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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          <head xml:id="echoid-head625" xml:space="preserve" style="it">
            <pb o="22" file="0324" n="324" rhead="VITELLONIS OPTICAE"/>
          lum cum baſi continet, maior angulo ſuperiori alterius: & ſi minor, minor.</head>
          <p>
            <s xml:id="echoid-s21439" xml:space="preserve">Sint itẽ duo trianguli a b c & d e f, habentes baſes b c & e f æquales:</s>
            <s xml:id="echoid-s21440" xml:space="preserve"> diuidaturq́;</s>
            <s xml:id="echoid-s21441" xml:space="preserve"> baſis b c ք ęqua-
              <lb/>
            lia in puncto g, & baſis e f in
              <lb/>
              <figure xlink:label="fig-0324-01" xlink:href="fig-0324-01a" number="318">
                <variables xml:id="echoid-variables302" xml:space="preserve">k a n m b g c l</variables>
              </figure>
              <figure xlink:label="fig-0324-02" xlink:href="fig-0324-02a" number="319">
                <variables xml:id="echoid-variables303" xml:space="preserve">d e h f</variables>
              </figure>
            pũcto h:</s>
            <s xml:id="echoid-s21442" xml:space="preserve"> & ducãtur lineę a g,
              <lb/>
            d h, quę ſint ęquales, & utraq;</s>
            <s xml:id="echoid-s21443" xml:space="preserve">
              <lb/>
            ipſarum incidat obliquè ſuæ
              <lb/>
            baſi:</s>
            <s xml:id="echoid-s21444" xml:space="preserve"> ſit aũt angulus a g c ma-
              <lb/>
            ior angulo d h f.</s>
            <s xml:id="echoid-s21445" xml:space="preserve"> Dico, quòd ſi
              <lb/>
            maior ſit linea a g, ꝗ̃ linea g c:</s>
            <s xml:id="echoid-s21446" xml:space="preserve">
              <lb/>
            erit angulus b a c maior an
              <lb/>
            gulo e d f:</s>
            <s xml:id="echoid-s21447" xml:space="preserve"> & ſi linea a g ſit mi-
              <lb/>
            nor, ꝗ̃ linea g c, erit angulus
              <lb/>
            b a c minor angulo e d f.</s>
            <s xml:id="echoid-s21448" xml:space="preserve"> Cir-
              <lb/>
            cum ſcribatur enim per 5 p 4
              <lb/>
            trigono a b c circulus:</s>
            <s xml:id="echoid-s21449" xml:space="preserve"> & duca
              <lb/>
            tur à puncto g perpendicula-
              <lb/>
            ris ſuper lineã b c per 11 p 1:</s>
            <s xml:id="echoid-s21450" xml:space="preserve">
              <lb/>
            quæ producta ad circũferen-
              <lb/>
            tiam, ſit g k.</s>
            <s xml:id="echoid-s21451" xml:space="preserve"> Erit itaq;</s>
            <s xml:id="echoid-s21452" xml:space="preserve"> g k per 1 p 3 pars diametri circuli propoſiti, quę cõpleta, ſit k l.</s>
            <s xml:id="echoid-s21453" xml:space="preserve"> Sit itaq;</s>
            <s xml:id="echoid-s21454" xml:space="preserve">, ut prius,
              <lb/>
            linea a g maior ꝗ̃ linea g c:</s>
            <s xml:id="echoid-s21455" xml:space="preserve"> eſt aũt linea k g maior, ꝗ̃ linea g l per 48 huius.</s>
            <s xml:id="echoid-s21456" xml:space="preserve"> In linea ergo g k eſt centrũ
              <lb/>
            circuli:</s>
            <s xml:id="echoid-s21457" xml:space="preserve"> eſt ergo linea k g maior ꝗ̃ linea a g per 7 p 3:</s>
            <s xml:id="echoid-s21458" xml:space="preserve"> ergo & maior ꝗ̃ linea d h, quę eſt ęqualis ipſi a g ex
              <lb/>
            hypotheſi.</s>
            <s xml:id="echoid-s21459" xml:space="preserve"> Fiat itaq;</s>
            <s xml:id="echoid-s21460" xml:space="preserve"> per 23 p 1 ſuper punctũ g terminũ lineę g c, angulus ęqualis angulo d h f, qui ſit
              <lb/>
            m g c:</s>
            <s xml:id="echoid-s21461" xml:space="preserve"> cadatq́;</s>
