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-div757" type="section" level="0" n="0">
          <pb o="26" file="0328" n="328" rhead="VITELLONIS OPTICAE"/>
          <p>
            <s xml:id="echoid-s21728" xml:space="preserve">Eſto circulus a b c, cuius centrum ſit d, & eius diameter c e, quæ producatur indefinitè ad pun-
              <lb/>
            ctum f:</s>
            <s xml:id="echoid-s21729" xml:space="preserve"> & ab unaquaq;</s>
            <s xml:id="echoid-s21730" xml:space="preserve"> parte puncti e ſint a e & b e arcus æquales:</s>
            <s xml:id="echoid-s21731" xml:space="preserve"> & à punctis a & b ducantur lineæ
              <lb/>
            circulũ contingentes per 17 p 3.</s>
            <s xml:id="echoid-s21732" xml:space="preserve"> Dico, quòd illæ duę lineæ concurrẽt
              <lb/>
              <figure xlink:label="fig-0328-01" xlink:href="fig-0328-01a" number="329">
                <variables xml:id="echoid-variables313" xml:space="preserve">f h g a e b d c</variables>
              </figure>
            in uno puncto eductæ diametri e f.</s>
            <s xml:id="echoid-s21733" xml:space="preserve"> Quod ſi dicatur ipſas nõ concur-
              <lb/>
            rere in puncto uno diametri, concurrent tamen ambæ contingentes
              <lb/>
            cũ diametro d f:</s>
            <s xml:id="echoid-s21734" xml:space="preserve"> productis enim lineis d a, d b:</s>
            <s xml:id="echoid-s21735" xml:space="preserve"> erũtanguli ad puncta
              <lb/>
            a & b recti:</s>
            <s xml:id="echoid-s21736" xml:space="preserve"> ſed anguli e d a & e d b ſunt acuti per 33 p 6:</s>
            <s xml:id="echoid-s21737" xml:space="preserve"> arcus enim a
              <lb/>
            e, b e ſunt minores, quilibet, quarta circuli:</s>
            <s xml:id="echoid-s21738" xml:space="preserve"> ergo per 14 huius linearũ
              <lb/>
            contingentium utraq;</s>
            <s xml:id="echoid-s21739" xml:space="preserve"> concurret cum linea d f.</s>
            <s xml:id="echoid-s21740" xml:space="preserve"> Si itaq;</s>
            <s xml:id="echoid-s21741" xml:space="preserve"> non fit hoc in
              <lb/>
            eodem puncto:</s>
            <s xml:id="echoid-s21742" xml:space="preserve"> ſit, ut linea contingẽs ducta à puncto a, concurrat cũ
              <lb/>
            linea d f in puncto g:</s>
            <s xml:id="echoid-s21743" xml:space="preserve"> & contingens ducta à puncto b concurrat cum
              <lb/>
            d fin puncto h, quod ſit ultra punctum g:</s>
            <s xml:id="echoid-s21744" xml:space="preserve"> & ducatur linea a h:</s>
            <s xml:id="echoid-s21745" xml:space="preserve"> eritq́;</s>
            <s xml:id="echoid-s21746" xml:space="preserve">
              <lb/>
            per 27 p 3, & exhypotheſi angulus h d a æqualis angulo h d b:</s>
            <s xml:id="echoid-s21747" xml:space="preserve"> ergo
              <lb/>
            per 4 p 1 erit angulus h a d æqualis angulo h b a:</s>
            <s xml:id="echoid-s21748" xml:space="preserve"> & per 18 p 3 uterq;</s>
            <s xml:id="echoid-s21749" xml:space="preserve">
              <lb/>
            ipſorũ eſt rectus.</s>
            <s xml:id="echoid-s21750" xml:space="preserve"> Quia itaq;</s>
