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-div310" type="section" level="0" n="0">
          <pb o="137" file="0143" n="143" rhead="OPTICAE LIBER V."/>
        </div>
        <div xml:id="echoid-div312" type="section" level="0" n="0">
          <head xml:id="echoid-head325" xml:space="preserve" style="it">22. Si reflexio fiat à peripheria circuli (qui eſt communis ſectio ſuperficierum reflexionis &
            <lb/>
          ſpeculi ſphærici conuexi) inter rectam à uiſu ſpeculum tangentem, reflexionis puncto proxi-
            <lb/>
          mam, & lineam reflexionis æquãtem partem ſuam intra peripheriam eiuſdem ſemidiametro:
            <lb/>
          imago aliàs intra ſpeculum: aliàs in ſuperficie: aliàs extra uidebitur. 26 p 6. Item 27. 7 p 6.</head>
          <p>
            <s xml:id="echoid-s8075" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s8076" xml:space="preserve"> ſumpto quocunque puncto arcus h b:</s>
            <s xml:id="echoid-s8077" xml:space="preserve"> dico, quòd quędam eius imago erit intra ſpe
              <lb/>
            culum:</s>
            <s xml:id="echoid-s8078" xml:space="preserve"> quædam in ſuperficie ſpeculi:</s>
            <s xml:id="echoid-s8079" xml:space="preserve"> quædam extra ſpeculum.</s>
            <s xml:id="echoid-s8080" xml:space="preserve"> Sumatur aliquod eius pun-
              <lb/>
            ctum:</s>
            <s xml:id="echoid-s8081" xml:space="preserve"> & ſit n:</s>
            <s xml:id="echoid-s8082" xml:space="preserve"> & ducatur linea à pũcto g ſe-
              <lb/>
              <figure xlink:label="fig-0143-01" xlink:href="fig-0143-01a" number="52">
                <variables xml:id="echoid-variables42" xml:space="preserve">g z f h a b d c q e k ſ r</variables>
              </figure>
            cans circulum, quæ ſit g n q:</s>
            <s xml:id="echoid-s8083" xml:space="preserve"> & ducatur perpen-
              <lb/>
            dicularis d n f:</s>
            <s xml:id="echoid-s8084" xml:space="preserve"> & [per 23 p 1] protrahatur linea,
              <lb/>
            æqualem angulum tenens cum perpendiculari,
              <lb/>
            angulo f n g:</s>
            <s xml:id="echoid-s8085" xml:space="preserve"> & ſit e n.</s>
            <s xml:id="echoid-s8086" xml:space="preserve"> Quoniam linea n q minor
              <lb/>
            eſt k h [per 15 p 3] eſt etiam minor linea q d [nã
              <lb/>
            h k poſita eſt æqualis ſemidiametro ſphæræ] &
              <lb/>
            ita [per 18 p 1] q d n angulus minor eſt angu-
              <lb/>
            lo d n q:</s>
            <s xml:id="echoid-s8087" xml:space="preserve"> quare [per 15 p 1] minor angulo g
              <lb/>
            n f:</s>
            <s xml:id="echoid-s8088" xml:space="preserve"> quare etiam minor angulo e n f:</s>
            <s xml:id="echoid-s8089" xml:space="preserve"> Igitur e n
              <lb/>
            & d q concurrent [ad partes e & q per 11 ax.</s>
            <s xml:id="echoid-s8090" xml:space="preserve">]
              <lb/>
            Sit ergo concurſus in puncto e.</s>
            <s xml:id="echoid-s8091" xml:space="preserve"> Palàm [per 25
              <lb/>
            n 4] quòd linea e q d eſt perpendicularis ſuper
              <lb/>
            ſphæram:</s>
            <s xml:id="echoid-s8092" xml:space="preserve"> & ſecat lineam g n q, quæ eſt linea re-
              <lb/>
            flexionis, in puncto q, quod eſt punctum ſphæ-
              <lb/>
            ræ.</s>
            <s xml:id="echoid-s8093" xml:space="preserve"> Quare imago puncti e, cum fuerit reflexio ſu-
              <lb/>
            per punctum n, apparebit in puncto q:</s>
            <s xml:id="echoid-s8094" xml:space="preserve"> [per 3 n]
