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-div731" type="section" level="0" n="0">
          <p>
            <s xml:id="echoid-s21244" xml:space="preserve">
              <pb o="19" file="0321" n="321" rhead="LIBER PRIMVS."/>
            eſt æqualis arcui e g f per 26 p 3:</s>
            <s xml:id="echoid-s21245" xml:space="preserve"> erit ergo arcus a d c duplus arcui e g f:</s>
            <s xml:id="echoid-s21246" xml:space="preserve"> quod eſt propoſitum primũ.</s>
            <s xml:id="echoid-s21247" xml:space="preserve">
              <lb/>
            Quòd ſi circulus a b c d ſit minor circulo e g f, & angulus m g n ſit æ-
              <lb/>
              <figure xlink:label="fig-0321-01" xlink:href="fig-0321-01a" number="308">
                <variables xml:id="echoid-variables292" xml:space="preserve">h d l a c e g f p q b n d n a c g b</variables>
              </figure>
            qualis angulo a g c, facto angulo p b q ſuper centrum b, per 23 p 1 æ-
              <lb/>
            quali angulo a g c, & ductis lineis g p, g q, b p, b q:</s>
            <s xml:id="echoid-s21248" xml:space="preserve"> erit angulus p b q
              <lb/>
            duplus angulo p g q, per 20 p 3.</s>
            <s xml:id="echoid-s21249" xml:space="preserve"> Ergo angulus a g c eſt duplus angulo
              <lb/>
            p g q.</s>
            <s xml:id="echoid-s21250" xml:space="preserve"> Proportio itaq;</s>
            <s xml:id="echoid-s21251" xml:space="preserve"> arcus m f n ad ſui totam circumferentiã dupli-
              <lb/>
            catur reſpectu arcus a c ad totam ſui peripheriam.</s>
            <s xml:id="echoid-s21252" xml:space="preserve"> Quoniã enim an-
              <lb/>
            gulus m g n eſt duplus angulo p g q, erit per 33 p 6 arcus m f n duplus
              <lb/>
            arcui p f q:</s>
            <s xml:id="echoid-s21253" xml:space="preserve"> ſed arcus p f q eiuſdem eſt proportionis ad ſui peripheriã,
              <lb/>
            cuius eſt arcus a d c ad ſuam:</s>
            <s xml:id="echoid-s21254" xml:space="preserve"> arcus enim a d c ſi fuerit quinq;</s>
            <s xml:id="echoid-s21255" xml:space="preserve"> partiũ
              <lb/>
            reſpectu ſuæ circum ferentiæ:</s>
            <s xml:id="echoid-s21256" xml:space="preserve"> erit arcus m f n decem partium reſpe-
              <lb/>
            ctu ſuæ peripheriæ:</s>
            <s xml:id="echoid-s21257" xml:space="preserve"> & hoc eſt propoſitum.</s>
            <s xml:id="echoid-s21258" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div733" type="section" level="0" n="0">
          <head xml:id="echoid-head619" xml:space="preserve" style="it">45. À
            <unsure/>
          terminis lineæ intra circulum collocatæ partib. æqualib.
            <lb/>
          reſectis, & à punctis ſectionum perpendicularibus ſuper illam li-
            <lb/>
          neam ad circumferentiam productis: neceſſe eſt ductas perpen-
            <lb/>
          diculares æquales eſſe. Et ſi ductæ perpẽdiculares ſunt æquales: ne-
            <lb/>
          ceſſariũ eſt à terminis illius lineæ partes reſectas æquales eſſe.</head>
          <p>
            <s xml:id="echoid-s21259" xml:space="preserve">Sit circulus a k d, cuius cẽtrum r:</s>
            <s xml:id="echoid-s21260" xml:space="preserve"> in quo circulo collocata ſit linea
              <lb/>
            a d:</s>
            <s xml:id="echoid-s21261" xml:space="preserve"> à cuius terminis a & d reſecentur lineæ a b & d g æquales:</s>
            <s xml:id="echoid-s21262" xml:space="preserve"> & à
              <lb/>
            prædictis b & g erigantur duæ lineæ perpẽdiculares ſuper lineã d a,
              <lb/>
            quę productę ad circũferentiã, ſint g k & b c.</s>
            <s xml:id="echoid-s21263" xml:space="preserve"> Dico, quòd linea g k eſt ęqualis lineę b c.</s>
            <s xml:id="echoid-s21264" xml:space="preserve"> Ducatur enim
              <lb/>
            â centror linea æquidiſtans lineæ a d per 31 p 1, quæ ſit l m diameter:</s>
