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-div677" type="section" level="0" n="0">
          <head xml:id="echoid-head587" xml:space="preserve" style="it">
            <pb o="8" file="0310" n="310" rhead="VITELLONIS OPTICAE"/>
          lio Theonis ad 5 definit. 6 element. & commentarijs in 1 librum magnæ cõſtructionis Ptolemæi.
            <lb/>
          Item è commentarijs Eutocij in 8 theor. 2 de ſphæra & cylindro Archimedis.</head>
          <p>
            <s xml:id="echoid-s20615" xml:space="preserve">Sint extra gradus tres lineæ, quæ a, b, g, quarum prima (quæ eſt a) ſit maior quàm media (quæ
              <lb/>
            eſt b) & b ſit maior quàm tertia, quæ eſt g:</s>
            <s xml:id="echoid-s20616" xml:space="preserve"> ſit q́;</s>
            <s xml:id="echoid-s20617" xml:space="preserve"> ipſius b ad ambas extremas proportio nota.</s>
            <s xml:id="echoid-s20618" xml:space="preserve"> Dico,
              <lb/>
            quòd proportio lineæ a ad lineam g tertiam componitur ex proportione lineæ a ad lineam b, & ex
              <lb/>
            proportione lineæ b ad lineam g.</s>
            <s xml:id="echoid-s20619" xml:space="preserve"> Quoniam enim proportio lineæ a ad lineam b eſt nota:</s>
            <s xml:id="echoid-s20620" xml:space="preserve"> ſit quanti-
              <lb/>
            tas d denominatio illius proportionis:</s>
            <s xml:id="echoid-s20621" xml:space="preserve"> & ſimiliter quia proportio lineæ b ad lineam g eſt nota:</s>
            <s xml:id="echoid-s20622" xml:space="preserve"> ſit
              <lb/>
            denominatio illius proportionis quantitas e:</s>
            <s xml:id="echoid-s20623" xml:space="preserve"> & ſit quantitas z denominatio proportionis lineæ a
              <lb/>
            ad lineam g.</s>
            <s xml:id="echoid-s20624" xml:space="preserve"> Dico, quòd ex ductu e in d fit z.</s>
            <s xml:id="echoid-s20625" xml:space="preserve"> Quoniam enim per 15 definitionem huius ex ductu z
              <lb/>
            denominationis proportionis lineæ a ad lineam g in ipſam lineam g minorem, quàm ſit a, fit linea
              <lb/>
            a:</s>
            <s xml:id="echoid-s20626" xml:space="preserve"> & ſimiliter ex ductu d in lineam b fit linea a:</s>
            <s xml:id="echoid-s20627" xml:space="preserve">
              <lb/>
              <figure xlink:label="fig-0310-01" xlink:href="fig-0310-01a" number="270">
                <variables xml:id="echoid-variables256" xml:space="preserve">a b g d e z</variables>
              </figure>
            ponatur itaq;</s>
            <s xml:id="echoid-s20628" xml:space="preserve"> z primum & d ſecundum, linea b
              <lb/>
            tertiũ & linea g quartũ.</s>
            <s xml:id="echoid-s20629" xml:space="preserve"> Quia itaq;</s>
            <s xml:id="echoid-s20630" xml:space="preserve"> illud, quod
              <lb/>
            fit ex ductu primi in quartum, eſt ęquale ei, qđ
              <lb/>
            fit ex ductu ſecũdi in tertium:</s>
            <s xml:id="echoid-s20631" xml:space="preserve"> patet per 16 p 6
              <lb/>
            quoniam eſt proportio primi ad ſecundum, ſi-
              <lb/>
            cut tertij ad quartum:</s>
            <s xml:id="echoid-s20632" xml:space="preserve"> eſt ergo proportio z ad
              <lb/>
            d, ſicut lineæ b ad lineam g:</s>
            <s xml:id="echoid-s20633" xml:space="preserve"> ergo denominatio
              <lb/>
            proportionis z ad d ex 5 ſuppoſitione eſt eadẽ
