Vigesimal System of Numeration as Prevailed in the Bārelā Tribe
Dipak Jadhav Lecturer in Mathematics Govt. Boys Higher Secondary School, Anjad Distt. Barwani 451556 (M. P.) India [email protected]
Abstract
Bārelā is a tribe. It resides in the western part of India. In Bārelā system of numeration other numbers stand in relation to twenty. This paper finds that no other vigesimal system whether it was in India or from abroad has been influential in shaping Bārelā system of numeration if their corresponding terms are taken into consideration. Gunnīsa, uganīsa, and oganīsa are the alternative terms used by Bārelās for 19. This paper also offers an insight into the formation of each of them.
Key Words: Bārelā, Vigesimal system of numeration, Vīsa, Vīsu.
Introduction
Bārelā is an Indian tribe. Most of its population lives in Barwani District of
State of Madhya Pradesh in India. The surrounding region, where as well Bārelās dwell, to this District encompasses the neighboring Khargone District of Madhya
Pradesh and the three neighboring Districts, namely, Jalgoan, Dhule, and Nandurbar, of State of Maharashtra. The term used to identify Bārelā in Maharashtra is Pāvarā.1
The entire region of their dwelling, which is in the western part of India, falls in the forest of Satpura range. The dialect which Bārelās speak is called Bārelī.
Their system of numeration is based on twenty. It is observed to be mostly practised in Pati and Barwani Blocks of Barwani District. Bārelās dealing with retailers in its terms can be seen in weekly haat bazars2 conducted in this region
(Parihar et el 2000; Thakur 2005, p. 7). Since it is practised among Bārelā community,
1Bārelā colleagues of the present author including Mr. Subhash Solanki and Mr. Balram Pawar from Pansemal Block of Barwani District has apprised him so. 2 A haat bazaar, simply called haat or hat or hāṭa (if transliterated), is an open-air market. It serves as a trading venue, conducted on a regular basis, for local people in rural or forest areas. Journal of Mathematics and Culture 61 December 2019 13(1) ISSN-1558-5336
an identifiable cultural group, it is an ethnomathematics. See definition of ethnomathematics suggested by D‟Ambrosio (D‟Ambrosio, 1985). Very older Bārelās still remember an accurate vigesimal system.
Group leader Vikas Parihar and his four co-workers of Grade XI from J. N.
Govt. Model Higher Secondary (Residential) School, Barwani (now known as
Ekalavya Model Residential School, Barwani) undertook a project on “Number-
Language System of Bārelās: A Mathematical Study” under the guidance of the present author in the academic year 2000-2001. Vikas Parihar, after their project had been selected first at District level and then at State level, presented its report in Hindi at National level under focal theme „Indigenous scientific knowledge: For a better tomorrow‟ in National Children Science Congress, held during December 27-31, 2000 at Kolkata (Parihar et el, 2000). The project represented State of Madhya Pradesh in
Indian Science Congress, held during January 3-7, 2001 at New Delhi (Anonymous,
2001, p. 27). It was selected for participation in the “Exchange of Young Researchers” program taken up by National Council for Science and Technology Communication,
Department of Science and Technology, Government of India. Under this program a fellowship to visit various science and research institutions in Germany was awarded to Vikas and his co-worker Nilesh. They visited Germany in April, 2001.3 The project-report was submitted to Honorable Governor of Madhya Pradesh in Bhopal on
May 04, 2001 when Vikas and Nilesh paid a courtesy visit to Governor House.4 The project was selected for featuring in the Television Show Ignited Minds. Vikas
Parihar accompanied by his guide teacher participated in its shooting conducted
3See letter D.O.No. CO/FP/O11/2000 dated January 12, 2001 issued by Ministry of Science and Technology, Government of India. 4 See Hindi Daily Newspaper Naiduniya dated May 05, 2001. Journal of Mathematics and Culture 62 December 2019 13(1) ISSN-1558-5336
during November 3-6, 2004 at Noida.5 The related episode of Ignited Minds was televised on DD-National on December 17, 2005.
