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

Table of contents

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[91.] Notæ in Propoſ. XLV.
Page: 107 (69)
[92.] SECTIO VNDECIMA Continens Propoſ. LXVIII. LXIX. LXX. & LXXI. Apollonij. PROPOSITIO LXVIII. LXIX.
Page: 108 (70)
[93.] PROPOSITIO LXX.
Page: 109 (71)
[94.] PROPOSITIO LXXI.
Page: 109 (71)
[95.] Notæ in Propoſit. LXVIII. LXIX. LXX. & LXXI.
Page: 109 (71)
[96.] SECTIO DVODECIMA Continens XXIX. XXX. XXXI. Propoſ. Appollonij.
Page: 110 (72)
[97.] Notæ in Propoſit. XXIX. XXX. & XXXI.
Page: 111 (73)
[98.] SECTIO DECIMATERTIA Continens Propoſ. LXIV. LXV. LXVI. LXVII. & LXXII. Apollonij. PROPOSITIO LXIV. LXV.
Page: 112 (74)
[99.] PROPOSITIO LXVI.
Page: 113 (75)
[100.] PROPOSITIO LXVII.
Page: 114 (76)
[101.] PROPOSITIO LXXII.
Page: 115 (77)
[102.] MONITVM.
Page: 116 (78)
[103.] LEMMA IX.
Page: 116 (78)
[104.] LEMMA X.
Page: 116 (78)
[105.] LEMMA XI.
Page: 117 (79)
[106.] Notæ in Propoſ. LXIV. & LXV.
Page: 117 (79)
[107.] Notæ in Propoſ. LXVI.
Page: 120 (82)
[108.] Ex demonſtratione præmiſſa propoſitionum 64. & 65. deduci poteſt conſectarium, à quo notæ ſubſe-quentes breuiores reddantur. COROLLARIVM PROPOSIT. LXIV. & LXV.
Page: 120 (82)
[109.] Notæ in Propoſ. LXVII.
Page: 121 (83)
[110.] COROLLARIVM PROPOSIT. LXVII.
Page: 122 (84)
[111.] Notæ in Propoſit. LXXII.
Page: 123 (85)
[112.] SECTIO DECIMAQVARTA Continens Propoſ. LXXIII. LXXIV. LXXV. LXXVI. & LXXVII. PROPOSITIO LXXIII.
Page: 126 (88)
[113.] PROPOSITO LXXIV.
Page: 128 (90)
[114.] PROPOSITO LXXV.
Page: 128 (90)
[115.] PROPOSITIO LXXVI.
Page: 129 (91)
[116.] PROPOSITIO LXXVII.
Page: 130 (92)
[117.] Notæ in Propoſit. LXXIII.
Page: 130 (92)
[118.] LEMMA XII.
Page: 130 (92)
[119.] Notæ in Propoſ. LXXIV.
Page: 134 (96)
[120.] Notæ in Propoſit. LXXV.
Page: 135 (97)
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          <pb o="55" file="0093" n="93" rhead="Conicor. Lib. V."/>
          <p style="it">
            <s xml:id="echoid-s2489" xml:space="preserve">Sit coniſectio A B C, cuius axis A D, & </s>
            <s xml:id="echoid-s2490" xml:space="preserve">in hyperbola, & </s>
            <s xml:id="echoid-s2491" xml:space="preserve">ellipſi centrum
              <lb/>
            E; </s>
            <s xml:id="echoid-s2492" xml:space="preserve">& </s>
            <s xml:id="echoid-s2493" xml:space="preserve">ſumantur quælibet duo puncta B, & </s>
            <s xml:id="echoid-s2494" xml:space="preserve">C, quæ in ellipſi ſint in eodem eius
              <lb/>
            quadrante, & </s>
            <s xml:id="echoid-s2495" xml:space="preserve">ducantur B F, C H perpendiculares ad axim, & </s>
            <s xml:id="echoid-s2496" xml:space="preserve">in parabola,
              <lb/>
            fiant F G, & </s>
            <s xml:id="echoid-s2497" xml:space="preserve">H I æquales ſemiſsi lateris recti; </s>
            <s xml:id="echoid-s2498" xml:space="preserve">at in hyperbola, & </s>
            <s xml:id="echoid-s2499" xml:space="preserve">ellipſi fiat
              <lb/>
            E F ad F G, nec non E H ad H I, vt latus tranſuerſum ad rectum, coniun-
              <lb/>
            ganturq; </s>
            <s xml:id="echoid-s2500" xml:space="preserve">rectæ B G, & </s>
            <s xml:id="echoid-s2501" xml:space="preserve">C I. </s>
            <s xml:id="echoid-s2502" xml:space="preserve">Manifeſtum eſt B G, & </s>
            <s xml:id="echoid-s2503" xml:space="preserve">C I eſſe lineas breuiſsimas,
              <lb/>
            quæ ſi producantur vltra axim (ex 28. </s>
            <s xml:id="echoid-s2504" xml:space="preserve">propoſitione huius libri) conuenient
              <lb/>
              <note position="right" xlink:label="note-0093-01" xlink:href="note-0093-01a" xml:space="preserve">8. 9. 10.
