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-s14014" xml:space="preserve">
              <pb o="202" file="0208" n="208" rhead="ALHAZEN"/>
            punctum b.</s>
            <s xml:id="echoid-s14015" xml:space="preserve"> Et ita duo puncta in his ſpeculis reflectentur ad idem punctum ex eadem parte:</s>
            <s xml:id="echoid-s14016" xml:space="preserve"> quod
              <lb/>
            eſt impoſsibile [& contra 29 n 5.</s>
            <s xml:id="echoid-s14017" xml:space="preserve">] Reſtat, ut punctum a reflectatur ad k, ab aliquo puncto arcus z f.</s>
            <s xml:id="echoid-s14018" xml:space="preserve">
              <lb/>
            Si ab illo puncto ducatur contingens:</s>
            <s xml:id="echoid-s14019" xml:space="preserve"> ſecabit lineam a z, & cadet inter z & c:</s>
            <s xml:id="echoid-s14020" xml:space="preserve"> quoniam punctum f
              <lb/>
            demiſsius eſt quolibet puncto arcus z f:</s>
            <s xml:id="echoid-s14021" xml:space="preserve"> & ita contingens à puncto f altior alijs, à punctis arcus z f
              <lb/>
            ductis.</s>
            <s xml:id="echoid-s14022" xml:space="preserve"> Cadat ergo contingens illa in punctum n:</s>
            <s xml:id="echoid-s14023" xml:space="preserve"> & ducatur linea m n:</s>
            <s xml:id="echoid-s14024" xml:space="preserve"> quę quidem linea cum tran-
              <lb/>
            ſeat per acumen trianguli b m t, & producta diuidat angulum, neceſſariò ſecabit b t.</s>
            <s xml:id="echoid-s14025" xml:space="preserve"> Secet in pun-
              <lb/>
            cto q:</s>
            <s xml:id="echoid-s14026" xml:space="preserve"> & ducatur linea g q.</s>
            <s xml:id="echoid-s14027" xml:space="preserve"> Sit autem i imago puncti a:</s>
            <s xml:id="echoid-s14028" xml:space="preserve"> o ſit imago puncti b:</s>
            <s xml:id="echoid-s14029" xml:space="preserve"> r ſit imago puncti q.</s>
            <s xml:id="echoid-s14030" xml:space="preserve"> Pa-
              <lb/>
            làm, cum b ſit propinquius puncto g, quàm a:</s>
            <s xml:id="echoid-s14031" xml:space="preserve"> erit o remotior à puncto g, quàm c [per 7 n.</s>
            <s xml:id="echoid-s14032" xml:space="preserve">] Ducatur
              <lb/>
            ergo linea i o.</s>
            <s xml:id="echoid-s14033" xml:space="preserve"> Palàm etiam [per 18 n 5.</s>
            <s xml:id="echoid-s14034" xml:space="preserve">16 p 5] quòd proportio a g ad a n, ſicut g i ad i n:</s>
            <s xml:id="echoid-s14035" xml:space="preserve"> & proportio
              <lb/>
            b
              <gap/>
            g ad b m, ſicut g o ad o m.</s>
            <s xml:id="echoid-s14036" xml:space="preserve"> Cum ergo lineæ a g, b g diuidantur ſecundũ hanc proportionem, utraq;</s>
            <s xml:id="echoid-s14037" xml:space="preserve">
              <lb/>
            in duobus punctis, & à punctis diuiſionum ducantur lineæ, quarum duæ, ſcilicet a b, n m concur-
              <lb/>
            rant ad idem punctum, ſcilicet q:</s>
            <s xml:id="echoid-s14038" xml:space="preserve"> tertia neceſſariò concurret ad idem punctum [per 9 n.</s>
            <s xml:id="echoid-s14039" xml:space="preserve">] Igitur i o
              <lb/>
            ꝓducta cadet ſuper q.</s>
            <s xml:id="echoid-s14040" xml:space="preserve"> Quare i o q eſt recta linea.</s>
            <s xml:id="echoid-s14041" xml:space="preserve"> Igitur i o r nõ erit recta:</s>
            <s xml:id="echoid-s14042" xml:space="preserve"> ſed i o r eſt imago lineę a q.</s>
            <s xml:id="echoid-s14043" xml:space="preserve">
              <lb/>
            Quare imago lineę a q erit curua.</s>
            <s xml:id="echoid-s14044" xml:space="preserve"> Poſito autẽ puncto b loco pũcti q, & aliquo pũcto lineæ a b poſito
              <lb/>
            loco pũcti b:</s>
            <s xml:id="echoid-s14045" xml:space="preserve"> erit eodẽ penitus modo probare, quòd imago lineæ a b eſt curua.</s>
            <s xml:id="echoid-s14046" xml:space="preserve"> Et hoc eſt propoſitũ.</s>
