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<s>MARINI GHETALDI <lb></lb>PATRICII <lb></lb>RAGVSINI <lb></lb>PROMOTVS ARCHIMEDIS <lb></lb>SEV.De varijs corporum generibus <lb></lb>grauitate &amp; magnitudine <lb></lb>comparatis.</s></p><figure id="id.029.01.001.1.jpg" xlink:href="029/01/001/1.jpg"></figure><p type="head">

<s>ROMAE, <lb></lb>Apud Aloyſium Zannettum. </s>

<s>MDCIII. <lb></lb><emph type="italics"></emph>SVPERIORVM PERMISSV.<emph.end type="italics"></emph.end></s></p><pb xlink:href="029/01/002.jpg"></pb><p type="head">

<s>REVERENDISSIMO <lb></lb>SERAPHINO OLIVARIO <lb></lb>RAZZALIO. <lb></lb>PATRIARCH AE <lb></lb>ALEXANDRINO. <lb></lb>MARINUS GHETALDUS. S.P.D.</s></p><p type="main">

<s>Egregia ſanè Reuerendiſsime <lb></lb>PRÆSVL quod probe noſti, vete<lb></lb>rum ſapientum fœlicitas fuit. </s>

<s>Eam <lb></lb>enim cum ingenij præſtantia, tum <lb></lb>prærogativa temporis laudem occu<lb></lb>parunt, <expan abbr="quã">quam</expan> vel ſperare poſterioribus <lb></lb><expan abbr="temerariũ">temerarium</expan> ſit. </s>

<s>Et vero illis non ſolum nos plurimum de<lb></lb>bemus, quod plurima ipſi perfecere: verum etiam quod <lb></lb>quædam quaſi fundamenta iecere, quibus dum rerum <lb></lb><expan abbr="nouarũ">nouarum</expan> molem conamur imponere, nos quoque experi<lb></lb>ri noſtras vires, exercere induſtriam, remque ſapientiæ <lb></lb>publicam amplificare poſsimus. </s>

<s>Quo in genere magno<lb></lb>rum ego virorum ſtudium potius, quam gloriam æmu<lb></lb>latus ſuper vnum ex Archimedeis fundamentis, de ra<lb></lb>tione, qua varia corporum genera inter ſe grauitate &amp; <lb></lb>magnitudine comparantur, fabricatus nonnulla ſum: <pb xlink:href="029/01/003.jpg"></pb>quæ nunc omnium oculis expoſiturus, vt eam <expan abbr="ſuſtineã">ſuſtineam</expan>, <lb></lb>perſonam, quam ſemper recuſaui, patrocinium huiuſ<lb></lb>modi quærendum mihi exiſtimaui, quod &amp; imbecillita<lb></lb>tem meam contra obtrectatorum, ſi qui forte exiſterent, <lb></lb>calumnias ſuſtineret; &amp; imminentem famæ, exiſtimatio <lb></lb>niſque iacturam auerteret. </s>

<s>Vnus igitur tu occuriſti SE<lb></lb>RAPHINE qui &amp; <expan abbr="cõmodiſsime">commodiſsime</expan> mihi patrocinari poſ<lb></lb>ſes, &amp; quodam quaſi iure deberes. </s>

<s>Cum enim tu me ad <lb></lb>emittendum id opus hortatu tuo compuleris, videbatur <lb></lb>quodam iure ad tuam fidem eius tutela pertinere. </s>

<s><expan abbr="Tuq;">Tuque</expan> <lb></lb>is es, quem non modo rudes, ſed etiam docti ſuſpiciunt. </s>

<s><lb></lb>Habet noſtra hæc ætas, quos admitetur; habet quos ex<lb></lb>tollat præclaros viros, ſed quos tibi anteponat, non facile <lb></lb>inueniet. </s>

<s>Degis ea in Vrbe, quæ laudis mediocritatem <lb></lb>vel nunquam agnouit, vel ſemper contempſit: neque in <lb></lb>tanta maieſtate, tua deficit virtus, ſed bono in lumine po<lb></lb>ſita collucet magis. </s>

<s>In primis enim tua vitæ integritas <lb></lb>eiuſmodi eſt, vt non contenta domeſtico præcomo latiſ<lb></lb>ſime peruagetur. </s>

<s>Habent omnes quod prædicent, &amp; imi<lb></lb>tentur; habet quod excipiat gratiſsima memoria poſte<lb></lb>ritas vniuerſa. </s>

<s>Doctrinæ vero ea excellentia es, vt ea ſatis <lb></lb>omnibus clarus, &amp; illuſtris, non ſatis tibi, recum aſsidue <lb></lb>certes. </s>

<s>nec mirum, qui vel à primis incunabulis ad ſum<lb></lb>ma contendebas, ſi prouecta iam ætate vix habeas, quo <lb></lb>altius contendas. </s>

<s>Præclarum ſanè &amp; eximium vno do<lb></lb>ctrinæ genere; ſublime, atque admirabile multiplici ex<lb></lb>cellere. </s>

<s>Tu vero în omni genere laudem egregiam aſſe<lb></lb>cutus es abſolutam videlicet tibî gloriam propoſitam <lb></lb>voluiſti; <expan abbr="quiq;">quique</expan> intelligebas hominem ad honeſta omnia <pb xlink:href="029/01/004.jpg"></pb>genitum, nullam tibi rerum glorioſarum partem con<lb></lb>temnendam putaſti. </s>

<s>Placet incredibili rerum humana<lb></lb>rum vſui diuinarum rerum cognitionem adiungere; vt <lb></lb>habeat animus à caducis ad æterna, ſe conferendo, vnde <lb></lb>oblectamentum capiat, &amp; admirationem. </s>

<s>Philoſophiam <lb></lb>ita tenes, vt qui in maximis negotijs aſsidue verſatus es, <lb></lb>videaris ſemper fuiſſe otioſus. </s>

<s>Quid dicam de ſingulari <lb></lb><expan abbr="caq;">caque</expan> eximia rerum cœleſtium, <expan abbr="totiuſq;">totiuſque</expan> mundi cognitio<lb></lb>ne quam tu tanta cum auiditate ex reconditiſsimis Ma<lb></lb>thematicorum fontibus hauſiſti, vt illud aſſecutus in eo <lb></lb>genere iam ſis, quod alij in maxima tranquillitate, in <lb></lb>ſummo otio vix auſint optare? </s>

<s>Exitum tuorum laborum <lb></lb>feliciſsimum vides: gloria multiplici frueris, neque illa <lb></lb>precaria ſed tua &amp; quibuſdam quaſi gradibus ad ampliſ<lb></lb>ſimos: honores euehendus, in ea conſtitueris dignitate; in <lb></lb>qua pro ſacroſancta Eccleſia nunquam non excubando, <lb></lb>in peramplo tot illuſtrium vitorum Theatro non alienæ <lb></lb>gloriæ ſpectator, ſed actor tuæ conſiſtas. </s>

<s>Tu vero, quod <lb></lb>rarum eſt, laudem ſapientiæ, quæ vix vllos habet at ter<lb></lb>minos, humanitatis tuæ terminis circumſcribis; &amp; expo<lb></lb>nis omnibus; vt ex tanto ſonte perennes ad omnium or<lb></lb>dinum homines fluuii deducantur. </s>

<s>Felix qui ſolidæ <lb></lb>felicitatis cauſam &amp; initium in re conſtitutum ita foues, <lb></lb>vt cum alijs illam communicando, non imminuas, ſed <lb></lb>amplifices, prægrande videlicet non ſuccreſcentis, ſed <lb></lb>adultæ iam virtutis fœnus honorem ex honore, laudem <lb></lb>ex laude conſequi vberiorem, hæc illa ſapienti viro non <lb></lb>indigna liberalitas, quæ rerum preſtantiſsimarum poſ<lb></lb>ſeſsione non imminuta, in copia tenuitatem non inqui<pb xlink:href="029/01/005.jpg"></pb>rens vbertatis ipſa ſuæ dominæ nunquam debilitatur, <lb></lb>nunquam deficit. </s>

<s>Quin etiam iſto loco conſtitutus bo<lb></lb>narum literarum ſtudioſos complecteris, ac tueris. </s>

<s>Hæc <lb></lb>eſt vera germanæ nota ſapientiæ, cum, quas vires nanci<lb></lb>ſcitur, ijs ſapientiam alit, tuus animus &amp; tuæ ſapientiæ, &amp; <lb></lb>alienæ par eſt. </s>

<s>Hæc ſunt firmiſsima &amp; ſolidiſsima funda<lb></lb>menta ad æternam poſteritatis memoriam, quam licet <lb></lb>proficiſci iam videas ex iis quibus abundas animi orna<lb></lb>mentis, neſcio tamen quo pacto gratior nobis accidit, <lb></lb>cum ex aliorum etiam prædonio ſuſcipit in crementum. </s>

<s><lb></lb>Hinc domus tua floret doctiſsimorum familiaritatibus; <lb></lb>hinc nulli ad tuam conſuetudinem præcluditur aditus; <lb></lb>hinc plurimorum ſtudia commouentur; hinc illa ſapien<lb></lb>tum æmulatio &amp; admiratio: hinc omnium omnino or<lb></lb>dinum ad te concurſus; tanquam ad ſapientiſsimum hu<lb></lb>mani <expan abbr="diuiniq;">diuinique</expan> iuris patronum, &amp; interpretem. </s>

<s>Quid ego <lb></lb>igitur quem tibi ſexcentis eximiæ beneuolentiæ argu<lb></lb>mentis obligatum voluſti, perexigiuum hunc ingenij <lb></lb>mei partum expoſiturus, vnum te nominis &amp; exiſtima<lb></lb>tionis meæ patronum non ſuſcipiam equidem in am<lb></lb>pliſsimi Theatri lucem ſapientiſsimorum &lt;32&gt; hominum <lb></lb>conſpectum, quibus abundat hæc noſtra ætas inferre me <lb></lb>non magnopere cogitabam: Tu hortatus es dubitan<lb></lb>tem impuliſti vel reluctantem: tuere igitur obſequen<lb></lb>tem. </s>

<s>iam nunc mihi videor in exigua vel nulla ſpe lau<lb></lb>dis ex ingenio meo; ſempiternam nominis immortali<lb></lb>tatem ex tuo patrocinio conſecutus. </s>

<s>Vale. </s></p><p type="main">

<s>Romæ VII. Kal. Maij. MDCIII. </s>


</p><pb xlink:href="029/01/006.jpg"></pb><p type="head">

<s>BENEVOLO <lb></lb>LECTORI.</s></p><figure id="id.029.01.006.1.jpg" xlink:href="029/01/006/1.jpg"></figure><p type="main">

<s>D<emph type="italics"></emph>iversa. corporum genera duplici ra<lb></lb>tione comparari inter ſe à Mathematicis <lb></lb>poſſunt mole ac pondere. </s>


<s>Pondere compara<lb></lb>tio fit, cum inter corpora diuerſi generis mo <lb></lb>le aqualia, quæritur, quę ſit ratio ponderis: <lb></lb>quanto videlicet, vnum altero grauius, aut leuius ſit. </s>

<s>Ma<lb></lb>gnitudine autem ſit collatio, cum poſita pari grauitate, quæ<lb></lb>ritur, que ſit ratio magnitudinis; quanto ſit alterum altero <lb></lb>maius, aut minus. </s>

<s>Quæ comparatio mihi cum videretur &amp; <lb></lb>iucunda cognitu, &amp; vſum nonnullum habere, nec fuſe à quo<lb></lb>piam explicata, non ita pridem ſuper ea non nihil cœpi me <lb></lb>liris ſed nihil de luce ac publico cogitabam. Io enim ego ſum, <lb></lb>qui malim ſcire, quam neſci diſcere, quam docere. </s>

<s>Sed ta<lb></lb>men cum Michael Sægnerus in rebus. </s>

<s>Mathematicis ex<lb></lb>cellens vir, ac Magiſter meus, cui ego plurimum debere me <lb></lb>fateor, ab eo enim prima elementa habui, repoſcere à me pu<lb></lb>blicum aliquem doctrina ſua fructum videretur ac Fede<lb></lb>ricus Saminiatus euius suorum ſuauitatem, &amp; beneuolęn<lb></lb>tiam erga me diu, dum ſimul hiſce studijs condiſcipuli ope<lb></lb>ram dedimus, expertus ſum, me ut aliquid auderem tum <lb></lb>oratione, tum exemplo ſuo excitaret, cœpi minus ab ea cogi<lb></lb>tatione alienus eſſe. </s>

<s>Deinde vero ſummos vixos habui <pb xlink:href="029/01/007.jpg"></pb>hortatores. </s>

<s>Etenim cum Clauium, quod tam diu cupiebam, <lb></lb>vidiſſem <gap></gap> minorem tanta scientia, &amp; fama viri beni<lb></lb>gnitatem <gap></gap>periſſem. </s>

<s>Ab eo ſimul ac Theodoſio. </s>

<s>Rubeo ho<lb></lb>mine mihi tum ex ſtudiorum ſimilitudine, tum præcipue ex <lb></lb>eius humanitate amiciſsimo, ad Reuerendiſsimum Sera<lb></lb>phinum deductus ſum. </s>

<s>Iſqs me tantus Præſul non ſolum <lb></lb>humaniſsime complexus est, verum etiam ita hortatus ad <lb></lb>eualganda, que ſcripſeram; plane vt mihi nefas putauerim <lb></lb>non parere. </s>

<s>Accipe igitur &amp; tu Lactor optime amico ac be<lb></lb>nigno animo laborem hunc, quem à me talium virorum <lb></lb>ſumma benignitas expreſsit. </s>

<s>Argumentum quidem, vt <lb></lb>dicebam non iniucundum eſt, nec ab vſu alienam. </s>

<s>Huiuſ<lb></lb>modi enim comparatione Archimedes mixtionem argemi <lb></lb>in aure deprehendit, &amp; furtum Aurificis in corona aurea <lb></lb>patefecit. </s>

<s>de quæ re ſuo loco ego tractabo, &amp; facilem mon<lb></lb>strabo viam, quæ vel argentum in aure, vel quod vis me<lb></lb>tallum in quolibet admistum deprehendi queat, &amp; alte<lb></lb>rum ab alieno diſcerni &amp; ſimul explicabo, quo pacto ex au<lb></lb>ri grauitate <gap></gap> qualitas, at perfectio intelligi poſsit. </s>

<s><lb></lb>Tati vere opuſculo nomen ab Archimede, quem Ducem ſe<lb></lb>quor, impoſui Nam cum ille, vt erat ſummus Magiſter, ſe<lb></lb>tis habuiſſet hunc, totum quasi fabricum poſito fundamen<lb></lb>to delineare in primo lib.

vbi agit de ijs quæ vehuntur in <lb></lb>aqua. </s>

<s>Opus ego pro<gap></gap> <expan abbr="eiq;">eique</expan> fundamento molem inijce<lb></lb>re conatus ſum paribus ſuis et elaboratam, atque distin<lb></lb>ctam. </s>

<s>Qua in re ſi quid aſſequutus ſum, quod publice pro<lb></lb>betur, gaudeo cauſa &amp; mea &amp; publica illud quidem ſpe<lb></lb>rofere vt conatus non diſpliceat.<emph.end type="italics"></emph.end></s></p><pb xlink:href="029/01/008.jpg" pagenum="1"></pb><p type="head">

<s>MARINI GHETALDI <lb></lb>PROMOTVS ARCHIMEDES <lb></lb>SEU.<emph type="italics"></emph>DE VARIIS CORPORVM GENERIBVS <lb></lb>Grauitate, &amp; magnitudine comparatis.<emph.end type="italics"></emph.end></s></p><p type="head">

<s>THEOREMA I. PROPOS. I.</s></p><p type="main">

<s>Si duorum Grauium Corporum eiuſdem ge<lb></lb>neris alterum alterius fuerit multiplex, quo<lb></lb>tuplex maius fuerit minoris, totuplex erit <lb></lb>maioris grauitas, grauitatis minoris. </s></p><p type="main">

<s>SINT duo corpora eiuſdem generis ABC, D, quorum grauita<lb></lb>tes, EFG, ipſius ABC, &amp; H, ip<lb></lb>ſius D, ſit autem corpus ABC, <lb></lb><figure id="id.029.01.008.1.jpg" xlink:href="029/01/008/1.jpg"></figure><lb></lb>multiplex corporis D. </s>

<s>Dico quo <lb></lb>tuplex eſt corpus ABC, corporis <lb></lb>D, totuplicem eſſe grauitatem, <lb></lb>EFG, grauitatis H, diuidatur <lb></lb>enim corpus ABC, in partes ip<lb></lb>ſi D, æquales, quæ ſint A, B, C, <lb></lb>quoniam igitur corpus A, æqua<lb></lb>le eſt corpori D, magnitudine, <lb></lb>&amp; ſunt eiuſdem generis, erit grauitas vnius æqualis grauitati alterius. </s>

<s><lb></lb>Sumatur grauitas E, æqualis grauitati H, erit igitur corporis A, gra<lb></lb>uitas E, &amp; reliqui corporis BC, grauitas FG. </s>

<s>Rurſus quoniam cor<lb></lb>pora B, D, ſunt magnitudine æqualia, erunt æquè grauia, ſumatur <lb></lb>grauitati H, æqualis grauitas F, erit igitur corporis B, grauitas F, &amp; <lb></lb>reliqui corporis C, grauitas G, &amp; ſic fiat, donec perueniatur ad vlti<lb></lb>mam partem corporis ABC, æqualem ipſi D, ſit ea vltima pars C, quo <lb></lb>niam igitur corpus C, æquatur magnitudine ipſi D, æquabitur, &amp; gra<lb></lb>uitate, quare grauitas G, æqualis erit grauitati H, ſequitur igitur <lb></lb>quot partes ſunt in corpore ABC, æquales ipſi D, tot eſſe partes in <lb></lb>grauitate EFG, æquales ipſi H, quoties enim ſumpſimus in corpore <lb></lb>ABC, corpus ipſi D æquale, toties &amp; in grauitate EFG, ſumpſimus <pb xlink:href="029/01/009.jpg" pagenum="2"></pb>grauitatem æqualem ipſi H. </s>

<s>Si duorum igitur grauium corporum <lb></lb>eiuldem generis, &amp; c. quod erat demonſtrandum. </s>

</p><p type="head">

<s>THEOREMA II. PROPOS. II.</s></p><p type="main">

<s>Corpora grauia eiuſdem generis magnitudine com<lb></lb>menſurabilia, eandem in grauitate rationem ha<lb></lb>bent, quam in magnitudine. </s></p><p type="main">

<s>SINT corpora commenſurabilia eiuſdem generis A, B, quorum <lb></lb>grauitates C, ipſius A, &amp; D, ipſius B, Dico eſſe vt A, ad B, ita C, ad D, <lb></lb>quoniam enim, A, B, commenſura<lb></lb>bilia ſunt, metietur ipſa aliquod <lb></lb>corpus, metiatur, &amp; ſit E, cuius <lb></lb><figure id="id.029.01.009.1.jpg" xlink:href="029/01/009/1.jpg"></figure><lb></lb><arrow.to.target n="note1"></arrow.to.target> grauitas F, ſitque corpus E, <expan abbr="eiuſdẽ">eiuſdem</expan> <lb></lb>generis cum corporibus A, B, *ergo <lb></lb>quotuplex eſt corpus A, ipſius E, <lb></lb><arrow.to.target n="note2"></arrow.to.target> totuplex erit grauitas C, grauitatis <lb></lb>F, &amp; quotuplex B, ipſius E, *totuplex <lb></lb>D, ipſius F, ſi igitur diuidantur cor<lb></lb>pora A, B, in partes æquales ipſi E, <lb></lb>&amp; grauitates quoque C, D, in partes æquales ipſi F, erit vt corporis <lb></lb>A, pars vna, ad corpus E, ita pars vna grauitatis C, ad grauitatem F, <lb></lb>æquale videlicet ad æquale, &amp; æque multiplicatis antecedentibus <lb></lb>erit vt A, ad E, ita C, ad F, ſunt enim antecedentium, hoc eſt, illarum <lb></lb>partium æque multiplicia A, C, eadem ratione, vt B, ad E, ita erit D, <lb></lb>ad F, &amp; conuertendo vt E, ad B, ita F, ad D. quoniam igitur vt A, ad <lb></lb><arrow.to.target n="note3"></arrow.to.target> E, ita eſt C, ad F, &amp; vt E, ad B, ita F, ad D, * erit ex æquali vt A, ad <lb></lb>B, ita C, ad D. corpora igitur commenſurabilia eiuſdem generis ean<lb></lb>dem in grauitate rationem habent, quam in magnitudine, quod erat <lb></lb>demonſtrandum. </s></p><p type="margin">

<s><margin.target id="note1"></margin.target><emph type="italics"></emph>Ex an<lb></lb>teced.<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note2"></margin.target><emph type="italics"></emph>Ex an<lb></lb>teced.<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note3"></margin.target>22. 5. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="head">

<s>THEOREMA III. PROPOS. III.</s></p><p type="main">

<s>Et incommenſurabilia corpora eiuſdem generis <lb></lb>eandem in grauitate rationem habent, quam in <lb></lb>magnitudine. </s></p><p type="main">

<s>SINT incommenſurabilia corpora A, BC, quorum grauitates <lb></lb>D, ipſius A, &amp; EF, ipſius BC. </s>

<s>Dico eſſe vt A, ad BC, ita D, ad EF, ſi <pb xlink:href="029/01/010.jpg" pagenum="3"></pb>enim non eſt vt A, ad BC, ita D, ad EF, erit vt A, ad BC, ita D, vel ad <lb></lb><figure id="id.029.01.010.1.jpg" xlink:href="029/01/010/1.jpg"></figure><lb></lb>minorem quam EF, <lb></lb>vel ad maiorem, ſit <lb></lb>primum ad minoré, <lb></lb>pempe ad EG, &amp; ex<lb></lb>ponatur aliquod cor <lb></lb>pus K, eiuſdem gene<lb></lb>ris cum corporibus <lb></lb>A, B C, cuius graui<lb></lb>tas ſit æqualis ipſi <lb></lb>GF, &amp; à corpore BC, <lb></lb>auferatur aliqua <lb></lb>pars HC, quæ ſit mi<lb></lb>nor corpore K, ita vt reliquæ pars. </s>

<s>BL, ſit commenſurabilis ipſi A. &amp; <lb></lb>ſit partis HC, grauitas IF, ergo reliquæ partis BL, grauitas erit EI. </s>

<s><lb></lb>Quoniam igitur corpus A, commenſurabile eſt ipſi BL,* erit vt A, ad<arrow.to.target n="note4"></arrow.to.target><lb></lb>BL, ita D, ad EI, ſed vt A, ad BC, ita eſt D, ad EG, atque A, primus, <lb></lb>proportionalium terminus in ſerie prima, * maiorem habet<arrow.to.target n="note5"></arrow.to.target> ratio<lb></lb>nem ad BL, ſecundum terminum, quam A, primus terminus in ſerie <lb></lb>ſecunda ad BC, ſecundum; ergo &amp; D, tertius terminus in ſerie prima <lb></lb>ad EI, quartum, maiorem habebit rationem quam D, tertius termi<lb></lb>nus in ſerie ſecunda ad EG, quartum, quoniam igitur D, maiorem<arrow.to.target n="note6"></arrow.to.target><lb></lb>habet rationem ad EI, quam ad EG,* erit EI, minor quam EG, quod <lb></lb>eſt abſurdum. </s>

<s>non igitur eſt vt A, ad BC, ita D, ad minorem quam EF. </s></p><p type="margin">

<s><margin.target id="note4"></margin.target>E<emph type="italics"></emph>x <expan abbr="an-tecedẽ-te">an<lb></lb>teceden<lb></lb>te.</expan><emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note5"></margin.target>2. 5. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note6"></margin.target>10. 5. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Deinde ſit vt A, ad BC, ita D, ad maiorem quam EF, nempe ad <lb></lb><figure id="id.029.01.010.2.jpg" xlink:href="029/01/010/2.jpg"></figure><lb></lb>EG, &amp; expoſito cor<lb></lb>pore K, vt dictum <lb></lb>eſt, cuius grauitas, <lb></lb>ſit æqualis grauita<lb></lb>ti FG, addatur cor<lb></lb>pori BC, aliquod <lb></lb>corpus CH, quod ſit <lb></lb>minus corpore K, &amp; <lb></lb>eiuſdem generis cú <lb></lb>corporibus A, BC, <lb></lb>ita vt totum corpus <lb></lb>BL, ſit commenſurabile ipſi A, &amp; ſit ipſius CH, grauitas FI, ergo to<lb></lb>tius corporis BL, grauitas erit EI; Quoniam igitur corpori A, com<lb></lb>menſurabile eſt corpus BL,* erit vt A, ad BL, ita D, ad EI, ſed vt A, ad<arrow.to.target n="note7"></arrow.to.target><lb></lb>BC, ita eſt D, ad EG, atque A, primus proportionalium terminus in <lb></lb>ſerie prima, * minorem habet rationem ad BL, ſecundum terminum, <pb xlink:href="029/01/011.jpg" pagenum="4"></pb>quam A, primus terminus in ſerie ſecunda ad BC, ſecundum, ergo, &amp; <lb></lb>D, tertius terminus in ſerie prima ad EI, quartum, minorem habebit<lb></lb>rationem quam D, tertius terminus in ſerie ſecunda ad EG, quartum. </s>

<s><lb></lb>Quoniam igitur D, minorem habet rationem ad EI, quam ad EG, <lb></lb><arrow.to.target n="note8"></arrow.to.target> erit * EI, maior quam EG, quod eſt abſurdum. </s>

<s>Non igitur eſt vt A, <lb></lb>ad BC, ita D, ad maiorem quam EF, oſtenſum autem eſt neque ad mi<lb></lb>norem; quare vt A, ad BC, ita erit D, ad EF. &amp; incommenſurabilia igi<lb></lb>tur corpora eiuſdem generis eandem in grauitate rationem habent, <lb></lb>quam in magnitudine, quod erat demonſtrandum. </s></p><p type="margin">

<s><margin.target id="note7"></margin.target><emph type="italics"></emph>Ex an<lb></lb>teced..<emph.end type="italics"></emph.end><lb></lb>8. 5.<emph type="italics"></emph>Ele<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note8"></margin.target>10. 5. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>ID QVOD nos duobus præcedentibus Theorematis de<lb></lb>monſtrauimus, <expan abbr="nõnulli">nonnulli</expan>, vt per ſe notum, &amp; vt commune quod<lb></lb>dam axioma ſupponunt, quam bene &amp; rationabiliter ipſi vide<lb></lb>rint; melius enim Euclides propoſitionem 20. primi libri Ele<lb></lb>mentorum ſuppoſuiſſet vt pronuntiatum; vnicuique enim no<lb></lb>tius eſt duo trianguli latera reliquo eſſe maiora (cum &amp; Aſino <lb></lb>illud ſit notum) quam corpora grauia eiuſdem generis eandem <lb></lb>in grauitate rationem habere, quam in magnitudine, &amp; tamen <lb></lb>illam propoſitionem demonſtrat Euclides, non ſupponit, non <lb></lb>igitur hæc, quæ minus ad principij rationem accedit, ſuppo<lb></lb>nenda fuit, ſed demonſtranda. </s></p><p type="head">

<s>THEOREMA IV. PROPOS. IV.</s></p><p type="main">

<s>Si quatuor corporum grauium primum ad <expan abbr="ſecundũ">ſecundum</expan> <lb></lb>eandem in magnitudine rationem habeat, quam <lb></lb>tertium ad quartum, primum autem, &amp; ſecundum ſint <lb></lb>eiuſdem generis, itidem tertium, &amp; quartum; &amp; in gra<lb></lb>uitate primum ad ſecundum eandem rationem habebit, <lb></lb>quam tertium ad quartum. </s></p><p type="main">

<s>PRIMVM enim A, ad ſecundum B, eandem in magnitudine ra<lb></lb>tionem habeat, quam tertium C, ad quartum D, ſint autem A, B, <lb></lb>eiuſdem generis, itidem C, D. Dico. </s>

<s>&amp; in grauitate primum A, ad <lb></lb>ſecundum B, eandem rationem habere, quam tertium C, ad D, quar<lb></lb>tum. </s>

<s>Sint enim earum grauitates E, ipſius A, &amp; F, ipſius B, ipſius <lb></lb>vero C, ſit grauitas G, &amp; ipſius D, grauitas H, quoniam igitur cor-<pb xlink:href="029/01/012.jpg" pagenum="5"></pb><figure id="id.029.01.012.1.jpg" xlink:href="029/01/012/1.jpg"></figure><lb></lb>pora A, B, eiuſdem ſunt generis, <lb></lb>ſimiliter, &amp; corpora C, D,* erit <lb></lb>vt A, ad B, ita E, ad F,* &amp; vt<arrow.to.target n="note9"></arrow.to.target><lb></lb>C, ad D, ita G, ad H. </s>

<s>Sed poni<lb></lb>tur vt A, ad B, ita eſſe C, ad D, <lb></lb>ergo vt E, ad F, ita erit G, ad <lb></lb>H. </s>

<s>Si igitur quatuor corporum <lb></lb>grauium, primum ad ſecundum <lb></lb>eandem in magnitudine ratio<lb></lb>nem habeat: etcæt. </s>

<s>quod demon<lb></lb>ſtrare oportebat. </s></p><p type="margin">

<s><margin.target id="note9"></margin.target>1. &amp; <gap></gap>. <lb></lb><emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="head">

<s>THEOREMA V. PROPOS. V.</s></p><p type="main">

<s>Solida corpora liquido grauiora demiſſa in liquidum <lb></lb>ferentur deoríum, donec deſcendant, &amp; erunt in li<lb></lb>quido tanto leuiora, quanta eſt grauitas liquidi magni<lb></lb>tudinem habentis ſolido corpori æqualem. </s></p><p type="main">

<s>HOC autem demonſtratum eſt ab Archimede propoſ.7. primi li<lb></lb>bri de ijs, quæ vehuntur in aqua. </s></p><p type="head">

<s>THEOREMA VI. PROPOS. VI.</s></p><p type="main">

<s>Si quatuor grauium corporum primum, &amp; ſecundum <lb></lb>fuerint magnitudine æqualia, tertium vero, &amp; quar<lb></lb>tum æque grauia, fuerint autem primum, &amp; tertium <lb></lb>eiuſdem generis, itidem ſecundum, &amp; quartum; erit, vt <lb></lb>grauitas corporis primi, ad grauitatem ſecundi, ita gra<lb></lb>uitas liquidi æqualis magnitudine corpori quarto, ad gra<lb></lb>uitatem liquidi tertio corpori æqualis. </s></p><p type="main">

<s>SINT quatuor corpora A, B, C, D, quorum A, primum, &amp; B, ſe<lb></lb>cundum ſint magnitudine æqualia, tertium vero C, &amp; D, quartum <lb></lb>æque grauia, ſint autem A, &amp; C, eiuſdem generis, itidem B, &amp; D. </s>

<s>Di<lb></lb>co vt grauitas corporis A, ad grauitatem corporis B, ita eſſe grauita<lb></lb>tem liquidi æqualis magnitudine corpori D, ad grauitatem liquidi <lb></lb>magnitudine corpori C, æqualis. </s>

<s>Accipiantur enim tria eiuſdem ge<pb xlink:href="029/01/013.jpg" pagenum="6"></pb>neris liquidi corpora E, F, G, quorum E, ſit æquale corpori A, vel B, <lb></lb><figure id="id.029.01.013.1.jpg" xlink:href="029/01/013/1.jpg"></figure><lb></lb>magnitudine, ipſum vero F, æqua <lb></lb>le corpori C, &amp; ipſum G, æquale <lb></lb>corpori D. </s>

