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-s8713" xml:space="preserve">
              <pb o="145" file="0151" n="151" rhead="OPTICAE LIBER V."/>
            ducatur æquidiſtans lineæ h m:</s>
            <s xml:id="echoid-s8714" xml:space="preserve"> quæ ſit z n:</s>
            <s xml:id="echoid-s8715" xml:space="preserve"> quæ quidem ſecabit h l [per lemma Procli ad 29 p 1] &
              <lb/>
            à puncto z ducatur æquidiſtans h l:</s>
            <s xml:id="echoid-s8716" xml:space="preserve"> quæ ſit z t:</s>
            <s xml:id="echoid-s8717" xml:space="preserve"> & ſecet h m in puncto t:</s>
            <s xml:id="echoid-s8718" xml:space="preserve"> & à puncto t ducatur ſectio
              <lb/>
            pyramidis t p, quam aſsignauit Apollonius in libro pyramidum [4 th 2:</s>
            <s xml:id="echoid-s8719" xml:space="preserve">] quæ quidem ſectio non
              <lb/>
            continget aliquam linearum z n, h l, inter quas iacet.</s>
            <s xml:id="echoid-s8720" xml:space="preserve"> Similiter fiat ſectio pyramidis ei oppoſita
              <lb/>
            inter eaſdem lineas:</s>
            <s xml:id="echoid-s8721" xml:space="preserve"> quæ ſit c u.</s>
            <s xml:id="echoid-s8722" xml:space="preserve"> Cum igitur li-
              <lb/>
              <figure xlink:label="fig-0151-01" xlink:href="fig-0151-01a" number="69">
                <variables xml:id="echoid-variables59" xml:space="preserve">a g e b d</variables>
              </figure>
              <figure xlink:label="fig-0151-02" xlink:href="fig-0151-02a" number="70">
                <variables xml:id="echoid-variables60" xml:space="preserve">h n t f x q c u p m z ſ</variables>
              </figure>
            nea minima ex lineis à puncto t ad ſectionem
              <lb/>
            c u ductis, fuerit æqualis diametro b g:</s>
            <s xml:id="echoid-s8723" xml:space="preserve"> circu-
              <lb/>
            lus factus ſecundum hanc minimam lineam,
              <lb/>
            poſito pede circini ſuper punctum t:</s>
            <s xml:id="echoid-s8724" xml:space="preserve"> contin-
              <lb/>
            get ſectionem c u.</s>
            <s xml:id="echoid-s8725" xml:space="preserve"> Si uerò minima ex lineis à
              <lb/>
            puncto t ad ſectionem c u ductis, fuerit minor
              <lb/>
            diametro b g:</s>
            <s xml:id="echoid-s8726" xml:space="preserve"> circulus factus modo prædicto
              <lb/>
            ſecundum quãtitatem b g, ſecabit ſectionem
              <lb/>
            in duobus punctis.</s>
            <s xml:id="echoid-s8727" xml:space="preserve"> Sit ergo t c minima, & æ-
              <lb/>
            qualis diametro b g:</s>
            <s xml:id="echoid-s8728" xml:space="preserve"> quæ quidem ſecabit z n
              <lb/>
            & h l:</s>
            <s xml:id="echoid-s8729" xml:space="preserve"> cum ducatur ad ſectionem, quæ inter
              <lb/>
            eas interiacet:</s>
            <s xml:id="echoid-s8730" xml:space="preserve"> & [per 31 p 1] ducatur à puncto
              <lb/>
            z æ quidiſtans huic:</s>
            <s xml:id="echoid-s8731" xml:space="preserve"> quæ quidem ſecabit h m,
              <lb/>
            h l [per lemma Procli ad 29 p 1] ſicut ſua ęqui
              <lb/>
            diſtans t c.</s>
            <s xml:id="echoid-s8732" xml:space="preserve"> Secet ergo in punctis m l:</s>
            <s xml:id="echoid-s8733" xml:space="preserve"> & ſit m z
              <lb/>
            l:</s>
            <s xml:id="echoid-s8734" xml:space="preserve"> & punctum ſectionis, in quo t c ſecat z n, ſit
              <lb/>
            q:</s>
            <s xml:id="echoid-s8735" xml:space="preserve"> & [per 23 p 1] ſuper diametrum g b fiat an-
              <lb/>
            gulus ęqualis angulo h l m:</s>
            <s xml:id="echoid-s8736" xml:space="preserve"> qui ſit d g b, & du-
              <lb/>
            cantur lineę duę a d, d b.</s>
            <s xml:id="echoid-s8737" xml:space="preserve"> Palàm ergo, cũ [per
