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Colored diff for /texts/archimedes/xml/Attic/gheta_promo_01_la_1603.xml between version 1.12 and 1.13

version 1.12, 2002/08/06 17:20:05 version 1.13, 2002/08/07 23:57:51
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 <s>de qu&aelig; re &longs;uo leco ego tractabo, &amp; facilem mon&shy;<lb/>strabo viam, qu&aelig; vel argentum in aure, vel quod visme&shy;<lb/>tallum in quolibet admistum diprebeudi queat, &amp; alte <lb/><gap/> ab aliero di&longs;rerni &amp; &longs;imu<gap/>, quo pacto ex au&shy;<lb/>ri grauitate <gap/> qualiras, <gap/>perfectio intelligi po&longs;sit. </s> <s>de qu&aelig; re &longs;uo leco ego tractabo, &amp; facilem mon&shy;<lb/>strabo viam, qu&aelig; vel argentum in aure, vel quod visme&shy;<lb/>tallum in quolibet admistum diprebeudi queat, &amp; alte <lb/><gap/> ab aliero di&longs;rerni &amp; &longs;imu<gap/>, quo pacto ex au&shy;<lb/>ri grauitate <gap/> qualiras, <gap/>perfectio intelligi po&longs;sit. </s>
  
 <s><lb/>Tati vere apu&longs;culo nomen ab Arthimede, quem Ducem &longs;e&shy;<lb/>quer impo&longs;ui Nam cum ille, vt erat &longs;ummus Magi&longs;ter, &longs;e&shy;<lb/>tis haliui&longs;&longs;et hunc, totem qu<gap/> fabr<gap/> pe&longs;ito fundumen&shy;<lb/>to delineare in primo lib. </s> <s><lb/>Tati vere apu&longs;culo nomen ab Arthimede, quem Ducem &longs;e&shy;<lb/>quer impo&longs;ui Nam cum ille, vt erat &longs;ummus Magi&longs;ter, &longs;e&shy;<lb/>tis haliui&longs;&longs;et hunc, totem qu<gap/> fabr<gap/> pe&longs;ito fundumen&shy;<lb/>to delineare in primo lib.
  
 <s>vbi agit de i&longs;s qu&aelig; vt huntur in <lb/>a qu&aelig;. </s> vbi agit de i&longs;s qu&aelig; vt huntur in <lb/>a qu&aelig;. </s>
  
 <s>Opus egapra<gap/> <expan abbr="eiq;">eique</expan> fundamenta molem inyce&shy;<lb/>re conatus &longs;um p<gap/>ribus &longs;uis etaberatam, atque dictin&shy;<lb/>ctam. </s> <s>Opus egapra<gap/> <expan abbr="eiq;">eique</expan> fundamenta molem inyce&shy;<lb/>re conatus &longs;um p<gap/>ribus &longs;uis etaberatam, atque dictin&shy;<lb/>ctam. </s>
  
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 <s>Deinde in&shy;<lb/>teligatur aqua con&longs;i&longs;tens, &amp; <lb/>manens, eiu&longs;que &longs;uper&longs;i&shy;<lb/>cies &longs;ph&aelig;rica C D E, cuius <lb/>&longs;ph&aelig;r&aelig; centrum K, aqu&aelig; <lb/>enim con&longs;i&longs;tentis, atque <lb/>manentis &longs;uper&longs;icies &longs;ph&aelig;&shy;<lb/>rica e&longs;t, cui <gap/> &longs;ph&aelig;r&aelig; cen&shy;<lb/>trum idem <gap/>, quod <expan abbr="centr&utilde;">centrum</expan> <lb/>terr&aelig;, hoc autem demon&longs;tratum e&longs;t ab Archimede Prop. </s> <s>Deinde in&shy;<lb/>teligatur aqua con&longs;i&longs;tens, &amp; <lb/>manens, eiu&longs;que &longs;uper&longs;i&shy;<lb/>cies &longs;ph&aelig;rica C D E, cuius <lb/>&longs;ph&aelig;r&aelig; centrum K, aqu&aelig; <lb/>enim con&longs;i&longs;tentis, atque <lb/>manentis &longs;uper&longs;icies &longs;ph&aelig;&shy;<lb/>rica e&longs;t, cui <gap/> &longs;ph&aelig;r&aelig; cen&shy;<lb/>trum idem <gap/>, quod <expan abbr="centr&utilde;">centrum</expan> <lb/>terr&aelig;, hoc autem demon&longs;tratum e&longs;t ab Archimede Prop. </s>
  
 <s>2. lib. </s> <s>2. lib.
  
 <s>1. de <lb/>ijs, qu&aelig; vehuntur in aqua. </s> 1. de <lb/>ijs, qu&aelig; vehuntur in aqua. </s>
  
