Maths history

51 - Li 56 - Lvi 52 - Lii 57 - Lvii 53 - Liii 58 - Lviii 54 - Liv 59 - Lix 55 - Lv 60 - Lx 61 - Lxi 71 - Lxxi 62 - Lxii 72 - Lxxii 63 - Lxiii 73 - Lxxiii 64 - Lxiv 74 - Lxxiv 65 - Lxv 75 - Lxxv 66 - Lxvi 76 - Lxxvi 67 - Lxvii 77 - Lxxvii 68 - Lxviii 78 - Lxxviii 69 - Lxix 79 - Lxxix 70 - Lxx 80 - Lxxx 81 - Lxxxi 82 - Lxxxii 83 - Lxxxiii 84 - Lxxxiv 85 - Lxxxv 86 - Lxxxvi 87 - Lxxxvii 88 - Lxxxviii 89 - Lxxxix 90 - XC 91 - XCi 50

92 - XCii 97 - XCvii 93 - XCiii 98 - XCviii 94 - XCiv 99 - XCix 95 - XCv 100 - C 96 - XCvi 200 - CC 2000 - MM 300 - CCC 3000 - MMM 400 - CD 4000 - MV 500 - D 5000 - V 600 - DC 6000 - VM 700 - DCC 7000 - VMM 800 - DCCC 8000 - VMMM 900 - CM 10000 - M 9000 - MX 10,000 - X 50,000 - L 10,000,000- X̿ 100,000- C 500,000- D 50,000,000- L̿ 1,000,000- M 5,000,000- V̿ 100,000,000- C̿ 500,000,000- D̿ 1,000,000,000- M̿ vd FwpaPLfs; ,yFthf;fg;gLfpd;wd. cNuhkd; vz;fs; $l;ly>; fopj;jy; nra;iffs; %yk; ngwg;gLfpd;wd. ,t;ntz; Kiwapy; rpy tpjpfSld; vz;FwpaPLfs; mikf;fg;gLfpd;wd. 4jlitfs; njhlu;r;rpahf xNu FwpaPL ,lk;ngwyhfhJ. 5> 50> 500 vDk; vz;fSf;Fupa FwpaPLfisf; fopj;J vz;FwpaPL Mf;f ,ayhJ> 1> 10> 100> 1000vDk; vz;fSf;Fupa FwpaPLfisf; fopj;J vz;ngwg;gLk; NghJ ,t;ntOq;fpy; Kjy; vz;Zf;Fupa FwpaPLfis kl;LNk fopf;fyhk;. 51

1000f;Fupa FwpaPl;bypUe;J 100f;Fupa FwpaPl;il khj;jpuNk fopf;f KbAk;. 10w;Fupa FwpaPl;il my;yJ 1w;Fupa FwpaPl;il fopf;f ,ayhJ. vdNt ,tn; tz;Kiw khw;wq;fSf;Flg; l;l ,yj;jPd; vz;KiwahFk;. fpNuf;fk; INuhg;gpa ehfupfq;fspy; fhyj;jhy; kpf Kej; pajhff; fpNuf;f ehfupfNk nfhs;sg;gLfpwJ. N`hkupd; fpNuf;f fhtpaq;fshd ,ypal;, xbrp vd;gd tlnkhopapy; ,aw;wg;gl;l kfhghujk;, ,uhkhazk; vd;gtw;wpw;F xj;jjhFk;. ,it cUthf;fg;gl;l fhyk; fp.K 7k,; 8k; E}w;whz;Lfshf ,Uf;fyhk; vd ek;gg;gLfpd;wJ. fp.K 1000 Mz;lstpy; fpNuf;fk; KOtJk; “Nlhupad;];” vDk; kf;fl; $l;lj;jhy; ifg;gw;wg;gl;L Xu; Ml;rpapd; fPo; nfhz;L tug;gl;lJ. kf;fshl;rpj; jj;Jtk; Kjd; Kjypy; fpNuf;fj;jpNyNa cUthdJ. Xypk;gpf,; kujd; Xl;lk; vd;gd fpNuf;fj;jpypUe;J cyfpw;F mwpKfkhditahFk;. ,tt; pjkhf INuhg;ghtpy; rpwe;J tpsq;fpa fpNuf;fk; fp.K 4k; E}w;whz;lstpy; krpNlhdpa kd;ddhfpa gpypg; vd;gtdhy; ifg;gw;wg;gl;lJ. gpypg; kd;ddpd; kfdhd mnyf;rhe;jupd; Ml;rpf;fhyj;jpy; ,e;jpah tiu Ml;rp tp];jupf;fg;gl;lJ. fpNuf;fk; XupU E}w;whz;Lfs; krpNlhdpa kd;du;fspd; Ml;rpf;Fl;gl;L gpd; cNuhkg; Nguurpd; vOr;rpAld; cNuhkpd; Ml;rpf;Fl;gl;Ltpl;lJ. fpNuf;fu; jj;Jtk,; fiy, tpQ;Qhdk;, kUj;Jtk;, fzpjk;, thdpay; Nghd;w Jiwfspy; rpwe;J tpsq;fpdhu;fs;. Nrhf;fpuB],; gpNsl;Nlh, mup];Nlhl;by; Nghd;w jj;JtQhdpfSk; ,A+f;fpspl;, igjfu]; Nghd;w fzpj mwpQu;fSk;, Nkiyj;Nja kUj;Jtj;jpd; je;ijahf; fUjg;gLk; `pg;Nghf;fpwRk;, ,yq;ifia Kjd; Kjypy; cyfg;glj;jpy; Fwpj;j FNshba]; njhykpAk; fpNuf;f Njrj;jtu;fshthu;. fpNuf;fu; g+kp xNu mr;rpy; RoYfpd;wJ vd;gij mwpe;jpUe;jdu;. xU tUlj;ij 365 ehl;fshfg; gpupj;J ehl;fhl;biaAk; Mf;fpapUe;jdu;. Gtpapd; Rw;wsT Vwj;jho 28000 iky;fs; vd fzpj;jpUe;jdu;. mj;Jld; #upa re;jpu fpufzq;fs; epfog;NghFk; fhyj;ijAk; Kd;$l;bNa mwpe;jpUe;jdu;. ,tu;fs; jkJ nkhopapd; tuptbtq;fis vz;FwpaPLfshff; nfhz;bUe;jdu;. fpNuf;fupd; vz;zy; Kiwapy; xw;iwg;gil vz;fs; xd;gjpw;Fk;, xd;gJ FwpaPLfSk; mtw;wpd; gj;jpd; klq;FfSf;Fk,; E}wpd; klq;FfSf;Fk;, Mapuj;jpd; klq;FfSf;Fk;, xd;gJ xd;gjhf ntt;NtW FwpaPLfSk; nfhLf;fg;gl;bUe;jd. ,tn; tz;fs; $l;ly; KiwapNyNa mike;jpUe;jd. tpQ;Qhd, fzpj, thdpay; Jiwfspy; kpfTk; KdN; dwpapUe;j fpNuf;fupd; vz;zy; Kiw gpd;tUkhW mike;jpUe;jJ. 52

