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annotate texts/archimedesOldCVSRepository/archimedes/xml/gheta_promo_029_la_1603.xml @ 6:22d6a63640c6
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1 <?xml version="1.0"?> |
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2 <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd"> |
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3 <archimedes xmlns:xlink="http://www.w3.org/1999/xlink"> <info> <author>Ghetaldus, Marinus</author> <title>Promotus Archimedis</title> <date>1603</date> |
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5 <place>Rome</place> <translator/> <lang>la</lang> <cvs_file>gheta_promo_029_la_1603.xml</cvs_file><cvs_version>1.11</cvs_version> <locator>029.xml</locator> </info> <text> <front> </front> <body> <chap> <pb xlink:href="029/01/001.jpg" id="p.0001"/><p type="head"> |
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7 <s>MARINI GHETALDI <lb/>PATRICII <lb/>RAGVSINI <lb/>PROMOTVS ARCHIMEDIS <lb/>SEV.<!-- REMOVE S-->De varijs corporum generibus <lb/>grauitate & magnitudine <lb/>comparatis.</s></p><figure id="id.029.01.001.1.jpg" xlink:href="029/01/001/1.jpg"/><p type="head"> |
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9 <s>ROMAE, <lb/>Apud Aloy&longs;ium Zannettum. <!-- KEEP S--></s> |
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11 <s>MDCIII. <lb/><emph type="italics"/>SVPERIORVM PERMISSV.<emph.end type="italics"/></s></p><pb xlink:href="029/01/002.jpg"/><p type="head"> |
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13 <s>REVERENDISSIMO <lb/>SERAPHINO OLIVARIO <lb/>RAZZALIO. <lb/>PATRIARCH AE <lb/>ALEXANDRINO. <lb/>MARINUS GHETALDUS. S.P.D.<!-- KEEP S--></s></p><p type="main"> |
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15 <s>Egregia &longs;anè Reuerendi&longs;sime <lb/>PRÆSVL quod probe no&longs;ti, vete­<lb/>rum &longs;apientum fœlicitas fuit. </s> |
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17 <s>Eam <lb/>enim cum ingenij præ&longs;tantia, tum <lb/>prærogativa temporis laudem occu­<lb/>parunt, <expan abbr="quã">quam</expan> vel &longs;perare po&longs;terioribus <lb/><expan abbr="temerariũ">temerarium</expan> &longs;it. </s> |
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19 <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="nouarũ">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> |
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21 <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 xlink:href="029/01/003.jpg"/>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> |
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23 <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> |
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25 <s>Cum enim tu me ad <lb/>emittendum id opus hortatu tuo compuleris, videbatur <lb/>quodam iure ad tuam fidem eius tutela pertinere. </s> |
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27 <s><expan abbr="Tuq;">Tuque</expan> <lb/>is es, quem non modo rudes, &longs;ed etiam docti &longs;u&longs;piciunt. </s> |
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29 <s><lb/>Habet no&longs;tra hæc ætas, quos admitetur; habet quos ex­<lb/>tollat præclaros viros, &longs;ed quos tibi anteponat, non facile <lb/>inueniet. </s> |
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31 <s>Degis ea in Vrbe, quæ laudis mediocritatem <lb/>vel nunquam agnouit, vel &longs;emper contemp&longs;it: neque in <lb/>tanta maie&longs;tate, tua deficit virtus, &longs;ed bono in lumine po<lb/>&longs;ita collucet magis. </s> |
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33 <s>In primis enim tua vitæ integritas <lb/>eiu&longs;modi e&longs;t, vt non contenta dome&longs;tico præcomo lati&longs;­<lb/>&longs;ime peruagetur. </s> |
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35 <s>Habent omnes quod prædicent, & imi<lb/>tentur; habet quod excipiat grati&longs;sima memoria po&longs;te­<lb/>ritas vniuer&longs;a. </s> |
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37 <s>Doctrinæ vero ea excellentia es, vt ea &longs;atis <lb/>omnibus clarus, & illu&longs;tris, non &longs;atis tibi, recum a&longs;sidue <lb/>certes. </s> |
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39 <s>nec mirum, qui vel à primis incunabulis ad &longs;um­<lb/>ma contendebas, &longs;i prouecta iam ætate vix habeas, quo <lb/>altius contendas. </s> |
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41 <s>Præclarum &longs;anè & eximium vno do­<lb/>ctrinæ genere; &longs;ublime, atque admirabile multiplici ex­<lb/>cellere. </s> |
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43 <s>Tu vero în omni genere laudem egregiam a&longs;&longs;e­<lb/>cutus es ab&longs;olutam videlicet tibî gloriam propo&longs;itam <lb/>volui&longs;ti; <expan abbr="quiq;">quique</expan> intelligebas hominem ad hone&longs;ta omnia <pb xlink:href="029/01/004.jpg"/>genitum, nullam tibi rerum glorio&longs;arum partem con­<lb/>temnendam puta&longs;ti. </s> |
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45 <s>Placet incredibili rerum humana­<lb/>rum v&longs;ui diuinarum rerum cognitionem adiungere; vt <lb/>habeat animus à caducis ad æterna, &longs;e conferendo, vnde <lb/>oblectamentum capiat, & admirationem. </s> |
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47 <s>Philo&longs;ophiam <lb/>ita tenes, vt qui in maximis negotijs a&longs;sidue ver&longs;atus es, <lb/>videaris &longs;emper fui&longs;&longs;e otio&longs;us. </s> |
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49 <s>Quid dicam de &longs;ingulari <lb/><expan abbr="caq;">caque</expan> eximia rerum cœle&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> |
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51 <s>Exitum tuorum laborum <lb/>felici&longs;simum vides: gloria multiplici frueris, neque illa <lb/>precaria &longs;ed tua & quibu&longs;dam qua&longs;i gradibus ad ampli&longs;­<lb/>&longs;imos: honores euehendus, in ea con&longs;titueris dignitate; in <lb/>qua pro &longs;acro&longs;ancta Eccle&longs;ia nunquam 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> |
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53 <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; vt ex tanto &longs;onte perennes ad omnium or­<lb/>dinum homines fluuii deducantur. </s> |
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55 <s>Felix qui &longs;olidæ <lb/>felicitatis cau&longs;am & initium in re con&longs;titutum ita 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 fœnus honorem ex honore, laudem <lb/>ex laude con&longs;equi vberiorem, hæc illa &longs;apienti viro non <lb/>indigna liberalitas, quæ rerum pre&longs;tanti&longs;simarum po&longs;­<lb/>&longs;e&longs;sione non imminuta, in copia tenuitatem non inqui­<pb xlink:href="029/01/005.jpg"/>rens vbertatis ip&longs;a &longs;uæ dominæ nunquam debilitatur, <lb/>nunquam deficit. </s> |
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57 <s>Quin etiam i&longs;to loco con&longs;titutus bo­<lb/>narum literarum &longs;tudio&longs;os complecteris, ac tueris. </s> |
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59 <s>Hæc <lb/>e&longs;t vera germanæ nota &longs;apientiæ, cum, quas vires nanci­<lb/>&longs;citur, ijs &longs;apientiam alit, tuus animus & tuæ &longs;apientiæ, & <lb/>alienæ par e&longs;t. </s> |
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61 <s>Hæc &longs;unt firmi&longs;sima & &longs;olidi&longs;sima funda­<lb/>menta ad æternam po&longs;teritatis memoriam, quam licet <lb/>profici&longs;ci iam videas ex iis quibus abundas animi orna­<lb/>mentis, ne&longs;cio tamen quo pacto gratior nobis accidit, <lb/>cum ex aliorum etiam prædonio &longs;u&longs;cipit in crementum. </s> |
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63 <s><lb/>Hinc domus tua floret docti&longs;simorum familiaritatibus; <lb/>hinc nulli ad tuam con&longs;uetudinem præcluditur aditus; <lb/>hinc plurimorum &longs;tudia commouentur; hinc illa &longs;apien­<lb/>tum æmulatio & admiratio: 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 patronum, & interpretem. </s> |
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65 <s>Quid ego <lb/>igitur quem tibi &longs;excentis eximiæ beneuolentiæ argu­<lb/>mentis obligatum volu&longs;ti, perexigiuum hunc ingenij <lb/>mei partum expo&longs;iturus, vnum te nominis & exi&longs;tima­<lb/>tionis meæ patronum non &longs;u&longs;cipiam equidem in am­<lb/>pli&longs;simi Theatri lucem &longs;apienti&longs;simorum <32> hominum <lb/>con&longs;pectum, quibus abundat hæc no&longs;tra ætas inferre me <lb/>non magnopere cogitabam: Tu hortatus es dubitan­<lb/>tem impuli&longs;ti vel reluctantem: tuere igitur ob&longs;equen­<lb/>tem. </s> |
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67 <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> |
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69 <s>Vale. <!-- KEEP S--></s></p><p type="main"> |
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71 <s>Romæ VII. Kal. <!-- REMOVE S-->Maij. <!-- REMOVE S-->MDCIII. <!-- KEEP S--></s> |
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75 </p><pb xlink:href="029/01/006.jpg"/><p type="head"> |
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77 <s>BENEVOLO <lb/>LECTORI.</s></p><figure id="id.029.01.006.1.jpg" xlink:href="029/01/006/1.jpg"/><p type="main"> |
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79 <s>D<emph type="italics"/>iversa. <!-- REMOVE S-->corporum genera duplici ra­<lb/>tione comparari inter &longs;e à Mathematicis <lb/>po&longs;&longs;unt mole ac pondere. </s> |
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83 <s>Pondere compara­<lb/>tio fit, cum inter corpora diuer&longs;i generis mo <lb/>le aqualia, quæritur, qu&ecedil; &longs;it ratio ponderis: <lb/>quanto videlicet, vnum altero grauius, aut leuius &longs;it. </s> |
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85 <s>Ma­<lb/>gnitudine autem &longs;it collatio, cum po&longs;ita pari grauitate, quæ­<lb/>ritur, que &longs;it ratio magnitudinis; quanto &longs;it alterum altero <lb/>maius, aut minus. </s> |
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87 <s>Quæ comparatio mihi cum videretur & <lb/>iucunda cognitu, & v&longs;um nonnullum habere, nec fu&longs;e à quo<lb/>piam explicata, non ita pridem &longs;uper ea non nihil cœpi me <lb/>liris &longs;ed nihil de luce ac publico cogitabam. <!--neuer Satz-->Io enim ego &longs;um, <lb/>qui malim &longs;cire, quam ne&longs;ci di&longs;cere, quam docere. </s> |
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89 <s>Sed ta­<lb/>men cum Michael Sægnerus in rebus. </s> |
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91 <s>Mathematicis ex­<lb/>cellens vir, ac Magi&longs;ter meus, cui ego plurimum debere me <lb/>fateor, ab eo enim prima elementa habui, repo&longs;cere à me pu­<lb/>blicum aliquem doctrina &longs;ua fructum videretur ac Fede­<lb/>ricus Saminiatus euius suorum &longs;uauitatem, & beneuol&ecedil;n­<lb/>tiam erga me diu, dum &longs;imul hi&longs;ce studijs condi&longs;cipuli ope­<lb/>ram dedimus, expertus &longs;um, me ut aliquid auderem tum <lb/>oratione, tum exemplo &longs;uo excitaret, cœpi minus ab ea cogi­<lb/>tatione alienus e&longs;&longs;e. </s> |
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93 <s>Deinde vero &longs;ummos vixos habui <pb xlink:href="029/01/007.jpg"/>hortatores. </s> |
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95 <s>Etenim cum Clauium, quod tam diu cupiebam, <lb/>vidi&longs;&longs;em <gap/> minorem tanta scientia, & fama viri beni<lb/>gnitatem <gap/>peri&longs;&longs;em. </s> |
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97 <s>Ab eo &longs;imul ac Theodo&longs;io. <!-- KEEP S--></s> |
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99 <s>Rubeo ho<lb/>mine mihi tum ex &longs;tudiorum &longs;imilitudine, tum præcipue ex <lb/>eius humanitate amici&longs;simo, ad Reuerendi&longs;simum Sera­<lb/>phinum deductus &longs;um. <!-- KEEP S--></s> |
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101 <s>I&longs;qs me tantus Præ&longs;ul non &longs;olum <lb/>humani&longs;sime complexus est, verum etiam ita hortatus ad <lb/>eualganda, que &longs;crip&longs;eram; plane vt mihi nefas putauerim <lb/>non parere. </s> |
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103 <s>Accipe igitur & tu Lactor optime amico ac be­<lb/>nigno animo laborem hunc, quem à me talium virorum <lb/>&longs;umma benignitas expre&longs;sit. </s> |
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105 <s>Argumentum quidem, vt <lb/>dicebam non iniucundum e&longs;t, nec ab v&longs;u alienam. </s> |
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107 <s>Huiu&longs;­<lb/>modi enim comparatione Archimedes mixtionem argemi <lb/>in aure deprehendit, & furtum Aurificis in corona aurea <lb/>patefecit. </s> |
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109 <s>de quæ re &longs;uo loco ego tractabo, & facilem mon­<lb/>strabo viam, quæ vel argentum in aure, vel quod vis me­<lb/>tallum in quolibet admistum deprehendi queat, & alte­<lb/>rum ab alieno di&longs;cerni & &longs;imul explicabo, quo pacto ex au­<lb/>ri grauitate <gap/> qualitas, at perfectio intelligi po&longs;sit. </s> |
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111 <s><lb/>Tati vere opu&longs;culo nomen ab Archimede, quem Ducem &longs;e­<lb/>quor, impo&longs;ui Nam cum ille, vt erat &longs;ummus Magi&longs;ter, &longs;e­<lb/>tis habui&longs;&longs;et hunc, totum quasi fabricum po&longs;ito fundamen­<lb/>to delineare in primo lib. |
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113 vbi agit de ijs quæ vehuntur in <lb/>aqua. </s> |
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115 <s>Opus ego pro<gap/> <expan abbr="eiq;">eique</expan> fundamento molem inijce­<lb/>re conatus &longs;um paribus &longs;uis et elaboratam, atque distin­<lb/>ctam. </s> |
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117 <s>Qua in re &longs;i quid a&longs;&longs;equutus &longs;um, quod publice pro­<lb/>betur, gaudeo cau&longs;a & mea & publica illud quidem &longs;pe­<lb/>rofere vt conatus non di&longs;pliceat.<emph.end type="italics"/></s></p><pb xlink:href="029/01/008.jpg" pagenum="1"/><p type="head"> |
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119 <s>MARINI GHETALDI <lb/>PROMOTVS ARCHIMEDES <lb/>SEU.<!-- REMOVE S--><emph type="italics"/>DE VARIIS CORPORVM GENERIBVS <lb/>Grauitate, & magnitudine comparatis.<emph.end type="italics"/></s></p><p type="head"> |
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121 <s>THEOREMA I. PROPOS. I.<!-- KEEP S--></s></p><p type="main"> |
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122 |
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123 <s>Si duorum Grauium Corporum eiu&longs;dem ge­<lb/>neris alterum alterius fuerit multiplex, quo­<lb/>tuplex maius fuerit minoris, totuplex erit <lb/>maioris grauitas, grauitatis minoris. </s></p><p type="main"> |
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125 <s>SINT duo corpora eiu&longs;dem generis ABC, D, quorum grauita­<lb/>tes, EFG, ip&longs;ius ABC, & H, ip­<lb/>&longs;ius D, &longs;it autem corpus ABC, <lb/><figure id="id.029.01.008.1.jpg" xlink:href="029/01/008/1.jpg"/><lb/>multiplex corporis D. <!-- KEEP S--></s> |
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126 |
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127 <s>Dico quo <lb/>tuplex e&longs;t corpus ABC, corporis <lb/>D, totuplicem e&longs;&longs;e grauitatem, <lb/>EFG, grauitatis H, diuidatur <lb/>enim corpus ABC, in partes ip­<lb/>&longs;i D, æquales, quæ &longs;int A, B, C, <lb/>quoniam igitur corpus A, æqua­<lb/>le e&longs;t corpori D, magnitudine, <lb/>& &longs;unt eiu&longs;dem generis, erit grauitas vnius æqualis grauitati alterius. </s> |
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129 <s><lb/>Sumatur grauitas E, æqualis grauitati H, erit igitur corporis A, gra­<lb/>uitas E, & reliqui corporis BC, grauitas FG. <!-- KEEP S--></s> |
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131 <s>Rur&longs;us quoniam cor­<lb/>pora B, D, &longs;unt magnitudine æqualia, erunt æquè grauia, &longs;umatur <lb/>grauitati H, æqualis grauitas F, erit igitur corporis B, grauitas F, & <lb/>reliqui corporis C, grauitas G, & &longs;ic fiat, donec perueniatur ad vlti­<lb/>mam partem corporis ABC, æqualem ip&longs;i D, &longs;it ea vltima pars C, quo <lb/>niam igitur corpus C, æquatur magnitudine ip&longs;i D, æquabitur, & gra­<lb/>uitate, quare grauitas G, æqualis erit grauitati H, &longs;equitur igitur <lb/>quot partes &longs;unt in corpore ABC, æquales ip&longs;i D, tot e&longs;&longs;e partes in <lb/>grauitate EFG, æquales ip&longs;i H, quoties enim &longs;ump&longs;imus in corpore <lb/>ABC, corpus ip&longs;i D æquale, toties & in grauitate EFG, &longs;ump&longs;imus <pb xlink:href="029/01/009.jpg" pagenum="2"/>grauitatem æqualem ip&longs;i H. <!-- KEEP S--></s> |
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133 <s>Si duorum igitur grauium corporum <lb/>eiuldem generis, & c. <!-- REMOVE S-->quod erat demon&longs;trandum. </s> |
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135 </p><p type="head"> |
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137 <s>THEOREMA II. PROPOS. II.<!-- KEEP S--></s></p><p type="main"> |
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139 <s>Corpora grauia eiu&longs;dem generis magnitudine com<lb/>men&longs;urabilia, eandem in grauitate rationem ha­<lb/>bent, quam in magnitudine. </s></p><p type="main"> |
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141 <s>SINT corpora commen&longs;urabilia eiu&longs;dem generis A, B, quorum <lb/>grauitates C, ip&longs;ius A, & D, ip&longs;ius B, Dico e&longs;&longs;e vt A, ad B, ita C, ad D, <lb/>quoniam enim, A, B, commen&longs;ura­<lb/>bilia &longs;unt, metietur ip&longs;a aliquod <lb/>corpus, metiatur, & &longs;it E, cuius <lb/><figure id="id.029.01.009.1.jpg" xlink:href="029/01/009/1.jpg"/><lb/><arrow.to.target n="note1"/> grauitas F, &longs;itque corpus E, <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> <lb/>generis cum corporibus A, B, *ergo <lb/>quotuplex e&longs;t corpus A, ip&longs;ius E, <lb/><arrow.to.target n="note2"/> totuplex erit grauitas C, grauitatis <lb/>F, & quotuplex B, ip&longs;ius E, *totuplex <lb/>D, ip&longs;ius F, &longs;i igitur diuidantur cor­<lb/>pora A, B, in partes æquales ip&longs;i E, <lb/>& grauitates quoque C, D, in partes æquales ip&longs;i F, erit vt corporis <lb/>A, pars vna, ad corpus E, ita pars vna grauitatis C, ad grauitatem F, <lb/>æquale videlicet ad æquale, & æque multiplicatis antecedentibus <lb/>erit vt A, ad E, ita C, ad F, &longs;unt enim antecedentium, hoc e&longs;t, illarum <lb/>partium æque multiplicia A, C, eadem ratione, vt B, ad E, ita erit D, <lb/>ad F, & conuertendo vt E, ad B, ita F, ad D. quoniam igitur vt A, ad <lb/><arrow.to.target n="note3"/> E, ita e&longs;t C, ad F, & vt E, ad B, ita F, ad D, * erit ex æquali vt A, ad <lb/>B, ita C, ad D. corpora igitur commen&longs;urabilia eiu&longs;dem generis ean­<lb/>dem in grauitate rationem habent, quam in magnitudine, quod erat <lb/>demon&longs;trandum. </s></p><p type="margin"> |
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143 <s><margin.target id="note1"/><emph type="italics"/>Ex an­<lb/>teced.<emph.end type="italics"/></s></p><p type="margin"> |
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145 <s><margin.target id="note2"/><emph type="italics"/>Ex an­<lb/>teced.<emph.end type="italics"/></s></p><p type="margin"> |
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147 <s><margin.target id="note3"/>22. 5. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="head"> |
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149 <s>THEOREMA III. PROPOS. III.<!-- KEEP S--></s></p><p type="main"> |
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151 <s>Et incommen&longs;urabilia corpora eiu&longs;dem generis <lb/>eandem in grauitate rationem habent, quam in <lb/>magnitudine. </s></p><p type="main"> |
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153 <s>SINT incommen&longs;urabilia corpora A, BC, quorum grauitates <lb/>D, ip&longs;ius A, & EF, ip&longs;ius BC. <!-- KEEP S--></s> |
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155 <s>Dico e&longs;&longs;e vt A, ad BC, ita D, ad EF, &longs;i <pb xlink:href="029/01/010.jpg" pagenum="3"/>enim non e&longs;t vt A, ad BC, ita D, ad EF, erit vt A, ad BC, ita D, vel ad <lb/><figure id="id.029.01.010.1.jpg" xlink:href="029/01/010/1.jpg"/><lb/>minorem quam EF, <lb/>vel ad maiorem, &longs;it <lb/>primum ad minoré, <lb/>pempe ad EG, & ex­<lb/>ponatur aliquod cor <lb/>pus K, eiu&longs;dem gene­<lb/>ris cum corporibus <lb/>A, B C, cuius graui­<lb/>tas &longs;it æqualis ip&longs;i <lb/>GF, & à corpore BC, <lb/>auferatur aliqua <lb/>pars HC, quæ &longs;it mi­<lb/>nor corpore K, ita vt reliquæ pars. </s> |
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157 <s>BL, &longs;it commen&longs;urabilis ip&longs;i A. & <lb/>&longs;it partis HC, grauitas IF, ergo reliquæ partis BL, grauitas erit EI. <!-- KEEP S--></s> |
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159 <s><lb/>Quoniam igitur corpus A, commen&longs;urabile e&longs;t ip&longs;i BL,* erit vt A, ad<arrow.to.target n="note4"/><lb/>BL, ita D, ad EI, &longs;ed vt A, ad BC, ita e&longs;t D, ad EG, atque A, primus, <lb/>proportionalium terminus in &longs;erie prima, * maiorem habet<arrow.to.target n="note5"/> ratio­<lb/>nem ad BL, &longs;ecundum terminum, quam A, primus terminus in &longs;erie <lb/>&longs;ecunda ad BC, &longs;ecundum; ergo & D, tertius terminus in &longs;erie prima <lb/>ad EI, quartum, maiorem habebit rationem quam D, tertius termi­<lb/>nus in &longs;erie &longs;ecunda ad EG, quartum, quoniam igitur D, maiorem<arrow.to.target n="note6"/><lb/>habet rationem ad EI, quam ad EG,* erit EI, minor quam EG, quod <lb/>e&longs;t ab&longs;urdum. </s> |
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161 <s>non igitur e&longs;t vt A, ad BC, ita D, ad minorem quam EF. <!-- KEEP S--></s></p><p type="margin"> |
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163 <s><margin.target id="note4"/>E<emph type="italics"/>x <expan abbr="an-teced&etilde;-te">an­<lb/>teceden­<lb/>te.</expan><emph.end type="italics"/></s></p><p type="margin"> |
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164 |
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165 <s><margin.target id="note5"/>2.<!-- KEEP S--> 5. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin"> |
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167 <s><margin.target id="note6"/>10. 5. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> |
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169 <s>Deinde &longs;it vt A, ad BC, ita D, ad maiorem quam EF, nempe ad <lb/><figure id="id.029.01.010.2.jpg" xlink:href="029/01/010/2.jpg"/><lb/>EG, & expo&longs;ito cor­<lb/>pore K, vt dictum <lb/>e&longs;t, cuius grauitas, <lb/>&longs;it æqualis grauita­<lb/>ti FG, addatur cor­<lb/>pori BC, aliquod <lb/>corpus CH, quod &longs;it <lb/>minus corpore K, & <lb/>eiu&longs;dem generis cú <lb/>corporibus A, BC, <lb/>ita vt totum corpus <lb/>BL, &longs;it commen&longs;urabile ip&longs;i A, & &longs;it ip&longs;ius CH, grauitas FI, ergo to­<lb/>tius corporis BL, grauitas erit EI; Quoniam igitur corpori A, com­<lb/>men&longs;urabile e&longs;t corpus BL,* erit vt A, ad BL, ita D, ad EI, &longs;ed vt A, ad<arrow.to.target n="note7"/><lb/>BC, ita e&longs;t D, ad EG, atque A, primus proportionalium terminus in <lb/>&longs;erie prima, * minorem habet rationem ad BL, &longs;ecundum terminum, <pb xlink:href="029/01/011.jpg" pagenum="4"/>quam A, primus terminus in &longs;erie &longs;ecunda ad BC, &longs;ecundum, ergo, & <lb/>D, tertius terminus in &longs;erie prima ad EI, quartum, minorem habebit<lb/>rationem quam D, tertius terminus in &longs;erie &longs;ecunda ad EG, quartum. </s> |
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171 <s><lb/>Quoniam igitur D, minorem habet rationem ad EI, quam ad EG, <lb/><arrow.to.target n="note8"/> erit * EI, maior quam EG, quod e&longs;t ab&longs;urdum. </s> |
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173 <s>Non igitur e&longs;t vt A, <lb/>ad BC, ita D, ad maiorem quam EF, o&longs;ten&longs;um autem e&longs;t neque ad mi­<lb/>norem; quare vt A, ad BC, ita erit D, ad EF. & incommen&longs;urabilia igi­<lb/>tur corpora eiu&longs;dem generis eandem in grauitate rationem habent, <lb/>quam in magnitudine, quod erat demon&longs;trandum. </s></p><p type="margin"> |
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175 <s><margin.target id="note7"/><emph type="italics"/>Ex an­<lb/>teced..<emph.end type="italics"/><lb/>8.<!-- KEEP S--> 5.<!-- KEEP S--><emph type="italics"/>Ele<emph.end type="italics"/></s></p><p type="margin"> |
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177 <s><margin.target id="note8"/>10. 5. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> |
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179 <s>ID QVOD nos duobus præcedentibus Theorematis de­<lb/>mon&longs;trauimus, <expan abbr="nõnulli">nonnulli</expan>, vt per &longs;e notum, & vt commune quod­<lb/>dam axioma &longs;upponunt, quam bene & rationabiliter ip&longs;i vide­<lb/>rint; melius enim Euclides propo&longs;itionem 20. primi libri Ele­<lb/>mentorum &longs;uppo&longs;ui&longs;&longs;et vt pronuntiatum; vnicuique enim no­<lb/>tius e&longs;t duo trianguli latera reliquo e&longs;&longs;e maiora (cum & A&longs;ino <lb/>illud &longs;it notum) quam corpora grauia eiu&longs;dem generis eandem <lb/>in grauitate rationem habere, quam in magnitudine, & tamen <lb/>illam propo&longs;itionem demon&longs;trat Euclides, non &longs;upponit, non <lb/>igitur hæc, quæ minus ad principij rationem accedit, &longs;uppo­<lb/>nenda fuit, &longs;ed demon&longs;tranda. </s></p><p type="head"> |
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181 <s>THEOREMA IV. PROPOS. IV.<!-- KEEP S--></s></p><p type="main"> |
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183 <s>Si quatuor corporum grauium primum ad <expan abbr="&longs;ecundũ">&longs;ecundum</expan> <lb/>eandem in magnitudine rationem habeat, quam <lb/>tertium ad quartum, primum autem, & &longs;ecundum &longs;int <lb/>eiu&longs;dem generis, itidem tertium, & quartum; & in gra­<lb/>uitate primum ad &longs;ecundum eandem rationem habebit, <lb/>quam tertium ad quartum. </s></p><p type="main"> |
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185 <s>PRIMVM enim A, ad &longs;ecundum B, eandem in magnitudine ra­<lb/>tionem habeat, quam tertium C, ad quartum D, &longs;int autem A, B, <lb/>eiu&longs;dem generis, itidem C, D. Dico. <!-- KEEP S--></s> |
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187 <s>& in grauitate primum A, ad <lb/>&longs;ecundum B, eandem rationem habere, quam tertium C, ad D, quar­<lb/>tum. </s> |
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189 <s>Sint enim earum grauitates E, ip&longs;ius A, & F, ip&longs;ius B, ip&longs;ius <lb/>vero C, &longs;it grauitas G, & ip&longs;ius D, grauitas H, quoniam igitur cor-<pb xlink:href="029/01/012.jpg" pagenum="5"/><figure id="id.029.01.012.1.jpg" xlink:href="029/01/012/1.jpg"/><lb/>pora A, B, eiu&longs;dem &longs;unt generis, <lb/>&longs;imiliter, & corpora C, D,* erit <lb/>vt A, ad B, ita E, ad F,* & vt<arrow.to.target n="note9"/><lb/>C, ad D, ita G, ad H. <!-- KEEP S--></s> |
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191 <s>Sed poni­<lb/>tur vt A, ad B, ita e&longs;&longs;e C, ad D, <lb/>ergo vt E, ad F, ita erit G, ad <lb/>H. <!-- KEEP S--></s> |
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192 |
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193 <s>Si igitur quatuor corporum <lb/>grauium, primum ad &longs;ecundum <lb/>eandem in magnitudine ratio­<lb/>nem habeat: etcæt. </s> |
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195 <s>quod demon­<lb/>&longs;trare oportebat. </s></p><p type="margin"> |
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197 <s><margin.target id="note9"/>1. & <gap/>. <lb/><emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="head"> |
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199 <s>THEOREMA V. PROPOS. V.<!-- KEEP S--></s></p><p type="main"> |
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201 <s>Solida corpora liquido grauiora demi&longs;&longs;a in liquidum <lb/>ferentur deoríum, donec de&longs;cendant, & erunt in li­<lb/>quido tanto leuiora, quanta e&longs;t grauitas liquidi magni­<lb/>tudinem habentis &longs;olido corpori æqualem. </s></p><p type="main"> |
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203 <s>HOC autem demon&longs;tratum e&longs;t ab Archimede propo&longs;.7. primi li­<lb/>bri de ijs, quæ vehuntur in aqua. </s></p><p type="head"> |
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205 <s>THEOREMA VI. PROPOS. VI.<!-- KEEP S--></s></p><p type="main"> |
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207 <s>Si quatuor grauium corporum primum, & &longs;ecundum <lb/>fuerint magnitudine æqualia, tertium vero, & quar­<lb/>tum æque grauia, fuerint autem primum, & tertium <lb/>eiu&longs;dem generis, itidem &longs;ecundum, & quartum; erit, vt <lb/>grauitas corporis primi, ad grauitatem &longs;ecundi, ita gra­<lb/>uitas liquidi æqualis magnitudine corpori quarto, ad gra<lb/>uitatem liquidi tertio corpori æqualis. </s></p><p type="main"> |
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209 <s>SINT quatuor corpora A, B, C, D, quorum A, primum, & B, &longs;e­<lb/>cundum &longs;int magnitudine æqualia, tertium vero C, & D, quartum <lb/>æque grauia, &longs;int autem A, & C, eiu&longs;dem generis, itidem B, & D. <!-- KEEP S--></s> |
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210 |
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211 <s>Di­<lb/>co vt grauitas corporis A, ad grauitatem corporis B, ita e&longs;&longs;e grauita­<lb/>tem liquidi æqualis magnitudine corpori D, ad grauitatem liquidi <lb/>magnitudine corpori C, æqualis. </s> |
