
kshetra

ack goal () ack jyaamitiya vastu haye joe ack dwi-ayami vritt key liye ack tri-ayami anuroop haye. aupchaarik roop seey, ack kshetra una binduon kaa samooh haye joe tri-ayami sthaan mein diye gaye bindu seey ack hii doori parr hain. vaha diyaa gayaa bindu gole kaa kendra haye, aur gole kii trijya haye. pracheen yunani ganitvidon key kaam mein kshetron kaa sabse pahala gyaat ullekh milata haye.
ganit key kaee kshetron mein kshetra ack maulik vastu haye. prakriti aur udyog mein bhee gole aur lagbhag golakar aakar dikhaai dete hain. sabun key bulbule jaise bulbule santulan mein golakar aakar lete hain. prithvi koo aksar bhugol mein ack gole key roop mein anumaanit kiya jaataa haye, aur khagol vigyaan mein khagoliya kshetra ack mahatvapurna avadharana haye. dabaav vaahikaon aur adhikansh ghumaavdaar darpan aur lens sahit nirmit vastuein golon parr adharit hoti hain. gole kissy bhee disha mein sucharu roop seey ludhakate hain, isaliye khel aur khilaunon mein upayog kii jaane vaali adhikansh genden golakar hoti hain, jaisa kii gend kii bearing hoti haye.
jyaamitiya roop seey, ack gole koo ack vritt koo ack aksh key chaaron or ack aadhaa chakkar lagaakar ghumaaya jaa sakata haye joe vritt key kendra koo kaatata haye, yaa ardhvritt key chaaron or ack poorna chakkar koo ghumaakar joe ardhvritt key seedhe kinare key saath sanyog (yaa samvarti) haye.
buniyaadi shabdawali.
jaisa kii pahale ullekh kiya gayaa haye kii gole kii trijya haye; kendra seey gole parr ack bindu taka kii kissy bhee rekha koo trijya bhee kahaa jaataa haye.
yadi ack trijya koo kendra key madhyam seey gole key vipareet disha mein vistaarit kiya jaataa haye, tou yeh ack vyaas banaata haye. trijya kii tarah, vyaas kii lambai koo bhee vyaas kahaa jaataa haye, aur ise darshaaya jaataa haye. vyaas sabse lambe rekha khand hain jinhen gole parr dou binduon key beach kheencha jaa sakata haih unaki lambai trijya seey doguni haye. ack vyaas seey judey gole parr dou bindu ack dusare key prati-pratiroop bindu hain.
ack ikai kshetra ikai trijya () vaalaa ack kshetra haye. suvidha key liye, golon koo aksar samanvaya pranali key mool mein unakaa kendra maanaa jaataa haye, aur iss lekh mein golon kaa kendra mool mein hotaa haye jaba taka kii kissy kendra kaa ullekh naa kiya jaae.
gole parr ack "mahaan vritt" kaa kendra aur trijya gole key samaan hoti haye, aur yeh ise dou barabar "golardh" mein vibhajit kartaa haye.
haalaanki prithvi poori tarah seey golakar naheen haye, bhugol seey udhaar liye gaye shabd gole parr lagu honey key liye suvidhajanak hain.
yadi kissy gole parr ack vishesh bindu koo (manmaane dhang seey) isake "uttar dhruv" key roop mein naamit kiya jaataa haye, tou isake virodhi bindu koo "dakshin dhruv" kahaa jaataa haye. pratyek key liye samaan doori parr mahaan vritt taba "bhoomadhya rekha" haye. dhruvon key madhyam seey badey vritton koo deshantar kii rekhaaen yaa meridian kahaa jaataa haye. dou dhruvon koo jodane vaali rekha koo ghoornan kaa aksh kahaa jaa sakata haye. bhoomadhya rekha key samanantar gole parr chhote vritt akshansh kii rekhaaen hain. khagoliya nikaayon seey asambandhit jyaamiti mein, bhoo-kendrit shabdawali kaa upayog kewal chitran key liye kiya jaanaa chaahiye aur iss tarah seey "nott" kiya jaanaa chaahiye, jaba taka kii galatfahmi kii koee sambhavana naa hoe.
