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Colored diff for /texts/archimedes/xml/Attic/comma_centr_01_la_1565.xml between version 1.11 and 1.16

version 1.11, 2002/08/18 16:13:04 version 1.16, 2002/08/18 16:43:32
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 <s>quadri&shy;<lb/>lateri enim a b c d cen&shy;<lb/>trum e&longs;t k, ex decima e&shy;<lb/>iu&longs;dem libri Archime&shy;<lb/>dis, quippe <expan abbr="c&utilde;">cum</expan> in eo om <lb/>nes diametri <expan abbr="c&otilde;ueni&atilde;t">conueniant</expan>. </s><lb/> <s>quadri&shy;<lb/>lateri enim a b c d cen&shy;<lb/>trum e&longs;t k, ex decima e&shy;<lb/>iu&longs;dem libri Archime&shy;<lb/>dis, quippe <expan abbr="c&utilde;">cum</expan> in eo om <lb/>nes diametri <expan abbr="c&otilde;ueni&atilde;t">conueniant</expan>. </s><lb/>
  
 <s>Sed in figura a l b m c n <lb/><arrow.to.target n="marg19"></arrow.to.target><lb/>do, quoniam trianguli <lb/>a l b centrum grauitatis <lb/><arrow.to.target n="marg20"></arrow.to.target><lb/>e&longs;t in linea l e: <expan abbr="trapezij&qacute;">trapezijque</expan>; a b m o centrum in linea e k: trape <lb/>zij o m c d in k g: &amp; trianguli c n d in ip&longs;a g n: erit magnitu <lb/>dinis ex his omnibus con&longs;tantis, uidelicet totius figur&aelig; cen <lb/>trum grauitatis in linea l n: &amp; o b candem cau&longs;&longs;am in linea <lb/>o m. </s> <s>Sed in figura a l b m c n <lb/><arrow.to.target n="marg19"></arrow.to.target><lb/>do, quoniam trianguli <lb/>a l b centrum grauitatis <lb/><arrow.to.target n="marg20"></arrow.to.target><lb/>e&longs;t in linea l e: <expan abbr="trapezij&qacute;">trapezijque</expan>; a b m o centrum in linea e k: trape <lb/>zij o m c d in k g: &amp; trianguli c n d in ip&longs;a g n: erit magnitu <lb/>dinis ex his omnibus con&longs;tantis, uidelicet totius figur&aelig; cen <lb/>trum grauitatis in linea l n: &amp; o b eandem cau&longs;&longs;am in linea <lb/>o m. </s>
  
 <s>e&longs;t enim trianguli a o d centrum in linea o h: trapezij <lb/>a l n d in h k: trapezij l b c n in k f: &amp; trianguli b m c in fm. </s><lb/> <s>e&longs;t enim trianguli a o d centrum in linea o h: trapezij <lb/>a l n d in h k: trapezij l b c n in k f: &amp; trianguli b m c in fm. </s><lb/>
  
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 <s>erit igitur a centrum grauitatis <lb/>ip&longs;ius figur&aelig;, quod proxime <expan abbr="o&longs;t&etilde;">o&longs;tem</expan> <lb/>dimus. </s> <s>erit igitur a centrum grauitatis <lb/>ip&longs;ius figur&aelig;, quod proxime <expan abbr="o&longs;t&etilde;">o&longs;tem</expan> <lb/>dimus. </s>
  
