tag:blogger.com,1999:blog-309608762024-02-19T16:47:54.445+00:00JignashiChandra Doshihttp://www.blogger.com/profile/12676853087431948250noreply@blogger.comBlogger10125tag:blogger.com,1999:blog-30960876.post-25355827346769985222023-02-20T15:47:00.000+00:002023-02-20T15:47:07.774+00:00<p> </p><p><br /></p><p align="center" class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm; text-align: center;"><span style="font-family: "Times New Roman", serif; font-size: 20pt;">Times Tables<o:p></o:p></span></p><p align="center" class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm; text-align: center;"><span style="font-family: "Times New Roman", serif; font-size: 20pt;"> </span></p><p align="center" class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm; text-align: center;"><span style="font-family: "Times New Roman", serif; font-size: 16pt;">by<o:p></o:p></span></p><p align="center" class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm; text-align: center;"><span style="font-family: "Times New Roman", serif; font-size: 16pt;"> </span></p><p align="center" class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm; text-align: center;"><span style="font-family: "Times New Roman", serif; font-size: 20pt;">Chandrakant Doshi<o:p></o:p></span></p><p align="center" class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm; text-align: center;"><span style="font-family: "Times New Roman", serif; font-size: 20pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 16pt;">Email: </span><a><span style="font-family: "Times New Roman", serif; font-size: 16pt;">chandra@agnani.co.uk</span></a><span style="font-family: "Times New Roman", serif; font-size: 16pt;"><o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm; text-indent: 36pt;"><span style="font-family: "Times New Roman", serif; font-size: 16pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm; text-indent: 36pt;"><br /></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 16pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><i><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Abstract :</span></i><i><span style="font-family: "Times New Roman", serif; font-size: 16pt;"><o:p></o:p></span></i></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 16pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><i><span style="font-family: "Times New Roman", serif; font-size: 14pt;">This note outlines a novel approach to help the children learn the Times Tables.<o:p></o:p></span></i></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Comic Sans MS"; font-size: 14pt;"> <o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;">One topic in mathematics that turns children off the subject is the learning of times or multiplication tables. Many approaches have been devised to help children in memorising these tables. As they progress through the school, recalling multiplication of two numbers is required for manytopics in maths. <o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Another approach to help the children learn these times tables is presented here. Parents and grandparents can play an important part here. Children love to play games and they definitely love to show off. What better way then to invite a child/grandchild to test their parents/grandparents knowledge of times table? This becomes a game and there is plenty of opportunity to show off. Incentive to learn the tables comes automatically with the game.<o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Multiplication tables run from 2 to 10 or 12. Generally they are printed as columns but they can also be printed in a grid format. The latter are for use by the older children. To help children learn their times tables with greater interest a partially filled grid of the times table could also be utilised. This should engage their minds in working out the missing steps.<o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Once a child has learnt the times table then applying that knowledge involves the child running through, in her/his mind, that particular table starting from one. They carry on till the required multiplier is reached. <o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Here the times 5 table provides some advantage. Multiplication of any two numbers, where at least one number is greater than 5, can be performed with increasing ease than the usual method of running through, in their mind or by the use of fingers, the times table of one of the number, starting from 1. <o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Children could be shown how to utilise their knowledge of times 5 table to run through the multipliers higher than 5. With right sort of examples their knowledge of times tables can be reinforced.<o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Let us take a specific example. We wish to know the result of multiplying 7 x 3. We could run through the times table for 3, starting from 1 till we arrive at 7. <o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Another way is to go straight to the times 5 table. We have 3 times 5 = 15, then running forward we have 3 times 6 = 18 and finally 3 times 7 = 21. <o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;">In both cases the result is 7 x 3 = 21. <o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;">This requires mastering times 5 table thoroughly. Children learn times 10 table very quickly, once they realise that all they need to do is put 0 at the end of the number being multiplied. Times 5 table has a similar ease of obtaining the result: multiplication by 5 results in a number ending in either 5 or 0. This should prove a great help to mastering the times 5 table. Children will quickly pick up the fact that multiplying 5 by an even number always results in the number ending in 0. Even 0, odd 5 becomes the rule. This proves an easy introduction to odd or even numbers.<o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Times tables can also be printed in a grid format (see Table A1). Familiarising its use helps a child to see the multiplication in reverse order, ie, 3*7=7*3. A challenge for a child is to start using a partially filled multiplication table so the child gets to reinforce his knowledge of the table in small step. Table A2 is an example of a partially filled multiplication table. Similar tables can be prepared as necessary. <o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Children, and perhaps even the parents, may need to be taught how to use these Tables. One way to proceed is to let the child ask a few questions using the table. Then gradually permit the child to pick some questions from the tables and encourage the child to ask more questions from memory only. Proceeding this way the child should be able to come up with more questions from memory to test their parent/grandparent. <o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;">Parents are busy people and not all may find time to practice such exercises with their children. Grandparents, if living near, could prove helpful if they are prepared to undertake the task. Some parents may need to learn their times tables but their experience and maturity should make this task fairly easy. <o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><o:p> </o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><o:p> </o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><o:p> </o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><o:p> </o:p></p><br /><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><o:p> </o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><o:p> </o:p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9G3DHhfUYSpCdKr-WsmpQfjLYBCx1vXAyXVwffhh7BDC1bQ5xOqOggG0MzcRTN4WiYzgYFHxHCfgx0u7m1boljbaXHoRYsOnKAJ60_rO4Fp8NDksXsdExiYI75bDOpAIBf2XepC3Vo_vt3GdcG9Ctkk8kWFvuGIfiD7zvraDhgBSvygZ0Rw/s1813/44D87229-988E-4E60-AE0D-E2727E468A5E.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1422" data-original-width="1813" height="398" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9G3DHhfUYSpCdKr-WsmpQfjLYBCx1vXAyXVwffhh7BDC1bQ5xOqOggG0MzcRTN4WiYzgYFHxHCfgx0u7m1boljbaXHoRYsOnKAJ60_rO4Fp8NDksXsdExiYI75bDOpAIBf2XepC3Vo_vt3GdcG9Ctkk8kWFvuGIfiD7zvraDhgBSvygZ0Rw/w508-h398/44D87229-988E-4E60-AE0D-E2727E468A5E.jpeg" width="508" /></a></div><div><br /></div><div><br /></div><div><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVL4Io38yvbZxfu67cGv-N6sEQ-65YKetdfZXjFFsahl-QqNvVReciZ-eTgBt53oWsMKE2XvBous2m7X0e8KwMwLn8O2VHpGgWW9oedffGJ1YJYTV4ozvn8NX9C7a5MVNgOa-A7oLPmeQWNpuikxmVFvCKfVzZK9wpT8EcMwQ_HHMOBHETgQ/s1557/140421D9-251B-4CD4-BDD8-DFA9F70D30CA.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1424" data-original-width="1557" height="431" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVL4Io38yvbZxfu67cGv-N6sEQ-65YKetdfZXjFFsahl-QqNvVReciZ-eTgBt53oWsMKE2XvBous2m7X0e8KwMwLn8O2VHpGgWW9oedffGJ1YJYTV4ozvn8NX9C7a5MVNgOa-A7oLPmeQWNpuikxmVFvCKfVzZK9wpT8EcMwQ_HHMOBHETgQ/w471-h431/140421D9-251B-4CD4-BDD8-DFA9F70D30CA.jpeg" width="471" /></a></div><div><br /></div><div><br /></div><br /><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><br /></p>Chandra Doshihttp://www.blogger.com/profile/12676853087431948250noreply@blogger.comtag:blogger.com,1999:blog-30960876.post-49506551305746478212023-02-08T11:35:00.015+00:002023-02-08T11:42:19.367+00:00<p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span></p><p align="center" class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm; text-align: center;"><span style="font-family: "Times New Roman", serif; font-size: 20pt;">Papers on Academia<o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;"><span style="font-size: 18.666666px; text-indent: 48px;"><br /></span></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 14pt;"><span style="font-size: 18.666666px; text-indent: 48px;">Email: chandra@agnani.co.uk</span></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="-webkit-text-size-adjust: auto; font-family: "Times New Roman", serif; font-size: 12pt; text-indent: 36pt;"><br /></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; margin: 0cm;"><span style="font-size: medium;"><span style="-webkit-text-size-adjust: auto; font-family: "Times New Roman", serif; text-indent: 36pt;">8</span><sup style="-webkit-text-size-adjust: auto; font-family: "Times New Roman", serif; text-indent: 36pt;">th</sup><span style="-webkit-text-size-adjust: auto; font-family: "Times New Roman", serif; text-indent: 36pt;">. February, 2023 </span></span><span style="-webkit-text-size-adjust: auto; font-family: "Times New Roman", serif; font-size: 12pt; text-indent: 36pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; margin: 0cm;"><span style="font-family: "Comic Sans MS";"><i><span style="font-size: large;">Some of my early papers and notes are to be found on this blog but the later works are uploaded to Academia. <o:p></o:p></span></i></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; margin: 0cm;"><span style="font-family: "Comic Sans MS";"><i><span style="font-size: large;"> </span></i></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; margin: 0cm;"><span style="font-family: "Comic Sans MS";"><i><span style="font-size: large;">Below is the list of my work on Academia, including the links to access them. <o:p></o:p></span></i></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; margin: 0cm;"><span style="font-family: "Comic Sans MS";"><i><span style="font-size: large;"> </span></i></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; margin: 0cm;"><span style="font-family: "Comic Sans MS"; font-size: large;"><i><span>This list will be updated as new papers are added to Academia.</span></i><span><o:p></o:p></span></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; margin: 0cm;"><o:p><span style="font-size: large;"> </span></o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><o:p> </o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><b><span style="color: #00b050; font-family: "Times New Roman", serif; font-size: 12pt;">Harappan Linear Measures<o:p></o:p></span></b></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><a href="https://academia.edu/resource/work/91385667" style="color: #954f72;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;">https://academia.edu/resource/work/91385667</span></a><span style="font-family: "Times New Roman", serif; font-size: 12pt;"><o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><b><span style="color: #00b050; font-family: "Times New Roman", serif; font-size: 12pt;">Accuracy of Ancient Weights<o:p></o:p></span></b></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><a href="https://academia.edu/resource/work/55221784" style="color: #954f72;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;">https://academia.edu/resource/work/55221784</span></a><span style="font-family: "Times New Roman", serif; font-size: 12pt;"><o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><b><span style="color: #00b050; font-family: "Times New Roman", serif; font-size: 12pt;">Ancient Weights: Two Cultures<o:p></o:p></span></b></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><a href="https://academia.edu/resource/work/42941631" style="color: #954f72;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;">https://academia.edu/resource/work/42941631</span></a><span style="font-family: "Times New Roman", serif; font-size: 12pt;"><o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><b><span style="color: #00b050; font-family: "Times New Roman", serif; font-size: 12pt;">Comparison Ancient Weight Systems<o:p></o:p></span></b></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><a href="https://academia.edu/resource/work/30143220" style="color: #954f72;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;">https://academia.edu/resource/work/30143220</span></a><span style="font-family: "Times New Roman", serif; font-size: 12pt;"><o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><b><span style="color: #00b050; font-family: "Times New Roman", serif; font-size: 12pt;">Ancient Egyptian Weight Systems<o:p></o:p></span></b></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><o:p> </o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><a href="https://academia.edu/resource/work/4846620" style="color: #954f72;">https://academia.edu/resource/work/4846620</a><o:p></o:p></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><b><span style="color: #00b050; font-family: "Times New Roman", serif; font-size: 12pt;">An Externally Pivoted Balance<o:p></o:p></span></b></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><a href="https://academia.edu/resource/work/8066648" style="color: #954f72;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;">https://academia.edu/resource/work/8066648</span></a><span style="font-family: "Times New Roman", serif; font-size: 12pt;"><o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><b><span style="color: #00b050; font-family: "Times New Roman", serif; font-size: 12pt;">Ancient Weights from South Asian Civilization<o:p></o:p></span></b></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><a href="https://academia.edu/resource/work/4846613" style="color: #954f72;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;">https://academia.edu/resource/work/4846613</span></a><span style="font-family: "Times New Roman", serif; font-size: 12pt;"><o:p></o:p></span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p><p class="MsoNormal" style="-webkit-text-size-adjust: auto; font-family: Calibri, sans-serif; font-size: 11pt; margin: 0cm;"><span style="font-family: "Times New Roman", serif; font-size: 12pt;"> </span></p>Chandra Doshihttp://www.blogger.com/profile/12676853087431948250noreply@blogger.comtag:blogger.com,1999:blog-30960876.post-23433908567300452902021-01-27T21:24:00.000+00:002021-01-27T22:25:07.590+00:00Ancient Weights:2 cultures<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-family: "times new roman" , serif; font-size: 20pt;">Ancient Weights: 2 Cultures<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 16pt;">by<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 20pt;">Chandrakant Doshi<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">email: </span><a href="mailto:jignashi@yahoo.co.uk" style="color: #954f72;"><span style="font-family: "times new roman" , serif; font-size: 14pt;">jignashi@yahoo.co.uk</span></a><span style="font-family: "times new roman" , serif; font-size: 14pt;"><o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">Key words: weights, ancient, Harappan, Indus, Mesopotamian, Ur, Nippur, Mohenjo Daro, Dholavira, stone, chert, hematite.</span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">Mesopotamian and Harappan civilizations both had their own weight systems. This note examines these systems, recognising that the recovered weights from each site are a mix of two grades of weights (details below), one accurate and one less so and demonstrates that the accurate grades from these bronze age cultures have similar accuracy. These two grades will be referred to here as Class A and Class B weights, specimens of Class A being the more accurate of the two Classes. <o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">For this analysis weights from </span><span style="font-family: "times new roman" , serif; font-size: 13.5pt;">Mohenjo Daro<sup>1</sup>, Dholavira<sup>2</sup>, Nippur<sup>3</sup> and Ur<sup>4 </sup>are used. In all these weight collections the details of the materials of weights are available. This is important, as it will become clear further down.</span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">Harappan weights were first analysed by A. S. Hemmy in late 1920s and he </span><span style="font-family: "times new roman" , serif; font-size: 14pt;">noted the difference in accuracy of weights of different materials when analysing Mohenjo Daro specimens. </span><br />
<i><span style="color: #1a3ff9; font-family: "times new roman" , serif; font-size: 14pt;">“The weights not made of chert are on the whole not so accurate as those that are”,</span></i><span style="font-family: "times new roman" , serif; font-size: 14pt;"> he wrote in Further Excavations at Mohenjo Daro<sup>1</sup>, (MjD FEM), page 605.</span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">A. S. Hemmy<sup>5</sup></span><span style="font-family: "times new roman" , serif; font-size: 14pt;"> also</span><span style="font-family: "times new roman" , serif;"> </span><span style="font-family: "times new roman" , serif; font-size: 14pt;">analysed weight systems of the Harappan and Mesopotamian civilizations and concluded that the Harappan weights were by far the more accurate of the two. For comparison, Hemmy used Mohenjo Daro weights on the one hand and a collection of weights from Ur and Susa on the other. No references were given for the sources of these data. But he did observe that</span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<i><span style="color: #1a3ff9; font-family: "times new roman" , serif; font-size: 14pt;">“In the Indus system, the great majority conform to one standard (13.625 gm or 210.2 gr.), and may be divided into two grades, one very accurate, one much less so.”<o:p></o:p></span></i></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">W. B. Hafford<sup>6</sup> also noted two different grades in the weight collection from Nippur:<o:p></o:p></span></div>
