15 November, 2010

Artistry at Girnar


Chandrakant Doshi

Abstract: This note examines certain artistic rendering of Brahmi script at Girnar and postulates its influence on modern Gujarati script.

KEY WORDS: Brahmi, ancient, script, inscriptions, Asoka, Girnar, Gujarati

Asoka, who ruled in India in the third century BCE, left a series of inscriptions all over his empire. These inscriptions comprise his famous edicts, found on rocks, pillars and in caves. A set of these edicts can be seen on a boulder near Girnar, in Saurashtra, Gujarat.

These edicts could be read in modern times following their decipherment by James Prinsep. Alexander Cunningham, who published a collection of these inscriptions1 in 1877, noted variations in the rendering of specific letters in the Girnar script2 when comparing with the Delhi-Topra and Dhauli edicts. He made no mention of variations in medial or diacritic marks attached to the glyph of a syllable. This note examines some of these variations observed in Girnar edicts.

The inscriptions are in the ancient Brahmi script. List of Brahmi script, along with Romanized Devanagari transliteration are available on various web sites, some of which are given in the bibliography3. I have prepared a table of Brahmi, Devanagari and Romanized Devanagari using the Kyoto-Harvard transliteration scheme and a copy is reproduced in the Appendix (Image 12). The Brahmi characters are hand written and generally follow the examples found at Girnar. I prefer Kyoto-Harvard transliteration scheme for its compactness and use it throughout this note, printing such occurrences in red.

Eugene Hultzsch4 published a fresh compilation of these edicts in 1926. He worked with fresh mechanical copies furnished by Archaeological Survey of India.

A comparison of Cunningham's copy of the rock edicts at Girnar with the estampages of Hultzsch shows that in Cunningham's reproduction the Brahmi characters are schematics, closely following the calligraphic quality of Delhi-Topra5 edict. A section of the North Face of this pillar edict is reproduced for reference, Image 13, in the Appendix. The first edict is delimited by green arrows. A copy of the Devanagari transliteration of the same is given below it (Image 14). Both these examples are from Hultzsch's corpus.

Asoka's first edict from Girnar is widely used as an illustration on many web sites. It is based on Cunningham’s copy and shown on a pink background but true colour of the rock is grey. This can be seen in the photographs of the boulder (Images 7 and 8), reproduced further down.

The Girnar edicts on the rock face are arranged in two columns, separated by a vertical line. Each edict is also separated by a horizontal line. The left hand column carries the first five edicts followed by the 13th at the bottom. The right hand column carries the edicts 6 to 12 and the 14th at the bottom. The characters in the edicts are 1.2 inches6 high.

The image of the first five edicts from Cunningham's corpus7 can be seen in the Appendix, Image 15. The first two edicts from a plate in Hultzsch’s corpus8 are also reproduced to illustrate the variations in script, Image 16.

For the close comparison of Brahmi characters in the two images, specially enlarged and annotated images of the first edict from each source are reproduced below, Images 1 and 2. A Devanagari transliteration9 of this edict from Hultzsch is also included, Image 3.

The diacritic or medial mark for the short i is a vertical line above the glyph of a syllable. In most cases it is attached to the glyph via a horizontal line, the two lines being normally at right angles. The long i is marked by two vertical lines. An example of both these can be seen in the first line of Girnar edict. In the reproduction from Cunningham, Image 1, these two occurrences are marked by blue arrows.

Examples of immaculate rendering of these diacritic marks can be seen in the first line of the Delhi-Topra edict, Image 13.

An examination of the same first line in Hultzsch's estampage, Image 2, shows a completely different situation. The short and long i in the first line, pointed by the blue arrows, are free flowing, more cursive rather than angular. Looking at the pointed glyph on the fifth line, which is the syllable mhi, the short i is rendered as an arc atop the glyph of the syllable ma. This is done in a manner completely different from the first two examples. Script of Girnar edicts represents an artistic flair not seen in other places.

This individuality of expression is clearly seen in five examples of the syllable pri, pointed by the red arrows in both the above images.

