bhaarateeya ganit

ganiteeya gaveshana ka mahatvapoorn bhaag bhaarateeya upamahaadveep mein utpann hua hai. sankhya, shoonya, sthaaneeya maan, ankaganit, jyaamiti, beejaganit, kailakulas aadi ka praarambhik kaarya Bhaarat mein sampann hua. ganit-vigyaan na keval audyogik kraanti ka balki paravarti kaal mein huin vaigyaanik unnati ka bhi kendra bindu raha hai. bina ganit ke vigyaan ki koi bhi shaakha poorn naheen ho sakti. Bhaarat ne audyogik kraanti ke liye na keval aarthik pooainji pradaan ki varan vigyaan ki neenv ke jeevant tatv bhi pradaan kiye jiske bina maanavata vigyaan aur uchch takaneeki ke is aadhunik daur mein pravesh naheen kar paati. videshi vidvaanon ne bhi ganit ke kshetr mein Bhaarat ke yogadaan ki muktakanth se saraahana ki hai.

anukram

'ganit' shabd ka itihaas

vishv ke praacheenatam granth ved sanhitaaon se ganit tatha jyotish ko alag-alag shaastron ke roop mein maanyata praapt ho chuki thi. yajurved mein khagolashaastr (jyotish) ke vidvaan ke liye 'nakshatradarsh' ka prayog kiya hai tatha yeh salaah di hai ki uttam pratibha praapt karne ke liye uske paas jaana chaahiye (prajnyaaanaaya nakshatradarsham). ved mein shaastr ke roop mein 'ganit' shabd ka naamat: ullekh to naheen kiya hai par yeh kaha hai ki jal ke vividh roopon ka lekha-jokha rakhane ke liye 'ganak' ki sahaayata li jaani chaahiye.

shaastr ke roop mein 'ganit' ka praacheenatam prayog lagadh rishi dvaara prokt vedaang jyotish naamak granth ka ek shlok mein maana jaata hai. par isse bhi poorv chhaandogya upanishd mein sanatkumaar ke poochhane par naarad ne jo 18 adheet vidyaaon ki soochi prastut ki hai, usamein jyotish ke liye 'nakshatr vidya' tatha ganit ke liye 'raashi vidya' naam pradaan kiya hai. isse bhi prakat hai ki us samay in shaastron ki tatha inke vidvaanon ki alag-alag prasiddhi ho chali thi.

aage chalakar is shaastr ke liye anek naam viksit hote rahe. sarvapratham brahmagupt ne paat ya paati ka prayog kiya. baad mein shreedharaachaarya ne 'paati ganit' naam se mahaneeya granth likha. tab se yeh naam lokapriya ho gaya. paati ya takhti par khadiya dvaara sankriyaaeain karne se yeh naam samaaj mein chalne laga. arab mein bhi ganit ki is paddhati ko apnaane se is naam ke vajan par 'ilm hisaab al takht' naam prachalit hua.

paribhaasha

bhaarateeya parampara mein ganesh daivajnya ne apne granth buddhivilaasini mein ganit ki paribhaasha nimnavat ki hai-

ganyate sankhyaayate tadganitm. tatpratipaadakatven tatsanjnyaan shaastran uchyate.
(jo pariklan karta aur ginta hai, vah ganit hai tatha vah vigyaan jo iska aadhaar hai vah bhi ganit kahalaata hai.)

ganit do prakaar ka hai-

  • vyaktaganit ya paateeganit - ismein vyakt raashiyon ka upayog kiya jaata hai.
  • avyaktaganit ya beejaganit - ismein avyakt ya aajnyaaat raashiyon ka upayog kiya jaata hai. avyakt sankhyaaon ko 'varn' bhi kehte hain. inhein 'ya', 'ka', 'ni' aadi se niroopit kiya jaata hai. (jaise aajkal roman aksharon x, y, z ka prayog kiya jaata hai. (ka = kaalak, ni = neelak, ya = yaavat, ta = taavat)

bhaarateeya granthon mein ganit ki mahatta ka prakaashan

vedaang jyotish mein ganit ka sthaan sarvopari (moodhanya) bataaya gaya hai -

yatha shikha mayooraanaan naagaanaan manayo yatha.
tadvad vedaangashaastraanaan ganitan moordhni sansthitm.. (vedaang jyotish - 5)

