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-s14314" xml:space="preserve">
              <pb o="206" file="0212" n="212" rhead="ALHAZEN"/>
            diſtans eſt & æqualis f m:</s>
            <s xml:id="echoid-s14315" xml:space="preserve"> [Nam cum axis k d & recta f m ſint perpendiculares circulo b t o:</s>
            <s xml:id="echoid-s14316" xml:space="preserve"> ille per
              <lb/>
            21 d 11, hæc per fabricationem:</s>
            <s xml:id="echoid-s14317" xml:space="preserve"> erunt ipſæ inter ſe parallelæ per 6 p 11:</s>
            <s xml:id="echoid-s14318" xml:space="preserve"> & æquales per 34 p 1:</s>
            <s xml:id="echoid-s14319" xml:space="preserve"> quia circu
              <lb/>
            li b t o, e s p ſunt paralleli] & ita [per 33 p 1] k f æquidiſtans & æqualis d m.</s>
            <s xml:id="echoid-s14320" xml:space="preserve"> Similiter f m æquidiſtans
              <lb/>
            & æqualis e t:</s>
            <s xml:id="echoid-s14321" xml:space="preserve"> [per 30 p 1:</s>
            <s xml:id="echoid-s14322" xml:space="preserve"> quia e t latus cylindraceum parallelum eſt axi k d per 21 d 11] & k e æqualis
              <lb/>
            & æquidiſtans d t:</s>
            <s xml:id="echoid-s14323" xml:space="preserve"> & ita e f erit æquidiſtans & æqualis t m [per 33 p 1.</s>
            <s xml:id="echoid-s14324" xml:space="preserve">] Verùm ſuperficies k d l eſt or-
              <lb/>
            thogonalis ſuper ſuperficiem ſectionis b e t:</s>
            <s xml:id="echoid-s14325" xml:space="preserve"> [quia per axem ducitur, & angulus g d b in e llipſis pla-
              <lb/>
            no rectus eſt ex theſi] & eſt orthogonalis ſuper ſuperficiem circuli e s p [per 18 p 11:</s>
            <s xml:id="echoid-s14326" xml:space="preserve"> quia tranſit per
              <lb/>
            axem, perpendicularem circulo per 21 d 11.</s>
            <s xml:id="echoid-s14327" xml:space="preserve">] Ergo eſt perpendicularis ſuper lineam, communem ſe-
              <lb/>
            ctioni & circulo [per 19 p 11] quæ eſt e f.</s>
            <s xml:id="echoid-s14328" xml:space="preserve"> Igitur [per 3 d 11] angulus e f k rectus.</s>
            <s xml:id="echoid-s14329" xml:space="preserve"> Similiter angulus t m
              <lb/>
            d rectus [per 10 p 11:</s>
            <s xml:id="echoid-s14330" xml:space="preserve"> ſunt enim e f, f k parallelæ ipſis t m, m d, ut patuit, & in circulis parallelis.</s>
            <s xml:id="echoid-s14331" xml:space="preserve">] Cũ
              <lb/>
            igitur angulus d m t ſit rectus:</s>
            <s xml:id="echoid-s14332" xml:space="preserve"> & g t d rectus:</s>
            <s xml:id="echoid-s14333" xml:space="preserve"> [per 18 p 3] multiplicatio d m in m g erit, ſicut t m in
              <lb/>
            ſe.</s>
            <s xml:id="echoid-s14334" xml:space="preserve"> [Nam quia ab angulo g t d recto ducta eſt t m, perpẽdicularis baſi g d:</s>
            <s xml:id="echoid-s14335" xml:space="preserve"> erit per 8 p 6, ut d m ad m t,
              <lb/>
            ſic m t ad m g.</s>
            <s xml:id="echoid-s14336" xml:space="preserve"> Ita que per 17 p 6 rectangulum comprehenſum ſub extremis d m, g m æquatur qua-
              <lb/>
            drato mediæ t m.</s>
            <s xml:id="echoid-s14337" xml:space="preserve">] Sed quoniam f m æquidiſtat g l:</s>
            <s xml:id="echoid-s14338" xml:space="preserve"> [Nam cum g l ſit communis ſectio duorum pla-
              <lb/>
            norum, quorum alterum l e t g ſpeculum tangit, reliquum h d g l per axem ſecat:</s>
