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

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            <s xml:id="echoid-s8374" xml:space="preserve">
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            G L cum eadem G I haud rectos efficiet, vnde producta hinc inde ad alte-
              <lb/>
            ram partem cadet intra circulum G L I, eius peripheriæ occurrens in L.
              <lb/>
            </s>
            <s xml:id="echoid-s8375" xml:space="preserve">Cum ergo G L ſit tota intra circulum, circulus verò totus intra ſolidum,
              <lb/>
            erit quoquè G L tota intra ſolidum: </s>
            <s xml:id="echoid-s8376" xml:space="preserve">quare planum, quod per A F, & </s>
            <s xml:id="echoid-s8377" xml:space="preserve">G L
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            ductum fuit, fecabit omnino interius ſolidum G H I, de quo aliquam ter-
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            minatam portionem abſcindet (cum idem planum vndique productum de
              <lb/>
            exteriori ſolido ponatur quoque portionem quandam auferre) cuius con-
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            uexa ſuperficies tota erit intra portionem exterioris ſolidi ab eodem plano
              <lb/>
            abſciſſam.</s>
            <s xml:id="echoid-s8378" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8379" xml:space="preserve">Si verò punctum G (quod nuper oſtẽſum fuit eſſe cõtactum plani per A F
              <lb/>
            ducti, ad planum per axem A B C recti, cum interioris ſolidi G H I ſuper-
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            ficie) fuerit in ipſo axis vertice H, vt in hac tertia figura, oſtendetur etiam
              <lb/>
            quodlibet aliud planum A L F per rectam A F ductum, ſed ad planum per
              <lb/>
            axem A B C inclinatum, quodque de exteriori ſolido aliquam portionem
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            abſcindat, omnino ſecare interius ſolidum, ideoque de ipſo quandam por-
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            tionem terminatam auferre.</s>
            <s xml:id="echoid-s8380" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8381" xml:space="preserve">Nam, in prædicto contingente plano A
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              <figure xlink:label="fig-0301-01" xlink:href="fig-0301-01a" number="244">
                <image file="0301-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0301-01"/>
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            E F, ducta per G quacumq; </s>
            <s xml:id="echoid-s8382" xml:space="preserve">recta G E cũ
              <lb/>
            G A quemlibet angulum conſtituente, & </s>
            <s xml:id="echoid-s8383" xml:space="preserve">
              <lb/>
            per rectam G E, ac per axim G D ducto
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            alio plano, id in interiori ſolido deſcribet
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            genitricem ſectionem L G M, quam
              <note symbol="a" position="right" xlink:label="note-0301-01" xlink:href="note-0301-01a" xml:space="preserve">12. Ar-
                <lb/>
              chim. de
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              Conoid.
                <lb/>
              &c.</note>
            tinget in G recta G E eorundem plano-
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            rum communis ſectio, cum hæc ponatur
              <lb/>
            eſſe in plano contingente vniuerſam ſolid
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            ſuperficiem, ſed planum inclinatum A L
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            F vndiq; </s>
            <s xml:id="echoid-s8384" xml:space="preserve">productum ad alteram partium,
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            vtputa ad E, cadit infra contingens pla-
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            num, cum eo commune habens tantùm
              <lb/>
            rectam A F, ergo & </s>
            <s xml:id="echoid-s8385" xml:space="preserve">communis ſectio ipſius plani inclinati cum ſectione L
              <lb/>
            G M, nempe recta G L cadet infra idem planum contingens, ac ideo infra
              <lb/>
            rectam G E; </s>
            <s xml:id="echoid-s8386" xml:space="preserve">ſed G L, & </s>
            <s xml:id="echoid-s8387" xml:space="preserve">G E ſunt in plano L G M, atque G E ipſam ſe-
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            ctionem contingit, vt modò oſtendimus, quare G L, quæ cadit infra G E
              <lb/>
            cadet omnino intra ſectionem L G M, ſiue intra ſolidum, ac
              <note symbol="b" position="right" xlink:label="note-0301-02" xlink:href="note-0301-02a" xml:space="preserve">32. pri-
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              mi conic.</note>
            planum inclinatum, quod per A F, & </s>
            <s xml:id="echoid-s8388" xml:space="preserve">G L ducitur, ſecabit omnino interius
              <lb/>
            ſolidum, ac de ipſo quandam terminatam portionem auferet, cum idem
              <lb/>
            planum inclinatum ponatur de exteriori terminatam portionem abſcindere.</s>
            <s xml:id="echoid-s8389" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8390" xml:space="preserve">Itaque, cum in vtroque caſu demonſtratum ſit, planum inclinatum tran-
              <lb/>
            ſiens per A F, & </s>
            <s xml:id="echoid-s8391" xml:space="preserve">G L, de interiori ſolido G H I aliquam portionem ſecare,
              <lb/>
            poſſibile erit ipſi plano, hoc eſt baſibus vtriuſque portionis, aliud
              <note symbol="c" position="right" xlink:label="note-0301-03" xlink:href="note-0301-03a" xml:space="preserve">69. h.</note>
            æquidiſtans ducere, quod interioris portionis ſuperficiem contingat: </s>
            <s xml:id="echoid-s8392" xml:space="preserve">quare
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            ſi mente concipiatur iam hoc ductum eſſe, ac vndique productum, patet hoc
              <lb/>
            ipſum planum contingens, de prædicta exteriori portione dempta à plano
              <lb/>
            per A F, & </s>
            <s xml:id="echoid-s8393" xml:space="preserve">G L ducto, aliam portionem abſcindere, ſed illa omninò mi-
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            norem (pars enim ſuo toto minor eſt) at hęc minor portio æqualis eſt
              <note symbol="d" position="right" xlink:label="note-0301-04" xlink:href="note-0301-04a" xml:space="preserve">ex Sch.
                <lb/>
              Prop. 80.
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              huius.</note>
            tioni A B F abſciſſæ à plano, quod per A F ductum fuit ad planum per axem
              <lb/>
            A B C rectum (vtraque enim talium portionum terminatur à planis </s>
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