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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          <p>
            <s xml:id="echoid-s10989" xml:space="preserve">
              <pb o="167" file="0173" n="173" rhead="OPTICAE LIBER V."/>
            minor h g:</s>
            <s xml:id="echoid-s10990" xml:space="preserve"> [per 7 p 3] non ergo erit ꝓportio e c ad h c, ſicut e d ad d h:</s>
            <s xml:id="echoid-s10991" xml:space="preserve"> & ita [ք 3 p 6] d c nó diuidet an
              <lb/>
            gulũ e c h ք ęqualia.</s>
            <s xml:id="echoid-s10992" xml:space="preserve"> Similiter, ſic ſumatur inter g & z, poterit improbari.</s>
            <s xml:id="echoid-s10993" xml:space="preserve"> Et ita patet ꝓpoſitũ.</s>
            <s xml:id="echoid-s10994" xml:space="preserve"> Notã-
              <lb/>
            dum tñ, quòd e eſt punctum intellectuale:</s>
            <s xml:id="echoid-s10995" xml:space="preserve"> & circulus ille (cuius e eſt polus) eſt circulus intellectua-
              <lb/>
            lis:</s>
            <s xml:id="echoid-s10996" xml:space="preserve"> & h punctũ intellectuale.</s>
            <s xml:id="echoid-s10997" xml:space="preserve"> Vn-
              <lb/>
              <figure xlink:label="fig-0173-01" xlink:href="fig-0173-01a" number="110">
                <variables xml:id="echoid-variables100" xml:space="preserve">g c f q a h d e z b</variables>
              </figure>
            de, quod dictũ eſt, ſecundũ geo-
              <lb/>
            metricã demõſtrationẽ eſt intel-
              <lb/>
            ligendum, nõ ſecundum uiſus ꝓ
              <lb/>
            bationẽ:</s>
            <s xml:id="echoid-s10998" xml:space="preserve"> cũ intellectualia uiſum
              <lb/>
            lateant.</s>
            <s xml:id="echoid-s10999" xml:space="preserve"> Sed quoniã forma h con
              <lb/>
            tinua uidetur formis aliorũ pun
              <lb/>
            ctorũ:</s>
            <s xml:id="echoid-s11000" xml:space="preserve"> uidebitur quidẽ à uiſu for
              <lb/>
            ma, cuius punctũ medium h:</s>
            <s xml:id="echoid-s11001" xml:space="preserve"> &
              <lb/>
            locus puncti medij illius formæ
              <lb/>
            erit e:</s>
            <s xml:id="echoid-s11002" xml:space="preserve"> & reflectetur h forma à lo-
              <lb/>
            co ſpeculi circulari, cuiusmediũ
              <lb/>
            erit circulus p̃dictus, & e polus
              <lb/>
            eius.</s>
            <s xml:id="echoid-s11003" xml:space="preserve"> Cum aũt e d fuerit maior d
              <lb/>
            h:</s>
            <s xml:id="echoid-s11004" xml:space="preserve"> in tãtum poterit eſſe maior, ut
              <lb/>
            nõ reflectatur h ad e à puncto g.</s>
            <s xml:id="echoid-s11005" xml:space="preserve">
              <lb/>
            Sciendum, quòd, niſi fuerit pro-
              <lb/>
            portio e a ad a h maior, quã e d
              <lb/>
            ad d h:</s>
            <s xml:id="echoid-s11006" xml:space="preserve"> nõ poterit h reflecti ad e.</s>
            <s xml:id="echoid-s11007" xml:space="preserve">
              <lb/>
            Si enim poteſt reflecti:</s>
            <s xml:id="echoid-s11008" xml:space="preserve"> reflectatur à puncto:</s>
