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-s12372" xml:space="preserve">
              <pb o="184" file="0190" n="190" rhead="ALHAZEN"/>
            t l, t m, h l, h m.</s>
            <s xml:id="echoid-s12373" xml:space="preserve"> Palàm, quòd quatuor angulorum ſuper q quilibet eſt rectus [per 3 d 11:</s>
            <s xml:id="echoid-s12374" xml:space="preserve"> quia axis pe
              <gap/>
              <gap/>
              <lb/>
            pendicularis eſt circulo l m per 21 d 11] & t q æqualis q h
              <lb/>
              <figure xlink:label="fig-0190-01" xlink:href="fig-0190-01a" number="147">
                <variables xml:id="echoid-variables137" xml:space="preserve">d z b t m l q r p h k f g e a</variables>
              </figure>
            [per fabricationem] & q l æqualis q m:</s>
            <s xml:id="echoid-s12375" xml:space="preserve"> [per 15 d 1] erũt
              <lb/>
            illa triangula ſimilia [per 4 p 1.</s>
            <s xml:id="echoid-s12376" xml:space="preserve"> 4 p.</s>
            <s xml:id="echoid-s12377" xml:space="preserve"> 1 d 6] & anguli t l q, q l
              <lb/>
            h æquales:</s>
            <s xml:id="echoid-s12378" xml:space="preserve"> ſimiliter anguli t m q, q m h æquales.</s>
            <s xml:id="echoid-s12379" xml:space="preserve"> Siergo
              <lb/>
            fuerit t centrum uiſus:</s>
            <s xml:id="echoid-s12380" xml:space="preserve"> reflectetur quidem h ad punctum
              <lb/>
            t à punctol:</s>
            <s xml:id="echoid-s12381" xml:space="preserve"> & ſimiliter à puncto m [per 12 n 4.</s>
            <s xml:id="echoid-s12382" xml:space="preserve">] Si ergo
              <lb/>
            moueatur triangulũ t l h, immoto axe t h:</s>
            <s xml:id="echoid-s12383" xml:space="preserve"> deſcribet pun-
              <lb/>
            ctum l circulũ:</s>
            <s xml:id="echoid-s12384" xml:space="preserve"> & ſemper duo anguli t l q, q l h manebunt
              <lb/>
            æquales:</s>
            <s xml:id="echoid-s12385" xml:space="preserve"> & ſemper in hoc motu reflectetur h ad t.</s>
            <s xml:id="echoid-s12386" xml:space="preserve"> Pro-
              <lb/>
            ducatur autem linea p h k, donec cõcurrat cum linea t l:</s>
            <s xml:id="echoid-s12387" xml:space="preserve">
              <lb/>
            [concurret autẽ per lemma Procli ad 29 p 1:</s>
            <s xml:id="echoid-s12388" xml:space="preserve"> quia m l, c k
              <lb/>
            ſunt parallelę per 29 p 1] & ſit cõcurſus f.</s>
            <s xml:id="echoid-s12389" xml:space="preserve"> Palàm [per 7 n]
              <lb/>
            quòd ferit locus imaginis.</s>
            <s xml:id="echoid-s12390" xml:space="preserve"> Et motu trianguli t l h, mo-
              <lb/>
            uebitur triangulum t f h:</s>
            <s xml:id="echoid-s12391" xml:space="preserve"> & hoc motu punctum f deſcri-
              <lb/>
            bet circulum extra columnam:</s>
            <s xml:id="echoid-s12392" xml:space="preserve"> & totus ille circulus erit
              <lb/>
            locus imaginis.</s>
            <s xml:id="echoid-s12393" xml:space="preserve"> Et hoc eſt propoſitum Idẽ erit probandi
              <lb/>
            modus, ſumptis quibuslibet duobus punctis in axe.</s>
            <s xml:id="echoid-s12394" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div440" type="section" level="0" n="0">
          <head xml:id="echoid-head398" xml:space="preserve" style="it">95. Si communis ſectio ſuperficierum, reflexionis
            <lb/>
          & ſpeculi cylindracei caui fuerit circulus, uelellipſis:
            <lb/>
          reflexio fiet aliâs ab uno: aliâs à duobus: aliâs àtri-
