ඳ೩ϣ Bodhi Field !ܟဂࡆվౢߎй́Ĉ How Large is One Asamkhyeya?

ͦͷ!ܔϕ!͚0ᛋ WRITTEN AND TRANSLATED BY BHIKSHU JIN YONG

Δ samkhyeya is a word that appears often in theဲټ෼ऱත֮נቖચਢ۵ඒᆖࠢխᆖൄ ऱ ABuddhist texts. For example, Shakyamuni Buddhaܖ۵ਢᆖመԿՕॳቖચ؍ۋΚᤩ૫ڕॳᤝ ጩૠΔྥۖΔ،ਢ is said to have practiced for three great asamkhyeya kalpasױ۵ऱΖॳቖચრ᤟լګଥ۩թ ;’టऱՕࠩྤऄૠጩႯΛԭ壂অΕ۵٠֗Soothill/ before becoming a Buddha. Asamkhyeya means ‘incalculableܡ however, is it really too large to calculate? Ding Fubao (ԭ ࠢઃᎅԫॳቖચ࿛࣍ԫՏᆄᆄᆄڗHodous\2^࿛۵ᖂ and Soothill/Hodous’ [1] Buddhist ,(壂অ), Foguang (۵٠ 2158*ΔۖBuddhism A to Z\3^!ঞᎅ( !ᆄᆄᆄᆄᆄ٢ Dictionaries all state that one asamkhyeya is equal to 10 to the 6: ,ΖThomas Cleary! power of 47 (1047), while A to Z [2], among others* 21( !ֱڻ:ԫॳቖચ࿛࣍21ऱ6 խ֮៬᤟੡૎֮ऱFlower Ornament states 10 to the power of 59 (1059). The Flower OrnamentطലԶԼဎᣤ ԫॳቖચ࿛࣍21ऱ3/14y! Scripture [3] (Avatamsaka ) translated from Chinese intoנૠጩהScripture\4^Δ (Ζߒຶᓴ English by Thomas Cleary states an incalculable (asamkhyeya*591-931-387-762-621-416-748-331-696-97:-21313( !ֱڻ2143 ਢᓴॺࡋΛ is equal to 10 to the power of 2.03 x 1032 (10202,986,585,220,637,305,510 ?୛Օஃխ᤟ऱόॳḛሒᗣଟ ,651,276,820,480). Which one is correct thenخ׈ᘣဆ៳ထΕ [4] [5] ԫྤ塒ᑇ ’s Abhidharmakosha , * talks about 60ڶॐᓵύ\5^รԿ։ܑ׈঴ᓵ֗քԼᑇΚό ;੡ՏΔԼՏ੡ numbers from 1 up, with 10-fold consecutive increasesۍΔԼۍࡨ੡ԫΔԫԼ੡ԼΔԼԼ੡ however, eight numbers are missing from the list. The Chinese ᆄΔԼᆄ੡੖ՐΔԼ੖Ր੡৫੖ՐΔԼ৫੖Ր੡ version translated by the Venerable Xuan Zang makes it very ᑇխݱ؈塒ڼଟિΞԼՕၐ᧤ᣰ੡ॳቖٞળΖ࣍ clear that asamkhyeya is the largest one in the list of 52 numbers [ΔԼᆄ੡੖ՐΔॳቖચਢ [6], one asamkhyeya is thus equal to 1051. Mahavyutpatti [vۯքԼᑇ੡ԼၞڼԶΖύ ࢬ٨ऱնԼԲᑇխऱ່ՕᑇΔਚԫॳቖચ੡2162Ζ further lists these eight missing numbers (also see նԼԲᑇin ᆠ Foguang Dictionary). However, one asamkhyeya may be equalټᓵΔݱ؈ऱԶᑇຟֺॳቖચՕΔ៬᤟ڼ௅ᖕ .[ጠΰߠ۵٠۵ᖂ᢯ࠢμն to 1059, if all eight missing numbers are smaller than it [5],[8ټԶᑇऱڼנ٨ޓ^Օႃ\8 ,ݱ؈ऱԶᑇຟֺॳቖચ՛\6^-\9^Δ Or one asamkhyeya may be somewhere between 1051 and 1059ڕԼԲᑇνයαΖ .ঞԫॳቖચ੡216:ΖངߢհΔԫॳቖચտ࣍2162֗ depending on where the eight missing numbers are located [9] հၴΔီݱ؈ԶᑇऱՕ՛ۖࡳΖ On the other hand, Roll 10 of the Gandavyuha:216 translated from Sanskrit into Chinese by Prajna (768 C.E.) ៬᤟ऱ؄Լဎ !*ڣΔ౳ૉऄஃ! )۫ց879ۖྥ lists 144 numbers and asamkhyeya is the 125th one. It states: Δॳቖચਢร236ՕᑇΖڗ255ଡᑇڶᣤ\:^รԼ࠴٨ “100,000 makes one laksha, 100 lakshas makes one koti. ੖ՐۍՏ੡ԫ੖ՐΙԫۍՀΚόࢬᘯΚڕᆖ֮ᆏᙕ A koti of kotis makes one ayuta. An ayuta of ayutas makes ੡ one nayuta…” Hence 1 laksha equals to105, 1 koti equalsڍॳൌڍΙॳൌڍ੡ԫଟિΙଟિଟિ੡ԫॳൌ ԫ੖Ր࿛࣍216-!ԫଟ to107, 1 ayuta equals to1014, 1 nayuta equals to 1028…, and1נૠጩױڼطΞΞύהطԫ߷

