jyaamiti

brahmagupt ka prameya, iske anusaar AF = FD.

jyaamiti ya rekhaaganit (en:Geometry) ganit ki teen vishaal shaakhaaon mein se ek hai. ismein binduon, rekhaaon, talon aur thos cheejon ke gunasvabhaav, maapan aur unke antariksh mein saapekshik sthiti ka adhyayan kiya jaata hai. jyaamiti, gyaan ki sabse praacheen shaakhaaon mein se ek hai.

jyaamiti ganit ki vah shaakha hai jismein binduon, rekhaaon, vakron, samatalon ityaadi ka adhyayan hota hai. bhoomi ke naap sambandhi kaaryon se is vigyaan ki utpatti hui, isaliye is ganit ko bhoomiti bhi kehte hain. aarambh mein yeh adhyayan rekhaaon tatha rekhaaon se ghire kshetron ke gunon tak hi seemit raha, jiske kaaran jyaamiti ka naam rekhaaganit bhi hai.

Bhaarat mein yajnyaavediyon ke nirmaan kaarya mein ganitjnyaon ka dhyaan jyaamiti ke adhyayan ki aur aakrusht kiya, unke adhyayan mein kshetrasamiti ka put adhik tha. itihaasayajnyaon ka mat hai ki bhaaratavaasi isa se 1,000 varsh poorv aise sambandh jaise 32 + 42 = 52 (32 + 42 = 52) jaante the, parantu aise hi katipya chhutaput sameekaranon ke atirikt unhonne aise sambandhon ka kisi vyaapak roop se adhyayan naheen kiya. isa se lagbhag 600 varsh poorv rom ke ganitjnya pithaagorais ne is sambandh ka bade tarkapoorn dhang se adhyayan kiya aur yeh bataaya ki ek samakon tribhuj mein karn par ka varg anya bhujaaon ke oopar vargon ke yogafal ke baraabar hota hai.

vaise to jyaamiti ka adhyayan sabhi puraane sabhya deshon, jaise misr, baibiloniya, cheen, Bhaarat tatha yoonaan, mein lagbhag saath hi saath aarambh hua, parantu jitni unnati is vigyaan mein yoonaan ne ki utani kisi aur desh ne naheen ki. isa se lagbhag 300 varsh poorv yoonaan ke ek ganitj yooklid ne us samay tak jitne tathya gyaat the un sabko bade tarkapoorn dhang se kramabaddh kiya. gyaat tathyon ke aadhaar par usane anya tathya siddh karne ka prayatn kiya. is prakaar tathyon ko kramabaddh karne par vah kuchh aise praarambhik tathyon par pahuaincha jinko siddh karna kathin hai. vaise ve bilkul spasht prateet hote hain. ye tathya itne saral hain ki yooklid ne inhein svayansiddh maan liya aur inhein swayam tathya kaha hai. inheen tathyon par jyaamiti ke prameyon ka pramaan nirbhar hai. ve tathya nimnalikhit hain :

1. ve vastueain, jo ek hi vastu ke baraabar hon, aapas mein bhi baraabar hoti hain.

2. yadi baraabar vastuon mein baraabar vastueain jod di jaayain to yogafal baraabar hote hain.

3. yadi baraabar vastuon mein se baraabar vastueain ghata di jaayain to sheshafal baraabar hote hain.

4. baraabar vastuon ke samaan gune baraabar hote hain.

5. yadi do rekhaaon ko teesari rekha kaate aur ek or ke ant:konon ka yog do samakon se kam ho to jidhr jod kam hai udhar hi donon rekhaaeain badhaai jaane par ek bindu par mileingi.

6. isi prakaar rachanaakaarya mein bhi ek rachana se doosari rachana kar sakte hain, parantu ant mein kuchh aisi rachanaaon par pahuainchate hain jinka prayog doosare prayogon par nirbhar naheen karta. in rachanaaon ko bhi swayam prayog maankar hi aage badh sakte hain. ve hain :

1. kisi bhi bindu se ek rekha kheenchi ja sakti hai.

2. seemit rekhaaeain donon or badhaai ja sakti hai.

