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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          <head xml:id="echoid-head339" xml:space="preserve" style="it">
            <pb o="147" file="0153" n="153" rhead="OPTICAE LIBER V."/>
          duab{us} rectis, à dictis punctis ad punctum tact{us} ductis. 135 p 1.</head>
          <p>
            <s xml:id="echoid-s8877" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s8878" xml:space="preserve"> duobus punctis datis, ſcilicet e, d, & dato circulo:</s>
            <s xml:id="echoid-s8879" xml:space="preserve"> eſt inuenire punctum in eo, ut an-
              <lb/>
            gulum contentum à lineis, à punctis prædictis ad illud punctum ductis, diuidat per æqualia,
              <lb/>
            linea circulum contingens in illo puncto.</s>
            <s xml:id="echoid-s8880" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s8881" xml:space="preserve"> ducatur à puncto e ad centrum circu
              <lb/>
            li dati, linea e g:</s>
            <s xml:id="echoid-s8882" xml:space="preserve"> & producatur uſq;</s>
            <s xml:id="echoid-s8883" xml:space="preserve"> ad circumferentiam:</s>
            <s xml:id="echoid-s8884" xml:space="preserve"> & ſit e s:</s>
            <s xml:id="echoid-s8885" xml:space="preserve"> deinde ducatur linea g d.</s>
            <s xml:id="echoid-s8886" xml:space="preserve"> Et ſit [per
              <lb/>
            10 p 6] m i linea diuiſa in puncto c, ut ſit proportio i c ad e m, ſicut e g ad g d:</s>
            <s xml:id="echoid-s8887" xml:space="preserve"> & [per 10 p 1] diuidatur
              <lb/>
            m i per æqualia in puncto n:</s>
            <s xml:id="echoid-s8888" xml:space="preserve"> & [per 11 p 1] ducatur perpendicularis n o:</s>
            <s xml:id="echoid-s8889" xml:space="preserve"> & ſuper punctum m fiat an-
              <lb/>
            gulus ęqualis medietati anguli d g s [per 9 & 23 p 1] per lineã m o.</s>
            <s xml:id="echoid-s8890" xml:space="preserve"> Palàm, quòd erit minor recto.</s>
            <s xml:id="echoid-s8891" xml:space="preserve"> [Nã
              <lb/>
            anguli ad g deinceps æquantur duobus rectis per 13 p 1:</s>
            <s xml:id="echoid-s8892" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s8893" xml:space="preserve"> d g s ijſdem eſt minor:</s>
            <s xml:id="echoid-s8894" xml:space="preserve"> quare d g s dimi
              <lb/>
            diatus minor eſt recto] & o n m rectus:</s>
            <s xml:id="echoid-s8895" xml:space="preserve"> igitur [per 11 ax.</s>
            <s xml:id="echoid-s8896" xml:space="preserve">] cõcurret cum n o:</s>
            <s xml:id="echoid-s8897" xml:space="preserve"> concurrat aũt in puncto
              <lb/>
            o:</s>
            <s xml:id="echoid-s8898" xml:space="preserve"> & ducatur à puncto c [per præcedentem numerum] linea ad triangulũ:</s>
            <s xml:id="echoid-s8899" xml:space="preserve"> quę ſit c k f:</s>
            <s xml:id="echoid-s8900" xml:space="preserve"> ita ut propor
              <lb/>
            tio k f ad m f ſit, ſicut proportio e g ad g s:</s>
            <s xml:id="echoid-s8901" xml:space="preserve"> & [per 23 p 1] ſuper punctum g [terminum lineæ e g] fiat
              <lb/>
            angulus æqualis angulo m f k, per lineam uſq;</s>
            <s xml:id="echoid-s8902" xml:space="preserve"> ad circulum ductam:</s>
            <s xml:id="echoid-s8903" xml:space="preserve"> quæ ſit a g:</s>
            <s xml:id="echoid-s8904" xml:space="preserve"> & ſit angulus a g e:</s>
            <s xml:id="echoid-s8905" xml:space="preserve"> &
              <lb/>
            ducantur lineæ a g, a d.</s>