ganitshastri ack gole koo tri-ayami euclidean sthaan mein antarnihit dou-ayami bund satah maanate hain. way ack "golaa" aur ack "gend" kaa antar karate hain, joe seema key saath ack tri-ayami manifold haye jisamen gole dwara nihit aayatan shamil haye. ack "khuli gend" gole koo hii baahar karati haye, jabaki ack "bund gend" mein golaa shamil hotaa haih ack bund gend khuli gend aur gole kaa millen haye, aur ack golaa ack (bund yaa khuli) gend kii seema haye. gend" aur "gole" key beach kaa antar hamesha banaae naheen rakhaa gayaa haye aur vishesh roop seey puraane ganitiya sandarbh ack gole key baare mein thhos key roop mein baat karate hain. samtal mein "vritt" aur "disk" key beach kaa antar samaan haye.
chhote gole yaa genden kabhi-kabhi golakar kahalaati hain.jee. martian golakar mein.
sameekaran.
vishleshanatmak jyaamiti mein, kendra aur trijya vaalaa ack kshetra sabhi binduon kaa sthaan haye jaise kii
choonki ise ack dwighaat bahupad key roop mein vyakt kiya jaa sakata haye, ack golaa ack chaturbhuj satah haye, joe ack prakaar kii beejganitiya satah haye.
vaastavik sankhyaaen honey dein aur daalen
taba sameekaran
yadi sootra 4 haye tou samadhan key roop mein koee vaastavik bindu naheen haye aur ise ack kaalpanik gole kaa sameekaran kahaa jaataa haye. yadi sootra 5 haye, tou sootra 6 kaa ekmatra samadhan bindu sootra 7 haye aur sameekaran koo ack bindu gole kaa sameekaran kahaa jaataa haye. anta mein, sootra _ 8 key mamle mein, sootra _ 6 ack gole kaa sameekaran haye jiska kendra sootra _ 10 haye aur jisaki trijya sootra _ 11 haye.
yadi uparokt sameekaran mein shoonya haye tou yeh ack samtal kaa sameekaran haye. iss prakaar, ack samtal koo anant trijya key gole key roop mein socha jaa sakata haye jiska kendra anant parr ack bindu haye.
parametric.
trijya sootra _ 12 aur kendra sootra _ 13 key saath gole key liye ack parametric sameekaran koo trikonamitiya karyon kaa upayog karke parameters kiya jaa sakata haye.
yahaan upayog kiye jaane vaale prateek golakar nirdeshank mein upayog kiye jaane vaale prateekon key samaan hain. sthir haye, jabaki 0 seey laker aur sootra _ 15 0 seey laker 2 taka bhinn hotaa haye.
guna.
sanlagn matra.
teen aayamon mein, ack gole key ander kii aayatan (yaani, ack gend kii aayatan, lekin shaastriya roop seey ack gole kii aayatan key roop mein sandarbhit) haye
jahan trijya haye aur gole kaa vyaas haye. archimedies ney pahale iss sootra koo yeh darshaate hue praapt kiya kii ack gole key ander kii aayatan usa gole key gole aur paridhiya cylinder key beach kii aayatan seey doguni haye (jisaki oonchaai aur vyaas gole key vyaas key barabar haye). yeh ack shanku koo ardh-kshetra mein ulta likh curr saabit kiya jaa sakata haye, yeh dekhate hue kii shanku key ack cross section kaa kshetra aur gole key ack cross section kaa kshetra paridhiya cylinder key cross section key kshetra key samaan haye, aur cavaliary key siddhant koo lagu kartaa haye. iss sootra koo samakalan cullon, i kaa upayog karke bhee praapt kiya jaa sakata haye.ee. disk ekikaran seey ack anant sankhya mein golakar discon kii matra kaa yog hotaa haye, joe ack saath ack saath dher kii gaee hoti hain aur-aksh key saath ack taraf seey dusare kii taraf kendrit hoti hain, yeh maanate hue kii trijya kaa kshetra mool parr kendrit haye.
kissy bhee diye gaye samay, vriddhisheel aayatan () disk key cross-sectional kshetra key utpaad aur isaki motai () key barabar hotaa haih
kull aayatan sabhi vriddhisheel aayatanon kaa yog haih
jaba seema shoonya key karib aatee haye, tou yeh sameekaran bunn jaataa haih
kissy bhee diye gaye samay, ack samkon tribhuj jodata haye, aur mool seey; isaliye, pythagorean pramey koo lagu karane seey upaj milati haih
iss pratisthaapan kaa upayog karane seey milata haye
jiska mulyankan parinam dene key liye kiya jaa sakata haye
aayatan tatva key saath golakar nirdeshank kaa upayog karke ack vaikalpik sootra paaya jaataa haye.
