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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            <s xml:id="echoid-s11872" xml:space="preserve">
              <pb o="177" file="0183" n="183" rhead="OPTICAE LIBER V."/>
            æqualis eſt angulo o d a [per ſabricationem:</s>
            <s xml:id="echoid-s11873" xml:space="preserve">] igitur e t d medietas anguli o d a:</s>
            <s xml:id="echoid-s11874" xml:space="preserve"> ſed angulus o d a cũ
              <lb/>
            angulo o d f ualet duos rectos [per 13 p 1] & [per 32 p 1] tres anguli trianguli e t d duos rectos:</s>
            <s xml:id="echoid-s11875" xml:space="preserve"> ab-
              <lb/>
            lato e d t cõmuni:</s>
            <s xml:id="echoid-s11876" xml:space="preserve"> reſtat angulus t e d æqualis medietati anguli o d a, & angulo o d n [nam poſt ſub-
              <lb/>
            ductionem communis anguli t d e, relinquũtur anguli d t e, t e d æquales angulis o d t, o d a:</s>
            <s xml:id="echoid-s11877" xml:space="preserve"> ſed d t e
              <lb/>
            æquatur dimidiato angulo o d a, ut patuit:</s>
            <s xml:id="echoid-s11878" xml:space="preserve"> reliquus igitur t e d ęquatur dimidiato angulo o d a & an
              <lb/>
            gulo o d n ſimul utriq;</s>
            <s xml:id="echoid-s11879" xml:space="preserve">.] Sed angulus o d p cũ medietate anguli o d a eſt rectus:</s>
            <s xml:id="echoid-s11880" xml:space="preserve"> [ꝗ a enim anguli o d
              <lb/>
            k, o d a æquãtur duobus rectis per 13 p 1:</s>
            <s xml:id="echoid-s11881" xml:space="preserve"> & angulus o d u eſt dimidius anguli o d k per fabricationẽ:</s>
            <s xml:id="echoid-s11882" xml:space="preserve">
              <lb/>
            duo igitur dimidiati anguli duorũ rectorũ æquãtur uni recto] igitur angulus t e d eſt acutus:</s>
            <s xml:id="echoid-s11883" xml:space="preserve"> [quia
              <lb/>
            enim angulus o d p cum dimidiato angulo o d a æquatur unirecto ex concluſo:</s>
            <s xml:id="echoid-s11884" xml:space="preserve"> & maior eſt angulo
              <lb/>
            o d n:</s>
            <s xml:id="echoid-s11885" xml:space="preserve"> quia, ut patuit, n cadit inter p & o:</s>
            <s xml:id="echoid-s11886" xml:space="preserve"> ergo angulus t e d æqualis angulo o d n, & dimidiato o d a,
              <lb/>
            erit minor recto:</s>
            <s xml:id="echoid-s11887" xml:space="preserve"> ideoq́;</s>
            <s xml:id="echoid-s11888" xml:space="preserve"> acutus] quare ei contrapoſitus eſt acutus [per 15 p 1.</s>
            <s xml:id="echoid-s11889" xml:space="preserve">] Igitur ſi à puncto k
              <lb/>
            ducatur perpendicularis ad t z:</s>
            <s xml:id="echoid-s11890" xml:space="preserve"> [per 12 p 1] cadet inter e & z.</s>
            <s xml:id="echoid-s11891" xml:space="preserve"> Si enim ſupra e ceciderit, cum angulus
              <lb/>
            t e k ſit obtuſus:</s>
            <s xml:id="echoid-s11892" xml:space="preserve"> [per 13 p 1:</s>
            <s xml:id="echoid-s11893" xml:space="preserve"> acutus enim concluſus eſt t e d] accidet triangulũ habere duos angulos
              <lb/>
            rectum & obtuſum [contra 32 p 1.</s>
            <s xml:id="echoid-s11894" xml:space="preserve">] Sit ergo perpendicularis k q.</s>
            <s xml:id="echoid-s11895" xml:space="preserve"> Dico, quo d k t ſe habet ad t f, ſicut
              <lb/>
            k d ad d o, t o enim aut eſt æquidiſtans k d:</s>
            <s xml:id="echoid-s11896" xml:space="preserve"> aut concurrit cum ea.</s>
