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-div806" type="section" level="0" n="0">
          <pb o="36" file="0338" n="338" rhead="VITELLONIS OPTICAE"/>
        </div>
        <div xml:id="echoid-div807" type="section" level="0" n="0">
          <head xml:id="echoid-head667" xml:space="preserve" style="it">93. Omnis ſuperficiei planæ ſecantis columnam rotundam ſecundum axis longitudinem &
            <lb/>
          ſuperficiei columnæ communis ſectio eſt rectangulum ſub duab{us} lineis longitudinis columnæ,
            <lb/>
          & duab{us} diametris baſium contentum. Ex quo patet, quoniam illa ſuperficies per æqualia diui
            <lb/>
          dit columnam. È
            <unsure/>
          21 defin. 11. element.</head>
          <p>
            <s xml:id="echoid-s22386" xml:space="preserve">Columna rotunda ſit, cuius axis e f:</s>
            <s xml:id="echoid-s22387" xml:space="preserve"> ſecetq́;</s>
            <s xml:id="echoid-s22388" xml:space="preserve"> ipſam per e f ſuperfi-
              <lb/>
              <figure xlink:label="fig-0338-01" xlink:href="fig-0338-01a" number="351">
                <variables xml:id="echoid-variables335" xml:space="preserve">g a m e n b h i c p f o d k l</variables>
              </figure>
            cies plana, ſitq́;</s>
            <s xml:id="echoid-s22389" xml:space="preserve"> communis ſectio ſecundum puncta a, b, c, d.</s>
            <s xml:id="echoid-s22390" xml:space="preserve"> Dico,
              <lb/>
            quòd ſectio a b c d eſt quadrangula rectangula ſub lineis longitudi-
              <lb/>
            nis columnæ, & duabus diametris baſium contenta.</s>
            <s xml:id="echoid-s22391" xml:space="preserve"> Ducatur enim
              <lb/>
            linea e a in baſi columnæ & in ſuperficie ſecante:</s>
            <s xml:id="echoid-s22392" xml:space="preserve"> hæc eſt ergo ſemi-
              <lb/>
            diameter circuli baſis columnæ.</s>
            <s xml:id="echoid-s22393" xml:space="preserve"> Producatur itaq;</s>
            <s xml:id="echoid-s22394" xml:space="preserve"> taliter, ut linea e g
              <lb/>
            compleat diametrum baſis columnæ, cadetq́;</s>
            <s xml:id="echoid-s22395" xml:space="preserve"> linea e g in ſuperficie
              <lb/>
            plana columnam ſecante.</s>
            <s xml:id="echoid-s22396" xml:space="preserve"> Si enim linea e g nõ eſt ducta in ſuperficie
              <lb/>
            plana columnam ſecante:</s>
            <s xml:id="echoid-s22397" xml:space="preserve"> ducatur linea b e in illa ſuperficie ſecante.</s>
            <s xml:id="echoid-s22398" xml:space="preserve">
              <lb/>
            Lineæ ergo b e & e a ſunt linea una:</s>
            <s xml:id="echoid-s22399" xml:space="preserve"> quoniam ſunt in una ſuperficie
              <lb/>
            productæ ambæ orthogonaliter ſuper axem e f cõtinuè:</s>
            <s xml:id="echoid-s22400" xml:space="preserve"> ſimiliterq́;</s>
            <s xml:id="echoid-s22401" xml:space="preserve">
              <lb/>
            quia linea e g complet diametrum a e, non in ſuperficie ſecante, ſed
              <lb/>
            alia:</s>
            <s xml:id="echoid-s22402" xml:space="preserve"> erit ergo lineæ a g pars in plano, pars in ſublimi:</s>
            <s xml:id="echoid-s22403" xml:space="preserve"> quod eſt con-
              <lb/>
            tra 1 p 11.</s>
            <s xml:id="echoid-s22404" xml:space="preserve"> Palàm itaq;</s>
            <s xml:id="echoid-s22405" xml:space="preserve">, quoniam linea a b eſt diameter baſis, & quòd
              <lb/>
            punctus g cadit ſuper punctum b.</s>
            <s xml:id="echoid-s22406" xml:space="preserve"> Similiterq́;</s>
            <s xml:id="echoid-s22407" xml:space="preserve"> declarandum de linea
              <lb/>
            c d, quoniam eſt diameter alterius baſis.</s>
            <s xml:id="echoid-s22408" xml:space="preserve"> Lineæ quoq;</s>