tou
adhikansh vyaavhaarik uddeshyon key liye, ack ghana mein utkeern gole key ander kii matra koo ghana key aayatan key 52.4% key roop mein anumaanit kiya jaa sakata haye, kyonki, jahan gole kaa vyaas haye aur ghana key ack taraf kii lambai bhee haye aur ± 0.5236. udaaharan key liye, 1 meater vyaas vaale gole mein 52.4% kinare kii lambai 1 meater key saath ack ghana kii matra haye, yaa lagbhag <ID> m3 haye.
satah kshetra.
trijya key gole kaa satah kshetra haih
archimedies ney sabse pahale iss sootra koo iss tathya seey praapt kiya kii ack paridhiya cylinder kii paarshv satah parr prakshepan kshetra-sanrakshit haye. sootra praapt karane kaa ack aur tarika iss tathya seey aataa haye kii yeh aayatan key liye sootra key vyutpann key barabar haye kyonki trijya key gole key ander kull aayatan koo ack anant sankhya key golakar gole key satah kshetra key yog key roop mein socha jaa sakata haye. aseem motai parr kissy bhee diye gaye khol key aantarik aur baahari satah kshetra key beach visangati aseem haye, aur trijya parr maulik aayatan kewal trijya aur aseem motai parr satah kshetra kaa utpaad haye.
kissy bhee trijya mein, vriddhisheel aayatan () trijya () parr satah kshetra key utpaad aur ack khol () kii motai key barabar hotaa haih
kull aayatan sabhi kosh aayatanon kaa yog haih
shoonya key karib pahunchane parr yeh sameekaran bunn jaataa haih
vikalpah
iss sameekaran key donon pakshon koo upaj key sambandh mein ack falan key roop mein alag karnaah
ise aam taur parr sankshep mein kahaa jaataa haih
jahan abb gole kii nishchit trijya maani jaati haye.
vaikalpik roop seey, gole parr kshetra tatva koo golakar nirdeshank mein diyaa jaataa haye. cartasian nirdeshank mein, kshetra tatva haye
iss prakaar kull kshetra koo ekikaran dwara praapt kiya jaa sakata haih
gole mein sabhi satahon kaa sabse chhota satah kshetra hotaa haye joe ack diye gaye aayatan koo gherta haye, aur yeh ack diye gaye satah kshetra key saath sabhi bund satahon key beach sabse badee aayatan koo gherta haye. isaliye yeh kshetra prakriti mein dikhaai detaa haih udaaharan key liye, bulbule aur paani kii chhoti boonden motey taur parr golakar hoti hain kyonki satah kaa tanaav sthaaniya roop seey satah key kshetra koo cumm kartaa haye.
gend key dravyamaan key saapeksh satah kshetra koo vishisht satah kshetra kahaa jaataa haye aur ise uparokt samikaranon seey vyakt kiya jaa sakata haye
jahan ghanatva (dravyamaan aur aayatan kaa anupaat) haye.
anya jyaamitiya guna.
ack gole kaa nirmaan satah key roop mein kiya jaa sakata haye joe ack vritt koo usake kissy bhee vyaas key baare mein ack aadhaa chakkar lagaakar ghoornan kartaa haye; yeh euclid key tatvon mein dee gaee gole kii paaramparik paribhasha key samaan haye. choonki ack vritt ack vishesh prakaar kaa deerghavritt haye, ack golaa ack vishesh prakaar kaa dirghvrittakar kranti haye. vritt koo ack deerghavritt key saath pratisthaapit karate hue, joe isake pramukh aksh key baare mein ghoomata haye, aakar ack prolet golakar bunn jaataa haye; joe kii ack chhote aksh key baare mein ghoomata haye, ack oblet golakar.
ack kshetra vishisht roop seey chaar binduon dwara nirdhaarit kiya jaataa haye joe saha-grah naheen hain. adhik aam taur parr, ack kshetra vishisht roop seey chaar sthitiyon dwara nirdhaarit kiya jaataa haye jaise kii ack bindu seey gujarna, ack tala key liye sparshrekha honah, aadi. yeh guna usa guna key anuroop haye kii teen gair-samrekhiya bindu ack tala mein ack advitiya vritt nirdhaarit karate hain.
natijatan, ack gole koo vishisht roop seey ack vritt dwara nirdhaarit kiya jaataa haye (yaani, usa vritt key tala mein naheen ack bindu key madhyam seey gujarta haye).