            <s xml:id="echoid-s11897" xml:space="preserve"> Sit æquidiſtans:</s>
            <s xml:id="echoid-s11898" xml:space="preserve"> erit ergo [per 29
              <lb/>
              <figure xlink:label="fig-0183-01" xlink:href="fig-0183-01a" number="129">
                <variables xml:id="echoid-variables119" xml:space="preserve">b t o u p n g k e f d a q z m</variables>
              </figure>
              <figure xlink:label="fig-0183-02" xlink:href="fig-0183-02a" number="130">
                <variables xml:id="echoid-variables120" xml:space="preserve">b u t o p n g k e f d a q z m</variables>
              </figure>
            p 1] angulus o d a æqualis angulo t o d:</s>
            <s xml:id="echoid-s11899" xml:space="preserve"> & ita t o d æqualis angulo o t f [æquatus enim eſt o t f ipſi
              <lb/>
            o d a.</s>
            <s xml:id="echoid-s11900" xml:space="preserve">] Et o d, t ſaut ſunt æquidiſtãtes:</s>
            <s xml:id="echoid-s11901" xml:space="preserve"> aut cõcurrunt.</s>
            <s xml:id="echoid-s11902" xml:space="preserve"> Si æquidiſtantes, cũ cadant inter æquidiſtan-
              <lb/>
            tes [k d, t o] erũt [per 34 p 1] æquales.</s>
            <s xml:id="echoid-s11903" xml:space="preserve"> Si uerò cõcurrunt:</s>
            <s xml:id="echoid-s11904" xml:space="preserve"> faciẽt triangulũ, cuius latera æqualia [per
              <lb/>
            6 p 1] quia reſpiciunt æquales angulos:</s>
            <s xml:id="echoid-s11905" xml:space="preserve"> [f t o, & d o t] & f d ſecat illa latera æquidiſtanter baſi.</s>
            <s xml:id="echoid-s11906" xml:space="preserve"> Erit
              <lb/>
            ergo [per 2 p 6.</s>
            <s xml:id="echoid-s11907" xml:space="preserve">18 p 5] proportio unius laterum ad d o, ſicut alterius ad f t:</s>
            <s xml:id="echoid-s11908" xml:space="preserve"> & ita t f æqualis d o [per
              <lb/>
            9 p 5.</s>
            <s xml:id="echoid-s11909" xml:space="preserve">] Ethoc dico, ſilineæ illæ concurrant ſub k d.</s>
            <s xml:id="echoid-s11910" xml:space="preserve"> Et ſi cõcurrant ſub t o:</s>
            <s xml:id="echoid-s11911" xml:space="preserve"> eadem erit probatio:</s>
            <s xml:id="echoid-s11912" xml:space="preserve"> quia
              <lb/>
            ſiet triangulum, cuius unũ latus eſt t o, & alia duo latera æqualia:</s>
            <s xml:id="echoid-s11913" xml:space="preserve"> [per 6 p 1] & erit [per 2 p 6.</s>
            <s xml:id="echoid-s11914" xml:space="preserve">18 p 5]
              <lb/>
            proportio unius laterum ad d o, ſicut alterius a d t f:</s>
            <s xml:id="echoid-s11915" xml:space="preserve"> & ita [per 9 p 5] t ſ æqualis d o.</s>
            <s xml:id="echoid-s11916" xml:space="preserve"> Item angulus t d
              <lb/>
            k eſt æqualis angulo d t o [per 29 p 1] quia d tinter æquidiſtantes:</s>
            <s xml:id="echoid-s11917" xml:space="preserve"> [ex theſi:</s>
            <s xml:id="echoid-s11918" xml:space="preserve"> nempe k d, t o] igitur
              <lb/>
            eſt æqualis angulo d t k:</s>
            <s xml:id="echoid-s11919" xml:space="preserve"> [qui ex theſi & 12 n 4 æquatur angulo d t o] quare [per 6 p 1] d k æqualis
              <lb/>
            eſt t k.</s>
            <s xml:id="echoid-s11920" xml:space="preserve"> Igitur [per 7 p 5] proportio tkad t f, ſicut k d ad d o.</s>
            <s xml:id="echoid-s11921" xml:space="preserve"> Siuero to concurrit cum k d:</s>
            <s xml:id="echoid-s11922" xml:space="preserve"> concurrat
              <lb/>
            ex parte a in puncto l.</s>
            <s xml:id="echoid-s11923" xml:space="preserve"> Scimus [è demõſtratis à Theo