            <s xml:id="echoid-s14047" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div480" type="section" level="0" n="0">
          <figure number="172">
            <variables xml:id="echoid-variables162" xml:space="preserve">ſ k x b a s
              <gap/>
            t c q f m
              <gap/>
            o h z i g p d</variables>
          </figure>
          <head xml:id="echoid-head430" xml:space="preserve" style="it">17. Si uiſ{us} ſit extra ſuperficem incidentiæ: imago lineæ rectæ infinitæ, ſecantis inæquabili-
            <lb/>
          ter peripheriam circuli (qui eſt communis ſectio
            <lb/>
          ſuperficierum, reflexionis & ſpeculi ſphærici con- uexi) uidebitur curua. 52 p 6.</head>
          <p>
            <s xml:id="echoid-s14048" xml:space="preserve">SI uerò a b ſecet circulum:</s>
            <s xml:id="echoid-s14049" xml:space="preserve"> ſecet in puncto e:</s>
            <s xml:id="echoid-s14050" xml:space="preserve"> m
              <lb/>
            finis contingentiæ lineæ contingentis circulũ
              <lb/>
            e h z, à puncto f productæ ad lineã b g:</s>
            <s xml:id="echoid-s14051" xml:space="preserve"> b igitur
              <lb/>
            reflectitur ad d ab aliquo pũcto arcus h p.</s>
            <s xml:id="echoid-s14052" xml:space="preserve"> Arcus ab
              <lb/>
            illo puncto reflexionis uſq;</s>
            <s xml:id="echoid-s14053" xml:space="preserve"> ad h, aut eſt æqualis ar-
              <lb/>
            cui h e:</s>
            <s xml:id="echoid-s14054" xml:space="preserve"> aut maior:</s>
            <s xml:id="echoid-s14055" xml:space="preserve"> aut minor.</s>
            <s xml:id="echoid-s14056" xml:space="preserve"> Si æqualis:</s>
            <s xml:id="echoid-s14057" xml:space="preserve"> palàm
              <lb/>
            quòd arcus ille eſt æqualis arcui h f [ut patuit præ-
              <lb/>
            cedente numero.</s>
            <s xml:id="echoid-s14058" xml:space="preserve">] Sit q punctum circuli, in quod
              <lb/>
            cadit contingẽs ducta à puncto m exparte e.</s>
            <s xml:id="echoid-s14059" xml:space="preserve"> Igitur
              <lb/>
            a e tranſit per punctũ q:</s>
            <s xml:id="echoid-s14060" xml:space="preserve"> & ita m q ſecat a e per pun-
              <lb/>
            ctum e [quia in hoc caſu q & e coniungũtur, unũq́;</s>
            <s xml:id="echoid-s14061" xml:space="preserve">
              <lb/>
            punctum fiũt.</s>
            <s xml:id="echoid-s14062" xml:space="preserve">] Si uerò arcus ille minor eſt arcu h e:</s>
            <s xml:id="echoid-s14063" xml:space="preserve">
              <lb/>
            ſecabit quidem m q lineam a e ultra punctum q:</s>
            <s xml:id="echoid-s14064" xml:space="preserve"> ſe-
              <lb/>
            cet in t, ut efficiatur triangulum e q t.</s>
            <s xml:id="echoid-s14065" xml:space="preserve"> Si uerò arcus ille fuerit maior arcu h e:</s>
            <s xml:id="echoid-s14066" xml:space="preserve"> ſecabit quidem linea in
              <lb/>
            q lineam a e citra punctum q.</s>
            <s xml:id="echoid-s14067" xml:space="preserve"> Siue hoc, ſiue illud fuerit:</s>
            <s xml:id="echoid-s14068" xml:space="preserve"> iteretur probatio, & eodem penitus modo
              <lb/>
            probabitur, quòd imago lineæ a b eſt curua.</s>
            <s xml:id="echoid-s14069" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s14070" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div481" type="section" level="0" n="0">
          <figure number="173">
            <variables xml:id="echoid-variables163" xml:space="preserve">d a b e h z g</variables>
          </figure>
          <head xml:id="echoid-head431" xml:space="preserve" style="it">18. Si uiſ{us} ſit in ſuperficie incidentiæ, extra rectam lineam infinitam per centrum circuli
            <lb/>
          (qui eſt communis ſectio ſuperficierum, reflexionis & ſpeculi ſphæ-
            <lb/>
          riciconuexi) trãſeuntis: imago illi{us} lineæ uidebitur recta. 53 p 6.</head>
