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-div771" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s21947" xml:space="preserve">
              <pb o="29" file="0331" n="331" rhead="LIBER PRIMVS."/>
            los per definitionẽ poli.</s>
            <s xml:id="echoid-s21948" xml:space="preserve"> Et ſi aliqui circuli eoſdẽ habent polos, patet per 14 p 11, quòd ipſi ſunt æqui-
              <lb/>
            diſtantes:</s>
            <s xml:id="echoid-s21949" xml:space="preserve"> & hoc proponebatur.</s>
            <s xml:id="echoid-s21950" xml:space="preserve"> Quòd ſi etiã reliquus circulorũ æquidiſtantium eſſet circulus ma-
              <lb/>
            gnus, eadem eſſet demonſtratio.</s>
            <s xml:id="echoid-s21951" xml:space="preserve"> Duo uerò circuli magni eiuſdem ſphęræ ſibi inuicem æquidiſtare
              <lb/>
            non poſſunt:</s>
            <s xml:id="echoid-s21952" xml:space="preserve"> quoniam amborum illorum eſt idem centrum, quod eſt centrum ſphæræ.</s>
            <s xml:id="echoid-s21953" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div772" type="section" level="0" n="0">
          <head xml:id="echoid-head643" xml:space="preserve" style="it">69. Si plana ſuperficies ſecet ſphærã, cõmunis ſectio erit circulus. Ex quo patet, quoniã à quo-
            <lb/>
          libet puncto in diametro uel ſuperficie ſphærica dato, eſt poſsibile totali ſuperficiei ſphæricæ circu-
            <lb/>
          lumcircumduci, alij etiam circulo illius æquidiſtantem. 1 th. 1 ſphær. Theodoſy.</head>
          <p>
            <s xml:id="echoid-s21954" xml:space="preserve">Sit ſphęra, cuius centrũ a, ſeceturq́;</s>
            <s xml:id="echoid-s21955" xml:space="preserve"> per planam ſuperficiẽ.</s>
            <s xml:id="echoid-s21956" xml:space="preserve"> Dico, quòd cõmunis ſectio ſuperficiei
              <lb/>
            ſphęricæ & planæ eſt circulus.</s>
            <s xml:id="echoid-s21957" xml:space="preserve"> Si enim fiat ſectio ք centrũ
              <lb/>
              <figure xlink:label="fig-0331-01" xlink:href="fig-0331-01a" number="338">
                <variables xml:id="echoid-variables322" xml:space="preserve">d f b c e d</variables>
              </figure>
            a:</s>
            <s xml:id="echoid-s21958" xml:space="preserve"> tũc patet, quòd oẽs lineæ ductæ à cẽtro a ad ſphæræ ſu-
              <lb/>
            perficiẽ, quę ſunt in illa plana ſuքficie ſecãte, & terminan-
              <lb/>
            tur ad cõmunem terminũ illorũ, ſunt æquales per defini-
              <lb/>
            tionẽ ſphęræ:</s>
            <s xml:id="echoid-s21959" xml:space="preserve"> ergo per definitionẽ circuli, illa cõmunis ſe-
              <lb/>
            ctio eſt circulus.</s>
            <s xml:id="echoid-s21960" xml:space="preserve"> Si aũt ſuperficies plana ſecet ſphærã datã
              <lb/>
            nõ per centrũ a:</s>
            <s xml:id="echoid-s21961" xml:space="preserve"> ducatur per 11 p 11 à centro a perpẽdicula-
              <lb/>
            ris ſuper ſuperficiẽ ſecantẽ, quę ſit a b, & cõtinuẽtur lineæ
              <lb/>
            a c, a d, a e, a f, & quot quis uoluerit ad illã ſectionem com-
              <lb/>
            munem à cẽtro ipſius ſphęræ:</s>
            <s xml:id="echoid-s21962" xml:space="preserve"> ducãtur quoq;</s>
            <s xml:id="echoid-s21963" xml:space="preserve"> lineę c b, d b,
              <lb/>
            e b, f b, in ipſa ſuperficie ſecãte, ad puncta, quibus incidũt
              <lb/>
            lineę ex centro ſphęræ ductæ.</s>
            <s xml:id="echoid-s21964" xml:space="preserve"> Palàm ergo per 47 p 1, quo-
              <lb/>
            niã quadratũ lineæ a c eſt ęquale duobus quadratis linea-
              <lb/>
            rum a b & b c:</s>
            <s xml:id="echoid-s21965" xml:space="preserve"> & ſimiliter quadratum lineę a d eſt æquale
              <lb/>
            duob.</s>
            <s xml:id="echoid-s21966" xml:space="preserve"> quadratis linearũ a b & b d:</s>
            <s xml:id="echoid-s21967" xml:space="preserve"> ſed quadratũ lineæ a c
              <lb/>
            eſt æquale quadrato lineæ a d:</s>
