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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      <text xml:lang="lat" type="free">
        <div xml:id="echoid-div777" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s22020" xml:space="preserve">
              <pb o="30" file="0332" n="332" rhead="VITELLONIS OPTICAE"/>
            tingit ſphęrã ք definitionẽ ſuքficiei planę ſphęrã cõtingẽtis.</s>
            <s xml:id="echoid-s22021" xml:space="preserve"> Ex hoc itaq;</s>
            <s xml:id="echoid-s22022" xml:space="preserve"> patet, quoniã omnis linea
              <lb/>
            à cẽtro ſphęræ ducta, ſit erecta ſuք planã ſuքficiẽ, ſphęrã ipſam in pũ-
              <lb/>
              <figure xlink:label="fig-0332-01" xlink:href="fig-0332-01a" number="340">
                <variables xml:id="echoid-variables324" xml:space="preserve">g h e b f d a</variables>
              </figure>
            cto ſuæ incidẽtię cõtingentẽ, & anguli incidẽtiæ ſint æquales:</s>
            <s xml:id="echoid-s22023" xml:space="preserve"> quoniã
              <lb/>
            ipſa eſt perpẽdicularis ſuք ſphęrę ſuperficiẽ, ք definitionẽ perpẽdicu-
              <lb/>
            laris:</s>
            <s xml:id="echoid-s22024" xml:space="preserve"> anguli enim ſemicirculorũ oẽs ſunt æquales ք 43 huius.</s>
            <s xml:id="echoid-s22025" xml:space="preserve"> Et quo-
              <lb/>
            niã linea ab ꝓducta ad punctũ g, eſt adhuc erecta ſuք ſuքficiẽ planã,
              <lb/>
            ſphęrã cõtingentẽ in pũcto b:</s>
            <s xml:id="echoid-s22026" xml:space="preserve"> palã, ք a linea g b, & quęcũq;</s>
            <s xml:id="echoid-s22027" xml:space="preserve"> alia քpẽdi-
              <lb/>
            cularis erigi poteſt ſuք ſuքficiẽ planã in pũcto b, cõtingẽtẽ ſphęrã, trã-
              <lb/>
            ſit cẽtrũ ſphęræ a:</s>
            <s xml:id="echoid-s22028" xml:space="preserve"> ꝗ a ſi à pũcto b poſsit alia linea erigi ſuք ſuքficiẽ cõ-
              <lb/>
            tingẽtẽ, nõ trãſiẽs cetrũ ſphærę a:</s>
            <s xml:id="echoid-s22029" xml:space="preserve"> ſit illa h b d, & ſit angul
              <emph style="sub">9</emph>
            h b e rectus:</s>
            <s xml:id="echoid-s22030" xml:space="preserve">
              <lb/>
            ſed angul
              <emph style="sub">9</emph>
            g b e eſt rectus ք 13 p 1, cũ angul
              <emph style="sub">9</emph>
            a b e ſit rect
              <emph style="sub">9</emph>
            ex hypotheſi:</s>
            <s xml:id="echoid-s22031" xml:space="preserve">
              <lb/>
            erit itaq;</s>
            <s xml:id="echoid-s22032" xml:space="preserve"> rectus maior recto:</s>
            <s xml:id="echoid-s22033" xml:space="preserve"> qđ eſt impoſsibile:</s>
            <s xml:id="echoid-s22034" xml:space="preserve"> patet ergo ꝓpoſitũ.</s>
            <s xml:id="echoid-s22035" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div779" type="section" level="0" n="0">
          <head xml:id="echoid-head647" xml:space="preserve" style="it">73. Omnium ſphærarum, quarum conuexæ ſuperficies æquidi-
            <lb/>
          ſtant, uel ſecundũ ſe tot{as} ſe contingunt, neceſſariò eſt idẽ centrum.</head>
          <p>
            <s xml:id="echoid-s22036" xml:space="preserve">Sint duę ſphęræ, quarũ cõuexæ ſuքficies æquidiſtẽt, ſectæ ք æqua-
              <lb/>
            lia ք unã planã ſuքficiẽ:</s>
