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Colored diff for /texts/archimedes/xml/Attic/monte_aeque_01_la_1588.xml between version 1.5 and 1.6

version 1.5, 2003/03/28 20:43:41 version 1.6, 2003/06/26 15:02:39
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 <s>Ob&longs;eruandum autem occurrit in demon&longs;trationibus, ab <lb/>Archimede allatis; qu&ograve;d in prima demon&longs;tratione &longs;upponit <lb/>Archimedes, HFGI e&longs;&longs;e parallelogrammum. qu&ograve;d vt &longs;it pa&shy;<lb/>rallelogrammum, nece&longs;&longs;e e&longs;t &longs;upponere centra grauitatis HI <lb/>&longs;ecare lineas KF LG in partes inuicem proportionales. quod <lb/>tamen &longs;upponi po&longs;&longs;e minim&egrave; videtur. Et &longs;i quis ex quinto <lb/>po&longs;tulato obijceret, centragrauitatis in &aelig;qualibus, &longs;imilibu&longs;&shy;<lb/>qu&egrave; figuris e&longs;&longs;e &aelig;qualiter po&longs;ita; admitti quidem pote&longs;t; quo- <s>Ob&longs;eruandum autem occurrit in demon&longs;trationibus, ab <lb/>Archimede allatis; qu&ograve;d in prima demon&longs;tratione &longs;upponit <lb/>Archimedes, HFGI e&longs;&longs;e parallelogrammum. qu&ograve;d vt &longs;it pa&shy;<lb/>rallelogrammum, nece&longs;&longs;e e&longs;t &longs;upponere centra grauitatis HI <lb/>&longs;ecare lineas KF LG in partes inuicem proportionales. quod <lb/>tamen &longs;upponi po&longs;&longs;e minim&egrave; videtur. Et &longs;i quis ex quinto <lb/>po&longs;tulato obijceret, centragrauitatis in &aelig;qualibus, &longs;imilibu&longs;&shy;<lb/>qu&egrave; figuris e&longs;&longs;e &aelig;qualiter po&longs;ita; admitti quidem pote&longs;t; quo-
 <pb pagenum="161"/>niam figur&aelig;, ipforum qu&egrave; centra inter&longs;e coaptari po&longs;&longs;unt. vt <lb/>omnibus figuris rectilineis &ecedil;qualibus, &amp; &longs;imilib^{9} accidere po&shy;<lb/>te&longs;t. Hoc tam&eacute; contingere po&longs;&longs;e in parabolis, vt AKB BLC, vi <lb/>detur in <expan abbr="c&otilde;ueni&eacute;s">conueni&eacute;s</expan>. <expan abbr="N&atilde;">Nam</expan> quamuis AKB BLC &longs;int &aelig;quales, &amp; &longs;int <lb/><expan abbr="eti&atilde;">etiam</expan> &longs;imiles; non &longs;unt tamen &longs;imiles ea &longs;i militudine, vt &longs;untre <lb/>ctiline&aelig; figur&aelig;; vtantea diximus. Quod etiam <expan abbr="per&longs;picu&utilde;">per&longs;picuum</expan> fit ex <lb/>hoc, quia non &longs;emper coaptari porei&longs;t portio AKB <expan abbr="c&utilde;">cum</expan> portio&shy;<lb/>ne BLC. <expan abbr="n&otilde;">non</expan>. n.