Metrological Claims Based on Delhi Iron Pillar – A Closer Look
K. Chandra Hariƒ
Abstract
Candra Iron Pillar at Mehrauli, Delhi had been the subject of many metallurgy oriented studies
in the past and the revelations on its peculiar characteristics have given it a pride of place
among our national symbols of ancient science and technology. New scholarship had its focus
on fixing the original location, astronomical significance, mathematical significance of the
segments identifiable from the design etc. Present work offers a different look in understanding
the dimensions of the Pillar and in its original installation at Udayagiri as Vis̟ n̟ udhvajā.
Analysis of the segmental lengths of the Iron Pillar has shown that in the traditional unit of
aňgulam ≈ 1.90 cm, the Iron Pillar had a height of 324 aňgulams above the ground such that it
did cast a shadow of 10 aňgulams on the Ās̟ ad̟ ha śukl̟ a ekādaśī of 402 AD. Height of the Pillar
above ground level, 324 aňgulams also reflected the number of years of the Śaka kāla ( 324)
and the Guptakāla of 82nd year its 82:242:82 dimension of length covering the originally buried
part, stem and the decorative top. 242 aňgulams of the stem reflected the Śaka year for the
beginning of Guptakāla. The split sections as decorative bell capital and the cakra with the
division 82 = 55 + 27, reflected the Guptakāla of 55 years that preceded Candragupta’s
coronation and the 27 aňgulam height of the Cakra marked the regnal years of Candragupta-II
in 402 AD when the Vis̟ n̟ udhvajā got installed on the day of the Sanakanika inscription. 27
motifs of the Naks̟ atra Cakra at the top also can be interpreted as representing the regnal years
of Candragupta-II at the time of consecration of the Pillar. Hindu chronological and Siddhāntic
astronomical aspects of the establishment of Kr̟ ta-samvat can be explained with the possibility
of retrofitting 27x17 = 459 years to AD 402 or 27x5 years to the beginning of Śaka era in 78
AD in the same manner as 402 AD was Śaka 324 = 27x12 years.
Effort has been made to set right the discussions on the magnitude of aňgulam, yojana etc visa-vis notions on Indian metrology so that future studies may evolve against the background of
correct perspectives. Siddhāntic astronomical reasons behind the postulation of differing
earth’s circumference values in Āryabhat̟ iya and Sūryasiddhānta has been explained without
resorting to any assumptions of metrological complications arising from Kaut̟ ilya’s
Arthaśāstra.
ƒ
Chief Geophysicist (Wells), Centre of Excellence in Well Logging Technology, ONGC,
Baroda-9. [email protected]
1
I.
Introduction
The Candra Iron Pillar at Mehrauli, Delhi had been the subject of many studies since 1961 when
the National Metallurgical Laboratory undertook the first major study coinciding with the
centenary year of the Archaeological Survey of India. Anantharaman and Balasubrahmaniam1
have published books also covering various aspects of the Candra Iron Pillar.
Balasubrahmaniam has discussed the identity of Candra 2 , historical context1, engineering
design, original location vis-à-vis astronomical significance3,4 and metallurgical aspects5 of the
Pillar in his various papers. Schematic drawings having dimensions and photos of the Pillar
have also been brought out in the above works. Present author, a rather late entry into this field
of study have also studied the astronomical significance of the Iron Pillar in the context of the
Sanakānika inscription and has shown that the inscription had a date 29 May 402 CE of great
astronomical significance to Udayagiri6. Further, the possibility that Vis̟ n̟ upadagiri may have
been the cradle of Indian computational Astronomy (Siddhāntic) got discussed in the
subsequent two works 7 , 8 . Present discussion pertains to few recent publications in Current
Science in respect of the metrological tradition vis-a-vis the light thrown by Harappan studies
and the analysis of the dimensions of the Delhi Iron Pillar. Interesting facts such as the
constancy of aňgulam equal to 1.763 cm, astronomical use of Hasta equal to 42 aňgulams in
Āryabhat̟ a tradition, role of a standard of Dhanus equal to 108 aňgulam of 1.763 cm etc demand
a closer look to put the scientific developments of ancient India in a correct perspective.
Identification of the correct metrological unit and its value also leads us also into some
interesting astronomical and chronological content of the Candra Iron Pillar i.e. conclusions that
can be drawn from the dimensions of the Pillar and its installational geometry at Vis̟ n̟ upadagiri,
now known as Udayagiri (23N30, 77E45).
As we will be referring to the height and width of the Pillar and also the different proportions in
the following discussion, relevant pictures have been provided as Fig.19 and 2.
II.
