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-div855" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s23251" xml:space="preserve">
              <pb o="50" file="0352" n="352" rhead="VITELLONIS OPTICAE"/>
            mam concurrent in puncto uno:</s>
            <s xml:id="echoid-s23252" xml:space="preserve"> ſit punctus concurſus z:</s>
            <s xml:id="echoid-s23253" xml:space="preserve"> & lineæ productę ſint b z, c z, d z:</s>
            <s xml:id="echoid-s23254" xml:space="preserve"> ſitq́ue an
              <lb/>
            gulus b z c ęqualis angulo c z d:</s>
            <s xml:id="echoid-s23255" xml:space="preserve"> & ducatur linea z k.</s>
            <s xml:id="echoid-s23256" xml:space="preserve"> Dico, quòd angulus c z k eſt rectus.</s>
            <s xml:id="echoid-s23257" xml:space="preserve"> A puncto
              <lb/>
              <figure xlink:label="fig-0352-01" xlink:href="fig-0352-01a" number="377">
                <variables xml:id="echoid-variables361" xml:space="preserve">z b g c d k h</variables>
              </figure>
            enim c ducatur per 31 p 1 linea ęquidiſtãs lineę
              <lb/>
            z k, quæ ſit c h:</s>
            <s xml:id="echoid-s23258" xml:space="preserve"> quæ producta ſecabit lineam
              <lb/>
            z b per 2 huius:</s>
            <s xml:id="echoid-s23259" xml:space="preserve"> ſecet ergo ipſam in puncto g:</s>
            <s xml:id="echoid-s23260" xml:space="preserve">
              <lb/>
            & producatur linea z d, donec concurrat cum
              <lb/>
            linea g c h (concurret autem per 2 huius) & ſit
              <lb/>
            cõcurſus punctus h.</s>
            <s xml:id="echoid-s23261" xml:space="preserve"> Quia igitur ex hypotheſi
              <lb/>
            eſt proportio lineę b k ad lineam k d, ſicut li-
              <lb/>
            neę b c ad lineam c d, erit per 16 p 5 permuta-
              <lb/>
            tim proportio lineę b k ad lineam b c, ſicut li-
              <lb/>
            neę k d ad lineam d c:</s>
            <s xml:id="echoid-s23262" xml:space="preserve"> ſed per 29 p 1 trigona b z
              <lb/>
            k & b g c ſunt ęquiangula:</s>
            <s xml:id="echoid-s23263" xml:space="preserve"> ergo per 4 p 6 eſt proportio lineę b k ad lineam b c, quę eſt lineę z k ad li-
              <lb/>
            neam g c:</s>
            <s xml:id="echoid-s23264" xml:space="preserve"> ergo per 11 p 5 erit proportio lineæ z k ad lineam g c, ſicut lineæ k d ad lineam d c:</s>
            <s xml:id="echoid-s23265" xml:space="preserve"> ſed quæ
              <lb/>
            eſt proportio lineæ k d ad lineam d c, eadem eſt lineæ k z ad lineam c h per 15 & 29 p 1 & per 4 p 6:</s>
            <s xml:id="echoid-s23266" xml:space="preserve"> ꝗa
              <lb/>
            trigona k d z & c d h ſunt æquiangula.</s>
            <s xml:id="echoid-s23267" xml:space="preserve"> Habet itaque linea z k ad ambas lineas g c & h c eandem pro-
              <lb/>
            portionem:</s>
            <s xml:id="echoid-s23268" xml:space="preserve"> ergo per 9 p 5 linea g c eſt æqualis lineæ c h:</s>
            <s xml:id="echoid-s23269" xml:space="preserve"> ſed per 3 p 6 eſt proportio lineæ g c ad lineã
              <lb/>
            c h, ſicut lineæ g z ad lineam z h, cum linea z c diuidat angulum g z h per æqualia.</s>
            <s xml:id="echoid-s23270" xml:space="preserve"> Eſt ergo linea g z
              <lb/>
            æqualis lineę z h.</s>
            <s xml:id="echoid-s23271" xml:space="preserve"> Et quoniam linea g c eſt æqualis lineæ c h, & linea g z ęqualis lineæ z h, & latus c z
              <lb/>
            eſt commune ambobus trigonis g z c & h z c:</s>
            <s xml:id="echoid-s23272" xml:space="preserve"> erit per 8 p 1 angulus z c h æqualis angulo z c g:</s>
            <s xml:id="echoid-s23273" xml:space="preserve"> uter-
              <lb/>
            que ergo ipſorũ eſt rectus.</s>
            <s xml:id="echoid-s23274" xml:space="preserve"> Ergo per 29 p 1 angulus k z c eſt rectus:</s>
            <s xml:id="echoid-s23275" xml:space="preserve"> lineæ enim z k & c h ſunt æquidi-
