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-div602" type="section" level="1" n="202">
          <p style="it">
            <s xml:id="echoid-s6275" xml:space="preserve">
              <pb o="162" file="0200" n="200" rhead="Apollonij Pergæi"/>
            quando anguli vnius inæquales ſint angulis alterius, aut aliquaudo latera circa
              <lb/>
            angulos æquales non ſint proportionalia; </s>
            <s xml:id="echoid-s6276" xml:space="preserve">ita in definitione Mydorgiana, quia co-
              <lb/>
            niſectiones dicuntur ſimiles in quibus omnes axium abſcißæ, quæ proportionales
              <lb/>
            ſunt inter ſe in ijsdem ſunt rationibus ad conterminas potentiales, igitur eidem
              <lb/>
            ſubiecto deſinito, ideſt in duabus ſectionibus conicis ſimilibus, eſt impoſſibile, vt
              <lb/>
            reperiatur ſeries aliqua infinitarum ſimilium abſciſſarum in axibus, quæ ad con-
              <lb/>
            terminas potentiales non ſint in ijſdem rationibus, & </s>
            <s xml:id="echoid-s6277" xml:space="preserve">ſiquidem duæ paſſiones op-
              <lb/>
            poſitæ eidem ſubiecto definito conueniant nulla earum erit eius paſſio eſſentialis,
              <lb/>
            & </s>
            <s xml:id="echoid-s6278" xml:space="preserve">ideo definitio bona non erit: </s>
            <s xml:id="echoid-s6279" xml:space="preserve">vt exempli gratia quia in duobus ſimilibus cir-
              <lb/>
            culorum ſegmentis duo triangula inſcripta poſſunt eſſe æquiangula, & </s>
            <s xml:id="echoid-s6280" xml:space="preserve">etiam non
              <lb/>
            æquiangula; </s>
            <s xml:id="echoid-s6281" xml:space="preserve">ergo ſimilitudo inſcriptorum triangulorum non eſt paſſio eſſentialis
              <lb/>
            ſegmentorum circularium ſimilium inter ſe, & </s>
            <s xml:id="echoid-s6282" xml:space="preserve">ideo non erit bæc bona definitio:
              <lb/>
            </s>
            <s xml:id="echoid-s6283" xml:space="preserve">Similia circulorũ ſegmenta ſunt in quibus deſcribi poſſunt duo triangula ſi-
              <lb/>
            milia, & </s>
            <s xml:id="echoid-s6284" xml:space="preserve">ratio eſt, quia per definitionem nedum natura rei declaratur, & </s>
            <s xml:id="echoid-s6285" xml:space="preserve">indi-
              <lb/>
            catur, ſed etiam diftinguitur, & </s>
            <s xml:id="echoid-s6286" xml:space="preserve">diuerſificatur à qualibet alia; </s>
            <s xml:id="echoid-s6287" xml:space="preserve">& </s>
            <s xml:id="echoid-s6288" xml:space="preserve">quoniam in
              <lb/>
              <note position="left" xlink:label="note-0200-01" xlink:href="note-0200-01a" xml:space="preserve">Coroll.
                <lb/>
              Lem. 2.
                <lb/>
              huius.</note>
            ſectionibus ſimilibus reperiuntur duæ ſeries ſimilium abſciſſarum, quæ ad con-
              <lb/>
            terminas potentiales non ſunt in ijſdem rationibus; </s>
            <s xml:id="echoid-s6289" xml:space="preserve">& </s>
            <s xml:id="echoid-s6290" xml:space="preserve">è contra ex definitione,
              <lb/>
            Mydorgij duæ ſeries ſimilium abſciſſarum, quæ ad conterminas potentiales ſunt
              <lb/>
            in ijſdem rationibus, eſſentialiter conueniunt definito; </s>
            <s xml:id="echoid-s6291" xml:space="preserve">igitur hæ duæ oppoſitæ
              <lb/>
            paſſiones conueniunt eidem ſubiecto definito, ſcilicet ſectionibus ſimilibus iu-
              <lb/>
            xta Mydorgij ſententiam : </s>
            <s xml:id="echoid-s6292" xml:space="preserve">quapropter tradita definitio ſectionum ſimilium vi-
              <lb/>
            tioſa erit, & </s>
            <s xml:id="echoid-s6293" xml:space="preserve">manca.</s>
            <s xml:id="echoid-s6294" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s6295" xml:space="preserve">Vt autem hoc clarius pateat ex-
              <lb/>
              <figure xlink:label="fig-0200-01" xlink:href="fig-0200-01a" number="217">
