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-div686" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s20709" xml:space="preserve">
              <pb o="10" file="0312" n="312" rhead="VITELLONIS OPTICAE"/>
            arcum a b c in puncto a per 17 p 3:</s>
            <s xml:id="echoid-s20710" xml:space="preserve"> anguli ergo contingentiæ, qui ſunt e a c & f a b ſunt æquales
              <lb/>
            per 16 p 3:</s>
            <s xml:id="echoid-s20711" xml:space="preserve"> ſed anguli g a c & d a b ſunt æquales ex hypotheſi:</s>
            <s xml:id="echoid-s20712" xml:space="preserve"> erunt ergo anguli g a e & d a f æqua-
              <lb/>
            les.</s>
            <s xml:id="echoid-s20713" xml:space="preserve"> Et ad punctum, ubi linea g b ſecat lineam e f(quod ſit z) ducatur linea d z:</s>
            <s xml:id="echoid-s20714" xml:space="preserve"> ergo per præceden
              <lb/>
            tem ambæ lineæ g a & d a ſunt breuiores ambabus lineis g z & d z:</s>
            <s xml:id="echoid-s20715" xml:space="preserve"> cum angulus g z a ſit minor an-
              <lb/>
            gulo g a e, & angulus d z f ſit maior angulo d a f per 16 p 1.</s>
            <s xml:id="echoid-s20716" xml:space="preserve"> Sed linea g b eſt maior quàm linea
              <lb/>
            g z, ut totum parte, & linea d b eſt maior quàm linea d z per 19 p 1, quoniam angulus d z b eſt
              <lb/>
            maior angulus ſui trigoni.</s>
            <s xml:id="echoid-s20717" xml:space="preserve"> Patet ergo propoſitum in arcu circuli conuexo:</s>
            <s xml:id="echoid-s20718" xml:space="preserve"> & eodem modo demon
              <lb/>
            ſtrandum in quacunque alia columnali uel pyramidali ſectione ſecũdum ipſius conuexum:</s>
            <s xml:id="echoid-s20719" xml:space="preserve"> patet
              <lb/>
            ergo propoſitum.</s>
            <s xml:id="echoid-s20720" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div688" type="section" level="0" n="0">
          <head xml:id="echoid-head593" xml:space="preserve" style="it">19. Vna linea recta in duabus ſuperficiebus planis exiſtente, neceſſe est, ut illæ duæ ſuperſi-
            <lb/>
          cies ſecundum illam lineam ſe ſecent. E' 3 p 11 element.</head>
          <p>
            <s xml:id="echoid-s20721" xml:space="preserve">Sint duæ ſuperficies planæ a b c d & c d e f:</s>
            <s xml:id="echoid-s20722" xml:space="preserve"> in quarum utraque ſit linea c d.</s>
            <s xml:id="echoid-s20723" xml:space="preserve"> Dico, quòd illæ
              <lb/>
            duæ ſuperficies ſecant ſe ſuper lineam e d.</s>
            <s xml:id="echoid-s20724" xml:space="preserve"> Si enim illæ duæ ſuperfici-
              <lb/>
              <figure xlink:label="fig-0312-01" xlink:href="fig-0312-01a" number="277">
                <variables xml:id="echoid-variables262" xml:space="preserve">a b c d e f</variables>
              </figure>
            es ad lineam c d, ut ad communem terminum per modum unius ſu-
              <lb/>
            perficiei continuè copulentur:</s>
            <s xml:id="echoid-s20725" xml:space="preserve"> tunc patet, quòd ipſæ ſunt partes uni-
              <lb/>
            us ſuperficiei, & non duæ ſuperficies:</s>
            <s xml:id="echoid-s20726" xml:space="preserve"> quod eſt contra hypotheſim.</s>
            <s xml:id="echoid-s20727" xml:space="preserve">
              <lb/>
            Quòd ſi ipſæ ſuperficies datam lineam c d pertranſeant, nec ad ipſam,
              <lb/>
            ut ad communem terminum copulentur:</s>
