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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          <p>
            <s xml:id="echoid-s15830" xml:space="preserve">
              <pb o="223" file="0229" n="229" rhead="OPTICAE LIBER VI."/>
            31 p 3.</s>
            <s xml:id="echoid-s15831" xml:space="preserve"> Quare ſemiperipheria l f d æquatur ſemiperipheriæ l z d:</s>
            <s xml:id="echoid-s15832" xml:space="preserve"> & peripheria d m æquatur periphe-
              <lb/>
            riæ d h per 28 p 3, quia d m, d h æquatæ ſunt:</s>
            <s xml:id="echoid-s15833" xml:space="preserve"> reliqua igitur l f m æquatur reliquę l z h] & arcus
              <lb/>
            m f eſt æqualis arcui z h.</s>
            <s xml:id="echoid-s15834" xml:space="preserve"> [Nam propter æqualitatem ſemidiametrorum d f, & d z, ęquantur periphe
              <lb/>
            riæ d m f, d h z per 28 p 3:</s>
            <s xml:id="echoid-s15835" xml:space="preserve"> & per eandem peripheriæ d m & d h ęquales concluſæ ſunt:</s>
            <s xml:id="echoid-s15836" xml:space="preserve"> reliqua igitur
              <lb/>
            m f æquatur reliquæ z h.</s>
            <s xml:id="echoid-s15837" xml:space="preserve">] Ergo arcus l f eſt ęqualis arcui l z [per 3 ax:</s>
            <s xml:id="echoid-s15838" xml:space="preserve"> quare per 27 p 3 anguli l d f, l d
              <lb/>
            z ęquabuntur.</s>
            <s xml:id="echoid-s15839" xml:space="preserve">] Et continuemus lineas h b, h f, f m, b m, f z, f b.</s>
            <s xml:id="echoid-s15840" xml:space="preserve"> Angulus ergo b h d eſt acutus [quia
              <lb/>
            l h d rectus eſt concluſus] & angulus g d h rectus [per fabricationem.</s>
            <s xml:id="echoid-s15841" xml:space="preserve">] Ergo linea h b concurret cũ
              <lb/>
            linea d g extra circulum [per 11 ax.</s>
            <s xml:id="echoid-s15842" xml:space="preserve">] Concurrant ergo in q:</s>
            <s xml:id="echoid-s15843" xml:space="preserve"> h f ergo concurret etiam cum d g extra
              <lb/>
            circulum [eadem de cauſſa.</s>
            <s xml:id="echoid-s15844" xml:space="preserve">] Concurrant ergo in n.</s>
            <s xml:id="echoid-s15845" xml:space="preserve"> Et extrahamus f b, quouſque ſecet arcum l z:</s>
            <s xml:id="echoid-s15846" xml:space="preserve">
              <lb/>
            ſecet ergo in r:</s>
            <s xml:id="echoid-s15847" xml:space="preserve"> & continuemus r m:</s>
            <s xml:id="echoid-s15848" xml:space="preserve"> angulus ergo f r m, qui eſt in circumferentia, reſpicit arcum f m:</s>
            <s xml:id="echoid-s15849" xml:space="preserve">
              <lb/>
            & [per 16 p 1] angulus f b m eſt maior angulo f r m:</s>
            <s xml:id="echoid-s15850" xml:space="preserve"> & angulus f b m eſt in circumferentia a b g.</s>
            <s xml:id="echoid-s15851" xml:space="preserve"> Ergo
              <lb/>
            ſi b m linea extrahatur ex parte m:</s>
            <s xml:id="echoid-s15852" xml:space="preserve"> abſcindet de circulo a b g arcum maiorem ſimili arcui f m circuli
              <lb/>
            l h d [per 33 p 6] & arcus f m eſt ſimilis duplo arcus f e:</s>
            <s xml:id="echoid-s15853" xml:space="preserve"> [angulus enim duplus anguli f d e in periphe
              <lb/>
            ria circuli a b g conſtituti, inſiſtit in peripheriam duplam peripheriæ f e per 33 p 6] & arcus f e eſt æ-
              <lb/>
            qualis arcui a z:</s>
            <s xml:id="echoid-s15854" xml:space="preserve"> [quia enim anguli a d b, e d b ęquantur propter angulum a d e per rectam b d bifa-
              <lb/>
            riam ſectum:</s>
            <s xml:id="echoid-s15855" xml:space="preserve"> & z d b, f d b per concluſionem:</s>