            <s xml:id="echoid-s11009" xml:space="preserve"> quod ſit g:</s>
            <s xml:id="echoid-s11010" xml:space="preserve"> erit quidem g d h minor recto, cũ reſpiciat ſe-
              <lb/>
            ctionem minorẽ quarta.</s>
            <s xml:id="echoid-s11011" xml:space="preserve"> [quadrans enim peripherię ab angulo recto in cẽtro ſubtẽditur per 33 p 6.</s>
            <s xml:id="echoid-s11012" xml:space="preserve">
              <lb/>
            Vel angulus g d h minor eſt recto, quia ſemidiametro q d & recta g d cõprehenditur, ut demõſtratũ
              <lb/>
            eſt 60 n.</s>
            <s xml:id="echoid-s11013" xml:space="preserve">] Ducatur à puncto g cõtingens [per 17 p 3] quę neceſſariò cõcurret cũ e a:</s>
            <s xml:id="echoid-s11014" xml:space="preserve"> [per 11 ax:</s>
            <s xml:id="echoid-s11015" xml:space="preserve"> quia
              <lb/>
            anguli interiores ad g & d ſunt minores duobus rectis:</s>
            <s xml:id="echoid-s11016" xml:space="preserve"> cum angulus ad g ſit rectus per 18 p 3, ad d ue
              <lb/>
            rò acutus] ſit cõcurſus f.</s>
            <s xml:id="echoid-s11017" xml:space="preserve"> Erit quidẽ proportio e f ad f h, ſicut e d ad d h:</s>
            <s xml:id="echoid-s11018" xml:space="preserve"> [eſt enim per 64 n d h ad d e,
              <lb/>
            ſicut h fad e f:</s>
            <s xml:id="echoid-s11019" xml:space="preserve"> & per cõſectariũ 4 p 5, ut e f ad f h, ſic e d ad d h] ſed maior eſt proportio e a ad a h, quã
              <lb/>
            e f ad fh.</s>
            <s xml:id="echoid-s11020" xml:space="preserve"> [Quia enim a h minor eſt h f:</s>
            <s xml:id="echoid-s11021" xml:space="preserve"> erit ratio e h ad a h maior, quã ad h f per 8 p 5:</s>
            <s xml:id="echoid-s11022" xml:space="preserve"> & per 18 p 5, e a
              <lb/>
            ad a h maior, quã e f ad h f.</s>
            <s xml:id="echoid-s11023" xml:space="preserve">] Igitur maior eſt e a ad a h, ꝗ̃ e d ad d h:</s>
            <s xml:id="echoid-s11024" xml:space="preserve"> & ita neceſſariò:</s>
            <s xml:id="echoid-s11025" xml:space="preserve"> ſi h reflectitur ad
              <lb/>
            e:</s>
            <s xml:id="echoid-s11026" xml:space="preserve"> erit proportio e a ad a h maior, quàm e d ad d h.</s>
            <s xml:id="echoid-s11027" xml:space="preserve"> Patent ergo, quæ dicta ſunt:</s>
            <s xml:id="echoid-s11028" xml:space="preserve"> cum centrum uiſus &
              <lb/>
            punctum uiſum fuerint in eadem diametro.</s>
            <s xml:id="echoid-s11029" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div395" type="section" level="0" n="0">
          <head xml:id="echoid-head373" xml:space="preserve" style="it">70. Viſu & uiſibili extra circulum (qui eſt cõmunis ſectio ſuperficierũ, reflexionis & ſpeculi
            <lb/>
          ſphærici caui) ſitis in diuerſis diametris: ab uno puncto fit reflexio, et una uidetur imago. 24 p 8.</head>
          <p>
            <s xml:id="echoid-s11030" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s11031" xml:space="preserve"> cum punctũ uiſum & centrũ uiſus non fuerint in eadẽ diametro, & fuerint extra ſpe
              <lb/>
            culum:</s>
            <s xml:id="echoid-s11032" xml:space="preserve"> non reflectetur punctũ uiſum ad centrũ uiſus, niſi ab uno tantùm ſpeculi puncto.</s>
