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Colored diff for /texts/archimedes/xml/gheta_promo_029_la_1603.xml between version 1.2 and 1.9

version 1.2, 2003/11/11 18:49:36

version 1.9, 2003/11/17 17:54:45

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<s>Et vero illis non &longs;olum nos plurimum de<lb/>bemus, quod plurima ip&longs;i perfecere: verum etiam quod <lb/>quædam qua&longs;i fundamenta iecere, quibus dum rerum <lb/><expan abbr="nouar&utilde;">nouarum</expan> molem conamur imponere, nos quoque experi<lb/>ri no&longs;tras vires, exercere indu&longs;triam, remque &longs;apientiæ <lb/>publicam amplificare po&longs;simus. </s>

<s>Quo in genere magno<lb/>rum ego virorum &longs;tudium potius, quam gloriam æmu<lb/>latus &longs;uper vnum ex Archimedeis fundamentis, de ra<lb/>tione, qua varia corporum genera inter &longs;e grauitate & <lb/>magnitudine comparantur, fabricatus nonnulla &longs;um: <pb/>quæ nunc omnium oculis expo&longs;iturus, vt eam <expan abbr="&longs;u&longs;tineã">&longs;u&longs;tineam</expan>, <lb/>per&longs;onam, quam &longs;emper recu&longs;aui, patrocinium huiu&longs;<lb/>modi quærendum miniexi&longs;timaui, quod & &longs;mbecillita <lb/>tem meam contra obtrectatorum, &longs;i qui forte exi&longs;terent, <lb/>calumnias &longs;u&longs;tineret; & imminentem famæ, exi&longs;timatio <lb/>ni&longs;que iacturam auerteret. </s>

<s>Quo in genere magno<lb/>rum ego virorum &longs;tudium potius, quam gloriam æmu<lb/>latus &longs;uper vnum ex Archimedeis fundamentis, de ra<lb/>tione, qua varia corporum genera inter &longs;e grauitate & <lb/>magnitudine comparantur, fabricatus nonnulla &longs;um: <pb/>quæ nunc omnium oculis expo&longs;iturus, vt eam <expan abbr="&longs;u&longs;tineã">&longs;u&longs;tineam</expan>, <lb/>per&longs;onam, quam &longs;emper recu&longs;aui, patrocinium huiu&longs;<lb/>modi quærendum mihi exi&longs;timaui, quod & imbecillita<lb/>tem meam contra obtrectatorum, &longs;i qui forte exi&longs;terent, <lb/>calumnias &longs;u&longs;tineret; & imminentem famæ, exi&longs;timatio <lb/>ni&longs;que iacturam auerteret. </s>

<s>Vnus igitur tu occuri&longs;ti SE<lb/>RAPHINE qui & <expan abbr="cõmodi&longs;sime">commodi&longs;sime</expan> mihi patrocinari po&longs;<lb/>&longs;es, & quodam qua&longs;i iure deberes. </s>

Line 48

<s>Quid dicam de &longs;ingulari <lb/><expan abbr="caq;">caque</expan> eximia rerum c&oelig;le&longs;tium, <expan abbr="totiu&longs;q;">totiu&longs;que</expan> mundi cognitio<lb/>ne quam tu tanta cum auiditate ex reconditi&longs;simis Ma<lb/>thematicorum fontibus hau&longs;i&longs;ti, vt illud a&longs;&longs;ecutus in eo <lb/>genere iam &longs;is, quod alij in maxima tranquillitate, in <lb/>&longs;ummo otio vix au&longs;int optare? </s>

<s>Exitum tuorum labor um <lb/>f&oelig;lici&longs;simum vides: gloria multiplici frueris, <gap/>eque illa <lb/>precaria &longs;od tua & quibu&longs;dam qua&longs;i gradibus ad ampli&longs;<lb/>&longs;<gap/>mos: honores euehendus, in ea con&longs;titueris dignitate; in <lb/>qua pro &longs;acro&longs;anta Eccle&longs;ia <gap/>quam non excubando, <lb/>in peramplo tot illu&longs;trium vitorum Theatro non alienæ <lb/>gloriæ &longs;pectator, &longs;ed actor tuæ con&longs;i&longs;tas. </s>

<s>Exitum tuorum labor um <lb/>f&oelig;lici&longs;simum vides: gloria multiplici frueris, <gap/>eque illa <lb/>precaria &longs;ed tua & quibu&longs;dam qua&longs;i gradibus ad ampli&longs;<lb/>&longs;<gap/>mos: honores euehendus, in ea con&longs;titueris dignitate; in <lb/>qua pro &longs;acro&longs;ancta Eccle&longs;ia <gap/>quam non excubando, <lb/>in peramplo tot illu&longs;trium vitorum Theatro non alienæ <lb/>gloriæ &longs;pectator, &longs;ed actor tuæ con&longs;i&longs;tas. </s>

<s>Tu vero, quod <lb/>rarum e&longs;t, laudem &longs;a pientiæ, quæ vix vllos habeb at ter<lb/>minos, humanitatis tuæ terminis circum&longs;cribis; & expo<lb/>nis omnibus<gap/> vt extanto &longs;onte perennes ad omnium or<lb/>dinum homines fiuuli deducantur. </s>

<s>Tu vero, quod <lb/>rarum e&longs;t, laudem &longs;apientiæ, quæ vix vllos habet at ter<lb/>minos, humanitatis tuæ terminis circum&longs;cribis; & expo<lb/>nis omnibus<gap/> vt extanto &longs;onte perennes ad omnium or<lb/>dinum homines fiuuli deducantur. </s>

<s>F&oelig;lix qui &longs;olidæ <lb/>f&oelig;licitatis car&longs;am & in&longs;tium in re con&longs;titutumita foues, <lb/>vt cum alijs illam communicando, non imminuas, &longs;ed <lb/>amplifices, prægrande videlicet non &longs;uccre&longs;centis, &longs;ed <lb/>adultæ iam virtutis f&oelig;nus honorem ex honore, laudem <lb/>ex laude con&longs;equi vberiorem, hæcilla &longs;apienti viro non <lb/>indigna liberalitas, quæ terum pre&longs;tanti&longs;simarum po&longs;<lb/>&longs;e&longs;sione non imminuta, in copia tenuitatem non inqui-<pb/>rens vbertatis ip&longs;a&longs;uæ dominæ nunquam debilitatur, <lb/>nunquam deficit. </s>

Line 62

<s><lb/>Hinc domus tua floret docti&longs;simorum familiaritatibus; <lb/>hinc nulli ad tuam con&longs;uetudinom præcluditur aditus; <lb/>hinc plurimorum &longs;tudia commouentur; hinc illa &longs;apien <lb/>tum æmulatio & admiratio<gap/> hinc omnium omnino or<lb/>dinum ad te concur&longs;us; tanquam ad &longs;apienti&longs;simum hu<lb/>mani <expan abbr="diuiniq;">diuinique</expan> iuris paponum, & interpetem. </s>

<s>Quid ego <lb/>igitur quem tibi &longs;ex&oelig;ntis eximiæ beneuolenti&oelig; argu<lb/>mentis obligatum vo<gap/>&longs;ti, perexigitum hunc ingenij <lb/>mei partum exp<gap/>&longs;uiuus, vnum te nominis & exi&longs;tima<lb/>tionis meæ patronum non &longs;u&longs;eipiam<gap/> equide<gap/> in am<lb/>pli&longs;simi Theatri lucern &longs;apienti&longs;simorum<32> hominum <lb/>con&longs;pectum, quibus abundat hæd no&longs;tra ætas inferre me <lb/>non magnopere cogitabam <gap/>Tu hortatus es dubitan<lb/>tem impuli&longs;ti vel reluctamem<gap/> tuereigitur ob&longs;equen<lb/>tem. </s>

<s>Quid ego <lb/>igitur quem tibi &longs;excentis eximiæ beneuolenti&oelig; argu<lb/>mentis obligatum vo<gap/>&longs;ti, perexigitum hunc ingenij <lb/>mei partum exp<gap/>&longs;uiuus, vnum te nominis & exi&longs;tima<lb/>tionis meæ patronum non &longs;u&longs;eipiam<gap/> equide<gap/> in am<lb/>pli&longs;simi Theatri lucern &longs;apienti&longs;simorum<32> hominum <lb/>con&longs;pectum, quibus abundat hæd no&longs;tra ætas inferre me <lb/>non magnopere cogitabam <gap/>Tu hortatus es dubitan<lb/>tem impuli&longs;ti vel reluctamem<gap/> tuereigitur ob&longs;equen<lb/>tem. </s>

<s>iam nunc mihi videor in exigua vel nulla &longs;pe lau<lb/>dis ex ingenio meo; &longs;empiternam nominis immortali<lb/>tatem ex tuo patrocinio con&longs;ecutus. </s>

Line 98

<s>Rubeo ba <lb/>mine mihi tum ex &longs;tudiorum &longs;imilitudine, tum pracipue ex <lb/>eius humanitate amici&longs;simo, ad Reuerendi&longs;simum Sera<lb/>phinum deductus &longs;um. </s>

<s>I&longs;qs me tantus Præ&longs;ul non &longs;olum <lb/>humani&longs;sime complexus est, verum etiam ita bo<gap/>tus ad <lb/>eualganda, que &longs;crip&longs;eram; plane vt mihi nefas putauerim <lb/>non parere. </s>

<s>I&longs;qs me tantus Præ&longs;ul non &longs;olum <lb/>humani&longs;sime complexus est, ver <gap/> ets am ita bo<gap/>tus ad <lb/>eualganda, que &longs;crip&longs;eram; plane vt mihi nefas putauerim <lb/>non parere. </s>

<s>Aecipeigitur & tu Lactor optime amico ac be<lb/>uigno auimo laborem hunc, quem à me tali<gap/><lb/>&longs;umma benignitas expre&longs;sit. </s>

<s>Argumentum quidem, vt <lb/>dicebam non iniucundum e&longs;t, nec ab v&longs;u alienam. </s>

<s>Argumentum quidem, vt <lb/>dicob am non iniucundum e&longs;t, nec ab v&longs;u alienam. </s>

<s>Huiu&longs;<lb/>medi enim comparatione Archimides mixtionem argemi <lb/>in aure deprebeudit, & furtum Auri&longs;itis in corona aurea <lb/>patefecit. </s>

<s>de quæ re &longs;uo leco ego tractabo, & facilem mon<lb/>strabo viam, quæ vel argentum in aure, vel quod visme<lb/>tallum in quolibet admistum diprebeudi queat, & alte <lb/><gap/> ab aliero di&longs;rerni & &longs;imu<gap/>, quo pacto ex au<lb/>ri grauitate <gap/> qualiras, <gap/>perfectio intelligi po&longs;sit. </s>

<s>de quæ re &longs;uo leco ego tractabo, & facilem mon<lb/>strabo viam, quæ vel argentum in aure, vel quod visme<lb/>tallum in quolibet admistum diprebeudi queat, & alte <lb/><gap/> ab aliero di&longs;rerni & &longs;imul explicabo, quo pacto ex au<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<lb/>quer impo&longs;ui Nam cum ille, vt erat &longs;ummus Magi&longs;ter, &longs;e<lb/>tis haliui&longs;&longs;et hunc, totem qu<gap/> fabr<gap/> pe&longs;ito fundumen<lb/>to delineare in primo lib.

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<s>Qvidam proponit aliquod corpus &longs;olidum notæ <lb/>magnitudinis, & vult &longs;cire quanta erit magnitudo <lb/>alicuius liquidi, grauitatem habentis propo&longs;ito corpori <lb/>&longs;olido æqualem. </s>

<s><emph type="italics"/>Sit propo&longs;itum aliquod corpus plumbeum A, cuius magnitudo &longs;it<emph.end type="italics"/><lb/>10<emph type="italics"/>, & oporteat &longs;cire quanta erit magnitudo aquæ grauitatem baben<lb/>tis æqualem propo&longs;ito plumbo A, accipiatur aliquod corpus <expan abbr="plumbe&utilde;">plumbeum</expan> <lb/>D, cuius grauitas<emph.end type="italics"/> 23<emph type="italics"/>, deinde aquæ magnitudinem habentis æqualem <lb/>plumbo D, inueniatur grauitas, vt in exemplo propo&longs;. </s>

<s><emph type="italics"/>Sit propo&longs;itum aliquod corpus plumbeum A, cuius magnitudo &longs;it<emph.end type="italics"/><lb/>10<emph type="italics"/>, & oporteat &longs;cire quanta erit magnitudo aquæ grauitatem haben<lb/>tis æqualem propo&longs;ito plumbo A, accipiatur aliquod corpus <expan abbr="plumbe&utilde;">plumbeum</expan> <lb/>D, cuius grauitas<emph.end type="italics"/> 23<emph type="italics"/>, deinde aquæ magnitudinem habentis æqualem <lb/>plumbo D, inueniatur grauitas, vt in exemplo propo&longs;. </s>

<s>8. dictum e&longs;t, <lb/>quæ &longs;it<emph.end type="italics"/> 2<emph type="italics"/>, & fiat vt<emph.end type="italics"/> 2<emph type="italics"/>, ad<emph.end type="italics"/> 23<emph type="italics"/>, ita<emph.end type="italics"/> 10<emph type="italics"/>, ad alium numerum qui&longs;it 115, is <lb/>igitur indicabit quanta erit magnitudo aquæ grauitatem habentis <lb/>æqualem propo&longs;ito plumbo A.<emph.end type="italics"/></s>

