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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        <div xml:id="echoid-div487" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s14232" xml:space="preserve">
              <pb o="205" file="0211" n="211" rhead="OPTICAE LIBER VI."/>
            te d:</s>
            <s xml:id="echoid-s14233" xml:space="preserve"> aut ex parte h g.</s>
            <s xml:id="echoid-s14234" xml:space="preserve"> Sit ex parte d:</s>
            <s xml:id="echoid-s14235" xml:space="preserve"> & ſit concurſus in puncto c:</s>
            <s xml:id="echoid-s14236" xml:space="preserve"> erit z q t linea recta:</s>
            <s xml:id="echoid-s14237" xml:space="preserve"> quare z q r erit
              <lb/>
            curua.</s>
            <s xml:id="echoid-s14238" xml:space="preserve"> Et ita imago lineæ h e curua.</s>
            <s xml:id="echoid-s14239" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s14240" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div489" type="section" level="0" n="0">
          <head xml:id="echoid-head436" xml:space="preserve" style="it">23. Imago peripheriæ cum uiſu in eodem planoſitæ, intra ſpeculum ſphæricum conuexum ſen
            <lb/>
          ſiliter uiſa, curua uidetur. 58. 62 p 6.</head>
          <p>
            <s xml:id="echoid-s14241" xml:space="preserve">SI uerò proponatur arcus extra ſpeculum:</s>
            <s xml:id="echoid-s14242" xml:space="preserve"> erit probare de eo, quòd imago ſit curua, ſicut proba
              <lb/>
            tum eſt, uiſu non exiſtente in eadẽ ſuperficie cũ arcu & centro ſpeculi.</s>
            <s xml:id="echoid-s14243" xml:space="preserve"> Et hoc eſt propoſitum.</s>
            <s xml:id="echoid-s14244" xml:space="preserve">
              <lb/>
            Igitur in his ſpeculis lineæ rectæ apparent curuæ, & ſimiliter curuæ apparẽt ſimiliter curuę.</s>
            <s xml:id="echoid-s14245" xml:space="preserve"> Si
              <lb/>
            autem proponatur uiſui in his ſpeculis corpus curuũ, ſed longũ, modicam habens latitudinem:</s>
            <s xml:id="echoid-s14246" xml:space="preserve"> ap-
              <lb/>
            parebit quidem corporis illius curuitas manifeſtè, cũ ipſa diſcerni poſsit per ea, quæ ſupra corpus
              <lb/>
            ſunt, aut infra.</s>
            <s xml:id="echoid-s14247" xml:space="preserve"> Non enim planè diſcernitur curuitas, niſi magna, ubi occultæ fuerint extremitates lõ
              <lb/>
            gitudinis & latitudinis.</s>
            <s xml:id="echoid-s14248" xml:space="preserve"> Vnde propoſito uiſui corpore conuexitatis modicæ & quantitatis magnæ,
              <lb/>
            nõ planè diſcernitur eius conuexitas, licet imago ipſius ſit cõuexa, cũ non appareant termini cor-
              <lb/>
            poris in longitudine uel latitudine.</s>
            <s xml:id="echoid-s14249" xml:space="preserve"> Amplius:</s>
            <s xml:id="echoid-s14250" xml:space="preserve"> errores in ſpeculis planis accidentes, omnes accidunt
              <lb/>
            & in his:</s>
            <s xml:id="echoid-s14251" xml:space="preserve"> & præterillos, accidit imagines linearum rectarum eſſe curuas:</s>
            <s xml:id="echoid-s14252" xml:space="preserve"> quod à ſpeculis planis eſt
              <lb/>
            remotum.</s>
            <s xml:id="echoid-s14253" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div490" type="section" level="0" n="0">
          <head xml:id="echoid-head437" xml:space="preserve">DE ERRORIBVS, QVI ACCIDVNT IN SPECVLIS CO-
            <lb/>
          lumnaribus conuexis. Cap. V.</head>
          <head xml:id="echoid-head438" xml:space="preserve" style="it">24. Si à duob{us} ellipſis cylindraceæ punctis ſint duæ perpendiculares: prima axi, continens
            <lb/>
          cum recta à ſecundo puncto, ad idem axis punctum ducta acutum angulum: ſecunda rectæ el-
            <lb/>
          lipſin in ſecundo puncto tangenti: ultra axem & dictum acutum angulum concurrent. 114
            <lb/>
          p 1. 44 p 7.</head>
          <p>
            <s xml:id="echoid-s14254" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s14255" xml:space="preserve"> in ſpeculis columnaribus exterioribus errores accidunt ijdem, qui in ſpeculis ſphę-
