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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          <pb o="162" file="0168" n="168" rhead="ALHAZEN"/>
        </div>
        <div xml:id="echoid-div379" type="section" level="0" n="0">
          <head xml:id="echoid-head363" xml:space="preserve" style="it">60. In ſpeculo ſphærico cauo, imago uidetur aliâs in reflexionis puncto: aliâs in uiſu: aliâs ul-
            <lb/>
          tra: aliâs citra ſpeculum: aliâs inter uiſum & ſpeculum. 11 p 8.</head>
          <p>
            <s xml:id="echoid-s10498" xml:space="preserve">IN ſpeculis ſphæricis concauis aliquãdo perpendicularis à pũcto uiſo ducta ſecat lineã reflexio-
              <lb/>
            nis:</s>
            <s xml:id="echoid-s10499" xml:space="preserve"> aliquãdo eſt æquidiſtãs ei.</s>
            <s xml:id="echoid-s10500" xml:space="preserve"> Quãdo ſecat:</s>
            <s xml:id="echoid-s10501" xml:space="preserve"> erit locus formæ aliquãdo in ſpeculo:</s>
            <s xml:id="echoid-s10502" xml:space="preserve"> aliquãdo ul-
              <lb/>
            tra ſpeculũ:</s>
            <s xml:id="echoid-s10503" xml:space="preserve"> aliquãdo citra.</s>
            <s xml:id="echoid-s10504" xml:space="preserve"> Et cũ fuerit locus formæ citra ſpeculũ:</s>
            <s xml:id="echoid-s10505" xml:space="preserve"> aliquando erit inter uiſum &
              <lb/>
            ſpeculũ:</s>
            <s xml:id="echoid-s10506" xml:space="preserve"> aliquãdo in cẽtro uiſus:</s>
            <s xml:id="echoid-s10507" xml:space="preserve"> aliquãdo citra cẽtrũ uiſus.</s>
            <s xml:id="echoid-s10508" xml:space="preserve"> Et nos h æc demõſtrabimus.</s>
            <s xml:id="echoid-s10509" xml:space="preserve"> Sit a cẽtrũ
              <lb/>
            uiſus:</s>
            <s xml:id="echoid-s10510" xml:space="preserve"> d cẽtrũ ſpeculi:</s>
            <s xml:id="echoid-s10511" xml:space="preserve"> & fiat ſuperficies ſuper hæc puncta:</s>
            <s xml:id="echoid-s10512" xml:space="preserve"> quæ ſecabit ſpeculũ ſuք circulũ [per 1 th.</s>
            <s xml:id="echoid-s10513" xml:space="preserve"> 1
              <lb/>
            ſphær.</s>
            <s xml:id="echoid-s10514" xml:space="preserve"> Theodo.</s>
            <s xml:id="echoid-s10515" xml:space="preserve">] ꝗ circulus ſit h b f g:</s>
            <s xml:id="echoid-s10516" xml:space="preserve"> erit qúidẽ ſuքficies hęc, ſuքficies reflexionis:</s>
            <s xml:id="echoid-s10517" xml:space="preserve"> quoniã eſt ortho
              <lb/>
            gonalis ſuք quãlibet ſuperficiẽ, cõtingentẽ circulũ:</s>
            <s xml:id="echoid-s10518" xml:space="preserve"> [ք 13 n 4] & ducatur linea a d:</s>
            <s xml:id="echoid-s10519" xml:space="preserve"> & à pũcto a duca
              <lb/>
            tur linea ad circulũ maior ꝗ̃ a d:</s>
            <s xml:id="echoid-s10520" xml:space="preserve"> quę ſit a e:</s>
            <s xml:id="echoid-s10521" xml:space="preserve"> & à pũcto d ducatur ad circulũ æquidiſtãs a e:</s>
            <s xml:id="echoid-s10522" xml:space="preserve"> q̃ ſit d h:</s>
            <s xml:id="echoid-s10523" xml:space="preserve"> &
              <lb/>
            ꝓducatur a d uſq;</s>
            <s xml:id="echoid-s10524" xml:space="preserve"> in pũcta b, i:</s>
            <s xml:id="echoid-s10525" xml:space="preserve"> [qñ nimirũ uiſus fuerit intra circulũ h b f g:</s>
            <s xml:id="echoid-s10526" xml:space="preserve"> qđ ſi fuerit in peripheria
              <lb/>
            uel extra:</s>
            <s xml:id="echoid-s10527" xml:space="preserve"> linea a d ab una tãtùm parte in peripheriã cõtinuabitur:</s>