            <s xml:id="echoid-s21265" xml:space="preserve"> & diuidatur linea d a in duo æ-
              <lb/>
            qualia in puncto e per 10 p 1, & à puncto e, ducatur per-
              <lb/>
              <figure xlink:label="fig-0321-02" xlink:href="fig-0321-02a" number="309">
                <variables xml:id="echoid-variables293" xml:space="preserve">k c d g e b a l n r f m</variables>
              </figure>
            pendicularis ſuper l m per 12 p 1:</s>
            <s xml:id="echoid-s21266" xml:space="preserve"> hęc ergo per 1 p 3 tran-
              <lb/>
            ſibit cẽtrum circuli, quod eſt punctũ r:</s>
            <s xml:id="echoid-s21267" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s21268" xml:space="preserve"> linea e r.</s>
            <s xml:id="echoid-s21269" xml:space="preserve">
              <lb/>
            Educatur aũt linea k g ultra punctum g ad diametrum
              <lb/>
            l m in punctũ n, & linea c b in punctũ f, & copulẽtur li-
              <lb/>
            neę k r & c r.</s>
            <s xml:id="echoid-s21270" xml:space="preserve"> Quia ita q;</s>
            <s xml:id="echoid-s21271" xml:space="preserve"> linea d e eſt ęqualis lineæ a e, &
              <lb/>
            lineę d g & b a ex hypotheſi ſunt ęquales:</s>
            <s xml:id="echoid-s21272" xml:space="preserve"> remanet ergo
              <lb/>
            linea g e æqualis lineę e b:</s>
            <s xml:id="echoid-s21273" xml:space="preserve"> ſed per 34 p 1, linea g e eſt æ-
              <lb/>
            qualis lineæ n r, & linea e b ęqualis lineę r f:</s>
            <s xml:id="echoid-s21274" xml:space="preserve"> ſunt ergo
              <lb/>
            lineæ n r & r f æquales:</s>
            <s xml:id="echoid-s21275" xml:space="preserve"> ſed per 47 p 1, quadratum lineę
              <lb/>
            r k ualet duo quadrata linearum k n & r n:</s>
            <s xml:id="echoid-s21276" xml:space="preserve"> quia ex præ-
              <lb/>
            miſsis angulus k n r eſt rectus:</s>
            <s xml:id="echoid-s21277" xml:space="preserve"> & ſimiliter quadratum
              <lb/>
            lineę c r ualet duo quadrata linearũ c f & r f:</s>
            <s xml:id="echoid-s21278" xml:space="preserve"> eſt aũt qua
              <lb/>
            dratum lineę k r ęquale quadrato lineæ c r, quoniã li-
              <lb/>
            nea k r eſt ęqualis lineæ c r per definitionem circuli:</s>
            <s xml:id="echoid-s21279" xml:space="preserve"> &
              <lb/>
            quadratũ lineæ n r eſt ęquale quadrato lineæ f r.</s>
            <s xml:id="echoid-s21280" xml:space="preserve"> Relin
              <lb/>
            quitur ergo quadratũ lineæ k n ęquale quadrato lineæ c f.</s>
            <s xml:id="echoid-s21281" xml:space="preserve"> E ſt ergo linea k n æqualis lineę c f:</s>
            <s xml:id="echoid-s21282" xml:space="preserve"> ſed per
              <lb/>
            25 huius linea g n eſt æqualis b f.</s>
            <s xml:id="echoid-s21283" xml:space="preserve"> Relinquitur ergo linea k g ęqualis lineę c b:</s>
            <s xml:id="echoid-s21284" xml:space="preserve"> quod eſt primũ propo-
              <lb/>
            ſitũ.</s>
            <s xml:id="echoid-s21285" xml:space="preserve"> Conuerſa etiã patet, manente totali diſpoſitione, ut prius.</s>
            <s xml:id="echoid-s21286" xml:space="preserve"> Quia enim linea g n eſt æqualis lineæ
              <lb/>
            b f per 34 p 1, & linea k g æqualis lineæ c b ex hypotheſi:</s>
            <s xml:id="echoid-s21287" xml:space="preserve"> erit tota linea k n ęqualis toti lineę c f.</s>
            <s xml:id="echoid-s21288" xml:space="preserve"> Ergo
              <lb/>
            per 47 p 1 erit linea n r ęqualis lineę r f.</s>
            <s xml:id="echoid-s21289" xml:space="preserve"> Ergo & linea g e ipſi lineę e b ęqualis erit, & linea d g ipſi li-
              <lb/>
            neę b a:</s>
            <s xml:id="echoid-s21290" xml:space="preserve"> quod eſt propoſitum ſecundum.</s>
