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-s6891" xml:space="preserve">
              <pb o="117" file="0123" n="123" rhead="OPTICAE LIBER IIII."/>
            columnæ huic oppoſitam:</s>
            <s xml:id="echoid-s6892" xml:space="preserve"> & circulus ſectionis trãſit per has duas lineas longitudinis:</s>
            <s xml:id="echoid-s6893" xml:space="preserve"> & linea con-
              <lb/>
            tingens circulum ſectionis:</s>
            <s xml:id="echoid-s6894" xml:space="preserve"> cum ſit in ſuperficie aliqua:</s>
            <s xml:id="echoid-s6895" xml:space="preserve"> ſecat columnam ſuper aliquas longitudinis
              <lb/>
            lineas, ſibi inuicem æquidiſtantes:</s>
            <s xml:id="echoid-s6896" xml:space="preserve"> & ſi tranſit per unam earum, tran-
              <lb/>
              <figure xlink:label="fig-0123-01" xlink:href="fig-0123-01a" number="27">
                <variables xml:id="echoid-variables17" xml:space="preserve">a e g c b d h f</variables>
              </figure>
            ſibit per alteram, & ad paritatem angulorum.</s>
            <s xml:id="echoid-s6897" xml:space="preserve"> Cum ergo tranſeat per
              <lb/>
            punctum, in quo circulus ſectionis ſecat primã longιtudinis lineam:</s>
            <s xml:id="echoid-s6898" xml:space="preserve">
              <lb/>
            tranſibit etiam per punctũ, in quo alia longitudinis linea tangit hunc
              <lb/>
            circulum:</s>
            <s xml:id="echoid-s6899" xml:space="preserve"> & ita ſecat circulum.</s>
            <s xml:id="echoid-s6900" xml:space="preserve"> Quare non contingit, quod eſt contra
              <lb/>
            hypotheſin.</s>
            <s xml:id="echoid-s6901" xml:space="preserve"> Palàm ergo, quòd duæ illæ ſuperficies cõtingunt ſpecu-
              <lb/>
            lum, & quod inter illas cadit ex ſuperficie ſpeculi, eſt, quod apparet
              <lb/>
            uiſui.</s>
            <s xml:id="echoid-s6902" xml:space="preserve"> Cum autem duarum illarum ſuperficierum ſit concurſus in cen
              <lb/>
            tro uiſus, ſecabunt ſe, & linea ſectionis communis tranſibit per cen-
              <lb/>
            trum uiſus:</s>
            <s xml:id="echoid-s6903" xml:space="preserve"> & erit æquidiftans axi columnæ.</s>
            <s xml:id="echoid-s6904" xml:space="preserve"> Quoniam enim axis co-
              <lb/>
            lumnæ orthogonalis eſt ſuper circulum ſectionis [per conuerſam 14
              <lb/>
            p 11] & lineæ longitudinis columnæ orthogonales ſuper eundẽ cir-
              <lb/>
            culum [per 8 p 11:</s>
            <s xml:id="echoid-s6905" xml:space="preserve"> latera enim cylindri parallela ſunt axi, perpendicu-
              <lb/>
            lari ad circulum ſectionis per 21 d 11] etiam ſuperficies tangẽtes co-
              <lb/>
            lumnam ſecũdum lineas has:</s>
            <s xml:id="echoid-s6906" xml:space="preserve"> orthogonales erunt ſuper circulũ eun-
              <lb/>
            dem [per 18 p 11:</s>
            <s xml:id="echoid-s6907" xml:space="preserve">] ergo & ſuper ſuperficiem ſecantẽ columnam in illo
              <lb/>
            circulo.</s>
            <s xml:id="echoid-s6908" xml:space="preserve"> Quare linea communis harum ſuperficierum eſt orthogona
              <lb/>
            lis ſuper eandem ſuperficiem [per 19 p 11] quare æquidiſtans axi co-
              <lb/>
            lumnæ [per 6 p 11.</s>
            <s xml:id="echoid-s6909" xml:space="preserve">]</s>
          </p>
        </div>
        <div xml:id="echoid-div245" type="section" level="0" n="0">
          <head xml:id="echoid-head274" xml:space="preserve" style="it">27. Si linea recta à cẽtro uiſ{us}, ducta ad punctũ cõſpicuæ ſuper-
