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Colored diff for /texts/archimedes/xml/gheta_promo_029_la_1603.xml between version 1.21 and 1.22

version 1.21, 2006/05/03 18:18:27 version 1.22, 2006/05/10 10:13:55
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 <s><emph type="italics"/>Accepto, vt diximus aliquo corpore plumbeo D, cuius grauitas<emph.end type="italics"/> 23, <lb/><emph type="italics"/>&amp; aqu&aelig; magnitudinem habentis &aelig;qualem plumbo D, inuenta graui&shy;<emph.end type="italics"/><pb xlink:href="029/01/024.jpg" 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&shy;<lb/>rum qui &longs;it<emph.end type="italics"/> 86 22/23&lt;31&gt; <emph type="italics"/>is igitur numerus erit cubus diametris &longs;ph&aelig;r&aelig; ex <lb/>plumbo, grauitatem &aelig;qualem habentis propo&longs;ite ex aqua &longs;ph&aelig;r&aelig; B, <lb/>quare eius latus cubicum, quod est<emph.end type="italics"/> 4 41/100&lt;31&gt; <emph type="italics"/>fer&egrave;, indicabit ip&longs;am <lb/>diametrum.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Accepto, vt diximus aliquo corpore plumbeo D, cuius grauitas<emph.end type="italics"/> 23, <lb/><emph type="italics"/>&amp; aqu&aelig; magnitudinem habentis &aelig;qualem plumbo D, inuenta graui&shy;<emph.end type="italics"/><pb xlink:href="029/01/024.jpg" 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&shy;<lb/>rum qui &longs;it<emph.end type="italics"/> 86 22/23&lt;31&gt; <emph type="italics"/>is igitur numerus erit cubus diametris &longs;ph&aelig;r&aelig; ex <lb/>plumbo, grauitatem &aelig;qualem habentis propo&longs;ite ex aqua &longs;ph&aelig;r&aelig; B, <lb/>quare eius latus cubicum, quod est<emph.end type="italics"/> 4 41/100&lt;31&gt; <emph type="italics"/>fer&egrave;, indicabit ip&longs;am <lb/>diametrum.<emph.end type="italics"/></s></p><p type="main">
  
 <s><emph type="italics"/>Similiter &longs;i propo&longs;itum corpus aqueum B, fuerit cubicum, vel alicu<lb/>ius alterius form&aelig; regularis, eadem ratione vtemur ad inueniendum <lb/>latus cubi ex plumbo, grauitatem habentis &aelig;qualem 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/>&lt;31&gt; <emph type="italics"/>cubus ex plumbo &aelig;qualis grauitate propo&longs;ito ex aqua cubo B, <lb/>quare latus cubicum numeri<emph.end type="italics"/> 86 22/2<gap/>&lt;31&gt; <emph type="italics"/>quod e&longs;t<emph.end type="italics"/> 4 41/100&lt;31&gt; <emph type="italics"/>fer&egrave;, indicabit <lb/>qu&aelig;&longs;itum latus cubi ex plumbo.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Similiter &longs;i propo&longs;itum corpus aqueum B, fuerit cubicum, vel alicu&shy;<lb/>ius alterius form&aelig; regularis, eadem ratione vtemur ad inueniendum <lb/>latus cubi ex plumbo, grauitatem habentis &aelig;qualem propo&longs;ito ex aqua <lb/>tubo 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/>&lt;31&gt; <emph type="italics"/>cubus ex plumbo &aelig;qualis grauitate propo&longs;ito ex aqua cubo B, <lb/>quare latus cubicum numeri<emph.end type="italics"/> 86 22/2<gap/>&lt;31&gt; <emph type="italics"/>quod e&longs;t<emph.end type="italics"/> 4 41/100&lt;31&gt; <emph type="italics"/>fer&egrave;, indicabit <lb/>qu&aelig;&longs;itum latus cubi ex plumbo.<emph.end type="italics"/></s></p><p type="main">
  
 <s>Neque di&longs;&longs;imili ratione inuenienda erit magnitudo auri, <lb/>argenti, cer&aelig;, aut cuiu&longs;cunque &longs;olidi, grauitatem habentis <lb/>propo&longs;ito corpori liquido &aelig;qualem. </s></p><p type="head"> <s>Neque di&longs;&longs;imili ratione inuenienda erit magnitudo auri, <lb/>argenti, cer&aelig;, aut cuiu&longs;cunque &longs;olidi, grauitatem habentis <lb/>propo&longs;ito corpori liquido &aelig;qualem. </s></p><p type="head">
  
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 <s><emph type="italics"/>Contra, qu&aelig;ro quomodo &longs;e habent in magnitudine aurum, &amp; plumbum. </s> <s><emph type="italics"/>Contra, qu&aelig;ro quomodo &longs;e habent in magnitudine aurum, &amp; plumbum. </s>
  
 <s>inteligatur <lb/>aurum, quoniam grauius e&longs;t plumbo, magnitudinem habere 1, &amp; in linea plumbi, &longs;ub ti&shy;<lb/>tulo auri, qu&aelig;ratur plumbi magnitudo, ea erit 1 15/23, aurum igitur ad plumbum &longs;e ba-<emph.end type="italics"/><pb xlink:href="029/01/040.jpg" pagenum="33"/><emph type="italics"/><gap/>bit in magnitudine vt 1, ad 1 19/21, &longs;i enim &longs;umantur duo corpora &aelig;que grauia, vnum <lb/>aureum, alterum plumbeum, &longs;it autem corporis aurei magnitudo 1, plumbei erit 1 1<gap/>/21, <lb/>quare corpus aureum ad corpus plumbeum eiu&longs;dem grauitatis &longs;e habebit in magnitudi&shy;<lb/>ne vt 1, ad 1 19/21, comparantur autem inter &longs;e genera diuer&longs;a magnitudine, in corpori&shy;<lb/>bus &aelig;que grauibus.<emph.end type="italics"/></s></p><p type="main"> <s>inteligatur <lb/>aurum, quoniam grauius e&longs;t plumbo, magnitudinem habere 1, &amp; in linea plumbi, &longs;ub ti&shy;<lb/>tulo auri, qu&aelig;ratur plumbi magnitudo, ea erit 1 15/23, aurum igitur ad plumbum &longs;e ba-<emph.end type="italics"/><pb xlink:href="029/01/040.jpg" pagenum="33"/><emph type="italics"/>bebit in magnitudine vt 1, ad 1 19/21, &longs;i enim &longs;umantur duo corpora &aelig;que grauia, vnum <lb/>aureum, alterum plumbeum, &longs;it autem corporis aurei magnitudo 1, plumbei erit 1 1<gap/>/21, <lb/>quare corpus aureum ad corpus plumbeum eiu&longs;dem grauitatis &longs;e habebit in magnitudi&shy;<lb/>ne vt 1, ad 1 19/21, comparantur autem inter &longs;e genera diuer&longs;a magnitudine, in corpori&shy;<lb/>bus &aelig;que grauibus.<emph.end type="italics"/></s></p><p type="main">
  
 <s><emph type="italics"/>Qu&aelig;ro denique quomodo &longs;e habent in magnitudine ferrum, &amp; aqua, ponatur ferrum, <lb/>vt grauius aqua, magnitudinem habere 1, &amp; in linea aqu&aelig;, &longs;ub titulo ferri, qu&aelig;ratur <lb/>aqu&aelig; magnitudo, ea erit<emph.end type="italics"/> 8, <emph type="italics"/>ferrum igitur ad aquam &longs;e habebit in magnitudine vt 1, ad<emph.end type="italics"/> 8. </s></p><p type="head"> <s><emph type="italics"/>Qu&aelig;ro denique quomodo &longs;e habent in magnitudine ferrum, &amp; aqua, ponatur ferrum, <lb/>vt grauius aqua, magnitudinem habere 1, &amp; in linea aqu&aelig;, &longs;ub titulo ferri, qu&aelig;ratur <lb/>aqu&aelig; magnitudo, ea erit<emph.end type="italics"/> 8, <emph type="italics"/>ferrum igitur ad aquam &longs;e habebit in magnitudine vt 1, ad<emph.end type="italics"/> 8. </s></p><p type="head">
  
