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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          <p>
            <s xml:id="echoid-s13708" xml:space="preserve">
              <pb o="198" file="0204" n="204" rhead="ALHAZEN"/>
            Quare imago puncti remotioris ab e remotior eſt à centro, imagine propinquioris:</s>
            <s xml:id="echoid-s13709" xml:space="preserve"> & finis contin-
              <lb/>
            gentiæ remotioris remotior à centro, fine propinquioris.</s>
            <s xml:id="echoid-s13710" xml:space="preserve"> Quod erat propoſitum.</s>
            <s xml:id="echoid-s13711" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div465" type="section" level="0" n="0">
          <head xml:id="echoid-head421" xml:space="preserve" style="it">8. Si data recta in duob{us} punctis ſecta, ſit ad alterũ extremorũ ſegmentorũ, ut reliquũ ex-
            <lb/>
          tremum ad intermediũ: & ab altero ipſi{us} termino, ſectionum́ punctis tres rectæ in eodẽ pun
            <lb/>
          cto cõcurrant: recta à reliquo termino ſecãs cõcurrentes, ſecabitur proportionaliter datæ. 123 p 1.</head>
          <p>
            <s xml:id="echoid-s13712" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s13713" xml:space="preserve"> propoſita linea a b, & diuiſa in punctis g, d, ut ſit proportio a b ad b d, ſicut a g ad g d:</s>
            <s xml:id="echoid-s13714" xml:space="preserve">
              <lb/>
            ſi à punctis ſectionũ ducantur tres lineæ concurrentes in punctum unum, ſcilicet g e, d e, b e:</s>
            <s xml:id="echoid-s13715" xml:space="preserve">
              <lb/>
            & à puncto a ducatur linea ſecans illas tres lineas:</s>
            <s xml:id="echoid-s13716" xml:space="preserve"> Dico, quòd linea illa diuiſa erit ſecundum
              <lb/>
            prædictam proportionẽ.</s>
            <s xml:id="echoid-s13717" xml:space="preserve"> Ducatur linea a c ſecans tria latera g e, d e, b e in tribus punctis z, h, c.</s>
            <s xml:id="echoid-s13718" xml:space="preserve"> Dico
              <lb/>
            quòd proportio a c ad c h, ſicut a z ad z h.</s>
            <s xml:id="echoid-s13719" xml:space="preserve"> Ducatur [per 31 p 1] à puncto h æquidiſtans a b:</s>
            <s xml:id="echoid-s13720" xml:space="preserve"> quæ ſit h q.</s>
            <s xml:id="echoid-s13721" xml:space="preserve">
              <lb/>
            Palàm [è demonſtratis à Theone ad 5 d 6] quòd proportio a b ad b d, conſtat ex proportionibus a b
              <lb/>
            ad h q, & h q ad b d.</s>
            <s xml:id="echoid-s13722" xml:space="preserve"> Sed quoniã q h æquidiſtat a b:</s>
            <s xml:id="echoid-s13723" xml:space="preserve"> erit triangulũ c q h ſimile triangulo c a b:</s>
            <s xml:id="echoid-s13724" xml:space="preserve">] per 29 p
              <lb/>
            1.</s>
            <s xml:id="echoid-s13725" xml:space="preserve">4 p.</s>
            <s xml:id="echoid-s13726" xml:space="preserve"> 1 d 6] & erit proportio a b ad q h, ſicut a c ad c h.</s>
            <s xml:id="echoid-s13727" xml:space="preserve"> Similiter triangulũ q e h ſimile triangulo b e d:</s>
            <s xml:id="echoid-s13728" xml:space="preserve">
              <lb/>
            igitur erit porportio q h ad b d, ſicut h e ad e d.</s>
            <s xml:id="echoid-s13729" xml:space="preserve"> Ergo
              <lb/>
              <figure xlink:label="fig-0204-01" xlink:href="fig-0204-01a" number="163">
                <variables xml:id="echoid-variables153" xml:space="preserve">e c
                  <gap/>
                h m z b d
                  <gap/>
                a</variables>
              </figure>
            proportio a b ad b d, cõſtat ex proportionibus a c ad
              <lb/>
            c h & h e ad e d.</s>
            <s xml:id="echoid-s13730" xml:space="preserve"> Producatur q h, uſq;</s>
            <s xml:id="echoid-s13731" xml:space="preserve"> dum cadat ſuper
              <lb/>
            e g in puncto m.</s>
