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-div783" type="section" level="0" n="0">
          <pb o="31" file="0333" n="333" rhead="LIBER PRIMVS."/>
        </div>
        <div xml:id="echoid-div784" type="section" level="0" n="0">
          <head xml:id="echoid-head651" xml:space="preserve" style="it">77. Sphærarum ſe contingentium, centra diuerſa eſſe eſt neceſſe.</head>
          <p>
            <s xml:id="echoid-s22106" xml:space="preserve">Signentur enim in utralibet ſphærarum à puncto contactus duo circuli maiores per 69 huius,
              <lb/>
            ſecantes eorum ſuperficiebus planis ſphæras per ſua centra, & per puncta contactuum.</s>
            <s xml:id="echoid-s22107" xml:space="preserve"> Et quia cen
              <lb/>
            tra horum circulorum ſunt centra ſphærarum ſuarum per definitionem circulorum magnorũ:</s>
            <s xml:id="echoid-s22108" xml:space="preserve"> hos
              <lb/>
            autem circulos centra diuerſa habere eſt concluſio 6 p 3.</s>
            <s xml:id="echoid-s22109" xml:space="preserve"> Patet ergo propoſitum.</s>
            <s xml:id="echoid-s22110" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div785" type="section" level="0" n="0">
          <head xml:id="echoid-head652" xml:space="preserve" style="it">78. Centrorum, ſphærarum ſe extrinſecus contingentium, diſtantiam ſecundum lineam com
            <lb/>
          poſitam ex ambarum ſphærarum ſemidiametris. intrinſecus uerò ſe contingentium, ſecundum
            <lb/>
          exceſſum ſemidiametri maioris ad ſemidiametrum minoris eſſe, palàm est.</head>
          <p>
            <s xml:id="echoid-s22111" xml:space="preserve">Hoc patet per 76 huius.</s>
            <s xml:id="echoid-s22112" xml:space="preserve"> Quoniam enim contactus ſphærarum fit ſecundum unum tantùm pun-
              <lb/>
            ctum:</s>
            <s xml:id="echoid-s22113" xml:space="preserve"> punctus uerò eſt, cui pars nõ eſt:</s>
            <s xml:id="echoid-s22114" xml:space="preserve"> tunc euidẽs eſt, quòd punctus ille cõmunis in utraq;</s>
            <s xml:id="echoid-s22115" xml:space="preserve"> interſe
              <lb/>
            ctione nihil adimit de diametrorum quantitate:</s>
            <s xml:id="echoid-s22116" xml:space="preserve"> indiuiſibile enim (cum non ſit pars quanti) nec ad-
              <lb/>
            dit nec minuit aliquid de quanto.</s>
            <s xml:id="echoid-s22117" xml:space="preserve"> Et ſic patet propoſitum.</s>
            <s xml:id="echoid-s22118" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div786" type="section" level="0" n="0">
          <head xml:id="echoid-head653" xml:space="preserve" style="it">79. Si concauũ alicuius ſphæræ, ſuperficiem aliquam ſecundum eam totam contingat: neceſſe
            <lb/>
          eſt ſuperficiem contactam partem ſphæræ minoris eſſe.</head>
          <p>
            <s xml:id="echoid-s22119" xml:space="preserve">Sit, ut aliqua ſphæra ſecundũ ſuum concauũ contingat aliquã ſuperficiem ſecundũ oẽs illius par
              <lb/>
            tes, ſicut uas ſphæricũ ſuperficiem aquę contentę.</s>
            <s xml:id="echoid-s22120" xml:space="preserve"> Dico, quòd uerũ eſt quod proponitur.</s>
            <s xml:id="echoid-s22121" xml:space="preserve"> Ducantur
              <lb/>
            enim lineę plurimę à centro ſphærę ad locum contactus ſui cum illa ſuperficie.</s>
            <s xml:id="echoid-s22122" xml:space="preserve"> Et quia omnes lineę
              <lb/>
            productæ ad cõcauũ ſphærę ſunt æquales inter ſe ex definitione ſphæræ, & ſunt æquales productis
              <lb/>
            lineis ad conuexũ ſuperficiei cõtactę:</s>
            <s xml:id="echoid-s22123" xml:space="preserve"> patet ex dicta definitiõe, quoniã illa ſuքficies eſt pars ſphærę:</s>
            <s xml:id="echoid-s22124" xml:space="preserve">
              <lb/>
            & quælibet intellecta exten di ſecundũ cõcauũ ambientis ſphærę, ſphærã minorẽ cõplebit.</s>
            <s xml:id="echoid-s22125" xml:space="preserve"> Eſt ergo
              <lb/>
            pars minoris ſphærę.</s>
            <s xml:id="echoid-s22126" xml:space="preserve"> Linea quoq;</s>
            <s xml:id="echoid-s22127" xml:space="preserve"> in illa ſuperficie ſignata, eſt pars circuli ex 9 p 3, idem habens cen
