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-div414" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s11661" xml:space="preserve">
              <pb o="175" file="0181" n="181" rhead="OPTICAE LIBER V."/>
            ceſſariò, ut g p diuidat k o propter arcum a b, qué diuidit ex circulo a b t linea g t per æqualia:</s>
            <s xml:id="echoid-s11662" xml:space="preserve"> [peri-
              <lb/>
            pheria enim b o æquatur peripheriæ c a ex concluſo:</s>
            <s xml:id="echoid-s11663" xml:space="preserve">] & ſimiliter linea k o.</s>
            <s xml:id="echoid-s11664" xml:space="preserve"> Sit ergo punctum con-
              <lb/>
            curſus lineæ g p cum k o, punctum l:</s>
            <s xml:id="echoid-s11665" xml:space="preserve"> & ducatur linea t p.</s>
            <s xml:id="echoid-s11666" xml:space="preserve"> Cum igitur duæ lineæ g p, g t ſint æquales:</s>
            <s xml:id="echoid-s11667" xml:space="preserve">
              <lb/>
            [per15 d 1] erũt [per 5 p 1] duo anguli g p t, g t p æquales:</s>
            <s xml:id="echoid-s11668" xml:space="preserve"> & [per 32 p 1] uterq;</s>
            <s xml:id="echoid-s11669" xml:space="preserve"> acutus.</s>
            <s xml:id="echoid-s11670" xml:space="preserve"> Ductaigitur
              <lb/>
            perpendiculari ſuper g t à punctot:</s>
            <s xml:id="echoid-s11671" xml:space="preserve"> [per 11 p 1] cõtingetcirculum ſpeculi [per conſectarium 16 p 3]
              <lb/>
            & producta, cadet ſuper terminum diametri minoris circuli:</s>
            <s xml:id="echoid-s11672" xml:space="preserve"> cum angulus, quem efficit cum g t, re-
              <lb/>
            ipiciat arcum ſemicirculi minoris circuli:</s>
            <s xml:id="echoid-s11673" xml:space="preserve"> [per 31 p 3] & cũ to cadatſuprako, & k o producta tran-
              <lb/>
            ſeat per cẽtrum minoris circuli:</s>
            <s xml:id="echoid-s11674" xml:space="preserve"> [per conſectarium 1 p 3, quia recta linea o k bifariam, & ad angulos
              <lb/>
            rectos ſecat rectam a b] neceſſario illa perpendicularis cadet ſuper terminum k o producta:</s>
            <s xml:id="echoid-s11675" xml:space="preserve"> [per 31
              <lb/>
            p 3] & p t eſt inſerior illa perpẽdiculari, habito reſpectu ad n.</s>
            <s xml:id="echoid-s11676" xml:space="preserve"> Igitur quæcung;</s>
            <s xml:id="echoid-s11677" xml:space="preserve"> linea ducatur à pun-
              <lb/>
            cto g ad lineam t p, ſecans diametrum illius circuli, quæ eſt o k:</s>
            <s xml:id="echoid-s11678" xml:space="preserve"> cadet in punctum aliquod lineæ t p,
              <lb/>
            citra illam perpendicularem.</s>
            <s xml:id="echoid-s11679" xml:space="preserve"> Cum igitur g p cadat in p, & ſecet o k:</s>
            <s xml:id="echoid-s11680" xml:space="preserve"> erit p citra perpendicularem, &
              <lb/>
            inſra arcum illius perpendicularis.</s>
            <s xml:id="echoid-s11681" xml:space="preserve"> Facto igitur circulo tranſeunte per tria puncta a, b, p:</s>
            <s xml:id="echoid-s11682" xml:space="preserve"> tranſibit
              <lb/>
            quidem per l, & ſecabit circulum a b t in duobus punctis a, b:</s>
            <s xml:id="echoid-s11683" xml:space="preserve"> & cum exeat à puncto b, & iterum re:</s>
            <s xml:id="echoid-s11684" xml:space="preserve">
              <lb/>
            deat in punctum p, inferius punctot, cum p ſit citra illum circulum:</s>
            <s xml:id="echoid-s11685" xml:space="preserve"> neceſſariò ſecabit illum in ter-
              <lb/>