Line 376

<s>SINT propo&longs;ita duo corpora æ<lb/>quæ grauia, A, quidem &longs;olidum, B, <lb/>vero liquidum, &longs;it autem, liquidi B; <lb/><figure id="fig10"/><lb/>data magnitudo F, & oporteat &longs;olidi <lb/>A, magnitudinem inuenire. </s>

<s>Accipia<lb/>tur aliquod corpus &longs;olidum D, <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> <lb/>generis cum corpore &longs;olido A, cuius <lb/>grauitas &longs;it G, deinde liquidi quod &longs;it <lb/>E, eiu&longs;dem generis cum corpore liqui <lb/>do B, magnitudinem æqualem<arrow.to.target n="note24"/> haben<lb/>tis &longs;olido D, *inueniatur grauitas, <lb/>quæ &longs;it H, & fiat vt grauitas G; ad gra<lb/>uitatem H, ita F, magnitudo, ad aliam magnitudinem, quæ &longs;it C; <lb/>quoniam igitur &longs;unt quatuor corpora D, E, A, B, quorum primum D, <lb/>& &longs;ecundum E, &longs;unt magnitudine æqualia, tertium vero A, & <expan abbr="quart&utilde;">quartum</expan> <lb/>B, æquæ grauia, & &longs;unt eiu&longs;dem generis &longs;olida D, A, &longs;imiliter, & liqui<lb/>da E, B,* erit vt grauitas G, ad grauitatem H, ita F, magnitudo ad ma<arrow.to.target n="note25"/><lb/>gnitudinem &longs;olidi A, &longs;ed vt grauitas G, ad grauitatem H, ita e&longs;t ma<lb/>gnitudo F, ad C, magnitudinem, ergo magnitudo C, æqualis erit ma<lb/>gnitudini corporis &longs;olidi A, inuenta igitur e&longs;t corporis &longs;olidi A, ma<lb/>gnitudo C, quod erat &longs;aciendum. </s>

<s><margin.target id="note24"/>8.<emph type="italics"/>huius<emph.end type="italics"/></s>

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<s>Qvod &longs;i propo&longs;ita duo corpora æque grauia A, B, fue <lb/>rint regularia vtpote &longs;phærica, fuerit autem liqui-<pb pagenum="16"/>dæ &longs;phæræ B, data diameter F, & oporteat inuenire quan <lb/>ta erit diameter &longs;olidæ &longs;phæræ A, ita faciendum erit. </s>

<s>Accepto vt &longs;upra corpore &longs;olido D, & liquid E, inuenta grauitate, <lb/>vt dictum e&longs;t, fiat vt grauitas G, ad prauitatem H, ita cubus ex F, ad <lb/>alium cubum, cuius latus &longs;it C, Quoniam igitur eadem ratione qua <lb/>fupra o&longs;tendetut, vt grauitas G, ad grauitatem H, ita e&longs;&longs;e magnitudi<lb/>nem &longs;phæræ B, ad &longs;phæræ A, magnitudinem, &longs;ed magnitudo &longs;phæræ B, <lb/><arrow.to.target n="note26"/> ad magnitudinem &longs;phæræ A, *triplicatam rationem habet cius, quam <lb/>F, diameter &longs;phæræ B, ad diametrum &longs;phæræ A, &longs;imiliter, & cubus ex <lb/><arrow.to.target n="note27"/> F, ad cubum ex diametro &longs;phæræ A, <expan abbr="triplicatã*">triplicatan*</expan> rationem liabet eius, <lb/>quam F, ad diametrum &longs;phæræ A, ergo, vt grauitas G, ad <expan abbr="grauitat&etilde;">grauitatem</expan> <lb/>H, ita erit cubus ex F, ad cubum ex diametro &longs;phæræ A, &longs;ed vt graui<lb/>tas G, ad grauitatem H, ita e&longs;t cubus ex F, ad cubum ex C, ergo cubus <lb/>ex C, æqualis erit eubo diametri &longs;phæræ A, quare, & latus C, æquabi<lb/>tur ip&longs;ius &longs;phæræ A, diametro, inuenta igitur e&longs;t quantitas diametri <lb/>&longs;olidæ &longs;phæræ A, quod facere oportebat. </s>

<s>Accepto vt &longs;upra corpore &longs;olido D, & liquid E, inuenta grauitate, <lb/>vt dictum e&longs;t, fiat vt grauitas G, ad prauitatem H, ita cubus ex F, ad <lb/>alium cubum, cuius latus &longs;it C, Quoniam igitur eadem ratione qua <lb/>fupra o&longs;tendetut, vt grauitas G, ad grauitatem H, ita e&longs;&longs;e magnitudi<lb/>nem &longs;phæræ B, ad &longs;phæræ A, magnitudinem, &longs;ed magnitudo &longs;phæræ B, <lb/><arrow.to.target n="note26"/> ad magnitudinem &longs;phæræ A, *triplicatam rationem habet eius, quam <lb/>F, diameter &longs;phæræ B, ad diametrum &longs;phæræ A, &longs;imiliter, & cubus ex <lb/><arrow.to.target n="note27"/> F, ad cubum ex diametro &longs;phæræ A, <expan abbr="triplicatã*">triplicatan*</expan> rationem liabet eius, <lb/>quam F, ad diametrum &longs;phæræ A, ergo, vt grauitas G, ad <expan abbr="grauitat&etilde;">grauitatem</expan> <lb/>H, ita erit cubus ex F, ad cubum ex diametro &longs;phæræ A, &longs;ed vt graui<lb/>tas G, ad grauitatem H, ita e&longs;t cubus ex F, ad cubum ex C, ergo cubus <lb/>ex C, æqualis erit eubo diametri &longs;phæræ A, quare, & latus C, æquabi<lb/>tur ip&longs;ius &longs;phæræ A, diametro, inuenta igitur e&longs;t quantitas diametri <lb/>&longs;olidæ &longs;phæræ A, quod facere oportebat. </s>

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<s>Quod &longs;i propo&longs;itum corpus aqueum B, &longs;it &longs;phæricum, cuius <lb/>&longs;phæræ diameter &longs;it 10, & oporteat inuenire quanta erit dia<lb/>meter &longs;phæræ ex plumbo, grauitatem habentis æqualem pro<lb/>po&longs;itæ &longs;phæræ B, ita faciendum erit. </s>

<s><emph type="italics"/>Accepto, vt diximus aliquo corpore plumbeo D, cuius grauitas<emph.end type="italics"/> 23, <lb/><emph type="italics"/>& aquæ magnitudinem habentis æqualem plumbo D, inuenta graui-<emph.end type="italics"/><pb pagenum="17"/><emph type="italics"/>tate 2, fiat vt<emph.end type="italics"/> 23<emph type="italics"/>, ad<emph.end type="italics"/> 2<emph type="italics"/>, ita cubus ex<emph.end type="italics"/> 10<emph type="italics"/>, hoc est<emph.end type="italics"/> 1000<emph type="italics"/>, ad alium nume<lb/>rum qui &longs;it<emph.end type="italics"/> 86 22/2<gap/><31> <emph type="italics"/>is igitur numerus erit cubus diametrs &longs;phæræ ex <lb/>plumbo, grauitatem æqualem habentis propo&longs;ite ex aqua &longs;phæræ B, <lb/>quare eius latus cubicum, quod est<emph.end type="italics"/> 4 41/100<31> <emph type="italics"/>ferè, indicabit ip&longs;am <lb/>diametrum.<emph.end type="italics"/></s>

<s><emph type="italics"/>Accepto, vt diximus aliquo corpore plumbeo D, cuius grauitas<emph.end type="italics"/> 23, <lb/><emph type="italics"/>& aquæ magnitudinem habentis æqualem plumbo D, inuenta graui<emph.end type="italics"/><pb pagenum="17"/><emph type="italics"/>tate 2, fiat vt<emph.end type="italics"/> 23<emph type="italics"/>, ad<emph.end type="italics"/> 2<emph type="italics"/>, ita cubus ex<emph.end type="italics"/> 10<emph type="italics"/>, hoc est<emph.end type="italics"/> 1000<emph type="italics"/>, ad alium nume<lb/>rum qui &longs;it<emph.end type="italics"/> 86 22/2<gap/><31> <emph type="italics"/>is igitur numerus erit cubus diametrs &longs;phæræ ex <lb/>plumbo, grauitatem æqualem habentis propo&longs;ite ex aqua &longs;phæræ B, <lb/>quare eius latus cubicum, quod est<emph.end type="italics"/> 4 41/100<31> <emph type="italics"/>ferè, indicabit ip&longs;am <lb/>diametrum.<emph.end type="italics"/></s>

<s><emph type="italics"/>Similiter &longs;i propo&longs;itum corpus aqueum B, fuerit cubicum, vel alicu <lb/>las alterius formæ regularis, eadem ratione vtemur ad inueniendum <lb/>latus cubi ex plumbo, grauitatem habentis æqualem propo&longs;ito ex aqua <lb/><gap/>bo B, nam &longs;i ex aqua cubi B, datum &longs;it latus<emph.end type="italics"/> 10<emph type="italics"/>, erit numerus<emph.end type="italics"/><lb/>86 22/2<gap/><31> <emph type="italics"/>cubus ex plumbo æqualis grauitate propo&longs;ito ex aqua cubo B, <lb/>quare latus cubicum numeri<emph.end type="italics"/> 86 22/2<gap/><31> <emph type="italics"/>quod e&longs;t<emph.end type="italics"/> 4 41/100<31> <emph type="italics"/>ferè, indicabit <lb/>quæ&longs;itum latus cubi ex plumbo.<emph.end type="italics"/></s>

Line 476

<s><emph type="italics"/>SIT propo&longs;itum aliquod corpus plumbeum A, cuius magnitudo<emph.end type="italics"/><lb/>740, <emph type="italics"/>& oporteat inuenire quanta erit magnitudo &longs;tanni, grauita<lb/>tem habentis æqualem propo&longs;ito plumbo A. </s>

<s>Accipiatur aliquod cor<lb/>pus plumbeum D, cuius grauitas &longs;it 115, deinde stanni, magnitudi<lb/>ne æqualis plumbo D, inueniatur grauitas, quæ &longs;it<emph.end type="italics"/> 74, <emph type="italics"/>quod quomo<lb/>do fieri oportent, dictum est in antecedentis Problematis exemplo, & <lb/>fiat vt<emph.end type="italics"/> 74, <emph type="italics"/>ad<emph.end type="italics"/> 115. <emph type="italics"/>ita<emph.end type="italics"/> 740, <emph type="italics"/>ad alium numerum qui &longs;it<emph.end type="italics"/> 1150, <emph type="italics"/>is igitur <lb/>numcrus indicabit quanta erit magnitudo &longs;tanni, grauitatem baben <lb/>tis æqualem propo&longs;ito plumbo A.<emph.end type="italics"/></s>

<s>Accipiatur aliquod cor<lb/>pus plumbeum D, cuius grauitas &longs;it 115, deinde stanni, magnitudi<lb/>ne æqualis plumbo D, inueniatur grauitas, quæ &longs;it<emph.end type="italics"/> 74, <emph type="italics"/>quod quomo<lb/>do fieri oportent, dictum est in antecedentis Problematis exemplo, & <lb/>fiat vt<emph.end type="italics"/> 74, <emph type="italics"/>ad<emph.end type="italics"/> 115. <emph type="italics"/>ita<emph.end type="italics"/> 740, <emph type="italics"/>ad alium numerum qui &longs;it<emph.end type="italics"/> 1150, <emph type="italics"/>is igitur <lb/>numcrus indicabit quanta erit magnitudo &longs;tanni, grauitatem haben<lb/>tis æqualem propo&longs;ito plumbo A.<emph.end type="italics"/></s>

<s>Quod &longs;i propo&longs;itum corpus plumbeum A, &longs;it &longs;phæricum, cu<lb/>ius &longs;phæræ diameter &longs;it 10, & oporteat inuenire quanta erit <lb/>diameter &longs;phæræ ex &longs;tanno, grauitatem habentis æqualem <lb/>propo&longs;itæ &longs;phæræ A, ita faciendum erit. </s>

Line 508

<s>Qvidam proponit aliquod corpus liquidum notæ <lb/>grauitatis, & vult &longs;cire, quanta erit grauitas alte<lb/>rius liquidi, magnitudinem habentis propo&longs;ito corpori <lb/>liquido æqualem. </s>

<s><emph type="italics"/>Sit propo&longs;itum aliquod olei corpus A, cuius grauitas<emph.end type="italics"/> 550, <emph type="italics"/>& <lb/>oporteat inuenire, quanta erit grauitas aquæ, magnitudinem haben-<emph.end type="italics"/><pb pagenum="23"/><emph type="italics"/>tis æqualem propo&longs;ito oleo A, Accipiatur aliquod corpus &longs;olidum C, <lb/>vtpote plumbeum, & aquæ, magnitudinem habentis æqualem plumb<gap/><lb/>C, inueniatur grauitas, quæ &longs;it 12, vt in exemplo propo&longs;. </s>