xw;iwg;gil vz;fs; gj;Jf;fs; E}Wfs; 53

,tn; tz; Kiwapy; xw;iwg;gil vz;fSld; rpW NfhL ,Ltjd; %yk; Mapuq;fs; Fwpf;fg;gLk;. vz;fs; $l;ly; KiwapNyNa FwpaPLfshy; Mf;fg;gl;bUf;Fk;. = 10 + 3 = 13 = 800 + 40 + 3 = 843 MFk; ,t;tpjk; Njitfn; fhg;g Vw;w FwpaPLfisg; gadg; Lj;jp vz;ziy Nkw;nfhz;ldu;. vdpDk; ,k;Kiw gadg; hlb; w;F fbdkhdJ. Vnddpy; 27 mbg;gil FwpaPLfis epidtpy; itj;jpUf;f Ntz;Lk;. mj;Jld; ,k;Kiwapy; tiuaiwf;Fl;gl;l vz;fisNa Fwpf;fKbAk;. ,yj;jPd; vz;fs kf;fsplk; jpzpf;fg;glhtpl;lhYk; gutyhf fpilj;j cau; fUj;Jff; shy; fpNuf;f rhk;uh[a; toP ;rr; piaj; njhlu;e;J XupU E}w;whz;Lfs; krpNlhdpah kd;du;fspd; Ml;rpapypUe;J INuhg;ghtpd; njd;gFjp fp.K. 2k; E}w;whz;L gbg;gbahf cNuhkupd; nry;thfF; f;F cl;gl;lJ. Muk;gj;jpy; ijgu; ejpf;fiuapYk; cNuhk; efiur; R+oTk; tho;e;j kpff; Fiwthd kf;fs; $l;lj;jhy; Ngrg;gl;L te;j ,yj;jPd; nkhop cNuhkg; Nguurpd; vOr;rpAld; mjd; Ml;rp nkhopahfpaJ. cNuhkg; Nguurpd; tpupthf;fj;Jld; ,yj;jPd; nkhop INuhg;ghtpy; gutyhff; fhzg;gl;l fiy> gz;ghl;L rkaf; fUj;Jff; is cst; hq;fpf; nfhz;lJ. Vida Ml;rp nkhopfisg; Nghy ,yj;jPd; nkhop cNuhkg; NguuRf;Fl;gl;l gpuNjr nkhopahf jd;idr; nrk;ikg;gLj;jpf; nfhz;lJ. fp.K. 3k; E}w;whz;by; N`hkupd; fpNuf;f fhtpakhd xbrp> ,ypypa]; Md;Nuhdpf;f]; (Livivus Andronicus) vd;gtuhy; ,yj;jPd; nkhopapy; nkhopngau;ff; g;gl;lJ. ,JNt ,yj;jPd; nkhopapy; vOjg;gl;l Kjy; ,yf;fpakhFk;. gpd; fp.gp. 4k; E}w;whz;lstpy; tptpypa Ntjk; vgpNua nkhopapypUe;J ,yj;jPd; nkhopf;F nkhop ngau;f;fg;glL; fhyg;Nghf;fpy; fpwp];jtu;fSila topghl;L nkhopahf;fg;gl;Ltpl;lJ. vdpDk; ,yj;jPd; nkhop ,t;tpjk; cau;j;jg;gl;likf;F Ml;rpahsupd; MjuTk; Kf;fpa fhuzkhFk;. cNuhkg; Nguuir Ml;rp nra;j kd;ddhd ghy]; (Aemilius Paullus) vd;gtd; fp.K. 168 ,y; krpNlhdpa kd;ddhd ngu;rpA+ir (Perseus) ntd;w NghJ jpiwahf mtDila E}y; epiyaj;ijf; Nfl;lhdhk;. 54

,tt; pjk; jd;idr; nrk;ikg;gLj;jpf; nfhz;l ,yj;jPd; nkhop fpNuf;f nkhopapypUe;J cst; hq;fpf; nfhz;l mwptpay; mk;rq;fs; mNefkhdit. mjdhy; ,yj;jPd; vz;fSk; fpNuf;f vz; KiwapypUe;J tpUj;jpaile;jNj MFk;. ,yj;jPd; nkhopapy; vz;fs; gpd;tUkhW mikAk;. vz;fs; ,yj;jPd; vz; vz;fs; ,yj;jPd; vz; 1 I 50 L 2 II 100 C 3 III 400 4 IIII 500 CCCC 5 V 600 IƆ 6 VI 700 IϽС 7 VII 800 IϽСС 8 900 9 VIII IϽССС 10 VIIII 1000 IϽСССС 2000 X СIϽ СIϽСIϽ vgpNua vz;fs; fpwp];Jfhyk; vgpNua nkhopapd; nry;thf;Ff;Fl;gl;lJ. tptpypa Ntjk; Kjypy; vgpNua nkhopapy; vOjg;gl;L gpd; fp.gp 4k; Ehw;whz;lstpy; ,yj;jPd; nkhopf;F nkhopngau;f;fg;gl;lJ. Mjyhy; vgpNua nkhopapd; Njhw;wk; fpwp];J fhyj;jpw;F Kw;gl;ljhFk;. vgpNua nkhopapYk; vz;FwpaPLfs; $l;ly; KiwapNyNa mikf;fg;gl;Ls;sd. vdpDk; nghJthd Nkw;F ehLfspd; vz;zy; KiwfspypUe;J vgpNua vz;zy; Kiw NtWgl;lJ. Vida vz;zy; Kiwfs; ,lkpUe;J tykhf jdpj;jdp FwpaPLfSf;fhd ngWkjpfs; ,wq;F tupirapy; mikAk;. Mdhy; vgpNua vz;fspy; FwpaPLfspd; ngWkjp ,lkpUe;J tykhf VWtupirapy; mikAk;. ,tn; tz;zy; Kiwapd; vz;FwpaPLfs; fPNo fhl;lg;gl;Ls;sJ. ,k;Kiwapy; 5784 vDk; vz; (4+30+50+300+400+5000) vDk; fUj;jpy; vd vOjg;gLk;. ,k;Kiwapy; xw;iwg;gil vz;fs;, gj;Jff; s;, EhWf;fs;, Mapuq;fs; vd xt;nthd;wpw;Fk; xd;gJ xd;gJ FwpaPLfshf gy FwpaPLfs; gadg; Lj;jg;gLk;. mjdhy; ,f;FwpaPLfisg; gad;gLj;Jk; NghJ epidtpy; epWj;JtJ kpff; fbdkhf ,Uf;Fk;. mj;Jld; ,k;KiwapYk; ,lg;ngWkhdk; gad;gLj;jg;glhJ $l;ly; nra;if %yk; FwpaPLfs; Fwpf;fg;gLtjhy; vz; gad;ghL vy;iyg;gLj;jg;gLfpd;wJ. 55

rk];fpUj vz;fs; ,e;jpa fiy ,yf;fpa gz;ghl;lk;rq;fs; cd;dj epiyf;F cau;j;jg;gl;likf;F Mupa ehfupfNk fhuzk; vd 19 Mk; E}w;whz;bYk; 20 Mk; E}w;whz;bd; Kw;gFjpapYk; $wg;gl;L te;jJ. vdpdpDk; 1920 fspy; rpe;Jntsp ehfupfk; fz;Lgpbf;fg;gl;l gpd; mJ njhlu;ghf khWgl;l fUj;Jff; s; Vw;glj; njhlq;fptpl;ld. vt;tpjkhf ,Ug;gpDk; ,e;jpa ehfupfj;jpy; Mupar; nry;thfi; f Fiwj;J kjpg;gpLjy; Vw;ff;$ba xd;whfj; Njhd;wtpy;iy. Mupau; tUif fp.K. 1500 – 1000 Mz;Lfs; vdf; nfhss; g;gl;lhYk; fpwp];J fhyj;jpw;F Kd;Ng tlnkhopapy; vz;zy; njhlu;ghf Fwpg;gplf;$ba tsu;r;rp Vw;gl;bUe;jjhff; nfhs;syhk;. tlnkhoapy; gytpj vz;zy; Kiwfs; cgNahfpf;fg;glL; te;Jss; d. mtw;Ws; mÑug;gs;sp> G+jrq;fpia> flaghjp> ghuhf;fB> mq;fg;gs;sp vd;gd Kf;fpakhditahFk;. mÑug;gs;sp Kiwapy; rpy Fwpg;gpl;l vOj;Jf;fSf;F vz;kjpg;G ,l;L ,yf;fq;fSf;Fg; gjpyhf me;j vOj;Jf;fs; vOjg;gLk;. xd;W Kjy; xd;gJtiu xd;gJ vOj;Jf;fSk;> gj;J Kjy; njhz;Z}W tiu xtn; thU gj;Jf;Fk;> E}W> Mapuk; Mfpatw;Wf;Fk; jdpj;jdp vOj;Jf;fs; cz;L. 56