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<i><span style="color: #1a3ff9; font-family: "times new roman" , serif; font-size: 14pt;">“Were the well-made hematite and bronze pieces the official or royal weights used by an upper echelon of merchants and bureaucrats, while the crude types of less dense stone and less regular form were those of the more local people?”<o:p></o:p></span></i></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">When a wide range of values need to be displayed on a chart then converting each number that represents the weight of a specimen to its logarithm makes it possible to display it on a standard size chart. Such a chart is referred to as a log chart. Logarithm or log of a number is expressed with respect to another number, called the base. Commonly used base is 10. Charts used here are all to base 2, which has the advantage that values which are halves or doubles of another value have log values that differ by 1. Since most weight systems have values which are halves or doubles of previous values, specially at the lower end, then the choice of 2 is an advantage. With suitably scaled axis, this is easily seen. Divergence from the binary scale also shows clearly. And the use of 2 as a base provides a better resolution.<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">To prepare this special chart, the weight list is sorted by the material of the specimens and for each material, by the value of the specimens, lowest value first. The numerical value of each weight is converted to log to base 2. These log values are then plotted on a chart. As can be seen in the accompanying charts, vertical axis represents log(W) while the horizontal axis represents the item number on the list. The vertical axis is suitably adjusted to have the value of the unit weight fall on a grid line.<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">Sorting the weights by material and then value results in the weights on the chart successively moving up as well as to the right. Once the weights of the first material are exhausted then the plotted point moves down and to the right. So, jumps from top to bottom are due to the change of material. This way, weights of different materials can be identified on the chart. <o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">An examination of the chart reveals that for one material the display is a staircase ( implying many specimens are close together in value) while for the rest of the materials, the display comprises of mostly scattered points, with no systematic structure evident. A full sequence of specimens of this single material, from the lowest value to the highest, is present in the collection. This is not the case for the rest of the materials.</span><span style="font-family: "times new roman" , serif;"></span><span style="color: #af87f9; font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">Chart D below displays these weights from Dholavira arranged in alphabetical order by material and within each material by numerical order, lowest first. Weights made of chert are over-plotted in red and shows the typical staircase representing a weight system. The flats are made by a large number of specimens close in value. Some of the other materials also show a few specimens on a flat but generally there are few specimens that make up the flat and the range of nominal values covered by the staircase is rather small. So, there are indeed two grades of weights present in the recovered collection from Dholavira. <o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">A similar situation is observed with weights recovered from Mohenjo Daro by Ernest Mackay<sup>1</sup>. Chart A displays these weights.<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">There are two papers by W. B. Hafford available online and which include a list of weights from the Mesopotamian sites of Ur<sup>2</sup> and Nippur<sup>3</sup>. Each list includes, besides the weight of the specimen, the material it is made of, the shape, ratio of the specimen and the derived unit of the system. Charts for each of these two weight lists, along similar line to Chart D reveal two grades of weights in each. Hematite, a hard stone, is the material for one of the grades of weights and is over-plotted in red in the charts below. This can be seen in Chart 1 for Ur and Chart 2 for Nippur.<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">Mesopotamian weights are scattered across various museums and institutions and Hafford has put considerable effort to collect them together. As he notes, there are many more of these in the museums in Iraq which were not accessible to him.<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">Looking at the four charts above it is clear that weight collection from each site can be separated into two categories. The criteria that partitions these weights is the type of stone used in manufacturing them: chert for Harappan weights and hematite for Mesopotamian weights. Both of these are hard stones and the weights made of these materials are referred to as Class A weights. To establish that these Class A weights are the more accurate of the collection, we need to examine their frequency distribution charts. <o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">A frequency distribution chart displays the number of specimens in a given interval of weight. Easiest way to achieve this is to normalise the whole weight list first. Each specimen allocated to a ratio or nominal weight is divided by that nominal weight. A specimen weighing exactly the nominal weight will return a value of 1.00, while values below or above the nominal value will return a value less than or greater than 1.00, respectively. <o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">The range of normalised values is divided into intervals of 0.01, starting with the interval at the nominal value. This is symmetrically arranged about it, so the interval covers the range 0.995 to 1.005 of the normalised value. The entire range of the normalised values is similarly covered. <o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">The frequency distribution is the number of weight specimens occurring within any given interval. A plot of these numbers is the frequency distribution of that weight system. These distributions are based on a unit weight of 13.625 grams for the Harappan system and 8.40 grams for the Mesopotamian system. <o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">The plots display the distribution of Class A and Class B weights on the same chart for each of the four sites. The Harappan weights, Chart B and Chart E show the peak frequency for Class A at the normalised value of 1.00. Very few of Class B specimens are present in Chart B whereas very large number are present in Chart E. <o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">Among the Mesopotamian weights, the peak of Class A weights at Ur (Chart 3) occurs at the normalised value of 0.97 with the frequency at normalised value of 1.00 is only slightly lower and is the second highest peak in the distribution. For Nippur the peak for Class A weights does occur at the normalised value of 1.00.<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">Frequency distribution charts of the Class A weights for both the cultures are mostly symmetrical about the nominal weights. This confirms that the weights of chert and hematite are more accurate than the rest of the collection, as already noted by Hemmy<sup>1,5</sup> and Hafford<sup>3</sup>. Since a specimen of hard stone is less prone to chipping, it is likely that these weights were issued by some central authority. Class B weights cannot have been made by the same authority because, if they were, they could easily have manufactured them using the hard stones. It is likely that they were made by some regional authority which could handle the softer materials used in their manufacture. It is also possible that they were made by individuals for their specific use.<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">An examination of the frequency distribution of Class B weights from the four sites show that they differ widely. This would support the view that many of these were made by individuals for the activity they were engaged in, such as mixing spices, medicinal products or alloys like bronze. <o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">The frequency distribution of Class B weights from MjD FEM is hardly distinguishable from that of Class A material (Chart B). This was the only Harappan weight collection with material data available to Hemmy when he first analysed weights from this culture. It is no surprise therefore at the conclusion he reached, knowing the chert weights were more accurate than other materials. None of the Harappan weights he subsequently analysed had material information recorded. </span><span style="color: red; font-family: "times new roman" , serif; font-size: 14pt;"><o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">Frequency distribution presents a qualitative picture of the weight system. Standard deviation would represent a quantitative measure of a weight system. A normalised value of a weight specimen represents its deviation from its nominal weight, which has a normalised value of 1.00. The whole list of normalised values of a system can be used to calculate the standard deviation of that collection. <o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">In Table 1 below are listed standard deviations of weights from 4 sites under 2 headings. Class (A + B) specimens are a mix of all the materials whereas Class A specimens are made of only a specific stone, different for each culture. <o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">Standard deviations of MjD FEM, Nippur and Ur are close in value for Class A collection of weights. Compare this with the same for Class (A + B). This clearly demonstrates that Class A specimens, made of hard stone, are the officially authorised weights. All the others should be considered unauthorised specimens, made of different materials and may not have been closely calibrated to the authorised specimens. A mixture of these two sources of weights do not correctly reflect the accuracy of weight systems of each culture. <o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">Dholavira weights present a unique picture. </span><span style="font-family: "times new roman" , serif; font-size: 14pt;">First the whole collection needs to be separated into 3 different standards R. S. Bisht has identified. Only then the weights of Harappan standard can be examined. As it is, the Class A weights have their peak at the normalised value of 1.00 but the distribution is not symmetrical about it.<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">More weight data from new sites in both cultures need to be examined to see if they support the view that the Class A weights have similar accuracy in both these cultures. In that respect Dholavira weights need to be re-examined. </span><span style="font-family: "times new roman" , serif; font-size: 14pt;">W. B. Hafford has collected new Mesopotamian weight material (email, 15 August, 2019) that he is examining, and it would be interesting to see how the result of this new material fits with the scheme described above. </span><span style="font-family: "times new roman" , serif; font-size: 14pt;"><o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">The analysis presented here demonstrates that both Harappan and Mesopotamian civilisations had achieved similar accuracy with their weight systems. It was the mix of two grades of weights that masked the true accuracy of the Mesopotamian weight system. It may be that this mix of two grades of weights was a general practice among these two as well as other cultures. <o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">References:<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">1/ </span><span style="font-family: "times new roman" , serif; font-size: 13.5pt;">Further Excavations at Mohenjo Daro, Ernest Mackay (1938), pp 603</span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">2/ </span><span style="font-family: "times new roman" , serif; font-size: 13.5pt;">Dholavira Excavation Report - R. S. Bisht (2015). This is available online and can be accessed here: </span><a href="https://l.facebook.com/l.php?u=http://asi.nic.in/pdf_data/dholavira_excavation_report_new.pdf&h=ATOBACQ84bmP5mQRPyHPFyZtkF1qFnis2Aochw7W3cGCBC_kyjs8IgANy2pw646QQ7vDMxgbjxVfRRwfBvhw7TFNoR7OagBS3hogKV0Nc6xJQEUAJ9YZbDoxkQH_TEfx0Q&enc=AZNEEAlw2-8hWVByaQCuqUweN0ZHxU8kLbWgb_dHVuo7unGVywNIz2-H9g_5GQqMDu7jAsOdbWqZHhn24lVP56MdrIyB3ViTODnPR-mvDWKxCajYpKEB4Ptgq3_rv9hzcdXtl4JiyteezE9yvfCTUng7FeGS3_lVF2QdqD_aZTuMVLEjSrFapA-Dyxr8D3kxmXU&s=1" style="color: #954f72;"><span style="font-family: "times new roman" , serif; font-size: 14pt;">http://asi.nic.in/pdf_d…/dholavira_excavation_report_new.pdf</span></a><span style="font-family: "times new roman" , serif; font-size: 13.5pt;"> </span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">3/ </span><span style="font-family: "times new roman" , serif; font-size: 13.5pt;">Balance Pan Weights from Nippur by W. B. Hafford</span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<a href="https://www.academia.edu/2916303/Mesopotamian_Mensuration_Balance_Pan_Weights_from_Nippur" style="color: #954f72;"><span style="font-family: "times new roman" , serif; font-size: 13.5pt;">https://www.academia.edu/2916303/Mesopotamian_Mensuration_Balance_Pan_Weights_from_Nippur</span></a><u><span style="color: blue; font-family: "times new roman" , serif; font-size: 13.5pt;"><o:p></o:p></span></u></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">4/ </span><span style="font-family: "times new roman" , serif; font-size: 13.5pt;">The Balance Pan Weights from Ur by William B. Hafford, in Akkadica 133 (2012)</span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<a href="https://www.academia.edu/2916278/Weighing_in_Mesopotamia_The_Balance_Pan_Weights_from_Ur" style="color: #954f72;"><span style="font-family: "times new roman" , serif; font-size: 13.5pt;">https://www.academia.edu/2916278/Weighing_in_Mesopotamia_The_Balance_Pan_Weights_from_Ur</span></a><u><span style="color: blue; font-family: "times new roman" , serif; font-size: 13.5pt;"><o:p></o:p></span></u></div>
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<span style="font-family: "times new roman" , serif;">5/ </span><span style="font-family: "times new roman" , serif; font-size: 14pt;">The Statistical Treatment of Ancient Weights by A. S. Hemmy, Ancient Egypt, December 1935 pp 83</span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif; font-size: 14pt;">6/ </span><span style="font-family: "times new roman" , serif; font-size: 13.5pt;">Balance Pan Weights from Nippur by W. B. Hafford: page 34<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , serif; font-size: 14pt;">© 2020 Chandrakant Doshi<o:p></o:p></span></div>
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Chandra Doshihttp://www.blogger.com/profile/12676853087431948250noreply@blogger.comtag:blogger.com,1999:blog-30960876.post-62725747091601008442019-05-30T22:28:00.000+01:002020-04-15T21:52:29.183+01:00Dholavira Weights<div dir="ltr" style="text-align: left;" trbidi="on">
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<span style="font-size: x-large;">Dholavira weights</span><span style="font-family: "times new roman" , serif; font-size: 16pt;"><o:p></o:p></span><br />
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<span style="font-family: "times new roman" , serif;"><span style="font-size: medium;"><br /></span></span><span style="font-family: "times new roman" , serif;"><span style="font-size: medium;">Chandrakant Doshi</span></span></h3>
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<span lang="EN-US"><span style="font-size: large;">email: <a href="mailto:jignashi@yahoo.co.uk" style="color: purple;">jignashi@yahoo.co.uk</a><o:p></o:p></span></span></div>
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<span lang="EN-US"><span style="font-size: large;">Key words: Harappa, Indus, Dholavira, Gujarat, ancient weights<o:p></o:p></span></span></div>
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<span style="font-size: large;"><span style="font-family: "times new roman" , serif;">Weights have been recovered from most major sites of the Harappan civilization, including Dholavira, a Harappan site in Kutch, Gujarat. It was excavated from 1989 by R. S. Bisht and in </span><span style="font-family: "times new roman" , serif;">the excavation report Bisht </span><span style="font-family: "times new roman" , serif;">lists 996 weight<sup>1</sup>specimens. Dholavira flourished in third millennium BCE and has revealed a few unique features not identified elsewhere in the Harappan Civilization, including the heaviest weight recovered from any site. Bisht has identified 3 different weight systems. One of these is the standard Harappan system with the unit weight of 13.625 grams. This note examines the weights that conform to the standard Harappan system.</span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></span></div>
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<span style="font-family: "times new roman" , serif;"><span style="font-size: large;">Weight collection at Dholavira consists of 20 different shapes and 44 different materials. Many of the shapes and materials have not been seen among the recovered weights from earlier excavated sites at Harappa<sup>2</sup>, Mohenjo Daro<sup>2</sup>and Chanhu Daro<sup>2</sup>. These shapes and materials are listed in tables below. <o:p></o:p></span></span></div>
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<span style="font-family: "times new roman" , serif;"><span style="font-size: large;">One useful way of studying the relations between the weights and highlight the systematic nature of the system is by an appropriate chart. A weight system would show the characteristic staircase where the flats or rungs are produced by specimens with values in close proximity to each other. A log chart is an appropriate way of displaying this systematic nature as it can accommodate the wide range of values on a single chart. <o:p></o:p></span></span></div>
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<span style="font-family: "times new roman" , serif;"><span style="font-size: large;">A chart of all the 996 specimens is shown in Image 1. The weights are all sorted in ascending order and converted to log<b><sub>2, </sub></b>as this provides<sub></sub>high resolution and highlights the binary steps of the weight system. The converted values are plotted on the chart. The specimens within 5% of the Harappan weight system are over plotted in red on the chart. <o:p></o:p></span></span></div>
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<span style="font-family: "times new roman" , serif;"><span style="font-size: large;">Table 1 lists the weight ratios and the nominal weights of the Harappan system as well as the frequencies, within 5% of the nominal weights, of the Dholavira collection. </span></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , serif;"><span style="font-size: large;">Table 2 lists all the shapes and their frequencies. The largest number of weights are listed as cuboids (387), followed by spheroids (124). <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , serif;"><span style="font-size: large;">The chart below, Image 2, is a plot of all the 996 weights. For this chart, the weights are all arranged in alphabetical order and then, for each shape, the weights are sorted in ascending order. This way all the shapes can be identified with the plot of each shape starting at the bottom of the chart and reaching towards the top. The plot of the cuboid specimens, which distinctly shows the characteristic staircase of a weight system, is over plotted in red for easy identification. <o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "times new roman" , serif;"><span style="font-size: large;">Weights are made of 44 different materials. These are listed in Table 3.<o:p></o:p></span></span></div>
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<span style="font-family: "times new roman" , serif;"><span style="font-size: large;">A few of the weights with sufficient number of specimens are plotted in the chart below, Image 3. First, the weights are separated by material and, for each material, the weight list is sorted in ascending order and log of each weight to base 2 is plotted on the chart below.<sub></sub>The weights of 6 different materials are displayed on the chart. <o:p></o:p></span></span></div>
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<span style="font-size: large;">The vertical scale of the chart is adjusted to show the binary nature of the system at lower weight values. 3.77 on the vertical axis corresponds to 13.625 grams. Weights of chert, agate and sandstone show distinct staircase, indicating many specimens at the nominal values of the weight system. That is not the case with the other three materials. <o:p></o:p></span></div>
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<span style="font-size: large;">Another way of characterising a weight system is by plotting its frequency distribution. This is shown in Image 4. As most of the weight specimens follow a binary scale, the range of the distribution is from 0.75 to 1.50 normalised values. Height of the columns in the chart represent the frequency. The columns in blue comprise all the specimens within a given range, irrespective of their condition of preservation. Red columns represent only perfect, meaning undamaged, specimens.<o:p></o:p></span></div>
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<span style="font-size: large;">The peak frequency is at the nominal values, i.e. 1.00 on the chart. R. S. Bisht has identified 3 different weight systems, which all contribute to the distribution. Frequency distribution of only the chert specimens gives a better representation of the Harappan weight system. This is shown in Image 5.<o:p></o:p></span></div>
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<span style="font-family: "times new roman" , serif;"><span style="font-size: large;">A. S. Hemmy analysed the Harappan weight system and established the unit of the system at 13.625<sup>3</sup>grams. Weights below the unit formed a binary sequence of successive halves. Heavier weights followed the sequence of 1, 2, 4, 10 for each decade starting from the unit weight. At the top end the sequence was 100, 200, 400, 800 as no weights at ratio 1000 were recovered at that time. Only two damaged specimens support Hemmy’s top ratio of 800. <o:p></o:p></span></span></div>
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<span style="font-family: "times new roman" , serif;"><span style="font-size: large;">M. S. Vats carried out excavation at Harappa in the 1930s, although the excavation report was not published until 1940. In the report Vats mentions a heavy weight of 26535.6 grams<sup>4</sup>, though this is not included in the official weight<sup>5</sup>list. No reason is given for this omission. <o:p></o:p></span></span></div>
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<a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="about:invalid#zClosurez" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><span style="font-family: "times new roman" , serif;"><span style="font-size: large;">Among the Dholavira weights is one perfect specimen of limestone, weighing 13720 grams, which is close to the ratio 1000. (Acc No. 54404). The heaviest weight is also a limestone specimen weighing 35700 grams, (Acc No. 54550). This is 31% higher than the nearest attainable nominal value of 27250 grams. The condition of preservation of this last piece is not recorded.<o:p></o:p></span></span></div>
