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-s21315" xml:space="preserve">
              <pb o="20" file="0322" n="322" rhead="VITELLONIS OPTICAE"/>
            ergo per 5 p 6 angulus b k c eſt ęqualis angulo f l g:</s>
            <s xml:id="echoid-s21316" xml:space="preserve"> & angulus k b c æqualis angulo l f g:</s>
            <s xml:id="echoid-s21317" xml:space="preserve"> ſed exprę-
              <lb/>
              <figure xlink:label="fig-0322-01" xlink:href="fig-0322-01a" number="310">
                <variables xml:id="echoid-variables294" xml:space="preserve">
                  <gap/>
                b c a m n d</variables>
              </figure>
              <figure xlink:label="fig-0322-02" xlink:href="fig-0322-02a" number="311">
                <variables xml:id="echoid-variables295" xml:space="preserve">l f g e o p h</variables>
              </figure>
            miſsis anguli d b c & h f g ſunt ęqua
              <lb/>
            les:</s>
            <s xml:id="echoid-s21318" xml:space="preserve"> eſt ergo angulus d b k æqualis
              <lb/>
            angulo h f l.</s>
            <s xml:id="echoid-s21319" xml:space="preserve"> Ducãtur ergo lineæ d k
              <lb/>
            & h l.</s>
            <s xml:id="echoid-s21320" xml:space="preserve"> Quia itaq;</s>
            <s xml:id="echoid-s21321" xml:space="preserve"> in trigonis b d k &
              <lb/>
            f h l anguli ęquales (qui d b k & h f l)
              <lb/>
            ſunt laterib.</s>
            <s xml:id="echoid-s21322" xml:space="preserve"> ꝓportiõalib.</s>
            <s xml:id="echoid-s21323" xml:space="preserve"> cõtẽti, pa-
              <lb/>
            tet ք 6 p 6, quoniã illa trigona ſunt
              <lb/>
            æquiangula:</s>
            <s xml:id="echoid-s21324" xml:space="preserve"> ergo angulus b k d eſt
              <lb/>
            ęqualis angulo fl h, & angulus b d k
              <lb/>
            ęqualis angulo f h l:</s>
            <s xml:id="echoid-s21325" xml:space="preserve"> ſed angulus a d
              <lb/>
            b eſt æqualis angulo e h f ex hypo-
              <lb/>
            theſi, propter ſimilitudinem arcuũ
              <lb/>
            a b & e f.</s>
            <s xml:id="echoid-s21326" xml:space="preserve"> Totus ergo angulus m d k
              <lb/>
            eſt æqualis toti angulo o h l:</s>
            <s xml:id="echoid-s21327" xml:space="preserve"> ergo ք
              <lb/>
            32 p 1 trigona d k m & h l o ſunt ęqui
              <lb/>
            angula, & angulus k m d eſt ęqualis
              <lb/>
            angulo l o h:</s>
            <s xml:id="echoid-s21328" xml:space="preserve"> ergo per 4 p 6 erit pro
              <lb/>
            portio lineę m k ad lineã o l, ſicut lineę k d ad lineã l h:</s>
            <s xml:id="echoid-s21329" xml:space="preserve"> ergo ք 11 p 5 ſicut lineę a d ad lineam e h.</s>
            <s xml:id="echoid-s21330" xml:space="preserve"> Quia
              <lb/>
            itaq;</s>
            <s xml:id="echoid-s21331" xml:space="preserve"> ex pręmiſsis angulus m k n eſt ęqualis angulo o l p, & angulus k m n ęqualis angulo l o p:</s>
            <s xml:id="echoid-s21332" xml:space="preserve"> patet
              <lb/>
            per 32 p 1, quoniã trigona k m n & l o p ſunt ęquiangula:</s>
            <s xml:id="echoid-s21333" xml:space="preserve"> ergo per 4 p 6 eſt proportio lineę m n ad li-
              <lb/>
            neam o p, ſicut lineæ m k ad lineã o l:</s>
            <s xml:id="echoid-s21334" xml:space="preserve"> ergo per 11 p 5, ſicut lineæ a d ad lineã e h.</s>
