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Appendix Version 2009-02-14
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Thu, 02 May 2013 11:12:52 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0" encoding="utf-8"?><echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" xmlns:dcterms="http://purl.org/dc/terms" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.0RC"> <metadata> <dcterms:identifier>ECHO:FQPFR8XP.xml</dcterms:identifier> <dcterms:creator identifier="GND:123521041">Ghetaldi, Marino</dcterms:creator> <dcterms:title xml:lang="la">Marini Ghetaldi Promotus Archimedis seu de variis corporum generibus gravitate et magnitudine comparatis</dcterms:title> <dcterms:date xsi:type="dcterms:W3CDTF">1603</dcterms:date> <dcterms:language xsi:type="dcterms:ISO639-3">lat</dcterms:language> <dcterms:rights>CC-BY-SA</dcterms:rights> <dcterms:license xlink:href="http://creativecommons.org/licenses/by-sa/3.0/">CC-BY-SA</dcterms:license> <dcterms:rightsHolder xlink:href="http://www.mpiwg-berlin.mpg.de">Max Planck Institute for the History of Science, Library</dcterms:rightsHolder> <parameters>despecs = 1.1.2</parameters> <log>valid, but has some problems around line 1700 (div and p)</log> </metadata> <text xml:lang="la" type="free"> <div xml:id="echoid-div1" type="section" level="1" n="1"><pb file="0001" n="1"/> <pb file="0002" n="2"/> <pb file="0003" n="3"/> <pb file="0004" n="4"/> <handwritten/> <handwritten/> <handwritten/> <pb file="0005" n="5"/> </div> <div xml:id="echoid-div2" type="section" level="1" n="2"> <head xml:id="echoid-head1" xml:space="preserve">MARINI GHETALDI <lb/>PATRICII</head> <head xml:id="echoid-head2" xml:space="preserve">RAGVSINI <lb/>PROMOTVS ARCHIMEDIS <lb/>SEV</head> <head xml:id="echoid-head3" xml:space="preserve">De varijs corporum generibus <lb/>grauitate & magnitudine <lb/>comparatis.</head> <figure> <image file="0005-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0005-01"/> </figure> </div> <div xml:id="echoid-div3" type="section" level="1" n="3"> <head xml:id="echoid-head4" xml:space="preserve">ROM AE, <lb/>Apud Aloyſium Zannettum. MDCIII. <lb/>SVPERIORV M PERMISSV.</head> <pb file="0006" n="6"/> <pb file="0007" n="7"/> </div> <div xml:id="echoid-div4" type="section" level="1" n="4"> <head xml:id="echoid-head5" xml:space="preserve">REVERENDISSIMO <lb/>SERAPHINO OLIVARIO <lb/>RAZZALIO. <lb/>PATRIARCH AE <lb/>ALEXANDRINO.</head> <head xml:id="echoid-head6" xml:space="preserve">Marinus Ghetaldus. S. P. D.</head> <p> <s xml:id="echoid-s1" xml:space="preserve">EGREGIA ſanè Reuerendiſsime <lb/>PRÆSVL quod probe noſti, vete-<lb/>rum ſapientum ſoelicitas fuit. </s> <s xml:id="echoid-s2" xml:space="preserve">Eam <lb/>enim cum ingenij præſtantia, tum <lb/>prærogatiua temporis laudem occu-<lb/>parunt, quã vel ſperare poſterioribus <lb/>temerariũ ſit. </s> <s xml:id="echoid-s3" xml:space="preserve">Et vero illis non ſolum nos plurimum de-<lb/>bemus, quod plurima ipſi perfecere: </s> <s xml:id="echoid-s4" xml:space="preserve">verum etiam quod <lb/>quædam quaſi fundamenta iecere, quibus dum rerum <lb/>nouarũ molem conamur imponere, nos quoque experi-<lb/>r<unsure/>i noſtras vires, exercere induſtriam, remque ſapientiæ <lb/>publicam amplificare poſsimus. </s> <s xml:id="echoid-s5" xml:space="preserve">Quo in genere magno-<lb/>rum ego virorum ſtudium potius quam gloriam æmu-<lb/>latus ſuper vnum ex Archimedeis fundamentis, de ra-<lb/>tione, qua varia corporum genera inter ſe grauitate & </s> <s xml:id="echoid-s6" xml:space="preserve"><lb/>magnitudine comparantur, fabricatus nonnulla ſum:</s> <s xml:id="echoid-s7" xml:space="preserve"> <pb file="0008" n="8"/> quæ nunc omnium oculis expoſiturus, vt eam ſuſtineã, <lb/>perſonam, quam ſemper recuſaui, patrocinium huiuſ-<lb/>modi quærendum mihi exiſtimaui, quod & </s> <s xml:id="echoid-s8" xml:space="preserve">imbecillita <lb/>tem meam contra obtrectatorum, ſi qui forte exiſterent, <lb/>calumnias ſuſtineret; </s> <s xml:id="echoid-s9" xml:space="preserve">& </s> <s xml:id="echoid-s10" xml:space="preserve">imminentem famæ, exiſtimatio <lb/>niſque iacturam auerteret. </s> <s xml:id="echoid-s11" xml:space="preserve">Vnus igitur tu occuriſti SE-<lb/>RAPHINE qui & </s> <s xml:id="echoid-s12" xml:space="preserve">cõmodiſsime mihi patrocinari poſ-<lb/>ſes, & </s> <s xml:id="echoid-s13" xml:space="preserve">quodam quaſi iure deberes. </s> <s xml:id="echoid-s14" xml:space="preserve">Cum enim tu me ad <lb/>emittendum id opus hortatu tuo compuleris, videbatur <lb/>quodam iure ad tuam fidem eius tutela pertinere. </s> <s xml:id="echoid-s15" xml:space="preserve">Tuq; <lb/></s> <s xml:id="echoid-s16" xml:space="preserve">is es, quem non modo rudes, ſed etiam docti ſuſpiciunt. </s> <s xml:id="echoid-s17" xml:space="preserve"><lb/>Habet noſtra hæc ætas, quos admiretur; </s> <s xml:id="echoid-s18" xml:space="preserve">habet quos ex-<lb/>tollat præclaros viros, ſed quos tibi anteponat, non facile <lb/>inueniet. </s> <s xml:id="echoid-s19" xml:space="preserve">Degis ea in Vrbe, quæ laudis mediocritatem <lb/>vel nunquam agnouit, vel ſemper contempſit: </s> <s xml:id="echoid-s20" xml:space="preserve">neque in <lb/>tanta maieſtate, tua deficit virtus, ſed bono in lumine po <lb/>ſita collucet magis. </s> <s xml:id="echoid-s21" xml:space="preserve">In primis enim tua vitæ integritas <lb/>eiuſmodi eſt, vt non contenta domeſtico præconio latiſ-<lb/>ſime peruagetur. </s> <s xml:id="echoid-s22" xml:space="preserve">Habent omnes quod prædicent, & </s> <s xml:id="echoid-s23" xml:space="preserve">imi <lb/>tentur; </s> <s xml:id="echoid-s24" xml:space="preserve">habet quod excipiat gratiſsima memoria poſte-<lb/>ritas vniuerſa. </s> <s xml:id="echoid-s25" xml:space="preserve">Doctrinæ vero ea excellentia es, vt ea ſatis <lb/>omnibus clarus, & </s> <s xml:id="echoid-s26" xml:space="preserve">illuſtris, non ſatis tibi, tecum aſsidue <lb/>certes. </s> <s xml:id="echoid-s27" xml:space="preserve">nec mirum, qui vel à primis incunabulis ad ſum-<lb/>ma contendebas, ſi prouecta iam ætate vix habeas, quo <lb/>altius contendas. </s> <s xml:id="echoid-s28" xml:space="preserve">Præclarum ſanè & </s> <s xml:id="echoid-s29" xml:space="preserve">eximium vno do-<lb/>ctrinæ genere; </s> <s xml:id="echoid-s30" xml:space="preserve">ſublime, atque admirabile multiplici ex-<lb/>cellere. </s> <s xml:id="echoid-s31" xml:space="preserve">Tu vero in omni genere laudem egregiam aſſe-<lb/>cutus es abſolutam videlicet tibi gloriam propoſitam <lb/>voluiſti; </s> <s xml:id="echoid-s32" xml:space="preserve">quiq; </s> <s xml:id="echoid-s33" xml:space="preserve">intelligebas hominem ad honeſta omnia <pb file="0009" n="9"/> genitum, nullam tibi rerum glorioſarum partem con-<lb/>temnendam putaſti. </s> <s xml:id="echoid-s34" xml:space="preserve">Placet incredibili rerum humana-<lb/>rum vſui diuinarum rerum cognitionem adiungere; </s> <s xml:id="echoid-s35" xml:space="preserve">vt <lb/>habeat animus à caducis ad æterna, ſe conferendo, vnde <lb/>oblectamentum capiat, & </s> <s xml:id="echoid-s36" xml:space="preserve">admirationem. </s> <s xml:id="echoid-s37" xml:space="preserve">Philoſophiam <lb/>ita tenes, vt qui in maximis negocijs aſsidue verſatus es, <lb/>videaris ſemper fuiſſe otioſus. </s> <s xml:id="echoid-s38" xml:space="preserve">Quid dicam de ſingulari <lb/>eaq; </s> <s xml:id="echoid-s39" xml:space="preserve">eximia rerum cœleſtium, totiuſq; </s> <s xml:id="echoid-s40" xml:space="preserve">mundi cognitio-<lb/>ne? </s> <s xml:id="echoid-s41" xml:space="preserve">quam tu tanta cum auiditate ex reconditiſsimis Ma-<lb/>thematicorum fontibus hauſiſti, vt illud aſſecutus in eo <lb/>genere iam ſis, quod alij in maxima tranquillitate, in <lb/>ſummo otio vix auſint optare? </s> <s xml:id="echoid-s42" xml:space="preserve">Exitum tuorum labor um <lb/>fœliciſsimum vides: </s> <s xml:id="echoid-s43" xml:space="preserve">gloria multiplici frueris, neque illa <lb/>precaria ſed tua. </s> <s xml:id="echoid-s44" xml:space="preserve">& </s> <s xml:id="echoid-s45" xml:space="preserve">quibuſdam quaſi gradibus ad ampliſ-<lb/>ſimos honores euehendus, in ea conſtitueris dignitate; </s> <s xml:id="echoid-s46" xml:space="preserve">in <lb/>qua pro ſacroſanta Eccleſia nunquam non excubando, <lb/>in peramplo tot illuſtrium virorum Theatro non alienæ <lb/>gloriæ ſpectator, ſed actor tuæ conſiſtas. </s> <s xml:id="echoid-s47" xml:space="preserve">Tu vero, quod <lb/>rarum eſt, laudem ſapientiæ, quæ vix vllos habeb at ter-<lb/>minos, humanitatis tuæ terminis circumſcribis; </s> <s xml:id="echoid-s48" xml:space="preserve">& </s> <s xml:id="echoid-s49" xml:space="preserve">expo-<lb/>nis omnibus: </s> <s xml:id="echoid-s50" xml:space="preserve">vt extanto fonte perennes ad omnium or-<lb/>dinum homines riuuli deducantur. </s> <s xml:id="echoid-s51" xml:space="preserve">Fœlix qui ſolidæ <lb/>fœlicitatis cauſam & </s> <s xml:id="echoid-s52" xml:space="preserve">initium in te conſtitutum ita foues, <lb/>vt cum alijs illam communicando, non imminuas, ſed <lb/>amplifices. </s> <s xml:id="echoid-s53" xml:space="preserve">prægrande videlicet non ſuccreſcentis, ſed <lb/>adultæ iam virtutis fœnus honorem ex honore, laudem <lb/>ex laude conſequi vberiorem, hæc illa ſapienti viro non <lb/>indigna liberalitas, quæ rerum preſtantiſsimarum poſ-<lb/>ſeſsione non imminuta, in copia tenuitatem non inqui- <pb file="0010" n="10"/> rens vbertatis ipſa ſuæ domina nunquam debilitatur, <lb/>nunquam deficit. </s> <s xml:id="echoid-s54" xml:space="preserve">Quin etiam iſto loco conſtitutus bo-<lb/>narum literarum ſtudioſos complecteris, ac tueris. </s> <s xml:id="echoid-s55" xml:space="preserve">Hæc <lb/>eſt vera germanæ nota ſapientiæ, cum, quas vires nanci-<lb/>ſcitur, ijs ſapientiam alit, tuus animus & </s> <s xml:id="echoid-s56" xml:space="preserve">tuæ ſapientiæ, & </s> <s xml:id="echoid-s57" xml:space="preserve"><lb/>alienæ par eſt. </s> <s xml:id="echoid-s58" xml:space="preserve">Hæc ſunt firmiſsima & </s> <s xml:id="echoid-s59" xml:space="preserve">ſolidiſsima funda-<lb/>menta ad æternam poſteritatis memoriam, quam licet <lb/>proficiſci iam videas ex ijs quibus abundas animi orna-<lb/>mentis, neſcio tamen quo pacto gratior nobis acccidit, <lb/>cum ex aliorum etiam præconio ſuſcipit incrementum. <lb/></s> <s xml:id="echoid-s60" xml:space="preserve">Hinc domus tua floret doctiſsimorum familiaritatibus; </s> <s xml:id="echoid-s61" xml:space="preserve"><lb/>hinc nulli ad tuam conſuetudinem præcluditur aditus; </s> <s xml:id="echoid-s62" xml:space="preserve"><lb/>hinc plurimorum ſtudia commouentur; </s> <s xml:id="echoid-s63" xml:space="preserve">hinc illa ſapien <lb/>tum æmulatio & </s> <s xml:id="echoid-s64" xml:space="preserve">admiratio: </s> <s xml:id="echoid-s65" xml:space="preserve">hinc omnium omnino or-<lb/>dinum ad te concurſus tanquam ad ſapientiſsimum hu-<lb/>mani diuiniq; </s> <s xml:id="echoid-s66" xml:space="preserve">iuris patronum, & </s> <s xml:id="echoid-s67" xml:space="preserve">interpetem. </s> <s xml:id="echoid-s68" xml:space="preserve">Quid ego <lb/>igitur quem tibi ſexcentis eximiæ beneuolentiæ argu-<lb/>mentis obligatum voluiſti, perexiguum hunc ingenij <lb/>mei partum expoſiturus, vnum te nominis & </s> <s xml:id="echoid-s69" xml:space="preserve">exiſtima-<lb/>tionis meæ patronum non ſuſcipiam? </s> <s xml:id="echoid-s70" xml:space="preserve">equidem in am-<lb/>pliſsimi Theatri lucem ſapientiſsimorumq; </s> <s xml:id="echoid-s71" xml:space="preserve">hominum <lb/>conſpectum quibus abundat hæc noſtra ætas inferre me <lb/>non magnopere cogitabam: </s> <s xml:id="echoid-s72" xml:space="preserve">Tu hortatus es dubitan-<lb/>tem impuliſti vel reluctantem: </s> <s xml:id="echoid-s73" xml:space="preserve">tuere igitur obſequen-<lb/>tem iam nunc mihi videor in exigua vel nulla ſpe lau-<lb/>dis ex ingenio meo, fempiternam nominis immortali-<lb/>tatem ex tuo patrocinio conſecutus. </s> <s xml:id="echoid-s74" xml:space="preserve">Vale.</s> <s xml:id="echoid-s75" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s76" xml:space="preserve">Romæ VII. </s> <s xml:id="echoid-s77" xml:space="preserve">Kal. </s> <s xml:id="echoid-s78" xml:space="preserve">Maij. </s> <s xml:id="echoid-s79" xml:space="preserve">MDCIII.</s> <s xml:id="echoid-s80" xml:space="preserve"/> </p> <pb file="0011" n="11"/> </div> <div xml:id="echoid-div5" type="section" level="1" n="5"> <head xml:id="echoid-head7" xml:space="preserve">BENEVOLO <lb/>LECTORI.</head> <p style="it"> <s xml:id="echoid-s81" xml:space="preserve">DIVERSA corporum genera duplici ra-<lb/>tione comparari inter ſe à Mathematicis <lb/>poſſunt mole ac pondere. </s> <s xml:id="echoid-s82" xml:space="preserve">Pondere compara-<lb/>tio fit, cum inter corpora diuer ſi generis mo <lb/>le æqualia, quæritur, quę ſit ratio ponderis: <lb/></s> <s xml:id="echoid-s83" xml:space="preserve">quanto videlicet, vnum altero grauius, aut leuius ſit. </s> <s xml:id="echoid-s84" xml:space="preserve">Ma-<lb/>gnitudine autem fit collatio, cum poſita pari grauitate, quæ-<lb/>ritur, quæ ſit ratio magnitudinis; </s> <s xml:id="echoid-s85" xml:space="preserve">quanto ſit alterum altero <lb/>maius, aut minus. </s> <s xml:id="echoid-s86" xml:space="preserve">Quæ comparatio mihi cum videretur & </s> <s xml:id="echoid-s87" xml:space="preserve"><lb/>iucunda cognitu, & </s> <s xml:id="echoid-s88" xml:space="preserve">vſum nonnullum habere, nec fuſe à quo <lb/>piam explicata, non ita pridem ſuper ea non nihil cœpi mo-<lb/>liri; </s> <s xml:id="echoid-s89" xml:space="preserve">ſed nibil de luce ac publico cogitabam. </s> <s xml:id="echoid-s90" xml:space="preserve">Is enim ego ſum, <lb/>qui malim ſcire, quam noſci; </s> <s xml:id="echoid-s91" xml:space="preserve">diſcere, quam docere. </s> <s xml:id="echoid-s92" xml:space="preserve">Sed ta-<lb/>men cum Michael Coignetus in rebus Mathematicis ex-<lb/>cellens vir, ac Magiſter meus, cui ego plurimum debere me <lb/>fateor, ab eo enim prima elementa habui, repoſcere à me pu-<lb/>blicum aliquem doctrinæ ſuæ fructum videretur. </s> <s xml:id="echoid-s93" xml:space="preserve">ac Fede-<lb/>ricus Saminiatus cuius morum ſuauitatem, & </s> <s xml:id="echoid-s94" xml:space="preserve">beneuolen-<lb/>tiam erga me diu, dum ſimul hiſce studijs condiſcipuli ope-<lb/>ram dedimus, expertus ſum, me vt aliquid auderem tum <lb/>oratione, tum exemplo ſuo excitaret, cœpi minus ab ea cogi-<lb/>tatione alienus eſſe. </s> <s xml:id="echoid-s95" xml:space="preserve">Deinde vero ſummos viros habui <pb file="0012" n="12"/> hortatores. </s> <s xml:id="echoid-s96" xml:space="preserve">Etenim cum Clauium, quod iam diu cupiebam, <lb/>vidiſſem, nec minorem tanta ſcientia, & </s> <s xml:id="echoid-s97" xml:space="preserve">fama viri beni-<lb/>gnitatem comperiſſem. </s> <s xml:id="echoid-s98" xml:space="preserve">Ab eo ſimul ac Theodoſio Rubeo ho-<lb/>mine mihi tum ex ſtudior um ſimilitudine, tum præcipue ex <lb/>eius humanitate amiciſsimo, ad Reuerendiſsimum Sera-<lb/>phinum deductus ſum. </s> <s xml:id="echoid-s99" xml:space="preserve">Iſq; </s> <s xml:id="echoid-s100" xml:space="preserve">me tantus Præſul non ſolum <lb/>bumaniſsime complexus est, verum etiam ita hortatus ad <lb/>euulganda, quæ ſcripſeram; </s> <s xml:id="echoid-s101" xml:space="preserve">plane vt mihi nefas putauerim <lb/>non parere. </s> <s xml:id="echoid-s102" xml:space="preserve">Accipe igitur & </s> <s xml:id="echoid-s103" xml:space="preserve">tu Lector optime amico ac be-<lb/>nigno animo laborem hunc, quem à me talium virorum <lb/>ſumma benignitas expreſsit. </s> <s xml:id="echoid-s104" xml:space="preserve">Argumentum quidem, v@ <lb/>dicebam non iniucundum eſt, nec ab vſu alienum. </s> <s xml:id="echoid-s105" xml:space="preserve">Huiuſ-<lb/>modi enim comparatione Archimedes mixtionem argenti <lb/>in auro deprehendit, & </s> <s xml:id="echoid-s106" xml:space="preserve">furtum Aurificis in corona aurea <lb/>pateſecit. </s> <s xml:id="echoid-s107" xml:space="preserve">de quare ſuo loco ego tractabo, & </s> <s xml:id="echoid-s108" xml:space="preserve">facilem mon-<lb/>strabo viam, qua vel argentum in auro, vel q́uod vis me-<lb/>tallum in quolibet admistum deprehendi queat, & </s> <s xml:id="echoid-s109" xml:space="preserve">alte <lb/>rum ab altero diſcerni; </s> <s xml:id="echoid-s110" xml:space="preserve">& </s> <s xml:id="echoid-s111" xml:space="preserve">ſimul explicabo, quo pacto ex au-<lb/>ri grauitate eius qualitas, nota, ac perfectio intelligi poſsit. <lb/></s> <s xml:id="echoid-s112" xml:space="preserve">Toti vero opuſculo nomen ab Archimede, quem Ducem ſe-<lb/>quor, impoſui. </s> <s xml:id="echoid-s113" xml:space="preserve">Nam cum ille, vt erat ſummus Magiſter, ſa-<lb/>tis habuiſſet hanc totam quaſi fabricam, poſito fundamen-<lb/>to delineare in primo lib. </s> <s xml:id="echoid-s114" xml:space="preserve">vbi agit de ijs quæ vehuntur in <lb/>aqua. </s> <s xml:id="echoid-s115" xml:space="preserve">Opus ego promouere; </s> <s xml:id="echoid-s116" xml:space="preserve">eiq; </s> <s xml:id="echoid-s117" xml:space="preserve">fundamento molem inijce-<lb/>re conatus ſum partibus ſuis elaboratam, atque distin-<lb/>ctam. </s> <s xml:id="echoid-s118" xml:space="preserve">Qua in re ſi quid aſſequutus ſum, quod publice pro-<lb/>beiur, gaudeo cauſa & </s> <s xml:id="echoid-s119" xml:space="preserve">mea & </s> <s xml:id="echoid-s120" xml:space="preserve">publica: </s> <s xml:id="echoid-s121" xml:space="preserve">illud quidem ſpe-<lb/>ro fore vt conatus non diſpliceat.</s> <s xml:id="echoid-s122" xml:space="preserve"/> </p> <pb o="1" file="0013" n="13" rhead="MARINI GHETALDI PROMOTVS ARCHIMEDES"/> </div> <div xml:id="echoid-div6" type="section" level="1" n="6"> <head xml:id="echoid-head8" xml:space="preserve">Seu, <lb/>DE VARIIS CORPORVM GENERIBVS <lb/>Grauitate, & magnitudine comparatis.</head> <head xml:id="echoid-head9" xml:space="preserve">THEOREMA I. PROPOS. I.</head> <p> <s xml:id="echoid-s123" xml:space="preserve">SI duorum Grauium Corporum eiuſdem ge-<lb/>neris alterum alterius fuerit multiplex, quo-<lb/>tuplex maius fuerit minoris, totuplex erit <lb/>maioris grauitas, grauitatis minoris.</s> <s xml:id="echoid-s124" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s125" xml:space="preserve">SINT duo corpora eiuſdem generis ABC, D, quorum grauita-<lb/>tes, EFG, ipſius ABC, & </s> <s xml:id="echoid-s126" xml:space="preserve">H, ip-<lb/> <anchor type="figure" xlink:label="fig-0013-01a" xlink:href="fig-0013-01"/> ſius D, ſit autem corpus ABC, <lb/>multiplex corporis D. </s> <s xml:id="echoid-s127" xml:space="preserve">Dico quo <lb/>tuplex eſt corpus ABC, corporis <lb/>D, totuplicem eſſe grauitatem <lb/>EFG, grauitatis H, diuidatur <lb/>enim corpus ABC, in partes ip-<lb/>ſi D, æquales, quæ ſint A, B, C, <lb/>quoniam igitur corpus A, æqua <lb/>le eſt corpori D, magnitudine, <lb/>& </s> <s xml:id="echoid-s128" xml:space="preserve">ſunt eiuſdem generis, erit grauitas vnius æqualis grauitati alterius. <lb/></s> <s xml:id="echoid-s129" xml:space="preserve">Sumatur grauitas E, æqualis grauitati H, erit igitur corporis A, gra-<lb/>uitas E, & </s> <s xml:id="echoid-s130" xml:space="preserve">reliqui corporis BC, grauitas FG. </s> <s xml:id="echoid-s131" xml:space="preserve">Rurſus quoniam cor-<lb/>pora B, D, ſunt magnitudine æqualia, erunt æquè grauia, ſumatur <lb/>grauitati H, æqualis grauitas F, erit igitur corporis B, grauitas F, & </s> <s xml:id="echoid-s132" xml:space="preserve"><lb/>reliqui corporis C, grauitas G, & </s> <s xml:id="echoid-s133" xml:space="preserve">ſic fiat, donec perueniatur ad vlti-<lb/>mam partem corporis ABC, æqualem ipſi D, ſit ea vltima pars C, quo <lb/>niam igitur corpus C, æquatur magnitudine ipſi D, æquabitur, & </s> <s xml:id="echoid-s134" xml:space="preserve">gra-<lb/>uitate, quare grauitas G, æqualis erit grauitati H, ſequitur igitur <lb/>quot partes ſunt in corpore ABC, æquales ipſi D, tot eſſe partes in. </s> <s xml:id="echoid-s135" xml:space="preserve"><lb/>grauitate EFG, æquales ipſi H, quoties enim ſumpſimus in corpore. </s> <s xml:id="echoid-s136" xml:space="preserve"><lb/>ABC, corpus ipſi D æquale, toties & </s> <s xml:id="echoid-s137" xml:space="preserve">in grauitate EFG, ſumpſimus <pb o="2" file="0014" n="14" rhead="PROMOTVS"/> grauitatem æqualem ipſi H. </s> <s xml:id="echoid-s138" xml:space="preserve">Si duorum igitur grauium corporum <lb/>eiuldem generis, & </s> <s xml:id="echoid-s139" xml:space="preserve">c. </s> <s xml:id="echoid-s140" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s141" xml:space="preserve"/> </p> <div xml:id="echoid-div6" type="float" level="2" n="1"> <figure xlink:label="fig-0013-01" xlink:href="fig-0013-01a"> <image file="0013-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0013-01"/> </figure> </div> </div> <div xml:id="echoid-div8" type="section" level="1" n="7"> <head xml:id="echoid-head10" xml:space="preserve">THEOREMA II. PROPOS. II.</head> <p> <s xml:id="echoid-s142" xml:space="preserve">COrpora grauia eiuſdem generis magnitudine com <lb/>menſurabilia, eandem in grauitate rationem ha-<lb/>bent, quam in magnitudine.</s> <s xml:id="echoid-s143" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s144" xml:space="preserve">SINT corpora commenſurabilia eiuſdem generis A, B, quorum <lb/>grauitates C, ipſius A, & </s> <s xml:id="echoid-s145" xml:space="preserve">D, ipſius B, Dico eſſe vt A, ad B, ita C, ad D, <lb/>quoniam enim, A, B, commenſura-<lb/> <anchor type="figure" xlink:label="fig-0014-01a" xlink:href="fig-0014-01"/> bilia ſunt, metietur ipſa aliquod <lb/>corpus, metiatur, & </s> <s xml:id="echoid-s146" xml:space="preserve">ſit E, cuius <lb/>grauitas F, ſitque corpus E, eiuſdĕ <lb/> <anchor type="note" xlink:label="note-0014-01a" xlink:href="note-0014-01"/> generis cum corporibus A, B, <anchor type="note" xlink:href="" symbol="*"/>ergo quotuplex eſt corpus A, ipſius E, <lb/>totuplex erit grauitas C, grauitatis <lb/> <anchor type="note" xlink:label="note-0014-02a" xlink:href="note-0014-02"/> F, & </s> <s xml:id="echoid-s147" xml:space="preserve">quotuplex B, ipſius E, <anchor type="note" xlink:href="" symbol="*"/>totuplex D, ipſius F, ſi igitur diuidantur cor-<lb/>pora A, B, in partes æquales ipſi E, <lb/>& </s> <s xml:id="echoid-s148" xml:space="preserve">grauitates quoque C, D, in partes æquales ipſi F, erit vt corporis <lb/>A, pars vna, ad corpus E, ita pars vna grauitatis C, ad grauitatem F, <lb/>æquale videlicet ad æquale, & </s> <s xml:id="echoid-s149" xml:space="preserve">æque multiplicatis antecedentibus <lb/>erit vt A, ad E, ita C, ad F, ſunt enim antecedentium, hoc eſt, illarum <lb/>partium æque multiplicia A, C, eadem ratione, vt B, ad E, ita erit D, <lb/>ad F, & </s> <s xml:id="echoid-s150" xml:space="preserve">conuertendo vt E, ad B, ita F, ad D. </s> <s xml:id="echoid-s151" xml:space="preserve">quoniam igitur vt A, ad <lb/>E, ita eſt C, ad F, & </s> <s xml:id="echoid-s152" xml:space="preserve">vt E, ad B, ita F, ad D, <anchor type="note" xlink:href="" symbol="*"/> erit ex æquali vt A, ad <anchor type="note" xlink:label="note-0014-03a" xlink:href="note-0014-03"/> B, ita C, ad D. </s> <s xml:id="echoid-s153" xml:space="preserve">corpora igitur commenſurabilia eiuſdem generis ean-<lb/>dem in grauitate rationem habent, quam in magnitudine, quod erat <lb/>demonſtrandum.</s> <s xml:id="echoid-s154" xml:space="preserve"/> </p> <div xml:id="echoid-div8" type="float" level="2" n="1"> <figure xlink:label="fig-0014-01" xlink:href="fig-0014-01a"> <image file="0014-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0014-01"/> </figure> <note position="left" xlink:label="note-0014-01" xlink:href="note-0014-01a" xml:space="preserve">Ex an-<lb/>teced.</note> <note position="left" xlink:label="note-0014-02" xlink:href="note-0014-02a" xml:space="preserve">Ex an-<lb/>teced.</note> <note position="left" xlink:label="note-0014-03" xlink:href="note-0014-03a" xml:space="preserve">22. 5. <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div10" type="section" level="1" n="8"> <head xml:id="echoid-head11" xml:space="preserve">THEOREMA III. PROPOS. III.</head> <p> <s xml:id="echoid-s155" xml:space="preserve">ET incommenſurabilia corpora eiuſdem generis <lb/>eandem in grauitate rationem habent, quam in <lb/>magnitudìne.</s> <s xml:id="echoid-s156" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s157" xml:space="preserve">SINT incommenſurabilia corpora A, BC, quorum grauitates <lb/>D, ipſius A, & </s> <s xml:id="echoid-s158" xml:space="preserve">EF, ipſius BC. </s> <s xml:id="echoid-s159" xml:space="preserve">Dico eſſe vt A, ad BC, ita D, ad EF, ſi <pb o="3" file="0015" n="15" rhead="ARCHIMEDES."/> enim non eſt vt A, adBC, ita D, ad EF, erit vt A, ad BC, ita D, vel ad <lb/>minorem quam EF, <lb/> <anchor type="figure" xlink:label="fig-0015-01a" xlink:href="fig-0015-01"/> vel ad maiorem, ſit <lb/>primum ad minorĕ, <lb/>nempe ad EG, & </s> <s xml:id="echoid-s160" xml:space="preserve">ex-<lb/>ponatur aliquod cor <lb/>pus K, eiuſdem gene-<lb/>ris cum corporibus <lb/>A, B C, cuius graui-<lb/>tas ſit æqualis ipſi <lb/>GF, & </s> <s xml:id="echoid-s161" xml:space="preserve">à corpore BC, <lb/>auferatur aliqua <lb/>pars HC, quæ ſit mi-<lb/>nor corpore K, ita vt reliqua pars BL, ſit commenſurabilis ipſi A. </s> <s xml:id="echoid-s162" xml:space="preserve">& </s> <s xml:id="echoid-s163" xml:space="preserve"><lb/>ſit partis HC, grauitas IF, ergo reliquæ partis BL, grauitas erit EI. <lb/></s> <s xml:id="echoid-s164" xml:space="preserve">Quoniam igitur corpus A, commenſurabile eſt ipſi BL,<anchor type="note" xlink:href="" symbol="*"/> erit vt A, ad <anchor type="note" xlink:label="note-0015-01a" xlink:href="note-0015-01"/> BL, ita D, ad EI, ſed vt A, ad BC, ita eſt D, ad EG, atque A, primus, <lb/>proportionalium terminus in ſerie prima, <anchor type="note" xlink:href="" symbol="*"/>maiorem habet ratio nem ad BL, ſecundum terminum, quam A, primus terminus in ſerie <lb/> <anchor type="note" xlink:label="note-0015-02a" xlink:href="note-0015-02"/> ſecunda ad BC, ſecundum; </s> <s xml:id="echoid-s165" xml:space="preserve">ergo & </s> <s xml:id="echoid-s166" xml:space="preserve">D, tertius terminus in ſerie prima <lb/>ad EI, quartum, maiorem habebit rationem quam D, tertius termi-<lb/>nus in ſerie ſecunda ad I G, quartum, quoniam igitur D, maiorem <lb/>habet rationem ad EI, quam ad EG, <anchor type="note" xlink:href="" symbol="*"/> erit EI, minor quam EG, quod <anchor type="note" xlink:label="note-0015-03a" xlink:href="note-0015-03"/> eſt abſurdum. </s> <s xml:id="echoid-s167" xml:space="preserve">non igitur eſt vt A, ad BC, ita D, ad minorem quam E F.</s> <s xml:id="echoid-s168" xml:space="preserve"/> </p> <div xml:id="echoid-div10" type="float" level="2" n="1"> <figure xlink:label="fig-0015-01" xlink:href="fig-0015-01a"> <image file="0015-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0015-01"/> </figure> <note position="right" xlink:label="note-0015-01" xlink:href="note-0015-01a" xml:space="preserve">Ex an-<lb/>tecedĕ-<lb/>te.</note> <note position="right" xlink:label="note-0015-02" xlink:href="note-0015-02a" xml:space="preserve">8. 5. <lb/>Elem.</note> <note position="right" xlink:label="note-0015-03" xlink:href="note-0015-03a" xml:space="preserve">10. 5. <lb/>Elem.</note> </div> <p> <s xml:id="echoid-s169" xml:space="preserve">Deinde ſit vt A, ad BC, ita D, ad maiorem quam EF, nempe ad <lb/>EG, & </s> <s xml:id="echoid-s170" xml:space="preserve">expoſito cor-<lb/> <anchor type="figure" xlink:label="fig-0015-02a" xlink:href="fig-0015-02"/> pore K, vt dictum <lb/>eſt, cuius grauitas, <lb/>ſit æqualis grauita-<lb/>ti FG, addatur cor-<lb/>pori BC, aliquod <lb/>corpus CH, quod ſit <lb/>minus corpore K, & </s> <s xml:id="echoid-s171" xml:space="preserve"><lb/>eiuſdem generis cũ <lb/>corporibus A, BC, <lb/>ita vt totum corpus <lb/>BL, ſit commenſurabile ipſi A, & </s> <s xml:id="echoid-s172" xml:space="preserve">ſit ipſius CH, grauitas FI, ergo to <lb/>tius corporis BL, grauitas erit EI; </s> <s xml:id="echoid-s173" xml:space="preserve">Quoniam igitur corpori A, com-<lb/>menſurabile eſt corpus BL, <anchor type="note" xlink:href="" symbol="*"/> erit vt A, ad BL, ita D, ad EI, ſed vt A, ad <anchor type="note" xlink:label="note-0015-04a" xlink:href="note-0015-04"/> BC, ita eſt D, ad EG, atque A, primus proportionalium terminus in <lb/>ſerie prima, <anchor type="note" xlink:href="" symbol="*"/> minorem habet rationem ad BL, ſecundum terminum, <anchor type="note" xlink:label="note-0015-05a" xlink:href="note-0015-05"/> <pb o="4" file="0016" n="16" rhead="PROMOTVS"/> quam A, primus terminus in ſerie ſecunda ad BC, ſecundum, ergo, & </s> <s xml:id="echoid-s174" xml:space="preserve"><lb/>D, tertius terminus in ſerie prima ad EI, quartum, minorem habebit <lb/>rationem quam D, tertius terminus in ſerie ſecunda ad EG, quartum. <lb/></s> <s xml:id="echoid-s175" xml:space="preserve">Quoniam igitur D, minorem habet rationem ad EI, quam ad EG, <lb/>erit <anchor type="note" xlink:href="" symbol="*"/> EI, maior quam EG, quod eſt abſurdum. </s> <s xml:id="echoid-s176" xml:space="preserve">Non igitur eſt vt A, <anchor type="note" xlink:label="note-0016-01a" xlink:href="note-0016-01"/> ad BC, ita D, ad maiorem quam EF, oſtenſum autem eſt neque ad mi-<lb/>norem; </s> <s xml:id="echoid-s177" xml:space="preserve">quare vt A, ad BC, ita erit D, ad EF. </s> <s xml:id="echoid-s178" xml:space="preserve">& </s> <s xml:id="echoid-s179" xml:space="preserve">incommenſurabilia igi-<lb/>tur corpora eiuſdem generis eandem in grauitate rationem habent, <lb/>quam in magnitudine, quod erat demonſtrandum.</s> <s xml:id="echoid-s180" xml:space="preserve"/> </p> <div xml:id="echoid-div11" type="float" level="2" n="2"> <figure xlink:label="fig-0015-02" xlink:href="fig-0015-02a"> <image file="0015-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0015-02"/> </figure> <note position="right" xlink:label="note-0015-04" xlink:href="note-0015-04a" xml:space="preserve">Ex an <lb/>teced.</note> <note position="right" xlink:label="note-0015-05" xlink:href="note-0015-05a" xml:space="preserve">8. 5. Ele</note> <note position="left" xlink:label="note-0016-01" xlink:href="note-0016-01a" xml:space="preserve">10. 5. <lb/>Elem.</note> </div> <p> <s xml:id="echoid-s181" xml:space="preserve">ID QVOD nos duobus præcedentibus Theorematis de-<lb/>monſtrauimus, nõnulli, vt per ſe notum, & </s> <s xml:id="echoid-s182" xml:space="preserve">vt commune quod-<lb/>dam axioma ſupponunt, quam bene & </s> <s xml:id="echoid-s183" xml:space="preserve">rationabiliter ipſi vide-<lb/>rint; </s> <s xml:id="echoid-s184" xml:space="preserve">melius enim Euclides propoſitionem 20. </s> <s xml:id="echoid-s185" xml:space="preserve">primi libri Ele-<lb/>mentorum ſuppoſuiſſet vt pronunciatum; </s> <s xml:id="echoid-s186" xml:space="preserve">vnicuique enim no-<lb/>tius eſt duo trianguli latera reliquo eſſe maiora (cum & </s> <s xml:id="echoid-s187" xml:space="preserve">Aſino <lb/>illud ſit notum) quam corpora grauia eiuſdem generis eandem <lb/>in grauitate rationem habere, quam in magnitudine, & </s> <s xml:id="echoid-s188" xml:space="preserve">tamen <lb/>illam propoſitionem demonſtrat Euclides, non ſupponit, non <lb/>igitur hæc, quæ minus ad principij rationem accedit, ſuppo-<lb/>nenda fuit, ſed demonſtranda.</s> <s xml:id="echoid-s189" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div13" type="section" level="1" n="9"> <head xml:id="echoid-head12" xml:space="preserve">THEOREMA IV. PROPOS. IV.</head> <p> <s xml:id="echoid-s190" xml:space="preserve">SI quatuor corporum grauium primum ad ſecundũ <lb/>eandem in magnitudine rationem habeat, quam <lb/>tertium ad quartum, primum autem, & </s> <s xml:id="echoid-s191" xml:space="preserve">ſecundum ſint <lb/>eiuſdem generis, itidem tertium, & </s> <s xml:id="echoid-s192" xml:space="preserve">quartum; </s> <s xml:id="echoid-s193" xml:space="preserve">& </s> <s xml:id="echoid-s194" xml:space="preserve">in gra-<lb/>uitate primum ad ſecundum eandem rationem habebit, <lb/>quam tertium ad quartum.</s> <s xml:id="echoid-s195" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s196" xml:space="preserve">PRIMVM enim A, ad ſecundum B, eandem in magnitudine ra-<lb/>tionem habeat, quam tertium C, ad quartum D, ſint autem A, B, <lb/>eiuſdem generis, itidem C, D. </s> <s xml:id="echoid-s197" xml:space="preserve">Dico & </s> <s xml:id="echoid-s198" xml:space="preserve">in grauitate primum A, ad <lb/>ſecundum B, eandem rationem habere, quam tertium C, ad D, quar-<lb/>tum. </s> <s xml:id="echoid-s199" xml:space="preserve">Sint enim earum grauitates E, ipſius A, & </s> <s xml:id="echoid-s200" xml:space="preserve">F, ipſius B, ipſius <lb/>vero C, ſit grauitas G, & </s> <s xml:id="echoid-s201" xml:space="preserve">ipſius D, grauitas H, quoniam igitur cor- <pb o="5" file="0017" n="17" rhead="ARCHIMEDES."/> pora A, B, eiuſdem ſunt generis, <lb/> <anchor type="figure" xlink:label="fig-0017-01a" xlink:href="fig-0017-01"/> ſimiliter, & </s> <s xml:id="echoid-s202" xml:space="preserve">corpora C, D, <anchor type="note" xlink:href="" symbol="*"/> erit vt A, ad B, ita E, ad F, <anchor type="note" xlink:href="" symbol="*"/> & </s> <s xml:id="echoid-s203" xml:space="preserve">vt C, ad D, ita G, ad H. </s> <s xml:id="echoid-s204" xml:space="preserve">Sed poni-<lb/>tur vt A, ad B, ita eſſe C, ad D, <lb/>ergo vt E, ad F, ita erit G, ad <lb/>H. </s> <s xml:id="echoid-s205" xml:space="preserve">Si igitur quatuor corporum <lb/>grauium, primum ad ſecundum <lb/>eandem in magnitudine ratio-<lb/>nem habeat: </s> <s xml:id="echoid-s206" xml:space="preserve">etcæt. </s> <s xml:id="echoid-s207" xml:space="preserve">quod demon-<lb/>ſtrare oportebat.</s> <s xml:id="echoid-s208" xml:space="preserve"/> </p> <div xml:id="echoid-div13" type="float" level="2" n="1"> <figure xlink:label="fig-0017-01" xlink:href="fig-0017-01a"> <image file="0017-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0017-01"/> </figure> </div> </div> <div xml:id="echoid-div15" type="section" level="1" n="10"> <head xml:id="echoid-head13" xml:space="preserve">THEOREMA V. PROPOS. V.</head> <p> <s xml:id="echoid-s209" xml:space="preserve">SOlida corpora liquido grauiora demiſſa in liquidum <lb/>ferentur deorſum, donec deſcendant, & </s> <s xml:id="echoid-s210" xml:space="preserve">erunt in li-<lb/>quido tanto leuiora, quanta eſt grauitas liquidi magni-<lb/>tudinem habentis ſolido corpori æqualem.</s> <s xml:id="echoid-s211" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s212" xml:space="preserve">HOC autem demonſtratum eſt ab Archimede propoſ. </s> <s xml:id="echoid-s213" xml:space="preserve">7. </s> <s xml:id="echoid-s214" xml:space="preserve">primi li-<lb/>bri de ijs, quæ vehuntur in aqua.</s> <s xml:id="echoid-s215" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div16" type="section" level="1" n="11"> <head xml:id="echoid-head14" xml:space="preserve">THEOREMA VI. PROPOS. VI.</head> <p> <s xml:id="echoid-s216" xml:space="preserve">SI quatuor grauium corporum primum, & </s> <s xml:id="echoid-s217" xml:space="preserve">ſecundum <lb/>fuerint magnitudine æqualia, tertium vero, & </s> <s xml:id="echoid-s218" xml:space="preserve">quar-<lb/>tum æque grauia, fuerint autem primum, & </s> <s xml:id="echoid-s219" xml:space="preserve">tertium <lb/>eiuſdem generis, itidem ſecundum, & </s> <s xml:id="echoid-s220" xml:space="preserve">quartum; </s> <s xml:id="echoid-s221" xml:space="preserve">erit, vt <lb/>grauitas corporis primi, ad grauitatem ſecundi, ita gra-<lb/>uitas liquidi æqualis magnitudine corpori quarto, ad gra <lb/>uitatem liquidi tertio corpori æqualis.