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-s9846" xml:space="preserve">
              <pb o="156" file="0162" n="162" rhead="ALHAZEN"/>
            flectetur ad punctum a.</s>
            <s xml:id="echoid-s9847" xml:space="preserve"> [per 12 n 4.</s>
            <s xml:id="echoid-s9848" xml:space="preserve">] Si ergo dicatur, quòd ab alio puncto, quàm à puncto g, poteſt
              <lb/>
            forma b reflecti ad a:</s>
            <s xml:id="echoid-s9849" xml:space="preserve"> illud aliud punctũ aut erit in linea lõgitudinis, quæ eſt g z:</s>
            <s xml:id="echoid-s9850" xml:space="preserve"> aut in alia.</s>
            <s xml:id="echoid-s9851" xml:space="preserve"> Si eſt in li
              <lb/>
            nea g z:</s>
            <s xml:id="echoid-s9852" xml:space="preserve"> ducatur ab eo perpẽdicularis:</s>
            <s xml:id="echoid-s9853" xml:space="preserve"> quę neceſſariò ſecabit lineã a k [quia ſecar angulũ lineis inci
              <lb/>
            dẽtię & reflexionis cõprehenſum, ut patet per 13 n 4] & [per 28 p 1] erit æquidiſtãs lineę a m:</s>
            <s xml:id="echoid-s9854" xml:space="preserve"> & li-
              <lb/>
            nea ducta à puncto b ad illud punctũ neceſſariò cõcurret cũ a m:</s>
            <s xml:id="echoid-s9855" xml:space="preserve"> [per lemma Procli ad 29 p 1] & e-
              <lb/>
            rit punctũ illud, & punctũ m in eadẽ ſuperficie:</s>
            <s xml:id="echoid-s9856" xml:space="preserve"> & linea illa aut cadet ſuper pũctũ m:</s>
            <s xml:id="echoid-s9857" xml:space="preserve"> autſuper aliud.</s>
            <s xml:id="echoid-s9858" xml:space="preserve">
              <lb/>
            Si ſuper punctũ m:</s>
            <s xml:id="echoid-s9859" xml:space="preserve"> erit ducere à puncto b ad punctũ m duas lineas rectas:</s>
            <s xml:id="echoid-s9860" xml:space="preserve"> quod eſt impoſsibile.</s>
            <s xml:id="echoid-s9861" xml:space="preserve"> [ſic
              <lb/>
            enim duę rectę lineę ſpatiũ cõprehenderẽt cõtra 12 ax.</s>
            <s xml:id="echoid-s9862" xml:space="preserve">] Si aũtad aliud punctũ lineę a m:</s>
            <s xml:id="echoid-s9863" xml:space="preserve"> ducatur à
              <lb/>
            puncto illo linea ad punctũ z:</s>
            <s xml:id="echoid-s9864" xml:space="preserve"> & probabitur, quòd hęc linea cũ h z facit lineã rectã, ſicut probatũ eſt
              <lb/>
            de linea z m:</s>
            <s xml:id="echoid-s9865" xml:space="preserve"> & ita à puncto h erit ducere duas lineas rectas, per punctũ z trãſeuntes in diuerſa pun-
              <lb/>
            cta lineę a m cadẽtes:</s>
            <s xml:id="echoid-s9866" xml:space="preserve"> quod eſt impoſsibile [& cõtra 1 p 11:</s>
            <s xml:id="echoid-s9867" xml:space="preserve"> hocq́;</s>
            <s xml:id="echoid-s9868" xml:space="preserve"> modo duarü rectarũ linearũ eſſet
              <lb/>
            cõmune ſegmentum contra lineę rectę definitionẽ.</s>
            <s xml:id="echoid-s9869" xml:space="preserve">] Palàm ergo, quòd à nullo puncto lineę g z, niſi
              <lb/>
            à g, poteſt b reflecti ad a.</s>
            <s xml:id="echoid-s9870" xml:space="preserve"> Si dicatur, quòd à puncto extra hãc lineam ſumpto:</s>
            <s xml:id="echoid-s9871" xml:space="preserve"> ducatur ſuper punctũ
              <lb/>
            illud linea longitudinis ſpeculi:</s>
            <s xml:id="echoid-s9872" xml:space="preserve"> [per 7 th.</s>
            <s xml:id="echoid-s9873" xml:space="preserve"> Sereni de ſectione cylindri] & à puncto circuli e z i, in
              <lb/>
            quod cadit hęc linea, probabitur h reflecti ad a ſecundũ ſuprà dictã probationẽ:</s>