Important Metrological Aspects
Balasubrahmaniam has brought out the mathematical significance of the dimensions of the
Delhi Iron Pillar in a recent study and the important conclusions can be summed up as below:
1. Dimensions of the DIP are based on the basic unit of measurement aňgulam = 1.763 cm
2. Aňgulam 1.763 cm leads to significant mathematical ratios in the relative dimensions of the
identiable sections of the Pillar.
2
3. Author defines Dhanus as equal to 108 angulams in contradiction of the well known
tradition of 96 angulams.
4. Analysis point towards a metrological tradition extending from the ancient Harappan times
to the Gangetic times of 400 AD.
5. An earlier work of the same author had also established that the 227 Brahmi characters of
the inscriptions on the Iron Pillar had followed dimensions derived from the Indian angulam
of 1.905 cm.
6. Attempt has been made also to explain Aryabhata’s value of the earth’s circumference 3299
Yojanas as based on Hasta = 42 angulams.
These results have been the outcome of researches limited in their ambit and conditioned by the
local circumstances of the contrast of Harappan standard with the different sections of the Delhi
Iron Pillar. Further verification of these results demand investigation on a broader perspective
of the origin of the anthropomorphic units in India and the genesis of metrology in different
parts of the world.
Historical Review of the Length Measurement Units
Ancient units of length had their origin in the limbs of human body that served a comparison
with the objects measured.10 As for example of a unit of universal prevalence, we have the cubit
(Hasta in Sanskrit) taken to be the length of an arm from the elbow to the finger tip of the
middle finger. Units of length grew up in number as people resorted to shorter limbs as standard
in measuring shorter objects. Since the limbs varied in length from individual to invidual,
standardization of the units with appropriate sub-divisions emerged in course of time and
historical information suggests that the standard royal cubit of Egypt heralded the beginning of
metrology in third millennium BC.
Basic unit of length for Babylonians (1700 BC) was cubit (530 mm) longer than the Egyptian
cubit that measured 524 mm in modern units. 30 divisions called kus of the Babylonian cubit
was close to the digit and the Babylonian foot was 2/3rd of the cubit. Sub-divisions in a standard
scale depended on the number systems in vogue and except in Harappa/Indus where a decimal
system is known to have flourished, the divisions had been based on mostly 12 and its
convenient fractions. Indus inch had been 1.32 modern inches and 10 Indus inches became their
foot 33.5 cm. They also had scales with divisions of 0.367 inches and a hundred of these
divisions equalled a stride of 93cm. Greek system had a digit of 19.3mm, 16 digits in a foot
(30.88cm), and 24 fingers in a cubit. Greek system had been borrowed from the Babylonians
and Egyptians and Romans evolved their system based on the Greek. English foot and inch had
3
their origin from Olympic cubit which measured in modern terms 18.24 inches. Measurements
on the Great Pyramid stated to be of 500 cubit in length according to Petrie’s measurement
equalled 9068.8 inches and therefore the Egyptian cubit turns out to be 18.14in or 46.075 cm.
In different civilizations, there have been different standards such as Royal Cubits, Trade Cubits etc
and related smaller and larger divisions employed for specific purposes.11 Whitelaw presents the
contrast of the Roman and Greek system as follows:
“The Romans were strongly influenced by the Greek system and adopted many of the same units,
although the definitions were not always the same. For eg. the digitus at the root of the Roman
system is slightly smaller than the Greek daktylos and therefore the Roman foot (pes) is slightly
smaller than the Greek. The pes is also smaller than the English foot...”
Early state of the Indian tradition can be seen in appendix-1 which gives the length units as
enumerated in Arthaśāstra of Kautilya. Balasubrahmaniam has quoted the study of Damino to
suggest that the Kalibangan terracotta scale had divisions of both 17.63 mm and 19.04 mm in such a
way that 17.63*108 = 19.04*100 = 1904mm. Digit or breadth of the thumb had been the smallest
unit in common use and being an anthropomorphic unit of pre-historic origins, it cannot be
expected to follow any miniature standard such as 1.763 or 1.875 or 1.904 cm precisely in
applications. A longer units such as Hasta or Nara was a must as a standard with divisions and
sub-divisions for practical use. Man’s average height fixed locally would have been the
standard in different parts of the globe and this value would have been six feet and as
historically evolved it could have been 180 cm or six feet computed as we have inherited in
English tradition, 6x12x2.54 = 182.88 cm or somewhat near values. Greek unit of daktylos is
known to be just over 1.9275 cm and in a similar manner the angulam can also be taken as
around 3/4th of the modern inch and no standards of prevalence over the subcontinent has been
obtained in archaeological excavations to vouchsafe that the unit of Nara, Hasta or Aňgulam is
precisely so many millimeters or centimeters. When we consider the Chaldean, Egyptian, Greek
and Roman traditions, it is easy to realize the folly of defining a miniature standard like
aňgulam equal to 1.763 cm. Alternatively, if we are to define Dhanus as equal to 108x1.763
=190.4cm, it violates the well known Indian tradition of Dhanu equal to 96 aňgulams or 4
Hastas as attested in the Arthaśāstra of Kaut̟ ilya (300 BC) and 800 years after in Āryabhat̟ īya
and his later followers. In fact the proposed 108 aňgulam turns out to be an odd value of
190.4cm in contrast to the 96 aňgulams (=182.784) which could be average height or the
‘fathom’ measured from finger tip to finger tip with the hands outstretched.
