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-s41805" xml:space="preserve">
              <pb o="336" file="0638" n="638" rhead="VITELLONIS OPTICAE"/>
            ſemicirculũ minoris:</s>
            <s xml:id="echoid-s41806" xml:space="preserve"> linea enim t o cadit ſuper lineã k o, fitq́;</s>
            <s xml:id="echoid-s41807" xml:space="preserve"> angulus t o k minor recto per 42 th.</s>
            <s xml:id="echoid-s41808" xml:space="preserve"> 1
              <lb/>
            huius:</s>
            <s xml:id="echoid-s41809" xml:space="preserve"> linea enim o k eſt pars diametri circuli minoris, propter hoc quòd angulus o k b eſt rectus:</s>
            <s xml:id="echoid-s41810" xml:space="preserve">
              <lb/>
            & linea k o producta ſecat circulum minorẽ, tranſiens per eius centrũ per 1 p 3:</s>
            <s xml:id="echoid-s41811" xml:space="preserve"> ideo quòd ipſa ſecãs
              <lb/>
            lineam b a orthogonaliter, & per æqualia ſecat ipſam neceſſariò:</s>
            <s xml:id="echoid-s41812" xml:space="preserve"> ergo illa perpendicularis produ-
              <lb/>
            cta concurret cum linea k o per 14 th.</s>
            <s xml:id="echoid-s41813" xml:space="preserve"> 1 huius:</s>
            <s xml:id="echoid-s41814" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s41815" xml:space="preserve"> punctus concurſus in puncto termini diametri
              <lb/>
            circuli minoris per 31 p 3:</s>
            <s xml:id="echoid-s41816" xml:space="preserve"> cum ille angulus in ſemicirculo ſit rectus, qui fit ſuper punctum t terminũ
              <lb/>
            lineæ g t:</s>
            <s xml:id="echoid-s41817" xml:space="preserve"> ſed linea t p eſt inferior illa perpendiculari ex parte puncti n.</s>
            <s xml:id="echoid-s41818" xml:space="preserve"> Igitur quæcunq;</s>
            <s xml:id="echoid-s41819" xml:space="preserve"> linea duca-
              <lb/>
            tur à puncto g centro ſpeculi ad lineam t p, ſecans diametrum o k:</s>
            <s xml:id="echoid-s41820" xml:space="preserve"> illa cadet neceſſariò in aliquod
              <lb/>
            punctum lineæ t p citra perpendicularem.</s>
            <s xml:id="echoid-s41821" xml:space="preserve"> Cum igitur linea g p cadat in punctum p, & ſecet lineam
              <lb/>
            o k:</s>
            <s xml:id="echoid-s41822" xml:space="preserve"> erit punctus p citra illam perpendicularem, & infra arcum minoris circuli, cui ſubtenditur illa
              <lb/>
            perpendicularis.</s>
            <s xml:id="echoid-s41823" xml:space="preserve"> Facto igitur circulo trãſeunte per tria puncta, quę ſunt a, b, p, tranſibit quidem ille
              <lb/>
            circulus per punctum l:</s>
            <s xml:id="echoid-s41824" xml:space="preserve"> quoniam linea p l ſecabit illum circulum, ſicuti priorem circulum a b t ſeca
              <lb/>
            bat linea t o.</s>
            <s xml:id="echoid-s41825" xml:space="preserve"> Circulus itaq;</s>
            <s xml:id="echoid-s41826" xml:space="preserve"> a b p ſecabit circulum a b t in duobus punctis a & b:</s>
            <s xml:id="echoid-s41827" xml:space="preserve"> & cum exeat à pun-
              <lb/>
            cto b, & iterum redeat in punctum p inferiorem puncto t (cum ſit citra illum circulum uerſus pun-
              <lb/>