Instances of the immaculately rendered short or long i in the first edict are more an exception than a rule. Two such examples occur in the tenth line. One can be seen in the word dhaMmalipI, though both medial marks on the glyph of the syllable pa are not clearly visible on the estampage. Even Cunningham shows only one medial mark but Hultzsch’s Devanagari transliteration (Image 3) has the long i. Damage to the rock makes it difficult to make out both the diacritic marks. The other instance can be seen towards the end, where tI has two vertical marks for the long i clearly visible.

In the second edict, Image 4, both the perfectly vertical and cursive form of the diacritic mark for i are observed. However, slackness has entered on the very last word, mainly in regard to the glyph of the syllable sa.

The word is pasumanusAnaM and is enclosed in a red rectangle in the image of the estampage below.

On the left, the blue arrows point to sa (top) and pa (bottom). They both have an upright limb ending in a semicircle at the bottom. What distinguishes them is the little wing to the left of the limb attached to sa but not pa. This is clearly seen for the marked characters and can also be observed in the table of scripts, Image 12.

An examination of pasumanusAnaM shows that while pa has its limb upright, that for sa is slanted. This is observed twice in this last word but nowhere else in this edict. The improved engraving in the second edict indicates the presence of some supervision which seems to have disappeared at the last moment. The experimentation with the glyph of sa in that last word is not repeated elsewhere.

Girnar is located in Gujarat, where the regional language is Gujarati. Its script is based on Devanagari, with two essential differences. This is best illustrated by an example.

The horizontal line on top of each character is absent in Gujarati. In Devanagari, the vertical limb in a character or a diacritic is a rigidly straight line. In Gujarati such lines end in neat little loops.

The Brahmi script of the Delhi-Topra pillar edicts with its perfect calligraphy is also observed in some of the other pillar edicts as well as many rock edicts. The artistic flair observed in the Brahmi script of Girnar edicts is not found elsewhere.

The Devanagari script reflects the calligraphic quality of the Delhi-Topra edict while Gujarati reflects the cursive nature of characters found in the Girnar edicts. The rendering of the Brahmi in the Asokan edicts at Girnar, with their artistic flair, makes its appearance in the adaptation of Gujarati script from Devanagari.

Asokan edicts are over 2200 years old while modern Gujarati script has been in use for only a few hundred years. Examples of the loops in which the vertical limbs end, as noted above, can be seen in the centuries old copper plate inscriptions in modern Gujarati. One such example10 is reproduced in the Appendix, Image 17. It is dated Vikram Samvat (VS) 1711 (=1655 CE) and confirms a land grant originally made in VS 1611.

Photos of the Girnar rock.

It is rare to see many photos of the rock or pillar with the inscription readable. As luck would have it, while searching the web for material about Girnar, I chanced upon two photographs of the boulder with the Asokan edicts. With the kind permission of the photographer, Manish Khamesra, these photos are reproduced here.

The first photo, Image 7, shows the full height of the boulder with the right column in view. Edicts 6 to 12 and 14 are engraved on this face, though it is difficult to read the edicts at the top of the nearly conical rock. However parts of other edicts are possible to read. Horizontal lines separating each edict are also clearly visible.

Second photo, Image 8, is a close up of the lower part of the rock and shows the three lower edicts. Two horizontal lines separate the three edicts. The top edict, which has only a few lines showing in the photo, is the eleventh edict. Below it, between the horizontal lines, is the twelfth and at the bottom is the fourteenth edict. A part of the left hand side of the inscription is not visible in this photograph but can be read from the first, Image 7.

What is most striking about the photo is the crystal clear quality of the inscription. The estampage does not do justice to this beautiful work of art. A close examination of the photo shows areas where the rock has flaked. The area is lighter in shade and the incision of the character is shallow. These photos demonstrate the need to have all these edicts photographed.

A copy of the Devanagari transliteration of the fourteenth edict along with Hultzsch’s estampage of the same is reproduced here (Images 9 and 10). Blue arrows on the image of the Devanagari transliteration mark the start position of each line of the inscription visible in the photograph above. Hultzsch’s translatation12 of the fourteenth edict is also included (Image 11).

