Measurements & Numbers

 

 

Measurements

 

 

 

Note:  This somewhat disordered file, little more than a compilation of various sources, is part of an attempt to understand the various Tibetan systems of measurement and their history (but also bits about numbers and numeric codes, iconometry, economics and the like).  Input by Dan Martin, Jerusalem.  Diacritic Sanskrit and Tibetan script are both now done in a unicode font, but beware of the diacritics and verify that they are correct.  Someday someone will make a perfect study that will supply full coverage of all these things.  Until then this may be of some use to some people some of the time.  I realize that the bibliographical references are often presented in an inadequate form, and hope to repair this problem in the near future.

 

Notice the high numbers and measurements in the account of the historical Buddha's childhood schooling as told in Lalitavistara, ch. 12 (English translation, pp. 223-227).

 

Dpa'-bo Gtsug-lag-phreng-ba attributed the introduction of standard weights and measures to the grandfather of Srong-btsan-sgam-po, Stag-ri-gnyan-gzigs.  On this point, see cited text & translation in Eric Haarh, Yar-lu Dynasty, p. 125.  " 'di dus khri nor snang btsun mong gi bus // bre phul srang bcos 'bru mar tshad kyis bcal // dga' gnyis tshong byas mthun gnyis la lung bsdebs // de gong bod na tshong dang bre srang med //"

 

A passage in the Sba-bzhed, echoed in the Rgyal-rabs Gsal-ba'i Me-long, attributes the introduction of standard Magadha weights of bre, srang and zho, to the reign of Ral-pa-can (see Sørensen, Tibetan Buddhist Historiography, p. 412).

 

Source:  Brandon Dotson's dissertation, pp. 322-323.

 

As CHAB-SPEL (1989: 128-29) notes in his treatment of this passage, the introduction of weights and measures such as bre, srang and phul is attributed to a minister during the reign of Srong-btsan Sgam-po’s grandfather, Stag-bu Snya-gzigs (KhG: 171). This would date their introduction to the mid to late sixth century.

 

Considering the tradition of weights and measures in the present catalogue, CHABSPEL

(1989: 129-30) notes the following related traditions of weights and measures.

 

Weights and Measures for Grain:

Tradition One:

1 phul = 3.5 ya-khyor;

1 bre = 21 ya-khyor = 6 phul;

20 bre = 1 khal.

Tradition Two:

1 phul = 3 ya-khyor;

1 bre = 21 ya-khyor = 7 phul;

20 bre = 1 khal.

 

Weights and Measures for Meat, Butter and so forth:

20 se = 1 zho;

10 zho = 1 srang;

4 srang or 4 spor = 1 nyag;

20 nyag = 80 srang = 1 khal.

 

Weights and Measures for Silver, Gold and so forth:

7 sran = 1 nam; 7 nam = 1 zho; 10 zho = 1 srang.

 

- - -


From: Dung-dkar Rin-po-che, History, pp. 60-61.

 

"Some historians say that horses were priced at four  ཁལ་་[khal] of grain each."

 

Source:  Dung-dkar 184:  bre srang sogs kyi 'jal tshad rgya gar ma gha dha'i tshad gzhi dang mthun par gcig gyur byed dgos pa de yin / de'i dgos pa ni / skabs der bod sa gnas su phul drug la bre gang du rtsis pa'i drug phul ma dang phul bdun la bre gang du rtsis pa'i bdun phul ma zhes mi 'dra ba gnyis yod pa'i thog bod bre gang la drug phul rtsis na ma gha dha'i bre phyed las med cing / bdun phul yin na ma ga dha'i bre gang la bod bre do dang khyor ba gang yod pa bcas khyad par chen po yod pa dang / de bzhin ma gha dha'i gser zho brgyad la gser srang gang / gser srang bcu drug la gser bre gang du rtsis kyi yod pas / ma gha dha'i gser srang gang la bod kyi gser srang do yod pa bcas kyi he bag chen po des bod dang rgya gar bar tshong 'brel byed skabs stabs bde yong ched / 'jal gshor 'degs gsum gyi tshad gcig gyur gyi gtan 'bebs mdzad pa yin.

 

 

Source:  Topgyal in Tibet Journal XXIII no 3 (1998) 36:  "two goats are equal to one sheep, five or six sheeps to one 'bri or yak and four or five yaks to one horse.  Two yaks are the equal of one mdzo.  It is customary to regard horses and mules as of equal value.  There are a few variations in this value system between the different areas."

 

 

Source:  Precious Deposits V 164-5.  Picture of a measuring box with an inscription on the side.  The label says, "95. Wooden Sheng, Qing Dynasty, Measuring apparatus, 4.8 cm high, 18.6 cm. long.  Wooden dou and sheng originally appeared in the period of Tubo bTsan-po sTag-ru-gnya-gzigs.  In the reign of Srong-btsan-sgam-po (grandson of sTag-ru-gnya-gzigs), dou was standardized.  One dou equals to twenty sheng, one sheng amounts to six phul.  Grains of one dou weigh about thirteen kilograms.

 

Length Measurements (in Iconometry)

 

Source:  Martelli in Tibet Journal 26 nos 3-4 (Aut/Wint 2001) 151, but notice that practically the same information may be found in Lo Bue in Skorupski, ed., Indo-Tibetan Studies (Tring 1990), p. 190, etc.:

 

atoms (paramāṇu or trasareṇu, rdul phra rab)

8 atoms                       =          1 point of a hair (aṇu, skra'i rtse mo)

8 hair tips                   =          1 nit (likṣa or likṣā, sro ma)

8 nits                            =          1 louse (yākā or yūka, shig)

8 lice                             =          1 grain of barley (yava, nas)

8 barley grains          =          1 finger (agula, sor or sor mo)

12 fingers                    =          1 tāla

 

But these measurements aren't actually used by Tibetan artists.  The measurements they most frequently use are the following (p. 158).

 

4 nas                 =          1 rkang

4 rkang             =          1 sor (sor mo)

                           =          1 cha phran

4 sor mo          =          1 chag (fist width)

12 sor mo        =          1 thal mo (or, mthil, tāla) or                  

                           =          1 mtho (or mtho zhal, span) or

                           =          1 zhal (or, zhal tshad) or

                           =          1 gdong (size of face) or

                           =          1 cha chen

24 sor mo        =          1 khru (hasta, cubit)

 

On Human measurements in art:

 

Source:  Lokesh Chandra, Buddhist Colossi and the Avataṃsaka Sūtras, contained in: Lokesh Chandra, Cultural Horizons of India, International Academy of Indian Culture (New Delhi 1993), vol. 6, pp. 32-51.