            <s xml:id="echoid-s14339" xml:space="preserve"> utrũque uerò per-
              <lb/>
            pendiculare eſt circulo b t o per 21 d.</s>
            <s xml:id="echoid-s14340" xml:space="preserve"> 18 p 11:</s>
            <s xml:id="echoid-s14341" xml:space="preserve"> erit ipſa g l eidem circulo perpendicularis per 19 p 11.</s>
            <s xml:id="echoid-s14342" xml:space="preserve">
              <lb/>
            Quare per 6 p 11 erit parallela axi:</s>
            <s xml:id="echoid-s14343" xml:space="preserve"> ideoque per 30 p 1 ipſi f m] erit [per 2 p 6] proportio d f ad f l, ſi-
              <lb/>
            cut d m ad m g.</s>
            <s xml:id="echoid-s14344" xml:space="preserve"> Sed d f maior d m [per 19 p 1:</s>
            <s xml:id="echoid-s14345" xml:space="preserve"> quia angulus ad m rectus eſt perfabricationem.</s>
            <s xml:id="echoid-s14346" xml:space="preserve">] Igi-
              <lb/>
            tur fl maior m g [per 14 p 5.</s>
            <s xml:id="echoid-s14347" xml:space="preserve">] Igitur maior eſt multiplicatio d f in f l, quàm d m in m g:</s>
            <s xml:id="echoid-s14348" xml:space="preserve"> ergo maior ꝗ̃
              <lb/>
            t m in ſe.</s>
            <s xml:id="echoid-s14349" xml:space="preserve"> Quare cum t m ſit æqualis e f [ex concluſo] erit multiplicatio d f in f l maior ductu lineæ e
              <lb/>
            fin ſe.</s>
            <s xml:id="echoid-s14350" xml:space="preserve"> Quare angulus l e d maior recto.</s>
            <s xml:id="echoid-s14351" xml:space="preserve"> Si enim rectus eſſet, cum linea e f ſit perpendicularis ſuper
              <lb/>
            l d [rectus enim demonſtratus eſt angulus e f k] eſſet ductus d fin fl æqualis quadrato e f [per 8.</s>
            <s xml:id="echoid-s14352" xml:space="preserve">17
              <lb/>
            p 6.</s>
            <s xml:id="echoid-s14353" xml:space="preserve">] Reſtat ergo [per 13 p 1] ut angulus d e q ſit acutus.</s>
            <s xml:id="echoid-s14354" xml:space="preserve"> Igitur orthogonalis ducta à puncto e, ortho
              <lb/>
            gonalis, inquam, ſuper contingentem q l, cadet ſub linea e d, & concurret cum perpendiculari b d
              <lb/>
            ſub puncto d.</s>
            <s xml:id="echoid-s14355" xml:space="preserve"> [Quòd enim perpendicularis illa & b d concurrant, patet per 11 ax:</s>
            <s xml:id="echoid-s14356" xml:space="preserve"> quia anguli, e d b
              <lb/>
            & comprehenſus ab e d & dicta perpendiculari, ſunt acuti:</s>
            <s xml:id="echoid-s14357" xml:space="preserve"> ille per theſin, hic, quia pars eſt recti, cõ-
              <lb/>
            prehenſi à tangente e q & dicta perpendiculari.</s>
            <s xml:id="echoid-s14358" xml:space="preserve">] Quod eſt propoſitum.</s>
            <s xml:id="echoid-s14359" xml:space="preserve"> His præmiſsis accedẽdum
              <lb/>
            eſt ad propoſitum.</s>
            <s xml:id="echoid-s14360" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div492" type="section" level="0" n="0">
          <head xml:id="echoid-head439" xml:space="preserve" style="it">25. Si uiſ{us}, & linea recta, axi ſpeculi cylindracei conuexi parallela, fuerint in eodem plana:
            <lb/>
          à toto cylindri latere ad uiſum reflecti poteſt: & imago uidetur linea recta, æqualis par alle-
            <lb/>
          læ. 50 p 7.</head>
          <p>
            <s xml:id="echoid-s14361" xml:space="preserve">PRoponatur columna:</s>
            <s xml:id="echoid-s14362" xml:space="preserve"> [ut in ſequente numero] linea æquidiſtans axi ſit t h.</s>
