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-div703" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s20860" xml:space="preserve">
              <pb o="13" file="0315" n="315" rhead="LIBER I."/>
            recto:</s>
            <s xml:id="echoid-s20861" xml:space="preserve"> in uniuerſaliori ſcientia, ut in ea, quę de elementatis concluſionibus, uniuerſaliorem dignã
              <lb/>
            propoſitione exiſtimantes.</s>
            <s xml:id="echoid-s20862" xml:space="preserve"> Sit ita que angu-
              <lb/>
              <figure xlink:label="fig-0315-01" xlink:href="fig-0315-01a" number="287">
                <variables xml:id="echoid-variables271" xml:space="preserve">b a h c ſ d g e</variables>
              </figure>
            lus rectus a b c, quem in partes tres ęquales uo
              <lb/>
            lumus diuidere:</s>
            <s xml:id="echoid-s20863" xml:space="preserve"> aſſumatur ergo linea quęcun-
              <lb/>
            que, & ſit d e:</s>
            <s xml:id="echoid-s20864" xml:space="preserve"> ſuper quam conſtituatur trigo nũ
              <lb/>
            ęquilaterum per 1 p 1:</s>
            <s xml:id="echoid-s20865" xml:space="preserve"> quòd ſit d f e, cuius angu-
              <lb/>
            lus d f e diuidatur per ęqualia per 9 p 1 ducta li
              <lb/>
            nea f g:</s>
            <s xml:id="echoid-s20866" xml:space="preserve"> erit ergo angulus d f g tertia pars unius
              <lb/>
            recti, cum ipſe ſit ſexta pars duorum rectorum
              <lb/>
            per 32 p 1:</s>
            <s xml:id="echoid-s20867" xml:space="preserve"> ergo per pręcedentem ab angulo re-
              <lb/>
            cto a b c reſecetur angulus a b h ęqualis angu-
              <lb/>
            lo d f g, & diuidatur angulus h b c per ęqualia per 9 p 1:</s>
            <s xml:id="echoid-s20868" xml:space="preserve"> patet ergo propoſitum.</s>
            <s xml:id="echoid-s20869" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div705" type="section" level="0" n="0">
          <head xml:id="echoid-head603" xml:space="preserve" style="it">29. Linea diuidens angulum alicuius trigoni, producta, baſim ſubtenſam illi angulo neceſſa
            <lb/>
          riò ſecabit: & ſi linea ſecans baſim, ad punctum concurſ{us} laterum trigoni producatur: illa an-
            <lb/>
          gulum baſi oppoſitum ſecabit.</head>
          <p>
            <s xml:id="echoid-s20870" xml:space="preserve">Sit, ut linea b d ſecet angulum a b c trigoni a b c.</s>
            <s xml:id="echoid-s20871" xml:space="preserve"> Dico, quòd eadem linea b d producta, neceſſa-
              <lb/>
            riò ſecabit baſim a c illi angulo ſubtenſam.</s>
            <s xml:id="echoid-s20872" xml:space="preserve"> Si enim non ſecabit baſim a c, concurret tamen cũ pro-
              <lb/>
            ducta a c per 14 huius:</s>
            <s xml:id="echoid-s20873" xml:space="preserve"> ideo quia anguli b a c & a b f ſunt
              <lb/>
              <figure xlink:label="fig-0315-02" xlink:href="fig-0315-02a" number="288">
                <variables xml:id="echoid-variables272" xml:space="preserve">b a d c f</variables>
              </figure>
            minores duobus rectis ex hypotheſi & per 32 p 1:</s>
            <s xml:id="echoid-s20874" xml:space="preserve"> ſit ergo
              <lb/>
            concurſus in puncto fultra punctum c.</s>
            <s xml:id="echoid-s20875" xml:space="preserve"> Eſt ergo trigono-
              <lb/>
            rum a b c & a b f angulus b a c cõmunis, & angulus b c a
              <lb/>
            maior angulo b f c per 16 p 1:</s>
            <s xml:id="echoid-s20876" xml:space="preserve"> erit ergo per 32 p 1 angulus a
              <lb/>
