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-s11085" xml:space="preserve">
              <pb o="168" file="0174" n="174" rhead="ALHAZEN"/>
            maior, ꝗ̃ h l ad l t, ut cõſtat ex 8 p 5.</s>
            <s xml:id="echoid-s11086" xml:space="preserve">] Palã igitur, quòd ſit & h ęqualiter diſtẽt à cẽtro, & fuerint ſuper
              <lb/>
            contingentẽ:</s>
            <s xml:id="echoid-s11087" xml:space="preserve"> non reflectetur t ad h, niſi ab uno ſpeculi puncto tãtùm:</s>
            <s xml:id="echoid-s11088" xml:space="preserve"> & unicus erit eius imaginis lo
              <lb/>
            cus.</s>
            <s xml:id="echoid-s11089" xml:space="preserve"> Si uerò t d, h d ſunt inæquales:</s>
            <s xml:id="echoid-s11090" xml:space="preserve"> ſecentur ad æqualitatẽ [per 3 p 1] & fiat demonſtratio, ut antea.</s>
            <s xml:id="echoid-s11091" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div397" type="section" level="0" n="0">
          <head xml:id="echoid-head374" xml:space="preserve" style="it">71. Si angulum comprebẽſum à duab{us} diametris, in centro circuli (qui eſt communis ſectio
            <lb/>
          ſuperficierum, reflexionis & ſpeculi ſphærici caui) tertia bifariã ſecet: & ab eius termino in pe-
            <lb/>
          ripheria dicto angulo ſubtenſa, ſint perpendiculares ſuper dictas diametros: puncta diametro-
            <lb/>
          rum, tum in quæ perpendiculares cadunt: tũ citr a hæc, à ſpeculi centro æquabiliter diſtãtia, à
            <lb/>
          ſecantis diametri terminis tantùm inter ſe mutuò reflectẽtur: duas́ babebũt imagines. 25 p 8.</head>
          <p>
            <s xml:id="echoid-s11092" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s11093" xml:space="preserve"> b d q, a d g ſint duæ diametri ſphæræ:</s>
            <s xml:id="echoid-s11094" xml:space="preserve"> & diameter e d z diuidat angulũ b d g per ęqua-
              <lb/>
            lia:</s>
            <s xml:id="echoid-s11095" xml:space="preserve"> & à puncto e ducãtur duæ perpendiculares, ſuper duas diametros b d, d g:</s>
            <s xml:id="echoid-s11096" xml:space="preserve"> [per 12 p 1] quę
              <lb/>
            ſint e t, e h.</s>
            <s xml:id="echoid-s11097" xml:space="preserve"> Palàm [per 26 p 1] quòd triangulũ e t d æquale eſt triangulo e h d, & angulus t e d
              <lb/>
            angulo h e d, latusq́;</s>
            <s xml:id="echoid-s11098" xml:space="preserve"> t d lateri h d, & latus e t lateri e h:</s>
            <s xml:id="echoid-s11099" xml:space="preserve"> cũ e d ſit cõmunis utriq;</s>
            <s xml:id="echoid-s11100" xml:space="preserve">. tigitur reflectetur ad
              <lb/>
            h à puncto e.</s>
            <s xml:id="echoid-s11101" xml:space="preserve"> [per 12 n 4.</s>
            <s xml:id="echoid-s11102" xml:space="preserve">] Eodẽ modo à puncto z [Quia enim angulus b d g bifariã ſectus eſt per li-
              <lb/>
            neã e d:</s>
            <s xml:id="echoid-s11103" xml:space="preserve"> erit angulus t d z æqualis angulo h d z per 13 p 1, & t d æquatur ex cõcluſo ipſi h d, latusq́;</s>
            <s xml:id="echoid-s11104" xml:space="preserve"> d z
              <lb/>
            cõmune:</s>
            <s xml:id="echoid-s11105" xml:space="preserve"> angulus igitur t z d æquatur angulo h z d per 4 p 1.</s>