            <s xml:id="echoid-s9874" xml:space="preserve"> ſed iã probatũ eſt.</s>
            <s xml:id="echoid-s9875" xml:space="preserve">
              <lb/>
            quòd h à puncto z reflectitur ad a.</s>
            <s xml:id="echoid-s9876" xml:space="preserve"> Etita impoſsibile:</s>
            <s xml:id="echoid-s9877" xml:space="preserve"> [quia ita à duobus ſpeculi punctis forma e-
              <lb/>
            iuſdem uiſibilis ad eundem uiſum reflecteretur, contra 51 n 4, & 29 n.</s>
            <s xml:id="echoid-s9878" xml:space="preserve">] Reſtat ergo ut à ſolo puncto
              <lb/>
            ſpeculi reflectatur b ad a.</s>
            <s xml:id="echoid-s9879" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s9880" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div361" type="section" level="0" n="0">
          <head xml:id="echoid-head351" xml:space="preserve" style="it">48. Si communis ſectio ſuperſicierum, reflexionis & ſpeculi cylindracei conuexi fuerit elli-
            <lb/>
          pſis: uiſu & uiſibili datis, punctum reflexionis inucnire. 29 p 7.</head>
          <p>
            <s xml:id="echoid-s9881" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s9882" xml:space="preserve"> dato pũcto b, quod reflectatur ad a:</s>
            <s xml:id="echoid-s9883" xml:space="preserve"> erit in uenire punctũ reflexionis:</s>
            <s xml:id="echoid-s9884" xml:space="preserve"> & hoc patebit
              <lb/>
            per reuolutionẽ prędictę probationis.</s>
            <s xml:id="echoid-s9885" xml:space="preserve"> Ducatur à puncto a ſuperficies æquidiſtãs baſi colu-
              <lb/>
            mnę:</s>
            <s xml:id="echoid-s9886" xml:space="preserve"> quę quidẽ ſecabit columnã ſuper circulũ:</s>
            <s xml:id="echoid-s9887" xml:space="preserve"> [per 5 th.</s>
            <s xml:id="echoid-s9888" xml:space="preserve"> Sereni de ſectione cylindri] qui ſit
              <lb/>
            e z i:</s>
            <s xml:id="echoid-s9889" xml:space="preserve"> & ducatur à puncto b perpẽdicularis ſuperhãc ſuperficiẽ:</s>
            <s xml:id="echoid-s9890" xml:space="preserve"> quę ſit b h:</s>
            <s xml:id="echoid-s9891" xml:space="preserve"> & inueniatur in hac ſu-
              <lb/>
            perficie punctũ, à quo fit reflexio h ad a:</s>
            <s xml:id="echoid-s9892" xml:space="preserve"> [ut traditũ eſt 31 uel 39 n] quod ſit z:</s>
            <s xml:id="echoid-s9893" xml:space="preserve"> & à puncto z ducatur
              <lb/>
            linea longitudinis:</s>
            <s xml:id="echoid-s9894" xml:space="preserve"> [per 7 th.</s>
            <s xml:id="echoid-s9895" xml:space="preserve"> Sereni de ſectione cylindri] quę ſit z g:</s>
            <s xml:id="echoid-s9896" xml:space="preserve"> & à pũcto z perpẽdicularis z
              <lb/>
            l:</s>
            <s xml:id="echoid-s9897" xml:space="preserve"> & huic æquidiſtãs à pũcto a:</s>
            <s xml:id="echoid-s9898" xml:space="preserve"> quę ſit a m:</s>
            <s xml:id="echoid-s9899" xml:space="preserve"> & etiã linea h z producatur, quouſq;</s>
            <s xml:id="echoid-s9900" xml:space="preserve"> cõcurrat cũea:</s>
            <s xml:id="echoid-s9901" xml:space="preserve"> [con
              <lb/>
            curret uerò per lemma Procli ad 29 p 1] & ſit cõcurſus in pũcto m:</s>
            <s xml:id="echoid-s9902" xml:space="preserve"> & à pũcto m ducatur linea ad b:</s>
            <s xml:id="echoid-s9903" xml:space="preserve">
              <lb/>
            quę neceſſariò ſecabit lineã z g:</s>
            <s xml:id="echoid-s9904" xml:space="preserve"> cũ ſit in eadẽ ſuperficie cũ ea:</s>
