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-s9213" xml:space="preserve">
              <pb o="150" file="0156" n="156" rhead="ALHAZEN"/>
            cationẽ] n l ęqualis h:</s>
            <s xml:id="echoid-s9214" xml:space="preserve"> & a d ad h, ſicut e ad z.</s>
            <s xml:id="echoid-s9215" xml:space="preserve"> Igitur [ք 11 p 5] t q ad q g, ſicut e ad z.</s>
            <s xml:id="echoid-s9216" xml:space="preserve"> Qđ eſt ꝓpoſitum.</s>
            <s xml:id="echoid-s9217" xml:space="preserve">
              <lb/>
            Põt aũt cõtĩgere:</s>
            <s xml:id="echoid-s9218" xml:space="preserve"> quòd à pũcto c erit ducere lineas duas, ſimiles c l n:</s>
            <s xml:id="echoid-s9219" xml:space="preserve"> & tũc erit ducere duas lineas à
              <lb/>
            puncto d, ſimiles t q, ut utriuſq;</s>
            <s xml:id="echoid-s9220" xml:space="preserve"> ad partẽ, ꝗ̃ ſecat ex a g, ſit ꝓportio, ſicut e ad z:</s>
            <s xml:id="echoid-s9221" xml:space="preserve"> & erit eadẽ probatio.</s>
            <s xml:id="echoid-s9222" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div345" type="section" level="0" n="0">
          <head xml:id="echoid-head342" xml:space="preserve" style="it">39. Viſu & uiſibili à centro ſpeculi ſphærici conuexi inæquabiliter diſtantib{us}, punctum re-
            <lb/>
          flexionis inuenire. 22 p 6.</head>
          <p>
            <s xml:id="echoid-s9223" xml:space="preserve">PRędictis habitis, dato ſpeculo ſphærico:</s>
            <s xml:id="echoid-s9224" xml:space="preserve"> erit inuenire punctũ reflexionis.</s>
            <s xml:id="echoid-s9225" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s9226" xml:space="preserve"> ſit a cẽ-
              <lb/>
            trũ uiſus:</s>
            <s xml:id="echoid-s9227" xml:space="preserve"> b punctũ uiſum:</s>
            <s xml:id="echoid-s9228" xml:space="preserve"> g centrũ ſphærę:</s>
            <s xml:id="echoid-s9229" xml:space="preserve"> & ducantur lineę a g, b g:</s>
            <s xml:id="echoid-s9230" xml:space="preserve"> & ſumatur ſuperficies, in
              <lb/>
            qua ſunt hę duę lineæ [ſunt enim in eadẽ ſuքficie ք 23 n 4] & ſumatur circulus, cõmunis huic
              <lb/>
            ſuperficiei & ſpeculo.</s>
            <s xml:id="echoid-s9231" xml:space="preserve"> Inuenietur ergo punctũ reflexionis in hoc circulo.</s>
            <s xml:id="echoid-s9232" xml:space="preserve"> Et ſumatur linea alia m k:</s>
            <s xml:id="echoid-s9233" xml:space="preserve">
              <lb/>
            & [ք 10 p 6] diuidatur in pũcto f, ut m f ſe habeat ad f k, ſicut b g ad g a:</s>
            <s xml:id="echoid-s9234" xml:space="preserve"> & [per 10 p 1] diuidatur m k ք
              <lb/>
            æqualia in puncto o:</s>
            <s xml:id="echoid-s9235" xml:space="preserve"> & [per 11 p 1] ducatur à puncto o perpẽdicularis:</s>
            <s xml:id="echoid-s9236" xml:space="preserve"> quę ſit c o:</s>
            <s xml:id="echoid-s9237" xml:space="preserve"> & ducatur à pũcto
              <lb/>
            k linea ad c o, tenens cũ ea angulũ æqualẽ medietati anguli b g a:</s>
            <s xml:id="echoid-s9238" xml:space="preserve"> [hoc aũt fiet:</s>
            <s xml:id="echoid-s9239" xml:space="preserve"> ſi linea e g bifariã ſe-
              <lb/>
            cans angulũ b g a, & o c infinitę intelligãtur, educta à puncto b perpendiculari ſuper e g, fiat angu-
              <lb/>
            lus o k c ęqualis angulo e b g:</s>
            <s xml:id="echoid-s9240" xml:space="preserve"> tũc enim (quia anguli ad o & e ſunt recti) ęquabitur angulus o c k an-
              <lb/>
            gulo e g b per 32 p 1] quę ſit k c:</s>
