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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[161] l u r c z o d t m g b n k q f a s p x e s
Page: 202
[162] d t e h s n q b l q m f p a g
Page: 203
[163] e c h m z b d a
Page: 204
[164] e n c z b d g a
Page: 204
[165] c h z b d g a
Page: 205
[166] b e a d h z m g
Page: 205
[167] p o b c e l m t n a q k f d g
Page: 206
[168] b d a e h t z g f
Page: 206
[169] e b f a d m h t z g
Page: 207
[170] q e a b d m h z
Page: 207
[171] l k x s y e t q b a f u r m h o m z g p d
Page: 207
[172] ſ k x b a s t c q f m o h z i g p d
Page: 208
[173] d a b e h z g
Page: 208
[174] d a b e h z g
Page: 209
[175] a d b b g
Page: 209
[176] a d f b ſ m e c z g
Page: 209
[177] h e m c u t s k o b z ſ q r f g a d
Page: 210
[178] h e m c u s t b o q z r f g a d
Page: 210
[179] i h e m c t z u s b o k q r f g a d
Page: 210
[180] n q e ſ g t f m o K d h c a s u p z b
Page: 211
[181] t n q z g m b ſ f h r a d e k o
Page: 213
[182] t i y n q g z x m b c ſ f h s r a d p e k o u
Page: 213
[183] f d b g t e h e
Page: 214
[184] e c s ſ o f i g m b k z d t q p h y n r u a x
Page: 215
[185] CIN EMATH EQUE FRANCAISE BIBLIOTHEQUE MUSEE
Page: 215
[186] a e t o f z h g d j c p k b q r
Page: 216
[187] a o u m h z t s n d ſ e q f p
Page: 217
[188] a o u p m h z t x b n y c q s l d g e K f r
Page: 218
[189] f u q b m t n e o z a
Page: 221
[190] f q b u g m c n K p a
Page: 221
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          <p>
            <s xml:id="echoid-s10069" xml:space="preserve">
              <pb o="158" file="0164" n="164" rhead="ALHAZEN"/>
            al:</s>
            <s xml:id="echoid-s10070" xml:space="preserve"> ſed ſecãt ſe ſuper pũctũ b:</s>
            <s xml:id="echoid-s10071" xml:space="preserve"> [quia uiſibile eſt in qualibet reflexionis ſuperficie ք 23 n 4] qđ eſt im-
              <lb/>
            poſsibile.</s>
            <s xml:id="echoid-s10072" xml:space="preserve"> Quoniã b nõ eſt in linea a l:</s>
            <s xml:id="echoid-s10073" xml:space="preserve"> qđ patet ք hoc:</s>
            <s xml:id="echoid-s10074" xml:space="preserve"> quoniã fl æquidiſtat b m.</s>
            <s xml:id="echoid-s10075" xml:space="preserve"> [ut patu it proximo
              <lb/>
            numero per fabricationẽ & 30 p 1.</s>
            <s xml:id="echoid-s10076" xml:space="preserve">] Reſtat ergo, ut à nullo pũcto lineæ e g, pręterquã à g, poſsit re-
              <lb/>
            flecti b a d a.</s>
            <s xml:id="echoid-s10077" xml:space="preserve"> Si aũt ab aliquo pũcto extra lineã e g:</s>
            <s xml:id="echoid-s10078" xml:space="preserve"> ſit illud u:</s>
            <s xml:id="echoid-s10079" xml:space="preserve"> & ducatur linea lõgitudinis e u o:</s>
            <s xml:id="echoid-s10080" xml:space="preserve"> & ſu-
              <lb/>
            matur ſuperficies æquidiſtãs baſi, trãſiẽs ք pũctũ u.</s>
            <s xml:id="echoid-s10081" xml:space="preserve"> [ut dictũ eſt proximo numero.</s>
            <s xml:id="echoid-s10082" xml:space="preserve">] Palã, quòd a n
              <lb/>
            ſecabit hãc ſuperficiẽ:</s>
            <s xml:id="echoid-s10083" xml:space="preserve"> [quia e g parallela ipſi a n, eandẽ ſecat] ſit punctũ ſectionis y.</s>
