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-s20918" xml:space="preserve">
              <pb o="14" file="0316" n="316" rhead="VITELLONIS OPTICAE"/>
            ſim b c perpendicularis a d.</s>
            <s xml:id="echoid-s20919" xml:space="preserve"> Dico, quòd propoſitus iſoſceles diuiſus eſt in duos trigonos par-
              <lb/>
            tiales ſimiles.</s>
            <s xml:id="echoid-s20920" xml:space="preserve"> Quoniam enim per 5 p 1 angulus a b d eſt æqualis angulo a c d, ſed & per definitio-
              <lb/>
            nem perpendicularis 10 defin.</s>
            <s xml:id="echoid-s20921" xml:space="preserve"> 1.</s>
            <s xml:id="echoid-s20922" xml:space="preserve"> elem.</s>
            <s xml:id="echoid-s20923" xml:space="preserve"> anguli a d b & a d c ſunt æqua-
              <lb/>
              <figure xlink:label="fig-0316-01" xlink:href="fig-0316-01a" number="293">
                <variables xml:id="echoid-variables277" xml:space="preserve">a b d c</variables>
              </figure>
            les, quia recti:</s>
            <s xml:id="echoid-s20924" xml:space="preserve"> patet per 32 p 1, quòd anguli b a d & c a d ſunt æquales.</s>
            <s xml:id="echoid-s20925" xml:space="preserve">
              <lb/>
            Ergo trigoni a b d & a c d ſunt æquianguli:</s>
            <s xml:id="echoid-s20926" xml:space="preserve"> ergo per 4 p 6 latera illo-
              <lb/>
            rum trigonorũ æquos angulos reſpicientia, ſunt proportionalia:</s>
            <s xml:id="echoid-s20927" xml:space="preserve"> ſunt
              <lb/>
            ergo illa trigona partialia, quæ a b d & a c d ſimilia per definitionem
              <lb/>
            ſimilium trigonorum:</s>
            <s xml:id="echoid-s20928" xml:space="preserve"> patet ergo propoſitum primum.</s>
            <s xml:id="echoid-s20929" xml:space="preserve"> Et quoniam
              <lb/>
            illa trigona a b d & a c d ſunt ſimilia, & eorum latera a b & a c ſunt æ-
              <lb/>
            qualia, & latus a d cõmune:</s>
            <s xml:id="echoid-s20930" xml:space="preserve"> patet, quòd etiam latera c d & b d ſunt æ-
              <lb/>
            qualia.</s>
            <s xml:id="echoid-s20931" xml:space="preserve"> Linea ergo քpendicularis, quę a d, neceſſariò pertingit ad me-
              <lb/>
            dium punctum lineæ b c:</s>
            <s xml:id="echoid-s20932" xml:space="preserve"> quod eſt propoſitum ſecundum.</s>
            <s xml:id="echoid-s20933" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div711" type="section" level="0" n="0">
          <head xml:id="echoid-head606" xml:space="preserve" style="it">32. Linea ducta à quocun puncto unius lateris trigoni produ-
            <lb/>
          cti, ultr a trigonum ſecans latus ab illo puncto remotius, & propin-
            <lb/>
          quius illi neceſſariò ſecabit.</head>
          <p>
            <s xml:id="echoid-s20934" xml:space="preserve">Sit trigonum a b c, cuius latus a b producatur ultra punctum b ad
              <lb/>
            punctum d:</s>
            <s xml:id="echoid-s20935" xml:space="preserve"> & à puncto d ducatur linea d e ſecans latus trigoni a c in puncto e.</s>
            <s xml:id="echoid-s20936" xml:space="preserve"> Dico, quòd d e ne-
              <lb/>
            ceſſariò ſecabit latus b c.</s>
            <s xml:id="echoid-s20937" xml:space="preserve"> Si enim non ſecabit latus b c, ſed ſolum latus
              <lb/>
              <figure xlink:label="fig-0316-02" xlink:href="fig-0316-02a" number="294">
                <variables xml:id="echoid-variables278" xml:space="preserve">d b a e c f f</variables>
              </figure>
            a c, ducatur linea d c, & producatur in continuum & directum:</s>
            <s xml:id="echoid-s20938" xml:space="preserve"> ſecabit
              <lb/>
            itaq;</s>
            <s xml:id="echoid-s20939" xml:space="preserve"> linea d c in aliquo puncto lineam d e:</s>
            <s xml:id="echoid-s20940" xml:space="preserve"> quoniam cum linea d c exeat
              <lb/>
            â puncto d, à quo exit etiam linea d e, & terminetur ad pũctum c inter-
              <lb/>
            iacens punctum e, neceſſariò illam ſecabit:</s>
            <s xml:id="echoid-s20941" xml:space="preserve"> ſit punctus ſectionis f.</s>
            <s xml:id="echoid-s20942" xml:space="preserve"> Pa-
              <lb/>
            làm itaq;</s>
            <s xml:id="echoid-s20943" xml:space="preserve">, quoniam duæ rectæ lineæ, quæ ſunt d f & d e f includunt ſu-
              <lb/>
            perficiem:</s>
            <s xml:id="echoid-s20944" xml:space="preserve"> quod eſt impoſsibile.</s>
            <s xml:id="echoid-s20945" xml:space="preserve"> Idem quoque accidit, ſi linea d e duca-
              <lb/>
            tur extra lineam b c ultra punctum a:</s>
            <s xml:id="echoid-s20946" xml:space="preserve"> quod eſt propoſitum.</s>
            <s xml:id="echoid-s20947" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div713" type="section" level="0" n="0">
          <head xml:id="echoid-head607" xml:space="preserve" style="it">33. Si à punctis terminalibus unius lateris trianguli duæ rectæ