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 <s><emph type="italics"/>Porro ad inueniendas grauitates &longs;ph&aelig;rarum ex reliquis metallis <lb/>vel ex quacunque alia materia, h&aelig;c erit ratio.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Porro ad inueniendas grauitates &longs;ph&aelig;rarum ex reliquis metallis <lb/>vel ex quacunque alia materia, h&aelig;c erit ratio.<emph.end type="italics"/></s></p><p type="main">
  
 <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 de grauitate &longs;ph&aelig;r&aelig;<lb/><expan abbr="pl&umacr;be&aelig;">plumbe&aelig;</expan> qu&aelig;ritur cuius diameter est<emph.end type="italics"/> 1/4, <emph type="italics"/>vnci&aelig;) ita 19. grauitas vi<gap/><lb/>licet &longs;ph&aelig;r&aelig; stanne&aelig; <expan abbr="diametr&umacr;">diametrum</expan> habentis<emph.end type="italics"/> 1/4, <emph type="italics"/>vnci&aelig;, ad <expan abbr="ali&umacr;">alium</expan> numerum<lb/>qui &longs;it<emph.end type="italics"/> 29 19/74, <emph type="italics"/>grauitas igitur &longs;ph&aelig;r&aelig; plumbe&aelig;, diametrum haben<lb/>tis<emph.end type="italics"/> 1/4, <emph type="italics"/>vnci&aelig;, 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, &amp; magnitudine, appo&longs;uimus.<emph.end type="italics"/></s></p><p type="main"> <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 de grauitate &longs;ph&aelig;r&aelig;<lb/><expan abbr="pl&umacr;be&aelig;">plumbe&aelig;</expan> qu&aelig;ritur cuius diameter est<emph.end type="italics"/> 1/4, <emph type="italics"/>vnci&aelig;) ita 19. grauitas vide&shy;<lb/>licet &longs;ph&aelig;r&aelig; stanne&aelig; <expan abbr="diametr&umacr;">diametrum</expan> habentis<emph.end type="italics"/> 1/4, <emph type="italics"/>vnci&aelig;, ad <expan abbr="ali&umacr;">alium</expan> numerum<lb/>qui &longs;it<emph.end type="italics"/> 29 19/74, <emph type="italics"/>grauitas igitur &longs;ph&aelig;r&aelig; plumbe&aelig;, diametrum haben<lb/>tis<emph.end type="italics"/> 1/4, <emph type="italics"/>vnci&aelig;, erit gran.<emph.end type="italics"/> 29 39/74. <emph type="italics"/>&longs;tannum enim ad plumbum rationem <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&shy;<lb/>te, &amp; magnitudine, appo&longs;uimus.<emph.end type="italics"/></s></p><p type="main">
  
 <s><emph type="italics"/>Si vero qu&aelig;ratur de grauitate &longs;ph&aelig;r&aelig; plumbe&aelig;, 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 grauia<emph.end type="italics"/></s></p><pb xlink:href="029/01/048.jpg" pagenum="41"/><p type="table"> <s><emph type="italics"/>Si vero qu&aelig;ratur de grauitate &longs;ph&aelig;r&aelig; plumbe&aelig;, 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 grauia<emph.end type="italics"/></s></p><pb xlink:href="029/01/048.jpg" pagenum="41"/><p type="table">
  
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 quare radix cubica numeri<emph.end type="italics"/> 5 49/190, <emph type="italics"/>dabit qu&aelig;&longs;itam dia&shy;<lb/>metrum, &amp; quoniam numerus<emph.end type="italics"/> 5 49/190, <emph type="italics"/>non e&longs;t pr&aelig;ci&longs;e cubus, &amp; ideo <lb/>non explicabitur eius radix accurate, multiplicetur per 1000000, &amp; <lb/>ex producto<emph.end type="italics"/> 5257894 14/19, <emph type="italics"/>neglecto fracto<emph.end type="italics"/> 14/19, <emph type="italics"/>eruatur radix, tan&shy;<lb/>quam ex accurato numero cubo, ea erit<emph.end type="italics"/> 174: <emph type="italics"/>fer&egrave;, &amp; erit centupla ra&shy;<lb/>dicis numeri<emph.end type="italics"/> 5 49/190, <emph type="italics"/>quia numerus<emph.end type="italics"/> 5 49/190, <emph type="italics"/>multiplicatus fuit per <lb/>cubum ex 100; diameter igitur &longs;ph&aelig;r&aelig; ferre&aelig;, grauitatem habentis <lb/>1, lib.  quare radix cubica numeri<emph.end type="italics"/> 5 49/190, <emph type="italics"/>dabit qu&aelig;&longs;itam dia&shy;<lb/>metrum, &amp; quoniam numerus<emph.end type="italics"/> 5 49/190, <emph type="italics"/>non e&longs;t pr&aelig;ci&longs;e cubus, &amp; ideo <lb/>non explicabitur eius radix accurate, multiplicetur per 1000000, &amp; <lb/>ex producto<emph.end type="italics"/> 5257894 14/19, <emph type="italics"/>neglecto fracto<emph.end type="italics"/> 14/19, <emph type="italics"/>eruatur radix, tan&shy;<lb/>quam ex accurato numero cubo, ea erit<emph.end type="italics"/> 174: <emph type="italics"/>fer&egrave;, &amp; erit centupla ra&shy;<lb/>dicis numeri<emph.end type="italics"/> 5 49/190, <emph type="italics"/>quia numerus<emph.end type="italics"/> 5 49/190, <emph type="italics"/>multiplicatus fuit per <lb/>cubum ex 100; diameter igitur &longs;ph&aelig;r&aelig; ferre&aelig;, grauitatem habentis <lb/>1, lib.
  
 erit<emph.end type="italics"/> 1 74/100: <emph type="italics"/>deinde &longs;i duplicetur<emph.end type="italics"/> 5257894 24/19, <emph type="italics"/>&amp; ex ita du&shy;<lb/>plicato eruatur radix cubica<emph.end type="italics"/> 219, <emph type="italics"/>ea dabit centuplum diametri &longs;ph&aelig;&shy;<lb/>ra ferre&aelig;, grauitatem habentis<emph.end type="italics"/> 2, <emph type="italics"/>lib.  
  
 &amp; &longs;i triplicetur, triplicati radix <lb/>cubica<emph.end type="italics"/> 250<gap/> <emph type="italics"/>dabit centuplum diametri &longs;ph&aelig;r&aelig; ferre&aelig;, cuius grauitas <lb/>crit<emph.end type="italics"/> 3, <emph type="italics"/>lib.  
  
