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-s7898" xml:space="preserve">
              <pb o="134" file="0140" n="140" rhead="ALHAZEN"/>
            at, ad ſunt in eadem ſuperficie orthogonali ſuper ſuperficiem ſpeculi.</s>
            <s xml:id="echoid-s7899" xml:space="preserve"> Similiter g z, a z, a d ſunt in
              <lb/>
            eadem ſuperficie orthogonali:</s>
            <s xml:id="echoid-s7900" xml:space="preserve"> & linea d t communis ſuperficiei a d t b & ſuperficiei ſpeculi:</s>
            <s xml:id="echoid-s7901" xml:space="preserve"> & d z
              <lb/>
            linea communis ſuperficiei a d z g & ſuperficiei ſpeculi.</s>
            <s xml:id="echoid-s7902" xml:space="preserve"> Si iam b t, g z fuerint in eadem ſuperficie
              <lb/>
            orthogonali, erit [per 3 p 11] t d z linea una recta:</s>
            <s xml:id="echoid-s7903" xml:space="preserve"> & perpendicula-
              <lb/>
              <figure xlink:label="fig-0140-01" xlink:href="fig-0140-01a" number="45">
                <variables xml:id="echoid-variables35" xml:space="preserve">b g a t z d h</variables>
              </figure>
            ris a d aut erit inter duas perpendiculares productas ad ſuperficiem
              <lb/>
            ſpeculi à duobus uiſibus:</s>
            <s xml:id="echoid-s7904" xml:space="preserve"> aut extra.</s>
            <s xml:id="echoid-s7905" xml:space="preserve"> Vtrumlibet ſit:</s>
            <s xml:id="echoid-s7906" xml:space="preserve"> linea b t ſecabit
              <lb/>
            ex perpendiculari a d ultra ſpeculum partem, æqualem parti, quæ eſt
              <lb/>
            a d [per 11 n.</s>
            <s xml:id="echoid-s7907" xml:space="preserve">] Similiter g z ſecabit ex eadem perpendiculari partem
              <lb/>
            ultra ſpeculum, æqualem illi parti.</s>
            <s xml:id="echoid-s7908" xml:space="preserve"> Illæ igitur duæ lineæ reflexionis
              <lb/>
            ſecabunt perpendicularem ultra ſpeculum in eodem puncto.</s>
            <s xml:id="echoid-s7909" xml:space="preserve"> Ergo
              <lb/>
            imago puncti a in eodem perpendicularis puncto percipietur ab u-
              <lb/>
            troque uiſu.</s>
            <s xml:id="echoid-s7910" xml:space="preserve"> Quare unica tantùm erit imago & eadem:</s>
            <s xml:id="echoid-s7911" xml:space="preserve"> & in eodem
              <lb/>
            loco:</s>
            <s xml:id="echoid-s7912" xml:space="preserve"> quæ eſſet uno tantùm uiſu adhibito.</s>
            <s xml:id="echoid-s7913" xml:space="preserve"> Si uerò puncta t, z non
              <lb/>
            fuerint in eadem ſuperficie reflexionis orthogonali ſuper ſpeculum:</s>
            <s xml:id="echoid-s7914" xml:space="preserve">
              <lb/>
            eadem tamen erit probatio:</s>
            <s xml:id="echoid-s7915" xml:space="preserve"> quòd utraque linea reflexionis ſecet ex
              <lb/>
            perpendiculari partem, æqualẽ parti ſuperiori:</s>
            <s xml:id="echoid-s7916" xml:space="preserve"> & erit ſectio linearũ
              <lb/>
            reflexionis cum perpendiculari in eodem puncto.</s>
            <s xml:id="echoid-s7917" xml:space="preserve"> Quare patet pro-
              <lb/>
            poſitum.</s>
            <s xml:id="echoid-s7918" xml:space="preserve"> Si uerò fuerit punctum a in perpendiculari ducta ab uno
              <lb/>
            uiſu ad ſuperficiem ſpeculi tantùm, ſecundum eundem uiſum com-
              <lb/>
            prehendetur [per 11 n 4] ultra ſpeculum in puncto perpẽdicularis,
              <lb/>
            tãtùm elõgato à ſuperficie ſpeculi, quantũ diſtat a ab eadẽ [per 11 n.</s>
