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-s16011" xml:space="preserve">
              <pb o="225" file="0231" n="231" rhead="OPTICAE LIBER VI."/>
            linea z o e conuexa, comprehendetur concaua:</s>
            <s xml:id="echoid-s16012" xml:space="preserve"> & r u f concaua:</s>
            <s xml:id="echoid-s16013" xml:space="preserve"> conuexa.</s>
            <s xml:id="echoid-s16014" xml:space="preserve"> Si ergo unaquęque linea
              <lb/>
            rum z u e, z o e, r u f habuerit unam imaginem:</s>
            <s xml:id="echoid-s16015" xml:space="preserve">
              <lb/>
              <figure xlink:label="fig-0231-01" xlink:href="fig-0231-01a" number="201">
                <variables xml:id="echoid-variables190" xml:space="preserve">k q p
                  <gap/>
                t ſ n g b o r f e u m z d h a</variables>
              </figure>
            tunc forma illarum linearum erit eodem modo,
              <lb/>
            quo declarauimus:</s>
            <s xml:id="echoid-s16016" xml:space="preserve"> & ſi habuerit alias imagines:</s>
            <s xml:id="echoid-s16017" xml:space="preserve">
              <lb/>
            fortè erunt ſimiles alijs imaginibus, & fortè di-
              <lb/>
            uerſæ.</s>
            <s xml:id="echoid-s16018" xml:space="preserve"> Patet ergo ex iſtis figuris, quòd lineę re-
              <lb/>
            ctæ in ſpeculis concauis quandoque compre-
              <lb/>
            henduntur rectæ:</s>
            <s xml:id="echoid-s16019" xml:space="preserve"> quandoque conuexæ:</s>
            <s xml:id="echoid-s16020" xml:space="preserve"> quan-
              <lb/>
            doque concauæ:</s>
            <s xml:id="echoid-s16021" xml:space="preserve"> & lineæ conuexæ quandoque
              <lb/>
            comprehenduntur conuexæ:</s>
            <s xml:id="echoid-s16022" xml:space="preserve"> quandoque con-
              <lb/>
            cauę:</s>
            <s xml:id="echoid-s16023" xml:space="preserve"> & concauę quandoque comprehendun-
              <lb/>
            tur conuexę:</s>
            <s xml:id="echoid-s16024" xml:space="preserve"> quandoque concauę.</s>
            <s xml:id="echoid-s16025" xml:space="preserve"> Formę ergo
              <lb/>
            ſuperficierum uiſibilium comprehenduntur a-
              <lb/>
            liter, quàm ſunt, in huiuſmodi ſpeculis.</s>
            <s xml:id="echoid-s16026" xml:space="preserve"> Nam li-
              <lb/>
            neę rectę non ſunt, niſi in ſuperficiebus rectis:</s>
            <s xml:id="echoid-s16027" xml:space="preserve"> &
              <lb/>
            cum linea recta, quę exiſtit in ſuperficie plana,
              <lb/>
            comprehenditur conuexa aut concaua:</s>
            <s xml:id="echoid-s16028" xml:space="preserve"> tunc ſu-
              <lb/>
            perficies, in qua ipſa linea eſt, comprehendetur
              <lb/>
            conuexa aut concaua.</s>
            <s xml:id="echoid-s16029" xml:space="preserve"> Cum ergo uiſus compre-
              <lb/>
            hendat lineas conuexas & concauas, & rectas
              <lb/>
            aliter, quàm ſint:</s>
            <s xml:id="echoid-s16030" xml:space="preserve"> comprehendet ſuperficies, in
              <lb/>
            quibus ſunt, aliter, quàm ſint.</s>
            <s xml:id="echoid-s16031" xml:space="preserve"> Patet ergo ex prę
              <lb/>
            dictis, quòd in omnibus, quæ in ſpeculis con-
              <lb/>
            cauis comprehenduntur, accidit fallacia:</s>
            <s xml:id="echoid-s16032" xml:space="preserve"> ſed in
              <lb/>
            quibuſdam accidit ſemper, & in omni poſitio-
              <lb/>
            ne, in quibuſdam accidit in aliqua poſitione.</s>
            <s xml:id="echoid-s16033" xml:space="preserve"> Fal
              <lb/>
            lacię autem compoſitæ accidunt in his ſpeculis
              <lb/>
            eo modo, quo incompoſitæ.</s>