            <s xml:id="echoid-s11106" xml:space="preserve"> Quare per 12 n 4 t & h reflectẽtur inter
              <lb/>
            ſe à puncto z.</s>
            <s xml:id="echoid-s11107" xml:space="preserve">] Et palã [per 66 n] quòd t non reflectetur ad h, ab aliquo puncto arcus a b, uel arcus g
              <lb/>
            q:</s>
            <s xml:id="echoid-s11108" xml:space="preserve"> nec reflectetur ab alio puncto arcus a q, ꝗ̃ à puncto z ſecundũ ſupradictam probationẽ:</s>
            <s xml:id="echoid-s11109" xml:space="preserve"> [numero
              <lb/>
            præcedẽte.</s>
            <s xml:id="echoid-s11110" xml:space="preserve">] Verũ quòd ab alio puncto arcus b g, ꝗ̃ à puncto e, nõ poſsit reflecti:</s>
            <s xml:id="echoid-s11111" xml:space="preserve"> patebit ſic.</s>
            <s xml:id="echoid-s11112" xml:space="preserve"> Detur o
              <lb/>
            punctũ:</s>
            <s xml:id="echoid-s11113" xml:space="preserve"> & ducantur lineę o d, h o, t o:</s>
            <s xml:id="echoid-s11114" xml:space="preserve"> fiatq́;</s>
            <s xml:id="echoid-s11115" xml:space="preserve"> circulus ad quantitatẽ lineæ d e, trãſiens per tria puncta,
              <lb/>
            t, d, h:</s>
            <s xml:id="echoid-s11116" xml:space="preserve"> [tranſibit aũt per conuerſionẽ 31 p 3 demonſtratã à Theone in cõm entarijs in 3 librũ magnę cõ
              <lb/>
            ſtructionis Ptolemęi, & à Cãpano ad 31 p 13:</s>
            <s xml:id="echoid-s11117" xml:space="preserve">] cuius quidẽ circuli linea d e erit diameter:</s>
            <s xml:id="echoid-s11118" xml:space="preserve"> cũ angulus e
              <lb/>
            t d, quẽ reſpicit, ſit rectus Igitur circulus ille tranſibit per punctũ e.</s>
            <s xml:id="echoid-s11119" xml:space="preserve"> Cum igitur e ſit cõmunis utriq;</s>
            <s xml:id="echoid-s11120" xml:space="preserve">
              <lb/>
            circulo, & ſit ſuper eandẽ diametrum:</s>
            <s xml:id="echoid-s11121" xml:space="preserve"> continget circulus minor maiorem in puncto e:</s>
            <s xml:id="echoid-s11122" xml:space="preserve"> ſicut probat
              <lb/>
            Euclidis [13 p 3.</s>
            <s xml:id="echoid-s11123" xml:space="preserve">] Igitur circulus iſte ſecabit lineam d o, [ſecus tangeret maiorem circulũ in puncto
              <lb/>
            o:</s>
            <s xml:id="echoid-s11124" xml:space="preserve"> ſicq́;</s>
            <s xml:id="echoid-s11125" xml:space="preserve"> in duobus punctis e & o tangeret contra 13 p 3] ſecet in puncto l:</s>
            <s xml:id="echoid-s11126" xml:space="preserve"> & ducantur lineæ t l, h l.</s>
            <s xml:id="echoid-s11127" xml:space="preserve"> Iam
              <lb/>
            patet [è ſuperioribus] quod t d eſt ęqualis h d [ergo per 28 p 3 peripheria t d æquatur peripheriæ h
              <lb/>
            d.</s>
            <s xml:id="echoid-s11128" xml:space="preserve">] Igitur angulus t l d æqualis angulo d l h [per 27 p
              <lb/>
              <figure xlink:label="fig-0174-01" xlink:href="fig-0174-01a" number="112">
                <variables xml:id="echoid-variables102" xml:space="preserve">e p o l g h n d m t b q a z</variables>
              </figure>
            3] quia ſuper æquales arcus.</s>
            <s xml:id="echoid-s11129" xml:space="preserve"> Reſtat [per 13 p 1] t l o æ-
              <lb/>
