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-s12033" xml:space="preserve">
              <pb o="179" file="0185" n="185" rhead="OPTICAE LIBER V."/>
            ãngulus e t d æqualis medietati anguli o d a:</s>
            <s xml:id="echoid-s12034" xml:space="preserve"> erit quidem acutus:</s>
            <s xml:id="echoid-s12035" xml:space="preserve"> [quia æquatur angulo rectilineo
              <lb/>
            dimidiato, ut oſtenſum eſt 36 n] igitur t d concur-
              <lb/>
              <figure xlink:label="fig-0185-01" xlink:href="fig-0185-01a" number="134">
                <variables xml:id="echoid-variables124" xml:space="preserve">
                  <gap/>
                k e d q h z</variables>
              </figure>
              <figure xlink:label="fig-0185-02" xlink:href="fig-0185-02a" number="135">
                <variables xml:id="echoid-variables125" xml:space="preserve">l b k d o</variables>
              </figure>
            ret cum perpendiculari [per 11 ax.</s>
            <s xml:id="echoid-s12036" xml:space="preserve">] Sit concurſus
              <lb/>
            in puncto h:</s>
            <s xml:id="echoid-s12037" xml:space="preserve"> & [per 38 n] ducatur linea d e k, ut ſit
              <lb/>
            proportio k d ad d t, ſicut k d ad ſemidiametrum
              <lb/>
            ſphæræ [erit igitur ſemidiameter ſphæræ æqualis
              <lb/>
            d t per 9 p 5.</s>
            <s xml:id="echoid-s12038" xml:space="preserve">] Etangulo, quem habemus k d t fiat
              <lb/>
            [per 23 p 1] in ſpeculo angulus æqualis, ſcilicet k d
              <lb/>
            t.</s>
            <s xml:id="echoid-s12039" xml:space="preserve"> Dico, quò d t eſt punctum reflexionis, Et ſi prædi-
              <lb/>
            ctam probationẽ replicaueris, manifeſtè uidebis.</s>
            <s xml:id="echoid-s12040" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div422" type="section" level="0" n="0">
          <head xml:id="echoid-head387" xml:space="preserve" style="it">84. Siduo puncta extra circulum (quieſt com-
            <lb/>
          munis ſectio ſuperficierum reflexionis & ſpecu-
            <lb/>
          li ſphæricicaui) uel alterum intra, reliquum ex-
            <lb/>
          tra, in diuerſis diametris, à centro inæquabiliter
            <lb/>
          diſtantia, reflectantur à peripheria comprehen-
            <lb/>
          ſa inter ſemidiametros, extra quas ipſa ſunt: ab
            <lb/>
          uno puncto tantùm reflectentur. 38 p 8.</head>
          <p>
            <s xml:id="echoid-s12041" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s12042" xml:space="preserve"> ſumptis duobus punctis in diuerſis diametris, quæ puncta inæqualis ſint longitu.</s>
            <s xml:id="echoid-s12043" xml:space="preserve">
              <lb/>
            dinis à centro:</s>
            <s xml:id="echoid-s12044" xml:space="preserve"> ſi fuerint extra circulum, & reflectantur ab a-
              <lb/>
              <figure xlink:label="fig-0185-03" xlink:href="fig-0185-03a" number="136">
                <variables xml:id="echoid-variables126" xml:space="preserve">a b n m
                  <gap/>
                k l q g d h
                  <gap/>
                e</variables>
              </figure>
            liquo puncto arcus oppoſiti diametris:</s>
            <s xml:id="echoid-s12045" xml:space="preserve"> non reflectentur ab
              <lb/>
            alio eiuſdem arcus.</s>
            <s xml:id="echoid-s12046" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s12047" xml:space="preserve"> ſint a, b puncta in diuerſis diame-
              <lb/>
            tris, extra circulum:</s>
            <s xml:id="echoid-s12048" xml:space="preserve"> g centrum:</s>
            <s xml:id="echoid-s12049" xml:space="preserve"> t punctum reflexionis:</s>
            <s xml:id="echoid-s12050" xml:space="preserve"> & ducantur
              <lb/>
            b t, a t, t g:</s>
            <s xml:id="echoid-s12051" xml:space="preserve"> b t ſecabit arcum circuli:</s>
            <s xml:id="echoid-s12052" xml:space="preserve"> ſit punctum ſectionis q:</s>
            <s xml:id="echoid-s12053" xml:space="preserve"> a t ſeca-
              <lb/>
            bit ſimiliter arcum circuli:</s>
            <s xml:id="echoid-s12054" xml:space="preserve"> ſit punctum ſectionis m.</s>
            <s xml:id="echoid-s12055" xml:space="preserve"> Quoniam angu-
              <lb/>
            lus b t g ęqualis eſt angulo a t g:</s>
            <s xml:id="echoid-s12056" xml:space="preserve"> [per 12 n 4:</s>
            <s xml:id="echoid-s12057" xml:space="preserve"> quia t eſt reflexionis pun
              <lb/>
            ctum ex theſi] cadent in arcus circuli æquales:</s>
            <s xml:id="echoid-s12058" xml:space="preserve"> [per 26 p 3] quod p
              <gap/>
              <lb/>
            tebit producta ſemidiametro t g in p.</s>
            <s xml:id="echoid-s12059" xml:space="preserve"> Erit ergo arcus q p æqualis ar-
              <lb/>
            cui m p.</s>
            <s xml:id="echoid-s12060" xml:space="preserve"> Si igitur b reflectitur ab alio puncto:</s>
            <s xml:id="echoid-s12061" xml:space="preserve"> ſit illud h:</s>