<s>Quoniam igitur eſt vt <lb></lb>D, ad G, ita B, ad E, æquale vi<lb></lb>delicet ad æquale, erit permutan<lb></lb>do vt D, ad B, ita G, ad E, &amp; quo<lb></lb>niam ſunt eiuſdem generis corpo<lb></lb>ra D, B, ſimiliter &amp; corpora G, E, <lb></lb><arrow.to.target n="note10"></arrow.to.target> erit * vt grauitas corporis D, hoc <lb></lb>eſt ipſius C, ponuntur enim æque <lb></lb>grauia corpora C, D, ad grauita<lb></lb>tem corporis B, ita liquidi G, gra<lb></lb>uitas ad grauitatem liquidi E. </s>

<s><lb></lb>Similiter quoniam eſt vt A, ad E, <lb></lb>ita C, ad F, æquale videlicet ad <lb></lb>æquale, erit permutando vt A, ad <lb></lb>C, ita E, ad F, &amp; quoniam ponuntur eiuſdem generis corpora A, C, <lb></lb><arrow.to.target n="note11"></arrow.to.target> itidem E, F,* erit vt grauitas corporis A, ad grauitatem ipſius C, ita <lb></lb>liquidi E, grauitas ad grauitatem liquidi F, ſed vt grauitas corporis <lb></lb>C, ad grauitatem corporis B; ita eſt grauitas liquidi G, ad grauita<lb></lb><arrow.to.target n="note12"></arrow.to.target>tem liquidi E, vt eſt demonſtratum, ergo * in perturbata proportione <lb></lb>erit vt grauitas corporis A, ad ipſius corporis B, grauitatem, ita liqui <lb></lb>di G, grauitas, ad grauitatem liquidi F. ſi igitur quatuor grauium <lb></lb>corporum primum, &amp; ſecundum, &amp; c. quod erat demonſtrandum. </s>

</p><p type="margin">

<s><margin.target id="note10"></margin.target>2. &amp; 3. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note11"></margin.target>2. &amp; 3. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note12"></margin.target>23. 5. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="head">

<s>THEOREMA VII. PROPOS. VII.</s></p><p type="main">

<s>Si quatuor grauium corporum primum, &amp; <expan abbr="ſecundũ">ſecundum</expan>, <lb></lb>fuerint magnitudine æqualia, tertium vero, &amp; quar<lb></lb>tum æque grauia, fuerint autem primum, &amp; tertium <lb></lb>eiuſdem generis, itidem ſecundum, &amp; quartum; primum <lb></lb>ad ſecundum eandem in grauitate rationem habebit, <lb></lb>quam habet in magnitudine quartum ad tertium. </s></p><p type="main">

<s>SINT quatuor grauia corpora A, B, C, D, quorum A, primum, <lb></lb>&amp; B, ſecundum ſint magnitudine æqualia, tertium vero C, &amp; D, quar<lb></lb>tum æque grauia, ſint autem A, &amp; C, eiuſdem generis, itidem B, &amp; <lb></lb>D. </s>

<s>Dico corpus A, eandem in grauitate rationem habere ad corpus <pb xlink:href="029/01/014.jpg" pagenum="7"></pb>B, quam corpus D, habet in magnitudine ad C, corpus. </s>

<s>Sit enim li<lb></lb><figure id="id.029.01.014.1.jpg" xlink:href="029/01/014/1.jpg"></figure><lb></lb>quidi magnitudine æqualis corpo<lb></lb>ri C, grauitas E, ſimiliter, &amp; liqui<lb></lb>di æqualis magnitudine corpori D, <lb></lb>grauitas F, quoniam igitur gra<lb></lb>uia corpora eiuſdem generis, ean<lb></lb>dem in magnitudine rationem*<arrow.to.target n="note13"></arrow.to.target> ha<lb></lb>bent, quam in grauitate, erit vt ma<lb></lb>gnitudo liquidi æqualis corpori D, <lb></lb>ad magnitudinem liquidi æqualis <lb></lb>corpori C, hoc eſt, vt magnitudo <lb></lb>corporis D, ad magnitudinem cor<lb></lb>poris C, ita grauitas F, ad grauita<lb></lb>tem E, ſed vt grauitas F, ad grauitatem E, * ita eſt grauitas corporis<arrow.to.target n="note14"></arrow.to.target><lb></lb>A, ad grauitatem corporis B,* ergo vt grauitas corporis A, ad graui<lb></lb>tatem corporis B, ita erit magnitudo corporis D, ad corporis C, ma<lb></lb>gnitudinem. </s>

<s>Si quatuor igitur grauium corporum primum, &amp; ſecun<lb></lb>dum, &amp; c. quod erat demonſtrandum. </s>

</p><p type="margin">

<s><margin.target id="note13"></margin.target>2. &amp; 3. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note14"></margin.target><emph type="italics"></emph>Ex an<lb></lb>teced.<emph.end type="italics"></emph.end><lb></lb>11. 5. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="head">

<s>PROBLEMA I. PROPOS. VIII.</s></p><p type="main">

<s>Propoſitis duobus corporibus magnitudine æquali<lb></lb>bus, vno ſolido, altero liquido, data ſolidi corporis <lb></lb>grauitate, grauitatem liquidi inuenire. </s></p><figure id="id.029.01.014.2.jpg" xlink:href="029/01/014/2.jpg"></figure><p type="main">

<s>SINT duo propoſita corpo<lb></lb>ra magnitudine æqualia A, B, <lb></lb>quorum A, ſit ſolidum, B, vero <lb></lb>liquidum, &amp; ſit ſolidi data graui<lb></lb>tas CD, Oportet inuenire quan<lb></lb>ta erit grauitas liquidi B. </s>

<s>Si ſo<lb></lb>lidum A, grauius ſit liquido, de<lb></lb>mittatur in <expan abbr="liquidũ">liquidum</expan>, &amp; habeat in <lb></lb>liquido grauitatem ED, quoniam <lb></lb>igitur ſolidum A, grauius eſt li<lb></lb>quido, demiſſum in liquidum <lb></lb>erit* in liquido tanto leuius,<arrow.to.target n="note15"></arrow.to.target> <expan abbr="quã-ta">quan<lb></lb>ta</expan> eſt grauitas liquidi magnitu<lb></lb>dine æqualis ſolido A, ſed <expan abbr="ſolidũ">ſolidum</expan> <lb></lb>A, leuius eſt in liquido, quanta <pb xlink:href="029/01/015.jpg" pagenum="8"></pb>eſt grauitas CE, ergo grauitas liquidi magnitudine æqualis ſolido A, <lb></lb>erit CE. </s></p><p type="margin">

<s><margin.target id="note15"></margin.target>huius</s></p><p type="main">

<s>Si vero ſolidum A, ſit leuius liquido, accipiatur aliquod aliud cor<lb></lb>pus ſolidum F, grauius liquido, ita vt ſolidum conſtans ex vtriſque <lb></lb>ſolidis A, F, demiſſum in liquidum feratur deorſum, &amp; ſit ſolidi F, <lb></lb>grauitas DG, item eiuſdem ſolidi F, in liquido videlicet exiſtentis ſit <lb></lb><arrow.to.target n="note16"></arrow.to.target> grauitas HG,* ergo liquidi magnitudine æqualis ſolido F, erit gra<lb></lb>uitas DH. </s></p><p type="margin">

<s><margin.target id="note16"></margin.target>5.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Et quoniam ſolidi A, grauitas eſt CD, ſolidi vero F, grauitas DG, <lb></lb>erit vtrorumque ſolidorum A, F, grauitas CG. coniungantur ſolida <lb></lb>A, F, &amp; ſolidum ex vtriſque conſtans demittatur in liquidum, &amp; ha<lb></lb>beat in liquido grauitatem GI, (habebit autem in liquido minorem <lb></lb>grauitatem, quam ſolum ſolidum F, quoniam ſolidum F, grauius li<lb></lb>quido fertur deorſum nullo prohibente, &amp; coniunctum cum ſolido A, <lb></lb>leuiori liquido ab eo ſuſtinetur, ne deorſum feratur <expan abbr="tãta">tanta</expan> vi, qua ſeiun<lb></lb>ctum) quoniam igitur ſolidi, quod conſtat ex vtriſque ſolidis A, F, <lb></lb><arrow.to.target n="note17"></arrow.to.target> grauitas eſt CG, in liquido vero exiſtentis grauitas GI, * erit liquidi <lb></lb>habentis magnitudinem æqualem vtriſque ſolidis A, F, grauitas CI, <lb></lb>ſed grauitas liquidi æqualis magnitudine ſolido F, eſt DH, ergo reli<lb></lb>qui liquidi æqualis ſolido A, erit grauitas CD, IH, ſed liquidum B, <lb></lb>æquatur magnitudine ſolido A, ergo grauitas liquidi B, erit CD, IH, <lb></lb>inuenta igitur eſt liquidi corporis B, grauitas CD, IH, de qua quæ<lb></lb>rebatur. </s></p><p type="margin">

<s><margin.target id="note17"></margin.target>5.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Placet huic Problemati exemplum apponere, vt vnicuique <lb></lb>etiam diſciplinæ Mathematicæ experto ad vſum pateat adi<lb></lb>tus; quare etiam ſequentibus Problematis apponemus ſimilia <lb></lb>exempla. </s></p><p type="head">

<s>Exemplum.</s></p><p type="main">

<s>Qvidam proponit aliquod corpus ſolidum notæ <lb></lb>grauitatis, &amp; vult ſcire quanta erit grauitas liqui<lb></lb>di, magnitudinem habentis propoſito Corpori ſolido æ<lb></lb>qualem. </s></p><p type="main">

<s><emph type="italics"></emph>Sit primum propoſitum aliquod corpus plumbeum A, euius graui<lb></lb>tas ſit<emph.end type="italics"></emph.end> 23. <emph type="italics"></emph>&amp; oporteat ſcire quanta erit grauitas aquæ magnitudinem <lb></lb>habentis æqualem propoſito plumbo A, ponderetur plumbum A, in <lb></lb>aqua (modum quo ponderanda ſint corpora ſolida in aqua, inferius <lb></lb>apponemus) &amp; habeat grauitatem<emph.end type="italics"></emph.end> 21. <emph type="italics"></emph>quoniam igitur numerus<emph.end type="italics"></emph.end> 23. <pb xlink:href="029/01/016.jpg" pagenum="9"></pb><emph type="italics"></emph>ſuperat numerum<emph.end type="italics"></emph.end> 21, <emph type="italics"></emph>numero 2, erit grauitàs aquæ magnitudinem <lb></lb>habentis æqualem plumbo A, 2.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Eadem via omnium liquidorum inuenitur grauitas, quan<lb></lb>do nimirum corpus ſolidum ſit grauius liquido, cuius liquidi <lb></lb>quærenda eſt grauitas, hoc eſt quando corpus ſolidum demiſ<lb></lb>ſum in liquidum feratur deorſum. </s></p><p type="main">

<s>Quando vero corpus ſolidum fuerit leuius liquido, hoc eſt <lb></lb>demiſſum in liquidum non deſcendat, per adiectionem alicuius <lb></lb>alius ſolidi corporis liquido grauioris, quæſita liquidi grauitas <lb></lb>inuenietur. </s></p><p type="main">

<s><emph type="italics"></emph>Sit igitur propoſitum aliquod cereum corpus A, cuius grauitas ſit<emph.end type="italics"></emph.end><lb></lb>21. <emph type="italics"></emph>&amp; oporteat ſcire quanta erit grauitas aquæ magnitudinem haben<lb></lb>tis æqualem ceræ A. </s>

<s>Quoniam cera leuior est, quam aqua, ſi demitta<lb></lb>tur in aquam non feretur deorſum, accipiatur aliquod corpus ſolidum <lb></lb>F, grauius quam aqua, ita vt corpus constans ex vtriſque corporibus <lb></lb>A, F, demiſſum in aquam feratur deorſum, ſit igitur corpus F, plum<lb></lb>beum, cuius grauitas ſit v.g.<emph.end type="italics"></emph.end> 23, <emph type="italics"></emph>&amp; eiuſdem in aqua ponderati<emph.end type="italics"></emph.end> 21, <lb></lb><emph type="italics"></emph>ergo aquæ magnitudinem habentis <expan abbr="æqualẽ">æqualem</expan> plumbo F, erit grauitas<emph.end type="italics"></emph.end> 2, </s></p><p type="main">

<s><emph type="italics"></emph>Et quoniam ceræ A, grauitas est<emph.end type="italics"></emph.end> 21, <emph type="italics"></emph>plumbi vero F,<emph.end type="italics"></emph.end> 23, <emph type="italics"></emph>erit vtro<lb></lb>rumque corporum A, F, ceræ nimirum, &amp; plumbi grauitas<emph.end type="italics"></emph.end> 44, <emph type="italics"></emph>coniun<lb></lb>gatur cera, &amp; plumbum, &amp; ita coniuncta ponderentur in aqua, &amp; <lb></lb>habeant grauitatem<emph.end type="italics"></emph.end> 20, <emph type="italics"></emph>quoniam igitur numerus<emph.end type="italics"></emph.end> 44, <emph type="italics"></emph>ſuperat nume<lb></lb>rum<emph.end type="italics"></emph.end> 20, <emph type="italics"></emph>numero<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>erit grauitas aquæ habentis magnitudinem æqua<lb></lb>lem vtriſque corporibus ceræ &amp; plumbi<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>ſed grauitas aquæ magni<lb></lb>tudinem habentis æqualem plumbo F, est<emph.end type="italics"></emph.end> 2, <emph type="italics"></emph>ergo reliquum quod est<emph.end type="italics"></emph.end><lb></lb>22, <emph type="italics"></emph>erit grauitas, aquæ magnitudine æqualis propoſitæ ceræ A.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>At vero ſi propoſitum fuerit aliquod corpus ſolidum ma<lb></lb>gni ponderis, ita vt difficile poſſit ponderari in aqua, hac via <lb></lb>inuenietur aquæ quæſita grauitas. </s></p><p type="main">

<s><emph type="italics"></emph>Sit aliquod corpus <expan abbr="plumbeũ">plumbeum</expan> A, cuius grauitas<emph.end type="italics"></emph.end> 2300, <emph type="italics"></emph>&amp; oporteat in <lb></lb>uenire grauitatem aquæ magnitudinem habentis <expan abbr="æqualẽ">æqualem</expan> plumbo A, <lb></lb>accipiatur aliquod paruum plumbi corpus F, cuius grauitas ſit v.g.<emph.end type="italics"></emph.end><lb></lb>23, <emph type="italics"></emph>&amp; inueniatur grauitas aquæ magnitudine æqualis plumbo F, vt <lb></lb>dictum est, quæ ſit.<emph.end type="italics"></emph.end> 2, <emph type="italics"></emph>&amp; fiat vt<emph.end type="italics"></emph.end> 23, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 2, <emph type="italics"></emph>ita<emph.end type="italics"></emph.end> 2300, <emph type="italics"></emph>ad alium numerum <lb></lb>qui ſit<emph.end type="italics"></emph.end> 200. <emph type="italics"></emph>grauitas igitur aquæ magnitudinem habentis æqualem <lb></lb>plumbo A, erit<emph.end type="italics"></emph.end> 200. </s></p><p type="main">

<s><emph type="italics"></emph>Similiter ſit aliquod cereum corpus A, cuius grauitas<emph.end type="italics"></emph.end> 2100, <emph type="italics"></emph>&amp; opor<lb></lb>teat facere, quod imperatum eſt. </s>

<s>accipiatur aliquod paruum ceræ cor<lb></lb>pus F, cuius grauitas ſit v.g.<emph.end type="italics"></emph.end> 21, <emph type="italics"></emph>&amp; inuenta grauitate aquæ magni-<emph.end type="italics"></emph.end><pb xlink:href="029/01/017.jpg" pagenum="10"></pb><emph type="italics"></emph>tudinem habentis æqualem ceræ F, quæ ſit<emph.end type="italics"></emph.end> 12, <emph type="italics"></emph>fiat vt<emph.end type="italics"></emph.end> 21, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 22, <emph type="italics"></emph>ita<emph.end type="italics"></emph.end><lb></lb>2100, <emph type="italics"></emph>ad alium numerum qui ſit<emph.end type="italics"></emph.end> 2200; <emph type="italics"></emph>erit igitur grauitas aquæ ma <lb></lb>gnitudinem habentis æqualem ceræ A,<emph.end type="italics"></emph.end> 2200. </s></p><p type="main">

<s>Neque neceſſe eſt, vt illud corpus ſolidum magni ponderis <lb></lb>reipſa proponatur, ſufficit enim vt eius grauitas notificetur <lb></lb>numero tantum. </s></p><p type="main">

<s>Si autem propoſitum fuerit inuenire quanta erit graui<lb></lb>tas argenti viui magnitudine æqualis propoſito corpori ſo<lb></lb>lido A; ratione qua ſupra, non inuenietur ipſa grauitas, quo<lb></lb>niam nullum corpus demiſſum in argentum viuum fertur <lb></lb>deorſum, niſi aurum, aurum vero in ipſo argento viuo perrum<lb></lb>pitur, ſed qua ratione inuenienda ſit ipſa argenti viui graui<lb></lb>tas, dicemus ad finem exempli propoſitionis decimæ quartæ. </s></p><p type="main">

<s>Quomodo ponderanda ſint corpora ſolida in aqua. </s></p><p type="main">

<s>C<emph type="italics"></emph>orpus quod ponderandum proponitur ſeta equina ex altera li<lb></lb>bræ lance appendatur, in altera lance ponantur pondera, &amp; cor<lb></lb>pus appenſum demittatur in aquam, ita vt in aqua libere pendeat, ne<lb></lb>que lancem, cui appenſum est corpus, neque aliam in qua ſunt <expan abbr="ponderã">pondera</expan> <lb></lb>aqua contingat, &amp; ita ponderetur propoſitum corpus, ac ſi in aere pen <lb></lb>deret.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Dixi ſeta e quina corpus ponderandum debere appendi, quia fere <lb></lb>æque grauis eſt atque aqua, &amp; ideo nihil addet, vel minuet grauita<lb></lb>tis in ipſo corpore ponderando.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Quod ſi corpus ponderandum fuerit, tam graue, vt ſeta ſimplici ſu<lb></lb>stineri nequeat, appendatur pluribus ſimul iunctis ſetis, &amp; ne aliquid <lb></lb>grauitatis ſetarum coniunctio addat corpori ponderando, ponantur <lb></lb>in altera lance totidem ſetæ æquales eis, quæ ex lance, cui appenſum eſt <lb></lb>corpus pendent, vſque ad corpus appenſum, hac igitur ſetarum addi<lb></lb>tione æque ponderabunt lances, &amp; quamuis illæ ſetæ, quibus appen<lb></lb>ſum est corpus, ſint longiores, quam aliæ alteri lanci addita,  longitudi<lb></lb>ne partium, quibus ligatum est corpus, tamen quoniam illæ partes <lb></lb>æque graues ſunt, atque aqua, exiſtentes cum ipſo corpore in aqua, nul<lb></lb>lam grauitatem habebunt, &amp; ideo illæ ſetæ, quæ alias ſuperant dictis <lb></lb>partibus, &amp; ſi longiores, non erunt grauiores quam aliæ, existenti<lb></lb>bus, nempe, vt dictum eſt, illis partibus cum ipſo corpore in aqua. </s>

<s>Sic <lb></lb>igitur in aqua ponderanda erunt ſolida corpora, quod animaduertiſſe <lb></lb>fuit operæ pretium.<emph.end type="italics"></emph.end></s></p><pb xlink:href="029/01/018.jpg" pagenum="11"></pb><p type="head">

<s>PROBLEMA II. PROPOS. IX.</s></p><p type="main">

<s>Propoſitis duobus corporibus magnitudine æquali<lb></lb>bus, vno ſolido, altero liquido, data corporis li<lb></lb>quidi grauitate, grauitatem ſolidi inuenire. </s></p><p type="main">

<s>SINT duo propoſita corpora <lb></lb>magnitudine æqualia, A, quidem ſo<lb></lb><figure id="id.029.01.018.1.jpg" xlink:href="029/01/018/1.jpg"></figure><lb></lb>lidum, B, vero liquidum, ſit autem li<lb></lb>quidi B, data grauitas F, &amp; oporteat <lb></lb>inuenire grauitatem ſolidi A, accipia<lb></lb>tur aliquod corpus ſolidum D, <expan abbr="eiuſdẽ">eiuſdem</expan> <lb></lb>generis, cum ſolido A, cuius grauitas <lb></lb>ſit H, deinde liquidi eiuſdem generis <lb></lb>cum liquido B, magnitudine æqualis <lb></lb>ſolido D,* inueniatur grauitas quæ ſit<arrow.to.target n="note18"></arrow.to.target><lb></lb>G, &amp; fiat vt G, ad H, ita F, ad aliam <lb></lb>grauitatem, quæ ſit C. </s>

<s>Dico ſolidi A, <lb></lb>grauitatem eſſe C, accipiatur enim aliquod corpus liquidum E, eiuſ<lb></lb>dem generis cum liquido B, grauitatem habens æqualem ſolido D. </s>

<s><lb></lb>Quoniam igitur ſunt quatuor corpora grauia B, A, E, D, quorum pri<lb></lb>mum B, &amp; ſecundum A, ſunt magnitudine æqualia, tertium vero E, <lb></lb>&amp; quartum D, æque grauia, &amp; ſunt eiuſdem generis corpora B, E, ſi<lb></lb>militer, &amp; corpora A, D,*erit vt grauitas liquidi æqualis<arrow.to.target n="note19"></arrow.to.target> magnitudi<lb></lb>ne ſolido D, hoc eſt vt G, ad grauitatem liquidi E; hoc eſt ad H, <expan abbr="ponũ-tnr">ponun<lb></lb>tnr</expan> enim æque grauia corpora D, E, ita grauitas F, ad ſolidi A, graui<lb></lb>tatem, ſed vt grauitas G, ad grauitatem H, ita eſt grauitas F, ad C, <lb></lb>grauitatem, ergo grauitas C, æqualis erit grauitati ſolidi A. </s>

<s>Inuenta <lb></lb>igitur eſt ſolidi A, grauitas C, quod facere oportebat. </s></p><p type="margin">

<s><margin.target id="note18"></margin.target>8.<emph type="italics"></emph>hu<lb></lb>ius.<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note19"></margin.target>6.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="head">

<s>Exemplum.</s></p><p type="main">

<s>Qvidam proponit aliquod corpus liquidum notæ <lb></lb>grauitatis, &amp; vult ſcire quanta erit grauitas alicu<lb></lb>ius ſolidi, magnitudinem habentis propoſito Corpori li<lb></lb>quido æqualem. </s></p><p type="main">

<s><emph type="italics"></emph>Sit propoſitum aliquod corpus aqueum B, euius grauitas ſit 100. &amp;<emph.end type="italics"></emph.end><pb xlink:href="029/01/019.jpg" pagenum="12"></pb><emph type="italics"></emph>oporteat ſcire quanta erit grauitas plumbi magnitudinem habentis <lb></lb>æqualem propoſitæ aquæ B, verbi gratia ſit vas aliquod plenum aqua, <lb></lb>cuius aquæ grauitas ſit 100, &amp; oporteat ſcire, ſi illud <expan abbr="idẽ">idem</expan> vas replea<lb></lb>tur plumbo, quanta illius plumbi erit grauitas. </s>

<s>Accipiatur aliquod <lb></lb>plumbeum corpus D, cuius grauitas ſit<emph.end type="italics"></emph.end> 23<emph type="italics"></emph>, deinde aquæ <expan abbr="magnitudinẽ">magnitudinem</expan> <lb></lb>habentis æqualem plumbo D, inueniatur grauitas, quod quomodo fie<lb></lb>ri oporteat iam dictum eſt in antecedentis problematis exemplo: ſit igi<lb></lb>tur ea inuenta grauitas<emph.end type="italics"></emph.end> 2<emph type="italics"></emph>, &amp; fiat vt<emph.end type="italics"></emph.end> 2<emph type="italics"></emph>, ad<emph.end type="italics"></emph.end> 23<emph type="italics"></emph>, ita<emph.end type="italics"></emph.end> 100<emph type="italics"></emph>, ad alium nume<lb></lb>rum qui ſit<emph.end type="italics"></emph.end> 1150<emph type="italics"></emph>, is igitur numerus erit grauitas plumbi magnitu<lb></lb>dinem habentis propoſitæ aquæ B, æqualem, hoc est illius plumbi, quod <lb></lb>in vaſe continetur.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>At vero ſi <expan abbr="propoſitũ">propoſitum</expan> fuerit inuenire <expan abbr="quãta">quanta</expan> erit grauitas ceræ, <lb></lb>aut ligni, aut <expan abbr="cuiuſcũque">cuiuſcunque</expan> ſolidi leuioris quam aqua, nihil diuer<lb></lb>ſi in opere accidet, niſi quod ratio inueniendi grauitatem <lb></lb>aquæ magnitudinem habentis æqualem corpori ſolido leuio<lb></lb>ri, quàm aqua, differt in aliquo à ratione, qua inuenitur graui<lb></lb>tas aquæ magnitudinem habentis æqualem ſolido corpori <lb></lb>grauiori, quam aqua, ſed vtramque rationem exemplo ante<lb></lb>cedentis Problematis illuſtrauimus, in eo enim ſatis explica<lb></lb>tum eſt de vtraque. </s></p><p type="main">

<s><emph type="italics"></emph>Sed ne exemplorum inopia laborare videamur, ſit inuenienda gra<lb></lb>uitas ceræ magnitudinem habentis æqualem propoſitæ aquæ B, acci<lb></lb>piatur aliquod cereum corpus D, cuius grauitas ſit<emph.end type="italics"></emph.end> 21, <emph type="italics"></emph>deinde aquæ <lb></lb>magnitudinem habentis æqualem ceræ D, inueniatur grauitas, vt in <lb></lb>antecedentis Problematis exemplo dictum est, quæ grauitas ſit<emph.end type="italics"></emph.end> 22, <emph type="italics"></emph>&amp; <lb></lb>fiat vt<emph.end type="italics"></emph.end> 22<emph type="italics"></emph>, ad<emph.end type="italics"></emph.end> 21<emph type="italics"></emph>, ita<emph.end type="italics"></emph.end> 100<emph type="italics"></emph>, hoc est grauitas aquæ B, ad alium numerum <lb></lb>qui ſit<emph.end type="italics"></emph.end> 95 5/11. <emph type="italics"></emph>is igitur numerus indicabit quanta erit grauitas ceræ <lb></lb>magnitudinem habentis æqualem propoſitæ aquæ B.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Similiter ſi propoſitum liquidum corpus B, fuerit olei, aut <lb></lb>vini, aut cuiuſcumque liquidi, præter argenti viui, eadem om<lb></lb>nino via, qua ante, inuenietur quæſita corporis ſolidi grauitas, <lb></lb>ſed de argento viuo tractabimus ad finem propoſitionis deci<lb></lb>mæ quartæ. </s></p><p type="head">

<s>PROBLEMA III. PROPOS. X.</s></p><p type="main">

<s>Propoſitis duobus corporibus æque grauibus, vno <lb></lb>ſolido, altero liquido, data ſolidi corporis magnitu<pb xlink:href="029/01/020.jpg" pagenum="13"></pb>dine, magnitudinem liquidi inuenire. </s></p><p type="main">

<s>SINT duo propoſita corpora <lb></lb>æque grauia, A, quidem ſolidum B, <lb></lb>vero liquidum, ſit autem ſolidi A, da<lb></lb><figure id="id.029.01.020.1.jpg" xlink:href="029/01/020/1.jpg"></figure><lb></lb>ta magnitudo C, &amp; oporteat inuenire <lb></lb>quanta erit magnitudo liquidi B, Ac<lb></lb>cipiatur aliquod corpus ſolidum D, <lb></lb>eiuſdem generis cum ſolido A, &amp; ſit <lb></lb>eius grauitas G, &amp; liquidi, quod ſit E, <lb></lb>eiuſdem gener is cum liquido B, ma<lb></lb>gnitudinem habentis æqualem ſolido <lb></lb>D,* inueniatur grauitas quæ ſit H, &amp;<arrow.to.target n="note20"></arrow.to.target><lb></lb>fiat vt grauitas H, ad grauitatem G, <lb></lb>ita magnitudo C, ad aliam magnitudinem quæ ſit F. </s>

<s>Quoniam igitur <lb></lb>ſunt quatuor corpora grauia E, D, B, A, quorum primum E, &amp; ſecun<lb></lb>dum D, ſunt æqualia magnitudine, tertium vero B, &amp; quartum A, <lb></lb>æque grauia, &amp; ſunt eiuſdem generis corpora E, B, ſimiliter, &amp; corpo<lb></lb>ra D, A, *erit ut grauitas H, ad grauitatem G, ita magnitudo C, ad<arrow.to.target n="note21"></arrow.to.target> li<lb></lb>quidi B, magnitudinem, ſed vt grauitas H, ad grauitatem G, ita eſt <lb></lb>magnitudo C, ad magnitudinem F, ergo magnitudo F, æqualis erit <lb></lb>magnitudini liquidi B, inuenta igitur eſt liquidi corporis B, magni<lb></lb>tudo F, quod facere oportebat. </s></p><p type="margin">

<s><margin.target id="note20"></margin.target>8.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note21"></margin.target>7.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Sed quoniam corporum regularium magnitudo quo<lb></lb>que exprimitur latere eiuſdem corporis, vel diametro, ſi <lb></lb>propoſita duo corpora A, B, fuerint regularia, vtpote ſphę <lb></lb>rica, fuerit autem ſphæræ A, data diameter C, &amp; oporteat <lb></lb>inuenire quanta erit diameter ſphæræ B. ita faciendum <lb></lb>erit. </s></p><p type="main">

<s>Accepto, vt diximus, aliquo corpore ſolido D, eiuſdem generis <expan abbr="cũ">cum</expan> <lb></lb>ſphæra A, &amp; inuenta grauitate liquidi E, vt ſupra, fiat vt grauitas H, <lb></lb>ad grauitatem G, ita cubus ex C, ad alium cubum, cuius latus ſit F, <lb></lb>dico ipſum latus F, æquale eſſe diametro ſphæræ B. </s>

<s>Quoniam enim <lb></lb>eadem ratione qua ſupra demonſtrabitur, vt grauitas H, ad grauita<lb></lb>tem G, ita eſſe magnitudinem ſphæræ A, ad ſphæræ B, magnitudinem, <lb></lb>ſed magnitudo ſphæræ A, ad magnitudinem ſphæræ B,* triplicatam<arrow.to.target n="note22"></arrow.to.target><lb></lb>rationem habet eius, quam C, diameter ſphæræ A, ad diametrum <lb></lb>ſphæræ B, ſimiliter &amp; cubus ex C, ad cubum ex diametro ſphæræ B, <pb xlink:href="029/01/021.jpg" pagenum="14"></pb><arrow.to.target n="note23"></arrow.to.target> tripiicatam* rationem habet eius, quam C, ad diametrum ſphæræ, <lb></lb>B, ergo vt grauitas H, ad grauitatem G, ita erit cubus ex C, ad cubum <lb></lb>ex diametro ſphæræ B, ſed vt grauitas H, ad grauitatem G, ſta eſt cu<lb></lb>bus ex C, ad cubum ex F, ergo cubus ex F, æqualis erit cubo diame<lb></lb>triſphæræ B; quare &amp; latus F, æquabitur ſphæræ B, diametro. </s>

<s>inuen<lb></lb>ta igitur eſt quantitas diametri liquidæ ſphæræ, B, quod facere opor<lb></lb>tebat. </s></p><p type="margin">

<s><margin.target id="note22"></margin.target>18.12. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note23"></margin.target>33.12. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="head">

<s>Exemplum.</s></p><p type="main">

<s>Qvidam proponit aliquod corpus ſolidum notæ <lb></lb>magnitudinis, &amp; vult ſcire quanta erit magnitudo <lb></lb>alicuius liquidi, grauitatem habentis propoſito corpori <lb></lb>ſolido æqualem. </s></p><p type="main">

