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-s13783" xml:space="preserve">
              <pb o="199" file="0205" n="205" rhead="OPTICAE LIBER VI."/>
            Similiter ſi ponatur, quòd g z, b c concurrant ad punctum e:</s>
            <s xml:id="echoid-s13784" xml:space="preserve"> probabitur, quòd d h concurret
              <lb/>
            ad idem.</s>
            <s xml:id="echoid-s13785" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div469" type="section" level="0" n="0">
          <figure number="165">
            <variables xml:id="echoid-variables155" xml:space="preserve">c h z b d g a</variables>
          </figure>
          <head xml:id="echoid-head423" xml:space="preserve" style="it">10. Si data recta in duob{us} punctis ſecta, ſit ad alterum extremorum ſegmẽtorum, ſicut re-
            <lb/>
          liquum extremum ad intermedium: & ab altero
            <lb/>
          ipſi{us} termino, ſectionum́ punctis tres rectæ li- neæ ſint parallelæ: recta à reliquo termino ſecan s parallel{as}, ſecabitur proportionaliter datæ. 122 p 1.</head>
          <p>
            <s xml:id="echoid-s13786" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s13787" xml:space="preserve"> diuiſa ab ſecundum hanc proportio-
              <lb/>
            nem:</s>
            <s xml:id="echoid-s13788" xml:space="preserve"> ſi fuerint lineæ g z, d h, b c æquidiſtãtes:</s>
            <s xml:id="echoid-s13789" xml:space="preserve">
              <lb/>
            & ducatur ac diuidens illas:</s>
            <s xml:id="echoid-s13790" xml:space="preserve"> erit ac diuiſa ſe-
              <lb/>
            cundum hanc proportionem.</s>
            <s xml:id="echoid-s13791" xml:space="preserve"> Cum d h ſit æquidi-
              <lb/>
            ſtans g z:</s>
            <s xml:id="echoid-s13792" xml:space="preserve"> erit [per 2 p 6] proportio a z ad z h, ſicut a g
              <lb/>
            ad g d:</s>
            <s xml:id="echoid-s13793" xml:space="preserve"> & cum b c ſit æquidiſtãs d h:</s>
            <s xml:id="echoid-s13794" xml:space="preserve"> erit [per 2 p 6.</s>
            <s xml:id="echoid-s13795" xml:space="preserve"> 18
              <lb/>
            p 5] a b ad b d, ſicut a c ad c h:</s>
            <s xml:id="echoid-s13796" xml:space="preserve"> ſed [ex theſi] a b ad b d,
              <lb/>
            ſicut a g ad g d:</s>
            <s xml:id="echoid-s13797" xml:space="preserve"> erit [per 11 p 5] a c ad ch, ſicut a z ad
              <lb/>
            z h.</s>
            <s xml:id="echoid-s13798" xml:space="preserve"> Et ita patet propoſitum.</s>
            <s xml:id="echoid-s13799" xml:space="preserve"> His præmiſsis, acceda-
              <lb/>
            mus ad propoſitum.</s>
            <s xml:id="echoid-s13800" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div470" type="section" level="0" n="0">
          <head xml:id="echoid-head424" xml:space="preserve" style="it">11. Sirecta linea à uiſu ſit perpendicularis ſu-
            <lb/>
          perficiei incidentiæ: imago perιpheriæ concentricæ
            <lb/>
          peripheriæ circuli (qui eſt communis ſectio ſuperficierum reflexionis & ſpeculi ſphærici cõuexi)
            <lb/>
          uidebitur curua, & par allela ipſi peripheriæ concentricæ. 46 p 6.</head>
          <p>
            <s xml:id="echoid-s13801" xml:space="preserve">PRimum de arcu declaremus, qnomodo in his ſpeculis imago
              <lb/>
              <figure xlink:label="fig-0205-02" xlink:href="fig-0205-02a" number="166">
                <variables xml:id="echoid-variables156" xml:space="preserve">b e a d h
                  <gap/>
                z
                  <gap/>
                m
                  <gap/>
                g</variables>
              </figure>
            eius ſit curua, curuitate quidem ſpeculum non reſpiciente, ſed
              <lb/>
            centrũ.</s>
