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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            <s xml:id="echoid-s18022" xml:space="preserve">
              <pb o="259" file="0265" n="265" rhead="OPTICAE LIBER VII."/>
            in circulo:</s>
            <s xml:id="echoid-s18023" xml:space="preserve"> angulus ſectionis erit æqualis angulo, qui eſt apud circumferentiam, quam chordant
              <lb/>
            duo arcus, quos diſtinguunt illæ duæ chordæ.</s>
            <s xml:id="echoid-s18024" xml:space="preserve"> Et ſi duæ hneæ ſecuerint circulum, & ſecuerint ſe
              <lb/>
            extra circulum:</s>
            <s xml:id="echoid-s18025" xml:space="preserve"> angulus ſectionis erit æqualis angulo,
              <lb/>
              <figure xlink:label="fig-0265-01" xlink:href="fig-0265-01a" number="225">
                <variables xml:id="echoid-variables212" xml:space="preserve">h a b g e f d e z</variables>
              </figure>
            qui eſt apud circũferentiam, quã chordat exceſſus ma
              <lb/>
            ioris illorum duorũ arcuũ, quos diſtinguunt illæ duæ
              <lb/>
            lineæ, ſupra reliquũ.</s>
            <s xml:id="echoid-s18026" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s18027" xml:space="preserve"> in circulo a b c d ſe-
              <lb/>
            cent ſe duæ chordę a c, b d in e.</s>
            <s xml:id="echoid-s18028" xml:space="preserve"> Dico igitur, quòd angu
              <lb/>
            lus a e b eſt æqualis angulo, qui eſt apud circumferen-
              <lb/>
            rẽtiam, quam reſpiciunt duo arcus a b, c d:</s>
            <s xml:id="echoid-s18029" xml:space="preserve"> & quòd an-
              <lb/>
            gulus b e c eſt æqualis angulo in circumferẽtia, quam
              <lb/>
            reſpiciũt duo arcus d g a, b z c.</s>
            <s xml:id="echoid-s18030" xml:space="preserve"> Extrahamus enim ex b
              <lb/>
            lineam b z æquidiſtantẽ lineæ a c [ք 31 p 1] arcus ergo
              <lb/>
            c z eſt æqualis arcui a b [Ducta enim recta a z:</s>
            <s xml:id="echoid-s18031" xml:space="preserve"> æquabi
              <lb/>
            tur angulus c a z angulo a z b ք 29 p 1:</s>
            <s xml:id="echoid-s18032" xml:space="preserve"> ideoq́;</s>
            <s xml:id="echoid-s18033" xml:space="preserve"> periphe-
              <lb/>
            ria c z peripherię a b ք 26 p 3:</s>
            <s xml:id="echoid-s18034" xml:space="preserve">] & arcusc d eſt cõmunis:</s>
            <s xml:id="echoid-s18035" xml:space="preserve">
              <lb/>
            ergo arcus d z eſt æqualis duobus arcubus, a b, c d:</s>
            <s xml:id="echoid-s18036" xml:space="preserve"> ſed
              <lb/>
            arcus d z reſpicit angulũ d b z [ք 8 d 3] ergo d z reſpicit
              <lb/>
            arcus æquales duob.</s>
            <s xml:id="echoid-s18037" xml:space="preserve"> arcubus a b, c d:</s>
            <s xml:id="echoid-s18038" xml:space="preserve"> & [ք 29 p 1] an-
              <lb/>
            gulus d b z eſt æqualis angulo a e b:</s>
            <s xml:id="echoid-s18039" xml:space="preserve"> ergo angulus a e b
              <lb/>
            eſt æqualis angulo, qui eſt in circum ferẽtia, quã reſpi-
              <lb/>
            ciunt duo arcus a b, c d.</s>
            <s xml:id="echoid-s18040" xml:space="preserve"> Et hoc eſt quod uoluimus.</s>
            <s xml:id="echoid-s18041" xml:space="preserve"> Itẽ continuemus d z:</s>
            <s xml:id="echoid-s18042" xml:space="preserve"> & producamus z b in h:</s>
            <s xml:id="echoid-s18043" xml:space="preserve"> e-
              <lb/>