            <s xml:id="echoid-s16034" xml:space="preserve"> Et hoc uoluimus
              <lb/>
            declarare.</s>
            <s xml:id="echoid-s16035" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div536" type="section" level="0" n="0">
          <head xml:id="echoid-head467" xml:space="preserve">DE ERRORIBVS, QVI ACCI-
            <lb/>
          dunt in ſpeculis columnaribus
            <lb/>
          concauis. Cap. VIII.</head>
          <p>
            <s xml:id="echoid-s16036" xml:space="preserve">IN his autem accidunt ſimiles eis, qui accidũt in ſphęricis concauis.</s>
            <s xml:id="echoid-s16037" xml:space="preserve"> Accidunt enim fallaciæ, quę
              <lb/>
            proueniunt ex reflexione, ſcilicet debilitas lucis & coloris:</s>
            <s xml:id="echoid-s16038" xml:space="preserve"> & diuerſitas ſitus, & remotionis, quę
              <lb/>
            accidunt omnibus ſpeculis.</s>
            <s xml:id="echoid-s16039" xml:space="preserve"> Accidit autem eis ex diuerſitate quantitatis ſimile illi, quod accidit
              <lb/>
            in ſpeculis ſphęricis concauis.</s>
            <s xml:id="echoid-s16040" xml:space="preserve"> Et uidetur etiam unum uiſibile, unum:</s>
            <s xml:id="echoid-s16041" xml:space="preserve"> & duo:</s>
            <s xml:id="echoid-s16042" xml:space="preserve"> & tria:</s>
            <s xml:id="echoid-s16043" xml:space="preserve"> & quatuor:</s>
            <s xml:id="echoid-s16044" xml:space="preserve"> &
              <lb/>
            rectum & conuexum ſecundum diuerſos ſitus:</s>
            <s xml:id="echoid-s16045" xml:space="preserve"> & planum uidetur concauum & conuexum.</s>
            <s xml:id="echoid-s16046" xml:space="preserve"> Oſten
              <lb/>
            demus ergo qualiter in his ſpeculis diuerſatur quantitas & numerus rei uiſæ:</s>
            <s xml:id="echoid-s16047" xml:space="preserve"> & qualiter apparet re
              <lb/>
            ctum & conuerſum eo modo, quo in ſpeculis ſphęricis concauis declarauimus.</s>
            <s xml:id="echoid-s16048" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div537" type="section" level="0" n="0">
          <head xml:id="echoid-head468" xml:space="preserve" style="it">51. Siuiſ{us} ſit extra planũ lineærectæ, parallelæ axi ſpeculi cylindraceicaui: imago aliàs ui-
            <lb/>
          debitur recta & maior ipſa linea: aliâs caua: aliâs cõuexa: aliâs ſimplex: aliâs multiplex. 25 p 9.</head>
          <p>
            <s xml:id="echoid-s16049" xml:space="preserve">ITeremus ergo primam figuram ex duabus figuris pręmiſsis in fallacijs ſpeculorum columnariũ
              <lb/>
            conuexorum, & ijſdem literis.</s>
            <s xml:id="echoid-s16050" xml:space="preserve"> In illa autem figura [quę eſt 26 n] patuit, quòd lineę e g, g t, e b, q b,
              <lb/>
            e a, a h reflectuntur ſecundum angulos æquales:</s>
            <s xml:id="echoid-s16051" xml:space="preserve"> & quòd lineę e k, h a, q b, t g coniunguntur in o:</s>
            <s xml:id="echoid-s16052" xml:space="preserve">
              <lb/>
            & quòd linea a b g eſt linea recta extenſa in longitudine ſpeculi:</s>
            <s xml:id="echoid-s16053" xml:space="preserve"> & quòd lineę g z, b l, a d ſunt perpẽ-
              <lb/>
            diculares ſuper ſuperficiẽ, contingentẽ ſuperficiem, quæ trãſit per lineã a b g:</s>
            <s xml:id="echoid-s16054" xml:space="preserve"> & quòd linea a b g eſt
              <lb/>
            perpẽdicularis ſuք ſuperficiẽ, in qua eſt triãgulũ e b o:</s>
            <s xml:id="echoid-s16055" xml:space="preserve"> & quòd linea t q eſt æqualis q h, & a b ęqualis
              <lb/>
            b g:</s>
            <s xml:id="echoid-s16056" xml:space="preserve"> & quòd s c, i ſunt imagines h, q, t:</s>
            <s xml:id="echoid-s16057" xml:space="preserve"> & quòd c eſt propinquius puncto e, quàm linea s i:</s>
            <s xml:id="echoid-s16058" xml:space="preserve"> & quòd li-
              <lb/>
