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-s41395" xml:space="preserve">
              <pb o="330" file="0632" n="632" rhead="VITELLONIS OPTICAE"/>
            th 1 huius:</s>
            <s xml:id="echoid-s41396" xml:space="preserve"> manifeſtum eſt quia non eſt poſsibile in propoſito arcu inueniri aliud punctum pręmiſ-
              <lb/>
            ſæ reflexionis.</s>
            <s xml:id="echoid-s41397" xml:space="preserve"> Patet ergo, quod proponebatur.</s>
            <s xml:id="echoid-s41398" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1647" type="section" level="0" n="0">
          <head xml:id="echoid-head1238" xml:space="preserve" style="it">28. Si angulum à duabus diametris circuli magni ſpeculi ſphærici concaui contentum diui-
            <lb/>
          dat alia diameter per æqualia: ab omni puncto arcus interiacentis ſemidiametros primas, in
            <lb/>
          quibus punct a reflexanõ conſiſtunt (præter punctum, cui incidit diameter angulum diuidens)
            <lb/>
          infinit a punctorum paria inæqualiter à centro circuli diſtantiũ reflectuntur. Alhaz. 74 n 5.</head>
          <p>
            <s xml:id="echoid-s41399" xml:space="preserve">Sit diſpoſitio figuræ præ cedentis:</s>
            <s xml:id="echoid-s41400" xml:space="preserve"> ſecentq́;</s>
            <s xml:id="echoid-s41401" xml:space="preserve"> circulum (qui eſt communis ſectio ſuperficiei refle-
              <lb/>
            xionis & ſuperficiei ſpeculi ſphærici concaui) duæ diametri, quæ ſint b q & a g, ſuper centrum d:</s>
            <s xml:id="echoid-s41402" xml:space="preserve">
              <lb/>
            diuidatq́;</s>
            <s xml:id="echoid-s41403" xml:space="preserve"> diameter e d z angulum b d g per æqualia.</s>
            <s xml:id="echoid-s41404" xml:space="preserve"> Dico quòd quicunq;</s>
            <s xml:id="echoid-s41405" xml:space="preserve"> punctus ſumatur in arcu
              <lb/>
            a q, pręter punctum z, ab illo poſſunt reflecti infinita paria punctorum inæqualiter à centro diſtan-
              <lb/>
            tium.</s>
            <s xml:id="echoid-s41406" xml:space="preserve"> Sumatur enim in arcu a q punctus h:</s>
            <s xml:id="echoid-s41407" xml:space="preserve"> & ſumatur in ſemidiametro d g punctus l:</s>
            <s xml:id="echoid-s41408" xml:space="preserve"> & à ſemidia-
              <lb/>
            metro b d ſecetur linea m d æqualis lineæ l d:</s>
            <s xml:id="echoid-s41409" xml:space="preserve"> & ducantur lineæ l m, l h, m h, d h:</s>
            <s xml:id="echoid-s41410" xml:space="preserve"> ſecabitq́;</s>
            <s xml:id="echoid-s41411" xml:space="preserve"> diameter
              <lb/>
            e z lineam m l per 29 th.</s>
            <s xml:id="echoid-s41412" xml:space="preserve"> 1 huius, quia ſecat angulum b d g, cui ſubtenditur linea l m:</s>
            <s xml:id="echoid-s41413" xml:space="preserve"> ſit ergo punctus
              <lb/>
            ſectionis f:</s>
            <s xml:id="echoid-s41414" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s41415" xml:space="preserve"> per 4 p 1 & ex hypotheſi linea m f æqualis lineæ fl.</s>
            <s xml:id="echoid-s41416" xml:space="preserve"> Producatur quoq;</s>
            <s xml:id="echoid-s41417" xml:space="preserve"> linea h d,
              <lb/>
            quouſq;</s>
            <s xml:id="echoid-s41418" xml:space="preserve"> cadat ſuper lineam m l:</s>
            <s xml:id="echoid-s41419" xml:space="preserve"> cadet autem per 29 th.</s>
            <s xml:id="echoid-s41420" xml:space="preserve"> 1 huius:</s>
            <s xml:id="echoid-s41421" xml:space="preserve"> ſitq́;</s>
            <s xml:id="echoid-s41422" xml:space="preserve"> punctus ſectionis n:</s>
            <s xml:id="echoid-s41423" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s41424" xml:space="preserve"> linea