            <lb/>
          exeuntes, intr a trigonum ad punctum unum conueniant: erit angu
            <lb/>
          lus inferior æqualis ſuperiori, & duobus angulis inter lineas duct as
            <lb/>
          ad alia duo later a trigoni contentis.</head>
          <p>
            <s xml:id="echoid-s20948" xml:space="preserve">Sit trigonum a b c, à cuius unius laterum a b punctis terminalibus,
              <lb/>
            quæ ſunt a & b, ducantur lineæ taliter, ut intra trigonum a b c concur-
              <lb/>
            rant in puncto d.</s>
            <s xml:id="echoid-s20949" xml:space="preserve"> Dico, quòd angulus a d b eſt æqualis angulo a c b, &
              <lb/>
            inſuper duobus angulis c a d & c b d.</s>
            <s xml:id="echoid-s20950" xml:space="preserve"> Quòd enim angulus a d b ſit maior angulo a c b, hoc patet per
              <lb/>
            21 p 1.</s>
            <s xml:id="echoid-s20951" xml:space="preserve"> Producatur itaq;</s>
            <s xml:id="echoid-s20952" xml:space="preserve"> linea c d ultra punctum d uſq;</s>
            <s xml:id="echoid-s20953" xml:space="preserve"> ad punctum e.</s>
            <s xml:id="echoid-s20954" xml:space="preserve">
              <lb/>
              <figure xlink:label="fig-0316-03" xlink:href="fig-0316-03a" number="295">
                <variables xml:id="echoid-variables279" xml:space="preserve">c d e a b</variables>
              </figure>
            Eſt itaq;</s>
            <s xml:id="echoid-s20955" xml:space="preserve"> per 32 p 1 angulus e d a æqualis duobus angulis d c a & d a c:</s>
            <s xml:id="echoid-s20956" xml:space="preserve">
              <lb/>
            & ſimiliter angulus e d b æqualis eſt duobus angulis d b c & d c b.</s>
            <s xml:id="echoid-s20957" xml:space="preserve"> To-
              <lb/>
            tus ergo angulus a d b ęqualis eſt angulo a c b, & angulis d a c & d b c:</s>
            <s xml:id="echoid-s20958" xml:space="preserve">
              <lb/>
            quod eſt propoſitum.</s>
            <s xml:id="echoid-s20959" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div715" type="section" level="0" n="0">
          <head xml:id="echoid-head608" xml:space="preserve" style="it">34. Linea æqualis & æquidiſtans baſi alicuius trigoni, uicini-
            <lb/>
          or angulo ſupremo, maiori angulo neceſſariò ſubtenditur.</head>
          <p>
            <s xml:id="echoid-s20960" xml:space="preserve">Eſto trigonum a b c, cuius baſi a c:</s>
            <s xml:id="echoid-s20961" xml:space="preserve"> uicinior angulo a b c duca-
              <lb/>
            tur linea æqualis & æquidiſtans, quæ ſit d e.</s>
            <s xml:id="echoid-s20962" xml:space="preserve"> Dico, quòd ſi à puncto
              <lb/>
            b ducantur lineæ b d & b e, quòd angulus d b e eſt maior angulo a b
              <lb/>
            c.</s>
            <s xml:id="echoid-s20963" xml:space="preserve"> Quia enim linea d e eſt æqualis lineæ a c, palàm, quòd ipſa ſic pro-
              <lb/>
            ducta ſecat lineas a b & b c, argumento 16 huius:</s>
            <s xml:id="echoid-s20964" xml:space="preserve"> quòd etiã patet ex a-
              <lb/>
            lijs.</s>
            <s xml:id="echoid-s20965" xml:space="preserve"> Nam omnis linea cadens intra trigonum ſecans latera eius & æ-
              <lb/>
            quidiſtans, eſt minor baſi per 29 p 1 & 4 p 6.</s>
            <s xml:id="echoid-s20966" xml:space="preserve"> Secet ergo linea d e latus
              <lb/>
            b a in puncto f, & latus b c in puncto g.</s>
            <s xml:id="echoid-s20967" xml:space="preserve"> Quia ita que per 16 p 1 angulus b g f eſt maior angulo b e
              <lb/>
            g:</s>
            <s xml:id="echoid-s20968" xml:space="preserve"> erit per 29 p 1 angulus b c a maior angulo b e d:</s>
            <s xml:id="echoid-s20969" xml:space="preserve"> & ea-
              <lb/>
              <figure xlink:label="fig-0316-04" xlink:href="fig-0316-04a" number="296">
                <variables xml:id="echoid-variables280" xml:space="preserve">b d f g e a c</variables>
              </figure>
            dem ratione angulus b a c eſt maior angulo b d e:</s>
            <s xml:id="echoid-s20970" xml:space="preserve"> ne-
              <lb/>
            ceſſariò ergo per 32 p 1 erit angulus d b e cum angulis mi-
              <lb/>
            noribus ualens duos rectos, maior angulo a b c, ualente
              <lb/>
            cum duobus angulis maioribus duos rectos:</s>
            <s xml:id="echoid-s20971" xml:space="preserve"> patet ergo
              <lb/>
            propoſitum.</s>
            <s xml:id="echoid-s20972" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div717" type="section" level="0" n="0">
          <head xml:id="echoid-head609" xml:space="preserve" style="it">35. In trigono orthogonio ab uno reliquorum an-
            <lb/>
          gulorum producta linea ad baſim: erit remotioris an-
            <lb/>
          guli ad propinquiorem recto minor proportio, quàm
            <lb/>
          </head>
        </div>
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