            <s xml:id="echoid-s15856" xml:space="preserve"> ęquabitur reliquus a d z reliquo f d e:</s>
            <s xml:id="echoid-s15857" xml:space="preserve"> ideoq́ue peri-
              <lb/>
            pherię a z peripherię f e per 26 p 3] & arcus a z eſt ęqualis arcui e g [per 26 p 3:</s>
            <s xml:id="echoid-s15858" xml:space="preserve"> quia angulus a d z
              <lb/>
            ęquatus eſt angulo e d g.</s>
            <s xml:id="echoid-s15859" xml:space="preserve">] Ergo arcus f e eſt ęqualis arcui e g:</s>
            <s xml:id="echoid-s15860" xml:space="preserve"> ergo arcus g f eſt duplus arcus g e:</s>
            <s xml:id="echoid-s15861" xml:space="preserve"> er
              <lb/>
            go arcus g f eſt ſimilis arcui f m.</s>
            <s xml:id="echoid-s15862" xml:space="preserve"> Si ergo b m extrahatur rectè in partem m:</s>
            <s xml:id="echoid-s15863" xml:space="preserve"> abſcindet de circulo a b g
              <lb/>
            arcum ultra punctum g, maiorem arcu f g.</s>
            <s xml:id="echoid-s15864" xml:space="preserve"> Linea ergo b m ſecabit lineam d g inter duo puncta g, d.</s>
            <s xml:id="echoid-s15865" xml:space="preserve">
              <lb/>
            Secet ergo in o:</s>
            <s xml:id="echoid-s15866" xml:space="preserve"> & extrahamus lineam f m:</s>
            <s xml:id="echoid-s15867" xml:space="preserve"> & ſecet d o in u:</s>
            <s xml:id="echoid-s15868" xml:space="preserve"> [ſecabit autem:</s>
            <s xml:id="echoid-s15869" xml:space="preserve"> quia ſecat angulum
              <lb/>
            d m o à baſi d o ſubtenſum] & extrahamus b m in parte b:</s>
            <s xml:id="echoid-s15870" xml:space="preserve"> & ſecet arcum l r in c:</s>
            <s xml:id="echoid-s15871" xml:space="preserve"> & continuemus
              <lb/>
            c d.</s>
            <s xml:id="echoid-s15872" xml:space="preserve"> Quia ergo angulus b f z eſt in circumferentia a b g:</s>
            <s xml:id="echoid-s15873" xml:space="preserve"> erit [per 20 p 3] angulus b f z dimidius angu
              <lb/>
            li b d z:</s>
            <s xml:id="echoid-s15874" xml:space="preserve"> ſed angulus b d z eſt multiplus anguli z d a:</s>
            <s xml:id="echoid-s15875" xml:space="preserve"> [è fabricatione] ergo angulus b f z eſt multi-
              <lb/>
            plus anguli z d h:</s>
            <s xml:id="echoid-s15876" xml:space="preserve"> ergo [per 33 p 6] arcus r z eſt multiplus arcus z h:</s>
            <s xml:id="echoid-s15877" xml:space="preserve"> & arcus c z eſt maior arcu r z
              <lb/>
            [per 9 axiom.</s>
            <s xml:id="echoid-s15878" xml:space="preserve">] ergo arcus c z eſt multiplus arcus z h.</s>
            <s xml:id="echoid-s15879" xml:space="preserve"> Et continuemus c h:</s>
            <s xml:id="echoid-s15880" xml:space="preserve"> angulus ergo c h d cum
              <lb/>
            angulo c m d, eſt æqualis duobus rectis:</s>
            <s xml:id="echoid-s15881" xml:space="preserve"> [per 22 p 3] ergo angulus c h d eſt æqualis angulo b m e.</s>
            <s xml:id="echoid-s15882" xml:space="preserve">
              <lb/>
            [Nam per 13 p 1 anguli c m d, c m e ęquantur duobus rectis, quibus etiam ęquantur per proximam
              <lb/>
            concluſionem c h d, c m d:</s>
            <s xml:id="echoid-s15883" xml:space="preserve"> communi igitur c m d ſubducto, reliquus c h d æquabitur reliquo c m e
              <lb/>
            ſeu b m e.</s>
            <s xml:id="echoid-s15884" xml:space="preserve">] Sed angulus z h d addit ſuper angulum c h d, angulum c h z, qui eſt æqualis angulo c
              <lb/>
            d z:</s>
            <s xml:id="echoid-s15885" xml:space="preserve"> [per 27 p 3:</s>
