Prarambhik Jain Grantho me Bijganit

Author(s): Mukutbiharilal Agarwal
Publisher: Z_Deshbhushanji_Maharaj_Abhinandan_Granth_012045.pdf
Catalog link: https://jainqq.org/explore/211448/1

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prArambhika jaina granthoM meM bIjagaNita DaoN0 mukuTabihArI lAla agravAla 'sthaanaaNg-suutr'| (300 I0 pU0 lagabhaga) meM ajJAta rAzi ke lie 'yAvat-tAvat' zabda prayoga kiyA hai| 'uttarAdhyayanasUtra (lagabhaga 300 I0 pU0) meM jJAta athavA ajJAta rAzi kI ghAta ke lie prAcInatama hindU nAma upalabdha hote haiM / isameM dUsarI ghAta (arthAt ) ke lie 'varga' tIsarI ghAta (arthAt as) ke lie 'dhana', cauthI ghAta (arthAt a') ke lie 'varga-varga' [jisakA artha hai varga kA varga arthAt (a2)], chaThI ghAta (arthAt a) ke lie 'ghana varga' [arthAt (a)] tathA bArahavIM ghAta (arthAt al) ke lie 'ghana-varga-varga' [arthAt {(a)}] zabda prayoga kiye gaye haiN| ina zabdoM kI racanA meM siddhAnta (am) = amxm kA prayoga kiyA gayA hai / isa granya meM tIna se adhika viSama ghAta ke lie koI zabda nahIM milatA / parantu bAda ke granthoM meM pAMcavIM ghAta (arthAt a) ke lie 'varga ghama ghAta' (arthAt a.xas), sAtavIM ghAta (arthAt ') ke lie 'varga-varga ghana ghAta' (arthAt a.xaxas) Adi zabda milate haiN| isameM ghAta-siddhAnta (arthAt anxa = am+n) kA prayoga hai| isase spaSTa hai ki usa samaya nimna ghAta siddhAnta jJAta the / (1) (am) =amxm (2) axa=am+m 'anuyogadvArasUtra' meM, jo IsA-pUrva meM likhA huA grantha hai, ucca ghAtoM ke lie, cAhe ve pUrNAMka hoM athavA bhinnAtmaka, vizeSa zabda milate haiM / isa grantha meM kisI rAzi a ke prathama varga kA Azaya se hai, a ke dvitIya varga se Azaya (ar)=a' aura a ke tRtIya varga kA Azaya [(a)"] ==as se hai| isI prakAra aura Age kI ghAtoM ke lie hai| samAnyata: a ke veM varga kA Azaya alxaxp. . . . . . . . . // bAra =an hai| isI prakAra ke prathama vargamUla kA Azaya hai| a ke dvitIya vargamUla kA Azaya Vira)= hai| sAmAnyata: a kA n vA vargamUla al" hai / cihnoM ke niyama-gaNitasArasaMgraha' meM ghana aura RNa-cihnoM ke viSaya meM niyama isa prakAra milatA hai|" : "ghanAtmaka aura RNAtmaka rAzi ke jor3ane para prApta phala inakA antara hotA hai| parantu do RNAtmaka athavA do ghanAtmaka rAziyoM kA yoga kramazaH RNAtmaka aura ghanAtmaka rAzi hotA hai|" ghaTAne ke samaya cihnoM ke bAre meM gaNitasArasaMgraha' meM niyama isa prakAra haiM-gakimI dI haI saMkhyA meM se ghanAtmaka rAzi ghaTAne ke lie use RNAtmaka kara dete haiM, aura RNAtmaka rAzi ghaTAne ke lie use ghanAtmaka kara dete haiN| isake bAda donoM ko jor3a lete haiN|" guNA karate samaya cihnoM ke bAre meM isa grantha meM niyama isa prakAra hai-'do RNAtmaka athavA do ghanAtmaka rAziyAM, ekadUsare se guNita karane para, ghanAtmaka rAzi utpanna karatI haiM, parantu do rAziyA~, jinameM eka ghanAtmaka tathA dUsarI RNAtmaka ho, ekadUsare se guNA karane para RNAtmaka rAzi utpanna karatI haiN|" 1. sthAnAMga sUtra , sUtra 747 2 uttarAdhyayana sUtra, adhyAya 30, sUtra 10-11 3. anuyogadArasava, sUtra 142 4. gaNitasArasaMgraha, adhyAya 1, gAthA 50-51 5. bahI, adhyAya 1, gAthA 0 (ii) 6. vahI, adhyAya 1, gAthA 51 7. vahI, adhyAya 1, gAthA 50 (i) jaina prAcya vidyAe~

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bhAga ke sambandha meM, mahAvIrAcArya ne 'gaNitasArasaMgraha' meM cihnoM ke bAre meM nimnalikhita niyama diyA hai-"do RNAtmaka athavA do ghanAtmaka rAziyA~ eka-dUsare se bhAjita hone para ghanAtmaka rAzi utpanna karatI haiM, parantu do rAziyA~, jinameM eka ghanAtmaka aura dUsarI RNAtmaka ho, eka-dUsare se bhAjita karane para RNAtmaka rAzi utpanna karatI haiN|" varga tathA vargamUla jJAta karate samaya cihnoM ke viSaya meM AcArya mahAvIra nimnalikhita niyama kA ullekha karate haiM"dhanAtmaka athavA RNAtmaka rAzi kA varga ghanAtmaka hotA hai, tathA usa varga rAzi ke vargamUla kramaza: ghanAtmaka aura RNAtmaka hote haiM / cUMki RNAtmaka rAzi dekhane meM hI avarga hai, isalie RNAtmaka rAzi kA koI vargamUla nahIM hotA / " samIkaraNa ke prakAra - samIkaraNoM ko cAra