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-div800" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s22342" xml:space="preserve">
              <pb o="35" file="0337" n="337" rhead="LIBER PRIMVS."/>
            tis, axe exiſtente cõmuni, omnes anguli ad centrum b cõſtituti ſunt æquales.</s>
            <s xml:id="echoid-s22343" xml:space="preserve"> Patet ergo propoſitũ.</s>
            <s xml:id="echoid-s22344" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div802" type="section" level="0" n="0">
          <head xml:id="echoid-head664" xml:space="preserve" style="it">90. Omnis ſuperficiei planæ ſecantis pyramidem rotundam uel lateratam ſecundum a-
            <lb/>
          xis longitudinem & ſuperficiei conicæ communis ſectio eſt trigonum duab{us} lineis longitudi-
            <lb/>
          nis pyramid{ιs} & diametro baſis contentũ. Ex quo patet, quoniam illa ſuperficies dιuidit pyra-
            <lb/>
          midem per æqualia: & quòd ſuperficies, quæpyramidem ſecundum lineam longitudinis per æ-
            <lb/>
          qualia ſecuerit, ſecundum axem neceſſariò ſecabit. È
            <unsure/>
          18 defin. 11 element. item 3. theor. 1 Co-
            <lb/>
          nicorum Apollonij.</head>
          <p>
            <s xml:id="echoid-s22345" xml:space="preserve">Eſto pyramis rotunda a b c, cuius uertex a:</s>
            <s xml:id="echoid-s22346" xml:space="preserve"> & diameter baſis b c:</s>
            <s xml:id="echoid-s22347" xml:space="preserve"> & ſit centrum baſis d.</s>
            <s xml:id="echoid-s22348" xml:space="preserve"> Et palàm
              <lb/>
            per pręmiſſam, quoniã linea a d eſt axis illius pyramidis.</s>
            <s xml:id="echoid-s22349" xml:space="preserve"> Superficies
              <lb/>
              <figure xlink:label="fig-0337-01" xlink:href="fig-0337-01a" number="349">
                <variables xml:id="echoid-variables333" xml:space="preserve">a b d c</variables>
              </figure>
            itaq;</s>
            <s xml:id="echoid-s22350" xml:space="preserve"> plana ſecans pyramidem rotundam ſecundum axis longitudi-
              <lb/>
            nem, pertranſit puncta a & d:</s>
            <s xml:id="echoid-s22351" xml:space="preserve"> erit itaq;</s>
            <s xml:id="echoid-s22352" xml:space="preserve"> illa ſuperficies plana orthogo-
              <lb/>
            naliter erecta ſuper baſim pyramidis per 18 p 11.</s>
            <s xml:id="echoid-s22353" xml:space="preserve"> Communis itaq;</s>
            <s xml:id="echoid-s22354" xml:space="preserve"> ſe-
              <lb/>
            ctio baſis pyramidis & illιus ſuperficiei planę eſt linea recta per 3 p 11,
              <lb/>
            quæ eſt diameter baſis:</s>
            <s xml:id="echoid-s22355" xml:space="preserve"> & ſit hæc b c.</s>
            <s xml:id="echoid-s22356" xml:space="preserve"> Trigonũ itaq;</s>
            <s xml:id="echoid-s22357" xml:space="preserve"> a b c eſt in ſuper-
              <lb/>
            ficie ſecante:</s>
            <s xml:id="echoid-s22358" xml:space="preserve"> ſed & idem trigonum eſt in ſuperficie conica pyr mi-
              <lb/>
            dis.</s>
            <s xml:id="echoid-s22359" xml:space="preserve"> Et quoniam trigonum orthogonium b a d eſt illud, ex cuius per-
              <lb/>
            tranſitu deſcribιtur pyramis a b c, & trigonum a b c eſt duplum illi
              <lb/>
            per 1 p 6, patet illud, quod primò proponitur de pyramide rotunda.</s>
            <s xml:id="echoid-s22360" xml:space="preserve">
              <lb/>
            Patet etiam, quòd illa ſuperficies taliter pyramidem ſecans, diuidit
              <lb/>
            ipſam per æqualia:</s>
            <s xml:id="echoid-s22361" xml:space="preserve"> quoniam tranſiens uerticem & concluſa diame-
              <lb/>
            tro, per æqualia diuidit & baſim.</s>
            <s xml:id="echoid-s22362" xml:space="preserve"> In laterata uerò pyramide, aut ſu-
              <lb/>
            perficies plana ſecans tranſit latus aut angulum:</s>
            <s xml:id="echoid-s22363" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s22364" xml:space="preserve"> productis li-
              <lb/>
            neis ad terminum axis pyramidis, illa communis ſectio ſemper trigo
              <lb/>
            nus maior uel minor.</s>
            <s xml:id="echoid-s22365" xml:space="preserve"> Patet ergo propoſitum:</s>
            <s xml:id="echoid-s22366" xml:space="preserve"> quoniam & conuerſa
              <lb/>
            per ſe & ex præmιſsis patet.</s>
            <s xml:id="echoid-s22367" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div804" type="section" level="0" n="0">
          <head xml:id="echoid-head665" xml:space="preserve" style="it">91. Omnis pyramidis rotundæ uel lateratæ lineæ lõgitudinis ſu
