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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            <s xml:id="echoid-s7962" xml:space="preserve">
              <pb o="135" file="0141" n="141" rhead="OPTICAE LIBER V."/>
            [per 31 p 1] & concurrat cum linea h n in puncto h [cõcurret autem per lemma Procli ad 29 p 1.</s>
            <s xml:id="echoid-s7963" xml:space="preserve">]
              <lb/>
            Erit igitur [per 29 p 1] angulus n g d æqualis angulo g h a:</s>
            <s xml:id="echoid-s7964" xml:space="preserve"> ſed an-
              <lb/>
              <figure xlink:label="fig-0141-01" xlink:href="fig-0141-01a" number="47">
                <variables xml:id="echoid-variables37" xml:space="preserve">h a b e g p d z n q</variables>
              </figure>
            gulus n g d æqualis eſt angulo a g h [ergo per 1 ax angulus g h a
              <lb/>
            æqualis eſt angulo a g h.</s>
            <s xml:id="echoid-s7965" xml:space="preserve">] Quare [per 6 p 1] duo latera a g, h a
              <lb/>
            ſunt æqualia.</s>
            <s xml:id="echoid-s7966" xml:space="preserve"> Igitur [per 7 p 5] proportio a h ad g d, ſicut a g ad
              <lb/>
            eandem.</s>
            <s xml:id="echoid-s7967" xml:space="preserve"> Sed proportio a h ad g d, ſicut a n ad d n [per 4 p 6:</s>
            <s xml:id="echoid-s7968" xml:space="preserve">
              <lb/>
            ſunt enim triangula a h n, d g n æquiangula per 29 p 1, & quia an-
              <lb/>
            gulus ad n communis eſt utrique triangulo.</s>
            <s xml:id="echoid-s7969" xml:space="preserve">] Quare [per 11 p 5]
              <lb/>
            a n ad d n, ſicut a g ad g d:</s>
            <s xml:id="echoid-s7970" xml:space="preserve"> Igitur [per 16 p 5] proportio a n ad
              <lb/>
            a g:</s>
            <s xml:id="echoid-s7971" xml:space="preserve"> ſicut d n ad d g:</s>
            <s xml:id="echoid-s7972" xml:space="preserve"> Sed a n eſt maior a g:</s>
            <s xml:id="echoid-s7973" xml:space="preserve"> [per 19 p 1] quia reſpicit
              <lb/>
            angulum maiorem recto in triangulo a g n [rectus enim eſt, ut pa-
              <lb/>
            tuit, e g n.</s>
            <s xml:id="echoid-s7974" xml:space="preserve">] Igitur d n maior d g:</s>
            <s xml:id="echoid-s7975" xml:space="preserve"> quod eſt propoſitum.</s>
            <s xml:id="echoid-s7976" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div304" type="section" level="0" n="0">
          <head xml:id="echoid-head321" xml:space="preserve" style="it">18. Si in ſpeculo ſphærico conuexo perpendicularis incidentiæ
            <lb/>
          ſecetur à lineis reflexionis: & ſpeculum in reflexionis puncto tan-
            <lb/>
          gente: erit, ut tota perpendicularis ad inferum ſegmentum: ſic ſu-
            <lb/>
          perum ad intermedium. Et pars perpendicularis inter punctum
            <lb/>
          contingentiæ, & peripheriam, communem ſectionem ſuperficie-
            <lb/>
          rum reflexionis, & ſpeculi, erit minor eiuſdem peripheriæ ſemidia
            <lb/>
          metro. 12. 14 p 6.</head>
          <p>
            <s xml:id="echoid-s7977" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s7978" xml:space="preserve"> dico quòd linea ducta à fine contingentiæ, qui eſt e, uſque ad ſphæram perpendicu
              <lb/>
            lariter, id eſt e f, pars lineæ e n minor eſt ſemidiametro.</s>
            <s xml:id="echoid-s7979" xml:space="preserve"> Sit f punctum, in quo a n ſecat ſu-
              <lb/>
            perficiem ſphæræ.</s>
            <s xml:id="echoid-s7980" xml:space="preserve"> Dico ergo, quòd e f minor eſt n f.</s>
            <s xml:id="echoid-s7981" xml:space="preserve"> Quo
              <lb/>
              <figure xlink:label="fig-0141-02" xlink:href="fig-0141-02a" number="48">
                <variables xml:id="echoid-variables38" xml:space="preserve">a h b e g p f d z n q</variables>
              </figure>
            niam ut dictum eſt [proximo numero] proportio a g ad g d, ſicut
              <lb/>
            a e ad e d:</s>
            <s xml:id="echoid-s7982" xml:space="preserve"> ſed a n ad d n, ſicut a g ad g d:</s>
            <s xml:id="echoid-s7983" xml:space="preserve"> Igitur [per 11 p 5] a n
              <lb/>
            ad d n, ſicut a e ad e d:</s>
            <s xml:id="echoid-s7984" xml:space="preserve"> Igitur [per 16 p 5] a n ad a e, ſicut d n ad
              <lb/>
            d e:</s>
            <s xml:id="echoid-s7985" xml:space="preserve"> ſed [per 9 ax] a n maior a e.</s>
            <s xml:id="echoid-s7986" xml:space="preserve"> Quare d n maior d e:</s>
            <s xml:id="echoid-s7987" xml:space="preserve"> quare
              <lb/>
            d n maior d f:</s>
