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 contents

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[201.] PROOEMIVM LIBRI. CAP. I. 1. Viſio fit trifariam: rectè: reflexè: & refr actè. In præf. 1. 3. 10 Libr.
Page: 108 (102)
[202.] QVOD LVCI ACCIDAT REFLEXIO À POLITIS corporibus. Cap. II. 2. Lux & color reflectuntur à quolibet politæ ſuperficiei puncto, lineis rectis. 1 p 5.
Page: 108 (102)
[203.] 3. Lux & color à quolibet ſuperficiei coloratæ puncto ad quodlibet ſuperficiei politæ oppoſitæ punctum permixti confluunt. 2 p 5.
Page: 109 (103)
[204.] 4. Reflexio debilit at lucem & colorem: & omnino totam uiſibilis ſpeciem. 3 p 5.
Page: 109 (103)
[205.] 5. Lux & color reflexi ſunt debiliores luce & colore primis: fortiores autem ſecundis, cum quibus ab eodem ortu æquabiliter diſtant. 4 p 5.
Page: 109 (103)
[206.] DE MODO REFLEXIONIS FORMARVM À POLI-tis corporibus. Cap. III. 6. Lenitatis: politæ ſuperficiei: & perpendicularis incidentiæ definitiones. In def. 5 libr.
Page: 110 (104)
[207.] 7. Fabricatio & uſus organi reflexionis. 9 p 5.
Page: 110 (104)
[208.] 8. Fabricatio ſeptem ſpeculorum regularium. 8 p 5.
Page: 112 (106)
[209.] 9. Sit{us} & collocatio ſpeculorum regulariũ in reflexionis organo.10.12.13.14.15.16.17 p 5.
Page: 113 (107)
[210.] 10. Radi{us} ſpeculo plano obliqu{us}, in oppoſitam partem reflectitur: & æquat angulos inci-dentiæ & reflexionis. 10 p 5.
Page: 114 (108)
[211.] 11. Radi{us} ſpeculo perpendicularis, reflectitur in ſeipſum. 11.12 p 5.
Page: 114 (108)
[212.] 12. In ſpeculis, conuexis, cauis: ſphærico, conico cylindraceo, anguli incidentiæ & reflexio-nis æquantur. 12.13.14.15.16.17.20 p 5.
Page: 114 (108)
[213.] 13. Superficies reflexionis eſt perpendicularis plano ſpeculum in reflexionis puncto tan-genti. 25 p 5.
Page: 115 (109)
[214.] 14. Inter uiſibile & ſpeculũ innumer abiles pyramides fiũt alternis baſib. & uerticib{us}. 22 p 5.
Page: 117 (111)
[215.] 15. Lux à ſuperficie polita longinquiore reflexa, trifariam debilitatur.
Page: 117 (111)
[216.] 16. Lux & color reflectuntur per line{as} phyſic{as}, latitudine quadam prędit{as}. 3 p 2.6 p 5.
Page: 118 (112)
[217.] 17. Reflexio lucis & coloris à ſuperficie aſper a facta, plerun fugit uiſum. 1 p 5.
Page: 118 (112)
[218.] 18. Radij incidentiæ & reflexionis, ſit{us} ſimilitudine conueniunt. Ita anguli incidentiæ & reflexionis æquantur. 20 p 5.
Page: 118 (112)
[219.] 19. Colorem luci permiſtum reflecti, reflexionis organo ostenditur. 3 p 5.
Page: 119 (113)
[220.] QVOÒD COMPREHENSIO FORMARVM È CORPORIBVS politis fiat reflexione. Cap. 1111. 20. Falſa eſt utra opinio: & radios à uiſu ad ſpeculum miſſos, inde́ ad uiſibile reflexos, ima ginem percipere: & imaginẽ in ſpeculo iam antè impreſſam inde ad uiſum manare. 23. 24 p 5.
Page: 119 (113)
[221.] DE MODO COMPREHENSIONIS FORMARVM È COR-poribus politis. Cap. V. 21. Imago uiſibilis percipitur è reflexione formæ uiſibilis à ſpeculo ad uiſum facta. 24 p 5.
