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-div1115" type="section" level="1" n="354">
          <p style="it">
            <s xml:id="echoid-s13029" xml:space="preserve">
              <pb o="389" file="0427" n="428" rhead="Aſſumpt. Liber."/>
            pariterque proportio diametri ad circuli peripheriam ſatis compendioſe deduci
              <lb/>
            poteſt, quandoquidem inter figuram ordinatam eidem circulo inſcriptam, cuius
              <lb/>
            ſemilatus eſt E B, & </s>
            <s xml:id="echoid-s13030" xml:space="preserve">circumſcriptam duplo laterum numero, cuius duo ſemila-
              <lb/>
            tera ſunt C D B, circulus intermediat; </s>
            <s xml:id="echoid-s13031" xml:space="preserve">& </s>
            <s xml:id="echoid-s13032" xml:space="preserve">Perimeter circumſcriptæ figuræ ad
              <lb/>
            Perimetrum inſcriptæ eandem proportionem habet, quam diameter C A ad A E,
              <lb/>
            quæ proportio minui ſemper magis, ac magis poteſt in infinitum; </s>
            <s xml:id="echoid-s13033" xml:space="preserve">& </s>
            <s xml:id="echoid-s13034" xml:space="preserve">tandem ex
              <lb/>
            3. </s>
            <s xml:id="echoid-s13035" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s13036" xml:space="preserve">ſequenti, ex continua ſemipartitione quadrantis circuli elici poſſunt
              <lb/>
            ſubtenſæ ſucceſſiuè ſubdiuiſæ in infinitum, & </s>
            <s xml:id="echoid-s13037" xml:space="preserve">propterea dabitur proportio dia-
              <lb/>
            metri A C ad ſemiſubtenſam B E, ſed datur quadratum ipſius B E, igitur da-
              <lb/>
            tur rectangulum A E C ſub ſegmentis diametri, & </s>
            <s xml:id="echoid-s13038" xml:space="preserve">datur E C ex iam dicta 3.
              <lb/>
            </s>
            <s xml:id="echoid-s13039" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s13040" xml:space="preserve">igitur datur quoque E A; </s>
            <s xml:id="echoid-s13041" xml:space="preserve">eſtque B E ad C D B, vt E A ad diametrũ
              <lb/>
            A C, igitur quarta quantitas innoteſcet, ſcilicet rectæ C D B, quæ æqualia
              <lb/>
            ſunt vni lateri Poligoni circumſcripti duplo laterum numero, & </s>
            <s xml:id="echoid-s13042" xml:space="preserve">ideo habebitur
              <lb/>
            menſura totius Perimetri tum Poligoni inſcripti, cum circumſcripti, quare
              <lb/>
            menſura ipſius peripheriæ circuli, quæ intermedia eſt, facili negotio inueſtiga-
              <lb/>
            bitur.</s>
            <s xml:id="echoid-s13043" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1117" type="section" level="1" n="355">
          <head xml:id="echoid-head447" xml:space="preserve">PROPOSITIO III.</head>
          <p>
            <s xml:id="echoid-s13044" xml:space="preserve">S It C A ſegmentum circuli, & </s>
            <s xml:id="echoid-s13045" xml:space="preserve">B
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                <image file="0427-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0427-01"/>
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            punctum ſuper illud vbicumque,
              <lb/>
            & </s>
            <s xml:id="echoid-s13046" xml:space="preserve">B D perpendicularis ſuper A C, & </s>
            <s xml:id="echoid-s13047" xml:space="preserve">
              <lb/>
            ſegmentum D E æquale D A, & </s>
            <s xml:id="echoid-s13048" xml:space="preserve">arcus
              <lb/>
            B F æqualis arcui B A, vtique iuncta
              <lb/>
            C F erit æqualis ipſi C E.</s>
            <s xml:id="echoid-s13049" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s13050" xml:space="preserve">Demonſtratio. </s>
            <s xml:id="echoid-s13051" xml:space="preserve">Iungamus lineas A B, B F,
              <lb/>
            F E, E B; </s>
            <s xml:id="echoid-s13052" xml:space="preserve">& </s>
            <s xml:id="echoid-s13053" xml:space="preserve">quia arcus B A æqualis eſt arcui B F, erit A B æqualis
              <lb/>
            B F, & </s>
            <s xml:id="echoid-s13054" xml:space="preserve">quia A D æqualis eſt E D, & </s>
            <s xml:id="echoid-s13055" xml:space="preserve">duo anguli D ſunt recti, & </s>
            <s xml:id="echoid-s13056" xml:space="preserve">D B
              <lb/>
            communis, ergo A B æqualis eſt B E, & </s>
            <s xml:id="echoid-s13057" xml:space="preserve">propterea B F, B E ſunt æqua-
              <lb/>
            les; </s>
            <s xml:id="echoid-s13058" xml:space="preserve">& </s>
            <s xml:id="echoid-s13059" xml:space="preserve">duo anguli B F E, B E F ſunt æquales. </s>
            <s xml:id="echoid-s13060" xml:space="preserve">Et quia quadrilaterum.
              <lb/>
            </s>
            <s xml:id="echoid-s13061" xml:space="preserve">C F B A eſt in circulo, erit angulus C F B cum angulo C A B ipſi op-
              <lb/>
            poſito, immo cum angulo B E A, æqualis duobus rectis; </s>
            <s xml:id="echoid-s13062" xml:space="preserve">ſed angulus C
              <lb/>
            E B cum angulo B E A, æquales ſunt duobus rectis, ergo duo anguli C
              <lb/>
            F B, C E B ſunt æquales, & </s>
            <s xml:id="echoid-s13063" xml:space="preserve">remanent C F E, C E F æqualas; </s>
            <s xml:id="echoid-s13064" xml:space="preserve">ergo
              <lb/>
            C E æqualis eſt C F, & </s>
            <s xml:id="echoid-s13065" xml:space="preserve">hoc eſt quod voluimus.</s>
            <s xml:id="echoid-s13066" xml:space="preserve"/>
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        <div xml:id="echoid-div1119" type="section" level="1" n="356">
          <head xml:id="echoid-head448" xml:space="preserve">Notæ in Propoſit. III.</head>
          <p style="it">
            <s xml:id="echoid-s13067" xml:space="preserve">HAEc eſt propoſ. </s>
            <s xml:id="echoid-s13068" xml:space="preserve">5. </s>
            <s xml:id="echoid-s13069" xml:space="preserve">cap. </s>
            <s xml:id="echoid-s13070" xml:space="preserve">9. </s>
            <s xml:id="echoid-s13071" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s13072" xml:space="preserve">1. </s>
            <s xml:id="echoid-s13073" xml:space="preserve">Almag. </s>
            <s xml:id="echoid-s13074" xml:space="preserve">Ptol.</s>
            <s xml:id="echoid-s13075" xml:space="preserve">, ſed hic vniuerſalius pro-
              <lb/>
            nunciatur; </s>
            <s xml:id="echoid-s13076" xml:space="preserve">Ptolomeus enim ſupponit ſegmentum A B C ſemicirculum
              <lb/>
            eſſe, & </s>
            <s xml:id="echoid-s13077" xml:space="preserve">ex cognita circumferentia A F, & </s>
            <s xml:id="echoid-s13078" xml:space="preserve">corda F C, & </s>
            <s xml:id="echoid-s13079" xml:space="preserve">illius medietate A
              <lb/>
            B, quærit chordam A B; </s>
            <s xml:id="echoid-s13080" xml:space="preserve">eſt enim rectangulum ſub C A D æquale </s>
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