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-s21816" xml:space="preserve">
              <pb o="27" file="0329" n="329" rhead="LIBER PRIMVS."/>
            æquiangula per 32 p 1:</s>
            <s xml:id="echoid-s21817" xml:space="preserve"> ergo per 4 p 6 latera ſunt proportionalia:</s>
            <s xml:id="echoid-s21818" xml:space="preserve"> ſed latus d b eſt æquale ſibi:</s>
            <s xml:id="echoid-s21819" xml:space="preserve"> erit er-
              <lb/>
            go linea e b æqualis lineæ b f, & linea d e ęqualis
              <lb/>
              <figure xlink:label="fig-0329-01" xlink:href="fig-0329-01a" number="331">
                <variables xml:id="echoid-variables315" xml:space="preserve">a e g b d c f</variables>
              </figure>
            lineæ d f.</s>
            <s xml:id="echoid-s21820" xml:space="preserve"> Quod etiam ſic patere poteſt.</s>
            <s xml:id="echoid-s21821" xml:space="preserve"> Quia e-
              <lb/>
            nim à puncto e ducuntur duæ lineæ contingen-
              <lb/>
            tes circulum, ſcilicet e a & e b, patet per 58 huius,
              <lb/>
            quòd ipſæ ſunt æquales.</s>
            <s xml:id="echoid-s21822" xml:space="preserve"> Omnes ergo lineæ a e,
              <lb/>
            e b, b f, f c ſunt æquales.</s>
            <s xml:id="echoid-s21823" xml:space="preserve"> Ergo lineæ e d & f d
              <lb/>
            ſunt æquales:</s>
            <s xml:id="echoid-s21824" xml:space="preserve"> patet ergo propoſitum.</s>
            <s xml:id="echoid-s21825" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div763" type="section" level="0" n="0">
          <figure number="332">
            <variables xml:id="echoid-variables316" xml:space="preserve">g f h k b l a c e m d n</variables>
          </figure>
          <head xml:id="echoid-head636" xml:space="preserve" style="it">62. A duobus puuctis æqualiter diſtanti-
            <lb/>
          bus ab uno termino eductæ diametri, & à li-
            <lb/>
          nea circulum in termino propiore diametri cõ
            <lb/>
          tingente, duabus lineis ad alium terminũ dia-
            <lb/>
          metri productis: arcus interiacẽtes illarum line arum alter am & diametrum, ſunt æquales: il-
            <lb/>
          lis uerò ad alium punctum circumferentiæ produ-
            <lb/>
          ctis, arcus interiacent inæquales.</head>
          <p>
            <s xml:id="echoid-s21826" xml:space="preserve">Sit circulus a b c d, cuius centrum e:</s>
            <s xml:id="echoid-s21827" xml:space="preserve"> diameterq́;</s>
            <s xml:id="echoid-s21828" xml:space="preserve"> e-
              <lb/>
            ius d b educatur ad punctũ f:</s>
            <s xml:id="echoid-s21829" xml:space="preserve"> ſintq́;</s>
            <s xml:id="echoid-s21830" xml:space="preserve"> duo puncta g & h
              <lb/>
            ęqualiter diſtãtia à pũcto f eductę diametri:</s>
            <s xml:id="echoid-s21831" xml:space="preserve"> ducãtúr-
              <lb/>
            que duę lineę g d & h d adaliũ terminũ diametri ſecã-
              <lb/>
            tes circulũ:</s>
            <s xml:id="echoid-s21832" xml:space="preserve"> linea g d in pũcto a, & linea h d in pũcto c:</s>
            <s xml:id="echoid-s21833" xml:space="preserve">
              <lb/>
            & à puncto b ducatur linea cõtingens circulũ, quę ſit
              <lb/>
            k b l, à qua ęqualiter diſtẽt pũcta g & h.</s>
            <s xml:id="echoid-s21834" xml:space="preserve"> Dico, quòd ar-
              <lb/>