            <s xml:id="echoid-s10084" xml:space="preserve"> Similiter b m ſe-
              <lb/>
            cabit eandẽ:</s>
            <s xml:id="echoid-s10085" xml:space="preserve"> ſit punctũ ſectionis k:</s>
            <s xml:id="echoid-s10086" xml:space="preserve"> & ducãtur lineæ k u, y
              <lb/>
              <figure xlink:label="fig-0164-01" xlink:href="fig-0164-01a" number="90">
                <variables xml:id="echoid-variables80" xml:space="preserve">l d a e f
                  <gap/>
                x u y t k p r c z o h g M n q m i b s</variables>
              </figure>
            u, y k.</s>
            <s xml:id="echoid-s10087" xml:space="preserve"> Et cũ ſuperficies illa ſecet pyramidẽ ſuper circulũ,
              <lb/>
            trãſeuntẽ per u [per 4 th 1 coni.</s>
            <s xml:id="echoid-s10088" xml:space="preserve"> Apol.</s>
            <s xml:id="echoid-s10089" xml:space="preserve">] ducatur à pũcto u
              <lb/>
            linea ad cẽtrũ huius circuli, quę extra circulũ ꝓducta, ſit
              <lb/>
            r u:</s>
            <s xml:id="echoid-s10090" xml:space="preserve"> & ducãtur lineę e k, e y:</s>
            <s xml:id="echoid-s10091" xml:space="preserve"> quę quidẽ ſecabũt ſuperficiẽ
              <lb/>
            circuli p g:</s>
            <s xml:id="echoid-s10092" xml:space="preserve"> [quia ſecãt circulũ ipſi parallelũ, per u trãſeun
              <lb/>
            tẽ] & ſint pũcta ſectionũ s, i:</s>
            <s xml:id="echoid-s10093" xml:space="preserve"> & ducãtur lineæ i c, s c.</s>
            <s xml:id="echoid-s10094" xml:space="preserve"> Sicut
              <lb/>
            igitur probatũ eſt [proximo numero] de pũcto m:</s>
            <s xml:id="echoid-s10095" xml:space="preserve"> quòd,
              <lb/>
            nõ impediente pyramide, poteſt reflecti ad n à pũcto g:</s>
            <s xml:id="echoid-s10096" xml:space="preserve">
              <lb/>
            ita ꝓbabitur de pũcto k:</s>
            <s xml:id="echoid-s10097" xml:space="preserve"> qđ poteſt reflecti à puncto u ad
              <lb/>
            punctũ y:</s>
            <s xml:id="echoid-s10098" xml:space="preserve"> & eadẽ eſt ꝓbatio:</s>
            <s xml:id="echoid-s10099" xml:space="preserve"> & ita angulus r u y erit ęqua
              <lb/>
            lis angulo r u k [per 12 n 4.</s>
            <s xml:id="echoid-s10100" xml:space="preserve">] Palàm, quoniã b k eſt æquidi
              <lb/>
            ſtãs e g:</s>
            <s xml:id="echoid-s10101" xml:space="preserve"> [Nã b m parallela ipſi e g ք fabricationẽ, cõtinua-
              <lb/>
            ta eſt in pũctũ k] & linea, cõmunis ſuքficiei b g e k, & ſuք-
              <lb/>
            ficiei circùli p g, eſt linea m g.</s>
            <s xml:id="echoid-s10102" xml:space="preserve"> Igitur linea e k cũ ſit in hac
              <lb/>
            ſuperficie, & ſecet ſuperficiẽ circuli p g:</s>
            <s xml:id="echoid-s10103" xml:space="preserve"> [in pũcto s, ut pa
              <lb/>
            tuit] cadet ſuper lineã cõmunẽ, quę eſt m g.</s>
            <s xml:id="echoid-s10104" xml:space="preserve"> Erit igitur s
              <lb/>
            m g linea recta.</s>
            <s xml:id="echoid-s10105" xml:space="preserve"> Eodẽ modo cũ ſuperficies n y e g ſecet ſu