 

p. 34-35:

 

The breadth of the finger/aguli of the hand of the donor provides the proportions in portrait images.  The Śāriputra-śilpaśāstra, verse 21: "The wise man makes an image according to the size of the finger's breadth of the donor" (H. Ruelius, Śāriputra und Alekhyalakṣaṇa, 1974: 82-88, quoted in Schroeder 1981: 33).

 

(What follows is a quote from Lo Bue 1990: 192):

 

The need of linking images to the actual features of the human body is further revealed by the habit of basing iconometric units of measurement on those of real people. Apparently, as early as the 7th century, the Tibetan King Srong-brtsan-sgam-po commissioned a famous Newar artist to make eleven images of Avalokiteśvara of the same size as himself.

In the 14th century the prince of Gyantse had the silver image of Tārā in Indian style made in proportions having as their unit of measure his dead wife's thumb.  His grandson prince Rab brtan kun bzang 'phags pa (1389-1442) had a gilded copper statue of Vairocana measuring 21 spans (Tib. phyag mtho) made following his bodily measurements.

During the same century the Newar King Yakṣamalla commissioned an image of Viṣṇu to be made according to the bodily proportions of his deceased son."

----------

Source:  Heather Stoddard, Thang stong Rgyal po, Jesus and the Fifth Dalai Lama, Archiv Orientalni, vol. 71, no. 3 (2003), pp. 433-440, at p. 434, note 8, based on information from Rdo rje tshe ring.

 

tho [~mtho] = tip of thumb to tip of index finger.

khyid [~mkhyid] = knuckle span to the tip of the outstretched thumb.

pho dom [~pho 'dom] = from fingertip to fingertip of outstretched arms.

 

 

The marketing weights of both gold and silver:

 

1 བལ་སྲན་  [bal sran] (today called  དམར་རུ་མགོ་ནག་, dmar ru mgo nag, n. of a grass seed) == 1 སེ་བ་ [se ba].

8 སེ་བ་  [se ba] == 1 ཞོ་  [zho].

8 ཞོ་  [zho] == 1 སྲང་ [srang].

1 སྲང་  [srang] gold == 2 སྲང་  [srang] silver

4 སྲང་ [srang] gold == 8 སྲང་ [srang]   Tibetan silver == 1 བྲེ་  [bre] silver

16 སྲང་ [srang] gold  == 1 བྲེ་ [bre] gold == 2 བྲེ་ [bre] silver

50 སྲང་ [srang] silver == 1 བྲེ་ཆེན་  [bre chen] silver

25 སྲང་  [srang] gold == 1 ཤིང་སྲང་  [shing srang] (a སྲང་  [srang] made of wood).

 

Source:  Francke, Antiquities of Indian Tibet, vol. 2, p. 105, a note by Dr. K. Marx:  "one zho of gold is stated to weigh 1/4 tola, equivalent to almost 3 grammes.  Its value in silver is said to be equivalent to from 15 to 18 rupees.  This would correspond to the British guinea.  One zho of gold is the price charged, e.g., for large printed volumes like the Mdo-mang, which may be had at Leh lamasery, printed to order for this price."  See also Blue Annals, p. 383, which mentions the se ba measurement for gold. 

 

NOTE:  (Later) monetary units:

10 སྐར་ [skar] == 1 ཞོ་ [zho]

10 ཞོ་ [zho] == 1 སྲང་ [srang]

 

What are evidently also later monetary units are explained in Jampel Kaldhen, Interest Rates in Tibet, Tibet Journal 1 no 1 (1975) 109 as follows:

 

2 1/2 Skar ma [Skar]         =          1 Kha-gang.

2 Kha gang                           =          5 Skar ma.

3 Kha gang                           =          1 Phyed brgyad

10 Skar ma or 4 Kha gang=          1 Zho gang (Zho).

10 Zho gang                         =          1 Srang gang (Srang).

50 Srang gang                     =          1 Rdo tshad (or .Tam-mdo).

 

According to   Tsarong in Tibet Journal 23 no 3 (1998) 33, n. 6:  4 kha gang (or 10 skar ma) equals 1 zho gang.  10 zho gang equals 1 srang.  50 srang equals 1 rdo tshad.  In Pabongka, Liberation in Our Hands, pt. 1, p. 5, a srang is said to be equiv. to about ten or twenty dollars in modern currency.  In pt. 2, p. 83, it says that a kha gang is approximately a penny.

 

 

Thubten Kunga Chashab, A Study of Religious Expenditure in Tibet in the 19th Century Based on the Biography of Dharmabhadra, Acta Orientalia, vol. 63 (2002), p. 199:  Denominations of skar ma, kha, zho, am, ṭam srang, dngul srang, srang, rdo tshad and gser zho (although Kha, rdo tshad and gser zho are rarely mentioned).  He gives the weight of a srang as 37 grammes (which I calculate ought to be 1.295 ounces).

 

 

 

** ** **

 

 

Grain (and medicinal) measurements

{these are volume measures}.

 

1 ཁམ་ཚིག་ [kham tshig] == 2 large + 2 medium + 2 small barley corns

ཁམ་ཚིག་ [kham tshig] == 1 སྟར་ཕུལ་ [star phul]

སྟར་ཕུལ་ [star phul] == 1 བྱ་སྒོང་ [bya sgong]

6 བྱ་སྒོང་ [bya sgong] == 1 མཁར་ཕུལ་ [mkhar phul or phul] one phul being a 'handful'.

6 མཁར་ཕུལ་ [mkhar phul or phul] == 1 མཁར་བྲེ་ [mkhar bre or bre] one bre being variously described as a 'handsized bowl', 'equal to 2 quarts', 'about 2 pints', or 'the size of a cow's hoofprint'. (according to Pabongka, Liberation in Our Hands, pt. 1, p. 170, it is about a quart).

20 མཁར་བྲེ་ [mkhar bre or bre] == 1 མཁར་ཁལ་ [mkhar khal or khal].  According to Rebecca French, a khal is the same as a 'bo[1] (a wooden box approximately 10 by 10 by 6 inches, said to weigh approximately 30 pounds).

