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-div660" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s20515" xml:space="preserve">
              <pb o="6" file="0308" n="308" rhead="VITELLONIS OPTICAE"/>
            ad lineam c f maiorem, quàm ad lineam c g minorem:</s>
            <s xml:id="echoid-s20516" xml:space="preserve"> eſt ergo maior proportio lineæ a b ad linea m
              <lb/>
            c d, quàm lineę a e ad lineam c f:</s>
            <s xml:id="echoid-s20517" xml:space="preserve"> & hoc eſt propoſitum.</s>
            <s xml:id="echoid-s20518" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div661" type="section" level="0" n="0">
          <head xml:id="echoid-head579" xml:space="preserve" style="it">5. Cum fuerit proportio primi ad ſecundum, tanquam tertij ad quartũ: erit è contrario pro-
            <lb/>
          portio ſecundi ad primum, ſicut quarti ad tertium. E' 13 def. & conſectario 4 p 5 element.</head>
          <p>
            <s xml:id="echoid-s20519" xml:space="preserve">Sit enim a primum, & b ſecundum, & ctertium, & d quartum:</s>
            <s xml:id="echoid-s20520" xml:space="preserve"> & ſit
              <lb/>
            proportio a ad b, ſicut c ad d.</s>
            <s xml:id="echoid-s20521" xml:space="preserve"> Dico, quòd erit è contrario proportio b ad
              <lb/>
              <figure xlink:label="fig-0308-01" xlink:href="fig-0308-01a" number="262">
                <variables xml:id="echoid-variables248" xml:space="preserve">a b c d</variables>
              </figure>
            a, ſicut d ad c.</s>
            <s xml:id="echoid-s20522" xml:space="preserve"> Quoniam enim eſt proportio a ad b, ſicut c ad d:</s>
            <s xml:id="echoid-s20523" xml:space="preserve"> erit per 16
              <lb/>
            p 5 permutatim proportio a ad c, ſicut b ad d:</s>
            <s xml:id="echoid-s20524" xml:space="preserve"> eſt ergo proportio b ad d,
              <lb/>
            ſicut a ad c:</s>
            <s xml:id="echoid-s20525" xml:space="preserve"> ergo iterum per 16 p 5 erit permutatim proportio b ad a, ſi-
              <lb/>
            cut d ad c, ſecundi uidelicet ad primum, ſicut quarti ad tertium:</s>
            <s xml:id="echoid-s20526" xml:space="preserve"> quod eſt
              <lb/>
            propoſitum.</s>
            <s xml:id="echoid-s20527" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div663" type="section" level="0" n="0">
          <head xml:id="echoid-head580" xml:space="preserve" style="it">6. Cum fuerit quatuor quantitatum proportio primæ ad ſecundã
            <lb/>
          maior, quãtertiæ ad quartam: erit è contr ario minor proportio ſecun-
            <lb/>
          dæ ad primam, quàm quartæ ad tertiam. 26 p 5 element. in Campano.</head>
          <p>
            <s xml:id="echoid-s20528" xml:space="preserve">Eſto proportio lineæ a ad lineam b maior, quàm lineæ c ad lineam d.</s>
            <s xml:id="echoid-s20529" xml:space="preserve"> Dico, quôd erit è contrario
              <lb/>
            minor proportio lineæ b ad lineam a, quàm lineæ d ad li-
              <lb/>
              <figure xlink:label="fig-0308-02" xlink:href="fig-0308-02a" number="263">
                <variables xml:id="echoid-variables249" xml:space="preserve">a b e c d</variables>
              </figure>
            neam c.</s>
            <s xml:id="echoid-s20530" xml:space="preserve"> Sit enim per 3 huius, ut, quæ eſt proportio lineæ c
              <lb/>
            ad lineam d, eadem ſit lineæ e ad lineam b.</s>
            <s xml:id="echoid-s20531" xml:space="preserve"> Quia ergo ma-