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<span style="font-family: "times new roman" , serif;"><span style="font-size: large;">Two additional weight systems Bisht reports have not been analysed in this note. This will be undertaken at a later date. Any weight specimens conforming to these additional systems recovered from other sites will have to be examined at the same time. <o:p></o:p></span></span></div>
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<span style="font-family: "times new roman" , serif;"><span style="font-size: large;">The analysis of the weights that conform to the standard Harappan weight system confirms the remarkable accuracy of these weights as first observed by A. S. Hemmy<sup>6</sup>and Dholavira weights demonstrate that this standard was prevalent throughout the Harappans civilization.<o:p></o:p></span></span></div>
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<b><span lang="EN-US"><span style="font-size: large;">Acknowledgements<o:p></o:p></span></span></b></div>
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<span style="font-size: large;"><span lang="EN-US" style="font-family: "times new roman" , serif;">I would like to thank </span><span style="font-family: "times new roman" , serif;">Harappa.com for putting the link to the Dholavira Excavation Report on Tweeter (@Ancient_Indus). This note would not have been possible without access to the material found at Dholavira.<o:p></o:p></span></span></div>
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<span style="font-size: large;"><span style="font-family: "times new roman" , serif;">The link to the report: </span><span lang="EN-US"><span lang="EN-US" style="color: purple; font-family: "times new roman" , serif;"><span lang="EN-US"><a href="applewebdata://1435DE9B-A31B-449F-9600-0833D367BA0B/Fb.me/8z9Nkaucl" style="color: purple;">f<span lang="EN-US">b.me/8z9Nkaucl</span></a></span></span></span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></span></div>
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<span style="font-family: "times new roman" , serif;"><span style="font-size: large;">References:<o:p></o:p></span></span></div>
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<span style="font-size: large;"><span style="font-family: "times new roman" , serif;">1/. Weights and Measures, page 334 onwards, Dholavira Excavation Report, available Online: </span><span lang="EN-US"><a href="https://l.facebook.com/l.php?u=http://asi.nic.in/pdf_data/dholavira_excavation_report_new.pdf&h=ATOBACQ84bmP5mQRPyHPFyZtkF1qFnis2Aochw7W3cGCBC_kyjs8IgANy2pw646QQ7vDMxgbjxVfRRwfBvhw7TFNoR7OagBS3hogKV0Nc6xJQEUAJ9YZbDoxkQH_TEfx0Q&enc=AZNEEAlw2-8hWVByaQCuqUweN0ZHxU8kLbWgb_dHVuo7unGVywNIz2-H9g_5GQqMDu7jAsOdbWqZHhn24lVP56MdrIyB3ViTODnPR-mvDWKxCajYpKEB4Ptgq3_rv9hzcdXtl4JiyteezE9yvfCTUng7FeGS3_lVF2QdqD_aZTuMVLEjSrFapA-Dyxr8D3kxmXU&s=1" style="color: purple;"><span lang="EN-US" style="font-family: "times new roman" , serif;">http://asi.nic.in/pdf_d…/dholavira_excavation_report_new.pdf</span></a></span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></span></div>
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<span style="font-size: large;"><span style="font-family: "times new roman" , serif;"> The link was provided by </span><span lang="EN-US"><a href="mailto:Harappa.com@Ancient_Indus" style="color: purple;"><span lang="EN-US" style="font-family: "times new roman" , serif;">Harappa.com@Ancient_Indus</span></a></span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></span></div>
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<span style="font-size: large;"><span style="font-family: "times new roman" , serif;">2/. Weights from these sites were analysed in a previous paper, which can be viewed here: </span><span lang="EN-US"><a href="https://jignashi.blogspot.com/2010/01/" style="color: purple;"><span lang="EN-US" style="font-family: "times new roman" , serif;">https://jignashi.blogspot.com/2010/01/</span></a></span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></span></div>
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<span style="font-size: large;"><span style="font-family: "times new roman" , serif;">3/. </span><span style="font-family: "times new roman" , serif;">Chanhu Daro Excavations, 1943 pp 236</span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></span></div>
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<span style="font-size: large;"><span style="font-family: "times new roman" , serif;">4/. </span><span style="font-family: "times new roman" , serif;">Excavations at Harappa, M. S. Vats, 1940, p. 57, item 8 </span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></span></div>
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<span style="font-size: large;"><span style="font-family: "times new roman" , serif;">5/. </span><span style="font-family: "times new roman" , serif;">Excavations at Harappa, op cit, pages 363-365</span><span style="font-family: "times new roman" , serif;"><o:p></o:p></span></span></div>
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<span style="font-family: "times new roman" , serif;"><span style="font-size: large;">6/. Further Excavations at Mohenjo Daro, E. J. H. Mackay, </span></span><br />
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<span lang="EN-US" style="font-family: "times new roman" , serif;"><span style="font-size: medium;"><span style="font-size: large;">© 2019 Chandrakant Doshi</span><o:p></o:p></span></span></div>
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Chandra Doshihttp://www.blogger.com/profile/12676853087431948250noreply@blogger.comtag:blogger.com,1999:blog-30960876.post-24773296404190017412016-11-20T12:08:00.008+00:002020-04-15T21:56:36.114+01:00Comparison of Ancient Weight Systems<div dir="ltr" style="text-align: left;" trbidi="on">
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<b style="text-align: center;"><span style="font-size: 20.0pt; mso-ansi-language: EN-US;">Comparison of Ancient Weight Systems</span></b></div>
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<a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=30960876" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="about:invalid#zClosurez" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><o:p style="font-weight: normal;"><span style="font-size: small;">by </span></o:p></h3>
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<span style="mso-ansi-language: EN-US; mso-bidi-font-size: 10.0pt;"><o:p> Chandrakant Doshi</o:p></span></h3>
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<div class="MsoNormal" style="margin-bottom: 12.0pt;">
<span style="mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt; mso-bidi-language: HI;">email:
<a href="file:///jignashi@yahoo.co.uk"><span style="color: #0000e9;">jignashi@yahoo.co.uk</span></a></span><span style="mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;"><o:p></o:p></span></div>
</div>
<div>
<br /></div>
<div style="text-align: center;">
<i>Abstract: This paper presents a comparison of Harappan and Egyptian weight systems using a detailed analysis that reveals the reason for the differences among these systems. </i></div>
<div>
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<span style="mso-ansi-language: EN-US; mso-bidi-font-size: 10.0pt;"><o:p>Key words: Ancient, weights, Harappan, Egyptian, Peyem, Q</o:p></span>edet, Stater.</div>
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<span style="mso-ansi-language: EN-US; mso-bidi-font-size: 10.0pt;">In two earlier papers
the weights from the ancient civilizations of Harappa and Egypt were examined.
Harappan weights were shown to be more accurate than the Egyptian Peyem and
Qedet weight systems while the </span><span lang="EN-GB" style="mso-bidi-font-size: 10.0pt;">Staters showed comparatively similar accuracy to those of the</span><span style="mso-ansi-language: EN-US; mso-bidi-font-size: 10.0pt;"> Harappan weights. It
is the purpose of this paper to explore further these weight systems.</span></div>
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<span style="mso-ansi-language: EN-US; mso-bidi-font-size: 10.0pt;">The
Harappan weights from three excavation sites were analysed in "Ancient
Weights from South Asian Civilization<sup>1</sup>". The paper on the
Egyptian weights is titled "Ancient Egyptian Weight Systems<sup>2</sup>"
and is based on the collection of stone weights in Petrie’s Ancient Weights and
Measures<sup>3</sup>. </span></div>
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<span style="mso-ansi-language: EN-US; mso-bidi-font-size: 10.0pt;">For the purpose of carrying out a detailed
examination of these systems, a range of weights lying within 5% of the unit
weight and its multiples, ie. the nominal values, is selected. Frequency
distribution of the weights for each ratio is prepared and displayed on a
single chart. Such individual frequency distributions are best displayed using
line charts as the distribution for each ratio shows up clearly. The composite frequency
distribution, shown in previous papers, is a sum of individual distributions at
each normalised value. A composite distribution chart for the limited range
being examined in this paper is also included for easy reference. <o:p></o:p></span></div>
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<span style="mso-ansi-language: EN-US; mso-bidi-font-size: 10.0pt;">The charts of the four systems analysed are
given below. The horizontal axis represents normalised values. The vertical
axis represents the frequency. Individual distribution charts are plotted in
different colours to help identify the nominal values represented. Only the
nominal values with sufficient number of specimens to show up on the chart are plotted.<o:p></o:p></span></div>
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<span style="mso-ansi-language: EN-US; mso-bidi-font-size: 10.0pt;">Total collection of weight pieces in each
system is tabulated in Table 1. Included in the same table are the number of
specimens within 5% of nominal weights as well as the number that make up the charts
of the individual frequency distributions for each of the systems. <o:p></o:p></span></div>
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<span style="color: #1a3ff9; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">Harappan weight system<o:p></o:p></span></div>
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<span lang="EN-GB">Image 1A shows the composite frequency
distribution for the Harappan weights that falls within +/- 5% of nominal
weight values. A chart of the total collection of 697 specimens of Harappan
weights is to be found in the earlier paper<sup>1</sup>.<o:p></o:p></span></div>
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<span lang="EN-GB">The individual frequency distribution is
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<span style="color: #1a3ff9; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">Egyptian weight systems<o:p></o:p></span></div>
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<span lang="EN-GB">Three of the weight systems of ancient
Egypt, i.e. Peyem, Qedet, and Stater are examined similarly and the individual
and composite frequency distribution charts are given below. <o:p></o:p></span></div>
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<span lang="EN-GB">The composite distribution of 262 out of
269 Peyem weights is shown in Image 2A. The range of nominal weights in the
individual frequency distribution, shown in Image 2B, is from 30 to 12010
grains. <o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjk4etYwVORk6Qkqcuc5RtQ6EelNTZlT2srKtIsKcr-UlSU0WWRqsJQo-EtsTCwGlEfo5TIr1WJCmsvjM9_4i62IfScKvc8gvHuFvROBrnhO4mNEH-vgCPK-0n55Yxg319jOAqu/s1600/Image+2A.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="146" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjk4etYwVORk6Qkqcuc5RtQ6EelNTZlT2srKtIsKcr-UlSU0WWRqsJQo-EtsTCwGlEfo5TIr1WJCmsvjM9_4i62IfScKvc8gvHuFvROBrnhO4mNEH-vgCPK-0n55Yxg319jOAqu/s400/Image+2A.jpg" width="400" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4TkgpaU_saWPgIZF48AVJODorrw72dStoGqmUkD63XXSLGwy6r_dfZNfa_FX4vE9UTTb6xLthuUoDmW5ml2yBR5ekvJjB_zzT_mTskgp6Uz9J_W7P49GGuPBXAULfifQxpHww/s1600/Image+2B.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="150" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4TkgpaU_saWPgIZF48AVJODorrw72dStoGqmUkD63XXSLGwy6r_dfZNfa_FX4vE9UTTb6xLthuUoDmW5ml2yBR5ekvJjB_zzT_mTskgp6Uz9J_W7P49GGuPBXAULfifQxpHww/s400/Image+2B.jpg" width="400" /></a></div>
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<span lang="EN-GB">The composite distribution of 791 out of
861 Qedet weights is shown in Image 3A. The range of nominal weights in the individual
frequency distribution, shown in Image 3B, is from 72.25 to 7225 grains. <o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZ9JRWS7Ip7ZxwF88fGGKlNFjcJmWNZrN25wGZlnHxuBQEQTDd8Ukm50hPhKDzy-7nmPirUFVeodSfV8U2_U3L1LWSnd2gxWedzk0QncsMHpjU3SiPPMH2bvIWCYsUADirce_S/s1600/Image+3B.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="146" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZ9JRWS7Ip7ZxwF88fGGKlNFjcJmWNZrN25wGZlnHxuBQEQTDd8Ukm50hPhKDzy-7nmPirUFVeodSfV8U2_U3L1LWSnd2gxWedzk0QncsMHpjU3SiPPMH2bvIWCYsUADirce_S/s400/Image+3B.jpg" width="400" /></a></div>
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<span lang="EN-GB">The composite distribution of 374 out of
396 Stater weights is shown in Image 4A. The range of nominal weights in the
individual frequency distribution, shown in Image 4B, is from 67.7 to 6766.4
grains. <o:p></o:p></span></div>
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<span style="color: #1a3ff9; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">Harappan weight system<o:p></o:p></span></div>
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<br /></div>
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<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">Individual frequency distributions of the Harappan
weight system all peak at the nominal values. They are fairly symmetrical about
these peaks. The collection includes both the damaged and unfinished specimens,
as noted for some of the specimens. This probably accounts for the slight asymmetry
of the distribution observed in some cases. <o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span lang="EN-GB" style="color: black;">There are also two
small peaks, one on either side of the nominal values. The small peak on the
left of the nominal values is likely to be the result of damaged specimens,
whereas that to the right of the nominal values likely due to the unfinished
specimens. It is strange that the peaks on the left, at 0.97 of the normalised
values, are all coincident. Those to the right, occurring from 1.03 to 1.05 of
the normalised values, are spread out, as one would expect.<o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">It is certainly remarkable that the peaks of all the
individual weight ratios of the Harappan system are concurrent at the nominal
values considering the weights were recovered from three different locations,
all a great distance apart. <o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span style="color: #1a3ff9; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">Egyptian weight systems<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-GB">The Egyptian weight systems all show
multiple peaks. As many as four peaks are observed for the Qedet and Peyem
systems and these are distributed across +/- 5% of the nominal values. While
many peaks are concurrent at certain points, there is no single point where
peaks of all the ratios occur.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-GB">For the Stater weight system a few of the
ratios have a single peak and the rest have a double peak. More than 2 peaks
are not observed for any ratio. Single peaks are concurrent at the nominal
values and the double peaks are separately concurrent and appear on either side
of the nominal values.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span style="color: #1a3ff9; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">Named location weights<o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">In the table of weights, Petrie<sup>3</sup> has
recorded the find locations of some of the specimens. An examination of these
weights shows that a large majority are within 5% of the nominal values but
very few are exactly at the ratios of the system. Nor are multiples exact. None
of the specimens from a single location represent a complete set of nominal
weights for the system represented. <o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">Images 5, 6 and 7 </span><!--[if supportFields]><span
style='mso-bidi-font-size:14.0pt;color:black;mso-ansi-language:EN-US'><span
style='mso-element:field-begin'></span><span
style='mso-spacerun:yes'> </span>TOC \c "Figure" <span
style='mso-element:field-separator'></span></span><![endif]--><!--[if supportFields]><span
style='mso-bidi-font-size:14.0pt;color:black;mso-ansi-language:EN-US'><span
style='mso-element:field-end'></span></span><![endif]--><span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">display the information of these
specimens. The images comprise two charts locked together, sharing the title
and the horizontal axis. The upper chart displays the weight of each specimen,
expressed as log<sub>2</sub>(weight). The lower chart represents the normalized
value of each specimen. The horizontal axis represents the item number, on a
list, of these weights. The list is sorted first alphabetically and then by value.
<o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">The line in off-white in the upper chart connects all
the weights in the list while the segments in colour connect all the specimens
found at a single location. The legend on the right identifies the location.
Only the locations with sufficient number of specimens are so identified in the
chart. <o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">Individual points only are plotted in the lower chart.
All the specimens are plotted in off-white to start with and where sufficient
number of specimens are present at a named location, these are plotted over in
colours, identifying their location. <o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br />
<br />
<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">The charts are locked together such that a vertical
line through a point on one chart connects to the same specimen on the other
chart. <o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">An inspection of these charts shows that the
individual frequency distributions for some of the locations occupy the full
range of normalized values observed with the whole collection while it is much
reduced for others. However, such reduced spreads do not line up in a
systematic way to support any view of improved accuracy among these specimens. <o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEx-GUIcBo-13ZXJ5XjCP-a9Mn9wS3P3NClr_MVy-lMqPKXyHOys6uLLY6yIc95RNm_fbkmYK3i8DajMzZGcbtr4L0X3SInhyphenhyphenhmokNv-nk5WfWQx74BwYe4JF3_7rsgcy-di5K/s1600/Image+7.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="232" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgEx-GUIcBo-13ZXJ5XjCP-a9Mn9wS3P3NClr_MVy-lMqPKXyHOys6uLLY6yIc95RNm_fbkmYK3i8DajMzZGcbtr4L0X3SInhyphenhyphenhmokNv-nk5WfWQx74BwYe4JF3_7rsgcy-di5K/s400/Image+7.jpg" width="400" /></a></div>
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">An examination of the shapes of the weight specimens of
the two systems may provide some clues regarding the different levels of
accuracy.<o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">Harappan weights occur in five different shapes. These
are cubical, truncated sphere, barrel, cylinder and conical. Each of these
shapes has at least one flat face. The weight of any specimen can be adjusted
by grinding the flat face. <o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">The cubical shapes are mostly cuboid, with a few
cubes. With six faces all at right angles, it is an easy shape to achieve. <o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">There is one weight list from Mohenjo Daro<sup>5</sup>
that includes not only the shape but dimensions of each specimen and an
examination of these show that all the cubical shapes have at least one parallel
set of square faces. Either of these faces can be ground to adjust the weight
of the specimen. Shape of the square face would be preserved, only the separation
of the square faces would change as the weight of the piece is reduced. This is
not the case with the truncated sphere and barrel shapes. Grounding only one of
the faces would alter the symmetry of the shape. This can be tested by measurements
of these two shapes. If the flat faces are symmetrical about the equatorial
plane then a special effort must have been made in manufacturing these weight
pieces. <o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">Egyptian weights<sup>3</sup> come in a large number of
shapes and not all have a flat face. Those with a flat face could be ground
down to adjust the weight where necessary. If the manufacturing did not require
a close control of the final weight, then the shape of the piece may not matter
much. Perhaps this is the reason why there are many oddly shaped specimens.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="mso-tab-count: 3;"> </span><span style="mso-tab-count: 4;"> </span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">A system of weights with multiples based on a specific
unit value is expected to have the largest number of specimens at those values.