bhAgoM meM vibhakta kiyA gayA hai| (1) eka varNa samIkaraNa, jo kevala ekaghAtIya hote haiN| inheM 'yAvat-tAvat' bhI kahate haiN| dvighAtIya samIkaraNa, jinheM varga samIkaraNa kahate haiN| aneka varNa samIkaraNa, jinameM aneka varSoM kA prayoga hotA hai / bhAvita samIkaraNa, jisameM do varNoM ke guNana kA prayoga hotA hai| eka varNa samIkaraNa- aise samIkaraNoM ko jaina sAhitya meM yAvat-tAvat' ke nAma se pukArA hai| araba aura yoropa ke gaNitajJoM dvArA ina sarala samIkaraNoM ko 'Rule of false position' ke nAma se sambodhita kiyA gayA hai| isa prakAra ke prazna tathA hala karane kI vidhi kA varNana 'bakSAlIgaNita' meM milatA hai| AryabhaTTa prathama (499 I0) ne bhI isa prakAra ke prazna hala karane kA niyama diyA hai jo isa prakAra hai "jJAta rAziyoM ke antara ko ajJAta rAzi ke guNakoM ke antara se bhAga dene para ajJAta rAzi kA mAna jJAta ho jAtA hai|" yathAax+c=bx+d aAcArya mahAvIra ne bhI 'gaNitasArasaMgraha' meM isa vidhi para aneka udAharaNa evaM hala karane kI vidhi kA varNana kiyA hai, jo isa prakAra hai ___ yadi kisI rAzi kAkA kA hai aura / kAkA kA yoga / hai, to batalAo ki vaha ajJAta rAzi kyA hai ? isa prakAra ke prazna meM ajJAta rAzi jJAta karane ke lie AcArya ne nimnalikhita niyama diyA hai ajJAta rAzi ke sthAna para eka rakhakara, prazna ke anusAra phala jJAta karo aura phira prApta phala se die hue phala ko bhAga do| isa prakAra prApta bhajanaphala hI ajJAta saMkhyA kA mAna hogaa| - - d| - | - | 1 kA , kAs d kA 0 hanum-8-0- 5 ataH vaha ajJAta rAzi hai| 1. gaNita-sArasaMgraha, adhyAya 1, gAthA 50 2. vahI, madhyAya 1, gAthA 52 3. sthAnAMga sUtra, sUtra 747 -4. AryabhaTTIya 1,30 5. gaNitasArasaMgraha, adhyAya 3, gAthA 108 6. vahI, adhyAya 3, gAthA 107 AcAryaratna zrI dezabhUSaNa jI mahArAja abhinandana grantha

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(3) aneka varNa samIkaraNa - ekacAtIya yugapat samIkaraNa kA bhI mASArya mahAvIra ne ullekha kiyA hai| udAharaNoM ke sAtha-sAtha unako hala karane ke lie niyama bhI die haiN| yathA- "9 mAtuluMga aura 7 sugandhita kapittha phaloM kI kImata 107 hai| puna: 7 mAtuluMga aura 9 sugandhita kapittha phaloM kI kImata 101 hai | he gaNitajJa ! eka mAtuluMga aura eka sugandhita kapittha kA kImata alaga-alaga kyA hai ?" " mAnA ki eka mAtuluMga kI kImata x aura eka kapittha kI kImata y hai 9x+ 7 = 107 aura 7x + 9y101 samAnya rUpa se isako isa prakAra likha sakate haiM ax + by = m aura bx +ay=n isake lie mahAvIrAcArya ne nimna hala diyA hai-x+abyam aura 8x+aby in. ... (a2 - b2 ) x = ambn p lAbha hai / yA tathA .. yA isakA prayoga karane para uparyukta udAharaNa kA hala nimna prakAra hai- 8 9X107 - 7X 101 92 - 72 7X 107-9 X 101 72 92 X= y= am-bn a2- ba abx+b2y=bm aura abx +a2y=an (b2 - a2) y = bm--an bm-an y= b3-a3 ataH eka mAtuluMga kI kImata 8 aura eka kapittha kI 5 hai / udAharaNa 2--" yantra aura auSadhi kI zakti vAle kisI mahApuruSa ne murgoM kI lar3AI hotI huI dekhI aura murgoM ke svAmiyoM se alaga-alaga rahasyamayI bhASA meM mantraNA kii| usane eka se kahA - yadi tumhArA pakSI jItatA hai, to tuma mujhe dA~va meM lagAyA huA dhana de denA aura yadi tuma hAra jAoge, to tumheM lagAye hue dhana kA de duuNgaa| vaha phira dUsare murge ke svAmI ke pAsa gayA jahA~ usane unhIM dazAoM meM lagAye gaye dhana kA bhAga dene kI pratijJA kI pratyeka dazA meM use donoM se kevala 12 svarNa-tukar3e lAbha ke rUpa meM mile / batalAo ki pratyeka murge ke svAmI ke pAsa dA~va para lagAne ke lie kitanA kitanA dhana thA ?"3 maiM I uparyukta prazna kA hala nimna prakAra diyA gayA hai / " -P aura y= 0 (c+d) (c+d)b - (a + b)c yahA~ x aura y donoM murgoM ke svAmiyoM ke hAtha kI rakameM haiM / b X= x= 1. gaNitasArasaMgraha, adhyAya 6, gAthA 1402-1421 2. vahI, adhyAya 5, gAthA 1391/2 3. vahI, adhyAya 6, gAthA 270-272 1/2 4. vahI, pradhyAya 6, gAthA 2682 - 2692 jaina prAcya vidyAeM d(a + b)xp d(a + b) - ( c+d )a tathA unase liye gaye bhinnIya bhAga haiM aura d 21

