudakasvamin’s
commentary on Khan̟ d̟ akhādyaka. Pr̟ thudakasvāmi has illustrated the use of prime meridian in
fixing Deśāntara for Kuruks̟ etra by taking the example of a lunar eclipse. Time difference of 1.5
ghat̟ is at Kurusks̟ etra was used to find the difference of yojanas to be 4800*1.5/60 = 120 yojanas.
It is interesting to note here that Āryabhat̟ a gave no cognizance to the traditional string of places
and instead specified the meridian simply in terms of Ujjayinī at 1/16th of the circumference i.e
longitude of 75E45 at 22N30. Verses that place Kanyākumāri, Kānchīpuram etc on the prime
meridian are of later origin and based on hearsay about such places as falling in the longitude of
Ujjayinī.
Inaccuracies in the conception of the date line through Ujjayinī is a sure indication of the lack of
precision in observing the planetary phenomena and verifying the computations. Precise value of
earth’s circumference based on actual distance between two places was therefore not a subject of
much attention with the traditional astronomers who subscribed to the verses like kanci....
Placing Kanyākumāri and Kānchīpuram on the Ujjayinī meridian shows that the south Indian
astronomers had little knowledge about the actual location of Ujjayinī and little exchange of the
observational data with the astronomers of Ujjayinī.
V. Conclusions
1. Earth’s circumference as given in Indian astronomy is a derived value based on observations
of lunar parallax and mean motion of moon that fitted the computational model.
2. Āryabhat̟ a himself had been author of two different systems - the sunrise and midnight – in
which the dimensions of orbits and disk sizes differed by 1:1.5 ratio. Āryabhat̟ īya or the
audayika (sunrise) model defined lunar orbit as 1 minute = 10 yojana while Āryārdharātra
siddhānta (midnight) model 1 minute = 15 yojanas. All other dimensions changed
accordingly in the proportion 1:1.5.
3. It has been explained that the Sūryasiddhānta contains the elements of the Āryārdharātra
siddhānta. Further the different models have been based on same principles of equal speed of
9
planets and the size of orbits proportional to the periods of planets as can be found expressed
in Āryabhat̟ īyam.
4. Practical application of the earth’s circumference value of the models had to be in terms of
the observation of the lunar eclipse at different places and the difference of time could be
converted to yojanas of the treatise for appropriate deśāntara correction.
5. Prime meridian as stated in Indian astronomical texts by a string of places beginning with the
hypothetical Laňkā also has been examined. Lack of precision in detailing the prime
meridian in later times is illustrated quoting Pr̟ thudaka and also Parameśvara.
6. Astronomical reconfirmation of the earth’s circumference value from observations of the sort
conducted by Eratosthenes or metrological or cartographic efforts to fix the unit of yojana is
missing in Indian astronomy even though yojana is described in terms of smaller
anthropomorphic units like Krosa, Nara, Hasta etc. This lacunae can be credited to over
emphasis and reliance on measurement of lunar parallax and the mean motion of moon in
fixing the dimensions.
7. Any effort to correlate the astronomical values with metrological units derived from
archaeological structures shall obviously lack scientific content and may lead to erroneous
inferences.
VI.
1
References
Balasubramaniam,R., On the confirmation of the traditional unit of length measure in the
estimates of circumference of the earth, R. Balasubramaniam, CURRENT SCIENCE, VOL.
96, NO. 4, 25 FEBRUARY 2009, pp. 547-52.
2
Shukla, K. S., Āryabhat̟ īīya, Indian National Science Academy, New Delhi, 1976, p. 19.
3
Burgess, E., Sūryasiddhānta, Chowkhamba, Varanasi (1976), p. 44 and p. 143-47
4
Ibid., p. 44
5
Bhāskara-I, Mahābhāskarīya, Ch. VII, p. 205-12, Luknow University, 1960.
6
Sengupta, PC., Khan̟ d̟ khādyaka, Part II, pp.16-17, University of Calcutta.
7
Hari, Chandra, K., Alleged Mistake of Āryabhat̟ a – Light onto His Place of Observation,
Current Science, Vol.93.12, 25 December 2007, p.1871,
8
Bhāskara-I, Mahābhāskarīya, II.1 &2, p. 10, Luknow University, 1960.
9
Rai, RN., The Prime Meridian in Indian Astronomy, IJHS Vol. 9.1, INSA, New Delhi-2
10
10
Sarma, KV., Goladipika, IV.91, Grahan̟ anyāyadīpikā etc, Visveranand Institute, Hoshiarpur
11
Hari, Chandra, K., Eclipse Observations of Parameśvara, IJHS, 38.1 (2003), p.48, INSA,
N.Delhi-2
11