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Show 152 ARITHMETICORVM Pnoposxrxo 78*. Omni: rdtionalz's quaiztftas dim'fz in binomium,exbibet in qua firms refid/mm, wins nomimt commczqfiu‘abilzkz [mtg/3" propor. tiomlia if/fius binomij nominibm . Exempli graria , rationalis quantitas c. diuidatur per abinomium, & proucniatb. LIBRI SECVNDY; PARS II. 153 gefimnmquarmm huius , Imbalzun mcmbra inuiccm pro- orriomlia & comm enthmbiiiz , {Cuzic‘cr n b. ipfi d c. atquc EC. ipfi e F. quod cfi {-mpoiitum. Similircr in cgturis bino- nomiis, & comm radicibns confhbit id, quéd demon- flrandum proponitur. Aio, quéd b. refiduum cl}, cuius nomina Commenfumbin lia fun: & proportionalia ipfius a. binomij nominibus. Pxoposx'rxo 81". Nam cum diuifbr a. in quotientcm Improducat diuifam C. fitéue binomium ,& c. rationalis :iam , per 76"1 prmCe- . Refiduaquorum radéces babcnr inuicem proportz'omiz‘a (9v c5. dcntem,b. réfiduum erit nominum cgmmenfilrabilium 8c propoutionalium ipfius mbinomij nominibus. quod cl} mcnfumlflzia nomina,fc rtizmtm' diam proportionalia intcrfl: ct commcrgfumbilia nomim . (Luod in pmecedcnti dc binomijs corumélue radicibus , qurc funt bimsmbrcs quantimtcs, Propolltum. Pnoposx'rxo 79*. Omni: rational}: quantitas din'ifa in rcfiduum, alike: in 9140 tiente binomihm , cums nomimz irromwm urabiliafimt @« prqportéonalia ipfius wfiduinomz'nibus . Sicurprreccdcns per 76" Pratmiffamfita prxfcns 13::1'77i demonfh‘atur. ofienfum fuir -. hic fimilltcr pcnitus demonftrabitur derofiduis, eorumciue radicibus , qua: {uat Rciiduales quantirares. (Ligndoquidsm eadcm film: Rcfidualium , qua: Bi" membrium nomina; qua: coniunétakimcmbrcs; ablata V:- to minusz‘i maiori rcfidualcs quantitatcs faciunt. Pnovosxrxo Pnovosx'rxo Enomia , qtmrI-m radica babcnt fnuz‘ccm proporfiomlia av commenfurabz'féa 71077: '21.: , fortiunmr quoque proportionm'z'a in. » i. b c d c f fer/:52?~ confmcnfilfrabzz'm nommd. Sin t,excmpli gratiafi b c. ‘ "m d e hbmomm term, quorum major: membm :1 b.d amino. g h k -------... 82'. 803. I m n r>a_ verb bc_. (if. dcindc fumanturhornm binomiorum ra__ dlccsfitq; lpfig a b c. mchx g1) k &ipfius d r.- f. radix I m n. {61‘ 57*huius crunu'lue per candtm ghk.1m11. bimedia- i9. fecunéa: Sim crgo talium him cdinlium meu‘hmmmio- ra quidem {:1}. 1m. xvi-mm VH6 11 k. m {1- Et {upponarur vtgh. ipfi l m. Atquc 2' Yr. ipfi m n. comps mum, proportionalia fin: Sc com111691111rfi3i11'a. Dico hiuc, qubd 8: binomiorum :1 b 5. def: ipfiun mcmbrum a b.1'pfi d c. atquc ipfum be. ipfi cf. comparamm , prooortionalia {um 86 commmfurabilia; quod {1c oz'tcnditur.‘ (,Luoniam quantitas g k. l n. habem mcmbm commenfiltabxlia , x‘k 'proportlona‘xia , maius maiori, &mim‘~.s minorimrunr coniuné‘tim 8: totum tori proportionalia , 33: per quadmgcfimam 0&2!- uam huius, commcnfumbilia. Igicur, per quitqquasrcflmam fecundam huius , ipfbrum g h. l n. quadrata , {ciliéer , a c. dfi e§untficut numerus qmdratus ad numcrum quadraFum mtcr {e , 8c idea commcnfumbflia : 8; idcirco pct {exau gcfimam Omnis irrationx/z's bémembri: qmntita: multiplicam raj/z" m if? c0771dim/cm r/uantftatem corzmdcm,fiuc proportionaliu menfitrabilium nominmn , producit quantimrcm potentm rangli,n.bl- mlem, ('7 quzndoque rationale-41.1.Suntogmtia FXCI11P dum comedialc {ccundum : & I). refiduum medialc iccun m inni- rundcm,;iue proportionalium 8:comiucnfiu‘ablliu flb.& Promipfum ipia autcm icct mnlripl omm: mcmbr Cc‘m guanducat c, Aio, quéd C. cl} porcminh'teu littioualis , Flue us a. Slrxph purer. 11c 039d lis. rariona irer doquc fimplic 5831111 pm: n {31:u c. qmdmtum (1.6: ipfius b.quadmtum um tcrnum pct i115,a'. binomium tcrtium. Atquc e. rchdu b. habcm 615 fintzergo ex (Lin 6. quantitas f. Et quonigm- a m not mmcc per hyp. proportionalia & commcnfurablha m,tcm & antePrgced per c. tgd quadra comm ism 2 mina nfu- lm & co‘mmc pmmiifam habcbunr inter ic proporuona brcm, per 743, vcl7y‘hu1us,‘ d. [)1- l‘abilia naming. (Eamo quannmticm nomium multiplicans c. binomium,pr0duc1t put for-0:19.9th ids? cfl,& lis mtiona rurf mtionalemlgi g1}. nmhtcr tauonghs rium I 1°: huius,cft radix iplius fpotc & c.m.1gmmdmc Et (1 f filcrit quadrams numcrus , tunc blmxhtcfrxchpum. mnd onfl d fuir dcm mtionahs cm. quo ‘ ' um |