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

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          <p>
            <s xml:id="echoid-s3799" xml:space="preserve">
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            ptotalis DEF erit Hyperbolæ circumſcriptus, cum totus cadat extra, & </s>
            <s xml:id="echoid-s3800" xml:space="preserve">quę-
              <lb/>
            libet ſectionis diameter, eaſdem ipſi applicatas, ad latcra anguli productas,
              <lb/>
            bifariam ſecet: </s>
            <s xml:id="echoid-s3801" xml:space="preserve">eritque _MINIMV S_, nam
              <unsure/>
            quælibet alia linea, quæ per
              <note symbol="a" position="right" xlink:label="note-0137-01" xlink:href="note-0137-01a" xml:space="preserve">ex 8. 2.
                <lb/>
              conic.</note>
            vel per E (quod idem eſt) intra ipſum ducitur, minorem quidem cum altera
              <lb/>
              <note symbol="b" position="right" xlink:label="note-0137-02" xlink:href="note-0137-02a" xml:space="preserve">8. huius.</note>
            aſymptoto conſtituit angulum, ſed omnino ſecat Hyperbolen. </s>
            <s xml:id="echoid-s3802" xml:space="preserve">Si ſecun- dum, duci poterunt ex G Hyperbolen contingentes GA, GC, & </s>
            <s xml:id="echoid-s3803" xml:space="preserve">tunc
              <note symbol="c" position="right" xlink:label="note-0137-03" xlink:href="note-0137-03a" xml:space="preserve">49. ſec.
                <lb/>
              conic.</note>
            gulus AGC erit quæſitus circumſcriptus: </s>
            <s xml:id="echoid-s3804" xml:space="preserve">quoniam ſi iungatur AC, & </s>
            <s xml:id="echoid-s3805" xml:space="preserve">bifa-
              <lb/>
            riam ſecetur in N, iuncta GN diameter eſt ſectionis, ſimulque anguli; </s>
            <s xml:id="echoid-s3806" xml:space="preserve">
              <note symbol="d" position="right" xlink:label="note-0137-04" xlink:href="note-0137-04a" xml:space="preserve">29. ſec.
                <lb/>
              conic.</note>
            erit _MINIMV S_, vt per ſe patet, cum quæ ex G ducitur intra angulum AGC
              <lb/>
            ſecet omnino Hyperbolen. </s>
            <s xml:id="echoid-s3807" xml:space="preserve">Sitertium: </s>
            <s xml:id="echoid-s3808" xml:space="preserve">ducantur GL, GM aſymptotis ęqui-
              <lb/>
            diſtantes, & </s>
            <s xml:id="echoid-s3809" xml:space="preserve">angulus LGM erit Hyperbolæ ABC circumſcriptus, cum cir-
              <lb/>
            cumſcriptus ſit angulo aſymptotali DEF: </s>
            <s xml:id="echoid-s3810" xml:space="preserve">nam ducta GEN ſectionis diame-
              <lb/>
            tro, applicataque quacunque LDANCFM; </s>
            <s xml:id="echoid-s3811" xml:space="preserve">in triangulis LGN, MGN eſt
              <lb/>
            ND ad DL, vt NE ad EG, vel vt NF ad FM, ſuntq; </s>
            <s xml:id="echoid-s3812" xml:space="preserve"> ND, NF inter ſe
              <note symbol="e" position="right" xlink:label="note-0137-05" xlink:href="note-0137-05a" xml:space="preserve">ex 8. 2.