</s> <s xml:id="echoid-s222" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s223" xml:space="preserve">SINT quatuor corpora A, B, C, D, quorum A, primum, & </s> <s xml:id="echoid-s224" xml:space="preserve">B, ſe-<lb/>cundum ſint magnitudine æqualia, tertium vero C, & </s> <s xml:id="echoid-s225" xml:space="preserve">D, quartum, <lb/>æque grauia, ſint autem A, & </s> <s xml:id="echoid-s226" xml:space="preserve">C, eiuſdem generis, itidem B, & </s> <s xml:id="echoid-s227" xml:space="preserve">D. </s> <s xml:id="echoid-s228" xml:space="preserve">Di-<lb/>co vt grauitas corporis A, ad grauitatem corporis B, ita eſſe grauita-<lb/>tem liquidi æqualis magnitudine corpori D, ad grauitatem liquidi <lb/>magnitudine corpori C, æqualis. </s> <s xml:id="echoid-s229" xml:space="preserve">Accipiantur enim tria eiuſdem ge- <pb o="6" file="0018" n="18" rhead="PROMOTVS"/> neris liquidi corpora E, F, G, quorum E, ſit æquale corpori A, vel B, <lb/>magnitudine, ipſum vero F, æqua <lb/> <anchor type="figure" xlink:label="fig-0018-01a" xlink:href="fig-0018-01"/> le corpori C, & </s> <s xml:id="echoid-s230" xml:space="preserve">ipſum G, æquale <lb/>corpori D. </s> <s xml:id="echoid-s231" xml:space="preserve">Quoniam igitur eſt vt <lb/>D, ad G, ita B, ad E, æquale vi-<lb/>delicet ad æquale, erit permutan-<lb/>do vt D, ad B, ita G, ad E, & </s> <s xml:id="echoid-s232" xml:space="preserve">quo-<lb/>niam ſunt eiuſdem generis corpo <lb/>ra D, B, ſimiliter & </s> <s xml:id="echoid-s233" xml:space="preserve">corpora G, E, <lb/>erit <anchor type="note" xlink:href="" symbol="*"/> vt grauitas corporis D, hoc <anchor type="note" xlink:label="note-0018-01a" xlink:href="note-0018-01"/> eſt ipſius C, ponuntur enim æque <lb/>grauia corpora C, D, ad grauita-<lb/>tem corporis B, ita liquidi G, gra-<lb/>uitas ad grauitatem liquidi E. <lb/></s> <s xml:id="echoid-s234" xml:space="preserve">Similiter quoniam eſt vt A, ad E, <lb/>ita C, ad F, æquale videlicet ad <lb/>æquale, erit permutando vt A, ad <lb/>C, ita E, ad F, & </s> <s xml:id="echoid-s235" xml:space="preserve">quoniam ponun<emph style="sub">t</emph>ur eiuſdem generis corpora A, C, <lb/> <anchor type="note" xlink:label="note-0018-02a" xlink:href="note-0018-02"/> itidem E, F, <anchor type="note" xlink:href="" symbol="*"/> erit vt grauitas corporis A, ad grauitatem ipſius C, ita liquidi E, grauitas ad grauitatem liquidi F, ſed vt grauitas corporis <lb/>C, ad grauitatem corporis B, ita eſt grauitas liquidi G, ad grauita-<lb/>tem liquidi E, vt eſt demonſtratum, ergo <anchor type="note" xlink:href="" symbol="*"/> in perturbata proportione <anchor type="note" xlink:label="note-0018-03a" xlink:href="note-0018-03"/> erit vt grauitas corporis A, ad ipſius corporis B, grauitatem, ita liqui <lb/>di G, grauitas, ad grauitatem liquidi F. </s> <s xml:id="echoid-s236" xml:space="preserve">ſi igitur quatuor grauium <lb/>corporum primum, & </s> <s xml:id="echoid-s237" xml:space="preserve">ſecundum, & </s> <s xml:id="echoid-s238" xml:space="preserve">c. </s> <s xml:id="echoid-s239" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s240" xml:space="preserve"/> </p> <div xml:id="echoid-div16" type="float" level="2" n="1"> <figure xlink:label="fig-0018-01" xlink:href="fig-0018-01a"> <image file="0018-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0018-01"/> </figure> <note position="left" xlink:label="note-0018-01" xlink:href="note-0018-01a" xml:space="preserve">2. & 3. <lb/>huins.</note> <note position="left" xlink:label="note-0018-02" xlink:href="note-0018-02a" xml:space="preserve">2. & 3. <lb/>huius.</note> <note position="left" xlink:label="note-0018-03" xlink:href="note-0018-03a" xml:space="preserve">23. 5. <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div18" type="section" level="1" n="12"> <head xml:id="echoid-head15" xml:space="preserve">THEOREMA VII. PROPOS. VII.</head> <p> <s xml:id="echoid-s241" xml:space="preserve">SI quatuor grauium corporum primum, & </s> <s xml:id="echoid-s242" xml:space="preserve">ſecundũ, <lb/>fuerint magnitudine æqualia, tertium vero, & </s> <s xml:id="echoid-s243" xml:space="preserve">quar-<lb/>tum æque grauia, fuerint autem primum, & </s> <s xml:id="echoid-s244" xml:space="preserve">tertium <lb/>eiuſdem generis, itidem ſecundum, & </s> <s xml:id="echoid-s245" xml:space="preserve">quartum; </s> <s xml:id="echoid-s246" xml:space="preserve">primum <lb/>ad ſecundum eandem in grauitate rationem habebit, <lb/>quam habet in magnitudine quartum ad tertium.</s> <s xml:id="echoid-s247" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s248" xml:space="preserve">SINT quatuor grauia corpora A, B, C, D, quorum A, primum <lb/>& </s> <s xml:id="echoid-s249" xml:space="preserve">B, ſecundum ſint magnitudine æqualia, tertium vero C, & </s> <s xml:id="echoid-s250" xml:space="preserve">D, quar-<lb/>tum æque grauia, ſint autem A, & </s> <s xml:id="echoid-s251" xml:space="preserve">C, eiuſdem generis, itidem B, & </s> <s xml:id="echoid-s252" xml:space="preserve"><lb/>D. </s> <s xml:id="echoid-s253" xml:space="preserve">Dico corpus A, eandem in grauitate rationem habere ad corpus <pb o="7" file="0019" n="19" rhead="ARCHIMEDES."/> B, quam corpus D, habet in magnitudine ad C, corpus. </s> <s xml:id="echoid-s254" xml:space="preserve">Sit enim li-<lb/>quidi magnitudine æqualis corpo-<lb/> <anchor type="figure" xlink:label="fig-0019-01a" xlink:href="fig-0019-01"/> ri C, grauicas E, ſimiliter, & </s> <s xml:id="echoid-s255" xml:space="preserve">liqui-<lb/>di æqualis magnitudine corpori D, <lb/>grauitas F, quoniam igitur gra-<lb/>uia corpora eiuſdem generis, ean-<lb/>dem in magnitudine rationem<anchor type="note" xlink:href="" symbol="*"/> ha- bent, quam in grauitate, erit vt ma-<lb/>gnitudo liquidi æqualis corpori D, <lb/>ad magnitudinem liquidi æqualis <lb/>corpori C, hoc eſt, vt magnitudo <lb/>corporis D, ad magnitudinem cor-<lb/>poris C, ita grauitas F, ad grauita-<lb/>tem E, ſed vt grauitas F, ad grauitatem E, <anchor type="note" xlink:href="" symbol="*"/> ita eſt grauitas corporis <anchor type="note" xlink:label="note-0019-01a" xlink:href="note-0019-01"/> A, ad grauitatem corporis B, <anchor type="note" xlink:href="" symbol="*"/> ergo vt grauitas corporis A, ad graui- tatem corporis B, ita erit magnitudo corporis D, ad corporis C, ma-<lb/> <anchor type="note" xlink:label="note-0019-02a" xlink:href="note-0019-02"/> gnitudinem. </s> <s xml:id="echoid-s256" xml:space="preserve">Si quatuor igitur grauium corporum primum, & </s> <s xml:id="echoid-s257" xml:space="preserve">ſecun-<lb/>dum, & </s> <s xml:id="echoid-s258" xml:space="preserve">c. </s> <s xml:id="echoid-s259" xml:space="preserve">quod erat demonſtrandum.</s> <s xml:id="echoid-s260" xml:space="preserve"/> </p> <div xml:id="echoid-div18" type="float" level="2" n="1"> <figure xlink:label="fig-0019-01" xlink:href="fig-0019-01a"> <image file="0019-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0019-01"/> </figure> <note position="right" xlink:label="note-0019-01" xlink:href="note-0019-01a" xml:space="preserve">Ex an-<lb/>teced.</note> <note position="right" xlink:label="note-0019-02" xlink:href="note-0019-02a" xml:space="preserve">11. 5. <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div20" type="section" level="1" n="13"> <head xml:id="echoid-head16" xml:space="preserve">PROBLEMA I. PROPOS. VIII.</head> <p> <s xml:id="echoid-s261" xml:space="preserve">PRopoſitis duobus corporibus magnitudine æquali-<lb/>bus, vno ſolido, altero liquido, data ſolidi corporis <lb/>grauitate, grauitatem liquidi inuenire.</s> <s xml:id="echoid-s262" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s263" xml:space="preserve">SINT duo propoſita corpo-<lb/> <anchor type="figure" xlink:label="fig-0019-02a" xlink:href="fig-0019-02"/> ra magnitudine æqualia A, B, <lb/>quorum A, ſit ſolidum, B, vero <lb/>liquidum, & </s> <s xml:id="echoid-s264" xml:space="preserve">ſit ſolidi data graui-<lb/>tas CD, Oportet inuenire quan-<lb/>ta erit grauitas liquidi B. </s> <s xml:id="echoid-s265" xml:space="preserve">Si ſo-<lb/>lidum A, grauius ſit liquido, de-<lb/>mittatur in liquidũ, & </s> <s xml:id="echoid-s266" xml:space="preserve">habeat in <lb/>liquido grauitatem ED, quoniam <lb/>igitur ſolidum A, grauius eſt li-<lb/>quido, demiſſum in liquidum <lb/>erit<anchor type="note" xlink:href="" symbol="*"/> in liquido tanto leuius, quã- ta eſt grauitas liquidi magnitu-<lb/>dine æqualis ſolido A, ſed ſolidũ <lb/>A, leuius eſt in liquido, quanta <pb o="8" file="0020" n="20" rhead="PROMOTVS"/> eſt grauitas CE, ergo grauitas liquidi magnitudine æqualis folido A, <lb/>erit CE.</s> <s xml:id="echoid-s267" xml:space="preserve"/> </p> <div xml:id="echoid-div20" type="float" level="2" n="1"> <figure xlink:label="fig-0019-02" xlink:href="fig-0019-02a"> <image file="0019-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0019-02"/> </figure> </div> <p> <s xml:id="echoid-s268" xml:space="preserve">Si vero ſolidum A, ſit leuius liquido, accipiatur aliquod aliud cor-<lb/>pus ſolidum F, grauius liquido, ita vt ſolidum conſtans ex vtriſque, <lb/>ſolidis A, F, demiſſum in liquidum feratur deorſum, & </s> <s xml:id="echoid-s269" xml:space="preserve">ſit ſolidi F, <lb/>grauitas DG, item eiuſdem ſolidi F, in liquido videlicet exiſtentis ſit <lb/> <anchor type="note" xlink:label="note-0020-01a" xlink:href="note-0020-01"/> grauitas HG, <anchor type="note" xlink:href="" symbol="*"/> ergo liquidi magnitudine æqualis ſolido F, erit gra- uitas DH.</s> <s xml:id="echoid-s270" xml:space="preserve"/> </p> <div xml:id="echoid-div21" type="float" level="2" n="2"> <note position="left" xlink:label="note-0020-01" xlink:href="note-0020-01a" xml:space="preserve">5. huius</note> </div> <p> <s xml:id="echoid-s271" xml:space="preserve">Et quoniam ſolidi A, grauitas eſt CD, ſolidi vero F, grauitas DG, <lb/>erit vtrorumque ſolidorum A, F, grauitas CG. </s> <s xml:id="echoid-s272" xml:space="preserve">coniungantur ſolida <lb/>A, F, & </s> <s xml:id="echoid-s273" xml:space="preserve">ſolidum ex vtriſque conſtans demittatur in liquidum, & </s> <s xml:id="echoid-s274" xml:space="preserve">ha-<lb/>beat in liquido grauitatem GI, (habebit autem in liquido minorem <lb/>grauitatem, quam ſolum ſolidum F, quoniam ſolidum F, grauius li-<lb/>quido fertur deorſum nullo prohibente, & </s> <s xml:id="echoid-s275" xml:space="preserve">coniunctum cum ſolido A, <lb/>leuiori liqnido ab eo ſuſtinetut, ne deorſum feratur tãta vi, qua ſeiun-<lb/>ctum) quoniam igitur ſolidi, quod conſtat ex vtriſque ſolidis A, F, <lb/> <anchor type="note" xlink:label="note-0020-02a" xlink:href="note-0020-02"/> grauitas eſt CG, in liquido vero exiſtentis grauitas GI, <anchor type="note" xlink:href="" symbol="*"/> erit liquidi habentis magnitudinem æqualem vtriſque ſolidis A, F, grauitas CI, <lb/>ſed grauitas liquidi æqualis magnitudine ſolido F, eſt DH, ergo reli-<lb/>qui liquidi æqualis ſolido A, erit grauitas CD, IH, ſed liquidum B, <lb/>æquatur magnitudine folido A, ergo grauitas liquidi B, erit CD, IH, <lb/>inuenta igitur eſt liquidi corporis B, grauitas CD, IH, de qua quæ-<lb/>rebatur.</s> <s xml:id="echoid-s276" xml:space="preserve"/> </p> <div xml:id="echoid-div22" type="float" level="2" n="3"> <note position="left" xlink:label="note-0020-02" xlink:href="note-0020-02a" xml:space="preserve">5. huius</note> </div> <p> <s xml:id="echoid-s277" xml:space="preserve">Placet huic Problemati exemplum apponere, vt vnicuique <lb/>etiam diſciplinæ Mathematicæ experto ad vſum pateat adi-<lb/>tus; </s> <s xml:id="echoid-s278" xml:space="preserve">quare etiam ſequentibus Problematis apponemus ſimilia <lb/>exempla.</s> <s xml:id="echoid-s279" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div24" type="section" level="1" n="14"> <head xml:id="echoid-head17" xml:space="preserve">Exemplum.</head> <p> <s xml:id="echoid-s280" xml:space="preserve">QVidam proponit aliquod corpus ſolidum notæ <lb/>grauitatis, & </s> <s xml:id="echoid-s281" xml:space="preserve">vult ſcire quanta erit grauitas liqui-<lb/>di, magnitudinem habentis propoſito Corpori ſolido æ-<lb/>qualem.</s> <s xml:id="echoid-s282" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s283" xml:space="preserve">Sit primum propoſitum aliquod corpus plumbeum A, cuius graui-<lb/>tas ſit 23. </s> <s xml:id="echoid-s284" xml:space="preserve">& </s> <s xml:id="echoid-s285" xml:space="preserve">oporteat ſcire quanta erit grauitas aquæ magnitudinem <lb/>babentis æqualem propoſito plumbo A, ponderetur plumbum A, in <lb/>aqua (modum quo ponderanda ſint corpora ſolida in aqua, inferius <lb/>apponemus) & </s> <s xml:id="echoid-s286" xml:space="preserve">babeat grauitatem 21. </s> <s xml:id="echoid-s287" xml:space="preserve">quoniam igitur numerus 23.</s> <s xml:id="echoid-s288" xml:space="preserve"> <pb o="9" file="0021" n="21" rhead="ARCHIMEDES."/> juperat numerum 21, numero 2, erit grauitàs aquæ magnitudinem <lb/>habentis æqualem plumbo A, 2.</s> <s xml:id="echoid-s289" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s290" xml:space="preserve">Eadem via omnium liquidorum inuenitur grauitas, quan-<lb/>do nimirum corpus ſolidum ſit grauius liquido, cuius liquidi <lb/>quærenda eſt grauitas, hoc eſt quando corpus ſolidum demiſ-<lb/>ſum in liquidum feratur deorſum.</s> <s xml:id="echoid-s291" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s292" xml:space="preserve">Quando vero corpus ſolidum fuerit leuius liquido, hoc eſt <lb/>demiſſum in liquidum non deſcendat, per adiectionem alicuius <lb/>alius ſolidi corporis liquido grauioris, quæſita liquidi grauitas <lb/>inuenietur.</s> <s xml:id="echoid-s293" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s294" xml:space="preserve">Sit igitur propoſitum aliquod cereum corpus A, cuius grauitas ſit <lb/>21. </s> <s xml:id="echoid-s295" xml:space="preserve">& </s> <s xml:id="echoid-s296" xml:space="preserve">oporteat ſcire quanta erit grauitas aquæ magnitudinem haben-<lb/>tis æqualem ceræ A. </s> <s xml:id="echoid-s297" xml:space="preserve">Quoniam cera leuior est, quam aqua, ſi demitta-<lb/>tur in aquam non feretur deorſum, accipiatur aliquod corpus ſolidum <lb/>F, grauius quam aqua, ita vt corpus constans ex vtriſque corporibus <lb/>A, F, demiſſum in aquam feratur deorſum, ſit igitur corpus F, plum-<lb/>beum, cuius grauitas ſit v. </s> <s xml:id="echoid-s298" xml:space="preserve">g. </s> <s xml:id="echoid-s299" xml:space="preserve">23, & </s> <s xml:id="echoid-s300" xml:space="preserve">eiuſdem in aqua ponderati 21, <lb/>e<unsure/>rgo aquæ magnitudinem habentis æqualĕplumbo F, erit grauitas 2,</s> </p> <p style="it"> <s xml:id="echoid-s301" xml:space="preserve">Et quoniam ceræ A, grauitas est 21, plumbi vero F, 23, erit vtro-<lb/>rumque corporum A, F, ceræ nimirum, & </s> <s xml:id="echoid-s302" xml:space="preserve">plumbi grauitas 44, coniun <lb/>gatur cera, & </s> <s xml:id="echoid-s303" xml:space="preserve">plumbum, & </s> <s xml:id="echoid-s304" xml:space="preserve">ita coniuncta ponderentur in aqua, & </s> <s xml:id="echoid-s305" xml:space="preserve"><lb/>habeant grauitatem 20, quoniam igitur numerus 44, ſuperat nume-<lb/>rum 20, numero 24, erit grauitas aquæ habentis magnitudinem æqua-<lb/>lem vtriſque corporibus ceræ & </s> <s xml:id="echoid-s306" xml:space="preserve">plumbi 24, ſed grauitas aquæ magni-<lb/>tudinem habentis æqualem plumbo F, est 2, ergo reliquum quod est <lb/>22, erit grauitas, aquæ magnitudine æqualis propoſitæ cera A.</s> <s xml:id="echoid-s307" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s308" xml:space="preserve">At vero ſi propoſitum fuerit aliquod corpus ſolidum ma-<lb/>gni ponderis, ita vt difficile poſſit ponderari in aqua, hac via <lb/>inuenietur aquæ quæſita grauitas.</s> <s xml:id="echoid-s309" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s310" xml:space="preserve">Sit aliquod corpus plumbeũ A, cuius grauitas 2300, & </s> <s xml:id="echoid-s311" xml:space="preserve">oporteat in <lb/>uenire grauitatem aquæ magnitudinem habentis æqualĕ plumbo A, <lb/>accipiatur aliquod paruum plumbi corpus F, cuius grauitas ſit v. </s> <s xml:id="echoid-s312" xml:space="preserve">g. <lb/></s> <s xml:id="echoid-s313" xml:space="preserve">23, & </s> <s xml:id="echoid-s314" xml:space="preserve">inueniatur grauitas aquæ magnitudine æqualis plumbo F, vt <lb/>dictum est, quæ ſit. </s> <s xml:id="echoid-s315" xml:space="preserve">2, & </s> <s xml:id="echoid-s316" xml:space="preserve">fiat vt 23, ad 2, ita 2300, ad alium numerum <lb/>qui ſit 200. </s> <s xml:id="echoid-s317" xml:space="preserve">grauitas igitur aquæ magnitudinem habentis æqualem <lb/>plumbo A, erit 200.</s> <s xml:id="echoid-s318" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s319" xml:space="preserve">Similiter ſit aliquod cereum corpus A, cuius grauitas 2100, & </s> <s xml:id="echoid-s320" xml:space="preserve">opor <lb/>teat facere, quod imperatùm eſt. </s> <s xml:id="echoid-s321" xml:space="preserve">accipiatur aliquod paruum ceræ cor-<lb/>pus F, cuius grauitas ſit v. </s> <s xml:id="echoid-s322" xml:space="preserve">g. </s> <s xml:id="echoid-s323" xml:space="preserve">21, & </s> <s xml:id="echoid-s324" xml:space="preserve">inuenta grauitate aquæ magni- <pb o="10" file="0022" n="22" rhead="PROMOTVS"/> tudinem habentis æqualem ceræ F, quæ ſit 22, fiat vt 21, ad 22, ita <lb/>2100, ad alium numerum qui ſit 2200; </s> <s xml:id="echoid-s325" xml:space="preserve">erit igitur grauitas aquæ ma <lb/>gnitudinem habentis æqualem ceræ A, 2200.</s> <s xml:id="echoid-s326" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s327" xml:space="preserve">Neque neceſſe eſt, vt illud corpus ſolidum magni ponderis <lb/>reipſa proponatur, ſufficit enim vt eius grauitas notificetur <lb/>numero tantum.</s> <s xml:id="echoid-s328" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s329" xml:space="preserve">Si autem propoſitum fuerit inuenire quanta erit graui-<lb/>tas argenti viui magnitudine æqualis propoſito corpori ſo-<lb/>lido A; </s> <s xml:id="echoid-s330" xml:space="preserve">ratione qua ſupta, non inuenietur ipſa grauitas, quo-<lb/>niam nullum corpus demiſſum in argentum viuum fertur <lb/>deorſum, niſi aurum, aurum vero in ipſo argento viuo perrum-<lb/>pitur, ſed qua ratione inuenienda ſit ipſa argenti viui graui-<lb/>tas, dicemus ad finem exempli propoſitionis decimæquartæ.</s> <s xml:id="echoid-s331" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s332" xml:space="preserve">Quomodo ponderanda ſint corpora ſolida in aqua.</s> <s xml:id="echoid-s333" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s334" xml:space="preserve">COrpus quod ponderandum proponitur ſeta equina ex altera li-<lb/>bræ lance appendatur, in altera lance ponantur pondera, & </s> <s xml:id="echoid-s335" xml:space="preserve">cor-<lb/>pus appenſum demittatur in aquam, ita vt in aqua libere pendeat, ne-<lb/>que lancem, cui appenſum est corpus, neque aliam in qua ſunt pondera <lb/>aqua contingat, & </s> <s xml:id="echoid-s336" xml:space="preserve">ita ponderetur propoſitum corpus, ac ſi in aere pen <lb/>deret.</s> <s xml:id="echoid-s337" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s338" xml:space="preserve">Dixi ſeta equina corpus ponder andum debere appendi, quia fere <lb/>æque grauis eſt atque aqua, & </s> <s xml:id="echoid-s339" xml:space="preserve">ideo nihil addet, vel minuet grauita-<lb/>tis in ipſo corpore ponderando.</s> <s xml:id="echoid-s340" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s341" xml:space="preserve">Quod ſi corpus ponderandum fuerit, tam graue, vt ſeta ſimplici ſu-<lb/>stineri nequeat, appendatur pluribus ſimul iunctis ſetis, & </s> <s xml:id="echoid-s342" xml:space="preserve">ne aliquid <lb/>grauitatis ſetarum coniunctio addat corpori ponderando, ponantur <lb/>in altera lance totidem ſetæ æquales eis, quæ ex lance, cui appenſum eſt <lb/>corpus pendent, vſque ad corpus appenſum, hac igitur ſetarum addi-<lb/>tione æque ponderabunt lances, & </s> <s xml:id="echoid-s343" xml:space="preserve">quamuis illæ ſetæ, quibus appen-<lb/>ſum est corpus, ſint longiores, quam aliæ alteri lanci additæ, longitudi-<lb/>ne partium, quibus ligatum est corpus, tamen quoniam illæ partes <lb/>æque graues ſunt, atque aqua, exiſtentes cum ipſo corpore in aqua, nul-<lb/>lam grauitatem habebunt, & </s> <s xml:id="echoid-s344" xml:space="preserve">ideo illæ ſetæ quæ alias ſuperant dictis <lb/>partibus, & </s> <s xml:id="echoid-s345" xml:space="preserve">ſi longiores, non erunt grauiores quam aliæ, existenti-<lb/>bus, nempe, vt dictum eſt, illis partibus cum ipſo corpore in aqua Sic <lb/>igitur in aqua ponderanda erunt ſolida corpora, quod animaduertiſſe <lb/>fuit operæ pretium.</s> <s xml:id="echoid-s346" xml:space="preserve"/> </p> <pb o="11" file="0023" n="23" rhead="ARCHIMEDES."/> </div> <div xml:id="echoid-div25" type="section" level="1" n="15"> <head xml:id="echoid-head18" xml:space="preserve">PROBLEMA II. PROPOS. IX.</head> <p> <s xml:id="echoid-s347" xml:space="preserve">PRopoſitis duobus corporibus magnitudine æquali-<lb/>bus, vno ſolido, altero liquido, data corporis li-<lb/>quidi grauitate, grauitatem ſolidi inuenire.</s> <s xml:id="echoid-s348" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s349" xml:space="preserve">SINT duo propoſita corpora <lb/>magnitudine æqualia, A, quidem ſo-<lb/> <anchor type="figure" xlink:label="fig-0023-01a" xlink:href="fig-0023-01"/> lidum, B, vero liquidum, ſit autem li-<lb/>quidi B, data grauitas F, & </s> <s xml:id="echoid-s350" xml:space="preserve">oporteat <lb/>inuenire grauitatem ſolidi A, accipia <lb/>tur aliquod corpus ſolidum D, eiuſdĕ <lb/>generis, cum ſolido A, cuius grauitas <lb/>ſit H, deinde liquidi eiuſdem generis <lb/>cum liquido B, magnitudine æqualis <lb/>ſolido D,<anchor type="note" xlink:href="" symbol="*"/> inueniatur grauitas quæ ſit <anchor type="note" xlink:label="note-0023-01a" xlink:href="note-0023-01"/> G, & </s> <s xml:id="echoid-s351" xml:space="preserve">fiat vt G, ad H, ita F, ad aliam <lb/>grauitatem, quæ ſit C. </s> <s xml:id="echoid-s352" xml:space="preserve">Dico ſolidi A, <lb/>grauitatem eſſe C, accipiatur enim aliquod corpus liquidum E, eiuſ-<lb/>dem generis cum liquido B, grauitatem habens æqualem ſolido D, <lb/>Quoniam igitur ſunt quatuor corpora grauia B, A, E, D, quorum pri-<lb/>mum B, & </s> <s xml:id="echoid-s353" xml:space="preserve">ſecundum A, ſunt magnitudine æqualia, tertium vero E, <lb/>& </s> <s xml:id="echoid-s354" xml:space="preserve">quartum D, æquegrauia, & </s> <s xml:id="echoid-s355" xml:space="preserve">ſunt eiuſdem generis corpora B, E, ſi-<lb/>militer, & </s> <s xml:id="echoid-s356" xml:space="preserve">corpora A, D,<anchor type="note" xlink:href="" symbol="*"/> erit vt grauitas liquidi æqualis magnitudi- <anchor type="note" xlink:label="note-0023-02a" xlink:href="note-0023-02"/> ne ſolido D, hoc eſt vt G, ad grauitatem liquidi E; </s> <s xml:id="echoid-s357" xml:space="preserve">hoc eſt ad H, ponũ-<lb/>tnr enim æque grauia corpora D, E, ita grauitas F, ad ſolidi A, graui-<lb/>tatem, ſed vt grauitas G, ad grauitatem H, ita eſt grauitas F, ad C, <lb/>grauitatem, ergo grauitas C, æqualis erit grauitati ſolidi A. </s> <s xml:id="echoid-s358" xml:space="preserve">Inuenta <lb/>igitur eſt ſolidi A, grauitas C, quod facere oportebat.</s> <s xml:id="echoid-s359" xml:space="preserve"/> </p> <div xml:id="echoid-div25" type="float" level="2" n="1"> <figure xlink:label="fig-0023-01" xlink:href="fig-0023-01a"> <image file="0023-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0023-01"/> </figure> <note position="right" xlink:label="note-0023-01" xlink:href="note-0023-01a" xml:space="preserve">8. hu-<lb/>ius.</note> <note position="right" xlink:label="note-0023-02" xlink:href="note-0023-02a" xml:space="preserve">6. huius</note> </div> </div> <div xml:id="echoid-div27" type="section" level="1" n="16"> <head xml:id="echoid-head19" xml:space="preserve">Exemplum.</head> <p> <s xml:id="echoid-s360" xml:space="preserve">QVidam proponit aliquod corpus liquidum notæ <lb/>grauitatis, & </s> <s xml:id="echoid-s361" xml:space="preserve">vult ſcire quanta erit grauitas alicu-<lb/>ius ſolidi, magnitudinem habentis propoſito Corpori li-<lb/>quido æqualem.</s> <s xml:id="echoid-s362" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s363" xml:space="preserve">Sitpropoſitum aliquod corpus aqueum B, euius grauitas ſit 100. </s> <s xml:id="echoid-s364" xml:space="preserve">&</s> <s xml:id="echoid-s365" xml:space="preserve"> <pb o="12" file="0024" n="24" rhead="PROMOTVS"/> oporteat ſcire quanta erit grauitas plumbi magnitudinem habentis <lb/>æqualem propoſitæ aquæ B, verbi gratia ſit vas aliquod plenum aqua, <lb/>cuius aquæ grauitas ſit 100, & </s> <s xml:id="echoid-s366" xml:space="preserve">oporteat ſcire, ſi illud idĕ vas replea-<lb/>tur plumbo, quanta illius plumbi erit grauitas. </s> <s xml:id="echoid-s367" xml:space="preserve">Accipiatur aliquod <lb/>plumbeum corpus D, cuius grauitas ſit 23, deinde aquæ magnitudinĕ <lb/>habentis æqualem plumbo D, inueniatur grauitas, quod quomodo fie-<lb/>ri oporteat iam dictum eſt in antecedentis problematis exemplo: </s> <s xml:id="echoid-s368" xml:space="preserve">ſit igi <lb/>tur ea inuenta grauitas 2, & </s> <s xml:id="echoid-s369" xml:space="preserve">fiat vt 2, ad 23, ita 100, ad alium nume-<lb/>rum qui ſit 1150, is igitur numerus erit grauitas plumbi magnitu-: <lb/></s> <s xml:id="echoid-s370" xml:space="preserve">dinem habentis propoſitæ aquæ B, æqualem, hoc est illius plumbi, quod <lb/>in vaſe continetur.</s> <s xml:id="echoid-s371" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s372" xml:space="preserve">At vero ſi propoſitũ fuerit inuenire quãta erit grauitas ceræ, <lb/>aut ligni, aut cuiuſcũque ſolidi leuioris quam aqua, nihil diuer <lb/>ſi in opere accidet, niſi quod ratio inueniendi grauitatem <lb/>aquæ magnitudinem habentis æqualem corpori ſolido leuio-<lb/>ri, quàm aqua, differt in aliquo à ratione, qua inuenitur graui-<lb/>tas aquæ magnitudinem habentis æqualem ſolido corpori <lb/>grauiori, quam aqua, ſed vtramque rationem exemplo ante-<lb/>cedentis Problematis illuſtrauimus, in eo enim ſatis explica-<lb/>tum eſt de vtraque.</s> <s xml:id="echoid-s373" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s374" xml:space="preserve">Sed ne exemplorum inopia laborare videamur, ſit inuenienda gra-<lb/>uitas ceræ magnitudinem habentis æqualem propoſitæ aquæ B, acci-<lb/>piatur aliquod cereum corpus D, cuius grauitas ſit 21, deinde aquæ <lb/>magnitudinem habentis æqualem ceræ D, inueniatur grauitas, vt in <lb/>antecedentis Problematis exemplo dictum est, quæ grauitas ſit 22, & </s> <s xml:id="echoid-s375" xml:space="preserve"><lb/>fiat vt 22, ad 21, ita 100, hoc est grauitas aquæ B, ad alium numerum <lb/>qui ſit 95 {5/11}. </s> <s xml:id="echoid-s376" xml:space="preserve">is igitur numerus indicabit quanta erit grauitas ceræ <lb/>magnitudinem habentis æqualem propoſitæ aquæ B.</s> <s xml:id="echoid-s377" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s378" xml:space="preserve">Similiter ſi propoſitum liquidum corpus B, fuerit olei, aut <lb/>vini, aut cuiuſcumque liquidi, præter argenti viui, eadem om-<lb/>nino via, qua ante, inuenietur quæſita corporis ſolidi grauitas, <lb/>ſed de argento viuo tractabimus ad finem propoſitionis deci-<lb/>mæ quartæ.</s> <s xml:id="echoid-s379" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div28" type="section" level="1" n="17"> <head xml:id="echoid-head20" xml:space="preserve">PROBLEMA III. PROPOS. X.</head> <p> <s xml:id="echoid-s380" xml:space="preserve">PRopoſitis duobus corporibus æque grauibus, vno <lb/>ſolido, altero liquido, data ſolidi corporis magnitu- <pb o="13" file="0025" n="25" rhead="ARCHIMEDES."/> dine, magnitudinem liquidi inuenire.</s> <s xml:id="echoid-s381" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s382" xml:space="preserve">SINT duo propoſita corpora <lb/> <anchor type="figure" xlink:label="fig-0025-01a" xlink:href="fig-0025-01"/> æque grauia, A, quidem ſolidum B, <lb/>vero liquidum, ſit autem ſolidi A, da-<lb/>ta magnitudo C, & </s> <s xml:id="echoid-s383" xml:space="preserve">oporteat inuenire <lb/>quanta erit magnitudo liquidi B, Ac-<lb/>cipiatur aliquod corpus ſolidum D, <lb/>eiuſdem generis cum ſolido A, & </s> <s xml:id="echoid-s384" xml:space="preserve">ſit <lb/>eius grauitas G, & </s> <s xml:id="echoid-s385" xml:space="preserve">liquidi, quod ſit E, <lb/>eiuſdem generis cum liquido B, ma-<lb/>gnitudinem habentis æqualem ſolido <lb/>D,<anchor type="note" xlink:href="" symbol="*"/> inueniatur grauitas quæ ſit H, &</s> <s xml:id="echoid-s386" xml:space="preserve"> <anchor type="note" xlink:label="note-0025-01a" xlink:href="note-0025-01"/> fiat vt grauitas H, ad grauitatem G, <lb/>ita magnitudo C, ad aliam magnitudinem quæ ſit F. </s> <s xml:id="echoid-s387" xml:space="preserve">Quoniam igitur <lb/>ſunt quatuor corpora grauia E, D, B, A, quorum primum E, & </s> <s xml:id="echoid-s388" xml:space="preserve">ſecun-<lb/>dum D, ſunt æqualia magnitudine, tertium vero B, & </s> <s xml:id="echoid-s389" xml:space="preserve">quartum A, <lb/>æque grauia, & </s> <s xml:id="echoid-s390" xml:space="preserve">ſunt eiuſdem generis corpora E, B, ſimiliter, & </s> <s xml:id="echoid-s391" xml:space="preserve">corpo-<lb/>ra D, A, <anchor type="note" xlink:href="" symbol="*"/> erit vt grauitas H, ad grauitatem G, ita magnitudo C, ad li- <anchor type="note" xlink:label="note-0025-02a" xlink:href="note-0025-02"/> quidi B, magnitudinem, ſed vt grauitas H, ad grauitatem G, ita eſt <lb/>magnitudo C, ad magnitudinem F, ergo magnitudo F, æqualis erit <lb/>magnitudini liquidi B, inuenta igitur eſt liquidi corporis B, magni-<lb/>tudo F, quod facere oportebat.</s> <s xml:id="echoid-s392" xml:space="preserve"/> </p> <div xml:id="echoid-div28" type="float" level="2" n="1"> <figure xlink:label="fig-0025-01" xlink:href="fig-0025-01a"> <image file="0025-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0025-01"/> </figure> <note position="right" xlink:label="note-0025-01" xlink:href="note-0025-01a" xml:space="preserve">8. buius</note> <note position="right" xlink:label="note-0025-02" xlink:href="note-0025-02a" xml:space="preserve">7. huius</note> </div> <p> <s xml:id="echoid-s393" xml:space="preserve">Sed quoniam corporum regularium magnitudo quo-<lb/>que exprimitur latere eiuſdem corporis, vel diametro, ſi <lb/>propoſita duo corpora A, B, ſuerint regularia, vtpote ſphę <lb/>rica, fuerit autem ſphæræ A, data diameter C, & </s> <s xml:id="echoid-s394" xml:space="preserve">oporteat <lb/>inuenire quanta erit diameter ſphæræ B. </s> <s xml:id="echoid-s395" xml:space="preserve">ita faciendum <lb/>erit.</s> <s xml:id="echoid-s396" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s397" xml:space="preserve">Accepto, vt diximus, aliquo corpore ſolido D, eiuſdem generis cũ <lb/>ſphæra A, & </s> <s xml:id="echoid-s398" xml:space="preserve">inuenta grauitate liquidi E, vt ſupra, fiat vt grauitas H, <lb/>ad grauitatem G, ita cubus ex C, ad alium cubum, cuius latus ſit F, <lb/>dico ipſum latus F, æquale eſſe diametro ſphæræ B. </s> <s xml:id="echoid-s399" xml:space="preserve">Quoniam enim <lb/>eadem ratione qua ſupra demonſtrabitur, vt grauitas H, ad grauita-<lb/>tem G, ita eſſe magnitudinem ſphæræ A, ad ſphæræ B, magnitudinem, <lb/>ſed magnitudo ſphæræ A, ad magnitudinem ſphæræ B, <anchor type="note" xlink:href="" symbol="*"/> triplicatam <anchor type="note" xlink:label="note-0025-03a" xlink:href="note-0025-03"/> rationem habet eius, quam C, diameter ſphæræ A, ad diametrum <lb/>ſphæræ B, ſimiliter & </s> <s xml:id="echoid-s400" xml:space="preserve">cubus ex C, ad cubum ex diametro ſphæræ B, <pb o="14" file="0026" n="26" rhead="PROMOTVS"/> <anchor type="note" xlink:label="note-0026-01a" xlink:href="note-0026-01"/> triplicatam <anchor type="note" xlink:href="" symbol="*"/> rationem habet eius, quam C, ad diametrum ſphæræ B, ergo vt grauitas H, ad grauitatem G, ita erit cubus ex C, ad cubum <lb/>ex diametro ſphæræ B, ſed vt grauitas H, ad grauitatem G, ita eſt cu <lb/>bus ex C, ad cubum ex F, ergo cubus ex F, æqualis erit cubo diame-<lb/>tri ſphæræ B; </s> <s xml:id="echoid-s401" xml:space="preserve">quare & </s> <s xml:id="echoid-s402" xml:space="preserve">latus F, æquabitur ſphæræ B, diametro. </s> <s xml:id="echoid-s403" xml:space="preserve">inuen-<lb/>ta igitur eſt quantitas diametri liquidæ ſphæræ B, quod facere opor-<lb/>tebat.</s> <s xml:id="echoid-s404" xml:space="preserve"/> </p> <div xml:id="echoid-div29" type="float" level="2" n="2"> <note position="right" xlink:label="note-0025-03" xlink:href="note-0025-03a" xml:space="preserve">18. 12. <lb/>Elem.</note> <note position="left" xlink:label="note-0026-01" xlink:href="note-0026-01a" xml:space="preserve">13. 12. <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div31" type="section" level="1" n="18"> <head xml:id="echoid-head21" xml:space="preserve">Exemplum.</head> <p> <s xml:id="echoid-s405" xml:space="preserve">Q Vidam proponit aliquod corpus ſolidum notæ <lb/>magnitudinis, & </s> <s xml:id="echoid-s406" xml:space="preserve">vult ſcire quanta erit magnitudo <lb/>alicuius liquidi, grauitatem habentis propoſito corpori <lb/>ſolido æqualem.</s> <s xml:id="echoid-s407" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s408" xml:space="preserve">Sit propoſitum aliquod corpus plumbeum A, cuius magnitudo ſit <lb/>10, & </s> <s xml:id="echoid-s409" xml:space="preserve">oporteat ſcire quanta erit magnitudo aquæ grauitatem haben-<lb/>tis æqualem propoſito plumbo A, accipiatur aliquod corpus plumbeũ <lb/>D, cuius grauitas 23, deinde aquæ magnitudinem habentis æqualem <lb/>plumbo D, inueniatur grauitas, vt in exemplo propoſ. </s> <s xml:id="echoid-s410" xml:space="preserve">8. </s> <s xml:id="echoid-s411" xml:space="preserve">dictum eſt, <lb/>quæ ſit 2, & </s> <s xml:id="echoid-s412" xml:space="preserve">fiat vt 2, ad 23, ita 10, ad alium numerum qui ſit 115, is <lb/>igitur indicabit quanta erit magnitudo aquæ grauitatem habentis <lb/>æqualem propoſito plumbo A.</s> <s xml:id="echoid-s413" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s414" xml:space="preserve">Quod ſi propoſitũ corpus plumbeum A ſit regulare vt po-<lb/>te ſphæricum, cuius ſphæræ diameter ſit 10, & </s> <s xml:id="echoid-s415" xml:space="preserve">oporteat inue-<lb/>nire quanta erit diameter ſphæræ ex aqua, grauitatem haben-<lb/>tis æqualem propoſitæ ſphæræ A, ita faciendum erit.</s> <s xml:id="echoid-s416" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s417" xml:space="preserve">Accipiatur, vt diximus, aliquod corpus plumbeum D, cuius gra-<lb/>uitas 23, deinde aquæ habentis magnitudinem æqualem plumbo D, <lb/>inueniatur grauitas quæ ſit 2, & </s> <s xml:id="echoid-s418" xml:space="preserve">fiat vt 2, ad 23, ita cubus ex 10, qui <lb/>eſt 1000, ad alium numerum qui ſit 11500, is igitur numerus erit <lb/>cubus diametri ſpbæræ ex aqua grauitatem habentis æqualem propo-<lb/>ſitæ ſpbæræ A, quare eius latus cubicum, quod est 22 {57/100}, proximũ <lb/>vero indicabit ipſam diametrum.</s> <s xml:id="echoid-s419" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s420" xml:space="preserve">Similiter ſi propoſitum corpus plumbeum A, fuerit cubicum, vel <lb/>alicuius alterius formæ regularis, eadem ratione inueniemus latus cu <lb/>bi ex aqua grauitatem habentis æqualem propoſito cubo A, nam ſi cu-<lb/>bi A, datum ſit latus 10, erit numerus 11500, cubus ex aqua æqualis <lb/>grauitate propoſito cubo A, quare latus cubicum numeri 11500, quod <pb o="15" file="0027" n="27" rhead="ARCHIMEDES."/> est 22 {57/100}. </s> <s xml:id="echoid-s421" xml:space="preserve">proxìmũ vero indicabit quæſitum latus cubi ex aqua.</s> <s xml:id="echoid-s422" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s423" xml:space="preserve">Neque diſſimili ratione inuenietur magnitudo olei, aut ar-<lb/>genti viui, aut cuiuſcumque generis liquidi grauitatem habĕ-<lb/>tis propoſito corpori ſolido æqualem, ſed quomodo inuenien-<lb/>da ſit grauitas argenti viui magnitudinem habentis æqualem <lb/>corpori ſolido, docebimus poſt exemplum propoſitionis de-<lb/>cimæ quartæ.</s> <s xml:id="echoid-s424" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div32" type="section" level="1" n="19"> <head xml:id="echoid-head22" xml:space="preserve">PROBLEMA IV. PROPOS. XI.