(jis prakaar moron ke sir par shikha aur naagon ke sir mein mani sarvochch sthaan mein hote hain usi prakaar vedaangashaastron mein ganit ka sthaan sabse upar (moordhanya) hai.

isi prakaar,

bahubhirpralaapai: kim, trayaloke sacharaare.
yad kinchid vastu tatsarvam, ganiten bina na hi .mahaaveeraachaarya, ganitsaarasangrah mein

(bahut pralaap karne se kya laabh hai ? is charaachar jagat mein jo koi bhi vastu hai vah ganit ke bina naheen hai / usako ganit ke bina naheen samjha ja sakta)

anya shaastron mein ganit ki vivechana

Bhaarat mein anya shaastron ke vidvaan bhi ganit ki bhaavana se ot-prot rahe prateet hote hain. un shaastron mein prasangavash ganit vishyak jaanakaariyaaain bikhri padi hain. mahaan vaiyaakaran paanini ne ganit ke anek shabdon ki sookshm vivechana ki hai. unhonne us samay ki aavashyakataanusaar pratishat ke sthaan par maas mein deya byaaj ke liye ek 'pratidsh' anupaat ka ullekh kiya hai (kuseedadashaikaadashaat shthanshthachau (pa.sa. 4.4.31)). chakravruddhi byaaj dvaara sarvaadhik badhi hui rakam ko 'mahaapravruddh' bataaya hai. tol, maap, sikke, panya dravya ke saikado shabdon ke varnan ke antargat trairaashik niyam ki soochana di hai.

darshanashaastr mein vedaant mein adhyaarop apavaad ke siddhaant beejaganiteeya sameekaran ya ankaganit ke 'ishtakarm' ke samakaksh hain. nyaaya shaastr ki anumaan ya tarkavidya sarvatha ganiteeya niyamon se sanchaalit hai.

E. poo. doosari shati mein pingal virchit chhandashaastr mein chhandon ke vibhed ko varnit karne wala 'meruprastaar' paaskal ke tribhuj se tulaneeya banta hai. vedon ke kramapaath, ghanapaath aadi mein ganit ke shreni-vyavahaar ke tattv vartamaan hain.

yadi yeh jaanana ho ki samaaj mein 56 prakaar ke vyanjan ka prayog kis prakaar prachalit hai, to iske liye vaidyakashaastr mein varnit ganit ke 'ank-paash' ke antargat 'sanchaya' (Combination) ke niyamon ke aadhaar par kul 6 rason ke dvaara 63 tatha antat: 56 vibhedon ki sankalpana ka adhyayan apekshit hoga.

saahitya-shaastr mein bhi ganit ke aadhaar par manoram rachanaaeain praapt hoti hain. vahaan paanineeya vyaakaran ke ek pramukh udaaharan 'laakruti' ke aadhaar par sukh-dukh mein ek samaan rahane vaale sajjan tatha 9 sankhya ki manohaari samaanata bataai gayi hai. mahaakavi shreeharsh ne bataaya hai ki damayanti ke kaan aakhir kyon tatha kis prakaar sarvatha nae rache gaye. unhonne maana ki upanishdon mein varnit 18 vidyaaon mein se 9-9 vidyaaon ka anupravesh damayanti ke kaanon ke andar tak hua tha. ye nav ank hi kaanon mein pahuainch kar shabdasaamya se 'nav' ban gaye—

asya yadashtaadash sanvibhjya vidyaa: shrutee: dadhraturardhamardham.
karnaantarutkeernagabheerarekh: kin tasya sankhyaiv nava navaank:.. (naishadh, 7.63)

khagol-vigyaan ke saath to ganit ka anyonya sambandh maana gaya hai. bhaaskaraachaarya ka kehna hai ki khagol tatha ganit mein ek doosare se anbhigya purush usi prakaar mahattvaheen hai, jaise ghrut ke bina vyanjan, raaja ke bina raajya tatha achhe vakta ke bina sabha hoti hai—

bhojyan yata sarvarasan vinaajyan raajyan yatha raajavivrjitan ch.
sabha na bhaateev suvaktruheena golaanabhijnyao ganakastathaatr.. (siddhaantashiromani, golaadhyaaya, shlok 4)

ganit ke vibhinn kshetron mein Bhaarat ka yogadaan

praacheenakaal tatha madhyakaal ke bhaarateeya ganitjnyaon dvaara ganit ke kshetr mein kiye gaye kuchh pramukh yogadaan neeche diye gaye hain-