 <s>Itaque quoniam circulus <lb/>a ad circulum d, uel ellip&longs;is a ad <lb/>ellip&longs;im d candem <expan abbr="proportion&etilde;">proportionem</expan> <lb/>habet, quam linea c a ad a b: <lb/>portiones uero &longs;unt minores cir <lb/><arrow.to.target n="marg21"></arrow.to.target><lb/>culo uel ellip&longs;i d: habebit circu&shy;<lb/>lus, uel ellip&longs;is ad portiones ma&shy;<lb/>iorem proportionem, qu&agrave;m c a <lb/><arrow.to.target n="marg22"></arrow.to.target><lb/>ad a b: &amp; diuidendo figura recti&shy;<lb/>linea e f g h k l m n ad portiones <pb pagenum="6"/><figure id="fig9"></figure><lb/>habebit maiorem <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>quam c b ad b a. </s> <s>Itaque quoniam circulus <lb/>a ad circulum d, uel ellip&longs;is a ad <lb/>ellip&longs;im d eandem <expan abbr="proportion&etilde;">proportionem</expan> <lb/>habet, quam linea c a ad a b: <lb/>portiones uero &longs;unt minores cir <lb/><arrow.to.target n="marg21"></arrow.to.target><lb/>culo uel ellip&longs;i d: habebit circu&shy;<lb/>lus, uel ellip&longs;is ad portiones ma&shy;<lb/>iorem proportionem, qu&agrave;m c a <lb/><arrow.to.target n="marg22"></arrow.to.target><lb/>ad a b: &amp; diuidendo figura recti&shy;<lb/>linea e f g h k l m n ad portiones <pb pagenum="6"/><figure id="fig9"></figure><lb/>habebit maiorem <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>quam c b ad b a. </s>
  
 <s>fiat o b ad b a, <lb/>ut figura rectilinea ad portio&shy;<lb/>nes. </s> <s>fiat o b ad b a, <lb/>ut figura rectilinea ad portio&shy;<lb/>nes. </s>
  
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 <s>eadem quoque ratione demon&longs;trabimus <pb/>t u, x y ip&longs;i g h &aelig;quidi&longs;tare. </s> <s>eadem quoque ratione demon&longs;trabimus <pb/>t u, x y ip&longs;i g h &aelig;quidi&longs;tare. </s>
  
 <s>Et quoniam triangula, qu&aelig; <lb/>fiunt &agrave; lineis K y, y u, u s, s h &aelig;qualiz funt inter &longs;e, &amp; &longs;imilia <lb/><arrow.to.target n="marg38"></arrow.to.target><lb/>triangulo K m h: habebit triangulum K m h ad <expan abbr="triangul&utilde;">triangulum</expan> <lb/>K <foreign lang="greek">d</foreign> y duplam proportioncm eius, qu&aelig; e&longs;t line&aelig; <foreign lang="greek">k</foreign> h ad K y. </s><lb/> <s>Et quoniam triangula, qu&aelig; <lb/>fiunt &agrave; lineis K y, y u, u s, s h &aelig;qualiz &longs;unt inter &longs;e, &amp; &longs;imilia <lb/><arrow.to.target n="marg38"></arrow.to.target><lb/>triangulo K m h: habebit triangulum K m h ad <expan abbr="triangul&utilde;">triangulum</expan> <lb/>K <foreign lang="greek">d</foreign> y duplam proportioncm eius, qu&aelig; e&longs;t line&aelig; <foreign lang="greek">k</foreign> h ad K y. </s><lb/>
  
 <s>&longs;ed K h po&longs;ita e&longs;t quadrupla ip&longs;ius k y. </s> <s>&longs;ed K h po&longs;ita e&longs;t quadrupla ip&longs;ius k y. </s>
  
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 <s>Sed ut circulus <lb/>uel ellip&longs;is g h ad figuram rectilincam in ip&longs;a de&longs;cri&shy;<lb/>ptam, ita e&longs;t cylindrus uel cylindri portio c c ad pri&longs;ma, <lb/>quod rectilineam figuram pro ba&longs;i habet, &amp; altitudinem <lb/>&aelig;qualem; id, quod infra demon&longs;trabitur. </s> <s>Sed ut circulus <lb/>uel ellip&longs;is g h ad figuram rectilincam in ip&longs;a de&longs;cri&shy;<lb/>ptam, ita e&longs;t cylindrus uel cylindri portio c c ad pri&longs;ma, <lb/>quod rectilineam figuram pro ba&longs;i habet, &amp; altitudinem <lb/>&aelig;qualem; id, quod infra demon&longs;trabitur. </s>
  