Indian tradition of length units as mentioned by Bhāskara-II (1072 AD) in his work Līlāvati is
reproduced below to nullify any doubts that may arise in this regard:
4
8 Yavams constitute an Aňgulam, 24 Aňgulams make one Hasta, 4 Hastams make on Dan̟ d̟ a and
2000 Dan̟ d̟ as make one Krosa. And 4 Krośas make 1 Yojana... etc
Dimensions of the Delhi Iron Pillar
In the light of the above account on the origin and evolution of ancient anthropomorphic units
of length, a re-look can be made at the studies on the dimensional aspects of Delhi Iron Pillar
and their mathematical significance. Line of inquiry pursued by Balasubrahmaniam is evident
from the following observations:
“A noteworthy feature of Table 1 is the low error margin between the proposed and measured
dimensions, in most cases. The percentage error is defined as the deviation of the actual
measurement from the proposed measurement, expressed in terms of percentage of the proposed
measurement. It may be noted at this juncture that even if the angulam had been taken as 1.75 cm
based on the markings seen on the Kalibangan terracotta scale7 or 1.77 cm based on the markings
seen on the Lothal ivory scale8, the errors would have been similarly low. It is therefore clear that
the ancient Indian unit of measurement, dhanus, that was utilized in the Harappan civilization was
used in the design of the Delhi Iron Pillar”
Pursuing the lines of inquiry initiated by Balasubrahmaniam in his September 2008 study,
Table-2 below presents us with few interesting facts:
Table-2:
Length for
Actual
Integer
Integers A Length (cm)
Angulams (RB)
Li (cm)
L0
36
63.47
62.46
24
42.31
42.42
12
21.16
20.32
86.4
152.32
154.98
259.2
456.97
457.15
64.8
114.24
104.14
86.4
152.32
154.94
345.6
609.3
612.18
432
761.62
767.08
%
Error
1.59
-0.26
3.97
-1.75
-0.04
8.84
-1.72
-0.47
-0.72
Hypothetical
%
L0 + 1.763 =
Error
Lh
64.223
1.21
44.183
4.42
22.083
4.54
156.743
2.85
458.913
0.43
105.903
-8.01
156.703
2.83
613.943
0.76
768.843
0.94
5
Given the large segments of the Delhi Iron Pillar, integer aňgulam lengths Li as derived by
Balasubrahmaniam and the error percentage derived relative to the actual length L0 in fact does
not convey that the unit of 1.763 cm is implicit in the identifiable segments of Delhi Iron Pillar.
It is evident that even if the actual length L0 is increased by an aňgulam of 1.763 cm, relative
error remains insignificant in view of the small unit and the large lengths involved.
i.e. (L0-Li)*100/L0 ≈ [(L0±1.763)-Li)*100/L0. It therefore becomes apparent that the integer
number of aňgulams derived is no way unique and it could have been more or less by one or
two aňgulams and hence the integer angulams or the ratios derived in terms of Dhanus equal to
108 aňgulams have no evidence value that supports the Balasubrahmaniam analysis.
In contrast to the analysis of dimensions in Table-1 of Balasubrahmaniam (Sep 2008), Table-3
below presents the consequence of different values of aňgulam for the segments of the Pillar.
Table-3: Fixing the Value of Aňgulam used in Delhi Iron Pillar
Identifiable segments
of Delhi Iron Pillar
Actual
RB Notion
Length
No. of Aňgulams for different values
cm
A = 1.763
A = 1.904
A = 1.825
A=2
A = 2.13
A = 1.889
Total Height
767.08
435.10
402.88
420.32
383.54
360.13
Height of Main Body
612.18
347.24
321.52
335.44
306.09
287.41
Height – Smooth stem
457.15
259.30
240.10
250.49
228.58
214.62
Height of Cap + Chakra
154.94
87.88
81.38
84.90
77.47
72.74
406.08 406
324.00 324
242.01 242
82.02
82
55.13
55
82.04
82
33.07
33
22.46 22.45
11.20 11.20
Height of Cap
104.14
59.07
54.70
57.06
52.07
48.89
Rough Bottom
154.98
87.91
81.40
84.92
77.49
72.76
Diameter (Bottom)
62.46
35.43
32.80
34.22
31.23
29.32
Diameter (Smooth level)
42.42
24.06
22.28
23.24
21.21
19.92
Diameter - Top
21.16
12.00
11.11
11.59
10.58
9.93
It is apparent from the last column corresponding to A = 1.889 that the segment-heights of the
Pillar possibly have an integer aňgulams structure if the aňgulam is taken as 1.889 cm ≈ 1.9 ≈
1.904 as we have inherited from the well known traditions.
