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-s14487" xml:space="preserve">
              <pb o="208" file="0214" n="214" rhead="ALHAZEN"/>
            neæ t i, h s cõcurrent ſuper idem punctũ lineæ e o.</s>
            <s xml:id="echoid-s14488" xml:space="preserve"> Concurrant in puncto u.</s>
            <s xml:id="echoid-s14489" xml:space="preserve"> Erit ergo t u h triangu-
              <lb/>
            lum, & in ſuperficie huius trianguli erit linea i s.</s>
            <s xml:id="echoid-s14490" xml:space="preserve"> Axis autem non eſt in eadem ſuperficie:</s>
            <s xml:id="echoid-s14491" xml:space="preserve"> uerùm t h
              <lb/>
            eſt in eadem ſuperficie cum axe.</s>
            <s xml:id="echoid-s14492" xml:space="preserve"> [ex theſi.</s>
            <s xml:id="echoid-s14493" xml:space="preserve">] Igitur ſuperficies illa ſecat ſuperficiem trianguli, ſuper
              <lb/>
            lineam communem:</s>
            <s xml:id="echoid-s14494" xml:space="preserve"> quæ eſt t h, non ſuper aliam.</s>
            <s xml:id="echoid-s14495" xml:space="preserve"> Cum ergo punctum c ſit in ſuperficie lineæ t h &
              <lb/>
            axis, & non ſit in linea t h:</s>
            <s xml:id="echoid-s14496" xml:space="preserve"> non eſt in ſuperficie trianguli t u h:</s>
            <s xml:id="echoid-s14497" xml:space="preserve"> & duo puncta i, s ſunt in ſuperficie il-
              <lb/>
            lius trianguli.</s>
            <s xml:id="echoid-s14498" xml:space="preserve"> Quare linea i c s eſt linea curua:</s>
            <s xml:id="echoid-s14499" xml:space="preserve"> & imago lineæ t h erit curua.</s>
            <s xml:id="echoid-s14500" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s14501" xml:space="preserve">
              <lb/>
            Sed eius curuitas eſt modica:</s>
            <s xml:id="echoid-s14502" xml:space="preserve"> quia perpendicularis ducta à puncto c ad punctum ſectionis lineæ i s
              <lb/>
            & ſuperficiei circuli, eſt ualde parua.</s>
            <s xml:id="echoid-s14503" xml:space="preserve"> Et quantò maior fuerit linea uiſa, æquidiſtans lineæ longitudi
              <lb/>
            nis ſpeculi:</s>
            <s xml:id="echoid-s14504" xml:space="preserve"> tantò imago eius erit minus curua:</s>
            <s xml:id="echoid-s14505" xml:space="preserve"> & quantò minor, tantò magis.</s>
            <s xml:id="echoid-s14506" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div497" type="section" level="0" n="0">
          <figure number="183">
            <variables xml:id="echoid-variables173" xml:space="preserve">f d b g t e h e</variables>
          </figure>
          <head xml:id="echoid-head442" xml:space="preserve" style="it">28. Si uiſ{us} ſit in communi ſectione planorum, lineæ rectæ & axis ſpeculi cylindracei conuexi,
            <lb/>
          inter ſeperpendicularium: fiet reflexio à peripheria circuli, qui eſt
            <lb/>
          communis ſectio plani lineæ & ſuperficiei ſpeculi: & imago uidebi- tur curua. 52 p 7.</head>
          <p>
            <s xml:id="echoid-s14507" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s14508" xml:space="preserve"> ſi linea t h ſecet ſuperficiem, in qua ſunt centrum ui-
              <lb/>
            ſus & axis, & ſit orthogonalis ſuper eam.</s>
            <s xml:id="echoid-s14509" xml:space="preserve"> Viſus aut erit in illa
