Castle’s outer to inner lengths† 4:3 0.7
Middle town’s length to castle’s inner length† 3:1 0.4
Middle town’s length to castle’s outer length† 9:4 0.2
City’s length to middle town’s length† 9:4 0.6
Middle town’s length to 6:5 0.3
ceremonial ground’s length†
* proposed by R.S. Bisht
† proposed by Michel Danino
The middle town (340 x 290 m) has proportions in the ratio 7:6 (in other words, its length is seven-sixths of the width, or 16.7 per cent longer). The long ceremonial ground, which runs alongside the northern sides of the castle and the bailey, reflects a precise ratio of 6:1. There are a few more ratios at work (Fig. 9.3 and Table 9.2), the most important of which is 9:4 (or 2.25): as I found in my study of Dholavira’s geometry,37 it is not only the ratio of the middle town’s length to the castle’s length, but also of the overall city’s length to the middle town’s length. Such a precise repetition, again, discounts the possibility of chance.
Moreover, the average margin of error of Dholavira’s major ratios (Table 9.2) is just 0.6 per cent—a remarkable degree of precision, given the irregularities of the terrain and dimensions running into hundreds of metres.
Looking beyond Dholavira, I found similar ratios at work at other sites too.38 For instance, the ratio 5:4 happens to reflect the proportions of Lothal (whose overall dimensions are 280 x 225 m), of Harappa’s ‘granary’, a huge building measuring 50 x 40 m, and of a major building in Mohenjo-daro’s ‘HR’ area measuring 18.9 x 15.2 m. I could not help noting that the ‘assembly hall’, also called ‘pillared hall’, located on the southern part of the acropolis, had four rows of five pillars each (5 x 4), while its dimensions (about 23 x 27 m) were in the ratio 7:6, thus reflecting Dholavira’s main two ratios! Or, if we look at the ratio 9:4, we find it at Mohenjo-daro again in a long building located just north of the Great Bath, which measures 56.4 x 25 m. (Other ratios clearly identifiable at Mohenjo-daro’s acropolis include 3:1, 3:2, 7:4 and 7:5.)
The emerging pattern is that when it came to fortifications and major buildings, rather than leave it to chance or circumstance, Harappans preferred to follow specific ratios, which they must have regarded as particularly auspicious. Their motives may have been aesthetical or religious, or both—perhaps also cosmological, as Mckim Malville proposed. Whatever they were, we find the same love for ratios in much of Sanskrit literature, and sometimes the very same ratios.
Take the case of Dholavira’s main ratio, 5:4. The Shatapatha Brāhmana describes the trapezoidal sacrificial ground,39 the mahāvedi (No. 1 in Fig. 9.4), where the fire altars are placed: its western side is thirty steps long while the shorter eastern side is twenty-four steps—a proportion of 1.25 or 5:4. A few centuries later, India’s most ancient texts of geometry, the Shulbasūtras, giving minute instructions on the construction of multi-layered altars, repeat the same proportions for the mahāvedi but in terms of precise linear units rather than steps.40
We find the ratio 5:4 again a millennium later, prescribed in various traditions of Vāstushāstra, the Indian science of architecture. Thus, in his encyclopaedic Brihat Samhitā, Varāhamihira states that for a ‘king’s palace . . . [the] length is greater than the breadth by a quarter’.41 In other words, the length is equal to the width plus one-fourth of the width—that is, five-fourths of the width, or 5:4, as with Dholavira’s city and castle enclosures. Should we be tempted to dismiss this as a coincidence, Varāhamihira states in the very next verse that in the case of the house of a commanderin-chief ‘[the] length exceeds the width by a sixth’,42 that is to say, seven-sixths or 7: 6—the same proportion as Dholavira’s middle town!
It was in fact this double identity between Dholavira’s two main ratios and those of Varāhamihira three millennia later that drew my attention to Dholavira’s geometry: it would have been more ‘practical’ for the Dholavirian engineers to plan the city’s enclosures as perfect squares; that they chose to depart from this more utilitarian shape and build rectangular enclosures according to specific proportions betrays a clear intent, probably the same as Varāhamihira’s in the Brihat Samhitā : a desire to embed auspiciousness in the town plan. We have here a continuity not just between Harappan and classical town planning, but between the Harappan and Vedic concepts.
