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-head377" xml:space="preserve" style="it">
            <pb o="171" file="0177" n="177" rhead="OPTICAE LIBER V."/>
          metris à centro inæquabiliter diſtantia, reflectuntur à quolibet puncto peripheriæ inter ſemidia
            <lb/>
          metros, extra quas ſunt, comprehenſæ: excepto eo, in quo ſecans diameter terminatur. 28 p 8.</head>
          <p>
            <s xml:id="echoid-s11373" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s11374" xml:space="preserve"> ductis diametris b q, a g:</s>
            <s xml:id="echoid-s11375" xml:space="preserve"> & diametro e z diuidente angulum b d g per æqualia.</s>
            <s xml:id="echoid-s11376" xml:space="preserve"> Dico,
              <lb/>
            quòd quodcũq;</s>
            <s xml:id="echoid-s11377" xml:space="preserve"> punctũ ſumatur in arcu a q, pręter punctũ z:</s>
            <s xml:id="echoid-s11378" xml:space="preserve"> [à pũcto enim z reflectũtur tan
              <lb/>
            tùm pũcta diametrorũ à centro æquabiliter diſtania, ut conſtat è ſuperioribus numeris] ab
              <lb/>
            illo poterunt reflecti infinita paria punctorũ, inæqualiter à centro diſtantiũ.</s>
            <s xml:id="echoid-s11379" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s11380" xml:space="preserve"> ſumatur h
              <lb/>
            punctum:</s>
            <s xml:id="echoid-s11381" xml:space="preserve"> & ſumatur in ſemidiametro g d punctũ l:</s>
            <s xml:id="echoid-s11382" xml:space="preserve"> & à ſemidiametro b d ſecetur m d, æqualis l d:</s>
            <s xml:id="echoid-s11383" xml:space="preserve">
              <lb/>
            [per 3 p 1] & ducantur lineę l m, l h, m h, d h.</s>
            <s xml:id="echoid-s11384" xml:space="preserve"> Punctũ, in quo e z diuidit l m, ſit f:</s>
            <s xml:id="echoid-s11385" xml:space="preserve"> erit [per 4 p 1] f l æqua-
              <lb/>
            lis f m:</s>
            <s xml:id="echoid-s11386" xml:space="preserve"> & ducatur h d, quouſq;</s>
            <s xml:id="echoid-s11387" xml:space="preserve"> cadat ſuper l m in pun-
              <lb/>
              <figure xlink:label="fig-0177-01" xlink:href="fig-0177-01a" number="116">
                <variables xml:id="echoid-variables106" xml:space="preserve">b a m h e f t d z n p l g q</variables>
              </figure>
            cto n:</s>
            <s xml:id="echoid-s11388" xml:space="preserve"> erit igitur l n minor m n.</s>
            <s xml:id="echoid-s11389" xml:space="preserve"> Verùm cum angulus
              <lb/>
            m d f ſit æqualis f d l [ex theſi] & angulo q d z:</s>
            <s xml:id="echoid-s11390" xml:space="preserve"> [per
              <lb/>
            15 p 1] & angulus m d a æqualis angulo l d q:</s>
            <s xml:id="echoid-s11391" xml:space="preserve"> [per ean
              <lb/>
            dem] & angulus a d h æqualis angulo n d l:</s>
            <s xml:id="echoid-s11392" xml:space="preserve"> erit angu-
              <lb/>
            lus l d h maior angulo m d h:</s>
            <s xml:id="echoid-s11393" xml:space="preserve"> [Quia enim m d n, id eſt
              <lb/>
            per 15 p 1 h d q maior eſt n d l, id eſt a d h, & m d a æ-
              <lb/>
            quatur ipſi l d q:</s>
            <s xml:id="echoid-s11394" xml:space="preserve"> angulus igitur l d h maior eſt angulo
              <lb/>
            m d h] igitur [per 24 p 1] l h erit maior m h:</s>
            <s xml:id="echoid-s11395" xml:space="preserve"> cum m d,
              <lb/>
            d h æqualia ſint l d, d h.</s>
            <s xml:id="echoid-s11396" xml:space="preserve"> Erit ergo angulus d h l minor
              <lb/>
            angulo d h m.</s>
            <s xml:id="echoid-s11397" xml:space="preserve"> Si enim eſſet æqualis:</s>
            <s xml:id="echoid-s11398" xml:space="preserve"> eſſet proportio l
              <lb/>
            h ad h m, ſicut l n ad n m:</s>
            <s xml:id="echoid-s11399" xml:space="preserve"> [per 3 p 6] qđ eſt impoſsibi
              <lb/>
            le.</s>
            <s xml:id="echoid-s11400" xml:space="preserve"> [Sic enim maior l h ad minorẽ h m eandẽ haberet
              <lb/>
            rationẽ, quam minor l n ad maiorẽ n m.</s>