              <lb/>
            perficiẽ circuli p g, ſuper lineã n g:</s>
            <s xml:id="echoid-s10106" xml:space="preserve"> linea e y cõcurret cũ li
              <lb/>
            nea n g.</s>
            <s xml:id="echoid-s10107" xml:space="preserve"> [in pũcto i, ut patuit.</s>
            <s xml:id="echoid-s10108" xml:space="preserve">] Igitur i n g linea eſt recta.</s>
            <s xml:id="echoid-s10109" xml:space="preserve">
              <lb/>
            Palã etiã, quòd ſuքficies i e c ſecat ſuperficiẽ circuli p g,
              <lb/>
            ſuper lineã i c, & ſecat ſuperficiẽ huic æquidiſtãtem, quæ
              <lb/>
            trãſit ք u, ſuper lineã y u.</s>
            <s xml:id="echoid-s10110" xml:space="preserve"> Ergo [per 16 p 11] y u æquidiſtat
              <lb/>
            i c.</s>
            <s xml:id="echoid-s10111" xml:space="preserve"> Similiter ſuperficies s e c ſecat ſuperficies illas æqui-
              <lb/>
            diſtãtes, ſuper duas lineas s c, k u.</s>
            <s xml:id="echoid-s10112" xml:space="preserve"> Ergo [per 16 p 11] s c ę-
              <lb/>
            quidiſtat k u.</s>
            <s xml:id="echoid-s10113" xml:space="preserve"> Similiter ſi ſumatur ſuperficies, ſecãs ſpecu
              <lb/>
            lũ ſuper lineã lõgitudinis e c, in qua ſuքficie ſuntru, c M:</s>
            <s xml:id="echoid-s10114" xml:space="preserve">
              <lb/>
            ſecabit illas ſuքficies æquidiſtãtes [nẽpe circulos ք u & c eductos] ſuք duas lineas M c, r u.</s>
            <s xml:id="echoid-s10115" xml:space="preserve"> Igitur [ք
              <lb/>
            16 p 11] hę duę lineæ ſunt æquidiſtãtes.</s>
            <s xml:id="echoid-s10116" xml:space="preserve"> Igitur angulus s c M æqualis eſt angulo k u r, & angulus M c
              <lb/>
            i æqualis angulo r u y.</s>
            <s xml:id="echoid-s10117" xml:space="preserve"> [ք 10 p 11.</s>
            <s xml:id="echoid-s10118" xml:space="preserve">] Sed iã patuit, qđ angulus k u r æqualis eſt r u y.</s>
            <s xml:id="echoid-s10119" xml:space="preserve"> Igitur [ք 1 ax.</s>
            <s xml:id="echoid-s10120" xml:space="preserve">] an-
              <lb/>
            gulus s c M æqualis eſt angulo M c i.</s>
            <s xml:id="echoid-s10121" xml:space="preserve"> Quare pũctũ s poteſt reflecti ad i à puncto c, nõ impediente py
              <lb/>
            ramide:</s>
            <s xml:id="echoid-s10122" xml:space="preserve"> ſed iã probatũ eſt [proximo numero] qđ punctũ m reflecti põt ad i à pũcto g.</s>
            <s xml:id="echoid-s10123" xml:space="preserve"> [cadunt.</s>
            <s xml:id="echoid-s10124" xml:space="preserve"> n.</s>
            <s xml:id="echoid-s10125" xml:space="preserve">
              <lb/>
            pũcta i, n, g in eandẽ rectã lineã, ut mõſtratũ eſt.</s>
            <s xml:id="echoid-s10126" xml:space="preserve">] Igitur punctũ s reflectitur ad i à duob.</s>
            <s xml:id="echoid-s10127" xml:space="preserve"> punctis cir-
              <lb/>
            culi p g.</s>
            <s xml:id="echoid-s10128" xml:space="preserve"> [nimirũ g & c] qđ eſt impoſsibile [& cõtra 51 n 4.</s>
            <s xml:id="echoid-s10129" xml:space="preserve"> 29.</s>