            <s xml:id="echoid-s15886" xml:space="preserve"> quia uterque inſiſtit in eandem peripheriam c z] & angulus c d z eſt multiplus an-
              <lb/>
            guli z d a, [per 33 p 6:</s>
            <s xml:id="echoid-s15887" xml:space="preserve"> quia peripheria c z multiplex oſtenſa eſt peripheriæ z h.</s>
            <s xml:id="echoid-s15888" xml:space="preserve">] Ergo angulus c h z
              <lb/>
            eſt multiplus anguli e d g:</s>
            <s xml:id="echoid-s15889" xml:space="preserve"> [quia multiplex eſt ad angulum z d h, æqualem ipſi e d g.</s>
            <s xml:id="echoid-s15890" xml:space="preserve">] Ergo angu-
              <lb/>
            lus z h d excedit angulum c h d multiplo anguli e d g.</s>
            <s xml:id="echoid-s15891" xml:space="preserve"> Angulus ergo z h d eſt æqualis angulo f m d:</s>
            <s xml:id="echoid-s15892" xml:space="preserve">
              <lb/>
            quia arcus f m d eſt ęqualis arcui z h d [per concluſionem.</s>
            <s xml:id="echoid-s15893" xml:space="preserve"> Itaque per 2 ax.</s>
            <s xml:id="echoid-s15894" xml:space="preserve"> peripheria z f d, in quam
              <lb/>
            inſiſtit angulus z h d, ęquabitur peripheriæ f z d, in quam inſ
              <gap/>
            ſtit angulus f m d:</s>
            <s xml:id="echoid-s15895" xml:space="preserve"> & idcirco z h d æ-
              <lb/>
            quabitur f m d per 27 p 3] & angulus c h d, ut declarauimus, eſt ęqualis angulo b m e.</s>
            <s xml:id="echoid-s15896" xml:space="preserve"> Ergo angu-
              <lb/>
            lus f m d excedit angulum b m e multiplo anguli e d g:</s>
            <s xml:id="echoid-s15897" xml:space="preserve"> ergo angulus f m d excedit angulum o m d
              <lb/>
            multiplo anguli e d g:</s>
            <s xml:id="echoid-s15898" xml:space="preserve"> [quia angulus o m d ęquatur angulo b m e per 15 p 1] & angulus m o g exce-
              <lb/>
            dit angulum o m d angulo e d g [nam angulus m o g æquatur angulis o m d & e d g per 32 p 1.</s>
            <s xml:id="echoid-s15899" xml:space="preserve">] Er-
              <lb/>
            go angulus f m d excedit angulum m o g, multiplo anguli e d g:</s>
            <s xml:id="echoid-s15900" xml:space="preserve"> & angulus f m d excedit angulum
              <lb/>
            m u d, angulo e d g ſolo:</s>
            <s xml:id="echoid-s15901" xml:space="preserve"> [quia per 32 p 1 æquatur angulis m d u ſeu e d g & m u d] ergo angulus m u d
              <lb/>
            eſt maior angulo m o g:</s>
            <s xml:id="echoid-s15902" xml:space="preserve"> ergo angulus m o u eſt maior angulo m u o:</s>
            <s xml:id="echoid-s15903" xml:space="preserve"> [Nam quia anguli ad u dein-
              <lb/>
            ceps ęquantur angulis ad o deinceps per 13 p 1:</s>
            <s xml:id="echoid-s15904" xml:space="preserve"> & m u d maior concluſus m o g:</s>
            <s xml:id="echoid-s15905" xml:space="preserve"> reliquus igitur m o u
              <lb/>
            maior eſt reliquo m u o] ergo [per 19 p 1] linea m u eſt maior linea m o.</s>
            <s xml:id="echoid-s15906" xml:space="preserve"> Et quia arcus z h d eſt ęqua
              <lb/>
            lis arcui f m d:</s>
            <s xml:id="echoid-s15907" xml:space="preserve"> erunt duo anguli h f d, m f d æquales [per 27 p 3:</s>
            <s xml:id="echoid-s15908" xml:space="preserve"> quia peripheriæ h d, m d æquales
              <lb/>
            ſunt concluſæ.</s>
            <s xml:id="echoid-s15909" xml:space="preserve">] Duæ ergo lineæ h f, f u reflectentur æqualiter:</s>
            <s xml:id="echoid-s15910" xml:space="preserve"> & ſimiliter h b, b o reflectentur ęqua
              <lb/>
            liter [propter concluſam æqualitatem angulorum h b d, o b d] q ergo eſt imago o:</s>
