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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[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
[181] t n q z g m b ſ f h r a d e k o
[182] t i y n q g z x m b c ſ f h s r a d p e k o u
[183] f d b g t e h e
[184] e c s ſ o f i g m b k z d t q p h y n r u a x
[185] CIN EMATH EQUE FRANCAISE BIBLIOTHEQUE MUSEE
[186] a e t o f z h g d j c p k b q r
[187] a o u m h z t s n d ſ e q f p
[188] a o u p m h z t x b n y c q s l d g e K f r
[189] f u q b m t n e o z a
[190] f q b u g m c n K p a
[191] d g t K z b e a o ſ h
[192] d g t k n z u e b a o ſ h m r
[193] d g p i t k b e a o l f q h
[194] p d h t z f b g a ſ e k q
[195] t f h a ſ i k d r e z b c m o g
[196] q h f d u o g c r e a n m z b
[197] t f h a p k l i d e z b n r m o g q
[198] ſ m s q c d r b n p t a h e g u i f
[199] q s n p e f o x u m l b z k d h a
[200] k q t ſ n ſ g b o e u z d h a
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        <div xml:id="echoid-div399" type="section" level="0" n="0">
          <head xml:id="echoid-head375" xml:space="preserve" style="it">
            <pb o="169" file="0175" n="175" rhead="OPTICAE LIBER V."/>
          perpendicularium terminos à centro ſpeculi æquabiliter diſtantia, à quatuor peripheriæ pũctis
            <lb/>
          inter ſe mutuo reflectentur, & quatuor habebunt imagines. 26 p 8.</head>
          <p>
            <s xml:id="echoid-s11176" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s11177" xml:space="preserve"> ſumptis duabus diametris b q, a g:</s>
            <s xml:id="echoid-s11178" xml:space="preserve"> & e z diuidente angulum earum per æqualia:</s>
            <s xml:id="echoid-s11179" xml:space="preserve"> ſu-
              <lb/>
            matur in b d punctum t ſupra punctum, in quod cadit perpendicularis, ducta à puncto e:</s>
            <s xml:id="echoid-s11180" xml:space="preserve"> &
              <lb/>
            in d g ſumatur d h æqualis d t:</s>
            <s xml:id="echoid-s11181" xml:space="preserve"> [per 3 p 1] & ducanturte, h e.</s>
            <s xml:id="echoid-s11182" xml:space="preserve"> Reflectetur quidem t ad h à pun-
              <lb/>
            cto e, & ſimiliter à puncto z, non ab alio puncto arcus a q:</s>
            <s xml:id="echoid-s11183" xml:space="preserve"> nec ab aliquo puncto arcus a b uel g q [per
              <lb/>
            66 n.</s>
            <s xml:id="echoid-s11184" xml:space="preserve">] Deinde à puncto t ducatur perpẽdicularis ſuper t d:</s>
            <s xml:id="echoid-s11185" xml:space="preserve"> [per 11 p 1] quæ quidem concurret cũ d e
              <lb/>
            extra circulum ſphærę, cũ angulus b d e ſit acutus [ut oſtenſum eſt 36 n:</s>
            <s xml:id="echoid-s11186" xml:space="preserve"> quare d e & perpẽdicularis
              <lb/>
            ſuper t d per 11 ax:</s>
            <s xml:id="echoid-s11187" xml:space="preserve"> concurrent:</s>
            <s xml:id="echoid-s11188" xml:space="preserve"> & quidem extra circulum b z g.</s>
            <s xml:id="echoid-s11189" xml:space="preserve"> Quia cum hæc perpẽdicularis, & ea,
              <lb/>
            quæ à puncto e ſuper eandem ſemidiametrum d b ducitur, ſint parallelę per 28 p 1:</s>
            <s xml:id="echoid-s11190" xml:space="preserve"> nunquã cõcurrẽt
              <lb/>
            per 35 d 1.</s>
            <s xml:id="echoid-s11191" xml:space="preserve"> Quare perpendicularis à puncto t continuata, cadet extra circulũ ultra punctũ e.</s>
            <s xml:id="echoid-s11192" xml:space="preserve"> Itaq;</s>
            <s xml:id="echoid-s11193" xml:space="preserve"> cõ-
              <lb/>