<s><emph type="italics"></emph>Sit propoſitum aliquod corpus plumbeum A, cuius magnitudo ſit<emph.end type="italics"></emph.end><lb></lb>10<emph type="italics"></emph>, &amp; oporteat ſcire quanta erit magnitudo aquæ grauitatem haben<lb></lb>tis æqualem propoſito plumbo A, accipiatur aliquod corpus <expan abbr="plumbeũ">plumbeum</expan> <lb></lb>D, cuius grauitas<emph.end type="italics"></emph.end> 23<emph type="italics"></emph>, deinde aquæ magnitudinem habentis æqualem <lb></lb>plumbo D, inueniatur grauitas, vt in exemplo propoſ. </s>

<s>8. dictum eſt, <lb></lb>quæ ſit<emph.end type="italics"></emph.end> 2<emph type="italics"></emph>, &amp; fiat vt<emph.end type="italics"></emph.end> 2<emph type="italics"></emph>, ad<emph.end type="italics"></emph.end> 23<emph type="italics"></emph>, ita<emph.end type="italics"></emph.end> 10<emph type="italics"></emph>, ad alium numerum qui ſit 115, is <lb></lb>igitur indicabit quanta erit magnitudo aquæ grauitatem habentis <lb></lb>æqualem propoſito plumbo A.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Quod ſi <expan abbr="propoſitũ">propoſitum</expan> corpus plumbeum A ſit regulare vt po<lb></lb>te ſphæricum, cuius ſphæræ diameter ſit 10, &amp; oporteat inue<lb></lb>nire quanta erit diameter ſphæræ ex aqua, grauitatem haben<lb></lb>tis æqualem propoſitæ ſphæræ A, ita faciendum erit. </s></p><p type="main">

<s><emph type="italics"></emph>Accipiatur, vt diximus, aliquod corpus plumbeum D, cuius gra<lb></lb>uitas<emph.end type="italics"></emph.end> 23<emph type="italics"></emph>, deinde aquæ habentis magnitudinem æqualem plumbo D, <lb></lb>inueniatur grauitas quæ ſit<emph.end type="italics"></emph.end> 2<emph type="italics"></emph>, &amp; fiat vt<emph.end type="italics"></emph.end> 2<emph type="italics"></emph>, ad<emph.end type="italics"></emph.end> 23<emph type="italics"></emph>, ita cubus ex<emph.end type="italics"></emph.end> 10<emph type="italics"></emph>, qui <lb></lb>eſt 1000, ad alium numerum qui ſit<emph.end type="italics"></emph.end> 11500<emph type="italics"></emph>, is igitur numerus erit <lb></lb>cubus diametri ſphæræ ex aqua grauitatem habentis æqualem propo<lb></lb>ſitæ ſphæræ A, quare eius latus cubicum, quod est<emph.end type="italics"></emph.end> 22 57/100. <emph type="italics"></emph><expan abbr="proximũ">proximum</expan> <lb></lb>vero indicabit ipſam diametrum.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Similiter ſi propoſitum corpus plumbeum A, fuerit cubicum, vel <lb></lb>alicuius alterius formæ regularis, eadem ratione inueniemus latus cu<lb></lb>bi ex aqua grauitatem habentis æqualem propoſito cubo A, nam ſi cu<lb></lb>bi A, datum ſit latus<emph.end type="italics"></emph.end> 10<emph type="italics"></emph>, erit numerus 11500, cubus ex aqua æqualis <lb></lb>grauitate propoſito cubo A, quare latus cubicum numeri<emph.end type="italics"></emph.end> 11500<emph type="italics"></emph>, quod<emph.end type="italics"></emph.end><pb xlink:href="029/01/022.jpg" pagenum="15"></pb><emph type="italics"></emph>eſt<emph.end type="italics"></emph.end> 22 17/100. <emph type="italics"></emph><expan abbr="proximũ">proximum</expan> vero indicabit quæſitum latus cubi ex aqua.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Neque diſſimili ratione inuenietur magnitudo olei, aut ar<lb></lb>genti viui, aut cuiuſcumque generis liquidi grauitatem habe<lb></lb>tis propoſito corpori ſolido æqualem, ſed quomodo inuenien<lb></lb>da ſit grauitas argenti viui magnitudinem habentis æqualem <lb></lb>corpori ſolido, docebimus poſt exemplum propoſitionis de<lb></lb>cimæ quartæ. </s></p><p type="head">

<s>PROBLEMA IV. PROPOS. XI.</s></p><p type="main">

<s>Propoſitis duobus corporibus æque grauibus, vno ſo<lb></lb>lido, altero liquido, data liquidi corporis magnitu<lb></lb>dine, magnitudinem ſolidi inuenire. </s></p><p type="main">

<s>SINT propoſita duo corpora æ<lb></lb>quæ grauia, A, quidem ſolidum, B, <lb></lb>vero liquidum, ſit autem, liquidi B; <lb></lb><figure id="id.029.01.022.1.jpg" xlink:href="029/01/022/1.jpg"></figure><lb></lb>data magnitudo F, &amp; oporteat ſolidi <lb></lb>A, magnitudinem inuenire. </s>

<s>Accipia<lb></lb>tur aliquod corpus ſolidum D, <expan abbr="eiuſdẽ">eiuſdem</expan> <lb></lb>generis cum corpore ſolido A, cuius <lb></lb>grauitas ſit G, deinde liquidi quod ſit <lb></lb>E, eiuſdem generis cum corpore liqui <lb></lb>do B, magnitudinem æqualem<arrow.to.target n="note24"></arrow.to.target> haben<lb></lb>tis ſolido D, *inueniatur grauitas, <lb></lb>quæ ſit H, &amp; fiat vt grauitas G; ad gra<lb></lb>uitatem H, ita F, magnitudo, ad aliam magnitudinem, quæ ſit C; <lb></lb>quoniam igitur ſunt quatuor corpora D, E, A, B, quorum primum D, <lb></lb>&amp; ſecundum E, ſunt magnitudine æqualia, tertium vero A, &amp; <expan abbr="quartũ">quartum</expan> <lb></lb>B, æquæ grauia, &amp; ſunt eiuſdem generis ſolida D, A, ſimiliter, &amp; liqui<lb></lb>da E, B,* erit vt grauitas G, ad grauitatem H, ita F, magnitudo ad ma<arrow.to.target n="note25"></arrow.to.target><lb></lb>gnitudinem ſolidi A, ſed vt grauitas G, ad grauitatem H, ita eſt ma<lb></lb>gnitudo F, ad C, magnitudinem, ergo magnitudo C, æqualis erit ma<lb></lb>gnitudini corporis ſolidi A, inuenta igitur eſt corporis ſolidi A, ma<lb></lb>gnitudo C, quod erat faciendum. </s></p><p type="margin">

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

<s><margin.target id="note25"></margin.target>7.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Qvod ſi propoſita duo corpora æque grauia A, B, fue<lb></lb>rint regularia vtpote ſphærica, fuerit autem liqui<pb xlink:href="029/01/023.jpg" pagenum="16"></pb>dæ ſphæræ B, data diameter F, &amp; oporteat inuenire quan<lb></lb>ta erit diameter ſolidæ ſphæræ A, ita faciendum erit. </s></p><p type="main">

<s>Accepto vt ſupra corpore ſolido D, &amp; liquidi E, inuenta grauitate, <lb></lb>vt dictum eſt, fiat vt grauitas G, ad prauitatem H, ita cubus ex F, ad <lb></lb>alium cubum, cuius latus ſit C, Quoniam igitur eadem ratione qua <lb></lb>ſupra oſtendetur, vt grauitas G, ad grauitatem H, ita eſſe magnitudi<lb></lb>nem ſphæræ B, ad ſphæræ A, magnitudinem, ſed magnitudo ſphæræ B, <lb></lb><arrow.to.target n="note26"></arrow.to.target> ad magnitudinem ſphæræ A, *triplicatam rationem habet eius, quam <lb></lb>F, diameter ſphæræ B, ad diametrum ſphæræ A, ſimiliter, &amp; cubus ex <lb></lb><arrow.to.target n="note27"></arrow.to.target> F, ad cubum ex diametro ſphæræ A, <expan abbr="triplicatã*">triplicatam*</expan> rationem habet eius, <lb></lb>quam F, ad diametrum ſphæræ A, ergo, vt grauitas G, ad <expan abbr="grauitatẽ">grauitatem</expan> <lb></lb>H, ita erit cubus ex F, ad cubum ex diametro ſphæræ A, ſed vt graui<lb></lb>tas G, ad grauitatem H, ita eſt cubus ex F, ad cubum ex C, ergo cubus <lb></lb>ex C, æqualis erit cubo diametri ſphæræ A, quare, &amp; latus C, æquabi<lb></lb>tur ipſius ſphæræ A, diametro, inuenta igitur eſt quantitas diametri <lb></lb>ſolidæ ſphæræ A, quod facere oportebat. </s></p><p type="margin">

<s><margin.target id="note26"></margin.target>18.12. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note27"></margin.target>33.11. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="head">

<s>Exemplum.</s></p><p type="main">

<s>Qvidam proponit aliquod corpus liquidum notæ <lb></lb>magnitudinis, &amp; vult inuenire quanta erit magni<lb></lb>tudo alicuius ſolidi grauitatem habentis propoſi<lb></lb>to corpori liquido æqualem. </s></p><p type="main">

<s><emph type="italics"></emph>Sit propoſitum aliquod corpus aqueum B, cuius magnitudo ſit 115, <lb></lb>&amp; oporteat inuenire quanta erit magnitudo plumbi grauitatem ha<lb></lb>bentis æqualem propoſitæ aquæ B, accipiatur aliquod corpus <expan abbr="plumbeũ">plumbeum</expan> <lb></lb>D, cuius grauitas ſit verbi gratia<emph.end type="italics"></emph.end> 23<emph type="italics"></emph>, deinde aquæ magnitudinem ba<lb></lb>bentis æqualem plumbo D, inueniatur grauitas quæ ſit<emph.end type="italics"></emph.end> 2. <emph type="italics"></emph>id autem do<lb></lb>cuit propoſitionis octauæ exemplum, &amp; fiat vt<emph.end type="italics"></emph.end> 23<emph type="italics"></emph>, ad<emph.end type="italics"></emph.end> 2<emph type="italics"></emph>, ita 115, ad <lb></lb>alium numerum qui ſit<emph.end type="italics"></emph.end> 10<emph type="italics"></emph>, is igitur numerus indicabit quanta erit <lb></lb>magnitudo plumbi grauitatem habentis æqualem propoſitæ aquæ B.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Quod ſi propoſitum corpus aqueum B, ſit ſphæricum, cuius <lb></lb>ſphæræ diameter ſit 10, &amp; oporteat inuenire quanta erit dia<lb></lb>meter ſphæræ ex plumbo, grauitatem habentis æqualem pro<lb></lb>poſitæ ſphæræ B, ita faciendum erit. </s></p><p type="main">

<s><emph type="italics"></emph>Accepto, vt diximus aliquo corpore plumbeo D, cuius grauitas<emph.end type="italics"></emph.end> 23, <lb></lb><emph type="italics"></emph>&amp; aquæ magnitudinem habentis æqualem plumbo D, inuenta graui<emph.end type="italics"></emph.end><pb xlink:href="029/01/024.jpg" pagenum="17"></pb><emph type="italics"></emph>tate 2, fiat vt<emph.end type="italics"></emph.end> 23<emph type="italics"></emph>, ad<emph.end type="italics"></emph.end> 2<emph type="italics"></emph>, ita cubus ex<emph.end type="italics"></emph.end> 10<emph type="italics"></emph>, hoc est<emph.end type="italics"></emph.end> 1000<emph type="italics"></emph>, ad alium nume<lb></lb>rum qui ſit<emph.end type="italics"></emph.end> 86 22/23&lt;31&gt; <emph type="italics"></emph>is igitur numerus erit cubus diametris ſphæræ ex <lb></lb>plumbo, grauitatem æqualem habentis propoſite ex aqua ſphæræ B, <lb></lb>quare eius latus cubicum, quod est<emph.end type="italics"></emph.end> 4 41/100&lt;31&gt; <emph type="italics"></emph>ferè, indicabit ipſam <lb></lb>diametrum.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Similiter ſi propoſitum corpus aqueum B, fuerit cubicum, vel alicu<lb></lb>ius alterius formæ regularis, eadem ratione vtemur ad inueniendum <lb></lb>latus cubi ex plumbo, grauitatem habentis æqualem propoſito ex aqua <lb></lb>tubo B, nam ſi ex aqua cubi B, datum ſit latus<emph.end type="italics"></emph.end> 10<emph type="italics"></emph>, erit numerus<emph.end type="italics"></emph.end><lb></lb>86 22/2<gap></gap>&lt;31&gt; <emph type="italics"></emph>cubus ex plumbo æqualis grauitate propoſito ex aqua cubo B, <lb></lb>quare latus cubicum numeri<emph.end type="italics"></emph.end> 86 22/2<gap></gap>&lt;31&gt; <emph type="italics"></emph>quod eſt<emph.end type="italics"></emph.end> 4 41/100&lt;31&gt; <emph type="italics"></emph>ferè, indicabit <lb></lb>quæſitum latus cubi ex plumbo.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Neque diſſimili ratione inuenienda erit magnitudo auri, <lb></lb>argenti, ceræ, aut cuiuſcunque ſolidi, grauitatem habentis <lb></lb>propoſito corpori liquido æqualem. </s></p><p type="head">

<s>PROBLEMA V. PROPOS. XII.</s></p><p type="main">

<s>Propoſitis duobus ſolidis corporibus magnitudine <lb></lb>æqualibus, data grauitate vnius, grauitatem al<lb></lb>terius inuenire. </s></p><p type="main">

<s>SINT propoſita duo corpora ſo<lb></lb>lida magnitudine æqualia A, B, ſit au<lb></lb><figure id="id.029.01.024.1.jpg" xlink:href="029/01/024/1.jpg"></figure><lb></lb>tem vnius, vtpote ipſius A, data gra<lb></lb>uitas C, &amp; oporteat inuenire grauita<lb></lb>tem ipſius B. </s>

<s>Accipiatur aliquod ſoli<lb></lb>dum corpus D, eiuſdem generis cum <lb></lb>corpore ſolido A, cui æquale grauita<lb></lb>te accipiatur alterum E, eiuſdem ge<lb></lb>neris cum corpore B, deinde liquidi <lb></lb>magnitudine æqualis corpori D,<arrow.to.target n="note28"></arrow.to.target> *in<lb></lb>ueniatur grauitas, quæ ſit G, item li<lb></lb>quidi eiuſdem generis, æqualis ma<lb></lb>gnitudine corpori E,* inueniatur grauitas, quæ ſit H, &amp; fiat vt H, ad<arrow.to.target n="note29"></arrow.to.target><lb></lb>G, ita C, ad aliam grauitatem, quæ ſit F. </s>

<s>Quoniam igitur ſunt qua<lb></lb>tuor corpora A, B, D, E, quorum A, B, primum videlicet, &amp; ſecundum <lb></lb>ſunt æqualia magnitudine, tertium vero D, &amp; E, quartum æque gra<lb></lb>uia, &amp; ſunt <expan abbr="eiuſdẽ">eiuſdem</expan> generis ſolida A, D, itidem ſolida B, E, *erit vt<arrow.to.target n="note30"></arrow.to.target> gra<lb></lb>uitas C, ad ſolidi B, grauitatem, ita grauitas H, ad grauitatem G, ſed <pb xlink:href="029/01/025.jpg" pagenum="18"></pb>vt grauitas H, ad <expan abbr="grauitatẽ">grauitatem</expan> G, ita eſt grauitas C, ad F, grauitatem; er<lb></lb>go grauitas F, æqualis erit grauitati ſolidi B, inuenta igitur eſt cor<lb></lb>poris ſolidi B, grauitas F, quod facere oportebat. </s></p><p type="margin">

<s><margin.target id="note28"></margin.target>8.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note29"></margin.target>5.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note30"></margin.target>6.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Hoc Problema magni momenti eſt, pleriſque artificibus <lb></lb>maximo vſui eſſe poteſt. </s>

<s>in arte fuſoria propoſito operis modu<lb></lb>lo, ex illius grauitate, facile metalli ad opus faciendum, gra<lb></lb>uitatem inueniet, ſi enim hoc ignoret artifex, periculum eſt, ne <lb></lb>metallum, aut deficiat, vel ſi multum eſt, ob nimiam graui<lb></lb>tatem difficile tractetur. </s></p><p type="main">

<s>Neque tormenti bellici magiſtro inutile erit, is enim cogni<lb></lb>ta grauitate alicuius globi, exempli gratia ex plumbo, ſtatim <lb></lb>alterius globi eiuſdem magnitudinis, vel ſit ex lapide, vel ex <lb></lb>fcrro, vel ex quacunque alia materia, grauitatem inueniet. </s></p><p type="head">

<s>Exemplum.</s></p><p type="main">

<s>Qvidam proponit aliquod corpus ſolidum notæ <lb></lb>grauitatis, &amp; vult ſcire quanta erit grauitas alicu<lb></lb>ius ſolidi, alterius generis, magnitudinem habentis pro<lb></lb>poſito corpori ſolido æqualem. </s></p><p type="main">

<s><emph type="italics"></emph>Sit propoſitum aliquod corpus plumbeum A, cuius grauitas ſit<emph.end type="italics"></emph.end><lb></lb>1150<emph type="italics"></emph>, &amp; oporteat inuenire quanta erit grauitas stanni <expan abbr="magnitudinẽ">magnitudinem</expan> <lb></lb>habentis æqualem propoſito plumbo A. </s>

<s>Accipiantur duo corpora æque <lb></lb>grauia, D, plumbeum, E, stanneum, deinde duarum quantitatum, <lb></lb>aquæ, quarum una ſit æqualis magnitudine plumbo D, altera ſtanno <lb></lb>E, <expan abbr="inueniãtur">inueniantur</expan> grauitates, quæ ſint, primæ videlicet quantitatis aquæ<emph.end type="italics"></emph.end><lb></lb>74, <emph type="italics"></emph>ſecundæ vero<emph.end type="italics"></emph.end> 115<emph type="italics"></emph>, &amp; fiat vt<emph.end type="italics"></emph.end> 115<emph type="italics"></emph>, ad<emph.end type="italics"></emph.end> 74<emph type="italics"></emph>, ita<emph.end type="italics"></emph.end> 1150<emph type="italics"></emph>, ad alium nume<lb></lb>rum, qui ſit<emph.end type="italics"></emph.end> 740<emph type="italics"></emph>, is igitur numerus indicabit grauitatem stanni, ma<lb></lb>gnitudinem habentis propoſito plumbo A.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Etiam ſi non accipiantur duo corpora, plumbeum videlicet <lb></lb>&amp; ſtanneum, æque grauia, ſed grauitate quacunque, grauitas <lb></lb>ſtanni magnitudinem habentis æqualem propoſito plumbo <lb></lb>D, inuenietur ſic. </s></p><p type="main">

<s><emph type="italics"></emph>Accipiantur duo corpora D, plumbeum, E, ſtanneum grauitate <lb></lb>quacunque, ſit vcrbi gratia plumbi D, grauitas<emph.end type="italics"></emph.end> 23<emph type="italics"></emph>, ſtanni vero E, <lb></lb>grauitas<emph.end type="italics"></emph.end> 37<emph type="italics"></emph>, deinde duarum quantitatum aquæ, quarum vna ſit ma<lb></lb>gnitudine æqualis plumbo D, altera stanno E, inueniantur graui<lb></lb>tates, quæ ſint, primæ videlicet quantitatis<emph.end type="italics"></emph.end> 2<emph type="italics"></emph>, ſecundæ vero<emph.end type="italics"></emph.end> 5<emph type="italics"></emph>, &amp; fiat,<emph.end type="italics"></emph.end><pb xlink:href="029/01/026.jpg" pagenum="19"></pb><emph type="italics"></emph>ut 23, ad 2, ita 37, ad<emph.end type="italics"></emph.end> 3 5/21. <emph type="italics"></emph>grauitas igitur aquæ, magnitudinem <lb></lb>habentis æqualem plumbo, cuius grauitas est 37, erit<emph.end type="italics"></emph.end> 3 5/21. </s></p><p type="main">

<s><emph type="italics"></emph>Et quoniam aquæ, magnitudinem habentis æqualem stanno E, cu<lb></lb>ius grauitas est 37, eſt grauitas<emph.end type="italics"></emph.end> 5, <emph type="italics"></emph>erunt grauitates duarum quan<lb></lb>titatum aquæ<emph.end type="italics"></emph.end> 3 5/21, &amp; 5, <emph type="italics"></emph>quarum quantitatum prima eſt æqualis <lb></lb>magnitudine corpori plumbeo, ſecunda stanneo, que ſunt æque gra<lb></lb>uia, vtriuſque enim grauitas est<emph.end type="italics"></emph.end> 37. <emph type="italics"></emph>Fiat igitur vt<emph.end type="italics"></emph.end> 5, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 3 5/21, <emph type="italics"></emph>it a<emph.end type="italics"></emph.end><lb></lb>1150, <emph type="italics"></emph>ad alium numerum, qui ſit<emph.end type="italics"></emph.end> 740, <emph type="italics"></emph>tanta igitur erit grauitas <lb></lb>ſtanni, magnitudinem habentis æqualem propoſito plumbo A, quanta <lb></lb>etiam inueniebatur &amp; ſupra.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Quod ſi propoſitum ſit cereum corpus aliquod, aut cuiuſ<lb></lb>cunque generis ſolidi, ſiue leuioris quam aqua, ſiue grauio<lb></lb>ris, &amp; oporteat inuenire grauitatem alicuius ſolidi alterius <lb></lb>generis, magnitudine æqualis propoſito corpori ſolido. </s>

<s>Ea<lb></lb>dem ratione qua ſupra inuenietur quæſita ſolidi grauitas, ſed <lb></lb>hoc ſolum animaduertendum eſt, quod non eadem ratione <lb></lb>inuenitur grauitas aquæ, magnitudinem habentis æqualem <lb></lb>propoſito cuiuſcunque generis ſolido, alia enim tenenda eſt <lb></lb>ratio ad inueniendam grauitatem prædictæ aquæ, quando <lb></lb>propoſitum ſolidum ſit grauius quam aqua, alia vero quando <lb></lb>leuius, ſed ſiue ſit leuius, ſiue grauius, de inuentione huiuſ<lb></lb>modi grauitatis, in exemplo propoſitionis octauæ, ſatis eſt ex<lb></lb>plicatum. </s></p><p type="head">

<s>PROBLEMA VI. PROPOS. XIII.</s></p><p type="main">

<s>Propoſitis duobus ſolidis corporibus æque graui<lb></lb>bus, data magnitudine vnius, magnitudinem alte<lb></lb>rius inuenire. </s></p><p type="main">

<s>SINT propoſita duo corpora ſolida æque grauia A, B, ſit au<lb></lb>tem vnius, vtpote ipſius A, data magnitudo C, &amp; oporteat inueni<lb></lb>re magnitudinem ipſius B, Accipiatur aliquod ſolidum corpus D, <lb></lb>eiuſdem generis cum ſolido A, &amp; ſit eius grauitas G, deinde ſolidi <lb></lb>corporis quod ſit E, eiuſdem generis cum ſolido B, magnitudine <lb></lb>æqualis ipſi D, inueniatur grauitas, quæ ſit H, hoc autem, Proble<lb></lb>ma antecedens docuit, &amp; fiat vt grauitas H, ad grauitatem G, ita <lb></lb>magnitudo C, ad aliam magnitudinem, quæ ſit F. </s>

<s>Quoniam igitur <pb xlink:href="029/01/027.jpg" pagenum="20"></pb>ſunt quatuor corpora grauia E, D, B, <lb></lb>A, quorum E, D, primum videlicet, <lb></lb><figure id="id.029.01.027.1.jpg" xlink:href="029/01/027/1.jpg"></figure><lb></lb>&amp; ſecundum, ſunt æqualia magnitu<lb></lb>dine, tertium vero B, &amp; quartum A, <lb></lb>æquegrauia, &amp; ſunt eiuſdem generis <lb></lb>corpora E, B, ſimiliter &amp; corpora <lb></lb><arrow.to.target n="note31"></arrow.to.target> D, A, * erit vt grauitas H, ad graui<lb></lb>tatem G, ita magnitudo C, ad corpo<lb></lb>ris B, magnitudinem, ſed vt grauitas <lb></lb>H, ad grauitatem G, ita eſt magnitu<lb></lb>do C, ad F, magnitudinem, ergo ma<lb></lb>gnitudo F, æqualis erit magnitudini <lb></lb>corporis B. inuenta igitur eſt corporis B, magnitudo F, quod facere <lb></lb>oportebat. </s></p><p type="margin">

<s><margin.target id="note31"></margin.target>7.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Quod ſi propoſita duo corpora æque grauia A, B, fue<lb></lb>rint regularia, vtpote ſphærica, fuerit autem ſphæræ A, <lb></lb>data diameter C, &amp; oporteat inuenire, quanta erit dia<lb></lb>meter ſphærę B, ita faciendum erit. </s></p><p type="main">

<s>Accepto corpore ſolido D, &amp; inuenta ſolidi corporis E, grauita<lb></lb>te, vt ſupra dictum eſt, fiat vt grauitas H, ad grauitatem G, ita cu<lb></lb>bus ex C, ad alium cubum, cuius latus ſit F. </s>

<s>Quoniam igitur eadem <lb></lb>ratione, qua ſupra, demonſtrabitur, vt grauitas H, ad grauitatem G, <lb></lb>ita eſſe magnitudinem ſphæræ A, ad ſphæræ B, magnitudinem, ſed <lb></lb><arrow.to.target n="note32"></arrow.to.target> magnitudo ſphæræ A, ad ſphæræ B, magnitudinem * triplicatam ra<lb></lb>tionem habet eius, quam C, diameter ſphæræ A, ad diametrum ſphæ<lb></lb><arrow.to.target n="note33"></arrow.to.target>ræ B. </s>

<s>Similiter &amp; cubus ex C, ad cubum, ex diametro ſphæræ B, *tri<lb></lb>plicatam rationem habet eius, quam C, ad ſphæræ B, diametrum; er<lb></lb>go vt grauitas H, ad grauitatem G, ita erit cubus ex C, ad cubum ex <lb></lb>diametro ſphæræ B, ſed vt grauitas H, ad grauitatem G, ita eſt cubus <lb></lb>ex C, ad cubum ex F; ergo cubus ex F, æqualis erit cubo diametri <lb></lb>ſphæræ B; quare &amp; latus F, æquabitur ſphæræ B, diametro. </s>

<s>inuenta <lb></lb>igitur eſt quantitas diametri ſphæræ B, quod facere oportebat. </s></p><p type="margin">

<s><margin.target id="note32"></margin.target>18.12. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note33"></margin.target>33.11. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Neque hoc Problema inutile erit tormenti bellici magiſtro, <lb></lb>is enim cognita diametro alicuius globi, exempli gratia, ex <lb></lb>plumbo, ſtatim alterius globi eandem habentis grauitatem, <lb></lb>diametrum inueniet, ſit globus ille, vel ex lapide, vel ex fer<lb></lb>ro, vel ex quocunque alio ſolidorum genere. </s></p><pb xlink:href="029/01/028.jpg" pagenum="21"></pb><p type="head">

<s>Exemplum.</s></p><p type="main">

<s>Qvidam proponit aliquod corpus ſolidum notæ <lb></lb>magnitudinis, &amp; vult inuenire, quanta erit magni<lb></lb>tudo alicuius ſolidi alterius generis, grauitatem <lb></lb>habentis propoſito corpori ſolido æqualem. </s></p><p type="main">

<s><emph type="italics"></emph>SIT propoſitum aliquod corpus plumbeum A, cuius magnitudo<emph.end type="italics"></emph.end><lb></lb>740, <emph type="italics"></emph>&amp; oporteat inuenire quanta erit magnitudo ſtanni, grauita<lb></lb>tem habentis æqualem propoſito plumbo A. </s>

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

<s>Quod ſi propoſitum corpus plumbeum A, ſit ſphæricum, cu<lb></lb>ius ſphæræ diameter ſit 10, &amp; oporteat inuenire quanta erit <lb></lb>diameter ſphæræ ex ſtanno, grauitatem habentis æqualem <lb></lb>propoſitæ ſphæræ A, ita faciendum erit. </s></p><p type="main">

<s><emph type="italics"></emph>Accipiatur vt diximus aliquod corpus plumbeum D, cuius graui<lb></lb>tas ſit 115, &amp; ſtanni, magnitudinem habentis æqualem plumbo D, in<lb></lb>ueniatur grauitas, quæ ſit<emph.end type="italics"></emph.end> 74. <emph type="italics"></emph>&amp; fiat vt 74, ad<emph.end type="italics"></emph.end> 115, <emph type="italics"></emph>ita cubus ex 10, <lb></lb>qui eſt 1000, ad alium numerum qui ſit<emph.end type="italics"></emph.end> 1554 2/17, <emph type="italics"></emph>is igitur numerus <lb></lb>erit cubus diametri ſphæræ ex stanno, grauitatem habentis æqualem <lb></lb>propoſitæ ex plumbo ſphæræ A, quare eius latus cubicum, quod est<emph.end type="italics"></emph.end><lb></lb>11 52/100, <emph type="italics"></emph>vero proximum, indicabit ipſam diametrum.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Similiter ſi propoſitum corpus plumbeum A, fuerit cubicum, vel <lb></lb>alicuius alterius formæ regularis, eadem ratione inuenietur latus cu<lb></lb>bi ex stanno, grauitatem habentis æqualem propoſito plumbeo cu<lb></lb>bo A, ſi enim ipſius cubi plumbei A, datum ſit latus 10, erit numerus<emph.end type="italics"></emph.end><lb></lb>1554 2/17&lt;31&gt; <emph type="italics"></emph>cubus ex ſtanno æqualis grauitate propoſito plumbeo Cu<lb></lb>bo A, quare latus. </s>

<s>cubitum numeri<emph.end type="italics"></emph.end> 1554 2/17&lt;31&gt; <emph type="italics"></emph>quod est<emph.end type="italics"></emph.end> 11 52/100&lt;31&gt; <emph type="italics"></emph>pro<lb></lb>ximum vero; indicabit quæ ſitum latus.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Neque diſſimili ratione inuenienda erit magnitudo auri, <lb></lb>argenti, cæræ, aut cuiuſcumque ſolidi, grauitatem habentis <lb></lb>propoſito corpori ſolido æqualem. </s></p><pb xlink:href="029/01/029.jpg" pagenum="22"></pb><p type="head">

<s>PROBLEMA VII. PROPOS. XIV.</s></p><p type="main">

<s>Propoſitis duobus liquidis corporibus magnitudine <lb></lb>ęqualibus, data grauitate vnius, grauitatem alte<lb></lb>rius inuenire. </s></p><figure id="id.029.01.029.1.jpg" xlink:href="029/01/029/1.jpg"></figure><p type="main">

<s>SINT propoſita duo cor<lb></lb>ra liquida, magnitudine æqualia <lb></lb>A, B, ſit autem vnius, vtpote li<lb></lb>quidi A, data grauitas G, &amp; <lb></lb>oporteat alterius liquidi B, gra<lb></lb>uitatem inuenire. </s>

<s>Accipiatur <lb></lb>aliquod corpus ſolidum C, &amp; li<lb></lb>quidi, quod ſit H, eiuſdem ge<lb></lb>neris cum liquido A, magnitu<lb></lb><arrow.to.target n="note34"></arrow.to.target>dine æqualis ſolido C, *inuenia<lb></lb>tur grauitas, quæ ſit D, ſimiliter <lb></lb>&amp; liquidi, quod ſit I, eiuſdem <lb></lb>generis cum liquido B, magni<lb></lb>tudine æqualis eidem ſolido C, <lb></lb><arrow.to.target n="note35"></arrow.to.target> * inueniatur grauitas, quæ ſit E, <lb></lb>&amp; fiat vt D, ad E, ita G, ad aliam grauitatem, quæ ſit F. </s>

