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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          <pb o="177" file="0215" n="215" rhead="Conicor. Lib. VI."/>
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        <div xml:id="echoid-div637" type="section" level="1" n="212">
          <head xml:id="echoid-head269" xml:space="preserve">PROPOSITIO XVI.</head>
          <p>
            <s xml:id="echoid-s6785" xml:space="preserve">SI ſectiones A B, C D ſimiles inter ſe, quæ ſint prius para-
              <lb/>
            bolæ, tangant lineæ A E, C F terminatæ ad earum axes
              <lb/>
            E B, F D, & </s>
            <s xml:id="echoid-s6786" xml:space="preserve">contineant cum illis angulos æquales E, F, & </s>
            <s xml:id="echoid-s6787" xml:space="preserve">
              <lb/>
            in qualibet earum educantur ordinationes G H, I K ad diame-
              <lb/>
            tros L A M, N C O tranſeuntes per puncta contactus axibus
              <lb/>
              <figure xlink:label="fig-0215-01" xlink:href="fig-0215-01a" number="239">
                <image file="0215-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0215-01"/>
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            æquidiſtantes, & </s>
            <s xml:id="echoid-s6788" xml:space="preserve">fuerit proportio ſuarum abſciſſarum A M, C
              <lb/>
            O ad lineas tangentes A E, C F eadem; </s>
            <s xml:id="echoid-s6789" xml:space="preserve">vtique ordinationes
              <lb/>
            abſcindent ex ſectionibus ſimilia ſegmenta, & </s>
            <s xml:id="echoid-s6790" xml:space="preserve">ſimiliter poſita, vt
              <lb/>
            G A H, I C K. </s>
            <s xml:id="echoid-s6791" xml:space="preserve">Si verò ordinationes ſecuerint ſimilia ſegmen-
              <lb/>
            ta; </s>
            <s xml:id="echoid-s6792" xml:space="preserve">vtique ſectiones ſimiles erunt, & </s>
            <s xml:id="echoid-s6793" xml:space="preserve">abſciſſarum ad lineas tan-
              <lb/>
            gentes proportio erit eadem, atque lineæ tangentes continebunt
              <lb/>
            cum axibus angulos æquales.</s>
            <s xml:id="echoid-s6794" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6795" xml:space="preserve">Educamus enim duas B L, D N ſuper duos axes B E, F D perpendi-
              <lb/>
            culares, quæ tangent ſectiones in B, D: </s>
            <s xml:id="echoid-s6796" xml:space="preserve">& </s>
            <s xml:id="echoid-s6797" xml:space="preserve">ponamus A P ad duplam A
              <lb/>
              <note position="right" xlink:label="note-0215-01" xlink:href="note-0215-01a" xml:space="preserve">32. lib. 1.</note>
            E, vt R A aſſumpta ad A L ei ſimilem, nec non C Q ad duplam C F,
              <lb/>
            vt aſſumpta S C ad C N; </s>
            <s xml:id="echoid-s6798" xml:space="preserve">igitur P A, Q C ſunt erecti duarum diametro-
              <lb/>
            rum L M, N O (52. </s>
            <s xml:id="echoid-s6799" xml:space="preserve">ex 1.) </s>
            <s xml:id="echoid-s6800" xml:space="preserve">ergo G M poteſt P A in A M, (12. </s>
            <s xml:id="echoid-s6801" xml:space="preserve">ex 1.)
              <lb/>
            </s>
            <s xml:id="echoid-s6802" xml:space="preserve">
              <note position="right" xlink:label="note-0215-02" xlink:href="note-0215-02a" xml:space="preserve">49 lib. 1.</note>
            & </s>
            <s xml:id="echoid-s6803" xml:space="preserve">ſimiliter I O poteſt O C in C Q, (12. </s>
            <s xml:id="echoid-s6804" xml:space="preserve">ex 1.) </s>
            <s xml:id="echoid-s6805" xml:space="preserve">& </s>
            <s xml:id="echoid-s6806" xml:space="preserve">propter æquidiſtan-
              <lb/>
              <note position="right" xlink:label="note-0215-03" xlink:href="note-0215-03a" xml:space="preserve">11. lib. 1.
                <lb/>
              lbidem.</note>
            tiam E B, L A, atque F D, C N ſunt ſimilia E R B, R L A, atque D
              <lb/>
            S F, S N C; </s>
            <s xml:id="echoid-s6807" xml:space="preserve">& </s>
            <s xml:id="echoid-s6808" xml:space="preserve">duo anguli E, F ſuppoſiti ſunt æquales; </s>
            <s xml:id="echoid-s6809" xml:space="preserve">igitur angulus R
              <lb/>
            A L æqualis eſt S C N, & </s>
            <s xml:id="echoid-s6810" xml:space="preserve">N, L ſunt recti; </s>
            <s xml:id="echoid-s6811" xml:space="preserve">quare R A ad A L, nempe
              <lb/>
            P A ad duplam A E eſt, vt S C ad N C, nempe vt Q C ad duplam
              <lb/>
            C F, & </s>
            <s xml:id="echoid-s6812" xml:space="preserve">M A ad A E ſuppoſita eſt, vt O C ad C F: </s>
            <s xml:id="echoid-s6813" xml:space="preserve">ergo M A ad A P
              <lb/>
            eſt, vt O C ad C Q, & </s>
            <s xml:id="echoid-s6814" xml:space="preserve">angulus O æqualis eſt M. </s>
            <s xml:id="echoid-s6815" xml:space="preserve">Oſtendetur igitur (vt
              <lb/>
              <note position="left" xlink:label="note-0215-04" xlink:href="note-0215-04a" xml:space="preserve">a</note>
            </s>
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