Maths history

E}y; tpguk; gjpgG; - 2021.09.01 E}ypd; ngau; - fzf;fhW gf;fq;fs; - 85 gjpgG; - fzpj kd;wk; (aho;gg; hzk; Njrpa fy;tpapaw; fy;Y}up Nfhg;gha;.) gjpgG; upik - 19k; mzp fzpjk; -B gpupT Mrpupa ntspaPL khztu;fs; - aho;g;ghzk; Njrpa fyt; papaw; fyY; }up 1

Kd;Diu Nfhtpl;-19 ,lu;fhyj;jpYk; kdpj r%fk; mjpNtfj;jpy; KdN; dwpr; nry;fpd;wJ vd;why; mjw;F mbg;gilahf miktJ mwptpay; kw;Wk; njhopDl;g tsu;r;rpNaad;wp NtnwJTkpy;iy. rikayiw njhl;L re;jpukz;lyk; tiu mwptpd; cr;r epiyf;F nry;y mbj;jskpl;lJ fzpjkhFk;. vdNt fzpjj;jpd; rpwg;ig tsu;r;rpia mwpe;jpUf;f Ntz;baJ ,d;wpaikahjjhFk;. MW vd;why; ejp vd;W xU fUj;JKz;L vz; FwpaPl;ilAk; Fwpf;Fk;. ePuhdJ gha;e;J nry;tjw;F topia fhz;gpg;gjhy; ek; KdN; dhu;fs; ejpf;F MW vd;W ngaupl;ldu;. me;j tifapy; ,e; E}yhdJ fzpjk; njhd;W njhl;L tsu;e;J te;j topia $Wtjhy; ,jw;F vd;dhy; 'fzf;fhW\" vd;w ngau; #l;lg;gl;Ls;sJ. NkYk; ,e; E}yhdJ fzpj tsu;r;rpiaAk;> vz;FwpaPl;L Kiw tsu;r;rpiaAk; fijfs; kw;Wk; Gjpu;fspD}L ,yFthf vLj;jpak;GfpwJ. ehshe;jk; gy;NtW kpd; E}y;fs; ntspte;J nfhz;bUe;jhYk; jkpo; nkhop E}y;fspd; gpwg;G FiwthfNt css; J. ,izaj;jsj;jpy; gy;NtW E}y;fs; jkpopy; ntspte;J nfhz;bUe;jhYk; ,t;thwhd E}y;fis njhFj;J ifapUg;gpy; itgg; jd; %yk; vkf;Fk; vk;ik Nghd;w tUq;fhy re;jjpapdUf;Fk; gadj; Uk; vd;gjpy; Iakpy;iy. E}yhrpupau; 2

Mrpr; nra;jp tho;f;ifapd; Xu; mq;fkhfg; gpd;dpg; gpize;Js;s fzpjj;jpd; tsu;r;rpf;F ngUk; gq;fhw;wp tUk; fzpjkd;w Gj;jfj; njhFg;ghd “fzf;fhW” kyUf;F tho;j;Jr; nra;jp toq;Ftjpy; ngUkfpo;r;rpailfpd;Nwd;. fzpjj;jpd; Kf;fpaj;Jtk; vDk; NghJ gbj;j r%fj;jpdu; njhlf;fk; gbf;fhj r%fj;jpdUk; gad;gLj;j $ba vspa Kiwapid nfhz;ljhfTk; vkJ md;whl nraw;ghLfSld; njhlu;GilajhfTk; fhzg;gLfpwJ. fzpjj;jpd; Kf;fpaj;Jtj;jpid khztu;fspilNa nfhz;L Nrug; ;gjpy; Mrpupau;fspd; gq;F kpf Kf;fpakhd xd;whf tpsq;Ffpd;wJ. mtu;fs; jq;fs; khztu;fspd; fy;tp tsu;r;rpf;F JizGuptjw;F md;whl nraw;ghLfSld; ,ize;J khztu;fs; Gupe;Jnfhss; $ba tifapy; fw;gpj;jy; Mw;wy; cilatu;fshf jpfo;fpd;wdu;. khztu;fspd; mwpTg; grpf;F ,k;kyupy; css; fzpj tsu;r;rp gw;wpa jfty;fs>; Muk;g fhy vz;FwpaPl;L Kiwfs; kw;Wk; fzpj Gjpu;fs; Nghd;wd gaDs;sjhf mikAk; vd;gNjhL khztu;fspd; Mf;fj;jpwd; ntspg;ghL> rpe;jidf;fsk;> Njly; Nghd;wtw;Wf;F ,r; nrad;KiwahdJ toptFf;fpd;wJ. jw;Nghija Nfhtpl; - 19 fhyfll; j;jpy; ,iza topf;fy;tp Clhf jkJ fy;tpr; nraw;ghLfis midj;J khztu;fSk; Nkw;nfhz;L tUfpd;wdu;. mtu;fspd; NjlYf;F ,g; Gj;jfkhdJ ngUk; gq;fhw;wf; $bajhf tpsq;Ffpd;wJ. fzpjkd;w “fzf;fhW” vDk; Gj;jf njhFg;G ntapaPl;Lr; nraw;ghLfis Kdn; dLj;j tpupTiu ,izg;ghsu; jpU. j.jtNerd; mtu;fspd; topg;gLj;jy; cz;ikapNyNa fhyj;jpd; Njit fUjp Nkw;nfhss; g;gl;l nraw;ghlhfNt fUJfpd;Nwd;. mNjNtis fzpjkd;w cg fhg;ghsu; m.Ffd; mtu;fspd; gq;fspg;ghdJ nry;tp.Rgtu;zh jk;gpuhrh Nghd;w khzt MrpupauJ thz;ikia NkYk; tYT+lL; k; vd;gjpy; Iakpy;iy. ,tt; ifapy; fzpjj;jpd; tsu;r;rp kw;Wk; FwpaPl;L Kiwfs; rhu;ghd jftyf; isAk; css; lf;fp ntspaplg;gLk; ,k; kyuhdJ khztu;fspd; fy;tpr; nraw;ghLfSf;F cWJizahfTk; Mrpupau;fspd; fw;gpj;jy; nraw;ghLfSf;F cjtp GuptjhfTk; jpfOk; vd;gJld; khzt Mrpupau;fspd; gd;Kf MSik tpUj;jpf;Fk; mwpT>jpwd;>kdg;ghq;F tsu;r;rpf;Fk; fzpjkd;wr; nraw;ghLfspy; khzt Mrpupau;fspd; gq;fspg;G ghuhl;LjYf;FupaJld;> ,johrpupau; nry;tp.Rgtu;zh jk;gpuhrh kw;Wk; mtUld; ,ize;J gq;fspg;Gr; nra;j khzt Mrpupauf; sJ thz;ikia NkYk; tYt+l;Lk; vd;gjpy; Iakpy;iy. mtu;fsJ thz;ikj;JtKk; gd;Kf MSikAk; tpUj;jpaila tho;j;Jf;fisAk; ,iw MrpfisAk; njuptpj;J epw;fpd;Nwd;. njhlu;e;J Nfhtpl; - 19 ,lu;fhyfl;lj;jpYk; vkJ fy;Y}upapd; khzt Mrpupau;fs; rthy;fis re;ju;g;gq;fshf;fp NkYk; gy;NtW nraw;ghLfis Kdn; dLf;f ,iwahrpia Ntz;b epw;fpd;Nwd;. 3

tho;j;Jr; nra;jp jpU.jpUehde;jk; n[afhz;Bgd; cg glP hjpgjp (epjpAk; epu;thfKk;) aho;g;ghzk; Njrpa fy;tpapaw; fy;Y}up 'fzpjk; Xu; fiy\" fiyfs; vy;yhNk ,urpf;ff; $ba Vuhskhd tplaq;fs; cs;sd. Guhjd kdpj ehfupfq;fs; tsu;r;rpaile;j fhyk; njhl;L fiyfSld; ,ize;J fzpjKk; tsu;r;rpaile;J te;Jss; J. fzpjj;jpy; vz;fs; FwpaPLfshf tsu;r;rpaile;j tpjk; kpfr; RthuprpakhdJ. fzpj tuyhw;iw Nehf;Fk; NghJ ,J ed;F Gyg;gLfpwJ. gy;NtW fzpj vz;FwpaPl;L Kiwfs; Njhw;wk; ngw;W tsu;r;rpaile;j NghJk; ,e;Jff; Sk; mNugpau;fSk; ngUik nfhs;Sk; tifapy; cyfj;jhu; vy;NyhuhYk; Vw;Wf;nfhss; g;gl;L ,d;Wk; vdW; k; epiyj;jpUf;ff; $ba gz;Gfis nfhz;L fhzg;gLk; xNu xU vz; Kiwik ,e;J muhgpa vz;Kiwik MFk;. ,tt; hwhd vz;Kiwikfs; cUthfpa tpjk; njhlu;ghf fzpjk; fw;NghupilNa Rthurpaj;jd;ikia Vw;gLj;Jtjhf css; J. me;j tifapy; fzpj vz;Kiwik njhlu;gpy; Mrpupag; gapYdu;fspilNa ,uridia Vw;gLj;jp fzpjk; fw;wypd; kPjhd Mu;tj;ij Vw;gLj;Jk; kpfr;rpwe;j Kaw;rpahf fzf;fhW vDk; E}y; ntspaPl;bid ehd; fhz;fpNwd;. tpupTiu ,izg;ghsu; jpU.j.jtNerd; mtu;fspd; topfhl;lypd; fPo; ntsptUk; fzf;fhW vDk; ,e; E}Yf;F vdJ MrpfSk; tho;j;Jff; Sk; cupj;jhfl;Lk;. ,e; E}Yf;F jq;fs; Mf;fg+ut; khd xj;Jiog;ig toq;fpa midj;J Mrpupa khztu;fSf;Fk; vdJ tho;j;JfSk; MrpfSk; ghuhl;LfSk; cupajhFf. ,e;E}ypid cUthff; rpwg;ghf Mrpupa khztu;fis topfhl;b Cf;Ftpf;Fk; tpupTiu ,izg;ghsu; jpU.j.jtNerd; mtu;fisAk; ghuhl;LtjpYk; tho;j;JtjpYk; kfpo;tilfpd;Nwd;. ed;wp 4

tho;j;Jr; nra;jp nry;tp. nry;tkyu; nry;tuh[h cg glP hjpgjp (fy;tpAk; juNkk;ghLk;) aho;g;ghzk; Njrpa fy;tpapaw; fy;Y}up aho;g;ghzk; Njrpa fy;tpapaw; fy;Y}upapd; gj;njhd;gjhk; mzpapdupy; jkpo;nkhop %y fzpjg;gpupT B tFg;G Kfpo;epiy Mrpupauf; s; 'fzf;fhW\" vDk; kpd; E}ypid ntspapLtijapl;L fy;Y}upapd; cg gPlhjpgjp vd;w tifapy; vdJ ghuhl;Lf;fisAk; tho;j;Jf;fisAk; njuptpg;gjpy; ngUkfpo;r;rpailfpd;Nwd;. jdJ rKjhaj;jpd; tuyhw;iw kwe;j xUtd; epidthw;wiy ,oe;jtdhfpd;whd; vd;w tifapy; fzpjj;ij fw;gpf;fpd;w> fw;fpd;w midtUk; fzpjj;jpd; tuyhw;iwAk; tsu;rr; piaAk; mwpe;jtuf; shf ,Ug;gpd; kl;LNk fzpjk; njhlu;ghd Neu; kdg;ghq;fpid tpUj;jp nra;J nfhs;s KbAk;. Mrpupau; fy;tpf;fhd fzpj fiyj;jpl;lj;jpYk; $l fzpjj;jpd; tuyhWk; vz;fspd; Njhw;wKk; vz; FwpaPLfspd; tpUj;jpAk; ,lk;ngw;Ws;sik NkYk; ,jw;Fr; rhd;whFk;. kdpj ehfupfq;fs; Njhw;wk; ngw;W tsu;r;rp ngw;w fhyj;jpypUe;J vz; FwpaPl;L tbtq;fSk; tpUj;jp ngw;W te;Jss; d. gz;ila vz; FwpaPl;L tbtq;fspYs;s FiwghLfs; midj;ijAk; eptu;j;jp nra;atyy; jhd ,e;J mNugpa vz;KiwikNa ,d;Wk; tof;fpYs;sJ. ,e;j tifapy; jkpo;nkhop %y fzpjg;gpupT B tFg;gpidr; Nrue; ;j khzt Mrpupau;fsplkpUe;J fzpj tuyhW gw;wpa jfty;fis Nrfupj;J mtw;iwj; njhFj;J kpd; Ehyhf ntspapLtjw;fhd nraw;ghl;by; <Lgl;l khzt Mrpupia nry;tp. Rgtu;zh jk;gpuh[h mtu;fspd; Kaw;rp ghul;ljf;fnthd;whFk;. NkYk;> ,k; kpd; Ehypid ntspapLtjw;F Cf;fkspj;J topfhl;ba tpupTiu ,izg;ghsu; jpU. jq;fuh[h jtNerd; mtu;fSf;Fk; fzpj kd;w cgNgh\\fu; jpU. mUzhrr; yk; Ffd; mtu;fSf;Fk; vdJ kdkhu;e;j ed;wpfSk; tho;j;Jf;fSk; cupj;jhfl;Lk;. aho;g;ghzk; Njrpa fy;tpapaw; fy;Y}upapd; midj;J Kfpo;epiy Mrpupauf; spdJk; Gj;jhff; j;jpwd; tpUj;jpf;F ,t;thwhd ntspaPLfs; ce;Jrf;jpahf mikAk; vd;w ek;gpfi; fAld; ,k; Kaw;rpapy; <Lgl;l midtUf;Fk; vdJ tho;j;Jf;fs;. 5

Mrpr; nra;jp jpU.jqf; uh[h jtNerd; ,izg;ghsu; (fy;tpAk; ju Nkk;ghLk;) aho;g;ghzk; Njrpa fy;tpapaw; fy;Y}up aho;g;ghzk; Njrpa fy;tpapaw; fy;Y}upapd; gj;njhd;gjhk; mzp fzpj kd;wj;jpduhy; ,yj;jpudpay; E}yhf ntspaplg;gLfpd;w fzpj tuyhWk; vz;FwpaPl;L KiwiaAk; css; lf;fpa 'fzf;fhW\" vDk; E}ypw;F tho;j;Jr; nra;jp toq;Ftjpy; kdkfpo;r;rp milfpd;Nwd;. ,e;j covid-19 ,lu; fhyj;jpYk; aho;g;ghzk; Njrpa fy;tpapaw; fy;Y}upapy; fw;wy; fw;gpj;jy; nraw;ghLfs; kl;Lkd;wp ,izg;ghltpjhd nraw;ghLfSk; Vida fy;Y}upfSf;F KdD; jhuzkhf gPlhjpgjp jpU. Rg;gpukzpak; gukhde;jk; mtu;fspd; topfhl;lypYk; cg gPlhjpgjpfspd; newpg;gLj;jypYk; tpupTiuahsu;fspdhy; kpf rpwg;ghd Kiwapy; epfo; epiyahf Kdn; dLf;fg;gl;L tUfpd;wJ. ,jd; XH mq;fkhf fzpj kd;wj;jpduhy; ,e;j fzpj tuyhWk; vz;FwpaPl;L KiwiaAk; css; lff; pa 'fzf;fhW\" vd;w ,yj;jpudpay; E}y; ntspaplgg; Lfpd;wJ. fy;tpaw;fy;Y}upapd; fzpjj;Jiw, tpQ;Qhdj;Jiw Mrpupa khztu;fspd; fzpj ghl fiyj;jpl;lj;jpy; fzpj tuyhWk; vz;FwpaPl;L KiwAk; css; lf;fg;glL; s;sJ. vdNt ,e;j fzpj tuyhw;iwAk; vz;FwpaPl;L KiwiaAk; nfhz;l ,yj;jpudpay; E}y; Mrpupa khztu;fSf;F ngupJk; gaDilajhf miktJld; Vida fzpjk; fw;Fk; khztHfSfF; k; gaDilajhf ,Uf;Fk; vd;gjpy; vJtpj Iag;ghLk; ,y;iy. me;j tifapy; ,e;E}iy ntspapLtjw;F cjtpa fzpjk; B gpupitr; Nrue; ;j Mrpupa khztu;fSf;Fk; ,johrpupau; nry;tp.j.Rgtu;zh mtu;fSf;Fk; vdJ ghuhl;Lf;fis njuptpg;gJld; ,t; ,yj;jpudpay; E}y; ntsptUtjw;F vdJ tho;j;Jf;fisAk; njuptpj;Jf; nfhs;fpNwd;. 6

fzpj kd;wg; nghWg;G tpupTiuahsupd; cs;sj;jpypUe;J...... tpidj;jpwDld; fw;gpj;jiy Nkw;nfhst; jw;F ghlmwpT> fw;gpj;jy; jpwd;> nghWik vDk; gz;Gfs; Mrpupauplj;jpy; ,Uf;fNtz;Lnkd vjpu;ghu;f;fg;gLfpd;wd. fzpj ghlj;ijf; fw;gpfF; k; xU MrpupaUf;F fzpj tuyhWgw;wpa mwpT mtrpakhFk;. 'fzf;fhW\" vDk; ,e;E}y; ,j;Njitia XusT jPu;f;Fk; vd vz;Zfpd;Nwd;. Mrpupaj; njhopyy; njhd;Wnjhl;L ele;JtUk; xU njhopyhFk;. ,j;njhopy; njhlu;e;J eilngwhJ tpLkhapd; kdpj ehfupfk; nraypoe;JtpLk;. fzpjf; fw;gpj;jy; njhopyhdJ ghlrhiyf;Fg; Gwk;ghf tlP ;bYk; r%fj;jpYs;s %j;NjhupdhYk; Nkw;nfhss; g;gl;L tUfpd;wJ. Mdhy; fzpjj; Jiwrhu; Mrpupau;fs; ,j; njhopiy ghlrhiy vDk; Kiwrhu; mikg;gpy; xOq;fhf nra;JtUfpd;wdu;. fzpjj; Jiw Mrpupaupfs; vjid vt;thW khztu;fSf;F fw;gpff; Ntz;Lk; vd;gij mwpe;jpUg;gJld; jhk; fw;gpj;jtw;iw khztu;fs; ve;jstpw;F cst; hq;fpAs;shu;fs; vd;gij kjpg;gPL nra;af;$batu;fshfTk; ,Uf;fNtz;Lk;. ,j;jpwd;fis jd;dfj;Nj tsu;j;Jf;nfhs;sNtz;ba xU fzpj Mrpupaiug; nghWj;jtiuapy; 'fzf;fhW\" vDk; ,e;E}y; ngupJk; cjTk; vd;gjpy; vtt; pj IaKkpy;iy. ,e;E}y; fzpjj;jpYs;s Mokhd fUj;Jff; is tpsf;Ftjw;F Kw;gltpy;iy. khwhf fzpjj;Jiw Mrpupauhf Mrpupaj; njhopypy; Gjpjhf ,ize;Jnfhs;gtu;fSf;F fzpj tuyhWgw;wp mwpKfk; nra;tij Kf;fpa Nehff; khff; nfhz;L njhFf;fg;gl;Ls;sJ. ,d;W fy;tpapaw; fy;Y}upfspd; fy;tpr; #oy; kpfNkhrkhd neUf;fbfisAk; gpur;rpidfisAk; Kfq;nfhLj;J tUfpd;wJ. 'nfhNuhdh tho;T\" ,ay;ghd fy;tpapaw; fy;Y}upr; nraw;ghLfis ngUksT Klf;fpAs;sJ. ,Ug;gpDk; fy;tpapaw; fy;Y}up vDk; epWtdj;jpD}lhfNt Kfpo;epiy Mrpupa khztu;fspd; fy;tpr; nraw;ghLfs; epiwTngw Ntz;bAs;sJ. ,j; Njitapd; xU gFjpia ,e;E}y; epiwTnra;Ankd ek;Gfpd;Nwd;. fy;tpapaw; fy;Y}upfspy; fw;Fk; fzpjj;Jiw khztu;fSf;Fk;> tpQ;Qhdj;Jiw khztu;fSf;Fk; fzpj tuyhWk; fzpj vz;FwpaPl;L KiwfSk; vd;gJ xU Nju;r;rpahf ,Ug;gjdhy; ,f; fzpj tuyhw;Wj; njhFg;ghdJ> mtu;fspd; fw;wy; Njitia eptu;j;jp nra;Ak; tifapy; ,t; ,yj;jpudpay; Gj;jfkhd 'fzf;fhW\" cUthf;fg;gl;Ls;sJ. NkYk; fy;tpapaw; fy;Y}upfspy; ,Wjpg; guPl;iria vjpu;Nehff; tpUf;Fk; fzpj tpQ;Qhd Mrpupa khztu;fspd; Rafw;wYf;fhf ,e;E}iyg; gadg; Lj;jf; $bajhf Mf;fg;gl;Ls;sik ,e;E}ypd; rpwg;gk;rq;fspy; xd;whFk;. ,e;E}y; NkYk; rpwg;Gw miktjw;F cjTk; tifapy; rpwe;j MNyhridfisAk; topfhl;ly;fisAk; toq;fpa vk; fy;Y}upapd; kjpg;ghu;e;j gPlhjpgjp Rg;gpukzpak; gukhde;jk; Iahtpw;Fk; fzpjj; Jiwiar; rhu;e;j cg gPlhjpgjpfshd n[afhz;Bgd; Iahtpw;Fk; nry;tuh[h nry;tkyu; mk;kzpf;Fk; fzpj kd;wk; rhu;ghd tho;j;Jf;fisAk; ghuhl;Lf;fisAk; njuptpj;Jfn; fhs;fpd;Nwhk;. NkYk; ,e;E}ypd; cUthf;fj;jpy; ngupJk; Jizepd;w fzpjj; Jiw rpNu];u tpupTiuahsUk; tpupTiu ,izg;ghsUkhfpa jpU jq;fuhrh jtNerd; mtu;fSf;Fk; fzpj kd;wk; rhug; hd tho;j;Jf;fisAk; ghuhl;Lf;fisAk; njuptpj;Jf;nfhsf; pd;Nwhk;. ,tt; ifapy; fzpjk; rhu;e;j nraw;ghLfisAk; mJ rhu;ghd jfty;fisAk; css; lf;fp ntspaplg;gLk; ,k; kyuhdJ khztu;fspd; fy;tpr;nraw;ghLfSf;F cWJizahfTk; 7

Mrpupau;fspd; fw;gpj;jy; nraw;ghLfSf;F cjtp GuptjhfTk; jpfOk; vd;gjpy; Iak; ,y;iy. vdNt NkYk; ,J Nghd;w kyu;fis fzpjkd;wk; ntspapl ,iwtd; JizGupthuhf. jpU.mUzhryk; Ffd; BSc.(cey), PGDE(Merit), MSc.(Phy of Mat), MATE(International), PGDEM(Merit) fzpjkd;w nghWg;G tpupTiuahsu; aho;g;ghzk; Njrpa fy;tpapaw; fy;Y}up 8