<s><emph type="italics"/>Sit propo&longs;itum aliquod olei corpus A, cuius grauitas<emph.end type="italics"/> 550, <emph type="italics"/>& <lb/>oporteat inuenire, quanta erit grauitas aquæ, magnitudinem haben<emph.end type="italics"/><pb pagenum="23"/><emph type="italics"/>tis æqualem propo&longs;ito oleo A, Accipiatur aliquod corpus &longs;olidum C, <lb/>vtpote plumbeum, & aquæ, magnitudinem habentis æqualem plumb<gap/><lb/>C, inueniatur grauitas, quæ &longs;it 12, vt in exemplo propo&longs;. </s>

<s>8. <expan abbr="dict&utilde;">dictum</expan> e&longs;t. </s>

Line 588

<s>habentis propo&longs;ito corpori liquido æqualem.</s>

<s><emph type="italics"/>Sit propo&longs;itum aliquod olei corpus A, cuius magnitudo<emph.end type="italics"/> 600. <emph type="italics"/>& <lb/>oporteat inuenire quanta srit magnitudo aquæ, grauitatem habentis <lb/>æqualem propo&longs;ito oleo A, accipiatur aliquod &longs;olidum corpus C, vt <lb/>pote plumbeum, & aquæ magnitudinem habentis æqualem plumbo <lb/>C, inueniatur grauitas, vt in exemplo prop.<emph.end type="italics"/> 8, <emph type="italics"/>dictum e&longs;t, quæ &longs;it<emph.end type="italics"/> 12. <lb/><emph type="italics"/>&longs;imiliter & olei æqudlem habentis magnitudinem eidem plumbo C, <lb/>inueniatur grauitas quæ &longs;it 11, & fiat vt 12, ad<emph.end type="italics"/> 11, <emph type="italics"/>ita<emph.end type="italics"/> 600, <emph type="italics"/>ad alium <lb/><expan abbr="numer&utilde;">numerum</expan> qui &longs;it<emph.end type="italics"/> 550. <emph type="italics"/>is igitur numerus indicabit quanta erit magni<lb/>tudo aquæ grauitatem habentis æqualem propo&longs;ito oleo A.<emph.end type="italics"/></s>

<s><emph type="italics"/>Sit propo&longs;itum aliquod olei corpus A, cuius magnitudo<emph.end type="italics"/> 600. <emph type="italics"/>& <lb/>oporteat inuenire quanta srit magnitudo aquæ, grauitatem habentis <lb/>æqualem propo&longs;ito oleo A, accipiatur aliquod &longs;olidum corpus C, vt <lb/>pote plumbeum, & aquæ magnitudinem habentis æqualem plumbo <lb/>C, inueniatur grauitas, vt in exemplo prop.<emph.end type="italics"/> 8, <emph type="italics"/>dictum e&longs;t, quæ &longs;it<emph.end type="italics"/> 12. <lb/><emph type="italics"/>&longs;imiliter & olei æqualem habentis magnitudinem eidem plumbo C, <lb/>inueniatur grauitas quæ &longs;it 11, & fiat vt 12, ad<emph.end type="italics"/> 11, <emph type="italics"/>ita<emph.end type="italics"/> 600, <emph type="italics"/>ad alium <lb/><expan abbr="numer&utilde;">numerum</expan> qui &longs;it<emph.end type="italics"/> 550. <emph type="italics"/>is igitur numerus indicabit quanta erit magni<lb/>tudo aquæ grauitatem habentis æqualem propo&longs;ito oleo A.<emph.end type="italics"/></s>

<s><emph type="italics"/>Similiter &longs;i <expan abbr="propo&longs;it&utilde;">propo&longs;itum</expan> &longs;it aliquod corpus aqueum A, cuius magnitu <lb/>do<emph.end type="italics"/> 5700, <emph type="italics"/>& oporteat inuenire, quanta erit magnitudo argenti viui, <lb/>grauitatem habentis æqualem propo&longs;itæ aquæ A. </s>

Line 608

<s>Hoc argumentum <lb/>licet primo a&longs;pectu probabile videatur, tamen fal&longs;o concludit. </s>

<s><lb/>verum e&longs;t quod aqua &longs;u&longs;tentat magis corpus planum, quam <lb/>conum, ip&longs;um tamen &longs;u&longs;tentat, netanta velocitate feratur <lb/>deor&longs;um, non ideo ip&longs;ius grauitati aliquid detrahit, neque <lb/>enim ex velociori motu &longs;impliciter inferri pote&longs;t maior gra<lb/>uitas, illud enim valeret etiam in aere, quod e&longs;t fal&longs;um, &longs;ed <lb/>ne huiu&longs;modi dubitatio veritatis &longs;pccie aliquem decipiat, &longs;e<lb/>quenti Theoremate eam de&longs;truere agrediar. </s>

<s><lb/>verum e&longs;t quod aqua &longs;u&longs;tentat magis corpus planum, quam <lb/>conum, ip&longs;um tamen &longs;u&longs;tentat, netanta velocitate feratur <lb/>deor&longs;um, non ideo ip&longs;ius grauitati aliquid detrahit, neque <lb/>enim ex velociori motu &longs;impliciter inferri pote&longs;t maior gra<lb/>uitas, illud enim valeret etiam in aere, quod e&longs;t fal&longs;um, &longs;ed <lb/>ne huiu&longs;modi dubitatio veritatis &longs;pecie aliquem decipiat, &longs;e<lb/>quenti Theoremate eam de&longs;truere agrediar. </s>

<s>THEOR EMA VIII. PROPOS. XVI.</s>

Line 618

<s>&longs;it enim &longs;i &longs;ieri pote&longs;t corpus A, leuius corpore B, <pb pagenum="29"/>& accipiatur aliquod corpus L, leuius quam aqua, ita vt cum ip&longs;i <lb/>corpori L, appendatur, corpus B, & ambo &longs;imul demittantur in <expan abbr="aquã">aquam</expan>, <lb/>&longs;int æque grauia atque aqua, neque &longs;ur&longs;um, neque deor&longs;um &longs;eran<lb/>tur, &longs;imiliter accipiatur alterum corpus M, eiu&longs;dem generis cum <lb/>corpore L, ip&longs;ique &longs;imile, & <lb/><figure id="fig13"/><lb/>æquale, & corpori M, appen <lb/>datur corpus A. </s>

<s>Deinde in<lb/>teligatur aqua con&longs;i&longs;tens, & <lb/>manens, eiu&longs;que &longs;uperfi<lb/>cies &longs;phærica C D E, cuius <lb/>&longs;phæræ centrum K, aquæ <lb/>enim con&longs;i&longs;tentis, atque <lb/>manentis &longs;uperficies &longs;phæ<lb/>rica e&longs;t, cui <gap/> &longs;phæræ cen<lb/>trum idem <gap/>, quod <expan abbr="centr&utilde;">centrum</expan> <lb/>terræ, hoc autem demon&longs;tratum e&longs;t ab Archimede Prop. 2. lib.

<s>Deinde in<lb/>teligatur aqua con&longs;i&longs;tens, & <lb/>manens, eiu&longs;que &longs;uper&longs;i<lb/>cies &longs;phærica C D E, cuius <lb/>&longs;phæræ centrum K, aquæ <lb/>enim con&longs;i&longs;tentis, atque <lb/>manentis &longs;uper&longs;icies &longs;phæ<lb/>rica e&longs;t, cui <gap/> &longs;phæræ cen<lb/>trum idem <gap/>, quod <expan abbr="centr&utilde;">centrum</expan> <lb/>terræ, hoc autem demon&longs;tratum e&longs;t ab Archimede Prop. 2. lib.

1. de <lb/>ijs, quæ vehuntur in aqua. </s>

<s>Inteligantur etiam duæ pyramides con<lb/>iunctæ, & continuatæ, æquales, & &longs;imiles KCD, KDE, pro ba&longs;ibus <lb/>habentes in &longs;uperficie aquæ parallelogramma, vertices autem pun<lb/>ctum K, & corpora L, B, comprehendantur à pyramide KDE, corpo<lb/>ra vero M, A, à pyramide KCD, & &longs;ub corporibus L, B, de&longs;cribatur <lb/>quædam alterius &longs;phæræ &longs;uperficies FGH, in aqua, circa centrum K, <lb/>poterit autem huiu&longs;modi &longs;uperficies &longs;ub corporibus L, B, de&longs;cribi, <lb/>quoniam & &longs;i ip&longs;i corpora demerguntur tota, non ideo feruntur <lb/>deor&longs;um, ponuntur enim æque grauia atque aqua. </s>

<s>Inteligantur etiam duæ pyramides con<lb/>iunctæ, & continuatæ, æquales, & &longs;imiles KCD, KDE, pro ba&longs;ibus <lb/>habentes in &longs;uperficie aquæ parallelogramma, vertices autem pun<lb/>ctum K, & corpora L, B, comprehendantur à pyramide KDE, corpo<lb/>ra vero M, A, à pyramide KCD, & &longs;ub corporibus L, B, de&longs;cribatur <lb/>quædam alterius &longs;phæræ &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 & &longs;i ip&longs;i corpora demerguntur tota, non ideo feruntur <lb/>deor&longs;um, ponuntur enim æque grauia atque aqua. </s>

<s>Quoniam <lb/>igitur eiu&longs;dem generis ponuntur corpora M, L, & æqualia, & &longs;imilia, <lb/>erunt æque grauia, cum in aqua, tum in aere, & quoniam corpus <lb/>A, leuins e&longs;t in aqua, corpore B, erunt corpora M, A, &longs;imul, in aqua <lb/>leuiora corporibus L, B, &longs;ed corpora L, B, &longs;imul, æque grauia &longs;unt at<lb/>que aqua, ergo corpora M, A, &longs;imul, leuiora erunt quam aqua; quare <lb/>corpus M, non demergetur totum, &longs;ed aliqua pars ip&longs;ius ex aquæ &longs;u<lb/>per&longs;icie extabit. </s>

Line 698

<s>Ad inueniendas igitur &longs;phærarum grauitates ex <lb/>data diametrorum magnitudine, bæc erit ratio.<emph.end type="italics"/></s>

<s><emph type="italics"/>Quæris grauitatem &longs;phæræ plumbeæ, diametrum habentis<emph.end type="italics"/> 3, <emph type="italics"/>vn<lb/>ciarum, in&longs;pice tabulam, & in linea trium vnciarum, &longs;ub titulo gra<lb/>uitatis plumbeæ &longs;phæræ, deprebendes ip&longs;am &longs;pbæram grauitætem ba<lb/>bere lib.<emph.end type="italics"/> 7, <emph type="italics"/>vne.<emph.end type="italics"/> 4, <emph type="italics"/>&longs;eru.<emph.end type="italics"/> 13, <emph type="italics"/>gran.<emph.end type="italics"/> 22 29/57. </s>

<s><emph type="italics"/>Quæris grauitatem &longs;phæræ plumbeæ, diametrum habentis<emph.end type="italics"/> 3, <emph type="italics"/>vn<lb/>ciarum, in&longs;pice tabulam, & in linea trium vnciarum, &longs;ub titulo gra<lb/>uitatis plumbeæ &longs;phæræ, deprebendes ip&longs;am &longs;phæram grauitætem ba<lb/>bere lib.<emph.end type="italics"/> 7, <emph type="italics"/>vnc.<emph.end type="italics"/> 4, <emph type="italics"/>&longs;eru.<emph.end type="italics"/> 13, <emph type="italics"/>gran.<emph.end type="italics"/> 22 29/57. </s>

<s><emph type="italics"/>Rur&longs;us, quæris grauitatem &longs;phæræ aureæ, diametrum habentis<emph.end type="italics"/> 6, <lb/><emph type="italics"/>vnciarum. </s>

Line 716

<s><emph type="italics"/>Primum inueniendam curauimus grauitatem alicuius &longs;phæræ, da<lb/>tem habentis diametrum, & ad hoc faciendum, oportebat aliquam <lb/>&longs;pbæram efficere, &longs;ed quoniam ad illam efficiendam, exactam buman<gap/><lb/>diligentia non &longs;ufficit, fieri curauimus Cylindrum ex &longs;tanno, altitu<lb/>dine aqualem diametro circuli, qui ba&longs;is e&longs;t ip&longs;ius Cylindri, is enim <lb/>torno &longs;ieri pote&longs;t multo exactior quam &longs;pbæra, & facilius. </s>

<s>huius au<lb/>tem Cylindri altitudo, vel diameter ip&longs;ius ba&longs;is, erat duarum vncia<lb/>rum prædicti pedis Romani, grauitas vero duarum librarum, cum <lb/>vna vncia, & octo &longs;crupulis, &longs;iue vt hoc pondus ad grana reducamus, <lb/>Cylindri grauitas erat Gran. 14592. abstulimns ab bah Cylindri <lb/>grauitate partem tertiam. </s>

Line 758

<s><emph type="italics"/>Porro ad inueniendas grauitates &longs;phærarum ex reliquis metallis <lb/>vel ex quacunquæ alia materia, hæc erit ratio.<emph.end type="italics"/></s>

