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Show 132 i ARITHMETICO‘R'VM . Partium bxotics & c. quantitates lmenf‘umt ipfom b. St'd c; metitur ipfam aigitur per diffin.commen(Lmbilium quantitatum ,'ip£‘c a b. quantitatcs inuiccm commcnfurabilcs. QLLOd filit refiduum propofiti. COROLLARIVM. Vndc maniflfium cf! , qubd durc quantitatcs inm'ccm in» communfumbilcs non fun: adinuiam ficut numcrus aliquis ad numerum aliqucm. Ircmc'luc, quéd dua' magnitu- dincs ,qux non {untadmuicun ficur numerus quitpiam ad numcrum quampiam , fun: inuiccm incémcnfnrabilcs. chuunrur ha‘c ex prmmifl‘a a dclh‘ué‘cionc contlaxiorum. PROPOSITIO 4+". 0mm; dim magnimzfine: 1'27? commmflambilesifimt izzuivcm tommmfurabilcs. Dunc quantirntcs (1n: ab. qux fingulx {mt ipfi c.commcnfi:r1';files. Aio, qubd 3C ipincg a 17, {um ad I. I BRI SECV'NDI, PARS II. 133‘ rcliquz commcnfumbilis. 8c iPCrintcrih commenfurabiles. Hm conclufioms facilé com‘tnnr cx hnc communi fentcntia : quoniam quantims,qurc metirur parrcs,1nctirur 8c totfi. Enqua: mctitur totum & ablamm,mctitur 3k rclié‘tfi. Pnopos 1T1 o 47". Omnium duamm quantitamm fizuicem incommenfiflnbilium congerie: ('9‘ excg/fizgflmr inter [6 d" ip/Z: inuiccm incommrnfie rabiles. Etfi congcries 7/712 ear/1m fit incommenfzflabilés, crit fir reliquda imammenfurabilix. 8t ipfx inter fl? incommcnfurabiles. Nam fi (ccus 61TH, tunc, per pmccdemcm, fequcrcmr con- trarifi filppofiti.0mnin01gitur vcm fun: qug proponuntur. PROPOSITIO 4,8". Omne: dm quantitates proportionalcs duabu: quantitatibus quoquo modo commmfurabililms, [mt codm mode commen- furabile: . Et proportionale: dimbm' aliquo modoincommen- inuiCc‘m comm cnfurrhw . Nam cum .1 c. Gut commen- furalzilibu: , fimt codem modo incemmenfumbilex. Excmpli fumbilcs erunt, per pr: _ dcntcm, ficut numc‘rus ad nume- gratin, fin: quantitatcs a b. ipfis c d. quanytimtibus inter {c mm: & {imiljtcly quoninm c b. commcnflu'abilcs Hunt, 86 commenfumbilibus proportionalcsthoc 60: fit 3. ad b. {icut c. ad d. Aio,qubd a b.6runt inuicem commenfi1rabilcs.Nam {i c dfunr commcnfurabiles,erunt per 45" huius, ficut numerus ad numcmm. Igirur erit and blicut numerus ad nu merumzqunrc per fecundam partem diéh'c 43‘- :1 b. fimt inter fc Commenfumbilcs.@d G c d.fint commcnfurabilcs, aio , qubd & a b. inter {e Incommcnfilrabilcs erunt. Nam (unqpcr corollar.4 5" huius, c d. non erurlt ficut numerus {icnt humerus ad numcrum. Sunmntui igfim‘ pm- quarmm c/mui Eucl, (res numeri d cf: Continunnus dms mtioncs ('i‘xicct, Vt Fcut eff 3. ad cfic fir numerus «Lad numcrum c. :3. [cut c. ad b. fic 3r, nmum‘us U. ad nnmcrum F. & tunc ex «3: H111cricflcutHummus d. ::d numcrum f. ficquantitas a. quintimtcm b. IngI‘prr{:cundam pnrum pgmccdmtis , quantitas (} 1). fun: ad inuium Ccnnncuiurabxles. V quod cfi propohtum. _ Pnoposr'er45". Efl alt01mm (11442 qlmmimtquunrum mm commenfmabifis adinmtcm mi mtiwwliqua verb eidcm incommc‘fiflabilis, fmzt ab.V111 gratiam‘lagnitudinum imcmmt'nfurtbiles . Excmpli hing?" Verb reliqua c. ipfi commcnfumbilis fit fcilicet a. b. muiccm 1n- mcnfumbilis cidem C. Aio tunc, quéd ipfce a commenfil- b. :1 cum: cnim Secus fimt. contmenfixrabilcs per prqhyp.c6mcnfin‘abilc‘s.igitur ope: a ipfa: {ed rabilcs: miflhm cum! I) c. inuiccm commenfiu‘ablles: quod aft {upcommcn u‘ poiixo conn‘arium. N6 igitur fun: a b. inuiccm rabilcs. ergo incommenfurabilcs. quod CR propofitum.‘ ‘Pnoposx'rxo 4.6". ‘ Omnium duarum qzmntimmm inuicem incommcnfurabilmm tongu‘écs 67* exctflhxfuntintt'rfe (y ipfix inmcem incommcnfurabilcs. Etfl congcrics 'vni carumfit incommcnfirabflis, erit (3' ‘ reliqua ad numerum , & idco ncquc a. ad b. erit hcu: numcrus ad numerum:& perinde pct {ccundam partcm diai corollarij a b. tuncincémcnfurabilcs inrcr {c crunt ficut proponitur. Item {1 c d. Pomntur nut potentia tantfiln , nut cubo tantfi, aut quadraro {ccundo tann‘xm commenfixrabilcs : codem pcnitus modo & ipfic ab. commenfurabilcs erunt. Si nurem cd. aliquo diétorum modorum ponanturincomxpcnfurabilcszeodem fimilitcr modo & ipi}; a b.1'ncommenlum- bilcs crunt: Canoniam {cilicct quantitatum proportiona& quim lium proportionalcs fun: ti qgadram, quim cubi, fecunda qmdram.Et idcirco (cquitut corum commcnfiu-a- bilims , v21 incommcnfurabilitas : quippc qua: comimtu: Proportioncm, adduéta 4 1' 8: c'ius corollario. 49'. PROPOSITIO Own}? quantim rutzonnl'ix multiplicmz: aliqrmm quantita.a‘gv Lomtcm, proauu; .I'M.;ntémtcm multiplimte wgnunfinmz , Ec 5 :".‘.*."ii_;';u‘.;bi3;7'11. |
