DAWN OF GEOMETRIC IDEA IN THE INDUS VALLEY CIVILIZATION Art_Dawn_of_Geo… · The civilization...
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328 SCIENCE AND CULTURE, SEPTEMBER-OCTOBER, 2018 T he present status of human civilization stands on the firm base of its past attainments and the geometrical knowledge in the 20 th century is the outcome of what the human race achieved in the past irrespective of geographical barrier among different countries in the world. As a monumental task it is, to collect all the research works from world-wide journals in the last century, we shall focus only on those papers published in India during this period. This is because, Indian civilization has some speciality over the modern Western one. The modern Indian civilization is the successor of the ancient Indian Vedic civilization having the origin in the Mohenjodaro-times. It has reached the present stage through a series of evolutionary stages we had to cross in course of time. In a word, the waves of influence of different branches of learning including educational system of ancient Egypt, Babylon, China and Greece have a great impact on it. Actually all the above- mentioned civilizations were well-connected primarily with trade relation which is usually followed by some sort of cultural exchange. 1,2 Let us glance at the traces of geometry in practical life of inhabitants of the ancient Indian civilization. This will help us to evaluate the status we have been able to reach over that we had in the past. The civilization mentioned above is called the Indus Valley civilization originated in the Indus Valley during 3250-2750 B.C. Archaeologists guessed that the civilization attained its highest phase during 2350-1770 B.C. 3 Excavations during the first half of the 20 th century at different places of this civilization, mainly Harappa and Mohenjodaro (both of them are now in Pakistan), reveal marvellous specimens of figures and solid bodies hinting at what we call basic geometric concepts. A bit elaboration will make the matter explicit. The city plannings of Harappa and Mohenjodaro, the two main Figure 1 : The Great Bath ARTICLE DAWN OF GEOMETRIC IDEA IN THE INDUS VALLEY CIVILIZATION SANATAN KOLEY* It is indeed difficult to ascertain when and where the beginning of geometric way of thinking occurred in ancient India. For an extensive search we have to go back to a civilization which according to scholiasts originated in the Indus valley during 3250-2750 B.C. and attained its peak during 2350-1770 B.C. It was discovered in the northwestern region of the undivided India during an archaeological excavation in the first half of the 20 th century. The city plannings of Harappa and Mohenjodaro, the two main centres of this civilization, having the evidences of definite applications of some practical knowledge of basic geometric figures appear to be the beginning of geometric sense in that age. *. Retd. Headmaster, Jagachha High School (H.S.), Howrah; Postal Address: KADAMBARI, BL-1, Flat -1A, 144 Mohiary Road, Jagachha, Howrah-711112. W.B. E-mail: [email protected]
Transcript of DAWN OF GEOMETRIC IDEA IN THE INDUS VALLEY CIVILIZATION Art_Dawn_of_Geo… · The civilization...
328 SCIENCE AND CULTURE, SEPTEMBER-OCTOBER, 2018
The present status of human civilization stands onthe firm base of its past attainments and thegeometrical knowledge in the 20th century is the
outcome of what the human race achieved in the pastirrespective of geographical barrier among differentcountries in the world. As a monumental task it is, tocollect all the research works from world-wide journals inthe last century, we shall focus only on those paperspublished in India during this period. This is because,Indian civilization has some speciality over the modernWestern one. The modern Indian civilization is thesuccessor of the ancient Indian Vedic civilization havingthe origin in the Mohenjodaro-times. It has reached thepresent stage through a series of evolutionary stages wehad to cross in course of time. In a word, the waves ofinfluence of different branches of learning includingeducational system of ancient Egypt, Babylon, China andGreece have a great impact on it. Actually all the above-mentioned civilizations were well-connected primarily withtrade relation which is usually followed by some sort ofcultural exchange.1,2 Let us glance at the traces ofgeometry in practical life of inhabitants of the ancientIndian civilization. This will help us to evaluate the status
we have been able to reach over that we had in thepast.
The civilization mentioned above is called the IndusValley civilization originated in the Indus Valley during3250-2750 B.C. Archaeologists guessed that the civilizationattained its highest phase during 2350-1770 B.C.3
Excavations during the first half of the 20th century atdifferent places of this civilization, mainly Harappa andMohenjodaro (both of them are now in Pakistan), revealmarvellous specimens of figures and solid bodies hintingat what we call basic geometric concepts.
