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-div1669" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s42352" xml:space="preserve">
              <pb o="344" file="0646" n="646" rhead="VITELLONIS OPTICAE"/>
            fiet reflexio:</s>
            <s xml:id="echoid-s42353" xml:space="preserve"> ſed & per 27 uel 29 huius ab uno tantũ puncto arcus n d fiet reflexio.</s>
            <s xml:id="echoid-s42354" xml:space="preserve"> Fiet itaq:</s>
            <s xml:id="echoid-s42355" xml:space="preserve"> in hoc
              <lb/>
            ſitu reflexio quádoq;</s>
            <s xml:id="echoid-s42356" xml:space="preserve"> à tribus punctis:</s>
            <s xml:id="echoid-s42357" xml:space="preserve"> quandoq;</s>
            <s xml:id="echoid-s42358" xml:space="preserve"> à quatuor, & non â pluribus.</s>
            <s xml:id="echoid-s42359" xml:space="preserve"> Quòd ſi pun cto b
              <lb/>
            exiſtente in peripheria circuli ſpeculi, punctus a ſit extra illum circulum:</s>
            <s xml:id="echoid-s42360" xml:space="preserve"> tunc patet quòd circulus
              <lb/>
            a b g nunquam ſecabit circulum ſpeculi ſecundum arcum l m:</s>
            <s xml:id="echoid-s42361" xml:space="preserve"> quoniam ſemidiameter g m, & peri-
              <lb/>
            pheriæ circuli communis ſectio eſt punctus m, in quo eſt punctus b:</s>
            <s xml:id="echoid-s42362" xml:space="preserve"> ſemidiameter uerò glproce-
              <lb/>
            dens ad punctum a extra circulum ſecat arcum t b.</s>
            <s xml:id="echoid-s42363" xml:space="preserve"> Omnes itaq;</s>
            <s xml:id="echoid-s42364" xml:space="preserve"> anguli arcus l m ſunt maiores an-
              <lb/>
            gulo b g d, ut patet ex præmiſsis:</s>
            <s xml:id="echoid-s42365" xml:space="preserve"> ergo per 34 huius ab uno tantùm pũcto uel forſan â duobus pun-
              <lb/>
            ctis arcus l m poteſt fieri reflexio punctorum a & b adinuicem:</s>
            <s xml:id="echoid-s42366" xml:space="preserve"> & ſimiliter ab uno pũcto arcus n d.</s>
            <s xml:id="echoid-s42367" xml:space="preserve">
              <lb/>
            Fiet itaq;</s>
            <s xml:id="echoid-s42368" xml:space="preserve"> in hoc ſitu reflexio à duob.</s>
            <s xml:id="echoid-s42369" xml:space="preserve"> aut à tribus pũctis ſpeculi, & nõ à pluribus.</s>
            <s xml:id="echoid-s42370" xml:space="preserve"> Palàm ergo quòd
              <lb/>
            puncta inæqualiter diſtantia à centro ſpeculi aliquando ab uno tantùm puncto ſpeculi:</s>
            <s xml:id="echoid-s42371" xml:space="preserve"> aliquan-
              <lb/>
            do à duobus:</s>
            <s xml:id="echoid-s42372" xml:space="preserve"> aliquando à tribus:</s>
            <s xml:id="echoid-s42373" xml:space="preserve"> aliquando à quatuor:</s>
            <s xml:id="echoid-s42374" xml:space="preserve"> nunquam à pluribus reflectuntur:</s>
            <s xml:id="echoid-s42375" xml:space="preserve"> ſecun-
              <lb/>
            dum hęc quoq;</s>
            <s xml:id="echoid-s42376" xml:space="preserve"> loca imaginum numerantur, quemadmodum patuit iam pluries in præmiſsis.</s>
            <s xml:id="echoid-s42377" xml:space="preserve"> Et
              <lb/>
            hoc eſt, quod ꝓponebatur declarandum.</s>
            <s xml:id="echoid-s42378" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1671" type="section" level="0" n="0">
          <head xml:id="echoid-head1251" xml:space="preserve" style="it">41. Exiſtentibus duobus punctis in diuerſis diametris circuli ſpeculi ſphærici concaui, &
            <lb/>
          æqualiter diſtantibus à centro, ſi linea coutinuans illa puncta ſecet circulum: poßibile eſt
            <lb/>