            <lb/>
          ficiei ſpeculi cylindr acei cõuexi, cõtinuetur: ſecabit ſpeculũ. 4.5 p 7.</head>
          <p>
            <s xml:id="echoid-s6910" xml:space="preserve">DIco ergo, quòd quocunq;</s>
            <s xml:id="echoid-s6911" xml:space="preserve"> puncto in ſectione ſpeculi apparen
              <lb/>
            te ſumpto:</s>
            <s xml:id="echoid-s6912" xml:space="preserve"> linea à centro uiſus ad punctum producta, ſecabit
              <lb/>
            ſpeculum.</s>
            <s xml:id="echoid-s6913" xml:space="preserve"> Quoniam intellecta linea longitudinis columnæ à
              <lb/>
            puncto ſumpto, tranſibit per circulum ſectionis, & tanget ipſum in
              <lb/>
            puncto:</s>
            <s xml:id="echoid-s6914" xml:space="preserve"> ad quod punctum ſi ducatur linea à centro uiſus:</s>
            <s xml:id="echoid-s6915" xml:space="preserve"> ſecabit ſpe-
              <lb/>
            culum:</s>
            <s xml:id="echoid-s6916" xml:space="preserve"> quia cadit inter lineas contingẽtes hunc circulum:</s>
            <s xml:id="echoid-s6917" xml:space="preserve"> ergo & ſu-
              <lb/>
            perficies à centro uiſus procedens, in qua fuerit hæc linea, ſecabit ſpe
              <lb/>
            culum.</s>
            <s xml:id="echoid-s6918" xml:space="preserve"> Cum ergo in eadem ſuperficie ſit linea à centro uiſus, ad pun-
              <lb/>
            ctum ſumptum ducta:</s>
            <s xml:id="echoid-s6919" xml:space="preserve"> ſecabit linea illa ſpeculum:</s>
            <s xml:id="echoid-s6920" xml:space="preserve"> & ita quælibet linea à centro uiſus, ad portionem
              <lb/>
            ſpeculi intellecta, ſecat ſpeculum.</s>
            <s xml:id="echoid-s6921" xml:space="preserve"> Eodẽ modo quælibet linea à linea cõmuni, per centrum uiſus in-
              <lb/>
            tellecta, ad hãc portionẽ ducta, ſecat ſpeculũ.</s>
            <s xml:id="echoid-s6922" xml:space="preserve"> Vnde quælibet ſuperficies tangens ſpeculũ in aliqua
              <lb/>
            portionis apparentis linea, ſecat ſuperficies, quę contingũt portionis extremitates:</s>
            <s xml:id="echoid-s6923" xml:space="preserve"> & nulla omniũ
              <lb/>
            ſuperficierum portionẽ tangentiũ, peruenit ad uiſus centrũ, ſed inter uiſum extẽditur & ſpeculum.</s>
            <s xml:id="echoid-s6924" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div246" type="section" level="0" n="0">
          <head xml:id="echoid-head275" xml:space="preserve" style="it">28. In ſpeculo cylindraceo conuexo, à quolibet conſpicuæ ſuperficiei puncto poteſt ad uiſum
            <lb/>
          reflexio fieri. 25 p 7.</head>
          <p>
            <s xml:id="echoid-s6925" xml:space="preserve">DIco ergo, quòd à quolibet puncto portionis huius poteſt fieri reflexio lucis.</s>
            <s xml:id="echoid-s6926" xml:space="preserve"> Dato enim pun-
              <lb/>
            cto, fiat ſuper ipſum circulus æquidiſtans columnæ baſibus:</s>
            <s xml:id="echoid-s6927" xml:space="preserve"> ſi ergo ſuperficies à cẽtro uiſus
              <lb/>
            procedens, & columnę ſuperficiem æquidiſtanter baſi ſecans, ſecet eam ſuper hunc circulũ:</s>
            <s xml:id="echoid-s6928" xml:space="preserve">
              <lb/>
            & linea à centro uiſus ad circuli centrũ ducta, tranſeat per punctum datũ:</s>
            <s xml:id="echoid-s6929" xml:space="preserve"> fiet reflexio ſormæ illius
              <lb/>
            puncti per eandem lineam ad lineæ ortum [per 11 n] quia linea illa eſt axis uiſus ſuper axem colu-
              <lb/>
            mnæ perpendicularis [per 21 d 11, 29 p 1.</s>
            <s xml:id="echoid-s6930" xml:space="preserve">] Sumpto autem puncto quocunq;</s>
