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-s11270" xml:space="preserve">
              <pb o="170" file="0176" n="176" rhead="ALHAZEN"/>
            ducatur r x æqualis lineę ri:</s>
            <s xml:id="echoid-s11271" xml:space="preserve"> & ducantur lineę h x, h i.</s>
            <s xml:id="echoid-s11272" xml:space="preserve"> Palàm ſecundum prædicta, quòd à puncto m
              <lb/>
            non poteſt linea duci ad latus fl, diuidens ipſum eo modo, quo diuidit lineam m c k, præter hanc ſo
              <lb/>
            lam lineam m c k.</s>
            <s xml:id="echoid-s11273" xml:space="preserve"> Si enim poſsit:</s>
            <s xml:id="echoid-s11274" xml:space="preserve"> ſit m p o.</s>
            <s xml:id="echoid-s11275" xml:space="preserve"> Palàm, quòd p o minor erit c k:</s>
            <s xml:id="echoid-s11276" xml:space="preserve"> quod quidẽ patebit ducta
              <lb/>
            linea p q æquidiſtante c k:</s>
            <s xml:id="echoid-s11277" xml:space="preserve"> quę erit minor c k:</s>
            <s xml:id="echoid-s11278" xml:space="preserve"> [cũ enim triangula k c f, q p f ſint æquiangula per 29.</s>
            <s xml:id="echoid-s11279" xml:space="preserve">
              <lb/>
            32 p 1:</s>
            <s xml:id="echoid-s11280" xml:space="preserve"> erit per 4 p 6 ut k f ad q f, ſic k c ad q p:</s>
            <s xml:id="echoid-s11281" xml:space="preserve"> ſed k f maior eſt q f per 9 ax.</s>
            <s xml:id="echoid-s11282" xml:space="preserve"> ergo k c maior eſt q p] &
              <lb/>
            maior p o:</s>
            <s xml:id="echoid-s11283" xml:space="preserve"> [quia maior eſt q p, quæ per 19 p 1 maior eſt o p, cũ angulus p o q ſit obtuſus per 32.</s>
            <s xml:id="echoid-s11284" xml:space="preserve"> 13 p 1]
              <lb/>
            & p l maior c l [per 9 ax:</s>
            <s xml:id="echoid-s11285" xml:space="preserve">] Igitur non erit proportio p o ad p l, ſicut k c ad c l.</s>
            <s xml:id="echoid-s11286" xml:space="preserve"> [Si enim ſit ut k c ad cl,
              <lb/>
            ſic o p ad p l:</s>
            <s xml:id="echoid-s11287" xml:space="preserve"> erit per 14 p 5 c l maior l p, contra 9 ax:</s>
            <s xml:id="echoid-s11288" xml:space="preserve"> quia k c maior eſt o p.</s>
            <s xml:id="echoid-s11289" xml:space="preserve">] Quare non erit propor-
              <lb/>
            tio p o ad p l, ſicut h d ad d t [per 11 p 5.</s>
            <s xml:id="echoid-s11290" xml:space="preserve">] Reſtat ergo ut à puncto m non ducatur alia, quàm m c k, ſimi
              <lb/>
            lis ei.</s>
            <s xml:id="echoid-s11291" xml:space="preserve"> Verùm cũ o d i ſit ęqualis angulo l c m, & angulus o i d ęqualis angulo c l m:</s>
            <s xml:id="echoid-s11292" xml:space="preserve"> [per fabricationẽ]
              <lb/>
            erit triãgulum c l m ſimile triangulo i o d [per 32 p 1.</s>
            <s xml:id="echoid-s11293" xml:space="preserve"> 4 p.</s>
            <s xml:id="echoid-s11294" xml:space="preserve"> 1 d 6.</s>
            <s xml:id="echoid-s11295" xml:space="preserve">] Igitur angulus i o d erit æqualis angu
              <lb/>
            lo l m c:</s>
            <s xml:id="echoid-s11296" xml:space="preserve"> reſtat [per 13 p 1] angulus r o h æqualis angulo k m n:</s>
            <s xml:id="echoid-s11297" xml:space="preserve"> & angulus h r o rectus ęqualis erit an-
              <lb/>
            gulo k n m:</s>
            <s xml:id="echoid-s11298" xml:space="preserve"> reſtat [per 32 p 1] angulus n k m æqualis angulo r h o.</s>
