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-s14414" xml:space="preserve">
              <pb o="207" file="0213" n="213" rhead="OPTICAE LIBER VI."/>
            gulus g b e eſt rectus.</s>
            <s xml:id="echoid-s14415" xml:space="preserve"> Ergo [per 47 p 1] quadratum lineæ g o ualet quadratum lineæ b g & qua-
              <lb/>
            dratum lineæ b o.</s>
            <s xml:id="echoid-s14416" xml:space="preserve"> Similiter quadratum g e ua
              <lb/>
              <figure xlink:label="fig-0213-01" xlink:href="fig-0213-01a" number="181">
                <variables xml:id="echoid-variables171" xml:space="preserve">t n q z g m b ſ f h r a d e k o</variables>
              </figure>
            let quadrata g b & b e.</s>
            <s xml:id="echoid-s14417" xml:space="preserve"> Et quoniam b e & b o
              <lb/>
            ſunt æquales:</s>
            <s xml:id="echoid-s14418" xml:space="preserve"> [per concluſionem] & g b com
              <lb/>
            munis:</s>
            <s xml:id="echoid-s14419" xml:space="preserve"> erit g o ęqualis g e [quia ipſarum qua-
              <lb/>
            drata æqualia.</s>
            <s xml:id="echoid-s14420" xml:space="preserve">] Igitur [per 5 p 1] angulus g o
              <lb/>
            e ęqualis angulo g e o.</s>
            <s xml:id="echoid-s14421" xml:space="preserve"> Ducta autem perpen-
              <lb/>
            diculari ſuper axem z g n:</s>
            <s xml:id="echoid-s14422" xml:space="preserve"> æquidiſtãs erit e o:</s>
            <s xml:id="echoid-s14423" xml:space="preserve">
              <lb/>
            [per 30 p 1] cum ſit æquidiſtans m b l.</s>
            <s xml:id="echoid-s14424" xml:space="preserve"> Igitur
              <lb/>
            [per 29 p 1] angulus t g n æqualis angulo g o
              <lb/>
            e:</s>
            <s xml:id="echoid-s14425" xml:space="preserve"> & angulus n g e æqualis angulo g e o:</s>
            <s xml:id="echoid-s14426" xml:space="preserve"> quare
              <lb/>
            angulus t g n æqualis n g e.</s>
            <s xml:id="echoid-s14427" xml:space="preserve"> Cum autem t g o,
              <lb/>
            n g z ſint in eadem ſuperficie, in qua g.</s>
            <s xml:id="echoid-s14428" xml:space="preserve"> Ergo
              <lb/>
            puncta o, g, terunt in eadẽ ſuperficie:</s>
            <s xml:id="echoid-s14429" xml:space="preserve"> & ita in
              <lb/>
            eadẽ ſuperficie ſunt lineę e g, o g t g [ք 1 p 11.</s>
            <s xml:id="echoid-s14430" xml:space="preserve">]
              <lb/>
            Igitur t reflectitur ad e à pũcto g.</s>
            <s xml:id="echoid-s14431" xml:space="preserve"> Sumpto aũt
              <lb/>
            in linea th puncto h eiuſdem longitudinis à puncto q, cuius eſt punctũ t, & linea ducta h o:</s>
            <s xml:id="echoid-s14432" xml:space="preserve"> tranſibit
              <lb/>
            quidẽ per punctũ lineæ a g:</s>
            <s xml:id="echoid-s14433" xml:space="preserve"> tranſeat per punctũ a:</s>
            <s xml:id="echoid-s14434" xml:space="preserve"> ductaq́;</s>
            <s xml:id="echoid-s14435" xml:space="preserve"> à puncto a ſuper axẽ perpendiculari d a,
              <lb/>
            & linea e a:</s>
            <s xml:id="echoid-s14436" xml:space="preserve"> erit, ſicut prius, probare:</s>
            <s xml:id="echoid-s14437" xml:space="preserve"> quòd duo anguli a b o, a b e recti:</s>
            <s xml:id="echoid-s14438" xml:space="preserve"> & duo latera a o, a e æqualia:</s>