</head> <p> <s xml:id="echoid-s425" xml:space="preserve">P Ropoſitis duobus corporibus æque grauibus, vno ſo <lb/>lido, altero liquido, data liquidi corporis magnitu-<lb/>dine, magnitudinem ſolidi inuenire.</s> <s xml:id="echoid-s426" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s427" xml:space="preserve">SINT propoſita duo corpora æ-<lb/> <anchor type="figure" xlink:label="fig-0027-01a" xlink:href="fig-0027-01"/> quæ grauia, A, quidem ſolidum, B, <lb/>vero liquidum, ſit autem liquidi B; <lb/></s> <s xml:id="echoid-s428" xml:space="preserve">data magnitudo F, & </s> <s xml:id="echoid-s429" xml:space="preserve">oporteat ſolidi <lb/>A, magnitudinem inuenire. </s> <s xml:id="echoid-s430" xml:space="preserve">Accipia-<lb/>tur aliquod corpus ſolidum D, eiuſdĕ <lb/>generis cum corpore ſolido A, cuius <lb/>grauitas ſit G, deinde liquidi quod ſit <lb/>E, eiuſdem generis cum corpore liqui <lb/>do B, magnitudinem æ qualem haben-<lb/>tis ſolido D, <anchor type="note" xlink:href="" symbol="*"/> inueniatur grauitas, <anchor type="note" xlink:label="note-0027-01a" xlink:href="note-0027-01"/> quæ ſit H, & </s> <s xml:id="echoid-s431" xml:space="preserve">fiat vt grauitas G; </s> <s xml:id="echoid-s432" xml:space="preserve">ad gra-<lb/>uitatem H, ita F, magnitudo, ad aliam magnitudinem, quæ ſit C; <lb/></s> <s xml:id="echoid-s433" xml:space="preserve">quoniam igitur ſunt quatuor corpora D, E, A, B, quorum primum D, <lb/>& </s> <s xml:id="echoid-s434" xml:space="preserve">ſecundum E, ſunt magnitudine æqualia, tertium vero A, & </s> <s xml:id="echoid-s435" xml:space="preserve">quartũ <lb/>B, æquæ grauia, & </s> <s xml:id="echoid-s436" xml:space="preserve">ſunt eiuſdem generis ſolida D, A, ſimiliter, & </s> <s xml:id="echoid-s437" xml:space="preserve">liqui-<lb/>da E, B, <anchor type="note" xlink:href="" symbol="*"/> erit vt grauitas G, ad grauitatem H, ita F, magnitudo ad ma <anchor type="note" xlink:label="note-0027-02a" xlink:href="note-0027-02"/> gnitudinem ſolidi A, ſed vt grauitas G, ad grauitatem H, ita eſt ma-<lb/>gnitudo F, ad C, magnitudinem, ergo magnitudo C, æqualis erit ma-<lb/>gnitudini corporis ſolidi A, inuenta igitur eſt corporis folidi A, ma-<lb/>gnitudo C, quod erat faciendum.</s> <s xml:id="echoid-s438" xml:space="preserve"/> </p> <div xml:id="echoid-div32" type="float" level="2" n="1"> <figure xlink:label="fig-0027-01" xlink:href="fig-0027-01a"> <image file="0027-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0027-01"/> </figure> <note position="right" xlink:label="note-0027-01" xlink:href="note-0027-01a" xml:space="preserve">8. huius</note> <note position="right" xlink:label="note-0027-02" xlink:href="note-0027-02a" xml:space="preserve">7. huius</note> </div> <p> <s xml:id="echoid-s439" xml:space="preserve">Q Vod ſi propoſita duo corpora æque grauia A, B, fue <lb/>rint regularia vtpote ſphærica, fuerit autem liqui- <pb o="16" file="0028" n="28" rhead="PROMOTVS"/> dæ ſphæræ B, data diameter F, & </s> <s xml:id="echoid-s440" xml:space="preserve">oporteat inuenire quan <lb/>ta erit diameter ſolidæ ſphæræ A, ita faciendum erit.</s> <s xml:id="echoid-s441" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s442" xml:space="preserve">Accepto vt ſupra corpore ſolido D, & </s> <s xml:id="echoid-s443" xml:space="preserve">liquidi E, inuenta grauitate, <lb/>vt dictum eſt, fiat vt grauitas G, ad prauitatem H, ita cubus ex F, ad <lb/>alium cubum, cuius latus ſit C, Quoniam igitur eadem ratione qua <lb/>ſupra oſtendetur, vt grauitas G, ad grauitatem H, ita eſſe magnitudi-<lb/>nem ſphæræ B, ad ſphæræ A, magnitudinem, ſed magnitudo ſphæræ B, <lb/>ad magnitudinem ſphæræ A, <anchor type="note" xlink:href="" symbol="*"/> triplicatam rationem habet eius, quam <anchor type="note" xlink:label="note-0028-01a" xlink:href="note-0028-01"/> F, diameter ſphæræ B, ad diametrum ſphæræ A, ſimiliter, & </s> <s xml:id="echoid-s444" xml:space="preserve">cubus ex <lb/>F, ad cubum ex diametro<unsure/> ſphæræ A, triplicatã <anchor type="note" xlink:href="" symbol="*"/> rationem habet eius, <anchor type="note" xlink:label="note-0028-02a" xlink:href="note-0028-02"/> quam F, ad diametrum ſphæræ A, ergo, vt grauitas G, ad grauitatĕ <lb/>H, ita erit cubus ex F, ad cubum ex diametro ſphæræ A, ſed vt graui-<lb/>tas G, ad grauitatem H, ita eſt cubus ex F, ad cubum ex C, ergo cubus <lb/>ex C, æqualis erit cubo diametri ſphæræ A, quare, & </s> <s xml:id="echoid-s445" xml:space="preserve">latus C, æquabi-<lb/>tur ipſius ſphæræ A, diametro, inuenta igitur eſt quantitas diametri <lb/>ſolidæ ſphæræ A, quod facere oportebat.</s> <s xml:id="echoid-s446" xml:space="preserve"/> </p> <div xml:id="echoid-div33" type="float" level="2" n="2"> <note position="left" xlink:label="note-0028-01" xlink:href="note-0028-01a" xml:space="preserve">18. 12. <lb/>Elem.</note> <note position="left" xlink:label="note-0028-02" xlink:href="note-0028-02a" xml:space="preserve">33. 11. <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div35" type="section" level="1" n="20"> <head xml:id="echoid-head23" xml:space="preserve">Exemplum.</head> <p> <s xml:id="echoid-s447" xml:space="preserve">Q Vidam proponit aliquod corpus liquidum notæ <lb/>magnitudinis, & </s> <s xml:id="echoid-s448" xml:space="preserve">vult inuenire quanta erit magni-<lb/>tudo alicuius ſolidi grauitatem habentis propoſi-<lb/>to corpori liquido æqualem.</s> <s xml:id="echoid-s449" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s450" xml:space="preserve">Sit propoſitum aliquod corpus aqueum B, cuius magnitudo ſit 115, <lb/>& </s> <s xml:id="echoid-s451" xml:space="preserve">oporteat inuenire quanta erit magnitudo plumbi grauitatem ha-<lb/>bentis æqualem propo ſitæ aquæ B, accipiatur aliquod corpus plumbeũ <lb/>D, cuius grauitas ſit verbi gratia 23, deinde aquæ magnitudinem ha-<lb/>bentis æqualem plumbo D, inueniatur grauitas quæ ſit 2. </s> <s xml:id="echoid-s452" xml:space="preserve">id autem do-<lb/>cuit propoſitionis octauæ exemplum, & </s> <s xml:id="echoid-s453" xml:space="preserve">fiat vi 23, ad 2, ita 115, ad <lb/>alium numerum qui ſit 10, is igitur numerus indicabit quanta erit <lb/>magnitudo plumbi grauitatem habentis æqualem propoſitæ aquæ B.</s> <s xml:id="echoid-s454" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s455" xml:space="preserve">Quod ſi propoſitum corpus aqueum B, ſit ſphæricum, cuius <lb/>ſphæræ diameter ſit 10, & </s> <s xml:id="echoid-s456" xml:space="preserve">oporteat inuenire quanta erit dia-<lb/>meter ſphæræ ex plumbo, grauitatem habentis æqualem pro-<lb/>poſitæ ſphæræ B, ita faciendum erit.</s> <s xml:id="echoid-s457" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s458" xml:space="preserve">Accepto, vt diximus aliquo corpore plumbeo D, cuius grauitas 23, <lb/>& </s> <s xml:id="echoid-s459" xml:space="preserve">aquæ magnitudinem habentis æqualem plumbo D, inuenta graui- <pb o="17" file="0029" n="29" rhead="ARCHIMEDES."/> tate 2, fiat vt 23, ad 2, ita cubus ex 10, boc est 1000, ad alium nume-<lb/>rum qui ſit 86 {22/23}. </s> <s xml:id="echoid-s460" xml:space="preserve">is igitur numerus erit cubus diametri ſpbæræ ex <lb/>plumbo, grauitatem æqualem babentis propoſitæ ex aqua ſpbæræ B, <lb/>quare eius latus cubicum, quod est 4 {43/100}. </s> <s xml:id="echoid-s461" xml:space="preserve">ferè, indicabit ipſam <lb/>diametrum.</s> <s xml:id="echoid-s462" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s463" xml:space="preserve">Similiter ſi propoſitum corpus aqueum B, fuerit cubicum, vel alicu <lb/>ius alterius formæ regularis, eadem ratione vtemur ad inueniendum <lb/>latus cubi ex plumbo, grauitatem babentis æqualem propoſito ex aqua <lb/>cubo B, nam ſi ex aqua cubi B, datum ſit latus 10, erit numerus <lb/>86 {22/23}. </s> <s xml:id="echoid-s464" xml:space="preserve">cubus ex plumbo æqualis grauitate propoſito ex aqua cubo B, <lb/>quare latus cubicum numeri 86 {22/23}. </s> <s xml:id="echoid-s465" xml:space="preserve">quod eſt 4 {43/100}. </s> <s xml:id="echoid-s466" xml:space="preserve">ferè, indicabit <lb/>quæſitum latus cubi ex plumbo.</s> <s xml:id="echoid-s467" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s468" xml:space="preserve">Neque diſſimili ratione inuenienda erit magnitudo auri, <lb/>argenti, ceræ, aut cuiuſcunque ſolidi, grauitatem habentis <lb/>propoſito corpori liquido æqualem.</s> <s xml:id="echoid-s469" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div36" type="section" level="1" n="21"> <head xml:id="echoid-head24" xml:space="preserve">PROBLEMAV. PROPOS. XII.</head> <p> <s xml:id="echoid-s470" xml:space="preserve">PRopoſitis duobus ſolidis corporibus magnitudine <lb/>æqualibus, data grauitate vnius, grauitatem al-<lb/>terius inuenire.</s> <s xml:id="echoid-s471" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s472" xml:space="preserve">SINT propoſita duo corpora ſo-<lb/> <anchor type="figure" xlink:label="fig-0029-01a" xlink:href="fig-0029-01"/> lida magnitudine æqualia A, B, ſit au-<lb/>tem vnius, vtpote ipſius A, data gra-<lb/>uitas C, & </s> <s xml:id="echoid-s473" xml:space="preserve">oporteat inuenire grauita-<lb/>tem ipſius B. </s> <s xml:id="echoid-s474" xml:space="preserve">Accipiatur aliquod ſoli-<lb/>dum corpus D, eiuſdem generis cum <lb/>corpore ſolido A, cuiæquale grauita-<lb/>te accipiatur alterum E, eiuſdem ge-<lb/>neris cum corpore B, deinde liquidi <lb/>magnitudine æqualis corpori D, <anchor type="note" xlink:href="" symbol="*"/> in- <anchor type="note" xlink:label="note-0029-01a" xlink:href="note-0029-01"/> ueniatur grauitas, quæ ſit G, item li-<lb/>quidi eiuſdem generis, æqualis ma-<lb/>gnitudine corpori E,<anchor type="note" xlink:href="" symbol="*"/> inueniatur grauitas, quæ ſit H, & </s> <s xml:id="echoid-s475" xml:space="preserve">fiat vt H, ad <anchor type="note" xlink:label="note-0029-02a" xlink:href="note-0029-02"/> G, ita C, ad aliam grauitatem, quæ ſit F. </s> <s xml:id="echoid-s476" xml:space="preserve">Quoniam igitur ſunt qua-<lb/>tuor corpora A, B, D, E, quorum A, B, primum videlicet, & </s> <s xml:id="echoid-s477" xml:space="preserve">ſecundum <lb/>ſunt æqualia magnitudine, tertium vero D, & </s> <s xml:id="echoid-s478" xml:space="preserve">E, quartum æque gra-<lb/>uia, & </s> <s xml:id="echoid-s479" xml:space="preserve">ſunt eiuſdĕ generis ſolida A, D, itidem ſolida B, E, <anchor type="note" xlink:href="" symbol="*"/> erit vt gra- <anchor type="note" xlink:label="note-0029-03a" xlink:href="note-0029-03"/> uitas C, ad ſolidi B, grauitatem, ita grauitas H, ad grauitatem G, ſed <pb o="18" file="0030" n="30" rhead="PROMOTVS"/> vt grauitas H, ad grauitatĕ G, ita eſt grauitas C, ad F, grauitatem; </s> <s xml:id="echoid-s480" xml:space="preserve">er-<lb/>go grauitas F, æqualis erit grauitati ſolidi B, inuenta igitur eſt cor-<lb/>poris ſolidi B, grauitas F, quod ſacere oportebat.</s> <s xml:id="echoid-s481" xml:space="preserve"/> </p> <div xml:id="echoid-div36" type="float" level="2" n="1"> <figure xlink:label="fig-0029-01" xlink:href="fig-0029-01a"> <image file="0029-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0029-01"/> </figure> <note position="right" xlink:label="note-0029-01" xlink:href="note-0029-01a" xml:space="preserve">8. buius</note> <note position="right" xlink:label="note-0029-02" xlink:href="note-0029-02a" xml:space="preserve">8. buius</note> <note position="right" xlink:label="note-0029-03" xlink:href="note-0029-03a" xml:space="preserve">6. buius</note> </div> <p> <s xml:id="echoid-s482" xml:space="preserve">Hoc Problema magni momenti eſt, pleriſque artificibus <lb/>maximo vſui eſſe poteſt. </s> <s xml:id="echoid-s483" xml:space="preserve">in arte fuſoria propoſito operis modu <lb/>lo, ex illius grauitate, facile metalli ad opus faciendum, gra-<lb/>uitatem inueniet, ſi enim hoc ignoret artifex, periculum eſt, ne <lb/>metallum, aut deficiat, vel ſi multum eſt, ob nimiam graui-<lb/>tatem difficile tractetur.</s> <s xml:id="echoid-s484" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s485" xml:space="preserve">Neque tormenti bellici magiſtro inutile erit, is enim cogni-<lb/>ta grauitate alicuius globi, exempli gratia ex plumbo, ſtatim <lb/>alterius globi eiuſdem magnitudinis, vel ſit ex lapide, vel ex <lb/>ferro, vel ex qua cunque alia materia, grauitatem inueniet.</s> <s xml:id="echoid-s486" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div38" type="section" level="1" n="22"> <head xml:id="echoid-head25" xml:space="preserve">Exemplum.</head> <p> <s xml:id="echoid-s487" xml:space="preserve">QVidam proponit aliquod corpus ſolidum notæ <lb/>grauitatis, & </s> <s xml:id="echoid-s488" xml:space="preserve">vult ſcire quanta erit grauitas alicu-<lb/>ius ſolidi, alterius generis, magnitudinem habentis pro-<lb/>poſito corpori ſolido æqualem.</s> <s xml:id="echoid-s489" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s490" xml:space="preserve">Sit propoſitum aliquod corpus plumbeum A, cuius grauitas ſit <lb/>1150, & </s> <s xml:id="echoid-s491" xml:space="preserve">oporteat inuenire quanta erit grauitas stanni magnitudinĕ <lb/>babentis æqualem propoſito plumbo A. </s> <s xml:id="echoid-s492" xml:space="preserve">Accipiantur duo corpora æque <lb/>grauia, D, plumbeum, E, stanneum, deinde duarum quantitatum, <lb/>aquæ, quarum vna ſit æqualis magnitudine plumbo D, altera ſtanno <lb/>E, inueniãtur grauitates, quæ ſint, primæ videlicet quantitatis aquæ <lb/>74, ſecundæ vero 115, & </s> <s xml:id="echoid-s493" xml:space="preserve">fiat vt 115, ad 74, ita 1150, ad alium nume-<lb/>rum, qui ſit 740, is igitur numerus indicabit grauitatem stanni, ma-<lb/>gnitudinem habentis propoſito plumbo A.</s> <s xml:id="echoid-s494" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s495" xml:space="preserve">Etiam ſi non accipiantur duo corpora, plumbeum videlicet <lb/>& </s> <s xml:id="echoid-s496" xml:space="preserve">ſtanneum, æque grauia, ſed grauitate quacunque, grauitas <lb/>ſtanni magnitudinem habentis æqualem propoſito plumbo <lb/>D, inuenietur ſic.</s> <s xml:id="echoid-s497" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s498" xml:space="preserve">Accipiantur duo corpora D, plumbeum, E, ſtanneum grauitate <lb/>quacunque, ſit vcrbi gratia plumbi D, grauitas 23, ſtanni vero E, <lb/>grauitas 37, deinde duarum quantitatum aquæ, quarum vna ſit ma-<lb/>gnitudine æqualis plumbo D, altera stanno E, inueniantur graui-<lb/>tates, quæ ſint, primæ videlicet quantitatis 2, ſecundæ vero 5, & </s> <s xml:id="echoid-s499" xml:space="preserve">fiat, <pb o="19" file="0031" n="31" rhead="ARCHIMEDES."/> vt 23, ad 2, ita 37, ad 3 {5/23}. </s> <s xml:id="echoid-s500" xml:space="preserve">grauitas igitur aquæ, magnitudinem <lb/>habentis æqualem plumbo, cuius grauitas est 37, erit 3 {5/23}.</s> <s xml:id="echoid-s501" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s502" xml:space="preserve">Et quoniam aquæ, magnitudinem habentis æqualem stanno E, cu-<lb/>ius grauitas est 37, eft grauitas 5, erunt grauitates duarum quan-<lb/>titatum aquæ 3 {5/23}, & </s> <s xml:id="echoid-s503" xml:space="preserve">5, quarum quantitatum prima eſt æqualis <lb/>magnitudine corpori plumbeo, ſecunda stanneo, quæ ſunt æque gra-<lb/>uia, vtriuſque enim grauitas ect 37. </s> <s xml:id="echoid-s504" xml:space="preserve">Fiat igitur vt 5, ad 3 {5/23}, ita <lb/>1150, ad alium numerum, qui ſit 740, tanta igitur erit grauitas <lb/>ſtanni, magnitudinem habentis æqualem propoſito plumbo A, quanta <lb/>etiam inueniebatur & </s> <s xml:id="echoid-s505" xml:space="preserve">ſupra.</s> <s xml:id="echoid-s506" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s507" xml:space="preserve">Quod ſi propoſitum ſit cereum corpus aliquod, aut cuiuſ-<lb/>cunque generis ſolidi, ſiue leuioris quam aqua, ſiue grauio-<lb/>ris, & </s> <s xml:id="echoid-s508" xml:space="preserve">oporteat inuenire grauitatem alicuius ſolidi alterius <lb/>generis, magnitudine æqualis propoſito corpori ſolido. </s> <s xml:id="echoid-s509" xml:space="preserve">Ea-<lb/>dem ratione qua ſupra inuenietur quæſita ſolidi grauitas, ſed <lb/>hoc ſolum animaduertendum eſt, quod non eadem ratione <lb/>inuenitur grauitas aquæ, magnitudinem habentis æqualem <lb/>propoſito cuiuſcunque generis ſolido, alia enim tenenda eſt <lb/>ratio ad inueniendam grauitatem prædictæ aquæ, quando <lb/>propoſitum ſolidum ſit grauius quam aqua, alia vero quando <lb/>leuius, ſed ſiue ſit leuius, ſiue grauius, de inuentione huiuſ-<lb/>modi grauitatis, in exemplo propoſitionis octauæ, ſatis eſt ex-<lb/>plicatum.</s> <s xml:id="echoid-s510" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div39" type="section" level="1" n="23"> <head xml:id="echoid-head26" xml:space="preserve">PROBLEMA VI. PROPOS. XIII.</head> <p> <s xml:id="echoid-s511" xml:space="preserve">PRopoſitis duobus ſolidis corporibus æque graui-<lb/>bus, data magnitudine vnius, magnitudinem alte-<lb/>rius inuenire.</s> <s xml:id="echoid-s512" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s513" xml:space="preserve">SINT propoſita duo corpora ſolida æque grauia A, B, ſit au-<lb/>tem vnius, vtpote ipſius A, data magnitudo C, & </s> <s xml:id="echoid-s514" xml:space="preserve">oporteat inueni-<lb/>re magnitudinem ipſius B, Accipiatur aliquod ſolidum corpus D, <lb/>eiuſdem generis cum ſolido A, & </s> <s xml:id="echoid-s515" xml:space="preserve">ſit eius grauitas G, deinde ſolidi <lb/>corporis quod ſit E, eiuſdem generis cum ſolido B, magnitudine <lb/>æqualis ipſi D, inueniatur grauitas, quæ ſit H, hoc autem, Proble-<lb/>ma antecedens docuit, & </s> <s xml:id="echoid-s516" xml:space="preserve">fiat vt grauitas H, ad grauitatem G, ita <lb/>magnitudo C, ad aliam magnitudinem, quæ ſit F. </s> <s xml:id="echoid-s517" xml:space="preserve">Quoniam igitur <pb o="20" file="0032" n="32" rhead="PROMOTVS"/> ſunt quatuor corpora grauia E, D, B, <lb/> <anchor type="figure" xlink:label="fig-0032-01a" xlink:href="fig-0032-01"/> A, quorum E, D, primum videlicet, <lb/>& </s> <s xml:id="echoid-s518" xml:space="preserve">ſecundum, ſunt æqualia magnitu-<lb/>dine, tertium vero B, & </s> <s xml:id="echoid-s519" xml:space="preserve">quartum A, <lb/>æquegrauia, & </s> <s xml:id="echoid-s520" xml:space="preserve">ſunt eiuſdem generis <lb/>corpora E, B, ſimiliter & </s> <s xml:id="echoid-s521" xml:space="preserve">corpora <lb/> <anchor type="note" xlink:label="note-0032-01a" xlink:href="note-0032-01"/> D, A, <anchor type="note" xlink:href="" symbol="*"/> erit vt grauitas H, ad graui- tatem G, ita magnitudo C, ad corpo-<lb/>ris B, magnitudinem, ſed vt grauitas <lb/>H, ad grauitatem G, ita eſt magnitu-<lb/>do C, ad F, magnitudinem, ergo ma-<lb/>gnitudo F, æqualis erit magnitudini <lb/>corporis B. </s> <s xml:id="echoid-s522" xml:space="preserve">inuenta igitur eſt corporis B, magnitudo F, quod facere <lb/>oportebat.</s> <s xml:id="echoid-s523" xml:space="preserve"/> </p> <div xml:id="echoid-div39" type="float" level="2" n="1"> <figure xlink:label="fig-0032-01" xlink:href="fig-0032-01a"> <image file="0032-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0032-01"/> </figure> <note position="left" xlink:label="note-0032-01" xlink:href="note-0032-01a" xml:space="preserve">7. buius</note> </div> <p> <s xml:id="echoid-s524" xml:space="preserve">Quod ſi propoſita duo corpora æque grauia A, B, fue <lb/>rint regularia, vtpote ſphærica, fuerit autem ſphæræ A, <lb/>data diameter C, & </s> <s xml:id="echoid-s525" xml:space="preserve">oporteat inuenire, quanta erit dia-<lb/>meter ſphærę B, ita faciendum erit.</s> <s xml:id="echoid-s526" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s527" xml:space="preserve">Accepto corpore ſolido D, & </s> <s xml:id="echoid-s528" xml:space="preserve">inuenta ſolidi corporis E, grauita-<lb/>te, vt ſupra dictum eſt, fiat vt grauitas H, ad grauitatem G, ita cu-<lb/>bus ex C, ad alium cubum, cuius latus ſit F. </s> <s xml:id="echoid-s529" xml:space="preserve">Quoniam igitur eadem <lb/>ratione, qua ſupra, demonſtrabitur, vt grauitas H, ad grauitatem G, <lb/>ita eſſe magnitudinem ſphæræ A, ad ſphæræ B, magnitudinem, ſed <lb/> <anchor type="note" xlink:label="note-0032-02a" xlink:href="note-0032-02"/> magnitudo ſphæræ A, ad ſphæræ B, magnitudinem <anchor type="note" xlink:href="" symbol="*"/> triplicatam ra- tionem habet eius, quam C, diameter ſphæræ A, ad diametrum ſphæ-<lb/> <anchor type="note" xlink:label="note-0032-03a" xlink:href="note-0032-03"/> ræ B. </s> <s xml:id="echoid-s530" xml:space="preserve">Similiter & </s> <s xml:id="echoid-s531" xml:space="preserve">cubus ex C, ad cubum, ex diametro ſphæræ B, <anchor type="note" xlink:href="" symbol="*"/> tri- plicatam rationem habet eius, quam C, ad ſphæræ B, diametrum; </s> <s xml:id="echoid-s532" xml:space="preserve">er-<lb/>go vt grauitas H, ad grauitatem G, ita erit cubus ex C, ad cubum ex <lb/>diametro ſphæræ B, ſed vt grauitas H, ad grauitatem G, ita eſt cubus <lb/>ex C, ad cubum ex F; </s> <s xml:id="echoid-s533" xml:space="preserve">ergo cubus ex F, æqualis erit cubo diametri <lb/>ſphæræ B; </s> <s xml:id="echoid-s534" xml:space="preserve">quare & </s> <s xml:id="echoid-s535" xml:space="preserve">latus F, æquabitur ſphæræ B, diametro. </s> <s xml:id="echoid-s536" xml:space="preserve">inuenta <lb/>igitur eſt quantitas diametri ſphæræ B, quod facere oportebat.</s> <s xml:id="echoid-s537" xml:space="preserve"/> </p> <div xml:id="echoid-div40" type="float" level="2" n="2"> <note position="left" xlink:label="note-0032-02" xlink:href="note-0032-02a" xml:space="preserve">18. 12. <lb/>Elem.</note> <note position="left" xlink:label="note-0032-03" xlink:href="note-0032-03a" xml:space="preserve">33. 11. <lb/>Elem.</note> </div> <p> <s xml:id="echoid-s538" xml:space="preserve">Neque hoc Problema inutile erit tormenti bellici magiſtro, <lb/>is enim cognita diametro alicuius globi, exempli gratia, ex <lb/>plumbo, ſtatim alterius globi eandem habentis grauitatem, <lb/>diametrum inueniet, ſit globus ille, vel ex lapide, vel ex fer-<lb/>ro, vel ex quocunque alio ſolidorum genere.</s> <s xml:id="echoid-s539" xml:space="preserve"/> </p> <pb o="21" file="0033" n="33" rhead="ARCHIMEDES."/> </div> <div xml:id="echoid-div42" type="section" level="1" n="24"> <head xml:id="echoid-head27" xml:space="preserve">Exemplum.</head> <p> <s xml:id="echoid-s540" xml:space="preserve">QVidam proponit aliquod corpus ſolidum notæ <lb/>magnitudinis, & </s> <s xml:id="echoid-s541" xml:space="preserve">vult inuenire, quanta erit magni <lb/>tudo alicuius ſolidi alterius generis, grauitatem <lb/>habentis propoſito corpori ſolido æqualem.</s> <s xml:id="echoid-s542" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s543" xml:space="preserve">S I T propoſitum aliquod corpus plumbeum A, cuius magnitudo <lb/>740, & </s> <s xml:id="echoid-s544" xml:space="preserve">oporteat inuenire quanta erit magnitudo ſtanni, grauita-<lb/>tem babentis æqualem propoſito plumbo A. </s> <s xml:id="echoid-s545" xml:space="preserve">Accipiatur aliquod cor-<lb/>pus plumbeum D, cuius grauitas ſit 115, deinde stanni, magnitudi-<lb/>ne æqualis plumbo D, inueniatur grauitas, quæ ſit 74, quod quomo-<lb/>do fieri oporteat, dictum est in antecedentis Problematis exemplo, & </s> <s xml:id="echoid-s546" xml:space="preserve"><lb/>fiat vt 74, ad 115. </s> <s xml:id="echoid-s547" xml:space="preserve">ita 740, ad alium numerum qui ſit 1150, is igitur <lb/>numerus indicabit quanta erit magnitudo ſtanni, grauitatem baben <lb/>tis æquatem propoſito plumbo A.</s> <s xml:id="echoid-s548" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s549" xml:space="preserve">Quod ſi propoſitum corpus plumbeum A, ſit ſphæricum, cu <lb/>ius ſphæræ diameter ſit 10, & </s> <s xml:id="echoid-s550" xml:space="preserve">oporteat inuenire quanta erit <lb/>diameter ſphæræ ex ſtanno, grauitatem habentis æqualem <lb/>propoſitæ ſphæræ A, ita faciendum erit.</s> <s xml:id="echoid-s551" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s552" xml:space="preserve">Accipiatur vt diximus aliquod corpus plumbeum D, cuius graui-<lb/>tas ſit 115, & </s> <s xml:id="echoid-s553" xml:space="preserve">ſtanni, magnitudinem habentis æqualem plumbo D, in-<lb/>ueniatur grauitas, quæ ſit 74. </s> <s xml:id="echoid-s554" xml:space="preserve">& </s> <s xml:id="echoid-s555" xml:space="preserve">fiat vt 74, ad 115, ita cubus ex 10, <lb/>qui eſt 1000, ad alium numerum qui ſit 1554 {2/37}, is igitur numerus <lb/>erit cubus diametri ſphæræ ex stanno, grauitatem babentis æqualem <lb/>propoſitæ ex plumbo ſpbæræ A, quare eius latus eubicum, quod est <lb/>11 {58/100}, vero proximum, indicabit ipſam diametrum.</s> <s xml:id="echoid-s556" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s557" xml:space="preserve">Similiter ſipropoſitum corpus plumbeum A, fuerit cubicum, vel <lb/>alicuius alterius formæ regularis, eadem ratione inuenietur latus cu <lb/>biex stanno, grauitatem babentis æqualem propoſito plumbeo cu-<lb/>bo A, ſi enim ipſius cubi plumbei A, datum ſit latus 10, erit numerus <lb/>1554 {2/37} cubus ex ſtanno æqualis grauitate propoſito plumbeo Cu <lb/>bo A, quare latus cubicum numeri 1554 {2/37} quod est 11 {58/100} pro <lb/>ximum vero, indicabit quæſitum latus.</s> <s xml:id="echoid-s558" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s559" xml:space="preserve">Neque diſſimili ratione inuenienda erit magnitudo auri, <lb/>argenti, cæræ, aut cuiuſcumque ſolidi, grauitatem habentis <lb/>propoſito corpori ſolido æqualem.</s> <s xml:id="echoid-s560" xml:space="preserve"/> </p> <pb o="22" file="0034" n="34" rhead="PROMOTVS"/> </div> <div xml:id="echoid-div43" type="section" level="1" n="25"> <head xml:id="echoid-head28" xml:space="preserve">PROBLEMA VII. PROPOS. XIV.</head> <p> <s xml:id="echoid-s561" xml:space="preserve">PRopoſitis duobus liquidis corporibus magnitudine <lb/>ęqualibus, data grauitate vnius, grauitatem alte-<lb/>rius inuenire.</s> <s xml:id="echoid-s562" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s563" xml:space="preserve">SINT propoſita duo cor-<lb/> <anchor type="figure" xlink:label="fig-0034-01a" xlink:href="fig-0034-01"/> ra liquida, magnitudine æqualia <lb/>A, B, ſit autem vnius, vtpote li-<lb/>quidi A, data grauitas G, & </s> <s xml:id="echoid-s564" xml:space="preserve"><lb/>oporteat alterius liquidi B, gra-<lb/>uitatem inuenire. </s> <s xml:id="echoid-s565" xml:space="preserve">Accipiatur <lb/>aliquod corpus ſolidum C, & </s> <s xml:id="echoid-s566" xml:space="preserve">li-<lb/>quidi, quod ſit H, eiuſdem ge-<lb/>neris cum liquido A, magnitu-<lb/> <anchor type="note" xlink:label="note-0034-01a" xlink:href="note-0034-01"/> dine æqualis ſolido C, <anchor type="note" xlink:href="" symbol="*"/> inuenia- tur grauitas, quæ ſit D, ſimiliter <lb/>& </s> <s xml:id="echoid-s567" xml:space="preserve">liquidi, quod ſit I, eiuſdem <lb/>generis cum liquido B, magni-<lb/>tudine æqualis eidem ſolido C, <lb/> <anchor type="note" xlink:label="note-0034-02a" xlink:href="note-0034-02"/> <anchor type="note" xlink:href="" symbol="*"/>inueniatur grauitas, quæ ſit E, & </s> <s xml:id="echoid-s568" xml:space="preserve">fiat vt D, ad E, ita G, ad aliam grauitatem, quæ ſit F. </s> <s xml:id="echoid-s569" xml:space="preserve">Quoniam <lb/>igitur eſt vt A, ad B, ita H, ad I, æquale videlicet ad æquale, erit per-<lb/>mutando vt A, ad H, ita B, ad I, & </s> <s xml:id="echoid-s570" xml:space="preserve">quoniam eiuſdem ſunt generis cor-<lb/> <anchor type="note" xlink:label="note-0034-03a" xlink:href="note-0034-03"/> pora A, H, ſimiliter & </s> <s xml:id="echoid-s571" xml:space="preserve">corpora B, I, <anchor type="note" xlink:href="" symbol="*"/> erit vt grauitas G, ad grauita- tem D, ita liquidi B, grauitas, ad grauitatem E, & </s> <s xml:id="echoid-s572" xml:space="preserve">permutando vt <lb/>grauitas G, ad grauitatem liquidi B, ita D, grauitas, ad grauitatem <lb/>E, ſed vt grauitas D, ad grauitatem E, ita eſt grauitas G, ad graui-<lb/>tatem F; </s> <s xml:id="echoid-s573" xml:space="preserve">ergo grauitas F, æqualis erit grauitati liquidi B. </s> <s xml:id="echoid-s574" xml:space="preserve">inuenta <lb/>igitur eſt liquidi corporis B, grauitas F, quod facere oportebat.</s> <s xml:id="echoid-s575" xml:space="preserve"/> </p> <div xml:id="echoid-div43" type="float" level="2" n="1"> <figure xlink:label="fig-0034-01" xlink:href="fig-0034-01a"> <image file="0034-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0034-01"/> </figure> <note position="left" xlink:label="note-0034-01" xlink:href="note-0034-01a" xml:space="preserve">8<unsure/>. huius</note> <note position="left" xlink:label="note-0034-02" xlink:href="note-0034-02a" xml:space="preserve">8. huius</note> <note position="left" xlink:label="note-0034-03" xlink:href="note-0034-03a" xml:space="preserve">4. buius</note> </div> </div> <div xml:id="echoid-div45" type="section" level="1" n="26"> <head xml:id="echoid-head29" xml:space="preserve">Exemplum.</head> <p> <s xml:id="echoid-s576" xml:space="preserve">QVidam proponit aliquod corpus liquidum notæ <lb/>grauitatis, & </s> <s xml:id="echoid-s577" xml:space="preserve">vult ſcire, quanta erit grauitas alte-<lb/>rius liquidi, magnitudinem habentis propoſito corpori <lb/>liquido æqualem.</s> <s xml:id="echoid-s578" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s579" xml:space="preserve">Sit propoſitum aliquod olei corpus A, cuius grauitas 550, & </s> <s xml:id="echoid-s580" xml:space="preserve"><lb/>oporteat inuenire, quanta erit grauitas aquæ, magnitudinem baben- <pb o="23" file="0035" n="35" rhead="ARCHIMEDES."/> tis æqualem propoſito oleo A, Accipiatur aliquod corpus ſolidum C, <lb/>vtpote plumbeum, & </s> <s xml:id="echoid-s581" xml:space="preserve">aquæ, magnitudinem babentis æqualem plumbo <lb/>C, inueniatur grauitas, quæ ſit 12, vt in exemplo propoſ. </s> <s xml:id="echoid-s582" xml:space="preserve">8. </s> <s xml:id="echoid-s583" xml:space="preserve">dictũ eſt. <lb/></s> <s xml:id="echoid-s584" xml:space="preserve">Similiter & </s> <s xml:id="echoid-s585" xml:space="preserve">olei, magnitudinem æqualem babentis, eidem plumbo C, <lb/>inueniatur grauitas, quæ ſit 11, & </s> <s xml:id="echoid-s586" xml:space="preserve">fiat, vt 11, ad 12, ita 550, ad aliũ <lb/>numerum qui ſit 600. </s> <s xml:id="echoid-s587" xml:space="preserve">is igitur numerus indicabit quanta erit graui-<lb/>tas aquæ, magnitudinem babentis æqualem propoſito oleo A.</s> <s xml:id="echoid-s588" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s589" xml:space="preserve">Si vero propoſitum ſit aliquod argenti viui corpus A, cuius <lb/>grauitas 95, & </s> <s xml:id="echoid-s590" xml:space="preserve">oporteat inuenire, quanta erit grauitas aquæ, magni-<lb/>tudinem babentis æqualem propoſito argento viuo A. </s> <s xml:id="echoid-s591" xml:space="preserve">Accipiatur ali-<lb/>quod vas vitreum mundum, & </s> <s xml:id="echoid-s592" xml:space="preserve">politum, cuius grauitas ſit v. </s> <s xml:id="echoid-s593" xml:space="preserve">g. </s> <s xml:id="echoid-s594" xml:space="preserve">91. <lb/></s> <s xml:id="echoid-s595" xml:space="preserve">ipſumq; </s> <s xml:id="echoid-s596" xml:space="preserve">vas plenum aqua ponderetur in aqua, & </s> <s xml:id="echoid-s597" xml:space="preserve">habeat grauitatem <lb/>55, quoniam igitur numerus 91, ſuper at numerũ 55, numero 36, <anchor type="note" xlink:href="" symbol="*"/>erit <anchor type="note" xlink:label="note-0035-01a" xlink:href="note-0035-01"/> grauitas aquæ, magnitudinem babentis æqualem ipſivaſi, boc est ſoli-<lb/>ditati ipſius vaſis 36, ponatur deinde in ipſum vas propoſitum argen-<lb/>tum viuũ A, nibil interest, vt vas ſit plenum, vel non, & </s> <s xml:id="echoid-s598" xml:space="preserve">quoniam ar <lb/>genti viui A, grauitas est 95, & </s> <s xml:id="echoid-s599" xml:space="preserve">vaſis vitrei grauitas 91, erit argenti <lb/>viui ſimul cum ipſo vaſe, grauitas 186. </s> <s xml:id="echoid-s600" xml:space="preserve">ponderetur itaque ipſum vas, <lb/>ſimulcum argento viuo A, in aqua, ita vt aqua repleat vaſis partem <lb/>vacuam, & </s> <s xml:id="echoid-s601" xml:space="preserve">ſit vaſis grauitas in aqua ſimul cum argento viuo 143, <lb/>quoniam igitur numerus 186, ſuperat numerum 143, numero 43, <anchor type="note" xlink:href="" symbol="*"/> erit <anchor type="note" xlink:label="note-0035-02a" xlink:href="note-0035-02"/> grauitas aquæ, magnitudinĕ babĕtis æqualem argento viuo, ſimul cũ <lb/>vaſe 43, ſed grauitas aquæ babentis magnitudinem æqualem vaſi est <lb/>36, ergo reliquum quod est 7, erit grauitas aquæ, magnitudinem ba-<lb/>bentis æqualem propoſito argento viuo A.</s> <s xml:id="echoid-s602" xml:space="preserve"/> </p> <div xml:id="echoid-div45" type="float" level="2" n="1"> <note position="right" xlink:label="note-0035-01" xlink:href="note-0035-01a" xml:space="preserve">5. buius</note> <note position="right" xlink:label="note-0035-02" xlink:href="note-0035-02a" xml:space="preserve">5. buius</note> </div> <p> <s xml:id="echoid-s603" xml:space="preserve">Sed ſi propoſitum fuerit aliquod magnum argenti viui cor <lb/>pus, ita vt difficile poſſit ponderari in aqua, hac via inuenietur <lb/>aquæ quæſita grauitas.</s> <s xml:id="echoid-s604" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s605" xml:space="preserve">Sit propoſitum aliquod magnum argenti viui corpus A, cuius gra-<lb/>uitas 5700. </s> <s xml:id="echoid-s606" xml:space="preserve">& </s> <s xml:id="echoid-s607" xml:space="preserve">oporteat facere, quod imperatum eſt. </s> <s xml:id="echoid-s608" xml:space="preserve">Accipiatur ali-<lb/>quodparuum argenti viui corpus C, cuius grauitas ſit 95, & </s> <s xml:id="echoid-s609" xml:space="preserve">aquæ, <lb/>magnitudinem babentis æqualem argento viuo C, inueniatur graui-<lb/>tas, eo modo quo dictum est, quæ ſit 7, & </s> <s xml:id="echoid-s610" xml:space="preserve">fiat vt 95, ad 7, ita 5700, ad <lb/>alium numerum, qui ſit 420, is igitur numerus indicabit quanta erit <lb/>grauitas aquæ, magnitudinem babentis æqualem propoſito argento <lb/>viuo A.</s> <s xml:id="echoid-s611" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s612" xml:space="preserve">Contra, ſit propoſitum aliquod corpus aqueum A, cuius grauitas <lb/>420, & </s> <s xml:id="echoid-s613" xml:space="preserve">oporteat inuenire quanta erit grauitas argenti viui, magni-<lb/>tudine æqualis propoſitæ aquæ A. </s> <s xml:id="echoid-s614" xml:space="preserve">facto, vt ſupra, & </s> <s xml:id="echoid-s615" xml:space="preserve">inuenta graui-<lb/>tate 7, aquæ ſcilicet magnitudinem babentis æqualem argĕto viuo C, <pb o="24" file="0036" n="36" rhead="PROMOTVS"/> ſiat vt 7, ad 95, ita 420, ad alium numerum, qui ſit 5700, is igi-<lb/>tur indicabit quanta erit grauitas argenti viui, magnitudine æqua-<lb/>lis propoſitæ aquæ A.</s> <s xml:id="echoid-s616" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s617" xml:space="preserve">Inueniemus etiam aliter, & </s> <s xml:id="echoid-s618" xml:space="preserve">expeditius grauitatem <lb/>aquæ, magnitudinem habentis æqualem propoſito ar-<lb/>gento viuo A.</s> <s xml:id="echoid-s619" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s620" xml:space="preserve">Accipiatur enim aliquod corpus aureum, cui ſuperinducatur ce-<lb/>rea tunica tenuiſſima, ne fiat argento viuo leuius, neue ab eodĕ diſſol-<lb/>uatur, deinde aquæ, magnitudinem babentis æqualem ipſi corpori au <lb/>reo inueniatur grauitas, vt dictum est in propoſ. </s> <s xml:id="echoid-s621" xml:space="preserve">8. </s> <s xml:id="echoid-s622" xml:space="preserve">exemplo, quæ ſit 7, <lb/>ſimiliter & </s> <s xml:id="echoid-s623" xml:space="preserve">argenti viui, vt aquæ, magnitudinem babentis æqualem <lb/>eidem corpori aureo, inueniatur grauitas, quæ ſit 95, & </s> <s xml:id="echoid-s624" xml:space="preserve">fiat vt 95, ad <lb/>7, ita 5700, ad 420, grauitas igitur aquæ, magnitudinem babentis æ-<lb/>qualem argento viuo A, erit 420.