42 ܛࣣඳ೩ঔ!!˟ŒŒˣѐȈ˘͡ ඳ೩ϣ Bodhi Field

8 25 39 37 74,436,800,0 ੡21 ΞΞ-ԫ asamkhyeya equals to 10 to the power of 7.44 x 10 (10הط੡!21 -!ԫ߷ڍિ੡21 -!ԫॳൌ .Ϟ, far greater than Kosha’s numbers(00,000,000,000,000,000,000,000,000,000 1-111-111-111-911-547-2185(!ֱڻॳቖચ࿛࣍21ऱ8/55!y!2148 [10] 11-111-111-111-111-111-111*Ϟ-!੷Օ࣍ଟॐᓵऱᑇଖΖ Chapter 30 (Roll 45) of the with 80 ៬᤟ऱԶԼဎ rolls translated from Sanskrit into Chinese by Shikshananda*ڣኔՐᣄॲऄஃ)۫ց763.821 (652-710 C.E.) lists 124 numbers+ and asamkhyeya is the 105th Δॳቖચ,ڗ235ଡᑇڶᣤ\21^խऱรԿԼॳቖચ঴٨ one. It starts the same as Gandhavyuha: “100 lakshas makes ੖Ր੡ԫଟۍՀΚόԫڕਢร216ՕᑇΖᆖ֮ᆏᙕ one koti. A koti of kotis makes one ayuta. An ayuta of ayutas makes one nayuta…” except it does not specify how ط੡ԫ߷ڍॳൌڍΔॳൌڍિΔଟિଟિ੡ԫॳൌ ࠢ࿛ large a laksha is. If we also take 1 laksha as 100,000 (FoguangڗΞΞύ௅ᖕ؄ԼဎᣤΕଟॐᓵΕ۵٠۵ᖂה ,؄ԼဎᣤΔԫଟિ࿛࣍ Buddhist Dictionary, Kosha[5]*, [6]), then 1 koti equals to107ٵڕԫ੖ՐਢԼᆄ)216*Δঞ ࿛࣍2139ΞΞۖԫॳ 1 ayuta equals to1014…, and 1 asamkhyeya equals to 10 to theהط࿛࣍2125-ԫ߷ڍ218-ԫॳൌ power of 7.10 x 1031 (1070,988,433,612,780,800,000,000,000,000,000). Cleary -111-911-891-723-544-99:-2181(!ֱڻቖચঞ࿛࣍21ऱ8/21y2142 ՛࣍!Cleary!ऱૠጩଖΔ translated this Sutra into English; however, he took 1 laksha asאຍᑇଖհࢬ!/*111-111-111-111 10 20 ੡! Cleary! ࠷ԫ੖Ր੡2121-! ׊ԫଟિ੡ό੖Ր 10 , and 1 laksha of lakshas as 1 koti or 10 . Hence, Clearyڂਢ came up with 10 to the power of 2.03 x1032 (10202,986,585,220,637,305 ੖Րύܛ2131Ζ ,510,651,276,820,480) for an incalculable (asamkhyeya), greater than the ៬᤟ऱքԼ*ڣ:۵቎ၐॲᢅऄஃ)۫ց46:.53 present calculation. ΖࠡรԲԼն֨׆ءऱխ᤟ڰဎᣤ\22^এဎᣤᆖ່ Chapter 25 (Roll 29) of the Avatamsaka Sutra[11] ΔፖԶԼဎᣤԼ with 60 rolls, the