            <s xml:id="echoid-s10528" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s10529" xml:space="preserve"> eadẽ demõſtrãdi ratio:</s>
            <s xml:id="echoid-s10530" xml:space="preserve"> ꝗa
              <lb/>
            a e ſemք maior eſſe debet a d] & ducatur linea d e.</s>
            <s xml:id="echoid-s10531" xml:space="preserve"> Palã, qđ angulus a e d eſt minor recto:</s>
            <s xml:id="echoid-s10532" xml:space="preserve"> [cõnexis
              <lb/>
            enim rectis e i, e b:</s>
            <s xml:id="echoid-s10533" xml:space="preserve"> erit angulus i e b rectus ք 31 p 3, ideoq́;</s>
            <s xml:id="echoid-s10534" xml:space="preserve"> a e d acutus] quoniã e d ſemidiameter:</s>
            <s xml:id="echoid-s10535" xml:space="preserve"> &
              <lb/>
            quęlibet linea in circulo cũ diametro facit angulũ acutũ [ut patet ք 31 p 3.</s>
            <s xml:id="echoid-s10536" xml:space="preserve"> 32 p 1 uel 9 ax.</s>
            <s xml:id="echoid-s10537" xml:space="preserve">] Et ſuք pun
              <lb/>
            ctũ e fiat angulus æqualis angulo a e d [ք 23 p 1] ꝗ ſit d e t.</s>
            <s xml:id="echoid-s10538" xml:space="preserve"> Palã, qđ e t cadet intra circulũ:</s>
            <s xml:id="echoid-s10539" xml:space="preserve"> [ſi.</s>
            <s xml:id="echoid-s10540" xml:space="preserve"> n.</s>
            <s xml:id="echoid-s10541" xml:space="preserve"> cade
              <lb/>
            ret extra:</s>
            <s xml:id="echoid-s10542" xml:space="preserve"> uel tãgeret peripheriã, efficeretq́;</s>
            <s xml:id="echoid-s10543" xml:space="preserve"> cũ ſemidiametro de angulũ rectũ ք 18 p 3:</s>
            <s xml:id="echoid-s10544" xml:space="preserve"> uel ſecaret, &
              <lb/>
            efficeret obtuſum:</s>
            <s xml:id="echoid-s10545" xml:space="preserve"> quorũ uterq;</s>
            <s xml:id="echoid-s10546" xml:space="preserve"> cũ acuto a e d maior ſit ք 11 12 d 1:</s>
            <s xml:id="echoid-s10547" xml:space="preserve"> mãdato ſatis factũ nõ eſſet] & ſe-
              <lb/>
            cabit lineã d h:</s>
            <s xml:id="echoid-s10548" xml:space="preserve"> [ք lẽma Procli ad 29 p 1:</s>
            <s xml:id="echoid-s10549" xml:space="preserve"> ꝗa ſecat a e ipſi parallelã] ſit pũctũ ſectiõis t.</s>
            <s xml:id="echoid-s10550" xml:space="preserve"> Palã etiã, qđ an
              <lb/>
            gulus a d e maior eſt angulo d e t:</s>
            <s xml:id="echoid-s10551" xml:space="preserve"> [ꝗa cũ a e maior ſit a d, maior erit ք
              <lb/>
              <figure xlink:label="fig-0168-01" xlink:href="fig-0168-01a" number="97">
                <variables xml:id="echoid-variables87" xml:space="preserve">l g e n h m t q u i a
                  <gap/>
                s z b k y f p o</variables>
              </figure>
            18 p 1 angulus a d e angulo a e d, cui ęquatus eſt angulus d e t] & ita et
              <lb/>
            ſecabit a b:</s>
            <s xml:id="echoid-s10552" xml:space="preserve"> [ꝗa.</s>
            <s xml:id="echoid-s10553" xml:space="preserve"> n.</s>
            <s xml:id="echoid-s10554" xml:space="preserve"> anguli a d e, e d b æquãtur duob.</s>
            <s xml:id="echoid-s10555" xml:space="preserve"> rectis per 13 p 1:</s>
            <s xml:id="echoid-s10556" xml:space="preserve"> &
              <lb/>
            angulus a d e maior eſt angulo d et:</s>
            <s xml:id="echoid-s10557" xml:space="preserve"> anguli igitur e d b, d e t minores
              <lb/>
            ſunt duob.</s>