&longs;emper recta linea BC erit &aelig;qualisip&longs;i BA; <expan abbr="ne&qacute;">neque</expan>; <lb/>&longs;ectionis linea BLC &longs;ectionis line&ecedil; BKA &ecedil;qualis exi&longs;tet. <expan abbr="C&utilde;">Cum</expan> <expan abbr="n&otilde;">non</expan> <lb/>&longs;emper AC, &amp; qu&aelig; &longs;untip&longs;i AC &aelig;quidi&longs;tates ad rectos &longs;int an <lb/>gulos diametro BD. &longs;i.n. &ecedil;quidi&longs;tantes line&ecedil; diametro fuerint <lb/>perpendiculares, tunc AB BC inter&longs;e &ecedil;quales e&longs;&longs;ent; <expan abbr="portio&qacute;">portioque</expan>; <lb/>AKB <expan abbr="c&utilde;">cum</expan> portione BLC coaptari po&longs;&longs;et: &longs;ec&ugrave;s autem minim&egrave;. <lb/>Quare centra grauiratis HI lineas KFLG in eadem proportio <lb/>ne &longs;ecare minim&egrave;&longs;upponi po&longs;&longs;e videtur; t&ugrave;m exijs, qu&aelig; dicta <lb/>&longs;unt; t&uacute; quia hoc o&longs;tendet Archimedes in &longs;eptima propo&longs;itio <lb/>ne. qu&ograve;d &longs;i adhuc non e&longs;t <expan abbr="dem&otilde;&longs;trat&uacute;">demon&longs;trat&uacute;</expan>, <expan abbr="n&otilde;">non</expan> pote&longs;t <expan abbr="quo&qacute;">quoque</expan>; &longs;uppo <lb/>ni; pr&aelig;&longs;ertim c&ugrave;m &longs;it demon&longs;trabile. ac propterea <expan abbr="dem&otilde;&longs;tra-tio">demon&longs;tra&shy;<lb/>tio</expan> nullam videturvim haberead <expan abbr="o&longs;tendend&utilde;">o&longs;tendendum</expan>, quod propo&longs;i&shy;<lb/>t&uacute; fuit. Huic <expan abbr="tam&etilde;">tamen</expan> occurri po&longs;&longs;evidetur <expan abbr="c&utilde;">cum</expan> Eutocio in exphca <lb/>tione huiusloci dicendo, hoc &longs;upponere Archimed&eacute;, quia por <lb/>tiones AKBBLC &longs;unt&ecedil;quales, quar&uacute; diametri KFLG &longs;unt &ecedil;&shy;<lb/>quales, &amp; <expan abbr="&ecedil;quidi&longs;t&atilde;tes">&ecedil;quidi&longs;tantes</expan>, qu&aelig; &longs;imiliter diuiduntur &agrave; punctis HI; <lb/>vnde erit kG ad HF, vt LI ad IG. ex quibus colligit HF ip&longs;i IG <lb/><expan abbr="&aelig;qual&etilde;">&aelig;qualem</expan> e&longs;&longs;e; ac propterea HG <expan abbr="parallelogr&atilde;m&utilde;">parallelogrammum</expan> exiltere. Qu&aelig; <expan abbr="t&ntilde;">tnm</expan> <lb/>re&longs;pon&longs;io <expan abbr="n&otilde;">non</expan> e&longs;t Eutocio digna. c&ugrave;m ex dictis <expan abbr="n&otilde;">non</expan> &longs;it omnin&ograve; <lb/>demon&longs;tratiua, vtres mathematic&ecedil; <expan abbr="requir&utilde;t">requirunt</expan>; quapropter omit <lb/>tenda e&longs;t.hac.n.ratione&longs;upponitur centra HI lineas KFLG in <lb/>eadem proportione &longs;ecare.quod nullo modo &longs;upponi pote&longs;t. <lb/>Quare dici poterit, &amp; forta&longs;le recti&ugrave;s, qu&ograve;d vis demon&longs;tratio&shy;<lb/>nis vidctur in hoc e&longs;&longs;e con&longs;tituta, vt &longs;upponatur puncta HI <expan abbr="v-bicun&qacute;">v&shy;<lb/>bicunque</expan>; e&longs;&longs;e po&longs;&longs;e in lineis KFLG; ita vt &longs;iue ducta HI fuerit, <lb/>&longs;iue etiam non fuerit ip&longs;i FG &aelig;quidi&longs;tans, demon&longs;tratio <expan abbr="tam&etilde;">tamen</expan> <lb/>&longs;uam &longs;emper habebit vim, <expan abbr="id&etilde;&qacute;">idenque</expan>; concludet. Nam ex <expan abbr="pr&aelig;ced&etilde;">pr&aelig;cedem</expan>. <lb/>ti patet centra grauitatis portionum AKB BLC e&longs;&longs;e in lineis <lb/>KF LG; hoce&longs;t inter puncta KF, &amp; LG. <expan abbr="&longs;uppon&atilde;turita&qacute;">&longs;upponanturitaque</expan>; <expan abbr="c&etilde;-tra">cen&shy;<lb/>tra</expan> grauitatis <expan abbr="portion&utilde;">portionum</expan> AKB BLC e&longs;&longs;e puncta HI <expan abbr="vbic&utilde;&qacute;">vbicunque</expan>; po&shy; <pb pagenum="161"/>niam figur&aelig;, ipforum qu&egrave; centra inter&longs;e coaptari po&longs;&longs;unt. vt <lb/>omnibus figuris rectilineis &ecedil;qualibus, &amp; &longs;imilib^{9} accidere po&shy;<lb/>te&longs;t. Hoc tam&eacute; contingere po&longs;&longs;e in parabolis, vt AKB BLC, vi <lb/>detur in <expan abbr="c&otilde;ueni&eacute;s">conueni&eacute;s</expan>. <expan abbr="N&atilde;">Nam</expan> quamuis AKB BLC &longs;int &aelig;quales, &amp; &longs;int <lb/><expan abbr="eti&atilde;">etiam</expan> &longs;imiles; non &longs;unt tamen &longs;imiles ea &longs;i militudine, vt &longs;untre <lb/>ctiline&aelig; figur&aelig;; vtantea diximus. Quod etiam <expan abbr="per&longs;picu&utilde;">per&longs;picuum</expan> fit ex <lb/>hoc, quia non &longs;emper coaptari porei&longs;t portio AKB <expan abbr="c&utilde;">cum</expan> portio&shy;<lb/>ne BLC. <expan abbr="n&otilde;">non</expan>. n.&longs;emper recta linea BC erit &aelig;qualisip&longs;i BA; <expan abbr="ne&qacute;">neque</expan>; <lb/>&longs;ectionis linea BLC &longs;ectionis line&ecedil; BKA &ecedil;qualis exi&longs;tet. <expan abbr="C&utilde;">Cum</expan> <expan abbr="n&otilde;">non</expan> <lb/>&longs;emper AC, &amp; qu&aelig; &longs;untip&longs;i AC &aelig;quidi&longs;tates ad rectos &longs;int an <lb/>gulos diametro BD. &longs;i.n. &ecedil;quidi&longs;tantes line&ecedil; diametro fuerint <lb/>perpendiculares, tunc AB BC inter&longs;e &ecedil;quales e&longs;&longs;ent; <expan abbr="portio&qacute;">portioque</expan>; <lb/>AKB <expan abbr="c&utilde;">cum</expan> portione BLC coaptari po&longs;&longs;et: &longs;ec&ugrave;s autem minim&egrave;. <lb/>Quare centra grauiratis HI lineas KFLG in eadem proportio <lb/>ne &longs;ecare minim&egrave;&longs;upponi po&longs;&longs;e videtur; t&ugrave;m exijs, qu&aelig; dicta <lb/>&longs;unt; t&uacute; quia hoc o&longs;tendet Archimedes in &longs;eptima propo&longs;itio <lb/>ne. qu&ograve;d &longs;i adhuc non e&longs;t <expan abbr="dem&otilde;&longs;trat&uacute;">demon&longs;trat&uacute;</expan>, <expan abbr="n&otilde;">non</expan> pote&longs;t <expan abbr="quo&qacute;">quoque</expan>; &longs;uppo <lb/>ni; pr&aelig;&longs;ertim c&ugrave;m &longs;it