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-div789" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s22179" xml:space="preserve">
              <pb o="32" file="0334" n="334" rhead="VITELLONIS OPTICAE"/>
            lus a b c non ſit circulus maior alicuius ſphærarũ ſe interſecantiũ, ſed minor:</s>
            <s xml:id="echoid-s22180" xml:space="preserve"> intelligatur in ipſo pro-
              <lb/>
            tracta diameter, quæ ſit l f per pũcta l & f, & utraq;</s>
            <s xml:id="echoid-s22181" xml:space="preserve"> ſphæra
              <lb/>
              <figure xlink:label="fig-0334-01" xlink:href="fig-0334-01a" number="342">
                <variables xml:id="echoid-variables326" xml:space="preserve">l h g b e c k a d f</variables>
              </figure>
            rum imaginetur ſecta per ſuperficiem planam trans cen-
              <lb/>
            trũ, & ք puncta f & l, quę ſunt in ſuperficie utriuſq;</s>
            <s xml:id="echoid-s22182" xml:space="preserve"> ſphæ
              <lb/>
            rę.</s>
            <s xml:id="echoid-s22183" xml:space="preserve"> Erit ergo per præmiſſa quilibet illorum circulorum
              <lb/>
            circulus maior in utraq;</s>
            <s xml:id="echoid-s22184" xml:space="preserve"> ſphærarum ſe interſecantiũ, ſe-
              <lb/>
            cabitq́;</s>
            <s xml:id="echoid-s22185" xml:space="preserve"> circulum a b c uterq;</s>
            <s xml:id="echoid-s22186" xml:space="preserve"> illorum circulorũ maiorum
              <lb/>
            per æqualia:</s>
            <s xml:id="echoid-s22187" xml:space="preserve"> quoniam arcus f l eſt medietas circumferen
              <lb/>
            tię circuli a b c:</s>
            <s xml:id="echoid-s22188" xml:space="preserve"> tranſeunt ergo ambo illi circuli maiores
              <lb/>
            per centrũ illius circuli a b c, quod eſt e.</s>
            <s xml:id="echoid-s22189" xml:space="preserve"> Imaginẽtur item
              <lb/>
            duo circuli alιj maiores in eiſdem ſphæris, quorum quili-
              <lb/>
            bet ſecet portionẽ circuli maioris ſuę ſphærę erectã ſuper
              <lb/>
            circulum a b c per æqualia:</s>
            <s xml:id="echoid-s22190" xml:space="preserve"> quod fieri poterit ex 30 p 3, di-
              <lb/>
            uiſo arcu f l utriuſq;</s>
            <s xml:id="echoid-s22191" xml:space="preserve"> circuli ſphærarum ſe interſecantium
              <lb/>
            per ęqualia, & à puncto ſectionis utriuſq;</s>
            <s xml:id="echoid-s22192" xml:space="preserve"> circuli imagina
              <lb/>
            ta ſuperficie plana tranſeunte centrum ſphærę utriuſq;</s>
            <s xml:id="echoid-s22193" xml:space="preserve">.