              <lb/>
            ricis exterioribus.</s>
            <s xml:id="echoid-s14256" xml:space="preserve"> Lineæ enim rectæ uidentur curuæ, & diminuta apparet rei quantitas:</s>
            <s xml:id="echoid-s14257" xml:space="preserve"> ſed
              <lb/>
            longè fortius in his, quàm in eis.</s>
            <s xml:id="echoid-s14258" xml:space="preserve"> Quoniam in ſphęricis res magna apparebit quidem minor,
              <lb/>
            ſed non multò minor:</s>
            <s xml:id="echoid-s14259" xml:space="preserve"> ſed in his res etiam maxima uidebitur minima.</s>
            <s xml:id="echoid-s14260" xml:space="preserve"> Similiter linea recta appare-
              <lb/>
            bit curua in ſpeculis ſphæricis, ſed modicæ curuitatis:</s>
            <s xml:id="echoid-s14261" xml:space="preserve"> in columnaribus-maximæ curuitatis.</s>
            <s xml:id="echoid-s14262" xml:space="preserve"> Vnde
              <lb/>
            multiplicantur errores columnaris ſpeculi ſuper errores ſphærici.</s>
            <s xml:id="echoid-s14263" xml:space="preserve"> Verùm in columnaribus aliquan
              <lb/>
            do fit reflexio à linea recta, ſcilicet à longitudine ſpeculi:</s>
            <s xml:id="echoid-s14264" xml:space="preserve"> aliquando à circulo:</s>
            <s xml:id="echoid-s14265" xml:space="preserve"> aliquando à ſectione.</s>
            <s xml:id="echoid-s14266" xml:space="preserve">
              <lb/>
            Quando linea uiſa fuerit æquidiſtans longitudini ſpeculi, fiet reflexio à linea longitudinis:</s>
            <s xml:id="echoid-s14267" xml:space="preserve"> & linea
              <lb/>
            uiſa apparebit recta, modicæ curuitatis.</s>
            <s xml:id="echoid-s14268" xml:space="preserve"> Et hæc quidem probabuntur:</s>
            <s xml:id="echoid-s14269" xml:space="preserve"> ad quorum probationẽ ne-
              <lb/>
            ceſſe quiddam præmitti:</s>
            <s xml:id="echoid-s14270" xml:space="preserve"> quod huiuſmodi eſt.</s>
            <s xml:id="echoid-s14271" xml:space="preserve"> Sumpta columnari ſectione, & ſumpto in ea puncto,
              <lb/>
            quod non ſit punctum reflexionis:</s>
            <s xml:id="echoid-s14272" xml:space="preserve"> ſi ab illo puncto ducatur linea ad perpendicularẽ, quæ eſt à pun-
              <lb/>
            cto reflexionis ad axem, & linea illa faciat angulum acutum cum perpendiculari:</s>
            <s xml:id="echoid-s14273" xml:space="preserve"> ſi ducatur à pun-
              <lb/>
            cto ſumpto linea, quæ ſit orthogonalis ſuper contingentem illud punctum:</s>
            <s xml:id="echoid-s14274" xml:space="preserve"> hæc linea concurret cũ
              <lb/>
            perpendiculari ſub axe, & ſub concurſu prioris lineæ cum perpendiculari.</s>
            <s xml:id="echoid-s14275" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s14276" xml:space="preserve"> ſit a e b ſe-
              <lb/>
            ctio:</s>
            <s xml:id="echoid-s14277" xml:space="preserve"> e punctum datum:</s>
            <s xml:id="echoid-s14278" xml:space="preserve"> n punctum uiſum:</s>
            <s xml:id="echoid-s14279" xml:space="preserve"> b punctum reflexionis:</s>
            <s xml:id="echoid-s14280" xml:space="preserve"> b d perpẽdicularis:</s>
            <s xml:id="echoid-s14281" xml:space="preserve"> e d b angulus
              <lb/>
            acutus:</s>
            <s xml:id="echoid-s14282" xml:space="preserve"> q e l contingens.</s>
            <s xml:id="echoid-s14283" xml:space="preserve"> Super b fiat circulus æquidiſtans baſi columnæ [ut oſtenſum eſt 47 n 5] ſci
              <lb/>
            licet b t o:</s>
            <s xml:id="echoid-s14284" xml:space="preserve"> & ducatur à puncto e linea longitudinis columnæ [ut eodem numero demonſtratũ eſt]
              <lb/>
            ſcilicet e t:</s>
            <s xml:id="echoid-s14285" xml:space="preserve"> ducatur axis d h:</s>
            <s xml:id="echoid-s14286" xml:space="preserve"> & [per 11 p 1] ducatur linea d g perpendicularis ſuper lineam b d, in ſu-
