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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[151] a g e u m q d o n z h p l
[152] a e u g d o p h q n k z i s t f
[153] f f e a z b h d g
[154] a f b m k q n e t h d z
[155] b a e p g d
[156] a b h z e p g d
[157] o z l h m n q t d a b e
[158] z i l m h n t d z a k g y c f b z r s u p a e x
[159] i u r c z h t m g b n q f a
[160] i u r k c z l b d t m g n q f a
[161] l u r c z o d t m g b n k q f a s p x e s
[162] d t e h s n q b l q m f p a g
[163] e c h m z b d a
[164] e n c z b d g a
[165] c h z b d g a
[166] b e a d h z m g
[167] p o b c e l m t n a q k f d g
[168] b d a e h t z g f
[169] e b f a d m h t z g
[170] q e a b d m h z
[171] l k x s y e t q b a f u r m h o m z g p d
[172] ſ k x b a s t c q f m o h z i g p d
[173] d a b e h z g
[174] d a b e h z g
[175] a d b b g
[176] a d f b ſ m e c z g
[177] h e m c u t s k o b z ſ q r f g a d
[178] h e m c u s t b o q z r f g a d
[179] i h e m c t z u s b o k q r f g a d
[180] n q e ſ g t f m o K d h c a s u p z b
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      <text xml:lang="lat" type="free">
        <div xml:id="echoid-div353" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s9660" xml:space="preserve">
              <pb o="154" file="0160" n="160" rhead="ALHAZEN"/>
            per perpendicularẽ, propter prædictã ibidẽ rationẽ.</s>
            <s xml:id="echoid-s9661" xml:space="preserve"> Et ſimiliter non poterit reflecti ab alio puncto
              <lb/>
            ſpeculi, quã à puncto perpendicularis huius:</s>
            <s xml:id="echoid-s9662" xml:space="preserve"> quia accideret duas perpẽdiculares cõcurrere, & effi-
              <lb/>
            ficere triangulum, cuius duo anguli recti, ſicut ſuprà patuit.</s>
            <s xml:id="echoid-s9663" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div355" type="section" level="0" n="0">
          <head xml:id="echoid-head348" xml:space="preserve" style="it">45. Si cõmunis ſectio ſuperficierũ, reflexionis & ſpeculi cylindracei cõuexi fuerit ellipſis: imago
            <lb/>
          uiſibilis obliquè reflexi, aliâs in ſuքficie ſpeculi: aliâs intra: aliâs extra ſpeculũ uidebitur. 49 p 7.</head>
          <p>
            <s xml:id="echoid-s9664" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s9665" xml:space="preserve"> ſumatur ſectio columnaris:</s>
            <s xml:id="echoid-s9666" xml:space="preserve"> & ſumatur in ea punctũ a:</s>
            <s xml:id="echoid-s9667" xml:space="preserve"> & ducatur contingens ſectio-
              <lb/>
            nẽ:</s>
            <s xml:id="echoid-s9668" xml:space="preserve"> quæ ſit e a t:</s>
            <s xml:id="echoid-s9669" xml:space="preserve"> & ſumatur perpendicularis ſuper a tintra ſpeculũ:</s>
            <s xml:id="echoid-s9670" xml:space="preserve"> quę ſit d a.</s>
            <s xml:id="echoid-s9671" xml:space="preserve"> Palàm, quòd d
              <lb/>
            a diuidit ſectionẽ in duas partes, in quarũ utraq;</s>
            <s xml:id="echoid-s9672" xml:space="preserve"> eſt punctũ unicũ, cuius puncti linea cõtin-
              <lb/>
            gens, erit æ quidiſtans a d.</s>
            <s xml:id="echoid-s9673" xml:space="preserve"> Sit ergo aliud punctũ g, cuius cõtingens cõcurrat cũ linea a d in puncto
              <lb/>
            h:</s>
            <s xml:id="echoid-s9674" xml:space="preserve"> & ducatur perpendicularis ſuper hãc cõtingentẽ:</s>
            <s xml:id="echoid-s9675" xml:space="preserve"> quæ ſit q g:</s>
            <s xml:id="echoid-s9676" xml:space="preserve"> & hæc quidẽ neceſſariò cõcurret cũ
              <lb/>
            h d, ſicut oſtenſum eſt in præcedente figura [eſt enim angulus q g h per fabricationẽ rectus:</s>
            <s xml:id="echoid-s9677" xml:space="preserve"> ergo q
              <lb/>
            g a maior eſt recto:</s>
            <s xml:id="echoid-s9678" xml:space="preserve"> & ob id h a g recto maior eſt.</s>