              <lb/>
            l n minor quàm linea n m:</s>
            <s xml:id="echoid-s41425" xml:space="preserve"> ideo, quia linea d n ſecat angulum f d l:</s>
            <s xml:id="echoid-s41426" xml:space="preserve"> quia angulus h d z (qui per 15 p 1
              <lb/>
            eſt æqualis angulo n d f) minor eſt angulo a d z.</s>
            <s xml:id="echoid-s41427" xml:space="preserve"> Ve-
              <lb/>
              <figure xlink:label="fig-0632-01" xlink:href="fig-0632-01a" number="751">
                <variables xml:id="echoid-variables728" xml:space="preserve">b a m h t e f d z p n l g q</variables>
              </figure>
            rùm eum angulus f d m ſit æqualis angulo ſ d l ex hy-
              <lb/>
            potheſi, & angulo q d z per 15 p 1, & angulus m d a ſit
              <lb/>
            æqualis angulo l d q:</s>
            <s xml:id="echoid-s41428" xml:space="preserve"> & angulus a d h æqualis angulo
              <lb/>
            n d l:</s>
            <s xml:id="echoid-s41429" xml:space="preserve"> angulus uerò m d n eſt maior angulo n d l, &
              <lb/>
            angulus h d q eſt maior angulo a d h:</s>
            <s xml:id="echoid-s41430" xml:space="preserve"> ergo totus an-
              <lb/>
            gulus l d h eſt maior toto angulo m d h:</s>
            <s xml:id="echoid-s41431" xml:space="preserve"> igitur per 24
              <lb/>
            p 1 linea l h eſt maior quàm linea h m, cum linea m d
              <lb/>
            ſit æqualis lineæ d l, & linea d h communis ambobus
              <lb/>
            trigonis m d h & l d h.</s>
            <s xml:id="echoid-s41432" xml:space="preserve"> Erit ergo angulus d h l minor
              <lb/>
            angulo d h m.</s>
            <s xml:id="echoid-s41433" xml:space="preserve"> Quoniã ſi detur, quòd ſit æqualis:</s>
            <s xml:id="echoid-s41434" xml:space="preserve"> tunc
              <lb/>
            erit proportio lineæ l h ad lineam m h, ſicut lineæ l n
              <lb/>
            ad lineam n m per 3 p 6:</s>
            <s xml:id="echoid-s41435" xml:space="preserve"> quod eſt impoſsibile per 8 p
              <lb/>
            5.</s>
            <s xml:id="echoid-s41436" xml:space="preserve"> Si uerò detur quòd angulus d h l ſit maior angulo
              <lb/>
            d h m:</s>
            <s xml:id="echoid-s41437" xml:space="preserve"> ergo per 27 th.</s>
            <s xml:id="echoid-s41438" xml:space="preserve"> 1 huius ſecetur ex angulo d h l
              <lb/>
            angulus æqualis angulo d h m:</s>
            <s xml:id="echoid-s41439" xml:space="preserve"> & ſequetur impoſsi-
              <lb/>
            bile, ut prius, producta illa linea ſecante, ad lineam l
              <lb/>
            n per 29 th.</s>
            <s xml:id="echoid-s41440" xml:space="preserve"> 1 huius.</s>
            <s xml:id="echoid-s41441" xml:space="preserve"> Eſt igitur angulus d h l minor an-
              <lb/>
            gulo d h m.</s>
            <s xml:id="echoid-s41442" xml:space="preserve"> Secetur igitur ab angulo m h d angulus æqualis angulo d h l, qui ſit angulus t h d.</s>
            <s xml:id="echoid-s41443" xml:space="preserve"> Ergo
              <lb/>
            forma puncti t per 20 th.</s>
            <s xml:id="echoid-s41444" xml:space="preserve"> 5 huius refle ctetur ad uiſum exiſtentem in puncto l à puncto ſpeculi, quod
              <lb/>
            eſt h:</s>
            <s xml:id="echoid-s41445" xml:space="preserve"> & linea t d eſt minor quàm linea l d:</s>
            <s xml:id="echoid-s41446" xml:space="preserve"> quoniam eſt minor quàm linea d m.</s>
            <s xml:id="echoid-s41447" xml:space="preserve"> Similiter ſi ſumantur
              <lb/>
            in ſemidiam etris b g & g d alia pũcta quàm l & m, æqualiter diſtantia à punctis l & m:</s>
            <s xml:id="echoid-s41448" xml:space="preserve"> ſimiliter pro-