  • (4) ganiteeya tarkashaastr (laujik): Formal grammars, formal language theory, the Panini-Backus form (paanini dekhein), Recursion (paanini dekhein)

bhaarateeya ganit ka itihaas

sabhi praacheen sabhyataaon mein ganit vidya ki pehli abhivyakti ganana pranaali ke roop mein pragat hoti hai. ati praarambhik samaajon mein sankhyaayein rekhaaon ke samooh dvaara pradarshit ki jaateen theen. yadyapi baad mein, vibhinn sankhyaaon ko vishisht sankhyaatmak naamon aur chihnon dvaara pradarshit kiya jaane laga, udaaharan svaroop Bhaarat mein aisa kiya gaya. rom jaise sthaanon mein unhein varnamaala ke aksharon dvaara pradarshit kiya gaya. yadyapi aaj ham apni dashamlav pranaali ke abhyast ho chuke hain, kintu sabhi praacheen sabhyataaon mein sankhyaaen dashamaadhaar pranaali par aadhaarit naheen theen. praacheen bebeelon mein 60 par aadhaarit sankhya-pranaali ka prachalan tha.

Bhaarat mein ganit ke itihaas ko mukhyata 5 kaalakhandon mein baanta gaya hai-

  • 1. aadi kaal (500 isvi poorv tak)
  • (k) vaidik kaal (1000 isvi poorv tak)- shunya aur dashamlav ki khoj
  • (kh) uttar vaidik kaal (1000 se 500 isvi poorv tak) is yug mein ganit ka Bhaarat mein adhik vikaas hua. isi yug mein bodhaayan shulv sootr ki khoj hui jise ham aaj paaithaagoras prameya ke naam se jaante hai.
  • 2. poorv madhya kaal – sine, cosine ki khoj hui.
  • 4. uttar-madhya kaal (1200 isvi se 1800 isvi tak) - neelakanth ne 1500 mein sin r ka maan nikaalne ka sootr diya jise ham gregari shreni ke naam se jaante hai.
  • 5. vartamaan kaal - raamaanujam aadi mahaan ganitjnya hue.

bhaarateeya ganit : ek sookshmaavalokan

ganit moolat: bhaarateeya upamahaadveep mein viksit hua. shoonya evam anant ki parikalpana, ankon ki dashamlav pranaali, rinaatmak sankhyaaen, ankaganit, beejaganit, jyaamiti evam trikonamiti ke vikaas ke liye sampoorn vishv Bhaarat ka krutajnya hai. ved vishv ki puraatan dharohar hai evam bhaarateeya ganit usase poornataya prabhaavit hai. vedaang jyotish mein ganit ki mahatta is prakaar vyakt ki gayi hai :

jis prakaar mayooron ki shikhaaen aur sarpon ki maniyaan shareer ke uchch sthaan mastishk par viraajamaan hain, usi prakaar sabhi vedaangon evam shaastron mein ganit ka sthaan sarvopari hai.

sindhu ghaati ki sabhyata bhaarateeya upamahaadveep ke pashchimottar bhaagon mein faili thi. itihaasakaar ise isa poorv 3300-1300 ka kaal maanate hain. mohanajodadon evam hadppa ki khudaai se praapt avasheshon evam shilaalekhon se us samay ki prayukt ganit ki jaankaari praapt hoti hai. us samay ki inton evam bhinn-bhinn bhaar ke parimaap ke vividh aakaaron se spasht hota hai ki praacheen bhaarateeyon ko jyaamiti ki praarambhik jaankaari thi. lanbaai ke parimaap ki vishisht vidhi thi jisse theek-theek oonchaai gyaat ho sake. inton ke nirmaan ki vidhi, shuddhamaap ke liye bhaar ke vividh aakaar evam lanbaai ke vividh parimaapon se spasht hai ki sindhu ghaati ki sabhyata parishkrut evam viksit thi. us samay ankaganit, jyaamiti evam praarambhik abhiyaantriki ka gyaan tha.