            <s xml:id="echoid-s21751" xml:space="preserve"> angulus d a g eſt rectus per 18 p 3:</s>
            <s xml:id="echoid-s21752" xml:space="preserve"> patet,
              <lb/>
            quòd ipſe eſt ęqualis angulo d a h recto, & angulus a d g eſt commu-
              <lb/>
            nis:</s>
            <s xml:id="echoid-s21753" xml:space="preserve"> erit ergo per 32 p 1 angulus a g d ęqualis angulo a h d, extrinſecus
              <lb/>
            ſcilicet intrinſeco in trigono a h g:</s>
            <s xml:id="echoid-s21754" xml:space="preserve"> quod eſt contra 16 p 1 & impoſsi-
              <lb/>
            bile.</s>
            <s xml:id="echoid-s21755" xml:space="preserve"> Patet ergo primum.</s>
            <s xml:id="echoid-s21756" xml:space="preserve"> Sed & ſi à puncto diametri h ducantur duæ
              <lb/>
            lineæ circulum contingentes in punctis a & b:</s>
            <s xml:id="echoid-s21757" xml:space="preserve"> erunt arcus a e & b e
              <lb/>
            æquales:</s>
            <s xml:id="echoid-s21758" xml:space="preserve"> trigona enim a h d & h b d ſunt æquilatera per præcedentẽ:</s>
            <s xml:id="echoid-s21759" xml:space="preserve">
              <lb/>
            ergo ſunt æquiangula per 8 p 1:</s>
            <s xml:id="echoid-s21760" xml:space="preserve"> eſt ergo angulus a d h æqualis angu-
              <lb/>
            lo b d h.</s>
            <s xml:id="echoid-s21761" xml:space="preserve"> Ergo per 26 p 3 arcus a e eſt æqual s arcui b e:</s>
            <s xml:id="echoid-s21762" xml:space="preserve"> quod eſt propoſitum:</s>
            <s xml:id="echoid-s21763" xml:space="preserve"> & patet corollarium.</s>
            <s xml:id="echoid-s21764" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div759" type="section" level="0" n="0">
          <head xml:id="echoid-head634" xml:space="preserve" style="it">60. Si intra duas lineas circulum contιngẽtes ab uno puncto ductas, aliæ duæ lineæ eundem
            <lb/>
          circulam contingentes ducantur: cadent puncta contingẽtiæ interiorum intra puncta contin-
            <lb/>
          gentiæ exteriorum: & ſiarcus hinc inde interiacentes puncta contingentiæ, fuerint æquales,
            <lb/>
          erit utrarum concurſus ſemper in eadẽ diametro circuli educta: interiores quo ad utram
            <lb/>
          partem productæ cum exterioribus neceſſariò concurrent.</head>
          <p>
            <s xml:id="echoid-s21765" xml:space="preserve">Eſto circulus a b c d e, cuius cẽtrũ k:</s>
            <s xml:id="echoid-s21766" xml:space="preserve"> & eius diameter e h educatur:</s>
            <s xml:id="echoid-s21767" xml:space="preserve"> & ſit, ut ab aliquo puncto ſuo,
              <lb/>
            quod ſit f, lineæ f a & f d contingentes circulũ ducantur:</s>
            <s xml:id="echoid-s21768" xml:space="preserve"> & inter lineas f a & f d ducantur ab aliquo
              <lb/>
            puncto ſuperficiei a f d, quod ſit g, lineæ g b & g c circulũ contingen-
              <lb/>
              <figure xlink:label="fig-0328-02" xlink:href="fig-0328-02a" number="330">
                <variables xml:id="echoid-variables314" xml:space="preserve">f g g m b p h c a k d b e</variables>
              </figure>
            tes in punctis b & c Dico, quòd puncta b & c cadent intra pũcta a &
              <lb/>
            d.</s>