          unũ illorum punctorum ad alterum reflecti ab uno tantumpuncto ſpeculi: uelà duobus: aut à
            <lb/>
          quatuor: ſed impoßibile ect à tribus: & ſecundum hæc loca imaginum numerantur. Alha-
            <lb/>
          zen 87 n 5.</head>
          <p>
            <s xml:id="echoid-s42379" xml:space="preserve">Sint, ut in præmiſſa, duo puncta a & b in diuerſis diametris circuli ſpeculi ſphærici concaui, quæ
              <lb/>
            ſintl d & m n, ita ut pũctus a ſit in diametro l d, & pũctus b in diametro m n:</s>
            <s xml:id="echoid-s42380" xml:space="preserve"> ſintq́;</s>
            <s xml:id="echoid-s42381" xml:space="preserve"> pũcta a & b æqua
              <lb/>
            liter diſtantia à centro ſpeculi, & linea a b ſit ducta ab uno illorũ pũctorũ ad alterũ ſecundũ circulũ
              <lb/>
            (qui eſt cõmunis ſectio ſuperficiei reflexionis & ſpeculi) cuius centrũ ſit g:</s>
            <s xml:id="echoid-s42382" xml:space="preserve"> dico quòd uerũ eſt qđ
              <lb/>
              <figure xlink:label="fig-0646-01" xlink:href="fig-0646-01a" number="777">
                <variables xml:id="echoid-variables754" xml:space="preserve">l m a b g n d</variables>
              </figure>
            proponitur.</s>
            <s xml:id="echoid-s42383" xml:space="preserve"> Quòd enim ab uno tantũ puncto ſpecu-
              <lb/>
            li quandoq;</s>
            <s xml:id="echoid-s42384" xml:space="preserve"> fiat illorũ pũctorum adinuicẽ mutua re-
              <lb/>
            flexio, patet per 19 huius:</s>
            <s xml:id="echoid-s42385" xml:space="preserve"> & etiã idẽ oſtẽdi poteſt per
              <lb/>
            modũ 24 huius:</s>
            <s xml:id="echoid-s42386" xml:space="preserve"> linearũ enim inæ qualitas in illo ſitu
              <lb/>
            naturã reflexionis nõ immutat, ut declaratum eſt in
              <lb/>
            20 th.</s>
            <s xml:id="echoid-s42387" xml:space="preserve"> 5 huius.</s>
            <s xml:id="echoid-s42388" xml:space="preserve"> Quãdoq;</s>
            <s xml:id="echoid-s42389" xml:space="preserve"> uerò fit mutua reflexio iſto-
              <lb/>
            rum pũctorum a & b à duobus tantũ pũctis ſpeculi,
              <lb/>
            ut patet per 25 huius.</s>
            <s xml:id="echoid-s42390" xml:space="preserve"> Quandoq;</s>
            <s xml:id="echoid-s42391" xml:space="preserve"> uerò fit reflexio mu
              <lb/>
            tua propoſitorũ pũctorum, quę ſunt a & b, à quatuor
              <lb/>
            pũctis circũferentiæ ipſius ſpeculi, ut patet per 26 hu
              <lb/>
            ius.</s>
            <s xml:id="echoid-s42392" xml:space="preserve"> A tribus uerò tantũ pũctis iſtorum ſpeculorum
              <lb/>
            formas pũctorum æqualiter diſtantium à centro ſpe
              <lb/>
            culi ad ſe mutuò reflecti eſt impoſsibile.</s>
            <s xml:id="echoid-s42393" xml:space="preserve"> Si enim ab
              <lb/>
            aliquibus duobus pũctis unius arcus fiat iſta mutua
              <lb/>
            reflexio, diuiſo arcu interiacente illa pũcta per æqua
              <lb/>
            lia, & ductis ad illud pũctum lineis, patet per 27 p 3 &
              <lb/>
            propter ęqualitatem laterum g a & g b, quoniam an-
              <lb/>
            guli conſtituti ſuper illud punctum fiunt æquales:</s>
            <s xml:id="echoid-s42394" xml:space="preserve"> ab
              <lb/>
            illo ergo pũcto fiet reflexio per 20 th.</s>
            <s xml:id="echoid-s42395" xml:space="preserve"> 5 huius:</s>
            <s xml:id="echoid-s42396" xml:space="preserve"> ſed &
              <lb/>
            fiet ab aliquo pũcto arcus oppoſiti illi arcui.</s>
