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Research Into Indic Mathematicians
#61
I proposed elsewhere that data for Groups (powers) of Tens or Dashatis in the Rgveda was expressed with two words: Pad and Hasta. Both words denote the limbs of human beings, each containing its group of tens, counted as the total of toes respectively fingers.

There is a Rc which exactly formulates what I've proposed. See below, under Niyuta in RV.



Previously, I thought that the highest number with a name in the Rgveda was Ayuta=10.000. This is true for the older layers off the Rgveda. I have found even higher named numbers, which belong to a later layer. See below, under Niyuta and ParArdha in RV.



Hasta

The searching started with this counting unit. Rc IV.58.3 of VAmadeva Gautama gives the sequence 432 and 7 Hastas or 10.000.000. I proposed a Vaidika counting along this line: 1 Hasta = 1x10, 2 hastas = 1 Hasta x 1 Hasta or 10x10=100, 3 Hastas = 2 Hasta x 1 Hasta or 100x10=1000, etc.

This Rc suggests the huge number 4.320.000.000 or a number identified later as a Kalpa.



Pad

Then I advanced with another Rc containing a concurring counting unit:

– Rc I.164.48 in the SUkta of RSHi DirghatamA Aucathya gives huge numbers starting with Ekapad or 10 to Navapad or 1.000.000.000 in Parame Vyoman or in the Highest Space. Here we see that GaurI has lowed or a shining object made noie and formed or generated multifold Salilas or waterfloods. A Gaura is the Horse-like speedy onager (Ger in ona-ger and Gur in Persian are related to the word Gaura which has its local form Khur in the Rann of Kutch – see the paper of prof. R.N. Iyengar). The female alien GaurI was joined with the (land of the) local Gaura.

Additional to this Rc, I found the next one:

– Rc X.117.8 of RSHi BhikSHu AngirA gives the ratio 4:3:2:1. The numbers are in Pads preceded with the sacred sequence 432. Thus, we have 432 x 10-100-1000-10.000 or 4320-43.200-432.000-4.320.000.



Huge numbers

The Yajurveda gives the groups of tens sequence from 10 to 1.000 billion or ParArdha. (While Shukla Yajurveda gives the sequence ayuta-niyuta-prayuta, KYV KS has the older sequence ayuta-prayuta-niyuta. It seems that the PaippalAda Atharvaveda supports the last with asking for Ayuta or 10.000 to become Prayuta or 100.000 (ayutam prayutam bhava).



Niyuta in RV

Now, I have found a new number in the Rgveda, that not only is higher than Ayuta, but even clearly proofs that higher number with the Pad counting, as I had proposed:

nákSHad dhávam aruNiíH puurvyáM raáT turó vishaám áÑgirasaam ánu dyuún |

tákSHad vájraM níyutaM tastámbhad dyaáM cátuSHpade náryaaya dvipaáde ||

Rc I.121.3 of RSHi KakSHIvAn Dairghatamasa, says that Dyo was fixed, forming a Niyuta-fold Vajra for the DvipAda CatuSHpad Narya. Here we have a juxtapositioning of niyuta+vajra versus dvipAda-catuShpad+narya, where the vajra has a connection with narya. As we are dealing here with counting and numbers, Niyuta must be a huge number equal to a DvipAda-CatuSHpad.

DvipAda as 10x10=100 and catuSHpad as 1000x10=10.000 gives 100 Ayuta or 1.000.000. And this is called Niyuta in Rk-pAda c.

Getting support of AV PaippalAda and KYV-KATHaka we know that after Ayuta or 10.000 comes Prayuta or 100.000. And KATHaka continues then with Niyuta or 1.000.000 (this is also the case in later literature, Br., Pur. As per MW dictionary). Thus, RV Niyuta = dvipAda-catuSHpad = 1.000.000!



