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-s8012" xml:space="preserve">
              <pb o="136" file="0142" n="142" rhead="ALHAZEN"/>
            per quem ſuperficies his lineis incluſa ſecat ſphæram:</s>
            <s xml:id="echoid-s8013" xml:space="preserve"> erit [per 25 n 4] a b portio apparens ex
              <lb/>
            hoc circulo.</s>
            <s xml:id="echoid-s8014" xml:space="preserve"> Dico ergo, quòd loca imaginum, quæ per reflexiones ab hac portione factas compre-
              <lb/>
            henduntur:</s>
            <s xml:id="echoid-s8015" xml:space="preserve"> quædam ſunt intra ſpeculum:</s>
            <s xml:id="echoid-s8016" xml:space="preserve"> quędam in ſu
              <lb/>
              <figure xlink:label="fig-0142-01" xlink:href="fig-0142-01a" number="50">
                <variables xml:id="echoid-variables40" xml:space="preserve">g m h z p b d a k</variables>
              </figure>
            perficie ſpeculi:</s>
            <s xml:id="echoid-s8017" xml:space="preserve"> quędam extra ſpeculũ.</s>
            <s xml:id="echoid-s8018" xml:space="preserve"> Et unumquod-
              <lb/>
            que horum eſt determinandum.</s>
            <s xml:id="echoid-s8019" xml:space="preserve"> Ducatur à puncto g li-
              <lb/>
            nea ſecans circulum, & pars eius, quæ eſt chorda arcus
              <lb/>
            circuli, ſit æqualis ſemidiametro circuli [id quod per 1
              <lb/>
            p 4 fieri poteſt:</s>
            <s xml:id="echoid-s8020" xml:space="preserve">] ſit linea illa g h k:</s>
            <s xml:id="echoid-s8021" xml:space="preserve"> & chorda æqualis
              <lb/>
            ſemidiametro ſit h k:</s>
            <s xml:id="echoid-s8022" xml:space="preserve"> & producatur à puncto h perpen-
              <lb/>
            dicularis, quæ ſit d h m.</s>
            <s xml:id="echoid-s8023" xml:space="preserve"> Dico, quòd formæ reflexę à pun
              <lb/>
            cto h locus eſt intra ſphęram.</s>
            <s xml:id="echoid-s8024" xml:space="preserve"> Ducatur [per 23 p 1] à pun
              <lb/>
            cto h linea æ qualem renens angulum cum m h, angulo
              <lb/>
            m h g:</s>
            <s xml:id="echoid-s8025" xml:space="preserve"> & ſit p h:</s>
            <s xml:id="echoid-s8026" xml:space="preserve"> reflectentur quidem puncta huius lineæ
              <lb/>
            à puncto h ad uiſum g, & nõ alterius [per 12 n 4.</s>
            <s xml:id="echoid-s8027" xml:space="preserve">] Suma
              <lb/>
            tur ergo aliquod eius punctum:</s>
            <s xml:id="echoid-s8028" xml:space="preserve"> & ſit p:</s>
            <s xml:id="echoid-s8029" xml:space="preserve"> & ducatur ab eo
              <lb/>
            linea ad centrum ſphærę quę ſit p d:</s>
            <s xml:id="echoid-s8030" xml:space="preserve"> erit [ut demonſtra
              <lb/>
            tum eſt 25 n 4] p d perpendicularis ſuper ſuperficiem,
              <lb/>
            contingentem ſphæram ſuper punctum eius, per quod
              <lb/>
            tranſit p d:</s>
            <s xml:id="echoid-s8031" xml:space="preserve"> & coniungatur d k.</s>
            <s xml:id="echoid-s8032" xml:space="preserve"> Verùm angulus p h m eſt
              <lb/>
            æqualis angulo m h g [ex fabricatione.</s>
            <s xml:id="echoid-s8033" xml:space="preserve">] Quare [per 15
              <lb/>
            p 1] ſimiliter æqualis eſt angulo contrapoſito k h d:</s>
            <s xml:id="echoid-s8034" xml:space="preserve"> ſed
              <lb/>
            [per hypotheſim & 5 p 1] k h d eſt æqualis k d h:</s>
            <s xml:id="echoid-s8035" xml:space="preserve"> quoni-
              <lb/>
            am reſpiciunt æqualia latera:</s>
            <s xml:id="echoid-s8036" xml:space="preserve"> Igitur [per 1 ax:</s>
            <s xml:id="echoid-s8037" xml:space="preserve">] angulus
              <lb/>
            p h m æ qualis eſt angulo k d m.</s>
            <s xml:id="echoid-s8038" xml:space="preserve"> Quare [per 28 p 1] lineę
              <lb/>
            k d, p h ſunt ęquidiſtantes:</s>
            <s xml:id="echoid-s8039" xml:space="preserve"> ergo [per 35 def 1] in infi-
              <lb/>
            nitum productę nun quam concurrent:</s>
            <s xml:id="echoid-s8040" xml:space="preserve"> & linea p d ſeca-
              <lb/>
            bit lineam, interiacentem inter k d, & p h [quia ſecat an-
              <lb/>
            gulum h d k ipſi h k ſubtenſum.</s>
            <s xml:id="echoid-s8041" xml:space="preserve">] Et ita quodcunq;</s>
            <s xml:id="echoid-s8042" xml:space="preserve"> pun-
              <lb/>
            ctum ſumatur in linea p h:</s>
            <s xml:id="echoid-s8043" xml:space="preserve"> linea ducta ab illo puncto, ad
              <lb/>
            punctum d, ſecabit lineam reflexionis intra ſphęram:</s>
            <s xml:id="echoid-s8044" xml:space="preserve"> quę quidem linea perpendicularis erit ſuper
              <lb/>
            ſphęram [per 25 n 4] ſicut eſt p d.</s>