            rit ergo [per 32 p 1] angulus h b d æqualis duob.</s>
            <s xml:id="echoid-s18044" xml:space="preserve"> angulis b d z, b z d, & [per 8 d 3] duo anguli b z d,
              <lb/>
            b d z reſpiciuntur à duobus arcubus b g d, b f z:</s>
            <s xml:id="echoid-s18045" xml:space="preserve"> angu-
              <lb/>
              <figure xlink:label="fig-0265-02" xlink:href="fig-0265-02a" number="226">
                <variables xml:id="echoid-variables213" xml:space="preserve">h a b e d c z</variables>
              </figure>
            lus ergo h b d eſt æqualis angulo, quem reſpicit arcus
              <lb/>
            d b z:</s>
            <s xml:id="echoid-s18046" xml:space="preserve"> & arcus a b eſt æqualis arcui z c, [ex cõcluſo:</s>
            <s xml:id="echoid-s18047" xml:space="preserve">] re
              <lb/>
            go arcus d b z eſt æqualis duobus arcubus d g a, b z c:</s>
            <s xml:id="echoid-s18048" xml:space="preserve">
              <lb/>
            ergo angulus h b e eſt æqualis angulo, quẽ reſpiciunt
              <lb/>
            duo arcus d g a, b z c:</s>
            <s xml:id="echoid-s18049" xml:space="preserve"> & [per 29 p 1] angulus h b e eſt ę-
              <lb/>
            qualis angulo b e c.</s>
            <s xml:id="echoid-s18050" xml:space="preserve"> Ergo angulus b e c eſt æqualis an-
              <lb/>
            gulo, qui eſt in circumferẽtia, quã reſpiciũt duo arcus
              <lb/>
            d g a, b z c.</s>
            <s xml:id="echoid-s18051" xml:space="preserve"> Et hoc eſt, quod uoluimus declarare.</s>
            <s xml:id="echoid-s18052" xml:space="preserve"> Et ſi li
              <lb/>
            nea h b z contingat circulum:</s>
            <s xml:id="echoid-s18053" xml:space="preserve"> tunc [per 32 p 3] angu-
              <lb/>
            lus e b z erit æqualis angulo cadẽti in portionẽ b a d:</s>
            <s xml:id="echoid-s18054" xml:space="preserve"> &
              <lb/>
            ſic arcus b c d reſpicit angulum apud circumferẽtiam,
              <lb/>
            æqualem angulo e b z:</s>
            <s xml:id="echoid-s18055" xml:space="preserve"> & [per 29 p 1] angulus e b z eſt
              <lb/>
            æqualis angulo b e a:</s>
            <s xml:id="echoid-s18056" xml:space="preserve"> ergo angulus b e a eſt æqualis an
              <lb/>
            gulo, qui eſt apud circumferentiam, quẽ reſpicit arcus
              <lb/>
            b c d:</s>
            <s xml:id="echoid-s18057" xml:space="preserve"> & arcus b c eſt æqualis arcui b a:</s>
            <s xml:id="echoid-s18058" xml:space="preserve"> quia diameter,
              <lb/>
            quæ exit ex b, eſt perpendicularis ſuper lineã a c:</s>
            <s xml:id="echoid-s18059" xml:space="preserve"> [Nã
              <lb/>
            diameter per punctũ b educta, eſt perpẽdicularis tan-
              <lb/>
            gẽti per 18 p 3:</s>
            <s xml:id="echoid-s18060" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s18061" xml:space="preserve"> per 29 p 1 eſt perpẽdicularis ipſi a c ad tangentẽ parallelę] quare [per 3 p 3] diui
              <lb/>
            ditipſam in duo æqualia:</s>
            <s xml:id="echoid-s18062" xml:space="preserve"> ergo arcus a b æqualis erit arcui b c:</s>
            <s xml:id="echoid-s18063" xml:space="preserve"> [ductis enim rectis a b, b c:</s>
            <s xml:id="echoid-s18064" xml:space="preserve"> erũt ipſæ
              <lb/>
            ք 4 p 1 æquales:</s>
            <s xml:id="echoid-s18065" xml:space="preserve"> ideoq́;</s>
            <s xml:id="echoid-s18066" xml:space="preserve"> peripheriæ a b, b c ipſis ſubtẽſæ, per 28 p 3:</s>