            tio puncto:</s>
            <s xml:id="echoid-s11686" xml:space="preserve"> quod eſt impoſsibile [& contra 10 p 3.</s>
            <s xml:id="echoid-s11687" xml:space="preserve">] Reſtat ergo, ut punctum a non reflectatur ad b à
              <lb/>
            duobus punctis arcus, interiacentis eorum diametros, id eſt arcus e z, ut uterq;</s>
            <s xml:id="echoid-s11688" xml:space="preserve"> angulus conſtans
              <lb/>
            ex angulo incidentiæ & reflexionis ſit minor angulo a g d.</s>
            <s xml:id="echoid-s11689" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div416" type="section" level="0" n="0">
          <head xml:id="echoid-head384" xml:space="preserve" style="it">81. Duo punctain diuerſis diametris circuli (qui eſt cõmunis ſectio ſuperficierum, reflexio-
            <lb/>
          nis, & ſpeculi ſphærici caui) à centro inæquabiliter diſtantia: à duobus punctis peripheriæ com-
            <lb/>
          prehenſæ inter ſemidiametros, in quibus ipſa ſunt, inter ſe mutuò reflecti poſſunt. 35 p 8.</head>
          <p>
            <s xml:id="echoid-s11690" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s11691" xml:space="preserve"> dico quòd poſſunt reflecti duo puncta ad ſe, inæqualis longitudinis à centro, à duo-
              <lb/>
            bus punctis arcus ipſa reſpicientis, id eſt diametros, in quibus ſunt puncta illa, interiacẽtis.</s>
            <s xml:id="echoid-s11692" xml:space="preserve">
              <lb/>
            Verbi gratia:</s>
            <s xml:id="echoid-s11693" xml:space="preserve"> ſumptis duabus ſemidiametris in circulo ſpheræ, ſcilicet b d, g d:</s>
            <s xml:id="echoid-s11694" xml:space="preserve"> diuidatur an-
              <lb/>
            gulus earũ p æqualia, perſemidiametrũ e d:</s>
            <s xml:id="echoid-s11695" xml:space="preserve"> [per 9 p 1] & in b d ſumatur punctũ m, ſupra punctũ,
              <lb/>
            in quod cadet perpendicularis ducta à puncto e ſuper b d:</s>
            <s xml:id="echoid-s11696" xml:space="preserve"> & ſumatur [per 3 p 1] n d æqualis m d:</s>
            <s xml:id="echoid-s11697" xml:space="preserve"> &
              <lb/>
            [per 5 p 4] fiat circulus tranſiens per tria puncta d, m, n:</s>
            <s xml:id="echoid-s11698" xml:space="preserve"> neceſſariò circulus ille tranſibit extra e.</s>
            <s xml:id="echoid-s11699" xml:space="preserve"> Si
              <lb/>
            enim per e:</s>
            <s xml:id="echoid-s11700" xml:space="preserve"> fieret quadrangulũ à quatuor punctis d, n, e, m:</s>
            <s xml:id="echoid-s11701" xml:space="preserve"> & duo anguli illius qua dranguli ſibi op-
              <lb/>
            poſiti ſunt æquales duobus rectis:</s>
            <s xml:id="echoid-s11702" xml:space="preserve"> [per 22 p 3] quod quidẽ non eſſet:</s>
            <s xml:id="echoid-s11703" xml:space="preserve"> cum linea e m ſit ſupra perpen
              <lb/>
            dicularem:</s>
            <s xml:id="echoid-s11704" xml:space="preserve"> & ideo angulus e m d acutus:</s>
            <s xml:id="echoid-s11705" xml:space="preserve"> [per 16 p.</s>
            <s xml:id="echoid-s11706" xml:space="preserve"> 12 d 1] & ſimiliter ei oppoſitus ſuper n, acutus:</s>
            <s xml:id="echoid-s11707" xml:space="preserve">
              <lb/>
            quia e n ſupra perpendicularem eſt.</s>
            <s xml:id="echoid-s11708" xml:space="preserve"> [Quare in quadrilatero circulo inſcripto oppoſiti anguli eſſent
              <lb/>
            minores duobus rectis contra 22 p 3.</s>
            <s xml:id="echoid-s11709" xml:space="preserve">] Similis erit improbatio:</s>
