Cardano, Girolamo, De subtilitate, 1663

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    <archimedes>
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            <p type="main">
              <s id="s.011124">
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              dixit, circuli quadraturam, cùm quę ſciri poſ­
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              ſunt, & nondum ſcita ſunt, refert: & ſi ſcita
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              non ſit, non obſtarem, quin ſciri non poſſit.
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              </s>
              <s id="s.011125">Non tamen dixit ſciri poſſe. </s>
              <s id="s.011126">Dupliciter au­
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              tem ſciri contingit illam: aut obſcuriore mo­
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              do, quàm cùm ignota eſt: velut per elicas li­
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              neas, quibus vtitur Archimedes, & æqualem
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              rectam circuli periferiæ deſcribit: aut per
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              tranſlationem, quam nemo adhuc tentare
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              auſus eſt, partim ob difficultatem, partim ob
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              demonſtrandi modum ignotum
                <emph type="italics"/>
              :
                <emph.end type="italics"/>
              alij quòd
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              confiſi ſint faciliori modo eam inuenire
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              poſſe: aliis quod ſcripta antiquorum huic
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              generi demonſtrationis neceſſaria deeſſent,
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              intacta fuit. </s>
              <s id="s.011127">Verùm cùm ad certam
                <expan abbr="notionẽ">notionem</expan>
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              non deueniat abſque demonſtrationis auxi­
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              lio, non licet illam percipere ante
                <expan abbr="demõſtra-tionem">demonſtra­
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                tionem</expan>
              , vt in quibuſdam arithmeticis quæ­
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              ſtionibus caſu quandoque contigit. </s>
              <s id="s.011128">Sed de
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              his hactenus, quæ ad demonſtrandi modos
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              pertinent, in quibus maximè artis Geome­
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              tricæ ſubtilitatem edocuimus. </s>
            </p>
            <p type="main">
              <s id="s.011129">Hanc proximè Arithmetica ſequitur, cu­
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              ius ſubtiliſſimum inuentum eſt ars,
                <expan abbr="quã">quam</expan>
              nos
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                <arrow.to.target n="marg1554"/>
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              Magnam vocauimus, à nobis inuenta, editá­
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              que, Algebraticam alij dixerunt, cuius eſt
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              multiplex vtilitas, Ingenium acuere, latera
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              quantitatum incognita inuenire, & explica­
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              re: eadémque lineis iuxta geometrica inſti­
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              tuta, vel planis aut ſolidis deſcribere: propo­
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              ſita ſoluere problemata,
                <expan abbr="ænigmatáq;">ænigmatáque</expan>
              & malè
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              ſoluta poſſe refellere, vt lateris heptagoni
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              magnitudinem à Bouillo
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              perperam, quæſitam, &
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                <expan abbr="æqualitatẽ">æqualitatem</expan>
              rectæ cum pe­
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              riferia circuli ex
                <expan abbr="libramẽ-to">libramen­
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                to</expan>
              à Nicolao Cuſa confi­
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              ctam, exploſam verò iure
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                <figure id="id.016.01.249.1.jpg" xlink:href="016/01/249/1.jpg" number="108"/>
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              à Ioanne Monteregio. </s>
              <s id="s.011130">Conſtant omnia ſim­
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                <arrow.to.target n="marg1555"/>
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              pliciſſima figura, quæ lineas, areas, & cor­
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              pora etiam oſtendit: ſed & poſt quartam ſe­
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              cundi Element. </s>
              <s id="s.011131">Euclidis, ſex proximas ſe­
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              quentes. </s>
              <s id="s.011132">Eſt verò & aliud ſolidi numeri cu­
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              bi compoſitionis genus Arithmeticæ pro­
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              prium, in quo genus reſolutionis, quod à
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              compoſitione ortum habet, fit manifeſtum.
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              </s>
              <s id="s.011133">Omnis enim cubus numerus componitur
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              ex quadrato ſui lateris, & duplo producti
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              ex latere in omnes antecedentes numeros
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              vſque ad vnitatem: velut capio 512. cu­
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              ius latus eſt 8. dico igitur, quòd 8. ductum
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              in ſe, & fit 64: & in duplum anteceden­
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              tium numerorum, qui ſunt 1. 2. 3. 4. 5. 6.
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              & 7 ab vnitate
                <emph type="italics"/>
              (
                <emph.end type="italics"/>
              vt vides ) incipiendo, &
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              eſt duplum horum 56. & productum ex 8.
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              in 56. eſt 448. faciunt ipſum cubum, id
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              eſt, 411. nam 448. & 64. faciunt iuncti 512.