            <s xml:id="echoid-s11033" xml:space="preserve"> Ver
              <lb/>
            bi gratia:</s>
            <s xml:id="echoid-s11034" xml:space="preserve"> ſit t punctũ uiſum:</s>
            <s xml:id="echoid-s11035" xml:space="preserve"> h centrũ uiſus:</s>
            <s xml:id="echoid-s11036" xml:space="preserve"> d centrũ ſphęræ:</s>
            <s xml:id="echoid-s11037" xml:space="preserve"> & ducãtur lineę h d, d t, h t.</s>
            <s xml:id="echoid-s11038" xml:space="preserve"> Suքfi
              <lb/>
            cies quidẽ h d t ſecat ſphærã ſuper circulum:</s>
            <s xml:id="echoid-s11039" xml:space="preserve"> [per 1 th.</s>
            <s xml:id="echoid-s11040" xml:space="preserve"> 1 ſphęr.</s>
            <s xml:id="echoid-s11041" xml:space="preserve">] qui ſit e b q g.</s>
            <s xml:id="echoid-s11042" xml:space="preserve"> Palàm, quòd t non refle
              <lb/>
            ctetur ad h, niſi ab aliquo puncto huius circuli:</s>
            <s xml:id="echoid-s11043" xml:space="preserve"> [quia ipſe eſt reflexionis ſuperficies.</s>
            <s xml:id="echoid-s11044" xml:space="preserve">] Producãtur er
              <lb/>
            go h d, t d uſq;</s>
            <s xml:id="echoid-s11045" xml:space="preserve"> ad circumferentiã circuli.</s>
            <s xml:id="echoid-s11046" xml:space="preserve"> Palã, quòd nõ reflectetur ab
              <lb/>
              <figure xlink:label="fig-0173-02" xlink:href="fig-0173-02a" number="111">
                <variables xml:id="echoid-variables101" xml:space="preserve">h l m t k g e b f d p q o z a</variables>
              </figure>
            arcu q g, uel b a ſecundum modũ prędictum [66 n.</s>
            <s xml:id="echoid-s11047" xml:space="preserve">] Reflectetur ergo
              <lb/>
            aut ab arcu g b:</s>
            <s xml:id="echoid-s11048" xml:space="preserve"> aut a q.</s>
            <s xml:id="echoid-s11049" xml:space="preserve"> Diuidatur [per 9 p 1] angulus t d h per ęqua-
              <lb/>
            lia, per lineã l e d z:</s>
            <s xml:id="echoid-s11050" xml:space="preserve"> & à puncto e ducatur contingens:</s>
            <s xml:id="echoid-s11051" xml:space="preserve"> [per 17 p 3] quę
              <lb/>
            ſit k e f.</s>
            <s xml:id="echoid-s11052" xml:space="preserve"> Si puncta t, h fuerint ſuper illam contingentẽ:</s>
            <s xml:id="echoid-s11053" xml:space="preserve"> nõ reflectetur
              <lb/>
            t ad h ab aliquo pũcto arcus b g.</s>
            <s xml:id="echoid-s11054" xml:space="preserve"> Cũ enim à puncto t ducetur linea ad
              <lb/>
            aliquod interius punctũ huius arcus:</s>
            <s xml:id="echoid-s11055" xml:space="preserve"> linea à puncto h ad idẽ punctũ
              <lb/>
            ducta, cadet ſuper ipſum exterius, interius nõ.</s>
            <s xml:id="echoid-s11056" xml:space="preserve"> Et ideo non erit refle-
              <lb/>
            xio [à caua ſpeculi ſuperficie.</s>
            <s xml:id="echoid-s11057" xml:space="preserve">] Et quòd ab uno puncto tãtùm arcus
              <lb/>
            a q fiat reflexio:</s>
            <s xml:id="echoid-s11058" xml:space="preserve"> palã erit ex hoc.</s>
            <s xml:id="echoid-s11059" xml:space="preserve"> Ducãtur enim lineæ t z, h z.</s>