E}yhrpupaupd; cs;sj;jpypUe;J..... aho;g;ghzk; Njrpa fy;tpapaw; fy;Yhupapd; ,uz;lhk; tUl Kfpo;epiy Mrpupa khztuhf fy;tp gapYk; ehd; vkJ fzpjk; B gpupT khztu;fspd; ; cjtpAld; fzpjtuyhW kw;Wk; vz;FwpaPl;L Kiw gw;wpa 'fzf;fhW\" vDk; E}y; tbtkhf xUq;fpizf;f fpilj;j tha;g;Gf;fhf kfpo;rr; pailfpd;Nwd;. mDkjpaspj;j kjpg;gpw;Fupa gPlhjpgjp cau;jpU.Rg;gpukzpak; gukhde;jk; [ah mtu;fSf;F ed;wpfisj; njuptpj;Jf; nfhz;L vd; kPJ ek;gpfi; f nfhz;l topfhl;ba tpupTiuahsu;fshd tpupTiu ,izg;ghsu; jpU. j. jtNerd; Iah kw;Wk; fzpjkd;w nghWg;G tpupTiuahsu; jpU.m.Ffd; Iah mtu;fSf;Fk; ed;wpfisj; njuptpj;Jf; nfhs;fpd;Nwd;. 'fzf;fhW\" vDk; ,e; Ehypy; vkJ fiyj;jpl;lj;jpy; css; lf;fg;gl;l fzpjtuyhWk; vz;FwpaPl;L KiwfSk; mlq;fpa jfty;fs; css; lff; g;gl;Ls;sJ. ,e; E}ypd; %ykhf ehk; khj;jpuky;ykhy; fy;tpapaw; fy;Y}upapy; gapytpUf;Fk; vjpu;fhy fzpj tpQ;Qhd khzt Mrpupau;fSk; kw;Wk; fzpjk; fw;gpj;Jf; nfhz;bUf;Fk; Mrpupau;fSk; fzpjj;jpy; Mu;tKs;NshUk; gadilthu;fs; vd ek;Gfpd;Nwd;. ,J Nghd;w E}y;fis fzpjj;Jiw kl;Lky;yhJ Vida Jiw khzt Mrpupau;fSk; vjpu;fhyj;jpy; ntspapl Ntz;Lk; vd;w vjpu;ghu;g;GlDk; ek;gpfi; fAlDk; ,e;E}iy ntspapLtjpy; kpf;f kfpo;tilfpd;Nwd;. nry;tp.j.Rgtu;zh fzpjk; B gpupT 19k; mzp aho;g;ghzk; Njrpa fy;tpaw; fy;Y}up 9

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nghUslf;fk; fzpj tuyhW rupj;jpuj;jpw;F Kw;gl;l fhyk; 1.fzpjj;jpd; Muk;gk; 2.Kfj;jy; msitfs; 3.kzpr;rl;lk; 4.Neuf;fzf;fPl;L Kiw 5.g+[;[pak; mwpKfk; #dpaj;jpypUe;J g+[;[pak; g+[;[paj;jpd; jsk; 6.tbtpay; mwpKfk; 7.Nfhzfzpjk; 8.π,d; kjpg;G fz;Lgpbg;G 9.gpjhfurpd; fz;Lgpbg;G 10.jkpof fzpj Nkij uhkhD[u; fzpj tsu;r;rp 1.ghgpNydpau; 2.vfpg;jpau; 3.,e;J muhgpau; 4.rpe;Jntsp ehfupfk; vz;fs; neLtupirapy; fzpjKk; tuyhWk; fzpjtpaYk; msitapaYk; fzpjtpay; kw;Wk; msitapay; gw;wpa ,e;jpa tuyhW 1.Ntjfhyj;jpw;F Ke;jpa fhyk; 2.Ntjfhyk; 3.Ntjfhyj;jpw;F gpe;jpa fhyk; CE 1200 tiu 4.13 k; E}w;whz;L Kjy; jw;fhyk; tiu 5.etPd fhyk; vz; FwpalP ;L Kiw ,aw;fzpjk; fzpjGjpu; 12

fzpj tuyhW rupj;jpuj;jpw;F Kw;gl;l fhyk; fzpjj;jpd; Muk;gk; fzpjk; vg;NghJ Muk;gkhdJ vd;gij rupahf $wKbahJ tpbDk; Mjp fhyj;jpy; tho;e;j kf;fs; nghUs;fis xg;gpl mwpe;j NghNj fzpjk; tsuj; njhlq;fp tpl;lJ. kpfg; goikahd NtLtu; fhyj;jpNyNa vz;Zjy; vDk; csuPjpahd nraw;ghL kdpjDs; Njhd;wpaJ. Mjp kf;fs; jq;fs; fhy; eilfisAk; kw;w nghUl;fisAk; fzf;fpLtjw;F fw;fs;> Fr;rpfs; Kjypatw;iw gadg; Lj;jpdu;. Kbr;Rf;fs; NghLjy;> milahsf;Fwpfs; ,Ljy; Nghd;w Kiwfis gpd;gw;wpdu;. ,itNa gbg;gbahf tsu;e;J ,d;W ehk; gadg; Lj;Jk; vz;fshf kyu;e;jd. Kfj;jy; msitfs; Mjpkf;fs; jq;fspd; cw;gj;jp nghUl;fis gupkhwpf;nfhs;s njhlq;fpa NghJ gz;lkhw;W Kiw eilKiwf;F te;jJ. ,jdhy; jq;fspd; nghUl;fis mse;J nfhLf;fj; njhlq;fpdu;. mg;NghJ njhlq;fpaJ jhd; Kfj;jy; msitfs;> Nrh;> MohfF; > gb> Kok; Nghd;w msitfs; mg;NghJ Njhd;wpaitNa. ,t;thNw jq;fsplk; css; ,lq;fspd; msit Fwpf;f fhzp> miuf;fhzp> trP k; Nghd;wtw;iw gadg; Lj;jpdu;. ,t;thwhf kf;fspd; tho;f;ifj;juk; gbg;gbahf cau Muk;gpj;j fhyj;jpy; gy mwpQh; ngUkf;fs; Njhd;wp gy fz;Lgpbg;Gf;fis je;jdh;. NtW topfshy; jPu;f;f Kbahj gpurpdq;fSf;F jPu;it Kd;itf;Fk; xU khu;f;fkhfNt fzpjk; Mjpfhyj;jpy; Muk;gkhdJ. 13

kzpr;rl;lk; Muk;g fhyj;jpy; vz;zpf;ifia fzf;fpl mwpe;j kf;fs; gpd;dh; $l;ly>; fopj;jy;> tFj;jy;> ngUf;fy; Kjypatw;iw nghUl;fisAk; fw;fisAk; Fr;rpfisAk; nfhz;L nra;a Muk;gpj;jdh;. ,e;j fhyfll; j;jpy; rdP h;fs; fk;gpfisf; nfhz;l xU kzpr;rl;lj;ij cUthf;fp mjd; %yk; $l;ly;> fopj;jy;> tFj;jy;> ngUf;fy; Kjypatw;iw nra;jdh;. mth;fs; cUthf;fpa rl;lj;jpy; fk;gpfF; 7 kzpfs; tjP k; ,Ue;jd. mjw;F ‘mgf];( Abacaus)’ vd ngaupll; dh;. ,ijNa ,U kzpfis Fiwj;J [g;ghdpah;fs; ‘Nrhughd;’ vd;W ngahpl;ldh;. ,t;thW fzpjtpaypd; mbg;gil nraypfs; mwpKfkhapd. ,d;iwa fzpg;nghwp gpwf;f ,e;j kzpr;rl;lNk mbg;gilahf ,Ue;jJ vd;W jw;NghJ mwpag;glL; s;sJ. gpd;dh; N[hd;Neg;gpah; vd;w mwpQh; NkYk; rpy khw;wq;fis nra;J xd;gJ rl;lq;fis nfhz;L xU fUtpia mikj;jhh;. ,jw;F Neg;gpah; rl;lq;fs; vd ngahpl;ldh;. ,jd; %yk; th;ff; %yk; (Roots) tiu fzf;fpl Kbe;jJ. ,thpd; ,e;j mupa fz;Lgpbg;G jhd; ,d;W ehk; rpy khw;wq;fSld; klf;if (Logarithms) vd;w ngahpy; gad;gLj;JfpNwhk;. Neuff; zf;fPL Kiw fhyj;ij Fwpg;gjw;fhf goq;fhyj;jpy; kzy; rhbfis gadg; Lj;jp kzy; nrhhpe;j msit nfhz;L Neuj;ij fzf;fpl;ldh;. gpd;dh; kzYf;F gjpy; ePiug; gadg; Lj;jpdh;. ,f; fhyfl;lj;jpy; thdtpay; gw;wp fypypNah> Mhpagl;lh; vd gyh; Muha;r;rpfis nra;J g+kp #upaid Rw;wp tUfpwJ vd;Wk;> g+kp jd;idj;jhNd xU Kiw Rw;wpf;nfhss; 23 kzp 56 epkplk;> 4.091 nehb vd $wpdh;. mit gpd;dh; tpQ;Qhd g+h;tkhf ep&gpf;fg;gl;L gpd;dh; mJNt xU ehisf;F 24 kzpNeuk; vd Fwpf;fg;gl;lJ. ehk; NgRtJ vijg;gw;wp vd;W njupahjJk; ehk; nrhy;tJ rupahdjh vd;W njupahjJkhd xU ghlk; fzpjk; vd ,dqf; hzg;glyhk;. - Ngu;ld; u];]y; 14

g+[;[pak; mwpKfk; ,jpfhrq;fspy; ntWik my;yJ ntw;wplk; vd;gijf; Fwpf;f “#d;ak;” vd;w nrhy; gad;gLj;jg;gl;lJ. gpd;dh; ‘Rd;dk;’ vd;w nrhy; gadg; Lj;jg;gl;lJ. ,ijNa fzpj Nkijfs; kjpg;gpy;yhjitfSf;F Gs;spiia (.) gadg; Lj;jpdh;. gpd;dh; Fog;gj;ij jtph;f;f Gs;spia Rw;wp tl;lkpl;ldu;. gpd;dh; O vd gadg; Lj;jpdh;. ,e;j #d;ak; vd;w nrhy; ntt;NtW nkhopfspy; khwp gpd;dh; gpnuQ;R nkhopapy; ZERO vd toq;fg;glL; mJNt epiyj;J tpl;lJ. ,ij fz;Lgpbj;J cyfpw;F toq;fpa ngUik ,e;jpahitNa rhUk;. ,ij toq;fpath; xU ngah; mwpag;glhj ,e;jpah; vd mwpag;glL; s;sJ. ,e;j 0 jhd; ,d;W fzpg;nghwpapy; jiyaha vz;zhf css; J. ; “fp.K ,uz;L my;yJ %d;whk; E}w;whz;LfSf;F Kdd; u; ,e;jpa ,ir tpj;thd; gpq;fyh igdup vz;fis gadg; Lj;jpapUf;fpwhu;” vd;W mnkupf;fhtpd; Xfy; N`hkh gy;fiyf;fofj;jpy; mwptpay; kw;Wk; thdpay; Nguhrpupau; rpj;jhuj; ff; $Wfpwhu;. ,it midj;JNk R+dpaj;jpy; ,Ue;J g+[;[paj;ij fz;lwpe;j ,e;jpahtpy; Njhd;wpaJ. ,e;jpa mwptpay; kw;Wk; fzpjtpayhsu;fshd Mu;agl;lh fp.gp 476 Mk; Mz;bYk;> gpuk;kFg;jh fp.gp 598 Mk; Mz;bYk; gpwej; du;. ,tu;fs; ,UtUk; etPd jrk ,l kjpg;gPl;L Kiwia Kjd;Kiwahf tptupj;j KdN; dhbfs; vd;Wk; g+[;[paj;jpd; gadg; hli; l epu;tfpf;Fk; jw;Nghija tpjpfis toq;fpatu;fshfTk; mwpag;gLfpd;wdu;. tl;l tbtpy; g+[;[paj;ij vOj;J tbtpy; Kjy;Kiwahf gjpT nra;jpUg;gJ Fthypau; Nfhl;ilapy; cs;s Myar; Rtupy; vd;W $wg;gl;lhYk;> %d;whk; my;yJ ehd;fhk; E}w;whz;ilr; Nrue; ;j gz;ila gÑhyp ifnaOj;J gpujpapy; (Bhakshali manuscript) g+[;[paj;jpd; tbtk; fhzg;gLtjhf $wg;gLfpwJ. ,Ue;j NghjpYk;>jl;rryP j;jpy; (jw;Nghija ghfp];jhdpy; css; J) fpilj;j goq;fhy ,e;jpa ghÑhyp ifbaOj;Jg; gpujpapy; g+[;[pak; vd;w vOj;J fhzg;gLfpwJ. jw;NghJ ,Jjhd; g+[;[paj;jpd; kpfg;gioa vOj;J tbtkhf fUjg;gLfpwJ. fUk;igg; Nghy; fzpjKk; Ritj;jtUf;F ,dpik jUk; Mf;];NghuL; gy;fiyf;fof fzpjtpay; Nguhrpupau; khu;f];L nrsNlha;> gy;fiyf;fof ,izaj;jsj;jpy; ,t;thW Fwpg;gpl;Ls;shu;. “fzpj tuyhw;wpy; kpfg;ngUk; rhjidahff; fUjg;gLk; g+[;[pak; vd;w vz; gÑhyp ifnaOj;J gpujpapy; Gs;spiag; Nghd;W fhzg;gLk; tbtj;jpy; ,Ue;J cUthdJ. ,e;jpahtpd; gz;ila fzpjtpayhsu;fs; toq;fpa g+[;[pak; 15

,d;iwa etdP cyfpd; mbj;jskhFk; tifapy; %d;whk; E}w;whz;bNyNa tpijf;fg;gl;lJ. gy E}w;whz;Lfshf ,e;jpa Jiz fz;lj;jpy; fzpjk; vt;thW Jy;ypakhdjhf ,Ue;jJ vd;Wk; fzpj gadg; hL toikAld; ,Ue;jJ vd;gijAk; ,e;j fz;Lgpbg;Gfs; Rl;bf;fhl;Lfpd;wd.”,e;jpahitj; jtpu gpw ehLfspy; Vd; g+[;[paj;jpd; fz;Lgpbg;G elff; tpy;iy?,jw;fhf $wg;gLk; rpy fhuzq;fs; Rthurpakhdit. rpy fyhrhuq;fspy; R+dpak; vd;gJ vjpu;kiw vz;zq;fis Njhw;Wtpg;gjhf fUjg;gl;lJ. cjhuzkhf> INuhg;ghtpy; g+[;[paj;ij gad;gLj;Jtjw;F fpwpj;Jt kjk; Muk;g fhyq;fspy; jil tpjpj;jJ. mjw;F fhuzk; vd;d njupAkh? “flTNs midj;Jkhf ,Ug;gjhy; R+dpak; mjhtJ xd;WNk ,y;iy vd;W nrhy;Yk; G+[;[pak; rhj;jhidf; Fwpf;Fk; FwpaPL”. vdNt ,JNghd;w fUj;Jf;fshy; R+dpaj;ij gw;wpa fz;Lgpbg;Gfs; gpw ehLfspy; g+[;[pakhf ,Ue;jNjh vd;dNth? Mdhy; Md;kPf mwpT Nkk;gl;bUe;j ,e;jpahtpy; jpahdKk; g+[;[paKk; fz;Lgpbf;fg;gl;ld. gz;ila ,e;jpah> fzpjj;Jiwf;F toq;fpa kw;nwhU Kf;fpakhd fz;Lgpbg;ghd igdup vz;fs; (,Uk vz;fs;) etPd cyfj;jld; Mokhd njhlu;G nfhz;lJ. ,e;j igdup vz;fs; etdP fzpg;gPLfs; kw;Wk; fzpdpfspd; mbg;gilahFk;. gpw ehfuPfq;fis tpl> ,e;jpa fyhrr; huj;jpy; g+[;[pak; njhlu;ghf tpupthd jj;Jtq;fs; fhzg;gLfpd;wd. ,e;jpa fyhrr; huj;jpy;> jpahdk; kw;wk; Nahfh Nghd;w Kiwfs; %yk; kdij R+dpakhf mjhtJ xd;Wkpy;yhy; ntWikahf itfF; k; Kiwfs; Njhd;wpd. ,e;J kw;Wk; ngsj;j rkaq;fspy;>g+[;[pak; kw;Wk; mjDld; njhlu;Gila Nfhl;ghLfs; Nghjpf;fg;gLfpd;wd. G+[;[paj;jpd; fz;Lgpbg;G fzpjj;Jiw tsu;r;rpf;F rpwe;j gq;fspg;ig toq;fpaJ. Ez;fzpjk;> ,aw;gpay;> nghwpapay; kw;Wk; etdP njhopy;El;gj;jpw;Fk; mbg;gilahdjhf jpfo;fpwJ g+[;[pak; vd;Dk; Ropak;. fzpjk; kw;Wk; tpQ;Qhdj;jpy;> g+[;[paj;jpd; fz;Lgpbg;G kpfTk; Mf;fg+u;tkhdJ. ,d;iwa cyfpy;> cyfpd; xtn; thU efu;Tf;Fk; mbg;gilahf jpfo;tJ g+[;[pak;.fzpjk;> Ez;fzpjk;> ,aw;gpay;> nghwpapay; kw;Wk; etdP njhopy;El;gj;jpd; njhlf;fKk; g+[;[paj;jpypUe;Nj rhj;jpakhdJ. etdP ,e;jpahtpw;Fk;> etdP cyfj;jpw;Fk; mbj;jskpl;l kpf Kf;fpakhd fz;Lgpbg;G ,e;jpa fyhrhuj;jpy; ve;j msT Kf;fpaj;Jtk; tha;e;jJ? #dpaj;jpy; ,Ue;J g+[;[pak; xU ,e;jpa njhd;kapy; tyY; eu; Njt;jj; gl;ehaf;> xU Kiw nrhd;d fij. Nguuru; mnyf;rhz;lu; ,e;jpahtpw;F te;jJ njhlu;Gila fij mJ. 16

Nghupy; gy ehLfis ntd;w gpwF ,e;jpah te;jile;j mnyf;rhz;lu;> ehfhNahfp xUtiu re;jpf;fpwhu;. epu;thzkhf ghiwapy; mku;e;J thdj;ij Nehff; p ghu;j;Jf; nfhz;bUe;jhu; mtu;. mtiu ghuj; ;J mnyf;rhz;lu; Nfl;lhu;> “vd;d nra;J nfhz;bUf;fpwPuf; s;?” “xd;Wkpy;yhjij czu;e;J nfhz;bUf;fpNwd;> eP vd;d nra;J nfhz;bUf;fpwha;” vd;W me;j Qhdp Nfl;lhu;. “ehd; cyfj;ij ntd;W nfhz;bUf;fpNwd;” vd gjpyspj;jhu; mnyf;rhz;lu;. ,UtUNk rpupj;jdu; vjpupy; ,Ug;gtu; igj;jpak;> Neuj;ij tzP bg;gjhf ,Utupd; kdjpYk; gu];guk; Fthypau; Myar; R+oypy; g+[;[pak; vd;w FwpaPL cUthff; g;gLtjw;F gy fhyk; Kdd; Nu ,e;j rk;gtk; ele;NjwpaJ. Mdhy; me;j Jwtp R+dpaj;ij gw;wp rpe;jpj;Jf; nfhz;bUe;jjw;Fk; xU njhlu;G ,Ue;jpUf;fyhk;. etdP b[pl;ly; fzpdpfs; ,uz;L rhj;jpaq;fspd; mbg;gilapy; ,aq;Ffpd;wd. ‘Md;’ kw;Wk; ‘M/g;’ (‘On’ & ‘Off ’) ‘On’ vd;gij ‘1’ vd;Wk; ‘Off ’ vd;gij ‘0’ mjhtJ g+[;[pak; vd;Wk; nfhz;L ,it ,aq;Ffpd;wd. “fp.K ,uz;L my;yJ %d;whk; E}w;whz;LfSf;F Kd;du; ,e;jpa ,ir tpj;thd; gpq;fyh igdup vz;fis gadg; Lj;jpapUf;fpwhu;” vd;W mnkupf;fhtpd; Xfy; N`hkh gy;fiyf;fofj;jpy; mwptpay; kw;Wk; thdpay; Nguhrpupau; rpj;jhu;j; ff; $Wfpwhu; ,it midj;JNk R+dpaj;jpy; ,Ue;J g+[;[paj;ij fz;lwpe;j ,e;jpahtpy; Njhd;wpaJ. G+[;[pak; vd;w vz;zpd; g+ut; Pfk; my;yJ Njhw;wj;ijg; gw;wp MuhAk; gzpapy; <Lgl;Ls;sJ neju;yhe;J ehl;il Nru;e;j [PNuh Mupf;,e;jpah (ZerOriglndia) vd;w mikg;G mjd; nrayhsu; lhf;lu; gPll; u; Nfhgl;]; g+[;aj;jpd; Njhw;wj;ijg; gw;wp xU flL; iuapy; ,t;thW Fwpg;gpLfpwhu;: “fzpjj;jpy; gadg; Lj;jg;gLk; g+[;[pak; vd;gJ ntWik njhlu;ghd kfhy jj;Jtkhd R+dpaj;jpy; ,Ue;J cUthfpapUf;fyhk;. R+dpaj;jpd; jj;Jtk; vd;gJ> xUtu; kdjpy; ,Ue;J gjpTfs; kw;Wk; vz;zq;fis ePf;fp ntWikahf;FtJ vd;w Gj;jkj Nfhl;ghLfspy; xd;whFk;. caukhd tpjhdq;fs; nfhz;l NfhGuq;fs;> mofpa Ntiyg;ghLfSld; $ba rpw;gq;fs; kw;Wk; tz;zkakhd Xtpaq;fs; nfhz;l xd;gjhk; E}w;whz;L Myak; xd;iwAk; ,f; Nfhl;ilapy; fhzyhk;. ,e;j Myak; ghiwfis nrJf;fp cUthf;fg;gl;lJ vd;gJ Fwpg;gplj;jf;fJ. G+[[; paj;jpd; jsk; rJu;G[ vd;W miof;fg;gLk; ,e;j Myak;> ,e;jpahtpd; gpw Guhjd Myaq;fis NghyNt tbtikf;fg;gl;bUe;jhYk; vOj;J tbtpy; g+[;[pak; mikf;fg;glb; Uf;Fk; Guhjd jsk; ,e;j Myak; vd;gJ ,jw;F jdpr;rpwg;G Nruf; ;fpwJ. 17

Myar; Rtupy; xd;gjhk; E}w;whz;by; nrJf;fg;gl;l vOj;Jf;fspy; ‘270’ vd;w vz; njspthfj; njupfpwJ. NkYk; gz;ila fhyj;jpy; ,Ue;Nj ,e;jpahtpy; fzpjj;jpy; kpfg;ngupa Mu;tk; ,Ue;jJ. gz;ila fpNuf;fu;fs; vz;fis 10>000 vd;gJ tiu fzf;fpl;l epiyapy; ,e;jpa fzpjtpayhsu;fs; bupy;ypad; Nghd;w kpfg;ngupa vz;fis gad;gLj;jj; njhlq;fptpl;ldu;. ,d;/gpdpl;b vd;W miof;fg;gLk; Kbtpypapd; gy;NtW tiffisAk; ,e;jpa fzpjtpayhsu;fs; gad;gLj;jpdhu;fs; vd;gJ fzpj;Jiwapy; ,e;jpau;fspd; KdN; dhbahf ,Ue;jhu;fs; vd;gij czu;j;JfpwJ. tbtpay; mwpKfk; gpNsl;Nlh vd;w jj;Jt Nkij fp.K. 387 ,y; [Nuhg;ghtpy; Kjy; Kjy; gy;fiyf;fofkhfpa vnjd;]; mfhlkpay; tbtq;fis msf;f rpy fUtpfis cUthff; pdhh;. mth; cUthf;fpait jhd; msTNfhy; (Scale) > fhk;g]; (Compass) Kjypad. mitjhd; ,d;W eilKiwapy; cs;sd. Nfhzfzpjk; gy fzpjNkijfs; cUthf;fpa mbg;gilj; jfty;fis nfhz;L Nfhg;ghe; pf;f]; vd;w Nghyhe;J ehl;L mwpQh; Nfhzfzpjk; vd;w gphpit toq;fpdh;. gpd;dh; ,thpd; fz;Lgpbg;Gf;fis gad;gLj;jp fypypNah Nghd;w thdptpay; mwpQh;fs; mz;lj;ijg; gw;wp gy cz;ikfis fz;lwpe;jdh;. ,af;fk;> tpir Nghd;w cz;ikfis fz;lwpa ,thpd; Nfhzfzpjk; jhd; gad;gl;lJ. ������- apd; kjpg;G fz;Lgpbg;G (1) fp.gp 475-550 fhyfll; j;jpy; ghld; htpy; tho;e;j ,e;jpa fzpjNkij “Mhpgghj;jpah” vd;w E}iy vOjpdhh;. mjpy; ,aw;fzpj tpjpfs; > Nfhztpjpfs; vd gy mupa fz;Lgpbg;Gf;fs; jug;gl;bUe;jd. 18

