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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            tet ergo ex hac figura, quòd linea recta in ſpeculis concauis comprehendatur concaua:</s>
            <s xml:id="echoid-s15771" xml:space="preserve"> & con-
              <lb/>
            uexa comprehendatur concaua:</s>
            <s xml:id="echoid-s15772" xml:space="preserve"> & quòd recta habet plures formas concauas.</s>
            <s xml:id="echoid-s15773" xml:space="preserve"/>
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        <div xml:id="echoid-div530" type="section" level="0" n="0">
          <head xml:id="echoid-head464" xml:space="preserve" style="it">48. Si duo uiſibilis puncta à duob{us} ſpeculi ſphærici caui punctis adunum uiſum reflexa,
            <lb/>
          in eadem ſpeculi diametro imagines ſu{as} habeant: recta inter centrum ſpeculi & imaginem
            <lb/>
          longinquiorem, ad rectam inter idem centrum & punctum uiſibilis à ſpeculi centro lon-
            <lb/>
          ginqui{us}, maiorem rationem habet: quàm recta inter ſpeculi centrum & imaginem pro-
            <lb/>
          pinquiorem, ad rectam inter idem centrum & punctum uiſibilis centro ſpeculi propin-
            <lb/>
          quius. 43 p 8.</head>
          <p>
            <s xml:id="echoid-s15774" xml:space="preserve">ITem:</s>
            <s xml:id="echoid-s15775" xml:space="preserve"> ſit ſpeculum concauum, per cuius centrum tranſeat plana ſuperficies:</s>
            <s xml:id="echoid-s15776" xml:space="preserve"> & faciat circu-
              <lb/>
            lum a b g [faciet autem per 1 th.</s>
            <s xml:id="echoid-s15777" xml:space="preserve"> 1 ſphær.</s>
            <s xml:id="echoid-s15778" xml:space="preserve">] & ſit centrum d:</s>
            <s xml:id="echoid-s15779" xml:space="preserve"> & extrahamus ex d lineam, quo-
              <lb/>
            cunque modo ſit:</s>
            <s xml:id="echoid-s15780" xml:space="preserve"> & ſit d g:</s>
            <s xml:id="echoid-s15781" xml:space="preserve"> & tranſeat extra circulum:</s>
            <s xml:id="echoid-s15782" xml:space="preserve"> & extrahamus ex d in ſuperficie huius
              <lb/>
            circuli lineam perpendicularem ſuper lineam d g [per 11 p 1] & ſit d a:</s>
            <s xml:id="echoid-s15783" xml:space="preserve"> & abſcindamus de angu-
              <lb/>
            lo a d g recto particulam paruam, quomodocunque ſit:</s>
            <s xml:id="echoid-s15784" xml:space="preserve"> & ſit angulus g d e, ita ut inter angu-
              <lb/>
            lum rectum & angulum a d e ſit multiplum anguli e d g:</s>
            <s xml:id="echoid-s15785" xml:space="preserve"> [id quod fieri poteſt continua anguli
              <lb/>
            recti biſſectione, donec angulus a d e ſit multiplex ad angulum e d g] & diuidamus angulum
              <lb/>
            a d e in duo æqualia, per lineam d b [per 9 p 1] & abſcindamus de angulo b d a æqualem an-
              <lb/>
            gulo e d g, per lineam z d:</s>
            <s xml:id="echoid-s15786" xml:space="preserve"> & extrahamus ex d lineam continentem cum b d angulum rectum:</s>
            <s xml:id="echoid-s15787" xml:space="preserve">
              <lb/>
            & ſit d x:</s>
            <s xml:id="echoid-s15788" xml:space="preserve"> & extrahamus a d in parte d:</s>
            <s xml:id="echoid-s15789" xml:space="preserve"> & ſit d k:</s>
            <s xml:id="echoid-s15790" xml:space="preserve"> & extrahamus ex z lineam continentem cum z d
              <lb/>
            angulum, æqualem angulo k d x:</s>
            <s xml:id="echoid-s15791" xml:space="preserve"> & ſit z h.</s>
            <s xml:id="echoid-s15792" xml:space="preserve"> Hæc ergo linea concurret cum d a:</s>
            <s xml:id="echoid-s15793" xml:space="preserve"> [per 11 ax.</s>
            <s xml:id="echoid-s15794" xml:space="preserve">] Nam
              <lb/>
            duo anguli k d x, a d z ſunt minores duobus rectis [ideoq́ue a d z, h z d ijſdem ſunt minores:</s>
            <s xml:id="echoid-s15795" xml:space="preserve"> quia
              <lb/>
            h z d æquatus eſt angulo k d x.</s>
            <s xml:id="echoid-s15796" xml:space="preserve">] Concurrant ergo in h.</s>
            <s xml:id="echoid-s15797" xml:space="preserve"> Angulus ergo z h d eſt æqualis angulo
              <lb/>
            z d x.</s>
            <s xml:id="echoid-s15798" xml:space="preserve"> [Quia enim tres anguli z d h, z d x, k d x æquantur duobus rectis per 13 p 1:</s>
            <s xml:id="echoid-s15799" xml:space="preserve"> quibus item
              <lb/>
            æquantur tres anguli trianguli z d h per 32 p 1:</s>
            <s xml:id="echoid-s15800" xml:space="preserve"> tres igitur illi tribus his æquantur.</s>
            <s xml:id="echoid-s15801" xml:space="preserve"> Itaque cum
              <lb/>
            z d h communis æquetur ſibi ipſi, & d z h æquatus ſit ipſi k d x:</s>
            <s xml:id="echoid-s15802" xml:space="preserve"> reliquus z h d æquabitur reli-
