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-s43749" xml:space="preserve">
              <pb o="359" file="0661" n="661" rhead="LIBER OCTAVVS."/>
            huius:</s>
            <s xml:id="echoid-s43750" xml:space="preserve"> & ex duobus punctis illius arcus ſimilibus duobus punctis b & f in 43 huius fit ab hoc arcu
              <lb/>
            illa reflexio ſormarum duorum punctorum, quæ ſunt u & o:</s>
            <s xml:id="echoid-s43751" xml:space="preserve"> erit ergo q imago puncti o, & n imago
              <lb/>
            puncti u.</s>
            <s xml:id="echoid-s43752" xml:space="preserve"> Ducatur ergo à puncto u in ſuperficie cir
              <lb/>
              <figure xlink:label="fig-0661-01" xlink:href="fig-0661-01a" number="794">
                <variables xml:id="echoid-variables771" xml:space="preserve">k q t l n f g b o l u z d h a</variables>
              </figure>
            culi a b g recta perpendicularis ſuper lineam d u:</s>
            <s xml:id="echoid-s43753" xml:space="preserve">
              <lb/>
            quæ ſit z u e:</s>
            <s xml:id="echoid-s43754" xml:space="preserve"> & à centro d ſecundum longitudinẽ
              <lb/>
            ſemidiametri d o fiat circulus:</s>
            <s xml:id="echoid-s43755" xml:space="preserve"> hic ergo circulus ſe
              <lb/>
            cabit lineã z u e in duobus pũctis per 2 p 3:</s>
            <s xml:id="echoid-s43756" xml:space="preserve"> ſecet er
              <lb/>
            go in punctis z & e:</s>
            <s xml:id="echoid-s43757" xml:space="preserve"> fiatq́;</s>
            <s xml:id="echoid-s43758" xml:space="preserve"> arcus circuli ſecundum
              <lb/>
            quantitatem lineæ d q à centro d:</s>
            <s xml:id="echoid-s43759" xml:space="preserve"> & ducantur à cẽ
              <lb/>
            tro ſpeculi d lineæ d z, d e:</s>
            <s xml:id="echoid-s43760" xml:space="preserve"> & producãtur extra ſpe
              <lb/>
            culum ad arcum circuli deſcripti à centro d ſecun
              <lb/>
            dum quantitatem ſemidiametri d q:</s>
            <s xml:id="echoid-s43761" xml:space="preserve"> & ſint d t, d k:</s>
            <s xml:id="echoid-s43762" xml:space="preserve">
              <lb/>
            & ducatur linea t k:</s>
            <s xml:id="echoid-s43763" xml:space="preserve"> ſecetq́;</s>
            <s xml:id="echoid-s43764" xml:space="preserve"> lineam d q in puncto l.</s>
            <s xml:id="echoid-s43765" xml:space="preserve">
              <lb/>
            Quia ergo linea h d eſt perpendicularis ſuper ſu-
              <lb/>
            perficiem circuli, palàm per definitionem lineæ e-
              <lb/>
            rectæ quoniam uterque angulus h d t, h d k eſt re-
              <lb/>
            ctus:</s>
            <s xml:id="echoid-s43766" xml:space="preserve"> & utraq;</s>
            <s xml:id="echoid-s43767" xml:space="preserve"> ſuperficies h d t & h d k in ſuperficie
              <lb/>
            ſpeculi continet arcũ interiacentẽ lineas h d & d t,
              <lb/>
            & lineas h d & d k per 69 th.</s>
            <s xml:id="echoid-s43768" xml:space="preserve"> 1 huius:</s>
            <s xml:id="echoid-s43769" xml:space="preserve"> quorũ arcuũ
              <lb/>
            quilibet eſt ęqualis arcui, qui eſt inter duas lineas
              <lb/>
            h d & d q:</s>
            <s xml:id="echoid-s43770" xml:space="preserve"> & utraq;</s>
            <s xml:id="echoid-s43771" xml:space="preserve"> linearum d z & d e eſt ęqualis
              <lb/>
            lineæ d o:</s>
            <s xml:id="echoid-s43772" xml:space="preserve"> quoniã omnes ſunt ſemidiametri eiuſdẽ
              <lb/>