 &amp; &longs;ic reliquarum &longs;ph&aelig;rarum in infinitum inuenientur <lb/>diametri. </s> &amp; &longs;ic reliquarum &longs;ph&aelig;rarum in infinitum inuenientur <lb/>diametri. </s>
  
 <s>multiplicia autem numeri<emph.end type="italics"/> 5257894 14/19, <emph type="italics"/>&longs;ola additione in&shy;<lb/>uenientur, vt dictum est &longs;upra de inuentione multiplicium numeri<emph.end type="italics"/><lb/>5684210 10/19. <emph type="italics"/>Atque hac ratione pr&aelig;dictam tabulam compo&longs;uimus.<emph.end type="italics"/></s></p><p type="margin"> <s>multiplicia autem numeri<emph.end type="italics"/> 5257894 14/19, <emph type="italics"/>&longs;ola additione in&shy;<lb/>uenientur, vt dictum est &longs;upra de inuentione multiplicium numeri<emph.end type="italics"/><lb/>5684210 10/19. <emph type="italics"/>Atque hac ratione pr&aelig;dictam tabulam compo&longs;uimus.<emph.end type="italics"/></s></p><p type="margin">
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 <s>SINT tria corpora &aelig;que grauia A, BC, D, quorum A, primum, <lb/>&amp; tertium D. &longs;int generis diuer&longs;i, portio vero &longs;ecundi B, &longs;it eiu&longs;dem <lb/>generis cum corpore A, &amp; portio C, eiu&longs;dem generis cum corpore <lb/><figure id="id.029.01.064.1.jpg" xlink:href="029/01/064/1.jpg"/><lb/>D, &longs;int etiam alia tria <lb/>corpora aquea P, OL, <lb/>&amp; Q, <expan abbr="quor&utilde;">quorum</expan> <gap/>, &longs;it &aelig;qua&shy;<lb/>le corpori A, magnitu&shy;<lb/>dine, ip&longs;um vero OL, <lb/>&aelig;quale corpori BC, &amp; <lb/>ip&longs;um Q, &aelig;quale cor&shy;<lb/>pori D, &amp; &longs;int earum <lb/>grauitates, G, ip&longs;ius P, <lb/>&amp; FV, ip&longs;ius OL, &amp; H, <lb/>ip&longs;ius <expan abbr="q.">que</expan> Dico vt diffe&shy;<lb/>rentia grauitatum G, <lb/>H, ad grauitatem cor&shy;<lb/>poris BC, ita e&longs;&longs;e diffe&shy;<lb/>rentiam grauitatum <lb/>G, FV, ad grauitatem <lb/>portionis C; &amp; ita differentiam grauitatum FV, H, ad portionis B, <lb/>grauitatem. </s> <s>SINT tria corpora &aelig;que grauia A, BC, D, quorum A, primum, <lb/>&amp; tertium D. &longs;int generis diuer&longs;i, portio vero &longs;ecundi B, &longs;it eiu&longs;dem <lb/>generis cum corpore A, &amp; portio C, eiu&longs;dem generis cum corpore <lb/><figure id="id.029.01.064.1.jpg" xlink:href="029/01/064/1.jpg"/><lb/>D, &longs;int etiam alia tria <lb/>corpora aquea P, OL, <lb/>&amp; Q, <expan abbr="quor&utilde;">quorum</expan> <gap/>, &longs;it &aelig;qua&shy;<lb/>le corpori A, magnitu&shy;<lb/>dine, ip&longs;um vero OL, <lb/>&aelig;quale corpori BC, &amp; <lb/>ip&longs;um Q, &aelig;quale cor&shy;<lb/>pori D, &amp; &longs;int earum <lb/>grauitates, G, ip&longs;ius P, <lb/>&amp; FV, ip&longs;ius OL, &amp; H, <lb/>ip&longs;ius <expan abbr="q.">que</expan> Dico vt diffe&shy;<lb/>rentia grauitatum G, <lb/>H, ad grauitatem cor&shy;<lb/>poris BC, ita e&longs;&longs;e diffe&shy;<lb/>rentiam grauitatum <lb/>G, FV, ad grauitatem <lb/>portionis C; &amp; ita differentiam grauitatum FV, H, ad portionis B, <lb/>grauitatem. </s>
  
 <s>Sit enim portionis B, grauitas E, &amp; portionis C, gra&shy;<lb/>uitas K; ergo totius corporis BC grauitas erit EK, &longs;i<gap/><expan abbr="q;">que</expan> portionis O, <lb/>qu&aelig; &longs;it &aelig;qualis portioni B, grauitas F, ergo reliqu&aelig; portionis L, <lb/><gap/>qualis portioni C, grauitas erit V, Quoniam igitur e&longs;t, vt A, ad P, <lb/>ita B, ad O, &aelig;quale videlicet ad &aelig;quale, erit permutando, vt A, ad B, <lb/>ita p.ad O, &amp; quoniam &longs;unt eiu&longs;dem generis A, B, &longs;imiliter &amp; P, O, <lb/>* erit vt grauitas corporis A, hoc e&longs;t vt EK, (ponuntur enim<arrow.to.target n="note50"/> cor&shy;<lb/>pora A, BC, D, &aelig;que grauia,) ad E, ita G, ad F, quod igitur fit ex EK, <lb/>&amp; F, nempe ex extremis, &aelig;quale erit ei, quod fit ex E, &amp; G, hoc e&longs;t <lb/>ex medijs. </s></p><p type="margin"> <s>Sit enim portionis B, grauitas E, &amp; portionis C, gra&shy;<lb/>uitas K; ergo totius corporis BC grauitas erit EK, <expan abbr="&longs;itq;">&longs;itque</expan> portionis O, <lb/>qu&aelig; &longs;it &aelig;qualis portioni B, grauitas F, ergo reliqu&aelig; portionis L, <lb/>&aelig;qualis portioni C, grauitas erit V, Quoniam igitur e&longs;t, vt A, ad P, <lb/>ita B, ad O, &aelig;quale videlicet ad &aelig;quale, erit permutando, vt A, ad B, <lb/>ita p.ad O, &amp; quoniam &longs;unt eiu&longs;dem generis A, B, &longs;imiliter &amp; P, O, <lb/>* erit vt grauitas corporis A, hoc e&longs;t vt EK, (ponuntur enim<arrow.to.target n="note50"/> cor&shy;<lb/>pora A, BC, D, &aelig;que grauia,) ad E, ita G, ad F, quod igitur fit ex EK, <lb/>&amp; F, nempe ex extremis, &aelig;quale erit ei, quod fit ex E, &amp; G, hoc e&longs;t <lb/>ex medijs. </s></p><p type="margin">
  
 <s><margin.target id="note50"/>4.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="note50"/>4.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main">
  
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 <s><margin.target id="note51"/>4.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="note51"/>4.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main">
  