3. ek bindu ko kendra maankar kisi trijya ka ek vrutt kheench sakte hain.

inke atirikt ve koi aur tath‌aaya bina siddh kiye hue sveekaar naheen karte. uparyukt paaainch swayam tathyon mein se chaar to itne saral tatha sapsht hain ki inhein siddh karna apne haath ko apna siddh karne ke baraabar hai, parantu paaainchavaaain svayantath‌aaya svayansiddh sa prateet naheen hota. ganitjnyaon ne is tathya ko svayansiddh maanane mein aapatti ki aur ise siddh karne ke bahut yatn kiye. inheen yatnon ke falasvaroop bade bade aavishkaar hue. isi prakaar jyaamiti mein nae nae paaribhaashik shabdon ka ullekh hota hai. ek shabd ki paribhaasha doosare shabdon ki paribhaasha par nirbhar karti hai. ant mein dekhte hain ki ye paribhaashaaeain bindu, rekha aur tal ki paribhaashaaon par aadhaarit hain. yooklid ke anusaar samatal vah hai jismein lanbaai chaudaai ho, parantu motaai na ho. bahut se log is paribhaasha par bhi sandeh karne lage hain, parantu thoda manan karne se yeh spasht ho jaaega ki paribhaasha theek hai. udaaharanaarth, yadi kaaainch ke ek baratan mein do aise taral padaarth bhar diye jaayain jo aapas mein na milte ho to jab ve sthir ho jaayain tab ham dekhagein ki ek tal donon padaarthon ko alag karta hai. usamein motaai naheen hai. yadi hoti to donon taralon ke beech aisa sthaan hota jismein na neeche ka padaarth hota na oopar ka, parantu aisa asambhav hai. is udaaharan se spasht ho gaya hoga ki tal mein motaai naheen hoti. ismein keval lanbaai aur chaudaai hi hoti hai. isi prakaar dhoop mein kisi samatal deevaar ki chhaaya dekhkar ham kah sakte hain ki rekha mein chaudaai naheen hoti. rekha tal mein sthit hai, at: tal ki motaai rekha ki motaai hui. isaliye rekha mein na motaai hoti hai na chaudaai, keval lanbaai hi hoti hai. rekhaaeain ek bindu par milti hain to rekha ki chaudaai bindu ki lanbaai hui, arthaat‌ bindu mein na lanbaai hoti hai, na chaudaai, motaai. keval sthaan hi hota hai.

sabhi is baat se parichit honge ki jyaamiti mein tribhuj, varg, vrutt, shanku, belan ityaadi ke gunon ka adhyayan hota hai1 puraane samay mein kuchh prashnon ne ganitjnyaon ko kaafi ulajhaae rakha. un prashnon ke halon ne bahut vichaaravardhan kiya, ismein koi shanka naheen, jaise aisa ghan banaana jiska ghanafal diye ghan ka duguna ho. us samay rachana ka arth patari aur parakaar ki sahaayata se hi rachana karna samjha jaata tha. doosra prashn tha aisa varg banaana jiska kshetrafal diye hue vrutt ke kshetrafal ke baraabar ho. teesara prashn tha ki ek diye hue kon ko teen baraabar bhaagon mein baaaintana. yeh kaam patari aur parakaar se asambhav hai, parantu anya upaayon se ho sakta hai. in prashnon ne shataabdiyon tak ganitjnyaon ko vyast rakha. inke vivechan se ganitjagat‌ ka bahut laabh pahuaincha, ismein koi sandeh naheen.

ek shanku ko kisi samatal se kaatne se jo deerghavrutt, paravalaya, tatha apirvalaya vakr bante hain unke gunon ka bhi yoonaaniyon ne adhyayan kiya. in adhyayanon ne kepalan ko apne niyam gyaat karne mein badi sahaayata di hogi.