            qualis angulo h l o:</s>
            <s xml:id="echoid-s11130" xml:space="preserve"> & angulus l o t ęqualis angulo l o
              <lb/>
            h ex hypotheſi:</s>
            <s xml:id="echoid-s11131" xml:space="preserve"> [quia ſunt anguli incidentiæ & refle-
              <lb/>
            xionis] & l o commune latus:</s>
            <s xml:id="echoid-s11132" xml:space="preserve"> erit [per 26 p 1] triangu
              <lb/>
            lum t l o æquale triangulo h l o:</s>
            <s xml:id="echoid-s11133" xml:space="preserve"> & erit t o ęqualis h o:</s>
            <s xml:id="echoid-s11134" xml:space="preserve">
              <lb/>
            quod eſt impoſsibile:</s>
            <s xml:id="echoid-s11135" xml:space="preserve"> quoniam [per 7 p 3] h o maior
              <lb/>
            h e, & t o minor t e:</s>
            <s xml:id="echoid-s11136" xml:space="preserve"> & t e, ſicut prius probatum eſt, æ-
              <lb/>
            qualis eſt h e:</s>
            <s xml:id="echoid-s11137" xml:space="preserve"> [linea igitur h o maior eſt linea t o.</s>
            <s xml:id="echoid-s11138" xml:space="preserve">] Re
              <lb/>
            ſtat ergo, ut t nõ reflectatur ad h, ab alio puncto, quã
              <lb/>
            ab e uel à z.</s>
            <s xml:id="echoid-s11139" xml:space="preserve"> Item à puncto e ducatur linea ſuper dia-
              <lb/>
            metrum t d:</s>
            <s xml:id="echoid-s11140" xml:space="preserve"> quæ ſit e m:</s>
            <s xml:id="echoid-s11141" xml:space="preserve"> & ſecetur à linea h d pars, æ-
              <lb/>
            qualis m d:</s>
            <s xml:id="echoid-s11142" xml:space="preserve"> quæ fit n d:</s>
            <s xml:id="echoid-s11143" xml:space="preserve"> & ducantur e m, e n.</s>
            <s xml:id="echoid-s11144" xml:space="preserve"> Palàm
              <lb/>
            [per 16 p 1] quòd e m d maior eſt recto:</s>
            <s xml:id="echoid-s11145" xml:space="preserve"> [quia angu-
              <lb/>
            lus e t d rectus eſt per fabricationem] ſecetur ex eo
              <lb/>
            æqualis recto per lineam p m [per 23 p 1] quæ cõcur-
              <lb/>
            ret cum d e:</s>
            <s xml:id="echoid-s11146" xml:space="preserve"> [per lemma Procli ad 29 p 1] ſit concur-
              <lb/>
            ſus punctum p:</s>
            <s xml:id="echoid-s11147" xml:space="preserve"> & ducatur n p:</s>
            <s xml:id="echoid-s11148" xml:space="preserve"> & fiat circulus ad quantitatem p d, tranſiens per tria puncta m, d, n.</s>
            <s xml:id="echoid-s11149" xml:space="preserve">
              <lb/>
            Cum p m d ſit rectus [ex fabricatione] erit p d diameter [per conſectarium 5 p 4] & tranſibit circu
              <lb/>
            lus per p, [ut oſtenſum eſt.</s>
            <s xml:id="echoid-s11150" xml:space="preserve">] Palàm ergo, quòd m reflectetur ad n à puncto e:</s>
            <s xml:id="echoid-s11151" xml:space="preserve"> [cum en:</s>
            <s xml:id="echoid-s11152" xml:space="preserve"> m per 4 p 1 tri
              <lb/>
            angulum d m p ſit æquilaterum & æquiangulum triangulo d n p:</s>
            <s xml:id="echoid-s11153" xml:space="preserve"> æquabitur m p ipſi n p, & angulus
              <lb/>
            d p m angulo d p n:</s>
            <s xml:id="echoid-s11154" xml:space="preserve"> ergo per 13 p 1.</s>