            curret cum ſemidiametro e d extra circulum b z g.</s>
            <s xml:id="echoid-s11194" xml:space="preserve">] Cõcurrat ergo in puncto o:</s>
            <s xml:id="echoid-s11195" xml:space="preserve"> & ducãtur lineę to,
              <lb/>
            h o.</s>
            <s xml:id="echoid-s11196" xml:space="preserve"> Et fiat circulus tranſiens per tria puncta t, d, h:</s>
            <s xml:id="echoid-s11197" xml:space="preserve"> qui neceſſariò tranſibit per punctum o [ut pręce-
              <lb/>
            dente numero demonſtratum eſt] & erit d o diameter eius:</s>
            <s xml:id="echoid-s11198" xml:space="preserve"> [per conſectarium 5 p 4] & ducatur li-
              <lb/>
            nea cõtingens circulũ b z g, in puncto e [per 17 p 3] quę ſit k e.</s>
            <s xml:id="echoid-s11199" xml:space="preserve"> Palàm, quòd ultimus circulus ſecabit
              <lb/>
            primum, ſcilicet b z g in duobus punctis:</s>
            <s xml:id="echoid-s11200" xml:space="preserve"> [per 10 p 3] ſint illa puncta l, m:</s>
            <s xml:id="echoid-s11201" xml:space="preserve"> & ducantur lineæ t l, h l,
              <lb/>
            l d, t m, d m, h m.</s>
            <s xml:id="echoid-s11202" xml:space="preserve"> Cũ ergo arcus t d ſit æqualis arcui h
              <lb/>
              <figure xlink:label="fig-0175-01" xlink:href="fig-0175-01a" number="113">
                <variables xml:id="echoid-variables103" xml:space="preserve">o e k m f l g h d t b q a z</variables>
              </figure>
            d:</s>
            <s xml:id="echoid-s11203" xml:space="preserve"> [per 28 p 3:</s>
            <s xml:id="echoid-s11204" xml:space="preserve"> quia rectę d t, d h ſunt ęquales per fa-
              <lb/>
            bricationem] erit [per 27 p 3] angulus t l d æqualis
              <lb/>
            angulo d l h.</s>
            <s xml:id="echoid-s11205" xml:space="preserve"> Et ita t reflectetur ad h à puncto l [per
              <lb/>
            12 n 4.</s>
            <s xml:id="echoid-s11206" xml:space="preserve">] Similiter angulus t m d æqualis angulo d m
              <lb/>
            h [per 27 p 3.</s>
            <s xml:id="echoid-s11207" xml:space="preserve">] Et ita t reflectetur ad h à puncto m.</s>
            <s xml:id="echoid-s11208" xml:space="preserve"> Pa
              <lb/>
            làm igitur, quòd t reflectitur à quatuor pũctis a d h:</s>
            <s xml:id="echoid-s11209" xml:space="preserve">
              <lb/>
            ſcilicet e, z, l, m:</s>
            <s xml:id="echoid-s11210" xml:space="preserve"> & quadruplex erit locus imaginis e-
              <lb/>
            ius.</s>
            <s xml:id="echoid-s11211" xml:space="preserve"> Et non poteſt t reflecti ad h ab alio puncto, quã
              <lb/>
            ab aliquo iſtorum.</s>
            <s xml:id="echoid-s11212" xml:space="preserve"> Detur enim f punctum:</s>
            <s xml:id="echoid-s11213" xml:space="preserve"> & ducan
              <lb/>
            tur lineæ t f, h f, d f:</s>
            <s xml:id="echoid-s11214" xml:space="preserve"> & producatur d f, quouſque cõ-
              <lb/>
            currat cum contingente k e:</s>
            <s xml:id="echoid-s11215" xml:space="preserve"> [concurret autem per
              <lb/>
            11 ax:</s>
            <s xml:id="echoid-s11216" xml:space="preserve"> quia angulus k e d rectus eſt per 18 p 3, & f d e
              <lb/>
            acutus, quia pars acuti b d e] & ſit concurſus k:</s>
            <s xml:id="echoid-s11217" xml:space="preserve"> & du
              <lb/>
            cantur lineæ t k, h k.</s>
            <s xml:id="echoid-s11218" xml:space="preserve"> Igitur angulus t f d æqualis an-