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kaI ajJAta rAziyoM vAle ekaghAtIya samIkaraNa ke bhI udAharaNa 'gaNitasArasaMgraha' meM milate haiN| yathA-"cAra vyApAriyoM ne milakara apane dhana ko vyApAra meM lgaayaa| mahasUla padAdhikArI ne una logoM meM se pratyeka se alaga-alaga vyApAra meM lagAyI gaI vastu ke mAna ke viSaya meM pUchA / unameM se eka zreSTha vaNika ne apanI lagAyI gaI rakama ko ghaTAkara 22 btlaayaa| dUsare te 23, tIsare ne 24 aura cauthe ne 27 batalAyA / isa prakAra kathana karane meM pratyeka ne apanI-apanI lagAyI huI rakamoM ko vastu ke kula mAna meM se ghaTA liyA thaa| batalAo ki pratyeka kA usa paNyadravya meM kitanA-kitanA hissA thA ?" uparyukta prazna kA hala nimna prakAra diyA gayA hai--"vastuoM ke saMyukta zeSoM ke mAnoM ke yoga ko eka kama manuSyoM kI saMkhyA dvArA bhAga dene para bhajanaphala, samasta vastuoM kA kula mAna hogaa| isa kula mAna meM se viziSTa mAnoM ko alaga-alaga ghaTAne para maMgata sAjhedAra kA hissA jJAta ho jAtA hai|" kalpanA kI ki cAra vyApAriyoM ke hisse kramazaH Xxx auratra haiN| .:. x+ +xa+xx _22+23+24+27 4-1 x=32-22=10 x=32-23=9 xs=32-24=8 xx=32-27-5 ataH una vyApAriyoM meM se pratyeka kA alaga-alaga hissA kramazaH 10, 9, 8 aura 5 hai| kaI ajJAta rAziyoM vAle ekaghAtIya samIkaraNa kA eka anya prakAra kA udAharaNa 'gaNitasArasaMgraha' meM upalabdha hotA hai| isakA nAmakaraNa AcArya mahAvIra ne 'vicitra kuTTIkAra vidhi' nAma se kiyA jisakA uddharaNa adhovaNita hai "tIna vyaktiyoM ne eka-dUsare se, unake pAsa kI rakamoM meM se, rakameM maaNgiiN| pahalA vyApArI dUsare se 4 aura tIsare se 5 mAMgakara zeSa ke kula dhana se duganA dhana vAlA bana jAtA hai / dUsarA vyApArI pahale se 4 aura tIsare se 6 mAMgakara zeSa ke kula dhana se tigunA dhana vAlA bana jAtA hai tIsarA syApArI pahale se 5 aura dUsare se 6 mAMgakara una donoM se pAMca gunA dhana vAlA bana jAtA hai| batalAo, usake sthoM kI rakameM kyA haiM ?"3. " na ko hala karane kA DhaMga nimna prakAra diyA gayA hai| """ mAMgI haiI rakamoM ke yoga ko, abhISTa vyakti ke apavartya meM eka jor3akara prApta rAzi se guNA karate haiN| ina guNanaphaloM se thailI kI rakama prApta karane vAle niyama dvArA, hAthoM kI rakameM prApta kara lete haiN|" thailI kI rakama prApta karane vAlA niyama isa prakAra hai - "jisa vyakti ke hAtha kA dhana nikAlanA ho, usake bhinna vAle bhAga meM usI kI apavartya rAzi ko anya vyaktiyoM ke bhinna vAle bhAga me guNA karake jor3a lete haiM, aura isa prakAra prApta yogoM meM kramaza: anya vyaktiyoM ke apavartya meM eka jor3akara yogaphala kA bhAga dete haiN| phira prApta labdhiyoM ko jor3akara yoga meM se, vyaktiyoM kI saMkhyA meM se 2 ghaTAkara isI vyakti ke bhinna vAle bhAga se guNA karake ghaTA dete haiN| aba prApta rAzi ko isake apavartya meM eka jor3a kara bhAga dete haiN|" aba prazna ko nimna prakAra hala kiyA gayA hai| 1. gaNitasArasaMgraha, pradhyAya 6, gAthA 160-162 2. vahI, adhyAya 6, gAthA 159 3. vahI, adhyAya 6, gAthA 2531--2551 4. vahI, adhyAya 6, gAthA 2511-2521 5. vahI, adhyAya 6, gAthA 241 22 AcAryaratna zrI dezabhUSaNa jI mahArAja abhinandana grantha "

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kalpanA kI ki prathama vyApArI para x, dUsare vyApArI para ", aura tIsare vyApArI ke hAtha meM 2 haiN| .:. x+4+5 = 2+2-4-5) ___y+4+6 = 3(z+x-4-6) +5+6 5(x+y-5-6) athavA 2(x+y+z) =27 3(x+y+z) 5(x+y+z) (x+y-+z) (x+y--2) (x+y+2) -2 =11 tInoM ko jor3ane para (x+y+z)-(x+y+2)=30 + (x+y+2)=30 athavA (3++b) 15 (x+y+:) ==30 ___x-+y+x uparokta tInoM samIkaraNoM meM x+y+2 kA mAna rakhane para =30 x 12 =24 2 = 9 ataH pahale vyApArI para 7, dUsare vyApArI para 8 aura tIsare vyApArI ke pAsa 9 haiN| byAja sambandhI kaI prazna bhI, jinameM aneka ajJAta rAzi ke yugapat samIkaraNa banate haiM, mahAvIrAcArya dvArA vaNita kiye gaye haiN| yathA-vibhinna byAja kI rAziyA~ nikAlane ke lie udAharaNa isa prakAra haiM "eka prazna meM diye gaye mUladhana 40, 30, 20 aura 50 haiM, aura mAsa kramazaH 5, 4, 3 aura 6 haiM / byAja kI rAziyoM kA yoga 34 hai| pratyeka byAja-rAzi nikaalo|" isakA hala isa prakAra diyA gayA hai| ICH yadi i+is+is+......... =[ ho to 1 = - C +C +Cat+......... jahA~ para 1.2ig.........vibhinna mUladhanoM para byAja, intents.........vibhinna avadhiyA~ tathA C, C, C,...vibhinna mUladhana haiN| vibhinna mUladhana nikAlane ke lie udAharaNa nimna prakAra diyA gayA hai "diye gaye vibhinna vyAja 10, 6, 3 aura 15 haiM tathA saMvAdI avadhiyA~ kramaza: 5, 4, 3 aura 6 mAsa haiN| vibhinna maladhanoM kI rakamoM kA yoga 140 hai| ye mUladhana kI rakameM kauna-kauna sI haiM ?"3 1. gaNitasArasaMgraha, adhyAya 6, gAthA 38 2. vahI, adhyAya 6, gAyA 37 3. vahI, pradhyAya 6, gAthA 37 jaina prAcya vidyAeM