demon&longs;trabile. ac propterea <expan abbr="dem&otilde;&longs;tra-tio">demon&longs;tra&shy;<lb/>tio</expan> nullam videturvim haberead <expan abbr="o&longs;tendend&utilde;">o&longs;tendendum</expan>, quod propo&longs;i&shy;<lb/>t&uacute; fuit. Huic <expan abbr="tam&etilde;">tamen</expan> occurri po&longs;&longs;evidetur <expan abbr="c&utilde;">cum</expan> Eutocio in exphca <lb/>tione huiusloci dicendo, hoc &longs;upponere Archimed&eacute;, quia por <lb/>tiones AKBBLC &longs;unt&ecedil;quales, quar&uacute; diametri KFLG &longs;unt &ecedil;&shy;<lb/>quales, &amp; <expan abbr="&ecedil;quidi&longs;t&atilde;tes">&ecedil;quidi&longs;tantes</expan>, qu&aelig; &longs;imiliter diuiduntur &agrave; punctis HI; <lb/>vnde erit kG ad HF, vt LI ad IG. ex quibus colligit HF ip&longs;i IG <lb/><expan abbr="&aelig;qual&etilde;">&aelig;qualem</expan> e&longs;&longs;e; ac propterea HG <expan abbr="parallelogr&atilde;m&utilde;">parallelogrammum</expan> exiltere. Qu&aelig; <expan abbr="t&ntilde;">tnm</expan> <lb/>re&longs;pon&longs;io <expan abbr="n&otilde;">non</expan> e&longs;t Eutocio digna. c&ugrave;m ex dictis <expan abbr="n&otilde;">non</expan> &longs;it omnin&ograve; <lb/>demon&longs;tratiua, vtres mathematic&ecedil; <expan abbr="requir&utilde;t">requirunt</expan>; quapropter omit <lb/>tenda e&longs;t.hac.n.ratione&longs;upponitur centra HI lineas KFLG in <lb/>eadem proportione &longs;ecare.quod nullo modo &longs;upponi pote&longs;t. <lb/>Quare dici poterit, &amp; forta&longs;le recti&ugrave;s, qu&ograve;d vis demon&longs;tratio&shy;<lb/>nis videtur in hoc e&longs;&longs;e con&longs;tituta, vt &longs;upponatur puncta HI <expan abbr="v-bicun&qacute;">v&shy;<lb/>bicunque</expan>; e&longs;&longs;e po&longs;&longs;e in lineis KFLG; ita vt &longs;iue ducta HI fuerit, <lb/>&longs;iue etiam non fuerit ip&longs;i FG &aelig;quidi&longs;tans, demon&longs;tratio <expan abbr="tam&etilde;">tamen</expan> <lb/>&longs;uam &longs;emper habebit vim, <expan abbr="id&etilde;&qacute;">idenque</expan>; concludet. Nam ex <expan abbr="pr&aelig;ced&etilde;">pr&aelig;cedem</expan>. <lb/>ti patet centra grauitatis portionum AKB BLC e&longs;&longs;e in lineis <lb/>KF LG; hoce&longs;t inter puncta KF, &amp; LG. <expan abbr="&longs;uppon&atilde;turita&qacute;">&longs;upponanturitaque</expan>; <expan abbr="c&etilde;-tra">cen&shy;<lb/>tra</expan> grauitatis <expan abbr="portion&utilde;">portionum</expan> AKB BLC e&longs;&longs;e puncta HI <expan abbr="vbic&utilde;&qacute;">vbicunque</expan>; po&shy;