 

According to Lobsang Gyatso, Memoirs of a Tibetan Lama, p. 206:  "The grain was measured in a big, square, measuring bucket called a bo ['bo].  There were nineteen buckets to a load (khel [i.e. khal]) so a bucket was about fifteen kilos worth of grain.  When you were measuring the grain, specific conventions governed allowable and forbidden leveling-off techniques."

 

According to Thubten Kunga Chashab, A Study of Religious Expenditure in Tibet in the 19th Century Based on the Biography of Dharmabhadra, Acta Orientalia, vol. 63 (2002), p. 201, citing Tshering the weight standards for such commodities as barley, butter and flour: 

One khal weighs 3.34 kg. while

one nya ga weighs 167 grams. 

Chashab also cites Goldstein's different opinion.

 

Dungkar Rinpoche says:  This is taken from commentaries written in those times on vinaya and medical texts, and represents a great advance on the method of measuring in handfuls in the time of Khri-srong-lde-btsan.  (Additional interpretations from R. French, The Golden Yoke, p. 230.)  There is a confusing explanation in Blue Annals, p. 402, "Mes tshon po presented him with several lumps of pigment and brown sugar with which one could cover a field which could be sown with one zho (1 zho = 1/10 of a 'bre) of seeds, and requested to be initiated."

 

* * *

 

 

Butter (and medicinal) market measures:

 

1 ཁམ་ཚིག་ [kham tshig, apricot kernel?] == 2 large + 2 medium + 2 small barley corns == 1 སེ་བ་ [se ba, rose hip?].

20 སེ་བ་ [se ba] == 1 ཞོ་ [zho]

10 ཞོ་ [zho] == 1 སྲང་ [srang] or 1 སྤོར་ [spor]

4 སྲང་ [srang] or 4 སྤོར་ [spor] == 1 ཉག་ [nyag or nya ga]

5 ཉག་ [nyag] == 1 ཁྱོར་བ་ [khyor ba]

4 ཁྱོར་བ་ [khyor ba] == 20 ཉག་ [nyag] == 1 ཁལ་ [khal]

 

According to   Tsarong in Tibet Journal 23 no 3 (1998) 33, n. 6:  20 gnya' ga of butter equals 1 khal.  12 khal equals one mar tang (approximately 40 kilograms).

In biography of 'Jig rten mgon po, as found in Lho-rong Chos-'byung, p. 355, it says that he had purchased a horse for 7 zho of brown sugar and offered it to the monastic community (sngar gyi rta de'i rin bu ram zho bdun lon pa dge 'dun la phul).

According to Precious Deposits V 102, one zho is about 5 grams, one nya ga (nyag) is 200 grams.

Thubten Kunga Chashab, A Study of Religious Expenditure in Tibet in the 19th Century Based on the Biography of Dharmabhadra, Acta Orientalia, vol. 63 (2002), p. 201:  The exchange rate between grain and money in the 1860's was 1 khal grain to 3 zho and 1 1/2 khal butter to 1 srang, but in another source a different exchange rate is given:  1 khal and 10 nya ga [i.e., 1 1/2 khal] of butter was worth 1 srang and 5 ṭam.  The government exchange rate for barley was 1 ṭam per khal (he's following Tsarong 1998 here).

 

* * *

 

 

Land measurement

 

Source:  Sumner Carnahan, In The Presence of My Enemies (1995), p. 15:  "A don is a measure of land equivalent to two kang; each kang is equivalent to sixty rukhay.  One rukhay amounts to the quantity of land that can be sown with a single bushel of barley."   p. 16:  "Generally, one don was equal to the amount of land that can be sown with 120 bushels of barley.  But in the lower central valley, in the Lhasa region — which was much more fertile — a different system was used."

 

Source:  Sarat Chandra Das, Journey to Lhasa & Central Tibet, p. 86:  "The ordinary kang  is a measure of land in which about 400 lbs of seed-grain can be sown.  The State tax on each kang is 50 srang (or ounces of silver) a year."

 

See 3-vol. dictionary under the word 'don, where it says that it is a basis for taxation, varying in size from place to place.  A large 'don could be as much as could be seeded with over 100 khal of barley.  A small 'don could be as little as what could be seeded with 20 or 30 khal.    Each 'don had either two or four rkang.

 

* * *

 

Abhidharma & sūtra systems of measurement.

 

These include minute size measurements, as well as measurements of length, both small and large. 

 

Source:  Mchims 'Jam-pa'i-dbyangs, Mdzod 'Grel Mngon-pa'i Rgyan, Khrung-go Bod-kyi Shes-rig Dpe-skrun-khang (Beijing 1989), pp. 320-321:

 

1.  7 phra rab kyi rdul

 = 1 rdul phran.

2.  7 rdul phran

 = 1 lcags rdul.

3.  7 lcags rdul

 = 1 chu rdul.

4.  7 chu rdul

 = 1 ri bong gi rdul.

5.  7 ri bong gi rdul

 = 1 lug gi rdul.

6.  7 lug gi rdul

 = 1 glang gi rdul.

7.  7 glang gi rdul

 = 1 nyi zer gyi rdul.

8.  7 nyi zer gyi rdul

 = 1 sro ma.

9.  7 sro ma

 = 1 sro ma de las byung ba'i shig.

10.  7 shig

 = 1 nas.

11.  7 nas

 = 1 sor mo'i tshigs.

 

In what follows we find a different Abhidharma system!

 

Source:  Lokaprajñāpti (Derge version), p. 18.7 ff. (with repetition at 20.1):

 

1.  49  nyi zer gyi rdul

 = 1 rdul phra rab.

2.  7 rdul phra rab

 = 1 rdul phra mo.

3.  7 rdul phra mo

 = 1 nyi zer gyi rdul.

4.  7 nyi zer gyi rdul

 = 1 ri bong gi rdul.

5.  7 ri bong gi rdul

 = 1 lug gi rdul.

6.  7 lug gi rdul

 = 1 glang rdul.

7.  7 glang rdul

 = 1 sro ma.

8.  7 sro ma

 = 1 a ba ga zha.

9.  7 a ba ga zha [a ba ga na]

 = 1 ba ti ka.

10.  7 ba ti ka

 = 1 nas 'bru.

11.  7 nas 'bru

 = 1 sor.

12.  24 sor

 = 1 khru.

