paravalayacha chaturthansh

parabolacha chaturthansh haah bhoomitivaril eka grunth aahe, joou archimediesne ii. sha. puu. kadhalelya shatkat lihila hota aanee tyachya alexandrian parichit dosithiusla sambodhit keela hota. tyat paravalayanshi sambandhit 24 prastav aahet, jyachi parinati done puravyanmadhye hote, jeey darshavtat qee paravalay vibhagache kshetrafal (paravalay aanee reshanni vedhalela pradesh) heey eka vishisht koralelya trikonache sootra aahe.
visheshtah thakvyachya paddhaticha aanee bhaumitik malikechya malikechya bhagat kushalatene vapar kelyabaddal, heey aarkimidijchya sarvat prasiddha kamanpaiki eka aahe. archemedies yaa kshetrache anant trikonanmadhye vibhajan karate, jyanche kshetra bhaumitik pragati tayar karate. tyanantar too parinami parinami bhaumitik malikechi berries mojato aanee heey parabolic segmentche kshetrafal aahe heey siddha karato. pracheen greek ganitatil 'reductio edd ubserdam' yuktivadacha haah sarvat atyadhunik vapar darshavito aanee 17 vya shatkat avibhajya ganitacha vikas hoiparyant archimediesche samadhan atulaneey rahile, tyanantar cavaliaryche chaturbhuj sootra aale.
mukhya pramey.
parabolic vibhag haah parabola aanee reshanni vedhalela pradesh aahe. parabolic khandache kshetrafal shodhanyasathi, archemedies eka vishisht koralelya trikonacha vichar karato. yaa trikonacha paya haah paravalayatil dilela taar aahe aanee tisra shirobindu haah paravalayavaril assa bindu aahe qee tyaa binduvaril paravalayala sparshika tee tarala samantar asate. kamachya prastav 1 madhye asey mhatale aahe qee akshala samantar kadhalelya kadhalelya shirobindutil resha taar samaan vibhaganmadhye vibhagate. mukhya pramey assa dawa karato qee paravalay khandache kshetrafal heey koralelya trikonache sootra _ 1 aahe.
majkurachi rachana.
eka shatkapurvichya menekamasamule parabola sarkhe shankuchya aakarache bhag aarkimidijchya kalat aadhich prasiddha hote. tathapi, vibhedak aanee avibhajya gananechya aagmanapurvi, shankuchya vibhagache kshetrafal shodhanyache konatehi sope marg navhate. parabola aanee tarane vedhalelya kshetravar visheshtah laksh kendrit karoon archemedies yaa samasyevar pahila pramanit upay pradan karato.
archemedies mukhya prameyache done purave detoh eka amoort yantrikicha vapar karoon aanee dusare shuddha bhoomitidware. pahilya puravyamadhye, archemedies gurutvakarshanachya kriyekhali samtol samtol livercha vichar karato, jyamadhye parbinduche bharit bhag aanee tribhuj heey fulakepasoon vishisht antaravar liverchya bahunvar latakalele asatat. jevha trikonache gurutvakarshanache kendra dnyat hote, tevha tribhujache santulan trikonachya kshetrafalachya drushtine paravalayache kshetrafal dete jyacha aadhar samaan aanee unchi samaan asate. yethe archemedies "samtol asalelya samatalanvar" samatalanvar" prakriyepasoon vichalit hotat, kaaran tyanchi gurutvakarshanachi kendre samtol patalichya khalee asatat. dusra aanee adhik prasiddha purava shuddha bhoomiticha vapar karato, visheshtah bhaumitik malikechi berries. 
chovis prastavanpaiki, pahilya teen prastavancha ullekh yuklidchya 'shankuchya ghatakan' (shankuchya bhaganvar euclidne kelele haravalele kaam) pasoon puravyashivay keela aahe. prastav 4 aanee 5 heey paravalayache prathamik gunadharm sthapit karatat. prastav 6-17 mukhya pramey yantrik purava detat; prastav 18-24 bhaumitik purava sadar karatat.
bhaumitik purava.
parabolic vibhagache vichhedan.
ujavya bajuchya aakrutimadhye darshavilyapramane, parabolic vibhagache anant trikonanmadhye vichhedan karane hii puravyabaddalchi mukhya kalpana aahe. nila trikon jyaa prakare mothya vibhagat korala aahe tyach prakare yaa pratyek trikonala tyachya swatachya parabolic vibhagat korale aahe.
trikonanche kshetra.
athara teey ekvis yaa prastavanmadhye, archemedies heey siddha karatat qee pratyek hiravya trikonache kshetrafal heey sootra 3 mhanje nilya trikonache kshetrafal aahe, jenekarun donhi hiravya trikonanchi ektrit berries sootra 4 mhanje nilya trikonache kshetrafal hoil. aadhunik drushtikonatun, yache kaaran asey qee hiravya trikonala sootra 5 rundi aanee sootra 4 nilya trikonachi unchi aaheh
tyach yuktivadanantar, 7 yaa sutratil pratyek pivalya trikonala sootra 3 asate, nilya trikonache kshetrafal kinwa sootra 9, sootra 10, nilya trikonache kshetrafal; 11 yaa sutratil pratyek laal trikonala sootra 3 asate, pivalya trikonache kshetrafal, sootra 13, nilya trikonache kshetrafal ii. thakwa paddhat vaparoon, heey asey aahe qee parabolic vibhagache ekoon kshetrafal khalilpramane dile jaateh
yethe 'tii' heey mothya nilya trikonache kshetrafal darshavate, dusari sandnya done hiravya trikonanche ekoon kshetrafal darshavate, tisari sandnya chaar pivalya trikonanche ekoon kshetrafal darshavate aanee ityadi. heey dene sope karate
malikechi berries.
purava purn karnyasathi, archemedies darshavito qee
vareel sootra heey bhaumitik malika aahe-pratyek salag sandnya mageel sandnyechya eka chaturthansh aahe. aadhunik ganitat, teey sootra heey bhaumitik malikesathichya berries sutrache eka vishesh udaharan aahe.
archemedies, jawalchya chitrat sachitra kelelya, purnapane bhaumitik paddhaticha vapar karoon bereejche moolyankan karato. heey chitra eka ekak varg darshavite jeey lahan chaurasanchya anantamadhye vichchhedit kele gele aahe. pratyek salag jambhalya rangachya chaurasamadhye mageel chauras kshetrachya eka chaturthansh kshetra asate, ekoon jambhalya kshetrachi berries asate.
tathapi, jambhalya rangache varg pivalya chaurasanchya donhi sanchanna ekroop asatat aanee tyamule ekak chauras kshetrafalachya sootra _ 18 laa jhakatat. asey aahe qee vareel malika sootra _ 1 madhye jamaa hote (kaaran 