 <pb pagenum="162"/>&longs;ita, <expan abbr="d&utilde;modo">dummodo</expan> &longs;int in lineis KF LG, veluti Archimedes ip&longs;e in <lb/>d mon&longs;tratione &longs;upponit. <expan abbr="Ducatur&qacute;">Ducaturque</expan>; HI; qu&aelig; vel ip&longs;i FG &aelig;&shy;<lb/>quidi&longs;tans erit, vel min&ugrave;s: &longs;i e&longs;t &aelig;quidi&longs;tans, <expan abbr="parallelogr&atilde;m&utilde;">parallelogrammum</expan> <lb/>e&longs;t HFGI, &amp; vera e&longs;t demon&longs;tratio Archimedis. &longs;i ver&ograve; <expan abbr="n&otilde;">non</expan> e&longs;t <lb/><expan abbr="&aelig;quidi&longs;t&atilde;s">&aelig;quidi&longs;tans</expan>, nihilominus veri&longs;&longs;ima e&longs;t eadem <expan abbr="dem&otilde;&longs;tratio">demon&longs;tratio</expan>. <expan abbr="N&atilde;">Nam</expan> <lb/>&longs;i HI ip&longs;i FG <expan abbr="n&otilde;">non</expan> e&longs;t <expan abbr="&ecedil;quidi&longs;t&atilde;s">&ecedil;quidi&longs;tans</expan>, patet in primis <expan abbr="p&utilde;ct&utilde;">punctum</expan> Qpropin <lb/>quius e&longs;&longs;e vertici B portionis ABC, <expan abbr="qu&atilde;">quam</expan> <expan abbr="punct&utilde;">punctum</expan> N, ac per con&shy;<lb/>&longs;equens, <expan abbr="qu&atilde;">quam</expan> punctum E centrum grauitatis trianguli ABC. <lb/>Etquoniam line&aelig; HI FG &agrave; lineis diuiduntur KF BN LG &ecedil; <lb/> <pb pagenum="162"/>&longs;ita, <expan abbr="d&utilde;modo">dummodo</expan> &longs;int in lineis KF LG, veluti Archimedes ip&longs;e in <lb/>d mon&longs;tratione &longs;upponit. <expan abbr="Ducatur&qacute;">Ducaturque</expan>; HI; qu&aelig; vel ip&longs;i FG &aelig;&shy;<lb/>quidi&longs;tans erit, vel min&ugrave;s: &longs;i e&longs;t &aelig;quidi&longs;tans, <expan abbr="parallelogr&atilde;m&utilde;">parallelogrammum</expan> <lb/>e&longs;t HFGI, &amp; vera e&longs;t demon&longs;tratio Archimedis. &longs;i ver&ograve; <expan abbr="n&otilde;">non</expan> e&longs;t <lb/><expan abbr="&aelig;quidi&longs;t&atilde;s">&aelig;quidi&longs;tans</expan>, nihilominus veri&longs;&longs;ima e&longs;t eadem <expan abbr="dem&otilde;&longs;tratio">demon&longs;tratio</expan>. <expan abbr="N&atilde;">Nam</expan> <lb/>&longs;i HI ip&longs;i FG <expan abbr="n&otilde;">non</expan> e&longs;t <expan abbr="&ecedil;quidi&longs;t&atilde;s">&ecedil;quidi&longs;tans</expan>, patet in primis <expan abbr="p&utilde;ct&utilde;">punctum</expan> Qpropin <lb/>quius e&longs;&longs;e vertici B portionis ABC, <expan abbr="qu&atilde;">quam</expan> <expan abbr="punct&utilde;">punctum</expan> N, ac per con&shy;<lb/>&longs;equens, <expan abbr="qu&atilde;">quam</expan> punctum E centrum grauitatis trianguli ABC. <lb/>Etquoniam line&aelig; HI FG &agrave; lineis diuiduntur KF BN LG &ecedil; <lb/>
 <arrow.to.target n="fig73"></arrow.to.target><lb/> <arrow.to.target n="fig73"></arrow.to.target><lb/>