            <lb/>
          bus: aliâs à quatuor ſpeculipũctis: totidem́ uidebun-
            <lb/>
          tur imagines. 14. 15 p 9.</head>
          <p>
            <s xml:id="echoid-s12395" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s12396" xml:space="preserve"> punctorum extra perpẽdicularem uiſus
              <lb/>
            ſumptorum quædam unicam habent imaginem:</s>
            <s xml:id="echoid-s12397" xml:space="preserve">
              <lb/>
            quædam duas:</s>
            <s xml:id="echoid-s12398" xml:space="preserve"> quædam tres:</s>
            <s xml:id="echoid-s12399" xml:space="preserve"> quædam quatuor:</s>
            <s xml:id="echoid-s12400" xml:space="preserve">
              <lb/>
            & non plures.</s>
            <s xml:id="echoid-s12401" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s12402" xml:space="preserve"> ſit a punctum uiſum extra
              <lb/>
            perpendicularem uiſus:</s>
            <s xml:id="echoid-s12403" xml:space="preserve"> & fiat ſuperficies tranſiens per a æquidiſtans baſibus ſpeculi:</s>
            <s xml:id="echoid-s12404" xml:space="preserve"> [ut oftẽſum
              <lb/>
            eſt 47 n] faciet quidem [per 5 th.</s>
            <s xml:id="echoid-s12405" xml:space="preserve"> Sereni de ſectione cylindri] circulum in columna.</s>
            <s xml:id="echoid-s12406" xml:space="preserve"> Sit centrum il-
              <lb/>
            lius circuli h:</s>
            <s xml:id="echoid-s12407" xml:space="preserve"> & ſumatur in ſuperficie circuli aliud
              <lb/>
              <figure xlink:label="fig-0190-02" xlink:href="fig-0190-02a" number="148">
                <variables xml:id="echoid-variables138" xml:space="preserve">s z o r x a h k g m u b d e t l f q p n</variables>
              </figure>
            punctum, quod ſit b:</s>
            <s xml:id="echoid-s12408" xml:space="preserve"> & ducãtur dιametri a h, b h.</s>
            <s xml:id="echoid-s12409" xml:space="preserve">
              <lb/>
            Palàm ex eis, quæ dicta ſunt in ſpeculis ſphæricis
              <lb/>
            concauis [86 n] quòd ab uno puncto arcus, quẽ
              <lb/>
            intercipiunt hæ duæ diametri, poteſt a reflecti ad
              <lb/>
            b:</s>
            <s xml:id="echoid-s12410" xml:space="preserve"> forſitan à duobus punctis, aut tribus, ſed non à
              <lb/>
            pluribus:</s>
            <s xml:id="echoid-s12411" xml:space="preserve"> ab arcu autem oppoſito, nõ niſi ab uno
              <lb/>
            puncto.</s>
            <s xml:id="echoid-s12412" xml:space="preserve"> Sit ergo, quòd a reflectatur ad b à tribus
              <lb/>
            punctιs interciſi arcus:</s>
            <s xml:id="echoid-s12413" xml:space="preserve"> & ſint puncta illa g, d, e:</s>
            <s xml:id="echoid-s12414" xml:space="preserve"> &
              <lb/>
            ducãtur lineæ a g, h g, b g, h d, b d, a d, a e, h e, b e:</s>
            <s xml:id="echoid-s12415" xml:space="preserve"> &
              <lb/>
            à puncto a ducantur in eadẽ ſuperficie tres lineæ
              <lb/>
            æquidiſtantes tribus diametris h g, h d, h e:</s>
            <s xml:id="echoid-s12416" xml:space="preserve"> quæ
              <lb/>
            ſint a k, a f, a n.</s>
            <s xml:id="echoid-s12417" xml:space="preserve"> Cum igitur a k ſit æquidiſtans h g:</s>
            <s xml:id="echoid-s12418" xml:space="preserve">
              <lb/>
            cõcurret b g cum a k:</s>
            <s xml:id="echoid-s12419" xml:space="preserve"> [per lemma Procli ad 29 p 1]
              <lb/>
            concurrat in puncto k.</s>