              <lb/>
            ſuperficie lineæ t h, ſecante orthogonaliter ſuperficiem axis
              <lb/>
            & uiſus:</s>
            <s xml:id="echoid-s14510" xml:space="preserve"> aut extra.</s>
            <s xml:id="echoid-s14511" xml:space="preserve"> Si fuerit in ſuperficie illa:</s>
            <s xml:id="echoid-s14512" xml:space="preserve"> aut ſupra lineam t h:</s>
            <s xml:id="echoid-s14513" xml:space="preserve"> aut
              <lb/>
            infra.</s>
            <s xml:id="echoid-s14514" xml:space="preserve"> Si ſupra, cum illa linea ſit corporalis, occultabit uiſui ſpeculũ:</s>
            <s xml:id="echoid-s14515" xml:space="preserve">
              <lb/>
            & ita non reflectetur, ſed forſan capita eius apparebunt & reflecten-
              <lb/>
            tur à circulo columnæ, qui communis eſt ſuperficiei lineæ t h, ſecanti
              <lb/>
            columnam, & columnæ.</s>
            <s xml:id="echoid-s14516" xml:space="preserve"> Et erit horum capitum imago, ſicut in ſphæ
              <lb/>
            ricis exterioribus [21 n.</s>
            <s xml:id="echoid-s14517" xml:space="preserve">] Similiter ſi uiſus fuerit ſub linea t h:</s>
            <s xml:id="echoid-s14518" xml:space="preserve"> occul-
              <lb/>
            tabitur pars eius propter caput, in quo eſt uiſus.</s>
            <s xml:id="echoid-s14519" xml:space="preserve"> Pars aũt lineæ uiſæ
              <lb/>
            reflectitur à circulo, eodẽ penitus modo, quo in exteriorib.</s>
            <s xml:id="echoid-s14520" xml:space="preserve"> ſphęricis.</s>
            <s xml:id="echoid-s14521" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div498" type="section" level="0" n="0">
          <head xml:id="echoid-head443" xml:space="preserve" style="it">29. Si uiſ{us} æquabiliter diſtans à terminis lineæ rectæ, ſit extra
            <lb/>
          eiuſdem planum, perpendiculare plano axis ſpeculi cylindracei cõ-
            <lb/>
          uexi: imago maximè curua uidebitur. 53 p 7.</head>
          <p>
            <s xml:id="echoid-s14522" xml:space="preserve">SI uerò uiſus fuerit extra ſuperficiem lineę t h, orthogonaliter ſe-
              <lb/>
            cantem ſuperficiem uiſus & axis:</s>
            <s xml:id="echoid-s14523" xml:space="preserve"> ſit e uiſus:</s>
            <s xml:id="echoid-s14524" xml:space="preserve"> & b g x columna:</s>
            <s xml:id="echoid-s14525" xml:space="preserve"> reflectetur h ad e ab aliquo pun-
              <lb/>
            cto columnæ:</s>
            <s xml:id="echoid-s14526" xml:space="preserve"> ſit à puncto b:</s>
            <s xml:id="echoid-s14527" xml:space="preserve"> & ſit t eiuſdem longitudinis à puncto e, cuius eſt h.</s>
            <s xml:id="echoid-s14528" xml:space="preserve"> Dico, quòd t
              <lb/>
            reflectetur ad e ab aliquo puncto columnæ.</s>
            <s xml:id="echoid-s14529" xml:space="preserve"> Et cum puncta h, t ſint eiuſdem ſitus & eiuſdem lon-
              <lb/>
            gitudinis à puncto e:</s>
            <s xml:id="echoid-s14530" xml:space="preserve"> erunt ſimiliter puncta reflexionum, ſcilicet b, g eiuſdem longitudinis & eiuſ-
              <lb/>
            dem ſitus à puncto e.</s>
            <s xml:id="echoid-s14531" xml:space="preserve"> Igitur duo puncta b, g erunt in circulo.</s>
