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

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        <div xml:id="echoid-div968" type="section" level="1" n="391">
          <head xml:id="echoid-head402" xml:space="preserve">PROBL. I. PROP. VI.</head>
          <p>
            <s xml:id="echoid-s9332" xml:space="preserve">Dato triangulo, cuius vnuſquiſq; </s>
            <s xml:id="echoid-s9333" xml:space="preserve">angulorum minor ſit gr. </s>
            <s xml:id="echoid-s9334" xml:space="preserve">120.
              <lb/>
            </s>
            <s xml:id="echoid-s9335" xml:space="preserve">punctum reperire, à quo ſi ad angulos tres rectę educantur, ipſarum
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            aggregatum ſit MINIMVM.</s>
            <s xml:id="echoid-s9336" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9337" xml:space="preserve">ESto triangulum A B C vt ponitur, & </s>
            <s xml:id="echoid-s9338" xml:space="preserve">inuenire oporteat punctum quale
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            imperatum eſt.</s>
            <s xml:id="echoid-s9339" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9340" xml:space="preserve">Super latera B A, B C ad partes baſis A C deſcribantur circuli portio-
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            nes A D B, C D B capientes angulos grad. </s>
            <s xml:id="echoid-s9341" xml:space="preserve">120. </s>
            <s xml:id="echoid-s9342" xml:space="preserve">ſiue æquales externo cuiuſ-
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            libet trianguli æquilateri, quarum portionum arcus omnino ſe mutuò
              <note symbol="*" position="left" xlink:label="note-0336-01" xlink:href="note-0336-01a" xml:space="preserve">5. App.</note>
            cabunt intra triangulum A B C, ſitque eorum interſectio punctum D. </s>
            <s xml:id="echoid-s9343" xml:space="preserve">Di-
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            co ipſum eſſe quæſitum.</s>
            <s xml:id="echoid-s9344" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9345" xml:space="preserve">Nam iunctis D A, D B, D C, erunt an-
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              <figure xlink:label="fig-0336-01" xlink:href="fig-0336-01a" number="267">
                <image file="0336-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0336-01"/>
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            guli A D B, C D B graduum 120. </s>
            <s xml:id="echoid-s9346" xml:space="preserve">vnde reli-
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            quus A D C, vſque ad quatuor rectorum cõ-
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            plementum item erit gr. </s>
            <s xml:id="echoid-s9347" xml:space="preserve">120. </s>
            <s xml:id="echoid-s9348" xml:space="preserve">Cum ergo tres
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            rectę D A, D B, D C ad punctum D coeun-
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            tes tres æquales angulos efficiant, cumque hi
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            ſimul ſumpti æquales ſint quatuor rectis, erit
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            ipſarum D A, D B, D C aggregatum _MINIMA_ quantitas. </s>
            <s xml:id="echoid-s9349" xml:space="preserve">Quare
              <note symbol="*" position="left" xlink:label="note-0336-02" xlink:href="note-0336-02a" xml:space="preserve">4. App.</note>
            uentum eſt punctum D, vti quærebatur. </s>
            <s xml:id="echoid-s9350" xml:space="preserve">Quod faciendum erat.</s>
            <s xml:id="echoid-s9351" xml:space="preserve"/>
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        <div xml:id="echoid-div971" type="section" level="1" n="392">
          <head xml:id="echoid-head403" xml:space="preserve">PROBL. II. PROP. VII.</head>
          <p>
            <s xml:id="echoid-s9352" xml:space="preserve">Datam rectam lineam terminatam ita diuidere, vt ſumpta par-
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            tium ipſius tertia proportionali, aggregatum extremarum ſit MI-
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            NIMA quantitas.</s>
            <s xml:id="echoid-s9353" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9354" xml:space="preserve">ESto data linea A B, quam ſecare oporteat, vt imperatum eſt.</s>
            <s xml:id="echoid-s9355" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9356" xml:space="preserve">Erigatur ex A ipſi A B perpendicularis, & </s>
            <s xml:id="echoid-s9357" xml:space="preserve">æqualis A D, iunctaq; </s>
            <s xml:id="echoid-s9358" xml:space="preserve">D
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            B ſecetur D E æqualis D A, & </s>
            <s xml:id="echoid-s9359" xml:space="preserve">ex E ſuper A B perpendicularis demitta-
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            tur E C. </s>
            <s xml:id="echoid-s9360" xml:space="preserve">Dico punctum C quæſitum ſoluere.</s>
            <s xml:id="echoid-s9361" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9362" xml:space="preserve">Nam bifariam ſecto angulo A D E per rectam D F ſecante A B in F, & </s>
            <s xml:id="echoid-s9363" xml:space="preserve">
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            iuncta F E: </s>
            <s xml:id="echoid-s9364" xml:space="preserve">cum ſit latus D A æquale D E, & </s>
            <s xml:id="echoid-s9365" xml:space="preserve">D F commune, & </s>
            <s xml:id="echoid-s9366" xml:space="preserve">anguli
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            A D F, E D F æquales, erunt baſes F A, F E æquales, & </s>
            <s xml:id="echoid-s9367" xml:space="preserve">reliquus angulus
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            F E D reliquo F A D æqualis ſiue rectus: </s>
            <s xml:id="echoid-s9368" xml:space="preserve">quare ſi cum centro F interuallo
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            F A circulus deſcribatur A E G, is tranſibit quoque per E, & </s>
            <s xml:id="echoid-s9369" xml:space="preserve">vtramque D.
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            </s>
            <s xml:id="echoid-s9370" xml:space="preserve">A, D B continget in A, E.</s>
            <s xml:id="echoid-s9371" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s9372" xml:space="preserve">Iam cum in ſemi-circulo ſit A C ad C E, vt C E ad C G, ſitque C B
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            æqualis C E (cum etiam A D ſit æqualis A B) erit A C ad C B, vt C B
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            ad C G. </s>
            <s xml:id="echoid-s9373" xml:space="preserve">Vnde aggregatum extremarum poſt ſegmenta A C, C B erit A
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            G; </s>
            <s xml:id="echoid-s9374" xml:space="preserve">quod eſſe _MINIMVM_ ſic demonſtrabitur.</s>
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