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-s42599" xml:space="preserve">
              <pb o="347" file="0649" n="649" rhead="LIBER OCTAVVS."/>
            æqualis angulo f r g, extrinſecus intrinſeco:</s>
            <s xml:id="echoid-s42600" xml:space="preserve"> quod eſt impoſsibile.</s>
            <s xml:id="echoid-s42601" xml:space="preserve"> Linea ergo b m non cadetin pun
              <lb/>
            ctum g, ſed ſecabit lineam d g inter duo puncta g & d:</s>
            <s xml:id="echoid-s42602" xml:space="preserve"> ſecet ergo in puncto o.</s>
            <s xml:id="echoid-s42603" xml:space="preserve"> Producatur quoq;</s>
            <s xml:id="echoid-s42604" xml:space="preserve"> li-
              <lb/>
            nea f m ultra punctum m:</s>
            <s xml:id="echoid-s42605" xml:space="preserve"> hæc ergo, quia ſecat angulum d m o, patet per 29 th.</s>
            <s xml:id="echoid-s42606" xml:space="preserve"> 1 huius quia ſecabit li
              <lb/>
            neam d o:</s>
            <s xml:id="echoid-s42607" xml:space="preserve"> ſecet illã in pũcto u:</s>
            <s xml:id="echoid-s42608" xml:space="preserve"> & producatur linea m b ultra punctũ b:</s>
            <s xml:id="echoid-s42609" xml:space="preserve"> ſecabitq́ arcũ l r:</s>
            <s xml:id="echoid-s42610" xml:space="preserve"> ſecet ipſum
              <lb/>
            in puncto c:</s>
            <s xml:id="echoid-s42611" xml:space="preserve"> & ducatur linea c d à puncto c ad centrũ ſpeculi.</s>
            <s xml:id="echoid-s42612" xml:space="preserve"> Quia ergo angulus b f z eſt in circum-
              <lb/>
            ferentia circuli a b g, erit angulus b f z medietas anguli b d z ք zo p 3:</s>
            <s xml:id="echoid-s42613" xml:space="preserve"> ſed angulus b d z eſt multiplus
              <lb/>
            anguli z d a:</s>
            <s xml:id="echoid-s42614" xml:space="preserve"> ergo angulus b f z multiplus angulo z d h:</s>
            <s xml:id="echoid-s42615" xml:space="preserve"> ergo & angulus r f z eſt multiplus eidẽ:</s>
            <s xml:id="echoid-s42616" xml:space="preserve"> ergo
              <lb/>
            per 33 p 6 arcus r z eſt multiplus arcui z h:</s>
            <s xml:id="echoid-s42617" xml:space="preserve"> arcus uerò c z eſt maior arcu r z, ut totũ ſua parte:</s>
            <s xml:id="echoid-s42618" xml:space="preserve"> ergo ar-
              <lb/>
            cus c z eſt multiplus arcus z h, uel maior multiplo.</s>
            <s xml:id="echoid-s42619" xml:space="preserve"> Ducatur itaq;</s>
            <s xml:id="echoid-s42620" xml:space="preserve"> linea c h:</s>
            <s xml:id="echoid-s42621" xml:space="preserve"> angulus ergo c h d & an-
              <lb/>
            gulus c m d ſunt æquales duobus rectis per 22 p 3:</s>
            <s xml:id="echoid-s42622" xml:space="preserve"> ſed angulus b m d cũ angulo b m e ualet duos re-
              <lb/>
            ctos per 13 p 1:</s>
            <s xml:id="echoid-s42623" xml:space="preserve"> relin quitur ergo ut angulus c h d ſit æqualis angulo b m e:</s>
            <s xml:id="echoid-s42624" xml:space="preserve"> ſed angulus z h d addit ſu-
              <lb/>
            per angulum c h d angulũ c h z, qui eſt per 27 p 3 æqualis angulo c d z:</s>
            <s xml:id="echoid-s42625" xml:space="preserve"> & angulus c d z eſt multiplus
              <lb/>
            anguli z d a per 33 p 6:</s>
