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-s21598" xml:space="preserve">
              <pb o="24" file="0326" n="326" rhead="VITELLONIS OPTICAE"/>
            qui eſt æqualis angulo a e b per 29 p 1:</s>
            <s xml:id="echoid-s21599" xml:space="preserve"> angulus itaque a e b eſt æqualis angulo in circumferentia, ca
              <lb/>
            denti in arcum æqualem duobus arcubus b a, & c d.</s>
            <s xml:id="echoid-s21600" xml:space="preserve"> Item d ucatur linea d z, & producatur linea z b
              <lb/>
            extra circulum in punctum h:</s>
            <s xml:id="echoid-s21601" xml:space="preserve"> erit ergo angulus h b d ext rinſecus æqualis duobus angulis intrinſe-
              <lb/>
            cis b d z, & b z d per 32 p 1:</s>
            <s xml:id="echoid-s21602" xml:space="preserve"> ſed duo anguli b z d & b d z re ſpiciuntur à duobus arcubus b g d, & b f z:</s>
            <s xml:id="echoid-s21603" xml:space="preserve">
              <lb/>
            angulus ergo h b d eſt æqualis angulo, quem reſpiciunt duo arcus b g d & b f z:</s>
            <s xml:id="echoid-s21604" xml:space="preserve"> hic autem eſt arcus
              <lb/>
            d a z:</s>
            <s xml:id="echoid-s21605" xml:space="preserve"> ſed arcus a b eſt æqualis arcui z c:</s>
            <s xml:id="echoid-s21606" xml:space="preserve"> arcus itaque d a z eſt æqualis duobus arcubus d g a & b z c.</s>
            <s xml:id="echoid-s21607" xml:space="preserve">
              <lb/>
            Cum itaque per 29 p 1 angulus h b e ſit æqualis angulo
              <lb/>
              <figure xlink:label="fig-0326-01" xlink:href="fig-0326-01a" number="324">
                <variables xml:id="echoid-variables308" xml:space="preserve">h a b e d z c</variables>
              </figure>
            b e c:</s>
            <s xml:id="echoid-s21608" xml:space="preserve"> patet, quia angulus b e c eſt æqualis angulo, quem
              <lb/>
            in circũferentia reſpiciunt duo arcus d g a & b z c.</s>
            <s xml:id="echoid-s21609" xml:space="preserve"> Quòd
              <lb/>
            ſi linea h b z contingit circulum, & non ſecat:</s>
            <s xml:id="echoid-s21610" xml:space="preserve"> tunc patet
              <lb/>
            per 32 p 3, quia angulus e b z eſt æqualis angulo cadenti
              <lb/>
            in portionem circuli, quę eſt b a d, & angulus e b h eſt ę-
              <lb/>
            qualis angulo cadenti in portionem circuli b c d:</s>
            <s xml:id="echoid-s21611" xml:space="preserve"> ſed an
              <lb/>
            gulus e b z eſt æqualis angulo b e a per 29 p 1.</s>
            <s xml:id="echoid-s21612" xml:space="preserve"> Angulus
              <lb/>
            itaque b e a eſt æqualis angulo, qui apud circumferen-
              <lb/>
            tiam cadit in arcum b c d:</s>
            <s xml:id="echoid-s21613" xml:space="preserve"> ſed arcus b c eſt æqualis arcui
              <lb/>
            b a per proximam pręcedentem:</s>
            <s xml:id="echoid-s21614" xml:space="preserve"> arcus ergo b c d eſt æ-
              <lb/>
            qualis duobus arcubus b a & c d.</s>
            <s xml:id="echoid-s21615" xml:space="preserve"> Angulus itaq;</s>
            <s xml:id="echoid-s21616" xml:space="preserve"> b e a eſt
              <lb/>
            æqualis angulo, qui apud circumferẽtiam reſpicit duos
              <lb/>
            arcus a b & c d:</s>
