Viviani, Vincenzo, De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

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            <s xml:id="echoid-s482" xml:space="preserve">
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            ctum FL, iſtaque rectangula æqualia oſtenfa funt, vnde latera quoq; </s>
            <s xml:id="echoid-s483" xml:space="preserve">HF,
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            FL æqualia crunt. </s>
            <s xml:id="echoid-s484" xml:space="preserve">Quod demonſtrandum erat.</s>
            <s xml:id="echoid-s485" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s486" xml:space="preserve">Sed quoniam eſt vt tranſuerſum HF ad rectum FL ita rectangulum.</s>
            <s xml:id="echoid-s487" xml:space="preserve"> HNF ad quadratum NM, atque hæc ipſa latera æqualia ſunt oſtenſa, ergo
              <lb/>
              <note symbol="a" position="right" xlink:label="note-0029-01" xlink:href="note-0029-01a" xml:space="preserve">21. pri-
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              mi conic.</note>
            rectangulum HNF æquabitur quadrato NM; </s>
            <s xml:id="echoid-s488" xml:space="preserve">quare in qualibet ſubcontra-
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            ria ſectione MFTH, deducta, vt in præcedenti, ex triangulo per axem coni
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            ſcaleni, quod tamen non ſit æquicrure, rectangula ſub ſegmentis diametri
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            ſunt ſemper æqualia quadratis eorum ordinatè applicatarum, quæ quando
              <lb/>
            cum diametro FH rectos angulos conſtituent, (quod eueniet cum commu-
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            nis ſectio DGE perpendicularis fuerit, non ſolùm baſi BGC trianguli per
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            axem, ſed etiam rectæ FHG communi ſectioni plani ſecantis cum prædicto
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            triangulo, hoc eſt quando triangulum per axem BAC rectum fuerit baſi co-
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            ni BC, nam tunc DGE communis ſectio plani ſecantis FH cum plano ba-
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            ſis coni BC, cum poſita ſit perpendicularis rectæ BGC, quæ eſt communis
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            ſectio trianguli per axem cum plano baſis coni, perpendicularis etiam erit plano trianguli BAC, vnde cum recta GHF rectos angulos faciet, ideoq;
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            </s>
            <s xml:id="echoid-s489" xml:space="preserve">
              <note symbol="b" position="right" xlink:label="note-0029-02" xlink:href="note-0029-02a" xml:space="preserve">4. def. lib.
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              II. Elem.</note>
            omnes in ſectione MFT ordinatim ductæ, ſiue ipſi DGE æquidiſtantes ei-
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            dem GFH erunt perpendiculares) Ellipſim efficient æqualium laterum cir-
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            ca axim FH, quæ eadem erit, ac circulus diametri FH. </s>
            <s xml:id="echoid-s490" xml:space="preserve">Si verò prædictæ
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            applicatæ ad obliquos angulos diametrum ſecabunt (quod accidet cum.
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            </s>
            <s xml:id="echoid-s491" xml:space="preserve">DGE obliquè ſecat rectam FHG) tunc ipſa ſectio erit pariter Ellipſis æqua-
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            lium laterum, ſed eius tranſuerſum latus, diameter erit non autem axis.</s>
            <s xml:id="echoid-s492" xml:space="preserve"/>
          </p>
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            <s xml:id="echoid-s493" xml:space="preserve">Non ſemper igitur ſubcontraria ſectione coni ſcaleni efficitur circulus, ſed
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            ſolùm cum triangulum per axem rectum eſt baſi coni, quo in caſu, vt viſum
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            eſt, ei debetur eadem proprietas, ac Ellipſi, æqualium tamen laterum circa.
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            </s>
            <s xml:id="echoid-s494" xml:space="preserve">axim. </s>
            <s xml:id="echoid-s495" xml:space="preserve">In ſectionibus autem ſubcontrarijs cuiuslibet alterius trianguli per
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            axem (dummodo non ſit triangulum æquicrure, quia tunc communis ſectio
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            plani ſecantis cum ipſo triangulo non conuenit cum baſi eiuſdem trianguli, ſed
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            ei æquidiſtat) oritur Ellipſis æqualium item laterum, ſed circa diametrum,
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            quæ oblquè ſecat applicatas. </s>
            <s xml:id="echoid-s496" xml:space="preserve">Hinc ergo liquidò conſtat in ſuperiori propoſitio-
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            ne opus non fuiſſe ſubcontrariam ſectionem reijcere, vti fit ab ipſo Apoll. </s>
            <s xml:id="echoid-s497" xml:space="preserve">in. </s>
            <s xml:id="echoid-s498" xml:space="preserve">
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            13. </s>
            <s xml:id="echoid-s499" xml:space="preserve">primi, atque ab alijs doctrinam conicam pertractantibus ſed hæc obiter
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            delibaſſe ſufficiat; </s>
            <s xml:id="echoid-s500" xml:space="preserve">quo etiam nomine liceat mihi inſequentes demonſtrationes
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            proferre, non tam vt deſiderio obſequar hominis mihi amiciſsimi, quam vt
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            alteri cuidam, quocum iam ab hinc multis annis illas, nec non plures alias
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            communicaui, in mentem redigam, eas, non eius, ſed quidquid ſunt ingenioli
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            mei eſſe inuenta; </s>
            <s xml:id="echoid-s501" xml:space="preserve">atque ita periculo occurram, ne ille, non dicam fidei, ſed
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            memoriæ forſan defectu ſibi eas aſciſcat. </s>
            <s xml:id="echoid-s502" xml:space="preserve">Hoc autem audentiùs faciam,
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            cum eæ non omnino ab inſtituto opere ſint alienæ, verſantur enim circà tan-
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            gentes coni-ſectionum ab Apoll. </s>
            <s xml:id="echoid-s503" xml:space="preserve">acutiſsimè quidem inuentas, ac negatiuè
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            oſtenſas in eius 33. </s>
            <s xml:id="echoid-s504" xml:space="preserve">ac 34. </s>
            <s xml:id="echoid-s505" xml:space="preserve">primi, à me autem neſcio anbreuiùs, euidentiùs
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            certè affirmatiuèque demonſtratas, ac Problematicè propoſitas, vt in ſe-
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            quentibus.</s>
            <s xml:id="echoid-s506" xml:space="preserve"/>
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