 <s>Inteligantur etiam du&aelig; pyramides con&shy;<lb/>iunct&aelig;, &amp; continuat&aelig;, &aelig;quales, &amp; &longs;imiles KCD, KDE, pro ba&longs;ibus <lb/>habentes in &longs;uper&longs;icie aqu&aelig; parallelogramma, vertices autem pun&shy;<lb/>ctum K, &amp; corpora L, B, comprehendantur &agrave; pyramide KDE, corpo&shy;<lb/>ra vero M, A, &agrave; pyramide KCD, &amp; &longs;ub corporibus L, B, de&longs;cribatur <lb/>qu&aelig;dam alterius &longs;ph&aelig;r&aelig; &longs;uper&longs;icies FGH, in aqua, circa centrum K, <lb/>poterit autem huiu&longs;modi &longs;uper&longs;icies &longs;ub corporibus L, B, de&longs;cribi, <lb/>quoniam &amp; &longs;i ip&longs;i corpora demerguntur tota, non ideo feruntur <lb/>deor&longs;um, ponuntur enim &aelig;que grauia atque aqua. </s> <s>Inteligantur etiam du&aelig; pyramides con&shy;<lb/>iunct&aelig;, &amp; continuat&aelig;, &aelig;quales, &amp; &longs;imiles KCD, KDE, pro ba&longs;ibus <lb/>habentes in &longs;uper&longs;icie aqu&aelig; parallelogramma, vertices autem pun&shy;<lb/>ctum K, &amp; corpora L, B, comprehendantur &agrave; pyramide KDE, corpo&shy;<lb/>ra vero M, A, &agrave; pyramide KCD, &amp; &longs;ub corporibus L, B, de&longs;cribatur <lb/>qu&aelig;dam alterius &longs;ph&aelig;r&aelig; &longs;uper&longs;icies FGH, in aqua, circa centrum K, <lb/>poterit autem huiu&longs;modi &longs;uper&longs;icies &longs;ub corporibus L, B, de&longs;cribi, <lb/>quoniam &amp; &longs;i ip&longs;i corpora demerguntur tota, non ideo feruntur <lb/>deor&longs;um, ponuntur enim &aelig;que grauia atque aqua. </s>
  
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 <s><emph type="italics"/>Qu&aelig;ro exempli gratia magnitudinem diametri &longs;ph&aelig;r&aelig; aure&aelig;, <lb/>grauitatem habentis 10, lib. </s> <s><emph type="italics"/>Qu&aelig;ro exempli gratia magnitudinem diametri &longs;ph&aelig;r&aelig; aure&aelig;, <lb/>grauitatem habentis 10, lib. </s>
  
 <s>Numeri in prima columna &longs;ub titulo <lb/>grauitatis denotant &longs;ph&aelig;rarum grauitates, reliqui vero in reliquis <lb/>columnis denotant diametrorum magnitudines; itaque in linea<emph.end type="italics"/> 10, <lb/><emph type="italics"/>lib. </s> <s>Numeri in prima columna &longs;ub titulo <lb/>grauitatis denotant &longs;ph&aelig;rarum grauitates, reliqui vero in reliquis <lb/>columnis denotant diametrorum magnitudines; itaque in linea<emph.end type="italics"/> 10, <lb/><emph type="italics"/>lib.
  
 <s>&longs;ub titulo magnitudinis diametri &longs;ph&aelig;r&aelig; aure&aelig;, datur qu&aelig;&longs;ita <lb/>diametri magnitudo partium<emph.end type="italics"/> 2 21/100: <emph type="italics"/>qualium pes vnus est<emph.end type="italics"/> 12. </s></p><p type="main"> &longs;ub titulo magnitudinis diametri &longs;ph&aelig;r&aelig; aure&aelig;, datur qu&aelig;&longs;ita <lb/>diametri magnitudo partium<emph.end type="italics"/> 2 21/100: <emph type="italics"/>qualium pes vnus est<emph.end type="italics"/> 12. </s></p><p type="main">
  
 <s><emph type="italics"/>Qu&aelig;ro magnitudinem diametri &longs;ph&aelig;r&aelig; ferre&aelig;, grauitatem haben&shy;<lb/>tis 50, lib. </s> <s><emph type="italics"/>Qu&aelig;ro magnitudinem diametri &longs;ph&aelig;r&aelig; ferre&aelig;, grauitatem haben&shy;<lb/>tis 50, lib.
  
 <s>in linea 50, lib. </s> in linea 50, lib. </s>
  
 <s>&longs;ub titulo magnitudinis diametri &longs;ph&aelig;r&aelig; <lb/>ferre&aelig;, datur qu&aelig;&longs;ita diametri magnitudo<emph.end type="italics"/> 6 40/100. </s></p><p type="main"> <s>&longs;ub titulo magnitudinis diametri &longs;ph&aelig;r&aelig; <lb/>ferre&aelig;, datur qu&aelig;&longs;ita diametri magnitudo<emph.end type="italics"/> 6 40/100. </s></p><p type="main">
  
 <s><emph type="italics"/>Qu&aelig;ro magnitudinem diametri &longs;ph&aelig;r&aelig; argente&aelig;, grauitatem ha&shy;<lb/>bentis 60, lib. </s> <s><emph type="italics"/>Qu&aelig;ro magnitudinem diametri &longs;ph&aelig;r&aelig; argente&aelig;, grauitatem ha&shy;<lb/>bentis 60, lib.
  
 <s>in linea 60, lib. </s> in linea 60, lib. </s>
  
 <s>&longs;ub titulo magnitudinis diametri &longs;ph&aelig;r&aelig; <lb/>argente&aelig;, datur ip&longs;a magnitudo<emph.end type="italics"/> 6 25/100. </s></p><pb pagenum="50"/><p type="main"> <s>&longs;ub titulo magnitudinis diametri &longs;ph&aelig;r&aelig; <lb/>argente&aelig;, datur ip&longs;a magnitudo<emph.end type="italics"/> 6 25/100. </s></p><pb pagenum="50"/><p type="main">
  
 <s><emph type="italics"/>Qu&aelig;ro denique magnitudinem diametri &longs;ph&aelig;r&aelig; stanne&aelig;, graui&shy;<lb/>tatem habentis 38, lib. </s> <s><emph type="italics"/>Qu&aelig;ro denique magnitudinem diametri &longs;ph&aelig;r&aelig; stanne&aelig;, graui&shy;<lb/>tatem habentis 38, lib.
  