              <lb/>
            Fiat itaq;</s>
            <s xml:id="echoid-s22194" xml:space="preserve"> ſectio arcus ſphęrę maioris in puncto g:</s>
            <s xml:id="echoid-s22195" xml:space="preserve"> & ſe-
              <lb/>
            ctio arcus ſphæræ minoris in puncto h:</s>
            <s xml:id="echoid-s22196" xml:space="preserve"> & ſiue hi cir-
              <lb/>
            culi maiores cum illis circulis, quos ſecãt, angulos æqua-
              <lb/>
            les ſphærales uel inæquales contineant, patet, cum à po-
              <lb/>
            lo circuli a b c per centra ſphærarum ambarum tranſeant, quoniam ambo ſecabunt circulum a b c
              <lb/>
            per æqualia.</s>
            <s xml:id="echoid-s22197" xml:space="preserve"> Tranſibunt ergo per centrum ipſius, quod eſt e.</s>
            <s xml:id="echoid-s22198" xml:space="preserve"> Linea ergo d g, quę per definitionem
              <lb/>
            maiorum circulorum, & 3 p 11 eſt communis ſectio duorum circulorum maiorũ in ſphęra maiori ſe
              <lb/>
            ſecantium, tranſit per centrum e:</s>
            <s xml:id="echoid-s22199" xml:space="preserve"> quoniã cum centrum e ſit in ſuperficie utriuſq;</s>
            <s xml:id="echoid-s22200" xml:space="preserve"> illorũ circulorum,
              <lb/>
            neceſſe eſt, ut ſit in linea cõmuni utriſq;</s>
            <s xml:id="echoid-s22201" xml:space="preserve">. Similiter etiã linea e h (quę eſt cõmunis ſectio circulorum
              <lb/>
            maiorũ in ſphæra minori ſe interſecantiũ) tranſit per centrũ e.</s>
            <s xml:id="echoid-s22202" xml:space="preserve"> Sed quia lineę e h, & lineę d g per defi
              <lb/>
            nitionem circulorũ ſe ſecantiũ, eſt aliqua linea recta cõmunis, ut e g, erit illa per 1 p 11 in eadẽ ſuperfi
              <lb/>
            cie cum illis:</s>
            <s xml:id="echoid-s22203" xml:space="preserve"> ergo erunt linea una.</s>
            <s xml:id="echoid-s22204" xml:space="preserve"> Tota ergo linea d e g h eſt linea una tranſiens per ambo centra
              <lb/>
            ſphærarum ſe interſecantiũ, & per centrum circuli, qui eſt cõmunis ſectio, cuius centrum eſt in peri
              <lb/>
            pheria cõmunis ſectionis ſuperficierum ſphęricarum ſe interſecantium.</s>
            <s xml:id="echoid-s22205" xml:space="preserve"> Patet ergo propoſitum pri
              <lb/>
            mum.</s>
            <s xml:id="echoid-s22206" xml:space="preserve"> Secundũ uerò patet ex pręmiſsis.</s>
            <s xml:id="echoid-s22207" xml:space="preserve"> Circuli enim maiores per ęqualia diuidentes circulum mi-
              <lb/>
            norem orthogonaliter eum ſecant, & eorum communis ſectio, ut linea d h per 19 p 11 ſuper eundem
              <lb/>
            circulum perpendicularis erit.</s>
            <s xml:id="echoid-s22208" xml:space="preserve"> Et hoc eſt propoſitum.</s>
            <s xml:id="echoid-s22209" xml:space="preserve"> Poteſt & idem per 66 & 67 huius facilius de-
              <lb/>
            monſtrari diligentiam adhibenti.</s>
            <s xml:id="echoid-s22210" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div791" type="section" level="0" n="0">
          <head xml:id="echoid-head657" xml:space="preserve" style="it">83. Si ſphæra ſphærã interſecet: lineã tranſeuntẽ centrũ circuli peripheriæ cõmunis ſectionis
            <lb/>
          perpendiculariter ſuper ipſius ſuperficiẽ inſiſtentẽ, ambarũ ſphærarũ centra tranſire neceſſe eſt.</head>
          <p>
            <s xml:id="echoid-s22211" xml:space="preserve">Hęc eſt cõuerſa pręcedẽtis, nec oportet in ipſius demonſtratiõe aliter immorari.</s>
            <s xml:id="echoid-s22212" xml:space="preserve"> Si enim ſit poſsi-
              <lb/>
            bile, ducatur linea per e centrũ circuli cõmunis ſectiõis ſphęrarũ, (qui eſt a b c) perpendiculariter ſu
              <lb/>
            per ipſius ſuperficiẽ ad aliũ aliquẽ punctũ, pręter centum ambarũ, uel alterius ſphęrarũ:</s>
            <s xml:id="echoid-s22213" xml:space="preserve"> & ſit linea
              <lb/>
            e k:</s>
            <s xml:id="echoid-s22214" xml:space="preserve"> & ducatur item per centra ambarũ ſphęrarũ alia linea, quę ſit d h.</s>