          <p>
            <s xml:id="echoid-s14071" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s14072" xml:space="preserve"> ſi in ſuperficie, in qua ſunt linea uiſa, & cẽtrum ſphæ-
              <lb/>
            ræ, fuerit uiſus:</s>
            <s xml:id="echoid-s14073" xml:space="preserve"> (ſuperiora enim dicta ſunt, non exiſtente uiſu
              <lb/>
            in illa ſuperficie) linea uiſa recta, aut concurret cum circulo
              <lb/>
            communi illi ſuperficiei & ſpeculo:</s>
            <s xml:id="echoid-s14074" xml:space="preserve"> aut non concurret.</s>
            <s xml:id="echoid-s14075" xml:space="preserve"> Si concurrat:</s>
            <s xml:id="echoid-s14076" xml:space="preserve">
              <lb/>
            angulus illarum linearum [quem nimirum efficiunt diameter opti-
              <lb/>
            ca g d & data recta a b continuata per centrum g] cadet ſuper centrũ
              <lb/>
            ſpeculi:</s>
            <s xml:id="echoid-s14077" xml:space="preserve"> quæ quidem linea uidebitur recta.</s>
            <s xml:id="echoid-s14078" xml:space="preserve"> Imago enim cuiuslibet
              <lb/>
            puncti illius lineæ apparet in ipſa linea [per 6 n 5.</s>
            <s xml:id="echoid-s14079" xml:space="preserve">] Et ita imago il-
              <lb/>
            lius lineæ eſt recta.</s>
            <s xml:id="echoid-s14080" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div482" type="section" level="0" n="0">
          <head xml:id="echoid-head432" xml:space="preserve" style="it">19. Si uiſ{us} ſit in ſuperficie incidẽtiæ: imago lineæ rectæ, infini-
            <lb/>
          tæ peripheriam circuli (qui eſt communis ſectio ſuperficierum, re-
            <lb/>
          flexionis & ſpeculi ſphærici conuexi) tangentis, & ad partem ui-
            <lb/>
          ſui oppoſitam obliquatæ, uidebitur punctum. 54 p 6.</head>
          <p>
            <s xml:id="echoid-s14081" xml:space="preserve">SI uerò linea ꝓpoſita declinata fuerit:</s>
            <s xml:id="echoid-s14082" xml:space="preserve"> aut erit declinatio ex par-
              <lb/>
            te uiſus:</s>
            <s xml:id="echoid-s14083" xml:space="preserve"> aut ex alia parte.</s>
            <s xml:id="echoid-s14084" xml:space="preserve"> Si ex alia parte:</s>
            <s xml:id="echoid-s14085" xml:space="preserve"> ſumatur punctum cir-
              <lb/>
            culi, à quo reflectatur aliquid uiſum:</s>
            <s xml:id="echoid-s14086" xml:space="preserve"> [per 39 n 5] & ſumatur li-
              <lb/>
            nea reflexionis aliqua.</s>
            <s xml:id="echoid-s14087" xml:space="preserve"> Aliqua linearum declinatarum cadet forſitan
              <lb/>
            ſuper hanc lineam reflexionis:</s>
            <s xml:id="echoid-s14088" xml:space="preserve"> quòd ſi fuerit:</s>
            <s xml:id="echoid-s14089" xml:space="preserve"> non uidebitur quidem hæc linea declinata, niſi ſecun-
              <lb/>
            dum unum punctum [ducta enim a g ſecante peripheriam circuli in puncto z:</s>
            <s xml:id="echoid-s14090" xml:space="preserve"> peripheria inter
              <lb/>
            punctum, à quo b reflectitur, & punctum z, continebit puncta reflexionis totius lineæ a b, ut pa-
              <lb/>
            tuit 16 n.</s>
            <s xml:id="echoid-s14091" xml:space="preserve">] Protracta igitur à centro uiſus ad centrum ſpeculi linea:</s>
            <s xml:id="echoid-s14092" xml:space="preserve"> ſumatur in arcu circuli citra
              <lb/>
            hanc lineam punctum, à quo reflectatur ad uiſum aliquod punctum lineæ declinatæ:</s>
            <s xml:id="echoid-s14093" xml:space="preserve"> ſed illud
              <lb/>
            punctum reflectitur à puncto prius aſsignato, quod eſt terminus lineæ reflexionis, cum li-
              <lb/>
            nea declinata ſit ſupra lineam reflexionis.</s>
            <s xml:id="echoid-s14094" xml:space="preserve"> Et ita illud punctum lineæ declinatæ reflectitur ad
              <lb/>
            </s>
          </p>
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