              <lb/>
            ior eſt proportio lineæ a ad lineam b, quàm lineæ c ad li-
              <lb/>
            neam d ex hypotheſi:</s>
            <s xml:id="echoid-s20532" xml:space="preserve"> patet, quòd minor eſt proportio li-
              <lb/>
            neæ e ad lineam b, quam lineę a ad lineã b:</s>
            <s xml:id="echoid-s20533" xml:space="preserve"> ergo per 10 p 5
              <lb/>
            linea a eſt maior quã linea e.</s>
            <s xml:id="echoid-s20534" xml:space="preserve"> Et quia eſt proportio lineæ e
              <lb/>
            ad lineam b, ſicut lineę c ad lineam d, erit per præmiſſam
              <lb/>
            eadem proportio lineę b ad lineã e, quę lineæ d ad lineam
              <lb/>
            c.</s>
            <s xml:id="echoid-s20535" xml:space="preserve"> Eſt autem per 8 p 5 minor proportio lineæ b ad lineam a,
              <lb/>
            quàm ad lineam e:</s>
            <s xml:id="echoid-s20536" xml:space="preserve"> eſt ergo minor proportio lineæ b ad lineam a, quàm lineę d ad lineam c:</s>
            <s xml:id="echoid-s20537" xml:space="preserve"> quod
              <lb/>
            eſt propoſitum.</s>
            <s xml:id="echoid-s20538" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div665" type="section" level="0" n="0">
          <head xml:id="echoid-head581" xml:space="preserve" style="it">7. Si quatuor quantitatum proportion alium prima fuerit maior quãſecunda, & tertia ma-
            <lb/>
          ior quã quarta: erit euerſim eadem proportio primæ ad augmentum ſui ſuper ſecundam, quæ ter
            <lb/>
          tiæ ad augmentum ſui ſuper quartam. E' 16 definit. & conſectario 19 p 5.</head>
          <p>
            <s xml:id="echoid-s20539" xml:space="preserve">Sint quatuor lineę proportionales a c prima:</s>
            <s xml:id="echoid-s20540" xml:space="preserve"> b c ſecunda:</s>
            <s xml:id="echoid-s20541" xml:space="preserve"> d ftertia:</s>
            <s xml:id="echoid-s20542" xml:space="preserve"> & e f quarta.</s>
            <s xml:id="echoid-s20543" xml:space="preserve"> Sitq́ue linea a b
              <lb/>
            maior quàm linea b c, & linea d f maior, quàm linea e f:</s>
            <s xml:id="echoid-s20544" xml:space="preserve"> ex-
              <lb/>
              <figure xlink:label="fig-0308-03" xlink:href="fig-0308-03a" number="264">
                <variables xml:id="echoid-variables250" xml:space="preserve">a b c d e f</variables>
              </figure>
            cedat quoque linea a c lineam b c, in linea a b, & linea d f
              <lb/>
            lineam e f, in linea d e.</s>
            <s xml:id="echoid-s20545" xml:space="preserve"> Dico, quòd eadem erit proportio
              <lb/>
            lineę a c ad lineam a b, quę lineę d f ad lineam d e.</s>
            <s xml:id="echoid-s20546" xml:space="preserve"> Quo-
              <lb/>
            niam enim eſt proportio lineę a c ad lineam b c, ſicut lineę
              <lb/>
            d f ad lineam e f:</s>
            <s xml:id="echoid-s20547" xml:space="preserve"> eſt ergo per 16 p 5 permutatim proportio
              <lb/>
            lineę a c ad lineam d f, ſicut lineę b c ad lineam e f:</s>
            <s xml:id="echoid-s20548" xml:space="preserve"> ergo per
              <lb/>
            19 p 5 erit proportio lineę a b ad lineam d e, ſicut lineę a c
              <lb/>
            ad lineam d f:</s>
            <s xml:id="echoid-s20549" xml:space="preserve"> ergo per 16 p 5 erit proportio lineę a b ad lineam a c.</s>
            <s xml:id="echoid-s20550" xml:space="preserve"> ſicut lineę d e ad lineam d f.</s>