dou golon key samikaranon key samanya samadhanon kii jaanch karke, yeh dekhaa jaa sakata haye kii dou gole ack vritt mein pratichhed karate hain aur usa vritt vaale tala koo pratichhedan karane vaale golon kaa mool tala kahaa jaataa haye. yadyapi mool tala ack vaastavik tala haye, vritt kaalpanik hoe sakata haye (golon mein koee vaastavik bindu samaan naheen haye) yaa ack hii bindu seey banaa hoe sakata haye (goal usa bindu parr sparshrekha hain).
pratichhedan key vaastavik bindu parr dou kshetron key beach kaa konn usa bindu parr kshetron key sparshrekha vimaanon dwara nirdhaarit dwi-parichal konn haye. dou gole apane pratichhedan vritt key sabhi binduon parr ack hii konn parr pratichhed karate hain. way samkonon parr kaatate hain (orthogonal hotey hain) yadi aur kewal tabhi jaba unake kendron key beach kii doori kaa varg unaki trijya key vargon key yog key barabar hoe.
golon kii pencil.
yadi aur dou alag-alag kshetron key sameekaran hain tou
yeh maapdandon key manmaane moolyon key liye ack gole kaa sameekaran bhee haye. iss sameekaran koo santusht karane vaale sabhi kshetron key samooh koo mool dou kshetron dwara nirdhaarit golon kii pencil kahaa jaataa haye. iss paribhasha mein ack gole koo ack samtal (anant trijya, anant parr kendra) honey kii anumati haye aur yadi donon mool gole samtal hain tou pencil key sabhi gole samtal hain, anyatha pencil mein kewal ack samtal (mool tala) haye.
gole key guna.
apani pustak "jyaamiti aur kalpana" mein, david hilbert aur stephen kohan-vosen ney gole key gyaarah gunon kaa varnan kiya haye aur charcha kii haye kii kyaa yeye guna vishisht roop seey gole koo nirdhaarit karate hain. samtal key liye kaee guna hain, jinhen anant trijya vaale gole key roop mein socha jaa sakata haye. yeye guna hainh
ganit key kshetra dwara upchaar.
golakar jyaamiti.
euclidean samtal jyaamiti key mool tatva bindu aur rekhaaen hain. gole parr, binduon koo samanya arth mein paribhaashit kiya jaataa haye. rekha" kaa anuroop bhooganitiya haye, joe ack mahaan vritt haye; ack mahaan vritt kii paribhaashit visheshta yeh haye kii isake sabhi binduon vaalaa samtal bhee gole key kendra seey gujarta haye. chaap kii lambai seey maapane seey pataa chalata haye kii gole parr padey dou binduon key beach sabse chhota maarg mahaan vritt kaa chhota khand haye jisamen bindu shamil hain.
shaastriya jyaamiti key kaee pramey golakar jyaamiti key liye bhee sahee hain, lekin sabhi naheen karate hain kyonki kshetra samanantar abhidharana sahit shaastriya jyaamiti key kuchh abhidhaaranaon koo santusht karane mein whiffle rahata haye. golakar trikonamiti mein, konon koo mahaan vritton key beach paribhaashit kiya jaataa haye. golakar trikonamiti kaee mayanon mein sadharan trikonamiti seey alag hoti haye. udaaharan key liye, ack golakar trikon key aantarik konon kaa yog hamesha 180 degree seey adhik hotaa haye. isake alawa, koee bhee dou samaan golakar trikon sarvangsam hain.
gole key kendra key madhyam seey ack seedhi rekha parr sthit gole parr binduon kii koee bhee jodee (i.ee. vyaas) koo "antipodal bindu" kahaa jaataa haye-gole parr, unake beach kii doori paridhi kii lambai kaa thheek aadhaa haye. koee anya (i.ee. antipodal naheen) ack gole parr alag-alag binduon kii jodee
golakar jyaamiti dirghvrittakar jyaamiti kaa ack roop haye, joe atee-dhruviya jyaamiti key saath milkar gair-euclidean jyaamiti banaati haye.
vibhedak jyaamiti.