              <lb/>
            gulo d f h ex hypotheſi:</s>
            <s xml:id="echoid-s11219" xml:space="preserve"> [& 12 n 4] reſtat [per 13 p 1]
              <lb/>
            angulus t f k æqualis angulo k fh.</s>
            <s xml:id="echoid-s11220" xml:space="preserve"> Sed angulus t k f
              <lb/>
            eſt æqualis angulo f k h [per 27 p.</s>
            <s xml:id="echoid-s11221" xml:space="preserve"> 3] quia ſuper ęqua
              <lb/>
            les arcus:</s>
            <s xml:id="echoid-s11222" xml:space="preserve"> & f k communis:</s>
            <s xml:id="echoid-s11223" xml:space="preserve"> erit [per 26 p 1] triangulum æquale triangulo:</s>
            <s xml:id="echoid-s11224" xml:space="preserve"> & ita t k æqualis k h:</s>
            <s xml:id="echoid-s11225" xml:space="preserve"> quod
              <lb/>
            eſt impoſsibile:</s>
            <s xml:id="echoid-s11226" xml:space="preserve"> quoniam h k maior h o, & t k minor to [per 7 p 3] & t o ęqualis h o.</s>
            <s xml:id="echoid-s11227" xml:space="preserve"> [Nam quia recta
              <lb/>
            d t æquatur ipſi d h per fabricationem, & angulus t d o ipſi h d o per theſim, & latus o d commune:</s>
            <s xml:id="echoid-s11228" xml:space="preserve">
              <lb/>
            ergo per 4 p 1 latus t o æquatur lateri h o:</s>
            <s xml:id="echoid-s11229" xml:space="preserve"> ideoq́;</s>
            <s xml:id="echoid-s11230" xml:space="preserve"> t k minor eſt h k.</s>
            <s xml:id="echoid-s11231" xml:space="preserve">] Palàm igitur, quòd non eſt refle-
              <lb/>
            xio ab aliquo puncto, quam à punctis quatuor.</s>
            <s xml:id="echoid-s11232" xml:space="preserve"> Igitur ſi in diuerſis diametris ſumantur duo puncta,
              <lb/>
            ſcilicet t, h, ęqualiter à centro diſtantia:</s>
            <s xml:id="echoid-s11233" xml:space="preserve"> ſi fuerint ſuper punctis diametrorum, in quę cadunt perpen
              <lb/>
            diculares, ductę à termino diametri diuidentis per æqualia angulum duarum diametrorũ:</s>
            <s xml:id="echoid-s11234" xml:space="preserve"> aut fue-
              <lb/>
            rint inter centrum & puncta illa, id eſt citra perpendiculares, dum æqualiter diſtent à centro:</s>
            <s xml:id="echoid-s11235" xml:space="preserve"> refle-
              <lb/>
            ctetur quidem t ad h à duobus punctis tantùm.</s>
            <s xml:id="echoid-s11236" xml:space="preserve"> Si uerò fuerint t & h à locis perpendicularium uſq;</s>
            <s xml:id="echoid-s11237" xml:space="preserve">
              <lb/>
            ad circulum:</s>
            <s xml:id="echoid-s11238" xml:space="preserve"> reflectetur quidem t ad h à quatuor punctis.</s>
            <s xml:id="echoid-s11239" xml:space="preserve"> Si uerò fuerint in circulo, uel extra:</s>
            <s xml:id="echoid-s11240" xml:space="preserve"> tamẽ
              <lb/>
            citra contingentem k e:</s>
            <s xml:id="echoid-s11241" xml:space="preserve"> reflectetur quidem t ad h à duobus punctis tantùm.</s>
            <s xml:id="echoid-s11242" xml:space="preserve"> Si uerò ſupra contingẽ
              <lb/>
            tem fuerint:</s>
            <s xml:id="echoid-s11243" xml:space="preserve"> reflectetur quidem t ad h ab uno puncto tantùm.</s>
            <s xml:id="echoid-s11244" xml:space="preserve"> Et hæc quidem accidunt, dum t ęqua-
              <lb/>
            liter diſtat à centro cum puncto h.</s>
            <s xml:id="echoid-s11245" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div401" type="section" level="0" n="0">