            <s xml:id="echoid-s9241" xml:space="preserve"> & à pũcto f ducatur linea ad c k:</s>
            <s xml:id="echoid-s9242" xml:space="preserve"> quę ſit f p:</s>
            <s xml:id="echoid-s9243" xml:space="preserve"> & cõcurrat cũ c o in pũcto
              <lb/>
            s, ita ut proportio s p ad p k ſit, ſicut b g ad ſemidiametrũ g d [per pręcedentẽ numerũ.</s>
            <s xml:id="echoid-s9244" xml:space="preserve">] Et [ք 23 p 1]
              <lb/>
            ex angulo b g a ſecetur ęqualis angulo f p k:</s>
            <s xml:id="echoid-s9245" xml:space="preserve"> [Id aũt fieri poſſe hinc cõſtat.</s>
            <s xml:id="echoid-s9246" xml:space="preserve"> Quia enim angulus s c p,
              <lb/>
            maior angulo c s p per 18 p 1 (cũ latus p s maius ſit latere c p:</s>
            <s xml:id="echoid-s9247" xml:space="preserve"> ſecus propoſitũ problema per lineã m k
              <lb/>
            expediri nõ poſſet) ęquetur per fabricationẽ dimidiato angulo b g a:</s>
            <s xml:id="echoid-s9248" xml:space="preserve"> ergo c s p eodẽ dimidiato mi-
              <lb/>
            nor eſt.</s>
            <s xml:id="echoid-s9249" xml:space="preserve"> Quare duo anguli s c p, c s p minores ſunt angulo b g a:</s>
            <s xml:id="echoid-s9250" xml:space="preserve"> at per 32 p 1 duob.</s>
            <s xml:id="echoid-s9251" xml:space="preserve"> angulis s c p, c s p
              <lb/>
            ęquatur angulus s p k:</s>
            <s xml:id="echoid-s9252" xml:space="preserve"> idcirco s p k minor eſt angulo b g a.</s>
            <s xml:id="echoid-s9253" xml:space="preserve"> Itaq;</s>
            <s xml:id="echoid-s9254" xml:space="preserve"> ab hoc ęqualis illi detrahi poteſt] ſci
              <lb/>
            licet d g b:</s>
            <s xml:id="echoid-s9255" xml:space="preserve"> & ducãtur lineę s k, b d:</s>
            <s xml:id="echoid-s9256" xml:space="preserve"> erit igitur [ք fabricationẽ] ꝓportio b g ad g d, ſicut s p ad p k:</s>
            <s xml:id="echoid-s9257" xml:space="preserve"> & i-
              <lb/>
            ta [per 6.</s>
            <s xml:id="echoid-s9258" xml:space="preserve"> 4 p.</s>
            <s xml:id="echoid-s9259" xml:space="preserve"> 1 d 6] triangulũ
              <lb/>
              <figure xlink:label="fig-0156-01" xlink:href="fig-0156-01a" number="80">
                <variables xml:id="echoid-variables70" xml:space="preserve">c p r m o f k y s</variables>
              </figure>
              <figure xlink:label="fig-0156-02" xlink:href="fig-0156-02a" number="81">
                <variables xml:id="echoid-variables71" xml:space="preserve">b f e m h u d a i z q c t y g ſ</variables>
              </figure>
            s p k ſimile triãgulo b g d:</s>
            <s xml:id="echoid-s9260" xml:space="preserve"> &
              <lb/>
            erit angulus s k p ęqualis an
              <lb/>
            gulo b d g.</s>
            <s xml:id="echoid-s9261" xml:space="preserve"> Sed forſan ſecun-
              <lb/>
            dũ prędicta [34.</s>
            <s xml:id="echoid-s9262" xml:space="preserve"> 38 n] poteri-
              <lb/>
            mus à puncto f ducere aliã li
              <lb/>
            neã ad c k, ſimilẽ s p:</s>
            <s xml:id="echoid-s9263" xml:space="preserve"> ut ſit ꝓ-
              <lb/>
            portio eius ad partẽ, ꝗ̃ ſeca-
              <lb/>
            bit ex c k, ſicut s p ad p k:</s>
            <s xml:id="echoid-s9264" xml:space="preserve"> &
              <lb/>
            tũc à pũcto k ad o s ducetur
              <lb/>
            alia linea ꝗ̃ s k, aliũ cũ c k an-
              <lb/>
            gulũ tenẽs maiorẽ uel mino
              <lb/>
            rem angulo c k s Si maior ex
              <lb/>
            his angulis non fuerit maior