<s>Quoniam <lb></lb>igitur eſt vt A, ad B, ita H, ad I, æquale videlicet ad æquale, erit per<lb></lb>mutando vt A, ad H, ita B, ad I, &amp; quoniam eiuſdem ſunt generis cor<lb></lb><arrow.to.target n="note36"></arrow.to.target>pora A, H, ſimiliter &amp; corpora B, I, * erit vt grauitas G, ad grauita<lb></lb>tem D, ita liquidi B, grauitas, ad grauitatem E, &amp; permutando vt <lb></lb>grauitas G, ad grauitatem liquidi B, ita D, grauitas, ad grauitatem <lb></lb>E, ſed vt grauitas D, ad grauitatem E, ita eſt grauitas G, ad graui<lb></lb>tatem F; ergo grauitas F, æqualis erit grauitati liquidi B. inuenta <lb></lb>igitur eſt liquidi corporis B, grauitas F, quod facere oportebat. </s></p><p type="margin">

<s><margin.target id="note34"></margin.target>2.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note35"></margin.target>3.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note36"></margin.target>4.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="head">

<s>Exemplum.</s></p><p type="main">

<s>Qvidam proponit aliquod corpus liquidum notæ <lb></lb>grauitatis, &amp; vult ſcire, quanta erit grauitas alte<lb></lb>rius liquidi, magnitudinem habentis propoſito corpori <lb></lb>liquido æqualem. </s></p><p type="main">

<s><emph type="italics"></emph>Sit propoſitum aliquod olei corpus A, cuius grauitas<emph.end type="italics"></emph.end> 550, <emph type="italics"></emph>&amp; <lb></lb>oporteat inuenire, quanta erit grauitas aquæ, magnitudinem haben<emph.end type="italics"></emph.end><pb xlink:href="029/01/030.jpg" pagenum="23"></pb><emph type="italics"></emph>tis æqualem propoſito oleo A, Accipiatur aliquod corpus ſolidum C, <lb></lb>vtpote plumbeum, &amp; aquæ, magnitudinem habentis æqualem plumbo<lb></lb>C, inueniatur grauitas, quæ ſit 12, vt in exemplo propoſ. </s>

<s>8. <expan abbr="dictũ">dictum</expan> eſt. </s>

<s><lb></lb>Similiter &amp; olei, magnitudinem æqualem habentis, eidem plumbo C, <lb></lb>inueniatur grauitas, quæ ſit 11, &amp; fiat, vt<emph.end type="italics"></emph.end> 11, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 12, <emph type="italics"></emph>ita 550, ad <expan abbr="aliũ">alium</expan> <lb></lb>numerum qui ſit<emph.end type="italics"></emph.end> 600. <emph type="italics"></emph>is igitur numerus indicabit quanta erit graui<lb></lb>tas aquæ, magnitudinem habentis æqualem propoſito oleo A.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Si vero propoſitum ſit aliquod argenti viui corpus A, cuius <lb></lb>grauitas<emph.end type="italics"></emph.end> 95, <emph type="italics"></emph>&amp; oporteat inuenire, quanta erit grauitas aquæ, magni<lb></lb>tudincm habentis æqualem propoſito argento viuo A. </s>

<s>Accipiatur ali<lb></lb>quod vas vitreum mundum, &amp; politum, cuius grauitas ſit v. g.<emph.end type="italics"></emph.end> 91. <lb></lb><emph type="italics"></emph><expan abbr="ipſumq;">ipſumque</expan> vas plenum aqua ponderetur in aqua, &amp; habeat grauitatem<emph.end type="italics"></emph.end><lb></lb>55, <emph type="italics"></emph>quoniam igitur numerus<emph.end type="italics"></emph.end> 91, <emph type="italics"></emph>ſuperat <expan abbr="numerũ">numerum</expan><emph.end type="italics"></emph.end> 55, <emph type="italics"></emph>numero<emph.end type="italics"></emph.end> 36, <emph type="italics"></emph>* erit<emph.end type="italics"></emph.end><arrow.to.target n="note37"></arrow.to.target><lb></lb><emph type="italics"></emph>grauitas aquæ, magnitudinem habentis æqualem ipſi vaſi, hoc est ſoli<lb></lb>ditati ipſius vaſis<emph.end type="italics"></emph.end> 36, <emph type="italics"></emph>ponatur deinde in ipſum vas propoſitum argen<lb></lb>tum <expan abbr="viuũ">viuum</expan> A, nihil interest, vt vas ſit plenum, vel non, &amp; quoniam ar<lb></lb>genti viui A, grauitas est<emph.end type="italics"></emph.end> 95, <emph type="italics"></emph>&amp; vaſis vitrei grauitas<emph.end type="italics"></emph.end> 91, <emph type="italics"></emph>erit argenti <lb></lb>viui ſimul cum ipſo vaſe, grauitas<emph.end type="italics"></emph.end> 186. <emph type="italics"></emph>ponderetur itaque ipſum vas, <lb></lb>ſimul cum argento viuo A, in aqua, ita vt aqua repleat vaſis partem <lb></lb>vacuam, &amp; ſit vaſis grauitas in aqua ſimul cum argento viuo<emph.end type="italics"></emph.end> 143, <lb></lb><emph type="italics"></emph>quoniam igitur numerus<emph.end type="italics"></emph.end> 186, <emph type="italics"></emph>ſuperat numerum<emph.end type="italics"></emph.end> 143, <emph type="italics"></emph>numero<emph.end type="italics"></emph.end> 43, <emph type="italics"></emph>* erit<emph.end type="italics"></emph.end><arrow.to.target n="note38"></arrow.to.target><lb></lb><emph type="italics"></emph>grauitas aquæ, <expan abbr="magnitudinẽ">magnitudinem</expan> <expan abbr="bibẽtis">bibentis</expan> æqualem argento viuo, ſimul <expan abbr="cũ">cum</expan> <lb></lb>vaſe<emph.end type="italics"></emph.end> 43, <emph type="italics"></emph>ſed grauitas aquæ habentis magnitudinem æqualem vaſi est<emph.end type="italics"></emph.end><lb></lb>36, <emph type="italics"></emph>ergo reliquum quod est<emph.end type="italics"></emph.end> 7, <emph type="italics"></emph>erit grauitas aquæ, magnitudinem ba<lb></lb>bentis æqualem propoſito argento viuo A.<emph.end type="italics"></emph.end></s>

</p><p type="margin">

<s><margin.target id="note37"></margin.target>5.<emph type="italics"></emph>huiu<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note38"></margin.target>5.<emph type="italics"></emph>huiu<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Sed ſi propoſitum fuerit aliquod magnum argenti viui cor <lb></lb>pus, ita vt difficile poſſit ponderari in aqua, hac via inuenietur <lb></lb>aquæ quæſita grauitas. </s></p><p type="main">

<s><emph type="italics"></emph>Sit propoſitum aliquod magnum argenti viui corpus A, cuius gra<lb></lb>uitas<emph.end type="italics"></emph.end> 5700. <emph type="italics"></emph>&amp; oporteat facere, quod imperatum eſt. </s>

<s>Accipiatur ali<lb></lb>quod paruum argenti viui corpus C, cuius grauitas ſit<emph.end type="italics"></emph.end> 95, <emph type="italics"></emph>&amp; aquæ, <lb></lb>magnitudinem habentis æqualem argento viuo C, inueniatur graui<lb></lb>tas, eo modo quo dictum est, quæ ſit<emph.end type="italics"></emph.end> 7, <emph type="italics"></emph>&amp; fiat vt<emph.end type="italics"></emph.end> 95, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 7, <emph type="italics"></emph>ita<emph.end type="italics"></emph.end> 5700, <emph type="italics"></emph>ad <lb></lb>alium numerum, qui ſit<emph.end type="italics"></emph.end> 420, <emph type="italics"></emph>is igitur numerus indicabit quanta erit <lb></lb>grauitas aquæ, magnitudinem habentis æqualem propoſito argento <lb></lb>viuo A.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Contra, ſit propoſitum aliquod corpus aqueum A, cuius grauitas<emph.end type="italics"></emph.end><lb></lb>420, <emph type="italics"></emph>&amp; oporteat inuenire quanta erit grauitas argenti viui, magni<lb></lb>tudine æqualis propoſitæ aquæ A. facto, vt ſupra, &amp; inuenta graui<lb></lb>tate<emph.end type="italics"></emph.end> 7, <emph type="italics"></emph>aquæ ſcilicet magnitudinem habentis æqualem <expan abbr="argẽto">argento</expan> viuo C,<emph.end type="italics"></emph.end><pb xlink:href="029/01/031.jpg" pagenum="24"></pb><emph type="italics"></emph>fiat vt<emph.end type="italics"></emph.end> 7, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 95, <emph type="italics"></emph>ita<emph.end type="italics"></emph.end> 420, <emph type="italics"></emph>ad alium numerum, qui ſit<emph.end type="italics"></emph.end> 5700, <emph type="italics"></emph>is igi<lb></lb>tur indicabit quanta erit grauitas argenti viui, magnitudine æqua<lb></lb>lis propoſitæ aquæ A.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Inueniemus etiam aliter, &amp; expeditius grauitatem <lb></lb>aquæ, magnitudinem habentis æqualem propoſito ar<lb></lb>gento viuo A. </s></p><p type="main">

<s><emph type="italics"></emph>Accipiatur enim aliquod corpus aureum, cui ſuperinducatur ce<lb></lb>rea tunica tenuiſſima, ne fiat argento viuo leuius, neue ab <expan abbr="eodẽ">eodem</expan> diſſol<lb></lb>uatur, deinde aquæ, magnitudinem habentis æqualem ipſi corpori an <lb></lb>reo inueniatur grauitas, vt dictum est in propoſ.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>exemplo, quæ ſit<emph.end type="italics"></emph.end> 7, <lb></lb><emph type="italics"></emph>ſimiliter &amp; argenti viui, vt aquæ, magnitudinem habentis æqualem <lb></lb>eidem corpori aureo, inueniatur grauitas, quæ ſit<emph.end type="italics"></emph.end> 95, <emph type="italics"></emph>&amp; fiat vt<emph.end type="italics"></emph.end> 95, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end><lb></lb>7, <emph type="italics"></emph>ita<emph.end type="italics"></emph.end> 5700, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 420, <emph type="italics"></emph>grauitas igitur aquæ, magnitudinem habentis æ<lb></lb>qualem argento viuo A, erit<emph.end type="italics"></emph.end> 420. </s></p><p type="main">

<s><emph type="italics"></emph>Contra. </s>

<s>ſit propoſitum aliquod corpus aqueum, cuius grauitas<emph.end type="italics"></emph.end><lb></lb>420, <emph type="italics"></emph>&amp; oporteat inuenire, quanta erit grauitas argenti viui, magni<lb></lb>tudine æqualis propoſitæ aquæ A. </s>

<s>Superinducta corpori aureo cerea <lb></lb>tunica, vt ſupra, &amp; inuentis grauitatibus<emph.end type="italics"></emph.end> 7, &amp; 95, <emph type="italics"></emph>aquæ nimirum, &amp; <lb></lb>argenti viut, magnitudine æqualium prædicto aureo corpori, fiat vt<emph.end type="italics"></emph.end><lb></lb>7, <emph type="italics"></emph>ad 95, ita<emph.end type="italics"></emph.end> 430, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 5700 <emph type="italics"></emph>grauitas igitur argenti viui, magnitudine<lb></lb>æqualis propoſito corpori aqueo A, erit<emph.end type="italics"></emph.end> 5700. </s></p><p type="main">

<s>Qua ratione inuenienda ſit grauitas argenti viui, ma<lb></lb>gnitudinem habentis propoſito cuicunque corpori ſoli<lb></lb>do æqualem. </s></p><p type="main">

<s><emph type="italics"></emph>Sit propoſitum aliquod corpus ſolidum, vtpote plumbeum A, cuius <lb></lb>grauitas<emph.end type="italics"></emph.end> 161, <emph type="italics"></emph>&amp; oporteat inuenire quanta erit grauitas argenti viui <lb></lb>magnitudine æqualis propoſito plumbo A. inueniatur grauitas aquæ <lb></lb>magnitudinem habentis æqualem plumbo A, vt in exemplo propoſi<lb></lb>tionis<emph.end type="italics"></emph.end> 8, <emph type="italics"></emph>dictum eſt, quæ ſit<emph.end type="italics"></emph.end> 14, <emph type="italics"></emph>&amp; inuenta grauitate argenti viui, ma<lb></lb>gnitudine æqualis ipſi aquæ, ea erit de qua quæritur, ſit enim inuen<lb></lb>ta argenti viui grauitas<emph.end type="italics"></emph.end> 190. <emph type="italics"></emph><expan abbr="Quoniã">Quoniam</expan> igitur argentum viuum, cuius <lb></lb>grauitas est<emph.end type="italics"></emph.end> 190, <emph type="italics"></emph>magnitudine æquatur aquæ, cuius grauitas eſt<emph.end type="italics"></emph.end> 14, <lb></lb><emph type="italics"></emph>ipſique aquæ æquatur magnitudine plumbum A, erit argentum vi<lb></lb>uum, cuius grauitas 190, magnitudine propoſito plumbo A, æquale; <lb></lb>quare inuenta est grauitas argenti viui, magnitudine æqualis propo<lb></lb>ſito plumbo A, quod facere oportebat.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Quomodo inuenienda ſit grauitas cuiuſcunque cor-<pb xlink:href="029/01/032.jpg" pagenum="25"></pb>poris ſolidi, magnitudinem habentis propoſito corpori <lb></lb>ex argento viuo æqualem. </s></p><p type="main">

<s><emph type="italics"></emph>Sit propoſitum aliquod corpus ex argento viuo A, cuius grauitas <lb></lb>190, &amp; oporteat inuenire quanta erit grauitas plumbi, magnitudine <lb></lb>æqualis propoſito argento viuo A. inueniatur grauitas aquæ, magni<lb></lb>tudinem habentis æqualem argento viuo A, quæ ſit<emph.end type="italics"></emph.end> 14, <emph type="italics"></emph>deinde inuen<lb></lb>ta grauitate plumbi, magnitudine æqualis ipſi aquæ, vt in exemplo <lb></lb>propoſ. </s>

<s>9. dictum est, ea erit de qua quæritur. </s>

<s>ſit enim inuenta plum<lb></lb>bigrauitas<emph.end type="italics"></emph.end> 161, <emph type="italics"></emph>quoniam igitur aqua, cuius grauitas est<emph.end type="italics"></emph.end> 14, <emph type="italics"></emph>æqua<lb></lb>tur magnitudine plumbo, cuius grauitas est<emph.end type="italics"></emph.end> 161, <emph type="italics"></emph>&amp; æquatur quoque <lb></lb>argento viuo A, plumbum cuius grauitas eſt<emph.end type="italics"></emph.end> 161, <emph type="italics"></emph>æquabitur magni<lb></lb>tudine argento viuo A. quare inuenta, eſt grauitas plumbi, magnitu<lb></lb>dine æqualis propoſito argento viuo A, quod facere oportebat.<emph.end type="italics"></emph.end></s></p><p type="head">

<s>PROBLEMA VIII. PROPOS. XV.</s></p><p type="main">

<s>Propoſitis duobus liquidis corporibus æquè graui<lb></lb>bus, data magnitudine vnius, magnitudinem alte<lb></lb>rius inuenire. </s></p><figure id="id.029.01.032.1.jpg" xlink:href="029/01/032/1.jpg"></figure><p type="main">

<s>SINT propoſita duo cor<lb></lb>pora liquida æquè grauia A, <lb></lb>B, ſit autem vnius vt pote li<lb></lb>quidi A, data magnitudo G, <lb></lb>&amp; oporteat inuenire quanta <lb></lb>erit magnitudo liquidi B. </s>

<s>Ac<lb></lb>cipiatur aliquod ſolidum cor <lb></lb>pus C, &amp; liquidi quod ſit H, <lb></lb>eiuſdem generis cum liquido <lb></lb>A, magnitudinem habentis <lb></lb>æqualem ſolido C, *<arrow.to.target n="note39"></arrow.to.target> inuenia<lb></lb>tur grauitas quæ ſit D, ſimili. </s>

<s><lb></lb>ter &amp; liquidi, quod ſit I, eiuſdé <lb></lb>generis cum liquido B, magni <lb></lb>tudinern habentis æqualem eidem ſolido C, * inueniatur grauitas,<arrow.to.target n="note40"></arrow.to.target><lb></lb>quæ ſit E, &amp; fiat vt grauitas E, ad grauitatem D, ita magnitudo G, ad <lb></lb>ad aliam <expan abbr="magnitadinẽ">magnitudinem</expan>, quæ ſie F. </s>

<s>Quoniam igitur ſunt quatuor cor<lb></lb>pora grauia I, H, B, A, quorum primum, &amp; ſecundum ſunt magnitu<lb></lb>dine æqualia, tertium vero, &amp; quartum æque grauia, &amp; ſunt eiuſdem <lb></lb>generis primum videlicet, &amp; tertium, ſimiliter eiuſdem generis ſe<pb xlink:href="029/01/033.jpg" pagenum="26"></pb><arrow.to.target n="note41"></arrow.to.target>cundum &amp; quartum, * Erit vt grauitas E, ad grauitatem D, ita ma<lb></lb>gnitudo G, ad liquidi B, magnitudinem, ſed vt grauitas E, ad graui<lb></lb>tatem D, ita eſt magnitudo G, ad F, magnitudinem; ergo magnitudo <lb></lb>F, æqualis erit magnitudini liquidi B. inuenta igitur eſt corporis li<lb></lb>quidi B, magnitudo F, quod facere oportebat. </s></p><p type="margin">

<s><margin.target id="note39"></margin.target>8.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note40"></margin.target>8.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note41"></margin.target>7.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Quod ſi propoſita duo corpora æque grauia fuerint <lb></lb>regularia, vtpote ſphærica, fuerit autem ſphęræ A, data <lb></lb>diameter G, &amp; oporteat inuenire, quanta erit diameter <lb></lb>ſphæræ B, ita faciendum erit. </s></p><figure id="id.029.01.033.1.jpg" xlink:href="029/01/033/1.jpg"></figure><p type="main">

<s>ACCEPTO aliquo cor <lb></lb>pore ſolido C, &amp; inuentis <lb></lb>grauitatibus D, E, <expan abbr="liquidorũ">liquidorum</expan> <lb></lb>H, I, vt ſupra, fiat vt grauitas <lb></lb>E, ad grauitatem D, ita cu<lb></lb>bus ex G, ad alium cubum, <lb></lb>cuius latus ſit F. </s>

<s>Quoniam <lb></lb>igitur eadem ratione, qua <lb></lb>ſupra oſtendetur, vt grauitas <lb></lb>E, ad grauitatem D, ita eſſe <lb></lb>magnitudinem ſphæræ A, ad <lb></lb>ſphæræ B, magnitudinem, ſed <lb></lb>magnitudo ſphæræ A, ad <lb></lb><arrow.to.target n="note42"></arrow.to.target> ſphæræ B, magnitudinem; * <lb></lb><expan abbr="triplicatã">triplicatam</expan> rationem habet eius, quam G, diameter ſphæræ A, ad dia<lb></lb>metrum ſphæræ B, ſimiliter &amp; cubus ex G, ad cubum diametri ſphæ<lb></lb><arrow.to.target n="note43"></arrow.to.target>ræ B, * triplicatam rationem habet eius, quam G, ad ſphæræ B, dia<lb></lb>metrum; ergo vt grauitas E, ad grauitatem D, ita erit cubus ex G, ad <lb></lb>cubum diametri ſphæræ B, ſed vt grauitas D, ita grauitatem D, ita <lb></lb>eſt cubus ex G, ad cubum ex F; ergo cubus ex F, æqualis erit cubo <lb></lb>diametri ſphæræ B; quare &amp; latus F, æquabitur diametro ipſius ſphæ<lb></lb>ræ B. inuenta igitur eſt quantitas diametri ſphæræ B, quod facere <lb></lb>oportebat. </s></p><p type="margin">

<s><margin.target id="note42"></margin.target>18.12. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note43"></margin.target>33.11. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="head">

<s>Exemplum.</s></p><p type="main">

<s>Qvidam proponit aliquod corpus liquidum notæ <lb></lb>magnitudinis, &amp; vult inuenire, quanta erit ma<lb></lb>gnitudo liquidi alterius generis, grauitatem </s></p><pb xlink:href="029/01/034.jpg" pagenum="27"></pb><p type="head">

<s>habentis propoſito corpori liquido æqualem.</s></p><p type="main">

<s><emph type="italics"></emph>Sit propoſitum aliquod olei corpus A, cuius magnitudo<emph.end type="italics"></emph.end> 600. <emph type="italics"></emph>&amp; <lb></lb>oporteat inuenire quanta erit magnitudo aquæ, grauitatem habentis <lb></lb>æqualem propoſito oleo A, accipiatur aliquod ſolidum corpus C, vt <lb></lb>pote plumbeum, &amp; aquæ magnitudinem habentis æqualem plumbo <lb></lb>C, inueniatur grauitas, vt in exemplo prop.<emph.end type="italics"></emph.end> 8, <emph type="italics"></emph>dictum eſt, quæ ſit<emph.end type="italics"></emph.end> 12. <lb></lb><emph type="italics"></emph>ſimiliter &amp; olei æqualem habentis magnitudinem eidem plumbo C, <lb></lb>inueniatur grauitas quæ ſit 11, &amp; fiat vt 12, ad<emph.end type="italics"></emph.end> 11, <emph type="italics"></emph>ita<emph.end type="italics"></emph.end> 600, <emph type="italics"></emph>ad alium <lb></lb><expan abbr="numerũ">numerum</expan> qui ſit<emph.end type="italics"></emph.end> 550. <emph type="italics"></emph>is igitur numerus indicabit quanta erit magni<lb></lb>tudo aquæ grauitatem habentis æqualem propoſito oleo A.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Similiter ſi <expan abbr="propoſitũ">propoſitum</expan> ſit aliquod corpus aqueum A, cuius magnitu<lb></lb>do<emph.end type="italics"></emph.end> 5700, <emph type="italics"></emph>&amp; oporteat inuenire, quanta erit magnitudo argenti viui, <lb></lb>grauitatem habentis æqualem propoſitæ aquæ A. </s>

<s>Accipiatur aliquod <lb></lb>corpus ſolidum C ſi aureum, ſuper inducatur ei cerea tunica propter <lb></lb>iam <expan abbr="dictã">dictam</expan> rationem, deinde argenti viui, magnitudine æqualis ipſi C, <lb></lb>inueniatur grauitas quæ ſit<emph.end type="italics"></emph.end> 95, <emph type="italics"></emph>ſimiliter &amp; aquæ <expan abbr="magnitudinẽ">magnitudinem</expan> ha<lb></lb>bentis æqualem eidem C, inueniatur grauitas quæ ſit<emph.end type="italics"></emph.end> 7, <emph type="italics"></emph>&amp; fiat vt<emph.end type="italics"></emph.end> 95, <lb></lb><emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 7, <emph type="italics"></emph>ita<emph.end type="italics"></emph.end> 5700, <emph type="italics"></emph>ad alium numerum, qui ſit<emph.end type="italics"></emph.end> 420, <emph type="italics"></emph>is igitur numerus in<lb></lb>dicabit quanta erit magnitudo argenti viui grauitatem habentis <lb></lb>æqualem propoſitæ aquæ A.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Quod ſi propoſitum corpus aqueum A, ſit ſphæricum, culus <lb></lb>ſphæræ diameter ſit 10, &amp; oporteat inuenire quanta erit dia<lb></lb>meter ſphæræ ex argento viuo, grauitatem habentis æqualem <lb></lb>propoſitæ ſphæræ A, ità faciendum erit. </s></p><p type="main">

<s><emph type="italics"></emph>Accepto vt diximus aliquo corpore ſolido C, &amp; inuentis grauita<lb></lb>tibus <expan abbr="liquidorũ">liquidorum</expan> aquæ ſcilicet &amp; argent viui magnitudinem æqua<lb></lb>lem habentium corpori C, quæ ſint<emph.end type="italics"></emph.end> 14, <emph type="italics"></emph>grauitas aquæ, &amp;<emph.end type="italics"></emph.end> 190, <emph type="italics"></emph>graui<lb></lb>tas argenti viui, fiat vt<emph.end type="italics"></emph.end> 190, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 14, <emph type="italics"></emph>ita cubus ex 10, hoc est ita 1000, <lb></lb>ad alium numerum, qui ſit<emph.end type="italics"></emph.end> 73 11/19, <emph type="italics"></emph>is igitur numerus erit cubus dia<lb></lb>metri ſphæræ ex argento viuo, grauitatem habentis æqualem propo<lb></lb>ſitæ ex aqua ſphæræ A: quare latus cubicum numeri<emph.end type="italics"></emph.end> 73 11/19, <emph type="italics"></emph>quod eſt<emph.end type="italics"></emph.end><lb></lb>4 19/100 <emph type="italics"></emph>proxime indicabit ipſam diametrum.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Similiter ſi propoſitum corpus aqueum A, fuerit cubicum, aut <lb></lb>alicuius alterius formæ regularis, eadem ratione, qua ſupra inuenit<lb></lb>tur latus cubi ex argento viuo, grauitate æqualis propoſito ex aqua <lb></lb>cubo A, nam ſi ipſius cabi A, <expan abbr="datũ">datum</expan> ſit latus<emph.end type="italics"></emph.end> 10, <emph type="italics"></emph>erit numerus<emph.end type="italics"></emph.end> 73 11/19, <lb></lb><emph type="italics"></emph>cubus ex argento viuo æqualis grauitate propoſito ex aqua cubo A; <lb></lb>quare latus cubicum numeri<emph.end type="italics"></emph.end> 73 11/19, <emph type="italics"></emph>quod est<emph.end type="italics"></emph.end> 4 19/100 <emph type="italics"></emph>proxime in<lb></lb>dicabit quæſitum latus cubi, ex argento viuo.<emph.end type="italics"></emph.end></s></p><pb xlink:href="029/01/035.jpg" pagenum="28"></pb><p type="main">

<s>Neque diſſimili ratione inuenietur magnitudo reliquo<lb></lb>rum omnium liquidorum, grauitate propoſito corpori cuiuſ<lb></lb>cumque generis liquidi, æqualium, quare dicta ſufficiant. </s></p><p type="main">

<s>Dvm adhuc Opuſculum ſub prælo eſſet, dubitandi anſam, <lb></lb>ex eo vir doctiſsimus, cui percurrendum illud tradi<lb></lb>deram, arripuit, quod ex grauitate, corporum in aqua <lb></lb>exiſtentium, non poſſet vera ratio, quam habent diuerſa <lb></lb>ipſorum corporum genera in grauitate, deprehendi, niſi <lb></lb>corpora fuerint ſimilia. </s>

<s>ſi enim (aiebat) accipiantur duo <lb></lb>corpora eiuſdem generis, &amp; grauitatis, quorum vnum ſit <lb></lb>planum, alterum conicam formam habens, &amp; ponderen<lb></lb>tur in aqua, ita vt coni vertex deorſum verſus pondeat, baſis <lb></lb>vero ipſius coni, &amp; latæ corporis plani ſuperficies æquidiſtent <lb></lb>horizonti conus in aqua maiorem habebit grauitatem, cor<lb></lb>pore plano, quia corpus planum magis ab aqua ſuſtentatur, <lb></lb>quam conus, &amp; hoc quidem manifeſtum eſt, quoniam ſi am<lb></lb>bo demittantur eodem tempore in aquam, conus citius ad <lb></lb>imum deſcendet, quam corpus planum. </s>

<s>Hoc argumentum <lb></lb>licet primo aſpectu probabile videatur, tamen falſo concludit. </s>

<s><lb></lb>verum eſt quod aqua ſuſtentat magis corpus planum, quam <lb></lb>conum, ipſum tamen ſuſtentat, ne tanta velocitate feratur <lb></lb>deorſum, non ideo ipſius grauitati aliquid detrahit, neque <lb></lb>enim ex velociori motu ſimpliciter inferri poteſt maior gra<lb></lb>uitas, illud enim valeret etiam in aere, quod eſt falſum, ſed <lb></lb>ne huiuſmodi dubitatio veritatis ſpecie aliquem decipiat, ſe<lb></lb>quenti Theoremate eam deſtruere agrediar. </s></p><p type="head">

<s>THEOREMA VIII. PROPOS. XVI.</s></p><p type="main">

<s>Corpora eiuſdem generis, &amp; grauitatis grauiora <lb></lb>quam aqua, etſi diſsimilia, æqualem in aqua <lb></lb>grauitatem habent. </s></p><p type="main">

<s>SINT duo eiuſdem generis, &amp; grauitatis corpora A, B, grauiora <lb></lb>quam aqua, &amp; ſint diſſimilia, dico ipſa corpora æqualem in aqua <lb></lb>grauitatem habere. </s>

<s>ſit enim ſi fieri poteſt corpus A, leuius corpore B, <pb xlink:href="029/01/036.jpg" pagenum="29"></pb>&amp; accipiatur aliquod corpus L, leuius quam aqua, ita vt cum ipſi <lb></lb>corpori L, appendatur, corpus B, &amp; ambo ſimul demittantur in <expan abbr="aquã">aquam</expan>, <lb></lb>ſint æque grauia atque aqua, neque ſurſum, neque deorſum ſeran<lb></lb>tur, ſimiliter accipiatur alterum corpus M, eiuſdem generis cum <lb></lb>corpore L, ipſique ſimile, &amp; <lb></lb><figure id="id.029.01.036.1.jpg" xlink:href="029/01/036/1.jpg"></figure><lb></lb>æquale, &amp; corpori M, appen<lb></lb>datur corpus A. </s>

<s>Deinde in<lb></lb>teligatur aqua conſiſtens, &amp; <lb></lb>manens, eiuſque ſuperfi<lb></lb>cies ſphærica C D E, cuius <lb></lb>ſphæræ centrum K, aquæ <lb></lb>enim conſiſtentis, atque <lb></lb>manentis ſuperficies ſphæ<lb></lb>rica eſt, cuius ſphæræ cen<lb></lb>trum idem eſt, quod <expan abbr="centrũ">centrum</expan> <lb></lb>terræ, hoc autem demonſtratum eſt ab Archimede Prop. 2. lib.

