Vitruvius, De architectura libri decem, 1567

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          <div xml:id="echoid-div713" type="section" level="2" n="134">
            <pb o="272" file="514.01.304" n="304" rhead="LIBER"/>
            <p style="it">
              <s xml:id="echoid-s20297" xml:space="preserve">Iucundas, & </s>
              <s xml:id="echoid-s20298" xml:space="preserve">gratas bominibus Architæ, & </s>
              <s xml:id="echoid-s20299" xml:space="preserve">Eratosthenis inuentiones fuiſſe, dicit Vitr. </s>
              <s xml:id="echoid-s20300" xml:space="preserve">Sed Cum ambo
                <lb/>
              eandem quæs̃tionem tractarent, nitereturq́; </s>
              <s xml:id="echoid-s20301" xml:space="preserve">alter, atque alter ratione diuerſa eam ſoluere, ambo ſuſpecti fue-
                <lb/>
              runt, non quia res eadem diuerſis modis inueniri non poßit, ſed quia, quineſcirent, cum uiderent Architam
                <lb/>
              ratione una, & </s>
              <s xml:id="echoid-s20302" xml:space="preserve">Eratoſthenem. </s>
              <s xml:id="echoid-s20303" xml:space="preserve">alia uti, ob eorum contentionem ſuſpicabantur, cogitantes ſtudioſe inter-ſe
                <lb/>
              contendere. </s>
              <s xml:id="echoid-s20304" xml:space="preserve">Quod perinde eſt, ac ſi quis turris altitudinem quadrante,alius radio, alius astrolabio, alius ſpe-
                <lb/>
              culo, alius alio instrumento metiretur, cum vulgus neſciat eandem eſſe omnium horum inſtrumentorum ra-
                <lb/>
              tionem, & </s>
              <s xml:id="echoid-s20305" xml:space="preserve">viam, quæ ab angulorum menſura diſceernitur, ſuſpicatur menſorum contertionem, atque æmu-
                <lb/>
              lationem uerum inuoluere, ac confundere inſtrumentorum diuerſitate. </s>
              <s xml:id="echoid-s20306" xml:space="preserve">Simile euenit ob Erathoſthenis, & </s>
              <s xml:id="echoid-s20307" xml:space="preserve">
                <lb/>
              Architæ contentiones. </s>
              <s xml:id="echoid-s20308" xml:space="preserve">Propoſita quæſtio erat, qua ratione cubus duplicari oporteret. </s>
              <s xml:id="echoid-s20309" xml:space="preserve">Cubus eſt, (ut dixi li-
                <lb/>
                <note position="left" xlink:label="note-514.01.304-01" xlink:href="note-514.01.304-01a" xml:space="preserve">10</note>
              bro quinto ) ſolidum corpus,planis ſex, & </s>
              <s xml:id="echoid-s20310" xml:space="preserve">lateribus conſtans æquaiibus Metiemur cubum, ſi duxerimus
                <lb/>
              aliquod unum ex ſuis lateribus in ſeipſum, & </s>
              <s xml:id="echoid-s20311" xml:space="preserve">eam ſummam quoque duxerimus per idem latus. </s>
              <s xml:id="echoid-s20312" xml:space="preserve">Detur cubus
                <lb/>
              cuius latera ſingula octo conſtet pedibus, duc octo in ſe,reddentur 64. </s>
              <s xml:id="echoid-s20313" xml:space="preserve">ducito rurſtus 64. </s>
              <s xml:id="echoid-s20314" xml:space="preserve">per octo,redden-
                <lb/>
              tur 5 12. </s>
              <s xml:id="echoid-s20315" xml:space="preserve">tot igitur quadratis perdibus conſtabit cubus. </s>
              <s xml:id="echoid-s20316" xml:space="preserve">Cum igitur cubusiam 512. </s>
              <s xml:id="echoid-s20317" xml:space="preserve">pedibus comſtare cogni-
                <lb/>
              tum ſit. </s>
              <s xml:id="echoid-s20318" xml:space="preserve">iuxta propoſitam quæstionem duplicationdus eſt. </s>
              <s xml:id="echoid-s20319" xml:space="preserve">Quod ut commode ſaciamus, neceſſario eſt ſcire, qua
                <lb/>
              ratione duabus propoſitis rectis. </s>
              <s xml:id="echoid-s20320" xml:space="preserve">& </s>
              <s xml:id="echoid-s20321" xml:space="preserve">inæqualibus lmeis. </s>