              <lb/>
            cum denominatione proportionis lineæ b ad
              <lb/>
            lineam g:</s>
            <s xml:id="echoid-s20634" xml:space="preserve"> ſed denominatio proportionis lineæ b ad lineam g eſt quantitas e:</s>
            <s xml:id="echoid-s20635" xml:space="preserve"> ergo denominatio ꝓ-
              <lb/>
            portionis z ad d eſt idẽ e:</s>
            <s xml:id="echoid-s20636" xml:space="preserve"> ergo ex ductu e in d fit z.</s>
            <s xml:id="echoid-s20637" xml:space="preserve"> Quia ergo denominatio proportionis lineę a ad
              <lb/>
            lineam g, quæ eſt z, producitur ex ductu denominationis proportionis lineæ a ad lineam b in de-
              <lb/>
            nominationem proportionis lineæ b ad lineam g:</s>
            <s xml:id="echoid-s20638" xml:space="preserve"> patet per 16 definitionem huius, quoniam pro-
              <lb/>
            portio lineæ a primæ ad lineam g tertiam componitur ex proportione lineæ a primæ ad lineam b
              <lb/>
            ſecundam, & ex proportione lineæ b ſecundæ ad lineam g tertiam:</s>
            <s xml:id="echoid-s20639" xml:space="preserve"> quod eſt propoſitum primum.</s>
            <s xml:id="echoid-s20640" xml:space="preserve">
              <lb/>
            Eodem quoq;</s>
            <s xml:id="echoid-s20641" xml:space="preserve"> modo poteſt faciliter demonſtrari de quotcunq;</s>
            <s xml:id="echoid-s20642" xml:space="preserve"> medijs inter quęlibet duo extrema
              <lb/>
            collocatis:</s>
            <s xml:id="echoid-s20643" xml:space="preserve"> ſemper enim proportio extremorum ad inuicem componitur ex omnibus proportioni
              <lb/>
            bus mediorum ad inuicem, & ad ipſa extrema.</s>
            <s xml:id="echoid-s20644" xml:space="preserve"> Similiter demonſtrandum uia diuiſionis, ſi mediam
              <lb/>
            contingat eſſe maiorem qualibet extremarum:</s>
            <s xml:id="echoid-s20645" xml:space="preserve"> patet ergo propoſitum.</s>
            <s xml:id="echoid-s20646" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div679" type="section" level="0" n="0">
          <head xml:id="echoid-head588" xml:space="preserve" style="it">14. Si linea recta ſuper duas rect{as} ceciderit, fecerit́ angulos coalternos inæquales, aut
            <lb/>
          duos intrinſecos minores duobus rectis, uel extrinſecum inæqualem intrinſeco: illas duas lineas
            <lb/>
          ad minorum angulorum partem concurrere eſt neceſſe, ad aliam uerò partem impoßibile: & ſi
            <lb/>
          lineæ concurrunt, neceſſe est dictos angulos aliquo propoſitorum modorum ſe habere. E' 27.28
            <lb/>
          p 1 element. Lemma Procli ad 16 p 1 elem.</head>
          <p>
            <s xml:id="echoid-s20647" xml:space="preserve">Sint duæ lineæ a b & c d, quas ſecet linea e fſecundum quod proponitur.</s>
            <s xml:id="echoid-s20648" xml:space="preserve"> Dico, quoniam lineæ
              <lb/>
            a b & c d concurrent.</s>
            <s xml:id="echoid-s20649" xml:space="preserve"> Si enim nõ concurrant, patet quòd ſunt æ quidiſtantes:</s>
            <s xml:id="echoid-s20650" xml:space="preserve"> ergo per 29 p 1 ſequi-
              <lb/>
            tur contrarium hypothe.</s>
            <s xml:id="echoid-s20651" xml:space="preserve"> quòd eſt inconueniens:</s>
            <s xml:id="echoid-s20652" xml:space="preserve"> concur
              <lb/>
              <figure xlink:label="fig-0310-02" xlink:href="fig-0310-02a" number="271">