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upayukta prazna ko nimna DhaMga se hala kiyA gayA hai| - yadi C+C+C+......... =C ho to C-- Cilti 1/1+in/totis/ts+............... vibhinna avadhiyAM jJAta karane kA udAharaNa isa prakAra hai isa prazna meM diye gaye mUladhana 40, 30, 20 aura 50 haiM tathA saMvAdI byAja-rAziyAM kramazaH 10, 6, 3 aura 15 haiN| vibhinna avadhiyoM kA mizrayoga 18 hai| batalAo ki ye avadhiyA~ kauna-kauna sI haiM ? isakA hala isa prakAra hai - .___tit/C yadi 11+2+to+......... =t ho to - + +....." (21 varga samIkaraNa - varga samIkaraNa kA niyama bahuta prAcIna hai| isakA prayoga vaidika racanAoM meM haA hai| sarala varga samIkaraNa 4x*...4dx= -C kA jyAmitIya hala 500 I0pU0 se 300 I0pU0 ke prAcIna jaina granthoM meM tathA umAsvAti (150 I0pa0) ke 'tattvArthAdhigama sUtra' meM isa isa prakAra diyA hai x= (d-d-c) 'bakSAlIhastalipi' (200 I0) meM bhI barga samIkaraNa kA ullekha milatA hai| 'gaNitasArasaMgraha' meM bhI varga samIkaraNa ke udAharaNa milate haiN| yathA "U~ToM ke jhaMDa kA bhAga vana meM dekhA gayA / usa jhuMDa ke vargamUla kA duganA bhAga parvata ke utAroM meM dekhA gyaa| ke tigune nadI ke tIra para dekhe gye| U~ToM kI kula saMkhyA kyA hai ?"5 yadi jhuMDa meM U~ToM kI saMkhyA : hai to, praznAnusAra 1 +2/x +15=x yA (1-1 ) -24x --15=0 isakA hala isa prakAra diyA gayA haiyadi samIkaraNa ( 1-4)x-Cvx =d ho, to varga samIkaraNa ke do mUla-varga samIkaraNa ke do mUla hote haiM, yaha bAta mahAvIrAcArya bhalI-bhAMti jAnate the| unake graMtha meM uddhRta udAharaNoM se yaha bilkula spaSTa hai / yathA - "jhaMDa ke 1veM bhAga dvArA guNita mayUroM ke jhuMDa kA 16 vAM bhAga Ama ke vRkSa para pAyA gyaa| zeSa ke bhAga dvArA guNita zeSa kA vA~ bhAga tathA zeSa 14 mayU roM ko tamAla ke vRkSa para dekhA gayA / batalAo, ve kula kitane haiM ?"? 1. gaNitasArasaMgraha, adhyAya 6, gAyA 39 2. vahI, adhyAya 6. gAthA 43 3. vahI, adhyAya 6,gAthA 42 4. Dalta, Geometry in the jain cosmography, Quellen and Studien zur Gas. d math Ab & Bd, (1931) pp. 224-254 5. gaNitasArasaMgraha, adhyAya 4, gAthA 34 6. vahI, adhyAya 4, gAthA 33 7. vahI, adhyAya 4, gAthA 59 AcAryaratna zrI dezabhUSaNa jI mahArAja abhinandana grantha

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hala-yadi mayUroM kI saMkhyA 4 hai, to praznAnusAra nimnalikhita varga samIkaraNa banatA haiix is + 16xy x lixy +14=x sarala karane para isakA sAmAnya rUpa isa prakAra hogA x-x+C=0 isako hala karane ke lie AcArya ne nimna niyama pratipAdita kiyA hai| *_a-V(:- 4c) bla 'gaNitasArasaMgraha' meM varga samIkaraNa ke anya prakAra ke udAharaNa bhI milate haiN| yathA (1) "kula jhaMDa ke 1 bhAga ke pUrNa varga se eka kama, bhaiMsoM kA jhuMDa vana meM krIr3A kara rahA hai| zeSa 15 parvata para ghAsa carate hue dikhAI de rahe haiN| to batAiye kula kitane bhaiMse haiM ?" (2) "kula jhuMDa ke 1 veM bhAga se do kama pramANa, usI pramANa dvArA guNita hone se labdha hasti jhuMDa-rAzi sallakI vana meM krIr3A kara rahA hai / zeSa hAthI, jo saMkhyA meM 6 kI varga rAzi-pramANa haiM, parvata para vicara rahe haiM / batalAo, ve kula kitane haiM ?"3 (3) "kula jhuMDa ke bhAga se 2 adhika rAzi ko sva dvArA guNita karane se prApta rAzi pramANa mayUra jambU vRkSa para manorama krIr3A kara khela rahe haiN| zeSa garvIle 22X5 mayUra Ama vRkSa para prasannatApUrvaka uchala rahe haiN| he mitra ! isa mayUra-jhuMDa ke kula mayUroM kI saMkhyA batAo !" upayukta praznoM se