The rest of the specimens should, in general, be equally distributed about those
values. The closeness of the distribution about the peak determines</span><span style="color: #1a3ff9; font-size: 18.0pt;"> </span><span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">the
accuracy of the manufactured weights. <o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">Of the four weight systems examined, the Harappan
system comes closest to the expected distribution. </span><span lang="EN-GB" style="color: black;">These weights could have been calibrated at some central
location, in which case all the specimens have to be transported there.
Alternatively, they could have been calibrated at the location they were found.
This would require, at each location, accurate sets of standard weights as well
as balances capable of weighing to high precision.<o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">Of the Egyptian systems examined, the Stater system
has a few of the ratios close to the expected distribution, the remainder
showing a split distribution. The Peyem and Qedet systems, on the other hand,
show multiple peaks.<o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<br /></div>
<div class="MsoNormal">
<span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">The Egyptian weight standard may have changed over the
long period of usage and if the recovered weights are a collection manufactured
under different standards then the observed distribution would be the result.
Both the Peyem and Qedet specimens are confined within 5% of the nominal
weights and this could be possibly because they were manufactured to no better
than 5% accuracy.</span><span style="color: #7030a0; mso-ansi-language: EN-US; mso-bidi-font-family: Arial; mso-bidi-font-size: 14.0pt; mso-bidi-language: HI;"> </span><span style="color: black; mso-ansi-language: EN-US; mso-bidi-font-size: 14.0pt;">A
specimen within this range could end at any value and the occurrence of peaks would
then be purely a matter of chance. A different collection would show different
arrangement of peaks. <o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-GB" style="color: black; mso-themecolor: text1;">This
detailed analysis of the weights points to an exercise of control in their
manufacture. Control implies organisation. It is strongest for the Harappan systems
and covers the whole area of the Harappan civilization. The Egyptian weight
systems demonstrate a varying degree of control in their manufacture. <o:p></o:p></span><br />
<span lang="EN-GB" style="color: black; mso-themecolor: text1;"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-GB" style="color: black; mso-themecolor: text1;">Harappan
civilization has left no textual documentation to learn more about this
organization. The Egyptian civilization has left plenty. A search through those
might prove beneficial regarding the organization pertaining to the manufacture
of their weight systems. <o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
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<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-GB">References:<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal" style="margin-left: 36.0pt; mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span class="MsoHyperlink"><span lang="EN-GB" style="text-decoration: none; text-underline: none;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">
</span></span></span></span><!--[endif]--><span lang="EN-GB">This paper can be viewed
here: <a href="https://jignashi.blogspot.co.uk/2010_01_01_archive.html" target="_blank">https://jignashi.blogspot.co.uk/<o:p></o:p></a></span></div>
<div class="MsoNormal" style="margin-left: 36.0pt;">
<span lang="EN-GB">A copy is also
available on Academia: <a href="https://www.academia.edu/4846613/Ancient_Weights_from_South_Asian_civilization">https://www.academia.edu/4846613/Ancient_Weights_from_South_Asian_civilization</a><o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 36.0pt;">
<br /></div>
<div class="MsoNormal" style="margin-left: 36.0pt; mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="EN-GB"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span lang="EN-GB">This paper can be viewed here: <a href="https://jignashi.blogspot.co.uk/search?updated-min=2013-01-01T00:00:00Z&updated-max=2014-01-01T00:00:00Z&max-results=1">https://jignashi.blogspot.co.uk/search?updated-min=2013-01-01T00:00:00Z&updated-max=2014-01-01T00:00:00Z&max-results=1</a><o:p></o:p></span><a href="https://www.academia.edu/4846620/Ancient_Egyptian_Weight_Systems" style="text-indent: 0px;">Ancient_Egyptian_Weight_Systems</a></div>
<div class="MsoNormal" style="margin-left: 36.0pt; mso-layout-grid-align: none; mso-pagination: none; text-autospace: none;">
<span lang="EN-GB">A copy is also available on
Academia: <a href="https://www.academia.edu/4846620/Ancient_Egyptian_Weight_Systems">https://www.academia.edu/4846620/Ancient_Egyptian_Weight_Systems</a></span><span style="color: #1a3ff9; font-size: 18.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal" style="margin-left: 36.0pt; mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="EN-GB"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span lang="EN-GB">Ancient Weights and Measures by
Sir Flinders Petrie <o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 36.0pt;">
<span lang="EN-GB">Department of
Egyptology, University College, London, 1926<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal" style="margin-left: 36.0pt; mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="EN-GB"><span style="mso-list: Ignore;">4.<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span lang="EN-GB">Ancient Weights from South
Asian Civilization, op cit<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 36.0pt;">
<span lang="EN-GB">Chart 1<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 36.0pt;">
<br /></div>
<div class="MsoNormal" style="margin-left: 36.0pt; mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="EN-GB"><span style="mso-list: Ignore;">5.<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span lang="EN-GB">Further Excavations at Mohenjo
Daro, E. J. H Mackay, 1938,<span style="mso-spacerun: yes;"> </span>pp
607-612<o:p></o:p></span></div>
<div class="MsoNormal" style="margin-left: 18.0pt;">
<br /></div>
<div class="MsoNormal" style="margin-left: 36.0pt; mso-list: l0 level1 lfo1; text-indent: -18.0pt;">
<!--[if !supportLists]--><span lang="EN-GB"><span style="mso-list: Ignore;">6.<span style="font: 7.0pt "Times New Roman";">
</span></span></span><!--[endif]--><span lang="EN-GB">Ancient Egyptian Weight
Systems, op cit, Charts 1b, 2b and 3b.<o:p></o:p></span></div>
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<span lang="EN-GB" style="font-family: "timesnewromanpsmt"; font-size: 16.0pt;">© 2016 Chandrakant Doshi<o:p></o:p></span></div>
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Chandra Doshihttp://www.blogger.com/profile/12676853087431948250noreply@blogger.comtag:blogger.com,1999:blog-30960876.post-12042572337133732422013-01-11T17:13:00.001+00:002013-01-11T17:13:53.406+00:00Ancient Egyptian Weight Systems<div dir="ltr" style="text-align: left;" trbidi="on">
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 18.0pt;"><b>Ancient Egyptian
Weight Systems<o:p></o:p></b></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: TimesNewRomanPSMT; font-size: 24.0pt;"><b>by</b></span><span lang="EN-US" style="font-family: Times-Roman; font-size: 16.0pt;"><o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: TimesNewRomanPSMT; font-size: 20.0pt;"><b>Chandrakant Doshi<o:p></o:p></b></span></div>
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<span lang="EN-US" style="font-family: TimesNewRomanPSMT; font-size: 14.0pt;">email:<span style="mso-spacerun: yes;"> </span>jignashi@yahoo.co.uk</span><span lang="EN-US" style="font-family: Times-Roman; font-size: 14.0pt;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: TimesNewRomanPSMT; font-size: 14.0pt;"><i>Abstract: </i></span><br />
<span lang="EN-US" style="font-family: TimesNewRomanPSMT; font-size: 14.0pt;"><i><br /></i></span>
<span lang="EN-US" style="font-family: TimesNewRomanPSMT; font-size: 14.0pt;"><i>This paper presents
an analysis of three of the ancient Egyptian<span style="mso-spacerun: yes;"> </span>weight systems.</i></span><span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><b><o:p></o:p></b></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Key words: Weights, Egyptian, ancient, Petrie,
Peyem, Qedet, Stater<o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">In
"Ancient Weights and Measures", Flinders Petrie<sup>1</sup> lists
stone and metal weights recovered from excavations in Egypt. This paper
presents an analysis of some of these stone weights which start with catalogue
number 2001. These are tabulated in plates 27 to 42 of Petrie’s publication.
The metal weights and the stone weights from earlier finds are not included in
this analysis.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">There
are over 2700 stone weight specimens listed in these tables. Of these, about
2100 pieces are purchased items. Just over 600 pieces have provenance<sup>2</sup>.
<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Petrie
classified this collection of stone weights under eight systems. <span style="color: black;">Information in the tables includes details about the
material of the specimens, form of each specimen, weight in grains, assigned
ratio and the unit value for each specimen. Some of the specimens are damaged.
Listed weight for such pieces is corrected by estimating the damage. The amount
added to the weight is listed separately in the last column headed
"Details". Corrected weights are used in this analysis.<o:p></o:p></span></span></div>
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<span lang="EN-US" style="color: black; font-family: "Times New Roman"; font-size: 14.0pt;">In the tables, the specimen</span><span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">s</span><span lang="EN-US" style="color: black; font-family: "Times New Roman"; font-size: 14.0pt;">
are sorted in an ascending order of the calculated unit value. In the first
column is the catalogue number for each specimen. This starts at 2001 to avoid
mix up with previously published lists of ancient Egyptian weights. Under each
weight system, a few specimen do not have a catalogue number. These appear as
an alternative fit with this weight system and full details of such specimen,
including the catalogue number, are given under the weight system they are
identified to belong. A note in the final column identifies that weight system.</span></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Analysis
of the ancient weight systems can be carried out by different methods and this
paper examines some of them. These methods are illustrated with three of the
ancient Egyptian weight systems. These are: Peyem, Qedet and Stater. The Peyem
is the first of the eight systems and has the lowest unit value, Qedet has the
most numerous specimens and the Stater weights show the best accuracy of the
eight systems. <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><b>Log
charts<o:p></o:p></b></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">A
log chart is a plot of the logarithm of the weight values. The weight list is
sorted in ascending order. This way the lowest weight appears on the left of
the chart, with heavier weights progressing towards the right. <o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">The
logarithm (log for short) of a number is the power to which another number,
called the base, is raised. The most common base is 10. Any number can be used
as a base. In the ancient weight systems, multiples and submultiples of 2 are
fairly standard. Hence<span style="mso-spacerun: yes;"> </span>the whole
list of weights is expressed as log to base 2 and plotted on a chart. With base
2, logarithms of 2, 4 and 8, for example, are 1, 2 and 3, respectively. A
number between the exact multiples of 2 will have its logarithm between the
logarithms of those multiples. For any two numbers, the logarithm of smaller
number is smaller than that of the larger number. <o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">The
log chart displays the weight list in a compressed form. This permits such a
large range to be displayed on a reasonable size chart.<span style="mso-spacerun: yes;"> </span>Hemmy used a log chart to display a
wide range of Harappan weights on a single chart<sup>3</sup>. He displayed the
number of specimens lying within a short range of nominal weight values for the
weights from 0.5 grams to 550 grams. This required some processing of weight
data before the charts could be prepared. In the charts presented here nothing
more than sorting the weight data in an ascending order is required.<o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">In
the charts, the vertical scale is
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<!--StartFragment--><span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-language: EN-US;">adjusted</span><!--EndFragment--> such that the grids are a unit
apart. This means that an increase of 1 <span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-language: EN-US;">on the
chart represents</span> a doubling of the
weight value. Similarly, a decrease of one <span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt; mso-ansi-language: EN-US; mso-fareast-language: EN-US;">represents</span><!--EndFragment--> halving of the value.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">A
log chart suitably prepared can identify the systematic nature of any weight
system and point to different multipliers used in the system. The frequency at
any particular value is also easily observable. Such information can be gleaned
without processing the weight data. In the end the raw data needs to be
processed to gather further information about the make up of the system.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br />
<br />
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><b>Table
of Ratios<o:p></o:p></b></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">A
table listing the ratios and frequencies of weights at those ratios is another
way of examining the weight system. Petrie assigned a ratio to every weight
specimen in each system and then using the weight of the piece, calculated the
unit value for each specimen. <o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Since
a specific value is not assigned to the unit of the system, an average of all
the unit values is used as the system's unit weight. The nominal weight for
each ratio is simply the product of this unit value and the assigned ratio. An
additional column lists the logarithm to base 2 of each nominal weight. This
should help to identify the plotted points on the log chart.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">The
frequencies are determined by counting the number of specimens that are
allocated to each nominal weight. This is accomplished as follows. Any weight
between two adjacent nominal weights A and B will be allocated to A if it is
less than (A+B)/2, otherwise it is allocated to B. Continuing this way, all the
weights between the lowest and the highest nominal weights are allocated to one
or the other weight ratio. The weights lighter than the lowest nominal value
are assigned to the lowest ratio. Similarly, those heavier than the highest
nominal value are assigned to the highest ratio.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">This
method of allocation of weights ensures that each specimen belongs to only one
ratio and that no specimens are left out. <o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br />
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><b>Histograms<o:p></o:p></b></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Another
way of displaying the characteristic of a weight system is by means of a
histogram or a frequency distribution chart. This displays the number of
specimens in a given interval of weight. To cover the entire range of the
system, the whole collection of weights is first normalised. To do this, each
specimen allocated to a ratio or nominal weight is divided by that nominal
weight. This will give a value of 1 for specimens that are exactly equal to the
nominal weight. Those specimens below nominal weight will return a value less
than 1 and those above greater than 1.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">The
number of pieces within 1% interval of the normalised value is counted. The
interval at the centre spans the range from 0.995 to 1.005 of the normalised
value and the rest of the range is similarly divided. The count of specimens in
each interval is displayed on a chart. This is the histogram or frequency
distribution of the weight system. <o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: center;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 20.0pt;"><b>Weight Systems<o:p></o:p></b></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><b>Peyem<o:p></o:p></b></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">There
are 269 specimens in this weight system. The unit value of these weights ranges from
112.0 grains to 125.2 grains and their average value is 119.6 grains. <o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Chart 1a
is a log plot of these Peyem weights. The binary scale
of the weight system, in parts, is apparent, as is the fact that the whole
system does not follow the binary scale. <o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">The
vertical axis is set to start at 3.90 = log<sub>2</sub>(15). The first six
steps in the chart, up to 8.90 on the vertical scale, represent a sequence of
binary multiples as they all lie on successive grid lines. The steps or flats
are made of weights very close in value.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">The
next flat is below the expected 9.90, indicating a multiplier less than 2 is
involved. An inspection of values in Table 1 shows that this multiplier is
1.25. Starting from this step, there are four flats that form another sequence.
There are a few odd groups that are not part of this sequence. Of these, two
are on the grid line and therefore belong to the sequence starting with the
lowest weight. Another single weight, also on the grid line, appears very near
the top. <o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: center;">
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o:title="Chart 1a"/>
</v:shape><![endif]--><!--[if !vml]--><span class="Apple-style-span" style="font-family: Times; font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixZGtcTOl7PbrNE3phtZK1EzvVRbTYMeIMDv-QvNJfdFK9s8UDj2gIUJkxDE2PlTRI7eFLe-lUgPdJs5j0YTv-kofJAutmeJSsOCrRK2BJkibENXJIIg3OvN8XWM9VdrVUcxu8/s1600/Chart+1a.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="267" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixZGtcTOl7PbrNE3phtZK1EzvVRbTYMeIMDv-QvNJfdFK9s8UDj2gIUJkxDE2PlTRI7eFLe-lUgPdJs5j0YTv-kofJAutmeJSsOCrRK2BJkibENXJIIg3OvN8XWM9VdrVUcxu8/s640/Chart+1a.jpg" width="640" /></a></span></span><span class="Apple-style-span" style="font-family: 'Times New Roman'; font-size: 19px;"> </span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">By
setting the vertical scale to different values, successive flats that form a
sequence can be identified. In the chart above, three separate groups that
follow a binary sequence can be seen.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Table
1 is constructed using 119.6 grains as the unit of the Peyem system. An
examination of the table shows the basic scale of 1, 2, 4, 5 repeating over the
next two decades. The expected ratio of 500 is absent and that at 1000 has only
one specimen. There are also three fractional ratios of a half, a quarter and
an eighth. <o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">There
are a few odd ratios: 3, 8, 12, 16, 60 and 1500. Their frequencies are low.
They do not form part of any sequence as their decimal multiples are absent.