            <s xml:id="echoid-s8906" xml:space="preserve"> Dico, quòd a eſt punctum, quod quęrimus.</s>
            <s xml:id="echoid-s8907" xml:space="preserve"> Ducatur linea e a.</s>
            <s xml:id="echoid-s8908" xml:space="preserve"> Cum igitur an
              <lb/>
            gulus m f k [per fabricationem] ſit æqualis angulo a g e:</s>
            <s xml:id="echoid-s8909" xml:space="preserve"> & [per fabricationem] proportio k f ad m
              <lb/>
            f, ſicut g e ad g a:</s>
            <s xml:id="echoid-s8910" xml:space="preserve"> cum [per 15 d 1] g a ſit æqualis g s:</s>
            <s xml:id="echoid-s8911" xml:space="preserve"> erit triangulum a g e ſimile triangulo m f k [per
              <lb/>
            6.</s>
            <s xml:id="echoid-s8912" xml:space="preserve"> 4 p.</s>
            <s xml:id="echoid-s8913" xml:space="preserve"> 1 d 6.</s>
            <s xml:id="echoid-s8914" xml:space="preserve">] Igitur angulus f m k eſt ęqualis angulo e a g, & angulus a e g æqualis angulo m k f.</s>
            <s xml:id="echoid-s8915" xml:space="preserve"> Iam
              <lb/>
            [per 23 p 1] à puncto a ducatur linea, tenens cum linea a e angulum æqualem angulo n m k:</s>
            <s xml:id="echoid-s8916" xml:space="preserve"> & ſit li-
              <lb/>
            nea a z:</s>
            <s xml:id="echoid-s8917" xml:space="preserve"> quę neceſſariò concurret cum linea e g.</s>
            <s xml:id="echoid-s8918" xml:space="preserve"> Quoniam, quæ eſt proportio k fad m f, ea eſt e g ad
              <lb/>
            g a, & angulus g a z ęqualis
              <lb/>
              <figure xlink:label="fig-0153-01" xlink:href="fig-0153-01a" number="74">
                <variables xml:id="echoid-variables64" xml:space="preserve">d a h ſ s u g e z t q</variables>
              </figure>
              <figure xlink:label="fig-0153-02" xlink:href="fig-0153-02a" number="75">
                <variables xml:id="echoid-variables65" xml:space="preserve">o k f i l
                  <unsure/>
                n m</variables>
              </figure>
            eſt angulo f m c.</s>
            <s xml:id="echoid-s8919" xml:space="preserve"> [ęqualis e-
              <lb/>
            nim concluſus eſt angulus
              <lb/>
            f m k angulo e a g] Igitur ſi-
              <lb/>
            cut linea m o concurrit cũ
              <lb/>
            f k in puncto f:</s>
            <s xml:id="echoid-s8920" xml:space="preserve"> ſic concur-
              <lb/>
            ret a z cum e g.</s>
            <s xml:id="echoid-s8921" xml:space="preserve"> Sit concur-
              <lb/>
            ſus in puncto z:</s>
            <s xml:id="echoid-s8922" xml:space="preserve"> & produ-
              <lb/>
            catur a z uſq;</s>
            <s xml:id="echoid-s8923" xml:space="preserve"> ad punctũ q:</s>
            <s xml:id="echoid-s8924" xml:space="preserve">
              <lb/>
            ita ut linea a z ſe habeat ad
              <lb/>
            z q, ſicut m c ad c i:</s>
            <s xml:id="echoid-s8925" xml:space="preserve"> [per 12 p
              <lb/>
            6] & ducatur linea e q.</s>
            <s xml:id="echoid-s8926" xml:space="preserve"> De-
              <lb/>
            inde [per 31 p 1] à puncto a
              <lb/>
            ducatur æquidiſtans e q:</s>
            <s xml:id="echoid-s8927" xml:space="preserve">
              <lb/>
            quę ſit a t:</s>
            <s xml:id="echoid-s8928" xml:space="preserve"> erit quidem [per
              <lb/>
            29 p 1] angulus a q e æqua-
              <lb/>
            lis angulo q a t.</s>
            <s xml:id="echoid-s8929" xml:space="preserve"> Et quoniam duo anguli z e a, e a t ſunt minores duobus rectis [quia per 29 p 1 angu-
              <lb/>
