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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[211.] SECTIO SEXTA Continens Propoſit. XV. XVI. & XVII. PROPOSITIO XV.
Page: 213 (175)
[212.] PROPOSITIO XVI.
Page: 215 (177)
[213.] PROPOSITIO XVII.
Page: 216 (178)
[214.] Notæ in Propoſit. XV.
Page: 219 (181)
[215.] MONITVM.
Page: 221 (183)
[216.] LEMMA VI.
Page: 221 (183)
[217.] LEMMA VII.
Page: 222 (184)
[218.] LEMMA VIII.
Page: 224 (186)
[219.] Notæ in Propoſit. XVI.
Page: 225 (187)
[220.] Notæ in Propoſit. XVII.
Page: 226 (188)
[221.] SECTIO SEPTIMA Continens Propoſit. XVIII. & XIX.
Page: 229 (191)
[222.] Notæ in Propoſit. XVIII. & XIX.
Page: 230 (192)
[223.] SECTIO OCTAVA Continens Propoſit. XX. & XXI. Apollonij. PROPOSITIO XX.
Page: 231 (193)
[224.] PROPOSITIO XXI.
Page: 233 (195)
[225.] PROPOSITIO XXII.
Page: 235 (197)
[226.] PROPOSITIO XXIII.
Page: 236 (198)
[227.] PROPOSITIO XXIV.
Page: 236 (198)
[228.] Notæ in Propoſit. XX.
Page: 237 (199)
[229.] Notæ in Propoſit. XXI.
Page: 241 (203)
[230.] Notæ in Propoſit. XXII.
Page: 243 (205)
[231.] Notæ in Propoſit. XXIII.
Page: 244 (206)
[232.] Notæ in Propoſit. XXIV.
Page: 245 (207)
[233.] SECTIO NONA Continens Propoſit. XXV.
Page: 245 (207)
[234.] Notæ in Propoſit. XXV.
Page: 246 (208)
[235.] LEMMA IX.
Page: 267 (229)
[236.] SECTIO DECIMA Continens Propoſit. XXVI. XXVII. & XXVIII. PROPOSITIO XXVI.
Page: 275 (237)
[237.] PROPOSITIO XXVII.
Page: 276 (238)
[238.] PROPOSITIO XXVIII.
Page: 278 (240)
[239.] Notæ in Propoſit. XXVI.
Page: 279 (241)
[240.] Notæ in Propoſit. XXVII.
Page: 280 (242)
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              <pb o="215" file="0253" n="253" rhead="Conicor. Lib. VI."/>
            æqualem eße O V, eo quod in perallelogrammis Q I, & </s>
            <s xml:id="echoid-s8056" xml:space="preserve">S K latera oppoſita ſunt
              <lb/>
            æqualia, & </s>
            <s xml:id="echoid-s8057" xml:space="preserve">ipſæ ordinatæ E K O I; </s>
            <s xml:id="echoid-s8058" xml:space="preserve">nec non M N, Q R æquales oſtenſæ ſunt:
              <lb/>
            </s>
            <s xml:id="echoid-s8059" xml:space="preserve">Deinde producantur, B E, O I ad ſectionem in C, P; </s>
            <s xml:id="echoid-s8060" xml:space="preserve">Et quia differentia qua-
              <lb/>
            dratorum B Z, L X, ſeu T Z, ideſt rectangulum B T C æquale eſt differentiæ
              <lb/>
              <note position="right" xlink:label="note-0253-01" xlink:href="note-0253-01a" xml:space="preserve">ex II.
                <lb/>
              lib. I.</note>
            rectangulorum Z G F, & </s>
            <s xml:id="echoid-s8061" xml:space="preserve">X G F ſeu rectangulo ſub abſciſſarum differentia X Z,
              <lb/>
            & </s>
            <s xml:id="echoid-s8062" xml:space="preserve">latere recto G F. </s>
            <s xml:id="echoid-s8063" xml:space="preserve">Simili modo rectangulum O V P æquale erit rectangulo ſub
              <lb/>
            abſciſſarum differentia R I, & </s>
            <s xml:id="echoid-s8064" xml:space="preserve">latere recto G F: </s>
            <s xml:id="echoid-s8065" xml:space="preserve">ſuntque rectangula contenta
              <lb/>
            ſub X Z, G F, & </s>
            <s xml:id="echoid-s8066" xml:space="preserve">ſub R I, G F æqualia, propterea quod later a X Z, R I æqua-
              <lb/>
            lia oſtenſa ſunt, & </s>
            <s xml:id="echoid-s8067" xml:space="preserve">latus rectum G F eſt commune; </s>
            <s xml:id="echoid-s8068" xml:space="preserve">igitur rectangula B T C, & </s>
            <s xml:id="echoid-s8069" xml:space="preserve">
              <lb/>
            O V P æqualia ſunt; </s>
            <s xml:id="echoid-s8070" xml:space="preserve">ideoque vt T C ad V P, ita reciprocè erit O V ad B T.