kiyahsj;jpy; ,k;Kiw Rtbfspd; VLfis ,yf;fkpLtjw;Fg; gad;gLj;jg;gLfpd;wJ. tlehl;by; ,k;Kiw Rtbfspd; VLfis vz;zpLtjw;F cgNahfpf;fg;gl;Ls;sJ. gioa Rtbfspy; css; cUtq;fs; mf;fhyj;jpy; toq;fpte;j ,yf;fq;fspdd; cUtq;fisNa ngupJk; xj;jpUf;fpd;wd. gpw;fhyj;Jr; Rtbfspy; ,t; vOj;Jf;fspd; tbtq;fSk;> mr;Rtbfs; vOjg;gl;l fhyj;J ,yf;fq;fspd; tbtq;fSk; NtWgl;Ls;sd. vz; kjpg;ig cila vOj;Jf;fspy; gpuNjrj;jpw;Fg; gpuNjrk; NtwghL cz;L. tl ehl;by; 4 vz;iz xU mÑuj;jhYk; njd;dhl;by; ,jid gpwpnjhU mÑuj;jhYk; Fwpj;Js;shu;fs;. mNjNghy 9 vDk; vz; tlehlby; xU vOj;jhYk; njd;dhl;by; NtnwhU vOj;jhYk; Fwpf;fg;gl;lJ. ,q;fdk; mÑug;gs;sp Kiwapy; vz; kjpg;Gila vOj;Jf;fs; gpuNjre;NjhWk; NtWgLfpd;wd. G+jrq;fpia Kiwapy; vz;fs; rpy Fwpg;gpl;l nrhw;fshy; Fwpg;gplg;gLk;. ,k;Kiw gioa E}y;fspYk; rhrdq;fspYk; fhzg;gLfpd;wJ. %Tyfk;> ehd;F Ntjq;fs; vd;gtw;wpw;F mika %d;W> ehd;F vd;gtw;iwf; Fwpf;f KiwNa Nyhfk;> Ntjk; vDk; nrhw;fs; gad;gLj;jg;gLk;. ,Nj Nghy ,k;Kiwapy; Vida vz;fSk; nrhw;fshy; Fwpg;gplg;gLk;. flaghjp Kiwapy; tlnkhop nka; vOj;Jff; Sf;F G+[;ak; Kjy; xd;gJ tiuapy; vz; kjpg;igf; nfhLj;J ,yf;fq;fSf;Fg; gjpyhf mtn; tOj;Jff; s; gad;gLj;jg;gLk;. ,k;Kiwapy; capu; vOj;Jff; Sf;Fk;> $l;nlOj;Jff; spy; Kjypy; tUtdtw;wpw;Fk; kjpg;Gf;fpilahJ. G+jrq;fpiaAk; ,k;KiwAk; tykpUe;J ,lkhf vz;zg;gLk;. ghuhff; B Kiwapy; f>fh Nghd;w vOj;Jf;fSfF; 1>2…. vd vz; kjpg;G ,lg;gl;Lg; gadg; Lj;jg;gLk;. ,k;Kiwapy; ffuk; Kbe;Jtpl;lhy; mjw;F mLj;j nka;naOj;ijg; gadg; Lj;Jtu;. rpyrkak; mfuk; Cu;e;j nka;naOj;Jff; is Kjypy; Kbj;Jtpl;L> mjw;Fg;gpwF Mfhuk; Cu;e;j nka;naOj;Jff; is cgNahfpg;gJk; cz;L. mq;fg;gs;sp Kiw ehk; ,g;nghOJ vOJtJ Nghy jrhk;r KiwahFk;. ,J fp.gp. 600 Mk; Mz;Lf;Fg; gpd; tl ehl;Lr; rhrdq;fspy; fhzg;gLtjhFk;. tlnkhop rhu;e;j vz;zy; Kiwapy; xw;iwg;gil vz;fs; xd;gJld; 20 Kjy; 90 tiu xtn; thU gj;jpw;Fk; xtn; thU jdpf;FwpaPlhf xd;gJ FwpaPLfs; cz;L. 200> 300 vDk; vz;fisf; Fwpf;Fk; NghJ 100 vDk; vz;zpd; cUtj;ij ,l;L mjd; tyJ Gwkhf KiwNa 1> 2 vDk; vz;fisf; Fwpg;ghu;fs;. ,NjNghy; 2000> 3000 vDk; vz;fisAk; 1000 vDk; vz;zpw;Fupa FwpaPl;il ,l;L mjd; tyJ Gwkhf KiwNa 1> 2 vDk; vz;fisg; NghLthu;fs;. vdpDk; 400> 4000 Nghd;w Vida vy;yh vz;fSk; 57

E}W> Mapuk; Mfpatw;wpd; tyJ Gwkhf Vw;w vz; FwpaPl;il ,Ltjd; %yk; ngw;Wf;nfhss; g;gl;lJ. ,k; Kiwapy; 100> 1000 vdg; gj;jpd; tYf;fSf;fhd FwpaPLfSld; xw;iwg;gil vz;fSf;fhd 9 FwpaPLfSk; gj;Jff; Sf;fhd 9 FwpaPLfSk; E}WfSf;fhd Nkyjpf ,uz;L FwpaPLfSk; (200> 300) Mapuq;fSf;fhd Nkyjpf ,uz;L FwpaPLfSkhf (2000> 3000) Mapuq;fs; tiu vz;z 24 FwpaPLfs; gadg; Lj;jg;gl;ld. 1 – 9 ᵉ 7– ɘ 2– 2 8– t 3– 3 9– e 4– 8 10 – 90 5– y 100 – 900 6 – ɛ 1000 – 9000 mj;Jld; rk];fpUjk; vd;gjd; fUj;J nrk;ikg;gLj;jg;gl;lJ vd;gjhFk;. fp.gp. 2Mk; E}w;whz;by; tlnkhop vz;fs; xUtpjkhfTk; gpw;fhyj;jpy; NtW tpjkhfTk; css; d. jkpo; vz;fs; jkpo; vz;fs; vd;gJ jkpopy; gadg; Lj;jg;gLk; vz;fis Fwpf;Fk;. ,it fy;ntl;Lf;fspYk; nrg;NgLfspYk; fhzg;gLk; fpue;j vOj;J Kiw MFk;. ,t; vz; tbtq;fs; gpw jkpo; vOj;Jf;fspd; tbtq;fis kpfTk; xj;J fhzg;gLk;. jkpo; vz;fSk; fpue;j vz;fSk; xNu vz; tbitf; nfhzb; Ug;gJ Fwpg;gplj;jf;fJ. njhy;fhg;gpau; vOj;jjpfhuj;jpy; vOj;Jf;fSf;F tiutpyf;fzk; toq;FtjhYk; jpUts;Stu; jdJ Kjy; Fwlg; hitNa “mfuKjy” vd njhlq;FtjhYk; jkpo; vOj;Jf;fs; 2500 Mz;LfSf;F Kd;Ng cUthfp tpl;ld vd;gJ njspthfptpl;lJ. ,d;W gof;fj;jpy; ,y;yhtpl;lhYk; jkpo; vd;fs; rpy fhyj;jpw;F Kd; gad;ghl;by; ,Ue;jd. gadg; hl;by; ,Ue;j> ,Uf;fpd;w cyfj;jpy; cs;s midj;J tup tbtq;fSf;Fk; FwpalP ;L vz;fs; toq;fg;gl NtzL; k; vd;gJ a+dpNfhL mikg;gpd; nfhs;if. ,jd; mbg;gilapy; 2000fspd; njhlf;fj;jpy; cj;jkk; mikg;gpd; gue;Jiuf;Nfw;g jkpo; vz;fSf;Fk; FwpaLP fs; toq;fg;gl;ld. ,d;iwa fzpdpfspYk; ifalf;ff; fUtpfspYk; ,ay;ghf cs;s jkpo; vOj;JUf;fspy; jkpo; vzf; Sk; Nru;fg;gl;Ls;sd. gad;gLj;Jk; Kiw – jkpo; vz;fspy; goq;fhyj;jpy; Ropak; (g+r;rpak;) ,y;yhky; NghapDk;> jw;fhyj;jpy; Ropak; jkpopy; vz;fis vOJk; NghJ gad;gLj;jg;gl;L tUtJ Fwpg;gpljf;fJ. 1825 Mk; Mz;L ntspte;j fzpj jPgpif vd;Dk; E}y; fzpj nraw;ghLfis vspikahf;Fk; nghUl;L jkpopy; Ropak; mwpKfg;gLj;jg;gLtjhf $WfpwJ. njhlf;fj;jpy; jkpo; vzf; s; ,lk; rhu;e;j Kiwapy; (positional system) vOjg; gad;gLj;jg;gltpy;iy. 10> 100> 1000 Mfpatw;Wf;F jdpj;jdp FwpaPfs; ,Ug;gijf; nfhz;L ,ij mwpayhk;. jkpo; vzf; s; vOj;jhy; vOjg;gLk; vzf; isr; 58