 <s>in linea 38, lib. </s> in linea 38, lib. </s>
  
 <s>&longs;ub titulo magnitudinis dia&shy;<lb/>metri &longs;ph&aelig;r&aelig; stanne&aelig;, datur qu&aelig;&longs;ita diametri magnitudo 6, ad <lb/>vnguem.<emph.end type="italics"/></s></p><p type="main"> <s>&longs;ub titulo magnitudinis dia&shy;<lb/>metri &longs;ph&aelig;r&aelig; stanne&aelig;, datur qu&aelig;&longs;ita diametri magnitudo 6, ad <lb/>vnguem.<emph.end type="italics"/></s></p><p type="main">
  
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 <s>De compo&longs;itione huius Tabul&aelig;. </s></p><p type="main"> <s>De compo&longs;itione huius Tabul&aelig;. </s></p><p type="main">
  
 <s><emph type="italics"/>Huius tabul&aelig; compo&longs;itio pendet ex pr&aelig;cedenti tabula, &amp; ex pro&shy;<lb/>po&longs;.<emph.end type="italics"/> 17, <emph type="italics"/>huius, &longs;i enim fiat vt grauitas &longs;ph&aelig;r&aelig; stanne&aelig;, diametrum <lb/>habentis vnius vnci&aelig;, id est, vt grana<emph.end type="italics"/> 1216, <emph type="italics"/>ad grauitatem &longs;ph&aelig;r&aelig; <lb/>vnius libr&aelig;, idest, adgrana<emph.end type="italics"/> 6912, <emph type="italics"/>ita cubus diametri vnius vnci&aelig;, <lb/>hoc e&longs;t, ita<emph.end type="italics"/> 1, <emph type="italics"/>ad alium numerum, qui &longs;it<emph.end type="italics"/> 5 11/19 <emph type="italics"/>is erit cubus diametri <lb/>&longs;ph&aelig;r&aelig; stanne&aelig;, grauitatem habentis 1, lib. </s> <s><emph type="italics"/>Huius tabul&aelig; compo&longs;itio pendet ex pr&aelig;cedenti tabula, &amp; ex pro&shy;<lb/>po&longs;.<emph.end type="italics"/> 17, <emph type="italics"/>huius, &longs;i enim fiat vt grauitas &longs;ph&aelig;r&aelig; stanne&aelig;, diametrum <lb/>habentis vnius vnci&aelig;, id est, vt grana<emph.end type="italics"/> 1216, <emph type="italics"/>ad grauitatem &longs;ph&aelig;r&aelig; <lb/>vnius libr&aelig;, idest, adgrana<emph.end type="italics"/> 6912, <emph type="italics"/>ita cubus diametri vnius vnci&aelig;, <lb/>hoc e&longs;t, ita<emph.end type="italics"/> 1, <emph type="italics"/>ad alium numerum, qui &longs;it<emph.end type="italics"/> 5 11/19 <emph type="italics"/>is erit cubus diametri <lb/>&longs;ph&aelig;r&aelig; stanne&aelig;, grauitatem habentis 1, lib.
  
 <s>demonstratum enim <lb/>est prop.<emph.end type="italics"/> 17, <emph type="italics"/>huius, &longs;ph&aelig;ras eiu&longs;dem generis inter &longs;e e&longs;&longs;e in grauitate, <lb/>vt diametrorum cubi in magnitudine; quare radix cubica numeri<emph.end type="italics"/><lb/>5 13/19, <emph type="italics"/>dabit ip&longs;am diametrum, &longs;ed quoniam numerus<emph.end type="italics"/> 5 13/19, <emph type="italics"/>non est <lb/>pr&aelig;ci&longs;e cubus, cius radix non explicabitur accurata, &longs;ed vt explicetur <lb/>ver&aelig; bene proxima, multiplicetur<emph.end type="italics"/> 5 13/19, <emph type="italics"/>per<emph.end type="italics"/> 1000000. <emph type="italics"/>&amp; ex producto<emph.end type="italics"/><lb/>5684210 10/19, <emph type="italics"/>neglecto fracto<emph.end type="italics"/> 10/19, <emph type="italics"/>eruatur radix, tanquam ex accu&shy;<lb/>rato numero cubo, ea erit<emph.end type="italics"/> 173. <emph type="italics"/>proxime, &amp; erit centupla radicis nu&shy;<lb/>meri<emph.end type="italics"/> 5 13/19, <emph type="italics"/>nam numerus<emph.end type="italics"/> 1000000, <emph type="italics"/>per quem fuit multiplicatus<emph.end type="italics"/><lb/>5 13/19, <emph type="italics"/>cubus e&longs;t ex<emph.end type="italics"/> 100<emph type="italics"/>; magnitudo igitur diametri &longs;ph&aelig;r&aelig; &longs;tanne&aelig;, <lb/>grauitatem habentis 1, lib. </s> demonstratum enim <lb/>est prop.<emph.end type="italics"/> 17, <emph type="italics"/>huius, &longs;ph&aelig;ras eiu&longs;dem generis inter &longs;e e&longs;&longs;e in grauitate, <lb/>vt diametrorum cubi in magnitudine; quare radix cubica numeri<emph.end type="italics"/><lb/>5 13/19, <emph type="italics"/>dabit ip&longs;am diametrum, &longs;ed quoniam numerus<emph.end type="italics"/> 5 13/19, <emph type="italics"/>non est <lb/>pr&aelig;ci&longs;e cubus, cius radix non explicabitur accurata, &longs;ed vt explicetur <lb/>ver&aelig; bene proxima, multiplicetur<emph.end type="italics"/> 5 13/19, <emph type="italics"/>per<emph.end type="italics"/> 1000000. <emph type="italics"/>&amp; ex producto<emph.end type="italics"/><lb/>5684210 10/19, <emph type="italics"/>neglecto fracto<emph.end type="italics"/> 10/19, <emph type="italics"/>eruatur radix, tanquam ex accu&shy;<lb/>rato numero cubo, ea erit<emph.end type="italics"/> 173. <emph type="italics"/>proxime, &amp; erit centupla radicis nu&shy;<lb/>meri<emph.end type="italics"/> 5 13/19, <emph type="italics"/>nam numerus<emph.end type="italics"/> 1000000, <emph type="italics"/>per quem fuit multiplicatus<emph.end type="italics"/><lb/>5 13/19, <emph type="italics"/>cubus e&longs;t ex<emph.end type="italics"/> 100<emph type="italics"/>; magnitudo igitur diametri &longs;ph&aelig;r&aelig; &longs;tanne&aelig;, <lb/>grauitatem habentis 1, lib.
  
