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-div345" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s9326" xml:space="preserve">
              <pb o="151" file="0157" n="157" rhead="OPTICAE LIBER V."/>
            lo o c k:</s>
            <s xml:id="echoid-s9327" xml:space="preserve"> & ita q d n medietas anguli b g a, & ita medietas anguli h d l [æquati angulo b g a.</s>
            <s xml:id="echoid-s9328" xml:space="preserve">] Sed [ք
              <lb/>
            3 p 6] angulus q d b eſt medietas anguli b d l:</s>
            <s xml:id="echoid-s9329" xml:space="preserve"> quoniã ꝓportio b q ad q l, ſicut b d ad d l:</s>
            <s xml:id="echoid-s9330" xml:space="preserve"> cũ triangulũ
              <lb/>
            d l q ſit ſimile triangulo b q i [ex cõcluſo] & b d æqualis b i, [ut patuit.</s>
            <s xml:id="echoid-s9331" xml:space="preserve">] Reſtat ergo, ut angulus n d
              <lb/>
            b ſit medietas anguli h d b:</s>
            <s xml:id="echoid-s9332" xml:space="preserve"> & ita b d n æqualis n d h.</s>
            <s xml:id="echoid-s9333" xml:space="preserve"> Producatur g d ultra d a d punctũ f.</s>
            <s xml:id="echoid-s9334" xml:space="preserve"> Quia igitur
              <lb/>
            [per 18 p 3] anguli f d n, g d n ſunt recti:</s>
            <s xml:id="echoid-s9335" xml:space="preserve"> ergo [ք 3 ax.</s>
            <s xml:id="echoid-s9336" xml:space="preserve">] reſtat b d f æqualis angulo h d g:</s>
            <s xml:id="echoid-s9337" xml:space="preserve"> Sed angulus
              <lb/>
            h d g æqualis angulo f d a contrà poſito [per 15 p 1.</s>
            <s xml:id="echoid-s9338" xml:space="preserve">] Quare b d f æqualis f d a.</s>
            <s xml:id="echoid-s9339" xml:space="preserve"> Et ita d eſt punctũ re-
              <lb/>
            flexionis [per 12 n 4.</s>
            <s xml:id="echoid-s9340" xml:space="preserve">] Ita dico:</s>
            <s xml:id="echoid-s9341" xml:space="preserve"> ſi a d cõcurrat cũ a g in pũcto a:</s>
            <s xml:id="echoid-s9342" xml:space="preserve"> quod quidẽ ſic patebit.</s>
            <s xml:id="echoid-s9343" xml:space="preserve"> Ducatur [per
              <lb/>
            31 p 1] linea h t æquidiſtãs b d.</s>
            <s xml:id="echoid-s9344" xml:space="preserve"> Palàm [è proximè demõſtratis] quòd angulus b d f æqualis eſt an-
              <lb/>
            gulo h d g:</s>
            <s xml:id="echoid-s9345" xml:space="preserve"> ſed [per 29 p 1] b d f eſt æqualis angulo h t d [ergo per 1 ax.</s>
            <s xml:id="echoid-s9346" xml:space="preserve"> h d g, h t d æquãtur.</s>
            <s xml:id="echoid-s9347" xml:space="preserve">] Quare
              <lb/>
            [per 6 p 1] h t erit æqualis h d.</s>
            <s xml:id="echoid-s9348" xml:space="preserve"> Sed proportio b d ad h t, ſicut b g ad g h.</s>
            <s xml:id="echoid-s9349" xml:space="preserve"> [ſunt enim triangula b d g, h
              <lb/>
            t g æquiangula:</s>
            <s xml:id="echoid-s9350" xml:space="preserve"> quãdoquidẽ angulus ad g cõmunis eſt, & g h t æquatur g b d per 29 p 1:</s>
            <s xml:id="echoid-s9351" xml:space="preserve"> ideoq́;</s>
            <s xml:id="echoid-s9352" xml:space="preserve"> per
              <lb/>
            32 p 1 tertius tertio.</s>
            <s xml:id="echoid-s9353" xml:space="preserve"> Quare per 4 p 6 habẽt latera æqualib.</s>
            <s xml:id="echoid-s9354" xml:space="preserve"> angulis oppoſita homologa.</s>
            <s xml:id="echoid-s9355" xml:space="preserve">] Igitur [per
              <lb/>
            7 p 5] proportio b d ad d h, ſicut b g ad g h.</s>
            <s xml:id="echoid-s9356" xml:space="preserve"> Sed h d producta cõcurret cũ g a [ut mõſtratũ eſt] & fiet
              <lb/>
            triangulũ ſimile triangulo h d l:</s>