The study in the above project was confined to the numbers from 1 to 100. It was aimed at understanding number-language system of Bārelās in mathematical terms, documenting it, and approximately defining a Bārelā number. In the next section of this paper, we shall see how the young researchers studied this peculiar system of numeration and looked at its underlying characteristics and laws. We will also see what the interesting features they found are.
This paper is the outcome of a variety of side trips. It is mainly aimed at making the essence of the above project-report available in English to the wider class of scholars in the field of Ethnomathematics. This will be done in the next section. It is also aimed at discovering the differences and similarities between Bārelā system and various other vigesimal systems of numeration prevailed in the other parts of
India and abroad. It will also discuss various aspects of Bārelā system related to
History of Mathematics.
Bārelā Number System
The young researchers collected the terms spoken by Bārelās for numbers from 1 to 100 from their Bārelā fellows of the residential school and divided them into two categories. The first is category of „generally spoken‟ and the second is that of „sometimes spoken‟. See Table 1. Table 1 is slightly enriched in information if compared with theirs.
5See letters, written by Pulse Media Pvt. Ltd. separately to the principal of the school and to the present author, received on October 18, 2004. Journal of Mathematics and Culture 63 December 2019 13(1) ISSN-1558-5336
From Table 1 they notice that ne stands for “plus”, koma or kama means
“minus” or “less”, and no term is used for “multiplied by”. They also observe that
Bārelā numbers are formed using five laws. Each law contains one or two of the three operations, namely, addition, subtraction, and multiplication. If two operations are contained in the law, one of them must be the operation of multiplication.
Below are shown those five laws using arrow. The Bārelā number can be read along the direction of and above the arrow. What is below the arrow shows the working of the law. Let us denote 20 by v , which is the first letter not only of vīsa
(twenty) or vīsu (twenty) but also of vigesimal.
Figure 1 shows how law of addition works.
vīsu ne duī
22
v + 2
Figure 1
Figure 2 displays how law of subtraction works.
vīsu koma tīna
17
v - 3
Figure 2
Figure 3 shows how law of multiplication works.
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cyāra vīsu
80
4 × v
Figure 3
Each of Figure 4 and Figure 5 shows how law of compound operation works. In Figure 4 it contains the operations of addition and multiplication while in
Figure 5 it contains the operations of subtraction and multiplication.
tīna vīsu ne duī
62
3 × v + 2
Figure 4
vīsu duī koma āṭha
32
v × 2 - 8
Figure 5
Each of Figure 6 and Figure 7 shows how law of flexibility works.
duī vīsu ne suḷe
56
2 × v + 16
Figure 6
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vīsu tīna koma cyāra
56
v × 3 - 4
Figure 7
56 is formed in Figure 6 using the operations of addition and multiplication while it is formed in Figure 7 using the operations of subtraction and multiplication.
This law provides facility to the Bārelās in numeration.
Observing Table 1, the young researchers find that there are cycles in Bārelā number system. They call those cycles vigesimal cycles (vīsu cakras). Each of them ends in multiples of twenty. There are five cycles in their study; first from 1 to 20, second from 21 to 40, third from 41 to 60, fourth from 61 to 80, and fifth from 81 to
100. See the third cycle in Figure 8. The operation of multiplication with that of subtraction is generally employed in the second half of each vigesimal cycle while the operation of multiplication with that of addition in its first half except for the first cycle.
2v + 5
First half
3v 2v + 10 Second half
3v - 5 Figure 8
Further, they find that Bārelā number has two parts. One is vigesimal part (vīsu bhāga) and the other is non-vigesimal part (avīsu bhāga). See Figure 9.
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43 = 2v + 3
vigesimal part non-vigesimal part
Figure 9
If z is a Bārelā number, z x y x, y where x v , 2v , 3v , 4v , 5v and y 10, -9, ..., 9, 10 . Assuming X-axis to be vigesimal axis (vīsu akṣa) and Y-axis to be non-vigesimal axis (avīsu akṣa), they demonstrate the entire column of „generally spoken‟ from Table 1 as shown in Figure 10.