15veen shataabdi tak jyaamiti mein praaya: naap sambandhi gunon ka hi adhyayan hota tha, parantu uske baad aise gunon ka bhi adhyayan hua jo naap par nirbhar naheen karte; jaise yadi do tribhujon ke sheershabindu ek teen bindugaami rekha par hon to sangat bhujaaeain ek rekha par mileingi. is saadhya ne ganitjnyaon ka dhyaan ek anya prakaar ki jyaamiti ki or aakrusht kiya jise prakshepeeya jyaamiti kehte hain. yadi ham kisi drushya ke chitr par dhyaan dein to anubhav karte hain ki use dekhkar drushya ka poora gyaan ho jaata hai. parantu chitr mein vrutt vrutt naheen rahata, na sabhi samaantar rekhaaeain samaantar rahati hai, na samakon samakon hi, balki kabhi samakon nyoon kon dikhaai deta hai, kabhi adhik kon; fir bhi drushya mein kuchh aise gun hai ki aakrutiyon ke badalne par bhi chitr se unka poora gyaan hota hai. ye gun nishchar kahalaate hain. aise hi gunon ka prakshepeeya jyaamiti mein adhyayan hota hai.

maan lein, ek bindu b aur ek chaturbhuj k kh g gh diya hua hai. yadi bindu b se chaturbhuj ke pratyek bindu ko milaanevaali rekhaaeain kheenchi jaayain aur unhein badha dein aur fir ek samatal se in rekhaaon ko kaatein to is tal par ek chitr banega. vah is chaturbhuj ka prakshep tatha yeh prayog bindu b ke saapeksh roopaantaran kahalaaega. isi prakaar doosra bindu lekar uske saapeksh is prakshep ka bhi prakshep nikaal sakte hain. jo gun naheen badalate unhein prakshep dvaara kisi saral bahubhuj mein badalkar adhyayan karte hain. ye gun mool bahubhuj ke liye bhi theek honge. saath hi kai roopaantaran milkar ek roopaantaran prayog ke samaan hote hain. in prayogon ka bhi adhyayan is jyaamiti ka ang hai.

yadi kisi gole ya vrutt ka kendra k ho tatha trijya tr ho aur yadi kisi bindu b ki kendra k se doori r ho aur yadi r' doori par rekha k b mein b' doosra bindu ho, jahaaain ra1 tra2 to b ke kisi bindupath ke sangat b' ka bhi path hoga. b' ka path b ke path ka pratiloman (inversion) kahalaata hai. pratyek kshetr pratiloman ka adhyayan hi is shaakha ka dhyeya hai.

yooklid ka 5vaaain svayansiddh tathya oopar diya ja chuka hai. ise svayansiddh maanane ke liye ganitjnya kabhi taiyaar naheen hue, balki unhonne ise siddh karne ke bade bade yatn kiye; parantu kaai santoshajanak uttar naheen mila. anusandhaan ke falasvaroop ganit ka bahut vikaas hua aur ek aisi jyaamiti ka aavishkaar hua jisne jyaamiti mein poorn kraanti utpann kar di. yooklid ne samatal par hi sab vivechan kiye, parantu ab har prakaar ke talon par alag alag vivechanaaeain hoti hain. iska vivechan kathin hai, at: iske liye paathak is vishay ki vishesh pustakon ka avlokan karein.

17veen shataabdi ke madhya mein fraanseesi ganitjnya dekaart (Descartes) ne jyaamiti mein beejaganit ka prayog kar ise bahut shaktishaali bana diya. usane pehle do kaatati hui rekhaaeain leen, jinhein aksh kehte hain. kisi bindu ki in rekhaaon ke samaantar naapi hui doori do sankhyaaon ya r se usaka sthaan nishchaya kiya. ye rekhaaeain bindu ke nirdeshaank kahalaati hain. in nirdeshaankon ki sahaayata se pratyek jyaamitiya tathya ko beejaganiteeya sameekaran dvaara pradarshit kiya ja sakta hai. is jyaamiti ka kai dishaaon mein vikaas hua.

pehli dasha mein to jyaamiti ka vyaapak roop saamane aaya, jaise ek ghaat ka sameekaran ek saral rekha pradarshit karta hai. isi prakaar do ghaat ka sameekaran ek shaankav (conic) pradarshit karta hai. isi prakaar teen, chaar aur uchchatar ghaaton ke sameekaranon ka adhyayan hone laga aur unke sangat vakron ke gunon ka vivechan pehle se bahut saral ho gaya. tal ke vakron tak hi naheen, avakaash (space) ke vakron ka bhi adhyayan sambhav ho gaya. iske liye ek bindugaami teen samatalon se kisi bindu ki dooriyon ya r l (x, y, z) na usaka sthaan nishchit karte hain aur pratyek bindupath ko ya, r, l (x, y, z) mein ek sameekaran dvaara pradarshit karte hain. in sameekaranon ke vivechan se talon or vakron ke gunon ka adhyayan saralata se hota hai.

