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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            <s xml:id="echoid-s8794" xml:space="preserve">
              <pb o="146" file="0152" n="152" rhead="ALHAZEN"/>
            pars eius interiacens inter latera a b, a g, ſit eius proportionis ad partem lateris a g, quæ eſt ab illa li-
              <lb/>
            nea uſq;</s>
            <s xml:id="echoid-s8795" xml:space="preserve"> ad punctum g, ſicut ſe habet e ad z, quæ ſunt datę lineæ.</s>
            <s xml:id="echoid-s8796" xml:space="preserve"> Sit punctum d in ipſo triangulo
              <lb/>
            a b g:</s>
            <s xml:id="echoid-s8797" xml:space="preserve"> & [per 31 p 1] ducatur ab eo linea æquidiſtans a b:</s>
            <s xml:id="echoid-s8798" xml:space="preserve"> quæ ſit d m:</s>
            <s xml:id="echoid-s8799" xml:space="preserve"> & [per 5 p 4] ſuper tria puncta
              <lb/>
            g, m, d fiat circulus:</s>
            <s xml:id="echoid-s8800" xml:space="preserve"> & protrahatur linea a d.</s>
            <s xml:id="echoid-s8801" xml:space="preserve"> Quoniam [per 29 p 1]
              <lb/>
              <figure xlink:label="fig-0152-01" xlink:href="fig-0152-01a" number="71">
                <variables xml:id="echoid-variables61" xml:space="preserve">q ſ a e z h a t d m c b d g n</variables>
              </figure>
            planum eſt, quòd angulus g m d eſt æqualis angulo g a b:</s>
            <s xml:id="echoid-s8802" xml:space="preserve"> erit [per
              <lb/>
            9 ax.</s>
            <s xml:id="echoid-s8803" xml:space="preserve">] maior g a d.</s>
            <s xml:id="echoid-s8804" xml:space="preserve"> Secetur ex eo æqualis per lineam m n:</s>
            <s xml:id="echoid-s8805" xml:space="preserve"> & ſit d n
              <lb/>
            m:</s>
            <s xml:id="echoid-s8806" xml:space="preserve"> & ſit [per 12 p 6] h linea, ad quam ſe habeat a d, ſicut ſe habet e
              <lb/>
            ad z:</s>
            <s xml:id="echoid-s8807" xml:space="preserve"> & à puncto n, quod eſt punctum circuli, ducatur linea ad dia-
              <lb/>
            metrum g m æqualis lineæ h, ſecundum ſuprà dicta [32 uel 33 n] &
              <lb/>
            ſit n l:</s>
            <s xml:id="echoid-s8808" xml:space="preserve"> [ita ut ſegmentum l c inter continuatam diametrum & peri
              <lb/>
            pheriam æquetur lineæ h] & punctum, in quo ſecat circulum, ſit c:</s>
            <s xml:id="echoid-s8809" xml:space="preserve">
              <lb/>
            & ducatur linea g c:</s>
            <s xml:id="echoid-s8810" xml:space="preserve"> & à puncto d ducatur linea ad punctum c:</s>
            <s xml:id="echoid-s8811" xml:space="preserve"> quę,
              <lb/>
            cum cadat inter duas æquidiſtantes, tenens angulum acutum cum
              <lb/>
            altera, ſi producatur, neceſſariò concurret cum alia [per lẽma Pro-
              <lb/>
            cli ad 29 p 1.</s>
            <s xml:id="echoid-s8812" xml:space="preserve">] Cõcurrat igitur:</s>
            <s xml:id="echoid-s8813" xml:space="preserve"> & ſit punctum concurſus q.</s>
            <s xml:id="echoid-s8814" xml:space="preserve"> Planum
              <lb/>
            [per 27 p 3] quòd angulus g m d eſt æqualis angulo g c d:</s>
            <s xml:id="echoid-s8815" xml:space="preserve"> quia ca-
              <lb/>
            dunt in eundem arcum:</s>
            <s xml:id="echoid-s8816" xml:space="preserve"> & [per 29 p 1] angulus g m d eſt æqualis
              <lb/>
            angulo g a b:</s>
            <s xml:id="echoid-s8817" xml:space="preserve"> reſtat igitur [per 1 ax.</s>
            <s xml:id="echoid-s8818" xml:space="preserve"> 13 p 1.</s>
            <s xml:id="echoid-s8819" xml:space="preserve"> 3 ax.</s>
            <s xml:id="echoid-s8820" xml:space="preserve">] ut angulus g c q ſit
              <lb/>
            æqualis angulo g a q.</s>
            <s xml:id="echoid-s8821" xml:space="preserve"> Sit t punctum, in quo d q ſecat a g:</s>
            <s xml:id="echoid-s8822" xml:space="preserve"> & [per 15
              <lb/>
            p 1] angulus g t c eſt æqualis angulo a t q:</s>