              <lb/>
            31 p 3] angulus g a b ſit rectus:</s>
            <s xml:id="echoid-s8738" xml:space="preserve"> alij duo anguli
              <lb/>
            trianguli a g b ualent rectũ [per 32 p 1.</s>
            <s xml:id="echoid-s8739" xml:space="preserve">] Quare
              <lb/>
            angulus l h m eſt rectus:</s>
            <s xml:id="echoid-s8740" xml:space="preserve"> [conſtat enim è duo-
              <lb/>
            bus angulis per fabricationẽ ęqualib.</s>
            <s xml:id="echoid-s8741" xml:space="preserve"> angulis
              <lb/>
            a g b, a b g rectũ ęquantibus] & eſt æqualis an
              <lb/>
            gulo g d b:</s>
            <s xml:id="echoid-s8742" xml:space="preserve"> & [per fabricationem] angulus h l
              <lb/>
            m eſt æqualis angulo d g b:</s>
            <s xml:id="echoid-s8743" xml:space="preserve"> Igitur [per 32 p 1]
              <lb/>
            tertius tertio:</s>
            <s xml:id="echoid-s8744" xml:space="preserve"> & triangulum ſimile triangulo
              <lb/>
            [per 4 p.</s>
            <s xml:id="echoid-s8745" xml:space="preserve"> 1 d 6.</s>
            <s xml:id="echoid-s8746" xml:space="preserve">] Quare ꝓportio b g ad b d eſt,
              <lb/>
            ſicut l m ad m h.</s>
            <s xml:id="echoid-s8747" xml:space="preserve"> Sed quoniam [per 27 p 3] angulus a d b æqualis eſt angulo b g a:</s>
            <s xml:id="echoid-s8748" xml:space="preserve"> quia cadunt in
              <lb/>
            eundem arcum:</s>
            <s xml:id="echoid-s8749" xml:space="preserve"> & angulus b g a æqualis angulo m h z:</s>
            <s xml:id="echoid-s8750" xml:space="preserve"> [per fabricationem] eſt ergo [per 1 ax.</s>
            <s xml:id="echoid-s8751" xml:space="preserve">] an-
              <lb/>
            gulus a d b æqualis angulo m h z.</s>
            <s xml:id="echoid-s8752" xml:space="preserve"> Et iam habemus, quòd angulus d b g eſt æqualis angulo h m z.</s>
            <s xml:id="echoid-s8753" xml:space="preserve"> I-
              <lb/>
            gitur [per 32 p 1] tertius tertio:</s>
            <s xml:id="echoid-s8754" xml:space="preserve"> & triangulum d e b ſimile triangulo m h z.</s>
            <s xml:id="echoid-s8755" xml:space="preserve"> [per 4 p.</s>
            <s xml:id="echoid-s8756" xml:space="preserve"> 1 d 6] Sit autem
              <lb/>
            e punctum, in quo linea a d ſecat diametrum b g.</s>
            <s xml:id="echoid-s8757" xml:space="preserve"> Igitur proportio b d ad d e, ſicut m h ad h z.</s>
            <s xml:id="echoid-s8758" xml:space="preserve"> Verùm
              <lb/>
            Apollonius [16 th 2] probat:</s>
            <s xml:id="echoid-s8759" xml:space="preserve"> quòd cum fuerint duæ ſectiones oppoſitæ, & producatur linea à ſe-
              <lb/>
            ctione ad aliam:</s>
            <s xml:id="echoid-s8760" xml:space="preserve"> pars eius, quæ interiacet inter unam ſectionẽ, & unam ex lineis, eſt ęqualis alij par-
              <lb/>
            ti, quæ interiacet inter aliam ſectionem, & aliam lineam.</s>
            <s xml:id="echoid-s8761" xml:space="preserve"> Quare q t æqualis eſt c f.</s>
            <s xml:id="echoid-s8762" xml:space="preserve"> Sed [per 34 p 1] t
              <lb/>
            q eſt æqualis m z:</s>
            <s xml:id="echoid-s8763" xml:space="preserve"> cum ſit illi æquidiſtans, inter duas æquidiſtantes.</s>
            <s xml:id="echoid-s8764" xml:space="preserve"> Igitur [per 1 ax.</s>
            <s xml:id="echoid-s8765" xml:space="preserve">] m z æqualis f
              <lb/>
            c:</s>
            <s xml:id="echoid-s8766" xml:space="preserve"> & [per 34 p 1] z l æqualis t f.</s>
            <s xml:id="echoid-s8767" xml:space="preserve"> Igitur [per 2 ax.</s>
            <s xml:id="echoid-s8768" xml:space="preserve">] m l æqualis t c.</s>
            <s xml:id="echoid-s8769" xml:space="preserve"> Quare proportio t c ad h z, ſicut m
              <lb/>
            l ad h z.</s>