            <s xml:id="echoid-s42626" xml:space="preserve"> quoniã, ut ſuprà patuit, arcus c z eſt multiplus arcui z h:</s>
            <s xml:id="echoid-s42627" xml:space="preserve"> ergo angulus c h z eſt
              <lb/>
            multiplus anguli e d g:</s>
            <s xml:id="echoid-s42628" xml:space="preserve"> angulus ergo d h z excedit angulum c h d in multiplo anguli e d g.</s>
            <s xml:id="echoid-s42629" xml:space="preserve"> Et quia ar
              <lb/>
            cus f m d eſt æqualis arcuι z h d per 64 th.</s>
            <s xml:id="echoid-s42630" xml:space="preserve"> 1 huius, remanet arcus f z d æqualis arcui z f d:</s>
            <s xml:id="echoid-s42631" xml:space="preserve"> ergo erit
              <lb/>
            per 27 p 3 angulus f m d æqualis angulo z h d:</s>
            <s xml:id="echoid-s42632" xml:space="preserve"> ſed angulus c h d eſt æqualis b m e:</s>
            <s xml:id="echoid-s42633" xml:space="preserve"> ergo angulus f m d
              <lb/>
            excedit angulum b m e in multiplo anguli e g d:</s>
            <s xml:id="echoid-s42634" xml:space="preserve"> ſed angulus o m d eſt æqualis angulo b m e per 15
              <lb/>
            p 1:</s>
            <s xml:id="echoid-s42635" xml:space="preserve"> ergo angulus f m d excedit angulum o m d in multiplo anguli e d g.</s>
            <s xml:id="echoid-s42636" xml:space="preserve"> Et quia angulus g o m ualet
              <lb/>
            angulum o m d, & angulũ o d m per 32 p 1:</s>
            <s xml:id="echoid-s42637" xml:space="preserve"> palàm quia angulus f m d excedit angulum m o g in mul-
              <lb/>
            tiplo anguli e d g:</s>
            <s xml:id="echoid-s42638" xml:space="preserve"> ſed angulus f m d per 32 p 1 excedit angulum m u d in ſolo angulo e d g:</s>
            <s xml:id="echoid-s42639" xml:space="preserve"> eſt ergo
              <lb/>
            angulus m u d maior angulo m o g:</s>
            <s xml:id="echoid-s42640" xml:space="preserve"> ergo angulus m o u eſt maior angulo m u o per 13 p 1 bιs ſumptá:</s>
            <s xml:id="echoid-s42641" xml:space="preserve">
              <lb/>
            ergo per 19 p 1 linea m u eſt maior quàm linea m o.</s>
            <s xml:id="echoid-s42642" xml:space="preserve"> Et quia arcus h d eſt æqualis arcuι m d per præ-
              <lb/>
            miſſa, erũt duo anguli h f d & m f d æquales per 27 p 3.</s>
            <s xml:id="echoid-s42643" xml:space="preserve"> Formæ ergo punctorum duarum linearum h
              <lb/>
            f & f u ad ſeinuicem reflectuntur:</s>
            <s xml:id="echoid-s42644" xml:space="preserve"> & ſimiliter formæ punctorum linearum h b & b o ad ſe inuicẽ re-
              <lb/>
            flectuntur:</s>
            <s xml:id="echoid-s42645" xml:space="preserve"> quoniã per præmiſſa angulus d b h eſt æqualis angulo d b m per 4 p 1 & per hypotheſes
              <lb/>
            præmiſſas.</s>
            <s xml:id="echoid-s42646" xml:space="preserve"> Duo ergo puncta, quæ ſunt o & u ad uiſum exiſtentem in puncto h reflectũtur à duobus
              <lb/>
            punctis ſpeculi, quæ ſunt b & f.</s>
            <s xml:id="echoid-s42647" xml:space="preserve"> Eſt ergo per 37 th.</s>
            <s xml:id="echoid-s42648" xml:space="preserve"> 5 huius punctus q imago puncti o, & punctus n
              <lb/>
            imago puncti u.</s>
            <s xml:id="echoid-s42649" xml:space="preserve"> Ducatur ergo expũcto m linea æquidiſtans lineæ h q per 31 p 1:</s>