doosari disha mein rachana sambandhi prashnon ka hal tatha kriyaaeain bahut saral ho gain. ye kriyaaeain keval kuchh sameekaranon ke hal par hi nirbhar hain, jismein bahut vyaapak prashn saralata se hal ho jaate hain; jaise yadi rekha (ax + by + c = o) kisi vakr (Ax² + By² + 2Hxy + 2Gx + 2F y + c) = o ko kaatati hai, to in donon sameekaranon ke hal unke kataan binduon ka sthaan nishchit kareinge. yadi in sameekaranon ke mool vaastavik hain, to rekha vakr ko kaatati hai. yadi baraabar hain to rekha vakr ko sparsh karti hai. yadi kaalpanik hain to rekha vakr ko naheen kaatati, parantu ham yeh kah sakte hain ki rekha vakr ko sadaiv do binduon par kaategi, chaahe bindu vaastavik ya sanpaati hon, athva kaalpanik hon. isi prakaar se tathya bade vyaapak roop mein diye ja sakte hain, jo saadhaaran jyaamiti mein sambhav naheen tha.

teesari disha mein nirdeshaank jyaamiti ne vimiti (dimension) ko vyaapak kiya. do sankhyaaeain ya, r (x, y) do vimitiyon (dimensions) mein tatha teen sankhyaaeain (ya, r, l) (x, y, z) teen vimitiyon mein kisi bindu ka sthaan nishchit karti hain. ab ganitjnyaon ke saamane yeh prashn utha ki chaar sankhyaaeain ya, r, l, va (x, y, z, t) ya paaainch sankhyaaeain ya, r, l, va, h (x, y, z, t, w) kya pradarshit kareingi. ganitjnyaon ne to amoort roop se apne mastishk mein badi aasaani se soch liya ki chaar sankhyaaeain chaar vimitiyon mein aur paaainch sankhyaaeain paaainch vimitiyon mein kisi bindu ka sthaan nishchit kareingi.

is prakaar unhonne s vimitiyon ka vichaar bhi achhi tarah soch liya. unhein isse koi matlab naheen ki paarthiv jagat‌ mein usaka koi udaaharan hai ya naheen. aainsataain ne avashya is vichaar ka apne saapeksh siddhaant mein upayog kiya aur vimiti ke vichaar ka spashteekaran kiya. ab is uchch vimiti ke vichaar ka aprayukt ganit mein kuchh kathin samasyaaon ko hal karne mein upayog karte hain. jaise kisi chal taral padaarth ke bhinn bhinn kanon ka sthaan, saat sankhyaaon se pradarshit karte hain. ve hain k, kh, g (a, b, c), usaka praarambhik sthaan, tatha teen veg, jo ya, r, l (x, y, z) aksh ke samaantar hon, tatha samay, yeh saat vimiti ka prashn samajhakar hal ho sakta hai.

chauthi disha mein nirdeshaank jyaamiti ne sankhyaaon ka vyaapakeekaran kiya aur kaalpanik sankhyaaon ka aavirbhaav hua. kalpanik bindu tatha kaalpanik vakr ityaadi vichaaron ne jyaamiti ko bahut mahatvashaali bana diya, jisse vyaapakeekaran mein aur adhik sahaayata mili, jaise anant par do kaalpanik binduon se jaanevaala shaankab vrutt hota hai, ityaadi.

iske atirikt jyaamiti ka vivechan bhinn bhinn prakaar ke nirdeshaankon ki sahaayata se hone laga, jaise samaghaateeya nirdeshaank, trikoneeya nirdeshaank, sparsheeya nirdeshaank ityaadi.

nirdeshaankon ke prayog ke lagbhag 50 varsh baad hi kalan (calculus) ka prayog bhi jyaamiti mein hone laga. is prayog ne jyaamiti mein nai nai vichaaradhaaraaeain utpann keen. inhein hi avakal jyaamiti kehte hain.

• prameya

• nirmeya

• bindu

• saral rekha

• tribhuj

• chaturbhuj

• vrutt

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