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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        <div xml:id="echoid-div1718" type="section" level="0" n="0">
          <pb o="365" file="0667" n="667" rhead="LIBER OCTAVVS."/>
        </div>
        <div xml:id="echoid-div1720" type="section" level="0" n="0">
          <head xml:id="echoid-head1276" xml:space="preserve" style="it">66. Imagines rerum inter ſpecula ſphærica concaua & uiſus apparentes, motis rebus, uiden-
            <lb/>
          tur ad partem contrariam moueri.</head>
          <p>
            <s xml:id="echoid-s44238" xml:space="preserve">Sit ſpeculi ſphærici concaui circulus a b g:</s>
            <s xml:id="echoid-s44239" xml:space="preserve"> cuius centrum ſit punctus d:</s>
            <s xml:id="echoid-s44240" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s44241" xml:space="preserve"> centrum uiſus e ci-
              <lb/>
            tra centrũ ſpeculi, quod eſt d:</s>
            <s xml:id="echoid-s44242" xml:space="preserve"> & ex lateribus aſpicientis ſint duo puncta rei uiſæ:</s>
            <s xml:id="echoid-s44243" xml:space="preserve"> quę ſint z & h:</s>
            <s xml:id="echoid-s44244" xml:space="preserve"> quæ
              <lb/>
              <figure xlink:label="fig-0667-01" xlink:href="fig-0667-01a" number="803">
                <variables xml:id="echoid-variables780" xml:space="preserve">a b g l c d k m h c z</variables>
              </figure>
            reflectantur ad uiſum à duobus punctis a & b:</s>
            <s xml:id="echoid-s44245" xml:space="preserve"> ſintq́;</s>
            <s xml:id="echoid-s44246" xml:space="preserve"> li
              <lb/>
            neæ reflexionum e a puncti z, & e b puncti h:</s>
            <s xml:id="echoid-s44247" xml:space="preserve"> ducan-
              <lb/>
            turq́;</s>
            <s xml:id="echoid-s44248" xml:space="preserve"> catheti incidentiæ z d c & h d k ſecantes lineas
              <lb/>
            reflexionum in punctis c & k:</s>
            <s xml:id="echoid-s44249" xml:space="preserve"> erunt ergo per 37 th.</s>
            <s xml:id="echoid-s44250" xml:space="preserve"> 5
              <lb/>
            huius puncta c & k loca imaginũ:</s>
            <s xml:id="echoid-s44251" xml:space="preserve"> c puncti z, & k pun-
              <lb/>
            cti h.</s>
            <s xml:id="echoid-s44252" xml:space="preserve"> Videbuntur itaq;</s>
            <s xml:id="echoid-s44253" xml:space="preserve"> formæ illorum punctorum in
              <lb/>
            diuerſis partibus alijs, quàm ſint res ipſæ per 49 hu-
              <lb/>
            ius.</s>
            <s xml:id="echoid-s44254" xml:space="preserve"> Quòd ſi punctus h rei uiſæ transferatur ad pun-
              <lb/>
            ctum l:</s>
            <s xml:id="echoid-s44255" xml:space="preserve"> & reflectatur à puncto ſpeculi g ad uiſum e:</s>
            <s xml:id="echoid-s44256" xml:space="preserve"> du
              <lb/>
            caturq́;</s>
            <s xml:id="echoid-s44257" xml:space="preserve"> linea reflexionis, quę ſit e g:</s>
            <s xml:id="echoid-s44258" xml:space="preserve"> & cathetus l d
              <lb/>
            m, ſecans lineam reflexionis, quæ eſt e g, in puncto m:</s>
            <s xml:id="echoid-s44259" xml:space="preserve">
              <lb/>
            erit per 37 th.</s>
            <s xml:id="echoid-s44260" xml:space="preserve"> 5 huius punctus m locus imaginis for-
              <lb/>
            mæ puncti l.</s>
            <s xml:id="echoid-s44261" xml:space="preserve"> Imago itaq;</s>
            <s xml:id="echoid-s44262" xml:space="preserve"> puncti h, quę eſt k, erit tranſ-
              <lb/>
            lata ad partem diuerſam illi, ad quam res uera tranſ-
              <lb/>
            lata eſt.</s>
