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Colored diff for /texts/archimedes/xml/carda_propo_015_la_1570.xml between version 1.25 and 1.26

version 1.25, 2004/04/06 17:28:10 version 1.26, 2004/04/06 17:37:36
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           <s id="id002695"><arrow.to.target n="marg520"/><lb/>4, &longs;ed iunctis 4 &amp; 3, &amp; abiecto 6 &longs;upere&longs;t 1, ergo 5 in 5 <expan abbr="duct&utilde;">ductum</expan>, &amp; diui<lb/>&longs;o producto relinquitur 1. Et &longs;imiliter capio 17, et <expan abbr="componi&ttilde;">componitur</expan> ex 12 &amp; <lb/>5 quadratum, ergo 17 componitur ex quadrato 12, in quo nihil &longs;u&shy;<lb/>pere&longs;t, &amp; duplo 5 in 12, in quo <expan abbr="eti&atilde;">etiam</expan> nihil &longs;upere&longs;t, &longs;i diuidatur per 6: <lb/>&amp; ex quadrato 5, in quo &longs;upere&longs;t 1, ergo in nullo numero <expan abbr="c&otilde;po&longs;ito">compo&longs;ito</expan> <lb/>ex 5 &amp; 6, uel compo&longs;itis ex 6, poterit produci numerus, qui diui&longs;us <lb/>per 6 relinquat 5, igitur neque talis numerus pot&eacute;rit <expan abbr="c&otilde;poni">componi</expan> ex duo&shy;<lb/>bus quadratis, in quib. </s>           <s id="id002695"><arrow.to.target n="marg520"/><lb/>4, &longs;ed iunctis 4 &amp; 3, &amp; abiecto 6 &longs;upere&longs;t 1, ergo 5 in 5 <expan abbr="duct&utilde;">ductum</expan>, &amp; diui<lb/>&longs;o producto relinquitur 1. Et &longs;imiliter capio 17, et <expan abbr="componi&ttilde;">componitur</expan> ex 12 &amp; <lb/>5 quadratum, ergo 17 componitur ex quadrato 12, in quo nihil &longs;u&shy;<lb/>pere&longs;t, &amp; duplo 5 in 12, in quo <expan abbr="eti&atilde;">etiam</expan> nihil &longs;upere&longs;t, &longs;i diuidatur per 6: <lb/>&amp; ex quadrato 5, in quo &longs;upere&longs;t 1, ergo in nullo numero <expan abbr="c&otilde;po&longs;ito">compo&longs;ito</expan> <lb/>ex 5 &amp; 6, uel compo&longs;itis ex 6, poterit produci numerus, qui diui&longs;us <lb/>per 6 relinquat 5, igitur neque talis numerus pot&eacute;rit <expan abbr="c&otilde;poni">componi</expan> ex duo&shy;<lb/>bus quadratis, in quib. </s>
           <s id="id002696">&longs;uper&longs;it 5 &amp; 1, quia nullus e&longs;t, in quo &longs;uper&shy;<lb/>&longs;it 5 facta diui&longs;ione per 6. Ex quo colligitur una regula: quod &longs;i quis <lb/>dicat multiplicaui 27 in &longs;e, et diui&longs;i per 13, uellem &longs;cire quid &longs;upere&longs;t, <lb/>dico quod &longs;ine multiplicatione et diui&longs;ione poteris hoc &longs;cire ex de&shy;<lb/>mon&longs;tratione dicta, diuide ergo 27 per 13, &amp; relinquitur 1, duc in &longs;e <lb/>fit 1: dices ergo, quod &longs;upererit 1, &amp; ita &longs;i ducerem 28 in &longs;e, &amp; diuide&shy;<lb/>rem per 11, dico quod &longs;upererit 3, nam diui&longs;o 28 per 11, relinquitur <lb/>6, duc in 6 fit 36, diuide per 11, relinquitur 3, ut dictum e&longs;t, &amp; tantum <lb/><expan abbr="relinqui&ttilde;">relinquitur</expan> ducto 28 in &longs;e &amp; fit 784, &amp; diui&longs;o per 11. Reuertendo ergo <lb/>ad propo&longs;itum, pater quod ex duobus tantum numeris imparibus <lb/>quadratis pote&longs;t conflari ille numerus, <expan abbr="quor&utilde;">quorum</expan> radices diui&longs;&aelig; per 6 <lb/>relinquunt 3. Sed de paribus uel &longs;upere&longs;t 2 uel 4 uel nihil, &longs;ed <expan abbr="&qtilde;dra&shy;tum">quadra&shy;<lb/>tum</expan> 2 e&longs;t 4, &amp; <expan abbr="&qtilde;dratum">quadratum</expan> 4 diui&longs;um per 6 etiam relinquit 4, ergo neque <lb/>ex duobus numeris, in quibus &longs;uper&longs;int 2, neque in quibus &longs;uper&longs;int <lb/>4, neque in quibus &longs;uper&longs;int in uno 2, in altero 4 <expan abbr="poter&utilde;t">poterunt</expan> quadrata, in <lb/>quibus &longs;emper &longs;upererit 4, &amp; iuncta faciunt 8, in quod ĚŠ&longs;upere&longs;t 2, <expan abbr="c&otilde;">con</expan>fla&shy;<lb/>re <expan abbr="numer&utilde;">numerum</expan> <expan abbr="dict&utilde;">dictum</expan> &longs;eu <expan abbr="qu&aelig;&longs;it&utilde;">qu&aelig;&longs;itum</expan>, qui