            <s xml:id="echoid-s21335" xml:space="preserve"> Quia itaq;</s>
            <s xml:id="echoid-s21336" xml:space="preserve"> a d ſemidia
              <lb/>
            meter maior eſt ſemidiametro e h:</s>
            <s xml:id="echoid-s21337" xml:space="preserve"> erit linea m n maior quã linea o p:</s>
            <s xml:id="echoid-s21338" xml:space="preserve"> patet ergo propoſitum.</s>
            <s xml:id="echoid-s21339" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div737" type="section" level="0" n="0">
          <head xml:id="echoid-head621" xml:space="preserve" style="it">47. À
            <unsure/>
          quocun puncto diametri circuli producta linea adperipheriam, ſi maior, quã illa,
            <lb/>
          fuerit una pars diametri: erit pars illa, maior reli-
            <lb/>
          qua ſui parte: & ſiminor, minor.</head>
          <figure number="312">
            <variables xml:id="echoid-variables296" xml:space="preserve">c a d b</variables>
          </figure>
          <p>
            <s xml:id="echoid-s21340" xml:space="preserve">Eſto circulus a b c, cuius diameter a b:</s>
            <s xml:id="echoid-s21341" xml:space="preserve"> in qua ſuma-
              <lb/>
            tur punctũ d, utcunq;</s>
            <s xml:id="echoid-s21342" xml:space="preserve"> cõtingit:</s>
            <s xml:id="echoid-s21343" xml:space="preserve"> & ducatur linea d c ad
              <lb/>
            circũferentiam, ita quòd pars diametri, quę eſt a d, ſit
              <lb/>
            maior ꝗ̃ linea d c.</s>
            <s xml:id="echoid-s21344" xml:space="preserve"> Dico, quòd linea a d eſt maior quã li
              <lb/>
            nea d b, quę eſt reliqua pars ipſius diametri:</s>
            <s xml:id="echoid-s21345" xml:space="preserve"> quod pa-
              <lb/>
            tet, ſi copulẽtur lineę a c & b c.</s>
            <s xml:id="echoid-s21346" xml:space="preserve"> Quia itaq;</s>
            <s xml:id="echoid-s21347" xml:space="preserve"> linea a d ma
              <lb/>
            ior eſt quã linea d c ex hypotheſi:</s>
            <s xml:id="echoid-s21348" xml:space="preserve"> ergo ք 18 p 1 angulus
              <lb/>
            a c d maior eſt angulo c a d, & angulus a c b eſt rectus
              <lb/>
            per 31 p 3:</s>
            <s xml:id="echoid-s21349" xml:space="preserve"> palã ergo per 32 p 1, quoniã angulus c b d ma
              <lb/>
            ior eſt angulo d c b.</s>
            <s xml:id="echoid-s21350" xml:space="preserve"> Quia enim angulus c b d cũ angu-
              <lb/>
            lo c a b ualet rectũ, & angulus d c b cũ angulo a c d, qui
              <lb/>
            eſt maior angulo c a d, ualet rectũ:</s>
            <s xml:id="echoid-s21351" xml:space="preserve"> patet, quòd angu-
              <lb/>
            lus c b d eſt maior angulo d c b:</s>
            <s xml:id="echoid-s21352" xml:space="preserve"> ergo per 19 p 1 erit la-
              <lb/>
            tus d c maius latere d b:</s>
            <s xml:id="echoid-s21353" xml:space="preserve"> ſed latus a d eſt maius latere d c.</s>
            <s xml:id="echoid-s21354" xml:space="preserve"> Ergo multo maius erit latus a d quã latus
              <lb/>
            d b.</s>
            <s xml:id="echoid-s21355" xml:space="preserve"> Et hoc eſt unum propoſitorum.</s>
            <s xml:id="echoid-s21356" xml:space="preserve"> Eodem quoq;</s>
            <s xml:id="echoid-s21357" xml:space="preserve"> modo demonſtrandum, ſi pars diametri, quæ eſt
              <lb/>