              <s xml:id="echoid-s20322" xml:space="preserve">duas medias inueniamus, quæ inter ſe & </s>
              <s xml:id="echoid-s20323" xml:space="preserve">cum primis
                <lb/>
              continua proportione reſpondeant. </s>
              <s xml:id="echoid-s20324" xml:space="preserve">Vt igitur inutniretur ea mediarum linearum comparatio, modi undecim
                <lb/>
              ab antiquis reperti ſunt. </s>
              <s xml:id="echoid-s20325" xml:space="preserve">Alij euidentibud matbematicorum rationibus uſi ſunt, alij. </s>
              <s xml:id="echoid-s20326" xml:space="preserve">& </s>
              <s xml:id="echoid-s20327" xml:space="preserve">præter demonſtra-
                <lb/>
              tiones inſtrument a ad id aptainuenere. </s>
              <s xml:id="echoid-s20328" xml:space="preserve">Archimedes organo uſus. </s>
              <s xml:id="echoid-s20329" xml:space="preserve">quod meſolabium, ideſt medicapium, ſi licet
                <lb/>
              ita dicere, numcupauit. </s>
              <s xml:id="echoid-s20330" xml:space="preserve">nam eo mediæ inter duas datas comparabiles inuenimus. </s>
              <s xml:id="echoid-s20331" xml:space="preserve">Plato quoque alio inſtrumẽto
                <lb/>
                <note position="left" xlink:label="note-514.01.304-02" xlink:href="note-514.01.304-02a" xml:space="preserve">20</note>
              uſus eſt, quod pari ratione dmeſolabium nocare poſſumus. </s>
              <s xml:id="echoid-s20332" xml:space="preserve">Architas demoſtrationes attulit lineis a dimidio
                <lb/>
              Cylindro ſumptis, quas in opus redigere numquam potuit. </s>
              <s xml:id="echoid-s20333" xml:space="preserve">Ego rationes, & </s>
              <s xml:id="echoid-s20334" xml:space="preserve">inſtrumenta explicabo, oſten-
                <lb/>
              damq́ quomodo ad duplicationem cubi, mediarum linearum inuentio nos ducat. </s>
              <s xml:id="echoid-s20335" xml:space="preserve">Quætionis primum afferam
                <lb/>
              occaſionem, e qua ſumma utilitas erit maniſeſta, quam habituri ſunt Architecti in egregijs eorum operibus
                <lb/>
              deformandis. </s>
              <s xml:id="echoid-s20336" xml:space="preserve">Legitur ergo Epiſtola Eratosthenis ad Potlemæum Regem in hunc modumperſcripta.</s>
              <s xml:id="echoid-s20337" xml:space="preserve"/>
            </p>
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          <div xml:id="echoid-div718" type="section" level="2" n="135">
            <head xml:id="echoid-head143" xml:space="preserve">PTOLEM AEO REGI ERATOSTHENES. S.</head>
            <p style="it">
              <s xml:id="echoid-s20338" xml:space="preserve">Dicitur ex antiquis tragædiarum compoſitoribus unum introducere Minoa Glauco ſepulchrum excitare
                <lb/>
              uolentem, cumq; </s>
              <s xml:id="echoid-s20339" xml:space="preserve">dictum fuiſſet illud quaqua uerſus eſſe pedes centum: </s>
              <s xml:id="echoid-s20340" xml:space="preserve">Dixit paruam fore arcam pro Regio
                <lb/>
              ſepulchro, duplicetur igitur, & </s>
              <s xml:id="echoid-s20341" xml:space="preserve">cubus non mutetur. </s>
              <s xml:id="echoid-s20342" xml:space="preserve">certe qui unum quodque latus duplicare uoluerit, non
                <lb/>
                <note position="left" xlink:label="note-514.01.304-03" xlink:href="note-514.01.304-03a" xml:space="preserve">30</note>
              erit erroris expers. </s>
              <s xml:id="echoid-s20343" xml:space="preserve">nam lateribus duplicatis planum quodlibet quadruplum efficietur, ipſum uero ſolidum
                <lb/>
              octuplum. </s>
              <s xml:id="echoid-s20344" xml:space="preserve">Quæſitum igitur est a geometris, qua ratione ſolido in eadem figura permanens duplum efficeretur.