 <s>crgo per conuer <lb/>&longs;ionem rationis, ut circulus, uel ellip&longs;is g h ad portioncs re <lb/>lictas, ita cylindrus, uel cylindri portio c e ad &longs;olidas por&shy;<lb/>tiones, quate cylindrus uel cylindri portio ad &longs;olidas por&shy;<lb/>tiones eandem proportionem habet, quam linea n <foreign lang="greek">k</foreign> ad k <lb/>&amp; diuidendo pri&longs;ma, cuius ba&longs;is e&longs;t rectilinea figura ad &longs;o&shy;<lb/>lidas portiones candem proportionem habet, quam n l ad <lb/>l k &amp; quoniam a cylindro uel cylindri portione, cuius gra&shy;<lb/>uitatis centrum e&longs;t l, aufertur pri&longs;ma ba&longs;im habens rectili&shy;<lb/>neam <expan abbr="figur&atilde;">figuram</expan>, cuius <expan abbr="centr&utilde;">centrum</expan> grauitatis e&longs;t K: re&longs;idu&aelig; magnitu <lb/>dinis ex &longs;olidis portionibus <expan abbr="c&otilde;po&longs;it&aelig;">compo&longs;it&aelig;</expan> grauitatis <expan abbr="c&etilde;tr&utilde;">centrum</expan> crit <lb/>in linea k l protracta, &amp; in puncto n; quod e&longs;t <expan abbr="ab&longs;urd&utilde;">ab&longs;urdum</expan>. </s> <s>crgo per conuer <lb/>&longs;ionem rationis, ut circulus, uel ellip&longs;is g h ad portioncs re <lb/>lictas, ita cylindrus, uel cylindri portio c e ad &longs;olidas por&shy;<lb/>tiones, quate cylindrus uel cylindri portio ad &longs;olidas por&shy;<lb/>tiones eandem proportionem habet, quam linea n <foreign lang="greek">k</foreign> ad k <lb/>&amp; diuidendo pri&longs;ma, cuius ba&longs;is e&longs;t rectilinea figura ad &longs;o&shy;<lb/>lidas portiones eandem proportionem habet, quam n l ad <lb/>l k &amp; quoniam a cylindro uel cylindri portione, cuius gra&shy;<lb/>uitatis centrum e&longs;t l, aufertur pri&longs;ma ba&longs;im habens rectili&shy;<lb/>neam <expan abbr="figur&atilde;">figuram</expan>, cuius <expan abbr="centr&utilde;">centrum</expan> grauitatis e&longs;t K: re&longs;idu&aelig; magnitu <lb/>dinis ex &longs;olidis portionibus <expan abbr="c&otilde;po&longs;it&aelig;">compo&longs;it&aelig;</expan> grauitatis <expan abbr="c&etilde;tr&utilde;">centrum</expan> crit <lb/>in linea k l protracta, &amp; in puncto n; quod e&longs;t <expan abbr="ab&longs;urd&utilde;">ab&longs;urdum</expan>. </s>
  
 <s>relin <lb/>quitur ergo, ut <expan abbr="c&etilde;trum">centrum</expan> grauitatis cylindri; uel cylindri por <lb/>tionis &longs;it <expan abbr="punct&utilde;">punctum</expan> k. </s> <s>relin <lb/>quitur ergo, ut <expan abbr="c&etilde;trum">centrum</expan> grauitatis cylindri; uel cylindri por <lb/>tionis &longs;it <expan abbr="punct&utilde;">punctum</expan> k. </s>
  
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 <s>Si enini <lb/>axes in eadem recta linea fuerint con&longs;tituti, b&aelig;c dao lo'i&shy;<lb/>da, in unum, atque idem &longs;olidum conuenient. </s> <s>Si enini <lb/>axes in eadem recta linea fuerint con&longs;tituti, b&aelig;c dao lo'i&shy;<lb/>da, in unum, atque idem &longs;olidum conuenient. </s>
  
 <s>quare <gap/>x <lb/>iis, qu&aelig; proxime tradita &longs;unt, habebit &longs;olidum a b ad &longs;o&shy;<lb/>lidum a c candem proportionem, quam axis d e ad e f <lb/>axem. </s> <s>quare <gap/>x <lb/>iis, qu&aelig; proxime tradita &longs;unt, habebit &longs;olidum a b ad &longs;o&shy;<lb/>lidum a c eandem proportionem, quam axis d e ad e f <lb/>axem. </s>
  