            <s xml:id="echoid-s12420" xml:space="preserve"> Similiter b d concurret cũ
              <lb/>
            a f:</s>
            <s xml:id="echoid-s12421" xml:space="preserve"> ſit cõcurſus in puncto f.</s>
            <s xml:id="echoid-s12422" xml:space="preserve"> Similiter b e cum a n:</s>
            <s xml:id="echoid-s12423" xml:space="preserve">
              <lb/>
            ſit concurſus in puncto n.</s>
            <s xml:id="echoid-s12424" xml:space="preserve"> Deinde à puncto h eri-
              <lb/>
            gatur axis:</s>
            <s xml:id="echoid-s12425" xml:space="preserve"> qui ſit h x:</s>
            <s xml:id="echoid-s12426" xml:space="preserve"> & à puncto b perpendicula-
              <lb/>
            ris ſuper ſuperficiem circuli:</s>
            <s xml:id="echoid-s12427" xml:space="preserve"> [per 12 p 11] quæ erit
              <lb/>
            æquidiſtans axi [per 6 p 11] quæ ſit b t:</s>
            <s xml:id="echoid-s12428" xml:space="preserve"> & ſumatur
              <lb/>
            in ea punctum quodcunq;</s>
            <s xml:id="echoid-s12429" xml:space="preserve">: quod ſit t:</s>
            <s xml:id="echoid-s12430" xml:space="preserve"> & ducantur
              <lb/>
            tres lineæ t k, t f, t n:</s>
            <s xml:id="echoid-s12431" xml:space="preserve"> & [per 12 p 11] à tribus punctis
              <lb/>
            g, d, e erigantur tres perpẽdiculares ſuper ſuper-
              <lb/>
            ficiem circuli:</s>
            <s xml:id="echoid-s12432" xml:space="preserve"> g m, d l, e q:</s>
            <s xml:id="echoid-s12433" xml:space="preserve"> erunt quidẽ [per 6 p 11]
              <lb/>
            æquidiſtãtes t b, e q:</s>
            <s xml:id="echoid-s12434" xml:space="preserve"> igitur erũt in ſuperficie trian
              <lb/>
            guli t b n:</s>
            <s xml:id="echoid-s12435" xml:space="preserve"> [per 35 d 1.</s>
            <s xml:id="echoid-s12436" xml:space="preserve"> 1 p 11] igitur e q ſecabit t n:</s>
            <s xml:id="echoid-s12437" xml:space="preserve">
              <lb/>
            [per lemma Procli ad 29 p 1] ſecet in puncto q:</s>
            <s xml:id="echoid-s12438" xml:space="preserve"> d l
              <lb/>
            ſecett fin puncto l:</s>
            <s xml:id="echoid-s12439" xml:space="preserve"> g m ſecet t k in puncto m.</s>
            <s xml:id="echoid-s12440" xml:space="preserve"> Et
              <lb/>
            erunt hæ tres perpẽdiculares, lineæ longitudinis
              <lb/>
            columnæ [ut patet è 21 d 11.</s>
            <s xml:id="echoid-s12441" xml:space="preserve">] À
              <unsure/>
            puncto q ducatur
              <lb/>
            æquidiſtans lineæ n a:</s>
            <s xml:id="echoid-s12442" xml:space="preserve"> [per 31 p 1] quæ quidẽ con-
              <lb/>
            curret cum axe x h:</s>
            <s xml:id="echoid-s12443" xml:space="preserve"> [per
              <gap/>
            lemma Procli
              <gap/>
            ad 29 p 1]
              <lb/>
            quoniam erit æquidiſtans e h:</s>
            <s xml:id="echoid-s12444" xml:space="preserve"> [per 30 p 1] ſit con
              <lb/>
            curſus in puncto u:</s>
            <s xml:id="echoid-s12445" xml:space="preserve"> & ducatur linea t a:</s>
            <s xml:id="echoid-s12446" xml:space="preserve"> quam ſeca
              <lb/>
            bit q u:</s>
            <s xml:id="echoid-s12447" xml:space="preserve"> quoniam q u ducitur à latere trianguli [tbn] & linea e q equidiſtãte baſi [t b.</s>
            <s xml:id="echoid-s12448" xml:space="preserve">] Sit punctum
              <lb/>
            </s>
          </p>
        </div>
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