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Colored diff for /texts/archimedes/xml/casat_terra_018_la_1658.xml between version 1.13 and 1.14

version 1.13, 2003/12/09 19:20:50

version 1.14, 2003/12/11 15:05:41

Line 2990

<s>Datis itaque va&longs;is &longs;imilibus inæqualibus, <lb/>& datâ Ratione &longs;uperficierum huiu&longs;modi va<lb/>&longs;orum, poterimus reperire &longs;uperficiem &longs;oli<lb/>di VY aut HT; quæ ex va&longs;orum &longs;uperficie<lb/><arrow.to.target n="note100"/><lb/>bus dempta reliquam aquæ &longs;uperficiem relin<lb/>quat in duplicatâ Ratione &longs;uperficierum va<lb/>&longs;orum. </s>

<s>Cum enim &longs;olidorum &longs;uperficies HT, <lb/>VY habeant Rationem æqualitatis, maior e&longs;t <lb/>Ratio totius &longs;uperficiei va&longs;is maioris ad totam <lb/>&longs;uperficiem minoris, quàm ablatæ VY ad <lb/>ablatam HT, ergo per 33 5. apud Clau. ma<lb/>ior e&longs;t Ratio reliquæ OM ad reliquam CD, <lb/>quàm <gap/>o<gap/> ad totam: quare pote&longs;t eria&mtail; <lb/>haberi Ratio duplicata Rationis totius ad to-<pb xlink:href="018/01/230.jpg" pagenum="216"/>tam. </s>

<s>Cum enim &longs;olidorum &longs;uperficies HT, <lb/>VY habeant Rationem æqualitatis, maior e&longs;t <lb/>Ratio totius &longs;uperficiei va&longs;is maioris ad totam <lb/>&longs;uperficiem minoris, quàm ablatæ VY ad <lb/>ablatam HT, ergo per 33 5. apud Clau. ma<lb/>ior e&longs;t Ratio reliquæ OM ad reliquam CD, <lb/>quàm totius ad totam: quare pote&longs;t etia&mtail; <lb/>haberi Ratio duplicata Rationis totius ad to<pb xlink:href="018/01/230.jpg" pagenum="216"/>tam. </s>

<s>Sic autem inue&longs;ligo &longs;uperficiem VY, <lb/>quæ &longs;it media proportionalis inter differen<lb/>tias illius & &longs;uperficiei va&longs;orum, hoc e&longs;t in<lb/>ter OM & CD. </s>

<s>Sic autem inue&longs;tigo &longs;uperficiem VY, <lb/>quæ &longs;it media proportionalis inter differen<lb/>tias illius & &longs;uperficiei va&longs;orum, hoc e&longs;t in<lb/>ter OM & CD. </s>

<s><margin.target id="note100"/><emph type="italics"/>C <lb/>Ex datis <lb/>duabus &longs;i<gap/><lb/>perficiebus <lb/>auferre ean<lb/>dem &longs;uper fi<lb/>cie<gap/>, quæ re <lb/>linquat re<lb/>sidua in Ra<lb/>tione dupli<lb/>cata data<lb/>rum.<emph.end type="italics"/></s>

<s><margin.target id="note100"/><emph type="italics"/>C <lb/>Ex datis <lb/>duabus &longs;u<lb/>perficiebus <lb/>auferre ean<lb/>dem &longs;uper fi<lb/>ciem, quæ re <lb/>linquat re<lb/>sidua in Ra<lb/>tione dupli<lb/>cata data<lb/>rum.<emph.end type="italics"/></s>

<s>Ratio &longs;uperficierum va&longs;orum &longs;it data 8 <lb/>ad 4; Ratio duplicata e&longs;t 8. ad 2. Pono &longs;u<lb/>perficiem &longs;olidi VY Algebricè 1&rx;. </s>

Line 3016

<s><emph type="italics"/>Gal.<emph.end type="italics"/></s><s> Eadem e&longs;&longs;e ratio omnino non po<lb/>te&longs;t; quia aquæ de&longs;cen&longs;us non ex &longs;olo &longs;patio, <lb/>quod ab eleuatâ terrâ relinqueretur, pende<lb/>ret, quemadmodum &longs;i globus ex aquâ i&ntail; <lb/>va&longs;e eximeretur, &longs;ed etiam ex diuersâ ip&longs;ius <lb/>aquæ in &longs;phæram conformatione. </s>

<s>Cum <lb/>verò aliam &longs;emper & aliam diuei&longs;aru&mtail; <lb/>&longs;phærarum portionem con&longs;titueret, &longs;eu po<lb/>tius Meni&longs;corum &longs;olidorum, donec demùm <lb/>in &longs;phæram integram aqueam di&longs;poneretur, <lb/>nulla certa & con&longs;tans Ratio afferri pote&longs;t; <lb/>&longs;ed dato certo &longs;patio, quod à centro terræ <pb xlink:href="018/01/232.jpg" pagenum="218"/>translato perficeretur in certâ ab vniuer&longs;i <lb/>centro di&longs;tantiâ, inue&longs;tigare oporteret, cuius <lb/>in &longs;phæræ &longs;uperficiem &longs;e data aquæ moles <lb/>di&longs;poneret in vtroque motûs termino, vt in<lb/>de colligeretur, quantum terræ ex aquis i&ntail; <lb/>motu illo emer&longs;i&longs;&longs;et. </s>