            <s xml:id="echoid-s7988" xml:space="preserve"> quare n f maior e f:</s>
            <s xml:id="echoid-s7989" xml:space="preserve"> quod eſt propoſitum.</s>
            <s xml:id="echoid-s7990" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div306" type="section" level="0" n="0">
          <head xml:id="echoid-head322" xml:space="preserve" style="it">19. Sirecta linea ab uno uiſu ſit perpendicularis ſpeculo ſphæ-
            <lb/>
          rico conuexo: unum ipſi{us} punctum, in quo uiſ{us} ſuperficiem ſe-
            <lb/>
          cat, ab uno ſpeculi puncto, in quod cadit, ad eundem uiſum refle-
            <lb/>
          ctetur. 10 p 6.</head>
          <p>
            <s xml:id="echoid-s7991" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s7992" xml:space="preserve"> ſit g centrum uiſus:</s>
            <s xml:id="echoid-s7993" xml:space="preserve"> d centrum ſphæræ:</s>
            <s xml:id="echoid-s7994" xml:space="preserve"> d z g per-
              <lb/>
            pendicularis à centro uiſus a d ſphæram.</s>
            <s xml:id="echoid-s7995" xml:space="preserve"> Dico, quòd nullius
              <lb/>
            puncti forma reflectitur per hãc perpendicularem, niſi pun-
              <lb/>
            cti eius, quod eſt in ſuperficie uiſus.</s>
            <s xml:id="echoid-s7996" xml:space="preserve"> Punctorum enim formæ poſt
              <lb/>
            centrum uiſus ſum ptorum non reflectuntur per eam, propter cauſ-
              <lb/>
            ſam ſupradictam [13 n.</s>
            <s xml:id="echoid-s7997" xml:space="preserve">] Similiter nec puncta inter ſuperficiem ui-
              <lb/>
            ſus & ſpeculum ſumpta.</s>
            <s xml:id="echoid-s7998" xml:space="preserve"> Dico etiam, quòd nullum punctum huius
              <lb/>
            perpendicularis reflectitur ab alio puncto ſpeculi.</s>
            <s xml:id="echoid-s7999" xml:space="preserve"> Si enim dicatur,
              <lb/>
            quòd ab alio puncto:</s>
            <s xml:id="echoid-s8000" xml:space="preserve"> ſit illud punctum a:</s>
            <s xml:id="echoid-s8001" xml:space="preserve"> erit
              <lb/>
              <figure xlink:label="fig-0141-03" xlink:href="fig-0141-03a" number="49">
                <variables xml:id="echoid-variables39" xml:space="preserve">x e g k z a d</variables>
              </figure>
            linea g a linea reflexionis:</s>
            <s xml:id="echoid-s8002" xml:space="preserve"> & à puncto illo in-
              <lb/>
            telligamus lineam ad a, quæ eſt linea, per quã
              <lb/>
            mouetur forma:</s>
            <s xml:id="echoid-s8003" xml:space="preserve"> & includunt hæ duæ lineæ
              <lb/>
            angulum ſuper a:</s>
            <s xml:id="echoid-s8004" xml:space="preserve"> quem quidem angulum ne-
              <lb/>
            ceſſariò diuidet per æqualia diameter d a, cum
              <lb/>
            ſit perpẽdicularis ſuper punctum a.</s>
            <s xml:id="echoid-s8005" xml:space="preserve"> Quia per-
              <lb/>
            pendicularis diuidit angulum ex linea motus
              <lb/>
            formę & linea reflexiõis, per ęqua [per 13 n 4.</s>
            <s xml:id="echoid-s8006" xml:space="preserve">]
              <lb/>
            Etita diameter d a concurret cum perpendicu
              <lb/>
            lari g d, inter punctum ſumptum & g.</s>
            <s xml:id="echoid-s8007" xml:space="preserve"> Et ita
              <lb/>
            duæ lineæ rectæ in duobus punctis concur-
              <lb/>
            rent, & ſuperficiem includent [contra 12 ax:</s>
            <s xml:id="echoid-s8008" xml:space="preserve">] Reſtat ergo, ut ſolius puncti, quod eſt in ſuperficie
              <lb/>
            uiſus, forma reflectatur à ſpeculo per perpendicularem, & uideatur in proprio imaginis loco, pro-
              <lb/>
            pter eius cum alijs punctis continuitatem.</s>
            <s xml:id="echoid-s8009" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div308" type="section" level="0" n="0">
          <head xml:id="echoid-head323" xml:space="preserve" style="it">20. Sipars lineæ reflexionis, intra peripheriam circuli (qui eſt communis ſectio ſuperficie-
            <lb/>
          rum reflexionis & ſpeculi ſphærici conuexi) continuatæ, æquetur ſemidiametro eiuſdem peri-
            <lb/>
          pheriæ: imago intra ſpeculum uidebitur. 24 p 6.</head>
          <p>
            <s xml:id="echoid-s8010" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s8011" xml:space="preserve"> g a, g b ſint lineæ à centro uiſus ductæ, contingentes ſphæram:</s>
            <s xml:id="echoid-s8012" xml:space="preserve"> & ſignetur circulus, ſu
              <lb/>
            </s>
          </p>
        </div>
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