            <s xml:id="echoid-s22409" xml:space="preserve"> a c & b d ſunt
              <lb/>
            lineæ longitudinis columnæ.</s>
            <s xml:id="echoid-s22410" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s22411" xml:space="preserve"> Ex hoc itaq;</s>
            <s xml:id="echoid-s22412" xml:space="preserve"> pa
              <lb/>
            tet, quoniã cum illa ſectio diuidat per æqualia baſes columnæ, quòd
              <lb/>
            etiam diuidit per æqualia columnam.</s>
            <s xml:id="echoid-s22413" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div809" type="section" level="0" n="0">
          <head xml:id="echoid-head668" xml:space="preserve" style="it">94. Superficiei ſecantis columnam rotundam æquidistanter ſuperficiei per axem ſecanti &
            <lb/>
          ſuperficiei columnaris, cõmunis ſectio eſt rectangulum ſub duab{us} lineis longitudinis columnæ,
            <lb/>
          & duab{us} lineis minorib{us} diametris baſium contentum. È
            <unsure/>
          21 defin. 11 elem.</head>
          <p>
            <s xml:id="echoid-s22414" xml:space="preserve">Sit, ut in præcedenti propoſitione, columna ſecta per planam ſuperficiem ſecundum ſectionem,
              <lb/>
            rectangula a b c d:</s>
            <s xml:id="echoid-s22415" xml:space="preserve"> cuius axis ſit e f:</s>
            <s xml:id="echoid-s22416" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s22417" xml:space="preserve"> nunc ſuperficies plana columnã ſecans, æquidiſtans ſuper-
              <lb/>
            ficiei a b c d, cuius communis ſectio cum ſuperficie columnæ ſit h i k l:</s>
            <s xml:id="echoid-s22418" xml:space="preserve"> ducanturq́;</s>
            <s xml:id="echoid-s22419" xml:space="preserve"> à punctis h & i li
              <lb/>
            neæ perpendiculares ſuper diametrum a b per 12 p 1, quæ ſint h m, i n.</s>
            <s xml:id="echoid-s22420" xml:space="preserve"> Erit itaq;</s>
            <s xml:id="echoid-s22421" xml:space="preserve"> linea m n æqualis li-
              <lb/>
            neæ h i, ut patet per 34 p 1:</s>
            <s xml:id="echoid-s22422" xml:space="preserve"> lineæ enim a b & h i ſunt æquidiſtantes ex hypotheſi, & lineæ h m & i n
              <lb/>
            ſunt æquidiſtantes per 28 p 1.</s>
            <s xml:id="echoid-s22423" xml:space="preserve"> Eſt ergo linea h i minor diametro a b.</s>
            <s xml:id="echoid-s22424" xml:space="preserve"> Similiter quoq;</s>
            <s xml:id="echoid-s22425" xml:space="preserve"> l k minor eſt dia
              <lb/>
            metro c d, ductis perpendicularibus lineis, quæ l o & k p:</s>
            <s xml:id="echoid-s22426" xml:space="preserve"> ſed lineæ h k & i l ſunt lineæ longitudinis
              <lb/>
            columnæ.</s>
            <s xml:id="echoid-s22427" xml:space="preserve"> Patet ergo propoſitum.</s>
            <s xml:id="echoid-s22428" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div810" type="section" level="0" n="0">
          <head xml:id="echoid-head669" xml:space="preserve" style="it">95. Omnis ſuperficies plana contingens pyramidem, uel columnam rotundam: ſecundum li-
            <lb/>
          neam longitudinis eſt contingens.</head>
          <p>
            <s xml:id="echoid-s22429" xml:space="preserve">Non enim ſecundum punctũ contingit ſuperficies plana propoſita corpora ſicut ſphæram:</s>
            <s xml:id="echoid-s22430" xml:space="preserve"> quo-
              <lb/>
            niam in ipſis eſt longitudo, quæ non eſt in ſphæra:</s>
            <s xml:id="echoid-s22431" xml:space="preserve"> ſed nec contingit ipſa ſecundũ ſuperficiem:</s>
            <s xml:id="echoid-s22432" xml:space="preserve"> quo-
              <lb/>
            niam cum in quolibet iſtorum corporũ ſint infiniti circuli ſuis baſibus æquidiſtantes & ipſæ baſes:</s>
            <s xml:id="echoid-s22433" xml:space="preserve">
