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

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          <head xml:id="echoid-head234" xml:space="preserve">THEOR. IV. PROP. VIII.</head>
          <p>
            <s xml:id="echoid-s5366" xml:space="preserve">MINIMA linearum ad vniuerſam Ellipſis peripheriam du-
              <lb/>
            cibilium, à puncto maioris axis, quod diſtet à vertice per in-
              <lb/>
            teruallum non maius dimidio recti lateris, eſt idem axis ſegmen-
              <lb/>
            tum, inter datum punctum, & </s>
            <s xml:id="echoid-s5367" xml:space="preserve">verticem interceptum.</s>
            <s xml:id="echoid-s5368" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5369" xml:space="preserve">Aliarum autem eductarum in minori portione Ellipſis, cuius
              <lb/>
            baſis, ſit applicata per datum punctum; </s>
            <s xml:id="echoid-s5370" xml:space="preserve">quæ cum MINIMA
              <lb/>
            minorem angulum conſtituit, minor eſt.</s>
            <s xml:id="echoid-s5371" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5372" xml:space="preserve">ESto Ellipſis A B C D, cuius axis maior A C, minor B D, centrum E,
              <lb/>
            & </s>
            <s xml:id="echoid-s5373" xml:space="preserve">latus rectum maioris axis C A ſit C F, & </s>
            <s xml:id="echoid-s5374" xml:space="preserve">regula A F: </s>
            <s xml:id="echoid-s5375" xml:space="preserve">ſegmentum
              <lb/>
            verò C G, ſit non mains
              <unsure/>
            dimidio C F. </s>
            <s xml:id="echoid-s5376" xml:space="preserve">Dico primùm G C eſſe _MINIMAM_
              <lb/>
            ducibilium ex G ad vniuerſam Ellipſis peripheriam A B C D.</s>
            <s xml:id="echoid-s5377" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5378" xml:space="preserve">Quod enim G C, licet ponatur
              <lb/>
              <figure xlink:label="fig-0191-01" xlink:href="fig-0191-01a" number="151">
                <image file="0191-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0191-01"/>
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            æqualis dimidio recti C E, ſit mi-
              <lb/>
            nor reliquo axis ſegmento G A, pa-
              <lb/>
            tet: </s>
            <s xml:id="echoid-s5379" xml:space="preserve">quoniam C A ad B D, eſt vt B D
              <lb/>
            ad C F, & </s>
            <s xml:id="echoid-s5380" xml:space="preserve">ſumptis ſubduplis, C E
              <lb/>
            ad E B, vt E B ad C G, eſtque C E
              <lb/>
            maior E B, quare E B quoque maior
              <lb/>
            eſt C G, & </s>
            <s xml:id="echoid-s5381" xml:space="preserve">eò magis A E, immò A
              <lb/>
            G maior G C.</s>
            <s xml:id="echoid-s5382" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5383" xml:space="preserve">Iam applicetur per G recta H G S,
              <lb/>
            regulæ occurrens in I. </s>
            <s xml:id="echoid-s5384" xml:space="preserve">Erit A E ad
              <lb/>
            ad A C, vt E L ad C F, ſed eſt A E
              <lb/>
            dimidia A C, quare E L recti C F
              <lb/>
            dimidia erit; </s>
            <s xml:id="echoid-s5385" xml:space="preserve">eſtque G I maior E L,
              <lb/>
            ergo G I maior eſt dimidio recti C F,
              <lb/>
            & </s>
            <s xml:id="echoid-s5386" xml:space="preserve">poſita eſt G C non maior dimidio
              <lb/>
            recti; </s>
            <s xml:id="echoid-s5387" xml:space="preserve">ergo G C erit omnino minor
              <lb/>
            G I, ſiue quadratum G C minus re-
              <lb/>
            ctangulo C G I, ſiue quadrato G H, hoc eſt linea G C minor ipſa G
              <note symbol="a" position="right" xlink:label="note-0191-01" xlink:href="note-0191-01a" xml:space="preserve">Coroll.
                <lb/>
              primę pri
                <lb/>
              mi huius.</note>
            ſed G H eſt _MINIMA_ ducibilium ex G ad peripheriam H A S, ergo GC eò ampliùs _MINIMA_ erit ad eandem maioris portionis peripheriam H A S.
              <lb/>
            </s>
            <s xml:id="echoid-s5388" xml:space="preserve"/>
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          <note symbol="b" position="right" xml:space="preserve">6. h.</note>
          <p>
            <s xml:id="echoid-s5389" xml:space="preserve">Ampliùs, ad peripheriam minoris portionis H C S ducatur quęcunque
              <lb/>
            G M, & </s>
            <s xml:id="echoid-s5390" xml:space="preserve">per M applicetur M N O. </s>
            <s xml:id="echoid-s5391" xml:space="preserve">Cum in triangulo rectangulo A C F
              <lb/>
            oſtenſa ſit C G minor quàm dimidium C A, ſed poſita ſit non maior di-
              <lb/>
            midio C F, & </s>
            <s xml:id="echoid-s5392" xml:space="preserve">ex puncto N in C G ſumpto, ducta ſit N O parallela ad
              <lb/>
            C F, erit N O maior aggregato C G cum G N, per primam partem 7. </s>
            <s xml:id="echoid-s5393" xml:space="preserve">hu-
              <lb/>
            ius; </s>
            <s xml:id="echoid-s5394" xml:space="preserve">ergo ſumpta communi altitudine N C, erit rectangulum O N C, ſiue
              <lb/>
              <note symbol="c" position="right" xlink:label="note-0191-03" xlink:href="note-0191-03a" xml:space="preserve">Coroll.
                <lb/>
              primę pri
                <lb/>
              mi huius.</note>
            quadratum M N maius rectangulo ſub C G cum G N in N C: </s>
            <s xml:id="echoid-s5395" xml:space="preserve">addito communi quadrato G N, erit quadratum M N cum quadrato N G, ſiue
              <lb/>
            vnicum quadratum G M, maius rectangulo ſub C G cum G N in N C, </s>
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