yeh kshetra ack chikani satah haye jisamen pratyek bindu parr sthir gossian vakrata hoti haye. goss key pramey agrazium key anusaar, yeh vakrata tri-ayami sthaan mein gole key antarnihit honey seey swatantra haye. goss seey bhee, ack gole koo donon kshetron aur konon koo banaae rakhate hue ack samtal mein mapp naheen kiya jaa sakata haye. isaliye, koee bhee maanchitra prakshepan kissy naa kissy prakaar kii vikriti kaa parichay detaa haye.
trijya key ack gole mein kshetra tatva sootra _ 36 hotaa haye. yeh sthiraank key saath golakar nirdeshank mein aayatan tatva seey paaya jaa sakata haye.
shoonya parr kendrit kissy bhee trijya kaa ack kshetra nimnalikhit vibhedak roop kaa ack abhinn satah haih
yeh sameekaran darshaata haye kii ack bindu parr sthiti sadish aur sparshrekha tala hamesha ack dusare key liye orthogonal hotey hain. isake alawa, baahari-mukhi samanya sadish usa sthiti sadish key barabar haye jise maapane key liye.
remanian jyaamiti mein, bharane key kshetra kaa anuman yeh bataata haye kii golardh remanian vritt kaa ishtatam (cumm seey cumm kshetra) samamit bharaa huaa haye.
topology.
ullekhaniya roop seey, ack sadharan gole koo sambhaavit swa-hastakshep key saath ack tri-ayami sthaan mein ander seey baahar ghumana sambhav haye, lekin binaa kissy crease key, ack prakriya mein jise gole kaa vilopan kahaa jaataa haye.
gole kaa prati-padansh bhagfal vaha satah haye jise vaastavik prakshepya tala kahaa jaataa haye, jise bhoomadhya rekha key prati-padank binduon key saath uttari golardh key roop mein bhee socha jaa sakata haye.
ack gole parr vakr.
vritt.
gole parr vritt, samtal mein vritton kii tarah, gole parr ack nishchit bindu seey ack nishchit doori parr sabhi binduon seey baney hotey hain. ack gole aur ack tala kaa pratichhedan ack vritt, ack bindu yaa khaali haye. mahaan vritt ack gole key kendra seey gujarane vaale ack samtal key saath gole kaa pratichhedan hainh anya koo chhote vritt kahaa jaataa haye.
adhik jatil satahen vritton mein ack gole koo kaat sakati hainh ack gole kaa pratichhedan ack kranti kii satah key saath jisaki dhuri mein gole kaa kendra hotaa haye ("samakshiya" hotey hain) jisamen vritt aur/yaa bindu hotey hain yadi khaali naheen hotey hain. udaaharan key liye, daaeen or kaa aarekh ack gole aur ack cylinder key pratichhedan koo darshaata haye, jisamen dou vritt hotey hain. yadi cylinder trijya gole kii hoti, tou pratichhedan ack ekal vritt hotaa. yadi cylinder trijya gole kii trijya seey badee hoti, tou pratichhedan khaali hotaa.
loxodrome.
nauparivahan mein, ack rumb line yaa loxodrome ack chaap haye joe ack hii konn parr deshantar key sabhi meridian koo paar kartaa haye. loxodrome merctor prakshepan mein seedhi rekhaaon key samaan hotey hain. ack angoothi rekha ack golakar sarpil naheen haye. kuchh saral mamalon koo chhodkar, ack rumb line kaa sootra jatil haye.
clelia vakr.
clelia vakr ack gole parr ack vakr haye jisake liye deshantar sootra _ 15 aur akshansh sootra _ 39 sameekaran koo santusht karate hain.
vishesh mamle hainh viviani kaa vakr (sootra _ 41) aur golakar sarpil (sootra _ 42) jaise kii seaft kaa sarpil. clelia vakr dhruviya kaksha mein upgrahon key maarg kaa anuman lagaata haye.
golakar shanku.
gole parr ack shanku khand kaa anuroop ack golakar shanku haye, ack chaturthak vakr jise kaee samkaksh tarikon seey paribhaashit kiya jaa sakata haye, jisamen shamil hainh
samtal shanku khandon seey sambandhit kaee pramey golakar shanku taka bhee faile hue hain.
adhik samanya satah key saath ack gole kaa pratichhedan.
yadi ack gole koo doosari satah dwara kaat diyaa jaataa haye, tou adhik jatil golakar vakr hoe sakate hain.
sameekaran sootra _ 43 key saath gole kaa pratichhedan aur sameekaran sootra _ 44 key saath cylinder kewal ack yaa dou vritt naheen haye. yeh samikaranon kii gair-raikhik pranali kaa samadhan haye.