 <s>erit<emph.end type="italics"/> 1 78/100. <emph type="italics"/>reliquarum autem ex &longs;tanno <lb/>&longs;ph&aelig;rarum, grauitatem habentium duplam prim&aelig;, triplam, quadru&shy;<lb/>plam &amp;c. </s> erit<emph.end type="italics"/> 1 78/100. <emph type="italics"/>reliquarum autem ex &longs;tanno <lb/>&longs;ph&aelig;rarum, grauitatem habentium duplam prim&aelig;, triplam, quadru&shy;<lb/>plam &amp;c. </s>
  
 <s>ita inuenientur diametri.<emph.end type="italics"/></s></p><p type="main"> <s>ita inuenientur diametri.<emph.end type="italics"/></s></p><p type="main">
  
 <s><emph type="italics"/>Duplum numeri<emph.end type="italics"/> 5684210 10/19, <emph type="italics"/>id est<emph.end type="italics"/> 11368421 1/19, <emph type="italics"/>erit cubus <lb/>centupli diametri &longs;pher&aelig; stanne&aelig;, grauitatem habentis duplam pri&shy;<lb/>m&aelig;, hoc est<emph.end type="italics"/> 2, <emph type="italics"/>lib. </s> <s><emph type="italics"/>Duplum numeri<emph.end type="italics"/> 5684210 10/19, <emph type="italics"/>id est<emph.end type="italics"/> 11368421 1/19, <emph type="italics"/>erit cubus <lb/>centupli diametri &longs;pher&aelig; stanne&aelig;, grauitatem habentis duplam pri&shy;<lb/>m&aelig;, hoc est<emph.end type="italics"/> 2, <emph type="italics"/>lib.
  
 <s>ex &longs;upra nominata enim prop.<emph.end type="italics"/> 17, <emph type="italics"/>huius, e&longs;t vt gra&shy;<lb/>uitas &longs;ph&aelig;r&aelig; vnius libr&aelig;, ad grauitatem &longs;ph&aelig;r&aelig; duarum librarum, <lb/>ita cubus diametri prim&aelig; &longs;ph&aelig;r&aelig;, ad cubum diametri &longs;ecund&aelig;. </s> ex &longs;upra nominata enim prop.<emph.end type="italics"/> 17, <emph type="italics"/>huius, e&longs;t vt gra&shy;<lb/>uitas &longs;ph&aelig;r&aelig; vnius libr&aelig;, ad grauitatem &longs;ph&aelig;r&aelig; duarum librarum, <lb/>ita cubus diametri prim&aelig; &longs;ph&aelig;r&aelig;, ad cubum diametri &longs;ecund&aelig;. </s>
  
 <s>Si <lb/>vero triplicetur numerus<emph.end type="italics"/> 5684210 10/19, <emph type="italics"/>eius triplum, quod est<emph.end type="italics"/><lb/>17052631 11/19, <emph type="italics"/>erit cubus centupli diametri &longs;ph&aelig;r&aelig; &longs;tanne&aelig;, graui&shy;<lb/>tatem habentis triplam prim&aelig;, idest<emph.end type="italics"/> 3, <emph type="italics"/>lib. </s> <s>Si <lb/>vero triplicetur numerus<emph.end type="italics"/> 5684210 10/19, <emph type="italics"/>eius triplum, quod est<emph.end type="italics"/><lb/>17052631 11/19, <emph type="italics"/>erit cubus centupli diametri &longs;ph&aelig;r&aelig; &longs;tanne&aelig;, graui&shy;<lb/>tatem habentis triplam prim&aelig;, idest<emph.end type="italics"/> 3, <emph type="italics"/>lib.
  
 <s>&amp; &longs;i quadruplicetur, eius<emph.end type="italics"/><pb pagenum="51"/><emph type="italics"/>quadruplum erit cubus centupli diametri &longs;ph&aelig;r&aelig; stanne&aelig;, grauita&shy;<lb/>tem habentis quadruplam prim&aelig;, &amp; &longs;ic deinceps. </s> &amp; &longs;i quadruplicetur, eius<emph.end type="italics"/><pb pagenum="51"/><emph type="italics"/>quadruplum erit cubus centupli diametri &longs;ph&aelig;r&aelig; stanne&aelig;, grauita&shy;<lb/>tem habentis quadruplam prim&aelig;, &amp; &longs;ic deinceps. </s>
  
 <s>itaque &longs;i ex eius <lb/>multiplicibus, neglectis fractis, <expan abbr="eru&atilde;tur">eruantur</expan> radices, tanquam ex accura&shy;<lb/>tis numeris cubis, ip&longs;&aelig; indicabunt diametrorum magnitudines in <lb/>ratione centupla. </s> <s>itaque &longs;i ex eius <lb/>multiplicibus, neglectis fractis, <expan abbr="eru&atilde;tur">eruantur</expan> radices, tanquam ex accura&shy;<lb/>tis numeris cubis, ip&longs;&aelig; indicabunt diametrorum magnitudines in <lb/>ratione centupla. </s>
  