nimna prakAra kA samIkaraNa banatA hai (x+d) + C= isa prakAra kA samIkaraNa hala karane kI vidhi AcArya ne isa prakAra batalAI hai - x = {{b/2a+d) + N (b/2a +dya-di- C A isake atirikta 'gaNitasArasaMgraha' meM aura bhI aneka udAharaNa milate haiM, jinase bilkula spaSTa hai ki varga samIkaraNoM ke do mUloM kI mahAvIrAcArya ko pUrNataH jAnakArI thii| parantu 'gaNitasArasaMgraha' meM kucha aise bhI prazna milate haiM, jinameM AcArya ne kevala eka hI mUla nikAlA hai| yathA "U~ToM ke jhaNDa kA bhAga vana meM dekhA gayA / usa jhuNDa ke vargamala kA duganA bhAga parvata ke utAroM para dekhA gyaa| 5 U~ToM ke tigune nadI ke kinAre para dekhe gye| UMToM kI kula saMkhyA kyA hai ?" isakA samIkaraNa isa prakAra banatA hai 1/4x+21 3/4x --2 +15=x --1530 yA 1. gaNitasArasaMgraha, adhyAya 4. gAthA 57 vahI, adhyAya 4, gAthA 62 3. vahI, adhyAya 4, gAthA 63 4. vahI, adhyAya 4, gAthA 64 5. vahI, adhyAya 4, gAthA 61 6. vahI, adhyAya 4, gAthA 34 jaina prAcya vidyAe~ 25

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yA / =4/34/(5) +4x15 = 6 yA =36 vargamUla kA mAna RNAtmaka nahIM ho sakatA hai| ataH vargamUla kI RNAtmaka rAzi ko chor3a diyA gayA hai| uccaghAtIya samIkaraNa-mahAvIrAcArya ne kucha ucca ghAtIya sarala samIkaraNoM kA bhI guNottara zreNI ke sambandha meM ullekha kiyA hai| ve samIkaraNa nimna prakAra haiM(1) ax -q (2) (-1), yahA~ para a guNottara zreNI kA prathama pada, 1 usakA guNadhana arthAt (n+1) vA pada hai, p usakA yoga tathA x ajJAta guNottara niSpatti hai| pahale samIkaraNa ko hala karane ke lie AcArya ne nimna niyama diyA hai "gaNaghana jaba prathama pada dvArA vibhAjita hotA hai, to bhAgaphala aisI svaguNita rAzi ke guNanaphala ke barAbara hotA hai, jisameM vaha rAzi, padoM kI saMkhyA bAra prakaTa hotI hai / arthAt x=ng dUsare prakAra kA samIkaraNa hala karane ke lie AcArya ne isa niyama kA ullekha kiyA hai-"vaha rAzi jisake dvArA zreNI ke yoga ko prathama pada dvArA vibhAjita karane se prApta huI rAzi meM se eka ghaTAne para utpanna rAzi meM kathita bhAjana sambhava ho (jabaki samaya-samaya para saba uttarottara bhajanaphaloM meM se eka ghaTAne ke bAda bhAga dene kI yaha vidhi kI jAtI ho), to vaha rAzi sAdhAraNa niSpatti hai|" Va yathA x-1 xn--1 x(x-1-1) tathA x-1 x-1 jo ki spaSTataH dvArA bhAjya hai / isake hala karane kI vidhi ko isa prakAra kaha sakate haiM-yoga ko prathama pada se bhAga dekara bhajana phala meM se eka ghttaao| phira kisI jA~ca-bhAjaka dvArA zeSa phala ko bhAga do| prApta bhajanaphala meM se puna: eka ghaTAkara phira usI jA~ca-bhAjaka se bhAga do| yaha kriyA bAra-bAra doharAne se yadi anta meM bhajanaphala eka A jAye, to jA~ca-bhAjaka hI guNa kA mAna hotA hai| ata: jA~ca-bhAjaka aisA cunanA cAhie ki anta meM bhajanaphala eka Ave / nimnalikhita udAharaNa dvArA upayukta vidhi saralatA se samajha meM A jaavegii| "yadi guNottara zreNI meM prathama pada 3, padoM kI saMkhyA 6, tathA zreNI kA yoga 4095 hai, to usakI sAdhAraNa niSpatti btaao|" hala 4095 : 3 = 1365 1365 - 1 = 1364 1. gaNitasArasaMgraha, adhyAya 2, gAthA 97 2. vahI, adhyAya 2, gAthA 101 3. vahI, adhyAya 2, gApA 102 26 AcAryaratna zrI dezabhUSaNa jI mahArAja abhinandana grantha

Page #9
aba jA~ca-bhAjaka 4 cunakara 1364-341, 341-1=340, 140 -85, 85-1 =84 84 = 21, 21-1=20, 20=5, 5-1=4,41 ata: abhISTa sAdhAraNa niSpatti 4 hai| mahAvIrAcArya ne nimna prakAra ke kucha samIkaraNoM kA bhI ullekha kiyA haia, Nbx +a Vb.(x-avbx)+a b{(x-1Nbix)-aVbox-aNb.x)}+...+R=x yA (x-AVbux)-avbg(x-OVER) -a Vbo{(x-01/bx)-a/box-a by-.."=R yadi bAI ora / pada ho to parimeyakaraNa karane para x kI 27 vIM ghAta kA samIkaraNa bana jAtA hai / ucita pratisthApana karane para uparokta samIkaraNa nimna prakAra ke eka sAdhAraNa varga samIkaraNa meM badala jAtA hai x-ABx =R isakA phala mahAvIrAcArya ne isa prakAra diyA hai__TA+NA2+4A/B . 2 isa phala ko AcArya ne 'sAra' kahA hai| uparokta samIkaraNa para AdhArita do prazna bhI 'gaNitasArasaMgraha' meM milate haiN| yathA (1) "hAthiyoM ke jhuNDa meM se, unakI saMkhyA ke bhAga ke vargamUla kA 9 guNA pramANa aura zeSa bhAga ke bhAga ke vargamUla kA 6 guNA pramANa aura anta meM zeSa 24 hAthI vana meM aise dekhe gaye, jinake caur3e gaNDasthaloM se mada jhara rahA thaa| batalAo. kula kitane hAthI haiM ?"2 hala-mAnA ki jhuNDa meM hAthiyoM kI saMkhyA hai| ataH diye hue praznAnusAra 1/3. +6/3 (-9/3 4 )+24=x y = x -9/2x rakhane para, to x =150, 24 jaba x -- 9 IT =60 aura jaba x-9 Vix = to x =3 (6143_1385) 1. gaNitasArasaMgraha, madhyAya 4, gAthA 52 2. vahI, madhyAya 4,gAthA 54-55 jaina prAcya vidyAe~