Altogether, 17 specimens out of a collection of 269 do not follow the basic 1,
2, 4, 5 scale.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><!--[if !supportEmptyParas]--> <!--[endif]--><o:p></o:p></span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6-wg0MIWQJHl_4_2DKocxKjCP1gZRqKGUftDLTYs56bnPphl13Czoc2OKalqviDoGjaljedvVqyktbyWOTEStq-64LdXSTxgsZMHoZcLEkijC-vz7g61Tux6knUeZr2TsJE3x/s1600/Table+1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6-wg0MIWQJHl_4_2DKocxKjCP1gZRqKGUftDLTYs56bnPphl13Czoc2OKalqviDoGjaljedvVqyktbyWOTEStq-64LdXSTxgsZMHoZcLEkijC-vz7g61Tux6knUeZr2TsJE3x/s640/Table+1.jpg" width="600" /></a></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: center;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><!--[if gte vml 1]><v:shape id="_x0000_i1028"
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<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Chart
1b is a histogram of the Peyem system. The set of nominal weights, as listed in
Table 1, occurs at the normalised value of 1.00 and there are four values where
the frequency is higher than at the nominal weights.<span style="mso-spacerun: yes;"> </span>The peak frequency occurs at 0.96 of the normalised value. <o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; text-align: center;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><!--[if !supportEmptyParas]--> </span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIwgrajy3bbXmjUHQNuMeYnXU88GGOPbvijKNPY6jgIYUgvUSYT0ZGunvSK374JnAlW1flSbadCdgmIX-PDmiil8nv9phNmd7kRId85S4ztyR2t39eDeKGwRF1uvTbOtf92NWa/s1600/Chart+1b.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="298" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIwgrajy3bbXmjUHQNuMeYnXU88GGOPbvijKNPY6jgIYUgvUSYT0ZGunvSK374JnAlW1flSbadCdgmIX-PDmiil8nv9phNmd7kRId85S4ztyR2t39eDeKGwRF1uvTbOtf92NWa/s640/Chart+1b.jpg" width="640" /></a><span class="Apple-style-span" style="font-family: 'Times New Roman'; font-size: 19px;"> </span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><b><br /></b></span>
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><b><br /></b></span>
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><b><br /></b></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><b>Qedet<o:p></o:p></b></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">The
collection of the Qedet system comprises 861 specimens. The unit value ranges
from 135.5 grains to 153.5 grains with the average equal to 144.5 grains. <o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Chart
2a displays the whole collection on a log scale to base 2. The unit value of
144.5 grains can be seen on the grid line at 7.17. The collections on the
following two succeeding grid lines form part of the binary sequence of 1, 2,
4. The presence of a large number of specimens just above the grid line at 9.17
shows a non-binary multiple of 5. Like Peyem, Qedet also follows a basic scale
of 1, 2, 4, 5.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">All
of these multiples can be seen in Table 2b, which also shows the frequencies at
different ratios as well as the value to log<sub>2</sub>(weight) for different
nominal weights.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Again
there are fractional ratios, this time a third and a sixth as well as a half,
with the fractional ratio of a sixth having a frequency of only one. The
collection on the grid line at 6.17 are weights at half the unit value.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Here
also there are odd ratios. These are the fractional ratios of a third and a
sixth and the ratios 2.5, 3, 6, 8, 25, 30, 150, 250. The frequency at ratio of
25, like that at one third, is surprisingly high. In all, there are 70 of these
odd ratios out of the total collection of 861 specimens.<o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><!--[if !supportEmptyParas]--> <!--[endif]--><o:p></o:p></span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5DrhAVZr4Sd4WzkG28YlEH7JmeWAF8hIxB0sdj1lQiQFJ6iZbRM1uXEgnrY1NDNtR4qSwDPZ5sXjMH_NSGmGdKOSy_POm5aNlW7aEp2n9WE-v6eDEFS664FAYGoXypd3zx4A2/s1600/Chart+2a.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="276" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5DrhAVZr4Sd4WzkG28YlEH7JmeWAF8hIxB0sdj1lQiQFJ6iZbRM1uXEgnrY1NDNtR4qSwDPZ5sXjMH_NSGmGdKOSy_POm5aNlW7aEp2n9WE-v6eDEFS664FAYGoXypd3zx4A2/s640/Chart+2a.jpg" width="640" /></a></div>
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<br /></div>
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<br /></div>
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<br /></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Chart
2b is a histogram of the Qedet system. The collection ranges from 0.93 to 1.06
of the normalised values of the weights. The peak frequency occurs at 0.98 of
the normalised value.<o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><b><!--[if !supportEmptyParas]--> </b></span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4fy-l8Vf6ytCyNgSoGpBhyJcdXWFaUCB86rKDsPFvdjK5T_ykFJg34ih_H-dOiC72LguPERIc9-gqVczbNjYBZOUMsOsC6od2SB0pW9FdGzwpBke3bGvLnLIxdnZdvA2-NleC/s1600/Chart+2b.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="266" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4fy-l8Vf6ytCyNgSoGpBhyJcdXWFaUCB86rKDsPFvdjK5T_ykFJg34ih_H-dOiC72LguPERIc9-gqVczbNjYBZOUMsOsC6od2SB0pW9FdGzwpBke3bGvLnLIxdnZdvA2-NleC/s640/Chart+2b.jpg" width="640" /></a><span class="Apple-style-span" style="font-family: 'Times New Roman'; font-size: 19px;"><b> </b></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><b><br /></b></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><b><br /></b></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><b>Stater<o:p></o:p></b></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">The
Stater system compromises 396 specimens. The unit value of weights ranges from
132.0 grains to 140.3 grains and their average value is 135.3 grains. <o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Chart
3a is a log plot of these weights. The vertical scale is adjusted so that the
value of 7.08 (= log<sub>2</sub>(135.3)) falls on a grid line. <o:p></o:p></span></div>
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<span class="Apple-style-span" style="font-family: 'Times New Roman';"><span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Table
3 shows the ratios of the Stater weight system. The scale of 1, 2, 5 better
describes the system as there are specimens with multiples of tens of this
scale up to a ratio of 1000. </span></span></div>
<span class="Apple-style-span" style="font-family: 'Times New Roman';">
</span><br />
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<span class="Apple-style-span" style="font-family: 'Times New Roman';"><span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">While
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multiples. All these ratios, along with the sixth, the third and 8 and 150 are
odd to the basic ratio of 1, 2, 5. In all, there are 62 of these odd ratios out
of the total collection of 396 specimens.<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">The
Stater system of weights is an odd one out of the three systems being analysed
in that its basic scale is different from the other two and it is also the most
accurate of the three. This can be seen from the histogram, Chart 3b. This
shows the distribution ranging from 0.98 to 1.04 of the normalised values. This
in spite of a large number of specimens with odd ratios.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><span class="Apple-style-span" style="font-family: Times; font-size: small;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAh9SpD3T-DEUbfTMC-Q-AHYwtFkQhouSfHFYs0R6SM9XH35Mp4hVbA0MWebfVElNtoX6ojyq-OMR06A-hcaScWc_14MMow0K_UoqlyEjdEgDng9pJyM5D9wZ3-YX86GzN6G0Z/s1600/Table+3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAh9SpD3T-DEUbfTMC-Q-AHYwtFkQhouSfHFYs0R6SM9XH35Mp4hVbA0MWebfVElNtoX6ojyq-OMR06A-hcaScWc_14MMow0K_UoqlyEjdEgDng9pJyM5D9wZ3-YX86GzN6G0Z/s640/Table+3.jpg" width="614" /></a></span></span></div>
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<br /></div>
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<br /></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><b>Comments<o:p></o:p></b></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">The
analysis of each weight system is based on Petrie's assignment of ratio to each
specimen from which a value of the unit of the system is derived. The histogram
or frequency distribution of the weight system is based on the average of the
derived values of the unit. The frequencies at each ratio listed in the Table
of Ratios are obtained from the weight tables. The Log Charts are based solely
on the weight of specimens which Petrie determined using a balance and set of grain
weights<sup>4</sup> specially manufactured for the purpose. <o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">A
set of weights manufactured to a specific value will display a spread about
that specific or intended value. The number of specimens at the intended value
is the highest and the distribution of the rest is symmetrical about it. The
distribution trails away from the peak. The distribution is shaped like a bell
and is known as the Normal distribution. <o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">The
frequency distribution for the Peyem and the Qedet systems, Charts 1b and 2b,
presents a challenge. The peak frequency is not at the centre of the
distribution. Frequencies away from the nominal value do not trail off as
expected for a Normal distribution. The frequency distribution of the Stater
system, however, is much closer to that expected for a Normal distribution. <o:p></o:p></span></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">These
histograms are fairly compact, the range of spread of the weights about the
nominal values is mostly less than 5%. The Stater system displays the narrowest
range, with most of the collection accommodated within 2% of the nominal
values. <o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Histograms
can be examined for slightly different values of the unit weight. For the Peyem
and Qedet systems, the distribution will roll along, with the frequencies
rippling along the top. No unit value can give a distribution with a single peak
frequency. But for the Stater system the matters are different. A slightly
higher value of the unit weight is found that produces an almost symmetrical
distribution with the peak frequency occurring at the nominal value. Chart 3b1
shows the distribution of the Stater system for two different values of the
unit weight: at the average value of 135.3 grains in blue, and at 135.8 grains
in red. <o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><br /></span></div>
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id="_x0000_i1026" type="#_x0000_t75" style='width:468pt;height:179pt'>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">All
the systems examined here include ternary and quinary ratios as well as the
usual binary and decimal ratios found in many of the ancient weight systems.
The quinary ratio can be explained as an intermediate stage of a decimal
scale. The ternary ratio is not so easily explained. Binary submultiples are a
standard feature of each weight system. Also, a few ternary submultiples are
present in two of the weight systems examined but none of the systems has any
quinary submultiple. <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Removing
the weights of the ternary scale alters the picture. Both total frequency and
distribution are affected. However, reassigning weights of the ternary scale to
the nearest ratio applicable alters the frequency distribution only. <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Examination
of the weight systems presented here is based on weights recovered nearly 100
years ago. Addition of newer finds could alter the picture. A larger collection
will give a better value of the unit of the weight system. It should also
provide a better view of the ratios that existed in different weight systems. <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><b>Acknowledgement</b><o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><b><br /></b></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">Special<span style="mso-spacerun: yes;"> </span>thanks are due to Norman J. Street for
providing access to certain books without which this paper would not have been
possible.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">References:<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">1.<span style="mso-tab-count: 1;"> </span>Ancient Weights
and Measures by Flinders Petrie, Department of Egyptology, University College,
London. 1926<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm; mso-outline-level: 1;">
<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">2.<span style="mso-tab-count: 1;"> </span>An
Analysis of the Petrie Collection of Weights by A. S. Hemmy in<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;"><span style="mso-spacerun: yes;"> </span><span style="mso-tab-count: 1;"> </span>The
Journal of Egyptian Archaeology, Vol. 23 No. 1 (June 1937), p 42 <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">3.<span style="mso-tab-count: 1;"> </span>The Statistical
Treatment of Ancient Weights by A. S. Hemmy, in Ancient Egypt, December 1935.
Page 86.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "Times New Roman"; font-size: 14.0pt;">4.<span style="mso-tab-count: 1;"> </span>Ancient Weights
and Measures, op cit, p3<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: TimesNewRomanPSMT; font-size: 16.0pt;">© 2012 Chandrakant Doshi<o:p></o:p></span></div>
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Chandra Doshihttp://www.blogger.com/profile/12676853087431948250noreply@blogger.comtag:blogger.com,1999:blog-30960876.post-42082049804825354962012-06-29T17:13:00.000+01:002020-05-29T18:47:07.720+01:00Minor Rock Edicts at Erragudi<div dir="ltr" style="text-align: left;" trbidi="on">
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 16pt;"><b>Minor Rock Edicts at Erragudi<o:p></o:p></b></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 16pt;"><b>by<o:p></o:p></b></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 16pt;"><b>Chandrakant Doshi<o:p></o:p></b></span><br />
<span lang="EN-US" style="font-family: "times new roman"; font-size: 16pt;"><b><br /></b></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">email: </span><span lang="EN-US" style="font-family: "comic sans ms"; font-size: 14pt;"><span style="font-family: "times new roman";"><a href="mailto:jignashi@yahoo.co.uk">jignashi@yahoo.co.uk</a></span></span><span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;"><o:p></o:p></span><br />
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 12pt;"><i>Abstract: This
note examines minor rock edicts at Erragudi and shows that the bidirectional
writing in the inscription is not boustrophedon style of writing.<o:p></o:p></i></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 12pt;"><i><br /></i></span></div>
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<span lang="EN-US">KEY WORDS: </span><span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">Brahmi,
bidirectional, ancient inscriptions, edicts, boustrophedon, Asoka.</span><span lang="EN-US"><o:p></o:p></span><br />
<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;"><br /></span>
<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;"><br /></span></div>
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</span><br />
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<span class="Apple-style-span" style="font-family: "times new roman"; font-size: medium;"><span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">Asoka's
edicts are found in many places. At Erragudi, two minor and 14 major rock
edicts are inscribed on 6 boulders. D. C. Sircar<sup>1</sup> has published
images of the rock faces as well as the text in his "Asokan Studies".
Not all the inscriptions have survived well and the best readable are the minor
edicts, found on boulder F. This note examines some of the notable features
present in these minor edicts including bidirectional writing and whether that
represents boustrophedon style of writing. <o:p></o:p></span></span></div>
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</span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">The
inscriptions are all in Brahmi. In the note Roman characters are used to
represent Brahmi using the Kyoto-Harvard transliteration scheme and are printed
in red.<o:p></o:p></span></div>
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<span class="Apple-style-span" style="font-family: "times new roman"; font-size: medium;"><span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">Plate
II in D. C. Sircar's "Asokan Studies" is based on the impressions
taken by N. P. Chakravarti<sup>2</sup> and is reproduced here as Image 1.<o:p></o:p></span></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">Image
2 is the upper half of Plate II.
It covers most of the first MRE. Only the last few characters at the end
are missing. It is this image that is used to trace the inscription, showing
the changing direction of writing. It relies totally on Sircar's reading of the
text regarding the direction of writing. <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">Most
of the text is engraved to be read from left to right with a few lines running
from right to left. In places a segment of text is detached from a line and
located elsewhere. To distinguish these differences, separate colours are used
in the tracing. <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">Normal
left to right text is in green, right to left in red and the detached segment
in purple. It so happens that the detached segments are all running left to
right. The flow of the text is indicated by the pencil lines, the direction
indicated by the arrows on these lines. All this can be seen in Image 3.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;"><o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">Writing
where alternate lines change direction is known as boustrophedon style of
writing. Characters orient with the changing direction. Same character on two
adjacent lines will appear as mirror image of each other. In the MREs at
Erragudi this does not happen. The characters do not orient themselves to the
changed direction of writing. The inscription in these MREs is not
boustrophedon style of writing in the conventional sense.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">In
Image 3a black lines point to same syllable on lines engraved in opposite
directions. Syllables marked out are <span style="color: red;">ha</span> and <span style="color: red;">sa</span>. <o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">An
examination of the image shows that the characters are of varying size. Their
alignment is not regular either. None of the lines runs strictly horizontal.
The first three lines are nearest to being horizontal for majority of their
run. There is certain amount of undulation, dictated, no doubt, by the surface
of the rock.<o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">The
first line of the engraving starts from left and at the end carries on the line
below, from right to left. The scheme of alternating directions breaks down
when the fourth line, starting at right edge, cuts off before reaching the left
edge. Here a segment of six characters (in purple) is engraved on the same
line, starting from left edge and running towards the point where the break
occurred. The reason for this is not obvious. Defects in the rock could not be
the cause as the six characters could easily have carried on to the left edge,
completing the line. <o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">Even
stranger is the positioning of the last character of this detached segment. The
character descends by almost a line height so that it is just above the
character on the continuance line below, which is engraved from left to right.
