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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        <div xml:id="echoid-div416" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s11764" xml:space="preserve">
              <pb o="176" file="0182" n="182" rhead="ALHAZEN"/>
            Totusigitur ſtr æquatur toti ft q] eſt acutus:</s>
            <s xml:id="echoid-s11765" xml:space="preserve"> & k f perpendicularis ſupert d.</s>
            <s xml:id="echoid-s11766" xml:space="preserve"> Quare [per 11 ax.</s>
            <s xml:id="echoid-s11767" xml:space="preserve">] k f
              <lb/>
            producta concurret cum t q:</s>
            <s xml:id="echoid-s11768" xml:space="preserve"> ſit concurſus s:</s>
            <s xml:id="echoid-s11769" xml:space="preserve"> & linea t s ducta à puncto t ad punctum concurſus, cu-
              <lb/>
            ius lineæ pars eſt t q:</s>
            <s xml:id="echoid-s11770" xml:space="preserve"> erit æqualis lineæ t r:</s>
            <s xml:id="echoid-s11771" xml:space="preserve"> [quia anguli ad ſſunt recti, & ft r, ft s æquantur, latusq́;</s>
            <s xml:id="echoid-s11772" xml:space="preserve"> t
              <lb/>
            fcommune:</s>
            <s xml:id="echoid-s11773" xml:space="preserve"> æquabitur r t ipſit s per 26 p 1] & ita t q minortz [quia minor eſt ipſatr, quæ pars eſt
              <lb/>
            ipſius t z.</s>
            <s xml:id="echoid-s11774" xml:space="preserve">] Quare [per 8 p 5] maior eſt proportio n t ad t q, quàm n t ad tz.</s>
            <s xml:id="echoid-s11775" xml:space="preserve"> Igitur maior eſt propor-
              <lb/>
            tio in ad m q, quàm in ad n z.</s>
            <s xml:id="echoid-s11776" xml:space="preserve"> Quare [per 10 p 5] m q minor eſt n z.</s>
            <s xml:id="echoid-s11777" xml:space="preserve"> Secetur igitur exn z æqualis ei
              <lb/>
            [per 3 p 1] quæ ſit n x.</s>
            <s xml:id="echoid-s11778" xml:space="preserve"> Quoniam [per 22 p 3] angulus l n d cum angulo l m d ualet duos rectos:</s>
            <s xml:id="echoid-s11779" xml:space="preserve"> erit
              <lb/>
            [per 13 p 1.</s>
            <s xml:id="echoid-s11780" xml:space="preserve"> 3 ax] angulus l n d æqualis q m d:</s>
            <s xml:id="echoid-s11781" xml:space="preserve"> & x n, n d, æqualia q m, m d.</s>
            <s xml:id="echoid-s11782" xml:space="preserve"> Igitur [per 4 p 1] q d æqua-
              <lb/>
            lis x d.</s>
            <s xml:id="echoid-s11783" xml:space="preserve"> Sed z d maior x d:</s>
            <s xml:id="echoid-s11784" xml:space="preserve"> quoniam angulus l n d cum angulo l m d ualet duos rectos:</s>
            <s xml:id="echoid-s11785" xml:space="preserve"> [per 22 p 3] ſed
              <lb/>
            angulus l m d acutus:</s>
            <s xml:id="echoid-s11786" xml:space="preserve"> cum angulus e m d ſit acutus [per 16 p.</s>
            <s xml:id="echoid-s11787" xml:space="preserve"> 12 d 1.</s>
            <s xml:id="echoid-s11788" xml:space="preserve">] Igitur angulus l n d maior eſt
              <lb/>
            recto:</s>
            <s xml:id="echoid-s11789" xml:space="preserve"> igitur z d maior x d [quia enim angulus l n d eſt obtuſus:</s>
            <s xml:id="echoid-s11790" xml:space="preserve"> erit per 32 p 1 n x d acutus, & per 13.</s>
            <s xml:id="echoid-s11791" xml:space="preserve">
              <lb/>
            32 p 1 z x d obtuſus, x z d acutus:</s>
            <s xml:id="echoid-s11792" xml:space="preserve"> quare per 19 p 1 z d maior eſt x d.</s>
            <s xml:id="echoid-s11793" xml:space="preserve">] Quare z d
              <unsure/>
