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-div512" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s15111" xml:space="preserve">
              <pb o="215" file="0221" n="221" rhead="OPTICAE LIBER VI."/>
            go t b concurret cum a m.</s>
            <s xml:id="echoid-s15112" xml:space="preserve"> [ſi enim ex trapezio a m b t fiat parallelogrammũ (æquato nẽpe latere
              <lb/>
            b m ipſi t a, cumq́ue eodem connexo) patebit per lemma Procli ad 29 p 1, a m concurrere cum t b:</s>
            <s xml:id="echoid-s15113" xml:space="preserve">
              <lb/>
            quia concurrit cum ipſius parallela.</s>
            <s xml:id="echoid-s15114" xml:space="preserve">] Concurrant ergo in f:</s>
            <s xml:id="echoid-s15115" xml:space="preserve"> fergo eſt imago m.</s>
            <s xml:id="echoid-s15116" xml:space="preserve"> [per 6 n 5.</s>
            <s xml:id="echoid-s15117" xml:space="preserve">] Et ſic
              <lb/>
            declarabitur, quòd t g concurret cum a n.</s>
            <s xml:id="echoid-s15118" xml:space="preserve"> Concurrat in q:</s>
            <s xml:id="echoid-s15119" xml:space="preserve"> q
              <lb/>
              <figure xlink:label="fig-0221-01" xlink:href="fig-0221-01a" number="189">
                <variables xml:id="echoid-variables178" xml:space="preserve">f u q b
                  <gap/>
                  <gap/>
                m t n e o z a</variables>
              </figure>
            ergo erit imago n.</s>
            <s xml:id="echoid-s15120" xml:space="preserve"> Et continuemus f q:</s>
            <s xml:id="echoid-s15121" xml:space="preserve"> quæ eſt diameter i-
              <lb/>
            maginis m b.</s>
            <s xml:id="echoid-s15122" xml:space="preserve"> Et quia t e, t z ſunt æquales:</s>
            <s xml:id="echoid-s15123" xml:space="preserve"> [per conſectariũ
              <lb/>
            Campani ad 36 p 3] erunt anguli t a e, t a z æquales [per 8
              <lb/>
            p 1:</s>
            <s xml:id="echoid-s15124" xml:space="preserve"> quia a e, a z æquantur per 15 d 1, & a t eſt cõmune latus]
              <lb/>
            & erunt lineæ t b, t g æquales [per 4 p 1:</s>
            <s xml:id="echoid-s15125" xml:space="preserve"> quia a b, a g æquan
              <lb/>
            tur per 15 d 1] & lineæ b m, g n æquales.</s>
            <s xml:id="echoid-s15126" xml:space="preserve"> [Quia enim b a, g a
              <lb/>
            æquantur per 15 d 1, & a t eſt cõmunis, angulusq́;</s>
            <s xml:id="echoid-s15127" xml:space="preserve"> b a t æqua
              <lb/>
            lis concluſus eſt angulo g a t:</s>
            <s xml:id="echoid-s15128" xml:space="preserve"> æquabitur per 4 p 1 angulus
              <lb/>
            b t a angulo g t a, ideoq́;</s>
            <s xml:id="echoid-s15129" xml:space="preserve"> per 13 p 1 angulus u t b angulo u t g.</s>
            <s xml:id="echoid-s15130" xml:space="preserve">
              <lb/>
            Quare cum anguli a d t deinceps recti ſint per fabricationẽ:</s>
            <s xml:id="echoid-s15131" xml:space="preserve">
              <lb/>
            æquabitur per 3 ax.</s>
            <s xml:id="echoid-s15132" xml:space="preserve"> angulus b t m angulo g t n, & anguli ad
              <lb/>
            m & n recti per 29 p 1, æquantur per 10 ax.</s>
