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

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        <div xml:id="echoid-div214" type="section" level="1" n="98">
          <p>
            <s xml:id="echoid-s2301" xml:space="preserve">
              <pb o="66" file="0090" n="90" rhead=""/>
            gentem ex vertice ſe mutuò ſecant, (extra tamen circumſcriptam) & </s>
            <s xml:id="echoid-s2302" xml:space="preserve">aſym-
              <lb/>
            ptotos inſcriptæ ſecat Hyperbolen circumſcriptam.</s>
            <s xml:id="echoid-s2303" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div215" type="section" level="1" n="99">
          <head xml:id="echoid-head104" xml:space="preserve">THEOR. XIII. PROP. XXXVII.</head>
          <p>
            <s xml:id="echoid-s2304" xml:space="preserve">Hyperbolæ concentricæ per eundem verticem ſimul adſcriptæ,
              <lb/>
            quarum recta latera ſint inæqualia, ſunt inter ſe nunquam coeuntes,
              <lb/>
            & </s>
            <s xml:id="echoid-s2305" xml:space="preserve">ſemper magis recedentes, & </s>
            <s xml:id="echoid-s2306" xml:space="preserve">in infinitum productæ, ad interual-
              <lb/>
            lum perueniunt maius quolibet dato interuallo, & </s>
            <s xml:id="echoid-s2307" xml:space="preserve">aſymptotos in-
              <lb/>
            ſcriptæ ſecat Hyperbolen circumſcriptam.</s>
            <s xml:id="echoid-s2308" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2309" xml:space="preserve">SInt duę Hyperbolę ABC, DBE per
              <lb/>
              <figure xlink:label="fig-0090-01" xlink:href="fig-0090-01a" number="60">
                <image file="0090-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0090-01"/>
              </figure>
            eundem verticem B ſimul adſcri-
              <lb/>
            pte, quarum idem centrum ſit F, idem-
              <lb/>
            que tranſuerſum BFG, ſed tamen Hy-
              <lb/>
            perbolæ ABC rectum latus ſit BH, ma-
              <lb/>
            ius recto BI Hyperbolæ DBE. </s>
            <s xml:id="echoid-s2310" xml:space="preserve">Dico
              <lb/>
            primùm eas ſimul eſſe non coeuntes.</s>
            <s xml:id="echoid-s2311" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2312" xml:space="preserve">Cum enim Hyperbolæ DBE, ABC
              <lb/>
            ſint per verticem ſimul adſcriptæ cum
              <lb/>
            eodem tranſuerſo BG, ipſa DBE, cuius
              <lb/>
            rectum minus eſt, inſcripta erit
              <note symbol="a" position="left" xlink:label="note-0090-01" xlink:href="note-0090-01a" xml:space="preserve">2. Co-
                <lb/>
              roll. 19. h.</note>
            perbolæ ABC, cuius rectum maius eſt,
              <lb/>
            hoc eſt, ſi iſtæ ſimul in infinitum produ-
              <lb/>
            cantur, erunt ſimul non coeuntes.</s>
            <s xml:id="echoid-s2313" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2314" xml:space="preserve">Iam dico, has etiam eſſe ſemper in-
              <lb/>
            ter ſe recedentes. </s>
            <s xml:id="echoid-s2315" xml:space="preserve">Ductis enim, & </s>
            <s xml:id="echoid-s2316" xml:space="preserve">pro-
              <lb/>
            tractis regulis; </s>
            <s xml:id="echoid-s2317" xml:space="preserve">GH, GI, & </s>
            <s xml:id="echoid-s2318" xml:space="preserve">applicatis
              <lb/>
            duabus vbicunque rectis ADL, MON; </s>
            <s xml:id="echoid-s2319" xml:space="preserve">quæ regulas ſecent in Q, S, T, V,
              <lb/>
            cum ſit vt quadratum MN ad quadratũ NO, ita recta VN ad NT, vel
              <note symbol="b" position="left" xlink:label="note-0090-02" xlink:href="note-0090-02a" xml:space="preserve">6. Co-
                <lb/>
              roll. 19. h.</note>
            SL ad SQ, vel quadratum AL ad LD, erit etiam recta MN ad NO, vt AL ad
              <lb/>
            LD, & </s>
            <s xml:id="echoid-s2320" xml:space="preserve">per conuerſionem rationis, & </s>
            <s xml:id="echoid-s2321" xml:space="preserve">permutando MN ad AL, vt MO ad
              <lb/>
            AD, ſed eſt MN maior AL, quare, & </s>
            <s xml:id="echoid-s2322" xml:space="preserve">MO erit maior AD; </s>
            <s xml:id="echoid-s2323" xml:space="preserve">ſimiliter
              <note symbol="c" position="left" xlink:label="note-0090-03" xlink:href="note-0090-03a" xml:space="preserve">32. h.</note>
            ſtrabitur quamlibet aliam interceptam applicatę portionem inter Hyperbo-
              <lb/>
            las infra MO, maiorem eſſe ipſa MO, & </s>
            <s xml:id="echoid-s2324" xml:space="preserve">hoc ſemper, quare huiuſmodi Hy-
              <lb/>
            perbolæ ſunt ſemper inter ſe recedentes. </s>
            <s xml:id="echoid-s2325" xml:space="preserve">Quod ſecundò, &</s>
            <s xml:id="echoid-s2326" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2327" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2328" xml:space="preserve">Ampliùs dico, has ſectiones in infinitum productas, aliquando perueni-
              <lb/>
            re, ad interuallum maius quolibet dato interuallo X. </s>
            <s xml:id="echoid-s2329" xml:space="preserve">Hoc autem, eadem
              <lb/>
            penitùs arte, ac in 33. </s>
            <s xml:id="echoid-s2330" xml:space="preserve">huius fieri poſſe demonſtrabitur. </s>
            <s xml:id="echoid-s2331" xml:space="preserve">Quod tertiò, &</s>
            <s xml:id="echoid-s2332" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2333" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2334" xml:space="preserve">Tandem ſit FP aſymptotos inſcriptæ DBC, & </s>
            <s xml:id="echoid-s2335" xml:space="preserve">FR aſymptotos circumſcri-
              <lb/>
            pte, & </s>
            <s xml:id="echoid-s2336" xml:space="preserve">contingens HB producatur, vtranque aſymptoton ſecans in P, R: </s>
            <s xml:id="echoid-s2337" xml:space="preserve">erit
              <lb/>
            ergo quadratum BP ęquale quartę parti figurę GBI, & </s>
            <s xml:id="echoid-s2338" xml:space="preserve">quadratũ BR
              <note symbol="d" position="left" xlink:label="note-0090-04" xlink:href="note-0090-04a" xml:space="preserve">8. huius.</note>
            parti figuræ GBH, ſed rectangulum GBI maius eſt rectangulo GBH, cum ſit
              <lb/>
            BI minor BH, ergo BP minor eſt BR; </s>
            <s xml:id="echoid-s2339" xml:space="preserve">hoc eſt FP aſymptoton inſcriptæ cadit
              <lb/>
            infra FR aſymptoton circumſcriptæ diuidens angulũ ab ipſius aſymptotis fa-
              <lb/>
            ctum, ex quo ipſa FP producta ſecabit Hyperbolen circumſcriptam ABC.</s>
            <s xml:id="echoid-s2340" xml:space="preserve">
              <note symbol="e" position="left" xlink:label="note-0090-05" xlink:href="note-0090-05a" xml:space="preserve">ibidem.</note>
            Quod erat vltimò, &</s>
            <s xml:id="echoid-s2341" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2342" xml:space="preserve"/>
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