            <s xml:id="echoid-s9679" xml:space="preserve"> Itaq;</s>
            <s xml:id="echoid-s9680" xml:space="preserve"> per 13 p 1 d g a, d a g ſunt minores duobus re-
              <lb/>
            ctis.</s>
            <s xml:id="echoid-s9681" xml:space="preserve"> Quare per 11 ax.</s>
            <s xml:id="echoid-s9682" xml:space="preserve"> q g, h a cotinuatæ cõcurrent.</s>
            <s xml:id="echoid-s9683" xml:space="preserve">] Sit concurſus in puncto d:</s>
            <s xml:id="echoid-s9684" xml:space="preserve"> & ducatur linea g a
              <lb/>
            uſq;</s>
            <s xml:id="echoid-s9685" xml:space="preserve"> ad p:</s>
            <s xml:id="echoid-s9686" xml:space="preserve"> & ducatur linea q a.</s>
            <s xml:id="echoid-s9687" xml:space="preserve"> Igitur angulus q a h aut eſt æqualis angulo h a p:</s>
            <s xml:id="echoid-s9688" xml:space="preserve"> aut maior:</s>
            <s xml:id="echoid-s9689" xml:space="preserve"> aut mi-
              <lb/>
            nor.</s>
            <s xml:id="echoid-s9690" xml:space="preserve"> Sit ęqualis.</s>
            <s xml:id="echoid-s9691" xml:space="preserve"> Procedet igitur forma puncti q ad
              <lb/>
              <figure xlink:label="fig-0160-01" xlink:href="fig-0160-01a" number="86">
                <variables xml:id="echoid-variables76" xml:space="preserve">s f h q n x r p l z u t m a b o e g k d</variables>
              </figure>
            a, & reflectetur ad p [per 12 n 4] quod uiſus ſit:</s>
            <s xml:id="echoid-s9692" xml:space="preserve"> &
              <lb/>
            locus imaginis erit punctũ ſectionis columnaris,
              <lb/>
            ſcilicet g [per 4 n.</s>
            <s xml:id="echoid-s9693" xml:space="preserve">] Si uerò ſupra punctum q ſuma
              <lb/>
            tur aliquod punctũ, ut punctũ f:</s>
            <s xml:id="echoid-s9694" xml:space="preserve"> erit quidẽ angulus
              <lb/>
            f a h minor angulo h a p [quia angulus h a p æqua-
              <lb/>
            tur h a q, qui per 9 ax.</s>
            <s xml:id="echoid-s9695" xml:space="preserve"> maior eſtangulo h a f.</s>
            <s xml:id="echoid-s9696" xml:space="preserve">] Fiat ei
              <lb/>
            æqualis n a h:</s>
            <s xml:id="echoid-s9697" xml:space="preserve"> cõcurret quidẽ n a cũg q [per 11 ax.</s>
            <s xml:id="echoid-s9698" xml:space="preserve">
              <lb/>
            ut antea] intra columnã.</s>
            <s xml:id="echoid-s9699" xml:space="preserve"> [quia punctũ n ſublimi-
              <lb/>
            us eſt puncto p.</s>
            <s xml:id="echoid-s9700" xml:space="preserve">] Sit in puncto k.</s>
            <s xml:id="echoid-s9701" xml:space="preserve"> Palã ergo, quòd
              <lb/>
            imago puncti f erit in puncto k [per 4 n] & imagi-
              <lb/>
            nes omniũ punctorũ lineæ q fultra punctũ q, intra
              <lb/>
            columnã.</s>
            <s xml:id="echoid-s9702" xml:space="preserve"> Si uerò inter q & t ſumatur punctum ali-
              <lb/>
            quod:</s>
            <s xml:id="echoid-s9703" xml:space="preserve"> ut punctũ r:</s>
            <s xml:id="echoid-s9704" xml:space="preserve"> erit angulus r a h maior angulo
              <lb/>
            h a p [quia h a p æquatur h a q, quo angulus h a r
              <lb/>
            maior eſt per 9 ax.</s>
            <s xml:id="echoid-s9705" xml:space="preserve">] Fiat ei ęqualis h a m.</s>
            <s xml:id="echoid-s9706" xml:space="preserve"> Palàm, qđ
              <lb/>
            m a cadet ſupra lineã g q, & extra ſectionem [cũ e-
              <lb/>
            nim linea p a (quæ cũ h a cõtinet angulũ æqualem
              <lb/>
            h a q) cõcurrat cũ ſectione in puncto g:</s>
            <s xml:id="echoid-s9707" xml:space="preserve"> & punctũ m ſit inferius puncto p:</s>
