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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          <p>
            <s xml:id="echoid-s21018" xml:space="preserve">
              <pb o="16" file="0318" n="318" rhead="VITELLONIS OPTICAE"/>
            à latere d a æquale lateri b a, quod ſit linea d f.</s>
            <s xml:id="echoid-s21019" xml:space="preserve"> Et quia linea d c eſt minor latere b c per 19 p 1:</s>
            <s xml:id="echoid-s21020" xml:space="preserve"> quo-
              <lb/>
            niã angulus b d c eſt rectus:</s>
            <s xml:id="echoid-s21021" xml:space="preserve"> protrahatur linea d c, & reſecetur in pũcto g taliter, ut ſit linea d g ęqua
              <lb/>
            lis lineæ b c.</s>
            <s xml:id="echoid-s21022" xml:space="preserve"> Quia ergo trigoni f d g duo latera f d & d g ſunt æqualia duobus lateribus a b & b c tri-
              <lb/>
            goni a b c, & angulus f d g æqualis eſt angulo a b c:</s>
            <s xml:id="echoid-s21023" xml:space="preserve"> quia uterq;</s>
            <s xml:id="echoid-s21024" xml:space="preserve"> rectus:</s>
            <s xml:id="echoid-s21025" xml:space="preserve"> erit per 4 p 1 baſis f g æqualis
              <lb/>
            baſi a c, & reliqui anguli reliquis angulis:</s>
            <s xml:id="echoid-s21026" xml:space="preserve"> angulus ergo f g d æqualis erit angulo a c b.</s>
            <s xml:id="echoid-s21027" xml:space="preserve"> Quia uerò
              <lb/>
            puncta a & fſunt in linea a d, & puncta c & g ſunt in linea d g:</s>
            <s xml:id="echoid-s21028" xml:space="preserve"> palàm, quia lineæ a c & f g ſunt in una
              <lb/>
            ſuperficie, quæ eſt a d g per 2 p 11:</s>
            <s xml:id="echoid-s21029" xml:space="preserve"> ergo interſecant ſe lineæ g f & c a:</s>
            <s xml:id="echoid-s21030" xml:space="preserve"> ſit earũ interſectio in puncto h.</s>
            <s xml:id="echoid-s21031" xml:space="preserve">
              <lb/>
            Quia uerò in trigono c h g latus g c protrahitur, palàm ex 16 p 1, quoniã angulus h c d maior eſt an-
              <lb/>
            gulo h g c:</s>
            <s xml:id="echoid-s21032" xml:space="preserve"> ergo & eius æquali, ſcilicet angulo a c b:</s>
            <s xml:id="echoid-s21033" xml:space="preserve"> angulus ergo a c d maior eſt angulo a c b:</s>
            <s xml:id="echoid-s21034" xml:space="preserve"> quod
              <lb/>
            eſt propoſitũ.</s>
            <s xml:id="echoid-s21035" xml:space="preserve"> Similiterq́;</s>
            <s xml:id="echoid-s21036" xml:space="preserve"> demonſtrandũ in alijs:</s>
            <s xml:id="echoid-s21037" xml:space="preserve"> ſi enim trigona propoſita fuerint in diuerſis locis
              <lb/>
            conſtituta, palàm, quia ipſis æqualia & æquiangula trigona ſic poſſunt ordinari, ut in figura diſpo-
              <lb/>
            nuntur, & demonſtratio facta de ijs ſe extendit ad alia.</s>
            <s xml:id="echoid-s21038" xml:space="preserve"> Patet ergo uniuerſaliter propoſitum.</s>
            <s xml:id="echoid-s21039" xml:space="preserve"> Et ex
              <lb/>
            hoc patet, quòd angulus b a c eſt maior angulo d a c per 32 p 1.</s>
            <s xml:id="echoid-s21040" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div722" type="section" level="0" n="0">
          <head xml:id="echoid-head612" xml:space="preserve" style="it">38. Omnium duorum trigonorum rectangulorũ, quorũ latus ſubtenſum recto angulo unius
            <lb/>
          ad minus latus eiuſdem proportionem habuerit maiorem, quàm latus ſubtenſum recto angulo
            <lb/>
          alterius ad minus latus eiuſdem: erit angulus linearum maioris proportionis maior angulo li-
            <lb/>
          nearum minoris proportionis: & econuerſo.</head>
          <p>