 <s>Sed o&longs;ten&longs;um e&longs;t id quod ex EK, fit &amp; F. &aelig;quale e&longs;&longs;e ei quod fit ex <lb/>G, &amp; E, ergo quo<gap/> fit ex EK, &amp; F, vna cum eo, quod ex BK, &amp; V, hoc <lb/>e&longs;t id quod fit ex EK, &amp; FV, &aelig;quale erit ei quod ex G, fit &amp; E, vna <lb/>eum eo quod ex H, &amp; K, &longs;ed quod ex G, fit &amp; E, &aelig;quale e&longs;t ei quod fit <lb/>ex G, &amp; EK, minus eo quod ex G, &amp; K, quod enim additur, idem &amp; <pb xlink:href="029/01/065.jpg" pagenum="58"/>minuitur; ergo quod fit ex EK, &amp; FV. &aelig;quale erit ei quod fit ex G, &amp; <lb/>EK, vna cum eo quod ex H, &amp; K, minus eo quod fit ex G, &amp; K. aufe&shy;<lb/>ratur vtrinque id quod fit ex G, &amp; EK, quod igitur fit ex FV, &amp; EK, <lb/><figure id="id.029.01.065.1.jpg" xlink:href="029/01/065/1.jpg"/><lb/>minus eo quod ex G, &amp; <lb/>EK, &aelig;quabitur ei quod <lb/>ex H, &amp; K, minus eo <lb/>quod fit ex G, &amp; K, &longs;ed <lb/>quod fit ex H, &amp; K, <lb/>minus eo quod fit ex <lb/>G, &amp; K, &aelig;quale e&longs;t ei <lb/>quod ex differentia ip&shy;<lb/>&longs;arum H, G, fit &amp; K, <lb/>&longs;imiliter, &amp; quod fit ex <lb/>FV, &amp; EK, minus eo <lb/>quod ex G, &amp; EK, <lb/>&aelig;quale e&longs;t ei quod ex <lb/>differentia ip&longs;arum <lb/>FV, G, fit &amp; EK, ergo <lb/>quod ex differentia <lb/>ip&longs;acum H, G, fit &amp; K, &aelig;quale erit ei quod ex differentia ip&longs;arum <lb/>FV, G, fit &amp; EK; &aelig;qualitatem ad proportionem reuocando, erit vt <lb/>differentia grauitatum. </s> <s>Sed o&longs;ten&longs;um e&longs;t id quod ex EK, fit &amp; F. &aelig;quale e&longs;&longs;e ei quod fit ex <lb/>G, &amp; E, ergo quod fit ex EK, &amp; F, vna cum eo, quod ex BK, &amp; V, hoc <lb/>e&longs;t id quod fit ex EK, &amp; FV, &aelig;quale erit ei quod ex G, fit &amp; E, vna <lb/>eum eo quod ex H, &amp; K, &longs;ed quod ex G, fit &amp; E, &aelig;quale e&longs;t ei quod fit <lb/>ex G, &amp; EK, minus eo quod ex G, &amp; K, quod enim additur, idem &amp; <pb xlink:href="029/01/065.jpg" pagenum="58"/>minuitur; ergo quod fit ex EK, &amp; FV. &aelig;quale erit ei quod fit ex G, &amp; <lb/>EK, vna cum eo quod ex H, &amp; K, minus eo quod fit ex G, &amp; K. aufe&shy;<lb/>ratur vtrinque id quod fit ex G, &amp; EK, quod igitur fit ex FV, &amp; EK, <lb/><figure id="id.029.01.065.1.jpg" xlink:href="029/01/065/1.jpg"/><lb/>minus eo quod ex G, &amp; <lb/>EK, &aelig;quabitur ei quod <lb/>ex H, &amp; K, minus eo <lb/>quod fit ex G, &amp; K, &longs;ed <lb/>quod fit ex H, &amp; K, <lb/>minus eo quod fit ex <lb/>G, &amp; K, &aelig;quale e&longs;t ei <lb/>quod ex differentia ip&shy;<lb/>&longs;arum H, G, fit &amp; K, <lb/>&longs;imiliter, &amp; quod fit ex <lb/>FV, &amp; EK, minus eo <lb/>quod ex G, &amp; EK, <lb/>&aelig;quale e&longs;t ei quod ex <lb/>differentia ip&longs;arum <lb/>FV, G, fit &amp; EK, ergo <lb/>quod ex differentia <lb/>ip&longs;acum H, G, fit &amp; K, &aelig;quale erit ei quod ex differentia ip&longs;arum <lb/>FV, G, fit &amp; EK; &aelig;qualitatem ad proportionem reuocando, erit vt <lb/>differentia grauitatum. </s>
  
 <s>H, G, ad grauitatem EK, ita differentia <lb/>grauitatum FV, G, ad graui<gap/> tom K, quod erat primo loco: demon&shy;<lb/>&longs;trandum. </s></p><p type="main"> <s>H, G, ad grauitatem EK, ita differentia <lb/>grauitatum FV, G, ad grauitatem K, quod erat primo loco: demon&shy;<lb/>&longs;trandum. </s></p><p type="main">
  
 <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. <!-- KEEP S--></s> <s>Dico quoque vt differentia grauitatum H, G, ad grauitatem EK, <lb/>ita e&longs;&longs;e differentiam grauitatum H, FV, ad grauitatem E. <!-- KEEP S--></s>
  
 <s>Quoniam <lb/>enim o&longs;ten&longs;um e&longs;t, quod fit ex EK, &amp; FV, &aelig;quale e&longs;&longs;e ei quod ex G. fit <lb/>&amp; E, vna cum eo quod ex H, &amp; K; quod autem fit ex H, &amp; K, &aelig;quat<gap/><lb/><gap/>i quod, ex H, fit &amp; EK, minus eo quod ex H, &amp; E, quod enim additur <lb/>idem &amp; minuitur: ergo quod fit ex EK, &amp; FV, &aelig;quale erit ei quod fit <lb/>ex H, &amp; EK, vna cum eo quod ex G, &amp; E, minus eo quod ex H, &amp; E. <lb/>addatur vtrin que quod ex H, fit &amp; E, &amp; &longs;ubducantur ea qu&aelig; fiunt ex <lb/>G, &amp; E, &amp; ex EK, &amp; FV; quod igitur fit ex H, &amp; E, minus eo quod ex <lb/>G, &amp; E, &aelig;quabitur ei quod ex H, fit &amp; EK, minus eo quod ex FV, &amp; <lb/>EK, &longs;ed quod fit ex H, &amp; E, minus eo quod ex G, &amp; E, &aelig;quale e&longs;t ei <lb/>quod ex, differentia ip&longs;arum H, G, fit &amp; E, &longs;imiliter, &amp; quod ex H, fit <lb/>&amp; EK, minus eo quod ex FV, &amp; EK, &aelig;quale e&longs;t ei quod ex differentia <lb/>ip&longs;arum H, FV, fit &amp; EK; ergo quod ex differentia ip&longs;arum H, G, fit <lb/>&amp; E, &aelig;quabitur ei quod ex differentia ip&longs;arum H, FV, fit &amp; EK; qua&shy;<lb/>re &aelig;qualitatem ad <expan abbr="proportion&etilde;">proportionem</expan> reuocando erit v<gap/> differentia graui<pb xlink:href="029/01/066.jpg" pagenum="59"/>tatum H, G, ad grauitatem EK, ita differentia grauitatum H, FV, <lb/>zd grauitatem E. quod &longs;ecundo loco fuit demon&longs;trandum. </s></p><p type="main"> <s>Quoniam <lb/>enim o&longs;ten&longs;um e&longs;t, quod fit ex EK, &amp; FV, &aelig;quale e&longs;&longs;e ei quod ex G. fit <lb/>&amp; E, vna cum eo quod ex H, &amp; K; quod autem fit ex H, &amp; K, &aelig;quatur&shy;<lb/>ri quod, ex H, fit &amp; EK, minus eo quod ex H, &amp; E, quod enim additur <lb/>idem &amp; minuitur: ergo quod fit ex EK, &amp; FV, &aelig;quale erit ei quod fit <lb/>ex H, &amp; EK, vna cum eo quod ex G, &amp; E, minus eo quod ex H, &amp; E. <lb/>addatur vtrin que quod ex H, fit &amp; E, &amp; &longs;ubducantur ea qu&aelig; fiunt ex <lb/>G, &amp; E, &amp; ex EK, &amp; FV; quod igitur fit ex H, &amp; E, minus eo quod ex <lb/>G, &amp; E, &aelig;quabitur ei quod ex H, fit &amp; EK, minus eo quod ex FV, &amp; <lb/>EK, &longs;ed quod fit ex H, &amp; E, minus eo quod ex G, &amp; E, &aelig;quale e&longs;t ei <lb/>quod ex, differentia ip&longs;arum H, G, fit &amp; E, &longs;imiliter, &amp; quod ex H, fit <lb/>&amp; EK, minus eo quod ex FV, &amp; EK, &aelig;quale e&longs;t ei quod ex differentia <lb/>ip&longs;arum H, FV, fit &amp; EK; ergo quod ex differentia ip&longs;arum H, G, fit <lb/>&amp; E, &aelig;quabitur ei quod ex differentia ip&longs;arum H, FV, fit &amp; EK; qua&shy;<lb/>re &aelig;qualitatem ad <expan abbr="proportion&etilde;">proportionem</expan> reuocando erit vt differentia graui<pb xlink:href="029/01/066.jpg" pagenum="59"/>tatum H, G, ad grauitatem EK, ita differentia grauitatum H, FV, <lb/>zd grauitatem E. quod &longs;ecundo loco fuit demon&longs;trandum. </s></p><p type="main">
  
 <s>Alia breuior Theorematis demon&longs;tratio. </s></p><p type="main"> <s>Alia breuior Theorematis demon&longs;tratio. </s></p><p type="main">
  