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 <s><emph type="italics"/>Pr&aelig;dicto numero<emph.end type="italics"/> 5684210 10/19, <emph type="italics"/>addatur eius duplum, id est,<emph.end type="italics"/><lb/>11368421 1/19, <emph type="italics"/>&longs;umma<emph.end type="italics"/> 17052631 11/19, <emph type="italics"/>dabit cius triplum, &longs;i vero ei <lb/>addatur eius triplum, id e&longs;t,<emph.end type="italics"/> 17052631 11/19, <emph type="italics"/>&longs;umma<emph.end type="italics"/> 22736842 2/19, <lb/><emph type="italics"/>dabit eius quadruplum, &amp; &longs;i eius quadruplum ei addatur, &longs;umma <lb/>dabit eius quintuplum, &amp; &longs;ic &longs;ola additione inuenientur eius quot&shy;<lb/>cunque multiplicia.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Pr&aelig;dicto numero<emph.end type="italics"/> 5684210 10/19, <emph type="italics"/>addatur eius duplum, id est,<emph.end type="italics"/><lb/>11368421 1/19, <emph type="italics"/>&longs;umma<emph.end type="italics"/> 17052631 11/19, <emph type="italics"/>dabit cius triplum, &longs;i vero ei <lb/>addatur eius triplum, id e&longs;t,<emph.end type="italics"/> 17052631 11/19, <emph type="italics"/>&longs;umma<emph.end type="italics"/> 22736842 2/19, <lb/><emph type="italics"/>dabit eius quadruplum, &amp; &longs;i eius quadruplum ei addatur, &longs;umma <lb/>dabit eius quintuplum, &amp; &longs;ic &longs;ola additione inuenientur eius quot&shy;<lb/>cunque multiplicia.<emph.end type="italics"/></s></p><p type="main">
  
 <s><emph type="italics"/>Eadem ratione inuenientur diametri &longs;ph&aelig;rarum ex quacunque <lb/>alia materia, &longs;i enim qu&aelig;ratur de magnitudine diametri verbi gra&shy;<lb/>tia &longs;ph&aelig;r&aelig; ferre&aelig;, grauitatem habentis 1, lib. </s> <s><emph type="italics"/>Eadem ratione inuenientur diametri &longs;ph&aelig;rarum ex quacunque <lb/>alia materia, &longs;i enim qu&aelig;ratur de magnitudine diametri verbi gra&shy;<lb/>tia &longs;ph&aelig;r&aelig; ferre&aelig;, grauitatem habentis 1, lib.
  
 <s>fiat vt grana<emph.end type="italics"/> 1314 22/37, <lb/><emph type="italics"/>id est vt grauitas &longs;ph&aelig;r&aelig; ferre&aelig;, cuius diameter e&longs;t vnius vnci&aelig;, ad <lb/>grauitatem vnius libr&aelig;, id est ad grana 6912, ita cubus diametri <lb/>vnius vnci&aelig;, hoc e&longs;t ita 1, ad alium numerum qui &longs;it<emph.end type="italics"/> 5 49/190, <emph type="italics"/>is igi&shy;<lb/>tur numerus * erit cubus diametri &longs;ph&aelig;r&aelig; ferre&aelig;, grauitatem haben-<emph.end type="italics"/><arrow.to.target n="note48"></arrow.to.target><lb/><emph type="italics"/>tis<emph.end type="italics"/> 1, <emph type="italics"/>lib. </s> fiat vt grana<emph.end type="italics"/> 1314 22/37, <lb/><emph type="italics"/>id est vt grauitas &longs;ph&aelig;r&aelig; ferre&aelig;, cuius diameter e&longs;t vnius vnci&aelig;, ad <lb/>grauitatem vnius libr&aelig;, id est ad grana 6912, ita cubus diametri <lb/>vnius vnci&aelig;, hoc e&longs;t ita 1, ad alium numerum qui &longs;it<emph.end type="italics"/> 5 49/190, <emph type="italics"/>is igi&shy;<lb/>tur numerus * erit cubus diametri &longs;ph&aelig;r&aelig; ferre&aelig;, grauitatem haben-<emph.end type="italics"/><arrow.to.target n="note48"></arrow.to.target><lb/><emph type="italics"/>tis<emph.end type="italics"/> 1, <emph type="italics"/>lib.
  
 <s>quare radix cubica numeri<emph.end type="italics"/> 5 49/190, <emph type="italics"/>dabit qu&aelig;&longs;itam dia&shy;<lb/>metrum, &amp; quoniam numerus<emph.end type="italics"/> 5 49/190, <emph type="italics"/>non e&longs;t pr&aelig;ci&longs;e cubus, &amp; ideo <lb/>non explicabitur cius radix accurate, multiplicetur per 1000000, &amp; <lb/>ex producto<emph.end type="italics"/> 5257894 14/19, <emph type="italics"/>neglecto fracto<emph.end type="italics"/> 14/19, <emph type="italics"/>eruatur radix, tan&shy;<lb/>quam ex accurato numero cubo, ea erit<emph.end type="italics"/> 174: <emph type="italics"/>fer&egrave;, &amp; erit centupla ra&shy;<lb/>dicis numeri<emph.end type="italics"/> 5 49/190, <emph type="italics"/>quia numerus<emph.end type="italics"/> 5 49/190, <emph type="italics"/>multiplicatus fuit per <lb/>cubum ex 100; diameter igitur &longs;ph&aelig;r&aelig; ferre&aelig;, grauitatem habentis <lb/>1, lib. </s> quare radix cubica numeri<emph.end type="italics"/> 5 49/190, <emph type="italics"/>dabit qu&aelig;&longs;itam dia&shy;<lb/>metrum, &amp; quoniam numerus<emph.end type="italics"/> 5 49/190, <emph type="italics"/>non e&longs;t pr&aelig;ci&longs;e cubus, &amp; ideo <lb/>non explicabitur cius radix accurate, multiplicetur per 1000000, &amp; <lb/>ex producto<emph.end type="italics"/> 5257894 14/19, <emph type="italics"/>neglecto fracto<emph.end type="italics"/> 14/19, <emph type="italics"/>eruatur radix, tan&shy;<lb/>quam ex accurato numero cubo, ea erit<emph.end type="italics"/> 174: <emph type="italics"/>fer&egrave;, &amp; erit centupla ra&shy;<lb/>dicis numeri<emph.end type="italics"/> 5 49/190, <emph type="italics"/>quia numerus<emph.end type="italics"/> 5 49/190, <emph type="italics"/>multiplicatus fuit per <lb/>cubum ex 100; diameter igitur &longs;ph&aelig;r&aelig; ferre&aelig;, grauitatem habentis <lb/>1, lib.
  