<s><emph type="italics"/>Fiat vt<emph.end type="italics"/> 1, <emph type="italics"/>ad<emph.end type="italics"/> 1 41/74, <emph type="italics"/>hoc e&longs;t vt<emph.end type="italics"/> 74, <emph type="italics"/>ad<emph.end type="italics"/> 115, (<emph type="italics"/>&longs;i degrauitate &longs;ph<gap/><lb/><expan abbr="pl&umacr;beæ">plumbeæ</expan> quæritur cuius diameter est<emph.end type="italics"/> 1/4, <emph type="italics"/>vnciæ) ita 19. grauitas vi<gap/><lb/>licet &longs;phæræ stanneæ <expan abbr="diametr&umacr;">diametrum</expan> habentis<emph.end type="italics"/> 1/4, <emph type="italics"/>vnciæ, ad <expan abbr="ali&umacr;">alium</expan> numer<gap/><lb/>qui &longs;it<emph.end type="italics"/> 29 19/74, <emph type="italics"/>grauitas igitur &longs;phæræ plumbeæ, diametrum hab<gap/><lb/>tis<emph.end type="italics"/> 1/4, <emph type="italics"/>vnciæ, erit gran.<emph.end type="italics"/> 29 39/74. <emph type="italics"/>&longs;tannum enim ad plumbum ration<gap/><lb/>habet in grauitate vt<emph.end type="italics"/> 1, <emph type="italics"/>ad<emph.end type="italics"/> 1 41/74, <emph type="italics"/>vt con&longs;picitur in prima tabelle <lb/>quam ad comparandum inter &longs;e duodecim corporum genera, grauit <lb/>te, & magnitudine, appo&longs;uimus.<emph.end type="italics"/></s>

<s><emph type="italics"/>Fiat vt<emph.end type="italics"/> 1, <emph type="italics"/>ad<emph.end type="italics"/> 1 41/74, <emph type="italics"/>hoc e&longs;t vt<emph.end type="italics"/> 74, <emph type="italics"/>ad<emph.end type="italics"/> 115, (<emph type="italics"/>&longs;i degrauitate &longs;phæræ<lb/><expan abbr="pl&umacr;beæ">plumbeæ</expan> quæritur cuius diameter est<emph.end type="italics"/> 1/4, <emph type="italics"/>vnciæ) ita 19. grauitas vi<gap/><lb/>licet &longs;phæræ stanneæ <expan abbr="diametr&umacr;">diametrum</expan> habentis<emph.end type="italics"/> 1/4, <emph type="italics"/>vnciæ, ad <expan abbr="ali&umacr;">alium</expan> numer<gap/><lb/>qui &longs;it<emph.end type="italics"/> 29 19/74, <emph type="italics"/>grauitas igitur &longs;phæræ plumbeæ, diametrum haben<lb/>tis<emph.end type="italics"/> 1/4, <emph type="italics"/>vnciæ, erit gran.<emph.end type="italics"/> 29 39/74. <emph type="italics"/>&longs;tannum enim ad plumbum ration<gap/><lb/>habet in grauitate vt<emph.end type="italics"/> 1, <emph type="italics"/>ad<emph.end type="italics"/> 1 41/74, <emph type="italics"/>vt con&longs;picitur in prima tabelle <lb/>quam ad comparandum inter &longs;e duodecim corporum genera, grauita<lb/>te, & magnitudine, appo&longs;uimus.<emph.end type="italics"/></s>

<s><emph type="italics"/>Si vero quæratur de grauitate &longs;phæræ plumbeæ, diametrum <lb/>habentis<emph.end type="italics"/> 2, <emph type="italics"/>vnciarum, fiat vt<emph.end type="italics"/> 74, <emph type="italics"/>ad<emph.end type="italics"/> 115, <emph type="italics"/>ita<emph.end type="italics"/> 9728, <emph type="italics"/>id est gra<gap/><emph.end type="italics"/></s><pb pagenum="41"/>

Line 818

<s><emph type="italics"/>Huius tabulæ compo&longs;itio pendet ex præcedenti tabula, & ex pro<lb/>po&longs;.<emph.end type="italics"/> 17, <emph type="italics"/>huius, &longs;i enim fiat vt grauitas &longs;phæræ stanneæ, diametrum <lb/>habentis vnius vnciæ, id est, vt grana<emph.end type="italics"/> 1216, <emph type="italics"/>ad grauitatem &longs;phæræ <lb/>vnius libræ, idest, adgrana<emph.end type="italics"/> 6912, <emph type="italics"/>ita cubus diametri vnius vnciæ, <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;phæræ stanneæ, grauitatem habentis 1, lib.

demonstratum enim <lb/>est prop.<emph.end type="italics"/> 17, <emph type="italics"/>huius, &longs;phæras 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/>præci&longs;e cubus, cius radix non explicabitur accurata, &longs;ed vt explicetur <lb/>veræ bene proxima, multiplicetur<emph.end type="italics"/> 5 13/19, <emph type="italics"/>per<emph.end type="italics"/> 1000000. <emph type="italics"/>& 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<lb/>rato numero cubo, ea erit<emph.end type="italics"/> 173. <emph type="italics"/>proxime, & erit centupla radicis nu<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;phæræ &longs;tanneæ, <lb/>grauitatem habentis 1, lib.

demonstratum enim <lb/>est prop.<emph.end type="italics"/> 17, <emph type="italics"/>huius, &longs;phæras 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/>præci&longs;e cubus, eius radix non explicabitur accurata, &longs;ed vt explicetur <lb/>veræ bene proxima, multiplicetur<emph.end type="italics"/> 5 13/19, <emph type="italics"/>per<emph.end type="italics"/> 1000000. <emph type="italics"/>& 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<lb/>rato numero cubo, ea erit<emph.end type="italics"/> 173. <emph type="italics"/>proxime, & erit centupla radicis nu<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;phæræ &longs;tanneæ, <lb/>grauitatem habentis 1, lib.

erit<emph.end type="italics"/> 1 78/100. <emph type="italics"/>reliquarum autem ex &longs;tanno <lb/>&longs;phærarum, grauitatem habentium duplam primæ, triplam, quadru<lb/>plam &c. </s>

Line 836

<s>Sed vt etiam euitetur labor multiplicandi prædi<lb/>ctum numerum<emph.end type="italics"/> 5684210 10/19, <emph type="italics"/>hac ratione inuenientur eius multi<lb/>plicia.<emph.end type="italics"/></s>

<s><emph type="italics"/>Prædicto 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, & &longs;i eius quadruplum ei addatur, &longs;umma <lb/>dabit eius quintuplum, & &longs;ic &longs;ola additione inuenientur eius quot<lb/>cunque multiplicia.<emph.end type="italics"/></s>

<s><emph type="italics"/>Prædicto 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 eius 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, & &longs;i eius quadruplum ei addatur, &longs;umma <lb/>dabit eius quintuplum, & &longs;ic &longs;ola additione inuenientur eius quot<lb/>cunque multiplicia.<emph.end type="italics"/></s>

<s><emph type="italics"/>Eadem ratione inuenientur diametri &longs;phærarum ex quacunque <lb/>alia materia, &longs;i enim quæratur de magnitudine diametri verbi gra<lb/>tia &longs;phæræ ferreæ, grauitatem habentis 1, lib.

fiat vt grana<emph.end type="italics"/> 1314 22/37, <lb/><emph type="italics"/>id est vt grauitas &longs;phæræ ferreæ, cuius diameter e&longs;t vnius vnciæ, ad <lb/>grauitatem vnius libræ, id est ad grana 6912, ita cubus diametri <lb/>vnius vnciæ, hoc e&longs;t ita 1, ad alium numerum qui &longs;it<emph.end type="italics"/> 5 49/190, <emph type="italics"/>is igi<lb/>tur numerus * erit cubus diametri &longs;phæræ ferreæ, grauitatem haben-<emph.end type="italics"/><arrow.to.target n="note48"/><lb/><emph type="italics"/>tis<emph.end type="italics"/> 1, <emph type="italics"/>lib.

fiat vt grana<emph.end type="italics"/> 1314 22/37, <lb/><emph type="italics"/>id est vt grauitas &longs;phæræ ferreæ, cuius diameter e&longs;t vnius vnciæ, ad <lb/>grauitatem vnius libræ, id est ad grana 6912, ita cubus diametri <lb/>vnius vnciæ, hoc e&longs;t ita 1, ad alium numerum qui &longs;it<emph.end type="italics"/> 5 49/190, <emph type="italics"/>is igi<lb/>tur numerus * erit cubus diametri &longs;phæræ ferreæ, grauitatem haben<emph.end type="italics"/><arrow.to.target n="note48"/><lb/><emph type="italics"/>tis<emph.end type="italics"/> 1, <emph type="italics"/>lib.

quare radix cubica numeri<emph.end type="italics"/> 5 49/190, <emph type="italics"/>dabit quæ&longs;itam dia<lb/>metrum, & quoniam numerus<emph.end type="italics"/> 5 49/190, <emph type="italics"/>non e&longs;t præci&longs;e cubus, & ideo <lb/>non explicabitur cius radix accurate, multiplicetur per 1000000, & <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<lb/>quam ex accurato numero cubo, ea erit<emph.end type="italics"/> 174: <emph type="italics"/>ferè, & erit centupla ra<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;phæræ ferreæ, grauitatem habentis <lb/>1, lib.

quare radix cubica numeri<emph.end type="italics"/> 5 49/190, <emph type="italics"/>dabit quæ&longs;itam dia<lb/>metrum, & quoniam numerus<emph.end type="italics"/> 5 49/190, <emph type="italics"/>non e&longs;t præci&longs;e cubus, & ideo <lb/>non explicabitur eius radix accurate, multiplicetur per 1000000, & <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<lb/>quam ex accurato numero cubo, ea erit<emph.end type="italics"/> 174: <emph type="italics"/>ferè, & erit centupla ra<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;phæræ ferreæ, grauitatem habentis <lb/>1, lib.

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"/>& ex ita du<lb/>plicato eruatur radix cubica<emph.end type="italics"/> 219, <emph type="italics"/>ea dabit centuplum diametri &longs;phæ<lb/>ra ferreæ, grauitatem habentis<emph.end type="italics"/> 2, <emph type="italics"/>lib.

Line 886

<s>Neque per collectionem quæ&longs;ita aquæ men&longs;ura inueniri <lb/>pote&longs;t: æquè enim impo&longs;&longs;ibile e&longs;t vniuer&longs;am illam æquam col<lb/>ligere, quæ extra vas effluit, quando corona, vel aurea ma&longs;&longs;a <lb/>vel argentea in ip&longs;o va&longs;e deprimitur, cum enim aqua è va&longs;e <lb/>effluat, pars ip&longs;ius aquæ va&longs;i, ex quo effluit, pars va&longs;i in quod <lb/>influit adhæret, & &longs;i vniuer&longs;a omnino &longs;emper non colligatur, <lb/>erit non parui erroris cau&longs;a, præter quam quod, non &longs;emper <lb/>adeo facile inuenitur par auri, argentique ma&longs;&longs;a, quando co<lb/>rona, vel alia auri ma&longs;&longs;a, quæ examinanda proponitur, medio<lb/>crem excederet magnitudinem. </s>

<s>Neque præterea pote&longs;t di&longs;cerni prædicta argenti portio in <lb/>aliqua auri parua ma&longs;&longs;a, differentiæ enim aquarum, quæ ex<lb/>tra vas effluunt, &longs;unt adeo exiguæ, vt ne cogno&longs;ci quidem <lb/>po&longs;&longs;int, quod &longs;i cogno&longs;cerentur, non &longs;emper erunt veræ, <lb/>&longs;iquidem non &longs;emper in va&longs;is medio in cumulum cre&longs;cens <lb/>æqualis aquæ co<gap/>ia remanet, &longs;ed maior interdum, inter<lb/>dum minor, vt con&longs;picitur. </s>

<s>Neque præterea pote&longs;t di&longs;cerni prædicta argenti portio in <lb/>aliqua auri parua ma&longs;&longs;a, differentiæ enim aquarum, quæ ex<lb/>tra vas effluunt, &longs;unt adeo exiguæ, vt ne cogno&longs;ci quidem <lb/>po&longs;&longs;int, quod &longs;i cogno&longs;cerentur, non &longs;emper erunt veræ, <lb/>&longs;iquidem non &longs;emper in va&longs;is medio in cumulum cre&longs;cens <lb/>æqualis aquæ copia remanet, &longs;ed maior interdum, inter<lb/>dum minor, vt con&longs;picitur. </s>

<s>fit enim vt aliquando cumulus <lb/>ille frangatur pluribus in locis, & ideo aqua diffundatur, vt <lb/>ferè nihilip&longs;ius cumuli &longs;uper&longs;it, aliquando vero frangatur <lb/>in vno tantum loco, & aqua colligens &longs;e in cumulum, parum <lb/>diffluat. </s>

Line 928

& oporteat facere quod imperatum e&longs;t. </s>

<s><lb/>Intelligantur duo corpora, vnum aureum, alterum argenteum, æque <lb/>grauia atque corona, deinde trium corporum ex aqua, magnitudine <lb/>æqualium, aureo &longs;eilicet corporivnum, alterum coronæ, tertium cor<lb/>pori argenteo, inueniantur grauitates, vt in exemplo prop.<emph.end type="italics"/> 8. <emph type="italics"/>dictum <lb/>est, quæ &longs;int primi nimirum corporis aquei<emph.end type="italics"/> 5, <emph type="italics"/>&longs;ecundi vero 6, & terty<emph.end type="italics"/><lb/>9 6/31, <emph type="italics"/>& fiat vt differentia inter<emph.end type="italics"/> 5, & 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 coronæ, ita differentia inter<emph.end type="italics"/> 5, & 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<lb/>na, qua detracta ex totali grauitate coronæ, reliquum<emph.end type="italics"/> 72 9/26, <emph type="italics"/>erit <lb/>grauitas portionis auri.<emph.end type="italics"/></s>