            <s xml:id="echoid-s21968" xml:space="preserve"> quoniã linea a c eſt æqualis
              <lb/>
            lineæ a d per definitionẽ ſphęræ, & quadratũ lineæ a b eſt ęquale ſibijpſi:</s>
            <s xml:id="echoid-s21969" xml:space="preserve"> relinquitur ergo quadratũ
              <lb/>
            lineæ c b æquale quadrato lineæ d b:</s>
            <s xml:id="echoid-s21970" xml:space="preserve"> eſt ergo linea c b æqualis lineæ d b:</s>
            <s xml:id="echoid-s21971" xml:space="preserve"> & ſimiliter erit linea d b
              <lb/>
            æqualis lineis e b & f b:</s>
            <s xml:id="echoid-s21972" xml:space="preserve"> eadẽ enim eſt demonſtratio, quotcunq;</s>
            <s xml:id="echoid-s21973" xml:space="preserve"> alijs lineis à cẽtro ſphærę a ad illam
              <lb/>
            communẽ ſectionem productis.</s>
            <s xml:id="echoid-s21974" xml:space="preserve"> Omnes itaq;</s>
            <s xml:id="echoid-s21975" xml:space="preserve"> lineæ à puncto b ad illã communem ſectionẽ ductæ,
              <lb/>
            ſunt æquales:</s>
            <s xml:id="echoid-s21976" xml:space="preserve"> ergo per 9 p 3 & per definitionẽ circuli, ut prius, punctũ b eſt centrũ circuli.</s>
            <s xml:id="echoid-s21977" xml:space="preserve"> Cõmunis
              <lb/>
            ergo ſectio iſtarũ ſuperficierũ eſt circulus:</s>
            <s xml:id="echoid-s21978" xml:space="preserve"> & hoc eſt propoſitũ.</s>
            <s xml:id="echoid-s21979" xml:space="preserve"> Patet etiã ex hoc corollariũ:</s>
            <s xml:id="echoid-s21980" xml:space="preserve"> quoniã
              <lb/>
            à pũcto dato per 12 p 1 producta perpẽdiculari ſuper diametrũ ſphęræ, imaginetur ſuperficies plana
              <lb/>
            ſecãs ſphærã ſecundũ illã perpendicularẽ:</s>
            <s xml:id="echoid-s21981" xml:space="preserve"> & patet propoſitũ per præmiſſa.</s>
            <s xml:id="echoid-s21982" xml:space="preserve"> Quòd ſi alicui circulo in
              <lb/>
            ſphęra ſignato æquidiſtãs duci debeat:</s>
            <s xml:id="echoid-s21983" xml:space="preserve"> à dato pũcto ducatur perpẽdicularis ſuper ſphęræ diametrũ
              <lb/>
            tranſeuntẽ circuli centrũ, cui æquidiſtãs debet duci circulus, & ꝓducatur in continuũ uſq;</s>
            <s xml:id="echoid-s21984" xml:space="preserve"> ad aliã
              <lb/>
            ſphęræ ſuperficiẽ, & ducatur alia linea à pũcto diametri utcũq;</s>
            <s xml:id="echoid-s21985" xml:space="preserve"> ſuք productã, & orthogonaliter ſu-
              <lb/>
            per diametrũ ſphęræ, imagineturq́;</s>
            <s xml:id="echoid-s21986" xml:space="preserve"> ſuperficies plana trãſiens terminos iſtarũ linearũ in ipſa ſuper-
              <lb/>
            ficie ſphęræ faciẽs ſectionẽ:</s>
            <s xml:id="echoid-s21987" xml:space="preserve"> quę per præmiſſa neceſſariò erit circulus:</s>
            <s xml:id="echoid-s21988" xml:space="preserve"> quia ք 4 p 11 diameter ſphęrę,
              <lb/>
            ſuper quã ducitur linea à pũcto dato, erit perpẽdicularis ſuper ſuperficiẽ in punctis illis, ut præmit-
              <lb/>
            titur, ſphæram ſecantem:</s>
            <s xml:id="echoid-s21989" xml:space="preserve"> unde à centro ſphæræ ductis lineis, ut prius, patet quod proponebatur.</s>
            <s xml:id="echoid-s21990" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div774" type="section" level="0" n="0">
          <head xml:id="echoid-head644" xml:space="preserve" style="it">70. À
            <unsure/>
          dato puncto ad datam ſphæram lineam contingentem ducere.</head>
          <p>
            <s xml:id="echoid-s21991" xml:space="preserve">Sit enim datũ punctũ a, & centrũ datę ſphę-
              <lb/>
              <figure xlink:label="fig-0331-02" xlink:href="fig-0331-02a" number="339">
                <variables xml:id="echoid-variables323" xml:space="preserve">c a d b</variables>
              </figure>
            ræ ſit punctũ b:</s>
            <s xml:id="echoid-s21992" xml:space="preserve"> & ducatur linea a b:</s>
            <s xml:id="echoid-s21993" xml:space="preserve"> & à cẽtro
              <lb/>
            ſphæræ, quod eſt b, ducatur linea b c, ut cõtin-