Non-vigesimal axis
10
cycle
5
First half First of second half First of third cycle half First of fourth cycle half First of fifth cycle First half First of firstcycle Vigesimal axis v 2v 3v 4v 5v
0
h f2e hal
cycle cycle
a con f of
d cycle
thir fifth
lf d fourth fou -5 o cyc rth
f le cyc
Second ofhalf Second ofhalf
Second cycle ofhalf first Second ofhalf fi Second ofhalf second cycle le -10 ft Figure 10
h From Figure 10 they infer that in Bārelā number system is found an upward move in the first half of each vigesimalc cycle and the move is maintained in its second half as well with a downward yjump just after the first half.
Finally, they approximatelyc define Bārelā number of n 5th cycle as follows
l
e Journal of Mathematics and Culture 67 December 2019 13(1) ISSN-1558-5336
n 1v y when 1 y 10 z nv y when -1 y 10 nv when y 0 and they remark that vigesimal system of Bārelās is based on decimal system.
Discussion
Vigesimal system of numeration is “a reminder of the barefoot days of,” writes
D. E. Smith, “the race, when men counted toes as well as fingers (Smith, 1958, p.
12)” totaling twenty.
Decimal place value notation is India‟s contribution to the world (Gupta,
1983, 23-38). It is based on the powers of ten viz. 10(ten), 102 (hundred), 103
(thousand), etc. Like it, vigesimal system runs into k of 20k where k is a positive integer. This kind of vigesimal system with place-value was developed in the Mayan civilization which was formed as early as 1500 BCE in Central America and Southern
Mexico and reached its height between 300 CE and 900 CE. The Mayan system included three basic symbols: dot for 1, dash or bar for 5, and a roughly half closed eye for 0 (Cajori, 1928, p. 43 and 1958, pp. 69-70). Although it was found in apparent isolation, a possible borrowing of zero from India is suspected as it was at its zenith during the period from the sixth century onwards (Menninger, 1969, p. 406; Gupta,
1983, 23). The Mayan system is used in some parts still. Currently it is still used in calendrics and divination.
The vigesimal system of Aztec, developed in Mexico, had two auxiliary bases
5 and 10 (Menninger, 1969, pp. 62-63). 1 was denoted by dot, 5 by block of five dots, and 10 by a lozenge (or rhombus). The number 20 was represented by a flag. Five times of it, yielding 100, was marked by quarter of the barbs of a feather. Half the
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barbs represented 200, three-fourths 300, and the entire feather 202. 202×20, yielding
203, was marked by a purse (Cajori, 1928, p. 41).
Unlike the above systems, besides Bārelās many cultures or ethnic groups in the world used or use in their number-language systems twenty as a base.
In India, the terms ekānna-viṃśati (= 20-1) and ekānna-catvāriṃśati (= 40-1), formed using law of subtraction, were spoken in Sanskrit for 19 and 39 respectively in
Vedic times. Both of the terms are contained in the Taittiriya Saṃhitā (vii.2.11)
(Datta and Singh, 1935, 14). It can be easily inferred that the spoken term ekānna- triṃśati (= 30 - 1) must have been prevalent in Sanskrit for 29 at that time. Here it may be noted that the term used for 29 in the Bakhshali Manuscript (c. 400 CE or 7th century CE) is ekonatriṃśa (= 30 - 1) (Datta and Singh, 1935, 61). Of our interest is that the alternative Sanskrit term nava-viṃśati (= 20+9) as well, formed using law of addition, was spoken for 29 in Vedic times. It is found in the Vājasaneyī Saṃhitā
(xiv.31) (Datta and Singh, 1935, 15). Similarly, the term ṇava-ya-vīsa or ṇava-vīsa
(20+9) was prevalent in the Prakrit literature. The use of this term can be seen in the
Gommaṭasāra composed by Nemicandra (981 CE) in Prakrit (Upadhye and Shastri, verse 544, p. 805). This shows that seeds of vigesimal system of numeration were sown, most likely in the process of naming numbers, in early India. The terminology of Number-Language System of Bārelās appears to be very close to Prakrit (Jadhav,
1998, p. 4).