The cursive diacritic for i, already observed, is still to be seen in the final edict, Image 10. Green arrows point to these in the first line of the 14th edict. Yet errors do creep in and can be seen at the very beginning of the first line, pointed by red arrows. The horizontal arm of the diacritic should be attached at the very top of the vertical limb of each glyph. After the first errors at the very beginning, the supervision is ever present, as pointed by the blue arrow; a missed glyph is being squeezed in. This is clearly visible in the photo above.

The technology of digital cameras has placed in the hands of researchers a tool that can provide copies of inscriptions quicker and of a better quality than those produced by earlier methods. They can augment the copies and estampages first produced nearly two centuries ago. A gallery of photos in public domain, such as Images 7 & 8 above, would surely inspire a few minds to study these ancient script and inscriptions.

This brief examination has revealed not only the artistry in the engraving of some characters but also occasional absence of supervision during engraving. The artistry in the glyphs points to a local tradition of the third century BCE that has shown itself in the modern Gujarati script.

A detailed examination of all the edicts at Girnar will help to identify the artistry mentioned in brief in this note but also enable us to understand the processes involved in engraving the edicts in the third century BCE.


I wish to express my thanks to Norman J. Street for providing access to the books not easily available. Many thanks also to Manish Khamesra for the kind permission to use the photos of the boulder at Girnar. Thanks also to my son Samir and his wife Susan for valuable comments and helpful discussions.



1/. Corpus Inscriptionum Indicarum Vol. I
Inscriptions of Asoka by Alexander Cunningham.
Office of the Superintendent of Government Printing, Calcutta, 1877

2/. Cunningham, op cit, page 14 - 15

3/. The following web sites have tables of Brahmi characters.

4/. Eugene Hultzsch
Corpus Inscriptionum Indicarum, Vol 1,
Inscriptions of Asoka, Government of India, 1925

5/. Hultzsch, op cit,
Image, p 122, inscription, p 119

6/. Cunningham, op cit, p 14

7/. Cunningham, op cit, Plate V

8/. Hultzsch, op cit, p 4

9/. Hultzsch, op cit, p 1

10/. The Cave-Temples of Western India by James Burgess and
Bhagwanlal Indraji Pandit, 1881, p112

11/. Manish Khamesra is a contributor on the website http://www.Ghumakkar.com
where the photos originally appeared.

They are part of a travel story "Junagadh through the ages", which can be accessed here: http://www.ghumakkar.com/2010/01/05/junagadh- %E2%80%93-a-journey-through-ages/

12/. Hultzsch, op cit, p26

© 2010 Chandrakant Doshi


23 April, 2010

An Externally Pivoted Balance


Chandrakant Doshi

email: jignashi@yahoo.co.uk

Abstract: This paper describes a balance in which friction is no hindrance to the accuracy of mass comparison.


The beam balance has a long history as an instrument of mass comparison. The earliest balances were suspended by a cord wrapped round the beam. Later versions had a central hole in the beam for the cord. In modern precision balances a knife-edge is used.

In a conventional balance as an equal mass comparator the point of suspension of the beam should be accurately located at the centre of the beam. In this paper I describe a method of suspending the beam that does not require the location of its centre.

The conventional balance has the point of suspension, or pivot, within the body of the beam and can be thought of as an internally pivoted balance.

The proposed balance has its pivot external to the beam and is simple to construct. In this paper, it is referred to as an externally pivoted balance.

The balance can be constructed using the materials excavated from various Harappan/Indus sites. While the balance was envisaged as a weighing instrument the ancients might have used, it is analysed here as an instrument in its own right.


The balance comprises a rigid beam suspended by means of a string tied to its ends over a rough pin so that it takes the shape of a triangle with the beam as its base (see Fig. A). Friction at the point of suspension is necessary for the arrangement to work as a balance.

The friction permits the beam to be inclined to the horizontal and still serve as a balance. The range of inclination about the horizontal is restricted and too large an inclination will drive the beam to a vertical position. Increasing the friction at the point of suspension will increase the usable range. An arrangement for clamping the string on the pin can also be useful.

So long as the beam inclination stays within the restricted range the arrangement behaves like a rigid body suspended from a frictionless pivot and hangs with its centre of mass directly below the point of suspension.