A bit elaboration will make the matter explicit. Thecity plannings of Harappa and Mohenjodaro, the two main
Figure 1 : The Great Bath
ARTICLE
DAWN OF GEOMETRIC IDEA IN THE INDUSVALLEY CIVILIZATION
SANATAN KOLEY*
It is indeed difficult to ascertain when and where the beginning of geometric way of thinkingoccurred in ancient India. For an extensive search we have to go back to a civilization whichaccording to scholiasts originated in the Indus valley during 3250-2750 B.C. and attained its peakduring 2350-1770 B.C. It was discovered in the northwestern region of the undivided India duringan archaeological excavation in the first half of the 20th century. The city plannings of Harappaand Mohenjodaro, the two main centres of this civilization, having the evidences of definiteapplications of some practical knowledge of basic geometric figures appear to be the beginningof geometric sense in that age.
*. Retd. Headmaster, Jagachha High School (H.S.), Howrah; PostalAddress: KADAMBARI, BL-1, Flat -1A, 144 Mohiary Road,Jagachha, Howrah-711112. W.B.E-mail: [email protected]
VOL. 84, NOS. 9–10 329
centres of this civilization, were extraordinary. All the mainstreets were straight and wide. A strikingly large structureknown as the Great Bath with a large swimming pool inthe centre (vide Fig. 1) has been found at Mohenjodaro.The drainage systems were exceptionally efficient. Houseswere made from burnt bricks. Each of the bricks resembled3-dimensional geometric configurations. Few were wedge-shaped, some were tetrahedral type, some were cuboidal,or right angled triangular shaped, also some were found tobe of pyramidal shaped. From these evidences, scholarsguessed that the dwellers of these two cities had a soundknowledge in geometry and mensuration. So, it would notbe irrelevant to guess that at least a first-hand geometricknowledge was common in the practical lives of theinhabitants of the said civilized cities in the Indus Valleycivilization.
From the relics excavated out of the ruins of about5000 years old civilization, we get more conclusiveevidences of pristine geometric knowledge of the citydwellers. These relics included the excavated pottery whichhad intricate geometrical patterns engraved on its surfaces.Some of the pottery possessed patterns of intersectingcircles (vide Fig. 2) inscribed on it.4 Some designs offigures like triangles, squares, rectangles etc. Besides, wenotice several rectilinear figures on the seals unearthed fromthe ruins.5 Clearly, it can be inferred that the dwellers haddistinct ideas of geometric figures like circle, triangle,square, rectangle etc. and also their related properties.
Figure 2 : Pattern of intersecting circles.
Knowledge about the construction of rightangled triangle and its properties appears to havebeen quite common to the dwellers of the cities.Use of a type of measuring scale made by theexoskeleton of some vertebrates was in theirpractice. They made use of a special instrumentto draw circles.6 Also, they were well acquaintedwith the use of stylus-like objects for engravingon the surface of pottery.7
It must be mentioned that at the early stage, thecivilization flourished mainly centring at two main cities,namely, Harappa and Mohenjodaro. But later it spreadover a vast territory of northwestern part of the then India.Of several towns developed, Chanhudaro, Lothal,Kalibangan, Dholavira are worth mentioning. From thereport of M.S.Vats (1953), we come to know that a bronze-made measuring scale was found in the ruins of Harappa.8
Also S.R. Rao’s report (1973) informs us that a smallmeasuring scale made of ivory and a beautiful instrumentbelieved to be used in measuring angles on plane surfaces,were discovered in the ruins of Lothal (in Gujarat ).9 Thediscovery of measuring scales in different Indus sitesindicate that the culture knew how to make accurate spatialmeasurements.10
One of the main features of the civilization was itsknowledge of brick-building technology. Relevantly, someof the ‘mantras’ expounded in the Yajurveda Samhita, theMaitrayani have been considered to be related withconsecration of bricks around 1000 B.C.11 However, thedwellers of Indus Valley knew the most ideal ratio oflength, breadth and thickness of cuboidal bricks which hadeffective binding strength. To make such bricks, generally,they considered the length as twice the breadth and thebreadth as twice the thickness. The contemporary size ofbricks varied in a range from 10.25 x 5 x 2.25 inches to20.5 x 8.5 x 2.25 inches. It is to be mentioned that thelarge sized bricks were utilised to cover city drains.12
Among several other things, style of construction of stairsin the towns is simply amazing even today.