,thpd; $w;Wg;gb 62832 ia 20000 My; tFf;f fpilg;gNj ������ vd;gjd; kjpg;G vd $wpdhh;. mjhtJ 3.1416 vd $wpdhh;. (2) Mf;fpkpb]; vd;gth; 3������ f;Fk; 3������������ f;Fk; ,ilg;gl;J jhd; ������ ,d; kjpg;G vdf; $wpdhh;. ������ ������������ (3) jkpof fzpjNkij ,uhkD[u; ������ apd; kjpg;ig fhz fPo;fhZk; tha;g;ghl;bid gad;gLj;jpdhh;. ,e;j tha;g;ghl;il gad;gLj;jp ,d;W fzpg;nghwp %yk; kpy;ypad; jrk ];jhdq;fs; tiu ������ apd; kjpg;G Jy;ypakhf fzf;fplg;glL; s;sJ. (4) fp.K 150 Mz;LfSf;F Kd; tho;e;j fhfi; fghbdpahh; vd;w Gyth; jdJ ghlypy; ������ d; kjpg;G gw;wp gpd;tUkhW $wpAs;shh;. “ tl;lj;jiuf; nfhz;L tpl;ljiu jhff; pd; rl;nldj; njhpAk; Fop ” ,jd; %yk; tl;lj;jpd; gug;ig fz;lwpa ,aYk;. ,itnay;yhk; Nrh;e;J jhd; fzpjk; Ntf eilNghlL; tsh;r;rpaile;jJ. ,e;j ������ jhd; ,d;W uhf;fl; Kjy; gy mwptpay; fz;Lgpbg;Gf;fSf;F fhuzkhf mike;jJ. gpjhfu]pd; fz;Lgpbg;G gpjhfu]pd; fz;Lgpbg;G gy khw;wq;fis cyfpy; cz;lhff; paJ. xU rkjsg;gug;ig ,ilntspapd;wp milf;f Ntz;Lkhdhy; gy rJuq;fs;> rkgf;f Kf;Nfhzq;fs;> xOq;fhd mWNfhzq;fs; Njit vd $wpdhh;. gpd;dh; nrq;Nfhz Kf;Nfhzk; gw;wp $wpa tpjp ,d;Wk; ngahpy; gpjhfu]; Njw;wk; vd toff; py; cs;sJ. jkpof fzpjNkij ,uhkD[h; <Nuhl;il rhh;e;j ,uhkD[u; vz;zpaypy; (Theory of Numbers) gy mupa fz;Lgpbg;Gf;fis toq;fpdhh;. ,thpd; fz;Lgpbg;Gf;fis fz;L tpae;j Nfk;gphpl;[; gy;fiyf;fof Nguhrphpau; Hardy ,tiu Nfk;gphpl;[f; ;F mioj;jhh;. mq;F ,uhkD[h; gy mjpra fz;Lgpbg;Gf;fis ntspapl;lhh;. gpd;dh; 1918 y; ,thpd; jpwikiaf; fz;L tpae;J ghuhl;b Fellow of Royal Society MfTk; Fellow of Trinity College MfTk; Nju;e;njLf;fg;glL; nfsutpf;fg;gl;lhh;. ,tt; hW gy;NtW fhyfl;lq;fspy; gy;NtW mwpQh;fspd; ciog;ghy; fzpjk; jioj;Njhq;fpaJ. fzpj tsh;rr; pahy; thdtpaYk;> mwptpaYk;> g+NyhftpaYk; tsh;r;rp mile;jd. nkhj;jj;jpy; etdP cyfpw;F fhuzkhf ,f; fzpjk; mike;jJ vd;why; kpifahfhJ. fzpj tsu;r;rp cyf ehLfs; midj;Jk; fzpj tsu;r;rpf;F cWJizahf ,Ue;jpUe;jhYk; rpy Fwpg;gpl;l ehLfNs rpwg;ghf JizGupe;Jss; d. 19

ghgpNyhdpau;fs; nkhrNlhkpah gFjpapy; (,d;iwa <uhf;) tho;e;j kf;fs; ghgpNydpauf; s;. fp.K. 539 Mk; Mz;L tiu ékpf;fbapy; GijAz;L fple;j 400 fspkz; jfLfs; fz;nlLf;fg;gl;ld. ghgpNydpauf; s; fspkz;zpy; vz;fis gjpj;J jfLfis cUthf;fpdu;. fp.K. 1800 Kjy; 1600 tiu cs;s fhyfll; j;jpy; jhd; mjpfkhd fspkz; jfLfs; fpilj;Jss; d. gpd;dq;fs; ,aw;fzpjk;> ,Ugb rkd;ghLfs;> gpjhfu]; Njw;wk;> jpupNfhzkpjp rhu;Gfs; Fwpj;j fzpj jfty;fs; ,j;jfLfspy; Fwpg;gplg;glb; Ue;jd. ghgpNyhdpauf; s; 60 I mbg;gilahff; nfhz;l vz;Kiwia gadg; Lj;jpdu;. ghgpNyhdpa vz;Kiwapy; ,Ue;Jjhd; ,d;W ehk; gadg; Lj;Jk; 1 epkplj;jpw;F 60 nehb> 1kzpf;F 60 epkplk;> tl;lj;jpd; Nfhz msT 3600(60ù6) Mfpait gof;fj;jpy; te;jd. 60-d; tFjpfshd 1>2>3>4>5>6>10>12>15>20 kw;Wk; 30 Mfpatw;iwg; gad;gLj;jp gpd;d tbt fzf;Ffs; kw;Wk; ,lkjpg;Gfisf; ifahz;ldu;. vdNt nkrgNlhgpah fzpjj;ij ghgpNyhdpauf; s; fhyk; vd Fwpg;gpLfpNwhk;. ghgpNyhdpauf; s;> vz;fspd; mbg;gil nraYf;F Kd;Ng fzf;fplg;gl;l tha;g;ghl;il gad;gLj;jpdu;. gug;gsT> fd msT fhz ghgpNyhdpauf; s; nghJ tpjpfis gad;gLj;jpdu;. NkYk; mtu;fs; fzpj fUj;Jf;fisAk; fz;Lgpbj;jdu;. mit tl;lj;jpd; Rw;wsT ® 3 Kiw tl;lj;jpd; tpl;lk;. tl;lj;jpd; gug;gsT ® 1Æ2(Rw;wsT)2. ,q;F π d; kjpg;G 3 vd gad;gLj;jpdu;. cUisapd; fdtsT ® mbg;gf;fg; gug;G ù cauk; Kjypad MFk;. vfpg;jpau;fs; 20

ghgpu]; ifnalOj;Jg; gpujpfs; %ykhfTk;> gok;ngUk; rpd;dq;fspd; %ykhfTk; vfpg;jpau;fspd; fzpj mwpitg; gw;wpa gy rhd;Wfs; ekf;F fpilj;jpUf;fpd;wd. vfpg;jpa fzpjkhdJ vfpg;jpa nkhopapy; vOjg;gl;lJ. Vw;fdNt fpNuf;f nkhopapy; nkhopapy; ,Ue;e fUj;Jfs; n`ydp]b; f; fhyj;jpy; ,Ue;J vfpg;jpa nkhopf;F khw;wg;gl;lJ. ,jdhy; vfpg;jpa fzpjkhdJ fpNuf;f kw;Wk; ghgpNyhdpa fzpjj;Jld; ,ize;J cUthfp mJ n`ydp]b; f; fzpjkhdJ. vfpg;J kf;fs; jq;fs; md;whl tho;f;iff;F Njitahd mbg;gil vz;fzpjj;ijAk;> tot fzpjj;ijAk;> Vw;gLj;jpf; nfhz;ldu;. ,tu;fs; KO vz;fisAk; gpd;dq;fisAk; $l;Ltw;Fk; fopg;gjw;Fk; mwpe;jpUe;jdu;. mtu;fspd; tzpf rk;ge;jkhd fzf;Ffisj; jPu;f;f ngUf;fiyAk;> tFj;jiyAk; ifahz;ldu;. XtN; thu; Mz;Lk; iey; ejpapd; gs;sj;jhf;Ffspd; epyg;gug;ig mse;J tup tpjpj;jdu;. ,e; epfo;T jhd; tbt fzpjj;jpd; gpwg;gplkhf mike;jJ. vfpg;jpau;fspd; vz;fs; 10 mbkhd vz;fs; MFk;. ,e;J mNugpau; fpNuf;fUf;Fg; gpd;du; fzpjk; tsu;r;rpAw;wjw;F ,e;Jff; Ns fhuzkhtu;. ,e;Jkjg; gpuhkzu;fs; kjj;ijAk; jj;Jtj;ijAk; kl;LNk jkJ tho;tpd; cau;jFjp vdf; fUJk; tof;fKilatu;fs;. fpNuf;fu;fs; vt;thW Nfj;jpufzpjj;Jiwapy; jkJ rpe;jidiar; nrYj;jpdhu;fNsh mtt; hW ,e;Jff; s; vz; fzpjj;jpYk; ml;rufzpjj;jpYk; jkJ rpe;jidiar; nrYj;jpdhu;fs;. ,JNt fpNuf;fu;fSf;Fk; ,e;Jf;fSf;FkpilNaAs;s xNunahU tpj;jpahrkhFk;. ,d;DnkhU Kiwapy; $wpdhy; fpNuf;fu;fs; cUtq;fis mbg;gilahf itj;J KdN; dwpdu;. ,e;Jff; s; vz;fis mbg;gilahf itj;J KdN; dwpdu;. ,e;Jf;fs; vz;fis mbg;gilahf itj;J KdN; dwpdu;. fpNuf;fu;fsplkpUe;J Nfj;jpufzpjk; Njhd;wpaJ Nghy; ,e;Jff; splkpUe;J ml;rufzpjk; Njhd;wpaJ. fp.K. 3000 Mz;lstpNyNa vz;fzpjj;ijg; Nghd;Nw thdrh]j; puKk; mjpfsT KdN; dw;wkile;jpUe;jJ vd Ntj Ehy;fs; tpsf;Ffpd;wd. fzpj ghlj;jpd; mj;jpthukhd 0.1.2.3 Nghd;w mbg;gil vz;fisAk; jrk vz;KiwiaAk; mwpKfg;gLj;jpa ngUik ,e;Jff; isNa rhUk;. fp.K. 430 ,y; nfsby;ypahthy; vOjg;gl;l ‘ mu;j;j rh];j;jpuk;’ vd;Dk; Ehypy; Fwpg;gpl;Ls;sjd; gb mg;NghJss; ,e;Jff; spd; fy;tpapy; vz;fzpjKk; ml;rufzpjKk; css; lff; g;gl;bUe;jJ vd mNrhff; fy;ntl;Lf;fs; rhd;Wgfpuf; pd;wd. ,e;Jff; spd; fzpj ghlj;jpd; KdN; dhbfSs; ,e;Jff; zpjtpayhsu; Mu;agl;lu; (fp.Gp. 476) gpukFg;ju; (fp.gp. 598) kfhtuP u; (fp.gp. 850) gh];fud;(fp.gp.1114) Mfpatuf; s; Kf;fpakhdtu;fshff; fUjg;gl;ltu;fshtu;. 21

vz;fzpjk;, ml;rufzpjk;, jpupNfhzfzpjk; njhlu;ghd Kf;fpa tplaq;fs; rpy ,tu;fspd; fz;Lgpbg;Gf;fshFk;. NkNy Fwpg;gpl;l fzpjg; gFjpfs; njhlu;ghf vOjg;gl;l Ehy;fspy; kfhtPudpd; “ fzpjk; - rhuk; - rq;fpue;jk; ” vd;Dk; EhYk; gh]f; udpd; “ rpj;jhe;j rpNuhd;kzp” vd;Dk; EhYk; cau;jhdj;jpy; itf;fg;glL; s;sd. gh];fudhy; vOjg;gl;l rpj;jhe;j rpNuhd;kzpapy; KOvz;fs; kl;Lkd;wp tpfpjKwh vz;fSk; css; lf;fg;gl;l ml;rufzpjr; rkd;ghLfs; fhzg;gLfpd;wd. Nfj;jpufzpjj;ij tpl jpupNfhz fzpjj;jpy; mjpf Cf;fk; fhl;lg;gl;Ls;sjhfj; njupfpwJ. ,d;W ehk; gad;gLj;Jk;, ������������������ ������������° = ������ vd;gJk; ������ ������������������ ������������° = √������ − ������ vd;w #j;jpuKk; ������ ,w;iwf;Fg; gj;J Ehw;whz;LfSf;F Kd;du; vOjg;gl;l “gQ;r rpj;jhe;jk;” vd;Dk; ,e;J E}ypy; fhzf;$bajhf ,Uf;fpwJ. ,e;Jff; Sf;Fk; mNugpauf; Sf;Fk; ,ilg;gl;l tu;j;jfj; njhlu;gpd; Ngwhf ,e;Jff; spd; ml;rufzpjk; mNuhgpahtpw;Fg; ngau;e;jJ. fp.gp. 777 ,y; ,e;jpahTf;F te;j mNuhgpa tpahghupfs; ‘ rpj;jhe;ju;’ vd;Dk; ,e;J fzpj Mf;fj;ij gf;jhj; efUf;F vLj;Jr;nrd;W mNuhgpa nkhopf;F khw;wp mq;F mr;rpl;L ntspapl;ldu.; ,tt; hW ,e;jpahtpypUe;J mNugpahTf;F ,lk;ngau;e;j ,e;Jf;fspd; ml;rufzpjk; ‘ Algebra’ vd;w ngaupy; mNugpaupd; Mf;fkhf ntspg;gLj;jg;gl;lJ. Al – khawarizmi fp.gp. 813 vd;w ngaUila mNugpa fzpjtpayhsNu ,tt; hW mr;rpl;L ntspapl;lhu;. jw;fhyj;jpy; ehk; ml;rufzpjg; ghlj;jpw;Fg; gad;gLj;Jk; Mq;fpyg; ngauhd ‘ Algebra’ vd;Dk; ml;rufzpjg; Gj;jfk; ntspaplg;gl;lJ. fzpjk; xU mwpthu;e;j tpisahl;L KbTfisf; nfhzUk; fiy ju;ff; nray;Kiw cs;Szu;T Kiw rpe;J rkntsp ehfupfk; 22

rpe;J rkntsp fhyj;jpy; jdpr;rpwg;G tha;e;jJ tPjpfSk; tbfhyikg;GfSk; MFk;. mit fzpjj;jpd; JizAlNdNa mikf;fg;gl;bUf;fpwJ. nrq;fy;: Muk;g fhyq;fspy; (Early Harappa) Rkhu; fp.K 3500> `ugg; h Kjy; Nyhj;jy; tiu nrq;fy; 1:2:3 vd;w tpfpjj;jpYk; Kjpu;e;j fhyq;fspy; (Mature Harappa) Rkhu; fp.K 2500 nrq;fy;fs; 1:2:4 (7:14:28 cm) tpfpjj;jpYk; ,Ue;jJ. gy;NtW ,lq;fspy; rPuhd tpfpjj;jpy; nrq;fy;fs; ,Ue;jpUf;fpwJ vd;why;> fzpjj;ij mf;fhy kf;fs; mwpe;jpUf;f NtzL; k;. msT (Lenth): Nyhj;jy; (,d;iwa FIuhj;) vd;w ,lj;jpy; je;jj;jpy; nra;ag;gl;l msTNfhy; xd;W njhy;nghUs; Ma;thsu;fshy; fz;nlLf;fg;gl;Ls;sJ. me;j msTNfhypy; 46mm mstpy; cs;s gFjp kl;LNk njspthf cs;sJ. me;j 46mm gFjp 27 NfhLfis nfhz;Ls;sJ. mjhtJ ,U NfhLfSf;F ,ilNa Md njhiyT 1.7mm. mjhtJ 10NfhLfs; = 1.7cm ,J nksupau;fs; fhyj;jpy; cs;s mu;j;jrh];jpuj;Jld; xj;Jg;NghfpwJ. tlnkhopapy; mq;Fy; (Angul) vd;w thu;j;ijia Nfs;tp gl;bUf;fpwuP ;fsh? mjw;F tpuy; vd;W mu;j;jk;. xU tpuypd; (eLtpuy;) mfyk; (width) 1.76cm. Mf 108 mq;Fyh = 190cm = 1 jD~; (Dhanus) ,jw;Fk; mNrhfh J}Zf;Fk; njhlu;G cs;sJ. ,e;j msTNfhy; mjDld; xj;Jg; NghfpwJ. vil (Weights): `ugg; h> nkhfd;[jhNuh kw;Wk; rhd;`P jhNuhtpy; 558 vilf;fw;fs; njhy;nghUs; Ma;thsu;fshy; fz;nlLf;fg;gl;Ls;sJ. Ie;J ntt;NtW mLf;Ffspy; ,Ue;J Njhzl; g;gl;l (xt;nthd;Wk; Rkhu; 1.5 kP Mok;) vilfSf;F ,ilapy; Fwpg;gplj;jf;f NtWghLfs; ,y;iy vd;W Ma;Tfs; $Wfpd;wJ. rpe;J rkntspapy; gad;gLj;jpa vilf;fy;fspy; 13.7g nfhz;l vilf;fy;Ny mjpfk; gad;gLj;jg;gl;lJ. nkl;upf; mikgG; f;F khw Kd;G rpe;Jrkntspapy; tof;fpy; ,Ue;jJ Rattis KiwahFk;. ,J xU Gunja Ratti (rptg;G Fz;Lkzp) vd;w xU tif tpijia mbg;gilahf nfhz;L tFf;fg;gl;lJ. xU Fz;Lkzp jq;fk; $l tPl;by; ,y;iy vd;w nrhy; ,d;Wk; tof;fpy; cs;sJ. gy Mapuk; Mz;LfSf;F Kd;Ng ,di; wa jkpofk; ,e;jpahtpd; xU mq;fk; vd;gjw;F ,J xU cjhuzk;. vz;fs; neLtupirapy; fzpjKk; tuyhWk; mwptpay; tsu;r;rpapd; epfo;jfTfs; cyfpd; vy;yh gFjpfspYk; Fwpg;gpl;l fhy ,ilntspfspy; ele;NjwpAs;sd. ,Jtiuapy; epfo;e;j kdpj mwpT tsu;r;rp vd;gJ kdk; jd;id jhNd mwpe;J nfhs;Sk; ju;ff; ujP pahd Kaw;rpia jtpu Ntwpy;iy vd;whu;. n`fy; r%fj;jpd; gbkhd tsu;r;rpapy; mwptpay; fz;Lgpbg;Gfs; xU gFjpahfpd;wd. Guhjd kdpjd; jd;idj; jhNd mwpe;J nfhs;Sk; Kaw;rpapd; xUgFjpahfNt fzpjtpay; cUthdJ. fzpjj;ij kdpjf; 23

fUj;jhf;fj;jpd; tiutpay; vd;wdu; fpNuf;fu;fs;. fpNuf;f jj;Jtthjpfshd a+f;ypl; vz;fisAk;> NfhLfisAk; mwpe;J nfhs;tjw;fhd Kiwia tsu;j;jhu;. a+f;ypl; ,e;j Kiwfis Elements vd;whu;. mtupd; vz;fSf;fhd FwpaPL tpj;jpahrkhf ,Ue;jJ. Gwtakhd fhl;rpg;gLj;jNy vz;fshf cUthfpd;wd vd;whu; a+f;ypl;. mtw;wpd; tpj;jpahrk; vd;gJ xd;wpypUe;J ,d;ndhd;whf tifawpjNy. 1+ 1 vd;gjpy; ,uz;LNk xNu khjpup Gyg;gl;lhYk; ,uz;Lk; NtwhdNj. kw;nwhUtuhd njhykp Nfhzq;fs; vt;thW njhlu;GWfpd;wd vd;gij Fwpj;J Muha;e;jhu;. muGyfpy; fzpj mwptpaypd; tsu;r;rp vd;gJ RNkupa> ghgpNyhdpa fhyj;ij KdN; dhf;FfpwJ. fp.K. ,uz;lhapuk; Mz;by; ghgpNyhdpau;fs; mWgij mbg;gilahf nfhz;l vz;khd Kiwia (Sexagesimal calculus) cUthf;fpdu;. (,e;epa #oypy; ngsj;jk; ,ij Kdn; dLj;jJ.) jw;Nghija gj;jbkhd Kiwf;F khw;whf mWgij mtu;fs; Kdi; tj;jhu;fs;. ,e;j KiwNa jw;NghJ Neu fzf;fPlb; y; gad;gLj;jg;gLfpwJ. (60 nehb> 60 epkplk;) ,J thdtpaypYk; jw;NghJ njhlu;fpwJ. gj;ij jtpu;j;J tpl;L mWgij mtu;fs; vLj;Jf; nfhz;l fhuzk; mWgjhdJ mNef vz;fshy; kPjpapd;wp tFgLfpwJ vd;gjhFk;(2>3>4>5>6>10>12>15>20>30>60). xU rkgf;f Kf;Nfhzj;jpd; xU Nfhzj;jpd; kjpg;G 60 MFk;. NkYk; nkhj;j Nfhzq;fspd; msT 360 MFk; NghJ xtn; thd;wpd; kjpg;G 60 Mf ,Uf;fpwJ. NkYk; Fwpg;gpl;l jFe;j tp~akhf ghgpNyhdpauf; s; vz;fis fzf;fpl fspkz;zhy; Md gyiffis cUthf;fp mij ml;ltizahff; fUjpdu;. A+g;gpuB]; ejpf;fiuapy; fz;nlLf;fg;gl;l ,e;j ml;ltiz gyifahdJ mtu;fs; vt;thW vz;khd fzf;fPlb; w;F cl;gl;bUe;jdu; vd;gij ntspg;gLj;JfpwJ. mtu;fs; 59 tiuAss; vz;fSf;F tu;f;fq;fisAk;> 32 tiuAs;s vz;fSf;F fdq;fisAk; fz;lwpe;jdu;. NkYk; vz;fspd; tu;f;fq;fspypUeJ; tu;f;f %yq;fisAk;> fdq;fspypUe;J fd%yq;fisAk; fz;lwpe;jdu;. mNj fhy fl;lj;jpy; fpNuf;fj;jpYk; fzpjtpay; tsu;r;rp ngw;wJ. fzpjtpaiy nghWj;jtiu ,];yhkpa fhy muGyfk; tuyhw;W gjpTfshf gq;fspg;Gfis nra;Js;sJ. Vohk; E}w;whz;L fhy Jtf;fj;jpy; ,Ue;J gjpd;%d;whk; E}w;whz;Lfs; tiu mjd; gq;fspg;G ePs;fpwJ. mg;gh]pl; tk;rk; fp.gp 766 y; kj;jpa fpof;fpy; Ml;rpia njhlq;FfpwJ. mjd; muruhd fyPgh my; kd;#u; ghf;jhij jiyik ,lkhff; nfhz;L mjpfhuj;ij njhlq;fpdhu;. mtUf;F gpd; te;j `h&d; my; u~Pj; ghf;jhj; kw;Wk; mjd; ntspg;gFjpfspy; E}yfq;fis njhlq;fpdhu;. ,tu; gy;NtW Jiwfs; njhlu;ghd Ma;Tikaq;fis njhlq;fp fzpjk;> Ntjpay;> thdtpay;> kUj;Jtk; Nghd;w gy;NtW Jiwfs; njhlu;ghd Ma;Tfis Nkw;nfhss; top Vw;gLj;jpdhu;. gpd;du; mtupd; kfdhd my; kh%d; fp.gp 817 y; ghf;jhij jiyik ,lkhff; nfhz;L mwpT juprd ,y;yk; (House of Wisdom) Muk;gpj;jhu;. ,jd; %yk; gy;NtWgl;l Ma;thus;fs;> tpQ;Qhdpfs; MfpNahu; ,jpy; ,izf;fg;gl;L mwptpy; tsu;r;rpf;fhd ePNuhl;lk; Vw;gLj;jg;gl;lJ. fpNuf;f> ,yj;jPd; nkhopfspy; css; mwptpay; E}y;fs; mugp nkhopf;Fk; mugp nkhop E}y;fs; fpNuf;f nkhopf;Fk; nkhopkhw;wk; nra;ag;gl;ld. md;W Kjy; fzpjk; muGyfpy; jdpj;j Jiwahf cUkhw;wk; mile;jJ. mg;gh]pl;fspd; mwpT juprd ,y;yk; fzpj Xl;lj;jpd; J}z;LNfhyhf ,Ue;jJ. fzpjj;jpd; mbg;gilahd nraNyhl;lk; vd;gJ Gjpu; kw;Wk; jPu;T ( Problem & Solution) Mfpatw;iw cs;slf;fp ,Uf;fpwJ. ,e;j jPu;Tfs; vd;WNk é[;[pa epiyf;F cl;gLtjpy;iy. mJ kPr;rpW vz;zhfNt ,Uf;Fk;. Cjhuzkhf 1I ghjpahf tFf;Fk; NghJ mjd; kjpg;G 0.5 MfpwJ. NkYk; tFf;Fk; NghJ 0.25 MfpwJ. ,jd; njhlu;r;rpapy; mjd; KbT 24

