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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        <div xml:id="echoid-div752" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s21654" xml:space="preserve">
              <pb o="25" file="0327" n="327" rhead="LIBER PRIMVS."/>
            micirculis hoc eſt poſsibile.</s>
            <s xml:id="echoid-s21655" xml:space="preserve"> Si enim ſemicirculi æquales fuerint:</s>
            <s xml:id="echoid-s21656" xml:space="preserve"> tunc in centro alterius ſemicirculi
              <lb/>
            ſuper ſemidiametrum conſtituto æquali angulo a c d, per 23 p 1, compleatur propoſitum ex 4 p 1, &
              <lb/>
            4 p 6.</s>
            <s xml:id="echoid-s21657" xml:space="preserve"> Quòd ſi alter ſemicirculus minor fuerit dato ſemicirculo:</s>
            <s xml:id="echoid-s21658" xml:space="preserve"> inſcribatur æqualis illi ſemicirculo
              <lb/>
            ad idem centrum:</s>
            <s xml:id="echoid-s21659" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s21660" xml:space="preserve"> æquidiſtans primo, & in punctum, ubi linea c d ipſum ſecabit, (quod ſit f)
              <lb/>
            ducatur linea à termino ſuæ ſemidiametri, quæ ſit e f:</s>
            <s xml:id="echoid-s21661" xml:space="preserve"> & patet propoſitum per definitionem circuli
              <lb/>
            & 29 p 1, & 4 p 6.</s>
            <s xml:id="echoid-s21662" xml:space="preserve"> Quòd ſi dato ſemicirculo alter fuerit maior, circumſcribatur æquidiſtãter eidem,
              <lb/>
            & producta linea à centro primi ſemicirculi ad datum punctum d, quouſq;</s>
            <s xml:id="echoid-s21663" xml:space="preserve"> tangat peripheriam al-
              <lb/>
            terius ſemicirculi, & coniungatur à puncto contactus alia linea ad terminum diametri:</s>
            <s xml:id="echoid-s21664" xml:space="preserve"> & deinde
              <lb/>
            compleatur, ut prius, demonſtratio:</s>
            <s xml:id="echoid-s21665" xml:space="preserve"> & patet propoſitum.</s>
            <s xml:id="echoid-s21666" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div754" type="section" level="0" n="0">
          <figure number="327">
            <variables xml:id="echoid-variables311" xml:space="preserve">d a g b f e c</variables>
          </figure>
          <head xml:id="echoid-head631" xml:space="preserve" style="it">57. À
            <unsure/>
          puncto uno ad datum ſemicir culum unam tantum lineam contingentẽ poßibile eſt
            <lb/>
          duci. Ex quo patet, quòd omnis linea ab eodẽ puncto ſub contingẽte ducta,
            <lb/>
          ſecat ſemicirculũ in uno pũcto ſupr a punctũ cõtingẽtiæ, & in alio ſub ipſo.</head>
          <p>
            <s xml:id="echoid-s21667" xml:space="preserve">Eſto datus ſemicirculus a b c, cuius cẽtrum e:</s>
            <s xml:id="echoid-s21668" xml:space="preserve"> & ſit extrà datus punctus d:</s>
            <s xml:id="echoid-s21669" xml:space="preserve"> à
              <lb/>
            quo ad ſemicirculũ ducatur linea contingẽs, quæ ſit d b.</s>
            <s xml:id="echoid-s21670" xml:space="preserve"> Dico, quòd à puncto
              <lb/>
            d ad ſemicirculum a b c, aliam contingẽtem;</s>
            <s xml:id="echoid-s21671" xml:space="preserve"> quàm lineam d b duci eſt impoſ-
              <lb/>
            ſibile.</s>
            <s xml:id="echoid-s21672" xml:space="preserve"> Si enim hoc ſit poſsibile, ducatur:</s>
            <s xml:id="echoid-s21673" xml:space="preserve"> hæc ergo contingens aut cadet ultra
              <lb/>
            punctum b, aut citra:</s>
            <s xml:id="echoid-s21674" xml:space="preserve"> ſit primò, ut cadat ultra punctum b, uerſus c in punctũ f,
              <lb/>
            & ſit d f:</s>
            <s xml:id="echoid-s21675" xml:space="preserve"> ducantur itaq;</s>
            <s xml:id="echoid-s21676" xml:space="preserve"> à centro e ad puncta contingentiæ, lineæ e f, e b, & pro
              <lb/>
            ducatur diameter c e a uſq;</s>