            <s xml:id="echoid-s14363" xml:space="preserve"> erit quidem æqui-
              <lb/>
            diſtans lineæ longitudinis columnæ [per 21 d 11.</s>
            <s xml:id="echoid-s14364" xml:space="preserve"> 30 p 1.</s>
            <s xml:id="echoid-s14365" xml:space="preserve">] Si ergo uiſus fuerit in eadem ſuperfi-
              <lb/>
            cie cum axe & linea t h:</s>
            <s xml:id="echoid-s14366" xml:space="preserve"> poterit quidem reflecti linea, & erit reflexio à linea longitudinis colu-
              <lb/>
            mnæ, quæ eſt linea communis ſuperficiei, in qua ſunt uiſus & axis, & ſuperficiei columnę, ſicut oſtẽ
              <lb/>
            ſum eſt in libro quinto [43.</s>
            <s xml:id="echoid-s14367" xml:space="preserve"> 89 n.</s>
            <s xml:id="echoid-s14368" xml:space="preserve">] Sicigitur uidebitur linea t h linea recta.</s>
            <s xml:id="echoid-s14369" xml:space="preserve"> Quoniam quælibet per-
              <lb/>
            pendicularis ducta à puncto lineæ t h, erit in eadem ſuperficie cum uiſu & axe.</s>
            <s xml:id="echoid-s14370" xml:space="preserve"> Et probabitur imagi
              <lb/>
            nem lineæ t h eſſe rectam, ſicut probatum eſt in ſpeculis planis de rectis lineis [2 n.</s>
            <s xml:id="echoid-s14371" xml:space="preserve">]</s>
          </p>
        </div>
        <div xml:id="echoid-div493" type="section" level="0" n="0">
          <head xml:id="echoid-head440" xml:space="preserve" style="it">26. Si uiſ{us} ſit extra planum lineæ rectæ, axi ſpeculi cylindracei conuexi parallelæ: à latere cy
            <lb/>
          lindri fit reflexio. 30 p 7.</head>
          <p>
            <s xml:id="echoid-s14372" xml:space="preserve">SI autem uiſus ſit extra ſuperficiem lineæ t h, & axis:</s>
            <s xml:id="echoid-s14373" xml:space="preserve"> & t h æquidiſtet axi:</s>
            <s xml:id="echoid-s14374" xml:space="preserve"> qui axis ſit z k:</s>
            <s xml:id="echoid-s14375" xml:space="preserve"> fiat ſu-
              <lb/>
            perficies per uiſum tranſiens, ſecans ſuperficiem columnæ æquidiſtanter baſi:</s>
            <s xml:id="echoid-s14376" xml:space="preserve"> [ut oſtenſum eſt
              <lb/>
            47 n 5] ſecabit quidem ſecundum circulum [per 5 th.</s>
            <s xml:id="echoid-s14377" xml:space="preserve"> Sereni de ſectione cylindri.</s>
            <s xml:id="echoid-s14378" xml:space="preserve">] Sit circu-
              <lb/>
            lus ille b f.</s>
            <s xml:id="echoid-s14379" xml:space="preserve"> Aliquod igitur punctum lineæ h t reflectitur ad uiſum, ab aliquo puncto huius circuli:</s>
            <s xml:id="echoid-s14380" xml:space="preserve"> ſit
              <lb/>
            punctum b:</s>
            <s xml:id="echoid-s14381" xml:space="preserve"> & uiſus ſit e:</s>
            <s xml:id="echoid-s14382" xml:space="preserve"> punctum illud lineæ t h, ſit q:</s>
            <s xml:id="echoid-s14383" xml:space="preserve"> & ducantur lineæ e b, q b, q e.</s>
            <s xml:id="echoid-s14384" xml:space="preserve"> Et ducatur à pũ
              <lb/>
            cto b linea longitudinis [ut monſtratum eſt 47 n 5] quæ ſit a b g:</s>
            <s xml:id="echoid-s14385" xml:space="preserve"> & ducatur à puncto b perpendicu-
              <lb/>
            laris, cadens ſuper axem in puncto l [cadet uerò per lẽma Procli ad 29 p 1:</s>
            <s xml:id="echoid-s14386" xml:space="preserve"> quia latus cylindraceũ &
              <lb/>
            axis ſunt paralleli per 21 d 11] quæ ſit m l:</s>