ved vishv ka sabse puraana granth hai. baal gangaadhar tilk ne khagoleeya ganana ke aadhaar par iska kaal isa poorv 6000-4500 varsh nirdhaarit kiya hai. rigved ki richaaon mein 10 par aadhaarit vividh ghaaton ki sankhyaaon ko alag-alag sanjnyaa di gayi hai, yatha ek (100 ), dash (101 ) shat (102 ) sahastr (103 ), aayut (104 ), laksh (105 ), prayut (106 ), koti (107 ), arbud (108 ), abj (109 ), kharb (1010 ), vikhrb (1011 ), mahaapadam (1012 ), shanku (1013 ), jaladhi (1014 ), antya (1015 ), madhya (1016 ) aur paraardh (1017 ).

in sankhyaaon se spasht hai ki vaidik kaal se hi ankon ki dashamlav pranaali prachalit hai. yajurved mein ganiteeya sankriyaaen-yog, antar, gunan, bhaag tatha bhinn aadi ka samaavesh hai, udaaharanaarth yajurved ki nimn richaaon par dhyaan dein.

eka ch me tistrashch me tistrashch me panyach ch me
panyach ch me sapt ch me sapt ch me nav ch me nav ch
mऽekaadash ch mein trayodash ch me trayodash ch me
panyaachadash ch me panyachadash ch me saptadash ch me saptadash
ch me navadash ch me navadash ch me ek vinshatishch me
trayaastranshachch me yajnyaen kalpantaam. 18.24

arthaat yagya ke falasvaroop hamaare nimitt ek-sankhyak stom (yagya karaane vaale), teen, paanch, saat, nau, gyaarah, terah, pandrah, satrah, unnees, ikkees, teis, pachchees, sattaais, unatees, ikatees aur taintees sankhyak stom sahaayak hokar abheesht praapt karaaen. is shlok mein visham sankhyaaon ki samaantar shreni prastut ki gayi hai-

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33

yagya ka arth sangatikran se hai. ankon ke ankon ki sangati se ank vidya banti hai. shlok mein pratyek sankhya ke saath 'ch' juda hai jiska arth 'aur' se hai. iska arth +1 jodne se sam athva visham raashiyaan ban jaati hain. isi se pahaada evam vargamool ke siddhaanton ka pratipaadan hota hai. is adhyaaya ka agala shlok (18.25) sam sankhyaaon ke samaantar shreni prastut karta hai.

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44.

at: nishchit roop se kaha ja sakta hai ki vaidik kaal mein-

(k) ek ankeeya sankhyaaen 1, 2, 3, ..9;
(kh) shoonya aur anant;
(g) kramaagat sankhyaaen evam bhinnaatmak sankhyaaen; tatha
(gh) ganiteeya sankriyaaon ka ullekhaneeya gyaan tha.

dhaarmik anushthaanon mein vediyon ki rachana ke liye jyaamiti ka aavishkaar hua. shatapath braahman evam taittareeya sanhita mein jyaamiti ki sankalpana prastut hai. par saamaanyatayaa: aisa vichaar hai ki vedaang jyotish ke shulvasootr se aadhunik jyaamiti ki neenv padi. vedaang jyotish ke anusaar soorya ki sankraanti evam vishuv ki sthitiyaan krutika nakshatr ke vasant vishuv ke aas-paas hain.

yeh sthiti isa poorv 1370 varsh ke lagbhag ki hai. at: vedaang jyotish ki rachana sanbhavat: isa poorv varsh 1300 ke aas-paas hui hogi. is yug ke mahaan ganitjnya baudhaayan, maanav, aapastamb, kaatyaayan rahe hain. in sabhi ne alag-alag soolv sootr ki rachana ki. bodhaayan ka soolv sootr is prakaar hai-

deergh chaturas syaakshvya rajju: paarshvamaani,
tiryaङ maani ch yatpruthbabhoote kuru hastedurbhaya karoti.
arthaat deergh chaturas (aayat) ke vikrn (rajjoo) ka kshetr (varg) ka maan aadhaar (paarshvamaani) evam triyangamaani (lanb) ke vargon ka yog hota hai. soolv sootr aadhunik kaal mein 'paaithaagoras ka sootr' ke naam se prachalit hai. paithaagoras ne isa poorv 535 mein misr ki yaatra ki thi. sambhav hai ki paithaagoras ko misr mein soolv sootr ki jaankaari praapt ho chuki ho.

bodhaayan ne aparimeya raashi 20.5 ka maan is prakaar diya hai:

20.5 = 1 + 1/3 + 1/3.4 - 1/3.4.3.4

maharshi lagadh ne rigved evam yajurved ki richaaon se vedaang jyotish sangraheet kiya. vedaang jyotish mein grahon ki sthiti, kaal evam gati ki ganana ke sootr diye gaye hain.

tithi me ka dashaamya staam parvamaansh samanvitaam.
vibhjya bhaj samuhen tithi nakshatramaadishet.
arthaat tithi ko 11 se guna kar usamein parv ke ansh jodein aur fir nakshatr sankhya se bhaag dein. is prakaar tithi ke nakshatr bataavein.

hamaare rishi, maharshiyon ko badi sankhyaaon mein apaar ruchi thi. isa poorv chhathi shataabdi mein Gautam buddh ki jeevani par aadhaarit 'lalitvistar' ki rachana hui. usamein Gautam buddh ke ganit kaushal ki pareeksha ka prasang aata hai. unase koti (107 ) se oopar sankhyaaon ke alag-alag naam ke baare mein poochha gaya. yuva siddhaarth (Gautam buddh ka bachapan ka naam) ne koti ke baad 1053 ki sankhyaaon ka alag-alag naam diya aur fir 1053 ko aadhaar maan kar 10421 tak ki sankhyaaon ko unke naamon se sambodhit kiya. Gautam buddh bauddh dharm ke pravartak the. unheen ke samakaksh mahaaveer svaami ka bhi padaarpan hua jinhonne jain dharm ki sthaapana ki. jain mahaapurushon ki ganit mein bhi ruchi thi. unki prasiddh rachanaaen- 'sooryaprajnyaapti sootr', 'vaishaali ganit', 'sthaanaang sootr', 'anuyogadvaar sootr' evam 'shatakhandaagam' hai. bhadravaahu evam umaasvati prasiddh jain ganitjnya the.

vaidik parampara mein guru apna gyaan maukhik roop se apne yogya shishya ko pradaan karta tha par isa poorv 5veen shataabdi mein braahmi lipi ka aavishkaar hua. ganit ki pustakon ki paandulipiyaan braahmi lipi mein taiyaar huin. 'bakhshaali paandulipi' pehli pustak thi jiske kuchh ansh peshaavar ke ek gaanv vakhshaali mein praapt hue. isa poorv 3 shataabdi ki likhi yeh pustak ek pramaanik granth hai. ismein ganiteeya sankriyaaon-dashamlav pranaali, bhinn, varg, dhan, byaaj, kraya evam vikraya aadi vishyon par vistrut charcha hui hai. aadhunik ganit ke truti sthiti (False Position) vidhi ka bhi yahaan samaavesh hai.

jyotish ki ek anya pustak 'soorya siddhaant' ki bhi rachana sanbhavat: isi dauraan hui. vaise iske lekhak ke baare mein koi jaankaari naheen hai. par mayaasur ko sooryadev ki aaraadhana ke falasvaroop yeh gyaan praapt hua tha. ni:sandeh yeh aaryon ki kruti naheen hai. soorya siddhaant mein badi se badi sankhyaaon ko vyakt karne ki vidhi varnit hai. ginti ke ankon ko sankhyaatmak shabdon mein vyakt kiya gaya hai, yatha rup (1), netr (2), agni (3), yug (4), indriya (5), ras (6), adri (7 - parvat shrunkhala), Basu (8), ank (9), rav (0). in shabdon ke paryaayavaachi shabd athva hindu devi-devataaon ke naam se bhi vyakt kiya gaya hai. pandrah ko tithi se tatha solah ko nishaakar se. ankon ko daaen se baaen ki taraf rakh kar badi se badi sankhya vyakt ki gayi hai. soorya siddhaant mein vividh ganiteeya sankriyaaon ka varnan hai. aadhunik trikonamiti ka aadhaar bhi soorya siddhaant ke teesare adhyaaya mein vidyamaan hai. jya, utkram jya evam koti jya paribhaashit kiya gaya hai. yahaan dhyaan dene yogya baat hai ki jya shabd arabi mein jaib se bana jiska laitin roopaantaran Sinus mein kiya gaya aur fir yeh vartamaan 'Sine' mein parivrtit hua. soorya siddhaant mein π ka maan 101/2 diya gaya hai.