po&longs;sit diuidi per 6: neque ex <expan abbr="&qtilde;d">quad</expan>. </s>           <s id="id002696">&longs;uper&longs;it 5 &amp; 1, quia nullus e&longs;t, in quo &longs;uper&shy;<lb/>&longs;it 5 facta diui&longs;ione per 6. Ex quo colligitur una regula: quod &longs;i quis <lb/>dicat multiplicaui 27 in &longs;e, et diui&longs;i per 13, uellem &longs;cire quid &longs;upere&longs;t, <lb/>dico quod &longs;ine multiplicatione et diui&longs;ione poteris hoc &longs;cire ex de&shy;<lb/>mon&longs;tratione dicta, diuide ergo 27 per 13, &amp; relinquitur 1, duc in &longs;e <lb/>fit 1: dices ergo, quod &longs;upererit 1, &amp; ita &longs;i ducerem 28 in &longs;e, &amp; diuide&shy;<lb/>rem per 11, dico quod &longs;upererit 3, nam diui&longs;o 28 per 11, relinquitur <lb/>6, duc in 6 fit 36, diuide per 11, relinquitur 3, ut dictum e&longs;t, &amp; tantum <lb/><expan abbr="relinqui&ttilde;">relinquitur</expan> ducto 28 in &longs;e &amp; fit 784, &amp; diui&longs;o per 11. Reuertendo ergo <lb/>ad propo&longs;itum, pater quod ex duobus tantum numeris imparibus <lb/>quadratis pote&longs;t conflari ille numerus, <expan abbr="quor&utilde;">quorum</expan> radices diui&longs;&aelig; per 6 <lb/>relinquunt 3. Sed de paribus uel &longs;upere&longs;t 2 uel 4 uel nihil, &longs;ed <expan abbr="&qtilde;dra&shy;tum">quadra&shy;<lb/>tum</expan> 2 e&longs;t 4, &amp; <expan abbr="&qtilde;dratum">quadratum</expan> 4 diui&longs;um per 6 etiam relinquit 4, ergo neque <lb/>ex duobus numeris, in quibus &longs;uper&longs;int 2, neque in quibus &longs;uper&longs;int <lb/>4, neque in quibus &longs;uper&longs;int in uno 2, in altero 4 <expan abbr="poter&utilde;t">poterunt</expan> quadrata, in <lb/>quibus &longs;emper &longs;upererit 4, &amp; iuncta faciunt 8, in quod &longs;upere&longs;t 2, <expan abbr="c&otilde;">con</expan>fla&shy;<lb/>re <expan abbr="numer&utilde;">numerum</expan> <expan abbr="dict&utilde;">dictum</expan> &longs;eu <expan abbr="qu&aelig;&longs;it&utilde;">qu&aelig;&longs;itum</expan>, qui po&longs;sit diuidi per 6: neque ex <expan abbr="&qtilde;d">quad</expan>. </s>
           <s id="id002697"><expan abbr="duo&shy;r&utilde;">duo&shy;<lb/>rum</expan> <expan abbr="num&etilde;ror&utilde;">numerorum</expan>, in <expan abbr="quor&utilde;">quorum</expan> altero nihil &longs;uper&longs;it in reliquo &longs;uper&longs;it 2 uel <lb/>4, quia in aggregato <expan abbr="&qtilde;drator&utilde;">quadratorum</expan> &longs;emper &longs;upererit 4. Ergo relinqui&shy;<lb/>tur quod ille numerus componetur ex duobus quadratis, uel impa<lb/>ribus, quorum latera diui&longs;a per 6 relinquunt 3, uel ex duobus pari&shy;<lb/>bus, quorum latera diui&longs;a per 6 nihil relinquant. </s>           <s id="id002697"><expan abbr="duo&shy;r&utilde;">duo&shy;<lb/>rum</expan> <expan abbr="num&etilde;ror&utilde;">numerorum</expan>, in <expan abbr="quor&utilde;">quorum</expan> altero nihil &longs;uper&longs;it in reliquo &longs;uper&longs;it 2 uel <lb/>4, quia in aggregato <expan abbr="&qtilde;drator&utilde;">quadratorum</expan> &longs;emper &longs;upererit 4. Ergo relinqui&shy;<lb/>tur quod ille numerus componetur ex duobus quadratis, uel impa<lb/>ribus, quorum latera diui&longs;a per 6 relinquunt 3, uel ex duobus pari&shy;<lb/>bus, quorum latera diui&longs;a per 6 nihil relinquant. </s>
           <s id="id002698">Oportet igitur <lb/>inuenire duos tales numeros quadratos numerorum imparium, in <lb/>quibus &longs;uper&longs;it 3, &longs;i diuidantur per 6, aut parium in quibus nihil &longs;u&shy;<lb/>per&longs;it, quorum aggregato diui&longs;o per 6 prodeat numerus <expan abbr="&qtilde;dratus&apos;">quadratus&apos;</expan>.</s>           <s id="id002698">Oportet igitur <lb/>inuenire duos tales numeros quadratos numerorum imparium, in <lb/>quibus &longs;uper&longs;it 3, &longs;i diuidantur per 6, aut parium in quibus nihil &longs;u&shy;<lb/>per&longs;it, quorum aggregato diui&longs;o per 6 prodeat numerus <expan abbr="&qtilde;dratus&apos;">quadratus&apos;</expan>.</s>
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