<s><lb/>Intelligantur duo corpora, vnum aureum, alterum argenteum, æque <lb/>grauia atque corona, deinde trium corporum ex aqua, magnitudine <lb/>æqualium, aureo &longs;cilicet corporivnum, alterum coronæ, tertium cor<lb/>pori argenteo, inueniantur grauitates, vt in exemplo prop.<emph.end type="italics"/> 8. <emph type="italics"/>dictum <lb/>est, quæ &longs;int primi nimirum corporis aquei<emph.end type="italics"/> 5, <emph type="italics"/>&longs;ecundi vero 6, & terty<emph.end type="italics"/><lb/>9 6/31, <emph type="italics"/>& fiat vt differentia inter<emph.end type="italics"/> 5, & 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 coronæ, ita differentia inter<emph.end type="italics"/> 5, & 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<lb/>na, qua detracta ex totali grauitate coronæ, reliquum<emph.end type="italics"/> 72 9/26, <emph type="italics"/>erit <lb/>grauitas portionis auri.<emph.end type="italics"/></s>

<s><emph type="italics"/>Vel &longs;i pro tertio proportionis termino &longs;umatur differentia inter<emph.end type="italics"/> 6, <lb/>& 9 6/31, <emph type="italics"/>quæ e&longs;t<emph.end type="italics"/> 3 6/31, <emph type="italics"/>quartus terminus<emph.end type="italics"/> 72 9/26, <emph type="italics"/>erit grauitas por<lb/>tionis auri, quæ &longs;i dematur ex totali grauitate coronæ, remanebit<emph.end type="italics"/><lb/>22 17/26, <emph type="italics"/>pro grauitate portionis argenti.<emph.end type="italics"/></s>

Line 942

& oporteat inuenire quanta erit portio æris in ip&longs;o corpore,<emph.end type="italics"/><pb pagenum="56"/><emph type="italics"/>& quanta auri. </s>

<s>Intelligantur duo corpo<gap/>i, vnum ex auro puro, al<lb/>terum ex ære, æque grauia atque corpus mistum, & trium corporum <lb/>ex aqua, quorum vnum &longs;it æquale corpori aureo magnitudine, alte<lb/>rum misto, tertium ærco, inueniantur grauitates, vt in exemplo pro<lb/>po&longs;.<emph.end type="italics"/>8.<emph type="italics"/>dictum est, quæ &longs;int<emph.end type="italics"/> 9, 11, <emph type="italics"/>& 19, & fiat vt differentia inter<emph.end type="italics"/> 9, & <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, & 11, <emph type="italics"/>ad<emph.end type="italics"/> 34 1/5, <emph type="italics"/>portio igitur corporis mi&longs;ti ærea grauitatem babebit<emph.end type="italics"/><lb/>34 1/5, <emph type="italics"/>quæ &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>

<s>Intelligantur duo corpori, vnum ex auro puro, al<lb/>terum ex ære, æque grauia atque corpus mistum, & trium corporum <lb/>ex aqua, quorum vnum &longs;it æquale corpori aureo magnitudine, alte<lb/>rum misto, tertium aureo, inueniantur grauitates, vt in exemplo pro<lb/>po&longs;.<emph.end type="italics"/>8.<emph type="italics"/>dictum est, quæ &longs;int<emph.end type="italics"/> 9, 11, <emph type="italics"/>& 19, & fiat vt differentia inter<emph.end type="italics"/> 9, & <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, & 11, <emph type="italics"/>ad<emph.end type="italics"/> 34 1/5, <emph type="italics"/>portio igitur corporis mi&longs;ti ærea grauitatem babebit<emph.end type="italics"/><lb/>34 1/5, <emph type="italics"/>quæ &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>

<s><emph type="italics"/>Vel &longs;i pro tertio proportionis termino &longs;umatur differentia inter<emph.end type="italics"/> 11, <lb/><emph type="italics"/>& 19, quartus terminus<emph.end type="italics"/> 136 4/5, <emph type="italics"/>erit grauitas portionis auri, qua ab<lb/>lata ex totali corporis mi&longs;ti grauitate, reliquum<emph.end type="italics"/> 34 1/5, <emph type="italics"/>dabit grauita<lb/>tem portionis æreæ.<emph.end type="italics"/></s>

Line 972

<s>Dico quoque vt differentia grauitatum H, G, ad grauitatem EK, <lb/>ita e&longs;&longs;e differenti <gap/> grauitatum H, FV, ad grauitatem E. </s>

<s>Quoniam <lb/>enim o&longs;ten&longs;um e&longs;t, quod fit ex EK, & FV, æquale e&longs;&longs;e ei quod ex G. fit <lb/>& E, vna cum eo quod ex H, & K; quod autem fit ex H, & K, æquat<gap/><lb/><gap/>i quod, ex H, fit & EK, minus eo quod ex H, & E, quod onim additur <lb/>idem & minuitur: ergo quod fit ex EK, & FV, æquale erit ei quod fit <lb/>ex H, & EK, vna cum eo quod ex G, & E, minus eo quod ex H, & E. <lb/>addatur vtrin que quod ex H, fit & E, & &longs;ubducantur ea quæ fiunt ex <lb/>G, & E, & ox EK, & FV; quod igitur fit ex H, & E, minus eo quod ex <lb/>G, & E, æquabitur ei quod ex H, fit & EK, minus eo quod ex FV, & <lb/>EK, &longs;ed quod fit ex H, & E, minus eo quod ex G, & E, æquale e&longs;t ei <lb/>quod ex, differ entia ip&longs;arum H, G, fit & E, &longs;imiliter, & quod ex H, fit <lb/>& EK, minus eo quod ex FV, & EK, æquale e&longs;t eiquod ex differentia <lb/>ip&longs;arum H, FV, fit & EK; ergo quod ex differentia ip&longs;arum H, G, fit <lb/>& E, æquabitur ei quod ex differentia ip&longs;arum H, FV, fit & EK; qua<lb/>re æqualitatem ad <expan abbr="proportion&etilde;">proportionem</expan> reuocando erit v<gap/> differen ti a grauis <pb pagenum="59"/>tatum H, G, ad grauitatem EK, ita differentia gra<gap/>tatum H, FV, <lb/>zd grauitatem E. quod &longs;ecundo loco fuit demon&longs;trandum. </s>

<s>Quoniam <lb/>enim o&longs;ten&longs;um e&longs;t, quod fit ex EK, & FV, æquale e&longs;&longs;e ei quod ex G. fit <lb/>& E, vna cum eo quod ex H, & K; quod autem fit ex H, & K, æquat<gap/><lb/><gap/>i quod, ex H, fit & EK, minus eo quod ex H, & E, quod onim additur <lb/>idem & minuitur: ergo quod fit ex EK, & FV, æquale erit ei quod fit <lb/>ex H, & EK, vna cum eo quod ex G, & E, minus eo quod ex H, & E. <lb/>addatur vtrin que quod ex H, fit & E, & &longs;ubducantur ea quæ fiunt ex <lb/>G, & E, & ox EK, & FV; quod igitur fit ex H, & E, minus eo quod ex <lb/>G, & E, æquabitur ei quod ex H, fit & EK, minus eo quod ex FV, & <lb/>EK, &longs;ed quod fit ex H, & E, minus eo quod ex G, & E, æquale e&longs;t ei <lb/>quod ex, differ entia ip&longs;arum H, G, fit & E, &longs;imiliter, & quod ex H, fit <lb/>& EK, minus eo quod ex FV, & EK, æquale e&longs;t eiquod ex differentia <lb/>ip&longs;arum H, FV, fit & EK; ergo quod ex differentia ip&longs;arum H, G, fit <lb/>& E, æquabitur ei quod ex differentia ip&longs;arum H, FV, fit & EK; qua<lb/>re æqualitatem ad <expan abbr="proportion&etilde;">proportionem</expan> reuocando erit v<gap/> differen tia graui<pb pagenum="59"/>tatum H, G, ad grauitatem EK, ita differentia gra<gap/>tatum H, FV, <lb/>zd grauitatem E. quod &longs;ecundo loco fuit demon&longs;trandum. </s>

<s>Alia breuior Theorematis demon&longs;tratio. </s>

Line 1006

<s>Qualitas enim auri in quouis cor <lb/>pore propo&longs;ito, exprimitur partibus auripuri, qua &longs;unt in ip&longs;o corpo<lb/>re, non in magnitudine, &longs;ed in grauitate &longs;umptis, qualibus totum cor<lb/>pus constat<emph.end type="italics"/> 24<emph type="italics"/>: vel quod idem est, auri qualitas exprimitur in ra<lb/>tione quam babent illæ partes in grauitate ad totum corpas: quod <lb/>exemplo clarius explicabitur in hunc modum.<emph.end type="italics"/></s>

<s><emph type="italics"/>Sit aliquod corpus aureum, exempli gratia<emph.end type="italics"/> 24. <emph type="italics"/>vnciarum, quod <lb/>expurgatum & ad aurum purum red<gap/>ctum, ami&longs;erit ex pri&longs;tina <lb/>grauitate nempe ex<emph.end type="italics"/> 24, <emph type="italics"/>vncys, quatuor vncias, ita vt reman&longs;erint <lb/>tantum<emph.end type="italics"/> 20, <emph type="italics"/>vnciæ auri puri, reliquum vero vel euanuerit in fumum, <lb/>vel fuerit alterius metalli. </s>

<s><emph type="italics"/>Sit aliquod corpus aureum, exempli gratia<emph.end type="italics"/> 24. <emph type="italics"/>vnciarum, quod <lb/>expurgatum & ad aurum purum reductum, ami&longs;erit ex pri&longs;tina <lb/>grauitate nempe ex<emph.end type="italics"/> 24, <emph type="italics"/>vncys, quatuor vncias, ita vt reman&longs;erint <lb/>tantum<emph.end type="italics"/> 20, <emph type="italics"/>vnciæ auri puri, reliquum vero vel euanuerit in fumum, <lb/>vel fuerit alterius metalli. </s>

<s>Totum igitur illud corpus aureum ab <lb/>initio propo&longs;itum, &longs;i adbuc intelligatur tale quale fuit ante expurga<lb/>tionem, appellabitur<emph.end type="italics"/> 20. <emph type="italics"/>partium, &longs;eu, (vt vulgo dicitur) di<emph.end type="italics"/> 20. <emph type="italics"/>ca<lb/>ratti. </s>

Line 1024

<s>Eius enim quali<lb/>tatem &longs;i forte aurifices inuestigare velint, detrabent ex eo particu<lb/>lam, verbi gratia, vnius vnciæ, vel qued idem est particulam<emph.end type="italics"/> 24. <pb pagenum="61"/><emph type="italics"/>&longs;<gap/>puforam; & banc particulam exeoquent ad qualitatem o&longs;que <lb/>c<gap/>ripuri. </s>

<s>Bt &longs;t quidem inuenerint, ex priori gr<gap/>uitate<emph.end type="italics"/> 24. <emph type="italics"/>&longs;or<gap/><lb/>pulorum, deperi&longs;&longs;e nibil: pronunciabunt aurum sllud, hoc est, non <lb/>&longs;olum partieulam illam excoctam, &longs;ed ctiam illud d quo fait dotr<gap/><lb/>sta, nec non & illud quod reman&longs;it post &longs;ubiractionim e&longs;&longs;e vel &longs;ui&longs;<gap/><lb/>&longs;e aurum prima qualitatis &longs;eu<emph.end type="italics"/> 24, <emph type="italics"/>partiam; vel quod idem est au<lb/>tum purum. </s>

<s>Bt &longs;t quidem inuenerint, ex priori grauitate<emph.end type="italics"/> 24. <emph type="italics"/>&longs;or<gap/><lb/>pulorum, deperi&longs;&longs;e nibil: pronunciabunt aurum sllud, hoc est, non <lb/>&longs;olum partieulam illam excoctam, &longs;ed etiam illud d quo fuit dotr<gap/><lb/>sta, nec non & illud quod reman&longs;it post &longs;ubiractionim e&longs;&longs;e vel &longs;ui&longs;<gap/><lb/>&longs;e aurum prima qualitatis &longs;eu<emph.end type="italics"/> 24, <emph type="italics"/>partiam; vel quod idem est au<lb/>tum purum. </s>

<s>Sivero deprebendirine gra<gap/>m diminuta<gap/>, ver<lb/>bi gratia, n<gap/>e e&longs;&longs;e<emph.end type="italics"/> 20. <emph type="italics"/>&longs;crupulorum, quæ ante defæcationem fait<emph.end type="italics"/> 24. <lb/><emph type="italics"/>dicturi &longs;unt aurum propo&longs;itam<emph.end type="italics"/> 24. <emph type="italics"/>vnciarum &longs;ni&longs;&longs;e<emph.end type="italics"/> 20. <emph type="italics"/>partium & <lb/>illud quod teman&longs;it e&longs;&longs;e<emph.end type="italics"/> 20. <emph type="italics"/>partium & d<gap/>nique particul<gap/> expur<lb/>gatam nunc quidem e&longs;&longs;e aurum purum, fui&longs;&longs;e vero particulam auri<emph.end type="italics"/><lb/>20. <emph type="italics"/>partium.<emph.end type="italics"/></s>