              <lb/>
            trum cum circulo, cui applicatur.</s>
            <s xml:id="echoid-s22128" xml:space="preserve"> Et ſic illa ſuperficies eſt pars minoris ſphærę.</s>
            <s xml:id="echoid-s22129" xml:space="preserve"> Quod eſt propoſitũ.</s>
            <s xml:id="echoid-s22130" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div787" type="section" level="0" n="0">
          <head xml:id="echoid-head654" xml:space="preserve" style="it">80. Si ſphæra ſphæram interſecet, communis ſectio ſuperficierum ſphæricarum ſe interſecan-
            <lb/>
          tium erit peripheria circuli.</head>
          <p>
            <s xml:id="echoid-s22131" xml:space="preserve">Quod hic proponitur, patet.</s>
            <s xml:id="echoid-s22132" xml:space="preserve"> Imaginetur enim ſuperficies ſecans ambas ſphæras ſecundum lineã
              <lb/>
            cõmunẽ ſectionis ſphærarũ, qualiſcũq;</s>
            <s xml:id="echoid-s22133" xml:space="preserve"> fuerit.</s>
            <s xml:id="echoid-s22134" xml:space="preserve"> Hæc ergo ſuperficies propter ſimilitudinẽ corporũ ſe
              <lb/>
            interſecantiũ plana erit:</s>
            <s xml:id="echoid-s22135" xml:space="preserve"> cõmunis ergo ſectio illius ſuperficiei & utriuſq;</s>
            <s xml:id="echoid-s22136" xml:space="preserve"> ſphærarũ erit circulus per
              <lb/>
            69 huius.</s>
            <s xml:id="echoid-s22137" xml:space="preserve"> Palàm ergo, quòd cõmunis linea interſectionis ſuperficierũ ſphærarum illarum erit peri-
              <lb/>
            pheria circuli, in qua incluſa ſuperficies, erit circulus communis illi ſectioni:</s>
            <s xml:id="echoid-s22138" xml:space="preserve"> quoniam aliàs corpus,
              <lb/>
            quo utræq;</s>
            <s xml:id="echoid-s22139" xml:space="preserve"> ſphærę communicant, eſt corpus cõmune ſphærarum interſectioni:</s>
            <s xml:id="echoid-s22140" xml:space="preserve"> & eſt corpus irregu
              <lb/>
            lare, duabus ſcilicet ſuperficiebus ſphæricis contentum & diuerſis, ſecundum diſpoſitionẽ ſe inter-
              <lb/>
            ſecantium ſphærarum.</s>
            <s xml:id="echoid-s22141" xml:space="preserve"> Patet ergo propoſitum.</s>
            <s xml:id="echoid-s22142" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div788" type="section" level="0" n="0">
          <head xml:id="echoid-head655" xml:space="preserve" style="it">81. Sphærarum ſe interſecantium, maiores circulos ſe inuicem ſecare palàm est. Ex quo patet
            <lb/>
          interſecantium ſe ſphærarum centra diuerſa eſſe.</head>
          <p>
            <s xml:id="echoid-s22143" xml:space="preserve">Primum patet ex definitione ſphærarum ſe interſecantium.</s>
            <s xml:id="echoid-s22144" xml:space="preserve"> Quoniam enim interſecantibus ſe
              <lb/>
            ſphæris, diameter unius per alteram abſcinditur, & maiorum circulorũ diametri ſunt etiam diame-
              <lb/>
            tri ſuarum ſphærarum (diuidunt enim circuli magni ſuas ſphæras per æqualia) tunc patet, quòd cir-
              <lb/>
            culis unius ſphæræ & alterius ſe interſecantium aliqua linea eſt cõmunis.</s>
            <s xml:id="echoid-s22145" xml:space="preserve"> Cum ergo unus circulus
              <lb/>
            aliũ non cõtineat, quia nec una ſphæra ſphæram aliam continet:</s>
            <s xml:id="echoid-s22146" xml:space="preserve"> palàm, quia tales circuli ſe inuicem
              <lb/>
            ſecant ex definitione taliũ circulorũ.</s>
            <s xml:id="echoid-s22147" xml:space="preserve"> Quia uerò ex 5 p 3 circulorũ ſe inuicem ſecantiũ centra eſſe di
              <lb/>
            uerſa neceſſe eſt, & idem eſt centrũ ſphærę, quod eſt centrũ circuli magni in illa ſphæra:</s>
            <s xml:id="echoid-s22148" xml:space="preserve"> patet corol-
              <lb/>
            arium, ſcilicet, quia interſecantium ſe ſphærarum centra ſunt diuerſa.</s>
            <s xml:id="echoid-s22149" xml:space="preserve"> Et hoc proponebatur.</s>
            <s xml:id="echoid-s22150" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div789" type="section" level="0" n="0">
          <head xml:id="echoid-head656" xml:space="preserve" style="it">82. Si ſphæra ſphæram interſecet: linea, quæ centra illarum ſphærarum tranſit, centrũ circuli