            b f maior angulo a b c:</s>
            <s xml:id="echoid-s20877" xml:space="preserve"> non ergo ſecat linea b f angulum
              <lb/>
            a b c:</s>
            <s xml:id="echoid-s20878" xml:space="preserve"> cadet itaq;</s>
            <s xml:id="echoid-s20879" xml:space="preserve"> neceſſariò inter puncta a & c:</s>
            <s xml:id="echoid-s20880" xml:space="preserve"> & ita ſeca-
              <lb/>
            bit baſim a c:</s>
            <s xml:id="echoid-s20881" xml:space="preserve"> quia ſi etiam caderet in punctũ a, uel in pun-
              <lb/>
            ctum c, non adhuc diuideret angulum a b c:</s>
            <s xml:id="echoid-s20882" xml:space="preserve"> patet ergo ꝓ-
              <lb/>
            poſitum primum.</s>
            <s xml:id="echoid-s20883" xml:space="preserve"> Patet etiã & reliquum propoſitorum:</s>
            <s xml:id="echoid-s20884" xml:space="preserve">
              <lb/>
            quoniam ſi linea b d ſecet baſim trigoni a b c, & applice-
              <lb/>
            tur puncto b, quod eſt punctus concurſus laterum a b & c b:</s>
            <s xml:id="echoid-s20885" xml:space="preserve"> patet, quòd linea b d ſecabit angulum
              <lb/>
            a b c:</s>
            <s xml:id="echoid-s20886" xml:space="preserve"> ſit enim per 16 p 1 angulus a d b maior angulo b a c b:</s>
            <s xml:id="echoid-s20887" xml:space="preserve"> ſed angulus a c eſt cõmunis ambobus tri
              <lb/>
            gonis a b c & a b d:</s>
            <s xml:id="echoid-s20888" xml:space="preserve"> ergo per 32 p 1 angulus a b d eſt minor angulo a b c.</s>
            <s xml:id="echoid-s20889" xml:space="preserve"> Eſt ergo ſectus angulus a b c
              <lb/>
            per lineam b d:</s>
            <s xml:id="echoid-s20890" xml:space="preserve"> quod eſt ſecundum propoſitorum.</s>
            <s xml:id="echoid-s20891" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div707" type="section" level="0" n="0">
          <head xml:id="echoid-head604" xml:space="preserve" style="it">30. Ab angulo dati trigoni linea perpendiculariter ad baſim producta, ſirectangulum ſub
            <lb/>
          partibus baſis contentum, maius fuerit quadrato perpendicularis: neceſſe est angulum (à quo
            <lb/>
          fit ductio) obtuſum eſſe: ſi minus, acutum: ſi æquale, rectum.</head>
          <p>
            <s xml:id="echoid-s20892" xml:space="preserve">Sit datus trigonus a b c, à cuius angulo b a c ducatur linea perpendicularis ſuper baſim b c:</s>
            <s xml:id="echoid-s20893" xml:space="preserve"> ſe-
              <lb/>
            cetq́;</s>
            <s xml:id="echoid-s20894" xml:space="preserve"> ipſam in puncto d:</s>
            <s xml:id="echoid-s20895" xml:space="preserve"> & ſit a d:</s>
            <s xml:id="echoid-s20896" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s20897" xml:space="preserve"> illud, quod fit ex ductu b d in d c maius quadrato lineæ a d.</s>
            <s xml:id="echoid-s20898" xml:space="preserve">
              <lb/>
            Dico, quòd angulus b a c eſt obtuſus.</s>
            <s xml:id="echoid-s20899" xml:space="preserve"> Patet e-
              <lb/>
              <figure xlink:label="fig-0315-03" xlink:href="fig-0315-03a" number="289">
                <variables xml:id="echoid-variables273" xml:space="preserve">a b d c</variables>
              </figure>
              <figure xlink:label="fig-0315-04" xlink:href="fig-0315-04a" number="290">
                <variables xml:id="echoid-variables274" xml:space="preserve">g e</variables>
              </figure>
              <figure xlink:label="fig-0315-05" xlink:href="fig-0315-05a" number="291">
                <variables xml:id="echoid-variables275" xml:space="preserve">a b d c</variables>