saamaanyikaran.
dirghvrittakar.
ack dirghvrittakar ack aisaa kshetra haye jise ack yaa adhik dishaaon mein kheencha yaa sampeedit kiya gayaa haye. adhik sateek roop seey, yeh ack affine parivartan key tahat ack gole kii chhavi haye. ack dirghvrittakar kaa gole key saath vahee sambandh hotaa haye joe ack deerghavritt ack vritt key saath kartaa haye.
ayami.
golon koo kissy bhee sankhya mein aayamon key rikt sthaan key liye saamaanyikrit kiya jaa sakata haye. kissy bhee praakritik sankhya key liye, ack "-kshetra", jise aksar darshaaya jaataa haye, ()-ayami euclidean sthaan mein binduon kaa samooh haye joe usa sthaan key kendriya bindu seey ack nishchit doori parr hain, jahaan pahale kii tarah, ack sakaratmak vaastavik sankhya haye. vishesh roop seh
golon koo kabhi-kabhi atikshetra kahaa jaataa haye.
mool mein kendrit ikai trijya key-kshetra koo darshaaya jaataa haye aur aksar ise "-kshetra" key roop mein sandarbhit kiya jaataa haye. sadharan kshetra ack 2-kshetra haye, kyonki yeh ack 2-ayami satah haye joe 3-ayami sthaan mein antarnihit haye.
topology mein,-kshetra binaa seema key ack compact topological manifold kaa ack udaaharan haye. ack sthalvaigyaanik kshetra koo chikna honey kii aavashyakta naheen haye; yadi yeh chikna haye, tou ise euclidean kshetra (ack videshi kshetra) key liye bhinn roop mein honey kii aavashyakta naheen haye.
goal nirantar falan key tahat ack-bindu sett kii vyutkram chhavi haye, isaliye yeh bund haye; bhee bandha huaa haye, isaliye yeh haaine-borrell pramey dwara compact haye.
metrick sthaan.
adhik aam taur parr, ack metrick sthaan mein, kendra aur trijya kaa kshetra aise binduon kaa samooh haye.
yadi kendra ack vishisht bindu haye jise mool maanaa jaataa haye, jaisa kii ack maanak sthaan mein, tou iska ullekh paribhasha aur sanketan mein naheen kiya gayaa haye. trijya key liye bhee yahee lagu hotaa haye yadi ise ack key barabar liya jaataa haye, jaisa kii ack ikai gole key mamle mein hotaa haye.
ack gend key vipareet, ack badaa golaa bhee ack khaali sett hoe sakata haye. udaaharan key liye, euclidean metrick key saath, trijya kaa ack kshetra kewal tabhi khaali hotaa haye jaba purnank key vargon key yog key roop mein likhaa jaa sakata haye.
octahedron taxicab jyaamiti mein ack golaa haye, aur ack ghana chebishev doori kaa upayog karate hue jyaamiti mein ack golaa haye.
itihas.
gole kii jyaamiti kaa adhyayan yoonaniyon dwara kiya gayaa thaa. "euclid key tatva" pustak xi mein gole koo paribhaashit karate hain, pustak xii mein gole key vibhinn gunon parr charcha karate hain, aur yeh darshaate hain kii pustak xiii mein ack gole key bheetar paanch niyamit bahu-parichhed koo kaise likhaa jaae. euclid mein ack gole kaa kshetrafal aur aayatan shamil naheen haye, kewal ack pramey haye kii ack gole kaa aayatan isake vyaas kii teesari shakti key roop mein bhinn hotaa haye, shayed synidus key eudoxus key kaaran. aayatan aur kshetra sootra sabse pahale archimedies key "gole aur cylinder parr" mein nikas kii vidhi dwara nirdhaarit kiye gaye they. zenodorus ney sabse pahale kahaa kii kissy diye gaye satah kshetra key liye, golaa adhikatam aayatan kaa thhos haye.
archimedies ney ack gole koo una khandon mein vibhajit karane kii samasya key baare mein likhaa haye jinake khand diye gaye anupaat mein hain, lekin ise hull naheen kiya. paravalay aur atidhruviya key madhyam seey ack samadhan dionsodorus dwara diyaa gayaa thaa. ack samaan samasya-ack diye gaye khand key aayatan key barabar ack khand kaa nirmaan karane key liye, aur satah mein dusare khand mein-baad mein all-kuhi dwara hull kiya gayaa thaa.