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

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            <s xml:id="echoid-s5010" xml:space="preserve">
              <pb o="152" file="0176" n="176" rhead=""/>
            atque ex N agatur N P O ipſi G L parallela, ex O verò recta O Q R paral-
              <lb/>
            lela ad L N, & </s>
            <s xml:id="echoid-s5011" xml:space="preserve">R S A ad N O, atque A D C ipſi O R, & </s>
            <s xml:id="echoid-s5012" xml:space="preserve">hoc fiat quoties li-
              <lb/>
            buerit: </s>
            <s xml:id="echoid-s5013" xml:space="preserve">patet, ſi per puncta I, P, S, interſectionum ipſarum diagona-
              <lb/>
            lium cum diametro, agantur applicatæ T V, X Y, Z K, &</s>
            <s xml:id="echoid-s5014" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5015" xml:space="preserve">& </s>
            <s xml:id="echoid-s5016" xml:space="preserve">circa diametri
              <lb/>
            ſegmenta F M, M Q, Q D, &</s>
            <s xml:id="echoid-s5017" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5018" xml:space="preserve">& </s>
            <s xml:id="echoid-s5019" xml:space="preserve">per extrema prædictarum applicatarum
              <lb/>
            deſcribantur Ellipſes T F V M, X M Y Q, Z Q K D, has omnes eſſe Para-
              <lb/>
            bolæ A B C inſcriptas, & </s>
            <s xml:id="echoid-s5020" xml:space="preserve">ſimiles inter ſe, ac ſe mutuò ſucceſſiuè
              <note symbol="a" position="left" xlink:label="note-0176-01" xlink:href="note-0176-01a" xml:space="preserve">@. h.</note>
            gentes. </s>
            <s xml:id="echoid-s5021" xml:space="preserve">Iam dico eaſdem Ellipſes, primæ B F ſimiles eſſe, atq; </s>
            <s xml:id="echoid-s5022" xml:space="preserve">inter ſe eam
              <lb/>
            rationem habere, ac numeri quadrati diſparium numerorum ab vnitate: </s>
            <s xml:id="echoid-s5023" xml:space="preserve">ni-
              <lb/>
            mirum eſſe in progreſſione numerorum 1. </s>
            <s xml:id="echoid-s5024" xml:space="preserve">9. </s>
            <s xml:id="echoid-s5025" xml:space="preserve">25. </s>
            <s xml:id="echoid-s5026" xml:space="preserve">49. </s>
            <s xml:id="echoid-s5027" xml:space="preserve">&</s>
            <s xml:id="echoid-s5028" xml:space="preserve">c.</s>
            <s xml:id="echoid-s5029" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5030" xml:space="preserve">Quoniam igitur eſt M B ad B I, vt B I
              <note symbol="b" position="left" xlink:label="note-0176-02" xlink:href="note-0176-02a" xml:space="preserve">1. Co-
                <lb/>
              roll. 13. h.</note>
              <figure xlink:label="fig-0176-01" xlink:href="fig-0176-01a" number="141">
                <image file="0176-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0176-01"/>
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            B F, erit diuidédo M I ad I B, vt I F ad F B,
              <lb/>
            ſed eſt I F æqualis F B, ex conſtructione,
              <lb/>
            quare M I ipſi I B æqualis erit, ac ideo in
              <lb/>
            Ellipſi T F V M, erit quadratum T I ad re-
              <lb/>
            ctangulum M I F, hoc eſt rectum eius
              <note symbol="c" position="left" xlink:label="note-0176-03" xlink:href="note-0176-03a" xml:space="preserve">21. pri-
                <lb/>
              mi Conic.</note>
            ad tranſuerſum, vt idem quadratum T I, vel
              <lb/>
            rectangulum ſub I B, & </s>
            <s xml:id="echoid-s5031" xml:space="preserve">recto B E, ad
              <note symbol="d" position="left" xlink:label="note-0176-04" xlink:href="note-0176-04a" xml:space="preserve">Coroll.