            ctum t) neceſſariò ſecabit illum circulum in tertio puncto, quod eſt contra 10 p 3 & impoſsibile.</s>
            <s xml:id="echoid-s41828" xml:space="preserve"> Re
              <lb/>
            ſtat igitur, ut forma puncti rei uiſæ, qui eſt a, non reflectatur ad uiſum exiſtentẽ in puncto b à duo-
              <lb/>
            bus punctis arcus z n:</s>
            <s xml:id="echoid-s41829" xml:space="preserve"> ita ut quilibet angulorum illorum ſit minor angulo a g d.</s>
            <s xml:id="echoid-s41830" xml:space="preserve"> Palàm ergo, quòd
              <lb/>
            impoſsibile eſt, ut forma puncti a reflectatur ad uiſum b à duobus punctis arcus interiacentis eo-
              <lb/>
            rum diametros, qui eſt e z, ita ut uter q;</s>
            <s xml:id="echoid-s41831" xml:space="preserve"> angulorum conſtantium ex angulis incidentiæ & reflexio-
              <lb/>
            nis ſit minor angulo a g d.</s>
            <s xml:id="echoid-s41832" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s41833" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1659" type="section" level="0" n="0">
          <head xml:id="echoid-head1245" xml:space="preserve" style="it">35. In ſpeculis ſphæricis cõcauis duos pũctos, qui in diuerſis diametris, & inæqualis diſtantiæ
            <lb/>
          à centro ſpeculi exiſtentes à duobus punctis ſpeculi arcus ſcilicet interiacentis ſemidiametros,
            <lb/>
          in quibus illi puncti conſiſtunt, ad ſe mutuò reflectantur, poßibile eſt inueniri. Alhazen 81 n 5.</head>
          <p>
            <s xml:id="echoid-s41834" xml:space="preserve">Sit circulus (qui eſt communis ſectio ſuperficiei reflexionis, & ſuperficiei ſpeculi ſphærici con-
              <lb/>
            caui) cuius centrum d:</s>
            <s xml:id="echoid-s41835" xml:space="preserve"> & ſumantur in ipſo duæ diametri, quæ ſint g a & b h, ſecantes ſe in centro d:</s>
            <s xml:id="echoid-s41836" xml:space="preserve">
              <lb/>
            dico quòd poſsibile eſt fieri, quod proponitur.</s>
            <s xml:id="echoid-s41837" xml:space="preserve"> Diuidatur enim angulus g d b per æqualia per ſemi
              <lb/>
            diametrum d e:</s>
            <s xml:id="echoid-s41838" xml:space="preserve"> & in ſemidiametro b d ſumatur punctus m ultra punctũ, in quem cadit perpendicu
              <lb/>
            laris ducta à puncto e ſuper diametrũ b d:</s>
            <s xml:id="echoid-s41839" xml:space="preserve"> & ſumatur linea n d in diametro d g æqualis lineæ m d:</s>
            <s xml:id="echoid-s41840" xml:space="preserve"> &
              <lb/>
            fiat per 5 p 4 circulus tranſiens per tria puncta m, d, n:</s>
            <s xml:id="echoid-s41841" xml:space="preserve"> hic ergo neceſſariò tranſibit ultra punctum e.</s>
            <s xml:id="echoid-s41842" xml:space="preserve">
              <lb/>
            Si enim detur, quòd ille circulus tranſeat punctum e, ducantur lineæ m e & n e:</s>
            <s xml:id="echoid-s41843" xml:space="preserve"> fietq́;</s>
            <s xml:id="echoid-s41844" xml:space="preserve"> quadrangulũ
              <lb/>
            d m e n intra circulũ:</s>
            <s xml:id="echoid-s41845" xml:space="preserve"> ergo per 22 p 3 duo anguli iſtius quadranguli ex aduerſo collocati, ut qui ſunt
              <lb/>
            ad punctos m & n, ſunt æquales duobus rectis:</s>