              <lb/>
            accideret illos ſecundum lineas in ſuperficie plana contingente ductas ad ipſorum contactum, non
              <lb/>
            contingi ſecundum punctũ, ſed ſecari:</s>
            <s xml:id="echoid-s22434" xml:space="preserve"> quod eſt contra 16 p 3, & impoſsibile.</s>
            <s xml:id="echoid-s22435" xml:space="preserve"> Non ergo continget ſu-
              <lb/>
            perficies plana propoſita corpora ſecundũ ſuperficiem.</s>
            <s xml:id="echoid-s22436" xml:space="preserve"> Reſtat ergo,
              <lb/>
              <figure xlink:label="fig-0338-02" xlink:href="fig-0338-02a" number="352">
                <variables xml:id="echoid-variables336" xml:space="preserve">a e d c g b</variables>
              </figure>
            ut ſecundũ lineam contingat.</s>
            <s xml:id="echoid-s22437" xml:space="preserve"> Et quia contingit in pyramide uerti-
              <lb/>
            cem & baſim & in columna ambas baſes:</s>
            <s xml:id="echoid-s22438" xml:space="preserve"> patet, quòd utrunq;</s>
            <s xml:id="echoid-s22439" xml:space="preserve"> illo-
              <lb/>
            rum ſecundum lineas ſuarum longitudinum eſt contingens.</s>
            <s xml:id="echoid-s22440" xml:space="preserve"> Patet
              <lb/>
            ergo propoſitum.</s>
            <s xml:id="echoid-s22441" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div812" type="section" level="0" n="0">
          <head xml:id="echoid-head670" xml:space="preserve" style="it">96. Omnis linea perpendicularis ſuper curuam ſuperficiem py
            <lb/>
          rami dis, uel columnæ rotundæ: neceſſariò trãſit per ipſarũ axem.</head>
          <p>
            <s xml:id="echoid-s22442" xml:space="preserve">Pyramis rotunda uel columna ſit, cuius linea longitudinis ſit a b:</s>
            <s xml:id="echoid-s22443" xml:space="preserve">
              <lb/>
            & eius axis a g:</s>
            <s xml:id="echoid-s22444" xml:space="preserve"> & ſit linea d e perpendicularis ſuper curuam illius ſu
              <lb/>
            perficiẽ.</s>
            <s xml:id="echoid-s22445" xml:space="preserve"> Dico, quòd linea e d tranſit per axem a g.</s>
            <s xml:id="echoid-s22446" xml:space="preserve"> Ducatur enim ſe-
              <lb/>
            midiameter baſis, quæ ſit b g.</s>
            <s xml:id="echoid-s22447" xml:space="preserve"> Quia ergo linea e d eſt perpendicula-
              <lb/>
            ris ſuper curuam ſuperficiem propoſitam:</s>
            <s xml:id="echoid-s22448" xml:space="preserve"> palàm per definitionem,
              <lb/>
            quoniã linea e d eſt perpendiculariter erecta ſuper ſuperficiem con-
              <lb/>
            tingentem pyramidem ſecundum aliquam lineam ſuę longitudinis:</s>
            <s xml:id="echoid-s22449" xml:space="preserve">
              <lb/>
            ſit hoc ſecundum lineam a b.</s>
            <s xml:id="echoid-s22450" xml:space="preserve"> Cadit ergo linea e d ſuper lineam a b.</s>
            <s xml:id="echoid-s22451" xml:space="preserve">
              <lb/>
            Palàm ergo per 2 p 11, quoniam lineę d e & a b ſunt in eadem ſuperfi-
              <lb/>
            cie.</s>
            <s xml:id="echoid-s22452" xml:space="preserve"> Et quia linea d e eſt perpendicularis ſuper curuam ſuperficiem
              <lb/>
            pyramidis:</s>
            <s xml:id="echoid-s22453" xml:space="preserve"> patet, quòd illa ſuperficies erit erecta ſuper ſuperficiem
              <lb/>
            conicam pyramidis, & in ipſa eſt linea a b.</s>
            <s xml:id="echoid-s22454" xml:space="preserve"> Producta ergo transpyra
              <lb/>
            midem, ſecabit ipſam ſecundũ lineam longitudinis a b per æqualia diuidens pyramidem, & tranſi-
              <lb/>
            </s>
          </p>
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