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

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            <s xml:id="echoid-s13505" xml:space="preserve">
              <pb o="196" file="0202" n="202" rhead="ALHAZEN"/>
            dico, quòd extra hanc totalẽ ſuperficiem licebit inuenire punctũ, ad quod reflectãtur duo pũcta i, n
              <lb/>
            à duobus ſpeculi punctis:</s>
            <s xml:id="echoid-s13506" xml:space="preserve"> & imago erit t q.</s>
            <s xml:id="echoid-s13507" xml:space="preserve"> Verbi gratia.</s>
            <s xml:id="echoid-s13508" xml:space="preserve"> Palàm, quòd angulus n a z duplus eſt ad an-
              <lb/>
            gulũ c a b:</s>
            <s xml:id="echoid-s13509" xml:space="preserve"> [oſtenſum enim eſt peripherias n u & u z ęquari:</s>
            <s xml:id="echoid-s13510" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s13511" xml:space="preserve"> n z dupla eſt ipſius u z, & per 33 p 6
              <lb/>
            angulus n a z duplus ad angulũ n a u, ideoq́;</s>
            <s xml:id="echoid-s13512" xml:space="preserve"> duplus ad æqualẽ i a b] & angulus i a o duplus ad angu
              <lb/>
            lum i a g, ſecundũ prædicta:</s>
            <s xml:id="echoid-s13513" xml:space="preserve"> [æquatus enim eſt o a g ipſi i a g:</s>
            <s xml:id="echoid-s13514" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s13515" xml:space="preserve"> totus i a o duplus eſt ad i a g] & an
              <lb/>
            gulus n a z nõ excedit angulũ i a o in angulo maiore angulo n a i.</s>
            <s xml:id="echoid-s13516" xml:space="preserve"> [Quia enim anguli n a z & i a o du-
              <lb/>
            pli ſunt angulorũ i a b & i a g:</s>
            <s xml:id="echoid-s13517" xml:space="preserve"> & i a b exuperat angulũ i a g, angulo g a b (qui per theſin minor eſt an-
              <lb/>
            gulo g a m) ergo angulus g a b minor eſt dimidiato angulo b a m (qui per 33 p 6 ęquatur angulo n a i,
              <lb/>
            ob peripherias f d & m b æquales) angulus igitur g a b minor eſt dimidiato angulo n a i.</s>
            <s xml:id="echoid-s13518" xml:space="preserve"> Quare angu
              <lb/>
            lus n a z exuperans angulũ i a o, duplo angulo g a b, nõ exuperat maiore angulo ꝗ̃ ſit n a i] & duo an-
              <lb/>
            guli i a o, i a n maiores tertio, qui eſt n a z:</s>
            <s xml:id="echoid-s13519" xml:space="preserve"> & duo z a n, n a i maiores tertio i a o:</s>
            <s xml:id="echoid-s13520" xml:space="preserve"> & duo n a z, i a o maio
              <lb/>
            res tertio n a i.</s>
            <s xml:id="echoid-s13521" xml:space="preserve"> Habemus ergo tres angulos [n a i, n a z, i a o] quorũ quilibet duo maiores ſunt tertio,
              <lb/>
            & oẽs ſimul quatuor rectis minores:</s>
            <s xml:id="echoid-s13522" xml:space="preserve"> [quia non totũ circa centrum a locum replent.</s>
            <s xml:id="echoid-s13523" xml:space="preserve">] Igitur [per 23 p
              <lb/>
            11] exillis licet facere angulũ corporalẽ.</s>
            <s xml:id="echoid-s13524" xml:space="preserve"> Fiat angulus ille ſuper a:</s>
            <s xml:id="echoid-s13525" xml:space="preserve"> & ſit linea s a erecta ſuper a:</s>
            <s xml:id="echoid-s13526" xml:space="preserve"> & angu
              <lb/>
            lus i a s ſit ęqualis angulo i a o:</s>
            <s xml:id="echoid-s13527" xml:space="preserve"> & angulus n a s ęqualis angulo n a z:</s>
            <s xml:id="echoid-s13528" xml:space="preserve"> angulus n a i manebit immotus:</s>