Vigesimal system of numeration is also used in the languages spoken in the region of East and Northeast India. That region is opposite in direction to and far in distance from the region of Bārelās. Muɳɖārī is mainly spoken in the State of
Jharkhand and in the adjoining region of the States of Orissa and West Bengal in
India. In it, 40 is spoken bar hisi (= 2×20), 60 api hisi (= 3×20), and 100 moɳe hisi (= Journal of Mathematics and Culture 69 December 2019 13(1) ISSN-1558-5336
5×20) where bar means 2 (bar-ia), api means 3 (api-a), moɳe means 5 (moɳe-a), and hisi means 20 (Osada, 2008, pp. 99, 114). In Santali, most widely spoken another
Muṃḍā language, 50 is spoken bār isī gäl (= 2×20+10) (Gvozdanović, 1999, p.223).
Atong is a Tibeto-Burman language spoken in the South Garo hills of Meghalaya
State in India and adjacent areas in Bangladesh. It contains numeration in vigesimal system (Breugel, 2014, pp. 178-203). In the State of Tripura in India, khol-pe is spoken for 20, khol-pe ci (= 20+10) for 30, khol-nui (= 20×2) for 40, khol-nui ci (=
20×2+10) for 50, khol-brui ci (= 20×4+10) for 90 (Mazaudon, 2008, p. 14). In these languages and Bārelī those seeds seem, although it cannot be said for certain, to have sprouted when and where social milieu favored them in India.
Numbers formed in Bārelī are somewhat similar to those formed in Magar as far as the use of laws of arithmetic operations is concerned. The latter is spoken in
Himalayas‟ section of Nepal. It made its vigesimal system less firm with Nepali loan roots. In it spoken are nis bis (= 2×20) for 40, nis bise das (= 2×20+10) for 50, som bis (= 3×20) for 60, pã:c bise nis (= 5×20+2) for 102 (as referred to in Mazaudon,
2008, p. 13). Bis is very close to vīsa of Bārelī as far as their pronunciation is concerned. Here it may be noted that 20 is spoken bīsa in Hindi, the national language of India.
Although the young researchers‟ study was for n 5 , it can be deduced that number-language system of Bārelās will not face, as we do not come across k 2, any hurdle for n 20. However, vigesimal system in Bārelī does not appear to be as full-fledged as that in Dzongkha, the national language of Bhutan, is. In Dzongkha, khe is spoken for 20, ɲiɕu (= 20 khe) for 202, kheche (= 20 ɲiɕu) for 203, and jãːche (=
20 kheche) for 204 (Mazaudon, 2008, pp. 3, 6-10).
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Bārelī appears to be more vigesimal than the French language is. The present form of the latter, partially vigesimal, except the French spoken in Belgium and
Switzerland, has vingt (= 20) instead of dix (= 10) as a base in the names of numbers from 80 to 99. For example, quatre-vingts (four twenties) is 80; quatre-vingt-sept
(four-twenty-seven) is 87; quatre-vingt-dix (four-twenty-ten) is 90; quatre-vingt-seize
(four-twenty-sixteen) is 96; and quatre-vingt-dix-neuf (four-twenty-ten-nine) is 99
(Laredo and Laredo, 2010, p. 58). In old French, the names for numbers from 30 to 79 as well were in either base 10 or base 20 (Einhorn, 1974, p. 110).
Yoruba, a language spoken in the South Western Nigeria, has as an elaborate vigesimal system of numeration as Bārelī has as far as the involvement of laws of addition, subtraction, and multiplication in forming numbers is concerned. It has 1, 2,
3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 200, and 400 as basic elements. They are not derived from other numbers. The basic word for twenty is ogún. Even decades are multiples of 20. 40 is ogójì (ogún-méjὶ, 20×2). Ọgọta (ogúnmẹ́ ta, 20×3) is 60. 80 is ọgọ́ rin
(ogúnmẹ́ rin, 20×4). 100 is ọgọ́ rùn-ún (ogúnmárùn-ún, 20×5). The odd decades are derived by subtracting ten from the next even decade. Those numbers of which units are in 1, 2, 3, 4 are formed by adding (lé nί …) 1, 2, 3, 4 respectively to each of the decades while of which units are in 5, 6, 7, 8, 9 are formed by subtracting (ó dίn or dίn nί …) 5, 4, 3, 2, 1 from the next decade respectively. Much higher numbers like
10000 are derived in Yoruba vigesimal system (Babarinde, 2013, pp. 78, 81-83, 86).