Before going to detailed discussions on the seals andthe designed pottery discovered in the Indus civilization,it is worthwhile to mention that the ruins of Dholavira (in
Figure 3 : Plan of Dholavira (adapted from R. S. Bisht, 1999)
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the Rann of Kachchh in Gujarat ) excavated under thesupervision of R.S.Bisht in the last decade of the 20th
century, revealed a large undisturbed town-plan and precisedelineation of multiple enclosures in a site measuring about48 hectares. A close observation of the town-plan ( videFig. 3) reveals a distinct geometric knowledge of the town-planners. Dholavira’s plan shows the distribution of thetownship in three strata namely, an uppermost stratumconsisted of a massive “castle” and an adjacent esplanade-like open level ground enclosed by boundary walls as isevident from the adjoined figure (this has been wronglytermed “bailey” in the figure), a middle town (whichincludes a huge ceremonial ground ), and a lower town, alarge part of which was occupied by several waterreservoirs.13,14,15
Now, detailed discussions may be made on the sealsand the designed pottery discovered. In this context, factsderived mainly from two books – one authored by JohnMarshall and another by Ernest Mackay must bementioned.16,17 (Both of them were a part of the team ofarchaeologists associated with the works of excavation in1922). Also, mention may be made of the conceptualknowledge of geometry of the dwellers and the relatedinferences the scholars had reached.18
It can be concluded from the discovered seals thatthe dwellers of Indus Valley civilization might have theknowledge regarding several rectilinear figures. In the book“Mohenjodaro and the Indus Civilization” by JohnMarshall19 some plates displaying some geometric figuresare seen. The plate no. 113 shows a rectangle divided intofour equal rectangles. The same plate shows a rectangledivided into twelve equal rectangles. Besides, it shows arhombus divided into four equal rhombuses. In plate no.114 it is shown that few concentric squares have beendisplayed (vide Fig. 4).
Figure 4 : Few Concentric Squares
Examining the designs on these seals and plates,scholars come to assumptions that the people of the
civilization might know the following postulates andgeometric constructions.20
Postulates
1. Each diagonal of a square or of a rectangle dividesit into two equal parts.
2. The diagonals of a square or rectangle are equalto each other.
3. The bisectors of the sides of a square areperpendicular to each other.
Constructions
a) To divide a given finite straight line into two equalparts.
b) To divide a given finite straight line into anynumber of equal parts.
c) To divide a given finite straight line into differentparts of known lengths.
d) To construct a square or a rectangle of givenside(s) and a (perfect) right angle.
e) To divide a square or a rectangle into a numberof small and equal squares or rectanglesrespectively.
f) To divide a rhombus into a number of small andequal rhombuses.
g) To draw two parallel straight lines.
As already mentioned, some interesting circles areseen engraved on some earthen containers. Again plateno. 17 in the book “Early Indus Civilizations” by ErnestMackay21 shows a circle divided into four equal parts byits two diameters. Also, in plate no. 18, four petals aredrawn in a circle (vide Fig. 5). To do the same, thecircumference might have been divided into 12 equal parts.
Figure 5 : Four petals are drawn in a circle
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The same plate displays eight petals drawn in a circle, forwhich it was necessary to divide the circumference into24 equal parts. In fact, the Indus civilization paid specialattention to a circle and its variants in the geometric designsmade on various artefacts.
Analysing these geometrical figures, scholars infer thatthe dwellers of Indus Valley might know the followingconstructions relating to circle.22
A) To construct a circle having a known diameter andto construct few concentric circles.
B) To divide a circle into few even number of equalparts by drawing diameters.
Considering these, it may be stated that the people ofthis civilization probably knew the following geometricalconstructions.23
i) To draw a perpendicular at a point on a givenstraight line.
ii) To bisect a given finite straight line with the helpof a compass.
iii) To bisect a given angle.
iv) If the radius of a circle is known then itscircumference can be divided into 6 equal parts.To divide the circumference of the circle into anynumber of equal parts which must essentially be 3or multiple of 3.
v) To inscribe the maximum square or hexagon orany regular polygon having odd or even numberof sides in a circle.
From the plates shown by Ernest Mackay,24 it canbe inferred that the construction of a circle through anytwo points, construction of tangent to a circle etc. wereknown to the dwellers of ancient Mohenjodaro andHarappa. Besides, they were conversant with the propertythat a tangent to a circle is perpendicular to its
corresponding radius and also with other related properties.
Some scholars opine that the dwellers of Indus Valleyknew the formulae of volume of solid figures, namely, cube,sphere, cylinder and they had the knowledge of thePythagoras’ triads.25
It will be beside the mark if we won’t mention oneof the signs that occurs in various inscriptions in theIndus valley civilization is in the form of two overlappingellipses (vide Fig. 6)26,27 and a star-like geometrical motifin ornament-like ivory objects found in Mohenjodaro ( videFig. 7).28,29 These speak of a fairly sophisticated knowledgeof geometry practised by the people of the civilization.