            <s xml:id="echoid-s9357" xml:space="preserve"> cũ habeant angulũ l h d cõmunẽ, & angulus h d l ſit æqualis angulo
              <lb/>
            h g a [per fabricationẽ:</s>
            <s xml:id="echoid-s9358" xml:space="preserve"> & per 32 p 1 reliquus reliquo.</s>
            <s xml:id="echoid-s9359" xml:space="preserve">] Igitur [per 4 p 6] proportio h d ad d l, ſicut h
              <lb/>
            g ad lineã, ꝗ̃ ſecat h d ex g a:</s>
            <s xml:id="echoid-s9360" xml:space="preserve"> & ꝓportio b d ad d l cõſtat ex ꝓportiõe b d ad d h, & d h ad d l [ratio.</s>
            <s xml:id="echoid-s9361" xml:space="preserve"> n.</s>
            <s xml:id="echoid-s9362" xml:space="preserve">
              <lb/>
            extremorũ cõponitur ex omnib.</s>
            <s xml:id="echoid-s9363" xml:space="preserve"> ratiõib.</s>
            <s xml:id="echoid-s9364" xml:space="preserve"> intermedijs, ut oſtẽdit Theon ad 5 d 6.</s>
            <s xml:id="echoid-s9365" xml:space="preserve">] Igitur cõſtat ex b g
              <lb/>
            ad g h, & g h ad lineã, ꝗ̃ ſecath d ex g a:</s>
            <s xml:id="echoid-s9366" xml:space="preserve"> ſed b d ad d l, ſicut b g ad g a [ut patuit.</s>
            <s xml:id="echoid-s9367" xml:space="preserve">] Igitur ꝓportio b g ad
              <lb/>
            g a cõſtat ex ꝓportionib.</s>
            <s xml:id="echoid-s9368" xml:space="preserve"> b g ad g h & g h ad lineã, ꝗ̃ ſecat h d ex g a:</s>
            <s xml:id="echoid-s9369" xml:space="preserve"> ſed cõſtat ex ꝓportionib.</s>
            <s xml:id="echoid-s9370" xml:space="preserve"> b g ad
              <lb/>
            g h, & g h ad g a.</s>
            <s xml:id="echoid-s9371" xml:space="preserve"> Igitur g a eſt linea ꝗ̃ ſecat h d ex g a:</s>
            <s xml:id="echoid-s9372" xml:space="preserve"> & ita cõcurret cũ ea in pũcto a.</s>
            <s xml:id="echoid-s9373" xml:space="preserve"> Qđ eſt ꝓpoſitũ.</s>
            <s xml:id="echoid-s9374" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div347" type="section" level="0" n="0">
          <head xml:id="echoid-head343" xml:space="preserve" style="it">40. Si radi{us} à uiſibili ſpeculo ſphærico cõuexo obliquè incidens, cum ſemidiametro eiuſdem an-
            <lb/>
          gulũ nõ maiorẽ recto coprehendat: non reflectetur ad uiſum ab illo incidẽtiæ puncto. 21. 22 p 6.</head>
          <p>
            <s xml:id="echoid-s9375" xml:space="preserve">SI uerò angulus c k s nõ fuerit maior recto.</s>
            <s xml:id="echoid-s9376" xml:space="preserve"> Dico, qđ nõ fiet reflexio ab aliquo pũcto ſpeculi ad
              <lb/>
            uiſum.</s>
            <s xml:id="echoid-s9377" xml:space="preserve"> Si enim dicatur, quòd poteſt:</s>
            <s xml:id="echoid-s9378" xml:space="preserve"> Sit d punctũ reflexionis:</s>
            <s xml:id="echoid-s9379" xml:space="preserve"> & producatur linea a d uſq;</s>
            <s xml:id="echoid-s9380" xml:space="preserve"> ad h
              <lb/>
            punctũ in diametro b g.</s>
            <s xml:id="echoid-s9381" xml:space="preserve"> Et [per 23 p 1] fiat angulus l d h æqualis angulo a g b:</s>
            <s xml:id="echoid-s9382" xml:space="preserve"> & producatur
              <lb/>
            cõtingens n d y:</s>
            <s xml:id="echoid-s9383" xml:space="preserve"> & fiat angu
              <lb/>
              <figure xlink:label="fig-0157-01" xlink:href="fig-0157-01a" number="82">
                <variables xml:id="echoid-variables72" xml:space="preserve">c p p m o f k s s</variables>
              </figure>
              <figure xlink:label="fig-0157-02" xlink:href="fig-0157-02a" number="83">
                <variables xml:id="echoid-variables73" xml:space="preserve">b e n h d a i z q u t y g ſ x</variables>
              </figure>
            lus q d n æqualis medietati
              <lb/>
            anguli a g b.</s>
            <s xml:id="echoid-s9384" xml:space="preserve"> Palàm, quòd
