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-s11948" xml:space="preserve">
              <pb o="178" file="0184" n="184" rhead="ALHAZEN"/>
            ſicut d k ad d s [eſt enim per 2 p 6, ut s t ad tr, ſic s k ad kd:</s>
            <s xml:id="echoid-s11949" xml:space="preserve"> & per 18 p 5, ut s r ad rt, ſic s d ad d k, &
              <lb/>
            per conſectarium 4 p 5, utrtadrs, ſic d k ad d s.</s>
            <s xml:id="echoid-s11950" xml:space="preserve">] Igitur [per 11 p 5] ktadts, ſicut d k ad d s.</s>
            <s xml:id="echoid-s11951" xml:space="preserve"> Sed que
              <lb/>
            niam angulus ft o æqualis eſt angulo o d a:</s>
            <s xml:id="echoid-s11952" xml:space="preserve"> [perfabrica-
              <lb/>
              <figure xlink:label="fig-0184-01" xlink:href="fig-0184-01a" number="132">
                <variables xml:id="echoid-variables122" xml:space="preserve" style="it">s g z k t e f d o b r a</variables>
              </figure>
            tionem] erit [per 13 p 1] angulus o d s ęqualis angulo fts
              <lb/>
            [& angulus ad s ęquatur ſibijpſi:</s>
            <s xml:id="echoid-s11953" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s11954" xml:space="preserve"> per 32 p 1 trian gula
              <lb/>
            s t f, d s o ſunt æquiangula.</s>
            <s xml:id="echoid-s11955" xml:space="preserve">] Igitur [ք 4 p 6] stad t f, ſicut
              <lb/>
            d s ad d o:</s>
            <s xml:id="echoid-s11956" xml:space="preserve"> & eſt k t ad t s, ſicut d k ad d s:</s>
            <s xml:id="echoid-s11957" xml:space="preserve"> & ts ad t f, ſicut
              <lb/>
            d s ad d o:</s>
            <s xml:id="echoid-s11958" xml:space="preserve"> quare [ք 22 p 5] ktadtſ, ſicut k d ad d o.</s>
            <s xml:id="echoid-s11959" xml:space="preserve"> Quod
              <lb/>
            eſt propoſitũ.</s>
            <s xml:id="echoid-s11960" xml:space="preserve"> Sed quoniã k z æquidiſtat t f:</s>
            <s xml:id="echoid-s11961" xml:space="preserve"> [per fabrica-
              <lb/>
            tionem] erit [per 29 p 1] angulus k z e ęqualis angulo e t
              <lb/>
            f:</s>
            <s xml:id="echoid-s11962" xml:space="preserve"> & ita triangulũ k z e ſimile triangulo e t ſ.</s>
            <s xml:id="echoid-s11963" xml:space="preserve"> [Nam anguli
              <lb/>
            ad e æquãtur per 15 p 1;</s>
            <s xml:id="echoid-s11964" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s11965" xml:space="preserve"> per 32 p 1.</s>
            <s xml:id="echoid-s11966" xml:space="preserve"> 4 p.</s>
            <s xml:id="echoid-s11967" xml:space="preserve"> 1 d 6 triangula
              <lb/>
            k z e, e t f ſunt ſimilia.</s>
            <s xml:id="echoid-s11968" xml:space="preserve">] Quare ꝓportio k e ad e ſ, ſicut k z
              <lb/>
            ad t f:</s>
            <s xml:id="echoid-s11969" xml:space="preserve"> ſed [per 3 p 6] k e ad e f, ſicut k t ad t f, propter angu-
              <lb/>
            lum ſuper t diuiſum ք æqualia.</s>
            <s xml:id="echoid-s11970" xml:space="preserve"> Igitur [ք 9 p 5] k z æqua-
              <lb/>
            lis eſt k t.</s>
            <s xml:id="echoid-s11971" xml:space="preserve"> Verùm quoniã k q eſt perpẽdicularis ſuper e z:</s>
            <s xml:id="echoid-s11972" xml:space="preserve">
              <lb/>
            [per ſabricationẽ] erũt omnes eius anguli recti:</s>
            <s xml:id="echoid-s11973" xml:space="preserve"> ſed an-
              <lb/>
            gulus e t d eſt acutus:</s>
            <s xml:id="echoid-s11974" xml:space="preserve"> quoniã eſt medietas anguli [fto, ut
              <lb/>
            patuit.</s>
            <s xml:id="echoid-s11975" xml:space="preserve">] Igitur k q cõcurret cũ t d [ք 11 ax.</s>
            <s xml:id="echoid-s11976" xml:space="preserve">] Sit cõcurſus
              <lb/>
            h:</s>
            <s xml:id="echoid-s11977" xml:space="preserve"> & ducatur linea e h:</s>