Figure 7 : Star-like geometrical motif
Conclusion
No material evidence of any mathematical theory orinformation has yet been obtained from the ruins of thecivilization, though measuring scale, tools for drawingcircles, stylus-like objects for engraving etc. have beendiscovered. The picture-form of writings on the seals havenot yet been fully deciphered, nothing can definitely besaid about the depth of their theoretical knowledge. Theopinions of archaeologists, engineers and mathematicianswell-acquainted with the detailed nature of the excavationsbased on the inferences they got at from the geometricalfigures incised on earthen containers, designs on seals areworth considering here. But these may not be regarded asconclusive or tenable to some of the present-day scholars,still these will help furthering the zeal of the reading publicfor unfolding the mathematical thoughts prevailing in thedays of yore.
Acknowledgement
The author expresses his sincere thanks to the learnedreferee for his valuable suggestions in the improvement ofthe paper.
References1. Suniti Kumar Chatterjee : India and Ethiopia (The Asiatic Society,
Kolkata, 2004), pp.9-14.2. R.C. Majumdar: The Age of Imperial Unity (Bharatiya Vidya
Bhavan, Mumbai, 2001), pp. 611-652.Figure 6 : Design of two overlapping ellipses
332 SCIENCE AND CULTURE, SEPTEMBER-OCTOBER, 2018
3. D.M. Bose, S.N. Sen and B.V. Subbarayappa (editors): A ConciseHistory of Science of India, (I.N.S.A., New Delhi, 1971), p.8.
4. Ernest Mackay: Early Indus Civilizations (Luzac & CompanyLtd., London, 1948), p.112.
5. ibid., p. 125.6. P.K. Majumdar: Prâcîna Bhârate Jyâmiti Carcâ (in Bengali)
(West Bengal State Book Board, Calcutta, 1992), pp.5-6.7. Gerard G. Emch, R. Sridharan and M.D. Srinivas (editors):
Contributions to the History of Indian Mathematics (HindustanBook Agency, New Delhi, 2005), p.2.
8. M.N. Deshpande: Archaeological sources for the reconstructionof the history of sciences of India, Ind. Jour .Hist. Sci., 6(1).10-11 (1971).
9. S.R.Rao: Lothal and the Indus Civilization ( Asia PublishingHouse, New Delhi, 1973).
10. B.Vij : Linear standard in the Indus Civilization, in Frontiers ofthe Indus Civilization ( editors B.B.Lal and S.P.Gupta), IndianArchaeology Society and Indian History & Culture Society ,New Delhi , 1984, pp.153-156.
11. S. Sinha, N. Yadav and M. Vahia: “In Square Circle : GeometricKnowledge of the Indus Civilization”, in Math Unlimited : Essaysin Mathematics (Editors – R. Sujatha, H. N. Ramaswamy andC. S. Yogananda), (Science Publishers and CRC Press, 2012),p.441.
12. Earnest Mackay, ibid., p.21.13. Michel Danino : New insights into Harappan town-planning ,
proportions ,and units , with special reference to Dholavira (Papersubmitted for publication to Man and Environment.), pp.1-2.
14. R. S. Bisht : Dholavira and Banawali - Two different paradigmsof the Harappan Urbs Format, Puratattva ( New Delhi ), 29.14-37 (1999).
15. Michel Danino : “Unravelling Dholavira’s Geometry”, in RecentResearches in Archaeology, History and Culture (Festschrift toProf. K. V. Raman), P. Chenna Reddy (editor), (Agam KalaPrakashan, Delhi, 2010), pp. 179-193.
16. John Marshall : Mohenjodaro and the Indus Civilization(Indological Book House, New Delhi, 1973).
17. Earnest Mackay, ibid.
18. R. P. Kulkarni: Geometry as known to the people of IndusCivilization, Ind. Jour. Hist. Sci., 13 (2).117-124 (1978).
19. John Marshall, ibid., plate nos. 113, 114.
20. R. P. Kulkarni, ibid., pp.119-120.
21. Ernest Mackay, ibid., plate nos. 17, 18.
22. R. P. Kulkarni, ibid., p. 121.
23. ibid., p.122.
24. Ernest Mackay, ibid., plate nos. 17, 18.
25. P. K. Majumdar, ibid., p.7.
26. S. Sinha, N. Yadav and M. Vahia, ibid., p.447.
27. E.J.H. Mackay : Further excavations at Mohenjodaro(Archaeological Survey of India, New Delhi, 1938), DK Area, Gsection, No.21 in plate LXVII.
28. S. Sinha, N. Yadav and M. Vahia, ibid., p.450.
29. E.J.H. Mackay, ibid., Nos.9-10 in plate CXLI.