          <head xml:id="echoid-head376" xml:space="preserve" style="it">73. Viſu & uiſibili in diuerſis diametris circuli (qui eſt communis ſectio ſuperficierum refle-
            <lb/>
          xionis & ſpeculi ſphæricicaui) à centro inæquabiliter diſtantibus: ab uno puncto peripheriæ in-
            <lb/>
          ter ſemidiametros, extra quas ſunt uiſus & uiſibile, reflexio fieripoteſt. 27 p 8. 120 p 1.</head>
          <p>
            <s xml:id="echoid-s11246" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s11247" xml:space="preserve"> t, h ſi fuerint in diuerſis diametris:</s>
            <s xml:id="echoid-s11248" xml:space="preserve"> & longitudo eorum à centro fuerit inęqualis:</s>
            <s xml:id="echoid-s11249" xml:space="preserve"> re-
              <lb/>
            flexio fiet ab uno puncto.</s>
            <s xml:id="echoid-s11250" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s11251" xml:space="preserve"> ducantur diametri a d g, b d q:</s>
            <s xml:id="echoid-s11252" xml:space="preserve"> & e z diuidat angulum
              <lb/>
            eorum per æqualia:</s>
            <s xml:id="echoid-s11253" xml:space="preserve"> & t propinquius ſit centro d, quàm h.</s>
            <s xml:id="echoid-s11254" xml:space="preserve"> Et ſumatur linea l y:</s>
            <s xml:id="echoid-s11255" xml:space="preserve"> & [per 10 p 6]
              <lb/>
            diuidatur in puncto m, ut ſit proportio y m ad m l, ſicut h d ad d t:</s>
            <s xml:id="echoid-s11256" xml:space="preserve"> & diuidatur l y in æqualia in pun-
              <lb/>
            cto n [per 10 p 1] & à puncto n ducatur perpendicularis n k:</s>
            <s xml:id="echoid-s11257" xml:space="preserve"> [per 11 p 1] & ſuper punctum l fiat angu
              <lb/>
            lus ęqualis medietati a d t per lineã f l:</s>
            <s xml:id="echoid-s11258" xml:space="preserve"> erit quidẽ angulus f l y acutus:</s>
            <s xml:id="echoid-s11259" xml:space="preserve"> [quia æquatus eſt dimidiato
              <lb/>
            angulo a d t acuto, ut oſtenſum eſt 36 n.</s>
            <s xml:id="echoid-s11260" xml:space="preserve">] Quare [per 11 ax] fl cõcurret cum n k:</s>
            <s xml:id="echoid-s11261" xml:space="preserve"> [quia l n k rectus eſt
              <lb/>
            per fabricationem] concurrant in puncto f:</s>
            <s xml:id="echoid-s11262" xml:space="preserve"> & [per 35 n] à puncto m ducatur linea ad latus fl, cõcur
              <lb/>
            rens cum latere n k in puncto, quod ſit k:</s>
            <s xml:id="echoid-s11263" xml:space="preserve"> & ſecet linea illa latus fl in puncto c, ut ſit proportio k c ad
              <lb/>
            c l, ſicut h d ad d z.</s>
            <s xml:id="echoid-s11264" xml:space="preserve"> Deinde ſuper pũctum d fiat angulus æqualis angulo l c m:</s>
            <s xml:id="echoid-s11265" xml:space="preserve"> [per 23 p 1] qui ſit i d a:</s>
            <s xml:id="echoid-s11266" xml:space="preserve">
              <lb/>
            & ſit i punctum circuli ſupra z, aut infra:</s>
            <s xml:id="echoid-s11267" xml:space="preserve"> & ſuper i punctũ fiat angulus ęqualis c l m:</s>
            <s xml:id="echoid-s11268" xml:space="preserve"> qui ſit o i d:</s>
            <s xml:id="echoid-s11269" xml:space="preserve"> & ſu
              <lb/>
            per hanc lineam o i [continuatã] ducatur perpendicularis à puncto h [per 12 p 1] quæ ſit h r:</s>
            <s xml:id="echoid-s11270" xml:space="preserve"> & pro-
              <lb/>
            </s>
          </p>
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