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

<s>Inteligantur etiam duæ pyramides con<lb></lb>iunctæ, &amp; continuatæ, æquales, &amp; ſimiles KCD, KDE, pro baſibus <lb></lb>habentes in ſuperficie aquæ parallelogramma, vertices autem pun<lb></lb>ctum K, &amp; corpora L, B, comprehendantur à pyramide KDE, corpo<lb></lb>ra vero M, A, à pyramide KCD, &amp; ſub corporibus L, B, deſcribatur <lb></lb>quædam alterius ſphæræ ſuperficies FGH, in aqua, circa centrum K, <lb></lb>poterit autem huiuſmodi ſuperficies ſub corporibus L, B, deſcribi, <lb></lb>quoniam &amp; ſi ipſi corpora demerguntur tota, non ideo feruntur <lb></lb>deorſum, ponuntur enim æque grauia atque aqua. </s>

<s>Quoniam <lb></lb>igitur eiuſdem generis ponuntur corpora M, L, &amp; æqualia, &amp; ſimilia, <lb></lb>erunt æque grauia, cum in aqua, tum in aere, &amp; quoniam corpus <lb></lb>A, leuius eſt in aqua, corpore B, erunt corpora M, A, ſimul, in aqua <lb></lb>leuiora corporibus L, B, ſed corpora L, B, ſimul, æque grauia ſunt at<lb></lb>que aqua, ergo corpora M, A, ſimul, leuiora erunt quam aqua; quare <lb></lb>corpus M, non demergetur totum, ſed aliqua pars ipſius ex aquæ ſu<lb></lb>perſicie extabit. </s></p><p type="main">

<s>Et quoniam eiuſdem generis, &amp; grauitatis ponuntur corpora A, B, <lb></lb>erunt magnitudine æqualia, &amp; per additionem æqualium æquali<lb></lb>bus, corpora M, A, erunt æqualia corporibus L, B, </s></p><p type="main">

<s>Quoniam igitur corpora M, A, æqualia ſunt corporibus L, B, pars <lb></lb>autem corporis M, extat ex aquæ ſuperficie, &amp; corpora L, B, tota de<lb></lb>merguntur, minus loci ocupabunt in aqua corpora M, A, quam cor<lb></lb>pora L, B, quare maior erit grauitas corporum M, A, &amp; aquæ conti<lb></lb>nentis ipſa corpora, quæ eſt in loco pyramidis CDGF, quam corpo<lb></lb>rum L, B, &amp; aquæ ipſa corpora <expan abbr="continẽtis">continentis</expan> in loco pyramidis DEHG, <pb xlink:href="029/01/037.jpg" pagenum="30"></pb>magis igitur aquæ pars premetur, quæ eſt ſub ſuperficie FG, quam <lb></lb>ca quæ eſt ſub ſuperficie GH; quare expellet partem minus preſſam, <lb></lb>(æqualiter enim &amp; continuatæ iacent inter ſeſe) &amp; <expan abbr="nõ">non</expan> manebit aqua, <lb></lb>quod eſt abſurdum, ponebatur enim manens. </s>

<s>non igitur corpus A, <lb></lb>leuius eſt in aqua corpore B. eadem ratione oſtendetur neque corpus <lb></lb>B, leuius eſſe in aqua corpore A, quare conſtat propoſitum. </s></p><p type="head">

<s>ALITER.</s></p><p type="main">

<s>Sint duo eiuſdem generis, &amp; grauitatis corpora A, B, grauiora <lb></lb>quam aqua, &amp; ſint diſſimilia. </s>

<s>oſtendendum eſt ipſa corpora æqualem <lb></lb>in aqua grauitatem habere, ſit enim corporis A, vel ipſius B, graui<lb></lb>tas CD, aquæ vero magni<lb></lb><figure id="id.029.01.037.1.jpg" xlink:href="029/01/037/1.jpg"></figure><lb></lb>tudinem habentis æqua<lb></lb>lem ipſi A, vel B, ſit graui<lb></lb>tas C, &amp; accipiatur ali<lb></lb>quod corpus L, leuius <expan abbr="quã">quam</expan> <lb></lb>aqua, cuius grauitas ſit <lb></lb>ipſi C, æqualis, aquæ ve<lb></lb>ro, magnitudinem haben<lb></lb>tis æqualem corpori L, ſit <lb></lb>grauitas æqualis ipſi CD, <lb></lb>itaque appenſo corpore <lb></lb>B, corpori L, corpus ex <lb></lb>vtriſque conſtans æque graue erit atque aqua, grauitas enim vtro<lb></lb>runque corporum B, L, eſt æqualis vtriſque grauitatibus CD, &amp; C, &amp; <lb></lb>grauitas aquæ, magnitudinem habentis æqualem <expan abbr="vtriſq;">vtriſque</expan> corporibus <lb></lb>L, B, æqualis eſt eiſdem grauitatibus CD, &amp; C, corpora igitur B, L, <lb></lb>demiſſa in aquam, neque ſurſum, neque deorſum ferentur, quia cor<lb></lb>pus B, grauius quam aqua fertur deorſum tanta vi, quanta à corpo<lb></lb>re <emph type="italics"></emph>L,<emph.end type="italics"></emph.end> ſurſum retrahitur. </s></p><p type="main">

<s>Rurſus accipiatur alterum corpus ſolidum M, eiuſdem generis <lb></lb>cum corpore L, ipſique ſimile, &amp; æquale, &amp; corpore A, appenſo ipſi <lb></lb>M, &amp; demiſſis ambobus in aquam, eadem ratione qua ſupra oſten<lb></lb>detur, corpora A, M, ſimul, eſſe æque grauia atque aqua, &amp; corpus <lb></lb>A, tanta vi deorſum ferri, quanta retrahitur ſurſum à corpore M, <lb></lb>ſed corpora M, L, æqualem vim habent retrahendi ſurſum, cum <lb></lb>ſint eiuſdem generis, &amp; æqualia, &amp; ſimilia, ergo æquali vi retra<lb></lb>hentur corpora A, B, ne deſcendant; quare conſtat ipſa corpora A, <lb></lb>B, æqualem in aqua grauitatem habere quod erat oſtendendum. </s></p><pb xlink:href="029/01/038.jpg" pagenum="31"></pb><p type="head">

<s>THEOREMA IX. PROPOS. XVII.</s></p><p type="main">

<s>Sphære eiuſdem generis inter ſe ſunt in grauitate, vt <lb></lb>diametrorum cubi in magnitudine. </s></p><p type="main">

<s>SINT ſphæræ eiuſdem gene<lb></lb><figure id="id.029.01.038.1.jpg" xlink:href="029/01/038/1.jpg"></figure><lb></lb>ris ABC, DEF, quarum diame<lb></lb>tri BC, EF. dico vt ſphęra ABC, <lb></lb>ſe habet in grauitate, ad ſphæram <lb></lb>DEF, ita ſe habere in maguitudi<lb></lb>ne cubum ex BC, ad cubum ex <lb></lb>EF, ſit enim ſphæræ ABC, graui<lb></lb>tas G, &amp; ſphæræ DEF, grauitas H, <lb></lb>quoniam igitur eiuſdem generis <lb></lb>ponuntur ſphæræ ABC, DEF, <lb></lb>erit^{*} vt ſphæra ABC, ad ſphæram DEF, ita grauitas G, ad H,<arrow.to.target n="note44"></arrow.to.target> graui<lb></lb>tatem, ſed ſphæra ABC, ad ſphæram DEF,^{*} triplicatam habet<arrow.to.target n="note45"></arrow.to.target> ra<lb></lb>tionem eius, quam diameter BC, ad EF, diametrum, ergo &amp; graui<lb></lb>tas G, ad grauitatem H, triplicatam habebit rationem eius, quam <lb></lb>habet BC, ad EF, ſed &amp; cubus ex BC, ad cubum ex EF,^{*} triplicatam<arrow.to.target n="note46"></arrow.to.target><lb></lb>rationem habet eius, quam BC, ad EF, ergo vt grauitas G, ad graui<lb></lb>tatem H, ita erit cubus ex BC, ad cubum ex EF. ſphæræ igitur eiuſ<lb></lb>dem generis inter ſe ſunt in grauitate, vt diametrorum cubi in ma<lb></lb>gnitudine, quod erat demonſtrandum. </s></p><pb xlink:href="029/01/039.jpg" pagenum="32"></pb><p type="margin">

<s><margin.target id="note44"></margin.target>2. &amp; 3. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note45"></margin.target>18. 12. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note46"></margin.target>33. 11. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="head">

<s>Ad comparandum inter ſe duodecim corporum genera <lb></lb>grauitate, &amp; magnitudine tabella.</s></p><p type="table">

<s>TABELLE WAR HIER</s></p><p type="main">

<s><emph type="italics"></emph>Quæro exempli gratia, quam habet rationem in grauitate plumbum ad aurum. </s>

<s>In<lb></lb>teligatur plumbum, quoniam leuius est auro, grauitatem habere 1, &amp; in linea plumbi, <lb></lb>in prima columna nominati ſub titulo auri, quæratur auri grauitas, ea erit<emph.end type="italics"></emph.end> 1 15/23. <emph type="italics"></emph>plum<lb></lb>bum igitur ad aurum rationem habebit in grauitate vt 1, ad 1 15/23, ſi enim ſumantur <lb></lb>duo corpora magnitudine æqualia, vnum plumbeum alterum aureum, ſit autem plum<lb></lb>bei corporis grauitas<emph.end type="italics"></emph.end> 1, <emph type="italics"></emph>aurei erit<emph.end type="italics"></emph.end> 1 15/23, <emph type="italics"></emph>quare corpus plumbeum ad corpus aureum <lb></lb>eiuſdem magnitudinis rationem habebit in grauitate vt 1, ad 1 15/23. comparantur au<lb></lb>tem inter ſe genera diuerſa grauitate, in corporibus magnitudine æqualibus.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Rurſus, quæro quam habet rationem in grauitate aqua ad argentum viuum. </s>

<s>inteli<lb></lb>gatur aqua, vt leuior argento viuo grauitatem habere 1, &amp; in linea aqua, ſubtitulo ar<lb></lb>genti viui, quæratur argenti viui grauitas, ea erit<emph.end type="italics"></emph.end> 13 4/7, <emph type="italics"></emph>aqua igitur ad argentum <expan abbr="viuũ">viuum</expan> <lb></lb>rationem habebit in grauitate vt 1, ad 13 4/7.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Contra, quæro quomodo ſe habent in magnitudine aurum, &amp; plumbum. </s>

<s>inteligatur <lb></lb>aurum, quoniam grauius eſt plumbo, magnitudinem habere 1, &amp; in linea plumbi, ſub ti<lb></lb>tulo auri, quæratur plumbi magnitudo, ea erit 1 15/23, aurum igitur ad plumbum ſe ba-<emph.end type="italics"></emph.end><pb xlink:href="029/01/040.jpg" pagenum="33"></pb><emph type="italics"></emph>bebit in magnitudine vt 1, ad 1 19/21, ſi enim ſumantur duo corpora æque grauia, vnum <lb></lb>aureum, alterum plumbeum, ſit autem corporis aurei magnitudo 1, plumbei erit 1 1<gap></gap>/21, <lb></lb>quare corpus aureum ad corpus plumbeum eiuſdem grauitatis ſe habebit in magnitudi<lb></lb>ne vt 1, ad 1 19/21, comparantur autem inter ſe genera diuerſa magnitudine, in corpori<lb></lb>bus æque grauibus.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Quæro denique quomodo ſe habent in magnitudine ferrum, &amp; aqua, ponatur ferrum, <lb></lb>vt grauius aqua, magnitudinem habere 1, &amp; in linea aquæ, ſub titulo ferri, quæratur <lb></lb>aquæ magnitudo, ea erit<emph.end type="italics"></emph.end> 8, <emph type="italics"></emph>ferrum igitur ad aquam ſe habebit in magnitudine vt 1, ad<emph.end type="italics"></emph.end> 8. </s></p><p type="head">

<s>Altera, ad comparandum inter ſe duodecim corporum genera, <lb></lb>grauitate, &amp; magnitudine, tabella.</s></p><p type="table">

<s>TABELLE WAR HIER</s></p><p type="main">

<s><emph type="italics"></emph>Quæro exempli gratia, quæ nam ſit ratio in grauitate, auri ad argentum. </s>

<s>intelliga<lb></lb>tur aurum, quoniam grauius est argento, grauitatem habere 100, &amp; in line a auri, ſub <lb></lb>titulo argenti, reperietur argenti grauitas<emph.end type="italics"></emph.end> 54 22/57, <emph type="italics"></emph>aurum igitur ad argentum rationem <lb></lb>habebit in grauitate vt<emph.end type="italics"></emph.end> 100, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 54 22/57, <emph type="italics"></emph>ſi enim ſumantur duo corpora, magnitudine <lb></lb>aqualia, vnum aureum, alterum argenteum, ſit autem aurei corporis grauitas<emph.end type="italics"></emph.end> 100, <emph type="italics"></emph>erit<emph.end type="italics"></emph.end><pb xlink:href="029/01/041.jpg" pagenum="34"></pb><emph type="italics"></emph>argentei<emph.end type="italics"></emph.end> 54 22/57, <emph type="italics"></emph>quare corpus aureum ad corpus argenteum eiuſdem <lb></lb>magnitudinis, rationem habebit in grauitate, vt<emph.end type="italics"></emph.end> 100, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 54 22/57. </s></p><p type="main">

<s><emph type="italics"></emph>Quæro, quomodo ſe habet in grauitate aqua ad vinum. </s>

<s>quoniam <lb></lb>æqua grauior est vino, intelligatur eius grauitas<emph.end type="italics"></emph.end> 100, <emph type="italics"></emph>&amp; quoniam in <lb></lb>linea aquæ, ſub titulo vint, datur vini grauitas<emph.end type="italics"></emph.end> 98 1/3, <emph type="italics"></emph>aqua ad vinum <lb></lb>ſe habebit in grauitate, vt<emph.end type="italics"></emph.end> 100, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 98 1/3. </s></p><p type="main">

<s><emph type="italics"></emph>Contra quæro quomodo ſe habent in magnitudine argentum, &amp; <lb></lb>aurum. </s>

<s>intelligatur argentum, vt leuius auro, magnitudinem balere<emph.end type="italics"></emph.end><lb></lb>100, <emph type="italics"></emph>&amp; in linea auri ſub titulo argenti, quæratur auri magnitudo, ea <lb></lb>erit<emph.end type="italics"></emph.end> 54 22/39, <emph type="italics"></emph>argentum igitur ad aurum ſe habebit in magnitudine, vt<emph.end type="italics"></emph.end><lb></lb>100, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 54 22/39, <emph type="italics"></emph>ſi enim ſumantur duo corpora æque grauia, vnum <lb></lb>argenteum, alterum aureum, ſit autem argentei corporis magnitudo<emph.end type="italics"></emph.end><lb></lb>100, <emph type="italics"></emph>erit aurei<emph.end type="italics"></emph.end> 54 22/39, <emph type="italics"></emph>quare corpus argenteum, ad corpus aureum <lb></lb>eiuſdem grauitatis, ſe habebit in magnitudine, vt 100, ad<emph.end type="italics"></emph.end> 54 22/39. </s></p><p type="main">

<s><emph type="italics"></emph>Quæro denique quomodo ſe habent in magnitudine aqua &amp; ar<lb></lb>gentum viuum. </s>

<s>quoniam aqua leuiora est argento viuo, intelligatur <lb></lb>eius magnitudo<emph.end type="italics"></emph.end> 100, <emph type="italics"></emph>&amp; in linea argenti viui, ſub titulo aquæ, quæ<lb></lb>ratur argenti viui magnitudo, &amp; reperietur<emph.end type="italics"></emph.end> 7 7/29, <emph type="italics"></emph>aqua igitur ad <lb></lb>argentum viuum ſe habebit in magnitudine, vt 100, ad<emph.end type="italics"></emph.end> 7 7/29. </s></p><p type="main">

<s>Hic ſequitur tabula, ad inueniendas ſphærarum grauita<lb></lb>tes, ex data diametrorum magnitudine, cuius hæc eſt <lb></lb>explicatio. </s></p><p type="main">

<s><emph type="italics"></emph>In dimetiendis ſphærarum diametris vtimur pede Romano anti<lb></lb>quo, cuius menſuram in margine appoſuimus, eaque reſpondet ad Ro<lb></lb>mani palmi, quo hodie vtimur, menſuram vt 4, ad 3, huiuſmodipe<lb></lb>dem diuidimus in duodecim partes æquales, ſeu vncias, quas inuenies <lb></lb>in prima Columna ſub titulo magnitudinis.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Ponderibus autem vtimur hac nostra tempeſtate vſitatis, libram <lb></lb>enim diuidimus in<emph.end type="italics"></emph.end> 12, <emph type="italics"></emph>vncias vnciam vero in<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>ſcrupula, &amp; ſcru<lb></lb>pulum in<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>grana. </s>

<s>Ad inueniendas igitur ſphærarum grauitates ex <lb></lb>data diametrorum magnitudine, hæc erit ratio.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Quæris grauitatem ſphæræ plumbeæ, diametrum habentis<emph.end type="italics"></emph.end> 3, <emph type="italics"></emph>vn<lb></lb>ciarum, inſpice tabulam, &amp; in linea trium vnciarum, ſub titulo gra<lb></lb>uitatis plumbeæ ſphæræ, deprehendes ipſam ſphæram grauitatem ba<lb></lb>bere lib.<emph.end type="italics"></emph.end> 7, <emph type="italics"></emph>vnc.<emph.end type="italics"></emph.end> 4, <emph type="italics"></emph>ſeru.<emph.end type="italics"></emph.end> 13, <emph type="italics"></emph>gran.<emph.end type="italics"></emph.end> 22 29/57. </s></p><p type="main">

<s><emph type="italics"></emph>Rurſus, quæris grauitatem ſphæræ aureæ, diametrum habentis<emph.end type="italics"></emph.end> 6, <lb></lb><emph type="italics"></emph>vnciarum. </s>

<s>in linea<emph.end type="italics"></emph.end> 6, <emph type="italics"></emph>vnciarum, ſub titulo grauitatis aureæ ſphæræ<emph.end type="italics"></emph.end><pb xlink:href="029/01/042.jpg" pagenum="35"></pb><emph type="italics"></emph>datur quæſita grauitas lib.<emph.end type="italics"></emph.end> 97, <emph type="italics"></emph>vnc.<emph.end type="italics"></emph.end> 6, <emph type="italics"></emph>ſcrup. </s>

<s>19, gran.<emph.end type="italics"></emph.end> 11 1/17. </s></p><p type="main">

<s><emph type="italics"></emph>Quæris denique grauitatem ſphæræ stanneæ, diametrum habentis <lb></lb>vnius pedis. </s>

<s>in linea vnius pedis, ſeu 12, vnciarum, ſub titulo graui<lb></lb>tatis ſphæræ ſtanneæ, datur quæſita ſphæræ grauitas lib.<emph.end type="italics"></emph.end> 304, <emph type="italics"></emph>adun<lb></lb>quem. </s>

<s>Atque ita reliquarum ſphærarum in tabula nominatarum, ex <lb></lb>data diametrorum magnitudine, grauitates inuenies.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Qua ratione hanc Tabulam compoſuimus. </s></p><p type="main">

<s><emph type="italics"></emph>Primum inueniendam curauimus grauitatem alicuius ſphæræ, da<lb></lb>tem habentis diametrum, &amp; ad hoc faciendum, oportebat aliquam <lb></lb>ſpbæram efficere, ſed quoniam ad illam efficiendam, exactam humana <lb></lb>diligentia non ſufficit, fieri curauimus Cylindrum ex ſtanno, altitu<lb></lb>dine aqualem diametro circuli, qui baſis eſt ipſius Cylindri, is enim <lb></lb>torno fieri poteſt multo exactior quam ſphæra, &amp; facilius. </s>

<s>huius au<lb></lb>tem Cylindri altitudo, vel diameter ipſius baſis, erat duarum vncia<lb></lb>rum prædicti pedis Romani, grauitas vero duarum librarum, cum <lb></lb>vna vncia, &amp; octo ſcrupulis, ſiue vt hoc pondus ad grana reducamus, <lb></lb>Cylindri grauitas erat Gran. 14592. abstulissent ab hac Cylindri <lb></lb>grauitate partem tertiam. </s>


<s>id est<emph.end type="italics"></emph.end> 4864, <emph type="italics"></emph>reliquum, quod est<emph.end type="italics"></emph.end> 9728. <emph type="italics"></emph>ſer<lb></lb>uauimus, pro grauitate ſphæræ, diametrum habentis æqualem altitu<lb></lb>dini Cylindri, oſtenſum enim est ab Archimede propoſ.<emph.end type="italics"></emph.end> 32, <emph type="italics"></emph>lib.<emph.end type="italics"></emph.end> 1, <emph type="italics"></emph>de <lb></lb>ſphæra, &amp; Cylindro, Cylindrum, qui baſim habeat maximo in ſphæra <lb></lb>circulo æqualem, &amp; altitudinem æqualem diametro ſphæræ, ad ipſam <lb></lb>ſpbæram ſeſquialterum eſſe; itaque grauitatem ſphæræ, diametrum <lb></lb>habentis duarum vnciarum inuenimus eſſe gran.<emph.end type="italics"></emph.end> 9728. </s></p><p type="main">

<s><emph type="italics"></emph>Inuenta igitur grauitate ſphæræ, cuius diameter est duarum vn<lb></lb>ciarum, facile inuenientur reliquarum <expan abbr="ſpbærarũ">ſphærarum</expan> grauitates, ſi enim <lb></lb>inuenienda ſit grauitas ſphæræ stannea habentis diametrum<emph.end type="italics"></emph.end> 1/4. <emph type="italics"></emph>vn<lb></lb>ciæ, fiat vt cubus ex<emph.end type="italics"></emph.end> 2, <emph type="italics"></emph>ad cubum ex<emph.end type="italics"></emph.end> 1/4, <emph type="italics"></emph>hoc est vt<emph.end type="italics"></emph.end> 512, <emph type="italics"></emph>ad 1, ita<emph.end type="italics"></emph.end> 9728, <lb></lb><emph type="italics"></emph>ad alium numerum, qui ſit<emph.end type="italics"></emph.end> 19, <emph type="italics"></emph>ſphæræ igitur cuius diameter eſt<emph.end type="italics"></emph.end> 1/4, <lb></lb><emph type="italics"></emph>vnciæ, grauitas erit gran. 19, ostenſum enim est prop.

17, huius, ſphæ<lb></lb>ras eiuſdem generis inter ſe eſſe in grauitate, vt diametrorum cubi in <lb></lb>magritudine.<emph.end type="italics"></emph.end></s>

</p><p type="main">

<s><emph type="italics"></emph>Rurſus ſit inuenienda grauitas ſphæræ stannæ habentis diame<lb></lb>trum<emph.end type="italics"></emph.end> 1/2, <emph type="italics"></emph>vnciæ, fiat vt cubus ex<emph.end type="italics"></emph.end> 1/4, <emph type="italics"></emph>ad cubum ex<emph.end type="italics"></emph.end> 1/2, <emph type="italics"></emph>hoc est vt<emph.end type="italics"></emph.end> 1, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 8, <lb></lb><emph type="italics"></emph>ita<emph.end type="italics"></emph.end> 19, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 152, <emph type="italics"></emph>ſphæra igitur, cuius diameter eſt<emph.end type="italics"></emph.end> 1/2, <emph type="italics"></emph>vnciæ, habebit gra<lb></lb>uitatem gran.<emph.end type="italics"></emph.end> 152. </s></p><p type="main">

<s><emph type="italics"></emph>Sit denique inuenienda grauitas ſphæræ stannæ, diametrum ha<lb></lb>bentis<emph.end type="italics"></emph.end> 1/4, <emph type="italics"></emph>vnciæ, fiat vt cubus ex<emph.end type="italics"></emph.end> 1/4, <emph type="italics"></emph>ad cubum ex<emph.end type="italics"></emph.end> 1/4, <emph type="italics"></emph>hoc eſt vt<emph.end type="italics"></emph.end> 1, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 27, <lb></lb><emph type="italics"></emph>ita<emph.end type="italics"></emph.end> 19, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 513, <emph type="italics"></emph>grauitas igitur ſphæræ habentis diametrum<emph.end type="italics"></emph.end> 1/4, <emph type="italics"></emph>vnciæ,<emph.end type="italics"></emph.end></s></p><pb xlink:href="029/01/043.jpg" pagenum="36"></pb><p type="main">

<s>Ad inueniendas ſphæra<lb></lb>diametrorum <lb></lb>T A B </s></p><p type="table">

<s>TABELLE WAR HIER</s></p><pb xlink:href="029/01/044.jpg" pagenum="37"></pb><p type="main">

<s>rum grauitates ex data <lb></lb>magnitudine <lb></lb>V L A. </s></p><p type="table">

<s>TABELLE WAR HIER</s></p><pb xlink:href="029/01/045.jpg"></pb><p type="table">

<s>TABELLE WAR HIER</s></p><pb xlink:href="029/01/046.jpg"></pb><p type="table">

<s>TABELLE WAR HIER</s></p><pb xlink:href="029/01/047.jpg" pagenum="40"></pb><p type="table">

<s>TABELLE WAR HIER<lb></lb><emph type="italics"></emph>erit gran. 513. &amp; ſic reliquarum ſphærarum ex ſtanno, diametros ha<lb></lb>bentium magnitudine quacunque, inuenientur grauitates.<emph.end type="italics"></emph.end></s>

</p><p type="main">

<s><emph type="italics"></emph>Aliter quoque &amp; expeditius reliquarum ſphærarum ex ſtanno in<lb></lb>uenientur grauitates.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Inuenta grauitate ſphæræ, diametrum habentis<emph.end type="italics"></emph.end> 1/4, <emph type="italics"></emph>vnciæ, mul<lb></lb>tiplicetur ipſa grauitas, per<emph.end type="italics"></emph.end> 8, <emph type="italics"></emph>hoc est per cubum ex<emph.end type="italics"></emph.end> 2, <emph type="italics"></emph>numerus pro <lb></lb>ductus dabit grauitatem ſphæræ, diametrum habentis<emph.end type="italics"></emph.end> 2/4, <emph type="italics"></emph>vnciæ, hoc<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note47"></arrow.to.target> <emph type="italics"></emph>eſt<emph.end type="italics"></emph.end> 1/2, <emph type="italics"></emph>ſphæræ {*} enim inter ſe in triplicata ſunt ratione ſuarum di<gap></gap><lb></lb>metrorum. </s>

<s>deinde ſi multiplicetur eadem grauitas per<emph.end type="italics"></emph.end> 27, <emph type="italics"></emph>hoc est per <lb></lb>cubum ex<emph.end type="italics"></emph.end> 3, <emph type="italics"></emph>numerus productus dabit grauitatem ſphæræ, habentis <lb></lb>diametrum<emph.end type="italics"></emph.end> 3/4, <emph type="italics"></emph>vnciæ, &amp; ſi multiplicetur per<emph.end type="italics"></emph.end> 64, <emph type="italics"></emph>hoc est per cubum <lb></lb>ex<emph.end type="italics"></emph.end> 4, <emph type="italics"></emph>numerus productus dabit grauitatem ſphæræ, cuius diameter eſt <emph.end type="italics"></emph.end><lb></lb>4/4, <emph type="italics"></emph>hoc est vnius vnciæ, &amp; eo deinceps continuo ordine,<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note47"></margin.target>18.12. <lb></lb><emph type="italics"></emph>Elem.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Porro ad inueniendas grauitates ſphærarum ex reliquis metallis <lb></lb>vel ex quacunque alia materia, hæc erit ratio.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Fiat vt<emph.end type="italics"></emph.end> 1, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 1 41/74, <emph type="italics"></emph>hoc eſt vt<emph.end type="italics"></emph.end> 74, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 115, (<emph type="italics"></emph>ſi de grauitate ſphæræ<lb></lb><expan abbr="plūbeæ">plumbeæ</expan> quæritur cuius diameter est<emph.end type="italics"></emph.end> 1/4, <emph type="italics"></emph>vnciæ) ita 19. grauitas vide<lb></lb>licet ſphæræ stanneæ <expan abbr="diametrū">diametrum</expan> habentis<emph.end type="italics"></emph.end> 1/4, <emph type="italics"></emph>vnciæ, ad <expan abbr="aliū">alium</expan> numerum<lb></lb>qui ſit<emph.end type="italics"></emph.end> 29 19/74, <emph type="italics"></emph>grauitas igitur ſphæræ plumbeæ, diametrum haben<lb></lb>tis<emph.end type="italics"></emph.end> 1/4, <emph type="italics"></emph>vnciæ, erit gran.<emph.end type="italics"></emph.end> 29 39/74. <emph type="italics"></emph>ſtannum enim ad plumbum rationem <lb></lb>habet in grauitate vt<emph.end type="italics"></emph.end> 1, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 1 41/74, <emph type="italics"></emph>vt conſpicitur in prima tabelle <lb></lb>quam ad comparandum inter ſe duodecim corporum genera, grauita<lb></lb>te, &amp; magnitudine, appoſuimus.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Si vero quæratur de grauitate ſphæræ plumbeæ, diametrum <lb></lb>habentis<emph.end type="italics"></emph.end> 2, <emph type="italics"></emph>vnciarum, fiat vt<emph.end type="italics"></emph.end> 74, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 115, <emph type="italics"></emph>ita<emph.end type="italics"></emph.end> 9728, <emph type="italics"></emph>id est grauia<emph.end type="italics"></emph.end></s></p><pb xlink:href="029/01/048.jpg" pagenum="41"></pb><p type="table">

<s>TABELLE WAR HIER<lb></lb><emph type="italics"></emph>ſphæræ stanneæ, cuius diameter est<emph.end type="italics"></emph.end> 2, <emph type="italics"></emph>vnciarum, ad alium numerum, <lb></lb>qui ſit<emph.end type="italics"></emph.end> 15 117 <gap></gap>, <emph type="italics"></emph>ſphæra igitur plumbea, cuius diameter est<emph.end type="italics"></emph.end> 2, <emph type="italics"></emph>vn<lb></lb>ciarum grauitatem habebit gran. 15 117 <gap></gap>, atque hæc erit obſer<lb></lb>uanda in reliquis ratio.<emph.end type="italics"></emph.end></s>

</p><p type="main">

<s><emph type="italics"></emph>Vel ſi ipſa grauitas<emph.end type="italics"></emph.end> 29 19/74, <emph type="italics"></emph>multiplicetur per ſingulos cubos, vt <lb></lb>dictum est de ſphera ſtannea, numeri producti dabunt grauitates <lb></lb>ſphærarum ex plumbo, ad quarum diametros latera cubica rationem <lb></lb>habebunt vt<emph.end type="italics"></emph.end> 4, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 1, <emph type="italics"></emph>quoniam<emph.end type="italics"></emph.end> 29 19/74, <emph type="italics"></emph>est grauitas ſphæræ plumbeæ, <lb></lb>diametrum habentis<emph.end type="italics"></emph.end> 1/4, <emph type="italics"></emph>vnciæ.<emph.end type="italics"></emph.end></s></p><p type="head">

<s>Sequitur, ad inueniendas diametrorum <lb></lb>magnitudines ex data ſphæ<lb></lb>rarum grauitate, <lb></lb>tabula.</s></p><pb xlink:href="029/01/049.jpg" pagenum="42"></pb><p type="table">

<s>TABELLE WAR HIER</s></p><pb xlink:href="029/01/050.jpg" pagenum="43"></pb><p type="table">

<s>TABELLE WAR HIER</s></p><pb xlink:href="029/01/051.jpg" pagenum="44"></pb><p type="table">

<s>TABELLE WAR HIER</s></p><pb xlink:href="029/01/052.jpg" pagenum="45"></pb><p type="table">

<s>TABELLE WAR HIER</s></p><pb xlink:href="029/01/053.jpg" pagenum="46"></pb><p type="table">

<s>TABELLE WAR HIER</s></p><pb xlink:href="029/01/054.jpg" pagenum="47"></pb><p type="table">

<s>TABELLE WAR HIER</s></p><pb xlink:href="029/01/055.jpg" pagenum="48"></pb><p type="table">

<s>TABELLE WAR HIER</s></p><pb xlink:href="029/01/056.jpg" pagenum="49"></pb><p type="table">

<s>TABELLE WAR HIER</s></p><p type="main">

<s>E<emph type="italics"></emph>st hæc tabula, quemadmodum &amp; eius vſus, præcedentis con<lb></lb>nuerſa, in ea enim inueniuntur ſphærarum grauitates ex data <lb></lb>diametrorum magnitudine, in hac vero deprehenduntur diametro<lb></lb>rum magnitudines ex data ſphærarum grauitate.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Quæro exempli gratia magnitudinem diametri ſphæræ aureæ, <lb></lb>grauitatem habentis 10, lib. </s>

<s>Numeri in prima columna ſub titulo <lb></lb>grauitatis denotant ſphærarum grauitates, reliqui vero in reliquis <lb></lb>columnis denotant diametrorum magnitudines; itaque in linea<emph.end type="italics"></emph.end> 10, <lb></lb><emph type="italics"></emph>lib.

