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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            <s xml:id="echoid-s8634" xml:space="preserve">
              <pb o="144" file="0150" n="150" rhead="ALHAZEN"/>
            tio [32 propoſitione] ſed [per 29 p 1] angulus n g d eſt æqualis angulo a b g:</s>
            <s xml:id="echoid-s8635" xml:space="preserve"> cum [per fabri-
              <lb/>
            cationem] n g ſit æquidiſtans a b.</s>
            <s xml:id="echoid-s8636" xml:space="preserve"> Igitur [per 1 ax] angulus n g d æqualis eſt angulo n a g, & an-
              <lb/>
            gulus n d g communis.</s>
            <s xml:id="echoid-s8637" xml:space="preserve"> Quare [per 32 p 1] tertius tertio eſt æqualis.</s>
            <s xml:id="echoid-s8638" xml:space="preserve"> Quare [per 4 p.</s>
            <s xml:id="echoid-s8639" xml:space="preserve"> 1 d 6] triangu-
              <lb/>
            lum n d g ſimile triangulo a d g.</s>
            <s xml:id="echoid-s8640" xml:space="preserve"> Igitur proportio a d ad d g, ſicut g d ad d n.</s>
            <s xml:id="echoid-s8641" xml:space="preserve"> Quare [per 17 p 6] quod
              <lb/>
            fit ex ductu a d in d n eſt æquale quadrato d g.</s>
            <s xml:id="echoid-s8642" xml:space="preserve"> Verùm quadratum a d eſt æquale ei, quod fit ex du-
              <lb/>
            ctu b d in d g:</s>
            <s xml:id="echoid-s8643" xml:space="preserve"> ſicut probat Euclides 36 propoſitione [libri tertij,] & quadratum a d eſt æquale ei,
              <lb/>
            quod fit ex ductu a d in d n, & ei quod fit ex ductu a d in n a [per 2 p 2:</s>
            <s xml:id="echoid-s8644" xml:space="preserve">] & illud, quod fit ex ductu b d
              <lb/>
            in d g, eſt æquale quadrato d g, & ei quod fit ex ductu b g in g d:</s>
            <s xml:id="echoid-s8645" xml:space="preserve"> ſicut probat Euclides [3 p 2.</s>
            <s xml:id="echoid-s8646" xml:space="preserve">] Abla-
              <lb/>
            tis ergo æqualibus [quadrato nempe d g & rectangulo a d n] reſtat [per 3 ax,] ut, quòd fit ex du-
              <lb/>
            ctu a d in a n, ſit æquale ei, quod fit ex b g in g d.</s>
            <s xml:id="echoid-s8647" xml:space="preserve"> Igitur [per 16 p 6] proportio primæ lineæ ad ſecun
              <lb/>
            dam, eſt ſicut tertiæ ad quartam [nempe ut a d ad g d, ſic b g ad a n:</s>
            <s xml:id="echoid-s8648" xml:space="preserve"> & alternè [per 16 p 5] ut a d ad
              <lb/>
            b g, ſic g d ad a n.</s>
            <s xml:id="echoid-s8649" xml:space="preserve">] Quare [per conſectarium 4 p 5] proportio a n ad g d, ſicut b g ad a d.</s>
            <s xml:id="echoid-s8650" xml:space="preserve"> Sediam
              <lb/>
            dictum eſt, quòd proportio a n ad g d eſt, ſicut b g ad e q.</s>
            <s xml:id="echoid-s8651" xml:space="preserve"> Igitur [per 9 p 5] e q eſt æqualis a d.</s>
            <s xml:id="echoid-s8652" xml:space="preserve"> Quod
              <lb/>
            eſt propoſitum.</s>
            <s xml:id="echoid-s8653" xml:space="preserve"> Quòd ſi a d non tetigerit circulum, ſed ſecuerit, &
              <lb/>
              <figure xlink:label="fig-0150-01" xlink:href="fig-0150-01a" number="67">
                <variables xml:id="echoid-variables57" xml:space="preserve">q d n e g h a b</variables>
              </figure>
            fuerit a g maior a b:</s>
            <s xml:id="echoid-s8654" xml:space="preserve"> ſecabit quidem arcũ a g.</s>
            <s xml:id="echoid-s8655" xml:space="preserve"> Secet ergo in puncto h:</s>
            <s xml:id="echoid-s8656" xml:space="preserve">
              <lb/>
            & ducatur linea h g.</s>
            <s xml:id="echoid-s8657" xml:space="preserve"> Palàm [per 22 p 3] quòd duo anguli a h g, a b g
              <lb/>
            ualent duos rectos:</s>
            <s xml:id="echoid-s8658" xml:space="preserve"> ſed angulus n g d æqualis eſt a b g [per 29 p 1:</s>
            <s xml:id="echoid-s8659" xml:space="preserve">
              <lb/>
            quia n g parallela ducta eſt ipſi a b.</s>
            <s xml:id="echoid-s8660" xml:space="preserve">] Igitur angulus a h g & angulus n