            <s xml:id="echoid-s22037" xml:space="preserve"> cõmunis ergo ſectio ſuperficierũ illarũ ſphæ
              <lb/>
            ricarũ & huius planæ erũt circuli:</s>
            <s xml:id="echoid-s22038" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s22039" xml:space="preserve"> magnus circulus maioris ſphęræ a b, & centrũ eius e:</s>
            <s xml:id="echoid-s22040" xml:space="preserve"> mino-
              <lb/>
            ris uerò ſphęrę circulus magnus ſit c d.</s>
            <s xml:id="echoid-s22041" xml:space="preserve"> Dico, quòd idẽ
              <lb/>
              <figure xlink:label="fig-0332-02" xlink:href="fig-0332-02a" number="341">
                <variables xml:id="echoid-variables325" xml:space="preserve">a c e h d b</variables>
              </figure>
            punctũ e etiã erit cẽtrũ circuli c d.</s>
            <s xml:id="echoid-s22042" xml:space="preserve"> Ducatur enim linea
              <lb/>
            a e b taliter, ut ſi e nõ ſit cẽtrũ amborũ circulorũ, linea
              <lb/>
            tñ a e b trãſeat ք ambo cẽtra (qđ poteſt fieri cõtinua-
              <lb/>
            tis cẽtris ք lineã rectã) & ꝓducta illa ad քipheriã ma-
              <lb/>
            ioris ſphęrę:</s>
            <s xml:id="echoid-s22043" xml:space="preserve"> hęc itaq;</s>
            <s xml:id="echoid-s22044" xml:space="preserve"> erit diameter circuli a b.</s>
            <s xml:id="echoid-s22045" xml:space="preserve"> Et quo-
              <lb/>
            niã circuli a b & c d ſunt in eadẽ ſuքficie:</s>
            <s xml:id="echoid-s22046" xml:space="preserve"> ſit ut diame-
              <lb/>
            ter a b ſecet քipheriã circuli c d in pũctis c & d:</s>
            <s xml:id="echoid-s22047" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s22048" xml:space="preserve">
              <lb/>
            recta c d diameter circuli c d.</s>
            <s xml:id="echoid-s22049" xml:space="preserve"> Quia ergo ꝓpter æqui-
              <lb/>
            diſtantiã circulorũ linea a c eſt æqualis lineæ b d, & li-
              <lb/>
            nea a e eſt æqualis lineæ e b:</s>
            <s xml:id="echoid-s22050" xml:space="preserve"> remanet linea c e æqualis
              <lb/>
            lineę e d.</s>
            <s xml:id="echoid-s22051" xml:space="preserve"> Et ꝗ a diameter c d diuiditur ք ęqualia in pũ-
              <lb/>
            cto e:</s>
            <s xml:id="echoid-s22052" xml:space="preserve"> patet, quòd pũctus e eſt cẽtrũ circuli c d.</s>
            <s xml:id="echoid-s22053" xml:space="preserve"> Si enim
              <lb/>
            nõ ſit pũctus e centrũ circuli c d:</s>
            <s xml:id="echoid-s22054" xml:space="preserve"> ſit cẽtrũ eius pũctus
              <lb/>
            h:</s>
            <s xml:id="echoid-s22055" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s22056" xml:space="preserve"> ք definitionẽ circuli linea h d æqualis lineæ a
              <lb/>
            c:</s>
            <s xml:id="echoid-s22057" xml:space="preserve"> erit ergo linea h a æqualis lineæ h b:</s>
            <s xml:id="echoid-s22058" xml:space="preserve"> ſed linea h a eſt
              <lb/>
            maior quàm linea a e:</s>
            <s xml:id="echoid-s22059" xml:space="preserve"> ergo h b eſt maior quã linea e b,
              <lb/>
            pars ſuo toto:</s>
            <s xml:id="echoid-s22060" xml:space="preserve"> quod eſt impoſsibile.</s>