            maior q d.</s>
            <s xml:id="echoid-s11794" xml:space="preserve"> Igitur q re-
              <lb/>
            flectitur ad z à duobus punctis t, l:</s>
            <s xml:id="echoid-s11795" xml:space="preserve"> & q & z ſuntinęqualis longitudinis à centro, & in diuerſis dia-
              <lb/>
            metris.</s>
            <s xml:id="echoid-s11796" xml:space="preserve"> Et quòd non ſint in eadem diametro, palàm:</s>
            <s xml:id="echoid-s11797" xml:space="preserve"> quoniam angulus x d n æqualis eſt angulo q d
              <lb/>
            m:</s>
            <s xml:id="echoid-s11798" xml:space="preserve"> addito ergo communiangulo x d m, erit angulus n d m æqualis angulo x d q:</s>
            <s xml:id="echoid-s11799" xml:space="preserve"> & minor duobus
              <lb/>
            rectis.</s>
            <s xml:id="echoid-s11800" xml:space="preserve"> Quare magis angulus z d q [pars anguli x d q] minor duobus rectis.</s>
            <s xml:id="echoid-s11801" xml:space="preserve"> Quare q & z non ſunt in
              <lb/>
            eadem diametro, ſed in diuerſis.</s>
            <s xml:id="echoid-s11802" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div418" type="section" level="0" n="0">
          <head xml:id="echoid-head385" xml:space="preserve" style="it">82. Siduo punctain diuerſis diametris circuli (qui eſt communis ſectio ſuperficierum, refle-
            <lb/>
          xionis & ſpeculi ſphærici caui) à centro inæquabiliter diſtantia, à duobus punctis peripheriæ
            <lb/>
          comprebenſæ inter ſemidiametros, in quibus ipſa ſunt, inter ſe mutuò reflect antur: à nullo alio
            <lb/>
          eiuſdem peripheriæ puncto reflecti poſſunt. 36 p 8.</head>
          <p>
            <s xml:id="echoid-s11803" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s11804" xml:space="preserve"> ſumptis duobus punctis, quæ ſint o, k, & inæqualiter diſtantibus à cẽtro:</s>
            <s xml:id="echoid-s11805" xml:space="preserve"> reflectetur
              <lb/>
            quidẽ unum ad aliud à duob pũctis arcus, reſpiciẽtis ſemidiametros, in quib.</s>
            <s xml:id="echoid-s11806" xml:space="preserve"> ſunt:</s>
            <s xml:id="echoid-s11807" xml:space="preserve"> ſed nõ ab
              <lb/>
            alio pũcto illius arcus, quàm ab illis duob.</s>
            <s xml:id="echoid-s11808" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s11809" xml:space="preserve"> d ſit centrũ:</s>
            <s xml:id="echoid-s11810" xml:space="preserve"> k remotius à d quàm o à
              <lb/>
            d:</s>
            <s xml:id="echoid-s11811" xml:space="preserve"> g d, b d ſemidiam etri:</s>
            <s xml:id="echoid-s11812" xml:space="preserve"> t punctũ unũ reflexionis.</s>
            <s xml:id="echoid-s11813" xml:space="preserve"> Palàm ex ſuperioribus, quod uterq;</s>
            <s xml:id="echoid-s11814" xml:space="preserve"> angulus con-
              <lb/>
            ſtans ex angulo incidẽtiæ & reflexionis, nõ erit min or angulo o d a:</s>
            <s xml:id="echoid-s11815" xml:space="preserve"> [ք 80 n] nec ęqualis [per 79 n]
              <lb/>
            alter ergo erit maior.</s>
            <s xml:id="echoid-s11816" xml:space="preserve"> Sit angulus cõſtans ex angulo incidẽtiæ & reflexionis, qui eſt ſuք t, maior an-
              <lb/>
            gulo o d a:</s>
            <s xml:id="echoid-s11817" xml:space="preserve"> & ducãtur lineæ o t, d t, k t:</s>
            <s xml:id="echoid-s11818" xml:space="preserve"> & ex angulo illo ſecetur angulus æqualis angulo o d a:</s>
            <s xml:id="echoid-s11819" xml:space="preserve"> [ք 23
              <lb/>
            p 1] quiſit o t ſ:</s>