            <s xml:id="echoid-s41846" xml:space="preserve"> quod eſt impoſsibile:</s>
            <s xml:id="echoid-s41847" xml:space="preserve"> quoniam duo anguli e m d &
              <lb/>
            e n d ambo ſunt acuti, minores duobus rectis:</s>
            <s xml:id="echoid-s41848" xml:space="preserve"> ideo quòd lineæ e m & e n cadunt ultra perpendicu-
              <lb/>
            lares ductas à pũcto e ſuper ſemidiametros b d & g d.</s>
            <s xml:id="echoid-s41849" xml:space="preserve"> Similis quoq;</s>
            <s xml:id="echoid-s41850" xml:space="preserve"> fiet deductio, ſi circulus trãſeat
              <lb/>
            citra punctum e:</s>
            <s xml:id="echoid-s41851" xml:space="preserve"> tunc enim anguli illius quadranguli cadentes ſuper punctum m & n, erũt iterum
              <lb/>
            minores duobus rectis.</s>
            <s xml:id="echoid-s41852" xml:space="preserve"> Tranſit igitur circulus d m n extra punctum e:</s>
            <s xml:id="echoid-s41853" xml:space="preserve"> ſecabit ergo circulum pro-
              <lb/>
            poſitum ipſius ſpeculi in duo bus punctis per 10 p 3:</s>
            <s xml:id="echoid-s41854" xml:space="preserve"> ſint illa duo puncta t & l:</s>
            <s xml:id="echoid-s41855" xml:space="preserve"> & ducantur lineæ n t,
              <lb/>
            m t, n l, d l, m l:</s>
            <s xml:id="echoid-s41856" xml:space="preserve"> & ducatur linea m n ſecans lineã t d in puncto f, & lineam e d in puncto p.</s>
            <s xml:id="echoid-s41857" xml:space="preserve"> Cum itaq;</s>
            <s xml:id="echoid-s41858" xml:space="preserve">,
              <lb/>
            ut patet ex præmiſsis, linea m d ſit æqualis lineæ n d, & li-
              <lb/>
              <figure xlink:label="fig-0638-01" xlink:href="fig-0638-01a" number="762">
                <variables xml:id="echoid-variables739" xml:space="preserve">k e l t r o z i g x b n p f m q d s n a</variables>
              </figure>
            nea p d cómunis ambobus trigonis p d m & p d n, & an-
              <lb/>
            gulus p d m æqualis angulo p d n:</s>
            <s xml:id="echoid-s41859" xml:space="preserve"> palàm per 4 p 1 quoniã
              <lb/>
            triangulus p m d æqualis eſt triangulo p n d:</s>
            <s xml:id="echoid-s41860" xml:space="preserve"> erit quoq;</s>
            <s xml:id="echoid-s41861" xml:space="preserve"> an
              <lb/>
            gulus f p d æqualis angulo n p d, & uterq;</s>
            <s xml:id="echoid-s41862" xml:space="preserve"> rectus:</s>
            <s xml:id="echoid-s41863" xml:space="preserve"> angulus
              <lb/>
            itaq;</s>
            <s xml:id="echoid-s41864" xml:space="preserve"> p f d eſt acutus per 32 p 1.</s>
            <s xml:id="echoid-s41865" xml:space="preserve"> Ducatur ergo à puncto ſ li-
              <lb/>
            nea perpendicularis ſuper lineam d t per 11 p 1, quæ produ
              <lb/>
            cta ad circunferentiam minoris circuli ſit linea f k.</s>
            <s xml:id="echoid-s41866" xml:space="preserve"> Hæc
              <lb/>
            itaq;</s>
            <s xml:id="echoid-s41867" xml:space="preserve"> ſecabit lineam l n:</s>
            <s xml:id="echoid-s41868" xml:space="preserve"> uel non ſecabit.</s>
            <s xml:id="echoid-s41869" xml:space="preserve"> Si non ſecet:</s>
            <s xml:id="echoid-s41870" xml:space="preserve"> erit