            cus a b & b c ſunt æquales.</s>
            <s xml:id="echoid-s21835" xml:space="preserve"> Ducatur enim linea g f h:</s>
            <s xml:id="echoid-s21836" xml:space="preserve">
              <lb/>
            erit ergo ex hypotheſi linea g f æqualis lineę h f:</s>
            <s xml:id="echoid-s21837" xml:space="preserve"> ideo,
              <lb/>
            quia puncta g & h ęqualiter diſtãt à puncto f:</s>
            <s xml:id="echoid-s21838" xml:space="preserve"> & ducã-
              <lb/>
            turlineę h l & g k perpẽdiculariter ſuper lineã k b l cõ
              <lb/>
            tingẽtẽ, ք 12 p 1:</s>
            <s xml:id="echoid-s21839" xml:space="preserve"> erũt ergo ex hypotheſi & illę ęquales:</s>
            <s xml:id="echoid-s21840" xml:space="preserve">
              <lb/>
            ergo ք 33 p 1 linea g h ęꝗdiſtat lineę k l.</s>
            <s xml:id="echoid-s21841" xml:space="preserve"> Ergo ք 18 p 3 &
              <lb/>
            29 p 1 anguli d f h & d f g ſunt recti:</s>
            <s xml:id="echoid-s21842" xml:space="preserve"> ergo ք 4 p 1 anguli
              <lb/>
            g d f & h d f ſunt ęquales.</s>
            <s xml:id="echoid-s21843" xml:space="preserve"> Ergo ք 26 p 3 arcus a b eſt ę-
              <lb/>
            qualis arcui b c.</s>
            <s xml:id="echoid-s21844" xml:space="preserve"> Patet quoq;</s>
            <s xml:id="echoid-s21845" xml:space="preserve"> manifeſtè, quòd ſi à pũctis g & h lineę ad aliud pũctũ circũferentię quã
              <lb/>
            ad pũctũ d ꝓducãtur, ut ad pũcta m ueln:</s>
            <s xml:id="echoid-s21846" xml:space="preserve"> quòd illę lineę arcus reſecabũt inęquales:</s>
            <s xml:id="echoid-s21847" xml:space="preserve"> quęlibet enim
              <lb/>
            illarũ, quę ſecat diametrũ, abſcindit minorẽ arcum, & alia maiorẽ:</s>
            <s xml:id="echoid-s21848" xml:space="preserve"> & hoc eſt, quod proponebatur.</s>
            <s xml:id="echoid-s21849" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div764" type="section" level="0" n="0">
          <figure number="333">
            <variables xml:id="echoid-variables317" xml:space="preserve">g b c a f d e</variables>
          </figure>
          <head xml:id="echoid-head637" xml:space="preserve" style="it">63. Diameter circuli diuidens hexagonum, eidẽ cir-
            <lb/>
          culo inſcriptum, ab oppoſitis angulis per æqualia, duob. lateribus medijs hexagoni erit æquidiſtans.</head>
          <p>
            <s xml:id="echoid-s21850" xml:space="preserve">Sit circulus, cuius centrũ ſit punctũ a:</s>
            <s xml:id="echoid-s21851" xml:space="preserve"> inſcriptus hexa-
              <lb/>
            gonus, qui b c d e f g:</s>
            <s xml:id="echoid-s21852" xml:space="preserve"> & ab oppoſitis angulis illius hexago
              <lb/>
            ni ducatur diameter b a e.</s>
            <s xml:id="echoid-s21853" xml:space="preserve"> Dico, quòd illa diameter æqui-
              <lb/>
            diſtat duobus medijs lateribus hexagoni, quæ ſunt c d &
              <lb/>
            g f.</s>
            <s xml:id="echoid-s21854" xml:space="preserve"> Ducantur enim lineæ a c & a d.</s>
            <s xml:id="echoid-s21855" xml:space="preserve"> Quia itaque lineę b c
              <lb/>
            & c d, (quę ſunt latera hexagoni) ſunt inter ſe ęqualia, &
              <lb/>
            utrunq;</s>
            <s xml:id="echoid-s21856" xml:space="preserve"> ipſorũ eſt ęquale ſemidiametro circuli per 15 p 4:</s>
            <s xml:id="echoid-s21857" xml:space="preserve">