            <s xml:id="echoid-s11401" xml:space="preserve">] Si autẽ fue-
              <lb/>
            rit maior:</s>
            <s xml:id="echoid-s11402" xml:space="preserve"> ſecetur ex e o æqualis:</s>
            <s xml:id="echoid-s11403" xml:space="preserve"> & improbabitur eo-
              <lb/>
            dẽ modo.</s>
            <s xml:id="echoid-s11404" xml:space="preserve"> Igitur eſt minor.</s>
            <s xml:id="echoid-s11405" xml:space="preserve"> Secetur igitur ab angulo
              <lb/>
            m h d ęqualis illi [d h l:</s>
            <s xml:id="echoid-s11406" xml:space="preserve">] qui ſit t h d.</s>
            <s xml:id="echoid-s11407" xml:space="preserve"> Igitur t reflectetur
              <lb/>
            ad l à puncto h [per 12 n 4.</s>
            <s xml:id="echoid-s11408" xml:space="preserve">] Et linea t d eſt minor l d.</s>
            <s xml:id="echoid-s11409" xml:space="preserve">
              <lb/>
            [quia minor eſt d m, pertheſin æquali ipſi d l.</s>
            <s xml:id="echoid-s11410" xml:space="preserve">] Similiter ſi ſumãtur in ſemidiametris b d, g d alia pun
              <lb/>
            cta, quàm l, m, æqualiter à puncto d diſtantia:</s>
            <s xml:id="echoid-s11411" xml:space="preserve"> probabitur ſimiliter, quòd à puncto h fit reflexio
              <gap/>
            pun-
              <lb/>
            ctorum ad inuicem, in æqualiter diſtantium à centro:</s>
            <s xml:id="echoid-s11412" xml:space="preserve"> & ita de infinitis punctis in his diametris ſum-
              <lb/>
            ptis ſimilis erit probatio:</s>
            <s xml:id="echoid-s11413" xml:space="preserve"> & à quocunq;</s>
            <s xml:id="echoid-s11414" xml:space="preserve"> puncto arcus a q ſumpto, præter quàm à puncto z.</s>
            <s xml:id="echoid-s11415" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div405" type="section" level="0" n="0">
          <head xml:id="echoid-head378" xml:space="preserve" style="it">75. Si uiſus & uiſibile in diuerſis diametris circuli (qui eſt cõmunis ſectio ſuperficierũ refle-
            <lb/>
          xiõis, et ſpeculι ſphærici caui) à cẽtro inæquabilιter diſtãtia, à pũcto aliquo peripheriæinter ſemi
            <lb/>
          diametros, extr a quas ſunt, inter ſe mutuò reflectãtur: ab uno tatùm puncto reflectẽtur. 29 p 8.</head>
          <p>
            <s xml:id="echoid-s11416" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s11417" xml:space="preserve"> ſumptis punctis t, l in diametris:</s>
            <s xml:id="echoid-s11418" xml:space="preserve"> quorũ inæqualis ſit lõgitudo à centro:</s>
            <s xml:id="echoid-s11419" xml:space="preserve"> reflectantur
              <lb/>
            ipla ad inuicẽ à puncto h:</s>
            <s xml:id="echoid-s11420" xml:space="preserve"> nõ poterit reflecti t ad l ab alio puncto arcus a q, quàm à puncto h.</s>
            <s xml:id="echoid-s11421" xml:space="preserve">
              <lb/>
            Si enim ab alio:</s>
            <s xml:id="echoid-s11422" xml:space="preserve"> ſit illud k:</s>
            <s xml:id="echoid-s11423" xml:space="preserve"> & ducantur, t k, l k,
              <lb/>
              <figure xlink:label="fig-0177-02" xlink:href="fig-0177-02a" number="117">
                <variables xml:id="echoid-variables107" xml:space="preserve">b a t h e p d z n l k g q</variables>
              </figure>
            d k, l t, t h, l h, n d h:</s>
            <s xml:id="echoid-s11424" xml:space="preserve"> & producatur d k;</s>
            <s xml:id="echoid-s11425" xml:space="preserve"> quouſq;</s>
            <s xml:id="echoid-s11426" xml:space="preserve"> con-
              <lb/>
            currat cum l t in puncto p [concurret autem, quia
              <lb/>
            ſecat angulũ t k l à baſi t l ſubtenſum.</s>
            <s xml:id="echoid-s11427" xml:space="preserve">] Palàm, quòd
              <lb/>
            proportio l h ad h t, ſicut l n ad n t.</s>
            <s xml:id="echoid-s11428" xml:space="preserve"> [per 3 p 6:</s>
            <s xml:id="echoid-s11429" xml:space="preserve"> quia e-
              <lb/>
            nim t ex theſi eſt reflexionis punctũ:</s>
            <s xml:id="echoid-s11430" xml:space="preserve"> æquabitur per