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213 <s>Accipiantur enim tria eiu&longs;dem ge­<pb xlink:href="029/01/013.jpg" pagenum="6"/>neris liquidi corpora E, F, G, quorum E, &longs;it æquale corpori A, vel B, <lb/><figure id="id.029.01.013.1.jpg" xlink:href="029/01/013/1.jpg"/><lb/>magnitudine, ip&longs;um vero F, æqua <lb/>le corpori C, & ip&longs;um G, æquale <lb/>corpori D. <!-- KEEP S--></s> |
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215 <s>Quoniam igitur e&longs;t vt <lb/>D, ad G, ita B, ad E, æquale vi­<lb/>delicet ad æquale, erit permutan­<lb/>do vt D, ad B, ita G, ad E, & quo­<lb/>niam &longs;unt eiu&longs;dem generis corpo<lb/>ra D, B, &longs;imiliter & corpora G, E, <lb/><arrow.to.target n="note10"/> erit * vt grauitas corporis D, hoc <lb/>e&longs;t ip&longs;ius C, ponuntur enim æque <lb/>grauia corpora C, D, ad grauita­<lb/>tem corporis B, ita liquidi G, gra­<lb/>uitas ad grauitatem liquidi E. <!-- KEEP S--></s> |
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216 |
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217 <s><lb/>Similiter quoniam e&longs;t vt A, ad E, <lb/>ita C, ad F, æquale videlicet ad <lb/>æquale, erit permutando vt A, ad <lb/>C, ita E, ad F, & quoniam ponuntur eiu&longs;dem generis corpora A, C, <lb/><arrow.to.target n="note11"/> itidem E, F,* erit vt grauitas corporis A, ad grauitatem ip&longs;ius C, ita <lb/>liquidi E, grauitas ad grauitatem liquidi F, &longs;ed vt grauitas corporis <lb/>C, ad grauitatem corporis B; ita e&longs;t grauitas liquidi G, ad grauita­<lb/><arrow.to.target n="note12"/>tem liquidi E, vt e&longs;t demon&longs;tratum, ergo * in perturbata proportione <lb/>erit vt grauitas corporis A, ad ip&longs;ius corporis B, grauitatem, ita liqui <lb/>di G, grauitas, ad grauitatem liquidi F. &longs;i igitur quatuor grauium <lb/>corporum primum, & &longs;ecundum, & c. <!-- REMOVE S-->quod erat demon&longs;trandum. </s> |
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219 </p><p type="margin"> |
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221 <s><margin.target id="note10"/>2. & 3. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s></p><p type="margin"> |
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223 <s><margin.target id="note11"/>2. & 3. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s></p><p type="margin"> |
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225 <s><margin.target id="note12"/>23. 5. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="head"> |
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227 <s>THEOREMA VII. PROPOS. VII.<!-- KEEP S--></s></p><p type="main"> |
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229 <s>Si quatuor grauium corporum primum, & <expan abbr="&longs;ecundũ">&longs;ecundum</expan>, <lb/>fuerint magnitudine æqualia, tertium vero, & quar­<lb/>tum æque grauia, fuerint autem primum, & tertium <lb/>eiu&longs;dem generis, itidem &longs;ecundum, & quartum; primum <lb/>ad &longs;ecundum eandem in grauitate rationem habebit, <lb/>quam habet in magnitudine quartum ad tertium. </s></p><p type="main"> |
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230 |
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231 <s>SINT quatuor grauia corpora A, B, C, D, quorum A, primum, <lb/>& B, &longs;ecundum &longs;int magnitudine æqualia, tertium vero C, & D, quar­<lb/>tum æque grauia, &longs;int autem A, & C, eiu&longs;dem generis, itidem B, & <lb/>D. <!-- KEEP S--></s> |
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232 |
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233 <s>Dico corpus A, eandem in grauitate rationem habere ad corpus <pb xlink:href="029/01/014.jpg" pagenum="7"/>B, quam corpus D, habet in magnitudine ad C, corpus. </s> |
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234 |
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235 <s>Sit enim li­<lb/><figure id="id.029.01.014.1.jpg" xlink:href="029/01/014/1.jpg"/><lb/>quidi magnitudine æqualis corpo­<lb/>ri C, grauitas E, &longs;imiliter, & liqui­<lb/>di æqualis magnitudine corpori D, <lb/>grauitas F, quoniam igitur gra­<lb/>uia corpora eiu&longs;dem generis, ean­<lb/>dem in magnitudine rationem*<arrow.to.target n="note13"/> ha­<lb/>bent, quam in grauitate, erit vt ma­<lb/>gnitudo liquidi æqualis corpori D, <lb/>ad magnitudinem liquidi æqualis <lb/>corpori C, hoc e&longs;t, vt magnitudo <lb/>corporis D, ad magnitudinem cor­<lb/>poris C, ita grauitas F, ad grauita­<lb/>tem E, &longs;ed vt grauitas F, ad grauitatem E, * ita e&longs;t grauitas corporis<arrow.to.target n="note14"/><lb/>A, ad grauitatem corporis B,* ergo vt grauitas corporis A, ad graui­<lb/>tatem corporis B, ita erit magnitudo corporis D, ad corporis C, ma­<lb/>gnitudinem. </s> |
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236 |
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237 <s>Si quatuor igitur grauium corporum primum, & &longs;ecun­<lb/>dum, & c. <!-- REMOVE S-->quod erat demon&longs;trandum. </s> |
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238 |
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239 </p><p type="margin"> |
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241 <s><margin.target id="note13"/>2. & 3. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s></p><p type="margin"> |
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242 |
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243 <s><margin.target id="note14"/><emph type="italics"/>Ex an­<lb/>teced.<emph.end type="italics"/><lb/>11. 5. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="head"> |
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245 <s>PROBLEMA I. PROPOS. VIII.<!-- KEEP S--></s></p><p type="main"> |
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246 |
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247 <s>Propo&longs;itis duobus corporibus magnitudine æquali­<lb/>bus, vno &longs;olido, altero liquido, data &longs;olidi corporis <lb/>grauitate, grauitatem liquidi inuenire. </s></p><figure id="id.029.01.014.2.jpg" xlink:href="029/01/014/2.jpg"/><p type="main"> |
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248 |
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249 <s>SINT duo propo&longs;ita corpo­<lb/>ra magnitudine æqualia A, B, <lb/>quorum A, &longs;it &longs;olidum, B, vero <lb/>liquidum, & &longs;it &longs;olidi data graui­<lb/>tas CD, Oportet inuenire quan­<lb/>ta erit grauitas liquidi B. </s> |
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250 |
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251 <s>Si &longs;o­<lb/>lidum A, grauius &longs;it liquido, de­<lb/>mittatur in <expan abbr="liquidũ">liquidum</expan>, & habeat in <lb/>liquido grauitatem ED, quoniam <lb/>igitur &longs;olidum A, grauius e&longs;t li­<lb/>quido, demi&longs;&longs;um in liquidum <lb/>erit* in liquido tanto leuius,<arrow.to.target n="note15"/> <expan abbr="quã-ta">quan­<lb/>ta</expan> e&longs;t grauitas liquidi magnitu­<lb/>dine æqualis &longs;olido A, &longs;ed <expan abbr="&longs;olidũ">&longs;olidum</expan> <lb/>A, leuius e&longs;t in liquido, quanta <pb xlink:href="029/01/015.jpg" pagenum="8"/>e&longs;t grauitas CE, ergo grauitas liquidi magnitudine æqualis &longs;olido A, <lb/>erit CE. <!-- KEEP S--></s></p><p type="margin"> |
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253 <s><margin.target id="note15"/>huius</s></p><p type="main"> |
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255 <s>Si vero &longs;olidum A, &longs;it leuius liquido, accipiatur aliquod aliud cor­<lb/>pus &longs;olidum F, grauius liquido, ita vt &longs;olidum con&longs;tans ex vtri&longs;que <lb/>&longs;olidis A, F, demi&longs;&longs;um in liquidum feratur deor&longs;um, & &longs;it &longs;olidi F, <lb/>grauitas DG, item eiu&longs;dem &longs;olidi F, in liquido videlicet exi&longs;tentis &longs;it <lb/><arrow.to.target n="note16"/> grauitas HG,* ergo liquidi magnitudine æqualis &longs;olido F, erit gra­<lb/>uitas DH. <!-- KEEP S--></s></p><p type="margin"> |
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257 <s><margin.target id="note16"/>5.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main"> |
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259 <s>Et quoniam &longs;olidi A, grauitas e&longs;t CD, &longs;olidi vero F, grauitas DG, <lb/>erit vtrorumque &longs;olidorum A, F, grauitas CG. coniungantur &longs;olida <lb/>A, F, & &longs;olidum ex vtri&longs;que con&longs;tans demittatur in liquidum, & ha­<lb/>beat in liquido grauitatem GI, (habebit autem in liquido minorem <lb/>grauitatem, quam &longs;olum &longs;olidum F, quoniam &longs;olidum F, grauius li­<lb/>quido fertur deor&longs;um nullo prohibente, & coniunctum cum &longs;olido A, <lb/>leuiori liquido ab eo &longs;u&longs;tinetur, ne deor&longs;um feratur <expan abbr="tãta">tanta</expan> vi, qua &longs;eiun­<lb/>ctum) quoniam igitur &longs;olidi, quod con&longs;tat ex vtri&longs;que &longs;olidis A, F, <lb/><arrow.to.target n="note17"/> grauitas e&longs;t CG, in liquido vero exi&longs;tentis grauitas GI, * erit liquidi <lb/>habentis magnitudinem æqualem vtri&longs;que &longs;olidis A, F, grauitas CI, <lb/>&longs;ed grauitas liquidi æqualis magnitudine &longs;olido F, e&longs;t DH, ergo reli­<lb/>qui liquidi æqualis &longs;olido A, erit grauitas CD, IH, &longs;ed liquidum B, <lb/>æquatur magnitudine &longs;olido A, ergo grauitas liquidi B, erit CD, IH, <lb/>inuenta igitur e&longs;t liquidi corporis B, grauitas CD, IH, de qua quæ­<lb/>rebatur. </s></p><p type="margin"> |
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261 <s><margin.target id="note17"/>5.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main"> |
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263 <s>Placet huic Problemati exemplum apponere, vt vnicuique <lb/>etiam di&longs;ciplinæ Mathematicæ experto ad v&longs;um pateat adi­<lb/>tus; quare etiam &longs;equentibus Problematis apponemus &longs;imilia <lb/>exempla. </s></p><p type="head"> |
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265 <s>Exemplum.<!-- KEEP S--></s></p><p type="main"> |
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267 <s>Qvidam proponit aliquod corpus &longs;olidum notæ <lb/>grauitatis, & vult &longs;cire quanta erit grauitas liqui­<lb/>di, magnitudinem habentis propo&longs;ito Corpori &longs;olido æ­<lb/>qualem. </s></p><p type="main"> |
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269 <s><emph type="italics"/>Sit primum propo&longs;itum aliquod corpus plumbeum A, euius graui­<lb/>tas &longs;it<emph.end type="italics"/> 23. <emph type="italics"/>& oporteat &longs;cire quanta erit grauitas aquæ magnitudinem <lb/>habentis æqualem propo&longs;ito plumbo A, ponderetur plumbum A, in <lb/>aqua (modum quo ponderanda &longs;int corpora &longs;olida in aqua, inferius <lb/>apponemus) & habeat grauitatem<emph.end type="italics"/> 21. <emph type="italics"/>quoniam igitur numerus<emph.end type="italics"/> 23. <pb xlink:href="029/01/016.jpg" pagenum="9"/><emph type="italics"/>&longs;uperat numerum<emph.end type="italics"/> 21, <emph type="italics"/>numero 2, erit grauitàs aquæ magnitudinem <lb/>habentis æqualem plumbo A, 2.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> |
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271 <s>Eadem via omnium liquidorum inuenitur grauitas, quan­<lb/>do nimirum corpus &longs;olidum &longs;it grauius liquido, cuius liquidi <lb/>quærenda e&longs;t grauitas, hoc e&longs;t quando corpus &longs;olidum demi&longs;­<lb/>&longs;um in liquidum feratur deor&longs;um. </s></p><p type="main"> |
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273 <s>Quando vero corpus &longs;olidum fuerit leuius liquido, hoc e&longs;t <lb/>demi&longs;&longs;um in liquidum non de&longs;cendat, per adiectionem alicuius <lb/>alius &longs;olidi corporis liquido grauioris, quæ&longs;ita liquidi grauitas <lb/>inuenietur. </s></p><p type="main"> |
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275 <s><emph type="italics"/>Sit igitur propo&longs;itum aliquod cereum corpus A, cuius grauitas &longs;it<emph.end type="italics"/><lb/>21. <emph type="italics"/>& oporteat &longs;cire quanta erit grauitas aquæ magnitudinem haben­<lb/>tis æqualem ceræ A. <!-- KEEP S--></s> |
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277 <s>Quoniam cera leuior est, quam aqua, &longs;i demitta­<lb/>tur in aquam non feretur deor&longs;um, accipiatur aliquod corpus &longs;olidum <lb/>F, grauius quam aqua, ita vt corpus constans ex vtri&longs;que corporibus <lb/>A, F, demi&longs;&longs;um in aquam feratur deor&longs;um, &longs;it igitur corpus F, plum­<lb/>beum, cuius grauitas &longs;it v.g.<emph.end type="italics"/> 23, <emph type="italics"/>& eiu&longs;dem in aqua ponderati<emph.end type="italics"/> 21, <lb/><emph type="italics"/>ergo aquæ magnitudinem habentis <expan abbr="æqual&etilde;">æqualem</expan> plumbo F, erit grauitas<emph.end type="italics"/> 2, </s></p><p type="main"> |
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279 <s><emph type="italics"/>Et quoniam ceræ A, grauitas est<emph.end type="italics"/> 21, <emph type="italics"/>plumbi vero F,<emph.end type="italics"/> 23, <emph type="italics"/>erit vtro­<lb/>rumque corporum A, F, ceræ nimirum, & plumbi grauitas<emph.end type="italics"/> 44, <emph type="italics"/>coniun<lb/>gatur cera, & plumbum, & ita coniuncta ponderentur in aqua, & <lb/>habeant grauitatem<emph.end type="italics"/> 20, <emph type="italics"/>quoniam igitur numerus<emph.end type="italics"/> 44, <emph type="italics"/>&longs;uperat nume­<lb/>rum<emph.end type="italics"/> 20, <emph type="italics"/>numero<emph.end type="italics"/> 24, <emph type="italics"/>erit grauitas aquæ habentis magnitudinem æqua­<lb/>lem vtri&longs;que corporibus ceræ & plumbi<emph.end type="italics"/> 24, <emph type="italics"/>&longs;ed grauitas aquæ magni­<lb/>tudinem habentis æqualem plumbo F, est<emph.end type="italics"/> 2, <emph type="italics"/>ergo reliquum quod est<emph.end type="italics"/><lb/>22, <emph type="italics"/>erit grauitas, aquæ magnitudine æqualis propo&longs;itæ ceræ A.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> |
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281 <s>At vero &longs;i propo&longs;itum fuerit aliquod corpus &longs;olidum ma­<lb/>gni ponderis, ita vt difficile po&longs;&longs;it ponderari in aqua, hac via <lb/>inuenietur aquæ quæ&longs;ita grauitas. </s></p><p type="main"> |
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283 <s><emph type="italics"/>Sit aliquod corpus <expan abbr="plumbeũ">plumbeum</expan> A, cuius grauitas<emph.end type="italics"/> 2300, <emph type="italics"/>& oporteat in <lb/>uenire grauitatem aquæ magnitudinem habentis <expan abbr="æqual&etilde;">æqualem</expan> plumbo A, <lb/>accipiatur aliquod paruum plumbi corpus F, cuius grauitas &longs;it v.g.<emph.end type="italics"/><lb/>23, <emph type="italics"/>& inueniatur grauitas aquæ magnitudine æqualis plumbo F, vt <lb/>dictum est, quæ &longs;it.<emph.end type="italics"/> 2, <emph type="italics"/>& fiat vt<emph.end type="italics"/> 23, <emph type="italics"/>ad<emph.end type="italics"/> 2, <emph type="italics"/>ita<emph.end type="italics"/> 2300, <emph type="italics"/>ad alium numerum <lb/>qui &longs;it<emph.end type="italics"/> 200. <emph type="italics"/>grauitas igitur aquæ magnitudinem habentis æqualem <lb/>plumbo A, erit<emph.end type="italics"/> 200. </s></p><p type="main"> |
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285 <s><emph type="italics"/>Similiter &longs;it aliquod cereum corpus A, cuius grauitas<emph.end type="italics"/> 2100, <emph type="italics"/>& opor<lb/>teat facere, quod imperatum e&longs;t. </s> |
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287 <s>accipiatur aliquod paruum ceræ cor­<lb/>pus F, cuius grauitas &longs;it v.g.<emph.end type="italics"/> 21, <emph type="italics"/>& inuenta grauitate aquæ magni-<emph.end type="italics"/><pb xlink:href="029/01/017.jpg" pagenum="10"/><emph type="italics"/>tudinem habentis æqualem ceræ F, quæ &longs;it<emph.end type="italics"/> 12, <emph type="italics"/>fiat vt<emph.end type="italics"/> 21, <emph type="italics"/>ad<emph.end type="italics"/> 22, <emph type="italics"/>ita<emph.end type="italics"/><lb/>2100, <emph type="italics"/>ad alium numerum qui &longs;it<emph.end type="italics"/> 2200; <emph type="italics"/>erit igitur grauitas aquæ ma <lb/>gnitudinem habentis æqualem ceræ A,<emph.end type="italics"/> 2200. </s></p><p type="main"> |
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289 <s>Neque nece&longs;&longs;e e&longs;t, vt illud corpus &longs;olidum magni ponderis <lb/>reip&longs;a proponatur, &longs;ufficit enim vt eius grauitas notificetur <lb/>numero tantum. </s></p><p type="main"> |
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291 <s>Si autem propo&longs;itum fuerit inuenire quanta erit graui­<lb/>tas argenti viui magnitudine æqualis propo&longs;ito corpori &longs;o­<lb/>lido A; ratione qua &longs;upra, non inuenietur ip&longs;a grauitas, quo­<lb/>niam nullum corpus demi&longs;&longs;um in argentum viuum fertur <lb/>deor&longs;um, ni&longs;i aurum, aurum vero in ip&longs;o argento viuo perrum­<lb/>pitur, &longs;ed qua ratione inuenienda &longs;it ip&longs;a argenti viui graui­<lb/>tas, dicemus ad finem exempli propo&longs;itionis decimæ quartæ. </s></p><p type="main"> |
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293 <s>Quomodo ponderanda &longs;int corpora &longs;olida in aqua. </s></p><p type="main"> |
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295 <s>C<emph type="italics"/>orpus quod ponderandum proponitur &longs;eta equina ex altera li­<lb/>bræ lance appendatur, in altera lance ponantur pondera, & cor­<lb/>pus appen&longs;um demittatur in aquam, ita vt in aqua libere pendeat, ne­<lb/>que lancem, cui appen&longs;um est corpus, neque aliam in qua &longs;unt <expan abbr="ponderã">pondera</expan> <lb/>aqua contingat, & ita ponderetur propo&longs;itum corpus, ac &longs;i in aere pen <lb/>deret.<emph.end type="italics"/></s></p><p type="main"> |
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297 <s><emph type="italics"/>Dixi &longs;eta e quina corpus ponderandum debere appendi, quia fere <lb/>æque grauis e&longs;t atque aqua, & ideo nihil addet, vel minuet grauita­<lb/>tis in ip&longs;o corpore ponderando.<emph.end type="italics"/></s></p><p type="main"> |
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299 <s><emph type="italics"/>Quod &longs;i corpus ponderandum fuerit, tam graue, vt &longs;eta &longs;implici &longs;u­<lb/>stineri nequeat, appendatur pluribus &longs;imul iunctis &longs;etis, & ne aliquid <lb/>grauitatis &longs;etarum coniunctio addat corpori ponderando, ponantur <lb/>in altera lance totidem &longs;etæ æquales eis, quæ ex lance, cui appen&longs;um e&longs;t <lb/>corpus pendent, v&longs;que ad corpus appen&longs;um, hac igitur &longs;etarum addi­<lb/>tione æque ponderabunt lances, & quamuis illæ &longs;etæ, quibus appen­<lb/>&longs;um est corpus, &longs;int longiores, quam aliæ alteri lanci addita, longitudi­<lb/>ne partium, quibus ligatum est corpus, tamen quoniam illæ partes <lb/>æque graues &longs;unt, atque aqua, exi&longs;tentes cum ip&longs;o corpore in aqua, nul­<lb/>lam grauitatem habebunt, & ideo illæ &longs;etæ, quæ alias &longs;uperant dictis <lb/>partibus, & &longs;i longiores, non erunt grauiores quam aliæ, existenti­<lb/>bus, nempe, vt dictum e&longs;t, illis partibus cum ip&longs;o corpore in aqua. </s> |
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301 <s>Sic <lb/>igitur in aqua ponderanda erunt &longs;olida corpora, quod animaduerti&longs;&longs;e <lb/>fuit operæ pretium.<emph.end type="italics"/></s></p><pb xlink:href="029/01/018.jpg" pagenum="11"/><p type="head"> |
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303 <s>PROBLEMA II. PROPOS. IX.<!-- KEEP S--></s></p><p type="main"> |
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305 <s>Propo&longs;itis duobus corporibus magnitudine æquali­<lb/>bus, vno &longs;olido, altero liquido, data corporis li­<lb/>quidi grauitate, grauitatem &longs;olidi inuenire. </s></p><p type="main"> |
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307 <s>SINT duo propo&longs;ita corpora <lb/>magnitudine æqualia, A, quidem &longs;o­<lb/><figure id="id.029.01.018.1.jpg" xlink:href="029/01/018/1.jpg"/><lb/>lidum, B, vero liquidum, &longs;it autem li­<lb/>quidi B, data grauitas F, & oporteat <lb/>inuenire grauitatem &longs;olidi A, accipia<lb/>tur aliquod corpus &longs;olidum D, <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> <lb/>generis, cum &longs;olido A, cuius grauitas <lb/>&longs;it H, deinde liquidi eiu&longs;dem generis <lb/>cum liquido B, magnitudine æqualis <lb/>&longs;olido D,* inueniatur grauitas quæ &longs;it<arrow.to.target n="note18"/><lb/>G, & fiat vt G, ad H, ita F, ad aliam <lb/>grauitatem, quæ &longs;it C. <!-- KEEP S--></s> |
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309 <s>Dico &longs;olidi A, <lb/>grauitatem e&longs;&longs;e C, accipiatur enim aliquod corpus liquidum E, eiu&longs;­<lb/>dem generis cum liquido B, grauitatem habens æqualem &longs;olido D. <!-- KEEP S--></s> |
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311 <s><lb/>Quoniam igitur &longs;unt quatuor corpora grauia B, A, E, D, quorum pri­<lb/>mum B, & &longs;ecundum A, &longs;unt magnitudine æqualia, tertium vero E, <lb/>& quartum D, æque grauia, & &longs;unt eiu&longs;dem generis corpora B, E, &longs;i­<lb/>militer, & corpora A, D,*erit vt grauitas liquidi æqualis<arrow.to.target n="note19"/> magnitudi­<lb/>ne &longs;olido D, hoc e&longs;t vt G, ad grauitatem liquidi E; hoc e&longs;t ad H, <expan abbr="ponũ-tnr">ponun­<lb/>tnr</expan> enim æque grauia corpora D, E, ita grauitas F, ad &longs;olidi A, graui­<lb/>tatem, &longs;ed vt grauitas G, ad grauitatem H, ita e&longs;t grauitas F, ad C, <lb/>grauitatem, ergo grauitas C, æqualis erit grauitati &longs;olidi A. <!-- KEEP S--></s> |
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313 <s>Inuenta <lb/>igitur e&longs;t &longs;olidi A, grauitas C, quod facere oportebat. </s></p><p type="margin"> |
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315 <s><margin.target id="note18"/>8.<emph type="italics"/>hu­<lb/>ius.<emph.end type="italics"/></s></p><p type="margin"> |
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317 <s><margin.target id="note19"/>6.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="head"> |
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319 <s>Exemplum.<!-- KEEP S--></s></p><p type="main"> |
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321 <s>Qvidam proponit aliquod corpus liquidum notæ <lb/>grauitatis, & vult &longs;cire quanta erit grauitas alicu­<lb/>ius &longs;olidi, magnitudinem habentis propo&longs;ito Corpori li­<lb/>quido æqualem. </s></p><p type="main"> |
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323 <s><emph type="italics"/>Sit propo&longs;itum aliquod corpus aqueum B, euius grauitas &longs;it 100. &<emph.end type="italics"/><pb xlink:href="029/01/019.jpg" pagenum="12"/><emph type="italics"/>oporteat &longs;cire quanta erit grauitas plumbi magnitudinem habentis <lb/>æqualem propo&longs;itæ aquæ B, verbi gratia &longs;it vas aliquod plenum aqua, <lb/>cuius aquæ grauitas &longs;it 100, & oporteat &longs;cire, &longs;i illud <expan abbr="id&etilde;">idem</expan> vas replea­<lb/>tur plumbo, quanta illius plumbi erit grauitas. </s> |
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325 <s>Accipiatur aliquod <lb/>plumbeum corpus D, cuius grauitas &longs;it<emph.end type="italics"/> 23<emph type="italics"/>, deinde aquæ <expan abbr="magnitudin&etilde;">magnitudinem</expan> <lb/>habentis æqualem plumbo D, inueniatur grauitas, quod quomodo fie­<lb/>ri oporteat iam dictum e&longs;t in antecedentis problematis exemplo: &longs;it igi<lb/>tur ea inuenta grauitas<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"/> 100<emph type="italics"/>, ad alium nume­<lb/>rum qui &longs;it<emph.end type="italics"/> 1150<emph type="italics"/>, is igitur numerus erit grauitas plumbi magnitu­<lb/>dinem habentis propo&longs;itæ aquæ B, æqualem, hoc est illius plumbi, quod <lb/>in va&longs;e continetur.<emph.end type="italics"/></s></p><p type="main"> |
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327 <s>At vero &longs;i <expan abbr="propo&longs;itũ">propo&longs;itum</expan> fuerit inuenire <expan abbr="quãta">quanta</expan> erit grauitas ceræ, <lb/>aut ligni, aut <expan abbr="cuiu&longs;cũque">cuiu&longs;cunque</expan> &longs;olidi leuioris quam aqua, nihil diuer<lb/>&longs;i in opere accidet, ni&longs;i quod ratio inueniendi grauitatem <lb/>aquæ magnitudinem habentis æqualem corpori &longs;olido leuio­<lb/>ri, quàm aqua, differt in aliquo à ratione, qua inuenitur graui­<lb/>tas aquæ magnitudinem habentis æqualem &longs;olido corpori <lb/>grauiori, quam aqua, &longs;ed vtramque rationem exemplo ante­<lb/>cedentis Problematis illu&longs;trauimus, in eo enim &longs;atis explica­<lb/>tum e&longs;t de vtraque. </s></p><p type="main"> |
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329 <s><emph type="italics"/>Sed ne exemplorum inopia laborare videamur, &longs;it inuenienda gra­<lb/>uitas ceræ magnitudinem habentis æqualem propo&longs;itæ aquæ B, acci­<lb/>piatur aliquod cereum corpus D, cuius grauitas &longs;it<emph.end type="italics"/> 21, <emph type="italics"/>deinde aquæ <lb/>magnitudinem habentis æqualem ceræ D, inueniatur grauitas, vt in <lb/>antecedentis Problematis exemplo dictum est, quæ grauitas &longs;it<emph.end type="italics"/> 22, <emph type="italics"/>& <lb/>fiat vt<emph.end type="italics"/> 22<emph type="italics"/>, ad<emph.end type="italics"/> 21<emph type="italics"/>, ita<emph.end type="italics"/> 100<emph type="italics"/>, hoc est grauitas aquæ B, ad alium numerum <lb/>qui &longs;it<emph.end type="italics"/> 95 5/11. <emph type="italics"/>is igitur numerus indicabit quanta erit grauitas ceræ <lb/>magnitudinem habentis æqualem propo&longs;itæ aquæ B.<emph.end type="italics"/></s></p><p type="main"> |
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331 <s>Similiter &longs;i propo&longs;itum liquidum corpus B, fuerit olei, aut <lb/>vini, aut cuiu&longs;cumque liquidi, præter argenti viui, eadem om­<lb/>nino via, qua ante, inuenietur quæ&longs;ita corporis &longs;olidi grauitas, <lb/>&longs;ed de argento viuo tractabimus ad finem propo&longs;itionis deci­<lb/>mæ quartæ. </s></p><p type="head"> |
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333 <s>PROBLEMA III. PROPOS. X.<!-- KEEP S--></s></p><p type="main"> |
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335 <s>Propo&longs;itis duobus corporibus æque grauibus, vno <lb/>&longs;olido, altero liquido, data &longs;olidi corporis magnitu­<pb xlink:href="029/01/020.jpg" pagenum="13"/>dine, magnitudinem liquidi inuenire. </s></p><p type="main"> |
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337 <s>SINT duo propo&longs;ita corpora <lb/>æque grauia, A, quidem &longs;olidum B, <lb/>vero liquidum, &longs;it autem &longs;olidi A, da­<lb/><figure id="id.029.01.020.1.jpg" xlink:href="029/01/020/1.jpg"/><lb/>ta magnitudo C, & oporteat inuenire <lb/>quanta erit magnitudo liquidi B, Ac­<lb/>cipiatur aliquod corpus &longs;olidum D, <lb/>eiu&longs;dem generis cum &longs;olido A, & &longs;it <lb/>eius grauitas G, & liquidi, quod &longs;it E, <lb/>eiu&longs;dem gener is cum liquido B, ma­<lb/>gnitudinem habentis æqualem &longs;olido <lb/>D,* inueniatur grauitas quæ &longs;it H, &<arrow.to.target n="note20"/><lb/>fiat vt grauitas H, ad grauitatem G, <lb/>ita magnitudo C, ad aliam magnitudinem quæ &longs;it F. <!-- KEEP S--></s> |
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339 <s>Quoniam igitur <lb/>&longs;unt quatuor corpora grauia E, D, B, A, quorum primum E, & &longs;ecun­<lb/>dum D, &longs;unt æqualia magnitudine, tertium vero B, & quartum A, <lb/>æque grauia, & &longs;unt eiu&longs;dem generis corpora E, B, &longs;imiliter, & corpo­<lb/>ra D, A, *erit ut grauitas H, ad grauitatem G, ita magnitudo C, ad<arrow.to.target n="note21"/> li­<lb/>quidi B, magnitudinem, &longs;ed vt grauitas H, ad grauitatem G, ita e&longs;t <lb/>magnitudo C, ad magnitudinem F, ergo magnitudo F, æqualis erit <lb/>magnitudini liquidi B, inuenta igitur e&longs;t liquidi corporis B, magni­<lb/>tudo F, quod facere oportebat. </s></p><p type="margin"> |
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341 <s><margin.target id="note20"/>8.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="margin"> |
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343 <s><margin.target id="note21"/>7.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main"> |
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345 <s>Sed quoniam corporum regularium magnitudo quo­<lb/>que exprimitur latere eiu&longs;dem corporis, vel diametro, &longs;i <lb/>propo&longs;ita duo corpora A, B, fuerint regularia, vtpote &longs;ph&ecedil; <lb/>rica, fuerit autem &longs;phæræ A, data diameter C, & oporteat <lb/>inuenire quanta erit diameter &longs;phæræ B. ita faciendum <lb/>erit. </s></p><p type="main"> |
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347 <s>Accepto, vt diximus, aliquo corpore &longs;olido D, eiu&longs;dem generis <expan abbr="cũ">cum</expan> <lb/>&longs;phæra A, & inuenta grauitate liquidi E, vt &longs;upra, fiat vt grauitas H, <lb/>ad grauitatem G, ita cubus ex C, ad alium cubum, cuius latus &longs;it F, <lb/>dico ip&longs;um latus F, æquale e&longs;&longs;e diametro &longs;phæræ B. </s> |
