
mohar-masteuroni pramey

ganitat, mohar-masheroni pramey asey sangato qee hokayantra aanee saral dharadvare kelee jaau shakanari konatihi bhaumitik rachana keval hokayantradvare kelee jaau shakate.
heey samajoon ghene aavashyak aahe qee "konatihi bhaumitik rachana" mhanje jyaa aakrutinmadhye saral resha nasatat, kaaran saral resheshivay saral resha kadhane spashtapane ashakya aahe. heey samajale jaate qee eka resha nishchit kelee jaate, jarr tyaa reshevar done vegale bindu dile kinwa bandhale gele asateel, jaree resheche konatehi drushya pratinidhitva nasel. heey pramey adhik tantotant asey mhatale jaau shakateh
jaree saral bajucha vapar bandhkamala lakshaniyaritya sope banavoo shakato, taree pramey darshavito qee bandhaleli aakruti purnapane paribhashit karananya binduncha konatahi sanch keval hokayantradvare nirdharit keela jaau shakato aanee saral bajoo vaparnyache ekmev kaaran mhanje saral resha pahanyachya saundaryasathi aahe, jee bandhkamachya uddeshane karyatmakdrushtya anavashyak aahe.
itihas.
haah parinam moolatah george mohar yanni 1672 madhye prakashit keela hota, parantu 1928 paryant tyacha purava aspashtatet rahila. 1797 madhye laurenzo masheroni yanni swatantrapane yaa prameyacha shodh lavala aanee mohar yanche karya punha shodhale jaiparyant teey 'masheroniche pramey' mhanoon olakhale jaat hote.
parinamache anek purave dnyat aahet. masheronicha 1797 chaa purava samanyatah pramukh sadhan mhanoon resheteel pratibimb vaparnyachya kalpanevar aadharit hota. moharcha upay vegala hota. 1890 madhye august edlerne ulatya parivartanacha vapar karoon purava prakashit keela.
bijganitiy drushtikon euclidean pratal aanee vastavik samanvay avkash sootra _ 1 yanchyatil samarupata vaparato. asha prakare, 1990 madhye siddha jhalelya siddhantachi eka majboot aavrutti. heey aarkimidijchya svayamsiddhatevar (jeey pratham-kram bhashet tayar kele jaau shakat nahee) prameyavar avalambun asalyache dekhil darshavate.
rachnatmak purava.
ruparesha.
heey pramey siddha karnyasathi, hokayantra aanee saral dharanchi pratyek moolbhut rachana keval hokayantra vaparoon shakya asalyache siddha karane aavashyak aahe, kaaran yaa itar serve bandhkamancha paya kinwa prathamik payya aahet. heey aaheth
heey samajale jaate qee saral resha saral dharanshivay kadhali jaau shakat nahee. eka resha konatyahi done bindunni dilee jaate asey manale jaate, kaaran ashee konatihi jodee eka adviteey resha paribhashit karate. aapan siddha karnyache uddisht asalelya siddhantachya uddeshanusar, vastavik resha kadhanyachi garaj nahee tarr saundaryatmak karanansathi aahe.
heey ektya kampasdware kele jaau shakate. yasathi saral bajuchi aavashyakta nahee.
heey bandhkam thet hokayantrane dekhil kele jaau shakate.
asha prakare, pramey siddha karnyasathi, #3 aanee #4 sathee keval kampas-fakt rachana dene aavashyak aahe.
sanket aanee tippanya.
pudhil sanketank yaa sampoorn lekhat vaparale jaatil. jyache kendra binduvar sthit aahe aanee jeey binduvarun jaate teey vartul yaa dware darshavile jaeel. kendra aanee trijya asalele vartul, kinwa reshecha vibhag, anukrame kinwa, dware darshavila jaeel.
sarvasadharan bandhkamanmadhye anekada anek badal asatat jeey samaan parinam detat. asha prakarat kelelya nivadi samanyata naan gamavata kelya jaau shakatat. tathapi, jevha kahitari kele jaau shakate heey siddha karnyasathi bandhkam vaparale jaat asel, tevha yaa serve vividh paryayanche varnan karane aavashyak nahee aanee pradarshanachya spashtatesathi, khalee fakt eka prakar deela jaeel. tathapi, anek rachana vegvegalya swarupat yetat, tyaa vartul ulatya vapartat qee nahee yavar avalambun asatat aanee shakya asalyas heey paryay dile jaatil.
heey lakshat ghene dekhil mahattvache aahe qee mohar-masheroni pramey siddha karananya kahee rachnansathi avkashat bindunche aniyantrit sthan nishchit karane aavashyak aahe, jase qee aadhich pradan naan kelele asatana vartulache kendra shodhane (khalee bandhkam pahaa). kahee bandhkam pratimananmadhye-jase qee rachnaksham sankhyechya bhaumitik vyakhyemadhye-heey pratibandhit kele jaau shakate. tathapi, asha pratimanat, udaharanarth, tyanchya kendranshivay vartule gruhitkadvare pradan kelee jaanar naheet, tyamule konatihi samasya nahee.
kahee prathamik bandhkame.