            <s xml:id="echoid-s8823" xml:space="preserve"> igitur [per 32 p 1] tertius
              <lb/>
            tertio.</s>
            <s xml:id="echoid-s8824" xml:space="preserve"> Quare triangulum a t q ſimile triangulo t c g [per 4 p.</s>
            <s xml:id="echoid-s8825" xml:space="preserve"> 1 d 6.</s>
            <s xml:id="echoid-s8826" xml:space="preserve">]
              <lb/>
            Igitur proportio q t ad t g, ſicut a t ad t c.</s>
            <s xml:id="echoid-s8827" xml:space="preserve"> Verùm [per fabricationẽ]
              <lb/>
            angulus n m d eſt æqualis angulo t a d:</s>
            <s xml:id="echoid-s8828" xml:space="preserve"> & [per 27 p 3] angulo n c d
              <lb/>
            [id eſt per 15 p 1 angulo l c t.</s>
            <s xml:id="echoid-s8829" xml:space="preserve">] Quare [per 1 ax.</s>
            <s xml:id="echoid-s8830" xml:space="preserve">] l c t æqualis t a d:</s>
            <s xml:id="echoid-s8831" xml:space="preserve"> & angulus c t l communis duobus
              <lb/>
            triangulis:</s>
            <s xml:id="echoid-s8832" xml:space="preserve"> quare [per 32 p 1] tertius tertio:</s>
            <s xml:id="echoid-s8833" xml:space="preserve"> & triangulum ſimile triangulo, ſcilicet t l c triangulot
              <lb/>
            a d [per 4 p.</s>
            <s xml:id="echoid-s8834" xml:space="preserve"> 1 d 6.</s>
            <s xml:id="echoid-s8835" xml:space="preserve">] Igitur proportio t a ad t c, ſicut proportio a d ad l c.</s>
            <s xml:id="echoid-s8836" xml:space="preserve"> Quare [per 11 p 5] proportio
              <lb/>
            a d ad l c, ſicut q t ad t g [patuit enim, ut q t ad t g, ſic a t ad t c.</s>
            <s xml:id="echoid-s8837" xml:space="preserve">] Sed [per fabricationem] l c eſt æqua
              <lb/>
            lis lineæ h:</s>
            <s xml:id="echoid-s8838" xml:space="preserve"> & [per fabricationem] proportio a d ad h, ſicut e ad z.</s>
            <s xml:id="echoid-s8839" xml:space="preserve"> Ergo [per 7.</s>
            <s xml:id="echoid-s8840" xml:space="preserve"> 11 p 5] proportio q t
              <lb/>
            ad t g, ſicut e ad z.</s>
            <s xml:id="echoid-s8841" xml:space="preserve"> Quod eſt propoſitum.</s>
            <s xml:id="echoid-s8842" xml:space="preserve"> Si uerò d ſumatur in illo latere extra triangulum:</s>
            <s xml:id="echoid-s8843" xml:space="preserve"> produ-
              <lb/>
            catur [per 31 p 1] à puncto d, æquidiſtans a b:</s>
            <s xml:id="echoid-s8844" xml:space="preserve"> & ſit d m:</s>
            <s xml:id="echoid-s8845" xml:space="preserve"> & ducatur a g, donec concurrat cum d m in
              <lb/>
            puncto m, [concurrat autem per lemma Procli ad 29 p 1.</s>
            <s xml:id="echoid-s8846" xml:space="preserve">] Et fiat circulus tranſiens per tria pun-
              <lb/>
            cta g, d, m:</s>
            <s xml:id="echoid-s8847" xml:space="preserve"> & ducatur linea a d:</s>
            <s xml:id="echoid-s8848" xml:space="preserve"> erit
              <lb/>
              <figure xlink:label="fig-0152-02" xlink:href="fig-0152-02a" number="72">
                <variables xml:id="echoid-variables62" xml:space="preserve">ſ a e z h d g c t b q a d n m</variables>
              </figure>
              <figure xlink:label="fig-0152-03" xlink:href="fig-0152-03a" number="73">
                <variables xml:id="echoid-variables63" xml:space="preserve">d b q a ſ e z h
                  <gap/>
                g c a m n d</variables>
              </figure>
            quidem [per 16 p 1] angulus g a d ma
              <lb/>
            ior angulo g m d:</s>
            <s xml:id="echoid-s8849" xml:space="preserve"> fiat [per 23 p 1] ei æ-
              <lb/>
            qualis:</s>
            <s xml:id="echoid-s8850" xml:space="preserve"> & ſit n m d:</s>
            <s xml:id="echoid-s8851" xml:space="preserve"> & à pũcto n, quod
              <lb/>
            eſt punctum circuli, ducatur linea æ-
              <lb/>
            qualis h lineæ [id uerò fiet per 33 uel
              <lb/>
            34 n:</s>