              </figure>
              <figure xlink:label="fig-0315-06" xlink:href="fig-0315-06a" number="292">
                <variables xml:id="echoid-variables276" xml:space="preserve">a b d f c</variables>
              </figure>
            nim per 17 p 6, quia non eſt proportio lineæ
              <lb/>
            b d ad lineam a d, quæ lineæ a d ad lineam d c.</s>
            <s xml:id="echoid-s20900" xml:space="preserve">
              <lb/>
            ſit ergo per 12 p 6, ut quæ eſt proportio lineæ
              <lb/>
            b d ad lineam a d, eadem ſit lineæ a d ad lineã
              <lb/>
            g e:</s>
            <s xml:id="echoid-s20901" xml:space="preserve"> erit ergo illud, quod fit ex ductu lineæ b d
              <lb/>
            in lineam g e æquale quadrato lineæ a d per
              <lb/>
            17 p 6:</s>
            <s xml:id="echoid-s20902" xml:space="preserve"> quia illud, quod fit ex ductu lineę b d in
              <lb/>
            lineam d c, eſt maius quadrato lineę a d:</s>
            <s xml:id="echoid-s20903" xml:space="preserve"> patet,
              <lb/>
            quòd linea g e eſt minor quàm linea d c per 1
              <lb/>
            p 6.</s>
            <s xml:id="echoid-s20904" xml:space="preserve"> Abſcindatur ergo à linea d c æqualis lineę
              <lb/>
            g e per 3 p 1, & ſit d f, ducaturq́;</s>
            <s xml:id="echoid-s20905" xml:space="preserve"> linea a f.</s>
            <s xml:id="echoid-s20906" xml:space="preserve"> Quia
              <lb/>
            itaq;</s>
            <s xml:id="echoid-s20907" xml:space="preserve"> illud, quod fit ex ductu lineæ b d in lineam d f, eſt æquale quadrato lineæ a d:</s>
            <s xml:id="echoid-s20908" xml:space="preserve"> patet per 17 p 6,
              <lb/>
            quoniam eſt proportio lineæ b d ad lineã a d, ſicut lineæ a d ad lineã d f:</s>
            <s xml:id="echoid-s20909" xml:space="preserve"> erit ergo per conuerſam 8
              <lb/>
            p 6 angulus b a f rectus.</s>
            <s xml:id="echoid-s20910" xml:space="preserve"> Ergo angulus b a c eſt eſt maior recto.</s>
            <s xml:id="echoid-s20911" xml:space="preserve"> Similiterq́;</s>
            <s xml:id="echoid-s20912" xml:space="preserve"> demonſtrandum, quòd ſi
              <lb/>
            illud, quòd fit ex ductu b d in d c ſit minus quadrato a d, quoniam angulus b a c eſt acutus:</s>
            <s xml:id="echoid-s20913" xml:space="preserve"> nam per
              <lb/>
            eadem fit demonftratio.</s>
            <s xml:id="echoid-s20914" xml:space="preserve"> Pater etiam per eandem conuerſam 8 p 6, quoniam ſi illud, quod fit ex du
              <lb/>
            ctu lineæ b d in lineam d c, ſit æquale quadrato lineæ a d, quoniam angulus b a c eſt rectus:</s>
            <s xml:id="echoid-s20915" xml:space="preserve"> patet
              <lb/>
            ergo propoſitum.</s>
            <s xml:id="echoid-s20916" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div709" type="section" level="0" n="0">
          <head xml:id="echoid-head605" xml:space="preserve" style="it">31. Abangulo iſoſcelis ducta perpendicularis ſuper baſim in duos partiales ſimiles trigo-
            <lb/>
          nos diuidit iſoſcelem. Ex quo patet, quòd linea perpendicularis ad medium punctum baſis ne-
            <lb/>
          ceſſariò pertingit.</head>
          <p>
            <s xml:id="echoid-s20917" xml:space="preserve">Sit iſoſceles a b c, cuius latera a b & a c ſint æqualia:</s>
            <s xml:id="echoid-s20918" xml:space="preserve"> & ab angulo b a c ducatur ſuper ba-
              <lb/>
            </s>
          </p>
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