earliest Chinese translation, translatedڗ233ଡᑇڶဆ៳ംॳቖચ঴ঞ٨ ։ઌ२Δႛ౒֟ร212່֗Օऱ235ႈΖॳቖચਢร by Buddhabhadra (359-429 C.E.) lists 122 numbers and ԫࣅරΔ asamkhyeya is the 104th one. The list is very similar to thatټՏۍՏۍՀΚόڕ215ՕᑇΖᆖ֮ᆏᙕ translated by Shikshananda except it is short of the 101st and the ڂΞΞύהطԫ߷ټ᧢ԫլ᧢Δլ᧢լټࣅරࣅර ࿛࣍2151Δ greatest (124th ) terms. It also takes 1 laksha as 105; however, 1הطΔԫࣅර! )ଟિ*࿛࣍2121Δԫ߷ڼ 10 laksha of lakshas is 1 koti or 10 . Hence 1 asamkhyeya equals 31-:23-:11-135-817-2161(ֱڻԫॳቖચঞ࿛࣍21ऱ6/18y2142 31 50,706,024,009,129,200,000,000,000,000,000 1-111-111-111-111-111*-!ฃ՛࣍ԶԼဎᣤऱᑇଖΖ to 10 to the power of 5.07 x 10 (10 ), smaller than the above calculations. ԲଡൄߠऱՕף।ԫ੡Ղ૪ಘᓵऱኴ૞Δ؆ Table 1 summarizes the above discussion plus a couple of ᑇΖ extra commonly seen numbers. ౨ऱ່՛ଖਢ2162Δ່ױጵՂࢬᓵΔԫॳቖચ In conclusion, one asamkhyeya may be as small as 1051 85-547-911-111-111-111-111-111-111-111-111-111-111 Օঞ੡21 Δီࠉᖕ or as large as 1074,436,000,000,000,000,000,000,000,000,000,000, or somewhere ऱᆖࠢۖࡳΖ؄Լဎᣤխऱᑇ٨່੡ᇡጐΔՂ૪ऱ in between, depending on the text source. The upper bound ᆖૠጩۖ൓Ζ is based on Gandavyuha whose sequence of numbers is mostڼਢᖕܛՂૻଖ extensive among all sources.

Notesု; * On p. 480 of Pruden’s translation, the sixteenth place should read the sixtieth place instead. Also on the same page, laksha (1000,000) should read 100,000 instead. Ϟ!This number is indeed too large to be calculated with a personal computer as rightly pointed out by Mochizuki (IABS 2008); however, its power can still be accurately computed. נᆖร65࠴ٍለ១ฃऱ٨ء .An abbreviated list of these numbers also appears in Roll 65 (Chap 39) of this Sutra + Ζڗຍࠄᑇ!!!