              <lb/>
            quo z d x.</s>
            <s xml:id="echoid-s15803" xml:space="preserve">] Et extrahamus ex z lineam conti-
              <lb/>
              <figure xlink:label="fig-0228-01" xlink:href="fig-0228-01a" number="199">
                <variables xml:id="echoid-variables188" xml:space="preserve">q s n p e f o
                  <gap/>
                x u m l b
                  <gap/>
                  <gap/>
                z k d h a</variables>
              </figure>
            nentem cum z h angulum, æqualem angulo b d
              <lb/>
            k obtuſo:</s>
            <s xml:id="echoid-s15804" xml:space="preserve"> & ſit z l.</s>
            <s xml:id="echoid-s15805" xml:space="preserve"> Duo ergo anguli l z d, b d z
              <lb/>
            ſunt minores duobus rectis.</s>
            <s xml:id="echoid-s15806" xml:space="preserve"> [Quia enim angu-
              <lb/>
            li b d k, b d a æquantur duobus rectis per 13 p 1:</s>
            <s xml:id="echoid-s15807" xml:space="preserve">
              <lb/>
            erunt anguli, b d k, id eſt, per fabricationem,
              <lb/>
            l z h, & b d z minores duobus rectis:</s>
            <s xml:id="echoid-s15808" xml:space="preserve"> ideoq́ue
              <lb/>
            l z d, b d z ijſdem multò minores erunt.</s>
            <s xml:id="echoid-s15809" xml:space="preserve">] Li-
              <lb/>
            nea ergo z l concurret cum d b [per 11 ax.</s>
            <s xml:id="echoid-s15810" xml:space="preserve">]
              <lb/>
            Concurrant ergo in l:</s>
            <s xml:id="echoid-s15811" xml:space="preserve"> & continuemus l h:</s>
            <s xml:id="echoid-s15812" xml:space="preserve"> & [per
              <lb/>
            5 p 4] circa triangulum h l d faciamus circu-
              <lb/>
            lum d h l:</s>
            <s xml:id="echoid-s15813" xml:space="preserve"> tranſibit ergo per z [per conuerſio-
              <lb/>
            nem 22 p 3] quia duo anguli l z h, l d h ſunt æ-
              <lb/>
            quales duobus rectis [quia æquantur duobus
              <lb/>
            angulis b d k, l d h æqualibus duobus rectis
              <lb/>
            per 13 p 1.</s>
            <s xml:id="echoid-s15814" xml:space="preserve">] Anguli ergo l h z, l d z ſunt æquales
              <lb/>
            [per 27 p 3] quia baſis eorum eſt idem arcus:</s>
            <s xml:id="echoid-s15815" xml:space="preserve">
              <lb/>
            [l z] ſed angulus z h d eſt æqualis angulo z d
              <lb/>
            x:</s>
            <s xml:id="echoid-s15816" xml:space="preserve"> [per concluſionem] remanet ergo angulus
              <lb/>
            l h d æqualis angulo l d x:</s>
            <s xml:id="echoid-s15817" xml:space="preserve"> & angulus l d x eſt
              <lb/>
            rectus:</s>
            <s xml:id="echoid-s15818" xml:space="preserve"> [per fabricationem] ergo angulus l h d
              <lb/>
            eſt rectus.</s>
            <s xml:id="echoid-s15819" xml:space="preserve"> Et abſcindamus exlinea d e lineam
              <lb/>
            d m, æqualem d h [per 3 p 1] & continuemus l m.</s>
            <s xml:id="echoid-s15820" xml:space="preserve">
              <lb/>
            Angulus ergo l m d eſt rectus.</s>
            <s xml:id="echoid-s15821" xml:space="preserve"> [quia per 4 p 1
              <lb/>
            æquatur angulo l h d recto concluſo:</s>
            <s xml:id="echoid-s15822" xml:space="preserve"> duo enim
              <lb/>
            latera h d, l d æquantur duobus lateribus m d,
              <lb/>
            l d, & angulus h d l angulo m d l per fabricatio-
              <lb/>
            nem.</s>
            <s xml:id="echoid-s15823" xml:space="preserve">] Circulus ergo l h d tranſit per m [per
              <lb/>
            conuerſionem 31 p 3 demonſtratam à Theone in
              <lb/>
            commentarijs in 3 librum magnæ conſtructio-
              <lb/>
            nis Ptolemæi] & ſecat arcum b e in compari pun
              <lb/>
            cto z.</s>
            <s xml:id="echoid-s15824" xml:space="preserve"> Secet ergo in f:</s>
            <s xml:id="echoid-s15825" xml:space="preserve"> & continuemus d f.</s>
            <s xml:id="echoid-s15826" xml:space="preserve"> An-
              <lb/>
            gulus ergo l d f erit æqualis angulo l d z:</s>
            <s xml:id="echoid-s15827" xml:space="preserve"> [per 27
              <lb/>
            p 3:</s>
            <s xml:id="echoid-s15828" xml:space="preserve"> quia arcus l m eſt æqualis arcui l h.</s>
            <s xml:id="echoid-s15829" xml:space="preserve"> [Quia
              <lb/>
            enim triangulo l m d circulus circumſcriptus
              <lb/>
            eſt, & angulus ad m rectus ex concluſo:</s>
            <s xml:id="echoid-s15830" xml:space="preserve"> erit l d diameter circuli per conſectarium 5 p 4, ſeu
              <lb/>
            </s>
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