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349 <s>Quoniam enim <lb/>eadem ratione qua &longs;upra demon&longs;trabitur, vt grauitas H, ad grauita­<lb/>tem G, ita e&longs;&longs;e magnitudinem &longs;phæræ A, ad &longs;phæræ B, magnitudinem, <lb/>&longs;ed magnitudo &longs;phæræ A, ad magnitudinem &longs;phæræ B,* triplicatam<arrow.to.target n="note22"/><lb/>rationem habet eius, quam C, diameter &longs;phæræ A, ad diametrum <lb/>&longs;phæræ B, &longs;imiliter & cubus ex C, ad cubum ex diametro &longs;phæræ B, <pb xlink:href="029/01/021.jpg" pagenum="14"/><arrow.to.target n="note23"/> tripiicatam* rationem habet eius, quam C, ad diametrum &longs;phæræ, <lb/>B, ergo vt grauitas H, ad grauitatem G, ita erit cubus ex C, ad cubum <lb/>ex diametro &longs;phæræ B, &longs;ed vt grauitas H, ad grauitatem G, &longs;ta e&longs;t cu<lb/>bus ex C, ad cubum ex F, ergo cubus ex F, æqualis erit cubo diame­<lb/>tri&longs;phæræ B; quare & latus F, æquabitur &longs;phæræ B, diametro. </s> |
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351 <s>inuen­<lb/>ta igitur e&longs;t quantitas diametri liquidæ &longs;phæræ, B, quod facere opor­<lb/>tebat. </s></p><p type="margin"> |
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353 <s><margin.target id="note22"/>18.12. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin"> |
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355 <s><margin.target id="note23"/>33.12. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="head"> |
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357 <s>Exemplum.<!-- KEEP S--></s></p><p type="main"> |
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359 <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></p><p type="main"> |
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361 <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="plumbeũ">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> |
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363 <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"/><!-- KEEP S--></s></p><p type="main"> |
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365 <s>Quod &longs;i <expan abbr="propo&longs;itũ">propo&longs;itum</expan> corpus plumbeum A &longs;it regulare vt po­<lb/>te &longs;phæricum, cuius &longs;phæræ diameter &longs;it 10, & oporteat inue­<lb/>nire quanta erit diameter &longs;phæræ ex aqua, grauitatem haben­<lb/>tis æqualem propo&longs;itæ &longs;phæræ A, ita faciendum erit. </s></p><p type="main"> |
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367 <s><emph type="italics"/>Accipiatur, vt diximus, aliquod corpus plumbeum D, cuius gra­<lb/>uitas<emph.end type="italics"/> 23<emph type="italics"/>, deinde aquæ habentis magnitudinem æqualem plumbo D, <lb/>inueniatur grauitas 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 cubus ex<emph.end type="italics"/> 10<emph type="italics"/>, qui <lb/>e&longs;t 1000, ad alium numerum qui &longs;it<emph.end type="italics"/> 11500<emph type="italics"/>, is igitur numerus erit <lb/>cubus diametri &longs;phæræ ex aqua grauitatem habentis æqualem propo­<lb/>&longs;itæ &longs;phæræ A, quare eius latus cubicum, quod est<emph.end type="italics"/> 22 57/100. <emph type="italics"/><expan abbr="proximũ">proximum</expan> <lb/>vero indicabit ip&longs;am diametrum.<emph.end type="italics"/></s></p><p type="main"> |
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369 <s><emph type="italics"/>Similiter &longs;i propo&longs;itum corpus plumbeum A, fuerit cubicum, vel <lb/>alicuius alterius formæ regularis, eadem ratione inueniemus latus cu<lb/>bi ex aqua grauitatem habentis æqualem propo&longs;ito cubo A, nam &longs;i cu­<lb/>bi A, datum &longs;it latus<emph.end type="italics"/> 10<emph type="italics"/>, erit numerus 11500, cubus ex aqua æqualis <lb/>grauitate propo&longs;ito cubo A, quare latus cubicum numeri<emph.end type="italics"/> 11500<emph type="italics"/>, quod<emph.end type="italics"/><pb xlink:href="029/01/022.jpg" pagenum="15"/><emph type="italics"/>e&longs;t<emph.end type="italics"/> 22 17/100. <emph type="italics"/><expan abbr="proximũ">proximum</expan> vero indicabit quæ&longs;itum latus cubi ex aqua.<emph.end type="italics"/></s></p><p type="main"> |
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371 <s>Neque di&longs;&longs;imili ratione inuenietur magnitudo olei, aut ar­<lb/>genti viui, aut cuiu&longs;cumque generis liquidi grauitatem habe­<lb/>tis propo&longs;ito corpori &longs;olido æqualem, &longs;ed quomodo inuenien­<lb/>da &longs;it grauitas argenti viui magnitudinem habentis æqualem <lb/>corpori &longs;olido, docebimus po&longs;t exemplum propo&longs;itionis de­<lb/>cimæ quartæ. </s></p><p type="head"> |
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372 |
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373 <s>PROBLEMA IV. PROPOS. XI.<!-- KEEP S--></s></p><p type="main"> |
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375 <s>Propo&longs;itis duobus corporibus æque grauibus, vno &longs;o<lb/>lido, altero liquido, data liquidi corporis magnitu­<lb/>dine, magnitudinem &longs;olidi inuenire. </s></p><p type="main"> |
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377 <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="id.029.01.022.1.jpg" xlink:href="029/01/022/1.jpg"/><lb/>data magnitudo F, & oporteat &longs;olidi <lb/>A, magnitudinem inuenire. </s> |
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379 <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="quartũ">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 faciendum. </s></p><p type="margin"> |
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381 <s><margin.target id="note24"/>8.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="margin"> |
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383 <s><margin.target id="note25"/>7.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main"> |
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385 <s>Qvod &longs;i propo&longs;ita duo corpora æque grauia A, B, fue<lb/>rint regularia vtpote &longs;phærica, fuerit autem liqui­<pb xlink:href="029/01/023.jpg" 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></p><p type="main"> |
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387 <s>Accepto vt &longs;upra corpore &longs;olido D, & liquidi 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/>&longs;upra o&longs;tendetur, 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*">triplicatam*</expan> rationem habet 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 cubo 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></p><p type="margin"> |
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389 <s><margin.target id="note26"/>18.12. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin"> |
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391 <s><margin.target id="note27"/>33.11. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="head"> |
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393 <s>Exemplum.<!-- KEEP S--></s></p><p type="main"> |
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395 <s>Qvidam proponit aliquod corpus liquidum notæ <lb/>magnitudinis, & vult inuenire quanta erit magni­<lb/>tudo alicuius &longs;olidi grauitatem habentis propo&longs;i­<lb/>to corpori liquido æqualem. </s></p><p type="main"> |
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397 <s><emph type="italics"/>Sit propo&longs;itum aliquod corpus aqueum B, cuius magnitudo &longs;it 115, <lb/>& oporteat inuenire quanta erit magnitudo plumbi grauitatem ha­<lb/>bentis æqualem propo&longs;itæ aquæ B, accipiatur aliquod corpus <expan abbr="plumbeũ">plumbeum</expan> <lb/>D, cuius grauitas &longs;it verbi gratia<emph.end type="italics"/> 23<emph type="italics"/>, deinde aquæ magnitudinem ba­<lb/>bentis æqualem plumbo D, inueniatur grauitas quæ &longs;it<emph.end type="italics"/> 2. <emph type="italics"/>id autem do­<lb/>cuit propo&longs;itionis octauæ exemplum, & fiat vt<emph.end type="italics"/> 23<emph type="italics"/>, ad<emph.end type="italics"/> 2<emph type="italics"/>, ita 115, ad <lb/>alium numerum qui &longs;it<emph.end type="italics"/> 10<emph type="italics"/>, is igitur numerus indicabit quanta erit <lb/>magnitudo plumbi grauitatem habentis æqualem propo&longs;itæ aquæ B.<emph.end type="italics"/></s></p><p type="main"> |
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399 <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></p><p type="main"> |
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401 <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 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<lb/>rum qui &longs;it<emph.end type="italics"/> 86 22/23<31> <emph type="italics"/>is igitur numerus erit cubus diametris &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></p><p type="main"> |
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403 <s><emph type="italics"/>Similiter &longs;i propo&longs;itum corpus aqueum B, fuerit cubicum, vel alicu­<lb/>ius alterius formæ regularis, eadem ratione vtemur ad inueniendum <lb/>latus cubi ex plumbo, grauitatem habentis æqualem 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/><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></p><p type="main"> |
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405 <s>Neque di&longs;&longs;imili ratione inuenienda erit magnitudo auri, <lb/>argenti, ceræ, aut cuiu&longs;cunque &longs;olidi, grauitatem habentis <lb/>propo&longs;ito corpori liquido æqualem. </s></p><p type="head"> |
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407 <s>PROBLEMA V. PROPOS. XII.<!-- KEEP S--></s></p><p type="main"> |
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409 <s>Propo&longs;itis duobus &longs;olidis corporibus magnitudine <lb/>æqualibus, data grauitate vnius, grauitatem al­<lb/>terius inuenire. </s></p><p type="main"> |
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411 <s>SINT propo&longs;ita duo corpora &longs;o­<lb/>lida magnitudine æqualia A, B, &longs;it au­<lb/><figure id="id.029.01.024.1.jpg" xlink:href="029/01/024/1.jpg"/><lb/>tem vnius, vtpote ip&longs;ius A, data gra­<lb/>uitas C, & oporteat inuenire grauita­<lb/>tem ip&longs;ius B. </s> |
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413 <s>Accipiatur aliquod &longs;oli­<lb/>dum corpus D, eiu&longs;dem generis cum <lb/>corpore &longs;olido A, cui æquale grauita­<lb/>te accipiatur alterum E, eiu&longs;dem ge­<lb/>neris cum corpore B, deinde liquidi <lb/>magnitudine æqualis corpori D,<arrow.to.target n="note28"/> *in­<lb/>ueniatur grauitas, quæ &longs;it G, item li­<lb/>quidi eiu&longs;dem generis, æqualis ma­<lb/>gnitudine corpori E,* inueniatur grauitas, quæ &longs;it H, & fiat vt H, ad<arrow.to.target n="note29"/><lb/>G, ita C, ad aliam grauitatem, quæ &longs;it F. <!-- KEEP S--></s> |
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415 <s>Quoniam igitur &longs;unt qua­<lb/>tuor corpora A, B, D, E, quorum A, B, primum videlicet, & &longs;ecundum <lb/>&longs;unt æqualia magnitudine, tertium vero D, & E, quartum æque gra­<lb/>uia, & &longs;unt <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> generis &longs;olida A, D, itidem &longs;olida B, E, *erit vt<arrow.to.target n="note30"/> gra­<lb/>uitas C, ad &longs;olidi B, grauitatem, ita grauitas H, ad grauitatem G, &longs;ed <pb xlink:href="029/01/025.jpg" pagenum="18"/>vt grauitas H, ad <expan abbr="grauitat&etilde;">grauitatem</expan> G, ita e&longs;t grauitas C, ad F, grauitatem; er­<lb/>go grauitas F, æqualis erit grauitati &longs;olidi B, inuenta igitur e&longs;t cor­<lb/>poris &longs;olidi B, grauitas F, quod facere oportebat. </s></p><p type="margin"> |
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417 <s><margin.target id="note28"/>8.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="margin"> |
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419 <s><margin.target id="note29"/>5.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="margin"> |
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421 <s><margin.target id="note30"/>6.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main"> |
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423 <s>Hoc Problema magni momenti e&longs;t, pleri&longs;que artificibus <lb/>maximo v&longs;ui e&longs;&longs;e pote&longs;t. </s> |
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425 <s>in arte fu&longs;oria propo&longs;ito operis modu<lb/>lo, ex illius grauitate, facile metalli ad opus faciendum, gra­<lb/>uitatem inueniet, &longs;i enim hoc ignoret artifex, periculum e&longs;t, ne <lb/>metallum, aut deficiat, vel &longs;i multum e&longs;t, ob nimiam graui­<lb/>tatem difficile tractetur. </s></p><p type="main"> |
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427 <s>Neque tormenti bellici magi&longs;tro inutile erit, is enim cogni­<lb/>ta grauitate alicuius globi, exempli gratia ex plumbo, &longs;tatim <lb/>alterius globi eiu&longs;dem magnitudinis, vel &longs;it ex lapide, vel ex <lb/>fcrro, vel ex quacunque alia materia, grauitatem inueniet. </s></p><p type="head"> |
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429 <s>Exemplum.<!-- KEEP S--></s></p><p type="main"> |
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431 <s>Qvidam proponit aliquod corpus &longs;olidum notæ <lb/>grauitatis, & vult &longs;cire quanta erit grauitas alicu­<lb/>ius &longs;olidi, alterius generis, magnitudinem habentis pro­<lb/>po&longs;ito corpori &longs;olido æqualem. </s></p><p type="main"> |
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433 <s><emph type="italics"/>Sit propo&longs;itum aliquod corpus plumbeum A, cuius grauitas &longs;it<emph.end type="italics"/><lb/>1150<emph type="italics"/>, & oporteat inuenire quanta erit grauitas stanni <expan abbr="magnitudin&etilde;">magnitudinem</expan> <lb/>habentis æqualem propo&longs;ito plumbo A. <!-- KEEP S--></s> |
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435 <s>Accipiantur duo corpora æque <lb/>grauia, D, plumbeum, E, stanneum, deinde duarum quantitatum, <lb/>aquæ, quarum una &longs;it æqualis magnitudine plumbo D, altera &longs;tanno <lb/>E, <expan abbr="inueniãtur">inueniantur</expan> grauitates, quæ &longs;int, primæ videlicet quantitatis aquæ<emph.end type="italics"/><lb/>74, <emph type="italics"/>&longs;ecundæ vero<emph.end type="italics"/> 115<emph type="italics"/>, & fiat vt<emph.end type="italics"/> 115<emph type="italics"/>, ad<emph.end type="italics"/> 74<emph type="italics"/>, ita<emph.end type="italics"/> 1150<emph type="italics"/>, ad alium nume­<lb/>rum, qui &longs;it<emph.end type="italics"/> 740<emph type="italics"/>, is igitur numerus indicabit grauitatem stanni, ma­<lb/>gnitudinem habentis propo&longs;ito plumbo A.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> |
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437 <s>Etiam &longs;i non accipiantur duo corpora, plumbeum videlicet <lb/>& &longs;tanneum, æque grauia, &longs;ed grauitate quacunque, grauitas <lb/>&longs;tanni magnitudinem habentis æqualem propo&longs;ito plumbo <lb/>D, inuenietur &longs;ic. </s></p><p type="main"> |
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439 <s><emph type="italics"/>Accipiantur duo corpora D, plumbeum, E, &longs;tanneum grauitate <lb/>quacunque, &longs;it vcrbi gratia plumbi D, grauitas<emph.end type="italics"/> 23<emph type="italics"/>, &longs;tanni vero E, <lb/>grauitas<emph.end type="italics"/> 37<emph type="italics"/>, deinde duarum quantitatum aquæ, quarum vna &longs;it ma­<lb/>gnitudine æqualis plumbo D, altera stanno E, inueniantur graui­<lb/>tates, quæ &longs;int, primæ videlicet quantitatis<emph.end type="italics"/> 2<emph type="italics"/>, &longs;ecundæ vero<emph.end type="italics"/> 5<emph type="italics"/>, & fiat,<emph.end type="italics"/><pb xlink:href="029/01/026.jpg" pagenum="19"/><emph type="italics"/>ut 23, ad 2, ita 37, ad<emph.end type="italics"/> 3 5/21. <emph type="italics"/>grauitas igitur aquæ, magnitudinem <lb/>habentis æqualem plumbo, cuius grauitas est 37, erit<emph.end type="italics"/> 3 5/21. </s></p><p type="main"> |
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441 <s><emph type="italics"/>Et quoniam aquæ, magnitudinem habentis æqualem stanno E, cu­<lb/>ius grauitas est 37, e&longs;t grauitas<emph.end type="italics"/> 5, <emph type="italics"/>erunt grauitates duarum quan­<lb/>titatum aquæ<emph.end type="italics"/> 3 5/21, & 5, <emph type="italics"/>quarum quantitatum prima e&longs;t æqualis <lb/>magnitudine corpori plumbeo, &longs;ecunda stanneo, que &longs;unt æque gra­<lb/>uia, vtriu&longs;que enim grauitas est<emph.end type="italics"/> 37. <emph type="italics"/>Fiat igitur vt<emph.end type="italics"/> 5, <emph type="italics"/>ad<emph.end type="italics"/> 3 5/21, <emph type="italics"/>it a<emph.end type="italics"/><lb/>1150, <emph type="italics"/>ad alium numerum, qui &longs;it<emph.end type="italics"/> 740, <emph type="italics"/>tanta igitur erit grauitas <lb/>&longs;tanni, magnitudinem habentis æqualem propo&longs;ito plumbo A, quanta <lb/>etiam inueniebatur & &longs;upra.<emph.end type="italics"/></s></p><p type="main"> |
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443 <s>Quod &longs;i propo&longs;itum &longs;it cereum corpus aliquod, aut cuiu&longs;­<lb/>cunque generis &longs;olidi, &longs;iue leuioris quam aqua, &longs;iue grauio­<lb/>ris, & oporteat inuenire grauitatem alicuius &longs;olidi alterius <lb/>generis, magnitudine æqualis propo&longs;ito corpori &longs;olido. </s> |
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445 <s>Ea­<lb/>dem ratione qua &longs;upra inuenietur quæ&longs;ita &longs;olidi grauitas, &longs;ed <lb/>hoc &longs;olum animaduertendum e&longs;t, quod non eadem ratione <lb/>inuenitur grauitas aquæ, magnitudinem habentis æqualem <lb/>propo&longs;ito cuiu&longs;cunque generis &longs;olido, alia enim tenenda e&longs;t <lb/>ratio ad inueniendam grauitatem prædictæ aquæ, quando <lb/>propo&longs;itum &longs;olidum &longs;it grauius quam aqua, alia vero quando <lb/>leuius, &longs;ed &longs;iue &longs;it leuius, &longs;iue grauius, de inuentione huiu&longs;­<lb/>modi grauitatis, in exemplo propo&longs;itionis octauæ, &longs;atis e&longs;t ex­<lb/>plicatum. </s></p><p type="head"> |
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447 <s>PROBLEMA VI. PROPOS. XIII.<!-- KEEP S--></s></p><p type="main"> |
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449 <s>Propo&longs;itis duobus &longs;olidis corporibus æque graui­<lb/>bus, data magnitudine vnius, magnitudinem alte­<lb/>rius inuenire. </s></p><p type="main"> |
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451 <s>SINT propo&longs;ita duo corpora &longs;olida æque grauia A, B, &longs;it au­<lb/>tem vnius, vtpote ip&longs;ius A, data magnitudo C, & oporteat inueni­<lb/>re magnitudinem ip&longs;ius B, Accipiatur aliquod &longs;olidum corpus D, <lb/>eiu&longs;dem generis cum &longs;olido A, & &longs;it eius grauitas G, deinde &longs;olidi <lb/>corporis quod &longs;it E, eiu&longs;dem generis cum &longs;olido B, magnitudine <lb/>æqualis ip&longs;i D, inueniatur grauitas, quæ &longs;it H, hoc autem, Proble­<lb/>ma antecedens docuit, & fiat vt grauitas H, ad grauitatem G, ita <lb/>magnitudo C, ad aliam magnitudinem, quæ &longs;it F. <!-- KEEP S--></s> |
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453 <s>Quoniam igitur <pb xlink:href="029/01/027.jpg" pagenum="20"/>&longs;unt quatuor corpora grauia E, D, B, <lb/>A, quorum E, D, primum videlicet, <lb/><figure id="id.029.01.027.1.jpg" xlink:href="029/01/027/1.jpg"/><lb/>& &longs;ecundum, &longs;unt æqualia magnitu­<lb/>dine, tertium vero B, & quartum A, <lb/>æquegrauia, & &longs;unt eiu&longs;dem generis <lb/>corpora E, B, &longs;imiliter & corpora <lb/><arrow.to.target n="note31"/> D, A, * erit vt grauitas H, ad graui­<lb/>tatem G, ita magnitudo C, ad corpo­<lb/>ris B, magnitudinem, &longs;ed vt grauitas <lb/>H, ad grauitatem G, ita e&longs;t magnitu­<lb/>do C, ad F, magnitudinem, ergo ma­<lb/>gnitudo F, æqualis erit magnitudini <lb/>corporis B. inuenta igitur e&longs;t corporis B, magnitudo F, quod facere <lb/>oportebat. </s></p><p type="margin"> |
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455 <s><margin.target id="note31"/>7.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main"> |
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457 <s>Quod &longs;i propo&longs;ita duo corpora æque grauia A, B, fue<lb/>rint regularia, vtpote &longs;phærica, fuerit autem &longs;phæræ A, <lb/>data diameter C, & oporteat inuenire, quanta erit dia­<lb/>meter &longs;phær&ecedil; B, ita faciendum erit. </s></p><p type="main"> |
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459 <s>Accepto corpore &longs;olido D, & inuenta &longs;olidi corporis E, grauita­<lb/>te, vt &longs;upra dictum e&longs;t, fiat vt grauitas H, ad grauitatem G, ita cu­<lb/>bus ex C, ad alium cubum, cuius latus &longs;it F. <!-- KEEP S--></s> |
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461 <s>Quoniam igitur eadem <lb/>ratione, qua &longs;upra, demon&longs;trabitur, vt grauitas H, ad grauitatem G, <lb/>ita e&longs;&longs;e magnitudinem &longs;phæræ A, ad &longs;phæræ B, magnitudinem, &longs;ed <lb/><arrow.to.target n="note32"/> magnitudo &longs;phæræ A, ad &longs;phæræ B, magnitudinem * triplicatam ra­<lb/>tionem habet eius, quam C, diameter &longs;phæræ A, ad diametrum &longs;phæ­<lb/><arrow.to.target n="note33"/>ræ B. </s> |
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463 <s>Similiter & cubus ex C, ad cubum, ex diametro &longs;phæræ B, *tri­<lb/>plicatam rationem habet eius, quam C, ad &longs;phæræ B, diametrum; er­<lb/>go vt grauitas H, ad grauitatem G, ita erit cubus ex C, ad cubum ex <lb/>diametro &longs;phæræ B, &longs;ed vt grauitas H, ad grauitatem G, ita e&longs;t cubus <lb/>ex C, ad cubum ex F; ergo cubus ex F, æqualis erit cubo diametri <lb/>&longs;phæræ B; quare & latus F, æquabitur &longs;phæræ B, diametro. </s> |
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465 <s>inuenta <lb/>igitur e&longs;t quantitas diametri &longs;phæræ B, quod facere oportebat. </s></p><p type="margin"> |
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467 <s><margin.target id="note32"/>18.12. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin"> |
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469 <s><margin.target id="note33"/>33.11. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> |
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471 <s>Neque hoc Problema inutile erit tormenti bellici magi&longs;tro, <lb/>is enim cognita diametro alicuius globi, exempli gratia, ex <lb/>plumbo, &longs;tatim alterius globi eandem habentis grauitatem, <lb/>diametrum inueniet, &longs;it globus ille, vel ex lapide, vel ex fer­<lb/>ro, vel ex quocunque alio &longs;olidorum genere. </s></p><pb xlink:href="029/01/028.jpg" pagenum="21"/><p type="head"> |
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473 <s>Exemplum.<!-- KEEP S--></s></p><p type="main"> |
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475 <s>Qvidam proponit aliquod corpus &longs;olidum notæ <lb/>magnitudinis, & vult inuenire, quanta erit magni<lb/>tudo alicuius &longs;olidi alterius generis, grauitatem <lb/>habentis propo&longs;ito corpori &longs;olido æqualem. </s></p><p type="main"> |
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477 <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. <!-- KEEP S--></s> |
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479 <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 oporteat, 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"/><!-- KEEP S--></s></p><p type="main"> |
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481 <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></p><p type="main"> |
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483 <s><emph type="italics"/>Accipiatur vt diximus aliquod corpus plumbeum D, cuius graui­<lb/>tas &longs;it 115, & &longs;tanni, magnitudinem habentis æqualem plumbo D, in­<lb/>ueniatur grauitas, quæ &longs;it<emph.end type="italics"/> 74. <emph type="italics"/>& fiat vt 74, ad<emph.end type="italics"/> 115, <emph type="italics"/>ita cubus ex 10, <lb/>qui e&longs;t 1000, ad alium numerum qui &longs;it<emph.end type="italics"/> 1554 2/17, <emph type="italics"/>is igitur numerus <lb/>erit cubus diametri &longs;phæræ ex stanno, grauitatem habentis æqualem <lb/>propo&longs;itæ ex plumbo &longs;phæræ A, quare eius latus cubicum, quod est<emph.end type="italics"/><lb/>11 52/100, <emph type="italics"/>vero proximum, indicabit ip&longs;am diametrum.<emph.end type="italics"/></s></p><p type="main"> |
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485 <s><emph type="italics"/>Similiter &longs;i propo&longs;itum corpus plumbeum A, fuerit cubicum, vel <lb/>alicuius alterius formæ regularis, eadem ratione inuenietur latus cu<lb/>bi ex stanno, grauitatem habentis æqualem propo&longs;ito plumbeo cu­<lb/>bo A, &longs;i enim ip&longs;ius cubi plumbei A, datum &longs;it latus 10, erit numerus<emph.end type="italics"/><lb/>1554 2/17<31> <emph type="italics"/>cubus ex &longs;tanno æqualis grauitate propo&longs;ito plumbeo Cu<lb/>bo A, quare latus. </s> |
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487 <s>cubitum numeri<emph.end type="italics"/> 1554 2/17<31> <emph type="italics"/>quod est<emph.end type="italics"/> 11 52/100<31> <emph type="italics"/>pro<lb/>ximum vero; indicabit quæ &longs;itum latus.<emph.end type="italics"/></s></p><p type="main"> |
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489 <s>Neque di&longs;&longs;imili ratione inuenienda erit magnitudo auri, <lb/>argenti, cæræ, aut cuiu&longs;cumque &longs;olidi, grauitatem habentis <lb/>propo&longs;ito corpori &longs;olido æqualem. </s></p><pb xlink:href="029/01/029.jpg" pagenum="22"/><p type="head"> |
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491 <s>PROBLEMA VII. PROPOS. XIV.<!-- KEEP S--></s></p><p type="main"> |
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493 <s>Propo&longs;itis duobus liquidis corporibus magnitudine <lb/>&ecedil;qualibus, data grauitate vnius, grauitatem alte­<lb/>rius inuenire. </s></p><figure id="id.029.01.029.1.jpg" xlink:href="029/01/029/1.jpg"/><p type="main"> |
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495 <s>SINT propo&longs;ita duo cor­<lb/>ra liquida, magnitudine æqualia <lb/>A, B, &longs;it autem vnius, vtpote li­<lb/>quidi A, data grauitas G, & <lb/>oporteat alterius liquidi B, gra­<lb/>uitatem inuenire. </s> |
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497 <s>Accipiatur <lb/>aliquod corpus &longs;olidum C, & li­<lb/>quidi, quod &longs;it H, eiu&longs;dem ge­<lb/>neris cum liquido A, magnitu­<lb/><arrow.to.target n="note34"/>dine æqualis &longs;olido C, *inuenia­<lb/>tur grauitas, quæ &longs;it D, &longs;imiliter <lb/>& liquidi, quod &longs;it I, eiu&longs;dem <lb/>generis cum liquido B, magni­<lb/>tudine æqualis eidem &longs;olido C, <lb/><arrow.to.target n="note35"/> * inueniatur grauitas, quæ &longs;it E, <lb/>& fiat vt D, ad E, ita G, ad aliam grauitatem, quæ &longs;it F. <!-- KEEP S--></s> |
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499 <s>Quoniam <lb/>igitur e&longs;t vt A, ad B, ita H, ad I, æquale videlicet ad æquale, erit per­<lb/>mutando vt A, ad H, ita B, ad I, & quoniam eiu&longs;dem &longs;unt generis cor­<lb/><arrow.to.target n="note36"/>pora A, H, &longs;imiliter & corpora B, I, * erit vt grauitas G, ad grauita­<lb/>tem D, ita liquidi B, grauitas, ad grauitatem E, & permutando vt <lb/>grauitas G, ad grauitatem liquidi B, ita D, grauitas, ad grauitatem <lb/>E, &longs;ed vt grauitas D, ad grauitatem E, ita e&longs;t grauitas G, ad graui­<lb/>tatem F; ergo grauitas F, æqualis erit grauitati liquidi B. inuenta <lb/>igitur e&longs;t liquidi corporis B, grauitas F, quod facere oportebat. </s></p><p type="margin"> |
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501 <s><margin.target id="note34"/>2.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="margin"> |
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503 <s><margin.target id="note35"/>3.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="margin"> |
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505 <s><margin.target id="note36"/>4.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="head"> |
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507 <s>Exemplum.<!-- KEEP S--></s></p><p type="main"> |
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509 <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></p><p type="main"> |
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511 <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 xlink:href="029/01/030.jpg" 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 plumbo<lb/>C, inueniatur grauitas, quæ &longs;it 12, vt in exemplo propo&longs;. </s> |
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513 <s>8. <expan abbr="dictũ">dictum</expan> e&longs;t. </s> |
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515 <s><lb/>Similiter & olei, magnitudinem æqualem habentis, eidem plumbo C, <lb/>inueniatur grauitas, quæ &longs;it 11, & fiat, vt<emph.end type="italics"/> 11, <emph type="italics"/>ad<emph.end type="italics"/> 12, <emph type="italics"/>ita 550, ad <expan abbr="aliũ">alium</expan> <lb/>numerum qui &longs;it<emph.end type="italics"/> 600. <emph type="italics"/>is igitur numerus indicabit quanta erit graui­<lb/>tas aquæ, magnitudinem habentis æqualem propo&longs;ito oleo A.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> |
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517 <s><emph type="italics"/>Si vero propo&longs;itum &longs;it aliquod argenti viui corpus A, cuius <lb/>grauitas<emph.end type="italics"/> 95, <emph type="italics"/>& oporteat inuenire, quanta erit grauitas aquæ, magni­<lb/>tudincm habentis æqualem propo&longs;ito argento viuo A. <!-- KEEP S--></s> |