            <s xml:id="echoid-s42650" xml:space="preserve"> quæ ſit linea m s:</s>
            <s xml:id="echoid-s42651" xml:space="preserve"> &
              <lb/>
            linea ęquidiſtans lineę h n, quę ſit m p.</s>
            <s xml:id="echoid-s42652" xml:space="preserve"> Quia ergo angulus h n d eſt maior angulo h q d per 16 p 1, erit
              <lb/>
            angulus m p o, qui per 29 p 1 eſt æqualis angulo h n d, maior angulo m s o, qui per 29 p 1 eſt æqualis
              <lb/>
            h q d:</s>
            <s xml:id="echoid-s42653" xml:space="preserve"> erit ergo punctum p inter duo puncta s & u per conuerſam 21 p 1.</s>
            <s xml:id="echoid-s42654" xml:space="preserve"> Et quia angulus h d n eſt re-
              <lb/>
            ctus:</s>
            <s xml:id="echoid-s42655" xml:space="preserve"> erit per 32 p 1 angulus h n d acutus:</s>
            <s xml:id="echoid-s42656" xml:space="preserve"> ergo angulus m p d eſt acutus:</s>
            <s xml:id="echoid-s42657" xml:space="preserve"> angulus ergo m p s eſt obtu-
              <lb/>
            ſus per 13 p 1:</s>
            <s xml:id="echoid-s42658" xml:space="preserve"> ergo linea m s eſt maior quàm linea m p per 19 p 1.</s>
            <s xml:id="echoid-s42659" xml:space="preserve"> Sed ex pręmiſsis linea m u eſt maior
              <lb/>
            quàm linea m o:</s>
            <s xml:id="echoid-s42660" xml:space="preserve"> ergo per 9 th.</s>
            <s xml:id="echoid-s42661" xml:space="preserve"> 1 huius maior eſt proportio lineæ m s ad lineam m o quàm lineæ m p
              <lb/>
            ad lineam m u:</s>
            <s xml:id="echoid-s42662" xml:space="preserve"> ſed proportio lineæ s m ad lineam m o eſt, ſicut proportio lineæ q b ad b o per 4 p 6:</s>
            <s xml:id="echoid-s42663" xml:space="preserve">
              <lb/>
            trigoni enim q b o & s m o ſunt æquianguli per 29 p 1:</s>
            <s xml:id="echoid-s42664" xml:space="preserve"> cum lineam s ſit æquidiſtans lineæ q b, & an-
              <lb/>
            gulus q o b ſit communis illis ambobus trigonis.</s>
            <s xml:id="echoid-s42665" xml:space="preserve"> Et ſimiliter proportio lineæ p m ad lineã m u eſt,
              <lb/>
            ſicut proportio lineæ n f ad lineam f u:</s>
            <s xml:id="echoid-s42666" xml:space="preserve"> per eadem ergo, quæ prius, & per 11 p 5 erit proportio lineæ
              <lb/>
            q b ad lineam b o maior proportione lineæ n f ad lineam f u:</s>
            <s xml:id="echoid-s42667" xml:space="preserve"> ſed proportio lineæ q b ad lineam b o
              <lb/>
            eſt, ſicut lineæ q d ad lineam d o:</s>
            <s xml:id="echoid-s42668" xml:space="preserve"> & proportio lineæ n f ad f u eſt, ſicut lineæ n d ad d u per ea, quæ
              <lb/>
            ſunt oſtenſa in 13 huius, quorum declarationem, cum manifeſta ſit, hic omittimus propter figuratio
              <lb/>
            nis multitudinem.</s>
            <s xml:id="echoid-s42669" xml:space="preserve"> Palàm ergo quòd proportio lineę q d ad lineam d o eſt maior proportione lineę
              <lb/>
            n d ad lineam d u.</s>
            <s xml:id="echoid-s42670" xml:space="preserve"> Et hoc eſt propoſitum.</s>
            <s xml:id="echoid-s42671" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1677" type="section" level="0" n="0">