            <s xml:id="echoid-s15133" xml:space="preserve"> Itaq;</s>
            <s xml:id="echoid-s15134" xml:space="preserve"> per 26 p 1 b
              <lb/>
            m æquatur g n:</s>
            <s xml:id="echoid-s15135" xml:space="preserve"> & m tipſi n t] & lineæ a m, a n æquales [per
              <lb/>
            4 p 1:</s>
            <s xml:id="echoid-s15136" xml:space="preserve"> quia latera m t, n t ęqualia concluſa ſunt, & commune
              <lb/>
            eſt a t, anguliq́;</s>
            <s xml:id="echoid-s15137" xml:space="preserve"> a d t deinceps recti] & proportio a f ad f m,
              <lb/>
            ſicut proportio a t ad m b [per 4 p 6:</s>
            <s xml:id="echoid-s15138" xml:space="preserve"> quia triangula a t f, m b f ſunt æquiangula per 29.</s>
            <s xml:id="echoid-s15139" xml:space="preserve"> 32 p 1.</s>
            <s xml:id="echoid-s15140" xml:space="preserve">] Et
              <lb/>
            proportio a q ad q n eſt, ſicut proportio a t ad n g.</s>
            <s xml:id="echoid-s15141" xml:space="preserve"> Ergo proportio a fad f m eſt, ſicut proportio a q
              <lb/>
            ad q n [per 7 p 5:</s>
            <s xml:id="echoid-s15142" xml:space="preserve"> quia ratio a t ad b m & ad g n eadem eſt, cum b m æqualis oſtenſa ſit ipſi g n] & a
              <lb/>
            m eſt ſicut a n [per concluſionem.</s>
            <s xml:id="echoid-s15143" xml:space="preserve">] Ergo a f eſt ſicut a q.</s>
            <s xml:id="echoid-s15144" xml:space="preserve"> [Quia enim per concluſionem eſt, ut a f ad
              <lb/>
            f m, ſic a q ad q n:</s>
            <s xml:id="echoid-s15145" xml:space="preserve"> erit per 16 p 5, ut f a ad a q, ſic f m ad q n:</s>
            <s xml:id="echoid-s15146" xml:space="preserve"> ergo per 19 p 5 ut a m ad a n, ſic a f ad a q:</s>
            <s xml:id="echoid-s15147" xml:space="preserve">
              <lb/>
            ſed a m æqualis oſtenſa eſt ipſi a n.</s>
            <s xml:id="echoid-s15148" xml:space="preserve"> Quare a f æqualis eſt a q.</s>
            <s xml:id="echoid-s15149" xml:space="preserve">] Ergo f q æquidiſtat n m [per proxi-
              <lb/>
            mam concluſionem & 2 p 6.</s>
            <s xml:id="echoid-s15150" xml:space="preserve">] Ergo f q eſt maior m n [per 4 p 6:</s>
            <s xml:id="echoid-s15151" xml:space="preserve"> quia a f ad a m, ſicut f q ad m n:</s>
            <s xml:id="echoid-s15152" xml:space="preserve"> ſed a f
              <lb/>
            maior eſt a m ք 9 ax:</s>
            <s xml:id="echoid-s15153" xml:space="preserve"> ergo f q maior eſt m n:</s>
            <s xml:id="echoid-s15154" xml:space="preserve"> ſed f q eſt diameter imaginis n m.</s>
            <s xml:id="echoid-s15155" xml:space="preserve"> Ergo ſi uiſus fuerit in
              <lb/>
            t, & linea m n fuerit in aliquo uiſibili:</s>
            <s xml:id="echoid-s15156" xml:space="preserve"> tunc uiſus comprehendet formam maiorem, quàm ſit.</s>
            <s xml:id="echoid-s15157" xml:space="preserve">]</s>
          </p>
        </div>
        <div xml:id="echoid-div514" type="section" level="0" n="0">
          <head xml:id="echoid-head456" xml:space="preserve" style="it">40. Si uiſ{us} fuerit ſublimior uiſibili intra ſpeculum ſphæricum cauum extremis ſuis à cen-
            <lb/>
          tro æquabiliter diſtante: imago uidebitur ultra ſpeculum, maior uiſibili. 47 p 8.</head>