            <s xml:id="echoid-s21462" xml:space="preserve"> pũctũ m in peripheriã circuli.</s>
            <s xml:id="echoid-s21463" xml:space="preserve"> E ſt itaq;</s>
            <s xml:id="echoid-s21464" xml:space="preserve"> ք 7 p 3 linea a g maior ꝗ̃ linea m g:</s>
            <s xml:id="echoid-s21465" xml:space="preserve"> ergo & linea
              <lb/>
            d h eſt maior ꝗ̃ linea m g.</s>
            <s xml:id="echoid-s21466" xml:space="preserve"> Producatur itaq;</s>
            <s xml:id="echoid-s21467" xml:space="preserve">, donec linea g m ſit ęqualis lineę d h:</s>
            <s xml:id="echoid-s21468" xml:space="preserve"> & ducãtur lineę n c
              <lb/>
            & n b.</s>
            <s xml:id="echoid-s21469" xml:space="preserve"> Erit itaq;</s>
            <s xml:id="echoid-s21470" xml:space="preserve"> angulus b n c ęqualis angulo e d f:</s>
            <s xml:id="echoid-s21471" xml:space="preserve"> ſed angulus b m c eſt maior angulo b n c:</s>
            <s xml:id="echoid-s21472" xml:space="preserve"> eſt ergo
              <lb/>
            angulus b a c maior angulo e d f per modum pręoſtẽſum.</s>
            <s xml:id="echoid-s21473" xml:space="preserve"> Similiter quoq;</s>
            <s xml:id="echoid-s21474" xml:space="preserve"> demonſtrandũ, ſi linea a g
              <lb/>
            ſit minor ꝗ̃ linea g c, quòd minor eſt angulus b a c angulo e d f:</s>
            <s xml:id="echoid-s21475" xml:space="preserve"> quod proponebatur demonſtrandũ.</s>
            <s xml:id="echoid-s21476" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div745" type="section" level="0" n="0">
          <figure number="320">
            <variables xml:id="echoid-variables304" xml:space="preserve">l n m d f e a g c h o k d f e b</variables>
          </figure>
          <head xml:id="echoid-head626" xml:space="preserve" style="it">52. Siduas lineas rectas ſecantes circulũ, æqua
            <lb/>
          les arcus interiaceant, illæ neceſſariò ſunt æquidi- ſtantes: idem́ accidit, ſi una earum fuerit ſecans& alia contingens.</head>
          <p>
            <s xml:id="echoid-s21477" xml:space="preserve">Sit circulus a b c, cuius centrum ſit punctum o:</s>
            <s xml:id="echoid-s21478" xml:space="preserve"> ſe-
              <lb/>
            centq́;</s>
            <s xml:id="echoid-s21479" xml:space="preserve"> duæ lineę a c & d e illum circulum taliter, ut ar
              <lb/>
            cus d a ſit ęqualis arcui e c.</s>
            <s xml:id="echoid-s21480" xml:space="preserve"> Dico, quòd lineæ a c & d e
              <lb/>
            ſunt ęquidiſtantes.</s>
            <s xml:id="echoid-s21481" xml:space="preserve"> Autitaq;</s>
            <s xml:id="echoid-s21482" xml:space="preserve"> o centrũ circuli eſt in al-
              <lb/>
            tera illarum linearum, aut in neurra:</s>
            <s xml:id="echoid-s21483" xml:space="preserve"> & tuncuel inter
              <lb/>
            utraſq;</s>
            <s xml:id="echoid-s21484" xml:space="preserve">, uel extra utraſq;</s>
            <s xml:id="echoid-s21485" xml:space="preserve">. Si ſit in altera ipſarum:</s>
            <s xml:id="echoid-s21486" xml:space="preserve"> eſto
              <lb/>
            quòd ſit in linea a c, & à centro o ducatur linea perpẽ
              <lb/>
            dicularis ſuper a c per 11 p 1, & producatur ad circũfe
              <lb/>
            rentiã, ſitq́;</s>
            <s xml:id="echoid-s21487" xml:space="preserve"> o b ſecans lineã d e in puncto f:</s>
            <s xml:id="echoid-s21488" xml:space="preserve"> & ducan-
              <lb/>
            tur lineę o d, o e, quę cum ſint ęquales, erunt per 5 p 1,
              <lb/>
            anguli o d f & o e f æquales:</s>
            <s xml:id="echoid-s21489" xml:space="preserve"> ſed angulus f o a eſt ęqua
              <lb/>
            lis angulo f o c, ꝗ a ſunt recti:</s>