                <lb/>
              1. huius.</note>
            gulum ſub eadem I B, & </s>
            <s xml:id="echoid-s5032" xml:space="preserve">ſub I F, hoc eſt vt
              <lb/>
            linea B E ad I F, (cum ſit I B communis re-
              <lb/>
            ctangulorum altitudo) vel ad ei æqualem B
              <lb/>
            F, nempe vt rectum ad tranſuerſum Ellipſis
              <lb/>
            BF: </s>
            <s xml:id="echoid-s5033" xml:space="preserve">quapropter Ellipſis B F ipſi T F V M eri
              <lb/>
            ſimilis, ſed vnaqueque aliarum inſcriptarum
              <lb/>
            Ellipſium circ? </s>
            <s xml:id="echoid-s5034" xml:space="preserve">diametri ſegmenta M Q, Q
              <lb/>
            D, &</s>
            <s xml:id="echoid-s5035" xml:space="preserve">c. </s>
            <s xml:id="echoid-s5036" xml:space="preserve">eidem T F V M eſt ſimilis, vt ſupra
              <lb/>
            monuimus, quare omnes huiuſm odi inſcri
              <lb/>
            ptæ Ellipſes erunt ſimiles inter ſe. </s>
            <s xml:id="echoid-s5037" xml:space="preserve">Et cum ſit
              <lb/>
            M I ęqualis I B, & </s>
            <s xml:id="echoid-s5038" xml:space="preserve">I B dupla F B, erit to-
              <lb/>
            ta M B quadrupla B F. </s>
            <s xml:id="echoid-s5039" xml:space="preserve">Si ergo B F conci-
              <lb/>
            piatur vt vnum, erit B I vt 2, & </s>
            <s xml:id="echoid-s5040" xml:space="preserve">M B vt 4,
              <lb/>
            atque M I vt 2, & </s>
            <s xml:id="echoid-s5041" xml:space="preserve">M F vt 3.</s>
            <s xml:id="echoid-s5042" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5043" xml:space="preserve">Cumq; </s>
            <s xml:id="echoid-s5044" xml:space="preserve">in triangulis P M N, I F G ſint anguli ad M, F inter ſe æquales,
              <lb/>
            ob æquidiſtantes applicatas M N, F G; </s>
            <s xml:id="echoid-s5045" xml:space="preserve">atque anguli ad P, I item ęquales,
              <lb/>
            ob parallelas diagonales N P, G I, erunt reliqui ad N, G pariter æquales,
              <lb/>
            ſiue ipſa triangula inter ſe ſimilia, vnde latus N M ad M P erit, vt latus G F
              <lb/>
            ad F I, & </s>
            <s xml:id="echoid-s5046" xml:space="preserve">permutando N M ad G F, vt M P ad F I, vel quadratum N M ad
              <lb/>
            G F, ſiue recta M B ad B F; </s>
            <s xml:id="echoid-s5047" xml:space="preserve">hoc eſt 4. </s>
            <s xml:id="echoid-s5048" xml:space="preserve">ad 1, vt quadratum M P ad
              <note symbol="e" position="left" xlink:label="note-0176-05" xlink:href="note-0176-05a" xml:space="preserve">20. pri-
                <lb/>
              mi Conic.</note>
            dratum F I; </s>
            <s xml:id="echoid-s5049" xml:space="preserve">vnde quadratum M P quadruplum erit quadrati F I, ſiue linea
              <lb/>
            M P dupla F I, ſiue dupla ad B F, ſed B F ponitur vt vnum, ergo M P erit
              <lb/>
            2; </s>
            <s xml:id="echoid-s5050" xml:space="preserve">eſtque B M 4, ergo B P erit 6, eſtque B M ad B P, vt eſt B P ad
              <note symbol="f" position="left" xlink:label="note-0176-06" xlink:href="note-0176-06a" xml:space="preserve">1. Co-
                <lb/>
              roll. 13. h.</note>
            Q, quare B Q erit vt 9, ſed B M eſt vt 4, ergo M Q erit 5. </s>
            <s xml:id="echoid-s5051" xml:space="preserve">Præterea, ea-
              <lb/>
            dem ratione, ac ſupra, oſtendetur triangulum R Q S ſimile triangulo G F I,
              <lb/>
            & </s>
            <s xml:id="echoid-s5052" xml:space="preserve">quadratum R Q ad G F eſſe vt quadratum Q S ad F I, ſed eſt
              <note symbol="g" position="left" xlink:label="note-0176-07" xlink:href="note-0176-07a" xml:space="preserve">20. pri-
                <lb/>
              mi Conic.</note>
            tum R Q nonuplum quadrati G F, cum ſit recta Q B nonupla B F, vt mo-
              <lb/>
            dò oſtendimus, ergo, & </s>
            <s xml:id="echoid-s5053" xml:space="preserve">quadratum Q S erit nonuplum quadrati F I, ſiue
              <lb/>
            quadrati B F, hoc eſt linea Q S tripla B F, quare tota B S erit vt 12; </s>
            <s xml:id="echoid-s5054" xml:space="preserve">eſtq;</s>
            <s xml:id="echoid-s5055" xml:space="preserve"/>
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