              <lb/>
            </s>
            <s xml:id="echoid-s8071" xml:space="preserve">Et primò quia diametri G Z, H K coincidunt, & </s>
            <s xml:id="echoid-s8072" xml:space="preserve">parabolæ H D compræhendi-
              <lb/>
            tur ab A G: </s>
            <s xml:id="echoid-s8073" xml:space="preserve">erit G Z maior quàm H K, ſeu quàm G I, & </s>
            <s xml:id="echoid-s8074" xml:space="preserve">B Z maior quàm
              <lb/>
            E K, & </s>
            <s xml:id="echoid-s8075" xml:space="preserve">L X quàm M N. </s>
            <s xml:id="echoid-s8076" xml:space="preserve">Si verò B E, L M parallelæ ſunt alicui rectæ lineæ
              <lb/>
            H Y diuidenti angulum G H K; </s>
            <s xml:id="echoid-s8077" xml:space="preserve">ergo Y Z, ſeu ei æqualis H K, vel G I minor
              <lb/>
            erit, quàm G Z. </s>
            <s xml:id="echoid-s8078" xml:space="preserve">Eadem ratione G X maior erit, quàm G R; </s>
            <s xml:id="echoid-s8079" xml:space="preserve">quare ordinatim
              <lb/>
            applicata B Z maior erit, quàm O I, & </s>
            <s xml:id="echoid-s8080" xml:space="preserve">Z C maior, quàm I P; </s>
            <s xml:id="echoid-s8081" xml:space="preserve">pariterque L
              <lb/>
            X, ſeu T Z maior erit, quàm Q R, ſeu V I; </s>
            <s xml:id="echoid-s8082" xml:space="preserve">ideoque T C maior erit, quàm
              <lb/>
            V P: </s>
            <s xml:id="echoid-s8083" xml:space="preserve">erat autem O V ad B T reciprocè, vt T C ad V P; </s>
            <s xml:id="echoid-s8084" xml:space="preserve">ergo O V, ſeu ei æqua-
              <lb/>
            lis S M maior erit, quàm B T: </s>
            <s xml:id="echoid-s8085" xml:space="preserve">ij verò addantur æquales L S, T E, quæ in
              <lb/>
            parallelogrammo S T ſunt latera oppoſita, igitur L M, maior erit quàm B E.</s>
            <s xml:id="echoid-s8086" xml:space="preserve"/>
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          <p style="it">
            <s xml:id="echoid-s8087" xml:space="preserve">Deinde quando diametri G I, H K ſibi mutuo congruunt ſit b minor qualibet
              <lb/>
            data recta linea, & </s>
            <s xml:id="echoid-s8088" xml:space="preserve">à vertice H ducatur H d cuius quadratũ æquale ſit rectangulo
              <lb/>
            H G F, & </s>
            <s xml:id="echoid-s8089" xml:space="preserve">fiat vt b ad H d, ita H d ad aliam rectam lineam æqualem C E; </s>
            <s xml:id="echoid-s8090" xml:space="preserve">atq;
              <lb/>
            </s>
            <s xml:id="echoid-s8091" xml:space="preserve">vt H d ad ſemiſſem sũmæ C E, & </s>
            <s xml:id="echoid-s8092" xml:space="preserve">b potentia, ita fiat longitudine H G ad G K,
              <lb/>
            ducaturque B K C ordinatim applicata ad diametrum G I. </s>
            <s xml:id="echoid-s8093" xml:space="preserve">Quoniam quadra-
              <lb/>
              <note position="right" xlink:label="note-0253-02" xlink:href="note-0253-02a" xml:space="preserve">II. lib. I.</note>
            tum E K æquale eſt parallelogrammo H K, G F (propterea quod parabolæ ſunt
              <lb/>
            æquales, & </s>
            <s xml:id="echoid-s8094" xml:space="preserve">diametri ſimiles) & </s>
            <s xml:id="echoid-s8095" xml:space="preserve">ijs adduntur inter ſe æqualia quadratum d H,
              <lb/>
            & </s>
            <s xml:id="echoid-s8096" xml:space="preserve">rectangulum H G F, erunt duo quadrata E K, & </s>
            <s xml:id="echoid-s8097" xml:space="preserve">d H ſimul ſumpta æqualia
              <lb/>
            rectãgulo K G F, ſeu quadrato B Z; </s>
            <s xml:id="echoid-s8098" xml:space="preserve">quare differentia quadratorũ B K, & </s>
            <s xml:id="echoid-s8099" xml:space="preserve">E K,
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            ideſt rectanguli B E C æqualis erit quadrato d H; </s>
            <s xml:id="echoid-s8100" xml:space="preserve">& </s>
            <s xml:id="echoid-s8101" xml:space="preserve">propterea d H media pro-
              <lb/>
            portionalis eſt inter C E, B E, ſed facta fuit media proportionalis inter C E,
              <lb/>
            & </s>
            <s xml:id="echoid-s8102" xml:space="preserve">b; </s>
            <s xml:id="echoid-s8103" xml:space="preserve">Ergo B E æqualis eſt b; </s>
            <s xml:id="echoid-s8104" xml:space="preserve">ideoque R E minor @@ qu@libet recta linea data.