            <s xml:id="echoid-s21617" xml:space="preserve"> quoniam angulus cadens in arcum b c d
              <lb/>
            eſt conſiſtens in portione circuli, quæ eſt b g d.</s>
            <s xml:id="echoid-s21618" xml:space="preserve"> Simi-
              <lb/>
            liter quoque poteſt declarari, quòd angulus b e c eſt
              <lb/>
            æqualis angulo apud circumferentiam, quem reſpiciũt
              <lb/>
            duo arcus b c & a d:</s>
            <s xml:id="echoid-s21619" xml:space="preserve"> quoniam angulus b e c eſt æqualis
              <lb/>
            angulo h b d, cuius ęqualitas per 32 p 3 cadit in portionem circuli b c d, hoc eſt in arcum b a d:</s>
            <s xml:id="echoid-s21620" xml:space="preserve"> eſt au-
              <lb/>
            tem ex præmiſsis arcus a b æqualis arcui b c:</s>
            <s xml:id="echoid-s21621" xml:space="preserve"> patet itaque propoſitum.</s>
            <s xml:id="echoid-s21622" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div751" type="section" level="0" n="0">
          <figure number="325">
            <variables xml:id="echoid-variables309" xml:space="preserve">e a b d f c</variables>
          </figure>
          <head xml:id="echoid-head629" xml:space="preserve" style="it">55. Angulus à duabus lineis ab uno puncto extra circulum dato, circulum ſecantibus con-
            <lb/>
          tentus, æqualis eſt angulo ſuper circumferẽtiam cadenti in arcũ,
            <lb/>
          quo maior arcuum inter illas duas lineas comprehenſus, excedit minorem. Alhazen 25 n 7.</head>
          <p>
            <s xml:id="echoid-s21623" xml:space="preserve">Eſto circulus a b c d, extra quem ſit datum punctum e:</s>
            <s xml:id="echoid-s21624" xml:space="preserve"> & ducan-
              <lb/>
            tur à puncto e duę lineę ſecantes circulum, quæ ſint e a d & e b c.</s>
            <s xml:id="echoid-s21625" xml:space="preserve"> Di-
              <lb/>
            co itaq;</s>
            <s xml:id="echoid-s21626" xml:space="preserve">, quòd angulus d e c eſt æqualis angulo, qui eſt apud circum-
              <lb/>
            ferentiam circuli, quem reſpicit arcus, in quo arcus d c excedit arcũ
              <lb/>
            a b.</s>
            <s xml:id="echoid-s21627" xml:space="preserve"> À
              <unsure/>
            pũcto enim a ducatur per circulum linea a f ęquidiſtans lineę
              <lb/>
            b c per 31 p 1:</s>
            <s xml:id="echoid-s21628" xml:space="preserve"> erit ergo per 53 huius arcus f c ęqualis arcui a b.</s>
            <s xml:id="echoid-s21629" xml:space="preserve"> Eſt itaq;</s>
            <s xml:id="echoid-s21630" xml:space="preserve">
              <lb/>
            arcus d f exceſſus arcus d c ſuper arcum a b:</s>
            <s xml:id="echoid-s21631" xml:space="preserve"> ſed angulus d a f apud
              <lb/>
            circumferentiã exiſtens cadit in arcum d f:</s>
            <s xml:id="echoid-s21632" xml:space="preserve"> & angulus d a f eſt æqua-
              <lb/>
            lis angulo d e c per 29 p 1.</s>
            <s xml:id="echoid-s21633" xml:space="preserve"> Ergo angulus d e c eſt æqualis angulo ca-
              <lb/>
            denti ſuper circumferentiam in arcum d f:</s>
            <s xml:id="echoid-s21634" xml:space="preserve"> quod eſt propoſitum.</s>
            <s xml:id="echoid-s21635" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div752" type="section" level="0" n="0">