            <s xml:id="echoid-s13529" xml:space="preserve">
              <lb/>
            & fiat linea a s æqualis lineæ a n uel a i:</s>
            <s xml:id="echoid-s13530" xml:space="preserve"> quæ oẽs ſunt æquales:</s>
            <s xml:id="echoid-s13531" xml:space="preserve"> & ꝓducãtur lineę t s, q s.</s>
            <s xml:id="echoid-s13532" xml:space="preserve"> Palàm, quo-
              <lb/>
            niã angulus t a s eſt æqualis angulo t a o [eſt enim t a pars lineæ i a [& duo latera [t a, & a o] lateribus
              <lb/>
            duob.</s>
            <s xml:id="echoid-s13533" xml:space="preserve"> [t a & a s] erit [per 4 p 1] baſis t s ęqualis baſi t o, & triangulũ triãgulo:</s>
            <s xml:id="echoid-s13534" xml:space="preserve"> & ita angulus g t a æqua
              <lb/>
            lis angulo s t a [ꝗ a g t pars eſt lineæ o t.</s>
            <s xml:id="echoid-s13535" xml:space="preserve">] Si-
              <lb/>
              <figure xlink:label="fig-0202-01" xlink:href="fig-0202-01a" number="161">
                <variables xml:id="echoid-variables151" xml:space="preserve">l u r c z o d t
                  <gap/>
                m g b n k q f a s p x e
                  <gap/>
                s</variables>
              </figure>
            militer angulus q a s ęqualis angulo q a z, &
              <lb/>
            latera [q a, a s] lateribus:</s>
            <s xml:id="echoid-s13536" xml:space="preserve"> [q a, a z] & [per 4 p
              <lb/>
            1] triangulũ æquale triangulo:</s>
            <s xml:id="echoid-s13537" xml:space="preserve"> & angulus m
              <lb/>
            q a æqualis angulo s q a [eſt enim m q pars
              <lb/>
            lineę z q.</s>
            <s xml:id="echoid-s13538" xml:space="preserve">] Diuidatur angulus t a s per æqua
              <lb/>
            lia per lineam a y [per 9 p 1.</s>
            <s xml:id="echoid-s13539" xml:space="preserve">] Sit y punctũ, in
              <lb/>
            quo linea illa ſecabit lineã t s.</s>
            <s xml:id="echoid-s13540" xml:space="preserve"> Palàm, cũ an-
              <lb/>
            gulus i a g ſit medietas angulii a o:</s>
            <s xml:id="echoid-s13541" xml:space="preserve"> erit an-
              <lb/>
            gulus t a g æqualis angulo t a y, & angulus
              <lb/>
            g t a æqualis y t a:</s>
            <s xml:id="echoid-s13542" xml:space="preserve"> & unũ latus cõmune, ſcili
              <lb/>
            cet t a:</s>
            <s xml:id="echoid-s13543" xml:space="preserve"> erit [per 26 p 1] t g æqualis t y, & trian
              <lb/>
            gulũ [y t a] triangulo:</s>
            <s xml:id="echoid-s13544" xml:space="preserve"> [g t a] & erit a y æqua
              <lb/>
            lis a g:</s>
            <s xml:id="echoid-s13545" xml:space="preserve"> & ita y in ſuperficie ſpeculi:</s>
            <s xml:id="echoid-s13546" xml:space="preserve"> [cũ enim
              <lb/>
            puncta g & y à centro a æquabiliter diſtent
              <lb/>
            per concluſionẽ proximam:</s>
            <s xml:id="echoid-s13547" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s13548" xml:space="preserve"> g ex theſi
              <lb/>
            in ſpeculi ſuperficie:</s>
            <s xml:id="echoid-s13549" xml:space="preserve"> erit y in eadẽ.</s>
            <s xml:id="echoid-s13550" xml:space="preserve">] Erit etiã
              <lb/>
            angulus i a g æqualis angulo i a y, & latera
              <lb/>
            [i a, a g] lateribus [i a, a y] & [per 4 p 1] trian-
              <lb/>
            gulum i a g triangulo [i a y] æquale:</s>