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 <s>hac enim reducta, non &longs;olum recte definiunt cuius <lb/>fuerit qualitatis particula illa purificata ante purificationem; ve&shy;<lb/>rum etiam cuius fuerit qualitatis, &amp; quot partium fuerit illud cor&shy;<lb/>pus, &agrave; quo eadem particula detracta fuit, &amp; illud, quod adhuc &longs;u&shy;<lb/>pere&longs;t, diminutum &longs;cilicet illa parte purificata, vt in eodem exemplo <lb/>propo&longs;ito, corporis aurei<emph.end type="italics"/> 24. <emph type="italics"/>vnciarum apparet. </s> <s>hac enim reducta, non &longs;olum recte definiunt cuius <lb/>fuerit qualitatis particula illa purificata ante purificationem; ve&shy;<lb/>rum etiam cuius fuerit qualitatis, &amp; quot partium fuerit illud cor&shy;<lb/>pus, &agrave; quo eadem particula detracta fuit, &amp; illud, quod adhuc &longs;u&shy;<lb/>pere&longs;t, diminutum &longs;cilicet illa parte purificata, vt in eodem exemplo <lb/>propo&longs;ito, corporis aurei<emph.end type="italics"/> 24. <emph type="italics"/>vnciarum apparet. </s>
  
 <s>Eius enim quali&shy;<lb/>tatem &longs;i forte aurifices inuestigare velint, detrahent ex eo particu&shy;<lb/>lam, verbi gratia, vnius vnci&aelig;, vel quod idem est particulam<emph.end type="italics"/> 24. <pb xlink:href="029/01/068.jpg" pagenum="61"/><emph type="italics"/>&longs;crupulorum; &amp; hanc particulam excoquent ad qualitatem o&longs;que <lb/>c<gap/>ripuri. </s> <s>Eius enim quali&shy;<lb/>tatem &longs;i forte aurifices inuestigare velint, detrahent ex eo particu&shy;<lb/>lam, verbi gratia, vnius vnci&aelig;, vel quod idem est particulam<emph.end type="italics"/> 24. <pb xlink:href="029/01/068.jpg" pagenum="61"/><emph type="italics"/>&longs;crupulorum; &amp; hanc particulam excoquent ad qualitatem o&longs;que&shy;<lb/>cori puri. </s>
  
 <s>Est &longs;i quidem inuenerint, ex priori grauitate<emph.end type="italics"/> 24. <emph type="italics"/>&longs;ru&shy;<lb/>pulorum, deperi&longs;&longs;e nihil: pronuntiabunt aurum illud, hoc est, non <lb/>&longs;olum particulam illam excoctam, &longs;ed etiam illud d quo fuit detra&shy;<lb/>cta, nec non &amp; illud quod reman&longs;it post &longs;ubtractionem 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&shy;<lb/>tum purum. </s> <s>Est &longs;i quidem inuenerint, ex priori grauitate<emph.end type="italics"/> 24. <emph type="italics"/>&longs;ru&shy;<lb/>pulorum, deperi&longs;&longs;e nihil: pronuntiabunt aurum illud, hoc est, non <lb/>&longs;olum particulam illam excoctam, &longs;ed etiam illud d quo fuit detra&shy;<lb/>cta, nec non &amp; illud quod reman&longs;it post &longs;ubtractionem e&longs;&longs;e vel fui&longs;&shy;<lb/>&longs;e aurum prima qualitatis &longs;eu<emph.end type="italics"/> 24, <emph type="italics"/>partiam; vel quod idem est au&shy;<lb/>tum purum. </s>
  
 <s>Si vero deprehenderine grauitatem diminutant, ver&shy;<lb/>bi gratia, nunc e&longs;&longs;e<emph.end type="italics"/> 20. <emph type="italics"/>&longs;crupulorum, qu&aelig; ante def&aelig;cationem 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 &amp; <lb/>illud quod reman&longs;it e&longs;&longs;e<emph.end type="italics"/> 20. <emph type="italics"/>partium &amp; denique particulam expur&shy;<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></p><p type="main"> <s>Si vero deprehenderine grauitatem diminutant, ver&shy;<lb/>bi gratia, nunc e&longs;&longs;e<emph.end type="italics"/> 20. <emph type="italics"/>&longs;crupulorum, qu&aelig; ante def&aelig;cationem 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 &amp; <lb/>illud quod reman&longs;it e&longs;&longs;e<emph.end type="italics"/> 20. <emph type="italics"/>partium &amp; denique particulam expur&shy;<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></p><p type="main">
  
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 <s>Nam cum in hac comparatione qualitatum, &longs;eor&longs;im habe a&shy;<lb/>tur ratio partium auri, &amp; &longs;eor&longs;im metallorum alligatorum; manife&shy;<lb/>stum est &longs;i grauitas totius corporis intelligatur diui&longs;a in<emph.end type="italics"/> 24. <emph type="italics"/>partes <lb/>&aelig;quales, ex quibus<emph.end type="italics"/> 20. <emph type="italics"/>&longs;int auri, du&aelig; argenti, &amp; du&aelig; &aelig;ris; quamli&shy;<lb/>bet partem auri cum qualibet parte argenti &amp; &aelig;ris collatam, magni&shy;<lb/>tudine e&longs;&longs;e minorem; &amp; &longs;imiliter partem argenti minorem parte <lb/>aris; propterea quod aurum omnia reliqua metalla &longs;uperet grauita<lb/>te quemadmodum &amp; argentum ip&longs;um as, vt constat experienties<lb/>atque hinc constat quam apte ac conuenienter Aurifices vtantur <lb/>vocabulo partium. </s> <s>Nam cum in hac comparatione qualitatum, &longs;eor&longs;im habe a&shy;<lb/>tur ratio partium auri, &amp; &longs;eor&longs;im metallorum alligatorum; manife&shy;<lb/>stum est &longs;i grauitas totius corporis intelligatur diui&longs;a in<emph.end type="italics"/> 24. <emph type="italics"/>partes <lb/>&aelig;quales, ex quibus<emph.end type="italics"/> 20. <emph type="italics"/>&longs;int auri, du&aelig; argenti, &amp; du&aelig; &aelig;ris; quamli&shy;<lb/>bet partem auri cum qualibet parte argenti &amp; &aelig;ris collatam, magni&shy;<lb/>tudine e&longs;&longs;e minorem; &amp; &longs;imiliter partem argenti minorem parte <lb/>aris; propterea quod aurum omnia reliqua metalla &longs;uperet grauita<lb/>te quemadmodum &amp; argentum ip&longs;um as, vt constat experienties<lb/>atque hinc constat quam apte ac conuenienter Aurifices vtantur <lb/>vocabulo partium. </s>
  
 <s>hac enim ratione <gap/>dem numero exprimunt vnam <lb/>quamque qualitatem auri cuiuslibet ma&longs;&longs;&aelig; propo&longs;it&aelig;. </s> <s>hac enim ratione eodem numero exprimunt vnam <lb/>quamque qualitatem auri cuiuslibet ma&longs;&longs;&aelig; propo&longs;it&aelig;. </s>
  
 <s>Sed nunc ad <lb/>&longs;ecundum veniamus &amp; modum alligationis. </s> <s>Sed nunc ad <lb/>&longs;ecundum veniamus &amp; modum alligationis. </s>
  
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 <s>Et quidem <lb/>&longs;ipartes auri excipias, aris atque argenti partes, qu&aelig; auro &longs;unt per&shy;<lb/>mi&longs;cend&aelig; &longs;emper volunte&longs;&longs;e &aelig;quales in grauitate: propterea quod <lb/>eadem experientia Magi&longs;tra didicerunt hunc e&longs;se mixtionis modum <lb/>longe optimum.<emph.end type="italics"/></s></p><p type="main"> <s>Et quidem <lb/>&longs;ipartes auri excipias, aris atque argenti partes, qu&aelig; auro &longs;unt per&shy;<lb/>mi&longs;cend&aelig; &longs;emper volunte&longs;&longs;e &aelig;quales in grauitate: propterea quod <lb/>eadem experientia Magi&longs;tra didicerunt hunc e&longs;se mixtionis modum <lb/>longe optimum.<emph.end type="italics"/></s></p><p type="main">
  