            <s xml:id="echoid-s11299" xml:space="preserve"> Ducta autem linea d i, donec con-
              <lb/>
            currat cum h r in puncto s:</s>
            <s xml:id="echoid-s11300" xml:space="preserve">
              <lb/>
              <figure xlink:label="fig-0176-01" xlink:href="fig-0176-01a" number="114">
                <variables xml:id="echoid-variables104" xml:space="preserve">b u a x r o i c p e d z s h g q</variables>
              </figure>
              <figure xlink:label="fig-0176-02" xlink:href="fig-0176-02a" number="115">
                <variables xml:id="echoid-variables105" xml:space="preserve">l m c k p q o f n y</variables>
              </figure>
            [concurret aũt per 11 ax:</s>
            <s xml:id="echoid-s11301" xml:space="preserve"> ꝗa
              <lb/>
            angulus ad r rectus eſt, ad i
              <lb/>
            uerò acutus] erit angulus s
              <lb/>
            d h æqualis angulo k c f [ք
              <lb/>
            15 p 1.</s>
            <s xml:id="echoid-s11302" xml:space="preserve"> 1 ax:</s>
            <s xml:id="echoid-s11303" xml:space="preserve">] & erit triangulũ
              <lb/>
            s d h ſimile triãgulo c k f [ք
              <lb/>
            32 p 1.</s>
            <s xml:id="echoid-s11304" xml:space="preserve"> 4 p.</s>
            <s xml:id="echoid-s11305" xml:space="preserve"> 1 d 6.</s>
            <s xml:id="echoid-s11306" xml:space="preserve">] Igitur pro-
              <lb/>
            portio s d ad d h, ſicut f c ad
              <lb/>
            c k:</s>
            <s xml:id="echoid-s11307" xml:space="preserve"> ſed [per 7 p 5] h d ad d i
              <lb/>
            [æqualẽ ipſi d z per 15 d 1]
              <lb/>
            ſicut k c ad c l [ք fabricatio
              <lb/>
            nẽ.</s>
            <s xml:id="echoid-s11308" xml:space="preserve">] Igitur [per 22 p 5] s d
              <lb/>
            ad d i, ſicut f c ad c l:</s>
            <s xml:id="echoid-s11309" xml:space="preserve"> igitur
              <lb/>
            [per 18 p 5] s i ad d i, ſicut fl
              <lb/>
            ad c l:</s>
            <s xml:id="echoid-s11310" xml:space="preserve"> ſed d i ad i o, ſicut c l
              <lb/>
            ad l m:</s>
            <s xml:id="echoid-s11311" xml:space="preserve"> cũ triangulũ d i o ſit
              <lb/>
            ſimile triangulo c l m.</s>
            <s xml:id="echoid-s11312" xml:space="preserve"> Igitur [per 22 p 5] s i ad i o, ſicut fl ad l m [& ք conſectariũ 4 p 5, ut i o ad s i, ſic
              <lb/>
            l m ad fl.</s>
            <s xml:id="echoid-s11313" xml:space="preserve">] Sed proportio s i ad i r, ſicut fl ad l n:</s>
            <s xml:id="echoid-s11314" xml:space="preserve"> quoniam triangulũ s i r ſimile eſt triangulo fl n [an-
              <lb/>
            gulus enim r i s æquatur angulo fl n per fabricationem, & s r i rectus fn l recto:</s>
            <s xml:id="echoid-s11315" xml:space="preserve"> ergo per 32 p 1.</s>
            <s xml:id="echoid-s11316" xml:space="preserve"> 4 p.</s>
            <s xml:id="echoid-s11317" xml:space="preserve"> 1
              <lb/>
            d 6 triangula s i r, fl n ſunt ſimilia.</s>
            <s xml:id="echoid-s11318" xml:space="preserve">] Igitur [per 22 p 5] proportio i o ad i r, ſicut l m ad l n.</s>
            <s xml:id="echoid-s11319" xml:space="preserve"> [& percon
              <lb/>
            ſectarium 4 p 5 ut i r ad i o, ſic l n ad l m.</s>
            <s xml:id="echoid-s11320" xml:space="preserve">] Igitur proportio y m ad l m, ſicut x o ad i o.</s>