Every Bārelā number 11 onwards is expressed in vigesimal form. See Table 1.
It is law of flexibility which makes Bārelā system vigesimally intensified. See Table
1. Using this law Bārelā expresses numbers in two modes. One is additive mode. See
Figure 6. The other is subtractive mode. See Figure 7. This law is no different from the law of compound operation as far as law is concerned but the former is different Journal of Mathematics and Culture 71 December 2019 13(1) ISSN-1558-5336
from the latter in the sense of application as both of the latter are applied in the former to express the same number.
The young researchers point out that vigesimal system of Bārelās is based on decimal system. The present author offers his interpretation for it as follows.
Bārelā system of numeration is a decimal-vigesimal system. That one auxiliary base or more auxiliary bases are used for the formation of numbers smaller than the main base in system of numeration when its main base is high like 20 is a usual phenomenon. The Mayan system of numeration is observed to be a decimal- vigesimal system while the Aztecs used a quinary-decimal-vigesimal system (as referred to in Gilsdorf, 2009, p. 89).
cycle
Second half of of cycle half first Second cycle second of Firsthalf second of half Second cycle third of Firsthalf third cycle of half Second cycle fourth of Firsthalf cycle of half fourth Second cycle fifth of Firsthalf of cycle half fifth Second First half of first cycle first of Firsthalf Number-line
Figure 11
In Figure 10, Bārelā numbers extending from 1 to 100 are shown on plane.
They form five upper segments and five lower segments. If they are put as shown in
Figure 11, a bigger segment, which is a part of number-line, is formed. This would not have happened if vigesimal part had been entirely separate from non-vigesimal part. This is not possible in case of complex numbers as the real part of every complex number is entirely separate from its imaginary part. Hence Bārelā system of numeration is a segment of number-line.
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In Sanskrit, 19 is spoken eka-ūna-viṃśati (= 20-1) where eka refers to one,
ūna to minus or less, and viṃśati to twenty. It is reduced to ekona-viṃśati. We have already noted that it was ekānna-viṃśati in Vedic times. In later times, ekānna was changed to ekona, and occasionally even the prefix eka was deleted and we have ūna- viṃśati (Datta and Singh, 1935, pp. 14-15). The scheme that what is to be subtracted is spoken first when numbers with respect to the main base or the adjacent base are formed using the law of subtraction is in practice right from Vedic times. Here we are able to observe that the same scheme is followed throughout number-language system of Bārelās.
Going by Table 1, we find that the alternative terms for 19 are gunnīsa, uganīsa, and oganīsa. The present author would like to explain how they would have been formed.
Eka-ūna-viṃśati will be iga-una-vīsa or iga-uṇa-vīsa when spoken in Prakrit.
Iga or ega refers to one, which is noticeable in igyāre (= 1+10) or gyāre. See Table 1.
It is also noticeable in gyāraha spoken in Hindi for 11. Before we offer our explanation, it should be noted that a and u becomes o when joined. We may obtain a number of combinations when iga-una-vīsa is joined to make a single word by deleting its some letters, as they will be shown in [], and inserting some letters, as they will be shown in ‹›, in it. Ga^unīsa ([i]ga-un[a]-[v]īsa) and gonīsa ([i]ga-un[a]-
[v]īsa) are two of them. Uganīsa (u-ga-nīsa) and oganīsa (o-g‹a›-nīsa) were obtained in Bārelī from ga^unīsa and gonīsa respectively when vowel-transposition (svara- viparyaya) was allowed. These are changes made at their morphological level but their semantic level remained unchanged. Attention-grabbing is that 19 is spoken ogaṇisa in Gujarātī, a regional language in India. It seems to have been derived from iga-uṇa-vīsa in the manner in which oganīsa was formed. As far as gunnīsa is Journal of Mathematics and Culture 73 December 2019 13(1) ISSN-1558-5336
concerned, we may suggest two options for its derivation. One is that it seems to have been derived from ekānna-viṃśati in some manner when the latter was prakritized.