                <lb/>
              huius.</note>
            alicubi, vt in K. </s>
            <s xml:id="echoid-s2505" xml:space="preserve">Dico, quod ex concurſu K nullus alius ramus breuiſecans
              <lb/>
            duci poteſt ad ſectionem A B C. </s>
            <s xml:id="echoid-s2506" xml:space="preserve">Extendatur ex K ſuper axim A D perpendi-
              <lb/>
            cularis K D, & </s>
            <s xml:id="echoid-s2507" xml:space="preserve">reperiatur ſectionis Trutina L competens menſuræ A D ipſius
              <lb/>
            concurſus K, vt in propoſitionibus 51. </s>
            <s xml:id="echoid-s2508" xml:space="preserve">& </s>
            <s xml:id="echoid-s2509" xml:space="preserve">52. </s>
            <s xml:id="echoid-s2510" xml:space="preserve">præcipitur. </s>
            <s xml:id="echoid-s2511" xml:space="preserve">Et certè perpendicu-
              <lb/>
            laris K D non erit maior, quàm L, aliàs duci non poſſet ramus vllus breui-
              <lb/>
              <note position="right" xlink:label="note-0093-02" xlink:href="note-0093-02a" xml:space="preserve">51. 52.
                <lb/>
              huius.</note>
            ſecans ex concurſu K ad ſectionem A B C, quod eſt falſum; </s>
            <s xml:id="echoid-s2512" xml:space="preserve">factæ enim fuerunt
              <lb/>
            K B, & </s>
            <s xml:id="echoid-s2513" xml:space="preserve">K C breuiſecantes; </s>
            <s xml:id="echoid-s2514" xml:space="preserve">Similiter K D non exit æqualis Trutinæ L, quan-
              <lb/>
            doquidem tunc vnica tantummodo breuiſecans ex K ad ſectionem A B C duci
              <lb/>
            poßet, quod rurſus falſum eſt, poſitæ enim fuerunt duæ breuiſecantes; </s>
            <s xml:id="echoid-s2515" xml:space="preserve">igitur per-
              <lb/>
            pendicularis K D neceſſario minor erit Trutina L, & </s>
            <s xml:id="echoid-s2516" xml:space="preserve">ideo ex concurſu K duæ
              <lb/>
              <note position="right" xlink:label="note-0093-03" xlink:href="note-0093-03a" xml:space="preserve">51. 52.