RUf;Ftjw;fhd FwpaPl;L KiwahfNt gadg; Lj;jg;gl;lJ. Ropak; mwpKfk; nra;ag;gl;l gpd;du; jhd; jkpo; vz;fs; ,lk; rhu;e;j Kiwapy; vOjg;gl Muk;gpj;jJ. cjhuzkhf - ,uz;lhapuj;J ehD}w;wp Ik;gj;jp %d;W vd;gJ gioa Kiwapd; gb vd;wthW vOjg;gLk;. எண்கள் : Ke;jpup> fhzp> xUkh vDk; gpd;dq;fis mbg;gilahff; nfhzL; gpd;dg; gad;ghL ,lk;ngw;wJ. ௨௰௧ =௩௰௧ =௪௰௧ = 21 31 41 ௨௰௨ =௩௰௨ =௪௰௨ = 22 32 42 ௧ = 1௰௧ = 11௨௰௩ =௩௰௩ =௪௰௩ ௨ = 2௰௨ = 1223 =௫௰௧ = 51 ௰௩ = 13 ௨௰௪ 33 43 ௫௰௨ = 52 ௩ = 3 ௰௪ = 14 24 =௩௰௪ =௪௰௪ = 53 ௪ = 4 =௫௰௩ = 54 ௰௫ = 15 ௨௰௫ 34 44 ௫௰௪ = 55 ௫= 5 ௰௬ = 16 25 =௩௰௫ =௪௰௫ = 56 ௬= 6 ௰௭ = 17 ௨௰௬ =௫௰௫ = 57 ௭= 7 ௰௮ 26 35 45 =௫௰௬ = 58 ௮= 8 18 = ௨௰௭ =௩௰௬ =௪௰௬ = 59 ௯= 9 ௰௯ = 27 ௫௰௭ ௰ = 10 ௨௰ = 19 ௨௰௮ 36 46 =௫௰௮ 20 =௩௰௭ =௪௰௭ ௫௰௯ 37 47 =௬௰ = 60 =௩௰௮ =௪௰௮ 28 38 48 ௨௰௯ =௩௰௯ =௪௰௯ = 29 39 49 ௩௰ = 30 ௪௰ = 40 ௫௰ = 50 jkpo; ,yf;fk; ngau; vz; msT ௨ந Kej; pup 1⁄320 Miuf;fhzp 1⁄160 Miuf;fhzp Kej; pup 3⁄320 Fhzp 1⁄80 Fhy; trP k; 1⁄64 Miukh 1⁄40 59

3Æ4 miutPrk; 1⁄32 Kf;fhzp 3⁄80 Kf;fhy; tPrk; 3⁄64 xUkh 1⁄64 Khfhzp (tPrk;) 1⁄16 ,Ukh 1⁄10 Miuf;fhy; 1⁄8 %d;Wkh 3⁄20 %d;W trP k; 3⁄16 ehYkh 1⁄5 Fhy; 1⁄4 Miu 1⁄2 Kf;fhy; 3⁄4 jkpopy; fPo;tha; ,yf;fq;fSk; gad;gLj;jg;gl;ld. jkpopy; fPo;tha; ,yf;fq;fSk; mtw;wpw;fhd FwpaLP fSk; gpd;tUkhW mikAk;. Kej; pupapd; Vopy; xU gq;F “,k;kp” vdTk;> ,k;kpapd; Vopy; xU gq;F “mZ” vdTk;> mZtpd; gjpndhd;wpy; xU gq;F “Kk;kp” vdTk;> Kk;kpapd; xd;gjpy; gq;F “Fdk;” vdTk; Fwpf;fg;gl;lJ. mjhtJ ,k;kp = ������⁄������������������ × ������⁄������ mZ = ������⁄������������������ × ������⁄������ × ������⁄������ Kk;kp = ������⁄������������������ × ������⁄������ × ������⁄������ × ������⁄������������ Fdk; = ������⁄������������������ × ������⁄������ × ������⁄������ x ������⁄������������ × ������⁄������ NkYk; jkpo; vz;Kiwapy; kpfg; ngupa vz;fSk; gadg; hl;by; ,Ue;jij rq;f ,yf;fpaq;fspy; ,Ue;J mwpaf;$bajhf css; J. 60

Iak; gy;nyd t&c kpWjp my; ngau; vz;zpZ khapadpiyAk; ( njhy; vO. 393 ) nghUl;ngau; my;yhj I> mk;> my; vd;gtw;iw ,Wjpahff; nfhz;l vz;Zg;ngau;fs; ‘rq;fk;’ (rq;F)> ‘tpe;jk;’ (kiy)> ‘FKjk;’ (Mk;gy;)> ‘gJkk;’ (jhkiu)> ‘ehL’ (Ftis)> ‘rKj;jpuk;’ (nea;jy;) nts;sk; vd;gd Ngnuz;fs; MFk;. ,it Nfhbapd; mbg;gilapy; mike;j Ngnuz;fs; MFk;. ‘rq;fnkd; klq;F nfhz;lJ tpe;jk;’ ‘tpe;jnkd; klq;F nfhz;lJ FKjk;’ ‘FKjnkd; klq;F nfhz;lJ gJkk;’ ‘gJknkd; klq;F nfhz;lJ ehL’ ‘ehnld; klq;F nfhz;lJ rKj;jpuk;’ ‘rKj;jpunkd; klq;F nfhz;lJ ntss; k;’ vDk; #j;jpuq;fs; fPo;tUk; nghUs; jUtdthFk;. rq;fk; Nfhb2 1014 tpe;jk; Nfhb3 1021 FKjk; Nfhb4 1028 gJkk; Nfhb5 1035 ehL Nfhb6 1042 rKj;jpuk; Nfhb7 1049 ntss; k; Nfhb8 1056 Nfhbf;F NkYs;s njhif ‘rq;fk;’ MFk;. rq;fpdJ cUtk; ,U Nfhb (,U Nfhzq;fSilaJ rq;F)> tpe;jj;jpd; cUtk; Kf;Nfhb (kiy Kf;NfhzKilaJ)> Mk;gypd; cUtk; ehw;Nfhb (ehY Nfhzk; - ehypjo;fisfAilaJ)> jhkiu cUtk; Iq;Nfhb (Ie;J ,jo;fis cilaJ)> gJkgej; k; Iapjohf tiuag;gLk;. ehl;bd; cUtKk;> Ftisapd; cUtKk; MW Nfhb MFk;. (MW Nfhzq;fis cilaJ ) rKj;jpuj;jpd; cUtk; VO Nfhb (VOfly;) ntss; j;jpd; cUtk; vz;Nfhb. (vl;Lj;jpf;fpYk; guTtJ). vd;W Nkw;$wpa njhiffspy; xyp tbtf; Fwpg;Gf;fs; vy;yhk; fhuzg;ngauhy; mike;jpUg;gjhf $wg;gLfpd;wJ. NkYk;> m̔ idj;Jky;y gy tLf;fyhk;gy; ,izj;njd vz; tuk;gwpah ahf;ifia̓ (gupghly;3 tup 44 – 45)vDk; ghly; tupfspYk; Kbtpy;yhj vz;fs; gw;wpf; Fwpg;glg;glLs;sJ. 61