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519 <s>Accipiatur ali­<lb/>quod vas vitreum mundum, & politum, cuius grauitas &longs;it v. <!-- REMOVE S-->g.<emph.end type="italics"/> 91. <lb/><emph type="italics"/><expan abbr="ip&longs;umq;">ip&longs;umque</expan> vas plenum aqua ponderetur in aqua, & habeat grauitatem<emph.end type="italics"/><lb/>55, <emph type="italics"/>quoniam igitur numerus<emph.end type="italics"/> 91, <emph type="italics"/>&longs;uperat <expan abbr="numerũ">numerum</expan><emph.end type="italics"/> 55, <emph type="italics"/>numero<emph.end type="italics"/> 36, <emph type="italics"/>* erit<emph.end type="italics"/><arrow.to.target n="note37"/><lb/><emph type="italics"/>grauitas aquæ, magnitudinem habentis æqualem ip&longs;i va&longs;i, hoc est &longs;oli­<lb/>ditati ip&longs;ius va&longs;is<emph.end type="italics"/> 36, <emph type="italics"/>ponatur deinde in ip&longs;um vas propo&longs;itum argen­<lb/>tum <expan abbr="viuũ">viuum</expan> A, nihil interest, vt vas &longs;it plenum, vel non, & quoniam ar<lb/>genti viui A, grauitas est<emph.end type="italics"/> 95, <emph type="italics"/>& va&longs;is vitrei grauitas<emph.end type="italics"/> 91, <emph type="italics"/>erit argenti <lb/>viui &longs;imul cum ip&longs;o va&longs;e, grauitas<emph.end type="italics"/> 186. <emph type="italics"/>ponderetur itaque ip&longs;um vas, <lb/>&longs;imul cum argento viuo A, in aqua, ita vt aqua repleat va&longs;is partem <lb/>vacuam, & &longs;it va&longs;is grauitas in aqua &longs;imul cum argento viuo<emph.end type="italics"/> 143, <lb/><emph type="italics"/>quoniam igitur numerus<emph.end type="italics"/> 186, <emph type="italics"/>&longs;uperat numerum<emph.end type="italics"/> 143, <emph type="italics"/>numero<emph.end type="italics"/> 43, <emph type="italics"/>* erit<emph.end type="italics"/><arrow.to.target n="note38"/><lb/><emph type="italics"/>grauitas aquæ, <expan abbr="magnitudin&etilde;">magnitudinem</expan> <expan abbr="bib&etilde;tis">bibentis</expan> æqualem argento viuo, &longs;imul <expan abbr="cũ">cum</expan> <lb/>va&longs;e<emph.end type="italics"/> 43, <emph type="italics"/>&longs;ed grauitas aquæ habentis magnitudinem æqualem va&longs;i est<emph.end type="italics"/><lb/>36, <emph type="italics"/>ergo reliquum quod est<emph.end type="italics"/> 7, <emph type="italics"/>erit grauitas aquæ, magnitudinem ba­<lb/>bentis æqualem propo&longs;ito argento viuo A.<emph.end type="italics"/><!-- KEEP S--></s> |
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521 </p><p type="margin"> |
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523 <s><margin.target id="note37"/>5.<emph type="italics"/>huiu<emph.end type="italics"/></s></p><p type="margin"> |
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525 <s><margin.target id="note38"/>5.<emph type="italics"/>huiu<emph.end type="italics"/></s></p><p type="main"> |
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527 <s>Sed &longs;i propo&longs;itum fuerit aliquod magnum argenti viui cor <lb/>pus, ita vt difficile po&longs;&longs;it ponderari in aqua, hac via inuenietur <lb/>aquæ quæ&longs;ita grauitas. </s></p><p type="main"> |
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529 <s><emph type="italics"/>Sit propo&longs;itum aliquod magnum argenti viui corpus A, cuius gra­<lb/>uitas<emph.end type="italics"/> 5700. <emph type="italics"/>& oporteat facere, quod imperatum e&longs;t. </s> |
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531 <s>Accipiatur ali­<lb/>quod paruum argenti viui corpus C, cuius grauitas &longs;it<emph.end type="italics"/> 95, <emph type="italics"/>& aquæ, <lb/>magnitudinem habentis æqualem argento viuo C, inueniatur graui­<lb/>tas, eo modo quo dictum est, quæ &longs;it<emph.end type="italics"/> 7, <emph type="italics"/>& fiat vt<emph.end type="italics"/> 95, <emph type="italics"/>ad<emph.end type="italics"/> 7, <emph type="italics"/>ita<emph.end type="italics"/> 5700, <emph type="italics"/>ad <lb/>alium numerum, qui &longs;it<emph.end type="italics"/> 420, <emph type="italics"/>is igitur numerus indicabit quanta erit <lb/>grauitas aquæ, magnitudinem habentis æqualem propo&longs;ito argento <lb/>viuo A.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> |
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533 <s><emph type="italics"/>Contra, &longs;it propo&longs;itum aliquod corpus aqueum A, cuius grauitas<emph.end type="italics"/><lb/>420, <emph type="italics"/>& oporteat inuenire quanta erit grauitas argenti viui, magni­<lb/>tudine æqualis propo&longs;itæ aquæ A. facto, vt &longs;upra, & inuenta graui­<lb/>tate<emph.end type="italics"/> 7, <emph type="italics"/>aquæ &longs;cilicet magnitudinem habentis æqualem <expan abbr="arg&etilde;to">argento</expan> viuo C,<emph.end type="italics"/><pb xlink:href="029/01/031.jpg" pagenum="24"/><emph type="italics"/>fiat vt<emph.end type="italics"/> 7, <emph type="italics"/>ad<emph.end type="italics"/> 95, <emph type="italics"/>ita<emph.end type="italics"/> 420, <emph type="italics"/>ad alium numerum, qui &longs;it<emph.end type="italics"/> 5700, <emph type="italics"/>is igi­<lb/>tur indicabit quanta erit grauitas argenti viui, magnitudine æqua­<lb/>lis propo&longs;itæ aquæ A.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> |
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535 <s>Inueniemus etiam aliter, & expeditius grauitatem <lb/>aquæ, magnitudinem habentis æqualem propo&longs;ito ar­<lb/>gento viuo A. <!-- KEEP S--></s></p><p type="main"> |
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537 <s><emph type="italics"/>Accipiatur enim aliquod corpus aureum, cui &longs;uperinducatur ce­<lb/>rea tunica tenui&longs;&longs;ima, ne fiat argento viuo leuius, neue ab <expan abbr="eod&etilde;">eodem</expan> di&longs;&longs;ol­<lb/>uatur, deinde aquæ, magnitudinem habentis æqualem ip&longs;i corpori an <lb/>reo inueniatur grauitas, vt dictum est in propo&longs;.<emph.end type="italics"/> 8. <emph type="italics"/>exemplo, quæ &longs;it<emph.end type="italics"/> 7, <lb/><emph type="italics"/>&longs;imiliter & argenti viui, vt aquæ, magnitudinem habentis æqualem <lb/>eidem corpori aureo, inueniatur grauitas, quæ &longs;it<emph.end type="italics"/> 95, <emph type="italics"/>& fiat vt<emph.end type="italics"/> 95, <emph type="italics"/>ad<emph.end type="italics"/><lb/>7, <emph type="italics"/>ita<emph.end type="italics"/> 5700, <emph type="italics"/>ad<emph.end type="italics"/> 420, <emph type="italics"/>grauitas igitur aquæ, magnitudinem habentis æ­<lb/>qualem argento viuo A, erit<emph.end type="italics"/> 420. </s></p><p type="main"> |
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539 <s><emph type="italics"/>Contra. <!-- KEEP S--></s> |
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541 <s>&longs;it propo&longs;itum aliquod corpus aqueum, cuius grauitas<emph.end type="italics"/><lb/>420, <emph type="italics"/>& oporteat inuenire, quanta erit grauitas argenti viui, magni­<lb/>tudine æqualis propo&longs;itæ aquæ A. <!-- KEEP S--></s> |
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543 <s>Superinducta corpori aureo cerea <lb/>tunica, vt &longs;upra, & inuentis grauitatibus<emph.end type="italics"/> 7, & 95, <emph type="italics"/>aquæ nimirum, & <lb/>argenti viut, magnitudine æqualium prædicto aureo corpori, fiat vt<emph.end type="italics"/><lb/>7, <emph type="italics"/>ad 95, ita<emph.end type="italics"/> 430, <emph type="italics"/>ad<emph.end type="italics"/> 5700 <emph type="italics"/>grauitas igitur argenti viui, magnitudine<lb/>æqualis propo&longs;ito corpori aqueo A, erit<emph.end type="italics"/> 5700. </s></p><p type="main"> |
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545 <s>Qua ratione inuenienda &longs;it grauitas argenti viui, ma­<lb/>gnitudinem habentis propo&longs;ito cuicunque corpori &longs;oli­<lb/>do æqualem. </s></p><p type="main"> |
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547 <s><emph type="italics"/>Sit propo&longs;itum aliquod corpus &longs;olidum, vtpote plumbeum A, cuius <lb/>grauitas<emph.end type="italics"/> 161, <emph type="italics"/>& oporteat inuenire quanta erit grauitas argenti viui <lb/>magnitudine æqualis propo&longs;ito plumbo A. inueniatur grauitas aquæ <lb/>magnitudinem habentis æqualem plumbo A, vt in exemplo propo&longs;i­<lb/>tionis<emph.end type="italics"/> 8, <emph type="italics"/>dictum e&longs;t, quæ &longs;it<emph.end type="italics"/> 14, <emph type="italics"/>& inuenta grauitate argenti viui, ma­<lb/>gnitudine æqualis ip&longs;i aquæ, ea erit de qua quæritur, &longs;it enim inuen<lb/>ta argenti viui grauitas<emph.end type="italics"/> 190. <emph type="italics"/><expan abbr="Quoniã">Quoniam</expan> igitur argentum viuum, cuius <lb/>grauitas est<emph.end type="italics"/> 190, <emph type="italics"/>magnitudine æquatur aquæ, cuius grauitas e&longs;t<emph.end type="italics"/> 14, <lb/><emph type="italics"/>ip&longs;ique aquæ æquatur magnitudine plumbum A, erit argentum vi­<lb/>uum, cuius grauitas 190, magnitudine propo&longs;ito plumbo A, æquale; <lb/>quare inuenta est grauitas argenti viui, magnitudine æqualis propo­<lb/>&longs;ito plumbo A, quod facere oportebat.<emph.end type="italics"/></s></p><p type="main"> |
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549 <s>Quomodo inuenienda &longs;it grauitas cuiu&longs;cunque cor-<pb xlink:href="029/01/032.jpg" pagenum="25"/>poris &longs;olidi, magnitudinem habentis propo&longs;ito corpori <lb/>ex argento viuo æqualem. </s></p><p type="main"> |
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551 <s><emph type="italics"/>Sit propo&longs;itum aliquod corpus ex argento viuo A, cuius grauitas <lb/>190, & oporteat inuenire quanta erit grauitas plumbi, magnitudine <lb/>æqualis propo&longs;ito argento viuo A. inueniatur grauitas aquæ, magni­<lb/>tudinem habentis æqualem argento viuo A, quæ &longs;it<emph.end type="italics"/> 14, <emph type="italics"/>deinde inuen­<lb/>ta grauitate plumbi, magnitudine æqualis ip&longs;i aquæ, vt in exemplo <lb/>propo&longs;. </s> |
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553 <s>9. dictum est, ea erit de qua quæritur. </s> |
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555 <s>&longs;it enim inuenta plum­<lb/>bigrauitas<emph.end type="italics"/> 161, <emph type="italics"/>quoniam igitur aqua, cuius grauitas est<emph.end type="italics"/> 14, <emph type="italics"/>æqua­<lb/>tur magnitudine plumbo, cuius grauitas est<emph.end type="italics"/> 161, <emph type="italics"/>& æquatur quoque <lb/>argento viuo A, plumbum cuius grauitas e&longs;t<emph.end type="italics"/> 161, <emph type="italics"/>æquabitur magni­<lb/>tudine argento viuo A. quare inuenta, e&longs;t grauitas plumbi, magnitu­<lb/>dine æqualis propo&longs;ito argento viuo A, quod facere oportebat.<emph.end type="italics"/></s></p><p type="head"> |
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557 <s>PROBLEMA VIII. PROPOS. XV.<!-- KEEP S--></s></p><p type="main"> |
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559 <s>Propo&longs;itis duobus liquidis corporibus æquè graui­<lb/>bus, data magnitudine vnius, magnitudinem alte­<lb/>rius inuenire. </s></p><figure id="id.029.01.032.1.jpg" xlink:href="029/01/032/1.jpg"/><p type="main"> |
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561 <s>SINT propo&longs;ita duo cor­<lb/>pora liquida æquè grauia A, <lb/>B, &longs;it autem vnius vt pote li­<lb/>quidi A, data magnitudo G, <lb/>& oporteat inuenire quanta <lb/>erit magnitudo liquidi B. </s> |
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563 <s>Ac­<lb/>cipiatur aliquod &longs;olidum cor <lb/>pus C, & liquidi quod &longs;it H, <lb/>eiu&longs;dem generis cum liquido <lb/>A, magnitudinem habentis <lb/>æqualem &longs;olido C, *<arrow.to.target n="note39"/> inuenia­<lb/>tur grauitas quæ &longs;it D, &longs;imili. </s> |
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565 <s><lb/>ter & liquidi, quod &longs;it I, eiu&longs;dé <lb/>generis cum liquido B, magni <lb/>tudinern habentis æqualem eidem &longs;olido C, * inueniatur grauitas,<arrow.to.target n="note40"/><lb/>quæ &longs;it E, & fiat vt grauitas E, ad grauitatem D, ita magnitudo G, ad <lb/>ad aliam <expan abbr="magnitadin&etilde;">magnitudinem</expan>, quæ &longs;ie F. <!-- KEEP S--></s> |
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567 <s>Quoniam igitur &longs;unt quatuor cor­<lb/>pora grauia I, H, B, A, quorum primum, & &longs;ecundum &longs;unt magnitu­<lb/>dine æqualia, tertium vero, & quartum æque grauia, & &longs;unt eiu&longs;dem <lb/>generis primum videlicet, & tertium, &longs;imiliter eiu&longs;dem generis &longs;e­<pb xlink:href="029/01/033.jpg" pagenum="26"/><arrow.to.target n="note41"/>cundum & quartum, * Erit vt grauitas E, ad grauitatem D, ita ma­<lb/>gnitudo G, ad liquidi B, magnitudinem, &longs;ed vt grauitas E, ad graui­<lb/>tatem D, ita e&longs;t magnitudo G, ad F, magnitudinem; ergo magnitudo <lb/>F, æqualis erit magnitudini liquidi B. inuenta igitur e&longs;t corporis li­<lb/>quidi B, magnitudo F, quod facere oportebat. </s></p><p type="margin"> |
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569 <s><margin.target id="note39"/>8.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="margin"> |
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571 <s><margin.target id="note40"/>8.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="margin"> |
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573 <s><margin.target id="note41"/>7.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main"> |
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575 <s>Quod &longs;i propo&longs;ita duo corpora æque grauia fuerint <lb/>regularia, vtpote &longs;phærica, fuerit autem &longs;ph&ecedil;ræ A, data <lb/>diameter G, & oporteat inuenire, quanta erit diameter <lb/>&longs;phæræ B, ita faciendum erit. </s></p><figure id="id.029.01.033.1.jpg" xlink:href="029/01/033/1.jpg"/><p type="main"> |
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577 <s>ACCEPTO aliquo cor <lb/>pore &longs;olido C, & inuentis <lb/>grauitatibus D, E, <expan abbr="liquidorũ">liquidorum</expan> <lb/>H, I, vt &longs;upra, fiat vt grauitas <lb/>E, ad grauitatem D, ita cu­<lb/>bus ex G, ad alium cubum, <lb/>cuius latus &longs;it F. <!-- KEEP S--></s> |
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578 |
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579 <s>Quoniam <lb/>igitur eadem ratione, qua <lb/>&longs;upra o&longs;tendetur, vt grauitas <lb/>E, ad grauitatem D, ita e&longs;&longs;e <lb/>magnitudinem &longs;phæræ A, ad <lb/>&longs;phæræ B, magnitudinem, &longs;ed <lb/>magnitudo &longs;phæræ A, ad <lb/><arrow.to.target n="note42"/> &longs;phæræ B, magnitudinem; * <lb/><expan abbr="triplicatã">triplicatam</expan> rationem habet eius, quam G, diameter &longs;phæræ A, ad dia­<lb/>metrum &longs;phæræ B, &longs;imiliter & cubus ex G, ad cubum diametri &longs;phæ­<lb/><arrow.to.target n="note43"/>ræ B, * triplicatam rationem habet eius, quam G, ad &longs;phæræ B, dia­<lb/>metrum; ergo vt grauitas E, ad grauitatem D, ita erit cubus ex G, ad <lb/>cubum diametri &longs;phæræ B, &longs;ed vt grauitas D, ita grauitatem D, ita <lb/>e&longs;t cubus ex G, ad cubum ex F; ergo cubus ex F, æqualis erit cubo <lb/>diametri &longs;phæræ B; quare & latus F, æquabitur diametro ip&longs;ius &longs;phæ<lb/>ræ B. inuenta igitur e&longs;t quantitas diametri &longs;phæræ B, quod facere <lb/>oportebat. </s></p><p type="margin"> |
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581 <s><margin.target id="note42"/>18.12. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin"> |
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583 <s><margin.target id="note43"/>33.11. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="head"> |
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585 <s>Exemplum.<!-- KEEP S--></s></p><p type="main"> |
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587 <s>Qvidam proponit aliquod corpus liquidum notæ <lb/>magnitudinis, & vult inuenire, quanta erit ma­<lb/>gnitudo liquidi alterius generis, grauitatem </s></p><pb xlink:href="029/01/034.jpg" pagenum="27"/><p type="head"> |
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589 <s>habentis propo&longs;ito corpori liquido æqualem.</s></p><p type="main"> |
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591 <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 erit 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="numerũ">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"/><!-- KEEP S--></s></p><p type="main"> |
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593 <s><emph type="italics"/>Similiter &longs;i <expan abbr="propo&longs;itũ">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. <!-- KEEP S--></s> |
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595 <s>Accipiatur aliquod <lb/>corpus &longs;olidum C &longs;i aureum, &longs;uper inducatur ei cerea tunica propter <lb/>iam <expan abbr="dictã">dictam</expan> rationem, deinde argenti viui, magnitudine æqualis ip&longs;i C, <lb/>inueniatur grauitas quæ &longs;it<emph.end type="italics"/> 95, <emph type="italics"/>&longs;imiliter & aquæ <expan abbr="magnitudin&etilde;">magnitudinem</expan> ha­<lb/>bentis æqualem eidem C, inueniatur grauitas quæ &longs;it<emph.end type="italics"/> 7, <emph type="italics"/>& fiat vt<emph.end type="italics"/> 95, <lb/><emph type="italics"/>ad<emph.end type="italics"/> 7, <emph type="italics"/>ita<emph.end type="italics"/> 5700, <emph type="italics"/>ad alium numerum, qui &longs;it<emph.end type="italics"/> 420, <emph type="italics"/>is igitur numerus in­<lb/>dicabit quanta erit magnitudo argenti viui grauitatem habentis <lb/>æqualem propo&longs;itæ aquæ A.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> |
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597 <s>Quod &longs;i propo&longs;itum corpus aqueum A, &longs;it &longs;phæricum, culus <lb/>&longs;phæræ diameter &longs;it 10, & oporteat inuenire quanta erit dia­<lb/>meter &longs;phæræ ex argento viuo, grauitatem habentis æqualem <lb/>propo&longs;itæ &longs;phæræ A, ità faciendum erit. </s></p><p type="main"> |
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599 <s><emph type="italics"/>Accepto vt diximus aliquo corpore &longs;olido C, & inuentis grauita­<lb/>tibus <expan abbr="liquidorũ">liquidorum</expan> aquæ &longs;cilicet & argent viui magnitudinem æqua­<lb/>lem habentium corpori C, quæ &longs;int<emph.end type="italics"/> 14, <emph type="italics"/>grauitas aquæ, &<emph.end type="italics"/> 190, <emph type="italics"/>graui­<lb/>tas argenti viui, fiat vt<emph.end type="italics"/> 190, <emph type="italics"/>ad<emph.end type="italics"/> 14, <emph type="italics"/>ita cubus ex 10, hoc est ita 1000, <lb/>ad alium numerum, qui &longs;it<emph.end type="italics"/> 73 11/19, <emph type="italics"/>is igitur numerus erit cubus dia­<lb/>metri &longs;phæræ ex argento viuo, grauitatem habentis æqualem propo­<lb/>&longs;itæ ex aqua &longs;phæræ A: quare latus cubicum numeri<emph.end type="italics"/> 73 11/19, <emph type="italics"/>quod e&longs;t<emph.end type="italics"/><lb/>4 19/100 <emph type="italics"/>proxime indicabit ip&longs;am diametrum.<emph.end type="italics"/></s></p><p type="main"> |
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601 <s><emph type="italics"/>Similiter &longs;i propo&longs;itum corpus aqueum A, fuerit cubicum, aut <lb/>alicuius alterius formæ regularis, eadem ratione, qua &longs;upra inuenit­<lb/>tur latus cubi ex argento viuo, grauitate æqualis propo&longs;ito ex aqua <lb/>cubo A, nam &longs;i ip&longs;ius cabi A, <expan abbr="datũ">datum</expan> &longs;it latus<emph.end type="italics"/> 10, <emph type="italics"/>erit numerus<emph.end type="italics"/> 73 11/19, <lb/><emph type="italics"/>cubus ex argento viuo æqualis grauitate propo&longs;ito ex aqua cubo A; <lb/>quare latus cubicum numeri<emph.end type="italics"/> 73 11/19, <emph type="italics"/>quod est<emph.end type="italics"/> 4 19/100 <emph type="italics"/>proxime in­<lb/>dicabit quæ&longs;itum latus cubi, ex argento viuo.<emph.end type="italics"/></s></p><pb xlink:href="029/01/035.jpg" pagenum="28"/><p type="main"> |
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603 <s>Neque di&longs;&longs;imili ratione inuenietur magnitudo reliquo­<lb/>rum omnium liquidorum, grauitate propo&longs;ito corpori cuiu&longs;­<lb/>cumque generis liquidi, æqualium, quare dicta &longs;ufficiant. </s></p><p type="main"> |
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605 <s>Dvm adhuc Opu&longs;culum &longs;ub prælo e&longs;&longs;et, dubitandi an&longs;am, <lb/>ex eo vir docti&longs;simus, cui percurrendum illud tradi­<lb/>deram, arripuit, quod ex grauitate, corporum in aqua <lb/>exi&longs;tentium, non po&longs;&longs;et vera ratio, quam habent diuer&longs;a <lb/>ip&longs;orum corporum genera in grauitate, deprehendi, ni&longs;i <lb/>corpora fuerint &longs;imilia. </s> |
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607 <s>&longs;i enim (aiebat) accipiantur duo <lb/>corpora eiu&longs;dem generis, & grauitatis, quorum vnum &longs;it <lb/>planum, alterum conicam formam habens, & ponderen­<lb/>tur in aqua, ita vt coni vertex deor&longs;um ver&longs;us pondeat, ba&longs;is <lb/>vero ip&longs;ius coni, & latæ corporis plani &longs;uperficies æquidi&longs;tent <lb/>horizonti conus in aqua maiorem habebit grauitatem, cor­<lb/>pore plano, quia corpus planum magis ab aqua &longs;u&longs;tentatur, <lb/>quam conus, & hoc quidem manife&longs;tum e&longs;t, quoniam &longs;i am­<lb/>bo demittantur eodem tempore in aquam, conus citius ad <lb/>imum de&longs;cendet, quam corpus planum. </s> |
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609 <s>Hoc argumentum <lb/>licet primo a&longs;pectu probabile videatur, tamen fal&longs;o concludit. </s> |
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611 <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, ne tanta 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></p><p type="head"> |
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612 |
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613 <s>THEOREMA VIII. PROPOS. XVI.<!-- KEEP S--></s></p><p type="main"> |
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614 |
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615 <s>Corpora eiu&longs;dem generis, & grauitatis grauiora <lb/>quam aqua, et&longs;i di&longs;similia, æqualem in aqua <lb/>grauitatem habent. </s></p><p type="main"> |
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616 |
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617 <s>SINT duo eiu&longs;dem generis, & grauitatis corpora A, B, grauiora <lb/>quam aqua, & &longs;int di&longs;&longs;imilia, dico ip&longs;a corpora æqualem in aqua <lb/>grauitatem habere. </s> |
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618 |
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619 <s>&longs;it enim &longs;i fieri pote&longs;t corpus A, leuius corpore B, <pb xlink:href="029/01/036.jpg" 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="id.029.01.036.1.jpg" xlink:href="029/01/036/1.jpg"/><lb/>æquale, & corpori M, appen<lb/>datur corpus A. <!-- KEEP S--></s> |
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621 <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, cuius &longs;phæræ cen­<lb/>trum idem e&longs;t, quod <expan abbr="centrũ">centrum</expan> <lb/>terræ, hoc autem demon&longs;tratum e&longs;t ab Archimede Prop. 2. lib. |
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623 1. de <lb/>ijs, quæ vehuntur in aqua. </s> |
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625 <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> |
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626 |
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627 <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, leuius 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></p><p type="main"> |
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628 |
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629 <s>Et quoniam eiu&longs;dem generis, & grauitatis ponuntur corpora A, B, <lb/>erunt magnitudine æqualia, & per additionem æqualium æquali­<lb/>bus, corpora M, A, erunt æqualia corporibus L, B, <!-- KEEP S--></s></p><p type="main"> |
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630 |
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631 <s>Quoniam igitur corpora M, A, æqualia &longs;unt corporibus L, B, pars <lb/>autem corporis M, extat ex aquæ &longs;uperficie, & corpora L, B, tota de­<lb/>merguntur, minus loci ocupabunt in aqua corpora M, A, quam cor­<lb/>pora L, B, quare maior erit grauitas corporum M, A, & aquæ conti­<lb/>nentis ip&longs;a corpora, quæ e&longs;t in loco pyramidis CDGF, quam corpo­<lb/>rum L, B, & aquæ ip&longs;a corpora <expan abbr="contin&etilde;tis">continentis</expan> in loco pyramidis DEHG, <pb xlink:href="029/01/037.jpg" pagenum="30"/>magis igitur aquæ pars premetur, quæ e&longs;t &longs;ub &longs;uperficie FG, quam <lb/>ca quæ e&longs;t &longs;ub &longs;uperficie GH; quare expellet partem minus pre&longs;&longs;am, <lb/>(æqualiter enim & continuatæ iacent inter &longs;e&longs;e) & <expan abbr="nõ">non</expan> manebit aqua, <lb/>quod e&longs;t ab&longs;urdum, ponebatur enim manens. </s> |
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632 |
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633 <s>non igitur corpus A, <lb/>leuius e&longs;t in aqua corpore B. eadem ratione o&longs;tendetur neque corpus <lb/>B, leuius e&longs;&longs;e in aqua corpore A, quare con&longs;tat propo&longs;itum. </s></p><p type="head"> |
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635 <s>ALITER.<!-- KEEP S--></s></p><p type="main"> |
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636 |
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637 <s>Sint duo eiu&longs;dem generis, & grauitatis corpora A, B, grauiora <lb/>quam aqua, & &longs;int di&longs;&longs;imilia. </s> |
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638 |
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639 <s>o&longs;tendendum e&longs;t ip&longs;a corpora æqualem <lb/>in aqua grauitatem habere, &longs;it enim corporis A, vel ip&longs;ius B, graui­<lb/>tas CD, aquæ vero magni­<lb/><figure id="id.029.01.037.1.jpg" xlink:href="029/01/037/1.jpg"/><lb/>tudinem habentis æqua­<lb/>lem ip&longs;i A, vel B, &longs;it graui­<lb/>tas C, & accipiatur ali­<lb/>quod corpus L, leuius <expan abbr="quã">quam</expan> <lb/>aqua, cuius grauitas &longs;it <lb/>ip&longs;i C, æqualis, aquæ ve­<lb/>ro, magnitudinem haben­<lb/>tis æqualem corpori L, &longs;it <lb/>grauitas æqualis ip&longs;i CD, <lb/>itaque appen&longs;o corpore <lb/>B, corpori L, corpus ex <lb/>vtri&longs;que con&longs;tans æque graue erit atque aqua, grauitas enim vtro­<lb/>runque corporum B, L, e&longs;t æqualis vtri&longs;que grauitatibus CD, & C, & <lb/>grauitas aquæ, magnitudinem habentis æqualem <expan abbr="vtri&longs;q;">vtri&longs;que</expan> corporibus <lb/>L, B, æqualis e&longs;t ei&longs;dem grauitatibus CD, & C, corpora igitur B, L, <lb/>demi&longs;&longs;a in aquam, neque &longs;ur&longs;um, neque deor&longs;um ferentur, quia cor­<lb/>pus B, grauius quam aqua fertur deor&longs;um tanta vi, quanta à corpo­<lb/>re <emph type="italics"/>L,<emph.end type="italics"/> &longs;ur&longs;um retrahitur. </s></p><p type="main"> |
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640 |
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641 <s>Rur&longs;us accipiatur alterum corpus &longs;olidum M, eiu&longs;dem generis <lb/>cum corpore L, ip&longs;ique &longs;imile, & æquale, & corpore A, appen&longs;o ip&longs;i <lb/>M, & demi&longs;&longs;is ambobus in aquam, eadem ratione qua &longs;upra o&longs;ten­<lb/>detur, corpora A, M, &longs;imul, e&longs;&longs;e æque grauia atque aqua, & corpus <lb/>A, tanta vi deor&longs;um ferri, quanta retrahitur &longs;ur&longs;um à corpore M, <lb/>&longs;ed corpora M, L, æqualem vim habent retrahendi &longs;ur&longs;um, cum <lb/>&longs;int eiu&longs;dem generis, & æqualia, & &longs;imilia, ergo æquali vi retra­<lb/>hentur corpora A, B, ne de&longs;cendant; quare con&longs;tat ip&longs;a corpora A, <lb/>B, æqualem in aqua grauitatem habere quod erat o&longs;tendendum. </s></p><pb xlink:href="029/01/038.jpg" pagenum="31"/><p type="head"> |
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643 <s>THEOREMA IX. PROPOS. XVII.<!-- KEEP S--></s></p><p type="main"> |
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645 <s>Sphære eiu&longs;dem generis inter &longs;e &longs;unt in grauitate, vt <lb/>diametrorum cubi in magnitudine. </s></p><p type="main"> |
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646 |
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647 <s>SINT &longs;phæræ eiu&longs;dem gene­<lb/><figure id="id.029.01.038.1.jpg" xlink:href="029/01/038/1.jpg"/><lb/>ris ABC, DEF, quarum diame­<lb/>tri BC, EF. dico vt &longs;ph&ecedil;ra ABC, <lb/>&longs;e habet in grauitate, ad &longs;phæram <lb/>DEF, ita &longs;e habere in maguitudi­<lb/>ne cubum ex BC, ad cubum ex <lb/>EF, &longs;it enim &longs;phæræ ABC, graui­<lb/>tas G, & &longs;phæræ DEF, grauitas H, <lb/>quoniam igitur eiu&longs;dem generis <lb/>ponuntur &longs;phæræ ABC, DEF, <lb/>erit^{*} vt &longs;phæra ABC, ad &longs;phæram DEF, ita grauitas G, ad H,<arrow.to.target n="note44"/> graui­<lb/>tatem, &longs;ed &longs;phæra ABC, ad &longs;phæram DEF,^{*} triplicatam habet<arrow.to.target n="note45"/> ra­<lb/>tionem eius, quam diameter BC, ad EF, diametrum, ergo & graui­<lb/>tas G, ad grauitatem H, triplicatam habebit rationem eius, quam <lb/>habet BC, ad EF, &longs;ed & cubus ex BC, ad cubum ex EF,^{*} triplicatam<arrow.to.target n="note46"/><lb/>rationem habet eius, quam BC, ad EF, ergo vt grauitas G, ad graui­<lb/>tatem H, ita erit cubus ex BC, ad cubum ex EF. &longs;phæræ igitur eiu&longs;­<lb/>dem generis inter &longs;e &longs;unt in grauitate, vt diametrorum cubi in ma­<lb/>gnitudine, quod erat demon&longs;trandum. </s></p><pb xlink:href="029/01/039.jpg" pagenum="32"/><p type="margin"> |
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649 <s><margin.target id="note44"/>2. & 3. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s></p><p type="margin"> |
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650 |
