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-s20765" xml:space="preserve">
              <pb o="11" file="0313" n="313" rhead="LIBER I."/>
            orthogonio) quoniam linea a e per 19 p 1 breuior eſt qualibet linearum a f, a g, a h, & etiam
              <lb/>
            aliarum quarumcunq;</s>
            <s xml:id="echoid-s20766" xml:space="preserve"> ſic productarum:</s>
            <s xml:id="echoid-s20767" xml:space="preserve"> patet ergo propoſitum in planis.</s>
            <s xml:id="echoid-s20768" xml:space="preserve"> Sed & in conuexis patet
              <lb/>
            idem:</s>
            <s xml:id="echoid-s20769" xml:space="preserve"> quoniam ſi perpendicularis ſuper conuexam
              <lb/>
              <figure xlink:label="fig-0313-01" xlink:href="fig-0313-01a" number="280">
                <variables xml:id="echoid-variables264" xml:space="preserve">a b c e f g h d i</variables>
              </figure>
            ſuperficiem ſit a e, & ſit b c d i ſuperficies plana con
              <lb/>
            tingens ſuperficiem conuexam ſecundum punctũ
              <lb/>
            e, ducanturq́;</s>
            <s xml:id="echoid-s20770" xml:space="preserve"> lineæ a f, a g, a h ſuper ſuperficiem pla
              <lb/>
            nam:</s>
            <s xml:id="echoid-s20771" xml:space="preserve"> erunt omnes illę maiores perpendiculari:</s>
            <s xml:id="echoid-s20772" xml:space="preserve"> er-
              <lb/>
            go eædem productæ ad ſuperficiem conuexã ſunt
              <lb/>
            multo maiores:</s>
            <s xml:id="echoid-s20773" xml:space="preserve"> patet ergo propoſitum.</s>
            <s xml:id="echoid-s20774" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div694" type="section" level="0" n="0">
          <head xml:id="echoid-head596" xml:space="preserve" style="it">22. Ducta linea à ſupremo termino lineæ ſu-
            <lb/>
          per ſuperficiem erectæ, ad lineam perpendicularẽ
            <lb/>
          cuicun lineæ à puncto incidẽtiæ lineæ erectæ in
            <lb/>
          ſubiecta ſuperficie protractæ: neceſſe eſt protractã
            <lb/>
          lineam ſuperiacenti perpendicularem eſſe. Lem-
            <lb/>
          ma ad 37 theorema opticorum Euclidis: item 42
            <lb/>
          theor. 6 libri μαθκματικυεμ συναγωγυεμ Pappi.</head>
          <p>
            <s xml:id="echoid-s20775" xml:space="preserve">Sit punctũ in aere datum, quod ſit a, à quo ad ſu-
              <lb/>
            perficiem planã ſubiectam, quæ ſit b c d, erigatur li-
              <lb/>
            nea per 12 p 11, quæ ſit a b, incidens datæ ſuperficiei in puncto b:</s>
            <s xml:id="echoid-s20776" xml:space="preserve"> & in ſuperficie b c d ducatur linea
              <lb/>
            d c, ut placuerit, & à puncto b ducatur perpendicularis ſuper lineam
              <lb/>
              <figure xlink:label="fig-0313-02" xlink:href="fig-0313-02a" number="281">
                <variables xml:id="echoid-variables265" xml:space="preserve">a c b d</variables>
              </figure>
            d c, quæ ſit b d:</s>
            <s xml:id="echoid-s20777" xml:space="preserve"> & copuletur linea a d.</s>
            <s xml:id="echoid-s20778" xml:space="preserve"> Dico, quòd a d eſt perpendi-
              <lb/>
            cularis ſuper lineã d c.</s>
            <s xml:id="echoid-s20779" xml:space="preserve"> Sumatur enim in linea d c quodcunq;</s>
            <s xml:id="echoid-s20780" xml:space="preserve"> punctũ,
              <lb/>
            ut c, & ducantur lineæ a c, b c.</s>
            <s xml:id="echoid-s20781" xml:space="preserve"> Quia itaq;</s>
            <s xml:id="echoid-s20782" xml:space="preserve"> linea a b eſt erecta ſuper ſu-
              <lb/>
            perficiẽ b c d, patet ք definitionẽ lineę erectę 3 defin.</s>
            <s xml:id="echoid-s20783" xml:space="preserve"> 11, quoniã angu-
              <lb/>
            lus a b c eſt rectus:</s>
            <s xml:id="echoid-s20784" xml:space="preserve"> ergo ք 47 p 1, quadratũ lineę a c eſt æquale duob.</s>
            <s xml:id="echoid-s20785" xml:space="preserve">
              <lb/>
            quadratis linearũ a b & b c:</s>
            <s xml:id="echoid-s20786" xml:space="preserve"> ſed & quadratũ lineę b c eſt æquale duob.</s>
            <s xml:id="echoid-s20787" xml:space="preserve">
              <lb/>
            quadratis c d & b d per 47 p 1, quia linea b d eſt perpẽdicularis ſuper
              <lb/>
            lineam c d ex hypotheſi.</s>
            <s xml:id="echoid-s20788" xml:space="preserve"> Quadratum itaq;</s>
            <s xml:id="echoid-s20789" xml:space="preserve"> lineæ a c eſt æquale tribus
              <lb/>