 <s>erit<emph.end type="italics"/> 1 74/100: <emph type="italics"/>deinde &longs;i duplicetur<emph.end type="italics"/> 5257894 24/19, <emph type="italics"/>&amp; ex ita du&shy;<lb/>plicato eruatur radix cubica<emph.end type="italics"/> 219, <emph type="italics"/>ea dabit centuplum diametri &longs;ph&aelig;&shy;<lb/>ra ferre&aelig;, grauitatem habentis<emph.end type="italics"/> 2, <emph type="italics"/>lib. </s> erit<emph.end type="italics"/> 1 74/100: <emph type="italics"/>deinde &longs;i duplicetur<emph.end type="italics"/> 5257894 24/19, <emph type="italics"/>&amp; ex ita du&shy;<lb/>plicato eruatur radix cubica<emph.end type="italics"/> 219, <emph type="italics"/>ea dabit centuplum diametri &longs;ph&aelig;&shy;<lb/>ra ferre&aelig;, grauitatem habentis<emph.end type="italics"/> 2, <emph type="italics"/>lib.
  
 <s>&amp; &longs;i triplicetur, triplicati radix <lb/>cubica<emph.end type="italics"/> 250<gap/> <emph type="italics"/>dabit centuplum diametri &longs;ph&aelig;r&aelig; ferre&aelig;, cuius grauitas <lb/>crit<emph.end type="italics"/> 3, <emph type="italics"/>lib. </s> &amp; &longs;i triplicetur, triplicati radix <lb/>cubica<emph.end type="italics"/> 250<gap/> <emph type="italics"/>dabit centuplum diametri &longs;ph&aelig;r&aelig; ferre&aelig;, cuius grauitas <lb/>crit<emph.end type="italics"/> 3, <emph type="italics"/>lib.
  
 <s>&amp; &longs;ic reliquarum &longs;ph&aelig;rarum in infinitum inuenientur <lb/>diametri. </s> &amp; &longs;ic reliquarum &longs;ph&aelig;rarum in infinitum inuenientur <lb/>diametri. </s>
  
 <s>multiplicia autem numeri<emph.end type="italics"/> 5257894 14/19, <emph type="italics"/>&longs;ola additione in&shy;<lb/>uenientur, vt dictum est &longs;upra de inuentione multiplicium numeri<emph.end type="italics"/><lb/>5684210 10/19. <emph type="italics"/>Atque hac ratione pr&aelig;dictam tabulam compo&longs;uimus.<emph.end type="italics"/></s></p><p type="margin"> <s>multiplicia autem numeri<emph.end type="italics"/> 5257894 14/19, <emph type="italics"/>&longs;ola additione in&shy;<lb/>uenientur, vt dictum est &longs;upra de inuentione multiplicium numeri<emph.end type="italics"/><lb/>5684210 10/19. <emph type="italics"/>Atque hac ratione pr&aelig;dictam tabulam compo&longs;uimus.<emph.end type="italics"/></s></p><p type="margin">
  
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 <s>I.</s></p><p type="main"> <s>I.</s></p><p type="main">
  
 <s><emph type="italics"/>Sit coron&aelig; grauitas<emph.end type="italics"/> 95, <emph type="italics"/>lib. </s> <s><emph type="italics"/>Sit coron&aelig; grauitas<emph.end type="italics"/> 95, <emph type="italics"/>lib.
  
 <s>&amp; oporteat facere quod imperatum e&longs;t. </s> &amp; oporteat facere quod imperatum e&longs;t. </s>
  