            li q e a, e a t æquantur duobus rectis] concurret a t neceſſariò cum e z [per 11 ax.</s>
            <s xml:id="echoid-s8930" xml:space="preserve">] Sit concurſus pun
              <lb/>
            ctum t.</s>
            <s xml:id="echoid-s8931" xml:space="preserve"> Palàm [ex prius demonſtratis] quòd angulus a e g eſt æqualis angulo m k f.</s>
            <s xml:id="echoid-s8932" xml:space="preserve"> Ducta autem à
              <lb/>
            puncto e linea perpẽdiculari ſuper a z:</s>
            <s xml:id="echoid-s8933" xml:space="preserve"> quæ ſit e l:</s>
            <s xml:id="echoid-s8934" xml:space="preserve"> erit [per 32 p 1] angulus a e l æqualis angulo m k n:</s>
            <s xml:id="echoid-s8935" xml:space="preserve">
              <lb/>
            cum [per fabricationem] angulus e a l ſit æqualis angulo k m n, & angulus a l e ęqualis m n k:</s>
            <s xml:id="echoid-s8936" xml:space="preserve"> quia
              <lb/>
            uterq;</s>
            <s xml:id="echoid-s8937" xml:space="preserve"> rectus:</s>
            <s xml:id="echoid-s8938" xml:space="preserve"> reſtat ergo [per 13 p 1.</s>
            <s xml:id="echoid-s8939" xml:space="preserve"> 3 ax.</s>
            <s xml:id="echoid-s8940" xml:space="preserve">] l e z æqualis angulo n k c:</s>
            <s xml:id="echoid-s8941" xml:space="preserve"> & angulus e l z rectus, ęqualis an
              <lb/>
            gulo k n c:</s>
            <s xml:id="echoid-s8942" xml:space="preserve"> reſtat [per 32 p 1] ut angulus e z l ſit ęqualis k c n:</s>
            <s xml:id="echoid-s8943" xml:space="preserve"> igitur [per 13 p 1.</s>
            <s xml:id="echoid-s8944" xml:space="preserve"> 3 ax.</s>
            <s xml:id="echoid-s8945" xml:space="preserve">] e z q æqualis angu
              <lb/>
            lo k c i.</s>
            <s xml:id="echoid-s8946" xml:space="preserve"> Palàm ergo [per 4 p.</s>
            <s xml:id="echoid-s8947" xml:space="preserve"> 1 d 6] quòd triangulum e a g ſimile eſt triangulo f m k:</s>
            <s xml:id="echoid-s8948" xml:space="preserve"> & triangulũ e a l
              <lb/>
            ſimile triangulo k m n:</s>
            <s xml:id="echoid-s8949" xml:space="preserve"> & triangulũ e l z ſimile k n c:</s>
            <s xml:id="echoid-s8950" xml:space="preserve"> & triangulũ e a z triangulo k m c [Nam ք fabri-
              <lb/>
            cationem angulus e a l æquatur angulo k m n, & angulus e z l ęqualis oſtenſus eſt angulo k c n:</s>
            <s xml:id="echoid-s8951" xml:space="preserve"> ergo
              <lb/>
            per 32 p 1 reliquus reliquo:</s>
            <s xml:id="echoid-s8952" xml:space="preserve"> ideoq́;</s>
            <s xml:id="echoid-s8953" xml:space="preserve"> per 4 p.</s>
            <s xml:id="echoid-s8954" xml:space="preserve"> 1 d 6 triangula e a z, k m c ſunt ſimilia] Ergo proportio a z
              <lb/>
            ad z e, ſicut m c ad c k, & [per fabricationem] proportio a z ad z q, ſicut m c ad c i:</s>
            <s xml:id="echoid-s8955" xml:space="preserve"> Igitur [per 22 p 5]
              <lb/>
            proportio q z ad e z, ſicut i c ad c k.</s>
            <s xml:id="echoid-s8956" xml:space="preserve"> Quare [per 6.</s>
            <s xml:id="echoid-s8957" xml:space="preserve"> 4 p.</s>
            <s xml:id="echoid-s8958" xml:space="preserve"> 1 d 6] triangulum q z e ſimile triangulo i c k:</s>
            <s xml:id="echoid-s8959" xml:space="preserve"> &
              <lb/>
            triangulũ q l e ſimile triangulo i n k [quia iam patuit triangulum e l z ſimile eſſe triangulo k n c:</s>
            <s xml:id="echoid-s8960" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s8961" xml:space="preserve">
              <lb/>