              <lb/>
              <figure xlink:label="fig-0183-03" xlink:href="fig-0183-03a" number="131">
                <variables xml:id="echoid-variables121" xml:space="preserve">u t b p n o g k e f d l
                  <unsure/>
                a q m z</variables>
              </figure>
            ne ad 5 d 6] quòd proportio k t ad t ſ compacta eſt ex
              <lb/>
            proportione k t ad tl, & tl ad t f:</s>
            <s xml:id="echoid-s11924" xml:space="preserve"> ſed [per 3 p 6] k t ad
              <lb/>
            tleſt, ſicut k d ad d l:</s>
            <s xml:id="echoid-s11925" xml:space="preserve"> quoniam d t diuidit angulum k
              <lb/>
            to per æqualia:</s>
            <s xml:id="echoid-s11926" xml:space="preserve"> & proportio tladtf, ſicut d l ad d o:</s>
            <s xml:id="echoid-s11927" xml:space="preserve">
              <lb/>
            quoniã angulus o d leſt æqualis angulo l t f [perſa-
              <lb/>
            bricationem] & angulus ſuperl communis:</s>
            <s xml:id="echoid-s11928" xml:space="preserve"> [trian-
              <lb/>
            gulis l t f, o d l] erit partiale triangulum ſimile totali
              <lb/>
            [per 32 p 1.</s>
            <s xml:id="echoid-s11929" xml:space="preserve"> 4 p.</s>
            <s xml:id="echoid-s11930" xml:space="preserve"> 1 d 6.</s>
            <s xml:id="echoid-s11931" xml:space="preserve">] Igitur proportio k t ad t f cõſtat
              <lb/>
            ex proportione k d ad d l, & proportione dl ad d o:</s>
            <s xml:id="echoid-s11932" xml:space="preserve">
              <lb/>
            ſed proportio k d ad d o conſtat exijſdem [aſſumpta
              <lb/>
            dlmedia interk d & d o.</s>
            <s xml:id="echoid-s11933" xml:space="preserve">] Quare proportio k t ad t f,
              <lb/>
            ſicut k d ad d o.</s>
            <s xml:id="echoid-s11934" xml:space="preserve"> Si uerò to concurrat cum k d ex par-
              <lb/>
            te g:</s>
            <s xml:id="echoid-s11935" xml:space="preserve"> ſit concurſus s.</s>
            <s xml:id="echoid-s11936" xml:space="preserve"> Et à puncto d ducatur æquidi-
              <lb/>
            ſtans lineæ k t:</s>
            <s xml:id="echoid-s11937" xml:space="preserve"> [per 31 p 1] quæ ſit d r, cõcurrens cum
              <lb/>
            to in puncto r:</s>
            <s xml:id="echoid-s11938" xml:space="preserve"> igitur [per 29 p 1] angulus k t d eſt
              <lb/>
            æqualis angulo t d r:</s>
            <s xml:id="echoid-s11939" xml:space="preserve"> ſed idẽ eſt æqualis angulo d t o
              <lb/>
            [pertheſin & 12 n 4.</s>
            <s xml:id="echoid-s11940" xml:space="preserve">] Quare [ք 6 p 1] d r eſt æqualis
              <lb/>
            tr.</s>
            <s xml:id="echoid-s11941" xml:space="preserve"> Sed quia triangulũ s t k ſimile eſt triangulo s r d:</s>
            <s xml:id="echoid-s11942" xml:space="preserve"> [per 29.</s>
            <s xml:id="echoid-s11943" xml:space="preserve"> 32 p 1.</s>
            <s xml:id="echoid-s11944" xml:space="preserve"> 4 p.</s>
            <s xml:id="echoid-s11945" xml:space="preserve"> 1 d 6] erit proportio d rad sr,
              <lb/>
            ſicut k t ad t s:</s>
            <s xml:id="echoid-s11946" xml:space="preserve"> & ita r t ad r s, ſicut kt ad ts:</s>
            <s xml:id="echoid-s11947" xml:space="preserve"> [ք 7 p 5:</s>
            <s xml:id="echoid-s11948" xml:space="preserve"> æqualis enim cõcluſa eſt tripſi d r] ſed r t ad r s,
              <lb/>
            </s>
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