            <s xml:id="echoid-s13732" xml:space="preserve"> [cadet aũt per lemma Procli ad 29 p
              <lb/>
            1] Proportio igitur a g ad g d, conſtat ex proportioni-
              <lb/>
            bus a g ad h m, & h m ad g d.</s>
            <s xml:id="echoid-s13733" xml:space="preserve"> Sed cum [per 29 p 1]
              <lb/>
            angulus e m h ſit æqualis angulo z g d:</s>
            <s xml:id="echoid-s13734" xml:space="preserve"> erit [per 13 p 1]
              <lb/>
            angulus h m zæqualis angulo z g a:</s>
            <s xml:id="echoid-s13735" xml:space="preserve"> & erit triangulũ
              <lb/>
            a z g ſimile triangulo h m z [quia enim anguli aduer-
              <lb/>
            ticem z æquantur per 15 p 1:</s>
            <s xml:id="echoid-s13736" xml:space="preserve"> æquabitur per 32 p 1 ter-
              <lb/>
            tius m h z tertio g a z.</s>
            <s xml:id="echoid-s13737" xml:space="preserve"> Quare per 4 p.</s>
            <s xml:id="echoid-s13738" xml:space="preserve">1 d 6 triangula h
              <lb/>
            m z, a g z ſunt ſimilia.</s>
            <s xml:id="echoid-s13739" xml:space="preserve">] Et erit proportio a z ad z h ſi-
              <lb/>
            cut a g ad h m.</s>
            <s xml:id="echoid-s13740" xml:space="preserve"> Sed [per 29 p 1.</s>
            <s xml:id="echoid-s13741" xml:space="preserve">4 p.</s>
            <s xml:id="echoid-s13742" xml:space="preserve">1 d 6] triangulum h
              <lb/>
            e m ſimile eſt triangulo g e d:</s>
            <s xml:id="echoid-s13743" xml:space="preserve"> erit igitur proportio h
              <lb/>
            m ad g d, ſicut h e ad e d.</s>
            <s xml:id="echoid-s13744" xml:space="preserve"> Igitur proportio a g ad g d,
              <lb/>
            conſtat ex proportione a z ad z h, & h e ad e d:</s>
            <s xml:id="echoid-s13745" xml:space="preserve"> & eadẽ
              <lb/>
            eſt a g ad g d, quæ a b ad b d [pertheſin.</s>
            <s xml:id="echoid-s13746" xml:space="preserve">] Igitur illa ea-
              <lb/>
            dem cõſtat ex proportionibus a z ad z h & h e ad e d.</s>
            <s xml:id="echoid-s13747" xml:space="preserve">
              <lb/>
            Igitur [ſubducta utrinq;</s>
            <s xml:id="echoid-s13748" xml:space="preserve"> ratione h c ad e d] eadẽ erit
              <lb/>
            proportio a c ad c h, quę eſt a z ad z h.</s>
            <s xml:id="echoid-s13749" xml:space="preserve"> Et ita eſt propoſitũ.</s>
            <s xml:id="echoid-s13750" xml:space="preserve"> Eadem erit probatio, quęcunq;</s>
            <s xml:id="echoid-s13751" xml:space="preserve"> linea duca
              <lb/>
            rur à puncto a, ſecans lineas illas tres concurrentes.</s>
            <s xml:id="echoid-s13752" xml:space="preserve"> Et ſi ducantur aliæ tres lineę à tribus punctis g,
              <lb/>
            d, b, ad aliud punctũ quàm e cõcurrentes, & à puncto a ducatur linea quæcũq;</s>
            <s xml:id="echoid-s13753" xml:space="preserve">, ſecans eas:</s>
            <s xml:id="echoid-s13754" xml:space="preserve"> diuidetur
              <lb/>
            ſecundum prædictã proportionẽ.</s>
            <s xml:id="echoid-s13755" xml:space="preserve"> Et ita quocunq;</s>
            <s xml:id="echoid-s13756" xml:space="preserve"> modo concurrãt tres lineę.</s>
            <s xml:id="echoid-s13757" xml:space="preserve"> Et ſi tres lineę e g, e d,
              <lb/>
            e b producantur ultra tria puncta b, d, g ex alia parte:</s>
            <s xml:id="echoid-s13758" xml:space="preserve"> & à puncto a ducantur lineæ, ſecantes eas ex il
              <lb/>
            la alia parte:</s>
            <s xml:id="echoid-s13759" xml:space="preserve"> nunquam illæ lineæ diuidentur ſecundum prædictam proportionem.</s>
            <s xml:id="echoid-s13760" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div467" type="section" level="0" n="0">