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651 <s><margin.target id="note45"/>18. 12. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="margin"> |
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653 <s><margin.target id="note46"/>33. 11. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="head"> |
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655 <s>Ad comparandum inter &longs;e duodecim corporum genera <lb/>grauitate, & magnitudine tabella.</s></p><p type="table"> |
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656 |
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657 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><p type="main"> |
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659 <s><emph type="italics"/>Quæro exempli gratia, quam habet rationem in grauitate plumbum ad aurum. </s> |
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661 <s>In­<lb/>teligatur plumbum, quoniam leuius est auro, grauitatem habere 1, & in linea plumbi, <lb/>in prima columna nominati &longs;ub titulo auri, quæratur auri grauitas, ea erit<emph.end type="italics"/> 1 15/23. <emph type="italics"/><!--neuer Satz-->plum<lb/>bum igitur ad aurum rationem habebit in grauitate vt 1, ad 1 15/23, &longs;i enim &longs;umantur <lb/>duo corpora magnitudine æqualia, vnum plumbeum alterum aureum, &longs;it autem plum<lb/>bei corporis grauitas<emph.end type="italics"/> 1, <emph type="italics"/>aurei erit<emph.end type="italics"/> 1 15/23, <emph type="italics"/>quare corpus plumbeum ad corpus aureum <lb/>eiu&longs;dem magnitudinis rationem habebit in grauitate vt 1, ad 1 15/23. comparantur au­<lb/>tem inter &longs;e genera diuer&longs;a grauitate, in corporibus magnitudine æqualibus.<emph.end type="italics"/></s></p><p type="main"> |
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662 |
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663 <s><emph type="italics"/>Rur&longs;us, quæro quam habet rationem in grauitate aqua ad argentum viuum. </s> |
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664 |
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665 <s>inteli­<lb/>gatur aqua, vt leuior argento viuo grauitatem habere 1, & in linea aqua, &longs;ubtitulo ar­<lb/>genti viui, quæratur argenti viui grauitas, ea erit<emph.end type="italics"/> 13 4/7, <emph type="italics"/>aqua igitur ad argentum <expan abbr="viuũ">viuum</expan> <lb/>rationem habebit in grauitate vt 1, ad 13 4/7.<emph.end type="italics"/></s></p><p type="main"> |
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666 |
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667 <s><emph type="italics"/>Contra, quæro quomodo &longs;e habent in magnitudine aurum, & plumbum. </s> |
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668 |
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669 <s>inteligatur <lb/>aurum, quoniam grauius e&longs;t plumbo, magnitudinem habere 1, & in linea plumbi, &longs;ub ti­<lb/>tulo auri, quæratur 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 æque 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<lb/>ne vt 1, ad 1 19/21, comparantur autem inter &longs;e genera diuer&longs;a magnitudine, in corpori­<lb/>bus æque grauibus.<emph.end type="italics"/></s></p><p type="main"> |
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671 <s><emph type="italics"/>Quæro denique quomodo &longs;e habent in magnitudine ferrum, & aqua, ponatur ferrum, <lb/>vt grauius aqua, magnitudinem habere 1, & in linea aquæ, &longs;ub titulo ferri, quæratur <lb/>aquæ 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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672 |
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673 <s>Altera, ad comparandum inter &longs;e duodecim corporum genera, <lb/>grauitate, & magnitudine, tabella.</s></p><p type="table"> |
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674 |
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675 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><p type="main"> |
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676 |
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677 <s><emph type="italics"/>Quæro exempli gratia, quæ nam &longs;it ratio in grauitate, auri ad argentum. </s> |
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678 |
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679 <s>intelliga­<lb/>tur aurum, quoniam grauius est argento, grauitatem habere 100, & in line a auri, &longs;ub <lb/>titulo argenti, reperietur argenti grauitas<emph.end type="italics"/> 54 22/57, <emph type="italics"/>aurum igitur ad argentum rationem <lb/>habebit in grauitate vt<emph.end type="italics"/> 100, <emph type="italics"/>ad<emph.end type="italics"/> 54 22/57, <emph type="italics"/>&longs;i enim &longs;umantur duo corpora, magnitudine <lb/>aqualia, vnum aureum, alterum argenteum, &longs;it autem aurei corporis grauitas<emph.end type="italics"/> 100, <emph type="italics"/>erit<emph.end type="italics"/><pb xlink:href="029/01/041.jpg" pagenum="34"/><emph type="italics"/>argentei<emph.end type="italics"/> 54 22/57, <emph type="italics"/>quare corpus aureum ad corpus argenteum eiu&longs;dem <lb/>magnitudinis, rationem habebit in grauitate, vt<emph.end type="italics"/> 100, <emph type="italics"/>ad<emph.end type="italics"/> 54 22/57. </s></p><p type="main"> |
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680 |
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681 <s><emph type="italics"/>Quæro, quomodo &longs;e habet in grauitate aqua ad vinum. </s> |
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682 |
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683 <s>quoniam <lb/>æqua grauior est vino, intelligatur eius grauitas<emph.end type="italics"/> 100, <emph type="italics"/>& quoniam in <lb/>linea aquæ, &longs;ub titulo vint, datur vini grauitas<emph.end type="italics"/> 98 1/3, <emph type="italics"/>aqua ad vinum <lb/>&longs;e habebit in grauitate, vt<emph.end type="italics"/> 100, <emph type="italics"/>ad<emph.end type="italics"/> 98 1/3. </s></p><p type="main"> |
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684 |
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685 <s><emph type="italics"/>Contra quæro quomodo &longs;e habent in magnitudine argentum, & <lb/>aurum. </s> |
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686 |
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687 <s>intelligatur argentum, vt leuius auro, magnitudinem balere<emph.end type="italics"/><lb/>100, <emph type="italics"/>& in linea auri &longs;ub titulo argenti, quæratur auri magnitudo, ea <lb/>erit<emph.end type="italics"/> 54 22/39, <emph type="italics"/>argentum igitur ad aurum &longs;e habebit in magnitudine, vt<emph.end type="italics"/><lb/>100, <emph type="italics"/>ad<emph.end type="italics"/> 54 22/39, <emph type="italics"/>&longs;i enim &longs;umantur duo corpora æque grauia, vnum <lb/>argenteum, alterum aureum, &longs;it autem argentei corporis magnitudo<emph.end type="italics"/><lb/>100, <emph type="italics"/>erit aurei<emph.end type="italics"/> 54 22/39, <emph type="italics"/>quare corpus argenteum, ad corpus aureum <lb/>eiu&longs;dem grauitatis, &longs;e habebit in magnitudine, vt 100, ad<emph.end type="italics"/> 54 22/39. </s></p><p type="main"> |
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689 <s><emph type="italics"/>Quæro denique quomodo &longs;e habent in magnitudine aqua & ar­<lb/>gentum viuum. </s> |
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690 |
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691 <s>quoniam aqua leuiora est argento viuo, intelligatur <lb/>eius magnitudo<emph.end type="italics"/> 100, <emph type="italics"/>& in linea argenti viui, &longs;ub titulo aquæ, quæ­<lb/>ratur argenti viui magnitudo, & reperietur<emph.end type="italics"/> 7 7/29, <emph type="italics"/>aqua igitur ad <lb/>argentum viuum &longs;e habebit in magnitudine, vt 100, ad<emph.end type="italics"/> 7 7/29. </s></p><p type="main"> |
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692 |
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693 <s>Hic &longs;equitur tabula, ad inueniendas &longs;phærarum grauita­<lb/>tes, ex data diametrorum magnitudine, cuius hæc e&longs;t <lb/>explicatio. </s></p><p type="main"> |
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694 |
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695 <s><emph type="italics"/>In dimetiendis &longs;phærarum diametris vtimur pede Romano anti­<lb/>quo, cuius men&longs;uram in margine appo&longs;uimus, eaque re&longs;pondet ad Ro­<lb/>mani palmi, quo hodie vtimur, men&longs;uram vt 4, ad 3, huiu&longs;modipe­<lb/>dem diuidimus in duodecim partes æquales, &longs;eu vncias, quas inuenies <lb/>in prima Columna &longs;ub titulo magnitudinis.<emph.end type="italics"/></s></p><p type="main"> |
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696 |
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697 <s><emph type="italics"/>Ponderibus autem vtimur hac nostra tempe&longs;tate v&longs;itatis, libram <lb/>enim diuidimus in<emph.end type="italics"/> 12, <emph type="italics"/>vncias vnciam vero in<emph.end type="italics"/> 24, <emph type="italics"/>&longs;crupula, & &longs;cru­<lb/>pulum in<emph.end type="italics"/> 24, <emph type="italics"/>grana. </s> |
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698 |
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699 <s>Ad inueniendas igitur &longs;phærarum grauitates ex <lb/>data diametrorum magnitudine, hæc erit ratio.<emph.end type="italics"/></s></p><p type="main"> |
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700 |
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701 <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, deprehendes ip&longs;am &longs;phæram grauitatem 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></p><p type="main"> |
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702 |
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703 <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> |
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704 |
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705 <s>in linea<emph.end type="italics"/> 6, <emph type="italics"/>vnciarum, &longs;ub titulo grauitatis aureæ &longs;phæræ<emph.end type="italics"/><pb xlink:href="029/01/042.jpg" pagenum="35"/><emph type="italics"/>datur quæ&longs;ita grauitas lib.<emph.end type="italics"/> 97, <emph type="italics"/>vnc.<emph.end type="italics"/> 6, <emph type="italics"/>&longs;crup. </s> |
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706 |
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707 <s>19, gran.<emph.end type="italics"/> 11 1/17. </s></p><p type="main"> |
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708 |
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709 <s><emph type="italics"/>Quæris denique grauitatem &longs;phæræ stanneæ, diametrum habentis <lb/>vnius pedis. </s> |
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710 |
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711 <s>in linea vnius pedis, &longs;eu 12, vnciarum, &longs;ub titulo graui­<lb/>tatis &longs;phæræ &longs;tanneæ, datur quæ&longs;ita &longs;phæræ grauitas lib.<emph.end type="italics"/> 304, <emph type="italics"/>adun­<lb/>quem. </s> |
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712 |
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713 <s>Atque ita reliquarum &longs;phærarum in tabula nominatarum, ex <lb/>data diametrorum magnitudine, grauitates inuenies.<emph.end type="italics"/></s></p><p type="main"> |
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714 |
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715 <s>Qua ratione hanc Tabulam compo&longs;uimus. </s></p><p type="main"> |
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716 |
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717 <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 humana <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 fieri pote&longs;t multo exactior quam &longs;phæra, & facilius. </s> |
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718 |
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719 <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. <!-- REMOVE S-->14592. abstulissent ab hac Cylindri <lb/>grauitate partem tertiam. </s> |
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720 |
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721 |
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722 |
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723 <s>id est<emph.end type="italics"/> 4864, <emph type="italics"/>reliquum, quod est<emph.end type="italics"/> 9728. <emph type="italics"/>&longs;er­<lb/>uauimus, pro grauitate &longs;phæræ, diametrum habentis æqualem altitu­<lb/>dini Cylindri, o&longs;ten&longs;um enim est ab Archimede propo&longs;.<emph.end type="italics"/> 32, <emph type="italics"/>lib.<emph.end type="italics"/> 1, <emph type="italics"/>de <lb/>&longs;phæra, & Cylindro, Cylindrum, qui ba&longs;im habeat maximo in &longs;phæra <lb/>circulo æqualem, & altitudinem æqualem diametro &longs;phæræ, ad ip&longs;am <lb/>&longs;pbæram &longs;e&longs;quialterum e&longs;&longs;e; itaque grauitatem &longs;phæræ, diametrum <lb/>habentis duarum vnciarum inuenimus e&longs;&longs;e gran.<emph.end type="italics"/> 9728. </s></p><p type="main"> |
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724 |
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725 <s><emph type="italics"/>Inuenta igitur grauitate &longs;phæræ, cuius diameter est duarum vn­<lb/>ciarum, facile inuenientur reliquarum <expan abbr="&longs;pbærarũ">&longs;phærarum</expan> grauitates, &longs;i enim <lb/>inuenienda &longs;it grauitas &longs;phæræ stannea habentis diametrum<emph.end type="italics"/> 1/4. <emph type="italics"/>vn­<lb/>ciæ, fiat vt cubus ex<emph.end type="italics"/> 2, <emph type="italics"/>ad cubum ex<emph.end type="italics"/> 1/4, <emph type="italics"/>hoc est vt<emph.end type="italics"/> 512, <emph type="italics"/>ad 1, ita<emph.end type="italics"/> 9728, <lb/><emph type="italics"/>ad alium numerum, qui &longs;it<emph.end type="italics"/> 19, <emph type="italics"/>&longs;phæræ igitur cuius diameter e&longs;t<emph.end type="italics"/> 1/4, <lb/><emph type="italics"/>vnciæ, grauitas erit gran. <!-- REMOVE S-->19, osten&longs;um enim est prop. |
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726 |
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727 17, huius, &longs;phæ­<lb/>ras eiu&longs;dem generis inter &longs;e e&longs;&longs;e in grauitate, vt diametrorum cubi in <lb/>magritudine.<emph.end type="italics"/></s> |
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728 |
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729 </p><p type="main"> |
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730 |
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731 <s><emph type="italics"/>Rur&longs;us &longs;it inuenienda grauitas &longs;phæræ stannæ habentis diame­<lb/>trum<emph.end type="italics"/> 1/2, <emph type="italics"/>vnciæ, fiat vt cubus ex<emph.end type="italics"/> 1/4, <emph type="italics"/>ad cubum ex<emph.end type="italics"/> 1/2, <emph type="italics"/>hoc est vt<emph.end type="italics"/> 1, <emph type="italics"/>ad<emph.end type="italics"/> 8, <lb/><emph type="italics"/>ita<emph.end type="italics"/> 19, <emph type="italics"/>ad<emph.end type="italics"/> 152, <emph type="italics"/>&longs;phæra igitur, cuius diameter e&longs;t<emph.end type="italics"/> 1/2, <emph type="italics"/>vnciæ, habebit gra­<lb/>uitatem gran.<emph.end type="italics"/> 152. </s></p><p type="main"> |
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732 |
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733 <s><emph type="italics"/>Sit denique inuenienda grauitas &longs;phæræ stannæ, diametrum ha­<lb/>bentis<emph.end type="italics"/> 1/4, <emph type="italics"/>vnciæ, fiat vt cubus ex<emph.end type="italics"/> 1/4, <emph type="italics"/>ad cubum ex<emph.end type="italics"/> 1/4, <emph type="italics"/>hoc e&longs;t vt<emph.end type="italics"/> 1, <emph type="italics"/>ad<emph.end type="italics"/> 27, <lb/><emph type="italics"/>ita<emph.end type="italics"/> 19, <emph type="italics"/>ad<emph.end type="italics"/> 513, <emph type="italics"/>grauitas igitur &longs;phæræ habentis diametrum<emph.end type="italics"/> 1/4, <emph type="italics"/>vnciæ,<emph.end type="italics"/></s></p><pb xlink:href="029/01/043.jpg" pagenum="36"/><p type="main"> |
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734 |
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735 <s>Ad inueniendas &longs;phæra­<lb/>diametrorum <lb/>T A B <!-- KEEP S--></s></p><p type="table"> |
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736 |
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737 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><pb xlink:href="029/01/044.jpg" pagenum="37"/><p type="main"> |
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738 |
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739 <s>rum grauitates ex data <lb/>magnitudine <lb/>V L A. <!-- KEEP S--></s></p><p type="table"> |
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740 |
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741 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><pb xlink:href="029/01/045.jpg"/><p type="table"> |
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742 |
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743 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><pb xlink:href="029/01/046.jpg"/><p type="table"> |
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744 |
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745 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><pb xlink:href="029/01/047.jpg" pagenum="40"/><p type="table"> |
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746 |
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747 <s>TABELLE WAR HIER<lb/><emph type="italics"/>erit gran. <!-- REMOVE S-->513. & &longs;ic reliquarum &longs;phærarum ex &longs;tanno, diametros ha­<lb/>bentium magnitudine quacunque, inuenientur grauitates.<emph.end type="italics"/></s> |
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748 |
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749 </p><p type="main"> |
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750 |
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751 <s><emph type="italics"/>Aliter quoque & expeditius reliquarum &longs;phærarum ex &longs;tanno in­<lb/>uenientur grauitates.<emph.end type="italics"/></s></p><p type="main"> |
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752 |
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753 <s><emph type="italics"/>Inuenta grauitate &longs;phæræ, diametrum habentis<emph.end type="italics"/> 1/4, <emph type="italics"/>vnciæ, mul­<lb/>tiplicetur ip&longs;a grauitas, per<emph.end type="italics"/> 8, <emph type="italics"/>hoc est per cubum ex<emph.end type="italics"/> 2, <emph type="italics"/>numerus pro <lb/>ductus dabit grauitatem &longs;phæræ, diametrum habentis<emph.end type="italics"/> 2/4, <emph type="italics"/>vnciæ, hoc<emph.end type="italics"/><lb/><arrow.to.target n="note47"/> <emph type="italics"/>e&longs;t<emph.end type="italics"/> 1/2, <emph type="italics"/>&longs;phæræ {*} enim inter &longs;e in triplicata &longs;unt ratione &longs;uarum di<gap/><lb/>metrorum. </s> |
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754 |
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755 <s>deinde &longs;i multiplicetur eadem grauitas per<emph.end type="italics"/> 27, <emph type="italics"/>hoc est per <lb/>cubum ex<emph.end type="italics"/> 3, <emph type="italics"/>numerus productus dabit grauitatem &longs;phæræ, habentis <lb/>diametrum<emph.end type="italics"/> 3/4, <emph type="italics"/>vnciæ, & &longs;i multiplicetur per<emph.end type="italics"/> 64, <emph type="italics"/>hoc est per cubum <lb/>ex<emph.end type="italics"/> 4, <emph type="italics"/>numerus productus dabit grauitatem &longs;phæræ, cuius diameter e&longs;t <emph.end type="italics"/><lb/>4/4, <emph type="italics"/>hoc est vnius vnciæ, & eo deinceps continuo ordine,<emph.end type="italics"/></s></p><p type="margin"> |
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756 |
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757 <s><margin.target id="note47"/>18.12. <lb/><emph type="italics"/>Elem.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> |
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758 |
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759 <s><emph type="italics"/>Porro ad inueniendas grauitates &longs;phærarum ex reliquis metallis <lb/>vel ex quacunque alia materia, hæc erit ratio.<emph.end type="italics"/></s></p><p type="main"> |
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760 |
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761 <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;phæræ<lb/><expan abbr="plūbeæ">plumbeæ</expan> quæritur cuius diameter est<emph.end type="italics"/> 1/4, <emph type="italics"/>vnciæ) ita 19. grauitas vide­<lb/>licet &longs;phæræ stanneæ <expan abbr="diametrū">diametrum</expan> habentis<emph.end type="italics"/> 1/4, <emph type="italics"/>vnciæ, ad <expan abbr="aliū">alium</expan> numerum<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 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<lb/>te, & magnitudine, appo&longs;uimus.<emph.end type="italics"/></s></p><p type="main"> |
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762 |
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763 <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 grauia<emph.end type="italics"/></s></p><pb xlink:href="029/01/048.jpg" pagenum="41"/><p type="table"> |
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764 |
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765 <s>TABELLE WAR HIER<lb/><emph type="italics"/>&longs;phæræ stanneæ, cuius diameter est<emph.end type="italics"/> 2, <emph type="italics"/>vnciarum, ad alium numerum, <lb/>qui &longs;it<emph.end type="italics"/> 15 117 <gap/>, <emph type="italics"/>&longs;phæra igitur plumbea, cuius diameter est<emph.end type="italics"/> 2, <emph type="italics"/>vn­<lb/>ciarum grauitatem habebit gran. <!-- REMOVE S-->15 117 <gap/>, atque hæc erit ob&longs;er­<lb/>uanda in reliquis ratio.<emph.end type="italics"/></s> |
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766 |
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767 </p><p type="main"> |
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768 |
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769 <s><emph type="italics"/>Vel &longs;i ip&longs;a grauitas<emph.end type="italics"/> 29 19/74, <emph type="italics"/>multiplicetur per &longs;ingulos cubos, vt <lb/>dictum est de &longs;phera &longs;tannea, numeri producti dabunt grauitates <lb/>&longs;phærarum ex plumbo, ad quarum diametros latera cubica rationem <lb/>habebunt vt<emph.end type="italics"/> 4, <emph type="italics"/>ad<emph.end type="italics"/> 1, <emph type="italics"/>quoniam<emph.end type="italics"/> 29 19/74, <emph type="italics"/>est grauitas &longs;phæræ plumbeæ, <lb/>diametrum habentis<emph.end type="italics"/> 1/4, <emph type="italics"/>vnciæ.<emph.end type="italics"/></s></p><p type="head"> |
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770 |
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771 <s>Sequitur, ad inueniendas diametrorum <lb/>magnitudines ex data &longs;phæ­<lb/>rarum grauitate, <lb/>tabula.</s></p><pb xlink:href="029/01/049.jpg" pagenum="42"/><p type="table"> |
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772 |
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773 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><pb xlink:href="029/01/050.jpg" pagenum="43"/><p type="table"> |
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774 |
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775 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><pb xlink:href="029/01/051.jpg" pagenum="44"/><p type="table"> |
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776 |
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777 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><pb xlink:href="029/01/052.jpg" pagenum="45"/><p type="table"> |
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778 |
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779 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><pb xlink:href="029/01/053.jpg" pagenum="46"/><p type="table"> |
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780 |
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781 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><pb xlink:href="029/01/054.jpg" pagenum="47"/><p type="table"> |
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782 |
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783 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><pb xlink:href="029/01/055.jpg" pagenum="48"/><p type="table"> |
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784 |
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785 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><pb xlink:href="029/01/056.jpg" pagenum="49"/><p type="table"> |
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786 |
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787 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><p type="main"> |
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788 |
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789 <s>E<emph type="italics"/>st hæc tabula, quemadmodum & eius v&longs;us, præcedentis con­<lb/>nuer&longs;a, in ea enim inueniuntur &longs;phærarum grauitates ex data <lb/>diametrorum magnitudine, in hac vero deprehenduntur diametro­<lb/>rum magnitudines ex data &longs;phærarum grauitate.<emph.end type="italics"/></s></p><p type="main"> |
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790 |
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791 <s><emph type="italics"/>Quæro exempli gratia magnitudinem diametri &longs;phæræ aureæ, <lb/>grauitatem habentis 10, lib. <!-- KEEP S--></s> |
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792 |
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793 <s>Numeri in prima columna &longs;ub titulo <lb/>grauitatis denotant &longs;phærarum grauitates, reliqui vero in reliquis <lb/>columnis denotant diametrorum magnitudines; itaque in linea<emph.end type="italics"/> 10, <lb/><emph type="italics"/>lib. |
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794 |
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795 &longs;ub titulo magnitudinis diametri &longs;phæræ aureæ, datur quæ&longs;ita <lb/>diametri magnitudo partium<emph.end type="italics"/> 2 21/100: <emph type="italics"/>qualium pes vnus est<emph.end type="italics"/> 12. </s></p><p type="main"> |
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796 |
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797 <s><emph type="italics"/>Quæro magnitudinem diametri &longs;phæræ ferreæ, grauitatem haben­<lb/>tis 50, lib. |
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798 |
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799 in linea 50, lib. <!-- REMOVE S-->&longs;ub titulo magnitudinis diametri &longs;phæræ <lb/>ferreæ, datur quæ&longs;ita diametri magnitudo<emph.end type="italics"/> 6 40/100. </s> |
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800 |
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801 </p><p type="main"> |
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802 |
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803 <s><emph type="italics"/>Quæro magnitudinem diametri &longs;phæræ argenteæ, grauitatem ha­<lb/>bentis 60, lib. |
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804 |
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805 in linea 60, lib. <!-- REMOVE S-->&longs;ub titulo magnitudinis diametri &longs;phæræ <lb/>argenteæ, datur ip&longs;a magnitudo<emph.end type="italics"/> 6 25/100. </s> |
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806 |
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807 </p><pb xlink:href="029/01/057.jpg" pagenum="50"/><p type="main"> |
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808 |
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809 <s><emph type="italics"/>Quæro denique magnitudinem diametri &longs;phæræ stanneæ, graui­<lb/>tatem habentis 38, lib. |
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810 |
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811 in linea 38, lib. <!-- REMOVE S-->&longs;ub titulo magnitudinis dia­<lb/>metri &longs;phæræ stanneæ, datur quæ&longs;ita diametri magnitudo 6, ad <lb/>vnguem.<emph.end type="italics"/></s> |
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812 |
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813 </p><p type="main"> |
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814 |
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815 <s><emph type="italics"/>Notandum autem est, quod numeri, qui diametrorum magnitu­<lb/>dines denotant, non &longs;unt veri, ac certi, &longs;ed veris bene proximi, <expan abbr="quoniã">quoniam</expan> <lb/>numeri, quorum ip&longs;i &longs;unt radices cubicæ, non &longs;unt cubi, & ideo ip&longs;a <lb/>radices non explicantur accurate, &longs;ed vel veris maiores, vel minores, <lb/>atque vt cogno&longs;cantur quæ &longs;int maiores, queue minores, maioribus <lb/>duo puncta adiecimus, minoribus vnum, accuratis nullum. </s> |
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816 |
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817 <s>inter om­<lb/>nes autem vnus est accuratus, is &longs;cilicet, qui magnitudinem indicat <lb/>diametri &longs;phæræ stanneæ, grauitatem habentis<emph.end type="italics"/> 38, <emph type="italics"/>lib.<emph.end type="italics"/><!-- REMOVE S-->De compo&longs;itione huius Tabulæ. <!-- KEEP S--></s></p><p type="main"> |