            <s xml:id="echoid-s7919" xml:space="preserve">]
              <lb/>
            Quia forma a uidetur continua cum formis aliorum punctorũ, quæ
              <lb/>
            quidem uidentur in locis ſimilibus:</s>
            <s xml:id="echoid-s7920" xml:space="preserve"> & ab alio uiſu comprehendetur
              <lb/>
            imago a in eodem perpendicularis puncto.</s>
            <s xml:id="echoid-s7921" xml:space="preserve"> Quare & ſic utriq;</s>
            <s xml:id="echoid-s7922" xml:space="preserve"> uiſui unica tantùm apparet image
              <lb/>
            puncti a, & in eodem eiuſdem perpendicularis puncto.</s>
            <s xml:id="echoid-s7923" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s7924" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div300" type="section" level="0" n="0">
          <head xml:id="echoid-head319" xml:space="preserve" style="it">16. In ſpeculo ſphærico conuexo linea reflexionis & perpendicularis incidentiæ concurrunt:
            <lb/>
          & imago uidetur in ipſarum concurſu. 9. 11 p 6. Idem 3 n.</head>
          <p>
            <s xml:id="echoid-s7925" xml:space="preserve">IN ſpeculis ſphæricis extrà politis patebit, quod diximus.</s>
            <s xml:id="echoid-s7926" xml:space="preserve"> Sit a punctum uiſum:</s>
            <s xml:id="echoid-s7927" xml:space="preserve"> b cẽtrum uiſus:</s>
            <s xml:id="echoid-s7928" xml:space="preserve">
              <lb/>
            g punctũ reflexionis.</s>
            <s xml:id="echoid-s7929" xml:space="preserve"> Palàm [per 23.</s>
            <s xml:id="echoid-s7930" xml:space="preserve"> 13 n 4] quòd b g, a g ſunt in eadem ſuperficie orthogona-
              <lb/>
            lι ſuper ſuperficiem ſphæram contingentẽ in puncto g:</s>
            <s xml:id="echoid-s7931" xml:space="preserve"> linea com
              <lb/>
              <figure xlink:label="fig-0140-02" xlink:href="fig-0140-02a" number="46">
                <variables xml:id="echoid-variables36" xml:space="preserve">a h b e g p d z n q</variables>
              </figure>
            munis ſuperficiei reflexionis & fuperficiei ſphæræ eſt circumferen-
              <lb/>
            tia [per 1 th 1 ſphær:</s>
            <s xml:id="echoid-s7932" xml:space="preserve"> uel 25.</s>
            <s xml:id="echoid-s7933" xml:space="preserve"> uel 45 n 4] & ſit z g q.</s>
            <s xml:id="echoid-s7934" xml:space="preserve"> Linea contingẽs
              <lb/>
            hunc circulum in puncto reflexionis ſit p g e:</s>
            <s xml:id="echoid-s7935" xml:space="preserve"> perpendicularis ſuper
              <lb/>
            hanc lineam ſit h g:</s>
            <s xml:id="echoid-s7936" xml:space="preserve"> planum, quòd h g perueniet ad centrum ſphæræ.</s>
            <s xml:id="echoid-s7937" xml:space="preserve">
              <lb/>
            Quod ſi non:</s>
            <s xml:id="echoid-s7938" xml:space="preserve"> cum linea à centro ſphæræ ducta ad punctum g, ſit e-
              <lb/>
            tiam perpendicularis ſuper lineam p g e [per 25 n 4 & 3 d 11:</s>
            <s xml:id="echoid-s7939" xml:space="preserve">] erit ab
              <lb/>
            eodem puncto in eandem partem ducere duas lineas perpendicula-
              <lb/>
            res ſuper unam lineam [& ſic pars æquaretur toti, contra 9 ax.</s>
            <s xml:id="echoid-s7940" xml:space="preserve">] Sit
              <lb/>
            autem centrum ſphæræ n:</s>
            <s xml:id="echoid-s7941" xml:space="preserve"> & ducatur linea à puncto uiſo ad centrum
              <lb/>
            ſphæræ, ſcilicet a n:</s>
            <s xml:id="echoid-s7942" xml:space="preserve"> quæ quidem erit perpendicularis ſuper ſuperfi-
              <lb/>