                <lb/>
              huius.</note>
            tantummodo breuiſecantes ad ſectionem A B C duci poſſunt, quæ ſunt B K, C K;
              <lb/>
            </s>
            <s xml:id="echoid-s2517" xml:space="preserve">& </s>
            <s xml:id="echoid-s2518" xml:space="preserve">propterea nullus alius ramus breuiſecans ex concurſu. </s>
            <s xml:id="echoid-s2519" xml:space="preserve">K ad ſectionem A B C
              <lb/>
            duci poteſt præter duos K B, & </s>
            <s xml:id="echoid-s2520" xml:space="preserve">K C; </s>
            <s xml:id="echoid-s2521" xml:space="preserve">quod erat primo loco oſtendendum.</s>
            <s xml:id="echoid-s2522" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2523" xml:space="preserve">Secundo ijſdem poſitis, dico, quod rami ducti inter K B, & </s>
            <s xml:id="echoid-s2524" xml:space="preserve">K C cadunt infra
              <lb/>
            lineas breuiſsimas ab eorom terminis ad axim ductas, & </s>
            <s xml:id="echoid-s2525" xml:space="preserve">quod rami producti ex
              <lb/>
            K ſupra breuiſccantem K B verſus A verticem ſectionis, vel infra ramum bre-
              <lb/>
            uiſecantem K C abſcindunt axis ſegmenta ex vertice minora, quàm abſcindant
              <lb/>
            lineæ breuiſsimæ ab eorum terminis ad axim ductæ. </s>
            <s xml:id="echoid-s2526" xml:space="preserve">Reperiatur denuo Trutina
              <lb/>
            L, oſtendetur, vt prius perpendicularis K D minor, quàm L, & </s>
            <s xml:id="echoid-s2527" xml:space="preserve">duæ tantummo-
              <lb/>
            do breuiſecantes K B, & </s>
            <s xml:id="echoid-s2528" xml:space="preserve">K C; </s>
            <s xml:id="echoid-s2529" xml:space="preserve">quare quilibet ramus ex K ad ſectionis punctum,
              <lb/>
              <note position="right" xlink:label="note-0093-04" xlink:href="note-0093-04a" xml:space="preserve">51. 52.
                <lb/>
              huius.</note>
            inter B, C poſitum extenſus, ſecat ſegmentum axis ex vertice A maius quàm ab-
              <lb/>
            ſcindat linea breniſsima ab eius termino ad axim ducta: </s>
            <s xml:id="echoid-s2530" xml:space="preserve">pariterque quilibet ra-
              <lb/>
            mus ex K ad punctum ſectionis ſupra B, poſitum, vel infra ramum K C exten-
              <lb/>
            ſus, abſcindet ſegmentum axis ex A minus, quàm ſecet linea breuiſsima ab
              <lb/>
            eius termino ad axim ducta; </s>
            <s xml:id="echoid-s2531" xml:space="preserve">quod erat oſtendendum.</s>
            <s xml:id="echoid-s2532" xml:space="preserve"/>
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        <div xml:id="echoid-div223" type="section" level="1" n="75">
          <head xml:id="echoid-head108" xml:space="preserve">Notæ in Propoſit. LVI.</head>
          <p>
            <s xml:id="echoid-s2533" xml:space="preserve">R Eperitur quidem in ramis aggregati ſecantis bifariam inclinatum,
              <lb/>
              <note position="left" xlink:label="note-0093-05" xlink:href="note-0093-05a" xml:space="preserve">a</note>
            ſuper quod non cadit perpendicularis, breuiſecans vna tantum, quo-
              <lb/>
            modocumque ſe habeant perpendicularis, & </s>
            <s xml:id="echoid-s2534" xml:space="preserve">menſura, &</s>
            <s xml:id="echoid-s2535" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2536" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2537" xml:space="preserve">Senſum huius propoſitionis nec Apollonius quidem ſi reuiuiſceret inſigni bar-
              <lb/>
            barie corruptum perciperet, cenſeo tamen, ſic reſtitui debere.</s>
            <s xml:id="echoid-s2538" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s2539" xml:space="preserve">In ellipſi ramorum ſecantium vtrumque axim à concur ſu vltra centrum po-
              <lb/>
            ſito egredientium, vnius tantùm portio inter axim maiorem, & </s>
            <s xml:id="echoid-s2540" xml:space="preserve">ſectionem inter-
              <lb/>
            cepta erit linea breuiſsima; </s>
            <s xml:id="echoid-s2541" xml:space="preserve">ſiue menſura ipſam comparatam, nec non perpendi-
              <lb/>
            cularis ipſam Trutinam ſuperet, æquet, vel ab ea deficiat.</s>
            <s xml:id="echoid-s2542" xml:space="preserve"/>
          </p>
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