            <s xml:id="echoid-s13551" xml:space="preserve"> & erit angulus a g i ęqualis angulo a y i:</s>
            <s xml:id="echoid-s13552" xml:space="preserve"> & linea i y ꝓducta, æqua
              <lb/>
            lis i g.</s>
            <s xml:id="echoid-s13553" xml:space="preserve"> Et producatur a y extra ſphęrã uſq;</s>
            <s xml:id="echoid-s13554" xml:space="preserve"> ad punctũ p:</s>
            <s xml:id="echoid-s13555" xml:space="preserve"> reſtabit angulus i g r æqualis angulo i y p [ք 13
              <lb/>
            p 1.</s>
            <s xml:id="echoid-s13556" xml:space="preserve">] Verũ cum t s ſit æqualis t o, & t y æqualis t g:</s>
            <s xml:id="echoid-s13557" xml:space="preserve"> [per cõcluſionẽ] reſtat g o æqualis y s.</s>
            <s xml:id="echoid-s13558" xml:space="preserve"> Igitur a y, y s
              <lb/>
            æqualia, a g, g o:</s>
            <s xml:id="echoid-s13559" xml:space="preserve"> & baſis a s æqualis baſi a o:</s>
            <s xml:id="echoid-s13560" xml:space="preserve"> erit [per 8 p 1] triangulũ [a y s] ęquale triangulo:</s>
            <s xml:id="echoid-s13561" xml:space="preserve"> [a g o] &
              <lb/>
            erit angulus a y s æqualis angulo a g o:</s>
            <s xml:id="echoid-s13562" xml:space="preserve"> reſtat [per 13 p 1] angulus s y p æqualis angulo o g r.</s>
            <s xml:id="echoid-s13563" xml:space="preserve"> Igitur duo
              <lb/>
            anguli i g r, o g r æquales ſunt duobus angulis i y p, s y p.</s>
            <s xml:id="echoid-s13564" xml:space="preserve"> Verùm linea a s ſecabit ſphęrã:</s>
            <s xml:id="echoid-s13565" xml:space="preserve"> ſit punctum
              <lb/>
            ſectionis e.</s>
            <s xml:id="echoid-s13566" xml:space="preserve"> Igitur tria pũcta e, y, d ſunt in ſuperficie ſphæræ.</s>
            <s xml:id="echoid-s13567" xml:space="preserve"> Quare linea e y d eſt pars circuli ſphærę:</s>
            <s xml:id="echoid-s13568" xml:space="preserve">
              <lb/>
            & eſt linea comunis ſuperficiei ſphærę & ſuperficiei reflexionis t s p.</s>
            <s xml:id="echoid-s13569" xml:space="preserve"> Quare punctũ i reflectitur ad
              <lb/>
            punctũ s à puncto y:</s>
            <s xml:id="echoid-s13570" xml:space="preserve"> & locus imaginis eſt t.</s>
            <s xml:id="echoid-s13571" xml:space="preserve"> Similiter diuiſo angulo n a s per æqualia per ax:</s>
            <s xml:id="echoid-s13572" xml:space="preserve"> probabi
              <lb/>
            tur modo prædicto, quòd q x æqualis eſt q m, & a x æqualis a m, & x s æqualis m z:</s>
            <s xml:id="echoid-s13573" xml:space="preserve"> & duo anguli n x
              <lb/>
            æ & s x æ æquales duobus angulis n m u, z m u.</s>
            <s xml:id="echoid-s13574" xml:space="preserve"> Et ita n reflectetur ad s à puncto x:</s>
            <s xml:id="echoid-s13575" xml:space="preserve"> & locus imaginis
              <lb/>
            q:</s>
            <s xml:id="echoid-s13576" xml:space="preserve"> & ita t q imago i n:</s>
            <s xml:id="echoid-s13577" xml:space="preserve"> [& ſic imago, ut prius, erit æqualis uiſibili:</s>
            <s xml:id="echoid-s13578" xml:space="preserve"> cum t q æqualis concluſa ſit ipſi i n.</s>
            <s xml:id="echoid-s13579" xml:space="preserve">]
              <lb/>
            Quod eſt propoſitũ.</s>
            <s xml:id="echoid-s13580" xml:space="preserve"> Amplius:</s>
            <s xml:id="echoid-s13581" xml:space="preserve"> ſi à puncto i ducatur perpendicularis ſuper n a:</s>
