Viviani, Vincenzo, De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei
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            <s xml:id="echoid-s3838" xml:space="preserve">
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            BGL erit diameter, tum datæ Hyperbolæ, tum deſcripti anguli; </s>
            <s xml:id="echoid-s3839" xml:space="preserve">& </s>
            <s xml:id="echoid-s3840" xml:space="preserve">cum GH,
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            GI, per punctum G, in Hyperbola ſumptum, ductæ ſint aſymptotis æquidi-
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            ſtantes, ipſæ, ad partes B productæ, Hyperbolæ occurrent, cum aſymptotos
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            ſecent, ſed ad partes H, I, nunquam cum ſectione conuenient, at
              <note symbol="a" position="left" xlink:label="note-0138-01" xlink:href="note-0138-01a" xml:space="preserve">Coroll.
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              11. huius.</note>
            que ducatur ex G extra angulum HGI, ſecabit producta alteram aſympto-
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            ton (cum ſecet in G ipſi parallelam GH) ac ideò priùs Hyperbolen datam:
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            </s>
            <s xml:id="echoid-s3841" xml:space="preserve">eſt igitur HGI _MAXIMV S_ inſcriptus angulus, vti quærebatur. </s>
            <s xml:id="echoid-s3842" xml:space="preserve">Quod primò
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            erat, &</s>
            <s xml:id="echoid-s3843" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3844" xml:space="preserve"/>
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            <s xml:id="echoid-s3845" xml:space="preserve">IAM ſit datus angulus HGI, & </s>
            <s xml:id="echoid-s3846" xml:space="preserve">datum punctum ſit B, in angulo tamen, qui
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            ei eſt ad verticem: </s>
            <s xml:id="echoid-s3847" xml:space="preserve">oportet per B _MINIMAM_ Hyperbolen circumſcribe-
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            re, cum dato ſemi- tranſuerſo R.</s>
            <s xml:id="echoid-s3848" xml:space="preserve"/>
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            <s xml:id="echoid-s3849" xml:space="preserve">Iungatur GB, & </s>
            <s xml:id="echoid-s3850" xml:space="preserve">producatur, ſu-
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              <figure xlink:label="fig-0138-01" xlink:href="fig-0138-01a" number="104">
                <image file="0138-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0138-01"/>
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            maturque B D æqualis R; </s>
            <s xml:id="echoid-s3851" xml:space="preserve">& </s>
            <s xml:id="echoid-s3852" xml:space="preserve">per D
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            agantur DE, DF, ipſis GH, GI pa-
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            rallelæ, & </s>
            <s xml:id="echoid-s3853" xml:space="preserve">per B cum afymptotis
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            DE, DF deſcribatur
              <note symbol="b" position="left" xlink:label="note-0138-02" xlink:href="note-0138-02a" xml:space="preserve">4. ſec.
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              conic.</note>
            ABC, Dico hanc eſſe _MINIMAM_
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            circumſcriptam quæſitam.</s>
            <s xml:id="echoid-s3854" xml:space="preserve"/>
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            <s xml:id="echoid-s3855" xml:space="preserve">Nam eadem ratione, vt ſupra.</s>
            <s xml:id="echoid-s3856" xml:space="preserve">,
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            oſtendetur DBGL eſſe diametrum
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            ſectionis, & </s>
            <s xml:id="echoid-s3857" xml:space="preserve">dati anguli, & </s>
            <s xml:id="echoid-s3858" xml:space="preserve">rectas
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            GH, GI, ad partes H, I productas (cum aſymptotis æquidiſtent) nunquam
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            cum ſectione conuenire; </s>
            <s xml:id="echoid-s3859" xml:space="preserve">ideoque Hyperbolen ABC dato angulo eſſe circũ-
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            ſcriptam: </s>