              <lb/>
            g d ſunt ęquales duobus rectis.</s>
            <s xml:id="echoid-s8661" xml:space="preserve"> Quare [per 13 p 1.</s>
            <s xml:id="echoid-s8662" xml:space="preserve"> 3 ax] angulus n g d
              <lb/>
            eſt ęqualis angulo n h g:</s>
            <s xml:id="echoid-s8663" xml:space="preserve"> & angulus n d g communis.</s>
            <s xml:id="echoid-s8664" xml:space="preserve"> Quare [per 32 p
              <lb/>
            1] tertius angulus tertio angulo eſt æqualis:</s>
            <s xml:id="echoid-s8665" xml:space="preserve"> & triangulum h g d ſimi
              <lb/>
            le triangulo n d g [per 4 p.</s>
            <s xml:id="echoid-s8666" xml:space="preserve"> 1 d 6.</s>
            <s xml:id="echoid-s8667" xml:space="preserve">] Igitur proportio h d ad d g eſt, ſicut
              <lb/>
            proportio d g ad d n.</s>
            <s xml:id="echoid-s8668" xml:space="preserve"> Quare [per 17 p 6] illud, quod fit ex ductu h d
              <lb/>
            in d n, eſt ęquale quadrato d g:</s>
            <s xml:id="echoid-s8669" xml:space="preserve"> ſed quod fit ex ductu a d in h d, eſt æ-
              <lb/>
            quale ei, quod fit ex ductu b d in d g, ſicut probat Euclides [cõſecta-
              <lb/>
            rio 36 p 3] & [per 1 p 2] illud, quod fit ex ductu a d in d h, eſt ęquale ei,
              <lb/>
            quod fit ex ductu d h in d n, & d h in a n:</s>
            <s xml:id="echoid-s8670" xml:space="preserve"> & [per 3 p 2] qđ fit ex ductu
              <lb/>
            b d in d g, eſt æquale ei, quod fit ex ductu b g in g d & quadrato d g.</s>
            <s xml:id="echoid-s8671" xml:space="preserve">
              <lb/>
            Ablatis igitur æqualibus, ſcilicet quadrato d g, & eo, quod fit ex du-
              <lb/>
            ctu d h in d n:</s>
            <s xml:id="echoid-s8672" xml:space="preserve"> reſtat [per 3 ax] ut illud, quod fit ex ductu d h in a n, ſit
              <lb/>
            ęquale ei, quod fit ex ductu b g in d g.</s>
            <s xml:id="echoid-s8673" xml:space="preserve"> Quare proportio ſecundę lineę
              <lb/>
            ad quartam, id eſt a n ad g d, ſicut tertiæ ad primã, id eſt b g ad d h [eſt
              <lb/>
            enim per 16 p 6 ut d h ad d g, ſic b g ad a n:</s>
            <s xml:id="echoid-s8674" xml:space="preserve"> & per 16 p 5, ut d h ad b g,
              <lb/>
            ſic d g ad a n, & per conſectarium 4 p 5, ut a n ad d g, ſic b g ad d h.</s>
            <s xml:id="echoid-s8675" xml:space="preserve">] Sed iam probatum eſt, quòd pro-
              <lb/>
            portio a n ad d g, ſicut b g ad e q.</s>
            <s xml:id="echoid-s8676" xml:space="preserve"> Igitur [per 9 p 5] e q eſt ęqualis d h.</s>
            <s xml:id="echoid-s8677" xml:space="preserve">
              <lb/>
              <figure xlink:label="fig-0150-02" xlink:href="fig-0150-02a" number="68">
                <variables xml:id="echoid-variables58" xml:space="preserve">d q n g a e h b</variables>
              </figure>
            Et ita eſt propoſitum.</s>
            <s xml:id="echoid-s8678" xml:space="preserve"> Si uerò a g ſit minor a b:</s>
            <s xml:id="echoid-s8679" xml:space="preserve"> & ſecet a d arcum a b:</s>
            <s xml:id="echoid-s8680" xml:space="preserve">
              <lb/>
            ſit ſectionis punctum h:</s>
            <s xml:id="echoid-s8681" xml:space="preserve"> & ducatur linea h g.</s>
            <s xml:id="echoid-s8682" xml:space="preserve"> Palàm [per fabricatio
              <lb/>
            nem primam & 29 p 1] quòd angulus n g d eſt æqualis angulo a b g:</s>
            <s xml:id="echoid-s8683" xml:space="preserve">
              <lb/>
            ſed [per 27 p 3] anguli a b g, a h g ſunt æquales:</s>
            <s xml:id="echoid-s8684" xml:space="preserve"> quia cadunt in eundẽ
              <lb/>
            arcum.</s>
            <s xml:id="echoid-s8685" xml:space="preserve"> Igitur [per 1 ax] angulus n g d eſt æqualis angulo a h g, & an-
              <lb/>
            gulus n d g communis.</s>
            <s xml:id="echoid-s8686" xml:space="preserve"> Quare [per 32 p 1] tertius tertio æqualis:</s>