Page: 120 (114)
[222.] 22. Si uiſibile & ſpeculum figuræ ſit{us}́ ſimilitudine conueniant: uera & distincta imago uidetur. 35 p 5.
Page: 120 (114)
[223.] 23. Superficies reflexionis quatuor habet puncta: uiſibilis: reflexionis: uiſ{us}: & terminũ per-pendicularis ductæ à puncto reflexionis ſuper planum in eodem puncto ſpeculum tangens. Ita perpendicularis hæc cõmunis eſt omnib{us} reflexionis ſuperficieb{us}. 27 p 5.6 p 6.24 p 7.3 p 8.3 p 9.
Page: 121 (115)
[224.] 24. Si uiſ{us} ſit extra ſuperficiem ſpeculi ſphærici conuexi, uelipſi continuam: communis ſe-ctio baſis pyramidis opticæ & ſuperficiei ſpeculi, erit peripheria minimi in ſphæra circuli. 3 p 6.
Page: 121 (115)
[225.] 25. Si duarum rectarum linearum à uiſu, alter a ſpeculum ſphæricum conuexum tangat, re-liqua per centrum ſecet: tangens circa ſecantem fixam cõuerſa, definiet ſegmentum ſuperficiei ſpeculι: à cui{us} puncto quolibet poteſt ad uiſum fieri reflexio. Et centra uiſ{us} & ſpeculi, puncta reflexionis & uiſibilis ſunt in reflexionis ſuperficie. 2.5.6 p 6.
Page: 122 (116)
[226.] 26. Siduo plana à cẽtro uiſiis, ducãtur ք later a cõſpicuam ſpeculi cylindracei cõuexi ſuperficiẽ terminãtia: tangẽt ſpeculũ: & facient in uiſu cõmunem ſectionẽ par allelã axiſpeculi. 2.3 p 7.
Page: 122 (116)
[227.] 27. Si linea recta à cẽtro uiſ{us}, ducta ad punctũ cõſpicuæ ſuper-ficiei ſpeculi cylindr acei cõuexi, cõtinuetur: ſecabit ſpeculũ. 4.5 p 7.
Page: 123 (117)
[228.] 28. In ſpeculo cylindraceo conuexo, à quolibet conſpicuæ ſuperficiei puncto poteſt ad uiſum reflexio fieri. 25 p 7.
Page: 123 (117)
[229.] 29. Si uiſ{us} ſit extra ſuperficiem ſpeculi cylindr acei conuexi, in plano uiſibilis per axem du-cto: cõm unis ſectio ſuperficier um reflexionis & ſpeculi, erit lat{us} cylindri: & unicum tantùm eſt in eadem conſpicua ſuperficie planum, à quo ad eundem uiſum reflexio fieri poteſt. 7.16 p 7.
Page: 123 (117)
[230.] 30. Si uiſ{us} ſit extrá ſuperficiem ſpeculi cylindracei cõuexi, in planò uiſibilis ad axem recto: communis ſectio ſuperficierum reflexionis & ſpeculi, erit circul{us}: & unic{us} tantùm eſt in ea-dem conſpicuà ſuperficie, à quo ad uiſum reflexio fieri poteſt. 9.17 p 7.