 <s><emph type="italics"/>Quando ergo aurifices volunt producere aurum cuiu&longs;cunque <lb/>qualitatis, accipiunt tot partes auri puri &aelig;quales, quot partium fu&shy;<lb/>turum e&longs;t aurum producendum, pauciores tamen partibus<emph.end type="italics"/> 24, &amp; <pb xlink:href="029/01/069.jpg" pagenum="62"/><emph type="italics"/>reliquas partes qu&aelig; d &longs;unt ad<emph.end type="italics"/> 24, <emph type="italics"/>explent argento &amp; <gap/>e, &longs;umendo <lb/>ex utroque metallo partes aquales in grauitate; atque bis rite inter &longs;e <lb/>permixtis componunt aurum de&longs;iderat&aelig; qualitatis: eamque deno&shy;<lb/>minant &agrave; partibus auri puri in mixtione a&longs;umptis. </s> <s><emph type="italics"/>Quando ergo aurifices volunt producere aurum cuiu&longs;cunque <lb/>qualitatis, accipiunt tot partes auri puri &aelig;quales, quot partium fu&shy;<lb/>turum e&longs;t aurum producendum, pauciores tamen partibus<emph.end type="italics"/> 24, &amp; <pb xlink:href="029/01/069.jpg" pagenum="62"/><emph type="italics"/>reliquas partes qu&aelig; d &longs;unt ad<emph.end type="italics"/> 24, <emph type="italics"/>explent argento &amp; &aelig;re, &longs;umendo <lb/>ex utroque metallo partes aquales in grauitate; atque bis rite inter &longs;e <lb/>permixtis componunt aurum de&longs;iderat&aelig; qualitatis: eamque deno&shy;<lb/>minant &agrave; partibus auri puri in mixtione a&longs;umptis. </s>
  
 <s>Et quoniam <lb/>non prodiret <gap/>ale pror&longs;us quale facere intendunt, &longs;ed paulo perfe&shy;<lb/>ctius; propterea quod <gap/> quidem partes in mixtione maneant, ex <lb/>argenta vero &amp; are aliquid deperdatur, &longs;olent Aurifices tanto plus <lb/>mi&longs;cere argenti &amp; aris quantum perdi po&longs;&longs;e deprehenderunt.<emph.end type="italics"/></s></p><p type="main"> <s>Et quoniam <lb/>non prodiret tale pror&longs;us quale facere intendunt, &longs;ed paulo perfe&shy;<lb/>ctius; propterea quod auri quidem partes in mixtione maneant, ex <lb/>argenta vero &amp; are aliquid deperdatur, &longs;olent Aurifices tanto plus <lb/>mi&longs;cere argenti &amp; aris quantum perdi po&longs;&longs;e deprehenderunt.<emph.end type="italics"/></s></p><p type="main">
  
 <s><emph type="italics"/>Verum nostra intentio non est omnia qu&aelig; ad eiu&longs;modi mixtione <lb/>pertinent hoc loco exponere; &longs;ed illud tantum vt receptum apud om<lb/>nes ad ferre voluimus, ex quo manifeste constat, qua metallorum <lb/>mixtio in &longs;ingulis qualitatum generibus statuatur: qua quidem e&longs;t <lb/>illa quam adduximus, nempe 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/>auri puri &amp; reliquam qua deest ad<emph.end type="italics"/> 24, <emph type="italics"/>partes con&longs;tare dimidia <lb/>parte argenti, &amp; dimidia <gap/>ris in grauitate. </s> <s><emph type="italics"/>Verum nostra intentio non est omnia qu&aelig; ad eiu&longs;modi mixtione <lb/>pertinent hoc loco exponere; &longs;ed illud tantum vt receptum apud om&shy;<lb/>nes ad ferre voluimus, ex quo manifeste constat, qua metallorum <lb/>mixtio in &longs;ingulis qualitatum generibus statuatur: qua quidem e&longs;t <lb/>illa quam adduximus, nempe 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/>auri puri &amp; reliquam qua deest ad<emph.end type="italics"/> 24, <emph type="italics"/>partes con&longs;tare dimidia <lb/>parte argenti, &amp; dimidia aris in grauitate. </s>
  
 <s>In auro vero<emph.end type="italics"/> 22, <emph type="italics"/>par&shy;<lb/>tium, auri e&longs;&longs;e<emph.end type="italics"/> 23, <emph type="italics"/>argenti vnam, &amp; &aelig;ris vnam, &longs;ie enim iterum <lb/>&longs;umma omnium partium est<emph.end type="italics"/> 24, <emph type="italics"/>eademque est ratio de reliquis ita <lb/>ut numerus partium auri, &longs;emper denominet qualitatem auri, &amp; <lb/>vna medietas reliquarum partium, qu&aelig; partibus a<gap/>ri de&longs;unt ad <lb/>complendas partes<emph.end type="italics"/> 24, <emph type="italics"/>&longs;it argenti, &amp; reliqua medietas &longs;it &aelig;ris. </s> <s>In auro vero<emph.end type="italics"/> 22, <emph type="italics"/>par&shy;<lb/>tium, auri e&longs;&longs;e<emph.end type="italics"/> 23, <emph type="italics"/>argenti vnam, &amp; &aelig;ris vnam, &longs;ie enim iterum <lb/>&longs;umma omnium partium est<emph.end type="italics"/> 24, <emph type="italics"/>eademque est ratio de reliquis ita <lb/>ut numerus partium auri, &longs;emper denominet qualitatem auri, &amp; <lb/>vna medietas reliquarum partium, qu&aelig; partibus auri de&longs;unt ad <lb/>complendas partes<emph.end type="italics"/> 24, <emph type="italics"/>&longs;it argenti, &amp; reliqua medietas &longs;it &aelig;ris. </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/>inuestigaturi &longs;umus auri qualitatem ex &longs;ola grauitate quam habet<lb/>in aere &amp; aqua, eamque qualitatem duplici via inuestigabimus, <lb/>vna per calculum, per tabellam altera: &amp; quia ad calculum &longs;pe&shy;<lb/>ctant ea, qu&aelig; &longs;uperi<gap/> inuenimus, de grauitate metallorum huc re&shy;<lb/>ferenda cen&longs;uimus qu&aelig; hic &longs;unt nece&longs;&longs;aria, cuiu&longs;modi &longs;unt auri, <lb/>argenti, atque &aelig;ris grauitas, quam obtinent in aere, &amp; aqua, quo<lb/>quidem ita &longs;e habet vt &longs;equitur.<emph.end type="italics"/></s></p><p type="main"> <s>hac <lb/>enim &longs;atis est &longs;uppo&longs;ui&longs;&longs;e, ad nouum illud artificium, quo paulo po&longs;t <lb/>inuestigaturi &longs;umus auri qualitatem ex &longs;ola grauitate quam habet<lb/>in aere &amp; aqua, eamque qualitatem duplici via inuestigabimus, <lb/>vna per calculum, per tabellam altera: &amp; quia ad calculum &longs;pe&shy;<lb/>ctant ea, qu&aelig; &longs;uperius inuenimus, de grauitate metallorum huc re&shy;<lb/>ferenda cen&longs;uimus qu&aelig; hic &longs;unt nece&longs;&longs;aria, cuiu&longs;modi &longs;unt auri, <lb/>argenti, atque &aelig;ris grauitas, quam obtinent in aere, &amp; aqua, quo&shy;<lb/>quidem ita &longs;e habet vt &longs;equitur.<emph.end type="italics"/></s></p><p type="main">
  