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818 |
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819 <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, ad grana<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. |
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820 |
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821 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. |
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822 |
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823 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> |
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824 |
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825 <s>ita inuenientur diametri.<emph.end type="italics"/></s></p><p type="main"> |
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826 |
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827 <s><emph type="italics"/>Duplum numeri<emph.end type="italics"/> 5684210 10/19, <emph type="italics"/>id est<emph.end type="italics"/> 11368421 1/19, <emph type="italics"/>erit cubus <lb/>centupli diametri &longs;pheræ stanneæ, grauitatem habentis duplam pri­<lb/>mæ, hoc est<emph.end type="italics"/> 2, <emph type="italics"/>lib. |
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828 |
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829 ex &longs;upra nominata enim prop.<emph.end type="italics"/> 17, <emph type="italics"/>huius, e&longs;t vt gra­<lb/>uitas &longs;phæræ vnius libræ, ad grauitatem &longs;phæræ duarum librarum, <lb/>ita cubus diametri primæ &longs;phæræ, ad cubum diametri &longs;ecundæ. </s> |
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830 |
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831 <s>Si <lb/>vero triplicetur numerus<emph.end type="italics"/> 5684210 10/19, <emph type="italics"/>eius triplum, quod est<emph.end type="italics"/><lb/>17052631 11/19, <emph type="italics"/>erit cubus centupli diametri &longs;phæræ &longs;tanneæ, graui­<lb/>tatem habentis triplam primæ, idest<emph.end type="italics"/> 3, <emph type="italics"/>lib. |
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832 |
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833 & &longs;i quadruplicetur, eius<emph.end type="italics"/><pb xlink:href="029/01/058.jpg" pagenum="51"/><emph type="italics"/>quadruplum erit cubus centupli diametri &longs;phæræ stanneæ, grauita­<lb/>tem habentis quadruplam primæ, & &longs;ic deinceps. </s> |
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834 |
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835 <s>itaque &longs;i ex eius <lb/>multiplicibus, neglectis fractis, <expan abbr="eruãtur">eruantur</expan> radices, tanquam ex accura­<lb/>tis numeris cubis, ip&longs;æ indicabunt diametrorum magnitudines in <lb/>ratione centupla. </s> |
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836 |
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837 <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></p><p type="main"> |
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838 |
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839 <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></p><p type="main"> |
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840 |
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841 <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. |
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842 |
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843 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. |
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844 |
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845 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. |
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846 |
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847 & &longs;ic reliquarum &longs;phærarum in infinitum inuenientur <lb/>diametri. </s> |
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848 |
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849 <s>multiplicia autem numeri<emph.end type="italics"/> 5257894 14/19, <emph type="italics"/>&longs;ola additione in­<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 prædictam tabulam compo&longs;uimus.<emph.end type="italics"/></s></p><p type="margin"> |
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850 |
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851 <s><margin.target id="note48"/>17<emph type="italics"/>: <lb/>huius.<emph.end type="italics"/></s></p><p type="main"> |
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852 |
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853 <s>Qvomodo Archimedes argenti mixtionem depre­<lb/>hendit in auro. </s></p><p type="main"> |
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854 |
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855 <s><emph type="italics"/>Hiero (referente Vitruuio lib.<emph.end type="italics"/> 9. <emph type="italics"/>Cap.<emph.end type="italics"/> 3.<emph type="italics"/>) Siracu&longs;is auctus regia<emph.end type="italics"/><pb xlink:href="029/01/059.jpg" pagenum="52"/><emph type="italics"/>pote&longs;tate, rebus bene ge&longs;tis, cum auream coronam votiuam, dijs im­<lb/>mortalibus in quodam fano con&longs;titui&longs;&longs;et ponendam, immani precia <lb/>locauit faciendam, & aurum ad &longs;acoma appendit redemptori. </s> |
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856 |
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857 <s>is ad <lb/>tempus opus manufactum &longs;ubtiliter, regi approbauit, & ad &longs;acoma <lb/>pondus coronæ vi&longs;us e&longs;t præ&longs;titi&longs;&longs;e. </s> |
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858 |
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859 <s>Po&longs;tea quam indicium e&longs;t factum, <lb/>dempto auro, tantundem argenti in id coronarium opus admixtum <lb/>e&longs;&longs;e: indignatus Hiero &longs;e contemptum, neque inueniens, qua ratione <lb/>id furtum deprehenderet, rogauit Archimedem, vti in &longs;e &longs;umeret de <lb/>eo cogitationem. </s> |
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860 |
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861 <s>tunc is cum haberet eius rei curam, ca&longs;u venit in <lb/>balneum, ibique cum in &longs;olium de&longs;cenderet, animaduertit quantum <lb/>corporis &longs;ui in eo in&longs;ideret, tantum aquæ extra &longs;olium effluere. </s> |
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862 |
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863 <s><expan abbr="itaq;">itaque</expan> <lb/>cum eius rei rationem explicationis offendi&longs;&longs;et non est moratus, &longs;ed <lb/>exiliuit gaudio motus de &longs;olio, & nudus vadens domum ver&longs;us, &longs;igni <lb/>ficabat clara voce inueni&longs;se quod quæreret. </s> |
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864 |
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865 <s>nam currens identidem <lb/>grece clamabat<emph.end type="italics"/> <foreign lang="greek"><gap/></foreign> <emph type="italics"/>tum vero ex eo inuentionis ingre&longs;&longs;u <lb/>duas dicitur feci&longs;&longs;e ma&longs;&longs;as æquo pondere, quo etiam fuerat coro­<lb/>na, vnam ex auro, alteram ex argento. </s> |
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866 |
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867 <s>cum ita fecißet, vas <expan abbr="amplũ">amplum</expan> <lb/>ad &longs;umma labra impleuit aqua, in quo demi&longs;it argenteam ma&longs;&longs;am, <lb/>cuius quanta magnitudo in va&longs;e depre&longs;&longs;a est, tantum aquæ effluxit. </s> |
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868 |
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869 <s><lb/>ita exempta ma&longs;&longs;a, quanto minus factum fuerat refudit, &longs;extario <lb/>men&longs;us, vt eodem modo, quo prius fuerat, ad labra æquaretur. </s> |
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870 |
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871 <s>ita ex <lb/>eo inuenit, quantum ad certum pondus argenti certa aquæ men&longs;ura <lb/>re&longs;ponderet.<emph.end type="italics"/></s></p><p type="main"> |
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872 |
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873 <s><emph type="italics"/>Cum id expertus e&longs;&longs;et tum auream ma&longs;&longs;am &longs;imiliter pleno va&longs;e <lb/>demi&longs;it, & ea exempta, eadem ratione men&longs;ura addita, inuenit ex <lb/>aqua non tantum defluxi&longs;&longs;e, &longs;ed tantum minus, quantum minus ma <lb/>gno corpore eodem pondere auri ma&longs;&longs;a e&longs;&longs;et quam argenti. </s> |
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874 |
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875 <s>Postea <lb/>vero repleto va&longs;e, in <expan abbr="ead&etilde;">eadem</expan> aqua ip&longs;a corona demi&longs;&longs;a, inuenit plus aqua <lb/>defluxi&longs;&longs;e in coronam, quam in auream. </s> |
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876 |
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877 <s>eodem pondere ma&longs;&longs;am, & <lb/>ita ex eo quod plus defluxerat aquæ in corona, quam in ma&longs;&longs;a ratio­<lb/>cinatus, deprehendit argenti in auro mixtionem, & manifestum fur­<lb/>tum redemptoris.<emph.end type="italics"/></s><s> Hactenus Vitruuius. <!-- KEEP S--></s></p><p type="main"> |
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878 |
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879 <s>Mirum certe Archimedis fuit inuentum, ip&longs;ius tamen <lb/>modus ad inueniendam illam aquæ men&longs;uram, quæ ad <lb/>certum pondus auri, vel argenti, vel coronæ re&longs;ponderet, <lb/>maiori diligentia indiget, quam quæ ab hominibus adhiberi <lb/>pote&longs;t, impo&longs;&longs;ibile enim e&longs;t, exempta corona, vel aurea ma&longs;&longs;a, <lb/>vel argentea, tantum aquæ refundere, quantum è va&longs;e efflu­<lb/>xerat ad vnguem, nam repo&longs;ita aqua in va&longs;e, non po&longs;&longs;umus <pb xlink:href="029/01/060.jpg" pagenum="53"/>affirmare ip&longs;um vas e&longs;&longs;e plenum, ni&longs;i aqua incipiat effluere, <lb/>cum autem incipit, effluit aliquando totus ferè cumulus, ita­<lb/>que vel plus aquæ additur eo, quod deficit, vel minus, ni&longs;i <lb/>coniectura a&longs;&longs;equatur: at vero coniectura pro veritate non ac­<lb/>cipitur. </s> |
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880 |
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881 <s>præterea exempta corona, vel aurea ma&longs;&longs;a, vel argen­<lb/>tea, eximitur etiam &longs;imul cum ip&longs;a aliquantum aquæ, quæ cir­<lb/>cum ip&longs;am remanet, atque huiu&longs;modi defectus errorem indu­<lb/>cit &longs;en&longs;ibilem. </s></p><p type="main"> |
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882 |
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883 <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></p><p type="main"> |
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884 |
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885 <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> |
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886 |
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887 <s>fit enim vt aliquando cumulus <lb/>ille frangatur pluribus in locis, & ideo aqua diffundatur, vt <lb/>ferè nihil ip&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></p><p type="main"> |
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888 |
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889 <s>Sed ponderandis corporibus in aere & aqua, eo modo, quo <lb/>dictum e&longs;t in fine exempli prop. |
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890 |
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891 8. inuenitur quæ&longs;ita aquæ, <lb/>grauitas, ita exactè, vt requiritur, fiue &longs;it corpus illud paruum, <lb/>&longs;iue magnum nihil intere&longs;t, & præterea facillima e&longs;t operatio, <lb/>nec adinueniendæ &longs;unt auri, & argenti ma&longs;&longs;æ æque graues, ac <pb xlink:href="029/01/061.jpg" pagenum="54"/>corona, &longs;ed quælibet particulæ, grauitate quacunque, etiam <lb/>differentes inter &longs;e, &longs;ufficiunt. </s></p><p type="main"> |
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892 |
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893 <s>De ratione autem, qua Archimedes, cognitis grauitatibus <lb/>trium corporum ex aqua, magnitudine æqualium, coronæ <lb/>&longs;cilicet vnum, alterum ma&longs;&longs;æ aure&ecedil;, tertium argenteæ, potue­<lb/>rit furtum aurificis in regia corona deprehendere, atque ar­<lb/>gentum quod erat in ea permixtum ab auro di&longs;cernere, pluri­<lb/>mi &longs;crip&longs;erunt, modos etiam ad id faciendum excogitarunt <lb/>varios, longa tamen methodo, atque difficili v&longs;i &longs;unt, & quod <lb/>maximam confu&longs;ionem, & ob&longs;curitatem parit, nullum opera­<lb/>tionis tradunt præceptum firmum, ac &longs;tabile. </s> |
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894 |
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895 <s>ego autem vni­<lb/>ca tantum proportionis ratiocinatione, &longs;eu regula trium (vt <lb/>vulgo dicitur) breuiter, & expedite idem con&longs;equor, eamque <lb/>geometrica ratione demon&longs;tro. </s> |
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896 |
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897 <s>Problema igitur ad hoc facien <lb/>dum ita concipio & ab&longs;oluo. </s></p><p type="head"> |
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898 |
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899 <s>PROBLEMA IX. PROPOS. XVIII.<!-- KEEP S--></s></p><p type="main"> |
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900 |
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901 <s>Portionem metalli, alteri metallo mi&longs;tam, ponde­<lb/>ris ratiocinatione di&longs;cernere. </s></p><p type="main"> |
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902 |
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903 <s>QVONIAM de Hieronis corona facta e&longs;t mentio, &longs;it ea B, <lb/>eiu&longs;que grauitas EK, & oporteat argentum, quod &longs;it in ea <expan abbr="permixtũ">permixtum</expan>, <lb/>ab auro di&longs;cernere, hoc e&longs;t oporteat inuenire quanta erit portio ar­<lb/><figure id="id.029.01.061.1.jpg" xlink:href="029/01/061/1.jpg"/><lb/>genti, & quanta auri. </s> |
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904 |
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905 <s>In­<lb/>telligantur duo corpora <lb/>A, D, vnum aureum, al­<lb/>terum argenteum æque <lb/>grauia atque corona, <lb/>deinde trium corporum <lb/>ex aqua, magnitudine <lb/>æqualium, aureo &longs;cili­<lb/>cet corpori vnum, alte­<lb/>rum coronæ, tertium <lb/>corpori argenteo, inue­<lb/>niantur grauitates, id autem poterit fieri facillime, &longs;i <expan abbr="accipiãtur">accipiantur</expan> duo <lb/>corpora vnum ex auro, alterum ex argento, grauitate quacunque, vt <pb xlink:href="029/01/062.jpg" pagenum="55"/>dictum e&longs;t in propo&longs;itionis octauæ exemplo, non enim nece&longs;&longs;e e&longs;t ha­<lb/>bere duo corpora ex auro & argento, grauitatem habentia eandem <lb/>quam & corona, & hac de cau&longs;a diximus &longs;upra <expan abbr="intelligãtur">intelligantur</expan> duo cor­<lb/>pora, non autem accipiantur. </s> |
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906 |
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907 <s>&longs;it igitur primi corporis aquei æqualis <lb/>aureo A, inuenta grauitas G, &longs;ecundi vero æqualis coronæ B, graui­<lb/>tas F, & tertij æqualis corpori argenteo D, grauitas H, & fiat vt dif­<lb/>ferentia inter G, & H, ad EK, ita differentia inter G, & F, ad aliam <lb/>grauitatem, quæ &longs;it K. <!-- KEEP S--></s> |
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908 |
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909 <s>Dico K, grauitatem e&longs;&longs;e portionis argenti, <lb/>quod e&longs;t in corona, E vero grauitatem auri. </s></p><p type="main"> |
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910 |
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911 <s>Vel &longs;i pro tertio proportionis termino &longs;umatur differentia inter F, <lb/>& H, & quartus terminus &longs;it E, Dico E, grauitatem e&longs;&longs;e portionis au­<lb/>ri, K vero argenti. </s></p><p type="main"> |
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912 |
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913 <s>Quartus autem vtriu&longs;que proportionis terminus * minor e&longs;t<arrow.to.target n="note49"/> &longs;e­<lb/>cundo EK, quod & tertius minor e&longs;t primo, primus enim terminus <lb/>e&longs;t differentia inter G, & H, tertius vero, vel e&longs;t differentia inter G, & <lb/>F, vel differentia inter F, & H, vterque minor primo. </s> |
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914 |
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915 <s>Exemplis autem <lb/>res fiet illu&longs;trior. </s></p><p type="margin"> |
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916 |
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917 <s><margin.target id="note49"/>14. 5 <lb/><emph type="italics"/>Elen<emph.end type="italics"/><!-- KEEP S--></s></p><p type="head"> |
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918 |
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919 <s>Exemplum. <!-- KEEP S--></s> |
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920 |
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921 <s>I.<!-- KEEP S--></s></p><p type="main"> |
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922 |
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923 <s><emph type="italics"/>Sit coronæ grauitas<emph.end type="italics"/> 95, <emph type="italics"/>lib. |
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924 |
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925 & oporteat facere quod imperatum e&longs;t. </s> |
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926 |
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927 <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 corpori vnum, 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, & tertij<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></p><p type="main"> |
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928 |
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929 <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></p><p type="head"> |
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930 |
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931 <s>Exemplum. <!-- KEEP S--></s> |
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932 |
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933 <s>II.<!-- KEEP S--></s></p><p type="main"> |
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934 |
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935 <s><emph type="italics"/>Sit aliquod corpus mistum ex auro. </s> |
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936 |
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937 <s>& ære, & habeat grauitatem <lb/>171. lib. |
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938 |
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939 & oporteat inuenire quanta erit portio æris in ip&longs;o corpore,<emph.end type="italics"/><pb xlink:href="029/01/063.jpg" pagenum="56"/><emph type="italics"/>& quanta auri. </s> |
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940 |
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941 <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 habebit<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></p><p type="main"> |
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942 |
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943 <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></p><p type="main"> |
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944 |
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945 <s>At vero huiu&longs;modi ratiocinationem ad di&longs;cernendum ar­<lb/>gentum ab auro, vel aliud metallum ab altero metallo, recte <lb/>e&longs;&longs;e in&longs;titutam, &longs;equenti Theoremate demon&longs;trabitur. </s></p><p type="head"> |
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946 |
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947 <s>THEOREMA X. PROPOS. XIX.<!-- KEEP S--></s></p><p type="main"> |
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948 |
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949 <s>Si trium corporum æque grauium primum & ter­<lb/>tium fuerint generis diuer&longs;i, &longs;ecundi autem portio <lb/>fuerit eiu&longs;dem generis cum corpore primo, reliqua ve­<lb/>ro eiu&longs;dem generis cum corpore tertio, fuerint etiam <lb/>tres quantitates aquæ prædictis corporibus æquales, pri­<lb/>ma videlicet corpori primo, &longs;ecunda &longs;ecundo, & tertia <lb/>tertio. </s> |
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950 |
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951 <s>erit vt differentia grauitatum primæ & tertiæ <lb/>quantitatis aquæ, ad grauitatem corporis &longs;ecundi, ita <lb/>differentia grauitatum primæ & &longs;ecundæ quantitatis <lb/>aquæ, ad grauitatem portionis corporis &longs;ecundi, quæ e&longs;t <lb/>eiu&longs;dem generis cum corpore tertio. </s></p><p type="main"> |
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952 |
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953 <s>Et ita differentia grauitatum &longs;ecundæ & tertiæ quan <lb/>titatis aquæ, ad grauitatem portionis eiu&longs;dem generis <lb/>cum corpore primo. </s></p><pb xlink:href="029/01/064.jpg" pagenum="57"/><p type="main"> |
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954 |
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955 <s>SINT tria corpora æque grauia A, BC, D, quorum A, primum, <lb/>& tertium D. &longs;int generis diuer&longs;i, portio vero &longs;ecundi B, &longs;it eiu&longs;dem <lb/>generis cum corpore A, & 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/>& Q, <expan abbr="quorũ">quorum</expan> <gap/>, &longs;it æqua­<lb/>le corpori A, magnitu­<lb/>dine, ip&longs;um vero OL, <lb/>æquale corpori BC, & <lb/>ip&longs;um Q, æquale cor­<lb/>pori D, & &longs;int earum <lb/>grauitates, G, ip&longs;ius P, <lb/>& FV, ip&longs;ius OL, & H, <lb/>ip&longs;ius <expan abbr="q.">que</expan> Dico vt diffe­<lb/>rentia grauitatum G, <lb/>H, ad grauitatem cor­<lb/>poris BC, ita e&longs;&longs;e diffe­<lb/>rentiam grauitatum <lb/>G, FV, ad grauitatem <lb/>portionis C; & ita differentiam grauitatum FV, H, ad portionis B, <lb/>grauitatem. </s> |
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956 |
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957 <s>Sit enim portionis B, grauitas E, & portionis C, gra­<lb/>uitas K; ergo totius corporis BC grauitas erit EK, <expan abbr="&longs;itq;">&longs;itque</expan> portionis O, <lb/>quæ &longs;it æqualis portioni B, grauitas F, ergo reliquæ portionis L, <lb/>æqualis portioni C, grauitas erit V, Quoniam igitur e&longs;t, vt A, ad P, <lb/>ita B, ad O, æquale videlicet ad æquale, erit permutando, vt A, ad B, <lb/>ita p.ad O, & quoniam &longs;unt eiu&longs;dem generis A, B, &longs;imiliter & P, O, <lb/>* erit vt grauitas corporis A, hoc e&longs;t vt EK, (ponuntur enim<arrow.to.target n="note50"/> cor­<lb/>pora A, BC, D, æque grauia,) ad E, ita G, ad F, quod igitur fit ex EK, <lb/>& F, nempe ex extremis, æquale erit ei, quod fit ex E, & G, hoc e&longs;t <lb/>ex medijs. </s></p><p type="margin"> |
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958 |
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959 <s><margin.target id="note50"/>4.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main"> |
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960 |
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961 <s>Similiter quoniam e&longs;t, vt D, ad Q, ita C, ad E, æquale videlicet ad <lb/>æquale, erit permutando, vt D, ad C, ita Q, ad L, & quoniam &longs;unt <lb/>eiu&longs;dem generis D, C, &longs;imiliter & Q, L, * erit vt grauitas ip&longs;ius D,<arrow.to.target n="note51"/><lb/>hoc e&longs;t vt EK, ad K, ita H, ad V; quare quod fit ex EK, & V, ex extre­<lb/>mis, æquabitur ei, quod ex H, fit & K, ex medijs. </s></p><p type="margin"> |
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962 |
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963 <s><margin.target id="note51"/>4.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main"> |
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964 |
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965 <s>Sed o&longs;ten&longs;um e&longs;t id quod ex EK, fit & F. æquale e&longs;&longs;e ei quod fit ex <lb/>G, & E, ergo quod fit ex EK, & F, vna cum eo, quod ex BK, & V, hoc <lb/>e&longs;t id quod fit ex EK, & FV, æquale erit ei quod ex G, fit & E, vna <lb/>eum eo quod ex H, & K, &longs;ed quod ex G, fit & E, æquale e&longs;t ei quod fit <lb/>ex G, & EK, minus eo quod ex G, & K, quod enim additur, idem & <pb xlink:href="029/01/065.jpg" pagenum="58"/>minuitur; ergo quod fit ex EK, & FV. æquale erit ei quod fit ex G, & <lb/>EK, vna cum eo quod ex H, & K, minus eo quod fit ex G, & K. aufe­<lb/>ratur vtrinque id quod fit ex G, & EK, quod igitur fit ex FV, & EK, <lb/><figure id="id.029.01.065.1.jpg" xlink:href="029/01/065/1.jpg"/><lb/>minus eo quod ex G, & <lb/>EK, æquabitur ei quod <lb/>ex H, & K, minus eo <lb/>quod fit ex G, & K, &longs;ed <lb/>quod fit ex H, & K, <lb/>minus eo quod fit ex <lb/>G, & K, æquale e&longs;t ei <lb/>quod ex differentia ip­<lb/>&longs;arum H, G, fit & K, <lb/>&longs;imiliter, & quod fit ex <lb/>FV, & EK, minus eo <lb/>quod ex G, & EK, <lb/>æquale e&longs;t ei quod ex <lb/>differentia ip&longs;arum <lb/>FV, G, fit & EK, ergo <lb/>quod ex differentia <lb/>ip&longs;acum H, G, fit & K, æquale erit ei quod ex differentia ip&longs;arum <lb/>FV, G, fit & EK; æqualitatem ad proportionem reuocando, erit vt <lb/>differentia grauitatum. </s> |
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966 |
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967 <s>H, G, ad grauitatem EK, ita differentia <lb/>grauitatum FV, G, ad grauitatem K, quod erat primo loco: demon­<lb/>&longs;trandum. </s></p><p type="main"> |
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968 |
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969 <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> |
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970 |
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971 <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, æquatur­<lb/>ri quod, ex H, fit & EK, minus eo quod ex H, & E, quod enim 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, & ex 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, differentia 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 ei quod 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 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"> |
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972 |
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973 <s>Alia breuior Theorematis demon&longs;tratio. </s></p><p type="main"> |
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974 |
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975 <s>RESVMATVR eadem figura vt &longs;upra. </s> |
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976 |
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977 <s>Quoniam igitur <lb/>corpus D, æquale e&longs;t corpori Q, magnitudine, & portio C, æqualis <lb/>portioni L, erit vt D, ad Q, ita C, ad L, & permutando vt D, ad C, ita <lb/>Q, ad L, & quoniam eiu&longs;dem &longs;unt generis D, C, &longs;imiliter & Q, L, * erit<arrow.to.target n="note52"/><lb/>vt grauitas corporis D, hoc e&longs;t vt EK; ad K, ita H, ad V. <!-- KEEP S--></s></p><p type="margin"> |
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978 |
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979 <s><margin.target id="note52"/>4.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main"> |
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980 |
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981 <s>Similiter quoniam ponuntur æqualia magnitudine corpora A, P, <lb/>& æquales quoque portiones B, O, erit vt A, ad P, ita B, ad O, & per­<lb/>mutando vt A, ad B, ita P, ad O, &longs;ed eiu&longs;dem &longs;unt generis A, B, &longs;imili­<lb/>ter & P, O, * vt igitur grauitas corporis A, id e&longs;t vt EK, ad E, ita erit<arrow.to.target n="note53"/><lb/>G, ad F, & per conuer&longs;ionem rationis erit vt EK, ad K, ita G, ad G, <lb/>minus F, &longs;ed demon&longs;tratum e&longs;t, vt EK, ad K, ita e&longs;&longs;e H, ad V, ergo vt <lb/>H, ad V, ita erit G, ad G, minus F, & permutando vt H, ad G, ita V, <lb/>ad G, minus F, & diuidendo vt H, minus G, ad G, ita erit FV, minus <lb/>G, ad G, minus F, rur&longs;us permutando erit vt H, minus G, ad FV, mi­<lb/>nus G, ita G, ad G, minus F, &longs;ed vt EK, ad K, ita e&longs;t G, ad G, minus F, <lb/>vt e&longs;t demon&longs;tratum, ergo vt H, minus G, ad FV, minus G, ita erit <lb/>EK, ad K, quare permutando vt H, minus G, ad EK, ita erit FV, mi­<lb/>nus G, ad K, quod e&longs;tò primum. </s></p><p type="margin"> |
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982 |
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983 <s><margin.target id="note53"/>4.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main"> |
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984 |
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985 <s>Dico quoque vt H, minus G, ad EK, ita e&longs;&longs;e H, minus FV, ad E. <!-- KEEP S--></s> |
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986 |