muhgpa vz;fs; muhgpa ejpf;fiu ehfupfq;fspd; njhlu;r;rpia cilaJ. mj;Jld; fpof;fpw;Fk; Nkw;Ff;Fkhd tu;j;jf ghijfs; muhgpahtpD}lfNt mike;jpUe;jd. Mdhy; Nkw;fpYs;s fpNuf;f> cNuhk> vgpNua nkhopfSk; fpof;fpy; jkpo; rk];fpuj nkhopfSk; tsr;rpaile;j fhyj;jpNyNa muhgpa nkhopAk;> tpUj;jpaile;jpUf;Fk;. mjdhy; muhgpa vz;fSk; ,q;F fUj;jpy; vLf;fg;gLfpd;wd. muhgpa vz;Fwp;aPLfs; ,U tpjkhdit. gioa Fwpa; PLfSk; mtw;wpw;fhd xyptbtq;fSk; Gjpa Fwp;aPLfSk; vd ,Utifg;gLk;. vz; gioa xyp gioa Gjpa Gjpa xyp tbtk; tbtk; Fwp;aLP thfpJ 1 mypg; (f) ,j;dhdp jyhjh 2 Ng (bay) mu;gM 3 fk;r`; 4 [Pk; (jeem) rpj;jh`; 5 jhy; rg; M 6 N` jkhdpa`; yht; jp]; M 7 ];N] m]w`; ,]Y; }d; 8 N`w jyh J}d; 9 Nj mu;g Cd; 10 Na k;#d; 20 fhg; KA,f rpj;J}d; rg; Cd; 30 yhk; jkhD}d; jp]; Cd; 40 kPk; kp mJd; kPmjhdp 50 E}d; jhJ kp mjpd; mu;g c kp mjpd; 60 ]Pd; fk;R kp mjpd; rpj;J kp mjpd; 70 mapd; rg;c kp mjpd; jkhdpA kp mjpd; 80 Ng (fay) jp]c; kp mjpd; my;Gd; 90 ]; ]hj; 100 Kaigf 200 Nw 300 \\Pd; 400 Nj TH 500 j;Nj THe 600 Nf Ghaye 700 jhy; Thall 800 o;No 900 ohj; 1000 MMapd; 62

,q;F gioa FwpaPLfs; fpNuf;f> rdP > vgpNua> vz; FwpaPLfisg; Nghy; xw;iwg;gil vz;fs;> gj;Jff; s;> E}Wf;fs; vd;gtw;wpw;F jdpj;jdpf; FwpaPLfs; css; d. vdNt ,k;Kiwapy; vz;fis mikf;Fk; NghJ mtn; tz;fisg; Nghy; $l;ly; nray; KiwNa ,lk;ngw Ntz;Lk;. vdpDk; Gjpa muG vz; FwpaPLfspy; xw;iwg;gil vz;fSf;F xd;gJ FwpaPLfSk; mtw;wpd; gj;jpd; E}wpd; klq;FfSf;F gj;jpd; tYf;fspw;F Vw;g Gs;spfSk; ,lk;ngWfpd;wd. 20If; Fwpf;Fk; NghJ ,uz;bw;fhd FwpaPl;il vOjp tyJGwk; xU Gs;sp ,lg;gLfpd;wd. ,NjNghy; Mapuj;ij Fwpf;f Fwpg;gpl;l vz;Zld; %d;W Gs;spfs; ,lg;gLk;. “vz;nzdg; Vid vOj;njdg; ,t;tpuz;Lk; fz;nzd;g thOk; capu;fF; .” jpUts;Stu; fzpjj;jpd; Kf;fpaj;jtj;ij 2000 tUlq;fSf;F KdN; g epiy epWj;jpAs;shu;. 63

khau; Kiw fpwp];jt fhyq;fspy; fhzg;gl;l ehfupfq;fspy; khah ehfupfKk; Kf;fpakhdjhFk;. ,e; ehfupfk; fupgpad; fliyr; #o;e;j kj;jpa mnkupf;f ehLfspy; epytp te;jJ. khah ehfupf kf;fs; jq;fk;> nrg;G Mfpatw;iw cgNahfpj;jdu;. mtu;fs; cUt vOj;ijg; gadg; Lj;jpdhu;fs;. vd;gij vOj;Jf;fs; nghwpf;fg;gl;l fw;rpyhrdq;fs; vLj;Jf;fhl;Lfpd;wd. fzpjk;> thdpay;> gQ;rhq;fk; Kjypad gw;wp mtu;fs; mwpe;jpUe;jdu;. 360 ehl;fis nfhz;l ehl;fhl;bnahd;iwAk; Kf;fpa rka eltbf;iffSf;fhf 260 ehl;fis nfhz;l ehl;fhl;bnahd;iwAk; mtu;fs; gad;gLj;jpdhu;fs;. cyfpy; Kjy; Kjypy; G+r;rpaj;ij gadg; Lj;jpatu;fs; khauf; Ns vdf;fUjg;gLfpd;wJ. khau;fs; epiyf;Fj;jhf ,lg;ngWkhdj;Jld; vz; FwpaPLfisg; gad;gLj;jpdu;. khau; vz;Fwp Kiwik vd;gJ Kjw;nfhyk;gpa khau; ehfupfj;jpy; gad;gLj;jg;gl;l ,Ugjpd;k vz;Fwp Kiwik MFk;. mtu;fs; mbg;gilahd 3 FwpaPLfSld; jkJ vz;ziy Nkw;nfhz;ldu;. Mdhy; ,tu;fSila vz;zy; Kiwapy; ,lg;ngWkhdk; kpfTk; rpf;fyhd Kiwapy; mike;jpUe;jJ. mtu;fSila mbg;gilf; FwpaPLfs; ,k; %d;W FwpaPLfisAk; nfhz;L ,lg;ngWkhdj;jpw;F xg;g Njitahd vz;fis mikj;Jf; nfhz;ldu;. ,k; Kiwapy; epiyf;Fj;jhf vz;ngWkhdk; xd;Wf;fs;> ,UgJfs;> Ke;E}w;W mWgJfs;> Vohapuj;J ,UE}Wfs; vd mjpfupf;fpd;wJ. khah ehfupf vz; FwpaPLfs; gpd;tUkhW mikAk;. 64

65

,q;F 20 vDk; vz;iz vOJk; NghJ ,lg;ngWkhdj;jpw;F xg;g epiyf;Fj;jhf G+r;rpaf; FwpaPl;il ,l;L mjd; Nky; xU Gs;sp ,lg;gLk;. ,t; ntz; Kiwapy; vDk; FwpaPL 5+1vDk; fUj;jpy; 6IAk; 5+20 vDk; fUj;jpy; 25 IAk; Fwpf;Fk;. mNjNghy; vDk; FwpaPL 5+2 vDk; fUj;jpy; 7 IAk; 5+(2×20) vDk; fUj;jpy; 45 Fwpf;Fk;. vdNt xU FwpaPL ,U vz;fisf; Fwpg;gJ ,e;j Kiwapd; Fiwghlhf cs;sJ. ,t;ntz;Kiw ehl;fhl;bfspd; NjitfSf;fhff; $Ljyhfg; gad;gLj;jg;gl;ljhy; ,lg;ngWkhd mjpfupg;G rpf;fyhdjhf cs;sJ. khah vz;fis gadg; Lj;jp 20f;Ff; fPNo vz;fisr; Nru;g;gJ kpfTk; vspJ. Xtn; thU kl;lj;jpYk; vz; rpd;dq;fis ,izg;gjd; %yk; $l;ly; nra;ag;gLfpd;wJ. ,t;ntz; Kiw rhjhuz fzpg;gPl;Lj; NjitfSf;F kpfTk; rpf;fyhdJ 15k; E}w;whz;by; nfhyk;g]; mj;jpyhe;jpf; rKj;jpuj;jpuj;ijf; flf;Fk; tiu cyfk; khauf; is mwpatpy;iy. vdpDk; Ie;jhf> ,Ugjhf vz;Zk; gz;G INuhg;gpau;fisg; Nghy; ,tu;fsplKk; ,Ue;Js;sJ. NkYk; tpIPrpky; mikgi; g ikakhff; nfhz;L khad; vz;Zk; Kiw if tpuy;fs; kw;Wk; fhy; tpuy;fisg; gad;gLj;jp vz;Zk; Nghf;fpypUe;J cUthf;fg;gl;lJ. kw;Wk; nkNrhmnkupf;f fyhr;rhuq;fspd; tuyhw;wpy; tsu;e;j gy mbg;gil Kiwfspy; xd;whFk;. 66