            a d, ſit minor quã linea d c:</s>
            <s xml:id="echoid-s21358" xml:space="preserve"> quoniã erit linea a d minor quã linea d b:</s>
            <s xml:id="echoid-s21359" xml:space="preserve"> & hoc proponebatur.</s>
            <s xml:id="echoid-s21360" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div738" type="section" level="0" n="0">
          <head xml:id="echoid-head622" xml:space="preserve" style="it">48. Si à quocun puncto diametri circuli duæ lineæ (quarum ſemper una ſit maior reliqua)
            <lb/>
          ad circuli peripheriã ducantur: erit pars diametri,
            <lb/>
          cuimaior linea propinquior ducitur, maior reliqua
            <lb/>
          ſui parte.</head>
          <figure number="313">
            <variables xml:id="echoid-variables297" xml:space="preserve">c g f e a h d b</variables>
          </figure>
          <p>
            <s xml:id="echoid-s21361" xml:space="preserve">Sit circulus a b e c, cuius diameter ſit a b:</s>
            <s xml:id="echoid-s21362" xml:space="preserve"> in qua ſu-
              <lb/>
            matur punctus d, ut libuerit:</s>
            <s xml:id="echoid-s21363" xml:space="preserve"> ducanturq́;</s>
            <s xml:id="echoid-s21364" xml:space="preserve"> à puncto d li-
              <lb/>
            neę, d c maior & d e minor:</s>
            <s xml:id="echoid-s21365" xml:space="preserve"> ſit aũt c ſuperior uerſus a,
              <lb/>
            & e inferior uerſus b.</s>
            <s xml:id="echoid-s21366" xml:space="preserve"> Dico, quòd pars diametri, quę eſt
              <lb/>
            a d, maior eſt quã d b.</s>
            <s xml:id="echoid-s21367" xml:space="preserve"> Ducatur enim linea c e, & ſuper
              <lb/>
            lineam c e ducatur à puncto d per 12 p 1 linea perpẽdi-
              <lb/>
            cularis, quę ſit d f.</s>
            <s xml:id="echoid-s21368" xml:space="preserve"> Quia itaq;</s>
            <s xml:id="echoid-s21369" xml:space="preserve"> quadratũ lineę d c per 47
              <lb/>
            p 1 ualet ambo quadrata linearũ d f & f c, & quadratũ
              <lb/>
            d e ualet ambo quadrata duarũ linearũ d f & f e, qua-
              <lb/>
            dratũ uerò lineę d c maius eſt quadrato lineę d e:</s>
            <s xml:id="echoid-s21370" xml:space="preserve"> i deo,
              <lb/>
            quia linea d c eſt maior ꝗ̃ linea d e:</s>
            <s xml:id="echoid-s21371" xml:space="preserve"> ablato itaq;</s>
            <s xml:id="echoid-s21372" xml:space="preserve"> quadra
              <lb/>
            to lineæ d f:</s>
            <s xml:id="echoid-s21373" xml:space="preserve"> relinquitur quadratũ lineæ c f, maius qua-
              <lb/>
            drato lineæ f e.</s>
            <s xml:id="echoid-s21374" xml:space="preserve"> Diuidatur itaq;</s>
            <s xml:id="echoid-s21375" xml:space="preserve"> linea c e in partes æqua
              <lb/>
            les in puncto g per 10 p 1, & ab illo puncto g ducatur
              <lb/>
            linea g h ad diametrum æquidiſtanter lineæ d f per 31 p 1:</s>
            <s xml:id="echoid-s21376" xml:space="preserve"> erititaque per 29 p 1 linea h g perpendicu-
              <lb/>
            laris ſuper lineam c e:</s>
            <s xml:id="echoid-s21377" xml:space="preserve"> ſecat autem h g ipſam c e in duo ęqualia:</s>
            <s xml:id="echoid-s21378" xml:space="preserve"> tranſit ergo linea h g ք centrũ circuli
              <lb/>
            </s>
          </p>
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