              <lb/>
            babitur quòd à puncto h fit reflexio punctorum in æqualiter diſtantium à centro adinuicem:</s>
            <s xml:id="echoid-s41449" xml:space="preserve"> & ita
              <lb/>
            de infinitis punctis in his diametris ſumptis ſemper ſimilis erit probatio:</s>
            <s xml:id="echoid-s41450" xml:space="preserve"> & à quocunq;</s>
            <s xml:id="echoid-s41451" xml:space="preserve"> puncto ar-
              <lb/>
            cus a q, præter quàm à puncto z, eadem eſt demonſtratio.</s>
            <s xml:id="echoid-s41452" xml:space="preserve"> A puncto uero z non eſt poſsibilis refle-
              <lb/>
            xio propter angulorum t z d & d z linæqualitatem:</s>
            <s xml:id="echoid-s41453" xml:space="preserve"> quæ patet per 4 p 1, reſecta per 3 p 1 linea l d in
              <lb/>
            puncto p ad æqualitatem lineæ d t, & copulata linea p z.</s>
            <s xml:id="echoid-s41454" xml:space="preserve"> Patet ergo propoſitum.</s>
            <s xml:id="echoid-s41455" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1649" type="section" level="0" n="0">
          <figure number="752">
            <variables xml:id="echoid-variables729" xml:space="preserve">b a t h e p d z n l k g q</variables>
          </figure>
          <head xml:id="echoid-head1239" xml:space="preserve" style="it">29. Puncto rei uiſæ & cẽtro uiſus intra ſpeculum in diuerſis diametris circuli magni ſpeculi
            <lb/>
          ſphærici concaui exiſtẽtibus, inæqualiter́ dιſtan-
            <lb/>
          tibus à centro: ſi ab aliquo puncto ſpeculi arcus ſci-
            <lb/>
          licet interiacentis ſemidiametros, in quibus illa punct a non conſiſtunt, fiat reflexio formarũ eiuſ- dem puncti ad eundem uiſum: ab alio puncto eiuſ- dem arcus eſt impoßibile reflecti. Alhaz. 75 n 5.</head>
          <p>
            <s xml:id="echoid-s41456" xml:space="preserve">Remaneat omnimoda diſpoſitio theorematis prę-
              <lb/>
            cedentis:</s>
            <s xml:id="echoid-s41457" xml:space="preserve"> & ſit, ut pũctus rei uiſæ, (qui eſt t) in ſemi-
              <lb/>
            diametro circuli d b à puncto arcus a q, (quod ſit h)
              <lb/>
            refle ctatur ad uiſum exiſtentẽ in pũcto l ſemidiame-
              <lb/>
            tri d g plus diſtantẽ à cẽtro ſpeculi, qđ eſt d, quã pũ-
              <lb/>
            ctus rei uiſæ, qui eſt t:</s>
            <s xml:id="echoid-s41458" xml:space="preserve"> ſintq́;</s>
            <s xml:id="echoid-s41459" xml:space="preserve"> puncta t & l ambo intra
              <lb/>
            ſpeculũ.</s>
            <s xml:id="echoid-s41460" xml:space="preserve"> Dico quòd formá pũcti t ad uiſum limpoſ-
              <lb/>
            ſibile eſt reflecti ab alio pũcto arc{us} a q, quàm à pũcto
              <lb/>
            h.</s>
            <s xml:id="echoid-s41461" xml:space="preserve"> Si enim ſit ipſum poſsibile ab alio puncto reflecti
              <lb/>
            ad uiſum l:</s>
            <s xml:id="echoid-s41462" xml:space="preserve"> ſit illud punctũ k:</s>
            <s xml:id="echoid-s41463" xml:space="preserve"> & ducátur lineæ t k, l k,
              <lb/>
            d k, l t, t h, l h:</s>
            <s xml:id="echoid-s41464" xml:space="preserve"> & linea n d h:</s>
            <s xml:id="echoid-s41465" xml:space="preserve"> & producatur linea k d,
              <lb/>
            quouſq;</s>
            <s xml:id="echoid-s41466" xml:space="preserve"> cadatin lineá l t in punctũ p:</s>
            <s xml:id="echoid-s41467" xml:space="preserve"> cadet autẽ per 29 th.</s>
            <s xml:id="echoid-s41468" xml:space="preserve"> 1 huius, ut in præmiſſa oſtendimus.</s>
            <s xml:id="echoid-s41469" xml:space="preserve"> </s>
          </p>
        </div>
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