 <s><lb/>Intelligantur duo corpora, vnum aureum, alterum argenteum, &aelig;que <lb/>grauia atque corona, deinde trium corporum ex aqua, magnitudine <lb/>&aelig;qualium, aureo &longs;eilicet corporivnum, alterum coron&aelig;, tertium cor&shy;<lb/>pori argenteo, inueniantur grauitates, vt in exemplo prop.<emph.end type="italics"/> 8. <emph type="italics"/>dictum <lb/>est, qu&aelig; &longs;int primi nimirum corporis aquei<emph.end type="italics"/> 5, <emph type="italics"/>&longs;ecundi vero 6, &amp; terty<emph.end type="italics"/><lb/>9 6/31, <emph type="italics"/>&amp; fiat vt differentia inter<emph.end type="italics"/> 5, &amp; 9 6/32, <emph type="italics"/>hoc e&longs;t vt<emph.end type="italics"/> 4 6/31, <emph type="italics"/>ad<emph.end type="italics"/> 95, <lb/><emph type="italics"/>grauitatem videlicet coron&aelig;, ita differentia inter<emph.end type="italics"/> 5, &amp; 6, <emph type="italics"/>hoc est 1, ad<emph.end type="italics"/><lb/>22 17/26, <emph type="italics"/>ergo<emph.end type="italics"/> 22 17/26, <emph type="italics"/>erit grauitas portionis argenti quod est in coro&shy;<lb/>na, qua detracta ex totali grauitate coron&aelig;, reliquum<emph.end type="italics"/> 72 9/26, <emph type="italics"/>erit <lb/>grauitas portionis auri.<emph.end type="italics"/></s></p><p type="main"> <s><lb/>Intelligantur duo corpora, vnum aureum, alterum argenteum, &aelig;que <lb/>grauia atque corona, deinde trium corporum ex aqua, magnitudine <lb/>&aelig;qualium, aureo &longs;eilicet corporivnum, alterum coron&aelig;, tertium cor&shy;<lb/>pori argenteo, inueniantur grauitates, vt in exemplo prop.<emph.end type="italics"/> 8. <emph type="italics"/>dictum <lb/>est, qu&aelig; &longs;int primi nimirum corporis aquei<emph.end type="italics"/> 5, <emph type="italics"/>&longs;ecundi vero 6, &amp; terty<emph.end type="italics"/><lb/>9 6/31, <emph type="italics"/>&amp; fiat vt differentia inter<emph.end type="italics"/> 5, &amp; 9 6/32, <emph type="italics"/>hoc e&longs;t vt<emph.end type="italics"/> 4 6/31, <emph type="italics"/>ad<emph.end type="italics"/> 95, <lb/><emph type="italics"/>grauitatem videlicet coron&aelig;, ita differentia inter<emph.end type="italics"/> 5, &amp; 6, <emph type="italics"/>hoc est 1, ad<emph.end type="italics"/><lb/>22 17/26, <emph type="italics"/>ergo<emph.end type="italics"/> 22 17/26, <emph type="italics"/>erit grauitas portionis argenti quod est in coro&shy;<lb/>na, qua detracta ex totali grauitate coron&aelig;, reliquum<emph.end type="italics"/> 72 9/26, <emph type="italics"/>erit <lb/>grauitas portionis auri.<emph.end type="italics"/></s></p><p type="main">
  
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 <s><emph type="italics"/>Sit aliquod corpus mistum ex auro. </s> <s><emph type="italics"/>Sit aliquod corpus mistum ex auro. </s>
  
 <s>&amp; &aelig;re, &amp; babeat grauitatem <lb/>171. lib. </s> <s>&amp; &aelig;re, &amp; babeat grauitatem <lb/>171. lib.
  
 <s>&amp; oporteat inuenire quanta erit portio &aelig;ris in ip&longs;o corpore,<emph.end type="italics"/><pb pagenum="56"/><emph type="italics"/>&amp; quanta auri. </s> &amp; oporteat inuenire quanta erit portio &aelig;ris in ip&longs;o corpore,<emph.end type="italics"/><pb pagenum="56"/><emph type="italics"/>&amp; quanta auri. </s>
  
 <s>Intelligantur duo corpo<gap/>i, vnum ex auro puro, al&shy;<lb/>terum ex &aelig;re, &aelig;que grauia atque corpus mistum, &amp; trium corporum <lb/>ex aqua, quorum vnum &longs;it &aelig;quale corpori aureo magnitudine, alte&shy;<lb/>rum misto, tertium &aelig;rco, inueniantur grauitates, vt in exemplo pro&shy;<lb/>po&longs;.<emph.end type="italics"/>8.<emph type="italics"/>dictum est, qu&aelig; &longs;int<emph.end type="italics"/> 9, 11, <emph type="italics"/>&amp; 19, &amp; fiat vt differentia inter<emph.end type="italics"/> 9, &amp; <lb/>19, <emph type="italics"/>ad<emph.end type="italics"/> 171, <emph type="italics"/>grauitatem videlicet corporis misti, ita differentia inter<emph.end type="italics"/><lb/>9, &amp; 11, <emph type="italics"/>ad<emph.end type="italics"/> 34 1/5, <emph type="italics"/>portio igitur corporis mi&longs;ti &aelig;rea grauitatem babebit<emph.end type="italics"/><lb/>34 1/5, <emph type="italics"/>qu&aelig; &longs;i auferatur ex totali corporis misti grauitate, remanebit<emph.end type="italics"/><lb/>136 4/5, <emph type="italics"/>pro grauitate portionis auri.<emph.end type="italics"/></s></p><p type="main"> <s>Intelligantur duo corpo<gap/>i, vnum ex auro puro, al&shy;<lb/>terum ex &aelig;re, &aelig;que grauia atque corpus mistum, &amp; trium corporum <lb/>ex aqua, quorum vnum &longs;it &aelig;quale corpori aureo magnitudine, alte&shy;<lb/>rum misto, tertium &aelig;rco, inueniantur grauitates, vt in exemplo pro&shy;<lb/>po&longs;.<emph.end type="italics"/>8.<emph type="italics"/>dictum est, qu&aelig; &longs;int<emph.end type="italics"/> 9, 11, <emph type="italics"/>&amp; 19, &amp; fiat vt differentia inter<emph.end type="italics"/> 9, &amp; <lb/>19, <emph type="italics"/>ad<emph.end type="italics"/> 171, <emph type="italics"/>grauitatem videlicet corporis misti, ita differentia inter<emph.end type="italics"/><lb/>9, &amp; 11, <emph type="italics"/>ad<emph.end type="italics"/> 34 1/5, <emph type="italics"/>portio igitur corporis mi&longs;ti &aelig;rea grauitatem babebit<emph.end type="italics"/><lb/>34 1/5, <emph type="italics"/>qu&aelig; &longs;i auferatur ex totali corporis misti grauitate, remanebit<emph.end type="italics"/><lb/>136 4/5, <emph type="italics"/>pro grauitate portionis auri.<emph.end type="italics"/></s></p><p type="main">
  
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 <s>Quartus enim ter&shy;<lb/>minus exhibebit grauilatem ip&longs;ius auri in aqua. </s> <s>Quartus enim ter&shy;<lb/>minus exhibebit grauilatem ip&longs;ius auri in aqua. </s>
  
 <s>Vt &longs;i propo&longs;itum <lb/><expan abbr="aur&utilde;">aurum</expan> &longs;it trium lib. </s> <s>Vt &longs;i propo&longs;itum <lb/><expan abbr="aur&utilde;">aurum</expan> &longs;it trium lib.
  