          <head xml:id="echoid-head422" xml:space="preserve" style="it">9. Si duæ rectæ facientes angulum, ſimiliter́ in duob{us} punctis ita ſectæ (ut tota ſit ad alterũ
            <lb/>
          extremorũ ſegmentorũ, ſicut reliquum extremum ad intermedium) baſi infinita cõnect antur:
            <lb/>
          rectæ per pũcta ſectionũ utriuſ, cũ baſi & inter ſe cõcurrẽtes, in eodẽ puncto cõcurrẽt. 124 p 1.</head>
          <p>
            <s xml:id="echoid-s13761" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s13762" xml:space="preserve"> data linea a b prædicto modo diuiſa:</s>
            <s xml:id="echoid-s13763" xml:space="preserve"> ſi à puncto a ducatur alia linea, uelut a c, quæ di
              <lb/>
            uidatur iuxta eandem proportionẽ:</s>
            <s xml:id="echoid-s13764" xml:space="preserve"> & à punctis diuiſionũ a b ducantur lineę ad puncta diui
              <lb/>
            ſionũ a c, quę quidẽ nõ ſint æquidiſtãtes:</s>
            <s xml:id="echoid-s13765" xml:space="preserve"> Dico quòd
              <lb/>
              <figure xlink:label="fig-0204-02" xlink:href="fig-0204-02a" number="164">
                <variables xml:id="echoid-variables154" xml:space="preserve">e n c
                  <gap/>
                z
                  <gap/>
                b d g a</variables>
              </figure>
            illæ tres concurrent in uno & eodẽ puncto.</s>
            <s xml:id="echoid-s13766" xml:space="preserve"> Sit pro-
              <lb/>
            portio ac ad c h, ſicut a z ad z h.</s>
            <s xml:id="echoid-s13767" xml:space="preserve"> Et quia b c, d h non
              <lb/>
            ſunt æquidiſtantes [ex theſi] igitur concurrent in ali
              <lb/>
            quo puncto:</s>
            <s xml:id="echoid-s13768" xml:space="preserve"> quod ſit e.</s>
            <s xml:id="echoid-s13769" xml:space="preserve"> Linea g z aut concurret ad
              <lb/>
            idem punctũ:</s>
            <s xml:id="echoid-s13770" xml:space="preserve"> aut non.</s>
            <s xml:id="echoid-s13771" xml:space="preserve"> Si ad idem:</s>
            <s xml:id="echoid-s13772" xml:space="preserve"> habemus propoſi
              <lb/>
            tum.</s>
            <s xml:id="echoid-s13773" xml:space="preserve"> Si nõ, ducatur linea e g:</s>
            <s xml:id="echoid-s13774" xml:space="preserve"> ſecabit quidem lineã a
              <lb/>
            c in alio puncto quàm z:</s>
            <s xml:id="echoid-s13775" xml:space="preserve"> ſit illud punctũ l.</s>
            <s xml:id="echoid-s13776" xml:space="preserve"> Erit ergo
              <lb/>
            proportio a c ad c h, ſicut a l ad l h iuxta priorẽ pro-
              <lb/>
            bationem [præcedentis numeri] ſed poſitum eſt a
              <lb/>
            c ad ch, ſicut a z ad z h.</s>
            <s xml:id="echoid-s13777" xml:space="preserve"> Et ita impoſsibile [nempe to-
              <lb/>
            tum æquari ſuæ parti.</s>
            <s xml:id="echoid-s13778" xml:space="preserve"> Quia enim per præcedentem
              <lb/>
            numerum eſt, ut a l ad l h, ſic a c ad c h, & ex theſi, ut
              <lb/>
            a c ad ch, ſic a z ad z h:</s>
            <s xml:id="echoid-s13779" xml:space="preserve"> erit per 11 p 5, ut a l ad l h, ſic a z
              <lb/>
            ad z h & per 18 p 5, ut a h ad h l, ſic a h ad h z.</s>
            <s xml:id="echoid-s13780" xml:space="preserve"> Quare
              <lb/>
            cum a h ad duas rectas h l, h z eandem habeat ratio-
              <lb/>
            nem, æquabuntur ipſæ inter ſe per 9 p 5:</s>
            <s xml:id="echoid-s13781" xml:space="preserve"> & ſic tota h
              <lb/>
            l erit æqualis parti h z.</s>
            <s xml:id="echoid-s13782" xml:space="preserve">] Similiter, ſi ponatur, quòd li
              <lb/>
            nea g z concurrat cum d h ad punctum e:</s>
            <s xml:id="echoid-s13783" xml:space="preserve"> probabitur hoc modo, quòd linea b c concurrat ad idem
              <lb/>
            </s>
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