                <lb/>
              </s>
              <s xml:id="echoid-s20345" xml:space="preserve">Quæſtio hæc cubi duplicatio nominata eſt. </s>
              <s xml:id="echoid-s20346" xml:space="preserve">nam propoſito cubo quærebant, qua uia alterum illi duplum effice-
                <lb/>
              rent. </s>
              <s xml:id="echoid-s20347" xml:space="preserve">Ambigentibus, & </s>
              <s xml:id="echoid-s20348" xml:space="preserve">labor antibus cæteris, prius extitit Hippocrates, qui indicauit id fieri poſſe,ſi con-
                <lb/>
              ſtitutis duabus lineis, quarum maior minoris eſſet dupla, duæ. </s>
              <s xml:id="echoid-s20349" xml:space="preserve">in continua proportione inuenirentur. </s>
              <s xml:id="echoid-s20350" xml:space="preserve">
                <lb/>
              Quare ea res dubia in maiorem difficultatem uerſa es̃t. </s>
              <s xml:id="echoid-s20351" xml:space="preserve">Aliquanto poſt Delij morbo laborantes, cum oraculo
                <lb/>
              Apollinis iuberentur aram ipſius duplicare, neque qua id fieri poſſet ratione ſatis uiderent,in eandem dubietatẽ
                <lb/>
              incidere, & </s>
              <s xml:id="echoid-s20352" xml:space="preserve">obiurgante Platone eos geometras, qui erant in Academia, ab ijs quæſitũ est, ut inuenirent quod
                <lb/>
              propoſitũfuerat. </s>
              <s xml:id="echoid-s20353" xml:space="preserve">ij cum lahori ſeſe dediſſent,& </s>
              <s xml:id="echoid-s20354" xml:space="preserve">conantes inuenter duas medias proportione reſpondentes dua-
                <lb/>
              bus propoſitis lineis: </s>
              <s xml:id="echoid-s20355" xml:space="preserve">dicitur Architam Tarentinum eas inueniſſe bemicylindrorum ratione. </s>
              <s xml:id="echoid-s20356" xml:space="preserve">Eudo xus uero fle-
                <lb/>
                <note position="left" xlink:label="note-514.01.304-04" xlink:href="note-514.01.304-04a" xml:space="preserve">40</note>
              xis quibuſdam lineis, cæterum uterque ſane approbatam harum rerum rationem inuenire, ſed neuter eas ad
                <lb/>
              uſum potuit accommodare, & </s>
              <s xml:id="echoid-s20357" xml:space="preserve">manibus experiri, excepto uno Menechmo, qui tamen parum fecit, & </s>
              <s xml:id="echoid-s20358" xml:space="preserve">id parũ
                <lb/>
              maxima cum difficultate.</s>
              <s xml:id="echoid-s20359" xml:space="preserve">Sed nos excogitauimus per organd facilem inuentionem,qua nõ tantum duas medias
                <lb/>
              proportionabiles duabus datis, ſed quotquot propoſitum fuerit, utinueniamus, & </s>
              <s xml:id="echoid-s20360" xml:space="preserve">eo inuento, poterimus de-
                <lb/>
              mum ad cubum reducere propoſitum ſolidum lineis æquè diſtantibus contentum, aut etiam ex una aliam
                <lb/>
              figuram formare, quæ aut æqualis, aut maior ſit ſeruata ſimilitudine. </s>
              <s xml:id="echoid-s20361" xml:space="preserve">Quoniam nullidubium eſt, quin huiuſ-
                <lb/>
              modi inſtrumento duplicari poßint aræ, ædificiaq́, & </s>
              <s xml:id="echoid-s20362" xml:space="preserve">ad cubum referri liquidonum, &</s>
              <s xml:id="echoid-s20363" xml:space="preserve">ſiccorum menſuræ, ut