              <lb/>
            recto:</s>
            <s xml:id="echoid-s9265" xml:space="preserve"> nõ licebit inuenire pũ
              <lb/>
            ctũ reflexionis [ut mox oſtẽ
              <lb/>
            detur.</s>
            <s xml:id="echoid-s9266" xml:space="preserve">] Sit ergo angulus c k s maior recto:</s>
            <s xml:id="echoid-s9267" xml:space="preserve"> erit angulus b d g [ꝗ illi ęqualis eſt oſtẽſus] maior recto.</s>
            <s xml:id="echoid-s9268" xml:space="preserve">
              <lb/>
            & inuenitur punctũ ſic.</s>
            <s xml:id="echoid-s9269" xml:space="preserve"> Ducatur [ք 17 p 3] cõtingens n d y.</s>
            <s xml:id="echoid-s9270" xml:space="preserve"> Et quia angulus p k o eſt minor recto [ք
              <lb/>
            32 p 1:</s>
            <s xml:id="echoid-s9271" xml:space="preserve"> ꝗ a c o k rectus eſt ք fabricationẽ] ſecetur [per 23 p 1] ex angulo b d g [ꝗ recto maior eſt ex con-
              <lb/>
            cluſione] æqualis ei:</s>
            <s xml:id="echoid-s9272" xml:space="preserve"> quι ſit q d g:</s>
            <s xml:id="echoid-s9273" xml:space="preserve"> eſt igitur triangulũ f p k ſimile triãgulo q g d [ęquatus.</s>
            <s xml:id="echoid-s9274" xml:space="preserve"> n.</s>
            <s xml:id="echoid-s9275" xml:space="preserve"> eſt angu-
              <lb/>
            lus s p k angulo b g d, & q d g angulo p k f:</s>
            <s xml:id="echoid-s9276" xml:space="preserve"> reliquus igitur ad f ęquatur reliquo ad q ք 32 p 1, & ք 4 p.</s>
            <s xml:id="echoid-s9277" xml:space="preserve"> 1
              <lb/>
            d 6 triãgula f p k, q d g ſunt ſimilia] & erit angulus d q b ęqualis angulo k f s [ք 13 p 1.</s>
            <s xml:id="echoid-s9278" xml:space="preserve"> 3 ax.</s>
            <s xml:id="echoid-s9279" xml:space="preserve">] & trιãgulũ
              <lb/>
            d q b ſimile triangulo k f s.</s>
            <s xml:id="echoid-s9280" xml:space="preserve"> [totus.</s>
            <s xml:id="echoid-s9281" xml:space="preserve"> n.</s>
            <s xml:id="echoid-s9282" xml:space="preserve"> angulus p k s ęquatur toti b d g, ut patuit:</s>
            <s xml:id="echoid-s9283" xml:space="preserve"> & p k f ęquatur ipſi q d
              <lb/>
            g ք proximã fabricationẽ:</s>
            <s xml:id="echoid-s9284" xml:space="preserve"> ergo per 3 ax.</s>
            <s xml:id="echoid-s9285" xml:space="preserve"> reliquus f k s ęquatur reliquo q d b:</s>
            <s xml:id="echoid-s9286" xml:space="preserve"> & ք 32 p 1 tertius tertio.</s>
            <s xml:id="echoid-s9287" xml:space="preserve">
              <lb/>
            Itaq;</s>
            <s xml:id="echoid-s9288" xml:space="preserve"> per 4 p.</s>
            <s xml:id="echoid-s9289" xml:space="preserve"> 1 d 6 d q b, k f s triangula ſunt ſimilia.</s>
            <s xml:id="echoid-s9290" xml:space="preserve">] Producatur d q, & [per 12 p 1] ducatur à puncto b
              <lb/>
            perpẽdicularis ſuք ipſam:</s>
            <s xml:id="echoid-s9291" xml:space="preserve"> quę ſit b z:</s>
            <s xml:id="echoid-s9292" xml:space="preserve"> erit [per 13 p 1] angulus b q z ęqualis angulo s f o & angulus b z
              <lb/>
            q rectus, ęqualis angulo s o f:</s>
            <s xml:id="echoid-s9293" xml:space="preserve"> & ita triãgulũ b q z ſimile triangulo s f o.</s>
            <s xml:id="echoid-s9294" xml:space="preserve"> Ducatur d z uſq;</s>
            <s xml:id="echoid-s9295" xml:space="preserve"> ad punctũ i:</s>
            <s xml:id="echoid-s9296" xml:space="preserve">
              <lb/>
            & ſit z i ęqualis z d [per 3 p 1.</s>
            <s xml:id="echoid-s9297" xml:space="preserve">] Palã [è triangulorũ z q b, s o f:</s>
