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-s8459" xml:space="preserve">
              <pb o="142" file="0148" n="148" rhead="ALHAZEN"/>
            & ſit t p linea, per quam forma mouetur ad ſpeculũ:</s>
            <s xml:id="echoid-s8460" xml:space="preserve"> & ducatur perpendicularis n t u:</s>
            <s xml:id="echoid-s8461" xml:space="preserve"> quę neceſſariò
              <lb/>
            diuidet angulum p t a per æqualia:</s>
            <s xml:id="echoid-s8462" xml:space="preserve"> [ut oſtenſum eſt 13 n 4] & ducatur perpendicularis n g k:</s>
            <s xml:id="echoid-s8463" xml:space="preserve"> erit [ք
              <lb/>
            21 p 1] angulus n t a maior n g a:</s>
            <s xml:id="echoid-s8464" xml:space="preserve"> reſtat ergo [per 13 p 1] angulus u t a minor angulo k g a.</s>
            <s xml:id="echoid-s8465" xml:space="preserve"> Quare angu-
              <lb/>
            lus p t u
              <gap/>
            minor angulo b g k:</s>
            <s xml:id="echoid-s8466" xml:space="preserve"> [angulus enim k g a æquatur angulo b g k, per 12 n 4.</s>
            <s xml:id="echoid-s8467" xml:space="preserve">] Sed [per 32 p
              <lb/>
            1] angulus p t u ualet angulum t n p, & t p n:</s>
            <s xml:id="echoid-s8468" xml:space="preserve"> quia exterior:</s>
            <s xml:id="echoid-s8469" xml:space="preserve"> & angulus b g k ualet angulũ g n b, & an-
              <lb/>
            gulum g b n Erunt ergo duo anguli t n p, t p n minores duobus angulis g b n, g n b:</s>
            <s xml:id="echoid-s8470" xml:space="preserve"> quod [per 9 ax]
              <lb/>
            eſt impoſsibile:</s>
            <s xml:id="echoid-s8471" xml:space="preserve"> cum angulus p n t contineat angulum g n b tanquam partem:</s>
            <s xml:id="echoid-s8472" xml:space="preserve"> & [per 16 p 1] angulus
              <lb/>
            t p n ſit maior g b n.</s>
            <s xml:id="echoid-s8473" xml:space="preserve"> Reſtat ergo, ut punctum p non reflectatur, niſi à punctis inter g & l intermedijs.</s>
            <s xml:id="echoid-s8474" xml:space="preserve">
              <lb/>
            Et omnes lineæ à puncto a per hæc puncta ductæ ad diametrum b n, cadunt in puncta ſphęræ à cen
              <lb/>
            tro uiſus magis elongata, puncto g.</s>
            <s xml:id="echoid-s8475" xml:space="preserve"> Et ita patet propoſitum.</s>
            <s xml:id="echoid-s8476" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div329" type="section" level="0" n="0">
          <head xml:id="echoid-head334" xml:space="preserve" style="it">31. Viſa & uiſibilia à centro ſpeculi ſphærici conuexi æquabiliter diſtantib{us}: punctum refle-
            <lb/>
          xionis inuenire. 20 p 6.</head>
          <p>
            <s xml:id="echoid-s8477" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s8478" xml:space="preserve"> dato ſpeculo, & dato puncto uiſo:</s>
            <s xml:id="echoid-s8479" xml:space="preserve"> eſt inuenire punctum reflexionis.</s>
            <s xml:id="echoid-s8480" xml:space="preserve"> Sit enim b pun-
              <lb/>
            ctum uiſum:</s>
            <s xml:id="echoid-s8481" xml:space="preserve"> a centrum uiſus:</s>
            <s xml:id="echoid-s8482" xml:space="preserve"> & ducantur ab eis duæ lineæ ad
              <lb/>
              <figure xlink:label="fig-0148-01" xlink:href="fig-0148-01a" number="61">
                <variables xml:id="echoid-variables51" xml:space="preserve">b d a f e g c</variables>
              </figure>
            centrum ſpeculi.</s>
            <s xml:id="echoid-s8483" xml:space="preserve"> Si fuerint duæ illæ lineæ æquales:</s>
            <s xml:id="echoid-s8484" xml:space="preserve"> erit facile
              <lb/>
            inuenire:</s>
            <s xml:id="echoid-s8485" xml:space="preserve"> quoniam ſumetur circulus ſphæræ in ſuperficie duarum il-
              <lb/>
            larum linearum.</s>