                <lb/>
              modiorum, & </s>
              <s xml:id="echoid-s20364" xml:space="preserve">ſimilium, quarum menſurarum larteribus uaſorum capacitas dignoſcitur.</s>
              <s xml:id="echoid-s20365" xml:space="preserve">& </s>
              <s xml:id="echoid-s20366" xml:space="preserve">ut ſummatim dicã,
                <lb/>
              quæſtionis buius cognitio utilies est uolentibus duplicare,aut maiorareddere organa, e quibus tela, ſaxa, aut
                <lb/>
              ferreæ pilæ mittuntur. </s>
              <s xml:id="echoid-s20367" xml:space="preserve">nam neceſſe eſt omnia in latum, & </s>
              <s xml:id="echoid-s20368" xml:space="preserve">in longum creſcere proportione quadam, ſiue fo-
                <lb/>
              ramina ſint, ſiue nerui, & </s>
              <s xml:id="echoid-s20369" xml:space="preserve">immiſſa alia, aut quicquid opus fuerit, ſi totum proportione quadam, ſiue fo-
                <lb/>
                <note position="left" xlink:label="note-514.01.304-05" xlink:href="note-514.01.304-05a" xml:space="preserve">50</note>
              fieri non poteſt ſine medij inuentione. </s>
              <s xml:id="echoid-s20370" xml:space="preserve">Demonſtrationem igitur, & </s>
              <s xml:id="echoid-s20371" xml:space="preserve">apparaiumprædicti organi tibi ſubſcripſi,
                <lb/>
              ac primum demonſtrationem.</s>
              <s xml:id="echoid-s20372" xml:space="preserve">#Proponantur duæ rectæ, & </s>
              <s xml:id="echoid-s20373" xml:space="preserve">inæquales lineæ, quarum altera ſit a b, al-
                <lb/>
              tera c, d, quæramus inter duas medias comparabiles ratione continua, ita quidem ut quemadmodum ſe ha
                <lb/>
              bet prima ad ſecundam, ita ſe habeat ſecunda ad tertiam, & </s>
              <s xml:id="echoid-s20374" xml:space="preserve">tertia ad quartam. </s>
              <s xml:id="echoid-s20375" xml:space="preserve">Detur utranque ex pro-
                <lb/>
              poſitis lineis ad perpendiculum cadere in rectam, quæ ſit b d, & </s>
              <s xml:id="echoid-s20376" xml:space="preserve">maior ſit a b, monor uero c d, ab. </s>
              <s xml:id="echoid-s20377" xml:space="preserve">a,
                <lb/>
              autem ad c, ducatur linea, quæ ulterius producta cadat in lineam b d, ibierit e. </s>
              <s xml:id="echoid-s20378" xml:space="preserve">Ducatur etiam ab a,
                <lb/>
              ſigno in lineam b d, linea, quæſit a f, & </s>
              <s xml:id="echoid-s20379" xml:space="preserve">ab f, rurſum ducatur linea a b, æquè distans, quæ ſitf g.
                <lb/>
              </s>
              <s xml:id="echoid-s20380" xml:space="preserve">quæ lineam a c, in puncto, g, tangat. </s>
              <s xml:id="echoid-s20381" xml:space="preserve">Similiter ducatur linea a ſigno g, æquidiſtans lineæ a f, & </s>
              <s xml:id="echoid-s20382" xml:space="preserve">
                <lb/>
              fit illa g h, quæ lineam b d, tangat, in ſigno b, a quo ſigno erigatur linea æquidistans ipſi a b, & </s>
              <s xml:id="echoid-s20383" xml:space="preserve">ſit
                <lb/>
                <note position="left" xlink:label="note-514.01.304-06" xlink:href="note-514.01.304-06a" xml:space="preserve">60</note>
              </s>
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