            <s xml:id="echoid-s14387" xml:space="preserve"> & ducatur à puncto e linea æquidiſtans l m:</s>
            <s xml:id="echoid-s14388" xml:space="preserve"> quæ ſit e o:</s>
            <s xml:id="echoid-s14389" xml:space="preserve"> &
              <lb/>
            ducatur q b, quouſque concurrat [concurret autem per allegatum Procli lemma] ſit concurſus in
              <lb/>
            puncto o.</s>
            <s xml:id="echoid-s14390" xml:space="preserve"> Palàm, quòd angulus q b m eſt æqualis angulo e b m:</s>
            <s xml:id="echoid-s14391" xml:space="preserve"> [anguli enim m b g, m b a recti per
              <lb/>
            fabricationem & 29 p 1, æquantur per 10 ax.</s>
            <s xml:id="echoid-s14392" xml:space="preserve"> itemq́ue q b g, e b a per 12 n 4:</s>
            <s xml:id="echoid-s14393" xml:space="preserve"> quare reliqui q b m, e b m
              <lb/>
            æquantur.</s>
            <s xml:id="echoid-s14394" xml:space="preserve">] Sed [per 29 p 1] angulus q b m æqualis eſt angulo b o e:</s>
            <s xml:id="echoid-s14395" xml:space="preserve"> quia l m æquidiſtans o e.</s>
            <s xml:id="echoid-s14396" xml:space="preserve"> Simi
              <lb/>
            liter [per eandem 29] angulus m b e æqualis angulo b e o:</s>
            <s xml:id="echoid-s14397" xml:space="preserve"> quia coalternus.</s>
            <s xml:id="echoid-s14398" xml:space="preserve"> Igitur angulus b o e æ-
              <lb/>
            qualis eſt angulo b e o.</s>
            <s xml:id="echoid-s14399" xml:space="preserve"> Quare [per 6 p 1] latera b o, b e æqualia.</s>
            <s xml:id="echoid-s14400" xml:space="preserve"> Sumaturautem aliud punctum in
              <lb/>
            linea t h:</s>
            <s xml:id="echoid-s14401" xml:space="preserve"> quod ſit t:</s>
            <s xml:id="echoid-s14402" xml:space="preserve"> & ducatur linea t o.</s>
            <s xml:id="echoid-s14403" xml:space="preserve"> Palàm, quòd linea t h æquidiſtat lineæ longitudinis, quæ eſt
              <lb/>
            a g [per 30 p 1:</s>
            <s xml:id="echoid-s14404" xml:space="preserve"> quia t h ex theſi parallela eſt axi, cui latus cylindraceum parallelum eſt per 21 d 11.</s>
            <s xml:id="echoid-s14405" xml:space="preserve">]
              <lb/>
            Ergo ſunt in eadem ſuperficie:</s>
            <s xml:id="echoid-s14406" xml:space="preserve"> [per 35 d 1] & in illa ſuperficie eſt linea q b o [per 7 p 11:</s>
            <s xml:id="echoid-s14407" xml:space="preserve"> quia conne-
              <lb/>
            ctit t h & a g.</s>
            <s xml:id="echoid-s14408" xml:space="preserve">] Quare in eadem erit linea t q [per 1 p 11.</s>
            <s xml:id="echoid-s14409" xml:space="preserve">] Secabit igitur lineam a g.</s>
            <s xml:id="echoid-s14410" xml:space="preserve"> Secet in puncto
              <lb/>
            g.</s>
            <s xml:id="echoid-s14411" xml:space="preserve"> Ducatur linea e g.</s>
            <s xml:id="echoid-s14412" xml:space="preserve"> Palàm etiam [per 8 p 11] quòd linea a g eſt perpendicularis ſuper ſuperficiem
              <lb/>
            circuli b f, ſicut axis, cui æquidiſtat, [per 21 d 11.</s>
            <s xml:id="echoid-s14413" xml:space="preserve">] Et ſuperficies illius circuli, eſt pars ſuperficiei, e
              <lb/>
            o b f, ſecans ſcilicet columnam æquidiſtanter baſi.</s>
            <s xml:id="echoid-s14414" xml:space="preserve"> Igitur [per 3 d 11] angulus g b o eſt rectus, & an-
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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