 

 

 

Quote from Abhidharmakosa (ACIP version):

 

[space and time measurements]

SKAD  CIG  PHRA  RAB  RDUL  DANG  NI  //

RDUL  PHRAN  DANG  NI  DE  BZHIN  DU  //

LCAGS  CHU  RI  BONG  LUG  DANG  GLANG  //

NYI  ZER  RDUL  DANG  SRO  MA  DANG  //

DE  LAS  BYUNG  DANG  DE  BZHIN  DU  //

SOR  TSHIGS  ZHES  BYA  GONG  BDUN  BSGYUR  //

SOR  MO  NYI  SHU  BZHI  LA  KHRU  //

KHRU  BZHI  LA  NI  GZHU  GANG  NGO/ 

DE  DAG  LNGA  BRGYA  RNAMS  LA  NI  //

RGYANG  GRAGS  DE  LA  DGON  PAR  'DOD  //

DE  BRGYAD  DPAG  TSHAD  CES  BYA'O  //

[time measurement]

SKAD  CIG  MA  BRGYA  NYI  SHU  LA  //

DE  YI  SKAD  CIG  DE  DAG  KYANG  //

DRUG  CU  LA  NI  THANG  CIG  GO  //

YUD  TSAM  NYIN  ZHAG  ZLA  GSUM  NI  //

GONG  NAS  GONG  DU  SUM  CU  'GYUR  //

ZHAG  MI  THUB  DANG  BCAS  PA  YI  //

ZLA  BA  BCU  GNYIS  LA  LO  GCIG  //

 

* * *

 

Discussion:  The Collected Works (Gsung-’bum) of the Great ’Jam-dbyangs-mkhyen-brtse’i-dbang-po, Gonpo Tsheten (Gangtok 1977+), vol. 3, p. 165:

 

til sbyang zhes pa / pā ha ste / 'bru 'jal ba'i tshad kyi bye brag yin zhing / grangs kyi bstan bcos las nyi shu pā ha ru bshad cing / khal nyi shu sbyang gcig yin par bsgyur ba ste /  'gyur rnying yang khungs che bas khal brgyad cur bshad pa'ang 'thad pa nyid du shes par bya'o / + /

 

lcags rdul ni / lcags rigs la yod pa'i bsags rdul phra shos dang / chu rdul ni rlangs pa la mngon pa dang / ri bong sogs gsum ni so so'i sha'i rdul rags zhib kyis gsungs par mngon no //

 

rdul phra rab ni / gzugs 'grib pa'i mtha' ni rdul gyi chung shos cha dbye bar mi nus pa'i rdzas rdul phra rab yin te / bsags rdul gyi gzugs 'grib pa'i mtha' shin tu phra ba dbang shes kyi yul ma yin pa de yin par blo brtan dang gang spel gnyis kas gsal bar bshad do //

 

de bdun rdul phran / de bdun lcags rdul sogs / + /

 

mdzod las /

rdul phran dang ni de bzhin du //

lcags chu ri bong lug dang glang //

nyi zer rdu dang sro ma dang //

de las shig dang de bzhin nas //

sor tshigs zhes bya gong bdun 'gyur //

 

rdul phran rab ni cha shas gtan med pas / zla med kyi rdul zhes pa shin tu phra ba zhig go //  [166] rdul phran ni rdul rdzas brgyad 'dus te rdzas rdul zhes pa sngar las rags pa zhig go //

 

lcags rdul ni lcags phyis pa na g.ya' lta bu'i rdul zhig dag 'gro rgyu yod pa de'o //

 

chu rdul ni chu'i khar rlung gis btab pa na rdul gyi dbyibs can phra mo zhig 'khyor rgyu yod pa de'o //

 

ri bong lug glang gsum ni de gsum gyi spu steng gi rdul /+/

 

* * *

Source:  Rospatt, Buddhist Doctrine of Momentariness, p. 99, etc. (see for further discussion!.  Abhidharma time measurements based onMahavibhasa Sūtra.

 

1 kṣana [skad cig] (=0.014 or 0.013 seconds, or 1/75th of a second) X 120 = 1 tatkṣana [de yi skad cig].

1 tatkṣana [de yi skad cig] X 60 = 1 lava [thang cig] (equiv. to 7200 kṣana).

1 lava [thang cig] X 30 = 1 muhūrta [yud tsam] (equiv. to 48 minutes, or 216,000 kṣana).

1 muhūrta [yud tsam] X 30 = 1 day [nyin zhag] (equiv. to 24 hours, or 6,480,000 kṣana).

 

Rospatt calculates the value of a kṣana to be more like 0.013 seconds.

 

Compare Abhidharmakosa (Pruden tr.), p. 475 (and further discussion in the notes), which is substantially identical. 

 

My calculation:  There are 86,400 seconds in a 24-hour day.  Hence a muhūrta, being 1/30th of that, is equivalent to 2880 seconds (48 minutes).  A lava is then 96 seconds (1.6 minutes, or, a minute and 36 seconds).  A tatkṣana, being 1/60th of that, is equivalent to 1.6 seconds.  A kṣana, being 1/120th of that, is equivalent to 0.01333333333 seconds.

 

* * *

 

Parallel from Mdzod-phug, chap. 6, end (note that there is a detailed discussion of this passage, and of measurements of space, verbalizations and time, in the work by Tre-ston as contained in Bonpo Grub mtha' Material, p. 354, line 4 ff.; and compare Sga-ston's commentary, p. 407 ff., where the animal names are explained because they are particles the size of dust that can settle on the ends of hairs of those animals, that the sun ray particles [floating dust?] are the first humans can see): 

 

ro kung ma ming lo tar ni /

slas dur gug ge tig mi min /

gzugs dang ming dang dus mtha' ni /

rdul dang yi ge skad cig ma /[2]

ti pra sla tur dar ce ji /

zangs ti 'bo la lug ti hab /

[1] phra rab, [2] rdul dang [3] rdul phran dang /

[4] lcags [5] chu [6] ri bong [7] lug dang [8] glang /

nyi 'khor sla tur pra min ci /

pra mo zag tsi sni tse bran /

[9] nyi zer rdul dang [10] sro ma dang / +

[35] [11] shig la [12] nas bskyed bdun 'gyur sor /

pra tse ne cu bi nga tsa /

tsa ring bi nga weg ti ci /

sor tshigs nyi shu rtsa bzhi khru /

khru bzhi la ni gzhu 'dom gang /

da dog nga ra 'u glang gyin /

de dag lnga brgya rgyang grags yin /

de ni ti gyad ge dhim gyin /

de dag brgyad la dpag tshad gcig /

tog ci ra nis cu tse li /

tog tig ma min da yun ci /

skad tsam brgya dang nyi shu la /

skad cig ma ni yin par 'dod /

de na drug cu ti tang ni /

ti tang ti yu dzag sla sum /

de dag drug cu thang gcig ste /

thang yud zhag zla sum cu 'gyur /

zla ri nyi chu la lod tig /

rgyu gang ti par ti pre gyin /

zla ba bcu gnyis lo gcig ste /

gang po 'phar ni phyed phyed rtsi /

nang bcud kyi lus tshad dang tshe tshad lo zla bshad pa / srid pa'i mdzod phugs kyi bcud bstan pa'i gnas drug pa'o // + ??