            <s xml:id="echoid-s9708" xml:space="preserve"> linea igitur m a cõtinua-
              <lb/>
            ta cõcurret cũ g q extra ſectionẽ.</s>
            <s xml:id="echoid-s9709" xml:space="preserve">] Sit in pũcto o.</s>
            <s xml:id="echoid-s9710" xml:space="preserve"> Erit igitur imago r in pũcto o [per 4 n.</s>
            <s xml:id="echoid-s9711" xml:space="preserve">] Et omniũ
              <lb/>
            punctorũ inter t, q interiacentiũ imagines, erũt extra ſectionẽ inter o & g.</s>
            <s xml:id="echoid-s9712" xml:space="preserve"> Siuerò angulus q a h fue
              <lb/>
            rit minor angulo h a p:</s>
            <s xml:id="echoid-s9713" xml:space="preserve"> ſecetur ex eo æqualis:</s>
            <s xml:id="echoid-s9714" xml:space="preserve"> & ſit h a n.</s>
            <s xml:id="echoid-s9715" xml:space="preserve"> Palàm, quòd imago q erit in puncto k:</s>
            <s xml:id="echoid-s9716" xml:space="preserve"> & o-
              <lb/>
            mniũ punctorũ ſuperiorũ imagines erũt intra ſectionẽ.</s>
            <s xml:id="echoid-s9717" xml:space="preserve"> Si uerò inferius ſumatur r punctũ, ut angu-
              <lb/>
            lus r a h ſit ęqualis angulo h a p:</s>
            <s xml:id="echoid-s9718" xml:space="preserve"> erit imago r in ſectione:</s>
            <s xml:id="echoid-s9719" xml:space="preserve"> & oẽs inter r & q intra:</s>
            <s xml:id="echoid-s9720" xml:space="preserve"> oẽs inter r & t extra.</s>
            <s xml:id="echoid-s9721" xml:space="preserve">
              <lb/>
            Si uerò angulus q a h fuerit maior angulo h a p:</s>
            <s xml:id="echoid-s9722" xml:space="preserve"> fiat ei æqualis h a m.</s>
            <s xml:id="echoid-s9723" xml:space="preserve"> Palàm, quòd m a ſecabit ſectio
              <lb/>
            nẽ:</s>
            <s xml:id="echoid-s9724" xml:space="preserve"> [quia e a t tangit] & ſecet in puncto b:</s>
            <s xml:id="echoid-s9725" xml:space="preserve"> & ducatur cõtingẽs ſuper punctũ b:</s>
            <s xml:id="echoid-s9726" xml:space="preserve"> quę cõcurret cũ d h,
              <lb/>
            utin puncto l [ducta enim recta d b:</s>
            <s xml:id="echoid-s9727" xml:space="preserve"> erit angulus d b l rectus, & b d lacutus:</s>
            <s xml:id="echoid-s9728" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s9729" xml:space="preserve"> tãgens ſectionẽ in
              <lb/>
            pũcto b cõcurret cũ d h per 11 ax.</s>
            <s xml:id="echoid-s9730" xml:space="preserve">] eritq́;</s>
            <s xml:id="echoid-s9731" xml:space="preserve"> [per 17 p 1] angulus d l b acutus, & angulus h l b obtuſus:</s>
            <s xml:id="echoid-s9732" xml:space="preserve">
              <lb/>
            [per 13 p 1] & l b cõcurrẽs cũ h g faciet cũ ea acutũ [per 32 p 1:</s>
            <s xml:id="echoid-s9733" xml:space="preserve"> quia angulus h l b eſt obtuſus.</s>
            <s xml:id="echoid-s9734" xml:space="preserve">] Duca
              <lb/>
            tur perpẽdicularis à pũcto b ſuper l b:</s>
            <s xml:id="echoid-s9735" xml:space="preserve"> quę ſit s b:</s>
            <s xml:id="echoid-s9736" xml:space="preserve"> ſecabit quidẽ h g, utin pũcto x:</s>
            <s xml:id="echoid-s9737" xml:space="preserve"> & faciet angulũ a-
              <lb/>
            cutũ cũ ea [per 15 p 1] quoniã angulus cõtrapoſitus ſimiliter erit acutus [ք 32 p 1:</s>
            <s xml:id="echoid-s9738" xml:space="preserve"> quia angulus ad b
              <lb/>
            rectus eſt] & h g ſecat q a:</s>
            <s xml:id="echoid-s9739" xml:space="preserve"> ſit punctũ ſectionis u:</s>
            <s xml:id="echoid-s9740" xml:space="preserve"> & facit acutũ angulũ cũ ea ſuper punctum u [cum
              <lb/>
            enim h g cõcurrat cum q a:</s>
            <s xml:id="echoid-s9741" xml:space="preserve"> & q a cum fd, & angulus h g q ſit rectus:</s>