</s> <s xml:id="echoid-s625" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s626" xml:space="preserve">Contra. </s> <s xml:id="echoid-s627" xml:space="preserve">ſit propoſitum aliquod corpus aqueum, cuius grauitas <lb/>420, & </s> <s xml:id="echoid-s628" xml:space="preserve">oporteat inuenire, quanta erit grauitas argenti viui, magni-<lb/>tudine æqualis propoſitæ aquæ A. </s> <s xml:id="echoid-s629" xml:space="preserve">Superinducta corpori aureo cerea <lb/>tunica, vt ſupra, & </s> <s xml:id="echoid-s630" xml:space="preserve">inuentis grauitatibus 7, & </s> <s xml:id="echoid-s631" xml:space="preserve">95, aquæ nimirum, & </s> <s xml:id="echoid-s632" xml:space="preserve"><lb/>argenti viui, magnitudine æqualium prædicto aureo corpori, fiat vt <lb/>7, ad 95, ita 420, ad 5700. </s> <s xml:id="echoid-s633" xml:space="preserve">grauitas igitur argenti viui, magnitudine <lb/>æqualis propoſito corpori aqueo A, erit 5700.</s> <s xml:id="echoid-s634" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s635" xml:space="preserve">Qua ratione inuenienda ſit grauitas argenti viui, ma-<lb/>gnitudinem habentis propoſito cuicunque corpori ſoli-<lb/>do æqualem.</s> <s xml:id="echoid-s636" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s637" xml:space="preserve">Sit propoſitum aliquod corpus ſolidum, vtpote plumbeum A, cuius <lb/>grauitas 161, & </s> <s xml:id="echoid-s638" xml:space="preserve">oporteat inuenire quanta erit grauitas argenti viui <lb/>magnitudine æqualis propoſito plumbo A. </s> <s xml:id="echoid-s639" xml:space="preserve">inueniatur grauitas aquæ <lb/>magnitudinem babentis æqualem plumbo A, vt in exemplo propoſi-<lb/>tionis 8, dictum eſt, quæ ſit 14, & </s> <s xml:id="echoid-s640" xml:space="preserve">inuenta grauitate argenti viui, ma-<lb/>gnitudine æqualis ipſi aquæ, ea erit de qua quæritur, ſit enim inuen <lb/>ta argenti viui grauitas 190. </s> <s xml:id="echoid-s641" xml:space="preserve">Q<unsure/>uoniã igitur argentum viuum, cuius <lb/>grauitas est 190, magnitudine æquatur aquæ, cuius grauitas eſt 14, <lb/>ipſique aquæ æquatur magnitudine plumbum A, erit argentum vi-<lb/>uum, cuius grauitas 190, magnitudine propoſito plumbo A, æquale; <lb/></s> <s xml:id="echoid-s642" xml:space="preserve">quare inuenta est grauitas argenti viui, magnitudine æqualis propo-<lb/>ſito plumbo A, quod facere oportebat.</s> <s xml:id="echoid-s643" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s644" xml:space="preserve">Quomodo inuenienda ſit grauitas cuiuſcunque cor- <pb o="25" file="0037" n="37" rhead="ARCHIMEDES."/> póris ſolidi, magnitudinem habentis propoſito corpori <lb/>ex argento viuo æqualem.</s> <s xml:id="echoid-s645" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s646" xml:space="preserve">Sit propoſitum aliquod corpus ex argento viuo A, euius grauitas <lb/>190, & </s> <s xml:id="echoid-s647" xml:space="preserve">oporteat inuenire quanta erit grauitas plumbi, magnitudine <lb/>æqualis propoſito argento viuo A. </s> <s xml:id="echoid-s648" xml:space="preserve">inueniatur grauitas aquæ, magni-<lb/>tudinem habentis æqualem argento viuo A, quæſit 14, deinde inuen-<lb/>ta grauitate plumbi, magnitudine æqualis ipſi aquæ, vt in exemplo <lb/>propoſ. </s> <s xml:id="echoid-s649" xml:space="preserve">9. </s> <s xml:id="echoid-s650" xml:space="preserve">dictum est, ea erit de qua quæritur. </s> <s xml:id="echoid-s651" xml:space="preserve">ſit enim inuenta plum-<lb/>bi grauitas 161, quoniam igitur aqua, cuius grauitas est 14, æqua-<lb/>tur magnitudine plumbo, cuius grauitas est 161, & </s> <s xml:id="echoid-s652" xml:space="preserve">æquatur quoque <lb/>argento viuo A, plumbum cuius grauitas eſt 161, æquabitur magni-<lb/>tudine argento viuo A. </s> <s xml:id="echoid-s653" xml:space="preserve">quare inuenia, eſt grauitas plumbi, magnitu-<lb/>dine æqualis propoſito argento viuo A, quod facere oportebat.</s> <s xml:id="echoid-s654" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div47" type="section" level="1" n="27"> <head xml:id="echoid-head30" xml:space="preserve">PROBLEMA VIII. PROPOS. XV.</head> <p> <s xml:id="echoid-s655" xml:space="preserve">PRopoſitis duobus liquidis corporibus æquè graui-<lb/>bus, data magnitudine vnius, magnitudinem alte-<lb/>rius inuenire.</s> <s xml:id="echoid-s656" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s657" xml:space="preserve">SINT propoſita duo cor-<lb/> <anchor type="figure" xlink:label="fig-0037-01a" xlink:href="fig-0037-01"/> pora liquida a què grauia A, <lb/>B, ſit autem vnius vt pote li-<lb/>quidi A, data magnitudo G, <lb/>& </s> <s xml:id="echoid-s658" xml:space="preserve">oporteat inuenire quanta <lb/>erit magnitudo liquidi B. </s> <s xml:id="echoid-s659" xml:space="preserve">Ac-<lb/>cipiatur aliquod ſolidum cor <lb/>pus C, & </s> <s xml:id="echoid-s660" xml:space="preserve">liquidi quod ſit H, <lb/>eiuſdem generis cum liquido <lb/>A, magnitudinem habentis <lb/>æqualem ſolido C,<anchor type="note" xlink:href="" symbol="*"/> inuenia- <anchor type="note" xlink:label="note-0037-01a" xlink:href="note-0037-01"/> tur grauitas quæ ſit D, ſimili-<lb/>ter & </s> <s xml:id="echoid-s661" xml:space="preserve">liquidi, quod ſitI, eiuſdĕ <lb/>generis cum liquido B, magni <lb/>tudinem habentis æqualem eidem ſolido C, <anchor type="note" xlink:href="" symbol="*"/> inueniatur grauitas, <anchor type="note" xlink:label="note-0037-02a" xlink:href="note-0037-02"/> quæ ſit E, & </s> <s xml:id="echoid-s662" xml:space="preserve">fiat vt grauitas E, ad grauitatem D, ita magnitudo G, ad <lb/>ad aliam magnitudinĕ, quæ ſit F. </s> <s xml:id="echoid-s663" xml:space="preserve">Quoniam igitur ſunt quatuor cor-<lb/>pora grauia I, H, B, A, quorum primum, & </s> <s xml:id="echoid-s664" xml:space="preserve">ſecundum ſunt magnitu-<lb/>dine æqualia, tertium vero, & </s> <s xml:id="echoid-s665" xml:space="preserve">quartum æque grauia, & </s> <s xml:id="echoid-s666" xml:space="preserve">ſunt eiuſdem <lb/>generis primum videlicet, & </s> <s xml:id="echoid-s667" xml:space="preserve">tertium, ſimiliter eiuſdem generis ſe- <pb o="26" file="0038" n="38" rhead="PROMOTVS"/> <anchor type="note" xlink:label="note-0038-01a" xlink:href="note-0038-01"/> cundum & </s> <s xml:id="echoid-s668" xml:space="preserve">quartum, <anchor type="note" xlink:href="" symbol="*"/> Erit vt grauitas E, ad grauitatem D, ita ma- gnitudo G, ad liquidi B, magnitudinem, ſed vt grauitas E, ad graui-<lb/>tatem D, ita eſt magnitudo G, ad F, magnitudinem; </s> <s xml:id="echoid-s669" xml:space="preserve">ergo magnitudo <lb/>F, æqualis erit magnitudini liquidi B. </s> <s xml:id="echoid-s670" xml:space="preserve">inuenta igitur eſt corporis li-<lb/>quidi B, magnitudo F, quod facere oportebat.</s> <s xml:id="echoid-s671" xml:space="preserve"/> </p> <div xml:id="echoid-div47" type="float" level="2" n="1"> <figure xlink:label="fig-0037-01" xlink:href="fig-0037-01a"> <image file="0037-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0037-01"/> </figure> <note position="right" xlink:label="note-0037-01" xlink:href="note-0037-01a" xml:space="preserve">8. huius</note> <note position="right" xlink:label="note-0037-02" xlink:href="note-0037-02a" xml:space="preserve">8. huius</note> <note position="left" xlink:label="note-0038-01" xlink:href="note-0038-01a" xml:space="preserve">7. buius</note> </div> <p> <s xml:id="echoid-s672" xml:space="preserve">Quod ſi propoſita duo corpora æque grauia fuerint <lb/>regularia, vtpote ſphærica, fuerit autem ſphęræ A, data <lb/>diameter G, & </s> <s xml:id="echoid-s673" xml:space="preserve">oporteat inuenire, quanta erit diameter <lb/>ſphæræ B, ita faciendum erit.</s> <s xml:id="echoid-s674" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s675" xml:space="preserve">ACCEPTO aliquo cor <lb/> <anchor type="figure" xlink:label="fig-0038-01a" xlink:href="fig-0038-01"/> pore ſolido C, & </s> <s xml:id="echoid-s676" xml:space="preserve">inuentis <lb/>grauitatibus D, E, liquidorũ <lb/>H, I, vt ſupra, fiat vt grauitas <lb/>E, ad grauitatem D, ita cu-<lb/>bus ex G, ad alium cubum, <lb/>cuius latus ſit F. </s> <s xml:id="echoid-s677" xml:space="preserve">Quoniam <lb/>igitur eadem ratione, qua <lb/>ſupra oſtendetur, vt grauitas <lb/>E, ad grauitatem D, ita eſſe <lb/>magnitudinem ſphæræ A, ad <lb/>ſphæræ B, magnitudinem, ſed <lb/>magnitudo ſphæræ A, ad <lb/> <anchor type="note" xlink:label="note-0038-02a" xlink:href="note-0038-02"/> ſphæræ B, magnitudinem, <anchor type="note" xlink:href="" symbol="*"/> triplicatã rationem habet eius, quam G, diameter ſphæræ A, ad dia-<lb/>metrum ſphæræ B, ſimiliter & </s> <s xml:id="echoid-s678" xml:space="preserve">cubus ex G, ad cubum diametri ſphæ-<lb/> <anchor type="note" xlink:label="note-0038-03a" xlink:href="note-0038-03"/> ræ B, <anchor type="note" xlink:href="" symbol="*"/> triplicatam rationem habet eius, quam G, ad ſphæræ B, dia- metrum; </s> <s xml:id="echoid-s679" xml:space="preserve">ergo vt grauitas E, ad grauitatem D, ita erit cubus ex G, ad <lb/>cubum diametri ſphæræ B, ſed vt grauitas D, ita grauitatem D, ita <lb/>eſt cubus ex G, ad cubum ex F; </s> <s xml:id="echoid-s680" xml:space="preserve">ergo cubus ex F, æqualis erit cubo <lb/>diametri ſphæræ B; </s> <s xml:id="echoid-s681" xml:space="preserve">quare & </s> <s xml:id="echoid-s682" xml:space="preserve">latus F, æquabitur diametro ipſius ſphæ <lb/>ræ B. </s> <s xml:id="echoid-s683" xml:space="preserve">inuenta igitur eſt quantitas diametri ſphæræ B, quod facere <lb/>oportebat.</s> <s xml:id="echoid-s684" xml:space="preserve"/> </p> <div xml:id="echoid-div48" type="float" level="2" n="2"> <figure xlink:label="fig-0038-01" xlink:href="fig-0038-01a"> <image file="0038-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0038-01"/> </figure> <note position="left" xlink:label="note-0038-02" xlink:href="note-0038-02a" xml:space="preserve">18. 12. <lb/>Elem.</note> <note position="left" xlink:label="note-0038-03" xlink:href="note-0038-03a" xml:space="preserve">33. 11. <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div50" type="section" level="1" n="28"> <head xml:id="echoid-head31" xml:space="preserve">Exemplum.</head> <p> <s xml:id="echoid-s685" xml:space="preserve">QVidam proponit aliquod corpus liquidum notæ <lb/>magnitudinis, & </s> <s xml:id="echoid-s686" xml:space="preserve">vult inuenire, quanta erit ma-<lb/>gnitudo liquidi alterius generis, grauitatem <pb o="27" file="0039" n="39" rhead="ARCHIMEDES."/> habentis propoſito corpori liquido æqualem.</s> <s xml:id="echoid-s687" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s688" xml:space="preserve">Sit propoſitum aliquod olei corpus A, cuius magnitudo 600. </s> <s xml:id="echoid-s689" xml:space="preserve">& </s> <s xml:id="echoid-s690" xml:space="preserve"><lb/>oporteat inuenire quanta erit magnitudo aquæ, grauitatem babentis <lb/>æqualem propoſito oleo A, accipiatur aliquod ſolidum corpus C, vt <lb/>pote plumbeum, & </s> <s xml:id="echoid-s691" xml:space="preserve">aquæ magnitudinem habentis æqualem plumbo <lb/>C, inueniatur grauitas, vt in exemplo prop. </s> <s xml:id="echoid-s692" xml:space="preserve">8, dictum eſt, quæ ſit 12. <lb/></s> <s xml:id="echoid-s693" xml:space="preserve">ſimiliter & </s> <s xml:id="echoid-s694" xml:space="preserve">olei æqualem habentis magnit udinem eidem plumbo C, <lb/>inueniatur grauitas quæ ſit 11, & </s> <s xml:id="echoid-s695" xml:space="preserve">fiat vt 12, ad 11, ita 600, ad alium <lb/>numerũ quiſit 550. </s> <s xml:id="echoid-s696" xml:space="preserve">is igitur numerus indicabit quanta erit magni-<lb/>tudo aquæ grauitatem habentis æqualem propoſito oleo A.</s> <s xml:id="echoid-s697" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s698" xml:space="preserve">Similiter ſi propoſitũ ſit aliquod corpus aqueum A, cuius magnita <lb/>do 5700, & </s> <s xml:id="echoid-s699" xml:space="preserve">oporteat inuenire, quanta erit magnitudo argecti viui, <lb/>grauitatem habentis æqualem propoſitæ aquæ A. </s> <s xml:id="echoid-s700" xml:space="preserve">Accipiatur aliquod <lb/>corpus ſolidum C, ſi aureum, ſuper inducatur ei cerea tunica propter <lb/>iam dictã rationem, deinde argenti viui, magnitudine æqualis ipſi C, <lb/>inueniatur grauitas quæ ſit 95, ſimiliter & </s> <s xml:id="echoid-s701" xml:space="preserve">aquæ magnitudinĕ ha. <lb/></s> <s xml:id="echoid-s702" xml:space="preserve">bentis equalem eidem C, inueniatur grauitas quæſit 7, & </s> <s xml:id="echoid-s703" xml:space="preserve">fiat vt 95, <lb/>ad 7, ita 5700, ad alium numerum, qui ſit 420, is igitur numerus in-<lb/>dicabit quanta erit magnitudo argenti viui grauitatem habentis <lb/>æqualem propoſitæ aquæ A.</s> <s xml:id="echoid-s704" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s705" xml:space="preserve">Quod ſi propoſitum corpus aqueum A, ſit ſphæricum, cuius <lb/>ſphæræ diameter ſit 10, & </s> <s xml:id="echoid-s706" xml:space="preserve">oporteat inuenire quanta erit dia-<lb/>meter ſphæræ ex argento viuo, grauitatem habentis æqualem <lb/>propoſitæ ſphæræ A, ita faciendum erit.</s> <s xml:id="echoid-s707" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s708" xml:space="preserve">Accepto vt diximus aliquo corpore ſolido C, & </s> <s xml:id="echoid-s709" xml:space="preserve">inuentis grauita-<lb/>tibus liquidorũ aquæ ſcilicet & </s> <s xml:id="echoid-s710" xml:space="preserve">argent viui magnitudinem æqua-<lb/>lem habentium corpori C, quæ ſint 14, grauitas aquæ, & </s> <s xml:id="echoid-s711" xml:space="preserve">190, graui-<lb/>tas argenti viui, fiat vt 190, ad 14, ita cubus ex 10, hoc est ita 1000, <lb/>ad alium numerum, qui ſit 73 {13/19}, is igitur numerus erit cubus dia-<lb/>metriſphæræ ex argento viuo, grauitatem habentis æqualem propo-<lb/>ſitæ ex aqua ſphæræ A: </s> <s xml:id="echoid-s712" xml:space="preserve">quare latus cubicum numeri 73 {13/19}, quod eſt <lb/>4 {19/100}. </s> <s xml:id="echoid-s713" xml:space="preserve">proxime indicabit ipſam diametrum.</s> <s xml:id="echoid-s714" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s715" xml:space="preserve">Similiter ſipropoſitum corpus aqueum A, fuerit cubicum, aut <lb/>alicuius alterius formæregularis, eadem ratione, qua ſupra inuenie-<lb/>tur latus cubi ex argento viuo, grauitate æqualis propoſito ex aqua <lb/>cubo A, nam ſi ipſius cubi A, datũ ſit latus 10, erit numerus 73 {13/19}, <lb/>cubus ex argento viuo æqualis grauitate propoſito ex aqua cubo A; <lb/></s> <s xml:id="echoid-s716" xml:space="preserve">quare latus cubicum numeri 73 {13/19}, quod est 4 {19/100}. </s> <s xml:id="echoid-s717" xml:space="preserve">proxime in-<lb/>dicabit quæſitum latus cubi, ex argento viuo.</s> <s xml:id="echoid-s718" xml:space="preserve"/> </p> <pb o="28" file="0040" n="40" rhead="PROMOTVS"/> <p> <s xml:id="echoid-s719" xml:space="preserve">Neque diſſimili ratione inuenietur magnitudo reliquo-<lb/>rum omnium liquidorum, grauitate propoſito corpori cuiuſ-<lb/>cumque generis liquidi, æqualium, quare dicta ſufficiant.</s> <s xml:id="echoid-s720" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s721" xml:space="preserve">DVm adhucOpuſculum ſub prælo eſſet, dubitandi anſam, <lb/>ex eo vir doctiſsimus, cui percurrendum illud tradi-<lb/>deram, arripuit, quod ex grauitate, corporum in aqua <lb/>exiſtentium, non poſſet vera ratio, quam habent diuerſa <lb/>ipſorum corporum genera in grauitate, deprehendi, niſi <lb/>corpora fuerint ſimilia. </s> <s xml:id="echoid-s722" xml:space="preserve">ſi enim (aiebat) accipiantur duo <lb/>corpora eiuſdem generis, & </s> <s xml:id="echoid-s723" xml:space="preserve">grauitatis, quorum vnum ſit <lb/>planum, alterum conicam formam habens, & </s> <s xml:id="echoid-s724" xml:space="preserve">ponderen-<lb/>turin aqua, ita vt coni vertex deorſum verſus pendeat, baſis <lb/>vero ipſius coni, & </s> <s xml:id="echoid-s725" xml:space="preserve">latæ corporis plani ſuperficies æquidiſtent <lb/>horizonti. </s> <s xml:id="echoid-s726" xml:space="preserve">conus in aqua maiorem habebit grauitatem, cor-<lb/>pore plano, quia corpus planum magis ab aqua ſuſtentatur, <lb/>quam conus, & </s> <s xml:id="echoid-s727" xml:space="preserve">hoc quidem manifeſtum eſt, quoniam ſi am-<lb/>bo demittantur eodem tempore in aquam, conus citius ad <lb/>imum deſcendet, quam corpus planum. </s> <s xml:id="echoid-s728" xml:space="preserve">Hoc argumentum <lb/>licet primo aſpectu probabilevideatur, tamen falſo concludit. <lb/></s> <s xml:id="echoid-s729" xml:space="preserve">verum eſt quod aqua ſuſtentat magis corpus planum, quam <lb/>conum, ipſum tamen ſuſtentat, netanta velocitate feratur <lb/>deorſum, non ideo ipſius grauitati aliquid detrahit, neque <lb/>enim ex velociori motu ſimpliciter inferri poteſt maior gra-<lb/>uitas, illud enim valeret etiam in aere, quod eſt falſum, ſed <lb/>ne huiuſmodi dubitatio veritatis ſpecie aliquem decipiat, ſe-<lb/>quenti Theoremate eam deſtruere agrediar.</s> <s xml:id="echoid-s730" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div51" type="section" level="1" n="29"> <head xml:id="echoid-head32" xml:space="preserve">THEOREMA VIII. PROPOS. XVI.</head> <p> <s xml:id="echoid-s731" xml:space="preserve">COrpora eiuſdem generis, & </s> <s xml:id="echoid-s732" xml:space="preserve">grauitatis grauiora <lb/>quam aqua, etſi diſsimilia, æqualem in aqua <lb/>grauitatem habent.</s> <s xml:id="echoid-s733" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s734" xml:space="preserve">SINT duo eiuſdem generis, & </s> <s xml:id="echoid-s735" xml:space="preserve">grauitatis corpora A, B, grauiora <lb/>quam aqua, & </s> <s xml:id="echoid-s736" xml:space="preserve">ſint diſſimilia, dico ipſa corpora æqualem in aqua <lb/>grauitatem habere. </s> <s xml:id="echoid-s737" xml:space="preserve">ſit enim ſi fieri poteſt corpus A, leuius corpore B, <pb o="29" file="0041" n="41" rhead="AR CHIMEDES."/> & </s> <s xml:id="echoid-s738" xml:space="preserve">accipiatur aliquod corpus L, leuius quam aqua, ita vt cum ipſi <lb/>corpori I.</s> <s xml:id="echoid-s739" xml:space="preserve">, appendatur corpus B, & </s> <s xml:id="echoid-s740" xml:space="preserve">ambo ſimul demittantur in aquã, <lb/>ſint æque grauia atque aqua, neque ſurſum, neque deorſum feran-<lb/>tur, ſimiliter accipiatur alterum corpus M, eiuſdem generis cum <lb/>corpore L, ipſique ſimile, & </s> <s xml:id="echoid-s741" xml:space="preserve"><lb/> <anchor type="figure" xlink:label="fig-0041-01a" xlink:href="fig-0041-01"/> æquale, & </s> <s xml:id="echoid-s742" xml:space="preserve">corpori M, appen <lb/>datur corpus A. </s> <s xml:id="echoid-s743" xml:space="preserve">Deinde in-<lb/>teligatur aqua conſiſtens, & </s> <s xml:id="echoid-s744" xml:space="preserve"><lb/>manens, eiuſque ſuperfi-<lb/>cies ſphærica C D E, cuius <lb/>ſphæræ centrum K, aquæ <lb/>enim conſiſtentis, atque<unsure/> <lb/>manentis ſuperficies ſphæ-<lb/>rica eſt, cuius ſphæræ cen-<lb/>trum idem eſt, quod centrũ <lb/>terræ, hoc autem demonſtratum eſt ab Archimede Prop. </s> <s xml:id="echoid-s745" xml:space="preserve">2. </s> <s xml:id="echoid-s746" xml:space="preserve">lib. </s> <s xml:id="echoid-s747" xml:space="preserve">1. </s> <s xml:id="echoid-s748" xml:space="preserve">de <lb/>ijs, quæ vehuntur in aqua. </s> <s xml:id="echoid-s749" xml:space="preserve">Inteligantur etiam duæ pyramides con-<lb/>iunctæ, & </s> <s xml:id="echoid-s750" xml:space="preserve">continuatæ, æquales, & </s> <s xml:id="echoid-s751" xml:space="preserve">ſimiles KCD, KDE, pro baſibus <lb/>habentes in ſuperficie aquæ parallelogramma, vertices autem pun-<lb/>ctum K, & </s> <s xml:id="echoid-s752" xml:space="preserve">corpora L, B, comprehendantur à pyramide KDE, corpo-<lb/>ra vero M, A, à pyramide KCD, & </s> <s xml:id="echoid-s753" xml:space="preserve">ſub corporibus L, B, deſcribatur <lb/>quædam alterius ſphæræ ſuperficies FGH, in a qua, circa centrum K, <lb/>poterit autem huiuſmodi ſuperficies ſub corporibus L, B, deſcribi, <lb/>quoniam & </s> <s xml:id="echoid-s754" xml:space="preserve">ſi ipſi corpora demerguntur tota, non ideo feruntur <lb/>deorſum, ponuntur enim æque grauia @atque aqua. </s> <s xml:id="echoid-s755" xml:space="preserve">Quoniam <lb/>igitur eiuſdem generis ponuntur corpora M, L, & </s> <s xml:id="echoid-s756" xml:space="preserve">æqualia, & </s> <s xml:id="echoid-s757" xml:space="preserve">ſimilia, <lb/>erunt æque grauia, tum in aqua, tum in aere, & </s> <s xml:id="echoid-s758" xml:space="preserve">quoniam corpus <lb/>A, leuius eſt in aqua, corpore B, erunt corpora M, A, ſimul, in aqua <lb/>leuiora corporibus L, B, ſed corpora L, B, ſimul, æque grauia ſunt at-<lb/>que aqua, ergo corpora M, A, ſimul, leuiora erunt quam aqua; </s> <s xml:id="echoid-s759" xml:space="preserve">quare <lb/>corpus M, non demergetur totum, ſed aliqua pars ipſius ex aquæ ſu-<lb/>perficie extabit.</s> <s xml:id="echoid-s760" xml:space="preserve"/> </p> <div xml:id="echoid-div51" type="float" level="2" n="1"> <figure xlink:label="fig-0041-01" xlink:href="fig-0041-01a"> <image file="0041-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0041-01"/> </figure> </div> <p> <s xml:id="echoid-s761" xml:space="preserve">Et quoniam eiuſdem generis, & </s> <s xml:id="echoid-s762" xml:space="preserve">grauitatis ponuntur corpora A, B, <lb/>erunt magnitudine æqualia, & </s> <s xml:id="echoid-s763" xml:space="preserve">per additionem æqualium æquali-<lb/>bus, corpora M, A, erunt æqualia corporibus L, B,</s> </p> <p> <s xml:id="echoid-s764" xml:space="preserve">Quoniam igitur corpora M, A, æqualia ſunt corporibus L, B, pars <lb/>autem corporis M, extat ex aquæ ſuperficie, & </s> <s xml:id="echoid-s765" xml:space="preserve">corpora L, B, tota de-<lb/>merguntur, minus loci ocupabunt in aqua corpora M, A, quam cor-<lb/>pora L, B, quare maior erit grauitas corporum M, A, & </s> <s xml:id="echoid-s766" xml:space="preserve">aquæ conti-<lb/>nentis ipſa corpora, quæ eſt in loco pyramidis CDGF, quam corpo-<lb/>rum L, B, & </s> <s xml:id="echoid-s767" xml:space="preserve">aquæ ipſa corpora continĕtis in loco pyramidis DEHG, <pb o="30" file="0042" n="42" rhead="PROMOTVS"/> magis igitur aquæ pars premetur, quæ eſt ſub ſuperficie FG, quam <lb/>ea quæ eſt ſub ſuperficie GH; </s> <s xml:id="echoid-s768" xml:space="preserve">quare expellet partem minus preſſam, <lb/>(æqualiter enim & </s> <s xml:id="echoid-s769" xml:space="preserve">continuatæ iacent inter ſeſe) & </s> <s xml:id="echoid-s770" xml:space="preserve">nõ manebit aqua, <lb/>quod eſt abſurdum, ponebatur enim manens. </s> <s xml:id="echoid-s771" xml:space="preserve">non igitur corpus A, <lb/>leuius eſt in aqua corpore B. </s> <s xml:id="echoid-s772" xml:space="preserve">eadem ratione oſtendetur neque corpus <lb/>B, leuius eſſe in aqua corpore A, quare conſtat propoſitum.</s> <s xml:id="echoid-s773" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div53" type="section" level="1" n="30"> <head xml:id="echoid-head33" xml:space="preserve">ALITER.</head> <p> <s xml:id="echoid-s774" xml:space="preserve">Sint duo eiuſdem generi@, & </s> <s xml:id="echoid-s775" xml:space="preserve">grauitatis corpora A, B, grauiora <lb/>quam aqua, & </s> <s xml:id="echoid-s776" xml:space="preserve">ſint diſſimilia. </s> <s xml:id="echoid-s777" xml:space="preserve">oſtendendum eſt ipſa corpora æqualem <lb/>in aqua grauitatem habere, ſit enim corporis A, vel ipſius B, graui-<lb/>tas CD, aquæ vero magni-<lb/> <anchor type="figure" xlink:label="fig-0042-01a" xlink:href="fig-0042-01"/> tudinem habentis æqua-<lb/>lem ipſi A, vel B, ſit graui-<lb/>tas C, & </s> <s xml:id="echoid-s778" xml:space="preserve">accipiatur ali-<lb/>quod corpus L, leuius quã <lb/>aqua, cuius grauitas ſit <lb/>ipſi C, æqualis, aquæ ve-<lb/>ro, magnitudinem haben-<lb/>tis æqualem corpori L, ſit <lb/>grauitas æqualis ipſi CD, <lb/>itaque appenſo corpore <lb/>B, corpori L, corpus ex <lb/>vtriſque conſtans æque graue erit atque aqua, grauitas enim vtro-<lb/>runque corporum B, L, eſt æqualis vtriſque grauitatibus CD, & </s> <s xml:id="echoid-s779" xml:space="preserve">C, & </s> <s xml:id="echoid-s780" xml:space="preserve"><lb/>grauitas aquæ, magnitudinem habentis æqualem vtriſq; </s> <s xml:id="echoid-s781" xml:space="preserve">corporibus <lb/>L, B, æqualis eſt eiſdem grauitatibus CD, & </s> <s xml:id="echoid-s782" xml:space="preserve">C, corpora igitur B, L, <lb/>demiſſa in aquam, neque ſurſum, neque deorſum ſerentur, quia cor-<lb/>pus B, grauius quam aqua fertur deorſum tanta vi, quanta à corpo-<lb/>re L, ſurſum retrahitur.</s> <s xml:id="echoid-s783" xml:space="preserve"/> </p> <div xml:id="echoid-div53" type="float" level="2" n="1"> <figure xlink:label="fig-0042-01" xlink:href="fig-0042-01a"> <image file="0042-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0042-01"/> </figure> </div> <p> <s xml:id="echoid-s784" xml:space="preserve">Rurſus accipiatur alterum corpus ſolidum M, eiuſdem generis <lb/>cum corpore L, ipſique ſimile, & </s> <s xml:id="echoid-s785" xml:space="preserve">æquale, & </s> <s xml:id="echoid-s786" xml:space="preserve">corpore A, appenſo ipſi <lb/>M, & </s> <s xml:id="echoid-s787" xml:space="preserve">demiſſis ambobus in aquam, eadem ratione qua ſupra oſten-<lb/>detur, corpora A, M, ſimul, eſſe æque grauia atque aqua, & </s> <s xml:id="echoid-s788" xml:space="preserve">corpus <lb/>A, tanta vi deorſum ferri, quanta retrahitur ſurſum à corpore M, <lb/>ſed corpora M, L, æqualem vim habent retrahendi ſurſum, cum <lb/>ſint eiuſdem generis, & </s> <s xml:id="echoid-s789" xml:space="preserve">æqualia, & </s> <s xml:id="echoid-s790" xml:space="preserve">ſimilia, ergo æquali vi retra-<lb/>hentur corpora A, B, ne deſcendant; </s> <s xml:id="echoid-s791" xml:space="preserve">quare conſtat ipſa corpora A, <lb/>B, æqualem in aqua grauitatem habere quod erat oſtendendum.</s> <s xml:id="echoid-s792" xml:space="preserve"/> </p> <pb o="31" file="0043" n="43" rhead="ARCHIMEDES."/> </div> <div xml:id="echoid-div55" type="section" level="1" n="31"> <head xml:id="echoid-head34" xml:space="preserve">THEOREMA IX. PROPOS. XVII.</head> <p> <s xml:id="echoid-s793" xml:space="preserve">SPhære eiuſdem generis inter ſe ſunt in grauitate, vt <lb/>diametrorum cubi in magnitudine.</s> <s xml:id="echoid-s794" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s795" xml:space="preserve">SINT ſphæræ eiuſdem gene-<lb/> <anchor type="figure" xlink:label="fig-0043-01a" xlink:href="fig-0043-01"/> ris ABC, DEF, quarum diame <lb/>tri BC, EF. </s> <s xml:id="echoid-s796" xml:space="preserve">dico vt ſphęra ABC, <lb/>ſe habet in grauitate, ad ſphæram <lb/>DEF, ita ſe habere in maguitudi-<lb/>ne cubum ex BC, ad cubum ex <lb/>EF, ſit enim ſphæræ ABC, graui-<lb/>tas G, & </s> <s xml:id="echoid-s797" xml:space="preserve">ſphæræ DEF, grauitas H, <lb/>quoniam igitur eiuſdem generis <lb/>ponuntur ſphæræ ABC, DEF, <lb/>erit <anchor type="note" xlink:href="" symbol="*"/> vt ſphæra ABC, ad ſphæram DEF, ita grauitas G, ad H, graui <anchor type="note" xlink:label="note-0043-01a" xlink:href="note-0043-01"/> tatem, ſed ſphæra ABC, ad ſphæram DEF, <anchor type="note" xlink:href="" symbol="*"/> triplicatam habet ra- <anchor type="note" xlink:label="note-0043-02a" xlink:href="note-0043-02"/> tionem eius, quam diameter BC, ad EF, diametrum, ergo & </s> <s xml:id="echoid-s798" xml:space="preserve">graui-<lb/>tas G, ad grauitatem H, triplicatam habebit rationem eius, quam <lb/>habet BC, ad EF, ſed & </s> <s xml:id="echoid-s799" xml:space="preserve">cubus ex BC, ad cubum ex EF, <anchor type="note" xlink:href="" symbol="*"/> triplicatam <anchor type="note" xlink:label="note-0043-03a" xlink:href="note-0043-03"/> rationem habet eius, quam BC, ad EF, ergo vt grauitas G, ad graui-<lb/>tatem H, ita erit cubus ex BC, ad cubum ex EF. </s> <s xml:id="echoid-s800" xml:space="preserve">ſphæræ igitur eiuſ-<lb/>dem generis inter ſe ſunt in grauitate, vt diametrorum cubi in ma-<lb/>gnitudine, quod erat demonſtrandum.</s> <s xml:id="echoid-s801" xml:space="preserve"/> </p> <div xml:id="echoid-div55" type="float" level="2" n="1"> <figure xlink:label="fig-0043-01" xlink:href="fig-0043-01a"> <image file="0043-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0043-01"/> </figure> <note position="right" xlink:label="note-0043-01" xlink:href="note-0043-01a" xml:space="preserve">2. & 3. <lb/>huius.</note> <note position="right" xlink:label="note-0043-02" xlink:href="note-0043-02a" xml:space="preserve">18. 12. <lb/>Elem.</note> <note position="right" xlink:label="note-0043-03" xlink:href="note-0043-03a" xml:space="preserve">33. 11. <lb/>Elem.</note> </div> <pb o="32" file="0044" n="44" rhead="PROMOTVS"/> </div> <div xml:id="echoid-div57" type="section" level="1" n="32"> <head xml:id="echoid-head35" xml:space="preserve">Ad comparandum inter ſe duodecim corporum genera <lb/>grauitate, & magnitudine tabella.</head> <note position="right" xml:space="preserve"> # Aurũ. # Ar. Vi. # Plum. # Arg. # Aes. # Ferrũ. # Stann. # Mel. # Aqua. # Vinũ. # Cera. # Ole@@ <lb/>Oleum. # 20 {8/11} # 14 {62/77} # 12 {6/11} # 11 {3/11} # 9 {9/11} # 8 {8/11} # 8 {4/55} # 1 {32/55} # 1 {1/11} # 1 {4/55} # 1 {5/121} # 1 <lb/>Cera. # 19 {19/21} # 14 {32/147} # 12 {1/21} # 10 {52/63} # 9 {9/21} # 8 {8/21} # 7 {89/105} # 1 {109/210} # 1 {1/21} # 1 {13/420} # 1 <lb/>Vinum. # 19 {19/59} # 13 {331/413} # 11 {41/59} # 10 {30/59} # 9 {9/59} # 8 {8/59} # 7 {31/59} # 1 {28/59} # 1 {1/59} # 1 <lb/>Aqua. # 19 # 13 {4/7} # 11 {1/2} # 10 {1/3} # 9 # 8 # 7 {2/5} # 1 {9/20} # 1 <lb/>Mel. # 13 {3/29} # 9 {73/203} # 7 {27/29} # 7 {11/87} # 6 {6/29} # 5 {15/29} # 5 {3/29} # 1 <lb/>Stannum. # 2 {21/37} # 1 {221/259} # 1 {41/74} # 1 {44/111} # 1 {8/37} # 1 {3/37} # 1 <lb/>Ferrum. # 2 {3/8} # 1 {39/56} # 1 {7/16} # 1 {7/24} # 1 {1/8} # 1 <lb/>Aes. # 2 {1/9} # 1 {32/63} # 1 {5/18} # 1 {4/27} # 1 <lb/>Argentum. # 1 {26/31} # 1 {68/217} # 1 {7/62} # 1 <lb/>Plumbum. # 1 {15/23} # 1 {29/161} # 1 <lb/>Arg. Viuũ. # 1 {38/95} # 1 <lb/>Aurum. # 1</note> <p style="it"> <s xml:id="echoid-s802" xml:space="preserve">Q<unsure/>uæro exempli gratia, quam habet rationem in grauitate plumbum ad aurum. </s> <s xml:id="echoid-s803" xml:space="preserve">In-<lb/>teligatur plumbum, quoniam leuius est auro, grauitatem habere 1, & </s> <s xml:id="echoid-s804" xml:space="preserve">in line a plumbi, <lb/>in prima columna nominati, ſub titulo auri, quæratur auri grauitas, ea erit 1 {15/23}. </s> <s xml:id="echoid-s805" xml:space="preserve">plum <lb/>bum igitur ad aurum rationem babebit in grauitate vt 1, ad 1 {85/23}, ſi enim ſumantur <lb/>duo corpora magnitudine æqualia, vnum plumbeum alterum aureum, ſit autem plum <lb/>bei corporis grauitas 1, aurei erit 1 {15/23}, quare corpus plumbeum ad corpus aureum <lb/>eiuſdem magnitudinis rationem habebit in grauitate vt 1, ad 1 {15/23}. </s> <s xml:id="echoid-s806" xml:space="preserve">comparantur au-<lb/>tem inter ſe genera diuerſa grauitate, in corporibus magnitudine æqualibus.</s> <s xml:id="echoid-s807" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s808" xml:space="preserve">Rurſus, quæro quam habet rationem in grauitate aqua ad argentum viuum. </s> <s xml:id="echoid-s809" xml:space="preserve">inteli-<lb/>gatur aqua, vt leuior argento viuo grauitatem habere 1, & </s> <s xml:id="echoid-s810" xml:space="preserve">in line a aquæ, ſubtitulo ar-<lb/>genti viui, quæratur argenti viui grauitas, ea erit 13 {4/7}, aqua igitur ad argentum viuũ <lb/>rationem habebit in grauitate vt 1, ad 13 {4/7}.</s> <s xml:id="echoid-s811" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s812" xml:space="preserve">Contra, quæro quomodo ſe habent in magnitudine aurum, & </s> <s xml:id="echoid-s813" xml:space="preserve">plumbum. </s> <s xml:id="echoid-s814" xml:space="preserve">inteligatur <lb/>aurum, quoniam grauius eſt plumbo, magnitudinem habere 1, & </s> <s xml:id="echoid-s815" xml:space="preserve">in linea plumbi, ſub ti-<lb/>tulo auri, quaratur plumbi magnitudo, ea erit 1 {15/23}, aurum igitur ad plumbum ſe ba- <pb o="33" file="0045" n="45" rhead="ARCHIMEDES."/> bebit in magnitudine vt 1, ad 1 {15/23}, ſi enim ſumantur duo corpora aque grauia, vnum <lb/>aureum, alterum plumbeum, ſit autem corporis aurei magnitudo 1, plumbei erit 1 {15/23}, <lb/>quare corpus aureum ad corpus plumbeum eiuſdem grauitatis ſe babebit in magnitudi-<lb/>ne vt 1, ad 1 {15/23}, comparantur autem inter ſe genera diuerſa magnitudine, in corpori-<lb/>bus æque grauibus.</s> <s xml:id="echoid-s816" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s817" xml:space="preserve">Quæro denique quomodo ſe babent in magnitudine ferrum, & </s> <s xml:id="echoid-s818" xml:space="preserve">aqua, ponatur ferrum, <lb/>vt grauius aqua, magnitudinem babere 1, & </s> <s xml:id="echoid-s819" xml:space="preserve">in linea aquæ, ſub titulo ferri, quæratur <lb/>aquæ magnitudo, ea erit 8, ferrumigitur ad aquam ſe babebit in magnitudine vt 1, ad 8.</s> <s xml:id="echoid-s820" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div58" type="section" level="1" n="33"> <head xml:id="echoid-head36" xml:space="preserve">Altera, ad comparandum inter ſe duodecim corporum genera, <lb/>grauitate, & magnitudine, tabella.</head> <note position="right" xml:space="preserve"> # Oleũ. # Cera. # Vinũ. # Aqua. # Mel. # Stann. # Ferrũ. # Aes. # Arg. # Plum. # Ar. Vi. # Aurũ. <lb/>Aurum. # 4 {47/57} # 5 {5/209} # 5 {10/57} # 5 {5/19} # 7 {12/19} # 38 {18/19} # 42 {2/19} # 47 {7/19} # 54 {22/57} # 60 {10/19} # 71 {3/7} # 100 <lb/>Arg. Viuũ # 6 {43/57} # 7 {7/209} # 7 {14/57} # 7 {7/19} # 10 {13/19} # 54 {10/19} # 58 {18/19} # 66 {6/19} # 76 {8/57} # 84 {14/19} # 100 <lb/>Plumbum. # 7 {67/69} # 8 {76/253} # 8 {38/69} # 8 {16/23} # 12 {19/23} # 64 {8/23} # 69 {13/23} # 78 {6/23} # 89 {59/69} # 100 <lb/>Argentum. # 8 {27/31} # 9 {81/341} # 9 {16/31} # 9 {21/31} # 14 {1/31} # 71 {19/31} # 77 {13/31} # 87 {3/31} # 100 <lb/>Aes. # 10 {5/27} # 10 {20/33} # 10 {25/27} # 11 {1/9} # 16 {1/9} # 82 {2/9} # 88 {8/9} # 100 <lb/>Ferrum. # 11 {11/24} # 11 {41/44} # 12 {7/24} # 12 {1/2} # 18 {1/8} # 92 {1/2} # 100 <lb/>Stannum. # 12 {43/111} # 12 {366/407} # 13 {32/111} # 13 {19/37} # 19 {27/37} # 100 <lb/>Mel. # 63 {19/87} # 65 {265/319} # 67 {71/87} # 68 {28/29} # 100 <lb/>Aqua. # 91 {2/3} # 95 {5/11} # 98 {1/3} # 100 <lb/>Vinum. # 93 {13/59} # 97 {47/649} # 100 <lb/>Cera. # 96 {2/63} # 100 <lb/>Oleum. # 100</note> <p style="it"> <s xml:id="echoid-s821" xml:space="preserve">Quæro exempli gratia, quæ nam ſit ratio in grauitate, auri ad argentum. </s> <s xml:id="echoid-s822" xml:space="preserve">intelliga-<lb/>tur aurum, quoniam grauius est argento, grauitatem babere 100, & </s> <s xml:id="echoid-s823" xml:space="preserve">in linea auri, ſub <lb/>titulo argenti, reperietur argenti grauitas 54 {22/57}, aurum igitur ad argentum rationem <lb/>babebit in grauitate vt 100, ad 54 {22/57}, ſi enim ſumantur duo corpora, magnitudine <lb/>æqualia, v<unsure/>num aureum, alterum argenteum, ſit autem aurei corporis grauitas 100, erit <pb o="34" file="0046" n="46" rhead="PROMOTVS"/> argentei 54 {22/57}, quare corpus aureum ad corpus argenteum eiuſdem <lb/>magnitudinis, rationem babebit in grauitate, vt 100, ad 54 {22/57}.</s> <s xml:id="echoid-s824" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s825" xml:space="preserve">Quæro, quomodo ſe babet in grauitate aqua ad vinum quoniam <lb/>aqua grauior est vino, intelligatur eius grauitas 100, & </s> <s xml:id="echoid-s826" xml:space="preserve">quoniam in <lb/>linea aquæ, ſub titulo vini, datur vini grauitas 98 {1/3}, aqua ad vinum <lb/>ſe babebit in gran<unsure/>itate, vt 100, ad 98 {1/3}.</s> <s xml:id="echoid-s827" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s828" xml:space="preserve">Contra quæro quomodo ſe babent in magnitudine argentum, & </s> <s xml:id="echoid-s829" xml:space="preserve"><lb/>aurum. </s> <s xml:id="echoid-s830" xml:space="preserve">intelligatur argentum, vt leuius auro, magni<unsure/>tudinem babere <lb/>100, & </s> <s xml:id="echoid-s831" xml:space="preserve">in linea auri, ſub titulo argenti, quæratur auri magnitudo, ea <lb/>erit 54 {22/57}, argentum igit<unsure/>r ad aurum ſe babebit in magnitudine, vt <lb/>100, ad 54 {22/57}, ſi enim ſumantur duo corpora æque grauia, vnum <lb/>argenteum, alterum aureum, ſit autem argentei corporis magnitudo <lb/>100, erit aurei 54 {22/57}, quare corpus argenteum, ad corpus aureum <lb/>eiuſdem grauitatis, ſe babebit in magnitudine, vt 100, ad 54 {22/57}.