Pans are easily attached at A and B by tying the harness to the loose ends of the string ACB. The arrangement may not be as good as the knife-edge pivots but is compatible with the suspension arrangement of the balance. Flexibility of the string would keep the pans hanging vertical.

Equilibrium can be detected by suspending a second beam parallel to the main beam to serve as a reference.


A comprehensive analysis of the conventional beam balance is given in Practical Physics1 . A modern treatment is available in Comprehensive Mass Metrology2 . History of the ancient balances can be read in Skinner3 and Kisch4.

AB is a rigid beam and ACB an inextensible string. The arrangement is suspended on a rough pin at C, as shown in Fig. A. For clarity, the pans and harness are not shown in the sketch.

Assume the beam to be uniform and the string to be of negligible mass.

Let the mass of the beam be M, acting at its centre of mass (CoM).

Let pans, each of mass P (including the harness), be attached at A and B, respectively. To calculate the sensitivity of the balance under load, add an appropriate mass T to each pan. Manually adjust the balance so that the beam, AB, is horizontal. The arrangement hangs so that the centre of mass of the assembly, D, is directly below C.

Let AD = l1 and BD = l2 .

When a small mass m is added on the pan at A the beam changes position, without the string slipping about C, so that the centre of mass is again below C.

In the new position the points A, B, and D move to A´, B´ and D´, respectively. A´H is drawn parallel to AB. Let angle(B´A´H) = θ. This is also the angle of inclination of A´B´ with AB. Let the point of intersection of A´B´ and CD be K, the centre of mass of the balance in the new position.

Taking moments about CD, we obtain

where A´K, B´K and KD´ are lengths on the beam, as marked on Fig. A, and g is acceleration due to gravity.

Dividing both sides by g⋅cosθ we obtain

Now, A´K = A´D´ - KD´ = l1 - CDtanϴ
since KD´ = CD´tanϴ and CD´ = CD

Also, B´K = l2 + CDtanϴ

Hence, (1) becomes

which simplifies to


as an expression for added mass in terms of ϴ and the balance constants.
Equation (2) shows that if the CoM of the beam is not in the middle of the beam an error will occur when making a mass comparison.

Putting l1 = l2 = l in equation (2) we obtain

Since m is much smaller than 2P + M it can be ignored in the RHS of the above equation. Also, for small θ, tanθ ≈ θ , and equation (3) simplifies to

Sensitivity of a balance is defined as radians per gram. The above equation can be written as

This is the sensitivity of the balance with a loading of 2T. By putting T = 0 we obtain the sensitivity of the unloaded balance, equation (5).

Equations (4) & (5) show that the sensitivity of the balance decreases with increasing load.


The externally pivoted balance offers some advantages over the conventional, centrally pivoted balance.

Friction at the point of suspension is essential for the operation of the balance but it does not introduce any errors when comparing masses. Also, unlike the conventional balance, knowledge of the location of the centre of the beam is not required.

Errors in measurement will occur if the centre of mass of the balance is not at the centre of the beam, as shown by equation (2). This can be corrected by the addition of weights at an appropriate place on the balance. A method of weighing known as substitution weighing eliminates this source of error. Accuracy of measurement also depends on the accurate detection of equilibrium.

Several Harappan sites have yielded materials, such as pans and beams, with which to construct a balance. Harappan weights have been analysed as more accurate than those of either of the contemporary civilizations in Egypt and Mesopotamia. The externally pivoted balance, with fewer sources of errors, could explain the achievement of such an accuracy. However, no actual example of such a balance has been reported from any Harappan site.


I would like to thank Dr. Stuart Davidson of National Physical Laboratory, Britain, for reading the paper and making valuable suggestions. I also thank my son Samir and his wife Susan for useful discussions and comments.
Many thanks to the librarians at the Keith Axon Centre of the Redbridge Libraries for obtaining the essential books.


1/. Practical Physics by R. T. Glazebrook and W. N. Shaw. Longmans, 1893 – pp. 99.