            <s xml:id="echoid-s21041" xml:space="preserve">Sint duo trigona rectangula a b c & d e f, quorũ anguli a b c & d e f ſint recti:</s>
            <s xml:id="echoid-s21042" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s21043" xml:space="preserve"> latus b c minus
              <lb/>
            latere a b, & latus e f minus latere d e:</s>
            <s xml:id="echoid-s21044" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s21045" xml:space="preserve"> maior proportio lineæ a c ad lineam f e.</s>
            <s xml:id="echoid-s21046" xml:space="preserve"> Dico, quòd an-
              <lb/>
            gulus a c b maior eſt angulo d f e.</s>
            <s xml:id="echoid-s21047" xml:space="preserve"> Quia enim maior eſt proportio lineæ a c ad lineã c b, quàm lineæ
              <lb/>
            d f ad lineam f e:</s>
            <s xml:id="echoid-s21048" xml:space="preserve"> ſed per 47 p 1 quadratũ lineæ
              <lb/>
              <figure xlink:label="fig-0318-01" xlink:href="fig-0318-01a" number="300">
                <variables xml:id="echoid-variables284" xml:space="preserve">a k b c</variables>
              </figure>
              <figure xlink:label="fig-0318-02" xlink:href="fig-0318-02a" number="301">
                <variables xml:id="echoid-variables285" xml:space="preserve">d e f</variables>
              </figure>
              <figure xlink:label="fig-0318-03" xlink:href="fig-0318-03a" number="302">
                <variables xml:id="echoid-variables286" xml:space="preserve">h g</variables>
              </figure>
            a c ualet quadrata duarum linearũ a b & c b:</s>
            <s xml:id="echoid-s21049" xml:space="preserve"> &
              <lb/>
            quadratũ lineæ d fualet quadrata duarũ linea
              <lb/>
            rum, quæ ſunt d e & f e:</s>
            <s xml:id="echoid-s21050" xml:space="preserve"> & quia per 20 p 6 pro-
              <lb/>
            portio quadratorũ eſt proportio duplicata la-
              <lb/>
            terũ:</s>
            <s xml:id="echoid-s21051" xml:space="preserve"> patet, quòd maior eſt proportio quadra.</s>
            <s xml:id="echoid-s21052" xml:space="preserve">
              <lb/>
            tia c ad quadratum c b, quàm quadrati d f ad
              <lb/>
            quadratũ f e:</s>
            <s xml:id="echoid-s21053" xml:space="preserve"> eſt ergo per 11 huius maior pro-
              <lb/>
            portio amborũ quadratorũ linearũ a b & b c
              <lb/>
            ad quadra tũ b c, quàm am borũ quadratorũ li-
              <lb/>
            nearũ d e & f e a d quadratũ f e:</s>
            <s xml:id="echoid-s21054" xml:space="preserve"> ergo per 12 hu-
              <lb/>
            ius maior eſt proportio quadrati a b ad qua-
              <lb/>
            dratum b c, quàm quadrati d e ad quadratũ e f:</s>
            <s xml:id="echoid-s21055" xml:space="preserve"> eſt ergo per 22 p 6 maior proportio lineę a b ad line-
              <lb/>
            am b c, quàm lineæ d e ad lineã e f.</s>
            <s xml:id="echoid-s21056" xml:space="preserve"> Eſto, ut, quæ eſt proportio lineæ d e ad lineã e f, eadẽ ſit alicuius
              <lb/>
            lineæ, ut g h ad lineam c b per 3 huius:</s>
            <s xml:id="echoid-s21057" xml:space="preserve"> erit ergo linea g h minor quàm linea a b per 10 p 5.</s>
            <s xml:id="echoid-s21058" xml:space="preserve"> Reſecetur
              <lb/>
            ergo per 3 p 1 ex linea a b æqualis lineæ g h:</s>
            <s xml:id="echoid-s21059" xml:space="preserve"> & ſit b k, & continuetur linea c k:</s>
            <s xml:id="echoid-s21060" xml:space="preserve"> erunt ergo per 6 p 6
              <lb/>
            trigona d e f & k b c æquiangula:</s>
            <s xml:id="echoid-s21061" xml:space="preserve"> angulus itaq;</s>
            <s xml:id="echoid-s21062" xml:space="preserve"> b c k eſt æqualis angulo e f d:</s>
            <s xml:id="echoid-s21063" xml:space="preserve"> ſed angulus b c a eſt