khalee samavisht kelelya vareel rachana #3 aanee #4 siddha karnyasathi, kahee aavashyak madhyasth rachana dekhil khalee spasht kelya aahet kaaran tyaa varanvar vaparlya jaatat aanee sandarbh dilya jaatat. hya dekhil keval hokayantra-bandhani aahet. khaleel serve rachana #1, #2, #5 aanee tyapurvi soochibaddha kelelya itar konatyahi bandhkamavar avalambun aahet.
kampas samatulya pramey (vartul anuvad).
vartulache naveen kendrat bhashantar karanyachi kinwa tyachi nakkal karanyachi kshamata yaa puravyanmadhye mahattvachi aahe aanee prameyatil satyata sthapit karnyasathi moolbhut aahe. pahilya trijyaevadhich trijya asalele, parantu vegalya binduvar kendrit asalele naveen vartul tayar karane heey kosalananya hokayantrala aadhunik, kathor hokayantrapasun vegale karnare mukhya vaishishtya aahe. kathor hokayantrat hii kshullakata aahe, parantu kosalananya hokayantrat too bandhkamachya shakyatecha prashna aahe. kosalananya hokayantra aanee kathor hokayantra yanchi samanata euclidne ('ghatakancha' pustak I prastav 2) saral dhaar aanee kosalananya hokayantrancha vapar karoon siddha kelee, jevha too moolatah vegalya kendrasah vartulachi prat tayar karato. hii samanata keval hokayantradvare dekhil sthapit kelee jaau shakate, jyacha purava mukhya lekhat aadhalu shakato.
reshechya bhagachi lambi vadhavane.
bindu shodhanyasathi aavashyak titkya vela yaa rachanechi punaravrutti kelee jaau shakate jenekarun konatyahi sakaratmak purnankachya reshechya bhagachi lambi = ω reshechya bhagachi lambi.
vartulat ulatya.
bindu assa aahe qee trijya trijyaevadhich aahe; kinwa.
vareel rachana ayashasvi jhalyas (mhanje, laal vartul aanee kaale vartul done bindunmadhye chhedat naheet), reshevar eka bindu shodha jenekarun reshechya vibhagachi lambi eka sakaratmak avibhajya gunank asel, samaja, lambichi aanee tyapeksha jaast asel (heey aarkimedchya svayamsiddhatedvare shakya aahe). varilpramane vartulat ulate shodha (laal aanee kalee vartule aata done bindunmadhye chhedali pahijet). aata bindu vadhavun prapt keela jaato jenekarun =.
resha aanee vartulache aantarchhed (bandhkam #4).
vartulache kendra resheshi samarekheey aahe qee nahee yavar avalambun resha aanee vartulachya chhedanbinduchi keval dishabhul rachana done prakarananmadhye vibhagali jaate. 
vartulache kendra resheshi samarekheey nasate.
vartulache kendra reshevar nahee asey samaja.
vartul ulat vaparoon paryayi rachana dekhil dilee jaau shakate.
vartulache kendra resheshi samarekheey aahe.
asha prakare asey disun aale aahe qee ekhadi resha aksharshah kadhali jaau shakat nahee parantu keval done bindunni paribhashit kelee jaau shakate heey samajoon ghetale tarr, ekhadi vyakti saral dhaar aanee hokayantrasaha karoo shakanari serve moolbhut rachana keval hokayantradvare kelee jaau shakate.
itar prakarchya pratibandhit bandhkamanche.
punarujjivan ganitshastradnya lodovico ferrari, gerolamo cardano aanee nikolo fontana tartaglia aanee itar 16 vya shatkat heey dakhavu shakale qee konatehi shasak aanee aachchadan bandhkam saral dhaar aanee nishchit rundichya hokayantrane (mhanje.eae. eka ganjalela kampas).
masheronichya nikalamule prerit houn, 1822 madhye jeane victor ponsletne tyach sankalpanevar eka farak asalyacha andaj vartavala. tyanni assa prastav mandala qee saral dhaar aanee hokayantradvare shakya asaleli konatihi bandhani keval saral dharane kelee jaau shakate. tathapi, eka aut ashee aahe qee tyache kendra olakhoon ekach vartul pradan karane aavashyak aahe. aata ponslet-steiner pramey mhanoon olakhale jaanare heey vidhan akra varshannantar jacob steinerne siddha kele. 
nantar 1904 madhye francesco savery yanni dilela purava eka purn vartul pradan karanyachi aavashyakta shithil karato aanee darshavito qee vartulachi konatihi lahan kamani, joparyant kendra adyap pradan kele aahe, toparyant ajunahi pureshi aahe.
yavyatirikt, jarr dusya ekagra kinwa chhedanare vartul, kinwa tisare vartul, kinwa eka bindu pradan kelele, naan chhedanare dusare vartul, asha puresha itar ekhadya goshtisathi badalale gele tarr kamanichya bhaganaivaji kendra swatach vagalale jaau shakate.
hokayantra samatulyata pramey darshavito qee varr namood kelelya serve bandhkamanmadhye, tyachya nishchit karnyayogya chhidrasah parichit aadhunik hokayantra, jeey antar hastantarit karnyasathi vaparale jaau shakate, teey 'sankuchit hokayantra' neey badalale jaau shakate, eka hokayantra jeey jevha jevha prushthavarun uchalale jaate tevha kosalate, jenekarun teey thet antar hastantarit karnyasathi vaparale jaau shakat nahee. kharey tarr, yuklidchya mool bandhkamanmadhye sankuchit honyajoga kampas vaparla jaato.