III.
Rationale of the Segmental Geometry of the Candra Iron Pillar
It is apparent from the above that the segments present the following integers as significant to the
ruler who paved the way for its fabrication and installation at Udayagiri.
1. Height of the main body = 324 aňgulams
It has been shown by Balasubrahmaniam that the original location of the Iron Pillar had been the
frontal area of Cave VI at Udayagiri where Candragupta’s presence on 29 May 402 AD is attested
6
by the Sanakanika inscription. Epoch in Indian tradition and Era had been Śaka 324 and therefore it
is possible that the main body of the Pillar was given a height equal to the years of the Śaka-kāla.
With the standard and generally accepted value of 1.903 cm, the 324 aňgulams have only less than
1% error and hence the rationale inferred and the magnitude are in reasonable agreement.
Height of the smooth portion is 457.15 cm and this is equal to 242.0 aňgulams when we take
aňgulam as 1.889 cm as indicated in table-3 above. It becomes therefore apparent that the
Viṣṇudhvajā was designed to reflect both the total number of years of the Śaka-kāla and also the
Gupta-kāla vis-a-vis the year of foundation of the dynasty viz., 242 of Śaka.
Arithmetic involved 324 = 242 + 82 = 81x4 = 27x12 and the symbolism of 27 and 12 viz., Nakṣatra
Cakra, got incorporated in view of their numerical representation and significance to the chronology
of the Guptas. Considering the Naks̟ atra Cakra of 27 motifs, we can see that 27x12 = 324 years
of Saka Era. 27x11 represented Saka 297 or 375 CE, year of coronation of Candragupta-II. 12 is
signified by the Cakra at the top of the Pillar. Guptakāla began in 321 CE or Saka 243 which is
27x9. And the Guptakāla was 81years elapsed = 27x3 in 402 CE.
2. Height of the burried rough bottom = 82 aňgulams
In a striking coincidence, we can find that the number 82 refers to Gupta-kāla i.e. dynastic era of the
Guptas had the 82nd year running. Burried bottom represented the past years of the Gupta dynasty.
3. Decorative bell capital + Cakra = 82 aňgulams = 55 + 27 (Cakra)
We meet with another remarkable coincidence here with the Guptakāla = 82 years which could be
split into 27 years of Candragupta-II and 55 years that preceded him. Cakra represented 27 and
these were the regnal years of Candragupta who ascended the throne in 375 AD (402 – 375 = 27).
If consider the elapsed years 81 of the Gupta-kala as 27x3 and extend the logic to Śaka-kala of
different epochs, we can findŚaka 324 = 27 x 12
Śaka 297 = 27 x 11 = Coronation of Candragupta
Śaka 243 = 27 x 9 = Gupta dynasty begins
This common factor of 27 can be extended back in time to understand the Hindu chronological and
siddāntic aspects of the epoch of Vikramābda or Vikram-samvat (Māl̟ avābda) as 402 AD – 27x17 =
57BC. The possibility arises that the Vikramābda got conceived by retrofitting 459 = 27x17 years to
the epoch 402 AD. 459 is the number of Mahāyugas in 6 Manvantaras (6x 72 = 432) and 27
Mahayugas of 7th Manvantara that preceded the present Mahyuga. Mahayuga consists of 4 padas
7
(kr̟ ta) and hence the retrofitted samvatsaras and the era were called Kr̟ ta-samvatsara. The gap
between 57 BC and 78 AD of Śaka-kāla too can be explained by the logic 27 x 5 years.
In choosing the symbol of Naks̟ atra cakra as the top of Viṣn̟ udhvajā, astronomers of Māl̟ wa had
sufficient chronological and historical reasons. Astronomical significance of the 402 AD epoch or
Śaka 324 has been explained in details in two works of the present author awaiting publication, viz
(1) 20 March 402 CE at Udayagiri: Candra Gupta-Vikramāditya Epoch of Indian Astronomy,
under publication (2) Epoch of 248 Vararuci Mnemonics of Moon and Other Hitherto Unknown
Signatures of Udayagiri in Indian Astronomy.
4. Astronomical aspect of the 324 aňgulam above the ground
Date of the Sanakānika inscription is 29 May 402 CE. 12 Significance of this date to
Vis̟ n̟ upadagiri (23N30) was that the sun rose exactly on the east and had a declination of 21.80.