Page #10
ata: * ke ina cAra mAnoM meM se kevala x=150 hI aisA mAna hai jo prazna kI pratyeka zarta ko pUrA karatA hai| ke anya mAna sambhava nahIM haiM / isalie AcArya ne mUla kA kevala dhanAtmaka cihna hI liyA hai| (2) "vArAhoM ke jhuNDa ke arddhabhAga ke vargamUla kI caugunI rAzi jaMgala meM gaI, jahA~ zera krIr3A kara rahe the| zeSa jhuNDa ke dasaveM bhAga ke vargamUla kI ATha gunI rAzi parvata para gii| zeSa ke arddhabhAga ke vargamUla kI nau gunI rAzi nadI ke kinAre-kinAre gaI aura anta meM 56 vArAha vana meM dekhe gye| batAo ki kula kitane vArAha the?"1 hala-kalpanA kI ki yadi jhuNDa meM vArAhoM kI saMkhyA x hai to, V +8 Videx-40x12) +9 V{(x-4472) - 8 Vix-4Vs/2) }+56=x aba y=x-4/7/2 rakhane para, y=8 Page #11
viSama saMkramaNa kA niyama-do vizeSa prakAra ke yugapat varga samIkaraNoM ko hala karane kI vidhi ko hindU gaNitajJoM ne 'viSamakarma' ke nAma se sambodhita kiyA hai| parantu mahAbIrAcArya ne isake lie 'viSama saMkramaNa' zabda kA prayoga kiyA hai| ye vizeSa prakAra ke yugapat varga samIkaraNa isa prakAra ke haiM x -y' = m tathA x --y-n......1 x2 - y2 = m x + y = p***** 2 inako hala karane ke lie AcArya ne isa prakAra niyama diyA hai| (1) x = (+n) aura y = (" - ") (2) x = Fo+m) aura y = (p-m) mahAvIrAcArya ne vyAja sambandhI kucha aise praznoM kA bhI ullekha kiyA hai, jinameM yugapat varga samIkaraNa kA prayoga hotA haiu+x = a, ur w=ax tathA +y = burw = ay _a-u b-u aura x=(-1), y =(25) aura =( b) ke uparyukta samIkaraNoM meM // ghana, tathA 5 avadhi ke lie, kramaza: x aura vyAja haiM tathA vyAja kI dara prati lie hai / isake atirikta aise prazna bhI haiM, jinameM nimnalikhita samIkaraNoM kA prayoga hotA hai +x = P ux w= am u + y = a, u v w =on yahA~ para x va y avadhiyA~ haiM / // mUladhana, w byAja kI dara prati aura m an vyAja kI rakameM haiN| q-u .: 5= mg-np _m-n pa aura x ; Im ) y = m -n P-9 . a(m-n) m-n)" " p-q)(mq-np) (4) bhAvita-xy=ax+by+C jaise samIkaraNa ko bhAvita kahate haiN| 'gaNitasArasaMgraha' meM ina samIkaraNoM kI carcA nahIM hai parantu brahmagupta aura bhAskara dvitIya meM ina samIkaraNoM ko hala karane kI vidhiyA~ varNita kI haiN| ekaghAta anirNIta samIkaraNa ___anirNIta samIkaraNoM kA adhyayana AryabhaTTa se prArambha ho gayA thA, aura unake bAda ke sabhI bhAratIya gaNitajJoM brahmagupta, mahAvIra, bhAskara Adi ne bhI isa viSaya kA vivecana kiyA hai / bhAratIya gaNitajJoM ne isa prakAra ke samIkaraNa 'kuTTaka', 'kuTTAkAra' 'kuTTIkAra' aura 'kuTTaka' ke nAma se sambodhita kiye haiN| bhAskara prathama (522 I0) ne isake lie 'kuTTAkAra' aura 'kuTTa' nAma diye / brahmagupta ne isake lie 'kuTTaka', 'kuTTAkAra' aura 'kuTTa' zabda prayoga kiye haiN| mahAvIra ne isako 'kuTTIkAra' ke nAma se sambodhita kiyA hai|' 1. gaNitasArasaMgraha. adhyAya 6, gAthA ) 2. vahI, adhyAya 6, gAthA 47 3. vahI, pradhyAya 6, gAthA 51 -4. gaNitasArasagraha, adhyAya 6, gAthA 791 jaina prAcya vidyAeM

Page #12