            <s xml:id="echoid-s3860" xml:space="preserve">ſed eſt quoque _MINIMA_, quoniam, quæ cum eodem tranſuerſo
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            adſcribitur, ſed cum recto maiori, maioreſt ipſa ABC; </s>
            <s xml:id="echoid-s3861" xml:space="preserve">quæ verò cum
              <note symbol="c" position="left" xlink:label="note-0138-03" xlink:href="note-0138-03a" xml:space="preserve">2. Co-
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              roll. 19. h.</note>
            cto minori, qualis eſt MBN, eſt quidem minor ABC, ſed omnino ſecat
              <note symbol="d" position="left" xlink:label="note-0138-04" xlink:href="note-0138-04a" xml:space="preserve">ibidem.</note>
            tera dati anguli: </s>
            <s xml:id="echoid-s3862" xml:space="preserve">quoniam ducta DO, quæ ſit aſymptotos inſcriptæ MBN,
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            ipſa cadet infra DE, ſed eam ſecat in D, quare producta, alteram
              <note symbol="e" position="left" xlink:label="note-0138-05" xlink:href="note-0138-05a" xml:space="preserve">ex 37. h.</note>
            lam ſecabit GH, ſed DO tota cadit extra BM, vnde occurſus DO cum GH
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            erit extra BM; </s>
            <s xml:id="echoid-s3863" xml:space="preserve">ſiue GH neceſſariò ſecabit priùs ſectionem BM. </s>
            <s xml:id="echoid-s3864" xml:space="preserve">Eſt igitur ſe-
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            ctio MBN _MINIMA_ circumſcripta quæſita dato angulo HGI, per datum
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            punctum B, & </s>
            <s xml:id="echoid-s3865" xml:space="preserve">cum dato ſemi-tranſuerſo R. </s>
            <s xml:id="echoid-s3866" xml:space="preserve">Quod vltimò faciendum erat.</s>
            <s xml:id="echoid-s3867" xml:space="preserve"/>
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          <head xml:id="echoid-head162" xml:space="preserve">PROBL. XXVIII. PROP. LXX.</head>
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            <s xml:id="echoid-s3868" xml:space="preserve">Dato angulo rectilineo, per punctum intra ipſum datum, cum
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            dato tranſuerſo latere, MAXIMAM Ellipſim inſcribere: </s>
            <s xml:id="echoid-s3869" xml:space="preserve">& </s>
            <s xml:id="echoid-s3870" xml:space="preserve">è con-
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            tra.</s>
            <s xml:id="echoid-s3871" xml:space="preserve"/>
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            <s xml:id="echoid-s3872" xml:space="preserve">SIt datus angulus ABC, & </s>
            <s xml:id="echoid-s3873" xml:space="preserve">punctum intra ipſum ſit D, per quod ei opor-
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            teat, cum dato tranſuerſo R, _MAXIMAM_ Ellipſim inſcribere.</s>
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            <s xml:id="echoid-s3875" xml:space="preserve">Sumatur DE ęqualis R, & </s>
            <s xml:id="echoid-s3876" xml:space="preserve">per D, & </s>
            <s xml:id="echoid-s3877" xml:space="preserve">E in angulo ABC applicentur
              <note symbol="a" position="left" xlink:label="note-0138-06" xlink:href="note-0138-06a" xml:space="preserve">66. h.</note>
            HEL, & </s>
            <s xml:id="echoid-s3878" xml:space="preserve">iungatur HG diametrum ſecans in M, per quod applicetur AMD,
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            & </s>
            <s xml:id="echoid-s3879" xml:space="preserve">circa diametrum DE, ac per terminos applicatæ AC, deſcribatur
              <note symbol="b" position="left" xlink:label="note-0138-07" xlink:href="note-0138-07a" xml:space="preserve">Coroll.
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              57. h.</note>
            DAEC. </s>
            <s xml:id="echoid-s3880" xml:space="preserve">Dico hanc eſſe _MAXIMAM_ quæſitam. </s>
            <s xml:id="echoid-s3881" xml:space="preserve">Quoniam, ipſa ADCE eſt
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            menſali HFGL, ſiue dato angulo inſcripta, & </s>
            <s xml:id="echoid-s3882" xml:space="preserve">quælibet alia Ellipſis
              <note symbol="c" position="left" xlink:label="note-0138-08" xlink:href="note-0138-08a" xml:space="preserve">Schol.
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              62. h.</note>
            </s>
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