Also, sun did cast a mid-day shadow of 10 Aňgulam and it can be inferred that the height of the
Pillar i.e. Vis̟ n̟ udhvajā (flagstaff of Vis̟ n̟ u) 324 aňgulams was designed for the purpose.
For a gnomon of height γ, the mid-day shadow ψ at a place of latitude φ is given as ψ = γ*Tan (φ − δ) where δ is the declination of the Sun. Mid-day declination of Sun at
Udayagiri on 29 May 402 CE was 21.76 degrees and yields the shadow for a gnomon of 324
aňgulams as 324*Tan (23.5 −21.76) = 9.84 ≈ 10 aňgulams. If we take the height above the
ground in cm as 612.18 cm, we get the shadow length as 18.6 cm or we get 18.6 = 10 aňgulam
or 1 aňgulam = 1.86 ≈ 1.9 cm. 324 aňgulams when converted to degrees became 27 which
represented the regnal years of Chandragupta-II. Nakṣatra Cakra at the top also had 27 motifs to
represent the Nakṣatras and the regnal years.
5. Diameters 33 at bottom, 22.45 at smooth/ground level and 11.2 at the top
It is easy to identify double the value of the top diameter at the ground level and thrice the value at
the bottom burried below ground. 33 may be in fact 33.6 and the offset of 0.6 aňgulam i.e. 1 cm we
find from three times the value of the top may be due to measurement error.
6. Segments viewed as fractions of the Dhanus = 96 Aňgulams
Balasubrahmaniam in his research study has pointed towards the possible representation of
standards like Dhanus in the design of the Pillar. Even though he had favored the unit of Dhanus
equal to 108 aňgulams of 1.763 cm, it may be interesting to look at the pillar with the traditional
definition of Dhanus as 96 aňgulams. Table-4 presents the relevant data:
8
Table-4: Dhanus = 108 A versus 96 A and Heights of Iron Pillar Segments
SegmentsA =1.763cm
(RB)
36
24
12
86.4
259.2
64.8
86.4
345.6
432
Fraction of
Dhanus
108 A
3/9
2/9
1/9
4/5
12/5
2/3x4/5
4/5
16/5
4
Segments
in A =
1.904 cm
32.80
22.28
10.67
81.40
240.10
54.70
81.38
321.52
402.88
Fraction
of Dhanus
96 A
3
4.5
9
12/10.
4/10.
18/10.
12/10.
3/10.
24/100.
Numbers
of A
% error
of fractions
32.00
21.33
10.67
80.00
240.00
53.33
80.00
320.00
400.00
2.45
4.25
0.05
1.72
0.04
2.49
1.69
0.47
0.71
It is apparent that any claim of Dhanus = 108 aňgulam as apparent in the design of the Delhi Iron
Pillar is superfluous. As already pointed out earlier in the discussion on table-2.
IV.
Assumed Constancy of Angulam = 1.763 cm
Research article appearing in a recent issue of Current Science13 had concluded that the aňgulam
had a constant value of 1.763 cm ever since the Harappan days of Indian civilization and on that
basis explanation was attempted for the value of earth’s circumference given by Āryabhat̟ a.
Crux of the discussion in the article is the different values of Hasta which find a mention in
Arthaśāstra of Kaut̟ ilya as variations of the units like Hasta for specific purposes. Despite the focus
on the metrological table of Kautilya, we miss the same in the article and what we see is an attempt
to define different Hastas as P-hasta (24 A) C-hasta (28 A) F-hasta (54 A) K-hasta (42 A) M-hasta
(32 A) based on a part of the enumerated table of units in a manner suited for the theory of the
author. Table-1 in appendix-1 presents the metrological data on length as seen in Arthaśāstra. 42
Aňgula unit finds mention as Kishku or Kamsa and not as Hasta14 and no fundamental role can be
seen in the enumeration by Kaut̟ ilya for Hasta. Author has used the mention of a 42 aňgulam unit
and interpreted it as Hasta to make his argument that the astronomers who had the circumference of
earth as 3300 Yojanas had been using the 42 aňgulam unit to make 1 Yojana equal to 16000 Hasta
equal to 11.85 kilo meter. All this exercise on an isolated mention of a measuring practice in
Arthaśāstra has been done to prove that India had a standardized value of aňgulam equal to 1.763
cm since Harappan times or the Bronze age. Fallacy of the notion that is getting concluded in the
paper may be understood from the fact that even in modern times we have the standard of 1 meter
and not of 1 cm or 1 mm as it is impossible to maintain and bring into practice a miniature standard
for practical reasons. Balasubramaniam’s argument for a miniature standard of 1.763 cm (1
9
Aňgulam) in contrast to the claim by Raju and Mainkar of the existence of a 50 cm (28 Angulam)
standard is interesting for the fallacy involved and therefore the same is quoted below:
“...In particular, they assumed that the hasta of 28 angulams was unchanged throughout history at a
value of 50 cm and therefore, went about analysing the other measures to fit their theory. It is not
surprising that a change in the angulam unit was inferred because of this assumption. The
procedure adopted by Raju and Mainkar is not precise because there is no supporting archaeological
evidence for assuming the constancy of the 50-cm measure through the ages, unlike several (dated)
archaeological material evidences9,10 offering strong support for the constancy of the angulam of
1.763 cm”
Whatever may be the archaeological evidence, can a small anthropomorphic unit such as Aňgulam
evolve as a standard without the prototype of a bigger unit like say the modern metre with which we
define cm and mm in terms of fractions?