            <s xml:id="echoid-s9905" xml:space="preserve"> quoniã cũ b h ſit æquidiſtãs g z:</s>
            <s xml:id="echoid-s9906" xml:space="preserve"> [per
              <lb/>
            6 p 11:</s>
            <s xml:id="echoid-s9907" xml:space="preserve"> eſt enim utraq;</s>
            <s xml:id="echoid-s9908" xml:space="preserve"> ipſarũ perpẽdicularis circulo e zi] erit h z m in ſuperficie illarũ:</s>
            <s xml:id="echoid-s9909" xml:space="preserve"> [per 7 p 11:</s>
            <s xml:id="echoid-s9910" xml:space="preserve">
              <lb/>
            quia cõnectit parallelas] & ita b m in eadẽ:</s>
            <s xml:id="echoid-s9911" xml:space="preserve"> quę, ſi ſecuerit z g in puncto g:</s>
            <s xml:id="echoid-s9912" xml:space="preserve"> erit g punctum reflexio-
              <lb/>
            nis:</s>
            <s xml:id="echoid-s9913" xml:space="preserve"> quod quidem, ſi reuoluas probationem prædictam, uidere poteris.</s>
            <s xml:id="echoid-s9914" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div362" type="section" level="0" n="0">
          <head xml:id="echoid-head352" xml:space="preserve" style="it">49. Si communis ſectio ſuperficierum, reflexionis & ſpeculi conici conuexi fuerit lat{us} coni:
            <lb/>
          locatum reflexionum tum imaginum eodem modo ſe habebunt, ut in ſpeculo plano. 42 p 7.</head>
          <p>
            <s xml:id="echoid-s9915" xml:space="preserve">IN ſpeculis exteriorib.</s>
            <s xml:id="echoid-s9916" xml:space="preserve"> pyramidalibus, ſi linea cõmunis ſupficiei reflexiõis & ſpeculi, fuerit linea
              <lb/>
            lõgitudinis ſpeculi:</s>
            <s xml:id="echoid-s9917" xml:space="preserve"> erit locus imaginis, ſicut aſsignatus eſt in ſpeculis planis.</s>
            <s xml:id="echoid-s9918" xml:space="preserve"> Et eadẽ eſt ꝓbatio.</s>
            <s xml:id="echoid-s9919" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div363" type="section" level="0" n="0">
          <head xml:id="echoid-head353" xml:space="preserve" style="it">50. Cõmunis ſectio ſuperficierũ, reflexiõis et ſpeculi conici cõuexi nõ eſt circul{us}. 12 p 7. Idẽ 41 n 4.</head>
          <p>
            <s xml:id="echoid-s9920" xml:space="preserve">QVòd aũt nõ poſsit eſſe linea cõmunis, circulus:</s>
            <s xml:id="echoid-s9921" xml:space="preserve"> palàm per hoc:</s>
            <s xml:id="echoid-s9922" xml:space="preserve"> q đ ſuperficies reflexionis or
              <lb/>
            thogonalis eſt ſuper ſuperficiẽ, cõtingentẽ ſpeculũ in pũcto reflexionis [per 13 n 4] & cir-
              <lb/>
            culus neceſſariò eſt æquidiſtans baſi.</s>
            <s xml:id="echoid-s9923" xml:space="preserve"> [per cõuerſionẽ 4 th 1 conicorũ Apollonij] Superfi-
              <lb/>
            cies ergo hęc æquidiſtãs baſi, nõ erit orthogonalis ſuper ſuperficiẽ, cõtingentẽ ſpeculũ.</s>
            <s xml:id="echoid-s9924" xml:space="preserve"> [Nam pla-
              <lb/>
            nũ tangẽs conũ, tangit in latere per 35 n 4, ad baſim & circulũ ipſi parallelũ obliquo:</s>
            <s xml:id="echoid-s9925" xml:space="preserve"> quia eſt latus
              <lb/>
            trianguli acutanguli facti à plano conũ per uerticẽ ſecante, per 3 th 1 conicorũ Apollonij.</s>
            <s xml:id="echoid-s9926" xml:space="preserve"> Quare cir
              <lb/>
            culus erit extra reflexionis ſuperficiem:</s>
            <s xml:id="echoid-s9927" xml:space="preserve"> neq;</s>
            <s xml:id="echoid-s9928" xml:space="preserve"> idcirco uiſibile ab ipſo ad uiſum reflectetur.</s>
            <s xml:id="echoid-s9929" xml:space="preserve">]</s>
          </p>
        </div>