é[;[pak; MfhJ. NkYk; vjpu;khwhf ngUf;fy; eilngWk; NghJ kPz;Lk; 1I mila KbAk;. Mf xU nghUs; rpije;J Ntnwhd;whf cUkhWfpwJ. kPz;Lk; kPSUthf;fk; mile;J Gjpa nghUshfpwJ. ,J ,af;ftpaypd; epiykhw;wk;> epiykWg;G Mfpatw;Wld; xj;jpirfpwJ. fzpjj;jpd; mbg;gilahd ngUf;fy;> tFj;jy; vd;gJ Xu; ,af;fNk. ,d;iwa fzpjj;jpd; Kf;fpa fpisahd Algebra vd;w ,aw;fzpjKk;> Algoritham vd;w xOq;fzpjKk; mg;gh]pl; fhyfl;lj;jpd; kpfg;ngUk; nfhilahFk;. Algebra vd;gJ kh%d; fhyj;jtuhd [hgu; K`k;kJ my;fthu];kp cUthf;fpajhFk;. mjd; gpwg;gplk; fpNuf;fk; jhd; vd;w fUj;jhly; Fwpj;J ,d;Wk; re;Njfk; epytp tUfpwJ. fthup];kp fp.gp 780 ,y; ,aw;fzpjj;jpd; Nfhl;ghLfis epWtpdhu;. mJ Hisab al-jabr w’al-muqabala vd;w ngaupy; E}yhf ntspte;jJ. ,e;j E}ypd; ngaupy; ,Ue;Jjhd; Algebra vd;w nrhy; cUthdJ. mjpy; vz;fis Fwpj;J Muha;fpwhu.; “ehd; kf;fs; vij fzf;fPlb; w;F gadg; Lj;Jfpwhu;fs; vd;gij fz;lwpAk; NghJ mJ vz;fshfpwJ. NkYk; xtn; thU vz;ZNk myFfis nfhz;bUf;fpwJ. NkYk; vy;yh vz;fSNk me;j myFfshy; tFf;fg;gLfpd;wd. NkYk; vz;fs; gj;J tiu ntspg;gLj;jg;gLfpd;wd. gj;J ,Uklq;F> Kk;klq;fhfpwJ. ,g;gbahf vz;fs; mjd; tupirapy; nry;fpd;wd. ,k;khjpupahd vz;fspd; mikg;G Kiwapy; ,Ue;J fthup]k; p jd; Ma;Tfisj; njhlq;fpdhu;. vz;fs; tu;f;fk; kw;Wk; tu;ff; %yq;fis (Square, Root) myFfshff; nfhz;bUf;fpd;wd vd;whu;. ,jd; top rkd;ghLfis cUthf;fpdhu;. ,JNt ,aw;fzpjj;jpw;fhd mbg;gil Neu;NfhL kw;Wk; ehw;fu rkd;ghLfis mtu; MW tifg;ghLfshfj; njhFj;jhu;. 1.tu;f;fk; tu;f;f %yj;jpw;F rkdhjy;. (Square equals to roots) 2.tu;f;fk; vz;zpw;F rkdhjy;. (Square equals to numbers) 3.tu;f;f %yk; vz;zpw;F rkdhjy; (Roots equals to numbers) 4.tu;f;fKk; tu;f;f%yKk; vz;zpw;F rkdhjy;. (Roots and Square equal to numbers) ù2 + 10x ® 39 5.tu;f;fKk; vz;Zk; tu;f;f %yj;jpw;F rkdhjy;. (square and numbers)ù2 + 21x ® 10x 6.tu;f;f %yKk; vz;Zk; tu;ff; j;jpw;F rkdhjy;.(Roots and numbers equal to Square)3x + 4 ® x2 Nkw;fz;l MWtpj topfis cUthff; pa gpd; fthup]k; p mjid KOik gLj;Jtjw;fhd topfis gw;wp Muha;e;jhu;. my; [gu; vd;gjw;F KOikg;gLj;jy; vd;gjhf mu;j;jk; nfhs;fpd;wdu;. me;E}ypy; rkd;ghLfis my;[gu; kw;Wk; tbt ,ay; (Geometry) topahf jPu;T fhZk; Kiwia Kdi; tff; pwhu;. gpd;tUk; cjhuzj;ij nfhLj;jhu;. ù2 + 10x ® 39 ve;j vz; ,ij KOikg;gLj;Jk;? ,jw;fhf tu;ff; j;jpd; kjpg;ghd 10 ,y; ghjpia vLj;jf;nfhs;s Ntz;Lk;. (5) mJ tu;f;fkhFk; NghJ mjd; kjpg;G 25 MfpwJ. ,jDld; 39 ia $l;Lk; NghJ 64 fpilff; pwJ. ,jd; tu;f;f%yk; 8. ,jd; kjpg;Gld; Kjy; kjpg;ghd 5If; fopf;Fk; NghJ jPu;thd 3 ekf;F fpilfF; k;. fthup];kp ,ij tbtpay; Kiwf;Fk; cl;gLj;jpdhu;. kjpg;Gfis rJuj;jpd; vy;iyf;F cl;gLj;jp Nkw;fz;lthW jPu;T fz;lhu;. NkYk; ,tUila fhyj;jpy; fpNuf;f fzpjtpayhsuhd a+f;ypl;bd; tbt ,ay; E}y;fs; mugpapy; nkhopngau;f;fg;gl;ld. ,tupd; tbt ,ay; Nfhl;ghLfspy; a+f;ypl;bd; jhff; Kk; cz;L. fthup];kp jd; ,ay; fzpj Kiwia vz;khd tpjpfspd; gb mikj;jhu;. NkYk; ,e;jpa mNugpa vz;fSf;fhd njhlu;Gfis Fwpj;J Muha;e;jhu;. ,e;jpau;fshy; fz;lwpag;gl;l g+[;[pak; 25

ghgpNyhdpauf; shYk; mwpag;gl;L ,Ue;jJ. mNj fhyfll; j;jpy; fpNuf;f ,yj;jPd; gFjpfspy; ntw;wplj;ij Fwpf;f “FwpaPL” gadg; Lj;jg;gl;lJ. my;[Pgpuh Fwpj;j fthup];kpapd; E}Yf;F gpwF me;j fzpjf; Nfhl;ghL Nkw;Fyfpw;Fk;> mz;il fpof;F ehLfSf;Fk; nrd;wile;jJ. fthup];kpapd; tbt ,ay; Nfhl;ghLfs; ,uz;lhk; E}w;whz;L a+j fzpjtpayhsuhd nefpkhtpd; Nfhl;ghLfis gupkhzg; gLj;jpajhFk;. nefpkhtpd; E}y; Mishnat ha middot vd;w ngaupy; ntspte;jJ. mjpy; mtu; tl;lj;jpd; gupjpahdJ mjd; tpl;lj;ij tpl 3 1Æ7 klq;fhdJ vd;whu;. ,ij kWj;j fthup];kp ,J ntWk; kughu;e;j NjhuhaNk. mJ ep&gpf;fg;gl Ntz;baJ mtrpak; vd;whu;. NkYk; ,jw;F khw;whf √10 kw;Wk; 62832Æ20000 I Kdi; tj;jhu;. ,J gpw;fhy thdtpayhsu;fshy; vLj;Jf; nfhs;sg;gll; J. NkYk; Kf;Nfhzj;jpd; gug;gsit rupahd Kiwapy; fz;lwpe;jJ. ,tupd; ngUk;Gss; papLk; rhjidahFk;. nefpkh mjid a= ½ (a+b+c) vd;whu.; fthup];kp mjid vspik gLj;jp rupahd topia fz;lwpAk; tpjkhf a = ½ bh vd;whu;. Nkw;fz;l Nfhl;ghLfspd; %yk; fthup];kp my;[Pgpuhtpd; je;ij vd Nkw;Fyfj;jhy; miof;fg;gl;lhu;. fpNuf;f> vfpg;jpa> RNkupa> ,e;jpa jhff; q;fspypUe;J thu;f;fg;gl;l fthup]k; papd; Nfhl;ghLfs; kWkyu;rr; p fhy [Nuhg;gh kw;Wk; mnkupf;fhtpy; ngUk; jhff; j;ij Vw;gLj;jpd. 18 Mk; E}w;whz;L gpnuQ;R jj;Jtthjpahd njfhu;Njapd; fzpj fz;Lgpbg;ghd tif Ez;fzpjk; kw;Wk; njhif Ez;fzpjk; (Differentiatiom and integration) Mfpatw;Wf;fhd J}z;LNfhyhfTk; ,Ue;jJ. fthup]k; pia njhlu;e;J mG fhkpy;> my; fuh[;> my;-Jrp> cku; fa;ahk;> ,g;Dy; fa;a+k;> ,g;Dy; gd;dh MfpNahu; my;[Pgpuhit tsu;njLj;jdu;. mG fuh[; fp.gp 953 y; ghfj; hjpy; gpwe;jtu.; fthup];kp Nghd;Nw mg;gh]pl;fspd; mwpT juprd ,y;yj;jpy; tsu;e;jtu;. ,tu; polynomial Nfhl;ghL Fwpj;J tpupthf Muha;e;jtu;. ,tupd; tpjpfSf;F vz;fis cjhuzq;fshf jej; NghJ epfOk; gupkhw;w fzkhf ,Ue;jJ ,tupd; tpjp. my; [Pgpuh kw;Wk; my;-fhupjk; Nghd;wtw;iw tsu;j;njLj;jtupy; my;-ng&dp Kf;fpakhdtu;. my;-fthup]k; papd; ,lj;ijr; Nru;e;j ,tu; fp.gp 908 ,y; gpwe;jtu.; ,e;jpahtpw;F tUif je;j Kjy; ghurPfu.; ,tupd; gq;fspg;G gy Jiwfisr; rhu;e;J tpuptilfpwJ. khDltpaypy; kj;jpa fpof;F rhu;e;j Kjy; rpe;jidahsu;. fzpjtpaypy; ,tupd; fzf;fPlhdJ ru;tNjr epiyapy; ftdk; ngw;wJ. Cjhuzkhf 1+2+3………….263 ,ij NfhJikapd; jhdpaq;fis vz;Zk; Kiwf;F xg;gpl;lhu;. xU nr]; (chess) gyifapy; Kjy; fll; j;jpy; xd;Wk; ,uz;lhk; fl;lj;jpy; ,uz;Lk;> %d;whk; fll; j;jpy; %d;Wk; ,Uf;Fk; epiyapy; mjd; nkhj;j kjpg;G ,tt; hW ,Uf;Fk;. (((162)2)2)2 – 1 ,jdpypUe;J ngwg;gLk; tpil 18>446>744>073>709>551>615. ,d;iwa fzpg;nghwp mwptpaypy; my-; ng&dpapd; ,e;j fzf;fPl;L KiwahdJ jdpg;gl;l $whf tpsq;Ffpd;wJ. NkYk; jpupNfhzkpjpapy; (Trigonometry) my;-ng&dpapd; ftdk; gpujpgypg;ig mile;j xd;W. Nfhzq;fs; vd;git Neuhf ,y;yhky; ntl;Lj; Njhw;wq;fisf; nfhz;bUf;fpd;wd vd;whu;. msTNfhy; kw;Wk; ftuhaq;fshy; jPu;fg;gl Kbahj fzpj rhu;ghd gpur;rpidfSf;F jPu;T fz;lhu;. Nfhzq;fspd; ,ilntspf;Fk; G+kpapd; Row;rpf;Fkhd njhlu;G Fwpj;J Muha;e;jhu;. thdtpaypy; ,tupd; gq;fspg;G Fwpg;gplj;jFe;jJ. Nfhs;fs; kw;Wk; G+kpapd; ,af;fk; gw;wpa ,tupd; rpe;jidfs; gpw;fhy thdtpayhsUf;F Kd; Cjhuzkhf ,Ue;jJ. 11 Mk; E}w;whz;bd; njhlf;fj;jpy; ,e;jpahtpe;F te;e my;-ng&dp ,e;jpa r%f fll; ikg;G Fwpj;J tpupthd Ma;T Nkw;nfhz;lhu;. NkYk; ,e;jpa jj;Jt E}y;fshd rhq;fpak; kw;Wk; gjQ;rypia mugpapy; nkhop ngau;j;jhu;. ,e;jpah gw;wpa ,tupd; E}y; ,e;jpa tuyhW (fpjhGy; `pe;j;) vd;w ngaupy; ntspte;jJ. my;-ng&dpapd; 75 Mz;L fhyk; gy;NtW Jiwfs; njhlu;ghd Njlyhf mike;jJ. fzpjj;jpy; tbt ,ay; kw;Wk; vz; Nfhl;ghLfis Fwpj;J Muha;e;j kw;nwhUtu; mG myp my; `rd; ,g;D ifjk;. fp.gp 965 y; <uhf;fpd; g];uhtpy; gpwe;jtu.; Mdhy; ,tupd; 26

ngUk;gFjp vfpg;jpy; fle;NjwpaJ. gj;jhk; E}w;whz;bd; ,ilg;gFjpapy; vfpg;jpy; ghj;jpkj; tk;rk; mjpfhuj;jpw;F te;jJ. mtu;fs; mg;gh]pl; tk;rj;ij Vw;f kWj;jdu;. fp.gp 969y; iey; gs;sj;jhf;F gFjpia ifg;gw;wp jq;fs; jiyefuhf nfa;Nuhit gpufldg;gLj;jpdu;. Kjy; muruhd mg;Jy; m]P]; jd; murikg;ig uhZt gyk; nghUe;jpajhf khw;wpdhu;. mtupd; uhZt nraw;ghLfs; rpupah tiu ePz;ld. ,tUf;Fg; gpd; mg+ `f;fPk; vd;gtu; gpd; njhlu;e;jhu;. mNj fhyfll; j;jpy; g];uhtpy; mikr;rf nghWg;G tfpj;j ,g;D ifjk; mij JweJ; tpl;L vfpg;ij te;jile;jhu;. ,tu; g];uhtpy; tpQ;Qhdpahf mwpag;glL; te;jhu;. ,sik fhyj;jpy; ,]y; hkpa kughu;e;j fy;tpia fw;w mtUf;F mjd; xU fll; k; rypg;ig Vw;gLj;jpaJ. mij tpl;L nysyPf Jiwfspy; ftdk; nrYj;j Muk;gpj;jhu;. ,]y; hkpd; nfhs;if mbg;gilapyhd gy gpupTfs; kPJ mtek;gpf;if nfhz;bUe;jhu;. mtupd; ftdk; ,aw;gpay;> thdtpay; kw;Wk; fzpjk; rhu;ghf ,Ue;eJ. vfpg;jpd; tUiff;F gpwF ,tupd; tpQ;Qhd epGzj;Jtj;ij fz;L iey; ejpapd; nts;s Xl;lj;ij fl;Lg;gLj;JtJ Fwpj;J Muha mikf;fg;gl;l FOtpd; jiytuhf mg+`f;fPk; ,tiu epakpj;jhu;. ,jd; gpwF me;j fkpl;b gy iey; ejpia Ma;T nra;J mJ njhopDl;g ujP pahf rhj;jpakw;wJ vd;W mwpf;if mspj;jJ. mg+`f;fPik nghWj;j tiu tpQ;Qhd tsu;r;rpapd; kPJ vt;tpj ek;gpf;ifAk; ,Uf;ftpy;iy. mtupd; nray;jpl;lk; ntWk; uhZt Af;jpahfNt ,Ue;jJ. mtu; fhyj;J gy tpQ;Qhdpfisf; nfhd;W mtu;fspd; fz;Lgpbg;Gfis nry;yhjjhff; pdhu;. ,e;j kNdhghtj;ij nfhz;l mg+`f;fPk; ,g;D ifjj;jpd; mwpf;ifia fz;L Vkhw;wkile;jhu;. ,jidj; njhlu;e;J mg+ `f;fPk; mtiu epu;thf gjtpapy; mku;j;jpdhu;. gpd;du; kpf mghafukhd kdpju; vd;gjhf epidj;J tpl;Lr; rpiwapy; itj;jhu;. ,J mg+ `f;fPkpd; kuzk; tiu njhlu;e;jJ. me;j fhyfll; j;jpYk; ,g;D ifjk; fzpjk; kw;Wk; ,aw;gpay; njhlu;ghd Ma;Tfspy; <Lgl;lhu;. rpiw thrk; Kbe;j gpd;du; gpe;ija fhyq;fspy; fzpj Ma;Tfspy; nrytpl;lhu;. xU Nfhs Mbapy; xspahdJ gl;L njwpf;Fk; NghJ Vw;gLk; vjpnuhsp Gs;spf;Fk; ghu;gtupd; ghui; tfF; khd J}uj;ij mstpl;lhu;. mij #upa cwT rhu;ghdjhf ek;gpdhu;. NkYk; jd;Dila xsp njhlu;ghd Ma;T vd;gJ mDgtg+u;tkhdJ vd;Wk; mUt Nfhl;ghL my;y vd;Wk; ntspg;gLj;jpdhu;. NkYk; vjpnuhspg;G kw;Wk; xsp tpyfy; (Refrection) Nfhl;ghL Fwpj;Jk; Muha;e;jhu;. vz; Nfhl;ghl;by; tpy;rd; Njw;wj;ij mbg;gilahf nfhz;L jPu;T nfhz;lhu;. 1+ (p-1)! kw;Wk; Kjd;ik vz;fis fz;lwpa topfis fz;lwpe;jhu;. Kjd;ik vz;iz fz;lwptjw;F k>1, Mf ,Ue;jhy; 2k-1b Kjd;ik vz;zhf ,Uf;Fk;.2k-1(2k-1) KO vz;zhf ,Uf;Fk;. ,tupd; xspapay; kw;Wk; fzpjk; njhlu;ghd Ma;Tfs; fpjhGy; kdhrpu; vd;w ngaupy; 7 ghfq;fis nfhz;l E}yhf ntspte;jJ. fzpjj;jpy; ,g;D ifjj;jpd; gq;fspg;G muGyfj;jpd; gjpakhf ,Uf;fpwJ. jpupNfhz kpjpapy; kpFe;j mjpu;Tfis Vw;gLj;jpatu; erUP j;jpd; my; Jrp. fp.gp 1201 y; <uhdpy; cs;s nfhurhdpy; gpwej; Jrp Nju;e;j rpe;jidahsu;. ,tupd; je;ij ~pah gpupit Nrue; ;jtu.; mg;gpuptpd; ePjpgjpahf ,Ue;jhu;. ,sikf; fhyj;jpy; kjf;fy;tpia gbf;Fk; fhyj;jpy; kw;w Jiwfs; rhu;e;J mtUf;F ehl;lk; te;jJ. ju;ff; tpay;> ,aw;gpay;> fzpjk; Nghd;w Jiwfs; Fwpj;j Mo;e;j gbg;G Nkw;nfhz;lhu;. fp.gp 1214 y; kq;Nfhypa jiytu; nrq;fp];fhdpd; gilfs; rPdhit ifg;gw;wpa gpd; <uhid Nehff; p te;jd. ,jdhy; my;-Jrp eprhg;g+Uf;F efu;e;jhu;. eprhg;g+u; mtupd; ,sikf;fhy nraw;ghLfSf;F cfe;jjhf ,Ue;jJ. ~uGj;jPd; Jrp vd;gtuplk; fzpjk; fw;whu;. mJ mtupd; Ma;TfSf;F xU njhlff; khf mike;jJ. kq;Nfhypa gilfs; nfhypF vd;gtupd; jiyikapy; 1258 y; <uhid ifg;gw;wpd. nfhypF mjd; kd;duhdhu;. tlNkw;F <uhdpYs;s kuNf efuj;ij jd; jiyefuhf mwptpj;jhu;. 27

gpd;du; my;-Jrp jd; Ma;T Kiwapd; tpupthff; j;jpw;fhf nfhypFld; ,ize;J nfhz;lhu;. nfhypF ,tupd; nraw;ghLfSf;F Mjutspj;jhu;. mq;F Jrpapd; Ntz;LNfhspd; gb thdtpay; Ma;T$lj;ij mikj;jhu;. me;j Ma;T$lj;ij gadg; Lj;jp Jrp gy thdtpay; rhjdq;fis tbtikj;jhu;. NkYk; fzpjk;> jj;Jtk;> thdpay; Nghd;w Jiwfspy; css; rpwe;j E}y;fis nfhz;l E}yfk; xd;W me;j $lj;jpy; cUthdJ. fzpjtpaypy; Jrpapd; rhjid jpupNfhzkpjpia (Trigonometry) fzpjj;jpd; jdpj;j gpupthf khw;wpaJ jhd;. mJtiu mJ thdtpaypd; ntWk; fUtpahf kl;LNk ,Ue;jJ. nrq;Nfhz Nfhs Kf;Nfhzj;jpd; MW gFjpfis mstpLk; Kiwia cUthf;fpdhu;. mjd; tha;g;ghL gpd; tUkhW mike;jJ. a/sin A = b/sinB = c/ sin C NkYk; KO vz;zpd; nth root fhZk; KiwAk; fz;lwpe;jhu;. Binomial Nfhl;ghl;by; ,tu; tFj;j KiwahdJ gpw;fhy fzpjtpayhsu;fSf;F cjtpfukhf ,Ue;jJ. a+f;ypl;bd; tbt ,ay; Kiwapy; FsWgbfis ntspg;gLj;jpa my;-Jrp tbt ,aYf;F Gjpa tpjpKiwfis tFj;jhu;. NkYk; fWg;G> ntsi; s epwq;fspd; fyitahy; cz;lhFk; khw;wq;fis Fwpj;J Muha;e;jhu;. ntspapd; (Space) gupzhkk; gw;wpa Nfs;tpfis vOg;gpdhu;. jj;Jtj;jpy; ,tupd; rpe;jidfs; ,g;D rPdhit gpd; njhlu;e;jd. ek; ,ay;G epiyia kPwpa kNfhd;dj mwpTepiy xd;W ,Uf;fpwJ vd;Wk; mij fz;lwpa Ntz;Lk; vd;Wk; Fwpg;gpl;lhu;. ,tupd; gy;NtW Jiwfs; rha;e;j Ma;Tiufs; E}y;fshf ntspte;jd. fpNuf;fk; kw;Wk; yj;jPd; nkhopfSf;F nkhopkhw;wKk; nra;ag;gl;ld. fzpjj;jpy; ngUk; mjpu;Tfis Vw;gLj;jpa my;-Jrp muG tuyhw;wpy; ,d;Wk; njhlu;fpwJ. my; [Pgpuhit tsu;j;njLj;jtu;fspy; ghurfP ftpQuhd cku;fa;ahk; Fwpg;gplj;jFe;jtu;. tuyhW cku;fa;ahik ntWk; ftpQuhfNt mwpKfg;gLj;JfpwJ. fp.gp 1048 y; <uhdpYs;s eprhg;g+upy; gpwej; tu;. fa;ahk; vd;w ghurPf nrhy;Yf;F $iu mikg;gtu; vd;gjhf mu;j;jk;. cku; fa;ahkpd; fhyk; ghurPf gFjpia nrYrpa JUf;fpau;fs; ifg;gw;wpa fhykhf ,Ue;jJ. 25k; tajpy; c];ngfp];jhdpd; rku;fz;l;bw;F nrd;w fa;ahk; mq;F ePjpgjpahf ,Ue;j mg+ jh`pupd; JizNahL jd; Ma;Tfis Nkw;nfhz;lhu;. mq;F jhd; mtupd; my;[Pgpuh gw;wpa Gj;jfkhd Treatise on Demonstration of Problems of Algebra it vOjpdhu;. nrY}rpa muruhd lhf;upy; ngf; <uhdpd; ,];g`hid jd; jiyefuhf mwptpj;jhu;. mtUf;F gpd; te;j thuprhd khypf; ~h ehl;by; khWjiy Vw;gLj;Jk; nghUl;L Gjpa Ma;T$lq;fis epWt KbT nra;jhu;. 28