Table-1 by Balasubrahmaniam depicts bigger units like Hasta, Danda, Goruta and Yojanam as
having variants with differing lengths but the common factor of a standard 1.763 cm aňgulam. But
we find no mention in the article as to how such a miniature standard could be maintained and
communicated across the 3600 years of unbroken tradition claimed. More over if there have been
different Hasta scales like 28A, 32A, 42A, 54A, 82A etc how the different standards could be in
circulation in a single name for use of the 42A scale by Āryabhat̟ a in 500 AD?
Discussion in the article on Āryabhat̟ īya to establish the argument on the invention of K-Hasta is
also noteworthy for the distorted content. To quote:
“The only clue to the measure of the yojanam that Aryabhatiya gives (in the same verse 7 of the
Gitikapada) is that one yojanam is equal to 8000 measures of nara, the height of an average person.
Using the value of yojanam as 11.847 km, according to the present article, the value of the nara can
be estimated as 1.48 m. The problem has to be analysed further by searching out clues for the length
of the nara defined in Aryabhatiya. This is beyond the scope of the present article”
Views as above distort the Āryabhat̟ a’s statement on length units seen in Āryabhat̟ īya. Verse 7 of
the Gītikā pāda which gives the value of Nr̟ or Nara as 8000 Nr̟ = 1 Yojana is not the only
information available in Ārybhat̟ īya. Āryabhat̟ a has clearly mentioned the units of Aňgula, Hasta
and their bearing to Nara in the next verse, i.e. verse 8 of Gītikā pāda. Translated by Shukla and
Sarma as – 15
“96 Aňgulas or 4 cubits (Hasta) make a Nr̟ (Nara)”
Āryabhat̟ a has therefore clearly mentioned his Hasta as of 24 Aňgulas in his work Āryabhat̟ īya
noted for its brevity in expression.
96 A = 4 Hasta = 1 Nara.
10
8000 Nara = 1 Yojana. That is, 1 Yojana = 8000 x 96 = 768000A.
Balasubrahmaniam’s discussion contradicts well known contents of Āryabhat̟ iya and unfortunately
he has also avoided literary evidence available in astronomical texts. As for example the verse
quoted above from Līlāvati of Bhāskara-II clearly spells out 8000 Dan̟ d̟ a = 1 Yojana = 14.50 km.
It may further be noted that the discussion on length units as given by Āryabhat̟ a and commented by
Bhāskara-I in tradition is in agreement with the units enumerated by Kaut̟ ilya in Arthaśāstra.
2 Vitasti = 24 Aňgula = 1 Aratni or Prajapatya Hasta
4 Aratnis = 4 Hasta = 96 Aňgulas = 1 Danda or Paurusha
2000 Danda = Goruta, 4 Goruta = 1 Yojana, 8000 Danda or Nara = 1 Yojana.
Kaut̟ ilya’s enumeration of units and the length units we find expressed in Āryabhat̟ īya are in
complete agreement. Bhāskara-I has stated that Purusha, Dhanu, Danda and Nara are synonymous
and hence one need not hesitate to accept the reading of 2000 Danda as 1 Goruta and 4 Goruta as 1
Yojana.
V.