<s>Sivero deprebendirine gra<gap/>m diminuta<gap/>, ver<lb/>bi gratia, n<gap/>e e&longs;&longs;e<emph.end type="italics"/> 20. <emph type="italics"/>&longs;crupulorum, quæ ante defæcationem fuit<emph.end type="italics"/> 24. <lb/><emph type="italics"/>dicturi &longs;unt aurum propo&longs;itam<emph.end type="italics"/> 24. <emph type="italics"/>vnciarum fui&longs;&longs;e<emph.end type="italics"/> 20. <emph type="italics"/>partium & <lb/>illud quod teman&longs;it e&longs;&longs;e<emph.end type="italics"/> 20. <emph type="italics"/>partium & d<gap/>nique particul<gap/> expur<lb/>gatam nunc quidem e&longs;&longs;e aurum purum, fui&longs;&longs;e vero particulam auri<emph.end type="italics"/><lb/>20. <emph type="italics"/>partium.<emph.end type="italics"/></s>

<s><emph type="italics"/>Et eodem modo pronunclabunt de quibafeunque alys auri quali<lb/>atibus, &longs;ecundum partes auri puri, quas in qualibet ma&longs;&longs;a auri in<lb/>uenerint, ea&longs;que vige&longs;imas quartai tutius grauitatis, non magnita<lb/>dinis. </s>

<s>Nam cum in hac comparatione qualitatum, &longs;epr&longs;im habe a<lb/>tar ratio partium auri, & &longs;eor&longs;im metallorum alligatorum; manife<lb/>stum est &longs;i grauitas totius corporis intelligatur diui&longs;a in<emph.end type="italics"/> 24. <emph type="italics"/>partes <lb/>equales, ex quibus<emph.end type="italics"/> 20. <emph type="italics"/>&longs;int auri, duæ argenti, & duæ æris; quumli<lb/>bet partem auri cum qualibet parte argenti & æris collatam, magni<lb/>tudine e&longs;&longs;e minorem; & &longs;imiliter partem argenti minorem parte <lb/>aris; propterea quod aurum omnia reliqua metalla &longs;uperet grauit<gap/><lb/>te quemadmodum & argentum ip&longs;uon as, vt constat experi<gap/>nt<gap/><lb/>atque hinc constat quam apte ac conuenienter Aurifices vtantur <lb/>voeabulo parlium. </s>

<s>Nam cum in hac comparatione qualitatum, &longs;eor&longs;im habe a<lb/>tar ratio partium auri, & &longs;eor&longs;im metallorum alligatorum; manife<lb/>stum est &longs;i grauitas totius corporis intelligatur diui&longs;a in<emph.end type="italics"/> 24. <emph type="italics"/>partes <lb/>equales, ex quibus<emph.end type="italics"/> 20. <emph type="italics"/>&longs;int auri, duæ argenti, & duæ æris; quumli<lb/>bet partem auri cum qualibet parte argenti & æris collatam, magni<lb/>tudine e&longs;&longs;e minorem; & &longs;imiliter partem argenti minorem parte <lb/>aris; propterea quod aurum omnia reliqua metalla &longs;uperet grauita<lb/>te quemadmodum & argentum ip&longs;uon as, vt constat experi<gap/>nt<gap/><lb/>atque hinc constat quam apte ac conuenienter Aurifices vtantur <lb/>voeabulo parlium. </s>

<s>bao enim ratione <gap/>dem numero exprimunt vnam <lb/>quamque qualitatem auri cuiuslibet ma&longs;&longs;æ propo&longs;itæ. </s>

Line 1038

<s>quem ydem ob&longs;eruant <lb/>breuiter adnotemus.<emph.end type="italics"/></s>

<s><emph type="italics"/>Inter varias autem & multiplices auri compo&longs;itiones quibus <lb/>cum alys metallis alligari potest, eam retinuere aurifices, quam diu<lb/>turna experientia deprebenderunt omnibus alys e&longs;&longs;e commodiorem, <lb/>eam nimirum quæ ab auri &longs;imilitudine vel minimum di&longs;cedat; qua<lb/>lis e&longs;t quæ &longs;olius argenti atque æris mixtione perficitur. </s>

<s><emph type="italics"/>Inter varias autem & multiplices auri compo&longs;itiones quibus <lb/>cum alys metallis alligari potest, eam retinuere aurifices, quam diu<lb/>turna experientia deprehenderunt omnibus alys e&longs;&longs;e commodiorem, <lb/>eam nimirum quæ ab auri &longs;imilitudine vel minimum di&longs;cedat; qua<lb/>lis e&longs;t quæ &longs;olius argenti atque æris mixtione perficitur. </s>

<s>Et quidem <lb/>&longs;ipartes auri excipias, aris atquo argenti partes, quæ auro &longs;unt per<lb/>mi&longs;cendæ &longs;emper volunte&longs;&longs;e æquales in grauitate: propterea quod <lb/>eadem experientia Magi&longs;tra didicerunt hunc e<32>e mixtionis modum <lb/>longe optimum.<emph.end type="italics"/></s>

<s><emph type="italics"/>Quando ergo aurifices volunt producere aurum cuiu&longs;cunque <lb/>qualitatis, accipiunt tot partes auri puri æquales, quot partium fu<lb/>turum e&longs;t aurum producendum, pauciores tamen partibus<emph.end type="italics"/> 24, & <pb pagenum="62"/><emph type="italics"/>veliquas partes quæ d &longs;unt ad<emph.end type="italics"/> 24, <emph type="italics"/>explent argento & <gap/>e, &longs;umendo <lb/>exvtreque metallo parles aquales in grauite; atque bis rite inter&longs;e <lb/>permixtis componunt aurum de&longs;ideratæ qualitatis: eamque deno<lb/>minant à partibus auripuri in mixtione a&longs;umptis. </s>

<s>Et quoniam <lb/>non prodiret <gap/>ale pror&longs;us quale &longs;acore intendunt, &longs;ed paulo perfe<lb/>ctius; propterea quod <gap/> quidem partes in wixtione <gap/>antant, ex <lb/>argenta vero & are aliquid deperdatur, &longs;olen<gap/> Aucifices tanto plue <lb/><gap/>i&longs;cere argenti & aris quantum perdi po&longs;&longs;e deprebenderunt.<emph.end type="italics"/></s>

<s>Et quoniam <lb/>non prodiret <gap/>ale pror&longs;us quale facere intendunt, &longs;ed paulo perfe<lb/>ctius; propterea quod <gap/> quidem partes in wixtione <gap/>antant, ex <lb/>argenta vero & are aliquid deperdatur, &longs;olen<gap/> Aucifices tanto plue <lb/><gap/>i&longs;cere argenti & aris quantum perdi po&longs;&longs;e deprehenderunt.<emph.end type="italics"/></s>

<s><emph type="italics"/>Veru<gap/> nostra iutentio non est omnia quæ ad ciu&longs;modi mixtione <lb/>pertinent hoc loco expon<gap/>e; &longs;ed illud tantum vt recept<gap/> apud o<gap/><lb/><gap/>es ad ferre voluimus, ex quo manifeste constat, qu<gap/> metallorum <lb/>mixtio in &longs;ingulis qualitatum generibus statuatur: quaquidem e&longs;t <lb/>illa quam add<gap/>ximus <gap/>mpe in auro<emph.end type="italics"/> 23, <emph type="italics"/>partium, partes<emph.end type="italics"/> 23, <emph type="italics"/>e&longs;&longs;e <lb/>auripuri & reliquamq <gap/>deest ad<emph.end type="italics"/> 24, <emph type="italics"/>partes con&longs;tare dimidia <lb/>parte arg<gap/>ti, & dimidia <gap/>ris in grauitate. </s>

<s>In auro vero<emph.end type="italics"/> 22, <emph type="italics"/>par<lb/>tium, auri e&longs;&longs;e<emph.end type="italics"/> 23, <emph type="italics"/>argenti vnam, & æris vnam, &longs;ie enim iterum <lb/>&longs;amma omnium partium est<emph.end type="italics"/> 240 <emph type="italics"/>cademqus est ratio de reliquis ita <lb/><gap/>t numerus partium auri, &longs;e<gap/>per denovinet qualitatem auri, & <lb/>vna medietas reliquarum partium, quæ partibus a<gap/>ri de&longs;unt ad <lb/><gap/>omplendas partes<emph.end type="italics"/> 24, <emph type="italics"/>&longs;it argenti, & reliqua medictas &longs;it æris. </s>

<s>hac <lb/>enim &longs;atis est &longs;uppo&longs;ui&longs;&longs;e, ad nouum illud artificium, quo paulo po&longs;t <lb/>inuestig aturi &longs;umus auri qualitat<gap/> ex &longs;ola grauitate quam babe<gap/><lb/>in aere & aqua, eamque qualitatem dupliei via inuestigabimus <gap/><lb/>vnape<gap/> calculum, par tabellam altera: & <gap/>ia ad calculum &longs;pe<lb/>stant ea, quæ &longs;uperi<gap/> inu<gap/>imus, de grauitate metallorum bucre<lb/>ferenda cen&longs;uimus quæ bic &longs;unt nece&longs;&longs;aria, cuiu&longs;modi &longs;unt auri, <lb/>argenti, atque æris grauitas, quam obtinent in aere, & aqua, qu<gap/><lb/>quidem ita &longs;e habet vt &longs;equitur.<emph.end type="italics"/></s>

<s>hac <lb/>enim &longs;atis est &longs;uppo&longs;ui&longs;&longs;e, ad nouum illud artificium, quo paulo po&longs;t <lb/>inuestig aturi &longs;umus auri qualitat<gap/> ex &longs;ola grauitate quam babe<gap/><lb/>in aere & aqua, eamque qualitatem dupliei via inuestigabimus <gap/><lb/>vnape<gap/> calculum, par tabellam altera: & quia ad calculum &longs;pe<lb/>stant ea, quæ &longs;uperi<gap/> inu<gap/>imus, de grauitate metallorum bucre<lb/>ferenda cen&longs;uimus quæ bic &longs;unt nece&longs;&longs;aria, cuiu&longs;modi &longs;unt auri, <lb/>argenti, atque æris grauitas, quam obtinent in aere, & aqua, qu<gap/><lb/>quidem ita &longs;e habet vt &longs;equitur.<emph.end type="italics"/></s>

<s><emph type="italics"/>Auripuri grauitas, quæ in aere e&longs;t<emph.end type="italics"/> 19, <emph type="italics"/>erit in aqua<emph.end type="italics"/> 18. </s>

Line 1066

<s><emph type="italics"/>Aes ad aquam &longs;e habet in grauitate vt<emph.end type="italics"/> 9, <emph type="italics"/>ad<emph.end type="italics"/> 1. </s>

<s><emph type="italics"/>Ex quibus clari&longs;&longs;ime colligitur, &longs;i aliquod corpus mistum con<lb/>stet partibus æqualibus argenti, & aeris in grauitate, quantam <lb/>grauitatem habeat in aqua & que &longs;it ratio in grauitate ip&longs;ius mi<lb/>sti ad aquam &longs;i enim grauitas aeris in aere &longs;it<emph.end type="italics"/> 9, <emph type="italics"/>eius grauitas in <lb/>aqua erit<emph.end type="italics"/> 8, <emph type="italics"/>& &longs;i grauitas argenti in aere &longs;it quoque<emph.end type="italics"/> 9, <emph type="italics"/>erit eius gra<gap/><emph.end type="italics"/><pb pagenum="63"/><emph type="italics"/>vitas in àqua<emph.end type="italics"/> 8 4/11, <emph type="italics"/>est enim vt 9, ad<emph.end type="italics"/> 8 4/11 <emph type="italics"/>vt<emph.end type="italics"/> 31, <emph type="italics"/>ad<emph.end type="italics"/> 28: <emph type="italics"/>Quare &longs;i <lb/>grauitas corporis mi&longs;ti ex argento, & ære iu<gap/>ta mixtionem prædi<lb/>ctam, quæ etiam &longs;ubinte<gap/>genda erit in &longs;iq<gap/>bus, in aere fuerit<emph.end type="italics"/><lb/>18, <emph type="italics"/>erit iu aqua<emph.end type="italics"/> 16<gap/> <emph type="italics"/>& còn&longs;eq<gap/> grauitas aquæ magnitu-<emph.end type="italics"/><arrow.to.target n="note54"/><lb/><emph type="italics"/>dinem habentis æqualem <gap/> eor por<gap/> misto<gap/><emph.end type="italics"/> <gap/>; <emph type="italics"/>quare corpus <lb/><gap/>stum ex argento & are ad carpui aqueum eia&longs;dem magnitudi<lb/><gap/>is, rationem habe<gap/>it <gap/> grauitate vt<emph.end type="italics"/> 18, <emph type="italics"/>ad<emph.end type="italics"/> 1 22/11, <emph type="italics"/>vel vt<emph.end type="italics"/> 1, <emph type="italics"/>ad<emph.end type="italics"/><lb/><gap/>, <emph type="italics"/>vet <gap/> nique in numerit integrinvt<emph.end type="italics"/> 299, <emph type="italics"/>ad<emph.end type="italics"/> 29, <emph type="italics"/>omnium enim <lb/>i&longs;torum numerorum eadem e&longs;t ratie.<emph.end type="italics"/></s>