            <s xml:id="echoid-s12062" xml:space="preserve"> & ducantur
              <lb/>
            lineæ b h, a h, g h.</s>
            <s xml:id="echoid-s12063" xml:space="preserve"> Secet b h circulum in puncto l:</s>
            <s xml:id="echoid-s12064" xml:space="preserve"> a h in puncto n:</s>
            <s xml:id="echoid-s12065" xml:space="preserve"> &
              <lb/>
            producatur h g in k.</s>
            <s xml:id="echoid-s12066" xml:space="preserve"> Secundum igitur prædictam probationem e-
              <lb/>
            rit l k æqualis n k:</s>
            <s xml:id="echoid-s12067" xml:space="preserve"> ſed iam habemus, quòd q p æqualis p m:</s>
            <s xml:id="echoid-s12068" xml:space="preserve"> quod eſt
              <lb/>
            impoſsibile [& contra 9 ax.</s>
            <s xml:id="echoid-s12069" xml:space="preserve">] Reſtat ut b non reflectatur ad a, à pun
              <lb/>
            cto h, uel ab alio puncto arcus oppoſiti diametris, præterquam à t.</s>
            <s xml:id="echoid-s12070" xml:space="preserve">
              <lb/>
            Similiter ſi fuerit alterum punctorum in circulo, alterum extra:</s>
            <s xml:id="echoid-s12071" xml:space="preserve"> ab
              <lb/>
            uno tantùm puncto arcus poterit reflecti ad aliud.</s>
            <s xml:id="echoid-s12072" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div424" type="section" level="0" n="0">
          <head xml:id="echoid-head388" xml:space="preserve" style="it">85. Sirecta linea connectens duo puncta in diuerſis diametris
            <lb/>
          circuli (qui eſt communis ſectio ſuperficierum reflexionis & ſpecu
            <lb/>
          li ſphæricicaui) à centro inæquabiliter diſtantia, tangat peripheriam dicti circuli, uelſit extra
            <lb/>
          ipſam: ab uno tantùm puncto reflexio fiet. 39 p 8.</head>
          <p>
            <s xml:id="echoid-s12073" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s12074" xml:space="preserve"> ſilinea ducta ab uno duorũ puncto-
              <lb/>
              <figure xlink:label="fig-0185-04" xlink:href="fig-0185-04a" number="137">
                <variables xml:id="echoid-variables127" xml:space="preserve">b a b a m
                  <gap/>
                f g d n</variables>
              </figure>
            rum, cõtingat circulũ, aut tota ſit extra:</s>
            <s xml:id="echoid-s12075" xml:space="preserve"> ſum-
              <lb/>
            pto quocũq;</s>
            <s xml:id="echoid-s12076" xml:space="preserve"> puncto in atcu oppoſito diame-
              <lb/>
            tris:</s>
            <s xml:id="echoid-s12077" xml:space="preserve"> [in quibus ſunt data puncta] altera linearum à
              <lb/>
            punctorum duorum altero, ad illud punctum ducta
              <lb/>
            rum, tota erit extra circulum:</s>
            <s xml:id="echoid-s12078" xml:space="preserve"> & ſic neutrum puncto
              <lb/>
            rum ad aliud reflectetur ab aliquo puncto illius ar-
              <lb/>
            cus:</s>
            <s xml:id="echoid-s12079" xml:space="preserve"> [m l] & ab uno ſolo puncto ſpeculi [in periphe-
              <lb/>
            ria d n ſumpto per 73 & præcedentem numeros.</s>
            <s xml:id="echoid-s12080" xml:space="preserve">]</s>
          </p>
        </div>
        <div xml:id="echoid-div426" type="section" level="0" n="0">
          <head xml:id="echoid-head389" xml:space="preserve" style="it">86. Sirecta linea connectens duo puncta in di-
            <lb/>
          uerſis diametris circuli (qui eſt communis ſectio
            <lb/>
          ſuperficierũ reflexiõis & ſpeculi ſphærici caui) à cẽ-
            <lb/>
          troinæquabiliter diſtantia, continuata eundem
            <lb/>
          ſecet: poſſunt dicta puncta ab uno, duobus, tri-
            <lb/>
          bus, aut quatuor punctis ſpeculi inter ſe reflecti.
            <lb/>
          40 p 8.</head>
          <p>
            <s xml:id="echoid-s12081" xml:space="preserve">SI uerò linea ducta ab uno pũcto ad aliud, ſecet circulũ:</s>
            <s xml:id="echoid-s12082" xml:space="preserve"> fiat circulus ք centrũ ſpeculi & illa duo
              <lb/>
            pũcta [ք 5 p 4.</s>
            <s xml:id="echoid-s12083" xml:space="preserve">] circulus ille aut tot
              <emph style="sub">9</emph>
            erit intra circulũ:</s>
            <s xml:id="echoid-s12084" xml:space="preserve"> aut cõtingetipſum intrinſecus:</s>
            <s xml:id="echoid-s12085" xml:space="preserve"> aut ſeca
              <lb/>
            bit.</s>
            <s xml:id="echoid-s12086" xml:space="preserve"> Sit tot
              <emph style="sub">9</emph>
            intra:</s>
            <s xml:id="echoid-s12087" xml:space="preserve"> & ducãtur duę
              <unsure/>
            lineę
              <unsure/>
            à duob.</s>
            <s xml:id="echoid-s12088" xml:space="preserve"> pũctis ad aliquod pũctũ arcus oppofiti:</s>
            <s xml:id="echoid-s12089" xml:space="preserve"> angul
              <emph style="sub">9</emph>
            ,
              <lb/>
            </s>
          </p>
        </div>
      </text>
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