Copies available on Internet Archive. Link to one such copy below:

2/.Comprehensive Mass Metrology

Edited by Manfred Kochsiek, Michael Glaser, Weinheim ; Chichester, 2000

3/. Weights and Measures:

their ancient origins and their development in Great Britain up to AD 1855 by F. G. Skinner, formerly Deputy Keeper in the Science Museum, London. HMSO 1967

4/. Scales and weights:

a historical outline by B. Z. Kisch, Yale University Press, 1965.

© 2010 Chandrakant Doshi


10 January, 2010

Ancient Weights from South Asian civilization


Chandrakant Doshi

email: jignashi@yahoo.co.uk

Key words: Harappa, Indus, weights


The ancient Harappan or Indus civilization, now widely known as the South Asian civilization, came to light with the excavations at Harappa in Punjab and Mohenjo Daro in Sindh, both now in Pakistan. The civilization flourished in the third millennium BCE. The first report on the excavations there listed, among various objects, a number of weights. These are exquisitely cut and polished pieces, weighing from about half a gram to over 11 kilograms. They are made of various materials: chert, a hard stone, being the most popular. They come in many different shapes: cubical, cylindrical, pyramidal, conical, barrel and truncated spheres.

A. S. Hemmy reported on these weights, defining the unit weight and scale of the system. The weights from the first four excavation reports are examined in this paper using tables and charts to show not only the systematic nature of the weight system but also the level of accuracy attained.


The excavation report of John Marshall1, “Mohenjo Daro and the Indus Civilization,” lists weights from both Mohenjo Daro and Harappa. Later excavations by M. S. Vats2 at Harappa and John Mackay3, 4 at Mohenjo Daro and Chanhu Daro have listed additional finds of weight specimens from these sites. Details of these are given in respective excavation reports.

Marshall`s report included weights from both Harappa and Mohenjo Daro. Further sessions at these sites produced additional reports. Details about the weights in each report vary considerably. For ease of reference these weights from the first four excavation reports are defined as Lists. Table 1 identifies each List. The details of the site, excavation report and the number of weight specimens are given for each List.

Information included for the weight specimens differs in each report. No information besides the weight is recorded in List 1 while List 2 makes mention of material and condition of some of the specimens. List 3 includes, besides the weight and material of each specimen, shape of each piece. List 4 provides the most comprehensive information: weight, condition of the specimen, shape, size, material and location of each piece. List 5 records only the weight and condition of each specimen.

In Lists 1 and 2, three damaged weights were corrected by completing the geometric shape. Fortunately, the original, uncorrected weight for each of these three pieces is also recorded in the reports and these are the values used here.


In a chapter included in ' Excavations at Chanhu Daro '5, published in 1943, Hemmy stated that... "From the much larger number of specimens collected at Mohenjo Daro and Harappa the value of the standard has been calculated as 13.625 gms."

Hemmy originally used the lowest weight as the unit of the weight system but later shifted to 13.625 grams as the unit, resulting in fractional ratios. I have used his earlier scheme in Table 2. The range of weight ratios6 runs from 1 to 12800. This set constitutes the nominal values of the Harappan system of weights and is referred to as such throughout this paper.

The accuracy of the weight system can be shown by a frequency distribution chart or histogram. The number of pieces within a narrow band or interval of weight values are tabulated. When the band of values is constant over the whole range then the frequency distribution chart is also a histogram.

Each band could be a small percentage of the nominal value. Simplest way to achieve this is to first normalise all the weight values.

Normalisation is the process of converting a set of data to a range centred around unity. This involves dividing each specimen by an appropriate nominal value from Table 2. The process of normalisation returns a value of 1 for each of the nominal values.

The binary scale runs, in segments, throughout the whole range of the weights, as can be seen from Table 2. The division should result, for most part, in normalised values between 0.75 and 1.5.

As it happens, 694 out of 697 specimens do indeed fall within this range. The three non-compliant weights occur in the range 54.5 grams and 136.25 grams. Rather than exclude them, their ratios to 54.5 grams, the attainable target value, are included in the complete list of normalised values.

The range of normalised values runs from 0.75 to 1.77, resulting in 102 equal steps, each spanning 1% of the nominal values. The first interval is centred around 1. Its boundaries are 0.995 and 1.005, giving a mean deviation of zero from the nominal values. The remaining steps are equally spaced either side of this central value.