            <s xml:id="echoid-s10130" xml:space="preserve"> 46 n.</s>
            <s xml:id="echoid-s10131" xml:space="preserve">] Reſtat ergo, ut primũ ſit impoſ
              <lb/>
            ſibile, ſcilicet, ut punctũ b reflectatur ad a ab aliquo puncto alio ſpeculi, ꝗ̃ à g.</s>
            <s xml:id="echoid-s10132" xml:space="preserve"> Quod eſt propoſitũ.</s>
            <s xml:id="echoid-s10133" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div369" type="section" level="0" n="0">
          <head xml:id="echoid-head357" xml:space="preserve" style="it">54. Viſu & uiſibili inter baſim ſpeculi conici conuexi, & planum per uerticem ductum, ba-
            <lb/>
          ſi́ parallelum poſitis: punctum reflexionis inuenire. 35 p 7.</head>
          <p>
            <s xml:id="echoid-s10134" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s10135" xml:space="preserve"> dato ſpeculo pyramidali:</s>
            <s xml:id="echoid-s10136" xml:space="preserve"> eſt inuenire punctũ reflexionis.</s>
            <s xml:id="echoid-s10137" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s10138" xml:space="preserve"> ſit g uertex
              <lb/>
            pyramidalis ſpeculi:</s>
            <s xml:id="echoid-s10139" xml:space="preserve"> & ſuper ipſum fiat ſuperficies æquidiſtãs baſi pyramidis:</s>
            <s xml:id="echoid-s10140" xml:space="preserve"> [ut oſtenfum
              <lb/>
            eſt 52 n] quę ſit m n g:</s>
            <s xml:id="echoid-s10141" xml:space="preserve"> a ſit pũctũ uiſum:</s>
            <s xml:id="echoid-s10142" xml:space="preserve"> b cẽtrũ uiſus.</s>
            <s xml:id="echoid-s10143" xml:space="preserve"> A & b aut erũt citra illã ſuperficiẽ:</s>
            <s xml:id="echoid-s10144" xml:space="preserve"> aut
              <lb/>
            ultra:</s>
            <s xml:id="echoid-s10145" xml:space="preserve"> aut in ipſa ſuperficie:</s>
            <s xml:id="echoid-s10146" xml:space="preserve"> aut unũ citra, aliud ultra:</s>
            <s xml:id="echoid-s10147" xml:space="preserve"> aut unũ in ſuperficie, aliud citra uel ultra.</s>
            <s xml:id="echoid-s10148" xml:space="preserve"> Sint
              <lb/>
            citra ſuperficiẽ:</s>
            <s xml:id="echoid-s10149" xml:space="preserve"> & à puncto a ducatur ſuperficies, ſecãs pyramidẽ ęquidiſtãter baſi:</s>
            <s xml:id="echoid-s10150" xml:space="preserve"> & ducatur à pun
              <lb/>
            cto g linea ad punctũ b:</s>
            <s xml:id="echoid-s10151" xml:space="preserve"> quę ꝓducta cadet in ſuperficiẽ ab a ductã, cũ ſit inter ſuperficies æquidiſtã-
              <lb/>
            tes:</s>
            <s xml:id="echoid-s10152" xml:space="preserve"> [quarũ una ք uerticẽ, altera ք uiſibile a ducitur] punctũ, in qđ cadit hęc linea, ſit h.</s>
            <s xml:id="echoid-s10153" xml:space="preserve"> Probatur
              <lb/>
            aũt modo ſuprà dicto [52 n] qđ a reflectitur ad h ab aliquo pũcto circuli, quẽ efficit ſuperficies, ſe-
              <lb/>
            cãs pyramidẽ, ducta à pũctis a, h:</s>
            <s xml:id="echoid-s10154" xml:space="preserve"> & inueniatur in circulo illo punctũ reflexionis:</s>
            <s xml:id="echoid-s10155" xml:space="preserve"> [ք 31 uel 39 n] &
              <lb/>
            ſit e:</s>