 <s>Si uero axes non &longs;int in eadem recta linea, demittan <lb/>tur a punctis d, &longs; perpendiculares ad ba&longs;is planum, d g, fh: <lb/>&amp; jungantur e g, e h. </s> <s>Si uero axes non &longs;int in eadem recta linea, demittan <lb/>tur a punctis d, &longs; perpendiculares ad ba&longs;is planum, d g, fh: <lb/>&amp; jungantur e g, e h. </s>
  
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 <s>ergo &amp; pri&longs;ma a e ad pri&longs;ma f l eandem propor&shy;<lb/>tionem habebit, quam altitudo ad altitudinem. </s> <s>ergo &amp; pri&longs;ma a e ad pri&longs;ma f l eandem propor&shy;<lb/>tionem habebit, quam altitudo ad altitudinem. </s>
  
 <s>&longs;equitur <lb/>igitur ut &amp; pyramides, qu&aelig; in &aelig;qualibus ba&longs;ibus <expan abbr="con&longs;titu&utilde;">con&longs;tituum</expan> <lb/>tur, candem inter &longs;e &longs;e, quam altitudines, proportionem <lb/>habeant.</s></p><p type="margin"> <s>&longs;equitur <lb/>igitur ut &amp; pyramides, qu&aelig; in &aelig;qualibus ba&longs;ibus <expan abbr="con&longs;titu&utilde;">con&longs;tituum</expan> <lb/>tur, eandem inter &longs;e &longs;e, quam altitudines, proportionem <lb/>habeant.</s></p><p type="margin">
  
 <s><margin.target id="marg63"></margin.target>6. du<gap/><lb/>cimi<gap/></s></p><p type="margin"> <s><margin.target id="marg63"></margin.target>6. du<gap/><lb/>cimi<gap/></s></p><p type="margin">
  
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 <s>Simili enim ratione, qua &longs;upra, demon&longs;trabi&shy;<lb/>tur quadratum a b ad quadratum f g ita e&longs;&longs;e, ut <expan abbr="quadrat&utilde;">quadratum</expan> <lb/><arrow.to.target n="marg81"></arrow.to.target><lb/>f g ad c d quadratum. </s> <s>Simili enim ratione, qua &longs;upra, demon&longs;trabi&shy;<lb/>tur quadratum a b ad quadratum f g ita e&longs;&longs;e, ut <expan abbr="quadrat&utilde;">quadratum</expan> <lb/><arrow.to.target n="marg81"></arrow.to.target><lb/>f g ad c d quadratum. </s>
  
 <s>Sed circuli inter &longs;e candem propor&shy;<lb/>tionem habent, quam diametrorum quadrata. </s> <s>Sed circuli inter &longs;e eandem propor&shy;<lb/>tionem habent, quam diametrorum quadrata. </s>
  
 <s>ellip&longs;es au&shy;<lb/>tem circa a b, f g, c d, qu&aelig; &longs;imiles &longs;unt, ut o&longs;tendimus in <expan abbr="c&otilde;-mentariis">con&shy;<lb/>mentariis</expan> in principium libri Archimedis de conoidibus, <lb/>&amp; &longs;ph&aelig;roidibus, eam <expan abbr="hab&etilde;t">habent</expan> proportionem, quam quadrar <lb/>ta diametrorum, qu&aelig; eiu&longs;dem rationis &longs;unt, ex corollaio&shy;<lb/><figure id="fig45"></figure><lb/>&longs;eptim&aelig; propo&longs;itionis eiu&longs;dem li&shy;<lb/>bri. </s> <s>ellip&longs;es au&shy;<lb/>tem circa a b, f g, c d, qu&aelig; &longs;imiles &longs;unt, ut o&longs;tendimus in <expan abbr="c&otilde;-mentariis">con&shy;<lb/>mentariis</expan> in principium libri Archimedis de conoidibus, <lb/>&amp; &longs;ph&aelig;roidibus, eam <expan abbr="hab&etilde;t">habent</expan> proportionem, quam quadrar <lb/>ta diametrorum, qu&aelig; eiu&longs;dem rationis &longs;unt, ex corollaio&shy;<lb/><figure id="fig45"></figure><lb/>&longs;eptim&aelig; propo&longs;itionis eiu&longs;dem li&shy;<lb/>bri. </s>
  