                <lb/>
              conic.</note>
            les, quare DL, FM ęquales erunt, & </s>
            <s xml:id="echoid-s3813" xml:space="preserve">totę NL, NM ęquales, ſiue GEN circũ-
              <lb/>
            ſcripti etiam anguli LGM diameter erit: </s>
            <s xml:id="echoid-s3814" xml:space="preserve">inſuper idem angulus LGM erit _MI-_
              <lb/>
            _NIMVS_: </s>
            <s xml:id="echoid-s3815" xml:space="preserve">nam recta, quę ex G intra ipſum ducitur, minorem angulum cum al-
              <lb/>
            tera nunc ductarum conſtituens, ſi producatur, ſecat vnam aſymptoton (cum
              <lb/>
            ei æquidiſtanter ductam ſecet in G) quare vlterius producta ſecabit
              <note symbol="f" position="right" xlink:label="note-0137-06" xlink:href="note-0137-06a" xml:space="preserve">35. h.</note>
            Hyperbolen. </s>
            <s xml:id="echoid-s3816" xml:space="preserve">Datę igitur Hyperbolę per datum extra ipſam pũctum in locis
              <lb/>
            poſſibilibus, circumſcriptus eſt _MINIMVS_ quæſitus angulus. </s>
            <s xml:id="echoid-s3817" xml:space="preserve">Quod, &</s>
            <s xml:id="echoid-s3818" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3819" xml:space="preserve"/>
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        <div xml:id="echoid-div368" type="section" level="1" n="156">
          <head xml:id="echoid-head161" xml:space="preserve">PROBL. XXVII. PROP. LXIX.</head>
          <p>
            <s xml:id="echoid-s3820" xml:space="preserve">Datę Hyperbolę, per punctum intra ipſam datum, MAXIMVM
              <lb/>
            angulum inſcribere. </s>
            <s xml:id="echoid-s3821" xml:space="preserve">Item.</s>
            <s xml:id="echoid-s3822" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3823" xml:space="preserve">Dato angulo, per punctum extra ipſum datum, cum dato ſemi-
              <lb/>
            tranſuerſo latere, MINIMAM Hyperbolen circumſcribere.</s>
            <s xml:id="echoid-s3824" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3825" xml:space="preserve">Oportet autem datum punctum eſſe in angulo, qui eſt ad verti-
              <lb/>
            cem dato.</s>
            <s xml:id="echoid-s3826" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3827" xml:space="preserve">SIt data Hyperbole ABC, cuius aſymptoti ſint DE, DF, & </s>
            <s xml:id="echoid-s3828" xml:space="preserve">punctum intra
              <lb/>
            ipſam ſit G, per quod ei oporteat _MAXIMV M_ angulum inſcribere.</s>
            <s xml:id="echoid-s3829" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3830" xml:space="preserve">Ducantur ex G rectæ GH, GI aſymptotis æquidiſtantes. </s>
            <s xml:id="echoid-s3831" xml:space="preserve">Dico angulum
              <lb/>
            HGI eſſe _MAXIMVM_ quæſitum.</s>
            <s xml:id="echoid-s3832" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s3833" xml:space="preserve">Nam iuncta DG, & </s>
            <s xml:id="echoid-s3834" xml:space="preserve">producta ad
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            L, ipſa GL neceſſariò diuidet angu-
              <lb/>
            lum HGI (vt ſatis patet) ſumptoque
              <lb/>
            in ea quolibet puncto L, & </s>
            <s xml:id="echoid-s3835" xml:space="preserve">applica-
              <lb/>
            ta in Hyperbola, ad diametrũ BL, or-
              <lb/>
            dinata ELF, Intera anguli HGI ſecã
              <lb/>
            in H, I; </s>
            <s xml:id="echoid-s3836" xml:space="preserve">erit ob triangulorum ſimili-
              <lb/>
            tudinem, DL ad LE, vt GL ad LH,
              <lb/>
            ſed DL ad LE eſt vt DL ad LF, cum
              <lb/>
            LE, LF ſint æquales, & </s>
            <s xml:id="echoid-s3837" xml:space="preserve">DL ad LF
              <lb/>
            eſt vt GL ad LI, quare GL ad LH erit vt GL ad LI, ſiue LH ęqualis LI: </s>
            <s xml:id="echoid-s3838" xml:space="preserve"/>
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