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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        <div xml:id="echoid-div792" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s22237" xml:space="preserve">
              <pb o="33" file="0335" n="335" rhead="LIBER PRIMVS."/>
            f e in linea a b:</s>
            <s xml:id="echoid-s22238" xml:space="preserve"> ſed linea f e eſt maior quàm linea e c:</s>
            <s xml:id="echoid-s22239" xml:space="preserve"> ergo a e, in qua linea a c excedit lineam e c, eſt
              <lb/>
            maior quàm linea a b.</s>
            <s xml:id="echoid-s22240" xml:space="preserve"> Plus ergo diſtant centra ſphærarum in interſectione, quàm in ſitu contactus:</s>
            <s xml:id="echoid-s22241" xml:space="preserve">
              <lb/>
            & ſecundum quòd peripheria circuli, quæ eſt communis ſectio ſuarum ſuperficierum, minoratu
              <gap/>
            ,
              <lb/>
            ſecundum hoc diſtantia centrorum augetur:</s>
            <s xml:id="echoid-s22242" xml:space="preserve"> & ſecundum quòd illa peripheria augetur, ſecundum
              <lb/>
            hoc diſtantia centrorum minuitur:</s>
            <s xml:id="echoid-s22243" xml:space="preserve"> & reſpectu partis uniuerſi, ad quam fit interſectio, plus profun-
              <lb/>
            datur centrum ſphæræ continentis, reſpectu contactus, in tanto, quantò linea a e fit maior quàm li
              <lb/>
            nea a b.</s>
            <s xml:id="echoid-s22244" xml:space="preserve"> Et hoc eſt, quod proponebatur.</s>
            <s xml:id="echoid-s22245" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div793" type="section" level="0" n="0">
          <head xml:id="echoid-head659" xml:space="preserve" style="it">85. Si duæ ſphæræ intra tertiam ſecundum circulũ æqualem circulo maiori ſphæræ, intra quã
            <lb/>
          fit interſectio, ſe interſecent: utra illarum ſphærarum ſphæram, intra quam fit interſectio, in-
            <lb/>
          terſecabit: et omniũ illarũ ſuperficierũ ſphæricarũ cõmunis ſectio erit peripheria circuli unius.</head>
          <p>
            <s xml:id="echoid-s22246" xml:space="preserve">Verbi gratia:</s>
            <s xml:id="echoid-s22247" xml:space="preserve"> ſit, ut ſphæra, cuius centrum a, interſecet ſphæram, cuius centrum ſit b, intra ſphæ-
              <lb/>
            ram, cuius centrum ſit c, ſecundum circulũ æqualẽ circulo maiori ſphę-
              <lb/>
              <figure xlink:label="fig-0335-01" xlink:href="fig-0335-01a" number="344">
                <variables xml:id="echoid-variables328" xml:space="preserve">b c a</variables>
              </figure>
            rę c.</s>
            <s xml:id="echoid-s22248" xml:space="preserve"> Dico, quòd ſphæra a & ſphæra b interſecabũt ſphæram c:</s>
            <s xml:id="echoid-s22249" xml:space="preserve"> & omniũ
              <lb/>
            ſuperficierum ſphæricarum illarum ſphærarum erit communis ſectio
              <lb/>
            peripheria circuli illius, ſecundum quẽ ſphærarum a & b fiebat interſe-
              <lb/>
            ctio, hoc eſt cuiuſdam circuli magni ſphæræ c.</s>
            <s xml:id="echoid-s22250" xml:space="preserve"> Quoniam enim circulus
              <lb/>
            maior diuidit ſphæram per æqualia, quia tranſit per centrũ eius ex defi-
              <lb/>
            nitione:</s>
            <s xml:id="echoid-s22251" xml:space="preserve"> tũc patet, quòd æqualis eidẽ, (undecunq;</s>
            <s xml:id="echoid-s22252" xml:space="preserve"> contingat eũ in ſphæ
              <lb/>
            ra produci) diuidet eã per æqualia:</s>