              <lb/>
            & eſt in ſuperficie ſphæræ.</s>
            <s xml:id="echoid-s8095" xml:space="preserve"> Si uerò in linea n e ſu-
              <lb/>
            matur punctum ultra e, utpoter:</s>
            <s xml:id="echoid-s8096" xml:space="preserve"> perpẽdicularis
              <lb/>
            ducta ab eo ad centrum ſphæræ, quæ ſit r d, ſeca
              <lb/>
            bit lineam g n q reflexionis, ultra punctum q:</s>
            <s xml:id="echoid-s8097" xml:space="preserve"> &
              <lb/>
            eſt extra ſphæram.</s>
            <s xml:id="echoid-s8098" xml:space="preserve"> Quare imago cuiuslibet pun-
              <lb/>
            cti lineæ e n ultra e ſumpti, erit extra ſuperficiem
              <lb/>
            ſpeculi.</s>
            <s xml:id="echoid-s8099" xml:space="preserve"> Si uerò in linea e n, citra punctum e ſu-
              <lb/>
            matur aliquod punctum:</s>
            <s xml:id="echoid-s8100" xml:space="preserve"> perpendicularis ab eo
              <lb/>
            ducta ad ſpeculum, ſecabit lineam g n q intra ſphæram:</s>
            <s xml:id="echoid-s8101" xml:space="preserve"> quoniam in puncto, quod eſt inter n & q.</s>
            <s xml:id="echoid-s8102" xml:space="preserve">
              <lb/>
            Quare imago cuiuslibet puncti lineæ e n inter e & n ſumpti, apparebit intra ſphæram.</s>
            <s xml:id="echoid-s8103" xml:space="preserve"> Eadem peni
              <lb/>
            tus erit probatio, ſumpto quocunque alio arcus b h puncto:</s>
            <s xml:id="echoid-s8104" xml:space="preserve"> & ita imago cuiuslibet puncti arcus
              <lb/>
            b h una ſola eſt imago in ſuperficie ſpeculi:</s>
            <s xml:id="echoid-s8105" xml:space="preserve"> aliarum quædam in ſpeculo:</s>
            <s xml:id="echoid-s8106" xml:space="preserve"> quædam extra.</s>
            <s xml:id="echoid-s8107" xml:space="preserve"> Et quod
              <lb/>
            demonſtratum eſt in arcu z b, eodem modo poteſt patére in arcu z a:</s>
            <s xml:id="echoid-s8108" xml:space="preserve"> & eadem penitus erit demon-
              <lb/>
            ſtratio, cuiuſcunque circuli ſphæræ ſumatur portio, uiſui oppoſita, à perpendiculari g d æqualiter
              <lb/>
            diuiſa.</s>
            <s xml:id="echoid-s8109" xml:space="preserve"> Vnde uiſu immoto, & perpendiculari g z d manente, ſi moueatur æquidiſtanter perpendi-
              <lb/>
            culari uiſus linea g h, ſecabit ex ſphæra motu ſuo portionem circularem:</s>
            <s xml:id="echoid-s8110" xml:space="preserve"> & cuiuslibet puncti hu-
              <lb/>
            ius portionis imago apparebit intra ſphæram.</s>
            <s xml:id="echoid-s8111" xml:space="preserve"> Si uerò linea g b contingens, moueatur æquidiſtan-
              <lb/>
            ter perpendiculari uiſus, ſecabit ex ſphæra portionem prædicta maiorem:</s>
            <s xml:id="echoid-s8112" xml:space="preserve"> & à quolibet puncto
              <lb/>
            excrementi unius portionis ſuper aliam reflectitur imago, cuius locus erit in ſuperficie ſphæ-
              <lb/>
            ræ:</s>
            <s xml:id="echoid-s8113" xml:space="preserve"> & aliarum quædam intra ſphæram:</s>
            <s xml:id="echoid-s8114" xml:space="preserve"> quædam extra.</s>
            <s xml:id="echoid-s8115" xml:space="preserve"> Scimus ex his, quòd in hoc ſpeculo quæli-
              <lb/>
            bet imago apparet in diametro ſphæræ:</s>
            <s xml:id="echoid-s8116" xml:space="preserve"> aut intra ſphærã:</s>
            <s xml:id="echoid-s8117" xml:space="preserve"> aut extra:</s>