            <lb/>
          peripheriæ cõmunis ſectionis tranſire, & ſuper ipſius ſuperficiem perpendicularẽ eſſe, neceſſe eſt.</head>
          <p>
            <s xml:id="echoid-s22151" xml:space="preserve">Circulus cõmunis ſectiõis ſphærarũ aut eſt circulus maior alterius ſpherarũ ſe interſecantiũ, aut
              <lb/>
            minor:</s>
            <s xml:id="echoid-s22152" xml:space="preserve"> ſi maior:</s>
            <s xml:id="echoid-s22153" xml:space="preserve"> hoc erit ſolũ, cũ maior ſphæra minorẽ interſecat.</s>
            <s xml:id="echoid-s22154" xml:space="preserve"> Si enim æquales ſphærę ſecundũ
              <lb/>
            circulũ maiorẽ ſe interſecarẽt, nõ eſſet ſphærarũ interſectio, ſed unius ſphærę ex duobus hemiſphæ
              <lb/>
            rijs æqualibus cõpoſitio.</s>
            <s xml:id="echoid-s22155" xml:space="preserve"> Si ergo circulus cõis ſectionis ſphęrarũ ſit circulus maior, nõ erit ille circu
              <lb/>
            lus maior, niſi in ſphæris inæqualibus ſe interſecãtibus, circulus ſphærę minoris:</s>
            <s xml:id="echoid-s22156" xml:space="preserve"> quoniã ipſum eſſe
              <lb/>
            circulũ maiorẽ ſphærę maioris eſt impoſsibile:</s>
            <s xml:id="echoid-s22157" xml:space="preserve"> quoniã maior circulus ſphærę maioris nõ poteſt ca-
              <lb/>
            dere in ſuperficiẽ ſphęrę minoris.</s>
            <s xml:id="echoid-s22158" xml:space="preserve"> Sit itaq;</s>
            <s xml:id="echoid-s22159" xml:space="preserve"> circulus talis a b c:</s>
            <s xml:id="echoid-s22160" xml:space="preserve"> & ſit centrũ maioris ſphærę d:</s>
            <s xml:id="echoid-s22161" xml:space="preserve"> ſphærę
              <lb/>
            uerò minoris e:</s>
            <s xml:id="echoid-s22162" xml:space="preserve"> erit quoq;</s>
            <s xml:id="echoid-s22163" xml:space="preserve"> e centrũ circuli a b c ex hypotheſi.</s>
            <s xml:id="echoid-s22164" xml:space="preserve"> Ducatur ergo linea d e:</s>
            <s xml:id="echoid-s22165" xml:space="preserve"> & patebit pro-
              <lb/>
            poſitum primum.</s>
            <s xml:id="echoid-s22166" xml:space="preserve"> Item ducantur lineę d a, b d, d c, & lineę a e, b e, c e:</s>
            <s xml:id="echoid-s22167" xml:space="preserve"> eruntq́;</s>
            <s xml:id="echoid-s22168" xml:space="preserve"> triangulorum d a e &
              <lb/>
            d b e latera æqualia:</s>
            <s xml:id="echoid-s22169" xml:space="preserve"> ideo, quoniam linea d e latus eſt commune, & latus d a æquale eſt lateri d b ex
              <lb/>
            definitione ſphærę:</s>
            <s xml:id="echoid-s22170" xml:space="preserve"> latus quoque a e ęquale eſt lateri b e ex definitione circuli:</s>
            <s xml:id="echoid-s22171" xml:space="preserve"> ergo per 8 p 1 anguli
              <lb/>
            ęquis lateribus contenti, erunt ęquales.</s>
            <s xml:id="echoid-s22172" xml:space="preserve"> Angulus ergo d e b ęqualis erit angulo d e a:</s>
            <s xml:id="echoid-s22173" xml:space="preserve"> ſimiliter an
              <lb/>
            gulus d e c erit ęqualis angulo d e b:</s>
            <s xml:id="echoid-s22174" xml:space="preserve"> & uniuerſaliter à quocunq;</s>
            <s xml:id="echoid-s22175" xml:space="preserve"> puncto circuli a b c ducantur lineę
              <lb/>
            ad e centrum ſphærę, anguli ſuper centrum e ſemper erunt æquales.</s>
            <s xml:id="echoid-s22176" xml:space="preserve"> Et quia ſuper eandem diame-
              <lb/>
            trum oppoſitis punctis ſignatis linea d e æquales angulos conſtituit:</s>
            <s xml:id="echoid-s22177" xml:space="preserve"> patet per definitionem per-
              <lb/>
            pendicularis, quoniam ipſa linea d e ſuper omnes diametros perpendicularis erit.</s>
            <s xml:id="echoid-s22178" xml:space="preserve"> Ergo per 4 p 11
              <lb/>
            linea d e ſuper ſuperficiem circuli a b c erecta eſt, & ſupeream perpendicularis.</s>
            <s xml:id="echoid-s22179" xml:space="preserve"> Si uerò circu-
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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