            <s xml:id="echoid-s11321" xml:space="preserve"> [Quia enim xi
              <lb/>
            dupla eſt ipſius i r, & y l dupla ipſius l n:</s>
            <s xml:id="echoid-s11322" xml:space="preserve"> erit igitur per 15 p 5 ut x i ad i o, ſic y l ad l m, & ք 17 p 5 ut x o
              <lb/>
            ad i o, ſic y m ad l m.</s>
            <s xml:id="echoid-s11323" xml:space="preserve">] Ducta autem à puncto i ęquidiſtante linea u i, lineę h x, & producta linea d a,
              <lb/>
            donec concurrat cum u i [concurret autem per lemma Procli ad 29 p 1] concurrat in puncto u:</s>
            <s xml:id="echoid-s11324" xml:space="preserve"> erit
              <lb/>
            triangulum o u i triangulo h o x ſimile [per 15.</s>
            <s xml:id="echoid-s11325" xml:space="preserve"> 29.</s>
            <s xml:id="echoid-s11326" xml:space="preserve"> 32 p 1.</s>
            <s xml:id="echoid-s11327" xml:space="preserve"> 4 p.</s>
            <s xml:id="echoid-s11328" xml:space="preserve"> 1 d 6.</s>
            <s xml:id="echoid-s11329" xml:space="preserve">] Igitur erit proportio h o ad o u, ſi-
              <lb/>
            cut y m ad m l:</s>
            <s xml:id="echoid-s11330" xml:space="preserve"> [Quia ob triãgulorum o u i, h o x oſtenſam ſimilitu dinẽ eſt, ut h o ad o u, ſic x o ad i o,
              <lb/>
            & ut x o ad i o, ſic y m ad m l ex cõcluſo:</s>
            <s xml:id="echoid-s11331" xml:space="preserve"> ergo per 11 p 5, ut h o ad o u, ſic y m ad m l] & ita h o ad o u,
              <lb/>
            ſicut h d ad d t.</s>
            <s xml:id="echoid-s11332" xml:space="preserve"> [Fuit enim per fabricationẽ h d ad d t, ſicut y m ad m l.</s>
            <s xml:id="echoid-s11333" xml:space="preserve">] Sed quoniã [per 4 p 1] triãgu
              <lb/>
            lũ h r i æquale eſt triangulo h r x:</s>
            <s xml:id="echoid-s11334" xml:space="preserve"> cũ h r ſit perpendicularis [per fabricationẽ:</s>
            <s xml:id="echoid-s11335" xml:space="preserve"> & x r æquetur ipſi r i,
              <lb/>
            latusq́;</s>
            <s xml:id="echoid-s11336" xml:space="preserve"> r h cõmune ſit.</s>
            <s xml:id="echoid-s11337" xml:space="preserve">] Igitur angulus h x r æqualis eſt angulo r i h:</s>
            <s xml:id="echoid-s11338" xml:space="preserve"> & ita r i h æqualis eſt angulo u i
              <lb/>
            o [quia u i o æquatur ipſι h x o propter ſimilitu dinem triangulorum u i o, h o x.</s>
            <s xml:id="echoid-s11339" xml:space="preserve">] Quare [per 3 p 6]
              <lb/>
            proportio h o ad o u, ſicut h i ad i u:</s>
            <s xml:id="echoid-s11340" xml:space="preserve"> & ita [per 11 p 5] h i ad i u, ſicut h d ad d t.</s>
            <s xml:id="echoid-s11341" xml:space="preserve"> Verùm angulus u i d ma
              <lb/>
            ior eſt angulo d i h:</s>
            <s xml:id="echoid-s11342" xml:space="preserve"> [quia æqualis concluſus eſt angulo o i h] ſecetur ab eo æqualis:</s>
            <s xml:id="echoid-s11343" xml:space="preserve"> & ſit p i d:</s>
            <s xml:id="echoid-s11344" xml:space="preserve"> & du-
              <lb/>
            catur linea p t:</s>
            <s xml:id="echoid-s11345" xml:space="preserve"> & p ſit punctum diametri d a.</s>
            <s xml:id="echoid-s11346" xml:space="preserve"> Palàm, quòd proportio h i ad u i cõſtat ex proportione
              <lb/>
            h i ad i p, & p i ad u i:</s>
            <s xml:id="echoid-s11347" xml:space="preserve"> [quia ratio extremorum componitur ex omnibus rationibus intermedijs, ut
              <lb/>