                <variables xml:id="echoid-variables257" xml:space="preserve">e a b c d f</variables>
              </figure>
            runt ergo.</s>
            <s xml:id="echoid-s20653" xml:space="preserve"> Ad partem uerò minorum angulorum cõcur-
              <lb/>
            rere eſt neceſſarium:</s>
            <s xml:id="echoid-s20654" xml:space="preserve"> quoniam ſi ad partem maiorum an-
              <lb/>
            gulorum concurrant, ſequetur angulum extrinſecum tri
              <lb/>
            goni contenti fieri minorẽ angulo intrinſeco:</s>
            <s xml:id="echoid-s20655" xml:space="preserve"> & eſt con-
              <lb/>
            tra 16 & 32 p 1.</s>
            <s xml:id="echoid-s20656" xml:space="preserve"> Et quia per præmiſſas probationes ad par-
              <lb/>
            tes minorum angulorum concurrunt:</s>
            <s xml:id="echoid-s20657" xml:space="preserve"> ſi ex conceſſo ad
              <lb/>
            partes maiorum angulorum concurrerent, ſequeretur
              <lb/>
            duas rectas lineas ſuperficiem includere:</s>
            <s xml:id="echoid-s20658" xml:space="preserve"> quod eſt impoſ
              <lb/>
            ſibile.</s>
            <s xml:id="echoid-s20659" xml:space="preserve"> Eſt ergo impoſsibile, ut ad partes maiorum angu-
              <lb/>
            lorum concurrant:</s>
            <s xml:id="echoid-s20660" xml:space="preserve"> quod eſt propoſitum primum.</s>
            <s xml:id="echoid-s20661" xml:space="preserve"> Sed &
              <lb/>
            ſi detur, quòd illæ lineæ concurrant, neceſſe eſt angulos aliquo propofitorum modorum ſe habere
              <lb/>
            per 32 p 1:</s>
            <s xml:id="echoid-s20662" xml:space="preserve"> patet ergo totum, quod proponebatur, ſeruata ſemper hypotheſi.</s>
            <s xml:id="echoid-s20663" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div681" type="section" level="0" n="0">
          <figure number="272">
            <variables xml:id="echoid-variables258" xml:space="preserve">a d e c b</variables>
          </figure>
          <head xml:id="echoid-head589" xml:space="preserve" style="it">15. Cumlineis, ſe inter duas line{as} æquidiſtantes, à
            <lb/>
          quarum terminis producuntur, ſecantibus, ex utra
            <lb/>
          parte ſectionis partes eiuſdẽ lineæ inter ſe fuerint æqua les: neceſſe eſt lineas, inter quas fit ſectio, æquales eſſe.</head>
          <p>
            <s xml:id="echoid-s20664" xml:space="preserve">Verbi gratia:</s>
            <s xml:id="echoid-s20665" xml:space="preserve"> ſit, ut duæ lineæ a b & c d inter duas line-
              <lb/>
            as æquidiſtantes, à quarũ terminis producũtur, quę ſint a
              <lb/>
            d & c b, ſecent ſe in puncto e, ita, quòd linea a e ſit æqualis
              <lb/>
            lineæ e b, & linea c e ſit æqualis ipſi e d.</s>
            <s xml:id="echoid-s20666" xml:space="preserve"> Dico, quòd linea
              <lb/>
            a d eſt æqualis lineæ c b.</s>
            <s xml:id="echoid-s20667" xml:space="preserve"> Quoniam enim per 15 p 1 angu-
              <lb/>
            lus a e d eſt æqualis angulo c e b, erit ex hypotheſi & per
              <lb/>
            4 p 1 linea a d æqualis lineæ c b:</s>
            <s xml:id="echoid-s20668" xml:space="preserve"> quod eſt propoſitum.</s>
            <s xml:id="echoid-s20669" xml:space="preserve"/>
          </p>
        </div>
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