Apollonius <Pergaeus>, Apollonii Pergaei Conicorvm Lib. V. VI. VII. paraphraste Abalphato Asphahanensi : nunc primum editi ; additvs in calce Archimedis assvmptorvm liber, ex codibvs arabicis mss Abrahamus Ecchellensis Maronita latinos reddidit, Jo. Alfonsvs Borellvs curam in geometricis versione contulit & [et] notas vberiores in vniuersum opus adiecit

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        <div xml:id="echoid-div610" type="section" level="1" n="204">
          <head xml:id="echoid-head258" xml:space="preserve">Notæ in Propoſit. XII.</head>
          <p style="it">
            <s xml:id="echoid-s6420" xml:space="preserve">SVpponamus itaque ſectiones A B, E F, earum inclinati, vel tran-
              <lb/>
              <note position="right" xlink:label="note-0204-01" xlink:href="note-0204-01a" xml:space="preserve">a</note>
            ſuerſi B a, F b, & </s>
            <s xml:id="echoid-s6421" xml:space="preserve">erecti eorum B D, F I ordinationes, & </s>
            <s xml:id="echoid-s6422" xml:space="preserve">propoſitio-
              <lb/>
            nes, vti diximus, &</s>
            <s xml:id="echoid-s6423" xml:space="preserve">c. </s>
            <s xml:id="echoid-s6424" xml:space="preserve">Ideſt. </s>
            <s xml:id="echoid-s6425" xml:space="preserve">Sint axes inclinati, ſiue tranſuerſi B a, F b, & </s>
            <s xml:id="echoid-s6426" xml:space="preserve">
              <lb/>
            maneant ſigna, ordinationes, & </s>
            <s xml:id="echoid-s6427" xml:space="preserve">proportiones eædem, quæ in præcedenti propoſi-
              <lb/>
            tione; </s>
            <s xml:id="echoid-s6428" xml:space="preserve">ſcilicet fiat C B ad B D, vt H F ad F I, & </s>
            <s xml:id="echoid-s6429" xml:space="preserve">quia D B ad B a eſt vt I
              <lb/>
            F ad F b ( propter ſimilitudinem figurarum D B a, I F b ) ergo ex æquali C
              <lb/>
            B ad B a erit vt H F ad F b; </s>
            <s xml:id="echoid-s6430" xml:space="preserve">& </s>
            <s xml:id="echoid-s6431" xml:space="preserve">comparando antecedentes ad ſummas termino-
              <lb/>
            rum in hyperbola, & </s>
            <s xml:id="echoid-s6432" xml:space="preserve">ad differentias in ellipſi erit B C ad C a vt F H ad H b:
              <lb/>
            </s>
            <s xml:id="echoid-s6433" xml:space="preserve">poſtea diuidantur tam B C, quàm F H in ijſdem rationibus in punctis K, L,
              <lb/>
            M, N, & </s>
            <s xml:id="echoid-s6434" xml:space="preserve">educantur ordinatim applicatæ, ſeu æquidiſtantes baſibus O P, Q R,
              <lb/>
            A S, T V, X r
              <unsure/>
            , E Z.</s>
            <s xml:id="echoid-s6435" xml:space="preserve"/>
          </p>
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            <s xml:id="echoid-s6436" xml:space="preserve">Quoniam figura ſectionis A B ſimilis eſt figuræ ſectionis E F erit qua-
              <lb/>
              <note position="right" xlink:label="note-0204-02" xlink:href="note-0204-02a" xml:space="preserve">b</note>
            dratum H E ad H b in H F, vt quadratum A C ad C a in C B, & </s>
            <s xml:id="echoid-s6437" xml:space="preserve">b H
              <lb/>
            in H F ad quadratum H F, vt C a in C B ad quadratnm C B ( nam po-
              <lb/>
            ſuimus H F ad F b, vt C B ad B a, &</s>
            <s xml:id="echoid-s6438" xml:space="preserve">c.) </s>
            <s xml:id="echoid-s6439" xml:space="preserve">Quouiam in figuris, ſeu rectan-
              <lb/>
            gulis ſimilibus D B a, & </s>
            <s xml:id="echoid-s6440" xml:space="preserve">I F b habet D B ad B a eandem proportionem, quàm
              <lb/>
              <note position="left" xlink:label="note-0204-03" xlink:href="note-0204-03a" xml:space="preserve">21. lib. I.</note>
            I F ad F b, & </s>