            <s xml:id="echoid-s21769" xml:space="preserve"> Si enim nõ caduntintra puncta a & d:</s>
            <s xml:id="echoid-s21770" xml:space="preserve"> aut cadũt in illa puncta aut
              <lb/>
            extra:</s>
            <s xml:id="echoid-s21771" xml:space="preserve"> ſi in illa, ducãtur lineæ k a & k d à cẽtro k ad puncta contingen
              <lb/>
            tiæ a & d:</s>
            <s xml:id="echoid-s21772" xml:space="preserve"> erit itaq;</s>
            <s xml:id="echoid-s21773" xml:space="preserve"> per 18 p 3 angulus k a frectus:</s>
            <s xml:id="echoid-s21774" xml:space="preserve"> & ſimiliter angulus
              <lb/>
            k a g rectus:</s>
            <s xml:id="echoid-s21775" xml:space="preserve"> & ſic rectus maior recto.</s>
            <s xml:id="echoid-s21776" xml:space="preserve"> Itẽ inter contingentẽ f a & cir-
              <lb/>
            culum, alia linea capitur, ut g a:</s>
            <s xml:id="echoid-s21777" xml:space="preserve"> hoc autẽ eſt cõtra 16 p 3.</s>
            <s xml:id="echoid-s21778" xml:space="preserve"> Palàm ergo,
              <lb/>
            quoniã impoſsibile.</s>
            <s xml:id="echoid-s21779" xml:space="preserve"> Si uerò detur, quòd puncta b & c cadant extra
              <lb/>
            pũcta a & d:</s>
            <s xml:id="echoid-s21780" xml:space="preserve"> ſit punctũ b ultra a punctũ, ſecabitq́;</s>
            <s xml:id="echoid-s21781" xml:space="preserve"> linea g b producta
              <lb/>
            lineam f a per 14 huius.</s>
            <s xml:id="echoid-s21782" xml:space="preserve"> Et quoniã eſt contingẽs ſolum in puncto b,
              <lb/>
            erit punctus ſectionis extra circulũ:</s>
            <s xml:id="echoid-s21783" xml:space="preserve"> ſit ille punctus m.</s>
            <s xml:id="echoid-s21784" xml:space="preserve"> Palàm itaq;</s>
            <s xml:id="echoid-s21785" xml:space="preserve">,
              <lb/>
            quoniã lineæ m a & m b ab uno pũcto m productæ ſemicirculũ con-
              <lb/>
            tingunt:</s>
            <s xml:id="echoid-s21786" xml:space="preserve"> quod eſt contra 57 huius.</s>
            <s xml:id="echoid-s21787" xml:space="preserve"> Non ergo cadit punctum b ultra
              <lb/>
            punctũ a, ſed intra.</s>
            <s xml:id="echoid-s21788" xml:space="preserve"> Similiterq́;</s>
            <s xml:id="echoid-s21789" xml:space="preserve"> demonſtrabitur, quia punctũ c cadit
              <lb/>
            intra punctum d.</s>
            <s xml:id="echoid-s21790" xml:space="preserve"> Cadũt ergo puncta contingẽtiæ interiorum intra
              <lb/>
            puncta contingẽtiæ exteriorũ.</s>
            <s xml:id="echoid-s21791" xml:space="preserve"> Sed & arcubus a b & c d exiſtẽtibus
              <lb/>
            ęqualibus, punctũ g neceſſariò cadit in diametro e h f.</s>
            <s xml:id="echoid-s21792" xml:space="preserve"> Si enim extra
              <lb/>
            illã, ducatur linea k g ſecãs circũferentiã in pũcto p.</s>
            <s xml:id="echoid-s21793" xml:space="preserve"> Quia ergo arcus
              <lb/>
            b p eſt æqualis arcui p c per præcedentẽ:</s>
            <s xml:id="echoid-s21794" xml:space="preserve"> arcus quoq;</s>
            <s xml:id="echoid-s21795" xml:space="preserve"> a b eſt æqualis
              <lb/>
