Ibn-al-Haitam, al-Hasan Ibn-al-Hasan; Witelo; Risner, Friedrich, Opticae thesavrvs Alhazeni Arabis libri septem, nunc primùm editi. Eivsdem liber De Crepvscvlis & Nubium ascensionibus. Item Vitellonis Thuvringopoloni Libri X. Omnes instaurati, figuris illustrati & aucti, adiectis etiam in Alhazenum commentarijs, a Federico Risnero, 1572
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            <pb o="219" file="0225" n="225" rhead="OPTICAE LIBER VI."/>
          rum, reflexionis & ſpeculi ſphærici caui) puncta dictæ rectæ intermedia à punctis dictæ peri-
            <lb/>
          pheriæ intermedijs reflectentur. 54. 42 p 8.</head>
          <p>
            <s xml:id="echoid-s15493" xml:space="preserve">SIt ergo ſpeculum ſphæricum concauum a b:</s>
            <s xml:id="echoid-s15494" xml:space="preserve"> & extrahamus in ipſo ſpeculo ſuperficiem planã,
              <lb/>
            tranſeuntem per centrũ:</s>
            <s xml:id="echoid-s15495" xml:space="preserve"> & faciat circulũ a b circa centrũ e [faciet autem per 1 th.</s>
            <s xml:id="echoid-s15496" xml:space="preserve"> 1 ſphær.</s>
            <s xml:id="echoid-s15497" xml:space="preserve">] &
              <lb/>
            extrahamus in hoc circulo duas diametros ſe ſecãtes a e o, b e d:</s>
            <s xml:id="echoid-s15498" xml:space="preserve"> & ſpeculum nõ excedat arcũ
              <lb/>
            b a d o:</s>
            <s xml:id="echoid-s15499" xml:space="preserve"> & ponamus in b e punctum z, quocunq;</s>
            <s xml:id="echoid-s15500" xml:space="preserve"> modo ſit:</s>
            <s xml:id="echoid-s15501" xml:space="preserve"> & ponamus in linea a e punctum k:</s>
            <s xml:id="echoid-s15502" xml:space="preserve"> & ſit
              <lb/>
            a k maior quàm k e:</s>
            <s xml:id="echoid-s15503" xml:space="preserve"> & continuemus z k:</s>
            <s xml:id="echoid-s15504" xml:space="preserve"> & tranſeat ad f:</s>
            <s xml:id="echoid-s15505" xml:space="preserve"> & continuemus e f:</s>
            <s xml:id="echoid-s15506" xml:space="preserve"> & ſit angulus g f e æqua
              <lb/>
            lis angulo z f e.</s>
            <s xml:id="echoid-s15507" xml:space="preserve"> [per 23 p 1.</s>
            <s xml:id="echoid-s15508" xml:space="preserve">] Quia igitur [per 7 p 3] f k eſt maior k a, & k a eſt maior quàm k e:</s>
            <s xml:id="echoid-s15509" xml:space="preserve"> [ex
              <lb/>
            theſi] erit f k maior quàm k e:</s>
            <s xml:id="echoid-s15510" xml:space="preserve"> angulus ergo f e k maior eſt angulo e f k:</s>
            <s xml:id="echoid-s15511" xml:space="preserve"> [per 18 p 1] ergo eſt maior
              <lb/>
            angulo e f g.</s>
            <s xml:id="echoid-s15512" xml:space="preserve"> Linea ergo f g concurret cũ linea k e.</s>
            <s xml:id="echoid-s15513" xml:space="preserve"> [ſi enim non concurrat:</s>
            <s xml:id="echoid-s15514" xml:space="preserve"> erit ad ipſam parallela:</s>
            <s xml:id="echoid-s15515" xml:space="preserve">
              <lb/>
            itaq;</s>
            <s xml:id="echoid-s15516" xml:space="preserve"> per 29 p 1 angulus e f g æquabitur angulo f e k, quo minor eſt concluſus.</s>
            <s xml:id="echoid-s15517" xml:space="preserve">] Concurrant ergo in
              <lb/>
            g.</s>
            <s xml:id="echoid-s15518" xml:space="preserve"> Duæ ergo lineæ z f, f g reflectuntur propter angu
              <lb/>
              <figure xlink:label="fig-0225-01" xlink:href="fig-0225-01a" number="195">
                <variables xml:id="echoid-variables184" xml:space="preserve">t f h a ſ i k d r e z b c m o g</variables>
              </figure>
            los æquales z f e, g f e:</s>
            <s xml:id="echoid-s15519" xml:space="preserve"> [per 12 n 4] k ergo eſt imago
              <lb/>
            g, ſi uiſus fuerit in z [per 6 n 5.</s>
            <s xml:id="echoid-s15520" xml:space="preserve">] Et extrahamus li-
              <lb/>
            neam z l h quocunq;</s>
            <s xml:id="echoid-s15521" xml:space="preserve"> modo ſit:</s>
            <s xml:id="echoid-s15522" xml:space="preserve"> & cõtinuemus e h,
              <lb/>
            h g, z g:</s>
            <s xml:id="echoid-s15523" xml:space="preserve"> & extrahamus f e uſq;</s>