gpd;du; c];ngfp];jhDf;F jd; gpujpepjpia mDg;gp ,q;F thdpay; Ma;T $lj;ij mikj;jhu;. mjd; gpd;du; mtupd; Kiwg;gbahd Mo;ik Jtff; k; ngw;wJ. fhyz;lu; Kiwia khw;wpaikg;gjpYk;> thdtpay; ml;ltizia cUthf;FtjpYk; fa;ahkpd; gq;fspg;G Fwpg;gplj;jFe;jJ. fa;ahk; xU tUlj;jpw;fhd ehl;fis Jy;ypakhf 365.24219858156 vd;whu;. mtu; mikj;j fyz;luhdJ cNuhkhdpauf; is jhz;Lk; tpjkhf mike;jJ. 1092 y; khyp~htpd; kiwTf;Fg;gpd;du; thupR Aj;jj;jhy; mg;gFjp epiyFiye;jJ. cku;fa;ahkpd; gilg;Gfs; vJTNk ntspf;nfhzug;gltpy;iy. ,jd; gpwF fa;ahk; Ju;Nkdp]j; hDf;F nrd;wile;jhu;. mq;F jhd; mtupd; Kiwg;gbahd fzpj Ma;Tfs; njhlf;fk; ngw;wJ. fd rkd;ghl;bw;F jPu;T fhz;gjw;fhf %d;W topfisf; fz;lwpe;jhu;. Jujp~;ltrkhf %d;whtJ top rupahf mikatpy;iy. NkYk; nrq;Nfhz Kf;Nfhzj;jpd; fu;z mbg;gilapyhd tha;g;ghl;il mikj;jhu;. ù3 + 200ù ® 20ù2 + 2000 vd;gJ mtu; mikj;j fd rkd;ghlhFk;. ,ij nrt;tf mjpgu tistpd; (Rectangular hyperbola) mikg;NghL nghUj;jpdhu;. Binomial Nfhl;ghl;bw;fhd juTfis cUthf;fpdhu;. fzpjk; gw;wpa mtupd; gy Ma;Tfs; fz;lwpag;glhky; ,Uf;fpd;wd. mtupd; ftpijfs; njhFf;fg;gl;L Uigaj; vd;w jiyg;gpy; ntspte;jJ. jd; fzpj Ma;Tfs; %yk; cku; fa;ahk; ghurfP tuyhw;wpd; jdpj;j gf;fkhf ,Uf;fpwhu;. kj;jpa fpof;fpd; fp.gp 750 Kjy; 1050 tiuapyhd fhyfl;lk; tuyhw;whrpupau;fshy; nghw;fhyk; vd;W miof;fg;gLfpwJ. mg;gh]pl; tk;rNk mjw;fhd ngaupd; ngUk;gFjpia milfpwJ. md;iwa mg;gh]pl; tk;rkhdJ vfpg;J> rpupah> ghy];jPd;> tl Mgpupf;fh> ];ngapd;> JUf;fp> <uhf; Mfpa gFjpfs; tiu tpuptile;jpUe;jJ. ghf;jhij jiyikaplkhff; nfhz;L mtu;fs; mikj;j mwpT juprd ,y;yk; tpQ;Qhd tsu;r;rpapd; ngUk; Cw;whf cUthdJ. gy fpNuf;f tpQ;Qhd E}y;fs; muG nkhopf;F nkhopkhw;wk; nra;ag;gl;ld. fp.gp 767 y; K`k;kJ gpd; m`;kj; mNugpa g+[;a Kiwia cUthff; paNjhL kj;jpa fpof;fpd; fzpj tuyhW njhlq;FfpwJ. my;-fthup];kp njhlq;fp itj;j my; [Pgpuh kw;Wk; my;fhupjk; ,d;iwa jfty; njhopDl;g cUthf;fj;jpd; gpd; fUtpahf ,Uf;fpwJ. my; fthup]k; pf;F gpwF ,e;jpahtpYk;> Nkw;FyfpYk; my;[Pgpuh gupzkpf;fg;gl;lJ. my;-fthup]k; p tupirapy; f[;[h[; ,g;D kj;ju;> gD} %]h> ,g;uh`pk; ,g;D rpdhd;> `idd; ,g;D <rhf;> jhgpj; ,g;D Fg;uh> `rd; c`;yprp Nghd;Nwhu; fzpjj;jpy; njhlUk; gbahd Ma;Tfis nra;jdu;. NkYk; rPdu;fshy; fz;Lgpbf;fg;gl;l fhfpjj;ij muGfs; fw;W mij cw;gj;jp nra;jdu;. fp.gp gjpndhd;whk; E}w;whz;by; ,ilg;gFjpapy; ghf;jhjpy; fhfpj njhopw;rhiy Vw;gLj;jg;gl;lJ. ,jid Kdi; tj;J mNef E}y;fs; cUthff; g;gl;ld. Mdhy; ,e;j tsu;rr; p epiyfs; vJTNk fhyj;ij jhz;ltpy;iy. gjp%d;whk; E}w;whz;Lld; epd;W NghdJ. me;E}w;whz;bd; ,Wjp gFjpapy; muGyfpy; ngUk; khWjy;fis Vw;gLj;jpa mg;gh]pl; tk;rk; gytPdkila njhlq;fpaJ. kq;Nfhypau;fs; 1256 y; <uhf;if ifg;gw;wpdu;. ghf;jhij ikag;gLj;jp mtu;fs; nra;j mopT nray;ghLfs; rpy E}w;whz;Lfspd; Njf;f epiyf;F njhlf;fk; Fwpj;jJ. kpfg; ngUk; Jaukhf mg;gh]pl;fshy; Vw;gLj;jg;gl;l mwpT juprd ,y;yk; epu;%ykhff; g;gl;lJ. mq;Fs;s nghUl;fs; midj;Jk; mopf;fg;gl;ld. Mtzq;fs; jPahf khwpaJ. E}yfq;fspy; css; Gj;jfq;fs; fpopf;fg;gl;L a+g;uB]; Mw;wpy; trP g;gl;ld. ,jdhy; a+g;uB]; ejp KOtJk; fUg;G ikahy; epug;gg;gl;lJ. NkYk; ,U kpy;ypad; kf;fs; nfhy;yg;gl;ldu;. Ma;T $lq;fs; jfu;ff; g;gl;ld. <uhff; pypUe;J kw;w gFjpfSf;F nry;Yk; rhiyfs; midj;Jk; Jz;bf;fg;gl;ld. gs;spfs; %lg;gl;ld. 29

kq;Nfhypauf; spd; ,e;j mopT nray;ghLfSf;F ntspNa kw;nwhU fhuzk; 11 kw;Wk; 13 Mk; E}w;whz;LfSf;F ,ilNa ele;j rpYit NghuhFk;. ,J NkYk; xU Njf;f epiyf;fhd Jtf;fkhf ,Ue;eJ. ];ngapdpd; toP ;rr; pahdJ muGfspd; mwptpay; Mf;fq;fs; vJTNk ntsptuhj fhupaj;jpw;fhd fhuzkhf ,Ue;jJ. Gj;jf tbtpy; ,Ue;jit midj;Jk; mopf;fg;gl;ld. gjpd;%d;whk; E}w;whz;by; muGyfpy; vOe;j Ky;yhaprk; kw;nwhU fhuzkhFk;. mtu;fs; vy;yhNk kj mbg;gilfspd; top eilngw Ntz;Lk; vd;wdu;. gpw;fhy cJkhdpa Nguurpd; ke;j epiyf;Fk; ,JNt fhuzk;. rkPg ,U E}w;whz;Lfspd; ,Wf;f epiyf;F rt+jpia ikag;gLj;jp cUthd t`;`hgpa ,af;fj;jpd; gq;fspg;G ehk; kWf;f Kbahj xd;W. ,jd; njhlu;rr; p gy E}w;whz;Lfspd; Njf;f epiyf;F njhlf;fkplyhk.; kj;jpa fpof;fpd; ehd;F E}w;whz;Lfshd nghw;fhyk; Nkw;fj;jpa Neu;ikahd tuyhw;whrpupau;fshy; ,d;Wk; Fwpf;fg;gLfpwJ. fzpjtpaYk; msitapaYk; mstpaypd; njhlf;fj;jpy; jhd; fzpjtpaYk; Njhd;wpaJ vd;gjw;F gy Mjhuq;fs; css; d. Ntshd; nray;fSf;fhf epyq;fis msT epu;zak; nra;Ak; NghJ fzpjtpay; Njhd;wpaJ. ,Nj Nghd;W tuyhw;wpy; njhlu;e;J thdpay; NjitfisAk; <Lnra;tjw;fhf fzpjtpay; tsu;e;jJ vd;gjw;Fk; gy Mjhuq;fs; css; d. fpwp];J gpwe;j gpd; ,aw;ifapd; ,ay;ghd epfo;Tfis tpsf;Ftjw;fhf fzpjtpay; NkYk; tsu;r;rpaile;jJ. fypypNah fzpjtpaypd; Kf;fpaj;Jtj;ij gpd;tUkhW Fwpg;gpl;lhu;. ,aw;ifiag; gw;wpa E}y; fzpjtpay; nkhopapy; vOjg;gl;lJ. jw;nghOJ fzpjtpay; nfhs;iffisAk; tptuq;fisAk; gad;gLj;jhj mwptpay; JiwNa ,y;iy vdyhk;. gpupl;ld; mUq;fhl;rpafj;jpy; fhzg;gLk; ,upz;l; njhFg;Gfspd; xU gFjpahf tpsq;Fk; f`{d; Nfhiuj;jhsf; s; jhd; vz;fzpjk; tbtf; fzpjk; tuyhW gw;wp ekf;Ff; fpilj;Js;s Kjy; Mtz Mjhuq;fs; MFk;. ,e;j Nfhiuj;jhs;fs; BCE 1900 Kjy; 1600f;Fs; vOjg;gl;bUf;f Ntz;Lk; vd;Wk; mf;Nkh]; vd;w Mrpupauhy; gbtk; vLf;fg;gl;lJ vd;Wk; ,tw;wpd; %yg;gbtk; BCE 2300f;Fk; Kw;gl;l kpfTk; goikahd vfpg;J ehl;L XII mur tk;r fhyj;ij Nrue; ;j Nfhiuj;jhs; cUis vd;Wk; njupa tUfpwJ. me;jf; Nfhiuj;jhspy; vz; fzpjk; gw;wpa nghJthd nra;Kiwfs; gw;wpAk; gpd;dq;fs; gw;wpAk; tptuq;fs; nfhLf;fg;gl;Ls;sd. ngUf;fy; njhlu;rpahd $l;lyf; spdhy; nra;ag;gl;lJ. msitapaYf;fhd tpjpfSk; nfhLf;fg;gl;Ls;sd. 30

,jpypUe;J 500 Kjy; 1000 tUlq;fSf;Fs; vz;zpf;if KiwAk; ngUf;fy; tFj;jy; tu;ff; k; tu;f;f%yk; fd rJuk; Nghd;wtw;wpw;fhd fzpjtpay; ml;ltidfSk; msj;jYf;fhd tpjpfSk; myFfSk; vfpg;jpau;fshy; ruP hf;fg;gl;Ltpl;ld. vfpg;jpau;fs; gj;J tiu css; vz;fis Fwpf;f nrq;Fj;Jf; NfhLfis cz;lhf;fpdu;. xd;iwf;Fwpf;f xU nrq;Fj;Jf; NfhLk; ,uz;ilfF; wpf;f ,uz;L NfhLfSk; gj;ijf; Fwpf;f 10 NfhLfSk; xd;wpd; gf;fj;jpy; xd;whf Nghll; du;. vfpg;jpau;fSf;F mbg;gil vz; 10 MFk;. vdpDk; vfpg;jpa vz; Kiw (mjpypUe;J Njhd;wpa cNuhkd; vz; KiwAk; ) ngUf;fiyAk; tFj;jiyAk; nra;tjw;F kpfTk; fbdkhf ,Ue;jJ. (vLj;Jf;fhl;lhf VIII I XXI My; ngUf;FtJ) ghgpNyhdpag; gFjpfspy; vfpg;jpa fhyj;jpw;F rkdhd (mjw;F rw;W Kdd; u; $l ,Uf;fyhk; ) fhyj;jpy; xNu rkaj;jpy; fzpjtpay; gupzkpj;jJ . 4000 BCEapd; NghJ fzpjtpay; fUj;JUf;fs; tsu Ntshz;ik kpfTk; Kf;fpa fhuzpahf jpfo;e;jJ. Ntshz; nray;fspy; <Lgl xU fyz;lu; Kiw Njitgg; l;lJ. ,jid fzpj mbg;gilapy; ghgpNyhdpau;fs; cUthf;fpdhu;. xU ehis fhyj;jpd; myfhf epu;zapg;gjw;F ,aw;if epfo;Tfs; ghgpNyhdpauf; is J}z;baJ. fhyj;jpd; ePz;l myfhf xU khjj;ijAk; ( month vd;w nrhy; moon vd;Wk; re;jpuid Fwpf;Fk; nrhy;ypypUe;J Njhd;wpaJ). xtn; thU khjKk; mkhthir md;W Muk;gpj;jJ. gpd;G gUtq;fspd; xU Row;rpapy; vtt; sT khjq;fs; tUfpd;wd vd;gij epu;zak; nra;Ak; Kaw;rpfs; Nkw;nfhss; g;gl;ld. Vwj;jho BCE 2000 tUlj;jpy; ghgpNyhdpa tUlk; 360 ehl;fs; css; lq;fpajhfTk; 12 khjq;fisf; nfhz;ljhfTk; epu;zak; MdJ. mtt; g;nghOJ Njitf;Nfw;g $Ljy; khjq;fs; ,ilr;nrUfg;gl;ld. xU ehs; kzpfshfTk; kzpfs; epkplq;fshfTk; epkplq;fs; tpdhbfshfTk; gpupf;fg;gll; d. gpd;G Neuj;ij mwpa #upa fbfhuq;fs; fz;Lgpbf;fg;gl;ld. ,jd; gpd;G 7 ehl;fisf; nfhz;l thuk; vd;w fhy myF epu;zapf;fg;gll; J. thuj;jpd; VO ehl;fSk; #upad; re;jpud; kw;Wk; me;j fhyj;jpy; mwpag;gll; 5 Nfhs;fshd nrt;tha;> Gjd>; tpahod;> ntss; p kw;Wk; rdpapd; ngau;fshy; mwpag;gll; d. BCE2500 tUlj;jpd; NghJ ghgpNyhdpauf; s; msitfSf;fhd epiyahd myFfisg; gadg; Lj;j Muk;gpj;Jtpl;ldu;. ghgpNyhdpauf; Sk; gd;dpnuz;nlkhd KiwiaAk; (,J gpd;dq;fis fz;Lgpbf;f kpfTk; cjTk; KiwahFk;) 10 iftpuy;fisg; gad;gLj;jp ngwg;gLk; jrk KiwAk; mwpe;jpUe;jdu;. epy msitapd; Muk;g fhyr; #j;jpuq;fspy; ,Ue;Jk; vz;fspy; ,Ue;Jk; tbtf;fzpjk; Njhd;wpaJ. ghgpNyhdpauf; isj; njhlu;e;J fpNuf;fu;fs; fzpjtpay; tsu;r;rpf;F cjtpAs;sdu;. kpy;ypl]; efuj;J NjyP]; vfpg;jpw;F tpa[k; nra;J tpl;L te;j gpd;G vfpg;jpa tbtf; fzpjj;jpd; mbg;gilapy; epy mstPL kw;Wk; ,lj;ijg; gw;wpa mwptpay; Nghd;wtw;iw ruP ;gLj;jpdhu;. gpj;jNfhu]; vd;gtu; fzpjtpaypd; Kf;fpaj;Jtj;ij czu;e;J gpd;tUkhW $wpdhu; . vy;yhg; nghUl;fSNk vz;fs; jhd; mtu; xU jiy rpwe;j jj;JtQhdpAk; kpfTk; rpwg;ghd Mdhy; tpNehjkhd fzpjtpay; epGzUkhthu;. fzf;fpay; kw;Wk; G+lftpay; vd;w Jiw nfhs;ifiag; nghJthd #j;jpuq;fshf Fiwj;J mtw;wpy; ,Ue;J jdpj;jdpahf xt;nthU fzpj tpdhTf;Fk; jPu;Tfhz Fwpg;gpl;l rkd;ghLfisg; ngw top Nfhypd. gpaup irkd; ,yhg;yh]; (1749- 1827) epAl;ldpd; ,ay;gpay; Ma;TfSf;F NkYk; gy fzpjtpay; jPu;Tfs; %yk; Mjhuq;fs; nfhLj;jhu;. mtUila E}yhd Mechanique Celeste (1799-1805) epAl;ldpd; vy;yh tpjj;jpYk; KOikahf;fpaJ. 31

19k; E}w;whz;by; fzpjtpaypy; gy Gjpa Jiwfs; Njhd;wpd. ,tw;wpy; kpf Kf;fpakhdit vz;nfhsi; f ( ,J /ngu;khlb; d; gilg;GfNshL njhlq;fpaJ.) tbtq;fs; kw;Wk; FOf;fs; gw;wpa nfhs;iffs; Kf;Nfhztpaypd; gd;kb fhythbr; rhu;gad;fs; gw;wpa nfhs;iffs; kw;Wk; rhu;gad;fs; gw;wpa nghJf;nfhsi; ffs; Nghd;witahFk;. njhFg;G kw;Wk; gFg;G Ma;T Kiwfs; xU Gjpa tbtff; zpjtpay; Jiwia cUthff; paJ. 1830 Mk; Mz;L A+f;ypl; jtpu;e;j tbtff; zpjk; njhlq;fg;gl;L 20 Mk; E}w;whz;L njhlff; j;Jf;Fs; ruP hff; g;gl;lJ. ,tw;wpy; Gjpa fzpjtpay; Jiwfs; ngUk;ghyhd ,ay;gpay; tpdhff; s; gytw;wpw;F jPu;T fhd cjtpd. 1822 Mk; Mz;L /G+upau; (glk; VII.2 ) jd;Dila Theorie analytique de la chalelur vd;w gilg;ig cUthf;fpdhu;. ,jpy; njhlu;r;rpahdNjh my;yJ njhlu;r;rpaw;wNjh Md xU khwpapd; xU rhu; gaid gd;kbj;njhluzp ePnljpuj; jfTfshf tpupthff; k; nra;ayhk; vdW; vLj;Jff; hl;bdhu;. ,yhf;uhq;Nf kw;Wk; ,yhg;yh]pd; gilg;Gf;fisf; fh]; vd;gtUk; NkYk; ruP hf;fpf; kpd;rhuk; gw;wpa jd;Dila Ma;T KbTfisg; ngw gad;gLj;jpdhu;. mstPl;by; css; jtWfs; gw;wpa nfhs;iffisAk; mtu; cUthf;fpdhu;. ,yhf;uhq;Nf ,af;ftpaypy; gadg; Lj;jpa tiff;nfO rkd;ghLfis `hkpy;ld; (1805- 1865) NkYk; tpupTgLj;jpdhu;. A+f;fpypbd; tbtfzpjj;ij rf;Nfup ,tghl;rPt;];fp (1826-1840) fh]; (1831-1846) Nghy;aaha; (1954) Nghd;Nwhu; NkYk; tpupTgLj;jpdhu;. A+f;fpypbd; jtpu;j;j tbtfzpjj;Jiwf;F ,uPkhd; (1844) Nfa;yP ngy;buhkp n`y;k; N`hy;l;]; fpisd; kw;Wk; tpl;]n; `hl; Nghd;Nwhu; NkYk; tptuq;fis Nruj; ;jdu;. A+f;fpypbd; jtpu;j;j ntspapd; gz;zgfis fzpjtpay; %ykhf epu;zapf;fyhk; vd;Wk; me;j tifahd ntsp ek;Kila Gyd;fSf;Fj; njupe;jhNyh njupahkNyh ,Ue;jhy; $l ,J rhj;jpakhdJ vd;Wk; Nkw;$wpatu;fspd; gilg;Gf;fs; vLj;Jff; hl;bd. ,e;j Ma;Tfs; midj;Jk; My;gh;l; [hd;];Bd; jd;Dila rhu;Gf; nfhs;ifia cUthf;f ngUk; gq;fhw;wpd vd;gJ Fwpg;gplj;jf;fJ. kdpj ,dk; vt;tsT goikahdNjh mt;tsT goikahdJ msitapaYk; ghgpNyhdpah vfpg;J fpup]; cNuhk; muhgpah kw;Wk; ,e;jpahtpy; msitapay; tsue; ;jpUe;jjw;fhd Mjhuq;fs; gy css; d. BCE2500 f;F Kd;Ng ghgpNyhdpauf; s; murf; fll; isg;gb Fwpg;gpl;l msitapay; myFfisg; gadg; Lj;jpdu;. mtu;fs; ePsj;jpw;fhf gad;gLj;jpa myF tpuy; (finger) MFk;. ,J 1.6 nr.kP my;yJ 2Æ3 mq;FykhFk;. xU mb (foot) 20 tpuy; nfhz;lJ. xU $gpl; (cubit) 30 tpuy; nfhz;lJ. xU (fop pole) 12$gpl;fSk; xU msitahspupd; fapW (curd) 120 $gpl;fSk; xU yPf; (league) 180 fapWfisAk; (mjhtJ 6.65 iky;fs;) nfhz;lJ vd;Wk; mtu;fSila msitapay; tpsq;FfpwJ. viliag; nghWj;jtiu xU jhdpak; 0.046fpuhk;fisAk; xU ~nP fy; (shekel) 8.416 fpuhk;fisAk; xU Nlyd;l; (talent) 30.5 fp.fp my;yJ 671Æ3 gTz;LfisAk; nfhz;lit. NkNy Fwpg;gpl;l ehLfspy; gad;gLj;jpa msitapay; KiwAk; myFfSk; ghgpNyhdpa; Kiwapy; ,Ue;J rw;Nw my;yJ kpfTk; NtWgl;ljhf ,Ue;jd. vilfspYk; msitfspYk; ehLfspilNa fhzg;gl;l mjpfsT NtWghLfs; mjpf ghjpg;Gf;fis Vw;gLj;jpajhy; (,d;Wk; Vw;gLj;jpf; nfhz;bUf;fpd;wd) xU nghJthd msit Kiw Njitg;gl;lJ. ,e;j mbg;gilapy; Njhw;wk; ngw;wJ jhd; gpuhd;]; ehl;L jrk my;yJ gjpd;khd Kiw . 1799 Mk; Mz;L gpuhd;]; ehl;bdhy; mq;fpfupff; g;hl;l ,k;Kiw 1812 Mk; Mz;L eilKiwf;F te;jJ. 1820k; Mz;L fll; hakhf;fg;gl;lJ. ,e;j Kiwg;gb ePsj;jpd; mbg;gil myF kPw;wu; MFk;. 32

ghup]; efuk; topahfr;nry;Yk; Gtpapd; ,U mr;Rf;fSk; xd;iw xd;W ntl;Ltjhy; Vw;gLk; ehd;F fhy;ntl;Lg; gFjpfspy; xd;wpd; gj;J kpy;ypad; gq;fpy; xU gq;F ePsk; 00c apy; xU cNyhff; fk;gpapy; ,uz;L milahsq;fspw;fpilapy; cs;s esP khff; Fwpf;fg;glL; mJNt kPl;lupd; mbg;gilahf itj;Jf; nfhs;sg;gl;lJ. G+kpapd; fhy; ntl;Lg; gFjpapd; msT gpd;dhy; rupahf epu;zak; nra;ag;gl;Lk; $l ,e;j cNyhf fk;gpapy; Fwpf;fg;gl;l ePs msT jhd; xU kPw;whf epiyj;J epd;wJ vd;gJ ,q;F Fwpg;gplj;jf;fJ. gupkhzj;jpd; myfhf yPll; u; Vw;Wf;nfhss; g;gl;lJ. Kjd;Kjy; xU yPl;lu; vd;gJ xU nlrp kPl;lu; gf;fthl;L ePsk; nfhz;l xU fd rJuj;jpw;F rkkhdJ vd;W epu;zapf;fg;gl;lhYk; ,jpy; css; ,lu;ghLfis kdjpy; nfhz;L 1901 Mk; Mz;L xU yPl;lu; vd;gJ xU ml;kh]g; pau; thA kz;ly mOj;jj;jpy; 40c ntg;gepiyapy; mjd; mjpfgl;r mlu;j;jpapy; xU fpNyhf;fpuhk; ePupd; nfhs;ssT vdf; nfhss; g;gl;lJ. nghUz;ikapd; myfhf fpNyhf;fpuhk; nfhs;sg;gl;lJ. 40c ntg;g epiyapy; xU fd nlrp kPl;lu; ePupd; nghUz;ik xU fpNyhf;fpuhk; MFk; . Neuj;jpd; mbg;gil myF tpdhb MFk;. ,J ruhrup #upa ehspd; 1/86400 gFjpahf Vw;Wf;nfhss; g;gl;lJ. 1822 Mk; Mz;L mbg;gil myFfspd; mbg;gilapy; ,uz;lhk; epiy my;yJ ngwg;gl;l myFfisg; /G+upau; epu;zapf;Fk; NghJ mit NtW gupkhzq;fisg; ngWk; vd;gijr; Rl;bf;fhl;bdhu;. vLj;Jf;fhl;lhf ehk; ePsj;ij L vd;Wk; nghUz;ikia M vd;Wk; Fwpg;gpl;lhy; jpirNtfj;jpd; V gupkhzk; mjhtJ xU myF Neuj;jpy; Nruf; ;fg;gll; ePsk; L/T my;yJ LT-1 MFk;. KLf;fj;jpd; mjhtJ xU myF Neuj;jpy; Nruf; ;fg;gl;l jpirNtfj;jpd; gupkhzk; V/T MFk;.  rPdhtpy; fp.K.200 iar; Nru;e;j Mik XL xd;wpy; khar;rJuk; fhzg;gl;lJ. Mdhy; gjpdhwhk; E}w;whz;by; jhd; khar;rJuk; mwpKfg;gLj;jg;gl;lJ.  “fzpjk; vd;gJ mwptpaypd; uhzp. vz; fzpjk; vd;gJ fzpjj;jpd; uhzp” vd;W $wpa fzpj Nkij fhu;y; nfs];.  “capUs;s capuw;witfspy; ehd; fzpjj;ijg; ghu;j;Njd;. fzpjk; cyif Ms;fpwJ.” vd;W $wpatu; kfh tPuu;(850 AD) fzpjtpay; kw;Wk; msitapay; gw;wpa ,e;jpa tuyhW Ntj fhyj;jpw;F Ke;jpa fhyk; rpe;J rkntsp ehfupf kf;fs; kpfr;rpwe;j msit Kiwfisg; ngw;wpUe;jdu; vd;gjw;F Mjhuq;fs; css; d. mtu;fSila efuikg;Gj; jpl;lkplYf;Fk;> flb; ltpay; jpwikfSf;Fk; rpwe;j msit Kiwfs; Njitgg; l;ld. mtu;fs; gad;gLj;jpajhf ek;gg;gLk; vilf;fhd myF cd;rpah (Uncia) (=27.2 fpuhk; vdg;gl;lJ). mtu;fs; ePsj;jpw;fhd myfhf rpe;J mq;Fyj;ij (indus inch) gad;gLj;jpdu;. ,J ,uz;L RNkupa R~pf;F (Sumerian shushi) rkkhFk;. 33