            cum partes partibus ſimiles ſint:</s>
            <s xml:id="echoid-s8962" xml:space="preserve"> totum triãgulum q l e toti i k n ſimile erit.</s>
            <s xml:id="echoid-s8963" xml:space="preserve"> Quare per 1 d 6, ut q l ad
              <lb/>
            l e, ſic i n ad n k:</s>
            <s xml:id="echoid-s8964" xml:space="preserve"> & ſimiliter ob triangulorum a e l, k m n ſimilitudinem eſt, ut e l ad a l, ſic k n ad m n]
              <lb/>
            erit ergo [per 22 p & conſectarium 4 p 5] proportio m n ad n i, ſicut a l ad l q:</s>
            <s xml:id="echoid-s8965" xml:space="preserve"> & ita a l æqualis l q [ꝗ̃a
              <lb/>
            m n æquata eſt ipſi n i] & [per 4 p 1] e q erit ęqualis e a:</s>
            <s xml:id="echoid-s8966" xml:space="preserve"> & angulus e q z æqualis angulo l a e:</s>
            <s xml:id="echoid-s8967" xml:space="preserve"> & [per
              <lb/>
            fabricationem & 29 p 1] angulus e q z ęqualis angulo z a t:</s>
            <s xml:id="echoid-s8968" xml:space="preserve"> igitur [per 15.</s>
            <s xml:id="echoid-s8969" xml:space="preserve"> 32 p 1] tertius tertio ęqualis.</s>
            <s xml:id="echoid-s8970" xml:space="preserve">
              <lb/>
            Quare [per 4 p 6] proportio q z ad z a, ſicut e z ad z t, & ſicut e q ad at:</s>
            <s xml:id="echoid-s8971" xml:space="preserve"> & [per 7 p 5] ſicut a e ad a t.</s>
            <s xml:id="echoid-s8972" xml:space="preserve">
              <lb/>
            Sed q z ad z a, ſicut e g ad d g [fuit enim per fabricationem e g ad g d, ſicut i c ad c m:</s>
            <s xml:id="echoid-s8973" xml:space="preserve"> item ut c m ad i
              <lb/>
            c, ſic a z a d z q, & per cõſectarium 4 p 5 ut i c ad c m, ſic z q ad a z:</s>
            <s xml:id="echoid-s8974" xml:space="preserve"> ergo per 11 p 5, ut e g ad g d, ſic z q ad
              <lb/>
            a z.</s>
            <s xml:id="echoid-s8975" xml:space="preserve">] Igitur [per 11 p 5] a e ad a t, ſicut e g ad g d.</s>
            <s xml:id="echoid-s8976" xml:space="preserve"> Fiat autem [per 23 p 1] ſuper punctum a angulus æ-
              <lb/>
            qualis angulo g a e:</s>
            <s xml:id="echoid-s8977" xml:space="preserve"> qui ſit u a g.</s>
            <s xml:id="echoid-s8978" xml:space="preserve"> Palàm, quòd angulus g a l eſt medietas anguli u a t:</s>
            <s xml:id="echoid-s8979" xml:space="preserve"> [Quia enim ex
              <lb/>
            concluſo anguli z a t, z a e æquantur eidem z q e:</s>
            <s xml:id="echoid-s8980" xml:space="preserve"> ipſi inter ſe æquantur.</s>
            <s xml:id="echoid-s8981" xml:space="preserve"> Itaq;</s>
            <s xml:id="echoid-s8982" xml:space="preserve"> ſi æqualib.</s>
            <s xml:id="echoid-s8983" xml:space="preserve"> æqualia ad-
              <lb/>
            dantur, æquabitur angulus g a l duobus angulis u a g, z a t.</s>
            <s xml:id="echoid-s8984" xml:space="preserve"> Quare totus u a t duplus erit anguli g a l]
              <lb/>
            Sed eſt medietas d g u:</s>
            <s xml:id="echoid-s8985" xml:space="preserve"> [quia angulus g a l ęqualis concluſus eſt angulo f m c:</s>
            <s xml:id="echoid-s8986" xml:space="preserve"> qui per fabricationem
              <lb/>
            eſt dimidius anguli d g u.</s>
            <s xml:id="echoid-s8987" xml:space="preserve">] Quare angulus u a t eſt ęqualis angulo d g u:</s>
            <s xml:id="echoid-s8988" xml:space="preserve"> [per 6 ax.</s>
            <s xml:id="echoid-s8989" xml:space="preserve">] ſed anguli u a t,
              <lb/>
            </s>
          </p>
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