            nea s i eſt in ſuperficie trianguli u h t:</s>
            <s xml:id="echoid-s16059" xml:space="preserve"> & quòd duæ lineæ u h, u t ſunt æquales:</s>
            <s xml:id="echoid-s16060" xml:space="preserve"> & quòd u s & u i ſunt
              <lb/>
            æquales:</s>
            <s xml:id="echoid-s16061" xml:space="preserve"> & quòd duæ lineæ e s, e i ſunt æquales.</s>
            <s xml:id="echoid-s16062" xml:space="preserve"> Et continuemus c u:</s>
            <s xml:id="echoid-s16063" xml:space="preserve"> & ſecet s i in æ:</s>
            <s xml:id="echoid-s16064" xml:space="preserve"> diuidet ergo i-
              <lb/>
            pſam in duo æqualia:</s>
            <s xml:id="echoid-s16065" xml:space="preserve"> nam h t eſt diuiſa in duo æqualia in q:</s>
            <s xml:id="echoid-s16066" xml:space="preserve"> [& linea i s parallela eſt ipſi t h:</s>
            <s xml:id="echoid-s16067" xml:space="preserve"> quia cũ
              <lb/>
            tota t u æqualis concluſa ſit toti h u, & pars i u parti s u:</s>
            <s xml:id="echoid-s16068" xml:space="preserve"> erit reliqua t i æqualis reliquæ h s:</s>
            <s xml:id="echoid-s16069" xml:space="preserve"> eſt igitur
              <lb/>
            per 7 p 5, ut u i ad i t, ſic u s ad s h:</s>
            <s xml:id="echoid-s16070" xml:space="preserve"> ergo per 2 p 6 h t & s i ſunt parallelæ.</s>
            <s xml:id="echoid-s16071" xml:space="preserve"> Itaque triangula t u q, i u æ:</s>
            <s xml:id="echoid-s16072" xml:space="preserve"> i-
              <lb/>
            tem q u h, æ u s ſunt æquiãgula per 29 p 1:</s>
            <s xml:id="echoid-s16073" xml:space="preserve"> & per 4 p 6, ut t q ad q u, ſic i æ ad æ u:</s>
            <s xml:id="echoid-s16074" xml:space="preserve"> & ut q u ad q h, ſic ę u
              <lb/>
            ad ę s:</s>
            <s xml:id="echoid-s16075" xml:space="preserve"> ergo per 22 p 5, ut t q ad q h, ſic i æ ad æ s.</s>
            <s xml:id="echoid-s16076" xml:space="preserve"> Quare cũ 26.</s>
            <s xml:id="echoid-s16077" xml:space="preserve"> 27 n, t q ęquata ſit ipſi q h:</s>
            <s xml:id="echoid-s16078" xml:space="preserve"> ęquabitur i æ
              <lb/>
            ipſi ę s] & erit c u in ſuperficie trianguli q u e, quæ eſt ſuperficies circuli b f, ęquidiſtantis baſi ſpecu-
              <lb/>
            li:</s>
            <s xml:id="echoid-s16079" xml:space="preserve"> ergo c erit in ſuperficie trianguli c u e:</s>
            <s xml:id="echoid-s16080" xml:space="preserve"> & eſt in ſuperficie trianguli c e i:</s>
            <s xml:id="echoid-s16081" xml:space="preserve"> ergo c eſt in linea, quæ eſt
              <lb/>
            differentia cõmunis his duabus ſuperficieb.</s>
            <s xml:id="echoid-s16082" xml:space="preserve"> ſed hęc differẽtia eſt linea e b:</s>
            <s xml:id="echoid-s16083" xml:space="preserve"> [ք 3 p 11] ergo c eſt in recti
              <lb/>
            tudine e b:</s>
            <s xml:id="echoid-s16084" xml:space="preserve"> & duę lineę h u, t u ſunt ſub duob.</s>
            <s xml:id="echoid-s16085" xml:space="preserve"> pũctis d, z:</s>
            <s xml:id="echoid-s16086" xml:space="preserve"> nã duę lineę h u, t u ſunt perpẽdiculares exe
              <lb/>
            untes ex h, t ſuper duas lineas, cõtingẽtes duas portiones, in quarũ circuferẽtia ſunt puncta a, g.</s>
            <s xml:id="echoid-s16087" xml:space="preserve"> Su-
              <lb/>
            perficies ergo triãguli u h t eſt ſub axe d l z.</s>
            <s xml:id="echoid-s16088" xml:space="preserve"> Sed nullũ pũctũ huius axis, quãuis exeat in infinitũ, erit
              <lb/>
            in ſuperficie trianguli u h t.</s>
            <s xml:id="echoid-s16089" xml:space="preserve"> Nam ſi eſſet:</s>
            <s xml:id="echoid-s16090" xml:space="preserve"> tunc ſi continuaretur cũ aliquo puncto lineæ h t linea re-
              <lb/>
            </s>
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