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-div408" type="section" level="1" n="125">
          <pb o="110" file="0148" n="148" rhead="Apollonij Pergæi"/>
        </div>
        <div xml:id="echoid-div411" type="section" level="1" n="126">
          <head xml:id="echoid-head170" xml:space="preserve">PROPOSITIO XXXXIII.</head>
          <p>
            <s xml:id="echoid-s4402" xml:space="preserve">SI inclinatus axis hyperboles erectum non excedit, nulla li-
              <lb/>
              <note position="right" xlink:label="note-0148-01" xlink:href="note-0148-01a" xml:space="preserve">a</note>
            nearum breuiſſimarum ſectioni ex altera parte occurret: </s>
            <s xml:id="echoid-s4403" xml:space="preserve">ſi
              <lb/>
            verò maior illo fuerit, tunc breuiſſimarum linearum aliquæ oc-
              <lb/>
            currunt ſectioni, aliquæ verò non occurrunt.</s>
            <s xml:id="echoid-s4404" xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">b</note>
          <p>
            <s xml:id="echoid-s4405" xml:space="preserve">Sit priùs D A non maior, quàm A G: </s>
            <s xml:id="echoid-s4406" xml:space="preserve">& </s>
            <s xml:id="echoid-s4407" xml:space="preserve">quia D A ad A G eandẽ pro-
              <lb/>
            portionem habet, quàm quadratum D A ad quadratum A E, erit D A
              <lb/>
            non maior quàm A E; </s>
            <s xml:id="echoid-s4408" xml:space="preserve">& </s>
            <s xml:id="echoid-s4409" xml:space="preserve">propterea angulus D E A non crit maior an-
              <lb/>
            gulo E D A: </s>
            <s xml:id="echoid-s4410" xml:space="preserve">ſed maior fuerat angulo B C H (41. </s>
            <s xml:id="echoid-s4411" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s4412" xml:space="preserve">ergo angulus
              <lb/>
            E D A, nempe A D F maior eſt, quàm B C D, & </s>
            <s xml:id="echoid-s4413" xml:space="preserve">propterea B C, D F
              <lb/>
            non conueniunt ad partes C, F; </s>
            <s xml:id="echoid-s4414" xml:space="preserve">igitur B C non occurrit ſectioni ad par-
              <lb/>
            tes K, A; </s>
            <s xml:id="echoid-s4415" xml:space="preserve">eo quod ſi illam ſecaret, etiam ipſi D F occurreret (8. </s>
            <s xml:id="echoid-s4416" xml:space="preserve">ex 2.)
              <lb/>
            </s>
            <s xml:id="echoid-s4417" xml:space="preserve">quare non occurrit ſectioni in duobus punctis.</s>
            <s xml:id="echoid-s4418" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4419" xml:space="preserve">Deinde ſit D A maior, quàm A
              <lb/>
              <figure xlink:label="fig-0148-01" xlink:href="fig-0148-01a" number="135">
                <image file="0148-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0148-01"/>
              </figure>
              <note position="right" xlink:label="note-0148-03" xlink:href="note-0148-03a" xml:space="preserve">c</note>
            G habebit E A ad A G maiorem
              <lb/>
            proportionẽ, quàm ad A D; </s>
            <s xml:id="echoid-s4420" xml:space="preserve">& </s>
            <s xml:id="echoid-s4421" xml:space="preserve">po-
              <lb/>
            natur I A ad A G, vt E A ad A
              <lb/>
            D; </s>
            <s xml:id="echoid-s4422" xml:space="preserve">ergo I A minor eſt, quàm E A,
              <lb/>
              <note position="left" xlink:label="note-0148-04" xlink:href="note-0148-04a" xml:space="preserve">Secunda
                <lb/>
              lib. 2.</note>
            quare recta D B, illam diuidens,
              <lb/>
            occurret ſectioni, & </s>
            <s xml:id="echoid-s4423" xml:space="preserve">cadat in B, du-