            <s xml:id="echoid-s44263" xml:space="preserve"> Et ſi puncta h & l fuerint ſurſum mota ſupra
              <lb/>
            uiſum:</s>
            <s xml:id="echoid-s44264" xml:space="preserve"> tunc imagines ipſorum, quæ ſunt k & m, uide-
              <lb/>
            buntur moueri deorſum.</s>
            <s xml:id="echoid-s44265" xml:space="preserve"> Et ſi puncta h & l ſuerint mo
              <lb/>
            ta ad dextram partem uiſus:</s>
            <s xml:id="echoid-s44266" xml:space="preserve"> formæ imaginum uide-
              <lb/>
            buntur moueri ad ſiniſtram:</s>
            <s xml:id="echoid-s44267" xml:space="preserve"> & ita ſemper mouentur imagines ad partem contrariam rebus.</s>
            <s xml:id="echoid-s44268" xml:space="preserve"> Patet
              <lb/>
            ergo propoſitum.</s>
            <s xml:id="echoid-s44269" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1722" type="section" level="0" n="0">
          <head xml:id="echoid-head1277" xml:space="preserve" style="it">67. Per ſpecula ſphærica concaua, quot libuerit, poßibile eſt formæ eiuſdem puncti imaginem
            <lb/>
          uideri. Euclides 15 th. catoptr. Ptolemæus 8 th. 2 catoptr.</head>
          <p>
            <s xml:id="echoid-s44270" xml:space="preserve">Fiat diſpoſitio, quæ in planis & conuexis ſphæricis ſpeculis:</s>
            <s xml:id="echoid-s44271" xml:space="preserve"> & ſit centrum uiſus a:</s>
            <s xml:id="echoid-s44272" xml:space="preserve"> & punctus
              <lb/>
            rei uiſæ ſit b:</s>
            <s xml:id="echoid-s44273" xml:space="preserve"> & ſecundum diſtantiam centri uiſus (quod eſt a) à puncto rei uiſæ, (quod eſt b) deſcri
              <lb/>
            batur polygonium æquilaterum & æquiangulum, quotcunq;</s>
            <s xml:id="echoid-s44274" xml:space="preserve"> angulorum placuerit:</s>
            <s xml:id="echoid-s44275" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s44276" xml:space="preserve">, exempli
              <lb/>
              <figure xlink:label="fig-0667-02" xlink:href="fig-0667-02a" number="804">
                <variables xml:id="echoid-variables781" xml:space="preserve">d l c e g z k a b</variables>
              </figure>
            cauſſa, pentagonum:</s>
            <s xml:id="echoid-s44277" xml:space="preserve"> quod ſit a b g d e:</s>
            <s xml:id="echoid-s44278" xml:space="preserve"> fiatq́;</s>
            <s xml:id="echoid-s44279" xml:space="preserve"> circu-
              <lb/>
            lus circumſcribens illud polygoniũ pentagonum
              <lb/>
            per 12 p 4:</s>
            <s xml:id="echoid-s44280" xml:space="preserve"> & ſuper illius pentagoni angulos ortho
              <lb/>
            gonaliter ſuper lineas à centro circuli circumſcri-
              <lb/>
            bentis polygoniũ productas ad circumferentiam
              <lb/>
            ſecundum ipſorum puncta media ſtatuantur ſpe-
              <lb/>
            cula ſphærica concaua, quæ ſint partes eiuſdem
              <lb/>
            ſphæræ & æquales portiones.</s>
            <s xml:id="echoid-s44281" xml:space="preserve"> Patet itaq;</s>
            <s xml:id="echoid-s44282" xml:space="preserve"> quoniam
              <lb/>
            ſuperficies plana pentagoni a b g d e ſecabit quod-
              <lb/>
            libet ſpeculorum ſecundum circulum per 69 th.</s>
            <s xml:id="echoid-s44283" xml:space="preserve"> 1
              <lb/>
            huius.</s>
            <s xml:id="echoid-s44284" xml:space="preserve"> Vnus itaq;</s>
            <s xml:id="echoid-s44285" xml:space="preserve"> arcus unius illorum circulorum
              <lb/>
            ſit z g c:</s>
            <s xml:id="echoid-s44286" xml:space="preserve"> ducanturq́;</s>
            <s xml:id="echoid-s44287" xml:space="preserve"> lineæ contingentes quemlibet