              <lb/>
            git, & copuletur linea a c:</s>
            <s xml:id="echoid-s21994" xml:space="preserve"> palamq́;</s>
            <s xml:id="echoid-s21995" xml:space="preserve"> ք 2 p 11, quo
              <lb/>
            niam trigonũ a b c eſt in una ſuperficie plana:</s>
            <s xml:id="echoid-s21996" xml:space="preserve">
              <lb/>
            hęcitaq;</s>
            <s xml:id="echoid-s21997" xml:space="preserve"> per præcedẽtem ſecabit ſphęrã ſecũ-
              <lb/>
            dũ circulũ, cui per 17 p 3 à pũcto a ducatur cõ-
              <lb/>
            tingẽs in pũcto d, quæ ſit a d:</s>
            <s xml:id="echoid-s21998" xml:space="preserve"> & patet ꝓpoſitũ.</s>
            <s xml:id="echoid-s21999" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div776" type="section" level="0" n="0">
          <head xml:id="echoid-head645" xml:space="preserve" style="it">71. Omnis ſuperficies plana contingens
            <lb/>
          ſphæram, ſecundũ unicum punctum eſt con-
            <lb/>
          tingens. 3 th. 1 ſphær. Theodoſij.</head>
          <p>
            <s xml:id="echoid-s22000" xml:space="preserve">Ducatur in plana ſuperficie contingente ſphæram, linea recta trans locum cõtactus, & in ſuper-
              <lb/>
            ficie ſphęræ circulus magnus.</s>
            <s xml:id="echoid-s22001" xml:space="preserve"> Si ergo ſuperficies plana contingit ſphæram ſecundum aliud quàm
              <lb/>
            ſecundum punctum, & linea recta continget circulum ſecundum idem:</s>
            <s xml:id="echoid-s22002" xml:space="preserve"> non ergo ſecundum pun-
              <lb/>
            ctum continget linea recta circulum:</s>
            <s xml:id="echoid-s22003" xml:space="preserve"> quod eſt contra 16 p 3:</s>
            <s xml:id="echoid-s22004" xml:space="preserve"> palàm ergo propoſitum.</s>
            <s xml:id="echoid-s22005" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div777" type="section" level="0" n="0">
          <head xml:id="echoid-head646" xml:space="preserve" style="it">72. À
            <unsure/>
          dato pũcto ſuքficiei ſphæricæ ſuքficiẽ planã cõtingentẽ ducere. Ex quo patet, ꝗ omnis
            <lb/>
          linea centrũ ſphæræ trãſiens, eſt perpẽdicularis ſuք eius ſuperficiẽ: & ſieſt perpendicularis ſuper
            <lb/>
          ſphæricam ſuperficiem, neceſſariò tranſit centrũ ſphæræ. È
            <unsure/>
          4 th. 1 ſphær. Theodoſy. Alh. 25 n 4.</head>
          <p>
            <s xml:id="echoid-s22006" xml:space="preserve">Eſto ſphęra, cuius centrũ ſit a, & circulus eius magnus b d c:</s>
            <s xml:id="echoid-s22007" xml:space="preserve"> ducaturq́;</s>
            <s xml:id="echoid-s22008" xml:space="preserve"> linea a b à cẽtro ad circũ-
              <lb/>
            ferentiã:</s>
            <s xml:id="echoid-s22009" xml:space="preserve"> & à pũcto b ducatur linea cõtingẽs circulũ, quę ſit f b e ք 17 p 3:</s>
            <s xml:id="echoid-s22010" xml:space="preserve"> erũt ergo anguli a b e & a b f
              <lb/>
            recti.</s>
            <s xml:id="echoid-s22011" xml:space="preserve"> Imaginatis quoq;</s>
            <s xml:id="echoid-s22012" xml:space="preserve"> ք 69 huius circulis quotcũq;</s>
            <s xml:id="echoid-s22013" xml:space="preserve"> in ſuքficie ſphęrę ſecantib.</s>
            <s xml:id="echoid-s22014" xml:space="preserve"> ſe in pũcto b, & du-
              <lb/>
            ctis lineis, cõtingentib.</s>
            <s xml:id="echoid-s22015" xml:space="preserve"> illos circulos in pũcto b:</s>
            <s xml:id="echoid-s22016" xml:space="preserve"> palàm ք 18 p 3, quoniã linea b a cũ omnib.</s>
            <s xml:id="echoid-s22017" xml:space="preserve"> illis lineis
              <lb/>
            cõtinetangulos rectos.</s>
            <s xml:id="echoid-s22018" xml:space="preserve"> Ergo oẽs illę lineæ ſunt in una ſuքficie plana ք 2 p 11.</s>
            <s xml:id="echoid-s22019" xml:space="preserve"> Illa itaq;</s>
            <s xml:id="echoid-s22020" xml:space="preserve"> ſuքficies con-
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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