            <s xml:id="echoid-s14532" xml:space="preserve"> Sit circulus b z g:</s>
            <s xml:id="echoid-s14533" xml:space="preserve"> eius centrum d:</s>
            <s xml:id="echoid-s14534" xml:space="preserve"> &
              <lb/>
            ducantur lineæ h b, b e, t g, g e:</s>
            <s xml:id="echoid-s14535" xml:space="preserve"> & à centro ducantur perpendiculares, ſuper contingentes circulum
              <lb/>
            in punctis b, g, ſcilicet d b o, d g s:</s>
            <s xml:id="echoid-s14536" xml:space="preserve"> & ducatur linea e d.</s>
            <s xml:id="echoid-s14537" xml:space="preserve"> Cum puncta h, e ſint eiuſdem ſitus & longitu
              <lb/>
            dinis, reſpectu e, & reſpectu d:</s>
            <s xml:id="echoid-s14538" xml:space="preserve"> & ſimiliter puncta b, g, eiuſdem ſitus, reſpectu e & reſpectu d:</s>
            <s xml:id="echoid-s14539" xml:space="preserve"> habe-
              <lb/>
            bunt lineæ h b, t g eundem ſitum, reſpectu lineæ e d.</s>
            <s xml:id="echoid-s14540" xml:space="preserve"> Et ita concurrent in idem punctum illius li-
              <lb/>
            neę.</s>
            <s xml:id="echoid-s14541" xml:space="preserve"> Sit concurſus in puncto l.</s>
            <s xml:id="echoid-s14542" xml:space="preserve"> Fiat linea longitudinis columnæ, [ut oſtenſum eſt 47 p 5] in qua
              <lb/>
            punctum z:</s>
            <s xml:id="echoid-s14543" xml:space="preserve"> & ſit hæc linea in ſuperficie uiſus & axis:</s>
            <s xml:id="echoid-s14544" xml:space="preserve"> quæ ſit a z:</s>
            <s xml:id="echoid-s14545" xml:space="preserve"> & ducantur lineæ l z n, d z c:</s>
            <s xml:id="echoid-s14546" xml:space="preserve"> q ſit
              <lb/>
            punctum lineæ t h, punctum ſcilicet, quod eſt in ſuperſicie uiſus & axis:</s>
            <s xml:id="echoid-s14547" xml:space="preserve"> & à puncto q ducatur linea
              <lb/>
            æquidiſtans lineę d z c [per 31 p 1] cadet quidem hęc linea ſuper axem:</s>
            <s xml:id="echoid-s14548" xml:space="preserve"> [per lemm a Procli ad 29
              <lb/>
            p 1] & l z n cadet in hanc lineam ſupra pũctum q.</s>
            <s xml:id="echoid-s14549" xml:space="preserve"> Cadat in punctum n.</s>
            <s xml:id="echoid-s14550" xml:space="preserve"> Palàm ex prædictis [12 n 4]
              <lb/>
            quòd angulus h b o ęqualis eſt o b e:</s>
            <s xml:id="echoid-s14551" xml:space="preserve"> ſed [per 15 p 1] angulus h b o æqualis eſt angulo l b d, per con-
              <lb/>
            trapoſitionem:</s>
            <s xml:id="echoid-s14552" xml:space="preserve"> & [per 32 p 1] angulus o b e æqualis eſt duobus angulis b e d, b d e:</s>
            <s xml:id="echoid-s14553" xml:space="preserve"> quia extrinſe-
              <lb/>
            cus.</s>
            <s xml:id="echoid-s14554" xml:space="preserve"> Ergo angulus l b d ęqualis eſt duobus angulis b e d, b d e.</s>
            <s xml:id="echoid-s14555" xml:space="preserve"> Fiat ergo angulus m b d æqualis
              <lb/>
            angulo b d e [per 23 p 1] remanet angulus m b l ęqualis angulo b e l.</s>
            <s xml:id="echoid-s14556" xml:space="preserve"> Quare ductus e m in m l æ-
              <lb/>
            qualis quadrato b m [triangula enim m e b, m b l ſunt ęquiangula:</s>