            <s xml:id="echoid-s18067" xml:space="preserve">] arcus ergo b c d eſt ęqualis duo-
              <lb/>
            bus arcubus a b, c d:</s>
            <s xml:id="echoid-s18068" xml:space="preserve"> ergo angulus b e a eſt æqualis angulo, ꝗ eſt apud circũferẽtiam, quẽ reſpiciunt
              <lb/>
            duo arcus a b, c d.</s>
            <s xml:id="echoid-s18069" xml:space="preserve"> Et ſimiliter declarabitur, quòd angulus b e c eſt æqualis angulo, qui eſt apud cir-
              <lb/>
            cumferentiam, quem reſpiciunt duo arcus b c, a d.</s>
            <s xml:id="echoid-s18070" xml:space="preserve"> Et hoc eſt quod uoluimus.</s>
            <s xml:id="echoid-s18071" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div589" type="section" level="0" n="0">
          <figure number="227">
            <variables xml:id="echoid-variables214" xml:space="preserve">e a b d f c</variables>
          </figure>
          <head xml:id="echoid-head508" xml:space="preserve" style="it">25. Si duæ rectæ lineæ circulo inſcriptæ, extrà cõtinuatæ cõcurrant: angulus concurſ{us} æqua-
            <lb/>
          tur angulo in peripheria, inſiſtenti in peripheriã, qua maior peri-
            <lb/>
          pheriarum inter inſcript{as} cõprehenſarũ exuperat minorẽ. 55 p 1.</head>
          <p>
            <s xml:id="echoid-s18072" xml:space="preserve">ITem:</s>
            <s xml:id="echoid-s18073" xml:space="preserve"> ſit e extra circulũ a b c d:</s>
            <s xml:id="echoid-s18074" xml:space="preserve"> & extrahamus ex e duas lineas ſe-
              <lb/>
            cantes circulũ a b c d:</s>
            <s xml:id="echoid-s18075" xml:space="preserve"> & ſint e a d, e b c.</s>
            <s xml:id="echoid-s18076" xml:space="preserve"> Dico ergo, quòd angulus
              <lb/>
            c e d eſt æqualis angulo, ꝗ eſt apud circũferẽtiã circuli, quẽ reſpi-
              <lb/>
            cit arcus exceſſus d c ſupera arcũ a b.</s>
            <s xml:id="echoid-s18077" xml:space="preserve"> Extrahamus enim lineã æquidi-
              <lb/>
            ſtantẽ lineæ b c [per 31 p 1] erit ergo [ut paulò antè oſtẽſum eſt] ar-
              <lb/>
            cus f c æqualis arcui a b:</s>
            <s xml:id="echoid-s18078" xml:space="preserve"> erit ergo arcus d f exceſſus arcus d c ſupra
              <lb/>
            arcũ a b:</s>
            <s xml:id="echoid-s18079" xml:space="preserve"> ſed [per 8 d 3] arcus d freſpicit angulũ d a f:</s>
            <s xml:id="echoid-s18080" xml:space="preserve"> & [per 29 p 1]
              <lb/>
            angulus d a f eſt æqualis angulo c e d:</s>
            <s xml:id="echoid-s18081" xml:space="preserve"> ergo c e d eſt æqualis angulo,
              <lb/>
            qui eſt apud circumferentiam d f.</s>
            <s xml:id="echoid-s18082" xml:space="preserve"> Et hoc eſt quod uoluimus.</s>
            <s xml:id="echoid-s18083" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div590" type="section" level="0" n="0">
          <head xml:id="echoid-head509" xml:space="preserve" style="it">26. Sι cõmunis ſectio ſuperficierũ refractionis & refractiui con
            <lb/>
          uexi fuerit peripheria: uiſibile in perpendiculari à uiſu ſuper re-
            <lb/>
          fractiuum duct a: rectè, & unum uidebitur. 22 p 10.</head>
          <p>
            <s xml:id="echoid-s18084" xml:space="preserve">HIs ergo declaratis, ſit uiſus punctũ a:</s>
            <s xml:id="echoid-s18085" xml:space="preserve"> & ſit pũctum b in aliquo
              <lb/>
            uiſo:</s>
            <s xml:id="echoid-s18086" xml:space="preserve"> & ſit ultra corpus diaphanũ groſsius corpore, qđ eſt in
              <lb/>
            </s>
          </p>
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