NOVEMBER 2008 VAJRA BODHI SEA 43 ඳ೩ϣ Bodhi Field

Table 1. Comparisons of Some Numbers from Kosha and Various Avatamsaka (numbers listed are powers of 10) *ֱڻኡ̙૰֌ᇳࣂ̜ͦྴ!)।٨ଖ੡21ऱۼĂයᚐག˫჋ᛋύ̓࣋˗ܐ ኡ ΰȇයᚐ ̰ȇයᚐ ˢȇයᣤۼ࣋ Kosha Gandavyuha 60 Rolls Avat. 80 Rolls Avat. ੖Ր!Laksha 5 5 5^ 5* ଟિ!Koti 7 7 10 7 Nayuta 11^^ 28 40 28 הط߷ ॳቖચ!Asamkhyeya 51-59 7.44 x 1037 5.07 x 1031 7.10 x 1031 ᎅ!Ineffable -- 4.88 x 1042 3.32 x 1036 4.65 x 1036ױլ ᎅ!!!!!! -- 1.95 x 1043 1.33 x 1037 ** 1.86 x 1037ױᎅլױլ Ineffable Ineffable ՏύۍՏڍCalled ߷ൌ^^ ᎅ᠏᠏ύױᎅ᠏᠏

:ᇷறەReferences ೶ 1. Soothill and Hodous’ Dictionary of Chinese Buddhist Terms. 2. Buddhism A to Z, compiled by Ronald B. Epstein, Burlingame, CA: Buddhist Text Translation Society, 2002. 3. The Flower Ornament Scripture, Book 30 The Incalculable, p.201, vol.2, translated by Thomas Cleary, Boston: Shambhala, 1986. 4. Abhidharmakosha, Roll 12 (Chapter 3), by Vasubandhu, translated into Chinese by Xuan Zang.πॳḛሒᗣଟॐ .୛Օஃխ᤟. CBETA, T29, no. 1558, p. 63, b13-c12خᓵρ࠴23τ։ܑ׈঴รԿυ-!׈ᘣဆ៳ທΕ!! 5. Abhidharmakoshabhasyam of Vasubandhu, vol. 2, Chap. 3, English translation by Leo M. Pruden, Berkeley, CA: Asian Humanities Press, 1991. 6. A Manual of by W.M. McGovern, pp.39-41, London: Kegan Paul, Trench, Trubner, 1923, reprinted by Chinese Materials Center, San Francisco, 1977. ࠇຟ֮ءֲ!-ᆠՕႃټBon-Zo-Kan-Wa Shiyaku taiko Honyaku myogi taishu, by Sakaki Ryozaburo et al., ៬᤟ .7 !!ઝՕᖂ៲ठ- No. 8041-8, Tokyo: Suzuki Gakujutsu Zaidan, 1962. 8. by W.R. Kloetzli, p.117, Delhi: Motilal Banarsidass, 1983. Gandavyuha, Roll 10, translated by Prajna. πՕֱᐖ۵ဎᣤᆖρ)؄Լဎᣤ*!࠴21!τԵլ৸ᤜᇞๅቼ੺ཏᔃ .9 .ᣋ঴υ/CBETA, T10, no. 293, p. 704, c28-p. 706, a12۩!! Avatamsaka Sutra, Roll 45 (Chapter 30), translated by Shikshananda. πՕֱᐖ۵ဎᣤᆖρ!)ԶԼဎᣤ*!࠴ .10 !!!56τॳቖચ঴รԿԼυ. CBETA, T10, no. 279, p. 237, b15-p. 238, b6. !!!:Avatamsaka Sutra, Roll 29 (Chapter 25), translated by Buddabhadra. πՕֱᐖ۵ဎᣤᆖρ!)քԼဎᣤ*!࠴3 .11 .τ֨׆ဆ៳ംॳቖચ঴รԲԼնυ. CBETA, T09, no. 278, p. 586, a13-c15!!

44 ܛࣣඳ೩ঔ!!˟ŒŒˣѐȈ˘͡