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

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        <div xml:id="echoid-div890" type="section" level="1" n="355">
          <p>
            <s xml:id="echoid-s8553" xml:space="preserve">
              <pb o="122" file="0308" n="308" rhead=""/>
            & </s>
            <s xml:id="echoid-s8554" xml:space="preserve">diameter B G erit æquo maior: </s>
            <s xml:id="echoid-s8555" xml:space="preserve">ſi igitur ipſa ad æquum reducatur in N,
              <lb/>
            ita vt, vel B N ſit æqualis ipſi E H, (dum ſolidum ſuerit Conoides Parabo-
              <lb/>
            licum,) vel ita vt B N, & </s>
            <s xml:id="echoid-s8556" xml:space="preserve">E H ad proprias ſemi- diametros ſint in eadem ra-
              <lb/>
            tione,) dum ſolidum ſuerit Hyperbolicum, vel Sphæra, aut Sphæroides;) </s>
            <s xml:id="echoid-s8557" xml:space="preserve">vel
              <lb/>
            ita vt eędem pertingant ad eandẽ ſimilem concentricam ſectionem inſcriptã;
              <lb/>
            </s>
            <s xml:id="echoid-s8558" xml:space="preserve">erit B N omnino minor B G, & </s>
            <s xml:id="echoid-s8559" xml:space="preserve">ſi per N agatur ipſi A C ęquidiſtans O N P,
              <lb/>
            quę ad eandem diametrum B G erit ordinatim ducta, atq; </s>
            <s xml:id="echoid-s8560" xml:space="preserve">minor ipſa A C,
              <lb/>
            ſiet portio, ſeu Canon O B P æqualis portioni, ſiue Canoni D E F, & </s>
            <s xml:id="echoid-s8561" xml:space="preserve">
              <note symbol="a" position="left" xlink:label="note-0308-01" xlink:href="note-0308-01a" xml:space="preserve">40. h. &
                <lb/>
              ex 45. h.</note>
            O P ſecabit B L in R, eritque B R altitudo Canonis O B P, cum ob paral-
              <lb/>
            lelas ſit angulus B R N rectus: </s>
            <s xml:id="echoid-s8562" xml:space="preserve">& </s>
            <s xml:id="echoid-s8563" xml:space="preserve">ſi per rectam O P ducatur planum O Q P,
              <lb/>
            quod baſi A I C ſit parallelum, ſiue rectum ad Canonem A B C, id abſcin-
              <lb/>
            det ex dato ſolido portionem
              <lb/>
            O B P, cuius altitudo erit B
              <lb/>
              <figure xlink:label="fig-0308-01" xlink:href="fig-0308-01a" number="249">
                <image file="0308-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0308-01"/>
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            R eadem atque Canonis O B
              <lb/>
            P. </s>
            <s xml:id="echoid-s8564" xml:space="preserve">Cumque Canon O B P
              <lb/>
            æqualis ſit Canoni D E F,
              <lb/>
            erit ſolida portio O B P
              <note symbol="b" position="left" xlink:label="note-0308-02" xlink:href="note-0308-02a" xml:space="preserve">78. h.</note>
            qualis ſolidæ portioni D E F,
              <lb/>
            ac ideo vt baſis O Q P ad ba-
              <lb/>
            ſim D K F, ita reciprocè
              <note symbol="c" position="left" xlink:label="note-0308-03" xlink:href="note-0308-03a" xml:space="preserve">ex pri-
                <lb/>
              ma parte
                <lb/>
              huius.</note>
            titudo E M ad altitudinem B
              <lb/>
            R, eſtque baſis D K F ad ba-
              <lb/>
            ſim A I C, ex hypotheſi, vt
              <lb/>
            altitudo B L ad altitudinem
              <lb/>
            E M, quare, ex æquali in ratione perturbata, erit baſis O Q P ad baſim A
              <lb/>
            I C, vt altitudo B L ad altitudinem B R, ſed eſt B L maior B R, ergo & </s>
            <s xml:id="echoid-s8565" xml:space="preserve">
              <lb/>
            baſis O Q P maior erit baſi A I C, quod eſt falſum, cum ſit minor, eò quod
              <lb/>
            O P diameter Ellipſis, aut circuli O Q P minor ſit homologa diametro A C
              <lb/>
            ſimilis Ellipſis, vel circuli A I C. </s>
            <s xml:id="echoid-s8566" xml:space="preserve">Non erit ergo Canonum A B C, D
              <note symbol="d" position="left" xlink:label="note-0308-04" xlink:href="note-0308-04a" xml:space="preserve">Coroll.
                <lb/>
              15. Arch.
                <lb/>
              de Co-
                <lb/>
              noid. &c.</note>
            F alter altero maior, quare inter ſe æquales eſſe neceſſe eſt: </s>
            <s xml:id="echoid-s8567" xml:space="preserve">ideoque, & </s>
            <s xml:id="echoid-s8568" xml:space="preserve">
              <lb/>
            portiones ſolidæ A B C, D E F ęquales erunt. </s>
            <s xml:id="echoid-s8569" xml:space="preserve">Quod ſecundò
              <note symbol="e" position="left" xlink:label="note-0308-05" xlink:href="note-0308-05a" xml:space="preserve">75. h.</note>
            propoſitum ſuit.</s>
            <s xml:id="echoid-s8570" xml:space="preserve"/>
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        <div xml:id="echoid-div895" type="section" level="1" n="356">
          <head xml:id="echoid-head365" xml:space="preserve">THEOR. LVIII. PROP. LXXXVIII.</head>
          <p>
            <s xml:id="echoid-s8571" xml:space="preserve">Æquales portiones ſolidæ de eodem quocunque Conoide, aut
              <lb/>
            Sphæra, aut Sphæroide ad ſibi inſcriptam Coni portionem, vel ad
              <lb/>
            circumſcriptum Cylindricum, vnam, eandemque ſimul habent
              <lb/>
            rationem.</s>
            <s xml:id="echoid-s8572" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8573" xml:space="preserve">ETenim huiuſmodi portiones habent baſes altitudinibus reciprocè pro-
              <lb/>
            portionales, vt in præcedenti, primo loco demonſtratum eſt, ſed ba-
              <lb/>
            ſes, & </s>
            <s xml:id="echoid-s8574" xml:space="preserve">altitudines portionum eædem ſunt, ac baſes, & </s>
            <s xml:id="echoid-s8575" xml:space="preserve">altitudines inſcripta-
              <lb/>
            rum Coniportionum, quare, & </s>
            <s xml:id="echoid-s8576" xml:space="preserve">Coni portionum baſes ipſarum altitudini-
              <lb/>
            bus erunt reciprocè proportionales, ſed eædem portiones Conorum </s>
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