            <s xml:id="echoid-s20551" xml:space="preserve"> Ergo
              <lb/>
            per 5 huius erit proportio lineę a c ad lineam a b, ſicut lineę d fad lineam d e:</s>
            <s xml:id="echoid-s20552" xml:space="preserve"> quod eſt propoſitum.</s>
            <s xml:id="echoid-s20553" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div667" type="section" level="0" n="0">
          <head xml:id="echoid-head582" xml:space="preserve" style="it">8. Si quatuor quantit atum prima fuerit maior ſecunda, & tertia maior quarta: erit maior
            <lb/>
          proportio primæ ad quartam, quàm ſecundæ ad tertiam. Conſectarium ex 8 p 5 element.</head>
          <p>
            <s xml:id="echoid-s20554" xml:space="preserve">Sint quatuor lineę a, b, c, d:</s>
            <s xml:id="echoid-s20555" xml:space="preserve"> & ſit a prima maior quàm b ſecũda, & ſit c tertia maior, quàm d quar-
              <lb/>
            ta.</s>
            <s xml:id="echoid-s20556" xml:space="preserve"> Dico, quòd maior eſt proportio lineæ a, ad lineam d, quàm
              <lb/>
              <figure xlink:label="fig-0308-04" xlink:href="fig-0308-04a" number="265">
                <variables xml:id="echoid-variables251" xml:space="preserve">a b c d</variables>
              </figure>
            lineę b ad lineam c.</s>
            <s xml:id="echoid-s20557" xml:space="preserve"> Quia enim linea c eſt maior quàm linea d
              <lb/>
            ex hypotheſi:</s>
            <s xml:id="echoid-s20558" xml:space="preserve"> patet per 8 p 5:</s>
            <s xml:id="echoid-s20559" xml:space="preserve"> quoniam maior eſt proportio li-
              <lb/>
            neę a ad lineam d, quàm ad lineam c:</s>
            <s xml:id="echoid-s20560" xml:space="preserve"> minor uero eſt proportio
              <lb/>
            lineę b ad lineã c, quàm lineę a, ad lineam c per eandem 8 p 5:</s>
            <s xml:id="echoid-s20561" xml:space="preserve">
              <lb/>
            quoniam ut pręmiſſum eſt linea a eſt maior quàm linea b.</s>
            <s xml:id="echoid-s20562" xml:space="preserve"> Et
              <lb/>
            quoniam quicquid eſt maius maiore, eſt maius minore:</s>
            <s xml:id="echoid-s20563" xml:space="preserve"> patet,
              <lb/>
            quòd maior eſt proportio lineę a ad lineam d, quàm lineę b ad
              <lb/>
            lineam c:</s>
            <s xml:id="echoid-s20564" xml:space="preserve"> patet ergo propoſitum.</s>
            <s xml:id="echoid-s20565" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div669" type="section" level="0" n="0">
          <head xml:id="echoid-head583" xml:space="preserve" style="it">9. Cum quatuor quantitatum prima fuerit maior quàm
            <lb/>
          tertia, & ſecunda minor quàm quarta: maior erit proportio primæ ad ſecundam, quàm tertiæ
            <lb/>
          ad quartam. Conſectarium ex 8 p 5 element.</head>
          <p>
            <s xml:id="echoid-s20566" xml:space="preserve">Sint quatuor lineę a prima:</s>
            <s xml:id="echoid-s20567" xml:space="preserve"> b ſecũda:</s>
            <s xml:id="echoid-s20568" xml:space="preserve"> c tertia:</s>
            <s xml:id="echoid-s20569" xml:space="preserve"> d quarta:</s>
            <s xml:id="echoid-s20570" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s20571" xml:space="preserve"> a maior quàm c, & ſit b minor quã d.</s>
            <s xml:id="echoid-s20572" xml:space="preserve">
              <lb/>
            </s>
          </p>
        </div>
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