            <s xml:id="echoid-s8118" xml:space="preserve"> aut in ſuperficie.</s>
            <s xml:id="echoid-s8119" xml:space="preserve"> Et omnis dia-
              <lb/>
            meter, in qua apparet imago aliqua in ſuperficie ſphæræ, aut extra, demiſsior eſt puncto ſphæræ,
              <lb/>
            quod tangit linea contingens à centro uiſus, ducta in ultimum punctum portionis apparentis.</s>
            <s xml:id="echoid-s8120" xml:space="preserve"> Sci
              <lb/>
            mus etiam, quòd quælibet linea reflexionis ſecat ſphæram in duobus punctis, in puncto reflexio-
              <lb/>
            nis, & in alio.</s>
            <s xml:id="echoid-s8121" xml:space="preserve"> Reſtatiam, ut loca imaginum certius determinemus.</s>
            <s xml:id="echoid-s8122" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div314" type="section" level="0" n="0">
          <head xml:id="echoid-head326" xml:space="preserve" style="it">23. Si linea reflexionis ſecans diametrum ſpeculi ſphærici conuexi: æquet ſegmentum ſuum
            <lb/>
          inter ſpeculi ſuperficiem & dictam diametrum, ſegmento eiuſdem diametri contermino centro
            <lb/>
          ſpeculi: erit hoc ſegmentum imaginum expers. 28 p 6.</head>
          <p>
            <s xml:id="echoid-s8123" xml:space="preserve">DIco, quòd ſumpta diametro, ſi ad ipſam ducatur linea ſecans ſphæram à centro uiſus, cuius
              <lb/>
            pars interiacens punctum ſectionis ſphæræ & punctum diametri, quam attingit, eſt æqua-
              <lb/>
            lis parti diametri, interiacenti inter punctum illud & centrum:</s>
            <s xml:id="echoid-s8124" xml:space="preserve"> punctum illud non eſt locus
              <lb/>
            alicuius imaginis.</s>
            <s xml:id="echoid-s8125" xml:space="preserve"> Verbi gratia.</s>
            <s xml:id="echoid-s8126" xml:space="preserve"> ſit a g circulus ſphæræ:</s>
            <s xml:id="echoid-s8127" xml:space="preserve"> h uiſus:</s>
            <s xml:id="echoid-s8128" xml:space="preserve"> e d ſemidiameter ſphæræ, ſiue per-
              <lb/>
            pendicularis:</s>
            <s xml:id="echoid-s8129" xml:space="preserve"> & h z ſit linea ſecans ſphæram ſuper punctum f, & concurrens cum e d in puncto z:</s>
            <s xml:id="echoid-s8130" xml:space="preserve"> &
              <lb/>
            ſit z f æqualis z d.</s>
            <s xml:id="echoid-s8131" xml:space="preserve"> Dico, quòd z non eſt locus alicuius imaginis.</s>
            <s xml:id="echoid-s8132" xml:space="preserve"> Palàm enim, quòd nõ eſt locus i-
              <lb/>
            maginis alterius, quàm alicuius puncti lineæ e d:</s>
            <s xml:id="echoid-s8133" xml:space="preserve"> quoniam imago cuiuslibet puncti eſt ſuper dia-
              <lb/>
            metrum, ab eo ad centrum ſphæræ ductam [per 10 n.</s>
            <s xml:id="echoid-s8134" xml:space="preserve">] Et quòd locus imaginis alicuius puncti e d
              <lb/>
            non ſit in z:</s>
            <s xml:id="echoid-s8135" xml:space="preserve"> ſic conſtabit.</s>
            <s xml:id="echoid-s8136" xml:space="preserve"> Ducatur perpendicularis à puncto d ſuper punctum f:</s>
            <s xml:id="echoid-s8137" xml:space="preserve"> & ſit d f n:</s>
            <s xml:id="echoid-s8138" xml:space="preserve"> & [per
              <lb/>
            23 p 1] ſuper punctum f fiat angulus æqualis angulo n fh:</s>
            <s xml:id="echoid-s8139" xml:space="preserve"> & ſit q f n.</s>
            <s xml:id="echoid-s8140" xml:space="preserve"> Palàm ergo [per 15 p 1.</s>
            <s xml:id="echoid-s8141" xml:space="preserve"> 1 ax]
              <lb/>
            </s>
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