            <s xml:id="echoid-s8770" xml:space="preserve"> [per 7 p 5] Quare proportio g b ad e d, ſicut t c ad h z.</s>
            <s xml:id="echoid-s8771" xml:space="preserve"> [demõſtratũ enim eſt, ut g b ad b d, ſic
              <lb/>
            l m ad m h:</s>
            <s xml:id="echoid-s8772" xml:space="preserve"> item ut b d ad d e, ſic m h ad h z:</s>
            <s xml:id="echoid-s8773" xml:space="preserve"> ergo per 22 p 5, ut g b ad d e, ſic l m ad h z:</s>
            <s xml:id="echoid-s8774" xml:space="preserve"> ſed ut l m ad h
              <lb/>
            z, ſic t c a d h z:</s>
            <s xml:id="echoid-s8775" xml:space="preserve"> quare per 11 p 5 ut g b ad d e, ſic t c ad h z.</s>
            <s xml:id="echoid-s8776" xml:space="preserve">] Et cum t c ſit æqualis b g [ex theſi] erit [per
              <lb/>
            14 p 5] e d æqualis h z.</s>
            <s xml:id="echoid-s8777" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s8778" xml:space="preserve"> Si autem lineat c ad ſectionem c u ducta, & minima:</s>
            <s xml:id="echoid-s8779" xml:space="preserve">
              <lb/>
            fuerit minor diametro b g:</s>
            <s xml:id="echoid-s8780" xml:space="preserve"> producatur ultra ſectionem, donec ſit æqualis, & ſecundum quantitatẽ
              <lb/>
            eius fiat circulus:</s>
            <s xml:id="echoid-s8781" xml:space="preserve"> qui quidem circulus ſecabit ſectionem in duobus punctis:</s>
            <s xml:id="echoid-s8782" xml:space="preserve"> à quibus lineæ ductæ
              <lb/>
            ad t, erunt æquales b g:</s>
            <s xml:id="echoid-s8783" xml:space="preserve"> [per 15 d 1] & à puncto z ducatur ęquidiſtãs utriq.</s>
            <s xml:id="echoid-s8784" xml:space="preserve"> Et tunc erit ducere à pun
              <lb/>
            cto a modo prædicto duas lineas, æquales lineæ datæ:</s>
            <s xml:id="echoid-s8785" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s8786" xml:space="preserve"> idem penitus probandi modus.</s>
            <s xml:id="echoid-s8787" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div337" type="section" level="0" n="0">
          <head xml:id="echoid-head338" xml:space="preserve" style="it">35. À
            <unsure/>
          puncto dato in altero laterum trianguli rectanguli angulum rectum continẽtium,
            <lb/>
          ducere per lat{us} angulo recto oppoſitum, rectam, cui{us} ſegmentum conterminum reliquo late-
            <lb/>
          ri infinito, habeat ad ſegmentum lateris angulo recto oppoſiti, conterminum primo lateri, ratio
            <lb/>
          nem in duab{us} rectis datam. 134 p 1.</head>
          <p>
            <s xml:id="echoid-s8788" xml:space="preserve">AMplius:</s>
            <s xml:id="echoid-s8789" xml:space="preserve"> dato triangulo orthogonio, & dato puncto in uno laterum angulum rectum conti-
              <lb/>
            nentium, eſt ducere à puncto illo lineam, ad aliud laterum continentium rectum, ſecantem
              <lb/>
            tertium oppoſitum recto, ita ut pars huius lineę interiacens interpunctum ſectionis & latus,
              <lb/>
            in quo non eſt punctum datum, ſe habeat ad partem lateris oppoſiti recto, quæ eſt de ſectione ad la
              <lb/>
            tus, in quo eſt punctum datum, ſicut data linea ad datam lineam.</s>
            <s xml:id="echoid-s8790" xml:space="preserve"> Verbi gratia:</s>
            <s xml:id="echoid-s8791" xml:space="preserve"> eſt triangulum datũ
              <lb/>
            a b g, cuius angulus a b g rectus:</s>
            <s xml:id="echoid-s8792" xml:space="preserve"> & in latere g b eſt punctum datum d, extra triangulum, aut intra.</s>
            <s xml:id="echoid-s8793" xml:space="preserve"> Di
              <lb/>
            co, quòd à puncto d eſt ducere lineam, ſecantem latus a g, & concurrentem cum latere a b:</s>
            <s xml:id="echoid-s8794" xml:space="preserve"> ita ut
              <lb/>
            </s>
          </p>
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