 <s><emph type="italics"/>Auri puri grauitas, qu&aelig; in aere e&longs;t<emph.end type="italics"/> 19, <emph type="italics"/>erit in aqua<emph.end type="italics"/> 18. </s></p><p type="main"> <s><emph type="italics"/>Auri puri grauitas, qu&aelig; in aere e&longs;t<emph.end type="italics"/> 19, <emph type="italics"/>erit in aqua<emph.end type="italics"/> 18. </s></p><p type="main">
  
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 <s>Sit exemp.: gratia propo&longs;ita aliqua ma&longs;&longs;a aurea, cuius gra&shy;<lb/>uitas in aere &longs;it vnc<emph.end type="italics"/> 24, <emph type="italics"/>&amp; oporteat invenire cuius qualitatis &longs;it ip&shy;<lb/>sum aurum. </s> <s>Sit exemp.: gratia propo&longs;ita aliqua ma&longs;&longs;a aurea, cuius gra&shy;<lb/>uitas in aere &longs;it vnc<emph.end type="italics"/> 24, <emph type="italics"/>&amp; oporteat invenire cuius qualitatis &longs;it ip&shy;<lb/>sum aurum. </s>
  
 <s>Ponderetur ea ma&longs;&longs;a in aqua &amp; habeat grauitatem <lb/>vnciarum<emph.end type="italics"/> 22 2<gap/>/5301, <emph type="italics"/>ergo * grauitas aqu&aelig; magnitudinem haben<emph.end type="italics"/><arrow.to.target n="note55"/><lb/><emph type="italics"/>tis <gap/>qualem propo&longs;it&aelig; ma&longs;&longs;&aelig; erit vnc.<emph.end type="italics"/> 1 2431/5301. </s></p><p type="margin"> <s>Ponderetur ea ma&longs;&longs;a in aqua &amp; habeat grauitatem <lb/>vnciarum<emph.end type="italics"/> 22 2<gap/>/5301, <emph type="italics"/>ergo * grauitas aqu&aelig; magnitudinem haben&shy;<emph.end type="italics"/><arrow.to.target n="note55"/><lb/><emph type="italics"/>tis &aelig;qualem propo&longs;it&aelig; ma&longs;&longs;&aelig; erit vnc.<emph.end type="italics"/> 1 2431/5301. </s></p><p type="margin">
  
 <s><margin.target id="note55"/>5.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="note55"/>5.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main">
  
 <s><emph type="italics"/>Deinde inueniatur grauitas aqu&aelig; magnitudine &aelig;qualis auro <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 aqu&aelig;.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="italics"/>Deinde inueniatur grauitas aqu&aelig; magnitudine &aelig;qualis auro <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 aqu&aelig;.<emph.end type="italics"/></s></p><p type="main">
  
 <s><emph type="italics"/>Fiat denique ut<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"/>vnti&aelig;, ad alium, numi&shy;<lb/>rus enim quartus, nempe vnc.<emph.end type="italics"/> 2<gap/> <emph type="italics"/>erit grauitas aqu&aelig;, magnitu&shy;<lb/>dine &aelig;qualis corpori mi&longs;to ex argento &amp; &aelig;re, euius grauitas est in <lb/>&aelig;r&egrave; vnc.<emph.end type="italics"/> 24, <emph type="italics"/>corpus enim ita mi&longs;tum, ad corpus aqueum eiu&longs;dem ma&shy;<lb/>gnitudinis rationem habet in grauitate vt<emph.end type="italics"/> 279, <emph type="italics"/>ad<emph.end type="italics"/> 29. </s></p><p type="main"> <s><emph type="italics"/>Fiat denique ut<emph.end type="italics"/> 279, <emph type="italics"/>ad<emph.end type="italics"/> 29, <emph type="italics"/>ita rur&longs;us<emph.end type="italics"/> 24, <emph type="italics"/>vnti&aelig;, ad alium, nume&shy;<lb/>rus enim quartus, nempe vnc.<emph.end type="italics"/> 2<gap/> <emph type="italics"/>erit grauitas aqu&aelig;, magnitu&shy;<lb/>dine &aelig;qualis corpori mi&longs;to ex argento &amp; &aelig;re, cuius grauitas est in <lb/>&aelig;r&egrave; vnc.<emph.end type="italics"/> 24, <emph type="italics"/>corpus enim ita mi&longs;tum, ad corpus aqueum eiu&longs;dem ma&shy;<lb/>gnitudinis rationem habet in grauitate vt<emph.end type="italics"/> 279, <emph type="italics"/>ad<emph.end type="italics"/> 29. </s></p><p type="main">
  
 <s><emph type="italics"/>Atque ita habebuntur tres gravitates trium aqu&aelig; quantitatum, <lb/>quarum prima &aelig;quatur auro puro<emph.end type="italics"/> 24, <emph type="italics"/>unciarum, &longs;ecunda ma&longs;&longs;a <lb/>propo&longs;it&aelig;<emph.end type="italics"/> 24, <emph type="italics"/>vnciarum, &amp; reliqua corpori mi&longs;to ex argento &amp; &aelig;re<lb/>&longs;imiliter<emph.end type="italics"/> 24, <emph type="italics"/><gap/>vnciarum qu&aelig; quidem tres grauitates in numeris di&shy;<lb/>&longs;ponantur eo ordine, quo &longs;equitur.<emph.end type="italics"/></s></p><p type="table"> <s><emph type="italics"/>Atque ita habebuntur tres gravitates trium aqu&aelig; quantitatum, <lb/>quarum prima &aelig;quatur auro puro<emph.end type="italics"/> 24, <emph type="italics"/>unciarum, &longs;ecunda ma&longs;&longs;a <lb/>propo&longs;it&aelig;<emph.end type="italics"/> 24, <emph type="italics"/>vnciarum, &amp; reliqua corpori mi&longs;to ex argento &amp; &aelig;re<lb/>&longs;imiliter<emph.end type="italics"/> 24, <emph type="italics"/>vnciarum qu&aelig; quidem tres grauitates in numeris di&shy;<lb/>&longs;ponantur eo ordine, quo &longs;equitur.<emph.end type="italics"/></s></p><p type="table">
  
 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><p type="main"> <s>TABELLE WAR HIER<!-- KEEP S--></s></p><p type="main">
  
 <s><emph type="italics"/>Deinde qu&aelig;ratur differentia inter primam &amp; tertiam aqu&aelig; gra&shy;<lb/>uitatem, qu&aelig; est vnc.<emph.end type="italics"/> 1 1227/5302, <emph type="italics"/>&amp; h&aelig;c differentia statuatur pro <lb/>primo proportionis termino, pro &longs;ecundo termino ponatur grauitas <lb/><gap/>a&longs;&longs;&aelig; propo&longs;it&aelig; idest vnc.<emph.end type="italics"/> 24, <emph type="italics"/>&amp; protertio denique termino ponatur <lb/>different<gap/> inter &longs;ecundam aqu&aelig; grauitatem &amp; tertiam, qu&aelig; est <lb/>vnc.<emph.end type="italics"/> 1 <gap/> <emph type="italics"/>quartus enim proportionalium terminus nempe<emph.end type="italics"/> 20, </s></p><pb xlink:href="029/01/071.jpg" pagenum="64"/><p type="table"> <s><emph type="italics"/>Deinde qu&aelig;ratur differentia inter primam &amp; tertiam aqu&aelig; gra&shy;<lb/>uitatem, qu&aelig; est vnc.<emph.end type="italics"/> 1 1227/5302, <emph type="italics"/>&amp; h&aelig;c differentia statuatur pro <lb/>primo proportionis termino, pro &longs;ecundo termino ponatur grauitas <lb/><gap/>a&longs;&longs;&aelig; propo&longs;it&aelig; idest vnc.<emph.end type="italics"/> 24, <emph type="italics"/>&amp; protertio denique termino ponatur <lb/>differentia inter &longs;ecundam aqu&aelig; grauitatem &amp; tertiam, qu&aelig; est <lb/>vnc.<emph.end type="italics"/> 1 <gap/> <emph type="italics"/>quartus enim proportionalium terminus nempe<emph.end type="italics"/> 20, </s></p><pb xlink:href="029/01/071.jpg" pagenum="64"/><p type="table">
  