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              Ab initio autem videbatur hoc non poſſe
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              demonſtrari, ſed via reſolutionis demonſtra­
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              uimus. </s>
              <s id="s.011134">Cùm enim quilibet numerus cubus
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              fiat ex quadrato lateris in latus ſuum, fiet
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              etiam ex quadrato lateris in vnitatem, &
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              eodem quadrato in latus vnitate dempta, ex
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              ſecundi Element. </s>
              <s id="s.011135">Euclidis primo theore­
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              mate. </s>
              <s id="s.011136">At productum quadrati in vnitatem
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              ſemper eſt æquale quadrato, ex demon­
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              ſtratis à nobis in principio ſexti operis perfe­
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              cti, idque etiam ſenſu ipſo percipitur pro­
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              ductum autem quadrati in latus, dempta
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              vnitate æquale producto lateris in duplum
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              aggregati omnium præcedentium numero­
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              rum: itaque hoc demonſtrato patet propo­
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              ſitum. </s>
              <s id="s.011137">Hoc autem rurſus reſolutione indi­
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              get: qualis enim proportio quadrati ad du­
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              plum illius aggregati, talis lateris ad ſeip­
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              ſum dempta vnitate. </s>
              <s id="s.011138">Igitur ex demonſtra­
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              tis ab Euclide in ſexto Elementorum, tan­
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              tùm fit ex latere in duplum illius aggregati
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              quantum ex quadrato in latus dempta vni­
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              tate. </s>
              <s id="s.011139">Oportet igitur rurſus illam propor­
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              tionem oſtendere, atque hæc eſt demon­
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              ſtratio: quia duplum illius aggregati ſem­
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              per eſt æquale producto maximi numeri in
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              maiorem vnitate, velut duplum aggregati
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              numerorum vſque ad 7. eſt 56. & hic fit
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              ex 7. maximo numero in 8. qui maximum
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              numerum vnitate excedit. </s>
              <s id="s.011140">Igitur cùm ex la­
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              tere in ſe ducto fiat quadratum ipſius lateris
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              & ex latere in ſeipſum detracta vnitate fiat
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              illud duplum, erit ex demonſtratis ab Eucli­
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              de proportio quadrati lateris ad duplum il­
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              lud qualis literis ad ſeipſum detracta vnita­
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              te, quod aſſumpſimus, demonſtrandum. </s>
              <s id="s.011141">Eſ­
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              ſet igitur iam hoc perfectè oſtenſum, niſi
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              quod nondum conſtat, quòd ex quolibet nu­
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              mero in vnitate minorem, fiat duplum ag­
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              gregati omnium antecedentium numero­
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              rum. </s>
              <s id="s.011142">Hoc verò ſic demonſtratur. </s>
            </p>
            <p type="margin">
              <s id="s.011143">
                <margin.target id="marg1554"/>
              Artis ma­
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              gnæ quintu­
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              plex vſus.</s>
            </p>
            <p type="margin">
              <s id="s.011144">
                <margin.target id="marg1555"/>
              Cubi numeri
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              pulchra
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              compoſitio,
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              & exem­
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              plum aliud
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              reſolutoriæ
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              methodi.</s>
            </p>
            <p type="main">
              <s id="s.011145">Quilibet autem duo numeri, æqualiter
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              à medio diſtantes iuncti, duplum medij nu­
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              meri efficiunt, igitur omnes numeri ab vni­
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              tate iuncti ſeriatim, tantum efficiunt, quan­
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              tum ſi medius numerus pro numero illorum
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              terminorum aſſumeretur. </s>
              <s id="s.011146">Sed maximus nu­
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              merus continet ad vnguem ordinem illo­
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              rum, igitur omnes numeri ſeriatim ab vni­
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              tate ſumpti: tantum iuncti faciunt, quan­
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              tum medius illorum productus in maiorem.
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              </s>
              <s id="s.011147">Igitur duplum aggregati talium numero­
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              rum, eſt æquale duplo producti medij in
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              maximum illorum. </s>
              <s id="s.011148">Sed numerus maximo
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              vnitate maior, duplus eſt medio, igitur ex
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              numero maximo in vnitate maiorem du­
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              plum fit aggregari omnium numerorum ab
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              vnitate ad maximum numerum. </s>
              <s id="s.011149">Eſt & me­
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              dius quidam compoſitionis modus. </s>
              <s id="s.011150">Sed in
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              iam declarato, licet per compoſitionem re­
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              ſoluta colligere, atque ſic demonſtrationem
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              theorematis afferre. </s>
              <s id="s.011151">Laudabimus & quæ­
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              dam ſingularia inuenta, vt Michaëlis Sti­
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              phelij laterum inuentionem, tranſlatam à
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                <arrow.to.target n="marg1556"/>
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              nobis in primum Operis perfecti librum. </s>
            </p>
            <p type="margin">
              <s id="s.011152">
                <margin.target id="marg1556"/>
              Michaëlis
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              Stiphelij in­
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              uentum in
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              Arithmeti­
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              ca.
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              </s>
              <s id="s.011153">Muſica ſub­
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              tilitatis in­
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              uentum.</s>
            </p>
            <p type="main">
              <s id="s.011154">Succedunt his muſica inuenta, triplicis
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              olim ordinis diateſſaron, è quibus ſolùm
                <expan abbr="vnũ">vnum</expan>
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              nunc diatonicum cognitum eſt, reliquos or­
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              dines ſeu incuria, ſeu difficultate amiſimus.
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              </s>
              <s id="s.011155">Nunc verò breuiſſimè illorum reſtitutionem
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              per nos factam atque in primo, & ſecundo
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              muſicæ tractatam doceamus. </s>
              <s id="s.011156">Chromaticum
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              fit per fictam muſicam, vbi non ſolùm in b
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                <arrow.to.target n="marg1557"/>
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              f a b mi, ſed & e la mi, & a la mi re hemi­
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              tonium inducitur, vt in cheli propriè, ſed &
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              organa eius rationis muſicæ ſunt capacia.
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              </s>
              <s id="s.011157">Dulciſſimus videtur hic modus diatonici ge­
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              neris comparatione, ob frequentia hemito­
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              nia: parua enim interualla, & notæ propor­
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              tiones, vocum ſuauitatem afferunt. </s>
              <s id="s.011158">Innititur
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              autem Chromaticum genus hemitonio, to-</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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