              <lb/>
            patetergo, quòd trigona a b c & a c d ſunt ęquilatera:</s>
            <s xml:id="echoid-s21858" xml:space="preserve"> er-
              <lb/>
            go per 8 p 1 ipſa ſunt ęquiangula:</s>
            <s xml:id="echoid-s21859" xml:space="preserve"> erit ergo angulus c a b ę-
              <lb/>
            qualis angulo a c d.</s>
            <s xml:id="echoid-s21860" xml:space="preserve"> Ergo per 27 p 1 lineæ a b & c d ęquidi
              <lb/>
            ſtant.</s>
            <s xml:id="echoid-s21861" xml:space="preserve"> Similiter quoq;</s>
            <s xml:id="echoid-s21862" xml:space="preserve"> poteſt demonſtrari de lineis a b &
              <lb/>
            f g.</s>
            <s xml:id="echoid-s21863" xml:space="preserve"> Patet ergo, quoniam diameter b e ęquidiſtat medijs la
              <lb/>
            teribus hexagoni:</s>
            <s xml:id="echoid-s21864" xml:space="preserve"> quod eſt propoſitum.</s>
            <s xml:id="echoid-s21865" xml:space="preserve"/>
          </p>
          <figure number="334">
            <variables xml:id="echoid-variables318" xml:space="preserve">g f c b d a</variables>
          </figure>
        </div>
        <div xml:id="echoid-div765" type="section" level="0" n="0">
          <head xml:id="echoid-head638" xml:space="preserve" style="it">64. Duobus circulis inæqualibus ſe ſecantibus, it a ut minor pertrã-
            <lb/>
          ſeat centrum maioris: arcum minor is interiacentem peripheriã ma-
            <lb/>
          ioris in centro maioris per æqualia diuidi eſt neceſſe.</head>
          <p>
            <s xml:id="echoid-s21866" xml:space="preserve">Sint duo circuli c f d maior, cuius centrũ ſit a:</s>
            <s xml:id="echoid-s21867" xml:space="preserve"> & c g d minor, cuius cen
              <lb/>
            trum ſit b:</s>
            <s xml:id="echoid-s21868" xml:space="preserve"> ſecentq́;</s>
            <s xml:id="echoid-s21869" xml:space="preserve"> ſe hi circuli in punctis c & d:</s>
            <s xml:id="echoid-s21870" xml:space="preserve"> tranſeatq́;</s>
            <s xml:id="echoid-s21871" xml:space="preserve"> minor (qui c
              <lb/>
            g d) per centrũ maioris, quod eſt a:</s>
            <s xml:id="echoid-s21872" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s21873" xml:space="preserve"> arcus c a d minoris circuli con
              <lb/>
            tentus intra peripheriam maioris.</s>
            <s xml:id="echoid-s21874" xml:space="preserve"> Dico, quòd arcus c a d diuiditur per
              <lb/>
            æqualia in puncto a.</s>
            <s xml:id="echoid-s21875" xml:space="preserve"> Ducatur enim linea copulans centra, quę ſit a b:</s>
            <s xml:id="echoid-s21876" xml:space="preserve"> &
              <lb/>
            hec producta compleat diametrũ minoris circuli, quæ ſit a b g:</s>
            <s xml:id="echoid-s21877" xml:space="preserve"> & ad pũ-
              <lb/>
            cta ſectionum c & d, ducantur lineæ a d, a c, b d, b c.</s>
            <s xml:id="echoid-s21878" xml:space="preserve"> Quia itaque in trigo-
              <lb/>
            nis a b c & a b d, duo latera a b & b c unius ſunt æqualia duobus laterib.</s>
            <s xml:id="echoid-s21879" xml:space="preserve">
              <lb/>
            a b & b d alterius:</s>
            <s xml:id="echoid-s21880" xml:space="preserve"> quoniam omnes ſunt rectę ex puncto b centro circuli
              <lb/>
            </s>
          </p>
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