              <lb/>
            </s>
            <s xml:id="echoid-s8105" xml:space="preserve">Quando verò diametri G Z, H K ſunt æquidiſtantes, ijsdem poſitis ducatur O
              <lb/>
            n parallela diametris ſecans B E in n. </s>
            <s xml:id="echoid-s8106" xml:space="preserve">Quia n Z eſt æqualis O I. </s>
            <s xml:id="echoid-s8107" xml:space="preserve">& </s>
            <s xml:id="echoid-s8108" xml:space="preserve">erat E K
              <lb/>
            æqualis O I, ergo n Z, & </s>
            <s xml:id="echoid-s8109" xml:space="preserve">E K æquales ſunt, & </s>
            <s xml:id="echoid-s8110" xml:space="preserve">addita, vel ablata comm@ni Z
              <lb/>
            E erit n E æqualis Z K; </s>
            <s xml:id="echoid-s8111" xml:space="preserve">& </s>
            <s xml:id="echoid-s8112" xml:space="preserve">propterea quælibet intercepta B E @@ior erit in
              <lb/>
            ſecundo caſu, & </s>
            <s xml:id="echoid-s8113" xml:space="preserve">minor in tertio, quàm n E, ſeu Z K à diametris compræben-
              <lb/>
            ſa.</s>
            <s xml:id="echoid-s8114" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8115" xml:space="preserve">Tertio quando B E, L M parallelæ ſunt alicui rectæ G a diuidenti angulum
              <lb/>
            H G I, erit K a, ſeu ei æqualis G Z minor, quàm H K, ſeu quàm G I, atq; </s>
            <s xml:id="echoid-s8116" xml:space="preserve">vt
              <lb/>
            prius rectangula B T C, & </s>
            <s xml:id="echoid-s8117" xml:space="preserve">O V P æqualia erunt, & </s>
            <s xml:id="echoid-s8118" xml:space="preserve">eorum latera reciprocè
              <lb/>
            proportionalia, eſtque S M æqualis minori O V, ergo S M minor erit quàm B
              <lb/>
            T; </s>
            <s xml:id="echoid-s8119" xml:space="preserve">& </s>
            <s xml:id="echoid-s8120" xml:space="preserve">additis æqualibus L S, & </s>
            <s xml:id="echoid-s8121" xml:space="preserve">T E, erit L M minor quàm B E.</s>
            <s xml:id="echoid-s8122" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s8123" xml:space="preserve">Tandem ſint interceptæ B E, L M parallelæ G V, H C portionibus interce-
              <lb/>
            ptarum diametrorum non congruentium, & </s>
            <s xml:id="echoid-s8124" xml:space="preserve">à terminis B, E, L, M, ducan-
              <lb/>
            tur ad diametros ordinatim applicatæ, eas ſecantes in Z, K, I, N, O, S, & </s>
            <s xml:id="echoid-s8125" xml:space="preserve">
              <lb/>
            ſectiones in P, & </s>
            <s xml:id="echoid-s8126" xml:space="preserve">R; </s>
            <s xml:id="echoid-s8127" xml:space="preserve">& </s>
            <s xml:id="echoid-s8128" xml:space="preserve">cadat B E inter duas diametros. </s>
            <s xml:id="echoid-s8129" xml:space="preserve">Quoniam </s>
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