Page: 124 (118)
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          <p>
            <s xml:id="echoid-s11517" xml:space="preserve">
              <pb o="173" file="0179" n="179" rhead="OPTICAE LIBER V."/>
            & o reflectũtur ad a à puncto e, & inæqualiter diſtant à centro cũ puncto a:</s>
            <s xml:id="echoid-s11518" xml:space="preserve"> & diameter o b cũ diame
              <lb/>
            tro a b g ex parte g facit angulũ maiorẽ angulo reflexionis & incidentiæ:</s>
            <s xml:id="echoid-s11519" xml:space="preserve"> & diameter n b minorẽ.</s>
            <s xml:id="echoid-s11520" xml:space="preserve"> Et
              <lb/>
            ita patet ꝓpoſitũ.</s>
            <s xml:id="echoid-s11521" xml:space="preserve"> Si uerò b a nõ fuerit perpẽdicularis ſuք e a:</s>
            <s xml:id="echoid-s11522" xml:space="preserve"> ducatur [per 12 p 1] perpẽdicularis:</s>
            <s xml:id="echoid-s11523" xml:space="preserve"> quę
              <lb/>
            ſit b k:</s>
            <s xml:id="echoid-s11524" xml:space="preserve"> quę quidẽ ſiue cadat ſupra a b, aut ſub:</s>
            <s xml:id="echoid-s11525" xml:space="preserve"> eadẽ erit ꝓbatio.</s>
            <s xml:id="echoid-s11526" xml:space="preserve"> Et b f ſit perpendicularis ſuper e o:</s>
            <s xml:id="echoid-s11527" xml:space="preserve"> &
              <lb/>
            ducatur f t æqualis a k:</s>
            <s xml:id="echoid-s11528" xml:space="preserve"> & ducatur b t.</s>
            <s xml:id="echoid-s11529" xml:space="preserve"> Palàm, quòd in triangulo k e b angulus e k b rectus, ęqualis eſt
              <lb/>
            angulo e f b, & [per 12 n 4] angulus k e b ęqualis angulo reflexiõis f e b:</s>
            <s xml:id="echoid-s11530" xml:space="preserve"> reſtat [per 32 p 1] tertius tertio
              <lb/>
            ęqualis:</s>
            <s xml:id="echoid-s11531" xml:space="preserve"> & cũ latus e b ſit cõmune utriq;</s>
            <s xml:id="echoid-s11532" xml:space="preserve"> triãgulo:</s>
            <s xml:id="echoid-s11533" xml:space="preserve"> erũt [per 26 p 1] triãgula æqualia:</s>
            <s xml:id="echoid-s11534" xml:space="preserve"> & erit f b æqualis
              <lb/>
            k b:</s>
            <s xml:id="echoid-s11535" xml:space="preserve"> ſed [ք fabricationẽ] a k eſt æqualis ft:</s>
            <s xml:id="echoid-s11536" xml:space="preserve"> erit ergo [per 4 p 1] a b æqualis b t, & angulus a b k æqualis
              <lb/>
            angulo f b t:</s>
            <s xml:id="echoid-s11537" xml:space="preserve"> addito igitur cõmuni angulo f b a:</s>
            <s xml:id="echoid-s11538" xml:space="preserve"> erit k b f æqualis t b a:</s>
            <s xml:id="echoid-s11539" xml:space="preserve"> Sed k b f & fe a ualent duos re-
              <lb/>
              <figure xlink:label="fig-0179-01" xlink:href="fig-0179-01a" number="121">
                <variables xml:id="echoid-variables111" xml:space="preserve">e o f t p d a b g k</variables>
              </figure>
              <figure xlink:label="fig-0179-02" xlink:href="fig-0179-02a" number="122">
                <variables xml:id="echoid-variables112" xml:space="preserve">e o f t p k d a b g</variables>
              </figure>
            ctos:</s>
            <s xml:id="echoid-s11540" xml:space="preserve"> [per 32 p 1:</s>
            <s xml:id="echoid-s11541" xml:space="preserve"> quia in quadrilatero e b anguli ad f & k recti ſunt.</s>
            <s xml:id="echoid-s11542" xml:space="preserve">] Quare t b a, t e a ualent duos re-
              <lb/>
            ctos:</s>
            <s xml:id="echoid-s11543" xml:space="preserve"> & ita t b g æqualis eſt angulo t e a:</s>
            <s xml:id="echoid-s11544" xml:space="preserve"> [quia t b g & t b a æquantur duobus rectis per 13 p 1] qui eſt
              <lb/>
            angulus conſtans ex angulo incidentiæ & reflexionis.</s>
            <s xml:id="echoid-s11545" xml:space="preserve"> Si igitur à puncto b ad lineam e t, ducatur li-
              <lb/>