            <s xml:id="echoid-s22253" xml:space="preserve"> & ſic interſecabit ſecun dũ illum cir
              <lb/>
            culum utraq;</s>
            <s xml:id="echoid-s22254" xml:space="preserve"> ſphærarum, ſcilicet a & b ſphæram c.</s>
            <s xml:id="echoid-s22255" xml:space="preserve"> Sphæra autem a in-
              <lb/>
            terſecante ſphæram b, communis ſectio eſt peripheria circuli per 80 hu
              <lb/>
            ius:</s>
            <s xml:id="echoid-s22256" xml:space="preserve"> diuidit autem iſte circulus ſphæram c per æqualia:</s>
            <s xml:id="echoid-s22257" xml:space="preserve"> ergo interſecat.</s>
            <s xml:id="echoid-s22258" xml:space="preserve">
              <lb/>
            Eſt ergo eius peripheria in ſuperficie ſphærę c:</s>
            <s xml:id="echoid-s22259" xml:space="preserve"> ſed & eadem peripheria
              <lb/>
            eſt in ſuperficiebus ſphærarum a & b.</s>
            <s xml:id="echoid-s22260" xml:space="preserve"> In omniũ ergo ſphærarum illarũ
              <lb/>
            triũ ſuperficieb.</s>
            <s xml:id="echoid-s22261" xml:space="preserve"> eſt illa circuli peripheria.</s>
            <s xml:id="echoid-s22262" xml:space="preserve"> Eſt ergo ipſa cõmunis ſectio
              <lb/>
            omnium ſuperficierum dictarum ſphærarum.</s>
            <s xml:id="echoid-s22263" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s22264" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div795" type="section" level="0" n="0">
          <head xml:id="echoid-head660" xml:space="preserve" style="it">86. Lineam à centro ſphæræ per centrum circuli ſphæram ſecantis, orthogonaliter ductam
            <gap/>
            <lb/>
          medio abſciſſæ portionis eſt neceſſarium applicari.</head>
          <p>
            <s xml:id="echoid-s22265" xml:space="preserve">Sit ſphæra, cuius centrum a, & ſit circulus b c d, cuius centrum ſit
              <lb/>
              <figure xlink:label="fig-0335-02" xlink:href="fig-0335-02a" number="345">
                <variables xml:id="echoid-variables329" xml:space="preserve">c f b e d a</variables>
              </figure>
            e, abſcindens portionem ſphærę:</s>
            <s xml:id="echoid-s22266" xml:space="preserve"> ducaturq́;</s>
            <s xml:id="echoid-s22267" xml:space="preserve"> linea a e, & producatur
              <lb/>
            uſq;</s>
            <s xml:id="echoid-s22268" xml:space="preserve"> ad ſuperficiem ſphæricam, cui incidat in puncto f.</s>
            <s xml:id="echoid-s22269" xml:space="preserve"> Dico, quòd li
              <lb/>
            nea a e neceſſariò applicatur puncto, qui eſt medium abſciſſę portio
              <lb/>
            nis ſphærę in conuexo uel concauo ipſius:</s>
            <s xml:id="echoid-s22270" xml:space="preserve"> & quòd hoc eſt punctum
              <lb/>
            f.</s>
            <s xml:id="echoid-s22271" xml:space="preserve"> Ducantur enim lineæ a b, a c & a d, & copulentur lineę e b, e c, e d:</s>
            <s xml:id="echoid-s22272" xml:space="preserve">
              <lb/>
            erunt itaq;</s>
            <s xml:id="echoid-s22273" xml:space="preserve"> trigona a e b, a e c & a e d omnia ſecundum latera ęquales
              <lb/>
            angulos reſpiciẽtia adinuicem proportionalia:</s>
            <s xml:id="echoid-s22274" xml:space="preserve"> quoniam illa ipſorũ
              <lb/>
            latera ſunt adinuicẽ æqualia, ut patet per ſphęrę & circuli definitio-
              <lb/>
            nes, & quia latus a e eſt omnibus commune:</s>
            <s xml:id="echoid-s22275" xml:space="preserve"> anguli itaq;</s>
