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

Table of figures

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[341] a c e h d b
Page: 332
[342] l h g b e c k a d f
Page: 334
[343] f c c l a
Page: 334
[344] b c a
Page: 335
[345] c f b e d a
Page: 335
[346] a b d c
Page: 335
[347] a b d c e
Page: 336
[348] a d b c
Page: 336
[349] a b d c
Page: 337
[350] d e a b c
Page: 337
[351] g a m e n b h i c p f o d k l
Page: 338
[352] a e d c g b
Page: 338
[353] d f f f g g b h h d c h e e c
Page: 339
[354] a e h f g b d c
Page: 340
[355] a f e g h b d c
Page: 341
[356] a k f l e m h g b d c
Page: 341
[357] a l f e h k g b d c
Page: 342
[358] d a b c
Page: 342
[359] a x e i b g d h c k f o l n m p
Page: 342
[360] a g g e b d c f
Page: 343
[361] a k b d c
Page: 343
[362] f e h g
Page: 343
[363] a m k n b d c
Page: 344
[364] f o l p p h g
Page: 344
[365] a e t g o f z h d c p y k b r q
Page: 345
[366] n q e t o l g f m d K d h c a s u p z b
Page: 346
[367] e b h a f c l m k d g
Page: 347
[368] b d a c e f g
Page: 348
[369] a b c d e f
Page: 348
[370] a h b z d g
Page: 348
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          <p>
            <s xml:id="echoid-s22669" xml:space="preserve">
              <pb o="40" file="0342" n="342" rhead="VITELLONIS OPTICAE"/>
            neæ in ipſis productę, ad unum communem terminum copulantur,
              <lb/>
              <figure xlink:label="fig-0342-01" xlink:href="fig-0342-01a" number="357">
                <variables xml:id="echoid-variables341" xml:space="preserve">a l f e h k g b d c</variables>
              </figure>
            & in illo ſe interſecant, ut in puncto e.</s>
            <s xml:id="echoid-s22670" xml:space="preserve"> Quòd enim illarum ſuperficie
              <lb/>
            rum communis ſectio ſit linea recta, patet per 3 p 11:</s>
            <s xml:id="echoid-s22671" xml:space="preserve"> quòd autem illa
              <lb/>
            linea (quæ eſt illarum linearum communis ſectio) ſit orthogonalis
              <lb/>
            ſuper axem pyramidis, qui eſt a d:</s>
            <s xml:id="echoid-s22672" xml:space="preserve"> patet:</s>
            <s xml:id="echoid-s22673" xml:space="preserve"> quoniam per 14 p 11 axis a d
              <lb/>
            eſt քpendicularis ſuper baſim pyramidis & ſuք ſuperficiẽ f g h:</s>
            <s xml:id="echoid-s22674" xml:space="preserve"> quo-
              <lb/>
            niam illæ ſuperficies ſunt ex hypotheſi æquidiſtantes.</s>
            <s xml:id="echoid-s22675" xml:space="preserve"> Ergo per defi
              <lb/>
            nitionem lineæ ſuper ſuperficiem erectæ, omnis linea ducta à pun-
              <lb/>
            cto axis e in ſuperficie f g h eſt perpendicularis ſuper axem a d.</s>
            <s xml:id="echoid-s22676" xml:space="preserve"> Li-
              <lb/>
            nea uerò, quę eſt communis ſectio iſtarum ſuperficierũ ſecantium,
              <lb/>
            neceſſariò cadit in ſuperficie f g h:</s>
            <s xml:id="echoid-s22677" xml:space="preserve"> alioquin nõ eſſet cõmunis ſectio.</s>
            <s xml:id="echoid-s22678" xml:space="preserve">
              <lb/>
            Palàm ergo propoſitum primum:</s>
            <s xml:id="echoid-s22679" xml:space="preserve"> quoniam communis ſectio ſuper-
              <lb/>
            ficierum taliter, ut proponitur, pyramidem ſecantium, eſt orthogo-
              <lb/>
            nalis ſuper axem pyramidis.</s>
            <s xml:id="echoid-s22680" xml:space="preserve"> Et eodem modo demonſtrando, idem
              <lb/>
            patet in columnis rotundis.</s>