<s>Cum <lb/>verò aliam &longs;emper & aliam diuer&longs;aru&mtail; <lb/>&longs;phærarum portionem con&longs;titueret, &longs;eu po<lb/>tius Meni&longs;corum &longs;olidorum, donec demùm <lb/>in &longs;phæram integram aqueam di&longs;poneretur, <lb/>nulla certa & con&longs;tans Ratio afferri pote&longs;t; <lb/>&longs;ed dato certo &longs;patio, quod à centro terræ <pb xlink:href="018/01/232.jpg" pagenum="218"/>translato perficeretur in certâ ab vniuer&longs;i <lb/>centro di&longs;tantiâ, inue&longs;tigare oporteret, cuius <lb/>in &longs;phæræ &longs;uperficiem &longs;e data aquæ moles <lb/>di&longs;poneret in vtroque motûs termino, vt in<lb/>de colligeretur, quantum terræ ex aquis i&ntail; <lb/>motu illo emer&longs;i&longs;&longs;et. </s>

<s><emph type="italics"/>Guld.<emph.end type="italics"/></s><s> Ab&longs;tineamus nunc, &longs;i placet, ab <lb/>hoc labore: quamuis datâ &longs;ectione lunulari, <lb/>& inuentis dimidiæ &longs;ectionis centro grauita<lb/>tis, ac viâ rotationis, po&longs;&longs;emus &longs;oliditatem <lb/>Meni&longs;coidis &longs;phærici inuenire; &longs;atis nunc <lb/>nobis e&longs;&longs;e puto inquirere; quantam in alti<lb/>tudinem &longs;u&longs;tolli terram oporteret, vt nul<lb/>lum amplius &longs;ub&longs;idium afferret aqua circum<lb/>fu&longs;a minuens terræ grauitatem. </s>

Line 3042

<s>Cæterùm fieri non pote&longs;t, <lb/>quin plures requirantur operationes, cu&mtail; <lb/>præter cubum primi lateris oporteat inueni<lb/>re &longs;olidum ex triplo quadrato primi lateris in <lb/>latus &longs;ecundum, & &longs;olidum ex triplo later&etail; <lb/>primo in quadratum lateris &longs;ecundi, & de<lb/>mum ip&longs;ius lateris &longs;ecundi cubum. </s>

<s>Quar&etail; <lb/>hic in &longs;chedula, quoniam &longs;ic placet, dato-<pb xlink:href="018/01/234.jpg" pagenum="220"/>rum cuborum radicem extraham in v&longs;u&mtail; <lb/>no&longs;træ quæ&longs;tionis. </s>

<s>Quar&etail; <lb/>hic in &longs;chedula, quoniam &longs;ic placet, dato<pb xlink:href="018/01/234.jpg" pagenum="220"/>rum cuborum radicem extraham in v&longs;u&mtail; <lb/>no&longs;træ quæ&longs;tionis. </s>

<s>Hæc autem mihi e&longs;t re<lb/>gul&atail;. </s>

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<s><lb/>Summam demùm ex propo&longs;ito numero &longs;ub<lb/>duco, & &longs;ic deinceps. </s>

<s>Quoniam it aque numeri hìc propo&longs;iti &longs;unt <lb/>18 ciphrarum, & &longs;unt &longs;ex puncta, vt tempo<lb/>ri parcam & labori, prima quatuor punct&atail; <lb/>accipto pro primo puncto; & ex Tabulà, in <lb/>qua habeo myriadem cuborum, inuenio ma-<pb xlink:href="018/01/235.jpg" pagenum="221"/>ximum cubum C, cuius latus e&longs;t A: &longs;ubtra<lb/>ho C ex D, & remanet E, cui addo tres fi<lb/>guras ad &longs;equens punctum pertinentes. </s>

<s>Quoniam it aque numeri hìc propo&longs;iti &longs;unt <lb/>18 ciphrarum, & &longs;unt &longs;ex puncta, vt tempo<lb/>ri parcam & labori, prima quatuor punct&atail; <lb/>accipio pro primo puncto; & ex Tabulà, in <lb/>qua habeo myriadem cuborum, inuenio ma<pb xlink:href="018/01/235.jpg" pagenum="221"/>ximum cubum C, cuius latus e&longs;t A: &longs;ubtra<lb/>ho C ex D, & remanet E, cui addo tres fi<lb/>guras ad &longs;equens punctum pertinentes. </s>