But then, Dholavira’s planners could not have translated such ratios onto the land without some defined unit of length, just as we would today use the metre. Here we come to a straightforward mathematical problem: given all the ratios at play and the known dimensions, can we work out the unit of length used to measure the city’s enclosures? I proposed a simple method to do just that,43 and calculated that the unit of length yielding the simplest results was 1.9 m. Expressed in terms of such a unit, all the city’s dimensions took on unexpectedly felicitous expressions (Fig. 9.5); the average margin of error with respect to the actual dimensions was a negligible 0.6 per cent.
Applying the same unit to structures at other Harappan sites gave striking results.44 Fig. 9.6 summarizes the ratios and measurements found in the major buildings of Mohenjo-daro’s acropolis, but such whole multiples of Dholavira’s unit appeared elsewhere too, for instance at Harappa, Kalibangan and Lothal.
There is more to say about the unit of 1.9 m, but we must first take a leap into historical times and pay a brief visit to Dholavira’s ‘twin city’.
FROM DHOLAVIRA TO KĀMPILYA
In the 1990s, while Dholavira was being excavated in the Rann of Kachchh by R.S. Bisht and his team, an Italian team led by the Indologist Gian Giuseppe Filippi visited the sleepy village of Kampil in the Farrukhabad district of Uttar Pradesh in the mid-Ganges Valley. A. Cunningham had probably been the first to propose, in 1878, that this was the Mahābhārata’s Kāmpilya, the capital of South Pāñchāla, whose king was Drupada, the father of Draupadī; several Indian archaeologists followed suit.
But five kilometres away from the village, the Italian team found a rectangular settlement of 780 x 660 m with remains of fortifications enclosing the city and following the cardinal directions; the place is called ‘Drupad Kila’ by the local villagers, in one more association with the epic. Substantial excavations have not yet been carried out, but judging from potsherds, baked bricks and other artefacts, the site goes back several centuries BCE.
Irrespective of Drupad Kila’s association with the Mahābhārata, when G.G. Filippi presented its discovery to R.S. Bisht in January 1998 (when the latter was Director of Excavations with the Archaeological Survey of India), he was ‘surprised to find that the dimensions and the orientation of the Drupad Kila coincided exactly with those of Dholavira’.45† But then, in Filippi’s words:
The problem is that Dholavira was a town of the Indus-Sarasvatī civilization, 2,000 years older than Kāmpilya. This fact offered evidence of the continuity of only one urban model from the Indus-Sarasvatī to the Ganges civilizations in the time frame of two millennia.46
Filippi was therefore convinced that his team’s discovery provided ‘important support to the theory of continuity’47 between the two civilizations and militated against the old ‘Aryan Invasion Theory’.48
Pending excavations, Drupad Kila provides one more link between India’s two urban phases. Let us turn to a few others.
WEIGHTS AND MEASURES
The standardized Harappan system of weights greatly impressed the early excavators: it was ‘unique in the ancient world’.49 Shaped as cubes or truncated spheres, made of chert or semi-precious stone, the weights start below a gram and stop over 10 kg, with some 14 stages in between. Taking the smallest weight (about 0.86 g) as a unit, a first series grows geometrically, with the next weight being twice that unit, the next twice again, etc.; to be precise, the series runs 1, 2, 4, 8, 16, 32, 64. The fifth weight in the series, 13.6 g, is the most common in the archaeological record. Then, instead of going on doubling the weights (which would give 128, 256 and so on), the series switches to multiples of the lower weight
s: we thus have 160 units, followed by 200, 320, 640, 1600, 3200, 6400, 8000 and, finally, 12,800.
Such a double series is also at the root of the weight system described in the Arthashāstra, as Indian metrologist V.B. Mainkar highlighted, who in addition showed that the actual weights described there in terms of a tiny seed called gunja precisely match their Harappan counterparts.50
The match is not just textual but historical too: towards the middle of the first millennium BCE, coins made of silver began to be issued in the Indo-Gangetic region, each bearing several punch marks depicting various motifs. Carefully weighing thousands of these punch-marked coins, from Taxila in particular, the historian-cum-mathematician D.D. Kosambi noted in 1941 how there was ‘every likelihood of the earlier Taxila hoard being weighed on much the same kind of balances and by much the same sort of weights, as at Mohenjo-daro some two thousand years earlier’.51
The British Indologist John E. Mitchiner52 went a step further and established correlations between the traditional system of weights used in India till recent decades and Harappan weights (Table 9.3), with a difference smaller than 1.8 per cent. Such a close match between the two systems is beyond the realm of coincidence.