<s><emph type="italics"/>Ex quibus clari&longs;&longs;ime colligitur, &longs;i aliquod corpus mistum con<lb/>stet partibus æqualibus argenti, & aeris in grauitate, quantam <lb/>grauitatem habeat in aqua & que &longs;it ratio in grauitate ip&longs;ius mi<lb/>sti ad aquam &longs;i enim grauitas aeris in aere &longs;it<emph.end type="italics"/> 9, <emph type="italics"/>eius grauitas in <lb/>aqua erit<emph.end type="italics"/> 8, <emph type="italics"/>& &longs;i grauitas argenti in aere &longs;it quoque<emph.end type="italics"/> 9, <emph type="italics"/>erit eius gra<gap/><emph.end type="italics"/><pb pagenum="63"/><emph type="italics"/>vitas in àqua<emph.end type="italics"/> 8 4/11, <emph type="italics"/>est enim vt 9, ad<emph.end type="italics"/> 8 4/11 <emph type="italics"/>vt<emph.end type="italics"/> 31, <emph type="italics"/>ad<emph.end type="italics"/> 28: <emph type="italics"/>Quare &longs;i <lb/>grauitas corporis mi&longs;ti ex argento, & ære iuncta mixtionem prædi<lb/>ctam, quæ etiam &longs;ubinte<gap/>genda erit in &longs;equantibus, in aere fuerit<emph.end type="italics"/><lb/>18, <emph type="italics"/>erit id aqua<emph.end type="italics"/> 16<gap/> <emph type="italics"/>& còn&longs;eq<gap/> grauitas aquæ magnitu<emph.end type="italics"/><arrow.to.target n="note54"/><lb/><emph type="italics"/>dinem habentis æqualem <gap/> eor por<gap/> misto<gap/><emph.end type="italics"/> <gap/>; <emph type="italics"/>quare corpus <lb/><gap/>stum ex argento & are ad carpui aqueum eia&longs;dem magnitudi<lb/><gap/>is, rationem habe<gap/>it <gap/> grauitate vt<emph.end type="italics"/> 18, <emph type="italics"/>ad<emph.end type="italics"/> 1 22/11, <emph type="italics"/>vel vt<emph.end type="italics"/> 1, <emph type="italics"/>ad<emph.end type="italics"/><lb/><gap/>, <emph type="italics"/>vet <gap/> nique in numerit integrinvt<emph.end type="italics"/> 299, <emph type="italics"/>ad<emph.end type="italics"/> 29, <emph type="italics"/>omnium enim <lb/>i&longs;torum numerorum eadem e&longs;t ratie.<emph.end type="italics"/></s>

<s><margin.target id="note54"/>5. <emph type="italics"/>huius<emph.end type="italics"/></s>

Line 1078

<s><margin.target id="note55"/>5.<emph type="italics"/>huius<emph.end type="italics"/></s>

<s><emph type="italics"/>Deinde inueniatur grauitas aquæ magnitudine æqualis aur<gap/><lb/>puro<emph.end type="italics"/> 24, <emph type="italics"/>vnciarum: hoc e&longs;t vt 19, ad 1, ita fiat<emph.end type="italics"/> 24, <emph type="italics"/>ad alium, nempe ad <lb/>vnciam<emph.end type="italics"/> 1 9/19, <emph type="italics"/>hic eni<gap/> numerus erit grauitas illius aquæ.<emph.end type="italics"/></s>

<s><emph type="italics"/>Deinde inueniatur grauitas aquæ magnitudine æqualis aur<gap/><lb/>puro<emph.end type="italics"/> 24, <emph type="italics"/>vnciarum: hoc e&longs;t vt 19, ad 1, ita fiat<emph.end type="italics"/> 24, <emph type="italics"/>ad alium, nempe ad <lb/>vnciam<emph.end type="italics"/> 1 9/19, <emph type="italics"/>hic enim numerus erit grauitas illius aquæ.<emph.end type="italics"/></s>

<s><emph type="italics"/>Fiat denique ot<emph.end type="italics"/> 279, <emph type="italics"/>ad<emph.end type="italics"/> 29, <emph type="italics"/>ita <gap/>ur&longs;ue<emph.end type="italics"/> 24, <emph type="italics"/>vntiæ, ad alium, numi<lb/>rus enim quartus, nempe vnc.<emph.end type="italics"/> 2<gap/> <emph type="italics"/>erit grauitas aquæ, magnitu<lb/><gap/>ine æqualis corport mi&longs;to ex argento & ære, euius grauitas est in <lb/>ærè vnc.<emph.end type="italics"/> 24, <emph type="italics"/>corp<gap/>s enim ita mi&longs;tum, ad corpus aqueum eiu&longs;dem ma<lb/><gap/> nitudin<gap/> rationem babet in grauitate vt<emph.end type="italics"/> 279, <emph type="italics"/>ad<emph.end type="italics"/> 29. </s>

<s><emph type="italics"/>Atque ita habeb<gap/>tur tres gra<gap/>tate<gap/> aquæ quantitatum, <lb/>quarum prima æquatur auto pu<gap/>o<emph.end type="italics"/> 24, <emph type="italics"/>vuctarum, &longs;<gap/>unda ma&longs;&longs;<gap/><lb/>propo&longs;itæ<emph.end type="italics"/> 24, <emph type="italics"/>vnciarum, & reliqua corpori mi&longs;to ex argento & <gap/><lb/>&longs;imiliter<emph.end type="italics"/> 24, <emph type="italics"/><gap/>vnciarum quæ quidem tres grauitates in numeris di<lb/>&longs;ponantur co ordine, quo &longs;equitur.<emph.end type="italics"/></s>

<s><emph type="italics"/>Atque ita habebuntur tres gravitates <gap/> aquæ quantitatum, <lb/>quarum prima æquatur auro puro<emph.end type="italics"/> 24, <emph type="italics"/>vuctarum, &longs;<gap/>unda ma&longs;&longs;<gap/><lb/>propo&longs;itæ<emph.end type="italics"/> 24, <emph type="italics"/>vnciarum, & reliqua corpori mi&longs;to ex argento & <gap/><lb/>&longs;imiliter<emph.end type="italics"/> 24, <emph type="italics"/><gap/>vnciarum quæ quidem tres grauitates in numeris di<lb/>&longs;ponantur co ordine, quo &longs;equitur.<emph.end type="italics"/></s>

<s>TABELLE WAR HIER</s>

<s><emph type="italics"/>Deinde quteratur differentia inter primam & tertiam aquæ gra<lb/>uitatem, quæ est vnc.<emph.end type="italics"/> 1 1227/5302, <emph type="italics"/>& hæc differentia statuatur pro <lb/>primo proportionis termino, pro &longs;ecundo termino ponatur grauitas <lb/><gap/>a&longs;&longs;æ propo&longs;itæ idest vnc.<emph.end type="italics"/> 24, <emph type="italics"/>& protertio denique termino ponatur <lb/>different<gap/> inter &longs;ecundam aquæ grauitatem & tertiam, quæ est <lb/>vnc.<emph.end type="italics"/> 1 <gap/> <emph type="italics"/>qua<gap/> eni<gap/> proportionalium ter <gap/>inus nempe<emph.end type="italics"/> 20, </s><pb pagenum="64"/>

<s><emph type="italics"/>Deinde quteratur differentia inter primam & tertiam aquæ gra<lb/>uitatem, quæ est vnc.<emph.end type="italics"/> 1 1227/5302, <emph type="italics"/>& hæc differentia statuatur pro <lb/>primo proportionis termino, pro &longs;ecundo termino ponatur grauitas <lb/><gap/>a&longs;&longs;æ propo&longs;itæ idest vnc.<emph.end type="italics"/> 24, <emph type="italics"/>& protertio denique termino ponatur <lb/>different<gap/> inter &longs;ecundam aquæ grauitatem & tertiam, quæ est <lb/>vnc.<emph.end type="italics"/> 1 <gap/> <emph type="italics"/>quart<gap/> enim proportionalium terminus nempe<emph.end type="italics"/> 20, </s><pb pagenum="64"/>

<s>TABELLE WAR HIER<lb/><emph type="italics"/>erit de<gap/>ominator qualitatis auri de qua <gap/>uaritur quia ille termi<lb/><gap/>ut indicat partes auri pari i<gap/> grauitate, qualibus ma&longs;&longs;a propo&longs;ita <lb/>constat<emph.end type="italics"/> 24. <emph type="italics"/>Hoc autem de<gap/>onstratum e&longs;t prop.<emph.end type="italics"/> 19, <emph type="italics"/>huius.<emph.end type="italics"/></s>

Line 1096

<s>immediate denominent au<lb/>rum propo&longs;itum e<32>e<emph.end type="italics"/> 20, <emph type="italics"/>partium. </s>

<s>Quando vero grauitas totius ma&longs;<lb/>&longs;anon exprimitur per numerum<emph.end type="italics"/> 24, <emph type="italics"/>tunc opus erit inquirere quot <lb/>partes vige&longs;imas quartas totius grauitatis efficiat quartus ille pro<lb/>portionis terminus vt in &longs;equ<gap/>nti <gap/>mplo clarius appar<gap/>bit.<emph.end type="italics"/></s>

<s>Quando vero grauitas totius ma&longs;<lb/>&longs;anon exprimitur per numerum<emph.end type="italics"/> 24, <emph type="italics"/>tunc opus erit inquirere quot <lb/>partes vige&longs;imas quartas totius grauitatis efficiat quartus ille pro<lb/>portionis terminus vt in &longs;equenti <gap/>mplo clarius appar<gap/>bit.<emph.end type="italics"/></s>

<s><emph type="italics"/>Sit euim propo&longs;ita a lia auri ma&longs;&longs;a cuius grauitas in aere &longs;it<emph.end type="italics"/> 5301 <lb/><emph type="italics"/>in aqua vero<emph.end type="italics"/> 4988, <emph type="italics"/>&longs;i igitur hic <gap/>erus &longs;ubtrabatur ex numero to<emph.end type="italics"/><lb/><arrow.to.target n="note56"/> <emph type="italics"/>tius grauitatis<emph.end type="italics"/> 5301, <emph type="italics"/>r<gap/>liquus numerus<emph.end type="italics"/> 313, * <emph type="italics"/>erit grauitas aquæ <lb/>propo&longs;itæ ma&longs;&longs;æ magnitudine æqualis. </s>

<s>Inueniantur <expan abbr="quoq;">quoque</expan> duæ aliæ <lb/>grauitates aquæ, vna re&longs;pondentis auro puro magnitudine, altera <lb/>corpori mi&longs;to ex argento & ære, ita tamen vt grauitas tum auri pu<gap/><lb/>ri, tum corporis misti &longs;it eadem quæ ma&longs;&longs;æ propo&longs;itæ, non &longs;ecus ac in <lb/>præcedenti oxemplo factitatum est. </s>

<s>Inueniantur <expan abbr="quoq;">quoque</expan> duæ aliæ <lb/>grauitates aquæ, vna re&longs;pondentis auro puro magnitudine, altera <lb/>corpori mi&longs;to ex argento & ære, ita tamen vt grauitas tum auri pu<lb/>ri, tum corporis misti &longs;it eadem quæ ma&longs;&longs;æ propo&longs;itæ, non &longs;ecus ac in <lb/>præcedenti oxemplo factitatum est. </s>

<s>hoc e&longs;t primo fiat vt<emph.end type="italics"/> 19, <emph type="italics"/>ad<emph.end type="italics"/> 1, <emph type="italics"/>ita<emph.end type="italics"/><lb/>5301, <emph type="italics"/>ad<emph.end type="italics"/> 279, <emph type="italics"/>hic enim numerus erit grauitas aquæ magnitudinem <lb/>habentis æqualem auro puro, <gap/>uius grauitas e&longs;t<emph.end type="italics"/> 5301. <emph type="italics"/>Deinde fia<gap/><lb/>vt<emph.end type="italics"/> 279, <emph type="italics"/>ad<emph.end type="italics"/> 29, <emph type="italics"/>ita rur&longs;um grauitas<emph.end type="italics"/> 5301, <emph type="italics"/>ad aliam, hac enim ratione <lb/>producetur nu<gap/>erus<emph.end type="italics"/> 551, <emph type="italics"/>debitus grauitati aquæ, magnitudine <lb/>aqualis corpori mi&longs;to ex argento & ære grauitatem habenti eandem <lb/>cum eadem ma<32>a propo&longs;ita. </s>

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<s>Reuocetur primum <lb/>ptopo&longs;itæ ma&longs;&longs;æ grauitas<emph.end type="italics"/> 837, <emph type="italics"/>ad grauitatem<emph.end type="italics"/> 5301, <emph type="italics"/>hoc est intelli<lb/>gatur ip&longs;a ma&longs;&longs;a grauitatem babere<emph.end type="italics"/> 5301. <emph type="italics"/>deinde fiat vt<emph.end type="italics"/> 837, <emph type="italics"/>ad<emph.end type="italics"/> 53, <lb/><emph type="italics"/>grauitatem vidclicet aquæ ip&longs;i ma&longs;&longs;æ æqualis, ita<emph.end type="italics"/> 5301, <emph type="italics"/>ad<emph.end type="italics"/> 335 2/3, <lb/><emph type="italics"/>ergo<emph.end type="italics"/> 335 2/3, <emph type="italics"/>erit grauitas aquæ magnitudinem habentis æqualem <lb/>aureæ ma<32>æ, cuius grauitas<emph.end type="italics"/> 837, <emph type="italics"/>reuocata est ad grauitatem<emph.end type="italics"/> 5301; <lb/><emph type="italics"/>quare grauitas &longs;ecundæ aquæ erit<emph.end type="italics"/> 335 2/1, <emph type="italics"/>& con&longs;equenter differen-<emph.end type="italics"/></s>

<s>TABELLE WAR HIER<lb/><emph type="italics"/>tia inter ip&longs;am grauitatem &longs;ecundæ aquæ & grauitatem tertiæ<emph.end type="italics"/> 551, <lb/><emph type="italics"/>erit<emph.end type="italics"/> 215 1/1, <emph type="italics"/>&longs;ed ip&longs;a differentia ponitur pro tertio proportionis ter<lb/>mino; crgo<emph.end type="italics"/> 215 1/3, <emph type="italics"/>erit quæ&longs;itus terminus, nempe proportionis ter-<emph.end type="italics"/></s><pb pagenum="66"/>

<s>TABELLE WAR HIER<lb/><emph type="italics"/>tia inter ip&longs;am grauitatem &longs;ecundæ aquæ & grauitatem tertiæ<emph.end type="italics"/> 551, <lb/><emph type="italics"/>erit<emph.end type="italics"/> 215 1/1, <emph type="italics"/>&longs;ed ip&longs;a differentia ponitur pro tertio proportionis ter<lb/>mino; crgo<emph.end type="italics"/> 215 1/3, <emph type="italics"/>erit quæ&longs;itus terminus, nempe proportionis ter<emph.end type="italics"/></s><pb pagenum="66"/>

<s>TABELLE WAR HIER<lb/><emph type="italics"/>tius. </s>

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<s>fiat vt lib.