</s> <s xml:id="echoid-s832" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s833" xml:space="preserve">Quæro denique quomodo ſe babent in magnitudine aqua & </s> <s xml:id="echoid-s834" xml:space="preserve">ar-<lb/>gentum viuum. </s> <s xml:id="echoid-s835" xml:space="preserve">quoniam aqua leuior est argento viuo, intelligatur <lb/>eius magnitudo 100, & </s> <s xml:id="echoid-s836" xml:space="preserve">in linea argenti viui, ſub titulo aquæ, quæ-<lb/>ratur argenti viui magnitudo, & </s> <s xml:id="echoid-s837" xml:space="preserve">reperietur 7 {7/19}, aqua igitur ad <lb/>argentum viuum ſe babebit in magnitudine, vt 100, ad 7 {7/19}.</s> <s xml:id="echoid-s838" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div59" type="section" level="1" n="34"> <head xml:id="echoid-head37" xml:space="preserve">Hic ſequitur tabula, ad inueniendas ſphærarum grauita-<lb/>tes, ex data diametrorum magnitudine, cuius hæc eſt <lb/>explicatio.</head> <p style="it"> <s xml:id="echoid-s839" xml:space="preserve">In dimetiendis ſphærarum diametris vtimur pede Romano anti-<lb/>quo, cuius menſuram in margine appoſuimus, eaque reſpondet ad Ro-<lb/>mani palmi, quo bodie vtimur, menſuram vt 4, ad 3, buiuſmodi pe-<lb/>dem diuidimus in duodecim partes æquales, ſeu vncias, quas inuenies <lb/>in prima Columna ſub titulo magnitudinis.</s> <s xml:id="echoid-s840" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s841" xml:space="preserve">Ponderibus autem vtimur bac nostra tempeſtate vſitatis, libram <lb/>enim diuidimus in 12, vncias vnciam vero in 24, ſcrupula, & </s> <s xml:id="echoid-s842" xml:space="preserve">ſcru-<lb/>pulum in 24, grana. </s> <s xml:id="echoid-s843" xml:space="preserve">Ad inueniendas igitur ſpbærarum grauitates ex <lb/>data diametrorum magnitudine, bæc eritratio.</s> <s xml:id="echoid-s844" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s845" xml:space="preserve">Quæris grauitatem ſphæræ plumbeæ, diametrum babentis 3, vn-<lb/>ci<unsure/>arum, inſpice tabulam, & </s> <s xml:id="echoid-s846" xml:space="preserve">in linea trium vnciarum, ſub titulo gra-<lb/>uitatis plumbeæ ſphæræ, deprebendes ipſam ſphæram grauitatem ba-<lb/>bere lib. </s> <s xml:id="echoid-s847" xml:space="preserve">7, vnc. </s> <s xml:id="echoid-s848" xml:space="preserve">4, ſcru. </s> <s xml:id="echoid-s849" xml:space="preserve">13, gran. </s> <s xml:id="echoid-s850" xml:space="preserve">22 {26<unsure/>/37}.</s> <s xml:id="echoid-s851" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s852" xml:space="preserve">Rurſus, quæris grauitatem ſphæræ aureæ, diametrum babentis 6, <lb/>vnciarum. </s> <s xml:id="echoid-s853" xml:space="preserve">in linea 6, vnciarum, ſub titulo grauitatis aureæ ſphæræ <pb o="35" file="0047" n="47" rhead="ARCHIMEDES."/> datur quæſita grauitas lib. </s> <s xml:id="echoid-s854" xml:space="preserve">97, vnc. </s> <s xml:id="echoid-s855" xml:space="preserve">6, ſcrup. </s> <s xml:id="echoid-s856" xml:space="preserve">19, gran. </s> <s xml:id="echoid-s857" xml:space="preserve">11 {1/37}.</s> <s xml:id="echoid-s858" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s859" xml:space="preserve">Quæris denique grauitatem ſphæræ stanneæ, diametrum babentis <lb/>vnius pedis. </s> <s xml:id="echoid-s860" xml:space="preserve">in linea vnius pedis, ſeu 12, vnciarum, ſub titulo graui-<lb/>tatis ſphæræ ſtanneæ, datur quæſita ſphæræ grauitas lib. </s> <s xml:id="echoid-s861" xml:space="preserve">304, adun-<lb/>quem. </s> <s xml:id="echoid-s862" xml:space="preserve">Atque ita reliquarum ſphærarum in tabula nominatarum, ex <lb/>data diametrorum magnitudine, grauitates inuenies.</s> <s xml:id="echoid-s863" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div60" type="section" level="1" n="35"> <head xml:id="echoid-head38" xml:space="preserve">Qua ratione hanc Tabulam compoſuimus.</head> <p style="it"> <s xml:id="echoid-s864" xml:space="preserve">Primum inueniendam curauimus gra<unsure/>uitatem alicuius ſphæræ, da-<lb/>tam babentis diametrum, & </s> <s xml:id="echoid-s865" xml:space="preserve">ad boc faciendum, oportebat aliquam <lb/>ſphæram efficere, ſed quoniam ad ill am efficiendam, exactam bumana <lb/>diligentia non ſufficit, fieri curauimus Cylindrum ex ſtanno, altitu-<lb/>dine æqualem diametro circuli, qui baſis eſt ipſius Cylindri, is enim <lb/>torno fieri poteſt multo exactior quam ſphæra, & </s> <s xml:id="echoid-s866" xml:space="preserve">facilius. </s> <s xml:id="echoid-s867" xml:space="preserve">buius au-<lb/>tem Cylindri altitudo, vel diameter ipſius baſis, erat duarum vncia-<lb/>rum prædicti pedis Romani, grauitas vero duarum librarum, cum <lb/>vna vncia, & </s> <s xml:id="echoid-s868" xml:space="preserve">octo ſcrupulis, ſiue vt boc pondus ad grana reducamus, <lb/>Cylindri grauitas erat Gran. </s> <s xml:id="echoid-s869" xml:space="preserve">14592. </s> <s xml:id="echoid-s870" xml:space="preserve">abstulimns ab bac Cylindri <lb/>grauitate partem tertiam, id est 4864, reliquum, quod est 9728. </s> <s xml:id="echoid-s871" xml:space="preserve">ſer-<lb/>uauimus, pro grauitate ſphæræ, diametrum babentis æqualem altitu-<lb/>dini Cylindri, oſtenſum enim est ab Archimede propoſ 32, lib. </s> <s xml:id="echoid-s872" xml:space="preserve">1, de <lb/>ſphæra, & </s> <s xml:id="echoid-s873" xml:space="preserve">Cylindro, Cylindrum, qui baſim babeat maximo in ſphæra <lb/>circulo æqualem, & </s> <s xml:id="echoid-s874" xml:space="preserve">altitudinem æqualem diametro ſphæræ, ad ipſam <lb/>ſphæram ſeſquialterum eſſe; </s> <s xml:id="echoid-s875" xml:space="preserve">itaque grauitatem ſphæræ, diametrum <lb/>babentis duarum vnciarum inuenimus eſſe gran. </s> <s xml:id="echoid-s876" xml:space="preserve">9728.</s> <s xml:id="echoid-s877" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s878" xml:space="preserve">Inuenta igitur grauitate ſphæræ, cuius diameter est duarum vn-<lb/>ciarum, facile inuenientur reliquarum ſphærarũ grauitates, ſi enim <lb/>inuenienda ſit grauitas ſphæræ stannea babentis diametrum {1/4}. </s> <s xml:id="echoid-s879" xml:space="preserve">vn-<lb/>ciæ. </s> <s xml:id="echoid-s880" xml:space="preserve">fiat vt cubus ex 2, ad cubum ex {1/4}, boc est vt 512, ad 1, ita 9728, <lb/>ad alium numerum, qui ſit 19, ſphæræ igitur cuius diameter eſt {1/4}, <lb/>vnciæ, grauitas erit gran. </s> <s xml:id="echoid-s881" xml:space="preserve">19, ostenſum enim est prop. </s> <s xml:id="echoid-s882" xml:space="preserve">17, buius, ſphæ-<lb/>ras eiuſdem generis inter ſe eſſe in grauitate, vt diametrorum cubi in <lb/>magnitudine.</s> <s xml:id="echoid-s883" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s884" xml:space="preserve">Rurſus ſit inuenienda grauitas ſphæræ stannæ babentis diame-<lb/>trum {1/2}, vnciæ, fiat vt cubus ex {1/4}, ad cubum ex {1/2}, boc est vt 1, ad 8, <lb/>ita 19, ad 152, ſphæra igitur, cuius diameter eſt {1/2}, vnciæ, babebit gra-<lb/>uitatem gran. </s> <s xml:id="echoid-s885" xml:space="preserve">152.</s> <s xml:id="echoid-s886" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s887" xml:space="preserve">Sit denique inuenienda grauitas ſphæræ stannæ, diametrum ba-<lb/>bentis {3/4}, vnciæ, fiat vt cubus ex {1/4}, ad cubum ex {3/4}, boc eſt vt 1, ad 27, <lb/>ita 19, ad 513, grauitas igitur ſphæræ babentis diametrum {3/4}, vnciæ,</s> </p> <pb o="36" file="0048" n="48"/> </div> <div xml:id="echoid-div61" type="section" level="1" n="36"> <head xml:id="echoid-head39" xml:space="preserve">Ad inueniendas ſphæra-<lb/>diametrorum <lb/>TAB</head> <p><s xml:id="echoid-s888" xml:space="preserve"><anchor type="note" xlink:label="note-0048-01a" xlink:href="note-0048-01"/> <pb o="37" file="0049" n="49"/> rum grauitates ex data <lb/>magnitudine <lb/>V L A.</s> <s xml:id="echoid-s889" xml:space="preserve"> <anchor type="note" xlink:label="note-0049-01a" xlink:href="note-0049-01"/> <pb file="0050" n="50"/> <anchor type="note" xlink:label="note-0050-01a" xlink:href="note-0050-01"/> <pb file="0051" n="51"/> <anchor type="note" xlink:label="note-0051-01a" xlink:href="note-0051-01"/> <pb file="0052" n="52"/> <anchor type="note" xlink:label="note-0052-01a" xlink:href="note-0052-01"/> erit gran. </s> <s xml:id="echoid-s890" xml:space="preserve">513. </s> <s xml:id="echoid-s891" xml:space="preserve">& </s> <s xml:id="echoid-s892" xml:space="preserve">ſic reliquarum ſpbærarum ex ſtanno, diametros ba- bentium magnitudine quacunque, inuenientur grauitates.</s> <s xml:id="echoid-s893" xml:space="preserve"/> </p> <div xml:id="echoid-div61" type="float" level="2" n="1"> <note position="right" xlink:label="note-0048-01" xlink:href="note-0048-01a" xml:space="preserve">Diametri \\ magnitu- \\ do. #### Aureæ ſpheræ \\ grauitas. #### Plumbeæ Sphęræ \\ grauitas. #### Argentea Sphæræ \\ grauitas. <lb/> # Lib. # Vn. # Scru. # Gra. # Lib. # Vn. # Scr. # Gra. # Lib. # Vn. # Scr. # Gra. <lb/>{1/4} # 0 # 0 # 2 # {29/37} # 0 # 0 # 1 # 5 {39/74} # 0 # 0 # 1 # 2 {59/111} <lb/>{1/2} # 0 # 0 # 16 # 6 {10/37} # 0 # 0 # 9 # 20 {8/37} # 0 # 0 # 8 # 20 {28/111} <lb/>{3/4} # 0 # 2 # 6 # 21 {6/37} # 0 # 1 # 9 # 5 {17/74} # 0 # 1 # 5 # 20 {13/37} <lb/>1 # 0 # 5 # 10 # 2 {6/37} # 0 # 3 # 6 # 17 {27/37} # 0 # 2 # 22 # 18 {2/111} <lb/>1 {1/4} # 0 # 10 # 14 # 1 {36/37} # 0 # 6 # 9 # 18 {65/74} # 0 # 5 # 18 # 4 {49/111} <lb/>1 {1/2} # 1 # 6 # 7 # 1 {11/37} # 0 # 11 # 1 # 17 {31/37} # 0 # 9 # 22 # 18 {30/37} <lb/>1 {3/4} # 2 # 5 # 1 # 4 {31/37} # 1 # 5 # 13 # 23 {57/74} # 1 # 3 # 19 # 4 {35/111} <lb/>2 # 3 # 7 # 8 # 17 {5/37} # 2 # 2 # 5 # 21 {31/37} # 1 # 11 # 14 # 0 {16/111} <lb/>2 {1/4} # 5 # 1 # 17 # 19 {14/37} # 3 # 1 # 8 # 19 {63/74} # 2 # 9 # 13 # 21 {18/37} <lb/>2 {1/2} # 7 # 0 # 16 # 15 {29/37} # 4 # 3 # 6 # 7 {1/37} # 3 # 10 # 1 # 11 {59/111} <lb/>2 {3/4} # 9 # 4 # 17 # 11 {8/37} # 5 # 8 # 5 # 12 {35/74} # 5 # 1 # 7 # 9 {52/111} <lb/>3 # 12 # 2 # 8 # 10 {14/37} # 7 # 4 # 13 # 22 {26/37} # 6 # 7 # 14 # 6 {18/37} <lb/>3 {1/4} # 15 # 6 # 1 # 17 {36/37} # 9 # 4 # 14 # 22 {65/74} # 8 # 5 # 4 # 17 {76/111} <lb/>3 {1/2} # 19 # 4 # 9 # 14 {26/37} # 11 # 8 # 15 # 22 {6/37} # 10 # 6 # 9 # 10 {58/111} <lb/>3 {3/4} # 23 # 9 # 2 # 5 {10/37} # 14 # 5 # 0 # 5 {53/74} # 12 # 11 # 10 # 23 {34/37} <lb/>4 # 28 # 10 # 21 # 17 {3/37} # 17 # 5 # 23 # 6 {26/37} # 15 # 8 # 16 # 1 {13/111} <lb/>4 {1/4} # 34 # 8 # 2 # 10 {27/37} # 20 # 11 # 20 # 10 {21/74} # 18 # 10 # 7 # 5 {46/111} <lb/>4 {1/2} # 41 # 1 # 22 # 11 {1/37} # 24 # 10 # 22 # 14 {30/37} # 22 # 4 # 15 # 3 {33/37} <lb/>4 {3/4} # 48 # 4 # 21 # 23 {36/37} # 29 # 3 # 14 # 13 {65/74} # 26 # 3 # 22 # 11 {86/111}</note> <note position="right" xlink:label="note-0049-01" xlink:href="note-0049-01a" xml:space="preserve">Diametri \\ magnitu. #### Aereæ Sphæræ \\ grauitas. #### Ferreæ Sphæræ \\ grauitas. #### Stanneæ ſphęræ \\ grauitas. <lb/> # Lib. # Vn. # Scr. # Gran. # Lib. # Vn. # Scr. # Gra. # Lib. # Vn. # Scr. # Gra. <lb/>{1/4} # 0 # 0 # 0 # 23 {4/37} # 0 # 0 # 0 # 20 {20/37} # 0 # 0 # 0 # 19 <lb/>{1/2} # 0 # 0 # 7 # 16 {32/37} # 0 # 0 # 6 # 20 {12/37} # 0 # 0 # 6 # 8 <lb/>{3/4} # 0 # 1 # 1 # 23 {34/37} # 0 # 0 # 23 # 2 {22/37} # 0 # 0 # 21 # 9 <lb/>1 # 0 # 2 # 13 # 14 {34/37} # 0 # 2 # 6 # 18 {22/37} # 0 # 2 # 2 # 16 <lb/>1 {1/4} # 0 # 5 # 0 # 8 {19/37} # 0 # 4 # 10 # 23 {21/37} # 0 # 4 # 2 # 23 <lb/>1 {1/2} # 0 # 8 # 15 # 23 {13/37} # 0 # 7 # 16 # 20 {28/37} # 0 # 7 # 3 # 0 <lb/>1 {3/4} # 1 # 1 # 18 # 6 {3/37} # 1 # 0 # 5 # 13 {15/37} # 0 # 11 # 3 # 13 <lb/>2 # 1 # 8 # 12 # 23 {13/37} # 1 # 6 # 6 # 4 {28/37} # 1 # 4 # 21 # 8 <lb/>2 {1/4} # 2 # 5 # 5 # 21 {30/37} # 2 # 1 # 23 # 22 {2/37} # 2 # 0 # 1 # 3 <lb/>2 {1/2} # 3 # 4 # 2 # 20 {4/37} # 2 # 11 # 15 # 20 {20/37} # 2 # 8 # 23 # 16 <lb/>2 {3/4} # 4 # 5 # 9 # 12 {33/37} # 3 # 11 # 11 # 3 {17/37} # 3 # 7 # 21 # 17 <lb/>3 # 5 # 9 # 7 # 18 {30/37} # 5 # 1 # 14 # 22 {2/37} # 4 # 9 # 0 # 0 <lb/>3 {1/4} # 7 # 4 # 3 # 8 {19/37} # 6 # 11 # 0 # 22 {10/37} # 6 # 0 # 11 # 7 <lb/>3 {1/2} # 9 # 2 # 2 # 0 {24/37} # 8 # 1 # 2 # 11 {9/37} # 7 # 6 # 12 # 8 <lb/>3 {3/4} # 11 # 3 # 9 # 13 {32/37} # 10 # 0 # 8 # 12 {12/37} # 9 # 3 # 7 # 21 <lb/>4 # 13 # 8 # 7 # 18 {30/37} # 12 # 2 # 1 # 14 {2/37} # 11 # 3 # 2 # 16 <lb/>4 {1/4} # 16 # 5 # 2 # 10 {5/37} # 14 # 7 # 4 # 19 {25/37} # 13 # 6 # 1 # 11 <lb/>4 {1/2} # 19 # 5 # 23 # 6 {18/37} # 17 # 3 # 23 # 8 {16/37} # 16 # 0 # 9 # 0 <lb/>4 {3/4} # 22 # 11 # 4 # 29 {19/37} # 20 # 4 # 14 # 7 {21/37} # 18 # 10 # 6 # 1</note> <note position="right" xlink:label="note-0050-01" xlink:href="note-0050-01a" xml:space="preserve">Diametri \\ magnitu. #### AEreæ Sphæræ \\ grauitas. #### Ferreæ Sphæræ \\ grauitas. #### Stanneæ ſphęræ \\ grauitas. <lb/> # Lib. # Vn. # Scr. # Gran. # Lib. # Vn. # Scr. # Gra. # Lib. # Vn. # Scr. # Gra. <lb/>5 # 26 # 8 # 22 # 17 {32/37} # 23 # 9 # 6 # 20 {12/37} # 21 # 11 # 21 # 8 <lb/>5 {1/4} # 30 # 11 # 12 # 20 {7/37} # 27 # 6 # 6 # 1 {35/37} # 25 # 5 # 11 # 15 <lb/>5 {1/2} # 35 # 7 # 4 # 7 {5/37} # 31 # 7 # 17 # 2 {25/37} # 29 # 3 # 5 # 16 <lb/>5 {3/4} # 40 # 8 # 2 # 20 {13/37} # 36 # 1 # 21 # 4 {28/37} # 34 # 3 # 8 # 5 <lb/>6 # 46 # 2 # 14 # 6 {18/37} # 41 # 0 # 23 # 8 {16/37} # 38 # 0 # 0 # 0 <lb/>6 {1/4} # 52 # 2 # 20 # 8 {7/37} # 46 # 5 # 4 # 17 {35/37} # 42 # 11 # 9 # 19 <lb/>6 {1/2} # 58 # 9 # 2 # 20 {4/37} # 52 # 2 # 18 # 12 {20/37} # 48 # 3 # 18 # 8 <lb/>6 {3/4} # 65 # 9 # 15 # 12 {33/37} # 58 # 5 # 21 # 19 {17/37} # 54 # 3 # 0 # 0 <lb/>7 # 73 # 4 # 16 # 5 {7/37} # 65 # 2 # 19 # 17 {35/37} # 60 # 4 # 2 # 16 <lb/>7 {1/4} # 81 # 6 # 10 # 15 {24/37} # 72 # 5 # 17 # 11 {9/37} # 67 # 0 # 11 # 23 <lb/>7 {1/2} # 90 # 3 # 4 # 14 {34/37} # 80 # 2 # 20 # 2 {22/37} # 74 # 2 # 15 # 0 <lb/>7 {3/4} # 99 # 7 # 3 # 21 {24/37} # 88 # 6 # 8 # 19 {9/37} # 81 # 10 # 16 # 13 <lb/>8 # 109 # 6 # 14 # 6 {18/37} # 97 # 4 # 12 # 16 {16/37} # 90 # 0 # 21 # 8 <lb/>8 {1/4} # 120 # 1 # 17 # 12 {3/37} # 106 # 9 # 12 # 21 {15/37} # 98 # 9 # 10 # 3 <lb/>8 {1/2} # 131 # 4 # 19 # 8 {3/37} # 116 # 9 # 14 # 13 {15/37} # 108 # 0 # 11 # 16 <lb/>8 {3/4} # 143 # 4 # 1 # 16 {5/37} # 127 # 4 # 22 # 19 {25/37} # 117 # 10 # 6 # 17 <lb/>9 # 155 # 11 # 18 # 3 {33/37} # 138 # 7 # 18 # 19 {17/37} # 128 # 3 # 0 # 0 <lb/>9 {1/4} # 169 # 4 # 2 # 15 # 150 # 6 # 7 # 16 # 139 # 2 # 20 # 7 <lb/>9 {1/2} # 183 # 5 # 8 # 20 {4/37} # 163 # 0 # 18 # 12 {20/37} # 150 # 10 # 0 # 8 <lb/>9 {3/4} # 198 # 3 # 18 # 13 {32/37} # 176 # 3 # 8 # 12 {12/37} # 163 # 0 # 16 # 21 <lb/>10 # 213 # 11 # 13 # 14 {34/37} # 190 # 2 # 6 # 18 {22/37} # 175 # 11 # 2 # 16 <lb/>10 {1/4} # 230 # 4 # 23 # 17 {34/37} # 204 # 9 # 18 # 10 {22/37} # 189 # 5 # 10 # 11 <lb/>10 {1/2} # 247 # 8 # 6 # 17 {19/37} # 220 # 2 # 0 # 15 {21/37} # 203 # 7 # 21 # 0 <lb/>10 {3/4} # 265 # 9 # 16 # 8 {13/37} # 236 # 3 # 6 # 12 {28/37} # 218 # 6 # 15 # 1</note> <note position="right" xlink:label="note-0051-01" xlink:href="note-0051-01a" xml:space="preserve">Diametri \\ magnitu- \\ do. #### Aureæ ſphæræ \\ grauitas. #### Plumbeæ Sphæræ \\ grauitas. #### Argenteæ ſphęræ \\ grauitas. <lb/> # Lib. # Vn. # Scr. # Gran. # Lib. # Vn. # Scr. # Gra. # Lib. # Vn. # Scr. # Gra. <lb/>5 # 56 # 5 # 13 # 6 {10/37} # 34 # 2 # 2 # 8 {8/37} # 30 # 8 # 11 # 20 {28/111} <lb/>5 {1/4} # 65 # 4 # 8 # 10 {23/37} # 39 # 6 # 16 # 23 {59/74} # 35 # 6 # 13 # 20 {19/37} <lb/>5 {1/2} # 75 # 1 # 19 # 17 {27/37} # 45 # 5 # 20 # 3 {29/37} # 40 # 10 # 11 # 3 {33/111} <lb/>5 {3/4} # 85 # 10 # 11 # 8 {11/37} # 51 # 11 # 16 # 23 {25/74} # 46 # 8 # 10 # 9 {16/111} <lb/>6 # 97 # 6 # 19 # 11 {1/37} # 59 # 0 # 15 # 13 {23/37} # 53 # 0 # 18 # 3 {33/37} <lb/>6 {1/4} # 110 # 3 # 8 # 6 {23/37} # 66 # 8 # 23 # 7 {59/74} # 59 # 11 # 17 # 3 {20/111} <lb/>6 {1/2} # 124 # 0 # 13 # 23 {29/37} # 75 # 0 # 23 # 15 {1/37} # 67 # 5 # 13 # 21 {43/111} <lb/>6 {3/4} # 138 # 11 # 0 # 19 {8/37} # 84 # 0 # 22 # 7 {73/74} # 75 # 6 # 15 # 4 {5/37} <lb/>7 # 154 # 11 # 4 # 21 {13/37} # 93 # 9 # 7 # 9 {11/37} # 84 # 3 # 3 # 12 {20/111} <lb/>7 {1/4} # 172 # 1 # 14 # 11 {26/37} # 104 # 2 # 6 # 3 {7/74} # 93 # 7 # 9 # 13 {58/111} <lb/>7 {1/2} # 190 # 6 # 17 # 18 {6/37} # 115 # 4 # 1 # 21 {27/37} # 103 # 7 # 15 # 23 {13/37} <lb/>7 {3/4} # 210 # 3 # 2 # 21 {26/37} # 127 # 3 # 3 # 15 {49/74} # 114 # 4 # 5 # 8 {95/111} <lb/>8 # 231 # 3 # 5 # 16 {24/37} # 139 # 11 # 18 # 5 {23/37} # 125 # 9 # 8 # 8 {104/111} <lb/>8 {1/4} # 253 # 7 # 15 # 14 {31/37} # 153 # 5 # 4 # 18 {57/74} # 137 # 11 # 7 # 3 {24/37} <lb/>8 {1/2} # 277 # 4 # 19 # 13 {31/37} # 167 # 10 # 19 # 10 {1@/37} # 150 # 10 # 9 # 19 {55/111} <lb/>8 {3/4} # 302 # 7 # 6 # 4 {27/37} # 183 # 1 # 20 # 19 {21/74} # 164 # 6 # 21 # 11 {46/111} <lb/>9 # 329 # 3 # 11 # 16 {8/37} # 199 # 3 # 12 # 22 {18/37} # 179 # 1 # 1 # 7 {5/37} <lb/>9 {1/4} # 357 # 6 # 0 # 5 # 216 # 4 # 14 # 0 {1/2} # 194 # 5 # 3 # 21 {2/3} <lb/>9 {1/2} # 387 # 3 # 7 # 23 {29/37} # 234 # 4 # 20 # 15 {1/37} # 210 # 7 # 11 # 22 {22/111} <lb/>9 {3/4} # 418 # 7 # 23 # 5 {10/37} # 253 # 4 # 19 # 17 {53/74} # 227 # 8 # 7 # 21 {18/37} <lb/>10 # 451 # 8 # 10 # 2 {6/37} # 273 # 4 # 18 # 17 {27/37} # 245 # 7 # 22 # 18 {2/111} <lb/>10 {1/4} # 486 # 5 # 4 # 19 {6/37} # 294 # 5 # 1 # 0 {17/74}<unsure/> # 264 # 6 # 14 # 19 {76/111} <lb/>10 {1/2} # 522 # 10 # 19 # 12 {36/37} # 316 # 5 # 21 # 22 {14/37} # 284 # 4 # 14 # 20 {4/37} <lb/>10 {3/4} # 561 # 1 # 18 # 12 {11/@@} # 339 # 7 # 16 # 21 {25/@} # 305 # 2 # 5 # 10 {53/@}</note> <note position="right" xlink:label="note-0052-01" xlink:href="note-0052-01a" xml:space="preserve">Diametri \\ magnitu. #### Aureæ Sphæræ \\ grauitas. #### Plumbeæ Sphæræ \\ grauitas. #### Argĕteæ ſphęræ \\ grauitas. <lb/> # Lib. # Vn. # Scr. # Gran. # Lib. # Vn. # Scr. # Gra. # Lib. # Vn. # Scr. # Gra. <lb/>11 # 601 # 2 # 13 # 21 {31/37} # 363 # 10 # 17 # 6 {10/37} # 326 # 11 # 17 # 5 {29/111} <lb/>11 {1/4} # 643 # 1 # 17 # 22 {11/37} # 389 # 3 # 6 # 10 {25/74} # 349 # 9 # 8 # 21 {30/37} <lb/>11 {1/2} # 686 # 11 # 18 # 18 {14/37} # 415 # 9 # 15 # 18 {26/37} # 373 # 7 # 11 # 1 {17/111} <lb/>11 {3/4} # 732 # 9 # 4 # 14 {29/37} # 443 # 6 # 4 # 16 {39/74} # 398 # 6 # 6 # 7 {22/111} <lb/>12 # 780 # 6 # 11 # 16 {8/37} # 472 # 5 # 4 # 12 {36/37} # 424 # 6 # 1 # 7 {5/37}</note> </div> <p style="it"> <s xml:id="echoid-s894" xml:space="preserve">Aliter quoque & </s> <s xml:id="echoid-s895" xml:space="preserve">expeditius reliquarum ſphærarum ex ſtanno in-<lb/>uenientur grauitates.</s> <s xml:id="echoid-s896" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s897" xml:space="preserve">Inuenta grauitate ſpbæræ, diametrum babentis {1/4}, vnciæ, ſi mul-<lb/>tiplicetur ipſa grauitas, per 8, boc est per cubum ex 2, numerus pro-<lb/>ductus dabit grauitatem ſphæræ, diametrum babentis {2/4}, vnciæ, boc <lb/> <anchor type="note" xlink:label="note-0052-02a" xlink:href="note-0052-02"/> eſt {1/2}, ſpbæræ * enim inter ſe in triplicata ſunt ratione ſuarum dia-<lb/>metrorum. </s> <s xml:id="echoid-s898" xml:space="preserve">deinde ſi multiplicetur eadem grauitas per 27, boc est per <lb/>cubum ex 3, numerus productus dabit grauitatem ſpbæræ, babentis <lb/>diametrum {3/4}, vnciæ, & </s> <s xml:id="echoid-s899" xml:space="preserve">ſi multiplicetur per 64, boc est per cubum <lb/>ex 4, numerus productus dabit grauitatem ſpbæræ, cuius diameter eſt <lb/>@, boc est vnius vnciæ, & </s> <s xml:id="echoid-s900" xml:space="preserve">eo deinceps continuo ordine,</s> </p> <div xml:id="echoid-div62" type="float" level="2" n="2"> <note position="left" xlink:label="note-0052-02" xlink:href="note-0052-02a" xml:space="preserve">18. 12. <lb/>Elem.</note> </div> <p style="it"> <s xml:id="echoid-s901" xml:space="preserve">Porro ad inueniendas grauitates ſpbærarum ex reliquis metallis, <lb/>@ex quacunquæ alia materia, bæc erit ratio.</s> <s xml:id="echoid-s902" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s903" xml:space="preserve">Fiat vt 1, ad 1 {41/74}, boc eſt vt 74, ad 115, (ſi degrauitate ſpbæræ <lb/>plũbeæ quæritur cuius diameter est {1/4}, vnciæ) ita 19. </s> <s xml:id="echoid-s904" xml:space="preserve">grauitas vide-<lb/>licet ſpbæræ stanneæ diametrũ babentis {1/4}, vnciæ, ad aliũ numerum <lb/>qui ſit 29 {39/74}, grauitas igitur ſpbæræ plumbeæ, diametrum baben-<lb/>tis {1/4}, vnciæ, erit gran. </s> <s xml:id="echoid-s905" xml:space="preserve">29 {39/74}. </s> <s xml:id="echoid-s906" xml:space="preserve">ſtannum enim ad plumbum rationem <lb/>babet in grauitate vt 1, ad 1 {41/74}, vt conſpicitur in prima tabella. <lb/></s> <s xml:id="echoid-s907" xml:space="preserve">quam ad comparandum inter ſe duodecim corporum genera, grauita-<lb/>te, & </s> <s xml:id="echoid-s908" xml:space="preserve">magnitudine, appoſuimus.</s> <s xml:id="echoid-s909" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s910" xml:space="preserve">Si vero quæratur de grauitate ſpbæræ plumbeæ, diametrum. <lb/></s> <s xml:id="echoid-s911" xml:space="preserve">babentis 2, vnciarum, fiat vt 74, ad 115, ita 9728, id est grauitas <pb o="41" file="0053" n="53" rhead="ARCHIMEDES."/> <anchor type="note" xlink:label="note-0053-01a" xlink:href="note-0053-01"/> ſphæræ stanneæ, cuius diameter est 2, vnciarum, ad alium numerum, <lb/>qui ſit 15117 {31/37}, ſphæra igitur plumbea, cuius diameter est 2, un-<lb/>ciarum grauitatem habebit gran. </s> <s xml:id="echoid-s912" xml:space="preserve">15117 {31/37}, atque hæc erit obſer-<lb/>uanda in reliquis ratio.</s> <s xml:id="echoid-s913" xml:space="preserve"/> </p> <div xml:id="echoid-div63" type="float" level="2" n="3"> <note position="right" xlink:label="note-0053-01" xlink:href="note-0053-01a" xml:space="preserve">Diametri \\ magnitu. #### AEreæ Sphæræ \\ grauitas #### Ferreæ Sphęrę \\ grauitas #### Stanneæ Sphęræ \\ grauitas. <lb/> # Lib. # Vn. # Scr. # Gran. # Lib. # Vn. # Scr. # Gra. # Lib. # Vn. # Scr. # Gra. <lb/>11 # 284 # 9 # 10 # 9 {3/37} # 253 # 1 # 17 # 5 {15/37} # 234 # 1 # 21 # 8 <lb/>11 {1/4} # 304 # 7 # 18 # 14 {13/37} # 270 # 9 # 13 # 20 {28/37} # 250 # 5 # 2 # 15 <lb/>11 {1/2} # 325 # 4 # 22 # 18 {30/37} # 289 # 3 # 1 # 14 {2/37} # 266 # 1 # 9 # 0 <lb/>11 {3/4} # 347 # 1 # 4 # 17 {4/37} # 308 # 6 # 9 # 12 {20/37} # 285 # 4 # 17 # 5 <lb/>12 # 369 # 8 # 18 # 3 {33/37} # 328 # 7 # 18 # 19 {17/37} # 304 # 0 # 0 # 0 <lb/></note> </div> <p style="it"> <s xml:id="echoid-s914" xml:space="preserve">V el ſi ipſa grauitas 29 {39/74}, multiplicetur per ſingulos cubos, vt <lb/>dictum est de ſphera ſtannea, numeri producti dabunt grauitates <lb/>ſphærarum ex plumbo, ad quarum diametros latera cubica rationem <lb/>babebunt vt 4, ad 1, quoniam 29 {39/74}. </s> <s xml:id="echoid-s915" xml:space="preserve">est grauitas ſpbæræ plumbeæ, <lb/>diametrum habentis {1/4}, vnciæ.</s> <s xml:id="echoid-s916" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div65" type="section" level="1" n="37"> <head xml:id="echoid-head40" xml:space="preserve">Sequitur, ad inueniendas diametrorum <lb/>magnitudines ex data ſphæ-<lb/>rarum grauitate, <lb/>tabula.</head> <pb o="42" file="0054" n="54" rhead="PROMOTVS"/> <note position="right" xml:space="preserve">Graui- \\ tasſphę \\ ræ. # Magnitu- \\ do diame \\ tri ſphæ \\ ræ aureæ # Magnitu- \\ do diame \\ tri ſphæ- \\ ræ plum- \\ beæ. # Magnitu \\ do diame \\ tri ſphæ- \\ ræ argen \\ teæ. # Magnitu \\ do diame \\ tri ſphæ- \\ ræ æreæ. # Magnitu- \\ do diame \\ tri ſphæ- \\ ræ Fer- \\ reæ. # Magnitu \\ do diame \\ tri ſphæ \\ -ræ ſtan. \\ -neæ. <lb/>Lib. 1 # 1 {30./100} # 1 {54./100} # 1 {59./100} # 1 {67./100} # 1 {74:/100} # 1 {78./100} <lb/>2 # 1 {64./100} # 1 {94./100} # 2 {1./100} # 2 {11:/100} # 2 {19./100} # 2 {25:/100} <lb/>3 # 1 {88./100} # 2 {22./100} # 2 {30./100} # 2 {41./100} # 2 {50./100} # 2 {57:/100} <lb/>4 # 2 {7:/100} # 2 {45:/100} # 2 {53./100} # 2 {65./100} # 2 {76./100} # 2 {83./100} <lb/>5 # 2 {22./100} # 2 {63./100} # 2 {73./100} # 2 {86./100} # 2 {97./100} # 3 {5./100} <lb/>6 # 2 {36./100} # 2 {80:/100} # 2 {90./100} # 3 {4:/100} # 3 {16:/100} # 3 {24./100} <lb/>7 # 2 {49./100} # 2 {95:/100} # 3 {5./100} # 3 {20:/100} # 3 {32./100} # 3 {41./100} <lb/>8 # 2 {60./100} # 3 {8./100} # 3 {19./100} # 3 {34./100} # 3 {47./100} # 3 {57:/100} <lb/>9 # 2 {71./100} # 3 {21:/100} # 3 {32./100} # 3 {48:/100} # 3 {61./100} # 3 {71./100} <lb/>10 # 2 {81:/100} # 3 {31./100} # 3 {44:/100} # 3 {60./100} # 3 {74./100} # 3 {84./100} <lb/>11 # 2 {90:/100} # 3 {42./100} # 3 {55./100} # 3 {72./100} # 3 {86./100} # 3 {97:/100} <lb/>12 # 2 {98./100} # 3 {53:/100} # 3 {65./100} # 3 {83./100} # 3 {98./100} # 4 {9:/100} <lb/>13 # 3 {6./100} # 3 {62./100} # 3 {75./100} # 3 {93./100} # 4 {9:/100} # 4 {20:/100} <lb/>14 # 3 {14./100} # 3 {71./100} # 3 {85:/100} # 4 {3./100} # 4 {19./100} # 4 {30./100} <lb/>15 # 3 {21./100} # 3 {80:/100} # 3 {94:/100} # 4 {12./100} # 4 {29:/100} # 4 {40./100} <lb/>16 # 3 {28./100} # 3 {88./100} # 4 {2./100} # 4 {21./100} # 4 {38./100} # 4 {49./100} <lb/>17 # 3 {35./100} # 3 {96./100} # 4 {11:/100} # 4 {30:/100} # 4 {47./100} # 4 {59:/100} <lb/>18 # 3 {41./100} # 4 {4:/100} # 4 {18./100} # 4 {38./100} # 4 {55./100} # 4 {67./100} <lb/>19 # 3 {47./100} # 4 {11./100} # 4 {26./100} # 4 {46./102} # 4 {64./100} # 4 {76./100} <lb/>20 # 3 {53./100} # 4 {18./100} # 4 {33./100} # 4 {54:/100} # 4 {72./100} # 4 {84./100}</note> <pb o="43" file="0055" n="55" rhead="ARCHIMEDES."/> <note position="right" xml:space="preserve">Graui- \\ tasſphę \\ ræ. # Magnitu- \\ do diame \\ tri ſphæ \\ ræ aureæ # Magnitu- \\ do diame \\ tri ſphæ- \\ ræ plum- \\ beæ. # Magnitu \\ do diame \\ tri ſphæ- \\ ræ argen \\ teæ. # Magnitu \\ do diame \\ tri ſphæ- \\ ræ æreæ. # Magnitu- \\ do diame \\ tri ſphæ- \\ ræ Fer- \\ reæ. # Magnitu \\ do diame \\ tri ſphæ \\ -ræ ſtan. \\ -neæ. <lb/>21 # 3 {59./100} # 4 {25./100} # 4 {40./100} # 4 {61./100} # 4 {80:/100} # 4 {92./100} <lb/>22 # 3 {65./100} # 4 {32:/100} # 4 {47./100} # 4 {68./100} # 4 {87./100} # 5. <lb/>23 # 3 {70./100} # 4 {38./100} # 4 {54./100} # 4 {75./100} # 4 {94./100} # 5 {7./100} <lb/>24 # 3 {75./100} # 4 {44./100} # 4 {60./100} # 4 {82./100} # 5 {1./100} # 5 {15:/100} <lb/>25 # 3 {81./100} # 4 {50./100} # 4 {67:/100} # 4 {89:/100} # 5 {8./100} # 5 {22:/100} <lb/>26 # 3 {86./100} # 4 {56./100} # 4 {73./100} # 4 {95./100} # 5 {15./100} # 5 {28./100} <lb/>27 # 3 {91:/100} # 4 {62./100} # 4 {79./100} # 5 {1./100} # 5 {22:/100} # 5 {35./100} <lb/>28 # 3 {95./100} # 4 {68:/100} # 4 {85:/100} # 5 {8:/100} # 5 {28./100} # 5 {42:/100} <lb/>29 # 4. # 4 {73./100} # 4 {90./100} # 5 {14:/100} # 5 {34./100} # 5 {48./100} <lb/>30 # 4 {5:/100} # 4 {79:/100} # 4 {96./100} # 5 {20:/100} # 5 {40./100} # 5 {54./100} <lb/>31 # 4 {9./100} # 4 {84./100} # 5 {1./100} # 5 {25./100} # 5 {46./100} # 5 {60./100} <lb/>32 # 4 {13./100} # 4 {89./100} # 5 {7:/100} # 5 {31./100} # 5 {52./100} # 5 {66./100} <lb/>33 # 4 {18./100} # 4 {94./100} # 5 {12./100} # 5 {36./100} # 5 {58:/100} # 5 {72./100} <lb/>34 # 4 {22./100} # 4 {99./100} # 5 {17./100} # 5 {42:/100} # 5 {63./100} # 5 {78./100} <lb/>35 # 4 {26./100} # 5 {4:/100} # 5 {22./100} # 5 {47:/100} # 5 {69./100} # 5 {84:/100} <lb/>36 # 4 {30./100} # 5 {8./100} # 5 {27./100} # 5 {52./100} # 5 {74./100} # 5 {89./100} <lb/>37 # 4 {34./100} # 5 {13./100} # 5 {32./100} # 5 {57./100} # 5 {79./100} # 5 {95:/100} <lb/>38 # 4 {38./100} # 5 {18:/100} # 5 {37:/100} # 5 {6./100} # 5 {84./100} # 6 <lb/>39 # 4 {42:/100} # 5 {22./100} # 5 {41./100} # 5 {67:/100} # 5 {89./100} # 6 {5./100} <lb/>40 # 4 {46:/100} # 5 {26./100} # 5 {46./100} # 5 {72:/100} # 5 {94./100} # 6 {10./100}</note> <pb o="44" file="0056" n="56" rhead="PROMOTVS"/> <note position="right" xml:space="preserve">Graui- \\ tasſphę \\ ræ. # Magnitu- \\ do diame \\ tri ſphæ \\ ræ aureæ # Magnitu- \\ do diame \\ tri ſphæ- \\ ræ plum- \\ beæ. # Magnitu \\ do diame \\ tri ſphæ- \\ ræ argen \\ teæ. # Magnitu \\ do diame \\ tri ſphæ- \\ ræ æreæ. # Magnitu- \\ do diame \\ tri ſphæ- \\ ræ Fer- \\ reæ. # Magnitu \\ do diame \\ tri ſphæ \\ -ræ ſtan. \\ -neæ. <lb/>41 # 4 {49./100} # 5 {31./100} # 5 {50./100} # 5 {76./100} # 5 {99./100} # 6 {15./100} <lb/>42 # 4 {53./100} # 5 {35./100} # 5 {55./100} # 5 {81./100} # 6 {4./100} # 6 {20./100} <lb/>43 # 4 {56./100} # 5 {39./100} # 5 {59./100} # 5 {85./100} # 6 {9./100} # 6 {25./100} <lb/>44 # 4 {60./100} # 5 {43./100} # 5 {63./100} # 5 {90./100} # 6 {14:/100} # 6 {30./100} <lb/>45 # 4 {63./100} # 5 {48./100} # 5 {68:/100} # 5 {95:/100} # 6 {18./100} # 6 {35:/100} <lb/>46 # 4 {67:/100} # 5 {52./100} # 5 {72./100} # 5 {99./100} # 6 {23./100} # 6 {39./100} <lb/>47 # 4 {70./100} # 5 {56./100} # 5 {76./100} # 6 {3./100} # 6 {27./100} # 6 {44:/100} <lb/>48 # 4 {73./100} # 5 {59./100} # 5 {80./100} # 6 {8:/100} # 6 {32:/100} # 6 {49:/100} <lb/>49 # 4 {77:/100} # 5 {63./100} # 5 {84./100} # 6 {12:/100} # 6 {36./100} # 6 {53./100} <lb/>50 # 4 {80./100} # 5 {67./100} # 5 {88./100} # 6 {16:/100} # 6 {40./100} # 6 {57./100} <lb/>51 # 4 {83./100} # 5 {71./100} # 5 {92./100} # 6 {20./100} # 6 {45:/100} # 6 {61./100} <lb/>52 # 4 {86./100} # 5 {75./100} # 5 {96:/100} # 6 {24./100} # 6 {49./100} # 6 {66./100} <lb/>53 # 4 {89./100} # 5 {78./100} # 6: # 6 {28./100} # 6 {53./100} # 6 {70./100} <lb/>54 # 4 {92./100} # 5 {82./100} # 6 {3./100} # 6 {32:/100} # 6 {57./100} # 6 {74./100} <lb/>55 # 4 {95./100} # 5 {85./100} # 6 {7./100} # 6 {36:/100} # 6 {61./100} # 6 {79:/100} <lb/>56 # 4 {98./100} # 5 {89./100} # 6 {11:/100} # 6 {40:/100} # 6 {65./100} # 6 {83:/100} <lb/>57 # 5 {1./100} # 5 {92./100} # 6 {14./100} # 6 {43./100} # 6 {69./100} # 6 {87:/100} <lb/>58 # 5 {4./100} # 5 {96./100} # 6 {18./100} # 6 {47./100} # 6 {73./100} # 6 {91:/100} <lb/>59 # 5 {7./100} # 5 {99./100} # 6 {22:/100} # 6 {51:/100} # 6 {77:/100} # 6 {95:/100} <lb/>60 # 5 {10./100} # 6 {3./100} # 6 {25./100} # 6 {34./100} # 6 {81:/100} # 6 {99:/100}</note> <pb o="45" file="0057" n="57" rhead="ARCHIMEDES."/> <note position="right" xml:space="preserve">Graui- \\ tasſphę \\ ræ. # Magnitu- \\ do diame \\ tri ſphæ \\ ræ aureæ # Magnitu- \\ do diame \\ tri ſphæ- \\ ræ plum- \\ beæ. # Magnitu \\ do diame \\ tri ſphæ- \\ ræ argen \\ teæ. # Magnitu \\ do diame \\ tri ſphæ- \\ ræ æreæ. # Magnitu- \\ do diame \\ tri ſphæ- \\ ræ Fer- \\ reæ. # Magnitu \\ do diame \\ tri ſphæ \\ -ræ ſtan. \\ -neæ. <lb/>61 # 5 {13./100} # 6 {6./100} # 6 {28./100} # 6 {58./100} # 6 {84./100} # 7 {2./100} <lb/>62 # 5 {16:/100} # 6 {9./100} # 6 {32:/100} # 6 {62:/100} # 6 {88./100} # 7 {6./100} <lb/>63 # 5 {18./100} # 6 {13./100} # 6 {35./100} # 6 {65./100} # 6 {91./100} # 7 {10./100} <lb/>64 # 5 {21./100} # 6 {16./100} # 6 {38./100} # 6 {69:/100} # 6 {95./100} # 7 {14:/100} <lb/>65 # 5 {24./100} # 6 {19./100} # 6 {42:/100} # 6 {72./100} # 6 {99./100} # 7 {17./100} <lb/>66 # 5 {27:/100} # 6 {22./100} # 6 {45./100} # 6 {76:/100} # 7 {3:/100} # 7 {21./100} <lb/>67 # 5 {29./100} # 6 {25./100} # 6 {48./100} # 6 {79./100} # 7 {6./100} # 7 {25:/100} <lb/>68 # 5 {32;/100} # 6 {28./100} # 6 {52:/100} # 6 {82./100} # 7 {10;/100} # 7 {28./100} <lb/>69 # 5 {34./100} # 6 {31./100} # 6 {55:/100} # 6 {86:/100} # 7 {13./100} # 7 {32:/100} <lb/>70 # 5 {37./100} # 6 {35:/100} # 6 {58./100} # 6 {89./100} # 7 {16./100} # 7 {35/100} <lb/>71 # 5 {40:/100} # 6 {38./100} # 6 {61./100} # 6 {92./100} # 7 {20./100} # 7 {39:/100} <lb/>72 # 5 {42./100} # 6 {41./100} # 6 {64./100} # 6 {96:/100} # 7 {23./100} # 7 {42./100} <lb/>73 # 5 {45:/100} # 6 {44:/100} # 6 {67./100} # 6 {99:/100} # 7 {27:/100} # 7 {46:/100} <lb/>74 # 5 {47./100} # 6 {47:/100} # 6 {70./100} # 7 {2:/100} # 7 {30./100} # 7 {49./100} <lb/>75 # 5 {49./100} # 6 {50:/100} # 6 {73./100} # 7 {5./100} # 7 {33./100} # 7 {53:/100} <lb/>76 # 5 {52./100} # 6 {52./100} # 6 {76./100} # 7 {8./100} # 7 {36./100} # 7 {56:/100} <lb/>77 # 5 {54./100} # 6 {55./100} # 6 {79./100} # 7 {11./100} # 7 {39./100} # 7 {59./100} <lb/>78 # 5 {57:/100} # 6 {58./100} # 6 {82./100} # 7 {14./100} # 7 {43:/100} # 7 {62./100} <lb/>79 # 5 {59./100} # 6 {61./100} # 6 {85./100} # 7 {17./100} # 7 {46./100} # 7 {66:/100} <lb/>80 # 5 {61./100} # 6 {64:/100} # 6 {88:/100} # 7 {20./100} # 7 {49./100} # 7 {69:/100}</note> <pb o="46" file="0058" n="58" rhead="PROMOTVS"/> <note position="right" xml:space="preserve">Graui- \\ tasſphę \\ ræ. # Magnitu- \\ do diame \\ tri ſphæ \\ ræ aureæ # Magnitu- \\ do diame \\ tri ſphæ- \\ ræ plum- \\ beæ. # Magnitu \\ do diame \\ tri ſphæ- \\ ræ argen \\ teæ. # Magnitu \\ do diame \\ tri ſphæ- \\ ræ æreæ. # Magnitu- \\ do diame \\ tri ſphæ- \\ ræ Fer- \\ reæ. # Magnitu \\ do diame \\ tri ſphæ \\ -ræ ſtan. \\ -neæ. <lb/>81 # 5 {64:/100} # 6 {66./100} # 6 {91:/100} # 7 {23./100} # 7 {52./100} # 7 {72./100} <lb/>82 # 5 {66./100} # 6 {69:/100} # 6 {93./100} # 7 {26./100} # 7 {55./100} # 7 {75./100} <lb/>83 # 5 {68./100} # 6 {@2./100} # 6 {96./100} # 7 {29./100} # 7 {58./100} # 7 {78./100} <lb/>84 # 5 {71:/100} # 6 {74./100} # 6 {99./100} # 7 {32./100} # 7 {61./100} # 7 {81./100} <lb/>85 # 5 {73./100} # 6 {77./100} # 7 {2./100} # 7 {35./100} # 7 {64./100} # 7 {85:/100} <lb/>86 # 5 {75./100} # 6 {80./100} # 7 {5:/100} # 7 {38:/100} # 7 {67./100} # 7 {88:/100} <lb/>87 # 5 {77./100} # 6 {82./100} # 7 {7./100} # 7 {41:/100} # 7 {70./100} # 7 {91:/100} <lb/>88 # 5 {80;/100} # 6 {85./100} # 7 {10./100} # 7 {44:/100} # 7 {73./100} # 7 {94:/100} <lb/>89 # 5 {82:/100} # 6 {88:/100} # 7 {13:/100} # 7 {46./100} # 7 {76./100} # 7 {97./100} <lb/>90 # 5 {84./100} # 6 {90./100} # 7 {15./100} # 7 {49./100} # 7 {79./100} # 8: <lb/>91 # 5 {86./100} # 6 {93./100} # 7 {18./100} # 7 {52./100} # 7 {82./100} # 8 {3:/100} <lb/>92 # 5 {88./100} # 6 {95./100} # 7 {21:/100} # 7 {55:/100} # 7 {85:/100} # 8 {6:/100} <lb/>93 # 5 {90./100} # 6 {98./100} # 7 {23./100} # 7 {57./100} # 7 {88:/100} # 8 {9:/100} <lb/>94 # 5 {92./100} # 7. # 7 {26:/100} # 7 {60./100} # 7 {90./100} # 8 {11./100} <lb/>95 # 5 {94./100} # 7 {3./100} # 7 {28./100} # 7 {63:/100} # 7 {93./100} # 8 {14./100} <lb/>96 # 5 {97:/100} # 7 {5./100} # 7 {31./100} # 7 {65./100} # 7 {96./100} # 8 {17./100} <lb/>97 # 5 {99./100} # 7 {8:/100} # 7 {34:/100} # 7 {68./100} # 7 {99./100} # 8 {20./100} <lb/>98 # 6 {1:/100} # 7 {10./100} # 7 {36./100} # 7 {71;/100} # 8 {2:/100} # 8 {23:/100} <lb/>99 # 6 {3:/100} # 7 {13:/100} # 7 {39:/100} # 7 {74:/100} # 8 {4./100} # 8 {26:/100} <lb/>100 # 6 {5:/100} # 7 {15./100} # 7 {41./100} # 7 {76./100} # 8 {7./100} # 8 {28./100}</note> <pb o="47" file="0059" n="59" rhead="ARCHIMEDES."