            Theon demonſtrauit ad 5 d 6.</s>
            <s xml:id="echoid-s11348" xml:space="preserve">] & [per 3 p 6] proportio h i ad i p, ſicut d h ad d p:</s>
            <s xml:id="echoid-s11349" xml:space="preserve"> quoniam d i diuidit
              <lb/>
            angulum p i h per ęqualia.</s>
            <s xml:id="echoid-s11350" xml:space="preserve"> Igitur proportio h i ad u i (quæ eſt h d ad d t) conſtat ex proportione h d
              <lb/>
            ad d p, & d p ad d t.</s>
            <s xml:id="echoid-s11351" xml:space="preserve"> Igitur proportio d p ad d t, ſicut p i ad u i.</s>
            <s xml:id="echoid-s11352" xml:space="preserve"> Verùm angulus o i h eſt medietas angu
              <lb/>
            li u i h:</s>
            <s xml:id="echoid-s11353" xml:space="preserve"> [ex concluſo] ſed angulus d i h medietas eſt anguli p i h:</s>
            <s xml:id="echoid-s11354" xml:space="preserve"> reſtat angulus d i o medietas anguli
              <lb/>
            p i u.</s>
            <s xml:id="echoid-s11355" xml:space="preserve"> Sed angulus d i o eſt medietas anguli t d p:</s>
            <s xml:id="echoid-s11356" xml:space="preserve"> quia eſt æqualis angulo fl m [qui ęquatus eſt dimi-
              <lb/>
            diato angulo a d t ſeu p d t.</s>
            <s xml:id="echoid-s11357" xml:space="preserve">] Igitur angulus p i u eſt ęqualis angulo t d p:</s>
            <s xml:id="echoid-s11358" xml:space="preserve"> & proportio d p ad d t, ſicut
              <lb/>
            p i ad u i.</s>
            <s xml:id="echoid-s11359" xml:space="preserve"> Igitur triangulũ u i p ſimile triangulo t p d:</s>
            <s xml:id="echoid-s11360" xml:space="preserve"> [per 6.</s>
            <s xml:id="echoid-s11361" xml:space="preserve"> 4 p.</s>
            <s xml:id="echoid-s11362" xml:space="preserve"> 1 d 6] & angulus u p i æqualis t p d:</s>
            <s xml:id="echoid-s11363" xml:space="preserve">
              <lb/>
            erit igitur [per 14 p 1] t p i linea recta:</s>
            <s xml:id="echoid-s11364" xml:space="preserve"> quia angulus d p t cum angulo t p o ualet duos rectos:</s>
            <s xml:id="echoid-s11365" xml:space="preserve"> & ita an
              <lb/>
            gulus o p i cum angulo o p t ualet duos rectos.</s>
            <s xml:id="echoid-s11366" xml:space="preserve"> [Idem uerò patet per conuerſionem 15 p 1 à Proclo
              <lb/>
            demonſtratã.</s>
            <s xml:id="echoid-s11367" xml:space="preserve">] Et ita [per 12 n 4] treflectetur ad h à puncto i.</s>
            <s xml:id="echoid-s11368" xml:space="preserve"> [quia linea t p i eſt linea incidentię, &
              <lb/>
            anguli t i d, h i d ſunt æquales per fabricationem.</s>
            <s xml:id="echoid-s11369" xml:space="preserve">] Et eadem erit probatio, ſiue ſit t extra circulum,
              <lb/>
            ſiue intra.</s>
            <s xml:id="echoid-s11370" xml:space="preserve"> Et ſimiliter ſumpto puncto h extra uel intra:</s>
            <s xml:id="echoid-s11371" xml:space="preserve"> dum inęqualiter diſtent à centro.</s>
            <s xml:id="echoid-s11372" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div403" type="section" level="0" n="0">
          <head xml:id="echoid-head377" xml:space="preserve" style="it">74. Si angulum comprehenſum à duabus diametris in centro circuli (qui eſt cõmunis ſectio
            <lb/>
          ſuperficierum reflexionis & ſpeculi ſphærici caui) tertia bifariam ſecet: puncta in dictis dia-
            <lb/>
          </head>
        </div>
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