            <s xml:id="echoid-s42397" xml:space="preserve"> Palã ergo quòd à quatuor pũctis ſpeculi fiet reflexio,
              <lb/>
            & non à tribus.</s>
            <s xml:id="echoid-s42398" xml:space="preserve"> Et quoniam, ut patet per præmiſſam & ex plur bus propoſitionibus huius libri,
              <lb/>
            nunquam fit à tribus punctis ſpeculi reflexio aliquorum duorum punctorum adinuicem, niſi fiat à
              <lb/>
            duobus punctis unius arcus, & ab aliquo puncto arcus oppoſiti interiacente illas diametros:</s>
            <s xml:id="echoid-s42399" xml:space="preserve"> patet
              <lb/>
            ergo quòd in hac diſpoſitione reflexio fiet ſemper à quatuor punctis ſpeculi propoſiti, & nunquam
              <lb/>
            à tribus.</s>
            <s xml:id="echoid-s42400" xml:space="preserve"> Et hoc proponebatur.</s>
            <s xml:id="echoid-s42401" xml:space="preserve"> Et quoniam hęc duo præmiſſa theoremata diſpoſuimus ſecundum
              <lb/>
            modum epilogi plurimorum præmiſſorum theorematum, ęſtimamus ipſa memorię cõmendanda.</s>
            <s xml:id="echoid-s42402" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1673" type="section" level="0" n="0">
          <head xml:id="echoid-head1252" xml:space="preserve" style="it">42. Siab uno puncto arcus circuli ſpeculi ſphærici concaui formæ unius termini lineæ totali-
            <lb/>
          ter uiſæ, ab alio quo puncto eiuſdem arcus formæ alterius termini eiuſdem lineæ fiat reflexio:
            <lb/>
          neceſſe eſt omnia punct a media lineæ uiſæ abillius arcus punctis medijs reflecti: ex quo patet
            <lb/>
          quòd loca imaginum punct orum mediorum cadunt inter imagines punctorum extremorum.
            <lb/>
          Alhazen 45 n 6.</head>
          <p>
            <s xml:id="echoid-s42403" xml:space="preserve">Quod hic proponitur ſpecialiter, quantùm ad primam ſui partem, uniuerſaliter eſt pręmiſſum in
              <lb/>
            24 th.</s>
            <s xml:id="echoid-s42404" xml:space="preserve"> 5 huius.</s>
            <s xml:id="echoid-s42405" xml:space="preserve"> Eſto ergo arcus circuli ſpeculi ſphærici concaui a f h:</s>
            <s xml:id="echoid-s42406" xml:space="preserve"> cuius centrum e:</s>
            <s xml:id="echoid-s42407" xml:space="preserve"> & ſit z centrũ
              <lb/>
            uiſus:</s>
            <s xml:id="echoid-s42408" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s42409" xml:space="preserve"> g r linea uiſa:</s>
            <s xml:id="echoid-s42410" xml:space="preserve"> cuius unus terminus (quig) reflectatur à puncto ſpeculi, qui ſit f:</s>
            <s xml:id="echoid-s42411" xml:space="preserve"> & ille ſit
              <lb/>
            aliquis punctus arcus dati, qui eſt a f h:</s>
            <s xml:id="echoid-s42412" xml:space="preserve"> & alter terminus lineę (qui eſt r) reflectatur à puncto h ar-
              <lb/>
            cus a f h.</s>
            <s xml:id="echoid-s42413" xml:space="preserve"> Dico quòd omnia puncta media lineæ g r reflectentur à punctis medijs arcus h f.</s>
            <s xml:id="echoid-s42414" xml:space="preserve"> Coapte-
              <lb/>
            tur enim linea g r (exempli cauſſa) diametro ſpeculi, q̃ ſit o a cadatq́;</s>
            <s xml:id="echoid-s42415" xml:space="preserve"> in ſemidiametrũ o e:</s>
            <s xml:id="echoid-s42416" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s42417" xml:space="preserve"> pũct{us}
              <lb/>
            z, ꝗ eſt cẽtrũ uiſus, in alia diametro eiuſdẽ circuli, quæ ſit d b, cadẽs in ſemidiametrũ e b:</s>
            <s xml:id="echoid-s42418" xml:space="preserve"> & ducãtur
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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