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

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          <p>
            <s xml:id="echoid-s9281" xml:space="preserve">
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            I D, I E, I F, I G ex aſſumpto puncto I ad datarum terminos B, C, D, E,
              <lb/>
            F, G; </s>
            <s xml:id="echoid-s9282" xml:space="preserve">itaque aggregatum ductarum ex A æquales angulos inter ſe efficien-
              <lb/>
            tes, & </s>
            <s xml:id="echoid-s9283" xml:space="preserve">quatuor rectos ſimul complentes eſt _MINIMVM_. </s>
            <s xml:id="echoid-s9284" xml:space="preserve">Quod erat, &</s>
            <s xml:id="echoid-s9285" xml:space="preserve">c.</s>
            <s xml:id="echoid-s9286" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9287" xml:space="preserve">Hinc ſolutio Gallici Problematis, ſequenti Lemmate præoſtenſo.</s>
            <s xml:id="echoid-s9288" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div966" type="section" level="1" n="390">
          <head xml:id="echoid-head401" xml:space="preserve">LEMMA III. PROP. V.</head>
          <p>
            <s xml:id="echoid-s9289" xml:space="preserve">Si in triangulo A B C fuerit angulus A B C, minor grad. </s>
            <s xml:id="echoid-s9290" xml:space="preserve">120.
              <lb/>
            </s>
            <s xml:id="echoid-s9291" xml:space="preserve">& </s>
            <s xml:id="echoid-s9292" xml:space="preserve">ſuper latera B A, B C deſcribantur ad partes baſis A C ſimiles
              <lb/>
            circuli portiones A E B, C D B capientes angulos graduum 120. </s>
            <s xml:id="echoid-s9293" xml:space="preserve">
              <lb/>
            Dico ipſarum peripherias ſe mutuò ſecare, atque omnino intra
              <lb/>
            triangulum A B C.</s>
            <s xml:id="echoid-s9294" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9295" xml:space="preserve">NOn enim ſe contingunt in B: </s>
            <s xml:id="echoid-s9296" xml:space="preserve">quoniam ducta ex B recta F B G vnam
              <lb/>
            harum portionum peripheriam contingente, ipſa, & </s>
            <s xml:id="echoid-s9297" xml:space="preserve">alteram quoque
              <lb/>
            continget: </s>
            <s xml:id="echoid-s9298" xml:space="preserve">quare angulas G B A à contingente, & </s>
            <s xml:id="echoid-s9299" xml:space="preserve">ſecante confectus equa-
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            lis erit ei, qui ſit in alterna portione A E B, nempe erit gr. </s>
            <s xml:id="echoid-s9300" xml:space="preserve">120. </s>
            <s xml:id="echoid-s9301" xml:space="preserve">& </s>
            <s xml:id="echoid-s9302" xml:space="preserve">ob eandem
              <lb/>
            rationem angulus G B C erit grad. </s>
            <s xml:id="echoid-s9303" xml:space="preserve">120. </s>
            <s xml:id="echoid-s9304" xml:space="preserve">vnde reliquus A B C, è quatuor
              <lb/>
            rectis, erit pariter gr. </s>
            <s xml:id="echoid-s9305" xml:space="preserve">120. </s>
            <s xml:id="echoid-s9306" xml:space="preserve">quod eſt contra hypotheſim, cum ſit minor.</s>
            <s xml:id="echoid-s9307" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9308" xml:space="preserve">Nec autem ſe ſecant extra trian-
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              <figure xlink:label="fig-0335-01" xlink:href="fig-0335-01a" number="266">
                <image file="0335-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0335-01"/>
              </figure>
            gulum ad partes G, vt in G: </s>
            <s xml:id="echoid-s9309" xml:space="preserve">nam
              <lb/>
            ducta G B, eſſet angulus G B A mi-
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            noreo, qui fit à contingente ex B cũ
              <lb/>
            ſecante B A, ſiue minor facto in al-
              <lb/>
            terna portione A E B, qui eſt grad.
              <lb/>
            </s>
            <s xml:id="echoid-s9310" xml:space="preserve">120. </s>
            <s xml:id="echoid-s9311" xml:space="preserve">itemque G B C minor eſſet gr. </s>
            <s xml:id="echoid-s9312" xml:space="preserve">
              <lb/>
            120. </s>
            <s xml:id="echoid-s9313" xml:space="preserve">quare reliquus A B C è grad. </s>
            <s xml:id="echoid-s9314" xml:space="preserve">
              <lb/>
            360. </s>
            <s xml:id="echoid-s9315" xml:space="preserve">maior eſſet omnino 120. </s>
            <s xml:id="echoid-s9316" xml:space="preserve">quod
              <lb/>
            item eſt contra hypotheſim, cum ſit
              <lb/>
            minor; </s>
            <s xml:id="echoid-s9317" xml:space="preserve">quapropter huiuſmodi peri-
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            pherias ſe mutuò ſecare infra B ad
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            partes baſis A C neceſſe eſt.</s>
            <s xml:id="echoid-s9318" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9319" xml:space="preserve">Verùm ipſarum interſectio haud fiet in baſi A C, nec infra, quoniam
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            ſi in ipſa baſi A C, vt in F, eſſet, ex conſtructione, angulus A F B grad.
              <lb/>
            </s>
            <s xml:id="echoid-s9320" xml:space="preserve">120. </s>
            <s xml:id="echoid-s9321" xml:space="preserve">ſiue maior recto, & </s>
            <s xml:id="echoid-s9322" xml:space="preserve">C F B pariter maior recto; </s>
            <s xml:id="echoid-s9323" xml:space="preserve">ex quo duo ſimul A F
              <lb/>
            B, C F B eſſent duobus rectis maiores; </s>
            <s xml:id="echoid-s9324" xml:space="preserve">quod eſt abſurdum, cum duos re-
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            ctos adæquent.</s>
            <s xml:id="echoid-s9325" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9326" xml:space="preserve">Sitandem eædem peripheriæ ſe mutuò ſecarent infra baſim A C, vt in H;
              <lb/>
            </s>
            <s xml:id="echoid-s9327" xml:space="preserve">iunctis A H, C H, eſſent pariter, ex conſtructione, duo ſimul anguli A H B,
              <lb/>
            C H B, ſiue vnicus A H C maior duobus rectis; </s>
            <s xml:id="echoid-s9328" xml:space="preserve">quod eſt falſum cum ipſe à
              <lb/>
            duobus rectis deficiat per aggregatum duorum angulorum A C H, C A H. </s>
            <s xml:id="echoid-s9329" xml:space="preserve">
              <lb/>
            Quamobrem huiuſmodi ſimilium portionum peripherię neceſſariò ſe mutuò
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            ſecabunt, atque intra triangulum A B C|. </s>
            <s xml:id="echoid-s9330" xml:space="preserve">Quod demonſtrandum erat.</s>
            <s xml:id="echoid-s9331" xml:space="preserve"/>
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