ſub titulo magnitudinis diametri ſphæræ aureæ, datur quæſita <lb></lb>diametri magnitudo partium<emph.end type="italics"></emph.end> 2 21/100: <emph type="italics"></emph>qualium pes vnus est<emph.end type="italics"></emph.end> 12. </s></p><p type="main">

<s><emph type="italics"></emph>Quæro magnitudinem diametri ſphæræ ferreæ, grauitatem haben<lb></lb>tis 50, lib.

in linea 50, lib. ſub titulo magnitudinis diametri ſphæræ <lb></lb>ferreæ, datur quæſita diametri magnitudo<emph.end type="italics"></emph.end> 6 40/100. </s>

</p><p type="main">

<s><emph type="italics"></emph>Quæro magnitudinem diametri ſphæræ argenteæ, grauitatem ha<lb></lb>bentis 60, lib.

in linea 60, lib. ſub titulo magnitudinis diametri ſphæræ <lb></lb>argenteæ, datur ipſa magnitudo<emph.end type="italics"></emph.end> 6 25/100. </s>

</p><pb xlink:href="029/01/057.jpg" pagenum="50"></pb><p type="main">

<s><emph type="italics"></emph>Quæro denique magnitudinem diametri ſphæræ stanneæ, graui<lb></lb>tatem habentis 38, lib.

in linea 38, lib. ſub titulo magnitudinis dia<lb></lb>metri ſphæræ stanneæ, datur quæſita diametri magnitudo 6, ad <lb></lb>vnguem.<emph.end type="italics"></emph.end></s>

</p><p type="main">

<s><emph type="italics"></emph>Notandum autem est, quod numeri, qui diametrorum magnitu<lb></lb>dines denotant, non ſunt veri, ac certi, ſed veris bene proximi, <expan abbr="quoniã">quoniam</expan> <lb></lb>numeri, quorum ipſi ſunt radices cubicæ, non ſunt cubi, &amp; ideo ipſa <lb></lb>radices non explicantur accurate, ſed vel veris maiores, vel minores, <lb></lb>atque vt cognoſcantur quæ ſint maiores, queue minores, maioribus <lb></lb>duo puncta adiecimus, minoribus vnum, accuratis nullum. </s>

<s>inter om<lb></lb>nes autem vnus est accuratus, is ſcilicet, qui magnitudinem indicat <lb></lb>diametri ſphæræ stanneæ, grauitatem habentis<emph.end type="italics"></emph.end> 38, <emph type="italics"></emph>lib.<emph.end type="italics"></emph.end>De compoſitione huius Tabulæ. </s></p><p type="main">

<s><emph type="italics"></emph>Huius tabulæ compoſitio pendet ex præcedenti tabula, &amp; ex pro<lb></lb>poſ.<emph.end type="italics"></emph.end> 17, <emph type="italics"></emph>huius, ſi enim fiat vt grauitas ſphæræ stanneæ, diametrum <lb></lb>habentis vnius vnciæ, id est, vt grana<emph.end type="italics"></emph.end> 1216, <emph type="italics"></emph>ad grauitatem ſphæræ <lb></lb>vnius libræ, idest, ad grana<emph.end type="italics"></emph.end> 6912, <emph type="italics"></emph>ita cubus diametri vnius vnciæ, <lb></lb>hoc eſt, ita<emph.end type="italics"></emph.end> 1, <emph type="italics"></emph>ad alium numerum, qui ſit<emph.end type="italics"></emph.end> 5 11/19 <emph type="italics"></emph>is erit cubus diametri <lb></lb>ſphæræ stanneæ, grauitatem habentis 1, lib.

demonstratum enim <lb></lb>est prop.<emph.end type="italics"></emph.end> 17, <emph type="italics"></emph>huius, ſphæras eiuſdem generis inter ſe eſſe in grauitate, <lb></lb>vt diametrorum cubi in magnitudine; quare radix cubica numeri<emph.end type="italics"></emph.end><lb></lb>5 13/19, <emph type="italics"></emph>dabit ipſam diametrum, ſed quoniam numerus<emph.end type="italics"></emph.end> 5 13/19, <emph type="italics"></emph>non est <lb></lb>præciſe cubus, eius radix non explicabitur accurata, ſed vt explicetur <lb></lb>veræ bene proxima, multiplicetur<emph.end type="italics"></emph.end> 5 13/19, <emph type="italics"></emph>per<emph.end type="italics"></emph.end> 1000000. <emph type="italics"></emph>&amp; ex producto<emph.end type="italics"></emph.end><lb></lb>5684210 10/19, <emph type="italics"></emph>neglecto fracto<emph.end type="italics"></emph.end> 10/19, <emph type="italics"></emph>eruatur radix, tanquam ex accu<lb></lb>rato numero cubo, ea erit<emph.end type="italics"></emph.end> 173. <emph type="italics"></emph>proxime, &amp; erit centupla radicis nu<lb></lb>meri<emph.end type="italics"></emph.end> 5 13/19, <emph type="italics"></emph>nam numerus<emph.end type="italics"></emph.end> 1000000, <emph type="italics"></emph>per quem fuit multiplicatus<emph.end type="italics"></emph.end><lb></lb>5 13/19, <emph type="italics"></emph>cubus eſt ex<emph.end type="italics"></emph.end> 100<emph type="italics"></emph>; magnitudo igitur diametri ſphæræ ſtanneæ, <lb></lb>grauitatem habentis 1, lib.

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

<s>ita inuenientur diametri.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Duplum numeri<emph.end type="italics"></emph.end> 5684210 10/19, <emph type="italics"></emph>id est<emph.end type="italics"></emph.end> 11368421 1/19, <emph type="italics"></emph>erit cubus <lb></lb>centupli diametri ſpheræ stanneæ, grauitatem habentis duplam pri<lb></lb>mæ, hoc est<emph.end type="italics"></emph.end> 2, <emph type="italics"></emph>lib.

ex ſupra nominata enim prop.<emph.end type="italics"></emph.end> 17, <emph type="italics"></emph>huius, eſt vt gra<lb></lb>uitas ſphæræ vnius libræ, ad grauitatem ſphæræ duarum librarum, <lb></lb>ita cubus diametri primæ ſphæræ, ad cubum diametri ſecundæ. </s>

<s>Si <lb></lb>vero triplicetur numerus<emph.end type="italics"></emph.end> 5684210 10/19, <emph type="italics"></emph>eius triplum, quod est<emph.end type="italics"></emph.end><lb></lb>17052631 11/19, <emph type="italics"></emph>erit cubus centupli diametri ſphæræ ſtanneæ, graui<lb></lb>tatem habentis triplam primæ, idest<emph.end type="italics"></emph.end> 3, <emph type="italics"></emph>lib.

&amp; ſi quadruplicetur, eius<emph.end type="italics"></emph.end><pb xlink:href="029/01/058.jpg" pagenum="51"></pb><emph type="italics"></emph>quadruplum erit cubus centupli diametri ſphæræ stanneæ, grauita<lb></lb>tem habentis quadruplam primæ, &amp; ſic deinceps. </s>

<s>itaque ſi ex eius <lb></lb>multiplicibus, neglectis fractis, <expan abbr="eruãtur">eruantur</expan> radices, tanquam ex accura<lb></lb>tis numeris cubis, ipſæ indicabunt diametrorum magnitudines in <lb></lb>ratione centupla. </s>

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

<s><emph type="italics"></emph>Prædicto numero<emph.end type="italics"></emph.end> 5684210 10/19, <emph type="italics"></emph>addatur eius duplum, id est,<emph.end type="italics"></emph.end><lb></lb>11368421 1/19, <emph type="italics"></emph>ſumma<emph.end type="italics"></emph.end> 17052631 11/19, <emph type="italics"></emph>dabit eius triplum, ſi vero ei <lb></lb>addatur eius triplum, id eſt,<emph.end type="italics"></emph.end> 17052631 11/19, <emph type="italics"></emph>ſumma<emph.end type="italics"></emph.end> 22736842 2/19, <lb></lb><emph type="italics"></emph>dabit eius quadruplum, &amp; ſi eius quadruplum ei addatur, ſumma <lb></lb>dabit eius quintuplum, &amp; ſic ſola additione inuenientur eius quot<lb></lb>cunque multiplicia.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Eadem ratione inuenientur diametri ſphærarum ex quacunque <lb></lb>alia materia, ſi enim quæratur de magnitudine diametri verbi gra<lb></lb>tia ſphæræ ferreæ, grauitatem habentis 1, lib.

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

quare radix cubica numeri<emph.end type="italics"></emph.end> 5 49/190, <emph type="italics"></emph>dabit quæſitam dia<lb></lb>metrum, &amp; quoniam numerus<emph.end type="italics"></emph.end> 5 49/190, <emph type="italics"></emph>non eſt præciſe cubus, &amp; ideo <lb></lb>non explicabitur eius radix accurate, multiplicetur per 1000000, &amp; <lb></lb>ex producto<emph.end type="italics"></emph.end> 5257894 14/19, <emph type="italics"></emph>neglecto fracto<emph.end type="italics"></emph.end> 14/19, <emph type="italics"></emph>eruatur radix, tan<lb></lb>quam ex accurato numero cubo, ea erit<emph.end type="italics"></emph.end> 174: <emph type="italics"></emph>ferè, &amp; erit centupla ra<lb></lb>dicis numeri<emph.end type="italics"></emph.end> 5 49/190, <emph type="italics"></emph>quia numerus<emph.end type="italics"></emph.end> 5 49/190, <emph type="italics"></emph>multiplicatus fuit per <lb></lb>cubum ex 100; diameter igitur ſphæræ ferreæ, grauitatem habentis <lb></lb>1, lib.

&amp; ſic reliquarum ſphærarum in infinitum inuenientur <lb></lb>diametri. </s>

<s>multiplicia autem numeri<emph.end type="italics"></emph.end> 5257894 14/19, <emph type="italics"></emph>ſola additione in<lb></lb>uenientur, vt dictum est ſupra de inuentione multiplicium numeri<emph.end type="italics"></emph.end><lb></lb>5684210 10/19. <emph type="italics"></emph>Atque hac ratione prædictam tabulam compoſuimus.<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note48"></margin.target>17<emph type="italics"></emph>: <lb></lb>huius.<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Qvomodo Archimedes argenti mixtionem depre<lb></lb>hendit in auro. </s></p><p type="main">

<s><emph type="italics"></emph>Hiero (referente Vitruuio lib.<emph.end type="italics"></emph.end> 9. <emph type="italics"></emph>Cap.<emph.end type="italics"></emph.end> 3.<emph type="italics"></emph>) Siracuſis auctus regia<emph.end type="italics"></emph.end><pb xlink:href="029/01/059.jpg" pagenum="52"></pb><emph type="italics"></emph>poteſtate, rebus bene geſtis, cum auream coronam votiuam, dijs im<lb></lb>mortalibus in quodam fano conſtituiſſet ponendam, immani precia <lb></lb>locauit faciendam, &amp; aurum ad ſacoma appendit redemptori. </s>

<s>is ad <lb></lb>tempus opus manufactum ſubtiliter, regi approbauit, &amp; ad ſacoma <lb></lb>pondus coronæ viſus eſt præſtitiſſe. </s>

<s>Poſtea quam indicium eſt factum, <lb></lb>dempto auro, tantundem argenti in id coronarium opus admixtum <lb></lb>eſſe: indignatus Hiero ſe contemptum, neque inueniens, qua ratione <lb></lb>id furtum deprehenderet, rogauit Archimedem, vti in ſe ſumeret de <lb></lb>eo cogitationem. </s>

<s>tunc is cum haberet eius rei curam, caſu venit in <lb></lb>balneum, ibique cum in ſolium deſcenderet, animaduertit quantum <lb></lb>corporis ſui in eo inſideret, tantum aquæ extra ſolium effluere. </s>

<s><expan abbr="itaq;">itaque</expan> <lb></lb>cum eius rei rationem explicationis offendiſſet non est moratus, ſed <lb></lb>exiliuit gaudio motus de ſolio, &amp; nudus vadens domum verſus, ſigni <lb></lb>ficabat clara voce inueniſse quod quæreret. </s>

<s>nam currens identidem <lb></lb>grece clamabat<emph.end type="italics"></emph.end> <foreign lang="grc"><gap></gap></foreign> <emph type="italics"></emph>tum vero ex eo inuentionis ingreſſu <lb></lb>duas dicitur feciſſe maſſas æquo pondere, quo etiam fuerat coro<lb></lb>na, vnam ex auro, alteram ex argento. </s>

<s>cum ita fecißet, vas <expan abbr="amplũ">amplum</expan> <lb></lb>ad ſumma labra impleuit aqua, in quo demiſit argenteam maſſam, <lb></lb>cuius quanta magnitudo in vaſe depreſſa est, tantum aquæ effluxit. </s>

<s><lb></lb>ita exempta maſſa, quanto minus factum fuerat refudit, ſextario <lb></lb>menſus, vt eodem modo, quo prius fuerat, ad labra æquaretur. </s>

<s>ita ex <lb></lb>eo inuenit, quantum ad certum pondus argenti certa aquæ menſura <lb></lb>reſponderet.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Cum id expertus eſſet tum auream maſſam ſimiliter pleno vaſe <lb></lb>demiſit, &amp; ea exempta, eadem ratione menſura addita, inuenit ex <lb></lb>aqua non tantum defluxiſſe, ſed tantum minus, quantum minus ma <lb></lb>gno corpore eodem pondere auri maſſa eſſet quam argenti. </s>

<s>Postea <lb></lb>vero repleto vaſe, in <expan abbr="eadẽ">eadem</expan> aqua ipſa corona demiſſa, inuenit plus aqua <lb></lb>defluxiſſe in coronam, quam in auream. </s>

<s>eodem pondere maſſam, &amp; <lb></lb>ita ex eo quod plus defluxerat aquæ in corona, quam in maſſa ratio<lb></lb>cinatus, deprehendit argenti in auro mixtionem, &amp; manifestum fur<lb></lb>tum redemptoris.<emph.end type="italics"></emph.end></s><s> Hactenus Vitruuius. </s></p><p type="main">

<s>Mirum certe Archimedis fuit inuentum, ipſius tamen <lb></lb>modus ad inueniendam illam aquæ menſuram, quæ ad <lb></lb>certum pondus auri, vel argenti, vel coronæ reſponderet, <lb></lb>maiori diligentia indiget, quam quæ ab hominibus adhiberi <lb></lb>poteſt, impoſſibile enim eſt, exempta corona, vel aurea maſſa, <lb></lb>vel argentea, tantum aquæ refundere, quantum è vaſe efflu<lb></lb>xerat ad vnguem, nam repoſita aqua in vaſe, non poſſumus <pb xlink:href="029/01/060.jpg" pagenum="53"></pb>affirmare ipſum vas eſſe plenum, niſi aqua incipiat effluere, <lb></lb>cum autem incipit, effluit aliquando totus ferè cumulus, ita<lb></lb>que vel plus aquæ additur eo, quod deficit, vel minus, niſi <lb></lb>coniectura aſſequatur: at vero coniectura pro veritate non ac<lb></lb>cipitur. </s>

<s>præterea exempta corona, vel aurea maſſa, vel argen<lb></lb>tea, eximitur etiam ſimul cum ipſa aliquantum aquæ, quæ cir<lb></lb>cum ipſam remanet, atque huiuſmodi defectus errorem indu<lb></lb>cit ſenſibilem. </s></p><p type="main">

<s>Neque per collectionem quæſita aquæ menſura inueniri <lb></lb>poteſt: æquè enim impoſſibile eſt vniuerſam illam æquam col<lb></lb>ligere, quæ extra vas effluit, quando corona, vel aurea maſſa <lb></lb>vel argentea in ipſo vaſe deprimitur, cum enim aqua è vaſe <lb></lb>effluat, pars ipſius aquæ vaſi, ex quo effluit, pars vaſi in quod <lb></lb>influit adhæret, &amp; ſi vniuerſa omnino ſemper non colligatur, <lb></lb>erit non parui erroris cauſa, præter quam quod, non ſemper <lb></lb>adeo facile inuenitur par auri, argentique maſſa, quando co<lb></lb>rona, vel alia auri maſſa, quæ examinanda proponitur, medio<lb></lb>crem excederet magnitudinem. </s></p><p type="main">

<s>Neque præterea poteſt diſcerni prædicta argenti portio in <lb></lb>aliqua auri parua maſſa, differentiæ enim aquarum, quæ ex<lb></lb>tra vas effluunt, ſunt adeo exiguæ, vt ne cognoſci quidem <lb></lb>poſſint, quod ſi cognoſcerentur, non ſemper erunt veræ, <lb></lb>ſiquidem non ſemper in vaſis medio in cumulum creſcens <lb></lb>æqualis aquæ copia remanet, ſed maior interdum, inter<lb></lb>dum minor, vt conſpicitur. </s>

<s>fit enim vt aliquando cumulus <lb></lb>ille frangatur pluribus in locis, &amp; ideo aqua diffundatur, vt <lb></lb>ferè nihil ipſius cumuli ſuperſit, aliquando vero frangatur <lb></lb>in vno tantum loco, &amp; aqua colligens ſe in cumulum, parum <lb></lb>diffluat. </s></p><p type="main">

<s>Sed ponderandis corporibus in aere &amp; aqua, eo modo, quo <lb></lb>dictum eſt in fine exempli prop.

8. inuenitur quæſita aquæ, <lb></lb>grauitas, ita exactè, vt requiritur, fiue ſit corpus illud paruum, <lb></lb>ſiue magnum nihil intereſt, &amp; præterea facillima eſt operatio, <lb></lb>nec adinueniendæ ſunt auri, &amp; argenti maſſæ æque graues, ac <pb xlink:href="029/01/061.jpg" pagenum="54"></pb>corona, ſed quælibet particulæ, grauitate quacunque, etiam <lb></lb>differentes inter ſe, ſufficiunt. </s></p><p type="main">

<s>De ratione autem, qua Archimedes, cognitis grauitatibus <lb></lb>trium corporum ex aqua, magnitudine æqualium, coronæ <lb></lb>ſcilicet vnum, alterum maſſæ aureę, tertium argenteæ, potue<lb></lb>rit furtum aurificis in regia corona deprehendere, atque ar<lb></lb>gentum quod erat in ea permixtum ab auro diſcernere, pluri<lb></lb>mi ſcripſerunt, modos etiam ad id faciendum excogitarunt <lb></lb>varios, longa tamen methodo, atque difficili vſi ſunt, &amp; quod <lb></lb>maximam confuſionem, &amp; obſcuritatem parit, nullum opera<lb></lb>tionis tradunt præceptum firmum, ac ſtabile. </s>

<s>ego autem vni<lb></lb>ca tantum proportionis ratiocinatione, ſeu regula trium (vt <lb></lb>vulgo dicitur) breuiter, &amp; expedite idem conſequor, eamque <lb></lb>geometrica ratione demonſtro. </s>

<s>Problema igitur ad hoc facien <lb></lb>dum ita concipio &amp; abſoluo. </s></p><p type="head">

<s>PROBLEMA IX. PROPOS. XVIII.</s></p><p type="main">

<s>Portionem metalli, alteri metallo miſtam, ponde<lb></lb>ris ratiocinatione diſcernere. </s></p><p type="main">

<s>QVONIAM de Hieronis corona facta eſt mentio, ſit ea B, <lb></lb>eiuſque grauitas EK, &amp; oporteat argentum, quod ſit in ea <expan abbr="permixtũ">permixtum</expan>, <lb></lb>ab auro diſcernere, hoc eſt oporteat inuenire quanta erit portio ar<lb></lb><figure id="id.029.01.061.1.jpg" xlink:href="029/01/061/1.jpg"></figure><lb></lb>genti, &amp; quanta auri. </s>

<s>In<lb></lb>telligantur duo corpora <lb></lb>A, D, vnum aureum, al<lb></lb>terum argenteum æque <lb></lb>grauia atque corona, <lb></lb>deinde trium corporum <lb></lb>ex aqua, magnitudine <lb></lb>æqualium, aureo ſcili<lb></lb>cet corpori vnum, alte<lb></lb>rum coronæ, tertium <lb></lb>corpori argenteo, inue<lb></lb>niantur grauitates, id autem poterit fieri facillime, ſi <expan abbr="accipiãtur">accipiantur</expan> duo <lb></lb>corpora vnum ex auro, alterum ex argento, grauitate quacunque, vt <pb xlink:href="029/01/062.jpg" pagenum="55"></pb>dictum eſt in propoſitionis octauæ exemplo, non enim neceſſe eſt ha<lb></lb>bere duo corpora ex auro &amp; argento, grauitatem habentia eandem <lb></lb>quam &amp; corona, &amp; hac de cauſa diximus ſupra <expan abbr="intelligãtur">intelligantur</expan> duo cor<lb></lb>pora, non autem accipiantur. </s>

<s>ſit igitur primi corporis aquei æqualis <lb></lb>aureo A, inuenta grauitas G, ſecundi vero æqualis coronæ B, graui<lb></lb>tas F, &amp; tertij æqualis corpori argenteo D, grauitas H, &amp; fiat vt dif<lb></lb>ferentia inter G, &amp; H, ad EK, ita differentia inter G, &amp; F, ad aliam <lb></lb>grauitatem, quæ ſit K. </s>

<s>Dico K, grauitatem eſſe portionis argenti, <lb></lb>quod eſt in corona, E vero grauitatem auri. </s></p><p type="main">

<s>Vel ſi pro tertio proportionis termino ſumatur differentia inter F, <lb></lb>&amp; H, &amp; quartus terminus ſit E, Dico E, grauitatem eſſe portionis au<lb></lb>ri, K vero argenti. </s></p><p type="main">

<s>Quartus autem vtriuſque proportionis terminus * minor eſt<arrow.to.target n="note49"></arrow.to.target> ſe<lb></lb>cundo EK, quod &amp; tertius minor eſt primo, primus enim terminus <lb></lb>eſt differentia inter G, &amp; H, tertius vero, vel eſt differentia inter G, &amp; <lb></lb>F, vel differentia inter F, &amp; H, vterque minor primo. </s>

<s>Exemplis autem <lb></lb>res fiet illuſtrior. </s></p><p type="margin">

<s><margin.target id="note49"></margin.target>14. 5 <lb></lb><emph type="italics"></emph>Elen<emph.end type="italics"></emph.end></s></p><p type="head">

<s>Exemplum. </s>

<s>I.</s></p><p type="main">

<s><emph type="italics"></emph>Sit coronæ grauitas<emph.end type="italics"></emph.end> 95, <emph type="italics"></emph>lib.

&amp; oporteat facere quod imperatum eſt. </s>

<s><lb></lb>Intelligantur duo corpora, vnum aureum, alterum argenteum, æque <lb></lb>grauia atque corona, deinde trium corporum ex aqua, magnitudine <lb></lb>æqualium, aureo ſcilicet corpori vnum, alterum coronæ, tertium cor<lb></lb>pori argenteo, inueniantur grauitates, vt in exemplo prop.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>dictum <lb></lb>est, quæ ſint primi nimirum corporis aquei<emph.end type="italics"></emph.end> 5, <emph type="italics"></emph>ſecundi vero 6, &amp; tertij<emph.end type="italics"></emph.end><lb></lb>9 6/31, <emph type="italics"></emph>&amp; fiat vt differentia inter<emph.end type="italics"></emph.end> 5, &amp; 9 6/32, <emph type="italics"></emph>hoc eſt vt<emph.end type="italics"></emph.end> 4 6/31, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 95, <lb></lb><emph type="italics"></emph>grauitatem videlicet coronæ, ita differentia inter<emph.end type="italics"></emph.end> 5, &amp; 6, <emph type="italics"></emph>hoc est 1, ad<emph.end type="italics"></emph.end><lb></lb>22 17/26, <emph type="italics"></emph>ergo<emph.end type="italics"></emph.end> 22 17/26, <emph type="italics"></emph>erit grauitas portionis argenti quod est in coro<lb></lb>na, qua detracta ex totali grauitate coronæ, reliquum<emph.end type="italics"></emph.end> 72 9/26, <emph type="italics"></emph>erit <lb></lb>grauitas portionis auri.<emph.end type="italics"></emph.end></s></p><p type="main">

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

<s>Exemplum. </s>

<s>II.</s></p><p type="main">

<s><emph type="italics"></emph>Sit aliquod corpus mistum ex auro. </s>

<s>&amp; ære, &amp; habeat grauitatem <lb></lb>171. lib.

&amp; oporteat inuenire quanta erit portio æris in ipſo corpore,<emph.end type="italics"></emph.end><pb xlink:href="029/01/063.jpg" pagenum="56"></pb><emph type="italics"></emph>&amp; quanta auri. </s>

<s>Intelligantur duo corpori, vnum ex auro puro, al<lb></lb>terum ex ære, æque grauia atque corpus mistum, &amp; trium corporum <lb></lb>ex aqua, quorum vnum ſit æquale corpori aureo magnitudine, alte<lb></lb>rum misto, tertium aureo, inueniantur grauitates, vt in exemplo pro<lb></lb>poſ.<emph.end type="italics"></emph.end>8.<emph type="italics"></emph>dictum est, quæ ſint<emph.end type="italics"></emph.end> 9, 11, <emph type="italics"></emph>&amp; 19, &amp; fiat vt differentia inter<emph.end type="italics"></emph.end> 9, &amp; <lb></lb>19, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 171, <emph type="italics"></emph>grauitatem videlicet corporis misti, ita differentia inter<emph.end type="italics"></emph.end><lb></lb>9, &amp; 11, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 34 1/5, <emph type="italics"></emph>portio igitur corporis miſti ærea grauitatem habebit<emph.end type="italics"></emph.end><lb></lb>34 1/5, <emph type="italics"></emph>quæ ſi auferatur ex totali corporis misti grauitate, remanebit<emph.end type="italics"></emph.end><lb></lb>136 4/5, <emph type="italics"></emph>pro grauitate portionis auri.<emph.end type="italics"></emph.end></s></p><p type="main">

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

<s>At vero huiuſmodi ratiocinationem ad diſcernendum ar<lb></lb>gentum ab auro, vel aliud metallum ab altero metallo, recte <lb></lb>eſſe inſtitutam, ſequenti Theoremate demonſtrabitur. </s></p><p type="head">

<s>THEOREMA X. PROPOS. XIX.</s></p><p type="main">

<s>Si trium corporum æque grauium primum &amp; ter<lb></lb>tium fuerint generis diuerſi, ſecundi autem portio <lb></lb>fuerit eiuſdem generis cum corpore primo, reliqua ve<lb></lb>ro eiuſdem generis cum corpore tertio, fuerint etiam <lb></lb>tres quantitates aquæ prædictis corporibus æquales, pri<lb></lb>ma videlicet corpori primo, ſecunda ſecundo, &amp; tertia <lb></lb>tertio. </s>

<s>erit vt differentia grauitatum primæ &amp; tertiæ <lb></lb>quantitatis aquæ, ad grauitatem corporis ſecundi, ita <lb></lb>differentia grauitatum primæ &amp; ſecundæ quantitatis <lb></lb>aquæ, ad grauitatem portionis corporis ſecundi, quæ eſt <lb></lb>eiuſdem generis cum corpore tertio. </s></p><p type="main">

<s>Et ita differentia grauitatum ſecundæ &amp; tertiæ quan <lb></lb>titatis aquæ, ad grauitatem portionis eiuſdem generis <lb></lb>cum corpore primo. </s></p><pb xlink:href="029/01/064.jpg" pagenum="57"></pb><p type="main">

<s>SINT tria corpora æque grauia A, BC, D, quorum A, primum, <lb></lb>&amp; tertium D. ſint generis diuerſi, portio vero ſecundi B, ſit eiuſdem <lb></lb>generis cum corpore A, &amp; portio C, eiuſdem generis cum corpore <lb></lb><figure id="id.029.01.064.1.jpg" xlink:href="029/01/064/1.jpg"></figure><lb></lb>D, ſint etiam alia tria <lb></lb>corpora aquea P, OL, <lb></lb>&amp; Q, <expan abbr="quorũ">quorum</expan> <gap></gap>, ſit æqua<lb></lb>le corpori A, magnitu<lb></lb>dine, ipſum vero OL, <lb></lb>æquale corpori BC, &amp; <lb></lb>ipſum Q, æquale cor<lb></lb>pori D, &amp; ſint earum <lb></lb>grauitates, G, ipſius P, <lb></lb>&amp; FV, ipſius OL, &amp; H, <lb></lb>ipſius <expan abbr="q.">que</expan> Dico vt diffe<lb></lb>rentia grauitatum G, <lb></lb>H, ad grauitatem cor<lb></lb>poris BC, ita eſſe diffe<lb></lb>rentiam grauitatum <lb></lb>G, FV, ad grauitatem <lb></lb>portionis C; &amp; ita differentiam grauitatum FV, H, ad portionis B, <lb></lb>grauitatem. </s>

<s>Sit enim portionis B, grauitas E, &amp; portionis C, gra<lb></lb>uitas K; ergo totius corporis BC grauitas erit EK, <expan abbr="ſitq;">ſitque</expan> portionis O, <lb></lb>quæ ſit æqualis portioni B, grauitas F, ergo reliquæ portionis L, <lb></lb>æqualis portioni C, grauitas erit V, Quoniam igitur eſt, vt A, ad P, <lb></lb>ita B, ad O, æquale videlicet ad æquale, erit permutando, vt A, ad B, <lb></lb>ita p.ad O, &amp; quoniam ſunt eiuſdem generis A, B, ſimiliter &amp; P, O, <lb></lb>* erit vt grauitas corporis A, hoc eſt vt EK, (ponuntur enim<arrow.to.target n="note50"></arrow.to.target> cor<lb></lb>pora A, BC, D, æque grauia,) ad E, ita G, ad F, quod igitur fit ex EK, <lb></lb>&amp; F, nempe ex extremis, æquale erit ei, quod fit ex E, &amp; G, hoc eſt <lb></lb>ex medijs. </s></p><p type="margin">

<s><margin.target id="note50"></margin.target>4.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Similiter quoniam eſt, vt D, ad Q, ita C, ad E, æquale videlicet ad <lb></lb>æquale, erit permutando, vt D, ad C, ita Q, ad L, &amp; quoniam ſunt <lb></lb>eiuſdem generis D, C, ſimiliter &amp; Q, L, * erit vt grauitas ipſius D,<arrow.to.target n="note51"></arrow.to.target><lb></lb>hoc eſt vt EK, ad K, ita H, ad V; quare quod fit ex EK, &amp; V, ex extre<lb></lb>mis, æquabitur ei, quod ex H, fit &amp; K, ex medijs. </s></p><p type="margin">

<s><margin.target id="note51"></margin.target>4.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Sed oſtenſum eſt id quod ex EK, fit &amp; F. æquale eſſe ei quod fit ex <lb></lb>G, &amp; E, ergo quod fit ex EK, &amp; F, vna cum eo, quod ex BK, &amp; V, hoc <lb></lb>eſt id quod fit ex EK, &amp; FV, æquale erit ei quod ex G, fit &amp; E, vna <lb></lb>eum eo quod ex H, &amp; K, ſed quod ex G, fit &amp; E, æquale eſt ei quod fit <lb></lb>ex G, &amp; EK, minus eo quod ex G, &amp; K, quod enim additur, idem &amp; <pb xlink:href="029/01/065.jpg" pagenum="58"></pb>minuitur; ergo quod fit ex EK, &amp; FV. æquale erit ei quod fit ex G, &amp; <lb></lb>EK, vna cum eo quod ex H, &amp; K, minus eo quod fit ex G, &amp; K. aufe<lb></lb>ratur vtrinque id quod fit ex G, &amp; EK, quod igitur fit ex FV, &amp; EK, <lb></lb><figure id="id.029.01.065.1.jpg" xlink:href="029/01/065/1.jpg"></figure><lb></lb>minus eo quod ex G, &amp; <lb></lb>EK, æquabitur ei quod <lb></lb>ex H, &amp; K, minus eo <lb></lb>quod fit ex G, &amp; K, ſed <lb></lb>quod fit ex H, &amp; K, <lb></lb>minus eo quod fit ex <lb></lb>G, &amp; K, æquale eſt ei <lb></lb>quod ex differentia ip<lb></lb>ſarum H, G, fit &amp; K, <lb></lb>ſimiliter, &amp; quod fit ex <lb></lb>FV, &amp; EK, minus eo <lb></lb>quod ex G, &amp; EK, <lb></lb>æquale eſt ei quod ex <lb></lb>differentia ipſarum <lb></lb>FV, G, fit &amp; EK, ergo <lb></lb>quod ex differentia <lb></lb>ipſacum H, G, fit &amp; K, æquale erit ei quod ex differentia ipſarum <lb></lb>FV, G, fit &amp; EK; æqualitatem ad proportionem reuocando, erit vt <lb></lb>differentia grauitatum. </s>