25 rpe;J mq;Fyq;fs; Kfyha fhyj;jpy; gad;gLj;jpa xU f[j;jpw;F (yard) rkkhFk;. vz;fs; nrq;Fj;J NfhLfshf Fwpf;fg;gl;ld. Ntjfhyk; Ntjfhyj;jpy; fzpjtpay; kpfTk; rpwg;ghf tse;jJ. vdpDk; gy E}yhrpupauf; s; Ntj fzpjtpaYf;F (vedic mathematics) nrhe;jkpy;yhj gy fzpjtpay; fz;Lgpbg;Gf;fis vLj;Jf; $wpapUg;gJ cz;ikapNyNa Ntj fzpj fz;Lgpbg;Gf;fisAk; Iak; nfhs;Sk; gbahf mike;jpUg;gJ tUe;jj;jf;fJ. cz;ikahd Ntj fzpjtpay; gilg;Gfs; Shulva sutra (sulba sutra) vd;Wk; miof;fg;gLk;. Baudhayana sutra, Aphastambha sutra, Yajurveda samhita, Taittiriya samhita kw;Wk; Vajasneyi samihita Nghd;w Ehy;fspy; fhzg;gLfpd;wd. Shulva sutra (Vwj;jhs BCE 1500) vd;w E}y;> mjd; ngaUf;F Vw;g> msitfSf;fhd tpjpfis vLj;J $Wfpd;wJ. ePsq;fs; fapWfs; nfhz;L msf;fg;gl;ld. ,jd; fhuzkhfNt gpw;fhyj;jpy; fapWfSf;F Shulva vd;w ngau; nfhLf;fg;gl;lJ vd;gJ ,q;F Fwpg;gplj;jf;fJ. ,e;j #j;jpuq;fs; BCE 8tJ Ehw;whz;LfSf;F Kd;gpUe;Nj tof;fj;jpypUe;jpUf;f Ntz;Lk;. tbtf; fzpjk;> vz;fzpjk; Mfpatw;iwg; gw;wpa fzpjtpay; fUj;Jff; s; Gdpj rlq;Ffs; nra;tjw;fhd eilKiwr; nray;fspypUe;J> Fwpg;ghf ahfq;fs; nra;tjw;fhd Nkilfs; jahupg;Gr; nray;fspypUe;J Njhd;wpad vd;gjw;F gy Mjhuq;fs; css; d. A+f;fpspl;bd; Elements Nghd;W Shulva theorem tpy; gpj;jf;Nfhu]; Njw;wk; fhzg;gLfpwJ. Mdhy; mjw;F ep&gzk; nfhLf;fg;gltpy;iy. ,jd; fhuzkhf ,e;jj; Njw;wk; gpj;jf; Nfhu]; Njw;wk; vd;gjw;F gjpyhf fy;th Njw;wk; (Shulva theorem) vd;W rupahf miof;fg;gl Ntz;Lk;. rprNuh (Cicero) vd;gtu; jhd; Kjd;Kjypy; ,e;jj; Njw;wj;ij gpj;jf;Nfhu]; vd;gtu; vOjpdhu; vd;W $wpatu;> ,tu; ,t;thW $Wk;NghJ gpj;jf;Nfhu]; ,weJ; Ie;J Ehw;whz;Lfs; fle;Jtpl;ld vd;gJ ,q;F Fwpg;gplj;jf;fJ. Shulva sutra tpy; ,Ugbr; rkd;ghLfs;> KbTngwh rkd;ghLfs; (indeterminate equations)> gFglh vz;fs; > KbTngwh vz;fs; (irrational numbers) kw;Wk; mtw;wpd; Vwj;jhs kjpg;Gf;fs; Nghd;wtw;iwg; gw;wpAk; tpsf;fq;fs; nfhLf;fg;gl;Ls;sd. vLj;Jf;fhl;lhf> 19Mk; Ehw;whz;L tiu jPu;T fhzKbahj xU tl;lj;ijr; rJukhf;Fk; (squaring the circle) (,jpy; ,uz;bd; gug;gsTk; xd;whf ,Uf;f Ntz;Lk;) tpdh ,e;j Ehypy; Ma;Tf;F vLj;Jf;nfhss; g;gl;Ls;sJ Fwpg;gplj;jf;fJ. xU myF Muk; (radius) nfhz;l tl;lj;jpd; gug;G ������ (mjhtJ ������������������) vd;gjhy; ������ (mjhtJ ������������������) msT nfhz;l xU rJuj;ij cUthf;f Ntz;Lk;. Mdhy; ������ ,d; kjpg;G KbTngwhj vz;zhd 3.14159…. vd;gjhy;> rJukhf;Fjy; fbdk;. vdpDk; ������ kw;Wk; √������ Mfpa KbTngwh vz;fSf;fhd Vwj;jhs kjpg;Gf;fs;> ep&gzk; ,y;yh tpbDk; gz;ila ,e;jpa fzpjtpay; tyY; eu;fSf;Fj; njupe;jpUe;jJ. vLj;Jf;fhl;lhf Shulva sutra √������ ,d; kjpg;ghf 1ñ1Æ3ñ1Æ(3ù4ù34) vd;W Fwpg;gpl;bUg;gJ ,q;F epidT $uj;jf;fJ. Baudhyana sutra Tk; Aphastamba sutra Tk; fpU~;z a[Pu; Ntjj;ij Nrue; ;jit. jw;nghOJ lNah/Ngz;l; rkd;ghLfs; (Diophant equations) vd;W nghJthf miof;fg;gLk; Kjy;epiy KbTngwh rkd;ghLfs; gw;wp ,e;j E}y;fs; tpsf;Ffpd;wd. vdpDk; 34

,e;j fpNuf;f fz;Lgpbg;G ,e;jpa fz;Lgpbg;gpw;F ntFfhyk; fopj;J Njhd;wpaJ. ,Nj Nghd;W ‘G+[a; k;’ (Zero) fz;Lgpbg;G Ntjfhyj;ij rhu;e;jJ. vz;fisj; jrk Kiwg;gb (gjpd;k Kiwg;gb) vOJtJk; Ntjfhy fz;Lgpbg;Gjhd;. Yajurveda samhita, Taittiriya samhita kw;Wk; Vajasneyi samhita Nghd;w gilg;Gfs; kpfg;ngupa vz;fisg; gjpd;khd Kiwg;gb 10d; klq;Ffshf $wpAs;sd. ,tw;wpy; kpfg;ngupajhf 1019 tiu Fwpg;gplg;gl;Ls;sJ. 10 vd;gJ jrk;(dasa)> 102 rjk;(sada)> 103 rf];uk;(sahasra)> 104 MAjk;(ayuta)> 105 epA+jk;(niyuta) vd;Wk; Fwpg;gplg;gl;Ls;sd. rkz fzpjtpay; mwpQuf; s; 1027 tiuapYk;> Gj;j fzpjtpay; mwpQuf; s; 1058 tiuapYk; Fwpg;gpl;Ls;sdu;. 27 Nghd;w vz;fs; jdpahf ngaupl;L rg;jtpk;rjp (saptavimsati) vd;W miof;fg;gl;lhYk;> vz;fis gjpd;khd KiwapYk; vOjpdu;. vLj;Jf;fhl;lhf> 972 vd;w vz; 1000 ,w;Fk; 28 vz;fs; FiwT vd;W fUjp 103 ,y; 28 FiwT vd;W Rl;bf;fhl;bdu;. gpd;d vz;fs; gw;wpAk; rk;`pj Ehy;fspy; $wg;gl;Ls;sd. miu(1Æ2) vd;gJ mu;jh(ardha) vd;Wk;> fhy; (1Æ4) vd;gJ gjh(pada) vd;Wk;> miuf;fhy; (1Æ8) vd;gJ rhgh(sapa)> fhyiuf;fhy; (1Æ16) vd;gJ fhyh(kala) vd;Wk; Fwpg;gplg;gl;Ls;sd. Taittriya Samhita apy; 1>3>5>7>9……...> 2>4>6>8>10………> 4>8>12>16>…...> 5>10>15>20>….. Nghd;w gbKiw vz;tupirfs; gw;wpAk; tpsff; g;gl;Ls;sd. 1200 Mz;LfSf;F Kd;G (Vwj;jhs CE 800 Mk; Mz;L) ghf;jhj; efupd; mg;Jyy; h my;kd;#u;(Abdullah Al Mansur) vd;w ,uz;lhk; mg;ghrpl; fhyp/g;(Abbasid Caliph) Vw;ghL nra;j xU fzpjtpay; epGzu;> ,e;J (,e;jpa) vz;fs; gw;wp xU Ma;Tr; nrhw;nghopT elj;jpdhu;. mg;nghOJ ,e;J vz;fisg; gw;wp ntspehl;ltUf;F vJTk; njupahj epiy. rk;`pj E}y;fisj; jtpu> ek;kpilNa Bakshali Manuscript vd;w fzpjtpay; Fwpg;NgLfs; css; d. 70 gf;fq;fisf; nfhz;l ,e;j Fwpg;NgLfs; G+u;[gj;uq;fs; (Boorjapatra) vd;W Fwpg;gplg;gLk; NjtjhU kug;gl;ilfspy; vOjg;gl;lit> Frhd; muru;fs; fhyj;jit(BCE 200)> rhujh(Sharada) ypgpapYk;(script)> fjh(gatha) vd;w gpuhf;upj(prakrit) nkhopapd; gpuptpYk; vOjg;gl;lJ. fzpjtpaypd; Nkd;ikahd ,jpy; fhzg;gLfpd;wd. ,Ugbr;rkd;ghLfs;> rupahd ,Ugbahf ,y;yhj vz;fSf;F ,Ugb %yq;fs; fz;Lgpbj;jy;> vz;fzpj kw;Wk; tbtf;fzpj gbKiw tupirfs; (arithmetic and geometric progressions) Nghd;wit gw;wpa tpsf;fq;fs; ,e;jf; Fwpg;NgLfspy; fhzg;gLfpd;wd. ,e;jpa vz;fzpjk; kpfTk; rpwg;G tha;e;jJ. Vnddpy; Vwj;jhs BCE 3Mk; E}w;whz;bNyNa ehk; ,d;W gad;gLj;Jk; vz;fspd; KdN; dhbahd FwpaPl;L xOq;F ,Ue;jjw;fhd Mjhuq;fs; fpilj;Js;sd. xOq;fw;w cNuhkhdpa vz;fs; Kiwapd; ,lj;ij ,e;jpa FwpaPl;L Kiw gpbj;Jf; nfhz;lJ. ,e;jpa Kiwjhd; vz;fspd; Njhw;wj;jpw;F Kf;fpakhdjhf mike;jJ. Mdhy; ,e;j vz;fs; ,e;jpa vz;fs; vd;gjw;Fg; gjpyhf muhgpa vz;fs;(Arabic numerals) vd;W jtwhd jiyg;igg; ngw;Wtpl;lJ. Jujp~; ;ltrkhdJ. Id;];Bd; Fwpg;gpl;lJ Nghy ‘,e;jpah;fs; jhd; cyfj;jpw;F vg;gb vz;ZtJ vd;gijf; fw;gpj;jdu;’. Ntj fhyj;jpw;Fg; gpe;jpa fhyk; CE 1200 tiu Kjyhk; Mupag;gl;lh vd;gtu; mtUila E}yhd Aryabhatya tpy;> ghly;fs; tbtj;jpy; Kf;Nfhztpaypd; ePnljpuj; jfT(sine) rhu; gadf; s; gw;wpa xU ml;ltiziaf; nfhLj;J> ,tw;iw [pah(jya) vd;w tlnkhopr; nrhy;ypdhy; mioj;Js;shu;. 3Ÿ45' ,ilntspapy; mike;Js;s Nfhzq;fspd; ePnljpuj;jfTfs; ,e;j ml;ltizapy; Fwpg;gplg;gl;Ls;sd. Nfhs Kf;Nfhztpay; (spherical trigonometry) gw;wp mtu; ngw;wpUe;j mwptpd; Mw;wiy ,e;j ml;ltiz Rl;bf;fhl;Lfpd;wJ. jd;Dila kw;nwhU E}yhd Gitikapada tpy; kpfTk; kjpEl;gkhd kw;Wk; jdpj;jd;ik tha;e;j xU vz;tupir Kiwiaf; nfhLj;Jss; hu;> ,ijj;jtpu 35

,Ugb %yq;fs;> Kk;kb %yq;fs;> gy;NtW tbtq;fspd; gug;gsT fz;Lgpbj;jy;> nfhs;ssT fz;Lgpbj;jy;> ������ ,d; kjpg;G Nghd;witfSk; tpsff; g;gl;Ls;sd. gpuk;kFg;jh (Vwj;jhs CE 598) kpfr;rpwe;j thdpay; epGzUk; fzpjtpay; tyY; eUk; Mthu;. ngy;ypad; rkd;ghL gw;wpa jPu;it ntw;wpfukhff; nfhLj;j Kjy; mwpQu; ,tu;jhd;. jd;Dila Brahmasphuta siddhanta vd;w E}ypd; 12tJ kw;Wk; 18tJ mj;jpahaq;fshd Ganitadhayah kw;Wk; Kuttadhayah tpy; mtu; KiwNa vz;fzpjk; kw;Wk; tbtf; fzpjj;ijg; gw;wp tpsf;fpAs;shu;. Kjyhk; gh];fuh (Baskara 1) (Vwj;jhs CE 600) vd;gtu; Kjy;epiy KbTngwh rkd;ghLfs; gw;wpa gilg;Gf;F ngau; ngw;wtu;. kfhtPuh (CE 815-878) vd;w rkz kjj;J fzpjtpay; mwpQu>; fu;ehlf uh~;lu; $l kd;dd; mNkhftu;kd; epUgJq;fd; mur rigapy; ,Ue;J nfhz;L Ganita-Sara-samgraha vd;w E}iyg; gilj;jhu;. ,e;E}ypy; gy fzpjtpay; tpdhff; s; tptupf;fg;gl;Ls;sd. ,tw;wpy; kpf Kf;fpakhdJ 2a vd;w ngupa mr;irAk;> 2b vd;w rpwpa mr;irAk; nfhz;l xU ePs;tl;lj;jpd; (Maj;j tpupj;jh vd;W mtuhy; miof;fg;gl;lJ) gug;gsT fz;Lgpbj;jy; (������������������) MFk;. vdpDk;> mjd; Rw;wsTf;fhd #j;jpuk; √(4a2ñ6b2) jtwhdJ vd;whYk;> Vwj;jho rupahd tpilahf mike;jJ. ,tUila E}ypy; G+[;aj;jpd; gad;fs; gw;wpAk; tpsff; g;gl;Ls;sd. ,uz;lhk; Mupag;gl;lh (Vwj;jho CE 950) Mahasddhanta vd;w E}ypd; Mrpupau; Mthu;. ,e;E}ypd; 18tJ mj;jpahakhd Kuttakhayaya tpy; Kbtpy;yh rkd;ghLfs; gw;wp tpsf;fg;gl;Ls;sd. rpupjuh (Vwj;jhs CE 990) vd;gtu; Patiganitha kw;Wk; Trisatika vd;w jiyg;Gf;fspy; vz;fzpj E}y;fis ntspaplL; s;shu;. =gjp (Vwj;jho CE 1000) Ganita tilaka vd;w vz;fzpj E}iyg; gilj;Jss; hu;. ,uz;lhk; gh];fuh (Vwf;Fiwa CE 1114) ,e;jpahtpd; jiyrpwe;j fzpjtpay; mwpQu;fspy; xUtuhthu;. ,jw;F xU vLj;Jf;fhl;L: 1657Mk; Mz;L kpfr;rpwe;j gpuhd;]; ehl;L fzpjtpay; epGzuhd /ngu;khl; jd;Dila fzpjtpay; ez;guhd ng];]pf;F jPu;T fhz;gjw;fhf gpd;tUk; rkd;ghl;L tpdhit mDg;gpapUe;jhu;. 61x2+1=y2, ,q;F x-k;> y-k; Kw;nwz;fs; (KO vz;fs;) ,e;j tpdhTf;F1732Mk; Mz;L tiu jPu;T fhzg;gltpy;iy. me;j Mz;L A+yu; ,jw;Fj; jPu;T nfhLj;jhu;. Mdhy; ,uz;lhk; gh];fuhtpd; gilg;Gf;fspy; 1150Mk; Mz;bNyNa ,jw;fhd jPu;T fhzg;gl;lJ. ,e;jr; rkd;ghl;by; x-f;fhd kpfr;rpwpa vz;zhf 226153980 y-f;fhd kpfr;rpwpa vz;zhf 1766319049 vd;gijAk; Fwpg;gpl;Ls;shu; vd;gJ ekf;F tpag;igj; jUfpwJ. ,e;jj; jPu;T Kiw rf;uthyh (chakravala) vdg;gLfpwJ. ,e;jj; jPu;TKiw mtUila Shiddhanta siromai vd;w E}ypy; fhzg;gLfpwJ. ,uz;lhk; gh];fuhtpd; ,d;ndhU rpwe;j fzpjtpay; E}y; Lilavati MFk;. nra;As; tbtj;jpy; nfhLf;fg;gl;Ls;s gy fzpjtpay; tpdhf;fs; mtUila kfs; yPyhtjpaplk; jPu;T Nflg; J Nghy mikf;fg;gl;Ls;sd. Kidtu; n[ae;j; eu;ypfu; jd;Dila E}yhd The scientific Edge (2003)y; gh];fuhtpd; xU tpdhit vLj;Jf;fhl;bAs;shu;. ‘nkhj;jKs;s NjdPf;fspy; xU ghjpapd; ,Ukb %y vz;zpf;if msT NjdPf;fs; khyjp kuj;ij Nehff; pr; nrd;wd. ,tw;iwj; njhlule;J 36

nkhj;jj;jpu; 8Æ9 gq;F NjdPf;fs; khyjp kuj;ij Nehf;fp tpiue;jd. xU NjdP kl;Lk; jhkiu kyUf;Fs; rpf;fpf; nfhz;L tpl;lJ. ,jd; mtyf;Fuy; Nfll; N[hbj; NjdP Njbte;jJ. Va; ngz;Nz! nkhj;jj;jpy; vt;tsT NjdPf;fs; ,Ue;jd? gjpy; $Wtha;!’ ,uz;lhk; gh];fuh Ez;nzz; fzpjk; gw;wpa fUj;ij Vwf;Fiwa vl;btpl;lhu; vd;gjw;F Mjhukhf tifaPl;L Ez;nzz; fzpjj;jpy; ,Ue;J ngwg;gLk; #j;jpuj;ij (d sin x= cos x dx Nghd;w rkd;ghl;il) mtu; vOjpAs;sijf; Fwpg;gplyhk;. etdP tifaPl;L Ez;vz; fzpjj;jpd; fUj;Jf;fspy; xd;whd rhu; gadpd; KbtpNyh my;yJ Muk;gj;jpNyh ngWjp kiwe;J tpLfpwJ vd;w fUj;JUit epA+l;lNdh> ,nya;g;dpN]h $Wtjw;F KdG; ,tu; $wpapUg;gJ Fwpg;gplj;jf;fJ. gpj;jf;Nfhu]; Njw;wj;jpw;fhd tbtfzpj jPu;it ,uz;lhk; gh];fuh nfhLj;jpUg;gJk; ,q;F Fwpg;gplj;jf;fJ. jpuhtpl ehfupf kf;fshd jkpou;fs; gad;gLj;jpa vz;Kiw gw;wp ,q;F rpwg;ghf Fwpg;gplg;gl Ntz;Lk;. kw;w Kiwfspy; vz;fs; xd;wpypUe;J vz;zj;njhlq;fg; gLfpd;wd. Mdhy; jkpo; vz; Kiwapy; vz;fs; 1Æ320y; (Ke;jpup vd;W miof;fg;gLfpwJ) njhlq;fp> 1Æ180 (miuf;fhuzp)> 1Æ80 (fhuzp)> 1Æ20 (xU kh)> 1Æ4 (fhy;)> 1Æ2(miu) kw;Wk; 1(xd;W) topahf vz;zg;gLfpd;wd. ,e;j tptuq;fs; 16Mk; E}w;whz;ilr; Nrue; ;j Kanakkathikaram (= fzf;fjpfhuk;) vd;w fhupehadhu; (Karinayanar) vOjpa E}ypYk; fhzg;gLfpd;wd.  rkd; vDk; FwpaPl;il (=) 1957 Mk; Mz;L uhgu;l; nuf;fhu;l; (Robert Recot) vd;gtuhy; mwpKfk; nra;ag;gl;lJ.  goq;fhyg; Gj;jfkhd mf;dp Guhzj;jpy; vzf; s; vOj;Jf;fspYk;> ,lkjpg;GfspYk; Fwpf;fg;gl;Ls;sd. 13Mk; E}w;whz;L Kjy; jw;fhyk; tiu ,uz;lhk; gh];fuhit njhlu;e;j fhyq;fspy; fzpjtpaypd; tu;zidfs; (connentares) jhd; mjpfkhfg; gilff; g;gl;ld. vd;whYk; xUrpy Gjpa fzpjtpay; gilg;Gf;fSk; cz;lhff; g;gl;ld. Fwpg;ghf Nfushtpy; fzpjtpay; mjpfkhfj; Jspu;j;jJ. ehuhaz gz;bjh (Narayana pandita) (Vwf;Fiwa CE 1350) Ganitakamudi vd;w vz;fzf;F E}iyAk; Bijagaritabatamsa vd;w Fwpf;fzf;fpay; E}iyAk; vOjpdhu;. 15Mk; E}w;whz;by; gy fzpjtpay; gilg;Gfs; Njhd;wpd. ,tw;wpy; Kf;fpakhdit karanapaddati, ganitayuktibhasha kw;Wk; sadratnamela ,it %d;wpYk; ePnljpuj;jfT> ePlz;ikj;jfT (cosine)> mz;ik vjpuj;jfT (tan) kw;Wk; ������ Nghd;wtw;wpd; Kf;Nfhztpay; njhluzpfSf;fhd tpjpfs; nfhLf;fg;gl;Ls;sd. ePyfz;l Nrhk rj;Jtd; vOjpa Tantrasamgraha vd;w E}ypy; Kf;Nfhztpay; tpjpfs; tpsf;fg;gl;Ls;sd. Mupag;gl;lh> ,uz;lhk; gh];fuh kw;Wk; gy Kdd; hs; fzpjtpay; epGzhf; spd; gilg;Gf;fisg; gw;wpa tu;zidfSk; tpsff; q;fSk; Njhd;wpd. ,tw;wpd; Kf;fpa gq;F Mw;wpatu;fs; fq;fhjuh> tp~;Z gz;bju;> fNzrnja;thzh> ,uFehjh;> fpU~;zh kw;Wk; ePyfz;l N[hjplu; Nghd;wtuf; s; Mtu;. ,e;jpa fzpjtpay; gilg;Gfspy; gy kw;w nkhopfspy; nkhop ngau;f;fg;gl;ld. mf;gupd; fll; isg;gb /ig]p gh];fuhtpd; Lilavati I ngu;rpa nkhopapy; nkhop ngau;j;jhu;. gh];fhutpd; Bijaganita mlhTy;yh e~Pjp (Ata Ullah Rashidi) vd;gtuhy; ngu;rpa nkhopapy; 37