            <s xml:id="echoid-s14439" xml:space="preserve">
              <lb/>
            & duo anguli h a r, e a r æquales:</s>
            <s xml:id="echoid-s14440" xml:space="preserve"> & ita h reflectetur ad e à puncto a.</s>
            <s xml:id="echoid-s14441" xml:space="preserve"> Similiter ſumpto quocunq, pun
              <lb/>
            cto lineę t h:</s>
            <s xml:id="echoid-s14442" xml:space="preserve"> erit probare, quòd reflectatur ab aliquo puncto lineę a g.</s>
            <s xml:id="echoid-s14443" xml:space="preserve"> Quare linea th reflectetur à
              <lb/>
            linea longitudinis, quæ eſt a g.</s>
            <s xml:id="echoid-s14444" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div495" type="section" level="0" n="0">
          <head xml:id="echoid-head441" xml:space="preserve" style="it">27. Si uiſ{us} ſit extra planum lineæ rectæ, axi ſpeculi cylindracei conuexi parallelæ: imago ui-
            <lb/>
          debitur parum curua, & minor ipſaparallela. 51 p 7.</head>
          <p>
            <s xml:id="echoid-s14445" xml:space="preserve">REſtat probare imaginem lineę t h eſſe curuã.</s>
            <s xml:id="echoid-s14446" xml:space="preserve"> Palàm ex prædictis, quòd q reflectitur ad e à pun
              <lb/>
            cto b, quod eſt punctum circuli.</s>
            <s xml:id="echoid-s14447" xml:space="preserve"> Sed cum ſic reflectatur à circulo:</s>
            <s xml:id="echoid-s14448" xml:space="preserve"> ſi ducatur linea à puncto q,
              <lb/>
            ad centrum illius circuli:</s>
            <s xml:id="echoid-s14449" xml:space="preserve"> concurret cum perpendiculari ducta à puncto b:</s>
            <s xml:id="echoid-s14450" xml:space="preserve"> [quia perpendicu
              <lb/>
            laris illa tranſit per eiuſdem circuli centrum, ut oſtenſum eſt 16 n 5] & erit cõcurſus in puncto axis.</s>
            <s xml:id="echoid-s14451" xml:space="preserve">
              <lb/>
            Ducatur ergo q l, concurrens cum m l in puncto axis:</s>
            <s xml:id="echoid-s14452" xml:space="preserve"> quod eſt l:</s>
            <s xml:id="echoid-s14453" xml:space="preserve"> & eſt centrum circuli f b:</s>
            <s xml:id="echoid-s14454" xml:space="preserve"> & produ-
              <lb/>
            catur e b, quouſq;</s>
            <s xml:id="echoid-s14455" xml:space="preserve"> concurrat cum q l.</s>
            <s xml:id="echoid-s14456" xml:space="preserve"> Sit concurſus in puncto c.</s>
            <s xml:id="echoid-s14457" xml:space="preserve"> Erit c imago q:</s>
            <s xml:id="echoid-s14458" xml:space="preserve"> & eſt c in ſuperficie,
              <lb/>
            in qua ſunt lineæ q h, & axis, & linea longitudinis a g [per 1 p 11.</s>
            <s xml:id="echoid-s14459" xml:space="preserve">] Palàm etiam [è 31 n 4] quod t refle
              <lb/>
            ctitur ad e, à puncto ſectionis columnaris, ſcilicet à puncto g.</s>
            <s xml:id="echoid-s14460" xml:space="preserve"> Eſt autem à puncto t unam ducere per
              <lb/>
            pendicularem, ſuper lineam contingentem in aliquo puncto ſectionem:</s>
            <s xml:id="echoid-s14461" xml:space="preserve"> quæ quidem concurret cũ
              <lb/>
            perpendiculari ducta à puncto g:</s>
            <s xml:id="echoid-s14462" xml:space="preserve"> quæ eſt n g z, ſub axe, id eſt, ſub puncto z:</s>
            <s xml:id="echoid-s14463" xml:space="preserve"> quod eſt concurſus per-
              <lb/>