          <head xml:id="echoid-head630" xml:space="preserve" style="it">56. In dato ſemicirculo ad unum punctũ circumferentiæ, dua-
            <lb/>
          bus lineis: una à termino diametri, & alia à centro ductis: ab eiſ-
            <lb/>
          dem punctis ad aliud punctum quodcun ſemicirculi dati lineas
            <lb/>
          duas prioribus duabus proportionales duci eſt impoßibile: in diuerſis uerò ſemicirculis hoc eſt
            <lb/>
          poßibile.</head>
          <p>
            <s xml:id="echoid-s21636" xml:space="preserve">Eſto datus ſemicirculus a d b:</s>
            <s xml:id="echoid-s21637" xml:space="preserve"> cuius diameter a b:</s>
            <s xml:id="echoid-s21638" xml:space="preserve"> centrum uerò c:</s>
            <s xml:id="echoid-s21639" xml:space="preserve"> & ſit aliquod punctum circũ-
              <lb/>
            ferentiæ d:</s>
            <s xml:id="echoid-s21640" xml:space="preserve"> & ducatur à puncto a termino dia-
              <lb/>
              <figure xlink:label="fig-0326-03" xlink:href="fig-0326-03a" number="326">
                <variables xml:id="echoid-variables310" xml:space="preserve">g d f a e c h b</variables>
              </figure>
            metri ad punctum d linea a d:</s>
            <s xml:id="echoid-s21641" xml:space="preserve"> & à cẽtro c linea
              <lb/>
            c d.</s>
            <s xml:id="echoid-s21642" xml:space="preserve"> Dico, quòd ſi à punctis a & c duæ lineæ ad
              <lb/>
            aliud punctum ſemicirculi ducantur:</s>
            <s xml:id="echoid-s21643" xml:space="preserve"> quòd illę
              <lb/>
            duę ductę lineę duabus lineis a d & c d propor
              <lb/>
            tionales non erunt.</s>
            <s xml:id="echoid-s21644" xml:space="preserve"> Sit enim, ſi poſsibile eſt,
              <lb/>
            ut à punctis a & c ducantur ad punctum g duę
              <lb/>
            lineæ a g & c g, & quę eſt proportio lineę a d ad
              <lb/>
            lineam c d, eadem ſit lineæ a g ad lineam c g, e-
              <lb/>
            rit permutatim per 16 p 5 proportio lineæ a d
              <lb/>
            ad lineam a g, ſicut lineę c d ad lineam c g:</s>
            <s xml:id="echoid-s21645" xml:space="preserve"> ſe d li
              <lb/>
            nea c d eſt æqualis lineę c g:</s>
            <s xml:id="echoid-s21646" xml:space="preserve"> quoniã ambę ſunt
              <lb/>
            ex cẽtro ſemicirculi:</s>
            <s xml:id="echoid-s21647" xml:space="preserve"> ergo linea a d ęqualis erit lineę a g:</s>
            <s xml:id="echoid-s21648" xml:space="preserve"> hoc aũt eſt impoſsibile ex 7 p 3 & 19 p 1:</s>
            <s xml:id="echoid-s21649" xml:space="preserve"> ma-
              <lb/>
            iori enim angulo ſubtẽditur linea a d ꝗ̃ linea a g:</s>
            <s xml:id="echoid-s21650" xml:space="preserve"> & eſt uicinior diametro.</s>
            <s xml:id="echoid-s21651" xml:space="preserve"> Patet ergo ꝓpoſitũ primũ:</s>
            <s xml:id="echoid-s21652" xml:space="preserve">
              <lb/>
            quia à quocũq;</s>
            <s xml:id="echoid-s21653" xml:space="preserve"> pũcto alio dato idẽ accidit impoſsibile, & eodẽ modo deducẽdũ.</s>
            <s xml:id="echoid-s21654" xml:space="preserve"> In diuerſis uerò ſe-
              <lb/>
            </s>
          </p>
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