nkhopngau;f;fg;gl;lJ. A+f;fpypl;bd; Elements tlnkhopapy; Rekhaganita (1723-1727) vd;w jiyg;gpy; [fe;ehjh (Jekannatha) vd;gtuhy; n[a;rpq; rtha; vd;w n[a;G+u; rpw;wurupd; MjuTld; nkhopngau;f;fg;gl;lJ. jw;fhy ,e;jpahtpd; jiyrpwe;j fzpjtpay; tyY; eu; rdP pthr ,uhkhD[d; Mthu;. ,tUila fzpjtpay; Mw;wiyf; fz;Lgpbj;jtu; Nfk;gpupl;[; gy;fiyf;fofj;ijr; Nru;e;j Nguhrpupau; G.H.Hardy Mthu;. vz;nfhsi; s gpuptpw;F 1915 Kjy; 1919 tiu ,uhkhD[d; Mw;wpa gilg;Gfs; cyfg; Gfo; tha;e;jit. bupdpl;b fy;Y}upapy; mtu; tpl;L tpl;Lr; nrd;w Fwpg;Gg;Gj;jfj;jpy; jPu;T fhzKbahj gy vz;nfhs;is njhlu;hgd tpdhff; s; fhzg;gLfpd;wd. kP fyg;G vz;fs; (highly compositive numbers) njhlu;ghd mtUila gilg;Gfs; mj;jifa vz;fisg; gw;wpa gy Gjpa Ma;TfSf;F topNfhypd.  nr]; gyifapy; cs;s 64 kl;Lky;y…mjpYs;s rJuq;fspd; nkhj;j vz;zpf;if 204 MFk;.  e ,d; kjpg;G 2.718281828……...,jpy; 1828 vd;w vz; jpUk;g jpUk;g tUfpwJ. etPdfhyk; fp.gp 400,y; fpNuf;f ehfuPfk; toP ;rr; pAw Muk;gpj;jJ. Nkw;Fyfpy; mJtiu ,Ue;Jte;j kWkyh;r;rpAk; xU ,Uz;l fhyj;ij Nehf;fp mbnaLj;J itj;jJ. 15k; E}w;whz;L tiu mJ jiy epkputpy;iy. ,jdhy; fzpjk; njhlh;ghfTk; NkNy $wg;gl;l ,Uz;l fhyj;jpy; vtt; pj tsh;r;rpAk; Nkw;Fyfpy; eilngwtpy;iy vdj; njhpfpwJ. 15k; E}w;whz;by; tho;e;j thdrh]j; putpayhsh; nfhg;gdpf;f]; NghyNt 16k; E}w;whz;by; tho;e;j tpQ;Qhdpahd fypypNahTk; nra;j thdrh];jputpay; Ma;Tfspd; Ngwhf> mJtiu fhyKk; #upaidg; gw;wpAk; g+kp cl;gl fpufq;fs; gw;wpAk; kdpjdplk; ,Ue;J te;j %lek;gpfi; ffs; gy mfw;wg;gl;ld. tpQ;Qhd mwptpd; gadhf Gjpa fhyj;Jf;F kdpjd; mbnaLj;J itj;jhd;. 16k; E}w;whz;by; tho;e;j fypypNah> nfg;yh; Nghd;w tpQ;Qhdpfspd; fz;Lgpbg;GfspD}lf kdpjd; Gjpa Afj;jpw;Fs; mbnaLj;J itj;jhd;. ,t;thW etdP tpQ;Qhdj;jpy; Vw;gl;l Gul;rpf;F rkdhd Gul;rp fzpjj;jpYk; 16k; E}w;whz;by; Vw;gl;lJ. N[hd;Neg;gpah; vDk; fzpjtpayhsh; klf;if tha;g;ghl;il fz;Lgpbj;jhh;. nudp njf;fhl; (1637) vDk; fzpjtpayhsuhy; Ms;$w;W Nfj;jufzpjk; fz;Lgbf;fg;gl;lJ. ,Jtiu fhyKk; nttN; twhf tsh;r;rp mile;j ml;rufzpjk;> Nfj;jpfzpjk; Mfpa ,uz;L gFjpfisAk; xd;wpidj;J gFg;G Nfj;jpfzpjk; vDk; Gjpa ghlj;jplj; ;ij mwpKfg;gLj;jpdhh;. rpf;fy;vz; 38

njhlh;ghd Gjpa Nfhl;ghLfs; rpytw;iwAk; fz;Lgpbj;j %th; vDk; fzpjtpayhsh; jkJ fz;Lgpbg;Gf;fis 1720,y; ntspapl;lhh;. kf;Nyhhpd; vDk; fzpjtpayhsh; fp.gp1740 ,y; tifaPL njhlh;ghd Kf;fpa fUj;Jf;fis Kdi; tj;jhh;. fp.gp.1750,y; xapyh; vDk; fzpjtpayhsh; Nfj;jpufzpjk; ml;rufzpjk; njhlh;ghd Gjpa fUj;Jf;fs; rpytw;iw ntspg;gLj;jpdhh;. fp.gp.1780,y; yfpuhQ; vDk; fzpjtpayhsh; Ez;fzpj ghlj;jpy; css; lfF; k; njhifaPlL; r; rkd;ghl;L KiwfisAk; NtW Nfhl;ghLfs; rpytw;iwAk; fz;Lgpbj;jhh;. 18k; E}w;whz;bw;F chpatuhd fzpjtpayhsh; fT]; (fp;.gp.1700) gy;Nfhzp mikg;G> vz;nfhsi; f> njhifaPl;L Nfj;jpufzpjk;> ml;rufzpjj;jpd;> tpjpfs; njhlh;ghd Kf;fpa Nfhl;ghLfs; rpytw;iw cyFf;F chpikahf;fpdhh;. mNjNghy; ,d;DnkhU gpugy fzpjtpayhsuhd yh[y; h[; ,ae;jputpay;> epfo;jfT> tifaPl;L rkd;ghLfs; njhlh;ghd Gjpa fz;Lgpbg;Gf;fisAk; fp.gp.1805 ,y; ntspg;gLj;jpdhh;. rpf;fy; vz;fis Mfd; tupg;glj;jpy; fhl;l KbAk; vd;gij Kjd;Kjyhf vLj;Jff; hl;batu; Mh;fd;l; vd;gth; 19;k; E}w;whz;bd; Kw;gFjpapy; thoe;j fzpjtpayhsh; ,d;Dk; fzpjtpayhsuhd Nghhpahh; (Fourier) 1822,y; njhlh;fs; rk;ge;jkhd ,d;DnkhU Kiwia mwpKfg;gLj;jpdhh;. Nfht; vDk; fzpjtpayhsh; rpf;fy; mirT njhlh;ghd Kf;fpa NfhlghLfs; rpytw;iw Kd;itj;jhh;. epa+l;ld; 1680,y; jhk; fz;Lgpbj;j ghlj;ij cyFf;F ntspg;gLj;Jk; NghNj iygdpl;]kP ; jhk; jdpahf fz;Lgbj;j Ez;fzpj ghltplaj;ij 1678,y; E}y; %yk; ntspapl;lhh;. iyg;dpl;]; Ez;fzpjk; kl;Lky;y ntspg;gil cz;ikfs>; gy;YWg;Gj;Njw;wk; > FwpaPl;L jh;ff; rh];jpuk; Nghd;w Kf;fpa fzpj Nfhl;ghLfs; gytw;wpdJk; je;ij vd fUjg;gLfpwhh.; 17k; E}w;whz;by; tho;e;j fzpjtpayhsuhd khh;];fh Nghhg;gpl;lhy; fp.gp.1690 ,y; Ez;fzpjj;ij gpuNahfpg;gjw;F Njitahd Kf;fpa tpjpfs; rpytw;iw fz;Lgbj;jhh;. 18k; E}w;whz;bw;F chpatuhd n[fd;gh; E}yp (fp.gp.1700) [pNahgd;dp Nfth (fp;gp1; 700) vd;gth;fis tpNrlkhf Fwpg;gpl Ntz;Lk;. gh;Ndhyp vd;gth; Ez;fzpjk;> epfo;jfT > klfi; f njhlh;ghd Njw;wq;fs; rpytw;iw fz;Lgpbj;jhh;. Nfth vd;gth; jsNfj;jpu fzpjk; njhlh;ghd Kf;fpa Njw;wq;fis fz;Lgpbj;jhh;. tpy;ypak; N[k;]; vDk; fzpjtpayhsh; tl;lj;Jld; njhlh;Gila tpfpjj;ij 1706 ,y; Kjd;Kiwahf gadg; Lj;jpdhh;. Nlyh; vd;Dk; fzpjtpayhsh; njhlh;fzpjk; gw;wpa Kf;fpa Nfhl;ghl;il fp.gp.1715 ,y; fz;Lgpbj;jhh;. 17Mk; E}w;whz;by; tho;e;j ,d;DnkhU fzpjtpayhshf; shd gh;kl; (1635) `hb rhh;f]; ; (1637) Mfpa ,Utiug; gw;wpAk; tpNrlkhf Fwpg;gpl Ntz;Lk;. gh;kl; vd;gth; vz;fs; gw;wpa nfhs;if ( Theory of Numbers ) cahT; >,opT > epfo;jfT njhlh;ghd Kf;fpa jj;Jtq;fis mwpKfg;gLj;jpdhh;. brhh;f]; ; nray;jpl;l Nfj;jpfuzpjk; (Projective Geometry) njhlh;ghd Kf;fpa Njw;wq;fis fz;Lgpbj;jhh;. ,d;Wk; brhh;f]; ; Njw;wk; vd;W cahj; u J}afzpj ghlj;jpl;lj;jpy; css; lf;fg;gl;bUg;gJ mth; fz;Lgpbj;j Njw;wq;fspy; xd;whFk;. fp.gp.1650 ,y; gpwe;jtuhd g];fhy; vd;w fzpjtpayhsh; epfo;jfT njhlh;ghf gy Nfhl;ghLfis Kdi; tj;jJld; g];fhy; Kf;Nfhzj;ijAk; fzf;fpLk; fUtpiaAk; fz;Lgpbj;jhh;. 17Mk; E}w;whz;by; [Nuhg;ghtpy; tho;e;j rpwe;j fzpjtpayhsh; vd fUjg;gLgth;. 1648,y; gpwej; gphpj;jhdpa Njrj;jtuhd Nrh; [rf; ePa+l;ldhthh;. gpuNahf fzpjk;> ngsjpf tpQ;Qhdk;> thdrh]j; puk; Nghd;w ghlq;fSf;F nghUj;jkhd Nfhl;ghLfs; rpytw;iw mth; 1680 ,y; ntspapl;lhh;. Principia vDk; ngaUila gpugy E}ypy; ,J ntspaplg;gl;Ls;sJ. ,af;ftpay;> ePh;epiyapay;> ePh;,af;ftpay; > GtpaPh;g;G Nghd;w gpuNahf fzpjk; njhlh;ghd Nfhl;ghLfs; rpytw;iw fz;Lgpbj;j epa+l;ld; J}afzpjj;Jld; njhlh;Gila Ez;fzpj ghlg;gug;ig css; lfF; k; Kf;fpa Nfhl;ghLfs; rpytw;iwAk; fz;Lgpbj;jhh;. Ez;fzpjj;ij fz;Lgpbj;j ngUik [rhf;epa+l;lDf;F fpilg;gJ NghyNt N;[h;kd; ehl;ilr; Nrh;e;jtuhd iygdpl;]; vDk; fzpjtpayhsh; rpf;fy; mirT njhlh;ghd Nfhl;ghLfis Kd;itj;jhh;. 19k; E}w;whz;by; fzpjghlk; njhlh;ghd Kf;fpa NtWghL vd Fwpg;gplf;$bajhf toikahd a+f;spl; Nfj;jpufzpj;jpw;F gjpyhf Kg;ghpkhd ntspg;gpurpdq;fs; jPh;f;Fk; njhlh;ghd a+f;spl; my;yhj Nfj;jpu fzpjk; Njhd;wpaJ. 39

mtt; hwhd Nfj;jpufzpjk; Njhd;Wtjw;F KdN; dhbahf eltbfi; f vLg;gth; vd wpkhd; (fp;.gp. 1826-1866) vd;Dk; fzpjtpayhsiu Fwpg;gplyhk;. ,jdhy; a+f;spl; my;yhj Nfj;jpfzpjk; vd miof;fg;gLfpwJ. a+f;spl; Nfj;jpufzpjj;jpy; jd;dpr;irahf ek;gpapUe;j “ ntspapy; css; Gs;spf;$lhd jug;gl;l Neh; tiuf;F rkhe;jukhf tiuaf;$ba xNunahU Neh;tiujhd; cz;L” vd;gij wpkhd; Nfj;jpufzpjj;jpy; cz;ik vd ek;gtpy;iy. toikahd Nfj;jpufzpjj;ijj; jtpu toikahd ml;rufzpjj;jpYk; 19k; E}w;whz;by; nghpanjhU khWjy; Vw;gl;lJ. ,t;tsT fhyKk; nka;naz; njhFjp kl;Lg;gLj;jg;gl;l ml;rufzpjj;JfF; gjpyhf fUj;J epiyahd ml;rufzpjk; Njhd;wpaJ. ,jw;F KdN; dhbahf Mh;jh;Nfyp (1821-1891) vd;gtiuAk; `kpy;ld; (1805-1865) vd;gtiuAk; Fwpg;gplyhk;. ml;rufzpjj;jpy; css; ntspg;gil cz;ikfs; njhlh;ghd Nfhl;ghL ,e;jg; Gul;rpapd; Ngwhfj; Njhd;wpaJ. ,ej; pahtpw;F tUif je;j kpfg; ngupa fpNuf;f fzpj Nkij gpjhfuR Mthu;. 19k; E}w;whz;bd; ,Wjpg;Fjpapy; Vw;gl;l ngUk; Gul;rpf;F “ N[hh;[; fd;lh; ” Kdi; tj;j njhilfn; fhs;if fhuzkhFk;. njhilf; nfhsi; ff;F NkYk; cap&l;ba “N[hh;[;Gy;Nkhh;fd;” vd;gth;fspd; Gjpa fUj;Jff; is fzpjj;ij jhff; tpaYld; njhlh;GgLj;jp fzpj msitapay; Njhd;wpaJ. 1916,y; “my;Ngh;l; [d;];bd;” mwpKfg;gLj;jpa rhh;Gf;Nfhl;ghL Kf;fpa ,lj;ij tfpj;jJ. 1917; k; Mz;L ,uhkD[d; vd;gth; Kdi; tj;j njhFj;jwp vz;Nfhl;ghL ,e; E}w;whz;by; Vw;gl;l Gjpa jpwd;fSs; Kf;fpakhditahFk;. ,tw;iw jtpu kpfTk; mz;ikapy; tsh;r;rpaile;j ,ae;jputpaypYk; fzpjtpayhsh;fs; $ba ftdj;ij nrYj;jp tUfpwhh;fs;. 15k; E}w;whz;bd; ,Wjpapy; Muk;gkhd Gjpa tpQ;Qhdj;Jld; Nrh;e;J fzpjKk; gy gphpTfisg; gphpe;J jpwikAld; tsh;e;J te;eJ. mjd;gpd; 4 E}w;whz;LfSf;F tpNrl KdN; dw;wj;Jld; tpUj;jpaile;J fzpjk; cyfj;jpd; gy;NtW ehLfspYk; css; gy;NtW fzpjtpayhsh;fs; fz;Lgpbj;j Gjpa tplaq;fSld; Nrh;e;J gue;j ghlk; Mfpd;wJ. fzpjk; Vida ghlq;fis tpl jw;NghJ kpfTk; mtrpakhdjhFk;. mj;Jld; vq;Fk; tpahgpj;Jss; J. ,jd; fhuzkhfNt jw;Nghija r%fj;jpy; fzpjk; Kf;fpakhdnjhd;whf tpsq;Ffpd;wJ. me;j tifapy; vkJ ehl;by; fzpjj;Jf;F nfhLf;fg;gl;l Kf;fpaj;Jtj;ij ehk; Ghpe;J nfhs;tJ kpfTk; mtrpakhdjhFk;. NkYk; mjd; Kf;fpaj;JtKk; mtrpaKk; mjpfhpf;Fk; vd;gjpy; [akpy;iy. ed;wp 40

vz; FwpaPl;L Kiwfs; mwpKfk; vz; nrhy; Mfpa ,uz;Lk; kdpjdpd; Muk;gfhy ntspg;ghLfshFk;. vz; vd;gJ ek;ik Rw;wpAs;s cyifgw;wpa gy tplaq;fisAk; VNjh xU tifapy; tifg;gLj;jp kdjpy; rpe;jpg;gijf; Fwpf;Fk; mjdhy;jhNdh vd;dNth jkpopy; vz;zpw;F Kd;dpiy nfhLj;J 'vz;Zk; vOj;Jk; fz;nzdj; jFk;\" vdf; $wg;gl;Ls;sJ. Mjpapy; kdpjd; vz;njhlh;ghd jd; fUj;ij ntspapLtjw;F iriffisAk; FwpaPLfisAk; gad;gLj;jpdhd;. Muk;gfhy kdpjdpd; vz;njhlh;ghd fUj;J ntspg;gl;bw;Fk; ,d;W xU Foe;ij mf; fUj;ij ntspg;gLj;Jtjw;Fk; mjpf NtWghL ,Uf;fkhl;lhJ vdf; fUjg;gLfpd;wJ. md;W vz; njhlh;ghd fUj;J ntspg;ghl;bw;F cWJizahf mtDila tpuy;fs; mike;jJ. njhlu;e;J irifia mLj;J FwpaPLfis gad;gLj;jpapUf;f KbAk; vd ehk; Cfpj;J mwpaf;$bajhf css; J. jw;Nghija vz;FwpaPLfshd ,yf;fq;fisf; Fwpf;Fk; Digits vDk; Mq;fpyr; nrhy;ypd; %yr;nrhy; Digitus vd;Dk; ,yj;jpd; nrhy;yhFk; ,yj;jpd; nkhopapy; ,r; nrhy; tpuy;fisf; Fwpf;Fk; vz;gw;wpa fUj;ij ntspg;gLj;j tpuy;fSld; vz;rl;lk;> kzy; Nkir> fapw;W KbrR; fs; milahskpLjy; tpijfs; rpWkzpfs; Nghd;witAk; gad;gLj;jg;gl;ld. ,e;jpahtpy; Mjpapy; vz;fis Fwpf;f gy nrhw;fs; gadg; Lj;jg;gl;ld. ,uz;ilf; Fwpf;f Nrhb mtyaq;fshd fz;fs; fuq;fs; ,wf;iffs; vd;gd gadg; Lj;jg;gl;ld. Guhzq;fspy; ehY fly;fNs css; d vdf; $wg;gl;ljhy; ehiyf;Fwpf;ff; fly;fs; gad;gLj;jg;gl;ld. Vida vz;zy; Kiwfisg; Nghy; gj;ijf;Fwpf;f tpuy;fSk; Kg;gj;jpuz;ilf; Fwpf;fg; gw;fSk; E}iwf;Fwpf;f kdpj MASk; cgNahfg;gl;ld. khahh;fspd; vz;KiwapYk; iftpuy;fSk; fhy;tpuy;fSk; gadg; Lj;jg;gl;ld. 'f;tpr;\" vd;Dk; nkhopapy; ,UgJ vd;Dk; vz;izf;Fwpf;f '`Ptpdhr;\" vd;Dk; nrhy; gad;gLj;jg;gLfpd;wJ. ,k; nkhopapy; ,r; nrhy;ypd; nghUs; KO Ms; vd;gjhFk; Xh; Mspd; if> fhy; tpuy;fs; ,UgJNk ,q;F fUj;jpy; nfhs;sg;ghl;bUf;fyhk;. epA+ n`g;hBP ]; jPtpy; css; mgp vdg;gLk; nkhopapy; Y}dh vd;why; ifiaf;Fwpf;Fk; mk; nkhopapy; Ie;ijf;Fwpf;Fk; nrhy;Yk; Y}dh vd;gNj MFk;. ,j; jPtpy; ,uz;L vd;gJ Y}th vdf; Fwpf;fg;gLk; Mjyhy; Y}th Y}dh vd;gJ ,Uiffs; vDk; fUj;jpy; gj;ijf;Fwpf;Fk; INuhg;ghtpy; cNuhkh rhk;uh[[; paj;jpd; toP ;r;rpf;Fg;gpd; fp.gp 5k; E}w;whz;by; ,Ue;J fp.gp 9k; E}w;whz;Ltiu 'fzpjj;jpw;F mwpKfk;\" vd;Dk; E}iy mbg;gilahff; nfhz;L fzf;Fr; rl;l kzpfisg; gad;gLj;jp fzf;fpLtJ epfo;e;J te;jjJ. ,f;fhyf; fhyg;gFjpapy; ,k;Kiwf;F xj;jjhf tzf;fj;Jf;Fhpa gpNl vd;gth; vOjpa 'fhyk; gw;wp\" vDk; E}ypy; Fwpg;gpl;bUe;j gb tpuy;fisf; nfhz;L vz;Zk; KiwAk; ,Ue;J te;jjJ. Mgphpf;f fz;lj;jpd; ,yNrhj;jh ehl;by; thOk; Nrhj;jh ,dkf;fs; vq;fdk; mb gj;jhff; nfhz;L vz; rl;lj;jpd; vz;Fwpg;gplg;gLfpd;wNjh mNj Nghy; kdpjid epWj;jp vz; njhlh;ghd fUj;ij ntspg;gLj;jpdh;. xd;Wf;fisf; Fwpg;gjw;F xUtiuAk;> gj;Jff; s; Nrh;e;jhy; ,UtiuAk; E}Wfs; nfhz;l vz; vdpd; %tiuAk; Mfg; gj;jpd; tOf;fSf;F xg;g kdpjiu mUfUNf epWj;jp vz; nghWkhdj;ij ntspg;gLj;Jthh;fs;. 41