The other seems to have been somewhat like the following. Eka-ūna-viṃśati → iga- una-vīṃsa (when prakritized) → iga-una-ṃvīsa → [i]g[a]-un[a]-[ṃ]‹n›[v]īsa → gunnīsa where ṃ, a vowel modification called anusvāra (nasalization), seems to have been converted into n.
Conclusion
No other vigesimal system whether it was in India or from abroad has been found to have been influential in shaping Bārelā system of numeration if their corresponding terms are taken into consideration. Law of flexibility makes Bārelā system of numeration interesting and different from the others. The system seems to have sprouted or been fascinated from the seeds of vigesimal system of numeration sown in early India and developed in accordance with the social and economic requirements of Bārelās. The above discussion reaffirms that it was nourished from decimal system of numeration originated and developed in India.
Those Bārelās who are graduated in modern education system are no longer interested to use their own system of numeration or do not know it at all. In fact, their system of numeration has been usurped by modern system. Because of this it has come under threat of extinction. Before it be extinct, its broad survey is needed so that we may study how it works when n 5 , especially when n 20. However, the present author is sure about its smooth working for n 20.
Table 1: The terms spoken by Bārelās for numbers from 1 to 100 Number spoken generally sometimes 1 eka - 2 duī duya, do 3 tīna taṇa, traṇa
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4 cyāra cāra 5 pāca pāsa 6 chava chova, cha^u 7 sāta - 8 āṭha āṭa 9 nava nova 10 dosa dasa, daha 11 gyāre igyāre, nauva koma eka vīsu 20 9 12 bāre āṭha koma eka vīsu 20 8, daha ne duya 10 2 13 tere sāta koma eka vīsu 20 7 14 cavade cauvade, chava koma eka vīsu 20 6 15 paṃdare pondre, pāca koma eka vīsu 20 5 16 sule sole, suḷe, cāra koma eka vīsu 20 4 17 sotre satre, satare, tīna koma eka vīsu 20 3 18 aṭhāre aṭṭhāre, oṭṭhāre, duya koma eka vīsu 20 2 19 uganīsa oganīsa, gunnīsa, eka koma vīsu 20 1 20 vīsu bīsu, vīha, vīhu, vīsa
1 21 vīsu ne eka 20 1 nauva koma ḍeḍha vīsu 1 2 20 9 22 vīsu ne duī 20 2 vīha ne be 20 2, 1 āṭha koma ḍeḍha vīsu 1 2 20 8
1 23 vīsu ne tīna 20 3 sāta koma ḍeḍha vīsu 1 2 20 7
1 24 vīsu ne cyāra 20 4 chava koma ḍeḍha vīsu 1 2 20 6
1 25 vīsu ne pāca 20 5 pāca koma ḍeḍha vīsu 1 2 20 5
1 26 vīsu ne chava 20 6 cyāra koma ḍeḍha vīsu 1 2 20 4
1 27 vīsu ne sāta 20 7 tīna koma ḍeḍha vīsu 1 2 20 3
1 28 vīsu ne āṭha 20 8 duī koma ḍeḍha vīsu 1 2 20 2 1 29 vīsu ne nova 20 9 eka koma ḍeḍha vīsu 1 2 20 1, gyāre koma duya vīsu 220 11 30 1 vīsu ne dosa 20 10 ḍeḍha vīsu 1 2 20 31 nova koma duī vīsu 220 9 vīsa ne igyāre 20 11,