            <s xml:id="echoid-s11060" xml:space="preserve"> Cũ angu
              <lb/>
            lus t d h diuiſus ſit per ęqualia:</s>
            <s xml:id="echoid-s11061" xml:space="preserve"> erit t d z ęqualis angulo h d z.</s>
            <s xml:id="echoid-s11062" xml:space="preserve"> [per 13 p
              <lb/>
            1.</s>
            <s xml:id="echoid-s11063" xml:space="preserve">] Lineæ igitur t d, h d aut ſunt æquales:</s>
            <s xml:id="echoid-s11064" xml:space="preserve"> aut nõ ſunt æquales.</s>
            <s xml:id="echoid-s11065" xml:space="preserve"> Si ſunt
              <lb/>
            æquales, & d z cõmunis:</s>
            <s xml:id="echoid-s11066" xml:space="preserve"> erit [per 4 p 1] triãgulũ t z d æquale triangu
              <lb/>
            lo h z d:</s>
            <s xml:id="echoid-s11067" xml:space="preserve"> & angulus t z h diuiſus per ęqualia, per lineã d z.</s>
            <s xml:id="echoid-s11068" xml:space="preserve"> Et ita t refle-
              <lb/>
            cterur ad h à puncto z.</s>
            <s xml:id="echoid-s11069" xml:space="preserve"> [per 12 n.</s>
            <s xml:id="echoid-s11070" xml:space="preserve">] Quòd aũt ab alio puncto nõ poſsit:</s>
            <s xml:id="echoid-s11071" xml:space="preserve">
              <lb/>
            ſic cõſtabit.</s>
            <s xml:id="echoid-s11072" xml:space="preserve"> Sumatur punctũ o:</s>
            <s xml:id="echoid-s11073" xml:space="preserve"> & ducãtur lineæ t o, h o:</s>
            <s xml:id="echoid-s11074" xml:space="preserve"> & linea o d m
              <lb/>
            per cẽtrum d diuidat angulum illum per ęqualia.</s>
            <s xml:id="echoid-s11075" xml:space="preserve"> Planũ [per 8 p 3] qđ
              <lb/>
            t z minor eſt t o, & h o minor h z:</s>
            <s xml:id="echoid-s11076" xml:space="preserve"> & proportio t z ad h z, ſicut t l ad l h:</s>
            <s xml:id="echoid-s11077" xml:space="preserve">
              <lb/>
            [per 3 p 6:</s>
            <s xml:id="echoid-s11078" xml:space="preserve"> eſt enim angulus t z h bifariã ſectus à recta linea z l] & erit
              <lb/>
            [per eandẽ] proportio t o ad h o, ſicut t m ad m h:</s>
            <s xml:id="echoid-s11079" xml:space="preserve"> ſed minor eſt ꝓpor
              <lb/>
            tio h o ad t o, quã h z ad t z.</s>
            <s xml:id="echoid-s11080" xml:space="preserve"> [quia enim è quatuor lineis h o, t o, h z, t z prima minor eſt quã tertia, ſe-
              <lb/>
            cunda maior ꝗ̃ quarta:</s>
            <s xml:id="echoid-s11081" xml:space="preserve"> erit ratio primæ ad ſecundã minor, ꝗ̃ tertię ad quartã, ut patet ex 8 p 5] Ergo
              <lb/>
            [per 11 p 5] minor eſt proportio h m ad m t, ꝗ̃ h l ad l t:</s>
            <s xml:id="echoid-s11082" xml:space="preserve"> quod eſt impoſsibile.</s>
            <s xml:id="echoid-s11083" xml:space="preserve"> [Nã cum è quatuor lineis
              <lb/>
            h m, m t, h l, l t prima h m maior ſit, ꝗ̃ tertia h l:</s>
            <s xml:id="echoid-s11084" xml:space="preserve"> ſecũda uerò m t minor, ꝗ̃ quarta l t:</s>
            <s xml:id="echoid-s11085" xml:space="preserve"> erit ratio h m ad m t
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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