            circuli.</s>
            <s xml:id="echoid-s43773" xml:space="preserve"> Illi ergo duo arcus ſunt huiuſmodi, quòd
              <lb/>
            exillis poſsibile eſt fieri reflexionẽ formarũ duo-
              <lb/>
            rum punctorum, quæ ſunt z & e, ab aliquibus pun
              <lb/>
            ctis illorum arcuum, ut patet per 20 huius.</s>
            <s xml:id="echoid-s43774" xml:space="preserve"> Interia
              <lb/>
            cent enim illi arcus ſemidiametros ſpeculi, in qui-
              <lb/>
            bus conſiſtunt centrum uiſus, quod eſt in puncto
              <lb/>
            h, & puncta, quorum formæ reflectũtur, quæ ſunt
              <lb/>
            e & z:</s>
            <s xml:id="echoid-s43775" xml:space="preserve"> incidentq́;</s>
            <s xml:id="echoid-s43776" xml:space="preserve"> formæ eorum illis punctis illorũ
              <lb/>
            arcuum, & reflectentur ad uiſum in punctum h ſe-
              <lb/>
            cundum angulos ęquales à duobus punctis ſpecu
              <lb/>
            li:</s>
            <s xml:id="echoid-s43777" xml:space="preserve"> & duę lineæ d t & d k ſunt æ quales lineę d q:</s>
            <s xml:id="echoid-s43778" xml:space="preserve"> er-
              <lb/>
            go punctum t eſt locus imaginis puncti z, & pun-
              <lb/>
            ctum k eſt locus imaginis puncti e.</s>
            <s xml:id="echoid-s43779" xml:space="preserve"> Et quia lineæ
              <lb/>
            d t, d q, d k ſunt æ quales, & lineę d z, d o, d e æ quales, erit per 7 p 5 proportio lineæ d t ad lineã d z,
              <lb/>
            ſicut lineæ d q ad lineam d o, & ſicut lineæ k d ad lineam d e:</s>
            <s xml:id="echoid-s43780" xml:space="preserve"> ſed per 43 huius proportio lineę d q a d
              <lb/>
            lineam d o eſt maior proportione lineę d n ad lineam d u:</s>
            <s xml:id="echoid-s43781" xml:space="preserve"> ergo ſimiliter proportio lineę k d ad lineã
              <lb/>
            d e eſt maior proportione lineę n d ad lineã d u:</s>
            <s xml:id="echoid-s43782" xml:space="preserve"> & ſimiliter proportio lineę d t ad lineam d z eſt ma-
              <lb/>
            ior proportione lineæ d n a d lineam d u.</s>
            <s xml:id="echoid-s43783" xml:space="preserve"> Et quia duę lineę d e & z d ſunt æ quales, & duæ lineæ d t
              <lb/>
            & d k ſunt æquales:</s>
            <s xml:id="echoid-s43784" xml:space="preserve"> erit per 7 p 5 proportio lineę d t a d lineã d z, ſicut lineę d k ad lineã d e:</s>
            <s xml:id="echoid-s43785" xml:space="preserve"> ergo per
              <lb/>
            17 p 5 erit proportio lineæ t z ad lineam z d, ſicut lineæ k e ad lineam d e:</s>
            <s xml:id="echoid-s43786" xml:space="preserve"> ergo per 2 p 6 linea t k eſt ę-
              <lb/>
            quidiſtans lineę ez:</s>
            <s xml:id="echoid-s43787" xml:space="preserve"> erit ergo per eandem 2 p 6 & per 18 p 5 proportio lineę l d ad lineam d u, ſicut li-
              <lb/>
            neę d k ad lineam d e, & ſicut lineę d t ad lineam d z:</s>
            <s xml:id="echoid-s43788" xml:space="preserve"> proportio ergo lineę l d ad lineẽ d u eſt maior
              <lb/>
            proportione lineę n d ad lineam d u:</s>
            <s xml:id="echoid-s43789" xml:space="preserve"> ergo per 10 p 5 linea l d eſt maior quàm linea n d:</s>
            <s xml:id="echoid-s43790" xml:space="preserve"> ergo punctus
              <lb/>
            n eſt inter puncta l & u:</s>
            <s xml:id="echoid-s43791" xml:space="preserve"> ſed punctus n eſt imago puncti u:</s>