<s>De<lb/>inde triplico A, & e&longs;t F, quem &longs;ub penulti<lb/>mâ figurâ colloco, cui &longs;ub&longs;cribo A &longs;ub figu<lb/>râ antepenultimâ. </s>

<s>Duco A in F, & produ<lb/>citur G: & a&longs;&longs;umpto G tanquam diui&longs;ore nu<lb/>meri E, habeo Quotientem B, &longs;cilicet latus <lb/>&longs;ecundum: Tum B in F duco, & 'produci<lb/>tur H &longs;ub penultimâ figurâ. </s>

<s>Duco A in F, & produ<lb/>citur G: & a&longs;&longs;umpto G tanquam diui&longs;ore nu<lb/>meri E, habeo Quotientem B, &longs;cilicet latus <lb/>&longs;ecundum: Tum B in F duco, & produci<lb/>tur H &longs;ub penultimâ figurâ. </s>

<s>Additis G & H, <lb/>fit &longs;umma I: quæ per latus &longs;ecundum duct&atail; <lb/>producit K; cui &longs;ub puncto dati numeri &longs;ub<lb/>&longs;cribitur lateris &longs;ecundi B cubus L; atque ex <lb/>K & L fit &longs;umma M auferenda ex E: & qui re<lb/>linquitur numerus N, pertinet ad &longs;equens <lb/>punctum. </s>

<s>Quod &longs;pectat ad numerorum collocatio<lb/>nem &longs;ub penultimâ vel antepenultimá figu<lb/><arrow.to.target n="note103"/><lb/>râ, &longs;atis videtis id factum e&longs;&longs;e propter com<lb/>pendium, quo omittuntur ciphræ nullita<lb/>tis, vltimo loco addendæ; e&longs;t &longs;iquidem la<lb/>teris primi numerus decadicus, &longs;i cum late<lb/>re &longs;ecundo comparetur: quare dum triplica<lb/>tur, & fit F, vna ciphra nullitatis e&longs;&longs;et &longs;ub <lb/>puncto: & dum hoc triplum per ip&longs;um la<lb/>tus ducitur, in producto e&longs;&longs;ent vltimo loco <lb/>duæ ciphræ ideò &longs;ub antepenultimâ figurâ <lb/>collocatur numerus productus, vt relinqua-</s><pb xlink:href="018/01/236.jpg"/>

<s>Quod &longs;pectat ad numerorum collocatio<lb/>nem &longs;ub penultimâ vel antepenultimá figu<lb/><arrow.to.target n="note103"/><lb/>râ, &longs;atis videtis id factum e&longs;&longs;e propter com<lb/>pendium, quo omittuntur ciphræ nullita<lb/>tis, vltimo loco addendæ; e&longs;t &longs;iquidem la<lb/>teris primi numerus decadicus, &longs;i cum late<lb/>re &longs;ecundo comparetur: quare dum triplica<lb/>tur, & fit F, vna ciphra nullitatis e&longs;&longs;et &longs;ub <lb/>puncto: & dum hoc triplum per ip&longs;um la<lb/>tus ducitur, in producto e&longs;&longs;ent vltimo loco <lb/>duæ ciphræ ideò &longs;ub antepenultimâ figurâ <lb/>collocatur numerus productus, vt relinqua</s><pb xlink:href="018/01/236.jpg"/>

<s><margin.target id="note103"/><emph type="italics"/>CIII <lb/>Huius me<lb/>th<gap/>di r<gap/>tio <lb/>ostenditur.<emph.end type="italics"/></s>

<s><margin.target id="note103"/><emph type="italics"/>CIII <lb/>Huius me<lb/>thodi ratio <lb/>ostenditur.<emph.end type="italics"/></s>

<s>TABELLE WAR HIER<pb xlink:href="018/01/237.jpg" pagenum="223"/>tur locus ciphrarum: id quod vt certiùs fiat, <lb/>colloco A &longs;ub F ita, vt A &longs;it &longs;ub antepenulti<lb/>mâ figurâ. </s>

<s>TABELLE WAR HIER</s><pb xlink:href="018/01/237.jpg" pagenum="223"/>

<s>tur locus ciphrarum: id quod vt certiùs fiat, <lb/>colloco A &longs;ub F ita, vt A &longs;it &longs;ub antepenulti<lb/>mâ figurâ. </s>

<s>Quia verò quando B in F duci<lb/>tur, vnica tantùm e&longs;&longs;et ciphra nullitatis ad <lb/>ip&longs;um F pertinens, ideò productus H &longs;cribi<lb/>tur &longs;ub penultimâ figurâ. </s>

<s>Cubus demùm L <lb/>&longs;ub puncto collocatur, quia nullam habet <lb/>ciphram nullitatis, quæ omittatur. </s>

<s>Cubus demùm L <lb/>&longs;ub puncto collocatur, quia nullam habet <lb/>ciphram nullitatis, quæ omittatur. </s>

<s>Quod autem pertinet ad ip&longs;am metho<lb/>dum, res clara e&longs;t. </s>

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