            <s xml:id="echoid-s22681" xml:space="preserve"> Ex quo patet & corollarium:</s>
            <s xml:id="echoid-s22682" xml:space="preserve"> quoniam
              <lb/>
            communis ſectio talium ſuperficierum eſt linea recta.</s>
            <s xml:id="echoid-s22683" xml:space="preserve"> In duobus au-
              <lb/>
            tem tantùm punctis, qui ſunt termini illius lineæ, fiet interſectio il-
              <lb/>
            larum ſectionum, quamuis in pluribus punctis hoc ſit fieri poſsibi-
              <lb/>
            le, cum ſe interſecant in eadẽ plana ſuperficie.</s>
            <s xml:id="echoid-s22684" xml:space="preserve"> Patet ergo propoſitũ.</s>
            <s xml:id="echoid-s22685" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div826" type="section" level="0" n="0">
          <head xml:id="echoid-head679" xml:space="preserve" style="it">105. Ex aliquo puncto baſis peripheriæ columnæ rotundæ ſemicirculo in ſuperficie cõuexa uel
            <lb/>
          cõcaua columnari circumducto: neceſſe eſt lineam ſemicirculum il-
            <lb/>
          lum per æqualia diuidentem ad ſuperficiem baſis erect am eſſe.</head>
          <figure number="358">
            <variables xml:id="echoid-variables342" xml:space="preserve">d a b c</variables>
          </figure>
          <p>
            <s xml:id="echoid-s22686" xml:space="preserve">Sit, ut ex aliquo puncto peripheriæ baſis colũnæ rotundę, qđ ſit a,
              <lb/>
            circumducatur ſemicirculus in ſuperficie columnæ concaua uel con
              <lb/>
            uexa, qui ſit b c d, & eius centrum erit punctum a:</s>
            <s xml:id="echoid-s22687" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s22688" xml:space="preserve"> ita, ut linea a d
              <lb/>
            diuidat illum ſemicirculum per æqualia in puncto d.</s>
            <s xml:id="echoid-s22689" xml:space="preserve"> Dico, quòd linea
              <lb/>
            a d eſt erecta ſuper ſuperficiem baſis columnę.</s>
            <s xml:id="echoid-s22690" xml:space="preserve"> Quoniam enim arcus
              <lb/>
            d b eſt æqualis arcui d c:</s>
            <s xml:id="echoid-s22691" xml:space="preserve"> patet, quòd angulus d a b eſt æqualis angulo
              <lb/>
            d a c per 27 p 3.</s>
            <s xml:id="echoid-s22692" xml:space="preserve"> Eſt igitur linea a d pars unius linearũ longitudinis co-
              <lb/>
            lũnę.</s>
            <s xml:id="echoid-s22693" xml:space="preserve"> Eſt ergo erecta ſuper baſim per 92 huius.</s>
            <s xml:id="echoid-s22694" xml:space="preserve"> Patet ergo propoſitũ.</s>
            <s xml:id="echoid-s22695" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div827" type="section" level="0" n="0">
          <head xml:id="echoid-head680" xml:space="preserve" style="it">106. Datæ pyramidirotundæ pyramidem eiuſdem uel diuerſæ al
            <lb/>
          titudinis inſcribere. Ex quo patet inſcriptæ angulum ad baſim, an-
            <lb/>
          gulo circumſcribentis maiorẽ eſse: & ſi inſcripta pyramis ad aliam
            <lb/>
          baſim priori baſi æquidiſtantem producatur, anguli productæ ad
            <lb/>
          baſim, angulis datæ pyramidis maiores erunt: & quantum-
            <lb/>
          cun anguli ad baſim augment antur, tantum anguli ad uerti
            <lb/>
          cem minuuntur.</head>
          <figure number="359">
            <variables xml:id="echoid-variables343" xml:space="preserve">a x e i b g d h c k f o l n m p</variables>
          </figure>
          <p>