/> <note position="right" xml:space="preserve">Graui- \\ tasſphę \\ ræ. # Magnitu- \\ do diame \\ tri ſphæ \\ ræ aureæ # Magnitu- \\ do diame \\ tri ſphæ- \\ ræ plum- \\ beæ. # Magnitu \\ do diame \\ tri ſphæ- \\ ræ argen \\ teæ. # Magnitu \\ do diame \\ tri ſphæ- \\ ræ æreæ. # Magnitu- \\ do diame \\ tri ſphæ- \\ ræ Fer- \\ reæ. # Magnitu \\ do diame \\ tri ſphæ \\ -ræ ſtan. \\ -neæ. <lb/>101 # 6 {7:/100} # 7 {17./100} # 7 {44:/100} # 7 {79:/100} # 8 {10:/100} # 8 {31./100} <lb/>102 # 6 {9:/100} # 7 {19./100} # 7 {46./100} # 7 {81./100} # 8 {12./100} # 8 {34:/100} <lb/>103 # 6 {11:/100} # 7 {22./100} # 7 {48./100} # 7 {84;/100} # 8 {15./100} # 8 {37:/100} <lb/>104 # 6 {13:/100} # 7 {24./100} # 7 {51:/100} # 7 {86./100} # 8 {18:/100} # 8 {39./100} <lb/>105 # 6 {15:/100} # 7 {27:/100} # 7 {53./100} # 7 {89:/100} # 8 {20./100} # 8 {42:/100} <lb/>106 # 6 {17:/100} # 7 {29./100} # 7 {55./100} # 7 {91./100} # 8 {23:/100} # 8 {45:/100} <lb/>107 # 6 {19:/100} # 7 {31./100} # 7 {58./100} # 7 {94:/100} # 8 {25./100} # 8 {47./100} <lb/>108 # 6 {21:/100} # 7 {33./100} # 7 {61:/100} # 7 {96./100} # 8 {28./100} # 8 {50./100} <lb/>109 # 6 {23:/100} # 7 {36:/100} # 7 {63:/100} # 7 {99:/100} # 8 {31:/100} # 8 {53:/100} <lb/>110 # 6 {24./100} # 7 {38./100} # 7 {65./100} # 8 {1./100} # 8 {33./100} # 8 {55./100} <lb/>111 # 6 {26./100} # 7 {40./100} # 7 {67./100} # 8 {3./100} # 8 {36:/100} # 8 {58:/100} <lb/>112 # 6 {28./100} # 7 {42./100} # 7 {70:/100} # 8 {6:/100} # 8 {38./100} # 8 {60./100} <lb/>113 # 6 {30./100} # 7 {45:/100} # 7 {72:/100} # 8 {8./100} # 8 {41:/100} # 8 {63:/100} <lb/>114 # 6 {32:/100} # 7 {47./100} # 7 {74./100} # 8 {11:/100} # 8 {43./100} # 8 {65./100} <lb/>115 # 6 {34:/100} # 7 {49./100} # 7 {76./100} # 8 {13./100} # 8 {46:/100} # 8 {68:/100} <lb/>116 # 6 {36:/100} # 7 {51./100} # 7 {79:/100} # 8 {15./100} # 8 {48./100} # 8 {70./100} <lb/>117 # 6 {37./100} # 7 {53./100} # 7 {81:/100} # 8 {18:/100} # 8 {50./100} # 8 {73:/100} <lb/>118 # 6 {39./100} # 7 {55./100} # 7 {83./100} # 8 {20./100} # 8 {53:/100} # 8 {75./100} <lb/>119 # 6 {41./100} # 7 {57./100} # 7 {85./100} # 8 {22./100} # 8 {55./100} # 8 {78./100} <lb/>120 # 6 {43:/100} # 7 {60:/100} # 7 {87./100} # 8 {24./100} # 8 {57./100} # 8 {80./100}</note> <pb o="48" file="0060" n="60" rhead="PROMOTVS"/> <note position="right" xml:space="preserve">Graui- \\ tasſphę \\ ræ. # Magnitu- \\ do diame \\ tri ſphæ \\ ræ aureæ # Magnitu- \\ do diame \\ tri ſphæ- \\ ræ plum- \\ beæ. # Magnitu \\ do diame \\ tri ſphæ- \\ ræ argen \\ teæ. # Magnitu \\ do diame \\ tri ſphæ- \\ ræ æreæ. # Magnitu- \\ do diame \\ tri ſphæ- \\ ræ Fer- \\ reæ. # Magnitu \\ do diame \\ tri ſphæ \\ -ræ ſtan. \\ -neæ. <lb/>121 # 6 {44./100} # 7 {62./100} # 7 {90:/100} # 8 {27:/100} # 8 {60./100} # 8 {83:/100} <lb/>122 # 6 {46./100} # 7 {64./100} # 7 {92:/100} # 8 {29./100} # 8 {62./100} # 8 {85./100} <lb/>123 # 6 {48./100} # 7 {66./100} # 7 {94./100} # 8 {31./100} # 8 {65:/100} # 8 {88:/100} <lb/>124 # 6 {50:/100} # 7 {68./100} # 7 {96./100} # 8 {34:/100} # 8 {67./100} # 8 {90:/100} <lb/>125 # 6 {52:/100} # 7 {70./100} # 7 {98./100} # 8 {36:/100} # 8 {69./100} # 8 {92./100} <lb/>126 # 6 {53./100} # 7 {72./100} # 8. # 8 {38./100} # 8 {72:/100} # 8 {95:/100} <lb/>127 # 6 {55./100} # 7 {74./100} # 8 {3:/100} # 8 {40./100} # 8 {74./100} # 8 {97./100} <lb/>128 # 6 {57:/100} # 7 {76./100} # 8 {5:/100} # 8 {43:/100} # 8 {76./100} # 8 {99./100} <lb/>129 # 6 {58./100} # 7 {78./100} # 8 {7:/100} # 8 {45:/100} # 8 {79:/100} # 9 {2./100} <lb/>130 # 6 {60./100} # 7 {80./100} # 8 {9:/100} # 8 {47:/100} # 8 {81:/100} # 9 {4./100} <lb/>131 # 6 {62:/100} # 7 {82./100} # 8 {11:/100} # 8 {49./100} # 8 {83./100} # 9 {6./100} <lb/>132 # 6 {64:/100} # 7 {84./100} # 8 {13:/100} # 8 {51./100} # 8 {85./100} # 9 {9;/100} <lb/>133 # 6 {65./100} # 7 {86./100} # 8 {15./100} # 8 {53./100} # 8 {88:/100} # 9 {11:/100} <lb/>134 # 6 {67:/100} # 7 {88./100} # 8 {17./100} # 8 {56:/100} # 8 {90:/100} # 9 {13./100} <lb/>135 # 6 {69:/100} # 7 {90./100} # 8 {19./100} # 8 {58:/100} # 8 {92./100} # 9 {16:/100} <lb/>136 # 6 {70./100} # 7 {92./100} # 8 {21./100} # 8 {60:/100} # 8 {94./100} # 9 {18:/100} <lb/>137 # 6 {72:/100} # 7 {94./100} # 8 {23./100} # 8 {62:/100} # 8 {96./100} # 9 {20./100} <lb/>138 # 6 {74:/100} # 7 {96./100} # 8 {25./100} # 8 {64:/100} # 8 {99:/100} # 9 {22./100} <lb/>139 # 6 {75./100} # 7 {98./100} # 8 {27./100} # 8 {66./100} # 9 {1:/100} # 9 {24./100} <lb/>140 # 6 {77:/100} # 8. # 8 {29./100} # 8 {68./100} # 9 {3:/100} # 9 {27:/100}</note> <pb o="49" file="0061" n="61" rhead="ARCHIMEDES."/> <note position="right" xml:space="preserve">Graui- \\ tasſphę \\ ræ. # Magnitu- \\ do diame \\ tri ſphæ \\ ræ aureæ # Magnitu- \\ do diame \\ tri ſphæ- \\ ræ plum- \\ beæ. # Magnitu \\ do diame \\ tri ſphæ- \\ ræ argen \\ teæ. # Magnitu \\ do diame \\ tri ſphæ- \\ ræ æreæ. # Magnitu- \\ do diame \\ tri ſphæ- \\ ræ Fer- \\ reæ. # Magnitu \\ do diame \\ tri ſphæ \\ -ræ ſtan. \\ -neæ. <lb/>141 # 6 {78/100} # 8 {2/100} # 8 {31/100} # 8 {70/100} # 9 {5/100} # 9 {29/100} <lb/>142 # 6 {80/100} # 8 {4/100} # 8 {33/100} # 8 {72/100} # 9 {7/100} # 9 {31/100} <lb/>143 # 6 {81/100} # 8 {6/100} # 8 {35/100} # 8 {74/100} # 9 {9/100} # 9 {33/100} <lb/>144 # 6 {83/100} # 8 {8/100} # 8 {37/100} # 8 {76/100} # 9 {11/100} # 9 {35/100} <lb/>145 # 6 {85/100} # 8 {9/100} # 8 {39/100} # 8 {78/100} # 9 {14/100} # 9 {38/100} <lb/>146 # 6 {86/100} # 8 {11/100} # 8 {41/100} # 8 {80/100} # 9 {16/100} # 9 {40/100} <lb/>147 # 6 {88/100} # 8 {13/100} # 8 {43/100} # 8 {82/100} # 9 {18/100} # 9 {42/100} <lb/>148 # 6 {89/100} # 8 {15/100} # 8 {45/100} # 8 {84/100} # 9 {20/100} # 9 {44/100} <lb/>149 # 6 {91/100} # 8 {17/100} # 8 {46/100} # 8 {86/100} # 9 {22/100} # 9 {46/100} <lb/>150 # 6 {92/100} # 8 {19/100} # 8 {48/100} # 8 {88/100} # 9 {24/100} # 9 {48/100}</note> <p style="it"> <s xml:id="echoid-s917" xml:space="preserve">EST bœc tabula, quemadmodum & </s> <s xml:id="echoid-s918" xml:space="preserve">eius vſus, præcedentis con-<lb/>nuerſa, in ea enim inueniuntur ſphærarum grauitates ex data <lb/>diametrorum magnitudine, in bac vero deprebenduntur diametro-<lb/>rum magnitudines ex data ſpbærarum grauitate.</s> <s xml:id="echoid-s919" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s920" xml:space="preserve">Quæro exempli gratia magnitudinem diametri ſpbæræ aureæ, <lb/>grauitatem babentis 10, lib. </s> <s xml:id="echoid-s921" xml:space="preserve">Numeri in prima columna ſub titulo <lb/>grauitatis denotant ſpbærarum grauitates, reliqui vero in reliquis <lb/>columnis denotant diametrorum magnitudines; </s> <s xml:id="echoid-s922" xml:space="preserve">itaque in linea 10, <lb/>lib. </s> <s xml:id="echoid-s923" xml:space="preserve">ſub titulo magnitudinis diametri ſpbæræ aureæ, datur quæſita <lb/>diametri magnitudo partium 2 {81/100} qualium pes vnus ect 12.</s> <s xml:id="echoid-s924" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s925" xml:space="preserve">Quæro magnitudinem diametri ſphæræ ferreæ, grauitatem baben-<lb/>tis 50, lib. </s> <s xml:id="echoid-s926" xml:space="preserve">in linea 50, lib. </s> <s xml:id="echoid-s927" xml:space="preserve">ſub titulo magnitudinis diametri ſpbæræ <lb/>ferreæ, datur quæſita diametri magnitudo 6 {40/100}.</s> <s xml:id="echoid-s928" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s929" xml:space="preserve">Quæro magnitudinem diametri ſpbæræ argenteæ, grauitatem ba-<lb/>bentis 60, lib. </s> <s xml:id="echoid-s930" xml:space="preserve">in linea 60, lib ſub titulo magnitudinis diametri ſpbæræ <lb/>argenteæ, datur ipſa magnitudo 6 {25/100}.</s> <s xml:id="echoid-s931" xml:space="preserve"/> </p> <pb o="50" file="0062" n="62" rhead="PROMOTVS"/> <p style="it"> <s xml:id="echoid-s932" xml:space="preserve">Quæro denique magnitudinem diametri ſpbæræ ctanneæ, graui-<lb/>tatem babentis 38, lib. </s> <s xml:id="echoid-s933" xml:space="preserve">in linea 38, lib. </s> <s xml:id="echoid-s934" xml:space="preserve">ſub titulo magnitudinis dia-<lb/>metri ſpbæræ ctanneæ, datur quæſita diametri magnitudo 6, ad <lb/>vnguem.</s> <s xml:id="echoid-s935" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s936" xml:space="preserve">Notandum autem ect, quod numeri, qui diametrorum magnitu-<lb/>dines denotant, non ſunt veri, ac certi, ſed veris bene proximi, quoniã <lb/>numeri, quorum ipſi ſunt radices cubicæ, non ſunt cubi, & </s> <s xml:id="echoid-s937" xml:space="preserve">ideo ipſæ <lb/>radices non explicantur accurate, ſed vel veris maiores, velminores, <lb/>atque vt cognoſcantur quæ ſint maiores, queue minores, maioribus <lb/>duo puncta adiecimus, minoribus vnum, accuratis nullum. </s> <s xml:id="echoid-s938" xml:space="preserve">inter om-<lb/>nes autem vnus ect accuratus, is ſcilicet, qui magnitudinem indicat <lb/>diametri ſpbæræ ctanneæ, grauitatem babentis 38, lib.</s> <s xml:id="echoid-s939" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s940" xml:space="preserve">De compoſitione huius Tabulæ.</s> <s xml:id="echoid-s941" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s942" xml:space="preserve">Huius tabulæ compoſitio pendet ex præcedenti tabula, & </s> <s xml:id="echoid-s943" xml:space="preserve">ex pro-<lb/>poſ. </s> <s xml:id="echoid-s944" xml:space="preserve">17, buius, ſienim fiat vt grauitas ſphæræ ctanneæ, diametrum <lb/>babentis vnius vnciæ, id ect, vt grana 1216, ad grauitatem ſphæræ <lb/>vnius libræ, idect, ad grana 6912, ita cubus diametri vnius vnciæ, <lb/>boc eſt, ita 1, ad alium numerum, qui ſit 5 {13/19} is erit cubus diametri <lb/>ſpbæræ ctanneæ, grauitatem babentis 1, lib. </s> <s xml:id="echoid-s945" xml:space="preserve">demonctratum enim <lb/>ect prop. </s> <s xml:id="echoid-s946" xml:space="preserve">17, buius, ſpbæras eiuſdem generis inter ſe eſſe in grauitate, <lb/>vt diametrorum cubi in magnitudine; </s> <s xml:id="echoid-s947" xml:space="preserve">quare radix cubica numeri <lb/>5 {13/19}, dabit ipſam diametrum, ſed quoniam numerus 5 {13/19}, non ect <lb/>præciſe cubus, eius radix non explicabitur accurata, ſed vt explicetur <lb/>veræ bene proxima, multiplicetur 5 {13/19}, per 1000000. </s> <s xml:id="echoid-s948" xml:space="preserve">& </s> <s xml:id="echoid-s949" xml:space="preserve">ex producto <lb/>5684210 {10/19}, neglecto fracto {10/19}, eruatur radix, tanquam ex accu-<lb/>rato numero cubo, ea erit 178, proxime, & </s> <s xml:id="echoid-s950" xml:space="preserve">erit centupla radicis nu-<lb/>meri 5 {13/19}, nam numerus 1000000, per quem fuit multiplicatus <lb/>5 {13/19}, cubus eſt ex 100; </s> <s xml:id="echoid-s951" xml:space="preserve">magnitudo igitur diametri ſpbæræ ſtanneæ, <lb/>grauitatem babentis 1, lib. </s> <s xml:id="echoid-s952" xml:space="preserve">erit 1 {78/100}. </s> <s xml:id="echoid-s953" xml:space="preserve">reliquarum autem ex ſtanno <lb/>ſpbærarum, grauitatem babentium duplam primæ, triplam, quadru-<lb/>plam & </s> <s xml:id="echoid-s954" xml:space="preserve">c. </s> <s xml:id="echoid-s955" xml:space="preserve">ita inuenientur diametri.</s> <s xml:id="echoid-s956" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s957" xml:space="preserve">Duplum numeri 5684210 {10/19}, id ect 11368421 {1/19}, erit cubus <lb/>centupli diametri ſpberæ ctanneæ, grauitatem babentis duplam pri-<lb/>mæ, boc ect 2, lib. </s> <s xml:id="echoid-s958" xml:space="preserve">ex ſupra nominata enim prop. </s> <s xml:id="echoid-s959" xml:space="preserve">17, buius, eſt vt gra-<lb/>uitas ſpbæræ vnius libræ, ad grauitatem ſpbæræ duarum librarum, <lb/>ita cubus diametri primæ ſpbæræ, ad cubum diametri ſecundæ. </s> <s xml:id="echoid-s960" xml:space="preserve">Si <lb/>vero triplicetur numerus 5684210 {10/19}, eius triplum, quod ect <lb/>17052631 {11/19}, erit cubus centupli diametri ſpbæræ ſtanneæ, graui-<lb/>statem babentis triplam primæ, idect 3, lib. </s> <s xml:id="echoid-s961" xml:space="preserve">& </s> <s xml:id="echoid-s962" xml:space="preserve">ſi quadruplicetur, eius <pb o="51" file="0063" n="63" rhead="ARCHIMEDES."/> quadruplum erit cubus centapli diametri ſphæræ ctanneæ, grauita-<lb/>tem babentis quadruplam primæ, & </s> <s xml:id="echoid-s963" xml:space="preserve">ſic deinceps. </s> <s xml:id="echoid-s964" xml:space="preserve">itaque ſi ex eius <lb/>multiplicibus, neglectis fractis, eruãtur radices, tanquam ex accura-<lb/>tis numeris cubis, ipſæ indicabunt diametrorum magnitudines in <lb/>ratione centupla. </s> <s xml:id="echoid-s965" xml:space="preserve">Sed vt etiam euitetur labor multiplicandi prædi-<lb/>ctum numerum 5684210 {10/19}, bac ratione inuenientur eius multi-<lb/>plicia.</s> <s xml:id="echoid-s966" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s967" xml:space="preserve">Prædicto numero 5684210 {10/19}, addatur eius duplum, idect, <lb/>11368421 {1/19}, ſumma 17052631 {11/19}, dabit eius triplum, ſi vero ei <lb/>addatur eius triplum, id eſt, 17052631 {11/19}, ſumma 22736842 {2/19}, <lb/>dabit eius quadruplum, & </s> <s xml:id="echoid-s968" xml:space="preserve">ſieius quadruplum ei addatur, ſumma <lb/>dabit eius quintuplum, & </s> <s xml:id="echoid-s969" xml:space="preserve">ſic ſola additione inuenientur eius quot-<lb/>cunque multiplicia.</s> <s xml:id="echoid-s970" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s971" xml:space="preserve">Eadem ratione inuenientur diametri ſpbærarum ex quacunque<unsure/> <lb/>alia materia, ſi enim quæratur de magnitudine diametri verbi gra-<lb/>tia ſpbæræ ferreæ, grauitatem babentis 1, lib. </s> <s xml:id="echoid-s972" xml:space="preserve">fiat vt grana 1314 {22/37}, <lb/>id ect vt grauitas ſpbæræ ferreæ, cuius diameter eſt vnius vnciæ, ad <lb/>grauitatem vnius libræ, id ect ad grana 6912, ita cubus diametri <lb/>vnius vnciæ, boc eſt ita 1, ad alium numerum qui ſit 5 {49/190}, is igi-<lb/>tur numerus<emph style="sub">*</emph> erit cubus diametri ſpbæræ ferreæ, grauitatem baben-<lb/> <anchor type="note" xlink:label="note-0063-01a" xlink:href="note-0063-01"/> tis 1, lib, quare radix cubica numeri 5 {49/190}, dabit quæſitam dia-<lb/>metrum, & </s> <s xml:id="echoid-s973" xml:space="preserve">quoniam numerus 5 {49/190}, non eſt præciſe cubus, & </s> <s xml:id="echoid-s974" xml:space="preserve">ideo <lb/>non explicabitur eius radix accurate, multiplicetur per 1000000, & </s> <s xml:id="echoid-s975" xml:space="preserve"><lb/>ex producto 5257894 {14/19}, neglecto fracto {14/19}, eruatur radix, tan-<lb/>quam ex accurato numero cubo, ea erit 174: </s> <s xml:id="echoid-s976" xml:space="preserve">ferè, & </s> <s xml:id="echoid-s977" xml:space="preserve">erit centupla ra-<lb/>dicis numeri 5 {49/190}, quia numerus 5 {49/190}, multiplicatus fuit per <lb/>cubum ex 100; </s> <s xml:id="echoid-s978" xml:space="preserve">diameter igitur ſpbæræ ferreæ, grauitatem babentis <lb/>1, lib. </s> <s xml:id="echoid-s979" xml:space="preserve">erit 1 {74/100}: </s> <s xml:id="echoid-s980" xml:space="preserve">deinde ſi duplicetur 5257894 {14/19}, & </s> <s xml:id="echoid-s981" xml:space="preserve">ex ita du-<lb/>plicato eruatur radix cubica 219, ea dabit centuplum diametri ſpbæ-<lb/>ræ ferreæ, grauitatem babentis 2, lib. </s> <s xml:id="echoid-s982" xml:space="preserve">& </s> <s xml:id="echoid-s983" xml:space="preserve">ſi triplicetur, triplicati radix <lb/>cubica 250 dabit centuplum diametri ſpbæræ ferreæ, cuius grauitas <lb/>erit 3, lib. </s> <s xml:id="echoid-s984" xml:space="preserve">& </s> <s xml:id="echoid-s985" xml:space="preserve">ſic reliquarum ſpbærarum in infinitum inuenientur <lb/>diametri. </s> <s xml:id="echoid-s986" xml:space="preserve">multiplicia autem numeri 5257894 {14/19}, ſola additione in-<lb/>uenientur, vt dictum ect ſupra de inuentione multiplicium numeri <lb/>5684210 {10/19}. </s> <s xml:id="echoid-s987" xml:space="preserve">Atque bac ratione prædictam tabulam compoſuimus.</s> <s xml:id="echoid-s988" xml:space="preserve"/> </p> <div xml:id="echoid-div65" type="float" level="2" n="1"> <note position="right" xlink:label="note-0063-01" xlink:href="note-0063-01a" xml:space="preserve">17. <lb/>buius.</note> </div> <p> <s xml:id="echoid-s989" xml:space="preserve">QVomodo Archimedes argenti mixtionem depre-<lb/>hendit in auro.</s> <s xml:id="echoid-s990" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s991" xml:space="preserve">Hiero (referente V itruuio lib. </s> <s xml:id="echoid-s992" xml:space="preserve">9. </s> <s xml:id="echoid-s993" xml:space="preserve">Cap. </s> <s xml:id="echoid-s994" xml:space="preserve">3.) </s> <s xml:id="echoid-s995" xml:space="preserve">Siracuſis auctus regia <pb o="52" file="0064" n="64" rhead="PROMOTVS"/> poteſtate, rebus bene geſtis, cum auream coronam votiuam, dijs im-<lb/>mortalibus in quodam fano conſtituiſſet ponendam, immani precio <lb/>locauit faciendam, & </s> <s xml:id="echoid-s996" xml:space="preserve">aurum ad ſacoma appendit redemptori. </s> <s xml:id="echoid-s997" xml:space="preserve">is ad <lb/>tempus opus manufactum ſubtiliter, regi approbauit, & </s> <s xml:id="echoid-s998" xml:space="preserve">ad ſacoma <lb/>pondus coronæ viſus eſt præſtitiſſe. </s> <s xml:id="echoid-s999" xml:space="preserve">Poſtea quam inditium eſt factum, <lb/>dempto auro, tantundem argenti in id coronarium opus admixtum <lb/>eſſe: </s> <s xml:id="echoid-s1000" xml:space="preserve">indignatus Hiero ſe contemptum, neque inueniens, qua ratione <lb/>id furtum deprebenderet, rogauit Archimedem, vti in ſe ſumeret de <lb/>eo cogitationem. </s> <s xml:id="echoid-s1001" xml:space="preserve">tunc is cum baberet eius rei curam, caſu venit in <lb/>balneum, ibique cum in ſolium deſcenderet, animaduertit quantum <lb/>corporis ſui in eo inſideret, tantum aquæ extra ſolium effluere. </s> <s xml:id="echoid-s1002" xml:space="preserve">itaq; <lb/></s> <s xml:id="echoid-s1003" xml:space="preserve">e<unsure/>um eius rei rationem explicationis offendiſſet non ect moratus, ſed <lb/>exiliuit gaudio motus de ſolio, & </s> <s xml:id="echoid-s1004" xml:space="preserve">nudus vadens domum verſus, ſigni <lb/>ficabat clara voce inueniße quod quæreret. </s> <s xml:id="echoid-s1005" xml:space="preserve">nam currens identidem <lb/>grece clamabat {δι}”ρη{κο} {δι}”ρη{κο} tum vero ex eo inuentionis ingreſſu <lb/>duas dicitur feciſſe maſſas æquo pondere, quo etiam fuerat coro-<lb/>na, vnam ex auro, alteram ex argento. </s> <s xml:id="echoid-s1006" xml:space="preserve">cum ita fecißet, vas amplũ <lb/>ad ſumma labra impleuit aqua, in quo demiſit argenteam maſſam. </s> <s xml:id="echoid-s1007" xml:space="preserve"><lb/>cuius quanta magnitudo in vaſe depreſſa ect, tantum aquæ effluxit. </s> <s xml:id="echoid-s1008" xml:space="preserve"><lb/>ita exempta maſſa, quanto minus factum fuerat refudit, ſextario <lb/>menſus, vt eodem modo, quo prius fuerat, ad labra æquaretur. </s> <s xml:id="echoid-s1009" xml:space="preserve">ita ex <lb/>eo inuenit, quantum ad certum pondus argenti certa aquæ menſura <lb/>reſponderet.</s> <s xml:id="echoid-s1010" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1011" xml:space="preserve">Cum id expertus eſſet tum auream maſſam ſimiliter pleno vaſe <lb/>demiſit, & </s> <s xml:id="echoid-s1012" xml:space="preserve">ea exempta, eadem ratione menſura addita, inuenit ex <lb/>aqua non tantum defluxiſſe, ſed tantum minus, quantum minus ma <lb/>gno corpore eodem pondere auri maſſa eſſet quam argenti. </s> <s xml:id="echoid-s1013" xml:space="preserve">Poctea <lb/>vero repleto vaſe, in eadĕ aqua ipſa corona demiſſa, inuenit plus aquæ <lb/>defluxiſſe in coronam, quam in auream eodem pondere maſſam, & </s> <s xml:id="echoid-s1014" xml:space="preserve"><lb/>ita ex eo quod plus defluxerat aquæ in corona, quam in maſſa ratio-<lb/>cinatus, deprebendit argenti in auro mixtionem, & </s> <s xml:id="echoid-s1015" xml:space="preserve">manifectum fur-<lb/>tum redemptoris.</s> <s xml:id="echoid-s1016" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div67" type="section" level="1" n="38"> <head xml:id="echoid-head41" xml:space="preserve">Hactenus Vitruuius.</head> <p> <s xml:id="echoid-s1017" xml:space="preserve">Mirum certe Archimedis fuit inuentum, ipſius tamen <lb/>modus ad inueniendam illam aquæ menſuram, quæ ad <lb/>certum pondus auri, vel argenti, vel coronæ reſponderet, <lb/>maiori diligentia indiget, quam quæ ab hominibus adhiberi <lb/>poteſt, impoſſibile enim eſt, exempta corona, vel aurea maſſa, <lb/>vel argentea, tantum aquæ refundere, quantum è vaſe efflu-<lb/>xerat ad vnguem, nam repoſita aqua in vaſe, non poſſumus <pb o="53" file="0065" n="65"/> affirmare ipſum vas eſſe plenum, niſi aqua incipiat effluere, <lb/>cum autem incipit, effluit aliquando totus ferè cumulus, ita-<lb/>que vel plus aquæ additur eo, quod deficit, vel minus, niſi <lb/>coniectura aſſequatur: </s> <s xml:id="echoid-s1018" xml:space="preserve">at vero coniectura pro veritate non ac-<lb/>cipitur. </s> <s xml:id="echoid-s1019" xml:space="preserve">præterea exempta corona, vel aurea maſſa, vel argen-<lb/>tea, eximitur etiam ſimul cum ipſa aliquantum aquæ, quæ cir-<lb/>cum ipſam remanet, atque huiuſmodi defectus errorem indu-<lb/>cit ſenſibilem.</s> <s xml:id="echoid-s1020" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1021" xml:space="preserve">Neque per collectionem quæſita aquæ menſura inueniri <lb/>poteſt:</s> <s xml:id="echoid-s1022" xml:space="preserve">æquè enim impoſſibile eſt vniuerſam illam æquam col-<lb/>ligere, quæ extra vas effluit, quando corona, vel aurea maſſa <lb/>vel argentea in ipſo vaſe deprimitur, cum enim aqua è vaſe <lb/>effluat, pars ipſius aquæ vaſi, ex quo effluit, pars vaſi in quod <lb/>influit adhæret, & </s> <s xml:id="echoid-s1023" xml:space="preserve">ſi vniuerſa omnino ſemper non colligatur, <lb/>erit non parui erroris cauſa, præter quam quod, non ſemper <lb/>adeo facile inuenitur par auri, argentique maſſa, quando co-<lb/>rona, vel alia auri maſſa, quæ examinanda proponitur, medio-<lb/>crem excederet magnitudinem.</s> <s xml:id="echoid-s1024" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1025" xml:space="preserve">Neque præterea poteſt diſcerni prædicta argenti portio in <lb/>aliqua auri parua maſſa, differentiæ enim aquarum, quæ ex-<lb/>tra vas effluunt, ſunt adeo exiguæ, vt ne cognoſci quidem <lb/>poſſint, quod ſi cognoſcerentur, non ſemper erunt veræ, <lb/>ſiquidem non ſemper in vaſis medio in cumulum creſcens <lb/>æqualis aquæ copia remanet, ſed maior interdum, inter-<lb/>dum minor, vt conſpicitur. </s> <s xml:id="echoid-s1026" xml:space="preserve">fit enim vt aliquando cumulus <lb/>ille frangatur pluribus in locis, & </s> <s xml:id="echoid-s1027" xml:space="preserve">ideo aqua diffundatur, vt <lb/>ferè nihil ipfius cumuli ſuperſit, aliquando vero frangatur <lb/>in vno tantum loco, & </s> <s xml:id="echoid-s1028" xml:space="preserve">aqua colligens ſe in cumulum, parum <lb/>diffluat.</s> <s xml:id="echoid-s1029" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1030" xml:space="preserve">Sed ponderandis corporibus in aere & </s> <s xml:id="echoid-s1031" xml:space="preserve">aqua, eo modo, quo <lb/>dictum eſt in fine exempli prop. </s> <s xml:id="echoid-s1032" xml:space="preserve">8. </s> <s xml:id="echoid-s1033" xml:space="preserve">inuenitur quæſita aquæ, <lb/>grauitas, ita exactè, vt requiritur, ſiue ſit corpus illud paruum, <lb/>ſiue magnum nihil intereſt, & </s> <s xml:id="echoid-s1034" xml:space="preserve">præterea facillima eſt operatio, <lb/>nec adinueniendæ ſunt auri, & </s> <s xml:id="echoid-s1035" xml:space="preserve">argenti maſſæ æque graues, ac <pb o="54" file="0066" n="66" rhead="PROMOTVS"/> corona, ſed quælibet particulæ, grauitate quacunque, ctiam <lb/>differentes interſe, ſufficiunt.</s> <s xml:id="echoid-s1036" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1037" xml:space="preserve">Deratione autem, qua Archimedes, cognitis grauitatibus <lb/>trium corporum ex aqua, magnitudine æqualium, coronæ <lb/>ſcilicet vnum, alterum maſſæ auree, tertium argenteæ, potue-<lb/>rit furtum aurificis in regia corona deprehendere, atque ar-<lb/>gentum quod erat in ea permixtum ab auro diſcernere, pluri-<lb/>mi ſcripſerunt, modos etiam ad id faciendum excogitarunt <lb/>varios, longa tamen methodo, atque difficili vſi ſunt, & </s> <s xml:id="echoid-s1038" xml:space="preserve">quod <lb/>maximam confuſionem, & </s> <s xml:id="echoid-s1039" xml:space="preserve">obſcuritatem parit, nullum opera-<lb/>tionis tradunt præceptum firmum, ac ſtabile. </s> <s xml:id="echoid-s1040" xml:space="preserve">ego autem vni-<lb/>ca tantum proportionis ratiocinatione, ſeu regula trium (vt <lb/>vulgo dicitur) breuiter, & </s> <s xml:id="echoid-s1041" xml:space="preserve">expedite idem conſequor, eamque <lb/>geometrica ratione demonſtro. </s> <s xml:id="echoid-s1042" xml:space="preserve">Problema igitur ad hoc facien <lb/>dum ita concipio & </s> <s xml:id="echoid-s1043" xml:space="preserve">abſoluo.</s> <s xml:id="echoid-s1044" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div68" type="section" level="1" n="39"> <head xml:id="echoid-head42" xml:space="preserve">PROBLEMA IX. PROPOS. XVIII.</head> <p> <s xml:id="echoid-s1045" xml:space="preserve">POrtionem metalli, alterimetallo miſtam, ponde-<lb/>ris ratiocinatione diſcernere.</s> <s xml:id="echoid-s1046" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1047" xml:space="preserve">QVONIAM de Hieronis corona facta eſt mentio, ſit ea B, <lb/>eiuſque grauitas EK, & </s> <s xml:id="echoid-s1048" xml:space="preserve">oporteat argentum, quod ſit in ea permixtũ, <lb/>ab auro diſcernere, hoc eſt oporteat inuenire quanta erit portio ar-<lb/> <anchor type="figure" xlink:label="fig-0066-01a" xlink:href="fig-0066-01"/> genti, & </s> <s xml:id="echoid-s1049" xml:space="preserve">quanta auri. </s> <s xml:id="echoid-s1050" xml:space="preserve">In-<lb/>telligantur duo corpora <lb/>A, D, vnum aureum, al-<lb/>terum argenteum æque <lb/>grauia atque corona,, <lb/>deinde trium corporum <lb/>ex aqua, magnitudine, <lb/>æqualium, aureo ſcili-<lb/>cet corpori vnum, alte-<lb/>rum coronæ, tertium <lb/>corpori argenteo, inue-<lb/>niantur grauitates, id autem poterit fieri facillime, ſi accipiãtur duo <lb/>corpora vnum ex auro, alterum ex argento, grauitate quacunque, vt <pb o="55" file="0067" n="67" rhead="ARCHIMEDES."/> dictum eſt in propoſitionis octauæ exemplo, non enim neceſſe eſt ha-<lb/>bere duo corpora ex auro & </s> <s xml:id="echoid-s1051" xml:space="preserve">argento, grauitatem habentia eandem <lb/>quam & </s> <s xml:id="echoid-s1052" xml:space="preserve">corona, & </s> <s xml:id="echoid-s1053" xml:space="preserve">hac de cauſa diximus ſupra intelligãtur duo cor-<lb/>pora, non autem accipiantur. </s> <s xml:id="echoid-s1054" xml:space="preserve">ſit igitur primi corporis aquei æqualis <lb/>aureo A, inuenta grauitas G, ſecundi vero æqualis coronæ B, graui-<lb/>tas F, & </s> <s xml:id="echoid-s1055" xml:space="preserve">tertij æqualis corpori argenteo D, grauitas H, & </s> <s xml:id="echoid-s1056" xml:space="preserve">fiat vt dif-<lb/>ferentia inter G, & </s> <s xml:id="echoid-s1057" xml:space="preserve">H, ad EK, ita differentia inter G, & </s> <s xml:id="echoid-s1058" xml:space="preserve">F, ad aliam-, <lb/>grauitatem, quæ ſit K. </s> <s xml:id="echoid-s1059" xml:space="preserve">Dico K, grauitatem eſſe portionis argenti, <lb/>quod eſt in corona, E vero grauitatem auri.</s> <s xml:id="echoid-s1060" xml:space="preserve"/> </p> <div xml:id="echoid-div68" type="float" level="2" n="1"> <figure xlink:label="fig-0066-01" xlink:href="fig-0066-01a"> <image file="0066-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0066-01"/> </figure> </div> <p> <s xml:id="echoid-s1061" xml:space="preserve">Vel ſi pro tertio proportionis termino ſumatur differentia inter F, <lb/>& </s> <s xml:id="echoid-s1062" xml:space="preserve">H, & </s> <s xml:id="echoid-s1063" xml:space="preserve">quartus terminus ſit E, Dico E, grauitatem eſſe portionis au-<lb/>ri, K vero argenti.</s> <s xml:id="echoid-s1064" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1065" xml:space="preserve">Quartus autem vtriuſque proportionis terminus * minor eſt ſe-<lb/> <anchor type="note" xlink:label="note-0067-01a" xlink:href="note-0067-01"/> cundo EK, quod & </s> <s xml:id="echoid-s1066" xml:space="preserve">tertius minor eſt primo, primus enim terminus <lb/>eſt differentia inter G, & </s> <s xml:id="echoid-s1067" xml:space="preserve">H, tertius vero, vel eſt differentia inter G, & </s> <s xml:id="echoid-s1068" xml:space="preserve"><lb/>F, vel differentia inter F, & </s> <s xml:id="echoid-s1069" xml:space="preserve">H, vter que minor primo. </s> <s xml:id="echoid-s1070" xml:space="preserve">Exemplis autem <lb/>res fiet illuſtrior.</s> <s xml:id="echoid-s1071" xml:space="preserve"/> </p> <div xml:id="echoid-div69" type="float" level="2" n="2"> <note position="right" xlink:label="note-0067-01" xlink:href="note-0067-01a" xml:space="preserve">14. 5. <lb/>Elem.</note> </div> </div> <div xml:id="echoid-div71" type="section" level="1" n="40"> <head xml:id="echoid-head43" xml:space="preserve">Exemplum. I.</head> <p style="it"> <s xml:id="echoid-s1072" xml:space="preserve">Sit coronæ grauitas 95, lib. </s> <s xml:id="echoid-s1073" xml:space="preserve">& </s> <s xml:id="echoid-s1074" xml:space="preserve">oporteat facere quod imperatum eſt. <lb/></s> <s xml:id="echoid-s1075" xml:space="preserve">Intellig antur duo corpora, vnum aureum, alterum argenteum, æque <lb/>grauia atque corona, deinde trium corporum ex aqua, magnitudine <lb/>æqualium, aureo ſcilicet corpori vnum, alterum coronæ, tertium cor-<lb/>pori argenteo, inueniantur grauitates, vt in exemplo prop. </s> <s xml:id="echoid-s1076" xml:space="preserve">8. </s> <s xml:id="echoid-s1077" xml:space="preserve">dictum <lb/>est, quæ ſint primi nimirum corporis aquei @, ſecundi vero 6, & </s> <s xml:id="echoid-s1078" xml:space="preserve">terty <lb/>9 {8/31}, & </s> <s xml:id="echoid-s1079" xml:space="preserve">fiat vt differentia inter 5, & </s> <s xml:id="echoid-s1080" xml:space="preserve">9 {6/31}, boc eſt vt 4 {6/31}, ad 95, <lb/>grauitatem videlicet coronæ, ita differentia inter 5, & </s> <s xml:id="echoid-s1081" xml:space="preserve">6, boc est 1, ad <lb/>22 {17/26}, ergo 22 {17/26}, erit grauitas portionis argenti quod est in coro-<lb/>na, qua detracta ex totali grauitate coronæ, reliquum 72 {9/26}, erit <lb/>grauitas portionis auri.</s> <s xml:id="echoid-s1082" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1083" xml:space="preserve">Vel ſi pro tertio proportionis termino ſumatur differentia inter 6, <lb/>& </s> <s xml:id="echoid-s1084" xml:space="preserve">9 {6/31}, quæ eſt 3 {6/31}, quartus terminus 72 {9/26}, erit grauitas por-<lb/>tionis auri, quæ ſi dematur ex totali grauitate coronæ, remanebit <lb/>22 {17/26}, pro grauitate portionis argenti.</s> <s xml:id="echoid-s1085" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div72" type="section" level="1" n="41"> <head xml:id="echoid-head44" xml:space="preserve">Exemplum. II.</head> <p style="it"> <s xml:id="echoid-s1086" xml:space="preserve">Sit aliquod corpus mistum ex auro. </s> <s xml:id="echoid-s1087" xml:space="preserve">& </s> <s xml:id="echoid-s1088" xml:space="preserve">ære, & </s> <s xml:id="echoid-s1089" xml:space="preserve">babeat grauitatem <lb/>171. </s> <s xml:id="echoid-s1090" xml:space="preserve">lib. </s> <s xml:id="echoid-s1091" xml:space="preserve">& </s> <s xml:id="echoid-s1092" xml:space="preserve">oporteat inuenire quanta erit portio æris in ipſo corpore, <pb o="56" file="0068" n="68" rhead="PROMOTVS"/> & </s> <s xml:id="echoid-s1093" xml:space="preserve">quanta auri. </s> <s xml:id="echoid-s1094" xml:space="preserve">Intellig antur duo corpora, vnum ex auro puro, al-<lb/>terum ex are, æque grauia atque corpus mistum, & </s> <s xml:id="echoid-s1095" xml:space="preserve">trium corporum <lb/>ex aqua, quorum vnum ſit æquale corpori aureo magnitudine, alte-<lb/>rum misto, tertium æreo, inueniantur grauitates, vt in exemplo pro-<lb/>poſ. </s> <s xml:id="echoid-s1096" xml:space="preserve">8. </s> <s xml:id="echoid-s1097" xml:space="preserve">dictum est, quæ ſint 9, 11, & </s> <s xml:id="echoid-s1098" xml:space="preserve">19, & </s> <s xml:id="echoid-s1099" xml:space="preserve">fiat vt differentia inter 9, & </s> <s xml:id="echoid-s1100" xml:space="preserve"><lb/>19, ad 171, grauitatem videlicet corporis misti, ita differentia inter <lb/>9, & </s> <s xml:id="echoid-s1101" xml:space="preserve">11, ad 34 {1/5}, portio igitur corporis miſti ærea grauitatem babebit <lb/>34 {1/5}, quæ ſi auferatur ex totali corporis misti grauitate, remanebit <lb/>136 {4/5}, pro grauitate portionis auri.</s> <s xml:id="echoid-s1102" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1103" xml:space="preserve">Vel ſi pro tertio proportionis termino ſumatur differentia inter 11, <lb/>& </s> <s xml:id="echoid-s1104" xml:space="preserve">19, quartus terminus 136 {4/6}, erit grauitas portionis auri, qua ab-<lb/>lata ex totali corporis miſti grauitate, reliquum 34 {1/5}, dabit grauita-<lb/>tem portionis æreæ.</s> <s xml:id="echoid-s1105" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1106" xml:space="preserve">At vero huiuſmodi ratio cinationem ad diſcernendum ar-<lb/>gentum ab auro, vel aliud metallum ab altero metallo, recte <lb/>eſſe inſtitutam, ſequenti Theoremate demonſtrabitur.</s> <s xml:id="echoid-s1107" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div73" type="section" level="1" n="42"> <head xml:id="echoid-head45" xml:space="preserve">THEOREMA X. PROPOS. XIX.</head> <p> <s xml:id="echoid-s1108" xml:space="preserve">SI trium corporum æque grauium primum & </s> <s xml:id="echoid-s1109" xml:space="preserve">ter-<lb/>tium fuerint generis diuerſi, ſecundi autem portio <lb/>fuerit eiuſdem generis cum corpore primo, reliqua ve-<lb/>ro eiuſdem generis cum corpore tertio, fuerint etiam <lb/>tres quantitates aquæ prædictis corporibus æquales, pri-<lb/>ma videlicet corpori primo, ſecunda ſecundo, & </s> <s xml:id="echoid-s1110" xml:space="preserve">tertia <lb/>tertio. </s> <s xml:id="echoid-s1111" xml:space="preserve">erit vt differentia grauitatum primæ & </s> <s xml:id="echoid-s1112" xml:space="preserve">tertiæ <lb/>quantitatis aquæ, ad grauitatem corporis ſecundi, ita <lb/>differentia grauitatum primæ & </s> <s xml:id="echoid-s1113" xml:space="preserve">ſecundæ quantitatis <lb/>aquæ, ad grauitatem portionis corporis ſecundi, quæ eſt <lb/>eiuſdem generis cum corpore tertio.</s> <s xml:id="echoid-s1114" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1115" xml:space="preserve">Et ita differentia grauitatum ſecundæ & </s> <s xml:id="echoid-s1116" xml:space="preserve">tertiæ quan <lb/>titatis aquæ, ad grauitatem portionis eiuſdem generis <lb/>cum corpore primo.</s> <s xml:id="echoid-s1117" xml:space="preserve"/> </p> <pb o="57" file="0069" n="69" rhead="ARCHIMEDES"/> <p> <s xml:id="echoid-s1118" xml:space="preserve">SINT tria corpora æque grauia A, BC, D, quorum A, primum, <lb/>& </s> <s xml:id="echoid-s1119" xml:space="preserve">tertium D. </s> <s xml:id="echoid-s1120" xml:space="preserve">ſint generis diuerſi, portio vero ſecundi B, ſit eiuſdem <lb/>generis cum corpore A. </s> <s xml:id="echoid-s1121" xml:space="preserve">& </s> <s xml:id="echoid-s1122" xml:space="preserve">portio C, eiuſdem generis cum corpore <lb/> <anchor type="figure" xlink:label="fig-0069-01a" xlink:href="fig-0069-01"/> D, ſint etiam alia tria <lb/>corpora aquea P, OL, <lb/>& </s> <s xml:id="echoid-s1123" xml:space="preserve">Q, quorũ P, ſit æqua-<lb/>le corpori A, magnitu-<lb/>dine, ipſum vero OL, <lb/>æquale corpori BC, & </s> <s xml:id="echoid-s1124" xml:space="preserve"><lb/>ipſum Q, æquale cor-<lb/>pori D, & </s> <s xml:id="echoid-s1125" xml:space="preserve">ſint earum <lb/>grauitates, G, ipſius P, <lb/>& </s> <s xml:id="echoid-s1126" xml:space="preserve">FV, ipſius OL, & </s> <s xml:id="echoid-s1127" xml:space="preserve">H, <lb/>ipſius Q. </s> <s xml:id="echoid-s1128" xml:space="preserve">Dico vt diffe-<lb/>rentia grauitatum G, <lb/>H, ad grauitatem cor-<lb/>poris BC, ita eſſe diffe-<lb/>rentiam grauitatum <lb/>G, FV, ad grauitatem <lb/>portionis C; </s> <s xml:id="echoid-s1129" xml:space="preserve">& </s> <s xml:id="echoid-s1130" xml:space="preserve">ita differentiam grauitatum FV, H, ad portionis B, <lb/>grauitatem. </s> <s xml:id="echoid-s1131" xml:space="preserve">Sit enim portionis B, grauitas E, & </s> <s xml:id="echoid-s1132" xml:space="preserve">portionis C, gra-<lb/>uitas K; </s> <s xml:id="echoid-s1133" xml:space="preserve">ergo totius corporis BC, grauitas erit EK, ſitq; </s> <s xml:id="echoid-s1134" xml:space="preserve">portionis O, <lb/>quæ ſit æqualis portioni B, grauitas F, ergo reliquæ portionis L, <lb/>æqualis portioni C, grauitas erit V, Quoniam igitur eſt, vt A, ad P, <lb/>ita B, ad O, æquale videlicet ad æquale, erit permutando, vt A, ad B, <lb/>ita p. </s> <s xml:id="echoid-s1135" xml:space="preserve">ad O, & </s> <s xml:id="echoid-s1136" xml:space="preserve">quoniam ſunt eiuſdem generis A, B, ſimiliter & </s> <s xml:id="echoid-s1137" xml:space="preserve">P, O, <lb/>* erit vt grauitas corporis A, hoc eſt vt EK, (ponuntur enim cor-<lb/> <anchor type="note" xlink:label="note-0069-01a" xlink:href="note-0069-01"/> pora A, BC, D, æque grauia,) ad E, ita G, ad F, quod igitur fit ex EK, <lb/>& </s> <s xml:id="echoid-s1138" xml:space="preserve">F, nempe ex extremis, æquale erit ei, quod fit ex E, & </s> <s xml:id="echoid-s1139" xml:space="preserve">G, hoc eſt <lb/>ex medijs.