            <s xml:id="echoid-s10156" xml:space="preserve"> & ducatur linea a b:</s>
            <s xml:id="echoid-s10157" xml:space="preserve"> & linea lõgitudinis pyramidis g e:</s>
            <s xml:id="echoid-s10158" xml:space="preserve"> & axis pyramidis g t:</s>
            <s xml:id="echoid-s10159" xml:space="preserve"> & ducatur à pun-
              <lb/>
            cto e linea ad centrũ circuli:</s>
            <s xml:id="echoid-s10160" xml:space="preserve"> quę quidẽ cadet ſuper axem:</s>
            <s xml:id="echoid-s10161" xml:space="preserve"> [ք 4 th 1 coni.</s>
            <s xml:id="echoid-s10162" xml:space="preserve"> Apol.</s>
            <s xml:id="echoid-s10163" xml:space="preserve"> quia cẽtrũ circuli eſt
              <lb/>
            in axe] & ſit e t:</s>
            <s xml:id="echoid-s10164" xml:space="preserve"> & erit [ք 18 p 3] orthogonalis ſuper lineã, cõtingentẽ circulũ illũ in pũcto e:</s>
            <s xml:id="echoid-s10165" xml:space="preserve"> & du-
              <lb/>
            ctis lineis a e, h e:</s>
            <s xml:id="echoid-s10166" xml:space="preserve"> ſecabit angulũ earũ ք æqualia:</s>
            <s xml:id="echoid-s10167" xml:space="preserve"> [ut oſtẽſum eſt 13 n 4] & diuidet lineã a h:</s>
            <s xml:id="echoid-s10168" xml:space="preserve"> [քa ſecat
              <lb/>
            angulũ ipſi ſubtẽſum:</s>
            <s xml:id="echoid-s10169" xml:space="preserve"> ſunt enim e h, e a, h a in eadẽ reflexionis ſuperficie ք 23 n 4] ſit pũctũ diuiſio-
              <lb/>
            nis r.</s>
            <s xml:id="echoid-s10170" xml:space="preserve"> Palàm, quoniã g e, e t efficiũt ſuperficiẽ, ſecantẽ lineã a b:</s>
            <s xml:id="echoid-s10171" xml:space="preserve"> ſit pũctũ ſectionis f:</s>
            <s xml:id="echoid-s10172" xml:space="preserve"> & à pũcto f duca-
              <lb/>
            tur perpẽdicularis ſuper lineã g e [ք 12 p 1] & ſit f q:</s>
            <s xml:id="echoid-s10173" xml:space="preserve"> quę quidẽ erit orthogonalis ſuper ſuperficiẽ,
              <lb/>
            cõtingẽtẽ pyramidẽ ſuper lineã g e.</s>
            <s xml:id="echoid-s10174" xml:space="preserve"> [quia enim f q perpẽdicularis eſt duab.</s>
            <s xml:id="echoid-s10175" xml:space="preserve"> rectis interſectis, in cõ-
              <lb/>
            muni ipſarũ ſectione (qđ eſt punctũ q) lateri nẽpe conico e g ք fabricationẽ proximã, & rectę pe-
              <lb/>
            ripheriã circuli ք punctũ q deſcripti, in extrema diametro tãgenti ք 18 p 3:</s>
            <s xml:id="echoid-s10176" xml:space="preserve"> erit perpẽdicularis plano
              <lb/>
            ք ipſas ducto ք 4 p 11, id eſt plano in latere conũ tãgente per 35 n 4.</s>
            <s xml:id="echoid-s10177" xml:space="preserve">] Deinde à pũcto a ducatur ęqui
              <lb/>
            diſtãs lineæ f q:</s>
            <s xml:id="echoid-s10178" xml:space="preserve"> & ſit a l:</s>
            <s xml:id="echoid-s10179" xml:space="preserve"> f q aũt cõcurrat cũ axe in pũcto k:</s>
            <s xml:id="echoid-s10180" xml:space="preserve"> [qđ enim cõcurrat, patet ք 11 ax.</s>
            <s xml:id="echoid-s10181" xml:space="preserve"> quia per
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
Impressum