              <lb/>
            caturque linea breuiſſima B C, & </s>
            <s xml:id="echoid-s4424" xml:space="preserve">
              <lb/>
            B H perpendicularis ad D C; </s>
            <s xml:id="echoid-s4425" xml:space="preserve">erit
              <lb/>
            I A ad A D, vt B H ad H D; </s>
            <s xml:id="echoid-s4426" xml:space="preserve">eſt-
              <lb/>
            que D A ad A G, vt D H ad H C;
              <lb/>
            </s>
            <s xml:id="echoid-s4427" xml:space="preserve">
              <note position="left" xlink:label="note-0148-05" xlink:href="note-0148-05a" xml:space="preserve">14. huius.</note>
            ergo I A ad A G, nempe E A ad
              <lb/>
            A D eſt, vt B H ad H C, & </s>
            <s xml:id="echoid-s4428" xml:space="preserve">pro-
              <lb/>
            pterea duo triangula E A D, B H
              <lb/>
            C ſunt ſimilia; </s>
            <s xml:id="echoid-s4429" xml:space="preserve">igitur angulus B C
              <lb/>
            H æqualis eſt E D A, nempe F D A; </s>
            <s xml:id="echoid-s4430" xml:space="preserve">quare B C, D F ſunt parallelæ,
              <lb/>
            nec poſſunt ſe ſe mutuo ſecare; </s>
            <s xml:id="echoid-s4431" xml:space="preserve">ergo B C non occurret ſectioni K A.
              <lb/>
            </s>
            <s xml:id="echoid-s4432" xml:space="preserve">
              <note position="left" xlink:label="note-0148-06" xlink:href="note-0148-06a" xml:space="preserve">13. lib. 2.</note>
            Lineæ vero breuiſſimæ, quæ in peripheria A B cadunt, continent cum.
              <lb/>
            </s>
            <s xml:id="echoid-s4433" xml:space="preserve">C A angulos minores angulo B C D (26. </s>
            <s xml:id="echoid-s4434" xml:space="preserve">27. </s>
            <s xml:id="echoid-s4435" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s4436" xml:space="preserve">vnde non occurrent
              <lb/>
              <note position="left" xlink:label="note-0148-07" xlink:href="note-0148-07a" xml:space="preserve">Conuerſ.
                <lb/>
              8. lib. 2.</note>
            ipſi D F, & </s>
            <s xml:id="echoid-s4437" xml:space="preserve">propterea neque ſectioni occurrent. </s>
            <s xml:id="echoid-s4438" xml:space="preserve">At ille, qui cadit extra
              <lb/>
            hanc ſectionis peripheriam; </s>
            <s xml:id="echoid-s4439" xml:space="preserve">ſi producatur continet cum C D angulum.
              <lb/>
            </s>
            <s xml:id="echoid-s4440" xml:space="preserve">maiorem angulo B C D (26. </s>
            <s xml:id="echoid-s4441" xml:space="preserve">27. </s>
            <s xml:id="echoid-s4442" xml:space="preserve">ex 5.) </s>
            <s xml:id="echoid-s4443" xml:space="preserve">igitur productus occurrit D F,
              <lb/>
            & </s>
            <s xml:id="echoid-s4444" xml:space="preserve">occurrit ſectioni A K: </s>
            <s xml:id="echoid-s4445" xml:space="preserve">quod erat oſtendendum.</s>
            <s xml:id="echoid-s4446" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div414" type="section" level="1" n="127">
          <head xml:id="echoid-head171" xml:space="preserve">Notæ in Propoſ. XXXXI.</head>
          <p>
            <s xml:id="echoid-s4447" xml:space="preserve">ANgulus contentus à breuiſſima linea, & </s>
            <s xml:id="echoid-s4448" xml:space="preserve">menſura minor eſt angulo
              <lb/>
            contento à diſtante cum continente in ſectione, &</s>
            <s xml:id="echoid-s4449" xml:space="preserve">c. </s>
            <s xml:id="echoid-s4450" xml:space="preserve">Adijcio </s>
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