            <s xml:id="echoid-s8045" xml:space="preserve"> Quare imago cuiuſcunque puncti lineę p h apparebit intra ſphę
              <lb/>
            ram [per 3 n.</s>
            <s xml:id="echoid-s8046" xml:space="preserve">]</s>
          </p>
        </div>
        <div xml:id="echoid-div310" type="section" level="0" n="0">
          <head xml:id="echoid-head324" xml:space="preserve" style="it">21. Si reflexio fiat à peripheria circuli (qui eſt communis ſectio ſuperficierum, reflexionis &
            <lb/>
          ſpeculi ſphærici conuexi) inter rectam à uiſu ad ſpeculi centrum ductam, & lineam reflexionis,
            <lb/>
          æquantem partem ſuam intra peripheriam, eiuſdem ſemidiametro: imago intra ſpeculum ui-
            <lb/>
          debitur. 25 p 6.</head>
          <p>
            <s xml:id="echoid-s8047" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s8048" xml:space="preserve"> arcus circuli interiacens inter punctum h, & punctum, per quod tranſit perpendi-
              <lb/>
            cularis à centro uiſus ducta:</s>
            <s xml:id="echoid-s8049" xml:space="preserve"> eſto h z.</s>
            <s xml:id="echoid-s8050" xml:space="preserve"> Dico, quod
              <lb/>
              <figure xlink:label="fig-0142-02" xlink:href="fig-0142-02a" number="51">
                <variables xml:id="echoid-variables41" xml:space="preserve">t g p b h i z d a k s</variables>
              </figure>
            à quocunq, puncto huius arcus fiat reflexio:</s>
            <s xml:id="echoid-s8051" xml:space="preserve"> lo-
              <lb/>
            cus imaginis erit intra ſphæram.</s>
            <s xml:id="echoid-s8052" xml:space="preserve"> Sit i punctum ſumptũ:</s>
            <s xml:id="echoid-s8053" xml:space="preserve">
              <lb/>
            & ducatur linea à centro uiſus ſecans cιrculũ ſuper pun-
              <lb/>
            ctum illud, quę ſit g is:</s>
            <s xml:id="echoid-s8054" xml:space="preserve"> & ducatur perpendicularis per
              <lb/>
            punctum hoc, quę ſit d i t:</s>
            <s xml:id="echoid-s8055" xml:space="preserve"> & [per 23 p 1] fiat linea p i, æ-
              <lb/>
            qualem tenens angulum cum it angulo tig.</s>
            <s xml:id="echoid-s8056" xml:space="preserve"> Palàm [per
              <lb/>
            12 n 4] quòd ſola puncta lineę p i reflectuntur à puncto
              <lb/>
            iad uiſum.</s>
            <s xml:id="echoid-s8057" xml:space="preserve"> Palàm etiam [per 15 p 3] quòd linea i s ma-
              <lb/>
            ior eſt linea k h.</s>
            <s xml:id="echoid-s8058" xml:space="preserve"> Quare maior s d [eſt enim h k ex prima
              <lb/>
            hypotheſi ęqualis ſemidiametro s d.</s>
            <s xml:id="echoid-s8059" xml:space="preserve">] Igitur [per 18 p 1]
              <lb/>
            angulus s d i maior eſt angulo s i d:</s>
            <s xml:id="echoid-s8060" xml:space="preserve"> quare [per 15 p 1]
              <lb/>
            eſt maior angulo g i t:</s>
            <s xml:id="echoid-s8061" xml:space="preserve"> quare eſt maior angulo tip.</s>
            <s xml:id="echoid-s8062" xml:space="preserve"> Igitur
              <lb/>
            lineę p i & s d nunquam concurrent [ad partes p & s:</s>
            <s xml:id="echoid-s8063" xml:space="preserve">
              <lb/>
            ſecus ſpatium comprehenderent contra 12 ax.</s>
            <s xml:id="echoid-s8064" xml:space="preserve"> quia con-
              <lb/>
            currunt ad partes i & d per 11 ax.</s>
            <s xml:id="echoid-s8065" xml:space="preserve">] Et linea ducta à pun-
              <lb/>
            cto quocunque p i lineę, ad punctum d, ſecat lineam s i
              <lb/>
            intra ſphęram:</s>
            <s xml:id="echoid-s8066" xml:space="preserve"> quę s i eſt linea reflexionis:</s>
            <s xml:id="echoid-s8067" xml:space="preserve"> & omnis
              <lb/>
            linea ducta à quocunq;</s>
            <s xml:id="echoid-s8068" xml:space="preserve"> puncto p i lineę, ad punctum d:</s>
            <s xml:id="echoid-s8069" xml:space="preserve">
              <lb/>
            erit perpendicularis ſuper ſphęram [ut oſtenſum eſt 25
              <lb/>
            n 4,] ſicut eſt p d.</s>
            <s xml:id="echoid-s8070" xml:space="preserve"> Et cum locus imaginis ſit in concur-
              <lb/>
            ſu perpendicularis à puncto uiſo & lineę reflexionis:</s>
            <s xml:id="echoid-s8071" xml:space="preserve">
              <lb/>
            [per 3n] erit imago cuiuslibet puncti lineę p i intra
              <lb/>
            ſphę
              <gap/>
            a n.</s>
            <s xml:id="echoid-s8072" xml:space="preserve"> Palàm ergo, quòd omnium imaginum arcus
              <lb/>
            hz, locus proprius erit intra ſpeculum:</s>
            <s xml:id="echoid-s8073" xml:space="preserve"> Quod
              <lb/>
            eſt propoſitum.</s>
            <s xml:id="echoid-s8074" xml:space="preserve"/>
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