 

 

The Gdags-pa has a similar system, but the Metal Particles and Water Particles are missing, and the order differs.  It has the following order: 1. rdul phran.  2. nyi zer rdul.  3. ri bong rdul.  4.  lug rdul.  5. glang rdul (these 'animal' names are not to be explained by the sizes of their hairs, etc.).  6. sro ma.  7. a ba ga na.  8. ba ti ka.  9. sor mo.  These likewise increase 7-fold at each step.

 

The Rgya-che Rol-pa has the following: 

1.  phra rab.

2.  rdul phra mo.

3.  rdul chung ngu.

4.  nyi zer rdul.

5.  ri bong rdul.

6.  lug rdul.

7.  glang rdul.

8.  sro ma.

9.  yungs 'bru.

10.  nas.

11.  sor mo.

 In this system also, each one is 7-fold the size of the preceeding one.  However, the minutest particle, the phra rab, has no size!

 

1.  14 sor mo

 = 1 khru (khru gang).

2.  4 khru

 = 1 gzhu 'dom (gzhu 'dom gang).

3.  500 of these 'dom

 = 1 rgyang grags.

4.  8 rgyang grags

 = 1 dpag tshad.

 

It comments that monasteries ought to be one rgyang grags above the village.

The Rgya-che Rol-pa comments that: gzhu stong la yul ma ga dha'i rgyang grags gcig dang rgyang grags bzhi la dpag tshad gcig tu bshad do.

 

 

Source:  Phur-bu-tshe-ring, Mdo kun las btus pa'i nang don rig pa'i tshig mdzod mu tig phreng ba, Bod ljongs mi dmangs dpe skrun khang (Lhasa 1994), p. 909:  gzhal byed bcu drug

1. rdul phra rab.

2. rdul phran.

3. lcags rdul.

4. chu rdul.

5. ri bong gi rdul.

6. lug rdul.

7. glang gi rdul.

8. nyi zer gyi rdul.

9. sro ma.

10. shig nas.

11. sor mo.

12. zheng bcas rim bzhi du 'gyur ba dang.

13. sor tshigs nyer bzhi la khru gang.

14. khru bzhi la 'dom gang.

15. 'dom lnga brgya la rgyang grags gcig.

16. rgyang grangs brgyad la dpag tshad gcig bcas mngon par mdzod du gzugs phra rags 'jal byed kyi tshad gzhi'o.

 

More discussion of weights & measures & money in S. Beyer, Classical Tibetan Language, p. 228.

 

Various Indian systems of measurement compared and discussed in Nandasena Mudiyanse, An Introduction to the Vaijayanta-Tantra, Annals of the Bhandarkar Oriental Research Institute, vol. 57 (1976) 167-174. 

 

JAMPEL KALDHEN ('Jam-dpal-bskal-ldan), Interest Rates in Tibet.  Tibet Journal 1 no 1 (1975) 109-112. Money, measurements.

 

The Kālacakra system (Orofino 1996, p. 137, etc.), says — in common with other Indian yoga traditions —  that an ordinary person breathes 21,600 times a day, and each breath lasts four seconds.  These correspond on the macrocosmic scale with the 21,600 ghaṭikās (Tib. chu tshod) (in 24-hour day there are 60, multiplied by the 360 days of the year = 21,6000). 

 

Bell, Religion, p. 104, note 6:  "One Tibetan span is the measure from the tip of the thumb to the tip of the second finger at full stretch."

 

Source: K. Mimaki's article in S. Karmay, ed., New Horizons in Bon Studies, p. 90:  "We know that a Buddhist yojana [dpag tshad] approximately corresponds to 7.3 km, which is half of an ordinary north Indian yojana."

 

Source: Illus. in Po-ta-la (1996) 190 are two measuring sticks and a plumbline.  sngar pho brang po ta la sogs bod kyi khang pa dang / mchod rten bzhengs skabs rgyug mtho la bsten nas 'dzugs skrun byed pa re / thig rgyug der rgyug mtho brgyad yod / rgyug mtho gang la tshon phyed bdun dang / tshon gang la skar ma bzhi / skar ma rer hon gnyis kyi rtsis gzhi bzung ba red / rgyug mtho gang la spyi khre'i skar ma nyi shu rtsa lnga dang mtshungs / rgyu rag ser gyi 'phyong rdo de bzo bkod thad drang blta ba'i yo byad zhig red.

 

Hartzell, Dissertation 737:  "As the slowest (known) planet to move through the heavens, Jupiter's 60-year cycle became the arbiter of time calculations—a cycle that appears to underlie our current time cycle of 60 seconds in a minute, and 60 minutes in an hour.  The Kālacakra and Trika Tantric systems share this rather curious and interesting system of time calculation that, while familiar to Indian astrologers and astronomers, is not commonly known about by modern scholars, even though it apparently derives from a Babylonian system.  During the course of 60 years, there are considered to be 21,600 days, since the ideal year for the Indian calendars was 360 days (60 x 360 = 21,600).  In a built-in system of macro-microcosmic mapking, it was also considered that a human being takes 21,600 breaths— consisting of an inhalation  & exhalation—during the course of one 24-hour day.  It turns out that a full prāna (inhalation & exhalation, prāna + upāna) lasts four seconds.  ...  this temporal unit of a four-second prāna ... goes back to practices of breath control used for chanting the Vedic mantras...

 

The 24-minute hour was called ghaṭikā, one 60th of a day.

 


 

***

 

 

 

Numbers (low numbers and numeric codes):

 

Word-symbols used in place of numbers found in back of Bod Rgya Shan-sbyar-gyi Shes-bya'i Rnam-grangs Kun-btus Tshig-mdzod:

 

0.  mkha.  thig.  stong pa.

1.  gzugs.  zla.  'od dkar.  bse ru.