            <s xml:id="echoid-s20728" xml:space="preserve"> palàm per 3 p 11, cum ipſæ
              <lb/>
            ad inuicem ſe ſecent, quòd ipſis aliqua linea eſt communis.</s>
            <s xml:id="echoid-s20729" xml:space="preserve"> Aut ergo
              <lb/>
            ſecant ſe ſuper lineam c d:</s>
            <s xml:id="echoid-s20730" xml:space="preserve"> & habetur propoſitum:</s>
            <s xml:id="echoid-s20731" xml:space="preserve"> aut ſuper aliam
              <lb/>
            quamcunque datam:</s>
            <s xml:id="echoid-s20732" xml:space="preserve"> & tunc, cum illa ſit ambabus propoſitis ſuper-
              <lb/>
            ficiebus communis per prænominatam 3 p 11, & eiſdem ſit linea c d
              <lb/>
            communis ex hypotheſi:</s>
            <s xml:id="echoid-s20733" xml:space="preserve"> ſequetur, ut duæ planæ ſuperficies illas du-
              <lb/>
            as lineas interiacentes corpus includãt:</s>
            <s xml:id="echoid-s20734" xml:space="preserve"> quod eſt impoſsibile, & con-
              <lb/>
            tra 4 ſuppoſitionem huius:</s>
            <s xml:id="echoid-s20735" xml:space="preserve"> patet ergo propoſitum.</s>
            <s xml:id="echoid-s20736" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div690" type="section" level="0" n="0">
          <head xml:id="echoid-head594" xml:space="preserve" style="it">20. Ab uno puncto in aere dato, ſuper unamquam ſubſtratã
            <lb/>
          planam uel conuexam ſuperficiem, una tantũ perpendicularis du-
            <lb/>
          ci potest. E' 11 & 13 p 11 elem.</head>
          <p>
            <s xml:id="echoid-s20737" xml:space="preserve">Sit data ſuperficies plana a b c d, & datus in aere punctus e.</s>
            <s xml:id="echoid-s20738" xml:space="preserve"> Dico, quòd à puncto e ad ſubſtra-
              <lb/>
            tam ſuperficiem, unam tantùm perpendicularem duci eſt poſsibi-
              <lb/>
              <figure xlink:label="fig-0312-02" xlink:href="fig-0312-02a" number="278">
                <variables xml:id="echoid-variables263" xml:space="preserve">e a b k l f g h m c d</variables>
              </figure>
            le.</s>
            <s xml:id="echoid-s20739" xml:space="preserve"> Sienim poſsibile, ſit ut ſuper ſuperficiem planam datam, quæ a
              <lb/>
            b c d, ducantur à puncto e duæ perpendiculares, quæ ſint e f & e g.</s>
            <s xml:id="echoid-s20740" xml:space="preserve">
              <lb/>
            Quia itaq;</s>
            <s xml:id="echoid-s20741" xml:space="preserve"> lineę e f & e g angulariter cõiunguntur in puncto e, pa
              <lb/>
            tet per 2 p 11, quoniam illæ duæ lineæ ſunt in eadem ſuperficie:</s>
            <s xml:id="echoid-s20742" xml:space="preserve"> &
              <lb/>
            quoniam lineæ illæ ſunt perpendiculares ſuper ſuperficiem a b c d,
              <lb/>
            erit ſuperficies, in qua ſunt lineæ illæ, erecta ſuper ſuperficiem a b
              <lb/>
            c d.</s>
            <s xml:id="echoid-s20743" xml:space="preserve"> Huius itaq;</s>
            <s xml:id="echoid-s20744" xml:space="preserve"> ſuperficiei & ſuperficiei a b c d communis ſectio
              <lb/>
            eſt linea f g per præmiſſam:</s>
            <s xml:id="echoid-s20745" xml:space="preserve"> in trigono itaque e f g ſunt duo angu
              <lb/>
            li recti, ſcilicet e f g & e g f per definitionem lineæ erectæ ſuper ſu
              <lb/>
            perficiem 3 definit.</s>
            <s xml:id="echoid-s20746" xml:space="preserve"> 11:</s>
            <s xml:id="echoid-s20747" xml:space="preserve"> hoc autem eſt impoſsibile, & contra 32 p 1.</s>