Indeed, the survival of the Harappan weight system in different forms is one of the clearest cases of continuity with classical India. Although some scholars, such as R.S. Sharma, chose to see in it nothing but ‘accidental similarities’,53 most archaeologists have accepted it. ‘The Indus weight system is identical to that used in the first kingdoms of the Gangetic plains . . . and is still in use today in traditional markets throughout Pakistan and India,’ writes Kenoyer.54 (Of course, this is no longer quite true for India because of the widespread adoption of the metric standard, but elderly Indians will understand what he had in mind.)
Table 9.3. Mitchiner’s comparison between Harappan and traditional weights : the match is almost perfect.
Harappan Weights
Unity 1 2 4 8 16 32 64
Value in grams 0.8525 1.705 3.41 6.82 13.64 27.28 54.56
Traditional Indian Weights
‘Rattis’ 8 16 32 64 128 256 512
‘Karshas’ 1 2 4 8 16
Value in grams 0.8375 1.675 3.35 6.70 13.40 26.80 53.60
Do we find such a survival in the field of linear measures? The answer is admittedly less straightforward. The clearest connection, proposed by Mainkar again, rested on a piece of ivory found by S.R. Rao at Lothal, which bore twenty-seven slightly irregular graduation lines spanning 46 mm. This points to an average unit of 1.77 mm, and Mainkar proposed that this unit was precisely one-tenth of the traditional angula described in Kautilya’s Arthashāstra.55 Defined as the maximum width of the middle finger, or as eight grains of barley placed width-wise next to each other, the angula (literally ‘finger’) is the Indian equivalent of the digit found in civilizations as far apart as Egypt, Mesopotamia, China, Greece, Japan or the Roman Empire, where it varied from 1.6 to 1.9 cm. Early scholars attributed the last value to the Indian angula, but that was mainly because it amounted to a convenient 3/4 inch; more precise estimations by Mainkar and his colleague L. Raju56 led them to a value of 1.78 cm. Naturally enough, Mainkar suggested that this was very close to ten divisions of the Lothal scale (1.77 cm), a view tentatively accepted by the historian of Indian science Debiprasad Chattopadhyaya.57
Three other scale-like objects have come to light in the Harappan cities so far. The first, discovered by Ernest Mackay in his 1930-31 season at Mohenjo-daro, was a broken piece of shell bearing eight divisions of precisely 6.7056 mm each, with a dot and a circle five graduations apart, which suggests a decimal system. However, attempts by Mackay, its discoverer, to relate such a unit to dimensions in Mohenjo-daro were, by his own admission, not very successful, and he suspected the existence ‘of a second system of measurement’.58 The second scale, found at Harappa, was a piece of bronze rod with four graduation lines at an interval of 9.34 mm; nothing much could be made out of it. Finally in the 1960s, a rough 9 cm-long piece of terracotta with what seemed to be graduation lines came to light at Kalibangan; it remained half-forgotten until R. Balasubramaniam, an expert in metallurgy, noted for his work on the Delhi Iron Pillar, recently examined it: he found a unit of 1.75 cm to make perfect sense of the graduations.59 This establishes not only a close connection with the Lothal unit of 1.77 mm, but also a direct link with the Arthashāstra’s angula. The latter link is a powerful one, as Balasubramaniam showed in addition that most divisions on the terracotta scale were one-eighth of 1.75 cm, and we have just seen that one of the two definitions given for the angula in the Arthashāstra is based on eight grains of barley—each grain being therefore one-eighth of an angula.
The Kalibangan scale, rough though it may be, provides a direct connection with linear measures of the historical era. Can we get an independent confirmation of this important claim? I propose that we can.