            <s xml:id="echoid-s22696" xml:space="preserve">Eſto exempli gratia, ut pyramis, cui alia eiuſdem altitudinis de-
              <lb/>
            bet inſcribi, ſit orthogonia, & ſit a b, a c, a e, a f lineis ſuæ longit udi
              <lb/>
            nis ſignata:</s>
            <s xml:id="echoid-s22697" xml:space="preserve"> & axis eius ſit a d:</s>
            <s xml:id="echoid-s22698" xml:space="preserve"> abſcindatur itaq;</s>
            <s xml:id="echoid-s22699" xml:space="preserve"> ſemidiameter ba-
              <lb/>
            ſis, quæ eſt d c, ut libuerit, & ſit abſciſſa in puncto h:</s>
            <s xml:id="echoid-s22700" xml:space="preserve"> producaturq́;</s>
            <s xml:id="echoid-s22701" xml:space="preserve">
              <lb/>
            linea a h, & habetur triãgulus a d h, cuius latera a h, d h latere a d fi-
              <lb/>
            xo manente, reuoluantur ad locũ, unde moueri incœperũt, ꝓue-
              <lb/>
            nietq́;</s>
            <s xml:id="echoid-s22702" xml:space="preserve"> pyramisa g h i k, cuius axis a d.</s>
            <s xml:id="echoid-s22703" xml:space="preserve"> Et ſic poteſt fieri inſcriptio
              <lb/>
            ad quodcũq;</s>
            <s xml:id="echoid-s22704" xml:space="preserve"> punctũ lineæ d c.</s>
            <s xml:id="echoid-s22705" xml:space="preserve"> Et hoc eſt, qđ ꝓponebatur primũ.</s>
            <s xml:id="echoid-s22706" xml:space="preserve">
              <lb/>
            Quod ſi diuerſę altitudinis pyramidẽ ad baſim cõmunẽ inſcribere
              <lb/>
            placuerit ſimilem priori datæ:</s>
            <s xml:id="echoid-s22707" xml:space="preserve"> ſignato puncto, ubi uolueris, in li-
              <lb/>
            nea axis a d, uel extra:</s>
            <s xml:id="echoid-s22708" xml:space="preserve"> tum intra corpus pyramidis, quod ſit x, pro-
              <lb/>
            ducantur lineæ à puncto x ad totam peripheriam, ut x b, x c, x e, x
              <lb/>
            f.</s>
            <s xml:id="echoid-s22709" xml:space="preserve"> Et patet propoſitum.</s>
            <s xml:id="echoid-s22710" xml:space="preserve"> Similiter erit faciendum, ſi quis inſcribere
              <lb/>
            uoluerit pyramidem ad baſim minorem baſi pyramidis datæ.</s>
            <s xml:id="echoid-s22711" xml:space="preserve"> Pa-
              <lb/>
            tet autem ex præmiſsis, cum omnes anguli cuiuſcunq;</s>
            <s xml:id="echoid-s22712" xml:space="preserve"> pyramidis
              <lb/>
            ad baſim ſint æquales per 89 huius, quoniã ex motu anguli unius
              <lb/>
            trianguli, omnes illi anguli cauſſantur:</s>
            <s xml:id="echoid-s22713" xml:space="preserve"> palàm, quòd quicquid in
              <lb/>
            triangulo cauſſante maiorem pyramidem reſpectu trianguli cauſ-
              <lb/>
            ſantis minorem pyramidem proueniet, in omnibus ſimilibus &
              <lb/>
            æqualibus triangulis maioris pyramidis ad ſimiles triangulos mi
              <lb/>
            noris prouenire neceſſe eſt.</s>
            <s xml:id="echoid-s22714" xml:space="preserve"> Cum ergo in triangulo d h a angulus
              <lb/>
            a h d ſit per 16 p 1 maior angulo a c d trianguli d c a:</s>
            <s xml:id="echoid-s22715" xml:space="preserve"> quoniã eſt ex-
              <lb/>
            trinſecus:</s>
            <s xml:id="echoid-s22716" xml:space="preserve"> patet, quòd omnes anguli pyramidis a g h i k ad baſim
              <lb/>
            </s>
          </p>
        </div>
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