            <s xml:id="echoid-s20748" xml:space="preserve">
              <lb/>
            Hoc autem etiam patet in ſuperficiebus conuexis:</s>
            <s xml:id="echoid-s20749" xml:space="preserve"> quia enim, per
              <lb/>
            5 definitionem huius omnis linea perpendicularis ſuper quam cun
              <lb/>
            que ſuperficiem conuexam, eſt perpendicularis ſuper planam ſu-
              <lb/>
            perficiem ipſam conuexam ſuperficiem in puncto incidentię lineę
              <lb/>
            illius contingentem:</s>
            <s xml:id="echoid-s20750" xml:space="preserve"> patet, quia in omni ſuperficie conuexaidem
              <lb/>
            accidit impoſsibile.</s>
            <s xml:id="echoid-s20751" xml:space="preserve"> Si enim ſit ſuperficies ſphærica cõuexa, in qua
              <lb/>
            ſit arcus f g:</s>
            <s xml:id="echoid-s20752" xml:space="preserve"> ſit ut ipſam contingat in puncto fſuperficies plana, in
              <lb/>
            qua ducatur linea h f k, & in puncto g ſuperficies plana, in qua ſit li-
              <lb/>
            nea l g m.</s>
            <s xml:id="echoid-s20753" xml:space="preserve"> Palàm ergo ex pręmiſsis, quia anguli e f k & e g l ſunt re-
              <lb/>
            cti.</s>
            <s xml:id="echoid-s20754" xml:space="preserve"> Producta quoq;</s>
            <s xml:id="echoid-s20755" xml:space="preserve"> chorda f g:</s>
            <s xml:id="echoid-s20756" xml:space="preserve"> palàm quia anguli e f g & e g f ſunt maiores duobus rectis, quod eſt
              <lb/>
            impoſsibile.</s>
            <s xml:id="echoid-s20757" xml:space="preserve"> Non eſt ergo poſsibile ab uno puncto dato plus una perpendiculari duci ad ſuperficiẽ
              <lb/>
            planam uel conuexam.</s>
            <s xml:id="echoid-s20758" xml:space="preserve"> Patet ergo propoſitum:</s>
            <s xml:id="echoid-s20759" xml:space="preserve"> quoniam in quibuſcunque alijs conuexis ſuperfi-
              <lb/>
            ciebus eſt eodem modo demonſtrandum.</s>
            <s xml:id="echoid-s20760" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div692" type="section" level="0" n="0">
          <head xml:id="echoid-head595" xml:space="preserve" style="it">21. Omnium linearum ab eodem puncto adeandem ſuperficiem planamuel conuexam pro-
            <lb/>
          ductarum, minima eſt perpendicularis. Albazen 5 n 5.</head>
          <p>
            <s xml:id="echoid-s20761" xml:space="preserve">Eſto ſuperficies plana b c d i:</s>
            <s xml:id="echoid-s20762" xml:space="preserve"> & punctum extrà ſignatum a, à quo ducantur plurimæ lineæ ad ſu-
              <lb/>
            perficiem datam, ut contingit, ſcilicet a e, a f, a g, a h, ſola tamen a e ſit perpendicularis.</s>
            <s xml:id="echoid-s20763" xml:space="preserve"> Dico, quòd li
              <lb/>
            nea a e eſt omnium aliarum breuiſsima.</s>
            <s xml:id="echoid-s20764" xml:space="preserve"> Ducantur enim lineæ e f, e g, e h, & componantur tri-
              <lb/>
            gona orthogonia.</s>
            <s xml:id="echoid-s20765" xml:space="preserve"> Palàm itaque (cum per 32 p 1 angulus rectus ſit maior in qualibet trigono
              <lb/>
              <figure xlink:label="fig-0312-03" xlink:href="fig-0312-03a" number="279">
                <image file="0312-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/figures/0312-03"/>
              </figure>
            </s>
          </p>
        </div>
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