            <s xml:id="echoid-s22061" xml:space="preserve"> Eſt ergo pũctus e
              <lb/>
            neceſſariò cẽtrum circuli c d.</s>
            <s xml:id="echoid-s22062" xml:space="preserve"> Et quia circulus c d eſt magnus circulus ſuę ſphęræ, patet quòd æqui-
              <lb/>
            diſtantium ſphęrarum eſt idem cẽtrum:</s>
            <s xml:id="echoid-s22063" xml:space="preserve"> quod eſt propoſitum primum.</s>
            <s xml:id="echoid-s22064" xml:space="preserve"> Et eodem modo de ſphæris
              <lb/>
            ſecundum totas ſuas ſuperficies contingentibus, eſt demonſtrandum.</s>
            <s xml:id="echoid-s22065" xml:space="preserve"> Lineæ enim ductæ à centro
              <lb/>
            ad concauum maioris & ad cõuexum minoris, ſunt ęquales:</s>
            <s xml:id="echoid-s22066" xml:space="preserve"> patet ergo illud quod proponebatur.</s>
            <s xml:id="echoid-s22067" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div781" type="section" level="0" n="0">
          <head xml:id="echoid-head648" xml:space="preserve" style="it">74. Si duæ ſphæræ fuerint æquidiſtãtes, uel ſecundũ totas ſuքficies ſe cõtingẽtes: quæcũ lineæ
            <lb/>
          ſuք unius earũ ſuperficiẽ perpẽdicularis fuerit, ſuք alterius quo ſuperficiẽ perpẽdicularis erit.</head>
          <p>
            <s xml:id="echoid-s22068" xml:space="preserve">Iſtud faciliter patet.</s>
            <s xml:id="echoid-s22069" xml:space="preserve"> Quoniã enim ex præmiſſa tales ſphęræ idẽ centrum habere neceſſariò com-
              <lb/>
            probantur:</s>
            <s xml:id="echoid-s22070" xml:space="preserve"> ergo per 72 huius linea perpendicularis ſuper alteram iſtarum ſphęrarum, centrũ ipſius
              <lb/>
            tranſit:</s>
            <s xml:id="echoid-s22071" xml:space="preserve"> ſed centrum ipſius eſt cẽtrum alterius.</s>
            <s xml:id="echoid-s22072" xml:space="preserve"> Ergo per eandem 72 huius ſuper alterius etiã ſphæ-
              <lb/>
            ræ ſuperficiem illa linea perpendicularis erit:</s>
            <s xml:id="echoid-s22073" xml:space="preserve"> & hoc eſt propoſitum.</s>
            <s xml:id="echoid-s22074" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div782" type="section" level="0" n="0">
          <head xml:id="echoid-head649" xml:space="preserve" style="it">75. Si duæ ſphæræ cẽtra diuerſa habuerint: impoßibile eſt, ut lineæ քpẽdiculares ſuք unius ſu-
            <lb/>
          perficiẽ, ſint perpẽdiculares ſuper alterius ſuperficiẽ, niſi unatantũ, quæ trãſit cẽtra ambarum.</head>
          <p>
            <s xml:id="echoid-s22075" xml:space="preserve">Quocũq;</s>
            <s xml:id="echoid-s22076" xml:space="preserve"> modo ſe habẽtibus adinuicẽ ſphęris, ſiue extrinſecus ſiue intrinſecus ſe cõtingẽtibus,
              <lb/>
            uel etiam ſe nõ contingẽtibus, uel etiã ſe adinuicẽ ſecãtibus, ſemper patet ex 72 huius, quoniã linea
              <lb/>
            tranſiens per cẽtra ipſarũ, eſt perpẽdicularis ſuper ſuperficiẽ utriuſq;</s>
            <s xml:id="echoid-s22077" xml:space="preserve">;:</s>
            <s xml:id="echoid-s22078" xml:space="preserve"> aliã quoq;</s>
            <s xml:id="echoid-s22079" xml:space="preserve"> lineã ſuper utriuſq;</s>
            <s xml:id="echoid-s22080" xml:space="preserve">
              <lb/>
            ſuperficiẽ perpendicularẽ eſſe, eſt impoſsibile.</s>