              <lb/>
            triangulũ h d l ſimile eſt tri-
              <lb/>
            angulo h g a [quia enim an-
              <lb/>
            gulus h d l æquatus eſt an-
              <lb/>
            gulo h g a:</s>
            <s xml:id="echoid-s9385" xml:space="preserve"> & d h g eſt com-
              <lb/>
            munis:</s>
            <s xml:id="echoid-s9386" xml:space="preserve"> æquabitur per 32 p 1
              <lb/>
            tertius tertio:</s>
            <s xml:id="echoid-s9387" xml:space="preserve"> & per 4 p.</s>
            <s xml:id="echoid-s9388" xml:space="preserve"> 1 d
              <lb/>
            6 triangula erunt ſimilia.</s>
            <s xml:id="echoid-s9389" xml:space="preserve">]
              <lb/>
            Quare proportio d h ad d l,
              <lb/>
            ſicut h g ad g a:</s>
            <s xml:id="echoid-s9390" xml:space="preserve"> ſed b d ad d
              <lb/>
            h, ſicut b g ad g h:</s>
            <s xml:id="echoid-s9391" xml:space="preserve"> qđ pate-
              <lb/>
            bit per æquidiſtantẽ h t ipſi
              <lb/>
            b d.</s>
            <s xml:id="echoid-s9392" xml:space="preserve"> [ſic enim triangula b g
              <lb/>
            d, h g t fient æquiangula.</s>
            <s xml:id="echoid-s9393" xml:space="preserve"> Et
              <lb/>
            h d ęquatur ipſi h t.</s>
            <s xml:id="echoid-s9394" xml:space="preserve"> Nam quia d per theſin eſt punctum reflexionis, & e g perpendicularis plano ſpe
              <lb/>
            culũ in reflexionis puncto tãgenti per 25 n 4:</s>
            <s xml:id="echoid-s9395" xml:space="preserve"> ęquabitur angulus b d e angulo a d e per 12 n 4:</s>
            <s xml:id="echoid-s9396" xml:space="preserve"> & per
              <lb/>
            29 p 1 b d e, id eſt a d e, id eſt ք 15 p 1 h d t ęquatur ipſi h t d:</s>
            <s xml:id="echoid-s9397" xml:space="preserve"> Itaq;</s>
            <s xml:id="echoid-s9398" xml:space="preserve"> per 6 p 1 latus h d æquatur lateri h t.</s>
            <s xml:id="echoid-s9399" xml:space="preserve">]
              <lb/>
            Igitur b d ad d l, ſicut b g ad g a [quia enim ex cõcluſo eſt, ut b d ad d h, ſic b g ad g h:</s>
            <s xml:id="echoid-s9400" xml:space="preserve"> itẽ ut d h ad d l,
              <lb/>
            ſic g h ad g a:</s>
            <s xml:id="echoid-s9401" xml:space="preserve"> erit per 22 p 5, ut b d ad d l, ſic b g ad g a.</s>
            <s xml:id="echoid-s9402" xml:space="preserve">] Sed cũ angulus b d e ſit æqualis angulo h d g:</s>
            <s xml:id="echoid-s9403" xml:space="preserve">
              <lb/>
            [ex cõcluſo] erit angulus b d n medietas anguli b d h [nã angulι n d e, n d g recti per 18 p 3, æquãtur
              <lb/>
            per 10 ax:</s>
            <s xml:id="echoid-s9404" xml:space="preserve"> & b d e ipſi h d g:</s>
            <s xml:id="echoid-s9405" xml:space="preserve"> ergo per 3 ax.</s>
            <s xml:id="echoid-s9406" xml:space="preserve"> reliquus b d n reliquo h d n æquatur.</s>
            <s xml:id="echoid-s9407" xml:space="preserve"> Itaq;</s>
            <s xml:id="echoid-s9408" xml:space="preserve"> b d n dimidius
              <lb/>
            eſt ipſius b d h.</s>
            <s xml:id="echoid-s9409" xml:space="preserve">] Sed n d q eſt medietas anguli h d l [eſt enim per fabricationẽ dimidius anguli a g b,
              <lb/>
            cui æquatus eſt h d l.</s>
            <s xml:id="echoid-s9410" xml:space="preserve">] Igitur b d q medietas anguli b d l.</s>
            <s xml:id="echoid-s9411" xml:space="preserve"> Quare [per 3 p 6] proportio b q ad q l, ſicut
              <lb/>
            b d ad d l.</s>
            <s xml:id="echoid-s9412" xml:space="preserve"> Ducatur [ք 31 p 1] à pũcto b ęquidiſtãs d l:</s>