1 ḍeḍha vīsu ne eka 1 2 20 1 32 āṭha koma duī vīsu 220 8 vīsa ne bāre 20 12,
1 ḍeḍha vīsu ne duya 1 2 20 2 33 sāta koma duī vīsu 2 20 7 vīsa ne tere 20 13,
1 ḍeḍha vīsu ne tīna 1 2 20 3 34 chava koma duī vīsu 2 20 6 vīsa ne cavade 20 14,
1 ḍeḍha vīsu ne cyāra 1 2 20 4
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35 pāca koma duī vīsu 2 20 5 vīsa ne pondre 20 15,
1 ḍeḍha vīsu ne pāca 1 2 20 5 36 cyāra koma duī vīsu 2 20 4 vīsa ne sule 20 16,
1 ḍeḍha vīsu ne chava 1 2 20 6 37 tīna koma duī vīsu 2 20 3 vīsa ne satare 20 17,
1 ḍeḍha vīsu ne sāta 1 2 20 7 38 do koma duī vīsu 2 20 2 vīsa ne aṭṭhāre 20 18,
1 ḍeḍha vīsu ne āṭha 1 2 20 8 39 eka koma duī vīsu 2 20 1 vīsa ne uganīsa 20 19,
1 ḍeḍha vīsu ne nova 1 2 20 9 40 duī vīsu 2 20 - 41 duī vīsu ne eka 2 20 1 uganīsa koma tīna vīsu 3 20 19 42 duī vīsu ne duī 2 20 2 aṭhāre koma tīna vīsu 320 18 43 duī vīsu ne tīna 2 20 3 satare koma tīna vīsu 320 17 44 duī vīsu ne cyāra 2 20 4 suḷe koma tīna vīsu 320 16 45 duī vīsu ne pāca 2 20 5 paṃdare koma tīna vīsu 320 15 46 duī vīsu ne chava 2 20 6 cavade koma tīna vīsu 320 14 47 duī vīsu ne sāta 2 20 7 tere koma tīna vīsu 320 13 48 duī vīsu ne āṭha 2 20 8 bare koma tīna vīsu 320 12 49 duī vīsu ne nova 2 20 9 gyāre koma tīna vīsu 320 11 50 1 duī vīsu ne dosa 2 20 10 aḍhī vīsu or ḍhāī vīsu2 2 20, pocāsa 50 51 nova koma tīna vīsu 320 9 duī vīsu ne gyāre 2 20 11,
1 ḍhāī vīsu ne eka 2 2 20 1 52 āṭha koma tīna vīsu 3 20 8 duī vīsu ne bāre 2 20 12,
1 ḍhāī vīsu ne duī2 2 20 2 53 sāta koma tīna vīsu 3 20 7 duī vīsu ne tere 2 20 13,
1 ḍhāī vīsu ne tīna 2 2 20 3 54 chava koma tīna vīsu 3 20 6 duī vīsu ne cavade 2 20 14,
1 ḍhāī vīsu ne cyāra 2 2 20 4 55 pāca koma tīna vīsu 3 20 5 duī vīsu ne paṃdre 2 20 15,
1 ḍhāī vīsu ne pāca 2 2 20 5 56 cyāra koma tīna vīsu 3 20 4 duī vīsu ne sule2 20 16,
1 ḍhāī vīsu ne chava 2 2 20 6 57 tīna koma tīna vīsu 3 20 3 duī vīsu ne satare2 20 17,
1 ḍhāī vīsu ne sāta2 2 20 7 58 do koma tīna vīsu 3 20 2 duī vīsu ne aṭṭhāre2 20 18,
1 ḍhāī vīsu ne āṭha2 2 20 8 59 eka koma tīna vīsu 3 20 1 duī vīsu ne uganīsa2 20 19,
Journal of Mathematics and Culture 76 December 2019 13(1) ISSN-1558-5336
1 ḍhāī vīsu ne nova 2 2 20 9 60 tīna vīsu 3 20 - 61 tīna vīsu ne eka 3 20 1 uganīsa koma cyāra vīsu 420 19, 1 ḍhāī vīsu ne gyāre 2 2 20 11 62 tīna vīsu ne duī 3 20 2 aṭṭhāre koma cyāra vīsu 420 18, 1 ḍhāī vīsu ne bāre 2 2 20 12 63 tīna vīsu ne tīna 3 20 3 satare koma cyāra vīsu 420 17,
1 ḍhāī vīsu ne tere 2 2 20 13 64 tīna vīsu ne cyāra 3 20 4 suḷe koma cyāra vīsu 420 16,
1 ḍhāī vīsu ne cavade 2 2 20 14 65 tīna vīsu ne pāca 3 20 5 pondre koma cyāra vīsu 420 15,
1 ḍhāī vīsu ne pondre 2 2 20 15 66 tīna vīsu ne chava 3 20 6 cavade koma cyāra vīsu 420 14,