            <s xml:id="echoid-s11710" xml:space="preserve"> ſi tranſeat circulus citra e.</s>
            <s xml:id="echoid-s11711" xml:space="preserve"> Tranſibit
              <lb/>
            ergo extra, & [per 10 p 3] ſecabit circulũ ſphæræ in duobus punctis, ſicut t, l:</s>
            <s xml:id="echoid-s11712" xml:space="preserve"> & ducantur lineæ m t,
              <lb/>
            d t, n t, m i, d l, n l:</s>
            <s xml:id="echoid-s11713" xml:space="preserve"> & ducatur linea m n ſecans t d in puncto f, lineam e d in puncto p.</s>
            <s xml:id="echoid-s11714" xml:space="preserve"> Palàm, cum m d
              <lb/>
            ſit æqualis n d [per fabricationem] & p d cõmunis, & angulus n d p æqualis angulo m d p:</s>
            <s xml:id="echoid-s11715" xml:space="preserve"> [per fa-
              <lb/>
            bricationem] erit [per 4 p 1] triangulum æquale triangulo:</s>
            <s xml:id="echoid-s11716" xml:space="preserve"> & erit angulus f p d rectus:</s>
            <s xml:id="echoid-s11717" xml:space="preserve"> [per 10 d 1]
              <lb/>
            igitur angulus p f d acutus [per 32 p 1] Ducatur [per 11 p 1] à pũcto f perpendicularis ſupert d:</s>
            <s xml:id="echoid-s11718" xml:space="preserve"> quæ
              <lb/>
            ſit k f.</s>
            <s xml:id="echoid-s11719" xml:space="preserve"> Palàm, quòd aliquod punctũ lineę n l, erit infe-
              <lb/>
              <figure xlink:label="fig-0181-01" xlink:href="fig-0181-01a" number="127">
                <variables xml:id="echoid-variables117" xml:space="preserve">k e
                  <gap/>
                t o z r l g b x n p f m q d s n a</variables>
              </figure>
            rius pũcto k, ſumpta inferioritate reſpectu n:</s>
            <s xml:id="echoid-s11720" xml:space="preserve"> ſitillud
              <lb/>
            punctũ z:</s>
            <s xml:id="echoid-s11721" xml:space="preserve"> & ducatur t z linea uſq;</s>
            <s xml:id="echoid-s11722" xml:space="preserve">; ad circulũ, cadẽs in
              <lb/>
            punctũ circuli:</s>
            <s xml:id="echoid-s11723" xml:space="preserve"> quod ſit o.</s>
            <s xml:id="echoid-s11724" xml:space="preserve"> Arcus n o aut minor eſt ar-
              <lb/>
            cu tl:</s>
            <s xml:id="echoid-s11725" xml:space="preserve"> aut nõ Sinõ fuerit minor:</s>
            <s xml:id="echoid-s11726" xml:space="preserve"> ſumatur ex eo arcus
              <lb/>
            minor;</s>
            <s xml:id="echoid-s11727" xml:space="preserve"> & ad terminũ illius arcus ducatur linea à pun
              <lb/>
            cto t:</s>
            <s xml:id="echoid-s11728" xml:space="preserve"> & erit idẽ, ac ſi arcus n o eſſet minor arcutl.</s>
            <s xml:id="echoid-s11729" xml:space="preserve"> Sit
              <lb/>
            igitur n o minortl.</s>
            <s xml:id="echoid-s11730" xml:space="preserve"> Palàm [per 33 p 6] angulus t n l
              <lb/>
            erit maior angulo o t n, quia reſpicit maiorẽ arcum.</s>
            <s xml:id="echoid-s11731" xml:space="preserve">
              <lb/>
            Secetur ex eo æqualis:</s>
            <s xml:id="echoid-s11732" xml:space="preserve"> & ſit i n z:</s>
            <s xml:id="echoid-s11733" xml:space="preserve"> & ſuper punctum t
              <lb/>
            lineæ t m, fiat angulus, æqualis angulo o t n [ք 23 p 1]
              <lb/>
            qui ſit q t m.</s>
            <s xml:id="echoid-s11734" xml:space="preserve"> Cum igiturangulus t m l ſit maior angu-
              <lb/>
            lo m t q:</s>
            <s xml:id="echoid-s11735" xml:space="preserve"> [ք 33 p 6:</s>