          <p>
            <s xml:id="echoid-s15158" xml:space="preserve">ITem:</s>
            <s xml:id="echoid-s15159" xml:space="preserve"> iteremus circulum b g:</s>
            <s xml:id="echoid-s15160" xml:space="preserve"> & lineam a u:</s>
            <s xml:id="echoid-s15161" xml:space="preserve"> & lineas a b, a g, t b, t g:</s>
            <s xml:id="echoid-s15162" xml:space="preserve"> & ſuper punctum t ſit perpen-
              <lb/>
            dicularis ſuper ſuperficiem circuli b g [per 12 p 11] & ſit t k:</s>
            <s xml:id="echoid-s15163" xml:space="preserve"> continuemus k a, k b, k g.</s>
            <s xml:id="echoid-s15164" xml:space="preserve"> Superfici-
              <lb/>
            es ergo k b a, k g a ſecant ſphæram ſuper centrum ſuum perpendiculariter, & ſuperficies tangen
              <lb/>
            tes ipſam [per 18 p 11.</s>
            <s xml:id="echoid-s15165" xml:space="preserve">] Ex ipſis ergo reflectitur forma:</s>
            <s xml:id="echoid-s15166" xml:space="preserve">
              <lb/>
              <figure xlink:label="fig-0221-02" xlink:href="fig-0221-02a" number="190">
                <variables xml:id="echoid-variables179" xml:space="preserve">f q b u g m c n K p a</variables>
              </figure>
            & duæ differentiæ cõmunes inter has duas ſuperficies
              <lb/>
            & ſphærã, ſunt circuli magni [per 1 th 1 ſphęr.</s>
            <s xml:id="echoid-s15167" xml:space="preserve">] à quorũ
              <lb/>
            circũferentia reflectũtur formæ.</s>
            <s xml:id="echoid-s15168" xml:space="preserve"> Et extrah amus b m in
              <lb/>
            ſuperficie b k a æquidiſtantẽ a k:</s>
            <s xml:id="echoid-s15169" xml:space="preserve"> & ſit minor, quã a k:</s>
            <s xml:id="echoid-s15170" xml:space="preserve"> &
              <lb/>
            cõtinuemus a m, & extrahatur rectè:</s>
            <s xml:id="echoid-s15171" xml:space="preserve"> & extrahatur k b,
              <lb/>
            donec cõcnrrat cum a m in f [cõcurret aũt, ut proximo
              <lb/>
            numero oſtẽſum eſt:</s>
            <s xml:id="echoid-s15172" xml:space="preserve"> quia b m minor eſt a k per ſabrica-
              <lb/>
            tionẽ.</s>
            <s xml:id="echoid-s15173" xml:space="preserve">] Et extrahatur n g in ſuperficie k g a:</s>
            <s xml:id="echoid-s15174" xml:space="preserve"> & ſit æqui-
              <lb/>
            diſtãs a k:</s>
            <s xml:id="echoid-s15175" xml:space="preserve"> & ponatur æqualis b m:</s>
            <s xml:id="echoid-s15176" xml:space="preserve"> & cõtinuemus a n, &
              <lb/>
            extrahatur rectè, donec cõcurrat in q:</s>
            <s xml:id="echoid-s15177" xml:space="preserve"> & cõtinuemus m
              <lb/>
            n, f q.</s>
            <s xml:id="echoid-s15178" xml:space="preserve"> Quia ergo b t eſt ſicut t a [ut ſuperiore numero
              <lb/>
            demonſtratũ eſt] erit b k, ſicut k a [per 4 p 1:</s>
            <s xml:id="echoid-s15179" xml:space="preserve"> nã t k com
              <lb/>
            mune latus eſt utriuſq;</s>
            <s xml:id="echoid-s15180" xml:space="preserve"> trianguli b t k, a t k, & anguli ad
              <lb/>
            t recti per 3 d 11] & g k, ſicut k a:</s>
            <s xml:id="echoid-s15181" xml:space="preserve"> ergo b k eſt, ſicut g k:</s>