            <s xml:id="echoid-s8486" xml:space="preserve"> Et ſcimus [per 29 n] quòd ab unico ſolo puncto il
              <lb/>
            lius circuli fit unius puncti reflexio.</s>
            <s xml:id="echoid-s8487" xml:space="preserve"> Diuidatur ergo [per 9 p 1] angu-
              <lb/>
            lus, quem continent in centro duæ illæ lineę, per æqualia:</s>
            <s xml:id="echoid-s8488" xml:space="preserve"> & ducatur
              <lb/>
            linea diuidens angulum, extra ſphæram:</s>
            <s xml:id="echoid-s8489" xml:space="preserve"> erit quidem [per 18 p 3] per-
              <lb/>
            pendicularis ſuper lineam contingentem hunc circulum in puncto,
              <lb/>
            per quod tranſit.</s>
            <s xml:id="echoid-s8490" xml:space="preserve"> Et ſi ducantur ad illud punctum duę lineę:</s>
            <s xml:id="echoid-s8491" xml:space="preserve"> una à cen
              <lb/>
            tro uiſus:</s>
            <s xml:id="echoid-s8492" xml:space="preserve"> alia à punctu uiſo:</s>
            <s xml:id="echoid-s8493" xml:space="preserve"> efficient cum perpendiculari illa & dua-
              <lb/>
            bus primis lineis, duo triangula:</s>
            <s xml:id="echoid-s8494" xml:space="preserve"> quorum duo latera duobus laterib.</s>
            <s xml:id="echoid-s8495" xml:space="preserve">
              <lb/>
            æqualia, & angulus angulo [ideoq́;</s>
            <s xml:id="echoid-s8496" xml:space="preserve"> per 4.</s>
            <s xml:id="echoid-s8497" xml:space="preserve"> 13 p 1.</s>
            <s xml:id="echoid-s8498" xml:space="preserve"> 3 ax angulus a e d æ-
              <lb/>
            quatur angulo b e d.</s>
            <s xml:id="echoid-s8499" xml:space="preserve">] Et ita punctum circuli, per quod perpendicula
              <lb/>
            ris illa tranſit:</s>
            <s xml:id="echoid-s8500" xml:space="preserve"> eſt punctum reflexionis:</s>
            <s xml:id="echoid-s8501" xml:space="preserve"> [quia c e d bifariam ſecat an-
              <lb/>
            gulum ab incidentię & reflexionis lineis comprehenſum, ut patuit 13
              <lb/>
            n 4.</s>
            <s xml:id="echoid-s8502" xml:space="preserve">] Si nero linea à puncto uiſo ad centrum ſphærę ducta, fuerit inę
              <lb/>
            qualis lineę à centro uiſus ad idem centrum ductę:</s>
            <s xml:id="echoid-s8503" xml:space="preserve"> oportet nos quę-
              <lb/>
            dam antecedentia præponere:</s>
            <s xml:id="echoid-s8504" xml:space="preserve"> quorum unum eſt.</s>
            <s xml:id="echoid-s8505" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div331" type="section" level="0" n="0">
          <head xml:id="echoid-head335" xml:space="preserve" style="it">32. À
            <unsure/>
          puncto dimidiatæ peripheriæ medio, ducere lineam re-
            <lb/>
          ctam, ut ſegmentum ei{us} conterminum continuatæ diametro, æquetur datæ lineæ rectæ. 128 p 1.</head>
          <p>
            <s xml:id="echoid-s8506" xml:space="preserve">SVmpta circuli diametro, & ſumpto in circumferentia puncto:</s>
            <s xml:id="echoid-s8507" xml:space="preserve"> eſt ducere ab eo ad diametrũ ex-
              <lb/>
            tra productam, lineam, quę à puncto, in quo ſecat circulum, uſq;</s>
            <s xml:id="echoid-s8508" xml:space="preserve"> ad cõcurſum cum diametro, ſit
              <lb/>
            æqualis lineæ datæ Verbi gratia, ſit q e data linea:</s>
            <s xml:id="echoid-s8509" xml:space="preserve"> g b diameter circuli a b g:</s>
            <s xml:id="echoid-s8510" xml:space="preserve"> a punctũ datum.</s>
            <s xml:id="echoid-s8511" xml:space="preserve"> Di
              <lb/>
            co, quòd à puncto a ducam lineam, quæ à pũcto, in quo ſecuerit circulum, uſq;</s>
            <s xml:id="echoid-s8512" xml:space="preserve"> ad diametrum g b, ſit
              <lb/>
            æqualis lineæ q e:</s>