              <lb/>
            quilibet punctus lineę l n propinquior puncto n, quàm
              <lb/>
            punctus k.</s>
            <s xml:id="echoid-s41871" xml:space="preserve"> Si ſecet:</s>
            <s xml:id="echoid-s41872" xml:space="preserve"> palàm itaq;</s>
            <s xml:id="echoid-s41873" xml:space="preserve"> quoniam aliquis punctus
              <lb/>
            lineæ l n erit inſerior puncto k, plus approximans ad pun
              <lb/>
            ctum n quàm punctũ k:</s>
            <s xml:id="echoid-s41874" xml:space="preserve"> ſit ille punctus z:</s>
            <s xml:id="echoid-s41875" xml:space="preserve"> & ducatur linea
              <lb/>
            t z:</s>
            <s xml:id="echoid-s41876" xml:space="preserve"> quæ producatur uſq;</s>
            <s xml:id="echoid-s41877" xml:space="preserve"> ad circunferentiam circuli mino
              <lb/>
            ris, cadatq́;</s>
            <s xml:id="echoid-s41878" xml:space="preserve"> in punctum o.</s>
            <s xml:id="echoid-s41879" xml:space="preserve"> Arcus itaq;</s>
            <s xml:id="echoid-s41880" xml:space="preserve"> n o aut eſt minor ar
              <lb/>
            cu t l:</s>
            <s xml:id="echoid-s41881" xml:space="preserve"> aut non.</s>
            <s xml:id="echoid-s41882" xml:space="preserve"> si non fuerit minor, abſcindatur ex eo ar-
              <lb/>
            cus minor arcul t, & ducatur ad terminum illius arcus li-
              <lb/>
            nea à puncto t, & erit idem, ſicuti ſi arcus n o ſit min or ar-
              <lb/>
            cu l t.</s>
            <s xml:id="echoid-s41883" xml:space="preserve"> Sit ergo arcus n o minor quàm ſit arcus t l:</s>
            <s xml:id="echoid-s41884" xml:space="preserve"> ergo per 33 p 6 angulus t n l eſt maior angulo o t n.</s>
            <s xml:id="echoid-s41885" xml:space="preserve">
              <lb/>
            Secetur ergo ex angulo t n l angulus æqualis angulo o t n, qui ſit i n z:</s>
            <s xml:id="echoid-s41886" xml:space="preserve"> cadetq́;</s>
            <s xml:id="echoid-s41887" xml:space="preserve"> punctum i in lineam
              <lb/>
            t z per 29 th.</s>
            <s xml:id="echoid-s41888" xml:space="preserve"> 1 huius:</s>
            <s xml:id="echoid-s41889" xml:space="preserve"> & ſuper punctum t lineæ m t per 23 p 1 fiat angulus æqualis angulo o t n, qui ſit
              <lb/>
            angulus q t m.</s>
            <s xml:id="echoid-s41890" xml:space="preserve"> Cum itaq;</s>
            <s xml:id="echoid-s41891" xml:space="preserve"> angulus t m l ſit maior angulo m t q:</s>
            <s xml:id="echoid-s41892" xml:space="preserve"> quia arcus t l eſt maior arcu n o, ut pa-
              <lb/>
            tet ex præmiſsis:</s>
            <s xml:id="echoid-s41893" xml:space="preserve"> arcus uerò n o determinat quantitatem anguli m t q, qui eſt æqualis angulo o t n:</s>
            <s xml:id="echoid-s41894" xml:space="preserve">
              <lb/>
            palàm ergo per 14 th.</s>
            <s xml:id="echoid-s41895" xml:space="preserve"> 1 huius quoniá concurret linea t q cum linea l m:</s>
            <s xml:id="echoid-s41896" xml:space="preserve"> ſit itaq;</s>
            <s xml:id="echoid-s41897" xml:space="preserve"> concurſus in puncto
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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