              <lb/>
            ſtantes.</s>
            <s xml:id="echoid-s23276" xml:space="preserve"> Patet ergo propoſitum.</s>
            <s xml:id="echoid-s23277" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div857" type="section" level="0" n="0">
          <head xml:id="echoid-head700" xml:space="preserve" style="it">126. Diuiſa linea per inæqualia: poßibile est minoriſüæ parti lineam adiungi, ita, ut illud,
            <lb/>
          quod fit ex ductu toti{us} lineæ diuiſæ cum adiecta in ipſam adiectam, æquale ſit quadrato ei{us},
            <lb/>
          quæ constat ex minore & adiecta.</head>
          <p>
            <s xml:id="echoid-s23278" xml:space="preserve">Sit data linea a b diuiſa per inęqualia in puncto c:</s>
            <s xml:id="echoid-s23279" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s23280" xml:space="preserve"> linea a c maior quã linea b c.</s>
            <s xml:id="echoid-s23281" xml:space="preserve"> Dico, quòd eſt
              <lb/>
            poſsibile inuenire quandam lineam, quæ adiecta ipſi lineæ b c, id efficiat, ut hoc, quod fit ex ductu
              <lb/>
            lineę compoſitæ ex linea a b, & ex adiecta in ipſam adiectam ſit æquale quadrato lineæ, quæ conſtat
              <lb/>
            ex b c parte minore, & ex adiecta.</s>
            <s xml:id="echoid-s23282" xml:space="preserve"> Aſſumatur enim quædam alia linea æqualis, uel minor linea a b,
              <lb/>
            quæ ſit d e, & quæ eſt proportio lineæ a c ad lineam b c, eadem ſit proportio lineæ d e ad quandam
              <lb/>
              <figure xlink:label="fig-0352-02" xlink:href="fig-0352-02a" number="378">
                <variables xml:id="echoid-variables362" xml:space="preserve">e d f</variables>
              </figure>
              <figure xlink:label="fig-0352-03" xlink:href="fig-0352-03a" number="379">
                <variables xml:id="echoid-variables363" xml:space="preserve">g h a c b i</variables>
              </figure>
            aliam lineã per
              <lb/>
            3 huius:</s>
            <s xml:id="echoid-s23283" xml:space="preserve"> quæ ſit
              <lb/>
            e f:</s>
            <s xml:id="echoid-s23284" xml:space="preserve"> aſſumatúr-
              <lb/>
            que linea d f ę-
              <lb/>
            qualis lineæ a
              <lb/>
            b.</s>
            <s xml:id="echoid-s23285" xml:space="preserve"> Et quoniam
              <lb/>
            exlineis d e, e
              <lb/>
            f, d f quęcun q;</s>
            <s xml:id="echoid-s23286" xml:space="preserve">
              <lb/>
            duę ſimul iun-
              <lb/>
            ctæ maiores ſunt tertia, ut patet ex præmiſsis, poſsibile eſt conſtitui triangulum per 22 p 1.</s>
            <s xml:id="echoid-s23287" xml:space="preserve"> Conſti-
              <lb/>
            tuatur ergo, & ſit d e f.</s>
            <s xml:id="echoid-s23288" xml:space="preserve"> Super terminum itaque lineæ a b, qui eſt a, conſtituatur angulus æqualis an-
              <lb/>
            gulo e d f per 23 p 1, qui ſit g a b:</s>
            <s xml:id="echoid-s23289" xml:space="preserve"> & reſecetur linea a g ad ęqualitatem lineæ d e, & ducatur linea g b.</s>
            <s xml:id="echoid-s23290" xml:space="preserve"> Er
              <lb/>
            go per 4 p 1, cum linea d f ſit æqualis lineæ a b, & linea a g æqualis lineę d e, & angulus g a b ſit ęqua-
              <lb/>
            lis angulo e d f:</s>
            <s xml:id="echoid-s23291" xml:space="preserve"> erit linea g b æqualis lineæ e f, & reliqui anguli trigoni a g b æquales erunt reliquis
              <lb/>
            angulis trigoni d e f.</s>
            <s xml:id="echoid-s23292" xml:space="preserve"> Ducatur itaq;</s>
            <s xml:id="echoid-s23293" xml:space="preserve"> linea g c.</s>
            <s xml:id="echoid-s23294" xml:space="preserve"> Et quoniam proportio lineæ d e ad lineã e f, ſicut lineæ
              <lb/>
            a c ad lineam c b:</s>
            <s xml:id="echoid-s23295" xml:space="preserve"> erit proportio lineæ a g ad lineam g b, ſicut lineæ a c ad lineam c b per 7 p 5:</s>