            <lb/>
          per axem in uertice tantùm ſe interſecant: productæ quo aliam
            <lb/>
          ſimilem pyramidem principiant, cui{us} lineæ longitudinis ſecun-
            <lb/>
          dum poſitionem & ſitum priori pyramidi modo contrario ſe habent. È
            <unsure/>
          18 defin. 11 elemen. item
            <lb/>
          1 defin. 1 Conicorum Apollonij.</head>
          <p>
            <s xml:id="echoid-s22368" xml:space="preserve">Quòd omnes lineę longitudinis pyramidis cuiuſcunq;</s>
            <s xml:id="echoid-s22369" xml:space="preserve"> prod ctę ſe ſuper axem in uertice ſecent,
              <lb/>
            euidens eſt:</s>
            <s xml:id="echoid-s22370" xml:space="preserve"> quonιam concurrunt omnes in illo puncto uerticis.</s>
            <s xml:id="echoid-s22371" xml:space="preserve"> Et quonιam omnes ſunt æquales
              <lb/>
            per 89 huius:</s>
            <s xml:id="echoid-s22372" xml:space="preserve"> patet, quia citra uerticem nulla ipſarum aliam interſe-
              <lb/>
              <figure xlink:label="fig-0337-02" xlink:href="fig-0337-02a" number="350">
                <variables xml:id="echoid-variables334" xml:space="preserve">d e a b c</variables>
              </figure>
            cat.</s>
            <s xml:id="echoid-s22373" xml:space="preserve"> Quòd etiam product æ aliam pyramιdem priori ſimilem princi-
              <lb/>
            pient, patet.</s>
            <s xml:id="echoid-s22374" xml:space="preserve"> Secet enιm ſuperficies plana pyramidem ſecundũ axis
              <lb/>
            longitudinem:</s>
            <s xml:id="echoid-s22375" xml:space="preserve"> erit ergo per præcedentem communis ſectio iſtius
              <lb/>
            ſuperficiei & ſuperficiei conicę pyramidis, trigonum æquum duplo
              <lb/>
            trigoni rectanguli pyramidem cauſſantis:</s>
            <s xml:id="echoid-s22376" xml:space="preserve"> ſed palàm per 36 huius,
              <lb/>
            quòd latera cuiuslibet trigoni producta principiant alium trigonũ
              <lb/>
            priori ſimile, cuius latera poſitionem & ſitum prioris trigoni lateri-
              <lb/>
            bus contrarium habent.</s>
            <s xml:id="echoid-s22377" xml:space="preserve"> Et quoniam tot poſſunt imaginari planæ ſu
              <lb/>
            perficies trans axem pyramidem ſecantes, quot ſunt lineæ longitu-
              <lb/>
            dinis imaginabiles in medietate pyramidis, pater, quoniam omnes
              <lb/>
            lineæ longitudinis productæ, principiant aliam pyramidem priori
              <lb/>
            ſimilem, lineis longitudinis à dextro prioris prodeuntibus in ſini-
              <lb/>
            ſtrum poſterioris, & à ſiniſtro prioris in dextrũ poſterioris, & è con-
              <lb/>
            uerſo.</s>
            <s xml:id="echoid-s22378" xml:space="preserve"> Patet ergo propoſitum.</s>
            <s xml:id="echoid-s22379" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div806" type="section" level="0" n="0">
          <head xml:id="echoid-head666" xml:space="preserve" style="it">92. Omnes lineæ longitudinis uni{us} columnæ rotundæ ſunt æ-
            <lb/>
          quales, rectos angulos cum ſemidiametris ſuarum baſium conti-
            <lb/>
          nentes, & in eadem ſuperficie cum axe exiſtentes. Ex quo patet,
            <lb/>
          quoniam axis cui{us}lιbet columnæ rotundæ centris ſuaru baſium
            <lb/>
          orthogonaliter inſiſtit. È
            <unsure/>
          21 defin. 11 element.</head>
          <p>
            <s xml:id="echoid-s22380" xml:space="preserve">Hoc non indiget demonſtratione alia, niſi ſimili illi, quæ fit in 89 huius.</s>
            <s xml:id="echoid-s22381" xml:space="preserve"> Sicut enim trigonum
              <lb/>
            orthogonium altero laterum rectum angulum continentium fixo, per reuolutionem ſuam cauſ-
              <lb/>
            ſat pyramidem rotundum:</s>
            <s xml:id="echoid-s22382" xml:space="preserve"> ſic quadrilaterum rectangulum altero ſuorum laterum fixo manente,
              <lb/>
            alijs tribus, quouſque ad locum ſuum redeant, circumductis, cauſſat motu ſuo figuram colu-
              <lb/>
            mnarem rotundam.</s>
            <s xml:id="echoid-s22383" xml:space="preserve"> fiet ergo probatio omnium eorum, quæ proponunttur hîc, ut in illa:</s>
            <s xml:id="echoid-s22384" xml:space="preserve"> quia pa-
              <lb/>
            tet totum euidenter.</s>
            <s xml:id="echoid-s22385" xml:space="preserve"/>
          </p>
        </div>
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    </echo>
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