            <s xml:id="echoid-s13802" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s13803" xml:space="preserve"> ſit ab arcus oppoſitus ſpeculo:</s>
            <s xml:id="echoid-s13804" xml:space="preserve"> & ſit g cen-
              <lb/>
            trum illius arcus, & ſimiliter centrum ſpeculi:</s>
            <s xml:id="echoid-s13805" xml:space="preserve"> d cẽtrum uiſus:</s>
            <s xml:id="echoid-s13806" xml:space="preserve"> & du-
              <lb/>
            cantur lineæ d g, a g, b g:</s>
            <s xml:id="echoid-s13807" xml:space="preserve"> & ſumatur e in arcu a b quocunq;</s>
            <s xml:id="echoid-s13808" xml:space="preserve"> modo:</s>
            <s xml:id="echoid-s13809" xml:space="preserve"> &
              <lb/>
            ducatur linea e g.</s>
            <s xml:id="echoid-s13810" xml:space="preserve"> Linea uerò d g non ſit in ſuperficie a b g.</s>
            <s xml:id="echoid-s13811" xml:space="preserve"> Linea igi-
              <lb/>
            tur d g aut erit orthogonalis ſuper ſuperficiẽ a b g:</s>
            <s xml:id="echoid-s13812" xml:space="preserve"> aut declinata.</s>
            <s xml:id="echoid-s13813" xml:space="preserve"> Sit
              <lb/>
            orthogonalis:</s>
            <s xml:id="echoid-s13814" xml:space="preserve"> erunt anguli d g a, d g e, d g b æquales [quia per 3 d 11
              <lb/>
            recti ſunt] & [per 15 d 1] latera lateribus.</s>
            <s xml:id="echoid-s13815" xml:space="preserve"> Quare [per 4 p 1] baſes baſi-
              <lb/>
            bus.</s>
            <s xml:id="echoid-s13816" xml:space="preserve"> Igitur omnia puncta arcus a b eiuſdem longitudinis erũt à cen-
              <lb/>
            tro uiſus.</s>
            <s xml:id="echoid-s13817" xml:space="preserve"> Quare imagines omniũ punctorũ, eiuſdẽ longitudinis ſunt
              <lb/>
            â cẽtro:</s>
            <s xml:id="echoid-s13818" xml:space="preserve"> ſintq́;</s>
            <s xml:id="echoid-s13819" xml:space="preserve"> q, m, l imagines ipſorũ a, e, b.</s>
            <s xml:id="echoid-s13820" xml:space="preserve"> Erit igitur g q ęqualis g m,
              <lb/>
            g l.</s>
            <s xml:id="echoid-s13821" xml:space="preserve"> Quare q m l erit arcus:</s>
            <s xml:id="echoid-s13822" xml:space="preserve"> [per 9 p 3] & cõuexitas ipſius reſpectu cen
              <lb/>
            tri, non reſpectu ſpeculi, ſiue loci reflexionis.</s>
            <s xml:id="echoid-s13823" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s13824" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div472" type="section" level="0" n="0">
          <head xml:id="echoid-head425" xml:space="preserve" style="it">12. Si recta linea à uiſu ſit obliqua ſuperficiei incidentiæ: ima-
            <lb/>
          go peripheriæ concentricæ peripheriæ circuli (qui eſt communis ſe-
            <lb/>
          ctio ſuperficierum, reflexionis & ſpeculi ſphærici conucxi) uidebi-
            <lb/>
          tur curua, non parallela peripheriæ concentricæ. 47 p 6.</head>
          <p>
            <s xml:id="echoid-s13825" xml:space="preserve">SI uerò linea d g non fuerit perpẽdicularis ſuper ſuperficiem a b g:</s>
            <s xml:id="echoid-s13826" xml:space="preserve"> ducta perpendiculari à pun-
              <lb/>
            cto d ſuper hanc ſuperficiem:</s>
            <s xml:id="echoid-s13827" xml:space="preserve"> [per 11 p 11] cum [per 5 n 5] illa perpendicularis ſit minor omni-
              <lb/>
            bus lineis ductis à puncto d ad hanc ſuperficiem:</s>
            <s xml:id="echoid-s13828" xml:space="preserve"> erit angulus, quem continet hæc perpendi-
              <lb/>
            cularis uerſus g, minor quolibet angulo uerſus punctũ g intellecto, quem continet alia linea à pun-
              <lb/>
            cto d ad hanc ſuperficiem ducta [per 16 p 1.</s>
            <s xml:id="echoid-s13829" xml:space="preserve">] Et linea ducta à puncto d ad hanc ſuperficiem, quan-