            <s xml:id="echoid-s10558" xml:space="preserve"> rectis:</s>
            <s xml:id="echoid-s10559" xml:space="preserve"> quare e t, d b cõcurrẽt ք 11 ax.</s>
            <s xml:id="echoid-s10560" xml:space="preserve">] ſecet in pũcto z.</s>
            <s xml:id="echoid-s10561" xml:space="preserve"> De-
              <lb/>
            inde à pũcto a ducatur ad arcũ e h linea:</s>
            <s xml:id="echoid-s10562" xml:space="preserve"> q̃ ſit a n:</s>
            <s xml:id="echoid-s10563" xml:space="preserve"> & ducatur linea d n:</s>
            <s xml:id="echoid-s10564" xml:space="preserve">
              <lb/>
            & ſuper pũctũ n fiat angulus æqualis angulo d n a, ք lineã m n:</s>
            <s xml:id="echoid-s10565" xml:space="preserve"> q̃ ne-
              <lb/>
            ceſſariò cadet intra circulũ:</s>
            <s xml:id="echoid-s10566" xml:space="preserve"> [ob cauſſam ꝓximè expoſitã] & ſecabit
              <lb/>
            d h:</s>
            <s xml:id="echoid-s10567" xml:space="preserve"> [Nã cũ angulus a n d à ſemidiametro d n & recta linea a n cõpre
              <lb/>
            henſus, ſit acutus, ut patuit:</s>
            <s xml:id="echoid-s10568" xml:space="preserve"> erit angulus d n m ipſi æquatus, acutus:</s>
            <s xml:id="echoid-s10569" xml:space="preserve">
              <lb/>
            ſed & n d m eſt acutus, ꝗa pars eſt acuti e d t:</s>
            <s xml:id="echoid-s10570" xml:space="preserve"> anguli igitur d n m, n d
              <lb/>
            m ſunt minores duob.</s>
            <s xml:id="echoid-s10571" xml:space="preserve"> rectis.</s>
            <s xml:id="echoid-s10572" xml:space="preserve"> Quare n m, d h cõcurrent ք 11 ax.</s>
            <s xml:id="echoid-s10573" xml:space="preserve">] ſecet
              <lb/>
            in pũcto m.</s>
            <s xml:id="echoid-s10574" xml:space="preserve"> Palã etiã, qđ a n cõcurret cũ d h extra circulũ:</s>
            <s xml:id="echoid-s10575" xml:space="preserve"> [per lẽma
              <lb/>
            Procli ad 29 p 1] ſit cõcurſus in l.</s>
            <s xml:id="echoid-s10576" xml:space="preserve"> Ducatur etiã à pũcto a linea ad arcũ
              <lb/>
            e i f:</s>
            <s xml:id="echoid-s10577" xml:space="preserve"> q̃ ſit a g:</s>
            <s xml:id="echoid-s10578" xml:space="preserve"> & ducatur d g:</s>
            <s xml:id="echoid-s10579" xml:space="preserve"> & fiat angulus d g q æqualis angulo a g d.</s>
            <s xml:id="echoid-s10580" xml:space="preserve">
              <lb/>
            Palã, qđ q g ſecabit d h:</s>
            <s xml:id="echoid-s10581" xml:space="preserve"> [ut patuit] ſit pũctũ ſectionis q.</s>
            <s xml:id="echoid-s10582" xml:space="preserve"> Palã etiã, qđ
              <lb/>
            a g cõcurret cũ d h ex parte f:</s>
            <s xml:id="echoid-s10583" xml:space="preserve"> [Nã qđ cõcurrat, cõſtatè lẽmate Pro-
              <lb/>
            cli ad 29 p 1:</s>
            <s xml:id="echoid-s10584" xml:space="preserve"> qđ uerò uerſus f, è paulò antè demõſtratis քſpicuũ eſt]
              <lb/>
            ſit cõcurſus o.</s>
            <s xml:id="echoid-s10585" xml:space="preserve"> Qđ aũt g q cadat inter d & h, palã:</s>
            <s xml:id="echoid-s10586" xml:space="preserve"> cũ arcus, quẽ ſecat g
              <lb/>
            o ex circulo, ſit maior arcu g h:</s>
            <s xml:id="echoid-s10587" xml:space="preserve"> ſi.</s>
            <s xml:id="echoid-s10588" xml:space="preserve"> n.</s>
            <s xml:id="echoid-s10589" xml:space="preserve"> ducatur linea g h:</s>
            <s xml:id="echoid-s10590" xml:space="preserve"> angulus h g d
              <lb/>