            <s xml:id="echoid-s9413" xml:space="preserve"> & ſit b i:</s>
            <s xml:id="echoid-s9414" xml:space="preserve"> & cõcurrat d q cũ e a in pũcto i:</s>
            <s xml:id="echoid-s9415" xml:space="preserve"> [cõcur-
              <lb/>
            ret aũt per lẽma Procli ad 29 p 1] & [ք 10 p 1] diuidatur d i in æqualia in pũcto z:</s>
            <s xml:id="echoid-s9416" xml:space="preserve"> & ducatur b z:</s>
            <s xml:id="echoid-s9417" xml:space="preserve"> e-
              <lb/>
            rit [ք 29.</s>
            <s xml:id="echoid-s9418" xml:space="preserve"> 15.</s>
            <s xml:id="echoid-s9419" xml:space="preserve"> 32 p 1.</s>
            <s xml:id="echoid-s9420" xml:space="preserve"> 4 p.</s>
            <s xml:id="echoid-s9421" xml:space="preserve"> 1 d 6] triangulũ b q i ſimile triangulo q d l.</s>
            <s xml:id="echoid-s9422" xml:space="preserve"> Igitur ut b q ad q l, ſic b i ad d l [at
              <lb/>
            oſtẽſum eſt, ut b q ad q l, ſic b d ad d l:</s>
            <s xml:id="echoid-s9423" xml:space="preserve"> ergo per 11 p 5 ut b i ad d l, ſic b d ad d l] & ita [per 9 p 5] b i ę-
              <lb/>
            qualis b d:</s>
            <s xml:id="echoid-s9424" xml:space="preserve"> & i q ad q d, ſicut m f ad f k:</s>
            <s xml:id="echoid-s9425" xml:space="preserve"> [eſt enim ob triangulorum b q i, q d l ſimilitudinem, ut i q ad
              <lb/>
            q d, ſic b q ad q l:</s>
            <s xml:id="echoid-s9426" xml:space="preserve"> & ut b q ad q l, ſic b i, id eſt, b d ad d l:</s>
            <s xml:id="echoid-s9427" xml:space="preserve"> & ut b d ad d l, ſic b g ad g a ex cõcluſo:</s>
            <s xml:id="echoid-s9428" xml:space="preserve"> & ut b g
              <lb/>
            ad g a, ſic m f ad f k per fabricationẽ:</s>
            <s xml:id="echoid-s9429" xml:space="preserve"> ergo per 11 p 5, ut i q ad q d, ſic m f ad f k] & ita [per 18 p 5] i d ad
              <lb/>
            q d, ſicut m k ad f k:</s>
            <s xml:id="echoid-s9430" xml:space="preserve"> & ita [ſumendo antecedentiũ dimidia per 15 p 5] d z ad q d, ſicut o k ad f k:</s>
            <s xml:id="echoid-s9431" xml:space="preserve"> & ita
              <lb/>
            [per 17 p 5] z q ad q d, ſicut o f ad fk.</s>
            <s xml:id="echoid-s9432" xml:space="preserve"> Palàm, quòd b z eſt perpendicularis:</s>
            <s xml:id="echoid-s9433" xml:space="preserve"> [quia enim b i æquatur
              <lb/>
            b d ex concluſo, & i z ipſi z d per fabricationem, & b z communis eſt:</s>
            <s xml:id="echoid-s9434" xml:space="preserve"> erũt triangula i b z, d b z æqui-
              <lb/>
            angula ք 8 p 1:</s>
            <s xml:id="echoid-s9435" xml:space="preserve"> & angulus b z i æquabitur angulo b z d:</s>
            <s xml:id="echoid-s9436" xml:space="preserve"> ſuntq́;</s>
            <s xml:id="echoid-s9437" xml:space="preserve"> deinceps:</s>
            <s xml:id="echoid-s9438" xml:space="preserve"> Quare per 10 d 1 b z perpẽdi
              <lb/>
            cularis eſt i d] ꝓducatur, donec cõcurrat cũ d g in pũcto x:</s>
            <s xml:id="echoid-s9439" xml:space="preserve"> qđ quidẽ poſsibile eſt [per 11 ax.</s>
            <s xml:id="echoid-s9440" xml:space="preserve">] cũ an-
              <lb/>
            gulus d z x ſit rectus, z d x minor recto.</s>
            <s xml:id="echoid-s9441" xml:space="preserve"> Et palã, qđ ꝓportio b g ad g d, ſicut s p ad p k:</s>
            <s xml:id="echoid-s9442" xml:space="preserve"> [ք fabricatio-
              <lb/>
            nẽ.</s>
            <s xml:id="echoid-s9443" xml:space="preserve">] Cũ ergo angulus c k s dicatur nõ eſſe maior recto:</s>
            <s xml:id="echoid-s9444" xml:space="preserve"> dico, qđ ſuք pũctũ k fiet maior recto, ք lineã
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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