            nea ultra t:</s>
            <s xml:id="echoid-s11546" xml:space="preserve"> faciet cum b g ex parte g, angulum minorẽ angulo conſtante ex angulo incidentiæ & re-
              <lb/>
            flexionis:</s>
            <s xml:id="echoid-s11547" xml:space="preserve"> & erit linea illa maior a b:</s>
            <s xml:id="echoid-s11548" xml:space="preserve"> quoniã t b [qua illa per 19 p 1 maior eſt] æqualis eſt a b.</s>
            <s xml:id="echoid-s11549" xml:space="preserve"> Et quæli
              <lb/>
            bet linea à puncto b ad e t ducta citra t:</s>
            <s xml:id="echoid-s11550" xml:space="preserve"> faciet angulũ t b g ex parte g, maiorẽ angulo cõſtante ex an-
              <lb/>
            gulo incidẽtiæ & reflexionis:</s>
            <s xml:id="echoid-s11551" xml:space="preserve"> & erit minor a b [quia minor æquali b t per 19 p 1.</s>
            <s xml:id="echoid-s11552" xml:space="preserve">] Et ita eſt propoſitũ.</s>
            <s xml:id="echoid-s11553" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div412" type="section" level="0" n="0">
          <head xml:id="echoid-head382" xml:space="preserve" style="it">79. Si uiſus & uiſibile in diuerſis diametris circuli (qui eſt communis ſectio ſuperficierum,
            <lb/>
          reflexionis & ſpeculi ſphærici caui) à centro inæquabiliter diſtantia, inter ſe reflectantur: angu-
            <lb/>
          lus exterior à diametris uiſus & uiſibilis factus, eſt inæqualis angulo incidentiæ & reflexionis
            <lb/>
          ſimul utri. 33 p 8.</head>
          <p>
            <s xml:id="echoid-s11554" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s11555" xml:space="preserve"> ſit b centrum uiſus:</s>
            <s xml:id="echoid-s11556" xml:space="preserve"> g centrum ſphæræ:</s>
            <s xml:id="echoid-s11557" xml:space="preserve"> ducatur diameter z b g d:</s>
            <s xml:id="echoid-s11558" xml:space="preserve"> & ſumatur ſuperfi-
              <lb/>
            cies, in qua ſit diameter ſecans ſphęram ſuper circulũ [per 1 th 1 ſphæ.</s>
            <s xml:id="echoid-s11559" xml:space="preserve">] qui ſit e z h.</s>
            <s xml:id="echoid-s11560" xml:space="preserve"> Dico, quòd
              <lb/>
            ſi punctum a reflectitur ad b ab aliquo puncto circuli:</s>
            <s xml:id="echoid-s11561" xml:space="preserve"> & inæqualis eſt diſtantia puncti a à cen-
              <lb/>
              <figure xlink:label="fig-0179-03" xlink:href="fig-0179-03a" number="123">
                <variables xml:id="echoid-variables113" xml:space="preserve">t z e b a g h d</variables>
              </figure>
              <figure xlink:label="fig-0179-04" xlink:href="fig-0179-04a" number="124">
                <variables xml:id="echoid-variables114" xml:space="preserve">t z e b a g h d</variables>
              </figure>
            tro, & puncti b ab eodem:</s>
            <s xml:id="echoid-s11562" xml:space="preserve"> diameter a g cum diametro g d, ex parte d faciet angulũ, quem impoſsibi-
              <lb/>
            le eſt eſſe æqualẽ angulo conſtanti ex angulo incidentiæ & reflexionis.</s>
            <s xml:id="echoid-s11563" xml:space="preserve"> Sit enim æqualis:</s>
            <s xml:id="echoid-s11564" xml:space="preserve"> & t ſit pun
              <lb/>
            ctum reflexionis:</s>
            <s xml:id="echoid-s11565" xml:space="preserve"> & ſit a g inæqualis b g:</s>
            <s xml:id="echoid-s11566" xml:space="preserve"> & ducantur lineæ t a, t g, t b:</s>
            <s xml:id="echoid-s11567" xml:space="preserve"> & fiat circulus tranſiẽs per tria
              <lb/>
            puncta a, g, b:</s>
            <s xml:id="echoid-s11568" xml:space="preserve"> [per 5 p 4] qui neceſſariò tranſibit per punctũ t.</s>
            <s xml:id="echoid-s11569" xml:space="preserve"> Si enim cadit extra:</s>
            <s xml:id="echoid-s11570" xml:space="preserve"> ductis lineis à pun
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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