            <s xml:id="echoid-s8852" xml:space="preserve"> ita ut non tota linea à puncto n
              <lb/>
            ducta, ſed pars eius contermina dia-
              <lb/>
            metro extrà continuatæ, æquetur ipſi
              <lb/>
            h] ad quam lineã h ſe habeat a d, ſicut
              <lb/>
            e ad z, & ſit n c l [tota nimirum linea,
              <lb/>
            cuius pars c l æquetur lineæ h] ſuper
              <lb/>
            diametrum m g:</s>
            <s xml:id="echoid-s8853" xml:space="preserve"> & concurſus ſit l.</s>
            <s xml:id="echoid-s8854" xml:space="preserve"> Cũ
              <lb/>
            igitur [per 22 p 3] angulus n m d & an
              <lb/>
            gulus n c d ualeant duos rectos:</s>
            <s xml:id="echoid-s8855" xml:space="preserve"> &
              <lb/>
            [per fabricationem] angulus n m d
              <lb/>
            ſit ęqualis angulo t a d:</s>
            <s xml:id="echoid-s8856" xml:space="preserve"> erũt duo trian
              <lb/>
            gula t c l, t a d ſimilia.</s>
            <s xml:id="echoid-s8857" xml:space="preserve"> [Quia enim an-
              <lb/>
            guli n c d & l c d æquantur duobus re
              <lb/>
            ctis per 13 p 1, quibus item ex conclu-
              <lb/>
            ſo æquantur n c d & t a d:</s>
            <s xml:id="echoid-s8858" xml:space="preserve"> communi igitur n c d ſubducto, æquabitur reliquus l c d reliquo t a d:</s>
            <s xml:id="echoid-s8859" xml:space="preserve"> &
              <lb/>
            anguli ad uerticem t æquantur per 15 p 1, & per 32 p 1, tertius tertio.</s>
            <s xml:id="echoid-s8860" xml:space="preserve"> Quare triangula t c l, t a d ſunt ſi-
              <lb/>
            milia per 4 p.</s>
            <s xml:id="echoid-s8861" xml:space="preserve"> 1 d 6.</s>
            <s xml:id="echoid-s8862" xml:space="preserve">] Et cum [per 27 p 3] duo anguli g c d, g m d ſint æquales:</s>
            <s xml:id="echoid-s8863" xml:space="preserve"> erunt duo triangula gt
              <lb/>
            c, t a q ſimilia:</s>
            <s xml:id="echoid-s8864" xml:space="preserve"> [Nam cum angulus g m d æquetur angulo t a q per 29 p 1 (parallelæ enim ſunt d m
              <lb/>
            b a per fabricationem) æquabitur per 1 ax.</s>
            <s xml:id="echoid-s8865" xml:space="preserve"> angulus t a q angulo t c g, & ad uerticem t æquantur per
              <lb/>
            15 p 1:</s>
            <s xml:id="echoid-s8866" xml:space="preserve"> ideoq́;</s>
            <s xml:id="echoid-s8867" xml:space="preserve"> per 32 p 1 triangula g t c, t a q ſunt æquiangula, & per 4 p.</s>
            <s xml:id="echoid-s8868" xml:space="preserve"> 1 d 6 ſimilia] & erit proportio
              <lb/>
            a d ad c l (quæ eſt æqualis h) ſicut q t ad t g:</s>
            <s xml:id="echoid-s8869" xml:space="preserve"> & ita eſt e ad z, ſicut q t ad t g.</s>
            <s xml:id="echoid-s8870" xml:space="preserve"> [Quia enim triangula t a d,
              <lb/>
            t c l ſunt æquiangula:</s>
            <s xml:id="echoid-s8871" xml:space="preserve"> erit per 4 p 6, ut a d ad c l, ſic a t ad t c.</s>
            <s xml:id="echoid-s8872" xml:space="preserve"> Rurſus quia triangula a t q, t c g funt æ-
              <lb/>
            quiangula:</s>
            <s xml:id="echoid-s8873" xml:space="preserve"> erit per 4 p 6, a t ad t c, ſicut q t ad t g.</s>
            <s xml:id="echoid-s8874" xml:space="preserve"> Quare per 11 p 5 ut a d ad cl (id eſt e ad z) ſic q t
              <lb/>
            ad t g.</s>
            <s xml:id="echoid-s8875" xml:space="preserve">] Quod eſt propoſitum.</s>
            <s xml:id="echoid-s8876" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div339" type="section" level="0" n="0">
          <head xml:id="echoid-head339" xml:space="preserve" style="it">36. Duob{us} punctis extra circuli peripheriam, uel uno extra, reliquo intra datis: inuenire in
            <lb/>
          peripheria punctum, in quo recta linea ipſam tangẽs, bif ariam ſecet angulum comprehenſum
            <lb/>
          </head>
        </div>
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