            <s xml:id="echoid-s6441" xml:space="preserve">vt D B ad B a, ita eſt quadratum A C ad rectangulum B C a,
              <lb/>
            pariterque vt I F ad F b ita eſt quadratum E H ad rectangulũ F H b ſed ( ſi-
              <lb/>
            cut in præcedenti nota dictum eſt) C a ad C B, ſeu rectangulum B C a ad qua-
              <lb/>
            dratum C B eandem proportionem habet, quàm H b ad H F, ſeu quàm rectan-
              <lb/>
            gulum F H b ad quadratum F H; </s>
            <s xml:id="echoid-s6442" xml:space="preserve">igitur ex æqualitate quadratum A C ad qua-
              <lb/>
            dratum C B eandem proportionem habet, quàm quadratum E H ad quadratum
              <lb/>
            H F.</s>
            <s xml:id="echoid-s6443" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s6444" xml:space="preserve">Atque quadratum H F ad H F in H b eſt vt quadratum C B ad B C in
              <lb/>
              <note position="right" xlink:label="note-0204-04" xlink:href="note-0204-04a" xml:space="preserve">C</note>
            C a (eo quod H F ad F b poſita fuit C B ad B a), ergo ex æqualitate, &</s>
            <s xml:id="echoid-s6445" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s6446" xml:space="preserve">Ideſt ſumã tur axium abſcißæ C B, H F, quæ ſint proportionales lateribus rectis
              <lb/>
            B D, & </s>
            <s xml:id="echoid-s6447" xml:space="preserve">F I, ſeu proportionales ſint lateribus tranſuerſis B a, & </s>
            <s xml:id="echoid-s6448" xml:space="preserve">F b, & </s>
            <s xml:id="echoid-s6449" xml:space="preserve">ſecẽtur
              <lb/>
            abſciſſæ B C, & </s>
            <s xml:id="echoid-s6450" xml:space="preserve">F H proportionaliter in punctis K, L, M, N, & </s>
            <s xml:id="echoid-s6451" xml:space="preserve">per puncta
              <lb/>
            diuiſionum ducantur ordinatim applicatæ A C, Q L, E H, X N, & </s>
            <s xml:id="echoid-s6452" xml:space="preserve">c. </s>
            <s xml:id="echoid-s6453" xml:space="preserve">Quia ſe-
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            ctiones A B, E F ſupponuntur ſimiles; </s>
            <s xml:id="echoid-s6454" xml:space="preserve">ergo ex definitione 2. </s>
            <s xml:id="echoid-s6455" xml:space="preserve">huius A C ad C B
              <lb/>
            eandem proportionem habebit, quàm E H ad H F, nec non Q L ad L B erit vt
              <lb/>
            X N ad N F; </s>
            <s xml:id="echoid-s6456" xml:space="preserve">& </s>
            <s xml:id="echoid-s6457" xml:space="preserve">ideo quadratum A C ad quadratum C B eandem proportionẽ
              <lb/>
            habet, quàm quadratum E H ad quadratum H F; </s>
            <s xml:id="echoid-s6458" xml:space="preserve">& </s>
            <s xml:id="echoid-s6459" xml:space="preserve">quia ex conſtructione,
              <lb/>
            iuxta leges definitionis 2. </s>
            <s xml:id="echoid-s6460" xml:space="preserve">vt C B ad B a ita erat H F ad F b, & </s>
            <s xml:id="echoid-s6461" xml:space="preserve">comparando
              <lb/>
            antecedentes ad terminorũ ſummas in hyperbolis, & </s>
            <s xml:id="echoid-s6462" xml:space="preserve">ad differentias in ellipſibus,
              <lb/>
            habebit B C ad C a, ſeu quadratum B C ad rectangulum B C a eandẽ propor-
              <lb/>
            tionem quàm F H habet ad H b, ſeu quàm quadratum F H habet ad rectangu-
              <lb/>
            lum F H b; </s>
            <s xml:id="echoid-s6463" xml:space="preserve">ergo ex æqualitate quadratum A C ad rectangulum B C a eãdem
              <lb/>
            proportionem habet, quàm quadratum E H ad rectangulum F H b; </s>
            <s xml:id="echoid-s6464" xml:space="preserve">eſt verò la-
              <lb/>
            tus rectum D B ad latus tranſuerſum B a, vt quadratum A C ad </s>
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