            maiorẽ reſpiciet arcũ angulo a g d.</s>
            <s xml:id="echoid-s10591" xml:space="preserve"> [ideoq́;</s>
            <s xml:id="echoid-s10592" xml:space="preserve"> ք 33 p 6 angulus h g d e-
              <lb/>
            rit maior angulo a g d:</s>
            <s xml:id="echoid-s10593" xml:space="preserve"> at angulo a g d, æquatus eſt angulus q g d:</s>
            <s xml:id="echoid-s10594" xml:space="preserve"> an-
              <lb/>
            gulus igitur h g d maior eſt angulo q g d:</s>
            <s xml:id="echoid-s10595" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s10596" xml:space="preserve"> linea q g ſecans angu-
              <lb/>
            lũ q g d, ſecabit baſim h d angulo q g d ſubtẽſam.</s>
            <s xml:id="echoid-s10597" xml:space="preserve">] Iterũ à pũcto a du
              <lb/>
            catur ad arcũ fb, linea a k, ſecãs d h in pũcto s:</s>
            <s xml:id="echoid-s10598" xml:space="preserve"> [qđ.</s>
            <s xml:id="echoid-s10599" xml:space="preserve"> n.</s>
            <s xml:id="echoid-s10600" xml:space="preserve"> ſecet, patet è lẽ
              <lb/>
            mate Procli ad 29 p 1] ut ſit k s maior s d:</s>
            <s xml:id="echoid-s10601" xml:space="preserve"> [ſecta nẽpe d f bifariã ք 10 ք
              <lb/>
            1, & ab a ducta linea a k ք ſectionis pũctũ, uel ք quodcũq;</s>
            <s xml:id="echoid-s10602" xml:space="preserve"> aliud uer-
              <lb/>
            ſus d:</s>
            <s xml:id="echoid-s10603" xml:space="preserve"> utroq;</s>
            <s xml:id="echoid-s10604" xml:space="preserve">. n.</s>
            <s xml:id="echoid-s10605" xml:space="preserve"> modo erit s k maior fs ք 7 p 3:</s>
            <s xml:id="echoid-s10606" xml:space="preserve"> & ob id maior s d] & ducatur k d.</s>
            <s xml:id="echoid-s10607" xml:space="preserve"> Palã, qđ angulus d k
              <lb/>
            a eſt acutus.</s>
            <s xml:id="echoid-s10608" xml:space="preserve"> [ut in principio huius numeri oſtẽſum eſt] Fiat [ք 23 p 1] ei æqualis:</s>
            <s xml:id="echoid-s10609" xml:space="preserve"> ꝗ ſit d k u.</s>
            <s xml:id="echoid-s10610" xml:space="preserve"> Palã, qđ,
              <lb/>
            cũ angulus k d s ſit maior angulo d k s:</s>
            <s xml:id="echoid-s10611" xml:space="preserve"> [ք 18 p 1:</s>
            <s xml:id="echoid-s10612" xml:space="preserve"> ꝗ a s k maior eſt s d ք fabricationẽ]k u cõcurret cũ d
              <lb/>
            h:</s>
            <s xml:id="echoid-s10613" xml:space="preserve"> [Nã cũ anguli h d k, k d s æquẽtur duob rectis ք 13 p 1:</s>
            <s xml:id="echoid-s10614" xml:space="preserve"> & k d s ſit maior d k s è cõcluſione:</s>
            <s xml:id="echoid-s10615" xml:space="preserve"> erit k d s
              <lb/>
            etiã maior d k u æquali d k s:</s>
            <s xml:id="echoid-s10616" xml:space="preserve"> anguli igitur h d k, d k u ſũt minores duob.</s>
            <s xml:id="echoid-s10617" xml:space="preserve"> rectis.</s>
            <s xml:id="echoid-s10618" xml:space="preserve"> Quare h d, ku cõcur
              <lb/>
            rẽt ք 11 ax.</s>
            <s xml:id="echoid-s10619" xml:space="preserve">] ſit cõcurſus in pũcto u.</s>
            <s xml:id="echoid-s10620" xml:space="preserve"> Palã ſecũdũ ſupra dicta [& 12 n 4] qđ pũctũ t mouetur ad e, & re
              <lb/>
            flectitur ad a:</s>
            <s xml:id="echoid-s10621" xml:space="preserve"> & քpẽdicularis à pũcto t ducta, eſt t d:</s>
            <s xml:id="echoid-s10622" xml:space="preserve"> q̃ քpẽdicularis eſt ſuք ſuքficiẽ, cõtingẽtẽ ſpecu