            ciem, contingentem ſphæram in puncto ſphæræ, per quod tranſit
              <lb/>
            [per 25 n 4.</s>
            <s xml:id="echoid-s7943" xml:space="preserve">] Et quoniam planum eſt, quòd b g ſecat ſphęram:</s>
            <s xml:id="echoid-s7944" xml:space="preserve"> cum
              <lb/>
            ſit inter h g, g p, quæ continent rectum angulum:</s>
            <s xml:id="echoid-s7945" xml:space="preserve"> concurret cum li-
              <lb/>
            nea a n:</s>
            <s xml:id="echoid-s7946" xml:space="preserve"> Et cũ perpẽdicularis h g ſit in ſuքficie reflexiõis [per 23 n 4]
              <lb/>
            erit centrum ſphæræ in eadem [per 1 p 11:</s>
            <s xml:id="echoid-s7947" xml:space="preserve"> quia h g continuata cadit
              <lb/>
            in n centrum ſphæræ, ut patuit] & ita a n in eadem ſuperficie cum
              <lb/>
            h g.</s>
            <s xml:id="echoid-s7948" xml:space="preserve"> Sit ergo concurſus b g cum a n, punctum d.</s>
            <s xml:id="echoid-s7949" xml:space="preserve"> Planum [per 3 n]
              <lb/>
            quòd d erit locus imaginis.</s>
            <s xml:id="echoid-s7950" xml:space="preserve"> Et hæc quidem intelligenda ſunt, quan-
              <lb/>
            do linea ducta à puncto uiſo ad centrum uiſus, non fuerit perpendi-
              <lb/>
            cularis ſuper ſpeculum [uiſu enim & uiſibili in recta linea perpendiculari ſuper ſpeculum colloca-
              <lb/>
            tis, reflexio fit per eandem perpendicularem, per 11 n 4.</s>
            <s xml:id="echoid-s7951" xml:space="preserve">]</s>
          </p>
        </div>
        <div xml:id="echoid-div302" type="section" level="0" n="0">
          <head xml:id="echoid-head320" xml:space="preserve" style="it">17. Finis contingentiæ in ſpeculo ſphærico, eſt concurſ{us} rectæ ſpeculum in reflexionis puncto
            <lb/>
          tangentis, cum perpendiculari incidentiæ uel reflexionis. Et rect a à centro ſpeculi ſphærici
            <lb/>
          conuexi ad imaginem, maior est recta ab imagine ad reflexionis punctum ducta. In def. 13 p 6.</head>
          <p>
            <s xml:id="echoid-s7952" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s7953" xml:space="preserve"> linea p g e ſecat lineam a n:</s>
            <s xml:id="echoid-s7954" xml:space="preserve"> ſit punctum ſectionis e:</s>
            <s xml:id="echoid-s7955" xml:space="preserve"> & dicitur punctum iſtud finis
              <lb/>
            contingentiæ.</s>
            <s xml:id="echoid-s7956" xml:space="preserve"> Dico, quòd in hoc ſitu linea à centro ſphæræ a d locum imaginis ducta, ma-
              <lb/>
            ior eſt linea, à loco imaginis ducta ad locum reflexionis, id eſt d n maior d g.</s>
            <s xml:id="echoid-s7957" xml:space="preserve"> Quoniam e-
              <lb/>
            nim angulus b g h eſt æqualis angulo h g a [ut demonſtratum eſt 13 n] ſed [per 15 p 1] angulus
              <lb/>
            b g h æqualis eſt angulo n g d:</s>
            <s xml:id="echoid-s7958" xml:space="preserve"> ergo [per 1 ax] angulus h g a æqualis eſt eidem:</s>
            <s xml:id="echoid-s7959" xml:space="preserve"> & e g perpendicu-
              <lb/>
            laris ſuper h g n [per fabricationem.</s>
            <s xml:id="echoid-s7960" xml:space="preserve">] Quare [per 3 ax] angulus a g æqualis eſt angulo e g d.</s>
            <s xml:id="echoid-s7961" xml:space="preserve"> Igi-
              <lb/>
            tur [per 3 p 6] proportio a g ad g d, ſicut a e ad e d.</s>
            <s xml:id="echoid-s7962" xml:space="preserve"> Protrahatur à puncto a æquidiſtans ipſi d g
              <lb/>
            </s>
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