bhaarateeya itihaas mein gupt kaal 'svarn yug' ke roop mein maana jaata hai. mahaaraaja shreegupt dvaara sthaapit gupt saamraajya poore bhaarateeya upamahaadveep mein faila tha. san 320-550 ke madhya is saamraajya mein gyaan ki har vidya mein mahattvapoorn aavishkaar hue. is kaal mein aaryabhat (476) ka aavirbhaav hua. unke janm sthaan ka theek-theek pata naheen hai par unka kaaryakshetr kusumapur (vartamaan Patna) raha. 121 shlokon ki unki rachana aaryabhateeya ke chaar khand hain- gitika pad (13), ganit pad (33), kaalakrupa pad (25) aur gol pad (50). pratham khand mein ank vidya ka varnan hai tatha dviteeya evam truteeya khand mein beejaganit, trikonamiti, jyaamiti evam jyotish par vistaarapoorvak varnan kiya gaya hai. unhonne π ka 4 ankon tak shuddh maan gyaat kiya- π = 3.4161 . sankhyaaon ko vyakt karne ke liye unhonne devanaagari varnamaala ke pehle 25 akshar (k-m) tak 1-25, ya-h (30, 40, 50, ... 100) aur svar a-au tak 100, 1002 , ... 1008 se pradarshit kiya. udaaharan ke liye :

jal ghini jh su bhru sru kh
(8 + 50) (4 + 20) (9 + 70) (90 + 9) 2 = 299792458

yahaan bhi sankhyaaen daaen se baaen ki taraf likhi gayi hain. aadhunik beej lekh (Cryptolgy) ke liye isse achha udaaharan kya ho sakta hai. aaryabhat ki samruti mein Bhaarat sarkaar ne 19 apairal 1975 ko pratham bhaarateeya upagrah aaryabhat ko pruthvi ki nimn kaksha mein sthaapit kiya.

aaryabhat ke kaaryon ko bhaaskaraachaarya (600 E) ne aage badhaaya. unhonne mahaabhaaskareeya, aaryabhateeya bhaashya evam laghubhaaskareeya ki rachana ki. mahaabhaaskareeya mein kuttak (Indeterminate) sameekaranon ki vivechana ki gayi hai bhaaskaraachaarya ki smruti mein dviteeya bhaarateeya upagrah ka naam 'bhaaskar' rakha gaya.

bhaaskaraachaarya ke samakaaleen bhaarateeya ganitjnya brahmagupt (598 E) the. brahmagupt ki prasiddh kruti braahmasfutasiddhaant hai. ismein 25 adhyaaya hain. beejaganit ke sameekaranon ke hal ki vidhi evam dvighaateeya kuttak sameekaran, X2 = N.y2 + 1 ka hal ismein diya gaya hai. joshef luis lagareinj (san 1736 - 1813) ne kuttak sameekaran ka hal pun: gyaat kiya. bhaaskaraachaarya ne prijm evam shanku ke aayatan gyaat karne ki vidhi bataai tatha gunottar shreni ke yog ka sootr diya. ''kisi raashi ko shoonya se vibhaajit karne par anant praapt hota hai'', kehne vaale vah pratham ganitjnya the. mahaaveeraachaarya (san 850) ne sankhyaaon ke laghutam maan gyaat karne ki vidhi prastut ki. ganitsaarasangrah unki kruti hai.

shreedharaachaarya (san 850) ne dvighaati sameekaranon ke hal ki vidhi di jo aaj 'shreedharaachaarya vidhi' ke naam se gyaat hai. unki rachanaaen -'navashatika', 'trishtika', evam 'paateeganit' hain. 'paateeganit' ka arabi bhaasha mein anuvaad 'hijaabul taaraapt' sheershak se hua. aaryabhat dviteeya (san 920 -1000) ne mahaasiddhaant ki rachana ki jismein ankaganit evam beejaganit ka ullekh hai. unhonne π ka maan 22/7 nirdhaarit kiya. shreepati mishr (san 1039) ne 'siddhaantasheshar' evam 'ganittilk' ki rachana ki jismein kramachaya evam sanchaya ke liye niyam diye gaye hain.

nemichndr siddhaantachakravarti (san 1100) samuchchaya siddhaant ko pratipaadit karne vaale pratham ganitjnya the. unhonne saarvabhaumik samuchchaya evam sabhi prakaar ke maanachitran (Mapping) evam suvyavasthit siddhaanton ka pratipaadan kiya. gaileeliyo evam George kaintar ne is vidhi ka 'ek se ek' (van-too-van) maanachitran mein upayog kiya.