                <image file="0200-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0200-01"/>
              </figure>
            ponantur duæ ſectiones A B, E F
              <lb/>
            eiuſdem nominis, quarum axes B
              <lb/>
            C, F H, & </s>
            <s xml:id="echoid-s6296" xml:space="preserve">propoſitum primò ſit de-
              <lb/>
            monſtrare ſectiones illas eſſe ſimiles
              <lb/>
            inter ſe; </s>
            <s xml:id="echoid-s6297" xml:space="preserve">ergo oſtendendum eſt paſ-
              <lb/>
            ſionem definitionis traditæ conueni-
              <lb/>
            re ſectionibus A B, E F; </s>
            <s xml:id="echoid-s6298" xml:space="preserve">quod ni-
              <lb/>
            mirum ſimiles axium abſcißæ in,
              <lb/>
            ijſdem rationibus debent eſſe adcõ-
              <lb/>
            terminas potentiales, & </s>
            <s xml:id="echoid-s6299" xml:space="preserve">quia in,
              <lb/>
            definitione nulla cautio, vel determinatio adhibetur, igitur ſumi poſſunt quæ-
              <lb/>
            libet axium abſciſſæ B C, F H, & </s>
            <s xml:id="echoid-s6300" xml:space="preserve">hæc ſecari proportionaliter in R, V, & </s>
            <s xml:id="echoid-s6301" xml:space="preserve">à
              <lb/>
            punctis diuiſionum duci poßunt ad axes ordinatim applicatæ A C, E H, Q R,
              <lb/>
            T V; </s>
            <s xml:id="echoid-s6302" xml:space="preserve">& </s>
            <s xml:id="echoid-s6303" xml:space="preserve">ſupponamus demonſtratum eſſe, quod B C ad C A ſit vt F H ad H E,
              <lb/>
            pariterque vt B R ad R Q ſit vt F V ad V T, tunc quidem ex vi definitionis
              <lb/>
            deducitur, quod ſimiles ſint ſectiones A B, & </s>
            <s xml:id="echoid-s6304" xml:space="preserve">E F. </s>
            <s xml:id="echoid-s6305" xml:space="preserve">At quia demonſtrari poteſt
              <lb/>
              <note position="left" xlink:label="note-0200-02" xlink:href="note-0200-02a" xml:space="preserve">ex Lem. 2.
                <lb/>
              huius.</note>
            in ijſdem ſectionibus (ſumendo abſciſſas B C, F H ad libitum, & </s>
            <s xml:id="echoid-s6306" xml:space="preserve">proportiona-
              <lb/>
            liter diuidendo eas in R, & </s>
            <s xml:id="echoid-s6307" xml:space="preserve">V) quod B C ad C A habet maiorem proportionem,
              <lb/>
              <note position="left" xlink:label="note-0200-03" xlink:href="note-0200-03a" xml:space="preserve">Coroll. 2.
                <lb/>
              Lem. 5.
                <lb/>
              huius.</note>
            quàm F H ad H E; </s>
            <s xml:id="echoid-s6308" xml:space="preserve">pariterque B R ad R Q maiorem proportionẽ habeat, quàm
              <lb/>
              <note position="left" xlink:label="note-0200-04" xlink:href="note-0200-04a" xml:space="preserve">Coroll. 2.
                <lb/>
              Lem. 5.
                <lb/>
              huius.</note>
            F V ad V T, & </s>
            <s xml:id="echoid-s6309" xml:space="preserve">ſic ſemper; </s>
            <s xml:id="echoid-s6310" xml:space="preserve">ergo non poterit deduci ſimilitudo potius quàm non
              <lb/>
            ſimilitudo; </s>
            <s xml:id="echoid-s6311" xml:space="preserve">ideoque definitio ſimilium ſectionum erit vitioſa, quandoquidem ex
              <lb/>
            ea duæ contradictoriæ deducuntur.</s>
            <s xml:id="echoid-s6312" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s6313" xml:space="preserve">Secundo loco ſupponantur duæ ſectiones A B, & </s>
            <s xml:id="echoid-s6314" xml:space="preserve">E F ſimiles inter ſe, & </s>
            <s xml:id="echoid-s6315" xml:space="preserve">pro-
              <lb/>
            poſitum, ſit demonſtrare quod axium figuræ, ſeu rectangula G B D, & </s>
            <s xml:id="echoid-s6316" xml:space="preserve">K F </s>
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