mjhtJ gj;JfF; l;gl;l vz; vdpd; xUth; jd; iftpuy;fs; %yk; njhifia ntspg;gLj;Jthh; E}Wf;Fl;gl;l vz; vdpd; Kjyhk; ,lj;jpy; ,Ug;gth; xd;Wf;fisAk; ,uz;lhk; ,lj;jpy; ,Ug;gth; gj;Jff; isAk; Fwpg;ghh;. ,t; vz; Kiw ,lkjpg;ig mbg;gilahff; nfhz;lJ. nkhrhk;gpf; ehl;by; thOk; ,nl];th ,dj;jth; jkJ Foe;ijapd; taij Fwpg;gh; mbkuj;jpypUe;J xt;nthU ntl;lhf ,l;L Foe;ij tsh;e;J jd; taij jNd fzf;fpLk; tiu ,J epfOk;. Mgphpf;fh Njr thNfhz;Nl gphptpdh; vz;ziy Nkw;nfhss; Kbr;Rf;fapw;Wf;fis gad;gLj;jpdh;. taijf; fzpg;gpl ,U fapWfis gad;gLj;jpdh.; xU fapW khjj;ijAk; ,uz;lhJ fapW Mz;ilAk; Fwpj;jJ. xU fapw;wpy; gpwe;jjpypUe;J xtn; thU ngsh;zkp jpdjpYk; xtn; thU KbrR; Nghlg;gLk; ,t;thW gd;dpuz;L ngsh;zkpfspd; gpd; kw;iwa fapw;wpy; xU KbrR; ,lg;gLk;. ,tt; hW Foe;ijapd; taij Mz;L khj hjP pahf fzpj;Jfn; fhst; h;. ie[h>; tl ie[Phpahitr; Nrh;e;j kf;fs; jkJ ke;ijfspd; vz;zpf;ifia Fr;rpfisg; gad;gLj;jp ntspg;gLj;Jth;. V tbtj;jpy; Fr;rpfs; itff; g;gl;lhy; E}W ke;ijfs; cz;nld;Wk; X Fr;rpfs; itf;fg;gl;lhy; Ik;gJ ke;ijfs; cz;nld;Wk; mwpag;gLk;. fpilkl;lj;jpy; xU Fr;rpia itj;jhy; mJ gj;ijf; Fwpf;Fk; epiyf;Fj;jhf itfF; k; Fr;rpfs; xd;Wf;fisf; Fwpf;Fk;. Nkw;Fwpg;gpl;l vz;zy; Kiwfis Nkw;nfhsS; k; ehLfSld; ve;jtpj njhlh;Gkw;w ehLfspYk; ,tt; pj vz;zy; Kiwfs; tof;fpy; ,Ue;jjij fhzf;$bahj css; J. rPd ehfupfk; Mrpahf; fz;lj;jpd; fpof;Ff; fiuapy; mike;Js;s rPd Njrk; kpfg;Guhjdkhd ehfupf mk;rq;fisf; nfhz;lJ. ,e; ehfupfj;ijg; gw;wpa jfty;fs; vYk;Gfs;> MikNahLfs;> gl;Lj;Jzpfs;> %q;fpyf; s;> kug;gyiffs; Mfpatw;wpy; vOjg;gl;l tplaq;fspypUe;Jk;> gioa E}y;fspypUe;Jk; ngwf;$bajhf css; J. rdP u; jkJ fUj;Jff; is ntspapl rpj;jpuq;fis gadg; Lj;jpdu;. mj;Jld; ,tu;fSila vOj;J Kiw epiyf;Fj;jhf NkypUe;J fPo;Nehff; p mike;jpUf;Fk;. rPdhtpy; vz; cUthfpaJ njhlu;ghf Kf;fpak; tha;e;j cWjpahd Fwpg;Gf;fs; njhy;nghUs; Ma;T %ykhfg; ngwg;gl;Ls;sd. rdP htpy; `pshd; \\hq;fp khfhzq;fspy; 7000 Mz;LfSf;F Kd;G “ahq;\\hNth” gz;ghl;by; rpy kl;ghz;lj;Jz;Lfs; fz;Lgpbff; g;gl;Ls;sd. ,tw;wpy; fhzg;gl;l rpwpa epiyf;Fj;Jf; NfhLfs; gz;ila rPd vz;fspd; Muk;gfhy tbtq;fshff; fUjg;gl;ld. rdP htpy; fp.K14 Mk; E}w;whz;lstpy; xU tifapyhd ehl;Fwpg;NgL Njhd;wpaJ. ahq;\\hNth gz;ghl;L kl;ghz;lj; Jz;Lfspy; mq;Fkpq;Fkhfr; rpy FwpaPLfs; 42

,Ue;jNghjpYk; rpNahl;bd; efug;gFjpapy; \\hq; Ml;rpf;fhyj;jpy; fz;Lgpbf;fg;gl;l tuptbtNk rPdhtpd; kpfj;njhd;ikahd tuptbtkhFk;. Muk;g fhyj;jpy; vYk;Gfspy; Nfs;tpfisAk; gjpy;fisAk; tpdhff; isAk; vOjp %jhijaupd; Mtpfsplk; Fwp Nfl;L te;jjhfj; njupfpd;wJ. mjdhy; ,t; tuptbtk; “Fwp vYk;G tuptbtk;” vdg;gLfpwJ. ,tt; hwhf vYk;Gfspy; vOjg;gl;l Rkhu; 5000 vOj;Jf;fspy; rdP htpd; kpfj; njhd;ikahd vz;fSk; mlq;fpapUe;jd. Nghupy; gpbf;fg;gl;l ifjpfspd; vz;zpf;if> nfhy;yg;gl;l vjpupfspd; vz;zpf;if> Ntl;ilahba tpyq;Ffs;> gwitfspd; vz;zpf;if vd;gd ,t; vYk;Gfspy; Fwpf;fg;gl;bUej; d. Fwp vYk;G tuptbtq;fspy; xd;W njhlf;fk; Kg;gjpdhapuk; tiu vz;fs; Fwpf;fg;gl;bUe;jd. ,t;ntz; FwpaPLfs; gpd;tUkhW> Rkhu; fp.K 11 Mk; E}w;whz;bypUe;J fp.K 221 Mk; Mz;L tiu ntz;fyj;jpd; kPJ vOjg;gl;ld. ,J “ntz;fy tuptbtk;” my;yJ “kzpg;ghj;jpu tuptbtk;” vdg;gll; J. ,t; tup tbtKk; Fwp vYk;Gfs; tuptbtk; Nghd;wjhFk;. vdpDk; ,t; ntz;fy tup tbtj;jpd; $l;Lr; nrhw;fs; Fwp vYk;G tup tbtj;jpypUe;J NtWgl;lit. rdP u;fshy; vz;fisf; Fwpf;f fzf;Ff; Fr;rpfSk; gad;gLj;jg;gl;ld. ,it “nrs” vd miof;fg;gl;l rpwpa %q;fpy; Fr;rpfshFk;. ,f; fzf;Ff; Fr;rpfisg; gadg; Lj;Jk; NghJ xw;iwg;gil vz;fs; epiyf;Fj;jhfTk;> gj;Jff; s; fpilahfTk;> E}Wf;fs; epiyf;Fj;jhfTk;> Mapuq;fs; fpilahfTk; gj;jpd; tYf;fspw;F Vw;g khwp khwp epiyf;Fj;jhfTk;> fpilahfTk; Fr;rpfs; mike;jd. G+r;rpaj;ij Fwpf;f fhypaplq;fs; gadg; Lj;jg;gl;ld. fzf;Ff; Fr;rpfis xw;iwg;gil vz;> gj;J> E}W> Mapuk;> gj;jhapuk; vd ,lJ Gwk; Nehf;fp epiyf;Fj;jhfTk;> fpilahfTk; khwp> khwp mikj;J vz;fs; cUthf;fg;gl;ld. ,k; Kiw gpd;tUkhW mikAk;. jw;Nghja rdP vz; FwpaPLfs; gpd;tUkhW fhzg;gLfpd;wd. 43

gj;Jff; s; E}Wf;fs; Mapuq;fs; gj;jhapuq;fs; Xd;gJ xw;iwg;gil vz;fis gj;J> E}W> Mapuk;> gj;jhapuk; Mfpatw;wpd; FwpaPLfshy; ngUf;Ftjd; %yk; gj;jpd; tYf;fSf;Fupa ngWkhdq;fs; ngwg;gLfpd;wd. ,k; Kiwapy; 39672 vDk; vz;izf; Fwpg;gpl Ntz;Lnkdpy;> 3×10000+9×1000+6×100+7×10+2 vDk; top vd vOjg;gLk;. 14 Mk; E}w;whz;by; vOjg;gl;l rdP f; fzpj Ma;T E}ypNy ,k; Kiw mike;Js;sJ. vfpg;jpa ehfupfk; Mgpupf;fh fz;lj;jpy; iey;ejpahy; nropg;ghf;fg;gLk; vfpg;J Njrk; Guhjd ehfuPfq;fspy; Kf;fpakhdJ.,e;ehfuPfk; fp.K 6000 Mz;LfSf;F Kdd; u; cUthdjhf ek;gg;gLfpwJ.fp.K 3500k; Mz;lstpy; tlgFjpapy; cs;s khtl;lq;fs; xd;WNrue; ;jjhy; Nky; vfpg;J vDk; ,uhr;rpak; cUthdJ. njd;gFjp ,rhr;rpaq;fs; xd;WNru;e;J fPo; vfpg;J ,uhr;rpak; cuthfpaJ. fp.K 3000 Mz;lstpy; nkd]; vd;w kd;dd; ,tt; pU ,uhr;rpaq;fisAk; ,izj;jhd;. vfpg;jpa vOj;JKiw `aNuhfpspgpf;]; vdg;gLk;. Muk;gj;jpy; ,jpy; rpj;jpuq;fNs mlq;fpapUe;jd. xU rpj;jpuk; xU fUj;ij Fwpf;Fk;. rpj;jputiffs; gpd;du; vOj;Jff; shf khwpd. ,t;ntOj;Jf;fs; Mapuf;fzf;fhd Mz;Lfshf xUtuhYk; thrpj;J Gupe;Jnfhs;s Kbatpy;iy. fp.gp1796 ,y; neg;Nghypad; vfpg;ij jhf;fpdhd;. neg;Nghypadpd; gilapy; 44

tpQ;QhdpfSk; GijnghUs; Muha;r;rpahsUk; ,Ue;jdu;.vfpg;jpYs;s nuhrl;Nlh vd;w ,lj;jpy; nfhg;bf; nkhopngau;g;Gf;fSld; mike;j `aNuhfpspgpf;]; vOj;Jf;fspdhyhd rhrdk; xd;W ,tu;fSf;F fpilj;jJ. ,ij Muha;e;j gpuhd;]pa GijnghUs; Muha;r;rpahsuhd rk;gpad; vd;gtu; mr;rhrdj;jpYs;s fpNuf;f vOj;Jf;fspd; JizAld; vfpg;jpa vOj;Jf;fis thrpg;gjw;F fz;Lgpbj;jhu;. vfpg;jpau; Rtu;fspYk; iggpu]; jhsf; spYk; vOj;Jf;fis vOjpdu;. tpQ;Qhdj;JiwapYk; vfpg;jpau; Kd;NdwpapUe;jdu;. ntss; g;ngUf;fpdhy; mbff; b khWk; tay;fs; Njhl;lq;fs; Mfpatw;wpd; vy;iyfs; kPz;Lk; mikf;f Ntz;bapUe;jjhy; epy mstlP ;LKiwnahd;iw Vw;gLj;jpf;nfhz;ldu;.gpukpl;Lf;fisAk; fll; lq;fisAk; rpiyfisAk; NehfF; k;NghJ vfpg;jpaupd; fzpjmwpTk; tpUj;jpaile;jpUe;jik Gydhfpd;wJ. vfpg;jpa ehfuPfj;jpd; Muk;g vOj;JUtq;fisg; Nghd;w vz;fis Fwpf;fTk; gad;gLj;jg;gl;ld. xd;iwf;Fwpf;f epiyf;Fj;jhd Nfhy; cUtKk; gj;ijf;Fwpf;f vYk;GUtKk; E}iwf;Fwpf;f fapw;Wr;RUs; tbtKk; Mapuj;ijf; Fwpf;f jhikiug;g+r;rpj;jpuKk; gj;jhapuj;ijf; Fwpf;f RlL; k; tpuypd; mikg;Gk; E}whapuj;ijf;Fwpf;f kPdpd; cUtKk; xU kpy;ypaidf;Fwpf;f Mr;ru;ag;gLk; kdpj cUtKk; tiuag;gl;ld. ,t;ntz;Kiwapy; 4532 [f; Fwpf;f 4 jlit Mapuj;jpw;Fupa FwpaPLk; 5 jlit E}wpw;Fupa FwpaPLk; 3 jlit gj;jpw;Fupa FwpaPLk; 2 jlit xd;wpw;Fupa FwpaPLk; ,l;Lf;fhl;lg;gLk;. ,q;F vt;tpj ,lg;ngWkhdKk; gad;gLj;jg;glhjjhy; ngupa ngWkjpAila FwpaPLfis khwpAk; vOjyhk;. gz;ila vfpg;jpa RtNuhtpaq;fSf;fpilNa gutyhf ,t;ntz; FwpaPLfSk; fhzg;gLfpd;wd. Guhjd vfpg;jpa kf;fs; 1/2 , 1/3 , 1/4 Nghd;w myFg;gpd;dq;fis gad;gLj;jpAs;sdu;. ,t;tyFg;gpd;dj;jpw;F Gwk;ghd tifapy; 2/3 f;F kl;Lk; FwpaPL gad;gLj;jpAs;sdu;. ,t;tyFg;gpd;dq;fs; etdP g;gpd;dq;fspd; mbg;gilg;gz;Gfis nfhz;Ls;sd. RNkupa ehfupfk; rpd;dhrpahtpy; A+g;gpujP];, ijf;fpw]; vDk; ,U ejpfs; tsk;gLj;Jk; gpuNjrj;jpy; Njhd;wp tsu;e;j ,e;Jehfupfk; ,U ejpfSf;fpilNaahd gpuNjrk; vDk; fUj;jpy; nkhrgj;Njkpa ehfupfk; vdTk; miof;fg;gLk;. fp.K 2350k; Mz;lstpy; mf;fhbau; trkhfpaJ. fp.K 1900k; Mz;lstpy; ggpNyhdpau; ,g;gpuNjrj;ij ifg;gw;wpdu;. nrkpl;bf; nkhop Ngrpa mf;fhbau; epWtpa NguuR ghurPf tisFlhtpypUe;J rpupah, nygdhd; tiu gutpaJ. ,g; Nguurpd; fhyj;jpy; vOj;J KiwapYk; msitfspYk; mwpKfg;gLj;jg;gl;l Gjpa Kiwfspdhy; KdG; ,Ue;j msit myFfs; ruP ikf;fg;gl;L mtw;wpw;fpilNa njhlu;G Vw;gLj;jg;gl;lJ. jdpj;jdp nghUl;fSf;fhd vz;zy; Kiw ntt;NtW mstLP fSf;Fk; gad;gLj;jg;gl;ld. vdpDk; Muk;gfhy vz;zy; Kiwfs; mf;fhbau; Ml;rpf;fhy vz;zy; Kiwf;F tpUj;jpaila fpl;lj;jl;l 1000 Mz;Lfs; nrd;wd. nkhrgj;Njkpau; Muk;gj;jpy; jkJ fzf;Fg;gjpTfSf;F fspkz; FLitfisAk; fspkz; tpy;iyfisAk; gad;gLj;jpdu;. nghUl;fs;,vz;zpf;if mstpLk; myF vd;gtw;iwf; Fwpf;f nttN; tW mstpYk; fspkz;tpy;iyfs; gadg; Lj;jg;gl;ld. FLitapd; ntspg;Gwj;jpy; 45

Nghlg;gl;l Kj;jpiufs; cupikahsiu my;yJ xg;ge;jf;fhuiu Fwpj;jJ. gpd;du; FLitfF; s; NghLk; fspkz; tpy;iyfspd; milahsk; Flit kPJ ,lg;gl;lJ. gpd; fspkz;tpy;iyfs; iftplg;gl;L tpy;iyfspd; Kj;jpiufs; jl;ilahd fspkz;jl;Lf;fspy; gjpf;fg;gll; d. gpd; tpy;iyfSf;Fg;gjpyhf ehzy; gad;gLj;jg;gl;lJ. fp.K 3200k; Mz;lstpy; vOJk; Kiw ed;F tpUj;jpaile;jJ. ,f; fhyg;gFjpapy; vz;fis Fwpf;f 30 milahsf; FwpfSk;, vz;zg;gLfpd;w nghUl;fs;, ,lq;fs;, mjpfhuG+u;tkhd ngau;fisf; Fwpf;f 800 milahsf; FwpfSk; ,Ue;jd. fp.K 3200 njhlf;fk; fp.K 2800 tiuahd fhyg;gFjpapy; nkhrg;gj;Njkpahtpy; vz;Ztnjw;nfd ntt;Ntwhd Vuhskhd Kiwfs; gad;gLj;jg;gl;ld. jdpj;jdp nghUl;fis vz;Ztjw;Fk; ePsj;ijf; fzf;fplTk; xU Kiw gadg; Lj;jg;gl;lJ. gug;gsitf; Fwpg;gpl gpwpnjhU Kiw gadg; Lj;jg;gl;lJ. jhdpaq;fis msf;Fk; Kfj;jysitfF; NtW xU Kiw gad;gLj;jg;gl;lJ. ,q;F ntt;NtW jhdpaq;fs; ntt;NtW Kfj;jysit Kiwapy; msf;fg;gll; d. nghUl;fis vz;Ztw;F nttN; twhd gyKiwfs; ,Ue;jhYk; vz; milahsq;fs; kpff;FiwthfNt ,Ue;jd. ehzypd; ehd;F FwpaPLfshd ngupa tll; k;, rpwpa tl;lk;, ngupa ntl;LKfk;, rpwpa ntl;LKfk; Mfpatw;iwf; nfhz;L jhd; vy;yh vz; FwpaPLfSk; cUthf;fg;gll; d. ,it jdpj;jdpahfTk; Nruj; ;Jk; gad;gLj;jg;gl;ld. ,tt; ilahsq;fs; ntt;NtW vz;Zk; Kiwfspy; ntt;NtW tpjkhfg; gad;gLj;jg;gl;ld. jdpj;jdp nghUl;fis vz;Zk; Kiwapy; xU rpwpa tl;lj;jpd; kjpg;G gj;J rpwpa ntl;LKfq;fs; MFk;. ,NjNghy; Kfj;jysitapy; xU rpwpa tl;lk; MW rpwpa ntl;LKfq;fisf; Fwpf;Fk;. gug;gsitf; fzf;fpLk; NghJ xU rpwpa tl;lk; gjpndl;Lr; rpwpa ntl;L Kfq;fisf; Fwpf;Fk;. Mjyhy; ,k;Kiwapy; vz; Fwpf;Fk; milahsq;fSf;nfd Fwpg;gpl;l vz; kjpg;G ,y;iy. fp.K 3200 ,ypUe;J fp.K 2600 tiu vz;tsu;rr; papy; ngupa khw;wkpy;iy. fp.K 2600 ,y; RNkupa ,uhr;rpak; cUthdjhy; njd; nkhrgj;Njkpah KOtjpYk; Xu; vOj;J Kiw Njhd;wpaJ. ,J Mg;G tbt vOj;J KiwahFk;. ,yj;jPd; nkhopapy; Cuneiform vd;gJ Mzp my;yJ Mg;G vDk; nghUs;gLk;. Mg;G vOj;Jf;fs; Kd;gpUe;j vOj;Jf;fistpl vOJtjw;F ,yFthditahFk;. ,f;fhyj;ijr; Nrue; ;j 100>000 f;Fk; mjpfkhd rhd;Wfspd; gb 60I mbahff; nfhz;l ,lkjpg;G Kiw gadg; hl;by; te;jjhfj; njupfpd;wJ. ,k;Kiwapy; 10If; Fwpf;f milahsKk; g+r;rpaj;ijf; Fwpf;f fhypaplKk; gad;gLj;jg;gl;ld. ,k;Kiwapy; ,uz;ilAk; mWgj;njhd;iwAk; Fwpf;f xNu FwpaPLfs; gadg; Lj;jg;gl;ld. Mdhy; ,lkjpg;gpw;F Vw;g ,ilntspapl;L vOjg;gl;ld. ,k;Kiw njhlu;e;J te;j 2000 Mz;L fhyj;jpw;F khwhky; gadg; Lj;jg;gl;lJ. Mg;G tbt vOj;J KiwahYk; 60I mbahff; nfhz;l ,lkjpg;G KiwahYk; kpFe;j gyd; Vw;gl;lJ. mjhtJ vy;yh msit KiwfspYk; fzf;FfSf;Fg; gad;gLj;j xNu vz;FwpaPLfs; NghJkhditahf ,Ue;jd. ,t;tpjkhf fp.K 3k; E}w;whz;Lf; filrp thff; py; ve;j xU KiwAlDk; njhlu;gw;w Xu; vz; Kiw cUthfpaJ. vDk; vz;zpw;fhd FwpaPLfs; fhyj;jpw;Ff; fhyk; jpUj;jpaikf;fg;gl;lss; ij gpd;tUk; FwpaPLfs; fhl;Lfpd;wd. 46

fp.K 3000 – fp.K 1000 fp.K 1000 – fp.K 4 fp.K 4 ,ypUe;J ,q;F g+r;rpaq;fSf;Fg; gjpyhf ,U rupTf;NfhLfs; Fwpf;fg;gl;Ls;sd. fp.K 3000 njhlq;fp fp.gp 600 tiu ggpNyhdpauf; s; jkf;nfd 60I mbahff; nfhz;l xu; vz;FwpaPl;L Kiwia Mf;fpf;nfhz;ldu;. vdpDk; fp.gp 7k; E}w;whz;Lf;Fg;gpd; xd;W Kjy; xd;gJ tiuahd vz;fSf;F jdpj;jdp FwpaPLfs; Vw;gl;bUf;fyhk; vd;gij ehk; mwpaf;$bajhf css; J. mt;ntz; Kiw glj;jpy; fhl;lg;gl;lss; J. ey;yJ vg;NghJk; mofhfj;jhd; Njhd;Wk; mofhdNjh xUNghJk; tpfpjj;jpy; FiwTglhJ.-gpNsNlh 47

,f; FwpaPLfs; vfpg;jpypUe;J ngwg;gl;ljhfTk; $wg;gLfpd;wJ. vdpDk; vfpg;jpa vz;Kiwapy; 1-9 tiu 1 vDk; FwpaPl;il vz;zpd; ngWkjpf;F jFe;jthW vOjg;gLfpd;wJ. vz; cNuhk FwpaPLfs; cNuhkGupiar; Rw;wpYk; tho;e;j kf;fs; Muk;gj;jpy; ,yj;jPd; nkhopiaNa Ngrp te;jdu;. fp.gp 5k; E}w;whz;lstpy; ,yj;jPd; tof;nfhope;J Nghf mjdplj;ij cNuhk nkhop ngw;Wf;nfhz;lJ. vz; gad;ghl;bYk; ,yj;jPd; nkhopapypUe;J ngwg;gl;l vz;FwpaPLfis ,yFthff; p cNuhku; jkJ vz;FwpaPl;L Kiwia Mf;fpf; nfhz;ldu;. IIII vDk; vz;FwpaPL gpw;fhyj;jpy; IV vd ,yFthf;fg;gl;lJ. 48

,t;thwhd rpy cNuhk vz;FwpaPLfs; 01 – i 11 - xi 02 - ii 12 - xii 03 - iii 13 - xiii 04 - iv 14 - xiv 05 - v 15 - xv 06 - vi 16 - xvi 07 - vii 17 - xvii 08 - viii 18 - xviii 09 - ix 19 - xix 10 - x 20 - xx 21 - xxi 31 - xxxi 22 - xxii 32 - xxxii 23 - xxiii 33 - xxxiii 24 - xxiv 34 - xxxiv 25 - xxv 35 - xxxv 26 - xxvi 36 - xxxvi 27 - xxvii 37 - xxxvii 28 - xxviii 38 - xxxviii 29 - xxix 39 - xxxix 30 - xxx 40 - XL 44 - XLiv 41 - XLi 45 - XLv 42 - XLii 46 - XLvi 43 - XLiii 47 - XLvii 48 - XLviii 49 - XLix 50 - L 49