            <s xml:id="echoid-s15911" xml:space="preserve"> & n imago u [per
              <lb/>
            6 n 5.</s>
            <s xml:id="echoid-s15912" xml:space="preserve">] Et extrahamus ex m lineam æquidiſtantem lineæ h q [per 31 p 1] & ſit m s:</s>
            <s xml:id="echoid-s15913" xml:space="preserve"> & extrahamus ex
              <lb/>
            m etiam lineam æquidiſtantem lineæ h n:</s>
            <s xml:id="echoid-s15914" xml:space="preserve"> & ſit m p.</s>
            <s xml:id="echoid-s15915" xml:space="preserve"> Quia ergo [per 16 p 1] angulus h n d eſt maior
              <lb/>
            angulo h q d:</s>
            <s xml:id="echoid-s15916" xml:space="preserve"> erit angulus m p o maior angulo m s o.</s>
            <s xml:id="echoid-s15917" xml:space="preserve"> [nam per 29 p 1 angulus m s o æ quatur angu-
              <lb/>
            lo ad q, & angulus m p o æquatur angulo ad n] p ergo erit inter duo puncta s, u.</s>
            <s xml:id="echoid-s15918" xml:space="preserve"> Et quia angulus
              <lb/>
            h d n eſt rectus [ex theſi:</s>
            <s xml:id="echoid-s15919" xml:space="preserve">] erit angulus h n d acutus [per 32 p 1] ergo angulus m p d eſt acutus:</s>
            <s xml:id="echoid-s15920" xml:space="preserve"> ergo
              <lb/>
            [per 13 p 1] angulus m p s eſt obtuſus:</s>
            <s xml:id="echoid-s15921" xml:space="preserve"> ergo [per 19 p 1] linea m s eſt maior, quàm m p:</s>
            <s xml:id="echoid-s15922" xml:space="preserve"> ſed m u eſt ma-
              <lb/>
            ior, quàm m o, ut diximus:</s>
            <s xml:id="echoid-s15923" xml:space="preserve"> ergo proportio s m ad m o eſt maior, quàm proportio p m ad m u:</s>
            <s xml:id="echoid-s15924" xml:space="preserve"> [ut
              <lb/>
            patet per 8 p 5] & [per 29 p 1.</s>
            <s xml:id="echoid-s15925" xml:space="preserve">4 p 6] proportio s m ad m o eſt, ſicut proportio q b ad b o:</s>
            <s xml:id="echoid-s15926" xml:space="preserve"> quia m s eſt
              <lb/>
            æquidiſtans b q:</s>
            <s xml:id="echoid-s15927" xml:space="preserve"> & ſimiliter proportio p m ad m u eſt, ſicut proportio n f a d f u:</s>
            <s xml:id="echoid-s15928" xml:space="preserve"> ergo [per 11 p 5] pro-
              <lb/>
            portio q b ad b o eſt maior, quàm proportio n f ad f u:</s>
            <s xml:id="echoid-s15929" xml:space="preserve"> & proportio q b ad b o eſt, ſicut proportio q d
              <lb/>
            ad d o:</s>
            <s xml:id="echoid-s15930" xml:space="preserve"> & proportio n f ad f u eſt, ſicut proportio n d ad d u, ut declarauimus in capitulo de imagine
              <lb/>
            [64 n 5.</s>
            <s xml:id="echoid-s15931" xml:space="preserve">] Ergo proportio q d ad d o eſt maior, quàm proportio n d ad d u.</s>
            <s xml:id="echoid-s15932" xml:space="preserve"> [Eſt autem q, imago pun-
              <lb/>
            cti o, à centro ſpeculi d longinquior:</s>
            <s xml:id="echoid-s15933" xml:space="preserve"> & o punctum uiſibilis ab eodem centro eſt lon-
              <lb/>
            ginquius.</s>
            <s xml:id="echoid-s15934" xml:space="preserve"> n uerò, imago puncti u centro ſpeculi d eſt propinquior:</s>
            <s xml:id="echoid-s15935" xml:space="preserve"> & u
              <lb/>
            alterum uiſibilis punctum eodem centro d eſt propinquius.</s>
            <s xml:id="echoid-s15936" xml:space="preserve">]
              <lb/>
            Quare patet propoſitum.</s>
            <s xml:id="echoid-s15937" xml:space="preserve"/>
          </p>
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      </text>
    </echo>
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