            <s xml:id="echoid-s43792" xml:space="preserve"> & duo puncta t & k ſunt imagines duorũ
              <lb/>
            punctorũ z & e:</s>
            <s xml:id="echoid-s43793" xml:space="preserve"> ergo imago lineę z u e rectæ eſt linea tranſiens per tria puncta t, n, k:</s>
            <s xml:id="echoid-s43794" xml:space="preserve"> linea uerò per-
              <lb/>
            tranſiens hæc puncta eſt conuexa.</s>
            <s xml:id="echoid-s43795" xml:space="preserve"> Patet ergo quòd imago lineę z e rectę uidebitur in hoc ſitu con-
              <lb/>
            uexa.</s>
            <s xml:id="echoid-s43796" xml:space="preserve"> Et hoc eſt propoſitum.</s>
            <s xml:id="echoid-s43797" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1704" type="section" level="0" n="0">
          <head xml:id="echoid-head1268" xml:space="preserve" style="it">58. In quibuſdam ſitibus reflexione facta à ſpeculis ſphœricis concauis, uiſus comprehendet i-
            <lb/>
          maginem concauam reflexam ex linea concaua uel conuexa. Alhazen 50 n 6.</head>
          <p>
            <s xml:id="echoid-s43798" xml:space="preserve">Sit diſpoſitio omnino, quę in pręcedente.</s>
            <s xml:id="echoid-s43799" xml:space="preserve"> Quia itaq;</s>
            <s xml:id="echoid-s43800" xml:space="preserve">, ut patet in præmiſſa, imago formæ puncti o
              <lb/>
            eſt punctum q, & imago formę puncti z eſt punctum t, & imago ſormæ puncti e eſt punctũ k:</s>
            <s xml:id="echoid-s43801" xml:space="preserve"> erit er-
              <lb/>
            go linea concaua reſpectu uiſus, quæ eſt t q k, imago lineę curuę reſpectu uiſus, conuexę tamen re-
              <lb/>
            ſpectu ſpeculi, quæ eſt linea z o e.</s>
            <s xml:id="echoid-s43802" xml:space="preserve"> Similiter quoq;</s>
            <s xml:id="echoid-s43803" xml:space="preserve">, ſi in linea z u ſignetur punctum m, qualitercunq;</s>
            <s xml:id="echoid-s43804" xml:space="preserve">
              <lb/>
            hoc contingat:</s>
            <s xml:id="echoid-s43805" xml:space="preserve"> & circa centrum m ſecundum longitudinẽ ſemidiametri m u deſcribatur arcus par-
              <lb/>
            ui circuli, qui ſit r u ſ:</s>
            <s xml:id="echoid-s43806" xml:space="preserve"> hic ergo arcus ſecabit circulum z o e in duobus punctis per 10 p 3:</s>
            <s xml:id="echoid-s43807" xml:space="preserve"> ſint illa duo
              <lb/>
            puncta f & r:</s>
            <s xml:id="echoid-s43808" xml:space="preserve"> & ducantur lineæ d r & d f:</s>
            <s xml:id="echoid-s43809" xml:space="preserve"> quę protrahantur uſque ad arcum t q k eductum:</s>
            <s xml:id="echoid-s43810" xml:space="preserve"> incidatq́;</s>
            <s xml:id="echoid-s43811" xml:space="preserve">
              <lb/>
            linea d f in punctum i, & linea d r in punctum p.</s>
            <s xml:id="echoid-s43812" xml:space="preserve"> Superficies ergo duarum linearũ h d & d p ſecabit
              <lb/>
            ſpeculum ſecundum circulum, à cuius circum ſerentię puncto aliquo duci poterunt ſecundũ angu-
              <lb/>
            los ęquales & ęqualiter ſe habentes lineę ad punctum h, in quo eſt centrũ uiſus, & a d punctũ r, qui
              <lb/>
            eſt punctus lineę uiſæ.</s>
            <s xml:id="echoid-s43813" xml:space="preserve"> Et ſimiliter ſuperficies duarum linearum h d & d i ſaciet in ſpeculo circulũ, à
              <lb/>
            cuius circumſerentia reflectetur ad uiſum ſorma puncti ſarcus r u f.</s>
            <s xml:id="echoid-s43814" xml:space="preserve"> Eſt ergo punctus p imago for-
              <lb/>
            </s>
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