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

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            <s xml:id="echoid-s3346" xml:space="preserve">
              <pb o="100" file="0124" n="124" rhead=""/>
            occurret, ſi ergo ipſa DL producatur, omnino ſecabit Hyperbolen
              <note symbol="a" position="left" xlink:label="note-0124-01" xlink:href="note-0124-01a" xml:space="preserve">35. h.</note>
            ſed DL tota cadit extra ſectionem ABC, cum ſit eius aſymptotos, quare
              <lb/>
            occurſus rectæ DL, cum ſectione EN, cadet extra ABC, ac ideò EN ſecabit
              <lb/>
            priùs circumſcriptam ABC: </s>
            <s xml:id="echoid-s3347" xml:space="preserve">vnde ſectio HEK eſt _MAXIMA_ inſcripta quæſi-
              <lb/>
            ta, cum dato recto EF. </s>
            <s xml:id="echoid-s3348" xml:space="preserve">Quod primò erat, &</s>
            <s xml:id="echoid-s3349" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3350" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3351" xml:space="preserve">IAM oporteat datæ Hyperbolę HEK, cuius aſymptoti IM, IQ, per datum
              <lb/>
            extra ipſam punctum B, quod (per ea, quæ in 53. </s>
            <s xml:id="echoid-s3352" xml:space="preserve">huius) ſit vel in angulo
              <lb/>
            ad verticem aſymptotalis, vt in prima figura, vel in ipſo aſymptotali MIQ,
              <lb/>
            vt in ſecunda, cum dato recto latere _MINIM AM_ Hyperbolen circumſcri-
              <lb/>
            bere.</s>
            <s xml:id="echoid-s3353" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3354" xml:space="preserve">Iungatur BI, & </s>
            <s xml:id="echoid-s3355" xml:space="preserve">
              <lb/>
              <figure xlink:label="fig-0124-01" xlink:href="fig-0124-01a" number="89">
                <image file="0124-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0124-01"/>
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            producatur vſque oc-
              <lb/>
            currat datæ ſectioni
              <lb/>
            HEK in E; </s>
            <s xml:id="echoid-s3356" xml:space="preserve">erit I E,
              <lb/>
            ipſius ſemi-tranſuer-
              <lb/>
            ſum, cuius rectum la-
              <lb/>
            tus ſit EF, & </s>
            <s xml:id="echoid-s3357" xml:space="preserve">ex B cõ-
              <lb/>
            cipiatur adſcribi Hy-
              <lb/>
            perbole TBV ſimilis,
              <lb/>
            & </s>
            <s xml:id="echoid-s3358" xml:space="preserve">concentrica datæ
              <lb/>
            HEK, cuius rectum
              <lb/>
            ſit BS; </s>
            <s xml:id="echoid-s3359" xml:space="preserve">& </s>
            <s xml:id="echoid-s3360" xml:space="preserve">datũ rectum
              <lb/>
            BR, in caſu primæ fi-
              <lb/>
            guræ (in quo datum punctum B cadit in angulo ad verticem aſymptotalis
              <lb/>
            MIQ) ſit cuiuslibet longitudinis; </s>
            <s xml:id="echoid-s3361" xml:space="preserve">in ſecundo verò non ſit minus BS, & </s>
            <s xml:id="echoid-s3362" xml:space="preserve">per B
              <lb/>
            cum recto BR adſcribatur Hyperbole ABC ſimilis datæ HEK, quæ item ſi-
              <lb/>
            milis erit TBV, & </s>
            <s xml:id="echoid-s3363" xml:space="preserve">ſit eius centrum D: </s>
            <s xml:id="echoid-s3364" xml:space="preserve">erit ergo in ſecunda figura, ob Hyper-
              <lb/>
            bolarum ABC, TBV ſimilitudinem, rectum BR ad BS vt ſemi- tranſuerſum
              <lb/>