bhaaskaraachaarya dviteeya (san 1114) ne 'siddhaantashiromani', 'leelaavati', 'beejaganit' 'gol adhyaaya', 'grahaganitm' evam 'karanakautuhal' ki rachana ki. beejaganit ke kuttak sameekaranon ke hal ki chakravaal vidhi di. yeh vidhi jarman ganitjnya haraman heinkel (san 1839-73) ko bahut pasand I. heinkal ke anusaar lagareinj se bhi poorv sankhya siddhaant mein chakravaal vidhi ek ullekhaneeya khoj hai. peeyare di faramet (san 1601-1665) ne bhi kuttak sameekaranon ke hal ke liye chakravaal vidhi ka prayog kiya tha.

bhaaskaraachaarya dviteeya ke pashchaat ganit mein abhiruchi Kerala ke nambudari braahmanon ne prakat ki. 'aaryabhateeya' ki ek paandulipi malayalam bhaasha mein Kerala mein praapt hui. Kerala ke vidvaanon mein Narayan pandit (san 1356) ka vishesh yogadaan hai. unki rachana-'ganitkaumudi' mein kramachaya evam sanchaya, sankhyaaon ka vibhaageekaran tatha aindr jaalik (Magic) varg ki vivechana hai. Narayan pandit ke chhaatr parameshvar (san 1370 - 1460) ne madhyamaan (Mean Value) siddhaant sthaapit kiya tatha trikonamiteeya falan jya ka shreni hal diya :

jya (x) = x - x3/3 +

parameshvar ke chhaatr neelakanth somayaaji (san 1444-1544) ne 'tantrasangrah' ki rachana ki. unhonne vyutakram sparsh jya ka shreni hal prastut kiya :

tan\-1 (x) = x - x3/3 + x5/5

iske saath hi ganiteeya vishleshan, sankhya siddhaant, anant shreni, satat bhinn par bhi unka amoolya yogadaan hai. vyutakram sparsh jya ka unka shreni hal vartamaan mein greegareej shreni ke naam se prachalit hai.

sampoornaanand sanskrut vishvavidyaalaya ke praachaarya rahe sudhaakar Dwivedi (san 1860-1922) ne deerghavrutalakshan, goleeya rekhaaganit, sameekaran meemaansha evam chalan-kalan par maulik pustakein likheen. aadhunik ganitjnya raamaanujan (san 1887-1920) ne lagbhag 50 ganiteeya sootron ka pratipaadan kiya. svaami bhaarati teerath ji mahaaraaj (san 1884-1960) ne vaidik ganit ke maadhyam se guna, bhaag, vargamool evam ghanamool ki saral vidhi prastut ki. haal hi mein ameriki antariksh kendra ke vaigyaanik reek vrugs ke anusaar paanini ki ashtaadhyaayi vyaakaran computer aadhaarit bhaasha progaamar ke liye bahut hi upayukt hai. isa poorv 650 mein likhi is pustak mein 4000 beejaganit jaise sootr hain.

is prakaar yeh kaha ja sakta hai ki vividh aayaamon mein bhaarateeya ganit bahut hi samruddh hai. computer-bhaashaaon ke saath-saath aadhunik ganit praacheen bhaarateeya ganit ka rini hai.

bhaarateeya ganit : vidvaanon ke udgaar

Bhaarat aur vaigyaanik kraanti mein David gre likhte hain :

pashchim mein ganit ka adhyayan lambe samay se kuchh had tak raashtr keindrit poorvaagrah se prabhaavit raha hai, ek aisa poorvaagrah jo praay: badbole jaativaad ke roop mein naheen balki gairapashchimi sabhyataaon ke vaastavik yogadaan ko nakaarane ya mitaane ke prayaas ke roop mein parilkshit hota hai. pashchim anya sabhyataaon visheshakar Bhaarat ka rini raha hai. aur yeh rin ''pashchimi'' vaigyaanik parampara ke praacheenatam kaal - greek samyata ke yug se praarambh hokar aadhunik kaal ke praarambh, punarutthaan kaal tak jaari raha hai - jab Europe apne andh yug se jaag raha tha.