</s> <s xml:id="echoid-s1140" xml:space="preserve"/> </p> <div xml:id="echoid-div73" type="float" level="2" n="1"> <figure xlink:label="fig-0069-01" xlink:href="fig-0069-01a"> <image file="0069-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0069-01"/> </figure> <note position="right" xlink:label="note-0069-01" xlink:href="note-0069-01a" xml:space="preserve">4. huius</note> </div> <p> <s xml:id="echoid-s1141" xml:space="preserve">Similiter quoniam eſt, vt D, ad Q, ita C, ad L, æquale videlicet ad <lb/>æquale, erit permutando, vt D, ad C, ita Q, ad L, & </s> <s xml:id="echoid-s1142" xml:space="preserve">quoniam ſunt <lb/>eiuſdem generis D, C, ſimiliter & </s> <s xml:id="echoid-s1143" xml:space="preserve">Q, L, * erit vt grauitas ipſius D, <lb/> <anchor type="note" xlink:label="note-0069-02a" xlink:href="note-0069-02"/> hoc eſt vt EK, ad K, ita H, ad V; </s> <s xml:id="echoid-s1144" xml:space="preserve">quare quod fit ex EK, & </s> <s xml:id="echoid-s1145" xml:space="preserve">V, ex extre-<lb/>mis, æquabitur ei, quod ex H, fit & </s> <s xml:id="echoid-s1146" xml:space="preserve">K, ex medijs.</s> <s xml:id="echoid-s1147" xml:space="preserve"/> </p> <div xml:id="echoid-div74" type="float" level="2" n="2"> <note position="right" xlink:label="note-0069-02" xlink:href="note-0069-02a" xml:space="preserve">4. huius</note> </div> <p> <s xml:id="echoid-s1148" xml:space="preserve">Sed oſtenſum eſt id quod ex EK, fit & </s> <s xml:id="echoid-s1149" xml:space="preserve">F. </s> <s xml:id="echoid-s1150" xml:space="preserve">æquale eſſe ei quod fit ex <lb/>G, & </s> <s xml:id="echoid-s1151" xml:space="preserve">E, ergo quod fit ex EK, & </s> <s xml:id="echoid-s1152" xml:space="preserve">F, vna cum eo, quod ex EK, & </s> <s xml:id="echoid-s1153" xml:space="preserve">V, hoc <lb/>eſb id quod fit ex EK, & </s> <s xml:id="echoid-s1154" xml:space="preserve">FV, æquale erit ei quod ex G, fit & </s> <s xml:id="echoid-s1155" xml:space="preserve">E, vna <lb/>cum eo quod ex H, & </s> <s xml:id="echoid-s1156" xml:space="preserve">K, ſed quod ex G, fit & </s> <s xml:id="echoid-s1157" xml:space="preserve">E, æquale eſt ei quod fit <lb/>ex G, & </s> <s xml:id="echoid-s1158" xml:space="preserve">EK, minus eo quod ex G, & </s> <s xml:id="echoid-s1159" xml:space="preserve">K, quod enim additur, idem &</s> <s xml:id="echoid-s1160" xml:space="preserve"> <pb o="58" file="0070" n="70" rhead="PROMOTVS"/> minuitur; </s> <s xml:id="echoid-s1161" xml:space="preserve">ergo quod fit ex EK, & </s> <s xml:id="echoid-s1162" xml:space="preserve">FV, æquale erit ei quod fit ex G, & </s> <s xml:id="echoid-s1163" xml:space="preserve"><lb/>EK, vna cum eo quod ex H, & </s> <s xml:id="echoid-s1164" xml:space="preserve">K, minus eo quod fit ex G, & </s> <s xml:id="echoid-s1165" xml:space="preserve">K. </s> <s xml:id="echoid-s1166" xml:space="preserve">aufe-<lb/>ratur vtrinque id quod fit ex G, & </s> <s xml:id="echoid-s1167" xml:space="preserve">EK, quod igitur fit ex FV, & </s> <s xml:id="echoid-s1168" xml:space="preserve">EK, <lb/> <anchor type="figure" xlink:label="fig-0070-01a" xlink:href="fig-0070-01"/> minus eo quod ex G, & </s> <s xml:id="echoid-s1169" xml:space="preserve"><lb/>EK, æquabitur ei quod <lb/>ex H, & </s> <s xml:id="echoid-s1170" xml:space="preserve">K, minus eo <lb/>quod fit ex G, & </s> <s xml:id="echoid-s1171" xml:space="preserve">K, ſed <lb/>quod fit ex H, & </s> <s xml:id="echoid-s1172" xml:space="preserve">K, <lb/>minus eo quod fit ex <lb/>G, & </s> <s xml:id="echoid-s1173" xml:space="preserve">K, æquale eſt ei <lb/>quod ex differentia ip-<lb/>ſarum H, G, fit & </s> <s xml:id="echoid-s1174" xml:space="preserve">K, <lb/>ſimiliter, & </s> <s xml:id="echoid-s1175" xml:space="preserve">quod fit ex <lb/>FV, & </s> <s xml:id="echoid-s1176" xml:space="preserve">EK, minus eo <lb/>quod ex G, & </s> <s xml:id="echoid-s1177" xml:space="preserve">EK, <lb/>æquale eſt ei quod ex <lb/>differentia ipſarum <lb/>FV, G, fit & </s> <s xml:id="echoid-s1178" xml:space="preserve">EK, ergo <lb/>quod ex differentia <lb/>ipſarum H, G, fit & </s> <s xml:id="echoid-s1179" xml:space="preserve">K, æquale erit ei quod ex differentia ipſarum <lb/>FV, G, fit & </s> <s xml:id="echoid-s1180" xml:space="preserve">EK; </s> <s xml:id="echoid-s1181" xml:space="preserve">æqualitatem ad proportionem reuocando, erit vt <lb/>differentia grauitatum H, G, ad grauitatem EK, ita differentia <lb/>grauitatum FV, G, ad grauitatem K, quod erat primo loco: </s> <s xml:id="echoid-s1182" xml:space="preserve">demon-<lb/>ſtrandum.</s> <s xml:id="echoid-s1183" xml:space="preserve"/> </p> <div xml:id="echoid-div75" type="float" level="2" n="3"> <figure xlink:label="fig-0070-01" xlink:href="fig-0070-01a"> <image file="0070-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0070-01"/> </figure> </div> <p> <s xml:id="echoid-s1184" xml:space="preserve">Dico quoque vt differentia grauitatum H, G, ad grauitatem EK, <lb/>ita eſſe differentiam grauitatum H, FV, ad grauitatem E. </s> <s xml:id="echoid-s1185" xml:space="preserve">Quoniam <lb/>enim oſtenſum eſt, quod fit ex EK, & </s> <s xml:id="echoid-s1186" xml:space="preserve">FV, æquale eſſe ei quod ex G, fit <lb/>& </s> <s xml:id="echoid-s1187" xml:space="preserve">E, vna cum eo quod ex H, & </s> <s xml:id="echoid-s1188" xml:space="preserve">K, quod autem fit ex H, & </s> <s xml:id="echoid-s1189" xml:space="preserve">K, æquatur <lb/>ei quod ex H, fit & </s> <s xml:id="echoid-s1190" xml:space="preserve">EK, minus eo quod ex H, & </s> <s xml:id="echoid-s1191" xml:space="preserve">E, quod enim additur <lb/>idem & </s> <s xml:id="echoid-s1192" xml:space="preserve">minuitur: </s> <s xml:id="echoid-s1193" xml:space="preserve">ergo quod fit ex EK, & </s> <s xml:id="echoid-s1194" xml:space="preserve">FV, æquale erit ei quod fit <lb/>ex H, & </s> <s xml:id="echoid-s1195" xml:space="preserve">EK, vna cum eo quod ex G, & </s> <s xml:id="echoid-s1196" xml:space="preserve">E, minus eo quod ex H, & </s> <s xml:id="echoid-s1197" xml:space="preserve">E. <lb/></s> <s xml:id="echoid-s1198" xml:space="preserve">addatur vtrinque quod ex H, fit & </s> <s xml:id="echoid-s1199" xml:space="preserve">E, & </s> <s xml:id="echoid-s1200" xml:space="preserve">ſubducantur ea quæ fiunt ex <lb/>G, & </s> <s xml:id="echoid-s1201" xml:space="preserve">E, & </s> <s xml:id="echoid-s1202" xml:space="preserve">ex EK, & </s> <s xml:id="echoid-s1203" xml:space="preserve">FV; </s> <s xml:id="echoid-s1204" xml:space="preserve">quod igitur fit ex H, & </s> <s xml:id="echoid-s1205" xml:space="preserve">E, minus eo quod cx <lb/>G, & </s> <s xml:id="echoid-s1206" xml:space="preserve">E, æquabitur ei quod ex H, fit & </s> <s xml:id="echoid-s1207" xml:space="preserve">EK, minus eo quod ex FV, & </s> <s xml:id="echoid-s1208" xml:space="preserve"><lb/>EK, ſed quod fit ex H, & </s> <s xml:id="echoid-s1209" xml:space="preserve">E, minus eo quod ex G, & </s> <s xml:id="echoid-s1210" xml:space="preserve">E, æquale eſt ei <lb/>quod ex differentia ipſarum H, G, fit & </s> <s xml:id="echoid-s1211" xml:space="preserve">E, ſimiliter & </s> <s xml:id="echoid-s1212" xml:space="preserve">quod ex H, fit <lb/>& </s> <s xml:id="echoid-s1213" xml:space="preserve">EK, minus eo quod ex FV, & </s> <s xml:id="echoid-s1214" xml:space="preserve">EK, æquale eſt ei quod ex differentia <lb/>ipſarum H, FV, fit & </s> <s xml:id="echoid-s1215" xml:space="preserve">EK; </s> <s xml:id="echoid-s1216" xml:space="preserve">ergo quod ex differentia ipſarum H, G, fit <lb/>& </s> <s xml:id="echoid-s1217" xml:space="preserve">E, æquabitur ei quod ex differentia ipſarum H, FV, fit & </s> <s xml:id="echoid-s1218" xml:space="preserve">EK; </s> <s xml:id="echoid-s1219" xml:space="preserve">qua-<lb/>re æqualitatem ad proportionĕ reuocando erit vt differentia graui- <pb o="59" file="0071" n="71" rhead="ARCHIMEDES."/> tatum H, G, ad grauitatem EK, ita differentia grauitatum H, FV, <lb/>ad grauitatem E. </s> <s xml:id="echoid-s1220" xml:space="preserve">quod ſecundo loco fuit demonſtrandum.</s> <s xml:id="echoid-s1221" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div77" type="section" level="1" n="43"> <head xml:id="echoid-head46" xml:space="preserve">Alia breuior Theorematis demonſtratio.</head> <p> <s xml:id="echoid-s1222" xml:space="preserve">RESVMATVR eadem figura vt ſupra. </s> <s xml:id="echoid-s1223" xml:space="preserve">Quoniam igitur <lb/>corpus D, æquale eſt corpori Q, magnitudine, & </s> <s xml:id="echoid-s1224" xml:space="preserve">portio C, æqualis <lb/>portioni L, erit vt D, ad Q, ita C, ad L, & </s> <s xml:id="echoid-s1225" xml:space="preserve">permutando vt D, ad C, ita <lb/>Q, ad L, & </s> <s xml:id="echoid-s1226" xml:space="preserve">quoniam eiuſdem ſunt generis D, C, ſimiliter & </s> <s xml:id="echoid-s1227" xml:space="preserve">Q, L, * erit <lb/> <anchor type="note" xlink:label="note-0071-01a" xlink:href="note-0071-01"/> vt grauitas corporis D, hoc eſt vt EK, ad K, ita H, ad V.</s> <s xml:id="echoid-s1228" xml:space="preserve"/> </p> <div xml:id="echoid-div77" type="float" level="2" n="1"> <note position="right" xlink:label="note-0071-01" xlink:href="note-0071-01a" xml:space="preserve">4. huius</note> </div> <p> <s xml:id="echoid-s1229" xml:space="preserve">Similiter quoniam ponuntur æqualia magnitudine corpora A, P, <lb/>& </s> <s xml:id="echoid-s1230" xml:space="preserve">æquales quoque portiones B, O, erit vt A, ad P, ita B, ad O, & </s> <s xml:id="echoid-s1231" xml:space="preserve">per-<lb/>mutando vt A, ad B, ita P, ad O, ſed eiuſdem ſunt generis A, B, ſimili-<lb/>ter & </s> <s xml:id="echoid-s1232" xml:space="preserve">P, O, * vt igitur grauitas corporis A, id eſt vt EK, ad E, ita erit <lb/> <anchor type="note" xlink:label="note-0071-02a" xlink:href="note-0071-02"/> G, ad F, & </s> <s xml:id="echoid-s1233" xml:space="preserve">per conuerſionem rationis erit vt EK, ad K, ita G, ad G, <lb/>minus F, ſed demonſtratum eſt, vt EK, ad K, ita eſſe H, ad V, ergo vt <lb/>H, ad V, ita erit G, ad G, minus F, & </s> <s xml:id="echoid-s1234" xml:space="preserve">permutando vt H, ad G, ita V, <lb/>ad G, minus F, & </s> <s xml:id="echoid-s1235" xml:space="preserve">diuidendo vt H, minus G, ad G, ita erit FV, minus <lb/>G, ad G, minus F, rurſus permutando erit vt H, minus G, ad FV, mi-<lb/>nus G, ita G, ad G, minus F, ſed vt EK, ad K, ita eſt G, ad G, minus F, <lb/>vt eſt demonſtratum, ergo vt H, minus G, ad FV, minus G, ita erit <lb/>EK, ad K, quare permutando vt H, minus G, ad EK, ita erit FV, mi-<lb/>nus G, ad K, quod eſtò primum.</s> <s xml:id="echoid-s1236" xml:space="preserve"/> </p> <div xml:id="echoid-div78" type="float" level="2" n="2"> <note position="right" xlink:label="note-0071-02" xlink:href="note-0071-02a" xml:space="preserve">4. huius</note> </div> <p> <s xml:id="echoid-s1237" xml:space="preserve">Dico quoque vt H, minus G, ad EK, ita eſſe H, minus FV, ad E. <lb/></s> <s xml:id="echoid-s1238" xml:space="preserve">Quoniam enim oſtenſum eſt vt EK, ad K, ita eſſe H, ad V, erit per <lb/>conuerſionem rationis vt EK, ad E, ita H, ad H, minus V, ſed demon-<lb/>ſtratum eſt vt EK, ad E, ita eſſe G, ad F, ergo vt H, ad H, minus V, ita <lb/>erit G, ad F, & </s> <s xml:id="echoid-s1239" xml:space="preserve">permutando vt H, ad G, ita H, minus V, ad F, & </s> <s xml:id="echoid-s1240" xml:space="preserve">diui-<lb/>dendo vt H, minus G, ad G, ita erit H, minus FV, ad F, & </s> <s xml:id="echoid-s1241" xml:space="preserve">permutan-<lb/>do vt H, minus G, ad H, minus FV, ita G, ad F, ſed vt EK, ad E, ita eſt <lb/>G, ad F, vt eſt demonſtratum, ergo vt H, minus G, ad H, minus FV, <lb/>ita erit EK, ad E, quare permutando, erit vt H, minus G, ad EK, ita <lb/>H, minus FV, ad E, quod erat ſecundo loco demonſtrandum.</s> <s xml:id="echoid-s1242" xml:space="preserve"/> </p> <p> <s xml:id="echoid-s1243" xml:space="preserve">SVpereſt igitur vt dicamus, qua ratione ex grauitate auri <lb/>cognoſci poſſit eius qualitas; </s> <s xml:id="echoid-s1244" xml:space="preserve">id quod ex ijs, quæ dicta <lb/>ſunt facile colligitur; </s> <s xml:id="echoid-s1245" xml:space="preserve">ſi videlicet nota fiat cuiuſuis maſſæ auri <lb/>grauitas, quam habet tum in aere, tum in aqua. </s> <s xml:id="echoid-s1246" xml:space="preserve">Sed ante <lb/>omnia, duo nobis ſunt præmittenda, & </s> <s xml:id="echoid-s1247" xml:space="preserve">explicanda. </s> <s xml:id="echoid-s1248" xml:space="preserve">nimirum <lb/>quid ſit aurum 24. </s> <s xml:id="echoid-s1249" xml:space="preserve">partium, ſeu (vt vulgo dicitur) di 24. </s> <s xml:id="echoid-s1250" xml:space="preserve">ca- <pb o="60" file="0072" n="72" rhead="PROMOTVS"/> ratti, quidue pauciorum, hoc eſt penes quid attendatur di-<lb/>uerſa auri qualitas. </s> <s xml:id="echoid-s1251" xml:space="preserve">Deinde quomodo aurum alligent Auri-<lb/>fices, vel alij ad quos alligandi officium ſpectat. </s> <s xml:id="echoid-s1252" xml:space="preserve">His enim <lb/>cognitis, non erit difficile, id quod proponitur, certa aliqua <lb/>ratione, aſſequi.</s> <s xml:id="echoid-s1253" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1254" xml:space="preserve">Aurum igitur 24. </s> <s xml:id="echoid-s1255" xml:space="preserve">partium appellacur aurum purum, pauciorum <lb/>vero àicitur non purum, ſed aliquo alio metallo, vel pluribus affe-<lb/>ctum. </s> <s xml:id="echoid-s1256" xml:space="preserve">& </s> <s xml:id="echoid-s1257" xml:space="preserve">quia hæc affectio multiplex eſt, ideo etiam auri qualitas, <lb/>quæ ex varia mixtione naſcitur, varia ſit est neceſſe: </s> <s xml:id="echoid-s1258" xml:space="preserve">quamuis vna <lb/>tantumſit qualitas auri puri. </s> <s xml:id="echoid-s1259" xml:space="preserve">Qualitas enim auri in quouis cor-<lb/>pore propoſito, exprimitur partibus auripuri, quæ ſunt in ipſo corpo-<lb/>re, non in magnitudine, ſed in grauitate ſumptis, qualibus totum cor-<lb/>pus constat 24: </s> <s xml:id="echoid-s1260" xml:space="preserve">vel quod idem est, auri qualitas exprimitur in ra-<lb/>tione quam habent illæ partes in grauitate ad totum corpus: </s> <s xml:id="echoid-s1261" xml:space="preserve">quod <lb/>exemplo clarius explicabitur in bunc modum.</s> <s xml:id="echoid-s1262" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1263" xml:space="preserve">Sit aliquod corpus aureum, exempli gratia 24. </s> <s xml:id="echoid-s1264" xml:space="preserve">vnciarum, quod <lb/>expurgatum & </s> <s xml:id="echoid-s1265" xml:space="preserve">ad aurum purum reductum, amiſerit ex priſtina <lb/>grauitate nempe ex 24, vncijs, quatuor vncias, ita vt remanſerint <lb/>tantum 20, vnciæ auri puri, reliquum vero vel eu anuerit in fumum, <lb/>vel fuerit alterius metalli. </s> <s xml:id="echoid-s1266" xml:space="preserve">Totum igitur illud corpus aureum ab <lb/>initio propoſitum, ſi adbuc intelligatur tale quale fuit ante expurga-<lb/>tionem, appellabitur 20. </s> <s xml:id="echoid-s1267" xml:space="preserve">partium, ſeu, (vt vulgo dicitur) di 20. </s> <s xml:id="echoid-s1268" xml:space="preserve">ca-<lb/>ratti. </s> <s xml:id="echoid-s1269" xml:space="preserve">eo quod tota illa maßa mista, 20. </s> <s xml:id="echoid-s1270" xml:space="preserve">tantum vncias auri puri con-<lb/>tinuerit. </s> <s xml:id="echoid-s1271" xml:space="preserve">Immo non ſolum illa maſſa auri, ſed etiam illa cuius ipſa <lb/>fuiſſet pars, vel quæ ipſius fuiſſet quæcunque pars dicetur 20, par-<lb/>tium. </s> <s xml:id="echoid-s1272" xml:space="preserve">Neque enim in alligationibus metallorum, alia est alli-<lb/>gatio partium, alia totius, ſed vtrorunque vna eademque eſt qua-<lb/>litas.</s> <s xml:id="echoid-s1273" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1274" xml:space="preserve">Et hoc est quod Aurifices in inuestigatione qualitatis auri ob-<lb/>ſeruant. </s> <s xml:id="echoid-s1275" xml:space="preserve">Nonenim purificant totum corpus propoſitum, ſed ali-<lb/>quam eius particulam etiam perexiguam, quam ſolam ad aurum pu-<lb/>rum reducunt. </s> <s xml:id="echoid-s1276" xml:space="preserve">hac enim reducta, non ſolum recte definiunt cuius <lb/>fuerit qualitatis particula illa purificata ante purificationem; </s> <s xml:id="echoid-s1277" xml:space="preserve">ve-<lb/>rum etiam cuius fuerit qualitatis, & </s> <s xml:id="echoid-s1278" xml:space="preserve">quot partium fuerit illud cor-<lb/>pus, à quo eadem particula detracta fuit, & </s> <s xml:id="echoid-s1279" xml:space="preserve">illud, quod adbuc ſu-<lb/>pereſt, diminutum ſcilicet illa parte purificata, vt in eodem ex emplo <lb/>propoſito, corporis aurei 24. </s> <s xml:id="echoid-s1280" xml:space="preserve">vnciarum apparet. </s> <s xml:id="echoid-s1281" xml:space="preserve">Eius enim quali-<lb/>tatem ſi forte aurifices inuestigare velint, detrabent ex eo particu-<lb/>lam, verbi gratia, vnius vnciæ, vel quod idem est particulam 24.</s> <s xml:id="echoid-s1282" xml:space="preserve"> <pb o="61" file="0073" n="73" rhead="ARCHIMEDES."/> ſcrupulorum; </s> <s xml:id="echoid-s1283" xml:space="preserve">& </s> <s xml:id="echoid-s1284" xml:space="preserve">banc particulam excoquent ad qualitatem vſque <lb/>auripuri. </s> <s xml:id="echoid-s1285" xml:space="preserve">Et ſi quidem inuenerint, ex priori grauitate 24. </s> <s xml:id="echoid-s1286" xml:space="preserve">ſcru-<lb/>pulorum, deperiſſe nibil: </s> <s xml:id="echoid-s1287" xml:space="preserve">pronunciabunt aurum illud, boc est, non <lb/>ſolum particulam illam excoctam, ſed etiam illud à quo fuit detra-<lb/>cta, nec non & </s> <s xml:id="echoid-s1288" xml:space="preserve">illud quod reman ſit post ſubtractionem eſſe velfuiſ-<lb/>ſe aurum primæ qualitatis ſeu 24, partium, vel quod idem est au-<lb/>num purum. </s> <s xml:id="echoid-s1289" xml:space="preserve">Sivero deprebenderint grauitatem diminutam, ver-<lb/>bi gratia, nunc eſſe 20. </s> <s xml:id="echoid-s1290" xml:space="preserve">ſcrupulorum, quæ ante defæcationem fuit 24: <lb/></s> <s xml:id="echoid-s1291" xml:space="preserve">dicturi ſunt aurum propoſitum 24. </s> <s xml:id="echoid-s1292" xml:space="preserve">vnciarum fuiſſe 20. </s> <s xml:id="echoid-s1293" xml:space="preserve">partium & </s> <s xml:id="echoid-s1294" xml:space="preserve"><lb/>illud quod remanſit eſſe 20. </s> <s xml:id="echoid-s1295" xml:space="preserve">partium, & </s> <s xml:id="echoid-s1296" xml:space="preserve">denique particulam expur-<lb/>gatam nunc quidem eſſe aurum purum, fuiſſe vero particulam auri <lb/>20. </s> <s xml:id="echoid-s1297" xml:space="preserve">partium.</s> <s xml:id="echoid-s1298" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1299" xml:space="preserve">Et eodem modo pronunciabunt de quibuſcunque alys auri quali-<lb/>atibus, ſecundum partes auri puri, quas in qualibet maſſa auri in-<lb/>uenerint, eaſque vigeſimas quartas totius grauitatis, non magnitu-<lb/>dinis. </s> <s xml:id="echoid-s1300" xml:space="preserve">Nam cum in bac comparatione qualitatum, ſeorſim babea-<lb/>tur ratio partium auri, & </s> <s xml:id="echoid-s1301" xml:space="preserve">ſeorſim metallorum alligatorum; </s> <s xml:id="echoid-s1302" xml:space="preserve">manife-<lb/>stum est ſi grauitas totius corporis intelligatur diuiſa in 24. </s> <s xml:id="echoid-s1303" xml:space="preserve">partes <lb/>æquales, ex quibus 20. </s> <s xml:id="echoid-s1304" xml:space="preserve">ſint auri, duæ argenti, & </s> <s xml:id="echoid-s1305" xml:space="preserve">duæ æris; </s> <s xml:id="echoid-s1306" xml:space="preserve">quamli-<lb/>bet partem auri cum qualibet parte argenti & </s> <s xml:id="echoid-s1307" xml:space="preserve">æris collatam, magni-<lb/>tudine eſſe minorem; </s> <s xml:id="echoid-s1308" xml:space="preserve">& </s> <s xml:id="echoid-s1309" xml:space="preserve">ſimiliter partem argenti minorem parte, <lb/>æris; </s> <s xml:id="echoid-s1310" xml:space="preserve">propterea quod aurum omnia reliqua metalla ſuperet grauita-<lb/>te quemadmodum & </s> <s xml:id="echoid-s1311" xml:space="preserve">argentum ipſum æs, vt constat experientia. <lb/></s> <s xml:id="echoid-s1312" xml:space="preserve">atque binc constat quam apte ac conuenienter Aurifices vtantur <lb/>vocabulo partium. </s> <s xml:id="echoid-s1313" xml:space="preserve">bac enim ratione eodem numero exprimunt vnam <lb/>quamque qualitatem auri cuiuslibet maſſæpropoſitæ. </s> <s xml:id="echoid-s1314" xml:space="preserve">Sed nunc ad <lb/>ſecundum veniamus & </s> <s xml:id="echoid-s1315" xml:space="preserve">modum alligationis quem ijdem obſeruant <lb/>breuiter adnotemus.</s> <s xml:id="echoid-s1316" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1317" xml:space="preserve">Inter varias autem & </s> <s xml:id="echoid-s1318" xml:space="preserve">multiplices auri compoſitiones quibus <lb/>cum alijs metallis alligari potest, eam retinuere aurifices, quam diu-<lb/>turna experientia deprebenderunt omnibus alijs eſſe commodiorem, <lb/>eam nimirum quæ ab auri ſimilitudine vel minimum diſcedat; </s> <s xml:id="echoid-s1319" xml:space="preserve">qua-<lb/>lis eſt quæ ſolius argenti atque æris mixtione perficitur. </s> <s xml:id="echoid-s1320" xml:space="preserve">Et quidem <lb/>ſi partes auri excipias, æris atque argenti partes, quæ auro ſunt per-<lb/>miſcendæ ſemper volunt eſſe æquales in grauitate: </s> <s xml:id="echoid-s1321" xml:space="preserve">propterea quod <lb/>eadem experientia Magiſtra didicerunt bunc eſse mixtionis modum <lb/>longe optimum.</s> <s xml:id="echoid-s1322" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1323" xml:space="preserve">Quando ergo aurifices volunt producere aurum cuiuſcunque. <lb/></s> <s xml:id="echoid-s1324" xml:space="preserve">qualitatis, accipiunt tot partes auri puri æquales, quot partium fu-<lb/>turum eſt aurum producendum, pauciores tamen partibus 24, &</s> <s xml:id="echoid-s1325" xml:space="preserve"> <pb o="62" file="0074" n="74" rhead="PROMOTVS"/> reliquas partes quæ deſunt ad 24, explent argento & </s> <s xml:id="echoid-s1326" xml:space="preserve">ære, ſumendo <lb/>ex vtroque metallo partes æquales in grauitate: </s> <s xml:id="echoid-s1327" xml:space="preserve">atque bis rite interſe <lb/>permixtis componunt aurum deſideratæ qualitatis: </s> <s xml:id="echoid-s1328" xml:space="preserve">eamque deno-<lb/>minant à partibus auri puri in mixtione aſumptis. </s> <s xml:id="echoid-s1329" xml:space="preserve">Et quoniam. <lb/></s> <s xml:id="echoid-s1330" xml:space="preserve">non prodiret tale prorſus quale facere intendunt, ſed paulo perfe-<lb/>ctius; </s> <s xml:id="echoid-s1331" xml:space="preserve">propterea quod auri quidem partes in mixtione maneant, ex <lb/>argento vero & </s> <s xml:id="echoid-s1332" xml:space="preserve">ære ali quid deper datur, ſolent Aurifices tanto plus <lb/>miſcere argenti & </s> <s xml:id="echoid-s1333" xml:space="preserve">æris quantum perdi poſſe deprebenderunt.</s> <s xml:id="echoid-s1334" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1335" xml:space="preserve">V erum nostra intentio non est omnia quæ ad eiuſmodi mixtiones <lb/>pertinent boc loco exponere; </s> <s xml:id="echoid-s1336" xml:space="preserve">ſed illud tantum vt receptum apud om-<lb/>nes ad ferre voluimus, ex quo manifeste constat, quæ metallorum <lb/>mixtio in ſingulis qualitatum generibus statuatur: </s> <s xml:id="echoid-s1337" xml:space="preserve">quæquidem eſt <lb/>illa quam adduximus nempe in auro 23, partium, partes 23, eſſe <lb/>auripuri, & </s> <s xml:id="echoid-s1338" xml:space="preserve">reliquam quæ deest ad 24, partes conſtare dimidia <lb/>parte argenti, & </s> <s xml:id="echoid-s1339" xml:space="preserve">dimidia æris in grauitate. </s> <s xml:id="echoid-s1340" xml:space="preserve">In auro vero 22, par-<lb/>tium, auri eſſe 22, argenti vnam, & </s> <s xml:id="echoid-s1341" xml:space="preserve">æris vnam, ſic entm iterum <lb/>fumma omnium partium est 24, eademque est ratio de reliquis ita <lb/>vt numerus partium auri, ſemper denominet qualitatem auri, & </s> <s xml:id="echoid-s1342" xml:space="preserve"><lb/>vna medietas reliquarum partium, quæ partibus auri deſunt ad <lb/>complendas partes 24, ſit argenti, & </s> <s xml:id="echoid-s1343" xml:space="preserve">reliqua medietas ſit æris. </s> <s xml:id="echoid-s1344" xml:space="preserve">bæc <lb/>enim ſatis est ſuppoſuiſſe, ad nouum illud artificium, quo paulo poſt <lb/>inuestig aturi ſumus auri qualitatem ex ſola grauitate quam habet <lb/>in aere & </s> <s xml:id="echoid-s1345" xml:space="preserve">aqua, eamque qualitatem duplici via inuestigabimus, <lb/>vna per calculum, per tabellam altera: </s> <s xml:id="echoid-s1346" xml:space="preserve">& </s> <s xml:id="echoid-s1347" xml:space="preserve">quia ad calculum ſpe-<lb/>ctant ea, quæ ſuperius inuenimus de grauitate metallorum bucre-<lb/>ferenda cenſuimus quæ bic ſunt neceſſaria, cuiuſmodi ſunt auri, <lb/>argenti, atque æris grauitas, quam obtinent in aere, & </s> <s xml:id="echoid-s1348" xml:space="preserve">aqua, quæ <lb/>quidem ita ſe babet vt ſequitur.</s> <s xml:id="echoid-s1349" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1350" xml:space="preserve">Auri puri grauitas, quæ in aere eſt 19, erit in aqua 18.</s> <s xml:id="echoid-s1351" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1352" xml:space="preserve">Argenti grauitas, quæ in aere eſt 31, erit in aqua 28.</s> <s xml:id="echoid-s1353" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1354" xml:space="preserve">Aeris grauitas, quæ in aere est 9, erit in aqua 8.</s> <s xml:id="echoid-s1355" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div80" type="section" level="1" n="44"> <head xml:id="echoid-head47" style="it" xml:space="preserve">Item.</head> <p style="it"> <s xml:id="echoid-s1356" xml:space="preserve">Aurum ad aquam ſe babet in grauitate vt 19, ad I.</s> <s xml:id="echoid-s1357" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1358" xml:space="preserve">Argentum ad aquam ſe babet in grauitate vt 31, ad 3.</s> <s xml:id="echoid-s1359" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1360" xml:space="preserve">Aes ad aquam ſe babet in grauitate vt 9, ad 1.</s> <s xml:id="echoid-s1361" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1362" xml:space="preserve">Ex quibus clariſſime colligitur, ſi aliquod corpus mistum con-<lb/>stet partibus æqualibus argenti, & </s> <s xml:id="echoid-s1363" xml:space="preserve">aeris in grauitate, quantam <lb/>grauitatem babeat in aqua. </s> <s xml:id="echoid-s1364" xml:space="preserve">& </s> <s xml:id="echoid-s1365" xml:space="preserve">quæ ſit ratio in grauitate ipſius mi-<lb/>sti ad aquam ſi enim grauitas aeris in aere ſit 9, eius grauitas in <lb/>aquaerit 8, & </s> <s xml:id="echoid-s1366" xml:space="preserve">ſi grauitas argenti in aere ſit quoque 9, erit eius gra- <pb o="63" file="0075" n="75" rhead="ARCHIMEDES"/> vitas in aqua 8 {4/31}, est enim vt 9, ad 8 {4/31}, vt 31, ad 28: </s> <s xml:id="echoid-s1367" xml:space="preserve">Quare ſi <lb/>grauitas corporis miſti ex argento, & </s> <s xml:id="echoid-s1368" xml:space="preserve">ære iuxta mixtionem prædi-<lb/>ctam, quæetiam ſubintelligenda erit in ſequentibus, in aere fuerit <lb/>18, erit in aqua 16 {4/31}, & </s> <s xml:id="echoid-s1369" xml:space="preserve">conſequenter<emph style="sub">*</emph> grauitas aquæ magnitu-<lb/> <anchor type="note" xlink:label="note-0075-01a" xlink:href="note-0075-01"/> dinem babentis æqualem tali corpori misto erit 1 {27/31}; </s> <s xml:id="echoid-s1370" xml:space="preserve">quare corpus <lb/>mistum ex argento & </s> <s xml:id="echoid-s1371" xml:space="preserve">ære ad corpus aqueum eiuſdem magnitudi-<lb/>nis, rationem babebit in grauitats vt 18, ad 1 {27/31}, vel vt 1, ad <lb/>{29/279}, vel denique in numeris integris vt 279, ad 29, omnium enim <lb/>iſtorum numerorum eadem eſt ratio.</s> <s xml:id="echoid-s1372" xml:space="preserve"/> </p> <div xml:id="echoid-div80" type="float" level="2" n="1"> <note position="right" xlink:label="note-0075-01" xlink:href="note-0075-01a" xml:space="preserve">5. huius</note> </div> <p style="it"> <s xml:id="echoid-s1373" xml:space="preserve">Quibus ſic conſtitutis inuenietur qualitas auri cuiuſcumque boc <lb/>modo. </s> <s xml:id="echoid-s1374" xml:space="preserve">Sit exemp. </s> <s xml:id="echoid-s1375" xml:space="preserve">gratia propoſita aliqua maſſa aurea, cuius gra-<lb/>uitas in aere ſit vnc 24, & </s> <s xml:id="echoid-s1376" xml:space="preserve">oporteat inuenire cuius qualitatis ſit ip-<lb/>ſum aurum. </s> <s xml:id="echoid-s1377" xml:space="preserve">Ponderetur ea maſſain aqua & </s> <s xml:id="echoid-s1378" xml:space="preserve">babeat grauitatem <lb/>vnciarum 22 {2818/5301}, ergo<emph style="sub">*</emph> grauitas aquæ magnitudinem baben <lb/> <anchor type="note" xlink:label="note-0075-02a" xlink:href="note-0075-02"/> tis æqualem propoſit æ maſſæ erit vnc. </s> <s xml:id="echoid-s1379" xml:space="preserve">1 {2483/5301}.</s> <s xml:id="echoid-s1380" xml:space="preserve"/> </p> <div xml:id="echoid-div81" type="float" level="2" n="2"> <note position="right" xlink:label="note-0075-02" xlink:href="note-0075-02a" xml:space="preserve">5. buiuse</note> </div> <p style="it"> <s xml:id="echoid-s1381" xml:space="preserve">Deinde inueniatur grauitas aquæ magnitudine æqualis auro <lb/>puro 24, vnciarum: </s> <s xml:id="echoid-s1382" xml:space="preserve">boc eſt vt 19, ad 1, ita fiat 24, ad alium, nempe ad <lb/>vnciam 1 {5/19}, bic enim numerus erit grauitas illius aquæ</s> </p> <p style="it"> <s xml:id="echoid-s1383" xml:space="preserve">Fiat denique vt 279, ad 29, ita rurſus 24, vnciæ, ad alium, nume-<lb/>rus enim quartus, nempe vnc. </s> <s xml:id="echoid-s1384" xml:space="preserve">2 {138/279} erit grauitas aquæ, magnitu-<lb/>dine æqualis corpori miſto ex argento & </s> <s xml:id="echoid-s1385" xml:space="preserve">ære, cuius grauitas est in <lb/>ære vnc. </s> <s xml:id="echoid-s1386" xml:space="preserve">24, corpus enim ita miſtum, ad corpus aqueum eiuſdem ma-<lb/>gnitudinis rationem babet in grauitate vt 279, ad 29.</s> <s xml:id="echoid-s1387" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1388" xml:space="preserve">Atque ita babebuntur tres grauitates trium aquæ quantitatum, <lb/>quarum prima æquatur auro puro 24, vuciarum, ſecunda maſſæ <lb/>propoſitæ 24, vnciarum, & </s> <s xml:id="echoid-s1389" xml:space="preserve">reliqua corpori misto ex argento & </s> <s xml:id="echoid-s1390" xml:space="preserve">ære <lb/>ſimiliter 24, vnciarum quæ quidem tres grauitates in numeris di-<lb/>ſponantur eo ordine, quo ſequitur.</s> <s xml:id="echoid-s1391" xml:space="preserve"/> </p> <note style="it" position="right" xml:space="preserve">Grauitas aquæ magnitu- \\ dine & qualis auro puro. # Grauitas aqu & magnitu- \\ dine & qualis maſſæ propo- \\ ſite. # Grauitas aque magnitu- \\ dine & qualis corpori miſto \\ ex argento & ære. <lb/>Vnc. 1 {5/19} # Vnc. 1 {2483/5301} # Vnc. 2 {138/279}</note> </div> <div xml:id="echoid-div83" type="section" level="1" n="45"> <head xml:id="echoid-head48" style="it" xml:space="preserve">Velin eadem denominatione.</head> <note style="it" position="right" xml:space="preserve">Vnc. 1 {1395/5301} # Vnc. 1 {2483/5301} # Vnc. 2 {2622/5301}.</note> <p style="it"> <s xml:id="echoid-s1392" xml:space="preserve">Deinde quæratur differentia inter primam & </s> <s xml:id="echoid-s1393" xml:space="preserve">tertiam aquæ gra-<lb/>uitatem, quæ est vnc. </s> <s xml:id="echoid-s1394" xml:space="preserve">1 {1227/5301}, & </s> <s xml:id="echoid-s1395" xml:space="preserve">bæc differentia statuatur pro <lb/>primo proportionis termino, pro ſecundo termino ponatur grauitas <lb/>maſſæ propoſitæ, idest vnc. </s> <s xml:id="echoid-s1396" xml:space="preserve">24, & </s> <s xml:id="echoid-s1397" xml:space="preserve">protertio denique termino ponatur <lb/>differentia inter ſecundam aquæ grauitatem & </s> <s xml:id="echoid-s1398" xml:space="preserve">tertiam, quæ est <lb/>vnc. </s> <s xml:id="echoid-s1399" xml:space="preserve">1 {139/5301} quartus enim proportionalium terminus nempe 20, <pb o="64" file="0076" n="76" rhead="PROMOTVS"/> <anchor type="note" xlink:label="note-0076-01a" xlink:href="note-0076-01"/> erit denominator qualitatis auri de qua quæritur quia ille termi-<lb/>nus indicat partes auri puri in grauitate, qualibus maſſa propoſita <lb/>constat 24. </s> <s xml:id="echoid-s1400" xml:space="preserve">Hoc autem demonstratum eſt prop. </s> <s xml:id="echoid-s1401" xml:space="preserve">19, buius.</s> <s xml:id="echoid-s1402" xml:space="preserve"/> </p> <div xml:id="echoid-div83" type="float" level="2" n="1"> <note style="it" position="right" xlink:label="note-0076-01" xlink:href="note-0076-01a" xml:space="preserve">I. # II. # III. # IIII. <lb/>1 {1227/5301}, # 24, # 1 {139/5301}, # 20,</note> </div> <p style="it"> <s xml:id="echoid-s1403" xml:space="preserve">Et quia in propoſito exemplo bæ partes, nempe vnc. </s> <s xml:id="echoid-s1404" xml:space="preserve">20, ſunt par-<lb/>tes vigeſimæ quartæ 24, vnciarum, quæ constituunt grauitatem to-<lb/>tius maſſæ. </s> <s xml:id="echoid-s1405" xml:space="preserve">binc fit quod eædem 20, vnc. </s> <s xml:id="echoid-s1406" xml:space="preserve">immediate denominent au-<lb/>rum propoſitum eße 20, partium. </s> <s xml:id="echoid-s1407" xml:space="preserve">Quando vero grauitas totius maſ-<lb/>ſæ non exprimitur per numerum 24, tunc opus erit inquirere quot <lb/>partes vigeſimas quartas totius grauitatis efficiat quartus ille pro-<lb/>portionis terminus vt in ſequenti exemplo clarius apparebit.</s> <s xml:id="echoid-s1408" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1409" xml:space="preserve">Sit enim propoſita alia auri maſſa cuius grauitas in aere ſit 5301 <lb/>in aqua vero 4988, ſi igitur bic numerus ſubtrabatur ex numero to <lb/> <anchor type="note" xlink:label="note-0076-02a" xlink:href="note-0076-02"/> tius grauitatis 5301, rcliqaus numerus 313,<emph style="sub">*</emph> erit grauitas aquæ <lb/>propoſitæ maſſæ magnitudine æqualis. </s> <s xml:id="echoid-s1410" xml:space="preserve">Inueniantur quoq; </s> <s xml:id="echoid-s1411" xml:space="preserve">duæ aliæ <lb/>grauitates aquæ, vnare ſpondentis auro puro magnitudine, altera. <lb/></s> <s xml:id="echoid-s1412" xml:space="preserve">corpori miſto ex argento & </s> <s xml:id="echoid-s1413" xml:space="preserve">ære, ita tamen vt grauitas tum auri pu-<lb/>ri, tum corporis misti ſit eadem quæ maſſæpropoſitæ, non ſecus ac in <lb/>præcedenti exemplo factitatum est. </s> <s xml:id="echoid-s1414" xml:space="preserve">boc eſt primo fiat vt 19, ad 1, ita <lb/>5301, ad 279, bic enim numerus erit grauitas aquæ magnitudinem <lb/>babentis æqualem auro puro, cuius grauitas eſt 5301. </s> <s xml:id="echoid-s1415" xml:space="preserve">Deinde fiat <lb/>vt 279, ad 29, ita rurſum grauitas 5301, ad aliam, bac enim ratione <lb/>producetur numerus 551, debitus grauitati aquæ, magnitudine, <lb/>æqualis corpori miſto ex argento & </s> <s xml:id="echoid-s1416" xml:space="preserve">ære, grauitatem babenti eandem <lb/>cum eadem maßa propoſita. </s> <s xml:id="echoid-s1417" xml:space="preserve">Atque bæ tres grauitates aquæ ſcriban-<lb/>tur eo ordine quoſupra; </s> <s xml:id="echoid-s1418" xml:space="preserve">inuentiſque differentijs inter primam & </s> <s xml:id="echoid-s1419" xml:space="preserve"><lb/> <anchor type="note" xlink:label="note-0076-03a" xlink:href="note-0076-03"/> tertiam, nec non inter ſecundam & </s> <s xml:id="echoid-s1420" xml:space="preserve">tertiam, quæ ſunt 272, 238; </s> <s xml:id="echoid-s1421" xml:space="preserve">sta-<lb/>tuatur pro primo proportionis termino prior differentia 272, & </s> <s xml:id="echoid-s1422" xml:space="preserve">pro <lb/>tertio posterior 238. </s> <s xml:id="echoid-s1423" xml:space="preserve">grauitas vero maſſæ propoſitæ 5301, ponatur <lb/>pro ſecundo termino, & </s> <s xml:id="echoid-s1424" xml:space="preserve">quæratur terminus quartus, qui inpræ-<lb/> <anchor type="note" xlink:label="note-0076-04a" xlink:href="note-0076-04"/> ſenti exemplo eſt 4638 {51/136}, is<emph style="sub">*</emph> enim indicabit grauitatem auri <lb/> <anchor type="note" xlink:label="note-0076-05a" xlink:href="note-0076-05"/> puriin maſſa propoſita. </s> <s xml:id="echoid-s1425" xml:space="preserve">Sed quoniam bæc grauitas non est expreſ-<lb/>ſa in partibus vigeſimis quartis totius grauitatis, id quod ad ger- <pb o="65" file="0077" n="77" rhead="ARCHIMEDES."