            <s xml:id="echoid-s11155" xml:space="preserve"> 3 ax.</s>
            <s xml:id="echoid-s11156" xml:space="preserve"> angulus m p e æquatur angulo n p e, latusq́ue p e commune
              <lb/>
            eſt:</s>
            <s xml:id="echoid-s11157" xml:space="preserve"> angulus igitur m e p æquatur angulo n e p per 4 p 1.</s>
            <s xml:id="echoid-s11158" xml:space="preserve"> Quare per 12 n 4 m & n à puncto e inter ſe
              <lb/>
            mutuò reflectuntur] & ſimiliter à puncto z:</s>
            <s xml:id="echoid-s11159" xml:space="preserve"> & non ab aliquo puncto arcus a b, uel g q:</s>
            <s xml:id="echoid-s11160" xml:space="preserve"> [per 66 n.</s>
            <s xml:id="echoid-s11161" xml:space="preserve">]
              <lb/>
            Et palàm, quòd non ab alio puncto arcus a q, quã à puncto z:</s>
            <s xml:id="echoid-s11162" xml:space="preserve"> & quòd non ab alio puncto arcus b g,
              <lb/>
            quàm à puncto e ſecundum modum prædictum.</s>
            <s xml:id="echoid-s11163" xml:space="preserve"> Sumpto enim puncto, & ductis lineis à punctis t,
              <lb/>
            d, h:</s>
            <s xml:id="echoid-s11164" xml:space="preserve"> & ſumpto puncto, in quo circulus ultimus ſecabit diametrum:</s>
            <s xml:id="echoid-s11165" xml:space="preserve"> & à punctis ſectionis ductis li-
              <lb/>
            neis ad puncta t, h:</s>
            <s xml:id="echoid-s11166" xml:space="preserve"> eadem erit improbatio, quæ prius.</s>
            <s xml:id="echoid-s11167" xml:space="preserve"> Palàm ergo ex prædictis:</s>
            <s xml:id="echoid-s11168" xml:space="preserve"> quòd ſi angulum
              <lb/>
            contentum duabus diametris, per æqualia diuidat tertia diameter:</s>
            <s xml:id="echoid-s11169" xml:space="preserve"> & à termino illius diametri du-
              <lb/>
            cantur perpendiculares ad illas diametros:</s>
            <s xml:id="echoid-s11170" xml:space="preserve"> puncta diametrorum, in quæ cadunt, ad ſe inuicem re-
              <lb/>
            flectuntur à duobus punctis ſpeculi tantùm.</s>
            <s xml:id="echoid-s11171" xml:space="preserve"> P unctorum aũt diametrorum citra hos terminos per-
              <lb/>
            pen dicularium ſumptorum, id eſt uerſus centrum:</s>
            <s xml:id="echoid-s11172" xml:space="preserve"> reflectitur quodlibet à duobus punctis tantùm:</s>
            <s xml:id="echoid-s11173" xml:space="preserve">
              <lb/>
            & unũ reflectitur ad illud, quod æqualiter diſtat à cẽtro:</s>
            <s xml:id="echoid-s11174" xml:space="preserve"> & omniũ talium duplex eſt imaginis locus.</s>
            <s xml:id="echoid-s11175" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div399" type="section" level="0" n="0">
          <head xml:id="echoid-head375" xml:space="preserve" style="it">72. Si angulũ cõprehenſum à duabus diametris in cẽtro circuli (qui eſt cõmunis ſectio ſuper-
            <lb/>
          ficierũ, reflexionis & ſpeculi ſphæricicaui) tertia bifariã ſecet: et ab eius termino in peripheria
            <lb/>
          dicto angulo ſubtẽſa, ſint քpẽdiculares ſuք dict{as} diametros: pũcta diametrorũ inter քipheriã et
            <lb/>
          </head>
        </div>
      </text>
    </echo>
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