            <s xml:id="echoid-s9298" xml:space="preserve"> itẽ q d b, k f s ſimilitudine] quòd z q ad
              <lb/>
            q b, & q b ad q d, ſicut o f ad f s, & f s ad f k [ideoq́;</s>
            <s xml:id="echoid-s9299" xml:space="preserve"> per 22 p 5, ut z q ad q d, ſic o f ad f k] & ex hoc [per
              <lb/>
            18 p 5] z d ad q d, ſicut o k ad f k:</s>
            <s xml:id="echoid-s9300" xml:space="preserve"> & ita [ſumendo antecedentiũ dupla per 15 p 5] i d ad q d, ſicut m k
              <lb/>
            ad f k:</s>
            <s xml:id="echoid-s9301" xml:space="preserve"> & ita [per 17 p 5] i q ad q d, ſicut m f ad f k:</s>
            <s xml:id="echoid-s9302" xml:space="preserve"> & [per 11 p 5] i q ad q d, ſicut b g ad g a [eſt enim per
              <lb/>
            fabricationẽ m f ad f k, ſicut b g ad g a.</s>
            <s xml:id="echoid-s9303" xml:space="preserve">] Ducatur aũt linea b i:</s>
            <s xml:id="echoid-s9304" xml:space="preserve"> & ei æquidiſtãs d l:</s>
            <s xml:id="echoid-s9305" xml:space="preserve"> erit triangulũ l d q
              <lb/>
            ſimile triãgulo b q i:</s>
            <s xml:id="echoid-s9306" xml:space="preserve"> [Nã per 29 p 1 angulus q d l ęquatur angulo b i q, & per 15 p 1 d q lipſi b q i:</s>
            <s xml:id="echoid-s9307" xml:space="preserve"> itaq;</s>
            <s xml:id="echoid-s9308" xml:space="preserve">
              <lb/>
            per 32 p 1 reliquus reliquo:</s>
            <s xml:id="echoid-s9309" xml:space="preserve"> & per 4 p.</s>
            <s xml:id="echoid-s9310" xml:space="preserve"> 1 d 6 triãgula d q l, b q i erunt ſimilia] & ꝓportio i q ad q d, ſicut
              <lb/>
            i b ad d l.</s>
            <s xml:id="echoid-s9311" xml:space="preserve"> Et cum i z ſit ęqualis z d, & b z perpendicularis:</s>
            <s xml:id="echoid-s9312" xml:space="preserve"> erit [per 4 p 1] b d æqualis b i.</s>
            <s xml:id="echoid-s9313" xml:space="preserve"> Quare e-
              <lb/>
            rit [per 7.</s>
            <s xml:id="echoid-s9314" xml:space="preserve"> 11 p 5] b d ad d l, ſicut b g ad g a.</s>
            <s xml:id="echoid-s9315" xml:space="preserve"> Ducatur à puncto d linea:</s>
            <s xml:id="echoid-s9316" xml:space="preserve"> quę ſit d h, æqualem tenens angu
              <lb/>
            lum cũ linea l d, angulo b g a:</s>
            <s xml:id="echoid-s9317" xml:space="preserve"> & cũ h l & d l concurrant:</s>
            <s xml:id="echoid-s9318" xml:space="preserve"> erunt [per 17 p 1] l h d, l d h minores duobus
              <lb/>
            rectis:</s>
            <s xml:id="echoid-s9319" xml:space="preserve"> & ita duo anguli a g h, d h g, eis ęquales, ſunt minores duobus rectis:</s>
            <s xml:id="echoid-s9320" xml:space="preserve"> quare [ք 11 ax.</s>
            <s xml:id="echoid-s9321" xml:space="preserve">] h d cõcur
              <lb/>
            ret cũ g a.</s>
            <s xml:id="echoid-s9322" xml:space="preserve"> Dico quòd cõcurret in pũcto a.</s>
            <s xml:id="echoid-s9323" xml:space="preserve"> Palã [per 18 p 3] quòd angulus g d n rectus, eſt ęqualis duo
              <lb/>
            bus angulis o c k & o k c:</s>
            <s xml:id="echoid-s9324" xml:space="preserve"> [quia ęqualis eſt angulo m o c recto, ęquali eiſdẽ angulis per 32 p 1] & an-
              <lb/>
            gulus o k c ęqualis angulo g d q:</s>
            <s xml:id="echoid-s9325" xml:space="preserve"> per fabricationem] reſtat [per 3 ax.</s>
            <s xml:id="echoid-s9326" xml:space="preserve">] angulus q d n ęqualis angu-
              <lb/>
            </s>
          </p>
        </div>
      </text>
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