            arcui c d ex hypotheſi:</s>
            <s xml:id="echoid-s21796" xml:space="preserve"> remanet ergo arcus c h æqualis arcui h b:</s>
            <s xml:id="echoid-s21797" xml:space="preserve"> ſed
              <lb/>
            arcus h b eſt maior arcu p b:</s>
            <s xml:id="echoid-s21798" xml:space="preserve"> ergo arcus c h eſt maior arcu c p, pars ſuo toto:</s>
            <s xml:id="echoid-s21799" xml:space="preserve"> qđ eſt impoſsibile.</s>
            <s xml:id="echoid-s21800" xml:space="preserve"> Nõ
              <lb/>
            ergo cadit pũctũ g extra diametrũ e h f.</s>
            <s xml:id="echoid-s21801" xml:space="preserve"> Palàm etiã eſt ք 14 huius, quoniã linea g b ꝓducta ultra pũ-
              <lb/>
            ctũ b, neceſſariò cõcurret cũ linea f a, & linea c g ꝓducta ultra pũctũ c, cõcurret neceſſariò cũ linea
              <lb/>
            f d:</s>
            <s xml:id="echoid-s21802" xml:space="preserve"> linea enim k c rectũ angulũ cõtinẽs cũ linea a g, cõtinet acutũ cũ linea f d:</s>
            <s xml:id="echoid-s21803" xml:space="preserve"> patet ergo ꝓpoſitũ.</s>
            <s xml:id="echoid-s21804" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div761" type="section" level="0" n="0">
          <head xml:id="echoid-head635" xml:space="preserve" style="it">61. Si ad mediũ punctũ arcus interiacẽtis punct a contingẽtiæ duarũ linearũ, abuno puncto
            <lb/>
          ad circulũ productarũ, linea cõtingens circulũ ad alias contingẽtes producatur: illa in puncto
            <lb/>
          ſuæ contingentiæ per æqualia diuiditur: & ab alys lineis cõtingentib. partes abſcindit æquales.</head>
          <p>
            <s xml:id="echoid-s21805" xml:space="preserve">Sit circulus a b c, quẽ contingãt duæ lineæ d a & d c, à puncto d productæ:</s>
            <s xml:id="echoid-s21806" xml:space="preserve"> producatur ergo dia-
              <lb/>
            meter g b d:</s>
            <s xml:id="echoid-s21807" xml:space="preserve"> & palàm ք 59 huius, quoniã ipſa diuidit angulũ a d c, & arcũ a c per æqualia in pũcto b.</s>
            <s xml:id="echoid-s21808" xml:space="preserve">
              <lb/>
            À
              <unsure/>
            puncto itaq;</s>
            <s xml:id="echoid-s21809" xml:space="preserve"> b producatur linea contingens circulũ per 17 p 3:</s>
            <s xml:id="echoid-s21810" xml:space="preserve"> h æ c itaq;</s>
            <s xml:id="echoid-s21811" xml:space="preserve"> quoniã eſt orthogonalis
              <lb/>
            ſuper diametrum g b, ut patet per 18 p 3:</s>
            <s xml:id="echoid-s21812" xml:space="preserve"> palàm per 14 huius, quia ipſa producta ſecabit lineas d a &
              <lb/>
            d c:</s>
            <s xml:id="echoid-s21813" xml:space="preserve"> ſit ergo ut ſecet lineam d a in puncto e, & lineam d c in puncto f.</s>
            <s xml:id="echoid-s21814" xml:space="preserve"> Quia itaq;</s>
            <s xml:id="echoid-s21815" xml:space="preserve"> e d b & f d b anguli
              <lb/>
              <gap/>
            unt æquales per 59 huius, & anguli d b e & d b f ſunt recti:</s>
            <s xml:id="echoid-s21816" xml:space="preserve"> palàm, quia trigona e b d & f d b ſunt
              <lb/>
            </s>
          </p>
        </div>
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