            BD ad ſemi-tranſuerſum BI, eſtq; </s>
            <s xml:id="echoid-s3365" xml:space="preserve">BR non minus BS, quare BD erit non minus
              <lb/>
            BD; </s>
            <s xml:id="echoid-s3366" xml:space="preserve">ex quo centrum D ſectionis ABC, vel cadet in I, vel ſupra I centrum
              <lb/>
            ſimilis ſectionis HEK: </s>
            <s xml:id="echoid-s3367" xml:space="preserve">vnde ipſa ABC erit omnino datæ HEK
              <note symbol="b" position="left" xlink:label="note-0124-02" xlink:href="note-0124-02a" xml:space="preserve">48. h.</note>
            pta.</s>
            <s xml:id="echoid-s3368" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3369" xml:space="preserve">Dicotandem ipſam ABC eſſe _MINIM AM_ quæſitam: </s>
            <s xml:id="echoid-s3370" xml:space="preserve">Quoniam alia Hy-
              <lb/>
            perbole, quæ per B adſcribitur, cum eodem recto BR, ſed cum ſemi-tranſ-
              <lb/>
            uerſo, quod minus ſit BD, eſt maior ipſa ABC; </s>
            <s xml:id="echoid-s3371" xml:space="preserve">quæ verò cum eodem
              <note symbol="c" position="left" xlink:label="note-0124-03" xlink:href="note-0124-03a" xml:space="preserve">3. Co-
                <lb/>
              19. huius.</note>
            cto BR, & </s>
            <s xml:id="echoid-s3372" xml:space="preserve">cum ſemi-tranſuerſo BX, quod excedat BD, qualis dicatur eſſe
              <lb/>
            ſectio TBV, eſt quidem minor eadem ABC, ſed omnino ſecat datã KEH.</s>
            <s xml:id="echoid-s3373" xml:space="preserve">
              <note symbol="d" position="left" xlink:label="note-0124-04" xlink:href="note-0124-04a" xml:space="preserve">ibidem.</note>
            Ductis enim ſimilium Hyperbolarum ABC, HEK aſymptotis DL, IM; </s>
            <s xml:id="echoid-s3374" xml:space="preserve">ipſę
              <lb/>
            erunt inter ſe parallelæ; </s>
            <s xml:id="echoid-s3375" xml:space="preserve">ductaque XY aſymptoto ſectionis TBV; </s>
            <s xml:id="echoid-s3376" xml:space="preserve">cum ſint
              <lb/>
            Hyperbole ABC, TBV per eundem verticem B adſcriptæ, cum eodem re-
              <lb/>
            cto BR earum aſymptoti DL, XY infra contingentem ex vertice B ſe mutuò
              <lb/>
            ſecabunt, & </s>
            <s xml:id="echoid-s3377" xml:space="preserve">cum XY ſecet DL, & </s>
            <s xml:id="echoid-s3378" xml:space="preserve">alteram huic æquidiſtantem IM ſecabit;</s>
            <s xml:id="echoid-s3379" xml:space="preserve">
              <note symbol="e" position="left" xlink:label="note-0124-05" xlink:href="note-0124-05a" xml:space="preserve">Coroll.
                <lb/>
              36. huius.</note>
            ſed eſt IM aſymptotos HEK, vnde XY producta ſecabit quidem HEK,
              <note symbol="f" position="left" xlink:label="note-0124-06" xlink:href="note-0124-06a" xml:space="preserve">35. h.</note>
            XY tota cadit extra TBV, cũ ſit eius aſymptotos; </s>
            <s xml:id="echoid-s3380" xml:space="preserve">quare XY conueniet cum
              <lb/>
            ſectione HEK, extra Hyperbolen TBV, vnde ipſa TBV ſecabit priùs inſcri-
              <lb/>
            ptam ſectionem HEK. </s>
            <s xml:id="echoid-s3381" xml:space="preserve">Quapropter ſectio ABC eſt _MINIMA_ circumſcripta
              <lb/>
            quæſita: </s>
            <s xml:id="echoid-s3382" xml:space="preserve">cum dato recto BR. </s>
            <s xml:id="echoid-s3383" xml:space="preserve">Quod ſecundò faciendum, ac demonſtrandum
              <lb/>
            erat.</s>
            <s xml:id="echoid-s3384" xml:space="preserve"/>
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