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 <s>quare pyra&shy;<lb/>mis, uel conus, uel coni por&shy;<lb/>tio, cuius ba&longs;is e&longs;t tribus illis <lb/>ba&longs;ibus &aelig;qualis ad a g b eam <lb/>habet proportionem, quam <lb/>ba&longs;es a b, e f, c d ad a b ba&longs;im. </s> <s>quare pyra&shy;<lb/>mis, uel conus, uel coni por&shy;<lb/>tio, cuius ba&longs;is e&longs;t tribus illis <lb/>ba&longs;ibus &aelig;qualis ad a g b eam <lb/>habet proportionem, quam <lb/>ba&longs;es a b, e f, c d ad a b ba&longs;im. </s>
  
 <s>Fru&longs;tum igitur a d ad a g b <pb/>pyramidem, uel conuni, uel coni portionem candem pro&shy;<lb/>portionem habet, quam ba&longs;es a b, c d un&agrave; cum e f ad ba&shy;<lb/>&longs;im a b. </s> <s>Fru&longs;tum igitur a d ad a g b <pb/>pyramidem, uel conuni, uel coni portionem eandem pro&shy;<lb/>portionem habet, quam ba&longs;es a b, c d un&agrave; cum e f ad ba&shy;<lb/>&longs;im a b. </s>
  
 <s>quod demon&longs;trare uolebamus.</s></p><p type="margin"> <s>quod demon&longs;trare uolebamus.</s></p><p type="margin">
  
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 <s>Qu&ograve;d cum g p &longs;it tripla p n; <lb/>erit etiam d z ip&longs;ius z m tri&shy;<lb/>pla. </s> <s>Qu&ograve;d cum g p &longs;it tripla p n; <lb/>erit etiam d z ip&longs;ius z m tri&shy;<lb/>pla. </s>
  
 <s>atque ob candem cau&longs;&shy;<lb/>&longs;am punctuniz e&longs;t <expan abbr="centr&utilde;">centrum</expan> gra&shy;<lb/>uitatis pyramidis b c f e d. </s> <s>atque ob eandem cau&longs;&shy;<lb/>&longs;am punctuniz e&longs;t <expan abbr="centr&utilde;">centrum</expan> gra&shy;<lb/>uitatis pyramidis b c f e d. </s>
  
 <s>iun <lb/>ctaigitur z u, in ea erit <expan abbr="c&etilde;trum">centrum</expan> <pb pagenum="36"/>grauitatis magnitudinis, qu&aelig; ex utri&longs;que pyramidibus <expan abbr="c&otilde;">com</expan> <lb/>&longs;tat; hoc e&longs;t ip&longs;ius fru&longs;ti. </s> <s>iun <lb/>ctaigitur z u, in ea erit <expan abbr="c&etilde;trum">centrum</expan> <pb pagenum="36"/>grauitatis magnitudinis, qu&aelig; ex utri&longs;que pyramidibus <expan abbr="c&otilde;">com</expan> <lb/>&longs;tat; hoc e&longs;t ip&longs;ius fru&longs;ti. </s>
  
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 <s>ergo linea x s per p tran&longs;ibit: &amp; line&aelig; r u, s x, q t in&shy;<lb/>ter &longs;e &aelig;quidi&longs;tantes erunt. </s> <s>ergo linea x s per p tran&longs;ibit: &amp; line&aelig; r u, s x, q t in&shy;<lb/>ter &longs;e &aelig;quidi&longs;tantes erunt. </s>
  