            <s xml:id="echoid-s11978" xml:space="preserve"> & [per 31 p 1] à pũcto e ducature æ-
              <lb/>
            quidiſtans h k, ꝓducta uſq;</s>
            <s xml:id="echoid-s11979" xml:space="preserve"> ad d h:</s>
            <s xml:id="echoid-s11980" xml:space="preserve"> quæ ſit e x:</s>
            <s xml:id="echoid-s11981" xml:space="preserve"> & mutetur
              <lb/>
            figura propter intricationẽ linearũ:</s>
            <s xml:id="echoid-s11982" xml:space="preserve"> & [per 5 p 4] fiat cir-
              <lb/>
            culus, trãſiens per tria puncta x, t, e:</s>
            <s xml:id="echoid-s11983" xml:space="preserve"> & ꝓducatur k d uſq;</s>
            <s xml:id="echoid-s11984" xml:space="preserve">
              <lb/>
            in circulũ, cadens in punctũ m:</s>
            <s xml:id="echoid-s11985" xml:space="preserve"> & educatur m t:</s>
            <s xml:id="echoid-s11986" xml:space="preserve"> erit [per
              <lb/>
            27 p 3] angulus t m e æqualis angulo t x e:</s>
            <s xml:id="echoid-s11987" xml:space="preserve"> quia cadunt in
              <lb/>
            eundẽ arcũ:</s>
            <s xml:id="echoid-s11988" xml:space="preserve"> [e f t] & [ք 29 p 1] angulus t x e æqualis an
              <lb/>
            gulo t h k:</s>
            <s xml:id="echoid-s11989" xml:space="preserve"> erit t m e æqualis angulo t h k.</s>
            <s xml:id="echoid-s11990" xml:space="preserve"> Secetur ab an-
              <lb/>
            gulo t m e, æqualis angulo d h e:</s>
            <s xml:id="echoid-s11991" xml:space="preserve"> [id uerò fieri poteſt:</s>
            <s xml:id="echoid-s11992" xml:space="preserve"> ꝗa
              <lb/>
            angulus t h k maior eſt angulo d h e per 9 ax:</s>
            <s xml:id="echoid-s11993" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s11994" xml:space="preserve"> t m e
              <lb/>
            eodẽ maior eſt] ꝗ ſit ſ m d:</s>
            <s xml:id="echoid-s11995" xml:space="preserve"> & punctũ, in quo ſ m ſecat t x, ſit i.</s>
            <s xml:id="echoid-s11996" xml:space="preserve"> Palàm quòd triangulũ i m d ſimile eſt
              <lb/>
            triangulo e d h [quia enim angulus f m d æquatus eſt angulo d h e, & anguli ad d æquãtur per 15 p 1
              <gap/>
              <lb/>
            ergo ք 32 p 1 triangula ſunt æquiangula, & ք 4 p.</s>
            <s xml:id="echoid-s11997" xml:space="preserve"> 1 d
              <lb/>
              <figure xlink:label="fig-0184-02" xlink:href="fig-0184-02a" number="133">
                <variables xml:id="echoid-variables123" xml:space="preserve">t f i k e d m q z x h</variables>
              </figure>
            6 ſimilia.</s>
            <s xml:id="echoid-s11998" xml:space="preserve">] Quare proportio h d ad d m, ſicut e h ad i
              <lb/>
            m.</s>
            <s xml:id="echoid-s11999" xml:space="preserve"> Et ſimiliter triangulũ t m d ſimile triangulo k h d:</s>
            <s xml:id="echoid-s12000" xml:space="preserve">
              <lb/>
            [Nã angulus t m d æqualis cõcluſus eſt angulo th k:</s>
            <s xml:id="echoid-s12001" xml:space="preserve">
              <lb/>
            & anguli ad d ęquãtur ք 1 5 p 1.</s>
            <s xml:id="echoid-s12002" xml:space="preserve"> Quare ut prius trian-
              <lb/>
            gula ſunt ſimilia] & ꝓportio k d ad d t, ſicut h d ad d
              <lb/>
            m:</s>
            <s xml:id="echoid-s12003" xml:space="preserve"> & ita [ք 11 p 5] k d ad d t, ſicut e h ad im.</s>
            <s xml:id="echoid-s12004" xml:space="preserve"> Sed pro-
              <lb/>
            portio k d ad d t nota:</s>
            <s xml:id="echoid-s12005" xml:space="preserve"> quoniã ſemք una & eadẽ per-
              <lb/>
            manet, quodcũq;</s>
            <s xml:id="echoid-s12006" xml:space="preserve"> punctũ reflexionis ſit t in arcu b g:</s>
            <s xml:id="echoid-s12007" xml:space="preserve">
              <lb/>
            quia ſemper linea t d eſt una:</s>
            <s xml:id="echoid-s12008" xml:space="preserve"> [quia eſt ſemidiameter
