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

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          <head xml:id="echoid-head36" xml:space="preserve">PROBL. IV. PROP. VI.</head>
          <p>
            <s xml:id="echoid-s672" xml:space="preserve">Data in quodam plano recta linea terminata, quæ ad alteram
              <lb/>
              <note position="right" xlink:label="note-0037-01" xlink:href="note-0037-01a" xml:space="preserve">Prop. 53.
                <lb/>
              primico-
                <lb/>
              nic.</note>
            partem in infinitum producatur: </s>
            <s xml:id="echoid-s673" xml:space="preserve">inuenire in dato plano coni-ſe-
              <lb/>
            ctionem, quę dicitur Hyperbole, cuius diameter ſit producta linea,
              <lb/>
            vertex eius terminus, tranſuerſum latus ſit data linea terminata, re-
              <lb/>
            ctum verò ſit alia quæcunque data linea finita, & </s>
            <s xml:id="echoid-s674" xml:space="preserve">ad ipſius diametr@
              <lb/>
            ordinatim ductæ efficiant angulos dato angulo æquales.</s>
            <s xml:id="echoid-s675" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s676" xml:space="preserve">SInt datæ rectæ lineæ terminatæ AB, BC, quæ in ſubiecto plano ad angu-
              <lb/>
            lum ABC, dato angulo P æqualem conſtituantur, & </s>
            <s xml:id="echoid-s677" xml:space="preserve">harum altera AB
              <lb/>
            ſit vtcunque producta ad BD: </s>
            <s xml:id="echoid-s678" xml:space="preserve">oportet in ſubiecto plano Hyperbolen deſcri-
              <lb/>
            bere, cuius diameter ſit BD, vertex B, tranſuerſum latus AB rectum BC, & </s>
            <s xml:id="echoid-s679" xml:space="preserve">
              <lb/>
            ordinatim ductæ ad diametrō BD conſtituant angulos, dato, angulo P æquales.</s>
            <s xml:id="echoid-s680" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s681" xml:space="preserve">Iungatur AC, & </s>
            <s xml:id="echoid-s682" xml:space="preserve">producatur, ſu-
              <lb/>
              <figure xlink:label="fig-0037-01" xlink:href="fig-0037-01a" number="14">
                <image file="0037-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0037-01"/>
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            maturq; </s>
            <s xml:id="echoid-s683" xml:space="preserve">in BD quodlibet punctum
              <lb/>
            D, per quod agatur in ſubiecto pla-
              <lb/>
            no recta linea DE ipſi BC parallela,
              <lb/>
            à qua, hinc inde producta, deman-
              <lb/>
            tur partes DF, DG, quæ ſint mediæ
              <lb/>
            proportionales inter BD, & </s>
            <s xml:id="echoid-s684" xml:space="preserve">DE; </s>
            <s xml:id="echoid-s685" xml:space="preserve">& </s>
            <s xml:id="echoid-s686" xml:space="preserve">
              <lb/>
            per rectam FG intelligatur planum
              <lb/>
            IFHG, diuerſum à plano, quod per
              <lb/>
            AD, & </s>
            <s xml:id="echoid-s687" xml:space="preserve">FG tranſit, quorum cõmunis
              <lb/>
            ſectio ſit recta FG, cui per D in pla-
              <lb/>
            no IFHG perpendicularis ducatur
              <lb/>
            IDH, in qua, ad partes I, ſumptum
              <lb/>
            ſit quodcunque punctum I, & </s>
            <s xml:id="echoid-s688" xml:space="preserve">fiat vt
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            ID ad DF, ita DF ad DH; </s>
            <s xml:id="echoid-s689" xml:space="preserve">& </s>
            <s xml:id="echoid-s690" xml:space="preserve">erit re-
              <lb/>
            ctangulum IDH æquale quadrato DF, uel
              <gap/>
            quadrato DG, ſed rectæ IH, FG ſe
              <lb/>
            mutuò ſecant ad rectos angulos in D, quare ſi circa IH circulus deſeribatur,
              <lb/>
            tranſibit ipſe per puncta FG. </s>
            <s xml:id="echoid-s691" xml:space="preserve">Tandem iungatur HA, & </s>
            <s xml:id="echoid-s692" xml:space="preserve">IB producatur ſe-
              <lb/>
            cans AH in L, & </s>
            <s xml:id="echoid-s693" xml:space="preserve">intelligatur conus cuius vertex L, baſis circulus I H, & </s>
            <s xml:id="echoid-s694" xml:space="preserve">cõ-
              <lb/>
            munis ſectio ſuperficiei conicæ cum ſubiecto plano ſit linea FMBNG. </s>
            <s xml:id="echoid-s695" xml:space="preserve">Dico
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            hanc eſſe quæſitam Hyperbolen.</s>
            <s xml:id="echoid-s696" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s697" xml:space="preserve">Conus enim LIH, cuius vertex L, & </s>
            <s xml:id="echoid-s698" xml:space="preserve">diameter baſis, I H, plano per axem
              <lb/>
            ſecatur triangulum facient LIH, & </s>
            <s xml:id="echoid-s699" xml:space="preserve">ſecatur altero plano (quod eſt datum pla-
              <lb/>
            num ſubiectum) ſecante baſim coni ſecundum rectam lineam F G, quæ ad
              <lb/>
            IH baſim trianguli per axem, eſt perpendicularis, & </s>
            <s xml:id="echoid-s700" xml:space="preserve">communis ſectio ſubie-
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            cti plani, & </s>
            <s xml:id="echoid-s701" xml:space="preserve">trianguli per axem, hoc eſt DB, producta ad B conuenit cum al-
              <lb/>
            tero latere HL extra verticem producto in puncto A, erit, per primam hu-
              <lb/>
            ius, ſectio FBG Hyperbole, cuius vertex B, diameter BD, & </s>
            <s xml:id="echoid-s702" xml:space="preserve">ordinatim du-
              <lb/>
            ctæ FG cum diametro BD, ad angulum FDB, angulo CBA, ſeu dato P æ-
              <lb/>
            qualem applicantur, ex conſtructione. </s>
            <s xml:id="echoid-s703" xml:space="preserve">Cumque factum ſit vt BD, ad DF
              <lb/>
            ita DF ad DE, erit rectangulum EDB æquale quadrato DF, ſiue </s>
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