            <s xml:id="echoid-s15182" xml:space="preserve"> &
              <lb/>
            [per 5 p 1] angulus k a b eſt, ſicut angulus k b a:</s>
            <s xml:id="echoid-s15183" xml:space="preserve"> & ſimi-
              <lb/>
            liter angulus k g a eſt, ſicut angulus k a g.</s>
            <s xml:id="echoid-s15184" xml:space="preserve"> Ergo angulus
              <lb/>
            a b m eſt, ſicut angulus a b k [quia per 29 p 1 angulus a
              <lb/>
            b m æquatur angulo k a b, cui æqualis cõcluſus eſt a b k] & angulus a g n eſt, ſicut angulus a g k.</s>
            <s xml:id="echoid-s15185" xml:space="preserve"> [Nã
              <lb/>
            per 29 p 1 angulus a g n æquatur angulo k a g, cui æqualis oſtẽſus eſt angulus a g k.</s>
            <s xml:id="echoid-s15186" xml:space="preserve">] Ergo erit angu
              <lb/>
            lus a b m, ſicut angulus a g n.</s>
            <s xml:id="echoid-s15187" xml:space="preserve"> [Quia enim g k æqualis concluſa eſt ipſi b k:</s>
            <s xml:id="echoid-s15188" xml:space="preserve"> & a g, a b æquantur
              <lb/>
            per 15 d 1:</s>
            <s xml:id="echoid-s15189" xml:space="preserve"> & cõmmunis eſt a k:</s>
            <s xml:id="echoid-s15190" xml:space="preserve"> æquabũtur anguli a b k, a g k per 8 p 1:</s>
            <s xml:id="echoid-s15191" xml:space="preserve"> & his ęquãtur per proximã cõ
              <lb/>
            cluſionẽ a b m, a g n.</s>
            <s xml:id="echoid-s15192" xml:space="preserve"> Quare a b m, a g n æquãtur] & linea b m, ſicut linea g n:</s>
            <s xml:id="echoid-s15193" xml:space="preserve"> [ex fabricatione] tũc li
              <lb/>
            nea a m erit, ſicut linea a n:</s>
            <s xml:id="echoid-s15194" xml:space="preserve"> [ք 4 p 1:</s>
            <s xml:id="echoid-s15195" xml:space="preserve"> quia a b, b m ęquãtur ipſis a g, g n, & angulus a b m angulo a g n]
              <lb/>
            tũc duę lineæ f q, m n erũt æquidiſtãtes:</s>
            <s xml:id="echoid-s15196" xml:space="preserve"> [per 2 p 6, ut proximo numero demõſtratũ eſt] tũc f q erit
              <lb/>
            maior linea m n.</s>
            <s xml:id="echoid-s15197" xml:space="preserve"> Tunc quando uiſus fuerit ſuper punctum k, & fuerit linea m n in aliquo uiſibili in-
              <lb/>
            feriore:</s>
            <s xml:id="echoid-s15198" xml:space="preserve"> tunc forma m extendetur ſuper lineam m b, & reflectetur per lineam b k in ſuperficie circu
              <lb/>
            li, tranſeuntis per puncta b, a, k:</s>
            <s xml:id="echoid-s15199" xml:space="preserve"> & forma puncti n extendetur ſuper lineam n g, & reſlectetur ſuper
              <lb/>
            lineam g k in ſuperficie circuli, tranſeuntis per tria puncta g, a, k.</s>
            <s xml:id="echoid-s15200" xml:space="preserve"> Et erit imago puncti f punctum m:</s>
            <s xml:id="echoid-s15201" xml:space="preserve">
              <lb/>
            [per 6 n 5] & punctum q erit imago puncti n:</s>
            <s xml:id="echoid-s15202" xml:space="preserve"> & erit linea f q diameter imaginis n m.</s>
            <s xml:id="echoid-s15203" xml:space="preserve"> Etiam decla-
              <lb/>
            </s>
          </p>
        </div>
      </text>
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