            <s xml:id="echoid-s11736" xml:space="preserve"> quia peripheria t l ſubtenſa angulo
              <lb/>
            t m l, maior eſt extheſi, peripheria n o, ſubtẽſa angu-
              <lb/>
            lo n t o, cui æquatus eſt angulus m t q] cõcurret linea
              <lb/>
            t q cũ linea l m:</s>
            <s xml:id="echoid-s11737" xml:space="preserve"> cõcurrat in puncto q.</s>
            <s xml:id="echoid-s11738" xml:space="preserve"> Cum igitur an-
              <lb/>
            gulus l m t ſit æqualis duob.</s>
            <s xml:id="echoid-s11739" xml:space="preserve"> angulis m q t, m t q [per
              <lb/>
            32 p 1] & angulus l n t ſit ęqualis l m t [ք 27 p 3] ꝗa ſunt ſuք eũdẽ arcũ:</s>
            <s xml:id="echoid-s11740" xml:space="preserve"> [l t] & ang
              <gap/>
              <gap/>
            us in z ſit ęqualis
              <lb/>
            in t q:</s>
            <s xml:id="echoid-s11741" xml:space="preserve"> [ք ſabricationẽ] erit angulus int æqualis angulo m q t:</s>
            <s xml:id="echoid-s11742" xml:space="preserve"> & ita triangulũ m q t ſimile triangulo
              <lb/>
            int [eſt enim angulus m t q æquatus angulo o t n:</s>
            <s xml:id="echoid-s11743" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s11744" xml:space="preserve"> ք 32 p 1 triãgula m t q, i t n ſunt æquiangula:</s>
            <s xml:id="echoid-s11745" xml:space="preserve">
              <lb/>
            & ք 4 p.</s>
            <s xml:id="echoid-s11746" xml:space="preserve"> 1 d 6 ſimilia.</s>
            <s xml:id="echoid-s11747" xml:space="preserve">] Et ſimiliter triangulũ i n z eſt ſimile triãgulo t n z:</s>
            <s xml:id="echoid-s11748" xml:space="preserve"> [cõmunis enim eſt angulus
              <lb/>
            n z t:</s>
            <s xml:id="echoid-s11749" xml:space="preserve"> & z n i æquatus eſt ipſi o t n:</s>
            <s xml:id="echoid-s11750" xml:space="preserve"> ergo ք 32 p 1.</s>
            <s xml:id="echoid-s11751" xml:space="preserve">4 p.</s>
            <s xml:id="echoid-s11752" xml:space="preserve">1 d 6 triãgula ſunt ſimilia] & ita ꝓportio n t ad t q,
              <lb/>
            ſicut n i ad m q:</s>
            <s xml:id="echoid-s11753" xml:space="preserve"> & ſimiliter ꝓportio t n ad t z, ſicut in ad n z.</s>
            <s xml:id="echoid-s11754" xml:space="preserve"> Sed t z maior t q:</s>
            <s xml:id="echoid-s11755" xml:space="preserve"> qđ ſic patet.</s>
            <s xml:id="echoid-s11756" xml:space="preserve"> Sit r pun-
              <lb/>
            ctũ, in quo t z ſecat k f.</s>
            <s xml:id="echoid-s11757" xml:space="preserve"> Angulus t freſtrectus:</s>
            <s xml:id="echoid-s11758" xml:space="preserve"> [nã k f քpẽdicularis ducta eſt ſuք t d] quare [ք 32 p 1]
              <lb/>
            angul
              <emph style="sub">9</emph>
            ſtracutus.</s>
            <s xml:id="echoid-s11759" xml:space="preserve"> Igitur angul
              <emph style="sub">9</emph>
            qtfei ęqualis.</s>
            <s xml:id="echoid-s11760" xml:space="preserve"> [Quia enim ex theſi recta d m æquatur ipſi d n:</s>
            <s xml:id="echoid-s11761" xml:space="preserve"> æqua
              <lb/>
            bitur peripheria d m peripheriæ d n ք 28 p 3:</s>
            <s xml:id="echoid-s11762" xml:space="preserve"> & angulus d t m angulo d t n:</s>
            <s xml:id="echoid-s11763" xml:space="preserve"> & m t q æquatus eſt o t n.</s>
            <s xml:id="echoid-s11764" xml:space="preserve">
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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