            <s xml:id="echoid-s8513" xml:space="preserve"> quod ſic conſtabit.</s>
            <s xml:id="echoid-s8514" xml:space="preserve"> Ducantur duę lineæ a b, a g:</s>
            <s xml:id="echoid-s8515" xml:space="preserve"> quę aut erunt æ quales:</s>
            <s xml:id="echoid-s8516" xml:space="preserve"> aut inęqua
              <lb/>
            les.</s>
            <s xml:id="echoid-s8517" xml:space="preserve"> Sint ęquales:</s>
            <s xml:id="echoid-s8518" xml:space="preserve"> & adiungatur lineæ q e linea talis, ut illud, quod fit ex ductu totius cum adiuncta
              <lb/>
            in ad ũctam, ſit
              <lb/>
              <figure xlink:label="fig-0148-02" xlink:href="fig-0148-02a" number="62">
                <variables xml:id="echoid-variables52" xml:space="preserve">q a e g</variables>
              </figure>
              <figure xlink:label="fig-0148-03" xlink:href="fig-0148-03a" number="63">
                <variables xml:id="echoid-variables53" xml:space="preserve">a z g e b q</variables>
              </figure>
              <figure xlink:label="fig-0148-04" xlink:href="fig-0148-04a" number="64">
                <variables xml:id="echoid-variables54" xml:space="preserve">d
                  <gap/>
                q g h
                  <gap/>
                a z b</variables>
              </figure>
            ęquale quadra
              <lb/>
            to a g.</s>
            <s xml:id="echoid-s8519" xml:space="preserve"> [id uerò
              <lb/>
            expeditè fiet:</s>
            <s xml:id="echoid-s8520" xml:space="preserve"> ſi
              <lb/>
            linea q e fiat dia
              <lb/>
            meter circuli,
              <lb/>
            cuius periphe-
              <lb/>
            riam tangens re
              <lb/>
            cta linea æqua-
              <lb/>
            lis a g, cõcurrat
              <lb/>
            cum continua-
              <lb/>
            ta diametro q e:</s>
            <s xml:id="echoid-s8521" xml:space="preserve">
              <lb/>
            ſic enim oblongum cõprehenſum ſub continuata diametro & exte-
              <lb/>
            riore eius ſegmento ęquabitur quadrato lineę a g per 36 p 3.</s>
            <s xml:id="echoid-s8522" xml:space="preserve">] Et ſit li
              <lb/>
            nea adiũcta e z.</s>
            <s xml:id="echoid-s8523" xml:space="preserve"> Cũ igitur illud, qđ fit ex ductu q z in e z ſit ęquale ei,
              <lb/>
            quod fit ex ductu a g in ſe:</s>
            <s xml:id="echoid-s8524" xml:space="preserve"> erit q z maior a g, & e z minor eadem.</s>
            <s xml:id="echoid-s8525" xml:space="preserve"> Si e-
              <lb/>
            nim e z fuerit æqualis, aut maior a g:</s>
            <s xml:id="echoid-s8526" xml:space="preserve"> eſt impoſsibile, ut ductus q z in
              <lb/>
            e z ſit æqualis quadrato a g [ſic enim oblongum comprehẽſum ſub
              <lb/>
            q z & e z ſemper maius eſſet quadrato a g:</s>
            <s xml:id="echoid-s8527" xml:space="preserve"> quia linea q z eſſet maior
              <lb/>
            e z, ut totum ſua parte.</s>
            <s xml:id="echoid-s8528" xml:space="preserve">] Si autem minor:</s>
            <s xml:id="echoid-s8529" xml:space="preserve"> palàm, quòd q z eſt maior a
              <lb/>
            g.</s>
            <s xml:id="echoid-s8530" xml:space="preserve"> Producatur ergo ad ęqualitatem:</s>
            <s xml:id="echoid-s8531" xml:space="preserve"> & ſit a g t:</s>
            <s xml:id="echoid-s8532" xml:space="preserve"> & poſito pede circini ſuper a, fiat circulus ſecundum
              <lb/>
            quantitatem a g t:</s>
            <s xml:id="echoid-s8533" xml:space="preserve"> qui quidem circulus ſecabit diametrum b g:</s>
            <s xml:id="echoid-s8534" xml:space="preserve"> [infinitè uerſus t continuatam] &
              <lb/>
            </s>
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