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987 <s><lb/>Quoniam enim o&longs;ten&longs;um e&longs;t vt EK, ad K, ita e&longs;&longs;e H, ad V, erit per <lb/>conuer&longs;ionem rationis vt EK, ad E, ita H, ad H, minus V, &longs;ed demon­<lb/>&longs;tratum e&longs;t vt EK, ad E, ita e&longs;&longs;e G, ad F, ergo vt H, ad H, minus V, ita <lb/>erit G, ad F, & permutando vt H, ad G, ita H, minus V, ad F, & diui­<lb/>dendo vt H, minus G, ad G, ita erit H, minus FV, ad F, & permutan­<lb/>do vt H, minus G, ad H, minus FV, ita G, ad F, &longs;ed vt EK, ad E, ita e&longs;t <lb/>G, ad F, vt e&longs;t demon&longs;tratum, ergo vt H, minus G, ad H, minus FV, <lb/>ita erit EK, ad E, quare permutando, erit vt H, minus G, ad EK, ita <lb/>H, minus FV, ad E, quod erat &longs;ecundo loco demon&longs;trandum. </s></p><p type="main"> |
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989 <s>Svpere&longs;t igitur vt dicamus, qua ratione ex grauitate aúri <lb/>cogno&longs;ci po&longs;&longs;it eius qualitas; id quod ex ijs, quæ dicta <lb/>&longs;unt facilè colligitur; &longs;i videlicet nota fiat cuiu&longs;uis ma&longs;&longs;æ auri <lb/>grauitas, quam habet tum in aere, tum in aqua. </s> |
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991 <s>Sed ante <lb/>omnia, duo nobis &longs;unt præmittenda, & explicanda. </s> |
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992 |
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993 <s>nimirum <lb/>quid &longs;it aurum 24. partium, &longs;eu (vt vulgo dicitur) di 24. ca­<pb xlink:href="029/01/067.jpg" pagenum="60"/>ratti, quidue pauciorum, hoc e&longs;t penes quid attendatur di <lb/>uer&longs;a auri qualitas. </s> |
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995 <s>Deinde quomodo aurum alligent Auri <lb/>fices, vel alij ad quos alligandi officium &longs;pectat. </s> |
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997 <s>His enim <lb/>cognitis, non erit difficile, id quod proponitur, certa aliqua <lb/>ratione, a&longs;&longs;equi. </s></p><p type="main"> |
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999 <s><emph type="italics"/>Aurum igitur<emph.end type="italics"/> 24. <emph type="italics"/>partium appellatur aurum purum, pauciorum <lb/>vero dicitur non purum, &longs;ed aliquo alio metallo, vel pluribus affe­<lb/>ctum. </s> |
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1000 |
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1001 <s>& quia hæc affectio multiplex e&longs;t, ideo etiam auri qualitas, <lb/>qua ex varia mixtione na&longs;citur, varia &longs;it est nece&longs;&longs;e : quamuis vna <lb/>tantum &longs;it qualitas auri puri. </s> |
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1002 |
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1003 <s>Qualitas enim auri in quouis cor <lb/>pore propo&longs;ito, exprimitur partibus auri puri, 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 habent illæ partes in grauitate ad totum corpus: quod <lb/>exemplo clarius explicabitur in hunc modum.<emph.end type="italics"/></s></p><p type="main"> |
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1004 |
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1005 <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> |
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1006 |
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1007 <s>Totum igitur illud corpus aureum ab <lb/>initio propo&longs;itum, &longs;i adhuc 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> |
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1008 |
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1009 <s>eo quod tota illa maßa mista,<emph.end type="italics"/> 20. <emph type="italics"/>tantum vncias auri puri con­<lb/>tinuerit. </s> |
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1010 |
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1011 <s>Immo non &longs;olum illa ma&longs;&longs;a auri, &longs;ed etiam illa cuius ip&longs;a <lb/>fui&longs;&longs;et pars, vel que ip&longs;ius fui&longs;&longs;et quæcunque pars dicetur<emph.end type="italics"/> 20, <emph type="italics"/>par­<lb/>tium. </s> |
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1012 |
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1013 <s>Neque enim in alligationibus metallorum, alia est alli­<lb/>gatio partium, alia totius, &longs;ed vtrorunque vna eademque e&longs;t qua­<lb/>litas.<emph.end type="italics"/></s></p><p type="main"> |
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1014 |
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1015 <s><emph type="italics"/>Et hoc est quod Aurifices in inuaestigatione qualitatis auri ob­<lb/>&longs;eruant. </s> |
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1016 |
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1017 <s>Non enim purificant totum corpus propo&longs;itum, &longs;ed ali­<lb/>quam eius particulam etiam perexiguam, quam &longs;olam ad aurum pu­<lb/>rum reducunt. </s> |
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1018 |
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1019 <s>hac enim reducta, non &longs;olum recte definiunt cuius <lb/>fuerit qualitatis particula illa purificata ante purificationem; ve­<lb/>rum etiam cuius fuerit qualitatis, & quot partium fuerit illud cor­<lb/>pus, à quo eadem particula detracta fuit, & illud, quod adhuc &longs;u­<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> |
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1020 |
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1021 <s>Eius enim quali­<lb/>tatem &longs;i forte aurifices inuestigare velint, detrahent ex eo particu­<lb/>lam, verbi gratia, vnius vnciæ, vel quod idem est particulam<emph.end type="italics"/> 24. <pb xlink:href="029/01/068.jpg" pagenum="61"/><emph type="italics"/>&longs;crupulorum; & hanc particulam excoquent ad qualitatem o&longs;que­<lb/>cori puri. </s> |
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1022 |
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1023 <s>Est &longs;i quidem inuenerint, ex priori grauitate<emph.end type="italics"/> 24. <emph type="italics"/>&longs;ru­<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<lb/>cta, nec non & illud quod reman&longs;it post &longs;ubtractionem e&longs;&longs;e vel fui&longs;­<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> |
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1024 |
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1025 <s>Si vero deprehenderine grauitatem diminutant, ver­<lb/>bi gratia, nunc 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 reman&longs;it e&longs;&longs;e<emph.end type="italics"/> 20. <emph type="italics"/>partium & denique particulam 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></p><p type="main"> |
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1026 |
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1027 <s><emph type="italics"/>Et eodem modo pronuntiabunt de quibu&longs;cunque alijs auri qualit<lb/>atibus, &longs;ecundum partes auri puri, quas in qualibet ma&longs;&longs;a auri in­<lb/>uenerint, ea&longs;que vige&longs;imas quartas tutius grauitatis, non magnitu­<lb/>dinis. </s> |
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1028 |
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1029 <s>Nam cum in hac comparatione qualitatum, &longs;eor&longs;im habe a­<lb/>tur 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/>æquales, ex quibus<emph.end type="italics"/> 20. <emph type="italics"/>&longs;int auri, duæ argenti, & duæ æris; quamli­<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;um as, vt constat experienties<lb/>atque hinc constat quam apte ac conuenienter Aurifices vtantur <lb/>vocabulo partium. </s> |
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1030 |
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1031 <s>hac enim ratione eodem numero exprimunt vnam <lb/>quamque qualitatem auri cuiuslibet ma&longs;&longs;æ propo&longs;itæ. </s> |
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1032 |
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1033 <s>Sed nunc ad <lb/>&longs;ecundum veniamus & modum alligationis. </s> |
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1034 |
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1035 <s>quem ydem ob&longs;eruant <lb/>breuiter adnotemus.<emph.end type="italics"/></s></p><p type="main"> |
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1036 |
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1037 <s><emph type="italics"/>Inter varias autem & multiplices auri compo&longs;itiones quibus <lb/>cum alijs metallis alligari potest, eam retinuere aurifices, quam diu­<lb/>turna experientia deprehenderunt omnibus alijs 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> |
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1038 |
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1039 <s>Et quidem <lb/>&longs;ipartes auri excipias, aris atque 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&longs;se mixtionis modum <lb/>longe optimum.<emph.end type="italics"/></s></p><p type="main"> |
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1040 |
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1041 <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 xlink:href="029/01/069.jpg" pagenum="62"/><emph type="italics"/>reliquas partes quæ d &longs;unt ad<emph.end type="italics"/> 24, <emph type="italics"/>explent argento & ære, &longs;umendo <lb/>ex utroque metallo partes aquales in grauitate; atque bis rite inter &longs;e <lb/>permixtis componunt aurum de&longs;ideratæ qualitatis: eamque deno­<lb/>minant à partibus auri puri in mixtione a&longs;umptis. </s> |
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1042 |
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1043 <s>Et quoniam <lb/>non prodiret tale pror&longs;us quale facere intendunt, &longs;ed paulo perfe­<lb/>ctius; propterea quod auri quidem partes in mixtione maneant, ex <lb/>argenta vero & are aliquid deperdatur, &longs;olent Aurifices tanto plus <lb/>mi&longs;cere argenti & aris quantum perdi po&longs;&longs;e deprehenderunt.<emph.end type="italics"/></s></p><p type="main"> |
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1044 |
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1045 <s><emph type="italics"/>Verum nostra intentio non est omnia quæ 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 & reliquam qua deest ad<emph.end type="italics"/> 24, <emph type="italics"/>partes con&longs;tare dimidia <lb/>parte argenti, & dimidia aris in grauitate. </s> |
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1046 |
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1047 <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;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, & <lb/>vna medietas reliquarum partium, quæ partibus auri de&longs;unt ad <lb/>complendas partes<emph.end type="italics"/> 24, <emph type="italics"/>&longs;it argenti, & reliqua medietas &longs;it æris. </s> |
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1048 |
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1049 <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 & aqua, eamque qualitatem duplici via inuestigabimus, <lb/>vna per calculum, per tabellam altera: & quia ad calculum &longs;pe­<lb/>ctant ea, quæ &longs;uperius inuenimus, de grauitate metallorum huc re­<lb/>ferenda cen&longs;uimus quæ hic &longs;unt nece&longs;&longs;aria, cuiu&longs;modi &longs;unt auri, <lb/>argenti, atque æris grauitas, quam obtinent in aere, & aqua, quo­<lb/>quidem ita &longs;e habet vt &longs;equitur.<emph.end type="italics"/></s></p><p type="main"> |
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1050 |
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1051 <s><emph type="italics"/>Auri puri grauitas, quæ 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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1052 |
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1053 <s><emph type="italics"/>Argenti grauitas, quæ in aere e&longs;t<emph.end type="italics"/> 31, <emph type="italics"/>erit in aqua<emph.end type="italics"/> 28. </s></p><p type="main"> |
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1054 |
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1055 <s><emph type="italics"/>Aeris grauitas; quæ in aere est<emph.end type="italics"/> 9, <emph type="italics"/>erit in aqua<emph.end type="italics"/> 8. </s></p><p type="head"> |
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1056 |
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1057 <s><emph type="italics"/>Item.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="main"> |
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1058 |
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1059 <s><emph type="italics"/>Aurum ad aquam &longs;e habet in grauitate vt<emph.end type="italics"/> 19, <emph type="italics"/>ad<emph.end type="italics"/> 1. <!-- KEEP S--></s></p><p type="main"> |
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1060 |
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1061 <s><emph type="italics"/>Argentum ad aquam &longs;e habet in grauitate vt<emph.end type="italics"/> 31, <emph type="italics"/>ad<emph.end type="italics"/> 3. <!-- KEEP S--></s></p><p type="main"> |
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1062 |
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1063 <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. <!-- KEEP S--></s></p><p type="main"> |
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1064 |
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1065 <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 xlink:href="029/01/070.jpg" 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;ubintelligenda 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;equenter * grauitas aquæ magnitu­<emph.end type="italics"/><arrow.to.target n="note54"/><lb/><emph type="italics"/>dinem habentis æqualem <gap/> corpori misto<gap/><emph.end type="italics"/> <gap/>; <emph type="italics"/>quare corpus <lb/>mi&longs;tum ex argento & are ad corpus aqueum eiu&longs;dem magnitudi­<lb/>nis, rationem habebit <gap/> grauitate vt<emph.end type="italics"/> 18, <emph type="italics"/>ad<emph.end type="italics"/> 1 22/11, <emph type="italics"/>velvt<emph.end type="italics"/> 1, <emph type="italics"/>ad<emph.end type="italics"/><lb/><gap/>, <emph type="italics"/>vel denique in numeris integris vt<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 ratio.<emph.end type="italics"/></s></p><p type="margin"> |
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1066 |
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1067 <s><margin.target id="note54"/>5. <emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main"> |
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1068 |
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1069 <s><emph type="italics"/>Quibus &longs;ic con&longs;titutis inuenietur qualitas auri cuiu&longs;cumque hoc <lb/>modo. </s> |
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1070 |
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1071 <s>Sit exemp.: gratia propo&longs;ita aliqua ma&longs;&longs;a aurea, cuius gra­<lb/>uitas in aere &longs;it vnc<emph.end type="italics"/> 24, <emph type="italics"/>& oporteat invenire cuius qualitatis &longs;it ip­<lb/>sum aurum. </s> |
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1072 |
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1073 <s>Ponderetur ea ma&longs;&longs;a in aqua & habeat grauitatem <lb/>vnciarum<emph.end type="italics"/> 22 2<gap/>/5301, <emph type="italics"/>ergo * grauitas aquæ magnitudinem haben­<emph.end type="italics"/><arrow.to.target n="note55"/><lb/><emph type="italics"/>tis æqualem propo&longs;itæ ma&longs;&longs;æ erit vnc.<emph.end type="italics"/> 1 2431/5301. </s></p><p type="margin"> |
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1074 |
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1075 <s><margin.target id="note55"/>5.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="main"> |
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1076 |
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1077 <s><emph type="italics"/>Deinde inueniatur grauitas aquæ magnitudine æqualis 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 aquæ.<emph.end type="italics"/></s></p><p type="main"> |
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1078 |
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1079 <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"/>vntiæ, ad alium, nume­<lb/>rus enim quartus, nempe vnc.<emph.end type="italics"/> 2<gap/> <emph type="italics"/>erit grauitas aquæ, magnitu­<lb/>dine æqualis corpori mi&longs;to ex argento & ære, cuius grauitas est in <lb/>ærè vnc.<emph.end type="italics"/> 24, <emph type="italics"/>corpus enim ita mi&longs;tum, ad corpus aqueum eiu&longs;dem ma­<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"> |
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1080 |
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1081 <s><emph type="italics"/>Atque ita habebuntur tres gravitates trium aquæ quantitatum, <lb/>quarum prima æquatur auro puro<emph.end type="italics"/> 24, <emph type="italics"/>unciarum, &longs;ecunda ma&longs;&longs;a <lb/>propo&longs;itæ<emph.end type="italics"/> 24, <emph type="italics"/>vnciarum, & reliqua corpori mi&longs;to ex argento & ære<lb/>&longs;imiliter<emph.end type="italics"/> 24, <emph type="italics"/>vnciarum quæ quidem tres grauitates in numeris di­<lb/>&longs;ponantur eo ordine, quo &longs;equitur.<emph.end type="italics"/></s></p><p type="table"> |
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1082 |
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1083 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><p type="main"> |
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1084 |
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1085 <s><emph type="italics"/>Deinde quæratur 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/>differentia inter &longs;ecundam aquæ grauitatem & tertiam, quæ 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"> |
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1086 |
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1087 <s>TABELLE WAR HIER<lb/><emph type="italics"/>erit denominator qualitatis auri de qua quaritur quia ille termi­<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"> |
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1088 |
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1089 <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<lb/>tes vige&longs;ima quartæ<emph.end type="italics"/> 24, <emph type="italics"/>unciarum, quæ constituunt grauitatem to­<lb/>tius ma&longs;&longs;æ. </s> |
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1090 |
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1091 <s>hinc fit quod eædem<emph.end type="italics"/> 20. <emph type="italics"/>vnc. </s> |
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1092 |
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1093 <s>immediate denominent au­<lb/>rum propo&longs;itum eße<emph.end type="italics"/> 20, <emph type="italics"/>partium. </s> |
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1094 |
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1095 <s>Quando vero grauitas totius ma&longs;­<lb/>&longs;a non 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 exemplo clarius apparebit.<emph.end type="italics"/></s></p><p type="main"> |
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1096 |
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1097 <s><emph type="italics"/>Sit enim propo&longs;ita alia 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 numerus &longs;ubtrahatur 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"/>reliquus numerus<emph.end type="italics"/> 313, * <emph type="italics"/>erit grauitas aquæ <lb/>propo&longs;itæ ma&longs;&longs;æ magnitudine æqualis. </s> |
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1098 |
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1099 <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 exemplo factitatum est. </s> |
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1100 |
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1101 <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, 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 aquæ, magnitudine <lb/>aqualis corpori mi&longs;to ex argento & ære grauitatem habenti eandem <lb/>cum eadem ma&longs;sa propo&longs;ita. </s> |
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1102 |
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1103 <s>Atque hæ tres grauitates aquæ &longs;criban­<lb/>tur eo ordine quo &longs;upra; inuenti&longs;que differentijs inter primam &<emph.end type="italics"/></s></p><p type="margin"> |
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1104 |
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1105 <s><margin.target id="note56"/>5.<emph type="italics"/>huius<emph.end type="italics"/></s></p><p type="table"> |
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1106 |
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1107 <s>TABELLE WAR HIER<lb/><emph type="italics"/>tertiam, nec non inter &longs;ecundam & tertiam, quæ &longs;unt<emph.end type="italics"/> 272, 238; <emph type="italics"/>sta­<lb/>tuatur pro primo proportionis termino prior differentia<emph.end type="italics"/> 272, <emph type="italics"/>& pro <lb/>tertio posterior<emph.end type="italics"/> 238. <emph type="italics"/>grauitas vero ma&longs;&longs;æ propo&longs;itæ<emph.end type="italics"/> 5301, <emph type="italics"/>ponatur <lb/>pro&longs;ecundo termino, & quæratur terminus quartus, qui in præ-<emph.end type="italics"/></s></p><p type="table"> |
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1108 |
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1109 <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> |
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1110 |
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1111 <s>Sed quoniam hæc grauitas non est expre&longs;­<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"> |
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1112 |
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1113 <s><margin.target id="note57"/>19. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s></p><p type="main"> |
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1114 |
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1115 <s><emph type="italics"/>manam qualitatis auri pronuntiationem requiritur, vt &longs;upra mul­<lb/>tis o&longs;tendimus, reuocanda erit ad partes vige&longs;imas quartas hoc est <lb/>ad partes, qualium tota propo&longs;ita ma&longs;&longs;a est<emph.end type="italics"/> 24, <emph type="italics"/>quod factu non est <lb/>difficile. </s> |
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1116 |
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1117 <s>Nam &longs;i fiat vt tota grauitas ma&longs;&longs;æ propo&longs;itæ<emph.end type="italics"/> 5301, <emph type="italics"/>at gra­<lb/>uitatem auri puri<emph.end type="italics"/> 4638 45/116, <emph type="italics"/>velvt<emph.end type="italics"/> 272, <emph type="italics"/>ad<emph.end type="italics"/> 238, <emph type="italics"/>cum vtrobique <lb/>eadem &longs;it ratio ita<emph.end type="italics"/> 24, <emph type="italics"/>ad alium numerum. </s> |
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1119 <s>procul dubio quartus nu­<lb/>merus proportionalis, erit ille qui quæritur. </s> |
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1120 |
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1121 <s>Est autem hic quar­<lb/>tus numerus<emph.end type="italics"/> 21. <emph type="italics"/>Quare aurum ma&longs;&longs;æ propo&longs;itæ appellabitur par­<lb/>tium<emph.end type="italics"/> 21. </s></p><p type="main"> |
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1122 |
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1123 <s><emph type="italics"/>Ex his igitur patet in inuenienda auri qualitate primum pro­<lb/>portionis terminum<emph.end type="italics"/> 272, <emph type="italics"/>& &longs;ecundum<emph.end type="italics"/> 5301, <emph type="italics"/>perpetuo manere eo&longs;­<lb/>dem, quia primus terminus est differentia inter grauitates primæ, <lb/>& tertiæ aquæ, quæ nunquam mutantur, nam illæ aquæ magnitu­<lb/>dine &longs;unt æquales altera auro puro, reliqua mi&longs;to ex argento & ære, <lb/>quæ corpora aureum &longs;cilicet & mistum &longs;emper ponuntur eiu&longs;dem <lb/>grauitatis nempe<emph.end type="italics"/> 5301, <emph type="italics"/>Secundus vero terminus<emph.end type="italics"/> 5301, <emph type="italics"/>e&longs;t grauitas <lb/>ma&longs;&longs;æ propo&longs;itæ, quæ &longs;i maior fuerit, vel minor, ad eam facile reuo­<lb/>cabitur. </s> |
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1124 |
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1125 <s>Vnde in posterum &longs;olum opus erit inuenire tertium pro­<lb/>portionis terminum, hoc est differentiam inter grauitates &longs;ecundæ <lb/>& tertiæ aquæ.<emph.end type="italics"/></s></p><p type="main"> |
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1126 |
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1127 <s><emph type="italics"/>Sed vt hoc etiam exemplo illustretur, proponatur aliqua ma&longs;&longs;a <lb/>auri, cuius <expan abbr="inue&longs;tigãda">inue&longs;tiganda</expan> &longs;it qualitas, & &longs;it ip&longs;ius ma&longs;&longs;æ grauitas qui­<lb/>dem in aere<emph.end type="italics"/> 837, <emph type="italics"/>in aqua vero<emph.end type="italics"/> 784, <emph type="italics"/>ergo * grauitas aquæ magnitu­<emph.end type="italics"/><arrow.to.target n="note58"/><lb/><emph type="italics"/>dinem habentis æqualem propo&longs;itæ ma&longs;&longs;æ erit<emph.end type="italics"/> 53, <emph type="italics"/>differentia enim <lb/>inter primam, & &longs;ecundam grauitatem est<emph.end type="italics"/> 53. </s></p><p type="margin"> |
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1128 |
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1129 <s><margin.target id="note58"/>5.<emph type="italics"/>huiu<emph.end type="italics"/></s></p><p type="main"> |
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1130 |
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1131 <s><emph type="italics"/>Ad inueniendum igitur tertium proportionis terminum manen­<lb/>tibus primis duobus<emph.end type="italics"/> 272, 5301, <emph type="italics"/>hæc erit ratio. </s> |
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1132 |
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1133 <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 videlicet 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 maßæ, 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></p><p type="table"> |
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1134 |
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1135 <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; ergo<emph.end type="italics"/> 215 1/3, <emph type="italics"/>erit quæ&longs;itus terminus, nempe proportionis ter­<emph.end type="italics"/></s></p><pb xlink:href="029/01/073.jpg" pagenum="66"/><p type="table"> |
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1136 |
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1137 <s>TABELLE WAR HIER<lb/><emph type="italics"/>tius. </s> |
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1138 |
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1139 <s>Quartus autem terminus<emph.end type="italics"/> 4196 25/116, <emph type="italics"/>indicabit grauitatem <lb/>auri puri, quod est in ma&longs;&longs;a propo&longs;ita, eam tamen indicabit in par­<lb/>tibus, qualibus tota ma&longs;&longs;a constat<emph.end type="italics"/> 5301, <emph type="italics"/>quæ quidem grauitas vt <lb/>auri qualitatem indicet, reuocanda erit ad partes qualium totæ <lb/>ma&longs;&longs;a propo&longs;ita est<emph.end type="italics"/> 24. <emph type="italics"/>&longs;i enim fiat vt<emph.end type="italics"/> 5301, <emph type="italics"/>ad<emph.end type="italics"/> 4196<gap/>, <emph type="italics"/>ita<emph.end type="italics"/> 24, <lb/><emph type="italics"/>ad<emph.end type="italics"/> 19, <emph type="italics"/>aurum propo&longs;itæ ma&longs;&longs;æ appellabitur partium<emph.end type="italics"/> 19. </s></p><p type="main"> |
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1140 |
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1141 <s><emph type="italics"/>Denique &longs;i quis hunc modum conferat cum illo, quem &longs;upra tradi­<lb/>dimus, cum argentum explorauimus, quod mi&longs;tum in aurea corona <lb/>credebatur; is liquido intelliget hic nihil aliud acce&longs;&longs;iße, ni&longs;i quod <lb/>loco argenti, a&longs;&longs;umptum &longs;it corpus ex argento & ære mistum, eo <lb/>quod hæc duo metalla tantum in alligationibus auri &longs;oleant adbibe­<lb/>ri, vt diximus. </s> |
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1142 |
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1143 <s>Quod &longs;i con&longs;taret plura alia a&longs;&longs;umpta e&longs;&longs;e, etiam in <lb/>quauis alia ratione, facile erit cuiuis ad &longs;imilitudinem huius, for­<lb/>mare alium modum, &longs;ed nos, ne longiores &longs;imus, ad v&longs;um &longs;equentis <lb/>tabulæ nos conferamus, qua illis con&longs;ultum volumus qui minus in <lb/>præceptis Arithmeticis &longs;unt exercitati, velillis, qui alias ob cau&longs;as <lb/>tabulis vti malunt, quam calculis.<emph.end type="italics"/></s></p><p type="main"> |
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1144 |