              <lb/>
            12 n 4 angulus l h n angulo t h n.</s>
            <s xml:id="echoid-s11431" xml:space="preserve">] Et ſimiliter cũ an-
              <lb/>
            gulus t p k ſit æqualis l k p, exhypotheſi:</s>
            <s xml:id="echoid-s11432" xml:space="preserve"> erit [per 3 p
              <lb/>
            6] proportio l k ad k t, ſicut l p ad p t:</s>
            <s xml:id="echoid-s11433" xml:space="preserve"> Sed [per 7
              <lb/>
            p 3] l h maior l k, & t h minor t k:</s>
            <s xml:id="echoid-s11434" xml:space="preserve"> igitur maior eſt pro
              <lb/>
            portio l h ad t h, quàm l k ad t k.</s>
            <s xml:id="echoid-s11435" xml:space="preserve"> [ut patet per 8 p 5.</s>
            <s xml:id="echoid-s11436" xml:space="preserve">]
              <lb/>
            Quare maior erit proportio l n ad n t, quàm l p ad p
              <lb/>
            t:</s>
            <s xml:id="echoid-s11437" xml:space="preserve"> quod planè impoſsibile.</s>
            <s xml:id="echoid-s11438" xml:space="preserve"> [Quia enim l n minor eſt
              <lb/>
            l p per 9 ax:</s>
            <s xml:id="echoid-s11439" xml:space="preserve"> & n t maior p t:</s>
            <s xml:id="echoid-s11440" xml:space="preserve"> erit ratio l n ad n t mi-
              <lb/>
            nor, quàm ratio l p ad p t, ut conſtat ex 8 p 5.</s>
            <s xml:id="echoid-s11441" xml:space="preserve">] Re-
              <lb/>
            ſtat, ut ab alio puncto arcus a q, quàm à puncto h
              <lb/>
            non poſsit t reflecti ad l.</s>
            <s xml:id="echoid-s11442" xml:space="preserve"> Palàm igitur, quæ acci-
              <lb/>
            dunt in arcu a q.</s>
            <s xml:id="echoid-s11443" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div407" type="section" level="0" n="0">
          <head xml:id="echoid-head379" xml:space="preserve" style="it">76. Viſu in diametro circuli (qui eſt communis ſectio ſuperficierũ, reflexionis & ſpeculi ſphæ-
            <lb/>
          rici caui) intra peripheriam poſito: uiſibile cum uiſu à centro utlibet diſtans: à quolιbet ſemicir-
            <lb/>
          culi puncto ad ipſum reflecti poteſt. 30 p 8.</head>
          <p>
            <s xml:id="echoid-s11444" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s11445" xml:space="preserve"> ſit a centrũ uiſus:</s>
            <s xml:id="echoid-s11446" xml:space="preserve"> b centrum ſpeculi:</s>
            <s xml:id="echoid-s11447" xml:space="preserve"> & ducatur diameter d a b g:</s>
            <s xml:id="echoid-s11448" xml:space="preserve"> & ſumatur ſuperfi.</s>
            <s xml:id="echoid-s11449" xml:space="preserve">
              <lb/>
            cies, in qua ſit a b quocunq;</s>
            <s xml:id="echoid-s11450" xml:space="preserve"> modo:</s>
            <s xml:id="echoid-s11451" xml:space="preserve"> quæ ſecabit ſphærã ſuper circulum:</s>
            <s xml:id="echoid-s11452" xml:space="preserve"> [per 1 th.</s>
            <s xml:id="echoid-s11453" xml:space="preserve"> 1 ſphær.</s>
            <s xml:id="echoid-s11454" xml:space="preserve">] qui
              <lb/>
            ſit d l g.</s>
            <s xml:id="echoid-s11455" xml:space="preserve"> Dico, quòd à quolibet puncto ſemicirculi d l g reflectuntur puncta ad a, inæqualis
              <lb/>
            longitudinis à centro, cum eo.</s>
            <s xml:id="echoid-s11456" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s11457" xml:space="preserve"> ſumatur punctũ e:</s>
            <s xml:id="echoid-s11458" xml:space="preserve"> & ducantur lineę e a, e b.</s>
            <s xml:id="echoid-s11459" xml:space="preserve"> Palàm, quòd
              <lb/>
            angulus a e b erit acutus:</s>
            <s xml:id="echoid-s11460" xml:space="preserve"> quia cadet in minorem arcum ſemicirculo.</s>
            <s xml:id="echoid-s11461" xml:space="preserve"> [Nam angulus inſiſtens in pe-
              <lb/>
            ripheriam ſemicirculi rectus eſt per 31 p 3.</s>
            <s xml:id="echoid-s11462" xml:space="preserve"> Vel etiã angulus a e b acutus eſt per ea, quæ 60 n demon-
              <lb/>
            </s>
          </p>
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