ParArdha in RV

The Asya VAmasya SUkta provides us with even huger numbers than the Niyuta. In I.164.12 the RSHi of that Rc gives other units to provide us with huge numbers.

páñcapaadam pitáraM dvaádashaakRtiM | divá aahuH páre árdhe puriiSHíNam |

áthemé anyá úpare vicakSHaNáM | saptácakre SHáLara aahur árpitam ||

They speak of the PancapAda Father as DvAdasha-AkRti in the upper region of heaven reaching the marshy shores.

PancapAda = 100.000 and DvAdasha-AkRti = 1.000.000.000.000, which is exactly a ParArdha of the Yajurvedas. Here we get explained the name of the huge number.

The second Rgardha explains the gap below (upare) between PancapAda and Dvadasha-AkRti as Sapta-cakra, starting with SHaL-ara or 1.000.000.

Remember that Rc I.164.48 first mentions the lower Pads, then leaves the first region of Vyoman, containing

5-6-7 Pads, to reach Parame Vyoman with 8-9 Pads. The number for Dashapad is Madhya in Yajurveda and for EkAdashapad is Anta. The numbername Madhya or intermediate, is between a Samudra (Navapad) and a next region, which must be containing Anta and ParArdha. As the last is Para-Ardha, the previous is the ordinary Ardha. Thus, after Samudra comes Madhya and then Ardha= YV Anta and finally Para-Ardha.



Note: What amazes me, is that the RSHi starts with the many saline Salilas coming towards a single Samudra and then reaches ParArdha in the PurISHin shores. This means that the RSHi anticipates having knowledge of or supposes new regions beyond ParArdha in the remotest sky (Div).



We can safely conclude:

a. that the Rgveda counting was a decimal one, with names for each number. gave Eka to Dasha upto ParArdha or 1-10 to 1.000.000.000.000 at least identified till now.

b. the RSHis of that work also used a different counting, in groups or powers of tens, with the twofold unit names of Pad (var. PAda) and Hasta.

c. they also were very familiar with the 4:3:2:1 ratio – huge numbers upto a named Niyuta and Pad or Hasta counting were multiplied with a preceding sacred number 432.

I haven't yet touched the device of diverse ChandA syllables for counting purposes.



Caturyuga division

These huge numbers are also part of the later PaurANika and SaiddhAntika systems. The PurANas are evolved out of the Vaidika PuRANa-veda and the combined Vaidika ItihAsa-PurANa. The ratio 4:3:2:1 coupled to these huge numbers, combined with the Vaidika AkSHa game with VibhItakas (see Xth MaNDala) containing that ratio led to the Caturyuga system.

The RSHis were also fond of DvAdashatis or duodecimal counting: 12-24-36-48-60-72-84-96-108-120-.... 360, ….. 432, etc. it is particularly this counting which led them to their sacred numbers, and secondarily to their BArhaspatya or sexagesimal counting.

The Rgvaidika 432 sapta-hasta number is 360x12.000 or a Kalpa. The Rgvaidika 432 PancapAda (Ayuta) is exactly a MahAyuga.

For a fractional division of the Pad in four, see I.164.45 (about VAk):

A. catvaári vaák párimitaa padaáni taáni vidur braahmaNaá yé maniiSHíNaH |

a. catvaári vaák párimitaa padaáni | b. taáni viduH braahmaNaáH yé maniiSHíNaH |

B. gúhaa triíNi níhitaa néÑgayanti turiíyaM vaacó manuSHyaaaà vadanti ||

c. gúhaa triíNi níhitaa ná iÑgayanti | d. turiíyam vaacáH manuSHyaàH vadanti ||

Speech hath been measured out in four divisions, the Brahmans who have understanding know them.

Three kept in close concealment cause no motion; of speech, men speak only the fourth division.