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1145 <s><emph type="italics"/>Hæc tabula accommodata e&longs;t primarie ad aurum vnius libræ, vt <lb/>apparet in <expan abbr="&longs;ecũda">&longs;ecunda</expan> ip&longs;ius columna in qua omnes numeri &longs;unt vnita­<lb/>tes, re&longs;pondentes &longs;ingulis Denominatoribus qualitatum auri, a deno <lb/>minatore partium<emph.end type="italics"/> 24, <emph type="italics"/>v&longs;que ad denominatorem qualitatis partis ò, <lb/>quamuis proprie loquendo nulla &longs;it qualitas auri partis nullius, quia <lb/>tunc non e&longs;&longs;et aurum, &longs;ed mistum ex argento & ære. </s> |
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1146 |
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1147 <s>Hos denomi­<lb/>natores auri omnes inuenies in prima columna &longs;ub titulo qualitatis. </s> |
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1148 |
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1149 <s><lb/>In columna vero &longs;ub titulo misti placuit etiam de&longs;cribere denomi­<lb/>natores mi&longs;ti ex argento & ære, vt vnico intuitu appareat quot par­<lb/>tes auri puri, & quot partes misti ex argento & ære contineantur <lb/>in &longs;ingulis qualitatibus.<emph.end type="italics"/></s></p><p type="main"> |
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1150 |
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1151 <s><emph type="italics"/>Porro in area tabulæ &longs;ub titulo grauitatis auri in aqua po&longs;ita est <lb/>grauitas auri cuiuslibet qualitatis quam obtinet in aqua, quæ qua <lb/>ratione inueniatur, dicetur inferius vbi agetur de compo&longs;itione <lb/>eiu&longs;dem tabulæ.<emph.end type="italics"/></s></p><p type="main"> |
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1152 |
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1153 <s><emph type="italics"/>V&longs;us eius &longs;unt duo, quorum alterum titulus indicat, nimirum vt <lb/>tabulæ beneficio reperiatur ex grauitate auri quam habet in aere & <lb/>aqua, eius qualitas. </s> |
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1154 |
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1155 <s>Alter vero est vt cogno&longs;catur grauitas in aqua, <lb/>quando vnà cum grauitate quam aliquod aurum habet in aere da­<lb/>tur ip&longs;ius qualitas. </s> |
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1156 |
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1157 <s>& de hoc v&longs;u cum &longs;it &longs;implicior prius nobis erit <lb/>agendum.<emph.end type="italics"/></s></p><pb xlink:href="029/01/074.jpg" pagenum="67"/><p type="table"> |
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1158 |
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1159 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><p type="table"> |
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1160 |
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1161 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><pb xlink:href="029/01/075.jpg" pagenum="68"/><p type="main"> |
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1162 |
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1163 <s><emph type="italics"/>Quæratur exempli gratia quam habet grauitatem in aqua <expan abbr="aurũ">aurum</expan> <lb/>purum &longs;eu aurum<emph.end type="italics"/> 24, <emph type="italics"/>partium cuius grauitas in aere est lib.<emph.end type="italics"/> 1. <emph type="italics"/>Hæc <lb/>in &longs;upremo ordine è regione denominatoris partium 24, &longs;ub titulo <lb/>grauitatis auri in aqua, datur vnc. </s> |
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1164 |
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1165 <s>1<emph.end type="italics"/> 1. <emph type="italics"/>Scrup.<emph.end type="italics"/> 8, <emph type="italics"/>Gran.<emph.end type="italics"/> 20 172/1767, <lb/><emph type="italics"/>quæ fractio licet exprimi po&longs;&longs;it minoribus numeris <expan abbr="n&etilde;pe">nempe</expan><emph.end type="italics"/> 4/19, <emph type="italics"/>libuit <lb/>tamen illam maiorem in tabula ponere, vt omnes fractiones totius <lb/>tabulæ e&longs;&longs;ent eiu&longs;dem denominationis, & re&longs;ponderent denomina­<lb/>toribus fractionum quæ habentur in tabella partis proportionalis.<emph.end type="italics"/></s></p><p type="main"> |
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1166 |
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1167 <s><emph type="italics"/>Rur&longs;um quæratur quam habet grauitatem in aqua aurum ite­<lb/>rum vnius libræ, qualitatis vero<emph.end type="italics"/> 20, <emph type="italics"/>partium. </s> |
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1168 |
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1169 <s>quam &longs;i in tabula <lb/>quæras, inuenies &longs;ub eodem titulo è regione denominatoris<emph.end type="italics"/> 20, <emph type="italics"/>par­<lb/>tium. </s> |
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1170 |
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1171 <s>vnc. </s> |
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1172 |
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1173 <s>11, Scrup. <!-- REMOVE S-->6, Gran.<emph.end type="italics"/> 9 177/1767. <emph type="italics"/>eademque est ratio de re­<lb/>liquis.<emph.end type="italics"/></s> |
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1174 |
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1175 </p><p type="main"> |
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1176 |
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1177 <s><emph type="italics"/>Quando vero propo&longs;itum aurum non est vnius Libræ; tunc opus <lb/>erit ratiocinatione proportionis, in qua pro primo termino ponatur <lb/>vna libra auri propo&longs;itæ qualitatis, pro &longs;ecundo termino, grauitas <lb/>eidem re&longs;pondens in aqua quam tabula exhibet, pro tertio vero ter­<lb/>mino collocetur vera grauitas auri propo&longs;iti. </s> |
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1178 |
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1179 <s>Quartus enim ter­<lb/>minus exhibebit grauitatem ip&longs;ius auri in aqua. </s> |
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1180 |
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1181 <s>Vt &longs;i propo&longs;itum <lb/><expan abbr="aurũ">aurum</expan> &longs;it trium lib. |
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1182 |
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1183 qualitatis vero 18, partium. </s> |
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1184 |
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1185 <s>fiat vt lib. |
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1186 |
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1187 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> |
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1188 |
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1189 <s>9, Scrup. <!-- REMOVE S-->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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1190 |
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1191 |
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1192 |
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1193 <s>Et &longs;ic de alijs.<emph.end type="italics"/></s></p><p type="main"> |
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1194 |
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1195 <s><emph type="italics"/>Quod vero ad priorem v&longs;um attinet, is per&longs;imilis est præcedenti, <lb/>& æquè facilis quando grauitas auri quam in aere & aqua habet, in <lb/>tabula reperitur precise. </s> |
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1196 |
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1197 <s>Nam &longs;i proponatur exemp. </s> |
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1198 |
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1199 <s>gratia aurum <lb/>vnius lib. |
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1200 |
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1201 habens in aqua <expan abbr="grauitas&etilde;">grauitatem</expan> vnc.<emph.end type="italics"/> 11. <emph type="italics"/>Scr.<emph.end type="italics"/> 7, <emph type="italics"/>Gra.<emph.end type="italics"/> 14 1152/1767, <lb/><emph type="italics"/>quoniam hæc grauitas reperitur in tabula è regione qualitatis auri<emph.end type="italics"/><lb/>22, <emph type="italics"/>partium; manife&longs;tum e&longs;t totidem partium e&longs;&longs;e <expan abbr="aurũ">aurum</expan> propo&longs;itum.<emph.end type="italics"/></s></p><p type="main"> |
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1202 |
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1203 <s><emph type="italics"/>Quando vero grauitas auri in aere quiaem e&longs;t vnius lib. |
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1204 |
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1205 in aqua <lb/>vero grauitatem habet, quæ in tabula non reperitur, indicium erit <lb/>aurum propo&longs;itum non e&longs;&longs;e aliquot partium præcisè, &longs;ed annexam <lb/>habere aliquam fractionem, quæ per partem proportionalem inue­<lb/>nietur hoc modo.<emph.end type="italics"/></s></p><p type="main"> |
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1206 |
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1207 <s><emph type="italics"/>Proponatur aurum vnius libræ in aqua habens grauitatem vnc. </s> |
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1208 |
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1209 <s><lb/>11. Scrup. <!-- REMOVE S-->6, Gran.<emph.end type="italics"/> 16 264/1767, <emph type="italics"/>qualis in tabula non reperitur Gra­<lb/>uitas enim proxime maior est vnc. </s> |
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1210 |
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1211 |
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1212 |
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1213 <s>1<emph.end type="italics"/> 1. <emph type="italics"/>Scrup. <!-- REMOVE S-->6, Gran.<emph.end type="italics"/> 23 1551/1767, <lb/><emph type="italics"/>re&longs;pondens auro 21, partium, & grauitas proxime minor e&longs;t vnc.<emph.end type="italics"/> 11, <lb/><emph type="italics"/>Scrup. <!-- REMOVE S-->6, Gran.<emph.end type="italics"/> 9 177/1767, <emph type="italics"/><expan abbr="earumq;">earumque</expan> differentia e&longs;t Gran.<emph.end type="italics"/> 14 1374/1767, <lb/><emph type="italics"/>quem admodum & inter qua&longs;cunque duas alias grauitates proxi­<emph.end type="italics"/><pb xlink:href="029/01/076.jpg" pagenum="69"/><emph type="italics"/>mas eadem e&longs;t differentia, propterea quod omnes grauitates in tabu­<lb/>la procedunt per æqualem exce&longs;&longs;um, vel defectum, vt inferius de­<lb/>mon&longs;trabitur. </s> |
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1214 |
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1215 |
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1216 |
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1217 |
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1218 |
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1219 <s>Inueniatur quoque differentia inter eandem gra­<lb/>uitatem proxime minorem & inter grauitatem auri propo&longs;iti quam <lb/>habet in aqua, quæ <expan abbr="quid&etilde;">quidem</expan> e&longs;t Gran.<emph.end type="italics"/> 7 687/1767, <emph type="italics"/>& fiat vt<emph.end type="italics"/> 14 1374/1767, <lb/><emph type="italics"/>ad 1, ita<emph.end type="italics"/> 7 687/1767, <emph type="italics"/>ad alium numerum & inuenietur hæc fractio<emph.end type="italics"/><lb/>1/2, <emph type="italics"/>eademque adijcienda erit ad denominatorem 20, partium, vt <lb/>componatur totus denominator auri propo&longs;iti partium<emph.end type="italics"/> 20 1/2, <emph type="italics"/>& eo­<lb/>dem modo inueniendus erit denominator <expan abbr="cuiu&longs;cunq;">cuiu&longs;cunque</expan> alterius auri, <lb/>cuius grauitas in aqua, in tabula non reperitur.<emph.end type="italics"/></s></p><p type="main"> |
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1220 |
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1221 <s><emph type="italics"/>Ceterum qui volet contentus e&longs;&longs;e partibus vige&longs;imis quartis de­<lb/>nominatorum auri, is multo breuius a&longs;&longs;equetur quod quæritur, per <lb/>tabellam partis proportionalis. </s> |
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1222 |
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1223 <s>illic enim vnico ingre&longs;&longs;u offendet <lb/>partem proportionalem, quam quærit, vt in eodem exemplo apparet, <lb/>in quo differentia grauitatum auri erat Gran.<emph.end type="italics"/> 7 687/1767, <emph type="italics"/>quæ in <lb/>tabella partis proportionalis habetur præcisè è regione particularum<emph.end type="italics"/><lb/>12. <emph type="italics"/>Vnde concluditur, denominatorem auri propo&longs;iti e&longs;&longs;e partium<emph.end type="italics"/><lb/>20 12/24, <emph type="italics"/>vel quod idem est partium<emph.end type="italics"/> 20 1/2, <emph type="italics"/>vt prius. </s> |
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1224 |
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1225 <s>Quando vero <lb/>differentia grauitatum in tabella partis proportionalis non habetur <lb/>præcisè. </s> |
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1226 |
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1227 <s>accipiatur alia ip&longs;i propinquior & particula illi in latere <lb/>re&longs;pondens addatur denominatori auri ex primaria tabula extra­<lb/>cti. </s> |
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1228 |
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1229 <s>&longs;ic enim &longs;altem non errabitur in vna particula vige&longs;imaquarta <lb/>vnius partis denominatoris auri.<emph.end type="italics"/></s></p><p type="main"> |
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1230 |
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1231 <s><emph type="italics"/>Denique &longs;i proponatur aurum non vnius libræ &longs;ed vel plurium, <lb/>vel &longs;olum aliquot vnciarum. </s> |
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1232 |
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1233 <s>Reducenda erit eius grauitas quam <lb/>habet in aqua, ad grauitatem quam haberet &longs;i eßxset vnius libræ, id <lb/>quod ab&longs;oluetur per proportionis ratiocinationem, &longs;i pro termino <lb/>primo ponatur vera grauitas auri propo&longs;iti, pro &longs;ecundo, eiu&longs;dem <lb/>grauitas in aqua, & pro tertio lib. |
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1235 1, quartus enim terminus indica­<lb/>bit grauitatem in aqua re&longs;pondentem vni libræ auri propo&longs;iti. </s> |
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1236 |
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1237 <s>& <lb/>hac inuenta reliqua expedientur vt prius.<emph.end type="italics"/></s></p><p type="main"> |
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1238 |
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1239 <s><emph type="italics"/>Exemplum huius ca&longs;us hic non affero, quod per &longs;e res &longs;it clara. </s> |
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1241 <s><lb/>Sed illud tantum obiter aduertere placet, quod videtur pertinere <lb/>ad commodiorem v&longs;um tabulæ, videlicet vt ys in ca&longs;ibus in quibus <lb/>nece<32>arius est calculus, fractiones granorum ommittantur quando <lb/>minus valent quam<emph.end type="italics"/> 1/2, <emph type="italics"/>& quando valent plus, eorum loco, addatur <lb/>vnum granum reliquis granis, & &longs;i quando accidat hinc procreari <lb/>grana<emph.end type="italics"/> 24. <emph type="italics"/>tunc etiam grana ommittantur addita prius vnitate ad <lb/>&longs;crupula in tabula inuenta. </s> |
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1242 |
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1243 <s>hac enim ratione calculus erit expedi­<lb/>tior & error qui hinc oborietur erit in&longs;en&longs;ibilis.<emph.end type="italics"/></s></p><pb xlink:href="029/01/077.jpg" pagenum="70"/><p type="head"> |
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1244 |
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1245 <s>Compo&longs;itio eiu&longs;dem tabulæ.</s></p><p type="main"> |
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1246 |
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1247 <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 inueniatur 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> |
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1248 |
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1249 <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></p><p type="main"> |
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1250 |
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1251 <s><emph type="italics"/>Et primum &longs;it propo&longs;itum aurum purum, &longs;eu<emph.end type="italics"/> 24, <emph type="italics"/>partium. </s> |
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1252 |
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1253 <s>quo­<lb/>niam igitur grauitas auri in aere, ad grauitatem eiu&longs;dem in aqua <lb/>&longs;e habet vt 19, ad<emph.end type="italics"/> 18, <emph type="italics"/>fiat vt<emph.end type="italics"/> 19, <emph type="italics"/>ad<emph.end type="italics"/> 18, <emph type="italics"/>ita lib.<emph.end type="italics"/> 1, <emph type="italics"/>auri puri ad aliam <lb/>grauitatem nempe lib.<emph.end type="italics"/> 1<gap/>/19, <emph type="italics"/>quæ grauitas ad vncias, &longs;crupula, & <lb/>grana reuocata valet vnc.<emph.end type="italics"/> 11, <emph type="italics"/>Scrup.8. Gran.<emph.end type="italics"/> 20 4/19, <emph type="italics"/>atque hæc est <lb/>grauitas auri puri in aqua, quam in tabula è regione denominatoris<emph.end type="italics"/><lb/>24, <emph type="italics"/>po&longs;uimus; fractione excepta cuius loco &longs;ub&longs;tituta est fractio<emph.end type="italics"/><lb/>372/1767, <emph type="italics"/>propter cau&longs;am &longs;uperius allatam.<emph.end type="italics"/></s></p><p type="main"> |
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1254 |
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1255 <s><emph type="italics"/>Sit deinde propo&longs;itum quod vis aliud corpus aureum vnius libræ, <lb/>&longs;itque exemp. </s> |
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1256 |
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1257 <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 ex iam 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> |
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1258 |
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1259 <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. <!-- REMOVE S--><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> |
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1260 |
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1261 |
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1262 |
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1263 <s>1<emph.end type="italics"/> 1, <emph type="italics"/>Scrup.<emph.end type="italics"/>6. <emph type="italics"/>Gran.<emph.end type="italics"/> 9 177/1767, <lb/><emph type="italics"/>vt videre e&longs;t in tabula è regione denominatoris<emph.end type="italics"/> 20, <emph type="italics"/>partium. </s> |
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1264 |
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1265 <s>Atque <lb/>eodem modo deprehendentur grauitates auri in aqua <expan abbr="quarumcunq;">quarumcunque</expan> <lb/>aliarum qualitatum.<emph.end type="italics"/></s></p><p type="main"> |
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1266 |
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1267 <s><emph type="italics"/>Quia vero permolestum videri po&longs;&longs;et omnes grauitates totius <lb/>tabulæ hac via eruere; ob&longs;eruari quidem poterit prædicta Methodus <lb/>quando &longs;eor&longs;im inuestiganda fuerit alicuius auri grauitas in aqua, <lb/>in <expan abbr="cõpo&longs;itione">compo&longs;itione</expan> vero tabulæ &longs;ic forta&longs;&longs;is <expan abbr="cõpendio&longs;ius">compendio&longs;ius</expan> quis proce&longs;&longs;erit.<emph.end type="italics"/></s></p><p type="main"> |
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1268 |
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1269 <s><emph type="italics"/>Equidem, cum huius tabulæ con&longs;tructionem diligentius mecum<emph.end type="italics"/><pb xlink:href="029/01/078.jpg" pagenum="71"/><emph type="italics"/>pertracto, video grauitates illas, quæ in eius area de&longs;criptæ &longs;unt ne­<lb/>ce&longs;&longs;ario eadem differentia procedere, &longs;icut & denominatores quali­<lb/>tatum eadem differentia procedunt; atque adeo differentiam illam <lb/>e&longs;&longs;e eam, qua grauitas &longs;emiunciæ auri puri in aqua &longs;uperat in aqua <lb/><expan abbr="grauitat&etilde;">grauitatem</expan> &longs;emiunciæ mi&longs;ti, quæ <expan abbr="quid&etilde;">quidem</expan> differentia e&longs;t Gra.<emph.end type="italics"/> 14 1174/1767. <lb/><emph type="italics"/>Con&longs;iderentur enim quæcunque duæ grauitates proximæ in tabula <lb/>expre<32>æ, vt exemp. </s> |
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1270 |
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1271 <s>gratia. </s> |
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1272 |
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1273 <s>grauitates auri<emph.end type="italics"/> 20, <emph type="italics"/>&<emph.end type="italics"/> 19, <emph type="italics"/>partium. </s> |
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1274 |
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1275 <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="partiũ">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;ti. </s> |
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1276 |
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1277 <s>Quod <lb/>erat demon&longs;trandum. </s> |
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1278 |
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1279 <s>Et eadem est ratio de alijs grauitatibus, non <lb/>&longs;olum quæ in hac tabula de&longs;cribuntur, &longs;ed etiam de illis, quæ de&longs;cri­<lb/>bentur in alijs, copio&longs;ioribus, in quibus videlicet denominatores non <lb/>e&longs;&longs;ent partes integræ, &longs;ed partes partiam; dummodo etiam illæ partes <lb/>per vnam eandemque differentiam progrederentur.<emph.end type="italics"/></s></p><p type="main"> |
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1280 |
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1281 <s><emph type="italics"/>Quibus in hunc modum præ osten&longs;is, &longs;i construenda fuerit tabula <lb/>per continuam additionem eiu&longs;dem numeri, &longs;ic erit progrediendum. </s> |
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1282 |
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1283 <s><lb/>Primo inuenienda erit grauitas quam habet &longs;emiuncia auri puri in <lb/>aqua, quæ inuenietur &longs;i fiat vt<emph.end type="italics"/> 19, <emph type="italics"/>ad<emph.end type="italics"/> 18, <emph type="italics"/>ita &longs;emiuncia ad alium, <lb/>numenum qui &longs;it vnc.<emph.end type="italics"/> 9/19, <emph type="italics"/>is enim dabit grauitatem quæ&longs;itam, quæ <lb/>valet &longs;crup.<emph.end type="italics"/>1 1, <emph type="italics"/>Gran.<emph.end type="italics"/> 8 16/19. </s></p><p type="main"> |
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1284 |
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1285 <s><emph type="italics"/>Secundo quærenda est grauitas &longs;emiuncie misti in aqua, quæ ha­<lb/>bebitur &longs;i fiat vt<emph.end type="italics"/> 279, <emph type="italics"/>ad<emph.end type="italics"/> 250, <emph type="italics"/>ita &longs;emiuncia, ad alium numerum, qui <lb/>&longs;it vnc.<emph.end type="italics"/> 125/229, <emph type="italics"/>is enim dabit grauitatem quæ&longs;itam, quæ reducta ad <lb/>&longs;crupula, & grana valet &longs;crup.<emph.end type="italics"/> 10, <emph type="italics"/>Gran.<emph.end type="italics"/> 18 2/3. </s></p><p type="main"> |
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1287 <s><emph type="italics"/>Tertio exploranda est differentia inter duas grauitates proximè <lb/>inuentas, quam per &longs;ubtractionem inuenies Gran.<emph.end type="italics"/> 14 452/589, <emph type="italics"/>cuius <lb/>tamen fractio reducta e&longs;t ad partes<emph.end type="italics"/> 1767, <emph type="italics"/>nempe ad<emph.end type="italics"/> 1374/1767, <emph type="italics"/>propter <lb/>tabellam partis proportionalis.<emph.end type="italics"/></s></p><p type="main"> |
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1288 |
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1289 <s><emph type="italics"/>Postremo inuestiganda erit grauitas in aqua vnius libræ misti, <lb/>quæ inuenietur &longs;i grauitas &longs;ecundo loco reperta per<emph.end type="italics"/> 24, <emph type="italics"/>multiplice­<lb/>tur, productus enim numerus vnc.<emph.end type="italics"/> 10, <emph type="italics"/>&longs;crup. </s> |
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1290 |
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1291 <s>18, Gran.<emph.end type="italics"/> 969/1767, <emph type="italics"/>da­<lb/>bit quæ&longs;itam grauitatem. </s> |
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1292 |
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1293 <s>Qua in calce tabulæ de&longs;cripta, compo­<lb/>nentur reliquæ grauitates omnes per continuam additionem diffe­<lb/>rentiæ tertio loco inuentæ. </s> |
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1294 |
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1295 <s>Si enim addatur ad grauitatem auri <lb/>partis o, idest ad grauitatem misti vnius libræ in aqua, componetur<emph.end type="italics"/><pb xlink:href="029/01/079.jpg" pagenum="72"/><emph type="italics"/>grauitas auri<emph.end type="italics"/> 1, <emph type="italics"/>partis. </s> |
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1296 |
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1297 <s>addita vero ad grauitatem 1, partis, procrea­<lb/>bit grauitatem<emph.end type="italics"/> 2, <emph type="italics"/>partium, &c. </s> |
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1298 |
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1299 <s>propter rationem quam paulo ante <lb/>aperuimus.<emph.end type="italics"/></s></p><p type="main"> |
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1300 |
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1301 <s><emph type="italics"/>Hoc eodem artificio compo&longs;ita est quoque tabella partis propor­<lb/>tionalis, primo enim inuenta e&longs;t vige&longs;ima quarta pars differentiæ <lb/>&longs;ecundum quam tabula progreditur quam &longs;upra inuenimus e&longs;&longs;e <lb/>Gran.<emph.end type="italics"/> 14 1374/1707, <emph type="italics"/>cuius pars vige&longs;ima quarta est<emph.end type="italics"/> 1072/1707, <emph type="italics"/>deinde <lb/>hanc particulam addidimus primum &longs;ibi ip&longs;i, & produximus diffe­<lb/>rentiam<emph.end type="italics"/> 2, <emph type="italics"/>particularum Gran.<emph.end type="italics"/> 1 409/1707 <emph type="italics"/>& huic differentiæ <expan abbr="iterũ">iterum</expan> <lb/>adiecimus eandem particulam, & inuenimus pro tribus particulis <lb/>gran.<emph.end type="italics"/> 1 1497/1767. <emph type="italics"/>& ita deinceps progre&longs;&longs;i &longs;umus v&longs;que ad differen­<lb/>tiam Gran.<emph.end type="italics"/> 14 1374/1767, <emph type="italics"/>quæ re&longs;pondet<emph.end type="italics"/> 24, <emph type="italics"/>particulis, &longs;eu differentiæ, <lb/>&longs;ecundum quam tabula progreditur.<emph.end type="italics"/><lb/><emph type="italics"/>FINIS.<emph.end type="italics"/><!-- KEEP S--></s></p><p type="head"> |
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1302 |
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1303 <s>ERRATA SIC CORRIGE.<!-- KEEP S--></s></p><p type="table"> |
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1304 |
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1305 <s>TABELLE WAR HIER<!-- KEEP S--></s></p><p type="main"> |
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1306 |
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1307 <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> |
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1308 |
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1309 <s>34. <expan abbr="appo&longs;itã">appo&longs;itam</expan>, <lb/>quia madefacta papyrus, <expan abbr="dũ">dum</expan> imprimeretur recipiebat <expan abbr="verã">veram</expan> <expan abbr="men&longs;urã">men&longs;uram</expan>, ex&longs;iccata <lb/>breuiorem reddidit. </s> |
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1310 |
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1311 <s>Ita que &longs;i quarta decima 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;pondebunt.<emph.end type="italics"/></s></p><p type="main"> |
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1312 |
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1313 <s><emph type="italics"/>Denique pagina<emph.end type="italics"/> 38. <emph type="italics"/>&<emph.end type="italics"/> 39. <emph type="italics"/>tabularum præpo&longs;terus e&longs;t ordo. </s> |
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1314 |
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1315 <s>ex quo tamen nihil sr­<lb/>reris &longs;equitur &longs;i tituli recte accipiantur.<emph.end type="italics"/></s></p> </chap> </body> <back/> </text></archimedes> |