In this way the last two characters from the detached segment and the first character
of the next line form a staircase, as if to gently guide the reader down to the
point on the next line where to continue reading. These three characters are
pointed to by blue lines in Image 3a. <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;"><br /></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">This
is an amazing feature but it is not repeated elsewhere in the inscription. It
may be that an attempt was started on the next detached segment but then
abandoned. This can be seen in the second 6 character segment in the line below
(in purple). The first two characters are starting to form a staircase but the
third is pulled up and the remaining continue on a more or less horizontal
line. The continuance (in green) is at the left edge of the line below this
segment. Yellow lines in Image 3a point to these three characters.<o:p></o:p></span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">An
examination of the alignment of the characters shows that very few are
perfectly aligned to the vertical while many characters lean to the left and a
few lean to the right.<o:p></o:p></span></div>
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<br /></div>
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<span class="Apple-style-span" style="font-family: "times new roman"; font-size: medium;">
</span><br />
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span class="Apple-style-span" style="font-family: "times new roman"; font-size: medium;"><span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">Another
peculiarity observed with the MREs is the cursive rendering of the diacritic
for <span style="color: red;">i</span>, which was observed in the Girnar Rock
Edicts, discussed in the Note<sup>3 </sup>entitled “Artistry at Girnar”. This
means there are two places identified with this particular style of diacritic
for short <span style="color: red;">i</span>. Four such examples in MRE I are
pointed by red lines in Image 2a. <o:p></o:p></span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzeyMJ5exfVNO2wJOedtOH7peh4hyCu_x7dOVTkupkajjgd7imkReWYvGlInalQjJvpD3JzWXSpI8-5KkuN7vMd9eWOy8cbh8OrtqqHWRMfTAZjm3fb0oMdeDUigKZj4GSVa78/s1600/Image+2a.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="328" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzeyMJ5exfVNO2wJOedtOH7peh4hyCu_x7dOVTkupkajjgd7imkReWYvGlInalQjJvpD3JzWXSpI8-5KkuN7vMd9eWOy8cbh8OrtqqHWRMfTAZjm3fb0oMdeDUigKZj4GSVa78/s640/Image+2a.jpg" width="640" /></a></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">Image
4 is part of the 13th. Erragudi rock edict and shows the state of the rock
face. It is possible, despite the chipped face, to recognise some of the
characters. They appear regular, like those at Jaugada and Dhauli. In this they
represent considerable improvement from the characters engraved in the MREs. In
this image it is not possible to check whether the cursive rendering of
diacritic <span style="color: red;">i</span> is present. A modern digital image
would be a great help.<o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBzp-ivZXE4tWlhiOPPhOBDf4QuSGGCW313hDVAKtnCNmsAG4nmZNLIEoht4Lcf3wGux039ZrxM5KTMWHaFgDWkgeMepPSk36yYlt4U7Quyp8N8ZrIEyfuOnxIKjMDmWRxouWH/s1600/Image+4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="396" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBzp-ivZXE4tWlhiOPPhOBDf4QuSGGCW313hDVAKtnCNmsAG4nmZNLIEoht4Lcf3wGux039ZrxM5KTMWHaFgDWkgeMepPSk36yYlt4U7Quyp8N8ZrIEyfuOnxIKjMDmWRxouWH/s640/Image+4.jpg" width="640" /></a></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;"><br /></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">Image
5 is a trace of the second minor rock edict at the site. Bidirectional writing
and detached segments are identified using the same colour scheme employed in
MRE I. The black arrow marks the start of this edict. Some of the characters in
the first line are truncated. The engraving is fairly regular even though the
lines are not horizontal, particularly at the beginning.
<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">Not many</span> lines are
written from right to left. The detached segments are short, the longest having just two characters. There is no staircase. <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;"><o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">The
Erragudi MREs display a few interesting features. <o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">The
characters vary in size and shape. This is particularly true with MRE I. <o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">A
few of the lines are engraved to be read from right to left whereas the normal
direction is from left to right, as observed in MREs at other sites as well as
all the other edicts in Brahmi. These irregular changes of direction do not
make the writing boustrophedon in the conventional sense since the characters
do not alter their orientation.<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">A
staircase in MRE I, from line 4 to 5 can hardly be accidental but is not
repeated elsewhere in the two MREs. <o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0cm;">
<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">Lack
of familiarity with the script could explain the inconsistency in the shape and
size of the characters. More than one person could have been involved in the
engraving of these edicts. Haphazard changes in direction is a little difficult
to explain. Perhaps the engravings were not continuously supervised. The
staircase could be an experiment to put matters right. Some of these features
have been smoothed out in MRE II. They appear to be absent from the major rock
edicts at the site.<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">References<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">1.
Asokan Studies <o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">by D. C. Sircar, Indian Museum, Calcutta, 1979<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">2. D. C. Sircar, op
cit, page 2<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;">3. Artistry at
Girnar, 2010<o:p></o:p></span></div>
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<span lang="EN-US" style="font-family: "times new roman"; font-size: 14pt;"><a href="http://jignashi.blogspot.com/">http://jignashi.blogspot.com</a><o:p></o:p></span><br />
<br />
<br />
<br />
<br />
<div style="text-align: center;">
<span class="Apple-style-span" style="font-family: "georgia" , serif; font-size: 16px;">© 2012 Chandrakant Doshi</span></div>
<span class="Apple-style-span" style="font-family: "georgia" , serif;"><br /></span></div>
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<span class="Apple-style-span" style="color: white;">*</span><br />
<span class="Apple-style-span" style="color: white;"><br /></span>
<span class="Apple-style-span" style="color: white;">*</span></div>
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Chandra Doshihttp://www.blogger.com/profile/12676853087431948250noreply@blogger.comtag:blogger.com,1999:blog-30960876.post-3248917579545702032010-10-20T18:46:00.000+01:002021-01-29T17:22:02.766+00:00Artistry at Girnar<div dir="ltr" style="text-align: left;" trbidi="on">
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<span class="Apple-style-span" style="font-family: "times" , "times new roman" , serif;"><span class="Apple-style-span" style="font-size: x-large;">Artistry at Girnar</span><o:p></o:p></span></div>
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<span class="Apple-style-span" style="font-family: "times" , "times new roman" , serif;"><span class="Apple-style-span" style="font-size: x-large;"><br /></span></span></div>
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<br /></div>
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<span class="Apple-style-span" style="font-family: "times" , "times new roman" , serif;">by<o:p></o:p></span></div>
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<span class="Apple-style-span" style="font-family: "times" , "times new roman" , serif;"><br /></span></div>
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<br /></div>
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<span class="Apple-style-span" style="font-family: "times" , "times new roman" , serif;"><span class="Apple-style-span" style="font-size: large;">Chandrakant Doshi</span><o:p></o:p></span></div>
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<span class="Apple-style-span" style="font-family: "times" , "times new roman" , serif;"><span class="Apple-style-span" style="font-size: large;"><br /></span></span></div>
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<br /></div>
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email: <a href="mailto:jignashi@yahoo.co.uk">jignashi@yahoo.co.uk</a><o:p></o:p></div>
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<i><span class="Apple-style-span" style="font-size: medium;">Abstract:</span><span class="Apple-style-span" style="font-size: medium;"> </span><span class="Apple-style-span" style="font-size: medium;">This note examines certain artistic rendering of Brahmi </span><span class="Apple-style-span" style="font-size: medium;"> </span><span class="Apple-style-span" style="font-size: medium;"> </span><span class="Apple-style-span" style="font-size: medium;"> </span><span class="Apple-style-span" style="font-size: medium;">script at Girnar and postulates its influence on modern </span><span class="Apple-style-span" style="font-size: medium;"> </span><span class="Apple-style-span" style="font-size: medium;"> </span><span class="Apple-style-span" style="font-size: medium;"> </span><span class="Apple-style-span" style="font-size: medium;">Gujarati script. <o:p></o:p></span></i></div>
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KEY WORDS: Brahmi, ancient, script, inscriptions, Asoka, Girnar, Gujarati<o:p></o:p></div>
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<o:p></o:p></div>
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<span style="color: black;">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.</span><span style="color: #996633;"> <o:p></o:p></span></div>
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<span style="color: black;">These edicts could be read in modern times following their decipherment by James Prinsep. </span>Alexander Cunningham, who published a collection of these inscriptions<sup>1</sup> in 1877, noted variations in the rendering of specific letters in the Girnar script<sup>2</sup> 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.<o:p></o:p></div>
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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 bibliography<sup>3</sup>. <span style="color: black;">I</span><span style="color: blue;"> </span><span style="color: black;">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. <o:p></o:p></span></div>
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Eugene Hultzsch<sup>4</sup> published a fresh compilation of these edicts in 1926. He worked with fresh mechanical copies furnished by Archaeological Survey of India. <o:p></o:p></div>
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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-Topra<sup>5</sup> 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.<o:p></o:p></div>
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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.<o:p></o:p></div>
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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 inches<sup>6</sup> high. <o:p></o:p></div>
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The image of the first five edicts from Cunningham's corpus<sup>7</sup> can be seen in the Appendix, Image 15. The first two edicts from a plate in Hultzsch’s corpus<sup>8</sup> are also reproduced to illustrate the variations in script, Image 16.<o:p></o:p></div>
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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 transliteration<sup>9</sup> of this edict from Hultzsch is also included, Image 3.<o:p></o:p></div>
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The diacritic or medial mark for the short <span style="color: red;">i</span> 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 <span style="color: red;">i</span> 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.<o:p></o:p></div>
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Examples of immaculate rendering of these diacritic marks can be seen in the first line of the Delhi-Topra edict, Image 13. <o:p></o:p></div>
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An examination of the same first line in Hultzsch's estampage, Image 2, shows a completely different situation. The short and long <span style="color: red;">i</span> 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 <span style="color: red;">mhi</span>, the short <span style="color: red;">i</span> is rendered as an arc atop the glyph of the syllable <span style="color: red;">ma</span>. 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. <o:p></o:p></div>
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This individuality of expression is clearly seen in five examples of the syllable <span style="color: red;">pri</span>, pointed by the red arrows in both the above images. <o:p></o:p></div>
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Instances of the immaculately rendered short or long<span style="color: #339966;"> </span><span style="color: red;">i</span><span style="color: blue;"> </span>in the first edict are more an exception than a rule.<span style="color: blue;"> </span>Two such examples occur in the tenth line. One can be seen in the word <span style="color: red;">dhaMmalipI</span>, though both medial marks on<span style="color: blue;"> </span><span style="color: black;">the glyph of the syllable</span><span style="color: blue;"> </span><span style="color: red;">pa</span><span style="color: blue;"> </span>are not clearly visible on the estampage. Even Cunningham shows only one medial mark but Hultzsch’s Devanagari transliteration (Image 3) has the long <span style="color: red;">i</span>. Damage to the rock makes it difficult to make out both the diacritic marks. The other instance can be seen towards the end, where<span style="color: blue;"> </span><span style="color: red;">tI</span><span style="color: blue;"> </span>has two vertical marks for the long <span style="color: red;">i </span>clearly visible.<span style="color: blue;"><o:p></o:p></span></div>
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In the second edict, Image 4, both the perfectly vertical and cursive form of the diacritic mark for <span style="color: red;">i</span> are observed. However, slackness has entered on the very last word, mainly in regard to the glyph of the syllable <span style="color: red;">sa</span>. <o:p></o:p></div>
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The word is <span style="color: red;">pasumanusAnaM</span> and is enclosed in a red rectangle in the image of the estampage below. <o:p></o:p></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIlByQO_MHrc9YouwObTtRk6kSFc0wFqeTxBCBQTvF7qOboaTew_7CV7oUJEK9Coxy50GeVwTZlMOnenqrhTYBxK7S8syS7KOT-F0Qz5rg27mKjFUtRikXGwnmkOcJ0Ce_9uCc/s1600/Image+4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="282" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIlByQO_MHrc9YouwObTtRk6kSFc0wFqeTxBCBQTvF7qOboaTew_7CV7oUJEK9Coxy50GeVwTZlMOnenqrhTYBxK7S8syS7KOT-F0Qz5rg27mKjFUtRikXGwnmkOcJ0Ce_9uCc/s640/Image+4.jpg" width="640" /></a></div>
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On the left, the blue arrows point to <span style="color: red;">sa</span> (top) and <span style="color: red;">pa</span> (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 <span style="color: red;">sa</span> but not <span style="color: red;">pa</span>. This is clearly seen for the marked characters and can also be observed in the table of scripts, Image 12. <o:p></o:p></div>
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An examination of <span style="color: red;">pasumanusAnaM</span> shows that while <span style="color: red;">pa</span> has its limb upright, that for <span style="color: red;">sa</span> 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 <span style="color: red;">sa</span> in that last word is not repeated elsewhere.<o:p></o:p></div>
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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.<o:p></o:p></div>
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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. <o:p></o:p></div>
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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.<o:p></o:p></div>
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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. <o:p></o:p></div>
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<span style="color: black;">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 example<sup>10</sup> 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. <o:p></o:p></span></div>
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Photos of the Girnar rock.<o:p></o:p></div>
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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. <o:p></o:p></div>
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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. <o:p></o:p></div>
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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. <o:p></o:p></div>
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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.<o:p></o:p></div>
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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 translatation<sup>12</sup> of the fourteenth edict is also included (Image 11). <o:p></o:p></div>
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<span style="color: black;">The cursive diacritic for </span><span style="color: red;">i</span><span style="color: black;">, 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.<o:p></o:p></span></div>
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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. <o:p></o:p></div>
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This brief examination has revealed not only the artistry in the engraving of some characters but also occasional absence of supervision during engraving. <span style="color: black;">The artistry in the glyphs points to a local tradition of the third century BCE that has shown itself in the modern Gujarati script. <o:p></o:p></span></div>
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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. <o:p></o:p></div>
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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. <o:p></o:p></div>
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<span class="Apple-style-span">Bibliography</span><o:p></o:p></div>
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1/. <span class="Apple-tab-span" style="white-space: pre;"> </span>Corpus Inscriptionum Indicarum Vol. I <o:p></o:p></div>
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<span class="Apple-tab-span" style="white-space: pre;"> <span class="Apple-tab-span" style="white-space: pre;"> </span></span>Inscriptions of Asoka by Alexander Cunningham.</div>
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<span class="Apple-tab-span" style="white-space: pre;"> <span class="Apple-tab-span" style="white-space: pre;"> </span></span>Office of the Superintendent of Government Printing, Calcutta, 1877</div>
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2/. <span class="Apple-tab-span" style="white-space: pre;"> </span>Cunningham, op cit, page 14 - 15<o:p></o:p></div>
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3/. <span class="Apple-tab-span" style="white-space: pre;"> </span>The following web sites have tables of Brahmi characters. <o:p></o:p><br />
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<a href="http://www.ancientscripts.com/brahmi.html">http://www.ancientscripts.com/brahmi.html</a></div>
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<span class="Apple-tab-span" style="white-space: pre;"><a href="http://en.wikipedia.org/wiki/Br%C4%81hm%C4%AB_script"> </a></span><a href="http://en.wikipedia.org/wiki/Br%C4%81hm%C4%AB_script">http://en.wikipedia.org/wiki/Brāhmī_script</a></div>
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<a href="http://www.omniglot.com/writing/brahmi.htm">www.omniglot.com/writing/brahmi.htm</a><o:p></o:p></div>