            <s xml:id="echoid-s13582" xml:space="preserve"> cadet inter n & q, non
              <lb/>
            extra n:</s>
            <s xml:id="echoid-s13583" xml:space="preserve"> cũ angulus i n a ſit acutus:</s>
            <s xml:id="echoid-s13584" xml:space="preserve"> quoniá æqualis angulo n i a [ducta enim recta in, ęquabuntur an-
              <lb/>
            guli ad baſim i n per 5 p 1] & ſi caderet քpendicularis illa extra n:</s>
            <s xml:id="echoid-s13585" xml:space="preserve"> eſſet acutus maior recto [per 16 p 1.</s>
            <s xml:id="echoid-s13586" xml:space="preserve">]
              <lb/>
            Faciet ergo perpendicularis illa angulũ rectũ ſuper n q, quẽ angulũ reſpicit linea i n.</s>
            <s xml:id="echoid-s13587" xml:space="preserve"> Quare [ք 19 p 1]
              <lb/>
            linea in maior eſt illa perpendiculari.</s>
            <s xml:id="echoid-s13588" xml:space="preserve"> Quare perpendicularis illa minor t q [ęquali ipſi in per cõclu
              <lb/>
            ſionẽ.</s>
            <s xml:id="echoid-s13589" xml:space="preserve">] Punctũ igitur lineę n q, in quod cadit perpẽdicularis, reflectitur ad punctũ s:</s>
            <s xml:id="echoid-s13590" xml:space="preserve"> imago uerò eius
              <lb/>
            cadet in lineã n a [per 3 n 5] ſupra punctũ q.</s>
            <s xml:id="echoid-s13591" xml:space="preserve"> Quia quantò remotiora ſunt puncta, quę reflectũtur, tan
              <lb/>
            tò loca imaginũ magis accedunt ad centrũ circuli [per 30 n 5.</s>
            <s xml:id="echoid-s13592" xml:space="preserve">] Et quæcunq;</s>
            <s xml:id="echoid-s13593" xml:space="preserve"> linea ducetur à puncto t
              <lb/>
            [quod eſt imago puncti i, reflexi à puncto ſpeculi y] ad aliquod punctũ n q ſupra q:</s>
            <s xml:id="echoid-s13594" xml:space="preserve"> erit maior t q [per
              <lb/>
            19 p 1.</s>
            <s xml:id="echoid-s13595" xml:space="preserve">] Igitur imago perpendicularis erit maior ipſa perpendiculari.</s>
            <s xml:id="echoid-s13596" xml:space="preserve"> [Quia enim t q æquatur ipſi i n,
              <lb/>
            quę maior cõcluſa eſt perpendiculari:</s>
            <s xml:id="echoid-s13597" xml:space="preserve"> ergo t q imago perpendicularis eadẽ maior eſt.</s>
            <s xml:id="echoid-s13598" xml:space="preserve">] Eodẽ modo
              <lb/>
            quęcunq;</s>
            <s xml:id="echoid-s13599" xml:space="preserve"> linea ducetur à puncto i ad n q, inter hanc perpendicularẽ & in:</s>
            <s xml:id="echoid-s13600" xml:space="preserve"> erit imago ipſius maior i-
              <lb/>
            pſa.</s>
            <s xml:id="echoid-s13601" xml:space="preserve"> Verùm determinentur hęc certius.</s>
            <s xml:id="echoid-s13602" xml:space="preserve"> Punctũ n quia reflectitur ad z à puncto m:</s>
            <s xml:id="echoid-s13603" xml:space="preserve"> & locus imaginis
              <lb/>
            eſt q:</s>
            <s xml:id="echoid-s13604" xml:space="preserve"> linea z m q ſecat circulũ in puncto, quod eſt 3:</s>
            <s xml:id="echoid-s13605" xml:space="preserve"> cõtingens ergo ducta à pũcto z ad circulũ:</s>
            <s xml:id="echoid-s13606" xml:space="preserve"> cadet
              <lb/>
            ſuper punctũ aliquod arcus m 3 [ſi enim z m tangeret:</s>
            <s xml:id="echoid-s13607" xml:space="preserve"> angulus z m a eſſet rectus per 18 p 3:</s>
            <s xml:id="echoid-s13608" xml:space="preserve"> quare per
              <lb/>
            </s>
          </p>
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