Here we again have a 4:3:2:1 = 10 ratio within a Pad = 10. An additional factor is that the 4/4th, 3/4th and 2/4th Pad parts are placed versus the 1/4th part. Just like, analogous to this, in Caturyuga counting the 1/4th Kali-Yuga is placed versus the other 3/4th Yugas in a Dharma sense. The AkSHa game and counting according to the 4:3:2:1 ratio came together. Remember that AkSHa is not only a dice game using the ratio 4:3:2:1, but it is by its name also connected with Time cycli: the Wheel of Time of DIrghatamA with 360 spokes has a nave or nAbhi consisting of three parts: 1. the ring, the Kha hole and the AkSHa pole connecting this 360 days wheel with higher wheels or period cycles. (Perhaps this threefold nAbhi, especially the Kha hole is the origin of the Indian zero sign. You first have the the nAbhi, then come the numbers from 1-360. The three NAbhis also point to 360/3=120 or Three CAturmAsikAs.)



1. Using the 4:3:2:1 ratio with the Rgvaidika MahAyuga number gives the duration of each single Yuga => 1.728.00 : 1.296.000 : 864.000 : 432.000. The last is a 432 Tripad.

2. Using the ratio 4:3:2:1 with the number 12.000 gives the duration of a Caturyuga => 4800-3600-2400-1200. (4:3:2:1 ratio in DvAdashatis gives 48:36:24:12)



The number 12, thus, was very special to the Vaidika people, which explains their Twelve-Year Sattra. DIrghatamA gives in his Asya VAmasya SUkta the image 360 spokes and 12 fellies. This 360 is one year multiplied by 12 = 4320 days in 12 years.

The 4320 = 360x12 years. But it also comes to 365x11,8 or 366x11,8 minus approximately a CAturmAsikA.



Vaidika DvA-dashatis

The ancient RSHis arrived at different sacred number, like 60, 72, 108, 360, 432 and their multiplication with the decimal Pad or Hasta units, because they also counted in 12s, a duodecimal system. Combining the DvA-dashati numbers within a Dashati table gave this:



000-001-002-003-004—005—006-007-008-009 (Dashati table)

DvAdashati series in a Dashati counting

000-012-024-036-048—060--072-084-096-108

120-132-144-156-168—180--192-204-216-228

240-252-264-276-288—300--312-324-336-348

360-372-384-396-408—420--432-etc.



In this DvA-dashati we have all the basis ingredients for a CaturyugI system (the number 432=36x12, and further at positions 004-003-002-001 we have the numbers 48-36-24-12, there is a 72, 108), but also 360 for days, 12 for months of 30 days.

(Perhaps this was put on a Vaidika IriNa-board, evolving in a ten x ten Pad or PAda (Dashapad or -pAda board. This must have facilitated counting. In later times we have the Chinese and Indians using the board for counting, see Plofker)



A last word, on the Aucathyas

The Aucathyas composed SUktas in the Rgveda which amounts to 40, almost a Family MaNDala. I call these the Aucathya CatvAriMsha RkKANDa. A Rk-KANDa of the Aucathya Kula, consisting of 25 SUktas of DIrghatamA Aucathya, 12 of KakSHIvAn Dairghatamasa (11 in the Ist and 1 in the IXth MaNDala) and 3 of Ucathya Angirasa (in the IXth).

About SUkta I.164, the huge numbers are those composed in JagatI ChandA. The number Niyuta is of KakSHivAn, which means he is earlier in time. Therefore, I conclude that the TriSHTubh Rcs in I.164 are those of DIrghatamA, and the JagatI ones are those of an Aucathya descendant, later in time than KakSHIvAn. And the latest Aucathya descendant, at a time just before the inclusion of the Ist MaNDala, must have finished the extended AprI SUkta in AnuSHTubh.

RSHi DIrghatamA composed the Rcs in TriSHTubh containing amongst others four important ones containing DvA suparNA (20), Ekam sadvipra (46), and PrchAmi tvA param antam (34) and DvAdasha pradhayash cakram (48).

The decendants of RSHi DIrghatamA elaborated further on his concept, leading to measuring the limits of the sky (caused by Rc 164.34) towards ParArdha in Div or furthest sky giving the figure of 1.000 billion.



BhikSHU AngirA also used the Pad unit counting, whereas VAmadeva Gautama of another AngirA branch, distantly related to the Aucathyas, used a Hasta unit.
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