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4/. <span class="Apple-tab-span" style="white-space: pre;"> </span>Eugene Hultzsch<o:p></o:p></div>
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<span class="Apple-tab-span" style="white-space: pre;"> <span class="Apple-tab-span" style="white-space: pre;"> </span></span>Corpus Inscriptionum Indicarum, Vol 1, <o:p></o:p></div>
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<span class="Apple-tab-span" style="white-space: pre;"><span class="Apple-tab-span" style="white-space: pre;"> </span></span>Inscriptions of Asoka, Government of India, 1925<o:p></o:p></div>
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5/. <span class="Apple-tab-span" style="white-space: pre;"> </span>Hultzsch, op cit, <o:p></o:p></div>
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<span class="Apple-tab-span" style="white-space: pre;"> </span>Image, p 122, inscription, p 119<o:p></o:p></div>
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6/. <span class="Apple-tab-span" style="white-space: pre;"> </span>Cunningham, op cit, p 14<o:p></o:p></div>
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7/. <span class="Apple-tab-span" style="white-space: pre;"> </span>Cunningham, op cit, Plate V<o:p></o:p></div>
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8/. <span class="Apple-tab-span" style="white-space: pre;"> </span>Hultzsch, op cit, p 4<o:p></o:p></div>
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9/. <span class="Apple-tab-span" style="white-space: pre;"> </span>Hultzsch, op cit, p 1<o:p></o:p></div>
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10/. The Cave-Temples of Western India by James Burgess and </div>
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<span class="Apple-tab-span" style="white-space: pre;"> <span class="Apple-tab-span" style="white-space: pre;"> </span></span>Bhagwanlal Indraji Pandit, 1881, p112<o:p></o:p></div>
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11/. Manish Khamesra is a contributor on the website <span style="color: windowtext;"><a href="http://www.ghumakkar.com/"><span class="Apple-tab-span" style="white-space: pre;"> </span>http://www.Ghumakkar.com</a></span> </div>
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<span class="Apple-tab-span" style="white-space: pre;"> </span>where the photos<span class="Apple-tab-span" style="white-space: pre;"><span class="Apple-tab-span" style="white-space: pre;"> </span></span>originally appeared.</div>
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They are part of a travel story "Junagadh through the ages", which can be accessed here: <a href="http://www.ghumakkar.com/2010/01/05/junagadh-%09%09%E2%80%93-a-journey-through-ages/">http://www.ghumakkar.com/2010/01/05/junagadh-</a><a href="http://www.ghumakkar.com/2010/01/05/junagadh-%09%09%E2%80%93-a-journey-through-ages/"> </a><a href="http://www.ghumakkar.com/2010/01/05/junagadh-%09%09%E2%80%93-a-journey-through-ages/"> </a><a href="http://www.ghumakkar.com/2010/01/05/junagadh-%09%09%E2%80%93-a-journey-through-ages/">%E2%80%93-a-journey-through-ages/</a><o:p></o:p></div>
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12/. Hultzsch, op cit, p26<o:p></o:p></div>
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<span lang="EN-US">© 2010 Chandrakant Doshi</span><o:p></o:p></div>
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Chandra Doshihttp://www.blogger.com/profile/12676853087431948250noreply@blogger.comtag:blogger.com,1999:blog-30960876.post-32676797665533027192010-04-23T12:01:00.085+01:002020-05-29T18:46:03.163+01:00An Externally Pivoted Balance<div dir="ltr" style="text-align: left;" trbidi="on">
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<span class="Apple-style-span" style="font-weight: bold;"><span class="Apple-style-span" style="font-size: x-large;">An Externally Pivoted Balance</span></span></div>
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by</div>
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<b>Chandrakant Doshi</b></div>
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email:<span class="Apple-tab-span" style="white-space: pre;"> </span><span class="Apple-style-span" style="color: #3333ff;"> jignashi@yahoo.co.uk</span><br />
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<i>Abstract: This paper describes a balance in which friction is no hindrance to the accuracy of mass comparison.</i><br />
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KEY WORDS: BALANCE, ANCIENT, ACCURACY, FRICTION<br />
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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.<br />
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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.<br />
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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.<br />
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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.<br />
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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.<br />
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<span style="font-weight: bold;">Construction</span><br />
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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.<br />
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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.<br />
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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.<br />
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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.<br />
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Equilibrium can be detected by suspending a second beam parallel to the main beam to serve as a reference.<br />
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<span style="font-weight: bold;">Analysis</span><br />
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A comprehensive analysis of the conventional beam balance is given in Practical Physics<span style="color: black;"><sup>1 </sup></span>. A modern treatment is available in Comprehensive Mass Metrology<span style="color: black;"><sup>2 </sup></span>. History of the ancient balances can be read in Skinner<span style="color: black;"><sup>3 </sup></span> and Kisch<span style="color: black;"><sup>4</sup></span>.<br />
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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.<br />
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Assume the beam to be uniform and the string to be of negligible mass.<br />
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Let the mass of the beam be M, acting at its centre of mass (CoM).<br />
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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.<br />
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Let AD = l<span style="color: black;"><sub>1 </sub></span> and BD = l<span style="color: black;"><sub>2 </sub></span> .<br />
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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. <br />
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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.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgll_6WmrnlTM1BMIio8n6Nk8y7ccyjWPQkbH7Rnp05TKkqfYguZvI4_RHyq_CPczPrRw33__U4l9q_wtMv0i1GTnShU9ZTBnDcocx2tRZsk2Tgz8aypaNrYuJWi6sMYyHBB66K/s1600/Fig+A.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5463316367022144194" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgll_6WmrnlTM1BMIio8n6Nk8y7ccyjWPQkbH7Rnp05TKkqfYguZvI4_RHyq_CPczPrRw33__U4l9q_wtMv0i1GTnShU9ZTBnDcocx2tRZsk2Tgz8aypaNrYuJWi6sMYyHBB66K/s400/Fig+A.jpg" style="cursor: hand; cursor: pointer; display: block; height: 350px; margin: 0px auto 10px; text-align: center; width: 400px;" /></a><br />
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Taking moments about CD, we obtain<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWW27dJdMn9bgsFHoPeKzJfWTSgx1CBsI-AJ9DjiSAq309_G8ifs1s7yyLshtp_xesXC7Km4tIAGlOwY2RU-KeafxOg6fRaZLPLrtoUJYTd80pEx1kRR3ddzUe7s50ATrqW34K/s1600/equation+0.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5463317699547933314" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWW27dJdMn9bgsFHoPeKzJfWTSgx1CBsI-AJ9DjiSAq309_G8ifs1s7yyLshtp_xesXC7Km4tIAGlOwY2RU-KeafxOg6fRaZLPLrtoUJYTd80pEx1kRR3ddzUe7s50ATrqW34K/s400/equation+0.jpg" style="cursor: hand; cursor: pointer; display: block; height: 37px; margin: 0px auto 10px; text-align: center; width: 400px;" /></a><br />
<br />
where A´K, B´K and KD´ are lengths on the beam, as marked on Fig. A, and g is acceleration due to gravity.<br />
<br />
Dividing both sides by g⋅cosθ we obtain<br />
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsrhwBweltWLpuC6riyZLGcDKRgCUn-eVw9pZtBtpBaUM4SAOBNLadRIoYgbVFBoYa8HwXx6t2_zSEq70UiI6dnDL-P_-UnQuOqlbGlasNw4kdcsQ344zs6oGIvqhTCUdRMdGN/s1600/Equation+1.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5463317706147660242" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsrhwBweltWLpuC6riyZLGcDKRgCUn-eVw9pZtBtpBaUM4SAOBNLadRIoYgbVFBoYa8HwXx6t2_zSEq70UiI6dnDL-P_-UnQuOqlbGlasNw4kdcsQ344zs6oGIvqhTCUdRMdGN/s400/Equation+1.jpg" style="cursor: hand; cursor: pointer; display: block; height: 40px; margin: 0px auto 10px; text-align: center; width: 400px;" /></a><br />
Now, A´K = A´D´ - KD´ = l<span style="color: black;"><sub>1 </sub></span> - CDtanϴ<br />
since KD´ = CD´tanϴ and CD´ = CD<br />
<br />
Also, B´K = l<span style="color: black;"><sub>2 </sub></span> + CDtanϴ<br />
<br />
Hence, (1) becomes<br />
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1g4QJRa039TjiT3SeOTqwE6HMFkKNlEqj__UpWUYYPSU-bjZ_fHTmeIzPP7qOUlIqxCxia6ie1ak_ytyroSbWuz6XECKOGKSaKIfMFRHROaTi6RUXZOWO-xZPKHwEd-3jYqjM/s1600/equation+1a.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5463317711219018818" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1g4QJRa039TjiT3SeOTqwE6HMFkKNlEqj__UpWUYYPSU-bjZ_fHTmeIzPP7qOUlIqxCxia6ie1ak_ytyroSbWuz6XECKOGKSaKIfMFRHROaTi6RUXZOWO-xZPKHwEd-3jYqjM/s400/equation+1a.jpg" style="cursor: hand; cursor: pointer; display: block; height: 37px; margin: 0px auto 10px; text-align: center; width: 400px;" /></a><br />
which simplifies to<br />
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6eThYWGnmIN2FnmuDxBzRlAqozxzkmmxB3N-BP4dWQVGmeWQbqodCoHjPf2aq5OOrNhYLmow7cGJWJ4YM5N-_Upisp83-wbI4lX6mgwAZuOCdTSK43ecRr8Yw_vdeVtWkQLc_/s1600/equation+1b.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5463317716529892642" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6eThYWGnmIN2FnmuDxBzRlAqozxzkmmxB3N-BP4dWQVGmeWQbqodCoHjPf2aq5OOrNhYLmow7cGJWJ4YM5N-_Upisp83-wbI4lX6mgwAZuOCdTSK43ecRr8Yw_vdeVtWkQLc_/s400/equation+1b.jpg" style="cursor: hand; cursor: pointer; display: block; height: 40px; margin: 0px auto 10px; text-align: center; width: 400px;" /></a><br />
giving<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirmor774_GqlqyPR88qKwHthfvlgmr0wVDOgHboW4QzAfi6vkS8z_IoR9w_aj80EJIjILL2TyJdQTNMds-tG_i8vTQZx2gsPSJHsWVjQNjRLQbQt0TzOppkOuuKCwwHIEW3C_E/s1600/Equation+2.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5463318899289769378" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirmor774_GqlqyPR88qKwHthfvlgmr0wVDOgHboW4QzAfi6vkS8z_IoR9w_aj80EJIjILL2TyJdQTNMds-tG_i8vTQZx2gsPSJHsWVjQNjRLQbQt0TzOppkOuuKCwwHIEW3C_E/s400/Equation+2.jpg" style="cursor: hand; cursor: pointer; display: block; height: 58px; margin: 0px auto 10px; text-align: center; width: 400px;" /></a><br />
as an expression for added mass in terms of ϴ and the balance constants.<br />
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.<br />
<br />
Putting l<span style="color: black;"><sub>1 </sub></span> = l<span style="color: black;"><sub>2 </sub></span> = l in equation (2) we obtain<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8P4CwCf9OAt1oyPChCs1vWLVbChTtPBzXwyMxr47FLSojjLzkArQAnJjZTLrDrJPzyhuMckzrclDZTn-mOvDLZIbumMPaTN9tkdmcDKLXHhrZZtyHpCb0k2fuq_kv5s8x1Jj5/s1600/Equation+3.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5463318903997578242" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8P4CwCf9OAt1oyPChCs1vWLVbChTtPBzXwyMxr47FLSojjLzkArQAnJjZTLrDrJPzyhuMckzrclDZTn-mOvDLZIbumMPaTN9tkdmcDKLXHhrZZtyHpCb0k2fuq_kv5s8x1Jj5/s400/Equation+3.jpg" style="cursor: hand; cursor: pointer; display: block; height: 57px; margin: 0px auto 10px; text-align: center; width: 400px;" /></a><br />
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<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRlMbTjRdnBIc9R5SdESJf_6I7kTOVpDfzz2CbnOulCR2sOEBpVxUOP6YKvVBNubrCnJhuqX1CRqtUMmX2AQ6qA5q-71z5R3Vz4pHrSajQhpqjMQKYgXssDKE2wMrSO7u13nVt/s1600/equation3a.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5463333351574462258" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRlMbTjRdnBIc9R5SdESJf_6I7kTOVpDfzz2CbnOulCR2sOEBpVxUOP6YKvVBNubrCnJhuqX1CRqtUMmX2AQ6qA5q-71z5R3Vz4pHrSajQhpqjMQKYgXssDKE2wMrSO7u13nVt/s400/equation3a.jpg" style="cursor: hand; cursor: pointer; display: block; height: 60px; margin: 0px auto 10px; text-align: center; width: 400px;" /></a><br />
Sensitivity of a balance is defined as radians per gram. The above equation can be written as<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhN2IaYm2lwYxVdC3bQF4A2HYgEK4F6KHXhaMmghctIgb6EiAqlNpKsQ_IsZoxMMeNXumwtY1SFm2On57Xw5JjPA_BMzP9fyygdkIy0tHKO4coIcN3hv3VtOXyE4C4LU__GzrBL/s1600/Equation+4.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5463294396045295314" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhN2IaYm2lwYxVdC3bQF4A2HYgEK4F6KHXhaMmghctIgb6EiAqlNpKsQ_IsZoxMMeNXumwtY1SFm2On57Xw5JjPA_BMzP9fyygdkIy0tHKO4coIcN3hv3VtOXyE4C4LU__GzrBL/s400/Equation+4.jpg" style="cursor: hand; cursor: pointer; display: block; height: 74px; margin: 0px auto 10px; text-align: center; width: 400px;" /></a><br />
<br />
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).<br />
<br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAhmWKo-2kJFXt3ch3BotTpnkf7Za94EejGZfGKdP7eBbMdrKYdj-zpS8KkdoRl-zh-xx2cTWzt1wRmYjTTl9BrLdbMiSbtofGe1eaxnQFD9-p1G2ARqVliWOCjlLnLrkHwrTh/s1600/Equation+5.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5463291912754451442" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAhmWKo-2kJFXt3ch3BotTpnkf7Za94EejGZfGKdP7eBbMdrKYdj-zpS8KkdoRl-zh-xx2cTWzt1wRmYjTTl9BrLdbMiSbtofGe1eaxnQFD9-p1G2ARqVliWOCjlLnLrkHwrTh/s400/Equation+5.jpg" style="cursor: hand; cursor: pointer; display: block; height: 74px; margin: 0px auto 10px; text-align: center; width: 400px;" /></a><br />
<br />
Equations (4) & (5) show that the sensitivity of the balance decreases with increasing load.<br />
<br />
<span style="font-weight: bold;">Comments</span><br />
<br />
The externally pivoted balance offers some advantages over the conventional, centrally pivoted balance.<br />
<br />
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.<br />
<br />
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.<br />
<br />
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.<br />
<br />
<span style="font-weight: bold;">Acknowledgements</span><br />
<br />
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.<br />
Many thanks to the librarians at the Keith Axon Centre of the Redbridge Libraries for obtaining the essential books.<br />
<br />
<br />
<b>Bibliography<o:p></o:p></b><br />
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<o:p></o:p></div>
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1/. Practical Physics by R. T. Glazebrook and W. N. Shaw. Longmans, 1893 – pp. 99. <o:p></o:p></div>
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Copies available on Internet Archive. Link to one such copy below: <a href="http://www.blogger.com/post-edit.g?blogID=30960876&postID=3267679766553302719"><br />http://www.archive.org/details/practicalphysic02shawgoog</a></div>
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2/.Comprehensive Mass Metrology<o:p></o:p><br />
<div class="MsoNormal" style="margin-left: 36pt;">
<span lang="EN-US">Edited by Manfred Kochsiek, Michael Glaser, Weinheim ; Chichester, 2000<o:p></o:p></span></div>
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3/. Weights and Measures:<o:p></o:p><br />
<div class="MsoNormal" style="margin-left: 36pt;">
<span lang="EN-US">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<o:p></o:p></span></div>
<br />
4/. Scales and weights:<o:p></o:p><br />
<div class="MsoNormal" style="margin-left: 36pt;">
<span lang="EN-US">a historical outline by B. Z. Kisch, Yale University Press, 1965.<o:p></o:p></span></div>
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<br />
© 2010 Chandrakant Doshi<br />
<br />
<span class="Apple-style-span" style="color: white;"><br />*<br />*<br />*<br />*</span></div>
Chandra Doshihttp://www.blogger.com/profile/12676853087431948250noreply@blogger.comtag:blogger.com,1999:blog-30960876.post-15504098088850894482010-01-10T13:02:00.062+00:002020-05-29T18:44:28.029+01:00Ancient Weights from South Asian civilization<div dir="ltr" style="text-align: left;" trbidi="on">
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<span class="Apple-style-span" style="font-family: "times new roman" , serif; font-weight: bold;"><span class="Apple-style-span" style="font-size: x-large;">Ancient Weights from South Asian civilization</span></span></div>
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by<o:p></o:p></div>
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<b><span class="Apple-style-span" style="font-size: large;">Chandrakant Doshi</span><span class="Apple-style-span" style="font-size: large;"><o:p></o:p></span></b></div>
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<b><span class="Apple-style-span" style="font-size: large;"> </span><span class="Apple-style-span" style="font-size: large;"><o:p></o:p></span></b></div>
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email: <a href="mailto:jignashi@yahoo.co.uk">jignashi@yahoo.co.uk</a><o:p></o:p></div>
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Key words: Harappa, Indus, weights<o:p></o:p></div>
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<b>Introduction<o:p></o:p></b></div>
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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. <o:p></o:p></div>
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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.</div>
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<span class="Apple-style-span" style="font-weight: bold;">Sources</span></div>
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The excavation report of John Marshall<sup>1</sup>, “Mohenjo Daro and the Indus Civilization,” lists weights from both Mohenjo Daro and Harappa. Later excavations by M. S. Vats<sup><span class="Apple-style-span" style="font-size: small;">2</span></sup> at Harappa and John Mackay<sup>3, 4</sup> 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.</div>
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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. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifTgtdXlPXOsh_DMYbgJ69WFKWwzElkE8pwDfnQzlrUlvNzodpO_okQ8DSuPR3dpyxlsaQfPyz75EeiqHWOVv9ODnipR_iB5LgZesIp-xOQDULr8c6RNe3DaoIlWl0x9h6U4PB/s1600-h/Table+1.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="text-decoration: none;"><br /></a></div>
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<span class="Apple-style-span" style="color: rgb(85 , 26 , 139);"><span class="Apple-style-span" style="color: rgb(0 , 0 , 0);"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFfW77fVaNmrVLL9dzK3JgYDFLDHcNgBlYpA5eomXMd3jMrKMI1bS0zBpJLzpS3IVmc5QHbisMeZbRQk4dNDDctwsrV3ybPgnN4xvvIGi5MBHSyWKqzIXcIC8A86Ls31NIP-Fi/s1600-h/Table+1.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><br /></a></span></span></div>
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<span class="Apple-style-span" style="color: rgb(85 , 26 , 139);"><span class="Apple-style-span" style="text-decoration: underline;"><span class="Apple-style-span" style="color: rgb(0 , 0 , 0);"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCF1XkDD4ZV2WL5BpGWv4T0mMaTX8hc61dVReUiyn5hdT7g-sdY-U5cZmZyhK78ZN8KG3JR7yg2_UfJW715u1GTyg2KYZvTVlV799l-ifXM9GanDmng5HESMkOXRi59pD6oxrZ/s1600-h/Table+1.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5426354901373463794" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCF1XkDD4ZV2WL5BpGWv4T0mMaTX8hc61dVReUiyn5hdT7g-sdY-U5cZmZyhK78ZN8KG3JR7yg2_UfJW715u1GTyg2KYZvTVlV799l-ifXM9GanDmng5HESMkOXRi59pD6oxrZ/s400/Table+1.jpg" style="cursor: pointer; display: block; height: 231px; margin: 0px auto 10px; text-align: center; width: 400px;" /></a></span></span></span></div>