            <s xml:id="echoid-s9742" xml:space="preserve"> erit per 32 p 1 angulus q u g acu-
              <lb/>
            tus.</s>
            <s xml:id="echoid-s9743" xml:space="preserve">] Quare s b & q u concurrunt [quia enim angulis s x h, qu g acutis cõcluſis æquãtur anguli ad
              <lb/>
            uerticẽ per 15 p 1.</s>
            <s xml:id="echoid-s9744" xml:space="preserve"> Ergo per 11 ax.</s>
            <s xml:id="echoid-s9745" xml:space="preserve"> q u & s b cõcurrũt.</s>
            <s xml:id="echoid-s9746" xml:space="preserve">] Sit cõcurſus in z.</s>
            <s xml:id="echoid-s9747" xml:space="preserve"> Palàm ergo, quòd forma pun
              <lb/>
            cti z mouebitur ad ſpeculũ per z a, & reflectetur per a m:</s>
            <s xml:id="echoid-s9748" xml:space="preserve"> & locus imaginis, b:</s>
            <s xml:id="echoid-s9749" xml:space="preserve"> & imagines punctorũ
              <lb/>
            lineæ z s ultra z, erunt intra ſectionẽ:</s>
            <s xml:id="echoid-s9750" xml:space="preserve"> & punctorũ citra z, extra ſectionem.</s>
            <s xml:id="echoid-s9751" xml:space="preserve"> Quod fuit propoſitum.</s>
            <s xml:id="echoid-s9752" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div357" type="section" level="0" n="0">
          <head xml:id="echoid-head349" xml:space="preserve" style="it">46. Si cõmunis ſectio ſuperficierũ, reflexiõis & ſpeculi cylindracei conuexi, fuerit lat{us} cylindri,
            <lb/>
          uel circul{us} baſib. parallel9
            <unsure/>
          : ab uno pũcto unũ uiſibilis pũctũ ad unũ uisũ reflectetur. 26. 27 p 7.</head>
          <p>
            <s xml:id="echoid-s9753" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s9754" xml:space="preserve"> ab uno ſolo pũcto ſpeculi colũnaris fit reflexio ad cẽtrũ uiſus:</s>
            <s xml:id="echoid-s9755" xml:space="preserve"> utpote pũctũ b refle-
              <lb/>
            ctatur ad a à pũcto g.</s>
            <s xml:id="echoid-s9756" xml:space="preserve"> Dico, quòd nõ reflectetur ad ipſum ab alio puncto ſpeculi, quã à pũcto
              <lb/>
            g.</s>
            <s xml:id="echoid-s9757" xml:space="preserve"> Quoniã, ſi in ſuperficie reflexionis, quæ eſt a b g, ſit totus axis ſpeculi:</s>
            <s xml:id="echoid-s9758" xml:space="preserve"> erit linea cõmunis
              <lb/>
            ſuperficiei ſpeculi & ſuperficiei reflexionis linea lõgitudinis ſpeculi [per 29 n 4.</s>
            <s xml:id="echoid-s9759" xml:space="preserve">] Et cũ in ſuperfi-
              <lb/>
            cie reflexionis ſit cẽtrũ uiſus, pũctũ uiſum, punctũ reflexiõis, & punctũ axis, in qđ cadit perpẽdicu
              <lb/>
            laris:</s>
            <s xml:id="echoid-s9760" xml:space="preserve"> [per 23.</s>
            <s xml:id="echoid-s9761" xml:space="preserve"> 34 n 4] una ſola ſuperficies ſumi poteſt, in qua ſit linea illa longitudinis, axis, & pun-
              <lb/>
            cta a, b, g.</s>
            <s xml:id="echoid-s9762" xml:space="preserve"> Quare non poteſt ſieri reflexio ad a, niſi ab aliquo puncto lineę longitudinis:</s>
            <s xml:id="echoid-s9763" xml:space="preserve"> ſed iam pro-
              <lb/>
            batũ eſt [51 n 4 generatim de quolibet ſpeculo, & 14 n ſpeciatim de ſpeculo plano] quòd nõ poteſt
              <lb/>
            fieri reflexio ad a ab alio puncto, quã à puncto g.</s>
            <s xml:id="echoid-s9764" xml:space="preserve"> Quare in hoc ſitu ab uno ſolo pũcto ſpeculi fit ad
              <lb/>
            a reflexio.</s>
            <s xml:id="echoid-s9765" xml:space="preserve"> Si uerò ſuperficies a b g ſit æquidiſtans baſi colũnæ:</s>
            <s xml:id="echoid-s9766" xml:space="preserve"> erit linea cõmunis, circulus æquidi-
              <lb/>
            </s>
          </p>
        </div>
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