 <s>qualitatis vero 18, partium. </s> qualitatis vero 18, partium. </s>
  
 <s>fiat vt lib. </s> <s>fiat vt lib. </s>
  
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 <s>Nam &longs;i proponatur exemp. </s> <s>Nam &longs;i proponatur exemp. </s>
  
 <s>gratia aurum <lb/>vnius lib. </s> <s>gratia aurum <lb/>vnius lib.
  
 <s>habens in aqua <expan abbr="grauitas&etilde;">grauitasem</expan> vnc.<emph.end type="italics"/> 11. <emph type="italics"/>Scr.<emph.end type="italics"/> 7, <emph type="italics"/>Gra.<emph.end type="italics"/> 14 1152/1767, <lb/><emph type="italics"/>quoniam h&aelig;c grauitas reperitur in tabula &egrave; regione qualitatis auri<emph.end type="italics"/><lb/>22, <emph type="italics"/>partium; manife&longs;tum e&longs;t totidem partium e&longs;&longs;e <expan abbr="aur&utilde;">aurum</expan> propo&longs;itum.<emph.end type="italics"/></s></p><p type="main"> habens in aqua <expan abbr="grauitas&etilde;">grauitasem</expan> vnc.<emph.end type="italics"/> 11. <emph type="italics"/>Scr.<emph.end type="italics"/> 7, <emph type="italics"/>Gra.<emph.end type="italics"/> 14 1152/1767, <lb/><emph type="italics"/>quoniam h&aelig;c grauitas reperitur in tabula &egrave; regione qualitatis auri<emph.end type="italics"/><lb/>22, <emph type="italics"/>partium; manife&longs;tum e&longs;t totidem partium e&longs;&longs;e <expan abbr="aur&utilde;">aurum</expan> propo&longs;itum.<emph.end type="italics"/></s></p><p type="main">
  
 <s><emph type="italics"/>Quando vero grauitas auri in aere quiaem e&longs;t vnius lib. </s> <s><emph type="italics"/>Quando vero grauitas auri in aere quiaem e&longs;t vnius lib.
  
 <s>in aqua <lb/>vero grauitatem habet, qu&aelig; in tabula non reperuur, indicium erit <lb/>aurum propo&longs;itum non e&longs;&longs;e aliquot partium pr&aelig;cis&egrave;, &longs;ed annexam <lb/>habere aliquam fractionem, qu&aelig; per partem proportionalem inue&shy;<lb/>nietur boc modo.<emph.end type="italics"/></s></p><p type="main"> in aqua <lb/>vero grauitatem habet, qu&aelig; in tabula non reperuur, indicium erit <lb/>aurum propo&longs;itum non e&longs;&longs;e aliquot partium pr&aelig;cis&egrave;, &longs;ed annexam <lb/>habere aliquam fractionem, qu&aelig; per partem proportionalem inue&shy;<lb/>nietur boc modo.<emph.end type="italics"/></s></p><p type="main">
  
 <s><emph type="italics"/>Proponatur aurum vnius libr&aelig; in aqua habens grauitatem vnc. </s> <s><emph type="italics"/>Proponatur aurum vnius libr&aelig; in aqua habens grauitatem vnc. </s>
  
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 <s><emph type="italics"/>Denique &longs;i proponatur aurum non vnius libr&aelig; &longs;ed vel plurium, <lb/>vel &longs;olum aliquot vnciarum. </s> <s><emph type="italics"/>Denique &longs;i proponatur aurum non vnius libr&aelig; &longs;ed vel plurium, <lb/>vel &longs;olum aliquot vnciarum. </s>
  
 <s>Reducenda erit eius grauitas quam <lb/>babet maqua, ad grauitatem quam haberet &longs;i e&lt;32&gt;et vnius libr&aelig;, id <lb/>quod ab&longs;oluetur per proportionis ratiocinationem, &longs;i pro termino <lb/>primo ponatur vera grauitas auri propo&longs;iti, pro &longs;ecundo, e iu&longs;dem <lb/>grauitas in aqua, &amp; pro tertio lib. </s> <s>Reducenda erit eius grauitas quam <lb/>babet maqua, ad grauitatem quam haberet &longs;i e&lt;32&gt;et vnius libr&aelig;, id <lb/>quod ab&longs;oluetur per proportionis ratiocinationem, &longs;i pro termino <lb/>primo ponatur vera grauitas auri propo&longs;iti, pro &longs;ecundo, e iu&longs;dem <lb/>grauitas in aqua, &amp; pro tertio lib.
  
 <s>1, quartus enim terminus indica&shy;<lb/>bit grauitatem in aqua re&longs;pondentem vni libr&aelig; auri propo&longs;iti. </s> 1, quartus enim terminus indica&shy;<lb/>bit grauitatem in aqua re&longs;pondentem vni libr&aelig; auri propo&longs;iti. </s>
  
 <s>&amp; <lb/>bac inuenta reliqua expedientur vt prius.<emph.end type="italics"/></s></p><p type="main"> <s>&amp; <lb/>bac inuenta reliqua expedientur vt prius.<emph.end type="italics"/></s></p><p type="main">
  

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