Earth’s Circumference in Āryabhat̟ īya & Sūryasiddhānta
Generalization of the earth’s circumference value into a ‘binary model’ of 3300 and 5000 Yojanas
is also not correct and angulam of 1.763 cm offers no explanation. Āryabhat̟ a had fixed the earth’s
circumference was 3299 on the basis of his astronomical observations and the computational
verification of results such as lunar parallax depended only on the correct ratio of the earth’s
diameter, distance to moon etc. 3299 Yojanas made the circumference equal to 47860 km in modern
terms which is pretty higher than actual. Any comparison of 3299 yojanas with ≈5000 yojanas of
Sūryasiddhānta tradition is meaningless as the moon’s orbit is defined differently as 1 minute of arc
is 10 yojana in Āryabhat̟ īya and 15 yojana in Sūryasiddhānta. Radius as such will be 3438*10 =
34380 yojana in Āryabhat̟ īya and 3438*15 = 51570 yojanas. 2πR = 216000 yojanas for Āryabhat̟ a
and 324025 for Sūryasiddhānta. Only requirement of these dimensions for reasonably accurate lunar
parallax was that the ratio of the distance to moon to the earth’s radius had to be close to reality ≈
65. Given the above two definitions of the lunar orbit, equatorial radius of earth was to obviously
differ as 34380/65 ≈ 525 and 51570/65 ≈ 790≈800. This is what we see as differing earth’s
circumference in Āryabhat̟ iya and Sūryasiddhānta.
525
Ο
M
)θ
θ
34377
Parallax in the Āryabhat̟ a tradition is –
Earth
p = ATAN(525/34377) =52.5’.
11
Khagola circumference = Number of revolutions of moon x orbit of moon
= 57753336 x 21600,0 =2474720576 yojana.
Daily motion = Khagola circumference ÷Number of days in the Mahāyuga
= 2474720576÷ 1577917500 =7905.81yojana = 790.581’
Parallax p ≈ 790.5/15 ≈ Daily motion/15.
Āryabhat̟ a had defined the Khagola-kaks̟ ya different from Sūryasiddhānta by defining Candrakaks̟ ya as 1 minute of arc = 10 yojana and in the same proportion he had fixed the radius of earth
based on the horizontal parallax of moon. Author of Sūryasiddhānta took the radius to be 15 times
the value of horizontal parallax and fixed his orbital length accordingly as noted earlier.
Origin of the astronomical parameters must be sought in astronomical observations and the
planetary model available in works like Āryabhat̟ īya and Sūryasiddhānta. By redefining aňgulam or
Hasta we cannot explain the different computational methods by geniuses like Āryabhat̟ a.
Āryabhat̟ a had no reason to depart from the traditional metrological units and if at all there was any
departure from tradition, a mention of the same would have been there in his work or the
commentaries that came up later on.
The intricacy and astronomical content of the many different Indian values of earth’s circumference
have been already explained by the present author in his papers on Āryabhat̟ a and his works viz., (1)
Critical Evidence to Fix the Native Place of Āryabhat̟ a, Curr Sci Vol.93, 8, 25 Oct 2007 (2) Alleged
Mistake of Āryabhat̟ a –Light onto His Place of Observation, Curr Sci Vol.93, 12, 25 Dec 2007, pp.
1870-73.
Author missed such references where astronomical explanation has been given of the differing
values of earth’s circumference and the explicit mention of Āryabhat̟ a that the Hasta is only 24
aňgulams and thus was led to wrong inferences.
VI.
Conclusions
1. Analysis of the segmental lengths of the Iron Pillar has shown that in the traditional unit of
aňgulam ≈ 1.90 cm, the Iron Pillar had a height of 324 aňgulams above the ground such that
it did cast a shadow of 10 aňgulams on the Ās̟ ad̟ ha śukl̟ a ekādaśī of 402 AD, 29 May 402
AD.
2. Height of the Pillar above ground level, 324 aňgulams also reflected the number of years of
the Śaka kāla ( 324) and the Guptakāla of 82nd year its 82:242:82 dimension of length
covering the originally buried part, stem and the decorative top.
3. 242 aňgulams of the stem reflected the Śaka year for the beginning of Guptakāla. The split
sections as decorative bell capital and the cakra with the division 82 = 55 + 27, reflected the
12
Guptakāla of 55 years that preceded Candragupta’s coronation and the 27 aňgulam height of
the Cakra marked the regnal years of Candragupta-II in 402 AD when the Vis̟ n̟ udhvajā got
installed on the day of the Sanakanika inscription.
4. 27 motifs of the Naks̟ atra Cakra at the top also can be interpreted as representing the regnal
years of Candragupta-II at the time of consecration of the Pillar.
5. Differing values of earth’s circumference as seen in Āryabhat̟ īya and Sūryasiddhānta are
shown to be the results of different orbitals models employed rather than due to any
metrological complications in defining aňgulams or Hasta.
1.
Hindu chronological and the siddhāntic ideas that led to the establishment of Kr̟ ta-samvat
can be understood from the naks̟ atra-cakra symbolism involving the number 27. Retrofitting
of 459 Kr̟ tasamvatsara meant 17x27 = 459 years at the epoch of 402 CE and and 5x27 = 135
years for the Śaka epoch of 78 CE. Possibility arises that the Malawābda had its origin
during the reign of Candragupta-II alias Vikramaditya as has come to be concluded by
historical studies.