            <s xml:id="echoid-s22081" xml:space="preserve"> Si enim ſit poſsibile:</s>
            <s xml:id="echoid-s22082" xml:space="preserve"> ducatur aliqua alia perpẽdicu-
              <lb/>
            lariter ſuper utriuſq;</s>
            <s xml:id="echoid-s22083" xml:space="preserve"> ſphęræ ſuperficiẽ:</s>
            <s xml:id="echoid-s22084" xml:space="preserve"> palamq́;</s>
            <s xml:id="echoid-s22085" xml:space="preserve"> erit ex eadẽ 72 huius ipſam per utriuſq;</s>
            <s xml:id="echoid-s22086" xml:space="preserve"> centrũ trã-
              <lb/>
            ſire:</s>
            <s xml:id="echoid-s22087" xml:space="preserve"> quod eſt oppoſitũ hypotheſi.</s>
            <s xml:id="echoid-s22088" xml:space="preserve"> Patet ergo, quoniã nullã aliam lineã, præter eã, quę tranſit centra
              <lb/>
            ambarũ, perpẽdiculariter duci ſuք utriuſq;</s>
            <s xml:id="echoid-s22089" xml:space="preserve"> ſphęrarũ ſuperficies eſt poſsibile.</s>
            <s xml:id="echoid-s22090" xml:space="preserve"> Et hoc eſt propoſitũ.</s>
            <s xml:id="echoid-s22091" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div783" type="section" level="0" n="0">
          <head xml:id="echoid-head650" xml:space="preserve" style="it">76. Si ſphæra ſphærã intrinſec
            <emph style="sub">9</emph>
          aut extrinſec
            <emph style="sub">9</emph>
          cõtingat: in uno tãtũ pũcto cõtingere eſt neceſſe.</head>
          <p>
            <s xml:id="echoid-s22092" xml:space="preserve">Si enim ſphęræ contingẽtes ſe intrinſecus, nõ in puncto ſe contingant:</s>
            <s xml:id="echoid-s22093" xml:space="preserve"> neceſſe eſt circulos ſuos
              <lb/>
            maiores a dinuicem applicatos non ſe in puncto contingere:</s>
            <s xml:id="echoid-s22094" xml:space="preserve"> quod eſt contra 13 p 3, & impoſsibile.</s>
            <s xml:id="echoid-s22095" xml:space="preserve">
              <lb/>
            Quòd ſi ſphęræ extrinſecus ſe contingentes, non ſe contingant in puncto:</s>
            <s xml:id="echoid-s22096" xml:space="preserve"> etiam hoc eſt contra na-
              <lb/>
            turam circulorum extrinſecus ſe contingentium, & contra eandẽ 13 p 3.</s>
            <s xml:id="echoid-s22097" xml:space="preserve"> Poteſt & hoc aliter demon-
              <lb/>
            ſtrari.</s>
            <s xml:id="echoid-s22098" xml:space="preserve"> Si enim inter illas ſphęras, quę ſe extrinſecus contingunt, imaginata fuerit ſuperficies plana:</s>
            <s xml:id="echoid-s22099" xml:space="preserve">
              <lb/>
            palàm ex 71 huius, quoniam utraq;</s>
            <s xml:id="echoid-s22100" xml:space="preserve"> illarum ſphęrarum illã ſuperficiem planam contingit in puncto.</s>
            <s xml:id="echoid-s22101" xml:space="preserve">
              <lb/>
            Ergo & ſeinuicem in puncto contingẽt:</s>
            <s xml:id="echoid-s22102" xml:space="preserve"> & propinquior eſt utriq;</s>
            <s xml:id="echoid-s22103" xml:space="preserve"> ſphærarum ipſa plana ſuperficies
              <lb/>
            interpoſita, quàm ſphæræ inter ſe.</s>
            <s xml:id="echoid-s22104" xml:space="preserve"> Et hoc eſt propoſitum.</s>
            <s xml:id="echoid-s22105" xml:space="preserve"/>
          </p>
        </div>
      </text>
    </echo>
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