              <lb/>
            circuli, qui eſt cõmunis ſectio ſuperſicierũ reflexio-
              <lb/>
            nis & ſpeculi] & k d ſimiliter [quia eſt diſtãtia pũcti
              <lb/>
            reflexi à cẽtro ſpeculi.</s>
            <s xml:id="echoid-s12009" xml:space="preserve">] Linea etiam e h unà in qua-
              <lb/>
            cunq;</s>
            <s xml:id="echoid-s12010" xml:space="preserve"> reflexione permanet, & nõ mutatur eius quã-
              <lb/>
            titas [quia angulus o d a idẽ ſemper քmanet:</s>
            <s xml:id="echoid-s12011" xml:space="preserve"> eiusq́;</s>
            <s xml:id="echoid-s12012" xml:space="preserve">
              <lb/>
            dimidius eſt angulus e t d:</s>
            <s xml:id="echoid-s12013" xml:space="preserve"> ꝗ a, ut patuit, dimidius eſt
              <lb/>
            angulifto, æquati angulo o d a.</s>
            <s xml:id="echoid-s12014" xml:space="preserve">] Quare linea im
              <lb/>
            ſemper erit una:</s>
            <s xml:id="echoid-s12015" xml:space="preserve"> quare punctũ ſnotũ & determina-
              <lb/>
            tum [quia per lineam i m longitudine ſemper eandẽ, continuatam in peripheriam oſtenditur.</s>
            <s xml:id="echoid-s12016" xml:space="preserve">] Si
              <lb/>
            ergo à tribus punctis arcus b g fieri poſſet reflexio:</s>
            <s xml:id="echoid-s12017" xml:space="preserve"> eſſet ducere à puncto fad circulum x t e tres li-
              <lb/>
            neas æquales:</s>
            <s xml:id="echoid-s12018" xml:space="preserve"> quarum cuiuslibet pars interiacens diametrum tx & circumferentiam circuli eſſet
              <lb/>
            æqualis lineæ i m:</s>
            <s xml:id="echoid-s12019" xml:space="preserve"> quia ſemper erit proportio k d ad d t, ſicut e h ad quamlibet illarum.</s>
            <s xml:id="echoid-s12020" xml:space="preserve"> Et patet ex
              <lb/>
            ſuperioribus [34 n] quòd non, niſi duæ æquales poſſunt.</s>
            <s xml:id="echoid-s12021" xml:space="preserve"> Quare à duobus tantùm punctis fiet re-
              <lb/>
            flexio.</s>
            <s xml:id="echoid-s12022" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s12023" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div420" type="section" level="0" n="0">
          <head xml:id="echoid-head386" xml:space="preserve" style="it">83. Datis duobus punctis in diuerſis diametris circuli (quieſt cõmunis ſectio ſuperficierum,
            <lb/>
          reflexionis & ſpeculi ſphærici caui) à centro inæquabiliter diſtantibus: inuenire in peripberia
            <lb/>
          comprebenſa inter ſemidiametros, in quibus ipſa ſunt, duo reflexionis puncta. 37 p 8.</head>
          <p>
            <s xml:id="echoid-s12024" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s12025" xml:space="preserve"> datis duobus punctis k, o in diuerſis diametris, inæ qualiter diſtantibus à centro:</s>
            <s xml:id="echoid-s12026" xml:space="preserve"> eſt
              <lb/>
            inuenire punctum reflexionis.</s>
            <s xml:id="echoid-s12027" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s12028" xml:space="preserve"> ſumatur linea z t:</s>
            <s xml:id="echoid-s12029" xml:space="preserve"> & [per 10 p 6] diuidatur in
              <lb/>
            puncto e, ut ſit proportio z e ad et, ſicut k d ad d o [in primo diagrammate præcedentis
              <lb/>
            numeri.</s>
            <s xml:id="echoid-s12030" xml:space="preserve">] Quoniam k d maior d o [extheſi præcedẽtis numeri] erit z e maior e t:</s>
            <s xml:id="echoid-s12031" xml:space="preserve"> diuidatur z t per
              <lb/>
            æqualia in puncto q:</s>
            <s xml:id="echoid-s12032" xml:space="preserve"> [per 10 p 1] & à puncto q ducatur perpendicularis ſuper z t:</s>
            <s xml:id="echoid-s12033" xml:space="preserve"> [per 11 p 1] & fiat
              <lb/>
            </s>
          </p>
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