 <s>TABELLE WAR HIER<lb/><emph type="italics"/>erit denominator qualitatis auri de qua <gap/>uaritur quia ille termi&shy;<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 demonstratum e&longs;t prop.<emph.end type="italics"/> 19, <emph type="italics"/>huius.<emph.end type="italics"/></s></p><p type="main"> <s>TABELLE WAR HIER<lb/><emph type="italics"/>erit denominator qualitatis auri de qua quaritur quia ille termi&shy;<lb/>nus indicat partes auri pari in grauitate, qualibus ma&longs;&longs;a propo&longs;ita <lb/>constat<emph.end type="italics"/> 24. <emph type="italics"/><!--neuer Satz-->Hoc autem demonstratum e&longs;t prop.<emph.end type="italics"/> 19, <emph type="italics"/>huius.<emph.end type="italics"/></s></p><p type="main">
  
 <s><emph type="italics"/>Et quia inpropo&longs;ito exemplo <gap/> partes, nempe vnc.<emph.end type="italics"/> 20, <emph type="italics"/>&longs;unt par&shy;<lb/>tes vige&longs;<gap/> quart&aelig;<emph.end type="italics"/> 24, <emph type="italics"/>unciarum, qu&aelig; constituunt grauitatem to&shy;<lb/>tius ma&longs;&longs;&aelig;. </s> <s><emph type="italics"/>Et quia inpropo&longs;ito exemplo ha partes, nempe vnc.<emph.end type="italics"/> 20, <emph type="italics"/>&longs;unt par&shy;<lb/>tes vige&longs;ima quart&aelig;<emph.end type="italics"/> 24, <emph type="italics"/>unciarum, qu&aelig; constituunt grauitatem to&shy;<lb/>tius ma&longs;&longs;&aelig;. </s>
  
 <s>hinc fit quod e&aelig;dem<emph.end type="italics"/> 20. <emph type="italics"/>vnc. </s> <s>hinc fit quod e&aelig;dem<emph.end type="italics"/> 20. <emph type="italics"/>vnc. </s>
  
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 <s>Inueniantur <expan abbr="quoq;">quoque</expan> du&aelig; ali&aelig; <lb/>grauitates aqu&aelig;, vna re&longs;pondentis auro puro magnitudine, altera <lb/>corpori mi&longs;to ex argento &amp; &aelig;re, ita tamen vt grauitas tum auri pu&shy;<lb/>ri, tum corporis misti &longs;it eadem qu&aelig; ma&longs;&longs;&aelig; propo&longs;it&aelig;, non &longs;ecus ac in <lb/>pr&aelig;cedenti exemplo factitatum est. </s> <s>Inueniantur <expan abbr="quoq;">quoque</expan> du&aelig; ali&aelig; <lb/>grauitates aqu&aelig;, vna re&longs;pondentis auro puro magnitudine, altera <lb/>corpori mi&longs;to ex argento &amp; &aelig;re, ita tamen vt grauitas tum auri pu&shy;<lb/>ri, tum corporis misti &longs;it eadem qu&aelig; ma&longs;&longs;&aelig; propo&longs;it&aelig;, non &longs;ecus ac in <lb/>pr&aelig;cedenti exemplo 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 aqu&aelig; magnitudinem <lb/>habentis &aelig;qualem auro puro, <gap/>uius grauitas e&longs;t<emph.end type="italics"/> 5301. <emph type="italics"/>Deinde fiat <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 aqu&aelig;, magnitudine <lb/>aqualis corpori mi&longs;to ex argento &amp; &aelig;re grauitatem habenti eandem <lb/>cum eadem ma&longs;sa propo&longs;ita. </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 aqu&aelig; magnitudinem <lb/>habentis &aelig;qualem auro puro, cuius grauitas e&longs;t<emph.end type="italics"/> 5301. <emph type="italics"/><!--neuer Satz-->Deinde fiat <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 numerus<emph.end type="italics"/> 551, <emph type="italics"/>debitus grauitati aqu&aelig;, magnitudine <lb/>aqualis corpori mi&longs;to ex argento &amp; &aelig;re grauitatem habenti eandem <lb/>cum eadem ma&longs;sa propo&longs;ita. </s>
  
 <s>Atque h&aelig; tres grauitates aqu&aelig; &longs;criban&shy;<lb/>tur eo ordine quo &longs;upra; inuenti&longs;que differentijs inter primam &amp;<emph.end type="italics"/></s></p><p type="margin"> <s>Atque h&aelig; tres grauitates aqu&aelig; &longs;criban&shy;<lb/>tur eo ordine quo &longs;upra; inuenti&longs;que differentijs inter primam &amp;<emph.end type="italics"/></s></p><p type="margin">
  
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 <s>TABELLE WAR HIER<lb/><emph type="italics"/>tertiam, nec non inter &longs;ecundam &amp; tertiam, qu&aelig; &longs;unt<emph.end type="italics"/> 272, 238; <emph type="italics"/>sta&shy;<lb/>tuatur pro primo proportionis termino prior differentia<emph.end type="italics"/> 272, <emph type="italics"/>&amp; pro <lb/>tertio posterior<emph.end type="italics"/> 238. <emph type="italics"/>grauitas vero ma&longs;&longs;&aelig; propo&longs;it&aelig;<emph.end type="italics"/> 5301, <emph type="italics"/>ponatur <lb/>pro&longs;ecundo termino, &amp; qu&aelig;ratur terminus quartus, qui in pr&aelig;-<emph.end type="italics"/></s></p><p type="table"> <s>TABELLE WAR HIER<lb/><emph type="italics"/>tertiam, nec non inter &longs;ecundam &amp; tertiam, qu&aelig; &longs;unt<emph.end type="italics"/> 272, 238; <emph type="italics"/>sta&shy;<lb/>tuatur pro primo proportionis termino prior differentia<emph.end type="italics"/> 272, <emph type="italics"/>&amp; pro <lb/>tertio posterior<emph.end type="italics"/> 238. <emph type="italics"/>grauitas vero ma&longs;&longs;&aelig; propo&longs;it&aelig;<emph.end type="italics"/> 5301, <emph type="italics"/>ponatur <lb/>pro&longs;ecundo termino, &amp; qu&aelig;ratur terminus quartus, qui in pr&aelig;-<emph.end type="italics"/></s></p><p type="table">
  
 <s>TABELLE WAR HIER<lb/><arrow.to.target n="note57"/> <emph type="italics"/>&longs;enti examploo e&longs;t<emph.end type="italics"/> 4638 51/<gap/>, <emph type="italics"/>is enim indicabit grauitatem auri <lb/>puri in ma&longs;&longs;a propo&longs;ita. </s> <s>TABELLE WAR HIER<lb/><arrow.to.target n="note57"/> <emph type="italics"/>&longs;enti examplo e&longs;t<emph.end type="italics"/> 4638 51/<gap/>, <emph type="italics"/>is enim indicabit grauitatem auri <lb/>puri in ma&longs;&longs;a propo&longs;ita. </s>
  
 <s>Sed quoniam <gap/>c grauitas non est expre&longs;&shy;<lb/>&longs;a in partibus vige&longs;imis quartis totius grauitatis, id quod ad g<gap/>,<emph.end type="italics"/></s></p><pb xlink:href="029/01/072.jpg" pagenum="65"/><p type="margin"> <s>Sed quoniam h&aelig;c grauitas non est expre&longs;&shy;<lb/>&longs;a in partibus vige&longs;imis quartis totius grauitatis, id quod ad ger-<emph.end type="italics"/></s></p><pb xlink:href="029/01/072.jpg" pagenum="65"/><p type="margin">
  
 <s><margin.target id="note57"/>19. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s></p><p type="main"> <s><margin.target id="note57"/>19. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s></p><p type="main">
  

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