            <s xml:id="echoid-s21490" xml:space="preserve"> angulus uerò d o a ęqua
              <lb/>
            lis eſt angulo e o c per 27 p 3, cum ex hypotheſi arcus d a ſit æqualis arcui e c:</s>
            <s xml:id="echoid-s21491" xml:space="preserve"> erit ergo angulus d o f
              <lb/>
            æqualis angulo e o f:</s>
            <s xml:id="echoid-s21492" xml:space="preserve"> ergo per 32 p 1 erit angulus d f o ęqualis angulo e f o:</s>
            <s xml:id="echoid-s21493" xml:space="preserve"> eſt ergo linea of perpendi
              <lb/>
            cularis ſuper lineã d e.</s>
            <s xml:id="echoid-s21494" xml:space="preserve"> Erunt ergo per 28 p 1 lineę d e,
              <lb/>
              <figure xlink:label="fig-0324-04" xlink:href="fig-0324-04a" number="321">
                <variables xml:id="echoid-variables305" xml:space="preserve">a o c d f e b</variables>
              </figure>
            & a c ęquidiſtãtes.</s>
            <s xml:id="echoid-s21495" xml:space="preserve"> Si uerò centrũ o fuerit inter ipſas
              <lb/>
            lineas a c & d e:</s>
            <s xml:id="echoid-s21496" xml:space="preserve"> ductis lineis à centro perpẽdicularib.</s>
            <s xml:id="echoid-s21497" xml:space="preserve">
              <lb/>
            ſuper utranq;</s>
            <s xml:id="echoid-s21498" xml:space="preserve"> illarũ, quę ſint o f, & o g, & ductis lineis
              <lb/>
            ad terminos linearum a c & d e, à cẽtro o, quę ſint o a,
              <lb/>
            o c, o d, o e, & diametro h k:</s>
            <s xml:id="echoid-s21499" xml:space="preserve"> fient ex utraq;</s>
            <s xml:id="echoid-s21500" xml:space="preserve"> parte cen-
              <lb/>
            tri o quatuor anguli ęquales duobus rectis ideo quia
              <lb/>
            anguli circa centrum ualent quatuor rectos, quo, ex
              <lb/>
            ęquo diuidit quælibet diameter:</s>
            <s xml:id="echoid-s21501" xml:space="preserve"> ſed angulus e o c eſt
              <lb/>
            ęqualis angulo d o a per 27 p 3:</s>
            <s xml:id="echoid-s21502" xml:space="preserve"> remanet ergo angulus
              <lb/>
            d o e ęqualis angulo a o c:</s>
            <s xml:id="echoid-s21503" xml:space="preserve"> per definitionẽ ergo circu-
              <lb/>
            li & per 6 p 6 trianguli d o e & a o c ſunt inuicẽ ęquiã
              <lb/>
            guli:</s>
            <s xml:id="echoid-s21504" xml:space="preserve"> ergo erit angulus g c o æqualis angulo o d f:</s>
            <s xml:id="echoid-s21505" xml:space="preserve"> ſed
              <lb/>
            angulus o g c eſt ęqualis angulo o f d:</s>
            <s xml:id="echoid-s21506" xml:space="preserve"> quia uterq;</s>
            <s xml:id="echoid-s21507" xml:space="preserve"> re-
              <lb/>
            ctus ex pręmiſsis:</s>
            <s xml:id="echoid-s21508" xml:space="preserve"> ergo per 32 p 1 trigona g o c, d o f
              <lb/>
            ſunt æquiangula:</s>
            <s xml:id="echoid-s21509" xml:space="preserve"> ergo per 14 p 1 lineę d o & o c con-
              <lb/>
            iunctæ ſunt linea una:</s>
            <s xml:id="echoid-s21510" xml:space="preserve"> quia anguli c o h & d o h ex præmiſsis ſunt ęquales duobus rectis.</s>
            <s xml:id="echoid-s21511" xml:space="preserve"> Ergo
              <lb/>
            per 27 p 1 patet propoſitum.</s>
            <s xml:id="echoid-s21512" xml:space="preserve"> Quòd ſi centrum o fuerit extra utraſque:</s>
            <s xml:id="echoid-s21513" xml:space="preserve"> ducatur perpendicu-
              <lb/>
            laris à centro o ſuperipſarum alteram:</s>
            <s xml:id="echoid-s21514" xml:space="preserve"> & ſit linea o g perpendicularis ſuper lineam a c, quæ diuidet
              <lb/>
            </s>
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