 <arrow.to.target n="marg290"></arrow.to.target> quidi&longs;tantibus, erit HQ ad QI, vt FN ad NG. e&longs;t autem FN i&shy;<lb/>pGNG &ecedil;qualis, ergo HQ ip&longs;i QI &ecedil;qualis quoque erit. itaque <lb/>quoniam portiones AKBBLC &longs;unt &aelig;quales, erit magnitudi&shy;<lb/>nis ex vtri&longs;que AKB BLC portionibus compo&longs;it&ecedil; <expan abbr="centr&utilde;">centrum</expan> gra&shy;<lb/>uitatis in medio line&ecedil; HI. ergo eritpunctum <expan abbr="q.">que</expan> quo cognito <lb/>eadem demon&longs;tratio Archimedis o&longs;tendet centrum grauita&shy;<lb/>tis portionis ABC e&longs;&longs;e inter puncta <expan abbr="Eq.">Eque</expan> Nam ex verbis ip&longs;ius, <lb/>c&ugrave;m ait, <emph type="italics"/>Quoniam autem trianguli ABC centrum grauitatis est <lb/>punctum E magnitudinis ver&ograve; ex vtri&longs;que AkB BLC compo&longs;ic&aelig; <lb/>est punctum <expan abbr="q;">que</expan> constat totius portionis ABC centrum grauitatis <lb/>e&longs;&longs;e in in linea QE. hoc est inter puncta QE. Quare totius portionis <lb/>centrum grauitatis propinquius e&longs;t vertici portionis, qu&agrave;m trian&shy;<lb/>guli plan&egrave; in&longs;cripti.<emph.end type="italics"/> <expan abbr="manife&longs;t&utilde;">manife&longs;tum</expan> e&longs;t igitur centrum grauitatis por <lb/>tionis ABC, &longs;iu&egrave; &longs;it HI ip&longs;i FG &aelig;quidi&longs;tans, &longs;iue non &aelig;. <lb/>quidi&longs;tans, propinquius e&longs;&longs;e vertici B portionis, qu&agrave;m <expan abbr="c&etilde;trum">centrum</expan>  <arrow.to.target n="marg290"></arrow.to.target> quidi&longs;tantibus, erit HQ ad QI, vt FN ad NG. e&longs;t autem FN i&shy;<lb/>pGNG &ecedil;qualis, ergo HQ ip&longs;i QI &ecedil;qualis quoque erit. itaque <lb/>quoniam portiones AKBBLC &longs;unt &aelig;quales, erit magnitudi&shy;<lb/>nis ex vtri&longs;que AKB BLC portionibus compo&longs;it&ecedil; <expan abbr="centr&utilde;">centrum</expan> gra&shy;<lb/>uitatis in medio line&ecedil; HI. ergo eritpunctum <expan abbr="q.">que</expan> quo cognito <lb/>eadem demon&longs;tratio Archimedis o&longs;tendet centrum grauita&shy;<lb/>tis portionis ABC e&longs;&longs;e inter puncta <expan abbr="Eq.">Eque</expan> Nam ex verbis ip&longs;ius, <lb/>c&ugrave;m ait, <emph type="italics"/>Quoniam autem trianguli ABC centrum grauitatis est <lb/>punctum E magnitudinis ver&ograve; ex vtri&longs;que AkB BLC compo&longs;ic&aelig; <lb/>est punctum <expan abbr="q;">que</expan> constat totius portionis ABC centrum grauitatis <lb/>e&longs;&longs;e in in linea QE. hoc est inter puncta QE. Quare totius portionis <lb/>centrum grauitatis propinquius e&longs;t vertici portionis, qu&agrave;m trian&shy;<lb/>guli plan&egrave; in&longs;cripti.<emph.end type="italics"/> <expan abbr="manife&longs;t&utilde;">manife&longs;tum</expan> e&longs;t igitur centrum grauitatis por <lb/>tionis ABC, &longs;iu&egrave; &longs;it HI ip&longs;i FG &aelig;quidi&longs;tans, &longs;iue non &aelig;. <lb/>quidi&longs;tans, propinquius e&longs;&longs;e vertici B portionis, qu&agrave;m <expan abbr="c&etilde;trum">centrum</expan>

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