A histogram of the normalised values, plotting the number of specimens in each interval, can be seen in Chart 1. The horizontal axis represents normalised values. The vertical axis represents the number of weight specimens in the given interval.

The peak frequency is 175 and is centred around the nominal values.

A. S. Hemmy used a procedure similar to the normalisation described above but his objective was to arrive at a more accurate value of the standard for the system of weights. All specimen were assigned a ratio with respect to the (then determined) standard value of 13.64 grams and the specimens within 7% of the corresponding standard values were divided by the assigned ratio. The results were then plotted to show the frequency distribution of the weights.

A selection was applied when arriving at the plot. As he writes5 … “All doubtful specimens were rejected, including those with unlikely ratios, as well as all weights below 6 gms.”

In preparing Chart 1 no weight specimens are left out. This removes the subjective element present when excluding some weight specimens.

Close to the nominal values, the distribution is fairly symmetrical. Of the 697 pieces, 608 are within +/- 10% of the nominal values and 533 are within +/- 5%. Outside of the 10% boundary there are many pieces with a frequency of just one. For clarity a magnified central section of the above chart is displayed in Chart 1a.

In Lists 4 and 5 an important detail included is the condition of each specimen. In List 4 these are classified under four categories: perfect, slightly chipped, chipped, badly chipped. In List 5 there are seven categories: perfect, good, slightly chipped, badly chipped, unfinished, burnt, dubious. Criteria for the classification are not included in any of the reports.

Perfect specimen is an undamaged specimen. The other categories represent various degrees of damage and are referred to, collectively, as damaged pieces. There are 90 perfect weight specimens in List 4 and 12 in List 5. Chart 2 is a histogram of the perfect and damaged specimens from List 4.

The peaks for both the perfect and damaged pieces occur at the nominal values. The largest of the weight value, a perfect piece, occurs at 177% of the nominal weight value.

The range covered in the chart is wide, from 75% to 177%. This results in a crowded central region. Chart 2a displays a restricted range of normalised values to improve matters.

There are 90 perfect and 130 damaged specimens in List 4; within the 10% range centred on the nominal values, the figures are 84 and 123, respectively.

In List 5 the number of perfect specimen is only 12 out of a total of 118. Chart 3 is a histogram of the prefect and damaged specimens from List 5 with Chart 3a displaying the central section.

From charts 2 and 3 it is clear that some of the perfect pieces appear far removed from the nominal values, though mostly singly. Perfect pieces would have been shaped and polished. In the next stage the pieces would be trimmed to the required weight. An insight into the stages involved in manufacturing these specimens is provided by two photographs included in separate excavation reports.

The image below appears in Marshall’s excavation report7, where Ernest Mackay writes: “They were first flaked into shape and then ground, and finally polished.”

Another photo appears in the report by Vats8, Plate CXVIII, shown below. In the report, Vats writes: “… they were first roughly chipped into shape and then ground and polished. The first process of chipping into shape is illustrated by No. 12 (1095) in Plate CXVIII ….”

Damage to a few of the pieces is visible in the photo. While most of the items in the photo are cubical, the last three are examples of cylinder, truncated sphere and cone.

One of the perfect pieces, farthest from nominal values, weighs in at 177% of a nominal value. A lot of effort would be required to trim it down to the required value.

Two of the prefect pieces appear at 93% of the nominal values. This could easily represent the lower end of the range of accuracy to which these pieces were being manufactured. If that is not the case then they can only be trimmed to a lower value. In that case they are weighing in at 186% of the lower nominal values. In view of a piece at 177% of a nominal value that could easily be the case.


A determination of the range of accuracy achieved by the ancients would be a clear step forward in understanding their technology in the manufacture of weights as well as their metrology. The weights have been manufactured to a great accuracy, as the charts show.

A collection of damaged and perfect weight pieces does not make it easy to establish the exact level of accuracy of the weight system. Examining the perfect pieces, identified in Lists 4 and 5 only, does not help in establishing the accuracy band either as there are perfect pieces far higher than the nominal values. These can be trimmed to the nominal value and are incomplete pieces. The number of such perfect pieces is rather small, as can be seen from Charts 2 and 3. Examination of weights from later excavations should help clear the picture.