            pendicularis n z & axis [per 24 n.</s>
            <s xml:id="echoid-s14464" xml:space="preserve">] Quoniam ducta linea t z:</s>
            <s xml:id="echoid-s14465" xml:space="preserve"> erit angulus t z n acutus:</s>
            <s xml:id="echoid-s14466" xml:space="preserve"> [quia conti-
              <lb/>
            nuato axe k z ultra z in y:</s>
            <s xml:id="echoid-s14467" xml:space="preserve"> erit angulus n z y rectus per fabricationẽ & 29 p 1.</s>
            <s xml:id="echoid-s14468" xml:space="preserve">] Producatur n z ultra z
              <lb/>
            in x.</s>
            <s xml:id="echoid-s14469" xml:space="preserve"> Ducatur ergo t x, concurrens cum n z in puncto x:</s>
            <s xml:id="echoid-s14470" xml:space="preserve"> & producatur e g, donec concurrat cum
              <lb/>
            t x in puncto i.</s>
            <s xml:id="echoid-s14471" xml:space="preserve"> Erit i imago puncti t [per 4 n 5.</s>
            <s xml:id="echoid-s14472" xml:space="preserve">] Similiter ducta à puncto h linea, quæ ſit orthogona
              <lb/>
            lis ſuper lineam, contingentem ſpeculum in puncto aliquo ſectionis, à quo h reflectitur ad e:</s>
            <s xml:id="echoid-s14473" xml:space="preserve"> cõcur-
              <lb/>
            ret cum perpendiculari d a r, ſub puncto d, quod eſt punctum axis [per 24 n.</s>
            <s xml:id="echoid-s14474" xml:space="preserve">] Concurrat in puncto
              <lb/>
            p:</s>
            <s xml:id="echoid-s14475" xml:space="preserve"> & producatur e a, donec concurrat cũ h p in puncto s.</s>
            <s xml:id="echoid-s14476" xml:space="preserve"> Erit imago puncti h punctum s [per 4 n 5.</s>
            <s xml:id="echoid-s14477" xml:space="preserve">]
              <lb/>
              <figure xlink:label="fig-0213-02" xlink:href="fig-0213-02a" number="182">
                <variables xml:id="echoid-variables172" xml:space="preserve">t i y n q g z x m b c ſ f h s r a d p e k o u</variables>
              </figure>
            Ducatur autem linea s t.</s>
            <s xml:id="echoid-s14478" xml:space="preserve"> Palàm, cum linea t i concurrat cum perpendiculari n z, quæ eſt æquidiſtãs
              <lb/>
            lineę e o:</s>
            <s xml:id="echoid-s14479" xml:space="preserve"> concurret cum linea e o [per lemma Procli ad 29 p 1.</s>
            <s xml:id="echoid-s14480" xml:space="preserve">] Sit concurſus in u.</s>
            <s xml:id="echoid-s14481" xml:space="preserve"> Similiter linea h
              <lb/>
            s, quoniam concurrit cum perpendiculari d a r, quæ eſt æquidiſtans e o:</s>
            <s xml:id="echoid-s14482" xml:space="preserve"> cõcurret cum e o.</s>
            <s xml:id="echoid-s14483" xml:space="preserve"> Sed quo-
              <lb/>
            niam ſitus t, reſpectu puncti e, idem eſt cum ſitu h & eadem longitudo:</s>
            <s xml:id="echoid-s14484" xml:space="preserve"> [quia th parallela eſt axi ex
              <lb/>
            theſi.</s>
            <s xml:id="echoid-s14485" xml:space="preserve">] Similiter ſitus puncti t & puncti h ad punctum q idem [ut præcedente numero patuit] & pũ
              <lb/>
            ctorum i, s, reſpectu o, etiam eſt idem:</s>
            <s xml:id="echoid-s14486" xml:space="preserve"> erit idem ſitus linearum t i, h s, reſpectu lineæ e o.</s>
            <s xml:id="echoid-s14487" xml:space="preserve"> Igitur li-
              <lb/>
            </s>
          </p>
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