rpq;fs vz;fs; fpwp];jt fhy ehfupfq;fSld; rpq;fs ehfupfj;ijAk; Nruj; ;Jf; nfhs;syhk;. Vnddpy; fp.K 273 ,y; mNrhfr; rf;futu;j;jpapd; kfdhd kfpe;j Njuu; tl ,e;jpahtpypUe;J ,yq;iff;F te;J Njtek;gpajPr kd;dDf;F ngsj;j kjj;ij Nghjpj;jhu;. NkYk; fp.K 103 ,y; tyfk;gh kd;dDila Ml;rpfhyj;jpy; jpupgplfk; E}YUg; ngw;wJ. mjdhy; rpq;fstu; vOj;Jf;fSf;fhd FwpaPLfis fp.K 2k; E}w;whz;by; my;yJ mjw;F KdN; g cUthf;fpf; nfhz;lhu;fs;. Guhjd rpq;fs vz;FwpaPl;L Kiwapy; xd;W njhlf;fk; gj;J tiuAs;s vz;fSf;Fk;> E}W tiuahd gj;jpd; klq;FfSf;Fk; Mapuk; tiuahd E}wpd; klq;FfSf;Fk; ntt;Ntwhd vz; FwpaPLfs; ,Ue;jd. ,q;F 27 vDk; vz;Zf;F 20 f;Fupa FwpaPLk; 7f;Fupa FwpaPLk; ,lg;gLfpwJ. mNj Nghy 35 vDk; vz;iz 30 f;Fupa FwpaPl;Lld; 5f;Fupa FwpaPl;il ,izj;J vOJtjhy; Fwpf;fg;gLk;. 538 I Rl;l Ie;JfF; upa FwpaPl;il ,l;L mjid mLj;J E}Wf;Fupa FwpaPl;il ,l;L njhlu;e;J Kg;gJfF; upa FwpaPLk; vl;bw;Fupa FwpaPLk; ,lg;gLk;. mjhtJ 5I E}why; ngUf;fp 30 clDk; 8clDk; $l;l Ntz;Lk;. mNjNghy 1971 If; Fwpf;f Mapuj;jpw;Fupa FwpaPL xd;gjw;Fupa FwpaPL E}wpw;Fupa FwpaPL vOgjpw;Fupa FwpaPL xd;wpw;Fupa FwpaPL vd;gd ,t;nthOq;fpy; vOjpf; fhl;lg;gLk;. ,q;F gj;jpd; klq;Ffshd 10> 20> 30………f;F jdpj;jdp FwpaPLfs; ,Ug;gjhy; etdP jrkj;jdi; kia Gyg;gLj;j khl;lhJ. ,tn; tz;zy; Kiw rk];fpUjj;Jld; neUq;fpa njhlu;GilaJ. 67

,aw;fzpjk; (ml;rufzpjk;) ,aw;fzpjkhdJ fzpjj;jpd; xU Kf;fpa gpupthFk;. vz; Nfhl;ghL> tbtpay;> gFtpay; Mfpa gFjpfis css; lf;fpaJ. nghJthf ,aw;fzpjk; vd;gJ fzpj tbtq;fisg; gw;wpAk;> mtw;iw MSk; tpjpfisg; gw;wpAk; gbg;gjhFk;. fzpjk;> mwptpay;> nghUspay;> nghwpapay; kw;Wk; kUj;Jtk; Nghd;wtw;Wf;Fk; mbg;gil ,aw;fzpjk; mtrpakhFk;. ,aw;fzpjj;jpd; KdN; dhbfshf my;-fthup];kp kw;Wk; Xku; fa;ahk; Nghd;Nwhu; mwpag;gLfpd;wdu;. ,aw;fzpjk; vz;fis kl;Lk; mbg;gilahff; nfhz;L fzpg;gpLk; vz;fzpjj;jpw;F mLj;j gbahFk;. Kjypy; fzpjj;jpy; vz;fzpjNk fw;gpf;fg;gLfpd;wd. Mifahy; vz;fzpjNk cz;ikapy; fzpjj;jpd; mupr;RtbahFk;. vz;fzpjk; kw;Wk; ,aw;fzpjk; ,uz;bw;FKs;s Kf;fpa NtWghL ,aw;fzpjj;jpy; ifahsg;gLk; khwpfSk; nghJ tbtj;jpw;fhd vz;fspw;fhd khwpfSkhFk;. khwpfis cgNahfpj;J Ez;kkhf (abstract) rpe;jpj;J nra;ag;gLk; fzpg;Gfis mbg;gil ,aw;fzpjk; nfhz;Ls;sJ. vz;fzpjj;jpy; vz;fs; kw;Wk; mtw;iw nfhz;L nra;ag;gLk; mbg;gilr; nray;fs; tptupf;fg;gLfpd;wd. mbg;gil ,aw;fzpjj;jpy; vz;fSf;Fg; gjpyhf x,y Nghd;w khwpfSk; a,b Nghd;w khwpfSk; gadg; Lj;jg;gl;L fzpjr; nray;fs; Nkw;nfhss; g;gLfpd;wd. vLj;Jf;fhl;lhf: E=mc2 vd;w rkd;ghl;by; E,m vOj;Jf;fs; khwpfshFk;. c khwpypahFk;. NkYk; ,aw;fzpjk; Nkd;NkYk; cau;epiyf;Fr; nry;y ,J tpuptile;J gy;NtW ngau;fspy; gpupe;J nry;fpd;wd. ,aw;fzpjj;jpy; Ma;Tfis Nkw;nfhs;Sk; fzpjtpayhsu; ,aw;fzpjtpayhsu; (algebraist) vdg;gLfpwhu;. mbg;gil ,aw;fzpjk; ,J ,aw;fzpjj;jpd; Kf;fpa gpupthFk;. ,J ,aw;fzpjj;jpd; mbg;gilf; fUj;Jf;fis tptupf;fpwJ. vz;fzpjk; kw;Wk; ,aw;fzpjk; ,uz;bw;FKs;s Kf;fpa NtWghL ,aw;fzpjj;jpy; fhzg;gLk; khwpfs;jhd;. vz;fzpjj;jpy; vz;fs; kw;Wk; mtw;iwf; nfhz;L nra;ag;gLk; mbg;gilr; nraw;ghLfshd $l;ly;> fopj;jy;> ngUf;fy;> tFj;jy; Mfpait Fwpj;j fUj;Jff; s; tptupf;fg;gLfpd;wd. mbg;gil ,aw;fzpjj;jpy; x kw;Wk; y Nghd;w khwpfSk;> vz;fSf;Fg; gjpy; a kw;Wk; b Nghd;w khwpypfSk; gad;gLj;jg;gl;L fzpjr; nray;fs; Nkw;nfhss; g;gLfpd;wd. khwpfis cgNahfpj;J Ez;kkhf (abstract) rpe;jpj;J nra;ag;gLk; fzpg;Gf;fis ,aw;fzpjk; nfhz;Ls;sJ. ,f;fzpjg;gpupit mbg;gil ml;rufzpjk; my;yJ mbg;gil fzpjk; my;yJ mbg;gil FWf;fzf;fpay; vd;Wk; Fwpg;gpLtJz;L. nghJthf khztu;fs; Kjypy; vz;fzpjk; fw;W gpd;du; ,aw;fzpjj;jpd; %yk; NkYk; Ez;kkhfr; rpe;jpf;f ce;jg;gLfpwhu;fs.; ,aw;fzpjj;jpy; rkd;ghLfs; Kf;fpa gq;F tfpf;fpd;wd. 68