              <lb/>
            tò remotior erit à perpendiculari, tantò maior erit, & continebit maiorem angulum uerſus g [per
              <lb/>
            21 p 1.</s>
            <s xml:id="echoid-s13830" xml:space="preserve">] Si ergo hæc perpendicularis non cadat in arcum a e b, ſed ex parte una:</s>
            <s xml:id="echoid-s13831" xml:space="preserve"> erunt omnes lineæ
              <lb/>
            ductæ à puncto d ad hunc arcum, declinatæ ad partem unam:</s>
            <s xml:id="echoid-s13832" xml:space="preserve"> & remotiores maiores, & maiorem
              <lb/>
            angulum continentes uerſus g.</s>
            <s xml:id="echoid-s13833" xml:space="preserve"> Sit ergo:</s>
            <s xml:id="echoid-s13834" xml:space="preserve"> & ſumantur tria puncta in arcu, ſcilicet e, c, b:</s>
            <s xml:id="echoid-s13835" xml:space="preserve"> finis contin-
              <lb/>
            gentiæ puncti b ſit l:</s>
            <s xml:id="echoid-s13836" xml:space="preserve"> finis contingẽtiæ puncti c, ſit m.</s>
            <s xml:id="echoid-s13837" xml:space="preserve"> Quoniam igitur c propinquius d, quam b:</s>
            <s xml:id="echoid-s13838" xml:space="preserve"> erit
              <lb/>
            ιn propinquius g quàm l:</s>
            <s xml:id="echoid-s13839" xml:space="preserve"> [per 7 n] & ita c m maior b l [quia gc, g b ęquantur per 15 d 1] q ſit imago
              <lb/>
            c:</s>
            <s xml:id="echoid-s13840" xml:space="preserve"> timago b:</s>
            <s xml:id="echoid-s13841" xml:space="preserve"> & ducatur t q:</s>
            <s xml:id="echoid-s13842" xml:space="preserve"> & ducantur lineæ c b, m l:</s>
            <s xml:id="echoid-s13843" xml:space="preserve"> quæ quidem productæ concurrent.</s>
            <s xml:id="echoid-s13844" xml:space="preserve"> Si enim à
              <lb/>
            puncto m duceretur æquidiſtans c b, ſecaret ex g b lineam æqualem c m [eſſet enim per 2 p 6 18 p 5,
              <lb/>
            ut g c ad c m, ſic g b ad rectam, quam ſecat parallela à pũcto m ducta ex g b:</s>
            <s xml:id="echoid-s13845" xml:space="preserve"> itaq, cum g c, g b æquen-
              <lb/>
            tur per 15 d 1:</s>
            <s xml:id="echoid-s13846" xml:space="preserve"> æquaretur c m, ſectæ per parallelam ex g b:</s>
            <s xml:id="echoid-s13847" xml:space="preserve"> ſed c m, ut patuit, maior eſt b l:</s>
            <s xml:id="echoid-s13848" xml:space="preserve"> quare c b,
              <lb/>
            m l productæ concurrent.</s>
            <s xml:id="echoid-s13849" xml:space="preserve">] Concurrant in puncto o.</s>
            <s xml:id="echoid-s13850" xml:space="preserve"> Et quoniam proportio g c ad c m, ſicut g q ad
              <lb/>
            q m [eſt enim per 18 n 5, ut c g ad g q, ſic c m ad m q:</s>
            <s xml:id="echoid-s13851" xml:space="preserve"> ergo per 16 p 5, ut c g ad c m, ſic g q ad q m.</s>
            <s xml:id="echoid-s13852" xml:space="preserve">] Si-
              <lb/>
            militer g b ad b l, ſicut g t ad t l:</s>
            <s xml:id="echoid-s13853" xml:space="preserve"> ergo linea q t concurret cum lineis c b, m l [per 9 n.</s>
            <s xml:id="echoid-s13854" xml:space="preserve">] Sit con-
              <lb/>
            curſus in puncto o.</s>
            <s xml:id="echoid-s13855" xml:space="preserve"> Finis contingentiæ puncti e ſit n.</s>
            <s xml:id="echoid-s13856" xml:space="preserve"> Quoniam punctum n demiſsius eſt puncto
              <lb/>
            m:</s>
            <s xml:id="echoid-s13857" xml:space="preserve"> [per 7 n] erit e n maior c m:</s>
            <s xml:id="echoid-s13858" xml:space="preserve"> ductis ergo lineis e c, m n, concurrent [ut antea.</s>
            <s xml:id="echoid-s13859" xml:space="preserve">] Sit concurſus in
              <lb/>
            </s>
          </p>
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