2.  lag.  mig.  phyogs.  zung.  'khrig.  bgrod.  mtshem.  thabs shes.

3.  'jig rten.  yon tan.  me.  rtse.  tsha ba.  srid pa.

4.  mtsho.  rgya mtsho.  chu.  rkang.  rig byed.  bdud.  sde pa.  chu gter.

5.  'byung ba.  dbang po.  mda'.  phung po.  'dod yon.  nyer spyod.

6.  mtshams.  ro.  dus.  bro ba.  rgyan.  bzang po.

7.  thub pa.  drang srong.  ri.  res gza'.  rin chen.  rta.  sa 'dzin.

8.  klu.  sbrul.  gdengs can.  lho 'gro.  sreg pa.  bkra shis.  lha.  nor lha.  nyer sras.

9.  rtsa.  gter.  gza'.  bu ga (bug).  srin po.

10.  phyogs.  'jug pa.  stobs.  dge ba.  khro bo.  'byor ba.  sor mo.

11.  'phrog byed.  drag po.  bde 'byung.  dbang phyug.  byed pa.

12.  nyi ma.  khyim.  skye mched.  rten 'brel.

13.  'dod pa.  smyos byed.  myos byed.  lus med.  gdugs rim.  sna tshogs.

14.  yid.  ma nu.  srid pa.  shed bu.

15.  tshes.  nyin zhag.

16.  mi bdag.  rgyal po.

17.  -.

18.  nyes pa.  skyon.  khams.

24.  rgyal ba.  yul.

25.  de nyid.

27.  skar ma.  'khor lo.

32.  so.  gnyis skyes.

60.  chu tshod.  dbyug gu.

 

There is probably no need to stress the importance of "word numerals" for those working with Tantric texts.  

 

One early study of them is by Schlegel in "Mode of Expressing Numerals in the Sanskrit and Tibetan Languages,"  JASB 3 (Jan 1834) 1-8.  Numbers.


Here is a very convenient list from B.V. Subharayappa and K.V. Sarma, Indian Astronomy, A Sourcebook, Nehru Centre, Bombay 1985. Appendix V.

 

Bhūtasaṃkhyā - Word numerals used in Indian Mathematical texts

 

[0]

ananta, antarikṣa, abhra, ambara, ākāśa, kha, gagana, jaladharapatha, nabha, pūrṇa, bindu, randhra, viyat, viṣṇupada, vyoma, śūnya; all synonyms of 'Sky'.

 

[1]

abja, ādi, indu, ilā, urvarā, kalādhara, ku, kṣapākara, kṣiti, kṣmā, go, candra, jagati, tanu, dharaṇi, dharā, nāyaka, pitāmaha, pṛthvī, prāleyāṃśu, bhū, mahī, mṛgāṅka, rajanīkara, rūpa, vasudhā, vasundharā, vidhu, śaśadhara, śaśāṅka, śaśī, śītakara, śītaraśmi, śītāṃśu, śveta, sudhāṃśu, soma, himakara, himagu, himāṃśu; all synonyms of 'Earth' and 'Moon'.

 

[2]

akṣi, ambaka, ayana, aśvin, īkṣaṇa, oṣṭha, kara, karṇa, kuca, kuṭumba, gulpha, cakṣu, jaṅghā, jānu, dasra, dṛṣṭi, dvandva, dvaya, naya, nayana, nāsatya, netra, pakṣa, bāhu, bhuja, yama, yamala, yugala, yugma, ravicandrau, raviputra, locana; all synonyms of 'Eye' and 'Hand'.

 

[3]

agni, anala, kāla, kṛśānu, guṇa, gṛha, jvalana, tapana, trikāla, trigata, triguṇa, trijagat, trinetra, dahana, pāvaka, pura, bhuvana, ratna, rāma, loka, vaiśvānara, vahni, sahodarāḥ, śikhin, haranetra, hutabhuk, hutabhuj, hutāśa, hutāśana, hotṛ; all synonyms of 'Fire' and 'Worlds'.

 

[4]

abdhi, ambudhi, ambhodha, ambhodhi, ambhonidhi, arṇava, āya, āśrama, udadhi, kaṣāya, kṛta, kendra, koṣṭha, gati, ghana, caraṇa, jala, jaladhi, jalanidhi, turya, diś, payodhi, payonidhi, praṇimnageśa, bandhu, yuga, lavaṇoda, varṇa, vāridhi, viṣanidhi, veda, śruti, samudra, salilākara, sāgara, sukha; all synonyms of 'Ocean'.

 

[5]

akṣa, artha, indriya, iṣu, karaṇīya, tattva, parva, pavana, pāṇḍava, prāṇa, bāṇa, bhāva, bhūta, mahābhūta, rāga, ratna, viṣaya, vrata, śara, śastra, sāyaka; all synonyms of 'Arrow'.

 

[6]

aṅga, ari, ṛtu, kāya, kāraka, kumāravadana, khara, tarka, darśana, dravya, māsārdha, rasa, rāga, lekhya, ṣaṇmukha, śāstra.

 

[7]

aga, acala, atri, adri, aśva, ṛṣi, kalatra, giri, graha, chandaḥ, tattva, turaga, dvipa, dhātu, dhī, naga, pannaga, parvata, bhaya, bhūbhṛt, mātṛka, muni, yati, vāji, vāra, vyasana, śaila, svara, haya; all synonyms of 'Horse' and 'Mountain'.

 

[8]

anīka, anuṣṭubha, ahi, ibha, karman, kuñjara, gaja, takṣa, tanu, danti, dik, diggaja, durita, dvīpa, dvirada, dhī, nāga, puṣkarin, bhūti, maṅgala, mada, mātaṅga, mati, vasu, sarpa, siddhi, sindhura, hastin; all synonyms of 'Elephant' and 'Serpent'.

 

[9]

aṅka, anilāhva, upendra, keśava, gīr, go, graha, chidra, tārkṣyadhvaj, durgā, dvāra, nanda, nidhi, padārtha, randhra, labdha, labdhi.

 

[10]

avatāra, aṅgulī, āśā, kakubh, karman, dik, diś, diśā, paṅkti, rāvaṇaśira.

 

[11]

akṣauhiṇī, īśa, īśvara, bharga, bhava, mahādeva, mṛḍa, rudra, śaṅkara, śiva, śūlin, svargeśa, hara; all synonyms of god 'Śiva'.