In Chart 1 the peak frequency at the nominal values is impressive. However, as the Charts 2 and 3 show, both the perfect and damaged pieces are present in that peak. Also, in the same charts, the peaks for perfect and damaged pieces occur at the nominal values. Despite the collection being a mix of damaged and perfect pieces, the accuracy of the weight system is remarkable. What could be the reason for such accuracy? Did it happen routinely or did the ancients go to great lengths to achieve it?

Damage to some of the pieces is slight. This can be seen in Plate CXVIII, illustrated above, from Excavations at Harappa by M. S. Vats. In the records 35 pieces in List 4 are classified as slightly chipped while 53 pieces in List 5 are classified as good or slightly chipped (distinction between these two categories is not specified in the report). Specimens from these classifications contribute to the peak frequencies observed for the damaged pieces at the nominal values.

Since the majority of specimens are cubical in shape, an estimate of the weight of these damaged pieces, by completing the geometry, should be possible. However the exercise is unlikely to be easy or cheap.

In this collection of perfect and damaged weights statistical information like average and standard deviation may not mean much. However, as an exercise, this is calculated for the entire collection and for a restricted collection within a band of 10% of the nominal values.

The average of the normalised values of all 697 pieces is 1.0237 with a standard deviation of 0.1060. For 608 pieces within a band of 10% centred around the nominal values the average of the normalised values is 0.9997 while the standard deviation is 0.0308.

Most of the pieces are cubical in shape. This is not a difficult shape to manufacture. Rubbing on a flat surface will ultimately produce a cube, though the symmetry may be slightly lost when trimming to the target value. Cylindrical shape too cannot be difficult to achieve. It is the manufacture of shapes like conical, barrel and spherical that is fascinating. How does one shape a sphere out of a stone with the technology of the third millennium BCE?

Modern facilities afforded by the computer has made the analysis of the long published data an easy exercise. In Hemmy’s time the task could not have been that easy, though I suspect he may have had assistance in analysing all the data.

Hemmy used four of the five Lists in Table 1 to determine the systematic nature of the weight system, including the nominal values and their ratios given in Table 2. For the weights in List 3, Vats9 wrote, “ … I have followed Mr. Hemmy’s designations and limits within which those designations apply.” Subsequent finds have all followed the designations established by Hemmy.

The analysis presented here deals with only the earliest finds of the weight specimens. This presents us with the opportunity to check the results of this analysis against weight specimens from later finds. With many sites reporting findings of weights it should prove a useful exercise and I hope this happens soon.


I would like to thank my son Samir and his wife Susan for various discussions and suggestions, which have proved helpful. My thanks also go to the librarians at the Keith Axon centre, Redbridge Libraries, for cheerfully obtaining those excavation reports without which this paper would not have been possible.


1/. Mohenjo-Daro and the Indus Civilization.

Being an official account of archæological excavations ... carried out by the Government of India between the years 1922 and 1927. Edited by Sir J. Marshall, Arthur Probsthain, London, 1931, pp 596-8

2/. Excavations at Harappa. Vats, Madho Sarup

Being an account of archælogical excavations at Harappa, carried out between the Years 1920-21 and 1933-34, etc.

2 vol. Delhi, 1940. pp 363-365

3/. Further Excavations at Mohenjo-daro. Mackay, Ernest John Henry

Being an official account of archæological excavations at Mohenjo-daro carried out by the Government of India between the years 1927 and 1931 ... With chapters by A. S. Hemmy ... and by B. S. Guha ... and P. C. Basu, etc

2 vol. Delhi, 1938, 37, Vol. 1, pp 607-612

4/. Excavations at Chanhu Daro, Mackay, Ernest John Henry, 1935-36, New Haven, 1943, pp 238-239

5/. Excavations at Chanhu Daro, op cit., p 236-7

6/. Excavations at Chanhu Daro, op cit., p 241

7/. Mohenjo Daro and the Indus Civilization, op. cit., p 462

8/. Excavations at Harappa, op. cit., p 361

9/. Excavations at Harappa, op. cit., p 360


*© 2010 Chandrakant Doshi