<s>H, G, ad grauitatem EK, ita differentia <lb></lb>grauitatum FV, G, ad grauitatem K, quod erat primo loco: demon<lb></lb>ſtrandum. </s></p><p type="main">

<s>Dico quoque vt differentia grauitatum H, G, ad grauitatem EK, <lb></lb>ita eſſe differentiam grauitatum H, FV, ad grauitatem E. </s>

<s>Quoniam <lb></lb>enim oſtenſum eſt, quod fit ex EK, &amp; FV, æquale eſſe ei quod ex G. fit <lb></lb>&amp; E, vna cum eo quod ex H, &amp; K; quod autem fit ex H, &amp; K, æquatur<lb></lb>ri quod, ex H, fit &amp; EK, minus eo quod ex H, &amp; E, quod enim additur <lb></lb>idem &amp; minuitur: ergo quod fit ex EK, &amp; FV, æquale erit ei quod fit <lb></lb>ex H, &amp; EK, vna cum eo quod ex G, &amp; E, minus eo quod ex H, &amp; E. <lb></lb>addatur vtrin que quod ex H, fit &amp; E, &amp; ſubducantur ea quæ fiunt ex <lb></lb>G, &amp; E, &amp; ex EK, &amp; FV; quod igitur fit ex H, &amp; E, minus eo quod ex <lb></lb>G, &amp; E, æquabitur ei quod ex H, fit &amp; EK, minus eo quod ex FV, &amp; <lb></lb>EK, ſed quod fit ex H, &amp; E, minus eo quod ex G, &amp; E, æquale eſt ei <lb></lb>quod ex, differentia ipſarum H, G, fit &amp; E, ſimiliter, &amp; quod ex H, fit <lb></lb>&amp; EK, minus eo quod ex FV, &amp; EK, æquale eſt ei quod ex differentia <lb></lb>ipſarum H, FV, fit &amp; EK; ergo quod ex differentia ipſarum H, G, fit <lb></lb>&amp; E, æquabitur ei quod ex differentia ipſarum H, FV, fit &amp; EK; qua<lb></lb>re æqualitatem ad <expan abbr="proportionẽ">proportionem</expan> reuocando erit vt differentia graui<pb xlink:href="029/01/066.jpg" pagenum="59"></pb>tatum H, G, ad grauitatem EK, ita differentia grauitatum H, FV, <lb></lb>zd grauitatem E. quod ſecundo loco fuit demonſtrandum. </s></p><p type="main">

<s>Alia breuior Theorematis demonſtratio. </s></p><p type="main">

<s>RESVMATVR eadem figura vt ſupra. </s>

<s>Quoniam igitur <lb></lb>corpus D, æquale eſt corpori Q, magnitudine, &amp; portio C, æqualis <lb></lb>portioni L, erit vt D, ad Q, ita C, ad L, &amp; permutando vt D, ad C, ita <lb></lb>Q, ad L, &amp; quoniam eiuſdem ſunt generis D, C, ſimiliter &amp; Q, L, * erit<arrow.to.target n="note52"></arrow.to.target><lb></lb>vt grauitas corporis D, hoc eſt vt EK; ad K, ita H, ad V. </s></p><p type="margin">

<s><margin.target id="note52"></margin.target>4.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Similiter quoniam ponuntur æqualia magnitudine corpora A, P, <lb></lb>&amp; æquales quoque portiones B, O, erit vt A, ad P, ita B, ad O, &amp; per<lb></lb>mutando vt A, ad B, ita P, ad O, ſed eiuſdem ſunt generis A, B, ſimili<lb></lb>ter &amp; P, O, * vt igitur grauitas corporis A, id eſt vt EK, ad E, ita erit<arrow.to.target n="note53"></arrow.to.target><lb></lb>G, ad F, &amp; per conuerſionem rationis erit vt EK, ad K, ita G, ad G, <lb></lb>minus F, ſed demonſtratum eſt, vt EK, ad K, ita eſſe H, ad V, ergo vt <lb></lb>H, ad V, ita erit G, ad G, minus F, &amp; permutando vt H, ad G, ita V, <lb></lb>ad G, minus F, &amp; diuidendo vt H, minus G, ad G, ita erit FV, minus <lb></lb>G, ad G, minus F, rurſus permutando erit vt H, minus G, ad FV, mi<lb></lb>nus G, ita G, ad G, minus F, ſed vt EK, ad K, ita eſt G, ad G, minus F, <lb></lb>vt eſt demonſtratum, ergo vt H, minus G, ad FV, minus G, ita erit <lb></lb>EK, ad K, quare permutando vt H, minus G, ad EK, ita erit FV, mi<lb></lb>nus G, ad K, quod eſtò primum. </s></p><p type="margin">

<s><margin.target id="note53"></margin.target>4.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="main">

<s>Dico quoque vt H, minus G, ad EK, ita eſſe H, minus FV, ad E. </s>

<s><lb></lb>Quoniam enim oſtenſum eſt vt EK, ad K, ita eſſe H, ad V, erit per <lb></lb>conuerſionem rationis vt EK, ad E, ita H, ad H, minus V, ſed demon<lb></lb>ſtratum eſt vt EK, ad E, ita eſſe G, ad F, ergo vt H, ad H, minus V, ita <lb></lb>erit G, ad F, &amp; permutando vt H, ad G, ita H, minus V, ad F, &amp; diui<lb></lb>dendo vt H, minus G, ad G, ita erit H, minus FV, ad F, &amp; permutan<lb></lb>do vt H, minus G, ad H, minus FV, ita G, ad F, ſed vt EK, ad E, ita eſt <lb></lb>G, ad F, vt eſt demonſtratum, ergo vt H, minus G, ad H, minus FV, <lb></lb>ita erit EK, ad E, quare permutando, erit vt H, minus G, ad EK, ita <lb></lb>H, minus FV, ad E, quod erat ſecundo loco demonſtrandum. </s></p><p type="main">

<s>Svpereſt igitur vt dicamus, qua ratione ex grauitate aúri <lb></lb>cognoſci poſſit eius qualitas; id quod ex ijs, quæ dicta <lb></lb>ſunt facilè colligitur; ſi videlicet nota fiat cuiuſuis maſſæ auri <lb></lb>grauitas, quam habet tum in aere, tum in aqua. </s>

<s>Sed ante <lb></lb>omnia, duo nobis ſunt præmittenda, &amp; explicanda. </s>

<s>nimirum <lb></lb>quid ſit aurum 24. partium, ſeu (vt vulgo dicitur) di 24. ca<pb xlink:href="029/01/067.jpg" pagenum="60"></pb>ratti, quidue pauciorum, hoc eſt penes quid attendatur di <lb></lb>uerſa auri qualitas. </s>

<s>Deinde quomodo aurum alligent Auri <lb></lb>fices, vel alij ad quos alligandi officium ſpectat. </s>

<s>His enim <lb></lb>cognitis, non erit difficile, id quod proponitur, certa aliqua <lb></lb>ratione, aſſequi. </s></p><p type="main">

<s><emph type="italics"></emph>Aurum igitur<emph.end type="italics"></emph.end> 24. <emph type="italics"></emph>partium appellatur aurum purum, pauciorum <lb></lb>vero dicitur non purum, ſed aliquo alio metallo, vel pluribus affe<lb></lb>ctum. </s>

<s>&amp; quia hæc affectio multiplex eſt, ideo etiam auri qualitas, <lb></lb>qua ex varia mixtione naſcitur, varia ſit est neceſſe : quamuis vna <lb></lb>tantum ſit qualitas auri puri. </s>

<s>Qualitas enim auri in quouis cor <lb></lb>pore propoſito, exprimitur partibus auri puri, qua ſunt in ipſo corpo<lb></lb>re, non in magnitudine, ſed in grauitate ſumptis, qualibus totum cor<lb></lb>pus constat<emph.end type="italics"></emph.end> 24<emph type="italics"></emph>: vel quod idem est, auri qualitas exprimitur in ra<lb></lb>tione quam habent illæ partes in grauitate ad totum corpus: quod <lb></lb>exemplo clarius explicabitur in hunc modum.<emph.end type="italics"></emph.end></s></p><p type="main">

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

<s>Totum igitur illud corpus aureum ab <lb></lb>initio propoſitum, ſi adhuc intelligatur tale quale fuit ante expurga<lb></lb>tionem, appellabitur<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph>partium, ſeu, (vt vulgo dicitur) di<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph>ca<lb></lb>ratti. </s>

<s>eo quod tota illa maßa mista,<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph>tantum vncias auri puri con<lb></lb>tinuerit. </s>

<s>Immo non ſolum illa maſſa auri, ſed etiam illa cuius ipſa <lb></lb>fuiſſet pars, vel que ipſius fuiſſet quæcunque pars dicetur<emph.end type="italics"></emph.end> 20, <emph type="italics"></emph>par<lb></lb>tium. </s>

<s>Neque enim in alligationibus metallorum, alia est alli<lb></lb>gatio partium, alia totius, ſed vtrorunque vna eademque eſt qua<lb></lb>litas.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Et hoc est quod Aurifices in inuaestigatione qualitatis auri ob<lb></lb>ſeruant. </s>

<s>Non enim purificant totum corpus propoſitum, ſed ali<lb></lb>quam eius particulam etiam perexiguam, quam ſolam ad aurum pu<lb></lb>rum reducunt. </s>

<s>hac enim reducta, non ſolum recte definiunt cuius <lb></lb>fuerit qualitatis particula illa purificata ante purificationem; ve<lb></lb>rum etiam cuius fuerit qualitatis, &amp; quot partium fuerit illud cor<lb></lb>pus, à quo eadem particula detracta fuit, &amp; illud, quod adhuc ſu<lb></lb>pereſt, diminutum ſcilicet illa parte purificata, vt in eodem exemplo <lb></lb>propoſito, corporis aurei<emph.end type="italics"></emph.end> 24. <emph type="italics"></emph>vnciarum apparet. </s>

<s>Eius enim quali<lb></lb>tatem ſi forte aurifices inuestigare velint, detrahent ex eo particu<lb></lb>lam, verbi gratia, vnius vnciæ, vel quod idem est particulam<emph.end type="italics"></emph.end> 24. <pb xlink:href="029/01/068.jpg" pagenum="61"></pb><emph type="italics"></emph>ſcrupulorum; &amp; hanc particulam excoquent ad qualitatem oſque<lb></lb>cori puri. </s>

<s>Est ſi quidem inuenerint, ex priori grauitate<emph.end type="italics"></emph.end> 24. <emph type="italics"></emph>ſru<lb></lb>pulorum, deperiſſe nihil: pronuntiabunt aurum illud, hoc est, non <lb></lb>ſolum particulam illam excoctam, ſed etiam illud d quo fuit detra<lb></lb>cta, nec non &amp; illud quod remanſit post ſubtractionem eſſe vel fuiſ<lb></lb>ſe aurum prima qualitatis ſeu<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>partiam; vel quod idem est au<lb></lb>tum purum. </s>

<s>Si vero deprehenderine grauitatem diminutant, ver<lb></lb>bi gratia, nunc eſſe<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph>ſcrupulorum, quæ ante defæcationem fuit<emph.end type="italics"></emph.end> 24. <lb></lb><emph type="italics"></emph>dicturi ſunt aurum propoſitam<emph.end type="italics"></emph.end> 24. <emph type="italics"></emph>vnciarum fuiſſe<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph>partium &amp; <lb></lb>illud quod remanſit eſſe<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph>partium &amp; denique particulam expur<lb></lb>gatam nunc quidem eſſe aurum purum, fuiſſe vero particulam auri<emph.end type="italics"></emph.end><lb></lb>20. <emph type="italics"></emph>partium.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Et eodem modo pronuntiabunt de quibuſcunque alijs auri qualit<lb></lb>atibus, ſecundum partes auri puri, quas in qualibet maſſa auri in<lb></lb>uenerint, eaſque vigeſimas quartas tutius grauitatis, non magnitu<lb></lb>dinis. </s>

<s>Nam cum in hac comparatione qualitatum, ſeorſim habe a<lb></lb>tur ratio partium auri, &amp; ſeorſim metallorum alligatorum; manife<lb></lb>stum est ſi grauitas totius corporis intelligatur diuiſa in<emph.end type="italics"></emph.end> 24. <emph type="italics"></emph>partes <lb></lb>æquales, ex quibus<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph>ſint auri, duæ argenti, &amp; duæ æris; quamli<lb></lb>bet partem auri cum qualibet parte argenti &amp; æris collatam, magni<lb></lb>tudine eſſe minorem; &amp; ſimiliter partem argenti minorem parte <lb></lb>aris; propterea quod aurum omnia reliqua metalla ſuperet grauita<lb></lb>te quemadmodum &amp; argentum ipſum as, vt constat experienties<lb></lb>atque hinc constat quam apte ac conuenienter Aurifices vtantur <lb></lb>vocabulo partium. </s>

<s>hac enim ratione eodem numero exprimunt vnam <lb></lb>quamque qualitatem auri cuiuslibet maſſæ propoſitæ. </s>

<s>Sed nunc ad <lb></lb>ſecundum veniamus &amp; modum alligationis. </s>

<s>quem ydem obſeruant <lb></lb>breuiter adnotemus.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Inter varias autem &amp; multiplices auri compoſitiones quibus <lb></lb>cum alijs metallis alligari potest, eam retinuere aurifices, quam diu<lb></lb>turna experientia deprehenderunt omnibus alijs eſſe commodiorem, <lb></lb>eam nimirum quæ ab auri ſimilitudine vel minimum diſcedat; qua<lb></lb>lis eſt quæ ſolius argenti atque æris mixtione perficitur. </s>

<s>Et quidem <lb></lb>ſipartes auri excipias, aris atque argenti partes, quæ auro ſunt per<lb></lb>miſcendæ ſemper volunteſſe æquales in grauitate: propterea quod <lb></lb>eadem experientia Magiſtra didicerunt hunc eſse mixtionis modum <lb></lb>longe optimum.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Quando ergo aurifices volunt producere aurum cuiuſcunque <lb></lb>qualitatis, accipiunt tot partes auri puri æquales, quot partium fu<lb></lb>turum eſt aurum producendum, pauciores tamen partibus<emph.end type="italics"></emph.end> 24, &amp; <pb xlink:href="029/01/069.jpg" pagenum="62"></pb><emph type="italics"></emph>reliquas partes quæ d ſunt ad<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>explent argento &amp; ære, ſumendo <lb></lb>ex utroque metallo partes aquales in grauitate; atque bis rite inter ſe <lb></lb>permixtis componunt aurum deſideratæ qualitatis: eamque deno<lb></lb>minant à partibus auri puri in mixtione aſumptis. </s>

<s>Et quoniam <lb></lb>non prodiret tale prorſus quale facere intendunt, ſed paulo perfe<lb></lb>ctius; propterea quod auri quidem partes in mixtione maneant, ex <lb></lb>argenta vero &amp; are aliquid deperdatur, ſolent Aurifices tanto plus <lb></lb>miſcere argenti &amp; aris quantum perdi poſſe deprehenderunt.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Verum nostra intentio non est omnia quæ ad eiuſmodi mixtione <lb></lb>pertinent hoc loco exponere; ſed illud tantum vt receptum apud om<lb></lb>nes ad ferre voluimus, ex quo manifeste constat, qua metallorum <lb></lb>mixtio in ſingulis qualitatum generibus statuatur: qua quidem eſt <lb></lb>illa quam adduximus, nempe in auro<emph.end type="italics"></emph.end> 23, <emph type="italics"></emph>partium, partes<emph.end type="italics"></emph.end> 23, <emph type="italics"></emph>eſſe <lb></lb>auri puri &amp; reliquam qua deest ad<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>partes conſtare dimidia <lb></lb>parte argenti, &amp; dimidia aris in grauitate. </s>

<s>In auro vero<emph.end type="italics"></emph.end> 22, <emph type="italics"></emph>par<lb></lb>tium, auri eſſe<emph.end type="italics"></emph.end> 23, <emph type="italics"></emph>argenti vnam, &amp; æris vnam, ſie enim iterum <lb></lb>ſumma omnium partium est<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>eademque est ratio de reliquis ita <lb></lb>ut numerus partium auri, ſemper denominet qualitatem auri, &amp; <lb></lb>vna medietas reliquarum partium, quæ partibus auri deſunt ad <lb></lb>complendas partes<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>ſit argenti, &amp; reliqua medietas ſit æris. </s>

<s>hac <lb></lb>enim ſatis est ſuppoſuiſſe, ad nouum illud artificium, quo paulo poſt <lb></lb>inuestigaturi ſumus auri qualitatem ex ſola grauitate quam habet<lb></lb>in aere &amp; aqua, eamque qualitatem duplici via inuestigabimus, <lb></lb>vna per calculum, per tabellam altera: &amp; quia ad calculum ſpe<lb></lb>ctant ea, quæ ſuperius inuenimus, de grauitate metallorum huc re<lb></lb>ferenda cenſuimus quæ hic ſunt neceſſaria, cuiuſmodi ſunt auri, <lb></lb>argenti, atque æris grauitas, quam obtinent in aere, &amp; aqua, quo<lb></lb>quidem ita ſe habet vt ſequitur.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Auri puri grauitas, quæ in aere eſt<emph.end type="italics"></emph.end> 19, <emph type="italics"></emph>erit in aqua<emph.end type="italics"></emph.end> 18. </s></p><p type="main">

<s><emph type="italics"></emph>Argenti grauitas, quæ in aere eſt<emph.end type="italics"></emph.end> 31, <emph type="italics"></emph>erit in aqua<emph.end type="italics"></emph.end> 28. </s></p><p type="main">

<s><emph type="italics"></emph>Aeris grauitas; quæ in aere est<emph.end type="italics"></emph.end> 9, <emph type="italics"></emph>erit in aqua<emph.end type="italics"></emph.end> 8. </s></p><p type="head">

<s><emph type="italics"></emph>Item.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Aurum ad aquam ſe habet in grauitate vt<emph.end type="italics"></emph.end> 19, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 1. </s></p><p type="main">

<s><emph type="italics"></emph>Argentum ad aquam ſe habet in grauitate vt<emph.end type="italics"></emph.end> 31, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 3. </s></p><p type="main">

<s><emph type="italics"></emph>Aes ad aquam ſe habet in grauitate vt<emph.end type="italics"></emph.end> 9, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 1. </s></p><p type="main">

<s><emph type="italics"></emph>Ex quibus clariſſime colligitur, ſi aliquod corpus mistum con<lb></lb>stet partibus æqualibus argenti, &amp; aeris in grauitate, quantam <lb></lb>grauitatem habeat in aqua &amp; que ſit ratio in grauitate ipſius mi<lb></lb>sti ad aquam ſi enim grauitas aeris in aere ſit<emph.end type="italics"></emph.end> 9, <emph type="italics"></emph>eius grauitas in <lb></lb>aqua erit<emph.end type="italics"></emph.end> 8, <emph type="italics"></emph>&amp; ſi grauitas argenti in aere ſit quoque<emph.end type="italics"></emph.end> 9, <emph type="italics"></emph>erit eius gra<gap></gap><emph.end type="italics"></emph.end><pb xlink:href="029/01/070.jpg" pagenum="63"></pb><emph type="italics"></emph>vitas in àqua<emph.end type="italics"></emph.end> 8 4/11, <emph type="italics"></emph>est enim vt 9, ad<emph.end type="italics"></emph.end> 8 4/11 <emph type="italics"></emph>vt<emph.end type="italics"></emph.end> 31, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 28: <emph type="italics"></emph>Quare ſi <lb></lb>grauitas corporis miſti ex argento, &amp; ære iuncta mixtionem prædi<lb></lb>ctam, quæ etiam ſubintelligenda erit in ſequantibus, in aere fuerit<emph.end type="italics"></emph.end><lb></lb>18, <emph type="italics"></emph>erit id aqua<emph.end type="italics"></emph.end> 16<gap></gap> <emph type="italics"></emph>&amp; cònſequenter * grauitas aquæ magnitu<emph.end type="italics"></emph.end><arrow.to.target n="note54"></arrow.to.target><lb></lb><emph type="italics"></emph>dinem habentis æqualem <gap></gap> corpori misto<gap></gap><emph.end type="italics"></emph.end> <gap></gap>; <emph type="italics"></emph>quare corpus <lb></lb>miſtum ex argento &amp; are ad corpus aqueum eiuſdem magnitudi<lb></lb>nis, rationem habebit <gap></gap> grauitate vt<emph.end type="italics"></emph.end> 18, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 1 22/11, <emph type="italics"></emph>velvt<emph.end type="italics"></emph.end> 1, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end><lb></lb><gap></gap>, <emph type="italics"></emph>vel denique in numeris integris vt<emph.end type="italics"></emph.end> 299, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 29, <emph type="italics"></emph>omnium enim <lb></lb>iſtorum numerorum eadem eſt ratio.<emph.end type="italics"></emph.end></s></p><p type="margin">

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

<s><emph type="italics"></emph>Quibus ſic conſtitutis inuenietur qualitas auri cuiuſcumque hoc <lb></lb>modo. </s>

<s>Sit exemp.: gratia propoſita aliqua maſſa aurea, cuius gra<lb></lb>uitas in aere ſit vnc<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>&amp; oporteat invenire cuius qualitatis ſit ip<lb></lb>sum aurum. </s>

<s>Ponderetur ea maſſa in aqua &amp; habeat grauitatem <lb></lb>vnciarum<emph.end type="italics"></emph.end> 22 2<gap></gap>/5301, <emph type="italics"></emph>ergo * grauitas aquæ magnitudinem haben<emph.end type="italics"></emph.end><arrow.to.target n="note55"></arrow.to.target><lb></lb><emph type="italics"></emph>tis æqualem propoſitæ maſſæ erit vnc.<emph.end type="italics"></emph.end> 1 2431/5301. </s></p><p type="margin">

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

<s><emph type="italics"></emph>Deinde inueniatur grauitas aquæ magnitudine æqualis auro <lb></lb>puro<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>vnciarum: hoc eſt vt 19, ad 1, ita fiat<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>ad alium, nempe ad <lb></lb>vnciam<emph.end type="italics"></emph.end> 1 9/19, <emph type="italics"></emph>hic enim numerus erit grauitas illius aquæ.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Fiat denique ut<emph.end type="italics"></emph.end> 279, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 29, <emph type="italics"></emph>ita rurſus<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>vntiæ, ad alium, nume<lb></lb>rus enim quartus, nempe vnc.<emph.end type="italics"></emph.end> 2<gap></gap> <emph type="italics"></emph>erit grauitas aquæ, magnitu<lb></lb>dine æqualis corpori miſto ex argento &amp; ære, cuius grauitas est in <lb></lb>ærè vnc.<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>corpus enim ita miſtum, ad corpus aqueum eiuſdem ma<lb></lb>gnitudinis rationem habet in grauitate vt<emph.end type="italics"></emph.end> 279, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 29. </s></p><p type="main">

<s><emph type="italics"></emph>Atque ita habebuntur tres gravitates trium aquæ quantitatum, <lb></lb>quarum prima æquatur auro puro<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>unciarum, ſecunda maſſa <lb></lb>propoſitæ<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>vnciarum, &amp; reliqua corpori miſto ex argento &amp; ære<lb></lb>ſimiliter<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>vnciarum quæ quidem tres grauitates in numeris di<lb></lb>ſponantur eo ordine, quo ſequitur.<emph.end type="italics"></emph.end></s></p><p type="table">

<s>TABELLE WAR HIER</s></p><p type="main">

<s><emph type="italics"></emph>Deinde quæratur differentia inter primam &amp; tertiam aquæ gra<lb></lb>uitatem, quæ est vnc.<emph.end type="italics"></emph.end> 1 1227/5302, <emph type="italics"></emph>&amp; hæc differentia statuatur pro <lb></lb>primo proportionis termino, pro ſecundo termino ponatur grauitas <lb></lb><gap></gap>aſſæ propoſitæ idest vnc.<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>&amp; protertio denique termino ponatur <lb></lb>differentia inter ſecundam aquæ grauitatem &amp; tertiam, quæ est <lb></lb>vnc.<emph.end type="italics"></emph.end> 1 <gap></gap> <emph type="italics"></emph>quartus enim proportionalium terminus nempe<emph.end type="italics"></emph.end> 20, </s></p><pb xlink:href="029/01/071.jpg" pagenum="64"></pb><p type="table">

<s>TABELLE WAR HIER<lb></lb><emph type="italics"></emph>erit denominator qualitatis auri de qua quaritur quia ille termi<lb></lb>nus indicat partes auri pari in grauitate, qualibus maſſa propoſita <lb></lb>constat<emph.end type="italics"></emph.end> 24. <emph type="italics"></emph>Hoc autem demonstratum eſt prop.<emph.end type="italics"></emph.end> 19, <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Et quia inpropoſito exemplo ha partes, nempe vnc.<emph.end type="italics"></emph.end> 20, <emph type="italics"></emph>ſunt par<lb></lb>tes vigeſima quartæ<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>unciarum, quæ constituunt grauitatem to<lb></lb>tius maſſæ. </s>

<s>hinc fit quod eædem<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph>vnc. </s>

<s>immediate denominent au<lb></lb>rum propoſitum eße<emph.end type="italics"></emph.end> 20, <emph type="italics"></emph>partium. </s>

<s>Quando vero grauitas totius maſ<lb></lb>ſa non exprimitur per numerum<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>tunc opus erit inquirere quot <lb></lb>partes vigeſimas quartas totius grauitatis efficiat quartus ille pro<lb></lb>portionis terminus vt in ſequenti exemplo clarius apparebit.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Sit enim propoſita alia auri maſſa cuius grauitas in aere ſit<emph.end type="italics"></emph.end> 5301 <lb></lb><emph type="italics"></emph>in aqua vero<emph.end type="italics"></emph.end> 4988, <emph type="italics"></emph>ſi igitur hic numerus ſubtrahatur ex numero to<emph.end type="italics"></emph.end><lb></lb><arrow.to.target n="note56"></arrow.to.target><emph type="italics"></emph>tius grauitatis<emph.end type="italics"></emph.end> 5301, <emph type="italics"></emph>reliquus numerus<emph.end type="italics"></emph.end> 313, * <emph type="italics"></emph>erit grauitas aquæ <lb></lb>propoſitæ maſſæ magnitudine æqualis. </s>

<s>Inueniantur <expan abbr="quoq;">quoque</expan> duæ aliæ <lb></lb>grauitates aquæ, vna reſpondentis auro puro magnitudine, altera <lb></lb>corpori miſto ex argento &amp; ære, ita tamen vt grauitas tum auri pu<lb></lb>ri, tum corporis misti ſit eadem quæ maſſæ propoſitæ, non ſecus ac in <lb></lb>præcedenti exemplo factitatum est. </s>

<s>hoc eſt primo fiat vt<emph.end type="italics"></emph.end> 19, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 1, <emph type="italics"></emph>ita<emph.end type="italics"></emph.end><lb></lb>5301, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 279, <emph type="italics"></emph>hic enim numerus erit grauitas aquæ magnitudinem <lb></lb>habentis æqualem auro puro, cuius grauitas eſt<emph.end type="italics"></emph.end> 5301. <emph type="italics"></emph>Deinde fiat <lb></lb>vt<emph.end type="italics"></emph.end> 279, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 29, <emph type="italics"></emph>ita rurſum grauitas<emph.end type="italics"></emph.end> 5301, <emph type="italics"></emph>ad aliam, hac enim ratione <lb></lb>producetur numerus<emph.end type="italics"></emph.end> 551, <emph type="italics"></emph>debitus grauitati aquæ, magnitudine <lb></lb>aqualis corpori miſto ex argento &amp; ære grauitatem habenti eandem <lb></lb>cum eadem maſsa propoſita. </s>

<s>Atque hæ tres grauitates aquæ ſcriban<lb></lb>tur eo ordine quo ſupra; inuentiſque differentijs inter primam &amp;<emph.end type="italics"></emph.end></s></p><p type="margin">

<s><margin.target id="note56"></margin.target>5.<emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s></p><p type="table">

<s>TABELLE WAR HIER<lb></lb><emph type="italics"></emph>tertiam, nec non inter ſecundam &amp; tertiam, quæ ſunt<emph.end type="italics"></emph.end> 272, 238; <emph type="italics"></emph>sta<lb></lb>tuatur pro primo proportionis termino prior differentia<emph.end type="italics"></emph.end> 272, <emph type="italics"></emph>&amp; pro <lb></lb>tertio posterior<emph.end type="italics"></emph.end> 238. <emph type="italics"></emph>grauitas vero maſſæ propoſitæ<emph.end type="italics"></emph.end> 5301, <emph type="italics"></emph>ponatur <lb></lb>proſecundo termino, &amp; quæratur terminus quartus, qui in præ-<emph.end type="italics"></emph.end></s></p><p type="table">

<s>TABELLE WAR HIER<lb></lb><arrow.to.target n="note57"></arrow.to.target> <emph type="italics"></emph>ſenti examplo eſt<emph.end type="italics"></emph.end> 4638 51/<gap></gap>, <emph type="italics"></emph>is enim indicabit grauitatem auri <lb></lb>puri in maſſa propoſita. </s>

<s>Sed quoniam hæc grauitas non est expreſ<lb></lb>ſa in partibus vigeſimis quartis totius grauitatis, id quod ad ger-<emph.end type="italics"></emph.end></s></p><pb xlink:href="029/01/072.jpg" pagenum="65"></pb><p type="margin">

<s><margin.target id="note57"></margin.target>19. <lb></lb><emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>manam qualitatis auri pronuntiationem requiritur, vt ſupra mul<lb></lb>tis oſtendimus, reuocanda erit ad partes vigeſimas quartas hoc est <lb></lb>ad partes, qualium tota propoſita maſſa est<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>quod factu non est <lb></lb>difficile. </s>

<s>Nam ſi fiat vt tota grauitas maſſæ propoſitæ<emph.end type="italics"></emph.end> 5301, <emph type="italics"></emph>at gra<lb></lb>uitatem auri puri<emph.end type="italics"></emph.end> 4638 45/116, <emph type="italics"></emph>velvt<emph.end type="italics"></emph.end> 272, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 238, <emph type="italics"></emph>cum vtrobique <lb></lb>eadem ſit ratio ita<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>ad alium numerum. </s>

<s>procul dubio quartus nu<lb></lb>merus proportionalis, erit ille qui quæritur. </s>

<s>Est autem hic quar<lb></lb>tus numerus<emph.end type="italics"></emph.end> 21. <emph type="italics"></emph>Quare aurum maſſæ propoſitæ appellabitur par<lb></lb>tium<emph.end type="italics"></emph.end> 21. </s></p><p type="main">

<s><emph type="italics"></emph>Ex his igitur patet in inuenienda auri qualitate primum pro<lb></lb>portionis terminum<emph.end type="italics"></emph.end> 272, <emph type="italics"></emph>&amp; ſecundum<emph.end type="italics"></emph.end> 5301, <emph type="italics"></emph>perpetuo manere eoſ<lb></lb>dem, quia primus terminus est differentia inter grauitates primæ, <lb></lb>&amp; tertiæ aquæ, quæ nunquam mutantur, nam illæ aquæ magnitu<lb></lb>dine ſunt æquales altera auro puro, reliqua miſto ex argento &amp; ære, <lb></lb>quæ corpora aureum ſcilicet &amp; mistum ſemper ponuntur eiuſdem <lb></lb>grauitatis nempe<emph.end type="italics"></emph.end> 5301, <emph type="italics"></emph>Secundus vero terminus<emph.end type="italics"></emph.end> 5301, <emph type="italics"></emph>eſt grauitas <lb></lb>maſſæ propoſitæ, quæ ſi maior fuerit, vel minor, ad eam facile reuo<lb></lb>cabitur. </s>