            <s xml:id="echoid-s11820" xml:space="preserve"> & diuidatur angulus f t k per æqualia per lineã t e [per 9 p 1] & à puncto k ducatur
              <lb/>
            æquidiſtãs t f:</s>
            <s xml:id="echoid-s11821" xml:space="preserve"> [per 31 p 1] quæ quidẽ cõcurret cũ te:</s>
            <s xml:id="echoid-s11822" xml:space="preserve"> [per lemma Procli ad 29 p 1] cõcurrat in pũcto
              <lb/>
            z:</s>
            <s xml:id="echoid-s11823" xml:space="preserve"> & ducatur linea o k:</s>
            <s xml:id="echoid-s11824" xml:space="preserve"> & diuidatur angulus o d k peræqualia, per lineã d u, ſecantẽlineã o k in pun-
              <lb/>
            cto p:</s>
            <s xml:id="echoid-s11825" xml:space="preserve"> & eſt k d maior o d [extheſi.</s>
            <s xml:id="echoid-s11826" xml:space="preserve">] Cũ igitur [per 3 p 6] ſit ꝓportio k d ad d o, ſicut k p ad p o:</s>
            <s xml:id="echoid-s11827" xml:space="preserve"> erit
              <lb/>
            k p maior p o.</s>
            <s xml:id="echoid-s11828" xml:space="preserve"> Itẽlinead t ſecet lineã o k in puncton.</s>
            <s xml:id="echoid-s11829" xml:space="preserve"> Dico, quod p cadit inter n & k, nõ inter n & o,
              <lb/>
            quod ſic patebit.</s>
            <s xml:id="echoid-s11830" xml:space="preserve"> Angulus k p dualet duos angulos p d o, p o d:</s>
            <s xml:id="echoid-s11831" xml:space="preserve"> & angulus o p d ualet duos angulos
              <lb/>
            p k d & p d k [per 32 p 1.</s>
            <s xml:id="echoid-s11832" xml:space="preserve">] Sed angulus p d o æqualis eſt angulo p d k:</s>
            <s xml:id="echoid-s11833" xml:space="preserve"> [per fabricationem] & [per
              <lb/>
            the ſim & 18 p 1] angulus k o d maior angulo o k d:</s>
            <s xml:id="echoid-s11834" xml:space="preserve"> igitur angulus k p d maior angulo o p d:</s>
            <s xml:id="echoid-s11835" xml:space="preserve"> igitur
              <lb/>
            [ք 13 p 1] angulus k p d maior recto:</s>
            <s xml:id="echoid-s11836" xml:space="preserve"> & angulus k n d
              <lb/>
              <figure xlink:label="fig-0182-01" xlink:href="fig-0182-01a" number="128">
                <variables xml:id="echoid-variables118" xml:space="preserve">b
                  <gap/>
                o
                  <gap/>
                p n g k e f d a q l m</variables>
              </figure>
            acutus:</s>
            <s xml:id="echoid-s11837" xml:space="preserve"> quod ſic cõſtabit:</s>
            <s xml:id="echoid-s11838" xml:space="preserve"> ſi fiat circulus per tria pũcta
              <lb/>
            o, t, k:</s>
            <s xml:id="echoid-s11839" xml:space="preserve"> [per 5 p 4] tranſibit infra d.</s>
            <s xml:id="echoid-s11840" xml:space="preserve"> Quoniã ſi tranſeat
              <lb/>
            per d:</s>
            <s xml:id="echoid-s11841" xml:space="preserve"> cũ angulus o t k ſit maior angulo o d a:</s>
            <s xml:id="echoid-s11842" xml:space="preserve"> [ք the-
              <lb/>
            ſin] erũtduo anguli o t k, o d k maiores duobus rectis
              <lb/>
            [cõtra 22 p 3.</s>
            <s xml:id="echoid-s11843" xml:space="preserve">] Si tranſeat ſupra d:</s>
            <s xml:id="echoid-s11844" xml:space="preserve"> eadẽ eſt demõſtra-
              <lb/>
            tio.</s>
            <s xml:id="echoid-s11845" xml:space="preserve"> Et linea n d diuidet arcũ illius circuli, qui eſt o k, ք
              <lb/>
            æqualia infra d.</s>
            <s xml:id="echoid-s11846" xml:space="preserve"> [Quia cum t ſit reflexionis punctũ ex