I referred earlier to a unit of 1.9 m which, according to my calculations, is at the root of Dholavira’s town planning; if we take an average angula of 1.76 cm between the Lothal and the Kalibangan scales, we find that Dholavira’s unit is precisely 108 times that measurement: 190 ≈ 108 x 1.76 (the margin of error is a microscopic 0.04 per cent). Now, it so happens that the Arthashāstra lists several units of length besides the angula; one of them is the dhanus (or ‘bow’), which is defined thus: ‘108 angulas make a dhanus, a measure [used] for roads and city-walls . . .’60 The parallel between the Arthashāstra’s dhanus of 108 angulas and Dholavira’s unit of 108 times the Lothal-Kalibangan angula seems too striking to be a mere coincidence, especially when the Arthashāstra specifies that the dhanus is to be used ‘for roads and city-walls’. This naturally suggests that the angula-dhanus system is of Harappan origin. Moreover, from the humble baked brick to doorways, many dimensions can be expressed neatly in terms of an angula of 1.76 cm, as I showed recently.61
There is nothing surprising about a Harappan angula of 1.76 cm, since, as I mentioned earlier, it falls within the range of the traditional digit in other cultures (1.6-1.9 cm). But why create a unit of 108 rather than 100 angulas when the Harappans, as their system of weight demonstrates, commonly used multiples of ten? Varāhamihira gives us a clue in his Brihat Samhitā62 when he states that the height of a tall man is 108 angulas, that of a medium man ninety-six angulas, and that of a short man eighty-four angulas (the same heights apply to statues of various deities63); using our Harappan angula of 1.76 cm, we get 1.90, 1.69 and 1.48 m respectively, quite consistent with ‘tall’, ‘medium’ and ‘short’. This pragmatic definition based on the human body makes sense: the Harappan dhanus is the height of a tall man.
Astronomical considerations may also have however played a part: 108 happens to be the distance between the sun and the earth in terms of the sun’s diameter, as the Indologist and scientist Subhash Kak pointed out.64 Although this statement may sound too sophisticated for the protohistoric age, in reality it takes no more than a stick to verify it (Fig. 9.7): view the standing stick at a distance equal to 108 times its length and you will see it exactly as large as the sun or the moon (in mathematical language: its apparent height will be the exact apparent diameter of the sun or the moon).
Such an observation would have been well within the Harappans’ competence: the Finnish scholar Erkka Maula, studying small drilled holes on ring stones found at Mohenjo-daro,65 demonstrated that, like their contemporaries in Egypt and Mesopotamia, the Harappans devoted considerable attention to tracking the sun’s path through the year—the only way to plan the next sowing or, perhaps, prepare for a festival coinciding with the spring equinox.
Whatever the exact origin of number 108 may be (and it is sacred in many traditions from Japan to ancient Greece to northern Europe), its long tradition in classical Hinduism (108 Upanishads, dance postures, rosary beads . . .) does seem to have Harappan roots.
As regards the dhanus of 1.9 m, which I calculated at Dholavira, R. Balasubramaniam showed in a recent study66 that in combin
ation with an angula of 1.76 cm, it expressed all the dimensions of the Delhi Iron Pillar with unexpected harmony (Fig. 9.8): the pillar’s total length of 7.67 m, for instance, is precisely four dhanus; the pillar’s diameter, thirty-six angulas at the bottom, shrinks to twenty-four angulas at ground level, finally to taper off at twelve angulas at the very top. If this were not enough, the ratio between the pillar’s entire length (7.67 m) and the portion above the ground (6.12 m) is 5:4, verified to 0.2 per cent—again, Dholavira’s master ratio! This bears out what we have already concluded from the texts : Harappan ratios and linear units survived the collapse of the Indus cities and passed on to those of the Ganges Valley.
This survival found one more compelling confirmation in recent original research conducted by two architects, Indian Mohan Pant and Japanese Shuji Funo.67 Their starting point was Thimi, a town east of Kathmandu that has existed for some fifteen centuries or more. Measuring blocks defined by a succession of regularly spaced east-west streets, they found an average width of 38.42 m; besides, a pattern of divisions on a long nearby strip of fields yielded an average of 38.48 m: why the same average, and what did this number mean? Pant and Funo then jumped a millennium back into the past and turned their attention to the highly regular street pattern at Sirkap, one of Taxila’s three mounds, excavated by Marshall over many years (Fig. 9.9). A detailed study of Marshall’s plan established that the average distance between parallel streets was—38.4 m! Moreover, on the nearby Bhir mound, also excavated by Marshall, they found ‘a number of blocks [of houses] in contiguity with a width of 19.2 m’,68 which is, of course, just half of 38.4 m.
Pant and Funo, trying to make sense of such regular patterns, were led to correlate these dimensions with the Arthashāstra system of linear measures: they adopted a danda (‘stick’ or ‘rod’, a synonym for the dhanus69) of 108 angulas, and, as prescribed by the text, a rajju (or ‘rope’) of ten dandas and a paridesha of two rajjus. Their danda had a value of 1.92 m, so that the block dimensions of 19.2 m were equivalent to 1 rajju, and those of 38.4 m became a neat one paridesha. At the other end of the scale, the value of the angula was 1.78 cm (1.92 m divided by 108).
The Lost River: On The Trail of Saraswati Page 17