/> manam qualitatis auri pronunciationem requiritur, vt ſupra mul-<lb/>tis oſtendimus, reuocanda erit ad partes vigeſimas quartas boc est <lb/>ad partes, qualium tota propoſita maſſa est 24, quod factu non est <lb/>difficile. </s> <s xml:id="echoid-s1426" xml:space="preserve">N am ſi fiat vt tota grauit as maſſæ propoſitæ 5301, algra-<lb/>uitatem auri puri 4638 {45/136}, vel vt 272, ad 238, cum vtrobique <lb/>eadem ſit ratio ita 24, ad a lium numerum. </s> <s xml:id="echoid-s1427" xml:space="preserve">proculdubio quartus nu-<lb/>merus proportionalis, erit ille qui quæritur. </s> <s xml:id="echoid-s1428" xml:space="preserve">Est autem bic quar-<lb/>tus numerus 21. </s> <s xml:id="echoid-s1429" xml:space="preserve">Quare aurum maſſæ propoſitæ appellabitur par-<lb/>tium 21.</s> <s xml:id="echoid-s1430" xml:space="preserve"/> </p> <div xml:id="echoid-div84" type="float" level="2" n="2"> <note position="left" xlink:label="note-0076-02" xlink:href="note-0076-02a" xml:space="preserve">5. buius</note> <note style="it" position="right" xlink:label="note-0076-03" xlink:href="note-0076-03a" xml:space="preserve">Grauitas aqu & magnitu- \\ dine & qualis auro puro. # Grauitas agua & qualis \\ maſlæ propoſita. # Grauitas aquæ æqualis \\ corpori miſto. <lb/>279 # 313 # 551</note> <note style="it" position="right" xlink:label="note-0076-04" xlink:href="note-0076-04a" xml:space="preserve">I. # II. # III. # IIII. <lb/>272, # 5301, # 238, # 4638 {51/136}.</note> <note position="left" xlink:label="note-0076-05" xlink:href="note-0076-05a" xml:space="preserve">19. <lb/>buius.</note> </div> <p style="it"> <s xml:id="echoid-s1431" xml:space="preserve">Ex his igitur patet in inuenienda auri qualitate primum pro-<lb/>portionis terminum 272, & </s> <s xml:id="echoid-s1432" xml:space="preserve">ſecundum 5301, perpetuo manere eoſ-<lb/>dem, quia primus terminus est differentia inter grauitates primæ, <lb/>& </s> <s xml:id="echoid-s1433" xml:space="preserve">tertiæ aquæ, quæ nunquam mutantur, nam illæ aquæ magnitu-<lb/>dine ſunt æquales altera auro puro, reliqua miſto ex argento & </s> <s xml:id="echoid-s1434" xml:space="preserve">ære, <lb/>quæ corpora aureum ſcilicet & </s> <s xml:id="echoid-s1435" xml:space="preserve">mistum ſemper ponuntur eiuſdem <lb/>grauitatis nempe 5301, Secundus vero terminus 5301, eſt grauit as <lb/>maſſæ propoſitæ, quæ ſi maior fuerit, vel minor, ad eam facile reuo-<lb/>cabitur. </s> <s xml:id="echoid-s1436" xml:space="preserve">V nde in posterum ſolum opus erit inuenire tertium pro-<lb/>portionis terminum, boc est differentiam inter grauitates ſecundæ <lb/>& </s> <s xml:id="echoid-s1437" xml:space="preserve">tertiæ aquæ.</s> <s xml:id="echoid-s1438" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1439" xml:space="preserve">Sed vt boc etiam exemplo illustretur, proponatur aliqua maſſa <lb/>auri, cuius inue ſtigãda ſit qualitas, & </s> <s xml:id="echoid-s1440" xml:space="preserve">ſit ipſius maſſæ grauitas qui-<lb/>dem in aere 837, in aqua vero 784, ergo * grauitas aquæ magnitu-<lb/> <anchor type="note" xlink:label="note-0077-01a" xlink:href="note-0077-01"/> dinem babentis æqualem propoſitæ maſſæ erit 53, differentia enim <lb/>inter primam, & </s> <s xml:id="echoid-s1441" xml:space="preserve">ſecundam grauitatem est 53.</s> <s xml:id="echoid-s1442" xml:space="preserve"/> </p> <div xml:id="echoid-div85" type="float" level="2" n="3"> <note position="right" xlink:label="note-0077-01" xlink:href="note-0077-01a" xml:space="preserve">5. huins</note> </div> <p style="it"> <s xml:id="echoid-s1443" xml:space="preserve">Ad inueniendum igitur tertium proportionis terminum manen-<lb/>tibus primis duobus 272, 5301, bæc erit ratio. </s> <s xml:id="echoid-s1444" xml:space="preserve">Reuocetur primum <lb/>ptopoſitæ maſſæ grauitas 837, ad grauitatem 5301, boc est intelli-<lb/>gatur ipſa maſſa grauit atem babere 5301. </s> <s xml:id="echoid-s1445" xml:space="preserve">deinde fiat vt 837, ad 53, <lb/>grauitatem videlicet aquæ ipſi maſſæ æqualis, ita 5301, ad 335 {2/3}, <lb/>ergo 335 {2/3}, erit grauitas aquæ magnitudinem babentis æqualem <lb/>aureæ maßæ, cuius grauitas 837, reuocata est ad grauitatem 5301; <lb/></s> <s xml:id="echoid-s1446" xml:space="preserve">quare grauitas ſecundæ aquæ erit 335 {2/3}, & </s> <s xml:id="echoid-s1447" xml:space="preserve">conſequenter differen-<lb/> <anchor type="note" xlink:label="note-0077-02a" xlink:href="note-0077-02"/> tia inter ipſam grauitatem ſecundæ aquæ & </s> <s xml:id="echoid-s1448" xml:space="preserve">grauitatem tertiæ 551, <lb/>erit 215 {1/3}, ſed ipſa differentia ponitur pro tertio proportionis ter-<lb/>mino; </s> <s xml:id="echoid-s1449" xml:space="preserve">ergo 215 {1/2}, erit quæſitus terminus, nempe proportionis ter- <pb o="66" file="0078" n="78" rhead="PROMOTVS"/> <anchor type="note" xlink:label="note-0078-01a" xlink:href="note-0078-01"/> tius. </s> <s xml:id="echoid-s1450" xml:space="preserve">Quartus autem terminus 4196 {25/136}, indicabit grauitatcm <lb/>auri puri, quod est in maſſa propoſita, eam tamen indicabit in par-<lb/>tibus, qualibus tota maſſa constat 5301, quæ quidem grauitas vt <lb/>auri qualitatem indicet, reuocanda erit ad partes qualium tota <lb/>maſſa propoſita est 24. </s> <s xml:id="echoid-s1451" xml:space="preserve">ſi enim. </s> <s xml:id="echoid-s1452" xml:space="preserve">fiat vt 5301, ad 4196 {85/136}, ita 24, <lb/>ad 19, aurum propoſitæ maſſæ appellabitur partium 19.</s> <s xml:id="echoid-s1453" xml:space="preserve"/> </p> <div xml:id="echoid-div86" type="float" level="2" n="4"> <note style="it" position="right" xlink:label="note-0077-02" xlink:href="note-0077-02a" xml:space="preserve">Grauitas primæ aquæ. # Grauitas ſecundæ aquæ. # Grauitas tertiæ aquæ. <lb/>279, # 335 {2/3}, # 551.</note> <note style="it" position="right" xlink:label="note-0078-01" xlink:href="note-0078-01a" xml:space="preserve">I. # II. # III. # IIII. <lb/>272, # 5301, # 215 {1/3}, # 4196 {25/136},</note> </div> <p style="it"> <s xml:id="echoid-s1454" xml:space="preserve">Denique ſi quis bunc modum conferat cum illo, quem ſupra tradi-<lb/>dimus, cum argentum explorauimus, quod miſtum in aurea corona <lb/>credebatur; </s> <s xml:id="echoid-s1455" xml:space="preserve">is liquido intelliget bic nibil aliud acceſſiße, niſi quod <lb/>loco argenti, aſſumptum ſit corpus ex argento & </s> <s xml:id="echoid-s1456" xml:space="preserve">ære mistum, eo <lb/>quod bæc duo metalla tantum in alligationibus auri ſoleant adbibe-<lb/>ri, vt diximus. </s> <s xml:id="echoid-s1457" xml:space="preserve">Quod ſi conſtaret plura alia aſſumpta eſſe, etiam in <lb/>quauis alia ratione, facile erit cuiuis ad ſimilitudinem buius, for-<lb/>mare alium modum, ſed nos, ne longiores ſimus, ad vſum ſequentis <lb/>tabulæ nos conferamus, quaillis conſultum volumus qui minus in <lb/>præceptis Aritbmeticis ſunt exercitati, vel illis, qui alias ob cauſas <lb/>tabulis vti malunt, quam calculis.</s> <s xml:id="echoid-s1458" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1459" xml:space="preserve">Hæc tabula accommodata eſt primarie ad aurum vnius libræ, vt <lb/>apparet in ſecũda ipſius columna in qua omnes numeri ſunt unita-<lb/>tes, reſpon dentes ſingulis Denominatoribus qualitatum auri, à deno <lb/>minatore partium 24, vſque ad denominatorem qualitatis partis ò, <lb/>quamuis proprie loquendo nulla ſit qualitas auri partis nullius, quia <lb/>tunc non eſſet aurum, ſed mistum ex argento & </s> <s xml:id="echoid-s1460" xml:space="preserve">ære. </s> <s xml:id="echoid-s1461" xml:space="preserve">Hos denomi-<lb/>natores auri omnes inuenies in prima columna ſub titulo qualitatis. <lb/></s> <s xml:id="echoid-s1462" xml:space="preserve">In columna vero ſub titulo misti placuit etiam deſcribere denomi-<lb/>natores miſti ex argento & </s> <s xml:id="echoid-s1463" xml:space="preserve">ære, vt vnico intuitu appare at quot par-<lb/>tes auri puri, & </s> <s xml:id="echoid-s1464" xml:space="preserve">quot partes misti ex argento & </s> <s xml:id="echoid-s1465" xml:space="preserve">ære contineantur <lb/>in ſingulis qualitatibus.</s> <s xml:id="echoid-s1466" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1467" xml:space="preserve">Porro in area tabulæ ſub titulo grauitatis auri in aqua poſita est <lb/>grauitas auri cuiuslibet qualitatis quam obtinet in aqua, quæ qua <lb/>ratione inueniatur, dicetur inferius vbi agetur de compoſitione, <lb/>eiuſdem tabulæ.</s> <s xml:id="echoid-s1468" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1469" xml:space="preserve">V ſus eius ſunt duo, quorum alterum titulus indicat, nimirum vt <lb/>tabulæ beneficio reperiatur ex grauitate auri quam babet in aere & </s> <s xml:id="echoid-s1470" xml:space="preserve"><lb/>aqua, eius qualitas. </s> <s xml:id="echoid-s1471" xml:space="preserve">Alter vero est vt cognoſcatur grauitas in aqua, <lb/>quando vnà cum grauitate quam aliquod aurum babet in aere da-<lb/>tur ipſius qualitas. </s> <s xml:id="echoid-s1472" xml:space="preserve">& </s> <s xml:id="echoid-s1473" xml:space="preserve">de boc vfu cum ſit ſimplicior prius nobis erit <lb/>agendum.</s> <s xml:id="echoid-s1474" xml:space="preserve"/> </p> <pb o="67" file="0079" n="79" rhead="ARCHIMEDES."/> <note position="right" xml:space="preserve"> ####### Tabula ad inueniendam qualitatem \\ Auri, ex grauitate quam ha- \\ bet in aere & aqua. <lb/>Qualitas \\ Auri. # Grauitas Auri \\ in aere. #### Grauitas Auri in aqua. # Miſtũ ex Arg. \\ & ære. <lb/>Part. # Lib. # Vnc. # Scrup. # Gran. # Num. Fract. # Part. <lb/>24 # 1 # 11. # 8. # 20. # 372 # 0 <lb/>23 # 1 # 11. # 8. # 5. # 765 # 1 <lb/>22 # 1 # 11. # 7. # 14. # 1158 # 2 <lb/>21 # 1 # 11. # 6. # 23. # 1551 # 3 <lb/>20 # 1 # 11. # 6. # 9. # 177 # 4 <lb/>19 # 1 # 11. # 5. # 18. # 570 # 5 <lb/>18 # 1 # 11. # 5. # 3. # 963 # 6 <lb/>17 # 1 # 11. # 4. # 12. # 1356 # 7 <lb/>16 # 1 # 11. # 3. # 21. # 1749 # 8 <lb/>15 # 1 # 11. # 3. # 7. # 375 # 9 <lb/>14 # 1 # 11. # 2. # 16. # 768 # 10 <lb/>13 # 1 # 11. # 2. # 1. # 1161 # 11 <lb/>12 # 1 # 11. # 1. # 10. # 1554 # 12 <lb/>11 # 1 # 11. # 0. # 20. # 180 # 13 <lb/>10 # 1 # 11. # 0. # 5. # 573 # 14 <lb/>9 # 1 # 10. # 23. # 14. # 966 # 15 <lb/>8 # 1 # 10. # 22. # 23. # 1359 # 16 <lb/>7 # 1 # 10. # 22. # 8. # 1752 # 17 <lb/>6 # 1 # 10. # 21. # 18. # 378 # 18 <lb/>5 # 1 # 10. # 21. # 3. # 771 # 19 <lb/>4 # 1 # 10. # 20. # 12. # 1164 # 20 <lb/>3 # 1 # 10. # 19. # 21. # 1557 # 21 <lb/>2 # 1 # 10. # 19. # 7. # 183 # 22 <lb/>1 # 1 # 10. # 18. # 16. # 576 # 23 <lb/>0 # 1 # 10. # 18. # 1. # 969 # 24 <lb/>Part. # Lib. ### Communis Denomin. fract. # 1767 # Part.</note> <note position="right" xml:space="preserve"> ### Tabella Partis pro \\ portionalis Deno- \\ minatorum Auri. <lb/>Pars proportio \\ nalis Auri in \\ partibus. 24. ## Differĕtia Gra \\ uitatum Auri \\ in aqua. <lb/>Part. # Gran. # Num: Fract. <lb/>1 # 0. # 1088 <lb/>2 # 1. # 409 <lb/>3 # 1. # 1497 <lb/>4 # 2. # 818 <lb/>5 # 3. # 139 <lb/>6 # 3. # 1227 <lb/>7 # 4. # 548 <lb/>8 # 4. # 1636 <lb/>9 # 5. # 957 <lb/>10 # 6. # 278 <lb/>11 # 6. # 1366 <lb/>12 # 7. # 687 <lb/>13 # 8. # 8 <lb/>14 # 8. # 1096 <lb/>15 # 9. # 417 <lb/>16 # 9. # 1505 <lb/>17 # 10. # 826 <lb/>18 # 11. # 147 <lb/>19 # 11. # 1235 <lb/>20 # 12. # 556 <lb/>21 # 12. # 1644 <lb/>22 # 13. # 965 <lb/>23 # 14. # 286 <lb/>24 # 14. # 1374 <lb/>Part. # Denom. # Fract. com. <lb/> # ## 1767</note> <pb o="68" file="0080" n="80" rhead="PROMOTVS"/> <p style="it"> <s xml:id="echoid-s1475" xml:space="preserve">Quæratur exempli gratia quam babet grauitatem in aqua aurũ <lb/>purum ſeu aurum 24, partium cuius grauitas in aere est lib. </s> <s xml:id="echoid-s1476" xml:space="preserve">1. </s> <s xml:id="echoid-s1477" xml:space="preserve">Hæe <lb/>in ſupremo ordine è regione den<unsure/>ominatoris partium 24, ſub titulo <lb/>grauitatis auri in aqua, datur vnc. </s> <s xml:id="echoid-s1478" xml:space="preserve">11. </s> <s xml:id="echoid-s1479" xml:space="preserve">Scrup. </s> <s xml:id="echoid-s1480" xml:space="preserve">8, Gran. </s> <s xml:id="echoid-s1481" xml:space="preserve">20 {372/1767}, <lb/>quæ fractio licet exprimi poſſit minoribus numeris nĕpe {4/19}, libuit <lb/>tamen illam maiorem in tabula ponere, vt omnes fractiones totius <lb/>tabulæ eſſent eiuſdem denominationis, & </s> <s xml:id="echoid-s1482" xml:space="preserve">reſponderent denomina-<lb/>toribus fractionum quæ babentur in tabella partis proportionalis.</s> <s xml:id="echoid-s1483" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1484" xml:space="preserve">Rurſum guæratur quam babet grauitatem in aqua aurum ite-<lb/>rum vnius libræ, qualitatis vero 20, partium. </s> <s xml:id="echoid-s1485" xml:space="preserve">quam ſi in tabula <lb/>quæras, inuenies ſub eodem titulo è regione denominatoris 20, par-<lb/>tium. </s> <s xml:id="echoid-s1486" xml:space="preserve">vnc. </s> <s xml:id="echoid-s1487" xml:space="preserve">11, Scrup. </s> <s xml:id="echoid-s1488" xml:space="preserve">6, Gran. </s> <s xml:id="echoid-s1489" xml:space="preserve">9 {177/1767}. </s> <s xml:id="echoid-s1490" xml:space="preserve">eademque est ratio de re-<lb/>liquis.</s> <s xml:id="echoid-s1491" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1492" xml:space="preserve">Quando vero propoſitum aurum non est unius Libræ; </s> <s xml:id="echoid-s1493" xml:space="preserve">tunc opus <lb/>erit ratiocinatione proportionis, in qua pro primo termino ponatur <lb/>vna libra auri propoſitæ qualitatis, pro ſecundo termino, grauitas <lb/>eidem reſpondens in aqua quam tabula exbibet, pro tertio vero ter-<lb/>mino collocetur vera grauitas auri propoſiti. </s> <s xml:id="echoid-s1494" xml:space="preserve">Quartus enim ter-<lb/>minus exbibebit grauitatem ipſius auri in aqua. </s> <s xml:id="echoid-s1495" xml:space="preserve">Vt ſipropoſitum <lb/>aurũ ſit trium lib. </s> <s xml:id="echoid-s1496" xml:space="preserve">qualitatis vero 18, partium. </s> <s xml:id="echoid-s1497" xml:space="preserve">fiat vt lib. </s> <s xml:id="echoid-s1498" xml:space="preserve">1. </s> <s xml:id="echoid-s1499" xml:space="preserve">ad vne <lb/>11, Scrup. </s> <s xml:id="echoid-s1500" xml:space="preserve">5, Gran. </s> <s xml:id="echoid-s1501" xml:space="preserve">3 {963/1767}, ita lib. </s> <s xml:id="echoid-s1502" xml:space="preserve">3. </s> <s xml:id="echoid-s1503" xml:space="preserve">ad alium numerum, is erit <lb/>lib. </s> <s xml:id="echoid-s1504" xml:space="preserve">2, vnc. </s> <s xml:id="echoid-s1505" xml:space="preserve">9, Scrup. </s> <s xml:id="echoid-s1506" xml:space="preserve">15, Gran. </s> <s xml:id="echoid-s1507" xml:space="preserve">10 {1122/1767}. </s> <s xml:id="echoid-s1508" xml:space="preserve">& </s> <s xml:id="echoid-s1509" xml:space="preserve">tanta erit grauitas au-<lb/>ripropoſiti in aqua. </s> <s xml:id="echoid-s1510" xml:space="preserve">Et ſic de alijs.</s> <s xml:id="echoid-s1511" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1512" xml:space="preserve">Quod vero ad priorem vfum attinet, is perſimilis est præcedenti, <lb/>& </s> <s xml:id="echoid-s1513" xml:space="preserve">æquè facilis quando grauitas auri quam in aere & </s> <s xml:id="echoid-s1514" xml:space="preserve">aqua babet, in <lb/>tabula reperitur precisè. </s> <s xml:id="echoid-s1515" xml:space="preserve">Nam ſi proponatur exemp. </s> <s xml:id="echoid-s1516" xml:space="preserve">gratia aurum <lb/>vnius lib. </s> <s xml:id="echoid-s1517" xml:space="preserve">babens in aqua grauitatĕ vnc. </s> <s xml:id="echoid-s1518" xml:space="preserve">11. </s> <s xml:id="echoid-s1519" xml:space="preserve">Scr. </s> <s xml:id="echoid-s1520" xml:space="preserve">7, Gra. </s> <s xml:id="echoid-s1521" xml:space="preserve">14 {1158/1767}, <lb/>quoniam bæc grauitas reperitur in tabula è regione qualitatis auri <lb/>22, partium; </s> <s xml:id="echoid-s1522" xml:space="preserve">manifeſtum eſt totidem partium eſſe aurũ propoſitum.</s> <s xml:id="echoid-s1523" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1524" xml:space="preserve">Quando vero grauitas auri in aere quiaem eſt vnius lib. </s> <s xml:id="echoid-s1525" xml:space="preserve">in aqua <lb/>vero grauitatem babet, quæ in tabula non reperitur, indicium erit <lb/>aurum propoſitum non eſſe aliquot partium præcisè, ſed annexam <lb/>babere aliquam fractionem, quæ per partem proportion alem inue-<lb/>nietur boc modo.</s> <s xml:id="echoid-s1526" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1527" xml:space="preserve">Proponatur aurum vnius libræ in aqua babens grauitatem vnc. <lb/></s> <s xml:id="echoid-s1528" xml:space="preserve">11. </s> <s xml:id="echoid-s1529" xml:space="preserve">Scrup. </s> <s xml:id="echoid-s1530" xml:space="preserve">6, Gran. </s> <s xml:id="echoid-s1531" xml:space="preserve">16 {864/1767}, qualis in tabula non reperitur Gra-<lb/>uitas enim proxime maior est vnc. </s> <s xml:id="echoid-s1532" xml:space="preserve">11. </s> <s xml:id="echoid-s1533" xml:space="preserve">Scrup. </s> <s xml:id="echoid-s1534" xml:space="preserve">6, Gran. </s> <s xml:id="echoid-s1535" xml:space="preserve">23 {1551/1767}, <lb/>reſpondens auro 21, partium, & </s> <s xml:id="echoid-s1536" xml:space="preserve">grauitas proxime minor eſt vnc. </s> <s xml:id="echoid-s1537" xml:space="preserve">11, <lb/>Scrup. </s> <s xml:id="echoid-s1538" xml:space="preserve">6, Gran. </s> <s xml:id="echoid-s1539" xml:space="preserve">9 {177/1767}, earumq; </s> <s xml:id="echoid-s1540" xml:space="preserve">differentia eſt Gran. </s> <s xml:id="echoid-s1541" xml:space="preserve">14 {1374/1767}, <lb/>quem admodum & </s> <s xml:id="echoid-s1542" xml:space="preserve">inter quaſcunque duas alias grauitates proxi- <pb o="69" file="0081" n="81" rhead="ARCHIMEDES."/> mas eadem eſt differentia, propterea quod omnes grauitates in tabu-<lb/>la procedunt per æqualem exceſſum, vel defectum, vt inferius de-<lb/>monſtrabitur. </s> <s xml:id="echoid-s1543" xml:space="preserve">Inueniatur quoque differentia inter eandem gra-<lb/>uitatem proxime minorem & </s> <s xml:id="echoid-s1544" xml:space="preserve">inter grauitatem auri propoſiti quam <lb/>babet in aqua, quæ quidĕ eſt Gran. </s> <s xml:id="echoid-s1545" xml:space="preserve">7 {627/1767}, & </s> <s xml:id="echoid-s1546" xml:space="preserve">fiat vt 14 {1374/1767}, <lb/>ad 1, ita 7 {637/1767}, ad alium numerum & </s> <s xml:id="echoid-s1547" xml:space="preserve">inuenietur bæc fractio <lb/>{1/2}, eademque adijcienda erit ad denominatorem 20, partium, vt <lb/>componatur totus denominator auripropoſiti partium 20 {1/2}, & </s> <s xml:id="echoid-s1548" xml:space="preserve">eo-<lb/>dem modo inueniendus erit denominator cuiuſcunq; </s> <s xml:id="echoid-s1549" xml:space="preserve">alterius auri, <lb/>cuius grauitas in aqua, in tabula non reperitur.</s> <s xml:id="echoid-s1550" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1551" xml:space="preserve">Ceterum qui volet contentus eſſe partibus vigeſimis quartis de-<lb/>nominatorum auri, is multo breuius aſſequetur quod quæritur, per <lb/>tabellam partis proportionalis. </s> <s xml:id="echoid-s1552" xml:space="preserve">illic enim vnico ingreſſu offendet <lb/>partem proportionalem, quam quærit, vt in eodem exemplo apparet, <lb/>in quo differentia grauitatum auri erat Gran. </s> <s xml:id="echoid-s1553" xml:space="preserve">7 {687/1767}, quæ in <lb/>tabella partis proportionalis babetur præcisè è regione particularum <lb/>12. </s> <s xml:id="echoid-s1554" xml:space="preserve">V nde concluditur, denominatorem auri propoſiti eſſe partium <lb/>20 {12/24}, vel quod idem es<unsure/>t partium 20 {1/2}, vt prius. </s> <s xml:id="echoid-s1555" xml:space="preserve">Quando vero <lb/>differentia grauitatum in tabella partis proportionalis non babetur <lb/>præcisè. </s> <s xml:id="echoid-s1556" xml:space="preserve">accipiatur alia ipſi propinquior & </s> <s xml:id="echoid-s1557" xml:space="preserve">particula illi in latere<unsure/> <lb/>reſpondens addatur denominatori auri ex primaria tabula extr a-<lb/>cti. </s> <s xml:id="echoid-s1558" xml:space="preserve">ſic enim ſaltem non errabitur in vna particula vigeſimaquarta <lb/>vnius partis denominatoris auri.</s> <s xml:id="echoid-s1559" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1560" xml:space="preserve">Denique ſi proponatur aurum non vnius libræ ſed vel plurium, <lb/>vel ſolum aliquot vnciarum. </s> <s xml:id="echoid-s1561" xml:space="preserve">R educenda erit eius grauitas quam <lb/>babet in aqua, ad grauitatem quam haberet ſi eßet vnius libræ, id <lb/>quod abſoluetur per proportionis ratiocinationem, ſi pro termino <lb/>primo ponatur vera grauitas auri propoſiti, pro ſecundo, e iuſdem <lb/>grauitas in aqua, & </s> <s xml:id="echoid-s1562" xml:space="preserve">pro tertio lib. </s> <s xml:id="echoid-s1563" xml:space="preserve">1, quartus enim terminus indic a-<lb/>bit grauitatem in aqua reſpondentem vni libræ auri propoſiti. </s> <s xml:id="echoid-s1564" xml:space="preserve">& </s> <s xml:id="echoid-s1565" xml:space="preserve"><lb/>bac inuenta reliqua expedientur vt prius.</s> <s xml:id="echoid-s1566" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1567" xml:space="preserve">Exemplum buius caſus bic non affero, quod per ſe res ſit clara. <lb/></s> <s xml:id="echoid-s1568" xml:space="preserve">Sed illud tantum obiter aduertere placet, quod videtur pertinere. </s> <s xml:id="echoid-s1569" xml:space="preserve"><lb/>ad commodiorem vſum tabulæ, videlicet vt ijs in caſibus in quibus <lb/>neceßarius est calculus, fractiones granorum ommittantur quando <lb/>minus valent quam {1/2}, & </s> <s xml:id="echoid-s1570" xml:space="preserve">quando valent plus, eorum loco, addatur <lb/>vnum granum reliquis granis, & </s> <s xml:id="echoid-s1571" xml:space="preserve">ſi quando accidat binc procreari <lb/>grana 24. </s> <s xml:id="echoid-s1572" xml:space="preserve">tunc etiam grana ommittantur addita prius vnitate ad <lb/>ſcrupula in tabula inuenta. </s> <s xml:id="echoid-s1573" xml:space="preserve">bac enim ratione calculus erit expedi-<lb/>tior & </s> <s xml:id="echoid-s1574" xml:space="preserve">error qui binc oborietur erit inſenſibilis.</s> <s xml:id="echoid-s1575" xml:space="preserve"/> </p> <pb o="70" file="0082" n="82" rhead="PROMOTVS"/> </div> <div xml:id="echoid-div88" type="section" level="1" n="46"> <head xml:id="echoid-head49" xml:space="preserve">Compoſitio eiuſdem tabulæ.</head> <p style="it"> <s xml:id="echoid-s1576" xml:space="preserve">Si ea quæ bactenus ſunt dictarectè intelligantur liquido appare-<lb/>bit compoſitionem tabulæ in eo conſistere, vt inueniatur grauitas <lb/>quam aurum cuiuſuis qualitatis babet in aqua, boc eſt ſi intelligan-<lb/>tur propoſitæ plures maſſæ auri, ſingulæ ſingularum librarum, & </s> <s xml:id="echoid-s1577" xml:space="preserve"><lb/>alia ſit qualitatis 24, partium alia 23, alia 22, & </s> <s xml:id="echoid-s1578" xml:space="preserve">c. </s> <s xml:id="echoid-s1579" xml:space="preserve">quanta ſit gra-<lb/>uitas vniuſcuiuſque in aqua, id quod boc extremo loco inuestig are. <lb/></s> <s xml:id="echoid-s1580" xml:space="preserve">docebimus.</s> <s xml:id="echoid-s1581" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1582" xml:space="preserve">Et primum ſit propoſitum aurum purum, ſeu 24, partium. </s> <s xml:id="echoid-s1583" xml:space="preserve">quo-<lb/>niam igitur grauitas auri in aere, ad grauitatem eiuſdem in aqua <lb/>ſe babet vt 19, ad 18, fiat vt 19, ad 18, ita lib. </s> <s xml:id="echoid-s1584" xml:space="preserve">1, auri puri ad aliam <lb/>grauitatem nempe lib. </s> <s xml:id="echoid-s1585" xml:space="preserve">{18/19}, quæ grauitas ad vncias, ſcrupula, & </s> <s xml:id="echoid-s1586" xml:space="preserve"><lb/>grana reuocata valet vnc. </s> <s xml:id="echoid-s1587" xml:space="preserve">11, Scrup. </s> <s xml:id="echoid-s1588" xml:space="preserve">8. </s> <s xml:id="echoid-s1589" xml:space="preserve">Gran. </s> <s xml:id="echoid-s1590" xml:space="preserve">20 {4/19}, atque bæc est <lb/>grauitas auri puri in aqua, quam in tabula è regione denominatoris <lb/>24, poſuimus; </s> <s xml:id="echoid-s1591" xml:space="preserve">fractione excepta cuius loco ſubſtituta est fractio <lb/>{372/1767}, propter cauſam ſuperius allatam.</s> <s xml:id="echoid-s1592" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1593" xml:space="preserve">Sit deinde propoſitum quod vis aliud corpus aureum vnius libræ, <lb/>ſitque exemp. </s> <s xml:id="echoid-s1594" xml:space="preserve">gratia illud aurum 20, partium, patet igitur ex defi-<lb/>nitione qualitatis, ex 24, ſemiuncijs totius grauitatis, 20, ſemiuncias <lb/>eſſe auri puri, duas argenti, & </s> <s xml:id="echoid-s1595" xml:space="preserve">reliquas duas æris & </s> <s xml:id="echoid-s1596" xml:space="preserve">quoniam gra-<lb/>uitas misti in aere, ad grauitatem eiuſdem in aqua rationem babet <lb/>vt 279, ad 250, vt ex iam dictis patet, fiat vt 279, ad 250, ita qua-<lb/>tuor ſemiunciæ, vel potius 2, vnciæ misti quod componit qualitatem <lb/>auri 20, partium, paulo ante propoſitam ad alias vncias. </s> <s xml:id="echoid-s1597" xml:space="preserve">inuenien-<lb/>tur enim pro grauitate illius miſti in aqua vnc. </s> <s xml:id="echoid-s1598" xml:space="preserve">1 {221/279}. </s> <s xml:id="echoid-s1599" xml:space="preserve">Est autem <lb/>grauitas auri puri 20, ſemiunciarum vel 10, vnciarum in aqua vnc. <lb/></s> <s xml:id="echoid-s1600" xml:space="preserve">9 {9/19}, eo quod ita ſe babeant 10, ad 9 {9/19}, vt 19, ad 18, Quare ſi bæ <lb/>duæ grauitates inuentæ collig antur in vnam ſummam, inueniemus <lb/>totam maſſam auri propoſitam, cuius grauitas in aere ponebatur lib. </s> <s xml:id="echoid-s1601" xml:space="preserve"><lb/>1, in aqua babere grauitatem vnc. </s> <s xml:id="echoid-s1602" xml:space="preserve">11 {1409/5301}. </s> <s xml:id="echoid-s1603" xml:space="preserve">& </s> <s xml:id="echoid-s1604" xml:space="preserve">facta reductione <lb/>fractionis ad ſcrupula, & </s> <s xml:id="echoid-s1605" xml:space="preserve">grana, vnc. </s> <s xml:id="echoid-s1606" xml:space="preserve">11, Scrup. </s> <s xml:id="echoid-s1607" xml:space="preserve">6. </s> <s xml:id="echoid-s1608" xml:space="preserve">Gran. </s> <s xml:id="echoid-s1609" xml:space="preserve">9 {177/1767}, <lb/>vt videre eſt in tabula è regione denominatoris 20, partium. </s> <s xml:id="echoid-s1610" xml:space="preserve">Atqus <lb/>eodem modo deprebendentur grauitates auri in aqua quarumcunq; </s> <s xml:id="echoid-s1611" xml:space="preserve"><lb/>aliarum qualitatum.</s> <s xml:id="echoid-s1612" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1613" xml:space="preserve">Quia vero permolestum videri poſſet omnes grauitates totius <lb/>tabulæ bac via eruere; </s> <s xml:id="echoid-s1614" xml:space="preserve">obſeruari quidem poterit prædicta Metbodus <lb/>quando ſeorſim inuestiganda fuerit alicuius auri grauitas in aqua, <lb/>in cõpoſitione vero tabulæ ſic fortaſſis cõpendioſius quis proceſſerit.</s> <s xml:id="echoid-s1615" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1616" xml:space="preserve">Equidem, cum buius tabulæ conſtructionem diligentius mecum <pb o="71" file="0083" n="83" rhead="ARCHIMEDES"/> pertracto, video grauitates illas, quæ in eius area deſcriptæ ſunt ne-<lb/>ceſſario eadem differentia procedere, ſicut & </s> <s xml:id="echoid-s1617" xml:space="preserve">denominatores quali-<lb/>tatum eadem differentia procedunt; </s> <s xml:id="echoid-s1618" xml:space="preserve">atque adeo differentiam illam <lb/>eſſe eam, qua grauitas ſemiunciæ auri puri in aqua ſuperat in aqua <lb/>grauitatĕ ſemiunciæ miſti, quæ quidĕ differentia eſt Gra. </s> <s xml:id="echoid-s1619" xml:space="preserve">14 {1374/1767}. <lb/></s> <s xml:id="echoid-s1620" xml:space="preserve">Conſiderentur enim quæcunque duæ grauitates proximæ in tabula <lb/>expreßæ, vt exemp. </s> <s xml:id="echoid-s1621" xml:space="preserve">gratia, grauitates auri 20, & </s> <s xml:id="echoid-s1622" xml:space="preserve">19, partium. </s> <s xml:id="echoid-s1623" xml:space="preserve">Quo-<lb/>niam igitur in auro 20, partium ſunt auri puri 20. </s> <s xml:id="echoid-s1624" xml:space="preserve">ſemiuneiæ, misti <lb/>vero 4, & </s> <s xml:id="echoid-s1625" xml:space="preserve">in auro 19, partium ſunt' auri puri 19. </s> <s xml:id="echoid-s1626" xml:space="preserve">ſemiunciæ, misti <lb/>vero 5, erit in auro 20,<unsure/> partiũ vna ſemiuncia auri puri pluſquam in <lb/>auro 19, partium, in auro autem 19, partium erit vna ſemiuncia <lb/>misti plus quam in auro 20, partium; </s> <s xml:id="echoid-s1627" xml:space="preserve">quare grauitas auri 20, par-<lb/>tium in aqua ſuperabit in aqua grauitatem auri 19, partium gra-<lb/>uitate, qua ſemiuncia auri puri ſuperat ſemiunciam miſti. </s> <s xml:id="echoid-s1628" xml:space="preserve">Quod <lb/>erat demonſtrandum. </s> <s xml:id="echoid-s1629" xml:space="preserve">Et eadem est ratio de alijs gr auitatibus, non <lb/>ſolum quæ in bac tabula deſcribuntur, ſed etiam de illis, quæ deſeri-<lb/>bentur in alijs, copioſioribus, in quibus videlicet denominatores non <lb/>eſſent partes integræ, ſed partes partium; </s> <s xml:id="echoid-s1630" xml:space="preserve">dummodo etiam illæ partes <lb/>per vnam eandemque differentiam progrederentur.</s> <s xml:id="echoid-s1631" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1632" xml:space="preserve">Quibus in bunc modum præ ostenſis. </s> <s xml:id="echoid-s1633" xml:space="preserve">ſi construenda fuerit tabula <lb/>per continuam additionem eiuſdem numeri, ſic erit progrediendum. <lb/></s> <s xml:id="echoid-s1634" xml:space="preserve">Primo inuenienda erit grauitas quam babet ſemiuncia auri puri in <lb/>aqua, quæ inuenietur ſi fiat vt 19, ad 18, it a ſemiuncia ad alium <lb/>numerum qui ſit vnc. </s> <s xml:id="echoid-s1635" xml:space="preserve">{9/19}, is enim dabit grauitatem quæſitam, quæ <lb/>valet ſcrup. </s> <s xml:id="echoid-s1636" xml:space="preserve">11, Gran. </s> <s xml:id="echoid-s1637" xml:space="preserve">8 {16/19}.</s> <s xml:id="echoid-s1638" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1639" xml:space="preserve">Secundo quærenda est grauitas ſ<unsure/>emiuncie misti in aqua, quæ ba-<lb/>bebitur fi fiat vt 279, ad 250, it a ſemiuncia, ad alium numerum, qui <lb/>ſit vnc. </s> <s xml:id="echoid-s1640" xml:space="preserve">{125/279}, is enim dabit grauitatem quæſitam, quæ reducta ad <lb/>ſcrupula, & </s> <s xml:id="echoid-s1641" xml:space="preserve">grana valet ſcrup. </s> <s xml:id="echoid-s1642" xml:space="preserve">10, Gran. </s> <s xml:id="echoid-s1643" xml:space="preserve">18 {2/3}.</s> <s xml:id="echoid-s1644" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1645" xml:space="preserve">Tertio exploranda est differentia inter duas grauitates proximè <lb/>inuentas, quam per ſubtractionem inuenies Gran. </s> <s xml:id="echoid-s1646" xml:space="preserve">14 {458/589}, cuius <lb/>tamen fractio reducta eſt ad partes 1767, nempe ad {1374/1767}, propter <lb/>tabellam partis proportionalis.</s> <s xml:id="echoid-s1647" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1648" xml:space="preserve">Postremo inuestig anda erit grauitas in aqua vnius libræ misti, <lb/>quæ inuenietur ſi grauitas ſecundo loco reperta per 24, multiplice-<lb/>tur, productus enim numerus vnc. </s> <s xml:id="echoid-s1649" xml:space="preserve">10, ſcrup. </s> <s xml:id="echoid-s1650" xml:space="preserve">18, Gran. </s> <s xml:id="echoid-s1651" xml:space="preserve">{969/1767}, da <lb/>bit quæſitam grauitatem. </s> <s xml:id="echoid-s1652" xml:space="preserve">Qua in calce tabulæ deſcripta, compo-<lb/>nentur reliquæ grauitates omnes per continuam additionem diffe-<lb/>rentiæ tertio loco inuentæ. </s> <s xml:id="echoid-s1653" xml:space="preserve">Si enim addatur ad grauitatem auri <lb/>partis o, idest ad grauitatem misti vnius libræ in aqua, componetur <pb o="72" file="0084" n="84" rhead="PROMOTVS"/> grauitas auri 1, partis. </s> <s xml:id="echoid-s1654" xml:space="preserve">addita veroad grauitatem 1<unsure/>, partis, procrea-<lb/>bit grauitatem 2, partium, & </s> <s xml:id="echoid-s1655" xml:space="preserve">c. </s> <s xml:id="echoid-s1656" xml:space="preserve">propter rationem quam paulo ante <lb/>aperuimus.</s> <s xml:id="echoid-s1657" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1658" xml:space="preserve">Hoc eodem artificio compoſita est quoque tabella partis propor-<lb/>tionalis, primo enim inuenta e@ vigeſima quarta pars differentiæ <lb/>ſecundum quam tabula progreditur quam ſupra inuenimus eſſe. <lb/></s> <s xml:id="echoid-s1659" xml:space="preserve">Gran. </s> <s xml:id="echoid-s1660" xml:space="preserve">14 {1374/1767}, cuius pars vigeſima quarta est {10@8/1707}, deinde <lb/>banc particulam addidimus primum ſibi ipſi, & </s> <s xml:id="echoid-s1661" xml:space="preserve">produximus diffe-<lb/>rentiam 2, particularum Gran. </s> <s xml:id="echoid-s1662" xml:space="preserve">1 {409/1767} & </s> <s xml:id="echoid-s1663" xml:space="preserve">buic differentiæ iterũ <lb/>adiecimus eandem particulam, & </s> <s xml:id="echoid-s1664" xml:space="preserve">inuenimus pro tribus particulis <lb/>gran. </s> <s xml:id="echoid-s1665" xml:space="preserve">1 {1497/1767}. </s> <s xml:id="echoid-s1666" xml:space="preserve">& </s> <s xml:id="echoid-s1667" xml:space="preserve">ita deinceps progreſſi ſumus vſque ad differen-<lb/>tiam Gran. </s> <s xml:id="echoid-s1668" xml:space="preserve">14 {1374/1767}, quæ reſpondet 24, particulis, ſeu differentiæ, <lb/>ſecundum quam tabula progreditur.</s> <s xml:id="echoid-s1669" xml:space="preserve"/> </p> </div> <div xml:id="echoid-div89" type="section" level="1" n="47"> <head xml:id="echoid-head50" xml:space="preserve">FINIS.</head> <head xml:id="echoid-head51" xml:space="preserve">ERRATA SIC CORRIGE.</head> <note position="right" xml:space="preserve">Pagina # Linea # Errata # Correcta <lb/>6 # 10 # 2. &. 3. # 4. huius. <lb/>6 # 20 # 2. & 3. # 4. buius. <lb/>7 # 25 # ſolidi data # ſolidi A, data. <lb/>16 # 4 # prauitatem # grauitatem <lb/>26 # 28 # grauitas D, ita # grauitas E, ad <lb/>38. & 39. # deſunt ſui numeri. <lb/>58 # 21 # æqualitatem # quare & qualitætem <lb/>61 # 13 # qualiatibus # qualitatibus <lb/>62 # 4 # aſumptis # aſſumptis <lb/>62 # 38 # aeris # æris <lb/>63 # 1 # vt 31. ad # ita 31. ad <lb/>63 # 22 # ære # æere</note> <p style="it"> <s xml:id="echoid-s1670" xml:space="preserve">Contractiorĕ quoq. </s> <s xml:id="echoid-s1671" xml:space="preserve">ad inuenies dimidij pedis menſurã in margine pag. </s> <s xml:id="echoid-s1672" xml:space="preserve">34. </s> <s xml:id="echoid-s1673" xml:space="preserve">appoſitã, <lb/>quia madefacta papyrus, dũ imprimeretur recipiebat verã menſurã, exſiccatæ <lb/>breuiorem reddidit. </s> <s xml:id="echoid-s1674" xml:space="preserve">Itaque ſi quartadecima pars vnius vnciæ addatur ipſi meæ <lb/>ſura componetur dimidij pedis menſura; </s> <s xml:id="echoid-s1675" xml:space="preserve">vel ſi ipſa menſura duplicetur, & </s> <s xml:id="echoid-s1676" xml:space="preserve">ei ad-<lb/>datur ſeptima pars vnius vncia fiet menſura vnius pedis, ad cuius rationem one <lb/>æes calculi in tabulas optime reſpondebunt.</s> <s xml:id="echoid-s1677" xml:space="preserve"/> </p> <p style="it"> <s xml:id="echoid-s1678" xml:space="preserve">Denique pagina 38. </s> <s xml:id="echoid-s1679" xml:space="preserve">& </s> <s xml:id="echoid-s1680" xml:space="preserve">39. </s> <s xml:id="echoid-s1681" xml:space="preserve">tabularum præpoſterus eſt ordo. </s> <s xml:id="echoid-s1682" xml:space="preserve">ex quo tæmen nibil er-<lb/>reris ſequitur ſi tituli recte accipiantur.</s> <s xml:id="echoid-s1683" xml:space="preserve"/> </p> <pb file="0085" n="85"/> <pb file="0086" n="86"/> <pb file="0087" n="87"/> <pb file="0088" n="88"/> </div></text> </echo>