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

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        <div xml:id="echoid-div448" type="section" level="1" n="186">
          <p>
            <s xml:id="echoid-s4601" xml:space="preserve">
              <pb o="137" file="0161" n="161" rhead=""/>
            DA, & </s>
            <s xml:id="echoid-s4602" xml:space="preserve">in triangulo LAI rectangulum LIR æquale quadrato IA. </s>
            <s xml:id="echoid-s4603" xml:space="preserve">Quod
              <lb/>
            ſerua.</s>
            <s xml:id="echoid-s4604" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4605" xml:space="preserve">Iam ſi ſectio primæ figure ABC fuerit Parabole, cum AE ſit ei contingens
              <lb/>
            erit EB æqualis BF, ergo rectangulum EDF cum quadrato FB
              <note symbol="*" position="right" xlink:label="note-0161-01" xlink:href="note-0161-01a" xml:space="preserve">20. pr.
                <lb/>
              conic.</note>
            quadrato BD, quare
              <lb/>
              <figure xlink:label="fig-0161-01" xlink:href="fig-0161-01a" number="127">
                <image file="0161-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0161-01"/>
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            ſolum rectangulũ EDF,
              <lb/>
            ſiue quadratum DA mi-
              <lb/>
            nus erit quadrato DB,
              <lb/>
            ſiue linea D A minor
              <lb/>
            DB.</s>
            <s xml:id="echoid-s4606" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4607" xml:space="preserve">Siverò eadem figura
              <lb/>
            Hyperbolen reprefen-
              <lb/>
            tet, reperto eius centro
              <lb/>
            Q, erit rectangulum
              <lb/>
            FQE ęquale
              <note symbol="a" position="right" xlink:label="note-0161-02" xlink:href="note-0161-02a" xml:space="preserve">37. pri-
                <lb/>
              mi conic.</note>
            QB, ergo FQ ad QB, vt
              <lb/>
            QB ad QE, vel vt
              <note symbol="b" position="right" xlink:label="note-0161-03" xlink:href="note-0161-03a" xml:space="preserve">Coroll.
                <lb/>
              12. h.</note>
            ad BE, ſed FQ maior eſt QB, ergo FB erit maior BE, ſiue pluſquam dimi-
              <lb/>
            dium ipſa FE, diuiſa ergo FE bifariam in V, erit FV minor FB, eritque re-
              <lb/>
            ctangulum EDF cum quadrato FV æquale quadrato DV, igitur ſolum re-
              <lb/>
            ctangulum EDF, hoc eſt quadratum DA minus quadrato DV, ſeu linea DA
              <lb/>
            minor DV, & </s>
            <s xml:id="echoid-s4608" xml:space="preserve">eò minor ipſa DB.</s>
            <s xml:id="echoid-s4609" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4610" xml:space="preserve">Amplius in Ellipſi ſecundæ figuræ, dum perpendicularis AD conuenit
              <lb/>
            cum axe maiori, eſt rectangulum ENF æquale quadrato NB, & </s>
            <s xml:id="echoid-s4611" xml:space="preserve">à
              <note symbol="c" position="right" xlink:label="note-0161-04" xlink:href="note-0161-04a" xml:space="preserve">37. pri-
                <lb/>
              mi conic.</note>
            proportionali NF dempta eſt pars ND, ergo per Lemma præcedens erit re-
              <lb/>
            ctangulum EDF, ſiue quadratum DA minus quadrato DB, hoc eſt perpen-
              <lb/>
            dicularis DA maiori axi occurrens, minor eiuſdem axis portione DB.</s>
            <s xml:id="echoid-s4612" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4613" xml:space="preserve">Tandem rectangulum LNR æquatur quadrato NH, & </s>
            <s xml:id="echoid-s4614" xml:space="preserve">tertiæ proportio-
              <lb/>
            nali NR addita eſt NI, ergo per idem Lemma erit rectangulum LIR, ſiue
              <lb/>
            quadratum IA maius quadrato IH, ſiue perpendicularis AI minori axi oc-
              <lb/>
            currens maior eiuſdem axis portione HI. </s>
            <s xml:id="echoid-s4615" xml:space="preserve">Quod fuit, &</s>
            <s xml:id="echoid-s4616" xml:space="preserve">c.</s>
            <s xml:id="echoid-s4617" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div454" type="section" level="1" n="187">
          <head xml:id="echoid-head192" xml:space="preserve">THEOR. XLIV. PROP. XC.</head>
          <p>
            <s xml:id="echoid-s4618" xml:space="preserve">Si quamcunque coni-ſectionem recta linea contingat ad pun-
              <lb/>
            ctum, quod non ſit axis vertex, à quo ductæ ſint duæ rectæ lineæ,
              <lb/>
            altera contingenti, altera autem axi perpendicularis; </s>
            <s xml:id="echoid-s4619" xml:space="preserve">erit in Para-
              <lb/>
            bola ea axis portio inter perpendiculares inrercepta æqualis, in
              <lb/>
            Hyperbola verò maior, ſed in Ellipſi minor dimidio recti lateris
              <lb/>
            eius axis, cui perpendiculares occurrunt.</s>
            <s xml:id="echoid-s4620" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4621" xml:space="preserve">SIt quæcunque coni-ſectio ABC, cuius axis BD, vertex B, & </s>
            <s xml:id="echoid-s4622" xml:space="preserve">aliud in ea
              <lb/>
            punctum ſit A, à quo ducta ſit contingens AE cum axe
              <note symbol="d" position="right" xlink:label="note-0161-05" xlink:href="note-0161-05a" xml:space="preserve">2. 4. h.</note>
              <note symbol="e" position="right" xlink:label="note-0161-06" xlink:href="note-0161-06a" xml:space="preserve">24. 25.
                <lb/>
              pr. eonic.</note>
            in E, atque ex A erecta ſit AD ipſi AE perpendicularis (quæ cum axe con-
              <lb/>
            ueniet in D) & </s>
            <s xml:id="echoid-s4623" xml:space="preserve">AF perpendicularis ad axem. </s>
            <s xml:id="echoid-s4624" xml:space="preserve">Dico primùm in
              <note symbol="f" position="right" xlink:label="note-0161-07" xlink:href="note-0161-07a" xml:space="preserve">88. h.</note>
            primæ figuræ, interceptam axis portionem DF dimidio recti lateris æqua-
              <lb/>
            lem eſſe.</s>
            <s xml:id="echoid-s4625" xml:space="preserve"/>
          </p>
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