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<span style="color: black;">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. <o:p></o:p></span></div>
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<span style="color: black;">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.<o:p></o:p></span></div>
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<b>Analysis<o:p></o:p></b></div>
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<span style="color: black;">In a chapter included in ' Excavations at Chanhu Daro '<sup><span class="Apple-style-span" style="font-size: x-small;">5</span></sup>, published in 1943, Hemmy stated that... </span><span style="color: blue;">"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." </span><span style="color: black;"><o:p></o:p></span></div>
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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 ratios<sup>6</sup> 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.<o:p></o:p></div>
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<span style="color: black;"> <o:p></o:p></span></div>
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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. <o:p></o:p></div>
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<o:p></o:p></div>
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Each band could be a small percentage of the nominal value. Simplest way to achieve this is to first normalise all the weight values.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKURkxdhKZSV9dVYeXEiIuHraTAL6ZSJ5ZNi16hUJfEiHVnpWr3eHE_bQMKJ9wb4_RRcqYS2B4fxFmQ9hyQMAcCBtf8cSylqipG_wBHZQ-gY4woXnJj9acYvFW0wbej32Yx5VM/s1600-h/Table+2.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="text-decoration: none;"><br /></a></div>
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<span class="Apple-style-span" style="color: rgb(0 , 0 , 238);"><span class="Apple-style-span" style="color: rgb(0 , 0 , 0);"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbyIqu0XZvxATltIq03KwlKs7oIJBL83MDempENzME84TrhUNaCbgjzfg0rRkDCQnaP6RheJcC13vOswz_BKuPGOmAW6Jbas70jW6yuUhOknKRvTVZi8ROhEVvWGr1SdrmOSbz/s1600-h/Table2.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><br /></a></span></span></div>
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<span class="Apple-style-span" style="color: rgb(85 , 26 , 139);"><span class="Apple-style-span" style="text-decoration: underline;"><span class="Apple-style-span" style="color: rgb(0 , 0 , 0);"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFq7A0D6tnqgpy4ZiysB7oOkBwQAkAcGCXMyJH0dPNFvDKaVvO-Za41L3hFLZgvBqo3pRgI1d_H7MuEPWDPBITUU8ukiNbumigJGvhII8ETAORO8djy7Kl4kjidEGAbwlyp9sV/s1600-h/Table+2.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5426356612261811842" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFq7A0D6tnqgpy4ZiysB7oOkBwQAkAcGCXMyJH0dPNFvDKaVvO-Za41L3hFLZgvBqo3pRgI1d_H7MuEPWDPBITUU8ukiNbumigJGvhII8ETAORO8djy7Kl4kjidEGAbwlyp9sV/s400/Table+2.jpg" style="cursor: pointer; display: block; height: 381px; margin: 0px auto 10px; text-align: center; width: 400px;" /></a></span></span></span></div>
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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. <o:p></o:p></div>
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<o:p></o:p></div>
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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. <o:p></o:p></div>
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<o:p></o:p></div>
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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. <o:p></o:p></div>
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<o:p></o:p></div>
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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.<o:p></o:p></div>
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<o:p></o:p></div>
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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. <o:p></o:p></div>
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<o:p></o:p></div>
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The peak frequency is 175 and is centred around the nominal values. <o:p></o:p></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZMTp9o2drRTuqkql6-OIuKcOYMN0w5z5-6kOdfrB6AAXE5T7b8usm8dmFpJrpITJl7SeJYa4_TjZSBAP8Pq2cAXWgbCqnIBs0yqO2Fxr-1FWqhVheCgBxAvbEFsG4J7Fgggc2/s1600-h/Chart+1.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="text-decoration: none;"><br /></a></div>
<div style="text-align: center;">
<span class="Apple-style-span" style="color: rgb(85 , 26 , 139);"><span class="Apple-style-span" style="color: rgb(0 , 0 , 0);"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMgvezkYSmZSR4SYkA_9PWvFcHoPAjss21tzDdjyCMGGPCY1bsVZC7mNOR7QoHB5qotjo165jhhbL2YvNz2wSlA8h-QpfFLP9iDwxiq5SX5p3unGTE3DBgg23aHUhXLcouaF4k/s1600-h/Chart+1.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><br /></a></span></span></div>
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<span class="Apple-style-span" style="color: rgb(85 , 26 , 139);"><span class="Apple-style-span" style="text-decoration: underline;"><span class="Apple-style-span" style="color: rgb(0 , 0 , 0);"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNiSNj_pvnthqRNrf6dgixzY2CyvDrdEDD_XBAgcvPbMkBnWVX6hJcCHS-VtX0RCUXub2TFuPLTPqz8TDSNBk8IXwy0u-Z2pfmCFha_jV1GXaNVRMCAwauflDhqRKzFOg9nsFQ/s1600-h/Chart+1.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5426302386857664482" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNiSNj_pvnthqRNrf6dgixzY2CyvDrdEDD_XBAgcvPbMkBnWVX6hJcCHS-VtX0RCUXub2TFuPLTPqz8TDSNBk8IXwy0u-Z2pfmCFha_jV1GXaNVRMCAwauflDhqRKzFOg9nsFQ/s400/Chart+1.jpg" style="cursor: pointer; display: block; height: 191px; margin: 0px auto 10px; text-align: center; width: 400px;" /></a></span></span></span></div>
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<div class="MsoNormal">
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. <o:p></o:p></div>
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<o:p></o:p></div>
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A selection was applied when arriving at the plot. As he writes<span style="color: black;"><sup>5 </sup></span> <span style="color: blue;">… “All doubtful specimens were rejected, including those with unlikely ratios, as well as all weights below 6 gms.” <o:p></o:p></span></div>
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<o:p></o:p></div>
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In preparing <span style="color: black;">Chart 1</span> no weight specimens are left out. This removes the subjective element present when excluding some weight specimens. <o:p></o:p></div>
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<o:p></o:p></div>
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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. <o:p></o:p></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQXw4qfyTXk9cBOYmwUxQOuPaX9xWDnVbSkVPheu8dCvsG_02sMf0tEMiyzmlVbet2j7kb0JDHgcdXMAKty7-Cirp05lcfSYENZ2AMgzCjufOKMXC44enXqKU6fej_EK2nYpQ2/s1600-h/Chart+1a.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="text-decoration: none;"><br /></a></div>
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<span class="Apple-style-span" style="color: rgb(0 , 0 , 238);"><span class="Apple-style-span" style="color: rgb(0 , 0 , 0);"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgGEPj4A7q4YlEAZKkboHxspjAt3QurDHPBA86ZESTwyG-jfNnbCBbWfm08InwxswaESM-IkzNly7eP6OAt5u51X-Vxn4RoDe3XYgN63K5Ad3m7IYTTBq_7_XD-xTQCsIBJofBY/s1600-h/Chart+1a.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5426302423105519490" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgGEPj4A7q4YlEAZKkboHxspjAt3QurDHPBA86ZESTwyG-jfNnbCBbWfm08InwxswaESM-IkzNly7eP6OAt5u51X-Vxn4RoDe3XYgN63K5Ad3m7IYTTBq_7_XD-xTQCsIBJofBY/s400/Chart+1a.jpg" style="cursor: pointer; display: block; height: 191px; margin: 0px auto 10px; text-align: center; width: 400px;" /></a></span></span></div>
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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. <o:p></o:p></div>
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<o:p></o:p></div>
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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. <o:p></o:p></div>
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<o:p></o:p></div>
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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.</div>
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<o:p></o:p></div>
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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.</div>
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<o:p></o:p></div>
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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. <o:p></o:p></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEih3ZtepE9X8k9W9TeRyrKZ4091B7c5x6-pzckuCcyK5mrCD8B8TS5DjlDG8-t7sRfnMh4QaiOdXY7Neod35dtgcMKEWlkgsMwiOLkK9-SGy4nz6dEnTPFFeDNY78d18smW8sGe/s1600-h/Chart+2.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5425097389791907986" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEih3ZtepE9X8k9W9TeRyrKZ4091B7c5x6-pzckuCcyK5mrCD8B8TS5DjlDG8-t7sRfnMh4QaiOdXY7Neod35dtgcMKEWlkgsMwiOLkK9-SGy4nz6dEnTPFFeDNY78d18smW8sGe/s320/Chart+2.jpg" style="cursor: pointer; display: block; height: 130px; margin: 0px auto 10px; text-align: center; width: 320px;" /></a><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgraJC5SPqxso3cePVLwo7wNJfJei-9iu-71PosZX9-Ve7ktkf0iGQvGSVA6unBb0c_iav9-2LVHSfyO67okZ0sPgeKQWxM5dNi0B2B2ReSOQ6YdHUQPBTSguBviEuA4uJkyczP/s1600-h/Chart+2a.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="text-decoration: none;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5425097388865962354" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgraJC5SPqxso3cePVLwo7wNJfJei-9iu-71PosZX9-Ve7ktkf0iGQvGSVA6unBb0c_iav9-2LVHSfyO67okZ0sPgeKQWxM5dNi0B2B2ReSOQ6YdHUQPBTSguBviEuA4uJkyczP/s320/Chart+2a.jpg" style="cursor: pointer; display: block; height: 130px; margin: 0px auto 10px; text-align: center; width: 320px;" /></a><br />
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<div class="MsoNormal">
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. <o:p></o:p></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJ9K91nYOf3ErMra98ZrwJL7nL4VhHpL42Bv-mARPtAVZXZIuH5Fg7ZvS3sJbXOn356wnylagXS9aHi_M0VZ5Tuk9hFPjvO5bcXHFSYzd7BYuiWRmcJOsi3gxRvzj1wqSB4EsX/s1600-h/Chart+3.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5425097126707879330" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJ9K91nYOf3ErMra98ZrwJL7nL4VhHpL42Bv-mARPtAVZXZIuH5Fg7ZvS3sJbXOn356wnylagXS9aHi_M0VZ5Tuk9hFPjvO5bcXHFSYzd7BYuiWRmcJOsi3gxRvzj1wqSB4EsX/s320/Chart+3.jpg" style="cursor: pointer; display: block; height: 115px; margin: 0px auto 10px; text-align: center; width: 320px;" /></a><br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWeYCUYeIbgprojlOKY9LNAD5cbnuvt6eXp8KNcunWPA4-6GlFcHjthZFCT3R75oP910VeGG48LrD1a1x3q6sWmKjFGLGN6X-C77qG5O01nHBeTx6UzPYWw5ezNRA0J9bhfTKC/s1600-h/Chart+3a.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="text-decoration: none;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5425097119469771346" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWeYCUYeIbgprojlOKY9LNAD5cbnuvt6eXp8KNcunWPA4-6GlFcHjthZFCT3R75oP910VeGG48LrD1a1x3q6sWmKjFGLGN6X-C77qG5O01nHBeTx6UzPYWw5ezNRA0J9bhfTKC/s320/Chart+3a.jpg" style="cursor: pointer; display: block; height: 116px; margin: 0px auto 10px; text-align: center; width: 320px;" /></a><br />
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<span style="color: black;">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. <o:p></o:p></span></div>
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<span style="color: black;"> <o:p></o:p></span></div>
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<span style="color: black;">The image below appears in Marshall’s excavation report<sup>7</sup>, where Ernest Mackay writes: </span><span style="color: blue;">“They were first flaked into shape and then ground, and finally polished.”</span><span style="color: black;"> <o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWAIHTsd2ifxWFGqRAicBcJKWERt3OmBrz7GauhYQiPjpSMxM457fm6jBJIh5wcEE1ptZVm-TCM5pbJ1DJCZ4mvX4yG7Q1dC0y5IOS7hBy2dluZ1_Tj5CTSNX0rkJGUxjR1ioz/s1600-h/MjD-Plate+CXXXIII.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5425096658725477602" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWAIHTsd2ifxWFGqRAicBcJKWERt3OmBrz7GauhYQiPjpSMxM457fm6jBJIh5wcEE1ptZVm-TCM5pbJ1DJCZ4mvX4yG7Q1dC0y5IOS7hBy2dluZ1_Tj5CTSNX0rkJGUxjR1ioz/s320/MjD-Plate+CXXXIII.jpg" style="cursor: pointer; display: block; height: 270px; margin: 0px auto 10px; text-align: center; width: 246px;" /></a><br />
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<br /></div>
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<span style="color: black;">Another photo appears in the report by Vats<sup>8</sup>, Plate CXVIII, shown below. In the report, Vats writes: </span><span style="color: blue;">“… 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 ….”<o:p></o:p></span></div>
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<span style="color: blue;"> <o:p></o:p></span></div>
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<span style="color: black;">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. <o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-SuSpEEC3xK8w3q2hGAYoF4JNqlda8rQw7x7ekt7UK7wOcIRnWGV8y2dEPSPwvBx4CD3sKMmMM8UwdRSwFXitvrZ0BsSCQ316pzGruPodHo4zLgDppNU1NX0BvTQ23J4m9cBo/s1600-h/Harappa+plate+118.jpg" onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" style="text-decoration: none;"><img alt="" border="0" id="BLOGGER_PHOTO_ID_5425096358118197154" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-SuSpEEC3xK8w3q2hGAYoF4JNqlda8rQw7x7ekt7UK7wOcIRnWGV8y2dEPSPwvBx4CD3sKMmMM8UwdRSwFXitvrZ0BsSCQ316pzGruPodHo4zLgDppNU1NX0BvTQ23J4m9cBo/s320/Harappa+plate+118.jpg" style="cursor: pointer; display: block; height: 243px; margin: 0px auto 10px; text-align: center; width: 320px;" /></a><br />
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<span style="color: black;">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. <o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: black;"> <o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: black;">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. <o:p></o:p></span></div>
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<span style="color: black;"> <o:p></o:p></span></div>
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<b> <o:p></o:p></b></div>
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<b>Comments<o:p></o:p></b></div>
<div class="MsoNormal">
<span style="color: rgb(255 , 102 , 0);"><b> <o:p></o:p></b></span></div>
<div class="MsoNormal">
<span style="color: black;">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. <o:p></o:p></span></div>
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<span style="color: black;"> <o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: black;">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. <o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: black;"> <o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: black;">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?<o:p></o:p></span></div>
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<span style="color: black;"> <o:p></o:p></span></div>
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<span style="color: black;">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.<o:p></o:p></span></div>
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<span style="color: black;"> <o:p></o:p></span></div>
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<span style="color: black;">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. <o:p></o:p></span></div>
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<span style="color: black;"> <o:p></o:p></span></div>
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<span style="color: black;">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. <o:p></o:p></span></div>
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<span style="color: black;"> <o:p></o:p></span></div>
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<span style="color: black;"> <o:p></o:p></span></div>
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<span style="color: black;">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. <o:p></o:p></span></div>
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<span style="color: black;">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?<o:p></o:p></span></div>
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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. <o:p></o:p></div>
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<span style="color: black;">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, Vats<sup>9</sup> wrote, </span><span style="color: blue;">“ … I have followed Mr. Hemmy’s designations and limits within which those designations apply.”</span><span style="color: black;"> Subsequent finds have all followed the designations established by Hemmy. <o:p></o:p></span></div>
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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. <o:p></o:p></div>
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<b>Acknowledgements<o:p></o:p></b></div>
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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. <o:p></o:p></div>
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<b>Bibliography<o:p></o:p></b></div>
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<o:p></o:p></div>
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1/. Mohenjo-Daro and the Indus Civilization. <o:p></o:p></div>
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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</div>
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2/. Excavations at Harappa. <span lang="EN-US">Vats, Madho Sarup</span><o:p></o:p></div>
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Being an account of archælogical excavations at Harappa, carried out between the Years 1920-21 and 1933-34, etc. <o:p></o:p></div>
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2 vol. Delhi, 1940. pp 363-365</div>
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3/. <span lang="EN-US">Further Excavations at Mohenjo-daro. Mackay, Ernest John Henry<o:p></o:p></span></div>
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<span lang="EN-US">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<o:p></o:p></span></div>
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<span lang="EN-US"><span class="Apple-tab-span" style="white-space: pre;"> </span> 2 vol. Delhi, 1938, 37, Vol. 1, pp 607-612</span></div>
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<span lang="EN-US"> <o:p></o:p></span></div>
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<o:p></o:p></div>
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4/. Excavations at Chanhu Daro, <span lang="EN-US">Mackay, Ernest John Henry</span>, 1935-36, New Haven, 1943, pp 238-239</div>
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5/. <span lang="EN-US">Excavations at Chanhu Daro, op cit., p 236-7<o:p></o:p></span></div>
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<span lang="EN-US"> <o:p></o:p></span></div>
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<span lang="EN-US">6/. Excavations at Chanhu Daro, op cit., p 241<o:p></o:p></span></div>
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<span lang="EN-US"> <o:p></o:p></span></div>
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<span lang="EN-US">7/. Mohenjo Daro and the Indus Civilization, op. cit., p 462<o:p></o:p></span></div>
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<span lang="EN-US"> <o:p></o:p></span></div>
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<span lang="EN-US">8/. Excavations at Harappa, op. cit., p 361<o:p></o:p></span></div>
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<span lang="EN-US"> <o:p></o:p></span></div>
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<span lang="EN-US">9/. Excavations at Harappa, op. cit., p 360<o:p></o:p></span></div>
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<span class="Apple-style-span" style="color: rgb(255 , 255 , 255);"><span class="Apple-tab-span" style="white-space: pre;"> </span>*<span class="Apple-style-span" style="color: rgb(0 , 0 , 0);">© 2010 Chandrakant Doshi</span></span></div>
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