fpNuf;f fhyj;jpa ,aw;fzpjk; ,aw;fzpjk; vd;gJ xU nkhop. gw;gy FwpaPLfSk; mitfis xd;Wf;nfhd;W vt;thW gpuNahfpg;gJ vd;gjw;F tpjpfSk; css; d. Mdhy; ,t;thwhd nkhop xd;W gad;gLtjw;F mk; nkhopf;F rupahd FwpaPl;L Kiw ,Ue;jhf Ntz;Lk;. mq;Fjhd; fpNuf;f fzpjk; jtwpaJ. mtu;fSf;F vy;yhNk tbtpay; jhd;. tbtpaypy; mghukhd jpwik ngw;wpUe;jhu;fs;. vz;fs; $l mtu;fSf;F xU Neu;Nfhl;bd; msTfNs. mjdhy; ,aw;fzpj tof;fkhd ‘khwp’ vd;w fUj;J mtu;fSila vz;zq;fSld; xj;JgN; ghftpy;iy. MdhYk; ,aw;fzpj khwpfs; Kyk; njhlu;Gfis cUthf;fp me;j njhlu;Gfis rkhspf;f mtu;fsplk; Nehff; Nkh> rhjdNkh ,Uf;ftpy;iy. ,aw;fzpjj;jpy; mtu;fSila KdN; dw;wk; kpff; Fiwthf ,Ue;jjw;F ,d;ndhU fhuzKk; ,Ue;jJ. mJjhd; mtu;fSf;F Kbtpypfisg; gw;wp ,Ue;j mr;rk;. Mu;fpkpb]; ig(π) apDila kjpg;igf; fz;Lgpbg;gjw;F gad;gLj;jpa Kiwf;F ntspg;gLj;Jif Kiw (Mehode of exhaustaion) vd;W ngau;. jpUg;gp jpUg;gp xU gy;Nfhzj;jpd; gf;fq;fspd; vz;zpf;ifia mjpfg;gLj;jpf; nfhz;Nl Ngha; mjDila Rw;wsit tpl;lj;jpd; ePsj;jhy; xt;nthU KiwAk; tFj;J ig ,w;F kjpg;Gfs; fz;Lgpbj;J nfhz;L NghFk; Kiwjhd; mJ. Methode of exhaustion “nka; cyfj;jpy; gpur;rpidfs; njhlu;ghfg; ngwg;gl;l mgpg;gpuhaq;fs; njhFf;fg;gLk; NghJ tpdhf;fs; vOfpd;wd. ,t; tpdhf;fspD}lhfg; ngwf;$ba vz;zf;fUf;fs; njhlu;ghf jd;dpr;irahf rpy KbTfs; Cfpf;fg; gLfpd;wd. mk; KbTfisg; ngWtjw;fhd ca;j;jwp mikg;Gr; nray;top fzpjkhFk;.” - nkhwp]; fpshapd; 69

,e;jpahtpy; ,aw;fzpjk; vz;fis vOJtjpy; ,lkjpg;Gj; jpl;lj;ij cUthf;fp tUq;fhy fzpjf; FwpaPl;L Kiwf;F mbNfhypaJ gioafhy ,e;jpah. ];GIpj;tIu; (3k; E}w;whz;L) vOjpa ‘atdIhjfk;’ vd;w E}ypy; ,t;tplkjpg;Gj; jpl;lk; gadg; Lj;jg; gl;bUg;gij ghuf; ;fyhk;. me;E}Ny> fhzhky; Ngha;tpl;l fpNuf;f NIh]pa Kiwiag; gw;wp ,uz;lhtJ E}w;whz;by; ,e;jpahtpy; vOjg;gl;l xU ciueil E}ypd; nra;As; eilkhw;wk;jhd;. fpwp];J rfhg;jj;jpd; Kjy; rpy E}w;whz;Lfspy; vOjg;gl;ljhff; fUjg;gLk; ghÑyp ifnaOj;Jg; gpujp xd;W 1881 ,y; jw;NghJ ghfp];jhdpYs;s ng`htUf;fUNf fz;Lgpbf;fg;gl;lJ. mjdpy; rk ,lkjpg;Gj; jpl;lKk;> ropf;Fg;gjpy; xU Gs;spAk; ruskhfg; gad;gLj;jg;gl;bUf;fpwJ. gpd;dq;fs;> tu;f;f%yq;fs;> Neupay; xUq;fikr; rkd;ghL> ,Ugbar; rkd;ghLfs;> $l;Lj;njhlu;> ngUf;Fj; njhlu; ,it ,lk;ngWfpd;wd. ,d;Dk; ,e;j E}ypy; ,e;jpahtpy; ,Ue;J muhgpau;fs; vLj;Jr; nrd;W ‘jq;fkakhd tpjp’ (Golden Rule) vd;W mtu;fshy; ngau; #l;lg;gl;l mbg;gilf; fzpj tpjp tptupf;fg;gLfpwJ. nfhLg;gpid-gad;-,r;irtpjp vd;W ,jw;Fg; ngauplyhk;. ,ijj;jhd; Mq;fpyj;jpy; Rule of Three vd;W nrhy;fpwhu;fs;. mitahtd: ,tt; sT gbg;gpid ,t;tsT gaidf; nfhLj;jhy;> ,t;tsT ,r;ir vt;tsT gaidf; nfhLf;Fk;? ,jw;F tpil fhz tpjp: nfhLg;gpid-gad-; ,r;ir ,k;%d;iwAk; ,e;j tupirapy; vOJ. eLtpYs;sij filrpapYs;sjhy; ngUf;fp Kjypy; css; jhy; tF. ,Jjhd; tpil nfhLg;gpid> ,r;ir ,uz;Lk; xNu myifr; rhu;e;jJ. gad; NtW myifr; rhu;e;jjhf ,Uf;fyhk;. 70

Njutpayhr; rkd;ghLfs; (Indeterminate Equation) Kjd;Kjypy; ,e;jpaf; fzpjj;jpy; vOj;jpy; fhzg;gLtJ ,e;jg; ghÑhyp ifnaOj;Jg; gpujpapy; jhd;. ,r;rkd;ghLfis gw;wp fpNuf;f ehl;L lnah/ghz;l]; 4k; E}w;whz;by; Ma;Tfs; nra;jpUe;jhYk;> ,e;jpaf; fzpj epGzu;fs; gpuk;kFg;ju;(7 k; E}w;whz;L)> gh];fuu1; (600-680)> gh];fuu;II(1114-1185) Njutpayhr; rkd;ghLfs; gw;wp gw;gy jPu;Tfis fz;L gpbj;J vOjpAs;sdu;. gh];fuu;II ,d; rf;uths Kiw ,d;Wk; gadg; l;Lf; nfhz;bUf;fpwJ. /ngu;khtpDila filrpj; Njw;wk; lNah/ghz;lrpd; Ma;Tfisg; gw;wp ghn~ (Bachet) vd;gtu; xU E}y; vOjpdhu;. ,e;E}ypd; xU gpujp /ngu;kh (17 k; E}w;whz;L) fpilj;jJ. mjdpy; gf;f Xuq;fspy; /ngu;kh jd; tpku;rdq;fis vOJtJ tof;fk;. x2 = y2 + a2 vd;w rkd;ghl;il gw;wpa Ngr;R te;j gf;fj;jpd; Xuj;jpy; /ngu;kh ifg;gl vOjp itj;j Fwpg;G tuyhW gilj;jJ. khwhf xU fd mLf;if (cubic power) ,uz;L fd mLf;FfshfNth> xU ehw;gb mLf;if ,uz;L ehw;gb mLf;fhfNth nghJthf ,uz;Lf;F Nkw;gl;l ve;j mLf;ifAk; mNj mLf;Ffs; ,uz;lhfNth gpupf;f KbahJ. ,Jjhd; /ngu;khtpDila filrpj;Njw;wk; vd;W ngau; ngw;W ehd;F E}w;whz;Lfs; fzpj cyif Ml;bg; gilj;jNjhL kl;Lky;yhky; vz; Nfhl;ghl;bYk; KOf; fzpjj;jpYNk Jizg;gpupTfs; Vw;glf; fhuzkhapUe;jJ. fzpjk; vd;gJ FwpaPLfs; %ykhf fhll; f;$ba fiy fll; ikg;Gfs; gw;wpa fw;wyhFk; kw;Wk; FwpaPLfs; %yk; fhl;lf;$ba rfy njhFjpfsJk; tpupthf;fNk fzpjkhFk;. -nkf]; gpshf; 71