 <s>Itaque cum fru&longs;ti a g latera pro&shy;<pb pagenum="37"/>ducta &longs;uerint, ita ut in unum punctum y cocant, erunt tri&agrave; <lb/>gala u y l, x y p, t y k inter &longs;e &longs;imilia: &amp; &longs;imilia etiam triangu <lb/>la l y r, p y s, k y <expan abbr="q.">que</expan> quare ut in 19 huius, demon&longs;trabitur <lb/>x p, ad p s: <expan abbr="itemq;">itemque</expan> t <foreign lang="greek">k</foreign> ad k q candem habere <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>quam u l ad l r. </s> <s>Itaque cum fru&longs;ti a g latera pro&shy;<pb pagenum="37"/>ducta &longs;uerint, ita ut in unum punctum y cocant, erunt tri&agrave; <lb/>gala u y l, x y p, t y k inter &longs;e &longs;imilia: &amp; &longs;imilia etiam triangu <lb/>la l y r, p y s, k y <expan abbr="q.">que</expan> quare ut in 19 huius, demon&longs;trabitur <lb/>x p, ad p s: <expan abbr="itemq;">itemque</expan> t <foreign lang="greek">k</foreign> ad k q eandem habere <expan abbr="proportion&etilde;">proportionem</expan>, <lb/>quam u l ad l r. </s>
  
 <s>Sed ut u l ad l <gap/>, ita e&longs;t triangulum a b c ad <lb/>triangulum a c d: &amp; ut t k ad K q, ita triangulum e f g ad <lb/>triangulum e g h. </s> <s>Sed ut u l ad l <gap/>, ita e&longs;t triangulum a b c ad <lb/>triangulum a c d: &amp; ut t k ad K q, ita triangulum e f g ad <lb/>triangulum e g h. </s>
  
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 <s>ergo <lb/>ex &aelig;quali id, quod relinquitur ex duabus tertiis quadrati <lb/>b c, demptis ab ip&longs;is quadrati a d duabus tertiis, ad <expan abbr="terti&atilde;">tertiam</expan> <lb/>partem quadrati b c, ut k h ad h f: &amp; ad portionem <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> <lb/>terti&aelig; partis, ad quam un&agrave; cum ip&longs;a portione, duplam pro <lb/>portionem habeat eius, qu&aelig; e&longs;t quadrati b c ad <expan abbr="quadrat&utilde;">quadratum</expan> <lb/>a d, ut K l ad l h. </s> <s>ergo <lb/>ex &aelig;quali id, quod relinquitur ex duabus tertiis quadrati <lb/>b c, demptis ab ip&longs;is quadrati a d duabus tertiis, ad <expan abbr="terti&atilde;">tertiam</expan> <lb/>partem quadrati b c, ut k h ad h f: &amp; ad portionem <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> <lb/>terti&aelig; partis, ad quam un&agrave; cum ip&longs;a portione, duplam pro <lb/>portionem habeat eius, qu&aelig; e&longs;t quadrati b c ad <expan abbr="quadrat&utilde;">quadratum</expan> <lb/>a d, ut K l ad l h. </s>
  
 <s>habet enim K l ad l h candem proportio&shy;<lb/>nem, quam conoidis portio b g c ad portionem a g d: por&shy;<lb/>tio autem b g c ad portionem a g d duplam proportionem <lb/>habet eius, qu&aelig; e&longs;t ba&longs;is b c ad ba&longs;im a d: hoc e&longs;t quadrati <lb/><arrow.to.target n="marg118"></arrow.to.target><lb/>b c ad quadratum a d; ut proxime demon&longs;tratum e&longs;t. </s> <s>habet enim K l ad l h eandem proportio&shy;<lb/>nem, quam conoidis portio b g c ad portionem a g d: por&shy;<lb/>tio autem b g c ad portionem a g d duplam proportionem <lb/>habet eius, qu&aelig; e&longs;t ba&longs;is b c ad ba&longs;im a d: hoc e&longs;t quadrati <lb/><arrow.to.target n="marg118"></arrow.to.target><lb/>b c ad quadratum a d; ut proxime demon&longs;tratum e&longs;t. </s>
  
 <s>quare <lb/>dempto a d quadrato &agrave; duabus tertiis quadrati b c, erit id, <lb/>quod relinquitur un&agrave; cum dicta portione terti&aelig; partis ad <lb/>reliquam eiu&longs;dem portionem, ut e l ad l f. </s> <s>quare <lb/>dempto a d quadrato &agrave; duabus tertiis quadrati b c, erit id, <lb/>quod relinquitur un&agrave; cum dicta portione terti&aelig; partis ad <lb/>reliquam eiu&longs;dem portionem, ut e l ad l f. </s>
  

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