              <lb/>
            maior angulo b c k, totũ parte.</s>
            <s xml:id="echoid-s21064" xml:space="preserve"> Angulus itaq;</s>
            <s xml:id="echoid-s21065" xml:space="preserve"> a c b maior eſt angulo d f e:</s>
            <s xml:id="echoid-s21066" xml:space="preserve"> & hoc eſt ꝓpoſitũ:</s>
            <s xml:id="echoid-s21067" xml:space="preserve"> ex quo
              <lb/>
            etiã patet, quòd eius cõuerſa eſt uera:</s>
            <s xml:id="echoid-s21068" xml:space="preserve"> quoniã in talibus trigonis lineæ maiores angulos continen-
              <lb/>
            tes, maiorem habent ad ſeinuicem proportionem.</s>
            <s xml:id="echoid-s21069" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div724" type="section" level="0" n="0">
          <figure number="303">
            <variables xml:id="echoid-variables287" xml:space="preserve">a c e f b d</variables>
          </figure>
          <head xml:id="echoid-head613" xml:space="preserve" style="it">39. A puncto in aere dato ad ſubſtratam planãſuperficiẽ una linea perpendiculariter, alia
            <lb/>
          obliquè incidente, & linea recta inter pũcta incidentiæ in ipſa ſu
            <lb/>
          perficie protracta: erit angulus à non perpendiculari cũ iacẽte li- nea contentus, minimus omnium angulorum ſub illa obliqua & quacun linea in ſubſtrata ſuperſicie protracta contentorum: & omnis angulus illi propinquior, eſt minor remotiore: & duo ex utra parte æqualiter approximantes, ſunt æquales. Lemma ad 37 the. opticorum Euclidis. 43 theor 6 libri συναγωγυζμ μαθκμα- τικυζμ Pappi.</head>
          <p>
            <s xml:id="echoid-s21070" xml:space="preserve">Sit punctus in aere datus a, cui ſit ſub ſtrata ſuperficies plana, quę
              <lb/>
            b c d, fuper quã ab illo puncto ducatur obliquè linea a b, ducaturq́;</s>
            <s xml:id="echoid-s21071" xml:space="preserve">
              <lb/>
            perpendiculariter linea a c, & copuletur linea b c.</s>
            <s xml:id="echoid-s21072" xml:space="preserve"> Dico, quòd angu-
              <lb/>
            lus a b c eſt minimus omnium angulorũ contentorũ ſub linea obli-
              <lb/>
            qua a b, & ſub unaquaq;</s>
            <s xml:id="echoid-s21073" xml:space="preserve"> linearũ à puncto b ductarũ in ſuperficie b
              <lb/>
            c d:</s>
            <s xml:id="echoid-s21074" xml:space="preserve"> & quòd ſemper propinquior ipſi eſt minor quàm remotior:</s>
            <s xml:id="echoid-s21075" xml:space="preserve"> &
              <lb/>
            quòd duo anguli æquales ſolũ ex utraq;</s>
            <s xml:id="echoid-s21076" xml:space="preserve"> parte ipſius cõſiſtunt.</s>
            <s xml:id="echoid-s21077" xml:space="preserve"> Duca
              <lb/>
            tur enim in data plana ſuperficie, utcunq;</s>
            <s xml:id="echoid-s21078" xml:space="preserve"> contingit, linea b d, & à
              <lb/>
            puncto c ducatur in eadem ſuperficie linea perpendicularis ſuper lineam b d per 11 p 1, quæ ſit c d,
              <lb/>
            & copuletur à puncto a linea a d:</s>
            <s xml:id="echoid-s21079" xml:space="preserve"> eſt ita q;</s>
            <s xml:id="echoid-s21080" xml:space="preserve"> per 22 huius linea a d perpẽdicularis ſuper lineam b d.</s>
            <s xml:id="echoid-s21081" xml:space="preserve"> Et
              <lb/>
            quoniam angulus a c d eſt rectus, palàm per 19 p 1, quoniam obliqua linea a d maior eſt catheto a c:</s>
            <s xml:id="echoid-s21082" xml:space="preserve">
              <lb/>
            linea itaq;</s>
            <s xml:id="echoid-s21083" xml:space="preserve"> b a ad lineam a c maiorẽ habet proportionẽ quàm ad lineã a d per 8 p 5:</s>
            <s xml:id="echoid-s21084" xml:space="preserve"> & anguli b c a &
              <lb/>
            </s>
          </p>
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