            <s xml:id="echoid-s15524" xml:space="preserve"> ad m.</s>
            <s xml:id="echoid-s15525" xml:space="preserve"> Proportio er-
              <lb/>
            go z m ad m g eſt, ſicut ꝓportio z f ad f g [per 3 p 6:</s>
            <s xml:id="echoid-s15526" xml:space="preserve">
              <lb/>
            quia angulus g f z bifariam ſectus eſt per rectã e f]
              <lb/>
            & [per 7 p 3] z h eſt maior quàm z f, & g h eſt mi-
              <lb/>
            nor quàm g f.</s>
            <s xml:id="echoid-s15527" xml:space="preserve"> Ergo proportio z h ad g h eſt maior,
              <lb/>
            quàm proportio z f ad f g:</s>
            <s xml:id="echoid-s15528" xml:space="preserve"> [ut conſtat ex 8 p 5] eſt
              <lb/>
            ergo maior quàm proportio z m ad m g.</s>
            <s xml:id="echoid-s15529" xml:space="preserve"> Ergo [per
              <lb/>
            3 p 6] linea, quę diuidit angulũ z h g in duo æqua-
              <lb/>
            lia, ſecat lineã m g:</s>
            <s xml:id="echoid-s15530" xml:space="preserve"> ſecat ergo lineã e g.</s>
            <s xml:id="echoid-s15531" xml:space="preserve"> Secet ergo
              <lb/>
            lineam e g in r:</s>
            <s xml:id="echoid-s15532" xml:space="preserve"> ergo angulus g h e maior eſt angulo
              <lb/>
            z h e:</s>
            <s xml:id="echoid-s15533" xml:space="preserve"> & h z ſecet a e in l.</s>
            <s xml:id="echoid-s15534" xml:space="preserve"> Ergo duæ lineæ z h, h r re-
              <lb/>
            flectũtur propter angulos æquales:</s>
            <s xml:id="echoid-s15535" xml:space="preserve"> [r h e, z h e per
              <lb/>
            12 n 4] & erit l imago r.</s>
            <s xml:id="echoid-s15536" xml:space="preserve"> Dico ergo, quòd forma cu-
              <lb/>
            iuslibet puncti lineæ g r reflectitur ad uiſum z ex
              <lb/>
            puncto aliquo arcus f h, & non ex alio.</s>
            <s xml:id="echoid-s15537" xml:space="preserve"> Huius rei demonſtratio eſt, quoniam in capitulo de imagi-
              <lb/>
            ne, quinto tractatu in duabus figuris [66 n] dictum eſt, quòd duo arcus a b, d o non poſſunt eſſe ta
              <lb/>
            les, quòd ex illis reflectatur aliquid de linea e o ad z:</s>
            <s xml:id="echoid-s15538" xml:space="preserve"> & arcus e o non eſt de ſpeculo:</s>
            <s xml:id="echoid-s15539" xml:space="preserve"> [nam ex theſi
              <lb/>
            ab arcu ſpeculi b a d o fit reflexio, cũ ille tantùm ſub uiſum in diametro d b poſitum cadat] nõ ergo
              <lb/>
            remanet niſi arcus a d.</s>
            <s xml:id="echoid-s15540" xml:space="preserve"> Sed in triceſima quinta figura [66 n 5] dictum eſt, quòd forma cuiuslibet
              <lb/>
            pũcti diametri e o reflectitur ab aliquo puncto arcus a d.</s>
            <s xml:id="echoid-s15541" xml:space="preserve"> Et in triceſima ſexta, capitulo de imagine
              <lb/>
            [73 n 5] patuit, quòd nunquã reflectitur forma puncti lineæ g r ad z ex arcu a d, niſi ex ſolo puncto.</s>
            <s xml:id="echoid-s15542" xml:space="preserve">
              <lb/>
            Forma ergo cuiuslibet puncti lineæ g r reflectitur ad z ex uno ſolo puncto arcus a d.</s>
            <s xml:id="echoid-s15543" xml:space="preserve"> Et ponamus
              <lb/>
            in linea g r punctum c.</s>
            <s xml:id="echoid-s15544" xml:space="preserve"> Dico ergo, quòd illud punctũ
              <lb/>
              <figure xlink:label="fig-0225-02" xlink:href="fig-0225-02a" number="196">
                <variables xml:id="echoid-variables185" xml:space="preserve">q h f d u o g c r e a n m z b</variables>
              </figure>
            non erit, niſi in arcu fh.</s>
            <s xml:id="echoid-s15545" xml:space="preserve"> Sin autem reflectatur forma c
              <lb/>
            ad z ex u, quod eſt in arcu a f:</s>
            <s xml:id="echoid-s15546" xml:space="preserve"> & continuemus lineas
              <lb/>
            z u, e u, g u, c u.</s>
            <s xml:id="echoid-s15547" xml:space="preserve"> Linea ergo g u erit maior g f [per 7 p 3]
              <lb/>
            & z u eſt minor quàm z f.</s>
            <s xml:id="echoid-s15548" xml:space="preserve"> Ergo [ut cõſtat ex 8 p 5] ꝓ-
              <lb/>