            <s xml:id="echoid-s6931" xml:space="preserve"> per quod tranſeat axis,
              <lb/>
            perpendicularis ſuper axem columnæ:</s>
            <s xml:id="echoid-s6932" xml:space="preserve"> fiet reflexio illius puncti per eundẽ axem [per 11 n.</s>
            <s xml:id="echoid-s6933" xml:space="preserve">] Si ueró
              <lb/>
            prætereat axem punctum ſumptum, quæcunq;</s>
            <s xml:id="echoid-s6934" xml:space="preserve"> ſit linea à centro circuli, æquidiſtantis baſibus per
              <lb/>
            ipſum punctum ducti, ad ſuperficiem in linea longitudinis columnæ per punctũ illud tranſeuntis,
              <lb/>
            contingentem:</s>
            <s xml:id="echoid-s6935" xml:space="preserve"> erit ſuper axem orthogonalis [per 21 d 11, & conuerſam 14 p 11.</s>
            <s xml:id="echoid-s6936" xml:space="preserve">] Quare ſuper lineam
              <lb/>
            longitudinis per punctum illud trãſeuntem [per 29 p 1.</s>
            <s xml:id="echoid-s6937" xml:space="preserve">] Et quoniã uiſus eſt altior ſuperficie pun-
              <lb/>
            ctum contingẽte:</s>
            <s xml:id="echoid-s6938" xml:space="preserve"> linea à cẽtro uiſus ad punctum ſumptũ ducta, tenebit acutum angulũ cũ perpen-
              <lb/>
            diculari illa, à pũcto ad centrũ circuli ducta:</s>
            <s xml:id="echoid-s6939" xml:space="preserve"> & hic eſt ex parte exteriore, quia obtuſum habet ex in-
              <lb/>
            teriore:</s>
            <s xml:id="echoid-s6940" xml:space="preserve"> & ex angulo recto, quem illa perpendicularis tenet cum linea ſuperficiei contingentis cir-
              <lb/>
            culum [per 18 p 3] poterit abſcindi acutus huic æqualis:</s>
            <s xml:id="echoid-s6941" xml:space="preserve"> & perpendicularis illa cum cẽtro uiſus eſt
              <lb/>
            in eadem ſuperficie:</s>
            <s xml:id="echoid-s6942" xml:space="preserve"> quare etiam cum linea à cẽtro ad punctum ducta:</s>
            <s xml:id="echoid-s6943" xml:space="preserve"> & erit linea reflexa in eadem
              <lb/>
            ſuperficie:</s>
            <s xml:id="echoid-s6944" xml:space="preserve"> quare cum linea à centro ad punctum ducta.</s>
            <s xml:id="echoid-s6945" xml:space="preserve"> Et erit hæc ſuperficies orthogonalis ſuper
              <lb/>
            ſuperficiem, contingentem ſpeculum in puncto illo:</s>
            <s xml:id="echoid-s6946" xml:space="preserve"> quoniam perpendicularis orthogonaliter ca-
              <lb/>
            dit ſuper hanc ſuperficiem:</s>
            <s xml:id="echoid-s6947" xml:space="preserve"> & huiuſinodi erit reflexionis ſuperficies.</s>
            <s xml:id="echoid-s6948" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div247" type="section" level="0" n="0">
          <head xml:id="echoid-head276" xml:space="preserve" style="it">29. Si uiſ{us} ſit extra ſuperficiem ſpeculi cylindr acei conuexi, in plano uiſibilis per axem du-
            <lb/>
          cto: cõm unis ſectio ſuperficier um reflexionis & ſpeculi, erit lat{us} cylindri: & unicum tantùm
            <lb/>
          eſt in eadem conſpicua ſuperficie planum, à quo ad eundem uiſum reflexio fieri poteſt. 7.16 p 7.</head>
          <p>
            <s xml:id="echoid-s6949" xml:space="preserve">ESt autẽ diuerſitas inter lineas ſuperficiebus reflexionis & ſuperficiei columnæ cõmunes.</s>
            <s xml:id="echoid-s6950" xml:space="preserve"> Cũ
              <lb/>
            enim reflexio erit per eundẽ radium:</s>
            <s xml:id="echoid-s6951" xml:space="preserve"> cadet idẽ radius ille orthogonaliter ſuper axem, & linea
              <lb/>
            </s>
          </p>
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