1 ḍhāī vīsu ne sule2 2 20 16 67 tīna vīsu ne sāta 3 20 7 tere koma cyāra vīsu 420 13,
1 ḍhāī vīsu ne sotre2 2 20 17 68 tīna vīsu ne āṭha 3 20 8 bare koma cyāra vīsu 420 12,
1 ḍhāī vīsu ne oṭṭhāre2 2 20 18 69 tīna vīsu ne nova 3 20 9 gyāre koma cyāra vīsu 420 11,
1 ḍhāī vīsu ne uganīsa2 2 20 19 70 tīna vīsu ne dosa 3 20 10 dosa koma cyāra vīsu 420 10 71 nova koma cyāra vīsu 420 9 tīna vīsu ne gyāre 3 20 11 72 āṭha koma cyāra vīsu 4 20 8 tīna vīsu ne bāre3 20 12 73 sāta koma cyāra vīsu 4 20 7 tīna vīsu ne tere 3 20 13 74 chava koma cyāra vīsu 4 20 6 tīna vīsu ne cavade 3 20 14 75 pāca koma cyāra vīsu 4 20 5 tīna vīsu ne pondre3 20 15 76 cyāra koma cyāra vīsu 4 20 4 tīna vīsu ne sule3 20 16 77 tīna koma cyāra vīsu 4 20 3 tīna vīsu ne sotre3 20 17 78 duī koma cyāra vīsu 4 20 2 tīna vīsu ne oṭṭhāre3 20 18 79 eka koma cyāra vīsu 4 20 1 tīna vīsu ne uganīsa3 20 19 80 cyāra vīsu 4 20 - 81 cyāra vīsu ne eka 4 20 1 uganīsa koma pāca vīsu 5 20 19 82 cyāra vīsu ne duī 4 20 2 oṭṭhāre koma pāca vīsu 5 20 18 83 cyāra vīsu ne tīna 4 20 3 sotre koma pāca vīsu 5 20 17 84 cyāra vīsu ne cyāra 4 20 4 suḷe koma pāca vīsu 5 20 16 85 cyāra vīsu ne pāca 4 20 5 pondre koma pāca vīsu 5 20 15 86 cyāra vīsu ne chava 4 20 6 cavade koma pāca vīsu 5 20 14 87 cyāra vīsu ne sāta 4 20 7 tere koma pāca vīsu 5 20 13 88 cyāra vīsu ne āṭha 4 20 8 bāre koma pāca vīsu 5 20 12 89 cyāra vīsu ne nova 4 20 9 gyāre koma pāca vīsu 5 20 11 Journal of Mathematics and Culture 77 December 2019 13(1) ISSN-1558-5336
90 cyāra vīsu ne dosa 4 20 10 dosa koma pāca vīsu 5 20 10 91 nova koma pāca vīsu 520 9 cyāra vīsu ne gyāre 4 20 11 92 āṭha koma pāca vīsu 5 20 8 cyāra vīsu ne bāre 4 20 12 93 sāta koma pāca vīsu 5 20 7 cyāra vīsu ne tere 4 20 13 94 chava koma pāca vīsu 5 20 6 cyāra vīsu ne cavade 4 20 14 95 pāca koma pāca vīsu 5 20 5 cyāra vīsu ne pondre 4 20 15 96 cyāra koma pāca vīsu 5 20 4 cyāra vīsu ne suḷe 4 20 16 97 tīna koma pāca vīsu 5 20 3 cyāra vīsu ne sotre4 20 17 98 duī koma pāca vīsu 5 20 2 cyāra vīsu ne oṭṭhāre 4 20 18 99 eka koma pāca vīsu 5 20 1 cyāra vīsu ne uganīsa 4 20 19 100 pāca vīsu 5 20 hava, sava
Acknowledgments: The present author is grateful to the referees and the editor of this journal for the suggestions offered by them to improve this paper. He is also thankful to his dear students Porlal Kharte (India) and Vikas Parihar (USA). Except for a few changes this paper was presented at National Seminar on Janajātīya Paramparā meṃ Bhāratīya Darśana, held during October 29-30, 2017 at Ambapani Distt. Barwani (M. P.) India. The author takes this opportunity to thank the organizers of the seminar for inviting him.
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Journal of Mathematics and Culture 79 December 2019 13(1) ISSN-1558-5336