            <s xml:id="echoid-s22215" xml:space="preserve"> Patet aũt per pręcedentẽ, quo-
              <lb/>
            niam hęc erit tranſiens per centrũ e, & erit perpendicularis ſuper ſuperficiẽ circuli a b c.</s>
            <s xml:id="echoid-s22216" xml:space="preserve"> Ab eodem
              <lb/>
            ergo pũcto ſuperficiei circuli a b c, utpote centro e, duę exeũt perpendiculares ſuper eandẽ circuli
              <lb/>
            ſuperficiem a b c, quę ſunt e d & e k:</s>
            <s xml:id="echoid-s22217" xml:space="preserve"> quod eſt contra 13 p 11, & impoſsibile.</s>
            <s xml:id="echoid-s22218" xml:space="preserve"> Patet ergo propoſitum.</s>
            <s xml:id="echoid-s22219" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div792" type="section" level="0" n="0">
          <figure number="343">
            <variables xml:id="echoid-variables327" xml:space="preserve">f c c l a</variables>
          </figure>
          <head xml:id="echoid-head658" xml:space="preserve" style="it">84. Si ſphæra ſphærã intrinſecus interſecet: neceſſe eſt centra illarũ ſphærarũ, reſpectu ſitus ſui
            <lb/>
          contactus ſecundum quantitatẽ peripheriæ circuli, qui eſt cõmu-
            <lb/>
          nis ſectio ſuarum ſuperficierũ plus diſtare: centrum ſphæræ con- tinentis plus profundari.</head>
          <p>
            <s xml:id="echoid-s22220" xml:space="preserve">Sphærę datę interſecare ſe debẽtes, ſi ęquales fuerint, & taliter ad
              <lb/>
            inuicẽ collocentur, ut nõ ſe interſecẽt:</s>
            <s xml:id="echoid-s22221" xml:space="preserve"> tunc ipſarũ idẽ erit centrũ:</s>
            <s xml:id="echoid-s22222" xml:space="preserve"> fa-
              <lb/>
            cta uerò interſectiõe ipſarũ, cẽtra diuerſantur per 81 huius:</s>
            <s xml:id="echoid-s22223" xml:space="preserve"> & ſecun-
              <lb/>
            dũ quod circuli քipheria, quę eſt cõmunis ſectio illarũ ſuperficierũ
              <lb/>
            ſphęricarũ, fit maior uel minor:</s>
            <s xml:id="echoid-s22224" xml:space="preserve"> ſecũdũ hoc plus uel minus diſtabũt
              <lb/>
            centra.</s>
            <s xml:id="echoid-s22225" xml:space="preserve"> Quòd ſi ſphęrę fuerint inęquales, quarũ una alterã intrinſe-
              <lb/>
            cus cõtingere poterit:</s>
            <s xml:id="echoid-s22226" xml:space="preserve"> tunc in ſitu ſuę cõtingentię centrorũ ſuorũ di
              <lb/>
            ſtantia ք 78 huius eſt exceſſus ſemidiametri ſphęrę maioris ad ſemi
              <lb/>
            diametrum minoris.</s>
            <s xml:id="echoid-s22227" xml:space="preserve"> Demus ergo, quòd centrum maioris ſit a, cen-
              <lb/>
            trũ minoris b, punctus cõtactus ſit c.</s>
            <s xml:id="echoid-s22228" xml:space="preserve"> Et quia cõtactus fit in puncto
              <lb/>
            per 76 huius, interſectio uerò fit ſecundũ circulũ per 80 huius:</s>
            <s xml:id="echoid-s22229" xml:space="preserve"> palã,
              <lb/>
            quia facta interſectione ſphærarum, abſcindet ſphęra a diametrũ b c
              <lb/>
            in puncto alio quàm in termino ſuo, qui eſt punctus c.</s>
            <s xml:id="echoid-s22230" xml:space="preserve"> Sit ergo pun-
              <lb/>
            ctus, in quo ipſum abſcindit, punctus e:</s>
            <s xml:id="echoid-s22231" xml:space="preserve"> ponaturq́;</s>
            <s xml:id="echoid-s22232" xml:space="preserve">, ut linea f e ſit æ-
              <lb/>
            qualis diametro ſphęrę b.</s>
            <s xml:id="echoid-s22233" xml:space="preserve"> Quoniam itaq;</s>
            <s xml:id="echoid-s22234" xml:space="preserve"> linea a c excedit lineam b
              <lb/>
            c in linea a b:</s>
            <s xml:id="echoid-s22235" xml:space="preserve"> linea uerò f e eſt ęqualis ſemidiametro b c:</s>
            <s xml:id="echoid-s22236" xml:space="preserve"> quoniam
              <lb/>
            ſunt ſemidiametri eiuſdem ſphęrę.</s>
            <s xml:id="echoid-s22237" xml:space="preserve"> Linea ergo a c excedit lineam
              <lb/>
            </s>
          </p>
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