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-div774" type="section" level="1" n="240">
          <pb o="244" file="0282" n="282" rhead="Apollonij Pergæi"/>
          <figure number="329">
            <image file="0282-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0282-01"/>
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          <p style="it">
            <s xml:id="echoid-s9136" xml:space="preserve">Igitur duo plana tranſeuntia per K L, T V eleuata ſuper triangulum.
              <lb/>
            </s>
            <s xml:id="echoid-s9137" xml:space="preserve">
              <note position="right" xlink:label="note-0282-01" xlink:href="note-0282-01a" xml:space="preserve">d</note>
            H F I ad angulos rectos producunt in cono H F I duas ſectiones hypor-
              <lb/>
            bolicas, quarum axes L M, V X, & </s>
            <s xml:id="echoid-s9138" xml:space="preserve">inclinati ipſarum L K, V T, & </s>
            <s xml:id="echoid-s9139" xml:space="preserve">
              <lb/>
            ſingulì eorum ad ſuos erectos ſunt, vt D B ad B E; </s>
            <s xml:id="echoid-s9140" xml:space="preserve">ergo figuræ trium.
              <lb/>
            </s>
            <s xml:id="echoid-s9141" xml:space="preserve">ſectionum ſunt ſimiles, & </s>
            <s xml:id="echoid-s9142" xml:space="preserve">æquales; </s>
            <s xml:id="echoid-s9143" xml:space="preserve">& </s>
            <s xml:id="echoid-s9144" xml:space="preserve">propterea duæ ſectiones, qua-
              <lb/>
            rum axes ſunt L M, V X ſunt æquales ſectioni A B, &</s>
            <s xml:id="echoid-s9145" xml:space="preserve">c. </s>
            <s xml:id="echoid-s9146" xml:space="preserve">Ex textu men-
              <lb/>
            doſo expungi debent ſuperuacanea aliqua verba, ſicut in contextu habetur. </s>
            <s xml:id="echoid-s9147" xml:space="preserve">
              <lb/>
            Non enim verum eſt, quod duæ tantummodo hyperbole æquales eidem A B duci
              <lb/>
            poſſunt in cono recto H F I, vertices habentes in lateribus H F, & </s>
            <s xml:id="echoid-s9148" xml:space="preserve">F I, ſed
              <lb/>
            quatuor inter ſe æquales eße poßunt; </s>
            <s xml:id="echoid-s9149" xml:space="preserve">nam ſuper latus F H duci poſſunt duæ
              <lb/>
            hyperbole, quarum axes tranſuerſi K L æquales ſint ipſi B D, & </s>
            <s xml:id="echoid-s9150" xml:space="preserve">æquidiſtan-
              <lb/>
            tes ſint rectis lineis F N, & </s>
            <s xml:id="echoid-s9151" xml:space="preserve">F S. </s>
            <s xml:id="echoid-s9152" xml:space="preserve">Quod ſic oſtendetur. </s>
            <s xml:id="echoid-s9153" xml:space="preserve">Quoniam recta linea
              <lb/>
            Q R ducta eſt parallela ipſi H I erunt duo arcus circuli intercepti H Q, I R
              <lb/>
            æquales inter ſe; </s>
            <s xml:id="echoid-s9154" xml:space="preserve">& </s>
            <s xml:id="echoid-s9155" xml:space="preserve">ideo duo anguli ad peripheriam H F Q, & </s>
            <s xml:id="echoid-s9156" xml:space="preserve">I F R æquales
              <lb/>
            erunt inter ſe; </s>
            <s xml:id="echoid-s9157" xml:space="preserve">poſita autem fuit K L æqualis, & </s>
            <s xml:id="echoid-s9158" xml:space="preserve">parallela ipſi F N; </s>
            <s xml:id="echoid-s9159" xml:space="preserve">igitur
              <lb/>
            duo anguli alterni K L F, & </s>
            <s xml:id="echoid-s9160" xml:space="preserve">H F N æquales ſunt inter ſe: </s>
            <s xml:id="echoid-s9161" xml:space="preserve">pari ratione; </s>
            <s xml:id="echoid-s9162" xml:space="preserve">quia
              <lb/>
            reliqua K L ducta eſt parallela ipſi F S, erit angulus externus S F I æqualis
              <lb/>
            interno, & </s>
            <s xml:id="echoid-s9163" xml:space="preserve">oppoſito, & </s>
            <s xml:id="echoid-s9164" xml:space="preserve">ad eaſdem partes L K F; </s>
            <s xml:id="echoid-s9165" xml:space="preserve">& </s>
            <s xml:id="echoid-s9166" xml:space="preserve">ideo duo triangula L F K
              <lb/>
            habent angulum F, communem, & </s>
            <s xml:id="echoid-s9167" xml:space="preserve">duos angolos in ſingulis triangulis K, & </s>
            <s xml:id="echoid-s9168" xml:space="preserve">
              <lb/>
            L æquales; </s>
            <s xml:id="echoid-s9169" xml:space="preserve">igitur ſunt æquiangula, & </s>
            <s xml:id="echoid-s9170" xml:space="preserve">ſimilia, &</s>
            <s xml:id="echoid-s9171" xml:space="preserve">, vt antea dictum eſt, fieri
              <lb/>
            poſſunt duæ rectæ lineæ K L æquales eidem D B, & </s>
            <s xml:id="echoid-s9172" xml:space="preserve">inter ſe: </s>
            <s xml:id="echoid-s9173" xml:space="preserve">ſi igitur per duas
              <lb/>
            rectas lineas K L ducantur plana perpendicularia ad planum trianguli per axim
              <lb/>
            H F I, eſſicientur in cono recto duæ hyperbole, quarum bini axes tranſuerſi K L
              <lb/>
            ſunt æquales: </s>
            <s xml:id="echoid-s9174" xml:space="preserve">& </s>
            <s xml:id="echoid-s9175" xml:space="preserve">quia, propter parallelas H I, Q R, eſt F N ad N Q ſeu qua-
              <lb/>
            dratum F N ad rectangulum F N Q vt F S æd S R ſeu vt quadratum F S ad
              <lb/>
            rectangum F S R; </s>
            <s xml:id="echoid-s9176" xml:space="preserve">ſed rectangulum H N I æquale eſt rectangulo F N Q, & </s>
            <s xml:id="echoid-s9177" xml:space="preserve">
              <lb/>
            rectangulum H S I æquale eſt rectangulo F S R: </s>
            <s xml:id="echoid-s9178" xml:space="preserve">ergo quadratum F N ad re-
              <lb/>
            ctangulum H N I eandem proportionem habet, quàm quaàratum F S ad rectã-
              <lb/>
            gulum H S I; </s>
            <s xml:id="echoid-s9179" xml:space="preserve">eſtque latus tranſuerſum K L ad ſuum latus rectum, vt quadra-
              <lb/>
              <note position="left" xlink:label="note-0282-02" xlink:href="note-0282-02a" xml:space="preserve">12. lib. 1.</note>
            tum F N ad rectangulum H N I, pariterque latus tranſuerſum K L alterius
              <lb/>
            ſectionis ad ſuum latus rectum eſt vt quadratum F S ad rectangulum H S I:
              <lb/>
            </s>
            <s xml:id="echoid-s9180" xml:space="preserve">
              <note position="left" xlink:label="note-0282-03" xlink:href="note-0282-03a" xml:space="preserve">Ibidem.</note>
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