VII. References
1
Balasubramaniam, R., Delhi Iron Pillar: New Insights, Indian Institute of Advanced Study,
Shimla, 2002, pp.6-18.
2
Balasubrahmaniam, R., Delhi Iron Pillar, IIM Metal News, Vol.7 (2), April 2004, pp.11-17
3
Balasubramaniam, R. and Dass, M. I., On the astronomical significance of the Delhi Iron
Pillar. Curr. Sci., 2004, 86, 1134–1142.
4
Sharan, M. Anand, Balasubramaniam, R., Date of Sanakanika Inscription and its astronomical
significance for archaeological structures at Udayagiri, Curr. Sci., 2004, 87, 1562-1566.
5
Balasubrahmaniam, R., Delhi Iron Pillar, IIM Metal News, Vol.7 (3), June 2004, pp.5 - 13
6
Hari, Chandra., K., Astronomical Alignment of Iron Pillar and Passageway at Udayagiri and
Date of Sanakanika Inscription, Current Science, Vol. 95, No.1, 10 July 2008.
7
Hari, Chandra., K., 20 March 402 CE at Udayagiri: Candra Gupta-Vikramāditya Epoch of
Indian Astronomy, under publication.
8
Hari, Chandra., K., Epoch of 248 Vararuci Mnemonics of Moon and Other Hitherto Unknown
Signatures of Udayagiri in Indian Astronomy, under publication.
9
Balasubrahmanyam, R., Dass, MI, Raven, EM, The Original Image Atop the Delhi Iron Pillar,
Indian Journal of History of Science, 39.2 (2004), pp.177-203.
13
10
Glazerbrook, Richard., Standards of Measurement: Their History and Development, Nature 4
July 1931.
11
Whitelaw Ian, Measure of all Things, Quid Publishing, New York, 2007, p.13.
12
(6) above and Correspondence in Current Science Vol. 96 (3), 10 February 2009, p.330 & 331
(Replies to critical comments)
13
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.
14
Joshi, MC., Scientific ideas and elements during First Millennium B.C and later, paper submitted
at IIT Kanpur - HIST 2004. 42 Aňgula is mentioned as Kishku quoting Kaut̟ ilya. Same can be
seen in Arthaśāstra under publication with English translation.
15
Āryabhat̟ īya, INSA 1976, New Delhi, pp.16-18. Without seeing this text and the Nara, Yojana,
Aňgula and Hasta mentioned by Āryabhat̟ a, none may resort to write a research article on
Āryabhat̟ a’s Yojana and Hasta units.
14
Appendix-1: Units from Arthaśāstra
Table-1
8 atoms (paramánavah) are equal to
8 particles are equal to
8 likshás are equal to
8 yúkas are equal to
8 yavas are equal to
4 angulas are equal to
8 angulas are equal to
12 angulas are equal to
14 angulas are equal to
2 vitastis are equal to
2 vitastis plus 1 dhanurgraha are
equal to
2 vitastis plus 1 dhanurmusti
42 angulas are equal to
54 angulas are equal to
84 angulas are equal to
4 aratnis are equal to
108 angulas are equal to
The same (108 angulas) are equal to
6 kamsas or 192 angulas are equal to
10 dandas are equal to
2 rajjus are equal to
3 rajjus are equal to
The same (3 rajjus) plus 2 dandas on
one side only are equal to
1000 dhanus are equal to
4 gorutas are equal to
1 particle thrown off by the wheel of a chariot.
1 likshá.
the middle of a yúka (louse) or a yúka of medium size.
1 yava (barley) of middle size.
1 angula or the middlemost joint of the middle finger
of a man of medium size may be taken to be equal to
an angula.
1 dhanurgraha.
1 dhanurmushti.
1 vitasti, or 1 chháyápaurusha.
1 sama, sala, pariraya, or pada.
1 aratni or 1 prájápatya hasta
1 hasta used in measuring balances and cubic
measures, and pasture lands.
1 kishku or 1 kamsa.
1 kishku according to sawyers and blacksmiths and
used in measuring the grounds for the encampment of
the army, for forts and palaces.
1 hasta used in measuring timber forests.
1 vyáma, used in measuring ropes and the depth of
digging, in terms of a man's height.
1 danda, 1 dhanus, 1 nálika and 1 paurusha.
1 garhapatya dhanus (i.e., a measure used by
carpenters called grihapati). This measure is used in
measuring roads and fort-walls.
1 paurusha, a measure used in building sacrificial
altars.
1 danda, used in measuring such lands as are gifted to
Bráhmans.
1 rajju.
1 paridesa (square measure).
1 nivartana (square measure).
1 báhu (arm).
1 goruta (sound of a cow).
1 yojana.
15