            <s xml:id="echoid-s21291" xml:space="preserve"> Patet ergo, quod proponebatur.</s>
            <s xml:id="echoid-s21292" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div735" type="section" level="0" n="0">
          <head xml:id="echoid-head620" xml:space="preserve" style="it">46. In duobus circulis inæqualibus duobus ſimilib. arcubus ſumptis, productis́, præter illos,
            <lb/>
          ad arcus alios ſimiles, ſemidiametris: ſi à punctis extra circulos proportionaliter ſemidiametris
            <lb/>
          diſtantibus ab utriſ extremitatibus amborum arcuum, per terminos ſimilium arcuum, li-
            <lb/>
          neæ ad diametros ducantur: pars diametri interiacens lineas arcus circuli maioris eſt maior
            <lb/>
          parte interiacente lineas arcus circuli minoris.</head>
          <p>
            <s xml:id="echoid-s21293" xml:space="preserve">Sint duo circuli inæquales, quorum maior ſit a b c, & eius centrum d, & ſemidiameter d a:</s>
            <s xml:id="echoid-s21294" xml:space="preserve"> minor
              <lb/>
            uerò ſit e f g.</s>
            <s xml:id="echoid-s21295" xml:space="preserve"> cuius centrum h, & ſemidiameter h e:</s>
            <s xml:id="echoid-s21296" xml:space="preserve"> ſignenturq́;</s>
            <s xml:id="echoid-s21297" xml:space="preserve"> in ipſis arcus ſimiles, in maiori circu
              <lb/>
            lo arcus b c, & in minori arcus f g:</s>
            <s xml:id="echoid-s21298" xml:space="preserve"> ſitq́ue arcus a b ſimilis arcui e f:</s>
            <s xml:id="echoid-s21299" xml:space="preserve"> ſit q́;</s>
            <s xml:id="echoid-s21300" xml:space="preserve"> punctũ k extra circulũ maio-
              <lb/>
            rem, & punctum l extra circulum minorem, taliter data, utilla puncta ſecundum proportionem ſe-
              <lb/>
            midiametri d a, ad ſemidiametrum h e diſtent ab utriſque terminis dictorum arcuum:</s>
            <s xml:id="echoid-s21301" xml:space="preserve"> erit ergo pro-
              <lb/>
            portio lineę k b ad lineam l f, & lineæ k c ad lineam l g, ſicut ſemidiametri a d ad h e:</s>
            <s xml:id="echoid-s21302" xml:space="preserve"> & producãtur li
              <lb/>
            neę ad ſemidiametros, k b in punctum m, & k c in punctum n.</s>
            <s xml:id="echoid-s21303" xml:space="preserve"> Similiter quoq;</s>
            <s xml:id="echoid-s21304" xml:space="preserve"> producatur linea l f
              <lb/>
            in punctum o, & l g in punctum p.</s>
            <s xml:id="echoid-s21305" xml:space="preserve"> Dico, quòd linea m n, pars ſemidiametri a d, eſt maior quã linea
              <lb/>
            o p, pars ſemidiametri e h.</s>
            <s xml:id="echoid-s21306" xml:space="preserve"> Ducantur enim chordę b c & f g:</s>
            <s xml:id="echoid-s21307" xml:space="preserve"> & copulentur à centris lineæ d b, d c,
              <lb/>
            h f, h g:</s>
            <s xml:id="echoid-s21308" xml:space="preserve"> palamq́;</s>
            <s xml:id="echoid-s21309" xml:space="preserve"> propter inæqualitatem circulorum, quoniam linea d b eſt maior quã linea h f:</s>
            <s xml:id="echoid-s21310" xml:space="preserve"> ſed
              <lb/>
            propter ſimilitudinem arcuum angulus b d c eſt ęqualis angulo f h g:</s>
            <s xml:id="echoid-s21311" xml:space="preserve"> ergo per 5 p 1 trigona b c d &
              <lb/>
            f g h ſunt ęquiangula.</s>
            <s xml:id="echoid-s21312" xml:space="preserve"> Ergo per 4 p 6 latera ſunt proportionalia:</s>
            <s xml:id="echoid-s21313" xml:space="preserve"> eſt ergo proportio lineæ b c ad li-
              <lb/>
            neam f g, ſicut lineę b d ad lineam f h:</s>
            <s xml:id="echoid-s21314" xml:space="preserve"> ergo ex hypotheſi & per 11 p 5, ſicut k b ad l f, & ſicut k c ad l g:</s>
            <s xml:id="echoid-s21315" xml:space="preserve">
              <lb/>
            </s>
          </p>
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