              <lb/>
            lũ:</s>
            <s xml:id="echoid-s10623" xml:space="preserve"> [ք 25 n 4] & eſt æquidiſtãs lineæ reflexiõis, q̃ eſt a e:</s>
            <s xml:id="echoid-s10624" xml:space="preserve"> [ք fabricationẽ] unde nõ cõcurret cũ e a:</s>
            <s xml:id="echoid-s10625" xml:space="preserve"> [ք
              <lb/>
            35 d 1.</s>
            <s xml:id="echoid-s10626" xml:space="preserve"> Imago igitur pũcti tuidebitur in reflexiõis pũcto e] Pũctũ aũt z mouetur ad e, & reflectitur ad
              <lb/>
            a:</s>
            <s xml:id="echoid-s10627" xml:space="preserve"> & քpẽdicularis ducta à pũcto z, eſt a z:</s>
            <s xml:id="echoid-s10628" xml:space="preserve"> q̃ cõcurrit cũ a e in pũcto a.</s>
            <s xml:id="echoid-s10629" xml:space="preserve"> Vnde locus formę pũcti z erit a.</s>
            <s xml:id="echoid-s10630" xml:space="preserve">
              <lb/>
            [ք 3 n.</s>
            <s xml:id="echoid-s10631" xml:space="preserve">] Pũctũ uerò m mouetur ad n, & reflectitur ad a:</s>
            <s xml:id="echoid-s10632" xml:space="preserve"> & քpẽdicularis ducta à pũcto m, quę eſt m d.</s>
            <s xml:id="echoid-s10633" xml:space="preserve">
              <lb/>
            cocurrit cũ a n in pũcto l, qđ eſt ultra ſpeculũ:</s>
            <s xml:id="echoid-s10634" xml:space="preserve"> & locus formę pũcti m erit l.</s>
            <s xml:id="echoid-s10635" xml:space="preserve"> Forma uerò pũcti q m o
              <lb/>
            uetur ad g, & reflectitur ad a:</s>
            <s xml:id="echoid-s10636" xml:space="preserve"> & locus eius erit o:</s>
            <s xml:id="echoid-s10637" xml:space="preserve"> qui eſt ultra uiſum.</s>
            <s xml:id="echoid-s10638" xml:space="preserve"> Et forma puncti u mouetur ad
              <lb/>
            k, & reflectitur ad a:</s>
            <s xml:id="echoid-s10639" xml:space="preserve"> & perpendicularis ab eo, eſt k d:</s>
            <s xml:id="echoid-s10640" xml:space="preserve"> & locus imaginis s.</s>
            <s xml:id="echoid-s10641" xml:space="preserve"> [inter uiſum & ſpeculũ.</s>
            <s xml:id="echoid-s10642" xml:space="preserve">]
              <lb/>
            Palàm ergo ex prædictis, quòd imaginum quædam inter uiſum & ſpeculum:</s>
            <s xml:id="echoid-s10643" xml:space="preserve"> quædam in ipſo uiſus
              <lb/>
            quædam citra uiſum:</s>
            <s xml:id="echoid-s10644" xml:space="preserve"> quædam ultra uiſum apparent.</s>
            <s xml:id="echoid-s10645" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s10646" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div381" type="section" level="0" n="0">
          <head xml:id="echoid-head364" xml:space="preserve" style="it">61. In ſpeculo ſphærico cauo imago prouario eius ſitu at loco uariè uidetur. 12 p 8.</head>
          <p>
            <s xml:id="echoid-s10647" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s10648" xml:space="preserve"> palàm, quoniã uiſus քfectius acquirit formas ſibi oppoſitas.</s>
            <s xml:id="echoid-s10649" xml:space="preserve"> [per 21.</s>
            <s xml:id="echoid-s10650" xml:space="preserve"> 38 n 1.</s>
            <s xml:id="echoid-s10651" xml:space="preserve"> 17 n 3.</s>
            <s xml:id="echoid-s10652" xml:space="preserve">] Vn
              <lb/>
            de cũlocus imaginis fuerit ultra ſpeculũ [ut in puncto l] autinter uiſum & ſpeculum:</s>
            <s xml:id="echoid-s10653" xml:space="preserve"> [ut in
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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