            <s xml:id="echoid-s22276" xml:space="preserve"> b a e, c a e, d
              <lb/>
            a e omnes ſunt æquales per 5 p 6:</s>
            <s xml:id="echoid-s22277" xml:space="preserve"> ergo per 26 p 3 arcus b f, c f, d f ſunt
              <lb/>
            æquales.</s>
            <s xml:id="echoid-s22278" xml:space="preserve"> Et quoniam productis quibuslibet lineis à centro ſphæræ
              <lb/>
            a ad peripheriam circuli b c d, idem ſemper accidit:</s>
            <s xml:id="echoid-s22279" xml:space="preserve"> palàm, quia pun
              <lb/>
            ctus f eſt in medio portiõis abſciſſę de ſphęra.</s>
            <s xml:id="echoid-s22280" xml:space="preserve"> Et hoc proponebatur.</s>
            <s xml:id="echoid-s22281" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div797" type="section" level="0" n="0">
          <head xml:id="echoid-head661" xml:space="preserve" style="it">87. Proportionem partis ſuperficiei ſphæricæ ad totalem ſuperficiem ſuæ ſphæræ, ſicut anguli
            <lb/>
          ſolidi in ipſam à centro ſphæræ cadentis, ad octo rectos ſolidos neceſſe eſt eſſe. È
            <unsure/>
          Nicolao Caba-
            <lb/>
          ſilla in 3 librum magnæ conſtructionis Ptolemæi.</head>
          <figure number="346">
            <variables xml:id="echoid-variables330" xml:space="preserve">a b d c</variables>
          </figure>
          <p>
            <s xml:id="echoid-s22282" xml:space="preserve">Verbi gratia:</s>
            <s xml:id="echoid-s22283" xml:space="preserve"> ſit a b c pars ſuperficiei ſphæricę ali-
              <lb/>
            cuius ſphærę, cuius centum ſit d:</s>
            <s xml:id="echoid-s22284" xml:space="preserve"> & ducantur lineæ
              <lb/>
            a d, b d, c d:</s>
            <s xml:id="echoid-s22285" xml:space="preserve"> & in ipſa ſuperficie ducantur lineæ a b, b
              <lb/>
            c, a c:</s>
            <s xml:id="echoid-s22286" xml:space="preserve"> fietq́;</s>
            <s xml:id="echoid-s22287" xml:space="preserve"> pyramis, cuius uertex eſt punctum d, &
              <lb/>
            baſis a b c.</s>
            <s xml:id="echoid-s22288" xml:space="preserve"> Palàm quoq;</s>
            <s xml:id="echoid-s22289" xml:space="preserve">, quoniã angulus circa pun-
              <lb/>
            ctum d eſt ſolidus, tribus angulis ſuperficialibus cõ-
              <lb/>
            tentus.</s>
            <s xml:id="echoid-s22290" xml:space="preserve"> Dico, quòd quę eſt proportio illius anguli
              <lb/>
            ad 8 rectos angulos ſolidos, qui replent locum ſoli-
              <lb/>
            dum circa centrum d, eadem erit proportio ſuperfi-
              <lb/>
            ciei ſphæricæ, quæ eſt a b c, ad totam ſphæricam ſu-
              <lb/>
            perficiem ſuę ſphæræ.</s>
            <s xml:id="echoid-s22291" xml:space="preserve"> Imaginentur enim plurimi
              <lb/>
            circuli magni, tranſeuntes per omnia puncta illius
              <lb/>
            ſuperficiei, non ſecantes ſe ſuper illam.</s>
            <s xml:id="echoid-s22292" xml:space="preserve"> Patet itaq;</s>
            <s xml:id="echoid-s22293" xml:space="preserve">,
              <lb/>
            quoniã aliqui arcus illorum circulorũ determinãtur
              <lb/>
            per lineas terminales illius ſuperficiei:</s>
            <s xml:id="echoid-s22294" xml:space="preserve"> omniũ aũt il-
              <lb/>
            lorũ arcuũ partialiũ ad totos ſuos circulos eſt ꝓpor
              <lb/>
            </s>
          </p>
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