            <s xml:id="echoid-s21677" xml:space="preserve"> ad punctum d.</s>
            <s xml:id="echoid-s21678" xml:space="preserve"> Palàm ergo per 18 p 3, quoniam an-
              <lb/>
            gulus e b d eſt rectus:</s>
            <s xml:id="echoid-s21679" xml:space="preserve"> ſimiliter angulus e f d eſt rectus.</s>
            <s xml:id="echoid-s21680" xml:space="preserve"> Sunt itaq;</s>
            <s xml:id="echoid-s21681" xml:space="preserve"> æquales, &
              <lb/>
            cadunt in trigonum e f d:</s>
            <s xml:id="echoid-s21682" xml:space="preserve"> quod eſt contra 21 p 1.</s>
            <s xml:id="echoid-s21683" xml:space="preserve"> Idẽ quoq;</s>
            <s xml:id="echoid-s21684" xml:space="preserve"> accidit impoſsibile,
              <lb/>
            ſi linea contingẽs ducta à puncto d ad ſemicirculum a b c, cadat inter puncta
              <lb/>
            b & a:</s>
            <s xml:id="echoid-s21685" xml:space="preserve"> ut linea d g.</s>
            <s xml:id="echoid-s21686" xml:space="preserve"> Palàm ergo corollarium:</s>
            <s xml:id="echoid-s21687" xml:space="preserve"> quoniam enim linea d g non con-
              <lb/>
            tingit ſemicirculum:</s>
            <s xml:id="echoid-s21688" xml:space="preserve"> ergo ipſa producta ſecat ipſum:</s>
            <s xml:id="echoid-s21689" xml:space="preserve"> & hoc eſt propoſitum.</s>
            <s xml:id="echoid-s21690" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div755" type="section" level="0" n="0">
          <head xml:id="echoid-head632" xml:space="preserve" style="it">58. Quælibet duæ lineæ ab uno puncto productæ circulum contingẽtes,
            <lb/>
          ſunt æquales: & arcus interiacens puncta contingentiæ eſt minor ſemicir-
            <lb/>
          culo. Linea quo diuidens angulum illarum per æqualia: & arcum inter-
            <lb/>
          iacentem diuidit per æqualia: & linea per æqualia diuidens arcum, hæc
            <lb/>
          producta per æqualia diuidit & angulum à lineis contingentibus conten-
            <lb/>
          tum. Conſectarium ſecundum Campani ad 36 p 3.</head>
          <p>
            <s xml:id="echoid-s21691" xml:space="preserve">Sit circulus a b c, cuius centrum f:</s>
            <s xml:id="echoid-s21692" xml:space="preserve"> & ſit, ut à puncto e ducantur duę lineæ circulum contingen-
              <lb/>
            tes per 17 p 3, quæ ſint e a & e c.</s>
            <s xml:id="echoid-s21693" xml:space="preserve"> Dico, quòd lineæ e a, e c ſunt æqua-
              <lb/>
              <figure xlink:label="fig-0327-02" xlink:href="fig-0327-02a" number="328">
                <variables xml:id="echoid-variables312" xml:space="preserve">e b a g c f d</variables>
              </figure>
            les:</s>
            <s xml:id="echoid-s21694" xml:space="preserve"> & quòd arcus a b c interiacens puncta contingentiæ eſt mi-
              <lb/>
            nor ſemicirculo:</s>
            <s xml:id="echoid-s21695" xml:space="preserve"> & ſi producatur à puncto e linea e b, diuidẽs angu-
              <lb/>
            lum a e c per æqualia:</s>
            <s xml:id="echoid-s21696" xml:space="preserve"> dico;</s>
            <s xml:id="echoid-s21697" xml:space="preserve"> quòd linea e b in puncto b diuidet arcum
              <lb/>
            a c per æqualia:</s>
            <s xml:id="echoid-s21698" xml:space="preserve"> & ſi linea d e diuidat arcum a c per æqualia, diuidet
              <lb/>
            etiam angulum a e c per æqualia.</s>
            <s xml:id="echoid-s21699" xml:space="preserve"> Ducatur enim primò linea e f diui-
              <lb/>
            dens a e c, quæ producta ſecabit circulũ:</s>
            <s xml:id="echoid-s21700" xml:space="preserve"> ſecet ergo ipſum in punctis
              <lb/>
            b & d.</s>
            <s xml:id="echoid-s21701" xml:space="preserve"> Palàm itaq;</s>
            <s xml:id="echoid-s21702" xml:space="preserve"> per 36 p 3, quoniam illud, quod fit ex ductu lineæ
              <lb/>
            d e in lineam e b, æquale eſt quadrato lineæ a e:</s>