<s>Vnde in posterum ſolum opus erit inuenire tertium pro<lb></lb>portionis terminum, hoc est differentiam inter grauitates ſecundæ <lb></lb>&amp; tertiæ aquæ.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Sed vt hoc etiam exemplo illustretur, proponatur aliqua maſſa <lb></lb>auri, cuius <expan abbr="inueſtigãda">inueſtiganda</expan> ſit qualitas, &amp; ſit ipſius maſſæ grauitas qui<lb></lb>dem in aere<emph.end type="italics"></emph.end> 837, <emph type="italics"></emph>in aqua vero<emph.end type="italics"></emph.end> 784, <emph type="italics"></emph>ergo * grauitas aquæ magnitu<emph.end type="italics"></emph.end><arrow.to.target n="note58"></arrow.to.target><lb></lb><emph type="italics"></emph>dinem habentis æqualem propoſitæ maſſæ erit<emph.end type="italics"></emph.end> 53, <emph type="italics"></emph>differentia enim <lb></lb>inter primam, &amp; ſecundam grauitatem est<emph.end type="italics"></emph.end> 53. </s></p><p type="margin">

<s><margin.target id="note58"></margin.target>5.<emph type="italics"></emph>huiu<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Ad inueniendum igitur tertium proportionis terminum manen<lb></lb>tibus primis duobus<emph.end type="italics"></emph.end> 272, 5301, <emph type="italics"></emph>hæc erit ratio. </s>

<s>Reuocetur primum <lb></lb>ptopoſitæ maſſæ grauitas<emph.end type="italics"></emph.end> 837, <emph type="italics"></emph>ad grauitatem<emph.end type="italics"></emph.end> 5301, <emph type="italics"></emph>hoc est intelli<lb></lb>gatur ipſa maſſa grauitatem babere<emph.end type="italics"></emph.end> 5301. <emph type="italics"></emph>deinde fiat vt<emph.end type="italics"></emph.end> 837, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 53, <lb></lb><emph type="italics"></emph>grauitatem videlicet aquæ ipſi maſſæ æqualis, ita<emph.end type="italics"></emph.end> 5301, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 335 2/3, <lb></lb><emph type="italics"></emph>ergo<emph.end type="italics"></emph.end> 335 2/3, <emph type="italics"></emph>erit grauitas aquæ magnitudinem habentis æqualem <lb></lb>aureæ maßæ, cuius grauitas<emph.end type="italics"></emph.end> 837, <emph type="italics"></emph>reuocata est ad grauitatem<emph.end type="italics"></emph.end> 5301; <lb></lb><emph type="italics"></emph>quare grauitas ſecundæ aquæ erit<emph.end type="italics"></emph.end> 335 2/1, <emph type="italics"></emph>&amp; conſequenter differen<emph.end type="italics"></emph.end></s></p><p type="table">

<s>TABELLE WAR HIER<lb></lb><emph type="italics"></emph>tia inter ipſam grauitatem ſecundæ aquæ &amp; grauitatem tertiæ<emph.end type="italics"></emph.end> 551, <lb></lb><emph type="italics"></emph>erit<emph.end type="italics"></emph.end> 215 1/1, <emph type="italics"></emph>ſed ipſa differentia ponitur pro tertio proportionis ter<lb></lb>mino; ergo<emph.end type="italics"></emph.end> 215 1/3, <emph type="italics"></emph>erit quæſitus terminus, nempe proportionis ter<emph.end type="italics"></emph.end></s></p><pb xlink:href="029/01/073.jpg" pagenum="66"></pb><p type="table">

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

<s>Quartus autem terminus<emph.end type="italics"></emph.end> 4196 25/116, <emph type="italics"></emph>indicabit grauitatem <lb></lb>auri puri, quod est in maſſa propoſita, eam tamen indicabit in par<lb></lb>tibus, qualibus tota maſſa constat<emph.end type="italics"></emph.end> 5301, <emph type="italics"></emph>quæ quidem grauitas vt <lb></lb>auri qualitatem indicet, reuocanda erit ad partes qualium totæ <lb></lb>maſſa propoſita est<emph.end type="italics"></emph.end> 24. <emph type="italics"></emph>ſi enim fiat vt<emph.end type="italics"></emph.end> 5301, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 4196<gap></gap>, <emph type="italics"></emph>ita<emph.end type="italics"></emph.end> 24, <lb></lb><emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 19, <emph type="italics"></emph>aurum propoſitæ maſſæ appellabitur partium<emph.end type="italics"></emph.end> 19. </s></p><p type="main">

<s><emph type="italics"></emph>Denique ſi quis hunc modum conferat cum illo, quem ſupra tradi<lb></lb>dimus, cum argentum explorauimus, quod miſtum in aurea corona <lb></lb>credebatur; is liquido intelliget hic nihil aliud acceſſiße, niſi quod <lb></lb>loco argenti, aſſumptum ſit corpus ex argento &amp; ære mistum, eo <lb></lb>quod hæc duo metalla tantum in alligationibus auri ſoleant adbibe<lb></lb>ri, vt diximus. </s>

<s>Quod ſi conſtaret plura alia aſſumpta eſſe, etiam in <lb></lb>quauis alia ratione, facile erit cuiuis ad ſimilitudinem huius, for<lb></lb>mare alium modum, ſed nos, ne longiores ſimus, ad vſum ſequentis <lb></lb>tabulæ nos conferamus, qua illis conſultum volumus qui minus in <lb></lb>præceptis Arithmeticis ſunt exercitati, velillis, qui alias ob cauſas <lb></lb>tabulis vti malunt, quam calculis.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Hæc tabula accommodata eſt primarie ad aurum vnius libræ, vt <lb></lb>apparet in <expan abbr="ſecũda">ſecunda</expan> ipſius columna in qua omnes numeri ſunt vnita<lb></lb>tes, reſpondentes ſingulis Denominatoribus qualitatum auri, a deno <lb></lb>minatore partium<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>vſque ad denominatorem qualitatis partis ò, <lb></lb>quamuis proprie loquendo nulla ſit qualitas auri partis nullius, quia <lb></lb>tunc non eſſet aurum, ſed mistum ex argento &amp; ære. </s>

<s>Hos denomi<lb></lb>natores auri omnes inuenies in prima columna ſub titulo qualitatis. </s>

<s><lb></lb>In columna vero ſub titulo misti placuit etiam deſcribere denomi<lb></lb>natores miſti ex argento &amp; ære, vt vnico intuitu appareat quot par<lb></lb>tes auri puri, &amp; quot partes misti ex argento &amp; ære contineantur <lb></lb>in ſingulis qualitatibus.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Porro in area tabulæ ſub titulo grauitatis auri in aqua poſita est <lb></lb>grauitas auri cuiuslibet qualitatis quam obtinet in aqua, quæ qua <lb></lb>ratione inueniatur, dicetur inferius vbi agetur de compoſitione <lb></lb>eiuſdem tabulæ.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Vſus eius ſunt duo, quorum alterum titulus indicat, nimirum vt <lb></lb>tabulæ beneficio reperiatur ex grauitate auri quam habet in aere &amp; <lb></lb>aqua, eius qualitas. </s>

<s>Alter vero est vt cognoſcatur grauitas in aqua, <lb></lb>quando vnà cum grauitate quam aliquod aurum habet in aere da<lb></lb>tur ipſius qualitas. </s>

<s>&amp; de hoc vſu cum ſit ſimplicior prius nobis erit <lb></lb>agendum.<emph.end type="italics"></emph.end></s></p><pb xlink:href="029/01/074.jpg" pagenum="67"></pb><p type="table">

<s>TABELLE WAR HIER</s></p><p type="table">

<s>TABELLE WAR HIER</s></p><pb xlink:href="029/01/075.jpg" pagenum="68"></pb><p type="main">

<s><emph type="italics"></emph>Quæratur exempli gratia quam habet grauitatem in aqua <expan abbr="aurũ">aurum</expan> <lb></lb>purum ſeu aurum<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>partium cuius grauitas in aere est lib.<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>Hæc <lb></lb>in ſupremo ordine è regione denominatoris partium 24, ſub titulo <lb></lb>grauitatis auri in aqua, datur vnc. </s>

<s>1<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>Scrup.<emph.end type="italics"></emph.end> 8, <emph type="italics"></emph>Gran.<emph.end type="italics"></emph.end> 20 172/1767, <lb></lb><emph type="italics"></emph>quæ fractio licet exprimi poſſit minoribus numeris <expan abbr="nẽpe">nempe</expan><emph.end type="italics"></emph.end> 4/19, <emph type="italics"></emph>libuit <lb></lb>tamen illam maiorem in tabula ponere, vt omnes fractiones totius <lb></lb>tabulæ eſſent eiuſdem denominationis, &amp; reſponderent denomina<lb></lb>toribus fractionum quæ habentur in tabella partis proportionalis.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Rurſum quæratur quam habet grauitatem in aqua aurum ite<lb></lb>rum vnius libræ, qualitatis vero<emph.end type="italics"></emph.end> 20, <emph type="italics"></emph>partium. </s>

<s>quam ſi in tabula <lb></lb>quæras, inuenies ſub eodem titulo è regione denominatoris<emph.end type="italics"></emph.end> 20, <emph type="italics"></emph>par<lb></lb>tium. </s>

<s>vnc. </s>

<s>11, Scrup. 6, Gran.<emph.end type="italics"></emph.end> 9 177/1767. <emph type="italics"></emph>eademque est ratio de re<lb></lb>liquis.<emph.end type="italics"></emph.end></s>

</p><p type="main">

<s><emph type="italics"></emph>Quando vero propoſitum aurum non est vnius Libræ; tunc opus <lb></lb>erit ratiocinatione proportionis, in qua pro primo termino ponatur <lb></lb>vna libra auri propoſitæ qualitatis, pro ſecundo termino, grauitas <lb></lb>eidem reſpondens in aqua quam tabula exhibet, pro tertio vero ter<lb></lb>mino collocetur vera grauitas auri propoſiti. </s>

<s>Quartus enim ter<lb></lb>minus exhibebit grauitatem ipſius auri in aqua. </s>

<s>Vt ſi propoſitum <lb></lb><expan abbr="aurũ">aurum</expan> ſit trium lib.

qualitatis vero 18, partium. </s>

<s>fiat vt lib.

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

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


<s>Et ſic de alijs.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Quod vero ad priorem vſum attinet, is perſimilis est præcedenti, <lb></lb>&amp; æquè facilis quando grauitas auri quam in aere &amp; aqua habet, in <lb></lb>tabula reperitur precise. </s>

<s>Nam ſi proponatur exemp. </s>

<s>gratia aurum <lb></lb>vnius lib.

habens in aqua <expan abbr="grauitasẽ">grauitatem</expan> vnc.<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>Scr.<emph.end type="italics"></emph.end> 7, <emph type="italics"></emph>Gra.<emph.end type="italics"></emph.end> 14 1152/1767, <lb></lb><emph type="italics"></emph>quoniam hæc grauitas reperitur in tabula è regione qualitatis auri<emph.end type="italics"></emph.end><lb></lb>22, <emph type="italics"></emph>partium; manifeſtum eſt totidem partium eſſe <expan abbr="aurũ">aurum</expan> propoſitum.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Quando vero grauitas auri in aere quiaem eſt vnius lib.

in aqua <lb></lb>vero grauitatem habet, quæ in tabula non reperitur, indicium erit <lb></lb>aurum propoſitum non eſſe aliquot partium præcisè, ſed annexam <lb></lb>habere aliquam fractionem, quæ per partem proportionalem inue<lb></lb>nietur hoc modo.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Proponatur aurum vnius libræ in aqua habens grauitatem vnc. </s>

<s><lb></lb>11. Scrup. 6, Gran.<emph.end type="italics"></emph.end> 16 264/1767, <emph type="italics"></emph>qualis in tabula non reperitur Gra<lb></lb>uitas enim proxime maior est vnc. </s>


<s>1<emph.end type="italics"></emph.end> 1. <emph type="italics"></emph>Scrup. 6, Gran.<emph.end type="italics"></emph.end> 23 1551/1767, <lb></lb><emph type="italics"></emph>reſpondens auro 21, partium, &amp; grauitas proxime minor eſt vnc.<emph.end type="italics"></emph.end> 11, <lb></lb><emph type="italics"></emph>Scrup. 6, Gran.<emph.end type="italics"></emph.end> 9 177/1767, <emph type="italics"></emph><expan abbr="earumq;">earumque</expan> differentia eſt Gran.<emph.end type="italics"></emph.end> 14 1374/1767, <lb></lb><emph type="italics"></emph>quem admodum &amp; inter quaſcunque duas alias grauitates proxi<emph.end type="italics"></emph.end><pb xlink:href="029/01/076.jpg" pagenum="69"></pb><emph type="italics"></emph>mas eadem eſt differentia, propterea quod omnes grauitates in tabu<lb></lb>la procedunt per æqualem exceſſum, vel defectum, vt inferius de<lb></lb>monſtrabitur. </s>


<s>Inueniatur quoque differentia inter eandem gra<lb></lb>uitatem proxime minorem &amp; inter grauitatem auri propoſiti quam <lb></lb>habet in aqua, quæ <expan abbr="quidẽ">quidem</expan> eſt Gran.<emph.end type="italics"></emph.end> 7 687/1767, <emph type="italics"></emph>&amp; fiat vt<emph.end type="italics"></emph.end> 14 1374/1767, <lb></lb><emph type="italics"></emph>ad 1, ita<emph.end type="italics"></emph.end> 7 687/1767, <emph type="italics"></emph>ad alium numerum &amp; inuenietur hæc fractio<emph.end type="italics"></emph.end><lb></lb>1/2, <emph type="italics"></emph>eademque adijcienda erit ad denominatorem 20, partium, vt <lb></lb>componatur totus denominator auri propoſiti partium<emph.end type="italics"></emph.end> 20 1/2, <emph type="italics"></emph>&amp; eo<lb></lb>dem modo inueniendus erit denominator <expan abbr="cuiuſcunq;">cuiuſcunque</expan> alterius auri, <lb></lb>cuius grauitas in aqua, in tabula non reperitur.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Ceterum qui volet contentus eſſe partibus vigeſimis quartis de<lb></lb>nominatorum auri, is multo breuius aſſequetur quod quæritur, per <lb></lb>tabellam partis proportionalis. </s>

<s>illic enim vnico ingreſſu offendet <lb></lb>partem proportionalem, quam quærit, vt in eodem exemplo apparet, <lb></lb>in quo differentia grauitatum auri erat Gran.<emph.end type="italics"></emph.end> 7 687/1767, <emph type="italics"></emph>quæ in <lb></lb>tabella partis proportionalis habetur præcisè è regione particularum<emph.end type="italics"></emph.end><lb></lb>12. <emph type="italics"></emph>Vnde concluditur, denominatorem auri propoſiti eſſe partium<emph.end type="italics"></emph.end><lb></lb>20 12/24, <emph type="italics"></emph>vel quod idem est partium<emph.end type="italics"></emph.end> 20 1/2, <emph type="italics"></emph>vt prius. </s>

<s>Quando vero <lb></lb>differentia grauitatum in tabella partis proportionalis non habetur <lb></lb>præcisè. </s>

<s>accipiatur alia ipſi propinquior &amp; particula illi in latere <lb></lb>reſpondens addatur denominatori auri ex primaria tabula extra<lb></lb>cti. </s>

<s>ſic enim ſaltem non errabitur in vna particula vigeſimaquarta <lb></lb>vnius partis denominatoris auri.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Denique ſi proponatur aurum non vnius libræ ſed vel plurium, <lb></lb>vel ſolum aliquot vnciarum. </s>

<s>Reducenda erit eius grauitas quam <lb></lb>habet in aqua, ad grauitatem quam haberet ſi eßxset vnius libræ, id <lb></lb>quod abſoluetur per proportionis ratiocinationem, ſi pro termino <lb></lb>primo ponatur vera grauitas auri propoſiti, pro ſecundo, eiuſdem <lb></lb>grauitas in aqua, &amp; pro tertio lib.

1, quartus enim terminus indica<lb></lb>bit grauitatem in aqua reſpondentem vni libræ auri propoſiti. </s>

<s>&amp; <lb></lb>hac inuenta reliqua expedientur vt prius.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Exemplum huius caſus hic non affero, quod per ſe res ſit clara. </s>

<s><lb></lb>Sed illud tantum obiter aduertere placet, quod videtur pertinere <lb></lb>ad commodiorem vſum tabulæ, videlicet vt ys in caſibus in quibus <lb></lb>nece&lt;32&gt;arius est calculus, fractiones granorum ommittantur quando <lb></lb>minus valent quam<emph.end type="italics"></emph.end> 1/2, <emph type="italics"></emph>&amp; quando valent plus, eorum loco, addatur <lb></lb>vnum granum reliquis granis, &amp; ſi quando accidat hinc procreari <lb></lb>grana<emph.end type="italics"></emph.end> 24. <emph type="italics"></emph>tunc etiam grana ommittantur addita prius vnitate ad <lb></lb>ſcrupula in tabula inuenta. </s>

<s>hac enim ratione calculus erit expedi<lb></lb>tior &amp; error qui hinc oborietur erit inſenſibilis.<emph.end type="italics"></emph.end></s></p><pb xlink:href="029/01/077.jpg" pagenum="70"></pb><p type="head">

<s>Compoſitio eiuſdem tabulæ.</s></p><p type="main">

<s><emph type="italics"></emph>Si ea quæ hactenus ſunt dicta rectè intelligantur liquido appare<lb></lb>bit compoſitionem tabulæ in eo conſistere, vt inueniatur grauitas <lb></lb>quam aurum cuiuſuis qualitatis habet in aqua, hoc eſt ſi intelligan<lb></lb>tur propoſitæ plures maſſæ auri, ſingulæ ſingularum librarum, &amp; <lb></lb>alia ſit qualitatis<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>partium alia<emph.end type="italics"></emph.end> 23, <emph type="italics"></emph>alia<emph.end type="italics"></emph.end> 22, <emph type="italics"></emph>&amp;c. </s>

<s>quanta ſit gra<lb></lb>uitas vniuſcuiuſque in aqua, id quod hoc extremo loco inuestigare <lb></lb>docebimus.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Et primum ſit propoſitum aurum purum, ſeu<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>partium. </s>

<s>quo<lb></lb>niam igitur grauitas auri in aere, ad grauitatem eiuſdem in aqua <lb></lb>ſe habet vt 19, ad<emph.end type="italics"></emph.end> 18, <emph type="italics"></emph>fiat vt<emph.end type="italics"></emph.end> 19, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 18, <emph type="italics"></emph>ita lib.<emph.end type="italics"></emph.end> 1, <emph type="italics"></emph>auri puri ad aliam <lb></lb>grauitatem nempe lib.<emph.end type="italics"></emph.end> 1<gap></gap>/19, <emph type="italics"></emph>quæ grauitas ad vncias, ſcrupula, &amp; <lb></lb>grana reuocata valet vnc.<emph.end type="italics"></emph.end> 11, <emph type="italics"></emph>Scrup.8. Gran.<emph.end type="italics"></emph.end> 20 4/19, <emph type="italics"></emph>atque hæc est <lb></lb>grauitas auri puri in aqua, quam in tabula è regione denominatoris<emph.end type="italics"></emph.end><lb></lb>24, <emph type="italics"></emph>poſuimus; fractione excepta cuius loco ſubſtituta est fractio<emph.end type="italics"></emph.end><lb></lb>372/1767, <emph type="italics"></emph>propter cauſam ſuperius allatam.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Sit deinde propoſitum quod vis aliud corpus aureum vnius libræ, <lb></lb>ſitque exemp. </s>

<s>gratia illud aurum<emph.end type="italics"></emph.end> 20, <emph type="italics"></emph>partium, patet igitur ex defi<lb></lb>nitione qualitatis, ex<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>ſemiunciis totius grauitatis,<emph.end type="italics"></emph.end> 20, <emph type="italics"></emph>ſemiuncias <lb></lb>eſſe auri puri, duas argenti, &amp; reliquas duas æris &amp; quoniam gra<lb></lb>uitas misti in aere, ad grauitatem eiuſdem in aqua rationem habet <lb></lb>vt<emph.end type="italics"></emph.end> 279, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 250, <emph type="italics"></emph>vt ex iam dictis patet, fiat vt<emph.end type="italics"></emph.end> 279, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 250, <emph type="italics"></emph>ita qua<lb></lb>tuor ſemiunciæ, vel potius <gap></gap>, vnciæ misti quod componit qualitatem <lb></lb>auri<emph.end type="italics"></emph.end> 20, <emph type="italics"></emph>partium, paulo ante propoſitam ad alias vncias. </s>

<s>inuenien<lb></lb>tur enim pro grauitate illius miſti in aqua vnc.<emph.end type="italics"></emph.end> 1 221/274. <emph type="italics"></emph>Est autem <lb></lb>grauitas auri puri<emph.end type="italics"></emph.end> 20, <emph type="italics"></emph>ſemiunciarum vel<emph.end type="italics"></emph.end> 10, <emph type="italics"></emph>vnciarum in aqua vnc.<emph.end type="italics"></emph.end><lb></lb>9 9/19, <emph type="italics"></emph>eo quod ita ſe habeant<emph.end type="italics"></emph.end> 10, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 9 9/29, <emph type="italics"></emph>vt 19, ad<emph.end type="italics"></emph.end> 18, <emph type="italics"></emph>Quare ſi hæ <lb></lb>duæ grauitates inuentæ colligantur in vnam ſummam, inueniemus <lb></lb>totam maſſam auri propoſitam, cuius grauitas in aere ponebatur lib. <lb></lb>1, in aqua babere grauitatem vnc.<emph.end type="italics"></emph.end> 11 1409/5301. <emph type="italics"></emph>&amp; facta reductione <lb></lb>fractionis ad ſcrupula, &amp; grana, vnc. </s>


<s>1<emph.end type="italics"></emph.end> 1, <emph type="italics"></emph>Scrup.<emph.end type="italics"></emph.end>6. <emph type="italics"></emph>Gran.<emph.end type="italics"></emph.end> 9 177/1767, <lb></lb><emph type="italics"></emph>vt videre eſt in tabula è regione denominatoris<emph.end type="italics"></emph.end> 20, <emph type="italics"></emph>partium. </s>

<s>Atque <lb></lb>eodem modo deprehendentur grauitates auri in aqua <expan abbr="quarumcunq;">quarumcunque</expan> <lb></lb>aliarum qualitatum.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Quia vero permolestum videri poſſet omnes grauitates totius <lb></lb>tabulæ hac via eruere; obſeruari quidem poterit prædicta Methodus <lb></lb>quando ſeorſim inuestiganda fuerit alicuius auri grauitas in aqua, <lb></lb>in <expan abbr="cõpoſitione">compoſitione</expan> vero tabulæ ſic fortaſſis <expan abbr="cõpendioſius">compendioſius</expan> quis proceſſerit.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Equidem, cum huius tabulæ conſtructionem diligentius mecum<emph.end type="italics"></emph.end><pb xlink:href="029/01/078.jpg" pagenum="71"></pb><emph type="italics"></emph>pertracto, video grauitates illas, quæ in eius area deſcriptæ ſunt ne<lb></lb>ceſſario eadem differentia procedere, ſicut &amp; denominatores quali<lb></lb>tatum eadem differentia procedunt; atque adeo differentiam illam <lb></lb>eſſe eam, qua grauitas ſemiunciæ auri puri in aqua ſuperat in aqua <lb></lb><expan abbr="grauitatẽ">grauitatem</expan> ſemiunciæ miſti, quæ <expan abbr="quidẽ">quidem</expan> differentia eſt Gra.<emph.end type="italics"></emph.end> 14 1174/1767. <lb></lb><emph type="italics"></emph>Conſiderentur enim quæcunque duæ grauitates proximæ in tabula <lb></lb>expre&lt;32&gt;æ, vt exemp. </s>

<s>gratia. </s>

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

<s>Quo<lb></lb>niam igitur in auro<emph.end type="italics"></emph.end> 20, <emph type="italics"></emph>partium ſunt auri puri<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph>ſemiunciæ, misti <lb></lb>vero<emph.end type="italics"></emph.end> 4, <emph type="italics"></emph>&amp; in auro<emph.end type="italics"></emph.end> 29, <emph type="italics"></emph>partium ſunt auri puri<emph.end type="italics"></emph.end> 19. <emph type="italics"></emph>ſemiunciæ, misti <lb></lb>vero<emph.end type="italics"></emph.end> 5, <emph type="italics"></emph>erit in auro<emph.end type="italics"></emph.end> 20. <emph type="italics"></emph><expan abbr="partiũ">partium</expan> vna ſemiuncia auri puri pluſquam in <lb></lb>auro<emph.end type="italics"></emph.end> 19, <emph type="italics"></emph>partium, in auro autem 19, partium erit vna ſemiuncia <lb></lb>misti plus quam in auro<emph.end type="italics"></emph.end> 20, <emph type="italics"></emph>partium; quare grauitas auri<emph.end type="italics"></emph.end> 20, <emph type="italics"></emph>par<lb></lb>tium in aqua ſuperabit in aqua grauitatem auri<emph.end type="italics"></emph.end> 19, <emph type="italics"></emph>partium gra<lb></lb>uitate, qua ſemiuncia auri puri ſuperat ſemiunciam miſti. </s>

<s>Quod <lb></lb>erat demonſtrandum. </s>

<s>Et eadem est ratio de alijs grauitatibus, non <lb></lb>ſolum quæ in hac tabula deſcribuntur, ſed etiam de illis, quæ deſcri<lb></lb>bentur in alijs, copioſioribus, in quibus videlicet denominatores non <lb></lb>eſſent partes integræ, ſed partes partiam; dummodo etiam illæ partes <lb></lb>per vnam eandemque differentiam progrederentur.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Quibus in hunc modum præ ostenſis, ſi construenda fuerit tabula <lb></lb>per continuam additionem eiuſdem numeri, ſic erit progrediendum. </s>

<s><lb></lb>Primo inuenienda erit grauitas quam habet ſemiuncia auri puri in <lb></lb>aqua, quæ inuenietur ſi fiat vt<emph.end type="italics"></emph.end> 19, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 18, <emph type="italics"></emph>ita ſemiuncia ad alium, <lb></lb>numenum qui ſit vnc.<emph.end type="italics"></emph.end> 9/19, <emph type="italics"></emph>is enim dabit grauitatem quæſitam, quæ <lb></lb>valet ſcrup.<emph.end type="italics"></emph.end>1 1, <emph type="italics"></emph>Gran.<emph.end type="italics"></emph.end> 8 16/19. </s></p><p type="main">

<s><emph type="italics"></emph>Secundo quærenda est grauitas ſemiuncie misti in aqua, quæ ha<lb></lb>bebitur ſi fiat vt<emph.end type="italics"></emph.end> 279, <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 250, <emph type="italics"></emph>ita ſemiuncia, ad alium numerum, qui <lb></lb>ſit vnc.<emph.end type="italics"></emph.end> 125/229, <emph type="italics"></emph>is enim dabit grauitatem quæſitam, quæ reducta ad <lb></lb>ſcrupula, &amp; grana valet ſcrup.<emph.end type="italics"></emph.end> 10, <emph type="italics"></emph>Gran.<emph.end type="italics"></emph.end> 18 2/3. </s></p><p type="main">

<s><emph type="italics"></emph>Tertio exploranda est differentia inter duas grauitates proximè <lb></lb>inuentas, quam per ſubtractionem inuenies Gran.<emph.end type="italics"></emph.end> 14 452/589, <emph type="italics"></emph>cuius <lb></lb>tamen fractio reducta eſt ad partes<emph.end type="italics"></emph.end> 1767, <emph type="italics"></emph>nempe ad<emph.end type="italics"></emph.end> 1374/1767, <emph type="italics"></emph>propter <lb></lb>tabellam partis proportionalis.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Postremo inuestiganda erit grauitas in aqua vnius libræ misti, <lb></lb>quæ inuenietur ſi grauitas ſecundo loco reperta per<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>multiplice<lb></lb>tur, productus enim numerus vnc.<emph.end type="italics"></emph.end> 10, <emph type="italics"></emph>ſcrup. </s>

<s>18, Gran.<emph.end type="italics"></emph.end> 969/1767, <emph type="italics"></emph>da<lb></lb>bit quæſitam grauitatem. </s>

<s>Qua in calce tabulæ deſcripta, compo<lb></lb>nentur reliquæ grauitates omnes per continuam additionem diffe<lb></lb>rentiæ tertio loco inuentæ. </s>

<s>Si enim addatur ad grauitatem auri <lb></lb>partis o, idest ad grauitatem misti vnius libræ in aqua, componetur<emph.end type="italics"></emph.end><pb xlink:href="029/01/079.jpg" pagenum="72"></pb><emph type="italics"></emph>grauitas auri<emph.end type="italics"></emph.end> 1, <emph type="italics"></emph>partis. </s>

<s>addita vero ad grauitatem 1, partis, procrea<lb></lb>bit grauitatem<emph.end type="italics"></emph.end> 2, <emph type="italics"></emph>partium, &amp;c. </s>

<s>propter rationem quam paulo ante <lb></lb>aperuimus.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Hoc eodem artificio compoſita est quoque tabella partis propor<lb></lb>tionalis, primo enim inuenta eſt vigeſima quarta pars differentiæ <lb></lb>ſecundum quam tabula progreditur quam ſupra inuenimus eſſe <lb></lb>Gran.<emph.end type="italics"></emph.end> 14 1374/1707, <emph type="italics"></emph>cuius pars vigeſima quarta est<emph.end type="italics"></emph.end> 1072/1707, <emph type="italics"></emph>deinde <lb></lb>hanc particulam addidimus primum ſibi ipſi, &amp; produximus diffe<lb></lb>rentiam<emph.end type="italics"></emph.end> 2, <emph type="italics"></emph>particularum Gran.<emph.end type="italics"></emph.end> 1 409/1707 <emph type="italics"></emph>&amp; huic differentiæ <expan abbr="iterũ">iterum</expan> <lb></lb>adiecimus eandem particulam, &amp; inuenimus pro tribus particulis <lb></lb>gran.<emph.end type="italics"></emph.end> 1 1497/1767. <emph type="italics"></emph>&amp; ita deinceps progreſſi ſumus vſque ad differen<lb></lb>tiam Gran.<emph.end type="italics"></emph.end> 14 1374/1767, <emph type="italics"></emph>quæ reſpondet<emph.end type="italics"></emph.end> 24, <emph type="italics"></emph>particulis, ſeu differentiæ, <lb></lb>ſecundum quam tabula progreditur.<emph.end type="italics"></emph.end><lb></lb><emph type="italics"></emph>FINIS.<emph.end type="italics"></emph.end></s></p><p type="head">

<s>ERRATA SIC CORRIGE.</s></p><p type="table">

<s>TABELLE WAR HIER</s></p><p type="main">

<s><emph type="italics"></emph><expan abbr="Contractiorẽ">Contractiorem</expan> <expan abbr="quoq.">quoque</expan> ad inuenies dimidij pedis <expan abbr="menſurã">menſuram</expan> in margine pag. </s>

<s>34. <expan abbr="appoſitã">appoſitam</expan>, <lb></lb>quia madefacta papyrus, <expan abbr="dũ">dum</expan> imprimeretur recipiebat <expan abbr="verã">veram</expan> <expan abbr="menſurã">menſuram</expan>, exſiccata <lb></lb>breuiorem reddidit. </s>

<s>Ita que ſi quarta decima pars vnius vncia addatur ipſi men <lb></lb>ſura componetur dimidij pedis menſura; vel ſi ipſa menſura duplicetur, &amp; ei ad<lb></lb>datur ſeptima pars vnius vncia fiet menſura vnius pedis, ad cuius rationem em <lb></lb>nes calculi in tabulas optime reſpondebunt.<emph.end type="italics"></emph.end></s></p><p type="main">

<s><emph type="italics"></emph>Denique pagina<emph.end type="italics"></emph.end> 38. <emph type="italics"></emph>&amp;<emph.end type="italics"></emph.end> 39. <emph type="italics"></emph>tabularum præpoſterus eſt ordo. </s>

<s>ex quo tamen nihil sr<lb></lb>reris ſequitur ſi tituli recte accipiantur.<emph.end type="italics"></emph.end></s></p>			</chap>		</body>		<back></back>	</text></archimedes>
author	Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date	Thu, 02 May 2013 11:08:12 +0200
parents	22d6a63640c6
children