            <s xml:id="echoid-s8687" xml:space="preserve"> &
              <lb/>
            triangula ſimilia [per 4 p.</s>
            <s xml:id="echoid-s8688" xml:space="preserve"> 1 d 6.</s>
            <s xml:id="echoid-s8689" xml:space="preserve">] Igitur proportio h d ad d g, ſicut d g
              <lb/>
            ad d n.</s>
            <s xml:id="echoid-s8690" xml:space="preserve"> Quare [per 17 p 6] quod fit ex ductu h d in d n, eſt æquale qua
              <lb/>
            drato d g:</s>
            <s xml:id="echoid-s8691" xml:space="preserve"> ſed quod fit ex ductu h d in d a, eſt æquale ei, quod fit ex du
              <lb/>
            ctu b d in d g [per conſectarium Campani ad 36 p 3] & [per 1 p 2] qđ
              <lb/>
            fit ex ductu h d in d a, eſt æquale ei, quod fit ex ductu d n in h d & a n
              <lb/>
            in h d:</s>
            <s xml:id="echoid-s8692" xml:space="preserve"> & [per 3 p 2] ductus b d in d g ualet quadratum d g, & ductum
              <lb/>
            b g in d g.</s>
            <s xml:id="echoid-s8693" xml:space="preserve"> Igitur remotis æqualibus:</s>
            <s xml:id="echoid-s8694" xml:space="preserve"> [rectangulo nimirum h d n, &
              <lb/>
            quadrato d g] erit [per 3 ax] ductus h d in a n, ſicut b g in d g.</s>
            <s xml:id="echoid-s8695" xml:space="preserve"> Igitur
              <lb/>
            [per 16 p 6.</s>
            <s xml:id="echoid-s8696" xml:space="preserve"> 16 p.</s>
            <s xml:id="echoid-s8697" xml:space="preserve"> 13 d 5] proportio a n ad d g, ſicut b g ad h d.</s>
            <s xml:id="echoid-s8698" xml:space="preserve"> Sed iam
              <lb/>
            dictum eſt, quòd proportio a n ad d g eſt, ſicut b g ad e q.</s>
            <s xml:id="echoid-s8699" xml:space="preserve"> Igitur [per
              <lb/>
            9 p 5] e q eſt æqualis h d.</s>
            <s xml:id="echoid-s8700" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s8701" xml:space="preserve"> Quare à puncto a da
              <lb/>
            to, duximus lineam, ſecantem circulum, & à puncto ſectionis ad dia
              <lb/>
            metrum eſt æqualis lineæ datæ.</s>
            <s xml:id="echoid-s8702" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div335" type="section" level="0" n="0">
          <head xml:id="echoid-head337" xml:space="preserve" style="it">34. À
            <unsure/>
          puncto peripheriæ circuli extra datam diametrum dato, ducere lineam rectam, it æ
            <lb/>
          ſectam data diametro, ut ſegmentum inter diametrum & punctum peripheriæ dato puncto op
            <lb/>
          poſitum, æquetur datæ rectæ, minori circuli diametro. 133 p 1.</head>
          <p>
            <s xml:id="echoid-s8703" xml:space="preserve">AMplius à puncto dato in circulo, extra diametrum eius, eſt ducere lineam per diametrum ad
              <lb/>
            circulum, ut pars eius à diametro ad circulum, ſit æqualis lineę datæ.</s>
            <s xml:id="echoid-s8704" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s8705" xml:space="preserve"> a b g ſit da
              <lb/>
            tus circulus:</s>
            <s xml:id="echoid-s8706" xml:space="preserve"> b g diameter:</s>
            <s xml:id="echoid-s8707" xml:space="preserve"> a punctum datum:</s>
            <s xml:id="echoid-s8708" xml:space="preserve"> h z linea data.</s>
            <s xml:id="echoid-s8709" xml:space="preserve"> Dico, quòd à puncto a eſt duce
              <lb/>
            re lineam, tranſeuntem per diametrum b g, cuius pars à diametro ad circulum ſit æqualis lineæ h z.</s>
            <s xml:id="echoid-s8710" xml:space="preserve">
              <lb/>
            Ducantur lineæ a b, a g:</s>
            <s xml:id="echoid-s8711" xml:space="preserve"> & [per 23 p 1] ſuper punctum h fiat angulus ęqualis angulo a g b per lineam
              <lb/>
            m h:</s>
            <s xml:id="echoid-s8712" xml:space="preserve"> & ſuper idem punctum fiat angulus ęqualis angulo a b g per lineam h l:</s>
            <s xml:id="echoid-s8713" xml:space="preserve"> & [per 31 p 1] à puncto z
              <lb/>
            </s>
          </p>
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