            <s xml:id="echoid-s21703" xml:space="preserve"> & eadẽ ratione qua-
              <lb/>
            drato lineæ e c.</s>
            <s xml:id="echoid-s21704" xml:space="preserve"> Ergo quadratum lineæ a c eſt ęquale quadrato lineæ
              <lb/>
            e c.</s>
            <s xml:id="echoid-s21705" xml:space="preserve"> Ergo & linea a e eſt æqualis lineæ c e:</s>
            <s xml:id="echoid-s21706" xml:space="preserve"> & hoc eſt primum propoſi-
              <lb/>
            torũ.</s>
            <s xml:id="echoid-s21707" xml:space="preserve"> Sed quia ductis lineis f a & f c, erunt anguli f c e & f a e recti, per
              <lb/>
            18 p 3:</s>
            <s xml:id="echoid-s21708" xml:space="preserve"> ſunt ergo æquales:</s>
            <s xml:id="echoid-s21709" xml:space="preserve"> ergo per 4 p 1 linea f e diuidit angulum a e c
              <lb/>
            per æqualia.</s>
            <s xml:id="echoid-s21710" xml:space="preserve"> Et quia lineæ c e & a e concurrunt in puncto e:</s>
            <s xml:id="echoid-s21711" xml:space="preserve"> palàm
              <lb/>
            per 32 p 1, quoniã anguli e f c & e f a ſunt minores duobus rectis.</s>
            <s xml:id="echoid-s21712" xml:space="preserve"> Ar-
              <lb/>
            cus ergo a b c eſt minor ſemicirculo per 33 p 6:</s>
            <s xml:id="echoid-s21713" xml:space="preserve"> quod eſt ſecundum.</s>
            <s xml:id="echoid-s21714" xml:space="preserve">
              <lb/>
            Ducatur quoq;</s>
            <s xml:id="echoid-s21715" xml:space="preserve"> linea a c ſecans lineam e d in puncto g:</s>
            <s xml:id="echoid-s21716" xml:space="preserve"> & ducantur
              <lb/>
            lineæ a b & a c.</s>
            <s xml:id="echoid-s21717" xml:space="preserve"> Quia ergo linea e g ſecat angulum a e c per æqualia:</s>
            <s xml:id="echoid-s21718" xml:space="preserve">
              <lb/>
            patet per 4 p 1, cum linea a e ſit æqualis lineæ e c, & latus e g ſit com-
              <lb/>
            mune, quoniã linea a g eſt æqualis lineæ c g, & angulus e g a eſt æqua
              <lb/>
            lis angulo e g c.</s>
            <s xml:id="echoid-s21719" xml:space="preserve"> Sed & trigonis a b g & c b g latus b g eſt comune:</s>
            <s xml:id="echoid-s21720" xml:space="preserve"> ergo per 4 p 1 erit linea a b æqua-
              <lb/>
            lis lineæ b c:</s>
            <s xml:id="echoid-s21721" xml:space="preserve"> ergo per 28 p 3 arcus a b eſt æqualis arcui b c.</s>
            <s xml:id="echoid-s21722" xml:space="preserve"> Eodem quoq;</s>
            <s xml:id="echoid-s21723" xml:space="preserve"> modo patet, quòd ſi linea
              <lb/>
            g e ſecat arcum a c per æqualia in puncto b, quòd ipſa etiam diuidet per æqualia angulũ a e c.</s>
            <s xml:id="echoid-s21724" xml:space="preserve"> Quia
              <lb/>
            enim trigona a e b & c e b ſunt æquilatera, ut patet:</s>
            <s xml:id="echoid-s21725" xml:space="preserve"> palam ergo per 8 p 1, quoniam angulus a e b eſt
              <lb/>
            æqualis angulo c e b:</s>
            <s xml:id="echoid-s21726" xml:space="preserve"> & hoc eſt totum quod proponebatur.</s>
            <s xml:id="echoid-s21727" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div757" type="section" level="0" n="0">
          <head xml:id="echoid-head633" xml:space="preserve" style="it">59. Arcubus æqualibus, minoribus quolibet, quarta circuli ex utra parte diametri cir-
            <lb/>
          culi reſectis: à terminis illorũ arcuum ductas contingentes in uno puncto eductæ diametri con-
            <lb/>
          currere eſt neceſſe: & ab uno puncto diametri ductas contingẽtes in terminis æqualiũ arcuum
            <lb/>
          contingere eſt neceſſe. Ex quo patet, quoniam omnem angulum & arcum à lineis contingenti-
            <lb/>
          bus contentum diuidit diameter educta per æqualia.</head>
        </div>
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