 

[12]

arka, āditya, ina, tīkṣṇāṃśu, dinanātha, dinapa, divākara, dyumaṇi, bhānu, bhāskara, maṇḍala, mārtaṇḍa, māsa, ravi, rāśi, vyaya, sūrya; all synonyms of 'Sun'.

 

[13]

aghoṣa, atijagatī, karaṇa, kāma, viśva, viśvedevāḥ.

 

[14]

indra, manu, loka, vidyā, śakra, śarva; all synonyms of 'Indra'.

 

[15]

ahan, ghasra, tithi, dina, pakṣa.

 

[16]

aṣṭi, kalā, nṛpa, bhūpa, bhūpati.

 

[17]

atyaṣṭi.

 

[18]

dhṛti, purāṇa, vidyā.

 

[19]

atidhṛti.

 

[20]

kṛti, nakha.

 

[21]

utkṛti, prakṛti, mūrchanā, svarga.

 

[22]

kṛti, jāti.

 

[23]

vikṛti.

 

[24]

arhat, gāyatrī, jina, siddha.

 

[25]

tattva.

 

[26]

utkṛti.

 

[27]

uḍu, nakṣatra, bha; all synonyms of 'Star' and 'Asterism'.

 

[32]

danta, rada; all synonyms of 'Teeth'.

 

[33]

amara, tridaśa, deva, sura, surādhipa; all synonyms of 'Gods'.

 

[40]

naraka.

 

[48]

jagati.

 

[49]

tāna.

 

 

The following list is from Thor-bu website (Peter Szantos):

 

SATURDAY, MARCH 22, 2008

Bhūtasaṃkhyā in the Kālacakra

 

 

Another entry on this blog deals with a fairly comprehensive list of 'word-numerals'. Here is the list compiled by Biswanath Banerjee in his "Critical Edition of Śrī Kālacakratantra-rāja", AS Calcutta 1984. (Appendix I). Words not occurring within the previous list are followed by an asterisk.

 

 

[0]  śūnya, ākāśa, ambara, kha.

 

[1]  candra, indu, śaśī.

 

[2]  akṣi, nayana, netra, adhara*, kara, yuga*, ayana, gati*.

 

[3]  agni, śikhin, vahni, guṇa, kāla, loka.

 

[4]  veda, yuga, abdhi, jaladhi.

 

[5]  bhūta, iṣu, bāṇa.

 

[6]

ṛtu, rasa.

 

[7]

muni, vijana*, adri, giri, śaila.

 

[8]

vasu, ahi, phaṇi*, nāga, uraga*.

 

[9]

graha, randhra, mūla*.

 

[10]

dik.

 

[11]

rudra, īśa, hara.

 

[12]

āditya, ravi, sūrya, arka, rāśicakra*.

 

[13]

madana*.

 

[14]

bhuvana*, manu.

 

[15]

dina, tithi, śaśikalā*.

 

[16]

nṛpa.

 

[18]

doṣa*.

 

[24]

jina.

 

[25]

tattva.

 

[28]

bhūmi*.

 

[30]

māsa*.

 

[32]

danta.

 

[60]

ghaṭi*.

 

posted by PDSz | 

 

 

 

 

Note: The list found in Csoma de Koros, A Grammar of the Tibetan Language in English, Baptist Mission Press (Calcutta 1834; reprint New Delhi 1983), pp. 155-157 is not different enough from this list to warrant copying it out.  It says it is based on the Vaiḍūrya Dkar po.

 

The following (he makes reference to the collection of synonyms by Dpal-khang Lo-tsā-ba) are from Per Kværne, 'Chronology', p. 220, note. 11: 

11.  drag po.

12.  nyi, nyi ma.

13.  'dod, 'dod pa.

14.  yid.

15.  tshes.

16.  rgyal po.

 

The following based on the Chandoratnākara by Ratnākaraśānti (according to Michael Hahn, Sanskrit Metrics as Studied at Buddhist Universities, Adyar Library Bulletin, vol. 52 (1988), pp. 34-35):

 

1. rūpa.

2. yama.

3. agni (also, dahana).

4. veda, abdhi (also, udadhi, samudra, salilanidhi).

5. śara, indriya, viṣaya (also, akṣa, iṣu, karaṇa).

6. ṛtu, rasa (also, svāda).

7. loka, aśva, muni (Tib. version adds svara; also, turaga, haya).

8. vasu, aga, madana.

9. randhra, graha (also, kha).

10. diś (also, āśā).

11. rudra (also, hara).

12. arka (also uṣṇagu, taraṇi, ravi).

13. bhuvana.

14. manu.

 

Source:  Lienhard, Life & Religious Belief as Reflected in Nepalese Folksongs, Studia Asiatica, vol. 3 (2002) 11-18, at p, 18:

 

Newars use symbolic words/encodements for numbers in dating their songs (the order is right-to-left). 

 

forest = 5.

elephant = 8.

jewel = 9.

--------------

 

This list of gambling numbers in the language of the Ma-sang[s] is from N.Norbu, Drung De'u & Bön, p. 15, and the added note.  The real spellings are given second, if they differ.  (Since two dice are used, the minimum possible result is '2', so there is no '1')

2. para.  pa ra.

3. sug. 

4. dzig.

5. kha.

6. ndrug.  'brug.

7. ri.

8. sha.

10. chu.

11. thog, thoge.  thog.  tho ge.

12. njam.  'jam.  OR chala.  ca la.

 

Other source for dicing numbers, see TS7 II 1056:  par ra, sug, tshigs (zing, zig, tsig), kha, drug (lug), rig (ri, tig, ting), sha (skya), dgu (sgug), chu, rdog (thog), 'jang.

 

 

 

High Numbers

 

One source for the following high numbers is:  Mchims 'Jam-pa'i-dbyangs, Mdzod 'Grel Mngon-pa'i Rgyan, Khrung-go Bod-kyi Shes-rig Dpe-skrun-khang (Beijing 1989), p. 328. 

 

There is a very important discussion in the Pruden tr. of the Abhidharmakośa, vol. 2, pp. 543-544 (note no. 507).  It appears that Vasubandhu already recognized that there were 8 missing numbers in the list of 60.

 

Another source is Anton Schiefner, Über die hohen Zahlen der Buddhisten, Mélanges Asiatiques [St. Petersburg], vol. 4 (1862), pp. 629-648 (have xerox of copy in Kern Institute library). 