              <lb/>
            illorum arcuum in punctis g, d, e:</s>
            <s xml:id="echoid-s44288" xml:space="preserve"> contingatq́;</s>
            <s xml:id="echoid-s44289" xml:space="preserve"> ar-
              <lb/>
            cum z g c in puncto g linea l k.</s>
            <s xml:id="echoid-s44290" xml:space="preserve"> Quia itaq;</s>
            <s xml:id="echoid-s44291" xml:space="preserve"> per 43 th.</s>
            <s xml:id="echoid-s44292" xml:space="preserve">
              <lb/>
            1 huius angulus portionis, qui eſt b g z, eſt æqualis
              <lb/>
            angulo d g c:</s>
            <s xml:id="echoid-s44293" xml:space="preserve"> anguli quoq;</s>
            <s xml:id="echoid-s44294" xml:space="preserve"> contingentiæ, qui ſunt
              <lb/>
            k g z & l g c ſunt æquales:</s>
            <s xml:id="echoid-s44295" xml:space="preserve"> palàm ergo per 20 th.</s>
            <s xml:id="echoid-s44296" xml:space="preserve"> 5
              <lb/>
            huius quoniam fit reflexio formæ puncti b à puncto ſpeculi g ad punctũ ſpeculi alterius, quod eſt
              <lb/>
            d.</s>
            <s xml:id="echoid-s44297" xml:space="preserve"> Et ſimiliter per eandem demonſtrationem fiet reflexio à puncto d ad punctum ſpeculi alterius,
              <lb/>
            quod eſt e, & à puncto e ad centrum uiſus, quod eſt a.</s>
            <s xml:id="echoid-s44298" xml:space="preserve"> Palàm ergo propoſitum.</s>
            <s xml:id="echoid-s44299" xml:space="preserve"> Et ſic quotcunq;</s>
            <s xml:id="echoid-s44300" xml:space="preserve"> fue
              <lb/>
            rint anguli polygonij, tot aſſumantur ſpecula, & ſemper accidet illud, quod præmiſſum eſt.</s>
            <s xml:id="echoid-s44301" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1724" type="section" level="0" n="0">
          <head xml:id="echoid-head1278" xml:space="preserve" style="it">68. A ſpeculis ſphæricis concauis ſoli oppoſitis ignem poßibile est accendi. Euclides 31
            <lb/>
          th. catoptr.</head>
          <p>
            <s xml:id="echoid-s44302" xml:space="preserve">Eſto ſpeculum ſphæricum concauum ſoli oppoſitum:</s>
            <s xml:id="echoid-s44303" xml:space="preserve"> in quo ſignetur circulus k a b g x, cuius
              <lb/>
            centrum ſit c:</s>
            <s xml:id="echoid-s44304" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s44305" xml:space="preserve">, ut ſuperficies plana ſecans ſpeculum ſecundum hunc circulum ſecet etiam cor-
              <lb/>
            pus ſolis trans centrum:</s>
            <s xml:id="echoid-s44306" xml:space="preserve"> ergo per 69 th.</s>
            <s xml:id="echoid-s44307" xml:space="preserve"> 1 huius communis ſectio illius ſuperficiei planæ & ſolis e-
              <lb/>
            rit circulus magnus, qui ſit d e z:</s>
            <s xml:id="echoid-s44308" xml:space="preserve"> & ab aliquo puncto illius circuli ſolaris, ut à puncto d, ducatur li-
              <lb/>
            nea, ſecundum quam procedens radius ad centrum ſpeculi, quod eſt c, incidat in punctum ſpeculi,
              <lb/>
            quod ſit g:</s>
            <s xml:id="echoid-s44309" xml:space="preserve"> & à puncto circuli ſolis, quod ſit e, procedens radius ad centrum ſpeculi, quod eſt c, inci-
              <lb/>
            dat in punctum ſpeculi b:</s>
            <s xml:id="echoid-s44310" xml:space="preserve"> & à puncto ſolis, quod ſit z, incidens radius per centrum ſpeculi c, cadat
              <lb/>
            in punctum ſpeculi a.</s>
            <s xml:id="echoid-s44311" xml:space="preserve"> Quia ergo omnes radij tranſeuntes per centrum c ſunt perpendiculares ſu-
              <lb/>
            </s>
          </p>
        </div>
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