            <s xml:id="echoid-s23296" xml:space="preserve"> ergo ք
              <lb/>
            3 p 6 angulus a g b diuiſus eſt per æqualia:</s>
            <s xml:id="echoid-s23297" xml:space="preserve"> palã autem, quòd angulus g c b eſt acutus:</s>
            <s xml:id="echoid-s23298" xml:space="preserve"> ſienim ſit re-
              <lb/>
            ctus, tunc trianguli a g c & g c b æquianguli per 32 p 1, quoniam ad punctum g duo ipſorum anguli
              <lb/>
            ſunt æquales:</s>
            <s xml:id="echoid-s23299" xml:space="preserve"> ergo latera eorum ſunt proportionalia per 4 p 6:</s>
            <s xml:id="echoid-s23300" xml:space="preserve"> erit ergo proportio lateris a c ad c b,
              <lb/>
            ſicut lateris g c ad ſeipſum:</s>
            <s xml:id="echoid-s23301" xml:space="preserve"> æqualis eſt ergo linea a c lineę c b:</s>
            <s xml:id="echoid-s23302" xml:space="preserve"> quod eſt contra hypotheſin & impoſ-
              <lb/>
            ſibile.</s>
            <s xml:id="echoid-s23303" xml:space="preserve"> Si uerò angulus g c b detur eſſe obtuſus, maior angulo g c a, palã per 32 p 1, quoniam angulus
              <lb/>
            g b c eſt minor angulo g a c.</s>
            <s xml:id="echoid-s23304" xml:space="preserve"> Ergo per 19 p 1 in trigono a g b latus g b maius eſt latere a g.</s>
            <s xml:id="echoid-s23305" xml:space="preserve"> Et quia eſt
              <lb/>
            proportio lineę b g ad lineam g a, ſicut lineę b c ad lineam c a:</s>
            <s xml:id="echoid-s23306" xml:space="preserve"> erit per 5 huius, per proportionem ſci-
              <lb/>
            licet, è contrario latus b c maius quàm latus a c:</s>
            <s xml:id="echoid-s23307" xml:space="preserve"> quod eſt contra hypotheſim.</s>
            <s xml:id="echoid-s23308" xml:space="preserve"> Palàm ergo, quoniam
              <lb/>
            angulus g c b eſt acutus.</s>
            <s xml:id="echoid-s23309" xml:space="preserve"> Ducatur itaque per 31 p 1, à pũcto c linea c h ęquidiſtans lineę g a, ſecans li-
              <lb/>
            neam g b in puncto h:</s>
            <s xml:id="echoid-s23310" xml:space="preserve"> erit ergo per 29 p 1 angulus g c h æqualis angulo c g a:</s>
            <s xml:id="echoid-s23311" xml:space="preserve"> ergo & angulo c g h:</s>
            <s xml:id="echoid-s23312" xml:space="preserve"> e-
              <lb/>
            rit quoque angulus h c b ęqualis angulo g a c.</s>
            <s xml:id="echoid-s23313" xml:space="preserve"> Super punctum itaque g terminum lineę b g fiat per
              <lb/>
            23 p 1 angulus ęqualis angulo g a c:</s>
            <s xml:id="echoid-s23314" xml:space="preserve"> ergo & angulo h c b, qui ſit b g i.</s>
            <s xml:id="echoid-s23315" xml:space="preserve"> Et quia angulus g c b eſt æqua-
              <lb/>
            lis duobus angulis c g a & c a g, ut patet ex pręmiſsis, & per 32 p 1:</s>
            <s xml:id="echoid-s23316" xml:space="preserve"> erit angulùs i g c ęqualis angulo
              <lb/>
            g c b.</s>
            <s xml:id="echoid-s23317" xml:space="preserve"> Et quoniam angulus g c b eſt acutus:</s>
            <s xml:id="echoid-s23318" xml:space="preserve"> palã ergo per 14 huius, quoniam lineę g i & c b concurrẽt:</s>
            <s xml:id="echoid-s23319" xml:space="preserve">
              <lb/>
            ſit punctus concurſus i.</s>
            <s xml:id="echoid-s23320" xml:space="preserve"> Ergo per 6 p 1 erit latus g i ęquale lateri c i.</s>
            <s xml:id="echoid-s23321" xml:space="preserve"> Quia itaq;</s>
            <s xml:id="echoid-s23322" xml:space="preserve"> angulus b g i eſt ęqua-
              <lb/>
            lis angulo g a i, & angulus g i a communis ambobus trigonis a g i & b g i:</s>
            <s xml:id="echoid-s23323" xml:space="preserve"> erit per 32 p 1 angulus a g i
              <lb/>
            æqualis angulo g b i.</s>
            <s xml:id="echoid-s23324" xml:space="preserve"> Ergo per 4 p 6 erit proportio lineę a i ad lineam i g, ſicut lineæ i g ad lineam
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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