1. ad vne <lb/>11, Scrup.<emph.end type="italics"/> 5, <emph type="italics"/>Gran.<emph.end type="italics"/> 3 963/1767, <emph type="italics"/>ita lib.<emph.end type="italics"/> 3. <emph type="italics"/>ad alium numerum, is erit <lb/>lib.<emph.end type="italics"/> 2, <emph type="italics"/>vnc. </s>

1. ad vnc <lb/>11, Scrup.<emph.end type="italics"/> 5, <emph type="italics"/>Gran.<emph.end type="italics"/> 3 963/1767, <emph type="italics"/>ita lib.<emph.end type="italics"/> 3. <emph type="italics"/>ad alium numerum, is erit <lb/>lib.<emph.end type="italics"/> 2, <emph type="italics"/>vnc. </s>

<s>9, Scrup. 15, Gran.<emph.end type="italics"/> 10 1122/1767. <emph type="italics"/>& tanta erit grauitas au<lb/>ri propo &longs;iti in aqua. </s>

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<s>Compo&longs;itio eiu&longs;dem tabulæ.</s>

<s><emph type="italics"/>Si ea quæ hactenus &longs;unt dicta rectè intelligantur liquido appare<lb/>bit compo&longs;itionem tabulæ in eo con&longs;istere, vt inuoniatur grauitas <lb/>quam aurum cuiu&longs;uis qualitatis habet in aqua, hoc e&longs;t &longs;i intelligan<lb/>tur propo&longs;itæ plures ma&longs;&longs;æ auri, &longs;ingulæ &longs;ingularum librarum, & <lb/>alia &longs;it qualitatis<emph.end type="italics"/> 24, <emph type="italics"/>partium alia<emph.end type="italics"/> 23, <emph type="italics"/>alia<emph.end type="italics"/> 22, <emph type="italics"/>&c. </s>

<s><emph type="italics"/>Si ea quæ bactenus &longs;unt dicta rectè intelligantur liquido appare<lb/>bit compo&longs;itionem tabulæ in eo con&longs;istere, vt inuoniatur grauitas <lb/>quam aurum cuiu&longs;uis qualitatis habet in aqua, hoc e&longs;t &longs;i intelligan<lb/>tur propo&longs;itæ plures ma&longs;&longs;æ auri, &longs;ingulæ &longs;ingularum librarum, & <lb/>alia &longs;it qualitatis<emph.end type="italics"/> 24, <emph type="italics"/>partium alia<emph.end type="italics"/> 23, <emph type="italics"/>alia<emph.end type="italics"/> 22, <emph type="italics"/>&c. </s>

<s>quanta &longs;it gra<lb/>uitas vniu&longs;cuiu&longs;que in aqua, id quod hoc extremo loco inuestigare <lb/>docebimus.<emph.end type="italics"/></s>

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<s><emph type="italics"/>Sit deinde propo&longs;itum quod vis aliud corpus aureum vnius libræ, <lb/>&longs;itque exemp. </s>

<s>gratia illud aurum<emph.end type="italics"/> 20, <emph type="italics"/>partium, patet igitur ex defi<lb/>nitione qualitatis, ex<emph.end type="italics"/> 24, <emph type="italics"/>&longs;emiuncys totius grauitatis,<emph.end type="italics"/> 20, <emph type="italics"/>&longs;emiuncias <lb/>e&longs;&longs;e auri puri, duas argenti, & reliquas duas æris & quoniam gra<lb/>uitas misti in aere, ad grauitatem eiu&longs;dem in aqua rationem habet <lb/>vt<emph.end type="italics"/> 279, <emph type="italics"/>ad<emph.end type="italics"/> 250, <emph type="italics"/>vt exiam dictis patet, fiat vt<emph.end type="italics"/> 279, <emph type="italics"/>ad<emph.end type="italics"/> 250, <emph type="italics"/>ita qua<lb/>tuor &longs;emiunciæ, vel potius <gap/>, vnciæ misti quod componit qualitatem <lb/>auri<emph.end type="italics"/> 20, <emph type="italics"/>partium, paulo ante propo&longs;itam ad alias vncias. </s>

<s>gratia illud aurum<emph.end type="italics"/> 20, <emph type="italics"/>partium, patet igitur ex defi<lb/>nitione qualitatis, ex<emph.end type="italics"/> 24, <emph type="italics"/>&longs;emiunciis totius grauitatis,<emph.end type="italics"/> 20, <emph type="italics"/>&longs;emiuncias <lb/>e&longs;&longs;e auri puri, duas argenti, & reliquas duas æris & quoniam gra<lb/>uitas misti in aere, ad grauitatem eiu&longs;dem in aqua rationem habet <lb/>vt<emph.end type="italics"/> 279, <emph type="italics"/>ad<emph.end type="italics"/> 250, <emph type="italics"/>vt exiam dictis patet, fiat vt<emph.end type="italics"/> 279, <emph type="italics"/>ad<emph.end type="italics"/> 250, <emph type="italics"/>ita qua<lb/>tuor &longs;emiunciæ, vel potius <gap/>, vnciæ misti quod componit qualitatem <lb/>auri<emph.end type="italics"/> 20, <emph type="italics"/>partium, paulo ante propo&longs;itam ad alias vncias. </s>

<s>inuenien<lb/>tur enim pro grauitate illius mi&longs;ti in aqua vnc.<emph.end type="italics"/> 1 221/274. <emph type="italics"/>Est autem <lb/>grauitas auri puri<emph.end type="italics"/> 20, <emph type="italics"/>&longs;emiunciarum vel<emph.end type="italics"/> 10, <emph type="italics"/>vnciarum in aqua vnc.<emph.end type="italics"/><lb/>9 9/19, <emph type="italics"/>eo quod ita &longs;e habeant<emph.end type="italics"/> 10, <emph type="italics"/>ad<emph.end type="italics"/> 9 9/29, <emph type="italics"/>vt 19, ad<emph.end type="italics"/> 18, <emph type="italics"/>Quare &longs;i hæ <lb/>duæ grauitates inuentæ colligantur in vnam &longs;ummam, inueniemus <lb/>totam ma&longs;&longs;am auri propo&longs;itam, cuius grauitas in aere ponebatur lib. <lb/>1, in aqua babere grauitatem vnc.<emph.end type="italics"/> 11 1409/5301. <emph type="italics"/>& facta reductione <lb/>fractionis ad &longs;crupula, & grana, vnc. </s>

Line 1276

<s>grauitates auri<emph.end type="italics"/> 20, <emph type="italics"/>&<emph.end type="italics"/> 19, <emph type="italics"/>partium. </s>

<s>Quo<lb/>niam igitur in auro<emph.end type="italics"/> 20, <emph type="italics"/>partium &longs;unt auri puri<emph.end type="italics"/> 20. <emph type="italics"/>&longs;emiuneiæ, misti <lb/>vero<emph.end type="italics"/> 4, <emph type="italics"/>& in auro<emph.end type="italics"/> 29, <emph type="italics"/>partium &longs;unt auri puri<emph.end type="italics"/> 19. <emph type="italics"/>&longs;emiunciæ, misti <lb/>vero<emph.end type="italics"/> 5, <emph type="italics"/>erit in auro<emph.end type="italics"/> 20. <emph type="italics"/><expan abbr="parti&utilde;">partium</expan> vna &longs;emiuncia auri puri plu&longs;quam in <lb/>auro<emph.end type="italics"/> 19, <emph type="italics"/>partium, in auro autem 19, partium erit vna &longs;emiuncia <lb/>misti plus quam in auro<emph.end type="italics"/> 20, <emph type="italics"/>partium; quare grauitas auri<emph.end type="italics"/> 20, <emph type="italics"/>par<lb/>tium in aqua &longs;uperabit in aqua grauitatem auri<emph.end type="italics"/> 19, <emph type="italics"/>partium gra<lb/>uitate, qua &longs;emiuncia auri puri &longs;uperat &longs;emiunciam mi&longs;li. </s>

<s>Quo<lb/>niam igitur in auro<emph.end type="italics"/> 20, <emph type="italics"/>partium &longs;unt auri puri<emph.end type="italics"/> 20. <emph type="italics"/>&longs;emiunciæ, misti <lb/>vero<emph.end type="italics"/> 4, <emph type="italics"/>& in auro<emph.end type="italics"/> 29, <emph type="italics"/>partium &longs;unt auri puri<emph.end type="italics"/> 19. <emph type="italics"/>&longs;emiunciæ, misti <lb/>vero<emph.end type="italics"/> 5, <emph type="italics"/>erit in auro<emph.end type="italics"/> 20. <emph type="italics"/><expan abbr="parti&utilde;">partium</expan> vna &longs;emiuncia auri puri plu&longs;quam in <lb/>auro<emph.end type="italics"/> 19, <emph type="italics"/>partium, in auro autem 19, partium erit vna &longs;emiuncia <lb/>misti plus quam in auro<emph.end type="italics"/> 20, <emph type="italics"/>partium; quare grauitas auri<emph.end type="italics"/> 20, <emph type="italics"/>par<lb/>tium in aqua &longs;uperabit in aqua grauitatem auri<emph.end type="italics"/> 19, <emph type="italics"/>partium gra<lb/>uitate, qua &longs;emiuncia auri puri &longs;uperat &longs;emiunciam mi&longs;li. </s>

<s>Quod <lb/>erat demon&longs;trandum. </s>

Line 1308

<s>TABELLE WAR HIER</s>

<s><emph type="italics"/><expan abbr="Contractior&etilde;">Contractiorem</expan> <expan abbr="quoq.">quoque</expan> ad inuenies dimidy pedis <expan abbr="men&longs;urã">men&longs;uram</expan> in margine pag. </s>

<s><emph type="italics"/><expan abbr="Contractior&etilde;">Contractiorem</expan> <expan abbr="quoq.">quoque</expan> ad inuenies dimidij pedis <expan abbr="men&longs;urã">men&longs;uram</expan> in margine pag. </s>

<s>34. <expan abbr="appo&longs;itã">appo&longs;itam</expan>, <lb/>quia mad facta papyrus, <expan abbr="d&utilde;">dum</expan> imprimeretur recipiebat <expan abbr="verã">veram</expan> <expan abbr="men&longs;urã">men&longs;uram</expan>, ex&longs;iccata <lb/>breuiorem red didit. </s>

<s>Ita que &longs;i quartadecima pars vnius vncia addatur ip&longs;i men <lb/>&longs;ura componetur dimidy pedis men&longs;ura; vel &longs;i ip&longs;a men&longs;ura duplicetur, & ei ad<lb/>datur &longs;eptima pars vnius vncia fiet men&longs;ura vnius pedis, ad cuius rationem em <lb/>nes calculi in tabulas optime re&longs;pond ebunt.<emph.end type="italics"/></s>

<s>Ita que &longs;i quartadecima pars vnius vncia addatur ip&longs;i men <lb/>&longs;ura componetur dimidij pedis men&longs;ura; vel &longs;i ip&longs;a men&longs;ura duplicetur, & ei ad<lb/>datur &longs;eptima pars vnius vncia fiet men&longs;ura vnius pedis, ad cuius rationem em <lb/>nes calculi in tabulas optime re&longs;pond ebunt.<emph.end type="italics"/></s>

<s><emph type="italics"/>Denique pagina<emph.end type="italics"/> 38. <emph type="italics"/>&<emph.end type="italics"/> 39. <emph type="italics"/>tabularum prapo&longs;terus e&longs;t ordo. </s>

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