kuTTa, kuTTaka, kuTTAkAra ye samasta zabda kuTTa se bane haiM jisakA Azaya kUTanA yA kucalanA hai| mahAvIrAcArya ne eka sthAna para batalAyA hai ki vidvAnoM ke anusAra 'kuTTIkAra' zabda 'prakSepaka' kA hI dUsarA nAma hai, jisakA artha choTe-choTe bhAgoM meM vibhAjita karanA hai / / AryabhaTTa ne ekaghAta anirNIta samIkaraNa ax+c=by ko hala karane ke lie isa prakAra niyama diyA hai "adhika zeSa vAle bhAjaka ko kama zeSa vAle bhAjaka se vibhAjita kro| prApta zeSa se phira kama zeSa vAle bhAjaka ko vibhAjita kro| isa taraha anta meM jo zeSa bace usako mana se cunI huI aisI saMkhyA dvArA guNA karo ki guNanaphala meM yadi samIkaraNa meM sthirAMka jor3A jAve (jabaki bhAgaphaloM kI saMkhyA sama ho) athavA ghaTAyA jAve (jaba ki bhAgaphaloM kI saMkhyA viSama ho), to prApta rAzi antima bhAjaka dvArA pUrNataH vibhAjita ho jAye / isake bAda bhajanaphaloM ko eka-dUsare ke nIce eka stambha meM likho| usake nIce mana se cunI huI saMkhyA tathA sabase nIce antima kriyA meM prApta bhajanaphala likho| isa stambha meM antima saMkhyA se ThIka eka Upara kI saMkhyA ko usake Upara kI saMkhyA se guNA karake nIce kI saMkhyA jor3a dete haiM / (aura antima saMkhyA ko miTA dete haiM / ) isa kriyA kI punarAvRtti taba taka hotI hai jaba taka ki stambha meM kevala do pada nahIM raha jAte / yahI pada nIce se kramazaH x aura y ke mAna hote haiN| yaha kriyA nimna udAharaNa se spaSTa hai.udAharaNa-137x+10=60y 60) 137 (2 120 ataH bhajanaphaloM kA stambha isa prakAra bnaa| 17) 60 (3 51 9) 17 ( / 9 8)9 (1 __8 pahale bhajanaphala ko chor3ane para bhajanaphaloM kI saMkhyA 3 raha jAtI hai / ataH hamako aisI saMkhyA cunanI hai ki jisako antima zeSa arthAt eka se guNA karane para, tathA guNanaphala meM se 10 ghaTAne para, zeSaphala antima se eka pahale zeSa arthAt 8 se pUrNataH vibhAjita ho jAve / mAnA, vaha saMkhyA 18 cunI, tAki 1X18-10-8X1 aba prathama stambha meM nIce 8 aura phira usake nIce eka likhaa| aba 18 ko usase Upara kI saMkhyA arthAt eka se guNA kiyA aura guNanaphala meM 18 se nIce kI saMkhyA eka ko jor3A / isa prakAra 18X1+1=19, dUsare stambha kI nIce se dUsarI saMkhyA ho gii| dUsare stambha kI zeSa saMkhyA vahI hotI hai, jo prathama stambha kI nIce se tIna saMkhyAoM ko chor3akara hai / yahI kriyA doharAne para tIsare, cauthe aura pAMcaveM stambha kI nIce se dUsarI saMkhyAeM kramA : 1931+182337,37x3+19=130 aura 130X2+37=297 haI / pratyeka stambha kI anya saMkhyAoM ko likhane para nimna tAlikA banI 22297 130 130 1 37 37 19 19 18 1. baNitasArasaMgraha, adhyAya 6. gAthA 791 2. adhikAgrabhAgahAraM chidyAdUnAgrabhAgahariNA meSaparasparabhaktaM matiguNamagrantare kSipta athaupari gaNitamaMtyayugUnAgraccheda bhAjite zeSa adhikAgracchedaguNa dvicchedrAgramadhikAgrayutama-mAryabhaTTIya, gAthA 32-33 AcAryaratna zrI dezabhaSaNa jI mahArAja abhinandana grantha

Page #13
ataH die hue samIkaraNa kA hala nimna huA x=130, y-297 parantu 297=137x2+23 aura 130=60x2+10 .:. samIkaraNa kA sarala hala nimna huA x=10, y=23 aura samIkaraNa kA sAmAnya hala nimna huA--- x=10+60m aura y=23+137m mahAvIra kA hala-mahAvIrAcArya ne samIkaraNa HT=y ko hala karane ke lie nimna niyama diyA hai - "ajJAta rAzi ke gaNaka ko diye gaye bhAjaka dvArA vibhAjita karate haiN| phira prathama bhAgaphala ko alaga kara dete haiN| isake bAda vibhinna pariNAma-zeSoM dvArA vibhinna pariNAmI bhAjakoM ke uttarottara bhAga se prApta vibhinna bhajanaphaloM ko eka-dUsare ke nIce rakhate haiM / jaba zeSaphala bahuta choTI saMkhyA raha jAtI hai to usako mana se cunI huI eka saMkhyA dvArA guNA karate haiN| isa guNanaphala ko praznAnusAra dI gaI jJAta saMkhyA dvArA bar3hAte athavA hrAsita karate haiM aura taba uparyukta uttarottara bhAga kI vidhi meM prApta antima bhAjaka dvArA vibhAjita karate rakhate haiN| isa mata se cunI huI saMkhyA aura abhI prApta bhajanaphala ko bhI uparyukta bhajanaphaloM ke nIce likhate haiM / isa prakAra beli jaisI aMkoM kI zrRMkhalA prApta hotI hai| isa zRMkhalA kI nimnatama saMkhyA ko, isake ThIka Upara kI saMkhyA meM Upara ke ThIka Upara kI saMkhyA kA guNana karane se prApta guNanaphala meM jor3ate haiM / yaha rIti taba taka jArI rakhate haiM jaba taka ki pUrI zRkhalA samApta nahIM ho jAtI hai / isa yoga meM pahale hI diye hue bhAjaka kA bhAga dete haiM / jo zeSaphala milatA hai, vahI ajJAta rAzi kA mAna hotA hai|" uparyukta vidhi nimna udAharaNa se spaSTa ho jAyegI "keloM kI 63 DheriyAM aura 7 kele ke phala 23 vyaktiyoM meM barAbara-barAbara bA~Ta diye gae, jisase kucha bhI zeSa na bcaa| batAo, eka DherI meM kitane phala the ?" upayukta prazna kA samIkaraNa isa prakAra huA 63xxlay 23 aba niyamAnusAra ajJAta rAzi ke gaNaka 63 ko jJAta bhAjaka 23 dvArA vibhAjita karate haiM aura jisa prakAra do saMkhyAoM kA mahattama samApavartaka nikAlate haiM, usI prakAra kI bhAga-vidhi yahA~ jArI rakhate haiN| 23) 63 (2 46 17) 23 (1 6) 17 (2 12 :) 6 (1 1. gaNitasArasaMgraha, madhyAya 6, gAyA 111 2. vahI, adhyAya 6, gAthA 1171 jaina prAcya vidyAe~