            <s xml:id="echoid-s14557" xml:space="preserve"> quia angulus m b l ęqualis con-
              <lb/>
            cluſus eſt angulo m e b, & communis utriuſque trianguli eſt b m e:</s>
            <s xml:id="echoid-s14558" xml:space="preserve"> reliquus igitur m l b ęquatur
              <lb/>
            reliquo l b e per 32 p 1.</s>
            <s xml:id="echoid-s14559" xml:space="preserve"> Quare per 4 p 6 erit, ut e m ad m b, ſic m b ad m l.</s>
            <s xml:id="echoid-s14560" xml:space="preserve"> Ergo per 17 p 6 rectangu-
              <lb/>
            lum comprehenſum ſub extremis e m & m l, ęquatur quadrato medię m b.</s>
            <s xml:id="echoid-s14561" xml:space="preserve">] Ducatur linea m z.</s>
            <s xml:id="echoid-s14562" xml:space="preserve">
              <lb/>
            Quoniam igitur angulus b d m maior eſt angulo z d m:</s>
            <s xml:id="echoid-s14563" xml:space="preserve"> [Nam propter ſimilem ſitum punctorum
              <lb/>
            reflexionis b & g, ęquatur angulus s d e angulo o d e:</s>
            <s xml:id="echoid-s14564" xml:space="preserve"> ſed angulus s d e maior eſt angulo z d m per
              <lb/>
            9 ax.</s>
            <s xml:id="echoid-s14565" xml:space="preserve"> Quare angulus o d e, id eſt, b d m maior eſt angulo z d m] & duo latera z d, d m ęqualia duo-
              <lb/>
            bus lateribus b d, d m:</s>
            <s xml:id="echoid-s14566" xml:space="preserve"> [ęquantur enim z d, b d per 15 d 1:</s>
            <s xml:id="echoid-s14567" xml:space="preserve"> & d m eſt communis] erit [per 24 p 1]
              <lb/>
            m b maior m z:</s>
            <s xml:id="echoid-s14568" xml:space="preserve"> quare ductus e m in m l maior eſt quadrato z m.</s>
            <s xml:id="echoid-s14569" xml:space="preserve"> Sit ductus e m in m i æqualis
              <lb/>
            quadrato m z:</s>
            <s xml:id="echoid-s14570" xml:space="preserve"> [per 11 p 6, ut demonſtratum eſt 6 n] & ducantur lineę i b, i z.</s>
            <s xml:id="echoid-s14571" xml:space="preserve"> Erit ergo angulus
              <lb/>
            m z i ęqualis angulo z e i, [eſt enim per proximam fabricationem & 17 p 6, ut e m ad m z, ſic m z ad
              <lb/>
            m i.</s>
            <s xml:id="echoid-s14572" xml:space="preserve"> Sunt igitur duo triangula e m z, i m z lateribus circa communem angulum i m z propor-
              <lb/>
            tionalia:</s>
            <s xml:id="echoid-s14573" xml:space="preserve"> itaque per 6 p 6 ſunt ęquiangula, & angulus m z i ęquatur angulo z e i.</s>
            <s xml:id="echoid-s14574" xml:space="preserve">] Quare m z l ma-
              <lb/>
            ior angulo z e d.</s>
            <s xml:id="echoid-s14575" xml:space="preserve"> Sed quoniam angulus m b d poſitus eſt ęqualis augulo b d m:</s>
            <s xml:id="echoid-s14576" xml:space="preserve"> erit [per 6 p 1]
              <lb/>
            linea m b ęqualis lineę m d:</s>
            <s xml:id="echoid-s14577" xml:space="preserve"> ſed m b maior m z, [ut patuit.</s>
            <s xml:id="echoid-s14578" xml:space="preserve">] Quare m d maior m z.</s>
            <s xml:id="echoid-s14579" xml:space="preserve"> Igitur
              <lb/>
            </s>
          </p>
        </div>
      </text>
    </echo>
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