            portio g u ad z u eſt maior proportione g f ad f z:</s>
            <s xml:id="echoid-s15549" xml:space="preserve"> ergo
              <lb/>
            maior proportione g m ad m z [quia enim angulus
              <lb/>
            g f z bifariam ſectus eſt per rectam f m:</s>
            <s xml:id="echoid-s15550" xml:space="preserve"> erit per 3 p 6 g f
              <lb/>
            ad f z, ſicut g m ad m z.</s>
            <s xml:id="echoid-s15551" xml:space="preserve">] Linea ergo, q̃ diuidit angulũ
              <lb/>
            g u z per æqualia, ſecat lineam z m:</s>
            <s xml:id="echoid-s15552" xml:space="preserve"> ſecat ergo z e:</s>
            <s xml:id="echoid-s15553" xml:space="preserve"> angu
              <lb/>
            lus ergo g u e eſt minor angulo e u z:</s>
            <s xml:id="echoid-s15554" xml:space="preserve"> ergo angulus c u
              <lb/>
            e multò minor eſt angulo e u z.</s>
            <s xml:id="echoid-s15555" xml:space="preserve"> [Itaq;</s>
            <s xml:id="echoid-s15556" xml:space="preserve"> cum anguli inci-
              <lb/>
            dentiæ & reflexionis ſint inæquales:</s>
            <s xml:id="echoid-s15557" xml:space="preserve"> nulla à puncto u
              <lb/>
            ad uiſum z fiet reflexio, ut patet per 12 n 4.</s>
            <s xml:id="echoid-s15558" xml:space="preserve">] Et ſimili-
              <lb/>
            ter de quolibet puncto arcus a u.</s>
            <s xml:id="echoid-s15559" xml:space="preserve"> Forma ergo c non re-
              <lb/>
            flectitur ad z, niſi ex arcu h f.</s>
            <s xml:id="echoid-s15560" xml:space="preserve"> Et dico, quòd non po-
              <lb/>
            teſt reflecti ex arcu h d.</s>
            <s xml:id="echoid-s15561" xml:space="preserve"> Quod ſi fuerit poſsibile:</s>
            <s xml:id="echoid-s15562" xml:space="preserve"> refle-
              <lb/>
            ctatur ex q, quod eſt in arcu h d:</s>
            <s xml:id="echoid-s15563" xml:space="preserve"> & continuemus lineas z q, c q, r q, e q, z r:</s>
            <s xml:id="echoid-s15564" xml:space="preserve"> & extrahamus e h ad n.</s>
            <s xml:id="echoid-s15565" xml:space="preserve"> Li-
              <lb/>
            nea ergo z q eſt maior quã z h [per 7 p 3], & linea q r eſt minor quàm h r:</s>
            <s xml:id="echoid-s15566" xml:space="preserve"> ergo proportio z q ad q r
              <lb/>
            eſt maior proportione z h ad h r:</s>
            <s xml:id="echoid-s15567" xml:space="preserve"> [ut patet per 8 p 5] quæ eſt, ſicut proportio z n ad n r [per 3 p 6:</s>
            <s xml:id="echoid-s15568" xml:space="preserve">
              <lb/>
            quia angulus r h z bifariam ſectus eſt per rectam h n.</s>
            <s xml:id="echoid-s15569" xml:space="preserve">] Linea ergo, quæ diuidit angulum z q r in duo
              <lb/>
            æqualia, ſecat lineam n r:</s>
            <s xml:id="echoid-s15570" xml:space="preserve"> ſecat ergo lineam e r:</s>
            <s xml:id="echoid-s15571" xml:space="preserve"> angulus ergo r q e eſt maior angulo e q z:</s>
            <s xml:id="echoid-s15572" xml:space="preserve"> angulus er
              <lb/>
            go c q e eſt multò maior angulo e q z.</s>
            <s xml:id="echoid-s15573" xml:space="preserve"> Hoc idem ſequitur in omni puncto arcus h d.</s>
            <s xml:id="echoid-s15574" xml:space="preserve"> Forma ergo c
              <lb/>
            non reflectitur ad z ex arcu h d:</s>
            <s xml:id="echoid-s15575" xml:space="preserve"> neque ex arcu a f.</s>
            <s xml:id="echoid-s15576" xml:space="preserve"> Sed iam patuit, quòd omnino debet reflecti ex ar-
              <lb/>
            cu a d.</s>
            <s xml:id="echoid-s15577" xml:space="preserve"> Forma ergo c non reflectitur ad z, niſi ex aliquo puncto arcus f h [nam quòd à punctis h &
              <lb/>
            freflexio nulla fiat, patet per 74.</s>
            <s xml:id="echoid-s15578" xml:space="preserve"> 75 n 5.</s>
            <s xml:id="echoid-s15579" xml:space="preserve">] Reflectatur ergo ex t:</s>
            <s xml:id="echoid-s15580" xml:space="preserve"> & continuemus lineas c t, & z t.</s>
            <s xml:id="echoid-s15581" xml:space="preserve"> Quia
              <lb/>
            </s>
          </p>
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