iske baad dau॰ gre Bhaarat mein ghatit ganit ke sarvaadhik mahatvapoorn viksit upalabdhiyon ki soochi banaate hue bhaarateeya ganit ke chamakate sitaaron jaise aaryabhat, brahmagupt, mahaaveer, bhaaskar aur maadhav ke yogadaanon ka sankshep mein varnan karte hain. ant mein ve jor dekar kehte hain -

Europe mein vaigyaanik kraanti ke vikaas mein Bhaarat ka yogadaan keval haasiye par likhi jaane waali tippani naheen hai jise aasaani se aur ataarkik taur par Europe keindrit poorvaagrah ke aadambar mein chhipa diya gaya hai. aisa karna itihaas ko vikrut karna hai aur vaishvik sabhyata mein Bhaarat ke mahaanatam yogadaan ko nakaarana hai.

bhaarateeya ganit : yoorokendreeyata ka shikaar

ab yeh spasht roop se maana jaane laga hai ki ganit mein Bhaarat ke yogadaan ko suniyojit tareeke se kamtar bataaya gaya hai ya usaki upeksha ki gayi hai. bhaarateeya maneeshiyon dvaara ganit mein bahut se yogadaan (anusandhaan aur vikaas) tatkaaleen yooropiyon ko pata the jinko unhone thoda bahut her-fer karke apne mool anusandhaan ke roop mein prastut kar diya.

bhaarateeya ganit ki shabdaavali

  • paateeganit
  • beejaganit
  • avyakt ganit
  • karan - ganana karne ki vidhi ya vidhi bataane wala granth
  • trairaashikvyavahaar (the rool of three)
  • yaavat-taavat - bhaarateeya beejaganit mein agyaat-raashi ke liye 'yaavat-taavat' (jitne ki utani maatra mein) ka prayog hua hai.
  • varn (variable) - brahmagupt ne agyaat raashi ke liye 'varn' (rang, akshar) shabd ka prayog kiya. isliye kaalaantar mein agyaat raashi ke liye kaalak (ka), neelak (ni), peelak (pi) aadi ka prayog hota raha.
  • karani (surd)
  • parikrm (mathematical operation)
  • mishrak-vyavahaar - ismein byaaj, svarn ki milaavat aadi se sambandhit prashn aate hain.
  • kshetraganitvyavahaar: (Measurement of Areas)
  • khaatavyavahaar: (calculations regarding excavations)
  • chhaayaavyavahaar: (Calculations relating to shadows)
  • ardhachhed - kisi sankhya N ka ardhachhed vah sankhya hai jisko 2 ke upar ghaat lagaane se N milta hai. at: 32 ka ardhachhed 5 hai.
  • ardhajya ya jyaardh (half cord)

bhaarateeya ganitjnya

inhein bhi dekhein

tippaniyaaain

ganit aur sangeet : pingal ne 300 E mein chhandashaastr naamak granth ki rachana ki thi. unane saanyojiki (kaambeenetareej) aur sangeet siddhaant ke paraspar sambandh ki pareeksha ki jo marsin, 1588-1648, dvaara sangeet siddhaant par rachit ek mahatvapoorn granth ka agradoot hai.

ganit aur vaastushilp : ankaganiteeya aur jyaamiteeya shreniyon mein ruchi utpann hone ka kaaran bhaarateeya vaastu ke design jaise mandir shikhar, gopuram aur mandiron ki bheetari chhat ki tech hain. vaastav mein jyaamiti aur vaastu saajasajja ka paraspar sambandh uchchatam star par viksit hua tha muslim shaasakon dvaara poshit vibhinn smaarakon ke nirmaan mein jo madhya Asia, faaras, Turkey, arab aur Bhaarat ke vaastushilpiyon dvaara nirmit kiye gaye the.

bhaarateeya ank pranaali ka prasaar : bhaarateeya ank pranaali ke pashchim mein prasaar ke pramaan ''krest of peekauk'' ke lekhak Joseph dvaara is prakaar diye gaye hain :

seberas sibokht, 662 E. ne ek Syriai pustak mein bhaarateeya jyotirvidon ke ''goodh anusandhaanon'' ka varnan karte hue unhein ''yoonaani aur bebeelaaniyn jyotirvidon ki apeksha adhik praveen'' aur ''sanganana ke unke bahumoolya tareekon ko varnanaateet'' bataaya hai aur uske baad usane unki nau ankon ki pranaali ke prayog ki charcha ki hai.

baahari kadiyaaain

sandarbh

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