Page #14
yahAM prathama bhajanaphala 2 ko chor3akara anya bhajanaphala bAjU ke stambha meM eka paMkti meM likha liye gaye haiN| aba hamako eka saMkhyA aisI cunanI hai, jisako yadi antima zeSa eka dvArA guNA kareM aura phira 7 jor3eM, to yogaphala antima bhAjaka eka ke dvArA pUrNataH bhAjana ke yogya ho| mAnA, vaha saMkhyA 1 cunI tAki 1x1+7=1x8 isa cunI huI saMkhyA eka ko zRMkhalA ke antima aMka ke nIce likhate haiN| phira isa cunI huI saMkhyA ke nIce, cunI huI saMkhyA kI sahAyatA se upayukta bhAga meM prApta bhajanaphala likhA jAtA hai / isa prakAra prathama stambha ke aMkoM kI pUrNa zRMkhalA prApta ho jAtI hai| aba zrRMkhalA ke nIce se upa antima aMka arthAt eka ko usake Upara ke aMka 4 dvArA gaNA karate haiM aura guNanaphala meM eka ke nIce kI saMkhyA 8 ko jor3ate haiN| isa prakAra prApta pariNAmI 1x4+8-12 ko dUsare stambha meM isa prakAra likhate haiM ki vaha prathama stambha ke aMka 4 ke saMvAdI sthAna meM ho| isake bAda isa 12 ko prathama stambha meM 4 ke Upara ke aMka eka dvArA gaNA karake 4 ke nIce ke aMka ko jor3ate haiN| isa prakAra prApta pariNAmI 12x1+1=13 ko dUsare stambha me 12 ke Upara likhate haiN| isI prakAra kriyA jArI rakhane se hamako 38 aura 51 bhI prApta hote haiM jo kramaza: 2 aura 1 ke saMvAdI sthAna para rakhe jAte haiN| isa 51 meM bhI 23 dvArA bhAga diyA jAtA hai, zaSa 5 bacatA hai| yahI 5 eka DherI meM keloM kI abhISTa saMkhyA hai| stambha isa prakAra hai 1 - 51 2 - 38 1 - 13 isa prakAra x-5 yA 5+23m huaa| samIkaraNa meM x kA mAna rakhakara -14 prApta ho jAtA hai / niSkarSa-upayukta vibecanoparAnta yaha spaSTata: kahA jA sakatA hai ki jaina sAhitya meM nihita bIjagaNita apanI mahattA ko samAhita kie hue hai| jainAcAryoM ne bIjagaNita ke pratipAdana para pratyeka dRSTikoNa se gahanatama vicArAtmaka parivezoM ko prastuta karake usakI samRddhi kA mahattvAMkana kiyA hai / jainAcAryoM ke stutya prayAsoM se bIjagaNita kI prAcInatA to AtI hI hai, sAtha hI usakI, AdhunikatA bhI hamako suspaSTata: jJAta ho jAtI hai / antata: kahA jA sakatA hai ki jainAcAryoM ne bIjagaNita para vistRta vicAra prastuta karake bIja gaNita kI svarUpatA, saiddhAntikatA evaM vyAvahArikatA rUpI triveNI ko pravAhita kiyA hai| bhAratIya gaNita kI maulikatA evaM prAcInatA ensAiklopIDiyA Apha briTAnikA (jilda 17, pR0 626, navama saMskaraNa) meM likhA hai "isameM koI sandeha nahIM ki hamAre (aMgrejI se) vartamAna aMka-krama kI utpatti bhArata se hai| sambhavata: khagola saMbaMdhI una sAraNiyoM ke sAtha, jinako eka bhAratIya rAjadUta 773 I0 meM bagadAda meM lAyA thA, ina aMkoM kA praveza araba meM haa| phira IsavI sana 9vIM zatAbdI ke prArambhika kAla meM prasiddha abU japhara mohammada ala khAhijmI ne arabI meM ukta aMka-krama kA vivecana kiyA aura usI samaya se araba meM usakA pracAra bar3hane lgaa| yoropa meM zunya-sahita sampUrNa aMka-krama IsavI sana 12vIM zatAbdI meM arabI se liyA gayA aura isa krama se banA huA aMkagaNita 'algoriTamasa' nAma se prasiddha huaa| alabarUnI ne bhI apanI bhArata-yAtrA' meM yahAM ke gaNita evaM jyotiSa kI muktakaMTha se sarAhanA kI hai| usake anusAra, jina bhinna-bhinna jAtiyoM se merA samparka rahA, una sabakI bhASAoM meM saMkhyAsUcaka cakroM ke nAmoM (ikAI, dahAI, saikar3A Adi) kA maiMne adhyayana kiyA hai, jisase mAlUma huA hai koI bhI jAti eka hajAra se Age nahIM jaatii| araba loga bhI eka hajAra taka (nAma) jAnate haiN| hindU apane saMkhyA-sUcaka krama ko aThArahaveM sthAna taka le jAte haiM, jisako parArddha' kahate haiM / [DaoN. mukuTabihArI lAla ke zodha-prabandha gaNita evaM jyotiSa ke vikAsa meM jainAcAryoM kA yogadAna' ke AdhAra para]. AcAryaratna zrI dezabhUSaNa jI mahArAja abhinandana grantha