            quadratis trium linearum, quæ ſunt a b & b d & c d:</s>
            <s xml:id="echoid-s20790" xml:space="preserve"> ſed quadratum li-
              <lb/>
            neæ a d eſt æquale duobus quadratis duarum linearum a b & b d:</s>
            <s xml:id="echoid-s20791" xml:space="preserve">
              <lb/>
            quadratum ergo lineæ a c eſt æquale duobus quadratis duarum li-
              <lb/>
            nearum a d & d c.</s>
            <s xml:id="echoid-s20792" xml:space="preserve"> Ergo per 48 p 1 angulus a d c eſt rectus.</s>
            <s xml:id="echoid-s20793" xml:space="preserve"> Patet er-
              <lb/>
            go, quòd linea a d eſt perpendicularis ſuper lineam d c:</s>
            <s xml:id="echoid-s20794" xml:space="preserve"> quod eſt
              <lb/>
            propoſitum.</s>
            <s xml:id="echoid-s20795" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div696" type="section" level="0" n="0">
          <head xml:id="echoid-head597" xml:space="preserve" style="it">23. Duabus planis ſuperficiebus æquidiſtantibus, una linea rect a incidente, quæ ad alteram
            <lb/>
          earũ erit perpendicularis, erit quo ad reliquã perpendicularis. Conuerſa 14 p 11 elem.</head>
          <p>
            <s xml:id="echoid-s20796" xml:space="preserve">Sit, ut duabus ſuperficiebus planis & æquidiſtantibus incidatun a linea, quæ a b, uni ipſarum
              <lb/>
            in puncto a, & reliquæ in puncto b.</s>
            <s xml:id="echoid-s20797" xml:space="preserve"> Dico, quòd ſi linea a b fuerit
              <lb/>
              <figure xlink:label="fig-0313-03" xlink:href="fig-0313-03a" number="282">
                <variables xml:id="echoid-variables266" xml:space="preserve">c d a b</variables>
              </figure>
            perpendicularis ſuper unam iſtarum ſuperficierum, quòd erit per-
              <lb/>
            pendicularis & ſuper reliquam.</s>
            <s xml:id="echoid-s20798" xml:space="preserve"> Nam à puncto a ducatur in altera ſu-
              <lb/>
            perficierum illarum linea recta, quæ a c, & in reliqua à puncto b du-
              <lb/>
            catur linea b d.</s>
            <s xml:id="echoid-s20799" xml:space="preserve"> Palàm itaque, quoniam lineæ a c & b d æquidiſtant:</s>
            <s xml:id="echoid-s20800" xml:space="preserve">
              <lb/>
            in infinitum enim protractæ non concurrent, quia & ſuperficies in
              <lb/>
            quibus ſunt, non concurrunt.</s>
            <s xml:id="echoid-s20801" xml:space="preserve"> Si itaque alter angulorum, qui b a c
              <lb/>
            uel a b d fueritrectus:</s>
            <s xml:id="echoid-s20802" xml:space="preserve"> palàm ſemper per 29 p 1, quoniam & reli-
              <lb/>
            quus ipſorum erit rectus.</s>
            <s xml:id="echoid-s20803" xml:space="preserve"> Et quoniam eodem modo poteſt hoc de-
              <lb/>
            clarari de omnibus lineis in ſuperficiebus hinc inde ductis à punctis
              <lb/>
            a & b:</s>
            <s xml:id="echoid-s20804" xml:space="preserve"> patet, quòd linea a b cum ſingulis ſibi conterminalibus lineis
              <lb/>
            in utraque ſuperficierum illarum productis angulos rectos facit.</s>
            <s xml:id="echoid-s20805" xml:space="preserve"> Si
              <lb/>
            eſt ergo linea a b perpendicularis ſuper alteram ſuperficierum, pa-
              <lb/>
            làm, quia erit perpendicularis ſuper reliquam ipſarum:</s>
            <s xml:id="echoid-s20806" xml:space="preserve"> & hoc eſt
              <lb/>
            propoſitum.</s>
            <s xml:id="echoid-s20807" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div698" type="section" level="0" n="0">
          <head xml:id="echoid-head598" xml:space="preserve" style="it">24. Si duæ ſuperficies uni ſuperficiei æquidiſtantes fuerint, eædem inter ſe erunt æquidiſtan
            <lb/>
          tes: ſuperficies quoque concurrens cum una æquidiſtantium ſuperficierum & cum reliqua con-
            <lb/>
          curret. E' 30 p 1 & 9 p 11 elementorum.</head>
          <p>
            <s xml:id="echoid-s20808" xml:space="preserve">Sint duæ ſuperficies a b c & g h k æquidiſtantes uni ſuperficiei, quæ d e f.</s>
            <s xml:id="echoid-s20809" xml:space="preserve"> Dico, quòd
              <lb/>
            illæ duæ ſuperficies a b c & g h k neceſſariò adinuicem æquidiſtabunt.</s>
            <s xml:id="echoid-s20810" xml:space="preserve"> Educatur enim à pun-
              <lb/>
            cto l ſuperficiei a b c linea perpendicularis ſuper illam ſuperficiem per 12 p undecimi, quæ
              <lb/>
            </s>
          </p>
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