              <lb/>
            theſi:</s>
            <s xml:id="echoid-s11847" xml:space="preserve"> æquabuntur anguli k t d, d t o per 12 n 4, & peri-
              <lb/>
            pheriæ illis ſubtenſæ per 26 p 3.</s>
            <s xml:id="echoid-s11848" xml:space="preserve">] Si autẽ à pũcto diui-
              <lb/>
            ſionis ducatur linea ad mediũ punctũ lineæ o k:</s>
            <s xml:id="echoid-s11849" xml:space="preserve"> quæ
              <lb/>
            eſt chorda illius arcus:</s>
            <s xml:id="echoid-s11850" xml:space="preserve"> erit linea illa perpendicularis
              <lb/>
            ſuper o k:</s>
            <s xml:id="echoid-s11851" xml:space="preserve"> [rectę enim lineæ à puncto medio periphe-
              <lb/>
            riæ k o, ductæ ad puncta k & o, æquantur per 29 p3:</s>
            <s xml:id="echoid-s11852" xml:space="preserve"> &
              <lb/>
            recta, quę ab eodem puncto connectit medium rectæ
              <lb/>
            k o, æquatur ſibijpſi.</s>
            <s xml:id="echoid-s11853" xml:space="preserve"> Quare per 8 p.</s>
            <s xml:id="echoid-s11854" xml:space="preserve"> 10 d 1 ipſa perpen
              <lb/>
            dicularis eſt ad k o] & cadet inter p & k:</s>
            <s xml:id="echoid-s11855" xml:space="preserve"> cũ p k ſit ma-
              <lb/>
            ior p o:</s>
            <s xml:id="echoid-s11856" xml:space="preserve"> [excõcluſo] & angulus ſuper n à parte illius perpẽdicularis & ex parte p erit acutus:</s>
            <s xml:id="echoid-s11857" xml:space="preserve"> [per
              <lb/>
            32 p 1] & angulus ſuper p ex parte o eſt acutus [per 13 p 1:</s>
            <s xml:id="echoid-s11858" xml:space="preserve"> oſtenſum enim eſt angulum k p d eſſe ob-
              <lb/>
            tuſum.</s>
            <s xml:id="echoid-s11859" xml:space="preserve">] Si ergo p cadit inter n & o:</s>
            <s xml:id="echoid-s11860" xml:space="preserve"> impoſsibile erit perpẽdicularem illam cadere inter n & p:</s>
            <s xml:id="echoid-s11861" xml:space="preserve"> quia
              <lb/>
            ſecatet d p, & fieret triangulũ, cuius unus angulus rectus, alius obtuſus [contra 32 p 1.</s>
            <s xml:id="echoid-s11862" xml:space="preserve">] Cadet ergo
              <lb/>
            intern & k:</s>
            <s xml:id="echoid-s11863" xml:space="preserve"> & erit angulus n ex parte perpendicularis acutus:</s>
            <s xml:id="echoid-s11864" xml:space="preserve"> igitur ex parte o obtuſus [per 13 p 1:</s>
            <s xml:id="echoid-s11865" xml:space="preserve">]
              <lb/>
            ergo p non cadit inter n & o:</s>
            <s xml:id="echoid-s11866" xml:space="preserve"> quia ita erit triangulum, cuius duo anguli obtuſi [eſt enim angulus k
              <lb/>
            p d obtuſus concluſus.</s>
            <s xml:id="echoid-s11867" xml:space="preserve">] Palàm, quòd angulus k t d eſt medietas anguli k t o:</s>
            <s xml:id="echoid-s11868" xml:space="preserve"> [per theſin & 12 n 4:</s>
            <s xml:id="echoid-s11869" xml:space="preserve">
              <lb/>
            quia t eſt punctũ reflexionis:</s>
            <s xml:id="echoid-s11870" xml:space="preserve"> & d t perpendicularis eſt plano ſpeculum in puncto t tangenti per 25
              <lb/>
            n 4] ſed k t e eſtmedietas anguli k t f [per fabricationem.</s>
            <s xml:id="echoid-s11871" xml:space="preserve">] Reſtat e t d medietas angulift o:</s>
            <s xml:id="echoid-s11872" xml:space="preserve"> ſed fto
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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