        <div xml:id="echoid-div364" type="section" level="0" n="0">
          <head xml:id="echoid-head354" xml:space="preserve" style="it">51. Si cõmunis ſectio ſuperficierũ reflexiõis & ſpeculi conici cõuexi fuerit ellipſis: imago uiſibilis
            <lb/>
          obliquè reflexi, aliâs in ſuperficie ſpeculi: aliâs intra: aliâs extra ſpeculũ uidebitur. 49 p 7.</head>
          <p>
            <s xml:id="echoid-s9930" xml:space="preserve">SI uerò cõmunis linea fuerit ſectio pyramidalis:</s>
            <s xml:id="echoid-s9931" xml:space="preserve"> imagines quędam erunt in ſuperficie ſpeculi:</s>
            <s xml:id="echoid-s9932" xml:space="preserve">
              <lb/>
            quędã intra ſpeculũ:</s>
            <s xml:id="echoid-s9933" xml:space="preserve"> quędã extra.</s>
            <s xml:id="echoid-s9934" xml:space="preserve"> Etidẽ eſt aſsignationis modus, qui fuit in ſpeculo columna-
              <lb/>
            ri exteriore:</s>
            <s xml:id="echoid-s9935" xml:space="preserve"> [44 n] & eadẽ ꝓbatio.</s>
            <s xml:id="echoid-s9936" xml:space="preserve"> Et (ſicut eſt in colũnari exteriore) [44 n] penperpẽdi
              <lb/>
            cularẽ uiſualẽ nõ reflectetur forma ad oculũ, niſi pũcti ſuperficiei oculi tãtũ:</s>
            <s xml:id="echoid-s9937" xml:space="preserve"> & hoc ab uno ſolo ſpe-
              <lb/>
            culi pũcto:</s>
            <s xml:id="echoid-s9938" xml:space="preserve"> & locus imaginis eius erit cõtinuus locis aliarũ imaginũ, ſicut patuit ſuperius [44 n.</s>
            <s xml:id="echoid-s9939" xml:space="preserve">]</s>
          </p>
        </div>
        <div xml:id="echoid-div365" type="section" level="0" n="0">
          <head xml:id="echoid-head355" xml:space="preserve" style="it">52. Si à puncto in communi ſectione ſuperficierum, reflexionis & ſpeculi conici conuexi dato, re
            <lb/>
          flexio fiat: poſſunt uiſ{us} & uiſibile ſic collocari, ut ab eodem puncto, tanquam puncto circuli ba-
            <lb/>
          ſi paralleli ad uiſum reflexio fiat. 32 p 7.</head>
          <p>
            <s xml:id="echoid-s9940" xml:space="preserve">REſtat in his ſpeculis declarare:</s>
            <s xml:id="echoid-s9941" xml:space="preserve"> quòd ab uno ſolo puncto eius fiat reflexio:</s>
            <s xml:id="echoid-s9942" xml:space="preserve"> quod ſic patebit.</s>
            <s xml:id="echoid-s9943" xml:space="preserve">
              <lb/>
            Sit uiſus a:</s>
            <s xml:id="echoid-s9944" xml:space="preserve"> b punctũ uiſum:</s>
            <s xml:id="echoid-s9945" xml:space="preserve"> g punctũ reflexionis:</s>
            <s xml:id="echoid-s9946" xml:space="preserve"> & ducatur ſuper punctũ g ſuperficies æqui
              <lb/>
            diſtãs baſi:</s>
            <s xml:id="echoid-s9947" xml:space="preserve"> [ductis nimirũ duabus perpẽdicularibus ſuper axem ſe interſecãtibus:</s>
            <s xml:id="echoid-s9948" xml:space="preserve"> una qui-
              <lb/>
            dẽ à reflexionis puncto per 12 p 1:</s>
            <s xml:id="echoid-s9949" xml:space="preserve"> altera uerò ab axis puncto, in quod illa cadit, per 11 p 1.</s>
            <s xml:id="echoid-s9950" xml:space="preserve"> Sic enim
              <lb/>
            axis, qui per 18 d 11 perpendicularis eſt baſi:</s>
            <s xml:id="echoid-s9951" xml:space="preserve"> erit per 4 p 11 perpendicularis plano ductarũ perpendi
              <lb/>
            culariũ.</s>
            <s xml:id="echoid-s9952" xml:space="preserve"> Quare per 14 p 11 baſis & hoc planũ erunt parallela] quę quidẽ ſecabit pyramidẽ ſuper cir-
              <lb/>
            </s>
          </p>
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