fzpjg; Gjpu; kdpj tuyhw;wpNyNa kpfg; ngupa fzpjg;Gjpu; ‘IPNdh gulhf;]; mjhtJ Kuz;ghL’ vd;gJjhd;. vyh vd;fpw fpNuf;fj;ijr; Nru;e;j IPNdh vd;w jj;Jt Qhdp nrhy;ypr; nrd;w fzpjg; Gjpu; ,e;jg; Gjpiu khj;jp Nahrpj;jhy; epa+l;ldpd; fy;Fy]; Id;];Bdpd; rhu;gpay; Nfhl;ghL mjhtJ fzpjj;jpw;F fhyk; vd;gJ vt;tsT Kf;fpak; vd;gJ GupAk;. ,e;j glj;jpy; mk;G xd;W gha;fpwJ. ,e;j xU epkplj;jpy; mjd; lhu;fn; fli; l mila Ntz;Lk;. mtu; nrhy;fpwhu; ,e;j mk;G nrd;W milaNt nra;ahJ vg;gb ghjp Neuj;jpy; mJ ghjp J}uk; nrd;W mile;jpUf;Fk;> kPjpg;ghjpj; J}uj;ij kWgbAk; ghjp ghjpahf gpupf;f Ntz;Lk;. me;jf;fhy J}uj;ij kPjp ghjp Neuj;jpy; nrd;wile;jpUf;Fk;. kWgbAk; kPjKs;s rJuj;ij ghjpahf gpupf;f Ntz;Lk; mij me;j mk;G flf;f Ntz;Lk;> vt;tsT kpr;rk; ,Ue;jhYk; me;j kpr;r J}uj;jpy; ghjp J}uk; kpr;rkpUf;Fk; flg;gjw;F> me;j mk;G filrp tiu ,yf;if nrd;W mila KbahJ. ,Jjhd; IPNdhgulhf;];. ,ij NtWtpjkhf nrhd;dhy;> Muk;gpfF; k; NghJ Mik nfhQ;r J}uk; Kd;dhb nry;y Muk;gpf;fpwJ. Xl;lg; ge;ja tPud; me;j Mikia gpd;dhy; Juj;Jfpwhd;. mtd; ghjp J}uk; Xb te;J me;j Mikia njhLk; NghJ me;j Mik ghjpapy; ghjp fhy;thrp J}uk; KdN; dwp ,Uf;Fk;> mtd; fhy;thrp J}uk; Kd;NdWk;NghJ me;j Mik miuf; fhy;thrp J}uk; KdN; dwp ,Uf;Fk;> filrp tiu me;j Xl;lg; ge;ja tPud; me;j Mikia njhl KbahJ. ,Jjhd; IPNdh gulhf]; ; tpsf;fk; Nkk;Nghf;fhf rpe;jpj;jhy; ,J rhj;jpak; vd;W Njhd;Wk;> Mdhy; yhIpf; gb xU jtW css; J. mij Gupe;J nfhs;s epa+l;ld; kw;Wk; Id;];Bd; nrhy;Yk; fhynkDk; gupkhzj;ij Gupe;J nfhz;lhy; ,e;j Kuz;ghl;by; ,Ue;J ntspNa tuyhk; my;yJ ,e;j GjpUf;F tpil fz;L gpbf;fyhk;. 72

mk;G Xl;lg;gej; a tuP u; ,UtUk; xNu Ntfj;jpy; kzpf;F 100 fpNyh kPl;lu; Ntfk; vd;W itj;J fzf;Fg; ghuj; ;Njhkhdhy; me;j mk;G ,yf;if njhl;bUf;Fk; Xl;lg;ge;ja tPud; Mikia jhz;b ,Ug;ghd;. J}uj;ij ghjpahf;Fk; NghJ Ntfj;ijAk; ghjpahf;FfpNwhk;> Xl;lg;gej; a tuP d; Mik Ntfk; ,uz;Lk; xd;W vd;W MFk; NghJjhd; ,J elfF; k; Mdhy; fhyk; Fwpf;fg;glL; tpl;lJ> vDk; nghOJ me;j mk;G nrd;wile;j gpd;G fzf;F NghlL; ghu;ff; Ntz;Lk;. mg;NghJ fhyf; fzf;F rupahf ,Uf;Fk;> mk;G nrd;wilahky; tpOe;jhy; jhd; ,e;j Kuz;ghL rhj;jpak;. mk;G xNu ruP hf nry;Yk; NghJ njhl;LtpLk; ,d;Dk; Ntfk; Mdhy; ,yf;ifNa jhz;b tpLk; mNjNghy; jhd; Xl;lg;ge;ja tuP d; ghjp ghjpahf Fiwj;Jf; nfhz;Nl NghtJ vd;gJ ,d;/gpdpl;b Kbtpy;yhj xU njhlu; Mdhy; nkhj;j $l;ly; xU gpdpl; vz;zpf;if> xU gpdpl; vz;zpf;if ,d;gpdpl; Mf gpupf;fyhk; vd;w rhj;jpak; css; J. mNjNghy; fhyk; vd;gJ fzpjj;jpy; vtt; sT Kf;fpak; vd;gij czu;j;Jfpd;w jj;Jtk; jhd; ,e;j IPNdh guhlf]; ;. uhkDIj;jpd; ke;jpu rJuk; (Magic square) 22 12 18 87 88 17 9 25 10 24 89 16 19 86 23 11 NkNyAss; uhkhDIj;jpd; ke;jpu rJuj;jpy; ePq;fs; ve;j tupirapy; $l;bdhYk; kw;Wk; ve;j epiuapy; $l;bdhYk; ve;j %iytpl;lj;jpy; $l;bdhYk; mjd; $l;Lj;njhif 139 Mf ,Uf;Fk; 22 12 18 87 88 17 9 25 10 24 89 16 19 86 23 11 NkNyAss; xNu tz;zk; ,lg;gl;l vz;fisf; $l;bdhy; mJTk; 139 MftpUf;Fk;. 73

22 12 18 87 88 17 9 25 10 24 89 16 19 86 23 11 NkNy tz;zk; jPl;lg;gl;bUf;Fk; vz;fis $l;bdhy; mJTk; 139 MftpUf;Fk;. 22 12 18 87 88 17 9 25 10 24 89 16 19 86 23 11 NkNyAss; rJuj;jpy; xNu tz;zj;jpy; ,Uf;Fk; vz;fspd; $l;Lj;njhifAk; 139 MftpUf;Fk;. ,jpy; mw;Gjk; vd;dntd;why; 22 12 1887 vd;w Kjy; epiuapy; Fwpg;gplg;gl;bUg;gJ =dpthr uhkhDIdpd; gpwe;j jpdkhFk;. ,e;jf; $l;Lj;njhif vt;thW tUk; vd;gjw;fhd #j;jpuj;ijAk; mtu; nfhLj;Jss; hu;. DD MM CC YY YY+1 CC-1 MM-3 DD+3 MM-2 DD+2 YY+2 CC-2 CC+1 YY-1 DD+1 MM-1 74

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