              <lb/>
            perficie circuli.</s>
            <s xml:id="echoid-s14287" xml:space="preserve"> Palàm, quod ſuperficies h d g eſt orthogonalis ſuper ſuperficiem circuli [per 18 p 11:</s>
            <s xml:id="echoid-s14288" xml:space="preserve">
              <lb/>
            quia ducitur per axem perpendicularem circulo per 21 d 11.</s>
            <s xml:id="echoid-s14289" xml:space="preserve">] Superficies uerò contingens columnã
              <lb/>
            in puncto b, erit æquidiſtans huic ſuperficiei:</s>
            <s xml:id="echoid-s14290" xml:space="preserve"> quoniam linea longitudinis ducta à puncto b eſt ęqui
              <lb/>
            diſtans axi [per 21 d 11.</s>
            <s xml:id="echoid-s14291" xml:space="preserve">] Et contingens circulum ſuper b eſt æquidiſtans d g [per 28 p 1:</s>
            <s xml:id="echoid-s14292" xml:space="preserve"> recti enim
              <lb/>
            ſunt anguli g d b per fabricationem, & comprehenſus ſub tangente in puncto b & ſemidiametro cir
              <lb/>
            culi d b per 18 p 3.</s>
            <s xml:id="echoid-s14293" xml:space="preserve">] Igitur ſuperficies, in qua ſunt lineæ l e, e t non eſt æquidiſtans ſuperficiei h d g
              <lb/>
            [quia non eſt parallela ſuperficiei tangenti ellipſin in puncto b:</s>
            <s xml:id="echoid-s14294" xml:space="preserve"> cum angulus e d b ſit acutus ex the-
              <lb/>
            ſi.</s>
            <s xml:id="echoid-s14295" xml:space="preserve">] Concurretigitur cum ea.</s>
            <s xml:id="echoid-s14296" xml:space="preserve"> Concurrat in linea l g:</s>
            <s xml:id="echoid-s14297" xml:space="preserve"> & ducatur linea t g:</s>
            <s xml:id="echoid-s14298" xml:space="preserve"> quæ quidem erit contin-
              <lb/>
            gens:</s>
            <s xml:id="echoid-s14299" xml:space="preserve"> cum ſuperficies l e t ſit contingens.</s>
            <s xml:id="echoid-s14300" xml:space="preserve"> Du-
              <lb/>
              <figure xlink:label="fig-0211-01" xlink:href="fig-0211-01a" number="180">
                <variables xml:id="echoid-variables170" xml:space="preserve">n q e ſ g t f m o K d h c a s u p z b</variables>
              </figure>
            cta autem linea t d:</s>
            <s xml:id="echoid-s14301" xml:space="preserve"> erit angulus g t d rectus:</s>
            <s xml:id="echoid-s14302" xml:space="preserve">
              <lb/>
            [per 18 p 3] quoniam t d diameter, [& t g tan-
              <lb/>
            git peripheriam in ipſius termino t.</s>
            <s xml:id="echoid-s14303" xml:space="preserve">] Fiat au-
              <lb/>
            tem ſuper e circulus æquidiſtans baſi colu-
              <lb/>
            mnæ [ut demonſtratum eſt 47 n 5] ſcilicet e
              <lb/>
            s p:</s>
            <s xml:id="echoid-s14304" xml:space="preserve"> punctum axis in hoc circulo ſit k:</s>
            <s xml:id="echoid-s14305" xml:space="preserve"> & du-
              <lb/>
            catur linea k e.</s>
            <s xml:id="echoid-s14306" xml:space="preserve"> Ducatur etiam linea d l:</s>
            <s xml:id="echoid-s14307" xml:space="preserve"> quæ
              <lb/>
            quidem ſecabit ſuperficiem circuli e s p:</s>
            <s xml:id="echoid-s14308" xml:space="preserve"> ſe-
              <lb/>
            cet in puncto f:</s>
            <s xml:id="echoid-s14309" xml:space="preserve"> ubicunque ſit punctum extra
              <lb/>
            circumferentiam uel intra:</s>
            <s xml:id="echoid-s14310" xml:space="preserve"> & ducantur lineæ
              <lb/>
            k f, e f:</s>
            <s xml:id="echoid-s14311" xml:space="preserve"> & [per 11 p 11] à puncto f ducaturper-
              <lb/>
            pendicularis ſuper ſuperficiem circuli b t o:</s>
            <s xml:id="echoid-s14312" xml:space="preserve"> quæſit f m:</s>
            <s xml:id="echoid-s14313" xml:space="preserve"> & ducatur linea t m.</s>
            <s xml:id="echoid-s14314" xml:space="preserve"> Palàm, quòd k d æqui-
              <lb/>
            </s>
          </p>
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