          <head xml:id="echoid-head1254" xml:space="preserve" style="it">44. In ſpeculis ſphæricis concauis imagine retro ſpecu-
            <lb/>
          lum occurrente: maior erit diſtantia imaginis à ſpeculo quàm
            <lb/>
          reiuiſæ.</head>
          <figure number="780">
            <variables xml:id="echoid-variables757" xml:space="preserve">n t l m s h s b k d e z a</variables>
          </figure>
          <p>
            <s xml:id="echoid-s42672" xml:space="preserve">Eſto ſpeculi ſphærici concaui circulus, qui a b g d:</s>
            <s xml:id="echoid-s42673" xml:space="preserve"> cuius cẽtrum
              <lb/>
            ſit e:</s>
            <s xml:id="echoid-s42674" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s42675" xml:space="preserve"> centrum uiſus z:</s>
            <s xml:id="echoid-s42676" xml:space="preserve"> & punctus rei uiſæ h:</s>
            <s xml:id="echoid-s42677" xml:space="preserve"> fiatq́;</s>
            <s xml:id="echoid-s42678" xml:space="preserve"> reflexio for
              <lb/>
            mæ puncti h ad uiſum z à puncto ſpeculi b, appareatq́;</s>
            <s xml:id="echoid-s42679" xml:space="preserve"> imago retro
              <lb/>
            ſpeculum:</s>
            <s xml:id="echoid-s42680" xml:space="preserve"> dico quòd maior erit diſtantia imaginis à ſpeculi ſuperfi
              <lb/>
            cie quàm ipſius rei uiſæ.</s>
            <s xml:id="echoid-s42681" xml:space="preserve"> Ducantur enim lineæ h b incidentię, & z b
              <lb/>
            reflexionis:</s>
            <s xml:id="echoid-s42682" xml:space="preserve"> & ducatur cathetus incidentiæ, quæ ſit e h g t:</s>
            <s xml:id="echoid-s42683" xml:space="preserve"> produ-
              <lb/>
            catur quoque linea reflexionis, quæ z b, donec lineæ e h & z b
              <lb/>
            concurrant in puncto t:</s>
            <s xml:id="echoid-s42684" xml:space="preserve"> erit ergo per 37 th.</s>
            <s xml:id="echoid-s42685" xml:space="preserve"> 5 huius punctum t lo-
              <lb/>
            cus imaginis.</s>
            <s xml:id="echoid-s42686" xml:space="preserve"> Dico quòd linea t b (quæ eſt diſtantia imaginis à
              <lb/>
            ſpeculo) eſt maior quàm linea b h, quæ eſt diſtantia rei uiſæ à pun-
              <lb/>
            cto reflexionis.</s>
            <s xml:id="echoid-s42687" xml:space="preserve"> Et ſimiliter linea h g eſt minor quàm linea g t.</s>
            <s xml:id="echoid-s42688" xml:space="preserve"> Du-
              <lb/>
            catur enim linea e b:</s>
            <s xml:id="echoid-s42689" xml:space="preserve"> & à puncto b ducatur linea contingens cir-
              <lb/>
            culum in puncto b per 17 p 3:</s>
            <s xml:id="echoid-s42690" xml:space="preserve"> quæ ſit l b k.</s>
            <s xml:id="echoid-s42691" xml:space="preserve"> Quia itaque anguli
              <lb/>
            cõtingentiæ, qui ſunt a b k & g b l, ſunt æquales per 16 p 3:</s>
            <s xml:id="echoid-s42692" xml:space="preserve"> & anguli
              <lb/>
            z b a & h b g æquales per 20 th.</s>
            <s xml:id="echoid-s42693" xml:space="preserve"> 5 huius:</s>
            <s xml:id="echoid-s42694" xml:space="preserve"> fit ergo angulus k b z æqualis angulo l b h:</s>
            <s xml:id="echoid-s42695" xml:space="preserve"> ſed angulus t b l
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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