 

The 60th 'place' in the number system is that called Grangs-med, and this is the one that measures the amount of time in the phrase Grangs-med gsum (3 'limitless' Kalpas).

 

Another interesting source for high numbers is the Mdzod-phug comm. by Sga-ston, p. 225.  This Bon source has a complete list of 60.

 

High numbers (source is Bu-ston, Collected Works, vol. 24, pp. 726-7):

 

1. gcig

1

2. bcu

10

3. brgya

100

4. stong

1,000

5. khri

10,000

6. 'bum

100,000

7. sa ya

1,000,000

8. bye ba

10,000,000

9. dung phyur

100,000,000

10. ther 'bum

1,000,000,000

11. ther 'bum chen po

10,000,000,000

12. khrag khrig

100,000,000,000

13. khrag khrig chen po

1,000,000,000,000

14. rab bkram

10,000,000,000,000

15. rab bkram chen po

100,000,000,000,000

16. gtams

1,000,000,000,000,000

17. gtams chen po

10,000,000,000,000,000

18. dkrigs

100,000,000,000,000,000

19. dkrigs chen po

1,000,000,000,000,000,000

20. mi 'khrugs pa

10,000,000,000,000,000,000

21. mi 'khrugs pa chen po

100,000,000,000,000,000,000

22. khyad phyin

1,000,000,000,000,000,000,000

23. khyad phyin chen po

10,000,000,000,000,000,000,000

24. pang steng

100,000,000,000,000,000,000,000

25. pang steng chen po

1,000,000,000,000,000,000,000,000

26. ded 'dren

10,000,000,000,000,000,000,000,000

27. ded 'dren chen po

100,000,000,000,000,000,000,000,000

28. mtha' snang

1,000,000,000,000,000,000,000,000,000

29. mtha' snang chen po

10,000,000,000,000,000,000,000,000,000

30. rgyu rig

100,000,000,000,000,000,000,000,000,000

31. rgyu rig chen po

1,000,000,000,000,000,000,000,000,000,000

32. 'od mdzes

10,000,000,000,000,000,000,000,000,000,000

33. 'od mdzes chen po

100,000,000,000,000,000,000,000,000,000,000

34. dbang po

1,000,000,000,000,000,000,000,000,000,000,000

35. dbang po chen po

10,000,000,000,000,000,000,000,000,000,000,000

36. legs phyin

100,000,000,000,000,000,000,000,000,000,000,000

37. legs phyin chen po

1,000,000,000,000,000,000,000,000,000,000,000,000

38. rtogs 'gro

10,000,000,000,000,000,000,000,000,000,000,000,000

39. rtogs 'gro chen po

100,000,000,000,000,000,000,000,000,000,000,000,000

40. 'bying rdul[3]

1,000,000,000,000,000,000,000,000,000,000,000,000,000

41. 'bying rdul chen po

10,000,000,000,000,000,000,000,000,000,000,000,000,000

42. rgya rtags

100,000,000,000,000,000,000,000,000,000,000,000,000,000

43. rgya rtags chen po

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000

44. stobs 'khor

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000

45. stobs 'khor chen po

100,000,000,000,000,000,000,000,000,000,000,000,000,000,000

46. brda shes

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

47. brda shes chen po

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

48. rnam 'byung

100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

49. rnam 'byung chen po

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

50. stobs mig

10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

51. stobs mig chen po

100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

52[60?]. grangs med pa

1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

 


As Bu-ston notes, there seem to be 8 missing (since the total is supposed to be 60 ending with grangs med pa).  Mchims seems to have a similar problem.  Mchims suggests adding 8 more 'places' before the last 'place', Grangs med (anywhere you like after the Dung phyur!).  This suggests how little practical value Mchims, unlike modern astrophysicists, placed on high numbers.

 

Another longer list of high numbers, as given in the Avatamsaka Sutra, is found in the Mahāvyutpatti. nos. 7697-7820.

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Notes, links and bibliography to end with:

For more on the symbolic number-words, see now

Edward Henning, “Symbolic numbers in Kālacakra literature

For an excellent discussion of Tibetan measurements in German (and nicely illustrated), see this essay entitled, “Tibetische Maßeinheiten.”

For mathematics, see yet other works of Dieter Schuh.

Following are a couple of old but still interesting writings on the subjects of measurements and numbers.

DECOURDEMANCHE, J.A., Traité des monnaies, mesures et poids anciens et modernes de l'Inde et de la Chine (P. Leroux 1913).

FLEET, JOHN FAITHFULL, Dimensions of Indian Cities and Countries.  Journal of the Royal Asiatic Society (July 1907), pp. 641-656.

GUPTA, RADHA CHARAN Circumference of the Jambudvîpa in Jaina Cosmography.  Indian Journal of the History of Science, vol. 10, no. 1 (1975).

JACQUET, E., Mode d'expression symbolique des nombres employé par les Indiens, les Tibétaines et les Javanais.  Journal Asiatique, vol. 16 (1835), pp. 5-42.

KLAFKOWSKI, PIOTR, Hand and Finger Measurements in Tibetan.  Lingua Posnaniensis, vol. 26 (1983), pp. 85-97.  

MITRA, SARAT CHANDRA, A Note on the Tibetan Method of Computing Distance by Means of Tea-cups.  Journal of the Asiatic Society of Bengal, vol. 14 (1927-31), pp. 798-799.

OSMASTON, HENRY & Tashi Rabgyas, Weights and Measures Used in Ladakh.  Contained in: J. Crook & H. Osmaston, eds., Himalayan Buddhist Villages (Bristol 1994), pp. 121-138.

SCHLEGEL, GUSTAV, Mode of Expressing Numerals in the Sanskrit and Tibetan Languages.  Journal of the Asiatic Society of Bengal, vol. 3 (Jan 1834), pp. 1-8.

THOMAS, F.W., Ancient Indian Weights.  Journal of the Asiatic Society of Bengal, vol. 33, no. 3 (1864), pp. 251-266; vol. 34, nos. 1-2 (1865), pp. 14-27, 51-70.  Measurements.



[1]The 3-vol. dictionary defines 'bo as follows:  'bru rigs tshad 'jal byed shing bzos lcags shan can.  This refers to a wooden measuring box banded with metal.

[2]Abhidharmakosa (Pruden tr.), p. 474 [i.e, Chap. 3, verse 85b-c]:  an atom, a syllable, and an instant is the limit of matter, of words, and of time.

[3]byed rdul in Mchims.

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